AS HandMathFunc 2010
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A copy of the NIST Handbook of Mathematical Functions, edited by Olver, Lozier, Boisvert and Clark and published by Cambridge University Press in 2010. It has 36 chapters by many expert authors, covering Gamma, Bessel, Airy, hypergeometric, elliptic, theta and zeta functions, orthogonal polynomials, Painlevé transcendents and others. It is a published reference book, not Phil's own writing; the file name suggests it is a download kept for reference.
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NIST Handbook of Mathematical Functions
Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical
functions, from elementary trigonometric functions to the multitude of special functions. These functions appear
whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are per-
formed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requirespractitioners to have ready access to a reliable collection of their properties.
This handbook results from a 10-year project conducted by the National Institute of Standards and Technology
with an international group of expert authors and validators. Printed in full color, it is destined to replace itspredecessor, the classic but long-outdated Handbook of Mathematical Functions , edited by Abramowitz and Stegun.
Included with every copy of the book is a CD with a searchable PDF.
Frank W. J. Olver is Professor Emeritus in the Institute for Physical Science and Technology and the Department
of Mathematics at the University of Maryland. From 1961 to 1986 he was a Mathematician at the National Bureau
of Standards in Washington, D.C. Professor Olver has published 76 papers in refereed and leading mathematics
journals, and he is the author of Asymptotics and Special Functions (1974). He has served as editor of SIAM
Journal on Numerical Analysis ,SIAM Journal on Mathematical Analysis ,Mathematics of Computation ,Methods
and Applications of Analysis , and the NBS Journal of Research .
Daniel W. Lozier leads the Mathematical Software Group in the Mathematical and Computational Sciences Division
of NIST. He received his Ph.D. in applied mathematics from the University of Maryland in 1979 and has been at
NIST since 1970. He is an active member of the SIAM Activity Group on Orthogonal Polynomials and SpecialFunctions, having served two terms as chair and one as vice-chair, and currently is serving as secretary. He has been
an editor of Mathematics of Computation and the NIST Journal of Research .
Ronald F. Boisvert leads the Mathematical and Computational Sciences Division of the Information Technology
Laboratory at NIST. He received his Ph.D. in computer science from Purdue University in 1979 and has been at
NIST since then. He has served as editor-in-chief of the ACM Transactions on Mathematical Software . He is currently
co-chair of the Publications Board of the Association for Computing Machinery (ACM) and chair of the International
Federation for Information Processing (IFIP) Working Group 2.5 (Numerical Software).
Charles W. Clark received his Ph.D. in physics from the University of Chicago in 1979. He is a member of the U.S.
Senior Executive Service and Chief of the Electron and Optical Physics Division and acting Group Leader of theNIST Synchrotron Ultraviolet Radiation Facility (SURF III). Clark serves as Program Manager for Atomic and
Molecular Physics at the U.S. Office of Naval Research and is a Fellow of the Joint Quantum Institute of NIST and
the University of Maryland at College Park and a Visiting Professor at the National University of Singapore.
Rainbow over Woolsthorpe Manor
From the frontispiece of the Notes and Records of the Royal Society of London , v. 36 (1981{82), with permission. Photograph
by Dr. Roy L. Bishop, Physics Department, Acadia University, Nova Scotia, Canada, with permission.
Commentary
The faint line below the main colored arc is a supernumerary rainbow , produced by the interference of
di erent sun-rays traversing a raindrop and emerging in the same direction. For each color, the intensity
pro le across the rainbow is an Airy function. Airy invented his function in 1838 precisely to describe
this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary
rainbows require the wave theory of light and are impossible to explain with Newton's picture of light as
a stream of independent corpuscles. The house in the picture is Newton's birthplace.
Sir Michael V. Berry
H. H. Wills Physics Laboratory
Bristol, United Kingdom
NIST Handbook of Mathematical Functions
F r a n kW .J .O l v e r
Editor-in-Chief and Mathematics Editor
Daniel W. Lozier
General Editor
Ronald F. Boisvert
Information Technology Editor
Charles W. Clark
Physical Sciences Editor
and
cambridge university press
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c/circlecopyrtNational Institute of Standards and Technology 2010
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Contents
Foreword . . . . . . . . . . . . . . . . . . vii
Preface . . . . . . . . . . . . . . . . . . . ix
Mathematical Introduction . . . . . . . xiii
1Algebraic and Analytic Methods
R. Roy, F. W. J. Olver, R. A. Askey, R. Wong 1
2Asymptotic Approximations
F. W. J. Olver, R. Wong . . . . . . . . . . 41
3Numerical Methods
N. M. Temme . . . . . . . . . . . . . . . . 71
4Elementary Functions
R. Roy, F. W. J. Olver . . . . . . . . . . . . 103
5Gamma Function
R. A. Askey, R. Roy . . . . . . . . . . . . . 135
6Exponential, Logarithmic, Sine, and
Cosine Integrals
N. M. Temme . . . . . . . . . . . . . . . . 149
7Error Functions, Dawson's and Fresnel
Integrals
N. M. Temme . . . . . . . . . . . . . . . . 159
8Incomplete Gamma and Related
Functions
R. B. Paris . . . . . . . . . . . . . . . . . . 173
9Airy and Related Functions
F. W. J. Olver . . . . . . . . . . . . . . . . 193
10 Bessel Functions
F. W. J. Olver, L. C. Maximon . . . . . . . 215
11 Struve and Related Functions
R. B. Paris . . . . . . . . . . . . . . . . . . 287
12 Parabolic Cylinder Functions
N. M. Temme . . . . . . . . . . . . . . . . 303
13 Con
uent Hypergeometric Functions
A. B. Olde Daalhuis . . . . . . . . . . . . . 321
14 Legendre and Related Functions
T. M. Dunster . . . . . . . . . . . . . . . . 351
15 Hypergeometric Function
A. B. Olde Daalhuis . . . . . . . . . . . . . 383
16 Generalized Hypergeometric Functions
and MeijerG-Function
R. A. Askey, A. B. Olde Daalhuis . . . . . . 403
17q-Hypergeometric and Related Func-
tions
G. E. Andrews . . . . . . . . . . . . . . . . 41918 Orthogonal Polynomials
T. H. Koornwinder, R. Wong, R. Koekoek,
R. F. Swarttouw . . . . . . . . . . . . . . . 435
19 Elliptic Integrals
B. C. Carlson . . . . . . . . . . . . . . . . 485
20 Theta Functions
W. P. Reinhardt, P. L. Walker . . . . . . . 523
21 Multidimensional Theta Functions
B. Deconinck . . . . . . . . . . . . . . . . 537
22 Jacobian Elliptic Functions
W. P. Reinhardt, P. L. Walker . . . . . . . 549
23 Weierstrass Elliptic and Modular
Functions
W. P. Reinhardt, P. L. Walker . . . . . . . 569
24 Bernoulli and Euler Polynomials
K. Dilcher . . . . . . . . . . . . . . . . . . 587
25 Zeta and Related Functions
T. M. Apostol . . . . . . . . . . . . . . . . 601
26 Combinatorial Analysis
D. M. Bressoud . . . . . . . . . . . . . . . 617
27 Functions of Number Theory
T. M. Apostol . . . . . . . . . . . . . . . . 637
28 Mathieu Functions and Hill's Equation
G. Wolf . . . . . . . . . . . . . . . . . . . 651
29 Lam e Functions
H. Volkmer . . . . . . . . . . . . . . . . . . 683
30 Spheroidal Wave Functions
H. Volkmer . . . . . . . . . . . . . . . . . . 697
31 Heun Functions
B. D. Sleeman, V. B. Kuznetsov . . . . . . 709
32 Painlev e Transcendents
P. A. Clarkson . . . . . . . . . . . . . . . . 723
33 Coulomb Functions
I. J. Thompson . . . . . . . . . . . . . . . 741
343j;6j;9jSymbols
L. C. Maximon . . . . . . . . . . . . . . . . 757
35 Functions of Matrix Argument
D. St. P. Richards . . . . . . . . . . . . . . 767
36 Integrals with Coalescing Saddles
M. V. Berry, C. J. Howls . . . . . . . . . . 775
Bibliography . . . . . . . . . . . . . . . . 795
Notations . . . . . . . . . . . . . . . . . 873
Index . . . . . . . . . . . . . . . . . . . . 887
v
Foreword
In 1964 the National Institute of Standards and Technology1published the Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathematical Tables , edited by Milton Abramowitz
and Irene A. Stegun. That 1046-page tome proved to be an invaluable reference for the many scien-
tists and engineers who use the special functions of applied mathematics in their day-to-day work,
so much so that it became the most widely distributed and most highly cited NIST publication
in the rst 100 years of the institution's existence.2The success of the original handbook, widely
referred to as \Abramowitz and Stegun" (\A&S"), derived not only from the fact that it provided
critically useful scienti c data in a highly accessible format, but also because it served to standardize
de nitions and notations for special functions. The provision of standard reference data of this type
is a core function of NIST.
Much has changed in the years since A&S was published. Certainly, advances in applied mathe-
matics have continued unabated. However, we have also seen the birth of a new age of computing
technology, which has not only changed how we utilize special functions, but also how we commu-
nicate technical information. The document you are now holding, or the Web page you are now
reading, represents an e ort to extend the legacy of A&S well into the 21st century. The new
printed volume, the NIST Handbook of Mathematical Functions , serves a similar function as the
original A&S, though it is heavily updated and extended. The online version, the NIST Digital
Library of Mathematical Functions (DLMF) , presents the same technical information along with
extensions and innovative interactive features consistent with the new medium. The DLMF may
well serve as a model for the e ective presentation of highly mathematical reference material on the
Web.
The production of these new resources has been a very complex undertaking some 10 years in
the making. This could not have been done without the cooperation of many mathematicians,
information technologists, and physical scientists both within NIST and externally. Their unfailing
dedication is acknowledged deeply and gratefully. Particular attention is called to the generous
support of the National Science Foundation, which made possible the participation of experts from
academia and research institutes worldwide.
Dr. Patrick D. Gallagher
Director, NIST
November 20, 2009
Gaithersburg, Maryland
1Then known as the National Bureau of Standards.
2D. R. Lide (ed.), A Century of Excellence in Measurement, Standards, and Technology , CRC Press, 2001.
vii
Preface
The NIST Handbook of Mathematical Functions , to-
gether with its Web counterpart, the NIST Digital Li-
brary of Mathematical Functions (DLMF) , is the cul-
mination of a project that was conceived in 1996 at the
National Institute of Standards and Technology (NIST).
The project had two equally important goals: to develop
an authoritative replacement for the highly successful
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables , published in 1964 by
the National Bureau of Standards (M. Abramowitz and
I. A. Stegun, editors); and to disseminate essentially the
same information from a public Web site operated by
NIST. The new Handbook and DLMF are the work of
many hands: editors, associate editors, authors, valida-
tors, and numerous technical experts. A summary of
the responsibilities of these groups may help in under-
standing the structure and results of this project.
Executive responsibility was vested in the editors:
Frank W. J. Olver (University of Maryland, College
Park, and NIST), Daniel W. Lozier (NIST), Ronald F.
Boisvert (NIST), and Charles W. Clark (NIST). Olver
was responsible for organizing and editing the mathe-
matical content after receiving it from the authors; for
communicating with the associate editors, authors, val-
idators, and other technical experts; and for assembling
theNotations section and the Index . In addition,
Olver was author or co-author of ve chapters. Lozier
directed the NIST research, technical, and support sta
associated with the project, administered grants and
contracts, together with Boisvert compiled the Soft-
ware sections for the Web version of the chapters,
conducted editorial and sta meetings, represented the
project within NIST and at professional meetings in
the United States and abroad, and together with Olver
carried out the day-to-day development of the project.
Boisvert and Clark were responsible for advising and
assisting in matters related to the use of information
technology and applications of special functions in the
physical sciences (and elsewhere); they also participated
in the resolution of major administrative problems when
they arose.
The associate editors are eminent domain experts
who were recruited to advise the project on strategy, ex-
ecution, subject content, format, and presentation, and
to help identify and recruit suitable candidate authors
and validators. The associate editors were:
Richard A. Askey
University of Wisconsin, MadisonMichael V. Berry
University of Bristol
Walter Gautschi (resigned 2002)
Purdue University
Leonard C. Maximon
George Washington University
Morris Newman
University of California, Santa Barbara
Ingram Olkin
Stanford University
Peter Paule
Johannes Kepler University
William P. Reinhardt
University of Washington
Nico M. Temme
Centrum voor Wiskunde en Informatica
Jet Wimp (resigned 2001)
Drexel University
The technical information provided in the Hand-
book and DLMF was prepared by subject experts from
around the world. They are identi ed on the title pages
of the chapters for which they served as authors and in
the table of Contents.
The validators played a critical role in the project,
one that was absent in its 1964 counterpart: to provide
critical, independent reviews during the development of
each chapter, with attention to accuracy and appropri-
ateness of subject coverage. These reviews have con-
tributed greatly to the quality of the product. The val-
idators were:
T. M. Apostol
California Institute of Technology
A. R. Barnett
University of Waikato, New Zealand
A. I. Bobenko
Technische Universit at, Berlin
B. B. L. Braaksma
University of Groningen
D. M. Bressoud
Macalester College
ix
x Preface
B. C. Carlson
Iowa State University
B. Deconinck
University of Washington
T. M. Dunster
University of California, San Diego
A. Gil
Universidad de Cantabria
A. R. Its
Indiana University{Purdue University, Indianapo-
lis
B. R. Judd
Johns Hopkins University
R. Koekoek
Delft University of Technology
T. H. Koornwinder
University of Amsterdam
R. J. Muirhead
P zer Global R&D
E. Neuman
University of Illinois, Carbondale
A. B. Olde Daalhuis
University of Edinburgh
R. B. Paris
University of Abertay Dundee
R. Roy
Beloit College
S. N. M. Ruijsenaars
University of Leeds
J. Segura
Universidad de Cantabria
R. F. Swarttouw
Vrije Universiteit Amsterdam
N. M. Temme
Centrum voor Wiskunde en Informatica
H. Volkmer
University of Wisconsin, Milwaukee
G. Wolf
Universit at Duisberg-Essen
R. Wong
City University of Hong KongAll of the mathematical information contained in the
Handbook is also contained in the DLMF, along with
additional features such as more graphics, expanded ta-
bles, and higher members of some families of formulas;
in consequence, in the Handbook there are occasional
gaps in the numbering sequences of equations, tables,
and gures. The Web address where additional DLMF
content can be found is printed in blue at appropriate
places in the Handbook. The home page of the DLMF
is accessible at http://dlmf.nist.gov/ .
The DLMF has been constructed speci cally for
e ective Web usage and contains features unique to
Web presentation. The Web pages contain many ac-
tive links, for example, to the de nitions of symbols
within the DLMF, and to external sources of reviews,
full texts of articles, and items of mathematical soft-
ware. Advanced capabilities have been developed at
NIST for the DLMF, and also as part of a larger re-
search e ort intended to promote the use of the Web
as a tool for doing mathematics. Among these capabili-
ties are: a facility to allow users to download LaTeX and
MathML encodings of every formula into document pro-
cessors and software packages (eventually, a fully seman-
tic downloading capability may be possible); a search
engine that allows users to locate formulas based on
queries expressed in mathematical notation; and user-
manipulable 3-dimensional color graphics.
Production of the Handbook and DLMF was a mam-
moth undertaking, made possible by the dedicated lead-
ership of Bruce R. Miller (NIST), Bonita V. Saunders
(NIST), and Abdou S. Youssef (George Washington
University and NIST). Miller was responsible for infor-
mation architecture, specializing LaTeX for the needs of
the project, translation from LaTeX to MathML, and
the search interface. Saunders was responsible for mesh
generation for curves and surfaces, data computation
and validation, graphics production, and interactive
Web visualization. Youssef was responsible for mathe-
matics search indexing and query processing. They were
assisted by the following NIST sta : Marjorie A. Mc-
Clain (LaTeX, bibliography), Joyce E. Conlon (bibliog-
raphy), Gloria Wiersma (LaTeX), Qiming Wang (graph-
ics generation, graphics viewers), and Brian Antonishek
(graphics viewers).
The editors acknowledge the many other individuals
who contributed to the project in a variety of ways.
Among the research, technical, and support sta at
NIST these are B. K. Alpert, T. M. G. Arrington, R.
Bickel, B. Blaser, P. T. Boggs, S. Burley, G. Chu, A.
Dienstfrey, M. J. Donahue, K. R. Eberhardt, B. R.
Fabijonas, M. Fancher, S. Fletcher, J. Fowler, S. P.
Frechette, C. M. Furlani, K. B. Gebbie, C. R. Hagwood,
A. N. Heckert, M. Huber, P. K. Janert, R. N. Kacker,
R. F. Kayser, P. M. Ketcham, E. Kim, M. J. Lieber-
Preface xi
man, R. R. Lipman, M. S. Madsen, E. A. P. Mai, W.
Mehuron, P. J. Mohr, S. Olver, D. R. Penn, S. Phoha,
A. Possolo, S. P. Ressler, M. Rubin, J. Rumble, C. A.
Schanzle, B. I. Schneider, N. Sedransk, E. L. Shirley,
G. W. Stewart, C. P. Sturrock, G. Thakur, S. Wakid,
and S. F. Zevin. Individuals from outside NIST are S. S.
Antman, A. M. Ashton, C. M. Bender, J. J. Benedetto,
R. L. Bishop, J. M. Borwein, H. W. Braden, C. Brezin-
ski, F. Chyzak, J. N. L. Connor, R. Cools, A. Cuyt,
I. Daubechies, P. J. Davis, C. F. Dunkl, J. P. Goed-
bloed, B. Gordon, J. W. Jenkins, L. H. Kellogg, C. D.
Kemp, K. S. K olbig, S. G. Krantz, M. D. Kruskal, W.
Lay, D. A. Lutz, E. L. Mans eld, G. Marsaglia, B. M.
McCoy, W. Miller, Jr., M. E. Muldoon, S. P. Novikov,
P. J. Olver, W. C. Parke, M. Petkovsek, W. H. Reid, B.
Salvy, C. Schneider, M. J. Seaton, N. C. Severo, I. A.
Stegun, F. Stenger, M. Steuerwalt, W. G. Strang, P. R.
Turner, J. Van Deun, M. Vuorinen, E. J. Weniger, H.
Wiersma, R. C. Winther, D. B. Zagier, and M. Zelen.
Undoubtedly, the editors have overlooked some individ-
uals who contributed, as is inevitable in a large long-
lasting project. Any oversight is unintentional, and the
editors apologize in advance.
The project was funded in part by NSF Award
9980036, administered by the NSF's Knowledge and
Distributed Intelligence Program. Within NIST nan-
cial resources and sta were committed by the Informa-tion Technology Laboratory, Physics Laboratory, Sys-
tems Integration for Manufacturing Applications Pro-
gram of the Manufacturing Engineering Laboratory,
Standard Reference Data Program, and Advanced Tech-
nology Program.
Notwithstanding the great care that has been exer-
cised by the editors, authors, validators, and the NIST
sta , it is almost inevitable that in a work of the mag-
nitude and scope of the NIST Handbook and DLMF
errors will still be present. Users need to be aware that
none of these individuals nor the National Institute of
Standards and Technology can assume responsibility for
any possible consequences of such errors.
Lastly, the editors appreciate the skill, and long ex-
perience, that was brought to bear by the publisher,
Cambridge University Press, on the production and
publication of the new Handbook.
Frank W. J. Olver
Editor-in-Chief and Mathematics Editor
Daniel W. Lozier
General Editor
Ronald F. Boisvert
Information Technology Editor
Charles W. Clark
Physical Sciences Editor
Mathematical Introduction
Organization and Objective
The mathematical content of the NIST Handbook of
Mathematical Functions has been produced over a ten-
year period. This part of the project has been carried
out by a team comprising the mathematics editor, au-
thors, validators, and the NIST professional sta . Also,
valuable initial advice on all aspects of the project was
provided by ten external associate editors.
The NIST Handbook has essentially the same ob-
jective as the Handbook of Mathematical Functions that
was issued in 1964 by the National Bureau of Standards
as Number 55 in the NBS Applied Mathematics Series
(AMS). This objective is to provide a reference tool for
researchers and other users in applied mathematics, the
physical sciences, engineering, and elsewhere who en-
counter special functions in the course of their everyday
work.
The mathematical project team has endeavored to
take into account the hundreds of research papers and
numerous books on special functions that have appeared
since 1964. As a consequence, in addition to providing
more information about the special functions that were
covered in AMS 55, the NIST Handbook includes sev-
eral special functions that have appeared in the interim
in applied mathematics, the physical sciences, and en-
gineering, as well as in other areas. See, for example,
Chapters 16, 17, 18, 19, 21, 27, 29, 31, 32, 34, 35, and
36.
Two other ways in which this Handbook di ers from
AMS 55, and other handbooks, are as follows.
First, the editors instituted a validation process for
the whole technical content of each chapter. This pro-
cess greatly extended normal editorial checking proce-
dures. All chapters went through several drafts (nine in
some cases) before the authors, validators, and editors
were fully satis ed.
Secondly, as described in the Preface , a Web ver-
sion (the NIST DLMF) is also available.
Methodology
The rst three chapters of the NIST Handbook and
DLMF are methodology chapters that provide detailed
coverage of, and references for, mathematical topics that
are especially important in the theory, computation,
and application of special functions. (These chapters
can also serve as background material for universitygraduate courses in complex variables, classical anal-
ysis, and numerical analysis.)
Particular care is taken with topics that are not dealt
with suciently thoroughly from the standpoint of this
Handbook in the available literature. These include, for
example, multivalued functions of complex variables, for
which new de nitions of branch points and principal val-
ues are supplied ( xx1.10(vi), 4.2(i)); the Dirac delta (or
delta function), which is introduced in a more readily
comprehensible way for mathematicians ( x1.17); numer-
ically satisfactory solutions of di erential and di erence
equations (xx2.7(iv), 2.9(i)); and numerical analysis for
complex variables (Chapter 3).
In addition, there is a comprehensive account of the
great variety of analytical methods that are used for
deriving and applying the extremely important asymp-
totic properties of the special functions, including dou-
ble asymptotic properties (Chapter 2 and xx10.41(iv),
10.41(v)).
Notation for the Special Functions
The rst section in each of the special function chapters
(Chapters 5{36) lists notation that has been adopted
for the functions in that chapter. This section may also
include important alternative notations that have ap-
peared in the literature. With a few exceptions the
adopted notations are the same as those in standard
applied mathematics and physics literature.
The exceptions are ones for which the existing no-
tations have drawbacks. For example, for the hyperge-
ometric function we often use the notation F(a;b;c;z)
(x15.2(i)) in place of the more conventional 2F1(a;b;c;z)
orF(a;b;c;z). This is because Fis akin to the notation
used for Bessel functions ( x10.2(ii)), inasmuch as Fis an
entire function of each of its parameters a,b, andc: this
results in fewer restrictions and simpler equations. Sim-
ilarly in the case of con
uent hypergeometric functions
(x13.2(i)).
Other examples are: (a) the notation for the Fer-
rers functions|also known as associated Legendre func-
tions on the cut|for which existing notations can eas-
ily be confused with those for other associated Legendre
functions (x14.1); (b) the spherical Bessel functions for
which existing notations are unsymmetric and inelegant
(xx10.47(i) and 10.47(ii)); and (c) elliptic integrals for
which both Legendre's forms and the more recent sym-
metric forms are treated fully (Chapter 19).
xiii
xiv Mathematical Introduction
TheNotations section beginning on p. 873 includes
all the notations for the special functions adopted in this
Handbook. In the corresponding section for the DLMF
some of the alternative notations that appear in the rst
section of the special function chapters are also included.
Common Notations and De nitions
C complex plane (excluding in nity).
D decimal places.
det determinant.
j;korjk Kronecker delta: 0 if j6=k; 1 if
j=k.
(or x) forward di erence operator:
f(x) =f(x+ 1) f(x).
r(orrx) backward di erence operator:
rf(x) =f(x) f(x 1). (See also
del operator in the Notations
section.)
empty sums zero.
empty products unity.
2 element of.
=2 not an element of.
8 for every.
=) implies.
() is equivalent to.
n! factorial: 1 23nif
n= 1;2;3;:::; 1 ifn= 0.
n!! double factorial: 2 46nif
n= 2;4;6;:::; 135nif
n= 1;3;5;:::; 1 ifn= 0; 1.
bxc
oor or integer part: the integer
such thatx 1<bxcx, withx
real.
dxe ceiling: the integer such that
xdxe<x+ 1, withxreal.
f(z)jC= 0 f(z) is continuous at all points of a
simple closed contour CinC.
<1 is nite, or converges.
much greater than.
= imaginary part.
i if and only if.
inf greatest lower bound (in mum).
sup least upper bound (supremum).
\ intersection.
[ union.
(a;b) open interval in R, or open
straight-line segment joining aandb
inC.
[a;b] closed interval in R, or closed
straight-line segment joining aandb
inC.
(a;b] or [a;b) half-closed intervals. is contained in.
is, or is contained in.
lim inf least limit point.
[aj;k] or [ajk] matrix with ( j;k)th element aj;kor
ajk.
A 1inverse of matrix A.
trA trace of matrix A.
ATtranspose of matrix A.
I unit matrix.
mod or modulo mn(modp) meanspdivides
m n, wherem,n, andpare
positive integers with m>n .
N set of all positive integers.
( )nPochhammer's symbol:
( + 1)( + 2)( +n 1) if
n= 1;2;3;:::; 1 ifn= 0.
Q set of all rational numbers.
R real line (excluding in nity).
< real part.
res residue.
S signi cant gures.
signx 1 ifx<0; 0 ifx= 0; 1 ifx>0.
n set subtraction.
Z set of all integers.
nZ set of all integer multiples of n.
Graphics
Special functions with one real variable are depicted
graphically with conventional two-dimensional (2D) line
graphs. See, for example, Figures 10.3.1{10.3.4.
With two real variables, special functions are de-
picted as 3D surfaces, with vertical height correspond-
ing to the value of the function, and coloring added to
emphasize the 3D nature. See Figures 10.3.5{10.3.8 for
examples.
Special functions with a complex variable are de-
picted as colored 3D surfaces in a similar way to func-
tions of two real variables, but with the vertical height
corresponding to the modulus (absolute value) of the
function. See, for example, Figures 5.3.4{5.3.6. How-
ever, in many cases the coloring of the surface is chosen
instead to indicate the quadrant of the plane to which
the phase of the function belongs, thereby achieving a
4D e ect. In these cases the phase colors that corre-
spond to the 1st, 2nd, 3rd, and 4th quadrants are ar-
ranged in alphabetical order: blue, green, red, and yel-
low, respectively, and a \Quadrant Colors" icon appears
alongside the gure. See, for example, Figures 10.3.9{
10.3.16.
Lastly, users may notice some lack of smoothness in
the color boundaries of some of the 4D-type surfaces;
see, for example, Figure 10.3.9. This nonsmoothness
arises because the mesh that was used to generate the
Mathematical Introduction xv
gure was optimized only for smoothness of the surface,
and not for smoothness of the color boundaries.
Applications
All of the special function chapters include sections de-
voted to mathematical, physical, and sometimes other
applications of the main functions in the chapter. The
purpose of these sections is simply to illustrate the im-
portance of the functions in other disciplines; no at-
tempt is made to provide exhaustive coverage.
Computation
All of the special function chapters contain sections
that describe available methods for computing the main
functions in the chapter, and most also provide refer-
ences to numerical tables of, and approximations for,
these functions. In addition, the DLMF provides refer-
ences to research papers in which software is developed,
together with links to sites where the software can be
obtained.
In referring to the numerical tables and approxima-
tions we use notation typi ed by x= 0(:05)1, 8D or 8S.
This means that the variable xranges from 0 to 1 in
intervals of 0.05, and the corresponding function values
are tabulated to 8 decimal places or 8 signi cant gures.
Another numerical convention is that decimals fol-
lowed by dots are unrounded; without the dots they
are rounded. For example, to 4D is 3:1415:::(un-
rounded) and 3.1416 (rounded).Veri cation
For all equations and other technical information this
Handbook and the DLMF either provide references to
the literature for proof or describe steps that can be
followed to construct a proof. In the Handbook this in-
formation is grouped at the section level and appears
under the heading Sources in the References section.
In the DLMF this information is provided in pop-up
windows at the subsection level.
For equations or other technical information that ap-
peared previously in AMS 55, the DLMF usually in-
cludes the corresponding AMS 55 equation number, or
other form of reference, together with corrections, if
needed. However, none of these citations are to be re-
garded as supplying proofs.
Special Acknowledgment
I pay tribute to my friend and predecessor Milton
Abramowitz. His genius in the creation of the National
Bureau of Standards Handbook of Mathematical Func-
tions paid enormous dividends to the world's scienti c,
mathematical, and engineering communities, and paved
the way for the development of the NIST Handbook of
Mathematical Functions and NIST Digital Library of
Mathematical Functions .
Frank W. J. Olver, Mathematics Editor
Chapter 1
Algebraic and Analytic Methods
R. Roy1, F. W. J. Olver2, R. A. Askey3and R. Wong4
Notation 2
1.1 Special Notation . . . . . . . . . . . . . 2
Areas 2
1.2 Elementary Algebra . . . . . . . . . . . . 2
1.3 Determinants . . . . . . . . . . . . . . . 3
1.4 Calculus of One Variable . . . . . . . . . 4
1.5 Calculus of Two or More Variables . . . . 7
1.6 Vectors and Vector-Valued Functions . . 9
1.7 Inequalities . . . . . . . . . . . . . . . . 12
1.8 Fourier Series . . . . . . . . . . . . . . . 13
1.9 Calculus of a Complex Variable . . . . . . 141.10 Functions of a Complex Variable . . . . . 18
1.11 Zeros of Polynomials . . . . . . . . . . . 22
1.12 Continued Fractions . . . . . . . . . . . . 24
1.13 Di erential Equations . . . . . . . . . . . 25
1.14 Integral Transforms . . . . . . . . . . . . 27
1.15 Summability Methods . . . . . . . . . . . 33
1.16 Distributions . . . . . . . . . . . . . . . . 35
1.17 Integral and Series Representations of the
Dirac Delta . . . . . . . . . . . . . . . . 37
References 39
1Department of Mathematics and Computer Science, Beloit College, Beloit, Wisconsin.
2Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland.
3Department of Mathematics, University of Wisconsin, Madison, Wisconsin.
4Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Kowloon, Hong Kong.
Acknowledgments : The authors thank Leonard Maximon and William Parke for their assistance with the writing of x1.17.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
1
2 Algebraic and Analytic Methods
Notation
1.1 Special Notation
(For other notation see pp. xiv and 873.)
x;y real variables.
z real variable inxx1.5{1.6.
z;w complex variables in xx1.9{1.11.
j;k;` integers.
m;n nonnegative integers, unless speci ed
otherwise.
hf;gi distribution.
deg degree.
primes derivatives with respect to the variable,
except where indicated otherwise.
Areas
1.2 Elementary Algebra
1.2(i) Binomial Coecients
In (1.2.1){(1.2.5) kandnare nonnegative integers and
kn.
1.2.1n
k
=n!
(n k)!k!=n
n k
:
Binomial Theorem
1.2.2(a+b)n=an+n
1
an 1b+n
2
an 2b2
++n
n 1
abn 1+bn:
1.2.3n
0
+n
1
++n
n
= 2n:
1.2.4n
0
n
1
++ ( 1)nn
n
= 0:
1.2.5n
0
+n
2
+n
4
++n
k
= 2n 1;
wherekisnorn 1 according as nis even or odd.
In (1.2.6){(1.2.9) kandmare nonnegative integers
andnis unrestricted.
1.2.6n
k
=n(n 1)(n k+ 1)
k!
=( 1)k( n)k
k!= ( 1)kk n 1
k
:
1.2.7n+ 1
k
=n
k
+n
k 1
:
1.2.8mX
k=0n+k
k
=n+m+ 1
m
:1.2.9n
0
n
1
++( 1)mn
m
= ( 1)mn 1
m
:
1.2(ii) Finite Series
Arithmetic Progression
1.2.10a+ (a+d) + (a+ 2d) ++ (a+ (n 1)d)
=na+1
2n(n 1)d=1
2n(a+`);
where`= last term of the series = a+ (n 1)d.
Geometric Progression
1.2.11a+ax+ax2++axn 1
=a(1 xn)
1 x, x6= 1.
1.2(iii) Partial Fractions
Let 1; 2;:::; nbe distinct constants, and f(x) be a
polynomial of degree less than n. Then
1.2.12f(x)
(x 1)(x 2)(x n)
=A1
x 1+A2
x 2++An
x n;
where
1.2.13 Aj=f( j)Q
k6=j( j k):
Also,
1.2.14
f(x)
(x 1)n=B1
x 1+B2
(x 1)2++Bn
(x 1)n;
where
1.2.15 Bj=f(n j)( 1)
(n j)!;
andf(k)is thek-th derivative of f(x1.4(iii)).
Ifm1;m2;:::;mnare positive integers and deg f <Pn
j=1mj, then there exist polynomials fj(x), degfj<
mj, such that
1.2.16
f(x)
(x 1)m1(x 2)m2(x n)mn
=f1(x)
(x 1)m1+f2(x)
(x 2)m2++fn(x)
(x n)mn:
To nd the polynomials fj(x),j= 1;2;:::;n , multiply
both sides by the denominator of the left-hand side and
equate coecients. See Chrystal (1959, pp. 151{159).
1.3 Determinants 3
1.2(iv) Means
The arithmetic mean ofnnumbersa1;a2;:::;anis
1.2.17 A=a1+a2++an
n:
The geometric mean Gand harmonic mean Hofn
positive numbers a1;a2;:::;anare given by
1.2.18 G= (a1a2an)1=n;
1.2.191
H=1
n1
a1+1
a2++1
an
:
Ifris a nonzero real number, then the weighted mean
M(r) ofnnonnegative numbers a1;a2;:::;an, andn
positive numbers p1;p2;:::;pnwith
1.2.20 p1+p2++pn= 1;
is de ned by
1.2.21M(r) = (p1ar
1+p2ar
2++pnar
n)1=r;
with the exception
1.2.22 M(r) = 0,r<0 anda1a2:::an= 0.
1.2.23 lim
r!1M(r) = max(a1;a2;:::;an);
1.2.24 lim
r! 1M(r) = min(a1;a2;:::;an):
Forpj= 1=n,j= 1;2;:::;n ,
1.2.25 M(1) =A; M ( 1) =H;
and
1.2.26 lim
r!0M(r) =G:
The last two equations require aj>0 for allj.
1.3 Determinants
1.3(i) De nitions and Elementary Properties
1.3.1 det[ajk] = a11a12
a21a22 =a11a22 a12a21:
1.3.2
det[ajk]
= a11a12a13
a21a22a23
a31a32a33
=a11 a22a23
a32a33 a12 a21a23
a31a33 +a13 a21a22
a31a32
=a11a22a33 a11a23a32 a12a21a33
+a12a23a31+a13a21a32 a13a22a31:
Higher-order determinants are natural generalizations.
The minorMjkof the entry ajkin thenth-order de-
terminant det[ ajk] is the (n 1)th-order determinant
derived from det[ ajk] by deleting the jth row and the
kth column. The cofactorAjkofajkis
1.3.3 Ajk= ( 1)j+kMjk:Annth-order determinant expanded by its jth row is
given by
1.3.4 det[ajk] =nX
`=1aj`Aj`:
If two rows (or columns) of a determinant are inter-
changed, then the determinant changes sign. If two rows
(columns) of a determinant are identical, then the de-
terminant is zero. If all the elements of a row (column)
of a determinant are multiplied by an arbitrary factor
, then the result is a determinant which is times the
original. If times a row (column) of a determinant is
added to another row (column), then the value of the
determinant is unchanged.
1.3.5 det[ajk]T= det[ajk];
1.3.6 det[ajk] 1=1
det[ajk];
1.3.7 det([ajk][bjk]) = (det[ajk])(det[bjk]):
Hadamard's Inequality
For real-valued ajk,
1.3.8 a11a12
a21a22 2
(a2
11+a2
12)(a2
21+a2
22);
1.3.9 det[ajk]2 nX
k=1a2
1k! nX
k=1a2
2k!
::: nX
k=1a2
nk!
:
Compare also (1.3.7) for the left-hand side. Equality
holds i
1.3.10aj1ak1+aj2ak2++ajnakn= 0
for every distinct pair of j;k, or when one of the factorsPn
k=1a2
jkvanishes.
1.3(ii) Special Determinants
Analternant is a determinant function of nvariables
which changes sign when two of the variables are inter-
changed. Examples:
1.3.11 det[fk(xj)], j= 1;:::;n ;k= 1;:::;n ,
1.3.12 det[f(xj;yk)],j= 1;:::;n ;k= 1;:::;n .
Vandermonde Determinant or Vandermondian
1.3.13 1x1x2
1xn 1
1
1x2x2
2xn 1
2...............
1xnx2
nxn 1
n =Y
1j<kn(xk xj):
4 Algebraic and Analytic Methods
Cauchy Determinant
1.3.14
det1
aj bk
= ( 1)n(n 1)=2
Y
1j<kn(ak aj)(bk bj),nY
j;k=1(aj bk):
Circulant
1.3.15 a1a2an
ana1an 1
............
a2a3a1
=nY
k=1(a1+a2!k+a3!2
k++an!n 1
k);
where!1;!2;:::;!nare thenth roots of unity (1.11.21).
Krattenthaler's Formula
For
1.3.16tjk= (xj+an)(xj+an 1)(xj+ak+1)
(xj+bk)(xj+bk 1)(xj+b2);
1.3.17 det[tjk] =Y
1j<kn(xj xk)Y
2jkn(bj ak):
1.3(iii) In nite Determinants
Letaj;kbe de ned for all integer values of jandk, and
Dn[aj;k] denote the (2 n+ 1)(2n+ 1) determinant
1.3.18
Dn[aj;k] = a n; na n; n+1::: a n;n
a n+1; na n+1; n+1::: a n+1;n
............
an; nan; n+1::: an;n :
IfDn[aj;k] tends to a limit Lasn!1 , then we say
that the in nite determinant D1[aj;k]converges and
D1[aj;k] =L.
Of importance for special functions are in nite de-
terminants of Hill's type . These have the property that
the double series
1.3.191X
j;k= 1jaj;k j;kj
converges (x1.9(vii)). Here j;kis the Kronecker delta.
Hill-type determinants always converge.
For further information see Whittaker and Watson
(1927, pp. 36{40) and Magnus and Winkler (1966, x2.3).1.4 Calculus of One Variable
1.4(i) Monotonicity
Iff(x1)f(x2) for every pair x1,x2in an interval
Isuch thatx1< x 2, thenf(x) isnondecreasing onI.
If thesign is replaced by <, thenf(x) is increas-
ing(also called strictly increasing ) onI. Similarly for
nonincreasing anddecreasing (strictly decreasing ) func-
tions. Each of the preceding four cases is classi ed as
monotonic ; sometimes strictly monotonic is used for the
strictly increasing or strictly decreasing cases.
1.4(ii) Continuity
A function f(x) is continuous on the right (orfrom
above ) atx=cif
1.4.1 f(c+)lim
x!c+f(x) =f(c);
that is, for every arbitrarily small positive constant
there exists (>0) such that
1.4.2jf(c+ ) f(c)j<;
for all such that 0 < . Similarly, it is continuous
on the left (orfrom below ) atx=cif
1.4.3 f(c )lim
x!c f(x) =f(c):
Andf(x) is continuous at cwhen both (1.4.1) and
(1.4.3) apply.
Iff(x) is continuous at each point c2(a;b), then
f(x) iscontinuous on the interval (a;b) and we write
f2C(a;b). If alsof(x) is continuous on the right
atx=a, and continuous on the left at x=b, then
f(x) iscontinuous on the interval [a;b], and we write
f(x)2C[a;b].
Aremovable singularity off(x) atx=coccurs when
f(c+) =f(c ) butf(c) is unde ned. For example,
f(x) = (sinx)=xwithc= 0.
Asimple discontinuity off(x) atx=coccurs when
f(c+) andf(c ) exist, but f(c+)6=f(c ). Iff(x)
is continuous on an interval Isave for a nite number
of simple discontinuities, then f(x) ispiecewise (orsec-
tionally ) continuous on I. For an example, see Figure
1.4.1
Figure 1.4.1 : Piecewise continuous function on [ a;b).
1.4 Calculus of One Variable 5
1.4(iii) Derivatives
The derivativef0(x) off(x) is de ned by
1.4.4f0(x) =df
dx= lim
h!0f(x+h) f(x)
h:
When this limit exists fisdi erentiable atx.
1.4.5 (f+g)0(x) =f0(x) +g0(x);
1.4.6 (fg)0(x) =f0(x)g(x) +f(x)g0(x);
1.4.7f
g0
(x) =f0(x)g(x) f(x)g0(x)
(g(x))2:
Higher Derivatives
1.4.8 f(2)(x) =d2f
dx2=d
dxdf
dx
;
1.4.9 f(n)=f(n)(x) =d
dxf(n 1)(x):
Iff(n)exists and is continuous on an interval I, then
we writef2Cn(I). Whenn1,fiscontinuously dif-
ferentiable onI. Whennis unbounded, fisin nitely
di erentiable onIand we write f2C1(I).
Chain Rule
Forh(x) =f(g(x)),
1.4.10 h0(x) =f0(g(x))g0(x):
Maxima and Minima
A necessary condition that a di erentiable function f(x)
has a local maximum (minimum ) atx=c, that is,
f(x)f(c), (f(x)f(c)) in a neighborhood c
xc+(>0) ofc, isf0(c) = 0.
Mean Value Theorem
Iff(x) is continuous on [ a;b] and di erentiable on ( a;b),
then there exists a point c2(a;b) such that
1.4.11 f(b) f(a) = (b a)f0(c):
Iff0(x)0 (0) (= 0) for all x2(a;b), thenfis
nondecreasing (nonincreasing) (constant) on ( a;b).
Leibniz's Formula
1.4.12(fg)(n)=f(n)g+n
1
f(n 1)g0+
+n
k
f(n k)g(k)++fg(n):
Fa a Di Bruno's Formula
1.4.13dn
dxnf(g(x))
=Xn!
m1!m2!mn!
f(k)(g(x))
g0(x)
1!m1g00(x)
2!m2
:::g(n)(x)
n!mn
;
where the sum is over all nonnegative integers
m1;m2;:::;mnthat satisfy m1+ 2m2++nmn=n,
andk=m1+m2++mn.L'H^ opital's Rule
If
1.4.14 lim
x!af(x) = lim
x!ag(x) = 0 (or1);
then
1.4.15 lim
x!af(x)
g(x)= lim
x!af0(x)
g0(x);
when the last limit exists.
1.4(iv) Inde nite Integrals
IfF0(x) =f(x), thenR
fdx =F(x) +C, whereCis a
constant.
Integration by Parts
1.4.16Z
fgdx =Z
fdx
g ZZ
fdxdg
dxdx:
1.4.17Z
xndx=8
<
:xn+1
n+ 1+C; n6= 1;
lnjxj+C; n = 1:
For the function ln see x4.2(i).
Seexx4.10, 4.26(ii), 4.26(iv), 4.40(ii), and 4.40(iv)
for inde nite integrals involving the elementary func-
tions.
For extensive tables of integrals, see Apelblat (1983),
Bierens de Haan (1867), Gradshteyn and Ryzhik (2000),
Gr obner and Hofreiter (1949, 1950), and Prudnikov
et al. (1986a,b, 1990, 1992a,b).
1.4(v) De nite Integrals
Supposef(x) is de ned on [ a;b]. Leta=x0< x 1<
< xn=b, andjdenote any point in [ xj;xj+1],
j= 0;1;:::;n 1. Then
1.4.18Zb
af(x)dx= limn 1X
j=0f(j)(xj+1 xj)
as max(xj+1 xj)!0. Continuity, or piecewise conti-
nuity, off(x) on [a;b] is sucient for the limit to exist.
1.4.19Zb
a(cf(x) +dg(x))dx=cZb
af(x)dx+dZb
ag(x)dx;
canddconstants.
1.4.20Zb
af(x)dx= Za
bf(x)dx:
1.4.21Zb
af(x)dx=Zc
af(x)dx+Zb
cf(x)dx:
6 Algebraic and Analytic Methods
In nite Integrals
1.4.22Z1
af(x)dx= lim
b!1Zb
af(x)dx:
Similarly forRa
1. Next, iff(b) =1, then
1.4.23Zb
af(x)dx= lim
c!b Zc
af(x)dx:
Similarly when f(a) =1.
When the limits in (1.4.22) and (1.4.23) exist, the
integrals are said to be convergent . If the limits exist
withf(x) replaced byjf(x)j, then the integrals are ab-
solutely convergent . Absolute convergence also implies
convergence.
Cauchy Principal Values
Letc2(a;b) and assume thatRc
af(x)dxandRb
c+f(x)dxexist when 0 < < min(c a;b c), but
not necessarily when = 0. Then we de ne
1.4.24Zb
af(x)dx=PZb
af(x)dx
= lim
!0+ Zc
af(x)dx+Zb
c+f(x)dx!
;
when this limit exists.
Similarly, assume thatRb
bf(x)dxexists for all -
nite values of b(>0), but not necessarily when b=1.
Then we de ne
1.4.25Z1
1f(x)dx=PZ1
1f(x)dx= lim
b!1Zb
bf(x)dx;
when this limit exists.
Fundamental Theorem of Calculus
ForF0(x) =f(x) withf(x) continuous,
1.4.26Zb
af(x)dx=F(b) F(a);
1.4.27d
dxZx
af(t)dt=f(x):
Change of Variables
If 0(x) is continuous or piecewise continuous, then
1.4.28Zb
af( (x)) 0(x)dx=Z (b)
(a)f(t)dt:
First Mean Value Theorem
Forf(x) continuous and (x)0 and integrable on
[a;b], there exists c2[a;b], such that
1.4.29Zb
af(x) (x)dx=f(c)Zb
a (x)dx:Second Mean Value Theorem
Forf(x) monotonic and (x) integrable on [ a;b], there
existsc2[a;b], such that
1.4.30Zb
af(x) (x)dx=f(a)Zc
a (x)dx+f(b)Zb
c (x)dx:
Repeated Integrals
Iff(x) is continuous or piecewise continuous on [ a;b],
then
1.4.31Zb
adxnZxn
adxn 1Zx2
adx1Zx1
af(x)dx
=1
n!Zb
a(b x)nf(x)dx:
Square-Integrable Functions
A function f(x) issquare-integrable if
1.4.32kfk2
2Zb
ajf(x)j2dx<1:
Functions of Bounded Variation
Witha < b , the total variation off(x) on a nite or
in nite interval ( a;b) is
1.4.33Va;b(f) = supnX
j=1jf(xj) f(xj 1)j;
where the supremum is over all sets of points x0<
x1<< xnin the closure of (a;b), that is, ( a;b)
witha;badded when they are nite. If Va;b(f)<1,
thenf(x) is of bounded variation on (a;b). In this case,
g(x) =Va;x(f) andh(x) =Va;x(f) f(x) are nonde-
creasing bounded functions and f(x) =g(x) h(x).
Iff(x) is continuous on the closure of ( a;b) andf0(x)
is continuous on ( a;b), then
1.4.34Va;b(f) =Zb
ajf0(x)dxj;
whenever this integral exists.
Lastly, whether or not the real numbers aandbsat-
isfya<b , and whether or not they are nite, we de ne
Va;b(f) by (1.4.34) whenever this integral exists. This
de nition also applies when f(x) is a complex function
of the real variable x. For further information on total
variation see Olver (1997b, pp. 27{29).
1.4(vi) Taylor's Theorem for Real Variables
Iff(x)2Cn+1[a;b], then
1.4.35f(x) =nX
k=0f(k)(a)
k!(x a)k+Rn;
1.4.36 Rn=f(n+1)(c)
(n+ 1)!(x a)n+1,a<c<x ,
and
1.4.37 Rn=1
n!Zx
a(x t)nf(n+1)(t)dt:
1.5 Calculus of Two or More Variables 7
1.4(vii) Maxima and Minima
Iff(x) is twice-di erentiable, and if also f0(x0) = 0 and
f00(x0)<0 (>0), thenx=x0is a local maximum
(minimum) (x1.4(iii)) of f(x). The overall maximum
(minimum) of f(x) on [a;b] will either be at a local
maximum (minimum) or at one of the end points aor
b.
1.4(viii) Convex Functions
A function f(x) isconvex on (a;b) if
1.4.38f((1 t)c+td)(1 t)f(c) +tf(d)
for anyc;d2(a;b), andt2[0;1]. See Figure 1.4.2. A
similar de nition applies to closed intervals [ a;b].
Iff(x) is twice di erentiable, then f(x) is convex i
f00(x)0 on (a;b). A continuously di erentiable func-
tion is convex i the curve does not lie below its tangent
at any point.
Figure 1.4.2 : Convex function f(x).g(t) =f((1 t)c+
td),l(t) = (1 t)f(c) +tf(d),c;d2(a;b), 0t1.
1.5 Calculus of Two or More Variables
1.5(i) Partial Derivatives
A function f(x;y) iscontinuous at a point (a;b) if
1.5.1 lim
(x;y)!(a;b)f(x;y) =f(a;b);
that is, for every arbitrarily small positive constant
there exists (>0) such that
1.5.2jf(a+ ;b+ ) f(a;b)j<;
for all and that satisfyj j;j j<.
A function is continuous on a point set Dif it is
continuous at all points of D. A function f(x;y) is
piecewise continuous onI1I2, whereI1andI2are in-
tervals, if it is piecewise continuous in xfor eachy2I2
and piecewise continuous in yfor eachx2I1.
1.5.3@f
@x=Dxf=fx= lim
h!0f(x+h;y) f(x;y)
h;
1.5.4@f
@y=Dyf=fy= lim
h!0f(x;y+h) f(x;y)
h:1.5.5@2f
@x@y=@
@x@f
@y
;@2f
@y@x=@
@y@f
@x
:
The function f(x;y) is continuously di erentiable if
f,@f/@x, and@f/@yare continuous, and twice-
continuously di erentiable if also@2f
@x2,@2f
@y2,
@2f=@x@y , and@2f=@y@x are continuous. In the lat-
ter event
1.5.6@2f
@x@y=@2f
@y@x:
Chain Rule
1.5.7d
dtf(x(t);y(t)) =@f
@xdx
dt+@f
@ydy
dt;
1.5.8@
@uf(x(u;v);y(u;v)) =@f
@x@x
@u+@f
@y@y
@u;
1.5.9@
@vf(x(u;v);y(u;v);z(u;v))
=@f
@x@x
@v+@f
@y@y
@v+@f
@z@z
@v:
Implicit Function Theorem
IfF(x;y) is continuously di erentiable, F(a;b) = 0,
and@F/@y6= 0 at (a;b), then in a neighborhood of
(a;b), that is, an open disk centered at a;b, the equa-
tionF(x;y) = 0 de nes a continuously di erentiable
functiony=g(x) such that F(x;g(x)) = 0,b=g(a),
andg0(x) = Fx=Fy.
1.5(ii) Coordinate Systems
Polar Coordinates
With 0r<1, 0 2,
1.5.10 x=rcos ; y =rsin ;
1.5.11@
@x= cos @
@r sin
r@
@ ;
1.5.12@
@y= sin @
@r+cos
r@
@ :
The Laplacian is given by
1.5.13r2f=@2f
@x2+@2f
@y2=@2f
@r2+1
r@f
@r+1
r2@2f
@ 2:
Cylindrical Coordinates
With 0r<1, 0 2, 1<z<1,
1.5.14x=rcos ; y =rsin ; z =z:
Equations (1.5.11) and (1.5.12) still apply, but
1.5.15
r2f=@2f
@x2+@2f
@y2+@2f
@z2=@2f
@r2+1
r@f
@r+1
r2@2f
@ 2+@2f
@z2:
8 Algebraic and Analytic Methods
Spherical Coordinates
With 0<1, 0 2, 0,
1.5.16x=sincos ; y =sinsin ; z =cos:
The Laplacian is given by
1.5.17r2f=@2f
@x2+@2f
@y2+@2f
@z2
=1
2@
@
2@f
@
+1
2sin2@2f
@ 2
+1
2sin@
@
sin@f
@
:
For applications and other coordinate systems see
xx12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14.
See also Morse and Feshbach (1953a, pp. 655-666).
1.5(iii) Taylor's Theorem; Maxima and Minima
Iffisn+ 1 times continuously di erentiable, then
1.5.18f(a+;b+) =f+
@
@x+@
@y
f+
+1
n!
@
@x+@
@yn
f+Rn;
wherefand its partial derivatives on the right-hand
side are evaluated at ( a;b), andRn=(2+2)n=2!0
as (;)!(0;0).
f(x;y) has a local minimum (maximum ) at (a;b) if
1.5.19@f
@x=@f
@y= 0 at (a;b),
and the second-order term in (1.5.18) is positive de nite
(negative de nite) , that is,
1.5.20@2f
@x2>0 (<0) at (a;b),
and
1.5.21@2f
@x2@2f
@y2 @2f
@x@y2
>0 at (a;b):
1.5(iv) Leibniz's Theorem for Di erentiation of
Integrals
Finite Integrals
1.5.22
d
dxZ (x)
(x)f(x;y)dy=f(x; (x)) 0(x) f(x; (x)) 0(x)
+Z (x)
(x)@f
@xdy:
Sucient conditions for validity are: (a) fand@f/@x
are continuous on a rectangle axb,cyd;
(b) whenx2[a;b] both (x) and (x) are continuously
di erentiable and lie in [ c;d].In nite Integrals
Suppose that a;b;c are nite,dis nite or +1, and
f(x;y),@f/@xare continuous on the partly-closed rect-
angle or in nite strip [ a;b][c;d). Suppose also thatRd
cf(x;y)dyconverges andRd
c(@f/@x)dyconverges
uniformly onaxb, that is, given any positive num-
ber, however small, we can nd a number c02[c;d)
that is independent of xand is such that
1.5.23 Zd
c1(@f/@x)dy <;
for allc12[c0;d) and allx2[a;b]. Then
1.5.24d
dxZd
cf(x;y)dy=Zd
c@f
@xdy,a<x<b .
1.5(v) Multiple Integrals
Double Integrals
Letf(x;y) be de ned on a closed rectangle R= [a;b]
[c;d]. For
1.5.25 a=x0<x1<<xn=b;
1.5.26 c=y0<y1<<ym=d;
let (j;k) denote any point in the rectangle [ xj;xj+1]
[yk;yk+1],j= 0;:::;n 1,k= 0;:::;m 1. Then the
double integral off(x;y) overRis de ned by
1.5.27ZZ
Rf(x;y)dA
= limX
j;kf(j;k)(xj+1 xj)(yk+1 yk)
as max((xj+1 xj)+(yk+1 yk))!0. Sucient condi-
tions for the limit to exist are that f(x;y) is continuous,
or piecewise continuous, on R.
Forf(x;y) de ned on a point set Dcontained in a
rectangleR, let
1.5.28f(x;y) =(
f(x;y);if (x;y)2D;
0; if (x;y)2RnD.
Then
1.5.29ZZ
Df(x;y)dA=ZZ
Rf(x;y)dA;
provided the latter integral exists.
Iff(x;y) is continuous, and Dis the set
1.5.30 axb; 1(x)y 2(x);
with 1(x) and 2(x) continuous, then
1.5.31ZZ
Df(x;y)dA=Zb
aZ 2(x)
1(x)f(x;y)dydx;
where the right-hand side is interpreted as the repeated
integral
1.5.32Zb
a Z 2(x)
1(x)f(x;y)dy!
dx:
1.6 Vectors and Vector-Valued Functions 9
In particular, 1(x) and 2(x) can be constants.
Similarly, if Dis the set
1.5.33 cyd; 1(y)x 2(y);
with 1(y) and 2(y) continuous, then
1.5.34ZZ
Df(x;y)dA=Zd
cZ 2(y)
1(y)f(x;y)dxdy:
Change of Order of Integration
IfDcan be represented in both forms (1.5.30) and
(1.5.33), and f(x;y) is continuous on D, then
1.5.35Zb
aZ 2(x)
1(x)f(x;y)dydx =Zd
cZ 2(y)
1(y)f(x;y)dxdy:
In nite Double Integrals
In nite double integrals occur when f(x;y) becomes in-
nite at points in Dor whenDis unbounded. In the
cases (1.5.30) and (1.5.33) they are de ned by taking
limits in the repeated integrals (1.5.32) and (1.5.34) in
an analogous manner to (1.4.22){(1.4.23).
Moreover, if a;b;c;d are nite or in nite constants
andf(x;y) is piecewise continuous on the set ( a;b)
(c;d), then
1.5.36Zb
aZd
cf(x;y)dydx =Zd
cZb
af(x;y)dxdy;
whenever both repeated integrals exist and at least one
is absolutely convergent.
Triple Integrals
Finite and in nite integrals can be de ned in a similar
way. Often the ( x;y;z ) sets are of the form
1.5.37axb; 1(x)y 2(x);
1(x;y)z 2(x;y):
1.5(vi) Jacobians and Change of Variables
Jacobian
1.5.38@(f;g)
@(x;y)= @f/@x @f /@y
@g/@x @g /@y ;
1.5.39@(x;y)
@(r; )=r(polar coordinates) :
1.5.40@(f;g;h )
@(x;y;z )= @f/@x @f /@y @f /@z
@g/@x @g /@y @g /@z
@h/@x @h /@y @h /@z ;
1.5.41@(x;y;z )
@(;; )=2sin(spherical coordinates) :Change of Variables
1.5.42ZZ
Df(x;y)dxdy
=ZZ
Df(x(u;v);y(u;v)) @(x;y)
@(u;v) dudv;
whereDis the image of Dunder a mapping ( u;v)!
(x(u;v);y(u;v)) which is one-to-one except perhaps for
a set of points of area zero.
1.5.43ZZZ
Df(x;y;z )dxdydz
=ZZZ
Df(x(u;v;w );y(u;v;w );z(u;v;w ))
@(x;y;z )
@(u;v;w ) dudvdw:
Again the mapping is one-to-one except perhaps for a
set of points of volume zero.
1.6 Vectors and Vector-Valued Functions
1.6(i) Vectors
1.6.1 a= (a1;a2;a3);b= (b1;b2;b3):
Dot Product (or Scalar Product)
1.6.2 ab=a1b1+a2b2+a3b3:
Magnitude and Angle of Vector a
1.6.3 kak=paa;
1.6.4 cos=ab
kakkbk;
is the angle between aandb.
Unit Vectors
1.6.5 i= (1;0;0);j= (0;1;0);k= (0;0;1);
1.6.6 a=a1i+a2j+a3k:
Cross Product (or Vector Product)
1.6.7 ij=k;jk=i;ki=j;
1.6.8 ji= k;kj= i;ik= j:
1.6.9
ab= i j k
a1a2a3
b1b2b3
= (a2b3 a3b2)i+ (a3b1 a1b3)j+ (a1b2 a2b1)k
=kakkbk(sin)n;
where nis the unit vector normal to aandbwhose
direction is determined by the right-hand rule; see Fig-
ure 1.6.1.
10 Algebraic and Analytic Methods
Figure 1.6.1 : Vector notation. Right-hand rule for cross
products.
Area of parallelogram with vectors aandbas sides
=kabk.
Volume of a parallelepiped with vectors a,b, and c
as edges =ja(bc)j.
1.6.10 a(bc) =b(ac) c(ab);
1.6.11 (ab)c=b(ac) a(bc):
1.6(ii) Vectors: Alternative Notations
The following notations are often used in the physics
literature; see for example Lorentz et al. (1923, pp. 122{
123).
Einstein Summation Convention
Much vector algebra involves summation over suces of
products of vector components. In almost all cases of
repeated suces, we can suppress the summation no-
tation entirely, if it is understood that an implicit sum
is to be taken over any repeated sux. Thus pairs of
inde nite suces in an expression are resolved by being
summed over (or \traced" over).
Example
1.6.12 ajbj=3X
j=1ajbj=ab:
Next,
1.6.13e1= (1;0;0);e2= (0;1;0);e3= (0;0;1);
compare (1.6.5). Thus ajej=a.
Levi-Civita Symbol
1.6.14
jk`=8
><
>:+1;ifj;k;` is even permutation of 1 ;2;3;
1;ifj;k;` is odd permutation of 1 ;2;3;
0;otherwise:Examples
1.6.15123=312= 1; 213=321= 1; 221= 0:
1.6.16 jk``mn=j;mk;n j;nk;m;
wherej;kis the Kronecker delta.
1.6.17 ejek=jk`e`;
compare (1.6.8).
1.6.18 ajejbkek=jk`ajbke`;
compare (1.6.7){(1.6.8).
Lastly, the volume of a parallelepiped with vectors
a,b, and cas edges isjjk`ajbkc`j.
1.6(iii) Vector-Valued Functions
Del Operator
1.6.19r=i@
@x+j@
@y+k@
@z:
The gradient of a di erentiable scalar function
f(x;y;z ) is
1.6.20 gradf=rf=@f
@xi+@f
@yj+@f
@zk:
The divergence of a di erentiable vector-valued func-
tionF=F1i+F2j+F3kis
1.6.21 divF=rF=@F1
@x+@F2
@y+@F3
@z:
The curl ofFis
1.6.22curlF=rF= i j k
@
@x@
@y@
@z
F1F2F3
=@F3
@y @F2
@z
i+@F1
@z @F3
@x
j
+@F2
@x @F1
@y
k:
1.6.23r(fg) =frg+grf;
1.6.24r(f=g) = (grf frg)=g2;
1.6.25r(fF) =f(rF) +Frf;
1.6.26r(FG) =G(rF) F(rG);
1.6.27r(rF) = div curl F= 0;
1.6.28r(fF) =f(rF) + (rf)F;
1.6.29r(rf) = curl grad f= 0;
1.6.30 r2f=r(rf);
1.6.31r2(fg) =fr2g+gr2f+ 2(rfrg);
1.6.32 r(rfrg) = 0;
1.6.33r(frg grf) =fr2g gr2f;
1.6.34r(rF) = curl curl F=r(rF) r2F:
1.6 Vectors and Vector-Valued Functions 11
1.6(iv) Path and Line Integrals
Note: The terminology open andclosed sets andbound-
ary points in the (x;y) plane that is used in this sub-
section andx1.6(v) is analogous to that introduced for
the complex plane in x1.9(ii).
c(t) = (x(t);y(t);z(t)), withtranging over an inter-
val andx(t);y(t);z(t) di erentiable, de nes a path.
1.6.35 c0(t) = (x0(t);y0(t);z0(t)):
The length of a path for atbis
1.6.36Zb
akc0(t)kdt:
The path integral of a continuous function f(x;y;z ) is
1.6.37Z
cfds=Zb
af(x(t);y(t);z(t))kc0(t)kdt:
The line integral of a vector-valued function F=F1i+
F2j+F3kalong cis given by
1.6.38Z
cFds=Zb
aF(c(t))c0(t)dt
=Zb
a
F1dx
dt+F2dy
dt+F3dz
dt
dt
=Z
cF1dx+F2dy+F3dz:
A path c1(t),t2[a;b], is a reparametrization ofc(t0),
t02[a0;b0], ifc1(t) =c(t0) andt0=h(t) withh(t) di er-
entiable and monotonic. If h(a) =a0andh(b) =b0, then
the reparametrization is called orientation-preserving ,
and
1.6.39Z
cFds=Z
c1Fds:
Ifh(a) =b0andh(b) =a0, then the reparametrization
isorientation-reversing and
1.6.40Z
cFds= Z
c1Fds:
In either case
1.6.41Z
cfds=Z
c1fds;
whenfis continuous, and
1.6.42Z
crfds=f(c(b)) f(c(a));
whenfis continuously di erentiable.
The geometrical image Cof a path cis called a sim-
ple closed curve ifcis one-to-one, with the exception
c(a) =c(b). The curve Cispiecewise di erentiable ifc
is piecewise di erentiable. Note that Ccan be given an
orientation by means of c.Green's Theorem
Let
1.6.43 F(x;y) =F1(x;y)i+F2(x;y)j
andSbe the closed and bounded point set in the ( x;y)
plane having a simple closed curve Cas boundary. If C
is oriented in the positive (anticlockwise) sense, then
1.6.44ZZ
S@F2
@x @F1
@y
dA=Z
CFds=Z
CF1dx+F2dy:
Sucient conditions for this result to hold are that
F1(x;y) andF2(x;y) are continuously di erentiable on
S, andCis piecewise di erentiable.
The area of Scan be found from (1.6.44) by taking
F(x;y) = yi,xj, or 1
2yi+1
2xj.
1.6(v) Surfaces and Integrals over Surfaces
Aparametrized surface Sis de ned by
1.6.45 (u;v) = (x(u;v);y(u;v);z(u;v))
with (u;v)2D, an open set in the plane.
Forx,y, andzcontinuously di erentiable, the vec-
tors
1.6.46 Tu=@x
@u(u0;v0)i+@y
@u(u0;v0)j+@z
@u(u0;v0)k
and
1.6.47 Tv=@x
@v(u0;v0)i+@y
@v(u0;v0)j+@z
@v(u0;v0)k
are tangent to the surface at (u0;v0). The surface is
smooth at this point if TuTv6= 0. A surface is smooth
if it is smooth at every point. The vector TuTvat
(u0;v0) is normal to the surface at (u0;v0).
The areaA(S) of a parametrized smooth surface is
given by
1.6.48 A(S) =ZZ
DkTuTvkdudv;
and
1.6.49kTuTvk
=s@(x;y)
@(u;v)2
+@(y;z)
@(u;v)2
+@(x;z)
@(u;v)2
:
The area is independent of the parametrizations.
For a sphere x=sincos ,y=sinsin ,
z=cos,
1.6.50kTT k=2jsinj:
For a surface z=f(x;y),
1.6.51A(S) =ZZ
Ds
1 +@f
@x2
+@f
@y2
dA:
12 Algebraic and Analytic Methods
For a surface of revolution, y=f(x),x2[a;b],
about thex-axis,
1.6.52A(S) = 2Zb
ajf(x)jp
1 + (f0(x))2dx;
and about the y-axis,
1.6.53A(S) = 2Zb
ajxjp
1 + (f0(x))2dx:
The integral of a continuous function f(x;y;z ) over
a surfaceSis
1.6.54ZZ
Sf(x;y;z )dS=ZZ
Df((u;v))kTuTvkdudv:
For a vector-valued function F,
1.6.55ZZ
SFdS=ZZ
DF(TuTv)dudv;
wheredSis the surface element with an attached nor-
mal direction TuTv.
A surface is orientable if a continuously varying nor-
mal can be de ned at all points of the surface. An
orientable surface is oriented if suitable normals have
been chosen. A parametrization (u;v) of an oriented
surfaceSisorientation preserving ifTuTvhas the
same direction as the chosen normal at each point of S,
otherwise it is orientation reversing .
If1and2are both orientation preserving or both
orientation reversing parametrizations of Sde ned on
open setsD1andD2respectively, then
1.6.56ZZ
1(D1)FdS=ZZ
2(D2)FdS;
otherwise, one is the negative of the other.
Stokes's Theorem
SupposeSis an oriented surface with boundary @S
which is oriented so that its direction is clockwise rela-
tive to the normals of S. Then
1.6.57ZZ
S(rF)dS=Z
@SFds;
when Fis a continuously di erentiable vector-valued
function.
Gauss's (or Divergence) Theorem
SupposeSis a piecewise smooth surface which forms
the complete boundary of a bounded closed point set
V, andSis oriented by its normal being outwards from
V. Then
1.6.58ZZZ
V(rF)dV=ZZ
SFdS;
when Fis a continuously di erentiable vector-valued
function.Green's Theorem (for Volume)
Forfandgtwice-continuously di erentiable functions
1.6.59ZZZ
V(fr2g+rfrg)dV=ZZ
Sf@g
@ndA;
and
1.6.60ZZZ
V(fr2g gr2f)dV=ZZ
S
f@g
@n g@f
@n
dA;
where@g/@n=rgnis the derivative of gnormal to
the surface outwards from Vandnis the unit outer
normal vector.
1.7 Inequalities
1.7(i) Finite Sums
In this subsection AandBare positive constants.
Cauchy{Schwarz Inequality
1.7.10
@nX
j=1ajbj1
A2
0
@nX
j=1a2
j1
A0
@nX
j=1b2
j1
A:
Equality holds i aj=cbj,8j;c= constant.
Conversely, ifPn
j=1ajbj2
ABfor allbjsuch
thatPn
j=1b2
jB, thenPn
j=1a2
jA.
H older's Inequality
Forp>1,1
p+1
q= 1,aj0,bj0,
1.7.2nX
j=1ajbj0
@nX
j=1ap
j1
A1=p0
@nX
j=1bq
j1
A1=q
:
Equality holds i ap
j=cbq
j,8j;c= constant.
Conversely, ifPn
j=1ajbjA1=pB1=qfor allbjsuch
thatPn
j=1bq
jB, thenPn
j=1ap
jA.
Minkowski's Inequality
Forp>1,aj0,bj0,
1.7.30
@nX
j=1(aj+bj)p1
A1=p
0
@nX
j=1ap
j1
A1=p
+0
@nX
j=1bp
j1
A1=p
:
The direction of the inequality is reversed, that is, ,
when 0< p < 1. Equality holds i aj=cbj,8j;
c= constant.
1.7(ii) Integrals
In this subsection aandb(>a) are real constants that
can be1, provided that the corresponding integrals
converge. Also AandBare constants that are not si-
multaneously zero.
1.8 Fourier Series 13
Cauchy{Schwarz Inequality
1.7.4 Zb
af(x)g(x)dx!2
Zb
a(f(x))2dxZb
a(g(x))2dx:
Equality holds i Af(x) =Bg(x) for allx.
H older's Inequality
Forp>1,1
p+1
q= 1,f(x)0,g(x)0,
1.7.5Zb
af(x)g(x)dx
Zb
a(f(x))pdx!1=p Zb
a(g(x))qdx!1=q
:
Equality holds i A(f(x))p=B(g(x))qfor allx.
Minkowski's Inequality
Forp>1,f(x)0,g(x)0,
1.7.6 Zb
a(f(x) +g(x))pdx!1=p
Zb
a(f(x))pdx!1=p
+ Zb
a(g(x))pdx!1=p
:
The direction of the inequality is reversed, that is, ,
when 0<p< 1. Equality holds i Af(x) =Bg(x) for
allx.
1.7(iii) Means
For the notation, see x1.2(iv).
1.7.7 HGA;
with equality i a1=a2==an.
1.7.8 min(a1;a2;:::;an)M(r)max(a1;a2;:::;an);
with equality i a1=a2==an, orr<0 and some
aj= 0.
1.7.9 M(r)M(s), r<s;
with equality i a1=a2==an, ors0 and some
aj= 0.
1.7(iv) Jensen's Inequality
Forfintegrable on [0 ;1],a < f (x)< b, and convex
on (a;b) (x1.4(viii)),
1.7.10 Z1
0f(x)dx
Z1
0 (f(x))dx;
1.7.11 expZ1
0ln(f(x))dx
<Z1
0f(x)dx:
For exp and ln see x4.2.1.8 Fourier Series
1.8(i) De nitions and Elementary Properties
Formally,
1.8.1f(x) =1
2a0+1X
n=1(ancos(nx) +bnsin(nx));
1.8.2an=1
Z
f(x) cos(nx)dx,n= 0;1;2;:::,
bn=1
Z
f(x) sin(nx)dx,n= 1;2;:::.
The series (1.8.1) is called the Fourier series off(x),
andan;bnare the Fourier coecients off(x).
Iff( x) =f(x), thenbn= 0 for alln.
Iff( x) = f(x), thenan= 0 for alln.
Alternative Form
1.8.3 f(x) =1X
n= 1cneinx;
1.8.4 cn=1
2Z
f(x)e inxdx:
Bessel's Inequality
1.8.51
2a2
0+1X
n=1(a2
n+b2
n)1
Z
(f(x))2dx:
1.8.61X
n= 1jcnj21
2Z
jf(x)j2dx:
Asymptotic Estimates of Coecients
Iff(x) is of period 2 , andf(m)(x) is piecewise contin-
uous, then
1.8.7 an;bn;cn=o
n m
,n!1 .
Uniqueness of Fourier Series
Iff(x) andg(x) are continuous, have the same period
and same Fourier coecients, then f(x) =g(x) for all
x.
Lebesgue Constants
1.8.8 Ln=1
Z
0 sin
n+1
2
t
sin 1
2tdt,n= 0;1;:::.
Asn!1
1.8.9 Ln(4=2) lnn;
see Frenzen and Wong (1986).
14 Algebraic and Analytic Methods
Riemann{Lebesgue Lemma
Forf(x) piecewise continuous on [ a;b] and real,
1.8.10Zb
af(x)eixdx!0, as !1 .
(1.8.10) continues to apply if either aorbor both are
in nite and/or f(x) has nitely many singularities in
(a;b), provided that the integral converges uniformly
(x1.5(iv)) ata;b, and the singularities for all suciently
large.
1.8(ii) Convergence
Letf(x) be an absolutely integrable function of period
2, and continuous except at a nite number of points in
any bounded interval. Then the series (1.8.1) converges
to the sum
1.8.111
2f(x ) +1
2f(x+)
at every point at which f(x) has both a left-hand deriva-
tive (that is, (1.4.4) applies when h!0 ) and a right-
hand derivative (that is, (1.4.4) applies when h!0+).
The convergence is non-uniform, however, at points
wheref(x )6=f(x+); seex6.16(i).
For other tests for convergence see Titchmarsh
(1962, pp. 405{410).
1.8(iii) Integration and Di erentiation
Ifanandbnare the Fourier coecients of a piecewise
continuous function f(x) on [0;2], then
1.8.12Zx
0(f(t) 1
2a0)dt=1X
n=1ansin(nx) +bn(1 cos(nx))
n,
0x2.
If a function f(x)2C2[0;2] is periodic, with pe-
riod 2, then the series obtained by di erentiating the
Fourier series for f(x) term by term converges at every
point tof0(x).
1.8(iv) Transformations
Parseval's Formula
1.8.131
Z
f(x)g(x)dx=1
2a0a0
0+1X
n=1(ana0
n+bnb0
n);
whenf(x) andg(x) are square-integrable and an;bnand
a0
n;b0
nare their respective Fourier coecients.Poisson's Summation Formula
Suppose that f(x) is twice continuously di erentiable
andf(x) andjf00(x)jare integrable over ( 1;1).
Then
1.8.14
1X
n= 1f(x+n) =1X
n= 1e2inxZ1
1f(t)e 2intdt:
An alternative formulation is as follows. Suppose
thatf(x) is continuous and of bounded variation on
[0;1). Suppose also that f(x) is integrable on [0 ;1)
andf(x)!0 asx!1 . Then
1.8.15
1
2f(0) +1X
n=1f(n) =Z1
0f(x)dx
+ 21X
n=1Z1
0f(x) cos(2nx)dx:
As a special case
1.8.161X
n= 1e (n+x)2!
=r
!
1 + 21X
n=1e n22=!cos(2nx)!
,
<!>0.
1.8(v) Examples
For collections of Fourier-series expansions see Prud-
nikov et al. (1986a, v. 1, pp. 725{740), Gradshteyn and
Ryzhik (2000, pp. 45{49), and Oberhettinger (1973).
1.9 Calculus of a Complex Variable
1.9(i) Complex Numbers
1.9.1 z=x+iy, x;y2R:
Real and Imaginary Parts
1.9.2 <z=x;=z=y:
Polar Representation
1.9.3 x=rcos; y =rsin;
where
1.9.4 r= (x2+y2)1=2;
and whenz6= 0,
1.9.5=!; !; +!;or !;
according as zlies in the 1st, 2nd, 3rd, or 4th quadrants.
Here
1.9.6 != arctan(jy=xj)2
0;1
2
:
1.9 Calculus of a Complex Variable 15
Modulus and Phase
1.9.7jzj=r;phz=+ 2n,n2Z.
The principal value of phzcorresponds to n= 0, that
is, phz. It is single-valued on Cnf0g, except
on the interval ( 1;0) where it is discontinuous and
two-valued. Unless indicated otherwise , these principal
values are assumed throughout this Handbook. (How-
ever, if we require a principal value to be single-valued,
then we can restrict <phz.)
1.9.8j<zjjzj;j=zjjzj;
1.9.9 z=rei;
where
1.9.10 ei= cos+isin;
seex4.14.
Complex Conjugate
1.9.11 z=x iy;
1.9.12 jzj=jzj;
1.9.13 phz= phz:
Arithmetic Operations
Ifz1=x1+iy1,z2=x2+iy2, then
1.9.14 z1z2=x1x2+i(y1y2);
1.9.15z1z2=x1x2 y1y2+i(x1y2+x2y1);
1.9.16z1
z2=z1z2
jz2j2=x1x2+y1y2+i(x2y1 x1y2)
x2
2+y2
2;
provided that z26= 0. Also,
1.9.17 jz1z2j=jz1jjz2j;
1.9.18 ph(z1z2) = phz1+ phz2;
1.9.19 z1
z2 =jz1j
jz2j;
1.9.20 phz1
z2= phz1 phz2:
Equations (1.9.18) and (1.9.20) hold for general values
of the phases, but not necessarily for the principal val-
ues.
Powers
1.9.21zn=
xn n
2
xn 2y2+n
4
xn 4y4
+in
1
xn 1y n
3
xn 3y3+
,
n= 1;2;:::.
DeMoivre's Theorem
1.9.22 cosn+isinn= (cos+isin)n,n2Z.Triangle Inequality
1.9.23jjz1j jz2jjjz1+z2jjz1j+jz2j:
1.9(ii) Continuity, Point Sets, and
Di erentiation
Continuity
A function f(z) is continuous at a point z0if
lim
z!z0f(z) =f(z0). That is, given any positive num-
ber, however small, we can nd a positive number
such thatjf(z) f(z0)j< for allzin the open disk
jz z0j<.
A function of two complex variables f(z;w) iscon-
tinuous at (z0;w0) if lim
(z;w)!(z0;w0)f(z;w) =f(z0;w0);
compare (1.5.1) and (1.5.2).
Point Sets in C
Aneighborhood of a point z0is a diskjz z0j<. An
open set inCis one in which each point has a neighbor-
hood that is contained in the set.
A pointz0is a limit point (limiting point oraccu-
mulation point ) of a set of points SinC(orC[1)
if every neighborhood of z0contains a point of Sdis-
tinct from z0. (z0may or may not belong to S.) As
a consequence, every neighborhood of a limit point of
Scontains an in nite number of points of S. Also, the
union ofSand its limit points is the closure ofS.
AdomainD, say, is an open set in Cthat is con-
nected , that is, any two points can be joined by a polyg-
onal arc (a nite chain of straight-line segments) lying
in the set. Any point whose neighborhoods always con-
tain members and nonmembers of Dis aboundary point
ofD. When its boundary points are added the domain
is said to be closed , but unless speci ed otherwise a do-
main is assumed to be open.
Aregion is an open domain together with none,
some, or all of its boundary points. Points of a region
that are not boundary points are called interior points .
A function f(z) iscontinuous on a region Rif for
each pointz0inRand any given number (>0) we can
nd a neighborhood of z0such thatjf(z) f(z0)j<
for all points zin the intersection of the neighborhood
withR.
Di erentiation
A function f(z) isdi erentiable at a pointzif the fol-
lowing limit exists:
1.9.24f0(z) =df
dz= lim
h!0f(z+h) f(z)
h:
Di erentiability automatically implies continuity.
16 Algebraic and Analytic Methods
Cauchy{Riemann Equations
Iff0(z) exists atz=x+iyandf(z) =u(x;y)+iv(x;y),
then
1.9.25@u
@x=@v
@y;@u
@y= @v
@x
at (x;y).
Conversely, if at a given point ( x;y) the partial
derivatives @u/@x,@u/@y,@v/@x, and@v/@yexist,
are continuous, and satisfy (1.9.25), then f(z) is di er-
entiable at z=x+iy.
Analyticity
A function f(z) is said to be analytic (holomorphic ) at
z=z0if it is di erentiable in a neighborhood of z0.
A function f(z) isanalytic in a domain Dif it is an-
alytic at each point of D. A function analytic at every
point of Cis said to be entire .
Iff(z) is analytic in an open domain D, then each
of its derivatives f0(z),f00(z),:::exists and is analytic
inD.
Harmonic Functions
Iff(z) =u(x;y)+iv(x;y) is analytic in an open domain
D, thenuandvareharmonic inD, that is,
1.9.26@2u
@x2+@2u
@y2=@2v
@x2+@2v
@y2= 0;
or in polar form ((1.9.3)) uandvsatisfy
1.9.27@2u
@r2+1
r@u
@r+1
r2@2u
@2= 0
at all points of D.
1.9(iii) Integration
AnarcCis given by z(t) =x(t) +iy(t),atb,
wherexandyare continuously di erentiable. If x(t)
andy(t) are continuous and x0(t) andy0(t) are piece-
wise continuous, then z(t) de nes a contour .
A contour is simple if it contains no multiple points,
that is, for every pair of distinct values t1;t2oft,
z(t1)6=z(t2). A simple closed contour is a simple con-
tour, except that z(a) =z(b).
Next,
1.9.28Z
Cf(z)dz=Zb
af(z(t))(x0(t) +iy0(t))dt;
for a contour Candf(z(t)) continuous, atb. If
f(z(t0)) =1,at0b, then the integral is de ned
analogously to the in nite integrals in x1.4(v). Similarly
whena= 1 orb= +1.
Jordan Curve Theorem
Any simple closed contour Cdivides Cinto two open do-
mains that have Cas common boundary. One of these
domains is bounded and is called the interior domain
ofC; the other is unbounded and is called the exterior
domain of C.Cauchy's Theorem
Iff(z) is continuous within and on a simple closed con-
tourCand analytic within C, then
1.9.29Z
Cf(z)dz= 0:
Cauchy's Integral Formula
Iff(z) is continuous within and on a simple closed con-
tourCand analytic within C, and ifz0is a point within
C, then
1.9.30 f(z0) =1
2iZ
Cf(z)
z z0dz;
and
1.9.31
f(n)(z0) =n!
2iZ
Cf(z)
(z z0)n+1dz,n= 1;2;3;:::,
provided that in both cases Cis described in the posi-
tive rotational (anticlockwise) sense.
Liouville's Theorem
Any bounded entire function is a constant.
Winding Number
IfCis a closed contour, and z062C, then
1.9.321
2iZ
C1
z z0dz=N(C;z0);
whereN(C;z0) is an integer called the winding num-
ber ofCwith respect to z0. IfCis simple and oriented
in the positive rotational sense, then N(C;z0) is 1 or 0
depending whether z0is inside or outside C.
Mean Value Property
Foru(z) harmonic,
1.9.33 u(z) =1
2Z2
0u(z+rei )d :
Poisson Integral
Ifh(w) is continuous on jwj=R, then with z=rei
1.9.34u(rei) =1
2Z2
0(R2 r2)h(Rei )d
R2 2Rrcos( ) +r2
is harmonic injzj<R. Also withjwj=R, lim
z!wu(z) =
h(w) asz!wwithinjzj<R.
1.9(iv) Conformal Mapping
The extended complex plane ,C[f1g , consists of the
points of the complex plane Ctogether with an ideal
point1called the point at in nity . A system of open
disks around in nity is given by
1.9.35 Sr=fzjjzj>1=rg[f1g , 0<r<1.
EachSris aneighborhood of1. Also,
1.9.361z=z1 =1;
1.9 Calculus of a Complex Variable 17
1.9.37 1z=z1=1, z6= 0,
1.9.38 z=1= 0;
1.9.39 z=0 =1, z6= 0.
A function f(z) is analytic at1ifg(z) =f(1=z) is
analytic at z= 0, and we set f0(1) =g0(0).
Conformal Transformation
Supposef(z) is analytic in a domain DandC1;C2are
two arcs in Dpassing through z0. LetC0
1;C0
2be the
images ofC1andC2under the mapping w=f(z). The
angle between C1andC2atz0is the angle between the
tangents to the two arcs at z0, that is, the di erence of
the signed angles that the tangents make with the pos-
itive direction of the real axis. If f0(z0)6= 0, then the
angle between C1andC2equals the angle between C0
1
andC0
2both in magnitude and sense. We then say that
the mapping w=f(z) isconformal (angle-preserving)
atz0.
The linear transformation f(z) =az+b,a6= 0, has
f0(z) =aandw=f(z) maps Cconformally onto C.
Bilinear Transformation
1.9.40 w=f(z) =az+b
cz+d,ad bc6= 0,c6= 0.
1.9.41 f( d=c) =1; f(1) =a=c:
1.9.42 f0(z) =ad bc
(cz+d)2,z6= d=c.
1.9.43 f0(1) =bc ad
c2:
1.9.44 z=dw b
cw+a:
The transformation (1.9.40) is a one-to-one confor-
mal mapping of C[f1g onto itself.
The cross ratio ofz1;z2;z3;z42C[f1g is de ned
by
1.9.45(z1 z2)(z3 z4)
(z1 z4)(z3 z2);
or its limiting form, and is invariant under bilinear
transformations.
Other names for the bilinear transformation are frac-
tional linear transformation ,homographic transforma-
tion, and M obius transformation .
1.9(v) In nite Sequences and Series
A sequencefzngconverges tozif lim
n!1zn=z. For
zn=xn+iyn, the sequencefzngconverges i the se-
quencesfxngandfyngseparately converge. A seriesP1
n=0znconverges if the sequence sn=Pn
k=0zkcon-
verges. The series is divergent ifsndoes not converge.
The series converges absolutely ifP1
n=0jznjconverges.A seriesP1
n=0znconverges (diverges) absolutely when
lim
n!1jznj1=n<1 (>1), or when lim
n!1jzn+1/znj<1
(>1). Absolutely convergent series are also convergent.
Letffn(z)gbe a sequence of functions de ned on a
setS. This sequence converges pointwise to a function
f(z) if
1.9.46 f(z) = lim
n!1fn(z)
for eachz2S. The sequence converges uniformly onS,
if for every >0 there exists an integer N, independent
ofz, such that
1.9.47 jfn(z) f(z)j<
for allz2SandnN.
A seriesP1
n=0fn(z)converges uniformly onS, if the
sequencesn(z) =Pn
k=0fk(z) converges uniformly on S.
WeierstrassM-test
SupposefMngis a sequence of real numbers such thatP1
n=0Mnconverges andjfn(z)jMnfor allz2S
and alln0. Then the seriesP1
n=0fn(z) converges
uniformly on S.
A doubly-in nite seriesP1
n= 1fn(z) converges
(uniformly) on Si each of the seriesP1
n=0fn(z) andP1
n=1f n(z) converges (uniformly) on S.
1.9(vi) Power Series
For a seriesP1
n=0an(z z0)nthere is a number R,
0R1 , such that the series converges for all zin
jz z0j< R and diverges for zinjz z0j> R. The
circlejz z0j=Ris called the circle of convergence of
the series, and Ris the radius of convergence . Inside
the circle the sum of the series is an analytic function
f(z). Forzinjz z0j(< R), the convergence is
absolute and uniform. Moreover,
1.9.48 an=f(n)(z0)
n!;
and
1.9.49 R= lim inf
n!1janj 1=n:
For the converse of this result see x1.10(i).
Operations
WhenPanznandPbnznboth converge
1.9.501X
n=0(anbn)zn=1X
n=0anzn1X
n=0bnzn;
and
1.9.51 1X
n=0anzn! 1X
n=0bnzn!
=1X
n=0cnzn;
where
1.9.52 cn=nX
k=0akbn k:
Next, let
1.9.53 f(z) =a0+a1z+a2z2+,a06= 0.
18 Algebraic and Analytic Methods
Then the expansions (1.9.54), (1.9.57), and (1.9.60) hold
for all suciently small jzj.
1.9.541
f(z)=b0+b1z+b2z2+;
where
1.9.55b0= 1=a0; b 1= a1=a2
0; b 2= (a2
1 a0a2)=a3
0;
1.9.56
bn= (a1bn 1+a2bn 2++anb0)=a0,n1.
Witha0= 1,
1.9.57 lnf(z) =q1z+q2z2+q3z3+;
(principal value), where
1.9.58q1=a1; q 2= (2a2 a2
1)=2;
q3= (3a3 3a1a2+a3
1)=3;
and
1.9.59
qn= (nan (n 1)a1qn 1 (n 2)a2qn 2
an 1q1)=n,
n2.
Also,
1.9.60 (f(z))=p0+p1z+p2z2+;
(principal value), where 2C,
1.9.61p0= 1; p 1=a1; p 2=(( 1)a2
1+ 2a2)=2;
and
1.9.62
pn= (( n+ 1)a1pn 1+ (2 n+ 2)a2pn 2+
+ ((n 1) 1)an 1p1+nan)=n,
n1.
For the de nitions of the principal values of ln f(z) and
(f(z))seexx4.2(i) and 4.2(iv).
Lastly, a power series can be di erentiated any num-
ber of times within its circle of convergence:
1.9.63f(m)(z) =1X
n=0(n+ 1)man+m(z z0)n,
jz z0j<R,m= 0;1;2;:::.
1.9(vii) Inversion of Limits
Double Sequences and Series
A set of complex numbers fzm;ngwheremandntake
all positive integer values is called a double sequence . It
converges to zif for every >0, there is an integer N
such that
1.9.64 jzm;n zj<
for allm;nN. Supposefzm;ngconverges to zand
the repeated limits
1.9.65 lim
m!1
lim
n!1zm;n
;lim
n!1
lim
m!1zm;nexist. Then both repeated limits equal z.
Adouble series is the limit of the double sequence
1.9.66 zp;q=pX
m=0qX
n=0m;n:
If the limit exists, then the double series is convergent ;
otherwise it is divergent . The double series is absolutely
convergent if it is convergent when m;nis replaced by
jm;nj.
If a double series is absolutely convergent, then it
is also convergent and its sum is given by either of the
repeated sums
1.9.671X
m=0 1X
n=0m;n!
;1X
n=0 1X
m=0m;n!
:
Term-by-Term Integration
Suppose the seriesP1
n=0fn(z), wherefn(z) is contin-
uous, converges uniformly on every compact set of a
domainD, that is, every closed and bounded set in D.
Then
1.9.68Z
C1X
n=0fn(z)dz=1X
n=0Z
Cfn(z)dz
for any nite contour CinD.
Dominated Convergence Theorem
Let (a;b) be a nite or in nite interval, and
f0(t);f1(t);:::be real or complex continuous functions,
t2(a;b). SupposeP1
n=0fn(t) converges uniformly in
any compact interval in ( a;b), and at least one of the
following two conditions is satis ed:
1.9.69Zb
a1X
n=0jfn(t)jdt<1;
1.9.701X
n=0Zb
ajfn(t)jdt<1:
Then
1.9.71Zb
a1X
n=0fn(t)dt=1X
n=0Zb
afn(t)dt:
1.10 Functions of a Complex Variable
1.10(i) Taylor's Theorem for Complex Variables
Letf(z) be analytic on the disk jz z0j<R. Then
1.10.1 f(z) =1X
n=0f(n)(z0)
n!(z z0)n:
The right-hand side is the Taylor series for f(z)at
z=z0, and its radius of convergence is at least R.
1.10 Functions of a Complex Variable 19
Examples
1.10.2 ez= 1 +z
1!+z2
2!+,jzj<1,
1.10.3 ln(1 +z) =z z2
2+z3
3 ,jzj<1,
1.10.4
(1 z) = 1 + z+ ( + 1)
2!z2+ ( + 1)( + 2)
3!z3
+, jzj<1.
Again, in these examples ln(1 + z) and (1 z) have
their principal values; see xx4.2(i) and 4.2(iv).
Zeros
An analytic function f(z) has a zero of order (ormul-
tiplicity )m(1) atz0if the rst nonzero coecient in
its Taylor series at z0is that of (z z0)m. Whenm= 1
the zero is simple .
1.10(ii) Analytic Continuation
Letf1(z) be analytic in a domain D1. Iff2(z), analytic
inD2, equalsf1(z) on an arc in D=D1\D2, or on
just an in nite number of points with a limit point in
D, then they are equal throughout Dandf2(z) is called
ananalytic continuation off1(z). We write ( f1;D1),
(f2;D2) to signify this continuation.
Supposez(t) =x(t) +iy(t),atb, is an arc and
a=t0< t1<< tn=b. Suppose the subarc z(t),
t2[tj 1;tj] is contained in a domain Dj,j= 1;:::;n .
The function f1(z) onD1is said to be analytically con-
tinued along the path z(t),atb, if there is a chain
(f1;D1), (f2;D2);:::; (fn;Dn).
Analytic continuation is a powerful aid in establish-
ing transformations or functional equations for complex
variables, because it enables the problem to be reduced
to: (a) deriving the transformation (or functional equa-
tion) with real variables; followed by (b) nding the
domain on which the transformed function is analytic.
Schwarz Re
ection Principle
LetCbe a simple closed contour consisting of a seg-
ment ABof the real axis and a contour in the upper
half-plane joining the ends of AB. Also, letf(z) be an-
alytic within C, continuous within and on C, and real
onAB. Thenf(z) can be continued analytically across
ABbyre
ection , that is,
1.10.5 f(z) =f(z):
1.10(iii) Laurent Series
Supposef(z) is analytic in the annulusr1<jz z0j<
r2, 0r1<r21, andr2(r1;r2). Then
1.10.6 f(z) =1X
n= 1an(z z0)n;where
1.10.7an=1
2iZ
jz z0j=rf(z)
(z z0)n+1dz;
and the integration contour is described once in the pos-
itive sense. The series (1.10.6) converges uniformly and
absolutely on compact sets in the annulus.
Letr1= 0, so that the annulus becomes the punc-
tured neighborhood N: 0<jz z0j< r2, and assume
thatf(z) is analytic in N, but not at z0. Thenz=z0
is an isolated singularity off(z). This singularity is re-
movable ifan= 0 for all n < 0, and in this case the
Laurent series becomes the Taylor series. Next, z0is a
pole ifan6= 0 for at least one, but only nitely many,
negativen. If nis the rst negative integer (counting
from 1) witha n6= 0, thenz0is apole of order (or
multiplicity )n. Lastly, if an6= 0 for in nitely many
negativen, thenz0is an isolated essential singularity .
The singularities of f(z) at in nity are classi ed in
the same way as the singularities of f(1=z) atz= 0.
An isolated singularity z0is always removable when
limz!z0f(z) exists, for example (sin z)=zatz= 0.
The coecient a 1of (z z0) 1in the Laurent series
forf(z) is called the residue off(z) atz0, and denoted
by resz=z0[f(z)], res
z=z0[f(z)], or (when there is no ambi-
guity) res[f(z)].
A function whose only singularities, other than the
point at in nity, are poles is called a meromorphic func-
tion. If the poles are in nite in number, then the point
at in nity is called an essential singularity : it is the
limit point of the poles.
Picard's Theorem
In any neighborhood of an isolated essential singularity,
however small, an analytic function assumes every value
inCwith at most one exception.
1.10(iv) Residue Theorem
Iff(z) is analytic within a simple closed contour C, and
continuous within and on C|except in both instances
for a nite number of singularities within C|then
1.10.8
1
2iZ
Cf(z)dz= sum of the residues of f(z) withinC:
Here and elsewhere in this subsection the path Cis de-
scribed in the positive sense.
20 Algebraic and Analytic Methods
Phase (or Argument) Principle
If the singularities within Care poles and f(z) is ana-
lytic and nonvanishing on C, then
1.10.9N P=1
2iZ
Cf0(z)
f(z)dz=1
2C(phf(z));
whereNandPare respectively the numbers of zeros
and poles, counting multiplicity, of fwithinC, and
C(phf(z)) is the change in any continuous branch of
ph(f(z)) aszpasses once around Cin the positive sense.
For examples of applications see Olver (1997b, pp. 252{
254).
In addition,
1.10.10
1
2iZ
Czf0(z)
f(z)dz= (sum of locations of zeros)
(sum of locations of poles) ;
each location again being counted with multiplicity
equal to that of the corresponding zero or pole.
Rouch e's Theorem
Iff(z) andg(z) are analytic on and inside a simple
closed contour C, andjg(z)j<jf(z)jonC, thenf(z)
andf(z) +g(z) have the same number of zeros inside
C.
1.10(v) Maximum-Modulus Principle
Analytic Functions
Iff(z) is analytic in a domain D,z02Dandjf(z)j
jf(z0)jfor allz2D, thenf(z) is a constant in D.
LetDbe a bounded domain with boundary @Dand
letD=D[@D. Iff(z) is continuous on Dand analytic
inD, thenjf(z)jattains its maximum on @D.
Harmonic Functions
Ifu(z) is harmonic in D,z02D, andu(z)u(z0) for
allz2D, thenu(z) is constant in D. Moreover, if Dis
bounded and u(z) is continuous on Dand harmonic in
D, thenu(z) is maximum at some point on @D.
Schwarz's Lemma
Injzj<R, iff(z) is analytic,jf(z)jM, andf(0) = 0,
then
1.10.11jf(z)jMjzj
Randjf0(0)jM
R:
Equalities hold i f(z) =Az, whereAis a constant
such thatjAj=M=R .
1.10(vi) Multivalued Functions
Functions which have more than one value at a given
pointzare called multivalued (ormany-valued ) func-
tions. LetF(z) be a multivalued function and Dbe a
domain. If we can assign a unique value f(z) toF(z) at
each point of D, andf(z) is analytic on D, thenf(z) is
abranch ofF(z).Example
F(z) =pzis two-valued for z6= 0. IfD=Cn( 1;0]
andz=rei, then one branch isprei=2, the other
branch is prei=2, with < < in both cases.
Similarly if D=Cn[0;1), then one branch isprei=2,
the other branch is prei=2, with 0<< 2in both
cases.
Acut domain is one from which the points on nitely
many nonintersecting simple contours ( x1.9(iii)) have
been removed. Each contour is called a cut. Acut neigh-
borhood is formed by deleting a ray emanating from the
center. (Or more generally, a simple contour that starts
at the center and terminates on the boundary.)
SupposeF(z) is multivalued and ais a point such
that there exists a branch of F(z) in a cut neighborhood
ofa, but there does not exist a branch of F(z) in any
punctured neighborhood of a. Thenais abranch point
ofF(z). For example, z= 0 is a branch point ofpz.
Branches can be constructed in two ways:
(a) By introducing appropriate cuts from the branch
points and restricting F(z) to be single-valued in the cut
plane (or domain).
(b) By specifying the value of F(z) at a point z0(not
a branch point), and requiring F(z) to be continuous on
any path that begins at z0and does not pass through
any branch points or other singularities of F(z).
If the path circles a branch point at z=a ktimes
in the positive sense, and returns to z0without encir-
cling any other branch point, then its value is denoted
conventionally as F((z0 a)e2ki+a).
Example
Let and be real or complex numbers that are not
integers. The function F(z) = (1 z) (1+z) is many-
valued with branch points at 1. Branches of F(z) can
be de ned, for example, in the cut plane Dobtained
fromCby removing the real axis from 1 to 1and from
1 to 1; see Figure 1.10.1. One such branch is ob-
tained by assigning (1 z) and (1 +z) their principal
values (x4.2(iv)).
Figure 1.10.1 : DomainD.
Alternatively, take z0to be any point in Dand set
F(z0) =e ln(1 z0)e ln(1+z0)where the logarithms as-
sume their principal values. (Thus if z0is in the in-
terval ( 1;1), then the logarithms are real.) Then the
value ofF(z) at any other point is obtained by analytic
continuation.
1.10 Functions of a Complex Variable 21
Thus ifF(z) is continued along a path that circles
z= 1mtimes in the positive sense and returns to
z0without circling z= 1, thenF((z0 1)e2mi+
1) =e ln(1 z0)e ln(1+z0)e2im . If the path also circles
z= 1ntimes in the clockwise or negative sense be-
fore returning to z0, then the value of F(z0) becomes
e ln(1 z0)e ln(1+z0)e2im e 2in .
1.10(vii) Inverse Functions
Lagrange Inversion Theorem
Supposef(z) is analytic at z=z0,f0(z0)6= 0, and
f(z0) =w0. Then the equation
1.10.12 f(z) =w
has a unique solution z=F(w) analytic at w=w0, and
1.10.13 F(w) =z0+1X
n=1Fn(w w0)n
in a neighborhood of w0, wherenFnis the residue of
1=(f(z) f(z0))natz=z0. (In other words nFnis
the coecient of ( z z0) 1in the Laurent expansion
of 1=(f(z) f(z0))nin powers of ( z z0); compare
x1.10(iii).)
Furthermore, if g(z) is analytic at z0, then
1.10.14g(F(w)) =g(z0) +1X
n=1Gn(w w0)n;
wherenGnis the residue of g0(z)=(f(z) f(z0))nat
z=z0.
Extended Inversion Theorem
Suppose that
1.10.15f(z) =f(z0) +1X
n=0fn(z z0)+n;
where > 0,f06= 0, and the series converges in a
neighborhood of z0. (For example, when is an integer
f(z) f(z0) has a zero of order atz0.) Letw0=f(z0).
Then (1.10.12) has a solution z=F(w), where
1.10.16 F(w) =z0+1X
n=1Fn(w w0)n=
in a neighborhood of w0,nFnbeing the residue of
1=(f(z) f(z0))n=atz=z0.
It should be noted that di erent branches of ( w
w0)1=used in forming ( w w0)n=in (1.10.16) give rise
to di erent solutions of (1.10.12). Also, if in addition
g(z) is analytic at z0, then
1.10.17g(F(w)) =g(z0) +1X
n=1Gn(w w0)n=;
wherenGnis the residue of g0(z)=(f(z) f(z0))n=at
z=z0.1.10(viii) Functions De ned by Contour
Integrals
LetDbe a domain and [ a;b] be a closed nite segment
of the real axis. Assume that for each t2[a;b],f(z;t)
is an analytic function of zinD, and also that f(z;t)
is a continuous function of both variables. Then
1.10.18 F(z) =Zb
af(z;t)dt
is analytic in Dand its derivatives of all orders can be
found by di erentiating under the sign of integration.
This result is also true when b=1, or whenf(z;t)
has a singularity at t=b, with the following conditions.
For eacht2[a;b),f(z;t) is analytic in D;f(z;t) is a
continuous function of both variables when z2Dand
t2[a;b); the integral (1.10.18) converges at b, and this
convergence is uniform with respect to zin every com-
pact subset SofD.
The last condition means that given (>0) there
exists a number a02[a;b) that is independent of zand
is such that
1.10.19 Zb
a1f(z;t)dt <;
for alla12[a0;b) and allz2S; comparex1.5(iv).
M-test
Ifjf(z;t)jM(t) forz2SandRb
aM(t)dtconverges,
then the integral (1.10.18) converges uniformly and ab-
solutely inS.
1.10(ix) In nite Products
Letpk;m=Qm
n=k(1 +an). If for some k1,pk;m!
pk6= 0 asm!1 , then we say that the in nite prod-
uctQ1
n=1(1 +an)converges . (The integer kmay be
greater than one to allow for a nite number of zero
factors.) The convergence of the product is absolute ifQ1
n=1(1 +janj) converges. The productQ1
n=1(1 +an),
withan6= 1 for alln, converges i P1
n=1ln(1 +an)
converges; and it converges absolutely i P1
n=1janjcon-
verges.
Supposean=an(z),z2D, a domain. The conver-
gence of the in nite product is uniform if the sequence
of partial products converges uniformly.
M-test
Suppose that an(z) are analytic functions in D. If there
is anN, independent of z2D, such that
1.10.20jln(1 +an(z))jMn,nN,
and
1.10.211X
n=1Mn<1;
then the productQ1
n=1(1 +an(z)) converges uniformly
to an analytic function p(z) inD, andp(z) = 0 only
22 Algebraic and Analytic Methods
when at least one of the factors 1 + an(z) is zero in D.
This conclusion remains true if, in place of (1.10.20),
jan(z)jMnfor alln, and againP1
n=1Mn<1.
Weierstrass Product
Iffzngis a sequence such thatP1
n=1jz 2
njis convergent,
then
1.10.22 P(z) =1Y
n=1
1 z
zn
ez=zn
is an entire function with zeros at zn.
1.10(x) In nite Partial Fractions
SupposeDis a domain, and
1.10.23 F(z) =1Y
n=1an(z), z2D,
wherean(z) is analytic for all n1, and the conver-
gence of the product is uniform in any compact subset
ofD. ThenF(z) is analytic in D.
If, also,an(z)6= 0 when n1 andz2D, then
F(z)6= 0 onDand
1.10.24F0(z)
F(z)=1X
n=1a0
n(z)
an(z):
Mittag-Leer's Expansion
Iffangandfzngare sequences such that zm6=zn
(m6=n) andP1
n=1janz 2
njis convergent, then
1.10.25 f(z) =1X
n=1an1
z zn+1
zn
is analytic in C, except for simple poles at z=znof
residuean.
1.11 Zeros of Polynomials
1.11(i) Division Algorithm
Horner's Scheme
Let
1.11.1f(z) =anzn+an 1zn 1++a0:
Then
1.11.2f(z) = (z )(bnzn 1+bn 1zn 2++b1)+b0;
wherebn=an,
1.11.3 bk= bk+1+ak,k=n 1;n 2;:::; 0,
1.11.4 f( ) =b0:Extended Horner Scheme
Withbkas in (1.11.1){(1.11.3) let cn=anand
1.11.5 ck= ck+1+bk,k=n 1;n 2;:::; 1.
Then
1.11.6 f0( ) =c1:
More generally, for polynomials f(z) andg(z), there
are polynomials q(z) andr(z), found by equating coef-
cients, such that
1.11.7 f(z) =g(z)q(z) +r(z);
where 0degr(z)<degg(z).
1.11(ii) Elementary Properties
A polynomial of degree nwith real or complex coef-
cients has exactly nreal or complex zeros counting
multiplicity. Every monic (coecient of highest power
is one) polynomial of odd degree with real coecients
has at least one real zero with sign opposite to that of
the constant term. A monic polynomial of even degree
with real coecients has at least two zeros of opposite
signs when the constant term is negative.
Descartes' Rule of Signs
The number of positive zeros of a polynomial with real
coecients cannot exceed the number of times the co-
ecients change sign, and the two numbers have same
parity. A similar relation holds for the changes in sign
of the coecients of f( z), and hence for the number
of negative zeros of f(z).
Example
1.11.8f(z) =z8+ 10z3+z 4;
f( z) =z8 10z3 z 4:
Both polynomials have one change of sign; hence for
each polynomial there is one positive zero, one negative
zero, and six complex zeros.
Next, letf(z) =anzn+an 1zn 1++a0. The
zeros ofznf(1=z) =a0zn+a1zn 1++anare recip-
rocals of the zeros of f(z).
The discriminant off(z) is de ned by
1.11.9 D=a2n 2
nY
j<k(zj zk)2;
wherez1;z2;:::;znare the zeros of f(z). The elemen-
tary symmetric functions of the zeros are (with an6= 0)
1.11.10z1+z2++zn= an 1=an;
X
1j<knzjzk=an 2=an;
...
z1z2zn= ( 1)na0=an:
1.11 Zeros of Polynomials 23
1.11(iii) Polynomials of Degrees Two, Three,
and Four
Quadratic Equations
The roots of az2+bz+c= 0 are
1.11.11 bp
D
2a; D =b2 4ac:
The sum and product of the roots are respectively b=a
andc=a.
Cubic Equations
Setz=w 1
3ato reducef(z) =z3+az2+bz+c
tog(w) =w3+pw+q, withp= (3b a2)=3,q=
(2a3 9ab+ 27c)=27. The discriminant of g(w) is
1.11.12 D= 4p3 27q2:
Let
1.11.13A=3q
27
2q+3
2p
3D; B = 3p=A:
The roots of g(w) = 0 are
1.11.141
3(A+B);1
3(A+2B);1
3(2A+B);
with
1.11.15= 1
2+1
2p
3 =e2i=3; 2=e 2i=3:
Addition of 1
3ato each of these roots gives the roots
off(z) = 0.
Example
f(z) =z3 6z2+6z 2,g(w) =w3 6w 6,A= 33p
4,
B= 33p
2. Roots of f(z) = 0 are 2 +3p
4 +3p
2,
2 +3p
4+3p
22, 2 +3p
42+3p
2.
For another method see x4.43.
Quartic Equations
Setz=w 1
4ato reducef(z) =z4+az3+bz2+cz+d
to
1.11.16g(w) =w4+pw2+qw+r;
p= ( 3a2+ 8b)=8; q = (a3 4ab+ 8c)=8;
r= ( 3a4+ 16a2b 64ac+ 256d)=256:
The discriminant of g(w) is
1.11.17
D= 16p4r 4p3q2 128p2r2+ 144pq2r 27q4+ 256r3:
For the roots 1; 2; 3; 4ofg(w) = 0 and the roots
1;2;3of the resolvent cubic equation
1.11.18 z3 2pz2+ (p2 4r)z+q2= 0;
we have
1.11.192 1=p
1+p
2+p
3;
2 2=p
1 p
2 p
3;
2 3= p
1+p
2 p
3;
2 4= p
1 p
2+p
3:
The square roots are chosen so that
1.11.20p
1p
2p
3= q:
Add 1
4ato the roots of g(w) = 0 to get those of
f(z) = 0.Example
f(z) =z4 4z3+ 5z+ 2,g(w) =w4 6w2 3w+ 4.
Resolvent cubic is z3+ 12z2+ 20z+ 9 = 0 with roots
1= 1,2= 1
2(11 +p
85),3= 1
2(11 p
85),
andp 1= 1,p 2=1
2(p
17 +p
5),p 3=
1
2(p
17 p
5). So 2 1= 1 +p
17, 2 2= 1 p
17,
2 3= 1 +p
5, 2 4= 1 p
5, and the roots of
f(z) = 0 are1
2(3p
17),1
2(1p
5).
1.11(iv) Roots of Unity and of Other Constants
The roots of
1.11.21zn 1 = (z 1)(zn 1+zn 2++z+ 1) = 0
are 1,e2i=n,e4i=n;:::;e(2n 2)i=n, and ofzn+ 1 = 0
they areei=n;e3i=n;:::;e(2n 1)i=n.
The roots of
1.11.22 zn=a+ib, a;breal,
are
1.11.23np
R
cos + 2k
n
+isin + 2k
n
;
whereR= (a2+b2)1=2, = ph(a+ib), with the prin-
cipal value of phase ( x1.9(i)), and k= 0;1;:::;n 1.
1.11(v) Stable Polynomials
1.11.24 f(z) =a0+a1z++anzn;
with real coecients, is called stable if the real parts of
all the zeros are strictly negative.
Hurwitz Criterion
Let
1.11.25
D1=a1; D 2= a1a3
a0a2 ; D 3= a1a3a5
a0a2a4
0a1a3 ;
and
1.11.26 Dk= det[h(1)
k;h(3)
k;:::;h(2k 1)
k];
where the column vector h(m)
kconsists of the rst k
members of the sequence am;am 1;am 2;:::withaj=
0 ifj <0 orj >n .
Thenf(z), withan6= 0, is stable i a06= 0;
D2k>0,k= 1;:::;1
2n
; signD2k+1= signa0,
k= 0;1;:::;1
2n 1
2
.
24 Algebraic and Analytic Methods
1.12 Continued Fractions
1.12(i) Notation
The notation used throughout this Handbook for the
continued fraction
1.12.1b0+a1
b1+a2
b2+...
is
1.12.2 b0+a1
b1+a2
b2+:
1.12(ii) Convergents
1.12.3 C=b0+a1
b1+a2
b2+,an6= 0,
1.12.4Cn=b0+a1
b1+a2
b2+an
bn=An
Bn:
Cnis called the nthapproximant orconvergent to C.
AnandBnare called the nth(canonical) numerator
and denominator respectively.
Recurrence Relations
1.12.5Ak=bkAk 1+akAk 2,Bk=bkBk 1+akBk 2,
k= 1;2;3;:::,
1.12.6A 1= 1; A 0=b0; B 1= 0; B 0= 1:Determinant Formula
1.12.7
AnBn 1 BnAn 1= ( 1)n 1nY
k=1ak,n= 0;1;2;:::.
1.12.8Cn Cn 1=( 1)n 1Qn
k=1ak
Bn 1Bn,n= 1;2;3;:::,
1.12.9Cn=b0+a1
B0B1 + ( 1)n 1Qn
k=1ak
Bn 1Bn:
1.12.10an=An 1Bn AnBn 1
An 1Bn 2 An 2Bn 1,n= 1;2;3;:::,
1.12.11an=Bn
Bn 2Cn 1 Cn
Cn 1 Cn 2,n= 2;3;4;:::,
1.12.12bn=AnBn 2 An 2Bn
An 1Bn 2 An 2Bn 1,n= 1;2;3;:::,
1.12.13bn=Bn
Bn 1Cn Cn 2
Cn 1 Cn 2,n= 2;3;4;:::,
1.12.14b0=A0=C0; b 1=B1; a 1=A1 A0B1:
Equivalence
Two continued fractions are equivalent if they have the
same convergents.
b0+a1
b1+a2
b2+ is equivalent to b0
0+
a0
1
b0
1+a0
2
b0
2+if there is a sequence fdng1
n=0,d0= 1,
dn6= 0, such that
1.12.15 a0
n=dndn 1an,n= 1;2;3;:::,
and
1.12.16 b0
n=dnbn,n= 0;1;2;::::
Formally,
1.12.17b0+a1
b1+a2
b2+a3
b3+=b0+a1=b1
1 +a2=(b1b2)
1 +a3=(b2b3)
1 +an=(bn 1bn)
1 +
=b0+1
(1/a1)b1+1
(a1/a2)b2+1
(a2/(a1a3))b3+1
(a1a3/(a2a4))b4+:
Series
1.12.18 p0+nX
k=1p1p2pk=p0+p1
1 p2
1 +p2 p3
1 +p3 pn
1 +pn, n= 0;1;2;:::,
whenpk6= 0,k= 1;2;3;:::.
1.12.19nX
k=0ckxk=c0+c1x
1 (c2/c1)x
1 + (c2/c1)x (c3/c2)x
1 + (c3/c2)x (cn/cn 1)x
1 + (cn/cn 1)x,n= 0;1;2;:::,
whenck6= 0,k= 1;2;3;:::.
1.13 Differential Equations 25
Fractional Transformations
De ne
1.12.20 Cn(w) =b0+a1
b1+a2
b2+an
bn+w:
Then
1.12.21 Cn(w) =An+An 1w
Bn+Bn 1w; Cn(0) =Cn; Cn(1) =Cn 1=An 1
Bn 1:
1.12(iii) Existence of Convergents
A sequencefCngin the extended complex plane, C[f1g , can be a sequence of convergents of the continued fraction
(1.12.3) i
1.12.22 C06=1; Cn6=Cn 1, n= 1;2;3;:::.
1.12(iv) Contraction and Extension
Acontraction of a continued fraction Cis a continued fraction C0whose convergents fC0
ngform a subsequence of
the convergentsfCngofC. Conversely, Cis called an extension ofC0. IfC0
n=C2n,n= 0;1;2;:::, thenC0is called
theeven part ofC. The even part of Cexists i b2k6= 0,k= 1;2;:::, and up to equivalence is given by
1.12.23b0+a1b2
a2+b1b2 a2a3b4
a3b4+b2(a4+b3b4) a4a5b2b6
a5b6+b4(a6+b5b6) a6a7b4b8
a7b8+b6(a8+b7b8) :
IfC0
n=C2n+1,n= 0;1;2;:::, thenC0is called the odd part ofC. The odd part of Cexists i b2k+16= 0,
k= 0;1;2;:::, and up to equivalence is given by
1.12.24a1+b0b1
b1 a1a2b3=b1
a2b3+b1(a3+b2b3) a3a4b1b5
a4b5+b3(a5+b4b5) a5a6b3b7
a6b7+b5(a7+b6b7) :
1.12(v) Convergence
A continued fraction converges if the convergents Cn
tend to a nite limit as n!1 .
Pringsheim's Theorem
The continued fractiona1
b1+a2
b2+converges when
1.12.25 jbnjjanj+ 1,n= 1;2;3;:::.
With these conditions the convergents CnsatisfyjCnj<
1 andCn!CwithjCj1.
Van Vleck's Theorem
Let the elements of the continued fraction
1
b1+1
b2+satisfy
1.12.26 1
2+<phbn<1
2 ,n= 1;2;3;:::,
whereis an arbitrary small positive constant. Then
the convergents Cnsatisfy
1.12.27 1
2+<phCn<1
2 ,n= 1;2;3;:::,
and the even and odd parts of the continued fraction
converge to nite values. The continued fraction con-
verges i , in addition,
1.12.281X
n=1jbnj=1:In this casejphCj1
2.
1.12(vi) Applications
For analytical and numerical applictions of continued
fractions to special functions see x3.10.
1.13 Di erential Equations
1.13(i) Existence of Solutions
A domain in the complex plane is simply-connected if it
has no \holes"; more precisely, if its complement in the
extended plane C[f1g is connected.
The equation
1.13.1d2w
dz2+f(z)dw
dz+g(z)w= 0;
wherez2D, a simply-connected domain, and f(z),
g(z) are analytic in D, has an in nite number of an-
alytic solutions in D. A solution becomes unique, for
example, when wanddw/dzare prescribed at a point
inD.
26 Algebraic and Analytic Methods
Fundamental Pair
Two solutions w1(z) andw2(z) are called a fundamental
pair if any other solution w(z) is expressible as
1.13.2 w(z) =Aw1(z) +Bw2(z);
whereAandBare constants. A fundamental pair can
be obtained, for example, by taking any z02Dand
requiring that
1.13.3
w1(z0) = 1; w0
1(z0) = 0; w 2(z0) = 0; w0
2(z0) = 1:
Wronskian
The Wronskian ofw1(z) andw2(z) is de ned by
1.13.4Wfw1(z);w2(z)g=w1(z)w0
2(z) w2(z)w0
1(z):
Then
1.13.5 Wfw1(z);w2(z)g=ce R
f(z)dz;
wherecis independent of z. Iff(z) = 0, then the Wron-
skian is constant.
The following three statements are equivalent:
w1(z) andw2(z) comprise a fundamental pair in D;
Wfw1(z);w2(z)gdoes not vanish in D;w1(z) andw2(z)
arelinearly independent , that is, the only constants A
andBsuch that
1.13.6 Aw1(z) +Bw2(z) = 0,8z2D,
areA=B= 0.
1.13(ii) Equations with a Parameter
Assume that in the equation
1.13.7d2w
dz2+f(u;z)dw
dz+g(u;z)w= 0;
uandzbelong to domains UandDrespectively, the
coecients f(u;z) andg(u;z) are continuous functions
of both variables, and for each xed u( xedz) the two
functions are analytic in z(inu). Suppose also that at
(a xed)z02D,wand@w/@zare analytic functions
ofu. Then at each z2D,w,@w/@zand@2w
@z2are
analytic functions of u.
1.13(iii) Inhomogeneous Equations
The inhomogeneous (ornonhomogeneous ) equation
1.13.8d2w
dz2+f(z)dw
dz+g(z)w=r(z)
withf(z),g(z), andr(z) analytic in Dhas in nitely
many analytic solutions in D. Ifw0(z) is any one
solution, and w1(z),w2(z) are a fundamental pair of
solutions of the corresponding homogeneous equation
(1.13.1), then every solution of (1.13.8) can be expressed
as
1.13.9w(z) =w0(z) +Aw1(z) +Bw2(z);
whereAandBare constants.Variation of Parameters
With the notation of (1.13.8) and (1.13.9)
1.13.10w0(z) =w2(z)Zw1(z)r(z)
Wfw1(z);w2(z)gdz
w1(z)Zw2(z)r(z)
Wfw1(z);w2(z)gdz:
1.13(iv) Change of Variables
Transformation of the Point at In nity
The substitution = 1=zin (1.13.1) gives
1.13.11d2W
d2+F()dW
d+G()W= 0;
where
1.13.12W() =w1
;
F() =2
1
2f1
;
G() =1
4g1
:
Elimination of First Derivative by Change of Dependent
Variable
The substitution
1.13.13w(z) =W(z) exp
1
2Z
f(z)dz
in (1.13.1) gives
1.13.14d2W
dz2 H(z)W= 0;
where
1.13.15 H(z) =1
4f2(z) +1
2f0(z) g(z):
Elimination of First Derivative by Change of Independent
Variable
In (1.13.1) substitute
1.13.16 =Z
exp
Z
f(z)dz
dz:
Then
1.13.17d2w
d2+g(z) exp
2Z
f(z)dz
w= 0:
Liouville Transformation
LetW(z) satisfy (1.13.14), (z) be any thrice-
di erentiable function of z, and
1.13.18 U(z) = (0(z))1=2W(z):
Then
1.13.19d2U
d2=
_z2H(z) 1
2fz;g
U:
1.14 Integral Transforms 27
Here dots denote di erentiations with respect to , and
fz;gis the Schwarzian derivative :
1.13.20fz;g= 2 _z1/2d2
d2( _z 1/2) =...z
_z 3
2z
_z2
:
Cayley's Identity
For arbitrary and,
1.13.21fz;g= (d/d)2fz;g+f;g:
1.13.22fz;g= (dz/d)2f;zg:
1.13(v) Products of Solutions
The product of any two solutions of (1.13.1) satis es
1.13.23
d3w
dz3+ 3fd2w
dz2+ (2f2+f0+ 4g)dw
dz+ (4fg+ 2g0)w= 0:
IfU(z) andV(z) are respectively solutions of
1.13.24d2U
dz2+IU= 0;d2V
dz2+JV= 0;
thenW=UVis a solution of
1.13.25
d
dzW000+ 2(I+J)W0+ (I0+J0)W
I J
= (I J)W:
1.13(vi) Singularities
For classi cation of singularities of (1.13.1) and expan-
sions of solutions in the neighborhoods of singularities,
seex2.7.
1.13(vii) Closed-Form Solutions
For an extensive collection of solutions of di erential
equations of the rst, second, and higher orders see
Kamke (1977).
1.14 Integral Transforms
1.14(i) Fourier Transform
The Fourier transform of a real- or complex-valued
functionf(t) is de ned by
1.14.1 F(x) =1p
2Z1
1f(t)eixtdt:
(Some references replace ixtby ixt.)
Iff(t) is absolutely integrable on ( 1;1), then
F(x) is continuous, F(x)!0 asx!1 , and
1.14.2jF(x)j1p
2Z1
1jf(t)jdt:Inversion
Suppose that f(t) is absolutely integrable on ( 1;1)
and of bounded variation in a neighborhood of t=u
(x1.4(v)). Then
1.14.31
2(f(u+) +f(u )) =1p
2Z1
1F(x)e ixudx;
where the last integral denotes the Cauchy principal
value (1.4.25).
In many applications f(t) is absolutely integrable
andf0(t) is continuous on ( 1;1). Then
1.14.4 f(t) =1p
2Z1
1F(x)e ixtdx:
Convolution
For Fourier transforms, the convolution (fg)(t) of two
functionsf(t) andg(t) de ned on ( 1;1) is given by
1.14.5 (fg)(t) =1p
2Z1
1f(t s)g(s)ds:
Iff(t) andg(t) are absolutely integrable on ( 1;1),
then so is ( fg)(t), and its Fourier transform is
F(x)G(x), whereG(x) is the Fourier transform of g(t).
Parseval's Formula
Supposef(t) andg(t) are absolutely integrable on
( 1;1), andF(x) andG(x) are their respective
Fourier transforms. Then
1.14.6 (fg)(t) =1p
2Z1
1F(x)G(x)e itxdx;
1.14.7Z1
1F(x)G(x)dx=Z1
1f(t)g( t)dt;
1.14.8Z1
1jF(x)j2dx=Z1
1jf(t)j2dt:
(1.14.8) is Parseval's formula .
Uniqueness
Iff(t) andg(t) are continuous and absolutely inte-
grable on ( 1;1), andF(x) =G(x) for allx, then
f(t) =g(t) for allt.
1.14(ii) Fourier Cosine and Sine Transforms
These are de ned respectively by
1.14.9 Fc(x) =r
2
Z1
0f(t) cos(xt)dt;
1.14.10 Fs(x) =r
2
Z1
0f(t) sin(xt)dt:
28 Algebraic and Analytic Methods
Inversion
Iff(t) is absolutely integrable on [0 ;1) and of bounded
variation (x1.4(v)) in a neighborhood of t=u, then
1.14.111
2(f(u+) +f(u )) =r
2
Z1
0Fc(x) cos(ux)dx;
1.14.121
2(f(u+) +f(u )) =r
2
Z1
0Fs(x) sin(ux)dx:
Parseval's Formula
IfR1
0jf(t)jdt <1,g(t) is of bounded variation on
(0;1) andg(t)!0 ast!1 , then
1.14.13Z1
0Fc(x)Gc(x)dx=Z1
0f(t)g(t)dt;
1.14.14Z1
0Fs(x)Gs(x)dx=Z1
0f(t)g(t)dt;
1.14.15Z1
0(Fc(x))2dx=Z1
0(f(t))2dt;
1.14.16Z1
0(Fs(x))2dx=Z1
0(f(t))2dt;
whereGc(x) andGs(x) are respectively the cosine and
sine transforms of g(t).
1.14(iii) Laplace Transform
Supposef(t) is a real- or complex-valued function and s
is a real or complex parameter. The Laplace transform
offis de ned by
1.14.17 L(f(t);s) =Z1
0e stf(t)dt:
Alternative notations are L(f(t)),L(f;s), or even
L(f), when it is not important to display all the vari-
ables.
Convergence and Analyticity
Assume that on [0 ;1)f(t) is piecewise continuous and
ofexponential growth , that is, constants Mand exist
such that
1.14.18 jf(t)jMe t, 0t<1.
ThenL(f(t);s) is an analytic function of sfor<s> .
Moreover,
1.14.19 L(f(t);s)!0,<s!1 .
Throughout the remainder of this subsection we as-
sume (1.14.18) is satis ed and <s> .
Inversion
Iff(t) is continuous and f0(t) is piecewise continuous
on [0;1), then
1.14.20
f(t) =1
2ilim
T!1Z+iT
iTetsL(f(t);s)ds,> .
Moreover, if L(f(t);s) =O
s K
in some half-plane
<s
andK > 1, then (1.14.20) holds for >
.Translation
If<s>max(<(a+ ); ), then
1.14.21 L(f(t);s a) =L
eatf(t);s
:
Also, ifa0 then
1.14.22 L(H(t a)f(t a);s) =e asL(f(t);s);
whereHis the Heaviside function; see (1.16.13).
Di erentiation and Integration
Iff(t) is piecewise continuous, then
1.14.23
dn
dsnL(f(t);s) =L(( t)nf(t);s),n= 1;2;3;:::.
If also lim t!0+f(t)=texists, then
1.14.24Z1
sL(f(t);u)du=Lf(t)
t;s
:
Periodic Functions
Ifa>0 andf(t+a) =f(t) fort>0, then
1.14.25 L(f(t);s) =1
1 e asZa
0e stf(t)dt:
Alternatively if f(t+a) = f(t) fort>0, then
1.14.26 L(f(t);s) =1
1 +e asZa
0e stf(t)dt:
Derivatives
Iff(t) is continuous on [0 ;1) andf0(t) is piecewise
continuous on (0 ;1), then
1.14.27 L(f0(t);s) =sL(f(t);s) f(0+):
Iff(t) andf0(t) are piecewise continuous on [0 ;1)
with discontinuities at (0 =) t0<t1<<tn, then
1.14.28L(f0(t);s) =sL(f(t);s) f(0+)
nX
k=1e stk(f(tk+) f(tk )):
Next, assume f(t),f0(t),:::,f(n 1)(t) are contin-
uous and each satis es (1.14.18). Also assume that
f(n)(t) is piecewise continuous on [0 ;1). Then
1.14.29
L
f(n)(t);s
=snL(f(t);s) sn 1f(0+)
sn 2f0(0+) f(n 1)(0+):
Convolution
For Laplace transforms, the convolution of two functions
f(t) andg(t), de ned on [0 ;1), is
1.14.30 (fg)(t) =Zt
0f(u)g(t u)du:
Iff(t) andg(t) are piecewise continuous, then
1.14.31 L(fg) =L(f)L(g):
1.14 Integral Transforms 29
Uniqueness
Iff(t) andg(t) are continuous and L(f) =L(g), then
f(t) =g(t).
1.14(iv) Mellin Transform
The Mellin transform of a real- or complex-valued func-
tionf(x) is de ned by
1.14.32 M(f;s) =Z1
0xs 1f(x)dx:
Alternative notations for M(f;s) areM(f(x);s)
andM(f).
Ifx 1f(x) is integrable on (0 ;1) for allin
a < < b , then the integral (1.14.32) converges and
M(f;s) is an analytic function of sin the vertical strip
a<<s<b . Moreover, for a<<b ,
1.14.33 lim
t!1M(f;+it) = 0:
Note: Iff(x) is continuous and and are real
numbers such that f(x) =O(x ) asx!0+ and
f(x) =O
x
asx!1 , thenx 1f(x) is integrable
on (0;1) for all2( ; ).
Inversion
Suppose the integral (1.14.32) is absolutely convergent
on the line<s=andf(x) is of bounded variation in
a neighborhood of x=u. Then
1.14.34
1
2(f(u+) +f(u )) =1
2ilim
T!1Z+iT
iTu sM(f;s)ds:
Iff(x) is continuous on (0 ;1) andM(f;+it) is
integrable on ( 1;1), then
1.14.35f(x) =1
2iZ+i1
i1x sM(f;s)ds:
Parseval-type Formulas
Supposex f(x) andx 1g(x) are absolutely in-
tegrable on (0 ;1) and either M(g;+it) or
M(f; 1 it) is absolutely integrable on ( 1;1).
Then fory>0,
1.14.36Z1
0f(x)g(yx)dx
=1
2iZ+i1
i1y sM(f; 1 s)M(g;s)ds;
1.14.37Z1
0f(x)g(x)dx
=1
2iZ+i1
i1M(f; 1 s)M(g;s)ds:
Whenfis real and =1
2,
1.14.38Z1
0(f(x))2dx=1
2Z1
1 M
f;1
2+it 2dt:Convolution
Let
1.14.39 (fg)(x) =Z1
0f(y)gx
ydy
y:
Ifx 1f(x) andx 1g(x) are absolutely integrable on
(0;1), then fors=+it,
1.14.40Z1
0xs 1(fg)(x)dx=M(f;s)M(g;s):
1.14(v) Hilbert Transform
The Hilbert transform of a real-valued function f(t) is
de ned in the following equivalent ways:
1.14.41H(f;x) =H(f(t);x) =H(f) =1
Z1
1f(t)
t xdt;
1.14.42H(f;x) = lim
y!0+1
Z1
1t x
(t x)2+y2f(t)dt;
1.14.43H(f;x) = lim
!0+1
Z1
f(x+t) f(x t)
tdt:
Inversion
Supposef(t) is continuously di erentiable on ( 1;1)
and vanishes outside a bounded interval. Then
1.14.44 f(x) = 1
Z1
1H(f;u)
u xdu:
Inequalities
Ifjf(t)jp,p >1, is integrable on ( 1;1), then so is
jH(f;x)jpand
1.14.45Z1
1jH(f;x)jpdxApZ1
1jf(t)jpdt;
whereAp= tan 1
2=p
when 1< p2, or cot 1
2=p
whenp2. These bounds are sharp, and equality holds
whenp= 2.
Fourier Transform
Whenf(t) satis es the same conditions as those for
(1.14.44),
1.14.461p
2Z1
1H(f;t)eixtdt= i(signx)F(x);
whereF(x) is given by (1.14.1).
1.14(vi) Stieltjes Transform
The Stieltjes transform of a real-valued function f(t) is
de ned by
1.14.47S(f;s) =S(f(t);s) =S(f) =Z1
0f(t)
s+tdt:
Sucient conditions for the integral to converge are
thatsis a positive real number, and f(t) =O
t
ast!1 , where>0.
30 Algebraic and Analytic Methods
If the integral converges, then it converges uniformly
in any compact domain in the complex s-plane not con-
taining any point of the interval ( 1;0]. In this case,
S(f;s) represents an analytic function in the s-plane
cut along the negative real axis, and
1.14.48dm
dsmS(f;s) = ( 1)mm!Z1
0f(t)dt
(s+t)m+1,
m= 0;1;2;::::
Inversion
Iff(t) is absolutely integrable on [0 ;R] for every nite
R, and the integral (1.14.47) converges, then
1.14.49lim
t!0+S(f; it) S(f; +it)
2i
=1
2(f(+) +f( ));
for all values of the positive constant for which the
right-hand side exists.
Laplace Transform
Iff(t) is piecewise continuous on [0 ;1) and the integral
(1.14.47) converges, then
1.14.50 S(f) =L(L(f)):1.14(vii) Tables
Table 1.14.1 : Fourier transforms.
f(t)1p
2Z1
1f(t)eixtdt
(
1;jtj<a;
0;otherwiser
2
sin(ax)
x
e ajtjr
2
a
a2+x2,a>0
te ajtjr
2
2iax
(a2+x2)2,a>0
jtje ajtjr
2
a2 x2
(a2+x2)2,a>0
e ajtj
jtj1=2(a+ (a2+x2)1=2)1=2
(a2+x2)1=2,a>0
sinh(at)
sinh(t)1p
2sina
coshx+ cosa, <
a<
cosh(at)
cosh(t)r
2
cos 1
2a
cosh 1
2x
coshx+ cosa, <
a<
e at2 1p
2ae x2=(4a),a>0
sin
at2
1p
2asinx2
4a
4
,a>0
cos
at2 1p
2acosx2
4a
4
,a>0
1.14 Integral Transforms 31
Table 1.14.2 : Fourier cosine transforms.
f(t)r
2
Z1
0f(t) cos(xt)dt,x>0
(
1;0<ta;
0;otherwiser
2
sin(ax)
x
1
a2+t2r
2e ax
a,<a>0
1
(a2+t2)2r
2(1 +ax)e ax
2a3,<a>0
4a3
4a4+t4pe axsin
ax+1
4
,<a>0
e atr
2
a
a2+x2,<a>0
e at2 1p
2ae x2=(4a),<a>0
sin
at2
1p
2asinx2
4a
4
,a>0
cos
at2 1p
2acosx2
4a
4
,a>0
ln
1 +a2
t2p
21 e ax
x,<a>0
lna2+t2
b2+t2p
2e bx e ax
x,<a>0,
<b>0Table 1.14.3 : Fourier sine transforms.
f(t)r
2
Z1
0f(t) sin(xt)dt,x>0
t 1r
2
t 1=2x 1=2
t 3=22x1=2
t
a2+t2r
2e ax,<a>0
t
(a2+t2)2r
8x
ae ax,<a>0
1
t(a2+t2)r
21 e ax
a2,<a>0
e at
tr
2
arctanx
a
,<a>0
e atr
2
x
a2+x2,<a>0
te atr
2
2ax
(a2+x2)2,<a>0
te at2(2a) 3=2xe x2=(4a),jphaj<1
2
sin(at)
t1p
2ln x+a
x a ,a>0
arctant
ar
2e ax
x,a>0
ln t+a
t a p
2sin(ax)
x,a>0
32 Algebraic and Analytic Methods
Table 1.14.4 : Laplace transforms.
f(t)Z1
0e stf(t)dt
11
s,<s>0
tn
n!1
sn+1,<s>0
1p
t1ps,<s>0
e at1
s+a,<(s+a)>0
tne at
n!1
(s+a)n+1,<(s+a)>0
e at e bt
b a1
(s+a)(s+b),a6=b,
<s> <a,
<s> <b
sin(at)a
s2+a2,<s>j=aj
cos(at)s
s2+a2,<s>j=aj
sinh(at)a
s2 a2,<s>j<aj
cosh(at)s
s2 a2,<s>j<aj
tsin(at)2as
(s2+a2)2,<s>j=aj
tcos(at)s2 a2
(s2+a2)2,<s>j=aj
e bt e at
tlns+a
s+b
,<s> <a,
<s> <b
2(1 cosh(at))
tln
1 a2
s2
,<(s+a)>0
2(1 cos(at))
tln
1 +a2
s2
,<s>0
sin(at)
tarctana
s
,<s>0Table 1.14.5 : Mellin transforms.
f(x)Z1
0xs 1f(x)dx
(
1; x<a;
0; xaas
s,a0,<s>0
(
ln(a=x); x<a;
0; xaas
s2,a0,<s>1
1
1 xcot(s), 0<<s<1,
(Cauchy p. v.)
1
1 +xcsc(s), 0<<s<1
ln(1 +ax)csc(s)
sas,jphaj<,
1<<s<0
ln 1 +x
1 x tan 1
2s
s, 1<<s<1
ln(1 +x)
xcsc(s)
1 s, 0<<s<1
arctanx sec 1
2s
2s, 1<<s<0
arccotxsec 1
2s
2s, 0<<s<1
1 +xcos
1 + 2xcos+x2cos(s)
sin(s), << ,
0<<s<1
xsin
1 + 2xcos+x2sin(s)
sin(s), << ,
0<<s<1
1.14(viii) Compendia
For more extensive tables of the integral transforms
of this section and tables of other integral transforms,
see Erd elyi et al. (1954a,b), Gradshteyn and Ryzhik
(2000), Marichev (1983), Oberhettinger (1972, 1974,
1990), Oberhettinger and Badii (1973), Oberhettinger
and Higgins (1961), Prudnikov et al. (1986a,b, 1990,
1992a,b).
1.15 Summability Methods 33
1.15 Summability Methods
1.15(i) De nitions for Series
1.15.1 sn=nX
k=0ak:
Abel Summability
1.15.21X
n=0an=s(A);
if
1.15.3 lim
x!1 1X
n=0anxn=s:
Ces aro Summability
1.15.41X
n=0an=s(C,1);
if
1.15.5 lim
n!1s0+s1++sn
n+ 1=s:
General Ces aro Summability
For > 1,
1.15.61X
n=0an=s(C, );
if
1.15.7 lim
n!1n!
( + 1)nnX
k=0( + 1)k
k!an k=s:
Borel Summability
1.15.81X
n=0an=s(B);
if
1.15.9 lim
t!1e t1X
n=0sn
n!tn=s:
1.15(ii) Regularity
Methods of summation are regular if they are consis-
tent with conventional summation. All of the methods
described inx1.15(i) are regular. For example if
1.15.101X
n=0an=s;
then
1.15.111X
n=0an=s(A):1.15(iii) Summability of Fourier Series
Poisson Kernel
1.15.12
P(r;) =1 r2
1 2rcos+r2=1X
n= 1rjnjein, 0r<1,
1.15.131
2Z2
0P(r;)d= 1:
Asr!1
1.15.14 P(r;)!0;
uniformly for 2[;2 ]. (Here and elsewhere in this
subsectionis a constant such that 0 << .)
Fej er Kernel
Forn= 0;1;2;:::,
1.15.15Kn() =1
n+ 1
sin 1
2(n+ 1)
sin 1
2!2
;
1.15.161
2Z2
0Kn()d= 1:
Asn!1
1.15.17 Kn()!0;
uniformly for 2[;2 ].
Abel Means
1.15.18 A(r;) =1X
n= 1rjnjF(n)ein;
where
1.15.19 F(n) =1
2Z2
0f(t)e intdt:
A(r;) is a harmonic function in polar coordinates
((1.9.27)), and
1.15.20A(r;) =1
2Z2
0P(r; t)f(t)dt:
Ces aro (or (C,1)) Means
Let
1.15.21n() =s0() +s1() ++sn()
n+ 1;
n= 0;1;2;:::, where
1.15.22 sn() =nX
k= nF(k)eik:
Then
1.15.23n() =1
2Z2
0Kn( t)f(t)dt:
34 Algebraic and Analytic Methods
Convergence
Iff() is periodic and integrable on [0 ;2], then as
n!1 the Abel means A(r;) and the (C,1) means
n() converge to
1.15.241
2(f(+) +f( ))
at every point where both limits exist. If f() is also
continuous, then the convergence is uniform for all .
For real-valued f(), if
1.15.251X
n= 1F(n)ein
is the Fourier series of f(), then the series
1.15.26 F(0) + 21X
n=1F(n)ein
can be extended to the interior of the unit circle as an
analytic function
1.15.27G(z) =G(x+iy) =u(x;y) +iv(x;y)
=F(0) + 21X
n=1F(n)zn:
Hereu(x;y) =A(r;) is the Abel (orPoisson )sum of
f(), andv(x;y) has the series representation
1.15.28 1X
n= 1i(signn)F(n)rjnjein;
comparex1.15(v).
1.15(iv) De nitions for Integrals
Abel Summability
R1
1f(t)dtisAbel summable toL, or
1.15.29Z1
1f(t)dt=L(A);
when
1.15.30 lim
!0+Z1
1e jtjf(t)dt=L:
Ces aro Summability
R1
1f(t)dtis(C,1) summable toL, or
1.15.31Z1
1f(t)dt=L(C,1);
when
1.15.32 lim
R!1ZR
R
1 jtj
R
f(t)dt=L:
IfR1
1f(t)dtconverges and equals L, then the in-
tegral is Abel and Ces aro summable to L.1.15(v) Summability of Fourier Integrals
Poisson Kernel
1.15.33 P(x;y) =2y
x2+y2,y>0, 1<x<1.
1.15.341
2Z1
1P(x;y)dx= 1:
For each>0,
1.15.35Z
jxjP(x;y)dx!0, as y!0.
Let
1.15.36h(x;y) =1p
2Z1
1e yjtje ixtF(t)dt;
whereF(t) is the Fourier transform of f(x) (x1.14(i)).
Then
1.15.37h(x;y) =1
2Z1
1f(t)P(x t;y)dt
is the Poisson integral off(t).
Iff(x) is integrable on ( 1;1), then
1.15.38 lim
y!0+Z1
1jh(x;y) f(x)jdx= 0:
Suppose now f(x) is real-valued and integrable on
( 1;1). Let
1.15.39 (z) = (x+iy) =i
Z1
1f(t)1
(x t) +iydt;
wherey >0 and 1< x <1. Then (z) is an ana-
lytic function in the upper half-plane and its real part
is the Poisson integral h(x;y); compare (1.9.34). The
imaginary part
1.15.40=(x+iy) =1
Z1
1f(t)x t
(x t)2+y2dt
is the conjugate Poisson integral off(x). Moreover,
limy!0+=(x+iy) is the Hilbert transform of f(x)
(x1.14(v)).
Fej er Kernel
1.15.41 KR(s) =1
R1 cos(Rs)
s2;
1.15.42Z1
1KR(s)ds= 1:
For each>0,
1.15.43Z
jsjKR(s)ds!0, asR!1 .
Let
1.15.44R() =1p
2ZR
R
1 jtj
R
e itF(t)dt;
then
1.15.45R() =Z1
1f(t)KR( t)dt:
1.16 Distributions 35
Iff() is integrable on ( 1;1), then
1.15.46 lim
R!1Z1
1jR() f()jd= 0:
1.15(vi) Fractional Integrals
For< >0, the fractional integral operator of order
is de ned by
1.15.47I f(x) =1
( )Zx
0(x t) 1f(t)dt:
For ( ) seex5.2, and compare (1.4.31) in the case when
is a positive integer.
1.15.48 I I =I + ,< >0,< >0.
For extensions of (1.15.48) see Love (1972b).
If
1.15.49 f(x) =1X
k=0akxk;
then
1.15.50I f(x) =1X
k=0k!
(k+ + 1)akxk+ :
1.15(vii) Fractional Derivatives
For 0<< <n ,nan integer,
1.15.51 D f(x) =dn
dxnIn f(x);
1.15.52 DkI =DnI +n k,k= 1;2;:::;n .
When none of , , and + is an integer
1.15.53 D D =D + :
Note thatD1=2D6=D3=2. See also Love (1972b).
1.15(viii) Tauberian Theorems
If
1.15.541X
n=0an=s(A); an> K
n,n>0,K > 0,
then
1.15.551X
n=0an=s:
If
1.15.56 lim
x!1 (1 x)1X
n=0anxn=s;
and eitherjanjKoran0, then
1.15.57 lim
n!1a0+a1++an
n+ 1=s:1.16 Distributions
1.16(i) Test Functions
Let be a function de ned on an open interval I=
(a;b), which can be in nite. The closure of the set of
points where 6= 0 is called the support of . If the sup-
port of is a compact set ( x1.9(vii)), then is called
afunction of compact support . A test function is an
in nitely di erentiable function of compact support.
A sequencef ngof test functions converges to a test
function if the support of every nis contained in a
xed compact set Kand asn!1 the sequencef (k)
ng
converges uniformly on Kto (k)fork= 0;1;2;:::.
The linear space of all test functions with the above
de nition of convergence is called a test function space .
We denote it byD(I).
A mapping onD(I) is a linear functional if it takes
complex values and
1.16.1 ( 1 1+ 2 2) = 1( 1) + 2( 2);
where 1and 2are real or complex constants. :
D(I)!Cis called a distribution if it is a continuous
linear functional on D(I), that is, it is a linear functional
and for every n! inD(I),
1.16.2 lim
n!1( n) = ( ):
From here on we write h; ifor ( ). The space
of all distributions will be denoted by D(I). A distri-
bution is called regular if there is a function fonI,
which is absolutely integrable on every compact subset
ofI, such that
1.16.3h; i=Z
If(x) (x)dx:
We denote a regular distribution by f, or simply f,
wherefis the function giving rise to the distribution.
(If a distribution is not regular, it is called singular .)
De ne
1.16.4h1+ 2; i=h1; i+h2; i;
1.16.5hc; i=ch; i=h;c i;
wherecis a constant. More generally, if (x) is an
in nitely di erentiable function, then
1.16.6 h ; i=h; i:
We say that a sequence of distributions fngconverges
to a distribution in Dif
1.16.7 lim
n!1hn; i=h; i
for all 2D(I).
1.16(ii) Derivatives of a Distribution
The derivative 0of a distribution is de ned by
1.16.8 h0; i= h; 0i, 2D(I).
36 Algebraic and Analytic Methods
Similarly
1.16.9D
(k); E
= ( 1)kD
; (k)E
,k= 1;2;:::.
For any locally integrable function f, its distributional
derivative isDf= 0
f.
1.16(iii) Dirac Delta Distribution
1.16.10h; i= (0), 2D(I),
1.16.11hx0; i= (x0), 2D(I),
1.16.12D
(n)
x0; E
= ( 1)n (n)(x0), 2D(I).
The Dirac delta distribution is singular.
1.16(iv) Heaviside Function
1.16.13 H(x) =(
1; x> 0;
0; x0:
1.16.14 H(x x0) =(
1; x>x 0;
0; xx0:
1.16.15 DH=;
1.16.16 DH(x x0) =x0:
Supposef(x) is in nitely di erentiable except at x0,
where left and right derivatives of all orders exist, and
1.16.17 n=f(n)(x0+) f(n)(x0 ):
Then
1.16.18Dmf=f(m)+0(m 1)
x0+1(m 2)
x0+
+m 1x0, m= 1;2;:::.
For > 1,
1.16.19 x
+=x H(x) =(
x ; x> 0;
0; x0:
For >0,
1.16.20 Dx
+= x 1
+:
For < 1 and not an integer, de ne
1.16.21 x
+=1
( + 1)nDnx +n
+;
wherenis an integer such that +n> 1. Similarly,
we write
1.16.22 ln+x=H(x) lnx=(
lnx; x> 0;
0; x0;
and de ne
1.16.23 ( 1)nn!x 1 n
+ =D(n+1)ln+x,n= 0;1;2;:::.1.16(v) Tempered Distributions
The spaceT(R) of test functions for tempered dis-
tributions consists of all in nitely-di erentiable func-
tions such that the function and all its derivatives are
O
jxj N
asjxj!1 for allN.
A sequencef ngof functions inTis said to converge
to a function 2T asn!1 if the sequencef (k)
ng
converges uniformly to (k)on every nite interval and
if the constants ck;Nin the inequalities
1.16.24 jxN (k)
njck;N
do not depend on n.
Atempered distribution is a continuous linear func-
tional onT. (See the de nition of a distribution in
x1.16(i).) The set of tempered distributions is denoted
byT.
A sequence of tempered distributions nconverges
to inTif
1.16.25 lim
n!1hn; i=h; i;
for all 2T.
The derivatives of tempered distributions are de ned
in the same way as derivatives of distributions.
For a detailed discussion of tempered distributions
see Lighthill (1958).
1.16(vi) Distributions of Several Variables
LetD(Rn) =Dnbe the set of all in nitely di erentiable
functions in nvariables, (x1;x2;:::;xn), with compact
support in Rn. Ifk= (k1;:::;kn) is a multi-index and
x= (x1;:::;xn)2Rn, then we write xk=xk1
1xknn
and (k)(x) =@k =(@xk1
1@xknn). A sequencef mg
of functions inDnconverges to a function 2Dnif
the supports of mlie in a xed compact subset Kof
Rnand (k)
mconverges uniformly to (k)inKfor every
multi-index k= (k1;k2;:::;kn). A distribution inRnis
a continuous linear functional on Dn.
The partial derivatives of distributions in Rncan
be de ned as inx1.16(ii). A locally integrable function
f(x) =f(x1;x2;:::;xn) gives rise to a distribution f
de ned by
1.16.26hf; i=Z
Rnf(x) (x)dx, 2Dn.
The distributional derivative Dkfoffis de ned by
1.16.27
Dkf;
= ( 1)jkjZ
Rnf(x) (k)(x)dx, 2Dn,
wherekis a multi-index and jkj=k1+k2++kn.
For tempered distributions the space of test func-
tionsTnis the set of all in nitely-di erentiable functions
ofnvariables that satisfy
1.16.28 jxm (k)(x)jcm;k,x2Rn.
1.17 Integral and Series Representations of the Dirac Delta 37
Herem= (m1;m2;:::;mn) andk= (k1;k2;:::;kn) are
multi-indices, and cm;kare constants. Tempered distri-
butions are continuous linear functionals on this space
of test functions. The space of tempered distributions
is denoted byT
n.
1.16(vii) Fourier Transforms of Distributions
Suppose is a test function in Tn. Then its Fourier
transform is
1.16.29F(x) =F=1
(2)n=2Z
Rn (t)eixtdt;
where x= (x1;x2;:::;xn) and xt=x1t1++
xntn.F(x) is also inTn. For a multi-index =
( 1; 2;:::; n), setj j= 1+ 2++ nand
1.16.30D =i j jD =1
i@
@x1 1
1
i@
@xn n
;
1.16.31P(x) =P=X
c x =X
c x 1
1x n
n;
and
1.16.32 P(D) =X
c D :
Then
1.16.331
(2)n=2Z
Rn(P(D) )(t)eixtdt=P( x)F(x);
and
1.16.341
(2)n=2Z
RnP(t) (t)eixtdt=P(D)F(x):
Ifu2T
nis a tempered distribution, then its Fourier
transformF(u) is de ned by
1.16.35 hF(u); i=hu;Fi, 2Tn,
whereFis given by (1.16.29). The Fourier transform
F(u) of a tempered distribution is again a tempered
distribution, and
1.16.36F(P(D)u) =P( x)F(u);
1.16.37 F(Pu) =P(D)F(u):
In (1.16.36) and (1.16.37) the derivatives in P(D) are
understood to be in the sense of distributions.
1.17 Integral and Series Representations of
the Dirac Delta
1.17(i) Delta Sequences
In applications in physics and engineering, the Dirac
delta distribution ( x1.16(iii)) is historically and custom-
arily replaced by the Dirac delta (orDirac delta func-
tion)(x). This is an operator with the properties:
1.17.1 (x) = 0, x2R,x6= 0,and
1.17.2Z1
1(x a) (x)dx= (a),a2R,
subject to certain conditions on the function (x).
From the mathematical standpoint the left-hand side
of (1.17.2) can be interpreted as a generalized integral
in the sense that
1.17.3 lim
n!1Z1
1n(x a) (x)dx= (a);
for a suitably chosen sequence of functions n(x),n=
1;2;:::. Such a sequence is called a delta sequence and
we write, symbolically,
1.17.4 lim
n!1n(x) =(x), x2R.
An example of a delta sequence is provided by
1.17.5 n(x a) =rn
e n(x a)2:
In this case
1.17.6 lim
n!1rn
Z1
1e n(x a)2 (x)dx= (a);
for all functions (x) that are continuous when x2
( 1;1), and for each a,R1
1e n(x a)2 (x)dxcon-
verges absolutely for all suciently large values of n.
The last condition is satis ed, for example, when (x) =
O
e x2
asx!1 , where is a real constant.
More generally, assume (x) is piecewise continuous
(x1.4(ii)) when x2[ c;c] for any nite positive real
value ofc, and for each a,R1
1e n(x a)2 (x)dxcon-
verges absolutely for all suciently large values of n.
Then
1.17.7
lim
n!1rn
Z1
1e n(x a)2 (x)dx=1
2 (a ) +1
2 (a+):
1.17(ii) Integral Representations
Formal interchange of the order of integration in the
Fourier integral formula ((1.14.1) and (1.14.4)):
1.17.81
2Z1
1e iatZ1
1 (x)eitxdx
dt= (a)
yields
1.17.9Z1
11
2Z1
1ei(x a)tdt
(x)dx= (a):
The inner integral does not converge. However, for
n= 1;2;:::,
1.17.101
2Z1
1e t2=(4n)ei(x a)tdt=rn
e n(x a)2:
Hence comparison with (1.17.5) shows that (1.17.9) can
be interpreted as a generalized integral (1.17.3) with
1.17.11n(x a) =1
2Z1
1e t2=(4n)ei(x a)tdt;
38 Algebraic and Analytic Methods
provided that (x) is continuous when x2( 1;1),
and for each a,R1
1e n(x a)2 (x)dxconverges abso-
lutely for all suciently large values of n(as in the case
of (1.17.6)). Then comparison of (1.17.2) and (1.17.9)
yields the formal integral representation
1.17.12 (x a) =1
2Z1
1ei(x a)tdt:
Other similar integral representations of the Dirac
delta that appear in the physics literature include the
following:
Bessel Functions and Spherical Bessel Functions
(xx10.2(ii), 10.47(ii))
1.17.13(x a) =xZ1
0tJ(xt)J(at)dt,
< > 1,x>0,a>0,
1.17.14
(x a) =2xa
Z1
0t2j`(xt)j`(at)dt,x>0,a>0.
See Arfken and Weber (2005, Eq. (11.59)) and Konopin-
ski (1981, p. 242). For a generalization of (1.17.14) see
Maximon (1991).
Coulomb Functions ( x33.14(iv))
1.17.15
(x a) =Z1
0s(x;`;r)s(a;`;r)dr,a>0,x>0.
See Seaton (2002).
Airy Functions (x9.2)
1.17.16(x a) =Z1
1Ai(t x) Ai(t a)dt:
See Vall ee and Soares (2004, x3.5.3).
1.17(iii) Series Representations
Formal interchange of the order of summation and inte-
gration in the Fourier summation formula ((1.8.3) and
(1.8.4)):
1.17.171
21X
k= 1e ikaZ
(x)eikxdx
= (a);
yields
1.17.18Z
(x)
1
21X
k= 1eik(x a)!
dx= (a):
The sumP1
k= 1eik(x a)does not converge, but
(1.17.18) can be interpreted as a generalized integral
in the sense that
1.17.19 lim
n!1Z
n(x a) (x)dx= (a);where
1.17.20
n(x a)
=1
2nX
k= neik(x a)
=sin
(n+1
2)(x a)
2sin 1
2(x a)!
;
provided that (x) is continuous and of period 2 ; see
x1.8(ii).
By analogy with x1.17(ii) we have the formal series
representation
1.17.21 (x a) =1
21X
k= 1eik(x a):
Other similar series representations of the Dirac
delta that appear in the physics literature include the
following:
Legendre Polynomials ( xx14.7(i) and 18.3)
1.17.22(x a) =1X
k=0(k+1
2)Pk(x)Pk(a):
Laguerre Polynomials ( x18.3)
1.17.23(x a) =e (x+a)=21X
k=0Lk(x)Lk(a):
Hermite Polynomials ( x18.3)
1.17.24(x a) =e (x2+a2)=2
p1X
k=0Hk(x)Hk(a)
2kk!:
Spherical Harmonics ( x14.30)
1.17.25(cos1 cos2)( 1 2)
=1X
`=0`X
m= `Y`;m(1; 1)Y
`;m(2; 2):
(1.17.22){(1.17.24) are special cases of Morse and
Feshbach (1953a, Eq. (6.3.11)). For (1.17.25) see Ar-
fken and Weber (2005, p. 792).
1.17(iv) Mathematical De nitions
The references given in xx1.17(ii){1.17(iii) are from the
physics literature. For mathematical interpretations of
(1.17.13), (1.17.15), (1.17.16) and (1.17.22){(1.17.25)
that resemble those given in xx1.17(ii) and 1.17(iii) for
(1.17.12) and (1.17.21), see Li and Wong (2008). For
(1.17.14) combine (1.17.13) and (10.47.3).
References 39
References
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x1.2Chrystal (1959, pp. 62{70, 482{483, 489), Hardy
et al. (1967, pp. 12{15).
x1.3Vein and Dale (1999, pp. 3{12, 33{34, 51{52, 57,
79{81), For (1.3.17) see Bressoud (1999, p. 67).
x1.4Hardy (1952, Chapters 5{7, and pp. 234{235,
247{248, 258, 285{292, 327{328), Olver (1997b,
pp. 28, 73), Rudin (1976, Chapter 5), Hardy et al.
(1967, pp. 70{77). For (1.4.13) see Riordan (1958,
pp. 35{36) and Knuth (1968, p. 50). For (1.4.31)
integrate by parts.
x1.5Marsden and Tromba (1996, Chapters 2, 3, 5, 6,
and pp. 358{371), Davis and Snider (1987, Chap-
ter 5), Protter and Morrey (1991, pp. 288, 298)
For (1.5.36) see Love (1970, 1972a).
x1.6Marsden and Tromba (1996, Chapter 1 and
pp. 144{147, 273{283, 396{417, 421{459, 470, 485,
506). For (1.6.9) see Hubbard and Hubbard (2002,
pp. 82{84).
x1.7Hardy et al. (1967, pp. 1{32, 130{147, 151).
x1.8Protter and Morrey (1991, Chapter 10), Tolstov
(1962, Chapter 1 and p. 77), Titchmarsh (1962,
Chapter 13 and pp. 419, 421). For the Riemann{
Lebesgue lemma see Olver (1997b, p. 73). For
Poisson's summation formula see Rademacher
(1973, pp. 71{75), Titchmarsh (1986a, p. 61). For
(1.8.16) set f(x) =e !x2in (1.8.14).
x1.9Copson (1935, Chapters 1{3 and pp. 56{69, 92{
98), Levinson and Redhe er (1970, Chapters 1{
3, and pp. 259{277, 349{351, 360), Markushevich
(1983, pp. 14{18, 41{46, 131{135), Markushevich
(1985, vol. 1,x34), Ahlfors (1966, pp. 168{169).
For a proof of the Jordan Curve Theorem see, for
example, Dienes (1931, pp. 177{197). The theo-
rem is valid with less restrictive conditions than
those assumed here. For the operations on series,
see Henrici (1974, Chapter 1) or Olver (1997b,
pp. 19{22). For (1.9.69){(1.9.71), see Titchmarsh
(1962,x1.77).x1.10 Copson (1935, pp. 72{81, 106{113, 117{120,
192{193, 438{440), Levinson and Redhe er (1970,
pp. 64{77, 140{143, 162{170, 392{395, 398{
402), Markushevich (1983, pp. 106{121, 234{245),
Titchmarsh (1962, pp. 13{19, 165{169, 246{250).
For (1.10.13) and (1.10.14) see Copson (1935,
x6.23). See also Andrews et al. (1999, pp. 629{
631) and Henrici (1974, pp. 57{59). The Extended
Inversion Theorem is proved in a similar way.
x1.11 Burnside and Panton (1960, Chapter 2 and
pp. 80{81), Dummit and Foote (1999, pp. 300{
301, 591{595, 611{616), Henrici (1977, vol. 2,
pp. 555{559). For the Horner scheme, see Burn-
side and Panton (1960, pp. 8{9). The double
Horner scheme is derived similarly.
x1.12 Jones and Thron (1980, pp. 20, 31{37, 42{43, 88,
92), Lorentzen and Waadeland (1992, pp. 8{9, 30,
32, 84{85).
x1.13 Olver (1997b, pp. 141{142, 145{147, 190{191),
Temme (1996a, pp. 84, 103), Watson (1944,
pp. 145{146). For (1.13.10) see Simmons (1972,
pp. 90{92).
x1.14 Titchmarsh (1986a, pp. 3{15, 42, 50{60, 119{
132, and 176{210), Schi (1999, pp. 12{57, 91{93,
151{157, and 209{218), Paris and Kaminski (2001,
pp. 79{89), Wong (1989, pp. 147{152 and 192{
194), Henrici (1986, vol. 3, pp. 197{202), Wid-
der (1941, pp. 325{328, 340{341), Davies (1984,
pp. 11{13, 103{108, 152{153, 209{211), Pinkus
and Zafrany (1997, pp. 147{149). For (1.14.46)
see Sneddon (1972, p. 234).
x1.15 Hardy (1949, pp. 10, 154{155), Weiss (1965,
pp. 131{135, 143{148), Andrews et al. (1999,
pp. 111{114, 602{607), Wong (1989, pp. 197{
198), Widder (1941, Chapter 5). For (1.15.24)
see K orner (1989, Chapters 2, 27).
x1.16 Wong (1989, pp. 241{254, 261{279).
x1.17 (1.17.6) is a special case of Theorem 7.1 of
Olver (1997b, Chapter 3) when (a)6= 0. This
theorem also extends straightforwardly to cover
(a) = 0. (1.17.7) is proved in a similar man-
ner. For (1.17.10) complete the square in the to-
tal power of e, make the change of variable =
(t=(2pn) i(x a)pn, and useR1
1e 2d =p.
Chapter 2
Asymptotic Approximations
F. W. J. Olver1and R. Wong2
Areas 42
2.1 De nitions and Elementary Properties . . 42
2.2 Transcendental Equations . . . . . . . . . 43
2.3 Integrals of a Real Variable . . . . . . . . 43
2.4 Contour Integrals . . . . . . . . . . . . . 46
2.5 Mellin Transform Methods . . . . . . . . 48
2.6 Distributional Methods . . . . . . . . . . 512.7 Di erential Equations . . . . . . . . . . . 55
2.8 Di erential Equations with a Parameter . 58
2.9 Di erence Equations . . . . . . . . . . . 61
2.10 Sums and Sequences . . . . . . . . . . . 63
2.11 Remainder Terms; Stokes Phenomenon . 66
References 69
1Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland.
2Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Kowloon, Hong Kong.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
41
42 Asymptotic Approximations
Areas
2.1 De nitions and Elementary Properties
2.1(i) Asymptotic and Order Symbols
LetXbe a point set with a limit point c. Asx!cin
X
f(x) (x)()f(x)= (x)!1: 2.1.1
f(x) =o( (x))()f(x)= (x)!0: 2.1.2
f(x) =O( (x))()jf(x)= (x)jis bounded. 2.1.3
The symbol Ocan also apply to the whole set X, and
not just as x!c.
Examples
2.1.4 tanhxx,x!0 inC.
2.1.5 e x=o(1),x!+1inR.
2.1.6 sin
x+x 1
=O
x 1
,x!1 inZ.
2.1.7 eix=O(1), x2R.
In (2.1.5) Rcan be replaced by any xed ray in the
sectorjphxj<1
2, or by the whole of the sector
jphxj1
2 . (Here and elsewhere in this chapter
is an arbitrary small positive constant.) But (2.1.5)
does not hold as x!1 injphxj<1
2(for example,
setx= 1 +itand lett!1 .)
IfP1
s=0aszsconverges for all suciently small jzj,
then for each nonnegative integer n
2.1.81X
s=naszs=O(zn),z!0 inC.
Example
2.1.9 ez= 1 +z+O
z2
,z!0 inC.
The symbols oandOcan be used generically. For
example,
2.1.10o( ) =O( ) ,o( ) +o( ) =o( ) ,
it being understood that these equalities are not re-
versible. (In other words = here really means .)
2.1(ii) Integration and Di erentiation
Integration of asymptotic and order relations is permis-
sible, subject to obvious convergence conditions. For
example, suppose f(x) is continuous and f(x)xas
x!+1inR, where(2C) is a constant. Then
2.1.11Z1
xf(t)dt x+1
+ 1,< < 1,
2.1.12Z
f(x)dx8
><
>:a constant,< < 1;
lnx; = 1;
x+1=(+ 1);< > 1:Di erentiation requires extra conditions. For exam-
ple, iff(z) is analytic for all suciently large jzjin a
sector Sandf(z) =O(z) asz!1 inS,being real,
thenf0(z) =O
z 1
asz!1 in any closed sector
properly interior to Sand with the same vertex ( Ritt's
theorem ). This result also holds with both O's replaced
byo's.
2.1(iii) Asymptotic Expansions
LetPasx sbe a formal power series (convergent or
divergent) and for each positive integer n,
2.1.13 f(x) =n 1X
s=0asx s+O
x n
asx!1 in an unbounded set XinRorC. ThenPasx sis aPoincar e asymptotic expansion , or simply
asymptotic expansion , off(x) asx!1 inX. Symbol-
ically,
2.1.14f(x)a0+a1x 1+a2x 2+,x!1 inX.
Condition (2.1.13) is equivalent to
2.1.15xn
f(x) n 1X
s=0asx s!
!an,x!1 inX;
for eachn= 0;1;2;:::. IfPasx sconverges for all
suciently largejxj, then it is automatically the asymp-
totic expansion of its sum as x!1 inC.
Ifcis a nite limit point of X, then
2.1.16
f(x)a0+a1(x c) +a2(x c)2+,x!cinX;
means that for each n, the di erence between f(x)
and thenth partial sum on the right-hand side is
O((x c)n) asx!cinX.
Most operations on asymptotic expansions can be
carried out in exactly the same manner as for conver-
gent power series. These include addition, subtraction,
multiplication, and division. Substitution, logarithms,
and powers are also permissible; compare Olver (1997b,
pp. 19{22). Di erentiation, however, requires the kind
of extra conditions needed for the Osymbol (x2.1(ii)).
For reversion see x2.2.
Asymptotic expansions of the forms (2.1.14),
(2.1.16) are unique. But for any given set of coecients
a0;a1;a2;:::, and suitably restricted Xthere is an in-
nity of analytic functions f(x) such that (2.1.14) and
(2.1.16) apply. For (2.1.14) Xcan be the positive real
axis or any unbounded sector in Cof nite angle. As
an example, in the sector jphzj1
2 (<1
2) each
of the functions 0 ;e z, ande pz(principal value) has
the null asymptotic expansion
2.1.17 0 + 0z 1+ 0z 2+,z!1 .
2.2 Transcendental Equations 43
2.1(iv) Uniform Asymptotic Expansions
If the set Xinx2.1(iii) is a closed sector phx
, then by de nition the asymptotic property (2.1.13)
holds uniformly with respect to ph x2[ ; ] asjxj!
1. The asymptotic property may also hold uniformly
with respect to parameters. Suppose uis a parameter
(or set of parameters) ranging over a point set (or sets)
U, and for each nonnegative integer n xn
f(u;x) n 1X
s=0as(u)x s!
is bounded as x!1 inX, uniformly for u2U. (The
coecients as(u) may now depend on u.) Then
2.1.18 f(u;x)1X
s=0as(u)x s
asx!1 inX, uniformly with respect to u2U.
Similarly for nite limit point cin place of1.
2.1(v) Generalized Asymptotic Expansions
Let s(x),s= 0;1;2;:::, be a sequence of functions
de ned in Xsuch that for each s
2.1.19 s+1(x) =o( s(x)),x!cinX,
wherecis a nite, or in nite, limit point of X. Then
f s(x)gis an asymptotic sequence orscale. Suppose
also thatf(x) andfs(x) satisfy
2.1.20 f(x) =n 1X
s=0fs(x) +O( n(x)),x!cinX,
forn= 0;1;2;:::. ThenPfs(x) is a generalized
asymptotic expansion off(x)with respect to the scale
f s(x)g. Symbolically,
2.1.21 f(x)1X
s=0fs(x);f s(x)g,x!cinX:
As inx2.1(iv), generalized asymptotic expansions can
also have uniformity properties with respect to param-
eters. For an example see x14.15(i).
Care is needed in understanding and manipulating
generalized asymptotic expansions. Many properties
enjoyed by Poincar e expansions (for example, multipli-
cation) do not always carry over. It can even happen
that a generalized asymptotic expansion converges, but
its sum is not the function being represented asymptot-
ically; for an example see x18.15(iii).
2.2 Transcendental Equations
Letf(x) be continuous and strictly increasing when
a<x<1and
2.2.1 f(x)x, x!1 .
Then fory >f (a) the equation f(x) =yhas a unique
rootx=x(y) in (a;1), and
2.2.2 x(y)y, y!1 .Example
2.2.3 t2 lnt=y:
Withx=t2,f(x) =x 1
2lnx. We may take a=1
2.
From (2.2.2)
2.2.4 t=y1
2(1 +o(1)) , y!1 .
Higher approximations are obtainable by successive re-
substitutions. For example
2.2.5 t2=y+ lnt=y+1
2lny+o(1);
and hence
2.2.6 t=y1
2
1 +1
4y 1lny+o
y 1
,y!1 .
An important case is the reversion of asymptotic
expansions for zeros of special functions. In place of
(2.2.1) assume that
2.2.7f(x)x+f0+f1x 1+f2x 2+,x!1 .
Then
2.2.8xy F0 F1y 1 F2y 2 ,y!1 ,
whereF0=f0andsFs(s1) is the coecient of
x 1in the asymptotic expansion of ( f(x))s(Lagrange's
formula for the reversion of series ). Conditions for the
validity of the reversion process in Care derived in Olver
(1997b, pp. 14{16). Applications to real and complex
zeros of Airy functions are given in Fabijonas and Olver
(1999). For other examples see de Bruijn (1961, Chap-
ter 2).
2.3 Integrals of a Real Variable
2.3(i) Integration by Parts
Assume that the Laplace transform
2.3.1Z1
0e xtq(t)dt
converges for all suciently large x, andq(t) is in nitely
di erentiable in a neighborhood of the origin. Then
2.3.2Z1
0e xtq(t)dt1X
s=0q(s)(0)
xs+1,x!+1.
If, in addition, q(t) is in nitely di erentiable on
[0;1) and
2.3.3n= sup
(0;1)(t 1lnjq(n)(t)=q(n)(0)j)
is nite and bounded for n= 0;1;2;:::, then thenth
error term (that is, the di erence between the integral
andnth partial sum in (2.3.2)) is bounded in absolute
value byjq(n)(0)=(xn(x n))jwhenxexceeds both 0
andn.
44 Asymptotic Approximations
For the Fourier integral
Zb
aeixtq(t)dt
assumeaandbare nite, and q(t) is in nitely di eren-
tiable on [a;b]. Then
2.3.4Zb
aeixtq(t)dteiax1X
s=0q(s)(a)i
xs+1
eibx1X
s=0q(s)(b)i
xs+1
,
x!+1.
Alternatively, assume b=1,q(t) is in nitely di eren-
tiable on [a;1), and each of the integralsR
eixtq(s)(t)dt,
s= 0;1;2;:::, converges as t!1 uniformly for all suf-
ciently large x. Then
2.3.5Z1
aeixtq(t)dteiax1X
s=0q(s)(a)i
xs+1
,x!+1.
In both cases the nth error term is bounded in abso-
lute value by x nVa;b
q(n 1)(t)
, where the variational
operatorVa;bis de ned by
2.3.6Va;b(f(t)) =Zb
ajf0(t)dtj;
seex1.4(v). For other examples, see Wong (1989, Chap-
ter 1).
2.3(ii) Watson's Lemma
Assume again that the integral (2.3.1) converges for all
suciently large x, but now
2.3.7 q(t)1X
s=0ast(s+ )=,t!0+,
whereandare positive constants. Then the series
obtained by substituting (2.3.7) into (2.3.1) and inte-
grating formally term by term yields an asymptotic ex-
pansion:
2.3.8Z1
0e xtq(t)dt1X
s=0 s+
as
x(s+)=,x!+1.
For the function see x5.2(i).
This result is probably the most frequently used
method for deriving asymptotic expansions of special
functions. Since q(t) need not be continuous (as long as
the integral converges), the case of a nite integration
range is included.Other types of singular behavior in the integrand
can be treated in an analogous manner. For example,
2.3.9Z1
0e xtq(t) lntdt1X
s=0 0s+
as
x(s+)=
(lnx)1X
s=0 s+
as
x(s+)=;
provided that the integral on the left-hand side of (2.3.9)
converges for all suciently large values of x. (In other
words, di erentiation of (2.3.8) with respect to the pa-
rameter(or) is legitimate.)
Another extension is to more general factors than
the exponential function. In addition to (2.3.7) assume
thatf(t) andq(t) are piecewise continuous ( x1.4(ii)) on
(0;1), and
2.3.10jf(t)jAexp( at), 0t<1,
2.3.11 q(t) =O(exp(bt)), t!+1,
whereA;a;b; are positive constants. Then
2.3.12Z1
0f(xt)q(t)dt1X
s=0M
f;s+
as
x(s+)=,
x!+1,
whereM(f; ) is the Mellin transform off(t) (x2.5(i)).
For a more detailed treatment of the integral (2.3.12)
seexx2.5, 2.6.
2.3(iii) Laplace's Method
Whenp(t) is real and xis a large positive parameter,
the main contribution to the integral
2.3.13 I(x) =Zb
ae xp(t)q(t)dt
derives from the neighborhood of the minimum of p(t)
in the integration range. Without loss of generality, we
assume that this minimum is at the left endpoint a.
Furthermore:
(a)p0(t) andq(t) are continuous in a neighborhood of
a, save possibly at a, and the minimum of p(t) in
[a;b) is approached only at a.
(b) Ast!a+
2.3.14p(t)p(a) +1X
s=0ps(t a)s+;
q(t)1X
s=0qs(t a)s+ 1;
and the expansion for p(t) is di erentiable. Again
andare positive constants. Also p0>0 (con-
sistent with (a)).
(c) The integral (2.3.13) converges absolutely for all
suciently large x.
2.3 Integrals of a Real Variable 45
Then
2.3.15Zb
ae xp(t)q(t)dte xp(a)1X
s=0 s+
bs
x(s+)=,
x!+1,
where the coecients bsare de ned by the expansion
2.3.16q(t)
p0(t)1X
s=0bsv(s+ )=,v!0+,
in whichv=p(t) p(a). For example,
2.3.17b0=q0
p=
0;
b1=q1
(+ 1)p1q0
2p01
p(+1)=
0;
b2=q2
(+ 2)(p1q1+p2q0)
2p0
+(+ 2)(++ 2)p2
1q0
23p2
01
p(+2)=
0:
In general
2.3.18
bs=1
res
t=aq(t)
(p(t) p(a))(+s)=
,s= 0;1;2;:::.
Watson's lemma can be regarded as a special case of
this result.
For error bounds for Watson's lemma and Laplace's
method see Boyd (1993) and Olver (1997b, Chapter 3).
These references and Wong (1989, Chapter 2) also con-
tain examples.
2.3(iv) Method of Stationary Phase
When the parameter xis large the contributions from
the real and imaginary parts of the integrand in
2.3.19 I(x) =Zb
aeixp(t)q(t)dt
oscillate rapidly and cancel themselves over most of the
range. However, cancellation does not take place near
the endpoints, owing to lack of symmetry, nor in the
neighborhoods of zeros of p0(t) becausep(t) changes rel-
atively slowly at these stationary points.
The rst result is the analog of Watson's lemma
(x2.3(ii)). Assume that q(t) again has the expansion
(2.3.7) and this expansion is in nitely di erentiable, q(t)
is in nitely di erentiable on (0 ;1), and each of the inte-
gralsR
eixtq(s)(t)dt,s= 0;1;2;:::, converges at t=1,
uniformly for all suciently large x. Then
2.3.20Z1
0eixtq(t)dt
1X
s=0exp(s+)i
2
s+
as
x(s+)=,
x!+1,where the coecients asare given by (2.3.7).
For the more general integral (2.3.19) we assume,
without loss of generality, that the stationary point (if
any) is at the left endpoint. Furthermore:
(a) On (a;b),p(t) andq(t) are in nitely di erentiable
andp0(t)>0.
(b) Ast!a+ the asymptotic expansions (2.3.14) ap-
ply, and each is in nitely di erentiable. Again ,
, andp0are positive.
(c) If the limit p(b) ofp(t) ast!b is nite, then
each of the functions
2.3.21Ps(t) =1
p0(t)d
dtsq(t)
p0(t),s= 0;1;2;:::,
tends to a nite limit Ps(b).
(d) Ifp(b) =1, thenP0(b) = 0 and each of the inte-
grals
2.3.22Z
eixp(t)Ps(t)p0(t)dt,s= 0;1;2;:::,
converges at t=buniformly for all suciently
largex.
Ifp(b) is nite, then both endpoints contribute:
2.3.23Zb
aeixp(t)q(t)dt
eixp(a)1X
s=0exp(s+)i
2
s+
bs
x(s+)=
eixp(b)1X
s=0Ps(b)i
xs+1
, x!+1:
But if (d) applies, then the second sum is absent. The
coecients bsare de ned as in x2.3(iii).
For proofs of the results of this subsection, error
bounds, and an example, see Olver (1974). For other
estimates of the error term see Lyness (1971). For ex-
tensions to oscillatory integrals with logarithmic singu-
larities see Wong and Lin (1978).
2.3(v) Coalescing Peak and Endpoint:
Bleistein's Method
In the integral
2.3.24I( ;x) =Zk
0e xp( ;t)q( ;t)t 1dt
k(1 ) andare positive constants, is a variable
parameter in an interval 1 2with 10 and
0< 2k, andxis a large positive parameter. As-
sume also that @2p( ;t)
@t2andq( ;t) are continuous
in andt, and for each the minimum value of p( ;t)
46 Asymptotic Approximations
in [0;k) is att= , at which point @p( ;t)/@tvan-
ishes, but both @2p( ;t)
@t2andq( ;t) are nonzero.
Whenx!+1Laplace's method ( x2.3(iii)) applies,
but the form of the resulting approximation is discon-
tinuous at = 0. In consequence, the approximation is
nonuniform with respect to and deteriorates severely
as !0.
A uniform approximation can be constructed by
quadratic change of integration variable:
2.3.25 p( ;t) =1
2w2 aw+b;
whereaandbare functions of chosen in such a way
thatt= 0 corresponds to w= 0, and the stationary
pointst= andw=acorrespond. Thus
2.3.26a= (2p( ;0) 2p( ; ))1=2; b =p( ;0);
2.3.27
w= (2p( ;0) 2p( ; ))1=2(2p( ;t) 2p( ; ))1=2;
the upper or lower sign being taken according as t? .
The relationship between tandwis one-to-one, and
because
2.3.28dw
dt=1
(2p( ;t) 2p( ; ))1=2@p( ;t)
@t
it is free from singularity at t= .
The integral (2.3.24) transforms into
2.3.29
I( ;x) =e xp( ;0)
Z
0exp
x 1
2w2 aw
f( ;w)w 1dw;
where
2.3.30 f( ;w) =q( ;t)t
w 1dt
dw;
=( ) being the value of watt=k. We now expand
f( ;w) in a Taylor series centered at the peak value
w=aof the exponential factor in the integrand:
2.3.31 f( ;w) =1X
s=0 s( )(w a)s;
with the coecients s( ) continuous at = 0. The
desired uniform expansion is then obtained formally as
in Watson's lemma and Laplace's method. We replace
the limitby1and integrate term-by-term:
2.3.32I( ;x)e xp( ;0)
x=21X
s=0 s( )Fs(apx)
xs=2,x!1 ,
where
2.3.33Fs(y) =Z1
0exp
1
2 2+y
( y)s 1d :
For examples and proofs see Olver (1997b, Chapter
9), Bleistein (1966), Bleistein and Handelsman (1975,
Chapter 9), and Wong (1989, Chapter 7).2.4 Contour Integrals
2.4(i) Watson's Lemma
The result inx2.3(ii) carries over to a complex param-
eterz. Except that is now permitted to be complex,
with<>0, we assume the same conditions on q(t) and
also that the Laplace transform in (2.3.8) converges for
all suciently large values of <z. Then
2.4.1Z1
0e ztq(t)dt1X
s=0 s+
as
z(s+)=
asz!1 in the sectorjphzj1
2 (<1
2), with
z(s+)=assigned its principal value.
Ifq(t) is analytic in a sector 1<pht < 2con-
taining pht= 0, then the region of validity may be
increased by rotation of the integration paths. We as-
sume that in any closed sector with vertex t= 0 and
properly interior to 1<pht < 2, the expansion
(2.3.7) holds as t!0, andq(t) =O
ejtj
ast!1 ,
whereis a constant. Then (2.4.1) is valid in any
closed sector with vertex z= 0 and properly interior
to 2 1
2 < phz < 1+1
2. (The branches of
t(s+ )=andz(s+)=are extended by continuity.)
For examples and extensions (including uniformity
and loop integrals) see Olver (1997b, Chapter 4), Wong
(1989, Chapter 1), and Temme (1985).
2.4(ii) Inverse Laplace Transforms
On the interval 0 < t <1letq(t) be di erentiable
ande ctq(t) be absolutely integrable, where cis a real
constant. Then the Laplace transform
2.4.2 Q(z) =Z1
0e ztq(t)dt
is continuous in<zcand analytic in<z >c , and by
inversion (x1.14(iii))
2.4.3q(t) =1
2ilim
!1Z+i
ietzQ(z)dz, 0<t<1,
where(c) is a constant.
Now assume that c>0 and we are given a function
Q(z) that is both analytic and has the expansion
2.4.4 Q(z)1X
s=0 s+
as
z(s+)=,z!1 ,
in the half-plane <zc. Here< > 0, > 0, and
z(s+)=has its principal value. Assume also (2.4.4) is
di erentiable. Then by integration by parts the integral
2.4.5 q(t) =1
2iZ+i1
i1etzQ(z)dz, 0<t<1,
is seen to converge absolutely at each limit, and be in-
dependent of 2[c;1). Furthermore, as t!0+,q(t)
has the expansion (2.3.7).
2.4 Contour Integrals 47
For larget, the asymptotic expansion of q(t) may be
obtained from (2.4.3) by Haar's method. This depends
on the availability of a comparison function F(z) for
Q(z) that has an inverse transform
2.4.6f(t) =1
2ilim
!1Z+i
ietzF(z)dz
with known asymptotic behavior as t!+1. By sub-
traction from (2.4.3)
2.4.7
q(t) f(t) =et
2lim
!1Z
eit (Q(+i ) F(+i ))d :
If this integral converges uniformly at each limit for
all suciently large t, then by the Riemann{Lebesgue
lemma (x1.8(i))
2.4.8 q(t) =f(t) +o
ect
,t!+1.
If, in addition, the corresponding integrals with Qand
Freplaced by their derivatives Q(j)andF(j),j=
1;2;:::;m , converge uniformly, then by repeated inte-
grations by parts
2.4.9 q(t) =f(t) +o
t mect
,t!+1.
The most successful results are obtained on moving the
integration contour as far to the left as possible. For
examples see Olver (1997b, pp. 315{320).
2.4(iii) Laplace's Method
LetPdenote the path for the contour integral
2.4.10 I(z) =Zb
ae zp(t)q(t)dt;
in whichais nite,bis nite or in nite, and !is the an-
gle of slope of Pata, that is, lim(ph( t a)) ast!a
alongP. Assume that p(t) andq(t) are analytic on
an open domain Tthat contains P, with the possible
exceptions of t=aandt=b. Other assumptions are:
(a) In a neighborhood of a
2.4.11p(t) =p(a) +1X
s=0ps(t a)s+;
q(t) =1X
s=0qs(t a)s+ 1;
with<>0,>0,p06= 0, and the branches of
(t a)and (t a)continuous and constructed
with ph(t a)!!ast!aalongP.
(b)zranges along a ray or over an annular sector
12,jzjZ, where= phz,2 1<,
andZ > 0.I(z) converges at babsolutely and
uniformly with respect to z.(c) Excluding t=a,<
eip(t) eip(a)
is positive
whent2P, and is bounded away from zero uni-
formly with respect to 2[1;2] ast!balong
P.
Then
2.4.12I(z)e zp(a)1X
s=0 s+
bs
z(s+)=
asz!1 in the sector 1phz2. The coecients
bsare determined as in x2.3(iii), the branch of ph p0
being chosen to satisfy
2.4.13j+!+ php0j1
2:
For examples see Olver (1997b, Chapter 4). For er-
ror bounds see Boyd (1993).
2.4(iv) Saddle Points
Now suppose that in (2.4.10) the minimum of <(zp(t))
onPoccurs at an interior point t0. Temporarily as-
sume that(= phz) is xed, so that t0is independent
ofz. We may subdivide
2.4.14I(z) =Zb
t0e zp(t)q(t)dt Za
t0e zp(t)q(t)dt;
and apply the result of x2.4(iii) to each integral on the
right-hand side, the role of the series (2.4.11) being
played by the Taylor series of p(t) andq(t) att=t0.
Ifp0(t0)6= 0, then= 1,is a positive integer, and
the two resulting asymptotic expansions are identical.
Thus the right-hand side of (2.4.14) reduces to the er-
ror terms. However, if p0(t0) = 0, then 2 and
di erent branches of some of the fractional powers of p0
are used for the coecients bs; again seex2.3(iii). In
consequence, the asymptotic expansion obtained from
(2.4.14) is no longer null.
Zeros ofp0(t) are called saddle points (orcols) owing
to the shape of the surface jp(t)j,t2C, in their vicin-
ity. Cases in which p0(t0)6= 0 are usually handled by
deforming the integration path in such a way that the
minimum of<(zp(t)) is attained at a saddle point or at
an endpoint. Additionally, it may be advantageous to
arrange that=(zp(t)) is constant on the path: this will
usually lead to greater regions of validity and sharper er-
ror bounds. Paths on which =(zp(t)) is constant are also
the ones on which jexp( zp(t))jdecreases most rapidly.
For this reason the name method of steepest descents is
often used. However, for the purpose of simply deriving
the asymptotic expansions the use of steepest descent
paths is not essential.
In the commonest case the interior minimum t0of
<(zp(t)) is a simple zero of p0(t). The nal expansion
48 Asymptotic Approximations
then has the form
2.4.15Zb
ae zp(t)q(t)dt2e zp(t0)1X
s=0
s+1
2b2s
zs+(1=2);
in which
2.4.16
b0=q
(2p00)1=2;
b2=
2q00 2p000q0
p00+5(p000)2
6(p00)2 piv
2p00
q1
(2p00)3=2;
withp;qand their derivatives evaluated at t0. The
branch of!0= ph(p00(t0)) is the one satisfying j+2!+
!0j1
2, where!is the limiting value of ph( t t0) as
t!t0fromb.
Higher coecients b2sin (2.4.15) can be found from
(2.3.18) with = 1,= 2, andsreplaced by 2 s. For
integral representations of the b2sand their asymptotic
behavior as s!1 see Boyd (1995). The last reference
also includes examples, as do Olver (1997b, Chapter 4),
Wong (1989, Chapter 2), and Bleistein and Handelsman
(1975, Chapter 7).
2.4(v) Coalescing Saddle Points: Chester,
Friedman, and Ursell's Method
Consider the integral
2.4.17 I( ;z) =Z
Pe zp( ;t)q( ;t)dt
in whichzis a large real or complex parameter, p( ;t)
andq( ;t) are analytic functions of tand continuous in t
and a second parameter . Suppose that on the integra-
tion path Pthere are two simple zeros of @p( ;t)/@t
that coincide for a certain value b of . The problem of
obtaining an asymptotic approximation to I( ;z) that
is uniform with respect to in a region containing b
is similar to the problem of a coalescing endpoint and
saddle point outlined in x2.3(v).
The change of integration variable is given by
2.4.18 p( ;t) =1
3w3+aw2+bw+c;
withaandbchosen so that the zeros of @p( ;t)/@tcor-
respond to the zeros w1( );w2( ), say, of the quadratic
w2+ 2aw+b. Then
2.4.19
I( ;z)
=e czZ
Qexp
z 1
3w3+aw2+bw
f( ;w)dw;
whereQis thew-map of P, and
2.4.20f( ;w) =q( ;t)dt
dw=q( ;t)w2+ 2aw+b
@p( ;t)/@t:
The function f( ;w) is analytic at w=w1( ) and
w=w2( ) when 6=b , and at the con
uence of thesepoints when =b . For largejzj,I( ;z) is approx-
imated uniformly by the integral that corresponds to
(2.4.19) when f( ;w) is replaced by a constant. By
making a further change of variable
2.4.21 w=z 1=3v a;
and assigning an appropriate value to cto modify the
contour, the approximating integral is reducible to an
Airy function or a Scorer function ( xx9.2, 9.12).
For examples, proofs, and extensions see Olver
(1997b, Chapter 9), Wong (1989, Chapter 7), Olde
Daalhuis and Temme (1994), Chester et al. (1957), and
Bleistein and Handelsman (1975, Chapter 9).
For a symbolic method for evaluating the coecients
in the asymptotic expansions see Vid unas and Temme
(2002).
2.4(vi) Other Coalescing Critical Points
The problems sketched in xx2.3(v) and 2.4(v) involve
only two of many possibilities for the coalescence of end-
points, saddle points, and singularities in integrals asso-
ciated with the special functions. For a coalescing sad-
dle point and a pole see Wong (1989, Chapter 7) and van
der Waerden (1951); in this case the uniform approxi-
mants are complementary error functions. For a coalesc-
ing saddle point and endpoint see Olver (1997b, Chapter
9) and Wong (1989, Chapter 7); if the endpoint is an
algebraic singularity then the uniform approximants are
parabolic cylinder functions with xed parameter, and
if the endpoint is not a singularity then the uniform
approximants are complementary error functions.
For two coalescing saddle points and an endpoint see
Leubner and Ritsch (1986). For two coalescing saddle
points and an algebraic singularity see Temme (1986),
Jin and Wong (1998). For a coalescing saddle point,
a pole, and a branch point see Ciarkowski (1989). For
many coalescing saddle points see x36.12. For double
integrals with two coalescing stationary points see Qiu
and Wong (2000).
2.5 Mellin Transform Methods
2.5(i) Introduction
Letf(t) be a locally integrable function on (0 ;1), that
is,RT
f(t)dtexists for all andTsatisfying 0 <<
T <1. The Mellin transform off(t) is de ned by
2.5.1 M(f;z) =Z1
0tz 1f(t)dt;
when this integral converges. The domain of analytic-
ity ofM(f;z) is usually an in nite strip a <<z < b
parallel to the imaginary axis. The inversion formula is
given by
2.5.2f(t) =1
2iZc+i1
c i1t zM(f;z)dz;
2.5 Mellin Transform Methods 49
witha<c<b .
One of the two convolution integrals associated with
the Mellin transform is of the form
2.5.3 I(x) =Z1
0f(t)h(xt)dt,x>0,
and
2.5.4 M(I;z) =M(f; 1 z)M(h;z):
IfM(f; 1 z) andM(h;z) have a common strip of
analyticity a<<z<b , then
2.5.5I(x) =1
2iZc+i1
c i1x zM(f; 1 z)M(h;z)dz;
wherea < c < b . Whenx= 1, this identity is a
Parseval-type formula; compare x1.14(iv).
IfM(f; 1 z) andM(h;z) can be continued ana-
lytically to meromorphic functions in a left half-plane,
and if the contour <z=ccan be translated to <z=d
withd<c , then
2.5.6
I(x) =X
d<<z<cres
x zM(f; 1 z)M(h;z)
+E(x);
where
2.5.7E(x) =1
2iZd+i1
d i1x zM(f; 1 z)M(h;z)dz:
The sum in (2.5.6) is taken over all poles of
x zM(f; 1 z)M(h;z) in the strip d<<z <c , and
it provides the asymptotic expansion of I(x) for small
values ofx. Similarly, if M(f; 1 z) andM(h;z) can
be continued analytically to meromorphic functions in
a right half-plane, and if the vertical line of integra-
tion can be translated to the right, then we obtain an
asymptotic expansion for I(x) for large values of x.
Example
2.5.8 I(x) =Z1
0J2
(xt)
1 +tdt, > 1
2,
whereJdenotes the Bessel function ( x10.2(ii)), and
xis a large positive parameter. Let h(t) =J2
(t) and
f(t) = 1=(1 +t). Then from Table 1.14.5 and Watson
(1944, p. 403)
2.5.9 M(f; 1 z) =
sin(z), 0<<z<1,
2.5.10
M(h;z) =2z 1
+1
2z
2
1 1
2z
1 + 1
2z
(z)
sin(z),
2 <<z<1.
In the half-plane <z > max(0; 2), the product
M(f; 1 z)M(h;z) has a pole of order two at each
positive integer, and
2.5.11
res
z=n
x zM(f; 1 z)M(h;z)
= (anlnx+bn)x n;where
2.5.12an=2n 1
+1
2n
2
1 1
2n
1 + 1
2n
(n);
2.5.13bn= an
ln 2 +1
2
+1
2n
+
1 1
2n
+1
2
1 + 1
2n
(n)
;
and is the logarithmic derivative of the gamma func-
tion (x5.2(i)).
We now apply (2.5.5) with max(0 ; 2)< c < 1,
and then translate the integration contour to the right.
This is allowable in view of the asymptotic formula
2.5.14j (x+iy)j=p
2e jyj=2jyjx (1=2)(1 +o(1));
asy!1 , uniformly for bounded jxj; see (5.11.9).
Then as in (2.5.6) and (2.5.7), with d= 2n+ 1 (0<
<1), we obtain
2.5.15I(x) = 2nX
s=0(aslnx+bs)x s+O
x 2n 1+
,
n= 0;1;2;:::.
From (2.5.12) and (2.5.13), it is seen that as=bs= 0
whensis even. Hence
2.5.16I(x) =n 1X
s=0(cslnx+ds)x 2s 1+O
x 2n 1+
;
wherecs= a2s+1,ds= b2s+1.
2.5(ii) Extensions
Letf(t) andh(t) be locally integrable on (0 ;1) and
2.5.17 f(t)1X
s=0ast s, t!0+,
where< s>< s0fors > s0, and< s!+1as
s!1 . Also, let
2.5.18 h(t)exp(itp)1X
s=0bst s,t!+1,
whereis real,p > 0,< s>< s0fors > s0, and
< s!+1ass!1 . To ensure that the integral
(2.5.3) converges we assume that
2.5.19 f(t) =O
t b
, t!+1,
withb+< 0>1, and
2.5.20 h(t) =O(tc), t!0+,
withc+< 0> 1. To apply the Mellin transform
method outlined in x2.5(i), we require the transforms
M(f; 1 z) andM(h;z) to have a common strip of an-
alyticity. This, in turn, requires b<< 0, c<< 0,
and either c << 0+ 1 or 1 b << 0. Following
Handelsman and Lew (1970, 1971) we now give an ex-
tension of this method in which none of these conditions
is required.
50 Asymptotic Approximations
First, we introduce the truncated functions f1(t) and
f2(t) de ned by
2.5.21 f1(t) =(
f(t);0<t1;
0; 1<t<1;
2.5.22 f2(t) =f(t) f1(t):
Similarly,
2.5.23 h1(t) =(
h(t);0<t1;
0; 1<t<1;
2.5.24 h2(t) =h(t) h1(t):
With these de nitions and the conditions (2.5.17){
(2.5.20) the Mellin transforms converge absolutely and
de ne analytic functions in the half-planes shown in Ta-
ble 2.5.1.
Table 2.5.1 : Domains of convergence for Mellin trans-
forms.
Transform Domain of Convergence
M(f1;z)<z> < 0
M(f2;z)<z<b
M(h1;z)<z> c
M(h2;z)<z<< 0
Furthermore, M(f1;z) can be continued analyt-
ically to a meromorphic function on the entire z-
plane, whose singularities are simple poles at s,
s= 0;1;2;:::, with principal part
2.5.25 as=(z+ s):
By Table 2.5.1, M(f2;z) is an analytic function in
the half-plane<z <b . Hence we can extend the de ni-
tion of the Mellin transform of fby setting
2.5.26 M(f;z) =M(f1;z) +M(f2;z)
for<z < b . The extended transform M(f;z) has the
same properties as M(f1;z) in the half-plane <z<b .
Similarly, if = 0 in (2.5.18), then M(h2;z) can be
continued analytically to a meromorphic function on the
entirez-plane with simple poles at s,s= 0;1;2;:::,
with principal part
2.5.27 bs=(z s):
Alternatively, if 6= 0 in (2.5.18), then M(h2;z) can
be continued analytically to an entire function.
SinceM(h1;z) is analytic for <z > cby Table
2.5.1, the analytically-continued M(h2;z) allows us to
extend the Mellin transform of hvia
2.5.28 M(h;z) =M(h1;z) +M(h2;z)
in the same half-plane. From (2.5.26) and (2.5.28), it
follows that both M(f; 1 z) andM(h;z) are de ned
in the half-plane <z>max(1 b; c).We are now ready to derive the asymptotic expan-
sion of the integral I(x) in (2.5.3) as x!1 . First we
note that
2.5.29 I(x) =2X
j;k=1Ijk(x);
where
2.5.30 Ijk(x) =Z1
0fj(t)hk(xt)dt:
By direct computation
2.5.31 I21(x) = 0, for x1.
Next from Table 2.5.1 we observe that the integrals for
the transform pair M(fj; 1 z) andM(hk;z) are ab-
solutely convergent in the domain Djkspeci ed in Table
2.5.2, and these domains are nonempty as a consequence
of (2.5.19) and (2.5.20).
Table 2.5.2 : Domains of analyticity for Mellin trans-
forms.
Transform Pair Domain Djk
M(f1; 1 z);M(h1;z) c<<z<1 +< 0
M(f1; 1 z);M(h2;z)<z<min(1 +< 0;< 0)
M(f2; 1 z);M(h1;z) max( c;1 b)<<z
M(f2; 1 z);M(h2;z) 1 b<<z<< 0
For simplicity, write
2.5.32Gjk(z) =M(fj; 1 z)M(hk;z):
From Table 2.5.2, we see that each Gjk(z) is analytic
in the domain Djk. Furthermore, each Gjk(z) has an
analytic or meromorphic extension to a half-plane con-
tainingDjk. Now suppose that there is a real number
pjkinDjksuch that the Parseval formula (2.5.5) applies
and
2.5.33Ijk(x) =1
2iZpjk+i1
pjk i1x zGjk(z)dz:
If, in addition, there exists a number qjk> pjksuch
that
2.5.34 sup
pjkxqjkjGjk(x+iy)j!0,y!1 ,
then
2.5.35Ijk(x) =X
pjk<<z<qjkres
x zGjk(z)
+Ejk(x);
where
2.5.36Ejk(x) =1
2iZqjk+i1
qjk i1x zGjk(z)dz=o
x qjk
asx!+1. (The last order estimate follows from
the Riemann{Lebesgue lemma, x1.8(i).) The asymp-
totic expansion of I(x) is then obtained from (2.5.29).
For further discussion of this method and examples,
see Wong (1989, Chapter 3), Paris and Kaminski (2001,
Chapter 5), and Bleistein and Handelsman (1975, Chap-
ters 4 and 6). The rst reference also contains explicit
2.6 Distributional Methods 51
expressions for the error terms, as do Soni (1980) and
Carlson and Gustafson (1985).
The Mellin transform method can also be extended
to derive asymptotic expansions of multidimensional in-
tegrals having algebraic or logarithmic singularities, or
both; see Wong (1989, Chapter 3), Paris and Kaminski
(2001, Chapter 7), and McClure and Wong (1987). See
also Br uning (1984) for a di erent approach.
2.5(iii) Laplace Transforms with Small
Parameters
Leth(t) satisfy (2.5.18) and (2.5.20) with c> 1, and
consider the Laplace transform
2.5.37 L(h;) =Z1
0h(t)e tdt:
Putx= 1=and break the integration range at t= 1,
as in (2.5.23) and (2.5.24). Then
2.5.38 L(h;) =I1(x) +I2(x);
where
2.5.39 Ij(x) =Z1
0e thj(xt)dt,j= 1;2.
SinceM(e t;z) = (z), by the Parseval formula
(2.5.5), there are real numbers p1andp2such that
c<p 1<1,p2<min(1;< 0), and
2.5.40
Ij(x) =1
2iZpj+i1
pj i1x z (1 z)M(hj;z)dz,j= 1;2.
SinceM(h;z) is analytic for<z> c, by (2.5.14),
2.5.41I1(x) =M(h1; 1)x 1
+1
2iZ+i1
i1x z (1 z)M(h1;z)dz;
for anysatisfying 1 << 2. Similarly, since M(h2;z)
can be continued analytically to a meromorphic func-
tion (when = 0) or to an entire function (when
6= 0), we can choose so that M(h2;z) has no poles
in 1<<z<2. Thus
2.5.42I2(x) =X
< 0<z1res
x z (1 z)M(h2;z)
+1
2iZ+i1
i1x z (1 z)M(h2;z)dz:
On substituting (2.5.41) and (2.5.42) into (2.5.38),
we obtain
2.5.43
L(h;) =M(h1; 1)
+X
< 0<z1res
z 1 (1 z)M(h2;z)
+X
1<<z<lres
z 1 (1 z)M(h;z)
+1
2iZl +i1
l i1z 1 (1 z)M(h;z)dz;wherel(2) is an arbitrary integer and is an arbi-
trary small positive constant. The last term is clearly
O
l 1
as!0+.
If= 0 in (2.5.18) and c > 1 in (2.5.20), and if
none of the exponents in (2.5.18) are positive integers,
then the expansion (2.5.43) gives the following useful
result:
2.5.44L(h;)1X
n=0bn (1 n) n 1
+1X
n=0( )n
n!M(h;n+ 1),!0+.
Example
2.5.45 L(h;) =Z1
0e t
1 +tdt,< >0.
Withh(t) = 1=(1 +t), we have M(h;z) =csc(z) for
0<<z <1. In the notation of (2.5.18) and (2.5.20),
= 0, s=s+ 1, andc= 0. Straightforward calcula-
tion gives
2.5.46res
z=k
z 1 (1 z)csc(z)
= ( ln+ (k))k 1
(k 1)!;
where (z) = 0(z)= (z). From (2.5.28)
2.5.47res
z=1
z 1 (1 z)M(h2;z)
= ( ln
) M(h1; 1);
where
is Euler's constant ( x5.2(ii)). Insertion of these
results into (2.5.43) yields
2.5.48
L(h;)( ln)1X
k=0k
k!+1X
k=0 (k+ 1)k
k!,!0+.
To verify (2.5.48) we may use
2.5.49 L(h;) =eE1();
compare (6.2.2) and (6.6.3).
For examples in which the integral de ning the
Mellin transform M(h;z) does not exist for any value
ofz, see Wong (1989, Chapter 3), Bleistein and Han-
delsman (1975, Chapter 4), and Handelsman and Lew
(1970).
2.6 Distributional Methods
2.6(i) Divergent Integrals
Consider the integral
2.6.1 S(x) =Z1
01
(1 +t)1=3(x+t)dt;
wherex>0. Fort>1,
2.6.2 (1 +t) 1=3=1X
s=0 1
3
s
t s (1=3):
52 Asymptotic Approximations
Motivated by Watson's lemma ( x2.3(ii)), we substitute
(2.6.2) in (2.6.1), and integrate term by term. This leads
to integrals of the form
2.6.3Z1
0t s (1=3)
x+tdt,s= 1;2;3;:::.
Although divergent, these integrals may be interpreted
in a generalized sense. For instance, we have
2.6.4Z1
0t 1
(x+t) + dt= ( ) ( )
( + )1
x ,< >0,< >0.
But the right-hand side is meaningful for all values of
and , other than nonpositive integers. We may
therefore de ne the integral on the left-hand side of
(2.6.4) by the value on the right-hand side, except when
; = 0; 1; 2;:::. With this interpretation
2.6.5Z1
0t s (1=3)
x+tdt=2p
3( 1)s
xs+(1=3),s= 0;1;2;:::.
Inserting (2.6.2) into (2.6.1) and integrating formally
term-by-term, we obtain
2.6.6S(x)2p
31X
s=0( 1)s 1
3
s
x s (1=3),x!1 .
However this result is incorrect. The correct result is
given by
2.6.7S(x)2p
31X
s=0( 1)s 1
3
s
x s (1=3)
1X
s=13s(s 1)!
25(3s 1)x s;
seex2.6(ii).
The fact that expansion (2.6.6) misses all the terms
in the second series in (2.6.7) raises the question: what
went wrong with our process of reaching (2.6.6)? In the
following subsections, we use some elementary facts of
distribution theory ( x1.16) to study the proper use of
divergent integrals. An important asset of the distribu-
tion method is that it gives explicit expressions for the
remainder terms associated with the resulting asymp-
totic expansions.
For an introduction to distribution theory, see Wong
(1989, Chapter 5). For more advanced discussions, see
Gel'fand and Shilov (1964) and Rudin (1973).
2.6(ii) Stieltjes Transform
Letf(t) be locally integrable on [0 ;1). The Stieltjes
transform off(t) is de ned by
2.6.8S(f;z) =Z1
0f(t)
t+zdt:To derive an asymptotic expansion of S(f;z) for large
values ofjzj, withjphzj<, we assume that f(t) pos-
sesses an asymptotic expansion of the form
2.6.9 f(t)1X
s=0ast s ,t!+1;
with 0< 1. For each n= 1;2;3;:::, set
2.6.10 f(t) =n 1X
s=0ast s +fn(t):
To each function in this equation, we shall assign a tem-
pered distribution (i.e., a continuous linear functional)
on the spaceTof rapidly decreasing functions on R.
Sincef(t) is locally integrable on [0 ;1), it de nes a
distribution by
2.6.11hf; i=Z1
0f(t) (t)dt, 2T:
In particular,
2.6.12
t ;
=Z1
0t (t)dt, 2T;
when 0< < 1. Since the functions t s ,s=
1;2;:::, are not locally integrable on [0 ;1), we can-
not assign distributions to them in a similar manner.
However, they are multiples of the derivatives of t .
Motivated by the de nition of distributional derivatives,
we can assign them the distributions de ned by
2.6.13
t s ;
=1
( )sZ1
0t (s)(t)dt, 2T;
where ( )s= ( + 1)( +s 1). Similarly, in the
case = 1, we de ne
2.6.14
t s 1;
= 1
s!Z1
0(lnt) (s+1)(t)dt, 2T:
To assign a distribution to the function fn(t), we rst
letfn;n(t) denote the nth repeated integral ( x1.4(v)) of
fn:
2.6.15fn;n(t) =( 1)n
(n 1)!Z1
t( t)n 1fn( )d :
For 0< < 1, it is easily seen that fn;n(t) is bounded
on [0;R] for any positive constant R, and isO(t ) as
t!1 . For = 1, we have fn;n(t) =O
t 1
ast!1
andfn;n(t) =O(lnt) ast!0+. In either case, we
de ne the distribution associated with fn(t) by
2.6.16hfn; i= ( 1)nZ1
0fn;n(t) (n)(t)dt, 2T;
since thenth derivative of fn;nisfn.
We have now assigned a distribution to each func-
tion in (2.6.10). A natural question is: what is the
exact relation between these distributions? The answer
is provided by the identities (2.6.17) and (2.6.20) given
below.
2.6 Distributional Methods 53
For 0< < 1 andn1, we have
2.6.17
hf; i=n 1X
s=0as
t s ;
nX
s=1csD
(s 1); E
+hfn; i
for any 2T, where
2.6.18 cs=( 1)s
(s 1)!M(f;s);
M(f;z) being the Mellin transform of f(t) or its ana-
lytic continuation ( x2.5(ii)). The Dirac delta distribu-
tion in (2.6.17) is given by
2.6.19D
(s); E
= ( 1)s (s)(0),s= 0;1;2;:::;
comparex1.16(iii).
For = 1
2.6.20
hf; i=n 1X
s=0as
t s 1;
nX
s=1dsD
(s 1); E
+hfn; i
for any 2T, where
2.6.21( 1)s+1ds+1=as
s!sX
k=11
k+1
s!lim
z!s+1
M(f;z)
+as
z s 1
;
fors= 0;1;2;:::.
To apply the results (2.6.17) and (2.6.20) to the
Stieltjes transform (2.6.8), we take a speci c function
2T. Let"be a positive number, and
2.6.22 "(t) =e "t
t+z,t2(0;1):
From (2.6.13) and (2.6.14)
2.6.23 lim
"!0
t s ; "
=
sin( )( 1)s
zs+ ;
2.6.24 lim
"!0
t s 1; "
=( 1)s+1
zs+1sX
k=11
k+( 1)s
zs+1lnz;
withs= 0;1;2;:::. From (2.6.11) and (2.6.16), we also
have
2.6.25 lim
"!0hf; "i=S(f;z);
2.6.26 lim
"!0hfn; "i=n!Z1
0fn;n(t)
(t+z)n+1dt:
On substituting (2.6.15) into (2.6.26) and interchanging
the order of integration, the right-hand side of (2.6.26)
becomes( 1)n
znZ1
0 nfn( )
+zd :
To summarize,
2.6.27S(f;z) =
sin( )n 1X
s=0( 1)sas
zs+
nX
s=1(s 1)!cs
zs+Rn(z);if 2(0;1) in (2.6.9), or
2.6.28
S(f;z) = lnzn 1X
s=0( 1)sas
zs+1+n 1X
s=0( 1)seds
zs+1+Rn(z);
if = 1 in (2.6.9). Here csis given by (2.6.18),
2.6.29eds= lim
z!s+1
M(f;z) +as
z s 1
;
and
2.6.30 Rn(z) =( 1)n
znZ1
0 nfn( )
+zd :
The expansion (2.6.7) follows immediately from
(2.6.27) with z=xandf(t) = (1 +t) (1=3); its region
of validity isjphxj (< ). The distribution
method outlined here can be extended readily to func-
tionsf(t) having an asymptotic expansion of the form
2.6.31 f(t)eict1X
s=0ast s ,t!+1;
wherec(6= 0) is real, and 0 < 1. For a more de-
tailed discussion of the derivation of asymptotic expan-
sions of Stieltjes transforms by the distribution method,
see McClure and Wong (1978) and Wong (1989, Chapter
6). Corresponding results for the generalized Stieltjes
transform
2.6.32Z1
0f(t)
(t+z)dt, >0;
can be found in Wong (1979). An application has been
given by L opez (2000) to derive asymptotic expansions
of standard symmetric elliptic integrals, complete with
error bounds; see x19.27(vi).
2.6(iii) Fractional Integrals
The Riemann{Liouville fractional integral of orderis
de ned by
2.6.33If(x) =1
()Zx
0(x t) 1f(t)dt,>0;
seex1.15(vi). We again assume f(t) is locally integrable
on [0;1) and satis es (2.6.9). We now derive an asymp-
totic expansion of If(x) for large positive values of x.
In terms of the convolution product
2.6.34 (fg)(x) =Zx
0f(x t)g(t)dt
of two locally integrable functions on [0 ;1), (2.6.33)
can be written
2.6.35 If(x) =1
()(t 1f)(x):
54 Asymptotic Approximations
The replacement of f(t) by its asymptotic expansion
(2.6.9), followed by term-by-term integration leads to
convolution integrals of the form
2.6.36(t 1t s )(x) =Zx
0(x t) 1t s dt,
s= 0;1;2;::::
Of course, except when s= 0 and 0< < 1, none of
these integrals exists in the usual sense. However, the
left-hand side can be considered as the convolution of
the two distributions associated with the functions t 1
andt s , given by (2.6.12) and (2.6.13).
To de ne convolutions of distributions, we rst in-
troduce the space K+of all distributions of the form
Dnf, wherenis a nonnegative integer, fis a locally in-
tegrable function on Rwhich vanishes on ( 1;0], and
Dnfdenotes the nth derivative of the distribution as-
sociated with f. ForF=DnfandG=DmginK+,
we de ne
2.6.37 FG=Dn+m(fg):
It is easily seen that K+forms a commutative, asso-
ciative linear algebra. Furthermore, K+contains the
distributions H,, andt,t>0, for any real (or com-
plex) number , whereHis the distribution associated
with the Heaviside function H(t) (x1.16(iv)), and tis
the distribution de ned by (2.6.12){(2.6.14), depending
on the value of . Since=DH, it follows that for
6= 1;2;:::,
2.6.38t 1(s 1)= ()
(+ 1 s)t s,t>0:
Using (5.12.1), we can also show that when 6= 1;2;:::
and is not a nonnegative integer,
2.6.39
t 1t s = () (1 s )
(+ 1 s )t s ,t>0;
and
2.6.40
t 1t s 1=( 1)s
s!Ds+1(t(lnt
(+ 1))) ,
t>0;
where
is Euler's constant ( x5.2(ii)).
To derive the asymptotic expansion of If(x), we
recall equations (2.6.17) and (2.6.20). In the sense of
distributions, they can be written
2.6.41f=n 1X
s=0ast s nX
s=1cs(s 1)+fn;
and
2.6.42f=n 1X
s=0ast s 1 nX
s=1ds(s 1)+fn:Substituting into (2.6.35) and using (2.6.38){(2.6.40),
we obtain
2.6.43t 1f=n 1X
s=0as () (1 s )
(+ 1 s )t s
nX
s=1cs ()
( s+ 1)t s+t 1fn
when 0< < 1, or
2.6.44
t 1f=n 1X
s=0( 1)sas
s!Ds+1(t(lnt
(+ 1)))
nX
s=1ds ()
( s+ 1)t s+t 1fn
when = 1. These equations again hold only
in the sense of distributions. Since the function
t(lnt
(+ 1)) and all its derivatives are lo-
cally absolutely continuous in (0 ;1), the distributional
derivatives in the rst sum in (2.6.44) can be replaced by
the corresponding ordinary derivatives. Furthermore,
sincef(n)
n;n(t) =fn(t), it follows from (2.6.37) that the re-
mainder terms t 1fnin the last two equations can be
associated with a locally integrable function in (0 ;1).
On replacing the distributions by their corresponding
functions, (2.6.43) and (2.6.44) give
2.6.45If(x) =n 1X
s=0as (1 s )
(+ 1 s )x s
nX
s=1cs
(+ 1 s)x s+1
xnn(x);
when 0< < 1, or
2.6.46
If(x)
=n 1X
s=0( 1)sas
s! (+ 1)ds+1
dxs+1(x(lnx
(+ 1)))
nX
s=1ds
( s+ 1)x s+1
xnn(x);
when = 1, where
2.6.47n(x) =nX
j=0n
j (+ 1)
(+ 1 j)I
tn jfn;j
(x);
fn;j(t) being the jth repeated integral of fn; compare
(2.6.15).
Example
Letf(t) =t1 =(1 +t), 0< < 1. Then
2.6.48If(x) =1
()Zx
0(x t) 1t1 (1 +t) 1dt;
2.7 Differential Equations 55
where>0. For 0<t<1
2.6.49f(t) =n 1X
s=0( 1)st s + ( 1)nt1 n
1 +t:
In the notation of (2.6.10), as= ( 1)sand
2.6.50 fn(t) = ( 1)nt1 n
1 +t:
Since
2.6.51 M(f;s) = ( 1)s=sin( );
from (2.6.45) it follows that
2.6.52If(x) =n 1X
s=0( 1)s (1 s )
(+ 1 s )x s
sin( )nX
s=11
(+ 1 s)x s
(s 1)!
+1
xnn(x):
Moreover,
2.6.53jn(x)j (+ 1) (1 )
(+ 1 ) (n+ )
nX
j=0n
j (n+ j)
j (+ 1 j)jx
forx>0.
It may be noted that the integral (2.6.48) can
be expressed in terms of the hypergeometric function
2F1(1;2 ; 2 +; x); seex15.2(i).
For proofs and other examples, see McClure and
Wong (1979) and Wong (1989, Chapter 6). If both f
andgin (2.6.34) have asymptotic expansions of the form
(2.6.9), then the distribution method can also be used
to derive an asymptotic expansion of the convolution
fg; see Li and Wong (1994).
2.6(iv) Regularization
The method of distributions can be further extended to
derive asymptotic expansions for convolution integrals:
2.6.54 I(x) =Z1
0f(t)h(xt)dt:
We assume that for each n= 1;2;3;:::,
2.6.55 f(t) =n 1X
s=0asts+ 1+fn(t);
where 0< 1 andfn(t) =O
tn+ 1
ast!0+.
Also,
2.6.56 h(t) =n 1X
s=0bst s +hn(t);where 0< 1, andhn(t) =O
t n
ast!1 .
Multiplication of these expansions leads to
2.6.57
f(t)h(xt) =n 1X
j=0n 1X
k=0ajbktj+ 1 k x k
+n 1X
j=0ajtj+ 1hn(xt)
+n 1X
k=0bkx k t k fn(t) +fn(t)hn(xt):
On inserting this identity into (2.6.54), we immediately
encounter divergent integrals of the form
2.6.58Z1
0tdt, 2R:
However, in the theory of generalized functions (distri-
butions), there is a method, known as \regularization",
by which these integrals can be interpreted in a mean-
ingful manner. In this sense
2.6.59Z1
0tdt= 0, 2C:
From (2.6.55) and (2.6.59)
2.6.60 M(f;z) =M(fn;z);
whereM(f;z) is the Mellin transform of for its ana-
lytic continuation. Also, when 6= ,
2.6.61 M(hx;j+ ) =x j M(h;j+ );
wherehx(t) =h(xt). Inserting (2.6.57) into (2.6.54), we
obtain from (2.6.59){(2.6.61)
2.6.62I(x) =n 1X
j=0ajM(h;j+ )x j
+n 1X
k=0bkM(f; 1 k )x k +n(x)
when 6= , where
n(x) =Z1
0fn(t)hn(xt)dt:
There is a similar expansion, involving logarithmic
terms, when = . For rigorous derivations of these
results and also order estimates for n(x), see Wong
(1979) and Wong (1989, Chapter 6).
2.7 Di erential Equations
2.7(i) Regular Singularities: Fuchs{Frobenius
Theory
Anordinary point of the di erential equation
2.7.1d2w
dz2+f(z)dw
dz+g(z)w= 0
56 Asymptotic Approximations
is one at which the coecients f(z) andg(z) are ana-
lytic. All solutions are analytic at an ordinary point,
and their Taylor-series expansions are found by equat-
ing coecients.
Other points z0aresingularities of the di erential
equation. If both ( z z0)f(z) and (z z0)2g(z) are ana-
lytic atz0, thenz0is aregular singularity (orsingularity
of the rst kind ). All other singularities are classi ed
asirregular .
In a punctured neighborhood Nof a regular singu-
larityz0
2.7.2
f(z) =1X
s=0fs(z z0)s 1; g(z) =1X
s=0gs(z z0)s 2;
with at least one of the coecients f0,g0,g1nonzero.
Let 1, 2denote the indices orexponents , that is, the
roots of the indicial equation
2.7.3Q( ) ( 1) +f0 +g0= 0:
Provided that 1 2is not zero or an integer, equation
(2.7.1) has independent solutions wj(z),j= 1;2, such
that
2.7.4wj(z) = (z z0) j1X
s=0as;j(z z0)s,z2N;
witha0;j= 1, and
2.7.5Q( j+s)as;j= s 1X
r=0(( j+r)fs r+gs r)ar;j;
whens= 1;2;3;:::.
If 1 2= 0;1;2;:::, then (2.7.4) applies only in
the casej= 1. But there is an independent solution
2.7.6w2(z) = (z z0) 21X
s=0
s6= 1 2bs(z z0)s
+cw1(z) ln(z z0), z2N:
The coecients bsand constant care again determined
by equating coecients in the di erential equation, be-
ginning with c= 1 when 1 2= 0, or with b0= 1
when 1 2= 1;2;3;:::.
The radii of convergence of the series (2.7.4), (2.7.6)
are not less than the distance of the next nearest singu-
larity of the di erential equation from z0.
To include the point at in nity in the foregoing clas-
si cation scheme, we transform it into the origin by re-
placingzin (2.7.1) with 1 =z; see Olver (1997b, pp. 153{
154). For corresponding de nitions, together with ex-
amples, for linear di erential equations of arbitrary or-
der seexx16.8(i){16.8(ii).
2.7(ii) Irregular Singularities of Rank 1
If the singularities of f(z) andg(z) atz0are no worse
than poles, then z0hasrank` 1, where`is the leastinteger such that ( z z0)`f(z) and (z z0)2`g(z) are an-
alytic atz0. Thus a regular singularity has rank 0. The
most common type of irregular singularity for special
functions has rank 1 and is located at in nity. Then
2.7.7 f(z) =1X
s=0fs
zs; g(z) =1X
s=0gs
zs;
these series converging in an annulus jzj> a, with at
least one of f0,g0,g1nonzero.
Formal solutions are
2.7.8 ejzzj1X
s=0as;j
zs, j= 1;2;
where1,2are the roots of the characteristic equation
2.7.9 2+f0+g0= 0;
2.7.10 j= (f1j+g1)=(f0+ 2j);
a0;j= 1, and
2.7.11(f0+ 2j)sas;j= (s j)(s 1 j)as 1;j
+sX
r=1(jfr+1+gr+1
(s r j)fr)as r;j;
whens= 1;2;:::. The construction fails i 1=2,
that is, when f2
0= 4g0: this case is treated below.
For larges,
2.7.12as;11
(1 2)s
1X
j=0aj;2(1 2)j (s+2 1 j);
2.7.13as;22
(2 1)s
1X
j=0aj;1(2 1)j (s+1 2 j);
where 1and 2are constants, and the Jth remain-
der terms in the sums are O( (s+2 1 J)) and
O( (s+1 2 J)), respectively (Olver (1994a)).
Hence unless the series (2.7.8) terminate (in which case
the corresponding jis zero) they diverge. However,
there are unique and linearly independent solutions
wj(z),j= 1;2, such that
2.7.14wj(z)ejz((2 1)z)j1X
s=0as;j
zs
asz!1 in the sectors
2.7.15 3
2+ph((2 1)z)3
2 ,j= 1;
2.7.16 1
2+ph((2 1)z)5
2 ,j= 2;
being an arbitrary small positive constant.
Although the expansions (2.7.14) apply only in the
sectors (2.7.15) and (2.7.16), each solution wj(z) can
2.7 Differential Equations 57
be continued analytically into any other sector. Typical
connection formulas are
2.7.17w1(z) =e2i1w1(ze 2i) +C1w2(z);
w2(z) =e 2i2w2(ze2i) +C2w1(z);
in whichC1,C2are constants, the so-called Stokes mul-
tipliers . In combination with (2.7.14) these formulas
yield asymptotic expansions for w1(z) in1
2+
ph((2 1)z)5
2 , andw2(z) in 3
2+
ph((2 1)z)1
2 . Furthermore,
2.7.18 1= ie(2 1)iC1=(2);2=iC2=(2):
Note that the coecients in the expansions (2.7.12),
(2.7.13) for the \late" coecients, that is, as;1,as;2with
slarge, are the \early" coecients aj;2,aj;1withjsmall.
This phenomenon is an example of resurgence , a classi-
cation due to Ecalle (1981a,b). See x2.11(v) for other
examples.
The exceptional case f2
0= 4g0is handled by Fabry's
transformation :
2.7.19 w=e f0z=2W; t =z1=2:
The transformed di erential equation either has a reg-
ular singularity at t=1, or its characteristic equation
has unequal roots.
For error bounds for (2.7.14) see Olver (1997b,
Chapter 7). For the calculation of Stokes multipliers
see Olde Daalhuis and Olver (1995b). For extensions
to singularities of higher rank see Olver and Stenger
(1965). For extensions to higher-order di erential equa-
tions see Stenger (1966a,b), Olver (1997a, 1999), and
Olde Daalhuis and Olver (1998).
2.7(iii) Liouville{Green (WKBJ) Approximation
For irregular singularities of nonclassi able rank, a pow-
erful tool for nding the asymptotic behavior of solu-
tions, complete with error bounds, is as follows:
Liouville{Green Approximation Theorem
In a nite or in nite interval ( a1;a2) letf(x) be real,
positive, and twice-continuously di erentiable, and g(x)
be continuous. Then in ( a1;a2) the di erential equation
2.7.20d2w
dx2= (f(x) +g(x))w
has twice-continuously di erentiable solutions
2.7.21
w1(x) =f 1=4(x) expZ
f1=2(x)dx
(1 +1(x));
2.7.22
w2(x) =f 1=4(x) exp
Z
f1=2(x)dx
(1 +2(x));
such that
2.7.23jj(x)j;1
2f 1=2(x)j0
j(x)jexp 1
2Vaj;x(F)
1,
j= 1;2;provided thatVaj;x(F)<1. HereF(x) is the error-
control function
2.7.24F(x) =Z1
f1=4d2
dx21
f1=4
g
f1=2
dx;
andVdenotes the variational operator ( x2.3(i)). Thus
2.7.25
Vaj;x(F) =Zx
aj 1
f1=4(t)d2
dt21
f1=4(t)
g(t)
f1=2(t)
dt :
Assuming alsoVa1;a2(F)<1, we have
2.7.26
w1(x)f 1=4(x) expZ
f1=2(x)dx
,x!a1+;
2.7.27
w2(x)f 1=4(x) exp
Z
f1=2(x)dx
,x!a2 :
Suppose in addition jR
f1=2(x)dxjis unbounded as
x!a1+ andx!a2 . Then there are solutions w3(x),
w4(x), such that
2.7.28
w3(x)f 1=4(x) expZ
f1=2(x)dx
,x!a2 ;
2.7.29
w4(x)f 1=4(x) exp
Z
f1=2(x)dx
,x!a1+:
The solutions with the properties (2.7.26), (2.7.27) are
unique, but not those with the properties (2.7.28),
(2.7.29). In fact, since
2.7.30 w1(x)=w4(x)!0,x!a1+;
w1(x) is a recessive (orsubdominant ) solution as x!
a1+, andw4(x) is a dominant solution as x!a1+.
Similarly for w2(x) andw3(x) asx!a2 .
Example
2.7.31d2w
dx2= (x+ lnx)w, 0<x<1:
We cannot take f=xandg= lnxbecauseR
gf 1=2dx
would diverge as x!+1. Instead set f=x+ lnx,
g= 0. By approximating
2.7.32f1=2=x1=2+1
2x 1=2lnx+O
x 3=2(lnx)2
;
we arrive at
2.7.33w2(x)x (1=4) pxexp
2x1=2 2
3x3=2
;
2.7.34w3(x)x (1=4)+pxexp
2
3x3=2 2x1=2
;
asx!+1,w2(x) being recessive and w3(x) dominant.
For other examples, and also the corresponding re-
sults when f(x) is negative, see Olver (1997b, Chap-
ter 6), Olver (1980a), Taylor (1978, 1982), and Smith
(1986). The rst of these references includes extensions
to complex variables and reversions for zeros.
58 Asymptotic Approximations
2.7(iv) Numerically Satisfactory Solutions
One pair of independent solutions of the equation
2.7.35 d2w
dz2=w
isw1(z) =ez,w2(z) =e z. Another is w3(z) = coshz,
w4(z) = sinhz. In theory either pair may be used to
construct any other solution
2.7.36 w(z) =Aw1(z) +Bw2(z);
or
2.7.37 w(z) =Cw3(z) +Dw4(z);
whereA;B;C;D are constants. From the numerical
standpoint, however, the pair w3(z) andw4(z) has the
drawback that severe numerical cancellation can occur
with certain combinations of CandD, for example if
CandDare equal, or nearly equal, and z, or<z, is
large and negative. This kind of cancellation cannot
take place with w1(z) andw2(z), and for this reason,
and following Miller (1950), we call w1(z) andw2(z) a
numerically satisfactory pair of solutions.
The solutions w1(z) andw2(z) are respectively re-
cessive and dominant as <z! 1 , and vice versa as
<z!+1. This is characteristic of numerically satis-
factory pairs. In a neighborhood, or sectorial neighbor-
hood of a singularity, one member has to be recessive.
In consequence, if a di erential equation has more than
one singularity in the extended plane, then usually more
than two standard solutions need to be chosen in order
to have numerically satisfactory representations every-
where.
In oscillatory intervals, and again following Miller
(1950), we call a pair of solutions numerically satisfac-
tory if asymptotically they have the same amplitude and
are1
2out of phase.
2.8 Di erential Equations with a Parameter
2.8(i) Classi cation of Cases
Many special functions satisfy an equation of the form
2.8.1 d2w
dz2=
u2f(z) +g(z)
w;
in whichuis a real or complex parameter, and asymp-
totic solutions are needed for large jujthat are uniform
with respect to zin a point set DinRorC. For ex-
ample,ucan be the order of a Bessel function or degree
of an orthogonal polynomial. The form of the asymp-
totic expansion depends on the nature of the transition
points inD, that is, points at which f(z) has a zero or
singularity. Zeros of f(z) are also called turning points .
There are three main cases. In Case I there are no
transition points in Dandg(z) is analytic. In Case II
f(z) has a simple zero at z0andg(z) is analytic at z0.
In Case IIIf(z) has a simple pole at z0and (z z0)2g(z)
is analytic at z0.The same approach is used in all three cases. First
we apply the Liouville transformation (x1.13(iv)) to
(2.8.1). This introduces new variables Wand, related
by
2.8.2 W= _z 1=2w;
dots denoting di erentiations with respect to . Then
2.8.3d2W
d2=
u2_z2f(z) + ()
W;
where
2.8.4 () = _z2g(z) + _z1=2d2
d2( _z 1=2):
The transformation is now specialized in such a way
that: (a)andzare analytic functions of each other at
the transition point (if any); (b) the approximating dif-
ferential equation obtained by neglecting () (or part
of ()) has solutions that are functions of a single vari-
able. The actual choices are as follows:
2.8.5 _z2f(z) = 1; =Z
f1=2(z)dz;
for Case I,
2.8.6 _z2f(z) =;2
33=2=Zz
z0f1=2(t)dt;
for Case II,
2.8.7 _z2f(z) = 1=; 21=2=Zz
z0f1=2(t)dt;
for Case III.
The transformed equation has the form
2.8.8 d2W
d2=
u2m+ ()
W;
withm= 0 (Case I), m= 1 (Case II), m= 1
(Case III). In Cases I and II the asymptotic solutions
are in terms of the functions that satisfy (2.8.8) with
() = 0. These are elementary functions in Case I,
and Airy functions ( x9.2) in Case II. In Case III the
approximating equation is
2.8.9d2W
d2=u2
+
2
W;
where= lim(2 ()) as!0. Solutions are
Bessel functions, or modi ed Bessel functions, of order
(1 + 4)1=2(xx10.2, 10.25).
For another approach to these problems based on
convergent inverse factorial series expansions see Dun-
ster et al. (1993) and Dunster (2001a, 2004).
2.8 Differential Equations with a Parameter 59
2.8(ii) Case I: No Transition Points
The transformed di erential equation is
2.8.10 d2W
d2= (u2+ ())W;
in whichranges over a bounded or unbounded interval
or domain , and () isC1or analytic on . The
parameteruis assumed to be real and positive. Corre-
sponding to each positive integer nthere are solutions
Wn;j(u;),j= 1;2, that depend on arbitrarily chosen
reference points j, areC1or analytic on , and as
u!1
2.8.11
Wn;1(u;) =eu n 1X
s=0As()
us+O1
un!
,21( 1);
2.8.12
Wn;2(u;) =e u n 1X
s=0( 1)sAs()
us+O1
un!
,
22( 2);
withA0() = 1 and
2.8.13
As+1() = 1
2A0
s()+1
2Z
()As()d,s= 0;1;2;:::;(the constants of integration being arbitrary). The ex-
pansions (2.8.11) and (2.8.12) are both uniform and dif-
ferentiable with respect to . The regions of validity
j( j) comprise those points that can be joined to
jinby a path Qjalong which<vis nondecreasing
(j= 1) or nonincreasing ( j= 2) asvpasses from jto
. In addition,VQj(A1) andVQj(An) must be bounded
onj( j).
For error bounds, extensions to pure imaginary or
complexu, an extension to inhomogeneous di erential
equations, and examples, see Olver (1997b, Chapter 10).
This reference also supplies sucient conditions to en-
sure that the solutions Wn;1(u;) andWn;2(u;) having
the properties (2.8.11) and (2.8.12) are independent of
n.
2.8(iii) Case II: Simple Turning Point
The transformed di erential equation is
2.8.14 d2W
d2= (u2+ ())W;
and for simplicity is assumed to range over a nite or
in nite interval ( 1; 2) with 1<0, 2>0. Again,
u > 0 and () isC1on ( 1; 2). Corresponding to
each positive integer nthere are solutions Wn;j(u;),
j= 1;2, that are C1on ( 1; 2), and asu!1
2.8.15Wn;1(u;) = Ai
u2=3 n 1X
s=0As()
u2s+O1
u2n 1!
+ Ai0
u2=3 n 2X
s=0Bs()
u2s+(4=3)+O1
u2n 1!
;
2.8.16Wn;2(u;) = Bi
u2=3 n 1X
s=0As()
u2s+O1
u2n 1!
+ Bi0
u2=3 n 2X
s=0Bs()
u2s+(4=3)+O1
u2n 1!
:
HereA0() = 1,
2.8.17 Bs() =8
>><
>>:1
21=2Z
0( (v)As(v) A00
s(v))dv
v1=2; > 0;
1
2( )1=2Z0
( (v)As(v) A00
s(v))dv
( v)1=2; < 0;
and
2.8.18 As+1() = 1
2B0
s() +1
2Z
()Bs()d;
whens= 0;1;2;:::. For Ai and Bi see x9.2. The expan-
sions (2.8.15) and (2.8.16) are both uniform and di er-
entiable with respect to . These results are valid when
V 1; 2
jj1=2B0
andV 1; 2
jj1=2Bn 1
are nite.
An alternative way of representing the error terms in
(2.8.15) and (2.8.16) is as follows. Let c= 0:36604:::
be the real root of the equation
2.8.19 Ai(x) = Bi(x)
of smallest absolute value, and de ne the envelopes ofAi(x) and Bi(x) by
2.8.20 env Ai(x) = env Bi(x) =
Ai2(x) + Bi2(x)1=2,
1<xc,
2.8.21env Ai(x) =p
2 Ai(x);env Bi(x) =p
2 Bi(x),
cx<1.
These envelopes are continuous functions of x, and as
u!1
60 Asymptotic Approximations
2.8.22Wn;1(u;) = Ai
u2=3n 1X
s=0As()
u2s
+ Ai0
u2=3n 2X
s=0Bs()
u2s+(4=3)
+ env Ai
u2=3
O1
u2n 1
;
2.8.23Wn;2(u;) = Bi
u2=3n 1X
s=0As()
u2s
+ Bi0
u2=3n 2X
s=0Bs()
u2s+(4=3)
+ env Bi
u2=3
O1
u2n 1
;
uniformly with respect to 2( 1; 2).
For error bounds, more delicate error estimates, ex-
tensions to complex andu, zeros, connection formulas,
extensions to inhomogeneous equations, and examples,see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid
(1974a,b), Boyd (1987), and Baldwin (1991).
For other examples of uniform asymptotic approx-
imations and expansions of special functions in terms
of Airy functions see especially x10.20 andxx12.10(vii),
12.10(viii); also xx12.14(ix), 13.20(v), 13.21(iii),
13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii),
32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13.
2.8(iv) Case III: Simple Pole
The transformed equation (2.8.8) is renormalized as
2.8.24d2W
d2=u2
4+2 1
42+ ()
W:
We again assume 2( 1; 2) with 1 1<0,
0< 21. Also, () isC1on ( 1; 2), andu>0.
The constant (=p1 + 4) is real and nonnegative.
There are two cases: 2(0; 2) and2( 1;0).
In the former, corresponding to any positive integer n
there are solutions Wn;j(u;),j= 1;2, that are C1on
(0; 2), and asu!1
2.8.25Wn;1(u;) =1=2I
u1=2n 1X
s=0As()
u2s+I+1
u1=2n 2X
s=0Bs()
u2s+1+1=2I
u1=2
O1
u2n 1
;
2.8.26Wn;2(u;) =1=2K
u1=2n 1X
s=0As()
u2s K+1
u1=2n 2X
s=0Bs()
u2s+1+1=2K
u1=2
O1
u2n 1
:
HereA0() = 1,
2.8.27 Bs() = A0
s() +1
1=2Z
0
(v)As(v)
+1
2
A0
s(v)dv
v1=2;
2.8.28 As+1() =Bs() B0
s() +Z
()Bs()d;
s= 0;1;2;:::. ForIandKseex10.25(ii). The expansions (2.8.25) and (2.8.26) are both uniform and di erentiable
with respect to . These results are valid when V0; 2
1=2B0
andV0; 2
1=2Bn 1
are nite.
If2( 1;0), then there are solutions Wn;j(u;),j= 3;4, that are C1on ( 1;0), and asu!1
2.8.29Wn;3(u;) =jj1=2J
ujj1=2 n 1X
s=0As()
u2s+O1
u2n 1!
jjJ+1
ujj1=2 n 2X
s=0Bs()
u2s+1+O1
u2n 2!
;
2.8.30Wn;4(u;) =jj1=2Y
ujj1=2 n 1X
s=0As()
u2s+O1
u2n 1!
jjY+1
ujj1=2 n 2X
s=0Bs()
u2s+1+O1
u2n 2!
:
HereA0() = 1,
2.8.31 Bs() = A0
s() +1
jj1=2Z0
(v)As(v)
+1
2
A0
s(v)dv
jvj1=2;
s= 0;1;2;:::, and (2.8.28) again applies. For JandYseex10.2(ii). The expansions (2.8.29) and (2.8.30) are both
uniform and di erentiable with respect to . These results are valid when V 1;0
jj1=2B0
andV 1;0
jj1=2Bn 1
are
nite.
Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions.
Letx=Xbe the smallest positive root of the equation
2.8.32 J(x) +Y(x) = 0:
2.9 Difference Equations 61
De ne
2.8.33 envJ(x) =p
2J(x);envY(x) =p
2jY(x)j, 0 <xX,
2.8.34 envJ(x) = envY(x) =
J2
(x) +Y2
(x)1=2, Xx<1.
Then asu!1
2.8.35Wn;3(u;) =jj1=2J
ujj1=2n 1X
s=0As()
u2s jjJ+1
ujj1=2n 2X
s=0Bs()
u2s+1+jj1=2envJ
ujj1=2
O1
u2n 1
;
2.8.36Wn;4(u;) =jj1=2Y
ujj1=2n 1X
s=0As()
u2s jjY+1
ujj1=2n 2X
s=0Bs()
u2s+1+jj1=2envY
ujj1=2
O1
u2n 1
;
uniformly with respect to 2( 1;0).
For error bounds, more delicate error estimates, ex-
tensions to complex ,, andu, zeros, and examples see
Olver (1997b, Chapter 12), Boyd (1990a), and Dunster
(1990a).
For other examples of uniform asymptotic approx-
imations and expansions of special functions in terms
of Bessel functions or modi ed Bessel functions of
xed order seexx13.8(iii), 13.21(i), 13.21(iv), 14.15(i),
14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv),
18.24, 33.20(iv).
2.8(v) Multiple and Fractional Turning Points
The approach used in preceding subsections for equa-
tion (2.8.1) also succeeds when z0is amultiple orfrac-
tional turning point . For the former f(z) has a zero
of multiplicity = 2;3;4;::: andg(z) is analytic. For
the latter (z z0) f(z) andg(z) are both analytic at
z0,(> 2) being a real constant. In both cases uni-
form asymptotic approximations are obtained in terms
of Bessel functions of order 1 =(+ 2). More generally,
g(z) can have a simple or double pole at z0. (In the
case of the double pole the order of the approximating
Bessel functions is xed but no longer 1 =(+ 2).) How-
ever, in all cases with > 2 and6= 0 or1, only
uniform asymptotic approximations are available, not
uniform asymptotic expansions. For results, including
error bounds, see Olver (1977c).
For connection formulas for Liouville{Green ap-
proximations across these transition points see Olver
(1977b,a, 1978).
2.8(vi) Coalescing Transition Points
Corresponding to the problems for integrals outlined in
xx2.3(v), 2.4(v), and 2.4(vi), there are analogous prob-
lems for di erential equations.
For two coalescing turning points see Olver (1975a,
1976) and Dunster (1996a); in this case the uniformapproximants are parabolic cylinder functions. (For
envelope functions for parabolic cylinder functions see
x14.15(v)).
For a coalescing turning point and double pole see
Boyd and Dunster (1986) and Dunster (1990b); in this
case the uniform approximants are Bessel functions of
variable order.
For a coalescing turning point and simple pole see
Nestor (1984) and Dunster (1994b); in this case the uni-
form approximants are Whittaker functions ( x13.14(i))
with a xed value of the second parameter.
For further examples of uniform asymptotic approx-
imations in terms of parabolic cylinder functions see
xx13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24.
For further examples of uniform asymptotic approx-
imations in terms of Bessel functions or modi ed Bessel
functions of variable order see xx13.21(ii), 14.15(ii),
14.15(iv), 14.20(viii), 30.9(i), 30.9(ii).
For examples of uniform asymptotic approximations
in terms of Whittaker functions with xed second pa-
rameter seex18.15(i) andx28.8(iv).
Lastly, for an example of a fourth-order di erential
equation, see Wong and Zhang (2007).
2.9 Di erence Equations
2.9(i) Distinct Characteristic Values
Many special functions that depend on parameters sat-
isfy a three-term linear recurrence relation
2.9.1w(n+ 2) +f(n)w(n+ 1) +g(n)w(n) = 0,
n= 0;1;2;:::;
or equivalently the second-order homogeneous linear dif-
ference equation
2.9.22w(n) + (2 +f(n)) w(n)
+ (1 +f(n) +g(n))w(n) = 0,n= 0;1;2;:::;
in which is the forward di erence operator ( x3.6(i)).
62 Asymptotic Approximations
Oftenf(n) andg(n) can be expanded in series
2.9.3f(n)1X
s=0fs
ns; g(n)1X
s=0gs
ns,n!1 ,
withg06= 0. (For the case g0= 0 see the nal para-
graph ofx2.9(ii) with Qnegative.) This situation is
analogous to second-order homogeneous linear di eren-
tial equations with an irregular singularity of rank 1 atin nity (x2.7(ii)). Formal solutions are
2.9.4 n
jn j1X
s=0as;j
ns, j= 1;2;
where1;2are the roots of the characteristic equation
2.9.5 2+f0+g0= 0;
2.9.6 j= (f1j+g1)=(f0j+ 2g0);
a0;j= 1, and
2.9.7 j(f0+ 2j)sas;j=sX
r=1
2
j2r+1 j+r s
r+ 1
+jr+1X
q=0 j+r s
r+ 1 q
fq+gr+1!
as r;j;
s= 1;2;3;:::. The construction fails i 1=2, that
is, whenf2
0= 4g0.
Whenf2
06= 4g0, there are linearly independent so-
lutionswj(n),j= 1;2, such that
2.9.8 wj(n)n
jn j1X
s=0as;j
ns,n!1:
Ifj2j>j1j, or ifj2j=j1jand< 2>< 1, then
w1(n) is recessive and w2(n) is dominant as n!1 . As
in the case of di erential equations ( xx2.7(iii), 2.7(iv))
recessive solutions are unique and dominant solutions
are not; furthermore, one member of a numerically sat-
isfactory pair has to be recessive. When j2j=j1jand
< 2=< 1neither solution is dominant and both are
unique.
For proofs see Wong and Li (1992a). For error
bounds see Zhang et al. (1996). See also Olver (1967b).
For asymptotic expansions in inverse factorial series
see Olde Daalhuis (2004a).
2.9(ii) Coincident Characteristic Values
When the roots of (2.9.5) are equal we denote them
both by. Assume rst 2 g16=f0f1. Then (2.9.1) has
independent solutions wj(n),j= 1;2, such that
2.9.9wj(n)nexp
( 1)jpn
n 1X
s=0( 1)jscs
ns=2;
where
2.9.10pg0=p
2f0f1 4g1, 4g0 =g0+ 2g1;
c0= 1, and higher coecients are determined by formal
substitution.
Alternatively, suppose that 2 g1=f0f1. Then the
indices 1; 2are the roots of
2.9.11 2g0 2 (f0f1+ 2g0) + 2g2 f0f2= 0:Provided that 2 1is not zero or an integer, (2.9.1)
has independent solutions wj(n),j= 1;2, of the form
2.9.12 wj(n)nn j1X
s=0as;j
ns,n!1;
witha0;j= 1 and higher coecients given by (2.9.7) (in
the present case the coecients of as;jandas 1;jare
zero).
If 2 1= 0;1;2;:::, then (2.9.12) applies only in
the casej= 1. But there is an independent solution
2.9.13
w2(n)nn 21X
s=0
s6= 2 1bs
ns+cw1(n) lnn,n!1:
The coecients bsand constant care again determined
by formal substitution, beginning with c= 1 when
2 1= 0, or with b0= 1 when 2 1= 1;2;3;:::.
(Compare (2.7.6).)
For proofs and examples, see Wong and Li (1992a).
For error bounds see Zhang et al. (1996).
For analogous results for di erence equations of the
form
2.9.14w(n+ 2) +nPf(n)w(n+ 1) +nQg(n)w(n) = 0;
in whichPandQare any integers see Wong and Li
(1992b).
2.9(iii) Other Approximations
For asymptotic approximations to solutions of second-
order di erence equations analogous to the Liouville{
Green (WKBJ) approximation for di erential equations
(x2.7(iii)) see Spigler and Vianello (1992, 1997) and
Spigler et al. (1999). Error bounds and applications
are included.
2.10 Sums and Sequences 63
For discussions of turning points, transition points,
and uniform asymptotic expansions for solutions of lin-
ear di erence equations of the second order see Wang
and Wong (2003, 2005).
For an introduction to, and references for, the gen-
eral asymptotic theory of linear di erence equations of
arbitrary order, see Wimp (1984, Appendix B).
2.10 Sums and Sequences
2.10(i) Euler{Maclaurin Formula
As inx24.2, letBnandBn(x) denote the nth Bernoulli
number and polynomial, respectively, and eBn(x) the
nth Bernoulli periodic function Bn(x bxc).
Assume that a;m, andnare integers such that
n > a ,m > 0, andf(2m)(x) is absolutely integrable
over [a;n]. Then
2.10.1
nX
j=af(j) =Zn
af(x)dx+1
2f(a) +1
2f(n)
+m 1X
s=1B2s
(2s)!
f(2s 1)(n) f(2s 1)(a)
+Zn
aB2m eB2m(x)
(2m)!f(2m)(x)dx:
This is the Euler{Maclaurin formula . Another version
is the Abel{Plana formula :
2.10.2
nX
j=af(j) =Zn
af(x)dx+1
2f(a) +1
2f(n)
2Z1
0=(f(a+iy))
e2y 1dy
+mX
s=1B2s
(2s)!f(2s 1)(n)
+ 2( 1)m
(2m)!Z1
0=(f(2m)(n+i#ny))y2mdy
e2y 1;
#nbeing some number in the interval (0 ;1). Sucient
conditions for the validity of this second result are:
(a) On the strip a<zn,f(z) is analytic in its
interior,f(2m)(z) is continuous on its closure, and
f(z) =o
e2j=zj
as=z!1 , uniformly with
respect to<z2[a;n].
(b)f(z) is real when azn.
(c) The rst in nite integral in (2.10.2) converges.Example
2.10.3 S(n) =nX
j=1jlnj
for largen. From (2.10.1)
2.10.4
S(n) =1
2n2lnn 1
4n2+1
2nlnn+1
12lnn+C
+m 1X
s=2( B2s)
2s(2s 1)(2s 2)1
n2s 2+Rm(n);
wherem(2) is arbitrary, Cis a constant, and
2.10.5Rm(n) =Z1
neB2m(x) B2m
2m(2m 1)x2m 1dx:
Fromx24.12(i), (24.2.2), and (24.4.27), eB2m(x) B2mis
of constant sign ( 1)m. ThusRm(n) andRm+1(n) are
of opposite signs, and since their di erence is the term
corresponding to s=min (2.10.4), Rm(n) is bounded
in absolute value by this term and has the same sign.
Formula (2.10.2) is useful for evaluating the con-
stant term in expansions obtained from (2.10.1). In the
present example it leads to
2.10.6C=
+ ln(2)
12 0(2)
22=1
12 0( 1);
where
is Euler's constant ( x5.2(ii)) and 0is the deriva-
tive of the Riemann zeta function ( x25.2(i)).eCis some-
times called Glaisher's constant . For further informa-
tion onCseex5.17.
Other examples that can be veri ed in a similar way
are:
2.10.7
n 1X
j=1j ( ) +n +1
+ 11X
s=0 + 1
sBs
ns,n!1;
where (6= 1) is a real constant, and
2.10.8n 1X
j=11
jlnn+
1
2n 1X
s=1B2s
2s1
n2s,n!1:
In both expansions the remainder term is bounded in
absolute value by the rst neglected term in the sum,
and has the same sign, provided that in the case of
(2.10.7), truncation takes place at s= 2m 1, wherem
is any positive integer satisfying m1
2( + 1).
For extensions of the Euler{Maclaurin formula to
functionsf(x) with singularities at x=aorx=n(or
both) see Sidi (2004). See also Weniger (2007).
For an extension to integrals with Cauchy principal
values see Elliott (1998).
2.10(ii) Summation by Parts
The formula for summation by parts is
2.10.9n 1X
j=1ujvj=Un 1vn+n 1X
j=1Uj(vj vj+1);
64 Asymptotic Approximations
where
2.10.10 Uj=u1+u2++uj:
This identity can be used to nd asymptotic approxima-
tions for large nwhen the factor vjchanges slowly with
j, andujis oscillatory; compare the approximation of
Fourier integrals by integration by parts in x2.3(i).
Example
2.10.11 S( ; ;n ) =n 1X
j=1eij j ;
where and are real constants with ei 6= 1.
As a rst estimate for large n
2.10.12
jS( ; ;n )jn 1X
j=1j =O(1); O(lnn);orO
n +1
;
according as < 1, = 1, or > 1; see (2.10.7),
(2.10.8). With uj=eij ,vj=j ,
2.10.13 Uj=ei (eij 1)=(ei 1);
and
2.10.14S( ; ;n ) =ei
ei 10
@ei(n 1) n 1
+n 1X
j=1eij (j (j+ 1) )1
A:
Since
2.10.15j (j+ 1) = j 1+ ( 1)O
j 2
for any real constant and the set of all positive integers
j, we derive
2.10.16
S( ; ;n ) =ei
ei 1
ei(n 1) n S( 1; ;n )
+O
n 1
+O(1)
:
From this result and (2.10.12)
2.10.17 S( ; ;n ) =O(n ) +O(1):
Then replacing by 1 and resubstituting in
(2.10.16), we have
2.10.18
S( ; ;n ) =ein
ei 1n +O
n 1
+O(1),n!1;
which is a useful approximation when >0.
For extensions to 0, higher terms, and other
examples, see Olver (1997b, Chapter 8).
2.10(iii) Asymptotic Expansions of Entire
Functions
The asymptotic behavior of entire functions de ned by
Maclaurin series can be approached by converting the
sum into a contour integral by use of the residue theo-
rem and applying the methods of xx2.4 and 2.5.Example
Fromxx16.2(i){16.2(ii)
2.10.190F2( ; 1;1;x) =1X
j=0xj
(j!)3:
We seek the behavior as x!+1. From (1.10.8)
2.10.20n 1X
j=0xj
(j!)3=1
2iZ
Cxt
( (t+ 1))3cot(t)dt;
whereCcomprises the two semicircles and two parts of
the imaginary axis depicted in Figure 2.10.1.
Figure 2.10.1 :t-plane. Contour C.
From the identities
2.10.21cot(t)
2i= 1
2 1
e 2it 1=1
2+1
e2it 1;
and Cauchy's theorem, we have
2.10.22n 1X
j=0xj
(j!)3=Zn (1=2)
1=2xt
( (t+ 1))3dt
Z
C1xt
( (t+ 1))3dt
e 2it 1
+Z
C2xt
( (t+ 1))3dt
e2it 1;
whereC1;C2denote respectively the upper and lower
halves of C. (5.11.7) shows that the integrals around
the large quarter circles vanish as n!1 . Hence
2.10.23
0F2( ; 1;1;x) =Z1
1=2xt
( (t+ 1))3dt
+ 2<Zi1
1=2xt
( (t+ 1))3dt
e 2it 1
=Z1
0xt
( (t+ 1))3dt+O(1),
x!+1;
2.10 Sums and Sequences 65
the last step following from jxtj1 whentis on the
interval [ 1
2;0], the imaginary axis, or the small semi-
circle. By application of Laplace's method ( x2.3(iii))
and use again of (5.11.7), we obtain
2.10.24 0F2( ; 1;1;x)exp
3x1=3
231=2x1=3,x!+1:
For generalizations and other examples see Olver
(1997b, Chapter 8), Ford (1960), and Berndt and Evans
(1984). See also Paris and Kaminski (2001, Chapter 5)
andxx16.11(i){16.11(ii).
2.10(iv) Taylor and Laurent Coecients:
Darboux's Method
Letf(z) be analytic on the annulus 0 <jzj< r, with
Laurent expansion
2.10.25 f(z) =1X
n= 1fnzn, 0<jzj<r:
What is the asymptotic behavior of fnasn!1 or
n! 1 ? More specially, what is the behavior of the
higher coecients in a Taylor-series expansion?
These problems can be brought within the scope of
x2.4 by means of Cauchy's integral formula
2.10.26 fn=1
2iZ
Cf(z)
zn+1dz;
whereCis a simple closed contour in the annulus that
enclosesz= 0. For examples see Olver (1997b, Chap-
ters 8, 9).
However, if ris nite and f(z) has algebraic or loga-
rithmic singularities on jzj=r, then Darboux's method
is usually easier to apply. We need a \comparison func-
tion"g(z) with the properties:
(a)g(z) is analytic on 0 <jzj<r.
(b)f(z) g(z) is continuous on 0 <jzjr.
(c) The coecients in the Laurent expansion
2.10.27g(z) =1X
n= 1gnzn, 0<jzj<r;
have known asymptotic behavior as n!1 .
By allowing the contour in Cauchy's formula to ex-
pand, we nd that
2.10.28
fn gn=1
2iZ
jzj=rf(z) g(z)
zn+1dz
=1
2rnZ2
0
f
rei
g
rei
e nid:
Hence by the Riemann{Lebesgue lemma ( x1.8(i))
2.10.29 fn=gn+o
r n
,n!1:This result is re nable in two important ways. First,
the conditions can be weakened. It is unnecessary for
f(z) g(z) to be continuous on jzj=r: it suces that
the integrals in (2.10.28) converge uniformly. For exam-
ple, Condition (b) can be replaced by:
(b0) On the circlejzj=r, the function f(z) g(z)
has a nite number of singularities, and at each
singularity zj, say,
2.10.30f(z) g(z) =O
(z zj)j 1
,z!zj;
wherejis a positive constant.
Secondly, when f(z) g(z) ismtimes continuously
di erentiable on jzj=rthe result (2.10.29) can be
strengthened. In these circumstances the integrals in
(2.10.28) are integrable by parts mtimes, yielding
2.10.31 fn=gn+o
r njnj m
,n!1:
Furthermore, (2.10.31) remains valid with the weaker
condition
2.10.32f(m)(z) g(m)(z) =O
(z zj)j 1
;
in the neighborhood of each singularity zj, again with
j>0.
Example
Let be a constant in (0 ;2) andPndenote the Leg-
endre polynomial of degree n. Fromx14.7(iv)
2.10.33f(z)1
(1 2zcos +z2)1=2
=1X
n=0Pn(cos )zn,jzj<1:
The singularities of f(z) on the unit circle are branch
points atz=ei . To match the limiting behavior of
f(z) at these points we set
2.10.34g(z) =e i=4(2 sin ) 1=2
e i z 1=2
+ei=4(2 sin ) 1=2
ei z 1=2:
Here the branch of
e i z 1=2is continuous in
thez-plane cut along the outward-drawn ray through
z=e i and equals ei =2atz= 0. Similarly for
ei z 1=2. In Condition (c) we have
2.10.35
gn=2
sin 1=2
n+1
2
n!cos
n +1
2 1
4
;
and in the supplementary conditions we may set m= 1.
Then from (2.10.31) and (5.11.7)
2.10.36
Pn(cos ) =2
nsin 1=2
cos
n +1
2 1
4
+o
n 1
:
For higher terms see x18.15(iii).
66 Asymptotic Approximations
For uniform expansions when two singularities coa-
lesce on the circle of convergence see Wong and Zhao
(2005).
For other examples and extensions see Olver (1997b,
Chapter 8), Olver (1970), Wong (1989, Chapter 2),
and Wong and Wyman (1974). See also Flajolet and
Odlyzko (1990).
2.11 Remainder Terms; Stokes
Phenomenon
2.11(i) Numerical Use of Asymptotic
Expansions
When a rigorous bound or reliable estimate for the re-
mainder term is unavailable, it is unsafe to judge the
accuracy of an asymptotic expansion merely from the
numerical rate of decrease of the terms at the point of
truncation. Even when the series converges this is un-
wise: the tail needs to be majorized rigorously before
the result can be guaranteed. For divergent expansions
the situation is even more dicult. First, it is impossi-
ble to bound the tail by majorizing its terms. Secondly,
the asymptotic series represents an in nite class of func-
tions, and the remainder depends on which member we
have in mind.
As an example consider
2.11.1 I(m) =Z
0cos(mt)
t2+ 1dt;
withma large integer. By integration by parts ( x2.3(i))
2.11.2 I(m)( 1)m1X
s=1qs()
m2s,m!1;
with
2.11.3q1(t) = 2t
(t2+ 1)2; q 2(t) =24(t3 t)
(t2+ 1)4;
q3(t) = 240(3t5 10t3+ 3t)
(t2+ 1)6:
On rounding to 5D, we have q1() = 0:05318,q2() =
0:04791,q3() = 0:08985. Hence
2.11.4I(10) 0:00053 18 + 0 :00000 48 0:00000 01
= 0:00052 71:
But this answer is incorrect: to 7D I(10) =
0:00045 58. The error term is, in fact, approximately
700 times the last term obtained in (2.11.4). The ex-
planation is that (2.11.2) is a more accurate expansion
for the function I(m) 1
2e mthan it is for I(m); see
Olver (1997b, pp. 76{78).
In order to guard against this kind of error remaining
undetected, the wanted function may need to be com-
puted by another method (preferably nonasymptotic)
for the smallest value of the (large) asymptotic variablexthat is intended to be used. If the results agree within
Ssigni cant gures, then it is likely| but not certain |
that the truncated asymptotic series will yield at least
Scorrect signi cant gures for larger values of x. For
further discussion see Bosley (1996).
InCboth the modulus and phase of the asymptotic
variablezneed to be taken into account. Suppose an
asymptotic expansion holds as z!1 in any closed sec-
tor within <phz < , say, but not in phz .
Then numerical accuracy will disintegrate as the bound-
ary rays ph z= , phz= are approached. In conse-
quence, practical application needs to be con ned to a
sector 0phz 0well within the sector of validity,
and independent evaluations carried out on the bound-
aries for the smallest value of jzjintended to be used.
The choice of 0and 0is facilitated by a knowledge of
the relevant Stokes lines; see x2.11(iv) below.
However, regardless whether we can bound the re-
mainder, the accuracy achievable by direct numerical
summation of a divergent asymptotic series is always
limited. The rest of this section is devoted to general
methods for increasing this accuracy.
2.11(ii) Connection Formulas
Fromx8.19(i) the generalized exponential integral is
given by
2.11.5Ep(z) =e zzp 1
(p)Z1
0e zttp 1
1 +tdt
when<p > 0 andjphzj<1
2, and by analytic con-
tinuation for other values of pandz. Application of
Watson's lemma ( x2.4(i)) yields
2.11.6 Ep(z)e z
z1X
s=0( 1)s(p)s
zs
whenpis xed and z!1 in any closed sector within
jphzj<3
2. As noted inx2.11(i), poor accuracy is
yielded by this expansion as ph zapproaches3
2or 3
2.
However, on combining (2.11.6) with the connection for-
mula (8.19.18), with m= 1, we derive
2.11.7Ep(z)2ie pi
(p)zp 1+e z
z1X
s=0( 1)s(p)s
zs;
valid asz!1 in any closed sector within1
2<phz<
7
2; compare (8.20.3). Since the ray ph z=3
2is well
away from the new boundaries, the compound expan-
sion (2.11.7) yields much more accurate results when
phz!3
2. In e ect, (2.11.7) \corrects" (2.11.6) by in-
troducing a term that is relatively exponentially small in
the neighborhood of ph z=, is increasingly signi cant
as phzpasses from to3
2, and becomes the dominant
contribution after ph zpasses3
2. See alsox2.11(iv).
2.11 Remainder Terms; Stokes Phenomenon 67
2.11(iii) Exponentially-Improved Expansions
The procedure followed in x2.11(ii) enabled Ep(z) to
be computed with as much accuracy in the sector
phz3as the original expansion (2.11.6) in
jphzj. We now increase substantially the accuracy
of (2.11.6) injphzjby re-expanding the remainder
term.
Optimum truncation in (2.11.6) takes place at s=
n 1, withjp+n 1j=jzj, approximately. Thus
2.11.8 n= p+ ;
wherez=ei, andj jis bounded as n!1 . From
(2.11.5) and the identity
2.11.91
1 +t=n 1X
s=0( 1)sts+ ( 1)ntn
1 +t,t6= 1;
we have
2.11.10
Ep(z) =e z
zn 1X
s=0( 1)s(p)s
zs+ ( 1)n2
(p)zp 1Fn+p(z);
where
2.11.11
Fn+p(z) =e z
2Z1
0e zttn+p 1
1 +tdt= (n+p)
2En+p(z)
zn+p 1:
Withngiven by (2.11.8), we have
2.11.12
Fn+p(z) =e z
2Z1
0exp
tei lntt 1
1 +tdt:
For largethe integrand has a saddle point at t=
e i. Followingx2.4(iv), we rotate the integration path
through an angle , which is valid by analytic con-
tinuation when < < . Then by application of
Laplace's method ( xx2.4(iii) and 2.4(iv)), we have
2.11.13
Fn+p(z)e i(+ )
1 +e ie z
(2)1=21X
s=0a2s(; )
s,!1;
uniformly when 2[ +; ] ( >0) andj jis
bounded. The coecients are rational functions of
and 1 +ei, for example, a0(; ) = 1, and
2.11.14
a2(; ) =1
12(6 2 6 + 1)
1 +ei+1
(1 +ei)2:
Owing to the factor e , that is,e jzjin (2.11.13),
Fn+p(z) is uniformly exponentially small compared with
Ep(z). For this reason the expansion of Ep(z) in
jphzj supplied by (2.11.8), (2.11.10), and
(2.11.13) is said to be exponentially improved .
If we permit the use of nonelementary functions as
approximants, then even more powerful re-expansions
become available. One is uniformly valid for +phz3 with boundedj j, and achieves uniform
exponential improvement throughout 0 phz:
2.11.15Fn+p(z)( 1)nie pi
1
2erfcq
1
2c()
iei( )e z
(2)1=21X
s=0h2s(; )
s!
:
Here erfc is the complementary error function ( x7.2(i)),
and
2.11.16 c() =q
2(1 +ei+i( ));
the branch being continuous with c() as!.
Also,
2.11.17h2s(; ) =ei ( )
1 +e ia2s(; )
+ ( 1)s 1i135(2s 1)
(c())2s+1;
witha2s(; ) as in (2.11.13), (2.11.14). In particular,
2.11.18 h0(; ) =ei ( )
1 +e i i
c():
For the sector 3+phz the conjugate
result applies.
Further details for this example are supplied in Olver
(1991a, 1994b). See also Paris and Kaminski (2001,
Chapter 6), and Dunster (1996b, 1997).
2.11(iv) Stokes Phenomenon
Two di erent asymptotic expansions in terms of ele-
mentary functions, (2.11.6) and (2.11.7), are available
for the generalized exponential integral in the sector
1
2 < phz <3
2. That the change in their forms is
discontinuous, even though the function being approx-
imated is analytic, is an example of the Stokes phe-
nomenon . Where should the change-over take place?
Can it be accomplished smoothly?
Satisfactory answers to these questions were found
by Berry (1989); see also the survey by Paris and Wood
(1995). These answers are linked to the terms involving
the complementary error function in the more power-
ful expansions typi ed by the combination of (2.11.10)
and (2.11.15). Thus if 0 (< ), then
c() lies in the right half-plane. Hence from x7.12(i)
erfcq
1
2c()
is of the same exponentially-small or-
der of magnitude as the contribution from the other
terms in (2.11.15) when is large. On the other hand,
when+3 ,c() is in the left half-plane
and erfcq
1
2c()
di ers from 2 by an exponentially-
small quantity. In the transition through =,
erfcq
1
2c()
changes very rapidly, but smoothly,
from one form to the other; compare the graph of its
modulus in Figure 2.11.1 in the case = 100.
68 Asymptotic Approximations
Figure 2.11.1 : Graph ofjerfc p
50c()
j.
In particular, on the ray =greatest accuracy
is achieved by (a) taking the average of the expansions
(2.11.6) and (2.11.7), followed by (b) taking account of
the exponentially-small contributions arising from the
terms involving h2s(; ) in (2.11.15).
Rays (or curves) on which one contribution in a
compound asymptotic expansion achieves maximum
dominance over another are called Stokes lines (=
in the present example). As these lines are
crossed exponentially-small contributions, such as that
in (2.11.7), are \switched on" smoothly, in the manner
of the graph in Figure 2.11.1.
For higher-order Stokes phenomena see Olde Daal-
huis (2004b) and Howls et al. (2004).2.11(v) Exponentially-Improved Expansions
(continued)
Expansions similar to (2.11.15) can be constructed for
many other special functions. However, to enjoy the
resurgence property ( x2.7(ii)) we often seek instead
expansions in terms of the F-functions introduced in
x2.11(iii), leaving the connection of the error-function
type behavior as an implicit consequence of this prop-
erty of theF-functions. In this context the F-functions
are called terminants , a name introduced by Dingle
(1973).
For illustration, we give re-expansions of the re-
mainder terms in the expansions (2.7.8) arising in
di erential-equation theory. For notational convenience
assume that the original di erential equation (2.7.1) is
normalized so that 2 1= 1. (This means that, if
necessary,zis replaced by z=(2 1).) From (2.7.12),
(2.7.13) it is then seen that the optimum number of
terms,n, in (2.7.14) is approximately jzj. We set
2.11.19wj(z) =ejzzjn 1X
s=0as;j
zs+R(j)
n(z),j= 1;2;
and expand
2.11.20 R(1)
n(z) = ( 1)n 1ie(2 1)ie2zz2
C1m 1X
s=0( 1)sas;2Fn+2 1 s(z)
zs+R(1)
m;n(z)!
;
2.11.21 R(2)
n(z) = ( 1)nie(2 1)ie1zz1
C2m 1X
s=0( 1)sas;1Fn+1 2 s(ze i)
zs+R(2)
m;n(z)!
;
withm= 0;1;2;:::, andC1;C2as in (2.7.17). Then as z!1 , withjn jzjjbounded and m xed,
2.11.22 R(1)
m;n(z) =(
O
e jzj zz m
;jphzj;
O(z m); jphzj5
2 ;
2.11.23 R(2)
m;n(z) =(
O
e jzj+zz m
;0phz2;
O(z m); 3
2+phz0 and 2phz7
2 ;
uniformly with respect to ph zin each case.
The relevant Stokes lines are ph z=forw1(z),
and phz= 0;2forw2(z). In addition to achiev-
ing uniform exponential improvement, particularly in
jphzjforw1(z), and 0phz2forw2(z), the
re-expansions (2.11.20), (2.11.21) are resurgent.
For further details see Olde Daalhuis and Olver
(1994). For error bounds see Dunster (1996c). For other
examples see Boyd (1990b), Paris (1992a,b), and Wong
and Zhao (2002b).
Often the process of re-expansion can be repeated
any number of times. In this way we arrive at hy-perasymptotic expansions . For integrals, see Berry and
Howls (1991), Howls (1992), and Paris and Kaminski
(2001, Chapter 6). For second-order di erential equa-
tions, see Olde Daalhuis and Olver (1995a), Olde Daal-
huis (1995, 1996), and Murphy and Wood (1997).
For higher-order di erential equations, see Olde
Daalhuis (1998a,b). The rst of these two references
also provides an introduction to the powerful Borel
transform theory. In this connection see also Byatt-
Smith (2000).
For nonlinear di erential equations see Olde Daal-
huis (2005a,b).
2.11 Remainder Terms; Stokes Phenomenon 69
For another approach see Paris (2001a,b).
2.11(vi) Direct Numerical Transformations
The transformations in x3.9 for summing slowly conver-
gent series can also be very e ective when applied to
divergent asymptotic series.
A simple example is provided by Euler's transforma-
tion (x3.9(ii)) applied to the asymptotic expansion for
the exponential integral ( x6.12(i)):
2.11.24 exE1(x)1X
s=0( 1)ss!
xs+1,x!+1:
Takingx= 5 and rounding to 5D, we obtain
2.11.25
e5E1(5) = 0:20000 0:04000 + 0:01600 0:00960
+ 0:00768 0:00768 + 0:00922 0:01290
+ 0:02064 0:03716 + 0:07432 :
The numerically smallest terms are the 5th and 6th.
Truncation after 5 terms yields 0.17408, compared with
the correct value
2.11.26 e5E1(5) = 0:17042::::
We now compute the forward di erences j,j=
0;1;2;:::, of the moduli of the rounded values of the
rst 6 neglected terms:
2.11.270= 0:00768 , 1= 0:00154 ,
2= 0:00214 , 3= 0:00192 ,
4= 0:00280 , 5= 0:00434 .
Multiplying these di erences by ( 1)j2 j 1and sum-
ming, we obtain
2.11.280:00384 0:00038 + 0:00027 0:00012
+ 0:00009 0:00007 = 0:00363:
Subtraction of this result from the sum of the rst 5
terms in (2.11.25) yields 0.17045, which is much closer
to the true value.
The process just used is equivalent to re-expanding
the remainder term of the original asymptotic series
(2.11.24) in powers of 1 =(x+ 5) and truncating the new
series optimally. Further improvements in accuracy can
be realized by making a second application of the Euler
transformation; see Olver (1997b, pp. 540{543).
Similar improvements are achievable by Aitken's
2-process, Wynn's -algorithm, and other accelera-
tion transformations. For a comprehensive survey see
Weniger (1989).
The following example, based on Weniger (1996), il-
lustrates their power.
For largejzj, withjphzj3
2 (<3
2), the Whit-
taker function of the second kind has the asymptotic
expansion (x13.19)
2.11.29 W;(z)1X
n=0an;in which
2.11.30an=e z=2
zn n!
2 ( 1
2)2
2 ( 3
2)2
2 ( n+1
2)2
:
Withz= 1:0,= 2:3,= 0:5, the values of anto 8D
are supplied in the second column of Table 2.11.1.
Table 2.11.1 : Whittaker functions with Levin's transfor-
mation.
n a n sn dn
0 0:60653 066 0 :60653 066 0 :60653 066
1 1:81352 667 1:20699 601 0:91106 488
2 0:35363 770 0:85335 831 0:82413 405
3 0:02475 464 0:82860 367 0:83323 429
4 0:00736 451 0:83596 818 0:83303 750
5 0:00676 062 0:82920 756 0:83298 901
6 0:01125 643 0:84046 399 0:83299 429
7 0:02796 418 0:81249 981 0:83299 530
8 0:09364 504 0:90614 485 0:83299 504
9 0:39736 710 0:50877 775 0:83299 501
10 2:05001 686 2:55879 461 0:83299 503
The next column lists the partial sums sn=a0+a1+
+an. Optimum truncation occurs just prior to the
numerically smallest term, that is, at s4. Comparison
with the true value
2.11.31W2:3;0:5(1:0) = 0:83299 50268 27526
shows that this direct estimate is correct to almost 3D.
The fourth column of Table 2.11.1 gives the results
of applying the following variant of Levin's transforma-
tion:
2.11.32dn=Pn
j=0( 1)j n
j
(j+ 1)n 1sj
aj+1Pn
j=0( 1)j n
j
(j+ 1)n 11
aj+1:
Byn= 10 we already have 8 correct decimals. Further-
more, on proceeding to higher values of nwith higher
precision, much more accuracy is achievable. For exam-
ple, using double precision d20is found to agree with
(2.11.31) to 13D.
However, direct numerical transformations need to
be used with care. Their extrapolation is based on as-
sumed forms of remainder terms that may not always
be appropriate for asymptotic expansions. For example,
extrapolated values may converge to an accurate value
on one side of a Stokes line ( x2.11(iv)), and converge to
a quite inaccurate value on the other.
70 Asymptotic Approximations
References
General References
The main references used in writing this chapter are
Olver (1997b) and Wong (1989).
For additional bibliographic reading see Bender
(1974), Bleistein and Handelsman (1975), Copson
(1965), de Bruijn (1961), Dingle (1973), Erd elyi (1956),
Jones (1972, 1997), Lauwerier (1974), Odlyzko (1995),
Paris and Kaminski (2001), Slavyanov and Lay (2000),
Temme (1995c), and Wasow (1965).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x2.1Olver (1997b, Chapter 1).
x2.2Olver (1997b, pp. 11{16), Fabijonas and Olver
(1999).x2.3Olver (1997b, Chapter 3). For (2.3.9) see Wong
(1989,x2.2). For (2.3.12) use termwise integration
in an analogous manner to that used to prove Wat-
son's lemma (Olver (1997b, pp. 71{72). (2.3.18)
follows from (1.10.15) and (1.10.17) with f(t) =
p(t),g(t) =q(t)
p0(t)(p(t) p(a))(=) 1
, us-
ing Cauchy's integral formula for the residue, and
integrating by parts. See also Cicuta and Mon-
taldi (1975).
x2.4Olver (1997b, Chapter 4 and pp. 315{320), Wong
(1989, p. 31).
x2.5Wong (1989, Chapter 3), Doetsch (1955, x6.5).
x2.6Wong (1989, Chapter 6).
x2.7Olver (1997b, Chapters 5{7), Olver (1994a),
Olde Daalhuis and Olver (1994), Olde Daalhuis
(1998a).
x2.8Olver (1997b, Chapters 10{12).
x2.10 Olver (1997b, Chapter 8).
x2.11 Olver (1997b, pp. 76{78 and 540{543), Olver
(1991a), Weniger (1996). The computations in the
example inx2.11(vi) were carried out at NIST.
Chapter 3
Numerical Methods
N. M. Temme1
Areas 72
3.1 Arithmetics and Error Measures . . . . . 72
3.2 Linear Algebra . . . . . . . . . . . . . . . 73
3.3 Interpolation . . . . . . . . . . . . . . . . 75
3.4 Di erentiation . . . . . . . . . . . . . . . 77
3.5 Quadrature . . . . . . . . . . . . . . . . 78
3.6 Linear Di erence Equations . . . . . . . . 853.7 Ordinary Di erential Equations . . . . . . 88
3.8 Nonlinear Equations . . . . . . . . . . . . 90
3.9 Acceleration of Convergence . . . . . . . 93
3.10 Continued Fractions . . . . . . . . . . . . 94
3.11 Approximation Techniques . . . . . . . . 96
3.12 Mathematical Constants . . . . . . . . . 100
References 100
1Centrum voor Wiskunde en Informatica, Department MAS, Amsterdam, The Netherlands.
Acknowledgments : The author thanks W. Gautschi and C. Brezinski for their valuable contributions, and F. W. J. Olver for
assisting with the writing of xx3.6, 3.7, 3.8, and 3.11.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
71
72 Numerical Methods
Areas
3.1 Arithmetics and Error Measures
3.1(i) Floating-Point Arithmetic
Computer arithmetic is described for the binary based
system with base 2; another frequently used system is
thehexadecimal system with base 16.
A nonzero normalized binary
oating-point machine
numberxis represented as
3.1.1x= ( 1)s(b0:b1b2:::bp 1)2E,b0= 1,
wheresis equal to 1 or 0, each bj,j1, is either 0 or
1,b1is the most signi cant bit ,p(2N) is the number
of signi cant bits bj,bp 1is the least signi cant bit ,E
is an integer called the exponent ,b0:b1b2:::bp 1is the
signi cand , andf=:b1b2:::bp 1is the fractional part .
The set of machine numbers R
is the union of 0 and
the set
3.1.2 ( 1)s2Ep 1X
j=0bj2 j;
withb0= 1 and all allowable choices of E,p,s, andbj.
LetEminEEmaxwithEmin<0 andEmax>0.
For given values of Emin,Emax, andp, the format width
in bitsNof a computer word is the total number of bits:
the sign (one bit), the signi cant bits b1;b2;:::;bp 1
(p 1 bits), and the bits allocated to the exponent (the
remainingN pbits). The integers p,Emin, andEmax
are characteristics of the machine. The machine epsilon
M, that is, the distance between 1 and the next larger
machine number with E= 0 is given by M= 2 p+1.
The machine precision is1
2M= 2 p. The lower and
upper bounds for the absolute values of the nonzero ma-
chine numbers are given by
3.1.3Nmin2Eminjxj2Emax+1
1 2 p
Nmax:
Under
ow (over
ow) after computing x6= 0 occurs
whenjxjis smaller (larger) than Nmin(Nmax).
IEEE Standard
The current standard is the ANSI/IEEE Standard 754;
see IEEE (1985, xx1{4). In the case of normalized bi-
nary representation the memory positions for single pre-
cision (N= 32,p= 24,Emin= 126,Emax= 127)
and double precision (N= 64,p= 53,Emin= 1022,
Emax = 1023) are as in Figure 3.1.1. The respec-
tive machine precisions are1
2M= 0:59610 7and
1
2M= 0:11110 15.1
s8
E23 bits
fN= 32,
p= 24
1
s11
E52 bits
fN= 64,
p= 53
Figure 3.1.1 : Floating-point arithmetic. Memory posi-
tions in single and double precision, in the case of binary
representation.
Rounding
Letxbe any positive number with
3.1.4x= (1:b1b2:::bp 1bpbp+1:::)2E;
NminxNmax, and
3.1.5x = (1:b1b2:::bp 1)2E;
x+= ((1:b1b2:::bp 1) +M)2E:
Then rounding by chopping orrounding down ofx
givesx , with maximum relative error M.Symmet-
ric rounding orrounding to nearest ofxgivesx orx+,
whichever is nearer to x, with maximum relative error
equal to the machine precision1
2M= 2 p.
Negative numbers xare rounded in the same way as
x.
For further information see Goldberg (1991) and
Overton (2001).
3.1(ii) Interval Arithmetic
Interval arithmetic is intended for bounding the total
e ect of rounding errors of calculations with machine
numbers. With this arithmetic the computed result can
be proved to lie in a certain interval, which leads to vali-
dated computing with guaranteed and rigorous inclusion
regions for the results.
LetGbe the set of closed intervals f[a;b]g. The ele-
mentary arithmetical operations on intervals are de ned
as follows:
3.1.6 IJ=fxyjx2I;y2Jg,I;J2G,
where2f +; ;;=g, with appropriate roundings of
the end points of IJwhen machine numbers are be-
ing used. Division is possible only if the divisor interval
does not contain zero.
A basic text on interval arithmetic and analysis is
Alefeld and Herzberger (1983), and for applications and
further information see Moore (1979) and Petkovi c and
Petkovi c (1998). The last reference includes analogs for
arithmetic in the complex plane C.
3.1(iii) Rational Arithmetics
Computer algebra systems use exact rational arithmetic
with rational numbers p=q, wherepandqare multi-
length integers. During the calculations common di-
visors are removed from the rational numbers, and the
3.2 Linear Algebra 73
nal results can be converted to decimal representations
of arbitrary length. For further information see Matula
and Kornerup (1980).
3.1(iv) Level-Index Arithmetic
To eliminate over
ow or under
ow in nite-precision
arithmetic numbers are represented by using general-
ized logarithms ln`(x) given by
3.1.7 ln0(x) =x;ln`(x) = ln(ln` 1(x)),`= 1;2;:::,
withx0 and`the unique nonnegative integer such
thataln`(x)2[0;1). In level-index arithmetic x
is represented by `+a(or (`+a) for negative num-
bers). Also in this arithmetic generalized precision can
be de ned, which includes absolute error and relative
precision (x3.1(v)) as special cases.
For further information see Clenshaw and Olver
(1984) and Clenshaw et al. (1989). For applications see
Lozier (1993).
For further references on level-index arithmetic (and
also other arithmetics) see Anuta et al. (1996). See also
Hayes (2009).
3.1(v) Error Measures
Ifxis an approximation to a real or complex number
x, then the absolute error is
3.1.8 a=jx xj:
Ifx6= 0, the relative error is
3.1.9 r= x x
x =a
jxj:
The relative precision is
3.1.10 rp=jln(x/x)j;
wherexx>0 for real variables, and xx6= 0 for
complex variables (with the principal value of the loga-
rithm).
The molli ed error is
3.1.11 m=jx xj
max(jxj;1):
For error measures for complex arithmetic see Olver
(1983).
3.2 Linear Algebra
3.2(i) Gaussian Elimination
To solve the system
3.2.1 Ax=b;with Gaussian elimination, where Ais a nonsingular
nnmatrix and bis ann1 vector, we start with the
augmented matrix
3.2.22
64a11a1nb1
............
an1annbn3
75:
By repeatedly subtracting multiples of each row
from the subsequent rows we obtain a matrix of the
form
3.2.32
6664u11u12u1ny1
0u22u2ny2
...............
0 0unnyn3
7775:
During this reduction process we store the multipli-
ers`jkthat are used in each column to eliminate other
elements in that column. This yields a lower triangular
matrix of the form
3.2.4 L=2
66641 0 0
`21 1 0
............
`n1`n;n 113
7775:
If we denote by Uthe upper triangular matrix com-
prising the elements ujkin (3.2.3), then we have the
factorization, or triangular decomposition ,
3.2.5 A=LU:
With y= [y1;y2;:::;yn]Tthe process of solution can
then be regarded as rst solving the equation Ly=b
fory(forward elimination ), followed by the solution of
Ux=yforx(back substitution ).
For more details see Golub and Van Loan (1996,
pp. 87{100).
Example
3.2.62
41 2 3
2 3 1
3 1 23
5=2
41 0 0
2 1 0
3 5 13
52
41 2 3
0 1 5
0 0 183
5:
In solving Ax= [1;1;1]T, we obtain by forward elim-
ination y= [1; 1;3]T, and by back substitution x=
[1
6;1
6;1
6]T.
In practice, if any of the multipliers `jkare unduly
large in magnitude compared with unity, then Gaussian
elimination is unstable. To avoid instability the rows
are interchanged at each elimination step in such a way
that the absolute value of the element that is used as a
divisor, the pivot element , is not less than that of the
other available elements in its column. Then j`jkj1
in all cases. This modi cation is called Gaussian elimi-
nation with partial pivoting .
For more information on pivoting see Golub and
Van Loan (1996, pp. 109{123).
74 Numerical Methods
Iterative Re nement
When the factorization (3.2.5) is available, the accu-
racy of the computed solution xcan be improved with
little extra computation. Because of rounding errors,
theresidual vector r=b Axis nonzero as a rule. We
solve the system Ax=rforx, taking advantage of
the existing triangular decomposition of Ato obtain an
improved solution x+x.
3.2(ii) Gaussian Elimination for a Tridiagonal
Matrix
Tridiagonal matrices are ones in which the only nonzero
elements occur on the main diagonal and two adjacent
diagonals. Thus
3.2.7 A=2
666664b1c1 0
a2b2c2
.........
an 1bn 1cn 1
0 anbn3
777775:
Assume that Acan be factored as in (3.2.5), but with-
out partial pivoting. Then
3.2.8 L=2
6666641 0 0
`21 0
.........
`n 11 0
0 `n13
777775;
3.2.9 U=2
666664d1u1 0
0d2u2
.........
0dn 1un 1
0 0 dn3
777775;
whereuj=cj,j= 1;2;:::;n 1,d1=b1, and
3.2.10`j=aj=dj 1; dj=bj `jcj 1,j= 2;:::;n .
Forward elimination for solving Ax=fthen becomes
y1=f1,
3.2.11 yj=fj `jyj 1,j= 2;:::;n ,
and back substitution is xn=yn=dn, followed by
3.2.12 xj= (yj ujxj+1)=dj,j=n 1;:::; 1.
For more information on solving tridiagonal systems
see Golub and Van Loan (1996, pp. 152{160).
3.2(iii) Condition of Linear Systems
Thep-norm of a vector x= [x1;:::;xn]Tis given by
3.2.13kxkp=0
@nX
j=1jxjjp1
A1=p
,p= 1;2;:::;
kxk1= max
1jnjxjj:The Euclidean norm is the casep= 2.
Thep-norm of a matrix A= [ajk] is
3.2.14kAkp= max
x6=0kAxkp
kxkp:
The casesp= 1;2, and1are the most important:
3.2.15kAk1= max
1knnX
j=1jajkj;
kAk1= max
1jnnX
k=1jajkj;
kAk2=q
(AAT);
where(AAT) is the largest of the absolute values of
the eigenvalues of the matrix AAT; seex3.2(iv). (We
are assuming that the matrix Ais real; if not ATis re-
placed by AH, the transpose of the complex conjugate
ofA.)
The sensitivity of the solution vector xin (3.2.1) to
small perturbations in the matrix Aand the vector b
is measured by the condition number
3.2.16 (A) =kAkpkA 1kp;
wherekkpis one of the matrix norms. For any norm
(3.2.14) we have (A)1. The larger the value (A),
the more ill-conditioned the system.
Letxdenote a computed solution of the system
(3.2.1), with r=b Axagain denoting the residual.
Then we have the a posteriori error bound
3.2.17kx xkp
kxkp(A)krkp
kbkp:
For further information see Brezinski (1999) and
Trefethen and Bau (1997, Chapter 3).
3.2(iv) Eigenvalues and Eigenvectors
IfAis annnmatrix, then a real or complex number
is called an eigenvalue ofA, and a nonzero vector x
a corresponding ( right )eigenvector , if
3.2.18 Ax=x:
A nonzero vector yis called a left eigenvector ofA
corresponding to the eigenvalue ifyTA=yTor,
equivalently, ATy=y. Anormalized eigenvector has
Euclidean norm 1; compare (3.2.13) with p= 2.
The polynomial
3.2.19 pn() = det[I A]
is called the characteristic polynomial ofAand its zeros
are the eigenvalues of A. The multiplicity of an eigen-
value is its multiplicity as a zero of the characteristic
polynomial (x3.8(i)). To an eigenvalue of multiplicity
m, there correspond mlinearly independent eigenvec-
tors provided that Aisnondefective , that is, Ahas a
complete set of nlinearly independent eigenvectors.
3.3 Interpolation 75
3.2(v) Condition of Eigenvalues
IfAis nondefective and is a simple zero of pn(), then
the sensitivity of to small perturbations in the matrix
Ais measured by the condition number
3.2.20 () =1
jyTxj;
where xandyare the normalized right and left eigen-
vectors of Acorresponding to the eigenvalue . Because yTx =jcosj, whereis the angle between yTand
xwe always have ()1. When Ais a symmetric
matrix, the left and right eigenvectors coincide, yield-
ing() = 1, and the calculation of its eigenvalues is a
well-conditioned problem.
3.2(vi) Lanczos Tridiagonalization of a
Symmetric Matrix
De ne the Lanczos vectors vjbyv0=0, a nor-
malized vector v1(perhaps chosen randomly), and for
j= 1;2;:::;n 1,
3.2.21 j+1vj+1=Avj jvj jvj 1;
j=vT
jAvj; j+1=vT
j+1Avj:
Then all vj, 1jn, are normalized and vT
jvk= 0
forj;k= 1;2;:::;n ,j6=k. The tridiagonal matrix
3.2.22 B=2
666664 1 2 0
2 2 3
.........
n 1 n 1 n
0 n n3
777775
has the same eigenvalues as A. Its characteristic poly-
nomial can be obtained from the recursion
3.2.23pk+1() = ( k+1)pk() 2
k+1pk 1(),
k= 0;1;:::;n 1,
withp 1() = 0,p0() = 1.
For numerical information see Stewart (2001,
pp. 347{368).
3.2(vii) Computation of Eigenvalues
Many methods are available for computing eigenvalues;
see Golub and Van Loan (1996, Chapters 7, 8), Tre-
fethen and Bau (1997, Chapter 5), and Wilkinson (1988,
Chapters 8, 9).
3.3 Interpolation
3.3(i) Lagrange Interpolation
The nodes orabscissaszkare real or complex; function
values arefk=f(zk). Givenn+1 distinct points zkandn+1 corresponding function values fk, the Lagrange in-
terpolation polynomial is the unique polynomial Pn(z)
of degree not exceeding nsuch that Pn(zk) =fk,
k= 0;1;:::;n . It is given by
3.3.1Pn(z) =nX
k=0`k(z)fk=nX
k=0!n+1(z)
(z zk)!0
n+1(zk)fk;
where
3.3.2`k(z) =nY0
j=0z zj
zk zj; `k(zj) =k;j:
Here the prime signi es that the factor for j=kis to
be omitted, k;jis the Kronecker symbol, and !n+1is
thenodal polynomial
3.3.3 !n+1(z) =nY
k=0(z zk):
With an error term the Lagrange interpolation for-
mula forfis given by
3.3.4 f(z) =nX
k=0`k(z)fk+Rn(z):
Iff,x(=z), and the nodes xkare real, and f(n+1)is
continuous on the smallest closed interval Icontaining
x;x0;x1;:::;xn, then the error can be expressed
3.3.5 Rn(x) =f(n+1)()
(n+ 1)!!n+1(x);
for some2I. Iffis analytic in a simply-connected
domainD(x1.13(i)), then for z2D,
3.3.6Rn(z) =!n+1(z)
2iZ
Cf()
( z)!n+1()d;
whereCis a simple closed contour in Ddescribed in
the positive rotational sense and enclosing the points
z;z1;z2;:::;zn.
3.3(ii) Lagrange Interpolation with
Equally-Spaced Nodes
The (n+ 1)-point formula (3.3.4) can be written in the
form
3.3.7
ft=f(x0+th) =n1X
k=n0An
kfk+Rn;t,n0<t<n 1,
where the nodes xk=x0+kh(h >0) and function f
are real,
3.3.8n0= 1
2(n ); n 1=1
2(n+);
3.3.9 =1
2(1 ( 1)n);
andAn
kare the Lagrangian interpolation coecients de-
ned by
3.3.10An
k=( 1)n1+k
(k n0)! (n1 k)!(t k)n1Y
m=n0(t m):
76 Numerical Methods
The remainder is given by
3.3.11
Rn;t=Rn(x0+th) =hn+1
(n+ 1)!f(n+1)()n1Y
k=n0(t k);
whereis as inx3.3(i).
Letcnbe de ned by
3.3.12 cn=1
(n+ 1)!maxn1Y
k=n0jt kj;
where the maximum is taken over t-intervals given in
the formulas below. Then for these t-intervals,
3.3.13jRn;tjcnhn+1 f(n+1)() :
Linear Interpolation
3.3.14ft= (1 t)f0+tf1+R1;t, 0 <t< 1,
3.3.15c1=1
8, 0 <t< 1.
Three-Point Formula
3.3.16 ft=1X
k= 1A2
kfk+R2;t,jtj<1,
3.3.17A2
1=1
2t(t 1); A2
0= 1 t2; A2
1=1
2t(t+ 1);
3.3.18 c2= 1=(9p
3) = 0:0641:::,jtj<1.
For four-point to eight-point formulas see http:
//dlmf.nist.gov/3.3.ii .
3.3(iii) Divided Di erences
The divided di erences offrelative to a sequence of
distinct points z0;z1;z2;::: are de ned by
3.3.34[z0]f=f0;
[z0;z1]f= ([z1]f [z0]f)=(z1 z0);
[z0;z1;z2]f= ([z1;z2]f [z0;z1]f)=(z2 z0);
and so on. Explicitly, the divided di erence of order n
is given by
3.3.35
[z0;z1;:::;zn]f=nX
k=00
BB@f(zk),Y
0jn
j6=k(zk zj)1
CCA:
Iffand thezk(=xk) are real, and fisntimes contin-
uously di erentiable on a closed interval containing the
xk, then
3.3.36 [x0;x1;:::;xn]f=f(n)()
n!
and againis as inx3.3(i). Iffis analytic in a simply-
connected domain D, then forz2D,
3.3.37 [z0;z1;:::;zn]f=1
2iZ
Cf()
!n+1()d;
where!n+1() is given by (3.3.3), and Cis a simple
closed contour in Ddescribed in the positive rotational
sense and enclosing z0;z1;:::;zn.3.3(iv) Newton's Interpolation Formula
This represents the Lagrange interpolation polynomial
in terms of divided di erences:
3.3.38
f(z) = [z0]f+ (z z0)[z0;z1]f
+ (z z0)(z z1)[z0;z1;z2]f+
+ (z z0)(z z1)(z zn 1)[z0;z1;:::;zn]f
+Rn(z):
The interpolation error Rn(z) is as inx3.3(i). New-
ton's formula has the advantage of allowing easy up-
dating: incorporation of a new point zn+1requires only
addition of the term with [ z0;z1;:::;zn+1]fto (3.3.38),
plus the computation of this divided di erence. An-
other advantage is its robustness with respect to con-
uence of the set of points z0;z1;:::;zn. For example,
fork+ 1 coincident points the limiting form is given by
[z0;z0;:::;z 0]f=f(k)(z0)=k!.
3.3(v) Inverse Interpolation
In this method we interchange the roles of the points zk
and the function values fk. It can be used for solving a
nonlinear scalar equation f(z) = 0 approximately. An-
other approach is to combine the methods of x3.8 with
direct interpolation and x3.4.
Example
To compute the rst negative zero a1 =
2:33810 7410 :::of the Airy function f(x) = Ai(x)
(x9.2). The inverse interpolation polynomial is given by
3.3.39x(f) = [f0]x+ (f f0)[f0;f1]x
+ (f f0)(f f1)[f0;f1;f2]x;
compare (3.3.38). With x0= 2:2,x1= 2:3,x2=
2:4, we obtain
3.3.40
x= 2:2
+ 1:44011 1973( f 0:09614 53780) + 0 :08865 85832
(f 0:09614 53780)( f 0:02670 63331) ;
and withf= 0 we nd that x= 2:33823 2462, with
4 correct digits. By using this approximation to xas
a new point, x3=x, and evaluating [ f0;f1;f2;f3]x=
1:12388 6190, we nd that x= 2:33810 7409, with 9
correct digits.
For comparison, we use Newton's interpolation for-
mula (3.3.38)
3.3.41f(x) = 0:09614 53780 + 0 :69439 04495( x+ 2:1)
0:03007 14275( x+ 2:2)(x+ 2:3);
with the derivative
3.3.42f0(x) = 0:55906 90257 0:06014 28550 x;
3.4 Differentiation 77
and compute an approximation to a1by using New-
ton's rule (x3.8(ii)) with starting value x= 2:5. This
gives the new point x3= 2:33934 0514. Then by
usingx3in Newton's interpolation formula, evaluat-
ing [x0;x1;x2;x3]f= 0:26608 28233 and recomputing
f0(x), another application of Newton's rule with start-
ing valuex3gives the approximation x= 2:33810 7373,
with 8 correct digits.
3.3(vi) Other Interpolation Methods
For Hermite interpolation, trigonometric interpolation,
spline interpolation, rational interpolation (by using
continued fractions), interpolation based on Chebyshev
points, and bivariate interpolation, see Bulirsch and
Rutishauser (1968), Davis (1975, pp. 27{31), and Mason
and Handscomb (2003, Chapter 6). These references
also describe convergence properties of the interpolation
formulas.
For interpolation of a bounded function fonRthe
cardinal function offis de ned by
3.3.43C(f;h)(x) =1X
k= 1f(kh)S(k;h)(x);
where
3.3.44 S(k;h)(x) =sin((x kh)=h)
(x kh)=h;
is called the Sinc function . For theory and applications
see Stenger (1993, Chapter 3).
3.4 Di erentiation
3.4(i) Equally-Spaced Nodes
The Lagrange ( n+ 1)-point formula is
3.4.1
hf0
t=hf0(x0+th) =n1X
k=n0Bn
kfk+hR0
n;t,n0<t<n 1,
and follows from the di erentiated form of (3.3.4). The
Bn
kare the di erentiated Lagrangian interpolation coef-
cients :
3.4.2 Bn
k=dAn
k/dt;
whereAn
kis as in (3.3.10).
Iff(n+2)(x) is continuous on the interval Ide ned
inx3.3(i), then the remainder in (3.4.1) is given by
3.4.3hR0
n;t=hn+1
(n+ 1)!
f(n+1)(0)d
dtn1Y
k=n0(t k)
+f(n+2)(1)n1Y
k=n0(t k)!
;
where0and12I.For the values of n0andn1used in the formulas
below
3.4.4
h R0
n;t hn+1
cn f(n+2)(1) +1
n+ 1 f(n+1)(0)
,
n0<t<n 1,
wherecnis de ned by (3.3.12), with numerical values
as inx3.3(ii).
Two-Point Formula
3.4.5 hf0
t= f0+f1+hR0
1;t, 0<t< 1.
Three-Point Formula
3.4.6hf0
t= 1
2(1 2t)f 1 2tf0+1
2(1 + 2t)f1+hR0
2;t,
jtj<1.
For four-point to eight-point formulas see http:
//dlmf.nist.gov/3.4.i .
For corresponding formulas for second, third, and
fourth derivatives, with t= 0, see Collatz (1960, Ta-
ble III, pp. 538{539). For formulas for derivatives with
equally-spaced real nodes and based on Sinc approxi-
mations (x3.3(vi)), see Stenger (1993, x3.5).
3.4(ii) Analytic Functions
Iffcan be extended analytically into the complex plane,
then from Cauchy's integral formula ( x1.9(iii))
3.4.171
k!f(k)(x0) =1
2iZ
Cf()
( x0)k+1d;
whereCis a simple closed contour described in the pos-
itive rotational sense such that Cand its interior lie in
the domain of analyticity of f, andx0is interior to C.
TakingCto be a circle of radius rcentered at x0, we
obtain
3.4.181
k!f(k)(x0) =1
2rkZ2
0f(x0+rei)e ikd:
The integral on the right-hand side can be approximated
by the composite trapezoidal rule (3.5.2).
Example
f(z) =ez,x0= 0. The integral (3.4.18) becomes
3.4.191
k!=1
2rkZ2
0ercoscos(rsin k)d:
With the choice r=k(which is crucial when kis large
because of numerical cancellation) the integrand equals
ekat the dominant points = 0;2, and in combina-
tion with the factor k kin front of the integral sign this
gives a rough approximation to 1 =k!. The choice r=k
is motivated by saddle-point analysis; see x2.4(iv) or ex-
amples inx3.5(ix). As explained in xx3.5(i) and 3.5(ix)
the composite trapezoidal rule can be very ecient for
computing integrals with analytic periodic integrands.
78 Numerical Methods
3.4(iii) Partial Derivatives
First-Order
For partial derivatives we use the notation ut;s=u(x0+
th;y 0+sh).
3.4.20@u0;0
@x=1
2h(u1;0 u 1;0) +O
h2
;
3.4.21
@u0;0
@x=1
4h(u1;1 u 1;1+u1; 1 u 1; 1) +O
h2
:
Second-Order
3.4.22@2u0;0
@x2=1
h2(u1;0 2u0;0+u 1;0) +O
h2
;
3.4.23@2u0;0
@x2=1
12h2( u2;0+ 16u1;0 30u0;0
+ 16u 1;0 u 2;0) +O
h4
;
3.4.24
@2u0;0
@x2=1
3h2(u1;1 2u0;1+u 1;1+u1;0 2u0;0+u 1;0
+u1; 1 2u0; 1+u 1; 1) +O
h2
:
3.4.25
@2u0;0
@x@y=1
4h2(u1;1 u1; 1 u 1;1+u 1; 1) +O
h2
;
3.4.26@2u0;0
@x@y= 1
2h2(u1;0+u 1;0+u0;1+u0; 1 2u0;0
u1;1 u 1; 1) +O
h2
:
Laplacian
3.4.27r2u=@2u
@x2+@2u
@y2:
3.4.28r2u0;0=1
h2(u1;0+u0;1+u 1;0+u0; 1 4u0;0)
+O
h2
;
3.4.29
r2u0;0
=1
12h2( 60u0;0+ 16(u1;0+u0;1+u 1;0+u0; 1)
(u2;0+u0;2+u 2;0+u0; 2)) +O
h4
:
For fourth-order formulas and the biharmonic oper-
ator see http://dlmf.nist.gov/3.4.iii .
The results in this subsection for the partial deriva-
tives follow from Panow (1955, Table 10). Those for the
Laplacian and the biharmonic operator follow from the
formulas for the partial derivatives.
For additional formulas involving values of r2uand
r4uon square, triangular, and cubic grids, see Collatz
(1960, Table VI, pp. 542{546).3.5 Quadrature
3.5(i) Trapezoidal Rules
The elementary trapezoidal rule is given by
3.5.1Zb
af(x)dx=1
2h(f(a) +f(b)) 1
12h3f00();
whereh=b a,f2C2[a;b], anda<<b .
The composite trapezoidal rule is
3.5.2Zb
af(x)dx=h(1
2f0+f1++fn 1+1
2fn) +En(f);
whereh= (b a)=n,xk=a+kh,fk=f(xk),
k= 0;1;:::;n , and
3.5.3 En(f) = b a
12h2f00(),a<<b .
If in addition fis periodic, f2Ck(R), and the
integral is taken over a period, then
3.5.4 En(f) =O
hk
, h!0.
In particular, when k=1the error term is an
exponentially-small function of 1 =h, and in these cir-
cumstances the composite trapezoidal rule is exception-
ally ecient. For an example see x3.5(ix).
Similar results hold for the trapezoidal rule in the
form
3.5.5Z1
1f(t)dt=h1X
k= 1f(kh) +Eh(f);
with a function fthat is analytic in a strip containing
R. For further information and examples, see Good-
win (1949a). In Stenger (1993, Chapter 3) the rule
(3.5.5) is considered in the framework of Sinc approx-
imations (x3.3(vi)). See also Poisson's summation for-
mula (x1.8(iv)).
Ifkin (3.5.4) is not arbitrarily large, and if odd-order
derivatives of fare known at the end points aandb,
then the composite trapezoidal rule can be improved by
means of the Euler{Maclaurin formula ( x2.10(i)). See
Davis and Rabinowitz (1984, pp. 134{142) and Temme
(1996a, p. 25).
3.5(ii) Simpson's Rule
Leth=1
2(b a) andf2C4[a;b]. Then the elementary
Simpson's rule is
3.5.6Zb
af(x)dx=1
3h(f(a) + 4f(1
2(a+b)) +f(b))
1
90h5f(4)();
wherea<<b .
3.5 Quadrature 79
Now leth= (b a)=n,xk=a+kh, andfk=f(xk),
k= 0;1;:::;n . Then the composite Simpson's rule is
3.5.7Zb
af(x)dx=1
3h(f0+ 4f1+ 2f2+ 4f3+ 2f4+
+ 4fn 1+fn) +En(f);
wherenis even and
3.5.8 En(f) = b a
180h4f(4)(),a<<b .
Simpson's rule can be regarded as a combination of
two trapezoidal rules, one with step size hand one with
step sizeh=2 to re ne the error term.
3.5(iii) Romberg Integration
Further re nements are achieved by Romberg integra-
tion. Iff2C2m+2[a;b], then the remainder En(f) in
(3.5.2) can be expanded in the form
3.5.9En(f) =c1h2+c2h4++cmh2m+O
h2m+2
;
whereh= (b a)=n. As in Simpson's rule, by combin-
ing the rule for hwith that for h=2, the rst error term
c1h2in (3.5.9) can be eliminated. With the Romberg
scheme successive terms c1h2;c2h4;:::, in (3.5.9) are
eliminated, according to the formula
3.5.10
Gk(1
2h) =Gk 1(1
2h) +Gk 1(1
2h) Gk 1(h)
4k 1,k1,
beginning with
3.5.11G0(h) =h(1
2f0+f1++fn 1+1
2fn);
although we may also start with the elementary rule
withG0(h) =1
2h(f(a) +f(b)) andh=b a. To gener-
ateGk(h) the quantities G0(h);G0(h=2);:::;G 0(h=2k)
are needed. These can be found by means of the recur-
sion
3.5.12G0(1
2h) =1
2G0(h) +1
2hn 1X
k=0f
x0+ (k+1
2)h
;
which depends on function values computed previously.
Iff2C2k+2(a;b), then forj;k= 0;1;:::,
3.5.13Zb
af(x)dx Gkb a
2j
= (b a)2k+3
2k(k+1)4 j(k+1)
(2k+ 2)!jB2k+2jf(2k+2)();
for some2(a;b). For the Bernoulli numbers Bmsee
x24.2(i).
Whenf2C1, the Romberg method a ords a
means of obtaining high accuracy in many cases with
a relatively simple adaptive algorithm. However, as il-
lustrated by the next example, other methods may be
more ecient.Example
WithJ0(t) denoting the Bessel function ( x10.2(ii)) the
integral
3.5.14Z1
0e ptJ0(t)dt=1p
p2+ 1
is computed with p= 1 on the interval [0 ;30]. Using
(3.5.10) with h= 30=4 = 7:5 we obtain G7(h) with 14
correct digits. About 29= 512 function evaluations are
needed. (With the 20-point Gauss{Laguerre formula
(x3.5(v)) the same precision can be achieved with 15
function evaluations.) With j= 2 andk= 7, the coef-
cient of the derivative f(16)() in (3.5.13) is found to
be (0:14:::)10 13.
See Davis and Rabinowitz (1984, pp. 440{441) for
modi cations of the Romberg method when the func-
tionfis singular.
3.5(iv) Interpolatory Quadrature Rules
Aninterpolatory quadrature rule
3.5.15Zb
af(x)w(x)dx=nX
k=1wkf(xk) +En(f);
with weight function w(x), is one for which En(f) = 0
wheneverfis a polynomial of degree n 1. The
nodesx1;x2;:::;xnare prescribed, and the weightswk
anderror termEn(f) are found by integrating the prod-
uct of the Lagrange interpolation polynomial of degree
n 1 andw(x).
If the extreme members of the set of nodes
x1;x2;:::;xnare the endpoints aandb, then the
quadrature rule is said to be closed . Or if the set
x1;x2;:::;xnlies in the open interval ( a;b), then the
quadrature rule is said to be open.
Rules of closed type include the Newton{Cotes for-
mulas such as the trapezoidal rules and Simpson's
rule. Examples of open rules are the Gauss formu-
las (x3.5(v)), the midpoint rule , and Fej er's quadrature
rule. For the latter a= 1,b= 1, and the nodes
xkare the extrema of the Chebyshev polynomial Tn(x)
(x3.11(ii) andx18.3). If we add 1 and 1 to this set of
xk, then the resulting closed formula is the frequently-
used Clenshaw{Curtis formula , whose weights are pos-
itive and given by
3.5.16wk=gk
n0
@1 bn=2cX
j=1bj
4j2 1cos(2jk=n )1
A;
wherexk= cos(k=n );k= 0;1;:::;n , and
3.5.17gk=(
1; k = 0;n;
2;otherwise;bj=(
1; j=1
2n;
2;otherwise:
For further information, see Mason and Handscomb
(2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74{
92), and Clenshaw and Curtis (1960).
80 Numerical Methods
For a detailed comparison of the Clenshaw{Curtis
formula with Gauss quadrature ( x3.5(v)), see Trefethen
(2008).
3.5(v) Gauss Quadrature
Letfpngdenote the set of monic polynomials pnof de-
green(coecient of xnequal to 1) that are orthogonal
with respect to a positive weight function won a nite
or in nite interval ( a;b); comparex18.2(i). In Gauss
quadrature (also known as Gauss{Christo el quadra-
ture) we use (3.5.15) with nodes xkthe zeros of pn,
and weights wkgiven by
3.5.18wk=Zb
apn(x)
(x xk)p0n(xk)w(x)dx:
Thewkare also known as Christo el coecients or
Christo el numbers and they are all positive. The re-
mainder is given by
3.5.19 En(f) =
nf(2n)()=(2n)!;
where
3.5.20
n=Zb
ap2
n(x)w(x)dx;
andis some point in ( a;b). As a consequence, the rule
is exact for polynomials of degree 2n 1.
In practical applications the weight function w(x) is
chosen to simulate the asymptotic behavior of the inte-
grand as the endpoints are approached. For C1func-
tions Gauss quadrature can be very ecient. In adap-
tive algorithms the evaluation of the nodes and weights
may cause diculties, unless exact values are known.
For the derivation of Gauss quadrature formulas see
Gautschi (2004, pp. 22{32), Gil et al. (2007a,x5.3), and
Davis and Rabinowitz (1984, xx2.7 and 3.6). Stroud and
Secrest (1966) includes computational methods and ex-
tensive tables. For further extensions, applications, and
computation of orthogonal polynomials and Gauss-type
formulas, see Gautschi (1994, 1996, 2004). For e ec-
tive testing of Gaussian quadrature rules see Gautschi
(1983).
For the classical orthogonal polynomials related to
the following Gauss rules, see x18.3. The given quanti-
ties
nfollow from (18.2.5), (18.2.7), Table 18.3.1, and
the relation
n=hn
k2
n.Gauss{Legendre Formula
3.5.21
[a;b] = [ 1;1]; w (x) = 1;
n=22n+1
2n+ 1(n!)4
((2n)!)2:
The nodes xkand weights wkforn= 5, 10 are
shown in Tables 3.5.1 and 3.5.2. The pn(x) are the
monic Legendre polynomials, that is, the polynomials
Pn(x) (x18.3) scaled so that the coecient of the highest
power ofxin their explicit forms is unity.
Table 3.5.1 : Nodes and weights for the 5-point Gauss{
Legendre formula.
xk wk
0:00000 00000 00000 0 :56888 88888 88889
0:53846 93101 05683 0 :47862 86704 99366
0:90617 98459 38664 0 :23692 68850 56189
Table 3.5.2 : Nodes and weights for the 10-point Gauss{
Legendre formula.
xk wk
0.14887 43389 81631 211 0.29552 42247 14752 870
0.43339 53941 29247 191 0.26926 67193 09996 355
0.67940 95682 99024 406 0.21908 63625 15982 044
0.86506 33666 88984 511 0.14945 13491 50580 593
0.97390 65285 17171 720 0.06667 13443 08688 138
For corresponding results for n= 20;40;80; see
http://dlmf.nist.gov/3.5.v .
Gauss{Chebyshev Formula
3.5.22
[a;b] = [ 1;1]; w (x) = (1 x2) 1=2;
n=
22n 1:
The nodesxkand weights wkare known explicitly:
3.5.23
xk= cos2k 1
2n
; wk=
n,k= 1;2;:::;n .
Nodes and weights are also known explicitly for the
other three weight functions in the set w(x) = (1
x)1=2(1+x)1=2; see http://dlmf.nist.gov/3.5.v .
Gauss{Jacobi Formula
3.5.26[a;b] = [ 1;1]; w (x) = (1 x) (1 +x) ;
n= (n+ + 1) (n+ + 1) (n+ + + 1)
(2n+ + + 1)( (2n+ + + 1))222n+ + +1n!,
> 1, > 1.
Thepn(x) are the monic Jacobi polynomials P( ; )
n(x) (x18.3).
Gauss{Laguerre Formula
3.5.27 [a;b) = [0;1); w (x) =x e x;
n=n! (n+ + 1), > 1.
3.5 Quadrature 81
If 6= 0 this is called the generalized Gauss{Laguerre formula .
The nodes xkand weights wkfor = 0 andn= 5, 10 are shown in Tables 3.5.6 and 3.5.7. The pn(x) are the
monic Laguerre polynomials Ln(x) (x18.3).
Table 3.5.6 : Nodes and weights for the 5-point Gauss{Laguerre formula.
xk wk
0:26356 03197 18141 0 :52175 56105 82809
0:14134 03059 10652 1010:39866 68110 83176
0:35964 25771 04072 1010:75942 44968 17076 10 1
0:70858 10005 85884 1010:36117 58679 92205 10 2
0:12640 80084 42758 1020:23369 97238 57762 10 4
Table 3.5.7 : Nodes and weights for the 10-point Gauss{Laguerre formula.
xk wk
0:13779 34705 40492 431 0 :30844 11157 65020 141
0:72945 45495 03170 498 0 :40111 99291 55273 552
0:18083 42901 74031 605 1010:21806 82876 11809 422
0:34014 33697 85489 951 1010:62087 45609 86777 475 10 1
0:55524 96140 06380 363 1010:95015 16975 18110 055 10 2
0:83301 52746 76449 670 1010:75300 83885 87538 775 10 3
0:11843 78583 79000 656 1020:28259 23349 59956 557 10 4
0:16279 25783 13781 021 1020:42493 13984 96268 637 10 6
0:21996 58581 19807 620 1020:18395 64823 97963 078 10 8
0:29920 69701 22738 916 1020:99118 27219 60900 856 10 12
For the corresponding results for n= 15;20 see http://dlmf.nist.gov/3.5.v .
Gauss{Hermite Formula
3.5.28 (a;b) = ( 1;1); w (x) =e x2;
n=pn!
2n:
The nodes xkand weights wkforn= 5;10 are shown in Tables 3.5.10 and 3.5.11. The pn(x) are the monic
Hermite polynomials Hn(x) (x18.3).
Table 3.5.10 : Nodes and weights for the 5-point Gauss{Hermite formula.
xk wk
0:00000 00000 00000 0 :94530 87204 82942
0:95857 24646 13819 0 :39361 93231 52241
0:20201 82870 45609 1010:19953 24205 90459 10 1
Table 3.5.11 : Nodes and weights for the 10-point Gauss{Hermite formula.
xk wk
0:34290 13272 23704 609 0 :61086 26337 35325 799
0:10366 10829 78951 365 1010:24013 86110 82314 686
0:17566 83649 29988 177 1010:33874 39445 54810 631 10 1
0:25327 31674 23278 980 1010:13436 45746 78123 269 10 2
0:34361 59118 83773 760 1010:76404 32855 23262 063 10 5
For the corresponding results for n= 15;20 see http://dlmf.nist.gov/3.5.v .
Gauss Formula for a Logarithmic Weight Function
3.5.29 [a;b] = [0;1]; w (x) = ln(1=x):
The nodesxkand weights wkforn= 5;10 are shown in Tables 3.5.14 and 3.5.15.
82 Numerical Methods
Table 3.5.14 : Nodes and weights for the 5-point Gauss formula for the logarithmic weight function.
xk wk
0:29134 47215 19721 10 10:29789 34717 82894
0:17397 72133 20898 0 :34977 62265 13224
0:41170 25202 84902 0 :23448 82900 44052
0:67731 41745 82820 0 :98930 45951 66331 10 1
0:89477 13610 31008 0 :18911 55214 31958 10 1
Table 3.5.15 : Nodes and weights for the 10-point Gauss formula for the logarithmic weight function.
xk wk
0:90426 30962 19965 064 10 20:12095 51319 54570 515
0:53971 26622 25006 295 10 10:18636 35425 64071 870
0:13531 18246 39250 775 0 :19566 08732 77759 983
0:24705 24162 87159 824 0 :17357 71421 82906 921
0:38021 25396 09332 334 0 :13569 56729 95484 202
0:52379 23179 71843 201 0 :93646 75853 81105 260 10 1
0:66577 52055 16424 597 0 :55787 72735 14158 741 10 1
0:79419 04160 11966 217 0 :27159 81089 92333 311 10 1
0:89816 10912 19003 538 0 :95151 82602 84851 500 10 2
0:96884 79887 18633 539 0 :16381 57633 59826 325 10 2
For the corresponding results for n= 15;20 see http://dlmf.nist.gov/3.5.v .
3.5(vi) Eigenvalue/Eigenvector Characterization
of Gauss Quadrature Formulas
All the monic orthogonal polynomials fpngused with
Gauss quadrature satisfy a three-term recurrence rela-
tion (x18.2(iv)):
3.5.30
pk+1(x) = (x k)pk(x) kpk 1(x),k= 0;1;:::,
with k>0,p 1(x) = 0, andp0(x) = 1.
The Gauss nodes xk(the zeros of pn) are the eigen-
values of the (symmetric tridiagonal) Jacobi matrix of
ordernn:
3.5.31
Jn=2
66666664 0p 1 0
p 1 1p 2
.........
p
n 2 n 2p
n 1
0p
n 1 n 13
77777775:
Letvkdenote the normalized eigenvector of Jncor-
responding to the eigenvalue xk. Then the weights are
given by
3.5.32 wk= 0v2
k;1,k= 1;2;:::;n ,
where 0=Rb
aw(x)dxandvk;1is the rst element of
vk. Also, the error constant (3.5.20) is given by
3.5.33
n= 0 1 n:Tables 3.5.1, 3.5.2, 3.5.6, 3.5.7, 3.5.10, and 3.5.11
can be veri ed by application of the results given in the
present subsection. In these cases the coecients k
and kare obtainable explicitly from results given in
x18.9(i).
3.5(vii) Oscillatory Integrals
Integrals of the form
3.5.34Zb
af(x) cos(!x)dx;Zb
af(x) sin(!x)dx;
can be computed by Filon's rule . See Davis and Rabi-
nowitz (1984, pp. 146{168).
Oscillatory integral transforms are treated in Wong
(1982) by a method based on Gaussian quadrature. A
comparison of several methods, including an extension
of the Clenshaw{Curtis formula ( x3.5(iv)), is given in
Evans and Webster (1999).
For computing in nite oscillatory integrals, Long-
man's method may be used. The integral is written
as an alternating series of positive and negative subin-
tegrals that are computed individually; see Longman
(1956). Convergence acceleration schemes, for example
Levin's transformation ( x3.9(v)), can be used when eval-
uating the series. Further methods are given in Clen-
denin (1966) and Lyness (1985).
For a comprehensive survey of quadrature of highly
oscillatory integrals, including multidimensional inte-
grals, see Iserles et al. (2006).
3.5 Quadrature 83
3.5(viii) Complex Gauss Quadrature
For the Bromwich integral
3.5.35
I(f) =1
2iZc+i1
c i1e sf()d,s>0,c>c 0>0,
acomplex Gauss quadrature formula is available. Here
f() is assumed analytic in the half-plane < >c 0and
bounded as !1 injphj1
2. The quadrature rule
for (3.5.35) is
3.5.36 I(f) =nX
k=1wkf(k) +En(f);whereEn(f) = 0 iff() is a polynomial of degree
2n 1 in 1=.Complex orthogonal polynomials
pn(1=) of degree n= 0;1;2;:::, in 1=that satisfy
the orthogonality condition
3.5.37Zc+i1
c i1e spk(1=)p`(1=)d= 0,k6=`,
are related to Bessel polynomials ( xx10.49(ii) and
18.34). The complex Gauss nodes khave positive real
part for all s>0.
The nodes and weights of the 5-point complex Gauss
quadrature formula (3.5.36) for s= 1 are shown in Ta-
ble 3.5.18. Extensive tables of quadrature nodes and
weights can be found in Krylov and Skoblya (1985).
Table 3.5.18 : Nodes and weights for the 5-point complex Gauss quadrature formula with s= 1.
k wk
3:65569 4325+6 :54373 6899i 3:83966 1630 0:27357 03863 i
3:65569 4325 6:54373 6899i 3:83966 1630+0 :27357 03863 i
5:70095 3299+3 :21026 5600i 25:07945 221 +2 :18725 2294i
5:70095 3299 3:21026 5600i 25:07945 221 2:18725 2294i
6:28670 4752+0 :00000 0000i 43:47958 116 +0 :00000 0000i
Example. Laplace Transform Inversion
Fromx1.14(iii)
3.5.38 G(p) =Z1
0e ptg(t)dt;
3.5.39 g(t) =1
2iZ+i1
i1etpG(p)dp;
with appropriate conditions. The pair
3.5.40 g(t) =J0(t); G (p) =1p
p2+ 1;
whereJ0(t) is the Bessel function ( x10.2(ii)), satisfy
these conditions, provided that > 0. The integral
(3.5.39) has the form (3.5.35) if we set =tp,c=t,
andf() =t 1sG(=t). We choose s= 1 so that
f() =O(1) at in nity. Equation (3.5.36), without the
error term, becomes
3.5.41 g(t) =nX
k=1wkkp
2
k+t2;
approximately.
Using Table 3.5.18 we compute g(t) forn= 5. The
results are given in the middle column of Table 3.5.19,
accompanied by the actual 10D values in the last col-
umn. Agreement is very good for small values of t, but
not for larger values. For these cases the integration
path may need to be deformed; see x3.5(ix).Table 3.5.19 : Laplace transform inversion.
t g (t) J0(t)
0.0 1.00000 00000 1.00000 00000
0.5 0.93846 98072 0.93846 98072
1.0 0.76519 76866 0.76519 76865
2.0 0.22389 07791 0.22389 10326
5.0 0.17759 67713 0.17902 54097
10.0 0.24593 57645 0.07540 53543
3.5(ix) Other Contour Integrals
A frequent problem with contour integrals is heavy can-
cellation, which occurs especially when the value of the
integral is exponentially small compared with the maxi-
mum absolute value of the integrand. To avoid cancella-
tion we try to deform the path to pass through a saddle
point in such a way that the maximum contribution of
the integrand is derived from the neighborhood of the
saddle point. For example, steepest descent paths can
be used; seex2.4(iv).
Example
In (3.5.35) take s= 1 andf() =e 2p, with
> 0. Whenis large the integral becomes expo-
nentially small, and application of the quadrature rule
ofx3.5(viii) is useless. In fact from (7.14.4) and the
84 Numerical Methods
inversion formula for the Laplace transform ( x1.14(iii))
we have
3.5.42 erfc=1
2iZc+i1
c i1e 2pd
,c>0,
where erfczis the complementary error function, and
from (7.12.1) it follows that
3.5.43 erfce 2
p, !1 .
With the transformation =2t, (3.5.42) becomes
3.5.44 erfc=1
2iZc+i1
c i1e2(t 2p
t)dt
t,c>0,
with saddle point at t= 1, and when c= 1 the vertical
path intersects the real axis at the saddle point. The
steepest descent path is given by =(t 2p
t) = 0, or in
polar coordinates t=reiwe haver= sec2 1
2
. Thus
3.5.45 erfc=e 2
2Z
e 2tan2(1
2)d:
The integrand can be extended as a periodic C1func-
tion on Rwith period 2 and as noted inx3.5(i), the
trapezoidal rule is exceptionally ecient in this case.
Table 3.5.20 gives the results of applying the com-
posite trapezoidal rule (3.5.2) with step size h;nindi-
cates the number of function values in the rule that are
larger than 10 15(we exploit the fact that the integrand
is even). All digits shown in the approximation in the
nal row are correct.
Table 3.5.20 : Composite trapezoidal rule for the integral
(3.5.45) with = 10.
h erfc n
0:25 0:20949 49432 96679 10 445
0:20 0:20886 11645 34559 10 446
0:15 0:20884 87588 72946 10 448
0:10 0:20884 87583 76254 10 4411A second example is provided in Gil et al. (2001),
where the method of contour integration is used to eval-
uate Scorer functions of complex argument ( x9.12). See
also Gil et al. (2003b).
Iffis meromorphic, with poles near the saddle
point, then the foregoing method can be modi ed. A
special case is the rule for Hilbert transforms ( x1.14(v)):
3.5.46H(f;x) =1
Z1
1f(t)
t xdt,x2R,
where the integral is the Cauchy principal value. See
Kress and Martensen (1970).
Other contour integrals occur in standard inte-
gral transforms or their inverses, for example, Han-
kel transforms (x10.22(v)), Kontorovich{Lebedev trans-
forms (x10.43(v)), and Mellin transforms ( x1.14(iv)).
3.5(x) Cubature Formulas
Table 3.5.21 supplies cubature rules, including weights
wj, for the disk D, given byx2+y2h2:
3.5.471
h2ZZ
Df(x;y)dxdy =nX
j=1wjf(xj;yj) +R;
and the square S, given byjxjh,jyjh:
3.5.481
4h2ZZ
Sf(x;y)dxdy =nX
j=1wjf(xj;yj) +R:
For these results and further information on cuba-
ture formulas see Cools (2003).
For integrals in higher dimensions, Monte Carlo
methods are another|often the only|alternative. The
standard Monte Carlo method samples points uniformly
from the integration region to estimate the integral and
its error. In more advanced methods points are sampled
from a probability distribution, so that they are concen-
trated in regions that make the largest contribution to
the integral. With Nfunction values, the Monte Carlo
method aims at an error of order 1 =p
N, independently
of the dimension of the domain of integration. See Davis
and Rabinowitz (1984, pp. 384{417) and Sch urer (2004).
3.6 Linear Difference Equations 85
Table 3.5.21 : Cubature formulas for disk and square.
Diagram ( xj;yj) wjR
(0;0)1
2O
h4
(h;0)1
8
(0;h)1
8
(1
2h;1
2h)1
4O
h4
(0;0)1
6O
h6
(h;0), (0;h)1
24
(1
2h;1
2h)1
6
(0;0)1
4O
h6
(1
3p
6h;0)1
8
(1
6p
6h;1
2p
2h)1
8
(0;0)4
9O
h4
(h;0), (0;h)1
9
(h;h)1
36
(1
3p
3h;1
3p
3h)1
4O
h4
(0;0)16
81O
h6
(q
3
5h;0), (0;q
3
5h)10
81
(q
3
5h;0), (q
3
5h;0)25
324
3.6 Linear Di erence Equations
3.6(i) Introduction
Many special functions satisfy second-order recurrence
relations, or di erence equations, of the form
3.6.1 anwn+1 bnwn+cnwn 1=dn;or equivalently,
3.6.2an2wn 1
+ (2an bn) wn 1+ (an bn+cn)wn 1=dn;
where wn 1=wn wn 1, 2wn 1= wn wn 1,
andn2Z. Ifdn= 0,8n, then the di erence equation
ishomogeneous ; otherwise it is inhomogeneous .
Di erence equations are simple and attractive for
computation. In practice, however, problems of severe
instability often arise and in xx3.6(ii){3.6(vii) we show
how these diculties may be overcome.
3.6(ii) Homogeneous Equations
Given numerical values of w0andw1, the solution wn
of the equation
3.6.3 anwn+1 bnwn+cnwn 1= 0;
withan6= 0,8n, can be computed recursively for
n= 2;3;:::. Unless exact arithmetic is being used,
however, each step of the calculation introduces round-
ing errors. These errors have the e ect of perturbing
the solution by unwanted small multiples of wnand of
an independent solution gn, say. This is of little conse-
quence if the wanted solution is growing in magnitude
at least as fast as any other solution of (3.6.3), and the
recursion process is stable .
But suppose that wnis a nontrivial solution such
that
3.6.4 wn=gn!0, n!1 .
Thenwnis said to be a recessive (equivalently, min-
imal ordistinguished )solution asn! 1 , and it is
unique except for a constant factor. In this situation
the unwanted multiples of gngrow more rapidly than
the wanted solution, and the computations are unstable .
Stability can be restored, however, by backward recur-
sion, provided that cn6= 0,8n: starting from wNand
wN+1, withNlarge, equation (3.6.3) is applied to gen-
erate in succession wN 1;wN 2;:::;w 0. The unwanted
multiples of gnnow decay in comparison with wn, hence
are of little consequence.
The values of wNandwN+1needed to begin the
backward recursion may be available, for example, from
asymptotic expansions ( x2.9). However, there are alter-
native procedures that do not require wNandwN+1to
be known in advance. These are described in xx3.6(iii)
and 3.6(v).
3.6(iii) Miller's Algorithm
Because the recessive solution of a homogeneous equa-
tion is the fastest growing solution in the backward di-
rection, it occurred to J.C.P. Miller (Bickley et al. (1952,
pp. xvi{xvii)) that arbitrary \trial values" can be as-
signed towNandwN+1, for example, 1 and 0. A \trial
solution" is then computed by backward recursion, in
86 Numerical Methods
the course of which the original components of the un-
wanted solution gndie away. It therefore remains to
apply a normalizing factor . The process is then re-
peated with a higher value of N, and the normalized
solutions compared. If agreement is not within a pre-
scribed tolerance the cycle is continued.
The normalizing factor can be the true value of w0
divided by its trial value, or can be chosen to satisfy
a known property of the wanted solution of the form
3.6.51X
n=0nwn= 1;
where the's are constants. The latter method is usu-
ally superior when the true value of w0is zero or patho-
logically small.
For further information on Miller's algorithm, in-
cluding examples, convergence proofs, and error anal-
yses, see Wimp (1984, Chapter 4), Gautschi (1967,
1997a), and Olver (1964a). See also Gautschi (1967)
and Gil et al. (2007a, Chapter 4) for the computation
of recessive solutions via continued fractions.
3.6(iv) Inhomogeneous Equations
Similar principles apply to equation (3.6.1) when
ancn6= 0,8n, anddn6= 0 for some, or all, values ofn. If, asn!1 , the wanted solution wngrows (de-
cays) in magnitude at least as fast as any solution of
the corresponding homogeneous equation, then forward
(backward) recursion is stable.
A new problem arises, however, if, as n!1 , the
asymptotic behavior of wnis intermediate to those of
two independent solutions fnandgnof the correspond-
ing inhomogeneous equation (the complementary func-
tions). More precisely, assume that f06= 0,gn6= 0 for
all suciently large n, and asn!1
3.6.6fn=gn!0; wn=gn!0:
Then computation of wnby forward recursion is unsta-
ble. If it also happens that fn=wn!0 asn!1 , then
computation of wnby backward recursion is unstable
as well. However, wncan be computed successfully in
these circumstances by boundary-value methods , as fol-
lows.
Let us assume the normalizing condition is of the
formw0=, whereis a constant, and then solve the
following tridiagonal system of algebraic equations for
the unknowns w(N)
1;w(N)
2;:::;w(N)
N 1; seex3.2(ii). Here
Nis an arbitrary positive integer.
3.6.72
66666664 b1a1 0
c2 b2a2
.........
cN 2 bN 2aN 2
0 cN 1 bN 13
777777752
66666664w(N)
1
w(N)
2
...
w(N)
N 2
w(N)
N 13
77777775=2
66666664d1 c1
d2
...
dN 2
dN 13
77777775:
Then asN!1 withn xed,w(N)
n!wn.
3.6(v) Olver's Algorithm
To apply the method just described a succession of val-
ues can be prescribed for the arbitrary integer Nand
the results compared. However, a more powerful pro-
cedure combines the solution of the algebraic equations
with the determination of the optimum value of N. It
is applicable equally to the computation of the reces-
sive solution of the homogeneous equation (3.6.3) or the
computation of any solution wnof the inhomogeneous
equation (3.6.1) for which the conditions of x3.6(iv) are
satis ed.
Suppose again that f06= 0,w0is given, and we wish
to calculate w1;w2;:::;wMto a prescribed relative ac-
curacyfor a given value of M. We rst compute, by
forward recurrence, the solution pnof the homogeneous
equation (3.6.3) with initial values p0= 0,p1= 1. At
the same time we construct a sequence en,n= 0;1;:::,de ned by
3.6.8 anen=cnen 1 dnpn;
beginning with e0=w0. (This part of the process is
equivalent to forward elimination.) The computation is
continued until a value N(M) is reached for which
3.6.9 eN
pNpN+1 min
1nM en
pnpn+1 :
Thenwnis generated by backward recursion from
3.6.10 pn+1wn=pnwn+1+en;
starting with wN= 0. (This part of the process is back
substitution.)
An example is included in the next subsection.
For further information, including a more general form
of normalizing condition, other examples, convergence
proofs, and error analyses, see Olver (1967a), Olver and
Sookne (1972), and Wimp (1984, Chapter 6).
3.6 Linear Difference Equations 87
3.6(vi) Examples
Example 1. Bessel Functions
The di erence equation
3.6.11 wn+1 2nwn+wn 1= 0,n= 1;2;:::,
is satis ed by Jn(1) andYn(1), whereJn(x) andYn(x)
are the Bessel functions of the rst kind. For large n,
3.6.12 Jn(1)1
(2n)1=2e
2nn
;
3.6.13 Yn(1)2
n1=22n
en
;
(x10.19(i)). Thus Yn(1) is dominant and can be com-
puted by forward recursion, whereas Jn(1) is recessive
and has to be computed by backward recursion. The
backward recursion can be carried out using indepen-
dently computed values of JN(1) andJN+1(1) or by
use of Miller's algorithm ( x3.6(iii)) or Olver's algorithm
(x3.6(v)).
Example 2. Weber Function
The Weber function En(1) satis es
3.6.14wn+1 2nwn+wn 1= (2=)(1 ( 1)n);forn= 1;2;:::, and asn!1
3.6.15 E2n(1)2
(4n2 1);
3.6.16 E2n+1(1)2
(2n+ 1);
seex11.11(ii). Thus the asymptotic behavior of the
particular solution En(1) is intermediate to those of
the complementary functions Jn(1) andYn(1); more-
over, the conditions for Olver's algorithm are satis ed.
We apply the algorithm to compute En(1) to 8S for
the rangen= 1;2;:::; 10, beginning with the value
E0(1) = 0:56865 663 obtained from the Maclaurin
series expansion ( x11.10(iii)).
In the notation of x3.6(v) we have M= 10 and
=1
210 8. The least value of Nthat satis es
(3.6.9) is found to be 16. The results of the compu-
tations are displayed in Table 3.6.1. The values of wn
forn= 1;2;:::; 10 are the wanted values of En(1). (It
should be observed that for n > 10, however, the wn
are progressively poorer approximations to En(1): the
underlined digits are in error.)
Table 3.6.1 : Weber function wn=En(1) computed by Olver's algorithm.
n p n en en=(pnpn+1) wn
0 0:00000 000 0:56865 663 0:56865 663
1 0:10000 0001010:70458 291 0 :35229 146 0 :43816 243
2 0:20000 0001010:70458 291 0 :50327 35110 10:17174 195
3 0:70000 0001010:96172 5971010:34347 35610 10:24880 538
4 0:40000 0001020:96172 5971010:76815 17410 30:47850 79510 1
5 0:31300 0001030:40814 1241030:42199 53410 30:13400 098
6 0:30900 0001040:40814 1241030:35924 75410 50:18919 44310 1
7 0:36767 0001050:47221 3401050:25102 02910 50:93032 34310 1
8 0:51164 8001060:47221 3401050:11324 80410 70:10293 81110 1
9 0:81496 0101070:10423 6161080:87496 48510 80:71668 63810 1
10 0:14618 1171090:10423 6161080:24457 82410 100:65021 29210 2
11 0:29154 73810100:37225 20110100:19952 02610 100:58373 94610 1
12 0:63994 24210110:37225 20110100:37946 27910 130:44851 38710 2
13 0:15329 46310130:19555 30410130:32057 90910 130:49269 38310 1
14 0:39792 61110140:19555 30410130:44167 17410 160:32792 86110 2
15 0:11126 60210160:14186 38410160:38242 25010 160:42550 62810 1
16 0:33340 01210170:14186 38410160:39924 86110 190:00000 000
3.6(vii) Linear Di erence Equations of Other
Orders
Similar considerations apply to the rst-order equation
3.6.17 anwn+1 bnwn=dn:
Thus in the inhomogeneous case it may sometimes be
necessary to recur backwards to achieve stability. Foranalyses and examples see Gautschi (1997a).
For a di erence equation of order k(3),
3.6.18an;kwn+k+an;k 1wn+k 1++an;0wn=dn;
or for systems of k rst-order inhomogeneous equations,
boundary-value methods are the rule rather than the
exception. Typically k `conditions are prescribed at
the beginning of the range, and `conditions at the end.
88 Numerical Methods
Here`2[0;k], and its actual value depends on the
asymptotic behavior of the wanted solution in relation
to those of the other solutions. Within this framework
forward and backward recursion may be regarded as the
special cases `= 0 and`=k, respectively.
For further information see Wimp (1984, Chapters
7{8), Cash and Zahar (1994), and Lozier (1980).
3.7 Ordinary Di erential Equations
3.7(i) Introduction
Consideration will be limited to ordinary linear second-
order di erential equations
3.7.1d2w
dz2+f(z)dw
dz+g(z)w=h(z);
wheref,g, andhare analytic functions in a domain
DC. Ifh= 0 the di erential equation is homoge-
neous , otherwise it is inhomogeneous . For applications
to special functions f,g, andhare often simple rational
functions.
For general information on solutions of equation
(3.7.1) seex1.13. For classi cation of singularities of
(3.7.1) and expansions of solutions in the neighborhoods
of singularities, see x2.7. For an introduction to numer-
ical methods for ordinary di erential equations, see As-
cher and Petzold (1998), Hairer et al. (1993), and Iserles
(1996).
3.7(ii) Taylor-Series Method: Initial-Value
Problems
Assume that we wish to integrate (3.7.1) along a nite
pathPfromz=atoz=bin a domain D. The
path is partitioned at P+ 1 points labeled successively
z0;z1;:::;zP, withz0=a,zP=b.
By repeated di erentiation of (3.7.1) all derivatives
ofw(z) can be expressed in terms of w(z) andw0(z) as
follows. Write
3.7.2
w(s)(z) =fs(z)w(z)+gs(z)w0(z)+hs(z),s= 0;1;2;:::,
with
3.7.3f0(z) = 1; g 0(z) = 0; h 0(z) = 0;
f1(z) = 0; g 1(z) = 1; h 1(z) = 0:
Then fors= 2;3;:::,
3.7.4fs(z) =f0
s 1(z) g(z)gs 1(z);
gs(z) =fs 1(z) f(z)gs 1(z) +g0
s 1(z);
hs(z) =h(z)gs 1(z) +h0
s 1(z):Write j=zj+1 zj,j= 0;1;:::;P , expandw(z)
andw0(z) in Taylor series ( x1.10(i)) centered at z=zj,
and apply (3.7.2). Then
3.7.5w(zj+1)
w0(zj+1)
=A( j;zj)w(zj)
w0(zj)
+b( j;zj);
where A( ;z) is the matrix
3.7.6 A( ;z) =A11( ;z)A12( ;z)
A21( ;z)A22( ;z)
;
andb( ;z) is the vector
3.7.7 b( ;z) =b1( ;z)
b2( ;z)
;
with
3.7.8A11( ;z) =1X
s=0 s
s!fs(z);
A12( ;z) =1X
s=0 s
s!gs(z);
A21( ;z) =1X
s=0 s
s!fs+1(z);
A22( ;z) =1X
s=0 s
s!gs+1(z);
3.7.9b1( ;z) =1X
s=0 s
s!hs(z); b2( ;z) =1X
s=0 s
s!hs+1(z):
If the solution w(z) that we are seeking grows in
magnitude at least as fast as all other solutions of
(3.7.1) as we pass along Pfromatob, thenw(z)
andw0(z) may be computed in a stable manner for
z=z0;z1;:::;zPby successive application of (3.7.5) for
j= 0;1;:::;P 1, beginning with initial values w(a)
andw0(a).
Similarly, if w(z) is decaying at least as fast as all
other solutions along P, then we may reverse the label-
ing of thezjalongPand begin with initial values w(b)
andw0(b).
3.7(iii) Taylor-Series Method: Boundary-Value
Problems
Now suppose the path Pis such that the rate of growth
ofw(z) along Pis intermediate to that of two other
solutions. (This can happen only for inhomogeneous
equations.) Then to compute w(z) in a stable manner
we solve the set of equations (3.7.5) simultaneously for
j= 0;1;:::;P , as follows. Let Abe the (2P)(2P+2)
band matrix
3.7 Ordinary Differential Equations 89
3.7.10 A=2
66666664 A( 0;z0) I 0 0 0
0 A( 1;z1)I 0 0
..................
0 0 A( P 2;zP 2) I 0
0 0 0 A( P 1;zP 1)I3
77777775
(Iand0being the identity and zero matrices of order 2 2.) Also let wdenote the (2 P+ 2)1 vector
3.7.11 w= [w(z0);w0(z0);w(z1);w0(z1);:::;w (zP);w0(zP)]T;
andbthe (2P)1 vector
3.7.12 b= [b1( 0;z0);b2( 0;z0);b1( 1;z1);b2( 1;z1);:::;b 1( P 1;zP 1);b2( P 1;zP 1)]T:
Then
3.7.13 Aw=b:
This is a set of 2 Pequations for the 2 P+ 2 unknowns,
w(zj) andw0(zj),j= 0;1;:::;P . The remaining two
equations are supplied by boundary conditions of the
form
3.7.14 0w(z0) + 0w0(z0) =
0;
1w(zP) + 1w0(zP) =
1;
where the 's, 's, and
's are constants.
If, for example, 0= 1= 0, then on moving the
contributions of w(z0) andw(zP) to the right-hand side
of (3.7.13) the resulting system of equations is not tridi-
agonal, but can readily be made tridiagonal by annihi-
lating the elements of Athat lie below the main diago-
nal and its two adjacent diagonals. The equations can
then be solved by the method of x3.2(ii), if the di eren-
tial equation is homogeneous, or by Olver's algorithm
(x3.6(v)). The latter is especially useful if the endpoint
bofPis at1, or if the di erential equation is inho-
mogeneous.
It will be observed that the present formulation of
the Taylor-series method permits considerable paral-
lelism in the computation, both for initial-value and
boundary-value problems.
For further information and examples, see Olde
Daalhuis and Olver (1998, x7) and Lozier and Olver
(1993). General methods for boundary-value problems
for ordinary di erential equations are given in Ascher
et al. (1995).
3.7(iv) Sturm{Liouville Eigenvalue Problems
Let (a;b) be a nite or in nite interval and q(x) be a
real-valued continuous (or piecewise continuous) func-
tion on the closure of ( a;b). The Sturm{Liouville eigen-
value problem is the construction of a nontrivial solutionof the system
3.7.15d2wk
dx2+ (k q(x))wk= 0;
3.7.16 wk(a) =wk(b) = 0;
with limits taken in (3.7.16) when aorb, or both, are
in nite. The values kare the eigenvalues and the cor-
responding solutions wkof the di erential equation are
theeigenfunctions . The eigenvalues kare simple, that
is, there is only one corresponding eigenfunction (apart
from a normalization factor), and when ordered increas-
ingly the eigenvalues satisfy
3.7.17 1<2<3<, limk!1k=1.
Ifq(x) isC1on the closure of ( a;b), then the dis-
cretized form (3.7.13) of the di erential equation can
be used. This converts the problem into a tridiagonal
matrix problem in which the elements of the matrix are
polynomials in ; comparex3.2(vi). The larger the ab-
solute values of the eigenvalues kthat are being sought,
the smaller the integration steps j jjneed to be.
For further information, including other methods
and examples, see Pryce (1993, x2.5.1).
3.7(v) Runge{Kutta Method
The Runge{Kutta method applies to linear or nonlinear
di erential equations. The method consists of a set of
rules each of which is equivalent to a truncated Taylor-
series expansion, but the rules avoid the need for ana-
lytic di erentiations of the di erential equation.
First-Order Equations
Forw0=f(z;w) the standard fourth-order rule reads
3.7.18wn+1=wn+1
6(k1+ 2k2+ 2k3+k4) +O
h5
;
90 Numerical Methods
whereh=zn+1 znand
3.7.19k1=hf(zn;wn);
k2=hf(zn+1
2h;wn+1
2k1);
k3=hf(zn+1
2h;wn+1
2k2);
k4=hf(zn+h;wn+k3):
The order estimate O
h5
holds if the solution w(z) has
ve continuous derivatives.
Second-Order Equations
Forw00=f(z;w;w0) the standard fourth-order rule
reads
3.7.20wn+1=wn+1
6h(6w0
n+k1+k2+k3) +O
h5
;
w0
n+1=w0
n+1
6(k1+ 2k2+ 2k3+k4) +O
h5
;
where
3.7.21
k1=hf(zn;wn;w0
n);
k2=hf(zn+1
2h;wn+1
2hw0
n+1
8hk1;w0
n+1
2k1);
k3=hf(zn+1
2h;wn+1
2hw0
n+1
8hk2;w0
n+1
2k2);
k4=hf(zn+h;wn+hw0
n+1
2hk3;w0
n+k3):
The order estimates O
h5
hold if the solution w(z) has
ve continuous derivatives.
An extensive literature exists on the numerical solu-
tion of ordinary di erential equations by Runge{Kutta,
multistep, or other methods. See, for example, Butcher
(1987), Dekker and Verwer (1984, Chapter 3), Hairer
et al. (1993, Chapter 2), and Hairer and Wanner (1996,
Chapter 4).
3.8 Nonlinear Equations
3.8(i) Introduction
The equation to be solved is
3.8.1 f(z) = 0;
wherezis a real or complex variable and the function f
is nonlinear. Solutions are called roots of the equation,
orzeros off. Iff(z0) = 0 andf0(z0)6= 0, thenz0is a
simple zero off. Iff(z0) =f0(z0) ==f(m 1)(z0) =
0 andf(m)(z0)6= 0, thenz0is a zero of fofmultiplicity
m; comparex1.10(i).
Sometimes the equation takes the form
3.8.2 z= (z);
and the solutions are called xed points of .
Equations (3.8.1) and (3.8.2) are usually solved by
iterative methods. Let z1;z2;::: be a sequence of ap-
proximations to a root, or xed point, . If
3.8.3jzn+1 j<Ajzn jp
for allnsuciently large, where Aandpare indepen-
dent ofn, then the sequence is said to have convergenceof thepthorder . (More precisely, pis the largest of the
possible set of indices for (3.8.3).) If p= 1 andA< 1,
then the convergence is said to be linear orgeometric .
Ifp= 2, then the convergence is quadratic ; ifp= 3,
then the convergence is cubic , and so on.
An iterative method converges locally to a solution
if there exists a neighborhood Nofsuch thatzn!
whenever the initial approximation z0lies withinN.
3.8(ii) Newton's Rule
This is an iterative method for real twice-continuously
di erentiable, or complex analytic, functions:
3.8.4 zn+1=zn f(zn)
f0(zn),n= 0;1;:::.
Ifis a simple zero, then the iteration converges lo-
cally and quadratically. For multiple zeros the conver-
gence is linear, but if the multiplicity mis known then
quadratic convergence can be restored by multiplying
the ratiof(zn)=f0(zn) in (3.8.4) by m.
For real functions f(x) the sequence of approxima-
tions to a real zero will always converge (and converge
quadratically) if either:
(a)f(x0)f00(x0)>0 andf0(x),f00(x) do not change
sign between x0and(monotonic convergence).
(b)f(x0)f00(x0)<0,f0(x),f00(x) do not change sign
in the interval ( x0;x1), and2[x0;x1] (mono-
tonic convergence after the rst iteration).
Example
f(x) =x tanx. The rst positive zero of f(x) lies in
the interval ( ;3
2); see Figure 4.15.3. From this graph
we estimate an initial value x0= 4:65. Newton's rule is
given by
3.8.5xn+1= (xn); (x) =x+xcot2x cotx:
Results appear in Table 3.8.1. The choice of x0here
is critical. When x04:2875 orx04:7125, New-
ton's rule does not converge to the required zero. The
convergence is faster when we use instead the function
f(x) =xcosx sinx; in addition, the successful interval
for the starting value x0is larger.
Table 3.8.1 : Newton's rule for x tanx= 0.
n x n
0 4.65000 00000 000
1 4.60567 66065 900
2 4.55140 53475 751
3 4.50903 76975 617
4 4.49455 61600 185
5 4.49341 56569 391
6 4.49340 94580 903
7 4.49340 94579 091
8 4.49340 94579 091
3.8 Nonlinear Equations 91
3.8(iii) Other Methods
Bisection Method
Iff(a)f(b)<0 witha<b , then the interval [ a;b] con-
tains one or more zeros of f. Bisection of this interval is
used to decide where at least one zero is located. All ze-
ros offin the original interval [ a;b] can be computed to
any predetermined accuracy. Convergence is slow how-
ever; see Kaufman and Lenker (1986) and Nievergelt
(1995).
Regula Falsi
Letx0andx1be such that f0=f(x0) andf1=
f(x1) have opposite signs. Inverse linear interpolation
(x3.3(v)) is used to obtain the rst approximation:
3.8.6x2=x1 x1 x0
f1 f0f1=f1x0 f0x1
f1 f0:
We continue with x2and either x0orx1, depending
which off0andf1is of opposite sign to f(x2), and so
on. The convergence is linear, and again more than one
zero may occur in the original interval [ x0;x1].
Secant Method
Whether or not f0andf1have opposite signs, x2is
computed as in (3.8.6). If the wanted zero is simple,
then the method converges locally with order of conver-
gencep=1
2(1 +p
5) = 1:618:::. Because the method
requires only one function evaluation per iteration, its
numerical eciency is ultimately higher than that of
Newton's method. There is no guaranteed convergence:
the rst approximation x2may be outside [ x0;x1].
Ste ensen's Method
This iterative method for solving z= (z) is given by
3.8.7
zn+1=zn ( (zn) zn)2
( (zn)) 2 (zn) +zn,n= 0;1;2;:::.
It converges locally and quadratically for both RandC.
For other ecient derivative-free methods, see Le
(1985).
Eigenvalue Methods
For the computation of zeros of orthogonal polynomials
as eigenvalues of nite tridiagonal matrices ( x3.5(vi)),
see Gil et al. (2007a, pp. 205{207). For the computa-
tion of zeros of Bessel functions, Coulomb functions, and
conical functions as eigenvalues of nite parts of in nite
tridiagonal matrices, see Grad and Zakraj sek (1973),
Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil
et al. (2007a, pp. 205{213).
3.8(iv) Zeros of Polynomials
The polynomial
3.8.8p(z) =anzn+an 1zn 1++a0,an6= 0,hasnzeros in C, counting each zero according to its mul-
tiplicity. Explicit formulas for the zeros are available if
n4; seexx1.11(iii) and 4.43. No explicit general for-
mulas exist when n5.
After a zero has been computed, the factor z is
factored out of p(z) as a by-product of Horner's scheme
(x1.11(i)) for the computation of p(). In this way poly-
nomials of successively lower degree can be used to nd
the remaining zeros. (This process is called de
ation .)
However, to guard against the accumulation of round-
ing errors, a nal iteration for each zero should also be
performed on the original polynomial p(z).
Example
p(z) =z4 1. The zeros are 1 andi. Newton's
method is given by
3.8.9 zn+1= (zn); (z) =3z4+ 1
4z3:
The results for z0= 1:5 are given in Table 3.8.2.
Table 3.8.2 : Newton's rule for z4 1 = 0.
n z n
0 1.50000 00000 000
1 1.19907 40740 741
2 1.04431 68969 414
3 1.00274 20038 676
4 1.00001 12265 490
5 1.00000 00001 891
6 1.00000 00000 000
As in the case of Table 3.8.1 the quadratic nature
of convergence is clearly evident: as the zero is ap-
proached, the number of correct decimal places doubles
at each iteration.
Newton's rule can also be used for complex zeros of
p(z). However, when the coecients are all real, com-
plex arithmetic can be avoided by the following iterative
process.
Bairstow's Method
Letz2 sz tbe an approximation to the real quadratic
factor ofp(z) that corresponds to a pair of conjugate
complex zeros or to a pair of real zeros. We con-
struct sequences qjandrj,j=n+ 1;n;:::; 0, from
qn+1=rn+1= 0,qn=rn=an, and forjn 1,
3.8.10
qj=aj+sqj+1+tqj+2; rj=qj+srj+1+trj+2:
Then the next approximation to the quadratic factor is
z2 (s+ s)z (t+ t), where
3.8.11
s=r3q0 r2q1
r2
2 `r3;t=`q1 r2q0
r2
2 `r3; ` =sr2+tr3:
92 Numerical Methods
The method converges locally and quadratically, ex-
cept when the wanted quadratic factor is a multiple fac-
tor ofq(z). On the last iteration qnzn 2+qn 1zn 3+
+q2is the quotient on dividing p(z) byz2 sz t.
Example
p(z) =z4 2z2+ 1. With the starting values s0=7
4,
t0= 1
2, an approximation to the quadratic factor
z2 2z+ 1 = (z 1)2is computed ( s= 2,t= 1).
Table 3.8.3 gives the successive values of sandt. The
quadratic nature of the convergence is evident.
Table 3.8.3 : Bairstow's method for factoring z4 2z2+1.
n s n tn
0 1.75000 00000 000 0.50000 00000 000
1 2.13527 29454 109 1.21235 75284 943
2 2.01786 10488 956 1.02528 61401 539
3 2.00036 06329 466 1.00047 63067 522
4 2.00000 01474 803 1.00000 01858 298
5 2.00000 00000 000 1.00000 00000 000
This example illustrates the fact that the method
succeeds even if the two zeros of the wanted quadratic
factor are real and the same.
For further information on the computation of zeros
of polynomials see McNamee (2007).
3.8(v) Zeros of Analytic Functions
Newton's rule is the most frequently used iterative pro-
cess for accurate computation of real or complex zeros
of analytic functions f(z). Another iterative method is
Halley's rule :
3.8.12zn+1=zn f(zn)
f0(zn) (f00(zn)f(zn)=(2f0(zn))):
This is useful when f(z) satis es a second-order linear
di erential equation because of the ease of computing
f00(zn). The rule converges locally and is cubically con-
vergent.
Initial approximations to the zeros can often be
found from asymptotic or other approximations to f(z),
or by application of the phase principle or Rouch e's the-
orem; seex1.10(iv). These results are also useful in en-
suring that no zeros are overlooked when the complex
plane is being searched.
For an example involving the Airy functions, see
Fabijonas and Olver (1999).
For xed-point methods for computing zeros of spe-
cial functions, see Segura (2002), Gil and Segura (2003),
and Gil et al. (2007a, Chapter 7).3.8(vi) Conditioning of Zeros
Supposef(z) also depends on a parameter , denoted
byf(z; ). Then the sensitivity of a simple zero zto
changes in is given by
3.8.13dz
d = @f
@ @f
@z:
Thus iffis the polynomial (3.8.8) and is the co-
ecientaj, say, then
3.8.14dz
daj= zj
f0(z):
For moderate or large values of nit is not uncommon
for the magnitude of the right-hand side of (3.8.14) to
be very large compared with unity, signifying that the
computation of zeros of polynomials is often an ill-posed
problem.
Example. Wilkinson's Polynomial
The zeros of
3.8.15p(x) = (x 1)(x 2)(x 20)
are well separated but extremely ill-conditioned. Con-
siderx= 20 andj= 19. We have p0(20) = 19! and
a19= 1 + 2 ++ 20 = 210. The perturbation factor
(3.8.14) is given by
3.8.16dx
da19= 2019
19!= ( 4:30:::)107:
Corresponding numerical factors in this example
for other zeros and other values of jare obtained in
Gautschi (1984,x4).
3.8(vii) Systems of Nonlinear Equations
For xed-point iterations and Newton's method for solv-
ing systems of nonlinear equations, see Gautschi (1997b,
Chapter 4,x9) and Ortega and Rheinboldt (1970).
3.8(viii) Fixed-Point Iterations: Fractals
The convergence of iterative methods
3.8.17 zn+1= (zn),n= 0;1;:::,
for solving xed-point problems (3.8.2) cannot always
be predicted, especially in the complex plane.
Consider, for example, (3.8.9). Starting this itera-
tion in the neighborhood of one of the four zeros 1;i,
sequencesfzngare generated that converge to these ze-
ros. For an arbitrary starting point z02C, conver-
gence cannot be predicted, and the boundary of the set
of pointsz0that generate a sequence converging to a
particular zero has a very complicated structure. It is
called a Julia set . In general the Julia set of an an-
alytic function f(z) is a fractal , that is, a set that is
self-similar. See Julia (1918) and Devaney (1986).
3.9 Acceleration of Convergence 93
3.9 Acceleration of Convergence
3.9(i) Sequence Transformations
All sequences (series) in this section are sequences (se-
ries) of real or complex numbers.
A transformation of a convergent sequence fsngwith
limitinto a sequenceftngis called limit-preserving if
ftngconverges to the same limit .
The transformation is accelerating if it is limit-
preserving and if
3.9.1 lim
n!1tn
sn = 0:
Similarly for convergent series if we regard the sum as
the limit of the sequence of partial sums.
It should be borne in mind that a sequence (series)
transformation can be e ective for one type of sequence
(series) but may not accelerate convergence for another
type. It may even fail altogether by not being limit-
preserving.
3.9(ii) Euler's Transformation of Series
IfS=P1
k=0( 1)kakis a convergent series, then
3.9.2 S=1X
k=0( 1)k2 k 1ka0;
provided that the right-hand side converges. Here is
theforward di erence operator :
3.9.3 ka0= k 1a1 k 1a0,k= 1;2;:::.
Thus
3.9.4 ka0=kX
m=0( 1)mk
m
ak m:
Euler's transformation is usually applied to alter-
nating series. Examples are provided by the following
analytic transformations of slowly-convergent series into
rapidly convergent ones:
3.9.5
ln 2 = 1 1
2+1
3 1
4+=1
121+1
222+1
323+;
3.9.6
4= 1 1
3+1
5 1
7+
=1
2
1 +1!
13+2!
35+3!
357+
:3.9(iii) Aitken's 2-Process
3.9.7tn=sn (sn)2
2sn=sn (sn+1 sn)2
sn+2 2sn+1+sn:
This transformation is accelerating if fsngis alinearly
convergent sequence , i.e., a sequence for which
3.9.8 lim
n!1sn+1
sn =,jj<1.
When applied repeatedly, Aitken's process is known
as the iterated 2-process . See Brezinski and Redivo Za-
glia (1991, pp. 39{42).
3.9(iv) Shanks' Transformation
Shanks' transformation is a generalization of Aitken's
2-process. Let kbe a xed positive integer. Then
the transformation of the sequence fsnginto a sequence
ftn;2kgis given by
3.9.9 tn;2k=Hk+1(sn)
Hk(2sn),n= 0;1;2;:::,
whereHmis the Hankel determinant
3.9.10Hm(un) = unun+1un+m 1
un+1un+2un+m
............
un+m 1un+mun+2m 2 :
The ratio of the Hankel determinants in (3.9.9) can
be computed recursively by Wynn's epsilon algorithm :
3.9.11
"(n)
1= 0; "(n)
0=sn, n= 0;1;2;:::,
"(n)
m+1="(n+1)
m 1+1
"(n+1)
m "(n)
m,n;m = 0;1;2;:::.
Thentn;2k="(n)
2k. Aitken's 2-process is the case
k= 1.
Ifsnis thenth partial sum of a power series f,
thentn;2k="(n)
2kis the Pad e approximant [( n+k)=k]f
(x3.11(iv)).
For further information on the epsilon algorithm see
Brezinski and Redivo Zaglia (1991, pp. 78{95).
Example
In Table 3.9.1 values of the transforms tn;2kare supplied
for
3.9.12 sn=nX
j=1( 1)j+1
j2;
withs1=1
122= 0:82246 70334 24 :::.
94 Numerical Methods
Table 3.9.1 : Shanks' transformation for sn=Pn
j=1( 1)j+1j 2.
n t n;2 tn;4 tn;6 tn;8 tn;10
0 0.80000 00000 00 0.82182 62806 24 0.82244 84501 47 0.82246 64909 60 0.82246 70175 41
1 0.82692 30769 23 0.82259 02017 65 0.82247 05346 57 0.82246 71342 06 0.82246 70363 45
2 0.82111 11111 11 0.82243 44785 14 0.82246 61821 45 0.82246 70102 48 0.82246 70327 79
3 0.82300 13550 14 0.82247 78118 35 0.82246 72851 83 0.82246 70397 56 0.82246 70335 90
4 0.82221 76684 88 0.82246 28314 41 0.82246 69467 93 0.82246 70314 36 0.82246 70333 75
5 0.82259 80392 16 0.82246 88857 22 0.82246 70670 21 0.82246 70341 24 0.82246 70334 40
6 0.82239 19390 77 0.82246 61352 37 0.82246 70190 76 0.82246 70331 54 0.82246 70334 18
7 0.82251 30483 23 0.82246 75033 13 0.82246 70400 56 0.82246 70335 37 0.82246 70334 26
8 0.82243 73137 33 0.82246 67719 32 0.82246 70301 49 0.82246 70333 73 0.82246 70334 23
9 0.82248 70624 89 0.82246 71865 91 0.82246 70351 34 0.82246 70334 48 0.82246 70334 24
10 0.82245 30535 15 0.82246 69397 57 0.82246 70324 88 0.82246 70334 12 0.82246 70334 24
3.9(v) Levin's and Weniger's Transformations
We give a special form of Levin's transformation in
which the sequence s=fsngof partial sums sn=Pn
j=0ajis transformed into:
3.9.13
L(n)
k(s) =Pk
j=0( 1)j k
j
cj;k;nsn+j/an+j+1
Pk
j=0( 1)j k
j
cj;k;n=an+j+1;
wherekis a xed nonnegative integer, and
3.9.14 cj;k;n=(n+j+ 1)k 1
(n+k+ 1)k 1:
Sequences that are accelerated by Levin's transforma-
tion include logarithmically convergent sequences, i.e.,
sequencessnconverging to such that
3.9.15 lim
n!1sn+1
sn = 1:
For further information see Brezinski and Redivo Za-
glia (1991, pp. 39{42).
InWeniger's transformations the numbers cj;k;n in
(3.9.13) are chosen as follows:
3.9.16 cj;k;n=( +n+j)k 1
( +n+k)k 1;
or
3.9.17 cj;k;n=(
n j)k 1
(
n k)k 1;
where (a)0= 1 and (a)j=a(a+ 1)(a+j 1) are
Pochhammer symbols ( x5.2(iii)), and the constants
and
are chosen arbitrarily subject to certain condi-
tions. See Weniger (1989).
3.9(vi) Applications and Further
Transformations
For examples and other transformations for convergent
sequences and series, see Wimp (1981, pp. 156{199),
Brezinski and Redivo Zaglia (1991, pp. 55{72), and Sidi(2003, Chapters 6, 12{13, 15{16, 19{24, and pp. 483{
492).
For applications to asymptotic expansions, see
x2.11(vi), Olver (1997b, pp. 540{543), and Weniger
(1989, 2003).
3.10 Continued Fractions
3.10(i) Introduction
Seex1.12 for relevant properties of continued fractions,
including the following de nitions:
3.10.1 C=b0+a1
b1+a2
b2+,an6= 0,
3.10.2Cn=b0+a1
b1+a2
b2+an
bn=An
Bn:
Cnis thenthapproximant orconvergent toC.
3.10(ii) Relations to Power Series
Every convergent, asymptotic, or formal series
3.10.3 u0+u1+u2+
can be converted into a continued fraction Cof type
(3.10.1), and with the property that the nth convergent
Cn=An=BntoCis equal to the nth partial sum of the
series in (3.10.3), that is,
3.10.4An
Bn=u0+u1++un,n= 0;1;:::.
For instance, if none of the unvanish, then we can de ne
3.10.5b0=u0; b 1= 1; a 1=u1;
bn= 1 +un
un 1; an= un
un 1,n2.
However, other continued fractions with the same
limit may converge in a much larger domain of the
complex plane than the fraction given by (3.10.4) and
(3.10.5). For example, by converting the Maclaurin
3.10 Continued Fractions 95
expansion of arctan z(4.24.3), we obtain a continued
fraction with the same region of convergence ( jzj1,
z6=i), whereas the continued fraction (4.25.4) con-
verges for all z2Cexcept on the branch cuts from ito
i1and ito i1.
Stieltjes Fractions
A continued fraction of the form
3.10.6 C=a0
1 a1z
1 a2z
1
is called a Stieltjes fraction ( S-fraction) . We say that it
corresponds to the formal power series
3.10.7 f(z) =c0+c1z+c2z2+
if the expansion of its nth convergent Cnin ascending
powers ofzagrees with (3.10.7) up to and including the
term inzn 1,n= 1;2;3;:::.
Quotient-Di erence Algorithm
For several special functions the S-fractions are known
explicitly, but in any case the coecients ancan al-
ways be calculated from the power-series coecients by
means of the quotient-di erence algorithm ; see Table
3.10.1.
Table 3.10.1 : Quotient-di erence scheme.
e1
0
e2
0
e3
0
e4
0q0
1
q1
1
q2
1
q3
1
...e0
1
e1
1
e2
1
e3
1q0
2
q1
2
q2
2
...e0
2
e1
2
e2
2q0
3
q1
3
...e0
3
e1
3...
...
The rst two columns in this table are de ned by
3.10.8en
0= 0, n= 1;2;:::,
qn
1=cn+1=cn, n= 0;1;:::,
where thecn(6= 0) appear in (3.10.7). We continue by
means of the rhombus rule
3.10.9ek
j=ek+1
j 1+qk+1
j qk
j, j1,k0,
qk
j+1=qk+1
jek+1
j=ek
j, j1,k0.
Then the coecients anof theS-fraction (3.10.6) are
given by
3.10.10
a0=c0; a 1=q0
1; a 2=e0
1; a 3=q0
2; a 4=e0
2; ::::The quotient-di erence algorithm is frequently un-
stable and may require high-precision arithmetic or ex-
act arithmetic. A more stable version of the algorithm
is discussed in Stokes (1980). For applications to Bessel
functions and Whittaker functions (Chapters 10 and
13), see Gargantini and Henrici (1967).
Jacobi Fractions
A continued fraction of the form
3.10.11C= 0
1 0z 1z2
1 1z 2z2
1 2z
is called a Jacobi fraction ( J-fraction) . We say that it is
associated with the formal power series f(z) in (3.10.7)
if the expansion of its nth convergent Cnin ascend-
ing powers of z, agrees with (3.10.7) up to and includ-
ing the term in z2n 1,n= 1;2;3;:::. For the same
functionf(z), the convergent Cnof the Jacobi fraction
(3.10.11) equals the convergent C2nof the Stieltjes frac-
tion (3.10.6).
Examples of S- andJ-Fractions
For elementary functions, see xx4.9 and 4.35.
For special functions see x5.10 (gamma function),
x7.9 (error function), x8.9 (incomplete gamma func-
tions),x8.17(v) (incomplete beta function), x8.19(vii)
(generalized exponential integral), xx10.10 and 10.33
(quotients of Bessel functions), x13.6 (quotients of con-
uent hypergeometric functions), x13.19 (quotients of
Whittaker functions), and x15.7 (quotients of hyperge-
ometric functions).
For further information and examples see Lorentzen
and Waadeland (1992, pp. 292{330, 560{599) and Cuyt
et al. (2008).
3.10(iii) Numerical Evaluation of Continued
Fractions
Forward Recurrence Algorithm
TheAnandBnof (3.10.2) can be computed by means
of three-term recurrence relations (1.12.5). However,
this may be unstable; also over
ow and under
ow may
occur when evaluating AnandBn(making it necessary
to re-scale from time to time).
Backward Recurrence Algorithm
To compute the Cnof (3.10.2) we perform the iterated
divisions
3.10.12
un=bn; uk=bk+ak+1
uk+1,k=n 1;n 2;:::; 0.
Thenu0=Cn. To achieve a prescribed accuracy, either
a priori knowledge is needed of the value of n, ornis
determined by trial and error. In general this algorithm
is more stable than the forward algorithm; see Jones
and Thron (1974).
96 Numerical Methods
Forward Series Recurrence Algorithm
The continued fraction
3.10.13 C=a0
1 a1
1 a2
1
can be written in the form
3.10.14 C=1X
k=0tk;
where
3.10.15t0=a0; tk=ktk 1; 0= 0;
k=ak(1 +k 1)
1 ak(1 +k 1),k= 1;2;3;:::.
Thenth partial sum t0+t1++tn 1equals thenth
convergent of (3.10.13), n= 1;2;3;:::. In contrast to
the preceding algorithms in this subsection no scaling
problems arise and no a priori information is needed.
In Gautschi (1979b) the forward series algorithm is
used for the evaluation of a continued fraction of an
incomplete gamma function (see x8.9).
Steed's Algorithm
This forward algorithm achieves eciency and stabil-
ity in the computation of the convergents Cn=An=Bn,
and is related to the forward series recurrence algorithm.
Again, no scaling problems arise and no a priori infor-
mation is needed.
Let
3.10.16
C0=b0; D 1= 1=b1;rC1=a1D1; C 1=C0+rC1:
(ris the backward di erence operator .) Then for n2,
3.10.17Dn=1
Dn 1an+bn;
rCn= (bnDn 1)rCn 1;
Cn=Cn 1+rCn:
The recurrences are continued until ( rCn)=Cnis within
a prescribed relative precision.
For further information on the preceding algorithms,
including convergence in the complex plane and meth-
ods for accelerating convergence, see Blanch (1964) and
Lorentzen and Waadeland (1992, Chapter 3). For the
evaluation of special functions by using continued frac-
tions see Cuyt et al. (2008), Gautschi (1967, x1), Gil
et al. (2007a, Chapter 6), and Wimp (1984, Chapter 4,
x5). See alsoxx6.18(i), 7.22(i), 8.25(iv), 10.74(v), 14.32,
28.34(ii), 29.20(i), 30.16(i), 33.23(v).
3.11 Approximation Techniques
3.11(i) Minimax Polynomial Approximations
Letf(x) be continuous on a closed interval [ a;b].
Then there exists a unique nth degree polynomialpn(x), called the minimax (orbest uniform ) polyno-
mial approximation to f(x) on [a;b], that minimizes
maxaxbjn(x)j, wheren(x) =f(x) pn(x).
A sucient condition for pn(x) to be the minimax
polynomial is that jn(x)jattains its maximum at n+ 2
distinct points in [ a;b] andn(x) changes sign at these
consecutive maxima.
If we have a suciently close approximation
3.11.1pn(x) =anxn+an 1xn 1++a0
tof(x), then the coecients akcan be computed iter-
atively. Assume that f0(x) is continuous on [ a;b] and
letx0=a,xn+1=b, andx1;x2;:::;xnbe the zeros of
0
n(x) in (a;b) arranged so that
3.11.2x0<x1<x2<<xn<xn+1:
Also, let
3.11.3 mj= ( 1)jn(xj),j= 0;1;:::;n + 1.
(Thus themjare approximations to m, wheremis
the maximum value of jn(x)jon [a;b].)
Then (in general) a better approximation to pn(x)
is given by
3.11.4nX
k=0(ak+ak)xk;
where
3.11.5nX
k=0xk
jak= ( 1)j(mj m),j= 0;1;:::;n + 1.
This is a set of n+ 2 equations for the n+ 2 unknowns
a0;a1;:::;anandm.
The iterative process converges locally and quadrat-
ically (x3.8(i)).
A method for obtaining a suciently accurate rst
approximation is described in the next subsection.
For the theory of minimax approximations see
Meinardus (1967). For examples of minimax polynomial
approximations to elementary and special functions see
Hart et al. (1968). See also Cody (1970) and Ralston
(1965).
3.11(ii) Chebyshev-Series Expansions
The Chebyshev polynomials Tnare given by
3.11.6 Tn(x) = cos(narccosx), 1x1.
They satisfy the recurrence relation
3.11.7
Tn+1(x) 2xTn(x) +Tn 1(x) = 0,n= 1;2;:::,
with initial values T0(x) = 1,T1(x) =x. They enjoy an
orthogonal property with respect to integrals:
3.11.8Z1
1Tj(x)Tk(x)p
1 x2dx=8
><
>:; j =k= 0;
1
2; j =k6= 0;
0; j6=k;
3.11 Approximation Techniques 97
as well as an orthogonal property with respect to sums,
as follows. When n>0 and 0jn, 0kn,
3.11.9nX00
`=0Tj(x`)Tk(x`) =8
><
>:n; j =k= 0 orn;
1
2n; j =k6= 0 orn;
0; j6=k;
wherex`= cos(`=n ) and the double prime means that
the rst and last terms are to be halved.
For these and further properties of Chebyshev poly-
nomials, see Chapter 18, Gil et al. (2007a, Chapter 3),
and Mason and Handscomb (2003).
Chebyshev Expansions
Iffis continuously di erentiable on [ 1;1], then with
3.11.10cn=2
Z
0f(cos) cos(n)d,n= 0;1;2;:::,
the expansion
3.11.11 f(x) =1X0
n=0cnTn(x), 1x1,
converges uniformly. Here the single prime on the
summation symbol means that the rst term is to be
halved. In fact, (3.11.11) is the Fourier-series expansion
off(cos); compare (3.11.6) and x1.8(i).
Furthermore, if f2C1[ 1;1], then the conver-
gence of (3.11.11) is usually very rapid; compare (1.8.7)
withkarbitrary.
For general intervals [ a;b] we rescale:
3.11.12 f(x) =1X0
n=0dnTn2x a b
b a
:
Because the series (3.11.12) converges rapidly we ob-
tain a very good rst approximation to the minimax
polynomial pn(x) for [a;b] if we truncate (3.11.12) at
its (n+ 1)th term. This is because in the notation of
x3.11(i)
3.11.13n(x) =dn+1Tn+12x a b
b a
;
approximately, and the right-hand side enjoys exactly
those properties concerning its maxima and minima
that are required for the minimax approximation; com-
pare Figure 18.4.3.
More precisely, it is known that for the interval [ a;b],
the ratio of the maximum value of the remainder
3.11.14 1X
k=n+1dkTk2x a b
b a
to the maximum error of the minimax polynomial pn(x)
is bounded by 1+ Ln, whereLnis thenthLebesgue con-
stant for Fourier series; see x1.8(i). Since L0= 1,Ln
is a monotonically increasing function of n, and (for ex-
ample)L1000= 4:07:::, this means that in practice thegain in replacing a truncated Chebyshev-series expan-
sion by the corresponding minimax polynomial approx-
imation is hardly worthwhile. Moreover, the set of min-
imax approximations p0(x);p1(x);p2(x);:::;pn(x) re-
quires the calculation and storage of1
2(n+ 1)(n+ 2) co-
ecients, whereas the corresponding set of Chebyshev-
series approximations requires only n+ 1 coecients.
Calculation of Chebyshev Coecients
Thecnin (3.11.11) can be calculated from (3.11.10), but
in general it is more ecient to make use of the orthogo-
nal property (3.11.9). Also, in cases where f(x) satis es
a linear ordinary di erential equation with polynomial
coecients, the expansion (3.11.11) can be substituted
in the di erential equation to yield a recurrence relation
satis ed by the cn.
For details and examples of these methods, see Clen-
shaw (1957, 1962) and Miller (1966). See also Mason
and Handscomb (2003, Chapter 10) and Fox and Parker
(1968, Chapter 5).
Summation of Chebyshev Series: Clenshaw's Algorithm
For the expansion (3.11.11), numerical values of the
Chebyshev polynomials Tn(x) can be generated by ap-
plication of the recurrence relation (3.11.7). A more
ecient procedure is as follows. Let cnTn(x) be the
last term retained in the truncated series. Beginning
withun+1= 0,un=cn, we apply
3.11.15
uk= 2xuk+1 uk+2+ck,k=n 1;n 2;:::; 0.
Then the sum of the truncated expansion equals1
2(u0
u2). For error analysis and modi cations of Clenshaw's
algorithm, see Oliver (1977).
Complex Variables
Ifxis replaced by a complex variable zandf(z) is
analytic, then the expansion (3.11.11) converges within
an ellipse. However, in general (3.11.11) a ords no ad-
vantage in Cfor numerical purposes compared with the
Maclaurin expansion of f(z).
For further details on Chebyshev-series expansions
in the complex plane, see Mason and Handscomb (2003,
x5.10).
3.11(iii) Minimax Rational Approximations
Letfbe continuous on a closed interval [ a;b] andwbe
a continuous nonvanishing function on [ a;b]:wis called
aweight function . Then the minimax (orbest uniform )
rational approximation
3.11.16 Rk;`(x) =p0+p1x++pkxk
1 +q1x++q`x`
oftype[k;`] tofon [a;b] minimizes the maximum value
ofjk;`(x)jon [a;b], where
3.11.17 k;`(x) =Rk;`(x) f(x)
w(x):
98 Numerical Methods
The theory of polynomial minimax approximation
given inx3.11(i) can be extended to the case when pn(x)
is replaced by a rational function Rk;`(x). There exists
a unique solution of this minimax problem and there
are at least k+`+ 2 values xj,ax0< x 1<<
xk+`+1b, such that mj=m, where
3.11.18 mj= ( 1)jk;`(xj),j= 0;1;:::;k +`+ 1,
andmis the maximum of jk;`(x)jon [a;b].
A collection of minimax rational approximations to
elementary and special functions can be found in Hart
et al. (1968).
A widely implemented and used algorithm for calcu-
lating the coecients pjandqjin (3.11.16) is Remez's
second algorithm . See Remez (1957), Werner et al.
(1967), and Johnson and Blair (1973).
Example
Withw(x) = 1 and 14-digit computation, we obtain
the following rational approximation of type [3 ;3] to
the Bessel function J0(x) (x10.2(ii)) on the interval
0xj0;1, wherej0;1is the rst positive zero of
J0(x):
3.11.19R3;3(x) =p0+p1x+p2x2+p3x3
1 +q1x+q2x2+q3x3;
with coecients given in Table 3.11.1.
Table 3.11.1 : Coecients pj,qjfor the minimax rational
approximation R3;3(x).
j p j qj
0 0.99999 99891 7854
1 0.34038 93820 9347 0.34039 05233 8838
2 0.18915 48376 3222 0.06086 50162 9812
3 0.06658 31942 0166 0.01864 47680 9090
The error curve is shown in Figure 3.11.1.
Figure 3.11.1 : ErrorR3;3(x) J0(x) of the minimax
rational approximation R3;3(x) to the Bessel function
J0(x) for 0xj0;1(= 0:89357:::).
3.11(iv) Pad e Approximations
Let
3.11.20 f(z) =c0+c1z+c2z2+be a formal power series. The rational function
3.11.21Np;q(z)
Dp;q(z)=a0+a1z++apzp
b0+b1z++bqzq
is called a Pad e approximant at zero offif
3.11.22Np;q(z) f(z)Dp;q(z) =O
zp+q+1
,z!0.
It is denoted by [ p=q]f(z). Thus if b06= 0, then the
Maclaurin expansion of (3.11.21) agrees with (3.11.20)
up to, and including, the term in zp+q.
The requirement (3.11.22) implies
3.11.23a0=c0b0;
a1=c1b0+c0b1;
...
ap=cpb0+cp 1b1++cp qbq;
0 =cp+1b0+cpb1++cp q+1bq;
...
0 =cp+qb0+cp+q 1b1++cpbq;
wherecj= 0 ifj <0. Withb0= 1, the last qequations
giveb1;:::;bqas the solution of a system of linear equa-
tions. The rst p+ 1 equations then yield a0;:::;ap.
The array of Pad e approximants
3.11.24[0=0]f[0=1]f[0=2]f
[1=0]f[1=1]f[1=2]f
[2=0]f[2=1]f[2=2]f
............
is called a Pad e table . Approximants with the same de-
nominator degree are located in the same column of the
table.
For convergence results for Pad e approximants, and
the connection with continued fractions and Gaussian
quadrature, see Baker and Graves-Morris (1996, x4.7).
The Pad e approximants can be computed by Wynn's
cross rule . Any ve approximants arranged in the Pad e
table as
W
SCN
E
satisfy
3.11.25
(N C) 1+ (S C) 1= (W C) 1+ (E C) 1:
Starting with the rst column [ n=0]f,n= 0;1;2;:::,
and initializing the preceding column by [ n= 1]f=1,
n= 1;2;:::, we can compute the lower triangular part
of the table via (3.11.25). Similarly, the upper triangu-
lar part follows from the rst row [0 =n]f,n= 0;1;2;:::,
by initializing [ 1=n]f= 0,n= 1;2;:::.
3.11 Approximation Techniques 99
For the recursive computation of [ n+k=k]fby
Wynn's epsilon algorithm, see (3.9.11) and the subse-
quent text.
Laplace Transform Inversion
Numerical inversion of the Laplace transform ( x1.14(iii))
3.11.26F(s) =L(f;s) =Z1
0e stf(t)dt
requiresf=L 1Fto be obtained from numerical val-
ues ofF. A general procedure is to approximate Fby
a rational function R(vanishing at in nity) and then
approximate fbyr=L 1R. WhenFhas an explicit
power-series expansion a possible choice of Ris a Pad e
approximation to F. See Luke (1969b, x16.4) for several
examples involving special functions.
For further information on Pad e approximations, see
Baker and Graves-Morris (1996, x4.7), Brezinski (1980,
pp. 9{39 and 126{177), and Lorentzen and Waadeland
(1992, pp. 367{395).
3.11(v) Least Squares Approximations
Suppose a function f(x) is approximated by the poly-
nomial
3.11.27pn(x) =anxn+an 1xn 1++a0
that minimizes
3.11.28 S=JX
j=1(f(xj) pn(xj))2:
Herexj,j= 1;2;:::;J , is a given set of distinct real
points andJn+1. From the equations @S/@ak= 0,
k= 0;1;:::;n , we derive the normal equations
3.11.292
6664X0X1Xn
X1X2Xn+1
............
XnXn+1X2n3
77752
6664a0
a1
...
an3
7775=2
6664F0
F1
...
Fn3
7775;
where
3.11.30Xk=JX
j=1xk
j; Fk=JX
j=1f(xj)xk
j:
(3.11.29) is a system of n+1 linear equations for the
coecients a0;a1;:::;an. The matrix is symmetric and
positive de nite, but the system is ill-conditioned when
nis large because the lower rows of the matrix are ap-
proximately proportional to one another. If J=n+ 1,
thenpn(x) is the Lagrange interpolation polynomial for
the setx1;x2;:::;xJ(x3.3(i)).
More generally, let f(x) be approximated by a linear
combination
3.11.31 n(x) =an n(x)+an 1 n 1(x)++a0 0(x)of given functions k(x),k= 0;1;:::;n , that minimizes
3.11.32JX
j=1w(xj) (f(xj) n(xj))2;
w(x) being a given positive weight function , and again
Jn+ 1. Then (3.11.29) is replaced by
3.11.332
6664X00X01X0n
X10X11X1n
............
Xn0Xn1Xnn3
77752
6664a0
a1
...
an3
7775=2
6664F0
F1
...
Fn3
7775;
with
3.11.34 Xk`=JX
j=1w(xj) k(xj) `(xj);
and
3.11.35 Fk=JX
j=1w(xj)f(xj) k(xj):
SinceXk`=X`k, the matrix is again symmetric.
If the functions k(x) are linearly independent on
the setx1;x2;:::;xJ, that is, the only solution of the
system of equations
3.11.36nX
k=0ck k(xj) = 0,j= 1;2;:::;J ,
isc0=c1==cn= 0, then the approximation
n(x) is determined uniquely.
Now suppose that Xk`= 0 whenk6=`, that is, the
functions k(x)are orthogonal with respect to weighted
summation on the discrete set x1;x2;:::;xJ. Then the
system (3.11.33) is diagonal and hence well-conditioned.
A set of functions 0(x); 1(x);:::; n(x) that is lin-
early independent on the set x1;x2;:::;xJ(compare
(3.11.36)) can always be orthogonalized in the sense
given in the preceding paragraph by the Gram{Schmidt
procedure; see Gautschi (1997b).
Example. The Discrete Fourier Transform
We takencomplex exponentials k(x) =eikx,k=
0;1;:::;n 1, and approximate f(x) by the linear com-
bination (3.11.31). The functions k(x) are orthogonal
on the setx0;x1;:::;xn 1,xj= 2j=n , with respect to
the weight function w(x) = 1, in the sense that
3.11.37n 1X
j=0 k(xj) `(xj) =nk;`,k;`= 0;1;:::;n 1,
k;`being Kronecker's symbol and the bar denoting
complex conjugate. In consequence we can solve the
system
3.11.38 fj=n 1X
k=0ak k(xj),j= 0;1;:::;n 1,
100 Numerical Methods
and obtain
3.11.39 ak=1
nn 1X
j=0fj k(xj),k= 0;1;:::;n 1.
With this choice of akandfj=f(xj), the correspond-
ing sum (3.11.32) vanishes.
The pair of vectors ff;ag
3.11.40f= [f0;f1;:::;fn 1]T;
a= [a0;a1;:::;an 1]T;
is called a discrete Fourier transform pair .
The Fast Fourier Transform
The direct computation of the discrete Fourier trans-
form (3.11.38), that is, of
3.11.41
fj=n 1X
k=0ak!jk
n; !n=e2i=n,j= 0;1;:::;n 1,
requires approximately n2multiplications. The method
of the fast Fourier transform (FFT) exploits the struc-
ture of the matrix
with elements !jk
n,j;k =
0;1;:::;n 1. Ifn= 2m, then
can be factored into m
matrices, the rows of which contain only a few nonzero
entries and the nonzero entries are equal apart from
signs. In consequence of this structure the number of
operations can be reduced to nm=nlog2noperations.
The property
3.11.42 !2(k (n=2))
n =!k
n=2
is of fundamental importance in the FFT algorithm. If
nis not a power of 2, then modi cations are possible.
For the original reference see Cooley and Tukey (1965).
For further details and algorithms, see Van Loan (1992).
For further information on least squares approxima-
tions, including examples, see Gautschi (1997b, Chapter
2) and Bj orck (1996, Chapters 1 and 2).
3.11(vi) Splines
Splines are de ned piecewise and usually by low-degree
polynomials. Given n+ 1 distinct points xkin the real
interval [a;b], with (a=)x0<x1<<xn 1<xn(=
b), on each subinterval [ xk;xk+1],k= 0;1;:::;n 1, a
low-degree polynomial is de ned with coecients deter-
mined by, for example, values fkandf0
kof a function
fand its derivative at the nodes xkandxk+1. The
set of all the polynomials de nes a function, the spline ,
on [a;b]. By taking more derivatives into account, the
smoothness of the spline will increase.
For splines based on Bernoulli and Euler polynomi-
als, seex24.17(ii).
For many applications a spline function is a more
adaptable approximating tool than the Lagrange in-
terpolation polynomial involving a comparable numberof parameters; see x3.3(i), where a single polynomial
is used for interpolating f(x) on the complete interval
[a;b]. Multivariate functions can also be approximated
in terms of multivariate polynomial splines. See de Boor
(2001), Chui (1988), and Schumaker (1981) for further
information.
In computer graphics a special type of spline is used
which produces a B ezier curve . A cubic B ezier curve is
de ned by four points. Two are endpoints: ( x0;y0) and
(x3;y3); the other points ( x1;y1) and (x2;y2) are con-
trol points. The slope of the curve at ( x0;y0) is tangent
to the line between ( x0;y0) and (x1;y1); similarly the
slope at (x3;y3) is tangent to the line between x2;y2
andx3;y3. The curve is described by x(t) andy(t),
which are cubic polynomials with t2[0;1]. A complete
spline results by composing several B ezier curves. A
special applications area of B ezier curves is mathemati-
cal typography and the design of type fonts. See Knuth
(1986, pp. 116-136).
3.12 Mathematical Constants
The fundamental constant
3.12.1= 3:14159 26535 89793 23846 :::
can be de ned analytically in numerous ways, for exam-
ple,
3.12.2 = 4Z1
0dt
1 +t2:
Other constants that appear in this Handbook include
the baseeof natural logarithms
3.12.3e= 2:71828 18284 59045 23536 ::: ;
seex4.2(ii), and Euler's constant
3.12.4
= 0:57721 56649 01532 86060 :::;
seex5.2(ii).
For access to online high-precision numerical values
of mathematical constants see Sloane (2003). For his-
torical and other information see Finch (2003).
References
General References
Lozier and Olver (1994) gives an overview of the numer-
ical evaluation of special functions. For more detailed
information see Gautschi (1997b), Gil et al. (2007a),
Henrici (1974, 1977, 1986), Hildebrand (1974), Luke
(1969a,b).
References 101
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x3.2Young and Gregory (1988, pp. 741{743), Wilkin-
son (1988, Chapter 2, xx8{10, and pp. 394{395,
423).
x3.3Davis (1975, Chapters 2{4), National Bureau of
Standards (1944, pp. xv{xvii), Hildebrand (1974,
Chapter 2), Ostrowski (1973, pp. 18{26).
x3.4Hildebrand (1974, pp. 85{89). The coecients Bn
k
are obtained by di erentiation of the An
k; compare
(3.4.2).
x3.5Davis and Rabinowitz (1984, pp. 54{58, 118{
120, 137, 434{436), Bauer et al. (1963), Golub
and Welsch (1969), Salzer (1955). For (3.5.18){
(3.5.19) see Waldvogel (2006). For Table 3.5.21
see Stroud (1971, pp. 243{249, 278{279).
Inx3.5(v) all numerical values of the nodes xkand
corresponding weights wkthat appear in the ta-
bles in the text and on the Web site can be com-puted, for example, by means of the quadruple-
precision analogs of the softwares recur and
gauss given in Gautschi (1994), or in the case of
the tables for the logarithmic weight function with
recur replaced by cheb , also provided in Gautschi
(1994). The three softwares can be used for other
values ofn, and other values of the parameters
and that appear in some of the weight functions.
x3.6Olver (1967a).
x3.8Gautschi (1997b, pp. 217{225, 230{234), Os-
trowski (1973, Chapters 3{11), Traub (1964,
pp. 268{269), National Physical Laboratory
(1961, pp. 57{59), Hildebrand (1974, p. 582).
x3.9Knopp (1964, pp. 253{255).
x3.10 Blanch (1964), Rutishauser (1957), Wall (1948,
pp. 17{19), Barnett et al. (1974), Barnett (1981a).
x3.11 Powell (1967), Meinardus (1967, x3), Wynn
(1966).
x3.12 For more digits in (3.12.1), (3.12.3), and (3.12.4)
see OEIS Sequences A000796, A001113, and
A001620. See also Sloane (2003).
Chapter 4
Elementary Functions
R. Roy1and F. W. J. Olver2
Notation 104
4.1 Special Notation . . . . . . . . . . . . . 104
Logarithm, Exponential, Powers 104
4.2 De nitions . . . . . . . . . . . . . . . . . 104
4.3 Graphics . . . . . . . . . . . . . . . . . . 106
4.4 Special Values and Limits . . . . . . . . . 107
4.5 Inequalities . . . . . . . . . . . . . . . . 108
4.6 Power Series . . . . . . . . . . . . . . . . 108
4.7 Derivatives and Di erential Equations . . 108
4.8 Identities . . . . . . . . . . . . . . . . . 109
4.9 Continued Fractions . . . . . . . . . . . . 109
4.10 Integrals . . . . . . . . . . . . . . . . . . 110
4.11 Sums . . . . . . . . . . . . . . . . . . . 110
4.12 Generalized Logarithms and Exponentials 111
4.13 Lambert W-Function . . . . . . . . . . . 111
Trigonometric Functions 111
4.14 De nitions and Periodicity . . . . . . . . 112
4.15 Graphics . . . . . . . . . . . . . . . . . . 112
4.16 Elementary Properties . . . . . . . . . . 115
4.17 Special Values and Limits . . . . . . . . . 116
4.18 Inequalities . . . . . . . . . . . . . . . . 116
4.19 Maclaurin Series and Laurent Series . . . 116
4.20 Derivatives and Di erential Equations . . 117
4.21 Identities . . . . . . . . . . . . . . . . . 117
4.22 In nite Products and Partial Fractions . . 118
4.23 Inverse Trigonometric Functions . . . . . 118
4.24 Inverse Trigonometric Functions: Further
Properties . . . . . . . . . . . . . . . . . 121
4.25 Continued Fractions . . . . . . . . . . . . 1214.26 Integrals . . . . . . . . . . . . . . . . . . 122
4.27 Sums . . . . . . . . . . . . . . . . . . . 123
Hyperbolic Functions 123
4.28 De nitions and Periodicity . . . . . . . . 123
4.29 Graphics . . . . . . . . . . . . . . . . . . 123
4.30 Elementary Properties . . . . . . . . . . 124
4.31 Special Values and Limits . . . . . . . . . 125
4.32 Inequalities . . . . . . . . . . . . . . . . 125
4.33 Maclaurin Series and Laurent Series . . . 125
4.34 Derivatives and Di erential Equations . . 125
4.35 Identities . . . . . . . . . . . . . . . . . 125
4.36 In nite Products and Partial Fractions . . 126
4.37 Inverse Hyperbolic Functions . . . . . . . 127
4.38 Inverse Hyperbolic Functions: Further
Properties . . . . . . . . . . . . . . . . . 129
4.39 Continued Fractions . . . . . . . . . . . . 129
4.40 Integrals . . . . . . . . . . . . . . . . . . 129
4.41 Sums . . . . . . . . . . . . . . . . . . . 130
Applications 130
4.42 Solution of Triangles . . . . . . . . . . . 130
4.43 Cubic Equations . . . . . . . . . . . . . . 131
4.44 Other Applications . . . . . . . . . . . . 131
Computation 131
4.45 Methods of Computation . . . . . . . . . 131
4.46 Tables . . . . . . . . . . . . . . . . . . . 132
4.47 Approximations . . . . . . . . . . . . . . 132
4.48 Software . . . . . . . . . . . . . . . . . . 133
References 133
1Department of Mathematics and Computer Science, Beloit College, Beloit, Wisconsin.
2Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland.
Acknowledgments : The authors are grateful to Steven G. Krantz and Peter R. Turner for advice on early drafts of this chapter.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
103
104 Elementary Functions
Notation
4.1 Special Notation
(For other notation see pp. xiv and 873.)
k;m;n integers.
a;c real or complex constants.
x;y real variables.
z=x+iycomplex variable.
e base of natural logarithms.
It is assumed the user is familiar with the de nitions
and properties of elementary functions of real arguments
x. The main purpose of the present chapter is to extend
these de nitions and properties to complex arguments
z.
The main functions treated in this chapter are the
logarithm ln z, Lnz; the exponential exp z,ez; the cir-
cular trigonometric (or just trigonometric) functions
sinz, cosz, tanz, cscz, secz, cotz; the inverse trigono-
metric functions arcsin z, Arcsinz, etc.; the hyper-
bolic trigonometric (or just hyperbolic) functions sinh z,
coshz, tanhz, cschz, sechz, cothz; the inverse hyper-
bolic functions arcsinh z, Arcsinhz, etc.
Sometimes in the literature the meanings of ln and
Ln are interchanged; similarly for arcsin zand Arcsin z,
etc. Sometimes \arc" is replaced by the index \ 1",
e.g. sin 1zfor arcsinzand Sin 1zfor Arcsinz.
Logarithm, Exponential, Powers
4.2 De nitions
4.2(i) The Logarithm
The general logarithm function Lnzis de ned by
4.2.1 Lnz=Zz
1dt
t, z6= 0;
where the integration path does not intersect the origin.
This is a multivalued function of zwith branch point at
z= 0.
The principal value , or principal branch , is de ned
by
4.2.2 lnz=Zz
1dt
t;
where the path does not intersect ( 1;0]; see Fig-
ure 4.2.1. ln zis a single-valued analytic function on
Cn( 1;0] and real-valued when zranges over the pos-
itive real numbers.
Figure 4.2.1 :z-plane: Branch cut for ln zandz .
The real and imaginary parts of ln zare given by
4.2.3 lnz= lnjzj+iphz, <phz< .
For phzseex1.9(i).
The only zero of ln zis atz= 1.
Most texts extend the de nition of the principal
value to include the branch cut
4.2.4 z=x, 1<x< 0,
by replacing (4.2.3) with
4.2.5 lnz= lnjzj+iphz, <phz.
With this de nition the general logarithm is given by
4.2.6 Lnz= lnz+ 2ki;
wherekis the excess of the number of times the path
in (4.2.1) crosses the negative real axis in the positive
sense over the number of times in the negative sense.
In this Handbook we allow a further extension by
regarding the cut as representing two sets of points, one
set corresponding to the \upper side" and denoted by
z=x+i0, the other set corresponding to the \lower
side" and denoted by z=x i0. Again see Figure 4.2.1.
Then
4.2.7 ln(xi0) = lnjxji, 1<x< 0,
with either upper signs or lower signs taken throughout.
Consequently ln zis two-valued on the cut, and discon-
tinuous across the cut. We regard this as the closed
de nition of the principal value .
In contrast to (4.2.5) the closed de nition is symmet-
ric. As a consequence, it has the advantage of extending
regions of validity of properties of principal values. For
example, with the de nition (4.2.5) the identity (4.8.7)
is valid only when jphzj<, but with the closed de -
nition the identity (4.8.7) is valid when jphzj. For
another example see (4.2.37).
In this Handbook it is usually clear from the context
which de nition of principal value is being used. How-
ever, in the absence of any indication to the contrary
it is assumed that the de nition is the closed one. For
other examples in this chapter see xx4.23, 4.24, 4.37,
and 4.38.
4.2 Definitions 105
4.2(ii) Logarithms to a General Base a
Witha;b6= 0 or 1,
4.2.8 logaz= lnz/lna;
4.2.9 logaz=logbz
logba;
4.2.10 logab=1
logba:
Natural logarithms have as base the unique positive
number
4.2.11e= 2:71828 18284 59045 23536 :::
such that
4.2.12 lne= 1:
Equivalently,
4.2.13Ze
1dt
t= 1:
Thus
4.2.14 logez= lnz;
4.2.15 log10z= (lnz)/(ln 10) = (log10e) lnz;
4.2.16 lnz= (ln 10) log10z;
4.2.17 log10e= 0:43429 44819 03251 82765 :::;
4.2.18 ln 10 = 2:30258 50929 94045 68401 ::::
logex= lnxis also called the Napierian orhyperbolic
logarithm. log10xis the common orBriggs logarithm.
4.2(iii) The Exponential Function
4.2.19 expz= 1 +z
1!+z2
2!+z3
3!+:
The function exp is an entire function of z, with no real
or complex zeros. It has period 2 i:
4.2.20 exp(z+ 2i) = expz:
Also,
4.2.21 exp( z) = 1=exp(z):
4.2.22 jexpzj= exp(<z):
The general value of the phase is given by
4.2.23 ph(expz) ==z+ 2k,k2Z:
Ifz=x+iy, then
4.2.24 expz=excosy+iexsiny:
If6= 0 then
4.2.25 expz=()z= Ln:4.2(iv) Powers
Powers with General Bases
The general athpower ofzis de ned by
4.2.26 za= exp(aLnz), z6= 0:
In particular, z0= 1, and if a=n= 1;2;3;:::, then
4.2.27za=zzz|{z}
ntimes= 1=z a:
In all other cases, zais a multivalued function with
branch point at z= 0. The principal value is
4.2.28 za= exp(alnz):
This is an analytic function of zonCn( 1;0], and
is two-valued and discontinuous on the cut shown in
Figure 4.2.1, unless a2Z.
4.2.29jzaj=jzj<aexp( (=a) phz);
4.2.30 ph(za) = (<a) phz+ (=a) lnjzj;
where phz2[ ;] for the principal value of za, and
is unrestricted in the general case. When ais real
4.2.31jzaj=jzja;ph(za) =aphz:
Unless indicated otherwise , it is assumed throughout
this Handbook that a power assumes its principal value.
With this convention,
4.2.32 ez= expz;
but the general value of ezis
4.2.33 ez= (expz) exp(2kzi),k2Z.
Forz= 1
4.2.34 e= 1 +1
1!+1
2!+1
3!+:
Ifzahas its general value, with a6= 0, and if w6= 0,
then
4.2.35za=w()z= exp1
aLnw
:
This result is also valid when zahas its principal value,
provided that the branch of Ln wsatis es
4.2.36 =1
aLnw
:
Another example of a principal value is provided by
4.2.37p
z2=(
z;<z0;
z;<z0:
Again, without the closed de nition the andsigns
would have to be replaced by >and<, respectively.
106 Elementary Functions
4.3 Graphics
4.3(i) Real Arguments
Figure 4.3.1 : lnxandex.
4.3(ii) Complex Arguments: Conformal Maps
Figure 4.3.2 illustrates the conformal mapping of the strip <=z < onto the whole w-plane cut along the
negative real axis, where w=ezandz= lnw(principal value). Corresponding points share the same letters, with
bars signifying complex conjugates. Lines parallel to the real axis in the z-plane map onto rays in the w-plane, and
lines parallel to the imaginary axis in the z-plane map onto circles centered at the origin in the w-plane. In the
labeling of corresponding points ris a real parameter that can lie anywhere in the interval (0 ;1).
(i)z-plane (ii) w-plane
A B C C D D E E F
z0r r +i r i i i r+i r i r
w1er er+i0 er i0 1 +i0 1 i0 e r+i0 e r i0e r
Figure 4.3.2 : Conformal mapping of exponential and logarithm. w=ez,z= lnw.
4.4 Special Values and Limits 107
4.3(iii) Complex Arguments: Surfaces
In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the
phase. See also p. xiv.
Figure 4.3.3 : ln(x+iy) (principal value). There is a
branch cut along the negative real axis.
Figure 4.3.4 :ex+iy.
4.4 Special Values and Limits
4.4(i) Logarithms
4.4.1 ln 1 = 0;
4.4.2 ln( 1i0) =i;
4.4.3 ln(i) =1
2i:
4.4(ii) Powers
4.4.4 e0= 1;
4.4.5 ei= 1;
4.4.6 ei=2=i;
4.4.7 e2ki= 1, k2Z,
4.4.8 ei=3=1
2ip
3
2;
4.4.9 e2i=3= 1
2ip
3
2;
4.4.10 ei=4=1p
2i1p
2;4.4.11 e3i=4= 1p
2i1p
2;
4.4.12 ii=e=2:
4.4(iii) Limits
4.4.13 lim
x!1x alnx= 0,<a>0,
4.4.14 lim
x!0xalnx= 0,<a>0,
4.4.15 lim
x!1xae x= 0;
4.4.16 lim
z!1zae z= 0,jphzj1
2 (<1
2),
wherea(2C) and(2(0;1
2]) are constants.
4.4.17 lim
n!1
1 +z
nn
=ez,z= constant.
4.4.18 lim
n!1
1 +1
nn
=e:
4.4.19lim
n!1 nX
k=11
k!
lnn!
=
= 0:57721 56649 01532 86060 :::;
where
is Euler's constant; see (5.2.3).
108 Elementary Functions
4.5 Inequalities
4.5(i) Logarithms
4.5.1x
1 +x<ln(1 +x)<x,x> 1,x6= 0,
4.5.2 x< ln(1 x)<x
1 x,x<1,x6= 0,
4.5.3 jln(1 x)j<3
2x, 0<x0:5828:::,
4.5.4 lnxx 1, x>0,
4.5.5 lnxa(x1=a 1), a,x>0,
4.5.6jln(1 +z)j ln(1 jzj),jzj<1.
For more inequalities involving the logarithm func-
tion see Mitrinovi c (1964, pp. 75{77), Mitrinovi c (1970,
pp. 272{276), and Bullen (1998, pp. 159{160).
4.5(ii) Exponentials
In (4.5.7){(4.5.12) it is assumed that x6= 0. (When
x= 0 the inequalities become equalities.)
4.5.7 e x=(1 x)<1 x<e x,x<1,
4.5.8 1 +x<ex, 1<x<1,
4.5.9 ex<1
1 x, x<1,
4.5.10x
1 +x<1 e x<x,x> 1,
4.5.11 x<ex 1<x
1 x, x<1,
4.5.12 ex=(1+x)<1 +x, x> 1,4.5.13 exy=(x+y)<
1 +x
yy
<ex,x>0,y>0,
4.5.14 e x<1 1
2x, 0<x1:5936:::,
4.5.151
4jzj<jez 1j<7
4jzj, 0<jzj<1,
4.5.16jez 1jejzj 1jzjejzj,z2C.
For more inequalities involving the exponential func-
tion see Mitrinovi c (1964, pp. 73{77), Mitrinovi c (1970,
pp. 266{271), and Bullen (1998, pp. 81{83).
4.6 Power Series
4.6(i) Logarithms
4.6.1 ln(1 +z) =z 1
2z2+1
3z3 ,jzj1,z6= 1,
4.6.2lnz=z 1
z
+1
2z 1
z2
+1
3z 1
z3
+,
<z1
2,
4.6.3lnz= (z 1) 1
2(z 1)2+1
3(z 1)3 ,
jz 1j1,z6= 0,
4.6.4
lnz= 2 z 1
z+ 1
+1
3z 1
z+ 13
+1
5z 1
z+ 15
+!
,
<z0,z6= 0,
4.6.5
lnz+ 1
z 1
= 21
z+1
3z3+1
5z5+
,jzj1,z6=1,
4.6.6 ln(z+a) = lna+ 2 z
2a+z
+1
3z
2a+z3
+1
5z
2a+z5
+!
,a>0,<z a,z6= a.
4.6(ii) Powers
Binomial Expansion
4.6.7
(1+z)a= 1+a
1!z+a(a 1)
2!z2+a(a 1)(a 2)
3!z3+;
valid whenais any real or complex constant and jzj<1.
Ifa= 0;1;2;:::, then the series terminates and zis un-
restricted.4.7 Derivatives and Di erential Equations
4.7(i) Logarithms
4.7.1d
dzlnz=1
z;
4.7.2d
dzLnz=1
z;
4.7.3dn
dznlnz= ( 1)n 1(n 1)!z n;
4.7.4dn
dznLnz= ( 1)n 1(n 1)!z n:
4.8 Identities 109
For a nonvanishing analytic function f(z), the gen-
eral solution of the di erential equation
4.7.5dw
dz=f0(z)
f(z)
is
4.7.6 w(z) = Ln(f(z)) + constant :
4.7(ii) Exponentials and Powers
4.7.7d
dzez=ez;
4.7.8d
dzeaz=aeaz;
4.7.9d
dzaz=azlna, a6= 0:
Whenazis a general power, ln ais replaced by the
branch of Ln aused in constructing az.
4.7.10d
dzza=aza 1;
4.7.11dn
dznza=a(a 1)(a 2)(a n+ 1)za n:
The general solution of the di erential equation
4.7.12dw
dz=f(z)w
is
4.7.13w= expZ
f(z)dz
+ constant:
The general solution of the di erential equation
4.7.14d2w
dz2=aw, a6= 0,
is
4.7.15 w=Aepaz+Be paz;
whereAandBare arbitrary constants.
For other di erential equations see Kamke (1977,
pp. 396{413).
4.8 Identities
4.8(i) Logarithms
In (4.8.1){(4.8.4) z1z26= 0.
4.8.1 Ln(z1z2) = Lnz1+ Lnz2:
This is interpreted that every value of Ln( z1z2) is one
of the values of Ln z1+ Lnz2, and vice versa.
4.8.2 ln(z1z2) = lnz1+ lnz2, phz1+ phz2,
4.8.3 Lnz1
z2= Lnz1 Lnz2;4.8.4 lnz1
z2= lnz1 lnz2, phz1 phz2.
In (4.8.5){(4.8.7) and (4.8.10) z6= 0.
4.8.5 Ln(zn) =nLnz, n2Z,
4.8.6 ln(zn) =nlnz,n2Z, nphz,
4.8.7 ln1
z= lnz,jphzj:
4.8.8 Ln(expz) =z+ 2ki,k2Z,
4.8.9 ln(expz) =z, =z,
4.8.10 exp(lnz) = exp(Ln z) =z:
Ifa6= 0 andazhas its general value, then
4.8.11 Ln(az) =zLna+ 2ki,k2Z.
Ifa6= 0 andazhas its principal value, then
4.8.12 ln(az) =zlna+ 2ki;
where the integer kis chosen so that<( izlna)+2k2
[ ;].
4.8.13 ln(ax) =xlna, a>0.
4.8(ii) Powers
4.8.14az1az2=az1+z2;
4.8.15azbz= (ab)z, pha+ phb,
4.8.16ez1ez2=ez1+z2;
4.8.17 (ez1)z2=ez1z2, =z1.
The restriction on z1can be removed when z2is an
integer.
4.9 Continued Fractions
4.9(i) Logarithms
4.9.1ln(1 +z) =z
1 +z
2 +z
3 +4z
4 +4z
5 +9z
6 +9z
7+,
jph(1 +z)j<.
4.9.2 ln1 +z
1 z
=2z
1 z2
3 4z2
5 9z2
7 16z2
9 ;
valid when z2Cn( 1; 1][[1;1); see Figure
4.23.1(i).
For other continued fractions involving logarithms
see Lorentzen and Waadeland (1992, pp. 566{568). See
also Cuyt et al. (2008, pp. 196{200).
110 Elementary Functions
4.9(ii) Exponentials
Forz2C,
4.9.3ez=1
1 z
1 +z
2 z
3 +z
2 z
5 +z
2
= 1 +z
1 z
2 +z
3 z
2 +z
5 z
2 +z
7
= 1 +z
1 (z=2) +z2=(43)
1 +z2=(415)
1 +z2=(435)
1+z2=(4(4n2 1))
1+
4.9.4 ez en 1(z) =zn
n! n!z
(n+ 1) +z
(n+ 2) (n+ 1)z
(n+ 3) +2z
(n+ 4) (n+ 2)z
(n+ 5) +3z
(n+ 6) ;
where
4.9.5 en(z) =nX
k=0zk
k!:
For other continued fractions involving the exponential function see Lorentzen and Waadeland (1992, pp. 563{
564). See also Cuyt et al. (2008, pp. 193{195).
4.9(iii) Powers
See Cuyt et al. (2008, pp. 217{220).
4.10 Integrals
4.10(i) Logarithms
4.10.1Zdz
z= lnz;
4.10.2Z
lnzdz=zlnz z;
4.10.3Z
znlnzdz=zn+1
n+ 1lnz zn+1
(n+ 1)2,n6= 1,
4.10.4Zdz
zlnz= ln(lnz);
4.10.5Z1
0lnt
1 tdt= 2
6;
4.10.6Z1
0lnt
1 +tdt= 2
12;
4.10.7Zx
0dt
lnt= li(x), x>1.
The left-hand side of (4.10.7) is a Cauchy principal value
(x1.4(v)). For li( x) seex6.2(i).
4.10(ii) Exponentials
Fora;b6= 0,
4.10.8Z
eazdz=eaz
a;
4.10.9Zdz
eaz+b=1
ab(az ln(eaz+b));4.10.10Zeaz 1
eaz+ 1dz=2
aln
eaz=2+e az=2
;
4.10.11Z1
1e cx2dx=r
c,<c>0,
4.10.12Zln 2
0xex
ex 1dx=2
12;
4.10.13Z1
0dx
ex+ 1= ln 2:
4.10(iii) Compendia
Extensive compendia of inde nite and de nite integrals
of logarithms and exponentials include Apelblat (1983,
pp. 16{47), Bierens de Haan (1939), Gr obner and Hofre-
iter (1949, pp. 107{116), Gr obner and Hofreiter (1950,
pp. 52{90), Gradshteyn and Ryzhik (2000, Chapters 2{
4), and Prudnikov et al. (1986a,xx1.3, 1.6, 2.3, 2.6).
4.11 Sums
For in nite series involving logarithms and/or exponen-
tials, see Gradshteyn and Ryzhik (2000, Chapter 1),
Hansen (1975,x44), and Prudnikov et al. (1986a, Chap-
ter 5).
4.12 Generalized Logarithms and Exponentials 111
4.12 Generalized Logarithms and
Exponentials
Ageneralized exponential function (x) satis es the
equations
4.12.1 (x+ 1) =e (x), 1<x<1;
4.12.2 (0) = 0;
and is strictly increasing when 0 x1. Its inverse
(x) is called a generalized logarithm . It, too, is strictly
increasing when 0 x1, and
4.12.3 (ex) = 1 + (x), 1<x<1;
4.12.4 (0) = 0:
These functions are not unique. The simplest choice
is given by
4.12.5 (x) = (x) =x, 0x1:
Then
4.12.6 (x) = ln(x+ 1), 1<x< 0;
and
4.12.7 (x) = exp expexp(x bxc),x>1;
where the exponentiations are carried out bxctimes.
Correspondingly,
4.12.8 (x) =ex 1, 1<x< 0;
and
4.12.9 (x) =`+ ln(`)x, x>1;
where ln(`)xdenotes the `-th repeated logarithm of x,
and`is the positive integer determined by the condition
4.12.10 0ln(`)x<1:
Both (x) and (x) are continuously di erentiable.
For further information, see Clenshaw et al. (1986).
ForC1generalized logarithms, see Walker (1991). For
analytic generalized logarithms, see Kneser (1950).
4.13 Lambert W-Function
The Lambert W-functionW(x) is the solution of the
equation
4.13.1 WeW=x:
On thex-interval [0;1) there is one real solution,
and it is nonnegative and increasing. On the x-interval
( 1=e;0) there are two real solutions, one increasing
and the other decreasing. We call the solution for which
W(x)W( 1=e) the principal branch and denote it by
Wp(x). The other solution is denoted by Wm( x). See
Figure 4.13.1.
Figure 4.13.1 : Branches Wp( x) and Wm(x) of the Lam-
bertW-function.AandBdenote the points 1=eand
e, respectively, on the x-axis.
Properties include:
4.13.2Wp( 1=e) = Wm( 1=e) = 1;
Wp(0) = 0;Wp(e) = 1:
4.13.3U+ lnU=x; U =U(x) =W(ex):
4.13.4dW
dx=e W
1 +W, x6= 1
e.
4.13.5 Wp(x) =1X
n=1( 1)n 1nn 2
(n 1)!xn,jxj<1
e:
4.13.6
W
e 1 (t2=2)
=1X
n=0( 1)n 1cntn,jtj<2p;
wheret0 for Wp,t0 for Wm,
4.13.7c0= 1;c1= 1;c2=1
3;c3=1
36;c4= 1
270;
4.13.8cn=1
n+ 1
cn 1 n 1X
k=2kckcn+1 k!
,n2,
and
4.13.9 135(2n+ 1)c2n+1=gn;
wheregnis de ned inx5.11(i).
Asx!+1
4.13.10
Wp(x) = ln+ln
+(ln)2
22 ln
2+O(ln)3
3
;
where= lnx. Asx!0
4.13.11
Wm(x) = ln ln
(ln)2
22 ln
2+O(ln)3
3
;
where= ln( 1=x).
For the foregoing results and further information see
Borwein and Corless (1999), Corless et al. (1996), de
Bruijn (1961, pp. 25{28), Olver (1997b, pp. 12{13), and
Siewert and Burniston (1973).
For integral representations of all branches of the
LambertW-function see Khey ts (2004).
112 Elementary Functions
Trigonometric Functions
4.14 De nitions and Periodicity
4.14.1 sinz=eiz e iz
2i;
4.14.2 cosz=eiz+e iz
2;
4.14.3 coszisinz=eiz;
4.14.4 tanz=sinz
cosz;
4.14.5 cscz=1
sinz;4.14.6 secz=1
cosz;
4.14.7 cotz=cosz
sinz=1
tanz:
The functions sin zand coszare entire. In Cthe ze-
ros of sinzarez=k,k2Z; the zeros of cos zare
z=
k+1
2
,k2Z. The functions tan z, cscz, secz,
and cotzare meromorphic, and the locations of their
zeros and poles follow from (4.14.4) to (4.14.7).
Fork2Z
4.14.8 sin(z+ 2k) = sinz;
4.14.9 cos(z+ 2k) = cosz;
4.14.10 tan(z+k) = tanz:
4.15 Graphics
4.15(i) Real Arguments
Figure 4.15.1 : sinxand cosx.
Figure 4.15.2 : Arcsinxand Arccos x. Principal values
are shown with thickened lines.
Figure 4.15.3 : tanxand cotx.
Figure 4.15.4 : arctanxand arccotx. Only principal val-
ues are shown. arccot xis discontinuous at x= 0.
4.15 Graphics 113
Figure 4.15.5 : cscxand secx.
Figure 4.15.6 : arccscxand arcsec x. Only principal
values are shown. (Both functions are complex when
1<x< 1.)
4.15(ii) Complex Arguments: Conformal Maps
Figure 4.15.7 illustrates the conformal mapping of the strip 1
2 <<z <1
2onto the whole w-plane cut along the
real axis from 1 to 1 and 1 to1, wherew= sinzandz= arcsinw(principal value). Corresponding points
share the same letters, with bars signifying complex conjugates. Lines parallel to the real axis in the z-plane map
onto ellipses in the w-plane with foci at w=1, and lines parallel to the imaginary axis in the z-plane map onto
rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points ris a real parameter that
can lie anywhere in the interval (0 ;1).
(i)z-plane (ii) w-plane
A B C C D D E E F
z01
21
2+ir1
2 ir ir ir 1
2+ir 1
2 ir 1
2
w0 1 cosh r+i0 coshr i0isinhr isinhr coshr+i0 coshr i0 1
Figure 4.15.7 : Conformal mapping of sine and inverse sine. w= sinz,z= arcsinw.
4.15(iii) Complex Arguments: Surfaces
In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the
phase. See also p. xiv.
114 Elementary Functions
Figure 4.15.8 : sin(x+iy).
Figure 4.15.9 : arcsin(x+iy) (principal value). There
are branch cuts along the real axis from 1 to 1 and
1 to1.
Figure 4.15.10 : tan(x+iy).
Figure 4.15.11 : arctan(x+iy) (principal value). There
are branch cuts along the imaginary axis from i1to
ianditoi1.
Figure 4.15.12 : csc(x+iy).
Figure 4.15.13 : arccsc(x+iy) (principal value). There
is a branch cut along the real axis from 1 to 1.
4.16 Elementary Properties 115
The corresponding surfaces for cos( x+iy),
cot(x+iy), and sec(x+iy) are similar. In consequence
of the identities
4.15.1 cos(x+iy) = sin
x+1
2+iy
;
4.15.2 cot(x+iy) = tan
x+1
2+iy
;
4.15.3 sec(x+iy) = csc
x+1
2+iy
;they can be obtained by translating the surfaces shown
in Figures 4.15.8, 4.15.10, 4.15.12 by 1
2parallel to
thex-axis, and adjusting the phase coloring in the case
of Figure 4.15.10.
The corresponding surfaces for arccos( x+iy),
arccot(x+iy), arcsec(x+iy) can be visualized from
Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equa-
tions (4.23.16){(4.23.18).
4.16 Elementary Properties
Figure 4.16.1 : Quadrants for the angle .
Table 4.16.1 : Signs of the trigonometric functions in the
four quadrants.
Quadrant sin ;csccos;sectan;cot
I + + +
II +
III +
IV + Table 4.16.2 : Trigonometric functions: quarter periods
and change of sign.
x 1
2 3
22
sinx sin cossin cossin
cosx cossin cossin cos
tanx tancottancottan
cscx csc seccsc seccsc
secx seccsc seccsc sec
cotx cottancottancot
Table 4.16.3 : Trigonometric functions: interrelations. All square roots have their principal values when the functions
are real, nonnegative, and nite.
sin=a cos=a tan=a csc=a sec=a cot=a
sina (1 a2)1=2a(1 +a2) 1=2a 1a 1(a2 1)1=2(1 +a2) 1=2
cos (1 a2)1=2a (1 +a2) 1=2a 1(a2 1)1=2a 1a(1 +a2) 1=2
tana(1 a2) 1=2a 1(1 a2)1=2a (a2 1) 1=2(a2 1)1=2a 1
csca 1(1 a2) 1=2a 1(1 +a2)1=2a a (a2 1) 1=2(1 +a2)1=2
sec (1 a2) 1=2a 1(1 +a2)1=2a(a2 1) 1=2a a 1(1 +a2)1=2
cota 1(1 a2)1=2a(1 a2) 1=2a 1(a2 1)1=2(a2 1) 1=2a
116 Elementary Functions
4.17 Special Values and Limits
Table 4.17.1 : Trigonometric functions: values at multiples of1
12.
sin cos tan csc sec cot
0 0 1 0 1 11
=121
4p
2(p
3 1)1
4p
2(p
3 + 1) 2 p
3p
2(p
3 + 1)p
2(p
3 1) 2 +p
3
=61
21
2p
31
3p
3 22
3p
3p
3
=41
2p
21
2p
2 1p
2p
2 1
=31
2p
31
2p
32
3p
3 21
3p
3
5=121
4p
2(p
3 + 1)1
4p
2(p
3 1) 2 +p
3p
2(p
3 1)p
2(p
3 + 1) 2 p
3
=2 1 0 1 11 0
7=121
4p
2(p
3 + 1) 1
4p
2(p
3 1) (2 +p
3)p
2(p
3 1) p
2(p
3 + 1) (2 p
3)
2=31
2p
3 1
2 p
32
3p
3 2 1
3p
3
3=41
2p
2 1
2p
2 1p
2 p
2 1
5=61
2 1
2p
3 1
3p
3 2 2
3p
3 p
3
11=121
4p
2(p
3 1) 1
4p
2(p
3 + 1) (2 p
3)p
2(p
3 + 1) p
2(p
3 1) (2 +p
3)
0 1 0 1 11
4.17.1 lim
z!0sinz
z= 1;
4.17.2 lim
z!0tanz
z= 1:
4.17.3 lim
z!01 cosz
z2=1
2:
4.18 Inequalities
Jordan's Inequality
4.18.12x
sinxx, 0x1
2.
4.18.2 xtanx, 0x<1
2,
4.18.3 cosxsinx
x1, 0x,
4.18.4 <sin(x)
x(1 x)4, 0 <x< 1.
Withz=x+iy,
4.18.5jsinhyjjsinzjcoshy;
4.18.6jsinhyjjcoszjcoshy;
4.18.7 jcsczjcschjyj;
4.18.8 jcoszjcoshjzj;
4.18.9 jsinzjsinhjzj;4.18.10jcoszj<2;jsinzj6
5jzj,jzj<1.
For more inequalities see Mitrinovi c (1964, pp. 101{
111), Mitrinovi c (1970, pp. 235{265), and Bullen (1998,
pp. 250{254).
4.19 Maclaurin Series and Laurent Series
4.19.1 sinz=z z3
3!+z5
5! z7
7!+;
4.19.2 cosz= 1 z2
2!+z4
4! z6
6!+:
In (4.19.3){(4.19.9), Bnare the Bernoulli numbers
andEnare the Euler numbers ( xx24.2(i){24.2(ii)).
4.19.3tanz=z+z3
3+2
15z5+17
315z7+
+( 1)n 122n(22n 1)B2n
(2n)!z2n 1+,
jzj<1
2,
4.19.4cscz=1
z+z
6+7
360z3+31
15120z5+
+( 1)n 12(22n 1 1)B2n
(2n)!z2n 1+,
0<jzj<,
4.20 Derivatives and Differential Equations 117
4.19.5secz= 1 +z2
2+5
24z4+61
720z6+
+( 1)nE2n
(2n)!z2n+,jzj<1
2,
4.19.6
cotz=1
z z
3 z3
45 2
945z5
( 1)n 122nB2n
(2n)!z2n 1 , 0<jzj<,
4.19.7 lnsinz
z
=1X
n=1( 1)n22n 1B2n
n(2n)!z2n,jzj<,
4.19.8
ln(cosz) =1X
n=1( 1)n22n 1(22n 1)B2n
n(2n)!z2n,jzj<1
2,
4.19.9
lntanz
z
=1X
n=1( 1)n 122n(22n 1 1)B2n
n(2n)!z2n,
jzj<1
2.
4.20 Derivatives and Di erential Equations
4.20.1d
dzsinz= cosz;
4.20.2d
dzcosz= sinz;
4.20.3d
dztanz= sec2z;
4.20.4d
dzcscz= csczcotz;
4.20.5d
dzsecz= secztanz;
4.20.6d
dzcotz= csc2z;
4.20.7dn
dznsinz= sin
z+1
2n
;
4.20.8dn
dzncosz= cos
z+1
2n
:
Witha6= 0, the general solutions of the di erential
equations
4.20.9d2w
dz2+a2w= 0;
4.20.10dw
dz2
+a2w2= 1;
4.20.11dw
dz a2w2= 1;are respectively
4.20.12 w=Acos(az) +Bsin(az);
4.20.13 w= (1=a) sin(az+c);
4.20.14 w= (1=a) tan(az+c);
whereA;B;c are arbitrary constants.
For other di erential equations see Kamke (1977,
pp. 355{358 and 396{400).
4.21 Identities
4.21(i) Addition Formulas
4.21.1 sinucosu=p
2 sin
u1
4
=p
2 cos
u1
4
:
4.21.2 sin(uv) = sinucosvcosusinv;
4.21.3 cos(uv) = cosucosvsinusinv;
4.21.4 tan(uv) =tanutanv
1tanutanv;
4.21.5 cot(uv) =cotucotv 1
cotucotv:
4.21.6 sinu+ sinv= 2 sinu+v
2
cosu v
2
;
4.21.7 sinu sinv= 2 cosu+v
2
sinu v
2
;
4.21.8 cosu+ cosv= 2 cosu+v
2
cosu v
2
;
4.21.9 cosu cosv= 2 sinu+v
2
sinu v
2
:
4.21.10 tanutanv=sin(uv)
cosucosv;
4.21.11 cotucotv=sin(vu)
sinusinv:
4.21(ii) Squares and Products
4.21.12 sin2z+ cos2z= 1;
4.21.13 sec2z= 1 + tan2z;
4.21.14 csc2z= 1 + cot2z:
4.21.15 2 sinusinv= cos(u v) cos(u+v);
4.21.16 2 cosucosv= cos(u v) + cos(u+v);
4.21.17 2 sinucosv= sin(u v) + sin(u+v):
4.21.18 sin2u sin2v= sin(u+v) sin(u v);
4.21.19 cos2u cos2v= sin(u+v) sin(u v);
4.21.20 cos2u sin2v= cos(u+v) cos(u v):
118 Elementary Functions
4.21(iii) Multiples of the Argument
4.21.21 sinz
2=1 cosz
21=2
;
4.21.22 cosz
2=1 + cosz
21=2
;
4.21.23
tanz
2=1 cosz
1 + cosz1=2
=1 cosz
sinz=sinz
1 + cosz:
In (4.21.21){(4.21.23) Table 4.16.1 and analytic contin-
uation will assist in resolving sign ambiguities.
4.21.24 sin( z) = sinz;
4.21.25 cos( z) = cosz;
4.21.26 tan( z) = tanz:
4.21.27 sin(2z) = 2 sinzcosz=2 tanz
1 + tan2z;
4.21.28cos(2z) = 2 cos2z 1 = 1 2 sin2z
= cos2z sin2z=1 tan2z
1 + tan2z;
4.21.29
tan(2z) =2 tanz
1 tan2z=2 cotz
cot2z 1=2
cotz tanz:
4.21.30 sin(3z) = 3 sinz 4 sin3z;
4.21.31 cos(3z) = 3 cosz+ 4 cos3z;
4.21.32 sin(4z) = 8 cos3zsinz 4 coszsinz;
4.21.33 cos(4z) = 8 cos4z 8 cos2z+ 1:
De Moivre's Theorem
Whenn2Z
4.21.34 cos(nz) +isin(nz) = (cosz+isinz)n:
This result is also valid when nis fractional or complex,
provided that <z.
4.21.35
sin(nz) = 2n 1n 1Y
k=0sin
z+k
n
,n= 1;2;3;:::.
Ift= tan 1
2z
, then
4.21.36
sinz=2t
1 +t2;cosz=1 t2
1 +t2; dz =2
1 +t2dt:4.21(iv) Real and Imaginary Parts; Moduli
Withz=x+iy
4.21.37 sinz= sinxcoshy+icosxsinhy;
4.21.38 cosz= cosxcoshy isinxsinhy;
4.21.39 tanz=sin(2x) +isinh(2y)
cos(2x) + cosh(2y);
4.21.40 cotz=sin(2x) isinh(2y)
cosh(2y) cos(2x):
4.21.41
jsinzj
= (sin2x+ sinh2y)1=2= 1
2(cosh(2y) cos(2x))1=2;
4.21.42jcoszj= (cos2x+ sinh2y)1=2
= 1
2(cosh(2y) + cos(2x))1=2;
4.21.43jtanzj=cosh(2y) cos(2x)
cosh(2y) + cos(2x)1=2
:
4.22 In nite Products and Partial Fractions
4.22.1 sinz=z1Y
n=1
1 z2
n22
;
4.22.2 cosz=1Y
n=1
1 4z2
(2n 1)22
:
Whenz6=n,n2Z,
4.22.3 cotz=1
z+ 2z1X
n=11
z2 n22;
4.22.4 csc2z=1X
n= 11
(z n)2;
4.22.5 cscz=1
z+ 2z1X
n=1( 1)n
z2 n22:
4.23 Inverse Trigonometric Functions
4.23(i) General De nitions
The general values of the inverse trigonometric functions
are de ned by
4.23.1 Arcsinz=Zz
0dt
(1 t2)1=2;
4.23.2 Arccosz=Z1
zdt
(1 t2)1=2;
4.23.3 Arctanz=Zz
0dt
1 +t2, z6=i;
4.23.4 Arccscz= Arcsin(1=z);
4.23.5 Arcsecz= Arccos(1=z);
4.23.6 Arccotz= Arctan(1 =z):
4.23 Inverse Trigonometric Functions 119
In (4.23.1) and (4.23.2) the integration paths may not
pass through either of the points t=1. The function
(1 t2)1=2assumes its principal value when t2( 1;1);
elsewhere on the integration paths the branch is deter-
mined by continuity. In (4.23.3) the integration path
may not intersect i. Each of the six functions is a
multivalued function of z. Arctanzand Arccot zhave
branch points at z=i; the other four functions have
branch points at z=1.
4.23(ii) Principal Values
The principal values (orprincipal branches ) of the in-
verse sine, cosine, and tangent are obtained by introduc-
ing cuts in the z-plane as indicated in Figures 4.23.1(i)
and 4.23.1(ii), and requiring the integration paths in
(4.23.1){(4.23.3) not to cross these cuts. Compare theprincipal value of the logarithm ( x4.2(i)). The princi-
pal branches are denoted by arcsin z, arccosz, arctanz,
respectively. Each is two-valued on the corresponding
cuts, and each is real on the part of the real axis that
remains after deleting the intersections with the corre-
sponding cuts.
The principal values of the inverse cosecant, secant,
and cotangent are given by
4.23.7 arccscz= arcsin(1=z);
4.23.8 arcsecz= arccos(1=z):
4.23.9 arccotz= arctan(1=z), z6=i.
These functions are analytic in the cut plane depicted
in Figures 4.23.1(iii) and 4.23.1(iv).
Except where indicated otherwise , it is assumed
throughout this Handbook that the inverse trigonomet-
ric functions assume their principal values.
(i) arcsinzand arccosz (ii) arctanz (iii) arccsczand arcsecz (iv) arccotz
Figure 4.23.1 :z-plane. Branch cuts for the inverse trigonometric functions.
Graphs of the principal values for real arguments are given in x4.15. This section also includes conformal mappings,
and surface plots for complex arguments.
4.23(iii) Re
ection Formulas
4.23.10 arcsin( z) = arcsinz;
4.23.11 arccos( z) = arccosz:
4.23.12 arctan( z) = arctanz, z6=i.
4.23.13 arccsc( z) = arccscz;
4.23.14 arcsec( z) = arcsecz:
4.23.15 arccot( z) = arccotz, z6=i.
4.23.16 arccosz=1
2 arcsinz;
4.23.17 arcsecz=1
2 arccscz:
4.23.18 arccotz=1
2 arctanz,<z?0.4.23(iv) Logarithmic Forms
Throughout this subsection allquantities assume their
principal values.
Inverse Sine
4.23.19arcsinz= iln
(1 z2)1=2+iz
,
z2Cn( 1; 1)[(1;1);
compare Figure 4.23.1(i). On the cuts
4.23.20
arcsinx=1
2iln
(x2 1)1=2+x
,x2[1;1),
4.23.21
arcsinx= 1
2iln
(x2 1)1=2 x
,
x2( 1; 1],
120 Elementary Functions
upper signs being taken on upper sides, and lower signs
on lower sides.
Inverse Cosine
4.23.22arccosz=1
2+iln
(1 z2)1=2+iz
,
z2Cn( 1; 1)[(1;1);
compare Figure 4.23.1(i). An equivalent de nition is
4.23.23
arccosz= 2iln 1 +z
21=2
+i1 z
21=2!
,
z2Cn( 1; 1)[(1;1);
see Kahan (1987).
On the cuts
4.23.24 arccosx=iln
(x2 1)1=2+x
,x2[1;1),
4.23.25arccosx=iln
(x2 1)1=2 x
,
x2( 1; 1],
the upper/lower signs corresponding to the upper/lower
sides.
Inverse Tangent
4.23.26
arctanz=i
2lni+z
i z
,z=i2Cn( 1; 1][[1;1);
compare Figure 4.23.1(ii). On the cuts
4.23.27arctan(iy) =1
2+i
2lny+ 1
y 1
,
y2( 1; 1)[(1;1),
the upper/lower sign corresponding to the right/left
side.Other Inverse Functions
For the corresponding results for arccsc z, arcsecz, and
arccotz, use (4.23.7){(4.23.9). Care needs to be taken
on the cuts, for example, if 0 <x<1then 1=(x+i0) =
(1=x) i0.
4.23(v) Fundamental Property
Withk2Z, the general solutions of the equations
4.23.28 z= sinw;
4.23.29 z= cosw;
4.23.30 z= tanw;
are respectively
4.23.31w= Arcsinz= ( 1)karcsinz+k;
4.23.32w= Arccosz=arccosz+ 2k;
4.23.33w= Arctanz= arctanz+k,z6=i.
4.23(vi) Real and Imaginary Parts
4.23.34 arcsinz= arcsin +iln
+ ( 2 1)1=2
;
4.23.35 arccosz= arccos iln
+ ( 2 1)1=2
;
4.23.36arctanz=1
2arctan2x
1 x2 y2
+1
4ilnx2+ (y+ 1)2
x2+ (y 1)2
;
wherez=x+iyandx2[ 1;1] in (4.23.34) and
(4.23.35), andjzj<1 in (4.23.36). Also,
4.23.37 =1
2
(x+ 1)2+y21=2+1
2
(x 1)2+y21=2;
4.23.38 =1
2
(x+ 1)2+y21=2 1
2
(x 1)2+y21=2:
4.23(vii) Special Values and Interrelations
Table 4.23.1 : Inverse trigonometric functions: principal values at 0, 1,1.
x arcsinxarccosxarctanxarccscxarcsecxarccotx
1 { { 1
2 01
2 0
1 1
2 1
4 1
2 1
4
0 01
2 0 { { 1
2
11
2 01
41
2 01
4
1 { {1
2 01
2 0
4.24 Inverse Trigonometric Functions: Further Properties 121
For interrelations see Table 4.16.3. For example,
from the heading and last entry in the penultimate col-
umn we have arcsec a= arccot
(a2 1) 1=2
.
4.23(viii) Gudermannian Function
The Gudermannian gd(x) is de ned by
4.23.39 gd(x) =Zx
0sechtdt, 1<x<1.
Equivalently,
4.23.40gd(x) = 2 arctan( ex) 1
2
= arcsin(tanh x) = arccsc(coth x)
= arccos(sech x) = arcsec(cosh x)
= arctan(sinh x) = arccot(csch x):
The inverse Gudermannian function is given by
4.23.41 gd 1(x) =Zx
0sectdt, 1
2<x<1
2.
Equivalently, and again when 1
2<x<1
2,
4.23.42gd 1(x) = ln tan 1
2x+1
4
= ln(secx+ tanx)
= arcsinh(tan x) = arccsch(cot x)
= arccosh(sec x) = arcsech(cos x)
= arctanh(sin x) = arccoth(csc x):
4.24 Inverse Trigonometric Functions:
Further Properties
4.24(i) Power Series
4.24.1
arcsinz=z+1
2z3
3+13
24z5
5+135
246z7
7+,jzj1.
4.24.2
arccosz= (2(1 z))1=2
1 +1X
n=1135(2n 1)
22n(2n+ 1)n!(1 z)n!
,
j1 zj2.
4.24.3
arctanz=z z3
3+z5
5 z7
7+,jzj1,z6=i.
4.24.4
arctanz=
2 1
z+1
3z3 1
5z5+,<z?0,jzj1.
4.24.5
arctanz=z
z2+ 1
1 +2
3z2
1 +z2+24
35z2
1 +z22
+!
,
<(z2)> 1
2,which requires z(=x+iy) to lie between the two rect-
angular hyperbolas given by
4.24.6 x2 y2= 1
2:
4.24(ii) Derivatives
4.24.7d
dzarcsinz= (1 z2) 1=2;
4.24.8d
dzarccosz= (1 z2) 1=2;
4.24.9d
dzarctanz=1
1 +z2:
4.24.10d
dzarccscz=1
z(z2 1)1=2,<z?0.
4.24.11d
dzarcsecz=1
z(z2 1)1=2,<z?0.
4.24.12d
dzarccotz= 1
1 +z2:
4.24(iii) Addition Formulas
4.24.13ArcsinuArcsinv
= Arcsin
u(1 v2)1=2v(1 u2)1=2
;
4.24.14ArccosuArccosv
= Arccos
uv((1 u2)(1 v2))1=2
;
4.24.15 ArctanuArctanv= Arctanuv
1uv
;
4.24.16ArcsinuArccosv
= Arcsin
uv((1 u2)(1 v2))1=2
= Arccos
v(1 u2)1=2u(1 v2)1=2
;
4.24.17ArctanuArccotv= Arctanuv1
vu
= Arccotvu
uv1
:
The above equations are interpreted in the sense that
every value of the left-hand side is a value of the right-
hand side and vice versa. All square roots have either
possible value.
4.25 Continued Fractions
4.25.1
tanz=z
1 z2
3 z2
5 z2
7 ,z6=1
2,3
2,:::.
122 Elementary Functions
4.25.2 tan(az) =atanz
1 +(1 a2) tan2z
3 +(4 a2) tan2z
5 +(9 a2) tan2z
7+,j<zj<1
2,az6=1
2;3
2;::: .
4.25.3arcsinzp
1 z2=z
1 12z2
3 12z2
5 34z2
7 34z2
9 ;
valid when zlies in the open cut plane shown in Figure
4.23.1(i).
4.25.4 arctanz=z
1 +z2
3 +4z2
5 +9z2
7 +16z2
9+;
valid when zlies in the open cut plane shown in Figure
4.23.1(ii).
4.25.5
e2aarctan(1=z)= 1 +2a
z a+a2+ 1
3z+a2+ 4
5z+a2+ 9
7z+;
valid when zlies in the open cut plane shown in Figure
4.23.1(iv).
See Lorentzen and Waadeland (1992, pp. 560{571)
for other continued fractions involving inverse trigono-
metric functions. See also Cuyt et al. (2008, pp. 201{
203, 205{210).
4.26 Integrals
4.26(i) Introduction
Throughout this section the variables are assumed to
be real. The results in xx4.26(ii) and 4.26(iv) can be
extended to the complex plane by using continuous
branches and avoiding singularities.
4.26(ii) Inde nite Integrals
4.26.1Z
sinxdx = cosx;
4.26.2Z
cosxdx = sinx:
4.26.3Z
tanxdx = ln(cosx), 1
2<x<1
2.
4.26.4Z
cscxdx = ln
tan1
2x
, 0 <x< .
4.26.5Z
secxdx = gd 1(x), 1
2<x<1
2.
For the right-hand side see (4.23.41) and (4.23.42).
4.26.6Z
cotxdx = ln(sinx), 0<x< .
4.26.7Z
eaxsin(bx)dx=eax
a2+b2(asin(bx) bcos(bx));
4.26.8Z
eaxcos(bx)dx=eax
a2+b2(acos(bx)+bsin(bx)):4.26(iii) De nite Integrals
Throughout this subsection mandnare integers.
Orthogonality Properties
4.26.9Z
0sin(mt) sin(nt)dt= 0, m6=n,
4.26.10Z
0cos(mt) cos(nt)dt= 0, m6=n,
4.26.11Z
0sin2(nt)dt=Z
0cos2(nt)dt=1
2,n6= 0.
4.26.12Z1
0sin(mt)
tdt=8
><
>:1
2; m> 0;
0; m = 0;
1
2; m< 0:
4.26.13Z1
0sin
t2
dt=Z1
0cos
t2
dt=1
2r
2:
4.26(iv) Inverse Trigonometric Functions
4.26.14Z
arcsinxdx =xarcsinx+ (1 x2)1=2, 1<x< 1,
4.26.15Z
arccosxdx =xarccosx (1 x2)1=2, 1<x< 1.
4.26.16Z
arctanxdx =xarctanx 1
2ln
1 +x2
,
1<x<1,
4.26.17Z
arccscxdx =xarccscx+ ln
x+ (x2 1)1=2
,
1<x<1,
4.26.18Z
arcsecxdx =xarcsecx ln
x+ (x2 1)1=2
,
1<x<1,
4.26.19Z
arccotxdx =xarccotx+1
2ln
1 +x2
, 0<x<1.
4.26.20Z
xarcsinxdx =x2
2 1
4
arcsinx+x
4(1 x2)1=2,
1<x< 1,
4.26.21Z
xarccosxdx =x2
2 1
4
arccosx x
4(1 x2)1=2,
1<x< 1.
4.27 Sums 123
4.26(v) Compendia
Extensive compendia of inde nite and de nite integrals
of trigonometric and inverse trigonometric functions in-
clude Apelblat (1983, pp. 48{109), Bierens de Haan
(1939), Gradshteyn and Ryzhik (2000, Chapters 2{4),
Gr obner and Hofreiter (1949, pp. 116{139), Gr obner
and Hofreiter (1950, pp. 94{160), and Prudnikov et al.
(1986a,xx1.5, 1.7, 2.5, 2.7).
4.27 Sums
For sums of trigonometric and inverse trigonometric
functions see Gradshteyn and Ryzhik (2000, Chapter
1), Hansen (1975, xx14{42), Oberhettinger (1973), and
Prudnikov et al. (1986a, Chapter 5).
Hyperbolic Functions
4.28 De nitions and Periodicity
4.28.1 sinhz=ez e z
2;
4.28.2 coshz=ez+e z
2;4.28.3 coshzsinhz=ez;
4.28.4 tanhz=sinhz
coshz;
4.28.5 cschz=1
sinhz;
4.28.6 sechz=1
coshz;
4.28.7 cothz=1
tanhz:
Relations to Trigonometric Functions
4.28.8 sin(iz) =isinhz;
4.28.9 cos(iz) = coshz;
4.28.10 tan(iz) =itanhz;
4.28.11 csc(iz) = icschz;
4.28.12 sec(iz) = sechz;
4.28.13 cot(iz) = icothz:
As a consequence, many properties of the hyperbolic
functions follow immediately from the corresponding
properties of the trigonometric functions.
Periodicity and Zeros
The functions sinh zand coshzhave period 2 i, and
tanhzhas periodi. The zeros of sinh zand coshzare
z=ikandz=i
k+1
2
, respectively, k2Z.
4.29 Graphics
4.29(i) Real Arguments
Figure 4.29.1 : sinhxand coshx.
Figure 4.29.2 : Principal values of arcsinh xand
arccoshx. (arccoshxis complex when x<1.)
124 Elementary Functions
Figure 4.29.3 : tanhxand cothx.
Figure 4.29.4 : Principal values of arctanh xand
arccothx. (arctanhxis complex when x< 1 orx>1,
and arccoth xis complex when 1<x< 1.)
Figure 4.29.5 : cschxand sechx.
Figure 4.29.6 : Principal values of arccsch xand
arcsechx. (arcsechxis complex when x<0 andx>1.)
4.29(ii) Complex Arguments
The conformal mapping w= sinhzis obtainable from Figure 4.15.7 by rotating both the w-plane and the z-plane
through an angle1
2, compare (4.28.8).
The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in
x4.15(iii) for the trigonometric and inverse trigonometric functions. They can be visualized with the aid of equations
(4.28.8){(4.28.13).
4.30 Elementary Properties
Table 4.30.1 : Hyperbolic functions: interrelations. All square roots have their principal values when the functions
are real, nonnegative, and nite.
sinh=a cosh=a tanh=a csch=a sech=a coth=a
sinha (a2 1)1=2a(1 a2) 1=2a 1a 1(1 a2)1=2(a2 1) 1=2
cosh (1 +a2)1=2a (1 a2) 1=2a 1(1 +a2)1=2a 1a(a2 1) 1=2
tanha(1 +a2) 1=2a 1(a2 1)1=2a (1 +a2) 1=2(1 a2)1=2a 1
cscha 1(a2 1) 1=2a 1(1 a2)1=2a a (1 a2) 1=2(a2 1)1=2
sech (1 +a2) 1=2a 1(1 a2)1=2a(1 +a2) 1=2a a 1(a2 1)1=2
cotha 1(a2+ 1)1=2a(a2 1) 1=2a 1(1 +a2)1=2(1 a2) 1=2a
4.31 Special Values and Limits 125
4.31 Special Values and Limits
Table 4.31.1 : Hyperbolic functions: values at multiples
of1
2i.
z 01
2i i3
2i1
sinhz0i 0 i1
coshz1 0 1 01
tanhz01i0 1i1
cschz1 i1i 0
sechz11 11 0
cothz1 01 0 1
4.31.1 lim
z!0sinhz
z= 1;
4.31.2 lim
z!0tanhz
z= 1;
4.31.3 lim
z!0coshz 1
z2=1
2:
4.32 Inequalities
Forxreal,
4.32.1 coshxsinhx
x3
;
4.32.2 sinxcosx<tanhx<x ,x>0,
4.32.3
jcoshx coshyjjx yjp
sinhxsinhy,x>0,y>0,
4.32.4 arctanx1
2tanhx, x0.
For these and other inequalities involving hyperbolic
functions see Mitrinovi c (1964, pp. 61, 76, 159) and
Mitrinovi c (1970, p. 270).
4.33 Maclaurin Series and Laurent Series
4.33.1 sinhz=z+z3
3!+z5
5!+;
4.33.2 coshz= 1 +z2
2!+z4
4!+:
4.33.3tanhz=z z3
3+2
15z5 17
315z7+
+22n(22n 1)B2n
(2n)!z2n 1+,
jzj<1
2.
ForB2nseex24.2(i). For expansions that correspond
to (4.19.4){(4.19.9), change ztoizand use (4.28.8){
(4.28.13).4.34 Derivatives and Di erential Equations
4.34.1d
dzsinhz= coshz;
4.34.2d
dzcoshz= sinhz;
4.34.3d
dztanhz= sech2z;
4.34.4d
dzcschz= cschzcothz;
4.34.5d
dzsechz= sechztanhz;
4.34.6d
dzcothz= csch2z:
Witha6= 0, the general solutions of the di erential
equations
4.34.7d2w
dz2 a2w= 0;
4.34.8dw
dz2
a2w2= 1;
4.34.9dw
dz2
a2w2= 1;
4.34.10dw
dz+a2w2= 1;
are respectively
4.34.11 w=Acosh(az) +Bsinh(az);
4.34.12 w= (1=a) sinh(az+c);
4.34.13 w= (1=a) cosh(az+c);
4.34.14 w= (1=a) coth(az+c);
whereA;B;c are arbitrary constants.
For other di erential equations see Kamke (1977,
pp. 289{400).
4.35 Identities
4.35(i) Addition Formulas
4.35.1 sinh(uv) = sinhucoshvcoshusinhv;
4.35.2 cosh(uv) = coshucoshvsinhusinhv;
4.35.3 tanh(uv) =tanhutanhv
1tanhutanhv;
4.35.4 coth(uv) =cothucothv+ 1
cothucothv:
126 Elementary Functions
4.35.5 sinhu+ sinhv= 2 sinhu+v
2
coshu v
2
;
4.35.6 sinhu sinhv= 2 coshu+v
2
sinhu v
2
;
4.35.7 coshu+coshv= 2 coshu+v
2
coshu v
2
;
4.35.8 coshu coshv= 2 sinhu+v
2
sinhu v
2
;
4.35.9 tanhutanhv=sinh(uv)
coshucoshv;
4.35.10 cothucothv=sinh(vu)
sinhusinhv:
4.35(ii) Squares and Products
4.35.11 cosh2z sinh2z= 1;
4.35.12 sech2z= 1 tanh2z;
4.35.13 csch2z= coth2z 1:
4.35.14 2 sinhusinhv= cosh(u+v) cosh(u v);
4.35.15 2 coshucoshv= cosh(u+v) + cosh(u v);
4.35.16 2 sinhucoshv= sinh(u+v) + sinh(u v):
4.35.17 sinh2u sinh2v= sinh(u+v) sinh(u v);
4.35.18 cosh2u cosh2v= sinh(u+v) sinh(u v);
4.35.19 sinh2u+ cosh2v= cosh(u+v) cosh(u v):
4.35(iii) Multiples of the Argument
4.35.20 sinhz
2=coshz 1
21=2
;
4.35.21 coshz
2=coshz+ 1
21=2
;
4.35.22
tanhz
2=coshz 1
coshz+ 11=2
=coshz 1
sinhz=sinhz
coshz+ 1:
The square roots assume their principal value on the
positive real axis, and are determined by continuity else-
where.
4.35.23 sinh( z) = sinhz;
4.35.24 cosh( z) = coshz;
4.35.25 tanh( z) = tanhz:
4.35.26 sinh(2z) = 2 sinhzcoshz=2 tanhz
1 tanh2z;
4.35.27cosh(2z) = 2 cosh2z 1 = 2 sinh2z+ 1
= cosh2z+ sinh2z;4.35.28 tanh(2z) =2 tanhz
1 + tanh2z;
4.35.29 sinh(3z) = 3 sinhz+ 4 sinh3z;
4.35.30 cosh(3z) = 3 coshz+ 4 cosh3z;
4.35.31 sinh(4z) = 4 sinh3zcoshz+ 4 cosh3zsinhz;
4.35.32 cosh(4z) = cosh4z+ 6 sinh2zcosh2z+ sinh4z:
4.35.33
cosh(nz)sinh(nz) = (coshzsinhz)n,n2Z.
4.35(iv) Real and Imaginary Parts; Moduli
Withz=x+iy
4.35.34 sinhz= sinhxcosy+icoshxsiny;
4.35.35 coshz= coshxcosy+isinhxsiny;
4.35.36 tanhz=sinh(2x) +isin(2y)
cosh(2x) + cos(2y);
4.35.37 cothz=sinh(2x) isin(2y)
cosh(2x) cos(2y):
4.35.38jsinhzj= (sinh2x+ sin2y)1=2
= 1
2(cosh(2x) cos(2y))1=2;
4.35.39jcoshzj= (sinh2x+ cos2y)1=2
= 1
2(cosh(2x) + cos(2y))1=2;
4.35.40jtanhzj=cosh(2x) cos(2y)
cosh(2x) + cos(2y)1=2
:
4.36 In nite Products and Partial Fractions
4.36.1 sinhz=z1Y
n=1
1 +z2
n22
;
4.36.2 coshz=1Y
n=1
1 +4z2
(2n 1)22
:
Whenz6=ni,n2Z,
4.36.3 cothz=1
z+ 2z1X
n=11
z2+n22;
4.36.4 csch2z=1X
n= 11
(z ni)2;
4.36.5 cschz=1
z+ 2z1X
n=1( 1)n
z2+n22:
4.37 Inverse Hyperbolic Functions 127
4.37 Inverse Hyperbolic Functions
4.37(i) General De nitions
The general values of the inverse hyperbolic functions
are de ned by
4.37.1 Arcsinhz=Zz
0dt
(1 +t2)1=2;
4.37.2 Arccoshz=Zz
1dt
(t2 1)1=2;
4.37.3 Arctanhz=Zz
0dt
1 t2, z6=1;
4.37.4 Arccschz= Arcsinh(1 =z);
4.37.5 Arcsechz= Arccosh(1 =z);
4.37.6 Arccothz= Arctanh(1 =z):
In (4.37.1) the integration path may not pass through
either of the points t=i, and the function (1 + t2)1=2
assumes its principal value when tis real. In (4.37.2)
the integration path may not pass through either of the
points1, and the function ( t2 1)1=2assumes its prin-
cipal value when t2(1;1). Elsewhere on the inte-
gration paths in (4.37.1) and (4.37.2) the branches are
determined by continuity. In (4.37.3) the integration
path may not intersect 1. Each of the six functions
is a multivalued function of z. Arcsinhzand Arccsch zhave branch points at z=i; the other four functions
have branch points at z=1.
4.37(ii) Principal Values
The principal values (orprincipal branches ) of the in-
verse sinh, cosh, and tanh are obtained by introducing
cuts in the z-plane as indicated in Figure 4.37.1(i)-(iii),
and requiring the integration paths in (4.37.1){(4.37.3)
not to cross these cuts. Compare the principal value of
the logarithm (x4.2(i)). The principal branches are de-
noted by arcsinh, arccosh, arctanh respectively. Each is
two-valued on the corresponding cut(s), and each is real
on the part of the real axis that remains after deleting
the intersections with the corresponding cuts.
The principal values of the inverse hyperbolic cose-
cant, hyperbolic secant, and hyperbolic tangent are
given by
4.37.7 arccschz= arcsinh(1 =z);
4.37.8 arcsechz= arccosh(1 =z):
4.37.9 arccothz= arctanh(1 =z), z6=1.
These functions are analytic in the cut plane depicted
in Figure 4.37.1(iv), (v), (vi), respectively.
Except where indicated otherwise , it is assumed
throughout this Handbook that the inverse hyperbolic
functions assume their principal values.
(i) arcsinhz (ii) arccosh z (iii) arctanh z
(iv) arccsch z (v) arcsech z (vi) arccoth z
Figure 4.37.1 :z-plane. Branch cuts for the inverse hyperbolic functions.
Graphs of the principal values for real arguments are given in x4.29. This section also indicates conformal
mappings, and surface plots for complex arguments.
128 Elementary Functions
4.37(iii) Re
ection Formulas
4.37.10 arcsinh( z) = arcsinhz:
4.37.11 arccosh( z) =i+ arccoshz,=z?0.
4.37.12 arctanh( z) = arctanhz, z6=1.
4.37.13 arccsch( z) = arccschz:
4.37.14 arcsech( z) =i+ arcsechz,=z?0.
4.37.15 arccoth( z) = arccothz, z6=1.
4.37(iv) Logarithmic Forms
Throughout this subsection allquantities assume their
principal values.
Inverse Hyperbolic Sine
4.37.16arcsinhz= ln
(z2+ 1)1=2+z
,
z=i2Cn( 1; 1)[(1;1);
compare Figure 4.37.1(i). On the cuts
4.37.17
arcsinh(iy) =1
2iln
(y2 1)1=2+y
,y2[1;1),
4.37.18arcsinh(iy) = 1
2iln
(y2 1)1=2 y
,
y2( 1; 1],
the upper/lower signs corresponding to the right/left
sides.
Inverse Hyperbolic Cosine
4.37.19
arccoshz= ln
(z2 1)1=2+z
,z2Cn( 1;1),
the upper or lower sign being taken according as <z?0;
compare Figure 4.37.1(ii). Also,
4.37.20
arccosh(iy) =1
2i+ ln
(y2+ 1)1=2y
,y?0.
It should be noted that the imaginary axis is not a cut;
the function de ned by (4.37.19) and (4.37.20) is ana-
lytic everywhere except on ( 1;1]. Compare Figure
4.37.1(ii).
An equivalent de nition is
4.37.21arccoshz= 2 ln z+ 1
21=2
+z 1
21=2!
,
z2Cn( 1;1);
see Kahan (1987).
On the part of the cuts from 1 to 1
4.37.22
arccoshx=ln
i(1 x2)1=2+x
,x2( 1;1],
the upper/lower sign corresponding to the upper/lower
side.On the part of the cut from 1 to 1
4.37.23
arccoshx=i+ ln
(x2 1)1=2 x
,x2( 1; 1],
the upper/lower sign corresponding to the upper/lower
side.
Inverse Hyperbolic Tangent
4.37.24
arctanhz=1
2ln1 +z
1 z
,z2Cn( 1; 1][[1;1);
compare Figure 4.37.1(iii). On the cuts
4.37.25arctanhx=1
2i+1
2lnx+ 1
x 1
,
x2( 1; 1)[(1;1),
the upper/lower sign corresponding to the upper/lower
sides.
Other Inverse Functions
For the corresponding results for arccsch z, arcsechz,
and arccoth z, use (4.37.7){(4.37.9); compare x4.23(iv).
4.37(v) Fundamental Property
Withk2Z, the general solutions of the equations
4.37.26 z= sinhw;
4.37.27 z= coshw;
4.37.28 z= tanhw;
are respectively given by
4.37.29w= Arcsinhz= ( 1)karcsinhz+ki;
4.37.30w= Arccoshz=arccoshz+ 2ki;
4.37.31w= Arctanhz= arctanhz+ki,z6=1.
4.37(vi) Interrelations
Table 4.30.1 can also be used to nd interrelations
between inverse hyperbolic functions. For example,
arcsecha= arccoth
(1 a2) 1=2
.
4.38 Inverse Hyperbolic Functions: Further Properties 129
4.38 Inverse Hyperbolic Functions: Further
Properties
4.38(i) Power Series
4.38.1arcsinhz=z 1
2z3
3+13
24z5
5 135
246z7
7+,
jzj<1.
4.38.2
arcsinhz= ln(2z)+1
21
2z2 13
241
4z4+135
2461
6z6 ,
<z>0,jzj>1.
4.38.3arccoshz= ln(2z) 1
21
2z2 13
241
4z4
135
2461
6z6 ,jzj>1.
4.38.4
arccoshz
= (2(z 1))1=2
1 +1X
n=1( 1)n135(2n 1)
22nn!(2n+ 1)(z 1)n!
,
<z>0,jz 1j2.
4.38.5
arctanhz=z+z3
3+z5
5+z7
7+,jzj1,z6=1.
4.38.6
arctanhz=i
2+1
z+1
3z3+1
5z5+,=z?0,jzj1.
4.38.7
arctanhz=z
1 z2
1 +2
3z2
z2 1+24
35z2
z2 12
+!
,
<(z2)<1
2,
which requires z(=x+iy) to lie between the two rect-
angular hyperbolas given by
4.38.8 x2 y2=1
2:
4.38(ii) Derivatives
In the following equations square roots have their prin-
cipal values.
4.38.9d
dzarcsinhz= (1 +z2) 1=2:
4.38.10d
dzarccoshz=(z2 1) 1=2,<z?0.
4.38.11d
dzarctanhz=1
1 z2:
4.38.12d
dzarccschz=1
z(1 +z2)1=2,<z?0.4.38.13d
dzarcsechz= 1
z(1 z2)1=2:
4.38.14d
dzarccothz=1
1 z2:
4.38(iii) Addition Formulas
4.38.15ArcsinhuArcsinhv
= Arcsinh
u(1 +v2)1=2v(1 +u2)1=2
;
4.38.16ArccoshuArccoshv
= Arccosh
uv((u2 1)(v2 1))1=2
;
4.38.17 ArctanhuArctanhv= Arctanhuv
1uv
;
4.38.18ArcsinhuArccoshv
= Arcsinh
uv((1 +u2)(v2 1))1=2
= Arccosh
v(1 +u2)1=2u(v2 1)1=2
;
4.38.19ArctanhuArccothv= Arctanhuv1
vu
= Arccothvu
uv1
:
The above equations are interpreted in the sense that
every value of the left-hand side is a value of the right-
hand side and vice-versa. All square roots have either
possible value.
4.39 Continued Fractions
4.39.1
tanhz=z
1 +z2
3 +z2
5 +z2
7 +,z6=1
2i;3
2i;::: .
4.39.2
arcsinhzp
1 +z2=z
1 +12z2
3 +12z2
5 +34z2
7 +34z2
9 +;
wherezis in the open cut plane of Figure 4.37.1(i).
4.39.3 arctanhz=z
1 z2
3 4z2
5 9z2
7 ;
wherezis in the open cut plane of Figure 4.37.1(iii).
For these and other continued fractions involving in-
verse hyperbolic functions see Lorentzen and Waadeland
(1992, pp. 569{571). See also Cuyt et al. (2008, pp. 211{
217).
4.40 Integrals
4.40(i) Introduction
Throughout this section the variables are assumed to
be real. The results in xx4.40(ii) and 4.40(iv) can be
extended to the complex plane by using continuous
branches and avoiding singularities.
130 Elementary Functions
4.40(ii) Inde nite Integrals
4.40.1Z
sinhxdx = coshx;
4.40.2Z
coshxdx = sinhx;
4.40.3Z
tanhxdx = ln(coshx):
4.40.4Z
cschxdx = ln
tanh 1
2x
, 0<x<1.
4.40.5Z
sechxdx = gd(x):
For the right-hand side see (4.23.39) and (4.23.40).
4.40.6Z
cothxdx = ln(sinhx), 0<x<1:
4.40(iii) De nite Integrals
4.40.7Z1
0e xsin(ax)
sinhxdx=1
2coth 1
2a
1
a,a6= 0,
4.40.8Z1
0sinh(ax)
sinh(x)dx=1
2tan 1
2a
, <a<;
4.40.9Z1
1eax
cosh 1
2x2dx=4a
sin(a), 1<a< 1;
4.40.10Z1
0tanh(ax) tanh(bx)
xdx= lna
b
,a>0,b>0:
4.40(iv) Inverse Hyperbolic Functions
4.40.11Z
arcsinhxdx =xarcsinhx (1 +x2)1=2:
4.40.12Z
arccoshxdx =xarccoshx (x2 1)1=2, 1<x<1;
4.40.13Z
arctanhxdx =xarctanhx+1
2ln
1 x2
,
1<x< 1,
4.40.14Z
arccschxdx =xarccschx+ arcsinhx, 0<x<1,
4.40.15Z
arcsechxdx =xarcsechx+ arcsinx, 0<x< 1,
4.40.16Z
arccothxdx =xarccothx+1
2ln
x2 1
,
1<x<1.4.40(v) Compendia
Extensive compendia of inde nite and de nite integrals
of hyperbolic functions include Apelblat (1983, pp. 96{
109), Bierens de Haan (1939), Gr obner and Hofre-
iter (1949, pp. 139{160), Gr obner and Hofreiter (1950,
pp. 160{167), Gradshteyn and Ryzhik (2000, Chapters
2{4), and Prudnikov et al. (1986a,xx1.4, 1.8, 2.4, 2.8).
4.41 Sums
For sums of hyperbolic functions see Gradshteyn and
Ryzhik (2000, Chapter 1), Hansen (1975, x43), Prud-
nikov et al. (1986a,x5.3), and Zucker (1979).
Applications
4.42 Solution of Triangles
4.42(i) Planar Right Triangles
Figure 4.42.1 : Planar right triangle.
4.42.1 sinA=a
c=1
cscA;
4.42.2 cosA=b
c=1
secA;
4.42.3 tanA=a
b=1
cotA:
4.42(ii) Planar Triangles
Figure 4.42.2 : Planar triangle.
4.42.4a
sinA=b
sinB=c
sinC;
4.43 Cubic Equations 131
4.42.5 c2=a2+b2 2abcosC;
4.42.6 a=bcosC+ccosB
4.42.7 area =1
2bcsinA= (s(s a)(s b)(s c))1=2;
wheres=1
2(a+b+c) (the semiperimeter).
4.42(iii) Spherical Triangles
Figure 4.42.3 : Spherical triangle.
4.42.8 cosa= cosbcosc+ sinbsinccosA;
4.42.9sinA
sina=sinB
sinb=sinC
sinc;
4.42.10 sinacosB= cosbsinc sinbcosccosA;
4.42.11 cosacosC= sinacotb sinCcotB;
4.42.12 cosA= cosBcosC+ sinBsinCcosa:
For these and other formulas see Smart (1962, Chap-
ter 1).
4.43 Cubic Equations
Let
4.43.1A=
4
3p1=2; B = 4
3p1=2;
C=
27q2
4p31=2
; D = 27q2
4p31=2
;
wherep(6= 0) andqare real constants. The roots of
4.43.2 z3+pz+q= 0
are:
(a)Asina,Asin
a+2
3
, andAsin
a+4
3
, with
sin(3a) =C, whenp<0 andC1.
(b)Acosha,Acosh
a+2
3i
, andAcosh
a+4
3i
,
with cosh(3 a) =C, whenp<0 andC > 1.
(c)Bsinha,Bsinh
a+2
3i
, andBsinh
a+4
3i
,
with sinh(3 a) =D, whenp>0.
Note that in Case (a) all the roots are real, whereas
in Cases (b) and (c) there is one real root and a conju-
gate pair of complex roots. See also x1.11(iii).4.44 Other Applications
For applications of generalized exponentials and gener-
alized logarithms to computer arithmetic see x3.1(iv).
For an application of the Lambert W-function to
generalized Gaussian noise see Chapeau-Blondeau and
Monir (2002).
Computation
4.45 Methods of Computation
4.45(i) Real Variables
Logarithms
The function ln xcan always be computed from its as-
cending power series after preliminary scaling. Suppose
rst 1=10x10. Then we take square roots repeat-
edly untiljyjis suciently small, where
4.45.1 y=x2 m 1:
After computing ln(1 + y) from (4.6.1)
4.45.2 lnx= 2mln(1 +y):
For other values of xsetx= 10m, where 1=10
10 andm2Z. Then
4.45.3 lnx= ln+mln 10:
Exponentials
Letxhave any real value. First, rescale via
4.45.4m=x
ln 10+1
2
; y =x mln 10:
Then
4.45.5 ex= 10mey;
and sincejyj1
2ln 10 = 1:15:::,eycan be computed
straightforwardly from (4.2.19).
Trigonometric Functions
Letxhave any real value. We rst compute =x=,
followed by
4.45.6 m=
+1
2
; =( m):
Then
4.45.7 sinx= ( 1)msin;cosx= ( 1)mcos;
and sincejj1
2= 1:57:::, sinand coscan be
computed straightforwardly from (4.19.1) and (4.19.2).
The other trigonometric functions can be found from
the de nitions (4.14.4){(4.14.7).
132 Elementary Functions
Inverse Trigonometric Functions
The function arctan xcan always be computed from
its ascending power series after preliminary transfor-
mations to reduce the size of x. From (4.24.15) with
u=v= ((1 +x2)1=2 1)=x, we have
4.45.8
2 arctan(1 +x2)1=2 1
x= arctanx, 0<x<1.
Beginning with x0=x, generate the sequence
4.45.9 xn=(1 +x2
n 1)1=2 1
xn 1,n= 1;2;3;:::,
untilxnis suciently small. We then compute
arctanxnfrom (4.24.3), followed by
4.45.10 arctanx= 2narctanxn:
Another method, when xis large, is to sum
4.45.11 arctanx=
2 1
x+1
3x3 1
5x5+:::;
compare (4.24.4).
As an example, take x= 9:47376. Then
4.45.12x1= 0:90000:::; x 2= 0:38373:::;
x3= 0:18528:::; x 4= 0:09185::::
From (4.24.3) arctan x4= 0:09160:::. From
(4.45.10)
4.45.13 arctanx= 16 arctan x4= 1:46563::::
As a check, from (4.45.11)
4.45.14
arctanx= 1:57079::: 0:10555:::+ 0:00039:::
= 1:46563::::
For the remaining inverse trigonometric functions,
we may use the identities provided by the fourth
row of Table 4.16.3. For example, arcsin x=
arctan
x(1 x2) 1=2
.
Hyperbolic and Inverse Hyperbolic Functions
The hyperbolic functions can be computed directly from
the de nitions (4.28.1){(4.28.7). The inverses arcsinh,
arccosh, and arctanh can be computed from the loga-
rithmic forms given in x4.37(iv), with real arguments.
For arccsch, arcsech, and arccoth we have (4.37.7){
(4.37.9).
Other Methods
See Luther (1995), Ziv (1991), Cody and Waite (1980),
Rosenberg and McNamee (1976), Carlson (1972a). For
interval-arithmetic algorithms, see Markov (1981). For
Shift-and-Add and CORDIC algorithms, see Muller
(1997), Merrheim (1994), Schelin (1983). For multi-
precision methods, see Smith (1989), Brent (1976).4.45(ii) Complex Variables
For lnzandez
4.45.15 lnz= lnjzj+iphz, phz,
4.45.16ez=e<z(cos(=z) +isin(=z)):
Seex1.9(i) for the precise relationship of ph zto the
arctangent function.
The trigonometric functions may be computed from
the de nitions (4.14.1){(4.14.7), and their inverses from
the logarithmic forms in x4.23(iv), followed by (4.23.7){
(4.23.9). Similarly for the hyperbolic and inverse hy-
perbolic functions; compare (4.28.1){(4.28.7), x4.37(iv),
and (4.37.7){(4.37.9).
For other methods see Miel (1981).
4.45(iii) Lambert W-Function
Forx2[ 1=e;1) the principal branch Wp( x) can be
computed by solving the de ning equation WeW=x
numerically, for example, by Newton's rule ( x3.8(ii)).
Initial approximations are obtainable, for example, from
the power series (4.13.6) (with t0) whenxis close to
1=e, from the asymptotic expansion (4.13.10) when x
is large, and by numerical integration of the di erential
equation (4.13.4) ( x3.7) for other values of x.
Similarly for Wm( x) in the interval [ 1=e;0).
See also Barry et al. (1995) and Chapeau-Blondeau
and Monir (2002).
4.46 Tables
Extensive numerical tables of all the elementary func-
tions for real values of their arguments appear in
Abramowitz and Stegun (1964, Chapter 4). This hand-
book also includes lists of references for earlier tables,
as do Fletcher et al. (1962) and Lebedev and Fedorova
(1960).
For 40D values of the rst 500 roots of tan x=x, see
Robinson (1972). (These roots are zeros of the Bessel
functionJ3=2(x); seex10.21.)
For 10S values of the rst ve complex roots of
sinz=az, cosz=az, and coshz=az, for selected
positive values of a, see Fettis (1976).
See also Luther (1995).
4.47 Approximations
4.47(i) Chebyshev-Series Expansions
Clenshaw (1962) and Luke (1975, Chapter 3) give 20D
coecients for ln, exp, sin, cos, tan, cot, arcsin, arctan,
arcsinh. Schonfelder (1980) gives 40D coecients for
sin, cos, tan.
4.48 Software 133
4.47(ii) Rational Functions
Hart et al. (1968) give ln, exp, sin, cos, tan, cot, arcsin,
arccos, arctan, sinh, cosh, tanh, arcsinh, arccosh. Pre-
cision is variable.
4.47(iii) Pad e Approximations
Luke (1975, Chapter 3) supplies real and complex ap-
proximations for ln, exp, sin, cos, tan, arctan, arcsinh.
Precision is variable.
4.47(iv) Additional References
See Luke (1975, pp. 288{289) and Luke (1969b, pp.74{
76).
4.48 Software
Seehttp://dlmf.nist.gov/4.48 .
References
General References
The main references used in writing this chapter are
Levinson and Redhe er (1970), Hobson (1928), Wall
(1948), and Whittaker and Watson (1927). For addi-
tional bibliographic reading see Copson (1935) and Sil-
verman (1967).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x4.2Levinson and Redhe er (1970, pp. 62{67), Hobson
(1928, pp. 289{301).
x4.3These graphics were produced at NIST.
x4.4Levinson and Redhe er (1970, pp. 62{63, 69),
Hardy (1952, pp. 403{420).
x4.5(4.5.1) and (4.5.5) can be veri ed by the methods
of Hardy et al. (1967, pp. 106{107). (4.5.2) and
(4.5.4) follow from (4.5.1). (4.5.3) follows from
the fact that x= 0 andx= 0:5828:::are suc-
cessive zeros of3
2x+ ln(1 x). (4.5.6) is obtained
from the Maclaurin expansion of ln(1 + z). (4.5.7)
to (4.5.12) are obtained by exponentiating the in-
equalities (4.5.1) and (4.5.2). For (4.5.13), see
Hardy et al. (1967, p. 102). (4.5.14) follows from
the fact that 1 1
2x e xhas 0 and 1 :5936:::
as consecutive zeros. (4.5.15) and (4.5.16) can be
derived from the Maclaurin expansion of ez.x4.6For (4.6.1) see Hardy (1952, pp. 471{473).
(4.6.2){(4.6.6) are variations of this. For (4.6.7)
see Hardy (1952, pp. 476{477).
x4.7Levinson and Redhe er (1970, pp. 53{54, 62{69).
x4.8Levinson and Redhe er (1970, pp. 62{66), Hobson
(1928, pp. 297{299).
x4.10 (4.10.1){(4.10.4) and (4.10.8){(4.10.10) can be
veri ed by di erentiation. For (4.10.5) and
(4.10.6), expand by the geometric series and in-
tegrate term by term to get a series which can
be summed by Andrews et al. (1999, p. 12). For
(4.10.11) apply (5.4.6) and (5.9.1). To evaluate
(4.10.12) and (4.10.13), expand by the geometric
series and integrate term by term. The diloga-
rithm series which appears from (4.10.12) can be
summed by Andrews et al. (1999, p. 105).
x4.13 To verify the radius of convergence of the series
(4.13.6) map the plane of Wonto the plane of t
viat= ( 2v)1=2, wherev=W+ lnW+ 1 i.
ThenWis analytic at t= 0, and its nearest singu-
larities to the origin are located at t= 2pei=4.
Figure 4.13.1 was produced at NIST.
x4.14 Levinson and Redhe er (1970, pp. 55{57).
x4.15 These graphics were produced at NIST.
x4.16 Hobson (1928, pp. 19{24).
x4.17 Hobson (1928, pp. 29{32, 53{75).
x4.18 For (4.18.1) see Copson (1935, p. 136). (4.18.3)
follows by the same method and (4.18.2) is a
consequence. (4.18.5) to (4.18.9) are straightfor-
ward and (4.18.10) is obtained from the Maclau-
rin expansions of cos zand sinz. For the sec-
ond inequality in (4.18.4), it is sucient to show
f(x)4x(1 x) sin(x)0 for 0x
1=2. The function f(x) is zero at x= 0 and
x= 1=2 and it has no zeros in (0 ;1=2), because
f0(x) = 4(1 2x) cos(x) can have only one
zero in (0;1=2) wherey= 4(1 2x) intersects
y=cos(x). The rst inequality is proved sim-
ilarly.
x4.19 Hobson (1928, pp. 288{293, 360{367).
x4.20 Levinson and Redhe er (1970, pp. 53{60).
x4.21 Hobson (1928, Chapter 4 and pp. 19, 21, 45, 52{
53, 60, 63{69, 237{239, 331). For (4.21.35) see
Walker (1996,x1.9).
134 Elementary Functions
x4.22 Hobson (1928, Chapter 17), Levinson and Red-
he er (1970, pp. 387{389).
x4.23 Levinson and Redhe er (1970, pp. 68{70), Hob-
son (1928, pp. 32{33, 332{333), Fletcher et al.
(1962,xx12.1, 12.2). (4.23.10){(4.23.18) and
also Table 4.23.1 follow from the de nitions in
xx4.23(i), 4.23(ii). To verify (4.23.19), denote
the right-hand side by (z), and the domain
Cn( 1; 1][[1;1) byD. Ifz=x2( 1;1),
then 0(x) = (1 x2) 1=2and (0) = 0. Hence
(4.23.19) applies; compare (4.23.1) with Arcsin re-
placed by arcsin. We may now extend (4.23.19) to
the rest ofDsimply by showing that (z) is ana-
lytic onD; comparex1.10(ii). Since the principal
value of (1 z2)1=2is analytic on D, the only possi-
ble singularities of (z) occur on the branch cut of
the logarithm, that is, when (1 z2)1=2= iz t
witht2[0;1). By squaring the last equation we
see that (1 z2)1=2+izis real only when zlies on
the imaginary axis, and it is then positive. The
proofs of (4.23.22), (4.23.23), (4.23.26) are simi-
lar, or in the case of (4.23.22) we may simply refer
to (4.23.16). (4.23.40) and (4.23.42) may be ver-
i ed by di erentiation plus comparison of values
asx!0.
x4.24 Hobson (1928, pp. 54{55, 279{280, 321), Levin-
son and Redhe er (1970, pp. 68{70). For (4.24.10)
and (4.24.11) note that the principal value of
(z2 1)1=2is discontinuous on the imaginary axis,
hence we switch to the other branch when crossing
this axis. This accounts for the two signs.
x4.25 Jones and Thron (1980, pp. 202{203), Wall
(1948, pp. 343{349).
x4.26 (4.26.1){(4.26.8) and (4.26.14){(4.26.21) may be
veri ed by di erentiation. For (4.26.12) and
(4.26.13) see Copson (1935, pp. 137 and 227).
x4.28 Hobson (1928, pp. 322{326), Levinson and Red-
he er (1970, pp. 56{57).x4.29 These graphics were produced at NIST.
x4.30 Hobson (1928, pp. 323{326).
x4.31 Hobson (1928, p. 326), Levinson and Redhe er
(1970, p. 61).
x4.35 Hobson (1928, pp. 323{325, 331).
x4.36 For (4.36.1){(4.36.5) replace zbyizin (4.22.1){
(4.22.5) and apply (4.28.8){(4.28.13).
x4.37 Levinson and Redhe er (1970, pp. 68{69).
(4.37.11) follows from (4.37.19). The equations
inx4.37(iv) may be veri ed in a similar manner
to those ofx4.23(iv). The only new feature is that
in (4.37.19) the principal value of ( z2 1)1=2is
discontinuous on the imaginary axis, hence to con-
tinue (z2 1)1=2analytically we switch to the other
branch. This accounts for the sign in (4.37.19).
x4.38 For (4.38.1) expand (1 + z2) 1=2by the bino-
mial theorem and integrate term by term. For
(4.38.2), write (1+ z2) 1=2=z 1(1+(1=z2)) 1=2,
<z >0, and then expand and integrate. To nd
the constant of integration note that for large z,
arcsinhzbehaves like ln(2 z) and the constant is
ln 2. (4.38.3) is proved similarly. (4.38.4){(4.38.7)
follow from the corresponding series for inverse
trigonometric functions in x4.24. For (4.38.9){
(4.38.14) take the derivatives of the logarithmic
forms of inverse hyperbolic functions in x4.37(iv).
For (4.38.15){(4.38.19) use similar analysis to that
forx4.24(iii).
x4.40 (4.40.1){(4.40.6) and (4.40.11){(4.40.16) may be
veri ed by di erentiation. For (4.40.7){(4.40.10)
see Copson (1935, p. 155).
x4.42 Hobson (1928, p. 18 and Chapter 10).
x4.43 Hobson (1928, p. 335).
Chapter 5
Gamma Function
R. A. Askey1and R. Roy2
Notation 136
5.1 Special Notation . . . . . . . . . . . . . 136
Properties 136
5.2 De nitions . . . . . . . . . . . . . . . . . 136
5.3 Graphics . . . . . . . . . . . . . . . . . . 136
5.4 Special Values and Extrema . . . . . . . 137
5.5 Functional Relations . . . . . . . . . . . 138
5.6 Inequalities . . . . . . . . . . . . . . . . 138
5.7 Series Expansions . . . . . . . . . . . . . 139
5.8 In nite Products . . . . . . . . . . . . . 139
5.9 Integral Representations . . . . . . . . . 139
5.10 Continued Fractions . . . . . . . . . . . . 140
5.11 Asymptotic Expansions . . . . . . . . . . 140
5.12 Beta Function . . . . . . . . . . . . . . . 142
5.13 Integrals . . . . . . . . . . . . . . . . . . 143
5.14 Multidimensional Integrals . . . . . . . . 1435.15 Polygamma Functions . . . . . . . . . . . 144
5.16 Sums . . . . . . . . . . . . . . . . . . . 144
5.17 Barnes' G-Function (Double Gamma
Function) . . . . . . . . . . . . . . . . . 144
5.18q-Gamma and Beta Functions . . . . . . 145
Applications 145
5.19 Mathematical Applications . . . . . . . . 145
5.20 Physical Applications . . . . . . . . . . . 145
Computation 146
5.21 Methods of Computation . . . . . . . . . 146
5.22 Tables . . . . . . . . . . . . . . . . . . . 146
5.23 Approximations . . . . . . . . . . . . . . 146
5.24 Software . . . . . . . . . . . . . . . . . . 147
References 147
1Department of Mathematics, University of Wisconsin, Madison, Wisconsin.
2Department of Mathematics and Computer Science, Beloit College, Beloit, Wisconsin.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 6) by P. J. Davis.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
135
136 Gamma Function
Notation
5.1 Special Notation
(For other notation see pp. xiv and 873.)
j;m;n nonnegative integers.
k nonnegative integer, except in x5.20.
x;y real variables.
z=x+iycomplex variable.
a;b;q;s;w real or complex variables with jqj<1.
arbitrary small positive constant.
Euler's constant ( x5.2(ii)).
primes derivatives with respect to the variable.
The main functions treated in this chapter are the
gamma function ( z), the psi function (or digamma
function) (z), the beta function B( a;b), and the q-
gamma function q(z).
The notation ( z) is due to Legendre. Alternative
notations for this function are: ( z 1) (Gauss) and
(z 1)!. Alternative notations for the psi function are:
(z 1) (Gauss) Jahnke and Emde (1945); ( z) Davis
(1933); F(z 1) Pairman (1919).
Properties
5.2 De nitions
5.2(i) Gamma and Psi Functions
Euler's Integral
5.2.1 (z) =Z1
0e ttz 1dt,<z>0.
When<z0, (z) is de ned by analytic continuation.
It is a meromorphic function with no zeros, and with
simple poles of residue ( 1)n=n! atz= n. 1= (z) is
entire, with simple zeros at z= n.
5.2.2 (z) = 0(z)= (z),z6= 0; 1; 2;:::.
(z) is meromorphic with simple poles of residue 1 at
z= n.
5.2(ii) Euler's Constant
5.2.3
= lim
n!1
1 +1
2+1
3++1
n lnn
= 0:57721 56649 01532 86060 ::::
5.2(iii) Pochhammer's Symbol
5.2.4 (a)0= 1;(a)n=a(a+ 1)(a+ 2)(a+n 1);
5.2.5 (a)n= (a+n)= (a),a6= 0; 1; 2;:::.5.3 Graphics
5.3(i) Real Argument
Figure 5.3.1 : (x) and 1= (x).x0= 1:46:::, (x0) =
0:88:::; seex5.4(iii).
2 4 6x
/H5008124
0
Figure 5.3.2 : ln (x). This function is convex on (0 ;1);
comparex5.5(iv).
Figure 5.3.3 : (x).
5.3(ii) Complex Argument
In the graphics shown in this subsection, both the height
and color correspond to the absolute value of the func-
tion. See also p. xiv.
5.4 Special Values and Extrema 137
Figure 5.3.4 :j (x+iy)j.
Figure 5.3.5 : 1=j (x+iy)j.
Figure 5.3.6 :j (x+iy)j.
5.4 Special Values and Extrema
5.4(i) Gamma Function
5.4.1 (1) = 1; n! = (n+ 1):5.4.2n!! =(
21
2n 1
2n+ 1
; n even;
1
221
2n+1
2 1
2n+ 1
; n odd:
(The second line of Formula (5.4.2) also applies when
n= 1.)
5.4.3j (iy)j=
ysinh(y)1=2
;
5.4.4 1
2+iy
1
2 iy
= 1
2+iy 2=
cosh(y);
5.4.5 1
4+iy
3
4 iy
=p
2
cosh(y) +isinh(y):
5.4.6 1
2
=1=2
= 1:77245 38509 05516 02729 :::;
5.4.7 1
3
= 2:67893 85347 07747 63365 :::;
5.4.8 2
3
= 1:35411 79394 26400 41694 :::;
5.4.9 1
4
= 3:62560 99082 21908 31193 :::;
5.4.10 3
4
= 1:22541 67024 65177 64512 ::::
5.4.11 0(1) =
:
5.4(ii) Psi Function
5.4.12 (1) =
; 0(1) =1
62;
5.4.13 1
2
=
2 ln 2; 0 1
2
=1
22:
For higher derivatives of (z) atz= 1 andz=1
2, see
x5.15.
5.4.14 (n+ 1) =nX
k=11
k
;
5.4.15
n+1
2
=
2 ln 2 + 2
1 +1
3++1
2n 1
,
n= 1;2;::::
5.4.16= (iy) =1
2y+
2coth(y);
5.4.17= 1
2+iy
=
2tanh(y);
5.4.18= (1 +iy) = 1
2y+
2coth(y):
Ifp;qare integers with 0 <p<q , then
5.4.19
p
q
=
lnq
2cotp
q
+1
2q 1X
k=1cos2kp
q
ln
2 2 cos2k
q
:
138 Gamma Function
5.4(iii) Extrema
Table 5.4.1 : 0(xn) = (xn) = 0.
n x n (xn)
0 1:46163 21449 0 :88560 31944
1 0:50408 30083 3:54464 36112
2 1:57349 84732 2 :30240 72583
3 2:61072 08875 0:88813 63584
4 3:63529 33665 0 :24512 75398
5 4:65323 77626 0:05277 96396
6 5:66716 24513 0 :00932 45945
7 6:67841 82649 0:00139 73966
8 7:68778 83250 0 :00018 18784
9 8:69576 41633 0:00002 09253
10 9:70267 25406 0 :00000 21574
Compare Figure 5.3.1.
Asn!1 ,
5.4.20xn= n+1
arctan
lnn
+O1
n(lnn)2
:
For error bounds for this estimate see Walker (2007,
Theorem 5).
5.5 Functional Relations
5.5(i) Recurrence
5.5.1 (z+ 1) =z (z);
5.5.2 (z+ 1) = (z) +1
z:
5.5(ii) Re
ection
5.5.3 (z) (1 z) ==sin(z),z6= 0;1;:::,
5.5.4 (z) (1 z) = =tan(z),z6= 0;1;:::.
5.5(iii) Multiplication
Duplication Formula
For 2z6= 0; 1; 2;:::,
5.5.5 (2z) = 1=222z 1 (z)
z+1
2
:Gauss's Multiplication Formula
Fornz6= 0; 1; 2;:::,
5.5.6 (nz) = (2)(1 n)=2nnz (1=2)n 1Y
k=0
z+k
n
:
5.5.7n 1Y
k=1 k
n
= (2)(n 1)=2n 1=2:
5.5.8 (2z) =1
2
(z) +
z+1
2
+ ln 2;
5.5.9 (nz) =1
nn 1X
k=0
z+k
n
+ lnn:
5.5(iv) Bohr{Mollerup Theorem
If a positive function f(x) on (0;1) satis esf(x+ 1) =
xf(x),f(1) = 1, and ln f(x) is convex (seex1.4(viii)),
thenf(x) = (x).
5.6 Inequalities
5.6(i) Real Variables
Throughout this subsection x>0.
5.6.1 1<(2) 1=2x(1=2) xex (x)<e1=(12x);
5.6.21
(x)+1
(1=x)2;
5.6.31
( (x))2+1
( (1=x))22;
Gautschi's Inequality
5.6.4 x1 s< (x+ 1)
(x+s)<(x+ 1)1 s, 0<s< 1:
5.6.5exp
(1 s)
x+s1=2
(x+ 1)
(x+s)exp
(1 s)
x+1
2(s+ 1)
,
0<s< 1.
5.6(ii) Complex Variables
5.6.6j (x+iy)jj (x)j;
5.6.7j (x+iy)j(sech(y))1=2 (x),x1
2.
Forb a1,a0, andz=x+iywithx>0,
5.6.8 (z+a)
(z+b) 1
jzjb a:
Forx0,
5.6.9j (z)j(2)1=2jzjx (1=2)e jyj=2exp 1
6jzj 1
:
5.7 Series Expansions 139
5.7 Series Expansions
5.7(i) Maclaurin and Taylor Series
Throughout this subsection (k) is as in Chapter 25.
5.7.11
(z)=1X
k=1ckzk;
wherec1= 1,c2=
, and
5.7.2(k 1)ck=
ck 1 (2)ck 2+(3)ck 3
+ ( 1)k(k 1)c1, k3.
For 15D numerical values of cksee Abramowitz and
Stegun (1964, p. 256), and for 31D values see Wrench
(1968).
5.7.3ln (1 +z) = ln(1 +z)
+z(1
) +1X
k=2( 1)k((k) 1)zk
k,
jzj<2.
5.7.4 (1 +z) =
+1X
k=2( 1)k(k)zk 1,jzj<1,
5.7.5 (1 +z) =1
2z
2cot(z) +1
z2 1+ 1
1X
k=1((2k+ 1) 1)z2k,
jzj<2,z6= 0;1.
For 20D numerical values of the coecients of the
Maclaurin series for ( z+ 3) see Luke (1969b, p. 299).
5.7(ii) Other Series
Whenz6= 0; 1; 2;:::,
5.7.6 (z) =
1
z+1X
k=1z
k(k+z)
=
+1X
k=01
k+ 1 1
k+z
;
and
5.7.7 z+ 1
2
z
2
= 21X
k=0( 1)k
k+z:
Also,
5.7.8= (1 +iy) =1X
k=1y
k2+y2:
5.8 In nite Products
5.8.1
(z) = lim
k!1k!kz
z(z+ 1)(z+k),z6= 0; 1; 2;:::;5.8.21
(z)=ze
z1Y
k=1
1 +z
k
e z=k;
5.8.3 (x)
(x+iy) 2
=1Y
k=0
1 +y2
(x+k)2
,x6= 0; 1;::::
If
5.8.4mX
k=1ak=mX
k=1bk;
then
5.8.51Y
k=0(a1+k)(a2+k)(am+k)
(b1+k)(b2+k)(bm+k)
= (b1) (b2) (bm)
(a1) (a2) (am);
provided that none of the bkis zero or a negative integer.
5.9 Integral Representations
5.9(i) Gamma Function
5.9.11
1
z==Z1
0exp( zt)t 1dt;
< > 0, > 0, and<z > 0. (The fractional powers
have their principal values.)
Hankel's Loop Integral
5.9.21
(z)=1
2iZ(0+)
1ett zdt;
where the contour begins at 1, circles the origin once
in the positive direction, and returns to 1.t zhas
its principal value where tcrosses the positive real axis,
and is continuous. See Figure 5.9.1.
0
Figure 5.9.1 :t-plane. Contour for Hankel's loop inte-
gral.
5.9.3c z (z) =Z1
1jtj2z 1e ct2dt,c>0,<z>0,
where the path is the real axis.
5.9.4 (z) =Z1
1tz 1e tdt+1X
k=0( 1)k
(z+k)k!,
z6= 0; 1; 2;:::.
140 Gamma Function
5.9.5 (z) =Z1
0tz 1
e t nX
k=0( 1)ktk
k!!
dt,
n 1<<z< n.
5.9.6 (z) cos 1
2z
=Z1
0tz 1costdt, 0<<z<1,
5.9.7 (z) sin 1
2z
=Z1
0tz 1sintdt, 1<<z<1.
5.9.8
1 +1
n
cos
2n
=Z1
0cos(tn)dt,n= 2;3;4;:::,
5.9.9
1 +1
n
sin
2n
=Z1
0sin(tn)dt,n= 2;3;4;::::
Binet's Formula
5.9.10ln (z) =
z 1
2
lnz z+1
2ln(2)
+ 2Z1
0arctan(t=z)
e2t 1dt;
wherejphzj<= 2 and the inverse tangent has its prin-
cipal value.
5.9.11
ln (z+ 1) =
z 1
2iZ c+1i
c 1iz s
ssin(s)( s)ds;
wherejphzj (<), 1<c< 2, and(s) is as in
Chapter 25.
For additional representations see Whittaker and
Watson (1927,xx12.31{12.32).
5.9(ii) Psi Function, Euler's Constant, and
Derivatives
For<z>0,
5.9.12 (z) =Z1
0e t
t e zt
1 e t
dt;
5.9.13 (z) = lnz+Z1
01
t 1
1 e t
e tzdt;
5.9.14 (z) =Z1
0
e t 1
(1 +t)zdt
t;
5.9.15 (z) = lnz 1
2z 2Z1
0tdt
(t2+z2)(e2t 1):
5.9.16 (z) +
=Z1
0e t e zt
1 e tdt=Z1
01 tz 1
1 tdt:
5.9.17 (z+ 1) =
+1
2iZ c+1i
c 1iz s 1
sin(s)( s)ds;wherejphzj (<) and 1<c< 2.
5.9.18
= Z1
0e tlntdt=Z1
01
1 +t e tdt
t
=Z1
0(1 e t)dt
t Z1
1e tdt
t
=Z1
0e t
1 e t e t
t
dt:
5.9.19 (n)(z) =Z1
0(lnt)ne ttz 1dt,n0,<z>0.
5.10 Continued Fractions
For<z>0,
5.10.1ln (z) +z
z 1
2
lnz 1
2ln(2)
=a0
z+a1
z+a2
z+a3
z+a4
z+a5
z+;
where
5.10.2
a0=1
12; a 1=1
30; a 2=53
210; a 3=195
371;
a4=22999
22737; a 5=299 44523
197 33142; a 6=10 95352 41009
4 82642 75462:
For exact values of a7toa11and 40S values of a0
toa40, see Char (1980). Also see Cuyt et al. (2008,
pp. 223{228), Jones and Thron (1980, pp. 348{350),
and Lorentzen and Waadeland (1992, pp. 221{224) for
further information.
5.11 Asymptotic Expansions
5.11(i) Poincar e-Type Expansions
Asz!1 in the sectorjphzj (<),
5.11.1
ln (z)
z 1
2
lnz z+1
2ln(2) +1X
k=1B2k
2k(2k 1)z2k 1
and
5.11.2 (z)lnz 1
2z 1X
k=1B2k
2kz2k:
For the Bernoulli numbers B2k, seex24.2(i).
With the same conditions,
5.11.3 (z)e zzz2
z1=2 1X
k=0gk
zk!
;
where
5.11.4
g0= 1; g 1=1
12; g 2=1
288; g 3= 139
51840;
g4= 571
24 88320; g 5=1 63879
2090 18880; g 6=52 46819
7 52467 96800:
Also,
5.11.5 gk=p
2 1
2
ka2k;
5.11 Asymptotic Expansions 141
wherea0=1
2p
2 and
5.11.6a0ak+1
2a1ak 1+1
3a2ak 2++1
k+ 1aka0
=1
kak 1, k1.
Wrench (1968) gives exact values of gkup tog20. Spira
(1971) corrects errors in Wrench's results and also sup-
plies exact and 45D values of gkfork= 21;22;:::; 30.
For an asymptotic expansion of gkask!1 see Boyd
(1994).
Terminology
The expansion (5.11.1) is called Stirling's series (Whit-
taker and Watson (1927, x12.33)), whereas the expan-
sion (5.11.3), or sometimes just its leading term, is
known as Stirling's formula (Abramowitz and Stegun
(1964,x6.1), Olver (1997b, p. 88)).
Next, and again with the same conditions,
5.11.7 (az+b)p
2e az(az)az+b (1=2);
wherea(>0) andb(2C) are both xed, and
5.11.8ln (z+h)
z+h 1
2
lnz z+1
2ln(2)
+1X
k=2( 1)kBk(h)
k(k 1)zk 1;
whereh(2[0;1]) is xed, and Bk(h) is the Bernoulli
polynomial de ned in x24.2(i).
Lastly, asy!1 ,
5.11.9j (x+iy)jp
2jyjx (1=2)e jyj=2;
uniformly for bounded real values of x.
5.11(ii) Error Bounds and Exponential
Improvement
If the sums in the expansions (5.11.1) and (5.11.2) are
terminated at k=n 1 (k0) andzis real and pos-
itive, then the remainder terms are bounded in mag-
nitude by the rst neglected terms and have the same
sign. Ifzis complex, then the remainder terms are
bounded in magnitude by sec2n 1
2phz
for (5.11.1), and
sec2n+1 1
2phz
for (5.11.2), times the rst neglected
terms.
For the remainder term in (5.11.3) write
5.11.10 (z) =e zzz2
z1=2 K 1X
k=0gk
zk+RK(z)!
,
K= 1;2;3;:::.
Then
5.11.11
jRK(z)j(1 +(K)) (K)
2(2)K+1jzjK
1+min(sec(ph z);2K1
2)
,
jphzj1
2,where(K) is as in Chapter 25. For this result and a
similar bound for the sector1
2phzsee Boyd
(1994).
For further information see Olver (1997b, pp. 293{
295), and for other error bounds see Whittaker and Wat-
son (1927,x12.33), Spira (1971), and Sch afke and Fin-
sterer (1990).
For re-expansions of the remainder terms in (5.11.1)
and (5.11.3) in series of incomplete gamma functions
with exponential improvement ( x2.11(iii)) in the asymp-
totic expansions, see Berry (1991), Boyd (1994), and
Paris and Kaminski (2001, x6.4).
5.11(iii) Ratios
In this subsection a,b, andcare real or complex con-
stants.
Ifz!1 in the sectorjphzj (<), then
5.11.12 (z+a)
(z+b)za b;
5.11.13 (z+a)
(z+b)za b1X
k=0Gk(a;b)
zk:
Also, with the added condition <(b a)>0,
5.11.14
(z+a)
(z+b)
z+a+b 1
2a b1X
k=0Hk(a;b)
z+1
2(a+b 1)2k:
Here
5.11.15
G0(a;b) = 1; G 1(a;b) =1
2(a b)(a+b 1);
G2(a;b) =1
12a b
2
(3(a+b 1)2 (a b+ 1));
5.11.16
H0(a;b) = 1; H 1(a;b) = 1
12a b
2
(a b+ 1);
H2(a;b) =1
240a b
4
(2(a b+ 1) + 5(a b+ 1)2):
In terms of generalized Bernoulli polynomials B(`)
n(x)
(x24.16(i)), we have for k= 0;1;:::;
5.11.17Gk(a;b) =a b
k
B(a b+1)
k(a);
5.11.18Hk(a;b) =a b
2k
B(a b+1)
2ka b+ 1
2
:
Lastly, and again if z!1 in the sectorjphzj
142 Gamma Function
(<), then
5.11.19
(z+a) (z+b)
(z+c)
1X
k=0( 1)k(c a)k(c b)k
k! (a+b c+z k):
For the error term in (5.11.19) in the case z=x(>0)
andc= 1, see Olver (1995).
5.12 Beta Function
In this section all fractional powers have their princi-
pal values, except where noted otherwise. In (5.12.1){
(5.12.4) it is assumed <a>0 and<b>0.
Euler's Beta Integral
5.12.1 B(a;b) =Z1
0ta 1(1 t)b 1dt= (a) (b)
(a+b):
5.12.2Z=2
0sin2a 1cos2b 1d=1
2B(a;b):
5.12.3Z1
0ta 1dt
(1 +t)a+b= B(a;b):
5.12.4Z1
0ta 1(1 t)b 1
(t+z)a+bdt= B(a;b)(1+z) az b,jphzj<.
5.12.5Z=2
0(cost)a 1cos(bt)dt
=
2a1
aB 1
2(a+b+ 1);1
2(a b+ 1),<a>0.
5.12.6Z
0(sint)a 1eibtdt
=
2a 1eib=2
aB 1
2(a+b+ 1);1
2(a b+ 1),
<a>0.
5.12.7Z1
0cosh(2bt)
(cosht)2adt= 4a 1B(a+b;a b),<a>j<bj.
5.12.8
1
2Z1
1dt
(w+it)a(z it)b=(w+z)1 a b
(a+b 1) B(a;b),
<(a+b)>1,<w>0,<z>0:
In (5.12.8) the fractional powers have their principal
values when w > 0 andz > 0, and are continued via
continuity.
5.12.91
2iZc+1i
c 1it a(1 t) 1 bdt=1
bB(a;b),
0<c< 1,<(a+b)>0.
5.12.10
1
2iZ(1+)
0ta 1(t 1)b 1dt=sin(b)
B(a;b),<a>0,with the contour as shown in Figure 5.12.1.
1 0
Figure 5.12.1 :t-plane. Contour for rst loop integral for
the beta function.
In (5.12.11) and (5.12.12) the fractional powers are
continuous on the integration paths and take their prin-
cipal values at the beginning.
5.12.111
e2ia 1Z(0+)
1ta 1(1 +t) a bdt= B(a;b);
when<b >0,ais not an integer and the contour cuts
the real axis between 1 and the origin. See Figure
5.12.2.
/H50081 0
Figure 5.12.2 :t-plane. Contour for second loop integral
for the beta function.
Pochhammer's Integral
Whena;b2C
5.12.12Z(1+;0+;1 ;0 )
Pta 1(1 t)b 1dt
= 4ei(a+b)sin(a) sin(b) B(a;b);
where the contour starts from an arbitrary point Pin
the interval (0 ;1), circles 1 and then 0 in the positive
sense, circles 1 and then 0 in the negative sense, and re-
turns toP. It can always be deformed into the contour
shown in Figure 5.12.3.
Figure 5.12.3 :t-plane. Contour for Pochhammer's inte-
gral.
5.13 Integrals 143
5.13 Integrals
In (5.13.1) the integration path is a straight line parallel to the imaginary axis.
5.13.11
2iZc+i1
c i1 (s+a) (b s)z sds= (a+b)za
(1 +z)a+b,<(a+b)>0, <a<c<<b,jphzj<.
5.13.21
2Z1
1j (a+it)j2e(2b )tdt= (2a)
(2 sinb)2a, a>0, 0<b< .
Barnes' Beta Integral
5.13.31
2Z1
1 (a+it) (b+it) (c it) (d it)dt= (a+c) (a+d) (b+c) (b+d)
(a+b+c+d),<a;<b;<c;<d>0.
Ramanujan's Beta Integral
5.13.4Z1
1dt
(a+t) (b+t) (c t) (d t)= (a+b+c+d 3)
(a+c 1) (a+d 1) (b+c 1) (b+d 1),<(a+b+c+d)>3.
de Branges{Wilson Beta Integral
5.13.51
4Z1
1Q4
k=1 (ak+it) (ak it)
(2it) ( 2it)dt=Q
1j<k4 (aj+ak)
(a1+a2+a3+a4),<(ak)>0,k= 1;2;3;4.
For compendia of integrals of gamma functions see Apelblat (1983, pp. 124{127 and 129{130), Erd elyi et al.
(1954a,b), Gradshteyn and Ryzhik (2000, pp. 644{652), Oberhettinger (1974, pp. 191{204), Oberhettinger and Badii
(1973, pp. 307{316), Prudnikov et al. (1986b, pp. 57{64), Prudnikov et al. (1992a, pp. 127{130), and Prudnikov et al.
(1992b, pp. 113{123).
5.14 Multidimensional Integrals
LetVnbe the simplex: t1+t2++tn1,tk0. Then for<zk>0,k= 1;2;:::;n + 1,
5.14.1Z
Vntz1 1
1tz2 1
2tzn 1
ndt1dt2dtn= (z1) (z2) (zn)
(1 +z1+z2++zn);
5.14.2Z
Vn
1 nX
k=1tk!zn+1 1nY
k=1tzk 1
kdtk= (z1) (z2) (zn+1)
(z1+z2++zn+1):
Selberg-type Integrals
Let
5.14.3 (t1;t2;:::;tn) =Y
1j<kn(tj tk):
Then
5.14.4Z
[0;1]nt1t2tmj(t1;:::;tn)j2cnY
k=1ta 1
k(1 tk)b 1dtk=1
( (1 +c))nmY
k=1a+ (n k)c
a+b+ (2n k 1)c
nY
k=1 (a+ (n k)c) (b+ (n k)c) (1 +kc)
(a+b+ (2n k 1)c);
provided that<a,<b>0,<c> min(1=n;<a=(n 1);<b=(n 1)).
Secondly,
5.14.5Z
[0;1)nt1t2tmj(t1;:::;tn)j2cnY
k=1ta 1
ke tkdtk=mY
k=1(a+ (n k)c)Qn
k=1 (a+ (n k)c) (1 +kc)
( (1 +c))n;
when<a>0,<c> min(1=n;<a=(n 1)).
Thirdly,
5.14.61
(2)n=2Z
( 1;1)nj(t1;:::;tn)j2cnY
k=1exp
1
2t2
k
dtk=Qn
k=1 (1 +kc)
( (1 +c))n,<c> 1=n.
144 Gamma Function
Dyson's Integral
5.14.71
(2)nZ
[ ;]nY
1j<knjeij eikj2bd1dn= (1 +bn)
( (1 +b))n,<b> 1=n.
5.15 Polygamma Functions
The functions (n)(z),n= 1;2;:::, are called the
polygamma functions . In particular, 0(z) is the
trigamma function ; 00, (3), (4)are the tetra-, penta-,
andhexagamma functions respectively. Most properties
of these functions follow straightforwardly by di eren-
tiation of properties of the psi function. This includes
asymptotic expansions: compare xx2.1(ii){2.1(iii).
In (5.15.2){(5.15.7) n;m = 1;2;3;:::, and for
(n+ 1) seex25.6(i).
5.15.1 0(z) =1X
k=01
(k+z)2,z6= 0; 1; 2;:::,
5.15.2 (n)(1) = ( 1)n+1n!(n+ 1);
5.15.3 (n) 1
2
= ( 1)n+1n!(2n+1 1)(n+ 1);
5.15.4 0
n 1
2
=1
22 4n 1X
k=11
(2k 1)2;
5.15.5 (n)(z+ 1) = (n)(z) + ( 1)nn!z n 1;
5.15.6
(n)(1 z) + ( 1)n 1 (n)(z) = ( 1)ndn
dzncot(z);
5.15.7 (n)(mz) =1
mn+1m 1X
k=0 (n)
z+k
m
:
Asz!1 injphzj (<)
5.15.8 0(z)1
z+1
2z2+1X
k=1B2k
z2k+1:
ForB2kseex24.2(i).
For continued fractions for 0(z) and 00(z) see Cuyt
et al. (2008, pp. 231{238).
5.16 Sums
5.16.11X
k=1( 1)k 0(k) = 2
8;
5.16.21X
k=11
k 0(k+ 1) =(3) = 1
2 00(1):
For further sums involving the psi function see
Hansen (1975, pp. 360{367). For sums of gamma func-
tions see Andrews et al. (1999, Chapters 2 and 3) and
xx15.2(i), 16.2.For related sums involving nite eld analogs of the
gamma and beta functions (Gauss and Jacobi sums)
see Andrews et al. (1999, Chapter 1) and Terras (1999,
pp. 90, 149).
5.17 Barnes' G-Function (Double Gamma
Function)
5.17.1G(z+ 1) = (z)G(z); G (1) = 1;
5.17.2 G(n) = (n 2)!(n 3)!1!,n= 2;3;:::.
5.17.3G(z+ 1) = (2)z=2exp
1
2z(z+ 1) 1
2
z2
1Y
k=1
1 +z
kk
exp
z+z2
2k
:
5.17.4
LnG(z+ 1) =1
2zln(2) 1
2z(z+ 1)
+zLn (z+ 1) Zz
0Ln (t+ 1)dt:
In this equation (and in (5.17.5) below), the Ln's have
their principal values on the positive real axis and are
continued via continuity, as in x4.2(i).
Whenz!1 injphzj (<),
5.17.5
LnG(z+ 1)1
4z2+z (z+ 1) 1
2z(z+ 1) +1
12
Lnz
lnA+1X
k=1B2k+2
2k(2k+ 1)(2k+ 2)z2k;
see Ferreira and L opez (2001). This reference also pro-
vides bounds for the error term. Here B2k+2is the
Bernoulli number ( x24.2(i)), and AisGlaisher's con-
stant , given by
5.17.6A=eC= 1:28242 71291 00622 63687 :::;
where
5.17.7
C= lim
n!1 nX
k=1klnk 1
2n2+1
2n+1
12
lnn+1
4n2!
=
+ ln(2)
12 0(2)
22=1
12 0( 1);
and0is the derivative of the zeta function (Chapter
25).
For Glaisher's constant see also Greene and Knuth
(1982, p. 100) and x2.10(i).
5.18q-Gamma and Beta Functions 145
5.18q-Gamma and Beta Functions
5.18(i)q-Factorials
5.18.1 (a;q)n=n 1Y
k=0(1 aqk),n= 0;1;2;:::,
5.18.2
n!q= 1(1 +q)(1 +q++qn 1) = (q;q)n(1 q) n:
Whenjqj<1,
5.18.3 (a;q)1=1Y
k=0(1 aqk):
See alsox17.2(i).
5.18(ii)q-Gamma Function
When 0<q< 1,
5.18.4 q(z) = (q;q)1(1 q)1 z=(qz;q)1;
5.18.5 q(1) = q(2) = 1;
5.18.6 n!q= q(n+ 1);
5.18.7 q(z+ 1) =1 qz
1 q q(z):
Also, ln q(x) is convex for x>0, and the analog of the
Bohr-Mollerup theorem ( x5.5(iv)) holds.
If 0<q<r< 1, then
5.18.8 q(x)< r(x);
when 0<x< 1 or whenx>2, and
5.18.9 q(x)> r(x);
when 1<x< 2.
5.18.10 lim
q!1 q(z) = (z):
For generalized asymptotic expansions of ln q(z) as
jzj!1 see Olde Daalhuis (1994) and Moak (1984).
5.18(iii)q-Beta Function
5.18.11 Bq(a;b) = q(a) q(b)
q(a+b):
5.18.12Bq(a;b) =Z1
0ta 1(tq;q)1
(tqb;q)1dqt,
0<q< 1,<a>0,<b>0.
Forq-integrals seex17.2(v).Applications
5.19 Mathematical Applications
5.19(i) Summation of Rational Functions
As shown in Temme (1996a, x3.4), the results given in
x5.7(ii) can be used to sum in nite series of rational
functions.
Example
5.19.1S=1X
k=0ak; ak=k
(3k+ 2)(2k+ 1)(k+ 1):
By decomposition into partial fractions ( x1.2(iii))
5.19.2ak=2
k+2
3 1
k+1
2 1
k+ 1
=1
k+ 1 1
k+1
2
21
k+ 1 1
k+2
3
:
Hence from (5.7.6), (5.4.13), and (5.4.19)
5.19.3S= 1
2
2 2
3
= 3 ln 3 2 ln 2 1
3p
3:
5.19(ii) Mellin{Barnes Integrals
Many special functions f(z) can be represented as a
Mellin{Barnes integral , that is, an integral of a product
of gamma functions, reciprocals of gamma functions,
and a power of z, the integration contour being doubly-
in nite and eventually parallel to the imaginary axis at
both ends. The left-hand side of (5.13.1) is a typical
example. By translating the contour parallel to itself
and summing the residues of the integrand, asymptotic
expansions of f(z) for largejzj, or smalljzj, can be ob-
tained complete with an integral representation of the
error term. For further information and examples see
x2.5 and Paris and Kaminski (2001, Chapters 5, 6, and
8).
5.19(iii)n-Dimensional Sphere
The volume Vand surface area Sof then-dimensional
sphere of radius rare given by
5.19.4V=1
2nrn
(1
2n+ 1); S =21
2nrn 1
(1
2n)=n
rV:
5.20 Physical Applications
Rutherford Scattering
In nonrelativistic quantum mechanics, collisions be-
tween two charged particles are described with the aid of
the Coulomb phase shift ph ( `+ 1 +i); see (33.2.10)
and Clark (1979).
146 Gamma Function
Solvable Models of Statistical Mechanics
Suppose the potential energy of a gas of npoint charges
with positions x1;x2;:::;xnand free to move on the
in nite line 1<x<1, is given by
5.20.1W=1
2nX
`=1x2
` X
1`<jnlnjx` xjj:
The probability density of the positions when the gas is
in thermodynamic equilibrium is:
5.20.2P(x1;:::;xn) =Cexp( W=(kT));
wherekis the Boltzmann constant, Tthe temperature
andCa constant. Then the partition function (with
= 1=(kT)) is given by
5.20.3 n( ) =Z
Rne Wdx
= (2)n=2 (n=2) ( n(n 1)=4)
(
1 +1
2
) nnY
j=1
1 +1
2j
:
See (5.14.6).
Forncharges free to move on a circular wire of ra-
dius 1,
5.20.4 W= X
1`<jnlnjei` eijj;
and the partition function is given by
5.20.5 n( ) =1
(2)nZ
[ ;]ne Wd1dn
=
1 +1
2n
(
1 +1
2
) n:
See (5.14.7).
For further information see Mehta (2004).
Elementary Particles
Veneziano (1968) identi es relationships between par-
ticle scattering amplitudes described by the beta func-
tion, an important early development in string theory.
Carlitz (1972) describes the partition function of dense
hadronic matter in terms of a gamma function.
Computation
5.21 Methods of Computation
An e ective way of computing ( z) in the right half-
plane is backward recurrence, beginning with a value
generated from the asymptotic expansion (5.11.3). Or
we can use forward recurrence, with an initial value ob-
tained e.g. from (5.7.3). For the left half-plane we can
continue the backward recurrence or make use of the
re
ection formula (5.5.3).Similarly for ln ( z), (z), and the polygamma func-
tions.
Another approach is to apply numerical quadrature
(x3.5) to the integral (5.9.2), using paths of steepest
descent for the contour. See Schmelzer and Trefethen
(2007).
For a comprehensive survey see van der Laan and
Temme (1984, Chapter III). See also Borwein and
Zucker (1992).
5.22 Tables
5.22(i) Introduction
For early tables for both real and complex variables see
Fletcher et al. (1962), Lebedev and Fedorova (1960),
and Luke (1975, p. 21).
5.22(ii) Real Variables
Abramowitz and Stegun (1964, Chapter 6) tabulates
(x), ln (x), (x), and 0(x) forx= 1(:005)2 to
10D; 00(x) and (3)(x) forx= 1(:01)2 to 10D; ( n),
1/ (n),
n+1
2
, (n), log10 (n), log10
n+1
3
,
log10
n+1
2
, and log10
n+2
3
forn= 1(1)101
to 8{11S; ( n+ 1) forn= 100(100)1000 to 20S.
Zhang and Jin (1996, pp. 67{69 and 72) tabulates
(x), 1/ (x), ( x), ln (x), (x), ( x), 0(x),
and 0( x) forx= 0(:1)5 to 8D or 8S; ( n+ 1) for
n= 0(1)100(10)250(50)500(100)3000 to 51S.
5.22(iii) Complex Variables
Abramov (1960) tabulates ln ( x+iy) forx= 1 (:01)
2,y= 0 (:01) 4 to 6D. Abramowitz and Stegun (1964,
Chapter 6) tabulates ln ( x+iy) forx= 1 (:1) 2,
y= 0 (:1) 10 to 12D. This reference also includes
(x+iy) for the same arguments to 5D. Zhang and Jin
(1996, pp. 70, 71, and 73) tabulates the real and imagi-
nary parts of ( x+iy), ln (x+iy), and (x+iy) for
x= 0:5;1;5;10,y= 0(:5)10 to 8S.
5.23 Approximations
5.23(i) Rational Approximations
Cody and Hillstrom (1967) gives minimax rational ap-
proximations for ln ( x) for the ranges 0 :5x1:5,
1:5x4, 4x12; precision is variable.
Hart et al. (1968) gives minimax polynomial and ratio-
nal approximations to ( x) and ln (x) in the intervals
0x1, 8x1000, 12x1000; precision
is variable. Cody et al. (1973) gives minimax rational
approximations for (x) for the ranges 0 :5x3 and
3x<1; precision is variable.
For additional approximations see Hart et al. (1968,
Appendix B), Luke (1975, pp. 22{23), and Weniger
(2003).
5.24 Software 147
5.23(ii) Expansions in Chebyshev Series
Luke (1969b) gives the coecients to 20D for the
Chebyshev-series expansions of (1 + x), 1= (1 +x),
(x+ 3), ln (x+ 3), (x+ 3), and the rst six deriva-
tives of (x+ 3) for 0x1. These coecients are
reproduced in Luke (1975). Clenshaw (1962) also gives
20D Chebyshev-series coecients for (1 + x) and its
reciprocal for 0x1. See Luke (1975, pp. 22{23)
for additional expansions.
5.23(iii) Approximations in the Complex Plane
See Schmelzer and Trefethen (2007) for a survey of ra-
tional approximations to various scaled versions of ( z).
For rational approximations to (z) +
see Luke
(1975, pp. 13{16).
5.24 Software
Seehttp://dlmf.nist.gov/5.24 .
References
General References
The main references used in writing this chapter are
Andrews et al. (1999), Carlson (1977b), Erd elyi et al.
(1953a), Nielsen (1906a), Olver (1997b), Paris and
Kaminski (2001), Temme (1996a), and Whittaker and
Watson (1927) .
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x5.2Olver (1997b, Chapter 2, xx1 and 2), Temme
(1996a, Chapters 1 and 3).
x5.3These graphics were computed at NIST.
x5.4Olver (1997b, Chapter 2, xx1 and 2), Andrews
et al. (1999,x1.2). For (5.4.2) use (5.4.6)
and (5.5.1). For (5.4.20) use (5.11.2) to solve
(1 x) =cot(x) withx= n+uandn
large.
x5.5Olver (1997b, Chapter 2, xx1 and 2), Temme
(1996a, Chapter 3), Andrews et al. (1999,x1.9).
(5.5.9) follows from (5.5.6).x5.6Gautschi (1959b, 1974), Alzer (1997a), Laforgia
(1984), Kershaw (1983), Lorch (2002), Carlson
(1977b,x3.10), Paris and Kaminski (2001, x2.1.3).
For (5.6.1) seex5.11(ii).
x5.7Wrench (1968) (errors on p. 621 are corrected
here), Erd elyi et al. (1953a,x1.17), Olver (1997b,
Chapter 2,x2), Temme (1996a, x3.4). (5.7.7) fol-
lows from (5.7.6) and (5.5.2).
x5.8Olver (1997b, Chapter 2, x1), Whittaker and Wat-
son (1927,x12.13).
x5.9Olver (1997b, Chapter 2, xx1 and 2), Temme
(1996a, Chapter 3), Whittaker and Watson (1927,
xx12.3{12.32 andx13.6). (5.9.3) follows from
(5.2.1) by a change of variables. (5.9.8) and (5.9.9)
follow from (5.9.6) and (5.9.7). (5.9.15) and
(5.9.17) are the di erentiated forms of (5.9.10)
and (5.9.11). (5.9.19) is the di erentiated form
of (5.2.1).
x5.10 Wall (1948, Chapter 19).
x5.11 Olver (1997b, Chapter 3, x8, Chapter 4,x5, and
Chapter 8,x4), Temme (1996a, x3.6.2), Paris and
Kaminski (2001,x2.2.5). (5.11.7) and (5.11.9) are
derived from (5.11.3).
x5.12 Carlson (1977b,x4.2 and p. 70), Nielsen (1906a,
x64), Temme (1996a, x3.8: an error in Ex.3.13 is
corrected here), Olver (1997b, p. 38). (5.12.11)
follows from (5.12.3).
x5.13 Paris and Kaminski (2001, x3.3.4), Titchmarsh
(1986a, pp. 188 and 194), Andrews et al. (1999,
x3.6).
x5.14 Andrews et al. (1999,xx1.8, 8.1{8.3, and 8.7),
Mehta (2004, pp. 224{227).
x5.16 Jordan (1939, pp. 344{345).
x5.17 Whittaker and Watson (1927, p. 264), Olver
(1997b, Chapter 8, x3.3), and the di erentiated
form of (25.4.1).
x5.18 Andrews et al. (1999,xx10.1{10.3).
x5.19 Stein and Shakarchi (2003, pp. 208{209) and
Robnik (1980). The formula for Vcan also be
veri ed by setting tk= (xk=r)2andzk=1
2;k=
1;2;:::;n , in (5.14.1). The formula for Scan be
veri ed in a similar way from (5.14.2), or derived
by di erentiating the formula for V.
x5.20 Andrews et al. (1999,x8.2), Mehta (2004, Chap-
ters 4 and 11).
Chapter 6
Exponential, Logarithmic, Sine, and Cosine Integrals
N. M. Temme1
Notation 150
6.1 Special Notation . . . . . . . . . . . . . 150
Properties 150
6.2 De nitions and Interrelations . . . . . . . 150
6.3 Graphics . . . . . . . . . . . . . . . . . . 151
6.4 Analytic Continuation . . . . . . . . . . . 151
6.5 Further Interrelations . . . . . . . . . . . 151
6.6 Power Series . . . . . . . . . . . . . . . . 151
6.7 Integral Representations . . . . . . . . . 152
6.8 Inequalities . . . . . . . . . . . . . . . . 152
6.9 Continued Fraction . . . . . . . . . . . . 153
6.10 Other Series Expansions . . . . . . . . . 153
6.11 Relations to Other Functions . . . . . . . 153
6.12 Asymptotic Expansions . . . . . . . . . . 1536.13 Zeros . . . . . . . . . . . . . . . . . . . 154
6.14 Integrals . . . . . . . . . . . . . . . . . . 154
6.15 Sums . . . . . . . . . . . . . . . . . . . 154
Applications 154
6.16 Mathematical Applications . . . . . . . . 154
6.17 Physical Applications . . . . . . . . . . . 155
Computation 155
6.18 Methods of Computation . . . . . . . . . 155
6.19 Tables . . . . . . . . . . . . . . . . . . . 156
6.20 Approximations . . . . . . . . . . . . . . 156
6.21 Software . . . . . . . . . . . . . . . . . . 157
References 157
1Centrum voor Wiskunde en Informatica, Department MAS, Amsterdam, The Netherlands.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 5) by Walter Gautschi and William
F. Cahill. Walter Gautschi provided the author with a list of references and comments collected since the original publication.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
149
150 Exponential, Logarithmic, Sine, and Cosine Integrals
Notation
6.1 Special Notation
(For other notation see pp. xiv and 873.)
xreal variable.
zcomplex variable.
nnonnegative integer.
arbitrary small positive constant.
Euler's constant ( x5.2(ii)).
Unless otherwise noted, primes indicate derivatives
with respect to the argument.
The main functions treated in this chapter are the
exponential integrals Ei( x),E1(z), and Ein(z); the log-
arithmic integral li( x); the sine integrals Si( z) and si(z);
the cosine integrals Ci( z) and Cin(z).
Properties
6.2 De nitions and Interrelations
6.2(i) Exponential and Logarithmic Integrals
The principal value of the exponential integral E1(z) is
de ned by
6.2.1 E1(z) =Z1
ze t
tdt, z6= 0;
where the path does not cross the negative real axis or
pass through the origin. As in the case of the logarithm
(x4.2(i)) there is a cut along the interval ( 1;0] and
the principal value is two-valued on ( 1;0).
Unless indicated otherwise , it is assumed throughout
this Handbook that E1(z) assumes its principal value.
This is also true of the functions Ci( z) and Chi(z) de-
ned inx6.2(ii).
6.2.2 E1(z) =e zZ1
0e t
t+zdt,jphzj<:
6.2.3 Ein(z) =Zz
01 e t
tdt:
Ein(z) is sometimes called the complementary exponen-
tial integral . It is entire.
6.2.4 E1(z) = Ein(z) lnz
:
In the next three equations x>0.
6.2.5 Ei(x) = Z1
xe t
tdt=Zx
1et
tdt;
6.2.6 Ei( x) = Z1
xe t
tdt= E1(x);
6.2.7 Ei(x) = Ein(x) + lnx+
:(Ei(x) is unde ned when x= 0, or when xis not real.)
The logarithmic integral is de ned by
6.2.8 li(x) =Zx
0dt
lnt= Ei(lnx),x>1.
The generalized exponential integral Ep(z),p2C,
is treated in Chapter 8.
6.2(ii) Sine and Cosine Integrals
6.2.9 Si(z) =Zz
0sint
tdt:
Si(z) is an odd entire function.
6.2.10 si(z) = Z1
zsint
tdt= Si(z) 1
2:
6.2.11 Ci(z) = Z1
zcost
tdt;
where the path does not cross the negative real axis or
pass through the origin. This is the principal value ;
compare (6.2.1).
6.2.12 Cin(z) =Zz
01 cost
tdt:
Cin(z) is an even entire function.
6.2.13 Ci(z) = Cin(z) + lnz+
:
Values at In nity
6.2.14 lim
x!1Si(x) =1
2; lim
x!1Ci(x) = 0:
Hyperbolic Analogs of the Sine and Cosine Integrals
6.2.15 Shi(z) =Zz
0sinht
tdt;
6.2.16 Chi(z) =
+ lnz+Zz
0cosht 1
tdt:
6.2(iii) Auxiliary Functions
6.2.17 f(z) = Ci(z) sinz si(z) cosz;
6.2.18 g(z) = Ci(z) cosz si(z) sinz:
6.2.19 Si(z) =1
2 f(z) cosz g(z) sinz;
6.2.20 Ci(z) = f(z) sinz g(z) cosz:
6.2.21df(z)
dz= g(z);dg(z)
dz= f(z) 1
z:
6.3 Graphics 151
6.3 Graphics
6.3(i) Real Variable
Figure 6.3.1 : The exponential integrals E1(x) and Ei(x),
0<x2.
Figure 6.3.2 : The sine and cosine integrals Si( x);Ci(x),
0x15.
For a graph of li( x) see Figure 6.16.2.
6.3(ii) Complex Variable
Figure 6.3.3 :jE1(x+iy)j, 4x4, 4y4.
Principal value. There is a cut along the negative real
axis. Also,jE1(z)j!1 logarithmically as z!0.6.4 Analytic Continuation
Analytic continuation of the principal value of E1(z)
yields a multi-valued function with branch points at
z= 0 andz=1. The general value of E1(z) is given
by
6.4.1 E1(z) = Ein(z) Lnz
;
compare (6.2.4) and (4.2.6). Thus
6.4.2 E1
ze2mi
=E1(z) 2mi,m2Z,
and
6.4.3
E1
zei
= Ein( z) lnz
i,jphzj.
The general values of the other functions are de ned
in a similar manner, and
6.4.4 Ci
zei
=i+ Ci(z);
6.4.5 Chi
zei
=i+ Chi(z);
6.4.6 f
zei
=eiz f(z);
6.4.7 g
zei
=ieiz+ g(z):
Unless indicated otherwise , in the rest of this chap-
ter and elsewhere in this Handbook the functions E1(z),
Ci(z), Chi(z), f(z), and g(z) assume their principal val-
ues, that is, the branches that are real on the positive
real axis and two-valued on the negative real axis.
6.5 Further Interrelations
Whenx>0,
6.5.1 E1( xi0) = Ei(x)i;
6.5.2 Ei(x) = 1
2(E1( x+i0) +E1( x i0));
6.5.31
2(Ei(x) +E1(x)) = Shi(x) = iSi(ix);
6.5.41
2(Ei(x) E1(x)) = Chi(x) = Ci(ix) 1
2i:
Whenjphzj<1
2,
6.5.5 Si(z) =1
2i(E1( iz) E1(iz)) +1
2;
6.5.6 Ci(z) = 1
2(E1(iz) +E1( iz));
6.5.7 g(z)if(z) =E1(iz)eiz:
6.6 Power Series
6.6.1 Ei(x) =
+ lnx+1X
n=1xn
n!n,x>0.
6.6.2E1(z) =
lnz 1X
n=1( 1)nzn
n!n:
6.6.3E1(z) = lnz+e z1X
n=0zn
n! (n+ 1);
152 Exponential, Logarithmic, Sine, and Cosine Integrals
where denotes the logarithmic derivative of the
gamma function ( x5.2(i)).
6.6.4 Ein(z) =1X
n=1( 1)n 1zn
n!n;
6.6.5 Si(z) =1X
n=0( 1)nz2n+1
(2n+ 1)!(2n+ 1);
6.6.6 Ci(z) =
+ lnz+1X
n=1( 1)nz2n
(2n)!(2n):
The series in this section converge for all nite values of
xandjzj.
6.7 Integral Representations
6.7(i) Exponential Integrals
6.7.1Z1
0e at
t+bdt=Z1
0eiat
t+ibdt=eabE1(ab),a>0,b>0;
6.7.2exZ
0e xt
1 tdt= Ei(x) Ei((1 )x),
0 <1,x>0:
6.7.3Z1
xeit
a2+t2dt=i
2a
eaE1(a ix) e aE1( a ix)
,
a>0,x>0;
6.7.4Z1
xteit
a2+t2dt=1
2
eaE1(a ix) +e aE1( a ix)
,
a>0,x>0:
6.7.5Z1
xe t
a2+t2dt= 1
2ai
eiaE1(x+ia)
e iaE1(x ia)
,
a>0,x2R;
6.7.6Z1
xte t
a2+t2dt=1
2
eiaE1(x+ia) +e iaE1(x ia)
,
a>0,x2R:
6.7.7Z1
0e atsin(bt)
tdt==Ein(a+ib),a;b2R;
6.7.8Z1
0e at(1 cos(bt))
tdt=<Ein(a+ib) Ein(a),
a;b2R:
Many integrals with exponentials and rational func-
tions, for example, integrals of the typeR
ezR(z)dz,
whereR(z) is an arbitrary rational function, can be
represented in nite form in terms of the function E1(z)
and elementary functions; see Lebedev (1965, p. 42).6.7(ii) Sine and Cosine Integrals
Whenz2C
6.7.9 si(z) = Z=2
0e zcostcos(zsint)dt;
6.7.10 Ein(z) Cin(z) =Z=2
0e zcostsin(zsint)dt;
6.7.11Z1
0(1 e at) cos(bt)
tdt=<Ein(a+ib) Cin(b),
a;b2R:
6.7(iii) Auxiliary Functions
6.7.12 g(z) +if(z) =e izZ1
zeit
tdt,jphzj:
The path of integration does not cross the negative real
axis or pass through the origin.
6.7.13 f(z) =Z1
0sint
t+zdt=Z1
0e zt
t2+ 1dt;
6.7.14 g(z) =Z1
0cost
t+zdt=Z1
0te zt
t2+ 1dt:
The rst integrals on the right-hand sides apply when
jphzj< ; the second ones when <z0 and (in the
case of (6.7.14)) z6= 0.
Whenjphzj<
6.7.15 f(z) = 2Z1
0K0
2p
zt
costdt;
6.7.16 g(z) = 2Z1
0K0
2p
zt
sintdt:
ForK0seex10.25(ii).
6.7(iv) Compendia
For collections of integral representations see Bierens de
Haan (1939, pp. 56{59, 72{73, 82{84, 121, 133{136, 155,
179{181, 223, 225{227, 230, 259{260, 374, 377, 397{398,
408, 416, 424, 431, 438{439, 442{444, 488, 496{500,
567{571, 585, 602, 638, 675{677), Corrington (1961),
Erd elyi et al. (1954a, vol. 1, pp. 267{270), Geller and Ng
(1969), Nielsen (1906b), Oberhettinger (1974, pp. 244{
246), Oberhettinger and Badii (1973, pp. 364{371), and
Watrasiewicz (1967).
6.8 Inequalities
In this section x>0.
6.8.11
2ln
1 +2
x
<exE1(x)<ln
1 +1
x
;
6.8.2x
x+ 1<xexE1(x)<x+ 1
x+ 2;
6.8.3x(x+ 3)
x2+ 4x+ 2<xexE1(x)<x2+ 5x+ 2
x2+ 6x+ 6:
6.9 Continued Fraction 153
6.9 Continued Fraction
6.9.1E1(z) =e z
z+1
1 +1
z+2
1 +2
z+3
1 +3
z+,
jphzj<:
See also Cuyt et al. (2008, pp. 287{290).
6.10 Other Series Expansions
6.10(i) Inverse Factorial Series
6.10.1E1(z) =e zc0
z+c1
z(z+ 1)+2!c2
z(z+ 1)(z+ 2)
+3!c3
z(z+ 1)(z+ 2)(z+ 3)+
,
<z>0;
where
6.10.2c0= 1; c 1= 1; c 2=1
2; c 3= 1
3; c 4=1
6;
and
6.10.3 ck= k 1X
j=0cj
k j, k1.
For a more general result (incomplete gamma func-
tion), and also for a result for the logarithmic integral,
see Nielsen (1906a, p. 283: Formula (3) is incorrect).
6.10(ii) Expansions in Series of Spherical Bessel
Functions
For the notation see x10.47(ii).
6.10.4 Si(z) =z1X
n=0
jn 1
2z2;
6.10.5 Cin(z) =1X
n=1an
jn 1
2z2;
6.10.6Ei(x) =
+lnjxj+1X
n=0( 1)n(x an)
i(1)
n 1
2x2
,
x6= 0,
where
6.10.7an= (2n+ 1) (1 ( 1)n+ (n+ 1) (1));
and denotes the logarithmic derivative of the gamma
function (x5.2(i)).
6.10.8
Ein(z) =ze z=2
i(1)
0 1
2z
+1X
n=12n+ 1
n(n+ 1)i(1)
n 1
2z!
:
For (6.10.4){(6.10.8) and further results see Harris
(2000) and Luke (1969b, pp. 56{57). An expansion
forE1(z) can be obtained by combining (6.2.4) and
(6.10.8).6.11 Relations to Other Functions
For the notation see xx8.2(i) and 13.2(i).
Incomplete Gamma Function
6.11.1 E1(z) = (0;z):
Con
uent Hypergeometric Function
6.11.2 E1(z) =e zU(1;1;z);
6.11.3 g(z) +if(z) =U(1;1; iz):
6.12 Asymptotic Expansions
6.12(i) Exponential and Logarithmic Integrals
6.12.1E1(z)e z
z
1 1!
z+2!
z2 3!
z3+
,
z!1 ,jphzj3
2 (<3
2):
Whenjphzj1
2the remainder is bounded in magni-
tude by the rst neglected term, and has the same sign
when phz= 0. When1
2jphzj< the remainder
term is bounded in magnitude by csc( jphzj) times the
rst neglected term. For these and other error bounds
see Olver (1997b, pp. 109{112) with = 0.
For re-expansions of the remainder term leading to
larger sectors of validity, exponential improvement, and
a smooth interpretation of the Stokes phenomenon, see
xx2.11(ii){2.11(iv), with p= 1.
6.12.2
Ei(x)ex
x
1 +1!
x+2!
x2+3!
x3+
,x!+1.
If the expansion is terminated at the nth term, then the
remainder term is bounded by 1 + (n+ 1) times the
next term. For the function seex9.7(i).
The asymptotic expansion of li( x) asx!1 is ob-
tainable from (6.2.8) and (6.12.2).
6.12(ii) Sine and Cosine Integrals
The asymptotic expansions of Si( z) and Ci(z) are given
by (6.2.19), (6.2.20), together with
6.12.3 f(z)1
z
1 2!
z2+4!
z4 6!
z6+
;
6.12.4 g(z)1
z2
1 3!
z2+5!
z4 7!
z6+
;
asz!1 injphzj (<).
The remainder terms are given by
6.12.5 f(z) =1
zn 1X
m=0( 1)m(2m)!
z2m+R(f)
n(z);
6.12.6 g(z) =1
z2n 1X
m=0( 1)m(2m+ 1)!
z2m+R(g)
n(z);
154 Exponential, Logarithmic, Sine, and Cosine Integrals
where, forn= 0;1;2;:::,
6.12.7R(f)
n(z) = ( 1)nZ1
0e ztt2n
t2+ 1dt;
6.12.8R(g)
n(z) = ( 1)nZ1
0e ztt2n+1
t2+ 1dt:
Whenjphzj1
4, these remainders are bounded in
magnitude by the rst neglected terms in (6.12.3) and
(6.12.4), respectively, and have the same signs as these
terms when ph z= 0. When1
4jphzj<1
2the
remainders are bounded in magnitude by csc(2 jphzj)
times the rst neglected terms.
For other phase ranges use (6.4.6) and (6.4.7).
For exponentially-improved asymptotic expansions, use
(6.5.5), (6.5.6), and x6.12(i).
6.13 Zeros
The function Ei( x) has one real zero x0, given by
6.13.1x0= 0:37250 74107 81366 63446 19918 66580 ::::
Ci(x) and si(x) each have an in nite number of pos-
itive real zeros, which are denoted by ck,sk, respec-
tively, arranged in ascending order of absolute value for
k= 0;1;2;:::. Values of c1andc2to 30D are given by
MacLeod (1996).
Ask!1 ,
6.13.2
ck;sk +1
16
31
3+1673
151
5 5 07746
1051
7+;
where =kforck, and = (k+1
2)forsk.
For these results, together with the next three terms
in (6.13.2), see MacLeod (2002a). See also Riekstyn s
(1991, pp. 176{177).
6.14 Integrals
6.14(i) Laplace Transforms
6.14.1Z1
0e atE1(t)dt=1
aln(1 +a),<a> 1;
6.14.2Z1
0e atCi(t)dt= 1
2aln
1 +a2
,<a>0;
6.14.3Z1
0e atsi(t)dt= 1
aarctana,<a>0:
6.14(ii) Other Integrals
6.14.4Z1
0E2
1(t)dt= 2 ln 2;
6.14.5Z1
0costCi(t)dt=Z1
0sintsi(t)dt= 1
4;
6.14.6Z1
0Ci2(t)dt=Z1
0si2(t)dt=1
2;
6.14.7Z1
0Ci(t) si(t)dt= ln 2:6.14(iii) Compendia
For collections of integrals, see Apelblat (1983, pp. 110{
123), Bierens de Haan (1939, pp. 373{374, 409, 479,
571{572, 637, 664{673, 680{682, 685{697), Erd elyi et al.
(1954a, vol. 1, pp. 40{42, 96{98, 177{178, 325), Geller
and Ng (1969), Gradshteyn and Ryzhik (2000, xx5.2{5.3
and 6.2{6.27), Marichev (1983, pp. 182{184), Nielsen
(1906b), Oberhettinger (1974, pp. 139{141), Oberhet-
tinger (1990, pp. 53{55 and 158{160), Oberhettinger
and Badii (1973, pp. 172{179), Prudnikov et al. (1986b,
vol. 2, pp. 24{29 and 64{92), Prudnikov et al. (1992a,
xx3.4{3.6), Prudnikov et al. (1992b,xx3.4{3.6), and Wa-
trasiewicz (1967).
6.15 Sums
6.15.11X
n=1Ci(n) =1
2(ln 2
);
6.15.21X
n=1si(n)
n=1
2(ln 1);
6.15.31X
n=1( 1)nCi(2n) = 1 ln 2 1
2
;
6.15.41X
n=1( 1)nsi(2n)
n=(3
2ln 2 1):
For further sums see Fempl (1960), Hansen (1975,
pp. 423{424), Harris (2000), Prudnikov et al. (1986b,
vol. 2, pp. 649{650), and Slavi c (1974).
Applications
6.16 Mathematical Applications
6.16(i) The Gibbs Phenomenon
Consider the Fourier series
6.16.1sinx+1
3sin(3x) +1
5sin(5x) +
=8
><
>:1
4; 0<x<;
0; x = 0;
1
4; <x< 0:
Thenth partial sum is given by
6.16.2Sn(x) =n 1X
k=0sin((2k+ 1)x)
2k+ 1=1
2Zx
0sin(2nt)
sintdt
=1
2Si(2nx) +Rn(x);
6.17 Physical Applications 155
where
6.16.3Rn(x) =1
2Zx
01
sint 1
t
sin(2nt)dt:
By integration by parts
6.16.4Rn(x) =O
n 1
, n!1 ,
uniformly for x2[ ;]. Hence, if xis xed and
n! 1 , thenSn(x)!1
4, 0, or 1
4according as
0<x< ,x= 0, or <x< 0; compare (6.2.14).
These limits are not approached uniformly, however.
The rst maximum of1
2Si(x) for positive xoccurs at
x=and equals (1 :1789:::)1
4; compare Figure
6.3.2. Hence if x==(2n) andn!1 , then the lim-
iting value of Sn(x) overshoots1
4by approximately
18%. Similarly if x==n, then the limiting value of
Sn(x) undershoots1
4by approximately 10%, and so
on. Compare Figure 6.16.1.
This nonuniformity of convergence is an illustration
of the Gibbs phenomenon . It occurs with Fourier-series
expansions of all piecewise continuous functions. See
Carslaw (1930) for additional graphs and information.
Figure 6.16.1 : Graph ofSn(x),n= 250, 0:1x0:1,
illustrating the Gibbs phenomenon.
6.16(ii) Number-Theoretic Signi cance of li(x)
If we assume Riemann's hypothesis that all nonreal ze-
ros of(s) have real part of1
2(x25.10(i)), then
6.16.5li(x) (x) =O pxlnx
,x!1;
where(x) is the number of primes less than or equal
tox. Comparex27.12 and Figure 6.16.2. See also Bays
and Hudson (2000).
Figure 6.16.2 : The logarithmic integral li( x), together
with vertical bars indicating the value of (x) forx=
10;20;:::; 1000.
6.17 Physical Applications
Geller and Ng (1969) cites work with applications from
di usion theory, transport problems, the study of the
radiative equilibrium of stellar atmospheres, and the
evaluation of exchange integrals occurring in quantum
mechanics. For applications in astrophysics, see also van
de Hulst (1980). Lebedev (1965) gives an application to
electromagnetic theory (radiation of a linear half-wave
oscillator), in which sine and cosine integrals are used.
Computation
6.18 Methods of Computation
6.18(i) Main Functions
For small or moderate values of xandjzj, the expan-
sion in power series ( x6.6) or in series of spherical Bessel
functions (x6.10(ii)) can be used. For large xorjzjthese
series su er from slow convergence or cancellation (or
both). However, this problem is less severe for the se-
ries of spherical Bessel functions because of their more
rapid rate of convergence, and also (except in the case
of (6.10.6)) absence of cancellation when z=x(>0).
For largexandjzj, expansions in inverse factorial
series (x6.10(i)) or asymptotic expansions ( x6.12) are
available. The attainable accuracy of the asymptotic
expansions can be increased considerably by exponen-
tial improvement. Also, other ranges of ph zcan be
covered by use of the continuation formulas of x6.4.
Quadrature of the integral representations is another
e ective method. For example, the Gauss-Laguerre for-
mula (x3.5(v)) can be applied to (6.2.2); see Todd (1954)
and Tseng and Lee (1998). For an application of the
Gauss-Legendre formula ( x3.5(v)) see Tooper and Mark
(1968).
156 Exponential, Logarithmic, Sine, and Cosine Integrals
Lastly, the continued fraction (6.9.1) can be used if
jzjis bounded away from the origin. Convergence be-
comes slow when zis near the negative real axis, how-
ever.
6.18(ii) Auxiliary Functions
Power series, asymptotic expansions, and quadrature
can also be used to compute the functions f( z) and g(z).
In addition, Acton (1974) developed a recurrence pro-
cedure, as follows. For n= 0;1;2;:::, de ne
6.18.1An=Z1
0te zt
1 +t2t2
1 +t2n
dt;
Bn=Z1
0e zt
1 +t2t2
1 +t2n
dt;
Cn=Z1
0e ztt2
1 +t2n
dt:
Then f(z) =B0, g(z) =A0, and
6.18.2An 1=An+z
2nCn; Bn 1=2nBn+zAn 1
2n 1;
Cn 1=Cn+Bn 1, n= 1;2;3;::::
A0,B0, andC0can be computed by Miller's algorithm
(x3.6(iii)), starting with initial values ( AN;BN;CN) =
(1;0;0), say, where Nis an arbitrary large integer, and
normalizing via C0= 1=z.
6.18(iii) Zeros
Zeros of Ci( x) and si(x) can be computed to high pre-
cision by Newton's rule ( x3.8(ii)), using values supplied
by the asymptotic expansion (6.13.2) as initial approx-
imations.
6.18(iv) Other References
For a comprehensive survey of computational methods
for the functions treated in this chapter, see van der
Laan and Temme (1984, Ch. IV).
6.19 Tables
6.19(i) Introduction
Lebedev and Fedorova (1960) and Fletcher et al. (1962)
give comprehensive indexes of mathematical tables.
This section lists relevant tables that appeared later.
6.19(ii) Real Variables
Abramowitz and Stegun (1964, Chapter 5)
includesx 1Si(x), x 2Cin(x),x 1Ein(x),
x 1Ein( x),x= 0(:01)0:5; Si(x), Ci(x),
Ei(x),E1(x),x= 0:5(:01)2; Si(x), Ci(x),
xe xEi(x),xexE1(x),x= 2(:1)10;xf(x),
x2g(x),xe xEi(x),xexE1(x),x 1= 0(:005)0:1;
Si(x), Cin(x),x= 0(:1)10. Accuracy varies but
is within the range 8S{11S.Zhang and Jin (1996, pp. 652, 689) includes Si( x),
Ci(x),x= 0(:5)20(2)30, 8D; Ei( x),E1(x),x=
[0;100], 8S.
6.19(iii) Complex Variables, z=x+iy
Abramowitz and Stegun (1964, Chapter 5) in-
cludes the real and imaginary parts of zezE1(z),
x= 19(1)20,y= 0(1)20, 6D; ezE1(z),x=
4(:5) 2,y= 0(:2)1, 6D;E1(z) + lnz,x=
2(:5)2:5,y= 0(:2)1, 6D.
Zhang and Jin (1996, pp. 690{692) in-
cludes the real and imaginary parts of E1(z),
x = 0:5;1;3;5;10;15;20;50;100,y =
0(:5)1(1)5(5)30 ;50;100, 8S.
6.20 Approximations
6.20(i) Approximations in Terms of Elementary
Functions
Hastings (1955) gives several minimax polyno-
mial and rational approximations for E1(x)+lnx,
xexE1(x), and the auxiliary functions f( x) and
g(x). These are included in Abramowitz and Ste-
gun (1964, Ch. 5).
Cody and Thacher (1968) provides minimax ra-
tional approximations for E1(x), with accuracies
up to 20S.
Cody and Thacher (1969) provides minimax ra-
tional approximations for Ei( x), with accuracies
up to 20S.
MacLeod (1996) provides rational approximations
for the sine and cosine integrals and for the auxil-
iary functions f and g, with accuracies up to 20S.
6.20(ii) Expansions in Chebyshev Series
Clenshaw (1962) gives Chebyshev coecients for
E1(x) lnjxjfor 4x4 andexE1(x) for
x4 (20D).
Luke and Wimp (1963) covers Ei( x) forx 4
(20D), and Si( x) and Ci(x) forx4 (20D).
Luke (1969b, pp. 41{42) gives Chebyshev expan-
sions of Ein( ax), Si(ax), and Cin( ax) for 1
x1,a2C. The coecients are given in terms
of series of Bessel functions.
6.21 Software 157
Luke (1969b, pp. 321{322) covers Ein( x) and
Ein( x) for 0x8 (the Chebyshev coef-
cients are given to 20D); E1(x) forx5 (20D),
and Ei(x) forx8 (15D). Coecients for the sine
and cosine integrals are given on pp. 325{327.
Luke (1969b, p. 25) gives a Chebyshev expansion
near in nity for the con
uent hypergeometric U-
function (x13.2(i)) from which Chebyshev expan-
sions near in nity for E1(z), f(z), and g(z) follow
by using (6.11.2) and (6.11.3). Luke also includes
a recursion scheme for computing the coecients
in the expansions of the Ufunctions. Ifjphzj<
the scheme can be used in backward direction.
6.20(iii) Pad e-Type and Rational Expansions
Luke (1969b, pp. 402, 410, and 415{421) gives
main diagonal Pad e approximations for Ein( z),
Si(z), Cin(z) (valid near the origin), and E1(z)
(valid for largejzj); approximate errors are given
for a selection of z-values.
Luke (1969b, pp. 411{414) gives rational approx-
imations for Ein( z).
6.21 Software
Seehttp://dlmf.nist.gov/6.21 .
References
General References
For general bibliographic reading see Andrews et al.
(1999), Je reys and Je reys (1956), Lebedev (1965),
Olver (1997b), and Temme (1996a).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x6.2Olver (1997b, pp. 40{42).
x6.3These graphics were produced at NIST.
x6.4For (6.4.1) see Olver (1997b, p. 40). (6.4.3) follows
from (6.6.2) and (6.6.4). (6.4.4) and (6.4.5) follow
from (6.2.13) and (6.2.16). (6.4.6) and (6.4.7) fol-
low from (6.2.17), (6.2.18), and (6.4.4).x6.5For (6.5.1) and (6.5.2) see Olver (1997b, p. 41).
(6.5.3) and (6.5.4) follow from (6.6.1), (6.6.2),
(6.6.5), and (6.6.6). For (6.5.5) and (6.5.6) see
Olver (1997b, p. 42). (6.5.7) follows from (6.2.10),
(6.2.17), (6.2.18), (6.5.5), and (6.5.6).
x6.6Olver (1997b, pp. 40{43). (6.6.3) follows from
(6.11.2) and (13.2.9).
x6.7(6.7.1) and (6.7.2) follow from the de nitions
(x6.2(i)). (6.7.3){(6.7.6) follow from di erentia-
tion with respect to x. (6.7.7) and (6.7.8) fol-
low from replacing the trigonometric functions
by exponentials. For (6.7.9){(6.7.11) see Nielsen
(1906b, p. 13: there are sign errors in Eq. (27)).
(6.7.12){(6.7.14) follow from (6.5.7), (6.2.1), and
(6.2.2); for the second equations in (6.7.13) and
(6.7.14) see Temme (1996a, pp. 187{188). For
(6.7.15) and (6.7.16) use (10.32.10).
x6.8See Gautschi (1959b) for (6.8.1), and Luke (1969b,
p. 201) for (6.8.2) and (6.8.3).
x6.9Nielsen (1906b, pp. 42{44), or Lorentzen and
Waadeland (1992, p. 577).
x6.10 Nielsen (1906a, p. 283). (6.10.3) follows from
1=(1 ln(1 t)) =P1
k=0cktk.
x6.11 Temme (1996a, pp. 180 and 187). For (6.11.3)
use (6.5.7).
x6.12 For (6.12.2) see Olver (1997b, p. 227). (6.12.3)
and (6.12.4) follow from (6.7.13) and (6.7.14)
by applying Watson's lemma ( x2.4(i)). (6.12.5){
(6.12.8) follow from (6.7.13), (6.7.14), and
the identity ( t2+ 1) 1=Pn 1
m=0( 1)mt2m+
( 1)nt2n(t2+1) 1. The error bounds are obtained
by settingt=p in (6.12.7) and (6.12.8), rotat-
ing the integration path in the -plane through
an angle 2 phz, and then replacing j + 1jby its
minimum value on the path.
x6.13 See Cody and Thacher (1969) for x0in (6.13.1).
x6.14 Nielsen (1906b, pp. 48{50, 53, and 54: there is
a1
2missing in the formula that corresponds to
(6.14.2) and a sign error in the formula that cor-
responds to (6.14.7)).
x6.15 Slavi c (1974).
x6.16 Temme (1996a, pp. 181{182: the numerical value
1:089490:::on p. 182 should be replaced by
1:1789:::). Gibbs reported this phenomenon in a
letter to Nature ,59(1899, p. 606). Figures 6.16.1
and 6.16.2 were produced at NIST.
Chapter 7
Error Functions, Dawson's and Fresnel Integrals
N. M. Temme1
Notation 160
7.1 Special Notation . . . . . . . . . . . . . 160
Properties 160
7.2 De nitions . . . . . . . . . . . . . . . . . 160
7.3 Graphics . . . . . . . . . . . . . . . . . . 160
7.4 Symmetry . . . . . . . . . . . . . . . . . 161
7.5 Interrelations . . . . . . . . . . . . . . . 162
7.6 Series Expansions . . . . . . . . . . . . . 162
7.7 Integral Representations . . . . . . . . . 162
7.8 Inequalities . . . . . . . . . . . . . . . . 163
7.9 Continued Fractions . . . . . . . . . . . . 163
7.10 Derivatives . . . . . . . . . . . . . . . . 163
7.11 Relations to Other Functions . . . . . . . 164
7.12 Asymptotic Expansions . . . . . . . . . . 164
7.13 Zeros . . . . . . . . . . . . . . . . . . . 165
7.14 Integrals . . . . . . . . . . . . . . . . . . 1667.15 Sums . . . . . . . . . . . . . . . . . . . 166
7.16 Generalized Error Functions . . . . . . . . 166
7.17 Inverse Error Functions . . . . . . . . . . 166
7.18 Repeated Integrals of the Complementary
Error Function . . . . . . . . . . . . . . . 167
7.19 Voigt Functions . . . . . . . . . . . . . . 167
Applications 168
7.20 Mathematical Applications . . . . . . . . 168
7.21 Physical Applications . . . . . . . . . . . 169
Computation 169
7.22 Methods of Computation . . . . . . . . . 169
7.23 Tables . . . . . . . . . . . . . . . . . . . 169
7.24 Approximations . . . . . . . . . . . . . . 170
7.25 Software . . . . . . . . . . . . . . . . . . 170
References 170
1Centrum voor Wiskunde en Informatica, Department MAS, Amsterdam, The Netherlands.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 7) by Walter Gautschi. Walter
Gautschi provided the author with a list of references and comments collected since the original publication.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
159
160 Error Functions, Dawson's and Fresnel Integrals
Notation
7.1 Special Notation
(For other notation see pp. xiv and 873.)
xreal variable.
zcomplex variable.
nnonnegative integer.
arbitrary small positive constant.
Euler's constant ( x5.2(ii)).
Unless otherwise noted, primes indicate derivatives
with respect to the argument.
The main functions treated in this chapter are the
error function erf z; the complementary error functions
erfczandw(z); Dawson's integral F(z); the Fresnel in-
tegralsF(z),C(z), andS(z); the Goodwin{Staton inte-
gralG(z); the repeated integrals of the complementary
error function inerfc(z); the Voigt functions U(x;t) and
V(x;t).
Alternative notations are P(z) =1
2erfc
z=p
2
,
Q(z) = (z) =1
2erfc
z=p
2
, Erfz=1
2perfz,
Er z=ez2F(z),C1(z) =Cp
2=z
,S1(z) =
Sp
2=z
,C2(z) =Cp
2z=
,S2(z) =Sp
2z=
.
The notations P(z),Q(z), and (z) are used in
mathematical statistics, where these functions are called
thenormal orGaussian probability functions .
Properties
7.2 De nitions
7.2(i) Error Functions
7.2.1 erfz=2pZz
0e t2dt;
7.2.2 erfcz=2pZ1
ze t2dt= 1 erfz;
7.2.3
w(z) =e z2
1 +2ipZz
0et2dt
=e z2erfc( iz):
erfz, erfcz, andw(z) are entire functions of z, as is
F(z) in the next subsection.
Values at In nity
7.2.4lim
z!1erfz= 1;lim
z!1erfcz= 0,
jphzj1
4 (<1
4).7.2(ii) Dawson's Integral
7.2.5 F(z) =e z2Zz
0et2dt:
7.2(iii) Fresnel Integrals
7.2.6F(z) =Z1
ze1
2it2dt;
7.2.7 C(z) =Zz
0cos 1
2t2
dt;
7.2.8 S(z) =Zz
0sin 1
2t2
dt;
F(z),C(z), andS(z) are entire functions of z, as are
f(z) and g(z) in the next subsection.
Values at In nity
7.2.9 lim
x!1C(x) =1
2;lim
x!1S(x) =1
2:
7.2(iv) Auxiliary Functions
7.2.10
f(z) = 1
2 S(z)
cos 1
2z2
1
2 C(z)
sin 1
2z2
;
7.2.11
g(z) = 1
2 C(z)
cos 1
2z2
+ 1
2 S(z)
sin 1
2z2
:
7.2(v) Goodwin{Staton Integral
7.2.12 G(z) =Z1
0e t2
t+zdt,jphzj<:
7.3 Graphics
7.3(i) Real Variable
Figure 7.3.1 : Complementary error functions erfc xand
erfc(10x), 3x3.
7.4 Symmetry 161
Figure 7.3.2 : Dawson's integral F(x), 3:5x3:5.
Figure 7.3.3 : Fresnel integrals C(x) andS(x), 0x4.
Figure 7.3.4 :jF(x)j2, 8x8. Fresnel (1818) intro-
duced the integral F(x) in his study of the interference
pattern at the edge of a shadow. He observed that the
intensity distribution is given by jF(x)j2.7.3(ii) Complex Variable
Figure 7.3.5 :jerf(x+iy)j, 3x3, 3y3.
Comparex7.13(i).
Figure 7.3.6 :jerfc(x+iy)j, 3x3, 3y3.
Comparexx7.12(i) and 7.13(ii).
7.4 Symmetry
7.4.1 erf( z) = erf(z);
7.4.2 erfc( z) = 2 erfc(z);
7.4.3 w( z) = 2e z2 w(z):
7.4.4 F( z) = F(z):
7.4.5C( z) = C(z); S ( z) = S(z);
7.4.6C(iz) =iC(z); S (iz) = iS(z):
7.4.7f(iz) = (1=p
2)e1
4i 1
2iz2 if(z);
g(iz) = (1=p
2)e 1
4i 1
2iz2+ig(z):
162 Error Functions, Dawson's and Fresnel Integrals
7.4.8f( z) =p
2 cos 1
4+1
2z2
f(z);
g( z) =p
2 sin 1
4+1
2z2
g(z):
7.5 Interrelations
7.5.1F(z) =1
2ip
e z2 w(z)
= 1
2ipe z2erf(iz):
7.5.2 C(z) +iS(z) =1
2(1 +i) F(z):
7.5.3C(z) =1
2+ f(z) sin 1
2z2
g(z) cos 1
2z2
;
7.5.4S(z) =1
2 f(z) cos 1
2z2
g(z) sin 1
2z2
:
7.5.5 e 1
2iz2F(z) = g(z) +if(z):
7.5.6e1
2iz2(g(z)if(z)) =1
2(1i) (C(z)iS(z)):
In (7.5.8){(7.5.10)
7.5.7 =1
2p(1i)z;
and either all upper signs or all lower signs are taken
throughout.
7.5.8 C(z)iS(z) =1
2(1i) erf:
7.5.9C(z)iS(z) =1
2(1i)
1 e1
2iz2w(i)
:
7.5.10 g(z)if(z) =1
2(1i)e2erfc:
7.5.11jF(x)j2= f2(x) + g2(x),x0,
7.5.12jF(x)j2= 2 + f2( x) + g2( x)
2p
2 cos 1
4+1
2x2
f( x)
2p
2 cos 1
4 1
2x2
g( x),
x0.
See Figure 7.3.4.
7.5.13G(x) =pF(x) 1
2e x2Ei
x2
,x>0.
For Ei(x) seex6.2(i).
7.6 Series Expansions
7.6(i) Power Series
7.6.1 erfz=2p1X
n=0( 1)nz2n+1
n!(2n+ 1);
7.6.2 erfz=2pe z21X
n=02nz2n+1
13(2n+ 1);
7.6.3w(z) =1X
n=0(iz)n
1
2n+ 1:
7.6.4C(z) =1X
n=0( 1)n(1
2)2n
(2n)!(4n+ 1)z4n+1;7.6.5C(z) = cos 1
2z21X
n=0( 1)n2n
13(4n+ 1)z4n+1
+ sin 1
2z21X
n=0( 1)n2n+1
13(4n+ 3)z4n+3:
7.6.6S(z) =1X
n=0( 1)n(1
2)2n+1
(2n+ 1)!(4n+ 3)z4n+3;
7.6.7S(z) = cos 1
2z21X
n=0( 1)n2n+1
13(4n+ 3)z4n+3
+ sin 1
2z21X
n=0( 1)n2n
13(4n+ 1)z4n+1:
The series in this subsection and in x7.6(ii) converge for
all nite values of jzj.
7.6(ii) Expansions in Series of Spherical Bessel
Functions
For the notation see xx10.47(ii) and 18.3.
7.6.8 erfz=2zp1X
n=0( 1)n
i(1)
2n
z2
i(1)
2n+1
z2
;
7.6.9erf(az) =2zpe(1
2 a2)z21X
n=0T2n+1(a)i(1)
n 1
2z2
,
1a1.
7.6.10 C(z) =z1X
n=0j2n 1
2z2
;
7.6.11 S(z) =z1X
n=0j2n+1 1
2z2
:
For further results see Luke (1969b, pp. 57{58).
7.7 Integral Representations
7.7(i) Error Functions and Dawson's Integral
Integrals of the typeR
e z2R(z)dz, whereR(z) is an ar-
bitrary rational function, can be written in closed form
in terms of the error functions and elementary functions.
7.7.1 erfcz=2
e z2Z1
0e z2t2
t2+ 1dt,jphzj1
4;
7.7.2
w(z) =1
iZ1
1e t2dt
t z=2z
iZ1
0e t2dt
t2 z2,=z>0:
7.7.3Z1
0e at2+2iztdt=1
2r
ae z2=a+ipaFzpa
,
<a>0:
7.7.4Z1
0e at
p
t+z2dt=r
aeaz2erfc paz
,<a>0,<z>0:
7.8 Inequalities 163
7.7.5Z1
0e at2
t2+ 1dt=
4ea
1 (erfpa)2
,<a>0:
7.7.6Z1
xe (at2+2bt+c)dt
=1
2r
ae(b2 ac)=aerfcpax+bpa
,<a>0:
7.7.7Z1
xe a2t2 (b2=t2)dt=p
4a
e2aberfc(ax+ (b=x))
+e 2aberfc(ax (b=x))
,
x>0,jphaj<1
4.
7.7.8Z1
0e a2t2 (b2=t2)dt=p
2ae 2ab,
jphaj<1
4,jphbj<1
4.
7.7.9Zx
0erftdt=xerfx+1p
e x2 1
:
7.7(ii) Auxiliary Functions
7.7.10 f(z) =1
p
2Z1
0e z2t=2
p
t(t2+ 1)dt,jphzj1
4;
7.7.11 g(z) =1
p
2Z1
0p
te z2t=2
t2+ 1dt,jphzj1
4;
7.7.12 g(z) +if(z) =e iz2=2Z1
zeit2=2dt:
Mellin{Barnes Integrals
7.7.13f(z) =(2) 3=2
2iZc+i1
c i1 s (s)
s+1
2
s+3
4
1
4 s
ds;
7.7.14g(z) =(2) 3=2
2iZc+i1
c i1 s (s)
s+1
2
s+1
4
3
4 s
ds:
In (7.7.13) and (7.7.14) the integration paths are
straight lines, =1
162z4, andcis a constant such
that 0<c<1
4in (7.7.13), and 0 <c<3
4in (7.7.14).
7.7.15Z1
0e atcos
t2
dt=r
2fap
2
,<a>0,
7.7.16Z1
0e atsin
t2
dt=r
2gap
2
,<a>0.
7.7(iii) Compendia
For other integral representations see Erd elyi et al.
(1954a, vol. 1, pp. 265{267, 270), Ng and Geller (1969),
Oberhettinger (1974, pp. 246{248), and Oberhettinger
and Badii (1973, pp. 371{377).7.8 Inequalities
LetM(x) denote Mills' ratio :
7.8.1 M(x) =R1
xe t2dt
e x2 =ex2Z1
xe t2dt:
(Other notations are often used.) Then
7.8.2
1
x+p
x2+ 2<M(x)1
x+p
x2+ (4=),x0;
7.8.3p
2px+ 2M(x)<1
x+ 1,x0;
7.8.4 M(x)<2
3x+p
x2+ 4,x> 1
2p
2;
7.8.5x2
2x2+ 1x2(2x2+ 5)
4x4+ 12x2+ 3xM(x)
<2x4+ 9x2+ 4
4x4+ 20x2+ 15<x2+ 1
2x2+ 3,
x0:
Next,
7.8.6Zx
0eat2dt<1
3ax
2eax2+ax2 2
,a;x> 0.
7.8.7Zx
0et2dt<ex2 1
x, x>0:
7.9 Continued Fractions
7.9.1
pez2erfcz=z
z2+1
2
1 +1
z2+3
2
1 +2
z2+,
<z>0;
7.9.2
pez2erfcz=2z
2z2+ 1 12
2z2+ 5 34
2z2+ 9 ,
<z>0;
7.9.3
w(z) =ip1
z 1
2
z 1
z 3
2
z 2
z ,=z>0:
See also Cuyt et al. (2008, pp. 255{260, 263{267,
270{273).
7.10 Derivatives
7.10.1
dn+1erfz
dzn+1= ( 1)n2pHn(z)e z2,n= 0;1;2;::::
For the Hermite polynomial Hn(z) seex18.3.
7.10.2 w0(z) = 2zw(z) + (2i=p);
164 Error Functions, Dawson's and Fresnel Integrals
7.10.3
w(n+2)(z) + 2zw(n+1)(z) + 2(n+ 1)w(n)(z) = 0,
n= 0;1;2;:::.
7.10.4df(z)
dz= zg(z);dg(z)
dz=zf(z) 1:
7.11 Relations to Other Functions
Incomplete Gamma Functions and Generalized
Exponential Integral
For the notation see xx8.2(i) and 8.19(i).
7.11.1 erfz=1p
1
2;z2
;
7.11.2 erfcz=1p 1
2;z2
;
7.11.3 erfcz=zpE1
2
z2
:
Con
uent Hypergeometric Functions
For the notation see x13.2(i).
7.11.4 erfz=2zpM 1
2;3
2; z2
=2zpe z2M
1;3
2;z2
;
7.11.5
erfcz=1pe z2U 1
2;1
2;z2
=zpe z2U
1;3
2;z2
:
7.11.6C(z) +iS(z) =zM 1
2;3
2;1
2iz2
=zeiz2=2M
1;3
2; 1
2iz2
:
Generalized Hypergeometric Functions
For the notation see xx16.2(i) and 16.2(ii).
7.11.7C(z) =z1F2 1
4;5
4;1
2; 1
162z4
;
7.11.8S(z) =1
6z3
1F2 3
4;7
4;3
2; 1
162z4
:
7.12 Asymptotic Expansions
7.12(i) Complementary Error Function
Asz!1
7.12.1
erfcze z2
pz1X
m=0( 1)m135(2m 1)
(2z2)m;
erfc( z)2 e z2
pz1X
m=0( 1)m135(2m 1)
(2z2)m;
both expansions being valid when jphzj 3
4
(<3
4).
Whenjphzj1
4the remainder terms are bounded
in magnitude by the rst neglected terms, and have
the same sign as these terms when ph z= 0. When
1
4jphzj<1
2the remainder terms are bounded
in magnitude by csc(2 jphzj) times the rst neglectedterms. For these and other error bounds see Olver
(1997b, pp. 109{112), with =1
2andzreplaced by
z2; compare (7.11.2).
For re-expansions of the remainder terms leading to
larger sectors of validity, exponential improvement, and
a smooth interpretation of the Stokes phenomenon, see
xx2.11(ii){2.11(iv) and use (7.11.3). (Note that some of
these re-expansions themselves involve the complemen-
tary error function.)
7.12(ii) Fresnel Integrals
The asymptotic expansions of C(z) andS(z) are given
by (7.5.3), (7.5.4), and
7.12.2 f(z)1
z1X
m=0( 1)m135(4m 1)
(z2)2m;
7.12.3 g(z)1
2z31X
m=0( 1)m135(4m+ 1)
(z2)2m;
asz!1 injphzj1
2 (<1
2). The remainder
terms are given by
7.12.4 f(z) =1
zn 1X
m=0( 1)m13(4m 1)
(z2)2m+R(f)
n(z);
7.12.5
g(z) =1
2z3n 1X
m=0( 1)m13(4m+ 1)
(z2)2m+R(g)
n(z);
where, forn= 0;1;2;::: andjphzj<1
4,
7.12.6R(f)
n(z) =( 1)n
p
2Z1
0e z2t=2t2n (1=2)
t2+ 1dt;
7.12.7R(g)
n(z) =( 1)n
p
2Z1
0e z2t=2t2n+(1=2)
t2+ 1dt:
Whenjphzj1
8,R(f)
n(z) andR(g)
n(z) are bounded in
magnitude by the rst neglected terms in (7.12.2) and
(7.12.3), respectively, and have the same signs as these
terms when ph z= 0. They are bounded by jcsc(4 phz)j
times the rst neglected terms when1
8jphzj<1
4.
For other phase ranges use (7.4.7) and (7.4.8). For
exponentially-improved expansions use (7.5.7), (7.5.10),
andx7.12(i).
7.12(iii) Goodwin{Staton Integral
See Olver (1997b, p. 115) for an expansion of G(z) with
bounds for the remainder for real and complex values of
z.
7.13 Zeros 165
7.13 Zeros
7.13(i) Zeros of erfz
erfzhas a simple zero at z= 0, and in the rst quad-
rant of Cthere is an in nite set of zeros zn=xn+iyn,
n= 1;2;3;:::, arranged in order of increasing absolute
value. The other zeros of erf zare zn,zn, zn.
Table 7.13.1 gives 10D values of the rst ve xnand
yn. For graphical illustration see Figure 7.3.5.
Table 7.13.1 : Zerosxn+iynof erfz:
n x n yn
1 1.45061 61632 1.88094 30002
2 2.24465 92738 2.61657 51407
3 2.83974 10469 3.17562 80996
4 3.33546 07354 3.64617 43764
5 3.76900 55670 4.06069 72339
Asn!1
7.13.1xn 1
4 1+1
16(1 +1
22) 3 ;
yn+1
4 1+1
16(1 +1
22) 3+;
where
7.13.2=q
(n 1
8); = ln
p
2
:
7.13(ii) Zeros of erfcz
In the second quadrant of C, erfczhas an in nite set of
zeroszn=xn+iyn,n= 1;2;3;:::, arranged in order
of increasing absolute value. The other zeros of erfc z
arezn. The zeros of w(z) areiznandizn.
Table 7.13.2 gives 10D values of the rst ve xnand
yn. For graphical illustration see Figure 7.3.6.
Table 7.13.2 : Zerosxn+iynof erfcz:
n x n yn
1 1:35481 01281 1.99146 68428
2 2:17704 49061 2.69114 90243
3 2:78438 76132 3.23533 08684
4 3:28741 07894 3.69730 97025
5 3:72594 87194 4.10610 72847
Asn!1
7.13.3xn +1
4 1 1
16(1 +1
22) 3+;
yn+1
4 1+1
16(1 +1
22) 3+;
where
7.13.4=q
(n 1
8); = ln
2p
2
:7.13(iii) Zeros of the Fresnel Integrals
Atz= 0,C(z) has a simple zero and S(z) has a triple
zero. In the rst quadrant of CC(z) has an in nite set
of zeroszn=xn+iyn,n= 1;2;3;:::, arranged in order
of increasing absolute value. Similarly for S(z). Letzn
be a zero of one of the Fresnel integrals. Then zn,zn,
zn,izn, izn,izn, iznare also zeros of the same
integral.
Tables 7.13.3 and 7.13.4 give 10D values of the rst
vexnandynofC(z) andS(z), respectively.
Table 7.13.3 : Complex zeros xn+iynofC(z):
n x n yn
1 1.74366 74862 0.30573 50636
2 2.65145 95973 0.25290 39555
3 3.32035 93363 0.22395 34581
4 3.87573 44884 0.20474 74706
5 4.36106 35170 0.19066 97324
Asn!1 thexnandyncorresponding to the zeros
ofC(z) satisfy
7.13.5xn+ ( 4)
83+; yn
2+,
with
7.13.6=p
4n 1; = (2=) ln():
Table 7.13.4 : Complex zeros xn+iynofS(z).
n x n yn
1 2.00925 70118 0.28854 78973
2 2.83347 72325 0.24428 52408
3 3.46753 30835 0.21849 26805
4 4.00257 82433 0.20085 10251
5 4.47418 92952 0.18768 85891
Asn!1 thexnandyncorresponding to the zeros
ofS(z) satisfy (7.13.5) with
7.13.7 = 2pn; = (2=) ln():
7.13(iv) Zeros ofF(z)
In consequence of (7.5.5) and (7.5.10), zeros of F(z) are
related to zeros of erfc z. Thus ifznis a zero of erfc z
(x7.13(ii)), then (1 + i)zn=pis a zero ofF(z).
For an asymptotic expansion of the zeros ofRz
0exp 1
2it2
dt(=F(0) F(z) =C(z) +iS(z)) see
Tu^ zilin (1971).
166 Error Functions, Dawson's and Fresnel Integrals
7.14 Integrals
7.14(i) Error Functions
Fourier Transform
7.14.1Z1
0e2iaterfc(bt)dt=1
apFa
b
+i
2a
1 e (a=b)2
,
a2C,jphbj<1
4:
Whena= 0 the limit is taken.
Laplace Transforms
7.14.2Z1
0e aterf(bt)dt=1
aea2=(4b2)erfca
2b
,
<a>0,jphbj<1
4;
7.14.3Z1
0e aterfp
btdt =1
ar
b
a+b,<a>0,<b>0;
7.14.4Z1
0e(a b)terfcp
at+rc
t
dt
=e 2(pac+p
bc)
p
b(pa+p
b),jphaj<1
2,<b>0,<c0:
7.14(ii) Fresnel Integrals
Laplace Transforms
7.14.5Z1
0e atC(t)dt=1
afa
,<a>0;
7.14.6Z1
0e atS(t)dt=1
aga
,<a>0;
7.14.7Z1
0e atC r
2t
!
dt=(p
a2+ 1 +a)1
2
2ap
a2+ 1,<a>0;
7.14.8Z1
0e atS r
2t
!
dt=(p
a2+ 1 a)1
2
2ap
a2+ 1,<a>0:
7.14(iii) Compendia
For collections of integrals see Apelblat (1983, pp. 131{
146), Erd elyi et al. (1954a, vol. 1, pp. 40, 96, 176{177),
Geller and Ng (1971), Gradshteyn and Ryzhik (2000,
xx5.4 and 6.28{6.32), Marichev (1983, pp. 184{189), Ng
and Geller (1969), Oberhettinger (1974, pp. 138{139,
142{143), Oberhettinger (1990, pp. 48{52, 155{158),
Oberhettinger and Badii (1973, pp. 171{172, 179{181),
Prudnikov et al. (1986b, vol. 2, pp. 30{36, 93{143),
Prudnikov et al. (1992a,xx3.7{3.8), and Prudnikov et al.
(1992b,xx3.7{3.8). In a series of ten papers Had zi (1968,
1969, 1970, 1972, 1973, 1975a,b, 1976a,b, 1978) givesmany integrals containing error functions and Fresnel
integrals, also in combination with the hypergeometric
function, con
uent hypergeometric functions, and gen-
eralized hypergeometric functions.
7.15 Sums
For sums involving the error function see Hansen (1975,
p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650{
651).
7.16 Generalized Error Functions
Generalizations of the error function and Dawson's in-
tegral areRx
0e tpdtandRx
0etpdt. These functions can
be expressed in terms of the incomplete gamma function
(a;z) (x8.2(i)) by change of integration variable.
7.17 Inverse Error Functions
7.17(i) Notation
The inverses of the functions x= erfy,x= erfcy,
y2R, are denoted by
7.17.1 y= inverfx; y = inverfcx;
respectively.
7.17(ii) Power Series
Witht=1
2px,
7.17.2 inverfx=t+1
3t3+7
30t5+127
630t7+,jxj<1.
For 25S values of the rst 200 coecients see Strecok
(1968).
7.17(iii) Asymptotic Expansion of inverfcxfor
Smallx
Asx!0
7.17.3 inverfcxu 1=2+a2u3=2+a3u5=2+a4u7=2+;
where
7.17.4a2=1
8v; a 3= 1
32(v2+ 6v 6);
a4=1
384(4v3+ 27v2+ 108v 300);
7.17.5 u= 2=ln
x2ln(1=x)
;
and
7.17.6 v= ln(ln(1=x)) 2 + ln:
7.18 Repeated Integrals of the Complementary Error Function 167
7.18 Repeated Integrals of the
Complementary Error Function
7.18(i) De nition
7.18.1 i 1erfc(z) =2pe z2;i0erfc(z) = erfcz;
and forn= 0;1;2;:::,
7.18.2
inerfc(z)
=Z1
zin 1erfc(t)dt=2pZ1
z(t z)n
n!e t2dt:
7.18(ii) Graphics
Figure 7.18.1 : Repeated integrals of the scaled com-
plementary error function 2n 1
2n+ 1
inerfc(x),n=
0;1;2;4;8;16.
7.18(iii) Properties
7.18.3d
dzinerfc(z) = in 1erfc(z),n= 0;1;2;:::;
7.18.4dn
dzn
ez2erfcz
= ( 1)n2nn!ez2inerfc(z),
n= 0;1;2;::::
7.18.5d2W
dz2+ 2zdW
dz 2nW= 0,
W(z) =Ainerfc(z) +Binerfc( z);
wheren= 1;2;3;:::, andA,Bare arbitrary constants.
7.18.6 inerfc(z) =1X
k=0( 1)kzk
2n kk!
1 +1
2(n k):
7.18.7inerfc(z) = z
nin 1erfc(z) +1
2nin 2erfc(z),
n= 1;2;3;::::
7.18(iv) Relations to Other Functions
For the notation see xx18.3, 13.2(i), and 12.2.Hermite Polynomials
7.18.8 ( 1)ninerfc(z) + inerfc( z) =i n
2n 1n!Hn(iz):
Con
uent Hypergeometric Functions
7.18.9
inerfc(z) =e z2
1
2n 1
2n+ 1M 1
2n+1
2;1
2;z2
z
2n 1 1
2n+1
2M 1
2n+ 1;3
2;z2!
;
7.18.10 inerfc(z) =e z2
2npU 1
2n+1
2;1
2;z2
:
Parabolic Cylinder Functions
7.18.11 inerfc(z) =e z2=2
p
2n 1U
n+1
2;zp
2
:
Probability Functions
7.18.12 inerfc(z) =1p
2n 1Hhnp
2z
:
See Je reys and Je reys (1956, xx23.081{23.09).
7.18(v) Continued Fraction
7.18.13
inerfc(z)
in 1erfc(z)=1=2
z+(n+ 1)=2
z+(n+ 2)=2
z+,<z>0:
See also Cuyt et al. (2008, p. 269).
7.18(vi) Asymptotic Expansion
7.18.14inerfc(z)2pe z2
(2z)n+11X
m=0( 1)m(2m+n)!
n!m!(2z)2m,
z!1 ,jphzj3
4 (<3
4):
7.19 Voigt Functions
7.19(i) De nitions
Forx2Randt>0,
7.19.1 U(x;t) =1p
4tZ1
1e (x y)2=(4t)
1 +y2dy;
7.19.2 V(x;t) =1p
4tZ1
1ye (x y)2=(4t)
1 +y2dy:
7.19.3
U(x;t) +iV(x;t) =r
4tez2erfcz,z= (1 ix)=(2p
t):
7.19.4
H(a;u) =a
Z1
1e t2dt
(u t)2+a2=1
apUu
a;1
4a2
:
H(a;u) is sometimes called the line broadening func-
tion; see, for example, Finn and Mugglestone (1965).
168 Error Functions, Dawson's and Fresnel Integrals
7.19(ii) Graphics
Figure 7.19.1 : Voigt function U(x;t),t= 0:1, 2:5, 5, 10.
Figure 7.19.2 : Voigt function V(x;t),t= 0:1, 2:5, 5, 10.
7.19(iii) Properties
7.19.5 lim
t!0U(x;t) =1
1 +x2;lim
t!0V(x;t) =x
1 +x2:
7.19.6 U( x;t) =U(x;t);V( x;t) = V(x;t):
7.19.7 0<U(x;t)1; 1V(x;t)1:
7.19.8 V(x;t) =xU(x;t) + 2t@U(x;t)
@x;
7.19.9 U(x;t) = 1 xV(x;t) 2t@V(x;t)
@x:
7.19(iv) Other Integral Representations
7.19.10 Uu
a;1
4a2
=aZ1
0e at 1
4t2cos(ut)dt;
7.19.11 Vu
a;1
4a2
=aZ1
0e at 1
4t2sin(ut)dt:Applications
7.20 Mathematical Applications
7.20(i) Asymptotics
For applications of the complementary error function in
uniform asymptotic approximations of integrals|saddle
point coalescing with a pole or saddle point coalescing
with an endpoint|see Wong (1989, Chapter 7), Olver
(1997b, Chapter 9), and van der Waerden (1951).
The complementary error function also plays a
ubiquitous role in constructing exponentially-improved
asymptotic expansions and providing a smooth inter-
pretation of the Stokes phenomenon; see xx2.11(iii) and
2.11(iv).
7.20(ii) Cornu's Spiral
Let the setfx(t);y(t);tgbe de ned by x(t) =C(t),
y(t) =S(t),t0. Then the set fx(t);y(t)gis called
Cornu's spiral : it is the projection of the corkscrew
on thefx;yg-plane. See Figure 7.20.1. The spiral has
several special properties (see Temme (1996a, p. 184)).
LetP(t) =P(x(t);y(t)) be any point on the projected
spiral. Then the arc length between the origin and
P(t) equalst, and is directly proportional to the cur-
vature atP(t), which equals t. Furthermore, because
dy/dx= tan 1
2t2
, the angle between the x-axis and
the tangent to the spiral at P(t) is given by1
2t2.
Figure 7.20.1 : Cornu's spiral, formed from Fresnel inte-
grals, is de ned parametrically by x=C(t),y=S(t),
t2[0;1).
7.21 Physical Applications 169
7.20(iii) Statistics
The normal distribution function with mean mand
standard deviation is given by
7.20.1
1
p
2Zx
1e (t m)2=(22)dt
=1
2erfcm x
p
2
=Qm x
=Px m
:
For applications in statistics and probability theory,
also for the role of the normal distribution functions
(the error functions and probability integrals) in the
asymptotics of arbitrary probability density functions,
see Johnson et al. (1994, Chapter 13) and Patel and
Read (1982, Chapters 2 and 3).
7.21 Physical Applications
The error functions, Fresnel integrals, and related func-
tions occur in a variety of physical applications. Fresnel
integrals and Cornu's spiral occurred originally in the
analysis of the di raction of light; see Born and Wolf
(1999,x8.7). More recently, Cornu's spiral appears in
the design of highways and railroad tracks, robot tra-
jectory planning, and computer-aided design; see Meek
and Walton (1992).
Carslaw and Jaeger (1959) gives many applications
and points out the importance of the repeated integrals
of the complementary error function inerfc(z). Fried
and Conte (1961) mentions the role of w(z) in the theory
of linearized waves or oscillations in a hot plasma; w(z)
is called the plasma dispersion function orFaddeeva
function ; see Faddeeva and Terent'ev (1954). Ng and
Geller (1969) cites work with applications from atomic
physics and astrophysics.
Voigt functions can be regarded as the convolution
of a Gaussian and a Lorentzian, and appear when the
analysis of light (or particulate) absorption (or emis-
sion) involves thermal motion e ects. These applica-
tions include astrophysics, plasma diagnostics, neutron
di raction, laser spectroscopy, and surface scattering.
See Mitchell and Zemansky (1961, xIV.2), Armstrong
(1967), and Ahn et al. (2001). Dawson's integral ap-
pears in de-convolving even more complex motional ef-
fects; see Pratt (2007).
Computation
7.22 Methods of Computation
7.22(i) Main Functions
The methods available for computing the main func-
tions in this chapter are analogous to those described inxx6.18(i){6.18(iv) for the exponential integral and sine
and cosine integrals, and similar comments apply. Ad-
ditional references are Matta and Reichel (1971) for the
application of the trapezoidal rule, for example, to the
rst of (7.7.2), and Gautschi (1970) and Cuyt et al.
(2008) for continued fractions.
7.22(ii) Goodwin{Staton Integral
See Goodwin and Staton (1948).
7.22(iii) Repeated Integrals of the
Complementary Error Function
The recursion scheme given by (7.18.1) and (7.18.7) can
be used for computing inerfc(x). See Gautschi (1977a),
where forward and backward recursions are used; see
also Gautschi (1961).
7.22(iv) Voigt Functions
The computation of these functions can be based on
algorithms for the complementary error function with
complex argument; compare (7.19.3).
7.22(v) Other References
For a comprehensive survey of computational methods
for the functions treated in this chapter, see van der
Laan and Temme (1984, Ch. V).
7.23 Tables
7.23(i) Introduction
Lebedev and Fedorova (1960) and Fletcher et al. (1962)
give comprehensive indexes of mathematical tables.
This section lists relevant tables that appeared later.
7.23(ii) Real Variables
Abramowitz and Stegun (1964, Chapter 7) in-
cludes erfx, (2=p)e x2,x2[0;2], 10D;
(2=p)e x2,x2[2;10], 8S; xex2erfcx,
x 22[0;0:25], 7D; 2n 1
2n+ 1
inerfc(x),n=
1(1)6;10;11,x2[0;5], 6S;F(x),x2[0;2], 10D;
xF(x),x 22[0;0:25], 9D;C(x),S(x),x2[0;5],
7D; f(x), g(x),x2[0;1],x 12[0;1], 15D.
Abramowitz and Stegun (1964, Table 27.6) in-
cludes the Goodwin{Staton integral G(x),x=
1(:1)3(:5)8, 4D; also G(x) + lnx,x= 0(:05)1, 4D.
Finn and Mugglestone (1965) includes the Voigt
functionH(a;u),u2[0;22],a2[0;1], 6S.
Zhang and Jin (1996, pp. 637, 639) includes
(2=p)e x2, erfx,x= 0(:02)1(:04)3, 8D;C(x),
S(x),x= 0(:2)10(2)100(100)500, 8D.
170 Error Functions, Dawson's and Fresnel Integrals
7.23(iii) Complex Variables, z=x+iy
Abramowitz and Stegun (1964, Chapter 7) in-
cludesw(z),x= 0(:1)3:9,y= 0(:1)3, 6D.
Zhang and Jin (1996, pp. 638, 640{641) in-
cludes the real and imaginary parts of erf z,
x2[0;5],y= 0:5(:5)3, 7D and 8D, re-
spectively; the real and imaginary parts ofR1
xeit2dt, (1=p)ei(x2+(=4))R1
xeit2dt,x=
0(:5)20(1)25, 8D, together with the correspond-
ing modulus and phase to 8D and 6D (degrees),
respectively.
7.23(iv) Zeros
Fettis et al. (1973) gives the rst 100 zeros of erf z
andw(z) (the table on page 406 of this reference
is forw(z), not for erfc z), 11S.
Zhang and Jin (1996, p. 642) includes the rst 10
zeros of erfz, 9D; the rst 25 distinct zeros of C(z)
andS(z), 8S.
7.24 Approximations
7.24(i) Approximations in Terms of Elementary
Functions
Hastings (1955) gives several minimax polynomial
and rational approximations for erf x, erfcxand
the auxiliary functions f( x) and g(x).
Cody (1969) provides minimax rational approxi-
mations for erf xand erfcx. The maximum rela-
tive precision is about 20S.
Cody (1968) gives minimax rational approxima-
tions for the Fresnel integrals (maximum relative
precision 19S); for a Fortran algorithm and com-
ments see Snyder (1993).
Cody et al. (1970) gives minimax rational approxi-
mations to Dawson's integral F(x) (maximum rel-
ative precision 20S{22S).
7.24(ii) Expansions in Chebyshev Series
Luke (1969b, pp. 323{324) covers1
2perfxand
ex2F(x) for 3x3 (the Chebyshev co-
ecients are given to 20D);pxex2erfcxand
2xF(x) forx3 (the Chebyshev coecients are
given to 20D and 15D, respectively). Coecients
for the Fresnel integrals are given on pp. 328{330
(20D).Bulirsch (1967) provides Chebyshev coecients
for the auxiliary functions f( x) and g(x) forx3
(15D).
Schonfelder (1978) gives coecients of Cheby-
shev expansions for x 1erfxon 0x2, for
xex2erfcxon [2;1), and forex2erfcxon [0;1)
(30D).
Shepherd and Laframboise (1981) gives coe-
cients of Chebyshev series for (1 + 2 x)ex2erfcx
on (0;1) (22D).
7.24(iii) Pad e-Type Expansions
Luke (1969b, vol. 2, pp. 422{435) gives main di-
agonal Pad e approximations for F(z), erfz, erfcz,
C(z), andS(z); approximate errors are given for
a selection of z-values.
7.25 Software
Seehttp://dlmf.nist.gov/7.25 .
References
General References
For general bibliographic reading see Carslaw and
Jaeger (1959), Lebedev (1965), Olver (1997b), and
Temme (1996a).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x7.2Olver (1997b, pp. 43{44) and Temme (1996a,
pp. 180, 182{183, 275{276).
x7.3These graphics were produced at NIST.
x7.4(7.4.7) follows from (7.2.10), (7.2.11), and (7.4.6).
(7.4.8) follows from (7.4.7).
x7.5(7.5.1) follows from (7.2.1){(7.2.3). (7.5.2) fol-
lows from (7.2.6){(7.2.9). (7.5.3) and (7.5.4) fol-
low from (7.2.10) and (7.2.11). (7.5.5) and (7.5.6)
follow from (7.2.10), (7.2.11), and (7.5.2). (7.5.8)
follows from (7.2.1), (7.2.7), and (7.2.8). (7.5.9)
follows from (7.2.2), (7.2.3), (7.5.7), and (7.5.8).
(7.5.10) follows from (7.2.2), (7.5.6), and (7.5.8).
(7.5.11) follows from (7.5.5). (7.5.12) follows from
(7.4.8) and (7.5.11). For (7.5.13) see Olver (1997b,
p. 44).
References 171
x7.6For (7.6.1){(7.6.3) see van der Laan and Temme
(1984, pp. 185{186). (7.6.4) and (7.6.6) follow
from (7.2.7) and (7.2.8). (7.6.5) and (7.6.7) fol-
low from (7.5.8) and (7.6.2). For (7.6.8) di er-
entiate: use (7.2.1) for the left-hand side, and
(10.47.7) and the second of (10.29.1) for the right-
hand side. The same method can be used for
(7.6.10) and (7.6.11). For (7.6.9) write the co-
ecients in the Chebyshev-series expansion of
exp
a2z2
erf(az) as integrals (x3.11(ii)), then ap-
ply (5.12.5), (7.6.2), and (13.6.9).
x7.7(7.7.1), (7.7.2), and (7.7.4) are given in van der
Laan and Temme (1984, pp. 185{186). (7.7.3)
follows from integrating from 0 to iz=a and from
iz=a to1. (7.7.5) follows by di erentiating with
respect to a(after multiplying the equation by
e a). (7.7.6), (7.7.7), and (7.7.9) follow by dif-
ferentiating with respect to x. For (7.7.8) let
x!0+ in (7.7.7) and use (7.2.2), (7.2.4). For
(7.7.10) and (7.7.11) see van der Laan and Temme
(1984, Chapter V). (7.7.12) follows from (7.5.5)
and (7.2.6). (7.7.13) and (7.7.14) follow by tak-
ing Mellin transforms ( x1.14(iv)), and applying
(7.7.10), (7.7.11), (5.12.1).
x7.8See Mills (1926), Mitrinovi c (1970, p. 177), and
Gautschi (1959b) for (7.8.2); Kesavan and Vasude-
vamurthy (1985) for the lower bound in (7.8.3);
Laforgia and Sismondi (1988) for the upper bound
in (7.8.3); Gupta (1970) for (7.8.4); Luke (1969b,
p. 201) for (7.8.5); Marti c (1978) for (7.8.6);
Crstici and Tudor (1975) for (7.8.7). See also Wu
(1982).
x7.9Nielsen (1906a, p. 217) and Lorentzen and Waade-
land (1992, pp. 576{577). (7.9.2) is the even part
of (7.9.1) (compare x1.12(iv)).
x7.10 These results may be veri ed by di erentiation
of the de nitions given in x7.2.
x7.11 These results may be veri ed by comparing the
power-series expansions of both sides of each equa-
tion. For (7.11.5) use (7.7.1) and (13.4.4).x7.12 (7.12.2) and (7.12.3) follow from (7.7.10) and
(7.7.11) by applying Watson's lemma in its ex-
tended form (x2.4(i)). (7.12.4){(7.12.7) follow
from (7.7.10), (7.7.11), and the identity ( t2+
1) 1=Pn 1
m=0( 1)mt2m+( 1)nt2n(t2+1) 1. The
error bounds are obtained by setting t=p in
(7.12.6) and (7.12.7), rotating the integration path
in the -plane through an angle 4 phz, and then
replacingj +1jby its minimum value on the path.
x7.13 Fettis et al. (1973), Fettis and Caslin (1973), and
Kreyszig (1957).
x7.14 For (7.14.1) and (7.14.2) integrate by parts
and apply (7.7.3), (7.7.6). (7.14.3) follows from
(7.14.4) with c= 0. For (7.14.4) integrate by parts
and apply (10.32.10), (10.39.2). For (7.14.5) and
(7.14.6) integrate by parts, and use (7.7.15) and
(7.7.16). For (7.14.7) and (7.14.8) consider the in-
tegralsR1
0e at
Cp
2t=
iSp
2t=
dt
and integrate by parts. The results are 1.
ap
2(ai)
, from which (7.14.7) and (7.14.8)
follow.
x7.17 For (7.17.2) see Carlitz (1963). (7.17.3) follows
from Blair et al. (1976) after modi cations.
x7.18 Hartree (1936) and Lorentzen and Waadeland
(1992, p. 577). The graphs were produced at
NIST.
x7.19 (7.19.3) follows from (7.2.3) and (7.7.2). (7.19.5)
follows from the de nitions (7.19.1), (7.19.2), to-
gether with (1.17.6) or x2.3(iii). For the rst of
(7.19.7) use (7.19.1) for the lower bound, and
(7.19.10) for the upper bound. For the second
of (7.19.7) use (7.19.11). For (7.19.8) and (7.19.9)
again use the de nitions (7.19.1) and (7.19.2). For
(7.19.10) and (7.19.11) see Armstrong (1967). The
graphs were produced at NIST.
x7.20 The diagram was produced by the author.
Chapter 8
Incomplete Gamma and Related Functions
R. B. Paris1
Notation 174
8.1 Special Notation . . . . . . . . . . . . . 174
Incomplete Gamma Functions 174
8.2 De nitions and Basic Properties . . . . . 174
8.3 Graphics . . . . . . . . . . . . . . . . . . 175
8.4 Special Values . . . . . . . . . . . . . . . 176
8.5 Con
uent Hypergeometric Representations 177
8.6 Integral Representations . . . . . . . . . 177
8.7 Series Expansions . . . . . . . . . . . . . 178
8.8 Recurrence Relations and Derivatives . . 178
8.9 Continued Fractions . . . . . . . . . . . . 179
8.10 Inequalities . . . . . . . . . . . . . . . . 179
8.11 Asymptotic Approximations and Expansions 179
8.12 Uniform Asymptotic Expansions for Large
Parameter . . . . . . . . . . . . . . . . . 181
8.13 Zeros . . . . . . . . . . . . . . . . . . . 182
8.14 Integrals . . . . . . . . . . . . . . . . . . 182
8.15 Sums . . . . . . . . . . . . . . . . . . . 183
8.16 Generalizations . . . . . . . . . . . . . . 183Related Functions 183
8.17 Incomplete Beta Functions . . . . . . . . 183
8.18 Asymptotic Expansions of Ix(a;b) . . . . 184
8.19 Generalized Exponential Integral . . . . . 185
8.20 Asymptotic Expansions of Ep(z) . . . . . 187
8.21 Generalized Sine and Cosine Integrals . . 188
Applications 189
8.22 Mathematical Applications . . . . . . . . 189
8.23 Statistical Applications . . . . . . . . . . 189
8.24 Physical Applications . . . . . . . . . . . 189
Computation 190
8.25 Methods of Computation . . . . . . . . . 190
8.26 Tables . . . . . . . . . . . . . . . . . . . 190
8.27 Approximations . . . . . . . . . . . . . . 191
8.28 Software . . . . . . . . . . . . . . . . . . 191
References 191
1Division of Mathematical Sciences, University of Abertay Dundee, Dundee, United Kingdom.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapters 5 and 6), by W. Gautschi and
F. Cahill, and P. J. Davis, respectively.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
173
174 Incomplete Gamma and Related Functions
Notation
8.1 Special Notation
(For other notation see pp. xiv and 873.)
x real variable.
z complex variable.
a;p real or complex parameters.
k;n nonnegative integers.
arbitrary small positive constant.
(z) gamma function ( x5.2(i)).
(z) 0(z)= (z).
Unless otherwise indicated, primes denote deriva-
tives with respect to the argument.
The functions treated in this chapter are the incom-
plete gamma functions
(a;z), (a;z),
(a;z),P(a;z),
andQ(a;z); the incomplete beta functions B x(a;b) and
Ix(a;b); the generalized exponential integral Ep(z); the
generalized sine and cosine integrals si( a;z), ci(a;z),
Si(a;z), and Ci(a;z).
Alternative notations include: Prym's functions
Pz(a) =
(a;z),Qz(a) = (a;z), Nielsen (1906a,
pp. 25{26), Batchelder (1967, p. 63); ( a;z)! =
(a+ 1;z), [a;z]! = (a+ 1;z), Dingle (1973);
B(a;b;x ) = Bx(a;b),I(a;b;x ) =Ix(a;b), Magnus et al.
(1966); Si(a;x)!Si(1 a;x), Ci(a;x)!Ci(1 a;x),
Luke (1975).
Incomplete Gamma Functions
8.2 De nitions and Basic Properties
8.2(i) De nitions
The general values of the incomplete gamma functions
(a;z) and (a;z) are de ned by
8.2.1
(a;z) =Zz
0ta 1e tdt,<a>0,
8.2.2 (a;z) =Z1
zta 1e tdt;
without restrictions on the integration paths. However,
when the integration paths do not cross the negative
real axis, and in the case of (8.2.2) exclude the origin,
(a;z) and (a;z) take their principal values ; comparex4.2(i). Except where indicated otherwise in this Hand-
book these principal values are assumed. For example,
8.2.3
(a;z) + (a;z) = (a),a6= 0; 1; 2;::::
Normalized functions are:
8.2.4P(a;z) =
(a;z)
(a); Q (a;z) = (a;z)
(a);
8.2.5 P(a;z) +Q(a;z) = 1:
In addition,
8.2.6
(a;z) =z aP(a;z) =z a
(a)
(a;z):
8.2.7
(a;z) =1
(a)Z1
0ta 1e ztdt,<a>0.
8.2(ii) Analytic Continuation
In this subsection the functions
and have their gen-
eral values.
The function
(a;z) is entire in zanda. Whenz6=
0, (a;z) is an entire function of a, and
(a;z) is mero-
morphic with simple poles at a= n,n= 0;1;2;:::,
with residue ( 1)n=n!.
Form2Z,
8.2.8
a;ze2mi
=e2mia
(a;z),a6= 0; 1; 2;:::,
8.2.9
a;ze2mi
=e2mia (a;z) + (1 e2mia) (a):
(8.2.9) also holds when ais zero or a negative integer,
provided that the right-hand side is replaced by its lim-
iting value. For example, in the case m= 1 we have
8.2.10e ia
a;zei
eia
a;ze i
= 2i
(1 a);
without restriction on a.
Lastly,
8.2.11
a;zei
= (a)(1 zaeia
(a; z)):
8.2(iii) Di erential Equations
Ifw=
(a;z) or (a;z), then
8.2.12d2w
dz2+
1 +1 a
zdw
dz= 0:
Ifw=ezz1 a (a;z), then
8.2.13d2w
dz2
1 +1 a
zdw
dz+1 a
z2w= 0:
Also,
8.2.14zd2
dz2+ (a+ 1 +z)d
dz+a
= 0:
8.3 Graphics 175
8.3 Graphics
8.3(i) Real Variables
Figure 8.3.1 : (a;x),a= 0.25, 1, 2, 2.5, 3.
Figure 8.3.2 :
(a;x),a= 0.25, 0.5, 0.75, 1.
Figure 8.3.3 :
(a;x),a= 1, 2, 2.5, 3.
Figure 8.3.4 :
(a;x) (=x aP(a;x)),a= 0.25, 0.5,
0.75, 1, 2.
Figure 8.3.5 :x a
(a;x) (=x aQ(a;x)),a= 0.25,
0.5, 1, 2.
Figure 8.3.6 :
(a;x) (=x aP(a;x)), 4x4,
5a4.
176 Incomplete Gamma and Related Functions
Some monotonicity properties of
(a;x) and (a;x)
in the four quadrants of the ( a;x)-plane in Figure 8.3.6
are given in Erd elyi et al. (1953b,x9.6).
Figure 8.3.7 :x a
(a;x) (=x aQ(a;x)), 0x4,
5a5.8.3(ii) Complex Argument
In the graphics shown in this subsection, height corre-
sponds to the absolute value of the function and color
to the phase. See p. xiv.
Figure 8.3.8 : (0:25;x+iy), 3x3, 3y3.
Principal value. There is a cut along the negative real
axis. When x=y= 0, (0:25;0) = (0:25) = 3:625:::.
Figure 8.3.9 :
(0:25;x+iy), 3x3, 3y3.
Principal value. There is a cut along the negative real
axis.
Figure 8.3.10 :
(0:25;x+iy), 3x3, 3y3.
For additional graphics see http://dlmf.nist.gov/8.
3.ii .
8.4 Special Values
For erf(z), erfc(z), andF(z), seexx7.2(i), 7.2(ii). For
En(z) seex8.19(i).
8.4.1
1
2;z2
= 2Zz
0e t2dt=perf(z);8.4.2
(a;0) =1
(a+ 1);
8.4.3
1
2; z2
=2ez2
zpF(z):
8.4.4 (0;z) =Z1
zt 1e tdt=E1(z);
8.4.5 (1;z) =e z;
8.5 Confluent Hypergeometric Representations 177
8.4.6 1
2;z2
= 2Z1
ze t2dt=perfc(z):
Forn= 0;1;2;:::,
8.4.7
(n+ 1;z) =n!(1 e zen(z));
8.4.8 (n+ 1;z) =n!e zen(z);
8.4.9 P(n+ 1;z) = 1 e zen(z);
8.4.10 Q(n+ 1;z) =e zen(z);
where
8.4.11 en(z) =nX
k=0zk
k!:
Also
8.4.12
( n;z) =zn;
8.4.13 (1 n;z) =z1 nEn(z);
8.4.14Q
n+1
2;z2
= erfc(z) +e z2
pnX
k=1z2k 1
1
2
k;
8.4.15
( n;z) =( 1)n
n!
E1(z) e zn 1X
k=0( 1)kk!
zk+1!
=( 1)n
n!( (n+ 1) lnz) z n1X
k=0
k6=n( z)k
k!(k n):
8.5 Con
uent Hypergeometric
Representations
For the con
uent hypergeometric functions M,M,
U, and the Whittaker functions M;andW;, see
xx13.2(i) and 13.14(i).
8.5.1
(a;z) =a 1zae zM(1;1 +a;z)
=a 1zaM(a;1 +a; z),a6= 0; 1; 2;:::.
8.5.2
(a;z) =e zM(1;1 +a;z) =M(a;1 +a; z):
8.5.3
(a;z) =e zU(1 a;1 a;z) =zae zU(1;1 +a;z):
8.5.4
(a;z) =a 1z1
2a 1
2e 1
2zM1
2a 1
2;1
2a(z):
8.5.5 (a;z) =e 1
2zz1
2a 1
2W1
2a 1
2;1
2a(z):8.6 Integral Representations
8.6(i) Integrals Along the Real Line
For the Bessel function J(z) and modi ed Bessel func-
tionK(z), seexx10.2(ii) and 10.25(ii).
8.6.1
(a;z) =za
sin(a)Z
0ezcostcos(at+zsint)dt,
a =2Z,
8.6.2
(a;z) =z1
2aZ1
0e tt1
2a 1Ja
2p
zt
dt,<a>0.
8.6.3
(a;z) =zaZ1
0exp
at ze t
dt,<a>0.
8.6.4 (a;z) =zae z
(1 a)Z1
0t ae t
z+tdt,
jphzj<,<a<1,
8.6.5 (a;z) =zae zZ1
0e zt
(1 +t)1 adt,<z>0,
8.6.6 (a;z) =2z1
2ae z
(1 a)Z1
0e tt 1
2aKa
2p
zt
dt,
<a<1,
8.6.7 (a;z) =zaZ1
0exp
at zet
dt,<z>0.
8.6(ii) Contour Integrals
8.6.8
(a;z) = iza
2 sin(a)Z(0+)
1ta 1eztdt,z6= 0,a =2Z;
ta 1takes its principal value where the path intersects
the positive real axis, and is continuous elsewhere on
the path.
8.6.9
a;zei
=ezeia
(1 +a)Z1
0tae zt
t 1dt,
<z>0,<a> 1,
where the integration path passes above or below the
pole att= 1, according as upper or lower signs are
taken.
Mellin{Barnes Integrals
In (8.6.10){(8.6.12), cis a real constant and the path of
integration is indented (if necessary) so that it separates
the poles of the gamma function from the other pole in
the integrand, in the case of (8.6.10) and (8.6.11), and
from the poles at s= 0;1;2;::: in the case of (8.6.12).
8.6.10
(a;z) =1
2iZc+i1
c i1 (s)
a sza sds,
jphzj<1
2,a6= 0; 1; 2;:::,
178 Incomplete Gamma and Related Functions
8.6.11
(a;z) =1
2iZc+i1
c i1 (s+a)z s
sds,jphzj<1
2,
8.6.12
(a;z) = za 1e z
(1 a)
1
2iZc+i1
c i1 (s+ 1 a)z s
sin(s)ds,
jphzj<3
2,a6= 1;2;3;:::.
8.6(iii) Compendia
For collections of integral representations of
(a;z)
and (a;z) see Erd elyi et al. (1953b,x9.3), Ober-
hettinger (1972, pp. 68{69), Oberhettinger and Badii
(1973, pp. 309{312), Prudnikov et al. (1992b,x3.10),
and Temme (1996a, pp. 282{283).
8.7 Series Expansions
For the functions en(z),i(1)
n(z), andL( )
n(x) see (8.4.11),
xx10.47(ii), and 18.3, respectively.
8.7.1
(a;z) =e z1X
k=0zk
(a+k+ 1)=1
(a)1X
k=0( z)k
k!(a+k):
8.7.2
(a;x+y)
(a;x)
= (a;x) (a;x+y)
=e xxa 11X
n=0(1 a)n
( x)n(1 e yen(y)),
jyj<jxj.
8.7.3 (a;z) = (a) 1X
k=0( 1)kza+k
k!(a+k)
= (a)
1 zae z1X
k=0zk
(a+k+ 1)!
,
a6= 0; 1; 2;:::.
8.7.4
(a;x) = (a)x1
2ae x1X
n=0en( 1)x1
2nIn+a
2x1=2
,
a6= 0; 1; 2;:::.
8.7.5
(a;z) =e 1
2z1X
n=0(1 a)n
(n+a+ 1)(2n+ 1) i(1)
n 1
2z
:
8.7.6 (a;x) =xae x1X
n=0L(a)
n(x)
n+ 1,x>0.
For an expansion for
(a;ix) in series of Bessel func-
tionsJn(x) that converges rapidly when a > 0 andx
(0) is small or moderate in magnitude see Barakat
(1961).8.8 Recurrence Relations and Derivatives
8.8.1
(a+ 1;z) =a
(a;z) zae z;
8.8.2 (a+ 1;z) =a (a;z) +zae z:
Ifw(a;z) =
(a;z) or (a;z), then
8.8.3w(a+2;z) (a+1+z)w(a+1;z)+azw(a;z) = 0:
8.8.4z
(a+ 1;z) =
(a;z) e z
(a+ 1):
8.8.5P(a+ 1;z) =P(a;z) zae z
(a+ 1);
8.8.6Q(a+ 1;z) =Q(a;z) +zae z
(a+ 1):
Forn= 0;1;2;:::,
8.8.7
(a+n;z) = (a)n
(a;z) zae zn 1X
k=0 (a+n)
(a+k+ 1)zk;
8.8.8
(a;z)
= (a)
(a n)
(a n;z) za 1e zn 1X
k=0 (a)
(a k)z k;
8.8.9
(a+n;z) = (a)n (a;z) +zae zn 1X
k=0 (a+n)
(a+k+ 1)zk;
8.8.10
(a;z)
= (a)
(a n) (a n;z) +za 1e zn 1X
k=0 (a)
(a k)z k;
8.8.11P(a+n;z) =P(a;z) zae zn 1X
k=0zk
(a+k+ 1);
8.8.12Q(a+n;z) =Q(a;z) +zae zn 1X
k=0zk
(a+k+ 1):
8.8.13d
dz
(a;z) = d
dz (a;z) =za 1e z;
8.8.14@
@a
(a;z)
a=0= E1(z) lnz:
ForE1(z) seex8.19(i).
Forn= 0;1;2;:::,
8.8.15dn
dzn(z a
(a;z)) = ( 1)nz a n
(a+n;z);
8.8.16dn
dzn(z a (a;z)) = ( 1)nz a n (a+n;z);
8.8.17dn
dzn(ez
(a;z)) = ( 1)n(1 a)nez
(a n;z);
8.8.18dn
dzn(zaez
(a;z)) =za nez
(a n;z);
8.8.19dn
dzn(ez (a;z)) = ( 1)n(1 a)nez (a n;z):
8.9 Continued Fractions 179
8.9 Continued Fractions
8.9.1 (a+ 1)ez
(a;z) =1
1 z
a+ 1 +z
a+ 2 (a+ 1)z
a+ 3 +2z
a+ 4 (a+ 2)z
a+ 5 +3z
a+ 6 ,a6= 1; 2;:::,
8.9.2 z aez (a;z) =z 1
1 +(1 a)z 1
1 +z 1
1 +(2 a)z 1
1 +2z 1
1 +(3 a)z 1
1 +3z 1
1 +,jphzj<.
For these expansions and further information see Jones
and Thron (1985). See also Cuyt et al. (2008, pp. 240{
251).
8.10 Inequalities
8.10.1 x1 aex (a;x)1,x>0, 0<a1,
8.10.2
(a;x)xa 1
a(1 e x),x>0, 0<a1.
The inequalities in (8.10.1) and (8.10.2) are reversed
whena1. If#is de ned by
8.10.3 x1 aex (a;x) = 1 +a 1
x#;
then#!1 asx!1 , and
8.10.4 0<#1, x>0,a2.
For further inequalities of these types see Qi and Mei
(1999).
Pad e Approximants
Forn= 1;2;:::,
8.10.5 An<x1 aex (a;x)<Bn,x>0,a<1,
where
8.10.6A1=x
x+ 1 a; B 1=x+ 1
x+ 2 a;
A2=x(x+ 3 a)
x2+ 2(2 a)x+ (1 a)(2 a);
B2=x2+ (5 a)x+ 2
x2+ 2(3 a)x+ (2 a)(3 a):
For hypergeometric polynomial representations of An
andBn, see Luke (1969b, x14.6).
Next, de ne
8.10.7I=Zx
0ta 1etdt= (a)xa
(a; x),<a>0.
Then
8.10.8(a+ 1)(a+ 2) x
(a+ 1)(a+ 2 +x)<ax ae xI <a+ 1
a+ 1 +x,
x>0,a0.
Also, de ne
8.10.9ca= ( (1 +a))1=(a 1); da= ( (1 +a)) 1=a:Then
8.10.10x
2a
1 +2
xa
1
<x1 aex (a;x)
x
aca
1 +ca
xa
1
,
x0, 0<a< 1,
and
8.10.11
(1 e ax)aP(a;x)(1 e ax)a,x0,a>0,
where
8.10.12
a=(
1;0<a< 1;
da; a> 1; a=(
da;0<a< 1;
1; a> 1:
Equalities in (8.10.11) apply only when a= 1.
Lastly,
8.10.13 (n;n)
(n)<1
2< (n;n 1)
(n),n= 1;2;3;:::.
8.11 Asymptotic Approximations and
Expansions
8.11(i) Large z, Fixeda
De ne
8.11.1uk= ( 1)k(1 a)k= (a 1)(a 2)(a k);
8.11.2
(a;z) =za 1e z n 1X
k=0uk
zk+Rn(a;z)!
,n= 1;2;:::.
Then asz!1 withaandn xed
8.11.3 Rn(a;z) =O
z n
,jphzj3
2 ,
wheredenotes an arbitrary small positive constant.
Ifais real and z(=x) is positive, then Rn(a;x) is
bounded in absolute value by the rst neglected term
un=xnand has the same sign provided that na 1.
For bounds on Rn(a;z) whenais real andzis complex
see Olver (1997b, pp. 109{112). For an exponentially-
improved asymptotic expansion ( x2.11(iii)) see Olver
(1991a).
180 Incomplete Gamma and Related Functions
8.11(ii) Large a, Fixedz
8.11.4
(a;z) =zae z1X
k=0zk
(a)k+1,a6= 0; 1; 2;:::.
This expansion is absolutely convergent for all nite z,
and it can also be regarded as a generalized asymptotic
expansion (x2.1(v)) of
(a;z) asa! 1 injphaj
.
Also,
8.11.5P(a;z)zae z
(1 +a)(2a) 1
2ea z(z=a)a,
a!1 ,jphaj .
8.11(iii) Large a, Fixedx=a
Ifx=a, with xed, then as a!+1
8.11.6
(a;x) xae x1X
k=0( a)kbk()
(x a)2k+1, 0<< 1,
8.11.7 (a;x)xae x1X
k=0( a)kbk()
(x a)2k+1,>1,where
8.11.8b0() = 1; b 1() =; b 2() =(2+ 1);
and fork= 1;2;:::,
8.11.9bk() =(1 )b0
k 1() + (2k 1)bk 1():
The expansion (8.11.7) also applies when a! 1
with <0, and in this case Gautschi (1959a) supplies
numerical bounds for the remainders in the truncated
expansion (8.11.7). For extensions to complex variables
see Temme (1994a, x4), and also Mahler (1930), Tricomi
(1950b), and Paris (2002b).
8.11(iv) Large a, Bounded (x a)=(2a)1
2
Ifx=a+ (2a)1
2yanda!+1, then
8.11.10
P(a+ 1;x) =1
2erfc( y) 1
3r
2
a(1+y2)e y2+O
a 1
;
8.11.11
(1 a; x) =xa 1
cos(a) +sin(a)
2pF(y) +2
3r
2
a
1 y2!
ey2+O
a 1!
;
in both cases uniformly with respect to bounded real
values ofy. For Dawson's integral F(y) seex7.2(ii).
See Tricomi (1950b) for these approximations, together
with higher terms and extensions to complex variables.
For related expansions involving Hermite polynomials
see Pagurova (1965).
8.11(v) Other Approximations
Asz!1 ,
8.11.12
(z;z)zz 1e z r
2z1
2 1
3+p
2
24z1
2 4
135z
+p
2
576z3
2+8
2835z2+:::!
,
jphzj .
For the function en(z) de ned by (8.4.11),
8.11.13 lim
n!1en(nx)
enx=8
><
>:0; x> 1;
1
2; x = 1;
1;0x<1:
Withx= 1, an asymptotic expansion of en(nx)=enx
follows from (8.11.14) and (8.11.16).IfSn(x) is de ned by
8.11.14 enx=en(nx) +(nx)n
n!Sn(x);
then
8.11.15 Sn(x) =
(n+ 1;nx)
(nx)ne nx:
Asn!1
8.11.16
Sn(1) 1
2n!en
nn 2
3+4
135n 1 8
2835n 2 16
8505n 3+:::;
8.11.17
Sn( 1) 1
2+1
8n 1+1
32n 2 1
128n 3 13
512n 4+::::
Also,
8.11.18 Sn(x)1X
k=0dk(x)n k,n!1 ,
uniformly for x2( 1;1 ], with
8.11.19 dk(x) =( 1)kbk(x)
(1 x)2k+1,k= 0;1;2;:::,
andbk(x) as inx8.11(iii).
For (8.11.18) and extensions to complex values of x
see Buckholtz (1963). For a uniformly valid expansion
forn!1 andx2[;1], see Wong (1973b).
8.12 Uniform Asymptotic Expansions for Large Parameter 181
8.12 Uniform Asymptotic Expansions for
Large Parameter
De ne
8.12.1=z=a; = (2( 1 ln))1=2;
where the branch of the square root is continuous and
satis es() 1 as!1. Then
8.12.21
22= 1 ln;d
d= 1
:
Also, denote
8.12.3P(a;z) =1
2erfc
p
a=2
S(a;);
8.12.4Q(a;z) =1
2erfc
p
a=2
+S(a;);
8.12.5 (a+ 1)eia
2i
a;zei
=1
2erfc
ip
a=2
+iT(a;);
and
8.12.6
z a
( a; z)
= cos(a) 2 sin(a)
e1
2a2
pF
p
a=2
+T(a;)!
;
whereF(x) is Dawson's integral; see x7.2(ii). Then as
a!1 in the sectorjphaj (<),
8.12.7 S(a;)e 1
2a2
p
2a1X
k=0ck()a k;
8.12.8T(a;)e1
2a2
p
2a1X
k=0ck()( a) k;
in each case uniformly with respect to in the sector
jphj2 (<2).
With= 1, the coecients ck() are given by
8.12.9c0() =1
1
; c 1() =1
3 1
3 1
2 1
12;
8.12.10ck() =1
d
dck 1() + ( 1)kgk
,k= 1;2;:::,
wheregk,k= 0;1;2;:::, are the coecients that ap-
pear in the asymptotic expansion (5.11.3) of ( z). The
right-hand sides of equations (8.12.9), (8.12.10) have re-
movable singularities at = 0, and the Maclaurin series
expansion of ck() is given by
8.12.11 ck() =1X
n=0dk;nn,jj<2p,
whered0;0= 1
3,
8.12.12
d0;n= (n+ 2) n+2, n1;
dk;n= ( 1)kgkd0;n+ (n+ 2)dk 1;n+2,n0,k1,and 3; 4;::: are de ned by
8.12.13 1 =+1
32+1X
n=3 nn,jj<2p.
In particular,
8.12.14 3=1
36; 4= 1
270; 5=1
4320;
6=1
17010; 7= 139
54 43200; 8=1
2 04120:
For numerical values of dk;nto 30D for k= 0(1)9 and
n= 0(1)Nk, whereNk= 28 4bk=2c, see DiDonato
and Morris (1986).
Special cases are given by
8.12.15
Q(a;a)1
2+1p
2a1X
k=0ck(0)a k,jphaj ,
8.12.16
eia
2isin(a)Q
a;aei
1
2 ip
2a1X
k=0ck(0)( a) k,jphaj ,
where
8.12.17c0(0) = 1
3; c 1(0) = 1
540;
c2(0) =25
6048; c 3(0) =101
1 55520;
c4(0) = 31 84811
36951 55200; c 5(0) = 27 45493
81517 36320:
For error bounds for (8.12.7) see Paris (2002a). For
the asymptotic behavior of ck() ask!1 see Dunster
et al. (1998) and Olde Daalhuis (1998c). The last ref-
erence also includes an exponentially-improved version
(x2.11(iii)) of the expansions (8.12.4) and (8.12.7) for
Q(a;z).
A di erent type of uniform expansion with coe-
cients that do not possess a removable singularity at
z=ais given by
8.12.18
Q(a;z)
P(a;z)
za 1
2e z
(a)
d( )1X
k=0Ak( )
zk=21X
k=1Bk( )
zk=2!
;
forz!1 injphzj<1
2, with<(z a)0 forP(a;z)
and<(z a)0 forQ(a;z). Here
8.12.19
= (z a)=pz; d ( ) =q
1
2e 2=2erfc
=p
2
;
and
8.12.20
A0( ) = 1; A 1( ) =1
2 +1
6 3; B 1( ) =1
3+1
6 2:
Higher coecients Ak( ),Bk( ), up tok= 8, are given
in Paris (2002b).
Lastly, a uniform approximation for ( a;ax) for
largea, with error bounds, can be found in Dunster
(1996a).
For other uniform asymptotic approximations of the
incomplete gamma functions in terms of the function
erfc see Paris (2002b) and Dunster (1996a).
182 Incomplete Gamma and Related Functions
Inverse Function
For asymptotic expansions, as a!1 , of the inverse
functionx=x(a;q) that satis es the equation
8.12.21 Q(a;x) =q
see Temme (1992a). These expansions involve the in-
verse error function inverfc( x) (x7.17), and are uniform
with respect to q2[0;1]. As a special case,
8.12.22x(a;1
2)a 1
3+8
405a 1+184
25515a 2+2248
34 44525a 3
+, a!1 .
8.13 Zeros
8.13(i)x-Zeros of
(a;x)
The function
(a;x) has no real zeros for a0. For
a<0 andn= 1;2;3;:::, there exist:
(a) one negative zero x (a) and no positive zeros
when 1 2n<a< 2 2n;
(b) one negative zero x (a) and one positive zero
x+(a) when 2n<a< 1 2n.
The negative zero x (a) decreases monotonically in
the interval 1<a< 0, and satis es
8.13.1 1 +a 1<x (a)<lnjaj, 1<a< 0.
When 5a4 the behavior of the x-zeros as
functions of acan be seen by taking the slice
(a;x) =
0 of the surface depicted in Figure 8.3.6. Note that from
(8.4.12)
( n;0) = 0,n= 1;2;3;:::.
For asymptotic approximations for x+(a) andx (a)
asa! 1 see Tricomi (1950b), with corrections by
K olbig (1972b).
8.13(ii)-Zeros of
(a;a )and (a;a )
For information on the distribution and computation of
zeros of
(a;a ) and (a;a ) in the complex -plane for
large values of the positive real parameter asee Temme
(1995a).
8.13(iii)a-Zeros of
(a;x)
For xedxandn= 1;2;3;:::,
(a;x) has:
(a) two zeros in each of the intervals 2n<a< 2 2n
whenx<0;
(b) two zeros in each of the intervals 2n<a< 1 2n
when 0<xx
n;
(c) zeros at a= nwhenx= 0.Asxincreases the positive zeros coalesce to form a dou-
ble zero at ( a
n;x
n). The values of the rst six double
zeros are given to 5D in Table 8.13.1. For values up
ton= 10 see K olbig (1972b). Approximations to a
n,
x
nfor largencan be found in K olbig (1970). When
x > x
na pair of conjugate trajectories emanate from
the pointa=a
nin the complex a-plane. See K olbig
(1970, 1972b) for further information.
Table 8.13.1 : Double zeros ( a
n;x
n) of
(a;x).
n a
nx
n
1 1:64425 0:30809
2 3:63887 0:77997
3 5:63573 1:28634
4 7:63372 1:80754
5 9:63230 2:33692
6 11:63126 2:87150
8.14 Integrals
8.14.1Z1
0e ax
(b;x)
(b)dx=(1 +a) b
a,<a>0,<b> 1,
8.14.2Z1
0e ax (b;x)dx= (b)1 (1 +a) b
a,
<a> 1,<b> 1.
In (8.14.1) and (8.14.2) limiting values are used when
b= 0.
8.14.3Z1
0xa 1
(b;x)dx= (a+b)
a,
<a<0,<(a+b)>0,
8.14.4Z1
0xa 1 (b;x)dx= (a+b)
a,<a>0,<(a+b)>0,
8.14.5Z1
0xa 1e sx
(b;x)dx
= (a+b)
b(1 +s)a+bF(1;a+b; 1 +b; 1=(1 +s)),
<s>0,<(a+b)>0,
8.14.6Z1
0xa 1e sx (b;x)dx
= (a+b)
a(1 +s)a+bF(1;a+b; 1 +a;s=(1 +s)),
<s> 1,<(a+b)>0,<a>0:
For the hypergeometric function F(a;b;c;z) see
x15.2(i).
8.15 Sums 183
For additional integrals see Apelblat (1983, x8.2),
Erd elyi et al. (1953b,x9.3), Erd elyi et al. (1954a,b),
Gradshteyn and Ryzhik (2000, x6.45), Marichev (1983,
pp.189{190), Oberhettinger (1972, pp. 68{69), Prud-
nikov et al. (1986b,xx1.2, 2.10), and Prudnikov et al.
(1992a,x3.10).
8.15 Sums
8.15.1
(a;x ) =a1X
k=0
(a+k;x)(1 )k
k!:
For sums of in nite series whose terms include in-
complete gamma functions, see Prudnikov et al. (1986b,
x5.2).
8.16 Generalizations
For a generalization of the incomplete gamma function,
including asymptotic approximations, see Chaudhry
and Zubair (1994, 2001) and Chaudhry et al. (1996).
Other generalizations are considered in Guthmann
(1991) and Paris (2003).
Related Functions
8.17 Incomplete Beta Functions
8.17(i) De nitions and Basic Properties
Throughoutxx8.17 and 8.18 we assume that a > 0,
b >0, and 0x1. However, in the case of x8.17 it
is straightforward to continue most results analytically
to other real values of a,b, andx, and also to complex
values.
8.17.1 Bx(a;b) =Zx
0ta 1(1 t)b 1dt;
8.17.2 Ix(a;b) = Bx(a;b)=B(a;b);
where, as inx5.12, B(a;b) denotes the Beta function:
8.17.3 B(a;b) = (a) (b)
(a+b):
8.17.4 Ix(a;b) = 1 I1 x(b;a);
8.17.5Ix(m;n m+ 1) =nX
j=mn
j
xj(1 x)n j;
8.17.6 Ix(a;a) =1
2I4x(1 x)
a;1
2
, 0x1
2.
For a historical pro le of B x(a;b) see Dutka (1981).8.17(ii) Hypergeometric Representations
8.17.7 Bx(a;b) =xa
aF(a;1 b;a+ 1;x);
8.17.8 Bx(a;b) =xa(1 x)b
aF(a+b;1;a+ 1;x);
8.17.9 Bx(a;b) =xa(1 x)b 1
aF1;1 b
a+ 1;x
x 1
:
For the hypergeometric function F(a;b;c;z) see
x15.2(i).
8.17(iii) Integral Representation
Witha>0,b>0, and 0<x< 1,
8.17.10Ix(a;b) =xa(1 x)b
2iZc+i1
c i1s a(1 s) bds
s x;
wherex<c< 1 and the branches of s aand (1 s) b
are continuous on the path and assume their principal
values when s=c.
Further integral representations can be obtained by
combining the results given in x8.17(ii) withx15.6.
8.17(iv) Recurrence Relations
With
8.17.11 x0= 1 x; c =a+b 1;
8.17.12 Ix(a;b) =xIx(a 1;b) +x0Ix(a;b 1);
8.17.13 (a+b)Ix(a;b) =aIx(a+ 1;b) +bIx(a;b+ 1);
8.17.14
(a+bx)Ix(a;b) =xbIx(a 1;b+ 1) +aIx(a+ 1;b);
8.17.15
(b+ax0)Ix(a;b) =ax0Ix(a+ 1;b 1) +bIx(a;b+ 1);
8.17.16
aIx(a+ 1;b) = (a+cx)Ix(a;b) cxIx(a 1;b);
8.17.17
bIx(a;b+ 1) = (b+cx0)Ix(a;b) cx0Ix(a;b 1);
8.17.18Ix(a;b) =Ix(a+ 1;b 1) +xa(x0)b 1
aB(a;b);
8.17.19Ix(a;b) =Ix(a 1;b+ 1) xa 1(x0)b
bB(a;b);
8.17.20Ix(a;b) =Ix(a+ 1;b) +xa(x0)b
aB(a;b);
8.17.21Ix(a;b) =Ix(a;b+ 1) xa(x0)b
bB(a;b):
184 Incomplete Gamma and Related Functions
8.17(v) Continued Fraction
8.17.22Ix(a;b) =xa(1 x)b
aB(a;b)
1
1 +d1
1 +d2
1 +d3
1+!
;
where
8.17.23d2m=m(b m)x
(a+ 2m 1)(a+ 2m);
d2m+1= (a+m)(a+b+m)x
(a+ 2m)(a+ 2m+ 1):
The 4mand 4m+ 1 convergents are less than Ix(a;b),
and the 4m+2 and 4m+3 convergents are greater than
Ix(a;b).
See also Cuyt et al. (2008, pp. 385{389).
The expansion (8.17.22) converges rapidly for x <
(a+ 1)=(a+b+ 2). For x > (a+ 1)=(a+b+ 2) or
1 x <(b+ 1)=(a+b+ 2), more rapid convergence is
obtained by computing I1 x(b;a) and using (8.17.4).
8.17(vi) Sums
For sums of in nite series whose terms involve the in-
complete Beta function see Hansen (1975, x62).
8.18 Asymptotic Expansions of Ix(a;b)
8.18(i) Large Parameters, Fixed x
Ifbandxare xed, with b>0 and 0<x< 1, then as
a!1
8.18.1
Ix(a;b) = (a+b)xa(1 x)b 1
n 1X
k=01
(a+k+ 1) (b k)x
1 xk
+O1
(a+n+ 1)!
;
for eachn= 0;1;2;:::. Ifb= 1;2;3;::: andnb,
then theO-term can be omitted and the result is exact.
Ifb!1 andaandxare xed, with a > 0 and
0<x< 1, then (8.18.1), with aandbinterchanged and
xreplaced by 1 x, can be combined with (8.17.4).
8.18(ii) Large Parameters: Uniform Asymptotic
Expansions
Largea, Fixedb
Let
8.18.2 = lnx:
Then asa!1 , withb(>0) xed,
8.18.3 Ix(a;b) (a+b)
(a)1X
k=0dkFk;uniformly for x2(0;1). The functions Fkare de ned
by
8.18.4aFk+1= (k+b a)Fk+kFk 1;
with
8.18.5F0=a bQ(b;a); F 1=b a
aF0+be a
a (b);
andQ(a;z) as inx8.2(i). The coecients dkare de ned
by the generating function
8.18.61 e t
tb 1
=1X
k=0dk(t )k:
In particular,
8.18.7d0=1 x
b 1
; d 1=x+x 1
(1 x)(b 1)d0:
Compare alsox24.16(i).
Symmetric Case
Let
8.18.8 x0=a=(a+b):
Then asa+b!1 ,
8.18.9Ix(a;b)1
2erfc
p
b=2
+1p
2(a+b)
x
x0a1 x
1 x0b1X
k=0( 1)kck()
(a+b)k;
uniformly for x2(0;1) anda=(a+b),b=(a+b)2
[;1 ], whereagain denotes an arbitrary small pos-
itive constant. For erfc see x7.2(i). Also,
8.18.10 1
22=x0lnx
x0
+ (1 x0) ln1 x
1 x0
;
with=(x x0)>0, and
8.18.11 c0() =1
p
x0(1 x0)
x x0;
with limiting value
8.18.12 c0(0) =1 2x0
3p
x0(1 x0):
For this result, and for higher coecients ck() see
Temme (1996a,x11.3.3.2). All of the ck() are analytic
at= 0.
8.19 Generalized Exponential Integral 185
General Case
Lete (z) denote the scaled gamma function
8.18.13e (z) = (2) 1=2ezz(1=2) z (z);
=b=a, andx0again be as in (8.18.8). Then as a!1
8.18.14
Ix(a;b)Q(b;a)
(2b) 1=2
e (b)x
x0a1 x
1 x0b1X
k=0hk(;)
ak;
uniformly for b2(0;1) andx2(0;1). Here
8.18.15
ln = lnx+ln(1 x) + (1 +) ln(1 +) ;
with ( )=(x0 x)>0, and
8.18.16h0(;) =1
(1 +) 3=2
x0 x
;
with limiting value
8.18.17 h0(;) =1
31 p1 + 1
:
For this result and higher coecients hk(;) see
Temme (1996a,x11.3.3.3). All of the hk(;) are ana-
lytic at=(corresponding to x=x0).
Inverse Function
For asymptotic expansions for large values of aand/or
bof thex-solution of the equation
8.18.18 Ix(a;b) =p, 0p1,
see Temme (1992b).
8.19 Generalized Exponential Integral
8.19(i) De nition and Integral Representations
Forp;z2C
8.19.1 Ep(z) =zp 1 (1 p;z):
Most properties of Ep(z) follow straightforwardly from
those of ( a;z). For an extensive treatment of E1(z)
see Chapter 6.
8.19.2 Ep(z) =zp 1Z1
ze t
tpdt:When the path of integration excludes the origin and
does not cross the negative real axis (8.19.2) de nes the
principal value ofEp(z), and unless indicated otherwise
in this Handbook principal values are assumed.
Other Integral Representations
8.19.3Ep(z) =Z1
1e zt
tpdt,jphzj<1
2,
8.19.4Ep(z) =zp 1e z
(p)Z1
0tp 1e zt
1 +tdt,
jphzj<1
2,<p>0.
Integral representations of Mellin{Barnes type for
Ep(z) follow immediately from (8.6.11), (8.6.12), and
(8.19.1).
8.19(ii) Graphics
Figure 8.19.1 :Ep(x), 0x3, 0p8.
In Figures 8.19.2 and 8.19.3, height corresponds to the
absolute value of the function and color to the phase.
See p. xiv.
186 Incomplete Gamma and Related Functions
Figure 8.19.2 :E1
2(x+iy), 4x4, 4y
4. Principal value. There is a branch cut along the
negative real axis.
Figure 8.19.3 :E1(x+iy), 4x4, 4y
4. Principal value. There is a branch cut along the
negative real axis.
For additional graphics see http://dlmf.nist.gov/8.19.ii .
8.19(iii) Special Values
8.19.5 E0(z) =z 1e z, z6= 0,
8.19.6 Ep(0) =1
p 1,<p>1,
8.19.7
En(z) =( z)n 1
(n 1)!E1(z)+e z
(n 1)!n 2X
k=0(n k 2)!( z)k,
n= 2;3;:::.
8.19(iv) Series Expansions
Forn= 1;2;3;:::,
8.19.8
En(z) =( z)n 1
(n 1)!( (n) lnz) 1X
k=0
k6=n 1( z)k
k!(1 n+k);
and
8.19.9
En(z) =( 1)nzn 1
(n 1)!lnz+e z
(n 1)!n 1X
k=1( z)k 1 (n k)
+e z( z)n 1
(n 1)!1X
k=0zk
k! (k+ 1);
withjphzjin both equations. For (x) seex5.2(i).
Whenp2C
8.19.10Ep(z) =zp 1 (1 p) 1X
k=0( z)k
k!(1 p+k);8.19.11
Ep(z) = (1 p)
zp 1 e z1X
k=0zk
(2 p+k)!
;
again withjphzjin both equations. The right-
hand sides are replaced by their limiting forms when
p= 1;2;3;:::.
8.19(v) Recurrence Relation and Derivatives
8.19.12 pEp+1(z) +zEp(z) =e z:
8.19.13d
dzEp(z) = Ep 1(z);
8.19.14d
dz(ezEp(z)) =ezEp(z)
1 +p 1
z
1
z:
p-Derivatives
Forj= 1;2;3;:::,
8.19.15
@jEp(z)
@pj= ( 1)jZ1
1(lnt)jt pe ztdt,<z>0.
For properties and numerical tables see Milgram (1985),
and also (when p= 1) MacLeod (2002b).
8.19(vi) Relation to Con
uent Hypergeometric
Function
8.19.16 Ep(z) =zp 1e zU(p;p;z ):
ForU(a;b;z ) seex13.2(i).
8.20 Asymptotic Expansions of Ep(z) 187
8.19(vii) Continued Fraction
8.19.17Ep(z) =e z
1
z+p
1 +1
z+p+ 1
1 +2
z+!
,
jphzj<.
See also Cuyt et al. (2008, pp. 277{285).
8.19(viii) Analytic Continuation
The general function Ep(z) is attained by extending the
path in (8.19.2) across the negative real axis. Unless p
is a nonpositive integer, Ep(z) has a branch point at
z= 0. Forz6= 0 each branch of Ep(z) is an entire
function of p.
8.19.18Ep
ze2mi
=2iempi
(p)sin(mp)
sin(p)zp 1+Ep(z),
m2Z,z6= 0.
8.19(ix) Inequalities
Forn= 1;2;3;::: andx>0,
8.19.19n 1
nEn(x)<En+1(x)<En(x);
8.19.20 (En(x))2<En 1(x)En+1(x);
8.19.211
x+n<exEn(x)1
x+n 1;
8.19.22d
dxEn(x)
En 1(x)>0:
8.19(x) Integrals
8.19.23Z1
zEp 1(t)dt=Ep(z),jphzj<,
8.19.24Z1
0e atEn(t)dt
=( 1)n 1
an
ln(1 +a) +n 1X
k=1( 1)kak
k!
,
n= 1;2;:::,<a> 1,
8.19.25Z1
0e attb 1Ep(t)dt= (b)(1 +a) b
p+b 1
F(1;b;p+b;a=(1 +a)),
<a> 1,<(p+b)>1.
8.19.26Z1
0Ep(t)Eq(t)dt=L(p) +L(q)
p+q 1,
p>0,q>0,p+q>1,
where
8.19.27
L(p) =Z1
0e tEp(t)dt=1
2pF
1;1; 1 +p;1
2
,p>0.For the hypergeometric function F(a;b;c;z) see
x15.2(i). When p= 1;2;3;:::,L(p) can also be evalu-
ated via (8.19.24).
For collections of integrals involving Ep(z), espe-
cially for integer p, see Apelblat (1983, xx7.1{7.2) and
LeCaine (1945).
8.19(xi) Further Generalizations
For higher-order generalized exponential integrals see
Meijer and Baken (1987) and Milgram (1985).
8.20 Asymptotic Expansions of Ep(z)
8.20(i) Large z
8.20.1
Ep(z) =e z
z n 1X
k=0( 1)k(p)k
zk+( 1)n(p)nez
zn 1En+p(z)!
,
n= 1;2;3;:::.
Asz!1
8.20.2Ep(z)e z
z1X
k=0( 1)k(p)k
zk,jphzj3
2 ,
and
8.20.3Ep(z)2i
(p)epizp 1+e z
z1X
k=0( 1)k(p)k
zk,
1
2+phz7
2 ,
again denoting an arbitrary small positive constant.
Where the sectors of validity of (8.20.2) and (8.20.3)
overlap the contribution of the rst term on the right-
hand side of (8.20.3) is exponentially small compared to
the other contribution; compare x2.11(ii).
For an exponentially-improved asymptotic expan-
sion ofEp(z) seex2.11(iii).
8.20(ii) Large p
Forx0 andp>1 letx=pand de neA0() = 1,
8.20.4Ak+1() = (1 2k)Ak() +(+ 1)dAk()
d,
k= 0;1;2;:::,
so thatAk() is a polynomial in of degreek 1 when
k1. In particular,
8.20.5
A1() = 1; A 2() = 1 2; A 3() = 1 8+ 62:
Then asp!1
8.20.6Ep(p)e p
(+ 1)p1X
k=0Ak()
(+ 1)2k1
pk;
uniformly for 2[0;1).
For further information, including extensions to
complex values of xandp, see Temme (1994a, x4) and
Dunster (1996b, 1997).
188 Incomplete Gamma and Related Functions
8.21 Generalized Sine and Cosine Integrals
8.21(i) De nitions: General Values
With
and denoting here the general values of the
incomplete gamma functions ( x8.2(i)), we de ne
8.21.1 ci(a;z)isi(a;z) =e1
2ia
a;ze1
2i
;
8.21.2 Ci(a;z)iSi(a;z) =e1
2ia
a;ze1
2i
:
Fromxx8.2(i) and 8.2(ii) it follows that each of the
four functions si( a;z), ci(a;z), Si(a;z), and Ci(a;z) is a
multivalued function of zwith branch point at z= 0.
Furthermore, si( a;z) and ci(a;z) are entire functions
ofa, and Si(a;z) and Ci(a;z) are meromorphic func-
tions ofawith simple poles at a= 1; 3; 5;::: and
a= 0; 2; 4;:::, respectively.
8.21(ii) De nitions: Principal Values
When phz= 0 (and when a6= 1; 3; 5;:::, in the
case of Si(a;z), ora6= 0; 2; 4;:::, in the case of
Ci(a;z)) the principal values of si(a;z), ci(a;z), Si(a;z),
and Ci(a;z) are de ned by (8.21.1) and (8.21.2) with
the incomplete gamma functions assuming their princi-
pal values (x8.2(i)). Elsewhere in the sector jphzj
the principal values are de ned by analytic continuation
from phz= 0; comparex4.2(i).
From here on it is assumed that unless indicated
otherwise the functions si( a;z), ci(a;z), Si(a;z), and
Ci(a;z) have their principal values.
Properties of the four functions that are stated be-
low inxx8.21(iii) and 8.21(iv) follow directly from the
de nitions given above, together with properties of the
incomplete gamma functions given earlier in this chap-
ter. In the case of x8.21(iv) the equation
8.21.3Z1
0ta 1eitdt=e1
2ia (a), 0<<a<1,
(obtained from (5.2.1) by rotation of the integration
path) is also needed.
8.21(iii) Integral Representations
8.21.4 si(a;z) =Z1
zta 1sintdt,<a<1,
8.21.5 ci(a;z) =Z1
zta 1costdt,<a<1,
8.21.6 Si(a;z) =Zz
0ta 1sintdt,<a> 1,
8.21.7 Ci(a;z) =Zz
0ta 1costdt,<a>0.
In these representations the integration paths do not
cross the negative real axis, and in the case of (8.21.4)
and (8.21.5) the paths also exclude the origin.8.21(iv) Interrelations
8.21.8
Si(a;z) = (a) sin 1
2a
si(a;z),a6= 1; 3; 5;:::,
8.21.9
Ci(a;z) = (a) cos 1
2a
ci(a;z),a6= 0; 2; 4;:::.
8.21(v) Special Values
8.21.10 si(0;z) = si(z);ci(0;z) = Ci(z);
8.21.11 Si(0;z) = Si(z):
For the functions on the right-hand sides of (8.21.10)
and (8.21.11) see x6.2(ii).
8.21.12 Si(a;1) = (a) sin 1
2a
,a6= 1; 3; 5;:::,
8.21.13 Ci(a;1) = (a) cos 1
2a
,a6= 0; 2; 4;:::.
8.21(vi) Series Expansions
Power-Series Expansions
8.21.14Si(a;z) =za1X
k=0( 1)kz2k+1
(2k+a+ 1)(2k+ 1)!,
a6= 1; 3; 5;:::,
8.21.15
Ci(a;z) =za1X
k=0( 1)kz2k
(2k+a)(2k)!,a6= 0; 2; 4;:::.
Spherical-Bessel-Function Expansions
8.21.16Si(a;z) =za1X
k=0
2k+3
2
1 1
2a
k 1
2+1
2a
k+1j2k+1(z),
a6= 1; 3; 5;:::,
8.21.17Ci(a;z) =za1X
k=0
2k+1
2 1
2 1
2a
k 1
2a
k+1j2k(z),
a6= 0; 2; 4;:::.
For jn(z) seex10.47(ii). For (8.21.16), (8.21.17), and
further expansions in series of Bessel functions see Luke
(1969b, pp. 56{57).
Applications 189
8.21(vii) Auxiliary Functions
8.21.18f(a;z) = si(a;z) cosz ci(a;z) sinz;
8.21.19g(a;z) = si(a;z) sinz+ ci(a;z) cosz:
8.21.20 si(a;z) =f(a;z) cosz+g(a;z) sinz;
8.21.21 ci(a;z) = f(a;z) sinz+g(a;z) cosz:
Whenjphzj<and<a<1,
8.21.22 f(a;z) =Z1
0sint
(t+z)1 adt;
8.21.23 g(a;z) =Z1
0cost
(t+z)1 adt:
Whenjphzj<1
2,
8.21.24
f(a;z) =za
2Z1
0
(1 +it)a 1+ (1 it)a 1
e ztdt;
8.21.25
g(a;z) =za
2iZ1
0
(1 it)a 1 (1 +it)a 1
e ztdt:
8.21(viii) Asymptotic Expansions
Whenz!1 withjphzj (<),
8.21.26f(a;z)za 11X
k=0( 1)k(1 a)2k
z2k;
8.21.27g(a;z)za 11X
k=0( 1)k(1 a)2k+1
z2k+1:
For the corresponding expansions for si( a;z) and
ci(a;z) apply (8.21.20) and (8.21.21).
Applications
8.22 Mathematical Applications
8.22(i) Terminant Function
The so-called terminant function Fp(z), de ned by
8.22.1Fp(z) = (p)
2z1 pEp(z) = (p)
2 (1 p;z);
plays a fundamental role in re-expansions of re-
mainder terms in asymptotic expansions, including
exponentially-improved expansions and a smooth inter-
pretation of the Stokes phenomenon. See xx2.11(ii){
2.11(v) and the references supplied in these subsections.8.22(ii) Riemann Zeta Function and Incomplete
Riemann Zeta Function
The function ( a;z), withjphaj1
2and phz=1
2,
has an intimate connection with the Riemann zeta func-
tion(s) (x25.2(i)) on the critical line <s=1
2. See Paris
and Cang (1997).
Ifx(s) denotes the incomplete Riemann zeta func-
tion de ned by
8.22.2 x(s) =1
(s)Zx
0ts 1
et 1dt,<s>1,
so that lim x!1x(s) =(s), then
8.22.3 x(s) =1X
k=1k sP(s;kx),<s>1.
For further information on x(s), including zeros and
uniform asymptotic approximations, see K olbig (1970,
1972a) and Dunster (2006).
8.23 Statistical Applications
The functions P(a;x) andQ(a;x) are used extensively
in statistics as the probability integrals of the gamma
distribution; see Johnson et al. (1994, pp. 337{414).
Particular forms are the chi-square distribution func-
tions; see Johnson et al. (1994, pp. 415{493). The func-
tion Bx(a;b) and its normalization Ix(a;b) play a similar
role in statistics in connection with the beta distribu-
tion; see Johnson et al. (1995, pp. 210{275). In queue-
ing theory the Erlang loss function is used, which can
be expressed in terms of the reciprocal of Q(a;x); see
Jagerman (1974) and Cooper (1981, pp. 80, 316{319).
8.24 Physical Applications
8.24(i) Incomplete Gamma Functions
The function
(a;x) appears in: discussions of power-
law relaxation times in complex physical systems (Sor-
nette (1998)); logarithmic oscillations in relaxation
times for proteins (Metzler et al. (1999)); Gaussian
orbitals and exponential (Slater) orbitals in quantum
chemistry (Shavitt (1963), Shavitt and Karplus (1965));
population biology and ecological systems (Camacho
et al. (2002)).
8.24(ii) Incomplete Beta Functions
The function Ix(a;b) appears in: Monte Carlo sam-
pling in statistical mechanics (Kofke (2004)); analy-
sis of packings of soft or granular objects (Prellberg
and Owczarek (1995)); growth formulas in cosmology
(Hamilton (2001)).
190 Incomplete Gamma and Related Functions
8.24(iii) Generalized Exponential Integral
The function Ep(x), withp > 0, appears in theories
of transport and radiative equilibrium (Hopf (1934),
Kourgano (1952), Alta c (1996)).
With more general values of p,Ep(x) supplies fun-
damental auxiliary functions that are used in the com-
putation of molecular electronic integrals in quantum
chemistry (Harris (2002), Shavitt (1963)), and also wave
acoustics of overlapping sound beams (Ding (2000)).
Computation
8.25 Methods of Computation
8.25(i) Series Expansions
Although the series expansions in xx8.7, 8.19(iv), and
8.21(vi) converge for all nite values of z, they are cum-
bersome to use when jzjis large owing to slowness of
convergence and cancellation. For large jzjthe corre-
sponding asymptotic expansions (generally divergent)
are used instead. See also Luke (1975, pp. 101{102)
and Temme (1994a).
8.25(ii) Quadrature
See Allasia and Besenghi (1987a) for the numerical com-
putation of ( a;z) from (8.6.4) by means of the trape-
zoidal rule.
8.25(iii) Asymptotic Expansions
DiDonato and Morris (1986) describes an algorithm for
computing P(a;x) andQ(a;x) fora0,x0, and
a+x6= 0 from the uniform expansions in x8.12. The
algorithm supplies 14S accuracy. A numerical inver-
sion procedure is also given for calculating the value of
x(with 10S accuracy), when aandP(a;x) are speci-
ed, based on Newton's rule ( x3.8(ii)). See also Temme
(1987, 1994a).
8.25(iv) Continued Fractions
The computation of
(a;z) and (a;z) by means of con-
tinued fractions is described in Jones and Thron (1985)
and Gautschi (1979a, xx4.3, 5). See also Jacobsen et al.
(1986) and Temme (1996a, p. 280).8.25(v) Recurrence Relations
Expansions involving incomplete gamma functions of-
ten require the generation of sequences P(a+n;x),
Q(a+n;x), or
(a+n;x) for xed aandn=
0;1;2;:::. An ecient procedure, based partly on the
recurrence relations (8.8.5) and (8.8.6), is described in
Gautschi (1979a, 1999).
Stable recursive schemes for the computation of
Ep(x) are described in Miller (1960) for x>0 and inte-
gerp. Forx>0 and realpsee Amos (1980) and Chiccoli
et al. (1987, 1988). See also Chiccoli et al. (1990) and
Stegun and Zucker (1974).
8.26 Tables
8.26(i) Introduction
For tables published before 1961 see Fletcher et al.
(1962) and Lebedev and Fedorova (1960).
8.26(ii) Incomplete Gamma Functions
Khamis (1965) tabulates P(a;x) fora=
0:05(:05)10(:1)20(:25)70, 0:0001x250 to
10D.
Pagurova (1963) tabulates P(a;x) andQ(a;x)
(with di erent notation) for a= 0(:05)3,x=
0(:05)1 to 7D.
Pearson (1965) tabulates the function I(u;p) (=
P(p+ 1;u)) forp= 1(:05)0(:1)5(:2)50,u=
0(:1)upto 7D, where I(u;up) rounds o to 1 to
7D; alsoI(u;p) forp= 0:75(:01) 1,u= 0(:1)6
to 5D.
Zhang and Jin (1996, Table 3.8) tabulates
(a;x) fora= 0:5;1;3;5;10;25;50;100,x=
0(:1)1(1)3;5(5)30;50;100 to 8D or 8S.
8.26(iii) Incomplete Beta Functions
Pearson (1968) tabulates Ix(a;b) forx=
0:01(:01)1,a;b= 0:5(:5)11(1)50, with ba, to
7D.
Zhang and Jin (1996, Table 3.9) tabulates Ix(a;b)
forx= 0(:05)1,a= 0:5;1;3;5;10,b= 1;10 to
8D.
8.26(iv) Generalized Exponential Integral
Abramowitz and Stegun (1964, pp. 245{248) tab-
ulatesEn(x) forn= 2;3;4;10;20,x= 0(:01)2
to 7D; also ( x+n)exEn(x) forn= 2;3;4;10;20,
x 1= 0(:01)0:1(:05)0:5 to 6S.
8.27 Approximations 191
Chiccoli et al. (1988) presents a short table of
Ep(x) forp= 9
2(1) 1
2, 0x200 to 14S.
Pagurova (1961) tabulates En(x) forn= 0(1)20,
x= 0(:01)2(:1)10 to 4-9S; exEn(x) forn=
2(1)10,x= 10(:1)20 to 7D; exEp(x) forp=
0(:1)1,x= 0:01(:01)7(:05)12(:1)20 to 7S or 7D.
Stankiewicz (1968) tabulates En(x) forn=
1(1)10,x= 0:01(:01)5 to 7D.
Zhang and Jin (1996, Table 19.1) tabulates
En(x) forn= 1;2;3;5;10;15;20,x=
0(:1)1;1:5;2;3;5;10;20;30;50;100 to 7D or 8S.
8.27 Approximations
8.27(i) Incomplete Gamma Functions
DiDonato (1978) gives a simple approximation for
the function F(p;x) =x pex2=2R1
xe t2=2tpdt
(which is related to the incomplete gamma func-
tion by a change of variables) for real pand
large positive x. This takes the form F(p;x) =
4x=h(p;x), approximately, where h(p;x) = 3(x2
p) +p
(x2 p)2+ 8(x2+p) and is shown to pro-
duce an absolute error O
x 7
asx!1 .
Luke (1975,x4.3) gives Pad e approximation meth-
ods, combined with a detailed analysis of the er-
ror terms, valid for real and complex variables ex-
cept on the negative real z-axis. See also Temme
(1994a,x3).
Luke (1969b, pp. 25, 40{41) gives Chebyshev-
series expansions for ( a;!z ) (by specifying pa-
rameters) with 1 ! <1, and
(a;!z ) with
0!1; see also Temme (1994a, x3).
Luke (1969b, p. 186) gives hypergeometric poly-
nomial representations that converge uniformly on
compact subsets of the z-plane that exclude z= 0
and are valid for jphzj<.
8.27(ii) Generalized Exponential Integral
Luke (1975, p. 103) gives Chebyshev-series expan-
sions forE1(x) and related functions for x5.
Luke (1975, p. 106) gives rational and Pad e ap-
proximations, with remainders, for E1(z) and
z 1Rz
0t 1(1 e t)dtfor complex zwithjphzj
.
Verbeeck (1970) gives polynomial and rational ap-
proximations for Ep(x) = (e x=x)P(z), approxi-
mately, where P(z) denotes a quotient of polyno-
mials of equal degree in z=x 1.8.28 Software
Seehttp://dlmf.nist.gov/8.28 .
References
General References
The main references used in writing this chapter are
Erd elyi et al. (1953b), Luke (1969b), and Temme
(1996a). For additional bibliographic reading see
Gautschi (1998), Olver (1997b), and Wong (1989).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x8.2Olver (1997b, p. 45), Temme (1996b). (8.2.12){
(8.2.14) follow from the de nitions in x8.2(i).
x8.3The graphics were produced at NIST.
x8.4Erd elyi et al. (1953b, Chapter 9). (8.4.14) follows
from (8.4.6) and (8.8.6). (8.4.15) follows from the
rst series in (8.7.3) by combining ( a) with the
termk=nof the series, then taking the limit as
a! n.
x8.5Slater (1960,x5.6) andx13.2(vii). For (8.5.4) see
(13.18.4).
x8.6For (8.6.1) use (8.6.8), replacing tbyeitwith
t. For (8.6.2) substitute for Ja
2p
zt
by (10.2.2), integrate term by term and refer to
(8.7.1) and (8.2.6). (8.6.6) may be proved in
a similar manner with the aid also of (10.25.2),
(10.27.4), (8.2.3), and analytic continuation when
a= 2; 3; 4;:::. For (8.6.3) and (8.6.7) apply
(8.2.1) and (8.2.2), taking new integration vari-
ableszet. For (8.6.4) and (8.6.5) see Temme
(1996a,xx11.2.1{11.2.2). For (8.6.8) assume tem-
porarily<a > 0, collapse the integration path
onto the interval [ 1;0] and use (8.2.1). For
(8.6.9) see Temme (1996b). For (8.6.10){(8.6.12)
see Paris and Kaminski (2001, x3.4.3).
x8.7Erd elyi et al. (1953b, Chapter 9). (8.7.3) follows
from (8.7.1) and (8.2.3). For (8.7.5) use (8.5.2)
and (13.11.1).
x8.8Erd elyi et al. (1953b, Chapter 9). These results
also follow straightforwardly from x8.2(i).
192 Incomplete Gamma and Related Functions
x8.10 Olver (1997b, pp. 66{67), Luke (1969b, pp. 195,
201). For (8.10.10){(8.10.13), see Gautschi
(1959b), Alzer (1997b), and Vietoris (1983).
x8.11 Olver (1997b, pp. 66, 109{112), Temme (1996a,
p. 280). For (8.11.4) see (8.5.1) or (8.7.1). (8.11.5)
follows from the leading terms of (8.11.4) and
(5.11.3). For (8.11.6) and (8.11.7) see Gautschi
(1959a) and Temme (1994a). (8.11.12) can be
obtained from (8.12.15), (8.2.4), and (5.11.3).
(8.11.15) follows from (8.4.7) with z=nx. For
(8.11.16) see Ramanujan (1962, pp. 323{324). For
(8.11.17) see Copson (1933).
x8.12 Temme (1979b, 1992a, 1996b), Paris (2002b),
and Ferreira et al. (2005).
x8.13 Erd elyi et al. (1953b,x9.6), Tricomi (1950b),
Lew (1994), and K olbig (1972b).
x8.14 (8.14.1){(8.14.2) are obtained by term-by-term
integration using (8.7.1) and (8.7.3). (8.14.3){
(8.14.6) are obtained by specializing (13.10.10),
(13.10.11), (13.10.3), (13.10.4) by means of
(8.5.1){(8.5.3).
x8.15 Tricomi (1950b).
x8.17 Temme (1996a,xx11.3{11.3.2). For (8.17.5) com-
bine (8.17.8) and (15.8.1). For (8.17.6) combine
(8.17.8), (15.8.18), and (15.8.1). For the last para-
graph ofx8.17(v) see Zhang and Jin (1996, p. 65).
x8.18 Temme (1996a,xx11.3.3.1{11.3.3.3). For (8.18.1)
use (8.17.9) and apply x15.12(ii).
x8.19 For (8.19.1){(8.19.4) see Temme (1996a, p. 180).
For (8.19.5){(8.19.7) use (8.19.1), (8.19.3),
(8.4.15). (8.19.8) follows from (8.4.13) and
(8.4.15). (8.19.9) follows from (6.6.3), (8.19.1),
and (8.4.15). (8.19.10) and (8.19.11) follow from(8.19.1) and (8.7.3). For (8.19.12){(8.19.16) com-
bine (8.19.1) with (8.8.2), (8.8.16), (8.8.19), and
(8.5.3). For (8.19.17) combine (8.9.2) and (8.19.1).
For (8.19.18) see Olver (1994b). For (8.19.19){
(8.19.22) see Hopf (1934, pp. 26{27). For
(8.19.23) use (8.19.13). For (8.19.24){(8.19.27)
see Kourgano (1952, Appendix 1). The graph-
ics were produced at NIST.
x8.20 Olver (1991a), Gautschi (1959a).
x8.21 Forx8.21(iii) follow the prescription given in the
nal paragraph of x8.21(ii). Thus for (8.21.4)
and (8.21.5) replace zbyizwith phz= 0 in
(8.2.2), deform the path of integration to run
along the positive imaginary axis, and replace t
byit. Then extend to the sector jphzj
by analytic continuation. Similarly for (8.21.6)
and (8.21.7). For x8.21(iv) temporarily restrict
0<<a<1. Then (8.21.8) and (8.21.9) follow im-
mediately from (8.21.3){(8.21.7). Subsequently,
ease the restrictions on aby analytic continuation
with respect to a; comparex8.21(i). For (8.21.12)
and (8.21.13) use (8.21.8) and (8.21.9), and also
(8.21.4) and (8.21.5). (8.21.14) and (8.21.15) are
obtained by expansion of the trigonometric func-
tions in (8.21.6), (8.21.7), and termwise integra-
tion. See also Luke (1975, p. 115). (8.21.22) and
(8.21.23) follow from (8.21.4), (8.21.5), (8.21.18),
and (8.21.19). For (8.21.24) and (8.21.25) as-
sume phz= 0, and in the integrals for ci( a;z)
isi(a;z) obtained from (8.21.4) and (8.21.5) set
t= (1 + )z, rotate the integration paths in
the -plane through1
2, and apply (8.21.18)
and (8.21.19). The restriction ph z= 0 is eased
tojphzj<1
2by analytic continuation. For
(8.21.26) and (8.21.27) apply Watson's lemma to
(8.21.24) and (8.21.25), and then extend the sector
of validity fromjphzj1
2 tojphzj ;
seex2.4(i).
Chapter 9
Airy and Related Functions
F. W. J. Olver1
Notation 194
9.1 Special Notation . . . . . . . . . . . . . 194
Airy Functions 194
9.2 Di erential Equation . . . . . . . . . . . 194
9.3 Graphics . . . . . . . . . . . . . . . . . . 195
9.4 Maclaurin Series . . . . . . . . . . . . . . 196
9.5 Integral Representations . . . . . . . . . 196
9.6 Relations to Other Functions . . . . . . . 196
9.7 Asymptotic Expansions . . . . . . . . . . 198
9.8 Modulus and Phase . . . . . . . . . . . . 199
9.9 Zeros . . . . . . . . . . . . . . . . . . . 200
9.10 Integrals . . . . . . . . . . . . . . . . . . 202
9.11 Products . . . . . . . . . . . . . . . . . . 203Related Functions 204
9.12 Scorer Functions . . . . . . . . . . . . . 204
9.13 Generalized Airy Functions . . . . . . . . 206
9.14 Incomplete Airy Functions . . . . . . . . 208
Applications 208
9.15 Mathematical Applications . . . . . . . . 208
9.16 Physical Applications . . . . . . . . . . . 209
Computation 209
9.17 Methods of Computation . . . . . . . . . 209
9.18 Tables . . . . . . . . . . . . . . . . . . . 210
9.19 Approximations . . . . . . . . . . . . . . 211
9.20 Software . . . . . . . . . . . . . . . . . . 212
References 212
1Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 10) by H. A. Antosiewicz. The author
is pleased to acknowledge the assistance of Bruce R. Fabijonas for computing the numerical tables in x9.9, and of Leonard Maximon for
writingx9.16.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
193
194 Airy and Related Functions
Notation
9.1 Special Notation
(For other notation see pp. xiv and 873.)
k nonnegative integer, except in x9.9(iii).
x real variable.
z(=x+iy) complex variable.
arbitrary small positive constant.
primes derivatives with respect to argument.
The main functions treated in this chapter are the
Airy functions Ai( z) and Bi(z), and the Scorer func-
tions Gi(z) and Hi(z) (also known as inhomogeneous
Airy functions).
Other notations that have been used are as fol-
lows: Ai( x) and Bi( x) for Ai(x) and Bi(x) (Jef-
freys (1928), later changed to Ai( x) and Bi(x));U(x) =pBi(x),V(x) =pAi(x) (Fock (1945)); A(x) =
3 1/3Ai
3 1/3x
(Szeg o (1967,x1.81));e0(x) =
Hi( x),ee0(x) = Gi( x) (Tumarkin (1959)).
Airy Functions
9.2 Di erential Equation
9.2(i) Airy's Equation
9.2.1d2w
dz2=zw:
All solutions are entire functions of z.
Standard solutions are:
9.2.2 w= Ai(z);Bi(z);Ai
ze2i=3
:
9.2(ii) Initial Values
9.2.3 Ai(0) =1
32=3 2
3= 0:35502 80538 :::;
9.2.4 Ai0(0) = 1
31=3 1
3= 0:25881 94037 :::;
9.2.5 Bi(0) =1
31=6 2
3= 0:61492 66274 :::;
9.2.6 Bi0(0) =31=6
1
3= 0:44828 83573 ::::9.2(iii) Numerically Satisfactory Pairs of
Solutions
Table 9.2.1 lists numerically satisfactory pairs of solu-
tions of (9.2.1) for the stated regions; compare x2.7(iv).
Table 9.2.1 : Numerically satisfactory solutions of Airy's
equation.
Pair Region
Ai(x);Bi(x) 1<x<1
Ai(z);Bi(z)(
jphzj1
3
1<z0
Ai(z);Ai
ze 2i=3
1
3phz
Ai(z);Ai
ze2i=3
phz1
3
Ai
ze2i=3
jph( z)j2
3
9.2(iv) Wronskians
9.2.7 WfAi(z);Bi(z)g=1
;
9.2.8 Wn
Ai(z);Ai
ze2i=3o
=ei=6
2;
9.2.9 Wn
Ai
ze 2i=3
;Ai
ze2i=3o
=1
2i:
9.2(v) Connection Formulas
9.2.10 Bi(z) =e i=6Ai
ze 2i=3
+ei=6Ai
ze2i=3
:
9.2.11 Ai
ze2i=3
=1
2ei=3(Ai(z)iBi(z)):
9.2.12
Ai(z) +e 2i=3Ai
ze 2i=3
+e2i=3Ai
ze2i=3
= 0;
9.2.13
Bi(z) +e 2i=3Bi
ze 2i=3
+e2i=3Bi
ze2i=3
= 0:
9.2.14 Ai( z) =ei=3Ai
zei=3
+e i=3Ai
ze i=3
;
9.2.15 Bi( z) =e i=6Ai
zei=3
+ei=6Ai
ze i=3
:
9.2(vi) Riccati Form of Di erential Equation
9.2.16dW
dz+W2=z;
W= (1=w)dw/dz, wherewis any nontrivial solution
of (9.2.1). See also Smith (1990).
9.3 Graphics 195
9.3 Graphics
9.3(i) Real Variable
Figure 9.3.1 : Ai(x), Bi(x),M(x). ForM(x) seex9.8(i).
Figure 9.3.2 : Ai0(x), Bi0(x),N(x). ForN(x) seex9.8(i).
9.3(ii) Complex Variable
In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the
phase. See also p. xiv.
Figure 9.3.3 : Ai(x+iy).
Figure 9.3.4 : Bi(x+iy).
Figure 9.3.5 : Ai0(x+iy).
Figure 9.3.6 : Bi0(x+iy).
196 Airy and Related Functions
9.4 Maclaurin Series
Forz2C
9.4.1 Ai(z) = Ai(0)
1 +1
3!z3+14
6!z6+147
9!z9+
+ Ai0(0)
z+2
4!z4+25
7!z7+258
10!z10+
;
9.4.2 Ai0(z) = Ai0(0)
1 +2
3!z3+25
6!z6+258
9!z9+
+ Ai(0)1
2!z2+14
5!z5+147
8!z8+
;
9.4.3 Bi(z) = Bi(0)
1 +1
3!z3+14
6!z6+147
9!z9+
+ Bi0(0)
z+2
4!z4+25
7!z7+258
10!z10+
;
9.4.4 Bi0(z) = Bi0(0)
1 +2
3!z3+25
6!z6+258
9!z9+
+ Bi(0)1
2!z2+14
5!z5+147
8!z8+
:
9.5 Integral Representations
9.5(i) Real Variable
9.5.1 Ai(x) =1
Z1
0cos 1
3t3+xt
dt:
9.5.2
Ai( x) =x1/2
Z1
1cos
x3/2(1
3t3+t2 2
3)
dt,x>0.
9.5.3
Bi(x)
=1
Z1
0exp
1
3t3+xt
dt+1
Z1
0sin 1
3t3+xt
dt:
See also (9.10.19), (9.11.3), (36.9.2), and Vall ee and
Soares (2004,x2.1.3).
9.5(ii) Complex Variable
9.5.4 Ai(z) =1
2iZ1ei=3
1e i=3exp 1
3t3 zt
dt;
9.5.5Bi(z) =1
2Z1ei=3
1exp 1
3t3 zt
dt
+1
2Z1e i=3
1exp 1
3t3 zt
dt:
9.5.6 Ai(z) =p
3
2Z1
0exp
t3
3 z3
3t3
dt:
9.5.7
Ai(z) =e
Z1
0exp
z1/2t2
cos 1
3t3
dt,jphzj<.
9.5.8
Ai(z) =e 1/6
p(48)1/6 5
6Z1
0e tt 1/6
2+t
1/6
dt,
jphzj<2
3.
In (9.5.7) and (9.5.8) =2
3z3/2.
See also (9.10.18) and (9.11.4).9.6 Relations to Other Functions
9.6(i) Airy Functions as Bessel Functions,
Hankel Functions, and Modi ed Bessel
Functions
For the notation see xx10.2(ii) and 10.25(ii). With
9.6.1 =2
3z3=2;
9.6.2Ai(z) = 1p
z=3K1=3()
=1
3pz
I 1=3() I1=3()
=1
2p
z=3e2i=3H(1)
1=3
ei=2
=1
2p
z=3ei=3H(1)
1=3
ei=2
=1
2p
z=3e 2i=3H(2)
1=3
e i=2
=1
2p
z=3e i=3H(2)
1=3
e i=2
;
9.6.3Ai0(z) = 1(z=p
3)K2=3()
= (z=3)
I2=3() I 2=3()
=1
2(z=p
3)e i=6H(1)
2=3
ei=2
=1
2(z=p
3)e 5i=6H(1)
2=3
ei=2
=1
2(z=p
3)ei=6H(2)
2=3
e i=2
=1
2(z=p
3)e5i=6H(2)
2=3
e i=2
;
9.6.4Bi(z) =p
z=3
I1=3() +I 1=3()
=1
2p
z=3
ei=6H(1)
1=3
e i=2
+e i=6H(2)
1=3
ei=2
=1
2p
z=3
e i=6H(1)
1=3
e i=2
+ei=6H(2)
1=3
ei=2
;
9.6 Relations to Other Functions 197
9.6.5Bi0(z) = (z=p
3)
I2=3() +I 2=3()
=1
2(z=p
3)
ei=3H(1)
2=3
e i=2
+e i=3H(2)
2=3
ei=2
=1
2(z=p
3)
e i=3H(1)
2=3
e i=2
+ei=3H(2)
2=3
ei=2
;
9.6.6
Ai( z) = (pz=3)
J1=3() +J 1=3()
=1
2p
z=3
ei=6H(1)
1=3() +e i=6H(2)
1=3()
=1
2p
z=3
e i=6H(1)
1=3() +ei=6H(2)
1=3()
;
9.6.7
Ai0( z) = (z=3)
J2=3() J 2=3()
=1
2(z=p
3)
e i=6H(1)
2=3() +ei=6H(2)
2=3()
=1
2(z=p
3)
e 5i=6H(1)
2=3()
+e5i=6H(2)
2=3()
;
9.6.8
Bi( z) =p
z=3
J 1=3() J1=3()
=1
2p
z=3
e2i=3H(1)
1=3() +e 2i=3H(2)
1=3()
=1
2p
z=3
ei=3H(1)
1=3() +e i=3H(2)
1=3()
;
9.6.9
Bi0( z) = (z=p
3)
J 2=3() +J2=3()
=1
2(z=p
3)
ei=3H(1)
2=3() +e i=3H(2)
2=3()
=1
2(z=p
3)
e i=3H(1)
2=3()
+ei=3H(2)
2=3()
:
9.6(ii) Bessel Functions, Hankel Functions, and
Modi ed Bessel Functions as Airy
Functions
Again, for the notation see xx10.2(ii) and 10.25(ii).
With
9.6.10 z= (3
2)2=3;
9.6.11J1=3() =1
2p
3=zp
3 Ai( z)Bi( z)
;
9.6.12J2=3() =1
2(p
3=z)
p
3 Ai0( z) + Bi0( z)
;
9.6.13I1=3() =1
2p
3=z
p
3 Ai(z) + Bi(z)
;
9.6.14I2=3() =1
2(p
3=z)
p
3 Ai0(z) + Bi0(z)
;9.6.15 K1=3() =p
3=zAi(z);
9.6.16 K2=3() = (p
3=z) Ai0(z);
9.6.17H(1)
1=3() =e i=3H(1)
1=3()
=e i=6p
3=z(Ai( z) iBi( z));
9.6.18H(1)
2=3() =e 2i=3H(1)
2=3()
=ei=6(p
3=z)
Ai0( z) iBi0( z)
;
9.6.19H(2)
1=3() =ei=3H(2)
1=3()
=ei=6p
3=z(Ai( z) +iBi( z));
9.6.20H(2)
2=3() =e2i=3H(2)
2=3()
=e i=6(p
3=z)
Ai0( z) +iBi0( z)
:
9.6(iii) Airy Functions as Con
uent
Hypergeometric Functions
For the notation see xx13.1, 13.2, and 13.14(i). With
as in (9.6.1),
9.6.21Ai(z) =1
2 1=2z 1=4W0;1=3(2)
= 3 1=6 1=22=3e U 5
6;5
3;2
;
9.6.22Ai0(z) = 1
2 1=2z1=4W0;2=3(2)
= 31=6 1=24=3e U 7
6;7
3;2
;
9.6.23Bi(z) =1
21=3 2
3z 1=4M0; 1=3(2)
+3
25=3 1
3z 1=4M0;1=3(2);
9.6.24Bi0(z) =21=3
1
3z1=4M0; 2=3(2)
+3
210=3 2
3z1=4M0;2=3(2);
9.6.25Bi(z) =1
31=6 2
3e
1F1 1
6;1
3; 2
+35=6
22=3 1
32=3e
1F1 5
6;5
3; 2
;
9.6.26Bi0(z) =31=6
1
3e
1F1
1
6; 1
3; 2
+37=6
27=3 2
34=3e
1F1 7
6;7
3;
:
198 Airy and Related Functions
9.7 Asymptotic Expansions
9.7(i) Notation
Heredenotes an arbitrary small positive constant and
9.7.1 =2
3z3/2:
Alsou0=v0= 1 and for k= 1;2;:::;
9.7.2uk=(2k+ 1)(2k+ 3)(2k+ 5)(6k 1)
(216)k(k)!;
vk=6k+ 1
1 6kuk:
Lastly,
9.7.3 (n) =1/2 1
2n+ 1
= 1
2n+1
2
:
Numerical values of this function are given in Table 9.7.1
forn= 1(1)20 to 2D. For large n,
9.7.4 (n)(1
2n)1/2:Table 9.7.1 : (n).
n (n)n (n)n (n)n (n)
1 1.57 6 3.20 11 4.25 16 5.09
2 2.00 7 3.44 12 4.43 17 5.24
3 2.36 8 3.66 13 4.61 18 5.39
4 2.67 9 3.87 14 4.77 19 5.54
5 2.95 10 4.06 15 4.94 20 5.68
9.7(ii) Poincar e-Type Expansions
Asz!1 the following asymptotic expansions are valid
uniformly in the stated sectors.
9.7.5 Ai(z)e
2pz1=41X
k=0( 1)kuk
k,jphzj ;
9.7.6 Ai0(z) z1=4e
2p1X
k=0( 1)kvk
k,jphzj ;
9.7.7 Bi(z)e
pz1=41X
k=0uk
k,jphzj1
3 ;
9.7.8 Bi0(z)z1=4e
p1X
k=0vk
k,jphzj1
3 :
9.7.9 Ai( z)1pz1=4
cos
1
41X
k=0( 1)ku2k
2k+ sin
1
41X
k=0( 1)ku2k+1
2k+1!
,jphzj2
3 ;
9.7.10 Ai0( z)z1=4
p
sin
1
41X
k=0( 1)kv2k
2k cos
1
41X
k=0( 1)kv2k+1
2k+1!
,jphzj2
3 ;
9.7.11 Bi( z)1pz1=4
sin
1
41X
k=0( 1)ku2k
2k+ cos
1
41X
k=0( 1)ku2k+1
2k+1!
,jphzj2
3 ;
9.7.12 Bi0( z)z1=4
p
cos
1
41X
k=0( 1)kv2k
2k+ sin
1
41X
k=0( 1)kv2k+1
2k+1!
,jphzj2
3 :
9.7.13Bi
zei=3
r
2
ei=6
z1=4
cos
1
41
2iln 21X
k=0( 1)ku2k
2k+ sin
1
41
2iln 21X
k=0( 1)ku2k+1
2k+1!
,
jphzj2
3 ;
9.7.14
Bi0
zei=3
r
2
ei=6z1=4
sin
1
41
2iln 21X
k=0( 1)kv2k
2k+ cos
1
41
2iln 21X
k=0( 1)kv2k+1
2k+1!
,
jphzj2
3 :
9.8 Modulus and Phase 199
9.7(iii) Error Bounds for Real Variables
In (9.7.5) and (9.7.6) the nth error term, that is, the er-
ror on truncating the expansion at nterms, is bounded
in magnitude by the rst neglected term and has the
same sign, provided that the following term is of op-
posite sign, that is, if n0 for (9.7.5) and n1 for
(9.7.6).
In (9.7.7) and (9.7.8) the nth error term is bounded
in magnitude by the rst neglected term multiplied by
2 (n) exp (=(72)) where= 5 for (9.7.7) and = 7
for (9.7.8), provided that n1 in both cases.
In (9.7.9){(9.7.12) the nth error term in each in nite
series is bounded in magnitude by the rst neglected
term and has the same sign, provided that the following
term in the series is of opposite sign.
As special cases, when 0 <x<1
9.7.15
Ai(x)e
2px1=4,jAi0(x)jx1=4e
2p
1 +7
72
,
9.7.16Bi(x)e
px1=4
1 +5
72exp5
72
;
Bi0(x)x1=4e
p
1 +7
72exp7
72
;
where=2
3x3=2.
9.7(iv) Error Bounds for Complex Variables
Whenn1 thenth error term in (9.7.5) and (9.7.6) is
bounded in magnitude by the rst neglected term mul-
tiplied by
9.7.172 exp
36jj
, 2 (n) exp
72jj
or4 (n)
jcos(ph)jnexp
36j<j
,
according asjphzj 1
3,1
3 jphzj 2
3, or
2
3jphzj. Here= 5 for (9.7.5) and = 7
for (9.7.6).
Corresponding bounds for the errors in (9.7.7) to
(9.7.14) may be obtained by use of these results and
those ofx9.2(v) and their di erentiated forms.
For other error bounds see Boyd (1993).
9.7(v) Exponentially-Improved Expansions
In (9.7.5) and (9.7.6) let
9.7.18 Ai(z) =e
2pz1=4 n 1X
k=0( 1)kuk
k+Rn(z)!
;
9.7.19 Ai0(z) = z1=4e
2p n 1X
k=0( 1)kvk
k+Sn(z)!
;withn=b2jjc. Then
9.7.20
Rn(z) = ( 1)nm 1X
k=0( 1)kukGn k(2)
k+Rm;n(z);
9.7.21
Sn(z) = ( 1)n 1m 1X
k=0( 1)kvkGn k(2)
k+Sm;n(z);
where
9.7.22 Gp(z) =ez
2 (p) (1 p;z):
(For the notation see x8.2(i).) And as z!1 withm
xed
9.7.23
Rm;n(z);Sm;n(z) =O
e 2jj m
,jphzj2
3:
For re-expansions of the remainder terms in (9.7.7){
(9.7.14) combine the results of this section with those
ofx9.2(v) and their di erentiated forms, as in x9.7(iv).
For higher re-expansions of the remainder terms see
Olde Daalhuis (1995, 1996), and Olde Daalhuis and
Olver (1995a).
9.8 Modulus and Phase
9.8(i) De nitions
Throughout this section xis real and nonpositive.
9.8.1 Ai(x) =M(x) sin(x);
9.8.2 Bi(x) =M(x) cos(x);
9.8.3 M(x) =q
Ai2(x) + Bi2(x);
9.8.4 (x) = arctan(Ai( x)=Bi(x)):
9.8.5 Ai0(x) =N(x) sin (x);
9.8.6 Bi0(x) =N(x) cos (x);
9.8.7 N(x) =q
Ai02(x) + Bi02(x);
9.8.8 (x) = arctan
Ai0(x)=Bi0(x)
:
Graphs ofM(x) andN(x) are included in x9.3(i).
The branches of (x) and (x) are continuous and xed
by(0) = (0) =1
6. (These de nitions of (x) and
(x) di er from Abramowitz and Stegun (1964, Chap-
ter 10), and agree more closely with those used in Miller
(1946) and Olver (1997b, Chapter 11).)
In terms of Bessel functions, and with =2
3jxj3=2,
9.8.9jxj1=2M2(x) =1
2
J2
1=3() +Y2
1=3()
;
9.8.10jxj 1=2N2(x) =1
2
J2
2=3() +Y2
2=3()
;
9.8.11(x) =2
3+ arctan
Y1=3()=J1=3()
;
9.8.12 (x) =1
3+ arctan
Y2=3()=J2=3()
:
200 Airy and Related Functions
9.8(ii) Identities
Primes denote di erentiations with respect to x, which
is continued to be assumed real and nonpositive.
9.8.13M(x)N(x) sin((x) (x)) = 1;
9.8.14M2(x)0(x) = 1,N2(x) 0(x) = 1x,
N(x)N0(x) =xM(x)M0(x) ,
9.8.15N2(x) =M02(x) +M2(x)02(x)
=M02(x) + 2M 2(x);
9.8.16x2M2(x) =N02(x) +N2(x) 02(x)
=N02(x) + 2x2N 2(x);
9.8.17tan((x) (x)) = 1=(M(x)M0(x))
= M(x)0(x)=M0(x);
9.8.18M00(x) =xM(x) + 2M 3(x) ,
M2000(x) 4x
M20(x) 2M2(x) = 0;
9.8.1902(x)+1
2(000(x)=0(x)) 3
4(00(x)=0(x))2= x:
9.8(iii) Monotonicity
Asxincreases from 1to 0 each of the functions M(x),
M0(x),jxj 1=4N(x),M(x)N(x),0(x), 0(x) is increas-
ing, and each of the functions jxj1=4M(x),(x), (x) is
decreasing.
9.8(iv) Asymptotic Expansions
Asx! 1
9.8.20
M2(x)1
( x)1=21X
k=0135(6k 1)
k!(96)k1
x3k;
9.8.21
N2(x)( x)1=2
1X
k=0135(6k 1)
k!(96)k1 + 6k
1 6k1
x3k;
9.8.22
(x)
4+2
3( x)3=2
1 +5
321
x3+1105
61441
x6
+82825
655361
x9+12820 31525
587 202561
x12+
;
9.8.23
(x)
4+2
3( x)3=2
1 7
321
x3 1463
61441
x6
4 95271
3 276801
x9 2065 30429
83 886081
x12
:
In (9.8.20) and (9.8.21) the remainder after nterms
does not exceed the ( n+1)th term in absolute value and
is of the same sign, provided that n0 for (9.8.20) and
n1 for (9.8.21).For higher terms in (9.8.22) and (9.8.23) see Fabi-
jonas et al. (2004). Also, approximate values (25S) of
the coecients of the powers x 15,x 18,:::; x 56are
available in Sherry (1959).
9.9 Zeros
9.9(i) Distribution and Notation
On the real line, Ai( x), Ai0(x), Bi(x), Bi0(x) each have
an in nite number of zeros, all of which are negative.
They are denoted by ak,a0
k,bk,b0
k, respectively, ar-
ranged in ascending order of absolute value for k=
1;2;::::
Ai(z) and Ai0(z) have no other zeros. However, Bi( z)
and Bi0(z) each have an in nite number of complex ze-
ros. They lie in the sectors1
3 < phz <1
2and
1
2 < phz < 1
3, and are denoted by k, 0
k, re-
spectively, in the former sector, and by k, 0
k, in the
conjugate sector, again arranged in ascending order of
absolute value (modulus) for k= 1;2;:::: Seex9.3(ii)
for visualizations.
For the distribution in Cof the zeros of Ai0(z)
Ai(z), whereis an arbitrary complex constant, see
Murave (1976).
9.9(ii) Relation to Modulus and Phase
9.9.1 (ak) =
a0
k+1
=k;
9.9.2 (bk) = (b0
k) = (k 1
2):
9.9.3 Ai0(ak) =( 1)k 1
M(ak);Bi0(bk) =( 1)k 1
M(bk);
9.9.4 Ai(a0
k) =( 1)k 1
N(a0
k);Bi(b0
k) =( 1)k
N(b0
k):
9.9(iii) Derivatives With Respect to k
Ifkis regarded as a continuous variable, then
9.9.5Ai0(ak) = ( 1)k 1
dak
dk 1=2
;
Ai(a0
k) = ( 1)k 1
a0
kda0
k
dk 1=2
:
See Olver (1954, Appendix).
9.9 Zeros 201
9.9(iv) Asymptotic Expansions
For largek
9.9.6 ak= T 3
8(4k 1)
;
9.9.7 Ai0(ak) = ( 1)k 1V 3
8(4k 1)
;
9.9.8 a0
k= U 3
8(4k 3)
;
9.9.9 Ai(a0
k) = ( 1)k 1W 3
8(4k 3)
:
9.9.10 bk= T 3
8(4k 3)
;
9.9.11 Bi0(bk) = ( 1)k 1V 3
8(4k 3)
;9.9.12 b0
k= U 3
8(4k 1)
;
9.9.13 Bi(b0
k) = ( 1)kW 3
8(4k 1)
:
9.9.14
k=ei=3T 3
8(4k 1) +3
4iln 2
;
9.9.15
Bi0( k) = ( 1)kp
2e i=6V 3
8(4k 1) +3
4iln 2
;
9.9.16
0
k=ei=3U 3
8(4k 3) +3
4iln 2
;
9.9.17
Bi( 0
k) = ( 1)k 1p
2ei=6W 3
8(4k 3) +3
4iln 2
:
Here
9.9.18 T(t)t2=3
1 +5
48t 2 5
36t 4+77125
82944t 6 1080 56875
69 67296t 8+16 23755 96875
3344 30208t 10
;
9.9.19U(t)t2=3
1 7
48t 2+35
288t 4 1 81223
2 07360t 6+186 83371
12 44160t 8 9 11458 84361
1911 02976t 10+
;
9.9.20V(t) 1=2t1=6
1 +5
48t 2 1525
4608t 4+23 97875
6 63552t 6 7 48989 40625
8918 13888t 8+14419 83037 34375
4 28070 66624t 10
;
9.9.21
W(t) 1=2t 1=6
1 7
96t 2+1673
6144t 4 843 94709
265 42080t 6+78 02771 35421
1 01921 58720t 8 20444 90510 51945
6 52298 15808t 10+
:
For higher terms see Fabijonas and Olver (1999).
For error bounds for the asymptotic expansions of ak,bk,a0
k, andb0
ksee Pittaluga and Sacripante (1991), and a
conjecture given in Fabijonas and Olver (1999).
9.9(v) Tables
Tables 9.9.1 and 9.9.2 give 10D values of the rst ve real zeros of Ai, Ai0, Bi, Bi0, together with the associated values
of the derivative or the function. Tables 9.9.3 and 9.9.4 give the corresponding results for the rst ve complex zeros
of Bi and Bi0in the upper half plane.
For versions of Tables 9.9.1{9.9.4 that cover k= 1(1)10 see http://dlmf.nist.gov/9.9.v .
Table 9.9.1 : Zeros of Ai and Ai0.
k a k Ai0(ak) a0
k Ai(a0
k)
1 2:33810 74105 0 :70121 08227 1:01879 29716 0 :53565 66560
2 4:08794 94441 0:80311 13697 3:24819 75822 0:41901 54780
3 5:52055 98281 0 :86520 40259 4:82009 92112 0 :38040 64686
4 6:78670 80901 0:91085 07370 6:16330 73556 0:35790 79437
5 7:94413 35871 0 :94733 57094 7:37217 72550 0 :34230 12444
Table 9.9.2 : Real zeros of Bi and Bi0.
k b k Bi0(bk) b0
k Bi(b0
k)
1 1:17371 32227 0 :60195 78880 2:29443 96826 0:45494 43836
2 3:27109 33028 0:76031 01415 4:07315 50891 0 :39652 28361
3 4:83073 78417 0 :83699 10126 5:51239 57297 0:36796 91615
4 6:16985 21283 0:88947 99014 6:78129 44460 0 :34949 91168
5 7:37676 20794 0 :92998 36386 7:94017 86892 0:33602 62401
202 Airy and Related Functions
Table 9.9.3 : Complex zeros of Bi.
e i=3 k Bi0( k)
k modulus phase modulus phase
1 2:35387 33809 0 :09533 49591 0:99310 68457 2 :64060 02521
2 4:09328 73094 0 :04178 55604 1:13612 83345 0:51328 28720
3 5:52350 35011 0 :02668 05442 1:22374 37881 2 :62462 83591
4 6:78865 95301 0 :01958 69751 1:28822 92493 0:51871 63829
5 7:94555 90160 0 :01547 08228 1:33979 47726 2 :62185 44560
Table 9.9.4 : Complex zeros of Bi0.
e i=3 0
k Bi( 0
k)
k modulus phase modulus phase
1 1:12139 32942 0 :33072 66208 0:75004 14897 0 :46597 78930
2 3:25690 82266 0 :05938 99367 0:59221 66315 2:63235 40329
3 4:82400 26102 0 :03278 56423 0:53787 06321 0 :51549 32992
4 6:16568 66408 0 :02266 24588 0:50611 02160 2:62362 85920
5 7:37383 79870 0 :01731 96481 0:48406 00643 0 :51928 28169
9.10 Integrals
9.10(i) Inde nite Integrals
9.10.1Z1
zAi(t)dt=
Ai(z) Gi0(z) Ai0(z) Gi(z)
;
9.10.2Zz
1Ai(t)dt=
Ai(z) Hi0(z) Ai0(z) Hi(z)
;
9.10.3Zz
1Bi(t)dt=Zz
0Bi(t)dt
=
Bi0(z) Gi(z) Bi(z) Gi0(z)
=
Bi(z) Hi0(z) Bi0(z) Hi(z)
:
For the functions Gi and Hi see x9.12.
9.10(ii) Asymptotic Approximations
9.10.4Z1
xAi(t)dt1
2 1=2x 3=4exp
2
3x3=2
,x!1;
9.10.5Zx
0Bi(t)dt 1=2x 3=4exp
2
3x3=2
,x!1:
9.10.6Zx
1Ai(t)dt= 1=2( x) 3=4cos
2
3( x)3=2+1
4
+O
jxj 9=4
, x! 1;9.10.7Zx
1Bi(t)dt= 1=2( x) 3=4sin
2
3( x)3=2+1
4
+O
jxj 9=4
, x! 1:
For higher terms in (9.10.4){(9.10.7) see Vall ee and
Soares (2004,x3.1.3). For error bounds see Boyd (1993).
See also Muldoon (1970).
9.10(iii) Other Inde nite Integrals
Letw(z) be any solution of Airy's equation (9.2.1).
Then
9.10.8Z
zw(z)dz=w0(z);
9.10.9Z
z2w(z)dz=zw0(z) w(z);
9.10.10Z
zn+3w(z)dz=zn+2w0(z) (n+ 2)zn+1w(z)
+ (n+ 1)(n+ 2)Z
znw(z)dz,
n= 0;1;2;::::
See alsox9.11(iv).
9.10(iv) De nite Integrals
9.10.11Z1
0Ai(t)dt=1
3,Z0
1Ai(t)dt=2
3,
9.11 Products 203
9.10.12Z0
1Bi(t)dt= 0:
9.10(v) Laplace Transforms
9.10.13Z1
1eptAi(t)dt=ep3=3,<p>0:
9.10.14Z1
0e ptAi(t)dt=e p3=3
1
3 p1F1 1
3;4
3;1
3p3
34=3 4
3
+p21F1 2
3;5
3;1
3p3
35=3 5
3!
,
p2C.
9.10.15Z1
0e ptAi( t)dt
=1
3ep3=3
1
3;1
3p3
1
3+ 2
3;1
3p3
2
3!
,<p>0;
9.10.16Z1
0e ptBi( t)dt
=1p
3ep3=3
2
3;1
3p3
2
3 1
3;1
3p3
1
3!
,
<p>0:
For the con
uent hypergeometric function 1F1and the
incomplete gamma function see xx13.1, 13.2, and
8.2(i).
For Laplace transforms of products of Airy functions
see Shawagfeh (1992).
9.10(vi) Mellin Transform
9.10.17Z1
0t 1Ai(t)dt= ( )
3( +2)=3 1
3 +2
3,< >0:
9.10(vii) Stieltjes Transforms
9.10.18
Ai(z) =z5=4e (2=3)z3=2
27=2Z1
0t 1=2e (2=3)t3=2Ai(t)
z3=2+t3=2dt,
jphzj<2
3:
9.10.19
Bi(x) =x5=4e(2=3)x3=2
25=2Z1
0t 1=2e (2=3)t3=2Ai(t)
x3=2 t3=2dt,
x>0;
where the last integral is a Cauchy principal value
(x1.4(v)).9.10(viii) Repeated Integrals
9.10.20Zx
0Zv
0Ai(t)dtdv =xZx
0Ai(t)dt Ai0(x) + Ai0(0);
9.10.21Zx
0Zv
0Bi(t)dtdv =xZx
0Bi(t)dt Bi0(x) + Bi0(0);
9.10.22Z1
0Z1
tZ1
tAi( t)(dt)n=2 cos 1
3(n 1)
3(n+2)=3 1
3n+2
3,
n= 1;2;::::
9.10(ix) Compendia
For further integrals, including the Airy transform,
seex9.11(iv), Widder (1979), Prudnikov et al. (1990,
x1.8.1), Prudnikov et al. (1992a, pp. 405{413), Prud-
nikov et al. (1992b,x4.3.25), Vall ee and Soares (2004,
Chapters 3, 4).
9.11 Products
9.11(i) Di erential Equation
9.11.1d3w
dz3 4zdw
dz 2w= 0,w=w1w2;
wherew1andw2are any solutions of (9.2.1). For
example,w= Ai2(z), Ai(z) Bi(z), Ai(z) Ai
ze2i=3
,
M2(z). Numerically satisfactory triads of solutions can
be constructed where needed on RorCby inspection
of the asymptotic expansions supplied in x9.7.
9.11(ii) Wronskian
9.11.2 W
Ai2(z);Ai(z) Bi(z);Bi2(z)
= 2 3:
9.11(iii) Integral Representations
9.11.3 Ai2(x) =1
4p
3Z1
0J0 1
12t3+xt
tdt,x0;
whereJ0is the Bessel function ( x10.2(ii)).
9.11.4
Ai2(z) + Bi2(z) =1
3=2Z1
0exp
zt 1
12t3
t 1=2dt:
For an integral representation of the Dirac delta in-
volving a product of two Ai functions see x1.17(ii).
For further integral representations see Reid (1995,
1997a,b).
204 Airy and Related Functions
9.11(iv) Inde nite Integrals
Letw1;w2be any solutions of (9.2.1), not necessarily
distinct. Then
9.11.5Z
w1w2dz= w0
1w0
2+zw1w2;
9.11.6Z
w1w0
2dz=1
2(w1w2+zWfw1;w2g);
9.11.7Z
w0
1w0
2dz=1
3(w1w0
2+w0
1w2+zw0
1w0
2 z2w1w2);
9.11.8Z
zw1w2dz=1
6(w1w0
2+w0
1w2) 1
3(zw0
1w0
2 z2w1w2);
9.11.9Z
zw1w0
2dz=1
2w0
1w0
2+1
4z2Wfw1;w2g;
9.11.10Z
zw0
1w0
2dz=3
10( w1w2+zw1w0
2+zw0
1w2)
+1
5(z2w0
1w0
2 z3w1w2):
ForR
znw1w2dz,R
znw1w0
2dz,R
znw0
1w0
2dz, where
nis any positive integer, see Albright (1977). For re-
lated integrals see Gordon (1969, Appendix B).
For any continuously-di erentiable function f
9.11.11Z1
w2
1f0w2
w1
dz=1
Wfw1;w2gfw2
w1
:
Examples
9.11.12Zdz
Ai2(z)=Bi(z)
Ai(z);
9.11.13Zdz
Ai(z) Bi(z)=lnBi(z)
Ai(z)
;
9.11.14ZAi(z) Bi(z)
Ai2(z) + Bi2(z)2dz=
2Bi2(z)
Ai2(z) + Bi2(z):
9.11(v) De nite Integrals
9.11.15Z1
0t 1Ai2(t)dt=2 ( )
1=212(2 +5)=6 1
3 +5
6,
< >0.
9.11.16Z1
1Ai3(t)dt= 2 1
3
42;
9.11.17Z1
1Ai2(t) Bi(t)dt= 2 1
3
4p
32:
9.11.18Z1
0Ai4(t)dt=ln 3
242:
9.11.19Z1
0dt
Ai2(t) + Bi2(t)=Z1
0tdt
Ai02(t) + Bi02(t)=2
6:For further de nite integrals see Prudnikov et al.
(1990,x1.8.2), Laurenzi (1993), Reid (1995, 1997a,b),
and Vall ee and Soares (2004, Chapters 3, 4).
Related Functions
9.12 Scorer Functions
9.12(i) Di erential Equation
9.12.1d2w
dz2 zw=1
:
Solutions of this equation are the Scorer functions and
can be found by the method of variation of parameters
(x1.13(iii)). The general solution is given by
9.12.2w(z) =Aw1(z) +Bw2(z) +p(z);
whereAandBare arbitrary constants, w1(z) andw2(z)
are any two linearly independent solutions of Airy's
equation (9.2.1), and p(z) is any particular solution of
(9.12.1). Standard particular solutions are
9.12.3 Gi(z) , Hi(z) ,e2i=3Hi
ze2i=3
,
where
9.12.4 Gi(z) = Bi(z)Z1
zAi(t)dt+ Ai(z)Zz
0Bi(t)dt;
9.12.5 Hi(z) = Bi(z)Zz
1Ai(t)dt Ai(z)Zz
1Bi(t)dt:
Gi(z) and Hi(z) are entire functions of z.
9.12(ii) Graphs
See Figures 9.12.1 and 9.12.2.
9.12(iii) Initial Values
9.12.6Gi(0) =1
2Hi(0) =1
3Bi(0)
= 1.
37=6 2
3
= 0:20497 55424 :::;
9.12.7Gi0(0) =1
2Hi0(0) =1
3Bi0(0) = 1.
35=6 1
3
= 0:14942 94524 ::::
9.12(iv) Numerically Satisfactory Solutions
Gi(x) is a numerically satisfactory companion to the
complementary functions Ai( x) and Bi(x) on the inter-
val 0x<1. Hi(x) is a numerically satisfactory com-
panion to Ai( x) and Bi(x) on the interval 1<x0.
InC, numerically satisfactory sets of solutions are
given by
9.12.8 Gi(z);Ai(z);Bi(z),jphzj1
3;
9.12 Scorer Functions 205
Figure 9.12.1 : Gi(x), Gi0(x).
Figure 9.12.2 : Hi(x), Hi0(x).
9.12.9
Hi(z);Ai
ze 2i=3
;Ai
ze2i=3
,jph( z)j2
3;
and
9.12.10e2i=3Hi
ze2i=3
;Ai(z);Ai
ze2i=3
,
phz1
3:
9.12(v) Connection Formulas
9.12.11 Gi(z) + Hi(z) = Bi(z);
9.12.12
Gi(z) =1
2ei=3Hi
ze 2i=3
+1
2e i=3Hi
ze2i=3
;
9.12.13 Gi(z) =ei=3Hi
ze2i=3
iAi(z);
9.12.14
Hi(z) =e2i=3Hi
ze2i=3
+ 2ei=6Ai
ze2i=3
:
9.12(vi) Maclaurin Series
9.12.15
Gi(z) =3 2=3
1X
k=0cos2k 1
3
k+ 1
3(31=3z)k
k!;
9.12.16
Gi0(z) =3 1=3
1X
k=0cos2k+ 1
3
k+ 2
3(31=3z)k
k!:
9.12.17 Hi(z) =3 2=3
1X
k=0 k+ 1
3(31=3z)k
k!;
9.12.18 Hi0(z) =3 1=3
1X
k=0 k+ 2
3(31=3z)k
k!:9.12(vii) Integral Representations
9.12.19 Gi(x) =1
Z1
0sin 1
3t3+xt
dt,x2R.
9.12.20 Hi(z) =1
Z1
0exp
1
3t3+zt
dt;
9.12.21
Gi(z) = 1
Z1
0exp
1
3t3 1
2zt
cos
1
2p
3zt+2
3
dt:
If=2
3z3=2or2
3x3=2, andK1=3is the modi ed
Bessel function (x10.25(ii)), then
9.12.22 Hi( z) =4z2
33=22Z1
0K1=3(t)
2+t2dt,jphzj<1
3;
9.12.23 Gi(x) =4x2
33=22Z1
0K1=3(t)
2 t2dt,x>0;
where the last integral is a Cauchy principal value
(x1.4(v)).
Mellin{Barnes Type Integral
9.12.24
Hi(z) =3 2=3
22iZi1
i1 1
3+1
3t
( t)(31=3eiz)tdt;
where the integration contour separates the poles of
1
3+1
3t
from those of ( t).
9.12(viii) Asymptotic Expansions
Functions and Derivatives
Asz!1 , and withdenoting an arbitrary small pos-
itive constant,
9.12.25 Gi(z)1
z1X
k=0(3k)!
k!(3z3)k,jphzj1
3 ;
9.12.26 Gi0(z) 1
z21X
k=0(3k+ 1)!
k!(3z3)k,jphzj1
3 :
9.12.27
Hi(z) 1
z1X
k=0(3k)!
k!(3z3)k,jph( z)j2
3 ;
206 Airy and Related Functions
9.12.28
Hi0(z)1
z21X
k=0(3k+ 1)!
k!(3z3)k,jph( z)j2
3 :
For other phase ranges combine these results with
the connection formulas (9.12.11){(9.12.14) and the
asymptotic expansions given in x9.7. For example, with
the notation ofx9.7(i).
9.12.29Hi(z) 1
z1X
k=0(3k)!
k!(3z3)k+e
pz1=41X
k=0uk
k,
jphzj :
Integrals
9.12.30Zz
0Gi(t)dt1
lnz+2
+ ln 3
3 1
1X
k=1(3k 1)!
k!(3z3)k,
jphzj1
3 :
9.12.31Zz
0Hi( t)dt1
lnz+2
+ ln 3
3
+1
1X
k=1( 1)k 1(3k 1)!
k!(3z3)k,
jphzj2
3 ;
where
is Euler's constant ( x5.2(ii)).
9.12(ix) Zeros
All zeros, real or complex, of Gi( z) and Hi(z) are simple.
Neither Hi( z) nor Hi0(z) has real zeros.
Gi(z) has no nonnegative real zeros and Gi0(z)
has exactly one nonnegative real zero, given by z=
0:60907 54170 7 :::. Both Gi( z) and Gi0(z) have an
in nity of negative real zeros, and they are interlaced.
For the above properties and further results, includ-
ing the distribution of complex zeros, asymptotic ap-
proximations for the numerically large real or complex
zeros, and numerical tables see Gil et al. (2003c).
For graphical illustration of the real zeros see Fig-
ures 9.12.1 and 9.12.2.9.13 Generalized Airy Functions
9.13(i) Generalizations from the Di erential
Equation
Equations of the form
9.13.1d2w
dz2=znw,n= 1;2;3;:::;
are used in approximating solutions to di erential equa-
tions with multiple turning points; see x2.8(v). The
general solution of (9.13.1) is given by
9.13.2 w=z1=2Zp();
where
9.13.3p=1
n+ 2,=2
n+ 2z(n+2)=2= 2pz1=(2p),
andZpis any linear combination of the modi ed Bessel
functionsIpandepiKp(x10.25(ii)).
Swanson and Headley (1967) de ne independent so-
lutionsAn(z) andBn(z) of (9.13.1) by
9.13.4An(z) = (2p=) sin(p)z1=2Kp() ,
Bn(z) = (pz)1=2(I p() +Ip()) ,
whenzis real and positive, and by analytic continuation
elsewhere. (All solutions of (9.13.1) are entire functions
ofz.) Whenn= 1;An(z) andBn(z) become Ai( z) and
Bi(z), respectively.
Properties of An(z) andBn(z) follow from the cor-
responding properties of the modi ed Bessel functions.
They include:
9.13.5An(0) =p1=2Bn(0) =p1 p
(1 p);
A0
n(0) =p1=2B0
n(0) =pp
(p):
9.13.6An( z) =(
pz1=2(J p() +Jp()); n odd;
p1=2Bn(z); n even;
9.13.7Bn( z) =(
(pz)1=2(J p() Jp()); n odd;
p 1=2An(z); n even:
9.13.8 WfAn(z);Bn(z)g=2
p1=2sin(p):
Asz!1
9.13.9 An(z) =p
p/sin(p)z n=4e
1 +O
1
, jphzj3p ;
9.13.10An( z) =(
2p
p=cos 1
2p
z n=4
cos
1
4
+ej=jO
1
;jphzj2p ,nodd;p
p=z n=4e
1 +O
1
; jphzjp ,neven;
9.13.11 Bn(z) = 1=2z n=4e
1 +O
1
, jphzjp ;
9.13.12Bn( z) =(
(2/p) sin 1
2p
z n=4
sin
1
4
+ej=jO
1
;jphzj2p ;nodd;
(1/p) sin(p)z n=4e
1 +O
1
; jphzj3p ;neven:
9.13 Generalized Airy Functions 207
The distribution in Cand asymptotic properties of the
zeros ofAn(z),A0
n(z),Bn(z), andB0
n(z) are investi-
gated in Swanson and Headley (1967) and Headley and
Barwell (1975).
In Olver (1977a, 1978) a di erent normalization is
used. In place of (9.13.1) we have
9.13.13d2w
dt2=1
4m2tm 2w;
wherem= 3;4;5;:::: For real variables the solutions
of (9.13.13) are denoted by Um(t),Um( t) whenmis
even, and by Vm(t),Vm(t) whenmis odd. (The over-
bar has nothing to do with complex conjugates.) Their
relations to the functions An(z) andBn(z) are given by
9.13.14m=n+ 2 = 1=p,t= (1
2m) 2=mz=2=m,
9.13.15p
2 1
2m(m 1)=mcsc(/m)An(z)
=(
Um(t); m even;
Vm(t); m odd;
9.13.16p 1
2m(m 2)=(2m)csc(/m)Bn(z)
=(
Um( t); m even;
Vm(t); m odd:
Properties and graphs of Um(t),Vm(t),Vm(t) are
included in Olver (1977a) together with properties and
graphs of real solutions of the equation
9.13.17d2w
dt2= 1
4m2tm 2w,meven;
which are denoted by Wm(t),Wm( t).
InC, the solutions of (9.13.13) used in Olver (1978)
are
9.13.18 w=Um(te 2ji=m),j= 0;1;2;::::
The function on the right-hand side is recessive in the
sector (2j 1)=mphz(2j+1)=m, and is there-
fore an essential member of any numerically satisfactory
pair of solutions in this region.
Another normalization of (9.13.17) is used in
Smirnov (1960), given by
9.13.19d2w
dx2+x w= 0;
where > 2 andx >0. Solutions are w=U1(x; ),
U2(x; ), where
9.13.20
U1(x; )
=1
( + 2)1=( +2)
+ 1
+ 2
x1=2J 1=( +2)2
+ 2x( +2)=2
;9.13.21
U2(x; ) = ( + 2)1=( +2)
+ 3
+ 2
x1=2J1=( +2)2
+ 2x( +2)=2
;
andJdenotes the Bessel function ( x10.2(ii)).
When is a positive integer the relation of these
functions to Wm(t),Wm( t) is as follows:
9.13.22 =m 2 ,x= (m=2)2=mt,
9.13.23
U1(x; ) =1=2
2(m+2)=(2m) (1=m)(Wm(t) +Wm( t));
9.13.24
U2(x; ) =1=2m2=m
2(m+2)=(2m) ( 1=m)(Wm(t) Wm( t)):
For properties of the zeros of the functions de ned
in this subsection see Laforgia and Muldoon (1988) and
references given therein.
9.13(ii) Generalizations from Integral
Representations
Reid (1972) and Drazin and Reid (1981, Appendix) in-
troduce the following contour integrals in constructing
approximate solutions to the Orr{Sommerfeld equation
for
uid
ow:
9.13.25Ak(z;p) =1
2iZ
Lkt pexp
zt 1
3t3
dt,
k= 1;2;3,p2C;
9.13.26B0(z;p) =1
2iZ
L0t pexp
zt 1
3t3
dt,
p= 0;1;2;:::;
9.13.27Bk(z;p) =Z
Ikt pexp
zt 1
3t3
dt,
k= 1;2;3,p= 0;1;2;:::;
withz2Cin all cases. The integration paths L0,L1,
L2,L3are depicted in Figure 9.13.1. I1,I2,I3are
depicted in Figure 9.13.2. When pis not an integer
the branch of t pin (9.13.25) is usually chosen to be
exp( p(lnjtj+ipht)) with 0pht<2.
208 Airy and Related Functions
Figure 9.13.1 :t-plane. Paths L0,L1,L2,L3.
Figure 9.13.2 :t-plane. Paths I1,I2,I3.
Whenp= 0
9.13.28 A1(z;0) = Ai(z);
9.13.29A2(z;0) =e2i=3Ai
ze2i=3
,
A3(z;0) =e 2i=3Ai
ze 2i=3
,
and
9.13.30B0(z;0) = 0 ,B1(z;0) =Hi(z) .
Each of the functions Ak(z;p) andBk(z;p) satis es
the di erential equation
9.13.31d3w
dz3 zdw
dz+ (p 1)w= 0;
and the di erence equation
9.13.32f(p 3) zf(p 1) + (p 1)f(p) = 0:
TheAk(z;p) are related by
9.13.33A2(z;p) =e 2(p 1)i=3A1
ze2i=3;p
,
A3(z;p) =e2(p 1)i=3A1
ze 2i=3;p
.
Connection formulas for the solutions of (9.13.31) in-
clude
9.13.34A1(z;p) +A2(z;p) +A3(z;p) +B0(z;p) = 0;
9.13.35B2(z;p) B3(z;p) = 2iA 1(z;p);
9.13.36B3(z;p) B1(z;p) = 2iA 2(z;p);
9.13.37B1(z;p) B2(z;p) = 2iA 3(z;p):
Further properties of these functions, and also of
similar contour integrals containing an additional fac-
tor (lnt)q,q= 1;2;:::; in the integrand, are derivedin Reid (1972), Drazin and Reid (1981, Appendix), and
Baldwin (1985). These properties include Wronskians,
asymptotic expansions, and information on zeros.
For further generalizations via integral representa-
tions see Chin and Hedstrom (1978), Janson et al. (1993,
x10), and Kamimoto (1998).
9.14 Incomplete Airy Functions
Incomplete Airy functions are de ned by the con-
tour integral (9.5.4) when one of the integration lim-
its is replaced by a variable real or complex param-
eter. For information, including asymptotic approxi-
mations, computation, and applications, see Levey and
Felsen (1969), Constantinides and Marhefka (1993), and
Michaeli (1996).
Applications
9.15 Mathematical Applications
Airy functions play an indispensable role in the con-
struction of uniform asymptotic expansions for contour
integrals with coalescing saddle points, and for solu-
tions of linear second-order ordinary di erential equa-
tions with a simple turning point. For descriptions of,
and references to, the underlying theory see xx2.4(v)
and 2.8(iii).
9.16 Physical Applications 209
9.16 Physical Applications
Airy functions are applied in many branches of both
classical and quantum physics. The function Ai( x) rst
appears as an integral in two articles by G.B. Airy on
the intensity of light in the neighborhood of a caustic
(Airy (1838, 1849)). Details of the Airy theory are given
in van de Hulst (1957) in the chapter on the optics of a
raindrop. See also Berry (1966, 1969).
The frequent appearances of the Airy functions in
both classical and quantum physics is associated with
wave equations with turning points, for which asymp-
totic (WKBJ) solutions are exponential on one side and
oscillatory on the other. The Airy functions consti-
tute uniform approximations whose region of validity in-
cludes the turning point and its neighborhood. Within
classical physics, they appear prominently in physical
optics, electromagnetism, radiative transfer,
uid me-
chanics, and nonlinear wave propagation. Examples
dealing with the propagation of light and with radiation
of electromagnetic waves are given in Landau and Lif-
shitz (1962). Extensive use is made of Airy functions in
investigations in the theory of electromagnetic di rac-
tion and radiowave propagation (Fock (1965)). A quite
di erent application is made in the study of the di rac-
tion of sound pulses by a circular cylinder (Friedlander
(1958)).
In
uid dynamics, Airy functions enter several
topics. In the study of the stability of a two-
dimensional viscous
uid, the
ow is governed by the
Orr{Sommerfeld equation (a fourth-order di erential
equation). Again, the quest for asymptotic approxima-
tions that are uniformly valid solutions to this equa-
tion in the neighborhoods of critical points leads (af-
ter choosing solvable equations with similar asymptotic
properties) to Airy functions. Other applications ap-
pear in the study of instability of Couette
ow of an in-
viscid
uid. These examples of transitions to turbulence
are presented in detail in Drazin and Reid (1981) with
the problem of hydrodynamic stability. The investiga-
tion of the transition between subsonic and supersonic
of a two-dimensional gas
ow leads to the Euler{Tricomi
equation (Landau and Lifshitz (1987)). An application
of Airy functions to the solution of this equation is given
in Gramtche (1981).
Airy functions play a prominent role in problems de-
ned by nonlinear wave equations. These rst appeared
in connection with the equation governing the evolution
of long shallow water waves of permanent form, gener-
ally called solitons, and are predicted by the Korteweg{
de Vries (KdV) equation (a third-order nonlinear partial
di erential equation). The KdV equation and solitons
have applications in many branches of physics, including
plasma physics lattice dynamics, and quantum mechan-ics. (Ablowitz and Segur (1981), Ablowitz and Clarkson
(1991), and Whitham (1974).)
Reference to many of these applications as well as
to the theory of elasticity and to the heat equation
are given in Vall ee and Soares (2004): a book devoted
speci cally to the Airy and Scorer functions and their
applications in physics.
An example from quantum mechanics is given in
Landau and Lifshitz (1965), in which the exact solu-
tion of the Schr odinger equation for the motion of a
particle in a homogeneous external eld is expressed in
terms of Ai( x). Solutions of the Schr odinger equation
involving the Airy functions are given for other poten-
tials in Vall ee and Soares (2004). This reference pro-
vides several examples of applications to problems in
quantum mechanics in which Airy functions give uni-
form asymptotic approximations, valid in the neighbor-
hood of a turning point. A study of the semiclassical
description of quantum-mechanical scattering is given
in Ford and Wheeler (1959a,b). In the case of the rain-
bow, the scattering amplitude is expressed in terms of
Ai(x), the analysis being similar to that given originally
by Airy (1838) for the corresponding problem in optics.
An application of the Scorer functions is to the prob-
lem of the uniform loading of in nite plates (Rothman
(1954a,b)).
Computation
9.17 Methods of Computation
9.17(i) Maclaurin Expansions
Although the Maclaurin-series expansions of xx9.4 and
9.12(vi) converge for all nite values of z, they are cum-
bersome to use when jzjis large owing to slowness of
convergence and cancellation. For large jzjthe asymp-
totic expansions of xx9.7 and 9.12(viii) should be used
instead. Since these expansions diverge, the accuracy
they yield is limited by the magnitude of jzj. How-
ever, in the case of Ai( z) and Bi(z) this accuracy can
be increased considerably by use of the exponentially-
improved forms of expansion supplied in x9.7(v).
9.17(ii) Di erential Equations
A comprehensive and powerful approach is to integrate
the de ning di erential equation (9.2.1) by direct nu-
merical methods. As described in x3.7(ii), to ensure sta-
bility the integration path must be chosen in such a way
that as we proceed along it the wanted solution grows
at least as fast as all other solutions of the di erential
equation. In the case of Ai( z), for example, this means
210 Airy and Related Functions
that in the sectors1
3 <jphzj< we may integrate
along outward rays from the origin with initial values
obtained fromx9.2(ii). But when jphzj<1
3the inte-
gration has to be towards the origin, with starting values
of Ai(z) and Ai0(z) computed from their asymptotic ex-
pansions. On the remaining rays, given by ph z=1
3
and, integration can proceed in either direction.
For further information see Lozier and Olver (1993)
and Fabijonas et al. (2004). The former reference in-
cludes a parallelized version of the method.
In the case of the Scorer functions, integration of
the di erential equation (9.12.1) is more dicult than
(9.2.1), because in some regions stable directions of in-
tegration do not exist. An example is provided by Gi( x)
on the positive real axis. In these cases boundary-value
methods need to be used instead; see x3.7(iii).
9.17(iii) Integral Representations
Among the integral representations of the Airy func-
tions the Stieltjes transform (9.10.18) furnishes a way
of computing Ai( z) in the complex plane, once values
of this function can be generated on the positive real
axis. For details, including the application of a gener-
alized form of Gaussian quadrature, see Gordon (1969,
Appendix A) and Schulten et al. (1979).
Gilet al. (2002a) describes two methods for the
computation of Ai( z) and Ai0(z) forz2C. In the
rst method the integration path for the contour inte-
gral (9.5.4) is deformed to coincide with paths of steep-
est descent (x2.4(iv)). The trapezoidal rule ( x3.5(i)) is
then applied. The second method is to apply general-
ized Gauss{Laguerre quadrature ( x3.5(v)) to the inte-
gral (9.5.8). For the second method see also Gautschi
(2002a). The methods for Ai0(z) are similar.
For quadrature methods for Scorer functions see
Gilet al. (2001), Lee (1980), and Gordon (1970, Ap-
pendix A); but see also Gautschi (1983).
9.17(iv) Via Bessel Functions
In consequence of x9.6(i), algorithms for generating
Bessel functions, Hankel functions, and modi ed Bessel
functions (x10.74) can also be applied to Ai( z), Bi(z),
and their derivatives.
9.17(v) Zeros
Zeros of the Airy functions, and their derivatives, can be
computed to high precision via Newton's rule ( x3.8(ii))
or Halley's rule ( x3.8(v)), using values supplied by the
asymptotic expansions of x9.9(iv) as initial approxima-
tions. This method was used in the computation of the
tables inx9.9(v). See also Fabijonas et al. (2004).
For the computation of the zeros of the Scorer func-
tions and their derivatives see Gil et al. (2003c).9.18 Tables
9.18(i) Introduction
Additional listings of early tables of the functions
treated in this chapter are given in Fletcher et al. (1962)
and Lebedev and Fedorova (1960).
9.18(ii) Real Variables
Miller (1946) tabulates Ai( x), Ai0(x) forx=
20(:01)2; log10Ai(x), Ai0(x)=Ai(x) forx=
0(:1)25(1)75; Bi( x), Bi0(x) forx= 10(:1)2:5;
log10Bi(x), Bi0(x)=Bi(x) forx= 0(:1)10;M(x),
N(x),(x), (x) (respectively F(x),G(x), (x),
(x)) forx= 80(1) 30(:1)0. Precision is gen-
erally 8D; slightly less for some of the auxiliary
functions. Extracts from these tables are included
in Abramowitz and Stegun (1964, Chapter 10),
together with some auxiliary functions for large
arguments.
Fox (1960, Table 3) tabulates 2 1=2x1=4
exp(2
3x3=2) Ai(x), 21=2x 1=4exp(2
3x3=2) Ai0(x),
1=2x1=4exp( 2
3x3=2) Bi(x), and1=2x 1=4
exp( 2
3x3=2) Bi0(x) for3
2x 3=2= 0(:001)0:05, to-
gether with similar auxiliary functions for negative
values ofx. Precision is 10D.
Zhang and Jin (1996, p. 337) tabulates Ai( x),
Ai0(x), Bi(x), Bi0(x) forx= 0(1)20 to 8S and
forx= 20(1)0 to 9D.
Yakovleva (1969) tabulates Fock's functions
U(x)pBi(x),U0(x)pBi0(x),V(x)pAi(x),V0(x)pAi0(x) forx= 9(:001)9.
Precision is 7S.
9.18(iii) Complex Variables
Woodward and Woodward (1946) tabulates the
real and imaginary parts of Ai( z), Ai0(z), Bi(z),
Bi0(z) for<z= 2:4(:2)2:4,=z= 2:4(:2)0. Pre-
cision is 4D.
Harvard (1945) tabulates the real and imaginary
parts ofh1(z),h0
1(z),h2(z),h0
2(z) for x0
<zx0, 0 =zy0,jx0+iy0j<6:1,
with interval 0.1 in <zand=z. Precision is 8D.
Hereh1(z) = 24=331=6iAi
e i=3z
,h2(z) =
24=331=6iAi
ei=3z
.
9.19 Approximations 211
9.18(iv) Zeros
Miller (1946) tabulates ak, Ai0(ak),a0
k, Ai(a0
k),
k= 1(1)50;bk, Bi0(bk),b0
k, Bi(b0
k),k= 1(1)20.
Precision is 8D. Entries for k= 1(1)20 are repro-
duced in Abramowitz and Stegun (1964, Chap-
ter 10).
Sherry (1959) tabulates ak, Ai0(ak),a0
k, Ai(a0
k),
k= 1(1)50; 20S.
Zhang and Jin (1996, p. 339) tabulates ak, Ai0(ak),
a0
k, Ai(a0
k),bk, Bi0(bk),b0
k, Bi(b0
k),k= 1(1)20; 8D.
Corless et al. (1992) gives the real and imaginary
parts of kfork= 1(1)13; 14S.
See alsox9.9(v).
9.18(v) Integrals
Rothman (1954a) tabulatesRx
0Ai(t)dtandRx
0Bi(t)dtforx= 10(:1)1and 10(:1)2, re-
spectively; 7D. The entries in the columns headedRx
0Ai( x)dxandRx
0Bi( x)dxall have the wrong
sign. The tables are reproduced in Abramowitz
and Stegun (1964, Chapter 10), and the sign er-
rors are corrected in later reprintings.
NBS (1958) tabulatesRx
0Ai( t)dt andRx
0Rv
0Ai( t)dtdv (see (9.10.20)) for x=
2(:01)5 to 8D and 7D, respectively.
Zhang and Jin (1996, p. 338) tabulatesRx
0Ai(t)dt
andRx
0Bi(t)dtforx= 10(:2)10 to 8D or 8S.
9.18(vi) Scorer Functions
Scorer (1950) tabulates Gi( x) and Hi( x) forx=
0(:1)10; 7D.
Rothman (1954b) tabulatesRx
0Gi(t)dt, Gi0(x),Rx
0Hi( t)dt, Hi0( x) forx= 0(:1)10; 7D.
NBS (1958) tabulates A0(x)Hi( x) and
A0
0(x)Hi0( x) forx= 0(:01)1(:02)5(:05)11
and 1=x= 0:01(:01)0:1;Rx
0A0(t)dtforx=
0:5;1(1)11. Precision is 8D.
Nosova and Tumarkin (1965) tabulates e0(x)
Hi( x),e0
0(x) Hi0( x),ee0( x)
Gi(x),ee0
0( x)Gi0(x) forx= 1(:01)10;
7D. Also included are the real and imaginary parts
ofe0(z) andie0
0(z), wherez=iyandy= 0(:01)9;
6-7D.
Gilet al. (2003c) tabulates the only positive zero
of Gi0(z), the rst 10 negative real zeros of Gi( z)
and Gi0(z), and the rst 10 complex zeros of Gi( z),
Gi0(z), Hi(z), and Hi0(z). Precision is 11 or 12S.9.18(vii) Generalized Airy Functions
Smirnov (1960) tabulates U1(x; ),U2(x; ),
de ned by (9.13.20), (9.13.21), and also
@U1(x; )/@x,@U2(x; )/@x, for = 1,x=
6(:01)10 to 5D or 5S, and also for =1
4,1
3,
1
2,2
3,3
4,5
4,4
3,3
2,5
3,7
4, 2,x= 0(:01)6; 4D.
9.19 Approximations
9.19(i) Approximations in Terms of Elementary
Functions
Mart n et al. (1992) provides two simple formu-
las for approximating Ai( x) to graphical accuracy,
one for 1<x0, the other for 0 x<1.
Moshier (1989,x6.14) provides minimax ratio-
nal approximations for calculating Ai( x), Ai0(x),
Bi(x), Bi0(x). They are in terms of the variable ,
where=2
3x3=2whenxis positive, =2
3( x)3=2
whenxis negative, and = 0 whenx= 0. The
approximations apply when 2 <1, that is,
when 32=3x <1or 1< x 32=3. The
precision in the coecients is 21S.
9.19(ii) Expansions in Chebyshev Series
These expansions are for real arguments xand are sup-
plied in sets of four for each function, corresponding
to intervals 1< xa,ax0, 0xb,
bx <1. The constants aandbare chosen numer-
ically, with a view to equalizing the e ort required for
summing the series.
Prince (1975) covers Ai( x), Ai0(x), Bi(x), Bi0(x).
The Chebyshev coecients are given to 10-11D.
Fortran programs are included. See also Razaz
and Schonfelder (1981).
N emeth (1992, Chapter 8) covers Ai( x),
Ai0(x), Bi(x), Bi0(x), and integralsRx
0Ai(t)dt,Rx
0Bi(t)dt,Rx
0Rv
0Ai(t)dtdv ,Rx
0Rv
0Bi(t)dtdv
(see also (9.10.20) and (9.10.21)). The Cheby-
shev coecients are given to 15D. Chebyshev
coecients are also given for expansions of the
second and higher (real) zeros of Ai( x), Ai0(x),
Bi(x), Bi0(x), again to 15D.
Razaz and Schonfelder (1980) covers Ai( x),
Ai0(x), Bi(x), Bi0(x). The Chebyshev coecients
are given to 30D.
212 Airy and Related Functions
9.19(iii) Approximations in the Complex Plane
Corless et al. (1992) describe a method of approx-
imation based on subdividing Cinto a triangular
mesh, with values of Ai( z), Ai0(z) stored at the
nodes. Ai(z) and Ai0(z) are then computed from
Taylor-series expansions centered at one of the
nearest nodes. The Taylor coecients are gener-
ated by recursion, starting from the stored values
of Ai(z), Ai0(z) at the node. Similarly for Bi( z),
Bi0(z).
9.19(iv) Scorer Functions
MacLeod (1994) supplies Chebyshev-series expan-
sions to cover Gi( x) for 0x<1and Hi(x) for
1< x0. The Chebyshev coecients are
given to 20D.
9.20 Software
Seehttp://dlmf.nist.gov/9.20 .
References
General References
The main references used in writing this chapter are
Miller (1946) and Olver (1997b). For additional bibli-
ographic reading see Bleistein and Handelsman (1975),
Je reys and Je reys (1956), Lebedev (1965), Temme
(1996a), Wasow (1965, 1985), and Wong (1989).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x9.2Miller (1946), Olver (1997b, Chapter 11).
x9.3These graphics were produced by NIST.
x9.4Miller (1946, p. B17), Olver (1997b, p. 54).
x9.5Miller (1946, p. B17), Olver (1997b, p. 103). For
(9.5.4) see Olver (1997b, p. 53). For (9.5.5) com-
bine (9.2.10) and (9.5.4). For (9.5.6) see Reid
(1995). For (9.5.7) see Copson (1963). (9.5.8)
follows from the rst of (9.6.2) and (10.32.9).x9.6Miller (1946, p. B17), Olver (1997b, pp. 392{393).
For (9.6.21){(9.6.24) combine (9.6.1){(9.6.5) and
(10.39.8){(10.39.10). For (9.6.25), (9.6.26) com-
bine (9.6.23), (9.6.24) with (13.14.2) and refer to
x13.1.
x9.7For (9.7.4) and Table 9.7.1 see Olver (1997b,
p. 225). For (9.7.5){(9.7.14) and x9.7(iii) see Olver
(1997b, pp. 392{393 and 413{414). For x9.7(iv)
see (9.6.1){(9.6.5) and Olver (1997b, pp. 266{267).
For (9.7.18){(9.7.23) see Olver (1991b, 1993a).
x9.8For (9.8.9){(9.8.12) combine (9.6.17) and (9.6.18)
with (10.4.3). For (9.8.13) use (9.2.7). For
(9.8.14){(9.8.19) combine (9.8.9){(9.8.12) with
x10.18(ii); see also Olver (1997b, p. 404), Miller
(1946, p. B10). For x9.8(iii) see Olver (1997b,
p. 404). Forx9.8(iv) combine (9.8.9){(9.8.12) with
x10.18(iii); see also Miller (1946, p. B48).
x9.9Olver (1997b, pp. 404, 414{415), Miller (1946,
p. B48), Olver (1954, Appendix). For the com-
putation of Tables 9.9.1{9.9.4 see x9.17(v).
x9.10 For (9.10.1) combine (9.12.4) and its di erenti-
ated form with the rst of (9.10.11). For (9.10.2)
combine (9.12.11) and its di erentiated form with
(9.10.1), then apply (9.2.7) and (9.10.11). (9.10.3)
is proved in a similar manner. For (9.10.4){
(9.10.7) integrate the leading terms of the asymp-
totic expansions given in x9.7(ii) and use (9.10.11),
(9.10.12). To verify (9.10.8){(9.10.10)|and also
(9.10.20), (9.10.21)|di erentiate, and refer to
(9.2.1). For (9.10.11) and (9.10.12) see Olver
(1997b, p. 431). For (9.10.13) see Widder (1979).
For (9.10.14){(9.10.16) see Gibbs (1973, problem
72{21). For (9.10.17) see Olver (1997b, p. 338).
For (9.10.18) and (9.10.19) see Schulten et al.
(1979). To verify (9.10.20) and (9.10.21) di eren-
tiate, and refer to (9.2.1). For (9.10.22) see Olver
(1997b, pp. 342{344).
x9.11 For (9.11.1) seex1.13(v). For (9.11.2) use (9.2.1)
and (9.2.7). For (9.11.3) and (9.11.4) see Lebedev
(1965, p. 142) and Muldoon (1977). For (9.11.5){
(9.11.14) see Albright (1977) and Albright and
Gavathas (1986). For (9.11.15) see Reid (1995).
For (9.11.16) and (9.11.17) see Reid (1997a). For
(9.11.18) see Laurenzi (1993).
For (9.11.19) extend the de nitions of x9.8(i) to
positive values of x, obtain the inde nite inte-
grals of 1=M2(x) andx=N2(x) via the rst two
of (9.8.14), then combine the values of (0) and
(0) given inx9.8(i) with (+1) = (+1) = 0
obtained from (9.8.4), (9.8.8), and x9.7(ii). (Com-
municated by M.E. Muldoon.)
References 213
x9.12 For (9.12.1){(9.12.7) see Olver (1997b, pp. 430{
431). Forx9.12(iv) refer to the asymptotic expan-
sions given inxx9.7(ii) and 9.12(viii). (9.12.11){
(9.12.14) can be veri ed with the aid of xx9.2(ii)
and 9.12(iii). For (9.12.17) expand the integral in
(9.12.20) in powers of ztand integrate term-by-
term by means of (5.2.1). For (9.12.15) substi-
tute into (9.12.11) by means of (9.12.17), (9.4.3),
(9.2.5), (9.2.6), and use (5.5.3). For (9.12.16) and
(9.12.18) use di erentiation. (9.12.20) can be ver-
i ed by showing that the right-hand side satis es
the di erential equation (9.12.1) and the initial
conditions given in x9.12(iii). For (9.12.19) com-
bine (9.5.3) and (9.12.11). For (9.12.21), see Lee
(1980). For (9.12.22), (9.12.23) see Gordon (1970,
Appendix A). For (9.12.24) see Exton (1983). For
(9.12.25), (9.12.27) see Olver (1997b, pp. 431{
432). For (9.12.26), (9.12.28) refer to x2.1(ii).
Except for the constant term, (9.12.31) can beveri ed by termwise integration of (9.12.27). To
evaluate the constant term replace zby x(
0) in (9.12.20) and integrate ( x1.5(v)) to ob-
tainRx
0Hi( t)dt=R1
0(1 e xt)e 1
3t3t 1dt.
Next, integrate the right-hand side of this equa-
tion by parts|integrating the factor t 1and dif-
ferentiating the rest. As x! 1 the asymp-
totic expansions ofR1
0xe xte 1
3t3(lnt)dtandR1
0e xtt2e 1
3t3(lnt)dtfollow from (2.3.9). Also,R1
0t2e 1
3t3(lnt)dtcan be found by replacing1
3t3
bytand referring to the rst of (5.9.18). For
(9.12.30) integrate (9.12.25) and obtain the con-
stant term by combining (9.12.12) and (9.12.31).
(Equations (9.12.30) and (9.12.31) rst appeared
in Rothman (1954b). As noted in this reference
these results were derived by the author of the
present DLMF chapter, but the proof was not in-
cluded.) The graphs were produced by NIST.
Chapter 10
Bessel Functions
F. W. J. Olver1and L. C. Maximon2
Notation 217
10.1 Special Notation . . . . . . . . . . . . . 217
Bessel and Hankel Functions 217
10.2 De nitions . . . . . . . . . . . . . . . . . 217
10.3 Graphics . . . . . . . . . . . . . . . . . . 218
10.4 Connection Formulas . . . . . . . . . . . 222
10.5 Wronskians and Cross-Products . . . . . 222
10.6 Recurrence Relations and Derivatives . . 222
10.7 Limiting Forms . . . . . . . . . . . . . . 223
10.8 Power Series . . . . . . . . . . . . . . . . 223
10.9 Integral Representations . . . . . . . . . 223
10.10 Continued Fractions . . . . . . . . . . . . 226
10.11 Analytic Continuation . . . . . . . . . . . 226
10.12 Generating Function and Associated Series 226
10.13 Other Di erential Equations . . . . . . . 226
10.14 Inequalities; Monotonicity . . . . . . . . . 227
10.15 Derivatives with Respect to Order . . . . 227
10.16 Relations to Other Functions . . . . . . . 228
10.17 Asymptotic Expansions for Large Argument 228
10.18 Modulus and Phase Functions . . . . . . 230
10.19 Asymptotic Expansions for Large Order . 231
10.20 Uniform Asymptotic Expansions for Large
Order . . . . . . . . . . . . . . . . . . . 232
10.21 Zeros . . . . . . . . . . . . . . . . . . . 235
10.22 Integrals . . . . . . . . . . . . . . . . . . 240
10.23 Sums . . . . . . . . . . . . . . . . . . . 246
10.24 Functions of Imaginary Order . . . . . . . 248
Modi ed Bessel Functions 248
10.25 De nitions . . . . . . . . . . . . . . . . . 248
10.26 Graphics . . . . . . . . . . . . . . . . . . 249
10.27 Connection Formulas . . . . . . . . . . . 251
10.28 Wronskians and Cross-Products . . . . . 251
10.29 Recurrence Relations and Derivatives . . 251
10.30 Limiting Forms . . . . . . . . . . . . . . 252
10.31 Power Series . . . . . . . . . . . . . . . . 252
10.32 Integral Representations . . . . . . . . . 252
10.33 Continued Fractions . . . . . . . . . . . . 253
10.34 Analytic Continuation . . . . . . . . . . . 253
10.35 Generating Function and Associated Series 254
10.36 Other Di erential Equations . . . . . . . 254
10.37 Inequalities; Monotonicity . . . . . . . . . 254
10.38 Derivatives with Respect to Order . . . . 254
10.39 Relations to Other Functions . . . . . . . 25410.40 Asymptotic Expansions for Large Argument 255
10.41 Asymptotic Expansions for Large Order . 256
10.42 Zeros . . . . . . . . . . . . . . . . . . . 258
10.43 Integrals . . . . . . . . . . . . . . . . . . 258
10.44 Sums . . . . . . . . . . . . . . . . . . . 260
10.45 Functions of Imaginary Order . . . . . . . 261
10.46 Generalized and Incomplete Bessel Func-
tions; Mittag-Leer Function . . . . . . . 261
Spherical Bessel Functions 262
10.47 De nitions and Basic Properties . . . . . 262
10.48 Graphs . . . . . . . . . . . . . . . . . . . 262
10.49 Explicit Formulas . . . . . . . . . . . . . 264
10.50 Wronskians and Cross-Products . . . . . 265
10.51 Recurrence Relations and Derivatives . . 265
10.52 Limiting Forms . . . . . . . . . . . . . . 265
10.53 Power Series . . . . . . . . . . . . . . . . 265
10.54 Integral Representations . . . . . . . . . 266
10.55 Continued Fractions . . . . . . . . . . . . 266
10.56 Generating Functions . . . . . . . . . . . 266
10.57 Uniform Asymptotic Expansions for Large
Order . . . . . . . . . . . . . . . . . . . 266
10.58 Zeros . . . . . . . . . . . . . . . . . . . 266
10.59 Integrals . . . . . . . . . . . . . . . . . . 267
10.60 Sums . . . . . . . . . . . . . . . . . . . 267
Kelvin Functions 267
10.61 De nitions and Basic Properties . . . . . 267
10.62 Graphs . . . . . . . . . . . . . . . . . . . 268
10.63 Recurrence Relations and Derivatives . . 269
10.64 Integral Representations . . . . . . . . . 269
10.65 Power Series . . . . . . . . . . . . . . . . 269
10.66 Expansions in Series of Bessel Functions . 270
10.67 Asymptotic Expansions for Large Argument 271
10.68 Modulus and Phase Functions . . . . . . 272
10.69 Uniform Asymptotic Expansions for Large
Order . . . . . . . . . . . . . . . . . . . 273
10.70 Zeros . . . . . . . . . . . . . . . . . . . 273
10.71 Integrals . . . . . . . . . . . . . . . . . . 274
Applications 274
10.72 Mathematical Applications . . . . . . . . 274
10.73 Physical Applications . . . . . . . . . . . 275
1Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland.
2Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, D.C.
215
216
Computation 276
10.74 Methods of Computation . . . . . . . . . 276
10.75 Tables . . . . . . . . . . . . . . . . . . . 278
10.76 Approximations . . . . . . . . . . . . . . 28110.77 Software . . . . . . . . . . . . . . . . . . 281
References 281
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapters 9, 10, and 11) by F. W. J. Olver,
H. A. Antosiewicz, and Y. L. Luke, respectively. The authors are pleased to acknowledge assistance of Martin E. Muldoon with xx10.21
and 10.42, Adri Olde Daalhuis with the veri cation of Eqs. (10.15.6){(10.15.9), (10.38.6), (10.38.7), (10.60.7){(10.60.9), and (10.61.9){
(10.61.12), Peter Paule and Fr ed eric Chyzak for the veri cation of Eqs. (10.15.6){(10.15.9), (10.38.6), (10.38.7), (10.56.1){(10.56.5),
(10.60.4), (10.60.6), (10.60.10), and (10.60.11) by application of computer algebra, Nico Temme with the veri cation of Eqs. (10.15.6){
(10.15.9), (10.38.6), and (10.38.7), and Roderick Wong with the veri cation of xx10.22(v) and 10.43(v).
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
Notation 217
Notation
10.1 Special Notation
(For other notation see pp. xiv and 873.)
m;n integers. Inxx10.47{10.71 nis nonnegative.
k nonnegative integer (except in x10.73).
x;y real variables.
z complex variable.
real or complex parameter (the order).
arbitrary small positive constant.
# z (d/dz).
(x) 0(x)= (x): logarithmic derivative of the
gamma function ( x5.2(i)).
primes derivatives with respect to argument, except
where indicated otherwise.
The main functions treated in this chapter are the
Bessel functions J(z),Y(z); Hankel functions H(1)
(z),
H(2)
(z); modi ed Bessel functions I(z),K(z); spher-
ical Bessel functions jn(z),yn(z),h(1)
n(z),h(2)
n(z); mod-
i ed spherical Bessel functions i(1)
n(z),i(2)
n(z),kn(z);
Kelvin functions ber (x), bei(x), ker(x), kei(x). For
the spherical Bessel functions and modi ed spherical
Bessel functions the order nis a nonnegative integer.
For the other functions when the order is replaced by
n, it can be any integer. For the Kelvin functions the
orderis always assumed to be real.
A common alternative notation for Y(z) isN(z).
Other notations that have been used are as follows.
Abramowitz and Stegun (1964): jn(z),yn(z),
h(1)
n(z),h(2)
n(z), for jn(z),yn(z),h(1)
n(z),h(2)
n(z), respec-
tively, when n0.
Je reys and Je reys (1956): Hs (z) forH(1)
(z),
Hi(z) forH(2)
(z), Kh(z) for (2=)K(z).
Whittaker and Watson (1927): K(z) for
cos()K(z).
For older notations see British Association for the
Advancement of Science (1937, pp. xix{xx) and Watson
(1944, Chapters 1{3).
Bessel and Hankel Functions
10.2 De nitions
10.2(i) Bessel's Equation
10.2.1 z2d2w
dz2+zdw
dz+ (z2 2)w= 0:
This di erential equation has a regular singularity at
z= 0 with indices , and an irregular singularity at
z=1of rank 1; compare xx2.7(i) and 2.7(ii).10.2(ii) Standard Solutions
Bessel Function of the First Kind
10.2.2J(z) = (1
2z)1X
k=0( 1)k(1
4z2)k
k! (+k+ 1):
This solution of (10.2.1) is an analytic function of z2C,
except for a branch point at z= 0 whenis not an in-
teger. The principal branch ofJ(z) corresponds to the
principal value of (1
2z)(x4.2(iv)) and is analytic in the
z-plane cut along the interval ( 1;0].
When=n(2Z),J(z) is entire in z.
For xedz(6= 0) each branch of J(z) is entire in .
Bessel Function of the Second Kind (Weber's Function)
10.2.3 Y(z) =J(z) cos() J (z)
sin():
Whenis an integer the right-hand side is replaced by
its limiting value:
10.2.4Yn(z) =1
@J(z)
@
=n+( 1)n
@J(z)
@
= n,
n= 0;1;2;:::.
Whether or not is an integer Y(z) has a branch point
atz= 0. The principal branch corresponds to the prin-
cipal branches of J(z) in (10.2.3) and (10.2.4), with
a cut in the z-plane along the interval ( 1;0].
Except in the case of Jn(z), the principal branches
ofJ(z) andY(z) are two-valued and discontinuous on
the cut phz=; comparex4.2(i).
BothJ(z) andY(z) are real when is real and
phz= 0.
For xedz(6= 0) each branch of Y(z) is entire in .
Bessel Functions of the Third Kind (Hankel Functions)
These solutions of (10.2.1) are denoted by H(1)
(z) and
H(2)
(z), and their de ning properties are given by
10.2.5 H(1)
(z)p
2=(z)ei(z 1
2 1
4)
asz!1 in +phz2 , and
10.2.6H(2)
(z)p
2=(z)e i(z 1
2 1
4)
asz!1 in 2+phz , whereis an
arbitrary small positive constant. Each solution has a
branch point at z= 0 for all 2C. The principal
branches correspond to principal values of the square
roots in (10.2.5) and (10.2.6), again with a cut in the
z-plane along the interval ( 1;0].
The principal branches of H(1)
(z) andH(2)
(z) are
two-valued and discontinuous on the cut ph z=.
For xedz(6= 0) each branch of H(1)
(z) andH(2)
(z)
is entire in .
218 Bessel Functions
Branch Conventions
Except where indicated otherwise , it is assumed through-
out this Handbook that the symbols J(z),Y(z),
H(1)
(z), andH(2)
(z) denote the principal values of these
functions.
Cylinder Functions
The notation C(z) denotes J(z),Y(z),H(1)
(z),
H(2)
(z), or any nontrivial linear combination of these
functions, the coecients in which are independent of z
and.
10.2(iii) Numerically Satisfactory Pairs of
Solutions
Table 10.2.1 lists numerically satisfactory pairs of so-
lutions (x2.7(iv)) of (10.2.1) for the stated intervals orregions in the case <0. When< <0,is replaced
by throughout.
Table 10.2.1 : Numerically satisfactory pairs of solutions
of Bessel's equation.
Pair Interval or Region
J(x);Y(x) 0 <x<1
J(z);Y(z) neighborhood of 0 in jphzj
J(z);H(1)
(z) 0phz
J(z);H(2)
(z) phz0
H(1)
(z);H(2)
(z) neighborhood of 1injphzj
10.3 Graphics
10.3(i) Real Order and Variable
See Figures 10.3.1{10.3.8. For the modulus and phase functions M(x),(x),N(x), and (x) seex10.18.
Figure 10.3.1 :J0(x);Y0(x);J1(x);Y1(x);0x10.
Figure 10.3.2 :J5(x);Y5(x);M5(x);0x15.
Figure 10.3.3 :J0
5(x);Y0
5(x);N5(x);0x15.
Figure 10.3.4 :5(x); 5(x);0x15.
10.3 Graphics 219
Figure 10.3.5 :J(x);0x10;05.
Figure 10.3.6 :Y(x);0<x10;05.
Figure 10.3.7 :J0
(x);0x10;05.
Figure 10.3.8 :Y0
(x);0:2x10;05.
220 Bessel Functions
10.3(ii) Real Order, Complex Variable
See Figures 10.3.9{10.3.16. In these graphics, height corresponds to the absolute value of the function and color to
the phase. See also p. xiv.
Figure 10.3.9 :J0(x+iy); 10x10; 4y4.
Figure 10.3.10 :H(1)
0(x+iy), 10x5; 2:8y
4. Principal value. There is a cut along the negative
real axis.
Figure 10.3.11 :J1(x+iy); 10x10; 4y4.
Figure 10.3.12 :H(1)
1(x+iy), 10x5; 2:8y
4. Principal value. There is a cut along the negative
real axis.
10.3 Graphics 221
Figure 10.3.13 :J5(x+iy); 10x10; 4y4.
Figure 10.3.14 :H(1)
5(x+iy), 20x10; 4y
4. Principal value. There is a cut along the negative
real axis.
Figure 10.3.15 :J5:5(x+iy), 10x10; 4y4.
Principal value. There is a cut along the negative real
axis.
Figure 10.3.16 :H(1)
5:5(x+iy), 20x10; 4y
4. Principal value. There is a cut along the negative
real axis.
10.3(iii) Imaginary Order, Real Variable
See Figures 10.3.17{10.3.19. For the notation see x10.24.
Figure 10.3.17 :eJ1=2(x);eY1=2(x);0:01x10.
Figure 10.3.18 :eJ1(x);eY1(x);0:01x10.
222 Bessel Functions
Figure 10.3.19 :eJ5(x);eY5(x);0:01x10.
10.4 Connection Formulas
Other solutions of (10.2.1) include J (z),Y (z),
H(1)
(z), andH(2)
(z).
10.4.1J n(z) = ( 1)nJn(z);
Y n(z) = ( 1)nYn(z);
10.4.2H(1)
n(z) = ( 1)nH(1)
n(z);
H(2)
n(z) = ( 1)nH(2)
n(z):
10.4.3H(1)
(z) =J(z) +iY(z);
H(2)
(z) =J(z) iY(z);
10.4.4J(z) =1
2
H(1)
(z) +H(2)
(z)
;
Y(z) =1
2i
H(1)
(z) H(2)
(z)
:
10.4.5J(z) = csc() (Y (z) Y(z) cos()):
10.4.6H(1)
(z) =eiH(1)
(z);
H(2)
(z) =e iH(2)
(z):
10.4.7H(1)
(z) =icsc()
e iJ(z) J (z)
= csc()
Y (z) e iY(z)
;
10.4.8H(2)
(z) =icsc()
J (z) eiJ(z)
= csc()
Y (z) eiY(z)
:
In (10.4.5), (10.4.7), and (10.4.8) limiting values are
taken when =n; compare (10.2.3) and (10.2.4).
See alsox10.11.10.5 Wronskians and Cross-Products
10.5.1
WfJ(z);J (z)g=J+1(z)J (z) +J(z)J 1(z)
= 2 sin()=(z);
10.5.2
WfJ(z);Y(z)g=J+1(z)Y(z) J(z)Y+1(z)
= 2=(z);
10.5.3
WfJ(z);H(1)
(z)g=J+1(z)H(1)
(z) J(z)H(1)
+1(z)
= 2i=(z);
10.5.4
WfJ(z);H(2)
(z)g=J+1(z)H(2)
(z) J(z)H(2)
+1(z)
= 2i=(z);
10.5.5Wn
H(1)
(z);H(2)
(z)o
=H(1)
+1(z)H(2)
(z)
H(1)
(z)H(2)
+1(z)
= 4i=(z):
10.6 Recurrence Relations and Derivatives
10.6(i) Recurrence Relations
WithC(z) de ned as inx10.2(ii),
10.6.1C 1(z) +C+1(z) = (2=z)C(z);
C 1(z) C+1(z) = 2C0
(z):
10.6.2C0
(z) =C 1(z) (=z)C(z);
C0
(z) = C+1(z) + (=z)C(z):
10.6.3J0
0(z) = J1(z); Y0
0(z) = Y1(z);
H(1)
00(z) = H(1)
1(z); H(2)
00(z) = H(2)
1(z):
Iff(z) =zpC(zq), wherep;q, and(6= 0) are
real or complex constants, then
10.6.4f 1(z) +f+1(z) = (2=)z qf(z);
(p+q)f 1(z) + (p q)f+1(z)
= (2=)z1 qf0
(z):
10.6.5zf0
(z) =qzqf 1(z) + (p q)f(z);
zf0
(z) = qzqf+1(z) + (p+q)f(z):
10.6(ii) Derivatives
Fork= 0;1;2;:::,
10.6.61
zd
dzk
(zC(z)) =z kC k(z);
1
zd
dzk
(z C(z)) = ( 1)kz kC+k(z):
10.6.7 C(k)
(z) =1
2kkX
n=0( 1)nk
n
C k+2n(z):
10.7 Limiting Forms 223
10.6(iii) Cross-Products
Let
10.6.8p=J(a)Y(b) J(b)Y(a);
q=J(a)Y0
(b) J0
(b)Y(a);
r=J0
(a)Y(b) J(b)Y0
(a);
s=J0
(a)Y0
(b) J0
(b)Y0
(a);
whereaandbare independent of . Then
10.6.9p+1 p 1= 2
aq 2
br;
q+1+r=
ap + 1
bp+1;
r+1+q=
bp + 1
ap+1;
s=1
2p+1+1
2p 1 2
abp;
and
10.6.10 ps qr= 4=(2ab):
10.7 Limiting Forms
10.7(i)z!0
Whenis xed and z!0,
10.7.1 J0(z)!1; Y 0(z)(2=) lnz;
10.7.2H(1)
0(z) H(2)
0(z)(2i=) lnz;
10.7.3J(z)(1
2z)= (+ 1),6= 1; 2; 3;:::,
10.7.4
Y(z) (1=) ()(1
2z) ,
< >0 or= 1
2; 3
2; 5
2;:::,
10.7.5Y (z) (1=) cos() ()(1
2z) ,
< >0,6=1
2;3
2;5
2;:::,
10.7.6Yi(z) =icsch()
(1 i)(1
2z) i icoth()
(1 +i)(1
2z)i
+ejphzjo(1), 2Rand6= 0:
See alsox10.24 when z=x(>0).
10.7.7
H(1)
(z) H(2)
(z) (i=) ()(1
2z) ,< >0.
ForH(1)
(z) andH(2)
(z) when< > 0 combine
(10.4.6) and (10.7.7). For H(1)
i(z) andH(2)
i(z) when
2Rand6= 0 combine (10.4.3), (10.7.3), and
(10.7.6).10.7(ii)z!1
Whenis xed and z!1 ,
10.7.8
J(z) =p
2=(z)
cos
z 1
2 1
4
+ej=zjo(1)
;
Y(z) =p
2=(z)
sin
z 1
2 1
4
+ej=zjo(1)
,
jphzj (<):
For the corresponding results for H(1)
(z) and
H(2)
(z) see (10.2.5) and (10.2.6).
10.8 Power Series
ForJ(z) see (10.2.2) and (10.4.1). When is not an in-
teger the corresponding expansions for Y(z),H(1)
(z),
andH(2)
(z) are obtained by combining (10.2.2) with
(10.2.3), (10.4.7), and (10.4.8).
Whenn= 0;1;2;:::,
10.8.1Yn(z) = (1
2z) n
n 1X
k=0(n k 1)!
k! 1
4z2k
+2
ln 1
2z
Jn(z) (1
2z)n
1X
k=0( (k+ 1)
+ (n+k+ 1))( 1
4z2)k
k!(n+k)!;
where (x) = 0(x)= (x) (x5.2(i)). In particular,
10.8.2
Y0(z) =2
ln 1
2z
+
J0(z)+2
1
4z2
(1!)2 (1+1
2)(1
4z2)2
(2!)2
+ (1 +1
2+1
3)(1
4z2)3
(3!)2
;
where
is Euler's constant ( x5.2(ii)).
For negative values of nuse (10.4.1).
The corresponding results for H(1)
n(z) andH(2)
n(z)
are obtained via (10.4.3) with =n.
10.8.3
J(z)J(z) = (1
2z)+1X
k=0(++k+ 1)k( 1
4z2)k
k! (+k+ 1) (+k+ 1):
10.9 Integral Representations
10.9(i) Integrals along the Real Line
Bessel's Integral
10.9.1
J0(z) =1
Z
0cos(zsin)d=1
Z
0cos(zcos)d;
10.9.2Jn(z) =1
Z
0cos(zsin n)d
=i n
Z
0eizcoscos(n)d,n2Z.
224 Bessel Functions
Neumann's Integral
10.9.3
Y0(z) =4
2Z1
2
0cos(zcos)
+ ln
2zsin2
d;
where
is Euler's constant ( x5.2(ii)).
Poisson's and Related Integrals
10.9.4J(z) =(1
2z)
1
2
+1
2Z
0cos(zcos)(sin)2d
=2(1
2z)
1
2
+1
2Z1
0(1 t2) 1
2cos(zt)dt,
< > 1
2.
10.9.5Y(z) =2(1
2z)
1
2
+1
2Z1
0(1 t2) 1
2sin(zt)dt
Z1
0e zt(1 +t2) 1
2dt
,
< > 1
2;jphzj<1
2.
Schl a
i's and Related Integrals
10.9.6
J(z) =1
Z
0cos(zsin )d
sin()
Z1
0e zsinht tdt,jphzj<1
2,
10.9.7
Y(z) =1
Z
0sin(zsin )d
1
Z1
0
et+e tcos()
e zsinhtdt,
jphzj<1
2.
Mehler{Sonine and Related Integrals
10.9.8J(x) =2
Z1
0sin(xcosht 1
2) cosh(t)dt;
Y(x) = 2
Z1
0cos(xcosht 1
2) cosh(t)dt,
j<j<1;x> 0.
In particular,
10.9.9J0(x) =2
Z1
0sin(xcosht)dt, x>0,
Y0(x) = 2
Z1
0cos(xcosht)dt,x>0.
10.9.10
H(1)
(z) =e 1
2i
iZ1
1eizcosht tdt, 0<phz< ,
10.9.11
H(2)
(z) = e1
2i
iZ1
1e izcosht tdt, <phz<0.10.9.12J(x) =2(1
2x)
1
2 1
2 Z1
1sin(xt)dt
(t2 1)+1
2;
Y(x) = 2(1
2x)
1
2 1
2 Z1
1cos(xt)dt
(t2 1)+1
2,
j<j<1
2,x>0.
10.9.13z+
z 1
2
J
(z2 2)1
2
=1
Z
0ecoscos(zsin )d
sin()
Z1
0e cosht zsinht tdt,
<(z+)>0,
10.9.14z+
z 1
2
Y
(z2 2)1
2
=1
Z
0ecossin(zsin )d
1
Z1
0
et+cosht+e t coshtcos()
e zsinhtdt, <(z)>0.
10.9.15z+
z 1
2
H(1)
(z2 2)1
2
=1
ie 1
2iZ1
1eizcosht+isinht tdt,
=(z)>0,
10.9.16z+
z 1
2
H(2)
(z2 2)1
2
= 1
ie1
2iZ1
1e izcosht isinht tdt,
=(z)<0.
10.9(ii) Contour Integrals
Schl a
i{Sommerfeld Integrals
Whenjphzj<1
2,
10.9.17J(z) =1
2iZ1+i
1 iezsinht tdt;
and
10.9.18H(1)
(z) =1
iZ1+i
1ezsinht tdt;
H(2)
(z) = 1
iZ1 i
1ezsinht tdt:
10.9 Integral Representations 225
Schl a
i's Integral
10.9.19J(z) =(1
2z)
2iZ(0+)
1exp
t z2
4tdt
t+1;
where the integration path is a simple loop contour, and
t+1is continuous on the path and takes its principal
value at the intersection with the positive real axis.
Hankel's Integrals
In (10.9.20) and (10.9.21) the integration paths are sim-
ple loop contours not enclosing t= 1. Also, (t2 1) 1
2
is continuous on the path, and takes its principal value
at the intersection with the interval (1 ;1).
10.9.20
J(z) = 1
2
(1
2z)
3
2iZ(1+)
0cos(zt)(t2 1) 1
2dt,
6=1
2;3
2;:::.
10.9.21
H(1)
(z) = 1
2
(1
2z)
3
2iZ(1+)
1+i1eizt(t2 1) 1
2dt;
H(2)
(z) = 1
2
(1
2z)
3
2iZ(1+)
1 i1e izt(t2 1) 1
2dt,
6=1
2;3
2;:::;jphzj<1
2.Mellin{Barnes Type Integrals
10.9.22
J(x) =1
2iZi1
i1 ( t)(1
2x)+2t
(+t+ 1)dt,< >0,x>0,
where the integration path passes to the left of t=
0;1;2;:::.
10.9.23J(z) =1
2iZ 1+ic
1 ic (t)
( t+ 1)(1
2z) 2tdt;
wherecis a positive constant and the integration path
encloses the points t= 0; 1; 2;:::.
In (10.9.24) and (10.9.25) cis any constant exceed-
ing max(<;0).
10.9.24
H(1)
(z) = e 1
2i
22Zc+i1
c i1 (t) (t )( 1
2iz) 2tdt,
0<phz< ,
10.9.25
H(2)
(z) =e1
2i
22Zc+i1
c i1 (t) (t )(1
2iz) 2tdt,
<phz<0.
For (10.9.22){(10.9.25) and further integrals of this
type see Paris and Kaminski (2001, pp. 114{116).
10.9(iii) Products
10.9.26 J(z)J(z) =2
Z=2
0J+(2zcos) cos( )d, <(+)> 1.
10.9.27 J(z)J() =2
Z=2
0J2
2(z)1
2sin
cos ((z ) cos)d, < > 1
2,
where the square root has its principal value.
10.9.28 J(z)J() =1
2iZc+i1
c i1exp1
2t z2+2
2t
Iz
tdt
t, < > 1,
wherecis a positive constant. For the function Iseex10.25(ii).
Mellin{Barnes Type
10.9.29 J(x)J(x) =1
2iZi1
i1 ( t) (2t+++ 1)(1
2x)++2t
(t++ 1) (t++ 1) (t+++ 1)dt, x>0,
where the path of integration separates the poles of ( t) from those of (2 t+++ 1). See Paris and Kaminski
(2001, p. 116) for related results.
Nicholson's Integral
10.9.30 J2
(z) +Y2
(z) =8
2Z1
0cosh(2t)K0(2zsinht)dt, jphzj<1
2.
For the function K0seex10.25(ii).
226 Bessel Functions
10.9(iv) Compendia
For collections of integral representations of Bessel and
Hankel functions see Erd elyi et al. (1953b,xx7.3 and
7.12), Erd elyi et al. (1954a, pp. 43{48, 51{60, 99{105,
108{115, 123{124, 272{276, and 356{357), Gr obner and
Hofreiter (1950, pp. 189{192), Marichev (1983, pp. 191{
192 and 196{210), Magnus et al. (1966,x3.6), and Wat-
son (1944, Chapter 6).
10.10 Continued Fractions
AssumeJ 1(z)6= 0. Then
10.10.1
J(z)
J 1(z)
=1
2z 1 1
2(+ 1)z 1 1
2(+ 2)z 1 ,
z6= 0;
10.10.2
J(z)
J 1(z)
=1
2z=
1 1
4z2=((+ 1))
1 1
4z2=((+ 1)(+ 2))
1 ,
6= 0; 1; 2;:::.
See also Cuyt et al. (2008, pp. 349{356).
10.11 Analytic Continuation
Whenm2Z,
10.11.1 J
zemi
=emiJ(z);
10.11.2
Y
zemi
=e miY(z) + 2isin(m) cot()J(z):
10.11.3
sin()H(1)
zemi
= sin((m 1))H(1)
(z)
e isin(m)H(2)
(z);
10.11.4
sin()H(2)
zemi
=eisin(m)H(1)
(z)
+ sin((m+ 1))H(2)
(z):
10.11.5H(1)
zei
= e iH(2)
(z);
H(2)
ze i
= eiH(1)
(z):
If=n(2Z), then limiting values are taken in
(10.11.2){(10.11.4):
10.11.6Yn
zemi
= ( 1)mn(Yn(z) + 2imJn(z));
10.11.7
H(1)
n
zemi
= ( 1)mn 1((m 1)H(1)
n(z) +mH(2)
n(z));10.11.8
H(2)
n
zemi
= ( 1)mn(mH(1)
n(z) + (m+ 1)H(2)
n(z)):
For real,
10.11.9J(z) =J(z); Y(z) =Y(z);
H(1)
(z) =H(2)
(z); H(2)
(z) =H(1)
(z):
For complex replacebyon the right-hand sides.
10.12 Generating Function and Associated
Series
Forz2Candt2Cnf0g,
10.12.1 e1
2z(t t 1)=1X
m= 1tmJm(z):
Forz;2C,
10.12.2cos(zsin) =J0(z) + 21X
k=1J2k(z) cos(2k);
sin(zsin) = 21X
k=0J2k+1(z) sin((2k+ 1));
10.12.3
cos(zcos) =J0(z) + 21X
k=1( 1)kJ2k(z) cos(2k);
sin(zcos) = 21X
k=0( 1)kJ2k+1(z) cos((2k+ 1)):
10.12.4 1 =J0(z) + 2J2(z) + 2J4(z) + 2J6(z) +;
10.12.5cosz=J0(z) 2J2(z) + 2J4(z) 2J6(z) +;
sinz= 2J1(z) 2J3(z) + 2J5(z) ;
10.12.6
1
2zcosz=J1(z) 9J3(z) + 25J5(z) 49J7(z) +;
1
2zsinz= 4J2(z) 16J4(z) + 36J6(z) :
10.13 Other Di erential Equations
In the following equations ;;p;q , andrare real or
complex constants with 6= 0,p6= 0, andq6= 0.
10.13.1w00+
2 2 1
4
z2
w= 0,w=z1
2C(z);
10.13.2w00+2
4z 2 1
4z2
w= 0,w=z1
2C
z1
2
;
10.13.3w00+2zp 2w= 0,w=z1
2C1=p
2z1
2p=p
;
10.13.4 w00 2 1
zw0+2w= 0,w=zC(z);
10.13.5z2w00+ (1 2r)zw0+ (2q2z2q+r2 2q2)w
= 0, w=zrC(zq);
10.13.6 w00+ (2e2z 2)w= 0,w=C(ez);
10.14 Inequalities; Monotonicity 227
10.13.7z2(z2 2)w00+z(z2 32)w0
+ ((z2 2)2 (z2+2))w= 0,
w=C0
(z);
10.13.8
w(2n)= ( 1)n2nz nw,
w=z1
2nCn
2eki=nz1
2
,k= 0;1;:::; 2n 1.
In (10.13.9){(10.13.11) C(z),D(z) are any cylin-
der functions of orders ;, respectively, and #=
z(d/dz).
10.13.9
z2w000+ 3zw00+ (4z2+ 1 42)w0+ 4zw= 0,
w=C(z)D(z);
10.13.10
z3w000+z(4z2+ 1 42)w0+ (42 1)w= 0,
w=zC(z)D(z);
10.13.11
#4 2(2+2)#2+ (2 2)2
w
+ 4z2(#+ 1)(#+ 2)w= 0,w=C(z)D(z):
For further di erential equations see Kamke (1977,
pp. 440{451). See also Watson (1944, pp. 95{100).
10.14 Inequalities; Monotonicity
10.14.1jJ(x)j1, 0;x2R;
jJ(x)j2 1
2, 1;x2R.
10.14.2 0<J()<21
3
32
3 2
3
1
3, >0.
For monotonicity properties of J() andJ0
() see
Lorch (1992).
10.14.3jJn(z)jej=zj, n2Z:
10.14.4jJ(z)jj1
2zjej=zj
(+ 1), 1
2.
10.14.5
jJ(x)jxexp
(1 x2)1
2
1 + (1 x2)1
2,0;0<x1;
see Siegel (1953).
10.14.6jJ0
(x)j(1 +x2)1
4
x(2)1
2xexp
(1 x2)1
2
1 + (1 x2)1
2,
>0;0<x1;
see Watson (1944, p. 255). For a related bound for
Y(x) see Siegel and Sleator (1954).
10.14.7 1J(x)
xJ()e(1 x),0;0<x1;
see Paris (1984). For similar bounds for C(x)
(x10.2(ii)) see Laforgia (1986).Kapteyn's Inequality
10.14.8
jJn(nz)j znexp
n(1 z2)1
2
1 + (1 z2)1
2 n,n= 0;1;2;:::,
where (1 z2)1
2has its principal value.
10.14.9jJn(nz)j1,n= 0;1;2;:::;z2K,
where Kis de ned inx10.20(ii).
For inequalities for the function ( + 1)(2=x)J(x)
with > 1
2see Neuman (2004).
For further monotonicity properties see Landau
(1999, 2000).
10.15 Derivatives with Respect to Order
Noninteger Values of
10.15.1
@J(z)
@=J(z) ln 1
2z
(1
2z)1X
k=0( 1)k (+k+ 1)
(+k+ 1)(1
4z2)k
k!;
10.15.2@Y(z)
@= cot()@J(z)
@ Y(z)
csc()@J (z)
@ J(z):
Integer Values of
10.15.3
@J(z)
@
=n=
2Yn(z) +n!
2(1
2z)nn 1X
k=0(1
2z)kJk(z)
k!(n k);
10.15.4
@Y(z)
@
=n=
2Jn(z) +n!
2(1
2z)nn 1X
k=0(1
2z)kYk(z)
k!(n k);
10.15.5
@J(z)
@
=0=
2Y0(z);@Y(z)
@
=0=
2J0(z):
228 Bessel Functions
Half-Integer Values of
For the notations Ci and Si see x6.2(ii). When x>0,
10.15.6
@J(x)
@
=1
2=r
2
x(Ci(2x) sinx Si(2x) cosx);
10.15.7
@J(x)
@
= 1
2=r
2
x(Ci(2x) cosx+ Si(2x) sinx);
10.15.8
@Y(x)
@
=1
2=r
2
x(Ci(2x) cosx
+ (Si(2x) ) sinx);
10.15.9
@Y(x)
@
= 1
2= r
2
x(Ci(2x) sinx
(Si(2x) ) cosx):
For further results see Brychkov and Geddes (2005)
and Landau (1999, 2000).
10.16 Relations to Other Functions
Elementary Functions
10.16.1J1
2(z) =Y 1
2(z) =2
z1
2
sinz;
J 1
2(z) = Y1
2(z) =2
z1
2
cosz;
10.16.2H(1)
1
2(z) = iH(1)
1
2(z) = i2
z1
2
eiz;
H(2)
1
2(z) =iH(2)
1
2(z) =i2
z1
2
e iz:
For these and general results when is half an odd
integer seexx10.47(ii) and 10.49(i).
Airy Functions
Seexx9.6(i) and 9.6(ii).
Parabolic Cylinder Functions
With the notation of x12.14(i),
10.16.3
J1
4(z) = 2 1
4 1
2z 1
4
W
0;2z1
2
W
0; 2z1
2
;
J 1
4(z) = 2 1
4 1
2z 1
4
W
0;2z1
2
+W
0; 2z1
2
:
10.16.4
J3
4(z) = 2 1
4 1
2z 3
4
W0
0;2z1
2
W0
0; 2z1
2
;
J 3
4(z) = 2 1
4 1
2z 3
4
W0
0;2z1
2
+W0
0; 2z1
2
:Principal values on each side of these equations corre-
spond.
Con
uent Hypergeometric Functions
10.16.5J(z) =(1
2z)eiz
(+ 1)M
+1
2;2+ 1;2iz
;
10.16.6H(1)
(z)
H(2)
(z))
=2 1
2iei(2z)
eizU(+1
2;2+ 1;2iz):
For the functions MandUseex13.2(i).
10.16.7J(z) =e(2+1)i=4
22 (+ 1)(2z) 1
2M0;(2iz),
26= 1; 2 3;:::,
10.16.8H(1)
(z)
H(2)
(z))
=e(2+1)i=42
z1
2
W0;(2iz):
For the functions M0;andW0;seex13.14(i).
In all cases principal branches correspond at least
whenjphzj1
2.
Generalized Hypergeometric Functions
10.16.9J(z) =(1
2z)
(+ 1)0F1
;+ 1; 1
4z2
:
For 0F1see (16.2.1).
With Fas inx15.2(i), and with zand xed,
10.16.10J(z) = (1
2z)limF
;;+ 1; z2=(4)
;
asand!1 inC. For this result see Watson (1944,
x5.7).
10.17 Asymptotic Expansions for Large
Argument
10.17(i) Hankel's Expansions
De nea0() = 1,
10.17.1
ak() =(42 12)(42 32)(42 (2k 1)2)
k!8k,
k1;
10.17.2 !=z 1
2 1
4;
and letdenote an arbitrary small positive constant.
Then asz!1 , with xed,
10.17.3J(z)2
z1
2
cos!1X
k=0( 1)ka2k()
z2k
sin!1X
k=0( 1)ka2k+1()
z2k+1!
,
jphzj ,
10.17 Asymptotic Expansions for Large Argument 229
10.17.4Y(z)2
z1
2
sin!1X
k=0( 1)ka2k()
z2k
+ cos!1X
k=0( 1)ka2k+1()
z2k+1!
,
jphzj ,
10.17.5H(1)
(z)2
z1
2
ei!1X
k=0ikak()
zk,
+phz2 ;
10.17.6H(2)
(z)2
z1
2
e i!1X
k=0( i)kak()
zk,
2+phz ;where the branch of z1
2is determined by
10.17.7 z1
2= exp 1
2lnjzj+1
2iphz
:
Corresponding expansions for other ranges of ph z
can be obtained by combining (10.17.3), (10.17.5),
(10.17.6) with the continuation formulas (10.11.1),
(10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also
the connection formula given by the second of (10.4.4).
10.17(ii) Asymptotic Expansions of Derivatives
We continue to use the notation of x10.17(i). Also,
b0() = 1,b1() = (42+ 3)=8, and fork2,
10.17.8 bk() =
(42 12)(42 32)(42 (2k 3)2)
(42+ 4k2 1)
k!8k:
Then asz!1 with xed,
10.17.9 J0
(z) 2
z1
2
sin!1X
k=0( 1)kb2k()
z2k+ cos!1X
k=0( 1)kb2k+1()
z2k+1!
,jphzj ,
10.17.10 Y0
(z)2
z1
2
cos!1X
k=0( 1)kb2k()
z2k sin!1X
k=0( 1)kb2k+1()
z2k+1!
, jphzj ,
10.17.11 H(1)
0(z)i2
z1
2
ei!1X
k=0ikbk()
zk, +phz2 ;
10.17.12 H(2)
0(z) i2
z1
2
e i!1X
k=0( i)kbk()
zk, 2+phz :
10.17(iii) Error Bounds for Real Argument and
Order
In the expansions (10.17.3) and (10.17.4) assume that
0 andz > 0. Then the remainder associated
with the sumP` 1
k=0( 1)ka2k()z 2kdoes not exceed
the rst neglected term in absolute value and has the
same sign provided that `max(1
2 1
4;1). Simi-
larly forP` 1
k=0( 1)ka2k+1()z 2k 1, provided that `
max(1
2 3
4;1).
In the expansions (10.17.5) and (10.17.6) assume
that > 1
2andz > 0. If these expansions are ter-
minated when k=` 1, then the remainder term is
bounded in absolute value by the rst neglected term,
provided that `max( 1
2;1).10.17(iv) Error Bounds for Complex Argument
and Order
For (10.17.5) and (10.17.6) write
10.17.13
H(1)
(z)
H(2)
(z))
=2
z1
2
ei! ` 1X
k=0(i)kak()
zk+R
`(;z)!
,
`= 1;2;:::.
Then
10.17.14 R
`(;z) 2ja`()jVz;i1
t `
exp
j2 1
4jVz;i1
t `
;
whereVdenotes the variational operator (2.3.6), and
the paths of variation are subject to the condition that
j=tjchanges monotonically. Bounds for Vz;i1
t `
are
given by
230 Bessel Functions
10.17.15Vz;i1
t `
8
><
>:jzj `; 0phz;
(`)jzj `; 1
2phz0 orphz3
2,
2 (`)j=zj `; <phz 1
2or3
2phz<2,
where (`) =1
2 1
2`+ 1
= 1
2`+1
2
; seex9.7(i). The
bounds (10.17.15) also apply to Vz; i1
t `
in the con-
jugate sectors.
Corresponding error bounds for (10.17.3) and
(10.17.4) are obtainable by combining (10.17.13) and
(10.17.14) with (10.4.4).
10.17(v) Exponentially-Improved Expansions
As inx9.7(v) denote
10.17.16 Gp(z) =ez
2 (p) (1 p;z);
where (1 p;z) is the incomplete gamma function
(x8.2(i)). Then in (10.17.13) as z!1 withj` 2jzjj
bounded and m(0) xed,
10.17.17R
`(;z) = ( 1)`2 cos()
m 1X
k=0(i)kak()
zkG` k(2iz)
+R
m;`(;z)!
;
where
10.17.18
R
m;`(;z) =O
e 2jzjz m
,jph(ze1
2i)j:
For higher re-expansions of the remainder terms see
Olde Daalhuis and Olver (1995a) and Olde Daalhuis
(1995, 1996).
10.18 Modulus and Phase Functions
10.18(i) De nitions
For0 andx>0
10.18.1 M(x)ei(x)=H(1)
(x);
10.18.2 N(x)ei (x)=H(1)
0(x);
whereM(x) (>0),N(x) (>0),(x), and (x) are
continuous real functions of andx, with the branches
of(x) and (x) xed by
10.18.3 (x)! 1
2; (x)!1
2,x!0+.10.18(ii) Basic Properties
10.18.4J(x) =M(x) cos(x);
Y(x) =M(x) sin(x);
10.18.5J0
(x) =N(x) cos (x);
Y0
(x) =N(x) sin (x);
10.18.6M(x) =
J2
(x) +Y2
(x)1
2;
N(x) =
J0
2(x) +Y0
2(x)1
2;
10.18.7(x) = Arctan( Y(x)=J(x));
(x) = Arctan( Y0
(x)=J0
(x)):
10.18.8
M2
(x)0
(x) =2
x; N2
(x) 0
(x) =2(x2 2)
x3;
10.18.9
N2
(x)
=M0
2(x) +M2
(x)0
2(x) =M0
2(x) +4
(xM(x))2;
10.18.10
(x2 2)M(x)M0
(x) +x2N(x)N0
(x) +xN2
(x) = 0:
10.18.11
tan( (x) (x)) =M(x)0
(x)
M0(x)=2
xM(x)M0(x);
10.18.12M(x)N(x) sin( (x) (x)) =2
x:
10.18.13
x2M00
(x) +xM0
(x) + (x2 2)M(x) =4
2M3(x);
10.18.14
w00+
1 +1
4 2
x2
w=4
2w3,w=x1
2M(x),
10.18.15x3w000+x(4x2+ 1 42)w0+ (42 1)w
= 0, w=xM2
(x):
10.18.160
2(x) +1
2000
(x)
0(x) 3
400
(x)
0(x)2
= 1 2 1
4
x2:
10.19 Asymptotic Expansions for Large Order 231
10.18(iii) Asymptotic Expansions for Large Argument
Asx!1 , with xed and= 42,
10.18.17 M2
(x)2
x
1 +1
2 1
(2x)2+13
24( 1)( 9)
(2x)4+135
246( 1)( 9)( 25)
(2x)6+
;
10.18.18(x)x 1
2+1
4
+ 1
2(4x)+( 1)( 25)
6(4x)3+( 1)(2 114+ 1073)
5(4x)5
+( 1)(53 15352+ 54703 3 75733)
14(4x)7+:
Also,
10.18.19 N2
(x)2
x
1 1
2 3
(2x)2 1
24( 1)( 45)
(2x)4
;
the general term in this expansion being
10.18.20 (2k 3)!!
(2k)!!( 1)( 9)( (2k 3)2)( (2k+ 1)(2k 1)2)
(2x)2k, k2;
and
10.18.21 (x)x 1
2 1
4
++ 3
2(4x)+2+ 46 63
6(4x)3+3+ 1852 2053+ 1899
5(4x)5+:
The remainder after kterms in (10.18.17) does not exceed the ( k+ 1)th term in absolute value and is of the same
sign, provided that k> 1
2.
10.19 Asymptotic Expansions for Large
Order
10.19(i) Asymptotic Forms
If!1 through positive real values, with z(6= 0)
xed, then
10.19.1 J(z)1p
2ez
2
;
10.19.2
Y(z) iH(1)
(z)iH(2)
(z) r
2
ez
2
:
10.19(ii) Debye's Expansions
If!1 through positive real values with (>0)
xed, then
10.19.3
J(sech )e(tanh )
(2tanh )1
21X
k=0Uk(coth )
k;
Y(sech ) e( tanh )
(1
2tanh )1
21X
k=0( 1)kUk(coth )
k;
10.19.4
J0
(sech )sinh(2 )
41
2
e(tanh )1X
k=0Vk(coth )
k;
Y0
(sech )
sinh(2 )
1
2
e( tanh )1X
k=0( 1)kVk(coth )
k:If! 1 through positive real values with
2
0;1
2
xed, and
10.19.5 =(tan ) 1
4;
then
10.19.6
J(sec )2
tan 1
2
cos1X
k=0U2k(icot )
2k
isin1X
k=0U2k+1(icot )
2k+1!
;
Y(sec )2
tan 1
2
sin1X
k=0U2k(icot )
2k
+icos1X
k=0U2k+1(icot )
2k+1!
;
10.19.7
J0
(sec )sin(2 )
1
2
sin1X
k=0V2k(icot )
2k
icos1X
k=0V2k+1(icot )
2k+1!
;
Y0
(sec )sin(2 )
1
2
cos1X
k=0V2k(icot )
2k
isin1X
k=0V2k+1(icot )
2k+1!
:
232 Bessel Functions
In these expansions Uk(p) andVk(p) are the polynomials
inpof degree 3kde ned inx10.41(ii).
For error bounds for the rst of (10.19.3) see Olver
(1997b, p. 382).10.19(iii) Transition Region
As!1 , witha(2C) xed,
10.19.8J
+a1
3
21
3
1
3Ai
21
3a1X
k=0Pk(a)
2k=3+22
3
Ai0
21
3a1X
k=0Qk(a)
2k=3, jphj1
2 ,
Y
+a1
3
21
3
1
3Bi
21
3a1X
k=0Pk(a)
2k=3 22
3
Bi0
21
3a1X
k=0Qk(a)
2k=3,jphj1
2 .
Also,
10.19.9H(1)
+a1
3
H(2)
+a1
39
=
;24
3
1
3ei=3Ai
ei=321
3a1X
k=0Pk(a)
2k=3+25
3
ei=3Ai0
ei=321
3a1X
k=0Qk(a)
2k=3;
with sectors of validity 1
2+ ph3
2 .
Here Ai and Bi are the Airy functions ( x9.2), and
10.19.10
P0(a) = 1; P 1(a) = 1
5a; P 2(a) = 9
100a5+3
35a2;
P3(a) =957
7000a6 173
3150a3 1
225;
P4(a) =27
20000a10 23573
1 47000a7+5903
1 38600a4+947
3 46500a;
10.19.11Q0(a) =3
10a2; Q 1(a) = 17
70a3+1
70;
Q2(a) = 9
1000a7+611
3150a4 37
3150a;
Q3(a) = 549
28000a8 1 10767
6 93000a5+79
12375a2:
For corresponding expansions for derivatives see
http://dlmf.nist.gov/10.19.iii .
For proofs and also for the corresponding expansions
for second derivatives see Olver (1952).
For higher coecients in (10.19.8) in the case a= 0
(that is, in the expansions of J() andY()), see Wat-
son (1944,x8.21) and Temme (1997).
10.20 Uniform Asymptotic Expansions for
Large Order
10.20(i) Real Variables
De ne=(z) to be the solution of the di erential
equation
10.20.1d
dz2
=1 z2
z2that is in nitely di erentiable on the interval 0 < z <
1, including z= 1. Then
10.20.2
2
33
2=Z1
zp
1 t2
tdt= ln
1 +p
1 z2
z!
p
1 z2,
0<z1,
10.20.32
3( )3
2=Zz
1p
t2 1
tdt=p
z2 1 arcsecz,
1z<1,
all functions taking their principal values, with =
1;0; 1, corresponding to z= 0;1;1, respectively.
As!1 through positive real values
10.20.4J(z)4
1 z21
40
@Ai
2
3
1
31X
k=0Ak()
2k
+Ai0
2
3
5
31X
k=0Bk()
2k1
A;
10.20.5Y(z) 4
1 z21
40
@Bi
2
3
1
31X
k=0Ak()
2k
+Bi0
2
3
5
31X
k=0Bk()
2k1
A;
10.20.6H(1)
(z)
H(2)
(z))
2ei=34
1 z21
40
@Ai
e2i=32
3
1
31X
k=0Ak()
2k+e2i=3Ai0
e2i=32
3
5
31X
k=0Bk()
2k1
A;
10.20 Uniform Asymptotic Expansions for Large Order 233
10.20.7 J0
(z) 2
z1 z2
41
40
@Ai
2
3
4
31X
k=0Ck()
2k+Ai0
2
3
2
31X
k=0Dk()
2k1
A;
10.20.8 Y0
(z)2
z1 z2
41
40
@Bi
2
3
4
31X
k=0Ck()
2k+Bi0
2
3
2
31X
k=0Dk()
2k1
A;
10.20.9H(1)
0(z)
H(2)
0(z))
4e2i=3
z1 z2
41
40
@e2i=3Ai
e2i=32
3
4
31X
k=0Ck()
2k+Ai0
e2i=32
3
2
31X
k=0Dk()
2k1
A;
uniformly for z2(0;1) in all cases, where Ai and Bi
are the Airy functions ( x9.2).
In the following formulas for the coecients Ak(),
Bk(),Ck(), andDk(),uk,vkare the constants de-
ned inx9.7(i), andUk(p),Vk(p) are the polynomials in
pof degree 3kde ned inx10.41(ii).
Interval 0<z< 1
10.20.10Ak() =2kX
j=0(3
2)jvj 3j=2U2k j
(1 z2) 1
2
;
10.20.11
Bk() = 1
22k+1X
j=0(3
2)juj 3j=2U2k j+1
(1 z2) 1
2
;
10.20.12
Ck() = 1
22k+1X
j=0(3
2)jvj 3j=2V2k j+1
(1 z2) 1
2
;
10.20.13Dk() =2kX
j=0(3
2)juj 3j=2V2k j
(1 z2) 1
2
:
Interval 1<z<1
In formulas (10.20.10){(10.20.13) replace 1
2, 1
2,
3j=2, and (1 z2) 1
2by i( )1
2,i( ) 1
2,
i3j( ) 3j=2, andi(z2 1) 1
2, respectively.
Note: Another way of arranging the above formu-
las for the coecients Ak();Bk();Ck(), andDk()
would be by analogy with (12.10.42) and (12.10.46). In
this way there is less usage of many-valued functions.
Values at= 0
10.20.14A0(0) = 1; A 1(0) = 1
225;
A2(0) =1 51439
2182 95000; A 3(0) = 8872 78009
250 49351 25000;
B0(0) =1
7021
3; B 1(0) = 1213
10 2375021
3;
B2(0) =1 65425 37833
3774 32055 0000021
3;
B3(0) = 430 99056 39368 59253
5 68167 34399 42500 0000021
3:
Each of the coecients Ak(),Bk(),Ck(), and
Dk(),k= 0;1;2;:::, is real and in nitely di eren-
tiable on the interval 1< <1. For (10.20.14)and further information on the coecients see Temme
(1997).
For numerical tables of =(z), (4=(1 z2))1
4
andAk(),Bk(),Ck(), andDk() see Olver (1962,
pp. 28{42).
10.20(ii) Complex Variables
The function =(z) given by (10.20.2) and (10.20.3)
can be continued analytically to the z-plane cut along
the negative real axis. Corresponding points of the map-
ping are shown in Figures 10.20.1 and 10.20.2.
The equations of the curved boundaries D1E1and
D2E2in the-plane are given parametrically by
10.20.15 = (3
2)2
3( i)2
3, 0 <1;
respectively.
The curves BP1E1andBP2E2in thez-plane are
the inverse maps of the line segments
10.20.16 =ei=3 , 0 (3
2)2
3;
respectively. They are given parametrically by
10.20.17
z=( coth 2)1
2i( 2 tanh )1
2, 0 0;
where 0= 1:19968:::is the positive root of the equa-
tion = coth . The points P1;P2where these curves
intersect the imaginary axis are ic, where
10.20.18 c= ( 2
0 1)1
2= 0:66274::: :
The eye-shaped closed domain in the uncut z-plane
that is bounded by BP1E1andBP2E2is denoted by
K; see Figure 10.20.3.
As!1 through positive real values the expan-
sions (10.20.4){(10.20.9) apply uniformly for jphzj
, the coecients Ak(),Bk(),Ck(), andDk(),
being the analytic continuations of the functions de ned
inx10.20(i) when is real.
For proofs of the above results and for error bounds
and extensions of the regions of validity see Olver
(1997b, pp. 419{425). For extensions to complex see
Olver (1954). For resurgence properties of the coe-
cients (x2.7(ii)) see Howls and Olde Daalhuis (1999).
For further results see Dunster (2001a), Wang and Wong
(2002), and Paris (2004).
234 Bessel Functions
Figure 10.20.1 :z-plane.P1andP2are the pointsic.
c= 0:66274:::.
Figure 10.20.2 :-plane.E1andE2are the points
ei=3(3=2)2=3:
Figure 10.20.3 :z-plane. Domain K(unshaded). c= 0:66274:::.
10.21 Zeros 235
10.20(iii) Double Asymptotic Properties
For asymptotic properties of the expansions (10.20.4){
(10.20.6) with respect to large values of zseex10.41(v).
10.21 Zeros
10.21(i) Distribution
The zeros of any cylinder function or its derivative are
simple, with the possible exceptions of z= 0 in the
case of the functions, and z= 0;in the case of the
derivatives.
Ifis real, then J(z),J0
(z),Y(z), andY0
(z), each
have an in nite number of positive real zeros. All of
these zeros are simple, provided that 1 in the case
ofJ0
(z), and 1
2in the case of Y0
(z). When all of
their zeros are simple, the mth positive zeros of these
functions are denoted by j;m,j0
;m,y;m, andy0
;mre-
spectively, except that z= 0 is counted as the rst zero
ofJ0
0(z). SinceJ0
0(z) = J1(z) we have
10.21.1 j0
0;1= 0; j0
0;m=j1;m 1,m= 2;3;:::.
When0, the zeros interlace according to the
inequalities
10.21.2j;1<j+1;1<j;2<j+1;2<j;3<;
y;1<y+1;1<y;2<y+1;2<y;3<;
10.21.3
j0
;1<y;1<y0
;1<j;1<j0
;2<y;2<:
The positive zeros of any two real distinct cylinder
functions of the same order are interlaced, as are the
positive zeros of any real cylinder function C(z) and
the contiguous function C+1(z). See also Elbert and
Laforgia (1994).
When 1 the zeros of J(z) are all real. If
< 1 andis not an integer, then the number of
complex zeros of J(z) is 2b c. Ifb cis odd, then
two of these zeros lie on the imaginary axis.
If0, then the zeros of J0
(z) are all real.
For information on the real double zeros of J0
(z)
andY0
(z) when < 1 and < 1
2, respectively, see
D oring (1971) and Kerimov and Skorokhodov (1986).
The latter reference also has information on double zeros
of the second and third derivatives of J(z) andY(z).
No two of the functions J0(z),J1(z),J2(z);:::, have
any common zeros other than z= 0; see Watson (1944,
x15.28).
10.21(ii) Analytic Properties
Ifis a zero of the cylinder function
10.21.4 C(z) =J(z) cos(t) +Y(z) sin(t);
wheretis a parameter, then
10.21.5 C0
() =C 1() = C+1():Ifis a zero of C0
(z), then
10.21.6 C() =
C 1() =
C+1():
The parameter tmay be regarded as a continuous
variable and ,as functions (t),(t) oft. If
0 and these functions are xed by
10.21.7 (0) = 0; (0) =j0
;1;
then
10.21.8
j;m=(m); y;m=(m 1
2),m= 1;2;:::,
10.21.9j0
;m=(m 1); y0
;m=(m 1
2),
m= 1;2;:::.
10.21.10
C0
() =
2d
dt 1
2
;C() =2
2
2d
dt 1
2
;
10.21.1122
d
dtd3
dt3
32
d2
dt22
422
d
dt2
+ (42
+ 1 42)d
dt4
= 0:
The functions (t) and(t) are related to the in-
verses of the phase functions (x) and (x) de ned in
x10.18(i): if 0, then
10.21.12(j;m) = (m 1
2); (y;m) = (m 1),
m= 1;2;:::,
10.21.13
j0
;m
= (m 1
2);
y0
;m
=m,
m= 1;2;:::.
For sign properties of the forward di erences that
are de ned by
10.21.14(t) =(t+ 1) (t);
2(t) = (t+ 1) (t);:::;
whent= 1;2;3;:::, and similarly for (t), see Lorch
and Szego (1963, 1964), Lorch et al. (1970, 1972), and
Muldoon (1977).
10.21(iii) In nite Products
10.21.15J(z) =(1
2z)
(+ 1)1Y
k=1
1 z2
j2
;k!
,0,
10.21.16J0
(z) =(1
2z) 1
2 ()1Y
k=1
1 z2
j0
;k2!
, >0.
236 Bessel Functions
10.21(iv) Monotonicity Properties
Any positive zero cof the cylinder function C(x) and
any positive zero c0ofC0
(x) such that c0>jjare de-
nable as continuous and increasing functions of :
10.21.17dc
d= 2cZ1
0K0(2csinht)e 2tdt;
10.21.18dc0
d=2c0
c02 2Z1
0(c02cosh(2t) 2)
K0(2c0sinht)e 2tdt;
whereK0is de ned inx10.25(ii).
In particular, j;m,y;m,j0
;m, andy0
;mare increas-
ing functions of when0. It is also true that the
positive zeros j00
andj000
ofJ00
(x) andJ000
(x), respec-
tively, are increasing functions of when > 0, pro-
vided that in the latter case j000
>p
3 when 0< < 1.j;m=andj0
;m=are decreasing functions of
when >0 form= 1;2;3;:::.
For further monotonicity properties see Elbert
(2001), Lorch (1990, 1993, 1995), Lorch and Szego
(1990, 1995), and Muldoon (1981). For inequalities for
zeros arising from monotonicity properties see Laforgia
and Muldoon (1983).
10.21(v) Inequalities
For bounds for the smallest real or purely imaginary
zeros ofJ(x) whenis real see Ismail and Muldoon
(1995).
10.21(vi) McMahon's Asymptotic Expansions
for Large Zeros
If(0) is xed, = 42, andm!1 , then
10.21.19j;m;y;ma 1
8a 4( 1)(7 31)
3(8a)3 32( 1)(832 982+ 3779)
15(8a)5
64( 1)(69493 1 538552+ 15 85743 62 77237)
105(8a)7 ;
wherea= (m+1
2 1
4)forj;m,a= (m+1
2 3
4)fory;m. Witha= (t+1
2 1
4), the right-hand side is the
asymptotic expansion of (t) for larget.
10.21.20j0
;m;y0
;mb + 3
8b 4(72+ 82 9)
3(8b)3 32(833+ 20752 3039+ 3537)
15(8b)5
64(69494+ 2 964923 12 480022+ 74 14380 58 53627)
105(8b)7 ;
whereb= (m+1
2 3
4)forj0
;m,b= (m+1
2 1
4)
fory0
;m, andb= (t+1
2+1
4)for(t).
For the next three terms in (10.21.19) and the
next two terms in (10.21.20) see Bickley et al. (1952,
p. xxxvii) or Olver (1960, pp. xvii{xviii).
For error bounds see Wong and Lang (1990), Wong
(1995), and Elbert and Laforgia (2000). See also Lafor-
gia (1979).
For themth positive zero j00
;mofJ00
(x) Wong and
Lang (1990) gives the corresponding expansion
10.21.21j00
;mc + 7
8c 282+ 424+ 1724
3(8c)3 ;
wherec= (m+1
2 1
4)if 0< < 1, andc=
(m+1
2 5
4)if > 1. An error bound is included
for the case 3
2.
10.21(vii) Asymptotic Expansions for Large
Order
LetC(x),(t), and(t) be de ned as in x10.21(ii)
andM(x),(x),N(x), and (x) denote the modulusand phase functions for the Airy functions and their
derivatives as inx9.8.
As!1 witht(>0) xed,
10.21.22 (t)1X
k=0 k
2k=3;
10.21.23 C0
((t))(2=)2
3
M
21
3 1X
k=0 k
2k=3;
where is given by
10.21.24
21
3
=t;
and
10.21.25 0= 1; 1= ; 2=3
10 2;
3= 1
350 3+1
70; 4= 479
63000 4 1
3150 ;
5=20231
80 85000 5 551
1 61700 2;
10.21.26
0= 1; 1= 4
5 ; 2=18
35 2;
3= 88
315 3 11
1575; 4=79586
6 06375 4+9824
6 06375 :
10.21 Zeros 237
As!1 witht(> 1
6) xed,
10.21.27 (t)1X
k=0 0
k
2k=3;
10.21.28 C((t))(2=)1
3
N
21
3 01X
k=0 0
k
2k=3;
where 0is given by
10.21.29
21
3 0
=t;
and
10.21.30
0
0= 1; 0
1= 0; 0
2=3
10 02 1
10 0 1;
0
3= 1
350 03 1
25 1
200 0 3;
0
4= 479
63000 04+509
31500 0+1
1500 0 2 1
2000 0 5;
10.21.31
0
0= 1; 0
1= 1
5 0; 0
2=9
350 02+1
100 0 1;
0
3=89
15750 03 47
4500+1
3000 0 3:
In particular, with the notation as below,
10.21.32 j;m1X
k=0 k
2k=3;
10.21.33 y;m1X
k=0 k
2k=3;
10.21.34J0
(j;m)( 1)m(2=)2
3
M(am)1X
k=0 k
2k=3;10.21.35Y0
(y;m)( 1)m 1(2=)2
3
M(bm)1X
k=0 k
2k=3;
and
10.21.36 j0
;m1X
k=0 0
k
2k=3;
10.21.37 y0
;m1X
k=0 0
k
2k=3;
10.21.38J
j0
;m
( 1)m 1(2=)1
3
N(a0m)1X
k=0 0
k
2k=3;
10.21.39Y
y0
;m
( 1)m 1(2=)1
3
N(b0m)1X
k=0 0
k
2k=3:
Heream,bm,a0
m,b0
mare themth negative zeros of
Ai(x), Bi(x), Ai0(x), Bi0(x), respectively (x9.9), k, k,
0
k, 0
kare given by (10.21.25), (10.21.26), (10.21.30),
and (10.21.31), with = 2 1
3amin the case of
j;mandJ0
(j;m), = 2 1
3bmin the case of y;m
andY0
(y;m), 0= 2 1
3a0
min the case of j0
;m
andJ
j0
;m
, 0= 2 1
3b0
min the case of y0
;mand
Y
y0
;m
.
For error bounds for (10.21.32) see Qu and Wong
(1999); for (10.21.36) and (10.21.37) see Elbert and
Laforgia (1997). See also Spigler (1980).
For the rst zeros rounded numerical values of the
coecients are given by
10.21.40j;1+ 1:85575 711
3+ 1:03315 0 1
3 0:00397 1 0:0908 5
3+ 0:043 7
3+;
y;1+ 0:93157 681
3+ 0:26035 1 1
3+ 0:01198 1 0:0060 5
3 0:001 7
3+;
J0
(j;1) 1:11310 28 2
3(1 + 1:48460 6 2
3+ 0:43294 4
3 0:1943 2+ 0:019 8
3+);
Y0
(y;1)0:95554 86 2
3(1 + 0:74526 1 2
3+ 0:10910 4
3 0:0185 2 0:003 8
3+);
j0
;1+ 0:80861 651
3+ 0:07249 0 1
3 0:05097 1+ 0:0094 5
3+;
y0
;1+ 1:82109 801
3+ 0:94000 7 1
3 0:05808 1 0:0540 5
3+:
J
j0
;1
0:67488 51 1
3(1 0:16172 3 2
3+ 0:02918 4
3 0:0068 2+);
Y
y0
;1
0:57319 40 1
3(1 0:36422 0 2
3+ 0:09077 4
3+ 0:0237 2+):
For numerical coecients for m= 2;3;4;5 see Olver
(1951, Tables 3{6).
The expansions (10.21.32){(10.21.39) become pro-
gressively weaker as mincreases. The approximations
that follow inx10.21(viii) do not su er from this draw-
back.10.21(viii) Uniform Asymptotic Approximations
for Large Order
As!1 the following four approximations hold uni-
formly form= 1;2;::::
10.21.41j;m=z() +z()(h())2B0()
2+O1
3
,
= 2
3am,
238 Bessel Functions
10.21.42
J0
(j;m) = 2
2
3Ai0(am)
z()h()
1 +O1
2
,= 2
3am,
10.21.43
j0
;m=z()+z()(h())2C0()
2+O1
,= 2
3a0
m,
10.21.44
J
j0
;m
=h() Ai(a0
m)
1
3
1 +O1
4
3
,= 2
3a0
m.
Hereamanda0
mdenote respectively the zeros of the
Airy function Ai( z) and its derivative Ai0(z); seex9.9.
Next,z() is the inverse of the function =(z) de ned
by (10.20.3). B0() andC0() are de ned by (10.20.11)
and (10.20.12) with k= 0. Lastly,
10.21.45 h() =
4=(1 z2)1
4:
(Note: If the term z()(h())2C0()=(2) in (10.21.43)
is omitted, then the uniform character of the error term
O(1/) is destroyed.)
Corresponding uniform approximations for y;m,
Y0
(y;m),y0
;m, andY
y0
;m
, are obtained from
(10.21.41){(10.21.44) by changing the symbols j,J, Ai,
Ai0,am, anda0
mtoy,Y, Bi, Bi0,bm, andb0
m, re-
spectively.
For derivations and further information, including
extensions to uniform asymptotic expansions, see Olver
(1954, 1960). The latter reference includes numerical
tables of the rst few coecients in the uniform asymp-
totic expansions.
10.21(ix) Complex Zeros
This subsection describes the distribution in Cof the
zeros of the principal branches of the Bessel functions
of the second and third kinds, and their derivatives, in
the case when the order is a positive integer n. For fur-
ther information, including uniform asymptotic expan-
sions, extensions to other branches of the functions and
their derivatives, and extensions to half-integer values
of the order, see Olver (1954). (There is an inaccuracy
in Figures 11 and 14 in this reference. Each curve that
represents an in nite string of nonreal zeros should be
located on the opposite side of its straight line asymp-
tote. This inaccuracy was repeated in Abramowitz and
Stegun (1964, Figures 9.5 and 9.6). See Kerimov and
Skorokhodov (1985a,b) and Figures 10.21.3{10.21.6.)
See also Cruz and Sesma (1982); Cruz et al. (1991),
Kerimov and Skorokhodov (1984c, 1987, 1988), Kokolo-
giannaki et al. (1992), and references supplied in
x10.75(iii).
Zeros ofYn(nz)andY0
n(nz)
In Figures 10.21.1, 10.21.3, and 10.21.5 the two contin-
uous curves that join the points 1 are the boundariesofK, that is, the eye-shaped domain depicted in Figure
10.20.3. These curves therefore intersect the imaginary
axis at the points z=ic, wherec= 0:66274:::.
The rst set of zeros of the principal value of Yn(nz)
are the points z=yn;m=n,m= 1;2;:::, on the positive
real axis (x10.21(i)). Secondly, there is a conjugate pair
of in nite strings of zeros with asymptotes =z=ia=n,
where
10.21.46 a=1
2ln 3 = 0:54931::: :
Lastly, there are two conjugate sets, with nzeros in each
set, that are asymptotically close to the boundary of K
asn!1 . Figures 10.21.1, 10.21.3, and 10.21.5 plot
the actual zeros for n= 1;5, and 10, respectively.
The zeros of Y0
n(nz) have a similar pattern to those
ofYn(nz).
Zeros ofH(1)
n(nz),H(2)
n(nz),H(1)
n0(nz),H(2)
n0(nz)
In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous
curve that joins the points 1 is the lower boundary of
K.
The rst set of zeros of the principal value of
H(1)
n(nz) is an in nite string with asymptote =z=
id=n, where
10.21.47 d=1
2ln 2 = 0:34657::: :
The only other set comprises nzeros that are asymp-
totically close to the lower boundary of Kasn!1 .
Figures 10.21.2, 10.21.4, and 10.21.6 plot the actual ze-
ros forn= 1;5, and 10, respectively.
The zeros of H(1)
n0(nz) have a similar pattern to
those ofH(1)
n(nz). The zeros of H(2)
n(nz) andH(2)
n0(nz)
are the complex conjugates of the zeros of H(1)
n(nz) and
H(1)
n0(nz), respectively.
Zeros ofJ0(z) iJ1(z)andJn(z) iJn+1(z)
For information see Synolakis (1988), MacDonald (1989,
1997), and Ikebe et al. (1993).
10.21(x) Cross-Products
Throughout this subsection we assume 0,x > 0,
>1, and we denote 4 2by.
The zeros of the functions
10.21.48 J(x)Y(x) Y(x)J(x)
and
10.21.49 J0
(x)Y0
(x) Y0
(x)J0
(x)
are simple and the asymptotic expansion of the mth
positive zero as m!1 is given by
10.21.50 +p
+q p2
3+r 4pq+ 2p3
5+;
10.21 Zeros 239
Figure 10.21.1 : ZerosofYn(nz) injphzj. Case
n= 1, 1:6<z2:6.
Figure 10.21.2 : Zeros ofH(1)
n(nz) injphzj.
Casen= 1, 2:8<z1:4.
Figure 10.21.3 : ZerosofYn(nz) injphzj. Case
n= 5, 2:6<z1:6.
Figure 10.21.4 : Zeros ofH(1)
n(nz) injphzj.
Casen= 5, 2:6<z1:6.
Figure 10.21.5 : ZerosofYn(nz) injphzj. Case
n= 10, 2:3<z1:9.
Figure 10.21.6 : Zeros ofH(1)
n(nz) injphzj.
Casen= 10, 2:3<z1:9.
240 Bessel Functions
where, in the case of (10.21.48),
10.21.51
=m
1; p = 1
8; q =( 1)( 25)(3 1)
6(4)3( 1);
r=( 1)(2 114+ 1073)(5 1)
5(4)5( 1);and, in the case of (10.21.49),
10.21.52 =(m 1)
1; p =+ 3
8;
q=(2+ 46 63)(3 1)
6(4)3( 1);
r=(3+ 1852 2053+ 1899)(5 1)
5(4)5( 1):
The asymptotic expansion of the large positive zeros
(not necessarily the mth) of the function
10.21.53 J0
(x)Y(x) Y0
(x)J(x)
is given by (10.21.50), where
10.21.54 =(m 1
2)
1; p =(+ 3) ( 1)
8( 1); q =(2+ 46 63)3 ( 1)( 25)
6(4)3( 1);
r=(3+ 1852 2053+ 1899)5 ( 1)(2 114+ 1073)
5(4)5( 1):
Higher coecients in the asymptotic expansions in
this subsection can be obtained by expressing the cross-
products in terms of the modulus and phase functions
(x10.18), and then reverting the asymptotic expansion
for the di erence of the phase functions.
For further information see Cochran (1963, 1964,
1966a,b), Kal ahne (1907), Martinek et al. (1966), Mul-
doon (1979), and Salchev and Popov (1976).
10.21(xi) Riccati{Bessel Functions
The Riccati{Bessel functions are (1
2x)1
2J(x) and
(1
2x)1
2Y(x). Except possibly for x= 0 their ze-
ros are the same as those of J(x) andY(x), respec-
tively. For information on the zeros of the derivatives
of Riccati{Bessel functions, and also on zeros of their
cross-products, see Boyer (1969). This information in-
cludes asymptotic approximations analogous to those
given inxx10.21(vi), 10.21(vii), and 10.21(x).10.21(xii) Zeros of J(x) +xJ0
(x)
For properties of the positive zeros of the function
J(x)+xJ0
(x), with andreal, see Landau (1999).
10.21(xiii) Rayleigh Function
The Rayleigh function n() is de ned by
10.21.55 n() =1X
m=1(j;m) 2n,n= 1;2;3;:::.
For properties, computation, and generalizations see
Kapitsa (1951a), Kerimov (1999), and Gupta and Mul-
doon (2000). See also Watson (1944, xx15.5, 15.51).
10.21(xiv)-Zeros
For information on zeros of Bessel and Hankel functions
as functions of the order, see Cochran (1965), Cochran
and Ho spiegel (1970), Hethcote (1970), and Conde and
Kalla (1979).
10.22 Integrals
10.22(i) Inde nite Integrals
In this subsection C(z) andD(z) denote cylinder functions( x10.2(ii)) of orders and, respectively, not necessarily
distinct.
10.22.1Z
z+1C(z)dz=z+1C+1(z);Z
z +1C(z)dz= z +1C 1(z):
10.22.2Z
zC(z)dz=1
22 1
+1
2
z(C(z)H 1(z) C 1(z)H(z)) , 6= 1
2.
10.22 Integrals 241
For the Struve function H(z) seex11.2(i).
10.22.3Z
eizzC(z)dz=eizz+1
2+ 1(C(z) iC+1(z)), 6= 1
2,
Z
eizz C(z)dz=eizz +1
1 2(C(z) +iC 1(z)), 6=1
2.
Products
10.22.4Z
zC(az)D(bz)dz=z(aC+1(az)D(bz) bC(az)D+1(bz))
a2 b2, a26=b2,
10.22.5Z
zC(az)D(az)dz=1
4z2(2C(az)D(az) C 1(az)D+1(az) C+1(az)D 1(az));
10.22.6Z
C(az)D(az)dz
z= az(C+1(az)D(az) C(az)D+1(az))
2 2+C(az)D(az)
+, 26=2,
10.22.7Z
z++1C(az)D(az)dz=z++2
2(++ 1)(C(az)D(az) +C+1(az)D+1(az)) , +6= 1,
Z
z +1C(az)D(az)dz=z +2
2(1 )(C(az)D(az) +C 1(az)D 1(az)) , +6= 1.
10.22(ii) Integrals over Finite Intervals
Throughout this subsection x>0.
10.22.8Zx
0J(t)dt= 21X
k=0J+2k+1(x), < > 1.
10.22.9Zx
0J2n(t)dt=Zx
0J0(t)dt 2n 1X
k=0J2k+1(x);Zx
0J2n+1(t)dt= 1 J0(x) 2nX
k=1J2k(x),n= 0;1;:::.
10.22.10Zx
0tJ(t)dt=x 1
2+1
2+1
2
1
2 1
2+1
21X
k=0(+ 2k+ 1) 1
2 1
2+1
2+k
1
2+1
2+3
2+kJ+2k+1(x),<(++ 1)>0.
10.22.11Zx
01 J0(t)
tdt=1
21X
k=1 (k+ 1) (1)
k!(1
2x)kJk(x);
10.22.12xZx
01 J0(t)
tdt= 21X
k=0(2k+ 3)( (k+ 2) (1))J2k+3(x)
=x 2J1(x) + 21X
k=0(2k+ 5) ( (k+ 3) (1) 1)J2k+5(x);
where (x) = 0(x)= (x) (x5.2(i)). See also (10.22.39).
Trigonometric Arguments
10.22.13Z1
2
0J2(2zcos) cos(2)d=1
2J+(z)J (z), < > 1
2,
10.22.14Z
0J2(2zsin) cos(2)d=cos()J+(z)J (z), < > 1
2,
10.22.15Z
0J2(2zsin) sin(2)d=sin()J+(z)J (z), < > 1.
10.22.16Z1
2
0J0(2zsin) cos(2n)d=1
2J2
n(z), n= 0;1;2;:::.
242 Bessel Functions
10.22.17Z1
2
0Y2(2zcos) cos(2)d=1
2cot(2)J+(z)J (z) 1
2csc(2)J (z)J (z), 1
2<< <1
2,
10.22.18Z1
2
0Y0(2zsin) cos(2n)d=1
2Jn(z)Yn(z), n= 0;1;2;:::.
10.22.19Z1
2
0J(zsin)(sin)+1(cos)2+1d= 2 (+ 1)z 1J++1(z),<> 1,< > 1,
10.22.20Z1
2
0J(zsin)(sin)(cos)2d=1
22 1z
+1
2
J2
1
2z
, <> 1
2,
10.22.21Z1
2
0Y(zsin)(sin)(cos)2d=1
22 1z
+1
2
J 1
2z
Y 1
2z
, <> 1
2.
10.22.22Z1
2
0J
zsin2
J
zcos2
(sin)2+1(cos)2+1d=
+1
2
+1
2
J++1
2(z)
(8z)1
2 (++ 1),<> 1
2;< > 1
2.
10.22.23Z1
2
0J
zsin2
J
zcos2
(sin)2 1secd=(++ ) (+ )2 1
(+ 1)z J++ (z),
<(+ )>0,< >0.
10.22.24Z1
2
0J
zsin2
J
zcos2
cotd=1
2 1J+(z), <>0;< > 1.
10.22.25Z1
2
0J(zsin)I(zcos)(tan)+1d= 1
2 1
2
(1
2z)
2 1
2+1
2+ 1J(z),< ><> 1.
ForIseex10.25(ii).
10.22.26Z1
2
0J(zsin)J(cos)(sin)+1(cos)+1d=zJ++1p
2+z2
(2+z2)1
2(++1),<> 1;< > 1.
Products
10.22.27Zx
0tJ2
1(t)dt= 21X
k=0(+ 2k)J2
+2k(x), < >0,
10.22.28Zx
0t
J2
1(t) J2
+1(t)
dt= 2J2
(x), < >0,
10.22.29Zx
0tJ2
0(t)dt=1
2x2
J2
0(x) +J2
1(x)
:
10.22.30Zx
0Jn(t)Jn+1(t)dt=1
2
1 J2
0(x)
nX
k=1J2
k(x) =1X
k=n+1J2
k(x), n= 0;1;2;:::.
Convolutions
10.22.31Zx
0J(t)J(x t)dt= 21X
k=0( 1)kJ++2k+1(x),<> 1;< > 1.
10.22.32Zx
0J(t)J1 (x t)dt=J0(x) cosx, 1<< <2.
10.22.33Zx
0J(t)J (x t)dt= sinx, j<j<1.
10.22.34Zx
0t 1J(t)J(x t)dt=J+(x)
, <>0;< > 1.
10.22 Integrals 243
10.22.35Zx
0J(t)J(x t)dt
t(x t)=(+)J+(x)
x, <>0;< >0.
Fractional Integral
10.22.361
( )Zx
0(x t) 1J(t)dt= 2 1X
k=0( )k
k!J+ +2k(x),< >0;<0.
When =m= 1;2;3;:::the left-hand side of (10.22.36) is the mth repeated integral of J(x) (xx1.4(v) and 1.15(vi)).
Orthogonality
If > 1, then
10.22.37Z1
0tJ(j;`t)J(j;mt)dt=1
2`;m(J0
(j;`))2;
wherej;`andj;mare zeros of J(x) (x10.21(i)), and `;mis Kronecker's symbol.
Also, ifa;b; are real constants with b6= 0 and > 1, then
10.22.38Z1
0tJ( `t)J( mt)dt=`;ma2
b2+ 2
` 2(J( `))2
2 2
`;
where `and mare positive zeros of aJ(x) +bxJ0
(x). (Compare (10.22.55)).
10.22(iii) Integrals over the Interval (x;1)
Whenx>0
10.22.39Z1
xJ0(t)
tdt+
+ ln 1
2x
=Zx
01 J0(t)
tdt=1X
k=1( 1)k 1(1
2x)2k
2k(k!)2;
10.22.40Z1
xY0(t)
tdt= 1
ln 1
2x
+
2+
6+2
1X
k=1( 1)k
(k+ 1) +1
2k ln 1
2x(1
2x)2k
2k(k!)2;
where
is Euler's constant ( x5.2(ii)).
10.22(iv) Integrals over the Interval (0;1)
10.22.41Z1
0J(t)dt= 1, < > 1,
10.22.42Z1
0Y(t)dt= tan 1
2
, j<j<1.
10.22.43Z1
0tJ(t)dt= 2 1
2+1
2+1
2
1
2 1
2+1
2, <(+)> 1,<<1
2,
10.22.44Z1
0tY(t)dt=2
1
2+1
2+1
2
1
2 1
2+1
2
sin 1
2 1
2
,<()> 1,<<1
2.
10.22.45Z1
01 J0(t)
tdt= sec 1
2
2 2 1
2+1
2, 1 <<<3.
10.22.46Z1
0t+1J(at)
(t2+b2)+1dt=ab
2 (+ 1)K (ab),a>0,<b>0, 1<< <2<+3
2.
10.22.47Z1
0tY(at)
t2+b2dt= b 1K(ab), a>0;<b>0; 1
2<< <5
2.
ForKseex10.25(ii).
10.22.48Z1
0J(xcosh )(cosh )1 (sinh )2+1d = 2 (+ 1)x 1J 1(x),x>0;< > 1;<>2<+1
2.
244 Bessel Functions
10.22.49Z1
0t 1e atJ(bt)dt=(1
2b)
a+ (+)F+
2;++ 1
2;+ 1; b2
a2
,<(+)>0;<(aib)>0,
10.22.50Z1
0t 1e atY(bt)dt= cot()(1
2b) (+)
(a2+b2)1
2(+)F+
2;1 +
2;+ 1;b2
a2+b2
csc()(1
2b) ( )
(a2+b2)1
2( )F
2;1
2; 1 ;b2
a2+b2
,
<>j<j;<(aib)>0.
For the hypergeometric function Fseex15.2(i).
10.22.51Z1
0J(bt) exp
p2t2
t+1dt=b
(2p2)+1exp
b2
4p2
, < > 1,<(p2)>0,
10.22.52Z1
0J(bt) exp( p2t2)dt=p
2pexp
b2
8p2
I/2b2
8p2
, < > 1;<(p2)>0,
10.22.53Z1
0Y2(bt) exp
p2t2
dt= p
2pexp
b2
8p2
Ib2
8p2
tan() +1
Kb2
8p2
sec()
,j<j<1
2,<(p2)>0.
ForIandKseex10.25(ii).
10.22.54Z1
0J(bt) exp
p2t2
t 1dt=(1
2b=p) 1
2+1
2
2pexp
b2
4p2
M
1
2 1
2+ 1;+ 1;b2
4p2
,
<(+)>0,<(p2)>0.
For the con
uent hypergeometric function Mseex13.2(i).
Orthogonality
10.22.55Z1
0t 1J+2`+1(t)J+2m+1(t)dt=`;m
2(2`++ 1), +`+m> 1.
Weber{Schafheitlin Discontinuous Integrals, including Special Cases
10.22.56Z1
0J(at)J(bt)
tdt=a 1
2+1
2 1
2+1
2
2b +1 1
2 1
2+1
2+1
2F
1
2(+ + 1);1
2( + 1);+ 1;a2
b2
,
0<a<b ,<(++ 1)><> 1.
If 0<b<a , then interchange aandb, and alsoand. Ifb=a, then
10.22.57Z1
0J(at)J(at)
tdt=(1
2a) 1 1
2+1
2 1
2+1
2
()
2 1
2+1
2 1
2+1
2
1
2+1
2 1
2+1
2
1
2+1
2+1
2+1
2,
<(++ 1)><>0.
10.22.58Z1
0J(at)J(bt)
tdt=(ab)
1
2+1
2
2(a2+b2) 1
2+1
2 1
2+1
2F2+ 1
4;2+ 3
4;+ 1;4a2b2
(a2+b2)2
,
a6=b,<(2+ 1)><> 1.
When<> 1
10.22.59Z1
0eibtJ(at)dt=8
>>><
>>>:exp(iarcsin(b=a))
(a2 b2)1
2; 0b<a;
iaexp 1
2i
(b2 a2)1
2
b+ (b2 a2)1
2;0<a<b:
10.22.60Z1
0eibtY0(at)dt=8
<
:(2i=)(a2 b2) 1
2arcsin(b=a); 0b<a;
(b2 a2) 1
2
1 +2i
lna
b+ (b2 a2)1
2
;0<a<b:
10.22 Integrals 245
When<>0,
10.22.61Z1
0t 1eibtJ(at)dt=8
>><
>>:(1=) exp(iarcsin(b=a));0ba;
aexp 1
2i
b+ (b2 a2)1
2; 0<ab:
When< ><> 1,
10.22.62Z1
0t +1J(at)J(bt)dt=8
<
:0; 0<b<a;
2 +1a(b2 a2) 1
b ( );0<ab:
When<>0,
10.22.63Z1
0J(at)J 1(bt)dt=8
><
>:b 1a ;0<b<a;
(2b) 1; b =a(>0);
0; 0<a<b:
Whenn= 0;1;2;::: and<> n 1,
10.22.64Z1
0J+2n+1(at)J(bt)dt=8
>>><
>>>:b (+n+ 1)
a+1n!F
n;+n+ 1;+ 1;b2
a2
;0<b<a;
( 1)n=(2a); b =a(>0);
0; 0<a<b:
10.22.65Z1
0J0(at) (J0(bt) J0(ct))dt
t=(
0; 0b<a; 0<ca;
ln(c=a);0b<ac:
Other Double Products
In (10.22.66){(10.22.70) a;b;c are positive constants.
10.22.66Z1
0e atJ(bt)J(ct)dt=1
(bc)1
2Q 1
2a2+b2+c2
2bc
, < > 1
2.
10.22.67Z1
0texp( p2t2)J(at)J(bt)dt=1
2p2exp
a2+b2
4p2
Iab
2p2
,< > 1;<(p2)>0.
10.22.68Z1
0texp( p2t2)J0(at)Y0(at)dt= 1
2p2exp
a2
2p2
K0a2
2p2
, <(p2)>0.
For the associated Legendre function Qseex14.3(ii) with = 0. ForIandKseex10.25(ii).
10.22.69Z1
0J(at)J(bt)tdt
t2 z2=(
1
2iJ(bz)H(1)
(az); a>b
1
2iJ(az)H(1)
(bz); b>a)
, < > 1;=z>0.
10.22.70Z1
0Y(at)J+1(bt)tdt
t2 z2=1
2J+1(bz)H(1)
(az), ab>0,< > 3
2;=z>0.
Equation (10.22.70) also remains valid if the order + 1 of theJfunctions on both sides is replaced by + 2n 3,
n= 1;2;:::, and the constraint < > 3
2is replaced by< > n+1
2.
See alsox1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions.
Triple Products
In (10.22.71) and (10.22.72) a;b;c are positive constants.
10.22.71Z1
0J(at)J(bt)J(ct)t1 dt=(bc) 1(sin ) 1
2
(2)1
2aP1
2
1
2(cos ),
<> 1
2;< > 1;jb cj<a<b +c;cos = (b2+c2 a2)=(2bc).
10.22.72Z1
0J(at)J(bt)J(ct)t1 dt=(bc) 1cos()(sinh ) 1
2
(1
23)1
2aQ1
2
1
2(cosh ),
<> 1
2;< > 1;a>b +c;cosh = (a2 b2 c2)=(2bc).
246 Bessel Functions
For the Ferrers function Pand the associated Legendre function Q, seexx14.3(i) and 14.3(ii), respectively.
In (10.22.74) and (10.22.75), a;b;c are positive constants and
10.22.73 A=s(s a)(s b)(s c); s =1
2(a+b+c):
(Thus ifa;b;c are the sides of a triangle, then A1
2is the area of the triangle.)
If< > 1
2, then
10.22.74Z1
0J(at)J(bt)J(ct)t1 dt=8
><
>:2 1A 1
2
1
2(abc)
+1
2; A> 0;
0; A 0:
Ifjj<1
2, then
10.22.75Z1
0Y(at)J(bt)J(ct)t1+dt=8
>>>>><
>>>>>: (abc)( A) 1
2
1
22+1 1
2 ;0<a<jb cj;
0; jb cj<a<b +c;
(abc)( A) 1
2
1
22+1 1
2 ; a>b +c:
Additional in nite integrals over the product of three Bessel functions (including modi ed Bessel functions) are
given in Gervois and Navelet (1984, 1985a,b, 1986a,b).
10.22(v) Hankel Transform
The Hankel transform (orBessel transform ) of a func-
tionf(x) is de ned as
10.22.76 g(y) =Z1
0f(x)J(xy)(xy)1
2dx:
Hankel's inversion theorem is given by
10.22.77 f(y) =Z1
0g(x)J(xy)(xy)1
2dx:
Sucient conditions for the validity of (10.22.77) are
thatR1
0jf(x)jdx <1when 1
2, or thatR1
0jf(x)jdx <1andR1
0x+1
2jf(x)jdx <1when
1< < 1
2; see Titchmarsh (1986a, Theorem 135,
Chapter 8) and Akhiezer (1988, p. 62).
For asymptotic expansions of Hankel transforms see
Wong (1976, 1977) and Frenzen and Wong (1985).
For collections of Hankel transforms see Erd elyi et al.
(1954b, Chapter 8) and Oberhettinger (1972).
10.22(vi) Compendia
For collections of integrals of the functions J(z),Y(z),
H(1)
(z), andH(2)
(z), including integrals with respect
to the order, see Andrews et al. (1999, pp. 216{225),
Apelblat (1983, x12), Erd elyi et al. (1953b,xx7.7.1{
7.7.7 and 7.14{7.14.2), Erd elyi et al. (1954a,b), Grad-
shteyn and Ryzhik (2000, xx5.5 and 6.5{6.7), Gr obner
and Hofreiter (1950, pp. 196{204), Luke (1962), Mag-
nuset al. (1966,x3.8), Marichev (1983, pp. 191{216),
Oberhettinger (1974, xx1.10 and 2.7), Oberhettinger
(1990,xx1.13{1.16 and 2.13{2.16), Oberhettinger and
Badii (1973,xx1.14 and 2.12), Okui (1974, 1975), Prud-
nikov et al. (1986b,xx1.8{1.10, 2.12{2.14, 3.2.4{3.2.7,3.3.2, and 3.4.1), Prudnikov et al. (1992a,xx3.12{3.14),
Prudnikov et al. (1992b,xx3.12{3.14), Watson (1944,
Chapters 5, 12, 13, and 14), and Wheelon (1968).
10.23 Sums
10.23(i) Multiplication Theorem
10.23.1
C(z) =1X
k=0(1)k(2 1)k(1
2z)k
k!Ck(z),
j2 1j<1.
IfC=Jand the upper signs are taken, then the restric-
tion onis unnecessary.
10.23(ii) Addition Theorems
Neumann's Addition Theorem
10.23.2 C(uv) =1X
k= 1Ck(u)Jk(v),jvj<juj.
The restrictionjvj<jujis unnecessary when C=Jand
is an integer. Special cases are:
10.23.3 J2
0(z) + 21X
k=1J2
k(z) = 1;
10.23.42nX
k=0( 1)kJk(z)J2n k(z)
+ 21X
k=1Jk(z)J2n+k(z) = 0, n1,
10.23.5
nX
k=0Jk(z)Jn k(z) + 21X
k=1( 1)kJk(z)Jn+k(z) =Jn(2z):
10.23 Sums 247
Graf's and Gegenbauer's Addition Theorems
De ne
10.23.6w=p
u2+v2 2uvcos ;
u vcos =wcos ; v sin =wsin ;
the branches being continuous and chosen so that w!u
and !0 asv!0. Ifu,vare real and positive and
0 , thenwand are real and nonnegative, and
the geometrical relationship is shown in Figure 10.23.1.
Figure 10.23.1 : Graf's and Gegenbauer's addition theo-
rems.
10.23.7C(w)cos
sin( ) =1X
k= 1C+k(u)Jk(v)cos
sin(k ),
jvei j<juj.
10.23.8
C(w)
w= 2 ()
1X
k=0(+k)C+k(u)
uJ+k(v)
vC()
k(cos ),
6= 0; 1;:::,jvei j<juj,
whereC()
k(cos ) is Gegenbauer's polynomial ( x18.3).
The restrictionjvei j<jujis unnecessary in (10.23.7)
whenC=Jandis an integer, and in (10.23.8) when
C=J.
The degenerate form of (10.23.8) when u=1is
given by
10.23.9
eivcos = ()
(1
2v)1X
k=0(+k)ikJ+k(v)C()
k(cos ),
6= 0; 1;:::.
Partial Fractions
For expansions of products of Bessel functions of the
rst kind in partial fractions see Rogers (2005).10.23(iii) Series Expansions of Arbitrary
Functions
Neumann's Expansion
10.23.10 f(z) =a0J0(z) + 21X
k=1akJk(z),jzj<c,
wherecis the distance of the nearest singularity of the
analytic function f(z) fromz= 0,
10.23.11 ak=1
2iZ
jzj=c0f(t)Ok(t)dt, 0<c0<c,
andOk(t) isNeumann's polynomial , de ned by the gen-
erating function:
10.23.12
1
t z=J0(z)O0(t) + 21X
k=1Jk(z)Ok(t),jzj<jtj.
On(t) is a polynomial of degree n+1 in 1/t:O0(t) =
1=tand
10.23.13
On(t) =1
4bn=2cX
k=0(n k 1)!n
k!2
tn 2k+1
,n= 1;2;:::.
For the more general form of expansion
10.23.14zf(z) =a0J(z) + 21X
k=1akJ+k(z)
see Watson (1944, x16.13), and for further generaliza-
tions see Watson (1944, Chapter 16) and (Erd elyi et al. ,
1953b,x7.10.1).
Examples
10.23.15(1
2z)=1X
k=0(+ 2k) (+k)
k!J+2k(z),
6= 0; 1; 2;:::,
10.23.16
Y0(z) =2
ln 1
2z
+
J0(z) 4
1X
k=1( 1)kJ2k(z)
k;
10.23.17Yn(z) = n!(1
2z) n
n 1X
k=0(1
2z)kJk(z)
k!(n k)
+2
ln 1
2z
(n+ 1)
Jn(z)
2
1X
k=1( 1)k(n+ 2k)Jn+2k(z)
k(n+k);
where
is Euler's constant and (n+ 1) =
0(n+ 1)= (n+ 1) (x5.2).
Other examples are provided by (10.12.1){(10.12.6),
(10.23.2), and (10.23.7).
248 Bessel Functions
Fourier{Bessel Expansion
Assumef(t) satis es
10.23.18Z1
0t1
2jf(t)jdt<1;
and de ne
10.23.19
am=2
(J+1(j;m))2Z1
0tf(t)J(j;mt)dt, 1
2,
wherej;mis as inx10.21(i). If 0 <x< 1, then
10.23.201
2f(x ) +1
2f(x+) =1X
m=1amJ(j;mx);
provided that f(t) is of bounded variation ( x1.4(v)) on
an interval [ a;b] with 0< a < x < b < 1. This re-
sult is proved in Watson (1944, Chapter 18) and further
information is provided in this reference, including the
behavior of the series near x= 0 andx= 1.
As an example,
10.23.21x=1X
m=12J(j;mx)
j;mJ+1(j;m), >0;0x<1.
(Note that when x= 1 the left-hand side is 1 and the
right-hand side is 0.)
Other Series Expansions
For other types of expansions of arbitrary functions
in series of Bessel functions, see Watson (1944, Chap-
ters 17{19) and Erd elyi et al. (1953b,xx7.10.2{7.10.4).
See also Sch afke (1960, 1961b).
10.23(iv) Compendia
For collections of sums of series involving Bessel or Han-
kel functions see Erd elyi et al. (1953b,x7.15), Grad-
shteyn and Ryzhik (2000, xx8.51{8.53), Hansen (1975),
Luke (1969b,x9.4), Prudnikov et al. (1986b, pp. 651{
691 and 697{700), and Wheelon (1968, pp. 48{51).
10.24 Functions of Imaginary Order
Withz=xandreplaced by i, Bessel's equation
(10.2.1) becomes
10.24.1x2d2w
dx2+xdw
dx+ (x2+2)w= 0:
For2Randx2(0;1) de ne
10.24.2eJ(x) = sech 1
2
<(Ji(x));
eY(x) = sech 1
2
<(Yi(x));
and
10.24.3 (1 +i) =
sinh()1
2
ei
;where
is real and continuous with
0= 0; compare
(5.4.3). Then
10.24.4eJ (x) =eJ(x);eY (x) =eY(x);
andeJ(x),eY(x) are linearly independent solutions of
(10.24.1):
10.24.5 WfeJ(x);eY(x)g= 2=(x):
Asx!+1, with xed,
10.24.6eJ(x) =p
2=(x) cos
x 1
4
+O
x 3
2
;
eY(x) =p
2=(x) sin
x 1
4
+O
x 3
2
:
Asx!0+, with xed,
10.24.7eJ(x) =2 tanh(1
2)
1
2
cos
ln(1
2x)
+O(x2);
10.24.8eY(x) =2 coth(1
2)
1
2
sin
ln(1
2x)
+O(x2), >0;
and
10.24.9eY0(x) =Y0(x) =2
ln(1
2x) +
+O(x2lnx);
where
denotes Euler's constant x5.2(ii).
In consequence of (10.24.6), when xis largeeJ(x)
andeY(x) comprise a numerically satisfactory pair of
solutions of (10.24.1); compare x2.7(iv). Also, in con-
sequence of (10.24.7){(10.24.9), when xis small either
eJ(x) and tanh(1
2)eY(x) oreJ(x) andeY(x) comprise
a numerically satisfactory pair depending whether 6= 0
or= 0.
For graphs of eJ(x) andeY(x) seex10.3(iii).
For mathematical properties and applications of
eJ(x) andeY(x), including zeros and uniform asymp-
totic expansions for large , see Dunster (1990a). In
this reference eJ(x) andeY(x) are denoted respectively
byFi(x) andGi(x).
Modi ed Bessel Functions
10.25 De nitions
10.25(i) Modi ed Bessel's Equation
10.25.1 z2d2w
dz2+zdw
dz (z2+2)w= 0:
This equation is obtained from Bessel's equation
(10.2.1) on replacing zbyiz, and it has the same kinds
of singularities. Its solutions are called modi ed Bessel
functions orBessel functions of imaginary argument .
10.26 Graphics 249
10.25(ii) Standard Solutions
10.25.2I(z) = (1
2z)1X
k=0(1
4z2)k
k! (+k+ 1):
This solution has properties analogous to those of J(z),
de ned inx10.2(ii). In particular, the principal branch
ofI(z) is de ned in a similar way: it corresponds to the
principal value of (1
2z), is analytic in Cn( 1;0], and
two-valued and discontinuous on the cut ph z=.
The de ning property of the second standard solu-
tionK(z) of (10.25.1) is
10.25.3 K(z)p
=(2z)e z;
asz!1 injphzj3
2 (<3
2). It has a branch
point atz= 0 for all 2C. The principal branch
corresponds to the principal value of the square root in
(10.25.3), is analytic in Cn( 1;0], and two-valued and
discontinuous on the cut ph z=.
BothI(z) andK(z) are real when is real and
phz= 0.
For xedz(6= 0) each branch of I(z) andK(z) is
entire in.Branch Conventions
Except where indicated otherwise it is assumed through-
out this Handbook that the symbols I(z) andK(z)
denote the principal values of these functions.
Symbol Z(z)
Corresponding to the symbol Cintroduced inx10.2(ii),
we sometimes use Z(z) to denote I(z),eiK(z), or
any nontrivial linear combination of these functions, the
coecients in which are independent of zand.
10.25(iii) Numerically Satisfactory Pairs of
Solutions
Table 10.25.1 lists numerically satisfactory pairs of solu-
tions (x2.7(iv)) of (10.25.1). It is assumed that <0.
When< <0,I(z) is replaced by I (z).
Table 10.25.1 : Numerically satisfactory pairs of solu-
tions of the modi ed Bessel's equation.
Pair Region
I(z);K(z)jphzj1
2
I(z);K
zei1
2phz3
2
10.26 Graphics
10.26(i) Real Order and Variable
See Figures 10.26.1{10.26.6.
Figure 10.26.1 :I0(x),I1(x),K0(x),K1(x), 0x3.
Figure 10.26.2 :e xI0(x),e xI1(x),exK0(x),exK1(x),
0x10.
250 Bessel Functions
Figure 10.26.3 :I(x), 0x5;04.
Figure 10.26.4 :K(x), 0:1x5;04.
Figure 10.26.5 :I0
(x), 0x5;04.
Figure 10.26.6 :K0
(x), 0:3x5;04.
10.26(ii) Real Order, Complex Variable
Apply (10.27.6) and (10.27.8) to x10.3(ii).
10.26(iii) Imaginary Order, Real Variable
See Figures 10.26.7{10.26.10. For the notation, see x10.45.
Figure 10.26.7 :eI1=2(x);eK1=2(x);0:01x3.
Figure 10.26.8 :eI1(x);eK1(x);0:01x3.
10.27 Connection Formulas 251
Figure 10.26.9 :eI5(x);eK5(x);0:01x3.
Figure 10.26.10 :eK5(x);0:01x3.
10.27 Connection Formulas
Other solutions of (10.25.1) are I (z) andK (z).
10.27.1 I n(z) =In(z);
10.27.2I (z) =I(z) + (2=) sin()K(z);
10.27.3 K (z) =K(z):
10.27.4 K(z) =1
2I (z) I(z)
sin():
Whenis an integer limiting values are taken:
10.27.5
Kn(z) =( 1)n 1
2
@I(z)
@
=n+@I(z)
@
= n!
,
n= 0;1;2;:::.
In terms of the solutions of (10.2.1),
10.27.6
I(z) =ei=2J
zei=2
, phz1
2,
10.27.7
I(z) =1
2ei=2
H(1)
zei=2
+H(2)
zei=2
,
phz1
2.
10.27.8
K(z)
=(
1
2iei=2H(1)
zei=2
; phz1
2;
1
2ie i=2H(2)
ze i=2
; 1
2phz:
10.27.9iJ(z) =e i=2K
ze i=2
ei=2K
zei=2
,jphzj1
2.
10.27.10
Y(z) =e i=2K
ze i=2
+ei=2K
zei=2
,
jphzj1
2.10.27.11Y(z) =e(+1)i=2I
zei=2
(2=)ei=2K
zei=2
,
1
2phz.
See alsox10.34.
Many properties of modi ed Bessel functions follow
immediately from those of ordinary Bessel functions by
application of (10.27.6){(10.27.8).
10.28 Wronskians and Cross-Products
10.28.1
WfI(z);I (z)g=I(z)I 1(z) I+1(z)I (z)
= 2 sin()=(z);
10.28.2
WfK(z);I(z)g=I(z)K+1(z) +I+1(z)K(z)
= 1=z:
10.29 Recurrence Relations and Derivatives
10.29(i) Recurrence Relations
WithZ(z) de ned as inx10.25(ii),
10.29.1Z 1(z) Z+1(z) = (2=z)Z(z);
Z 1(z) +Z+1(z) = 2Z0
(z):
10.29.2Z0
(z) =Z 1(z) (=z)Z(z);
Z0
(z) =Z+1(z) + (=z)Z(z):
10.29.3I0
0(z) =I1(z); K0
0(z) = K1(z):
252 Bessel Functions
10.29(ii) Derivatives
Fork= 0;1;2;:::,
10.29.41
zd
dzk
(zZ(z)) =z kZ k(z);
1
zd
dzk
(z Z(z)) =z kZ+k(z):
10.29.5Z(k)
(z) =1
2k
Z k(z) +k
1
Z k+2(z)
+k
2
Z k+4(z)++Z+k(z)
:
10.30 Limiting Forms
10.30(i)z!0
Whenis xed and z!0,
10.30.1I(z)(1
2z)= (+ 1),6= 1; 2; 3;:::,
10.30.2 K(z)1
2 ()(1
2z) ,< >0,
10.30.3 K0(z) lnz:
ForK(x), whenis purely imaginary and x!0+,
see (10.45.2) and (10.45.7).
10.30(ii)z!1
Whenis xed and z!1 ,
10.30.4I(z)ez=p
2z,jphzj1
2 ,
10.30.5I(z)e(+1
2)ie z=p
2z,
1
2+phz3
2 .
ForK(z) see (10.25.3).
10.31 Power Series
ForI(z) see (10.25.2) and (10.27.1). When is not
an integer the corresponding expansion for K(z) is ob-
tained from (10.25.2) and (10.27.4).
Whenn= 0;1;2;:::,
10.31.1Kn(z) =1
2(1
2z) nn 1X
k=0(n k 1)!
k!( 1
4z2)k
+ ( 1)n+1ln 1
2z
In(z)
+ ( 1)n1
2(1
2z)n1X
k=0( (k+ 1)
+ (n+k+ 1))(1
4z2)k
k!(n+k)!;where (x) = 0(x)= (x) (x5.2(i)). In particular,
10.31.2
K0(z) =
ln 1
2z
+
I0(z) +1
4z2
(1!)2
+ (1 +1
2)(1
4z2)2
(2!)2+ (1 +1
2+1
3)(1
4z2)3
(3!)2+:
For negative values of nuse (10.27.3).
10.31.3
I(z)I(z) = (1
2z)+1X
k=0(++k+ 1)k(1
4z2)k
k! (+k+ 1) (+k+ 1):
10.32 Integral Representations
10.32(i) Integrals along the Real Line
10.32.1
I0(z) =1
Z
0ezcosd=1
Z
0cosh(zcos)d:
10.32.2I(z) =(1
2z)
1
2
+1
2Z
0ezcos(sin)2d
=(1
2z)
1
2
+1
2Z1
1(1 t2) 1
2eztdt,
< > 1
2.
10.32.3 In(z) =1
Z
0ezcoscos(n)d:
10.32.4
I(z) =1
Z
0ezcoscos()d
sin()
Z1
0e zcosht tdt,jphzj<1
2.
10.32.5
K0(z) = 1
Z
0ezcos
+ ln
2z(sin)2
d:
10.32.6
K0(x) =Z1
0cos(xsinht)dt=Z1
0cos(xt)p
t2+ 1dt,x>0.
10.32.7K(x) = sec 1
2Z1
0cos(xsinht) cosh(t)dt
= csc 1
2Z1
0sin(xsinht) sinh(t)dt,
j<j<1,x>0.
10.32.8K(z) =1
2(1
2z)
+1
2Z1
0e zcosht(sinht)2dt
=1
2(1
2z)
+1
2Z1
1e zt(t2 1) 1
2dt,
< > 1
2,jphzj<1
2.
10.32.9
K(z) =Z1
0e zcoshtcosh(t)dt,jphzj<1
2.
10.33 Continued Fractions 253
10.32.10
K(z) =1
2(1
2z)Z1
0exp
t z2
4tdt
t+1,jphzj<1
4.
Basset's Integral
10.32.11K(xz) =
+1
2
(2z)
1
2xZ1
0cos(xt)dt
(t2+z2)+1
2,
< > 1
2,x>0,jphzj<1
2.
10.32(ii) Contour Integrals
10.32.12
I(z) =1
2iZ1+i
1 iezcosht tdt,jphzj<1
2.
Mellin{Barnes Type
10.32.13K(z) =(1
2z)
4iZc+i1
c i1 (t) (t )(1
2z) 2tdt,
c>max(<;0);jphzj<.
10.32.14
K(z) =1
22i
2z1
2e zcos()
Zi1
i1 (t) 1
2 t
1
2 t+
(2z)tdt,
1
2=2Z;jphzj<3
2.In (10.32.14) the integration contour separates the poles
of (t) from the poles of 1
2 t
1
2 t+
.
10.32(iii) Products
10.32.15
I(z)I(z) =2
Z1
2
0I+(2zcos) cos(( ))d,
<(+)> 1.
10.32.16
I(x)K(x) =Z1
0J(2xsinht)e( )tdt,
<()> 1
2,<()> 1,x>0.
10.32.17
K(z)K(z) = 2Z1
0K(2zcosht) cosh(()t)dt,
jphzj<1
2.
10.32.18
K(z)K()
=1
2Z1
0exp
t
2 z2+2
2t
Kz
tdt
t,
jphzj<,jphj<,jph(z+)j<1
4.
Mellin{Barnes Type
10.32.19K(z)K(z) =1
8iZc+i1
c i1
t+1
2+1
2
t+1
2 1
2
t 1
2+1
2
t 1
2 1
2
(2t)(1
2z) 2tdt,
c>1
2(j<j+j<j);jphzj<1
2.
For similar integrals for J(z)K(z) andI(z)K(z) see
Paris and Kaminski (2001, p. 116).
10.32(iv) Compendia
For collections of integral representations of modi ed
Bessel functions, or products of modi ed Bessel func-
tions, see Erd elyi et al. (1953b,xx7.3, 7.12, and 7.14.2),
Erd elyi et al. (1954a, pp. 48{60, 105{115, 276{285, and
357{359), Gr obner and Hofreiter (1950, pp. 193{194),
Magnus et al. (1966,x3.7), Marichev (1983, pp. 191{
216), and Watson (1944, Chapters 6, 12, and 13).
10.33 Continued Fractions
AssumeI 1(z)6= 0. Then
10.33.1
I(z)
I 1(z)=1
2z 1+1
2(+ 1)z 1+1
2(+ 2)z 1+,
z6= 0,10.33.2
I(z)
I 1(z)
=1
2z=
1+1
4z2=((+ 1))
1+1
4z2=((+ 1)(+ 2))
1+,
6= 0; 1; 2;:::.
See also Cuyt et al. (2008, pp. 361{367).
10.34 Analytic Continuation
Whenm2Z,
10.34.1 I
zemi
=emiI(z);
10.34.2
K
zemi
=e miK(z) isin(m) csc()I(z):
10.34.3I
zemi
= (i=)
emiK
zei
e(m1)iK(z)
;
10.34.4K
zemi
= csc()
sin(m)K
zei
sin((m1))K(z)
:
254 Bessel Functions
If=n(2Z), then limiting values are taken in
(10.34.2) and (10.34.4):
10.34.5
Kn
zemi
= ( 1)mnKn(z) + ( 1)n(m 1) 1miIn(z);
10.34.6Kn
zemi
=( 1)n(m 1)mKn
zei
( 1)nm(m1)Kn(z):
For real,
10.34.7 I(z) =I(z); K(z) =K(z):
For complex replacebyon the right-hand sides.
10.35 Generating Function and Associated
Series
Forz2Candt2Cnf0g,
10.35.1 e1
2z(t+t 1)=1X
m= 1tmIm(z):
Forz;2C,
10.35.2ezcos=I0(z) + 21X
k=1Ik(z) cos(k);
10.35.3
ezsin=I0(z) + 21X
k=0( 1)kI2k+1(z) sin((2k+ 1))
+ 21X
k=1( 1)kI2k(z) cos(2k):
10.35.4 1 =I0(z) 2I2(z) + 2I4(z) 2I6(z) +;
10.35.5ez=I0(z)2I1(z) + 2I2(z)2I3(z) +;
10.35.6coshz=I0(z) + 2I2(z) + 2I4(z) + 2I6(z) +:::;
sinhz= 2I1(z) + 2I3(z) + 2I5(z) +::::
10.36 Other Di erential Equations
The quantity 2in (10.13.1){(10.13.6) and (10.13.8) can
be replaced by 2if at the same time the symbol Cin
the given solutions is replaced by Z. Also,
10.36.1z2(z2+2)w00+z(z2+ 32)w0
(z2+2)2+z2 2
w= 0,w=Z0
(z),
10.36.2
z2w00+z(12z)w0+ (z 2)w= 0,
w=ezZ(z).
Di erential equations for products can be obtained
from (10.13.9){(10.13.11) by replacing zbyiz.10.37 Inequalities; Monotonicity
If(0) is xed, then throughout the interval 0 <
x <1,I(x) is positive and increasing, and K(x) is
positive and decreasing.
Ifx(>0) is xed, then throughout the interval
0< <1,I(x) is decreasing, and K(x) is increasing.
For sharper inequalities when the variables are real
see Paris (1984) and Laforgia (1991).
If 0 < andjphzj<, then
10.37.1 jK(z)j<jK(z)j:
See also Pal0tsev (1999) and Petropoulou (2000).
10.38 Derivatives with Respect to Order
10.38.1
@I(z)
@=I(z) ln 1
2z
(1
2z)1X
k=0 (+k+ 1)
(+k+ 1)(1
4z2)k
k!;
10.38.2@K(z)
@=1
2csc()@I (z)
@ @I(z)
@
cot()K(z), =2Z.
Integer Values of
10.38.3
( 1)n@I(z)
@
=n= Kn(z)
+n!
2(1
2z)nn 1X
k=0( 1)k(1
2z)kIk(z)
k!(n k);
10.38.4@K(z)
@
=n=n!
2(1
2z)nn 1X
k=0(1
2z)kKk(z)
k!(n k):
10.38.5@I(z)
@
=0= K0(z);@K(z)
@
=0= 0:
Half-Integer Values of
For the notations E1and Ei seex6.2(i). When x>0,
10.38.6
@I(x)
@
=1
2= 1p
2x
E1(2x)exEi(2x)e x
;
10.38.7@K(x)
@
=1
2=r
2xE1(2x)ex:
For further results see Brychkov and Geddes (2005).
10.39 Relations to Other Functions
Elementary Functions
10.39.1
I1
2(z) =2
z1
2
sinhz; I 1
2(z) =2
z1
2
coshz;
10.39.2 K1
2(z) =K 1
2(z) =
2z1
2e z:
For these and general results when is half an odd in-
teger seexx10.47(ii) and 10.49(ii).
10.40 Asymptotic Expansions for Large Argument 255
Airy Functions
Seexx9.6(i) and 9.6(ii).
Parabolic Cylinder Functions
With the notation of x12.2(i),
10.39.3 K1
4(z) =1
2z 1
4U
0;2z1
2
;
10.39.4
K3
4(z) =1
21
2z 3
4
1
2U
1;2z1
2
+U
1;2z1
2
:
Principal values on each side of these equations corre-
spond. For these and further results see Miller (1955,
pp. 42{43 and 77{79).
Con
uent Hypergeometric Functions
10.39.5I(z) =(1
2z)ez
(+ 1)M
+1
2;2+ 1;2z
;
10.39.6K(z) =1
2(2z)e zU
+1
2;2+ 1;2z
;
10.39.7
I(z) =(2z) 1
2M0;(2z)
22 (+ 1), 26= 1; 2; 3;:::,
10.39.8 K(z) =
2z1
2W0;(2z):
For the functions M,U,M0;, andW0;seexx13.2(i)
and 13.14(i).
Generalized Hypergeometric Functions and
Hypergeometric Function
10.39.9I(z) =(1
2z)
(+ 1)0F1
;+ 1;1
4z2
;
10.39.10I(z) = (1
2z)limF
;;+ 1;z2=(4)
;
asand!1 inC, withzand xed. For the
functions 0F1andFsee (16.2.1) andx15.2(i).
10.40 Asymptotic Expansions for Large
Argument
10.40(i) Hankel's Expansions
With the notation of xx10.17(i) and 10.17(ii), as z!1
with xed,
10.40.1
I(z)ez
(2z)1
21X
k=0( 1)kak()
zk,jphzj1
2 ,
10.40.2
K(z)
2z1
2e z1X
k=0ak()
zk,jphzj3
2 ,
10.40.3
I0
(z)ez
(2z)1
21X
k=0( 1)kbk()
zk,jphzj1
2 ,10.40.4
K0
(z)
2z1
2e z1X
k=0bk()
zk,jphzj3
2 .
Corresponding expansions for I(z),K(z),I0
(z),
andK0
(z) for other ranges of ph zare obtainable by
combining (10.34.3), (10.34.4), (10.34.6), and their dif-
ferentiated forms, with (10.40.2) and (10.40.4). In par-
ticular, use of (10.34.3) with m= 0 yields the following
more general (and more accurate) version of (10.40.1):
10.40.5I(z)ez
(2z)1
21X
k=0( 1)kak()
zk
ieie z
(2z)1
21X
k=0ak()
zk,
1
2+phz3
2 .
Products
With= 42and xed,
10.40.6
I(z)K(z)1
2z
1 1
2 1
(2z)2+13
24( 1)( 9)
(2z)4
;
10.40.7
I0
(z)K0
(z) 1
2z
1 +1
2 3
(2z)2 1
24( 1)( 45)
(2z)4
+
;
asz!1 injphzj1
2 . The general terms in
(10.40.6) and (10.40.7) can be written down by analogy
with (10.18.17), (10.18.19), and (10.18.20).
-Derivative
For xed,
10.40.8@K(z)
@
2z1
2e z
z1X
k=0 k()
(8z)k;
asz!1 injphzj3
2 . Here 0() = 1 and
10.40.9
k() =(42 12)(42 32)(42 (2k+ 1)2)
(k+ 1)!
1
42 12+1
42 32+
+1
42 (2k+ 1)2
:
10.40(ii) Error Bounds for Real Argument and
Order
In the expansion (10.40.2) assume that z >0 and the
sum is truncated when k=` 1. Then the remain-
der term does not exceed the rst neglected term in
256 Bessel Functions
absolute value and has the same sign provided that
`max(jj 1
2;1).
For the error term in (10.40.1) see x10.40(iii).
10.40(iii) Error Bounds for Complex Argument
and Order
For (10.40.2) write
10.40.10K(z) =
2z1
2e z ` 1X
k=0ak()
zk+R`(;z)!
,
`= 1;2;:::.
Then
10.40.11
jR`(;z)j2ja`()jVz;1
t `
exp
j2 1
4jVz;1
t 1
;
whereVdenotes the variational operator ( x2.3(i)), and
the paths of variation are subject to the condition that
j<tjchanges monotonically. Bounds for Vz;1
t `
are
given by
10.40.12
Vz;1
t `
8
><
>:jzj `;jphzj1
2;
(`)jzj `;1
2jphzj;
2 (`)j<zj `; jphzj3
2;
where (`) =1
2 1
2`+ 1
= 1
2`+1
2
; seex9.7(i).
A similar result for (10.40.1) is obtained by combin-
ing (10.34.3), with m= 0, and (10.40.10){(10.40.12);
see Olver (1997b, p. 269).
10.40(iv) Exponentially-Improved Expansions
In (10.40.10)
10.40.13
R`(;z) = ( 1)`2 cos()
m 1X
k=0ak()
zkG` k(2z) +Rm;`(;z)!
;
whereGp(z) is given by (10.17.16). If z! 1 with
j` 2jzjjbounded and m(0) xed, then
10.40.14 Rm;`(;z) =O
e 2jzjz m
,jphzj.For higher re-expansions of the remainder term see
Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995,
1996), and Paris (2001a,b).
10.41 Asymptotic Expansions for Large
Order
10.41(i) Asymptotic Forms
If!1 through positive real values with z(6= 0) xed,
then
10.41.1 I(z)1p
2ez
2
;
10.41.2 K(z)r
2ez
2
:
10.41(ii) Uniform Expansions for Real Variable
As!1 through positive real values,
10.41.3I(z)e
(2)1
2(1 +z2)1
41X
k=0Uk(p)
k;
10.41.4K(z)
21
2e
(1 +z2)1
41X
k=0( 1)kUk(p)
k;
10.41.5I0
(z)(1 +z2)1
4e
(2)1
2z1X
k=0Vk(p)
k;
10.41.6
K0
(z)
21
2(1 +z2)1
4e
z1X
k=0( 1)kVk(p)
k;
uniformly for 0 <z<1. Here
10.41.7= (1 +z2)1
2+ lnz
1 + (1 +z2)1
2;
10.41.8 p= (1 +z2) 1
2;
where the branches assume their principal values. Also,
Uk(p) andVk(p) are polynomials in pof degree 3k, given
byU0(p) =V0(p) = 1, and
10.41.9Uk+1(p) =1
2p2(1 p2)U0
k(p) +1
8Zp
0(1 5t2)Uk(t)dt;
Vk+1(p) =Uk+1(p) 1
2p(1 p2)Uk(p) p2(1 p2)U0
k(p), k= 0;1;2;:::.
Fork= 1;2;3,
10.41.10U1(p) =1
24(3p 5p3); U 2(p) =1
1152(81p2 462p4+ 385p6);
U3(p) =1
4 14720(30375p3 3 69603p5+ 7 65765p7 4 25425p9);
10.41.11V1(p) =1
24( 9p+ 7p3); V 2(p) =1
1152( 135p2+ 594p4 455p6);
V3(p) =1
4 14720( 42525p3+ 4 51737p5 8 83575p7+ 4 75475p9):
10.41 Asymptotic Expansions for Large Order 257
ForU4(p),U5(p),U6(p), see Bickley et al. (1952,
p. xxxv).
For numerical tables of =(z) and the coecients
Uk(p),Vk(p), see Olver (1962, pp. 43{51).
10.41(iii) Uniform Expansions for Complex
Variable
The expansions (10.41.3){(10.41.6) also hold uniformly
in the sectorjphzj1
2 (<1
2), with the branches
of the fractional powers in (10.41.3){(10.41.8) extended
by continuity from the positive real z-axis.Figures 10.41.1 and 10.41.2 show corresponding
points of the mapping of the z-plane and the -plane.
The curveE1BE2in thez-plane is the upper boundary
of the domain Kdepicted in Figure 10.20.3 and rotated
through an angle 1
2. ThusBis the point z=c,
wherecis given by (10.20.18).
For derivations of the results in this subsection, and
also error bounds, see Olver (1997b, pp. 374{378). For
extensions of the regions of validity in the z-plane and
extensions to complex values of see Olver (1997b,
pp. 378{382).
Figure 10.41.1 :z-plane.
Figure 10.41.2 :-plane.
For expansions in inverse factorial series see Dunster
et al. (1993).
10.41(iv) Double Asymptotic Properties
The series (10.41.3){(10.41.6) can also be regarded as
generalized asymptotic expansions for large jzj. Thus
asz!1 with`(1) and(>0) both xed,
10.41.12
I(z) =e
(2)1
2(1 +z2)1
4 ` 1X
k=0Uk(p)
k+O1
z`!
,
jphzj1
2 ,
10.41.13K(z) =
21
2e
(1 +z2)1
4
` 1X
k=0( 1)kUk(p)
k+O1
z`!
,
jphzj3
2 .
Similarly for (10.41.5) and (10.41.6).In the case of (10.41.13) with positive real values of
zthe result is a consequence of the error bounds given
in Olver (1997b, pp. 377{378). Then by expanding the
quantities, (1 +z2) 1
4, andUk(p),k= 0;1;:::;` 1,
and rearranging, we arrive at an expansion of the
right-hand side of (10.41.13) in powers of z 1. More-
over, because of the uniqueness property of asymptotic
expansions (x2.1(iii)) this expansion must agree with
(10.40.2), with zreplaced by z, up to and including
the term in z (` 1). It also enjoys the same sector of
validity.
To establish (10.41.12) we substitute into (10.34.3),
withm= 0 andzreplaced by z, by means of (10.41.13)
observing that when jzjis large the e ect of replacing
zbyzeiis to replace , (1 +z2)1
4, andpby ,
i(1 +z2)1
4, and p, respectively.
258 Bessel Functions
10.41(v) Double Asymptotic Properties
(Continued)
Similar analysis can be developed for the uniform
asymptotic expansions in terms of Airy functions
given inx10.20. We rst prove that for the expan-
sions (10.20.6) for the Hankel functions H(1)
(z) and
H(2)
(z) thez-asymptotic property applies when z!
i1, respectively. This is a consequence of the er-
ror bounds associated with these expansions. We then
extend the validity of this property from z! i1
toz! 1 in the sector +phz2
in the case of H(1)
(z), and toz! 1 in the sec-
tor 2+phz in the case of H(2)
(z).
This is done by re-expansion with the aid of (10.20.10),
(10.20.11), andx10.41(ii), followed by comparison with
(10.17.5) and (10.17.6), with zreplaced by z. Lastly,
we substitute into (10.4.4), again with zreplaced by z.
The nal results are:
10.41.14
J(z)
=4
1 z21
40
@Ai
2
3
1
3 `X
k=0Ak()
2k+O1
3`+3!
+Ai0
2
3
5
3 ` 1X
k=0Bk()
2k+O1
3`+1!1
A;
10.41.15
Y(z)
= 4
1 z21
40
@Bi
2
3
1
3 `X
k=0Ak()
2k+O1
3`+3!
+Bi0
2
3
5
3 ` 1X
k=0Bk()
2k+O1
3`+1!1
A;
asz!1 injphzj , or equivalently as !1 in
jph( )j2
3 , for xed`(0) and xed (>0).
It needs to be noted that the results (10.41.14) and
(10.41.15) do notapply when z!0+ or equivalently
!+1. This is because Ak() and 1
2Bk();k=
0;1;:::, do not form an asymptotic scale ( x2.1(v)) as
!+1; see Olver (1997b, pp. 422{425).
10.42 Zeros
Properties of the zeros of I(z) andK(z) may be de-
duced from those of J(z) andH(1)
(z), respectively,
by application of the transformations (10.27.6) and
(10.27.8).
For example, if is real, then the zeros of I(z) are
all complex unless 2`< < (2` 1) for some positive
integer`, in which event I(z) has two real zeros.The distribution of the zeros of Kn(nz) in the sector
3
2phz1
2in the cases n= 1;5;10 is obtained on
rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively,
through an angle 1
2so that in each case the cut lies
along the positive imaginary axis. The zeros in the sec-
tor 1
2phz3
2are their conjugates.
Kn(z) has no zeros in the sector jphzj1
2; this
result remains true when nis replaced by any real num-
ber. For the number of zeros of K(z) in the sector
jphzj, whenis real, see Watson (1944, pp. 511{
513).
See also Kerimov and Skorokhodov (1984b,a).
10.43 Integrals
10.43(i) Inde nite Integrals
LetZ(z) be de ned as in x10.25(ii). Then
10.43.1Z
z+1Z(z)dz=z+1Z+1(z);
Z
z +1Z(z)dz=z +1Z 1(z):
10.43.2Z
zZ(z)dz=1
22 1
+1
2
z
(Z(z)L 1(z) Z 1(z)L(z)) ,
6= 1
2.
For the modi ed Struve function L(z) seex11.2(i).
10.43.3Z
ezzZ(z)dz=ezz+1
2+ 1(Z(z)Z+1(z)) ,
6= 1
2,Z
ezz Z(z)dz=ezz +1
1 2(Z(z)Z 1(z)) ,
6=1
2.
10.43(ii) Integrals over the Intervals (0;x)and
(x;1)
10.43.4Zx
0I0(t) 1
tdt
=1
21X
k=1( 1)k 1 (k+ 1) (1)
k!(1
2x)kIk(x)
=2
x1X
k=0( 1)k(2k+ 3)( (k+ 2) (1))I2k+3(x):
10.43.5Z1
xK0(t)
tdt=1
2
ln 1
2x
+
2+2
24 1X
k=1
(k+ 1)
+1
2k ln 1
2x(1
2x)2k
2k(k!)2;
10.43 Integrals 259
where = 0/ and
is Euler's constant ( x5.2).
10.43.6Zx
0e tIn(t)dt=xe x(I0(x)+I1(x))+n(e xI0(x) 1)
+ 2e xn 1X
k=1(n k)Ik(x),
n= 0;1;2;:::.
10.43.7Zx
0ettI(t)dt=exx+1
2+ 1(I(x)I+1(x)),
< > 1
2,
10.43.8Zx
0ett I(t)dt= exx +1
2 1(I(x)I 1(x))
2 +1
(2 1) (),6=1
2.
10.43.9Zx
0ettK(t)dt=exx+1
2+ 1(K(x)K+1(x))
2 (+ 1)
2+ 1,< > 1
2,
10.43.10Z1
xett K(t)dt=exx +1
2 1(K(x) +K 1(x)),
< >1
2.
10.43(iii) Fractional Integrals
The Bickley function Ki (x) is de ned by
10.43.11 Ki (x) =1
( )Z1
x(t x) 1K0(t)dt;
when< >0 andx >0, and by analytic continuation
elsewhere. Equivalently,
10.43.12 Ki (x) =Z1
0e xcosht
(cosht) dt,x>0.Properties
10.43.13 Ki (x) =Z1
xKi 1(t)dt;
10.43.14 Ki0(x) =K0(x);
10.43.15 Ki n(x) = ( 1)ndn
dxnK0(x),n= 1;2;3;:::.
10.43.16 Ki (0) =p 1
2
2 1
2 +1
2, 6= 0; 2; 4;:::.
10.43.17 Ki +1(x) +xKi (x)
+ (1 ) Ki 1(x) xKi 2(x) = 0:
For further properties of the Bickley function, in-
cluding asymptotic expansions and generalizations, see
Amos (1983, 1989) and Luke (1962, Chapter 8).
10.43(iv) Integrals over the Interval ( 0;1)
10.43.18Z1
0K(t)dt=1
2sec(1
2),j<j<1.
10.43.19Z1
0t 1K(t)dt= 2 2 1
2 1
2
1
2+1
2
,
j<j<<.
10.43.20Z1
0cos(at)K0(t)dt=
2(1 +a2)1
2,j=aj<1,
10.43.21Z1
0sin(at)K0(t)dt=arcsinha
(1 +a2)1
2,j=aj<1.
When<>j<j,
10.43.22Z1
0t 1e atK(t)dt=8
<
: 1
21
2 ( ) (+)(1 a2) 1
2+1
4P +1
2
1
2(a); 1<a< 1;
1
21
2 ( ) (+)(a2 1) 1
2+1
4P +1
2
1
2(a);<a0;a6= 1:
For the second equation there is a cut in the a-plane along the interval [0 ;1], and all quantities assume their principal
values (x4.2(i)). For the Ferrers function Pand the associated Legendre function P, seexx14.3(i) and 14.21(i).
10.43.23Z1
0t+1I(bt) exp( p2t2)dt=b
(2p2)+1expb2
4p2
, < > 1;<(p2)>0,
10.43.24Z1
0I(bt) exp
p2t2
dt=p
2pexpb2
8p2
I1
2b2
8p2
, < > 1,<(p2)>0,
10.43.25Z1
0K(bt) exp
p2t2
dt=p
4psec 1
2
expb2
8p2
K1
2b2
8p2
,j<j<1,<(p2)>0.
260 Bessel Functions
10.43.26Z1
0K(at)J(bt)
tdt=b 1
2 1
2+1
2+1
2
1
2 1
2 1
2+1
2
2+1a +1
F ++ 1
2; + 1
2;+ 1; b2
a2
,
<(+ 1 )>j<j;<a>j=bj.
For the hypergeometric function Fseex15.2(i).
10.43.27Z1
0t++1K(at)J(bt)dt=(2a)(2b) (++ 1)
(a2+b2)++1,<(+ 1)>j<j;<a>j=bj.
10.43.28Z1
0texp( p2t2)I(at)I(bt)dt=1
2p2expa2+b2
4p2
Iab
2p2
,< > 1;<(p2)>0,
10.43.29Z1
0texp( p2t2)I0(at)K0(at)dt=1
4p2expa2
2p2
K0a2
2p2
, <(p2)>0.
For in nite integrals of triple products of modi ed and unmodi ed Bessel functions, see Gervois and Navelet
(1984, 1985a,b, 1986a,b).
10.43(v) Kontorovich{Lebedev Transform
The Kontorovich{Lebedev transform of a function g(x)
is de ned as
10.43.30f(y) =2y
2sinh(y)Z1
0g(x)
xKiy(x)dx:
Then
10.43.31 g(x) =Z1
0f(y)Kiy(x)dy;
provided that either of the following sets of conditions
is satis ed:
(a) On the interval 0 < x <1,x 1g(x) is con-
tinuously di erentiable and each of xg(x) and
xd(x 1g(x))
dxis absolutely integrable.
(b)g(x) is piecewise continuous and of bounded varia-
tion on every compact interval in (0 ;1), and each
of the following integrals
10.43.32Z1
2
0g(x)
xln1
x
dx;Z1
1
2jg(x)j
x1
2dx;
converges.
For asymptotic expansions of the direct transform
(10.43.30) see Wong (1981), and for asymptotic ex-
pansions of the inverse transform (10.43.31) see Naylor
(1990, 1996).
For collections of the Kontorovich{Lebedev trans-
form, see Erd elyi et al. (1954b, Chapter 12), Prudnikov
et al. (1986b, pp. 404{412), and Oberhettinger (1972,
Chapter 5).
10.43(vi) Compendia
For collections of integrals of the functions I(z) and
K(z), including integrals with respect to the order, seeApelblat (1983,x12), Erd elyi et al. (1953b,xx7.7.1{7.7.7
and 7.14{7.14.2), Erd elyi et al. (1954a,b), Gradshteyn
and Ryzhik (2000, xx5.5, 6.5{6.7), Gr obner and Hofre-
iter (1950, pp. 197{203), Luke (1962), Magnus et al.
(1966,x3.8), Marichev (1983, pp. 191{216), Oberhet-
tinger (1972), Oberhettinger (1974, xx1.11 and 2.7),
Oberhettinger (1990, xx1.17{1.20 and 2.17{2.20), Ober-
hettinger and Badii (1973, xx1.15 and 2.13), Okui (1974,
1975), Prudnikov et al. (1986b,xx1.11{1.12, 2.15{2.16,
3.2.8{3.2.10, and 3.4.1), Prudnikov et al. (1992a,xx3.15,
3.16), Prudnikov et al. (1992b,xx3.15, 3.16), Watson
(1944, Chapter 13), and Wheelon (1968).
10.44 Sums
10.44(i) Multiplication Theorem
10.44.1Z(z) =1X
k=0(2 1)k(1
2z)k
k!Zk(z),
j2 1j<1.
IfZ=Iand the upper signs are taken, then the re-
striction on is unnecessary.
Examples
10.44.2
I(z) =1X
k=0zk
k!J+k(z); J(z) =1X
k=0( 1)kzk
k!I+k(z):
10.44(ii) Addition Theorems
Neumann's Addition Theorem
10.44.3
Z(uv) =1X
k= 1(1)kZ+k(u)Ik(v),jvj<juj.
The restrictionjvj<jujis unnecessary when Z=Iand
is an integer.
10.45 Functions of Imaginary Order 261
Graf's and Gegenbauer's Addition Theorems
For results analogous to (10.23.7) and (10.23.8) see Wat-
son (1944,xx11.3 and 11.41).
10.44(iii) Neumann-Type Expansions
10.44.4 1
2z=1X
k=0( 1)k(+ 2k) (+k)
k!I+2k(z),
6= 0; 1; 2;:::.
10.44.5K0(z) =
ln 1
2z
+
I0(z) + 21X
k=1I2k(z)
k;
10.44.6Kn(z) =n!(1
2z) n
2n 1X
k=0( 1)k(1
2z)kIk(z)
k!(n k)
+ ( 1)n 1
ln 1
2z
(n+ 1)
In(z)
+ ( 1)n1X
k=1(n+ 2k)In+2k(z)
k(n+k);
where
is Euler's constant and = 0/ (x5.2).
10.44(iv) Compendia
For collections of sums and series involving modi-
ed Bessel functions see Erd elyi et al. (1953b,x7.15),
Hansen (1975), and Prudnikov et al. (1986b, pp. 691{
700).
10.45 Functions of Imaginary Order
Withz=x, andreplaced by i, the modi ed Bessel's
equation (10.25.1) becomes
10.45.1x2d2w
dx2+xdw
dx+ (2 x2)w= 0:
For2Randx2(0;1) de ne
eI(x) =<(Ii(x));eK(x) =Ki(x): 10.45.2
Then
eI (x) =eI(x);eK (x) =eK(x); 10.45.3
andeI(x),eK(x) are real and linearly independent so-
lutions of (10.45.1):
10.45.4 WfeK(x);eI(x)g= 1=x:
Asx!+1
10.45.5eI(x) = (2x) 1
2ex
1 +O(x 1)
;
eK(x) = (=(2x))1
2e x
1 +O(x 1)
:
Asx!0+
10.45.6
eI(x) =sinh()
1
2
cos
ln 1
2x
+O
x2
;where
is as inx10.24. The corresponding result for
eK(x) is given by
10.45.7
eK(x) =
sinh()1
2
sin
ln 1
2x
+O
x2
;
when >0, and
10.45.8eK0(x) =K0(x) = ln(1
2x)
+O(x2lnx);
where
again denotes Euler's constant ( x5.2(ii)).
In consequence of (10.45.5){(10.45.7), eI(x) and
eK(x) comprise a numerically satisfactory pair of solu-
tions of (10.45.1) when xis large, and either eI(x) and
(1=) sinh()eK(x), oreI(x) andeK(x), comprise a
numerically satisfactory pair when xis small, depend-
ing whether 6= 0 or= 0.
For graphs of eI(x) andeK(x) seex10.26(iii).
For properties of eI(x) andeK(x), including uni-
form asymptotic expansions for large and zeros, see
Dunster (1990a). In this reference eI(x) is denoted by
(1=) sinh()Li(x). See also Gil et al. (2003a) and
Balogh (1967).
10.46 Generalized and Incomplete Bessel
Functions; Mittag-Leer Function
The function (; ;z) is de ned by
10.46.1 (; ;z) =1X
k=0zk
k! (k+ ),> 1.
From (10.25.2)
10.46.2 I(z) = 1
2z
1;+ 1;1
4z2
:
For asymptotic expansions of (; ;z) asz!1
in various sectors of the complex z-plane for xed real
values ofand xed real or complex values of , see
Wright (1935) when > 0, and Wright (1940b) when
1< < 0. For exponentially-improved asymp-
totic expansions in the same circumstances, together
with smooth interpretations of the corresponding Stokes
phenomenon (xx2.11(iii){2.11(v)) see Wong and Zhao
(1999a) when >0, and Wong and Zhao (1999b) when
1<< 0.
The Laplace transform of (; ;z) can be expressed
in terms of the Mittag-Leer function :
10.46.3 Ea;b(z) =1X
k=0zk
(ak+b),a>0:
See Paris (2002c). This reference includes
exponentially-improved asymptotic expansions for
Ea;b(z) whenjzj!1 , together with a smooth interpre-
tation of Stokes phenomena. See also Wong and Zhao
(2002a), and for further information on the Mittag-
Leer function see Erd elyi et al. (1955,x18.1) and
Paris and Kaminski (2001, x5.1.4).
262 Bessel Functions
For incomplete modi ed Bessel functions and Han-
kel functions, including applications, see Cicchetti and
Faraone (2004).
Spherical Bessel Functions
10.47 De nitions and Basic Properties
10.47(i) Di erential Equations
10.47.1z2d2w
dz2+ 2zdw
dz+
z2 n(n+ 1)
w= 0;
10.47.2z2d2w
dz2+ 2zdw
dz
z2+n(n+ 1)
w= 0:
Here, and throughout the remainder of xx10.47{10.60,
nis a nonnegative integer . (This is in contrast to
other treatments of spherical Bessel functions, including
Abramowitz and Stegun (1964, Chapter 10), in which
ncan be any integer. However, there is a gain in sym-
metry, without any loss of generality in applications, on
restrictingn0.)
Equations (10.47.1) and (10.47.2) each have a reg-
ular singularity at z= 0 with indices n, n 1, and
an irregular singularity at z=1of rank 1; compare
xx2.7(i){2.7(ii).
10.47(ii) Standard Solutions
Equation (10.47.1)
10.47.3
jn(z) =q
1
2=zJn+1
2(z) = ( 1)nq
1
2=zY n 1
2(z);
10.47.4
yn(z) =q
1
2=zYn+1
2(z) = ( 1)n+1q
1
2=zJ n 1
2(z);
10.47.5
h(1)
n(z)
=q
1
2=zH(1)
n+1
2(z) = ( 1)n+1iq
1
2=zH(1)
n 1
2(z);
10.47.6
h(2)
n(z) =q
1
2=zH(2)
n+1
2(z) = ( 1)niq
1
2=zH(2)
n 1
2(z):
jn(z) and yn(z) are the spherical Bessel functions of the
rst and second kinds , respectively; h(1)
n(z) and h(2)
n(z)
are the spherical Bessel functions of the third kind .
Equation (10.47.2)
10.47.7 i(1)
n(z) =q
1
2=zIn+1
2(z)
10.47.8 i(2)
n(z) =q
1
2=zI n 1
2(z)
10.47.9 kn(z) =q
1
2=zKn+1
2(z) =q
1
2=zK n 1
2(z):i(1)
n(z),i(2)
n(z), and kn(z) are the modi ed spherical
Bessel functions .
Many properties of jn(z),yn(z),h(1)
n(z),h(2)
n(z),
i(1)
n(z),i(2)
n(z), and kn(z) follow straightforwardly
from the above de nitions and results given in
preceding sections of this chapter. For exam-
ple,z njn(z),zn+1yn(z),zn+1h(1)
n(z),zn+1h(2)
n(z),
z ni(1)
n(z),zn+1i(2)
n(z), andzn+1kn(z) are all entire
functions of z.
10.47(iii) Numerically Satisfactory Pairs of
Solutions
For (10.47.1) numerically satisfactory pairs of solutions
are given by Table 10.2.1 with the symbols J,Y,H, and
replaced by j,y,h, andn, respectively.
For (10.47.2) numerically satisfactory pairs of solu-
tions are i(1)
n(z) and kn(z) in the right half of the z-plane,
and i(1)
n(z) and kn( z) in the left half of the z-plane.
10.47(iv) Interrelations
10.47.10
h(1)
n(z) =jn(z) +iyn(z);h(2)
n(z) =jn(z) iyn(z):
10.47.11 kn(z) = ( 1)n+11
2
i(1)
n(z) i(2)
n(z)
:
10.47.12 i(1)
n(z) =i njn(iz);i(2)
n(z) =i n 1yn(iz):
10.47.13 kn(z) = 1
2inh(1)
n(iz) = 1
2i nh(2)
n( iz):
10.47(v) Re
ection Formulas
jn( z) = ( 1)njn(z); yn( z) = ( 1)n+1yn(z);10.47.14
h(1)
n( z) = ( 1)nh(2)
n(z);h(2)
n( z) = ( 1)nh(1)
n(z):10.47.15
i(1)
n( z) = ( 1)ni(1)
n(z);i(2)
n( z) = ( 1)n+1i(2)
n(z);10.47.16
10.47.17 kn( z) = 1
2
i(1)
n(z) +i(2)
n(z)
:
10.48 Graphs
For unmodi ed spherical Bessel functions see Fig-
ures 10.48.1{10.48.4. For modi ed spherical Bessel
functions see Figures 10.48.5{10.48.7.
10.48 Graphs 263
Figure 10.48.1 :jn(x);n= 0(1)4;0x12.
Figure 10.48.2 :yn(x);n= 0(1)4;0<x12.
Figure 10.48.3 :j5(x),y5(x),p
j2
5(x) +y2
5(x), 0x12.
Figure 10.48.4 :j0
5(x),y0
5(x),q
j0
52(x) +y0
52(x), 0x
12.
Figure 10.48.5 :i(1)
0(x),i(2)
0(x),k0(x), 0x4.
Figure 10.48.6 :i(1)
1(x);i(2)
1(x);k1(x);0x4.
Figure 10.48.7 :i(1)
5(x),i(2)
5(x),k5(x), 0x8.
264 Bessel Functions
10.49 Explicit Formulas
10.49(i) Unmodi ed Functions
De neak() as in (10.17.1). Then
10.49.1
ak(n+1
2) =8
<
:(n+k)!
2kk!(n k)!; k = 0;1;:::;n;
0; k =n+ 1;n+ 2;::: :
10.49.2
jn(z) = sin
z 1
2nbn=2cX
k=0( 1)ka2k(n+1
2)
z2k+1
+ cos
z 1
2nb(n 1)=2cX
k=0( 1)ka2k+1(n+1
2)
z2k+2:
10.49.3j0(z) =sinz
z;j1(z) =sinz
z2 cosz
z;
j2(z) =
1
z+3
z3
sinz 3
z2cosz:
10.49.4
yn(z) = cos
z 1
2nbn=2cX
k=0( 1)ka2k(n+1
2)
z2k+1
+ sin
z 1
2nb(n 1)=2cX
k=0( 1)ka2k+1(n+1
2)
z2k+2:
10.49.5y0(z) = cosz
z;y1(z) = cosz
z2 sinz
z;
y2(z) =1
z 3
z3
cosz 3
z2sinz:
10.49.6 h(1)
n(z) =eiznX
k=0ik n 1ak(n+1
2)
zk+1;
10.49.7 h(2)
n(z) =e iznX
k=0( i)k n 1ak(n+1
2)
zk+1:
10.49(ii) Modi ed Functions
Again, with ak(n+1
2) as in (10.49.1),
10.49.8i(1)
n(z) =1
2eznX
k=0( 1)kak(n+1
2)
zk+1
+ ( 1)n+11
2e znX
k=0ak(n+1
2)
zk+1:
10.49.9i(1)
0(z) =sinhz
z;i(1)
1(z) = sinhz
z2+coshz
z;
i(1)
2(z) =1
z+3
z3
sinhz 3
z2coshz:10.49.10i(2)
n(z) =1
2eznX
k=0( 1)kak(n+1
2)
zk+1
+ ( 1)n1
2e znX
k=0ak(n+1
2)
zk+1:
10.49.11i(2)
0(z) =coshz
z;i(2)
1(z) = coshz
z2+sinhz
z;
i(2)
2(z) =1
z+3
z3
coshz 3
z2sinhz:
10.49.12 kn(z) =1
2e znX
k=0ak(n+1
2)
zk+1:
10.49.13k0(z) =1
2e z
z;k1(z) =1
2e z1
z+1
z2
;
k2(z) =1
2e z1
z+3
z2+3
z3
:
Pn
k=0ak(n+1
2)zn kis sometimes called the Bessel
polynomial of degree n. For a survey of properties of
these polynomials and their generalizations see Gross-
wald (1978). See also x18.34, de Bruin et al. (1981a,b),
and Dunster (2001c).
10.49(iii) Rayleigh's Formulas
10.49.14jn(z) =zn
1
zd
dznsinz
z;
yn(z) = zn
1
zd
dzncosz
z:
10.49.15i(1)
n(z) =zn1
zd
dznsinhz
z;
i(2)
n(z) =zn1
zd
dzncoshz
z:
10.49.16 kn(z) = ( 1)n1
2zn1
zd
dzne z
z:
10.49(iv) Sums or Di erences of Squares
Denote
10.49.17sk(n+1
2) =(2k)!(n+k)!
22k(k!)2(n k)!,k= 0;1;:::;n .
Then
10.49.18 j2
n(z) +y2
n(z) =nX
k=0sk(n+1
2)
z2k+2:
10.49.19
j2
0(z) +y2
0(z) =z 2;j2
1(z) +y2
1(z) =z 2+z 4;
j2
2(z) +y2
2(z) =z 2+ 3z 4+ 9z 6:
10.49.20
i(1)
n(z)2
i(2)
n(z)2
= ( 1)n+1nX
k=0( 1)ksk(n+1
2)
z2k+2:
10.50 Wronskians and Cross-Products 265
10.49.21
i(1)
0(z)2
i(2)
0(z)2
= z 2;
i(1)
1(z)2
i(2)
1(z)2
=z 2 z 4;
i(1)
2(z)2
i(2)
2(z)2
= z 2+ 3z 4 9z 6:
10.50 Wronskians and Cross-Products
10.50.1Wfjn(z);yn(z)g=z 2;
Wn
h(1)
n(z);h(2)
n(z)o
= 2iz 2:
10.50.2Wn
i(1)
n(z);i(2)
n(z)o
= ( 1)n+1z 2;
Wn
i(1)
n(z);kn(z)o
=Wn
i(2)
n(z);kn(z)o
= 1
2z 2:
10.50.3jn+1(z)yn(z) jn(z)yn+1(z) =z 2;
jn+2(z)yn(z) jn(z)yn+2(z) = (2n+ 3)z 3:
10.50.4j0(z)jn(z) +y0(z)yn(z)
= cos 1
2nbn=2cX
k=0( 1)ka2k(n+1
2)
z2k+2
+ sin 1
2nb(n 1)=2cX
k=0( 1)ka2k+1(n+1
2)
z2k+3;
whereak(n+1
2) is given by (10.49.1).
Results corresponding to (10.50.3) and (10.50.4) for
i(1)
n(z) and i(2)
n(z) are obtainable via (10.47.12).
10.51 Recurrence Relations and Derivatives
10.51(i) Unmodi ed Functions
Letfn(z) denote any of jn(z),yn(z),h(1)
n(z), or h(2)
n(z).
Then
10.51.1fn 1(z) +fn+1(z) = ((2n+ 1)=z)fn(z);
nfn 1(z) (n+ 1)fn+1(z) = (2n+ 1)f0
n(z),
n= 1;2;:::;
10.51.2f0
n(z) =fn 1(z) ((n+1)=z)fn(z),n= 1;2;:::;
f0
n(z) = fn+1(z) + (n=z)fn(z),n= 0;1;::::
10.51.31
zd
dzm
(zn+1fn(z)) =zn m+1fn m(z),
m= 0;1;:::;n;1
zd
dzm
(z nfn(z)) = ( 1)mz n mfn+m(z),
m= 0;1;::::10.51(ii) Modi ed Functions
Letgn(z) denote i(1)
n(z),i(2)
n(z), or ( 1)nkn(z). Then
10.51.4gn 1(z) gn+1(z) = ((2n+ 1)=z)gn(z)
ngn 1(z) + (n+ 1)gn+1(z) = (2n+ 1)g0
n(z),
n= 1;2;:::;
10.51.5g0
n(z) =gn 1(z) ((n+1)=z)gn(z),n= 1;2;:::;
g0
n(z) =gn+1(z) + (n=z)gn(z),n= 0;1;::: :
10.51.61
zd
dzm
(zn+1gn(z)) =zn m+1gn m(z),
m= 0;1;:::;n;1
zd
dzm
(z ngn(z)) =z n mgn+m(z),
m= 0;1;::: :
10.52 Limiting Forms
10.52(i)z!0
10.52.1 jn(z);i(1)
n(z)zn=(2n+ 1)!!;
10.52.2
yn(z);ih(1)
n(z); ih(2)
n(z);( 1)ni(2)
n(z);(2=)kn(z)
(2n 1)!!=zn+1:
10.52(ii)z!1
10.52.3jn(z) =z 1sin(z 1
2n) +ej=zjO(z 2);
yn(z) = z 1cos(z 1
2n) +ej=zjO(z 2);
10.52.4 h(1)
n(z)i n 1z 1eiz;h(2)
n(z)in+1z 1e iz;
10.52.5
i(1)
n(z)i(2)
n(z)1
2z 1ez,jphzj1
2 (<1
2);
10.52.6 kn(z)1
2z 1e z:
10.53 Power Series
10.53.1 jn(z) =zn1X
k=0( 1
2z2)k
k!(2n+ 2k+ 1)!!;
10.53.2yn(z) = 1
zn+1nX
k=0(2n 2k 1)!!(1
2z2)k
k!
+( 1)n+1
zn+11X
k=n+1( 1
2z2)k
k!(2k 2n 1)!!:
266 Bessel Functions
10.53.3 i(1)
n(z) =zn1X
k=0(1
2z2)k
k!(2n+ 2k+ 1)!!;
10.53.4i(2)
n(z) =( 1)n
zn+1nX
k=0(2n 2k 1)!!( 1
2z2)k
k!
+1
zn+11X
k=n+1(1
2z2)k
k!(2k 2n 1)!!:
Forh(1)
n(z) and h(2)
n(z) combine (10.47.10), (10.53.1),
and (10.53.2). For kn(z) combine (10.47.11), (10.53.3),
and (10.53.4).
10.54 Integral Representations
10.54.1 jn(z) =zn
2n+1n!Z
0cos(zcos)(sin)2n+1d:
10.54.2 jn(z) =( i)n
2Z
0eizcosPn(cos) sind:
10.54.3 kn(z) =
2Z1
1e ztPn(t)dt,jphzj<1
2:
10.54.4jn(z) =( i)n+1
2Z( 1+;1+)
i1eiztQn(t)dt,
jphzj<1
2:
10.54.5h(1)
n(z) =( i)n+1
Z(1+)
i1eiztQn(t)dt;
h(2)
n(z) =( i)n+1
Z( 1+)
i1eiztQn(t)dt,
jphzj<1
2:
For the Legendre polynomial Pnand the associated Leg-
endre function Qnseexx18.3 and 14.21(i), with = 0
and=n.
Additional integral representations can be obtained
by combining the de nitions (10.47.3){(10.47.9) with
the results given in x10.9 andx10.32.
10.55 Continued Fractions
For continued fractions for jn+1(z)=jn(z) and
i(1)
n+1(z)=i(1)
n(z) see Cuyt et al. (2008, pp. 350, 353,
362, 363, 367{369).
10.56 Generating Functions
When 2jtj<jzj,
10.56.1cosp
z2 2zt
z=cosz
z+1X
n=1tn
n!jn 1(z);
10.56.2sinp
z2 2zt
z=sinz
z+1X
n=1tn
n!yn 1(z):10.56.3coshp
z2+ 2izt
z=coshz
z+1X
n=1(it)n
n!i(1)
n 1(z);
10.56.4sinhp
z2+ 2izt
z=sinhz
z+1X
n=1(it)n
n!i(2)
n 1(z);
10.56.5
exp
p
z2+ 2izt
z=e z
z+2
1X
n=1( it)n
n!kn 1(z):
10.57 Uniform Asymptotic Expansions for
Large Order
Asymptotic expansions for jn
(n+1
2)z
,yn
(n+1
2)z
,
h(1)
n
(n+1
2)z
,h(2)
n
(n+1
2)z
,i(1)
n
(n+1
2)z
, and
kn
(n+1
2)z
asn!1 that are uniform with respect
tozcan be obtained from the results given in xx10.20
and 10.41 by use of the de nitions (10.47.3){(10.47.7)
and (10.47.9). Subsequently, for i(2)
n
(n+1
2)z
the con-
nection formula (10.47.11) is available.
For the corresponding expansion for j0
n
(n+1
2)z
use
10.57.1
j0
n
(n+1
2)z
=1
2
((2n+ 1)z)1
2J0
n+1
2
(n+1
2)z
1
2
((2n+ 1)z)3
2Jn+1
2
(n+1
2)z
:
Similarly for the expansions of the derivatives of the
other six functions.
10.58 Zeros
Forn0 themth positive zeros of jn(x),j0
n(x),yn(x),
and y0
n(x) are denoted by an;m,a0
n;m,bn;m, andb0
n;m,
respectively, except that for n= 0 we count x= 0 as
the rst zero of j0
0(x).
With the notation of x10.21(i),
10.58.1an;m=jn+1
2;m; bn;m=yn+1
2;m;
10.58.2j0
n(an;m) =s
2jn+1
2;mJ0
n+1
2
jn+1
2;m
;
y0
n(bn;m) =s
2yn+1
2;mY0
n+1
2
yn+1
2;m
:
Hence properties of an;m andbn;m are derivable
straightforwardly from results given in xx10.21(i){
10.21(iii), 10.21(vi){10.21(viii), and 10.21(x). However,
there are no simple relations that connect the zeros
of the derivatives. For some properties of a0
n;mand
b0
n;m, including asymptotic expansions, see Olver (1960,
pp. xix{xxi).
See also Davies (1973), de Bruin et al. (1981a,b),
and Gottlieb (1985).
10.59 Integrals 267
10.59 Integrals
10.59.1Z1
1eibtjn(t)dt=8
><
>:inPn(b); 1<b< 1;
1
2(i)n; b =1;
0;b>1;
wherePnis the Legendre polynomial ( x18.3).
For an integral representation of the Dirac delta in
terms of a product of spherical Bessel functions of the
rst kind seex1.17(ii), and for a generalization see Max-
imon (1991).
Additional integrals can be obtained by combining
the de nitions (10.47.3){(10.47.9) with the results given
inx10.22 andx10.43. For integrals of products see also
Mehrem et al. (1991).
10.60 Sums
10.60(i) Addition Theorems
De neu,v,w, and as inx10.23(ii). Then with Pn
again denoting the Legendre polynomial of degree n,
10.60.1cosw
w= 1X
n=0(2n+ 1) jn(v)yn(u)Pn(cos ),
jvei j<juj.
10.60.2sinw
w=1X
n=0(2n+ 1) jn(v)jn(u)Pn(cos ):
10.60.3e w
w=2
1X
n=0(2n+ 1) i(1)
n(v)kn(u)Pn(cos ),
jvei j<juj.
10.60(ii) Duplication Formulas
10.60.4
jn(2z) = n!zn+1nX
k=02n 2k+ 1
k!(2n k+ 1)!jn k(z)yn k(z);
10.60.5
yn(2z)
=n!zn+1nX
k=0n k+1
2
k!(2n k+ 1)!
j2
n k(z) y2
n k(z)
;
10.60.6
kn(2z) =1
n!zn+1nX
k=0( 1)k2n 2k+ 1
k!(2n k+ 1)!k2
n k(z):10.60(iii) Other Series
10.60.7eizcos =1X
n=0(2n+ 1)injn(z)Pn(cos );
10.60.8ezcos =1X
n=0(2n+ 1) i(1)
n(z)Pn(cos );
10.60.9e zcos =1X
n=0( 1)n(2n+ 1) i(1)
n(z)Pn(cos ):
10.60.10
J0(zsin ) =1X
n=0(4n+ 1)(2n)!
22n(n!)2j2n(z)P2n(cos ):
10.60.111X
n=0j2
n(z) =Si(2z)
2z:
For Si seex6.2(ii).
10.60.121X
n=0(2n+ 1) j2
n(z) = 1;
10.60.131X
n=0( 1)n(2n+ 1) j2
n(z) =sin(2z)
2z;
10.60.141X
n=0(2n+ 1)( j0
n(z))2=1
3:
For further sums of series of spherical Bessel
functions, or modi ed spherical Bessel functions, see
x6.10(ii), Luke (1969b, pp. 55{58), Vavreck and Thomp-
son (1984), Harris (2000), and Rottbrand (2000).
10.60(iv) Compendia
For collections of sums of series relevant to spherical
Bessel functions or Bessel functions of half odd integer
order see Erd elyi et al. (1953b, pp. 43{45 and 98{105),
Gradshteyn and Ryzhik (2000, xx8.51, 8.53), Hansen
(1975), Magnus et al. (1966, pp. 106{108 and 123{138),
and Prudnikov et al. (1986b, pp. 635{637 and 651{700).
See also Watson (1944, Chapters 11 and 16).
Kelvin Functions
10.61 De nitions and Basic Properties
10.61(i) De nitions
Throughoutxx10.61{x10.71 it is assumed that x0,
2R, andnis a nonnegative integer .
10.61.1
berx+ibeix=J
xe3i=4
=eiJ
xe i=4
=ei=2I
xei=4
=e3i=2I
xe 3i=4
;
268 Bessel Functions
10.61.2kerx+ikeix=e i=2K
xei=4
=1
2iH(1)
xe3i=4
= 1
2ie iH(2)
xe i=4
:
When= 0 suces on ber, bei, ker, and kei are usually
suppressed.
Most properties of ber x, beix, kerx, and keix
follow straightforwardly from the above de nitions and
results given in preceding sections of this chapter.
10.61(ii) Di erential Equations
10.61.3x2d2w
dx2+xdw
dx (ix2+2)w= 0,
w= berx+ibeix;ber x+ibei x
kerx+ikeix;ker x+ikei x:
10.61.4x4d4w
dx4+ 2x3d3w
dx3 (1 + 22)
x2d2w
dx2 xdw
dx
+ (4 42+x4)w= 0,
w= berx;beix;kerx;keix:
10.61(iii) Re
ection Formulas for Arguments
In general, Kelvin functions have a branch point at
x= 0 and functions with arguments xeiare com-
plex. The branch point is absent, however, in the case
of berand beiwhenis an integer. In particular,
10.61.5
bern( x) = ( 1)nbernx;bein( x) = ( 1)nbeinx:
10.61(iv) Re
ection Formulas for Orders
10.61.6ber x= cos() berx+ sin() beix
+ (2=) sin() kerx;
bei x= sin() berx+ cos() beix
+ (2=) sin() keix:10.61.7ker x= cos() kerx sin() keix;
kei x= sin() kerx+ cos() keix:
10.61.8ber nx= ( 1)nbernx;bei nx= ( 1)nbeinx;
ker nx= ( 1)nkernx;kei nx= ( 1)nkeinx:
10.61(v) Orders1
2
10.61.9ber 1
2
xp
2
=2 3
4px
excos
x+
8
e xcos
x
8
;
bei 1
2
xp
2
=2 3
4px
exsin
x+
8
+e xsin
x
8
:
10.61.10ber 1
2
xp
2
=2 3
4px
exsin
x+
8
e xsin
x
8
;
bei 1
2
xp
2
= 2 3
4px
excos
x+
8
+e xcos
x
8
:
10.61.11ker 1
2
xp
2
= kei 1
2
xp
2
= 2 3
4r
xe xsin
x
8
;
10.61.12kei1
2
xp
2
= ker 1
2
xp
2
= 2 3
4r
xe xcos
x
8
:
10.62 Graphs
See Figures 10.62.1{10.62.4. For the modulus functions M(x) andN(x) seex10.68(i) with = 0.
Figure 10.62.1 : berx;beix;ber0x;bei0x;0x8.
Figure 10.62.2 : kerx;keix;ker0x;kei0x;0x8.
10.63 Recurrence Relations and Derivatives 269
Figure 10.62.3 :e x=p
2berx,e x=p
2beix,
e x=p
2M(x), 0x8.
Figure 10.62.4 :ex=p
2kerx,ex=p
2keix,ex=p
2N(x), 0
x8.
10.63 Recurrence Relations and Derivatives
10.63(i) berx,beix,kerx,keix
Letf(x),g(x) denote any one of the ordered pairs:
10.63.1berx;beix; beix; berx;
kerx;keix; keix; kerx:
Then
10.63.2
f 1(x) +f+1(x) = (p
2=x) (f(x) g(x));
f+1(x) +g+1(x) f 1(x) g 1(x) = 2p
2f0
(x);
f0
(x) = (1=p
2) (f 1(x) +g 1(x)) (=x)f(x);
f0
(x) = (1=p
2) (f+1(x) +g+1(x)) + (=x)f(x):
10.63.3p
2 ber0x= ber 1x+ bei 1x;p
2 bei0x= ber1x+ bei 1x:
10.63.4p
2 ker0x= ker 1x+ kei 1x;p
2 kei0x= ker1x+ kei 1x:
10.63(ii) Cross-Products
Let
10.63.5
p= ber2
x+ bei2
x; q= berxbei0
x ber0
xbeix;
r= berxber0
x+ beixbei0
x;
s=
ber0
x2+
bei0
x2:
Then
10.63.6p+1=p 1 (4=x)r;
q+1= (=x)p+r= q 1+ 2r;
r+1= ((+ 1)=x)p+1+q;
s=1
2p+1+1
2p 1 (2=x2)p;and
10.63.7 ps=r2
+q2
:
Equations (10.63.6) and (10.63.7) also hold when the
symbols ber and bei in (10.63.5) are replaced through-
out by ker and kei, respectively.
10.64 Integral Representations
Schl a
i-Type Integrals
10.64.1
bern
xp
2
=( 1)n
Z
0cos(xsint nt) cosh(xsint)dt;
10.64.2
bein
xp
2
=( 1)n
Z
0sin(xsint nt) sinh(xsint)dt:
See Apelblat (1991) for these results, and also for simi-
lar representations for ber
xp
2
, bei
xp
2
, and their
-derivatives.
10.65 Power Series
10.65(i) berxandbeix
10.65.1berx= (1
2x)1X
k=0cos 3
4+1
2k
k! (+k+ 1)(1
4x2)k;
beix= (1
2x)1X
k=0sin 3
4+1
2k
k! (+k+ 1)(1
4x2)k:
10.65.2berx= 1 (1
4x2)2
(2!)2+(1
4x2)4
(4!)2 ;
beix=1
4x2 (1
4x2)3
(3!)2+(1
4x2)5
(5!)2 :
270 Bessel Functions
10.65(ii) kerxandkeix
Whenis not an integer combine (10.65.1) with (10.61.6). Also, with (x) = 0(x)= (x),
10.65.3kernx=1
2(1
2x) nn 1X
k=0(n k 1)!
k!cos 3
4n+1
2k
(1
4x2)k ln 1
2x
bernx
+1
4beinx+1
2(1
2x)n1X
k=0 (k+ 1) + (n+k+ 1)
k!(n+k)!cos 3
4n+1
2k
(1
4x2)k;
10.65.4keinx= 1
2(1
2x) nn 1X
k=0(n k 1)!
k!sin 3
4n+1
2k
(1
4x2)k ln 1
2x
beinx
1
4bernx+1
2(1
2x)n1X
k=0 (k+ 1) + (n+k+ 1)
k!(n+k)!sin 3
4n+1
2k
(1
4x2)k:
10.65.5kerx= ln 1
2x
berx+1
4beix+1X
k=0( 1)k (2k+ 1)
((2k)!)2(1
4x2)2k;
keix= ln 1
2x
beix 1
4berx+1X
k=0( 1)k (2k+ 2)
((2k+ 1)!)2(1
4x2)2k+1:
10.65(iii) Cross-Products and Sums of Squares
10.65.6 ber2
x+ bei2
x= (1
2x)21X
k=01
(+k+ 1) (+ 2k+ 1)(1
4x2)2k
k!;
10.65.7 berxbei0
x ber0
xbeix= (1
2x)2+11X
k=01
(+k+ 1) (+ 2k+ 2)(1
4x2)2k
k!;
10.65.8 berxber0
x+ beixbei0
x=1
2(1
2x)2 11X
k=01
(+k+ 1) (+ 2k)(1
4x2)2k
k!;
10.65.9
ber0
x2+
bei0
x2= (1
2x)2 21X
k=02k2+ 2k+1
42
(+k+ 1) (+ 2k+ 1)(1
4x2)2k
k!:
10.65(iv) Compendia
For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).
10.66 Expansions in Series of Bessel Functions
10.66.1 berx+ibeix=1X
k=0e(3+k)i=4xkJ+k(x)
2k=2k!=1X
k=0e(3+3k)i=4xkI+k(x)
2k=2k!:
10.66.2 bern
xp
2
=1X
k= 1( 1)n+kJn+2k(x)I2k(x);bein
xp
2
=1X
k= 1( 1)n+kJn+2k+1(x)I2k+1(x):
10.67 Asymptotic Expansions for Large Argument 271
10.67 Asymptotic Expansions for Large Argument
10.67(i) berx;beix;kerx;keix, and Derivatives
De neak() andbk() as inxx10.17(i) and 10.17(ii). Then as x!1 with xed,
10.67.1 kerxe x=p
2
2x1
21X
k=0ak()
xkcosxp
2+
2+k
4+1
8
;
10.67.2 keix e x=p
2
2x1
21X
k=0ak()
xksinxp
2+
2+k
4+1
8
:
10.67.3 berxex=p
2
(2x)1
21X
k=0ak()
xkcosxp
2+
2+3k
4 1
8
1
(sin(2) kerx+ cos(2) keix);
10.67.4 beixex=p
2
(2x)1
21X
k=0ak()
xksinxp
2+
2+3k
4 1
8
+1
(cos(2) kerx sin(2) keix):
10.67.5 ker0
x e x=p
2
2x1
21X
k=0bk()
xkcosxp
2+
2+k
4 1
8
;
10.67.6 kei0
xe x=p
2
2x1
21X
k=0bk()
xksinxp
2+
2+k
4 1
8
:
10.67.7 ber0
xex=p
2
(2x)1
21X
k=0bk()
xkcosxp
2+
2+3k
4+1
8
1
(sin(2) ker0
x+ cos(2) kei0
x);
10.67.8 bei0
xex=p
2
(2x)1
21X
k=0bk()
xksinxp
2+
2+3k
4+1
8
+1
(cos(2) ker0
x sin(2) kei0
x):
The contributions of the terms in ker x, keix, ker0
x, and kei0
xon the right-hand sides of (10.67.3), (10.67.4),
(10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the
sense of Poincar e asymptotic expansions ( x2.1(iii)). However, their inclusion improves numerical accuracy.
10.67(ii) Cross-Products and Sums of Squares in the Case = 0
Asx!1
10.67.9 ber2x+ bei2xexp
2
2x
1 +1
4p
21
x+1
641
x2 33
256p
21
x3 1797
81921
x4+
;
10.67.10 berxbei0x ber0xbeixexp
2
2x1p
2+1
81
x+9
64p
21
x2+39
5121
x3+75
8192p
21
x4+
;
10.67.11 berxber0x+ beixbei0xexp
2
2x1p
2 3
81
x 15
64p
21
x2 45
5121
x3+315
8192p
21
x4+
;
10.67.12
ber0x2+
bei0x2exp
2
2x
1 3
4p
21
x+9
641
x2+75
256p
21
x3+2475
81921
x4+
:
10.67.13 ker2x+ kei2x
2xe xp
2
1 1
4p
21
x+1
641
x2+33
256p
21
x3 1797
81921
x4+
;
10.67.14 kerxkei0x ker0xkeix
2xe xp
21p
2 1
81
x+9
64p
21
x2 39
5121
x3+75
8192p
21
x4+
;
10.67.15 kerxker0x+ keixkei0x
2xe xp
21p
2+3
81
x 15
64p
21
x2+45
5121
x3+315
8192p
21
x4+
;
10.67.16
ker0x2+
kei0x2
2xe xp
2
1 +3
4p
21
x+9
641
x2 75
256p
21
x3+2475
81921
x4+
:
272 Bessel Functions
10.68 Modulus and Phase Functions
10.68(i) De nitions
10.68.1 M(x)ei(x)= berx+ibeix;
10.68.2 N(x)ei (x)= kerx+ikeix;
whereM(x) (>0),N(x) (>0),(x), and (x) are continuous real functions of xand, with the branches of
(x) and (x) chosen to satisfy (10.68.18) and (10.68.21) as x!1 . (See alsox10.68(iv).)
10.68(ii) Basic Properties
10.68.3 berx=M(x) cos(x);beix=M(x) sin(x);
10.68.4 kerx=N(x) cos (x);keix=N(x) sin (x):
10.68.5 M(x) = (ber2
x+ bei2
x)1/2; N(x) = (ker2
x+ kei2
x)1/2;
10.68.6 (x) = Arctan(bei x=berx); (x) = Arctan(kei x=kerx):
10.68.7 M n(x) =Mn(x); n(x) =n(x) n:
With arguments ( x) suppressed,
10.68.8ber0
x=1
2M+1cos
+1 1
4
1
2M 1cos
1 1
4
= (=x)Mcos+M+1cos
+1 1
4
= (=x)Mcos M 1cos
1 1
4
;
10.68.9bei0
x=1
2M+1sin
+1 1
4
1
2M 1sin
1 1
4
= (=x)Msin+M+1sin
+1 1
4
= (=x)Msin M 1sin
1 1
4
:
10.68.10 ber0x=M1cos
1 1
4
;bei0x=M1sin
1 1
4
:
10.68.11 M0
= (=x)M+M+1cos
+1 1
4
= (=x)M M 1cos
1 1
4
;
10.68.12 0
= (M+1=M) sin
+1 1
4
= (M 1=M) sin
1 1
4
:
10.68.13 M0
0=M1cos
1 0 1
4
; 0
0= (M1=M0) sin
1 0 1
4
:
10.68.14 d(xM2
0
)
dx=xM2
; x2M00
+xM0
2M=x2M0
2:
Equations (10.68.8){(10.68.14) also hold with the symbols ber, bei, M, andreplaced throughout by ker, kei,
N, and , respectively. In place of (10.68.7),
10.68.15 N (x) =N(x); (x) = (x) +:
10.68(iii) Asymptotic Expansions for Large Argument
Whenis xed,= 42, andx!1
10.68.16 M(x) =ex=p
2
(2x)1
2
1 1
8p
21
x+( 1)2
2561
x2 ( 1)(2+ 14 399)
6144p
21
x3+O1
x4
;
10.68.17 lnM(x) =xp
2 1
2ln(2x) 1
8p
21
x ( 1)( 25)
384p
21
x3 ( 1)( 13)
1281
x4+O1
x5
;
10.68.18 (x) =xp
2+1
2 1
8
+ 1
8p
21
x+ 1
161
x2 ( 1)( 25)
384p
21
x3+O1
x5
:
10.68.19 N(x) =e x=p
2
2x1
2
1 + 1
8p
21
x+( 1)2
2561
x2+( 1)(2+ 14 399)
6144p
21
x3+O1
x4
;
10.68.20 lnN(x) = xp
2+1
2ln
2x
+ 1
8p
21
x+( 1)( 25)
384p
21
x3 ( 1)( 13)
1281
x4+O1
x5
;
10.68.21 (x) = xp
2 1
2+1
8
1
8p
21
x+ 1
161
x2+( 1)( 25)
384p
21
x3+O1
x5
:
10.69 Uniform Asymptotic Expansions for Large Order 273
10.68(iv) Further Properties
Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi{xv). However,
care needs to be exercised with the branches of the phases. Thus this reference gives 1(0) =5
4(Eq. (6.10)), and
limx!1( 1(x) + (x=p
2)) = 5
8(Eqs. (10.20) and (Eqs. (10.26b)). However, numerical tabulations show that if
the second of these equations applies and 1(x) is continuous, then 1(0) = 3
4; compare Abramowitz and Stegun
(1964, p. 433).
10.69 Uniform Asymptotic Expansions for Large Order
LetUk(p) andVk(p) be the polynomials de ned in x10.41(ii), and
10.69.1 = (1 +ix2)1/2:
Then as!+1,
10.69.2 ber(x) +ibei(x)e
(2)1/2xe3i=4
1 +1X
k=0Uk( 1)
k;
10.69.3 ker(x) +ikei(x)e
21/2xe3i=4
1 + 1X
k=0( 1)kUk( 1)
k;
10.69.4 ber0
(x) +ibei0
(x)e
x
21/2xe3i=4
1 +1X
k=0Vk( 1)
k;
10.69.5 ker0
(x) +ikei0
(x) e
x
21/2xe3i=4
1 + 1X
k=0( 1)kVk( 1)
k;
uniformly for x2(0;1). All fractional powers take their principal values.
All four expansions also enjoy the same kind of double asymptotic property described in x10.41(iv).
Accuracy in (10.69.2) and (10.69.4) can be increased by including exponentially-small contributions as in (10.67.3),
(10.67.4), (10.67.7), and (10.67.8) with xreplaced by x.
10.70 Zeros
Asymptotic approximations for large zeros are as follows. Let = 42andf(t) denote the formal series
10.70.1 1
16t+ 1
32t2+( 1)(5+ 19)
1536t3+3( 1)2
512t4+:
Ifmis a large positive integer, then
10.70.2zeros of ber xp
2(t f(t)), t= (m 1
2 3
8);
zeros of bei xp
2(t f(t)), t= (m 1
2+1
8);
zeros of ker xp
2(t+f( t)), t= (m 1
2 5
8);
zeros of kei xp
2(t+f( t)), t= (m 1
2 1
8):
In the case = 0, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of
(10.70.2) corresponds to the mth zero of the function on the left-hand side. For the next six terms in the series
(10.70.1) see MacLeod (2002a).
274 Bessel Functions
10.71 Integrals
10.71(i) Inde nite Integrals
In the following equations f;gis any one of the four ordered pairs given in (10.63.1), and bf;bgis either the same
ordered pair or any other ordered pair in (10.63.1).
10.71.1Z
x1+fdx= x1+
p
2(f+1 g+1) = x1+
xg g0
;
10.71.2Z
x1 fdx=x1
p
2(f 1 g 1) =x1
xg+g0
:
10.71.3Z
x(fbg gbf)dx=x
2p
2
bf(f+1+g+1) bg(f+1 g+1) f(bf+1+bg+1) +g(bf+1 bg+1)
=1
2x(f0
bf fbf0
+g0
bg gbg0
);
10.71.4Z
x(fbg+gbf)dx=1
4x2(2fbg f 1bg+1 f+1bg 1+ 2gbf g 1bf+1 g+1bf 1):
10.71.5Z
x(f2
+g2
)dx=x(fg0
f0
g) = xp
2(ff+1+gg+1 fg+1+f+1g);
10.71.6Z
xfgdx=1
4x2(2fg f 1g+1 f+1g 1);
10.71.7Z
x(f2
g2
)dx=1
2x2
f2
f 1f+1 g2
+g 1g+1
:
Examples
10.71.8Z
xM2
(x)dx=x(berxbei0
x ber0
xbeix);Z
xN2
(x)dx=x(kerxkei0
x ker0
xkeix);
whereM(x) andN(x) are the modulus functions introduced in x10.68(i).
10.71(ii) De nite Integrals
See Kerr (1978) and Glasser (1979).
10.71(iii) Compendia
For in nite double integrals involving Kelvin functions
see Prudnikov et al. (1986b, pp. 630{631).
For direct and inverse Laplace transforms of Kelvin
functions see Prudnikov et al. (1992a,x3.19) and Prud-
nikov et al. (1992b,x3.19).
Applications
10.72 Mathematical Applications
10.72(i) Di erential Equations with Turning
Points
Bessel functions and modi ed Bessel functions are of-
ten used as approximants in the construction of uniform
asymptotic approximations and expansions for solutions
of linear second-order di erential equations containing aparameter. The canonical form of di erential equation
for these problems is given by
10.72.1d2w
dz2=
u2f(z) +g(z)
w;
wherezis a real or complex variable and uis a large
real or complex parameter.
Simple Turning Points
In regions in which (10.72.1) has a simple turning point
z0, that is,f(z) andg(z) are analytic (or with weaker
conditions if z=xis a real variable) and z0is a sim-
ple zero off(z), asymptotic expansions of the solutions
wfor largeucan be constructed in terms of Airy func-
tions or equivalently Bessel functions or modi ed Bessel
functions of order1
3(x9.6(i)). These expansions are uni-
form with respect to z, including the turning point z0
and its neighborhood, and the region of validity often
includes cut neighborhoods ( x1.10(vi)) of other singu-
larities of the di erential equation, especially irregular
singularities.
For further information and references see xx2.8(i)
and 2.8(iii).
10.73 Physical Applications 275
Multiple or Fractional Turning Points
Iff(z) has a double zero z0, or more generally z0is a
zero of order m,m= 2;3;4;:::, then uniform asymp-
totic approximations (but notexpansions) can be con-
structed in terms of Bessel functions, or modi ed Bessel
functions, of order 1 =(m+ 2). The number mcan also
be replaced by any real constant (> 2) in the sense
that (z z0) f(z) is analytic and nonvanishing at
z0; moreover, g(z) is permitted to have a single or dou-
ble pole at z0. The order of the approximating Bessel
functions, or modi ed Bessel functions, is 1 =(+ 2), ex-
cept in the case when g(z) has a double pole at z0. See
x2.8(v) for references.
10.72(ii) Di erential Equations with Poles
In regions in which the function f(z) has a simple pole
atz=z0and (z z0)2g(z) is analytic at z=z0(the
case= 1 inx10.72(i)), asymptotic expansions of the
solutionswof (10.72.1) for large ucan be constructed
in terms of Bessel functions and modi ed Bessel func-
tions of orderp1 + 4, whereis the limiting value of
(z z0)2g(z) asz!z0. These asymptotic expansions
are uniform with respect to z, including cut neighbor-
hoods ofz0, and again the region of uniformity often
includes cut neighborhoods of other singularities of the
di erential equation.
For further information and references see xx2.8(i)
and 2.8(iv).
10.72(iii) Di erential Equations with a Double
Pole and a Movable Turning Point
In (10.72.1) assume f(z) =f(z; ) andg(z) =g(z; )
depend continuously on a real parameter ,f(z; ) has
a simple zero z=z0( ) and a double pole z= 0, ex-
cept for a critical value =a, wherez0(a) = 0. Assume
that whether or not =a,z2g(z; ) is analytic at z= 0.
Then for large uasymptotic approximations of the solu-
tionswcan be constructed in terms of Bessel functions,
or modi ed Bessel functions, of variable order (in fact
the order depends on uand ). These approximations
are uniform with respect to both zand , including
z=z0(a), the cut neighborhood of z= 0, and =a.
Seex2.8(vi) for references.
10.73 Physical Applications
10.73(i) Bessel and Modi ed Bessel Functions
Bessel functions rst appear in the investigation of a
physical problem in Daniel Bernoulli's analysis of the
small oscillations of a uniform heavy
exible chain. For
this problem and its further generalizations, see Ko-
renev (2002, Chapter 4, x37) and Gray et al. (1922,
Chapter I,x1, Chapter XVI,x4).Bessel functions of the rst kind, Jn(x), arise nat-
urally in applications having cylindrical symmetry in
which the physics is described either by Laplace's equa-
tionr2V= 0, or by the Helmholtz equation ( r2+
k2) = 0.
Laplace's equation governs problems in heat conduc-
tion, in the distribution of potential in an electrostatic
eld, and in hydrodynamics in the irrotational motion of
an incompressible
uid. See Jackson (1999, Chapter 3,
xx3.7, 3.8, 3.11, 3.13), Lamb (1932, Chapter V, xx100{
102; Chapter VIII, xx186, 191{193; Chapter X, xx303,
304), Happel and Brenner (1973, Chapter 3, x3.3; Chap-
ter 7,x7.3), Korenev (2002, Chapter 4, x43), and Gray
et al. (1922, Chapter XI). In cylindrical coordinates r,
,z, (x1.5(ii) we have
10.73.1r2V=1
r@
@r
r@V
@r
+1
r2@2V
@ 2+@2V
@z2= 0;
and on separation of variables we obtain solutions of the
formein ezJn(r), from which a solution satisfying
prescribed boundary conditions may be constructed.
The Helmholtz equation, ( r2+k2) = 0, follows
from the wave equation
10.73.2 r2 =1
c2@2
@t2;
on assuming a time dependence of the form eikt. This
equation governs problems in acoustic and electromag-
netic wave propagation. See Jackson (1999, Chap-
ter 9,x9.6), Jones (1986, Chapters 7, 8), and Lord
Rayleigh (1945, Vol. I, Chapter IX, xx200{211, 218, 219,
221a; Vol. II, Chapter XIII, x272a; Chapter XV, x302;
Chapter XVIII; Chapter XIX, x350; Chapter XX, x357;
Chapter XXI,x369). It is fundamental in the study
of electromagnetic wave transmission. Consequently,
Bessel functions Jn(x), and modi ed Bessel functions
In(x), are central to the analysis of microwave and
optical transmission in waveguides, including coaxial
and ber. See Krivoshlykov (1994, Chapter 2, x2.2.10;
Chapter 5,x5.2.2), Kapany and Burke (1972, Chap-
ters 4{6; Chapter 7, xA.1), and Slater (1942, Chapter 4,
xx20, 25).
Bessel functions enter in the study of the scatter-
ing of light and other electromagnetic radiation, not
only from cylindrical surfaces but also in the statisti-
cal analysis involved in scattering from rough surfaces.
See Smith (1997, Chapter 3, x3.7; Chapter 6, x6.4),
Beckmann and Spizzichino (1963, Chapter 4, xx4.2, 4.3;
Chapter 5,xx5.2, 5.3; Chapter 6, x6.1; Chapter 7,x7.1.),
Kerker (1969, Chapter 5, x5.6.4; Chapter 7, x7.5.6), and
Bayvel and Jones (1981, Chapter 1, xx1.6.5, 1.6.6).
More recently, Bessel functions appear in the in-
verse problem in wave propagation, with applications in
medicine, astronomy, and acoustic imaging. See Colton
and Kress (1998, Chapter 2, xx2.4, 2.5; Chapter 3, x3.4).
276 Bessel Functions
In the theory of plates and shells, the oscillations of a
circular plate are determined by the di erential equa-
tion
10.73.3 r4W+2@2W
@t2= 0:
See Korenev (2002). On separation of variables into
cylindrical coordinates, the Bessel functions Jn(x), and
modi ed Bessel functions In(x) andKn(x), all appear.
10.73(ii) Spherical Bessel Functions
The functions jn(x),yn(x),h(1)
n(x), and h(2)
n(x) arise
in the solution (again by separation of variables) of
the Helmholtz equation in spherical coordinates ;;
(x1.5(ii)):
10.73.4
(r2+k2)f=1
2@
@
2@f
@
+1
2sin@
@
sin@f
@
+1
2sin2@2f
@ 2+k2f:
With the spherical harmonic Y`;m(; ) de ned as
inx14.30(i), the solutions are of the form f=
g`(k)Y`;m(; ) withg`=j`,y`,h(1)
`, orh(2)
`, depending
on the boundary conditions. Accordingly, the spherical
Bessel functions appear in all problems in three dimen-
sions with spherical symmetry involving the scattering
of electromagnetic radiation. See Jackson (1999, Chap-
ter 9,x9.6), Bayvel and Jones (1981, Chapter 1, x1.5.1),
and Konopinski (1981, Chapter 9, x9.1). In quantum
mechanics the spherical Bessel functions arise in the so-
lution of the Schr odinger wave equation for a particle
in a central potential. See Messiah (1961, Chapter IX,
xx7{10).
10.73(iii) Kelvin Functions
The analysis of the current distribution in circular
conductors leads to the Kelvin functions ber x, beix,
kerx, and keix. See Relton (1965, Chapter X, xx10.2,
10.3), Bowman (1958, Chapter III, xx51{53), McLach-
lan (1961, Chapters VIII and IX), and Russell (1909).
The McLachlan reference also includes other applica-
tions of Kelvin functions.
10.73(iv) Bickley Functions
See Bickley (1935) and Alta c (1996).
10.73(v) Rayleigh Function
For applications of the Rayleigh function n()
(x10.21(xiii)) to problems of heat conduction and dif-
fusion in liquids see Kapitsa (1951b).Computation
10.74 Methods of Computation
10.74(i) Series Expansions
The power-series expansions given in xx10.2 and 10.8,
together with the connection formulas of x10.4, can be
used to compute the Bessel and Hankel functions when
the argument xorzis suciently small in absolute
value. In the case of the modi ed Bessel function K(z)
see especially Temme (1975).
In other circumstances the power series are prone to
slow convergence and heavy numerical cancellation.
Ifxorjzjis large compared with jj2, then the
asymptotic expansions of xx10.17(i){10.17(iv) are avail-
able. Furthermore, the attainable accuracy can be
increased substantially by use of the exponentially-
improved expansions given in x10.17(v), even more so
by application of the hyperasymptotic expansions to be
found in the references in that subsection.
For large positive real values of the uniform asymp-
totic expansions of xx10.20(i) and 10.20(ii) can be used.
Moreover, because of their double asymptotic properties
(x10.41(v)) these expansions can also be used for large
xorjzj, whether or not is large. It should be noted,
however, that there is a diculty in evaluating the co-
ecientsAk(),Bk(),Ck(), andDk(), from the ex-
plicit expressions (10.20.10){(10.20.13) when zis close
to 1 owing to severe cancellation. Temme (1997) shows
how to overcome this diculty by use of the Maclaurin
expansions for these coecients or by use of auxiliary
functions.
Similar observations apply to the computation of
modi ed Bessel functions, spherical Bessel functions,
and Kelvin functions. In the case of the spherical Bessel
functions the explicit formulas given in xx10.49(i) and
10.49(ii) are terminating cases of the asymptotic expan-
sions given inxx10.17(i) and 10.40(i) for the Bessel func-
tions and modi ed Bessel functions. And since there are
no error terms they could, in theory, be used for all val-
ues ofz; however, there may be severe cancellation when
jzjis not large compared with n2.
10.74(ii) Di erential Equations
A comprehensive and powerful approach is to integrate
the di erential equations (10.2.1) and (10.25.1) by di-
rect numerical methods. As described in x3.7(ii), to
insure stability the integration path must be chosen in
such a way that as we proceed along it the wanted so-
lution grows in magnitude at least as fast as all other
solutions of the di erential equation.
10.74 Methods of Computation 277
In the interval 0 < x < ,J(x) needs to be inte-
grated in the forward direction and Y(x) in the back-
ward direction, with initial values for the former ob-
tained from the power-series expansion (10.2.2) and for
the latter from asymptotic expansions ( xx10.17(i) and
10.20(i)). In the interval <x<1either direction of
integration can be used for both functions.
Similarly, to maintain stability in the interval 0 <
x <1the integration direction has to be forwards in
the case of I(x) and backwards in the case of K(x),
with initial values obtained in an analogous manner to
those forJ(x) andY(x).
Forz2Cthe function H(1)
(z), for example, can
always be computed in a stable manner in the sector
0phzby integrating along rays towards the
origin.
Similar considerations apply to the spherical Bessel
functions and Kelvin functions.
For further information, including parallel methods
for solving the di erential equations, see Lozier and
Olver (1993).
10.74(iii) Integral Representations
For evaluation of the Hankel functions H(1)
(z) and
H(2)
(z) for complex values of andzbased on the in-
tegral representations (10.9.18) see Remenets (1973).
For applications of generalized Gauss{Laguerre
quadrature (x3.5(v)) to the evaluation of the modi ed
Bessel functions K(z) for 0< < 1 and 0< x <1
see Gautschi (2002a). The integral representation used
is based on (10.32.8).
For evaluation of K(z) from (10.32.14) with =n
andzcomplex, see Mechel (1966).
10.74(iv) Recurrence Relations
If values of the Bessel functions J(z),Y(z), or the
other functions treated in this chapter, are needed for
integer-spaced ranges of values of the order , then a
simple and powerful procedure is provided by recurrence
relations typi ed by the rst of (10.6.1).
Suppose, for example, =n20;1;2;:::; and
x2(0;1). ThenJn(x) andYn(x) can be generated
by either forward or backward recurrence on nwhen
n<x , but ifn>x then to maintain stability Jn(x) has
to be generated by backward recurrence on n, andYn(x)
has to be generated by forward recurrence on n. In the
case ofJn(x), the need for initial values can be avoided
by application of Olver's algorithm ( x3.6(v)) in conjunc-
tion with Equation (10.12.4) used as a normalizing con-
dition, or in the case of noninteger orders, (10.23.15).
For further information see Gautschi (1967), Olver
and Sookne (1972), Temme (1975), Campbell (1980),
and Kerimov and Skorokhodov (1984a).10.74(v) Continued Fractions
For applications of the continued-fraction expansions
(10.10.1), (10.10.2), (10.33.1), and (10.33.2) to the com-
putation of Bessel functions and modi ed Bessel func-
tions see Gargantini and Henrici (1967), Amos (1974),
Gautschi and Slavik (1978), Tretter and Walster (1980),
Thompson and Barnett (1986), and Cuyt et al. (2008).
10.74(vi) Zeros and Associated Values
Newton's rule (x3.8(i)) or Halley's rule ( x3.8(v)) can be
used to compute to arbitrarily high accuracy the real or
complex zeros of all the functions treated in this chapter.
Necessary values of the rst derivatives of the functions
are obtained by the use of (10.6.2), for example. New-
ton's rule is quadratically convergent and Halley's rule
is cubically convergent. See also Segura (1998, 2001).
Methods for obtaining initial approximations to
the zeros include asymptotic expansions ( xx10.21(vi)-
10.21(ix)), graphical intersection of 2 Dgraphs in R
(e.g.,x10.3(i)) with the x-axis, or graphical intersection
of 3Dcomplex-variable surfaces (e.g., x10.3(ii)) with the
planez= 0.
To ensure that no zeros are overlooked, standard
tools are the phase principle and Rouch e's theorem; see
x1.10(iv).
Real Zeros
See Olver (1960, pp. xvi{xxix), Grad and Zakraj sek
(1973), Temme (1979a), Ikebe et al. (1991), Za ropou-
loset al. (1996), Vrahatis et al. (1997a), Ball (2000),
and Gil and Segura (2003).
Complex Zeros
See Leung and Ghaderpanah (1979), Kerimov and
Skorokhodov (1984b,c, 1985a,b), Skorokhodov (1985),
Modenov and Filonov (1986), and Vrahatis et al.
(1997b).
Multiple Zeros
See Kerimov and Skorokhodov (1985c, 1986, 1987,
1988).
10.74(vii) Integrals
Hankel Transform
See Cornille (1972), Johansen and S rensen (1979),
Gabutti (1979), Gabutti and Minetti (1981), Candel
(1981), Wong (1982), Lund (1985), Piessens and Bran-
ders (1985), Hansen (1985), Bezvoda et al. (1986), Pu-
oskari (1988), Christensen (1990), Campos (1995), Lu-
cas and Stone (1995), Barakat and Parshall (1996), Sidi
(1997), Secada (1999).
278 Bessel Functions
Fourier{Bessel Expansion
For the computation of the integral (10.23.19) see
Piessens and Branders (1983, 1985), Lewanowicz (1991),
and Zhile kin and Kukarkin (1995).
Spherical Bessel Transform
The spherical Bessel transform is the Hankel transform
(10.22.76) in the case when is half an odd positive
integer.
See Lehman et al. (1981), Puoskari (1988), and
Sharafeddin et al. (1992).
Kontorovich{Lebedev Transform
See Ehrenmark (1995).
Products
For in nite integrals involving products of two Bessel
functions of the rst kind, see Linz and Kropp (1973),
Gabutti (1980), Ikonomou et al. (1995), and Lucas
(1995).
10.74(viii) Functions of Imaginary Order
For the computation of the functions eI(x) andeK(x)
de ned by (10.45.2) see Temme (1994b) and Gil et al.
(2002b, 2003a, 2004a).
10.75 Tables
10.75(i) Introduction
Comprehensive listings and descriptions of tables of
the functions treated in this chapter are provided in
Bateman and Archibald (1944), Lebedev and Fedorova
(1960), Fletcher et al. (1962), and Luke (1975, x9.13.2).
Only a few of the more comprehensive of these early ta-
bles are included in the listings in the following subsec-
tions. Also, for additional listings of tables pertaining
to complex arguments see Babushkina et al. (1997).
10.75(ii) Bessel Functions and their Derivatives
British Association for the Advancement of
Science (1937) tabulates J0(x),J1(x),x=
0(:001)16(:01)25, 10D; Y0(x),Y1(x),x=
0:01(:01)25, 8{9S or 8D. Also included are aux-
iliary functions to facilitate interpolation of the
tables ofY0(x),Y1(x) for small values of x, as well
as auxiliary functions to compute all four func-
tions for large values of x.
Bickley et al. (1952) tabulates Jn(x),Yn(x) or
xnYn(x),n= 2(1)20,x= 0(:01 or:1) 10(:1)25,
8D (forJn(x)), 8S (forYn(x) orxnYn(x));Jn(x),
Yn(x),n= 0(1)20,x= 0 or 0:1(:1)25, 10D (for
Jn(x)), 10S (for Yn(x)).Olver (1962) provides tables for the uniform
asymptotic expansions given in x10.20(i), includ-
ingand (4
(1 x2))1
4as functions of x(=z)
and the coecients Ak(),Bk(),Ck(),Dk()
as functions of . These enable J(x),Y(x),
J0
(x),Y0
(x) to be computed to 10S when
15, except in the neighborhoods of zeros.
The main tables in Abramowitz and Stegun (1964,
Chapter 9) give J0(x) to 15D, J1(x),J2(x),
Y0(x),Y1(x) to 10D,Y2(x) to 8D,x= 0(:1)17:5;
Yn(x) (2=)Jn(x) lnx,n= 0;1,x= 0(:1)2,
8D;Jn(x),Yn(x),n= 3(1)9,x= 0(:2)20, 5D
or 5S;Jn(x),Yn(x),n= 0(1)20(10)50 ;100,x=
1;2;5;10;50;100, 10S; modulus and phase func-
tionspxMn(x),n(x) x,n= 0;1;2, 1/x=
0(:01)0:1, 8D.
Achenbach (1986) tabulates J0(x),J1(x),Y0(x),
Y1(x),x= 0(:1)8, 20D or 18{20S.
Zhang and Jin (1996, pp. 185{195) tab-
ulatesJn(x),J0
n(x),Yn(x),Y0
n(x),n=
0(1)10(10)50 ;100,x= 1, 5, 10, 25, 50,
100, 9S;Jn+ (x),J0
n+ (x),Yn+ (x),Y0
n+ (x),
n= 0(1)5;10;30;50;100, =1
4;1
3;1
2;2
3;3
4,
x= 1;5;10;50, 8S; real and imaginary parts
ofJn+ (z),J0
n+ (z),Yn+ (z),Y0
n+ (z),n=
0(1)15;20(10)50;100, = 0;1
2,z= 4+2i, 20+10i,
8S.
10.75(iii) Zeros and Associated Values of the
Bessel Functions, Hankel Functions,
and their Derivatives
Real Zeros
British Association for the Advancement of
Science (1937) tabulates j0;m,J1(j0;m),j1;m,
J0(j1;m),m= 1(1)150, 10D; y0;m,Y1(y0;m),y1;m,
Y0(y1;m),m= 1(1)50, 8D.
Olver (1960) tabulates jn;m,J0
n(jn;m),j0
n;m,
Jn
j0
n;m
,yn;m,Y0
n(yn;m),y0
n;m,Yn
y0
n;m
,n=
0(1
2)201
2,m= 1(1)50, 8D. Also included are ta-
bles of the coecients in the uniform asymptotic
expansions of these zeros and associated values as
n! 1 ; seex10.21(viii), and more fully Olver
(1954).
Morgenthaler and Reismann (1963) tabulates j0
n;m
forn= 21(1)51 and j0
n;m<100, 7-10S.
Abramowitz and Stegun (1964, Chapter 9) tabu-
latesjn;m,J0
n(jn;m),j0
n;m,Jn
j0
n;m
,n= 0(1)8,
m= 1(1)20, 5D (10D for n= 0),yn;m,Y0
n(yn;m),
y0
n;m,Yn
y0
n;m
,n= 0(1)8,m= 1(1)20, 5D
10.75 Tables 279
(8D forn= 0),J0(j0;mx),m= 1(1)5,x=
0(:02)1, 5D. Also included are the rst 5 zeros of
the functions xJ1(x) J0(x),J1(x) xJ 0(x),
J0(x)Y0(x) Y0(x)J0(x),J1(x)Y1(x)
Y1(x)J1(x),J1(x)Y0(x) Y1(x)J0(x) for vari-
ous values of and 1in the interval [0 ;1], 4{8D.
Abramowitz and Stegun (1964, Chapter 10)
tabulatesj;m,J0
(j;m),j0
;m,J
j0
;m
,y;m,
Y0
(y;m),y0
;m,Y
y0
;m
,=1
2(1)191
2,m=
1(1)m, wheremranges from 8 at =1
2down
to 1 at= 191
2, 6{7D.
Makinouchi (1966) tabulates all values of j;mand
y;min the interval (0 ;100), with at least 29S.
These are for = 0(1)5, 10, 20; =3
2,5
2;
=m=n withm= 1(1)n 1 andn= 3(1)8,
except for=1
2.
D oring (1971) tabulates the rst 100 values of
(>1) for which J0
(x) has the double zero x=,
10D.
Heller (1976) tabulates j0;m,J1(j0;m),j1;m,
J0(j1;m),j0
1;m,J1
j0
1;m
form= 1(1)100, 25D.
Wills et al. (1982) tabulates j0;m,j1;m,y0;m,y1;m
form= 1(1)30, 35D.
Kerimov and Skorokhodov (1985c) tabulates 201
double zeros of J00
(x), 10 double zeros of J000
(x),
101 double zeros of Y0
(x), 201 double zeros of
Y00
(x), and 10 double zeros of Y000
(x), all to 8 or
9D.
Zhang and Jin (1996, pp. 196{198) tabulates jn;m,
j0
n;m,yn;m,y0
n;m,n= 0(1)3,m= 1(1)10, 8D;
the rst ve zeros of Jn(x)Yn(x) Jn(x)Yn(x),
J0
n(x)Y0
n(x) J0
n(x)Y0
n(x),n= 0;1;2,=
1:1(:1)1:6;1:8;2(:5)5, 7D.
Complex Zeros
Abramowitz and Stegun (1964, p. 373) tabulates
the three smallest zeros of Y0(z),Y1(z),Y0
1(z) in
the sector 0 <phz, together with the cor-
responding values of Y1(z),Y0(z),Y1(z), respec-
tively, to 9D. (There is an error in the value of
Y0(z) at the 3rd zero of Y1(z): the last four digits
should be 2533; see Amos (1985).)
D oring (1966) tabulates all zeros of Y0(z),Y1(z),
H(1)
0(z),H(1)
1(z), that lie in the sector jzj<158,
jphzj, to 10D. Some of the smaller zeros of
Yn(z) andH(1)
n(z) forn= 2;3;4;5;15 are also
included.Kerimov and Skorokhodov (1985a) tabulates 5
(nonreal) complex conjugate pairs of zeros of the
principal branches of Yn(z) andY0
n(z) forn=
0(1)5, 8D.
Kerimov and Skorokhodov (1985b) tabulates 50
zeros of the principal branches of H(1)
0(z) and
H(1)
1(z), 8D.
Kerimov and Skorokhodov (1987) tabulates 100
complex double zeros ofY0
ze i
and
H(1)
0
ze i
, 8D.
MacDonald (1989) tabulates the rst 30 zeros, in
ascending order of absolute value in the fourth
quadrant, of the function J0(z) iJ1(z), 6D.
(Other zeros of this function can be obtained by
re
ection in the imaginary axis).
Zhang and Jin (1996, p. 199) tabulates the real
and imaginary parts of the rst 15 conjugate pairs
of complex zeros of Y0(z),Y1(z),Y0
1(z) and the
corresponding values of Y1(z),Y0(z),Y1(z), re-
spectively, 10D.
10.75(iv) Integrals of Bessel Functions
Abramowitz and Stegun (1964, Chapter 11) tab-
ulatesRx
0J0(t)dt,Rx
0Y0(t)dt,x= 0(:1)10, 10D;Rx
0t 1(1 J0(t))dt,R1
xt 1Y0(t)dt,x= 0(:1)5,
8D.
Zhang and Jin (1996, p. 270) tabulatesRx
0J0(t)dt,Rx
0t 1(1 J0(t))dt,Rx
0Y0(t)dt,R1
xt 1Y0(t)dt,
x= 0(:1)1(:5)20, 8D.
10.75(v) Modi ed Bessel Functions and their
Derivatives
British Association for the Advancement of Sci-
ence (1937) tabulates I0(x),I1(x),x= 0(:001)5,
7{8D;K0(x),K1(x),x= 0:01(:01)5, 7{10D;
e xI0(x),e xI1(x),exK0(x),exK1(x),x=
5(:01)10(:1)20, 8D. Also included are auxiliary
functions to facilitate interpolation of the tables
ofK0(x),K1(x) for small values of x.
Bickley et al. (1952) tabulates x nIn(x) or
e xIn(x),xnKn(x) orexKn(x),n= 2(1)20,
x= 0(.01 or .1) 10(.1) 20, 8S; In(x),Kn(x),
n= 0(1)20,x= 0 or 0:1(:1)20, 10S.
Olver (1962) provides tables for the uniform
asymptotic expansions given in x10.41(ii), includ-
ingand the coecients Uk(p),Vk(p) as func-
tions ofp= (1 +x2) 1
2. These enable I(x),
K(x),I0
(x),K0
(x) to be computed to 10S
when16.
280 Bessel Functions
The main tables in Abramowitz and Stegun (1964,
Chapter 9) give e xIn(x),exKn(x),n= 0;1;2,
x= 0(:1)10(:2)20, 8D{10D or 10S;pxe xIn(x),
(px=)exKn(x),n= 0;1;2, 1=x= 0(:002)0:05;
K0(x) +I0(x) lnx,x(K1(x) I1(x) lnx),x=
0(:1)2, 8D;e xIn(x),exKn(x),n= 3(1)9,
x= 0(:2)10(:5)20, 5S; In(x),Kn(x),n=
0(1)20(10)50 ;100,x= 1;2;5;10;50;100, 9{10S.
Achenbach (1986) tabulates I0(x),I1(x),K0(x),
K1(x),x= 0(:1)8, 19D or 19{21S.
Zhang and Jin (1996, pp. 240{250) tabu-
latesIn(x),I0
n(x),Kn(x),K0
n(x),n=
0(1)10(10)50 ;100,x= 1;5;10;25;50;100, 9S;
In+ (x),I0
n+ (x),Kn+ (x),K0
n+ (x),n= 0(1)5,
10, 30, 50, 100, =1
4,1
3,1
2,2
3,3
4,x= 1, 5, 10, 50,
8S; real and imaginary parts of In+ (z),I0
n+ (z),
Kn+ (z),K0
n+ (z),n= 0(1)15, 20(10)50, 100,
= 0;1
2,z= 4 + 2i;20 + 10i, 8S.
10.75(vi) Zeros of Modi ed Bessel Functions
and their Derivatives
Parnes (1972) tabulates all zeros of the principal
value ofKn(z), forn= 2(1)10, 9D.
Leung and Ghaderpanah (1979), tabulates all ze-
ros of the principal value of Kn(z), forn= 2(1)10,
29S.
Kerimov and Skorokhodov (1984b) tabulates all
zeros of the principal values of Kn(z) andK0
n(z),
forn= 2(1)20, 9S.
Kerimov and Skorokhodov (1984c) tabulates all
zeros ofI n 1
2(z) andI0
n 1
2(z) in the sector
0phz1
2forn= 1(1)20, 9S.
Kerimov and Skorokhodov (1985b) tabulates all
zeros ofKn(z) andK0
n(z) in the sector 1
2 <
phz3
2forn= 0(1)5, 8D.
10.75(vii) Integrals of Modi ed Bessel
Functions
Abramowitz and Stegun (1964, Chapter 11) tabu-
latese xRx
0I0(t)dt,exR1
xK0(t)dt,x= 0(:1)10,
7D;e xRx
0t 1(I0(t) 1)dt,xexR1
xt 1K0(t)dt,
x= 0(:1)5, 6D.
Bickley and Nayler (1935) tabulates Ki n(x)
(x10.43(iii)) for n= 1(1)16,x= 0(:05)0:2(:1) 2, 3,
9D.
Zhang and Jin (1996, p. 271) tabu-
latese xRx
0I0(t)dt,e xRx
0t 1(I0(t) 1)dt,
exR1
xK0(t)dt,xexR1
xt 1K0(t)dt,x =
0(:1)1(:5)20, 8D.10.75(viii) Modi ed Bessel Functions of
Imaginary or Complex Order
For the notation see x10.45.
Zurina and Karmazina (1967) tabulates eK(x) for
= 0:01(:01)10,x= 0:1(:1)10:2, 7S.
Rappoport (1979) tabulates the real and imag-
inary parts of K1
2+i (x) for = 0:01(:01)10,
x= 0:1(:2)9:5, 7S.
10.75(ix) Spherical Bessel Functions, Modi ed
Spherical Bessel Functions, and their
Derivatives
The main tables in Abramowitz and Stegun (1964,
Chapter 10) give jn(x),yn(x)n= 0(1)8,x=
0(:1)10, 5{8S; jn(x),yn(x)n= 0(1)20(10)50,
100,x= 1;2;5;10;50;100, 10S; i(1)
n(x),kn(x),
n= 0;1;2,x= 0(:1)5, 4{9D; i(1)
n(x),kn(x),
n= 0(1)20(10)50, 100, x= 1;2;5;10;50;100, 10S.
(For the notation see x10.1 andx10.47(ii).)
Zhang and Jin (1996, pp. 296{305) tabulates
jn(x),j0
n(x),yn(x),y0
n(x),i(1)
n(x),i(1)
n0(x),kn(x),
k0
n(x),n= 0(1)10(10)30, 50, 100, x= 1, 5, 10, 25,
50, 100, 8S; xjn(x), (xjn(x))0,xyn(x), (xyn(x))0
(Riccati{Bessel functions and their derivatives),
n= 0(1)10(10)30, 50, 100, x= 1, 5, 10, 25,
50, 100, 8S; real and imaginary parts of jn(z),
j0
n(z),yn(z),y0
n(z),i(1)
n(z),i(1)
n0(z),kn(z),k0
n(z),
n= 0(1)15, 20(10)50, 100, z= 4 + 2i, 20 + 10i,
8S. (For the notation replace j;y;i;k byj,y,i(1),
k, respectively.)
10.75(x) Zeros and Associated Values of
Derivatives of Spherical Bessel
Functions
For the notation see x10.58.
Olver (1960) tabulates a0
n;m,jn
a0
n;m
,b0
n;m,
yn
b0
n;m
,n= 1(1)20,m= 1(1)50, 8D. Also in-
cluded are tables of the coecients in the uniform
asymptotic expansions of these zeros and associ-
ated values as n!1 .
10.76 Approximations 281
10.75(xi) Kelvin Functions and their Derivatives
Young and Kirk (1964) tabulates ber nx, beinx,
kernx, keinx,n= 0;1,x= 0(:1)10, 15D; ber nx,
beinx, kernx, keinx, modulus and phase func-
tionsMn(x),n(x),Nn(x), n(x),n= 0;1;2,
x= 0(:01)2:5, 8S, and n= 0(1)10,x= 0(:1)10,
7S. Also included are auxiliary functions to facili-
tate interpolation of the tables for n= 0(1)10 for
small values of x. (Concerning the phase functions
seex10.68(iv).)
Abramowitz and Stegun (1964, Chapter 9) tab-
ulates ber nx, beinx, kernx, keinx,n= 0;1,
x= 0(:1)5, 9{10D; xn(kernx+ (bernx)(lnx)),
xn(keinx+ (beinx)(lnx)),n= 0;1,x=
0(:1)1, 9D; modulus and phase functions Mn(x),
n(x),Nn(x), n(x),n= 0;1,x=
0(:2)7, 6D;pxe x=p
2Mn(x),n(x) (x=p
2),pxex=p
2Nn(x), n(x) + (x=p
2),n= 0;1, 1=x=
0(:01)0:15, 5D.
Zhang and Jin (1996, p. 322) tabulates ber x,
ber0x, beix, bei0x, kerx, ker0x, keix, kei0x,x=
0(1)20, 7S.
10.75(xii) Zeros of Kelvin Functions and their
Derivatives
Zhang and Jin (1996, p. 323) tabulates the rst 20
real zeros of ber x, ber0x, beix, bei0x, kerx, ker0x,
keix, kei0x, 8D.
10.76 Approximations
10.76(i) Introduction
Because of the comprehensive nature of more recent
software packages ( x10.77), the following subsections in-
clude only references that give representative examples
of the kind of approximations that can be used to gen-
erate the functions that appear in the present chapter.
For references to other approximations, see for example,
Luke (1975,x9.13.3).
10.76(ii) Bessel Functions, Hankel Functions,
and Modi ed Bessel Functions
Real Variable and Order :Functions
Luke (1971a,b, 1972), Luke (1975, Tables 9.1, 9.2, 9.5,
9.6, 9.11{9.15, 9.17{9.21), Weniger and C zek (1990),
N emeth (1992, Chapters 4{6).
Real Variable and Order :Zeros
Piessens (1984, 1990), Piessens and Ahmed (1986),
N emeth (1992, Chapter 7).Real Variable and Order :Integrals
Luke (1975, Tables 9.3, 9.4, 9.7{9.9, 9.16, 9.22), N emeth
(1992, Chapter 10).
Complex Variable; Real Order
Luke (1975, Tables 9.23{9.28), Coleman and Monaghan
(1983), Coleman (1987), Zhang (1996), Zhang and Bel-
ward (1997).
Real Variable; Imaginary Order
Poqu erusse and Alexiou (1999).
10.76(iii) Other Functions
Bickley Functions
Blair et al. (1978).
Spherical Bessel Functions
Delic (1979).
Kelvin Functions
Luke (1975, Table 9.10), N emeth (1992, Chapter 9).
10.77 Software
Seehttp://dlmf.nist.gov/10.77 .
References
General References
The main references used in writing this chapter are
Watson (1944) and Olver (1997b).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x10.2 Olver (1997b, pp. 57, 237{238, 242{243) and
Watson (1944, pp. 38{45, 57{64, 196{198). The
conclusions inx10.2(iii) follow from x2.7(iv) and
the limiting forms of the solutions as z!0 and
asz!1 ; seex10.7.
x10.3 These graphics were produced at NIST.
x10.4 Olver (1997b, pp. 56, 238{239, 242{243) and
Watson (1944, pp. 74{75).
x10.5 For the Wronskians use (1.13.5) and the limiting
forms inx10.7. Then for the cross-products apply
(10.6.2).
282 Bessel Functions
x10.6 For (10.6.1) and (10.6.2) see Olver (1997b,
pp. 58{59, 240{242) or Watson (1944, pp. 45, 66,
73{74). (10.6.3) are special cases, and (10.6.4),
(10.6.5) follow by straightforward substitution.
For (10.6.6) see Watson (1944, pp. 46). For
(10.6.7) use induction combined with the second
of (10.6.1). For (10.6.8){(10.6.10) see Goodwin
(1949b).
x10.7 For (10.7.1) and (10.7.3) use (10.2.2) and
(10.8.2). For (10.7.2) use (10.4.3) and (10.7.1).
For (10.7.4) and (10.7.5) use (10.2.3) and (10.7.3)
whenis not an integer; (10.4.1), (10.8.1) other-
wise. For (10.7.6) use (10.2.3) and (10.7.3). For
(10.7.7) use (10.4.3), (10.7.3), and (10.7.4). For
(10.7.8) see (10.17.3) and (10.17.4).
x10.8 Olver (1997b, p. 243) and Watson (1944, p. 147).
x10.9 Watson (1944, pp. 19{21, 47{48, 68{71, 150,
160{170, 174{180, 436, 438, 441{444). For
(10.9.3) see Olver (1997b, p. 244) (with \Exer-
cises 2.2 and 9.5" corrected to \Exercises 2.3 and
9.5"). For (10.9.5), (10.9.10), (10.9.11), (10.9.13),
(10.9.14) see Erd elyi et al. (1953b, pp. 18, 21, 82).
(The condition<(z)>0 in (10.9.14) is weaker
than the corresponding condition in Erd elyi et al.
(1953b, p. 82, Eq. (18)).) (10.9.15), (10.9.16) fol-
low from (10.9.10), (10.9.11) by change of vari-
ablesz=cosh ,t!t ln tanh(1
2 ), > 0.
For (10.9.27) see Erd elyi et al. (1953b, p. 47). See
also Olver (1997b, pp. 340{341).
x10.10 Watson (1944,xx5.6, 9.65).
x10.11 For (10.11.1){(10.11.5) use (10.2.2), (10.2.3),
(10.4.3). For (10.11.6){(10.11.8) take limits. For
(10.11.9) use the Schwarz Re
ection Principle
(x1.10(ii)).
x10.12 For (10.12.1) see Olver (1997b, pp. 55{56). For
(10.12.2){(10.12.6) set t=eiandiei, and ap-
ply other straighforward substitutions, including
di erentiations with respect to in the case of
(10.12.6). See also Watson (1944, pp. 22{23).
x10.13 These results are obtainable from (10.2.1) by
straightforward substitutions. See also x1.13(v).
x10.14 Watson (1944, pp. 49, 258{259, 268{270, 406)
and Olver (1997b, pp. 59, 426).
x10.15 For (10.15.1) see Watson (1944, pp. 61{62) or
Olver (1997b, p. 243). For (10.15.2) use (10.2.3).
For (10.15.3){(10.15.5) see Olver (1997b, p. 244).
(10.15.6){(10.15.9) appear without proof in Mag-
nus et al. (1966,x3.3.3). To derive (10.15.6)the left-hand side satis es the di erential equa-
tionx2(d2W
dx2) +x(dW/dx) + (x2 1
4)W=p
2=(x) sinx, obtained by di erentiating (10.2.1)
with respect to , setting=1
2, and referring
to (10.16.1) for w. This inhomogeneous equation
forWcan be solved by variation of parameters
(x1.13(ii)), using the fact that independent so-
lutions of the corresponding homogeneous equa-
tion areJ1
2(x) andY1
2(x) with Wronskian 2 =(x),
and subsequently referring to (6.2.9) and (6.2.11).
Similarly for (10.15.7). (10.15.8) and (10.15.9)
follow from (10.15.2), (10.15.6), (10.15.7), and
(10.16.1).
x10.16 For (10.16.3), (10.16.4) see Miller (1955, p. 43).
For (10.16.5) and (10.16.6) see Olver (1997b,
pp. 255, 259) and apply (10.27.8). For (10.16.7)
and (10.16.8) apply (13.14.4) and (13.14.5). For
(10.16.9) combine (10.2.2) and (16.2.1).
x10.17 Olver (1997b, pp. 237{242, 266{269), Watson
(1944, pp. 205{206). (10.17.8){(10.17.12) follow
by di erentiation of the corresponding expansions
inx10.17(i); compare x2.1(iii). For (10.17.16){
(10.17.18) see Olver (1991b, Theorem 1) or Olver
(1993a, Theorem 1.1), and (10.16.6).
x10.18 For (10.18.3) see x10.7(i). (10.18.4){(10.18.16)
are veri able by straightforward substitutions.
For (10.18.17), and also the concluding para-
graph ofx10.18(iii), see Watson (1944, pp. 448{
449). For (10.18.19) substitute into N2
(x) =
H(1)
0(x)H(2)
0(x) by means of (10.17.11),
(10.17.12). The general term in (10.18.20) can
be veri ed via (10.18.10). For (10.18.18) the rst
two terms can be found from (10.18.7), (10.17.3),
(10.17.4), except for an arbitrary integer multiple
of. Higher terms can be calculated via (10.18.8),
(10.18.17). By continuity, the multiple of is in-
dependent of , hence it may be determined, e.g.
by setting=1
2and referring to (10.16.1). Sim-
ilar methods can be used for (10.18.21), together
with the interlacing properties of the zeros of
J1=2(z),Y1=2(z), and their derivatives ( x10.21(i)).
See also Bickley et al. (1952, p. xxxiv).
x10.19 (10.19.1), (10.19.2) follow from (10.2.2),
(10.2.3), (10.4.3), (10.8.1), (5.5.3), (5.11.3). For
(10.19.3) and (10.19.6) see Watson (1944, pp. 241{
245) and Bickley et al. (1952, p. xxxv). The ex-
pansions for the derivatives are established in a
similar manner, with the coecients calculated by
term-by-term di erentiation; compare x2.1(iii).
x10.20 Olver (1997b, pp. 419{425), Olver (1954).
References 283
x10.21 Forx10.21(i) see Watson (1944, pp. 477{487),
Olver (1997b, pp. 244{249), D oring (1971), and
Kerimov and Skorokhodov (1985a). For x10.21(ii)
see Watson (1944, pp. 508 and 510) and Olver
(1950). (In the latter reference tin (10.21.4) is re-
placed by t.) (10.21.5) and (10.21.6) follow from
(10.6.2). (10.21.12) and (10.21.13) follow from
(10.18.3), (10.21.2), (10.21.3), and the fact that
(x) is increasing when x > 0, whereas (x)
is decreasing when 0 < x < and increasing
whenx > ; compare (10.18.8). For (10.21.15),
(10.21.16) see Watson (1944, pp. 497{498). For
x10.21(iv) see Watson (1944, pp. 508{510), Lorch
(1990, 1995), Wong and Lang (1991), McCann
(1977), Lewis and Muldoon (1977), and Mercer
(1992). For (10.21.19) see Watson (1944, pp. 503{
507) or Olver (1997b, pp. 247{248). Similar meth-
ods can be used for (10.21.20). For (10.21.22){
(10.21.40) see Olver (1951, 1952). The zeros de-
picted in Figures 10.21.1{10.21.6 were computed
at NIST using methods referred to in x10.74(vi).
For (10.21.48){(10.21.54) see McMahon (9495),
Gray et al. (1922, p. 261), and Cochran (1964).
x10.22 For (10.22.1){(10.22.3) di erentiate and use
(10.6.2), (11.4.27), (11.4.28). For (10.22.4){
(10.22.7) see Watson (1944, pp. 132{136). For
(10.22.8){(10.22.12) see Luke (1962, pp. 51{53).
To verify (10.22.13) construct the expansion of
the left-hand side in powers of zby use of
(10.2.2), followed by term-by-term integration
with the aid of (5.12.5) and (5.12.1). Then com-
pare the result with the corresponding expan-
sion of the right-hand side obtained from (10.8.3).
Next, the resultR2
0J2(2zsin)e2id =
eiJ+(z)J (z),< > 1
2, is proved in
a similar manner with the aid of (5.12.6) in place
of (5.12.5)|from which (10.22.14) and (10.22.15)
both follow. (10.22.17) follows by combining
(10.22.13) and (10.2.3); (10.22.16) is a special
case of (10.22.14). For (10.22.18) replace by
1
2 and set=nin (10.22.17); then apply
(10.2.3) and let !0. For (10.22.19), (10.22.22),
(10.22.25), (10.22.26) see Watson (1944, Chap-
ter 12). (In the case of (10.22.25), page 374 of this
reference lacks a factor1
2on the right-hand side.)
The veri cation of (10.22.20) is similar to that of
(10.22.13), the role of (5.12.5) now being played
by (5.12.2). For (10.22.21) combine (10.2.3) and
(10.22.20). For (10.22.23) and (10.22.24) see
Luke (1962, p. 302 (36) and p. 303 (39), respec-
tively). For (10.22.27) see Watson (1944, p. 151).
For (10.22.28), (10.22.29) di erentiate and use
(10.6.2). For (10.22.30) with n1 it follows bydi erentiation and use of (10.6.2) that the left-
hand side equalsRx
0t 1J2
n(t)dt 1
2J2
n(x); appli-
cation of Watson (1944, p. 152) yields the second
result, then for the rst result refer to (10.23.3).
Some modi cations of the proof of (10.22.30) are
needed when n= 0. For (10.22.31){(10.22.35) see
Watson (1944, p. 380). For (10.22.36) replace t
byz t, substitute for t via (10.23.15) (with z
replaced by t, andreplaced by ), and then ap-
ply (10.22.34). For (10.22.37) use (10.22.4) and
(10.22.5); a similar proof applies to (10.22.38) af-
ter replacing C1(az) andD1(bz) byC0
(az)
andD0
(bz), respectively, by means of (10.6.2).
For the rst result in (10.22.39) use (10.22.43)
with= 0 andreplaced by 1, split the
integration range at t=xand take limits as
!0; for the second result substitute into the
rst result by (10.2.2) and integrate term by term.
(10.22.40), is proved in a similar manner, start-
ing from (10.22.44) and substituting by means of
(10.8.2) and (10.2.2) with = 0 for the term-
by-term integration. For (10.22.41){(10.22.45) see
Luke (1962, pp. 56{57). For (10.22.46) see Erd elyi
et al. (1953b, p. 96). (10.22.47) is the special
case of Eq. (6) of Watson (1944, x13.53) obtained
by setting =b= 0,=+ 1, and sub-
sequently replacing kbyb. For (10.22.48) see
Sneddon (1966, Eq. (2.1.32)). For (10.22.49){
(10.22.59) see Watson (1944, pp. 385, 394, 403{
405, 407; there is an error in Eq. (1), p. 407). For
(10.22.60) di erentiate (10.22.59) with respect to
and use (10.2.4) with n= 0. For (10.22.61) see
Watson (1944, p. 405). (10.22.62) follows from
(10.22.56) with = 1 and (15.4.6). For
(10.22.63), (10.22.64) see Watson (1944, p. 404).
For (10.22.65) apply (10.22.56) with == 0,
then let!1. For (10.22.66), (10.22.67) see
Watson (1944, pp. 389, 395). For (10.22.68) set
a=bin (10.22.67), di erentiate with respect to
and apply (10.2.4) and (10.27.5) with n= 0. For
(10.22.69), (10.22.70), see Watson (1944, p. 429,
Eqs. (3),(4), with =+1 in (3)). For (10.22.71),
(10.22.72) see Watson (1944, pp. 411, 412). For
(10.22.74), (10.22.75) see Watson (1944, pp. 411)
and Askey et al. (1986).
x10.23 Watson (1944,xx5.22, 11.3, 11.4, 16.11 and
pp. 64, 67, 71, 138). (10.23.2) is obtained from
(10.23.7) by taking = 0 and = 0;. For
(10.23.21) see Temme (1996a, p. 247).
x10.24 (10.24.6){(10.24.9) follow from (10.24.2){
(10.24.4) combined with (10.2.2), (10.2.3),
(10.8.2), (10.17.3), and (10.17.4). (10.24.5) can
be veri ed from (1.13.5) and either (10.24.6) or
284 Bessel Functions
(10.24.7){(10.24.9) and their di erentiated forms.
x10.25 Olver (1997b, pp. 60, 236{237, 250). The con-
clusions inx10.25(iii) follow from x2.7(iv) and the
limiting forms of the solutions as z!0 andz!
1; see (10.25.3) and x10.30. See also (10.27.3).
x10.26 These graphics were produced at NIST.
x10.27 For (10.27.1){(10.27.6) and (10.27.8) see Olver
(1997b, pp. 60{61 and 250{252), Watson (1944,
pp. 77{79), and (10.11.5). For (10.27.7),
(10.27.9),{(10.27.11) combine these results with
(10.4.4), and also use (10.34.2) with m= 1.
x10.28 For the Wronskians use (1.13.5) and the limit-
ing forms inx10.30. For the cross-products apply
(10.29.2).
x10.29 Watson (1944, p. 79). For (10.29.5) use induc-
tion combined with the second of (10.29.1).
x10.30 For (10.30.1) use (10.25.2). For (10.30.2) and
(10.30.3) use (10.27.4) when is not an inte-
ger; (10.27.3), (10.31.1) otherwise. For (10.30.4),
(10.30.5) use (10.40.1) and (10.34.1) with m=1.
x10.31 Olver (1997b, p. 253) or Watson (1944, p. 80).
For (10.31.3) combine (10.8.3) and (10.27.6).
x10.32 Watson (1944, pp. 79, 80, 172, 181{183, 191,
193, 439{441), Erd elyi et al. (1953b, p. 82, 97{
98), Paris and Kaminski (2001, p. 114). Also
use (10.27.8). For (10.32.16) see Dixon and Fer-
rar (1930). (An error in the conditions has been
corrected.) For (10.32.19) see Titchmarsh (1986a,
Eq. (7.10.2)).
x10.33 Combine (10.10.1), (10.10.2) with (10.27.6).
x10.34 Watson (1944, p. 80) and Olver (1997b, pp. 253,
381). For (10.34.3) take m=1 in (10.34.2), and
combine with (10.34.1).
x10.35 For (10.35.1) replace zandtin (10.12.1) by
izand it, respectively, and apply (10.27.6).
(10.35.2){(10.35.6) are obtained by setting t=ei,
t= iei, together with other straightforward
substitutions.
x10.37 Olver (1997b, pp. 251{252). For (10.37.1) see
Everitt and Jones (1977).
x10.38 (10.38.1) is obtained by di erentiation of
(10.25.2); compare (10.15.1). For (10.38.2) use
(10.27.4). (10.38.3){(10.38.5) are proved in a sim-
ilar way to (10.15.3){(10.15.5). (10.38.6) and
(10.38.7) are stated without proof and in a slightly
di erent notation in Magnus et al. (1966,x3.3.3).Both cases of (10.38.6) can be derived by a
method analogous to that used for (10.15.6) and
(10.15.7). (10.38.7) follows from (10.38.2) and
(10.38.6).
x10.39 For (10.39.5){(10.39.10) combine (10.16.5){
(10.16.10) with (10.27.6) and (10.27.8).
x10.40 Watson (1944, pp. 202{203, 206{207), Olver
(1997b, pp. 250{251, 266{269, 325). Also use
(10.27.8). (10.40.3) and (10.40.4) are obtained
by di erentiation of (10.40.1) and (10.40.2); com-
parex2.1(iii). (10.40.6) and (10.40.7) are ob-
tained by multiplication of (10.40.1){(10.40.4):
that the coecients are the same as in (10.18.17)
and (10.18.19) is a consequence of the fact that
I(x)K(x) andI0
(x)K0
(x) satisfy the same dif-
ferential equations as M2
(x) =jH(1)
(x)j2=
H(1)
(x)H(2)
(x) andN2
(x) =jH(1)
0(x)j2=
H(1)
0(x)H(2)
0(x), respectively, except for replace-
ment ofxbyix. For the statement concerning the
accuracy of (10.40.5) use the error bounds given
by (10.40.10){(10.40.12). For (10.40.14) see Olver
(1991b) together with (10.39.6).
x10.41 Olver (1997b, pp. 374{378). For (10.41.1),
(10.41.2) combine (10.19.1), (10.19.2) with
(10.27.6), (10.27.8).
x10.42 Watson (1944, pp. 511{513) and Olver (1997b,
p. 254).
x10.43 For (10.43.1){(10.43.3) di erentiate, apply
(10.29.2), and also (11.4.29) and (11.4.30) in the
case of (10.43.2). For (10.43.4) replace xbyix
in (10.22.11), (10.22.12) and use (10.27.6). For
(10.43.5) combine (10.22.39) and (10.22.40) by
means of (10.4.3) to obtain an expansion forR1
x(H(1)
0(t)=t)dt; then replace xbyixand use
(10.27.8). For (10.43.6){(10.43.10) di erentiate,
applyx10.29(i) and also verify the limiting behav-
ior asx!0 orx!1 . For (10.43.12) substitute
into (10.43.11) by means of (10.32.9) with = 0,
invert the order of integration and apply (5.2.1).
(10.43.13){(10.43.16) follow from (10.43.12), and
in the case of (10.43.16), (5.12.1). For (10.43.17)
see Bickley and Nayler (1935). For x10.43(iv) see
Watson (1944, pp. 388, 394{395, 410). For some
results it is necessary to use the connection formu-
las (10.27.6); for example, to obtain (10.43.23) set
a=ibin Watson (1944, p. 394, Eq. (4)). Equa-
tions (10.43.22) follow from Eq. (7) of Watson
(1944,x13.21). For (10.43.25) see Erd elyi et al.
(1953b, p. 51). For (10.43.29) combine (10.22.68),
References 285
(10.27.6), (10.27.10). In x10.43(v), for Condi-
tions (a) see Sneddon (1972, pp. 359{361). For
Conditions (b) see Lebedev et al. (1965, pp. 194{
196).
x10.44 For (10.44.1) combine (10.23.1) with (10.27.6)
or with (10.27.8). Equations (10.44.2) are
special cases of (10.23.1) and (10.44.1) with
=i. For (10.44.3) combine (10.23.2) and
(10.27.1) with (10.27.6) or with (10.27.8). For
(10.44.4){(10.44.6) combine (10.23.15){(10.23.17)
with (10.27.6), (10.27.8), and (10.4.3).
x10.45 Equations (10.45.5){(10.45.8) follow from
(10.25.2), (10.27.4), (10.31.2), (10.40.1), and
(10.40.2). The Wronskian (10.45.4) can be veri ed
from (1.13.5) and either (10.45.5) or (10.45.6){
(10.45.8) and their di erentiated forms.
x10.47 For (10.47.3){(10.47.9) use (10.2.3), (10.4.6),
(10.27.3). For x10.47(iii) usex10.52. For
(10.47.10){(10.47.13) use (10.4.3), (10.27.4),
(10.27.6), (10.27.8), and the de nitions (10.47.3){
(10.47.9). For (10.47.14){(10.47.16) use (10.11.1),
(10.11.2), (10.34.1), with m= 1 in each case, and
the de nitions (10.47.3){(10.47.9). For (10.47.17)
use (10.47.11) and (10.47.16).
x10.48 These graphs were produced at NIST.
x10.49 For (10.49.1){(10.49.7) observe that when =
n+1
2the asymptotic expansions (10.17.3){
(10.17.6) terminate, and as a consequence of the
error bounds ofx10.17(iv) they represent the left-
hand sides exactly. For (10.49.8){(10.49.13) use
the same method as for (10.49.1){(10.49.7), or
combine the results of x10.49(i) with (10.47.12)
and (10.47.13). For the rst of (10.49.14) combine
the second of (10.51.3), with n= 0 andm=n,
and the rst of (10.49.3). Similarly for the second
of (10.49.14) and also (10.49.15), (10.49.16). For
(10.49.18) observe that from (10.18.6), (10.47.3),
and (10.47.4), j2
n(z) +y2
n(z) = (=(2z))M2
n+1
2(z).
Then apply (10.18.17). To derive (10.49.20) com-
bine (10.47.12) and (10.49.18).
x10.50 That the Wronskians are constant multiples of
z 2follows from (1.13.5). The constants can be
found from the limiting forms (and their deriva-
tives) given inxx10.52(i) or 10.52(ii). For (10.50.3)
combine (10.50.1) with (10.51.1) and (10.51.2).
For (10.50.4) use (10.49.2){(10.49.5).
x10.51 For (10.51.1) and (10.51.2) combine (10.6.1)
and (10.6.2) with the de nitions (10.47.3){
(10.47.5). For (10.51.3) apply induction with theaid of (10.51.2). For (10.51.4) and (10.51.5) com-
bine (10.29.1) and (10.29.2) with the de nitions
(10.47.7) and (10.47.9). For (10.51.6) apply in-
duction with the aid of (10.51.5).
x10.52 For (10.52.1), (10.52.2) use x10.53. For
(10.52.3){(10.52.6) use (10.49.2), (10.49.4),
(10.49.6){(10.49.8), (10.49.10), and (10.49.12).
x10.53 Combine (10.2.2) and (10.25.2) with (10.47.3),
(10.47.4), and (10.47.7).
x10.54 Watson (1944, pp. 50 and 174{175). For
(10.54.1) use (10.9.4).
x10.56 To verify (10.56.1) and (10.56.2) show that
each side of both equations satis es the di eren-
tial equation (2 t z)(d2w
dt2) + (dw/dt) =zw
via the rst of (10.51.1) and (10.49.3), (10.49.5).
Then check the initial conditions at t= 0.
(10.56.3) and (10.56.4) follow from (10.56.1) and
(10.56.2) via (10.47.12); then (10.56.5) follows
from (10.47.11).
x10.57 For (10.57.1) use the di erentiated form of the
rst of (10.47.3).
x10.59 For (10.59.1) suppose rst b6= 0. The left-hand
side is 2iR1
0sin(bt)jn(t)dtor 2R1
0cos(bt)jn(t)dt
according as nis odd or even, see (10.47.14).
Next, apply (10.22.64) with a= 1,=1
2or
1
2, and subsequently replace 2 n+ 1 or 2nbyn.
ForJ( 1/2 ) (bt) andJn+( 1/2 ) (t) we have (10.16.1)
and (10.47.3); also the function 2F1is interpreted
as a Legendre polynomial for both odd and even
nvia (14.3.11), (14.3.13), and (14.3.14). When
b= 0, use (10.22.43), (10.47.3), and also Pn(0) =
( 1)1
2n 1
2
1
2n.
(1
2n)! or 0, according as the non-
negative integer nis even or odd; see (14.5.1) and
x5.5.
x10.60 For (10.60.1){(10.60.3) use (10.23.8) with =1
2
andC=Y;J;H(1); subsequently apply (10.47.12)
and (10.47.13) in the case of (10.60.3). For
(10.60.4) set C=Y,u=v=z,=
n 1
2, and =in (10.23.8). Then re-
fer to (10.47.3), (10.47.4), and also apply the
following results obtained from Table 18.6.1:
C( n 1
2)
k( 1) equals (2 n+ 1)!/(k!(2n+ 1 k)!)
whenk= 0;1;:::; 2n+ 1, and equals 0 when
k= 2n+ 2;2n+ 3;:::. For (10.60.5) use the same
procedure, but with C=J. (10.60.6) follows by
combining (10.60.4) and (10.60.5) with x10.47(iv).
For (10.60.7){(10.60.9) see Watson (1944, pp.368{
369). For (10.60.10) use Watson (1944, p. 370,
Eq. (9)) with =1
2, = , 0=1
2; also
286 Bessel Functions
Eq. (18.7.9). For (10.60.11) see Watson (1944,
p. 152). For (10.60.12) and (10.60.13) substitute
u=v=z, with = 0 and, into (10.60.2). For
(10.60.14) see Vavreck and Thompson (1984).
x10.61 For (10.61.3) set z=xe3i=4in (10.2.1).
(10.61.4) follows by taking real and imagi-
nary parts, and straightforward substitutions.
For (10.61.5){(10.61.8) see Whitehead (1911).
(10.61.11) and (10.61.12) follow from the termi-
nating forms of (10.67.1) and (10.67.2). Then
(10.61.9) and (10.61.10) follow from these re-
sults and the terminating forms of (10.67.3) and
(10.67.4). (Compare the derivation of the results
given inx10.49(i) from (10.17.3){(10.17.6).) The
version of (10.61.9){(10.61.10) given in Apelblat
(1991) contains two sign errors.
x10.62 These graphs were produced at NIST.
x10.63 For (10.63.1){(10.63.4) set z=xe3i=4in
(10.6.1) and (10.6.2). For (10.63.5){(10.63.7) set
a=xe3i=4. Then from (10.61.1) and (10.63.5)
J(a)J(a) =p,J0
(a)J0
(a) =s,J(a)J0
(a) =
e3i=4(r iq),J(a)J0
(a) =e 3i=4(r+iq).
Combine these results with (10.6.2) and eliminate
the derivatives. See also Petiau (1955, pp. 266{
267) (but this reference contains errors). For
the functions ker xand keixuse the second of
(10.61.2).
x10.65 Whitehead (1911). For (10.65.1), (10.65.2)
combine (10.2.2), (10.61.1). For (10.65.3){
(10.65.5) combine (10.31.1), (10.61.1), and
(10.61.2); see also Young and Kirk (1964, p. x).
x10.66 For (10.66.1) apply (10.23.1) with C=Jand
=e3i=4; also (10.44.1) with Z=Iand=ei=4. For (10.66.2) apply (10.23.2) with C=J,
=n,u= x,v=ix, and equate real and
imaginary parts.
x10.67 For (10.67.1){(10.67.8) combine (10.61.1),
(10.61.2), and their di erentiated forms with
(10.40.1){(10.40.4). To obtain the exponentially-
small terms in (10.67.3), (10.67.4), (10.67.7),
and (10.67.8), use the identity iI
xei=4
=
K
xe 3i=4
eiK
xei=4
, obtained from
(10.27.6) and (10.27.9). The nal sentence in
x10.67(i) is justi ed by error bounds obtained as in
x10.40(iii). For (10.67.9){(10.67.16), rst replace
the cos and sin functions in (10.67.1){(10.67.4)
by exponential functions by constructing the cor-
responding expansions for ber xibeixand
kerxikeixand discarding the exponentially-
small terms. Then set = 0 and apply straight-
forward manipulations.
x10.68 (10.68.3){(10.68.15) are derived from the de ni-
tionsx10.68(i), the di erential equation (10.61.3),
the re
ection formulas in x10.61(iv), and re-
currence relations in x10.63(i) by straightfor-
ward manipulations. For (10.68.16){(10.68.21)
combine (10.68.5) and (10.68.6) with (10.67.1){
(10.67.4), ignoring the exponentially-small terms
in (10.67.3) and (10.67.4). See also Whitehead
(1911) and Young and Kirk (1964, pp. xiv{xv).
x10.69 Combine the results given in xx10.41(ii) and
10.41(iii) with the de nitions (10.61.1) and
(10.61.2).
x10.70 Revert (10.68.18) and (10.68.21) ( x2.2).
x10.71 Di erentiate and use (10.63.2) and (10.68.5).
See also Young and Kirk (1964, pp. xvi{xvii).
Chapter 11
Struve and Related Functions
R. B. Paris1
Notation 288
11.1 Special Notation . . . . . . . . . . . . . 288
Struve and Modi ed Struve Functions 288
11.2 De nitions . . . . . . . . . . . . . . . . . 288
11.3 Graphics . . . . . . . . . . . . . . . . . . 289
11.4 Basic Properties . . . . . . . . . . . . . . 291
11.5 Integral Representations . . . . . . . . . 292
11.6 Asymptotic Expansions . . . . . . . . . . 293
11.7 Integrals and Sums . . . . . . . . . . . . 293
11.8 Analogs to Kelvin Functions . . . . . . . 294
Related Functions 294
11.9 Lommel Functions . . . . . . . . . . . . 29411.10 Anger{Weber Functions . . . . . . . . . 295
11.11 Asymptotic Expansions of Anger{Weber
Functions . . . . . . . . . . . . . . . . . 297
Applications 298
11.12 Physical Applications . . . . . . . . . . . 298
Computation 298
11.13 Methods of Computation . . . . . . . . . 299
11.14 Tables . . . . . . . . . . . . . . . . . . . 299
11.15 Approximations . . . . . . . . . . . . . . 300
11.16 Software . . . . . . . . . . . . . . . . . . 300
References 300
1Division of Mathematical Sciences, University of Abertay Dundee, Dundee, United Kingdom.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 12) by M. Abramowitz. The author
is indebted to Adri Olde Daalhuis for correcting a long-standing error in Eq. (11.10.23) in previous literature.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
287
288 Struve and Related Functions
Notation
11.1 Special Notation
(For other notation see pp. xiv and 873.)
xreal variable.
zcomplex variable.
real or complex order.
ninteger order.
knonnegative integer.
arbitrary small positive constant.
Unless indicated otherwise , primes denote deriva-
tives with respect to the argument. For the functions
J(z),Y(z),H(1)
(z),H(2)
(z),I(z), andK(z) see
xx10.2(ii), 10.25(ii).
The functions treated in this chapter are the Struve
functions H(z) and K(z), the modi ed Struve func-
tions L(z) and M(z), the Lommel functions s;(z)
andS;(z), the Anger function J(z), the Weber func-
tionE(z), and the associated Anger{Weber function
A(z).
Struve and Modi ed Struve
Functions
11.2 De nitions
11.2(i) Power-Series Expansions
11.2.1 H(z) = (1
2z)+11X
n=0( 1)n(1
2z)2n
n+3
2
n++3
2;
11.2.2L(z) = ie 1
2iH(iz)
= (1
2z)+11X
n=0(1
2z)2n
n+3
2
n++3
2:
Principal values correspond to principal values of
(1
2z)+1; comparex4.2(i).
The expansions (11.2.1) and (11.2.2) are absolutely
convergent for all nite values of z. The functions
z 1H(z) andz 1L(z) are entire functions of z
and.
11.2.3 H0(z) =2
z z3
1232+z5
123252
;
11.2.4 L0(z) =2
z+z3
1232+z5
123252+
:
11.2.5 K(z) =H(z) Y(z);
11.2.6 M(z) =L(z) I(z):Principal values of K(z) and M(z) correspond to
principal values of the functions on the right-hand sides
of (11.2.5) and (11.2.6).
Unless indicated otherwise ,H(z),K(z),L(z),
andM(z) assume their principal values throughout
this Handbook.
11.2(ii) Di erential Equations
Struve's Equation
11.2.7d2w
dz2+1
zdw
dz+
1 2
z2
w=(1
2z) 1
p
+1
2:
Particular solutions:
11.2.8 w=H(z);K(z):
Modi ed Struve's Equation
11.2.9d2w
dz2+1
zdw
dz
1 +2
z2
w=(1
2z) 1
p
+1
2:
Particular solutions:
11.2.10 w=L(z);M(z):
11.2(iii) Numerically Satisfactory Solutions
In this subsection AandBare arbitrary constants.
Whenz=x, 0< x <1, and<0, numerically
satisfactory general solutions of (11.2.7) are given by
11.2.11 w=H(x) +AJ(x) +BY(x);
11.2.12 w=K(x) +AJ(x) +BY(x):
(11.2.11) applies when xis bounded, and (11.2.12) ap-
plies when xis bounded away from the origin.
Whenz2Cand<0, numerically satisfactory
general solutions of (11.2.7) are given by
11.2.13w=H(z) +AJ(z) +BH(1)
(z);
11.2.14w=H(z) +AJ(z) +BH(2)
(z);
11.2.15w=K(z) +AH(1)
(z) +BH(2)
(z):
(11.2.13) applies when 0 phzandjzjis bounded.
(11.2.14) applies when phz0 andjzjis
bounded. (11.2.15) applies when jphzjandzis
bounded away from the origin.
When<0, numerically satisfactory general so-
lutions of (11.2.9) are given by
11.2.16w=L(z) +AK(z) +BI(z);
11.2.17w=M(z) +AK(z) +BI(z):
(11.2.16) applies when jphzj1
2withjzjbounded.
(11.2.17) applies when jphzj 1
2withzbounded
away from the origin.
11.3 Graphics 289
11.3 Graphics
11.3(i) Struve Functions
Figure 11.3.1 :H(x) for 0x12 and=
0;1
2;1;3
2;2;3.
Figure 11.3.2 :K(x) for 0< x16 and=
0;1
2;1;3
2;2;3.
Figure 11.3.3 :H(x) for 0x12 and=
3; 2; 3
2; 1; 1
2.
Figure 11.3.4 :K(x) for 0< x16 and=
4; 3; 2; 1;0. If= 1
2; 3
2, . . . , then K(x) is
identically zero.
Figure 11.3.5 :H(x) for 0x8 and 44.
Figure 11.3.6 :K(x) for 0x8 and 44.
290 Struve and Related Functions
Figure 11.3.7 :jH0(x+iy)jfor 8x8 and 3
y3.
Figure 11.3.8 :jK0(x+iy)j(principal value) for 8
x8 and 3y3. There is a cut along the
negative real axis.
For further graphics see http://dlmf.nist.gov/11.3.i .
11.3(ii) Modi ed Struve Functions
Figure 11.3.13 :L(x) for 0x < 4:38 and=
0;1
2;1;3
2;2;3.
Figure 11.3.14 :M(x) for 0x16 and=
0;1
2;1;3
2;2;3.
Figure 11.3.15 :L(x) for 0x < 4:25 and=
3; 2; 3
2; 1; 1
2.
Figure 11.3.16 :M(x) for 0< x16 and=
3; 2; 3
2; 1; 1
2.
11.4 Basic Properties 291
Figure 11.3.17 :L(x) for 0x5:6 and 44.
Figure 11.3.18 :M(x) for 0x8 and 44.
For further graphics see http://dlmf.nist.gov/11.3.ii .
11.4 Basic Properties
11.4(i) Half-Integer Orders
Forn= 0;1;2;:::,
11.4.1 Kn+1
2(z) =2
z1
2nX
m=0(2m)! 2 2m
m! (n m)!(1
2z)n 2m;
11.4.2
Ln+1
2(z) =I n 1
2(z)
2
z1
2nX
m=0( 1)m(2m)! 2 2m
m! (n m)!(1
2z)n 2m;
11.4.3 H n 1
2(z) = ( 1)nJn+1
2(z);
11.4.4 L n 1
2(z) =In+1
2(z):
11.4.5 H1
2(z) =2
z1
2
(1 cosz);
11.4.6 H 1
2(z) =2
z1
2
sinz;
11.4.7 L1
2(z) =2
z1
2
(coshz 1);
11.4.8 L 1
2(z) =2
z1
2
sinhz;
11.4.9
H3
2(z) =z
21
2
1 +2
z2
2
z1
2
sinz+cosz
z
;11.4.10 H 3
2(z) =2
z1
2
cosz sinz
z
;
11.4.11
L3
2(z) = z
21
2
1 2
z2
+2
z1
2
sinhz coshz
z
;
11.4.12 L 3
2(z) =2
z1
2
coshz sinhz
z
:
11.4(ii) Inequalities
11.4.13 H(x)0,x>0,1
2.
11.4.14
H(z) =2(1
2z)+1
p
+3
2(1 +#),6= 3
2; 5
2; 7
2;:::,
where
11.4.15j#j<2
3exp1
4jzj2
j0+3
2j 1
;
andj0+3
2jis the smallest of the numbers j+3
2j,j+5
2j,
j+9
2j;:::.
11.4(iii) Analytic Continuation
11.4.16 H
zemi
=emi(+1)H(z),m2Z,
11.4.17 L
zemi
=emi(+1)L(z),m2Z.
292 Struve and Related Functions
11.4(iv) Expansions in Series of Bessel
Functions
11.4.18
H(z) =4
1=2
+1
2
1X
k=0(2k++ 1) (k++ 1)
k!(2k+ 1)(2k+ 2+ 1)J2k++1(z),
6= 1; 2; 3;:::,
11.4.19 H(z) =z
21=21X
k=0(1
2z)k
k!(k+1
2)Jk++1
2(z);
11.4.20 H(z) =(1
2z)+1
2
+1
21X
k=0(1
2z)k
k!(k++1
2)Jk+1
2(z);
11.4.21
H0(z) =4
1X
k=0J2k+1(z)
2k+ 1= 21X
k=0( 1)kJ2
k+1
2 1
2z
;
11.4.22H1(z) =2
(1 J0(z)) +4
1X
k=1J2k(z)
4k2 1
= 41X
k=0J2k+1
2 1
2z
J2k+3
2 1
2z
:
For these and further results see Luke (1969b, x9.4.5),
andx10.23(iii).
11.4(v) Recurrence Relations and Derivatives
11.4.23
H 1(z) +H+1(z) =2
zH(z) +(1
2z)
p
+3
2;
11.4.24
H 1(z) H+1(z) = 2H0
(z) (1
2z)
p
+3
2;
11.4.25
L 1(z) L+1(z) =2
zL(z) +(1
2z)
p
+3
2;
11.4.26
L 1(z) +L+1(z) = 2L0
(z) (1
2z)
p
+3
2:
11.4.27d
dz(zH(z)) =zH 1(z);
11.4.28d
dz
z H(z)
=2
p
+3
2 z H+1(z);
11.4.29d
dz(zL(z)) =zL 1(z);
11.4.30d
dz
z L(z)
=2
p
+3
2+z L+1(z):
11.4.31
H m(z) =zm 1
zd
dzm
(zH(z)),m= 1;2;3;:::,whereH(z) denotes either H(z) orL(z).
11.4.32 H0
0(z) =2
H1(z);d
dz(zH1(z)) =zH0(z);
11.4.33 L0
0(z) =2
+L1(z);d
dz(zL1(z)) =zL0(z):
11.4(vi) Derivatives with Respect to Order
For derivatives with respect to the order , see Apelblat
(1989) and Brychkov and Geddes (2005).
11.4(vii) Zeros
For properties of zeros of H(x) see Steinig (1970).
For asymptotic expansions of zeros of H0(x) see
MacLeod (2002a).
11.5 Integral Representations
11.5(i) Integrals Along the Real Line
11.5.1
H(z) =2(1
2z)
p
+1
2Z1
0(1 t2) 1
2sin(zt)dt
=2(1
2z)
p
+1
2Z=2
0sin(zcos)(sin)2d,
< > 1
2,
11.5.2
K(z) =2(1
2z)
p
+1
2Z1
0e zt(1 +t2) 1
2dt,<z>0,
11.5.3 K0(z) =2
Z1
0e zsinhtdt,<z>0,
11.5.4M(z) = 2(1
2z)
p
+1
2Z1
0e zt(1 t2) 1
2dt,
< > 1
2,
11.5.5 M0(z) = 2
Z=2
0e zcosd;
11.5.6
L(z) =2(1
2z)
p
+1
2Z=2
0sinh(zcos)(sin)2d,
< > 1
2,
11.5.7I (x) L(x)
=2(1
2x)
p
+1
2Z1
0(1 +t2) 1
2sin(xt)dt,
x>0,< <1
2:
11.5(ii) Contour Integrals
For loop-integral versions of (11.5.1), (11.5.2), (11.5.4),
and (11.5.7) see Babister (1967, xx3.3 and 3.14).
11.6 Asymptotic Expansions 293
Mellin{Barnes Integrals
11.5.8
(1
2x) 1H(x)
= 1
2iZi1
i1csc(s)
3
2+s
3
2++s(1
4x2)sds,
x>0,< > 1,
11.5.9
(1
2z) 1L(z)
=1
2iZ(0+)
1csc(s)
3
2+s
3
2++s( 1
4z2)sds:
In (11.5.8) and (11.5.9) the path of integration separates
the poles of the integrand at s= 0;1;2;::: from those
ats= 1; 2; 3;:::.
11.5(iii) Compendia
For further integral representations see Babister (1967,
xx3.3, 3.14), Erd elyi et al. (1954a,xx5.17, 15.3), Mag-
nuset al. (1966, p. 114), Oberhettinger (1972), Ober-
hettinger (1974,x2.7), Oberhettinger and Badii (1973,
x2.14), and Watson (1944, pp. 330, 374, and 426).
11.6 Asymptotic Expansions
11.6(i) Largejzj, Fixed
11.6.1
K(z)1
1X
k=0
k+1
2
(1
2z) 2k 1
+1
2 k ,jphzj ,
whereis an arbitrary small positive constant. If the
series on the right-hand side of (11.6.1) is truncated
afterm(0) terms, then the remainder term Rm(z) is
O
z 2m 1
. Ifis real,zis positive, and m+1
2 0,
thenRm(z) is of the same sign and numerically less than
the rst neglected term.
11.6.2M(z)1
1X
k=0( 1)k+1
k+1
2
(1
2z) 2k 1
+1
2 k ,
jphzj1
2 .
For re-expansions of the remainder terms in (11.6.1)
and (11.6.2), see Dingle (1973, p. 445).
For the corresponding expansions for H(z) and
L(z) combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6),
(10.17.4), and (10.40.1).
11.6.3Zz
0K0(t)dt 2
(ln(2z) +
)
2
1X
k=1( 1)k+1(2k)!(2k 1)!
(k!)2(2z)2k,jphzj ,11.6.4Zz
0M0(t)dt+2
(ln(2z) +
)
2
1X
k=1(2k)!(2k 1)!
(k!)2(2z)2k,jphzj1
2 ,
where
is Euler's constant ( x5.2(ii)).
11.6(ii) Largejj, Fixedz
11.6.5 H(z);L(z)z
p
2ez
2
,jphj .
More fully, the series (11.2.1) and (11.2.2) can be re-
garded as generalized asymptotic expansions ( x2.1(v)).
11.6(iii) Largejj, Fixedz=
For xed(>1)
11.6.6
K()(1
2) 1
p
+1
21X
k=0k!ck()
k,jphj1
2 ,
and for xed (>0)
11.6.7M() (1
2) 1
p
+1
21X
k=0k!ck(i)
k,
jphj1
2 :
Here
11.6.8c0() = 1; c 1() = 2 2;
c2() = 6 4 1
2 2; c 3() = 20 6 4 4;
c4() = 70 8 45
2 6+3
8 4;
and for higher coecients ck() see Dingle (1973,
p. 203).
For the corresponding result for H() use (11.2.5)
and (10.19.6). See also Watson (1944, p. 336).
For xed(>0)
11.6.9 L()I(),jphj1
2 ,
and for an estimate of the relative error in this approx-
imation see Watson (1944, p. 336).
11.7 Integrals and Sums
11.7(i) Inde nite Integrals
11.7.1Z
zH 1(z)dz=zH(z);
11.7.2Z
z H+1(z)dz= z H(z) +2 zp
+3
2;
11.7.3Z
zL 1(z)dz=zL(z);
294 Struve and Related Functions
11.7.4Z
z L+1(z)dz=z L(z) 2 zp
+3
2:
If
11.7.5 f(z) =Zz
0tH(t)dt;
then
11.7.6f+1(z) = (2+ 1)f(z) z+1H(z)
+(1
2z2)+1
(+ 1)p
+3
2,< > 1.
11.7(ii) De nite Integrals
11.7.7Z=2
0H(zsin)(sin)+1
(cos)2d
=2
p 1
2
z 1(1 cosz), 3
2<< <1
2,
11.7.8Z1
0H0(t)dt
t=1
2;Z1
0H1(t)dt
t2=1
4;
11.7.9Z1
0H(t)dt= cot 1
2
, 2<< <0,
11.7.10Z1
0t 1H(t)dt=
2+1 (+ 1),< > 3
2,
11.7.11Z1
0t 1H(t)dt= 1
2
2 1tan 1
2
1
2+ 1 ,
j<j<1,< >< 3
2,
11.7.12Z1
0t H(t)H(t)dt
=p (+)
2+
++1
2
+1
2
+1
2,
<(+)>0.
For other integrals involving products of Struve func-
tions see Zanovello (1978, 1995). For integrals involving
products of M(t) functions, see Paris and Sy (1983,
Appendix).
11.7(iii) Laplace Transforms
The following Laplace transforms of H(t) require<a>
0 for convergence, while those of L(t) require<a>1.
11.7.13Z1
0e atH0(t)dt=2
p
1 +a2ln
1 +p
1 +a2
a!
;
11.7.14Z1
0e atH1(t)dt=2
a 2a
p
1 +a2ln
1 +p
1 +a2
a!
;
11.7.15Z1
0e atL0(t)dt=2
p
a2 1arcsin1
a
;
11.7.16Z1
0e atL1(t)dt=2a
p
a2 1arctan1p
a2 1
2
a:11.7(iv) Integrals with Respect to Order
For integrals of H(x) and L(x) with respect to the
order, see Apelblat (1989).
11.7(v) Compendia
For further integrals see Apelblat (1983, x12.16), Babis-
ter (1967, Chapter 3), Erd elyi et al. (1954a,xx4.19,
6.8, 8.15, 9.4, 10.3, 11.3, and 15.3), Luke (1962,
Chapters 9, 11), Gradshteyn and Ryzhik (2000, x6.8),
Marichev (1983, pp. 192{193 and 215{216), Oberhet-
tinger (1972), Oberhettinger (1974, x1.12), Oberhet-
tinger (1990,xx1.21 and 2.21), Oberhettinger and Badii
(1973,x1.16), Prudnikov et al. (1990,xx1.4 and 2.7),
Prudnikov et al. (1992a,x3.17), and Prudnikov et al.
(1992b,x3.17).
For sums of Struve functions see Hansen (1975,
p. 456) and Prudnikov et al. (1990,x6.4.1).
11.8 Analogs to Kelvin Functions
For properties of Struve functions of argument xe3i=4
see McLachlan and Meyers (1936).
Related Functions
11.9 Lommel Functions
11.9(i) De nitions
The inhomogeneous Bessel di erential equation
11.9.1d2w
dz2+1
zdw
dz+
1 2
z2
w=z 1
can be regarded as a generalization of (11.2.7). Pro-
vided that 6= 1; 3; 5;:::, (11.9.1) has the
general solution
11.9.2w=s;(z) +AJ(z) +BY(z);
whereA,Bare arbitrary constants, s;(z) is the Lom-
mel function de ned by
11.9.3s;(z) =z+11X
k=0( 1)kz2k
ak+1(;);
and
11.9.4
ak(;) =kY
m=1
(+ 2m 1)2 2
,k= 0;1;2;:::.
11.10 Anger{Weber Functions 295
Another solution of (11.9.1) that is de ned for all values of andisS;(z), where
11.9.5S;(z) =s;(z) + 2 1 1
2+1
2+1
2
1
2 1
2+1
2
sin 1
2( )
J(z) cos 1
2( )
Y(z)
;
the right-hand side being replaced by its limiting form when is an odd negative integer.
Re
ection Formulas
11.9.6 s; (z) =s;(z); S; (z) =S;(z):
For the foregoing results and further information see Watson (1944, xx10.7{10.73) and Babister (1967, x3.16).
11.9(ii) Expansions in Series of Bessel Functions
When6= 1; 2; 3;:::,
11.9.7 s;(z) = 2+11X
k=0(2k++ 1) (k++ 1)
k!(2k+ + 1)(2k+++ 1)J2k++1(z);
11.9.8 s;(z) = 2(+ 1)=2 1
2+1
2+1
2
z(+1 )=21X
k=0(1
2z)k
k!(2k+ + 1)Jk+1
2(++1)(z):
For these and further results see Luke (1969b, x9.4.5).
11.9(iii) Asymptotic Expansion
For xedand,
11.9.9S;(z)z 11X
k=0( 1)kak( ;)z 2k,
z!1 ,jphzj (<).
Forak(;) see (11.9.4). If either of equals an odd
positive integer, then the right-hand side of (11.9.9) ter-
minates and represents S;(z) exactly.
For uniform asymptotic expansions, for large and
xed= 1;0;1;2;:::, of solutions of the inhomoge-
neous modi ed Bessel di erential equation that corre-
sponds to (11.9.1) see Olver (1997b, pp. 388{390).
11.9(iv) References
For further information on Lommel functions see Wat-
son (1944,xx10.7{10.75) and Babister (1967, Chap-
ter 3). For descriptive properties of s;(x) see Steinig
(1972).
For collections of integral representations and inte-
grals see Apelblat (1983, x12.17), Babister (1967, p. 85),
Erd elyi et al. (1954a,xx4.19 and 5.17), Gradshteyn and
Ryzhik (2000,x6.86), Marichev (1983, p. 193), Ober-
hettinger (1972, pp. 127{128, 168{169, and 188{189),
Oberhettinger (1974, xx1.12 and 2.7), Oberhettinger
(1990, pp. 105{106 and 191{192), Oberhettinger and
Badii (1973,x2.14), Prudnikov et al. (1990,xx1.6 and
2.9), Prudnikov et al. (1992a,x3.34), and Prudnikov
et al. (1992b,x3.32).11.10 Anger{Weber Functions
11.10(i) De nitions
The Anger function J(z) and Weber function E(z)
are de ned by
11.10.1 J(z) =1
Z
0cos( zsin)d;
11.10.2 E(z) =1
Z
0sin( zsin)d:
Each is an entire function of zand. Also,
11.10.3
1
Z2
0cos( zsin)d= (1 + cos(2 ))J(z)
+ sin(2)E(z):
The associated Anger{Weber function A(z) is de-
ned by
11.10.4 A(z) =1
Z1
0e t zsinhtdt,<z>0.
(11.10.4) also applies when <z= 0 and< >0.
11.10(ii) Di erential Equations
The Anger and Weber functions satisfy the inhomoge-
neous Bessel di erential equation
11.10.5d2w
dz2+1
zdw
dz+
1 2
z2
w=f(;z);
where
11.10.6 f(;z) =(z )
z2sin(),w=J(z),
or
11.10.7
f(;z) = 1
z2(z++ (z ) cos()),w=E(z).
296 Struve and Related Functions
11.10(iii) Maclaurin Series
11.10.8 J(z) = cos 1
2
S1(;z) + sin 1
2
S2(;z);
11.10.9 E(z) = sin 1
2
S1(;z) cos 1
2
S2(;z);
where
11.10.10S1(;z) =1X
k=0( 1)k(1
2z)2k
k+1
2+ 1
k 1
2+1;11.10.11S2(;z) =1X
k=0( 1)k(1
2z)2k+1
k+1
2+3
2
k 1
2+3
2:
These expansions converge absolutely for all nite val-
ues ofz.
11.10(iv) Graphics
Figure 11.10.1 : Anger function J(x) for 8x8
and= 0;1
2;1;3
2.
Figure 11.10.2 : Weber function E(x) for 8x8
and= 0;1
2;1;3
2.
Figure 11.10.3 : Anger function J(x) for 10x10
and 05.
Figure 11.10.4 : Weber function E(x) for 10x10
and 05.
11.10(v) Interrelations
11.10.12 J( z) =J (z);E( z) = E (z):
11.10.13 sin()J(z) = cos()E(z) E (z);
11.10.14 sin()E(z) =J (z) cos()J(z):
11.10.15 J(z) =J(z) + sin()A(z);
11.10.16 E(z) = Y(z) cos()A(z) A (z):11.10(vi) Relations to Other Functions
11.10.17 J(z) =sin()
(s0;(z) s 1;(z));
11.10.18E(z) = 1
(1 + cos())s0;(z)
(1 cos())s 1;(z):
11.11 Asymptotic Expansions of Anger{Weber Functions 297
11.10.19J 1
2(z) =E1
2(z)
= (1
2z) 1
2(A+( ) cosz A ( ) sinz);
11.10.20J1
2(z) = E 1
2(z)
= (1
2z) 1
2(A+( ) sinz+A ( ) cosz);
where
11.10.21A( ) =C( )S( ); = (2z=)1
2:
For the Fresnel integrals CandSseex7.2(iii).
Forn= 1;2;3;:::,
11.10.22
En(z) = Hn(z) +1
m1X
k=0
k+1
2
n+1
2 k(1
2z)n 2k 1;
and
11.10.23
E n(z) = H n(z)
+( 1)n+1
m2X
k=0
n k 1
2
k+3
2(1
2z) n+2k+1;
where
11.10.24m1=1
2n 1
2
; m 2=1
2n 3
2
:
11.10(vii) Special Values
J(0) =sin()
;E(0) =1 cos()
: 11.10.25
E0(z) = H0(z);E1(z) =2
H1(z): 11.10.26
11.10.27@
@J(z)
=0=1
2H0(z);
11.10.28@
@E(z)
=0=1
2J0(z):
11.10.29 Jn(z) =Jn(z), n2Z.
11.10(viii) Expansions in Series of Products of
Bessel Functions
11.10.30
J(z) =
2 sin 1
21X
k=0( 1)kJk 1
2+1
2 1
2z
Jk+1
2+1
2 1
2z
+ 2 cos 1
21X0
k=0( 1)kJk 1
2 1
2z
Jk+1
2 1
2z
;
11.10.31
E(z) =
2 cos 1
21X
k=0( 1)kJk 1
2+1
2 1
2z
Jk+1
2+1
2 1
2z
+ 2 sin 1
21X0
k=0( 1)kJk 1
2 1
2z
Jk+1
2 1
2z
;
where the prime on the second summation symbols
means that the rst term is to be halved.11.10(ix) Recurrence Relations and Derivatives
11.10.32 J 1(z) +J+1(z) =2
zJ(z) 2
zsin();
11.10.33
E 1(z) +E+1(z) =2
zE(z) 2
z(1 cos()):
11.10.34 2J0
(z) =J 1(z) J+1(z);
11.10.35 2E0
(z) =E 1(z) E+1(z);
11.10.36zJ0
(z)J(z) =zJ1(z)sin()
;
11.10.37
zE0
(z)E(z) =zE1(z)(1 cos())
:
11.10(x) Integrals and Sums
For collections of integral representations and integrals
see Erd elyi et al. (1954a,xx4.19 and 5.17), Marichev
(1983, pp. 194{195 and 214{215), Oberhettinger (1972,
p. 128), Oberhettinger (1974, xx1.12 and 2.7), Oberhet-
tinger (1990, pp. 105 and 189{190), Prudnikov et al.
(1990,xx1.5 and 2.8), Prudnikov et al. (1992a,x3.18),
Prudnikov et al. (1992b,x3.18), and Zanovello (1977).
For sums see Hansen (1975, pp. 456{457) and Prud-
nikov et al. (1990,xx6.4.2{6.4.3).
11.11 Asymptotic Expansions of
Anger{Weber Functions
11.11(i) Largejzj, Fixed
LetF0() =G0() = 1, and for k= 1;2;3;:::,
11.11.1
Fk() = (2 12)(2 32)(2 (2k 1)2);
Gk() = (2 22)(2 42)(2 (2k)2):
Then asz!1 injphzj (<)
11.11.2
J(z)J(z)
+sin()
z 1X
k=0Fk()
z2k
z1X
k=0Gk()
z2k!
;
11.11.3E(z) Y(z) 1 + cos()
z1X
k=0Fk()
z2k
(1 cos())
z21X
k=0Gk()
z2k;
11.11.4 A(z)1
z1X
k=0Fk()
z2k
z21X
k=0Gk()
z2k:
298 Struve and Related Functions
11.11(ii) Largejj, Fixedz
Ifzis xed, and !1 injphjin such a way
thatis bounded away from the set of all integers, then
11.11.5 J(z) =sin()
1 z
2 1+O1
2
;
11.11.6E(z) =2
sin2 1
2
+z
2 1cos2 1
2
+O1
2
:
If=n(2Z), then (11.10.29) applies for Jn(z), and
11.11.7E2n(z)2z
(4n2 1);
E2n+1(z)2
(2n+ 1),n!1 .
11.11(iii) Large , Fixedz=
For xed(>0),
11.11.8A()1
1X
k=0(2k)!ak()
2k+1,
!1 ,jphj (<),
where
11.11.9a0=1
1 +; a 1=
2(1 +)4;
a2=92
24(1 +)7; a 3= 2253 542+
720(1 +)10:
For xed(>1),
11.11.10 A () 1
1X
k=0(2k)!ak( )
2k+1,!+1.
For xed, 0<< 1,
11.11.11
A ()r
2
e 1X
k=0(1
2)kbk()
k,!+1,
where
11.11.12=p
1 2 ln
1 +p
1 2
!
;
and
11.11.13b0() =1
(1 2)1=4,b1() =2 + 32
12(1 2)7=4;
b2() =4 + 3002+ 814
864(1 2)13=4.
In particular, as !+1,
11.11.14 A ()1
( 1),>1,11.11.15
A ()2
1=2
1 +p
1 2
!
e p
1 2
(1 2)1=4,
0<< 1.
Also, as!+1,
11.11.16 A ()24=3
37=6 2
3
1=3;
and
11.11.17
A
+a1=3
= 21=3 1=3Hi
21=3a
+O
1
;
uniformly for bounded real values of a. For the Scorer
function Hi seex9.12(i).
All of (11.11.10){(11.11.17) can be regarded as spe-
cial cases of two asymptotic expansions given in Olver
(1997b, pp. 352{357) for A () as!+1, one be-
ing uniform for 1, whereagain denotes an
arbitrary small positive constant, and the other being
uniform for 1 <1. (Note that Olver's de nition
ofA(z) omits the factor 1 =in (11.10.4).) See also
Watson (1944,x10.15).
Lastly, corresponding asymptotic approximations
and expansions for J() and E() follow from
(11.10.15) and (11.10.16) and the corresponding asymp-
totic expansions for the Bessel functions J(z) and
Y(z); seex10.19(ii). In particular,
11.11.18 J()21=3
32=3 2
3
1=3,!+1,
11.11.19 E()21=3
37=6 2
3
1=3,!+1.
Applications
11.12 Physical Applications
Applications of Struve functions occur in water-wave
and surface-wave problems (Hirata (1975) and Ahmadi
and Widnall (1985)), unsteady aerodynamics (Shaw
(1985) and Wehausen and Laitone (1960)), distribu-
tion of
uid pressure over a vibrating disk (McLachlan
(1934)), resistive MHD instability theory (Paris and Sy
(1983)), and optical di raction (Levine and Schwinger
(1948)). More recently Struve functions have appeared
in many particle quantum dynamical studies of spin
decoherence (Shao and H anggi (1998)) and nanotubes
(Pedersen (2003)).
Computation 299
Computation
11.13 Methods of Computation
11.13(i) Introduction
Subsequent subsections treat the computation of Struve
functions. The treatment of Lommel and Anger{Weber
functions is similar. For a review of methods for the
computation of H(z) see Zanovello (1975).
11.13(ii) Series Expansions
Although the power-series expansions (11.2.1) and
(11.2.2), and the Bessel-function expansions of x11.4(iv)
converge for all nite values of z, they are cumbersome
to use whenjzjis large owing to slowness of convergence
and cancellation. For large jzjand/orjjthe asymptotic
expansions given in x11.6 should be used instead.
11.13(iii) Quadrature
For numerical purposes the most convenient of the rep-
resentations given in x11.5, at least for real variables,
include the integrals (11.5.2){(11.5.5) for K(z) and
M(z). Subsequently H(z) and L(z) are obtainable
via (11.2.5) and (11.2.6). Other integrals that appear
inx11.5(i) have highly oscillatory integrands unless zis
small.
For complex variables the methods described in
xx3.5(viii) and 3.5(ix) are available.
11.13(iv) Di erential Equations
A comprehensive approach is to integrate the de n-
ing inhomogeneous di erential equations (11.2.7) and
(11.2.9) numerically, using methods described in x3.7.
To insure stability the integration path must be chosen
so that as we proceed along it the wanted solution grows
in magnitude at least as rapidly as the complementary
solutions.
Suppose0 andxis real and positive. Then from
the limiting forms for small argument ( xx11.2(i), 10.7(i),
10.30(i)), limiting forms for large argument ( xx11.6(i),
10.7(ii), 10.30(ii)), and the connection formulas (11.2.5)
and (11.2.6), it is seen that H(x) and L(x) can be
computed in a stable manner by integrating forwards,
that is, from the origin toward in nity. The solution
K(x) needs to be integrated backwards for small x,
and either forwards or backwards for large xdepending
whether or not exceeds1
2. For M(x) both forward
and backward integration are unstable, and boundary-
value methods are required ( x3.7(iii)).11.13(v) Di erence Equations
Sequences of values of H(z) and L(z), withz xed,
can be computed by application of the inhomogeneous
di erence equations (11.4.23) and (11.4.25). There are
similar problems to those described in x11.13(iv) con-
cerning stability. In consequence forward recurrence,
backward recurrence, or boundary-value methods may
be necessary. See x3.6 for implementation of these meth-
ods, and with the Weber function En(x) as an example.
11.14 Tables
11.14(i) Introduction
For tables before 1961 see Fletcher et al. (1962) and
Lebedev and Fedorova (1960). Tables listed in these
Indices are omitted from the subsections that follow.
11.14(ii) Struve Functions
Abramowitz and Stegun (1964, Chapter 12) tab-
ulates Hn(x),Hn(x) Yn(x), andIn(x) Ln(x)
forn= 0;1 andx= 0(:1)5,x 1= 0(:01)0:2 to 6D
or 7D.
Agrest et al. (1982) tabulates Hn(x)
ande xLn(x) forn= 0;1 andx=
0(:001)5(:005)15(:01)100 to 11D.
Barrett (1964) tabulates Ln(x) forn= 0;1 and
x= 0:2(:005)4(:05)10(:1)19:2 to 5 or 6S, x=
6(:25)59:5(:5)100 to 2S.
Zanovello (1975) tabulates Hn(x) forn= 4(1)15
andx= 0:5(:5)26 to 8D or 9S.
Zhang and Jin (1996) tabulates Hn(x) and Ln(x)
forn= 4(1)3 andx= 0(1)20 to 8D or 7S.
11.14(iii) Integrals
Abramowitz and Stegun (1964, Chap-
ter 12) tabulatesRx
0(I0(t) L0(t))dtand
(2=)R1
xt 1H0(t)dtforx= 0(:1)5 to 5D or
7D;Rx
0(H0(t) Y0(t))dt (2=) lnx,Rx
0(I0(t)
L0(t))dt (2=) lnx, andR1
xt 1(H0(t) Y0(t))dt
forx 1= 0(:01)0:2 to 6D.
Agrest et al. (1982) tabulatesRx
0H0(t)dtand
e xRx
0L0(t)dtforx= 0(:001)5(:005)15(:01)100
to 11D.
11.14(iv) Anger{Weber Functions
Bernard and Ishimaru (1962) tabulates J(x) and
E(x) for= 10(:1)10 andx= 0(:1)10 to 5D.
Jahnke and Emde (1945) tabulates En(x) forn=
1;2 andx= 0(:01)14:99 to 4D.
300 Struve and Related Functions
11.14(v) Incomplete Functions
Agrest and Maksimov (1971, Chapter 11) de nes
incomplete Struve, Anger, and Weber functions
and includes tables of an incomplete Struve func-
tionHn(x; ) forn= 0;1,x= 0(:2)10, and
= 0(:2)1:4;1
2, together with surface plots.
11.15 Approximations
11.15(i) Expansions in Chebyshev Series
Luke (1975, pp. 416{421) gives Chebyshev-series
expansions for Hn(x),Ln(x), 0 jxj 8,
andHn(x) Yn(x),x8, forn= 0;1;Rx
0t mH0(t)dt,Rx
0t mL0(t)dt, 0 jxj 8,
m= 0;1 andRx
0(H0(t) Y0(t))dt,R1
xt 1(H0(t)
Y0(t))dt,x8; the coecients are to 20D.
MacLeod (1993) gives Chebyshev-series expan-
sions for L0(x),L1(x), 0x16, andI0(x)
L0(x),I1(x) L1(x),x16; the coecients are
to 20D.
11.15(ii) Rational and Polynomial
Approximations
Newman (1984) gives polynomial approximations
forHn(x) forn= 0;1, 0x3, and rational-
fraction approximations for Hn(x) Yn(x) for
n= 0;1,x3. The maximum errors do not
exceed 1:210 8for the former and 2 :510 8for
the latter.
11.16 Software
Seehttp://dlmf.nist.gov/11.16 .
References
General References
The main references used in writing this chapter are
Babister (1967, Chapter 3) and Watson (1944, Chapter
10). For additional bibliographic reading see Erd elyi
et al. (1953b,x7.5), Luke (1969b), Luke (1975, Chapter
10), Magnus et al. (1966,x3.10), and Olver (1997b).Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x11.2 Watson (1944, pp. 328{329), Olver (1997b,
pp. 274{277). The notation M(z) is new and
this function has been introduced to play a simi-
lar role to L(z) that K(z) does to H(z). For
x11.2(iii) seex2.7(iv) and Olver (1997b, pp. 274{
277). The last reference restricts (11.2.15) to the
sectorjphzj1
2, and instead covers the sector
1
2phz3
2with another set of solutions.
(Similarly for the conjugate sector 3
2phz
1
2.)
x11.3 The graphics were produced at NIST.
x11.4 Watson (1944,xx10.4, 10.45). (11.4.1), (11.4.2)
both follow from Erd elyi et al. (1953b, p. 39,
Eq. (64)): for (11.4.1) set =zand
use (11.2.5); for (11.4.2) replace Yn+1
2() by
( 1)n+1J n 1
2() in consequence of (10.2.3),
then set=izand use (11.2.2), (10.27.6). For
(11.4.3), (11.4.4) see Babister (1967, pp. 64, 75).
For (11.4.5){(11.4.12) combine (11.4.3), (11.4.4)
with (10.47.3), (10.47.7), x10.49(i),x10.49(ii),
(11.4.23), (11.4.25). (11.4.16), (11.4.17) follow
from (11.2.1), (11.2.2). For (11.4.31) see Babis-
ter (1967, pp. 60, 74).
x11.5 For (11.5.1){(11.5.3), (11.5.6), and (11.5.7) see
Watson (1944, pp. 328,331,332), together with
(11.2.5) in the case of (11.5.2) and (11.5.3), and
(11.2.2) in the case of (11.5.6). For (11.5.4) and
(11.5.5) see Babister (1967, Eq. (3.102)) and col-
lapse the integration path on to the interval [0 ;1].
For (11.5.8) see Babister (1967, x3.7) with modi-
ed convergence conditions. For (11.5.9) deform
the integration path in (11.5.8) into a loop and
use (11.2.2).
x11.6 For (11.6.1) apply Watson's lemma ( x2.4(i))
to (11.5.2), or combine Watson (1944, p. 333,
Eq. (2)) with (11.2.5). See also the subsequent
text in this reference, and Olver (1997b, p. 277,
Ex. 15.5). For (11.6.2) convert (11.5.4) to a loop
integralR(1+)
0to remove the restriction < > 1
2,
extend the loop to pass through the point t=1
on the positive real axis, then apply Laplace's
method (x2.4(iii)) to each of the two integrals
with paths from t= 0 tot=1, one pass-
ing below t= 1 and the other passing above
t= 1. For (11.6.3) write the integrals over the
References 301
intervals [0;1) and [z;1); use (11.6.1) with the
rst term extracted, and a limiting procedure
on the integral over [0 ;1). For (11.6.4) replace
zbyizin (11.6.3) and apply (11.2.5), (11.2.6),
(10.27.11). For (11.6.5) apply (5.11.7) to (11.2.1),
(11.2.2). For (11.6.6) and (11.6.9) see Watson
(1944,x10.43): a similar method can be used for
(11.6.7), starting from (11.5.4).
x11.7 For (11.7.1){(11.7.6) use x11.4(v). For (11.7.7){
(11.7.12) see Babister (1967, pp. 68, 71{72), Wat-
son (1944, pp. 392, 397). For (11.7.13){(11.7.16)
see Babister (1967, xx3.13, 3.15).
x11.9 Watson (1944,x10.75).
x11.10 Watson (1944, pp. 308{312). The nota-
tionA(z), without the factor 1 =, was intro-
duced in Olver (1997b, p. 84). For (11.10.12)
use (11.10.1), (11.10.2). For (11.10.16) com-
bine (11.10.14), (11.10.15), and (10.2.3). For
(11.10.19), (11.10.20), use (11.10.8){(11.10.11)
with=1
2and identify the resulting sums
with those associated with the right-hand sides
via (7.6.5), (7.6.7). For (11.10.22), (11.10.23)
see Watson (1944, pp. 336{337) or Erd elyi et al.(1953b, p. 40). The upper summation limit
in (11.10.23) is given incorrectly in Watson
(1944, p. 337), and this error is reproduced in
Erd elyi et al. (1953b), as well as in later print-
ings of Abramowitz and Stegun (1964, Chapter
12)|earlier printings contained a di erent error.
(11.10.23) can be derived by combining (11.2.1)
with (11.10.12), (11.10.22). For (11.10.25)
use (11.10.1) and (11.10.2). For (11.10.26)
use (11.10.22). (11.10.27) and (11.10.28) can
be obtained by di erentiation of (11.10.1) and
(11.10.2), followed by straightforward manipula-
tion of the integrals and comparison with (11.5.1)
and (11.10.1). For (11.10.29) use (11.10.1) and
(10.9.2). For (11.10.30){(11.10.31) see Luke
(1969b, p. 55).
The graphics were produced at NIST.
x11.11 Watson (1944, xx10.14{10.15). (11.11.2),
(11.11.3) follow from (11.10.15), (11.10.16).
(11.11.5), (11.11.6) follow from (11.10.8){
(11.10.11). Eqs. (11.11.7) follow from (11.6.5).
For (11.11.11), see Dingle (1973, p. 388). For
(11.11.8){(11.11.19), see Olver (1997b, pp. 103
and 352).
Chapter 12
Parabolic Cylinder Functions
N. M. Temme1
Notation 304
12.1 Special Notation . . . . . . . . . . . . . 304
Properties 304
12.2 Di erential Equations . . . . . . . . . . . 304
12.3 Graphics . . . . . . . . . . . . . . . . . . 305
12.4 Power-Series Expansions . . . . . . . . . 307
12.5 Integral Representations . . . . . . . . . 307
12.6 Continued Fraction . . . . . . . . . . . . 308
12.7 Relations to Other Functions . . . . . . . 308
12.8 Recurrence Relations and Derivatives . . 309
12.9 Asymptotic Expansions for Large Variable 309
12.10 Uniform Asymptotic Expansions for Large
Parameter . . . . . . . . . . . . . . . . . 309
12.11 Zeros . . . . . . . . . . . . . . . . . . . 31212.12 Integrals . . . . . . . . . . . . . . . . . . 313
12.13 Sums . . . . . . . . . . . . . . . . . . . 313
12.14 The Function W(a;x) . . . . . . . . . . 314
12.15 Generalized Parabolic Cylinder Functions 317
Applications 317
12.16 Mathematical Applications . . . . . . . . 317
12.17 Physical Applications . . . . . . . . . . . 317
Computation 317
12.18 Methods of Computation . . . . . . . . . 318
12.19 Tables . . . . . . . . . . . . . . . . . . . 318
12.20 Approximations . . . . . . . . . . . . . . 318
12.21 Software . . . . . . . . . . . . . . . . . . 318
References 318
1Centrum voor Wiskunde en Informatica, Department MAS, Amsterdam, The Netherlands.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 19) by J. C. P. Miller.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
303
304 Parabolic Cylinder Functions
Notation
12.1 Special Notation
(For other notation see pp. xiv and 873.)
x;y real variables.
z complex variable.
n;s nonnegative integers.
a; real or complex parameters.
arbitrary small positive constant.
Unless otherwise noted, primes indicate derivatives
with respect to the variable, and fractional powers take
their principal values.
The main functions treated in this chapter are the
parabolic cylinder functions (PCFs), also known as
Weber parabolic cylinder functions: U(a;z),V(a;z),
U(a;z), andW(a;z). These notations are due to Miller
(1952, 1955). An older notation, due to Whittaker
(1902), for U(a;z) isD(z). The notations are related
byU(a;z) =D a 1
2(z). Whittaker's notation D(z) is
useful when is a nonnegative integer (Hermite poly-
nomial case).
Properties
12.2 Di erential Equations
12.2(i) Introduction
PCFs are solutions of the di erential equation
12.2.1d2w
dz2+
az2+bz+c
w= 0;
with three distinct standard forms
12.2.2d2w
dz2 1
4z2+a
w= 0;
12.2.3d2w
dz2+ 1
4z2 a
w= 0;
12.2.4d2w
dz2+
+1
2 1
4z2
w= 0:
Each of these equations is transformable into the
others. Standard solutions are U(a;z),V(a;z),
U(a;x) (not complex conjugate), U( a;iz) for
(12.2.2);W(a;x) for (12.2.3); D(z) for (12.2.4),
where
12.2.5 D(z) =U
1
2 ;z
:
All solutions are entire functions of zand entire func-
tions ofaor.
For real values of z(=x), numerically satisfactory
pairs of solutions ( x2.7(iv)) of (12.2.2) are U(a;x) andV(a;x) whenxis positive, or U(a; x) andV(a; x)
whenxis negative. For (12.2.3) W(a;x) andW(a; x)
comprise a numerically satisfactory pair, for all x2R.
The solutions W(a;x) are treated inx12.14.
InC, forj= 0;1;2;3; U
( 1)j 1a;( i)j 1z
and
U
( 1)ja;( i)jz
comprise a numerically satisfactory
pair of solutions in the half-plane1
4(2j 3)phz
1
4(2j+ 1).
12.2(ii) Values at z= 0
12.2.6U(a;0) =p
21
2a+1
4 3
4+1
2a;
12.2.7U0(a;0) = p
21
2a 1
4 1
4+1
2a;
12.2.8V(a;0) =21
2a+1
4
3
4 1
2a2 1
4+1
2a;
12.2.9V0(a;0) =21
2a+3
4
1
4 1
2a2 3
4+1
2a:
12.2(iii) Wronskians
12.2.10 WfU(a;z);V(a;z)g=p
2=;
12.2.11 WfU(a;z);U(a; z)g=p
2
1
2+a;
12.2.12 WfU(a;z);U( a;iz)g=iei(1
2a+1
4):
12.2(iv) Re
ection Formulas
Forn= 0;1;:::,
12.2.13U
n 1
2; z
= ( 1)nU
n 1
2;z
;
12.2.14V
n+1
2; z
= ( 1)nV
n+1
2;z
:
12.2(v) Connection Formulas
12.2.15
U(a; z) = sin(a)U(a;z) +
(1
2+a)V(a;z);
12.2.16V(a; z) =cos(a)
(1
2 a)U(a;z) + sin(a)V(a;z):
12.2.17p
2U( a;iz) = 1
2+a
ei(1
2a 1
4)U(a;z)
+ei(1
2a 1
4)U(a; z)
:
12.2.18p
2U(a;z) = 1
2 a
ei(1
2a+1
4)U( a;iz)
+ei(1
2a+1
4)U( a;iz)
;
12.3 Graphics 305
12.2.19U(a;z) =ieiaU(a; z)
+p
2
1
2+aei(1
2a 1
4)U( a;iz):
12.2.20
V(a;z)
=i
(1
2 a)U(a;z) +r
2
ei(1
2a 1
4)U( a;iz):
12.2(vi) Solution U(a;x); Modulus and Phase
Functions
Whenz(=x) is real the solution U(a;x) is de ned by
12.2.21 U(a;x) = (1
2 a)V(a;x);
unlessa=1
2;3
2;:::, in which case U(a;x) is unde ned.
Its importance is that when ais negative andjajis large,
U(a;x) andU(a;x) asymptotically have the same enve-
lope (modulus) and are1
2out of phase in the oscil-
latory interval 2p a < x < 2p a. Properties ofU(a;x) follow immediately from those of V(a;x) via
(12.2.21).
In the oscillatory interval we de ne
12.2.22U(a;x) +iU(a;x) =F(a;x)ei(a;x);
12.2.23U0(a;x) +iU0(a;x) = G(a;x)ei (a;x);
whereF(a;x) (>0),(a;x),G(a;x) (>0), and (a;x)
are real.ForGis the modulus andor is the corre-
sponding phase .
For properties of the modulus and phase func-
tions, including di erential equations, see Miller (1955,
pp. 72{73). For graphs of the modulus functions see
x12.3(i).
12.3 Graphics
12.3(i) Real Variables
Figure 12.3.1 :U(a;x),a= 0.5, 2, 3.5, 5, 8.
Figure 12.3.2 :V(a;x),a= 0.5, 2, 3.5, 5, 8.
Figure 12.3.3 :U(a;x),a= 0:5, 2, 3:5, 5.
Figure 12.3.4 :V(a;x),a= 0:5, 2, 3:5, 5.
306 Parabolic Cylinder Functions
Figure 12.3.5 :U( 8;x),U( 8;x),F( 8;x), 4p
2
x4p
2.
Figure 12.3.6 :U0( 8;x),U0( 8;x),G( 8;x), 4p
2
x4p
2.
Figure 12.3.7 :U(a;x), 2:5a2:5, 2:5x2:5.
Figure 12.3.8 :V(a;x), 2:5a2:5, 2:5x2:5.
12.3(ii) Complex Variables
In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the
phase. See also p. xiv.
12.4 Power-Series Expansions 307
Figure 12.3.9 :U(3:5;x+iy), 3:6x5, 5y
5.
Figure 12.3.10 :U( 3:5;x+iy), 5x5, 3:5
y3:5.
12.4 Power-Series Expansions
12.4.1U(a;z) =U(a;0)u1(a;z) +U0(a;0)u2(a;z);
12.4.2V(a;z) =V(a;0)u1(a;z) +V0(a;0)u2(a;z);
where the initial values are given by (12.2.6){(12.2.9),
andu1(a;z) andu2(a;z) are the even and odd solutions
of (12.2.2) given by
12.4.3u1(a;z) =e 1
4z2
1 + (a+1
2)z2
2!
+ (a+1
2)(a+5
2)z4
4!+
;
12.4.4u2(a;z) =e 1
4z2
z+ (a+3
2)z3
3!
+ (a+3
2)(a+7
2)z5
5!+
:
Equivalently,
12.4.5
u1(a;z) =e1
4z2
1+(a 1
2)z2
2!+(a 1
2)(a 5
2)z4
4!+
;
12.4.6
u2(a;z) =e1
4z2
z+(a 3
2)z3
3!+(a 3
2)(a 7
2)z5
5!+
:
These series converge for all values of z.
12.5 Integral Representations
12.5(i) Integrals Along the Real Line
12.5.1
U(a;z) =e 1
4z2
1
2+aZ1
0ta 1
2e 1
2t2 ztdt,<a> 1
2,12.5.2
U(a;z) =ze 1
4z2
1
4+1
2aZ1
0t1
2a 3
4e t
z2+ 2t 1
2a 3
4dt,
jphzj<1
2,<a> 1
2,
12.5.3
U(a;z) =e 1
4z2
3
4+1
2aZ1
0t1
2a 1
4e t
z2+ 2t 1
2a 1
4dt,
jphzj<1
2,<a> 3
2,
12.5.4
U(a;z) =r
2
e1
4z2
Z1
0t a 1
2e 1
2t2cos
zt+ 1
2a+1
4
dt,
<a<1
2.
12.5(ii) Contour Integrals
The following integrals correspond to those of x12.5(i).
12.5.5U(a;z) = 1
2 a
2ie 1
4z2Z(0+)
1ezt 1
2t2ta 1
2dt,
a6=1
2;3
2;5
2;:::, <pht<:
Restrictions on aare not needed in the following two
representations:
12.5.6U(a;z) =e1
4z2
ip
2Zc+i1
c i1e zt+1
2t2t a 1
2dt,
1
2<pht<1
2,c>0 ,
12.5.7
V(a;z) =e 1
4z2
2Z ic+1
ic 1+Zic+1
ic 1
ezt 1
2t2ta 1
2dt,
<pht< ,c>0.
308 Parabolic Cylinder Functions
For proofs and further results see Miller (1955, x4)
and Whittaker (1902).
12.5(iii) Mellin{Barnes Integrals
12.5.8
U(a;z) =e 1
4z2z a 1
2
2i 1
2+a
Zi1
i1 (t) 1
2+a 2t
2tz2tdt,
a6= 1
2; 3
2; 5
2;:::,jphzj<3
4,
where the contour separates the poles of ( t) from those
of 1
2+a 2t
.
12.5.9
V(a;z) =r
2
e1
4z2za 1
2
2i 1
2 a
Zi1
i1 (t) 1
2 a 2t
2tz2tcos (t)dt,
a6=1
2;3
2;5
2;:::,jphzj<1
4,
where the contour separates the poles of ( t) from those
of 1
2 a 2t
.
12.5(iv) Compendia
For further collections of integral representations see
Apelblat (1983, pp. 427-436), Erd elyi et al. (1953b,
v. 2, pp. 119{120), Erd elyi et al. (1954a, pp. 289{
291 and 362), Gradshteyn and Ryzhik (2000, xx9.24{
9.25), Magnus et al. (1966, pp. 328{330), Oberhet-
tinger (1974, pp. 251{252), and Oberhettinger and Badii
(1973, pp. 378{384).
12.6 Continued Fraction
For a continued-fraction expansion of the ratio
U(a;x)/U(a 1;x) see Cuyt et al. (2008, pp. 340{341).
12.7 Relations to Other Functions
12.7(i) Hermite Polynomials
For the notation see x18.3.
12.7.1 U
1
2;z
=D0(z) =e 1
4z2;
12.7.2U
n 1
2;z
=Dn(z) =e 1
4z2Hen(z)
= 2 n=2e 1
4z2Hn
z=p
2
,
n= 0;1;2;::: ,
12.7.3
V
n+1
2;z
=p
2=e1
4z2( i)nHen(iz)
=p
2=e1
4z2( i)n2 1
2nHn
iz=p
2
,
n= 0;1;2;:::.12.7(ii) Error Functions, Dawson's Integral, and
Probability Function
For the notation see xx7.2 and 7.18.
12.7.4V
1
2;z
= (2p)e1
4z2F
z=p
2
;
12.7.5U 1
2;z
=D 1(z) =q
1
2e1
4z2erfc
z=p
2
;
12.7.6
U
n+1
2;z
=D n 1(z)
=r
2( 1)n
n!e 1
4z2dn
e1
2z2erfc
z=p
2
dzn ,
n= 0;1;2;:::;
12.7.7U
n+1
2;z
=e1
4z2Hhn(z)
=p21
2(n 1)e1
4z2inerfc
z=p
2
,
n= 1;0;1;:::.
12.7(iii) Modi ed Bessel Functions
For the notation see x10.25(ii).
12.7.8
U( 2;z) =z5=2
4p
2
2K1
4 1
4z2
+3K3
4 1
4z2
K5
4 1
4z2
;
12.7.9U( 1;z) =z3=2
2p
2
K1
4 1
4z2
+K3
4 1
4z2
;
12.7.10U(0;z) =rz
2K1
4 1
4z2
;
12.7.11U(1;z) =z3=2
p
2
K3
4 1
4z2
K1
4 1
4z2
:
For these, the corresponding results for U(a;z) with
a= 2,3, 1
2, 3
2, 5
2, and the corresponding re-
sults forV(a;z) witha= 0,1,2,3,1
2,3
2,5
2, see
Miller (1955, pp. 42{43 and 77{79).
12.7(iv) Con
uent Hypergeometric Functions
For the notation see xx13.2(i) and 13.14(i).
The even and odd solutions of (12.2.2) (see (12.4.3){
(12.4.6)) are given by
12.7.12u1(a;z) =e 1
4z2M 1
2a+1
4;1
2;1
2z2
=e1
4z2M
1
2a+1
4;1
2; 1
2z2
;
12.7.13u2(a;z) =ze 1
4z2M 1
2a+3
4;3
2;1
2z2
=ze1
4z2M
1
2a+3
4;3
2; 1
2z2
:
Also,
12.7.14U(a;z) = 2 1
4 1
2ae 1
4z2U 1
2a+1
4;1
2;1
2z2
= 2 3
4 1
2aze 1
4z2U 1
2a+3
4;3
2;1
2z2
= 2 1
2az 1
2W 1
2a;1
4 1
2z2
:
12.8 Recurrence Relations and Derivatives 309
(It should be observed that the functions on the right-
hand sides of (12.7.14) are multivalued; hence, for ex-
ample,zcannot be replaced simply by z.)
12.8 Recurrence Relations and Derivatives
12.8(i) Recurrence Relations
12.8.1zU(a;z) U(a 1;z) + (a+1
2)U(a+ 1;z) = 0;
12.8.2U0(a;z) +1
2zU(a;z) + (a+1
2)U(a+ 1;z) = 0;
12.8.3 U0(a;z) 1
2zU(a;z) +U(a 1;z) = 0;
12.8.4 2U0(a;z) +U(a 1;z) + (a+1
2)U(a+ 1;z) = 0:
(12.8.1){(12.8.4) are also satis ed by U(a;z).
12.8.5zV(a;z) V(a+ 1;z) + (a 1
2)V(a 1;z) = 0;
12.8.6V0(a;z) 1
2zV(a;z) (a 1
2)V(a 1;z) = 0;
12.8.7 V0(a;z) +1
2zV(a;z) V(a+ 1;z) = 0;
12.8.8 2V0(a;z) V(a+ 1;z) (a 1
2)V(a 1;z) = 0:
12.8(ii) Derivatives
Form= 0;1;2;:::,
12.8.9
dm
dzm
e1
4z2U(a;z)
= ( 1)m 1
2+a
me1
4z2U(a+m;z);
12.8.10
dm
dzm
e 1
4z2U(a;z)
= ( 1)me 1
4z2U(a m;z);
12.8.11dm
dzm
e1
4z2V(a;z)
=e1
4z2V(a+m;z);
12.8.12dm
dzm
e 1
4z2V(a;z)
= ( 1)m 1
2 a
me 1
4z2V(a m;z):
12.9 Asymptotic Expansions for Large
Variable
12.9(i) Poincar e-Type Expansions
Throughout this subsection is an arbitrary small pos-
itive constant.
Asz!1
12.9.1U(a;z)e 1
4z2z a 1
21X
s=0( 1)s 1
2+a
2s
s!(2z2)s,
jphzj3
4 (<3
4) ,
12.9.2V(a;z)r
2
e1
4z2za 1
21X
s=0 1
2 a
2s
s!(2z2)s,
jphzj1
4 (<1
4) .12.9.3
U(a;z)e 1
4z2z a 1
21X
s=0( 1)s 1
2+a
2s
s!(2z2)s
ip
2
1
2+aeiae1
4z2za 1
21X
s=0 1
2 a
2s
s!(2z2)s,
1
4+phz5
4 ,
12.9.4
V(a;z)r
2
e1
4z2za 1
21X
s=0 1
2 a
2s
s!(2z2)s
i
1
2 ae 1
4z2z a 1
21X
s=0( 1)s 1
2+a
2s
s!(2z2)s,
1
4+phz3
4 .
12.9(ii) Bounds and Re-Expansions for the
Remainder Terms
Bounds and re-expansions for the error term in (12.9.1)
can be obtained by use of (12.7.14) and xx13.7(ii),
13.7(iii). Corresponding results for (12.9.2) can be ob-
tained via (12.2.20).
12.10 Uniform Asymptotic Expansions for
Large Parameter
12.10(i) Introduction
In this section we give asymptotic expansions of PCFs
for large values of the parameter athat are uniform with
respect to the variable z, when both aandz(=x) are
real. These expansions follow from Olver (1959), where
detailed information is also given for complex variables.
With the transformations
12.10.1 a=1
22; x =tp
2;
(12.2.2) becomes
12.10.2d2w
dt2=4(t21)w:
With the upper sign in (12.10.2), expansions can be
constructed for large in terms of elementary functions
that are uniform for t2( 1;1) (x2.8(ii)). With the
lower sign there are turning points at t=1, which
need to be excluded from the regions of validity. These
cases are treated in xx12.10(ii){12.10(vi).
The turning points can be included if expansions
in terms of Airy functions are used instead of elemen-
tary functions (x2.8(iii)). These cases are treated in
xx12.10(vii){12.10(viii).
Throughout this section the symbol again denotes
an arbitrary small positive constant.
310 Parabolic Cylinder Functions
12.10(ii) Negative a,2p a<x<1
Asa! 1
12.10.3U
1
22;tp
2
g()e 2
(t2 1)1
41X
s=0As(t)
2s;
12.10.4
U0
1
22;tp
2
p
2g()(t2 1)1
4e 21X
s=0Bs(t)
2s;
12.10.5V
1
22;tp
2
2g()
(1
2+1
22)e2
(t2 1)1
4
1X
s=0( 1)sAs(t)
2s;
12.10.6V0
1
22;tp
2
p
2g()
(1
2+1
22)(t2 1)1
4
e21X
s=0( 1)sBs(t)
2s;
uniformly for t2[1 +;1), where
12.10.7=1
2tp
t2 1 1
2ln
t+p
t2 1
:
The coecients are given by
12.10.8As(t) =us(t)
(t2 1)3
2s;Bs(t) =vs(t)
(t2 1)3
2s;
whereus(t) andvs(t) are polynomials in tof degree 3s,
(sodd), 3s 2 (seven,s2). Fors= 0;1;2,
12.10.9u0(t) = 1; u 1(t) =t(t2 6)
24;
u2(t) = 9t4+ 249t2+ 145
1152;
12.10.10v0(t) = 1; v 1(t) =t(t2+ 6)
24;
v2(t) =15t4 327t2 143
1152:Higher polynomials us(t) can be calculated from the re-
currence relation
12.10.11 (t2 1)u0
s(t) 3stus(t) =rs 1(t);
where
12.10.128rs(t) = (3t2+ 2)us(t) 12(s+ 1)trs 1(t)
+ 4(t2 1)r0
s 1(t);
and thevs(t) then follow from
12.10.13vs(t) =us(t) +1
2tus 1(t) rs 2(t):
Lastly, the function g() in (12.10.3) and (12.10.4)
has the asymptotic expansion:
12.10.14g()h()
1 +1
21X
s=1
s
(1
22)s!
;
where
12.10.15 h() = 2 1
42 1
4e 1
421
22 1
2;
and the coecients
sare de ned by
12.10.16 1
2+z
p
2e zzz1X
s=0
s
zs;
compare (5.11.8). For s4
12.10.17
0= 1;
1= 1
24;
2=1
1152;
3=1003
4 14720;
4= 4027
398 13120:
12.10(iii) Negative a, 1<x< 2p a
When! 1 , asymptotic expansions for the func-
tionsU
1
22; tp
2
andV
1
22; tp
2
that are
uniform for t2[1 +;1) are obtainable by substitu-
tion into (12.2.15) and (12.2.16) by means of (12.10.3)
and (12.10.5). Similarly for U0
1
22; tp
2
and
V0
1
22; tp
2
.
12.10(iv) Negative a, 2p a<x< 2p a
Asa! 1
12.10.18 U
1
22;tp
2
2g()
(1 t2)1
4
cos1X
s=0( 1)seA2s(t)
4s sin1X
s=0( 1)seA2s+1(t)
4s+2!
;
12.10.19U0
1
22;tp
2
p
2g()(1 t2)1
4
sin1X
s=0( 1)seB2s(t)
4s+ cos1X
s=0( 1)seB2s+1(t)
4s+2!
;
12.10.20 V
1
22;tp
2
2g()
1
2+1
22
(1 t2)1
4
cos 1X
s=0( 1)seA2s(t)
4s sin 1X
s=0( 1)seA2s+1(t)
4s+2!
;
12.10.21V0
1
22;tp
2
p
2g()(1 t2)1
4
1
2+1
22
sin 1X
s=0( 1)seB2s(t)
4s+ cos 1X
s=0( 1)seB2s+1(t)
4s+2!
;
12.10 Uniform Asymptotic Expansions for Large Parameter 311
uniformly for t2[ 1 +;1 ]. The quantities and
are de ned by
12.10.22 =2 1
4; =2+1
4;
where
12.10.23 =1
2arccost 1
2tp
1 t2;
and the coecients eAs(t) andeBs(t) are given by
12.10.24eAs(t) =us(t)
(1 t2)3
2s;eBs(t) =vs(t)
(1 t2)3
2s;
compare (12.10.8).
12.10(v) Positive a, 1<x<1
Asa!1
12.10.25
U
1
22;tp
2
g()e 2
(t2+ 1)1
41X
s=0us(t)
(t2+ 1)3
2s1
2s;
uniformly for t2R. Here bars do not denote complex
conjugates; instead
12.10.26=1
2tp
t2+ 1 +1
2ln
t+p
t2+ 1
;
12.10.27 us(t) =isus( it);
and the function g() has the asymptotic expansion
12.10.28g()1
p
2h()
1 +1
21X
s=1( 1)s
s
(1
22)s!
;
whereh() and
sare as inx12.10(ii).
With the same conditions
12.10.29
U0
1
22;tp
2
p
2g()(t2+ 1)1
4e 21X
s=0vs(t)
(t2+ 1)3
2s1
2s;
where
12.10.30 vs(t) =isvs( it):12.10(vi) Modi cations of Expansions in
Elementary Functions
In Temme (2000) modi cations are given of Olver's ex-
pansions. An example is the following modi cation of
(12.10.3)
12.10.31U
1
22;tp
2
h()e 2
(t2 1)1
41X
s=0As( )
2s;
whereandh() are as in (12.10.7) and (12.10.15) ,
12.10.32 =1
2tp
t2 1 1
;
and the coecients As( ) are the product of sand a
polynomial in of degree 2s. They satisfy the recursion
12.10.33As+1( ) = 4 2( + 1)2d
d As( )
1
4Z
0
20u2+ 20u+ 3
As(u)du,
s= 0;1;2;:::;
starting with Ao( ) = 1. Explicitly,
12.10.34A1( ) = 1
12 (20 2+ 30 + 9);
A2( ) =1
288 2(6160 4+ 18480 3+ 19404 2
+ 8028 + 945):
The modi ed expansion (12.10.31) shares the prop-
erty of (12.10.3) that it applies when !1 uniformly
with respect to t2[1 +;1). In addition, it enjoys a
double asymptotic property: it holds if either or both
andttend to in nity. Observe that if t!1 , then
As( ) =O
t 2s
, whereasAs(t) =O(1) orO
t 2
ac-
cording as sis even or odd. The proof of the double
asymptotic property then follows with the aid of error
bounds; compare x10.41(iv).
For additional information see Temme (2000). See
also Olver (1997b, pp. 206{208) and Jones (2006).
12.10(vii) Negative a, 2p a<x<1.
Expansions in Terms of Airy
Functions
The following expansions hold for large positive real val-
ues of, uniformly for t2[ 1 +;1). (For complex
values ofandtsee Olver (1959).)
12.10.35 U
1
22;tp
2
21
21
3g() ()0
@Ai
4
31X
s=0As()
4s+Ai0
4
3
8
31X
s=0Bs()
4s1
A;
12.10.36 U0
1
22;tp
2
(2)1
22
3g()
()0
@Ai
4
3
4
31X
s=0Cs()
4s+ Ai0
4
31X
s=0Ds()
4s1
A;
312 Parabolic Cylinder Functions
12.10.37 V
1
22;tp
2
21
21
3g() ()
1
2+1
220
@Bi
4
31X
s=0As()
4s+Bi0
4
3
8
31X
s=0Bs()
4s1
A;
12.10.38 V0
1
22;tp
2
(2)1
22
3g()
() 1
2+1
220
@Bi
4
3
4
31X
s=0Cs()
4s+ Bi0
4
31X
s=0Ds()
4s1
A:
The variable is de ned by
12.10.392
33
2=;1t;(0);
2
3( )3
2=; 1<t1;(0);
where;are given by (12.10.7), (12.10.23), respec-
tively, and
12.10.40 () =
t2 11
4
:
The function =(t) is real for t> 1 and analytic at
t= 1. Inversely, with w= 2 1
3,
12.10.41t= 1 +w 1
10w2+11
350w3 823
63000w4
+1 50653
242 55000w5+,jj< 3
42
3:
Forg() see (12.10.14). The coecients As() and
Bs() are given by
12.10.42
As() = 3s2sX
m=0 m( ())6(2s m)u2s m(t);
2Bs() = 3s2s+1X
m=0 m( ())6(2s m+1)u2s m+1(t);
where () is as in (12.10.40), uk(t) is as inx12.10(ii),
0= 1, and
12.10.43 m=(2m+ 1)(2m+ 3)(6m 1)
m!(144)m;
m= 6m+ 1
6m 1 m:
The coecients Cs() andDs() in (12.10.36) and
(12.10.38) are given by
12.10.44Cs() = ()As() +A0
s() +Bs();
Ds() =As() + ()Bs 1() +B0
s 1();
where
12.10.45 () = 0()
()=1 2t( ())6
4:
Explicitly,
12.10.46
Cs() = 3s2s+1X
m=0 m( ())6(2s m+1)v2s m+1(t);
Ds() = 3s2sX
m=0 m( ())6(2s m)v2s m(t);
wherevk(t) is as inx12.10(ii).Modi ed Expansions
The expansions (12.10.35){(12.10.38) can be modi ed,
again see Temme (2000), and the new expansions hold if
either or both andttend to in nity. This is provable
by the methods used in x10.41(v).
12.10(viii) Negative a, 1<x< 2p a.
Expansions in Terms of Airy
Functions
When! 1 , asymptotic expansions for
U
1
22; tp
2
andV
1
22; tp
2
that are uni-
form fort2[ 1 +;1) are obtained by substitution
into (12.2.15) and (12.2.16) by means of (12.10.35)
and (12.10.37). Similarly for U0
1
22; tp
2
and
V0
1
22; tp
2
.
12.11 Zeros
12.11(i) Distribution of Real Zeros
Ifa 1
2;thenU(a;x) has no real zeros. If 3
2<
a < 1
2, thenU(a;x) has no positive real zeros. If
2n 3
2< a < 2n+1
2,n= 1;2;:::, thenU(a;x)
hasnpositive real zeros. Lastly, when a= n 1
2,
n= 1;2;::: (Hermite polynomial case) U(a;x) hasn
zeros and they lie in the interval [ 2p a;2p a]. For
further information on these cases see Dean (1966).
Ifa > 1
2;thenV(a;x) has no positive real zeros,
and ifa=3
2 2n,n2Z, thenV(a;x) has a zero at
x= 0.
12.11(ii) Asymptotic Expansions of Large Zeros
Whena> 1
2,U(a;z) has a string of complex zeros that
approaches the ray ph z=3
4asz!1 , and a conju-
gate string. When a> 1
2the zeros are asymptotically
given byza;sand za;s, wheresis a large positive integer
and
12.11.1
za;s=e3
4ip
2 s
1 ias
2 s+2a22
s 8a2s+ 4a2+ 3
16 2s
+O
3
s 3
s
;
with
12.11.2 s=
2s+1
2 a
+iln
1
22 a 1
2 1
2+a
;
12.12 Integrals 313
and
12.11.3 s= ln s 1
2i:
Whena=1
2these zeros are the same as the zeros of
the complementary error function erfc( z=p
2); compare
(12.7.5). Numerical calculations in this case show that
z1
2;scorresponds to the sth zero on the string; compare
x7.13(ii).
12.11(iii) Asymptotic Expansions for Large
Parameter
For large negative values of athe real zeros of U(a;x),
U0(a;x),V(a;x), andV0(a;x) can be approximated
by reversion of the Airy-type asymptotic expansions of
xx12.10(vii) and 12.10(viii). For example, let the sth
real zeros of U(a;x) andU0(a;x), counted in descend-
ing order away from the point z= 2p a, be denoted
byua;sandu0
a;s, respectively. Then
12.11.4ua;s21
2
p0( ) +p1( )
4+p2( )
8+
;
as(=p 2a)!1 ,s xed. Here = 4
3as,as
denoting the sth negative zero of the function Ai (see
x9.9(i)). The rst two coecients are given by
12.11.5 p0() =t();
wheret() is the function inverse to (t), de ned by
(12.10.39) (see also (12.10.41)), and
12.11.6p1() =t3 6t
24(t2 1)2+5
48((t2 1)3)1
2:
Similarly, for the zeros of U0(a;x) we have
12.11.7u0
a;s21
2
q0( ) +q1( )
4+q2( )
8+
;
where = 4
3a0
s,a0
sdenoting the sth negative zero of
the function Ai0and
12.11.8 q0() =t():
For the rst zero of U(a;x) we also have
12.11.9
ua;121
2
1 1:85575 708 4=3 0:34438 34 8=3
0:16871 5 4 0:11414 16=3 0:0808 20=3
;
where the numerical coecients have been rounded o .
For further information, including associated func-
tions, see Olver (1959).
12.12 Integrals
12.12.1Z1
0e 1
4t2t 1U(a;t)dt=p2 1
2(+a+1
2) ()
1
2(+a+3
2),
<>0 ,12.12.2Z1
0e 3
4t2t a 3
2U(a;t)dt
= 21
4+1
2a
a 1
2
cos
(1
4a+1
8)
,<a< 1
2;
12.12.3Z1
0e 1
4t2t a 1
2(x2+t2) 1U(a;t)dt
=p
=2 1
2 a
x a 3
2e1
4x2U( a;x),
<a<1
2;x> 0:
Nicholson-type Integral
12.12.4
(U(a;z))2+ (U(a;z))2
=23
2
1
2 aZ1
0e2at+1
2z2tanht
p
sinh (2t)dt,<a<1
2.
Whenz(=x) is real the left-hand side equals ( F(a;x))2;
compare (12.2.22).
For further integrals see xx13.10, 13.23, and use
(12.7.14).
For compendia of integrals see Erd elyi et al. (1953b,
v. 2, pp. 121{122), Erd elyi et al. (1954a,b, v. 1, pp. 60{
61, 115, 210{211, and 336; v. 2, pp. 76{80, 115, 151,
171, and 395{398), Gradshteyn and Ryzhik (2000, x7.7),
Magnus et al. (1966, pp. 330{331), Marichev (1983,
pp. 190{191), Oberhettinger (1974, pp. 144{145), Ober-
hettinger (1990, pp. 106{108 and 192), Oberhettinger
and Badii (1973, pp. 181{185), Prudnikov et al. (1986b,
pp. 36{37, 155{168, 243{246, 289{290, 327{328, 419{
420, and 619), Prudnikov et al. (1992a,x3.11), and
Prudnikov et al. (1992b,x3.11).
See also Barr (1968) and Lowdon (1970).
12.13 Sums
12.13(i) Addition Theorems
12.13.1
U(a;x+y) =e1
2xy+1
4y21X
m=0( y)m
m!U(a m;x);
12.13.2U(a;x+y)
=e 1
2xy 1
4y21X
m=0 a 1
2
m
ymU(a+m;x);
12.13.3
V(a;x+y) =e1
2xy+1
4y21X
m=0a 1
2
m
ymV(a m;x);
12.13.4V(a;x+y) =e 1
2xy 1
4y21X
m=0ym
m!V(a+m;x):
314 Parabolic Cylinder Functions
12.13.5
U(a;xcost+ysint)
=e1
4(xsint ycost)2
1X
m=0 a 1
2
m
(tant)mU(m+a;x)U
m 1
2;y
,
<a 1
2;0t1
4.
12.13.6
n!U
n+1
2;z
=ine 1
2z2erfc(z=p
2)U
n 1
2;iz
+b1
2n+1
2cX
m=1U
2m n 1
2;z
,
n= 0;1;2;::::
For erfc seex7.2(i).
12.13(ii) Other Series
For other series see Dhar (1940), Hansen (1975, pp. 421{
422), Hillion (1997), Miller (1974), Prudnikov et al.
(1986b, p. 651), Shanker (1940b,a,c), and Varma (1941).12.14 The Function W(a;x)
12.14(i) Introduction
In this section solutions of equation (12.2.3) are consid-
ered. This equation is important when aandz(=x)
are real, and we shall assume this to be the case. In
other cases the general theory of (12.2.2) is available.
W(a;x) andW(a; x) form a numerically satisfactory
pair of solutions when 1<x<1.
12.14(ii) Values at z= 0 and Wronskian
12.14.1 W(a;0) = 2 3
4 1
4+1
2ia
3
4+1
2ia 1
2
;
12.14.2W0(a;0) = 2 1
4 3
4+1
2ia
1
4+1
2ia 1
2
:
12.14.3 WfW(a;x);W(a; x)g= 1:
12.14(iii) Graphs
For the modulus functions eF(a;x) andeG(a;x) see
x12.14(x).
Figure 12.14.1 :k 1/2W(3;x),k1/2W(3; x),eF(3;x),
0x8.
Figure 12.14.2 :k 1/2W0(3;x),k1/2W0(3; x),eG(3;x),
0x8.
Figure 12.14.3 :k 1/2W( 3;x),k1/2W( 3; x),
eF( 3;x), 0x8.
Figure 12.14.4 :k 1/2W0( 3;x);k1/2W0( 3; x),
eG( 3;x), 0x8.
12.14 The Function W(a;x) 315
12.14(iv) Connection Formula
12.14.4W(a;x) =p
k=2e1
4a
eiU
ia;xe i=4
+e iU
ia;xei=4
;
where
12.14.5k=p
1 +e2a ea;1=k=p
1 +e2a+ea;
12.14.6 =1
8+1
2 2;
12.14.7 2= ph 1
2+ia
;
the branch of ph being zero when a= 0 and de ned by
continuity elsewhere.
12.14(v) Power-Series Expansions
12.14.8W(a;x) =W(a;0)w1(a;x) +W0(a;0)w2(a;x):
Herew1(a;x) andw2(a;x) are the even and odd solu-
tions of (12.2.3):
12.14.9 w1(a;x) =1X
n=0 n(a)x2n
(2n)!;
12.14.10 w2(a;x) =1X
n=0 n(a)x2n+1
(2n+ 1)!;
where n(a) and n(a) satisfy the recursion relations
12.14.11 n+2=a n+1 1
2(n+ 1)(2n+ 1) n;
n+2=a n+1 1
2(n+ 1)(2n+ 3) n;
with
12.14.12
0(a) = 1; 1(a) =a; 0(a) = 1; 1(a) =a:
Other expansions, involving cos 1
4x2
and sin 1
4x2
,
can be obtained from (12.4.3) to (12.4.6) by replacing
aby iaandzbyxei/4; see Miller (1955, p. 80), and
also (12.14.15) and (12.14.16).
12.14(vi) Integral Representations
These follow from the contour integrals of x12.5(ii),
which are valid for general complex values of the ar-
gumentzand parameter a. See Miller (1955, p. 26).
12.14(vii) Relations to Other Functions
Bessel Functions
For the notation see x10.2(ii). When x>0
12.14.13W(0;x) = 2 5
4px
J 1
4 1
4x2
J1
4 1
4x2
;
12.14.14
d
dxW(0;x) = 2 9
4xpx
J3
4 1
4x2
J 3
4 1
4x2
:Con
uent Hypergeometric Functions
For the notation see x13.2(i).
The even and odd solutions of (12.2.3) (see
x12.14(v)) are given by
12.14.15w1(a;x) =e 1
4ix2M 1
4 1
2ia;1
2;1
2ix2
=e1
4ix2M 1
4+1
2ia;1
2; 1
2ix2
;
12.14.16w2(a;x) =xe 1
4ix2M 3
4 1
2ia;3
2;1
2ix2
=xe1
4ix2M 3
4+1
2ia;3
2; 1
2ix2
:
12.14(viii) Asymptotic Expansions for Large
Variable
Write
12.14.17W(a;x) =r
2k
x(s1(a;x) cos! s2(a;x) sin!);
12.14.18
W(a; x) =r
2
kx(s1(a;x) sin!+s2(a;x) cos!);
where
12.14.19 !=1
4x2 alnx+1
4+1
2 2;
with 2given by (12.14.7). Then as x!1
12.14.20
s1(a;x)1 +d2
1!2x2 c4
2!22x4 d6
3!23x6+c8
4!24x8+;
12.14.21
s2(a;x) c2
1!2x2 d4
2!22x4+c6
3!23x6+d8
4!24x8 :
The coecients c2randd2rare obtainable by equating
real and imaginary parts in
12.14.22 c2r+id2r=
2r+1
2+ia
1
2+ia:
Equivalently,
12.14.23s1(a;x) +is2(a;x)1X
r=0( i)r 1
2+ia
2r
2rr!x2r:
12.14(ix) Uniform Asymptotic Expansions for
Large Parameter
The di erential equation
12.14.24d2w
dt2=4(1 t2)w
follows from (12.2.3), and has solutions
W 1
22;tp
2
. For real andtoscillations occur
outside the t-interval [ 1;1]. Airy-type uniform asymp-
totic expansions can be used to include either one of
the turning points 1. In the following expansions,
obtained from Olver (1959), is large and positive, and
is again an arbitrary small positive constant.
316 Parabolic Cylinder Functions
Positivea,2pa<x<1
12.14.25W
1
22;tp
2
2 1
2e 1
42l()
(t2 1)1
4
cos1X
s=0( 1)sA2s(t)
4s
sin1X
s=0( 1)sA2s+1(t)
4s+2!
;
12.14.26
W
1
22; tp
2
21
2e1
42l()
(t2 1)1
4
sin1X
s=0( 1)sA2s(t)
4s
+ cos1X
s=0( 1)sA2s+1(t)
4s+2!
;
uniformly for t2[1 +;1). HereAs(t) is as in
x12.10(ii),is de ned by
12.14.27 =2+1
4;
withgiven by (12.10.7), and
12.14.28l() =p
2e1
82ei(1
2 2 1
8)g(e 1
4i);withg() as inx12.10(ii). The function l() has the
asymptotic expansion
12.14.29 l()21
4
1
21X
s=0ls
4s;
with
12.14.30l0= 1; l 1= 1
1152; l 2= 16123
398 13120:
Positivea, 2pa<x< 2pa
12.14.31
W
1
22;tp
2
l()e2
21
2e1
42(1 t2)1
41X
s=0( 1)seAs(t)
2s;
uniformly for t2[ 1 +;1 ], withgiven by
(12.10.23) and eAs(t) given by (12.10.24).
The expansions for the derivatives corresponding to
(12.14.25), (12.14.26), and (12.14.31) may be obtained
by formal term-by-term di erentiation with respect to
t; compare the analogous results in xx12.10(ii){12.10(v).
Airy-type Uniform Expansions
12.14.32W
1
22;tp
2
1
21
3l()
21
2e1
42 ()0
@Bi
4
31X
s=0( 1)sAs()
4s+Bi0
4
3
8
31X
s=0( 1)sBs()
4s1
A;
12.14.33W
1
22; tp
2
1
21
3l()
2 1
2e 1
42 ()0
@Ai
4
31X
s=0( 1)sAs()
4s+Ai0
4
3
8
31X
s=0( 1)sBs()
4s1
A;
uniformly for t2[ 1 +;1), with, (),As(), andBs() as inx12.10(vii). For the corresponding expansions for
the derivatives see Olver (1959).
Negativea, 1<x<1
In this case there are no real turning points, and the solutions of (12.2.3), with zreplaced by x, oscillate on the
entire realx-axis.
12.14.34W
1
22;tp
2
l()
(t2+ 1)1
4
cos1X
s=0( 1)su2s(t)
(t2+ 1)3s4s sin1X
s=0( 1)su2s+1(t)
(t2+ 1)3s+3
24s+2!
;
12.14.35W0
1
22;tp
2
p
2l()(t2+ 1)1
4
sin1X
s=0( 1)sv2s(t)
(t2+ 1)3s4s+ cos1X
s=0( 1)sv2s+1(t)
(t2+ 1)3s+3
24s+2!
;
uniformly for t2R, where
12.14.36 =2+1
4;
andand the coecients us(t) andvs(t) as inx12.10(v).
12.14(x) Modulus and Phase Functions
As noted inx12.14(ix), when ais negative the solutions
of (12.2.3), with zreplaced by x, are oscillatory on the
whole real line; also, when ais positive there is a cen-
tral interval 2pa < x < 2pain which the solutionsare exponential in character. In the oscillatory intervals
we write
12.14.37
k 1/2W(a;x) +ik1/2W(a; x) =eF(a;x)eie(a;x);
12.15 Generalized Parabolic Cylinder Functions 317
12.14.38
k 1/2W0(a;x) +ik1/2W0(a; x) = eG(a;x)eie (a;x);
wherekis de ned in (12.14.5), and eF(a;x) (>0),e(a;x),
eG(a;x) (>0), ande (a;x) are real.eForeGis the mod-
ulus andeore is the corresponding phase . Compare
x12.2(vi).
For properties of the modulus and phase functions,
including di erential equations and asymptotic expan-
sions for large x, see Miller (1955, pp. 87{88). For
graphs of the modulus functions see x12.14(iii).
12.14(xi) Zeros of W(a;x),W0(a;x)
For asymptotic expansions of the zeros of W(a;x) and
W0(a;x), see Olver (1959).
12.15 Generalized Parabolic Cylinder
Functions
The equation
12.15.1d2w
dz2+
+ 1 2z
w= 0
can be viewed as a generalization of (12.2.4). This
equation arises in the study of non-self-adjoint elliptic
boundary-value problems involving an inde nite weight
function. See Faierman (1992) for power series and
asymptotic expansions of a solution of (12.15.1).
Applications
12.16 Mathematical Applications
PCFs are used as basic approximating functions in
the theory of contour integrals with a coalescing sad-
dle point and an algebraic singularity, and in the the-
ory of di erential equations with two coalescing turn-
ing points; seexx2.4(vi) and 2.8(vi). For examples see
xx13.20(iii), 13.20(iv), 14.15(v), and 14.26.
Sleeman (1968b) considers certain orthogonality
properties of the PCFs and corresponding eigenvalues.
In Brazel et al. (1992) exponential asymptotics are con-
sidered in connection with an eigenvalue problem in-
volving PCFs.
PCFs are also used in integral transforms with re-
spect to the parameter, and inversion formulas exist for
kernels containing PCFs. See Erd elyi (1941a), Cherry
(1948), and Lowdon (1970). Integral transforms and
sampling expansions are considered in Jerri (1982).12.17 Physical Applications
The main applications of PCFs in mathematical physics
arise when solving the Helmholtz equation
12.17.1 r2w+k2w= 0;
wherekis a constant, and r2is the Laplacian
12.17.2r2=@2
@x2+@2
@y2+@2
@z2
in Cartesian coordinates x;y;z of three-dimensional
space (x1.5(ii)). By using instead coordinates of the
parabolic cylinder ;; , de ned by
12.17.3x=; y =1
22 1
22; z =;
(12.17.1) becomes
12.17.41
2+2@2w
@2+@2w
@2
+@2w
@2+k2w= 0:
Settingw=U()V()W() and separating variables,
we obtain
12.17.5d2U
d2+
2+
U= 0;
d2V
d2+
2
V= 0;
d2W
d2+
k2
W= 0;
with arbitrary constants ;. The rst two equations
can be transformed into (12.2.2) or (12.2.3).
In a similar manner coordinates of the paraboloid of
revolution transform the Helmholtz equation into equa-
tions related to the di erential equations considered in
this chapter. See Buchholz (1969, x4) and Morse and
Feshbach (1953a, pp. 515 and 553).
Buchholz (1969) collects many results on boundary-
value problems involving PCFs. Miller (1974) treats
separation of variables by group theoretic methods.
Dean (1966) describes the role of PCFs in quantum me-
chanical systems closely related to the one-dimensional
harmonic oscillator.
Problems on high-frequency scattering in homoge-
neous media by parabolic cylinders lead to asymptotic
methods for integrals involving PCFs. For this topic and
other boundary-value problems see Boyd (1973), Hillion
(1997), Magnus (1941), Morse and Feshbach (1953a,b),
M uller (1988), Ott (1985), Rice (1954), and Shanmugam
(1978).
Lastly, parabolic cylinder functions arise in the de-
scription of ultra cold atoms in harmonic trapping po-
tentials; see Busch et al. (1998) and Edwards et al.
(1999).
318 Parabolic Cylinder Functions
Computation
12.18 Methods of Computation
Because PCFs are special cases of con
uent hypergeo-
metric functions, the methods of computation described
inx13.29 are applicable to PCFs. These include the use
of power-series expansions, recursion, integral represen-
tations, di erential equations, asymptotic expansions,
and expansions in series of Bessel functions. See, espe-
cially, Temme (2000) and Gil et al. (2004b, 2006b,c).
12.19 Tables
Abramowitz and Stegun (1964, Chapter 19) in-
cludesU(a;x) andV(a;x) fora= 0(:1)1(:5)5,
x= 0(:1)5, 5S;W(a;x) fora= 0(:1)1(1)5,
x= 0(:1)5, 4-5D or 4-5S.
Miller (1955) includes W(a;x),W(a; x), and re-
duced derivatives for a= 10(1)10,x= 0(:1)10,
8D or 8S. Modulus and phase functions, and also
other auxiliary functions are tabulated.
Fox (1960) includes modulus and phase functions
forW(a;x) andW(a; x), and several auxiliary
functions for x 1= 0(:005)0:1,a= 10(1)10, 8S.
Kireyeva and Karpov (1961) includes Dp(x(1 +i))
forx= 0(:1)5,p= 0(:1)2, andx= 5(:01)10,
p= 0(:5)2, 7D.
Karpov and Cistova (1964) includes Dp(x) for
p= 2(:1)0,x= 0(:01)5;p= 2(:05)0,
x= 5(:01)10, 6D.
Karpov and Cistova (1968) includes
e 1
4x2Dp( x) ande 1
4x2Dp(ix) forx= 0(:01)5
andx 1= 0(.001 or .0001)5, p= 1(:1)1, 7D or
8S.
Murzewski and Sowa (1972) includes D n(x)
=U
n 1
2;x
forn= 1(1)20,x= 0(:05)3, 7S.
Zhang and Jin (1996, pp. 455{473) includes
U
n 1
2;x
,V
n 1
2;x
,U
1
2;x
,
V
1
2;x
, and derivatives, =n+1
2,
n= 0(1)10(10)30, x= 0:5;1;5;10;30;50, 8S;
W(a;x),W( a;x), and derivatives, a=
h(1)5 +h,x= 0:5;1 anda=h(1)5 +h,x= 5,
h= 0;0:5, 8S. Also, rst zeros of U(a;x),V(a;x),
and of derivatives, a= 6(:5) 1, 6D; rst three
zeros ofW(a; x) and of derivative, a= 0(:5)4,
6D; rst three zeros of W( a;x) and of deriva-
tive,a= 0:5(:5)5:5, 6D; real and imaginaryparts ofU(a;z),a= 1:5(1)1:5,z=x+iy,
x= 0:5;1;5;10,y= 0(:5)10, 8S.
For other tables prior to 1961 see Fletcher et al.
(1962) and Lebedev and Fedorova (1960).
12.20 Approximations
Luke (1969b, pp. 25 and 35) gives Chebyshev-series
expansions for the con
uent hypergeometric functions
U(a;b;x ) andM(a;b;x ) (x13.2(i)) whose regions of va-
lidity include intervals with endpoints x=1and
x= 0, respectively. As special cases of these results
a Chebyshev-series expansion for U(a;x) valid when
x<1follows from (12.7.14), and Chebyshev-series
expansions for U(a;x) andV(a;x) valid when 0x
follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13).
Heredenotes an arbitrary positive constant.
12.21 Software
Seehttp://dlmf.nist.gov/12.21 .
References
General References
The main references used in writing this chapter are
Erd elyi et al. (1953b, v. 2), Miller (1955), and Olver
(1959). For additional bibliographic reading see Buch-
holz (1969), Lebedev (1965), Magnus et al. (1966),
Olver (1997b), and Temme (1996a).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x12.2 See Miller (1955, pp. 9{10, 17, 63{64, and 72),
Miller (1952), Miller (1950), and Olver (1997b,
Chapter 5,x3.3).
x12.3 These graphics were produced at NIST.
x12.4 See Miller (1955, pp. 61{63).
x12.5 See Whittaker (1902) and Miller (1955, pp. 19
and 25{26). For (12.5.4) combine (12.2.18) and
(12.5.1). In Miller (1955, p. 26) the conditions on
agiven in Eqs. (12.5.8) and (12.5.9) are missing.
These conditions are needed to ensure that in each
integrand no poles of the two gamma functions co-
incide.
References 319
x12.7 See Miller (1955, pp. 40{43, 73{74, 76, and 77{
79). For (12.7.7) combine (7.18.11) and (7.18.12).
x12.8 See Miller (1955, p. 65). (12.8.9), (12.8.10),
(12.8.11), and (12.8.12) can be obtained from
(12.5.1), (12.5.6), (12.5.7), and (12.5.9), respec-
tively.
x12.9 (12.9.1) is obtained from (12.7.14) and (13.7.3).
(12.9.2){(12.9.4) follow from (12.2.18) and
(12.2.20). See also Whittaker (1902) and Whit-
taker and Watson (1927, pp. 348{349).
x12.10 See Olver (1959). Equations (12.10.42){
(12.10.46) are rearrangements of Olver's results
and have the advantage of avoiding the many-
valued functions in the explicit expressions for
As(),Bs(),Cs(), andDs().
x12.11 See Whittaker and Watson (1927, p. 354),Dean (1966), Riekstyn s (1991, p. 195), and
Olver (1959). (12.11.9) is obtained by truncat-
ing (12.11.4) at its second term, and applying
(12.10.41) with terms up to and including w5.
x12.12 See Erd elyi et al. (1953b, Chapter 8). For
(12.12.4) see Durand (1975).
x12.13 (12.13.1){(12.13.4) follow from the results in
x12.8(ii) and Taylor's theorem ( x1.10(i)). For
(12.13.5) see Shanker (1939) or Erd elyi et al.
(1953b, Chapter 8). For (12.13.6) see Lepe (1985).
x12.14 See Miller (1955, pp. 17{18, 26, 43, 80{82, 87,
89), Miller (1952), and Olver (1959). The graphs
were produced at NIST.
x12.17 See Je reys and Je reys (1956, xx18.04 and
23.08) and Morse and Feshbach (1953a,b, pp. 553,
1403{1405).
Chapter 13
Con
uent Hypergeometric Functions
A. B. Olde Daalhuis1
Notation 322
13.1 Special Notation . . . . . . . . . . . . . 322
Kummer Functions 322
13.2 De nitions and Basic Properties . . . . . 322
13.3 Recurrence Relations and Derivatives . . 325
13.4 Integral Representations . . . . . . . . . 326
13.5 Continued Fractions . . . . . . . . . . . . 327
13.6 Relations to Other Functions . . . . . . . 327
13.7 Asymptotic Expansions for Large Argument 328
13.8 Asymptotic Approximations for Large Pa-
rameters . . . . . . . . . . . . . . . . . . 330
13.9 Zeros . . . . . . . . . . . . . . . . . . . 331
13.10 Integrals . . . . . . . . . . . . . . . . . . 332
13.11 Series . . . . . . . . . . . . . . . . . . . 333
13.12 Products . . . . . . . . . . . . . . . . . . 333
13.13 Addition and Multiplication Theorems . . 333
Whittaker Functions 334
13.14 De nitions and Basic Properties . . . . . 334
13.15 Recurrence Relations and Derivatives . . 336
13.16 Integral Representations . . . . . . . . . 337
13.17 Continued Fractions . . . . . . . . . . . . 33813.18 Relations to Other Functions . . . . . . . 338
13.19 Asymptotic Expansions for Large Argument 339
13.20 Uniform Asymptotic Approximations for
Large. . . . . . . . . . . . . . . . . . 339
13.21 Uniform Asymptotic Approximations for
Large. . . . . . . . . . . . . . . . . . 341
13.22 Zeros . . . . . . . . . . . . . . . . . . . 342
13.23 Integrals . . . . . . . . . . . . . . . . . . 343
13.24 Series . . . . . . . . . . . . . . . . . . . 344
13.25 Products . . . . . . . . . . . . . . . . . . 345
13.26 Addition and Multiplication Theorems . . 345
Applications 345
13.27 Mathematical Applications . . . . . . . . 345
13.28 Physical Applications . . . . . . . . . . . 346
Computation 346
13.29 Methods of Computation . . . . . . . . . 346
13.30 Tables . . . . . . . . . . . . . . . . . . . 347
13.31 Approximations . . . . . . . . . . . . . . 347
13.32 Software . . . . . . . . . . . . . . . . . . 347
References 347
1School of Mathematics, Edinburgh University, Edinburgh, United Kingdom.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 13) by L.J. Slater. The author is
indebted to J. Wimp for several references.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
321
322 Confluent Hypergeometric Functions
Notation
13.1 Special Notation
(For other notation see pp. xiv and 873.)
m integer.
n,snonnegative integers.
x,yreal variables.
z complex variable.
arbitrary small positive constant.
Euler's constant ( x5.2(ii)).
(x) Gamma function ( x5.2(i)).
(x) 0(x)/ (x).
The main functions treated in this chapter are the
Kummer functions M(a;b;z ) andU(a;b;z ), Olver's
function M(a;b;z ), and the Whittaker functions
M;(z) andW;(z).
Other notations are: 1F1(a;b;z) (x16.2(i)) and
(a;b;z) (Humbert (1920)) for M(a;b;z ); (a;b;z)
(Erd elyi et al. (1953a,x6.5)) forU(a;b;z );V(b
a;b;z ) (Olver (1997b, p. 256)) for ezU(a;b; z);
(1 + 2)M;(Buchholz (1969, p. 12)) for M;(z).
For an historical account of notations see Slater
(1960, Chapter 1).
Kummer Functions
13.2 De nitions and Basic Properties
13.2(i) Di erential Equation
Kummer's Equation
13.2.1 zd2w
dz2+ (b z)dw
dz aw= 0:
This equation has a regular singularity at the origin
with indices 0 and 1 b, and an irregular singularity
at in nity of rank one. It can be regarded as the lim-
iting form of the hypergeometric di erential equation
(x15.10(i)) that is obtained on replacing zbyz/b, let-
tingb!1 , and subsequently replacing the symbol c
byb. In e ect, the regular singularities of the hypergeo-
metric di erential equation at band1coalesce into an
irregular singularity at 1.Standard Solutions
The rst two standard solutions are:
13.2.2
M(a;b;z ) =1X
s=0(a)s
(b)ss!zs= 1 +a
bz+a(a+ 1)
b(b+ 1)2!z2+;
and
13.2.3 M(a;b;z ) =1X
s=0(a)s
(b+s)s!zs;
except that M(a;b;z ) does not exist when bis a non-
positive integer. In other cases
13.2.4 M(a;b;z ) = (b)M(a;b;z ):
The series (13.2.2) and (13.2.3) converge for all
z2C.M(a;b;z ) is entire in zanda, and is a mero-
morphic function of b.M(a;b;z ) is entire in z,a, and
b.
AlthoughM(a;b;z ) does not exist when b= n,
n= 0;1;2;:::, many formulas containing M(a;b;z )
continue to apply in their limiting form. In particular,
13.2.5
lim
b! nM(a;b;z )
(b)=M(a; n;z)
=(a)n+1
(n+ 1)!zn+1M(a+n+ 1;n+ 2;z):
Whena= n,n= 0;1;2;:::,M(a;b;z ) is a poly-
nomial inzof degree not exceeding n; this is also true of
M(a;b;z ) provided that bis not a nonpositive integer.
Another standard solution of (13.2.1) is U(a;b;z ),
which is determined uniquely by the property
13.2.6 U(a;b;z )z a,z!1 ,jphzj3
2 ,
whereis an arbitrary small positive constant. In gen-
eral,U(a;b;z ) has a branch point at z= 0. The princi-
pal branch corresponds to the principal value ofz ain
(13.2.6), and has a cut in the z-plane along the interval
( 1;0]; comparex4.2(i).
Whena= n,n= 0;1;2;:::,U(a;b;z ) is a poly-
nomial inzof degreen:
13.2.7U( n;b;z ) = ( 1)nnX
s=0n
s
(b+s)n s( z)s:
Similarly, when a b+ 1 = n,n= 0;1;2;:::,
13.2.8U(a;a+n+ 1;z) =z anX
s=0n
s
(a)sz s:
13.2 Definitions and Basic Properties 323
Whenb=n+ 1,n= 0;1;2;:::,
13.2.9U(a;n+ 1;z) =( 1)n+1
n! (a n)1X
k=0(a)k
(n+ 1)kk!zk(lnz+ (a+k) (1 +k) (n+k+ 1))
+1
(a)nX
k=1(k 1)!(1 a+k)n k
(n k)!z k;
ifa6= 0; 1; 2;:::, or
13.2.10
U(a;n+ 1;z) = ( 1)a aX
k=0 a
k
(n+k+ 1) a k( z)k;
ifa= 0; 1; 2;:::.
Whenb= n,n= 0;1;2;:::, the following equa-
tion can be combined with (13.2.9) and (13.2.10):
13.2.11U(a; n;z) =zn+1U(a+n+ 1;n+ 2;z):
13.2(ii) Analytic Continuation
Whenm2Z,
13.2.12
U
a;b;ze2im
=2ie ibmsin(bm)
(1 +a b) sin(b)M(a;b;z )
+e 2ibmU(a;b;z ):
Except when z= 0 each branch of U(a;b;z ) is en-
tire inaandb. Unless speci ed otherwise, however,
U(a;b;z ) is assumed to have its principal value.
13.2(iii) Limiting Forms as z!0
13.2.13 M(a;b;z ) = 1 +O(z):
Next, in cases when a= nor n+b 1, wheren
is a nonnegative integer,
13.2.14 U( n;b;z ) = ( 1)n(b)n+O(z);
13.2.15
U( n+b 1;b;z) = ( 1)n(2 b)nz1 b+O
z2 b
:
In all other cases
13.2.16
U(a;b;z ) = (b 1)
(a)z1 b+O
z2 <b
,<b2,b6= 2;
13.2.17
U(a;2;z) =1
(a)z 1+O(lnz);13.2.18
U(a;b;z ) = (b 1)
(a)z1 b+ (1 b)
(a b+ 1)+O
z2 <b
,
1<b<2,b6= 1;
13.2.19
U(a;1;z) = 1
(a)(lnz+ (a) + 2
) +O(zlnz);
13.2.20
U(a;b;z ) = (1 b)
(a b+ 1)+O
z1 <b
, 0<<b<1;
13.2.21
U(a;0;z) =1
(a+ 1)+O(zlnz);
13.2.22
U(a;b;z ) = (1 b)
(a b+ 1)+O(z),<b0,b6= 0:
13.2(iv) Limiting Forms as z!1
Except when a= 0; 1;::: (polynomial cases),
13.2.23 M(a;b;z )ezza b
(a) ,jphzj1
2 ,
whereis an arbitrary small positive constant.
ForU(a;b;z ) see (13.2.6).
13.2(v) Numerically Satisfactory Solutions
Fundamental pairs of solutions of (13.2.1) that are nu-
merically satisfactory ( x2.7(iv)) in the neighborhood of
in nity are
13.2.24U(a;b;z ) ,ezU
b a;b;e iz
,
1
2phz3
2,
13.2.25U(a;b;z ) ,ezU
b a;b;eiz
,
3
2phz1
2.
A fundamental pair of solutions that is numerically
satisfactory near the origin is
13.2.26
M(a;b;z ); z1 bM(a b+ 1;2 b;z),b62Z.
324 Confluent Hypergeometric Functions
Whenb=n+ 1 = 1;2;3;:::, a fundamental pair that is numerically satisfactory near the origin is M(a;n+ 1;z)
and
13.2.27nX
k=1n!(k 1)!
(n k)!(1 a)kz k 1X
k=0(a)k
(n+ 1)kk!zk(lnz+ (a+k) (1 +k) (n+k+ 1));
ifa n6= 0; 1; 2;:::, orM(a;n+ 1;z) and
13.2.28nX
k=1n!(k 1)!
(n k)!(1 a)kz k
aX
k=0(a)k
(n+ 1)kk!zk(lnz+ (1 a k) (1 +k) (n+k+ 1))+( 1)1 a( a)!1X
k=1 a(k 1 +a)!
(n+ 1)kk!zk;
ifa= 0; 1; 2;:::, orM(a;n+ 1;z) and
13.2.29nX
k=a(k 1)!
(n k)!(k a)!z k;
ifa= 1;2;:::;n .
Whenb= n= 0; 1; 2;:::, a fundamental pair that is numerically satisfactory near the origin is zn+1
M(a+n+ 1;n+ 2;z) and
13.2.30
n+1X
k=1(n+ 1)!(k 1)!
(n k+ 1)!( a n)kzn k+1 1X
k=0(a+n+ 1)k
(n+ 2)kk!zn+k+1(lnz+ (a+n+k+ 1) (1 +k) (n+k+ 2));
ifa6= 0; 1; 2;:::, orzn+1M(a+n+ 1;n+ 2;z) and
13.2.31n+1X
k=1(n+ 1)!(k 1)!
(n k+ 1)!( a n)kzn k+1
a n 1X
k=0(a+n+ 1)k
(n+ 2)kk!zn+k+1(lnz+ ( a n k) (1 +k) (n+k+ 2))
+ ( 1)n a( a n 1)!1X
k= a n(k+a+n)!
(n+ 2)kk!zn+k+1;
ifa= n 1; n 2; n 3;:::, orzn+1M(a+n+ 1;n+ 2;z) and
13.2.32n+1X
k=a+n+1(k 1)!
(n k+ 1)!(k a n 1)!zn k+1;
ifa= 0; 1;:::; n.
13.2(vi) Wronskians
13.2.33 W
M(a;b;z );z1 bM(a b+ 1;2 b;z)
= sin(b)z bez=;
13.2.34 WfM(a;b;z );U(a;b;z )g= z bez
(a);
13.2.35 W
M(a;b;z );ezU
b a;b;eiz
=ebiz bez
(b a);
13.2.36 W
z1 bM(a b+ 1;2 b;z);U(a;b;z )
= z bez
(a b+ 1);
13.2.37 W
z1 bM(a b+ 1;2 b;z);ezU
b a;b;eiz
= z bez
(1 a);
13.2.38 W
U(a;b;z );ezU
b a;b;eiz
=e(a b)iz bez:
13.3 Recurrence Relations and Derivatives 325
13.2(vii) Connection Formulas
Kummer's Transformations
13.2.39 M(a;b;z ) =ezM(b a;b; z);
13.2.40 U(a;b;z ) =z1 bU(a b+ 1;2 b;z):
13.2.411
(b)M(a;b;z ) =eai
(b a)U(a;b;z ) +e(b a)i
(a)ezU
b a;b;eiz
:
Also, when bis not an integer
13.2.42 U(a;b;z ) = (1 b)
(a b+ 1)M(a;b;z ) + (b 1)
(a)z1 bM(a b+ 1;2 b;z):
13.3 Recurrence Relations and Derivatives
13.3(i) Recurrence Relations
13.3.1 (b a)M(a 1;b;z) + (2a b+z)M(a;b;z ) aM(a+ 1;b;z) = 0;
13.3.2 b(b 1)M(a;b 1;z) +b(1 b z)M(a;b;z ) +z(b a)M(a;b+ 1;z) = 0;
13.3.3 (a b+ 1)M(a;b;z ) aM(a+ 1;b;z) + (b 1)M(a;b 1;z) = 0;
13.3.4 bM(a;b;z ) bM(a 1;b;z) zM(a;b+ 1;z) = 0;
13.3.5 b(a+z)M(a;b;z ) +z(a b)M(a;b+ 1;z) abM (a+ 1;b;z) = 0;
13.3.6 (a 1 +z)M(a;b;z ) + (b a)M(a 1;b;z) + (1 b)M(a;b 1;z) = 0:
13.3.7 U(a 1;b;z) + (b 2a z)U(a;b;z ) +a(a b+ 1)U(a+ 1;b;z) = 0;
13.3.8 (b a 1)U(a;b 1;z) + (1 b z)U(a;b;z ) +zU(a;b+ 1;z) = 0;
13.3.9 U(a;b;z ) aU(a+ 1;b;z) U(a;b 1;z) = 0;
13.3.10 (b a)U(a;b;z ) +U(a 1;b;z) zU(a;b+ 1;z) = 0;
13.3.11 (a+z)U(a;b;z ) zU(a;b+ 1;z) +a(b a 1)U(a+ 1;b;z) = 0;
13.3.12 (a 1 +z)U(a;b;z ) U(a 1;b;z) + (a b+ 1)U(a;b 1;z) = 0:
Kummer's di erential equation (13.2.1) is equivalent to
13.3.13 (a+ 1)zM(a+ 2;b+ 2;z) + (b+ 1)(b z)M(a+ 1;b+ 1;z) b(b+ 1)M(a;b;z ) = 0;
and
13.3.14 (a+ 1)zU(a+ 2;b+ 2;z) + (z b)U(a+ 1;b+ 1;z) U(a;b;z ) = 0:
13.3(ii) Di erentiation Formulas
13.3.15d
dzM(a;b;z ) =a
bM(a+ 1;b+ 1;z);
13.3.16dn
dznM(a;b;z ) =(a)n
(b)nM(a+n;b+n;z);
13.3.17
zd
dzzn
za 1M(a;b;z )
= (a)nza+n 1M(a+n;b;z );
13.3.18
dn
dzn
zb 1M(a;b;z )
= (b n)nzb n 1M(a;b n;z);13.3.19
zd
dzzn
zb a 1e zM(a;b;z )
= (b a)nzb a+n 1e zM(a n;b;z );
13.3.20dn
dzn
e zM(a;b;z )
= ( 1)n(b a)n
(b)ne zM(a;b+n;z);
13.3.21dn
dzn
zb 1e zM(a;b;z )
= (b n)nzb n 1e zM(a n;b n;z):
326 Confluent Hypergeometric Functions
13.3.22d
dzU(a;b;z ) = aU(a+ 1;b+ 1;z);
13.3.23dn
dznU(a;b;z ) = ( 1)n(a)nU(a+n;b+n;z);
13.3.24
zd
dzzn
za 1U(a;b;z )
= (a)n(a b+ 1)nza+n 1U(a+n;b;z );
13.3.25dn
dzn
zb 1U(a;b;z )
= ( 1)n(a b+ 1)nzb n 1U(a;b n;z);
13.3.26
zd
dzzn
zb a 1e zU(a;b;z )
= ( 1)nzb a+n 1e zU(a n;b;z );
13.3.27dn
dzn
e zU(a;b;z )
= ( 1)ne zU(a;b+n;z);
13.3.28dn
dzn
zb 1e zU(a;b;z )
= ( 1)nzb n 1e zU(a n;b n;z):
Other versions of several of the identities in this sub-
section can be constructed with the aid of the operator
identity
13.3.29
zd
dzzn
=zndn
dznzn,n= 1;2;3;:::.
13.4 Integral Representations
13.4(i) Integrals Along the Real Line
13.4.1
M(a;b;z ) =1
(a) (b a)Z1
0eztta 1(1 t)b a 1dt,
<b><a>0;
13.4.2
M(a;b;z ) =1
(b c)Z1
0M(a;c;zt )tc 1(1 t)b c 1dt,
<b><c>0;
13.4.3
M(a;b; z) =z1
2 1
2b
(a)Z1
0e tta 1
2b 1
2Jb 1
2p
zt
dt,
<a>0:
For the function Jb 1seex10.2(ii).
13.4.4U(a;b;z ) =1
(a)Z1
0e ztta 1(1 +t)b a 1dt,
<a>0,jphzj<1
2;
13.4.5
U(a;b;z )
=z1 a
(a) (1 +a b)Z1
0U(b a;b;t )e tta 1
t+zdt,
jphzj<,<a>max (<b 1;0);13.4.6
U(a;b;z )
=( 1)nz1 b n
(1 +a b)Z1
0M(b a;b;t )e ttb+n 1
t+zdt,
jphzj<,n= 0;1;2;:::, <b<n< 1 +<(a b),
13.4.7U(a;b;z ) =2z1
2 1
2b
(a) (a b+ 1)
Z1
0e tta 1
2b 1
2Kb 1
2p
zt
dt,
<a>max (<b 1;0);
13.4.8
U(a;b;z ) =zc a
Z1
0e zttc 1
2F1(a;a b+ 1;c; t)dt,
jphzj<1
2;
wherecis arbitrary,<c >0. For the functions Kb 1
and 2F1seex10.25(ii) andxx15.1, 15.2(i).
13.4(ii) Contour Integrals
13.4.9
M(a;b;z ) = (1 +a b)
2i (a)Z(1+)
0eztta 1(t 1)b a 1dt,
b a6= 1;2;3;:::,<a>0.
13.4.10
M(a;b;z )
=e ai (1 a)
2i (b a)Z(0+)
1eztta 1(1 t)b a 1dt,
a6= 1;2;3;:::,<(b a)>0.
Figure 13.4.1 : Contour of integration in (13.4.11).
(Compare Figure 5.12.3.)
13.4.11
M(a;b;z )
=e bi (1 a) (1 +a b)
1
42Z(0+;1+;0 ;1 )
eztta 1(1 t)b a 1dt,
a;b a6= 1;2;3;:::.
The contour of integration starts and terminates at a
point on the real axis between 0 and 1. It encircles
t= 0 andt= 1 once in the positive sense, and then
once in the negative sense. See Figure 13.4.1. The frac-
tional powers are continuous and assume their principal
13.5 Continued Fractions 327
values att= . Similar conventions also apply to the
remaining integrals in this subsection.
13.4.12
M(a;c;z )
= (b)
2iz1 bZ(0+;1+)
1eztt b
2F1(a;b;c; 1/t)dt,
b6= 0; 1; 2;:::,jphzj<1
2.
At the point where the contour crosses the interval
(1;1),t band the 2F1function assume their princi-
pal values; compare xx15.1 and 15.2(i). A special caseis
13.4.13M(a;b;z ) =z1 b
2iZ(0+;1+)
1eztt b
1 1
t a
dt,
jphzj<1
2:
13.4.14
U(a;b;z )
=e ai (1 a)
2iZ(0+)
1e ztta 1(1 +t)b a 1dt,
a6= 1;2;3;:::,jphzj<1
2.
The contour cuts the real axis between 1 and 0. At this
point the fractional powers are determined by ph t=
and ph(1 + t) = 0.
13.4.15U(a;b;z )
(c) (c b+ 1)=z1 c
2iZ(0+)
1eztt c
2F1
a;c;a+c b+ 1; 1 1
t
dt,jphzj<1
2.
Again,t cand the 2F1function assume their princi-
pal values where the contour intersects the positive real
axis.
13.4(iii) Mellin{Barnes Integrals
Ifa6= 0; 1; 2;:::, then
13.4.16
M(a;b; z) =1
2i (a)Zi1
i1 (a+t) ( t)
(b+t)ztdt,
jphzj<1
2;
where the contour of integration separates the poles of
(a+t) from those of ( t).
Ifaanda b+ 16= 0; 1; 2;:::, then
13.4.17
U(a;b;z )
=z a
2iZi1
i1 (a+t) (1 +a b+t) ( t)
(a) (1 +a b)z tdt,
jphzj<3
2;
where the contour of integration separates the poles of
(a+t) (1 +a b+t) from those of ( t).
13.4.18
U(a;b;z ) =z1 bez
2iZi1
i1 (b 1 +t) (t)
(a+t)z tdt,
jphzj<1
2;
where the contour of integration passes all the poles of
(b 1 +t) (t) on the right-hand side.
13.5 Continued Fractions
Ifa;b2Csuch thata6= 1; 2; 3;:::, anda b6=
0;1;2;:::, then
13.5.1M(a;b;z )
M(a+ 1;b+ 1;z)= 1 +u1z
1 +u2z
1 +;where
13.5.2u2n+1=a b n
(b+ 2n)(b+ 2n+ 1),
u2n=a+n
(b+ 2n 1)(b+ 2n).
This continued fraction converges to the meromorphic
function of zon the left-hand side everywhere in C. For
more details on how a continued fraction converges to a
meromorphic function see Jones and Thron (1980).
Ifa;b2Csuch thata6= 0; 1; 2;:::, andb a6=
2;3;4;:::, then
13.5.3U(a;b;z )
U(a;b 1;z)= 1 +v1=z
1 +v2=z
1 +;
where
13.5.4v2n+1=a+n,v2n=a b+n+ 1 .
This continued fraction converges to the meromorphic
function of zon the left-hand side throughout the sector
jphzj<.
See also Cuyt et al. (2008, pp. 322{330).
13.6 Relations to Other Functions
13.6(i) Elementary Functions
13.6.1 M(a;a;z ) =ez;
13.6.2 M(1;2;2z) =ez
zsinhz;
13.6.3 M(0;b;z) =U(0;b;z) = 1;
13.6.4 U(a;a+ 1;z) =z a:
328 Confluent Hypergeometric Functions
13.6(ii) Incomplete Gamma Functions
For the notation see xx6.2(i), 7.2(i), 8.2(i), and 8.19(i).
Whena bis an integer or ais a positive integer
the Kummer functions can be expressed as incomplete
gamma functions (or generalized exponential integrals).
For example,
13.6.5
M(a;a+ 1; z) =e zM(1;a+ 1;z) =az a
(a;z);
13.6.6U(a;a;z ) =z1 aU(1;2 a;z)
=z1 aezEa(z) =ez (1 a;z):
Special cases are the error functions
13.6.7 M 1
2;3
2; z2
=p
2zerf(z);
13.6.8 U 1
2;1
2;z2
=pez2erfc(z):
13.6(iii) Modi ed Bessel Functions
Whenb= 2athe Kummer functions can be expressed
as modi ed Bessel functions. For the notation see
xx10.25(ii) and 9.2(i).
13.6.9
M
+1
2;2+ 1;2z
= (1 +)ez(z/2) I(z);
13.6.10
U
+1
2;2+ 1;2z
=1pez(2z) K(z);
13.6.11U
5
6;5
3;4
3z3=2
=p35=6exp 2
3z3=2
22=3zAi(z):
13.6(iv) Parabolic Cylinder Functions
For the notation see x12.2.
13.6.12U 1
2a+1
4;1
2;1
2z2
= 21
2a+1
4e1
4z2U(a;z);
13.6.13U 1
2a+3
4;3
2;1
2z2
= 21
2a+3
4e1
4z2
zU(a;z):
13.6.14M 1
2a+1
4;1
2;1
2z2
=21
2a 3
4 1
2a+3
4
e1
4z2
p
(U(a;z) +U(a; z));
13.6.15M 1
2a+3
4;3
2;1
2z2
=21
2a 5
4 1
2a+1
4
e1
4z2
zp
(U(a; z) U(a;z)):13.6(v) Orthogonal Polynomials
Special cases ofx13.6(iv) are as follows. For the nota-
tion seexx18.3, 18.19.
Hermite Polynomials
13.6.16 M
n;1
2;z2
= ( 1)nn!
(2n)!H2n(z);
13.6.17 M
n;3
2;z2
= ( 1)nn!
(2n+ 1)!2zH2n+1(z);
13.6.18U 1
2 1
2n;3
2;z2
= 2 nz 1Hn(z):
Laguerre Polynomials
13.6.19
U( n; + 1;z) = ( 1)n( + 1)nM( n; + 1;z)
= ( 1)nn!L( )
n(z):
Charlier Polynomials
13.6.20
U( n;z n+ 1;a) = ( z)nM( n;z n+ 1;a)
=anCn(z;a):
13.6(vi) Generalized Hypergeometric Functions
13.6.21U(a;b;z ) =z a
2F0
a;a b+ 1; ; z 1
:
For the de nition of 2F0
a;a b+ 1; ; z 1
when
neitheranora b+1 is a nonpositive integer see x16.5.
13.7 Asymptotic Expansions for Large
Argument
13.7(i) Poincar e-Type Expansions
Asx!1
13.7.1 M(a;b;x )exxa b
(a)1X
s=0(1 a)s(b a)s
s!x s;
provided that a6= 0; 1;:::.
Asz!1
13.7.2
M(a;b;z )ezza b
(a)1X
s=0(1 a)s(b a)s
s!z s+eiaz a
(b a)1X
s=0(a)s(a b+ 1)s
s!( z) s, 1
2+phz3
2 ,
unlessa= 0; 1;::: andb a= 0; 1;:::. Heredenotes an arbitrary small positive constant. Also,
13.7.3 U(a;b;z )z a1X
s=0(a)s(a b+ 1)s
s!( z) s, jphzj3
2 :
13.7 Asymptotic Expansions for Large Argument 329
13.7(ii) Error Bounds
Figure 13.7.1 : Regions R1,R2,R2,R3, and R3are
the closures of the indicated unshaded regions bounded
by the straight lines and circular arcs centered at the
origin, with r=jb 2aj.
13.7.4
U(a;b;z ) =z an 1X
s=0(a)s(a b+ 1)s
s!( z) s+"n(z);where
13.7.5j"n(z)j; 1j"0
n(z)j
2 Cn (a)n(a b+ 1)n
n!za+n exp2 C 1
jzj
;
and with the notation of Figure 13.7.1
13.7.6Cn= 1 , (n) ,
(n) +2n
n,
according as
13.7.7z2R1,z2R2[R2,z2R3[R3,
respectively, with
13.7.8
=j(b 2a)/zj,=
1
2+1
2p
1 42 1/2
,
(n) =p 1
2n+ 1
= 1
2n+1
2
.
Also, when z2R1[R2[R2
13.7.9 =1
1 , =1 2+jzj 1
2(1 ),
=1
2 2a2 2ab+b +(1 +1
4)
(1 )2,
and whenz2R3[R3is replaced by andjzj 1is
replaced by jzj 1everywhere in (13.7.9).
For numerical values of (n) see Table 9.7.1.
Corresponding error bounds for (13.7.2) can be con-
structed by combining (13.2.41) with (13.7.4){(13.7.9).
13.7(iii) Exponentially-Improved Expansion
Let
13.7.10 U(a;b;z ) =z an 1X
s=0(a)s(a b+ 1)s
s!( z) s+Rn(a;b;z );
and
13.7.11
Rn(a;b;z ) =( 1)n2za b
(a) (a b+ 1) m 1X
s=0(1 a)s(b a)s
s!( z) sGn+2a b s(z) + (1 a)m(b a)mRm;n(a;b;z )!
;
wheremis an arbitrary nonnegative integer, and
13.7.12 Gp(z) =ez
2 (p) (1 p;z):
(For the notation see x8.2(i).) Then as z!1 withjjzj njbounded and a;b;m xed
13.7.13 Rm;n(a;b;z ) =(
O
e jzjz m
;jphzj;
O(ezz m); jphzj5
2 :
For proofs see Olver (1991b, 1993a). For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver
(1995a).
330 Confluent Hypergeometric Functions
13.8 Asymptotic Approximations for Large
Parameters
13.8(i) Largejbj, Fixedaandz
Ifb!1 inCin such a way that jb+nj>0 for all
n= 0;1;2;:::, then
13.8.1M(a;b;z ) =n 1X
s=0(a)s
(b)ss!zs+O
jbj n
:
For xedaandzinC
13.8.2M(a;b;z ) (b)
(b a)1X
s=0(a)sqs(z;a)b s a;asb!1 injphbj , whereq0(z;a) = 1 and
13.8.3
et 1a 1exp
t+z(1 e t)
=1X
s=0qs(z;a)ts+a 1:
When the foregoing results are combined with Kum-
mer's transformation (13.2.39), an approximation is ob-
tained for the case when jbjis large, andjb ajandjzj
are bounded.
13.8(ii) Large bandz, Fixedaandb=z
Let=z=b > 0 and=p
2( 1 ln) with
sign() = sign( 1). Then
13.8.4M(a;b;z )b1
2ae1
42b0
@ 1
a 1
U
a 1
2; p
b
+
1
a 1
1a!U
a 3
2; p
b
p
b1
A
and
13.8.5U(a;b;z )b 1
2ae1
42b0
@ 1
a 1
U
a 1
2;p
b
1
a 1
1a!U
a 3
2;p
b
p
b1
A
asb!1 , uniformly in compact -intervals of (0 ;1) and compact real a-intervals. For the parabolic cylinder
functionUseex12.2, and for an extension to an asymptotic expansion see Temme (1978).
Special cases are
13.8.6 M(a;b;b ) =pb
21
2a
1
1
2(a+ 1)+(a+ 1)p
8=b
3 1
2a+O1
b!
;
and
13.8.7 U(a;b;b ) =p(2b) 1
2a
1
1
2(a+ 1) (a+ 1)p
8=b
3 1
2a+O1
b!
:
To obtain approximations for M(a;b;z ) andU(a;b;z ) that hold as b!1 , witha >1
2 bandz >0 combine
(13.14.4), (13.14.5) with x13.20(i).
Also, more complicated|but more powerful|uniform asymptotic approximations can be obtained by combining
(13.14.4), (13.14.5) with xx13.20(iii) and 13.20(iv).
13.8(iii) Large a
For the notation see xx10.2(ii), 10.25(ii), and 2.8(iv).
Whena!+1withb(1) xed,
13.8.8U(a;b;x ) =2e1
2x
(a) r2
tanhw
21 e w
b
1 bK1 b(2 a) +a 1a 1+
1 + 1 b
e 2 aO(1)!
;
wherew= arccosh
1 + (2a) 1x
, and = (w+ sinhw)/2. (13.8.8) holds uniformly with respect to x2[0;1). For
the caseb>1 the transformation (13.2.40) can be used.
For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related
results seex13.21(i).
Whena! 1 withb(1) xed,
13.8.9M(a;b;x ) = (b)e1
2x
(1
2b a)x1
2 1
2b
Jb 1p
2x(b 2a)
+ envJb 1p
2x(b 2a)
O
jaj 1
2
;
13.9 Zeros 331
and
13.8.10U(a;b;x ) = 1
2b a+1
2
e1
2xx1
2 1
2b
cos(a)Jb 1p
2x(b 2a)
sin(a)Yb 1p
2x(b 2a)
+ envYb 1p
2x(b 2a)
O
jaj 1
2
;
uniformly with respect to bounded positive values of xin each case.
For asymptotic approximations to M(a;b;x ) andU(a;b;x ) asa! 1 that hold uniformly with respect to
x2(0;1) and bounded positive values of ( b 1)=jaj, combine (13.14.4), (13.14.5) with xx13.21(ii), 13.21(iii).
13.9 Zeros
13.9(i) Zeros of M(a;b;z )
Ifaandb a6= 0; 1; 2;:::, thenM(a;b;z ) has in-
nitely many z-zeros in C. Whena;b2Rthe number of
real zeros is nite. Let p(a;b) be the number of positive
zeros. Then
13.9.1 p(a;b) =d ae, a<0,b0;
13.9.2 p(a;b) = 0, a0,b0;
13.9.3 p(a;b) = 1, a0, 1<b< 0;
13.9.4p(a;b) =
1
2b
1
2(b+ 1)
,a0,b 1:
13.9.5
p(a;b) =d ae d be,d aed be,a<0,b<0;
13.9.6
p(a;b) =1
2(d be d ae+ 1)
1
2(d be d ae)
,
d be>d ae>0:
The number of negative real zeros n(a;b) is given by
13.9.7 n(a;b) =p(b a;b):
Whena<0 andb>0 let r,r= 1;2;3;:::, be the
positive zeros of M(a;b;x ) arranged in increasing order
of magnitude, and let jb 1;rbe therth positive zero of
the Bessel function Jb 1(x) (x10.21(i)). Then
13.9.8
r=j2
b 1;r
2b 4a
1 +2b(b 2) +j2
b 1;r
3(2b 4a)2!
+O1
a5
;
asa! 1 withr xed.
Inequalities for rare given in Gatteschi (1990), and
identities involving in nite series of all of the complex
zeros ofM(a;b;x ) are given in Ahmed and Muldoon
(1980).
For xeda;b2Cthe largez-zeros ofM(a;b;z ) sat-
isfy
13.9.9z=(2n+a)i+ ln
(a)
(b a)(2ni)b 2a
+O
n 1lnn
;
wherenis a large positive integer, and the logarithm
takes its principal value ( x4.2(i)).LetP denote the closure of the domain that is
bounded by the parabola y2= 4 (x+ ) and con-
tains the origin. Then M(a;b;z ) has no zeros in the
regionsPb/a, if 0< ba;P1, if 1ab;P ,
where = (2a b+ab)/(a(a+ 1)), if 0 < a < 1
andab < 2a/(1 a). The same results apply for
thenth partial sums of the Maclaurin series (13.2.2) of
M(a;b;z ).
More information on the location of real zeros can
be found in Zarzo et al. (1995).
For xedbandzinCthe largea-zeros ofM(a;b;z )
are given by
13.9.10
a= 2
4z
n2+ (b 3
2)n
1
16z
(b 3
2)22+4
3z2 8b(z 1) 4b2 3
+O
n 1
;
wherenis a large positive integer.
For xedaandzinCthe function M(a;b;z ) has
only a nite number of b-zeros.
13.9(ii) Zeros of U(a;b;z )
For xedaandbinC,U(a;b;z ) has a nite number of
z-zeros in the sector jphzj3
2 (<3
2). LetT(a;b)
be the total number of zeros in the sector jphzj< ,
P(a;b) be the corresponding number of positive zeros,
anda,b, anda b+ 1 be nonintegers. For the case
b1
13.9.11
T(a;b) =b ac+ 1,a<0, (a) (a b+ 1)>0;
13.9.12T(a;b) =b ac,a<0, (a) (a b+ 1)<0;
13.9.13 T(a;b) = 0, a>0;
and
13.9.14 P(a;b) =db a 1e,a+ 1<b;
13.9.15 P(a;b) = 0, a+ 1b:
For the case b1 we can use T(a;b) =T(a b+1;2 b)
andP(a;b) =P(a b+ 1;2 b).
In Wimp (1965) it is shown that if a;b2Rand
2a b > 1, thenU(a;b;z ) has no zeros in the sector
jphzj1
2.
Inequalities for the zeros of U(a;b;x ) are given in
Gatteschi (1990).
332 Confluent Hypergeometric Functions
For xedbandzinCthe largea-zeros ofU(a;b;z )
are given by
13.9.16
a n 2
pzn 2z
2+1
2b+1
4
+z2 1
3 4 2
+z (b 1)2+1
4
4pzn+O1
n
;
wherenis a large positive integer.
For xedaandzinC,U(a;b;z ) has two in nite
strings ofb-zeros that are asymptotic to the imaginary
axis asjbj!1 .
13.10 Integrals
13.10(i) Inde nite Integrals
Whena6= 1,
13.10.1Z
M(a;b;z )dz=1
a 1M(a 1;b 1;z);
13.10.2Z
U(a;b;z )dz= 1
a 1U(a 1;b 1;z):
Other formulas of this kind can be constructed by in-
version of the di erentiation formulas given in x13.3(ii).
13.10(ii) Laplace Transforms
For the notation see xx15.1, 15.2(i), and 10.25(ii).
13.10.3Z1
0e zttb 1M(a;c;kt )dt= (b)z b
2F1(a;b;c;k/z),
<b>0,<z>max (<k;0);
13.10.4Z1
0e zttb 1M(a;b;t )dt=z b
1 1
z a
,
<b>0,<z>1;
13.10.5Z1
0e ttb 1M(a;c;t )dt= (b) (c a b)
(c a) (c b),
<(c a)><b>0;
13.10.6Z1
0e zt t2t2b 2M
a;b;t2
dt
=1
2 1
2
b 1
2
U
b 1
2;a+1
2;1
4z2
,
<b>1
2,<z>0;
13.10.7Z1
0e zttb 1U(a;c;t )dt
= (b) (b c+ 1)
z b
2F1
a;b;a+b c+ 1; 1 1
z
,
<b>max (<c 1;0),<z>0:Loop Integrals
13.10.81
2iZ(0+)
1etzt aM(a;b;y /t)dt
=1
(a)z1
2(2a b 1)y1
2(1 b)Ib 1(2pzy),
<z>0:
13.10.91
2iZ(0+)
1etzt aU(a;b;y /t)dt
=2z1
2(2a b 1)y1
2(1 b)
(a) (a b+ 1)Kb 1(2pzy),<z>0:
For additional Laplace transforms see Erd elyi et al.
(1954a,xx4.22, 5.20), Oberhettinger and Badii (1973,
x1.17), and Prudnikov et al. (1992a,xx3.34, 3.35). In-
verse Laplace transforms are given in Oberhettinger and
Badii (1973,x2.16) and Prudnikov et al. (1992b,xx3.33,
3.34).
13.10(iii) Mellin Transforms
13.10.10Z1
0t 1M(a;b; t)dt= () (a )
(a) (b ), 0<<<<a;
13.10.11Z1
0t 1U(a;b;t )dt= () (a ) ( b+ 1)
(a) (a b+ 1),
max (<b 1;0)<<<<a:
For additional Mellin transforms see Erd elyi et al.
(1954a,xx6.9, 7.5), Marichev (1983, pp. 283{287), and
Oberhettinger (1974, xx1.13, 2.8).
13.10(iv) Fourier Transforms
13.10.12Z1
0cos(2xt)M
a;b; t2
dt
=p
2 (a)x2a 1e x2U
b 1
2;a+1
2;x2
,
<a>0:
For additional Fourier transforms see Erd elyi et al.
(1954a,xx1.14, 2.14, 3.3) and Oberhettinger (1990,
xx1.22, 2.22).
13.10(v) Hankel Transforms
For the notation see x10.2(ii).
13.10.13Z1
0e ttb 1 1
2M(a;b;t )J
2p
xt
dt
=x a+1
2e xM( b+ 1; a+ 1;x),
x>0, 2<a<<+5
2,<b>0;
13.10.14Z1
0e tt1
2M(a;b;t )J
2p
xt
dt
=x1
2e x
(b a)U(a;a b++ 2;x),
x>0, 1<< <2<(b a) 1
2;
13.11 Series 333
13.10.15Z1
0t1
2U(a;b;t )J
2p
xt
dt
= ( b+ 2)
(a)x1
2U( b+ 2; a+ 2;x),
x>0, max (<b 2; 1)<< <2<a+1
2;
13.10.16Z1
0e tt1
2U(a;b;t )J
2p
xt
dt
= ( b+ 2)x1
2e xM(a;a b++ 2;x),
x>0, max (<b 2; 1)<<:
For additional Hankel transforms and also other
Bessel transforms see Erd elyi et al. (1954b,x8.18) and
Oberhettinger (1972, xx1.16 and 3.4.42{46, 4.4.45{47,
5.94{97).
13.10(vi) Other Integrals
For integral transforms in terms of Whittaker functions
seex13.23(iv). Additional integrals can be found in
Apelblat (1983, pp. 388{392), Erd elyi et al. (1954b),
Gradshteyn and Ryzhik (2000, x7.6), Magnus et al.
(1966,x6.1.2), Prudnikov et al. (1990,xx1.13, 1.14, 2.19,
4.2.2), Prudnikov et al. (1992a,xx3.35, 3.36), and Prud-
nikov et al. (1992b,xx3.33, 3.34). See also (13.4.2),
(13.4.5), (13.4.6).
13.11 Series
Forz2C,
13.11.1M(a;b;z ) =
a 1
2
e1
2z 1
4z1
2 a
1X
s=0(2a 1)s(2a b)s
(b)ss!
a 1
2+s
Ia 1
2+s 1
2z
,
a+1
2;b6= 0; 1; 2;:::.
(13.6.9) is a special case.
For additional expansions combine (13.14.4),
(13.14.5), andx13.24. For other series expansions see
Hansen (1975,xx66 and 87), Prudnikov et al. (1990,
x6.6), and Tricomi (1954, x1.8). See alsox13.13.
13.12 Products
13.12.1
M(a;b;z )M( a; b; z)
+a(a b)z2
b2(1 b2)M(1 +a;2 +b;z)M(1 a;2 b; z) = 1:
For generalizations of this quadratic relation see Ma-
jima et al. (2000).
For integral representations, integrals, and series
containing products of M(a;b;z ) andU(a;b;z ) see
Erd elyi et al. (1953a,x6.15.3).13.13 Addition and Multiplication
Theorems
13.13(i) Addition Theorems for M(a;b;z )
The function M(a;b;x +y) has the following expan-
sions:
13.13.11X
n=0(a)nyn
(b)nn!M(a+n;b+n;x);
13.13.2x+y
x1 b1X
n=0(1 b)n( y/x)n
n!M(a;b n;x),
jyj<jxj;
13.13.3x
x+ya1X
n=0(a)nyn
n!(x+y)nM(a+n;b;x ),<(y=x)> 1
2;
13.13.4ey1X
n=0(b a)n( y)n
(b)nn!M(a;b+n;x);
13.13.5eyx
x+yb a1X
n=0(b a)nyn
n!(x+y)n
M(a n;b;x ),<((y+x)=x)>1
2;
13.13.6eyx+y
x1 b1X
n=0(1 b)n( y)n
n!xn
M(a n;b n;x),jyj<jxj:
13.13(ii) Addition Theorems for U(a;b;z )
The function U(a;b;x +y) has the following expansions:
13.13.71X
n=0(a)n( y)n
n!U(a+n;b+n;x),jyj<jxj;
13.13.8x+y
x1 b1X
n=0(1 +a b)n( y/x)n
n!U(a;b n;x),
jyj<jxj;
13.13.9x
x+ya1X
n=0(a)n(1 +a b)nyn
n!(x+y)nU(a+n;b;x ),
<(y=x)> 1
2;
13.13.10 ey1X
n=0( y)n
n!U(a;b+n;x),jyj<jxj;
13.13.11eyx
x+yb a1X
n=0( y)n
n!(x+y)nU(a n;b;x ),
<(y=x)> 1
2;
13.13.12
eyx+y
x1 b1X
n=0( y)n
n!xnU(a n;b n;x),jyj<jxj:
334 Confluent Hypergeometric Functions
13.13(iii) Multiplication Theorems for
M(a;b;z )andU(a;b;z )
To obtain similar expansions for M(a;b;xy ) and
U(a;b;xy ), replaceyin the previous two subsections
by (y 1)x.
Whittaker Functions
13.14 De nitions and Basic Properties
13.14(i) Di erential Equation
Whittaker's Equation
13.14.1d2W
dz2+
1
4+
z+1
4 2
z2
W= 0:
This equation is obtained from Kummer's equation
(13.2.1) via the substitutions W=e 1
2zz1
2+w,=
1
2b a, and=1
2b 1
2. It has a regular singularity at
the origin with indices1
2, and an irregular singularity
at in nity of rank one.
Standard Solutions
Standard solutions are:
13.14.2M;(z) =e 1
2zz1
2+M 1
2+ ;1 + 2;z
;
13.14.3W;(z) =e 1
2zz1
2+U 1
2+ ;1 + 2;z
;
except that M;(z) does not exist when 2 =
1; 2; 3;:::.Conversely,
13.14.4M(a;b;z ) =e1
2zz 1
2bM1
2b a;1
2b 1
2(z);
13.14.5U(a;b;z ) =e1
2zz 1
2bW1
2b a;1
2b 1
2(z):
The series
13.14.6
M;(z) =e 1
2zz1
2+1X
s=0 1
2+
s
(1 + 2)ss!zs
=z1
2+1X
n=02F1 n;1
2+
1 + 2; 2
1
2zn
n!,
26= 1; 2; 3;:::,
converge for all z2C.
In general M;(z) andW;(z) are many-valued
functions of zwith branch points at z= 0 and
z=1. The principal branches correspond to
the principal branches of the functions z1
2+and
U 1
2+ ;1 + 2;z
on the right-hand sides of the
equations (13.14.2) and (13.14.3); compare x4.2(i).
AlthoughM;(z) does not exist when 2 =
1; 2; 3;:::, many formulas containing M;(z) con-
tinue to apply in their limiting form. For example, if
n= 0;1;2;:::, then
13.14.7
lim
2! n 1M;(z)
(2+ 1)=
1
2n
n+1
(n+ 1)!M;1
2(n+1)(z)
=e 1
2zz 1
2n1X
s=n+1
1
2n
s
(s n)s!zs:
If 2=n, wheren= 0;1;2;:::, then
13.14.8W;1
2n(z) =( 1)ne 1
2zz1
2n+1
2
n! 1
2 1
2n nX
k=1n!(k 1)!
(n k)!
+1
2 1
2n
kz k
1X
k=0 1
2n+1
2
k
(n+ 1)kk!zk
lnz+ 1
2n+1
2 +k
(1 +k) (n+ 1 +k)!
,
1
2n 1
26= 0;1;2;:::,
or
13.14.9
W;1
2n(z) = ( 1) 1
2n 1
2e 1
2zz1
2n+1
2 1
2n 1
2X
k=0 1
2n 1
2
k
(n+ 1 +k) k 1
2n 1
2( z)k, 1
2n 1
2= 0;1;2;:::.
13.14(ii) Analytic Continuation
13.14.10 M;
zei
=ieiM ;(z):
In (13.14.11){(13.14.13) mis any integer.
13.14.11 M;
ze2mi
= ( 1)me2miM;(z):
13.14 Definitions and Basic Properties 335
13.14.12 W;
ze2mi
=( 1)m+12isin(2m )
1
2
(1 + 2) sin(2)M;(z) + ( 1)me 2miW;(z):
13.14.13( 1)mW;
ze2mi
= e2isin(2m ) + sin((2m 2))
sin(2)W;(z)
sin(2m )2iei
sin(2) 1
2+
1
2 W ;
zei
:
Except when z= 0, each branch of the functions
M;(z)/ (2+ 1) andW;(z) is entire in and.
Also, unless speci ed otherwise M;(z) andW;(z)
are assumed to have their principal values.
13.14(iii) Limiting Forms as z!0
13.14.14
M;(z) =z+1
2(1 +O(z)) , 26= 1; 2; 3;:::.
In cases when1
2 = n, wherenis a nonneg-
ative integer,
13.14.15
W1
2+n;(z) = ( 1)n(12)nz1
2+O
z3
2
:
In all other cases
13.14.16W;(z) = (2)
1
2+ z1
2 +O
z3
2 <
,
<1
2,6=1
2;
13.14.17 W;1
2(z) =1
(1 )+O(zlnz);
13.14.18
W;(z) = (2)
1
2+ z1
2 + ( 2)
1
2 z1
2+
+O
z3
2 <
, 0<<1
2,6= 0;
13.14.19W;0(z) = pz
1
2
lnz+ 1
2
+ 2
+O
z3/2lnz
:
ForW;(z) with<<0 use (13.14.31).
13.14(iv) Limiting Forms as z!1
Except when = 1
2; 3
2;::: (polynomial cases),
13.14.20M;(z) (1 + 2)e1
2zz .
1
2+
,
jphzj1
2 ,
whereis an arbitrary small positive constant. Also,
13.14.21 W;(z)e 1
2zz,jphzj3
2 .
13.14(v) Numerically Satisfactory Solutions
Fundamental pairs of solutions of (13.14.1) that are nu-
merically satisfactory ( x2.7(iv)) in the neighborhood of
in nity are
13.14.22W;(z); W ;
e iz
, 1
2phz3
2,
13.14.23W;(z); W ;
eiz
, 3
2phz1
2.A fundamental pair of solutions that is numerically
satisfactory in the sector jphzjnear the origin is
13.14.24 M;(z); M; (z), 2 62Z.
When 2is an integer we may use the results of
x13.2(v) with the substitutions b= 2+1,a= +1
2,
andW=e 1
2zz1
2+w, whereWis the solution of
(13.14.1) corresponding to the solution wof (13.2.1).
13.14(vi) Wronskians
13.14.25 WfM;(z);M; (z)g= 2;
13.14.26 WfM;(z);W;(z)g= (1 + 2)
1
2+ ;
13.14.27
W
M;(z);W ;
eiz
= (1 + 2)
1
2++e(1
2+)i;
13.14.28 WfM; (z);W;(z)g= (1 2)
1
2 ;
13.14.29W
M; (z);W ;
eiz
= (1 2)
1
2 +e(1
2 )i;
13.14.30 W
W;(z);W ;
eiz
=ei:
13.14(vii) Connection Formulas
13.14.31 W;(z) =W; (z):
13.14.32
1
(1 + 2)M;(z) =e( 1
2)i
1
2++W;(z)
+ei
1
2+ W ;
eiz
:
When 2is not an integer
13.14.33W;(z) = ( 2)
1
2 M;(z)
+ (2)
1
2+ M; (z):
336 Confluent Hypergeometric Functions
13.15 Recurrence Relations and Derivatives
13.15(i) Recurrence Relations
13.15.1 ( 1
2)M 1;(z) + (z 2)M;(z) + (++1
2)M+1;(z) = 0;
13.15.2 2(1 + 2)pzM 1
2; 1
2(z) (z+ 2)(1 + 2)M;(z) + (++1
2)pzM+1
2;+1
2(z) = 0;
13.15.3 ( 1
2)M 1
2;+1
2(z) + (1 + 2)pzM;(z) (++1
2)M+1
2;+1
2(z) = 0;
13.15.4 2M 1
2; 1
2(z) 2M+1
2; 1
2(z) pzM;(z) = 0;
13.15.5 2(1 + 2)M;(z) 2(1 + 2)pzM 1
2; 1
2(z) ( 1
2)pzM 1
2;+1
2(z) = 0;
13.15.6 2(1 + 2)pzM+1
2; 1
2(z) + (z 2)(1 + 2)M;(z) + ( 1
2)pzM 1
2;+1
2(z) = 0;
13.15.7 2(1 + 2)pzM+1
2; 1
2(z) 2(1 + 2)M;(z) + (++1
2)pzM+1
2;+1
2(z) = 0:
13.15.8 W+1
2;+1
2(z) pzW;(z) + ( 1
2)W 1
2;+1
2(z) = 0;
13.15.9 W+1
2; 1
2(z) pzW;(z) + (+ 1
2)W 1
2; 1
2(z) = 0;
13.15.10 2W;(z) pzW+1
2;+1
2(z) +pzW+1
2; 1
2(z) = 0;
13.15.11 W+1;(z) + (2 z)W;(z) + ( 1
2)(+ 1
2)W 1;(z) = 0;
13.15.12 ( 1
2)pzW 1
2;+1
2(z) + 2W;(z) (+ 1
2)pzW 1
2; 1
2(z) = 0;
13.15.13 (+ 1
2)pzW 1
2; 1
2(z) (z+ 2)W;(z) +pzW+1
2;+1
2(z) = 0;
13.15.14 ( 1
2)pzW 1
2;+1
2(z) (z 2)W;(z) +pzW+1
2; 1
2(z) = 0:
13.15(ii) Di erentiation Formulas
13.15.15dn
dzn
e1
2zz 1
2M;(z)
= ( 1)n( 2)ne1
2zz 1
2(n+1)M 1
2n; 1
2n(z);
13.15.16dn
dzn
e1
2zz 1
2M;(z)
= 1
2+
n
(1 + 2)ne1
2zz 1
2(n+1)M 1
2n;+1
2n(z);
13.15.17
zd
dzzn
e1
2zz 1M;(z)
= 1
2+
ne1
2zzn 1M n;(z);
13.15.18dn
dzn
e 1
2zz 1
2M;(z)
= ( 1)n( 2)ne 1
2zz 1
2(n+1)M+1
2n; 1
2n(z);
13.15.19dn
dzn
e 1
2zz 1
2M;(z)
= ( 1)n 1
2++
n
(1 + 2)ne 1
2zz 1
2(n+1)M+1
2n;+1
2n(z);
13.15.20
zd
dzzn
e 1
2zz 1M;(z)
= 1
2++
ne 1
2zz+n 1M+n;(z):
13.15.21dn
dzn
e1
2zz 1
2W;(z)
= ( 1)n 1
2+
ne1
2zz 1
2(n+1)W 1
2n;+1
2n(z);
13.15.22dn
dzn
e1
2zz 1
2W;(z)
= ( 1)n 1
2
ne1
2zz 1
2(n+1)W 1
2n; 1
2n(z);
13.15.23
zd
dzzn
e1
2zz 1W;(z)
= 1
2+
n 1
2
ne1
2zzn 1W n;(z);
13.15.24dn
dzn
e 1
2zz 1
2W;(z)
= ( 1)ne 1
2zz 1
2(n+1)W+1
2n;+1
2n(z);
13.15.25dn
dzn
e 1
2zz 1
2W;(z)
= ( 1)ne 1
2zz 1
2(n+1)W+1
2n; 1
2n(z);
13.15.26
zd
dzzn
e 1
2zz 1W;(z)
= ( 1)ne 1
2zz+n 1W+n;(z):
Other versions of several of the identities in this subsection can be constructed by use of (13.3.29).
13.16 Integral Representations 337
13.16 Integral Representations
13.16(i) Integrals Along the Real Line
In this subsection see xx10.2(ii), 10.25(ii) for the functions J2,I2, andK2, andxx15.1, 15.2(i) for 2F1.
13.16.1M;(z) = (1 + 2)z+1
22 2
1
2+
1
2++Z1
1e1
2zt(1 +t) 1
2 (1 t) 1
2+dt,<+1
2>j<j;
13.16.2M;(z) = (1 + 2)z
(1 + 2 2) (2)Z1
0M ; (zt)e1
2z(t 1)t 1
2(1 t)2 1dt,<+1
2><>0;
13.16.31
(1 + 2)M;(z) =pze1
2z
1
2++Z1
0e tt 1
2J2
2p
zt
dt, <(+) +1
2>0;
13.16.41
(1 + 2)M;(z) =pze 1
2z
1
2+ Z1
0e tt 1
2I2
2p
zt
dt, <( ) 1
2>0:
13.16.5 W;(z) =z+1
22 2
1
2+ Z1
1e 1
2zt(t 1) 1
2 (t+ 1) 1
2+dt,<+1
2><,jphzj<1
2;
13.16.6
W;(z) =e 1
2zz+1
1
2+
1
2 Z1
0W ;(t)e 1
2tt 1
t+zdt,jphzj<,<(1
2+ )>max (2<;0);
13.16.7W;(z) =( 1)ne 1
2zz1
2 n
(1 + 2) 1
2 Z1
0M ;(t)e 1
2ttn+ 1
2
t+zdt,
jphzj<,n= 0;1;2;:::, <(1 + 2)<n<j<j+<<1
2,
13.16.8 W;(z) =2pze 1
2z
1
2+
1
2 Z1
0e tt 1
2K2
2p
zt
dt,<( ) +1
2>0;
13.16.9 W;(z) =e 1
2zz+cZ1
0e zttc 1
2F11
2+ ;1
2
c; t
dt,jphzj<1
2;
wherecis arbitrary,<c>0.
13.16(ii) Contour Integrals
For contour integral representations combine (13.14.2) and (13.14.3) with x13.4(ii). See Buchholz (1969, x2.3), Erd elyi
et al. (1953a,x6.11.3), and Slater (1960, Chapter 3). See also x13.16(iii).
13.16(iii) Mellin{Barnes Integrals
If1
2+ 6= 0; 1; 2;:::, then
13.16.101
(1 + 2)M;
eiz
=e1
2z(1
2+)i
2i 1
2+ Zi1
i1 (t ) 1
2+ t
1
2++tztdt,jphzj<1
2;
where the contour of integration separates the poles of ( t ) from those of 1
2+ t
.
If1
2 6= 0; 1; 2;:::, then
13.16.11 W;(z) =e 1
2z
2iZi1
i1 1
2++t
1
2 +t
( t)
1
2+
1
2 z tdt,jphzj<3
2;
where the contour of integration separates the poles of 1
2++t
1
2 +t
from those of ( t).
13.16.12 W;(z) =e1
2z
2iZi1
i1 1
2++t
1
2 +t
(1 +t)z tdt, jphzj<1
2;
where the contour of integration passes all the poles of 1
2++t
1
2 +t
on the right-hand side.
338 Confluent Hypergeometric Functions
13.17 Continued Fractions
If;2Csuch that( 1
2)6= 1; 2; 3;:::, then
13.17.1pzM;(z)
M 1
2;+1
2(z)= 1 +u1z
1 +u2z
1 +;
where
13.17.2u2n+1= 1
2+++n
(2+ 2n+ 1)(2+ 2n+ 2);
u2n=1
2+ +n
(2+ 2n)(2+ 2n+ 1):
This continued fraction converges to the meromorphic
function of zon the left-hand side for all z2C. For
more details on how a continued fraction converges to a
meromorphic function see Jones and Thron (1980).
If;2Csuch that+1
2(+1)6= 1; 2; 3;:::,
then
13.17.3W;(z)pzW 1
2; 1
2(z)= 1 +v1=z
1 +v2=z
1 +;
where
13.17.4v2n+1=1
2+ +n; v 2n=1
2 +n:
This continued fraction converges to the meromorphic
function of zon the left-hand side throughout the sector
jphzj<.
See also Cuyt et al. (2008, pp. 336{337).
13.18 Relations to Other Functions
13.18(i) Elementary Functions
13.18.1 M0;1
2(2z) = 2 sinhz;
13.18.2
M; 1
2(z) =W; 1
2(z) =W; +1
2(z) =e 1
2zz;
13.18.3 M; 1
2(z) =e1
2zz :
13.18(ii) Incomplete Gamma Functions
For the notation see xx6.2(i), 7.2(i), and 8.2(i). When
1
2 is an integer the Whittaker functions can be ex-
pressed as incomplete gamma functions (or generalized
exponential integrals). For example,
13.18.4M 1
2;(z) = 2e1
2zz1
2
(2;z);13.18.5 W 1
2;(z) =e1
2zz1
2 (2;z):
Special cases are the error functions
13.18.6 M 1
4;1
4
z2
=1
2e1
2z2pzerf(z);
13.18.7 W 1
4; 1
4
z2
=e1
2z2pzerfc(z):
13.18(iii) Modi ed Bessel Functions
When= 0 the Whittaker functions can be expressed
as modi ed Bessel functions. For the notation see
xx10.25(ii) and 9.2(i).
13.18.8M0;(2z) = 22+1
2 (1 +)pzI(z);
13.18.9 W0;(2z) =p
2z/K(z);
13.18.10 W0;1
3
4
3z3
2
= 2pz1
4Ai(z):
13.18(iv) Parabolic Cylinder Functions
For the notation see x12.2.
13.18.11W 1
2a;1
4 1
2z2
= 21
2apzU(a;z);
13.18.12M 1
2a; 1
4 1
2z2
= 21
2a 1 1
2a+3
4p
z/
(U(a;z) +U(a; z));
13.18.13M 1
2a;1
4 1
2z2
= 21
2a 2 1
2a+1
4p
z/
(U(a; z) U(a;z)):
13.18(v) Orthogonal Polynomials
Special cases ofx13.18(iv) are as follows. For the nota-
tion seex18.3.
Hermite Polynomials
13.18.14M1
4+n; 1
4
z2
= ( 1)nn!
(2n)!e 1
2z2pzH2n(z);
13.18.15
M3
4+n;1
4
z2
= ( 1)nn!
(2n+ 1)!e 1
2z2pz
2H2n+1(z);
13.18.16W1
4+1
2n;1
4
z2
= 2 ne 1
2z2pzHn(z):
Laguerre Polynomials
13.18.17
W1
2 +1
2+n;1
2 (z) = ( 1)n( + 1)nM1
2 +1
2+n;1
2 (z)
= ( 1)nn!e 1
2zz1
2 +1
2L( )
n(z):
13.19 Asymptotic Expansions for Large Argument 339
13.19 Asymptotic Expansions for Large Argument
Asx!1
13.19.1 M;(x) (1 + 2)
1
2+ e1
2xx 1X
s=0 1
2 +
s 1
2++
s
s!x s, 6= 1
2; 3
2;:::.
Asz!1
13.19.2M;(z) (1 + 2)
1
2+ e1
2zz 1X
s=0 1
2 +
s 1
2++
s
s!z s
+ (1 + 2)
1
2++e 1
2z(1
2+ )iz1X
s=0 1
2+
s 1
2
s
s!( z) s,
1
2+phz3
2 ,
provided that both 6= 1
2; 3
2;:::. Again,denotes an arbitrary small positive constant. Also,
13.19.3 W;(z)e 1
2zz1X
s=0 1
2+
s 1
2
s
s!( z) s,jphzj3
2 :
Error bounds and exponentially-improved expan-
sions are derivable by combining xx13.7(ii) and 13.7(iii)
with (13.14.2) and (13.14.3). See also Olver (1965).
For an asymptotic expansion of W;(z) asz!1
that is valid in the sector jphzj and where the
real parameters ,are subject to the growth condi-
tions=o(z),=o(pz), see Wong (1973a).
13.20 Uniform Asymptotic Approximations
for Large
13.20(i) Large , Fixed
When!1 in the sectorjphj1
2 (<1
2), with
(2C) xed
13.20.1 M;(z) =z+1
2
1 +O
1
;
uniformly for bounded values of jzj; also
13.20.2
W;(x) = 1
2 (+) 1
4x1
2
1 +O
1
;
uniformly for bounded positive values of x. For an
extension of (13.20.1) to an asymptotic expansion, to-
gether with error bounds, see Olver (1997b, Chapter 10,
Ex. 3.4).
13.20(ii) Large ,0(1 )
Let
13.20.3 X=p
42 4x+x2:
Then as!1
13.20.4
M;(x) =r
2x
X42x
22 x+X
2( )
X+x 2
e1
2X
1 +O1
;13.20.5
W;(x) =rx
X22 x+X
( )xX+x 2
2
e 1
2X
1 +O1
;
uniformly with respect to x2(0;1) and2[0;(1
)], whereagain denotes an arbitrary small positive
constant.
13.20(iii) Large , (1 )
Let
13.20.6 =p
2j j=;
13.20.7 X=p
jx2 4x+ 42j;
13.20.8 (;;x ) =22 2+ 22
x2 4x+ 421
4px;
with the variable de ned implicitly as follows:
(a) In the case <<
13.20.9
p
2+ 2+ 2arcsinh
=X
2
ln
X+x 2
2p
2 2!
2 ln
X+ 22 x
xp
2 2!
:
(b) In the case =
13.20.10 =s
x
2 2 lnx
2
;
the upper or lower sign being taken according as x?2.
(In both cases (a) and (b) the x-interval (0 ;1) is
mapped one-to-one onto the -interval ( 1;1), with
340 Confluent Hypergeometric Functions
x= 0 and1corresponding to = 1 and1, respec-
tively.) Then as !1
13.20.11
W;(x) = 1
2 1
4+
e1
2(+)
(;;x )
U
;p
2
1 +O
1ln
;
13.20.12
M;(x) = (8)1
42
e2e
+1
2(+)
(;;x )
U
; p
2
1 +O
1ln
;uniformly with respect to x2(0;1) and2[ (1
);]. For the parabolic cylinder function Useex12.2.
These results are proved in Olver (1980b). This ref-
erence also supplies error bounds and corresponding ap-
proximations when x,, andare replaced by ix,i,
andi, respectively.
13.20(iv) Large ,=
Again de ne ,X, and (;;x ) by (13.20.6){
(13.20.8), but with now de ned by
13.20.13
p
2 2 2arccosh
=X
2
ln
X+x 2
2p
2 2!
2 ln
x X 22
xp
2 2!
,x2+ 2p
2 2,
13.20.14
p
2 2+ 2arcsin
=X
+2
arctanx 2
X
2 arctanx 22
X
,
2 2p
2 2x2+ 2p
2 2,
13.20.15
p
2 2 2arccosh
= X
+2
ln
2 X x
2p
2 2!
+ 2 ln
X+ 22 x
xp
2 2!
,
0<x2 2p
2 2;
when < , and by (13.20.10) when =. (As inx13.20(iii)x= 0 and1correspond to = 1 and1,
respectively). Then as !1
13.20.16W;(x) = 1
2 1
4+
e1
2(+)
(;;x )
U
;p
2
+ envU
;p
2
O
2
3
;
13.20.17M;(x) = (8)1
42
e2e
+1
2(+)
(;;x )
U
; p
2
+ envU
;p
2
O
2
3
;
uniformly with respect to 2[0;1) and2[;= ].
Also,
13.20.18W;(x) = 1
2 1
4+
e1
2(+)
(;;x )
U
;p
2
+ envU
; p
2
O
2
3
;
13.20.19
M;(x) = (8)1
42
e2e
+1
2(+)
(;;x )
U
; p
2
+ envU
; p
2
O
2
3
;
uniformly with respect to 2( 1;0] and2[;= ].
For the parabolic cylinder functions UandUsee
x12.2, and for the env functions associated with Uand
Useex14.15(v).
These results are proved in Olver (1980b). Equa-
tions (13.20.17) and (13.20.18) are simpler than (6.10)
and (6.11) in this reference. Olver (1980b) also supplies
error bounds and corresponding approximations whenx,, andare replaced by ix,i, andi, respectively.
It should be noted that (13.20.11), (13.20.16), and
(13.20.18) di er only in the common error terms. Hence
without the error terms the approximation holds for
(1 )=. Similarly for (13.20.12),
(13.20.17), and (13.20.19).
13.21 Uniform Asymptotic Approximations for Large 341
13.20(v) Large , Other Expansions
For uniform approximations valid when is large,x=i2
(0;1), and=i2[0;=], see Olver (1997b, pp. 401{
403). These approximations are in terms of Airy func-
tions.
For uniform approximations of M;i(z) and
W;i(z),andreal, one or both large, see Dunster
(2003a).
13.21 Uniform Asymptotic Approximations
for Large
13.21(i) Large , Fixed
For the notation see xx10.2(ii), 10.25(ii), and 2.8(iv).
When!1 through positive real values with
(0) xed
13.21.1M;(x) =px (2+ 1)
J2
2px
+ envJ2
2px
O
1
2
;
13.21.2
W;(x) =px
+1
2
sin( )J2
2px
cos( )Y2
2px
+ envY2
2px
O
1
2
;
13.21.3W ;
xe i
=px
+1
2ei
H(1)
2
2px
+ envY2
2px
O
1
2
;
13.21.4W ;
xei
=px
+1
2e i
H(2)
2
2px
+ envY2
2px
O
1
2
;uniformly with respect to x2(0;A] in each case, where
Ais an arbitrary positive constant.
Other types of approximations when ! 1
through positive real values with (0) xed are as
follows. De ne
13.21.5 2p
=p
x+x2+ ln px+p
1 +x
:
Then
13.21.6
M ;(4x)
=2 (2+ 1)
1
2x
1 +x1
4
I2
41
2
1 +O
1
;
13.21.7
W ;(4x)
=p
8=e
1
2x
1 +x1
4
K2
41
2
1 +O
1
;
uniformly with respect to x2(0;1).
For (13.21.6), (13.21.7), and extensions to asymp-
totic expansions and error bounds, see Olver (1997b,
Chapter 12, Exs. 12.4.5, 12.4.6). For extensions to com-
plex values of xsee L opez (1999).
13.21(ii) Large ,0(1 )
Let
13.21.8c(;) =eiq
1
2
+1
2e2
2 21
2
;
13.21.9 X=p
jx2 4x+ 42j;
13.21.10 (;;x ) =42
x2 4x+ 421
4px;
with the variable de ned implicitly by
13.21.11p
42 ln
2+p
42
2 p
42 !
=1
2X+ln
xp
2 2
22 x+X!
+ln
2p
2 2
2 x X!
,
0<x2 2p
2 2,
and
13.21.12p
42 2arctan p
42
2!
=1
2(X ) arctanx 22
X
+arcsin
X
2p
2 2!
,
2 2p
2 2x<2+ 2p
2 2.
Then as!1
13.21.13
M;(x) = (2+ 1)e2
2 21
2
+1
2
(;;x )
J2p
+ envJ2p
O
1
;
13.21.14
W;(x) =e i
++1
2
+1
2
c(;) (;;x )
sin( )J2p
cos( )Y2p
+ envY2p
O
1
;
342 Confluent Hypergeometric Functions
13.21.15
W ;
xe i
=c(;) (;;x )
H(1)
2p
+ envY2p
O
1
;
13.21.16
W ;
xei
=e 2ic(;) (;;x )
H(2)
2p
+ envY2p
O
1
;
uniformly with respect to 2[0;(1 )] andx2
0;(1 )(2+ 2p
2 2)i
, whereagain denotes an
arbitrary small positive constant. For the functions J2,
Y2,H(1)
2, andH(2)
2seex10.2(ii), and for the env func-
tions associated with J2andY2seex2.8(iv).
These approximations are proved in Dunster (1989).This reference also includes error bounds and extensions
to asymptotic expansions and complex values of x.
13.21(iii) Large ,0(1 )
(Continued)
Let
13.21.17bc(;) =p
21
6
+1
2e2
2 21
2
;
13.21.18 X=p
jx2 4x+ 42j;
13.21.19b (;;x ) = b
x2 4x+ 42!1
4p
2x;
and de ne the variable bimplicitly by
13.21.20b=
3
2
1
2X+ 2arctan
x xp
2 2 22
X!
+arccos
x 2
2p
2 2!!! 2=3
,
2 2p
2 2<x2+ 2p
2 2,
and
13.21.21 b=
3
2
1
2X+ln
xp
2 2
x 22 X!
+ln
2p
2 2
x 2+X!!! 2=3
,x2+ 2p
2 2.
Then as!1
13.21.22M;(x) =1
2 (2+ 1)
+1
2
bc(;)b (;;x )
sin( ) Ai
2
3b
+ cos( ) Bi
2
3b
+ envBi
2
3b
O
1
;
13.21.23 W;(x) =p
21
6+
1
22 2
e21
2
b (;;x )
Ai
2
3b
+ envAi
2
3b
O
1
;
13.21.24 W ;
xe i
=e( 1
6)ibc(;)b (;;x )
Ai
2
3be 2
3i
+ envBi
2
3b
O
1
;
13.21.25 W ;
xei
=e ( 1
6)ibc(;)b (;;x )
Ai
2
3be2
3i
+ envBi
2
3b
O
1
;
uniformly with respect to 2[0;(1 )] andx2h
(1 +)(2 2p
2 2);1
. For the functions Ai
and Bi seex9.2(i), and for the env functions associated
with Ai and Bi see x2.8(iii).
These approximations are proved in Dunster (1989).
This reference also includes error bounds and extensions
to asymptotic expansions and complex values of x.
13.21(iv) Large , Other Expansions
For a uniform asymptotic expansion in terms of Airy
functions for W;(4x) whenis large and positive,
is real withjjbounded, and x2[;1) see Olver
(1997b, Chapter 11, Ex. 7.3). This expansion is simplerin form than the expansions of Dunster (1989) that cor-
respond to the approximations given in x13.21(iii), but
the conditions on are more restrictive.
For asymptotic expansions having double asymp-
totic properties see Skovgaard (1966).
See alsox13.20(v).
13.22 Zeros
From (13.14.2) and (13.14.3) M;(z) has the same zeros
asM 1
2+ ;1 + 2;z
andW;(z) has the same
zeros asU 1
2+ ;1 + 2;z
, hence the results given
inx13.9 can be adopted.
Asymptotic approximations to the zeros when the
parameters and/orare large can be found by rever-
13.23 Integrals 343
sion of the uniform approximations provided in xx13.20
and 13.21. For example, if (0) is xed and (>0) is
large, then the rth positive zero rofM;(z) is given
by
13.22.1 r=j2
2;r
4+j2;rO
3
2
;wherej2;ris therth positive zero of the Bessel func-
tionJ2(x) (x10.21(i)). (13.22.1) is a weaker version of
(13.9.8).
13.23 Integrals
13.23(i) Laplace and Mellin Transforms
For the notation see xx15.1, 15.2(i), and 10.25(ii).
13.23.1Z1
0e ztt 1M;(t)dt=
++1
2
z+1
2++1
22F11
2+ ;1
2++
1 + 2;1
z+1
2
,<(++1
2)>0,<z>1
2.
13.23.2Z1
0e ztt 1
2M;(t)dt= (2+ 1)
z+1
2 1
2
z 1
2 1
2,<> 1
2,<z>1
2;
13.23.31
(1 + 2)Z1
0e 1
2tt 1M;(t)dt=
++1
2
( )
1
2++
1
2+ , 1
2 <<< <<:
13.23.4Z1
0e ztt 1W;(t)dt= 1
2++
1
2 +
2F11
2 +;1
2++
+ 1;1
2 z
,
<(+1
2)>j<j,<z> 1
2;
13.23.5Z1
0e1
2tt 1W;(t)dt= 1
2++
1
2 +
( )
1
2+
1
2 ,j<j 1
2<< < <:
13.23.61
(1 + 2)2iZ(0+)
1ezt+1
2t 1tM;
t 1
dt=z 1
2
1
2+ I2
2pz
,<z>0.
13.23.71
2iZ(0+)
1ezt+1
2t 1tW;
t 1
dt=2z 1
2
1
2+
1
2 K2
2pz
,<z>0.
For additional Laplace and Mellin transforms see Erd elyi et al. (1954a,xx4.22, 5.20, 6.9, 7.5), Marichev (1983,
pp. 283{287), Oberhettinger and Badii (1973, x1.17), Oberhettinger (1974, xx1.13, 2.8), and Prudnikov et al. (1992a,
xx3.34, 3.35). Inverse Laplace transforms are given in Oberhettinger and Badii (1973, x2.16) and Prudnikov et al.
(1992b,xx3.33, 3.34).
13.23(ii) Fourier Transforms
13.23.81
(1 + 2)Z1
0cos(2xt)e 1
2t2t 2 1M;
t2
dt=pe 1
2x2x+ 1
2 1
2++W1
2 3
2;1
2+1
2
x2
,<(+)> 1
2:
For additional Fourier transforms see Erd elyi et al. (1954a,xx1.14, 2.14, 3.3) and Oberhettinger (1990, xx1.22,
2.22).
13.23(iii) Hankel Transforms
For the notation see x10.2(ii).
13.23.9Z1
0e 1
2tt 1
2(+1)M;(t)J
2p
xt
dt= (1 + 2)
1
2 ++e 1
2xx1
2( 3
2)M1
2(+3 +1
2);1
2( + 1
2)(x),
x>0, 1
2<<<<(+1
2) +3
4;
344 Confluent Hypergeometric Functions
13.23.101
(1 + 2)Z1
0e 1
2tt1
2( 1) M;(t)J
2p
xt
dt=e 1
2xx1
2(+ 3
2)
1
2++W1
2( 3++1
2);1
2(+ 1
2)(x),
x>0, 1<< <2<(+) +1
2:
13.23.11Z1
0e1
2tt1
2( 1) W;(t)J
2p
xt
dt= ( 2+ 1)
1
2+ e1
2xx1
2( 3
2)W1
2(+3 1
2);1
2( ++1
2)(x),
x>0, max(2< 1; 1)<< <2<( ) +3
2;
13.23.12Z1
0e 1
2tt1
2( 1) W;(t)J
2p
xt
dt= ( 2+ 1)
3
2 +e 1
2xx1
2(+ 3
2)M1
2( 3++1
2);1
2( +1
2)(x),
x>0, max(2< 1; 1)<<:
For additional Hankel transforms and also other Bessel transforms see Erd elyi et al. (1954b,x8.18) and Oberhet-
tinger (1972,x1.16 and 3.4.42{46, 4.4.45{47, 5.94{97).
13.23(iv) Integral Transforms in terms of Whittaker Functions
Letf(x) be absolutely integrable on the interval [ r;R] for all positive r < R ,f(x) =O(x0) asx!0+, and
f(x) =O(e 1x) asx!+1, where1>1
2. Then forin the half-plane <1>max
0;< 1
2
13.23.13 g() =1
(1 + 2)Z1
0f(x)x 3
2M;(x)dx;
13.23.14 f(x) =1
ipxZ1+i1
1 i1g() 1
2+
W;(x)d:
For additional integral transforms see Magnus et al. (1966, p. 189), Prudnikov et al. (1992b,xx4.3.39{4.3.42), and
Wimp (1964).
13.23(v) Other Integrals
Additional integrals involving con
uent hypergeometric functions can be found in Apelblat (1983, pp. 388{392),
Erd elyi et al. (1954b), Gradshteyn and Ryzhik (2000, x7.6), and Prudnikov et al. (1990,xx1.13, 1.14, 2.19, 4.2.2).
See also (13.16.2), (13.16.6), (13.16.7).
13.24 Series
13.24(i) Expansions in Series of Whittaker Functions
For expansions of arbitrary functions in series of M;(z) functions see Sch afke (1961b).
13.24(ii) Expansions in Series of Bessel Functions
Forz2C, and again with the notation of xx10.2(ii) and 10.25(ii),
13.24.1
M;(z) = (+)22+2z1
2 1X
s=0( 1)s(2+ 2)s(2)s
(1 + 2)ss!(++s)I++s 1
2z
, 2;+6= 1; 2; 3;:::,
and
13.24.21
(1 + 2)M;(z) = 22z+1
21X
s=0p()
s(z)
2pz 2 sJ2+s
2pz
;
wherep()
0(z) = 1,p()
1(z) =1
6z2, and higher polynomials p()
s(z) are de ned by
13.24.3 exp
1
2z
cotht 1
tt
sinht1 2
=1X
s=0p()
s(z)
t
zs
:
(13.18.8) is a special case of (13.24.1).
Additional expansions in terms of Bessel functions are given in Luke (1959). See also L opez (1999).
For other series expansions see Prudnikov et al. (1990,x6.6). See alsox13.26.
13.25 Products 345
13.25 Products
13.25.1
M;(z)M; 1(z)
+(1
2++)(1
2+ )
4(1 +)(1 + 2)2M;+1(z)M; (z) = 1:
For integral representations, integrals, and series
containing products of M;(z) andW;(z) see Erd elyi
et al. (1953a,x6.15.3).
13.26 Addition and Multiplication
Theorems
13.26(i) Addition Theorems for M;(z)
The function M;(x+y) has the following expansions:
13.26.1e 1
2yx
x+y 1
21X
n=0( 2)n
n! ypxn
M 1
2n; 1
2n(x),jyj<jxj;
13.26.2e 1
2yx+y
x+1
21X
n=0 1
2+
n
(1 + 2)nn!ypxn
M 1
2n;+1
2n(x);
13.26.3e 1
2yx+y
x1X
n=0 1
2+
nyn
n!(x+y)nM n;(x),
<(y=x)> 1
2;
13.26.4e1
2yx
x+y 1
21X
n=0( 2)n
n! ypxn
M+1
2n; 1
2n(x),jyj<jxj;
13.26.5e1
2yx+y
x+1
21X
n=0 1
2++
n
(1 + 2)nn! ypxn
M+1
2n;+1
2n(x);
13.26.6e1
2yx
x+y1X
n=0 1
2++
nyn
n!(x+y)nM+n;(x),
<((y+x)=x)>1
2:
13.26(ii) Addition Theorems for W;(z)
The function W;(x+y) has the following expansions:
13.26.7e 1
2yx
x+y 1
21X
n=0 1
2
n
n! ypxn
W 1
2n; 1
2n(x), jyj<jxj;
13.26.8e 1
2yx+y
x+1
21X
n=0 1
2+
n
n! ypxn
W 1
2n;+1
2n(x), jyj<jxj;13.26.9e 1
2yx+y
x1X
n=0 1
2+
n 1
2
n
n!
y
x+yn
W n;(x),<(y=x)> 1
2;
13.26.10e1
2yx
x+y 1
21X
n=01
n! ypxn
W+1
2n; 1
2n(x),jyj<jxj;
13.26.11e1
2yx+y
x+1
21X
n=01
n! ypxn
W+1
2n;+1
2n(x),jyj<jxj;
13.26.12e1
2yx
x+y1X
n=01
n! y
x+yn
W+n;(x),
<(y=x)> 1
2:
13.26(iii) Multiplication Theorems for M;(z)
andW;(z)
To obtain similar expansions for M;(xy) and
W;(xy), replaceyin the previous two subsections by
(y 1)x.
Applications
13.27 Mathematical Applications
Con
uent hypergeometric functions are connected with
representations of the group of third-order triangular
matrices. The elements of this group are of the form
13.27.1 g=0
@1
0
0 0 11
A;
where , ,
,are real numbers, and
>0. Vilenkin
(1968, Chapter 8) constructs irreducible representations
of this group, in which the diagonal matrices correspond
to operators of multiplication by an exponential func-
tion. The other group elements correspond to integral
operators whose kernels can be expressed in terms of
Whittaker functions. This identi cation can be used
to obtain various properties of the Whittaker functions,
including recurrence relations and derivatives.
For applications of Whittaker functions to the uni-
form asymptotic theory of di erential equations with a
coalescing turning point and simple pole see xx2.8(vi)
and 18.15(i).
346 Confluent Hypergeometric Functions
13.28 Physical Applications
13.28(i) Exact Solutions of the Wave Equation
The reduced wave equation r2w=k2win paraboloidal
coordinates, x= 2pcos ,y= 2psin ,z=
, can be solved via separation of variables w=
f1()f2()eip , where
13.28.1
f1() = 1
2V(1)
;1
2p(2ik) ,f2() = 1
2V(2)
;1
2p( 2ik) ,
andV(j)
;(z),j= 1;2, denotes any pair of solutions of
Whittaker's equation (13.14.1). See Hochstadt (1971,
Chapter 7).
For potentials in quantum mechanics that are solv-
able in terms of con
uent hypergeometric functions see
Negro et al. (2000).
13.28(ii) Coulomb Functions
See Chapter 33.
13.28(iii) Other Applications
For dynamics of many-body systems see Meden and
Sch onhammer (1992); for tomography see D'Ariano
et al. (1994); for generalized coherent states see Barut
and Girardello (1971); for relativistic cosmology see
Cris ostomo et al. (2004).
Computation
13.29 Methods of Computation
13.29(i) Series Expansions
Although the Maclaurin series expansion (13.2.2) con-
verges for all nite values of z, it is cumbersome to use
whenjzjis large owing to slowness of convergence and
cancellation. For large jzjthe asymptotic expansions
ofx13.7 should be used instead. Accuracy is limited
by the magnitude of jzj. However, this accuracy can
be increased considerably by use of the exponentially-
improved forms of expansion supplied by the combina-
tion of (13.7.10) and (13.7.11), or by use of the hyper-
asymptotic expansions given in Olde Daalhuis and Olver
(1995a). For large values of the parameters aandbthe
approximations in x13.8 are available.
Similarly for the Whittaker functions.13.29(ii) Di erential Equations
A comprehensive and powerful approach is to integrate
the di erential equations (13.2.1) and (13.14.1) by di-
rect numerical methods. As described in x3.7(ii), to
insure stability the integration path must be chosen in
such a way that as we proceed along it the wanted so-
lution grows in magnitude at least as fast as all other
solutions of the di erential equation.
ForM(a;b;z ) andM;(z) this means that in the
sectorjphzjwe may integrate along outward rays
from the origin with initial values obtained from (13.2.2)
and (13.14.2).
ForU(a;b;z ) andW;(z) we may integrate along
outward rays from the origin in the sectors1
2 <
jphzj<3
2, with initial values obtained from connec-
tion formulas in x13.2(vii),x13.14(vii). In the sector
jphzj<1
2the integration has to be towards the ori-
gin, with starting values computed from asymptotic ex-
pansions (xx13.7 and 13.19). On the rays ph z=1
2,
integration can proceed in either direction.
13.29(iii) Integral Representations
The integral representations (13.4.1) and (13.4.4) can be
used to compute the Kummer functions, and (13.16.1)
and (13.16.5) for the Whittaker functions. In Allasia
and Besenghi (1991) and Allasia and Besenghi (1987b)
the high accuracy of the trapezoidal rule for the compu-
tation of Kummer functions is described. Gauss quadra-
ture methods are discussed in Gautschi (2002b).
13.29(iv) Recurrence Relations
The recurrence relations in xx13.3(i) and 13.15(i) can
be used to compute the con
uent hypergeometric func-
tions in an ecient way. In the following two examples
Olver's algorithm ( x3.6(v)) can be used.
Example 1
We assume 2 6= 1; 2; 3;:::. Then we have
13.29.1z2(n+ 1
2)
(n++1
2)2 2
(n+)(n++1
2)(n++ 1)y(n+ 1)
+ 16
(n+)2 1
2z 1
4
y(n)
16
(n+)2 1
4
y(n 1) = 0;
with recessive solution
13.29.2 y(n) =z n 1
2M;n+(z);
normalizing relation
13.29.3e 1
2z=1X
s=0(2)s 1
2+
s
(2)2ss!( z)sy(s);
and estimate
13.29.4 y(n) = 1 +O
n 1
,n!1 .
13.30 Tables 347
Example 2
We assume a;a+ 1 b6= 0; 1; 2;:::. Then we have
13.29.5(n+a)w(n) (2(n+a+1)+z b)w(n+1)
+ (n+a b+ 2)w(n+ 2) = 0;
with recessive solution
13.29.6 w(n) = (a)nU(n+a;b;z );
normalizing relation
13.29.7 z a=1X
s=0(a b+ 1)s
s!w(s);
and estimate
13.29.8w(n)pe1
2zz1
4(4a 2b+1)
(a) (a+ 1 b)n1
4(4a 2b 3)e 2pnz;
asn!1 . See Temme (1983), and also Wimp (1984,
Chapter 5).
13.30 Tables
Zurina and Osipova (1964) tabulates M(a;b;x )
andU(a;b;x ) forb= 2,a= 0:98(:02)1:10,
x= 0(:01)4, 7D or 7S.
Slater (1960) tabulates M(a;b;x ) fora= 1(:1)1,
b= 0:1(:1)1, andx= 0:1(:1)10, 7{9S; M(a;b;1)
fora= 11(:2)2 andb= 4(:2)1, 7D; the
smallest positive x-zero ofM(a;b;x ) fora=
4(:1) 0:1 andb= 0:1(:1)2:5, 7D.
Abramowitz and Stegun (1964, Chapter 13) tab-
ulatesM(a;b;x ) fora= 1(:1)1,b= 0:1(:1)1,
andx= 0:1(:1)1(1)10, 8S. Also the smallest pos-
itivex-zero ofM(a;b;x ) fora= 1(:1) 0:1 and
b= 0:1(:1)1, 7D.
Zhang and Jin (1996, pp. 411{423) tabulates
M(a;b;x ) andU(a;b;x ) fora= 5(:5)5,b=
0:5(:5)5, andx= 0:1;1;5;10;20;30, 8S (for
M(a;b;x )) and 7S (for U(a;b;x )).
For other tables prior to 1961 see Fletcher et al.
(1962) and Lebedev and Fedorova (1960).
13.31 Approximations
13.31(i) Chebyshev-Series Expansions
Luke (1969b, pp. 35 and 25) provides Chebyshev-series
expansions of M(a;b;x ) andU(a;b;x ) that include the
intervals 0x and x<1, respectively, where
is an arbitrary positive constant.13.31(ii) Pad e Approximations
For a discussion of the convergence of the Pad e approxi-
mants that are related to the continued fraction (13.5.1)
see Wimp (1985).
13.31(iii) Rational Approximations
In Luke (1977a) the following rational approximation is
given, together with its rate of convergence. For the
notation seex16.2(i).
Leta;a+ 1 b6= 0; 1; 2;:::,jphzj<,
13.31.1
An(z) =nX
s=0( n)s(n+ 1)s(a)s(b)s
(a+ 1)s(b+ 1)s(n!)2
3F3 n+s;n+ 1 +s;1
1 +s;a+ 1 +s;b+ 1 +s; z
;
and
13.31.2 Bn(z) = 2F2 n;n+ 1
a+ 1;b+ 1; z
:
Then
13.31.3zaU(a;1 +a b;z) = lim
n!1An(z)
Bn(z):
13.32 Software
Seehttp://dlmf.nist.gov/13.32 .
References
General References
The main references used in writing this chapter are
Buchholz (1969), Erd elyi et al. (1953a), Olver (1997b),
Slater (1960), and Temme (1996a). For additional bib-
liographic reading see Andrews et al. (1999), Hochstadt
(1971), Luke (1969a,b), Wang and Guo (1989), and
Whittaker and Watson (1927).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the vari-
ous sections this chapter. These sources supplement the
references that are quoted in the text.
x13.2 Olver (1997b, Chapter 7, xx3, 9, 10), Slater
(1960,xx1.5, 1.5.1, 2.1.2), and Temme (1996a,
xx7.1, 7.2). (13.2.7) and (13.2.8) are terminat-
ing forms of the asymptotic expansion (13.7.3)
(that thesign can be replaced by = in these
circumstances follows from (13.7.4) and (13.7.5).)
348 Confluent Hypergeometric Functions
To verify (13.2.12) replace the Ufunctions by M
functions by means of (13.2.42) and (13.2.4), then
recall that each Mfunction is an entire function
ofz. For (13.2.13){(13.2.22) see Temme (1996a,
Ex. 7.6: an error in the equation that corresponds
to (13.2.19) has been corrected). For (13.2.23)
see (13.7.2). (13.2.27){(13.2.32) are obtained by
considering limiting forms of the connection for-
mulas inx13.2(vii); see Olver (1997b, Chapter 7,
Ex. 10.6) and Slater (1960, xx1.5{1.5.1).
x13.3 Slater (1960,xx2.1, 2.2). Note that in this ref-
erence (2.2.7) and (2.1.32) contain errors. The
correct versions are (13.3.13) and (13.3.28), re-
spectively. To see that (13.3.13) and (13.3.14) are
equivalent to (13.2.1) use (13.3.16) and (13.3.23).
For the operator identity (13.3.29) see Fleury and
Turbiner (1994).
x13.4 Buchholz (1969, x1.4), Erd elyi et al. (1953a,
x6.11), and Slater (1960, Chapter 3). For (13.4.5)
use (13.2.41); compare the proof of Lemma 3.1 in
Olde Daalhuis and Olver (1994). For (13.4.16){
(13.4.18) combine the results of Buchholz (1969,
x5.4) with (13.14.2), (13.14.3).
x13.5 Jones and Thron (1980, Theorems 6.3 and 6.5).
x13.6 See Temme (1996a, xx7.2{7.3 and p. 254) and
Buchholz (1969, x3.3). In the former refer-
ence each of the equations on p. 180 that cor-
respond to (13.6.7) and (13.6.8) contains an er-
ror. For (13.6.16){(13.6.18) combine x13.6(iv)
withx12.7(i) and (18.5.18). (The last equation
is needed to illustrate that xnHn(x) is an even
function of x.) For (13.6.19) see (18.11.2). For
(13.6.20) combine (18.20.8) with (16.2.3), replace
the 1F1notation by M(x13.1), and then use
(13.2.42) (in which the nal term vanishes). Alter-
natively, for (13.6.20) combine (16.2.3) with An-
drews et al. (1999, p. 347). When neither anor
a b+ 1 is a nonpositive integer (13.6.21) can be
veri ed by comparison of (13.4.17) and (16.5.1).
Ifais a nonpositive integer, then both sides of
(13.6.21) reduce to a polynomial in z(compare
xx13.2(i) and 16.2(iv)), and (13.6.21) follows by
comparing coecients. Similarly if a b+ 1 is a
nonpositive integer, then both sides of (13.6.21)
reduce toz1 btimes a polynomial in zwith iden-
tical coecients.
x13.7 Temme (1996a,x7.2) or Olver (1997b, pp. 256{
258), and Olver (1965).
x13.8 Slater (1960,x4.3), Temme (1978), and Temme
(1990b). For (13.8.2) and (13.8.3) use Watson'slemma for loop integrals (Olver (1997b, x4.5)) and
(13.4.10).
x13.9 Buchholz (1969, Chapter 17), Erd elyi et al.
(1953a,x6.16), Slater (1960, Chapter 6), and Tri-
comi (1950a). The proof of (13.9.8) is given in
Tricomi (1947). (13.9.9) follows from (13.7.2).
For the paragraph following (13.9.9) see Andrews
et al. (1999,x4.16). For (13.9.10) and (13.9.16) use
(13.8.9), (13.8.10), and the asymptotics of zeros of
Bessel functions ( x10.21(vi)). For the nal para-
graph ofx13.9(ii) apply Kummer's transformation
(13.2.39) to the nal term in the connection rela-
tion (13.2.42) and then use the asymptotic rela-
tion (13.8.1).
x13.10 Erd elyi et al. (1953a,xx6.10, 6.15.2) and Slater
(1960, Chapter 3). Also Buchholz (1969, x11.1),
including the references given there. (13.10.14)
and (13.10.16) are from Erd elyi et al. (1954b,
x8.18). For (13.10.14) substitute the integral
(13.4.1) for M(a;b;t ), interchange the order of
integration, then apply (13.4.3) and (13.6.1) fol-
lowed by (13.4.4). For (13.10.16) interchange the
roles of (13.4.1) and (13.4.4).
x13.11 Slater (1960,x2.7.3: Eq. (2.7.14) has errors).
x13.12 (13.12.1) follows from the fact that its left-
hand side is bounded at in nity: use (13.2.39),
(13.2.41), and (13.7.3).
x13.13 Erd elyi et al. (1953a,x6.14) and Slater (1960,
xx2.3{2.3.3). In the rst reference Equation (2)
needs the constraint j 1j<1 and Equation (6)
should have no constraint. In the second reference
Eq. (2.3.6) contains an error: ( x+y)nshould be
replaced by ( x+y) n.
x13.14 Olver (1997b, Chapter 7, xx9{11, and Ex. 11.2),
Buchholz (1969, x2.3a), Slater (1960, xx1.7.1,
2.4.2), Temme (1996a, x7.2). For (13.14.8) and
(13.14.9) take limiting values in (13.14.33), using
(13.14.2) and (13.2.2). For (13.14.14){(13.14.19)
combinex13.2(iii) and (13.14.4), (13.14.5). For
(13.14.20), (13.14.21) use (13.19.2), (13.19.3). For
(13.14.32) and (13.14.33) combine (13.2.41) and
(13.2.42) with (13.14.4) and (13.14.5). (13.14.31)
follows from (13.14.33).
x13.15 Slater (1960,xx2.4, 2.4.1, 2.5). Note that
(2.5.4) and (2.5.10) contain errors: the correct ver-
sions are (13.15.2) and (13.15.10), respectively.
x13.16 Buchholz (1969, x5.4), Erd elyi et al. (1953a,
x6.11), and Slater (1960, Chapter 3). For x13.16(i)
combinex13.4(i) with (13.14.4) and (13.14.5).
References 349
x13.17 Jones and Thron (1980, Theorems 6.3 and 6.5).
x13.18 Combinex13.6 with (13.14.4) and (13.14.5).
x13.19 Temme (1996a,x7.2).
x13.20 Olver (1997b, Chapter 7, x11.2). For (13.20.2)
use (13.14.3) and apply the method of steepest
descents (x2.4(iv)) to the integral representation
(13.4.14).
x13.21 Slater (1960,x4.4.3). The asymptotic approxi-
mations (13.21.2){(13.21.4) follow from x13.21(ii)
and (5.11.7).
x13.22 Olver (1997b, Chapter 12, (7.05)).x13.23 Buchholz (1969, xx10, 11.1), Erd elyi et al.
(1953a,xx6.10, 6.15.2), Slater (1960, Chapter
3), and Snow (1952, Chapter XI). (13.23.3) and
(13.23.5) can also be derived as limiting forms
of (13.23.1) and (13.23.4), respectively; com-
pare (15.4.20). For (13.23.9){(13.23.12) combine
x13.10(v) with (13.14.4) and (13.14.5).
x13.24 Slater (1960,x2.7.3) and Buchholz (1969, x7.4).
In the rst reference (2.7.16) contains an error.
x13.25 For (13.25.1) combine (13.12.1) with (13.14.4).
x13.26 Slater (1960,xx2.6{2.6.3). Note that (2.6.3)
and (2.6.6) contain errors; the correct versions are
(13.26.3) and (13.26.6), respectively.
Chapter 14
Legendre and Related Functions
T. M. Dunster1
Notation 352
14.1 Special Notation . . . . . . . . . . . . . 352
Real Arguments 352
14.2 Di erential Equations . . . . . . . . . . . 352
14.3 De nitions and Hypergeometric Represen-
tations . . . . . . . . . . . . . . . . . . . 353
14.4 Graphics . . . . . . . . . . . . . . . . . . 355
14.5 Special Values . . . . . . . . . . . . . . . 359
14.6 Integer Order . . . . . . . . . . . . . . . 360
14.7 Integer Degree and Order . . . . . . . . . 360
14.8 Behavior at Singularities . . . . . . . . . 361
14.9 Connection Formulas . . . . . . . . . . . 362
14.10 Recurrence Relations and Derivatives . . 362
14.11 Derivatives with Respect to Degree or Order 363
14.12 Integral Representations . . . . . . . . . 363
14.13 Trigonometric Expansions . . . . . . . . . 364
14.14 Continued Fractions . . . . . . . . . . . . 364
14.15 Uniform Asymptotic Approximations . . . 365
14.16 Zeros . . . . . . . . . . . . . . . . . . . 368
14.17 Integrals . . . . . . . . . . . . . . . . . . 368
14.18 Sums . . . . . . . . . . . . . . . . . . . 370
14.19 Toroidal (or Ring) Functions . . . . . . . 37114.20 Conical (or Mehler) Functions . . . . . . 372
Complex Arguments 375
14.21 De nitions and Basic Properties . . . . . 375
14.22 Graphics . . . . . . . . . . . . . . . . . . 375
14.23 Values on the Cut . . . . . . . . . . . . . 376
14.24 Analytic Continuation . . . . . . . . . . . 376
14.25 Integral Representations . . . . . . . . . 377
14.26 Uniform Asymptotic Expansions . . . . . 377
14.27 Zeros . . . . . . . . . . . . . . . . . . . 377
14.28 Sums . . . . . . . . . . . . . . . . . . . 377
14.29 Generalizations . . . . . . . . . . . . . . 377
Applications 378
14.30 Spherical and Spheroidal Harmonics . . . 378
14.31 Other Applications . . . . . . . . . . . . 379
Computation 379
14.32 Methods of Computation . . . . . . . . . 379
14.33 Tables . . . . . . . . . . . . . . . . . . . 380
14.34 Software . . . . . . . . . . . . . . . . . . 380
References 380
1Department of Mathematics and Statistics, San Diego State University, San Diego, California.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 8) by Irene A. Stegun. The author
is indebted to Richard Paris for correcting a long-standing error in Eq. (14.18.3) in previous literature.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
351
352 Legendre and Related Functions
Notation
14.1 Special Notation
(For other notation see pp. xiv and 873.)
x,y, real variables.
z=x+iy complex variable.
m,n nonnegative integers used for order and
degree, respectively.
, general order and degree, respectively.
1
2+i complex degree, 2R.
Euler's constant ( x5.2(ii)).
arbitrary small positive constant.
(x) logarithmic derivative of gamma function
(x5.2(i)).
0(x)d (x)/dx.
F(a;b;c;z) Olver's scaled hypergeometric function:
F(a;b;c;z)/ (c).
Multivalued functions take their principal values
(x4.2(i)) unless indicated otherwise.
The main functions treated in this chapter are the
Legendre functions P(x),Q(x),P(z),Q(z); Ferrers
functions P
(x),Q
(x) (also known as the Legendre
functions on the cut); associated Legendre functions
P
(z),Q
(z),Q
(z); conical functions P
1
2+i (x),
Q
1
2+i (x),bQ
1
2+i (x),P
1
2+i (x),Q
1
2+i (x) (also
known as Mehler functions).
Among other notations commonly used in the lit-
erature Erd elyi et al. (1953a) and Olver (1997b) de-
note P
(x) and Q
(x) by P
(x) and Q
(x), respectively.
Magnus et al. (1966) denotes P
(x),Q
(x),P
(z), and
Q
(z) byP
(x),Q
(x),P
(z), andQ
(z), respectively.
Hobson (1931) denotes both P
(x) andP
(x) byP
(x);
similarly for Q
(x) andQ
(x).
Real Arguments
14.2 Di erential Equations
14.2(i) Legendre's Equation
14.2.1
1 x2d2w
dx2 2xdw
dx+(+ 1)w= 0:
Standard solutions: P(x),Q(x),Q 1(x),
P(x),Q(x),Q 1(x).P(x) and Q(x) are
real when2Randx2( 1;1), andP(x) andQ(x)
are real when 2Randx2(1;1).14.2(ii) Associated Legendre Equation
14.2.2
1 x2d2w
dx2 2xdw
dx+
(+ 1) 2
1 x2
w= 0:
Standard solutions: P
(x),P
(x),Q
(x),
Q
1(x),P
(x),P
(x),Q
(x),Q
1(x).
(14.2.2) reduces to (14.2.1) when = 0. Fer-
rers functions and the associated Legendre functions
are related to the Legendre functions by the equa-
tions P0
(x) = P(x),Q0
(x) = Q(x),P0
(x) =P(x),
Q0
(x) =Q(x),Q0
(x) =Q(x) =Q(x)= (+ 1).
P
(x),P
1
2+i (x), and Q
(x) are real when ,,
and 2R, andx2( 1;1);P
(x),Q
(x), andQ
(x)
are real when and2R, andx2(1;1).
Unless stated otherwise in xx14.2{14.20 it is assumed
that the arguments of the functions P
(x) and Q
(x) lie
in the interval ( 1;1), and the arguments of the func-
tionsP
(x),Q
(x), andQ
(x) lie in the interval (1 ;1).
For extensions to complex arguments see xx14.21{14.28.
14.2(iii) Numerically Satisfactory Solutions
Equation (14.2.2) has regular singularities at x= 1, 1,
and1, with exponent pairs
1
2;1
2
,
1
2;1
2
,
andf 1;g, respectively; compare x2.7(i).
When 6= 0; 1; 2;:::, and+6=
1; 2; 3;:::,P
(x) and P
( x) are linearly in-
dependent, and when <0 they are recessive at
x= 1 andx= 1, respectively. Hence they comprise
a numerically satisfactory pair of solutions ( x2.7(iv)) of
(14.2.2) in the interval 1< x < 1. When =
0; 1; 2;:::, or+= 1; 2; 3;:::,P
(x) and
P
( x) are linearly dependent, and in these cases ei-
ther may be paired with almost any linearly indepen-
dent solution to form a numerically satisfactory pair.
When<0 and< 1
2,P
(x) andQ
(x)
are linearly independent, and recessive at x= 1 and
x=1, respectively. Hence they comprise a numeri-
cally satisfactory pair of solutions of (14.2.2) in the in-
terval 1<x<1. With the same conditions, P
( x)
andQ
( x) comprise a numerically satisfactory pair of
solutions in the interval 1<x< 1.
14.2(iv) Wronskians and Cross-Products
14.2.3W
P
(x);P
( x)
=2
( ) (++ 1) (1 x2);
14.2.4WfP
(x);Q
(x)g= (++ 1)
( + 1) (1 x2);
14.2.5 P
+1(x)Q
(x) P
(x)Q
+1(x) = (++ 1)
( + 2);
14.3 Definitions and Hypergeometric Representations 353
14.2.6 W
P
(x);Q
(x)
=cos()
1 x2;
14.2.7 W
P
(x);P
(x)
= 2 sin()
(x2 1);
14.2.8W
P
(x);Q
(x)
= 1
(++ 1) (x2 1);
14.2.9 W
Q
(x);Q
1(x)
=cos()
x2 1;14.2.10
WfP
(x);Q
(x)g= ei (++ 1)
( + 1) (x2 1);
14.2.11
P
+1(x)Q
(x) P
(x)Q
+1(x) =ei (++ 1)
( + 2):
14.3 De nitions and Hypergeometric Representations
14.3(i) Interval 1<x< 1
The following are real-valued solutions of (14.2.2) when ,2Randx2( 1;1).
Ferrers Function of the First Kind
14.3.1 P
(x) =1 +x
1 x=2
F
+ 1; ; 1 ;1
2 1
2x
:
Ferrers Function of the Second Kind
14.3.2Q
(x) =
2 sin()
cos()1 +x
1 x=2
F
+ 1; ; 1 ;1
2 1
2x
(++ 1)
( + 1)1 x
1 +x=2
F
+ 1; ; 1 +;1
2 1
2x!
:
Here and elsewhere in this chapter
14.3.3 F(a;b;c;x) =1
(c)F(a;b;c;x)
is Olver's hypergeometric function ( x15.1).
P
(x) exists for all values of and.Q
(x) is unde ned when += 1; 2; 3;:::.
When=m= 0;1;2;:::, (14.3.1) reduces to
14.3.4 Pm
(x) = ( 1)m (+m+ 1)
2m ( m+ 1)
1 x2m=2F
+m+ 1;m ;m+ 1;1
2 1
2x
;
equivalently,
14.3.5 Pm
(x) = ( 1)m (+m+ 1)
( m+ 1)1 x
1 +xm=2
F
+ 1; ;m+ 1;1
2 1
2x
:
When=m(2Z) (14.3.2) is replaced by its limiting value; see Hobson (1931, x132) for details. See also (14.3.12){
(14.3.14) for this case.
14.3(ii) Interval 1<x<1
Associated Legendre Function of the First Kind
14.3.6 P
(x) =x+ 1
x 1=2
F
+ 1; ; 1 ;1
2 1
2x
:
354 Legendre and Related Functions
Associated Legendre Function of the Second Kind
14.3.7
Q
(x) =ei1=2 (++ 1)
x2 1=2
2+1x++1F
1
2+1
2+ 1;1
2+1
2+1
2;+3
2;1
x2
,+6= 1; 2; 3;::::
When=m= 1;2;3;:::, (14.3.6) reduces to
14.3.8 Pm
(x) = (+m+ 1)
2m ( m+ 1)
x2 1m=2F
+m+ 1;m ;m+ 1;1
2 1
2x
:
As standard solutions of (14.2.2) we take the pair P
(x) andQ
(x), where
14.3.9 P
(x) =x 1
x+ 1=2
F
+ 1; ;+ 1;1
2 1
2x
;
and
14.3.10 Q
(x) =e iQ
(x)
(++ 1):
LikeP
(x), but unlike Q
(x),Q
(x) is real-valued when ,2Randx2(1;1),and is de ned for all values of
and. The notation Q
(x) is due to Olver (1997b, pp. 170 and 178).
14.3(iii) Alternative Hypergeometric Representations
14.3.11 P
(x) = cos 1
2(+)
w1(;;x ) + sin 1
2(+)
w2(;;x );
14.3.12 Q
(x) = 1
2sin 1
2(+)
w1(;;x ) +1
2cos 1
2(+)
w2(;;x );
where
14.3.13 w1(;;x ) =2 1
2+1
2+1
2
1
2 1
2+ 1
1 x2 =2F
1
2 1
2;1
2 1
2+1
2;1
2;x2
;
14.3.14 w2(;;x ) =2 1
2+1
2+ 1
1
2 1
2+1
2x
1 x2 =2F 1
2 1
2 1
2;1
2 1
2+ 1;3
2;x2
:
14.3.15 P
(x) = 2
x2 1=2F
;++ 1;+ 1;1
2 1
2x
;
14.3.16cos()P
(x) =21=2x
x2 1=2
(++ 1)F
1
2 1
2;1
2 1
2+1
2;1
2 ;1
x2
1=2
x2 1=2
2+1 ( )x++1F
1
2+1
2+ 1;1
2+1
2+1
2;+3
2;1
x2
;
14.3.17P
(x) =
x2 1=2
2
F 1
2 1
2;1
2+1
2+1
2;1
2;x2
1
2 1
2+1
2
1
2+1
2+ 1 xF 1
2 1
2+1
2;1
2+1
2+ 1;3
2;x2
1
2 1
2
1
2+1
2+1
2!
;
14.3.18 P
(x) = 2 x
x2 1=2F
1
2 1
2;1
2 1
2+1
2;+ 1; 1 1
x2
;
14.3.19 Q
(x) =2 (+ 1)(x+ 1)=2
(x 1)(=2)++1F
+ 1;++ 1; 2+ 2;2
1 x
;
14.3.202 sin()
Q
(x) =(x+ 1)=2
(++ 1)(x 1)=2F
+ 1; ; 1 ;1
2 1
2x
(x 1)=2
( + 1)(x+ 1)=2F
+ 1; ;+ 1;1
2 1
2x
:
For further hypergeometric representations of P
(x) andQ
(x) see Erd elyi et al. (1953a, pp. 123{139), Andrews
et al. (1999,x3.1), Magnus et al. (1966, pp. 153{163), and x15.8(iv).
14.4 Graphics 355
14.3(iv) Relations to Other Functions
In terms of the Gegenbauer function C( )
(x) and the Jacobi function ( ; )
(t) (xx15.9(iii), 15.9(ii)):
14.3.21 P
(x) =2 (1 2) (++ 1)
( + 1) (1 ) (1 x2)=2C(1
2 )
+(x):
14.3.22 P
(x) =2 (1 2) (++ 1)
( + 1) (1 ) (x2 1)=2C(1
2 )
+(x):
14.3.23 P
(x) =1
(1 )x+ 1
x 1=2
( ;)
i(2+1)
arcsinh
(1
2x 1
2)1/2
:
Compare also (18.11.1).
14.4 Graphics
14.4(i) Ferrers Functions: 2D Graphs
Figure 14.4.1 :P0
(x);= 0;1
2;1;2;4.
Figure 14.4.2 :Q0
(x);= 0;1
2;1;2;4.
Figure 14.4.3 :P 1=2
(x);= 0;1
2;1;2;4.
Figure 14.4.4 :Q1=2
(x);= 0;1
2;1;2;4.
For additional graphs see http://dlmf.nist.gov/14.4.i .
356 Legendre and Related Functions
Figure 14.4.7 :P
0(x);= 0;1
2;1;2;4.
Figure 14.4.8 :Q
0(x);= 0;1
2;1;2;4.
Figure 14.4.9 :P
1=2(x);= 0;1
2;1;2;4.
Figure 14.4.10 :Q
1=2(x);= 0;1
2;1;2;4.
For additional graphs see http://dlmf.nist.gov/14.4.i .
14.4(ii) Ferrers Functions: 3D Surfaces
Figure 14.4.13 :P0
(x);010; 1<x< 1.
Figure 14.4.14 :Q0
(x);010; 1<x< 1.
14.4 Graphics 357
Figure 14.4.15 :P
0(x);010; 1<x< 1.
Figure 14.4.16 :Q
0(x);06:2; 1<x< 1.
14.4(iii) Associated Legendre Functions: 2D Graphs
Figure 14.4.17 :P0
(x);= 0;1
2;1;2;4.
Figure 14.4.18 :Q0
(x);= 0;1
2;1;2;4.
Figure 14.4.19 :P 1=2
(x);= 0;1
2;1;2;4.
Figure 14.4.20 :Q1=2
(x);= 0;1
2;1;2;4.
For additional graphs see http://dlmf.nist.gov/14.4.iii .
358 Legendre and Related Functions
Figure 14.4.23 :P
0(x);= 0;1
2;1;2;4.
Figure 14.4.24 :Q
0(x);= 0;2;4;8.
Figure 14.4.25 :P
1=2(x);= 0;1
2;1;2;4.
Figure 14.4.26 :Q
1=2(x);= 0;2;4;8.
For additional graphs see http://dlmf.nist.gov/14.4.iii .
14.4(iv) Associated Legendre Functions: 3D Surfaces
Figure 14.4.29 :P0
(x);010;1<x< 10.
Figure 14.4.30 :Q0
(x);010;1<x< 10.
14.5 Special Values 359
Figure 14.4.31 :P
0(x);010;1<x< 10.
Figure 14.4.32 :Q
0(x);010;1<x< 10.
14.5 Special Values
14.5(i)x= 0
14.5.1 P
(0) =21=2
1
2 1
2+ 1
1
2 1
2 1
2;
14.5.2dP
(x)
dx
x=0= 2+11=2
1
2 1
2+1
2
1
2 1
2;
14.5.3
Q
(0) = 2 11=2sin 1
2(+)
1
2+1
2+1
2
1
2 1
2+ 1 ,
+6= 1; 3; 5;:::,
14.5.4
dQ
(x)
dx
x=0=21=2cos 1
2(+)
1
2+1
2+ 1
1
2 1
2+1
2 ,
+6= 2; 4; 6;:::.
14.5(ii)= 0,= 0;1
14.5.5 P0(x) =P0(x) = 1;
14.5.6 P1(x) =P1(x) =x:
14.5.7 Q0(x) =1
2ln1 +x
1 x
;
14.5.8 Q1(x) =x
2ln1 +x
1 x
1:
14.5.9 Q0(x) =1
2lnx+ 1
x 1
;
14.5.10 Q1(x) =x
2lnx+ 1
x 1
1:14.5(iii)=1
2
In this subsection and the next two, 0 < < and
>0.
14.5.11 P1=2
(cos) =2
sin1=2
cos
+1
2
;
14.5.12 P 1=2
(cos) =2
sin1=2sin
+1
2
+1
2;
14.5.13 Q1=2
(cos) =
2 sin1=2
sin
+1
2
;
14.5.14 Q 1=2
(cos) =
2 sin1=2cos
+1
2
+1
2:
14.5.15
P1=2
(cosh) =2
sinh1=2
cosh
+1
2
;
14.5.16
P 1=2
(cosh) =2
sinh1=2sinh
+1
2
+1
2;
14.5.17
Q1=2
(cosh) =
2 sinh1=2exp
+1
2
+3
2:
14.5(iv)=
14.5.18 P
(cos) =(sin)
2 (+ 1);
14.5.19 P
(cosh) =(sinh)
2 (+ 1):
360 Legendre and Related Functions
14.5(v)= 0;=1
2
In this subsection K(k) andE(k) denote the com-
plete elliptic integrals of the rst and second kinds; see
x19.2(ii).
14.5.20 P1
2(cos) =2
2E
sin 1
2
K
sin 1
2
;
14.5.21 P 1
2(cos) =2
K
sin 1
2
;
14.5.22 Q1
2(cos) =K
cos 1
2
2E
cos 1
2
;
14.5.23 Q 1
2(cos) =K
cos 1
2
:
14.5.24P1
2(cosh) =2
e=2E
1 e 21=2
;
14.5.25P 1
2(cosh) =2
cosh 1
2K
tanh 1
2
;
14.5.26
Q1
2(cosh) = 2 1=2coshsech 1
2
K
sech 1
2
4 1=2cosh 1
2
E
sech 1
2
;
14.5.27Q 1
2(cosh) = 2 1=2e =2K
e
:
14.6 Integer Order
14.6(i) Nonnegative Integer Orders
Form= 0;1;2;:::,
14.6.1 Pm
(x) = ( 1)m
1 x2m=2dmP(x)
dxm;
14.6.2 Qm
(x) = ( 1)m
1 x2m=2dmQ(x)
dxm:
14.6.3 Pm
(x) =
x2 1m=2dmP(x)
dxm;
14.6.4 Qm
(x) =
x2 1m=2dmQ(x)
dxm;
14.6.5 (+ 1)mQm
(x) = ( 1)m
x2 1m=2dmQ(x)
dxm:
14.6(ii) Negative Integer Orders
Form= 1;2;3;:::,
14.6.6 P m
(x) =
1 x2 m=2Z1
x:::Z1
xP(x) (dx)m:
14.6.7P m
(x) =
x2 1 m=2Zx
1:::Zx
1P(x) (dx)m;
14.6.8Q m
(x) = ( 1)m
x2 1 m=2
Z1
x:::Z1
xQ(x) (dx)m:
For connections between positive and negative inte-
ger orders see (14.9.3), (14.9.4), and (14.9.13).14.7 Integer Degree and Order
14.7(i)= 0
Forn= 0;1;2;:::,
14.7.1 P0
n(x) =Pn(x) =P0
n(x) =Pn(x),x2R,
wherePn(x) is the Legendre polynomial of degree n. For
additional properties of Pn(x) see Chapter 18.
14.7.2
Q0
n(x) =Qn(x) =1
2Pn(x) ln1 +x
1 x
Wn 1(x);
whereW 1(x) = 0, and for n1,
14.7.3
Wn 1(x) =n 1X
s=0(n+s)!( (n+ 1) (s+ 1))
2s(n s)!(s!)2(x 1)s;
equivalently,
14.7.4Wn 1(x) =nX
k=11
kPk 1(x)Pn k(x):
14.7.5W0(x) = 1; W 1(x) =3
2x; W 2(x) =5
2x2 2
3:
Next,
14.7.6Q0
n(x) =Qn(x) =n!Q0
n(x) =n!Qn(x);
where
14.7.7
Qn(x) =1
2Pn(x) lnx+ 1
x 1
Wn 1(x),n= 0;1;2;:::.
14.7(ii) Rodrigues-Type Formulas
Form= 0;1;2;:::, andn= 0;1;2;:::,
14.7.8 Pm
n(x) = ( 1)m
1 x2m=2dm
dxmPn(x);
14.7.9 Qm
n(x) = ( 1)m
1 x2m=2dm
dxmQn(x);
14.7.10
Pm
n(x) = ( 1)m+n
1 x2m=2
2nn!dm+n
dxm+n
1 x2n:
14.7.11Pm
n(x) =
x2 1m=2dm
dxmPn(x);
14.7.12Qm
n(x) =
x2 1m=2dm
dxmQn(x);
14.7.13Pn(x) =1
2nn!dn
dxn
x2 1n;
14.7.14Pm
n(x) =
x2 1m=2
2nn!dm+n
dxm+n
x2 1n;
14.7.15Pm
m(x) =(2m)!
2mm!
x2 1m=2:
Whenmis even and mn,Pm
n(x) andPm
n(x) are
polynomials of degree n. Also,
14.7.16 Pm
n(x) =Pm
n(x) = 0, m>n .
14.8 Behavior at Singularities 361
14.7(iii) Re
ection Formulas
14.7.17 Pm
n( x) = ( 1)n mPm
n(x);
14.7.18 Qm
n( x) = ( 1)n m 1Qm
n(x):
14.7(iv) Generating Functions
When 1<x< 1 andjhj<1,
14.7.191X
n=0Pn(x)hn=
1 2xh+h2 1=2;
14.7.20
1X
n=0Qn(x)hn=1
(1 2xh+h2)1=2
ln
x h+
1 2xh+h21=2
(1 x2)1=2!
:
When 1<x< 1 andjhj>1,
14.7.211X
n=0Pn(x)h n 1=
1 2xh+h2 1=2:
Whenx > 1, (14.7.19) applies with jhj< x
x2 11=2. Also, with the same conditions
14.7.22
1X
n=0Qn(x)hn=1
(1 2xh+h2)1=2
ln
x h+
1 2xh+h21=2
(x2 1)1=2!
:
Lastly, when x > 1, (14.7.21) applies with jhj>
x+
x2 11=2.
For other generating functions see Magnus et al.
(1966, pp. 232{233) and Rainville (1960, pp. 163{165,
168, 170{171, 184).
14.8 Behavior at Singularities
14.8(i)x!1 orx! 1+
Asx!1 ,
14.8.1
P
(x)1
(1 )2
1 x=2
,6= 1;2;3;:::,
14.8.2
Pm
(x)( 1)m( m+ 1)2m
m!1 x
2m=2
,
m= 1;2;3;:::,6=m 1;m 2;:::; m,
14.8.3
Q(x) =1
2ln2
1 x
(+ 1) +O(1 x),
6= 1; 2; 3;:::,where
is Euler's constant ( x5.2(ii)). In the next three
relations<>0.
14.8.4
Q
(x)1
2cos() ()2
1 x=2
,6=1
2;3
2;5
2;:::,
14.8.5
Q
(x)
( 1)+(1=2) (++ 1)
2 (+ 1) ( + 1)1 x
2=2
,
=1
2;3
2;5
2;:::,6= 1; 2; 3;:::,
14.8.6 Q
(x) () ( + 1)
2 (++ 1)2
1 x=2
,
6= 1; 2; 3;:::.
The behavior of P
(x) and Q
(x) asx! 1+ fol-
lows from the above results and the connection formulas
(14.9.8) and (14.9.10).
14.8(ii)x!1+
14.8.7
P
(x)1
(1 )2
x 1=2
,6= 1;2;3;:::,
14.8.8
Pm
(x) (+m+ 1)
m! ( m+ 1)x 1
2m=2
,
m= 1;2;3;:::,m6= 1; 2; 3;:::,
14.8.9
Q(x) = ln(x 1)
2 (+ 1)+1
2ln 2
(+ 1)
(+ 1)
+O(x 1), 6= 1; 2; 3;:::,
14.8.10Q n(x)!( 1)n+1(n 1)!,n= 1;2;3;:::,
14.8.11Q
(x) ()
2 (++ 1)2
x 1=2
,
<>0,+6= 1; 2; 3;:::.
14.8(iii)x!1
14.8.12P
(x)
+1
2
1=2 ( + 1)(2x),
< > 1
2, 6= 1;2;3;:::,
14.8.13P
(x)
1
2
1=2 ( )(2x)+1,
< < 1
2,+6= 0;1;2;:::,
14.8.14P
1=2(x)1
1
2 2
x1=2
lnx,
6=1
2;3
2;5
2;:::,
362 Legendre and Related Functions
14.8.15
Q
(x)1=2
+3
2
(2x)+1,6= 3
2; 5
2; 7
2;:::,
14.8.16Q
n (1=2)(x)1=2
+n+1
2
n!
n+1
2
(2x)n+(1=2),
n= 1;2;3;:::, n+1
26= 0; 1; 2;:::.
14.9 Connection Formulas
14.9(i) Connections Between P
(x),
P
1(x), Q
(x), Q
1(x)
14.9.1
sin()
2 ( + 1)P
(x) = 1
(++ 1)Q
(x)
+cos()
( + 1)Q
(x):
14.9.2
2 sin()
( + 1)Q
(x) =1
(++ 1)P
(x)
cos()
( + 1)P
(x);
14.9.3 P m
(x) = ( 1)m ( m+ 1)
(+m+ 1)Pm
(x);
14.9.4Q m
(x) = ( 1)m ( m+ 1)
(+m+ 1)Qm
(x),
6=m 1;m 2;:::.
14.9.5 P
1(x) =P
(x);P
1(x) =P
(x);
14.9.6
cos() cos()P
(x) = sin((+))Q
(x)
sin(( ))Q
1(x):
14.9(ii) Connections Between P
(x),
Q
(x), Q
(x)
14.9.7sin(( ))
(++ 1)P
(x) =sin()
( + 1)P
(x)
sin()
( + 1)P
( x);
14.9.8
1
2sin(( ))P
(x) = cos(( ))Q
(x)
Q
( x);
14.9.92
(++ 1) ( )Q
(x)
= cos()P
(x) + cos()P
( x);
14.9.10
(2=) sin(( ))Q
(x) = cos(( ))P
(x)
P
( x):14.9(iii) Connections Between P
(x),
P
1(x),Q
(x),Q
1(x)
14.9.11P
1(x) =P
(x); P
1(x) =P
(x);
14.9.12 cos()P
(x) = Q
(x)
( )+Q
1(x)
(++ 1):
14.9.13
P m
(x) = ( m+ 1)
(+m+ 1)Pm
(x),6=m 1;m 2;:::.
14.9.14 Q
(x) =Q
(x);
14.9.15
2 sin()
Q
(x) =P
(x)
(++ 1) P
(x)
( + 1):
14.9(iv) Whipple's Formula
14.9.16
Q
(x)
= 1
21=2
x2 1 1=4P (1=2)
(1=2)
x
x2 1 1=2
:
Equivalently,
14.9.17
P
(x)
= (2=)1=2
x2 1 1=4Q+(1=2)
(1=2)
x
x2 1 1=2
:
14.10 Recurrence Relations and Derivatives
14.10.1P+2
(x) + 2(+ 1)x
1 x2 1=2P+1
(x)
+ ( )(++ 1) P
(x) = 0;
14.10.2
1 x21=2P+1
(x) ( + 1) P
+1(x)
+ (++ 1)xP
(x) = 0;
14.10.3( + 2) P
+2(x) (2+ 3)xP
+1(x)
+ (++ 1) P
(x) = 0;
14.10.4
1 x2dP
(x)
dx
= ( 1)P
+1(x) + (+ 1)xP
(x);
14.10.5
1 x2dP
(x)
dx= (+)P
1(x) xP
(x):
Q
(x) also satis es (14.10.1){(14.10.5).
14.10.6P+2
(x) + 2(+ 1)x
x2 1 1=2P+1
(x)
( )(++ 1)P
(x) = 0;
14.10.7
x2 11=2P+1
(x) ( + 1)P
+1(x)
+ (++ 1)xP
(x) = 0:
Q
(x) also satis es (14.10.6) and (14.10.7). In addition,
P
(x) andQ
(x) satisfy (14.10.3){(14.10.5).
14.11 Derivatives with Respect to Degree or Order 363
14.11 Derivatives with Respect to Degree or Order
14.11.1@
@P
(x) =cot()P
(x) 1
A
(x);
14.11.2@
@Q
(x) = 1
22P
(x) +sin()
sin() sin((+))Q
(x) 1
2cot((+))A
(x) +1
2csc((+))A
( x);
where
14.11.3 A
(x) = sin()1 +x
1 x=21X
k=0 1
2 1
2xk (k ) (k++ 1)
k! (k + 1)( (k++ 1) (k )):
14.11.4@
@P
(x)
=0= ( ( ) cot())P(x) +Q(x);
14.11.5@
@Q
(x)
=0= 1
42P(x) + ( ( ) cot())Q(x):
(14.11.1) holds if P
(x) is replaced by P
(x), provided that the factor ((1 + x)/(1 x))=2in (14.11.3) is replaced
by ((x+ 1)/(x 1))=2. (14.11.4) holds if P
(x),P(x), and Q(x) are replaced by P
(x),P(x), andQ(x),
respectively.
For further results see Magnus et al. (1966, pp. 177{178).
14.12 Integral Representations
14.12(i) 1<x< 1
Mehler{Dirichlet Formula
14.12.1 P
(cos) =21=2(sin)
1=2 1
2 Z
0cos
+1
2
t
(cost cos)+(1=2)dt, 0 << ,<<1
2.
14.12.2 P
(x) =
1 x2 =2
()Z1
xP(t)(t x) 1dt, <>0;
compare (14.6.6).
14.12.3
Q
(cos) =1=2 (++ 1)(sin)
2+1
+1
2
( + 1)Z1
0(sinht)2
(cos+isincosht)++1dt+Z1
0(sinht)2
(cos isincosht)++1dt
,
0<< ,<> 1
2,<()> 1.
14.12(ii) 1<x<1
14.12.4P
(x) =21=2
+1
2
x2 1=2
1=2 (++ 1) ( )Z1
0cosh
+1
2
t
(x+ cosht)+(1=2)dt,+6= 1; 2; 3;:::,<( )>0.
14.12.5P
(x) =
x2 1 =2
()Zx
1P(t)(x t) 1dt, <>0.
14.12.6Q
(x) =1=2
x2 1=2
2
+1
2
( + 1)Z1
0(sinht)2
x+ (x2 1)1=2cosht++1dt,<(+ 1)><> 1
2.
14.12.7Pm
(x) =(+ 1)m
Z
0
x+
x2 11=2cos
cos(m )d ;
14.12.8Pm
n(x) =2mm!(n+m)!
x2 1m=2
(2m)!(n m)!Z
0
x+
x2 11=2cos n m
(sin )2md , nm.
364 Legendre and Related Functions
14.12.9 Qm
n(x) =1
n!Zu
0
x
x2 11=2coshtn
cosh(mt)dt;
where
14.12.10 u=1
2lnx+ 1
x 1
:
14.12.11 Qm
n(x) =
x2 1m=2
2n+1n!Z1
1
1 t2n
(x t)n+m+1dt;
14.12.12 Qm
n(x) =1
(n m)!Pm
n(x)Z1
xdt
(t2 1) (Pm
n(t))2, nm.
Neumann's Integral
14.12.13 Qn(x) =1
2(n!)Z1
1Pn(t)
x tdt:
Heine's Integral
14.12.14 Qn(x) =1
n!Z1
0dt
x+ (x2 1)1=2coshtn+1:
For further integral representations see Erd elyi et al. (1953a, pp. 158{159) and Magnus et al. (1966, pp. 184{190),
and for contour integrals and other representations see x14.25.
14.13 Trigonometric Expansions
When 0<< ,
14.13.1 P
(cos) =2+1(sin)
1=21X
k=0 (++k+ 1)
+k+3
2
+1
2
k
k!sin((++ 2k+ 1));
14.13.2 Q
(cos) =1=22(sin)1X
k=0 (++k+ 1)
+k+3
2
+1
2
k
k!cos((++ 2k+ 1));
14.13.3 Pn(cos) =22n+2(n!)2
(2n+ 1)!1X
k=013(2k 1)
k!(n+ 1)(n+ 2)(n+k)
(2n+ 3)(2n+ 5)(2n+ 2k+ 1)sin((n+ 2k+ 1));
14.13.4 Qn(cos) =22n+1(n!)2
(2n+ 1)!1X
k=013(2k 1)
k!(n+ 1)(n+ 2)(n+k)
(2n+ 3)(2n+ 5)(2n+ 2k+ 1)cos((n+ 2k+ 1)):
For these and other trigonometric expansions see Erd elyi et al. (1953a, pp. 146{147).
14.14 Continued Fractions
14.14.11
2
x2 11=2P
(x)
P 1
(x)=x0
y0+x1
y1+x2
y2+;
where
14.14.2 xk=1
4( k+ 1)(++k)
x2 1
; yk= (+k)x;
provided that xk+1andykdo not vanish simultaneously for any k= 0;1;2;:::.
14.14.3 ( )Q
(x)
Q
1(x)=x0
y0 x1
y1 x2
y2 , 6=,
where now
14.14.4 xk= (++k)( +k); yk= (2+ 2k+ 1)x;
again provided xk+1andykdo not vanish simultaneously for any k= 0;1;2;:::.
14.15 Uniform Asymptotic Approximations 365
14.15 Uniform Asymptotic Approximations
14.15(i) Large , Fixed
For the interval 1<x< 1 with xed , real, and arbitrary xed values of the nonnegative integer J,
14.15.1 P
(x) =1x
1x=20
@J 1X
j=0(+ 1)j( )j
j! (j+ 1 +)1x
2j
+O1
(J+ 1 +)1
A
as!1 , uniformly with respect to x. In other words, the convergent hypergeometric series expansions of P
(x)
are also generalized (and uniform) asymptotic expansions as !1 , with scale 1/ ( j+ 1 +),j= 0;1;2;:::;
comparex2.1(v).
Provided that =2Zthe corresponding expansions for P
(x) and Q
(x) can be obtained from the connection
formulas (14.9.7), (14.9.9), and (14.9.10).
For the interval 1 < x <1the following asymptotic approximations hold when !1 , with( 1
2) xed,
uniformly with respect to x:
14.15.2 P
(x) =1
(+ 1)2u
1=2
K+1
2(u)
1 +O1
;
14.15.3 Q
(x) =1
+(1=2)u
21=2
I+1
2(u)
1 +O1
;
whereuis given by (14.12.10). Here IandKare the modi ed Bessel functions ( x10.25(ii)).
For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000).
14.15(ii) Large ,0+1
2(1 )
In this and subsequent subsections denotes an arbitrary constant such that 0 << 1.
As!1 ,
14.15.4 P
(x) =1
(+ 1)
1 2 =21
1 + (=2)+(1=4)p
x1=2
e
1 +O1
;
uniformly with respect to x2( 1;1) and+1
22[0;(1 )], where
14.15.5 =+1
2
(<1);
14.15.6 p=x
( 2x2+ 1 2)1=2;
and
14.15.7 =1
2ln1 +p
1 p
+1
2 ln1 p
1 + p
:
With the same conditions, the corresponding approximation for P
( x) is obtained by replacing e byeon
the right-hand side of (14.15.4). Approximations for P
(x) and Q
(x) can then be achieved via (14.9.7), (14.9.9),
and (14.9.10).
Next,
14.15.8P
(x) =2
1=21
(+ 1)1
1 + (=2)+(1=4)
1 2 =2 2+2
2(x2 1) + 11=4
K+1
2()
1+O1
;
14.15.9Q
(x) =
21=2e
+(1=2)1
1 + =2
1 2 (=2) (1=4) 2+2
2(x2 1) + 11=4
I+1
2()
1 +O1
;
uniformly with respect to x2(1;1) and+1
22[0;(1 )]. Here is again given by (14.15.5), and is de ned
implicitly by
14.15.10 ln
2+21=2+
ln
2+21=2=1
2ln
1 + 2
x2+ 1 2 2x
2x2 2+ 11=2
(x2 1) (1 2)!
+1
2 ln
2
2x2 1
+ 1 + 2 x
2x2 2+ 11=2
1 2!
:
366 Legendre and Related Functions
The interval 1 < x <1is mapped one-to-one to the interval 0 < <1, with the points x= 1 andx=1
corresponding to =1and= 0, respectively. For asymptotic expansions and explicit error bounds, see Dunster
(2003b).
14.15(iii) Large , Fixed
For!1 and xed(0),
14.15.11 P
(cos) =1
sin1=2
J
+1
2
+O1
envJ
+1
2
;
14.15.12 Q
(cos) =
2
sin1=2
Y
+1
2
+O1
envY
+1
2
;
uniformly for 2(0; ]. For the Bessel functions JandYseex10.2(ii), and for the env functions associated with
JandYseex2.8(iv).
Next,
14.15.13 P
(cosh) =1
sinh1=2
I
+1
2
1 +O1
;
14.15.14 Q
(cosh) =
(++ 1)
sinh1=2
K
+1
2
1 +O1
;
uniformly for 2(0;1).
For asymptotic expansions and explicit error bounds, see Olver (1997b, Chapter 12, xx12, 13) and Jones (2001).
For convergent series expansions see Dunster (2004).
See also Olver (1997b, pp. 311{313) and x18.15(iii) for a generalized asymptotic expansion in terms of elementary
functions for Legendre polynomials Pn(cos) asn!1 with xed.
14.15(iv) Large ,0(1 )(+1
2)
As!1 ,
14.15.15 P
(x) = y 2
1 2 x21=4
J
+1
2
y1=2
+O1
envJ
+1
2
y1=2
;
14.15.16 Q
(x) =
2y 2
1 2 x21=4
Y
+1
2
y1=2
+O1
envY
+1
2
y1=2
;
uniformly with respect to x2[0;1) and2[0;(1 )(+1
2)]. For , , andysee below.
Next,
14.15.17 P
(x) = 2 y
x2 1 + 21=4
I
+1
2
jyj1=2
1 +O1
;
14.15.18 Q
(x) =1
(++ 1) 2 y
x2 1 + 21=4
K
+1
2
jyj1=2
1 +O1
;
uniformly with respect to x2(1;1) and2[0;(1 )(+1
2)]. In (14.15.15){(14.15.18)
14.15.19 =
+1
2(<1);
14.15.20 =e +1
2
++1
2(=2)+(1=4)
+1
22 2 =2
;
and the variable yis de ned implicitly by
14.15.21
y 21=2 arctan
y 21=2
!
= arccos
x
(1 2)1=2!
2arccos
1 + 2
x2 1 + 2
(1 2) (1 x2)!
,
x
1 21=2,y 2,
14.15 Uniform Asymptotic Approximations 367
and
14.15.22
2 y1=2+1
2 lnjyj ln
2 y1=2+
= ln
x+
x2 1 + 21=2
(1 2)1=2!
+
2ln
1 2 1 x2
(1 + 2)x2 1 + 2+ 2 x(x2 1 + 2)1=2!
,
x
1 21=2,y 2,
where the inverse trigonometric functions take their principal values ( x4.23(ii)). The points x=
1 21=2,x= 1,
andx=1are mapped to y= 2,y= 0, andy= 1, respectively. The interval 0 x<1is mapped one-to-one
to the interval 1<yy0, wherey=y0is the (positive) solution of (14.15.21) when x= 0.
For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986).
14.15(v) Large ,(+1
2)(+1
2)=
Here we introduce the envelopes of the parabolic cylinder functions U( c;x),U( c;x), which are de ned in x12.2.
Forf(x) =U( c;x) orU( c;x), withcandxnonnegative,
14.15.23 envf(x) =(
(U( c;x))2+ (U( c;x))21=2;0xXc;p
2f(x); X cx<1;
wherex=Xcdenotes the largest positive root of the equation U( c;x) =U( c;x).
As!1 ,
14.15.24P
(x) =1
+1
21=42(+)=2 1
2+1
2+3
42 2
x2 a21=4
U
1
2;(2+ 1)1=2
+O
2=3
envU
1
2;(2+ 1)1=2
;
14.15.25Q
(x) =
+1
21=42(++2)=2 1
2+1
2+3
42 2
x2 a21=4
U
1
2;(2+ 1)1=2
+O
2=3
envU
1
2;(2+ 1)1=2
;
uniformly with respect to x2[0;1) and2[(+1
2);+1
2]. Here
14.15.26 a=
++1
2 +1
2 1=2
+1
2; =
2 +1
2
+1
2!1=2
;
and the variable is de ned implicitly by
14.15.27
1
2
2 21=2 1
2 2arccosh
=
1 a21=2arctanh
1
xx2 a2
1 a21=2!
arccoshx
a
,ax<1, <1,
and
14.15.28
1
2 2arcsin
+1
2
2 21=2= arcsinx
a
1 a21=2arctan
x1 a2
a2 x21=2!
, axa, ,
whena>0, and
14.15.29 2= ln
1 x2
, 1<x< 1,
whena= 0. The inverse hyperbolic and trigonometric functions take their principal values ( xx4.23(ii), 4.37(ii)).
Whena>0 the interval ax<1 is mapped one-to-one to the interval <1, with the points x= a,
x=a, andx= 1 corresponding to = ,= , and=1, respectively. When a= 0 the interval 1<x< 1
is mapped one-to-one to the interval 1< <1, with the points x= 1, 0, and 1 corresponding to = 1, 0,
and1, respectively.
368 Legendre and Related Functions
Next, as!1 ,
14.15.30 P
(x) =1
+1
21=42(+)=2 1
2+1
2+3
42+ 2
x2+a21=4
U
1
2;(2+ 1)1=2
1+O
1ln
;
uniformly with respect to x2( 1;1) and2[+1
2;(1=)(+1
2)]. Hereis de ned implicitly by
14.15.311
2
2+ 21=2+1
2 2arcsinh
=
1 +a21=2arctanh
x1 +a2
x2+a21=2!
arcsinhx
a
,
1<x< 1, 1< <1,
whena>0, which maps the interval 1<x< 1 one-to-one to the interval 1< <1: the points x= 1 and
x= 1 correspond to = 1and=1, respectively. When a= 0 (14.15.29) again applies. (The inverse hyperbolic
functions again take their principal values.)
Since (14.15.30) holds for negative x, corresponding approximations for Q
(x), uniformly valid in the interval
1<x< 1, can be obtained from (14.9.9) and (14.9.10).
For error bounds and other extensions see Olver (1975b).
14.16 Zeros
14.16(i) Notation
Throughout this section we assume that andare
real, and when they are not integers we write
14.16.1 =m+; =n+;
wherem,n2Zand,2(0;1). For all cases con-
cerning P
(x) andP
(x) we assume that 1
2with-
out loss of generality (see (14.9.5) and (14.9.11)).
14.16(ii) Interval 1<x< 1
The number of zeros of P
(x) in the interval ( 1;1) is
max(d jje;0) if any of the following sets of conditions
hold:
(a)0.
(b)>0,nm, and>.
(c)>0,n<m , andm nis odd.
(d)= 0;1;2;3;:::.
The number of zeros of P
(x) in the interval ( 1;1)
is max(d jje;0) + 1 if either of the following sets of
conditions holds:
(a)>0,n>m , and.
(b)>0,n<m , andm nis even.
The zeros of Q
(x) in the interval ( 1;1) interlace
those of P
(x).Q
(x) has max(d jje;0) +kzeros
in the interval ( 1;1), wherekcan take one of the val-
ues 1, 0, 1, 2, subject to max( d jje;0) +kbeingeven or odd according as cos( ) and cos() have op-
posite signs or the same sign. In the special case = 0
and=n= 0;1;2;3;:::,Qn(x) hasn+ 1 zeros in the
interval 1<x< 1.
For uniform asymptotic approximations for the ze-
ros of P m
n(x) in the interval 1<x< 1 whenn!1
withm(0) xed, see Olver (1997b, p. 469).
14.16(iii) Interval 1<x<1
P
(x) has exactly one zero in the interval (1 ;1) if ei-
ther of the following sets of conditions holds:
(a) > 0, > , =2Z, and sin(( )) and
sin() have opposite signs.
(b), =2Z, andbcis odd.
For all other values of and(with 1
2)P
(x)
has no zeros in the interval (1 ;1).
Q
(x) has no zeros in the interval (1 ;1) when
> 1, and at most one zero in the interval (1 ;1)
when < 1.
14.17 Integrals
14.17(i) Inde nite Integrals
14.17.1Z
1 x2 =2P
(x)dx=
1 x2 ( 1)=2P 1
(x):
14.17.2
Z
1 x2=2P
(x)dx=
1 x2(+1)=2
( )(++ 1)P+1
(x),
6=or 1.
14.17 Integrals 369
14.17.3Z
xP
(x)Q
(x)dx=1
2(+ 1)
(2 (+ 1)(+x2))P
(x)Q
(x)
+ (+ 1)( + 1)x(P
(x)Q
+1(x) +P
+1(x)Q
(x)) ( + 1)2P
+1(x)Q
+1(x)
,
6= 0; 1.
14.17.4Zx
(1 x2)3=2P
(x)Q
(x)dx
=1
(1 42) (1 x2)1=2
(1 22+ 2(+ 1)) P
(x)Q
(x) + (2+ 1)( 1)x(P
(x)Q
+1(x) +P
+1(x)Q
(x))
+ 2( 1)2P
+1(x)Q
+1(x)
,
6=1
2.
In (14.17.1){(14.17.4), Pmay be replaced by Q, and in (14.17.3) and (14.17.4), Qmay be replaced by P.
For further results, see Maximon (1955) and Prudnikov et al. (1990, pp. 37{39). See also (14.12.2), (14.12.5), and
(14.12.12).
14.17(ii) Barnes' Integral
14.17.5Z1
0x
1 x2=2P
(x)dx= 1
2+1
2
1
2+ 1
2+1 1
2 1
2+1
2+ 1
1
2+1
2+1
2+3
2,<> 1,<> 1.
14.17(iii) Orthogonality Properties
Forl;m;n = 0;1;2;:::,
14.17.6Z1
1Pm
l(x)Pm
n(x)dx=l;n(n+m)!
(n m)!
n+1
2;
14.17.7Z1
1Pm
l(x)P m
n(x)dx= ( 1)ml;n1
l+1
2;
14.17.8Z1
1Pl
n(x)Pm
n(x)
1 x2dx=l;m(n+m)!
(n m)!m, m> 0,
14.17.9Z1
1Pl
n(x)P m
n(x)
1 x2dx= ( 1)ll;m1
l, l>0.
14.17(iv) De nite Integrals of Products
With (x) = 0(x)= (x) (x5.2(i)),
14.17.10Z1
1P(x)P(x)dx=2 (2 sin() sin() ( (+ 1) (+ 1)) +sin(( )))
2( )(++ 1),6=or 1.
14.17.11Z1
1(P(x))2dx=2 2 sin2() 0(+ 1)
2
+1
2 , 6= 1
2.
14.17.12Z1
1Q(x)Q(x)dx=
( (+ 1) (+ 1))(1 + cos( ) cos()) +1
2sin(( ))
( )(++ 1),
6=or 1,and6= 1; 2; 3;:::.
14.17.13Z1
1(Q(x))2dx=2 2
1 + cos2()
0(+ 1)
2(2+ 1),6= 1
2or 1; 2; 3;:::.
14.17.14Z1
1P(x)Q(x)dx=2 sin() cos() ( (+ 1) (+ 1)) +cos(( ))
( )(++ 1),<>0,< >0,6=.
370 Legendre and Related Functions
14.17.15Z1
1P(x)Q(x)dx= sin(2) 0(+ 1)
(2+ 1),< >0.
14.17.16Z1
1Pm
l(x)Qm
n(x)dx=
1 ( 1)l+n
(l+m)!
(l n)(l+n+ 1)(l m)!,
l;m;n = 0;1;2;:::,l6=n.
14.17.17Z
0Ql(cos)Pm(cos)Pn(cos) sind
= 0,l;m;n = 1;2;3;:::,jm nj<l<m +n.
(Whenl+m+nis even the condition jm nj< l <
m+nis not needed.) Next,
14.17.18Z1
1P(x)Q(x)dx=1
( )(++ 1),
<>< >0.
14.17.19Z1
1Q(x)Q(x)dx
= (+ 1) (+ 1)
( )(++ 1),
<(+)> 1,6=,and6= 1; 2; 3;:::.
14.17.20Z1
1(Q(x))2dx= 0(+ 1)
2+ 1,< > 1
2.For further results, see Prudnikov et al. (1990,
pp. 194{240); also (34.3.21).
14.17(v) Laplace Transforms
For Laplace transforms and inverse Laplace transforms
involving associated Legendre functions, see Erd elyi
et al. (1954a, pp. 179{181, 270{272), Oberhettinger and
Badii (1973, pp. 113{118, 317{324), Prudnikov et al.
(1992a,xx3.22, 3.32, and 3.33), and Prudnikov et al.
(1992b,xx3.20, 3.30, and 3.31).
14.17(vi) Mellin Transforms
For Mellin transforms involving associated Legendre
functions see Oberhettinger (1974, pp. 69{82) and
Marichev (1983, pp. 247{283), and for inverse trans-
forms see Oberhettinger (1974, pp. 205{215).
14.18 Sums
14.18(i) Expansion Theorem
For expansions of arbitrary functions in series of Leg-
endre polynomials see x18.18(i), and for expansions of
arbitrary functions in series of associated Legendre func-
tions see Sch afke (1961b).
14.18(ii) Addition Theorems
In (14.18.1) and (14.18.2), 1,2, and1+2all lie in [0;), and is real.
14.18.1 P(cos1cos2+ sin1sin2cos ) =P(cos1)P(cos2) + 21X
m=1( 1)mP m
(cos1)Pm
(cos2) cos(m );
14.18.2 Pn(cos1cos2+ sin1sin2cos ) =nX
m= n( 1)mP m
n(cos1)Pm
n(cos2) cos(m ):
In (14.18.3), 1lies in (0;1
2),2and1+2both lie in (0 ;),1<2, is real, and 6= 1; 2; 3;:::.
14.18.3 Q(cos1cos2+ sin1sin2cos ) =P(cos1)Q(cos2) + 21X
m=1( 1)mP m
(cos1)Qm
(cos2) cos(m ):
In (14.18.4) and (14.18.5), 1and2are positive, and is real; also in (14.18.5) 1<2and6= 1; 2; 3;:::.
14.18.4
P(cosh1cosh2 sinh1sinh2cos ) =P(cosh1)P(cosh2) + 21X
m=1( 1)mP m
(cosh1)Pm
(cosh2) cos(m );
14.18.5
Q(cosh1cosh2 sinh1sinh2cos ) =P(cosh1)Q(cosh2) + 21X
m=1( 1)mP m
(cosh1)Qm
(cosh2) cos(m ):
14.19 Toroidal (or Ring) Functions 371
14.18(iii) Other Sums
14.18.6 (x y)nX
k=0(2k+ 1)Pk(x)Pk(y) = (n+ 1) (Pn+1(x)Pn(y) Pn(x)Pn+1(y));
14.18.7 (x y)nX
k=0(2k+ 1)Pk(x)Qk(y) = (n+ 1) (Pn+1(x)Qn(y) Pn(x)Qn+1(y)) 1:
Zonal Harmonic Series
14.18.8 P( x) =sin()
1X
n=02n+ 1
( n)(+n+ 1)Pn(x), =2Z.
Dougall's Expansion
14.18.9 P
(x) =sin()
1X
n=0( 1)n 2n+ 1
( n)(+n+ 1)P
n(x), 1<x1,0, =2Z.
For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22).
14.18(iv) Compendia
For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367{377, 457{460, and 475),
Erd elyi et al. (1953a,x3.10), Gradshteyn and Ryzhik (2000, x8.92), Magnus et al. (1966, pp. 178{184), and Prudnikov
et al. (1990,xx5.2, 6.5). See also x18.18 and (34.3.19).
14.19 Toroidal (or Ring) Functions
14.19(i) Introduction
When=n 1
2,n= 0;1;2;:::,2R, andx2(1;1) solutions of (14.2.2) are known as toroidal orring functions .
This form of the di erential equation arises when Laplace's equation is transformed into toroidal coordinates (;; ),
which are related to Cartesian coordinates ( x;y;z ) by
14.19.1 x=csinhcos
cosh cos; y =csinhsin
cosh cos; z =csin
cosh cos;
where the constant cis a scaling factor. Most required properties of toroidal functions come directly from the results
forP
(x) andQ
(x). In particular, for = 0 and=1
2seex14.5(v).
14.19(ii) Hypergeometric Representations
With Fas inx14.3 and>0,
14.19.2 P
1
2(cosh) = (1 2)22
(1 ) (1 e 2)e(+(1=2))F 1
2 ;1
2+ ; 1 2;e 2
,6=1
2.
14.19.3 Q
1
2(cosh) =1=2
1 e 2
e(+(1=2))F
+1
2;++1
2;+ 1;e 2
:
14.19(iii) Integral Representations
With>0,
14.19.4Pm
n 1
2(cosh) =
n+m+1
2
(sinh)m
2m1=2
n m+1
2
m+1
2Z
0(sin )2m
(cosh+ cos sinh)n+m+(1=2)d ;
14.19.5Qm
n 1
2(cosh) =
n+1
2
n+m+1
2
n m+1
2Z1
0cosh(mt)
(cosh+ coshtsinh)n+(1=2)dt, m<n +1
2.
372 Legendre and Related Functions
14.19(iv) Sums
With>0,
14.19.6 Q
1
2(cosh) + 21X
n=1
+n+1
2
+1
2Q
n 1
2(cosh) cos(n ) = 1
21=2(sinh)
(cosh cos )+(1=2),<> 1
2.
14.19(v) Whipple's Formula for Toroidal Functions
With>0,
14.19.7 Pm
n 1
2(cosh) =
n+m+1
2
n m+1
22
sinh1=2
Qn
m 1
2(coth);
14.19.8 Qm
n 1
2(cosh) =
m n+1
2
m+n+1
2
2 sinh1=2
Pn
m 1
2(coth):
14.20 Conical (or Mehler) Functions
14.20(i) De nitions and Wronskians
Throughoutx14.20 we assume that = 1
2+i ,with0and 0. (14.2.2) takes the form
14.20.1
1 x2d2w
dx2 2xdw
dx
2+1
4+2
1 x2
w= 0:
Solutions are known as conical orMehler functions . For 1< x < 1 and >0, a numerically satisfactory pair of
real conical functions is P
1
2+i (x) and P
1
2+i ( x).
Another real-valued solution bQ
1
2+i (x) of (14.20.1) was introduced in Dunster (1991). This is de ned by
14.20.2 bQ
1
2+i (x) =<
eiQ
1
2+i (x)
1
2sin()P
1
2+i (x):
Equivalently,
14.20.3bQ
1
2+i (x) =e sin() sinh( )
2(cosh2( ) sin2())P
1
2+i (x) +(e cos2() + sinh( ))
2(cosh2( ) sin2())P
1
2+i ( x):
bQ
1
2+i (x) exists except when =1
2;3
2;::: and = 0; comparex14.3(i). It is an important companion solution to
P
1
2+i (x) when is large; compare xx14.20(vii), 14.20(viii), and 10.25(iii).
14.20.4 Wn
P
1
2+i (x);P
1
2+i ( x)o
=2
j
+1
2+i
j2(1 x2):
14.20.5 Wn
P
1
2+i (x);bQ
1
2+i (x)o
=(e cos2() + sinh( ))
j
+1
2+i
j2(cosh2( ) sin2())(1 x2);
provided that bQ
1
2+i (x) exists.
Lastly, for the range 1 <x<1,P
1
2+i (x) is a real-valued solution of (14.20.1); in terms of Q
1
2i (x) (which
are complex-valued in general):
14.20.6 P
1
2+i (x) =ie i
sinh( )
+1
2+i 2
Q
1
2+i (x) Q
1
2 i (x)
, 6= 0.
14.20 Conical (or Mehler) Functions 373
14.20(ii) Graphics
Figure 14.20.1 :P0
1
2+i (x); = 0;1;2;4;8.
Figure 14.20.2 :bQ0
1
2+i (x); = 0;1
2;1;2;4.
Figure 14.20.3 :P 1=2
1
2+i (x); = 0;1;2;4;8.
Figure 14.20.4 :bQ 1=2
1
2+i (x), =1
2;1;2;4. (This function
does not exist when = 0.)
For additional graphs see http://dlmf.nist.gov/14.20.ii .
14.20(iii) Behavior as x!1
The behavior of P
1
2+i (x) asx!1 is given in
x14.8(i). For >0 andx!1 ,
14.20.7bQ
1
2+i (x)1
2 ()2
1 x=2
;
14.20.8
bQ
1
2+i (x) ()(e cos2() + sinh( ))
2(cosh2( ) sin2())
+1
2+i 2
2
1 x=2
:
14.20(iv) Integral Representation
When 0<< ,
14.20.9 P 1
2+i (cos) =2
Z
0cosh( )p
2(cos cos)d :14.20(v) Trigonometric Expansion
14.20.10
P 1
2+i (cos) = 1 +4 2+ 12
22sin2 1
2
+
4 2+ 12
4 2+ 32
2242sin4 1
2
+, 0 .
From (14.20.9) or (14.20.10) it is evident that
P 1
2+i (cos) is positive for real .
14.20(vi) Generalized Mehler{Fock
Transformation
14.20.11
f( ) =
sinh( ) 1
2 +i
1
2 i Z1
1P
1
2+i (x)g(x)dx;
where
14.20.12 g(x) =Z1
0P
1
2+i (x)f( )d :
374 Legendre and Related Functions
Special cases:
14.20.13P 1
2+i (x) =cosh( )
Z1
1P 1
2+i (t)
x+tdt;
14.20.14
Z1
0 tanh( )
cosh( )P 1
2+i (x)P 1
2+i (y)d =1
y+x:
14.20(vii) Asymptotic Approximations: Large
, Fixed
For !1 and xed,
14.20.15P
1
2+i (cos) =1
sin1=2
I( )
(1 +O(1/ ));
14.20.16bQ
1
2+i (cos) =1
sin1=2
K( )
(1 +O(1/ ));
uniformly for 2(0; ], whereIandKare the
modi ed Bessel functions ( x10.25(ii)) and is an arbi-
trary constant such that 0 < < . For asymptotic
expansions and explicit error bounds, see Olver (1997b,
pp. 473{474). See also Zurina and Karmazina (1966).14.20(viii) Asymptotic Approximations: Large
,0A
In this subsection and x14.20(ix),Aanddenote arbi-
trary constants such that A>0 and 0<< 2.
As !1 ,
14.20.17
P
1
2+i (x) =(; ) 2+
1 + 2 x21=4
I
1=2
(1 +O(1/ ));
14.20.18
bQ
1
2+i (x) =(; ) 2+
1 + 2 x21=4
K
1=2
(1 +O(1/ ));
uniformly for x2[ 1 +;1) and2[0;A ]. Here
14.20.19 == ;
14.20.20 (; ) =exp( arctan )
(2+ 2)=2:
The variable is de ned implicitly by
14.20.21
2+1=2+1
2 ln ln
2+1=2+
= arccos
x
(1 + 2)1=2!
+
2ln
1 + 2+
2 1
x2 2 x
1 + 2 x21=2
(1 + 2) (1 x2)!
;
where the inverse trigonometric functions take their principal values. The interval 1<x< 1 is mapped one-to-one
to the interval 0 <<1, with the points x= 1 andx= 1 corresponding to =1and= 0, respectively.
For extensions to complex arguments (including the range 1 <x<1), asymptotic expansions, and explicit error
bounds, see Dunster (1991).
14.20(ix) Asymptotic Approximations: Large ,0 A
As!1 ,
14.20.22 P
1
2+i (x) = exp( arctan )
(+ 1) (1 + 2)=2e
(1 + 2 x2 2)1=4
1 +O1
;
uniformly for x2( 1;1) and 2[0;A]. Here
14.20.23 = =;
and the variable is de ned by
14.20.24=1
2ln
1 2
x2+ 1 + 2+ 2x
1 + 2 2x21=2
1 x2!
+ arctan
xp
1 + 2 2x2!
1
2ln
1 + 2
;
with the inverse tangent taking its principal value. The
interval 1<x< 1 is mapped one-to-one to the inter-
val 1<<1, with the points x= 1,x= 0, and
x= 1 corresponding to = 1,= 0, and=1,respectively.
With the same conditions, the corresponding ap-
proximation for P
1
2+i ( x) is obtainable by replacing
e byeon the right-hand side of (14.20.22). Ap-
Complex Arguments 375
proximations for P
1
2+i (x) andbQ
1
2+i (x) can then be
achieved via (14.9.7) and (14.20.3).
For extensions to complex arguments (including the
range 1<x<1), asymptotic expansions, and explicit
error bounds, see Dunster (1991).
14.20(x) Zeros and Integrals
For zeros of P 1
2+i (x) see Hobson (1931, x237).
For integrals with respect to involving P 1
2+i (x),
see Prudnikov et al. (1990, pp. 218{228).
Complex Arguments
14.21 De nitions and Basic Properties
14.21(i) Associated Legendre Equation
14.21.1
1 z2d2w
dz2 2zdw
dz+
(+ 1) 2
1 z2
w= 0:
Standard solutions: the associated Legendre functions
P
(z),P
(z),Q
(z), andQ
1(z).P
(z) and
Q
(z) exist for all values of ,, andz, except pos-
siblyz=1 and1, which are branch points (or
poles) of the functions, in general. When zis com-
plexP
(z),Q
(z), andQ
(z) are de ned by (14.3.6){
(14.3.10) with xreplaced by z: the principal branches
are obtained by taking the principal values of all the
multivalued functions appearing in these representa-
tions when z2(1;1), and by continuity elsewherein thez-plane with a cut along the interval ( 1;1];
comparex4.2(i). The principal branches of P
(z) and
Q
(z) are real when ,2Randz2(1;1).
14.21(ii) Numerically Satisfactory Solutions
When< 1
2and<0, a numerically satis-
factory pair of solutions of (14.21.1) in the half-plane
jphzj1
2is given by P
(z) andQ
(z).
14.21(iii) Properties
Many of the properties stated in preceding sections
extend immediately from the x-interval (1 ;1) to the
cutz-plane Cn( 1;1]. This includes, for example,
the Wronskian relations (14.2.7){(14.2.11); hypergeo-
metric representations (14.3.6){(14.3.10) and (14.3.15){
(14.3.20); results for integer orders (14.6.3){(14.6.5),
(14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11){
(14.7.16); behavior at singularities (14.8.7){(14.8.16);
connection formulas (14.9.11){(14.9.16); recurrence re-
lations (14.10.3){(14.10.7). The generating function
expansions (14.7.19) (with Preplaced by P) and
(14.7.22) apply when jhj<min z
z2 11=2 ;
(14.7.21) (with Preplaced by P) applies whenjhj>
max z
z2 11=2 .
14.22 Graphics
In the graphics shown in this section, height corresponds
to the absolute value of the function and color to the
phase. See also p. xiv.
Figure 14.22.1 :P0
1/2(x+iy); 5x5; 5y5.
There is a cut along the real axis from 1 to 1.
Figure 14.22.2 :P 1/2
1/2(x+iy); 5x5; 5y5.
There is a cut along the real axis from 1 to 1.
376 Legendre and Related Functions
Figure 14.22.3 :P 1
1/2(x+iy); 5x5; 5y5.
There is a cut along the real axis from 1 to 1.
Figure 14.22.4 :Q0
0(x+iy); 5x5; 5y5.
There is a cut along the real axis from 1 to 1.
14.23 Values on the Cut
When 1<x< 1,
14.23.1 P
(xi0) =ei=2P
(x);
14.23.2 Q
(xi0) =ei=2
(++ 1)
Q
(x)1
2iP
(x)
:
In terms of the hypergeometric function F(x14.3(i))
14.23.3
Q
(xi0) =ei=23=2
1 x2=2
2+1
xF 1
2 1
2+1
2;1
2+1
2+ 1;3
2;x2
1
2 1
2+1
2
1
2+1
2+1
2iF 1
2 1
2;1
2+1
2+1
2;1
2;x2
1
2 1
2+ 1
1
2+1
2+ 1!
:
Conversely,
14.23.4 P
(x) =ei=2P
(xi0);
14.23.5Q
(x) =1
2 (++ 1)
e i=2Q
(x+i0)
+ei=2Q
(x i0)
;
or equivalently,
14.23.6Q
(x) =ei=2 (++ 1)Q
(xi0)
1
2iei=2P
(xi0):
If cuts are introduced along the intervals ( 1; 1]
and [1;1), then (14.23.4) and (14.23.6) could be used
to extend the de nitions of P
(x) and Q
(x) to complex
x.
The conical function de ned by (14.20.2) can be rep-
resented similarly by
14.23.7bQ
1
2+i (x) =1
2e3i=2Q
1
2+i (x i0)
+1
2e 3i=2Q
1
2 i (x+i0):14.24 Analytic Continuation
Letsbe an arbitrary integer, and P
zesi
and
Q
zesi
denote the branches obtained from the prin-
cipal branches by making1
2scircuits, in the positive
sense, of the ellipse having 1 as foci and passing
throughz. Then
14.24.1
P
zesi
=esiP
(z)
+2isin
+1
2
s
e si=2
cos() ( )Q
(z);
14.24.2Q
zesi
= ( 1)se siQ
(z);
the limiting value being taken in (14.24.1) when 2 is
an odd integer.
Next, letP
;s(z) andQ
;s(z) denote the branches
obtained from the principal branches by encircling the
branch point 1 (but not the branch point 1)stimes
in the positive sense. Then
14.24.3P
;s(z) =esiP
(z);
14.25 Integral Representations 377
14.24.4Q
;s(z) =e siQ
(z)
isin(s)
sin() ( + 1)P
(z);
the limiting value being taken in (14.24.4) when 2Z.
For xedz, other than1 or1, each branch of
P
(z) andQ
(z) is an entire function of each param-
eterand.
The behavior of P
(z) andQ
(z) asz! 1 from
the left on the upper or lower side of the cut from 1
to 1 can be deduced from (14.8.7){(14.8.11), combined
with (14.24.1) and (14.24.2) with s=1.
14.25 Integral Representations
The principal values of P
(z) andQ
(z) (x14.21(i))
are given by
14.25.1
P
(z)
=
z2 1=2
2 ( ) (+ 1)Z1
0(sinht)2+1
(z+ cosht)++1dt,
<>< > 1,
14.25.2
Q
(z) =1=2
z2 1=2
2
+1
2
( + 1)
Z1
0(sinht)2
z+ (z2 1)1=2cosht++1dt,
<(+ 1)><> 1
2;
where the multivalued functions have their principal val-
ues when 1 <z<1and are continuous in Cn( 1;1].
For corresponding contour integrals, with less re-
strictions on and, see Olver (1997b, pp. 174{179),
and for further integral representations see Magnus
et al. (1966,x4.6.1).
14.26 Uniform Asymptotic Expansions
The uniform asymptotic approximations given in x14.15
forP
(x) andQ
(x) for 1< x <1are ex-
tended to domains in the complex plane in the following
references:xx14.15(i) and 14.15(ii), Dunster (2003b);
x14.15(iii), Olver (1997b, Chapter 12); x14.15(iv), Boyd
and Dunster (1986). For an extension of x14.15(iv) to
complex argument and imaginary parameters, see Dun-
ster (1990b).
See also Frenzen (1990), Gil et al. (2000), Shivaku-
mar and Wong (1988), Ursell (1984), and Wong (1989)
for uniform asymptotic approximations obtained from
integral representations.14.27 Zeros
P
(xi0) (either side of the cut) has exactly one zero
in the interval ( 1; 1) if either of the following sets
of conditions holds:
(a) < 0, =2Z,2Z, and sin(( )) and
sin() have opposite signs.
(b);2Z,+ <0, andis odd.
For all other values of the parameters P
(xi0) has
no zeros in the interval ( 1; 1).
For complex zeros of P
(z) see Hobson (1931, xx233,
234, and 238).
14.28 Sums
14.28(i) Addition Theorem
When<z1>0,<z2>0,jph(z1 1)j< , and
jph(z2 1)j<,
14.28.1P
z1z2
z2
1 11=2
z2
2 11=2cos
=P(z1)P(z2) + 21X
m=1( 1)m ( m+ 1)
(+m+ 1)
Pm
(z1)Pm
(z2) cos(m );
where the branches of the square roots have their prin-
cipal values when z1;z22(1;1) and are continuous
whenz1;z22Cn(0;1]. For this and similar results see
Erd elyi et al. (1953a,x3.11).
14.28(ii) Heine's Formula
14.28.2
1X
n=0(2n+ 1)Qn(z1)Pn(z2) =1
z1 z2,z12E1,z22E2,
whereE1andE2are ellipses with foci at 1,E2being
properly interior to E1. The series converges uniformly
forz1outside or onE1, andz2within or onE2.
14.28(iii) Other Sums
Seex14.18(iv).
14.29 Generalizations
Solutions of the equation
14.29.1
1 z2d2w
dz2 2zdw
dz
+
(+ 1) 2
1
2(1 z) 2
2
2(1 +z)
w
= 0
378 Legendre and Related Functions
are called Generalized Associated Legendre Functions .
As in the case of (14.21.1), the solutions are hyper-
geometric functions, and (14.29.1) reduces to (14.21.1)
when1=2=. For properties see Virchenko and
Fedotova (2001) and Braaksma and Meulenbeld (1967).
For inhomogeneous versions of the associated Leg-
endre equation, and properties of their solutions, see
Babister (1967, pp. 252{264).
Applications
14.30 Spherical and Spheroidal Harmonics
14.30(i) De nitions
Withlandmintegers such that 0 ml, andand
angles such that 0 , 0 2,
14.30.1
Yl;m(; ) =(l m)!(2l+ 1)
4(l+m)!1=2
eim Pm
l(cos);14.30.2
Ym
l(; ) = cos(m )Pm
l(cos) or sin(m )Pm
l(cos):
Yl;m(; ) are known as spherical harmonics .Ym
l(; )
are known as surface harmonics of the rst kind :
tesseral form < l and sectorial form=l. Sometimes
Yl;m(; ) is denoted by i lDlm(; ); also the de nition
ofYl;m(; ) can di er from (14.30.1), for example, by
inclusion of a factor ( 1)m.
Pm
n(x) andQm
n(x) (x > 1) are often referred to as
theprolate spheroidal harmonics of the rst and second
kinds , respectively. Pm
n(ix) andQm
n(ix) (x > 0) are
known as oblate spheroidal harmonics of the rst and
second kinds , respectively. Segura and Gil (1999) intro-
duced the scaled oblate spheroidal harmonics Rm
n(x) =
e in=2Pm
n(ix) andTm
n(x) =iein=2Qm
n(ix) which are
real whenx>0 andn= 0;1;2;:::.
14.30(ii) Basic Properties
Most mathematical properties of Yl;m(; ) can be de-
rived directly from (14.30.1) and the properties of the
Ferrers function of the rst kind given earlier in this
chapter.
Explicit Representation
14.30.3 Yl;m(; ) =( 1)l+m
2ll!(l m)!(2l+ 1)
4(l+m)!1=2
eim (sin)md
d(cos)l+m
(sin)2l:
Special Values
14.30.4 Yl;m(0; ) =8
><
>:2l+ 1
41=2
; m = 0;
0; m = 1;2;3;:::;
14.30.5 Yl;m 1
2;
=8
>><
>>:( 1)(l+m)=2eim
2l 1
2l 1
2m
! 1
2l+1
2m
!(l m)!(l+m)!(2l+ 1)
41=2
;1
2l+1
2m2Z;
0;1
2l+1
2m =2Z:
Symmetry
14.30.6 Yl; m(; ) = ( 1)mY
l;m(; ):
Parity Operation
14.30.7 Yl;m( ; +) = ( 1)lYl;m(; ):
Orthogonality
14.30.8Z2
0Z
0Y
l1;m1(; )Yl2;m2(; ) sindd =l1;l2m1;m2;
here and elsewhere in this section the asterisk (*) denotes complex conjugate.
See also (34.3.22), and for further related integrals see Askey et al. (1986).
14.31 Other Applications 379
14.30(iii) Sums
Distributional Completeness
For a series representation of the product of two Dirac
deltas in terms of products of spherical harmonics see
x1.17(iii).
Addition Theorem
14.30.9Pl(cos1cos2+ sin1sin2cos( 1 2))
=4
2l+ 1lX
m= lY
l;m(1; 1)Yl;m(2; 2):
See also (18.18.9) and (34.3.19).
14.30(iv) Applications
In general, spherical harmonics are de ned as the class
of homogeneous harmonic polynomials. See Andrews
et al. (1999, Chapter 9). The special class of spheri-
cal harmonics Yl;m(; ), de ned by (14.30.1), appear in
many physical applications. As an example, Laplace's
equationr2W= 0 in spherical coordinates ( x1.5(ii)):
14.30.101
2@
@
2@W
@
+1
2sin@
@
sin@W
@
+1
2sin2@2W
@ 2= 0;
has solutions W(;; ) =lYl;m(; ), which are ev-
erywhere one-valued and continuous.
In the quantization of angular momentum the spher-
ical harmonics Yl;m(; ) are normalized solutions of the
eigenvalue equation
14.30.11 L2Yl;m= h2l(l+ 1)Yl;m;
where his the reduced Planck's constant, and L2is the
angular momentum operator in spherical coordinates:
14.30.12 L2= h21
sin@
@
sin@
@
+1
sin2@2
@ 2
;
see Edmonds (1974, x2.5).
For applications in geophysics see Stacey (1977,
xx4.2, 6.3, and 8.1).
14.31 Other Applications
14.31(i) Toroidal Functions
Applications of toroidal functions include expansion of
vacuum magnetic elds in stellarators and tokamaks
(van Milligen and L opez Fraguas (1994)), analytic solu-
tions of Poisson's equation in channel-like geometries
(Hoyles et al. (1998)), and Dirichlet problems with
toroidal symmetry (Gil et al. (2000)).14.31(ii) Conical Functions
The conical functions Pm
1
2+i (x) appear in boundary-
value problems for the Laplace equation in toroidal co-
ordinates (x14.19(i)) for regions bounded by cones, by
two intersecting spheres, or by one or two confocal hy-
perboloids of revolution (K olbig (1981)). These func-
tions are also used in the Mehler{Fock integral trans-
form (x14.20(vi)) for problems in potential and heat the-
ory, and in elementary particle physics (Sneddon (1972,
Chapter 7) and Braaksma and Meulenbeld (1967)). The
conical functions and Mehler{Fock transform general-
ize to Jacobi functions and the Jacobi transform; see
Koornwinder (1984a) and references therein.
14.31(iii) Miscellaneous
Many additional physical applications of Legendre poly-
nomials and associated Legendre functions include so-
lution of the Helmholtz equation, as well as the
Laplace equation, in spherical coordinates (Temme
(1996a)), quantum mechanics (Edmonds (1974)), and
high-frequency scattering by a sphere (Nussenzveig
(1965)). See also x18.39.
Legendre functions P(x) of complex degree ap-
pear in the application of complex angular momentum
techniques to atomic and molecular scattering (Connor
and Mackay (1979)).
Computation
14.32 Methods of Computation
Essentially the same comments that are made in x15.19
concerning the computation of hypergeometric func-
tions apply to the functions described in the present
chapter. In particular, for small or moderate values
of the parameters andthe power-series expansions
of the various hypergeometric function representations
given inxx14.3(i){14.3(iii), 14.19(ii), and 14.20(i) can
be selected in such a way that convergence is stable,
and reasonably rapid, especially when the argument of
the functions is real. In other cases recurrence relations
(x14.10) provide a powerful method when applied in a
stable direction (x3.6); see Olver and Smith (1983) and
Gautschi (1967).
Other methods include:
Application of the uniform asymptotic expansions
for large values of the parameters given in xx14.15
and 14.20(vii){14.20(ix).
Numerical integration ( x3.7) of the de ning di er-
ential equations (14.2.2), (14.20.1), and (14.21.1).
380 Legendre and Related Functions
Quadrature (x3.5) of the integral representations
given inxx14.12, 14.19(iii), 14.20(iv), and 14.25;
see Segura and Gil (1999) and Gil et al. (2000).
Evaluation (x3.10) of the continued fractions given
inx14.14. See Gil and Segura (2000).
14.33 Tables
Abramowitz and Stegun (1964, Chapter 8) tabu-
lates Pn(x) forn= 0(1)3;9;10,x= 0(:01)1, 5{
8D; P0
n(x) forn= 1(1)4;9;10,x= 0(:01)1, 5{7D;
Qn(x) and Q0
n(x) forn= 0(1)3;9;10,x= 0(:01)1,
6{8D;Pn(x) andP0
n(x) forn= 0(1)5;9;10,x=
1(:2)10, 6S;Qn(x) andQ0
n(x) forn= 0(1)3;9;10,
x= 1(:2)10, 6S. (Here primes denote derivatives
with respect to x.)
Zhang and Jin (1996, Chapter 4) tabulates Pn(x)
forn= 2(1)5;10,x= 0(:1)1, 7D; Pn(cos)
forn= 1(1)4;10,= 0(5)90, 8D; Qn(x) for
n= 0(1)2;10,x= 0(:1)0:9, 8S; Qn(cos) for
n= 0(1)3;10,= 0(5)90, 8D; Pm
n(x) for
m= 1(1)4,n m= 0(1)2,n= 10,x= 0;0:5, 8S;
Qm
n(x) form= 1(1)4,n= 0(1)2;10, 8S; Pm
(cos)
form= 0(1)3,= 0(:25)5,= 0(15)90, 5D;
Pn(x) forn= 2(1)5;10,x= 1(1)10, 7S; Qn(x)
forn= 0(1)2;10,x= 2(1)10, 8S. Corresponding
values of the derivative of each function are also
included, as are 6D values of the rst 5 -zeros
ofPm
(cos) and of its derivative for m= 0(1)4,
= 10;30;150.
Belousov (1962) tabulates Pm
n(cos) (normalized)
form= 0(1)36,n m= 0(1)56,= 0(2:5)90,
6D.
Zurina and Karmazina (1964, 1965) tabulate the
conical functions P 1
2+i (x) for = 0(:01)50,
x= 0:9(:1)0:9, 7S;P 1
2+i (x) for = 0(:01)50,
x= 1:1(:1)2(:2)5(:5)10(10)60, 7D. Auxiliary ta-
bles are included to facilitate computation for
larger values of when 1<x< 1.
Zurina and Karmazina (1963) tabulates the con-
ical functions P1
1
2+i (x) for = 0(:01)25,x=
0:9(:1)0:9, 7S;P1
1
2+i (x) for = 0(:01)25,x=
1:1(:1)2(:2)5(:5)10(10)60, 7S. Auxiliary tables are
included to assist computation for larger values of
when 1<x< 1.
For tables prior to 1961 see Fletcher et al. (1962)
and Lebedev and Fedorova (1960).
14.34 Software
Seehttp://dlmf.nist.gov/14.34 .References
General References
The main reference used in writing this chapter is
Olver (1997b). For additional bibliographic reading see
Erd elyi et al. (1953a, Chapter III), Hobson (1931), Jef-
freys and Je reys (1956), MacRobert (1967), Magnus
et al. (1966), Robin (1957, 1958, 1959), Snow (1952),
Szeg o (1967), Temme (1996a), and Wong (1989).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x14.2 Forxx14.2(i), 14.2(ii) see Olver (1997b, pp. 169).
Forx14.2(iii) see Olver (1997b, p. 172) for the pair
P
(x) andQ
(x). The result for P
(x) and
P
( x) follows from the fact that when <0,
P
(x) is recessive as x!1 and P
( x) is
recessive as x! 1+; seex14.8(i). Forx14.2(iv)
see Olver (1997b, p. 172) for (14.2.4), (14.2.5); the
other results may be derived in a similar manner,
or by application of the connection formulas in
x14.9.
x14.3 For (14.3.1){(14.3.4) see Olver (1997b, pp. 159,
186). The version of (14.3.4) given in Hob-
son (1931, p. 386) has an error. For (14.3.5)
use (14.3.1) and (14.9.3). For (14.3.8) see Olver
(1997b, p. 159, Eq. (9.05)). For (14.3.11) and
(14.3.12) see Olver (1997b, p. 187). For (14.3.16){
(14.3.20) see Erd elyi et al. (1953a, pp. 123{139).
For (14.3.21) combine (14.3.22) and (14.23.4). For
(14.3.22) see Erd elyi et al. (1953a, p. 175). For
(14.3.23) see (14.3.6) and Olver (1997b, p. 167).
x14.4 These graphics were produced at NIST.
x14.5 (14.5.1){(14.5.4) may be derived from (14.3.11){
(14.3.14) and also (14.10.4) with Preplaced by Q.
For (14.5.5){(14.5.10) use x14.7(i). For (14.5.11){
(14.5.19) see Erd elyi et al. (1953a, p. 150).
(14.5.20){(14.5.27) are given in Magnus et al.
(1966, p. 173): to verify these compare the hyper-
geometric representations of the Legendre func-
tions and elliptic integrals ( xx14.3 and 19.5).
x14.6 See Erd elyi et al. (1953a, pp. 148{149). For
(14.6.5) combine (14.3.10) and (14.6.4).
References 381
x14.7 Olver (1997b, pp. 174, 181{182, 188). For
(14.7.17), (14.7.18) use (14.9.8), (14.9.10). For
(14.7.19){(14.7.22) see Erd elyi et al. (1953a,
p. 154) and Olver (1997b, pp. 51, 85).
x14.8 Olver (1997b, pp. 171, 173, 186), Erd elyi et al.
(1953a, p. 163). (14.8.5) may be derived from
(14.8.1), (14.9.2). (14.8.10) may be derived from
(14.8.7), (14.9.12). (14.8.16) may be derived from
(14.8.15), (14.9.12).
x14.9 Olver (1997b, pp. 171, 174, 186, 188). For
(14.9.3), (14.9.4) use (14.9.1), (14.9.2). For
(14.9.17) use (14.9.14), (14.9.16).
x14.10 Erd elyi et al. (1953a, pp. 160{161).
x14.11 (14.11.1) may be derived from (14.3.1).
(14.11.2) may be derived from (14.9.10) and
(14.11.1). (14.11.4) may be derived from (14.3.1)
and the hypergeometric expansion for Q(x)
(Hobson (1931,x132)). (14.11.5) may be derived
from (14.9.8), (14.9.10), and (14.11.4).
x14.12 Erd elyi et al. (1953a, pp. 155{159), Olver
(1997b, pp. 181{183, 185).
x14.13 Erd elyi et al. (1953a, pp. 146, 151).
x14.14 (14.14.1) follows from (14.10.6). (14.14.3)
follows from (14.10.3), with P
(x) replaced by
Q
(x). For further details see Gil et al. (2000).
x14.15 Dunster (2003b), Olver (1997b, pp. 463{469),
Boyd and Dunster (1986). (14.15.1) may be de-
rived from (14.3.1) and x15.12(ii). For (14.15.24){
(14.15.31) see Olver (1975b).
x14.16 Hobson (1931, pp. 386{389, 399{401).
x14.17 Erd elyi et al. (1953a, pp. 170{172), Olver
(1997b, pp. 188{189). (14.17.1){(14.17.4) may be
veri ed by di erentiation and using the recurrence
relations (x14.10). (14.17.7), (14.17.9) may be de-
rived from (14.9.3), (14.17.6), (14.17.8). The ver-
sion of (14.17.16) given in Erd elyi et al. (1953a,
p. 171, Eq. (18)) is incorrect. For (14.17.17) see
Din (1981).
x14.18 Erd elyi et al. (1953a, pp. 162, 167{169), Olver
(1997b, p. 183). Errors in Erd elyi et al. (1953a,pp. 168{169) have been corrected. (14.18.2) may
be derived from (14.7.16), (14.9.3), (14.18.1).
(14.18.8) may be derived from (14.7.17), (14.18.9).
x14.19 Erd elyi et al. (1953a, pp. 156{157, 166,
173). For (14.19.7), (14.19.8) combine (14.9.16),
(14.9.17) with (14.9.11){(14.9.13).
x14.20 (14.20.3) follows from (14.9.10), (14.20.2).
(14.20.4), (14.20.5) follow from (14.2.3), (14.20.3).
(14.20.6) follows from (14.3.10), (14.9.12).
(14.20.7), (14.20.8) follow from x14.8(i) and
(14.20.3). (14.20.9) follows from (14.12.1). For
(14.20.10) see Erd elyi et al. (1953a, p. 174).
For (14.20.11){(14.20.14) see Braaksma and
Meulenbeld (1967). For (14.20.15) see Olver
(1997b, p. 473). (14.20.16) may be derived from
(14.20.18). Forxx14.20(viii), 14.20(ix) see Dun-
ster (1991, Eqs. (5.11), (5.14) have been cor-
rected). The graphs were produced at NIST.
x14.21 Olver (1997b, pp. 169{185), Erd elyi et al.
(1953a, Chapter 3).
x14.22 These graphics were produced at NIST.
x14.23 For (14.23.1), (14.23.5) see Olver (1997b,
p. 185). For (14.23.7) see Dunster (1991).
(14.23.2) may be derived from (14.23.4), (14.24.2).
(14.23.3) may be derived from (14.3.11), (14.3.12),
(14.9.14). (14.23.4) may be derived from
(14.23.1). (14.23.6) may be derived from
(14.23.1), (14.23.2).
x14.24 Olver (1997b, p, 179).
x14.25 For (14.25.1) see Olver (1997b, p. 179). For
(14.25.2) see Erd elyi et al. (1953a, p. 155).
x14.27 Hobson (1931, pp. 391{399).
x14.28 Erd elyi et al. (1953a, p. 168) or Olver (1997b,
p. 473).
x14.30 Edmonds (1974, pp. 20{24, 63). (Note that
Edmonds'Pm
l(x) di ers by a factor ( 1)mfrom
Pm
l(x).) (14.30.4) may be derived from (14.8.1),
(14.8.2), (14.30.1). (14.30.5) may be derived from
(14.5.1), (14.30.1). (14.30.7) may be derived from
(14.7.17), (14.30.1). (14.30.3) also follows from
(14.30.1), (14.7.10).
Chapter 15
Hypergeometric Function
A. B. Olde Daalhuis1
Notation 384
15.1 Special Notation . . . . . . . . . . . . . 384
Properties 384
15.2 De nitions and Analytical Properties . . . 384
15.3 Graphics . . . . . . . . . . . . . . . . . . 385
15.4 Special Cases . . . . . . . . . . . . . . . 386
15.5 Derivatives and Contiguous Functions . . 387
15.6 Integral Representations . . . . . . . . . 388
15.7 Continued Fractions . . . . . . . . . . . . 389
15.8 Transformations of Variable . . . . . . . . 390
15.9 Relations to Other Functions . . . . . . . 393
15.10 Hypergeometric Di erential Equation . . 394
15.11 Riemann's Di erential Equation . . . . . 39615.12 Asymptotic Approximations . . . . . . . . 396
15.13 Zeros . . . . . . . . . . . . . . . . . . . 398
15.14 Integrals . . . . . . . . . . . . . . . . . . 398
15.15 Sums . . . . . . . . . . . . . . . . . . . 399
15.16 Products . . . . . . . . . . . . . . . . . . 399
Applications 399
15.17 Mathematical Applications . . . . . . . . 399
15.18 Physical Applications . . . . . . . . . . . 400
Computation 400
15.19 Methods of Computation . . . . . . . . . 400
15.20 Software . . . . . . . . . . . . . . . . . . 401
References 401
1School of Mathematics, Edinburgh University, Edinburgh, United Kingdom.
Acknowledgments : This chapter is based in part on Chapter 15 of Abramowitz and Stegun (1964) by Fritz Oberhettinger. The
author thanks Richard Askey and Simon Ruijsenaars for many helpful recommendations.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
383
384 Hypergeometric Function
Notation
15.1 Special Notation
(For other notation see pp. xiv and 873.)
x real variable.
z=x+iycomplex variable.
a;b;c real or complex parameters.
k;`;m;n integers.
s nonnegative integer.
arbitrary small positive constant.
(z) gamma function ( x5.2(i)).
(z) 0(z)/ (z).
Unless indicated otherwise primes denote derivatives
with respect to the variable.
We use the following notations for the hypergeomet-
ric function:
15.1.1 2F1(a;b;c;z) =F(a;b;c;z) =Fa;b
c;z
;
and also
15.1.2
F(a;b;c;z)
(c)=F(a;b;c;z) =Fa;b
c;z
=2F1(a;b;c;z);
(Olver (1997b, Chapter 5)).
Properties
15.2 De nitions and Analytical Properties
15.2(i) Gauss Series
The hypergeometric function F(a;b;c;z) is de ned by
theGauss series
15.2.1
F(a;b;c;z) =1X
s=0(a)s(b)s
(c)ss!zs
= 1 +ab
cz+a(a+ 1)b(b+ 1)
c(c+ 1)2!z2+
= (c)
(a) (b)1X
s=0 (a+s) (b+s)
(c+s)s!zs;
on the diskjzj<1, and by analytic continuation else-
where. In general, F(a;b;c;z) does not exist whenc= 0; 1; 2;:::. The branch obtained by introduc-
ing a cut from 1 to + 1on the real z-axis, that is, the
branch in the sector jph(1 z)j, is the principal
branch (orprincipal value ) ofF(a;b;c;z).
For all values of c
15.2.2 F(a;b;c;z) =1X
s=0(a)s(b)s
(c+s)s!zs,jzj<1,
again with analytic continuation for other values of z,
and with the principal branch de ned in a similar way.
Except where indicated otherwise principal branches
ofF(a;b;c;z) and F(a;b;c;z) are assumed throughout
this Handbook.
The di erence between the principal branches on the
two sides of the branch cut ( x4.2(i)) is given by
15.2.3
Fa;b
c;x+i0
Fa;b
c;x i0
=2i
(a) (b)(x 1)c a bFc a;c b
c a b+ 1; 1 x
,
x>1.
On the circle of convergence, jzj= 1, the Gauss se-
ries:
(a) Converges absolutely when <(c a b)>0.
(b) Converges conditionally when 1<
<(c a b)0 andz= 1 is excluded.
(c) Diverges when <(c a b) 1.
For the case z= 1 see alsox15.4(ii).
15.2(ii) Analytic Properties
The principal branch of F(a;b;c;z) is an entire func-
tion ofa,b, andc. The same is true of other branches,
provided that z= 0, 1, and1are excluded. As a multi-
valued function of z,F(a;b;c;z) is analytic everywhere
except for possible branch points at z= 0, 1, and1.
The same properties hold for F(a;b;c;z), except that
as a function of c,F(a;b;c;z) in general has poles at
c= 0; 1; 2;:::.
Because of the analytic properties with respect to a,
b, andc, it is usually legitimate to take limits in for-
mulas involving functions that are unde ned for certain
values of the parameters.
15.3 Graphics 385
For example, when a= m,m= 0;1;2;:::, and
c6= 0; 1; 2;:::,F(a;b;c;z) is a polynomial:
15.2.4F( m;b;c;z)
=mX
n=0( m)n(b)n
(c)nn!zn=mX
n=0( 1)nm
n(b)n
(c)nzn:
This formula is also valid when c= m `,`=
0;1;2;:::, provided that we use the interpretation
15.2.5F m;b
m `;z
= lim
c! m `
lim
a! mFa;b
c;z
;and not
15.2.6F m;b
m `;z
= lim
a! mFa;b
a `;z
;
which is sometimes used in the literature. (Both in-
terpretations give solutions of the hypergeometric dif-
ferential equation (15.10.1), as does F(a;b;c;z), which
is analytic at c= 0; 1; 2;:::.) For illustration see
Figures 15.3.6 and 15.3.7.
In the case c= mthe right-hand side of (15.2.4)
becomes the rst m+ 1 terms of the Maclaurin series
for (1 z) b.
15.3 Graphics
15.3(i) Graphs
Figure 15.3.1 :F 4
3;9
16;14
5;x
; 100x1.
Figure 15.3.2 :F(5; 10; 1;x); 0:023x1.
Figure 15.3.3 :F(1; 10; 10;x); 3x1.
Figure 15.3.4 :F(5;10; 1;x); 1x0:022.
386 Hypergeometric Function
15.3(ii) Surfaces
In Figures 15.3.5 and 15.3.6, height corresponds to the absolute value of the function and color to the phase. See
also p. xiv.
Figure 15.3.5 :F 4
3;9
16;14
5;x+iy
;0x2; 0:5
y0:5. (There is a cut along the real axis from 1 to
1.)
Figure 15.3.6 :F
3;3
5;u+iv;1
2
; 6u2; 2
v2. (Withc=u+ivthe only poles occur at c=
0; 1; 2; comparex15.2(ii).)
Figure 15.3.7 :jF
3;3
5;u+iv;1
2
j; 6u2; 2
v2.
15.4 Special Cases
15.4(i) Elementary Functions
The following results hold for principal branches when
jzj<1, and by analytic continuation elsewhere. Excep-
tions are (15.4.8) and (15.4.10), that hold for jzj< /4,
and (15.4.12), (15.4.14), and (15.4.16), that hold for
jzj< /2.
15.4.1 F(1;1; 2;z) = z 1ln(1 z);
15.4.2F 1
2;1;3
2;z2
=1
2zln1 +z
1 z
;
15.4.3F 1
2;1;3
2; z2
=z 1arctanz;15.4.4F 1
2;1
2;3
2;z2
=z 1arcsinz;
15.4.5F 1
2;1
2;3
2; z2
=z 1ln
z+p
1 +z2
:
15.4.6 F(a;b;b;z) = (1 z) a;
comparex15.2(ii).
15.4.7F
a;1
2+a;1
2;z2
=1
2
(1 +z) 2a+ (1 z) 2a
;
15.4.8F
a;1
2+a;1
2; tan2z
= (cosz)2acos(2az):
15.4.9F
a;1
2+a;3
2;z2
=1
(2 4a)z
(1 +z)1 2a (1 z)1 2a
;
15.4.10
F
a;1
2+a;3
2; tan2z
= (cosz)2asin((1 2a)z)
(1 2a) sinz:
15.4.11F
a;a;1
2; z2
=1
2p
1 +z2+z2a
+p
1 +z2 z2a
;
15.4.12 F
a;a;1
2; sin2z
= cos(2az):
15.4.13
F
a;1 a;1
2; z2
=1
2p
1 +z2p
1 +z2+z2a 1
+p
1 +z2 z2a 1
;
15.4.14F
a;1 a;1
2; sin2z
=cos((2a 1)z)
cosz:
15.5 Derivatives and Contiguous Functions 387
15.4.15
F
a;1 a;3
2; z2
=1
(2 4a)zp
1 +z2+z1 2a
p
1 +z2 z1 2a
;
15.4.16F
a;1 a;3
2; sin2z
=sin((2a 1)z)
(2a 1) sinz:
15.4.17F
a;1
2+a; 1 + 2a;z
= 1
2+1
2p
1 z 2a;
15.4.18F
a;1
2+a; 2a;z
=1p1 z 1
2+1
2p
1 z1 2a:
15.4.19F(a+ 1;b;a;z) = (1 (1 (b/a))z) (1 z) 1 b:
For an extensive list of elementary representations
see Prudnikov et al. (1990, pp. 468{488).
15.4(ii) Argument Unity
If<(c a b)>0, then
15.4.20 F(a;b;c; 1) = (c) (c a b)
(c a) (c b):
Ifc=a+b, then
15.4.21 lim
z!1 F(a;b;a+b;z)
ln(1 z)= (a+b)
(a) (b):
If<(c a b) = 0 andc6=a+b, then
15.4.22lim
z!1 (1 z)a+b c
F(a;b;c;z)
(c) (c a b)
(c a) (c b)
= (c) (a+b c)
(a) (b):
If<(c a b)<0, then
15.4.23 lim
z!1 F(a;b;c;z)
(1 z)c a b= (c) (a+b c)
(a) (b):
Chu{Vandermonde Identity
15.4.24 F( n;b;c; 1) =(c b)n
(c)n,n= 0;1;2;:::.
Dougall's Bilateral Sum
This is a generalization of (15.4.20). If a;bare not in-
tegers and<(c+d a b)>1, then
15.4.25
1X
n= 1 (a+n) (b+n)
(c+n) (d+n)
=2
sin(a) sin(b) (c+d a b 1)
(c a) (d a) (c b) (d b):15.4(iii) Other Arguments
15.4.26F(a;b;a b+ 1; 1) = (a b+ 1) 1
2a+ 1
(a+ 1) 1
2a b+ 1:
15.4.27F(1;a;a+ 1; 1) =1
2a
1
2a+1
2
1
2a
:
15.4.28
F
a;b;1
2a+1
2b+1
2;1
2
=p 1
2a+1
2b+1
2
1
2a+1
2
1
2b+1
2:
15.4.29
F
a;b;1
2a+1
2b+ 1;1
2
=2p
a b 1
2a+1
2b+ 1
1
1
2a
1
2b+1
2
1
1
2a+1
2
1
2b!
:
15.4.30F
a;1 a;b;1
2
=21 bp (b)
1
2a+1
2b
1
2b 1
2a+1
2:
15.4.31
F
a;1
2+a;3
2 2a; 1
3
=8
9 2a 4
3
3
2 2a
3
2
4
3 2a:
15.4.32F
a;1
2+a;5
6+2
3a;1
9
=p3
4a 5
6+2
3a
1
2+1
3a
5
6+1
3a:
15.4.33F
3a;1
3+a;2
3+ 2a;ei/3
=peia/216
27(3a+1)=6 5
6+a
2
3+a
2
3:
15.5 Derivatives and Contiguous Functions
15.5(i) Di erentiation Formulas
15.5.1d
dzF(a;b;c;z) =ab
cF(a+ 1;b+ 1;c+ 1;z);
15.5.2
dn
dznF(a;b;c;z) =(a)n(b)n
(c)nF(a+n;b+n;c+n;z):
15.5.3
zd
dzzn
za 1F(a;b;c;z)
= (a)nza+n 1F(a+n;b;c;z):
15.5.4dn
dzn
zc 1F(a;b;c;z)
= (c n)nzc n 1F(a;b;c n;z):
15.5.5
zd
dzzn
zc a 1(1 z)a+b cF(a;b;c;z)
= (c a)nzc a+n 1(1 z)a n+b cF(a n;b;c;z):
388 Hypergeometric Function
15.5.6
dn
dzn
(1 z)a+b cF(a;b;c;z)
=(c a)n(c b)n
(c)n(1 z)a+b c nF(a;b;c+n;z):
15.5.7
(1 z)d
dz(1 z)n
(1 z)a 1F(a;b;c;z)
= ( 1)n(a)n(c b)n
(c)n(1 z)a+n 1
F(a+n;b;c+n;z):
15.5.8
(1 z)d
dz(1 z)n
zc 1(1 z)b cF(a;b;c;z)
= (c n)nzc n 1(1 z)b c+nF(a n;b;c n;z):
15.5.9dn
dzn
zc 1(1 z)a+b cF(a;b;c;z)
= (c n)nzc n 1(1 z)a+b c n
F(a n;b n;c n;z):
Other versions of several of the identities in this sub-
section can be constructed with the aid of the operator
identity
15.5.10
zd
dzzn
=zndn
dznzn,n= 1;2;3;:::.
See Erd elyi et al. (1953a, pp. 102{103).
15.5(ii) Contiguous Functions
The six functions F(a1;b;c;z),F(a;b1;c;z),
F(a;b;c1;z) are said to be contiguous toF(a;b;c;z).
15.5.11(c a)F(a 1;b;c;z)
+ (2a c+ (b a)z)F(a;b;c;z)
+a(z 1)F(a+ 1;b;c;z) = 0;
15.5.12(b a)F(a;b;c;z) +aF(a+ 1;b;c;z)
bF(a;b+ 1;c;z) = 0;
15.5.13(c a b)F(a;b;c;z)
+a(1 z)F(a+ 1;b;c;z)
(c b)F(a;b 1;c;z) = 0;
15.5.14c(a+ (b c)z)F(a;b;c;z)
ac(1 z)F(a+ 1;b;c;z)
+ (c a)(c b)zF(a;b;c+ 1;z) = 0;
15.5.15(c a 1)F(a;b;c;z) +aF(a+ 1;b;c;z)
(c 1)F(a;b;c 1;z) = 0;
15.5.16c(1 z)F(a;b;c;z) cF(a 1;b;c;z)
+ (c b)zF(a;b;c+ 1;z) = 0;
15.5.17(a 1 + (b+ 1 c)z)F(a;b;c;z)
+ (c a)F(a 1;b;c;z)
(c 1)(1 z)F(a;b;c 1;z) = 0;15.5.18c(c 1)(z 1)F(a;b;c 1;z)
+c(c 1 (2c a b 1)z)F(a;b;c;z)
+ (c a)(c b)zF(a;b;c+ 1;z) = 0:
By repeated applications of (15.5.11){(15.5.18) any
functionF(a+k;b+`;c+m;z), in which k;`;m are
integers, can be expressed as a linear combination of
F(a;b;c;z) and any one of its contiguous functions, with
coecients that are rational functions of a;b;c , andz.
An equivalent equation to the hypergeometric di er-
ential equation (15.10.1) is
15.5.19
z(1 z)(a+ 1)(b+ 1)F(a+ 2;b+ 2;c+ 2;z)
+ (c (a+b+ 1)z)(c+ 1)F(a+ 1;b+ 1;c+ 1;z)
c(c+ 1)F(a;b;c;z) = 0:
Further contiguous relations include:
15.5.20
z(1 z) (dF(a;b;c;z)/dz)
= (c a)F(a 1;b;c;z) + (a c+bz)F(a;b;c;z)
= (c b)F(a;b 1;c;z) + (b c+az)F(a;b;c;z);
15.5.21
c(1 z) (dF(a;b;c;z)/dz)
= (c a)(c b)F(a;b;c+ 1;z)+c(a+b c)F(a;b;c;z):
15.6 Integral Representations
The function F(a;b;c;z) (notF(a;b;c;z)) has the fol-
lowing integral representations:
15.6.1
1
(b) (c b)Z1
0tb 1(1 t)c b 1
(1 zt)adt,<c><b>0.
15.6.2 (1 +b c)
2i (b)Z(1+)
0tb 1(t 1)c b 1
(1 zt)adt,
c b6= 1;2;3;:::,<b>0.
15.6.3e bi (1 b)
2i (c b)Z(0+)
1tb 1(t+ 1)a c
(t zt+ 1)adt,
b6= 1;2;3;:::,<(c b)>0.
15.6.4e bi (1 b)
2i (c b)Z(0+)
1tb 1(1 t)c b 1
(1 zt)adt,
b6= 1;2;3;:::,<(c b)>0.
15.6.5e ci (1 b) (1 +b c)
1
42Z(0+;1+;0 ;1 )
Atb 1(1 t)c b 1
(1 zt)adt,
b;c b6= 1;2;3;:::.
15.6.6
1
2i (a) (b)Zi1
i1 (a+t) (b+t) ( t)
(c+t)( z)tdt,
a;b6= 0; 1; 2;:::.
15.7 Continued Fractions 389
15.6.71
2i (a) (b) (c a) (c b)Zi1
i1 (a+t) (b+t) (c a b t) ( t)(1 z)tdt,
a;b;c a;c b6= 0; 1; 2;:::.
15.6.81
(c d)Z1
0F(a;b;d;zt)td 1(1 t)c d 1dt, <c><d>0.
15.6.9Z1
0td 1(1 t)c d 1
(1 zt)a+b F a; b
d;zt
Fa+b ; d
c d;(1 t)z
1 zt
dt,<c><d>0.
These representations are valid when jph(1 z)j< ,
except (15.6.6) which holds for jph( z)j< . In all
cases the integrands are continuous functions of ton
the integration paths, except possibly at the endpoints.
In addition:
In (15.6.1) all functions in the integrand assume
their principal values.
In (15.6.2) the point 1/ zlies outside the integration
contour,tb 1and (t 1)c b 1assume their principal
values where the contour cuts the interval (1 ;1), and
(1 zt)a= 1 att= 0.
In (15.6.3) the point 1/( z 1) lies outside the in-
tegration contour, the contour cuts the real axis be-
tweent= 1 and 0, at which point ph t=and
ph(1 +t) = 0.
In (15.6.4) the point 1/ zlies outside the integration
contour, and at the point where the contour cuts the
negative real axis ph t=and ph(1 t) = 0.
In (15.6.5) the integration contour starts and termi-
nates at a point Aon the real axis between 0 and 1.
It encircles t= 0 andt= 1 once in the positive di-
rection, and then once in the negative direction. See
Figure 15.6.1. At the starting point ph tand ph(1 t)
are zero. Compare Figure 5.12.3.
In (15.6.6) the integration contour separates the
poles of (a+t) and (b+t) from those of ( t), and
( z)thas its principal value.
In (15.6.7) the integration contour separates
the poles of ( a+t) and (b+t) from those of
(c a b t) and ( t), and (1 z)thas its prin-
cipal value.
In each of (15.6.8) and (15.6.9) all functions in the
integrand assume their principal values.
Figure 15.6.1 :t-plane. Contour of integration in
(15.6.5).
15.7 Continued Fractions
Ifjph(1 z)j<, then
15.7.1F(a;b;c;z)
F(a;b+ 1;c+ 1;z)=t0 u1z
t1 u2z
t2 u3z
t3 ;
where
15.7.2tn=c+n; u 2n+1= (a+n)(c b+n);
u2n= (b+n)(c a+n):
Ifjzj<1, then
15.7.3F(a;b;c;z)
F(a;b+ 1;c+ 1;z)=v0 w1
v1 w2
v2 w3
v3 ;
where
15.7.4vn=c+n+ (b a+n+ 1)z;
wn= (b+n)(c a+n)z:
If<z<1
2, then
15.7.5
F(a;b;c;z)
F(a+ 1;b+ 1;c+ 1;z)=x0+y1
x1+y2
x2+y3
x3+;
where
15.7.6xn=c+n (a+b+ 2n+ 1)z;
yn= (a+n)(b+n)z(1 z):
See also Cuyt et al. (2008, pp. 295{309).
390 Hypergeometric Function
15.8 Transformations of Variable
15.8(i) Linear Transformations
All functions in this subsection and x15.8(ii) assume their principal values.
15.8.1Fa;b
c;z
= (1 z) aFa;c b
c;z
z 1
= (1 z) bFc a;b
c;z
z 1
= (1 z)c a bFc a;c b
c;z
,
jph(1 z)j<.
15.8.2
sin((b a))
Fa;b
c;z
=( z) a
(b) (c a)Fa;a c+ 1
a b+ 1;1
z
( z) b
(a) (c b)Fb;b c+ 1
b a+ 1;1
z
,jph( z)j<.
15.8.3
sin((b a))
Fa;b
c;z
=(1 z) a
(b) (c a)Fa;c b
a b+ 1;1
1 z
(1 z) b
(a) (c b)Fb;c a
b a+ 1;1
1 z
,
jph( z)j<.
15.8.4
sin((c a b))
Fa;b
c;z
=1
(c a) (c b)Fa;b
a+b c+ 1; 1 z
(1 z)c a b
(a) (b)Fc a;c b
c a b+ 1; 1 z
,
jphzj<,jph(1 z)j<.
15.8.5
sin((c a b))
Fa;b
c;z
=z a
(c a) (c b)Fa;a c+ 1
a+b c+ 1; 1 1
z
(1 z)c a bza c
(a) (b)Fc a;1 a
c a b+ 1; 1 1
z
,jphzj<,jph(1 z)j<.
15.8(ii) Linear Transformations: Limiting Cases
Withm= 0;1;2;:::, polynomial cases of (15.8.2){(15.8.5) are given by
15.8.6F m;b
c;z
=(b)m
(c)m( z)mF m;1 c m
1 b m;1
z
=(b)m
(c)m(1 z)mF m;c b
1 b m;1
1 z
;
15.8.7F m;b
c;z
=(c b)m
(c)mF m;b
b c m+ 1; 1 z
=(c b)m
(c)mzmF m;1 c m
b c m+ 1; 1 1
z
;
with the understanding that if b= `,`= 0;1;2;:::, thenm`.
Whenb ais an integer limits are taken in (15.8.2) and (15.8.3) as follows.
Ifb ais a nonnegative integer, then
15.8.8
Fa;a+m
c;z
=( z) a
(a+m)m 1X
k=0(a)k(m k 1)!
k! (c a k)z k+( z) a
(a)1X
k=0(a+m)k
k!(k+m)! (c a k m)( 1)kz k m
(ln( z) + (k+ 1) + (k+m+ 1) (a+k+m) (c a k m)) ,
jzj>1;jph( z)j<,
15.8.9Fa;a+m
c;z
=(1 z) a
(a+m) (c a)m 1X
k=0(a)k(c a m)k(m k 1)!
k!(z 1) k
+( 1)m(1 z) a m
(a) (c a m)1X
k=0(a+m)k(c a)k
k!(k+m)!(1 z) k
(ln(1 z) + (k+ 1) + (k+m+ 1) (a+k+m) (c a+k)),
jz 1j>1;jph(1 z)j<.
In (15.8.8) when c a k mis a nonpositive integer (c a k m)/ (c a k m) is interpreted as
( 1)m+k+a c+1(m+k+a c)!. Also, ifais a nonpositive integer, then (15.8.6) applies.
15.8 Transformations of Variable 391
Alternatively, if b ais a negative integer, then we interchange aandbinF(a;b;c;z).
In a similar way, when c a bis an integer limits are taken in (15.8.4) and (15.8.5) as follows.
Ifc a bis a nonnegative integer, then
15.8.10
Fa;b
a+b+m;z
=1
(a+m) (b+m)m 1X
k=0(a)k(b)k(m k 1)!
k!(z 1)k (z 1)m
(a) (b)1X
k=0(a+m)k(b+m)k
k!(k+m)!(1 z)k
(ln(1 z) (k+ 1) (k+m+ 1) + (a+k+m) + (b+k+m)) ,
jz 1j<1;jph(1 z)j<,
15.8.11
Fa;b
a+b+m;z
=z a
(a+m)m 1X
k=0(a)k(m k 1)!
k! (b+m k)
1 1
zk
z a
(a)1X
k=0(a+m)k
k!(k+m)! (b k)( 1)k
1 1
zk+m
ln1 z
z
(k+ 1) (k+m+ 1) + (a+k+m) + (b k)
,
<z>1
2;jphzj<;jph(1 z)j<.
In (15.8.11) when b kis a nonpositive integer,
(b k)/ (b k) is interpreted as ( 1)k b+1(k b)!.
Also, ifaorbor both are nonpositive integers, then
(15.8.7) applies.
Lastly, ifc a bis a negative integer, then we rst
apply the transformation
15.8.12
F(a;b;a+b m;z) = (1 z) mF
~a;~b; ~a+~b+m;z
,
~a=a m;~b=b m.
15.8(iii) Quadratic Transformations
A quadratic transformation relates two hypergeometric
functions, with the variable in one a quadratic function
of the variable in the other, possibly combined with a
fractional linear transformation.
A necessary and sucient condition that there ex-
ists a quadratic transformation is that at least one of
the equations shown in Table 15.8.1 is satis ed.Table 15.8.1 : Quadratic transformations of the hyper-
geometric function.
Group 1 Group 2 Group 3 Group 4
c=a b+ 1a=b+1
2
c= 2a c =b a+ 1b=a+1
2c=1
2
c= 2b c =1
2(a+b+ 1)c=a+b+1
2c=3
2
a+b= 1c=a+b 1
2
The hypergeometric functions that correspond to
Groups 1 and 2 have zas variable. The hypergeomet-
ric functions that correspond to Groups 3 and 4 have
a nonlinear function of zas variable. The transforma-
tion formulas between two hypergeometric functions in
Group 2, or two hypergeometric functions in Group 3,
are the linear transformations (15.8.1).
In the equations that follow in this subsection all
functions take their principal values.
Group 1 ! Group 3
15.8.13 Fa;b
2b;z
=
1 1
2z aF
1
2a;1
2a+1
2
b+1
2;z
2 z2!
, jph(1 z)j<,
15.8.14 Fa;b
2b;z
= (1 z) a/2F
1
2a;b 1
2a
b+1
2;z2
4z 4!
, jph(1 z)j<.
Group 2 ! Group 3
15.8.15 Fa;b
a b+ 1;z
= (1 +z) aF1
2a;1
2a+1
2
a b+ 1;4z
(1 +z)2
, jzj<1,
15.8.16 Fa;b
a b+ 1;z
= (1 z) aF1
2a;1
2a b+1
2
a b+ 1; 4z
(1 z)2
, jzj<1.
392 Hypergeometric Function
15.8.17 Fa;b
1
2(a+b+ 1);z
= (1 2z) aF
1
2a;1
2a+1
2
1
2(a+b+ 1);4z(z 1)
(1 2z)2!
, <z<1
2,
15.8.18 Fa;b
1
2(a+b+ 1);z
=F
1
2a;1
2b
1
2(a+b+ 1); 4z(1 z)!
, <z<1
2.
15.8.19 Fa;1 a
c;z
= (1 2z)1 a c(1 z)c 1F1
2(a+c);1
2(a+c 1)
c;4z(z 1)
(1 2z)2
,<z<1
2,
15.8.20 Fa;1 a
c;z
= (1 z)c 1F1
2(c a);1
2(a+c 1)
c; 4z(1 z)
, <z<1
2.
Group 2 ! Group 1
15.8.21 Fa;b
a b+ 1;z
=
1 +pz 2aFa;a b+1
2
2a 2b+ 1;4pz
(1 +pz)2
, jphzj<,jzj<1.
15.8.22Fa;b
1
2(a+b+ 1);z
= p
1 z 1 1p
1 z 1+ 1!a
F
a;1
2(a+b)
a+b;4p
1 z 1
p
1 z 1+ 12!
,jph( z)j<,<z<1
2.
15.8.23
Fa;1 a
c;z
=p
1 z 1 11 ap
1 z 1+ 1a 2c+1
1 z 1c 1F
c a;c 1
2
2c 1;4p
1 z 1
p
1 z 1+ 12!
,
jph( z)j<,<z<1
2.
Group 2 ! Group 4
15.8.24Fa;b
a b+ 1;z
= (1 z) a (a b+ 1) 1
2
1
2a+1
2
1
2a b+ 1F
1
2a;1
2a b+1
2
1
2;z+ 1
z 12!
+ (1 +z)(1 z) a 1 (a b+ 1)
1
2
1
2a
1
2a b+1
2F
1
2a+1
2;1
2a b+ 1
3
2;z+ 1
z 12!
,
jph( z)j<.
15.8.25Fa;b
1
2(a+b+ 1);z
= 1
2(a+b+ 1)
1
2
1
2a+1
2
1
2b+1
2F
1
2a;1
2b
1
2; (1 2z)2!
+ (1 2z) 1
2(a+b+ 1)
1
2
1
2a
1
2bF
1
2a+1
2;1
2b+1
2
3
2; (1 2z)2!
,
jphzj<,jph(1 z)j<.
15.8.26Fa;1 a
c;z
= (1 z)c 1 (c) 1
2
1
2(c a+ 1)
1
2c+1
2aF
1
2c 1
2a;1
2c+1
2a 1
2
1
2; (1 2z)2!
+ (1 2z)(1 z)c 1 (c)
1
2
1
2c 1
2a
1
2(c+a 1)F
1
2c 1
2a+1
2;1
2c+1
2a
3
2; (1 2z)2!
,
jphzj<,jph(1 z)j<.
Group 4 ! Group 2
15.8.272 1
2
a+b+1
2
a+1
2
b+1
2F
a;b;1
2;z
=F
2a;2b;a+b+1
2;1
2 1
2pz
+F
2a;2b;a+b+1
2;1
2+1
2pz
,
jphzj<,jph(1 z)j<.
15.9 Relations to Other Functions 393
15.8.28
2pz
1
2
a+b 1
2
a 1
2
b 1
2F
a;b;3
2;z
=F
2a 1;2b 1;a+b 1
2;1
2 1
2pz
F
2a 1;2b 1;a+b 1
2;1
2+1
2pz
,jphzj<,jph(1 z)j<.
15.8(iv) Quadratic Transformations (Continued)
When the intersection of two groups in Table 15.8.1 is not empty there exist special quadratic transformations, with
only one free parameter, between two hypergeometric functions in the same group.
Examples
b=1
3a+1
3,c= 2b=a b+ 1 in Groups 1 and 2.
(15.8.21) becomes
15.8.29 F
a;1
3a+1
3
2
3a+2
3;z!
=
1 +pz 2aF
a;2
3a+1
6
4
3a+1
3;4pz
(1 +pz)2!
:
This is a quadratic transformation between two cases in Group 1.
We can also use (15.8.13), followed by the inverse of (15.8.15), and obtain
15.8.30
1 1
2z aF
1
2a;1
2a+1
2
1
3a+5
6;z
2 z2!
=F
a;1
3a+1
3
2
3a+2
3;z!
= (1 +z) aF
1
2a;1
2a+1
2
2
3a+2
3;4z
(1 +z)2!
;
which is a quadratic transformation between two cases in Group 3.
For further examples see Andrews et al. (1999, pp. 130{132 and 176{177).
15.8(v) Cubic Transformations
Examples
15.8.31 F
3a;3a+1
2
4a+2
3;z!
=
1 9
8z 2aF
a;a+1
2
2a+5
6;27z2(z 1)
(9z 8)2!
, <z<8
9.
With=e2i/3(1 z)=
z e4i/3
15.8.32
1 z3a
( z)3a
1
a+2
3
2
3F
a;a+1
3
2
3;z 3!
+e1
3i
z (a) 4
3F
a+1
3;a+2
3
4
3;z 3!!
=33
2a+1
2e1
2ai
a+1
3
(1 )a
2
2a+2
3
( )2aF
a+1
3;3a
2a+2
3; 1!
, jzj>1,jph( z)j<1
3.
Ramanujan's Cubic Transformation
15.8.33
F
1
3;2
3
1; 1 1 z
1 + 2z3!
= (1 + 2z)F1
3;2
3
1;z3
;
provided that zlies in the intersection of the
open disks z 1
41
4p
3i <1
2p
3, or equivalently,
jph((1 z)/(1 + 2z))j< = 3. This is used in a cubic
analog of the arithmetic-geometric mean. See Borwein
and Borwein (1991), and also Berndt et al. (1995).
For further examples and higher-order transforma-
tions see Goursat (1881), Watson (1910), and Vid unas
(2005); see also Erd elyi et al. (1953a, pp. 67 and 113{
114).15.9 Relations to Other Functions
15.9(i) Orthogonal Polynomials
For the notation see xx18.3 and 18.19.
Jacobi
15.9.1
P( ; )
n(x) =( + 1)n
n!F n;n+ + + 1
+ 1;1 x
2
:
Gegenbauer (or Ultraspherical)
15.9.2C()
n(x) =(2)n
n!F n;n+ 2
+1
2;1 x
2
:
15.9.3C()
n(x) = (2x)n()n
n!F 1
2n;1
2(1 n)
1 n;1
x2
:
394 Hypergeometric Function
15.9.4C()
n(cos) =eni()n
n!F n;
1 n;e 2i
:
Chebyshev
15.9.5Tn(x) =F n;n
1
2;1 x
2
:
15.9.6Un(x) = (n+ 1)F n;n+ 2
3
2;1 x
2
:
Legendre
15.9.7 Pn(x) =F n;n+ 1
1;1 x
2
:Krawtchouk
15.9.8
Kn(x;p;N) =F n; x
N;1
p
,n= 0;1;2;:::;N ;
compare alsox15.2(ii).
Meixner
15.9.9Mn(x; ;c) =F n; x
; 1 1
c
:
Meixner{Pollaczek
15.9.10
P()
n(x; ) =(2)n
n!eni F n;+ix
2; 1 e 2i
:
15.9(ii) Jacobi Function
This is a generalization of Jacobi polynomials ( x18.3) and has the representation
15.9.11 ( ; )
(t) =F1
2( + + 1 i);1
2( + + 1 +i)
+ 1; sinh2t
:
The Jacobi transform is de ned as
15.9.12 ef() =Z1
0f(t) ( ; )
(t)(2 sinht)2 +1(2 cosht)2 +1dt;
with inverse
15.9.13 f(t) =1
2iZi1
i1ef(i) ( ; )
i(t) 1
2( + + 1 +)
1
2( + 1 +)
( + 1) ()2 + +1 d;
where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and
15.9.14 ( ; )
(t) = (2 cosht)i 1F1
2( + + 1 i);1
2( + 1 i)
1 i; sech2t
:
For this result, together with restrictions on the functions f(t) andef(), see Koornwinder (1984a).
15.9(iii) Gegenbauer Function
This is a generalization of Gegenbauer (or ultraspheri-
cal) polynomials ( x18.3). It is de ned by:
15.9.15
C()
(z) = ( + 2)
(2) ( + 1)F ; + 2
+1
2;1 z
2
:
15.9(iv) Associated Legendre Functions; Ferrers
Functions
Any hypergeometric function for which a quadratic
transformation exists can be expressed in terms of as-
sociated Legendre functions or Ferrers functions. For
examples seexx14.3(i){14.3(iii) and 14.21(iii).
For further examples see http://dlmf.nist.gov/
15.9.iv .15.10 Hypergeometric Di erential Equation
15.10(i) Fundamental Solutions
15.10.1z(1 z)d2w
dz2+ (c (a+b+ 1)z)dw
dz abw= 0:
This is the hypergeometric di erential equation . It has
regular singularities at z= 0;1;1, with corresponding
exponent pairsf0;1 cg,f0;c a bg,fa;bg, respec-
tively. When none of the exponent pairs di er by an
integer, that is, when none of c,c a b,a bis an
integer, we have the following pairs f1(z),f2(z) of fun-
damental solutions. They are also numerically satisfac-
tory (x2.7(iv)) in the neighborhood of the corresponding
singularity.
15.10 Hypergeometric Differential Equation 395
Singularityz= 0
15.10.2f1(z) =Fa;b
c;z
;
f2(z) =z1 cFa c+ 1;b c+ 1
2 c;z
;
15.10.3 Wff1(z);f2(z)g= (1 c)z c(1 z)c a b 1:
Singularityz= 1
15.10.4f1(z) =Fa;b
a+b+ 1 c; 1 z
;
f2(z) = (1 z)c a bFc a;c b
c a b+ 1; 1 z
;
15.10.5 Wff1(z);f2(z)g= (a+b c)z c(1 z)c a b 1:Singularityz=1
15.10.6f1(z) =z aFa;a c+ 1
a b+ 1;1
z
;
f2(z) =z bFb;b c+ 1
b a+ 1;1
z
;
15.10.7 Wff1(z);f2(z)g= (a b)z c(z 1)c a b 1:
(a) Ifcequalsn= 1;2;3;:::, anda= 1;2;:::;n 1,
then fundamental solutions in the neighborhood of z=
0 are given by (15.10.2) with the interpretation (15.2.5)
forf2(z).
(b) Ifcequalsn= 1;2;3;:::, anda6= 1;2;:::;n 1,
then fundamental solutions in the neighborhood of z=
0 are given by F(a;b;n;z) and
15.10.8Fa;b
n;z
lnz n 1X
k=1(n 1)!(k 1)!
(n k 1)!(1 a)k(1 b)k( z) k
+1X
k=0(a)k(b)k
(n)kk!zk( (a+k) + (b+k) (1 +k) (n+k)) ,a;b6=n 1;n 2;:::; 0; 1; 2;:::,
or
15.10.9F m;b
n;z
lnz n 1X
k=1(n 1)!(k 1)!
(n k 1)!(m+ 1)k(1 b)k( z) k
+mX
k=0( m)k(b)k
(n)kk!zk( (1 +m k) + (b+k) (1 +k) (n+k))
+ ( 1)mm!1X
k=m+1(k 1 m)!(b)k
(n)kk!zk,a= m,m= 0;1;2;:::;b6=n 1;n 2;:::; 0; 1; 2;:::,
or
15.10.10F m; `
n;z
lnz n 1X
k=1(n 1)!(k 1)!
(n k 1)!(m+ 1)k(`+ 1)k( z) k
+`X
k=0( m)k( `)k
(n)kk!zk( (1 +m k) + (1 +` k) (1 +k) (n+k))
+ ( 1)``!mX
k=`+1(k 1 `)!( m)k
(n)kk!zk, a= m,m= 0;1;2;:::;b= `,`= 0;1;2;:::;m .
Moreover, in (15.10.9) and (15.10.10) the symbols aand
bare interchangeable.
(c) Ifcequals 2 n= 0; 1; 2;:::, then fundamen-
tal solutions in the neighborhood of z= 0 are given by
zn 1times those in (a) and (b) with aandbreplaced
bya+n 1 andb+n 1, respectively.
(d) Ifa+b+ 1 cequalsn= 1;2;3;:::, or
2 n= 0; 1; 2;:::, then fundamental solutions in
the neighborhood of z= 1 are given by those in (a),
(b), and (c) with zreplaced by 1 z.
(e) Finally, if a b+ 1 equals n= 1;2;3;:::, or
2 n= 0; 1; 2;:::, then fundamental solutions in theneighborhood of z=1are given by z atimes those in
(a), (b), and (c) with bandzreplaced by a c+ 1 and
1/z, respectively.
15.10(ii) Kummer's 24 Solutions and
Connection Formulas
The three pairs of fundamental solutions given by
(15.10.2), (15.10.4), and (15.10.6) can be transformed
into 18 other solutions by means of (15.8.1), leading to a
total of 24 solutions known as Kummer's solutions . See
http://dlmf.nist.gov/15.10.ii for Kummer's solu-
tions and their connection formulas.
396 Hypergeometric Function
15.11 Riemann's Di erential Equation
15.11(i) Equations with Three Singularities
The importance of (15.10.1) is that any homogeneous linear di erential equation of the second order with at most
three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). The most general
form is given by
15.11.1d2w
dz2+1 a1 a2
z +1 b1 b2
z +1 c1 c2
z
dw
dz
+( )(
)a1a2
z +( )(
)b1b2
z +(
)(
)c1c2
z
w
(z )(z )(z
)= 0;
with
15.11.2 a1+a2+b1+b2+c1+c2= 1:
Herefa1;a2g,fb1;b2g,fc1;c2gare the exponent pairs
at the points , ,
, respectively. Cases in which there
are fewer than three singularities are included automat-
ically by allowing the choice f0;1gfor exponent pairs.
Also, if any of , ,
, is at in nity, then we take the
corresponding limit in (15.11.1).
The complete set of solutions of (15.11.1) is denoted
byRiemann's P-symbol :
15.11.3 w=P8
<
:
a1b1c1z
a2b2c29
=
;:
In particular,
15.11.4w=P8
<
:0 11
0 0 a z
1 c c a b b9
=
;
denotes the set of solutions of (15.10.1).15.11(ii) Transformation Formulas
A conformal mapping of the extended complex plane
onto itself has the form
15.11.5 t= (z+)/(z+);
where,,,are real or complex constants such that
= 1. These constants can be chosen to map any
two sets of three distinct points f ; ;
gandfe ;e ;e
g
onto each other. Symbolically:
15.11.6P8
<
:
a1b1c1z
a2b2c29
=
;=P8
<
:e e e
a1b1c1t
a2b2c29
=
;:
The reduction of a general homogeneous linear di er-
ential equation of the second order with at most three
regular singularities to the hypergeometric di erential
equation is given by
15.11.7P8
<
:
a1b1c1z
a2b2c29
=
;=z
z
a1z
z
b1
P8
>><
>>:0 1 1
0 0 a1+b1+c1(z )(
)
(z
)( )
a2 a1b2 b1a1+b1+c29
>>=
>>;:
We also have
15.11.8 z(1 z)P8
<
:0 11
a1b1c1z
a2b2c29
=
;=P8
<
:0 1 1
a1+ b 1+ c 1 z
a2+ b 2+ c 2 9
=
;;
for arbitrary and.
15.12 Asymptotic Approximations
15.12(i) Large Variable
For the asymptotic behavior of F(a;b;c;z) asz!1
witha,b,c xed, combine (15.2.2) with (15.8.2) or
(15.8.8).15.12(ii) Large c
Letdenote an arbitrary small positive constant. Also
leta;b;z be real or complex and xed, and at least one
of the following conditions be satis ed:
(a)aand/orb2f0; 1; 2;:::g.
(b)<z<1
2andjc+njfor alln2f0;1;2;:::g.
15.12 Asymptotic Approximations 397
(c)<z=1
2andjphcj .
(d)<z >1
2and 1
2+phc ++1
2 ,
where
15.12.1 = arctanphz ph(1 z)
lnj1 z 1j
;
withzrestricted so that 2[0;1
2).
Then for xed m2f0;1;2;:::g,
15.12.2
F(a;b;c;z) =m 1X
s=0(a)s(b)s
(c)ss!zs+O
c m
,jcj!1 .
Similar results for other sectors are given in Wagner
(1988). For the more general case in which a2=o(c)
andb2=o(c) see Wagner (1990).
15.12(iii) Other Large Parameters
Again, throughout this subsection denotes an arbi-
trary small positive constant, and a;b;c;z are real orcomplex and xed.
As!1 ,
15.12.3
Fa;b
c+;z
(c+)
(c b+)1X
s=0qs(z)(b)s s b;
whereq0(z) = 1 andqs(z),s= 1;2;:::, are de ned by
the generating function
15.12.4
et 1
tb 1
et(1 c)
1 z+ze t a=1X
s=0qs(z)ts:
Ifjph(1 z)j<, then (15.12.3) applies when jphj
1
2 . If<z1
2, then (15.12.3) applies when
jphj .
Ifjph(z 1)j< , then as!1 withjphj
,
15.12.5
Fa+;b
c;1
2 1
2z
= 2(a+b 1)=2(z+ 1)(c a b 1)=2
(z 1)c=2p
sinh
+1
2a 1
2b1 c
Ic 1
(+1
2a 1
2b)
(1 +O( 2))
+Ic 2
(+1
2a 1
2b)
2+a b
c 1
2
c 3
21
coth
+1
2(2c a b 1)(a+b 1) tanh 1
2
+O( 2)!
;
where
15.12.6 = arccoshz:
ForI(z) seex10.25(ii). For this result and an extension to an asymptotic expansion with error bounds see Jones
(2001).
See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1).
Ifjphzj<, then as!1 withjphj ,
15.12.7Fa;b
c+; z
= 2b c+(1=2)z+ 1
2pz
a=2U
a 1
2; p
(1 +z)c a bz1 c
z 11 a
+O( 1)!
+(a 1)/2
U
a 3
2; p
(1 +z)c a bz1 c
z 11 a
2c b ( 1/2 )
z 1a
+O( 1)!!
;
where
15.12.8 =
2 ln
1 z 1
z+ 12!!1=2
;
with the branch chosen to be continuous and < > 0 when<((z 1)/(z+ 1))>0. ForU(a;z) seex12.2, and for
an extension to an asymptotic expansion see Olde Daalhuis (2003a).
398 Hypergeometric Function
Ifjphzj<, then as!1 withjphj1
2 ,
15.12.9
(z+ 1)3=2(2)c 1Fa+;b+ 2
c; z
= 1=3
ei(a c++(1=3))Ai
e 2i/32/3 2
+ei(c a (1=3))Ai
e2i/32/3 2
a0() +O( 1)
+ 2=3
ei(a c++(2=3))Ai0
e 2i/32/3 2
+ei(c a (2=3))Ai0
e2i/32/3 2
a1() +O( 1)
;
where
15.12.10 = arccosh 1
4z 1
;
15.12.11 =
3
2+9
4ln2 +e
2 +e 1=3
;
with the branch chosen to be continuous and >0 when >0. Also,
15.12.12 a0() =1
2G0( ) +1
2G0( ); a 1() = 1
2G0( ) 1
2G0( )
= ;
where
15.12.13 G0( ) =
2 +ec b ( 1/2 )
1 +ea c+( 1/2 )
z 1 e a+( 1/2 )r
e e :
For Ai(z) seex9.2, and for further information and an extension to an asymptotic expansion see Olde Daalhuis
(2003b). (Two errors in this reference are corrected in (15.12.9).)
By combination of the foregoing results of this subsection with the linear transformations of x15.8(i) and the con-
nection formulas of x15.10(ii), similar asymptotic approximations for F(a+e1;b+e2;c+e3;z) can be obtained
withej=1 or 0,j= 1;2;3. For more details see Olde Daalhuis (2010). For other extensions, see Wagner (1986)
and Temme (2003).
15.13 Zeros
LetN(a;b;c ) denote the number of zeros of F(a;b;c;z) in the sectorjph(1 z)j<. Ifa,b,care real,a,b,c,c a,
c b6= 0; 1; 2;:::, and, without loss of generality, ba,ca+b(compare (15.8.1)), then
15.13.1 N(a;b;c ) =8
><
>:0; a> 0;
b ac+1
2(1 +S); a< 0;c a>0;
b ac+1
2(1 +S) +ba c+ 1cS; a< 0;c a<0;
whereS= sign( (a) (b) (c a) (c b)).
Ifa,b,c,c a, orc b2f0; 1; 2;:::g, then
F(a;b;c;z) is not de ned, or reduces to a polynomial,
or reduces to (1 z)c a btimes a polynomial.
For further information on the location of real zeros
see Zarzo et al. (1995). A small table of zeros is given
in Conde and Kalla (1981).
15.14 Integrals
The Mellin transform of the hypergeometric function of
negative argument is given by
15.14.1Z1
0xs 1Fa;b
c; x
dx= (s) (a s) (b s)
(a) (b) (c s),
min(<a;<b)><s>0.
Integrals of the formR
x (x+t) F(a;b;c;x)dxand
more complicated forms are given in Apelblat (1983,pp. 370{387), Prudnikov et al. (1990,xx1.15 and 2.21),
and Gradshteyn and Ryzhik (2000, x7.5).
Fourier transforms of hypergeometric functions are
given in Erd elyi et al. (1954a,xx1.14 and 2.14). Laplace
transforms of hypergeometric functions are given in
Erd elyi et al. (1954a,x4.21), Oberhettinger and Badii
(1973,x1.19), and Prudnikov et al. (1992a,x3.37). In-
verse Laplace transforms of hypergeometric functions
are given in Erd elyi et al. (1954a,x5.19), Oberhet-
tinger and Badii (1973, x2.18), and Prudnikov et al.
(1992b,x3.35). Mellin transforms of hypergeometric
functions are given in Erd elyi et al. (1954a,x6.9), Ober-
hettinger (1974,x1.15), and Marichev (1983, pp. 288{
299). Inverse Mellin transforms are given in Erd elyi
et al. (1954a,x7.5). Hankel transforms of hypergeomet-
ric functions are given in Oberhettinger (1972, x1.17)
and Erd elyi et al. (1954b,x8.17).
15.15 Sums 399
For other integral transforms see Erd elyi et al.
(1954b), Prudnikov et al. (1992b,x4.3.43), and also
x15.9(ii).
15.15 Sums
15.15.1Fa;b
c;1
z
=
1 z0
z a1X
s=0(a)s
s!
F s;b
c;1
z0
1 z
z0 s
:
Herez0(6= 0) is an arbitrary complex constant and the
expansion converges when jz z0j>max(jz0j;jz0 1j).
For further information see B uhring (1987a) and Kalla
(1992).For compendia of nite sums and in nite series in-
volving hypergeometric functions see Prudnikov et al.
(1990,xx5.3 and 6.7) and Hansen (1975).
15.16 Products
15.16.1Fa;b
c 1
2;z
Fc a;c b
c+1
2;z
=1X
s=0(c)s
c+1
2
sAszs,jzj<1,
whereA0= 1 andAs,s= 1;2;:::, are de ned by the
generating function
15.16.2
(1 z)a+b cF(2a;2b; 2c 1;z) =1X
s=0Aszs,jzj<1.
Also,
15.16.3Fa;b
c;z
Fa;b
c;
=1X
s=0(a)s(b)s(c a)s(c b)s
(c)s(c)2ss!(z)sFa+s;b+s
c+ 2s;z+ z
,
jzj<1,jj<1,jz+ zj<1.
15.16.4 Fa;b
c;z
F a; b
c;z
+ab(a c)(b c)
c2(1 c2)z2F1 +a;1 +b
2 +c;z
F1 a;1 b
2 c;z
= 1:
Generalized Legendre's Relation
15.16.5F1
2+; 1
2
1 ++;z
F1
2 ;1
2+
1 ++; 1 z
+F1
2+;1
2
1 ++;z
F 1
2 ;1
2+
1 ++; 1 z
F1
2+;1
2
1 ++;z
F1
2 ;1
2+
1 ++; 1 z
= (1 ++) (1 ++)
+++3
2
1
2+,
jphzj<,jph(1 z)j<.
For further results of this kind, and also series of
products of hypergeometric functions, see Erd elyi et al.
(1953a,x2.5.2).
Applications
15.17 Mathematical Applications
15.17(i) Di erential Equations
This topic is treated in xx15.10 and 15.11.
The logarithmic derivatives of some hypergeomet-
ric functions for which quadratic transformations ex-
ist (x15.8(iii)) are solutions of Painlev e equations. See
x32.10(vi).15.17(ii) Conformal Mappings
The quotient of two solutions of (15.10.1) maps the
closed upper half-plane =z0 conformally onto a
curvilinear triangle. See Klein (1894) and Hochstadt
(1971). Hypergeometric functions, especially complete
elliptic integrals, also play an important role in quasi-
conformal mapping. See Anderson et al. (1997).
15.17(iii) Group Representations
For harmonic analysis it is more natural to represent hy-
pergeometric functions as a Jacobi function ( x15.9(ii)).
For special values of and there are many group-
theoretic interpretations. First, as spherical functions
on noncompact Riemannian symmetric spaces of rank
one, but also as associated spherical functions, inter-
twining functions, matrix elements of SL(2 ;R), and
spherical functions on certain nonsymmetric Gelfand
400 Hypergeometric Function
pairs. Harmonic analysis can be developed for the Ja-
cobi transform either as a generalization of the Fourier-
cosine transform ( x1.14(ii)) or as a specialization of a
group Fourier transform. For further information see
Koornwinder (1984a).
15.17(iv) Combinatorics
In combinatorics, hypergeometric identities classify sin-
gle sums of products of binomial coecients. See Ego-
rychev (1984,x2.3).
Quadratic transformations give insight into the re-
lation of elliptic integrals to the arithmetic-geometric
mean (x19.22(ii)). See Andrews et al. (1999,x3.2).
15.17(v) Monodromy Groups
The three singular points in Riemann's di erential equa-
tion (15.11.1) lead to an interesting Riemann sheet
structure. By considering, as a group, all analytic trans-
formations of a basis of solutions under analytic continu-
ation around all paths on the Riemann sheet, we obtain
the monodromy group. These monodromy groups are
nite i the solutions of Riemann's di erential equation
are all algebraic. For a survey of this topic see Gray
(2000).
15.18 Physical Applications
The hypergeometric function has allowed the develop-
ment of \solvable" models for one-dimensional quantum
scattering through and over barriers (Eckart (1930),
Bhattacharjie and Sudarshan (1962)), and generalized
to include position-dependent e ective masses (Dekar
et al. (1999)).
More varied applications include photon scattering
from atoms (Gavrila (1967)), energy distributions of
particles in plasmas (Mace and Hellberg (1995)), con-
formal eld theory of critical phenomena (Burkhardt
and Xue (1991)), quantum chromo-dynamics (Atkinson
and Johnson (1988)), and general parametrization of
the e ective potentials of interaction between atoms in
diatomic molecules (Herrick and O'Connor (1998)).
Computation
15.19 Methods of Computation
15.19(i) Maclaurin Expansions
The Gauss series (15.2.1) converges for jzj<1. For
z2Rit is always possible to apply one of the lineartransformations in x15.8(i) in such a way that the hy-
pergeometric function is expressed in terms of hyper-
geometric functions with an argument in the interval
[0;1
2].
Forz2Cit is possible to use the linear transforma-
tions in such a way that the new arguments lie within
the unit circle, except when z=ei=3. This is because
the linear transformations map the pair fei=3;e i=3g
onto itself. However, by appropriate choice of the con-
stantz0in (15.15.1) we can obtain an in nite series that
converges on a disk containing z=ei=3. Moreover, it
is also possible to accelerate convergence by appropriate
choice ofz0.
Large values ofjajorjbj, for example, delay conver-
gence of the Gauss series, and may also lead to severe
cancellation.
For further information see B uhring (1987a), Forrey
(1997), and Kalla (1992).
15.19(ii) Di erential Equation
A comprehensive and powerful approach is to integrate
the hypergeometric di erential equation (15.10.1) by di-
rect numerical methods. As noted in x3.7(ii), the inte-
gration path should be chosen so that the wanted so-
lution grows in magnitude at least as fast as all other
solutions. However, since the growth near the singular-
ities of the di erential equation is algebraic rather than
exponential, the resulting instabilities in the numerical
integration might be tolerable in some cases.
15.19(iii) Integral Representations
The representation (15.6.1) can be used to compute
the hypergeometric function in the sector jph(1 z)j<
. Gauss quadrature approximations are discussed in
Gautschi (2002b).
15.19(iv) Recurrence Relations
The relations in x15.5(ii) can be used to compute
F(a;b;c;z), provided that care is taken to apply these
relations in a stable manner; see x3.6(ii). Initial values
for moderate values of jajandjbjcan be obtained by
the methods ofx15.19(i), and for large values of jaj,jbj,
orjcjvia the asymptotic expansions of xx15.12(ii) and
15.12(iii).
For example, in the half-plane <z1
2we can use
(15.12.2) or (15.12.3) to compute F(a;b;c+N+ 1;z)
andF(a;b;c+N;z), whereNis a large positive inte-
ger, and then apply (15.5.18) in the backward direction.
When<z >1
2it is better to begin with one of the lin-
ear transformations (15.8.4), (15.8.7), or (15.8.8). For
further information see Gil et al. (2006a, 2007b).
15.20 Software 401
15.20 Software
Seehttp://dlmf.nist.gov/15.20 .
References
General References
The main references used in writing this chapter are
Andrews et al. (1999) and Temme (1996a). For addi-
tional bibliographic reading see Erd elyi et al. (1953a),
Hochstadt (1971), Luke (1969a), Olver (1997b), Slater
(1966), Wang and Guo (1989), and Whittaker and Wat-
son (1927).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x15.2 Andrews et al. (1999,x2.1), Olver (1997b,
Chapter 5, Theorem 9.1), Temme (1996a, x5.1).
(15.2.3) is a consequence of (15.8.4).
x15.3 These graphics were produced at NIST.
x15.4 Andrews et al. (1999,x2.2). For (15.4.1){
(15.4.6) see Temme (1996a, x5.1). For (15.4.7) use
(15.8.27), (15.4.6), and (5.5.5). For (15.4.9) use
(15.8.28), (15.4.6), and (5.5.5). For (15.4.11) use
(15.8.1) and (15.4.7). For (15.4.13) use (15.8.1)
and (15.4.11). For (15.4.15) use (15.8.1) and
(15.4.9). For (15.4.17) and (15.4.18) use (15.8.15)
and (15.4.6). For (15.4.19) use (15.5.11) and
(15.4.6). For (15.4.21) use (15.8.10). For (15.4.22)
and (15.4.23) use (15.8.4), (5.5.3), and (5.5.1). For
(15.4.25) see Dougall (1907) or Andrews et al.
(1999,x2.8). For (15.4.26) use (15.8.24) and
(5.5.5). For (15.4.27) use (15.2.1) and (5.7.7).
For (15.4.28) use (15.8.1), (15.4.26), and (5.5.5).
For (15.4.29) use (15.8.1), (15.5.15), (15.4.26),
(5.5.5), and (5.5.1). For (15.4.30) use (15.8.1)
and (15.4.28). For (15.4.31) use (15.8.1), Erd elyi
et al. (1953a, Eq. (2.11.41)), (15.4.20), and (5.5.5).
For (15.4.32) use (15.8.30), (15.4.28), (5.5.5),
and (5.5.6). (Note that Erd elyi et al. (1953a,
Eq. (2.8.54)) contains an error.) For (15.4.33) let
z! 1 in (15.8.32) and use (5.5.5).
x15.5 Andrews et al. (1999,x2.5). For (15.5.2){(15.5.9)
use induction. For (15.5.10) see Fleury and Tur-
biner (1994).x15.6 Andrews et al. (1999,xx2.2, 2.4, 2.9), Erd elyi
et al. (1953a,x2.1.3), and Whittaker and Wat-
son (1927, pp. 290{291). (15.6.3) follows from
(15.6.4).
x15.7 Andrews et al. (1999, pp. 94, 97{98, and Ex. 26
on p. 119), Lorentzen and Waadeland (1992, x6.1),
and Berndt (1989, pp. 134{137). These refer-
ences contain several restrictions on the param-
etersa,b, andc. This is because they use the
functionF(a;b;c;z). No restrictions are needed
forF(a;b;c;z).
x15.8 For (15.8.1){(15.8.4) see Olver (1997b, Chap-
ter 5,x10). For (15.8.5) combine (15.8.1) and
(15.8.2). (15.8.6) and (15.8.7) are obtained as
limits of (15.8.2){(15.8.5) as a! m, together
with (5.5.3). For (15.8.8) and (15.8.10) see Erd elyi
et al. (1953a,xx2.1.4 and 2.3.1). (15.8.9) and
(15.8.11) follow from (15.8.10) and (15.8.8), re-
spectively, via (15.8.1). For x15.8(iii) see Andrews
et al. (1999,x3.1). For (15.8.31) and (15.8.32)
see Goursat (1881, Eq. (110)) and Watson (1910).
The version of (15.8.31) given in Erd elyi et al.
(1953a, p. 114 (40)) contains a typographical er-
ror. For (15.8.33) see Chan (1998).
x15.9 For (15.9.1){(15.9.10) see xx18.5(ii) and 18.20(ii).
For (15.9.15) see Erd elyi et al. (1953a,xx3.15.1
and 3.15.2).
x15.10 Andrews et al. (1999,x2.3), Olver (1997b,
pp. 163{168), and Luke (1969a, Chapter III).
(15.10.9) is a corrected version of (2.3.20) in the
rst reference.
x15.11 Andrews et al. (1999,x2.3) or Olver (1997b,
pp. 156{158).
x15.12 For (15.12.2) see Wagner (1988). The region of
validity given in Luke (1969a, p. 235) is incorrect.
For (15.12.3) see Luke (1969a, x7.2) and Olver
(1997b, p. 162). The sector of validity given in
the rst reference is incorrect. See the third foot-
note in the second reference.
x15.13 Runckel (1971).
x15.14 Andrews et al. (1999,x2.4).
x15.16 Burchnall and Chaundy (1940, 1948) and El-
liott (1903). For (15.16.4) use (15.8.2) and
(15.8.4), combined with (15.8.1) to show that the
left-hand side of (15.16.4) is an entire function of
z. Then apply Liouville's theorem (1.9(iii)).
Chapter 16
Generalized Hypergeometric Functions and Meijer
G-Function
R. A. Askey1and A. B. Olde Daalhuis2
Notation 404
16.1 Special Notation . . . . . . . . . . . . . 404
Generalized Hypergeometric Functions 404
16.2 De nition and Analytic Properties . . . . 404
16.3 Derivatives and Contiguous Functions . . 405
16.4 Argument Unity . . . . . . . . . . . . . . 405
16.5 Integral Representations and Integrals . . 408
16.6 Transformations of Variable . . . . . . . . 408
16.7 Relations to Other Functions . . . . . . . 409
16.8 Di erential Equations . . . . . . . . . . . 409
16.9 Zeros . . . . . . . . . . . . . . . . . . . 410
16.10 Expansions in Series of pFqFunctions . . 410
16.11 Asymptotic Expansions . . . . . . . . . . 411
16.12 Products . . . . . . . . . . . . . . . . . . 412
Two-Variable Hypergeometric Functions 412
16.13 Appell Functions . . . . . . . . . . . . . 412
16.14 Partial Di erential Equations . . . . . . . 413
16.15 Integral Representations and Integrals . . 41416.16 Transformations of Variables . . . . . . . 414
MeijerG-Function 415
16.17 De nition . . . . . . . . . . . . . . . . . 415
16.18 Special Cases . . . . . . . . . . . . . . . 416
16.19 Identities . . . . . . . . . . . . . . . . . 416
16.20 Integrals and Series . . . . . . . . . . . . 416
16.21 Di erential Equation . . . . . . . . . . . 417
16.22 Asymptotic Expansions . . . . . . . . . . 417
Applications 417
16.23 Mathematical Applications . . . . . . . . 417
16.24 Physical Applications . . . . . . . . . . . 417
Computation 418
16.25 Methods of Computation . . . . . . . . . 418
16.26 Approximations . . . . . . . . . . . . . . 418
16.27 Software . . . . . . . . . . . . . . . . . . 418
References 418
1Department of Mathematics, University of Wisconsin, Madison, Wisconsin.
2School of Mathematics, Edinburgh University, Edinburgh, United Kingdom.
Acknowledgments : The authors are pleased to acknowledge the assistance of B. L. J. Braaksma with xx16.5 and 16.11.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
403
404 Generalized Hypergeometric Functions and Meijer G-Function
Notation
16.1 Special Notation
(For other notation see pp. xiv and 873.)
p;q nonnegative integers.
k;n nonnegative integers, unless
stated otherwise.
z complex variable.
a1;a2;:::;ap
b1;b2;:::;bq
real or complex parameters.
arbitrary small positive constant.
a vector (a1;a2;:::;ap).
b vector (b1;b2;:::;bq).
(a)k (a1)k(a2)k(ap)k.
(b)k(b1)k(b2)k(bq)k.
D d /dz.
# z d /dz.
The main functions treated in this chap-
ter are the generalized hypergeometric function
pFq
a1;:::;ap
b1;:::;bq;z
, the Appell (two-variable hypergeomet-
ric) functions F1( ; ; 0;
;x;y),F2( ; ; 0;
;
0;x;y),
F3( ; 0; ; 0;
;x;y),F4( ; ;
;
0;x;y), and the Mei-
jerG-functionGm;n
p;q
z;a1;:::;ap
b1;:::;bq
. Alternative no-
tations are pFq a
b;z
,pFq(a1;:::;ap;b1;:::;bq;z),
andpFq(a;b;z) for the generalized hypergeometric
function, F1( ; ; 0;
;x;y),F2( ; ; 0;
;
0;x;y),
F3( ; 0; ; 0;
;x;y),F4( ; ;
;
0;x;y), for the Ap-
pell functions, and Gm;n
p;q(z;a;b) for the Meijer G-
function.
Generalized Hypergeometric
Functions
16.2 De nition and Analytic Properties
16.2(i) Generalized Hypergeometric Series
Throughout this chapter it is assumed that none of the
bottom parameters b1,b2,:::,bqis a nonpositive inte-
ger, unless stated otherwise. Then formally
16.2.1pFqa1;:::;ap
b1;:::;bq;z
=1X
k=0(a1)k(ap)k
(b1)k(bq)kzk
k!:
Equivalently, the function is denoted by pFq a
b;z
or
pFq(a;b;z), and sometimes, for brevity, by pFq(z).
16.2(ii) Case pq
Whenpqthe series (16.2.1) converges for all nite
values ofzand de nes an entire function.16.2(iii) Case p=q+ 1
Suppose rst one or more of the top parameters ajis
a nonpositive integer. Then the series (16.2.1) termi-
nates and the generalized hypergeometric function is a
polynomial in z.
If none of the ajis a nonpositive integer, then the
radius of convergence of the series (16.2.1) is 1, and
outside the open disk jzj<1 the generalized hyper-
geometric function is de ned by analytic continuation
with respect to z. The branch obtained by introducing
a cut from 1 to + 1on the real axis, that is, the branch
in the sectorjph(1 z)j, is the principal branch
(orprincipal value ) ofq+1Fq(a;b;z); comparex4.2(i).
Elsewhere the generalized hypergeometric function is a
multivalued function that is analytic except for possible
branch points at z= 0;1, and1.Unless indicated oth-
erwise it is assumed that in this Handbook generalized
hypergeometric functions assume their principal values.
On the circlejzj= 1 the series (16.2.1) is absolutely
convergent if<
q>0, convergent except at z= 1 if
1<<
q0, and divergent if <
q 1, where
16.2.2
q= (b1++bq) (a1++aq+1):
16.2(iv) Case p>q + 1
Polynomials
In general the series (16.2.1) diverges for all nonzero
values ofz. However, when one or more of the top
parameters ajis a nonpositive integer the series termi-
nates and the generalized hypergeometric function is a
polynomial in z. Note that if mis the value of the nu-
merically largest ajthat is a nonpositive integer, then
the identity
16.2.3
p+1Fq m;a
b;z
=(a)m( z)m
(b)mq+1Fp m;1 m b
1 m a;( 1)p+q
z
can be used to interchange pandq.
Note also that any partial sum of the generalized hy-
pergeometric series can be represented as a generalized
hypergeometric function via
16.2.4
mX
k=0(a)k
(b)kzk
k!
=(a)mzm
(b)mm!q+2Fp m;1;1 m b
1 m a;( 1)p+q+1
z
:
Non-Polynomials
Seex16.5 for the de nition of pFq(a;b;z) as a contour
integral when p>q +1 and none of the akis a nonposi-
tive integer. (However, except where indicated otherwise
in this Handbook we assume that when p>q +1 at least
one of theakis a nonpositive integer.)
16.3 Derivatives and Contiguous Functions 405
16.2(v) Behavior with Respect to Parameters
Let
16.2.5 pFq(a;b;z) =pFqa1;:::;ap
b1;:::;bq;z
( (b1) (bq)) =1X
k=0(a1)k(ap)k
(b1+k) (bq+k)zk
k!;
compare (15.2.2) in the case p= 2,q= 1. When pq+ 1 andzis xed and not a branch point, any branch of
pFq(a;b;z) is an entire function of each of the parameters a1;:::;ap;b1;:::;bq.
16.3 Derivatives and Contiguous Functions
16.3(i) Di erentiation Formulas
16.3.1dn
dznpFqa1;:::;ap
b1;:::;bq;z
=(a)n
(b)npFqa1+n;:::;ap+n
b1+n;:::;bq+n;z
;
16.3.2dn
dzn
z
pFqa1;:::;ap
b1;:::;bq;z
= (
n+ 1)nz
n
p+1Fq+1
+ 1;a1;:::;ap
+ 1 n;b1;:::;bq;z
;
16.3.3
zd
dzzn
z
1
p+1Fq
;a1;:::;ap
b1;:::;bq;z
= (
)nz
+n 1
p+1Fq
+n;a1;:::;ap
b1;:::;bq;z
;
16.3.4dn
dzn
z
1
pFq+1a1;:::;ap
;b1;:::;bq;z
= (
n)nz
n 1
pFq+1a1;:::;ap
n;b1;:::;bq;z
:
Other versions of these identities can be constructed with the aid of the operator identity
16.3.5
zd
dzzn
=zndn
dznzn, n= 1;2;:::.
16.3(ii) Contiguous Functions
Two generalized hypergeometric functions pFq(a;b;z) are (generalized) contiguous if they have the same pair of
values ofpandq, and corresponding parameters di er by integers. If pq+ 1, then any q+ 2 distinct contiguous
functions are linearly related. Examples are provided by the following recurrence relations:
16.3.6 z0F1( ;b+ 1;z) +b(b 1)0F1( ;b;z) b(b 1)0F1( ;b 1;z) = 0;
16.3.73F2a1+ 2;a2;a3
b1;b2;z
a1(a1+1)(1 z)+3F2a1+ 1;a2;a3
b1;b2;z
a1(b1+b2 3a1 2+z(2a1 a2 a3+1))
+3F2a1;a2;a3
b1;b2;z
(2a1 b1)(2a1 b2) +a1 a2
1 z(a1 a2)(a1 a3)
3F2a1 1;a2;a3
b1;b2;z
(a1 b1)(a1 b2) = 0:
For further examples see xx13.3(i), 15.5(ii), and the following references: Rainville (1960, x48), Wimp (1968), and
Luke (1975,x5.13).
16.4 Argument Unity
16.4(i) Classi cation
The function q+1Fq(a;b;z) iswell-poised if
16.4.1 a1+b1==aq+bq=aq+1+ 1:
It is very well-poised if it is well-poised and a1=b1+ 1.
The special case q+1Fq(a;b; 1) isk-balanced ifaq+1is a nonpositive integer and
16.4.2 a1++aq+1+k=b1++bq:
Whenk= 1 the function is said to be balanced orSaalsch utzian .
406 Generalized Hypergeometric Functions and Meijer G-Function
16.4(ii) Examples
The function q+1Fqwith argument unity and general values of the parameters is discussed in B uhring (1992). Special
cases are as follows:
Pfa {Saalsch utz Balanced Sum
16.4.33F2 n;a;b
c;d; 1
=(c a)n(c b)n
(c)n(c a b)n;
whenc+d=a+b+ 1 n,n= 0;1;:::. See Erd elyi et al. (1953a,x4.4(4)) for a non-terminating balanced identity.
Dixon's Well-Poised Sum
16.4.43F2a;b;c
a b+ 1;a c+ 1; 1
= 1
2a+ 1
(a b+ 1) (a c+ 1) 1
2a b c+ 1
(a+ 1) 1
2a b+ 1
1
2a c+ 1
(a b c+ 1);
when<(a 2b 2c)> 2, or when the series terminates with a= n:
16.4.53F2 n;b;c
1 b n;1 c n; 1
=8
<
:0; n = 2k+ 1;
(2k)! (b+k) (c+k) (b+c+ 2k)
k! (b+ 2k) (c+ 2k) (b+c+k); n = 2k;
wherek= 0;1;:::.
Watson's Sum
16.4.63F2a;b;c
1
2(a+b+ 1);2c; 1
= 1
2
c+1
2
1
2(a+b+ 1)
c+1
2(1 a b)
1
2(a+ 1)
1
2(b+ 1)
c+1
2(1 a)
c+1
2(1 b);
when<(2c a b)> 1, or when the series terminates with a= n.
Whipple's Sum
16.4.73F2a;1 a;c
d;2c d+ 1; 1
= (d) (2c d+ 1)21 2c
c+1
2(a d+ 1)
c+ 1 1
2(a+d)
1
2(a+d)
1
2(d a+ 1);
when<c>0 or whenais an integer.
D zrbasjan's Sum
This is (16.4.7) in the case c= n:
16.4.83F2 n;a;1 a
d;1 d 2n; 1
= 1
2(a+d)
n 1
2(d a+ 1)
n 1
2d
n 1
2(d+ 1)
n, n= 0;1;:::.
Rogers{Dougall Very Well-Poised Sum
16.4.95F4
a;1
2a+ 1;b;c;d
1
2a;a b+ 1;a c+ 1;a d+ 1; 1!
= (a b+ 1) (a c+ 1) (a d+ 1) (a b c d+ 1)
(a+ 1) (a b c+ 1) (a b d+ 1) (a c d+ 1);
when<(b+c+d a)<1, or when the series terminates with d= n.
Dougall's Very Well-Poised Sum
16.4.107F6
a;1
2a+ 1;b;c;d;f; n
1
2a;a b+ 1;a c+ 1;a d+ 1;a f+ 1;a+n+ 1; 1!
=(a+ 1)n(a b c+ 1)n(a b d+ 1)n(a c d+ 1)n
(a b+ 1)n(a c+ 1)n(a d+ 1)n(a b c d+ 1)n, n= 0;1;:::,
when 2a+ 1 =b+c+d+f n. The last condition is equivalent to the sum of the top parameters plus 2 equals the
sum of the bottom parameters, that is, the series is 2-balanced.
16.4 Argument Unity 407
16.4(iii) Identities
16.4.11 3F2a;b;c
d;e; 1
= (e) (d+e a b c)
(e a) (d+e b c)3F2a;d b;d c
d;d+e b c; 1
;
when<(d+e a b c)>0 and<(e a)>0. The function 3F2(a;b;c ;d;e; 1) is analytic in the parameters
a;b;c;d;e when its series expansion converges and the bottom parameters are not negative integers or zero. (16.4.11)
provides a partial analytic continuation to the region when the only restrictions on the parameters are <(e a)>0,
andd;e, andd+e b c6= 0; 1;:::. A detailed treatment of analytic continuation in (16.4.11) and asymptotic
approximations as the variables a;b;c;d;e approach in nity is given by Aomoto (1987).
There are two types of three-term identities for 3F2's. The rst are recurrence relations that extend those for
2F1's; seex15.5(ii). Examples are (16.3.7) with z= 1. Also,
16.4.12(a d)(b d)(c d)
3F2a;b;c
d+ 1;e; 1
3F2a;b;c
d;e; 1
+abc3F2a;b;c
d;e; 1
=d(d 1)(a+b+c d e+ 1)
3F2a;b;c
d;e; 1
3F2a;b;c
d 1;e; 1
;
and
16.4.13 3F2a;b;c
d;e; 1
=c(e a)
de3F2a;b+ 1;c+ 1
d+ 1;e+ 1; 1
+d c
d3F2a;b+ 1;c
d+ 1;e; 1
:
Methods of deriving such identities are given by Bailey (1964), Rainville (1960), Raynal (1979), and Wilson (1978).
Lists are given by Raynal (1979) and Wilson (1978). See Raynal (1979) for a statement in terms of 3 jsymbols
(Chapter 34). Also see Wilf and Zeilberger (1992a,b) for information on the Wilf{Zeilberger algorithm which can be
used to nd such relations.
The other three-term relations are extensions of Kummer's relations for 2F1's given inx15.10(ii). See Bailey
(1964, pp. 19{22).
Balanced 4F3(1) series have transformation formulas and three-term relations. The basic transformation is given
by
16.4.14 4F3 n;a;b;c
d;e;f; 1
=(e a)n(f a)n
(e)n(f)n4F3 n;a;d b;d c
d;a e n+ 1;a f n+ 1; 1
;
whena+b+c n+ 1 =d+e+f. These series contain 6 jsymbols as special cases when the parameters are integers;
comparex34.4.
The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson
class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs.
Contiguous balanced series have parameters shifted by an integer but still balanced. One example of such a
three-term relation is the recurrence relation (18.26.16) for Racah polynomials. See Raynal (1979), Wilson (1978),
and Bailey (1964).
A di erent type of transformation is that of Whipple:
16.4.157F6
a;1
2a+ 1;b;c;d;e;f
1
2a;a b+ 1;a c+ 1;a d+ 1;a e+ 1;a f+ 1; 1!
= (a d+ 1) (a e+ 1) (a f+ 1) (a d e f+ 1)
(a+ 1) (a d e+ 1) (a d f+ 1) (a e f+ 1)4F3a b c+ 1;d;e;f
a b+ 1;a c+ 1;d+e+f a; 1
;
when the series on the right terminates and the series on the left converges. When the series on the right does not
terminate, a second term appears. See Bailey (1964, x4.4(4)).
Transformations for both balanced 4F3(1) and very well-poised 7F6(1) are included in Bailey (1964, pp. 56{63). A
similar theory is available for very well-poised 9F8(1)'s which are 2-balanced. See Bailey (1964, xx4.3(7) and 7.6(1))
for the transformation formulas and Wilson (1978) for contiguous relations.
Relations between three solutions of three-term recurrence relations are given by Masson (1991). See also
Lewanowicz (1985) (with corrections in Lewanowicz (1987)) for further examples of recurrence relations.
16.4(iv) Continued Fractions
For continued fractions for ratios of 3F2functions with argument unity, see Cuyt et al. (2008, pp. 315{317).
408 Generalized Hypergeometric Functions and Meijer G-Function
16.4(v) Bilateral Series
Denote, formally, the bilateral hypergeometric function
16.4.16 pHqa1;:::;ap
b1;:::;bq;z
=1X
k= 1(a1)k:::(ap)k
(b1)k:::(bq)kzk:
Then
16.4.17 2H2a;b
c;d; 1
= (c) (d) (1 a) (1 b) (c+d a b 1)
(c a) (d a) (c b) (d b),<(c+d a b)>1.
This is Dougall's bilateral sum ; see Andrews et al. (1999,x2.8).
16.5 Integral Representations and Integrals
Whenz6= 0 andak6= 0; 1; 2;:::,k= 1;2;:::;p ,
16.5.1pQ
k=1 (ak) qQ
k=1 (bk)
pFqa1;:::;ap
b1;:::;bq;z
=1
2iZ
LpQ
k=1 (ak+s) qQ
k=1 (bk+s)
( s)( z)sds;
where the contour of integration separates the poles of ( ak+s),k= 1;:::;p , from those of ( s).
Suppose rst that Lis a contour that starts at in nity on a line parallel to the positive real axis, encircles the
nonnegative integers in the negative sense, and ends at in nity on another line parallel to the positive real axis. Then
the integral converges when p<q + 1 provided that z6= 0, or when p=q+ 1 provided that 0 <jzj<1, and provides
an integral representation of the left-hand side with these conditions.
Secondly, suppose that Lis a contour from i1toi1. Then the integral converges when q < p + 1 and
jph( z)j<(p+ 1 q)=2. In the case p=qthe left-hand side of (16.5.1) is an entire function, and the right-hand
side supplies an integral representation valid when jph( z)j< = 2. In the case p=q+ 1 the right-hand side of
(16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector jph(1 z)j<;
comparex16.2(iii). Lastly, when p > q + 1 the right-hand side of (16.5.1) can be regarded as the de nition of the
(customarily unde ned) left-hand side. In this event, the formal power-series expansion of the left-hand side (obtained
from (16.2.1)) is the asymptotic expansion of the right-hand side as z!0 in the sectorjph( z)j(p+1 q )=2,
whereis an arbitrary small positive constant.
Next, when pq,
16.5.2p+1Fq+1a0;:::;ap
b0;:::;bq;z
= (b0)
(a0) (b0 a0)Z1
0ta0 1(1 t)b0 a0 1
pFqa1;:::;ap
b1;:::;bq;zt
dt,<b0><a0>0,
16.5.3 p+1Fqa0;:::;ap
b1;:::;bq;z
=1
(a0)Z1
0e tta0 1
pFqa1;:::;ap
b1;;bq;zt
dt,<z<1,<a0>0,
16.5.4 pFq+1a1;:::;ap
b0;:::;bq;z
= (b0)
2iZc+i1
c i1ett b0pFqa1;:::;ap
b1;:::;bq;z
t
dt,c>0,<b0>0.
In (16.5.2){(16.5.4) all many-valued functions in the integrands assume their principal values, and all integration
paths are straight lines.
(16.5.2) also holds when p=q+ 1, provided that jph(1 z)j< . In (16.5.3) the restriction <z < 1 can be
removed when p<q . (16.5.4) also holds when p=q+ 1, provided that max(0 ;<z)<c. Lastly, the restrictions on
the parameters can be eased by replacing the integration paths with loop contours; see Luke (1969a, x3.6).
Laplace transforms and inverse Laplace transforms of generalized hypergeometric functions are given in Prudnikov
et al. (1992a,x3.38) and Prudnikov et al. (1992b,x3.36). For further integral representations and integrals see
Apelblat (1983,x16), Erd elyi et al. (1953a,x4.6), Erd elyi et al. (1954a,xx6.9 and 7.5), Luke (1969a, x3.6), and
Prudnikov et al. (1990,xx2.22, 4.2.4, and 4.3.1).
16.6 Transformations of Variable
Quadratic
16.6.1 3F2a;b;c
a b+ 1;a c+ 1;z
= (1 z) a
3F2a b c+ 1;1
2a;1
2(a+ 1)
a b+ 1;a c+ 1; 4z
(1 z)2
:
16.7 Relations to Other Functions 409
Cubic
16.6.2 3F2a;2b a 1;2 2b+a
b;a b+3
2;z
4
= (1 z) a
3F2
1
3a;1
3a+1
3;1
3a+2
3
b;a b+3
2; 27z
4(1 z)3!
:
For Kummer-type transformations of 2F2functions see Miller (2003) and Paris (2005a), and for further transfor-
mations see Erd elyi et al. (1953a,x4.5).
16.7 Relations to Other Functions
For orthogonal polynomials see Chapter 18. For 3 j, 6j, 9jsymbols see Chapter 34. Further representations of special
functions in terms of pFqfunctions are given in Luke (1969a, xx6.2{6.3), and an extensive list of q+1Fqfunctions
with rational numbers as parameters is given in Krupnikov and K olbig (1997).
16.8 Di erential Equations
16.8(i) Classi cation of Singularities
Anordinary point of the di erential equation
16.8.1dnw
dzn+fn 1(z)dn 1w
dzn 1+fn 2(z)dn 2w
dzn 2++f1(z)dw
dz+f0(z)w= 0
is a valuez0ofzat which all the coecients fj(z),j= 0;1;:::;n 1, are analytic. If z0is not an ordinary point but
(z z0)n jfj(z),j= 0;1;:::;n 1, are analytic at z=z0, thenz0is aregular singularity . All other singularities
areirregular . Comparex2.7(i) in the case n= 2. Similar de nitions apply in the case z0=1: we transform1into
the origin by replacing zin (16.8.1) by 1 =z; again comparex2.7(i).
For further information see Hille (1976, pp. 360{370).
16.8(ii) The Generalized Hypergeometric Di erential Equation
With the notation
16.8.2 D=d
dz; # =zd
dz;
the function w=pFq(a;b;z) satis es the di erential equation
16.8.3 (#(#+b1 1)(#+bq 1) z(#+a1)(#+ap))w= 0:
Equivalently,
16.8.4 zqDq+1w+qX
j=1zj 1( jz+ j)Djw+ 0w= 0, pq;
or
16.8.5 zq(1 z)Dq+1w+qX
j=1zj 1( jz+ j)Djw+ 0w= 0, p=q+ 1;
where jand jare constants. Equation (16.8.4) has a regular singularity at z= 0, and an irregular singularity at
z=1, whereas (16.8.5) has regular singularities at z= 0, 1, and1. In each case there are no other singularities.
Equation (16.8.3) is of order max( p;q+ 1). In Letessier et al. (1994) examples are discussed in which the generalized
hypergeometric function satis es a di erential equation that is of order 1 or even 2 less than might be expected.
When nobjis an integer, and no two bjdi er by an integer, a fundamental set of solutions of (16.8.3) is given
by
16.8.6w0(z) =pFqa1;:::;ap
b1;:::;bq;z
; wj(z) =z1 bjpFq1 +a1 bj;:::; 1 +ap bj
2 bj;1 +b1 bj;::::::;1 +bq bj;z
,j= 1;:::;q ,
whereindicates that the entry 1 + bj bjis omitted. For other values of the bj, series solutions in powers of z
(possibly involving also ln z) can be constructed via a limiting process; compare x2.7(i) in the case of second-order
di erential equations. For details see Smith (1939a,b), and N rlund (1955).
Whenp=q+ 1, and no two ajdi er by an integer, another fundamental set of solutions of (16.8.3) is given by
16.8.7 ewj(z) = ( z) ajq+1Fqaj;1 b1+aj;:::; 1 bq+aj
1 a1+aj;::::::;1 aq+1+aj;1
z
,j= 1;:::;q + 1,
410 Generalized Hypergeometric Functions and Meijer G-Function
whereindicates that the entry 1 aj+ajis omitted. We have the connection formula
16.8.8 q+1Fqa1;:::;aq+1
b1;:::;bq;z
=q+1X
j=10
@q+1Q
k=1
k6=j (ak aj)
(ak),
qQ
k=1 (bk aj)
(bk)1
Aewj(z),jph( z)j:
More generally if z0(2C) is an arbitrary constant, jz z0j>max (jz0j;jz0 1j), andjph(z0 z)j<, then
16.8.9q+1Q
k=1 (ak) qQ
k=1 (bk)
q+1Fqa1;:::;aq+1
b1;:::;bq;z
=q+1X
j=1(z0 z) aj1X
n=0 (aj+n)
n!0
@q+1Q
k=1
k6=j (ak aj n),
qQ
k=1 (bk aj n)1
A
q+1Fqa1 aj n;:::;aq+1 aj n
b1 aj n;:::;bq aj n;z0
(z z0) n:
(Note that the generalized hypergeometric functions on the right-hand side are polynomials in z0.)
Whenp=q+ 1 and some of the ajdi er by an integer a limiting process can again be applied. For details see
N rlund (1955). In this reference it is also explained that in general when q>1 no simple representations in terms of
generalized hypergeometric functions are available for the fundamental solutions near z= 1. Analytical continuation
formulas for q+1Fq(a;b;z) nearz= 1 are given in B uhring (1987b) for the case q= 2, and in B uhring (1992) for the
general case.
16.8(iii) Con
uence of Singularities
Ifpq, then
16.8.10 lim
j j!1p+1Fqa1;:::;ap;
b1;:::;bq;z
=pFqa1;:::;ap
b1;:::;bq;z
:
Thus in the case p=qthe regular singularities of the function on the left-hand side at and1coalesce into an
irregular singularity at 1.
Next, ifpq+ 1 andjph j (<), then
16.8.11 lim
j j!1pFq+1a1;:::;ap
b1;:::;bq; ; z
=pFqa1;:::;ap
b1;:::;bq;z
;
provided that in the case p=q+1 we havejzj<1 whenjph j1
2, andjzj<jsin(ph )jwhen1
2jph j
(<).
16.9 Zeros
Assume that p=qand none of the ajis a nonpositive integer. Then pFp(a;b;z) has at most nitely many zeros if
and only if the ajcan be re-indexed for j= 1;:::;p in such a way that aj bjis a nonnegative integer.
Next, assume that p=qand that the ajand the quotients ( a)j=(b)jare all real. Then pFp(a;b;z) has at most
nitely many real zeros.
These results are proved in Ki and Kim (2000). For further information on zeros see Hille (1929).
16.10 Expansions in Series of pFqFunctions
The following expansion, with appropriate conditions and together with similar results, is given in Fields and Wimp
(1961):
16.10.1p+rFq+sa1;:::;ap;c1;:::;cr
b1;:::;bq;d1;:::;ds;z
=1X
k=0(a)k( )k( )k( z)k
(b)k(
+k)kk!p+2Fq+1 +k; +k;a1+k;:::;ap+k
+ 2k+ 1;b1+k;:::;bq+k;z
r+2Fs+2 k;
+k;c1;:::;cr
; ;d 1;:::;ds;
:
Here , , and
are free real or complex parameters.
16.11 Asymptotic Expansions 411
The next expansion is given in N rlund (1955, equation (1.21)):
16.10.2 p+1Fpa1;:::;ap+1
b1;:::;bp;z
= (1 z) a11X
k=0(a1)k
k!p+1Fp k;a2;:::;ap+1
b1;:::;bp;z
z 1k
:
Whenj 1j<1 the series on the right-hand side converges in the half-plane <z<1
2.
Expansions of the formP1
n=1(1)npFp+1
a;b; n2z2
are discussed in Miller (1997), and further series of gener-
alized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, xx5.10.2 and 5.11), and Prudnikov
et al. (1990,xx5.3, 6.8{6.9).
16.11 Asymptotic Expansions
16.11(i) Formal Series
For subsequent use we de ne two formal in nite series, Ep;q(z) andHp;q(z), as follows:
16.11.1 Ep;q(z) = (2)(p q)/2 ( 1/2 )ez1/1X
k=0ck
z1/ k
, p<q + 1,
16.11.2 Hp;q(z) =pX
m=11X
k=0( 1)k
k! (am+k)0
@pQ
`=1
`6=m (a` am k),
qQ
`=1 (b` am k)1
Az am k:
In (16.11.1)
16.11.3 =q p+ 1; =a1++ap b1 bq+1
2(q p);
and
16.11.4 c0= 1; ck= 1
kk 1X
m=0cmek;m, k1,
where
16.11.5 ek;m=q+1X
j=1(1 bj+m)+k m0
@pQ
`=1(a` bj),
q+1Q
`=1
`6=j(b` bj)1
A;
andbq+1= 1.
It may be observed that Hp;q(z) represents the sum of the residues of the poles of the integrand in (16.5.1) at
s= aj; aj 1;:::,j= 1;:::;p , provided that these poles are all simple, that is, no two of the ajdi er by an
integer. (If this condition is violated, then the de nition of Hp;q(z) has to be modi ed so that the residues are those
associated with the multiple poles. In consequence, logarithmic terms may appear. See (15.8.8) for an example.)
16.11(ii) Expansions for Large Variable
In this subsection we assume that none of a1;a2;:::;apis a nonpositive integer.
Casep=q+ 1
The formal series (16.11.2) for Hq+1;q(z) converges ifjzj>1, and
16.11.6q+1Q
`=1 (a`) qQ
`=1 (b`)
q+1Fqa1;:::;aq+1
b1;:::;bq;z
=Hq+1;q( z),jph( z)j;
compare (16.8.8).
Casep=q
Asz!1 injphzj,
16.11.7qQ
`=1 (a`) qQ
`=1 (b`)
qFqa1;:::;aq
b1;:::;bq;z
Hq;q(zei) +Eq;q(z);
where upper or lower signs are chosen according as zlies in the upper or lower half-plane. (Either sign may be used
when phz= 0 since the rst term on the right-hand side becomes exponentially small compared with the second
term.)
For the special case a1= 1,p=q= 2 explicit representations for the right-hand side of (16.11.7) in terms of
generalized hypergeometric functions are given in Kim (1972).
412 Generalized Hypergeometric Functions and Meijer G-Function
Casep=q 1
Asz!1 injphzj,
16.11.8q 1Q
`=1 (a`) qQ
`=1 (b`)
q 1Fqa1;:::;aq 1
b1;:::;bq; z
Hq 1;q(z) +Eq 1;q(ze i) +Eq 1;q(zei):
Casepq 2
Asz!1 injphzj,
16.11.9pQ
`=1 (a`) qQ
`=1 (b`)
pFqa1;:::;ap
b1;:::;bq; z
Ep;q(ze i) +Ep;q(zei):
16.11(iii) Expansions for Large Parameters
Ifzis xed andjph(1 z)j<, then for each nonnegative integer m
16.11.10
p+1Fpa1+r;:::;ak 1+r;ak;:::;ap+1
b1+r;:::;bk+r;bk+1;:::;bp;z
=m 1X
n=0(a1+r)n(ak 1+r)n(ak)n(ap+1)n
(b1+r)n(bk+r)n(bk+1)n(bp)nzn
n!+O1
rm
;
asr!+1. Herekcan have any integer value from 1 to p. Also ifp<q , then
16.11.11pFqa1+r;:::;ap+r
b1+r;:::;bq+r;z
=m 1X
n=0(a1+r)n(ap+r)n
(b1+r)n(bq+r)nzn
n!+O1
r(q p)m
;
again asr!+1. For these and other results see Knottnerus (1960). See also Luke (1969a, x7.3).
Asymptotic expansions for the polynomials p+2Fq( r;r+a0;a;b;z) asr!1 through integer values are given
in Fields and Luke (1963a,b) and Fields (1965).
16.12 Products
16.12.10F1( ;a;z)0F1( ;b;z) = 2F31
2(a+b);1
2(a+b 1)
a;b;a +b 1; 4z
:
16.12.2
2F1a;b
a+b+1
2;z2
=3F22a;2b;a+b
a+b+1
2;2a+ 2b;z
:
More generally,
16.12.3
2F1a;b
c;z2
=1X
k=0(2a)k(2b)k
c 1
2
k
(c)k(2c 1)kk!4F3
1
2k;1
2(1 k);a+b c+1
2;1
2
a+1
2;b+1
2;3
2 k c; 1!
zk,jzj<1.
For further identities see Goursat (1883) and Erd elyi et al. (1953a,x4.3).
Two-Variable Hypergeometric Functions
16.13 Appell Functions
The following four functions of two real or complex variables xandycannot be expressed as a product of two 2F1
functions, in general, but they satisfy partial di erential equations that resemble the hypergeometric di erential
16.14 Partial Differential Equations 413
equation (15.10.1):
16.13.1 F1( ; ; 0;
;x;y) =1X
m;n=0( )m+n( )m( 0)n
(
)m+nm!n!xmyn, max ( jxj;jyj)<1,
16.13.2 F2( ; ; 0;
;
0;x;y) =1X
m;n=0( )m+n( )m( 0)n
(
)m(
0)nm!n!xmyn, jxj+jyj<1,
16.13.3 F3( ; 0; ; 0;
;x;y) =1X
m;n=0( )m( 0)n( )m( 0)n
(
)m+nm!n!xmyn, max ( jxj;jyj)<1,
16.13.4 F4( ; ;
;
0;x;y) =1X
m;n=0( )m+n( )m+n
(
)m(
0)nm!n!xmyn,p
jxj+p
jyj<1.
Here and elsewhere it is assumed that neither of the bottom parameters
and
0is a nonpositive integer.
16.14 Partial Di erential Equations
16.14(i) Appell Functions
16.14.1x(1 x)@2F1
@x2+y(1 x)@2F1
@x@y+ (
( + + 1)x)@F1
@x y@F1
@y F 1= 0;
y(1 y)@2F1
@y2+x(1 y)@2F1
@x@y+ (
( + 0+ 1)y)@F1
@y 0x@F1
@x 0F1= 0;
16.14.2x(1 x)@2F2
@x2 xy@2F2
@x@y+ (
( + + 1)x)@F2
@x y@F2
@y F 2= 0;
y(1 y)@2F2
@y2 xy@2F2
@x@y+ (
0 ( + 0+ 1)y)@F2
@y 0x@F2
@x 0F2= 0;
16.14.3x(1 x)@2F3
@x2+y@2F3
@x@y+ (
( + + 1)x)@F3
@x F 3= 0;
y(1 y)@2F3
@y2+x@2F3
@x@y+ (
( 0+ 0+ 1)y)@F3
@y 0 0F3= 0;
16.14.4x(1 x)@2F4
@x2 2xy@2F4
@x@y y2@2F4
@y2+ (
( + + 1)x)@F4
@x ( + + 1)y@F4
@y F 4= 0;
y(1 y)@2F4
@y2 2xy@2F4
@x@y x2@2F4
@x2+ (
0 ( + + 1)y)@F4
@y ( + + 1)x@F4
@x F 4= 0:
16.14(ii) Other Functions
In addition to the four Appell functions there are 24 other sums of double series that cannot be expressed as a
product of two 2F1functions, and which satisfy pairs of linear partial di erential equations of the second order. Two
examples are provided by
16.14.5G2( ; 0; ; 0;x;y) =1X
m;n=0 ( +m) ( 0+n) ( +n m) ( 0+m n)
( ) ( 0) ( ) ( 0)xmyn
m!n!,jxj<1,jyj<1,
16.14.6 G3( ; 0;x;y) =1X
m;n=0 ( + 2n m) ( 0+ 2m n)
( ) ( 0)xmyn
m!n!, jxj+jyj<1
4.
(The region of convergence jxj+jyj<1
4is not quite maximal.) See Erd elyi et al. (1953a,xx5.7.1{5.7.2) for further
information.
414 Generalized Hypergeometric Functions and Meijer G-Function
16.15 Integral Representations and Integrals
16.15.1 F1( ; ; 0;
;x;y) = (
)
( ) (
)Z1
0u 1(1 u)
1
(1 ux) (1 uy) 0du,< >0,<(
)>0,
16.15.2F2( ; ; 0;
;
0;x;y) = (
) (
0)
( ) ( 0) (
) (
0 0)Z1
0Z1
0u 1v 0 1(1 u)
1(1 v)
0 0 1
(1 ux vy) dudv ,
<
>< >0,<
0>< 0>0,
16.15.3F3( ; 0; ; 0;
;x;y) = (
)
( ) ( 0) (
0)ZZ
u 1v 0 1(1 u v)
0 1
(1 ux) (1 vy) 0dudv ,
<(
0)>0,< >0,< 0>0,
where is the triangle de ned by u0,v0,u+v1.
16.15.4F4( ; ;
;
0;x(1 y);y(1 x))
= (
) (
0)
( ) ( ) (
) (
0 )Z1
0Z1
0u 1v 1(1 u)
1(1 v)
0 1
(1 ux)
+
0 1(1 vy)
+
0 1(1 ux vy) +
0+1dudv ,
<
>< >0,<
0>< >0.
For these and other formulas, including double Mellin{Barnes integrals, see Erd elyi et al. (1953a,x5.8). These
representations can be used to derive analytic continuations of the Appell functions, including convergent series
expansions for large x, largey, or both. For inverse Laplace transforms of Appell functions see Prudnikov et al.
(1992b,x3.40).
16.16 Transformations of Variables
16.16(i) Reduction Formulas
16.16.1 F1( ; ; 0; + 0;x;y) = (1 y)
2F1 ;
+ 0;x y
1 y
;
16.16.2 F2( ; ; 0;
; 0;x;y) = (1 y)
2F1 ;
;x
1 y
;
16.16.3 F2( ; ; 0;
; ;x;y) = (1 y) 0F1
; 0; 0;
;x;x
1 y
;
16.16.4 F3( ;
; ; 0;
;x;y) = (1 y) 0F1
; ; 0;
;x;y
y 1
;
16.16.5 F3( ;
; ;
;
;x;y) = (1 y) +
2F1 ;
;x+y xy
;
16.16.6 F4( ; ;
; +
+ 1;x(1 y);y(1 x)) = 2F1 ;
;x
2F1 ;
+
+ 1;y
:
See Erd elyi et al. (1953a,x5.10) for these and further reduction formulas. An extension of (16.16.6) is given by
16.16.7F4( ; ;
;
0;x(1 y);y(1 x))
=1X
k=0( )k( )k( +
0+ 1)k
(
)k(
0)kk!xkyk
2F1 +k; +k
+k;x
2F1 +k; +k
0+k;y
;
see Burchnall and Chaundy (1940, 1941).
16.16(ii) Other Transformations
16.16.8F1( ; ; 0;
;x;y) = (1 x) (1 y) 0F1
; ; 0;
;x
x 1;y
y 1
= (1 x) F1
;
0; 0;
;x
x 1;y x
1 x
;
MeijerG-Function 415
16.16.9 F2( ; ; 0;
;
0;x;y) = (1 x) F2
;
; 0;
;
0;x
x 1;y
1 x
;
16.16.10F4( ; ;
;
0;x;y) = (
0) ( )
(
0 ) ( )( y) F4
;
0+ 1;
; + 1;x
y;1
y
+ (
0) ( )
(
0 ) ( )( y) F4
;
0+ 1;
; + 1;x
y;1
y
:
For quadratic transformations of Appell functions see Carlson (1976).
MeijerG-Function
16.17 De nition
Again assume a1;a2;:::;apandb1;b2;:::;bqare real or complex parameters. Assume also that mandnare integers
such that 0mqand 0np, and none of ak bjis a positive integer when 1 knand 1jm. Then
theMeijerG-function is de ned via the Mellin{Barnes integral representation:
16.17.1Gm;n
p;q(z;a;b) =Gm;n
p;q
z;a1;:::;ap
b1;:::;bq
=1
2iZ
LmQ
`=1 (b` s)nQ
`=1 (1 a`+s) q 1Q
`=m (1 b`+1+s)p 1Q
`=n (a`+1 s)
zsds;
where the integration path Lseparates the poles of the factors ( b` s) from those of the factors (1 a`+s).
There are three possible choices for L, illustrated in Figure 16.17.1 in the case m= 1,n= 2:
(i)Lgoes from i1toi1. The integral converges if p+q<2(m+n) andjphzj<(m+n 1
2(p+q)).
(ii)Lis a loop that starts at in nity on a line parallel to the positive real axis, encircles the poles of the ( b` s)
once in the negative sense and returns to in nity on another line parallel to the positive real axis. The integral
converges for all z(6= 0) ifp<q , and for 0<jzj<1 ifp=q1.
(iii)Lis a loop that starts at in nity on a line parallel to the negative real axis, encircles the poles of the (1 a`+s)
once in the positive sense and returns to in nity on another line parallel to the negative real axis. The integral
converges for all zifp>q , and forjzj>1 ifp=q1.
Case (i) Case (ii) Case (iii)
Figure 16.17.1 :s-plane. Path Lfor the integral representation (16.17.1) of the Meijer G-function.
When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer G-function.
Assumepq, no two of the bottom parameters bj,j= 1;:::;m , di er by an integer, and aj bkis not a positive
integer when j= 1;2;:::;n andk= 1;2;:::;m . Then
16.17.2 Gm;n
p;q
z;a1;:::;ap
b1;:::;bq
=mX
k=1Am;n
p;q;k(z)pFq 11 +bk a1;:::; 1 +bk ap
1 +bk b1;::::::;1 +bk bq; ( 1)p m nz
;
416 Generalized Hypergeometric Functions and Meijer G-Function
whereindicates that the entry 1 + bk bkis omitted. Also,
16.17.3 Am;n
p;q;k(z) =mY
`=1
`6=k (b` bk)nY
`=1 (1 +bk a`)zbk, q 1Y
`=m (1 +bk b`+1)p 1Y
`=n (a`+1 bk)!
:
16.18 Special Cases
The 1F1and 2F1functions introduced in Chapters 13 and 15, as well as the more general pFqfunctions introduced
in the present chapter, are all special cases of the Meijer G-function. This is a consequence of the following relations:
16.18.1pFqa1;:::;ap
b1;:::;bq;z
=qQ
k=1 (bk) pQ
k=1 (ak)
G1;p
p;q+1
z;1 a1;:::; 1 ap
0;1 b1;:::; 1 bq
=qQ
k=1 (bk) pQ
k=1 (ak)
Gp;1
q+1;p
1
z;1;b1;:::;bq
a1;:::;ap
:
As a corollary, special cases of the 1F1and 2F1functions, including Airy functions, Bessel functions, parabolic cylinder
functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases
of the Meijer G-function. Representations of special functions in terms of the Meijer G-function are given in Erd elyi
et al. (1953a,x5.6), Luke (1969a, xx6.4{6.5), and Mathai (1993, x3.10).
16.19 Identities
16.19.1 Gm;n
p;q1
z;a1;:::;ap
b1;:::;bq
=Gn;m
q;p
z;1 b1;:::; 1 bq
1 a1;:::; 1 ap
;
16.19.2 zGm;n
p;q
z;a1;:::;ap
b1;:::;bq
=Gm;n
p;q
z;a1+;:::;ap+
b1+;:::;bq+
;
16.19.3 Gm;n+1
p+1;q+1
z;a0;:::;ap
b1;:::;bq;a0
=Gm;n
p;q
z;a1;:::;ap
b1;:::;bq
;
16.19.4Gm;n
p;q
z;a1;:::;ap
b1;:::;bq
=2p+1+b1++bq m n a1 ap
m+n 1
2(p+q)G2m;2n
2p;2q
22p 2qz2;1
2a1;1
2a1+1
2;:::;1
2ap;1
2ap+1
2
1
2b1;1
2b1+1
2;:::;1
2bq;1
2bq+1
2!
;
16.19.5 #Gm;n
p;q
z;a1;:::;ap
b1;:::;bq
=Gm;n
p;q
z;a1 1;a2;:::;ap
b1;:::;bq
+ (a1 1)Gm;n
p;q
z;a1;:::;ap
b1;:::;bq
;
16.19.6Z1
0t a0(1 t)a0 bq+1 1Gm;n
p;q
zt;a1;:::;ap
b1;:::;bq
dt= (a0 bq+1)Gm;n+1
p+1;q+1
z;a0;:::;ap
b1;:::;bq+1
;
where again #=z d/dz. For conditions for (16.19.6) see Luke (1969a, Chapter 5). This reference and Mathai (1993,
xx2.2 and 2.4) also supply additional identities.
16.20 Integrals and Series
Integrals of the Meijer G-function are given in Apelblat (1983, x19), Erd elyi et al. (1953a,x5.5.2), Erd elyi et al.
(1954a,xx6.9 and 7.5), Luke (1969a, x3.6), Luke (1975, x5.6), Mathai (1993, x3.10), and Prudnikov et al. (1990,
x2.24). Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer G-function are given in
Prudnikov et al. (1992a,x3.40) and Prudnikov et al. (1992b,x3.38).
Series of the Meijer G-function are given in Erd elyi et al. (1953a,x5.5.1), Luke (1975, x5.8), and Prudnikov et al.
(1990,x6.11).
16.21 Differential Equation 417
16.21 Di erential Equation
w=Gm;n
p;q(z;a;b) satis es the di erential equation
16.21.1
( 1)p m nz(# a1+ 1)(# ap+ 1) (# b1)(# bq)
w= 0;
where again #=z d/dz. This equation is of order max( p;q). In consequence of (16.19.1) we may assume, without
loss of generality, that pq. With the classi cation of x16.8(i), when p<q the only singularities of (16.21.1) are a
regular singularity at z= 0 and an irregular singularity at z=1. Whenp=qthe only singularities of (16.21.1)
are regular singularities at z= 0, ( 1)p m n, and1.
A fundamental set of solutions of (16.21.1) is given by
16.21.2 G1;p
p;q
ze(p m n 1)i;a1;:::;ap
bj;b1;:::;bj 1;bj+1;:::;bq
, j= 1;:::;q .
For other fundamental sets see Erd elyi et al. (1953a,x5.4) and Marichev (1984).
16.22 Asymptotic Expansions
Asymptotic expansions of Gm;n
p;q(z;a;b) for largezare given in Luke (1969a, xx5.7 and 5.10) and Luke (1975, x5.9).
For asymptotic expansions of Meijer G-functions with large parameters see Fields (1973, 1983).
Applications
16.23 Mathematical Applications
16.23(i) Di erential Equations
A variety of problems in classical mechanics and math-
ematical physics lead to Picard{Fuchs equations. These
equations are frequently solvable in terms of generalized
hypergeometric functions, and the monodromy of gen-
eralized hypergeometric functions plays an important
role in describing properties of the solutions. See, for
example, Berglund et al. (1994).
16.23(ii) Random Graphs
A substantial transition occurs in a random graph of
nvertices when the number of edges becomes approx-
imately1
2n. In Janson et al. (1993) limiting distribu-
tions are discussed for the sparse connected components
of these graphs, and the asymptotics of three 2F2func-
tions are applied to compute the expected value of the
excess.
16.23(iii) Conformal Mapping
The Bieberbach conjecture states that ifP1
n=0anznis
a conformal map of the unit disk to any complex do-
main, thenjanjnja1j. In the proof of this conjecture
de Branges (1985) uses the inequality
16.23.1 3F2
n;n+ + 2;1
2( + 1)
+ 1;1
2( + 3);x!
>0;when 0x<1, > 2, andn= 0;1;2;:::. The proof
of this inequality is given in Askey and Gasper (1976).
See also Kazarino (1988).
16.23(iv) Combinatorics and Number Theory
Many combinatorial identities, especially ones involving
binomial and related coecients, are special cases of hy-
pergeometric identities. In Petkov sek et al. (1996) tools
are given for automated proofs of these identities.
16.24 Physical Applications
16.24(i) Random Walks
Generalized hypergeometric functions and Appell func-
tions appear in the evaluation of the so-called Watson
integrals which characterize the simplest possible lattice
walks. They are also potentially useful for the solution
of more complicated restricted lattice walk problems,
and the 3D Ising model; see Barber and Ninham (1970,
pp. 147{148).
16.24(ii) Loop Integrals in Feynman Diagrams
Appell functions are used for the evaluation of one-loop
integrals in Feynman diagrams. See Cabral-Rosetti and
Sanchis-Lozano (2000).
For an extension to two-loop integrals see Moch et al.
(2002).
418 Generalized Hypergeometric Functions and Meijer G-Function
16.24(iii) 3j,6j, and 9jSymbols
The 3jsymbols, or Clebsch{Gordan coecients, play
an important role in the decomposition of reducible rep-
resentations of the rotation group into irreducible rep-
resentations. They can be expressed as 3F2functions
with unit argument. The coecients of transformations
between di erent coupling schemes of three angular mo-
menta are related to the Wigner 6 jsymbols. These are
balanced 4F3functions with unit argument. Lastly, spe-
cial cases of the 9 jsymbols are 5F4functions with unit
argument. For further information see Chapter 34 and
Varshalovich et al. (1988,xx8.2.5, 8.8, and 9.2.3).
Computation
16.25 Methods of Computation
Methods for computing the functions of the present
chapter include power series, asymptotic expansions, in-
tegral representations, di erential equations, and recur-
rence relations. They are similar to those described for
con
uent hypergeometric functions, and hypergeomet-
ric functions inxx13.29 and 15.19. There is, however,
an added feature in the numerical solution of di eren-
tial equations and di erence equations (recurrence re-
lations). This occurs when the wanted solution is in-
termediate in asymptotic growth compared with other
solutions. In these cases integration, or recurrence, in
either a forward or a backward direction is unstable. In-
stead a boundary-value problem needs to be formulated
and solved. Seexx3.6(vii), 3.7(iii), Olde Daalhuis and
Olver (1998), Lozier (1980), and Wimp (1984, Chap-
ters 7, 8).
16.26 Approximations
For discussions of the approximation of generalized hy-
pergeometric functions and the Meijer G-function in
terms of polynomials, rational functions, and Cheby-
shev polynomials see Luke (1975, xx5.12 - 5.13) and
Luke (1977b, Chapters 1 and 9).
16.27 Software
Seehttp://dlmf.nist.gov/16.27 .
References
General References
The main references used in writing this chapter are
Erd elyi et al. (1953a), Luke (1969a, 1975). For addi-
tional bibliographic reading see Andrews et al. (1999)
and Slater (1966).Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x16.2 Luke (1975, Chapter 5), Slater (1966, Chap-
ter 2). The statement that follows (16.2.5) follows
from the uniform convergence of (16.2.5) when
pq, and also when p=q+ 1 provided that
jzj<1. For other values of z, apply the straight-
forward generalization (to higher-order di erential
equations) of Theorem 3.2 in Olver (1997b, Chap-
ter 5).
x16.3 Luke (1975,x5.2.2), Rainville (1960, x48). For
(16.3.5) see Fleury and Turbiner (1994).
x16.4 Andrews et al. (1999, Chapters 2 and 3), Slater
(1966, Chapters 2 and 6). For (16.4.13) see Dono-
vanet al. (1999).
x16.5 Luke (1969a,x3.6). To justify the last sentence
in the third paragraph, translate the integration
contourLin (16.5.1) to the right, then apply the
residue theorem ( x1.10(iv)) making use of the es-
timate (5.11.9) for the gamma function; compare
x2.4(ii).
x16.6 For (16.6.1) see Whipple (1927). For (16.6.2) see
Bailey (1929).
x16.8 Luke (1969a,xx3.5 and 5.1). For (16.8.9) see
B uhring (1988).
x16.11 Paris and Kaminski (2001, x2.3), Wright
(1940a, p. 391), Meijer (1946, p. 1172), Luke
(1969a, Chapter 7).
x16.12 Erd elyi et al. (1953a,x4.3). For (16.12.1) see
Bailey (1928). For (16.12.2) see Clausen (1828).
For (16.12.3) see Chaundy (1969, Chapter 12,
Problem 12).
x16.13 Erd elyi et al. (1953a,x5.7).
x16.14 Erd elyi et al. (1953a,x5.9).
x16.16 Erd elyi et al. (1953a,x5.11).
x16.17 Luke (1969a, Chapter 5).
x16.18 Luke (1969a, Chapter 5).
x16.19 These results are straightforward consequences
of the de nition (16.17.1).
x16.21 Luke (1969a, Chapter 5).
Chapter 17
q-Hypergeometric and Related Functions
G. E. Andrews1
Notation 420
17.1 Special Notation . . . . . . . . . . . . . 420
Properties 420
17.2 Calculus . . . . . . . . . . . . . . . . . . 420
17.3q-Elementary and q-Special Functions . . 422
17.4 Basic Hypergeometric Functions . . . . . 423
17.5 0 0;1 0;1 1Functions . . . . . . . . . . 423
17.6 2 1Function . . . . . . . . . . . . . . . 424
17.7 Special Cases of Higher r sFunctions . . 426
17.8 Special Cases of r rFunctions . . . . . . 427
17.9 Transformations of Higher r rFunctions 428
17.10 Transformations of r rFunctions . . . . 429
17.11 Transformations of q-Appell Functions . . 43017.12 Bailey Pairs . . . . . . . . . . . . . . . . 430
17.13 Integrals . . . . . . . . . . . . . . . . . . 431
17.14 Constant Term Identities . . . . . . . . . 431
17.15 Generalizations . . . . . . . . . . . . . . 432
Applications 432
17.16 Mathematical Applications . . . . . . . . 432
17.17 Physical Applications . . . . . . . . . . . 432
Computation 432
17.18 Methods of Computation . . . . . . . . . 432
17.19 Software . . . . . . . . . . . . . . . . . . 432
References 432
1Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
419
420 q-Hypergeometric and Related Functions
Notation
17.1 Special Notation
(For other notation see pp. xiv and 873.)
k;j;m;n;r;s nonnegative integers.
z complex variable.
x real variable.
q(2C) base: unless stated otherwise jqj<1.
(a;q)nq-shifted factorial:
(1 a)(1 aq)
1 aqn 1
.
The main functions treated in this chapter are
the basic hypergeometric (or q-hypergeometric) func-
tionr s(a1;a2;:::;ar;b1;b2;:::;bs;q;z), the bilateral
basic hypergeometric (or bilateral q-hypergeometric)
function r s(a1;a2;:::;ar;b1;b2;:::;bs;q;z), and the
q-analogs of the Appell functions (1)(a;b;b0;c;x;y),
(2)(a;b;b0;c;c0;x;y), (3)(a;a0;b;b0;c;x;y), and
(4)(a;b;c;c0;x;y).
Another function notation used is the \idem" func-
tion:
f( 1; 2;:::; n) + idem( 1; 2;:::; n)
=nX
j=1f( j; 1; 2;:::; j 1; j+1;:::; n):
These notations agree with Gasper and Rahman
(2004) (except for the q-Appell functions which are not
considered in this reference). A slightly di erent no-
tation is that in Bailey (1935) and Slater (1966); see
x17.4(i). Fine (1988) uses F(a;b;t:q) for a particular
specialization of a 2 1function.
Properties
17.2 Calculus
17.2(i)q-Calculus
Forn= 0;1;2;:::,
17.2.1 (a;q)n= (1 a)(1 aq)(1 aqn 1);
17.2.2 (a;q) n=1
(aq n;q)n=( q=a)nq(n
2)
(q=a;q)n:
For2C
17.2.3 (a;q)=1Y
j=01 aqj
1 aq+j
;when this product converges.
17.2.4 (a;q)1=1Y
j=0(1 aqj);
17.2.5 (a1;a2;:::;ar;q)n=rY
j=1(aj;q)n;
17.2.6 (a1;a2;:::;ar;q)1=rY
j=1(aj;q)1:
17.2.7
a;q 1
n=
a 1;q
n( a)nq (n
2);
17.2.8
a;q 1
n
(b;q 1)n=
a 1;q
n
(b 1;q)na
bn
;
17.2.9 (a;q)n=
q1 n=a;q
n( a)nq(n
2);
17.2.10(a;q)n
(b;q)n=
q1 n=a;q
n
(q1 n=b;q)na
bn
;
17.2.11
aq n;q
n= (q=a;q)n
a
qn
q (n
2);
17.2.12(aq n;q)n
(bq n;q)n=(q=a;q)n
(q=b;q)na
bn
:
17.2.13 (a;q)n k=(a;q)n
(q1 n=a;q)k
q
ak
q(k
2) nk;
17.2.14(a;q)n k
(b;q)n k=(a;q)n
(b;q)n
q1 n=b;q
k
(q1 n=a;q)kb
ak
;
17.2.15
aq n;q
k=(a;q)k(q=a;q)n
(q1 k=a;q)nq nk;
17.2.16
aq n;q
n k=(q=a;q)n
(q=a;q)k
a
qn k
q(k
2) (n
2);
17.2.17 (aqn;q)k=(a;q)k
aqk;q
n
(a;q)n;
17.2.18
aqk;q
n k=(a;q)n
(a;q)k:
17.2.19 (a;q)2n=
a;aq;q2
n;
more generally,
17.2.20 (a;q)kn=
a;aq;:::;aqk 1;qk
n:
17.2.21
a2;q2
n= (a;q)n( a;q)n;
17.2.22
qa1
2; aq1
2;q
n
a1
2; a1
2;q
n=
aq2;q2
n
(a;q2)n=1 aq2n
1 a;
more generally,
17.2.23
aq1
k;q!ka1
k;:::;q!k 1
ka1
k;q
n
a1
k;!ka1
k;:::;!k 1
ka1
k;q
n
=
aqk;qk
n
(a;qk)n=1 aqkn
1 a;
17.2 Calculus 421
where!k=e2i=k.
17.2.24
lim
!0(a= ;q)n n= lim
!1(a;q)n n= ( a)nq(n
2);
17.2.25 lim
!0(a= ;q)n
(b= ;q)n= lim
!1(a;q)n
(b;q)n=a
bn
;
17.2.26 lim
!0(a= ;q)n(b= ;q)n
(c= 2;q)n= ( 1)nab
cn
q(n
2):
17.2(ii) Binomial Coecients
17.2.27n
m
q=(q;q)n
(q;q)m(q;q)n m
=(q n;q)m( 1)mqnm (m
2)
(q;q)m;
17.2.28 lim
q!1n
m
q=n
m
=n!
m!(n m)!;
17.2.29m+n
m
q=
qn+1;q
m
(q;q)m;
17.2.30 n
m
q=m+n 1
m
q( 1)mq mn (m
2);
17.2.31n
m
q=n 1
m 1
q+qmn 1
m
q;
17.2.32n
m
q=n 1
m
q+qn mn 1
m 1
q;
17.2.33
lim
n!1n
m
q=1
(q;q)m=1
(1 q)(1 q2)(1 qm);
17.2.34 lim
n!1rn+u
sn+t
q=1
(q;q)1=1Y
j=11
(1 qj);
provided that r>s .17.2(iii) Binomial Theorem
17.2.35
nX
j=0n
j
q( z)jq(j
2)= (z;q)n
= (1 z)(1 zq)(1 zqn 1):
In the limit as q!1, (17.2.35) reduces to the stan-
dard binomial theorem
17.2.36nX
j=0n
j
( z)j= (1 z)n:
Also,
17.2.371X
n=0(a;q)n
(q;q)nzn=(az;q)1
(z;q)1;
provided thatjzj<1. Whena=qm+1, wheremis a
nonnegative integer, (17.2.37) reduces to the q-binomial
series
17.2.381X
n=0n+m
n
qzn=1
(z;q)m+1:
17.2.39nX
j=0n
j
q2qj= ( q;q)n;
17.2.402nX
j=0( 1)j2n
j
q=
q;q2
n:
Whenn!1 in (17.2.35), and when m!1 in
(17.2.38), the results become convergent in nite series
and in nite products (see (17.5.1) and (17.5.4)).
17.2(iv) Derivatives
Theq-derivatives of f(z) are de ned by
17.2.41Dqf(z) =8
<
:f(z) f(zq)
(1 q)z; z6= 0;
f0(0); z = 0;
and
17.2.42 f[n](z) =Dn
qf(z) =8
><
>:z n(1 q) nPn
j=0q nj+(j+1
2)( 1)jn
j
qf(zqj); z6= 0;
f(n)(0) (q;q)n
n!(1 q)n; z = 0:
Whenq!1 theq-derivatives converge to the corre-
sponding ordinary derivatives.
Product Rule
17.2.43Dq(f(z)g(z)) =g(z)f[1](z) +f(zq)g[1](z):Leibniz Rule
17.2.44Dn
q(f(z)g(z)) =nX
j=0n
j
qf[n j](zqj)g[j](z):
q-di erential equations are considered in x17.6(iv).
422 q-Hypergeometric and Related Functions
17.2(v) Integrals
Iff(x) is continuous at x= 0, then
17.2.45Z1
0f(x)dqx= (1 q)1X
j=0f(qj)qj;
and more generally,
17.2.46Za
0f(x)dqx=a(1 q)1X
j=0f(aqj)qj:
Iff(x) is continuous on [0 ;a], then
17.2.47 lim
q!1 Za
0f(x)dqx=Za
0f(x)dx:
In nite Range
17.2.48Z1
0f(x)dqx
= lim
n!1Zq n
0f(x)dqx= (1 q)1X
j= 1f(qj)qj;
provided thatP1
j= 1f(qj)qjconverges.
17.2(vi) Rogers{Ramanujan Identities
17.2.491 +1X
n=1qn2
(1 q)(1 q2)(1 qn)
=1Y
n=01
(1 q5n+1)(1 q5n+4);
17.2.501 +1X
n=1qn2+n
(1 q)(1 q2)(1 qn)
=1Y
n=01
(1 q5n+2)(1 q5n+3):
These identities are the rst in a large collection of
similar results. See x17.14.
17.3q-Elementary and q-Special Functions
17.3(i) Elementary Functions
q-Exponential Functions
17.3.1eq(x) =1X
n=0(1 q)nxn
(q;q)n=1
((1 q)x;q)1;
17.3.2Eq(x) =1X
n=0(1 q)nq(n
2)xn
(q;q)n= ( (1 q)x;q)1:q-Sine Functions
17.3.3sinq(x) =1
2i(eq(ix) eq( ix))
=1X
n=0(1 q)2n+1( 1)nx2n+1
(q;q)2n+1;
17.3.4Sinq(x) =1
2i(Eq(ix) Eq( ix))
=1X
n=0(1 q)2n+1qn(2n+1)( 1)nx2n+1
(q;q)2n+1:
q-Cosine Functions
17.3.5
cosq(x) =1
2(eq(ix)+eq( ix)) =1X
n=0(1 q)2n( 1)nx2n
(q;q)2n;
17.3.6Cosq(x) =1
2(Eq(ix) +Eq( ix))
=1X
n=0(1 q)2nqn(2n 1)( 1)nx2n
(q;q)2n:
See also Suslov (2003).
17.3(ii) Gamma and Beta Functions
Seex5.18.
17.3(iii) Bernoulli Polynomials; Euler and
Stirling Numbers
q-Bernoulli Polynomials
17.3.7
n(x;q) = (1 q)1 nnX
r=0( 1)rn
rr+ 1
(1 qr+1)qrx:
q-Euler Numbers
17.3.8
Am;s(q)
=q(s m
2)+(s
2)sX
j=0( 1)jq(j
2)m+ 1
j
q(1 qs j)m
(1 q)m:
q-Stirling Numbers
17.3.9
am;s(q) =q (s
2)(1 q)s
(q;q)ssX
j=0( 1)jq(j
2)s
j
q(1 qs j)m
(1 q)m:
These were introduced in Carlitz (1954b, 1958). The
n(x;q) are, in fact, rational functions of q, and not
necessarily polynomials. The Am;s(q) are always poly-
nomials in q, and theam;s(q) are polynomials in qfor
0sm.
17.3(iv) Theta Functions
Seexx17.8 and 20.5.
17.3(v) Orthogonal Polynomials
Seexx18.27{18.29.
17.4 Basic Hypergeometric Functions 423
17.4 Basic Hypergeometric Functions
17.4(i)r sFunctions
17.4.1
r+1 sa0;a1;a2;:::;ar
b1;b2;:::;bs;q;z
=r+1 s(a0;a1;:::;ar;b1;b2;:::;bs;q;z)
=1X
n=0(a0;q)n(a1;q)n(ar;q)n
(q;q)n(b1;q)n(bs;q)n
( 1)nq(n
2)s r
zn:
Here and elsewhere it is assumed that the bjdo not take
any of the values q n. The in nite series converges for
allzwhens>r , and forjzj<1 whens=r.
17.4.2lim
q!1 r+1 rqa0;qa1;:::;qar
qb1;:::;qbr;q;z
=r+1Fra0;a1;:::;ar
b1;:::;br;z
:
For the function on the right-hand side see x16.2(i).
This notation is from Gasper and Rahman (2004).
It is slightly at variance with the notation in Bailey
(1935) and Slater (1966). In these references the factor
( 1)nq(n
2)s r
is not included in the sum. In practice
this discrepancy does not usually cause serious problems
because the case most often considered is r=s.
17.4(ii)r sFunctions
17.4.3
r sa1;a2;:::;ar
b1;b2;:::;bs;q;z
=r s(a1;a2;:::;ar;b1;b2;:::;bs;q;z)
=1X
n= 1(a1;a2;:::;ar;q)n( 1)(s r)nq(s r)(n
2)zn
(b1;b2;:::;bs;q)n
=1X
n=0(a1;a2;:::;ar;q)n( 1)(s r)nq(s r)(n
2)zn
(b1;b2;:::;bs;q)n
+1X
n=1(q=b1;q=b 2;:::;q=bs;q)n
(q=a1;q=a 2;:::;q=ar;q)nb1b2bs
a1a2arzn
:
Here and elsewhere the bjmust not take any of the
valuesq n, and theajmust not take any of the values
qn+1. The in nite series converge when srprovided
thatj(b1bs)=(a1arz)j<1 and also, in the case
s=r,jzj<1.
17.4.4lim
q!1 r rqa1;qa2;:::;qar
qb1;qb2;:::;qbr;q;z
=rHra1;a2;:::;ar
b1;b2;:::;br;z
:
For the function rHrseex16.4(v).17.4(iii) Appell Functions
The following de nitions apply when jxj<1 andjyj<
1:
17.4.5(1)(a;b;b0;c;x;y)
=X
m;n0(a;q)m+n(b;q)m(b0;q)nxmyn
(q;q)m(q;q)n(c;q)m+n;
17.4.6(2)(a;b;b0;c;c0;x;y)
=X
m;n0(a;q)m+n(b;q)m(b0;q)nxmyn
(q;q)m(q;q)n(c;q)m(c0;q)n;
17.4.7(3)(a;a0;b;b0;c;x;y)
=X
m;n0(a;b;q)m(a0;b0;q)nxmyn
(q;q)m(q;q)n(c;q)m+n;
17.4.8 (4)(a;b;c;c0;x;y) =X
m;n0(a;b;q)m+nxmyn
(q;c;q)m(q;c0;q)n:
17.4(iv) Classi cation
The series (17.4.1) is said to be balanced or
Saalsch utzian when it terminates, r=s,z=q, and
17.4.9 qa0a1as=b1b2bs:
The series (17.4.1) is said to be k-balanced when
r=sand
17.4.10 qka0a1as=b1b2bs:
The series (17.4.1) is said to be well-poised when
r=sand
17.4.11 a0q=a1b1=a2b2==asbs:
The series (17.4.1) is said to be very-well-poised
whenr=s, (17.4.11) is satis ed, and
17.4.12 b1= b2=pa0:
The series (17.4.1) is said to be nearly-poised when
r=sand
17.4.13a0q=a1b1=a2b2==as 1bs 1:
17.5 0 0;1 0;1 1Functions
Euler's Second Sum
17.5.1
0 0( ; ;q;z) =1X
n=0( 1)nq(n
2)zn
(q;q)n= (z;q)1,jzj<1;
compare (17.3.2).
q-Binomial Series
17.5.2 1 0(a; ;q;z) =(az;q)1
(z;q)1,jzj<1;
compare (17.2.37).
424 q-Hypergeometric and Related Functions
q-Binomial Theorem
17.5.31 0
q n; ;q;z
=
zq n;q
n:
This is (17.2.35) reformulated.
Euler's First Sum
17.5.41 0(0; ;q;z) =1X
n=0zn
(q;q)n=1
(z;q)1,jzj<1;
compare (17.3.1).
Cauchy's Sum
17.5.51 1a
c;q;c=a
=(c=a;q)1
(c;q)1,jcj<jaj.
17.6 2 1Function
17.6(i) Special Values
q-Gauss Sum
17.6.12 1a;b
c;q;c/(ab)
=(c=a;c=b ;q)1
(c;c=(ab);q)1:
Firstq-Chu{Vandermonde Sum
17.6.22 1a;q n
c;q;cqn/a
=(c=a;q)n
(c;q)n:
Secondq-Chu{Vandermonde Sum
This reverses the order of summation in (17.6.2):
17.6.32 1a;q n
c;q;q
=an(c=a;q)n
(c;q)n:Andrews{Askey Sum
17.6.4
2 1b2;b2
c
c;q2;cq
b2
=1
2
b2;q;q2
1
(c;cq=b2;q2)1(c=b;q)1
(b;q)1+( c=b;q)1
( b;q)1
,
jcqj<jb2j.
Bailey{Daum q-Kummer Sum
17.6.5
2 1a;b
aq=b;q; q=b
=( q;q)1
aq;aq2
b2;q2
1
( q=b;aq=b ;q)1,
jbj>jqj.
17.6(ii) 2 1Transformations
Heine's First Transformation
17.6.62 1a;b
c;q;z
=(b;az;q)1
(c;z;q)12 1c=b;z
az;q;b
,
jzj<1;jbj<1.
Heine's Second Tranformation
17.6.7
2 1a;b
c;q;z
=(c=b;bz ;q)1
(c;z;q)12 1abz/c;b
bz;q;c=b
,
jzj<1;jcj<jbj.
Heine's Third Transformation
17.6.82 1a;b
c;q;z
=(abz/c;q)1
(z;q)12 1c=a;c=b
c;q;abz /c
,
jzj<1;jabzj<jcj.
Fine's First Transformation
17.6.9 2 1q;aq
bq;q;z
= (1 b)(aq=b)
(1 (aq/b))1X
n=0(aq;azq=b ;q)nqn
(azq2=b;q)n+(aq;azq=b ;q)1
(aq=b;q)12 1q;0
bq;q;z
,jzj<1.
Fine's Second Transformation
17.6.10 (1 z)2 1q;aq
bq;q;z
=1X
n=0(b=a;q)n( az)nq(n2+n)=2
(bq;zq ;q)n, jzj<1.
Fine's Third Transformation
17.6.111 z
1 b2 1q;aq
bq;q;z
=1X
n=0(aq;q)n(azq=b ;q)2nbn
(zq;aq=b ;q)n aq1X
n=0(aq;q)n(azq=b ;q)2n+1(bq)n
(zq;q)n(aq=b;q)n+1,jzj<1;jbj<1.
Rogers{Fine Identity
17.6.12 (1 z)2 1q;aq
bq;q;z
=1X
n=0(aq;azq=b ;q)n
(bq;zq ;q)n(1 azq2n+1)(bz)nqn2, jzj<1.
17.6 2 1Function 425
Nonterminating Form of the q-Vandermonde Sum
17.6.13 2 1(a;b;c;q;q) +(q=c;a;b ;q)1
(c=q;aq=c;bq=c ;q)12 1
aq=c;bq=c ;q2=c;q;q
=(q=c;abq=c ;q)1
(aq=c;bq=c ;q)1;
17.6.141X
n=0(a;q)n
b;q2
nzn
(q;q)n(azb;q2)n=
az;bz ;q2
1
(z;azb ;q2)12 1a;b
bz;q2;zq
:
Three-Term 2 1Transformations
17.6.152 1a;b
c;q;z
=(abz=c;q=c ;q)1
(az=c;q=a ;q)12 1c=a;cq= (abz)
cq=(az);q;bq=c
b;q=c;c=a;az=q;q2=(az);q
1
(c=q;bq=c;q=a;az=c;cq= (az);q)12 1aq=c;bq=c
q2=c;q;z
,jzj<1;jbqj<jcj.
17.6.162 1a;b
c;q;z
=(b;c=a;az;q= (az);q)1
(c;b=a;z;q=z ;q)12 1a;aq=c
aq=b;q;cq= (abz)
+(a;c=b;bz;q= (bz);q)1
(c;a=b;z;q=z ;q)12 1b;bq=c
bq=a;q;cq= (abz)
,jzj<1,jabzj<jcqj.
17.6(iii) Contiguous Relations
Heine's Contiguous Relations
17.6.17 2 1a;b
c=q;q;z
2 1a;b
c;q;z
=cz(1 a)(1 b)
(q c)(1 c)2 1aq;bq
cq;q;z
;
17.6.18 2 1aq;b
c;q;z
2 1a;b
c;q;z
=az1 b
1 c2 1aq;bq
cq;q;z
;
17.6.19 2 1aq;b
cq;q;z
2 1a;b
c;q;z
=az(1 b)(1 (c=a))
(1 c)(1 cq)2 1aq;bq
cq2;q;z
;
17.6.20 2 1aq;b=q
c;q;z
2 1a;b
c;q;z
=az(1 b=(aq))
1 c2 1aq;b
cq;q;z
;
17.6.21 b(1 a)2 1aq;b
c;q;z
a(1 b)2 1a;bq
c;q;z
= (b a)2 1a;b
c;q;z
;
17.6.22a
1 b
c
2 1a;b=q
c;q;z
b
1 a
c
2 1a=q;b
c;q;z
= (a b)
1 abz
cq
2 1a;b
c;q;z
;
17.6.23
q
1 a
c
2 1a=q;b
c;q;z
+ (1 a)
1 abz
c
2 1aq;b
c;q;z
=
1 +q a aq
c+a2z
c abz
c
2 1a;b
c;q;z
;
17.6.24(1 c)(q c)(abz c)2 1a;b
c=q;q;z
+z(c a)(c b)2 1a;b
cq;q;z
= (c 1)(c(q c) +z(ca+cb ab abq))2 1a;b
c;q;z
:
17.6(iv) Di erential Equations
Iterations ofD
17.6.25 Dn
q2 1a;b
c;q;zd
=(a;b;q)ndn
(c;q)n(1 q)n2 1aqn;bqn
cqn;q;dz
;
17.6.26Dn
q(z;q)1
(abz=c ;q)12 1a;b
c;q;z
=(c=a;c=b ;q)n
(c;q)n(1 q)nab
cn(zqn;q)1
(abz=c ;q)12 1a;b
cqn;q;zqn
:
426 q-Hypergeometric and Related Functions
q-Di erential Equation
17.6.27z(c abqz)D2
q2 1a;b
c;q;z
+1 c
1 q+(1 a)(1 b) (1 abq)
1 qz
Dq2 1a;b
c;q;z
(1 a)(1 b)
(1 q)22 1a;b
c;q;z
= 0:
(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions a!qa,b!qb,c!qc, followed
by limq!1 .
17.6(v) Integral Representations
17.6.28 2 1q ;q
q
;q;z
= q(
)
q( ) q(
)Z1
0t 1(tq;q)
1
(xt;q) dqt:
17.6.29 2 1a;b
c;q;z
= 1
2i(a;b;q)1
(q;c;q)1Zi1
i1
q1+;cq;q
1
(aq;bq;q)1( z)
sin()d;
wherejzj<1,jph( z)j< , and the contour of integration separates the poles of
q1+;cq;q
1=sin() from
those of 1=
aq;bq;q
1, and the in mum of the distances of the poles from the contour is positive.
17.6(vi) Continued Fractions
For continued-fraction representations of the 2 1func-
tion, see Cuyt et al. (2008, pp. 395{399).
17.7 Special Cases of Higher r sFunctions
17.7(i) 2 2Functions
q-Analog of Bailey's 2F1( 1)Sum
17.7.12 2a;q=a
q;b;q; b
=
ab;bq=a ;q2
1
(b;q)1,jbj<1:
q-Analog of Gauss's 2F1( 1)Sum
17.7.22 2a2;b2
abq1
2; abq1
2;q; q
=
a2q;b2q;q2
1
(q;a2b2q;q2)1:
Sum Related to (17.6.4)
17.7.32 2c2
b2;b2
c;cq;q2;q
=1
2
b2;q;q2
1
(c;cq;q2)1(c=b;q)1
(b;q)1+( c=b;q)1
( b;q)1
:
17.7(ii) 3 2Functions
q-Pfa {Saalsch utz Sum
17.7.43 2a;b;q n
c;abq1 n=c;q;q
=(c=a;c=b ;q)n
(c;c=(ab);q)n:Nonterminating Form of the q-Saalsch utz Sum
17.7.53 2a;b;c
e;f;q;q
+(q=e;a;b;c;qf=e ;q)1
(e=q;aq=e;bq=e;cq=e;f ;q)1
3 2aq=e;bq=e;cq=e
q2=e;qf=e;q;q
=(q=e;f=a;f=b;f=c ;q)1
(aq=e;bq=e;cq=e;f ;q)1;
whereef=abcq.
F. H. Jackson's Terminating q-Analog of Dixon's Sum
17.7.6
3 2q 2n;b;c
q1 2n=b;q1 2n=c;q;q2 n
bc
=(b;c;q)n(q;bc;q)2n
(q;bc;q)n(b;c;q)2n:
Continued Fractions
For continued-fraction representations of a ratio of 3 2
functions, see Cuyt et al. (2008, pp. 399{400).
17.7(iii) Other r sFunctions
q-Analog of Dixon's 3F2(1)Sum
17.7.74 3
a; qa1
2;b;c
a1
2;aq=b;aq=c;q;qa1
2
bc!
=
aq;qa1
2=b;qa1
2=c;aq= (bc);q
1
aq=b;aq=c;qa1
2;qa1
2=(bc);q
1:
Gasper{Rahman q-Analog of Watson's 3F2Sum
17.7.8
8 7
;q1
2; q1
2;a;b;c; c;q=c2
1
2; 1
2;q=a;q=b;q=c; q=c;c2;q; q
ab!
=
q;c2=;q
1
aq;bq;c2q=a;c2q=b;q2
1
(q=a;q=b ;q)1(q;abq;c2q;c2q=(ab);q2)1;
17.8 Special Cases of r rFunctions 427
where= c(ab=q)1
2.
Andrews' Terminating q-Analog of (17.7.8)
17.7.94 3q n;aqn;c; c
(aq)1
2; (aq)1
2;c2;q;q
=8
><
>:0; n odd,
cn
q;aq=c2;q2
n=2
(aq;c2q;q2)n=2; n even.
Gasper{Rahman q-Analog of Whipple's 3F2Sum
17.7.10
8 7
c;q( c)1
2; q( c)1
2;a;q=a;c; d; q=d
( c)1
2; ( c)1
2; cq=a; ac; q;cq=d;cd;q;c!
=( c; cq;q)1
acd;acq=d;cdq=a;cq2=(ad);q2
1
(cd;cq=d; ac; cq=a;q)1:
Andrews' Terminating q-Analog
17.7.114 3q n;qn+1;c; c
e;c2q=e; q;q;q
=
eq n;eqn+1;c2q1 n=e;c2qn+2=e;q2
1
(e;c2q=e;q)1:
Firstq-Analog of Bailey's 4F3(1)Sum
17.7.12
4 3a;aq;b2q2n;q 2n
b;bq;a2q2;q2;q2
=an( q;b=a ;q)n
( aq;b;q)n:
Secondq-Analog of Bailey's 4F3(1)Sum
17.7.134 3a;aq;b2q2n 2;q 2n
b;bq;a2;q2;q2
=an( q;b=a ;q)n(1 bqn 1)
( a;b;q)n(1 bq2n 1):
F. H. Jackson's q-Analog of Dougall's 7F6(1)Sum
17.7.14
8 7
a;qa1
2; qa1
2;b;c;d;e;q n
a1
2; a1
2;aq=b;aq=c;aq=d;aq=e;aqn+1;q;q!
=(aq;aq= (bc);aq=(bd);aq=(cd);q)n
(aq=b;aq=c;aq=d;aq= (bcd);q)n;wherea2q=bcdeq n.
Limiting Cases of (17.7.14)
17.7.156 5
a;qa1
2; qa1
2;b;c;d
a1
2; a1
2;aq=b;aq=c;aq=d;q;aq
bcd!
=(aq;aq= (bc);aq=(bd);aq=(cd);q)1
(aq=b;aq=c;aq=d;aq= (bcd);q)1;
and whend=q n,
17.7.166 5
a;qa1
2; qa1
2;b;c;q n
a1
2; a1
2;aq=b;aq=c;aqn+1;q;aqn+1
bc!
=(aq;aq= (bc);q)n
(aq=b;aq=c ;q)n:
See http://dlmf.nist.gov/17.7.iii for additional
results.
17.8 Special Cases of r rFunctions
Jacobi's Triple Product
17.8.11X
n= 1( z)nqn(n 1)=2= (q;z;q=z ;q)1;
compare (20.5.9).
Ramanujan's 1 1Summation
17.8.21 1a
b;q;z
=(q;b=a;az;q= (az);q)1
(b;q=a;z;b= (az);q)1:
Quintuple Product Identity
17.8.31X
n= 1( 1)nqn(3n 1)=2z3n(1 +zqn)
= (q; z; q=z;q)1
qz2;q=z2;q2
1:
Bailey's Bilateral Summations
17.8.42 2(b;c;aq=b;aq=c ;q; aq=(bc)) =(aq=(bc);q)1
aq2=b2;aq2=c2;q2;aq;q=a ;q2
1
(aq=b;aq=c;q=b;q=c; aq=(bc);q)1;
17.8.53 3b;c;d
q=b;q=c;q=d;q;q
bcd
=(q;q=(bc);q=(bd);q=(cd);q)1
(q=b;q=c;q=d;q= (bcd);q)1;
17.8.64 4
qa1
2;b;c;d
a1
2;aq=b;aq=c;aq=d;q;qa3
2
bcd!
=
aq;aq= (bc);aq=(bd);aq=(cd);qa1
2=b;qa1
2=c;qa1
2=d;q;q=a ;q
1
aq=b;aq=c;aq=d;q=b;q=c;q=d;qa1
2;qa 1
2;qa3
2=(bcd);q
1;
428 q-Hypergeometric and Related Functions
17.8.7
6 6
qa1
2; qa1
2;b;c;d;e
a1
2; a1
2;aq=b;aq=c;aq=d;aq=e;q;qa2
bcde!
=(aq;aq= (bc);aq=(bd);aq=(be);aq=(cd);aq=(ce);aq=(de);q;q=a ;q)1
(aq=b;aq=c;aq=d;aq=e;q=b;q=c;q=d;q=e;qa2=(bcde);q)1:
17.9 Transformations of Higher r rFunctions
17.9(i) 2 1!2 2,3 1, or 3 2
F. H. Jackson's Transformations
17.9.1 2 1a;b
c;q;z
=(za;q)1
(z;q)12 2a;c=b
c;az;q;bz
;
17.9.2 2 1q n;b
c;q;z
=(c=b;q)n
(c;q)nbn
3 1q n;b;q=c
bq1 n=c;q;z=c
;
17.9.3 2 1a;b
c;q;z
=(abz=c ;q)1
(bz=c;q)13 2a;c=b; 0
c;cq=bz;q;q
;
17.9.4 2 1q n;b
c;q;z
=(c=b;q)n
(c;q)nbz
qn
3 2q n;q=z;q1 n=c
bq1 n=c;0;q;q
;
17.9.5 2 1q n;b
c;q;z
=(c=b;q)n
(c;q)n3 2q n;b;bzq n=c
bq1 n=c;0;q;q
:
17.9(ii) 3 2!3 2
Transformations of 3 2-Series
17.9.6 3 2a;b;c
d;e;q;de= (abc)
=(e=a;de= (bc);q)1
(e;de= (abc);q)13 2a;d=b;d=c
d;de= (bc);q;e=a
;
17.9.7 3 2a;b;c
d;e;q;de= (abc)
=(b;de= (ab);de=(bc);q)1
(d;e;de= (abc);q)13 2d=b;e=b;de= (abc)
de=(ab);de=(bc);q;b
;
17.9.8 3 2q n;b;c
d;e;q;q
=(de=(bc);q)n
(e;q)nbc
dn
3 2q n;d=b;d=c
d;de= (bc);q;q
;
17.9.9 3 2q n;b;c
d;e;q;q
=(e=c;q)n
(e;q)ncn
3 2q n;c;d=b
d;cq1 n=e;q;bq
e
;
17.9.10 3 2q n;b;c
d;e;q;deqn
bc
=(e=c;q)n
(e;q)n3 2q n;c;d=b
d;cq1 n=e;q;q
:
q-Sheppard Identity
17.9.11 3 2q n;b;c
d;e;q;q
=(e=c;d=c ;q)n
(e;d;q)ncn
3 2q n;c;cbq1 n
(de)
cq1 n/e;cq1 n/d;q;q
;
For further results see http://dlmf.nist.gov/17.9.ii .
17.9(iii) Further r sFunctions
Sears' Balanced 4 3Transformations
Withdef=abcq1 n
17.9.144 3q n;a;b;c
d;e;f;q;q
=(e=a;f=a ;q)n
(e;f;q)nan
4 3q n;a;d=b;d=c
d;aq1 n=e;aq1 n=f;q;q
=(a;ef= (ab);ef=(ac);q)n
(e;f;ef= (abc);q)n4 3q n;e=a;f=a;ef= (abc)
ef=(ab);ef=(ac);q1 n=a;q;q
:
17.10 Transformations of r rFunctions 429
Watson'sq-Analog of Whipple's Theorem
Withna nonnegative integer
17.9.15(aq;aq= (de);q)n
(aq=d;aq=e ;q)n4 3aq=(bc);d;e;q n
aq=b;aq=c;deq n=a;q;q
=8 7
a;qa1
2; qa1
2;b;c;d;e;q n
a1
2; a1
2;aq=b;aq=c;aq=d;aq=e;aqn+1;q;a2q2+n
bcde!
:
Bailey's Transformation of Very-Well-Poised 8 7
17.9.16
8 7
a;qa1
2; qa1
2;b;c;d;e;f
a1
2; a1
2;aq=b;aq=c;aq=d;aq=e;aq=f;q;a2q2
bcdef!
=(aq;aq= (de);aq=(df);aq=(ef);q)1
(aq=d;aq=e;aq=f;aq= (def);q)14 3aq=(bc);d;e;f
aq=b;aq=c;def=a;q;q
+
aq;aq= (bc);d;e;f;a2q2=(bdef);a2q2=(cdef);q
1
(aq=b;aq=c;aq=d;aq=e;aq=f;a2q2=(bcdef );def= (aq);q)14 3aq=(de);aq=(df);aq=(ef);a2q2=(bcdef )
a2q2=(bdef);a2q2=(cdef);aq2=(def);q;q
:
For additional results see http://dlmf.nist.gov/17.9.iii and Gasper and Rahman (2004, Appendix III and
Chapter 2).
17.9(iv) Bibasic Series
Mixed-Base Heine-Type Transformations
17.9.191X
n=0
a;q2
n(b;q)n
(q2;q2)n(c;q)nzn=(b;q)1
az;q2
1
(c;q)1(z;q2)11X
n=0(c=b;q)2n
z;q2
nb2n
(q;q)2n(az;q2)n
+(b;q)1
azq;q2
1
(c;q)1(zq;q2)11X
n=0(c=b;q)2n+1
zq;q2
nb2n+1
(q;q)2n+1(azq;q2)n:
17.9.201X
n=0
a;qk
n(b;q)knzn
(qk;qk)n(c;q)kn=(b;q)1
az;qk
1
(c;q)1(z;qk)11X
n=0(c=b;q)n
z;qk
nbn
(q;q)n(az;qk)n,k= 1;2;3;::::
17.10 Transformations of r rFunctions
Bailey's 2 2Transformations
17.10.1 2 2a;b
c;d;q;z
=(az;d=a;c=b;dq= (abz);q)1
(z;d;q=b;cd= (abz);q)12 2a;abz=d
az;c;q;d
a
;
17.10.2 2 2a;b
c;d;q;z
=(az;bz;cq= (abz);dq=(abz);q)1
(q=a;q=b;c;d ;q)12 2abz=c;abz=d
az;bz;q;cd
abz
:
Other Transformations
17.10.38 8
qa1
2; qa1
2;c;d;e;f;aq n;q n
a1
2; a1
2;aq=c;aq=d;aq=e;aq=f;qn+1;aqn+1;q;a2q2n+2
cdef!
=(aq;q=a;aq= (cd);aq=(ef);q)n
(q=c;q=d;aq=e;aq=f ;q)n4 4e;f;aqn+1=(cd);q n
aq=c;aq=d;qn+1;ef=(aqn);q;q
;
17.10.4
2 2e;f
aq=c;aq=d;q;aq
ef
=(q=c;q=d;aq=e;aq=f ;q)1
(aq;q=a;aq= (cd);aq=(ef);q)11X
n= 1(1 aq2n) (c;d;e;f ;q)n
(1 a) (aq=c;aq=d;aq=e;aq=f ;q)nqa3
cdefn
qn2:
17.10.5
(aq=b;aq=c;aq=d;aq=e;q= (ab);q=(ac);q=(ad);q=(ae);q)1
(fa;ga;f=a;g=a;qa2;q=a2;q)18 8qa; qa;ba;ca;da;ea;fa;ga
a; a;aq=b;aq=c;aq=d;aq=e;aq=f;aq=g;q;q2
bcdefg
=(q;q=(bf);q=(cf);q=(df);q=(ef);qf=b;qf=c;qf=d;qf=e ;q)1
(fa;q= (fa);aq=f;f=a;g=f;fg;qf2;q)1
8 7f2;qf; qf;fb;fc;fd;fe;fg
f; f;fq=b;fq=c;fq=d;fq=e;fq=g;q;q2
bcdefg
+ idem(f;g):
430 q-Hypergeometric and Related Functions
17.10.6(aq=b;aq=c;aq=d;aq=e;aq=f;q= (ab);q=(ac);q=(ad);q=(ae);q=(af);q)1
(ag;ah;ak;g=a;h=a;k=a;qa2;q=a2;q)1
10 10qa; qa;ba;ca;da;ea;fa;ga;ha;ka
a; a;aq=b;aq=c;aq=d;aq=e;aq=f;aq=g;aq=h;aq=k;q;q2
bcdefghk
=(q;q=(bg);q=(cg);q=(dg);q=(eg);q=(fg);qg=b;qg=c;qg=d;qg=e;qg=f ;q)1
(gh;gk;h=g;ag;q= (ag);g=a;aq=g;qg2;q)1
10 9g2;qg; qg;gb;gc;gd;ge;gf;gh;gk
g; g;qg=b;qg=c;qg=d;qg=e;qg=f;qg=h;qg=k;q;q2
bcdefghk
+ idem(g;h;k):
17.11 Transformations of q-Appell Functions
17.11.1 (1)(a;b;b0;c;x;y) =(a;bx;b0y;q)1
(c;x;y ;q)13 2c=a;x;y
bx;b0y;q;a
;
17.11.2 (2)(a;b;b0;c;c0;x;y) =(b;ax;q)1
(c;x;q)1X
n;r=0(a;b0;q)n(c=b;x ;q)rbryn
(q;c0;q)n(q)r(ax;q)n+r;
17.11.3 (3)(a;a0;b;b0;c;x;y) =(a;bx;q)1
(c;x;q)1X
n;r=0(a0;b0;q)n(x;q)r(c=a;q)n+raryn
(q;c=a ;q)n(q;bx;q)r:
Of (17.11.1){(17.11.3) only (17.11.1) has a natural generalization: the following sum reduces to (17.11.1) when
n= 2.
17.11.4X
m1;:::;mn=0(a;q)m1+m2++mn(b1;q)m1(b2;q)m2(bn;q)mnxm1
1xm2
2xmnn
(q;q)m1(q;q)m2(q;q)mn(c;q)m1+m2++mn
=(a;b1x1;b2x2;:::;bnxn;q)1
(c;x1;x2;:::;xn;q)1n+1 nc=a;x 1;x2;:::;xn
b1x1;b2x2;:::;bnxn;q;a
:
17.12 Bailey Pairs
Bailey Transform
17.12.11X
n=0 n
n=1X
n=0 nn;
where
17.12.2 n=nX
j=0 jun jvn+j;
n=1X
j=njuj nvj+n:
Bailey Pairs
A sequence of pairs of rational functions of several vari-
ables ( n; n),n= 0;1;2;:::, is called a Bailey pair
provided that for each n=0
17.12.3 n=nX
j=0 j
(q;q)n j(aq;q)n+j:
Weak Bailey Lemma
If ( n; n) is a Bailey pair, then
17.12.41X
n=0qn2an n=1
(aq;q)11X
n=0qn2an n:Strong Bailey Lemma
If ( n; n) is a Bailey pair, then so is ( 0
n; 0
n), where
17.12.5
aq
1;aq
2;q
n 0
n= (1;2;q)naq
12n
n
aq
1;aq
2;q
n 0
n
=nX
j=0(1;2;q)jaq
12;q
n jaq
12j j
(q;q)n j
When (17.12.5) is iterated the resulting in nite se-
quence of Bailey pairs is called a Bailey Chain .
The Bailey pair that implies the Rogers{Ramanujan
identitiesx17.2(vi) is:
17.12.6 n=(a;q)n(1 aq2n)( 1)nqn(3n 1)=2an
(q;q)n(1 a);
n=1
(q;q)n:
The Bailey pair and Bailey chain concepts have been
extended considerably. See Andrews (2000, 2001), An-
drews and Berkovich (1998), Andrews et al. (1999),
17.13 Integrals 431
Milne and Lilly (1992), Spiridonov (2002), and Warnaar
(1998).
17.13 Integrals
In this section, for the function qseex5.18(ii).
17.13.1Zd
c( qx=c;q)1(qx=d;q)1
( ax=c;q)1(bx=d;q)1dqx
=(1 q) (q;q)1(ab;q)1cd( c=d;q)1( d=c;q)1
(a;q)1(b;q)1(c+d) ( bc=d;q)1( ad=c;q)1;
or, when 0 <q< 1,
17.13.2Zd
c( qx=c;q)1(qx=d;q)1
( xq =c;q)1(xq =d;q)1dqx
= q( ) q( )
q( + )cd
c+d( c=d;q)1( d=c;q)1
( q c=d;q)1( q d=c;q)1:Ramanujan's Integrals
17.13.3
Z1
0t 1
tq + ;q
1
( t;q)1dqt= ( ) (1 ) q( )
q(1 ) q( + );
17.13.4Z1
0t 1
ctq + ;q
1
( ct;q)1dqt
= q( ) q( ) ( cq ;q)1
q1 =c;q
1
q( + ) ( c;q)1( q=c;q)1:
Askey (1980) conjectured extensions of the foregoing
integrals that are closely related to Macdonald (1982).
These conjectures are proved independently in Hab-
sieger (1988) and Kadell (1988).
17.14 Constant Term Identities
Zeilberger{Bressoud Theorem (Andrews' q-Dyson Conjecture)
17.14.1(q;q)a1+a2++an
(q;q)a1(q;q)a2(q;q)an= coe . of x0
1x0
2x0
ninY
1j<knxj
xk;q
ajqxk
xj;q
ak:
Rogers{Ramanujan Constant Term Identities
In the following, G(q) andH(q) denote the left-hand sides of (17.2.49) and (17.2.50), respectively.
17.14.21X
n=0qn(n+1)
(q2;q2)n( q;q2)n+1= coe . of z0in
zq;q2
1
z 1q;q2
1
q2;q2
1
(z 1q2;q2)1( q;q2)1(z 1q;q2)1
=1
( q;q2)1coe . ofz0in
zq;q2
1
z 1q;q2
1
q2;q2
1
(z 1q;q)1=H(q)
( q;q2)1;
17.14.31X
n=0qn(n+1)
(q2;q2)n( q;q2)n+1= coe . of z0in
zq;q2
1
z 1q;q2
1
q2;q2
1
(z 1;q2)1( q;q2)1(z 1q;q2)1
=1
( q;q2)1coe . ofz0in
zq;q2
1
z 1q;q2
1
q2;q2
1
(z 1;q)1=G(q)
( q;q2)1;
17.14.41X
n=0qn2
(q2;q2)n(q;q2)n= coe . of z0in
zq;q2
1
z 1q;q2
1
q2;q2
1
( z 1;q2)1(q;q2)1(z 1;q2)1
=1
(q;q2)1coe . ofz0in
zq;q2
1
z 1q;q2
1
q2;q2
1
(z 2;q4)1=G(q4)
(q;q2)1;
17.14.51X
n=0qn2+2n
(q2;q2)n(q;q2)n+1= coe . of z0in
zq;q2
1
z 1q;q2
1
q2;q2
1
( q2z 1;q2)1(q;q2)1(z 1q2;q2)1
=1
(q;q2)1coe . ofz0in
zq;q2
1
z 1q;q2
1
q2;q2
1
(q4z 2;q4)1=H(q4)
(q;q2)1:
432 q-Hypergeometric and Related Functions
Macdonald (1982) includes extensive conjectures on
generalizations of (17.14.1) to root systems. These con-
jectures were proved in Cherednik (1995), Habsieger
(1986), and Kadell (1994); see also Macdonald (1998).
For additional results of the type (17.14.2){(17.14.5) see
Andrews (1986, Chapter 4).
17.15 Generalizations
For higher-dimensional basic hypergometric functions,
see Milne (1985b,c,d,a, 1988, 1994, 1997) and Gustafson
(1987).
Applications
17.16 Mathematical Applications
Many special cases of q-series arise in the theory of
partitions, a topic treated in xx27.14(i) and 26.9. In
Lie algebras Lepowsky and Milne (1978) and Lepowsky
and Wilson (1982) laid foundations for extensive inter-
action with q-series. These and other applications are
described in the surveys Andrews (1974, 1986). More
recent applications are given in Gasper and Rahman
(2004, Chapter 8) and Fine (1988, Chapters 1 and 2).
17.17 Physical Applications
In exactly solved models in statistical mechanics (Bax-
ter (1981, 1982)) the methods and identities of x17.12
play a substantial role. See Berkovich and McCoy
(1998) and Bethuel (1998) for recent surveys.
Quantum groups also apply q-series extensively.
Quantum groups are really not groups at all but certain
Hopf algebras. They were given this name because they
play a role in quantum physics analogous to the role of
Lie groups and special functions in classical mechanics.
See Kassel (1995).
A substantial literature on q-deformed quantum-
mechanical Schr odinger equations has developed re-
cently. It involves q-generalizations of exponentials and
Laguerre polynomials, and has been applied to the prob-
lems of the harmonic oscillator and Coulomb potentials.
See Micu and Papp (2005), where many earlier refer-
ences are cited.Computation
17.18 Methods of Computation
The two main methods for computing basic hypergeo-
metric functions are: (1) numerical summation of the
de ning series given in xx17.4(i) and 17.4(ii); (2) modu-
lar transformations. Method (1) is applicable within the
circles of convergence of the de ning series, although it
is often cumbersome owing to slowness of convergence
and/or severe cancellation. Method (2) is very powerful
when applicable (Andrews (1976, Chapter 5)); however,
it is applicable only rarely. Lehner (1941) uses Method
(2) in connection with the Rogers{Ramanujan identi-
ties.
Method (1) can sometimes be improved by appli-
cation of convergence acceleration procedures; see x3.9.
Shanks (1955) applies such methods in several q-series
problems; see Andrews et al. (1986).
17.19 Software
Seehttp://dlmf.nist.gov/17.19 .
References
General References
The main reference used in writing this chapter is
Gasper and Rahman (2004). For additional biblio-
graphic reading see Andrews (1974, 1976, 1986), An-
drews et al. (1999), Bailey (1935), Fine (1988), Kac and
Cheung (2002), and Slater (1966).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x17.2 Andrews (1976, pp. 17, 36, 37, 49 and x7.3).
(17.2.43) is derived from (17.2.41); (17.2.42) and
(17.2.44) follow by induction.
x17.5 Andrews (1976, pp. 17, 19, 36), Gasper and Rah-
man (2004, pp. 25{26).
x17.6 Gasper and Rahman (2004, pp. 13{15, 18, 23,
26{28, 115, 356, 363{364). For (17.6.9){(17.6.12)
see Fine (1988, pp. 12{15), and for (17.6.14) see
Andrews (1966a).
References 433
x17.7 Gasper and Rahman (2004, pp. 17, 19, 28{
29, 42{44, 58, 61{62, 81{83, 355{356), Andrews
(1996). For (17.7.3) combine Gasper and Rahman
(2004, p. 359, Equation (III.4)) with (17.6.4).
x17.8 Gasper and Rahman (2004, pp. 15{16, 52, 140{
141, 146{147, 149{150, 153).
x17.9 Gasper and Rahman (2004, pp. 43, 50, 63{64,
70{73, 359, 361), Andrews (1966b). For (17.9.11)
apply (17.9.10) to (17.9.9) and interchange the
roles ofdande.x17.10 Gasper and Rahman (2004, pp. 147{150, 364{
366).
x17.11 Andrews (1972).
x17.12 Andrews (1984).
x17.13 Gasper and Rahman (2004, p. 52), Berndt
(1991, p. 29), Askey (1980).
x17.14 Zeilberger and Bressoud (1985), Andrews
(1986, p. 34).
Chapter 18
Orthogonal Polynomials
T. H. Koornwinder1, R. Wong2, R. Koekoek3and R. F. Swarttouw4
Notation 436
18.1 Notation . . . . . . . . . . . . . . . . . . 436
General Orthogonal Polynomials 437
18.2 General Orthogonal Polynomials . . . . . 437
Classical Orthogonal Polynomials 438
18.3 De nitions . . . . . . . . . . . . . . . . . 438
18.4 Graphics . . . . . . . . . . . . . . . . . . 440
18.5 Explicit Representations . . . . . . . . . . 442
18.6 Symmetry, Special Values, and Limits to
Monomials . . . . . . . . . . . . . . . . . 443
18.7 Interrelations and Limit Relations . . . . 444
18.8 Di erential Equations . . . . . . . . . . . 445
18.9 Recurrence Relations and Derivatives . . 446
18.10 Integral Representations . . . . . . . . . 447
18.11 Relations to Other Functions . . . . . . . 448
18.12 Generating Functions . . . . . . . . . . . 449
18.13 Continued Fractions . . . . . . . . . . . . 450
18.14 Inequalities . . . . . . . . . . . . . . . . 450
18.15 Asymptotic Approximations . . . . . . . . 451
18.16 Zeros . . . . . . . . . . . . . . . . . . . 454
18.17 Integrals . . . . . . . . . . . . . . . . . . 455
18.18 Sums . . . . . . . . . . . . . . . . . . . 459
Askey Scheme 462
18.19 Hahn Class: De nitions . . . . . . . . . . 462
18.20 Hahn Class: Explicit Representations . . . 462
18.21 Hahn Class: Interrelations . . . . . . . . 46318.22 Hahn Class: Recurrence Relations and Dif-
ferences . . . . . . . . . . . . . . . . . . 464
18.23 Hahn Class: Generating Functions . . . . 466
18.24 Hahn Class: Asymptotic Approximations . 466
18.25 Wilson Class: De nitions . . . . . . . . . 467
18.26 Wilson Class: Continued . . . . . . . . . 468
Other Orthogonal Polynomials 470
18.27q-Hahn Class . . . . . . . . . . . . . . . 470
18.28 Askey{Wilson Class . . . . . . . . . . . . 472
18.29 Asymptotic Approximations for q-Hahn
and Askey{Wilson Classes . . . . . . . . 474
18.30 Associated OP's . . . . . . . . . . . . . . 474
18.31 Bernstein{Szeg o Polynomials . . . . . . . 474
18.32 OP's with Respect to Freud Weights . . . 475
18.33 Polynomials Orthogonal on the Unit Circle 475
18.34 Bessel Polynomials . . . . . . . . . . . . 476
18.35 Pollaczek Polynomials . . . . . . . . . . . 476
18.36 Miscellaneous Polynomials . . . . . . . . 477
18.37 Classical OP's in Two or More Variables . 477
Applications 478
18.38 Mathematical Applications . . . . . . . . 478
18.39 Physical Applications . . . . . . . . . . . 479
Computation 479
18.40 Methods of Computation . . . . . . . . . 479
18.41 Tables . . . . . . . . . . . . . . . . . . . 480
18.42 Software . . . . . . . . . . . . . . . . . . 480
References 480
1University of Amsterdam, Korteweg{de Vries Institute, Amsterdam, The Netherlands.
2City University of Hong Kong, Liu Bie Ju Centre for Mathematical Sciences, Kowloon, Hong Kong.
3Delft University of Technology, Delft Institute of Applied Mathematics, Delft, The Netherlands.
4Vrije Universiteit Amsterdam, Department of Mathematics, Amsterdam, The Netherlands.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
435
436 Orthogonal Polynomials
Notation
18.1 Notation
18.1(i) Special Notation
(For other notation see pp. xiv and 873.)
x;y real variables.
z(=x+iy) complex variable.
q real variable such that 0 <q< 1, unless
stated otherwise.
`;m nonnegative integers.
n nonnegative integer, except in x18.30.
N positive integer.
(x a) Dirac delta ( x1.17).
arbitrary small positive constant.
pn(x) polynomial in xof degreen.
p 1(x) 0.
w(x) weight function ( 0) on an open
interval (a;b).
wx weights (>0) at points x2Xof a nite
or countably in nite subset of R.
OP's orthogonal polynomials.
x-Di erences
Forward di erences:
x(f(x)) =f(x+ 1) f(x);
n+1
x(f(x)) = x( n
x(f(x))):
Backward di erences:
rx(f(x)) =f(x) f(x 1);
rn+1
x(f(x)) =rx(rn
x(f(x))):
Central di erences in imaginary direction:
x(f(x)) =
f(x+1
2i) f(x 1
2i)
=i;
n+1
x(f(x)) =x(n
x(f(x))):
q-Pochhammer Symbol
(z;q)0= 1;(z;q)n= (1 z)(1 zq)(1 zqn 1);
(z1;:::;zk;q)n= (z1;q)n(zk;q)n:
In niteq-Product
(z;q)1=1Y
j=0(1 zqj);
(z1;:::;zk;q)1= (z1;q)1(zk;q)1:
18.1(ii) Main Functions
The main functions treated in this chapter are:Classical OP's
Jacobi:P( ; )
n(x).
Ultraspherical (or Gegenbauer): C()
n(x).
Chebyshev of rst, second, third, and fourth kinds:
Tn(x),Un(x),Vn(x),Wn(x).
Shifted Chebyshev of rst and second kinds: T
n(x),
U
n(x).
Legendre:Pn(x).
Shifted Legendre: P
n(x).
Laguerre:L( )
n(x) andLn(x) =L(0)
n(x). (L( )
n(x) with
6= 0 is also called Generalized Laguerre.)
Hermite:Hn(x),Hen(x).
Hahn Class OP's
Hahn:Qn(x; ; ;N ).
Krawtchouk: Kn(x;p;N).
Meixner:Mn(x; ;c).
Charlier:Cn(x;a).
Continuous Hahn: pn
x;a;b;a;b
.
Meixner{Pollaczek: P()
n(x; ).
Wilson Class OP's
Wilson:Wn(x;a;b;c;d ).
Racah:Rn(x; ; ;
; ).
Continuous Dual Hahn: Sn(x;a;b;c ).
Dual Hahn: Rn(x;
;;N ).
q-Hahn Class OP's
q-Hahn:Qn(x; ; ;N ;q).
Bigq-Jacobi:Pn(x;a;b;c ;q).
Littleq-Jacobi:pn(x;a;b;q).
q-Laguerre: L( )
n(x;q).
Stieltjes{Wigert: Sn(x;q).
Discreteq-Hermite I: hn(x;q).
Discreteq-Hermite II: ~hn(x;q).
Askey{Wilson Class OP's
Askey{Wilson: pn(x;a;b;c;djq).
Al-Salam{Chihara: Qn(x;a;bjq).
Continuous q-Ultraspherical: Cn(x; jq).
Continuous q-Hermite:Hn(xjq).
Continuous q 1-Hermite:hn(xjq)
q-Racah:Rn(x; ; ;
;jq).
General Orthogonal Polynomials 437
Other OP's
Bessel:yn(x;a).
Pollaczek:P()
n(x;a;b).
Classical OP's in Two Variables
Disk:R( )
m;n(z).
Triangle:P ; ;
m;n (x;y).
18.1(iii) Other Notations
In Szeg o (1975, x4.7) the ultraspherical polynomials
C()
n(x) are denoted by P()
n(x). The ultraspherical
polynomials will not be considered for = 0. They
are de ned in the literature by C(0)
0(x) = 1 and
18.1.1C(0)
n(x) =2
nTn(x) =2(n 1)! 1
2
nP( 1
2; 1
2)
n (x),
n= 1;2;3;:::.
Nor do we consider the shifted Jacobi polynomials:
18.1.2Gn(p;q;x ) =n!
(n+p)nP(p q;q 1)
n (2x 1);
or the dilated Chebyshev polynomials of the rst and
second kinds:
18.1.3Cn(x) = 2Tn 1
2x
; Sn(x) =Un 1
2x
:
In Koekoek and Swarttouw (1998) xdenotes the
operatorix.
General Orthogonal Polynomials
18.2 General Orthogonal Polynomials
18.2(i) De nition
Orthogonality on Intervals
Let (a;b) be a nite or in nite open interval in R. A sys-
tem (or set) of polynomials fpn(x)g,n= 0;1;2;:::; is
said to be orthogonal on (a;b)with respect to the weight
functionw(x) (0)if
18.2.1Zb
apn(x)pm(x)w(x)dx= 0,n6=m.
Herew(x) is continuous or piecewise continuous or in-
tegrable, and such that 0 <Rb
ax2nw(x)dx <1for all
n.
It is assumed throughout this chapter that for each
polynomial pn(x) that is orthogonal on an open interval
(a;b) the variable xis con ned to the closure of ( a;b)
unless indicated otherwise. (However, under appropri-
ate conditions almost all equations given in the chapter
can be continued analytically to various complex values
of the variables.)Orthogonality on Finite Point Sets
LetXbe a nite set of distinct points on R, or a count-
able in nite set of distinct points on R, andwx,x2X,
be a set of positive constants. Then a system of polyno-
mialsfpn(x)g,n= 0;1;2;:::; is said to be orthogonal
onXwith respect to the weightswxif
18.2.2X
x2Xpn(x)pm(x)wx= 0, n6=m,
whenXis in nite, or
18.2.3X
x2Xpn(x)pm(x)wx= 0,n;m = 0;1;:::;N ;n6=m,
whenXis a nite set of N+ 1 distinct points. In the
former case we also require
18.2.4X
x2Xx2nwx<1,n= 0;1;:::,
whereas in the latter case the system fpn(x)gis nite:
n= 0;1;:::;N .
More generally than (18.2.1){(18.2.3), w(x)dxmay
be replaced in (18.2.1) by a positive measure d (x),
where (x) is a bounded nondecreasing function on the
closure of (a;b) with an in nite number of points of in-
crease, and such that 0 <Rb
ax2nd (x)<1for all
n. See McDonald and Weiss (1999, Chapters 3, 4) and
Szeg o (1975,x1.4).
18.2(ii)x-Di erence Operators
If the orthogonality discrete set Xisf0;1;:::;Ngor
f0;1;2;:::g, then the role of the di erentiation op-
eratord/dxin the case of classical OP's ( x18.3) is
played by x, the forward-di erence operator, or by
rx, the backward-di erence operator; compare x18.1(i).
This happens, for example, with the Hahn class OP's
(x18.20(i)).
If the orthogonality interval is ( 1;1) or (0;1),
then the role of d/dxcan be played by x, the
central-di erence operator in the imaginary direction
(x18.1(i)). This happens, for example, with the contin-
uous Hahn polynomials and Meixner{Pollaczek polyno-
mials (x18.20(i)).
18.2(iii) Normalization
The orthogonality relations (18.2.1){(18.2.3) each de-
termine the polynomials pn(x) uniquely up to constant
factors, which may be xed by suitable normalization.
If we de ne
18.2.5hn=Zb
a(pn(x))2w(x)dxorX
x2X(pn(x))2wx;
438 Orthogonal Polynomials
18.2.6
~hn=Zb
ax(pn(x))2w(x)dxorX
x2Xx(pn(x))2wx;
and
18.2.7pn(x) =knxn+~knxn 1+~~knxn 2+;
then two special normalizations are: (i) orthonormal
OP's :hn= 1,kn>0; (ii) monic OP's :kn= 1.
18.2(iv) Recurrence Relations
As inx18.1(i) we assume that p 1(x)0.
First Form
18.2.8
pn+1(x) = (Anx+Bn)pn(x) Cnpn 1(x),n0.
HereAn,Bn(n0), andCn(n1) are real constants,
andAn 1AnCn>0 forn1. Then
18.2.9
An=kn+1
kn; Bn= ~kn+1
kn+1 ~kn
kn!
An= ~hn
hnAn;
Cn=An~~kn+Bn~kn ~~kn+1
kn 1=An
An 1hn
hn 1:
Second Form
18.2.10
xpn(x) =anpn+1(x) +bnpn(x) +cnpn 1(x),n0.
Herean,bn(n0),cn(n1) are real constants, and
an 1cn>0 (n1). Then
18.2.11an=kn
kn+1; bn=~kn
kn ~kn+1
kn+1=~hn
hn;
cn=~~kn an~~kn+1 bn~kn
kn 1=an 1hn
hn 1:
If the OP's are orthonormal, then cn=an 1(n1).
If the OP's are monic, then an= 1 (n0).
Conversely, if a system of polynomials fpn(x)gsat-
is es (18.2.10) with an 1cn>0 (n1), thenfpn(x)gis orthogonal with respect to some positive measure on
R(Favard's theorem). The measure is not necessarily
of the form w(x)dxnor is it necessarily unique.
18.2(v) Christo el{Darboux Formula
18.2.12
nX
`=0p`(x)p`(y)
h`=kn
hnkn+1pn+1(x)pn(y) pn(x)pn+1(y)
x y,
x6=y.
Con
uent Form
18.2.13
nX
`=0(p`(x))2
h`=kn
hnkn+1
p0
n+1(x)pn(x) p0
n(x)pn+1(x)
:
18.2(vi) Zeros
Allnzeros of an OP pn(x) are simple, and they are
located in the interval of orthogonality ( a;b). The zeros
ofpn(x) andpn+1(x) separate each other, and if m<n
then between any two zeros of pm(x) there is at least
one zero of pn(x).
For illustrations of these properties see Figures
18.4.1{18.4.7.
Classical Orthogonal Polynomials
18.3 De nitions
Table 18.3.1 provides the de nitions of Jacobi, Laguerre,
and Hermite polynomials via orthogonality and normal-
ization (xx18.2(i) and 18.2(iii)). This table also includes
the following special cases of Jacobi polynomials: ultra-
spherical, Chebyshev, and Legendre.
18.3 Definitions 439Table 18.3.1 : Orthogonality properties for classical OP's: intervals, weight functions, normalizations, leading coecients, and parameter constraints.
In the second row Adenotes 2 + +1 (n+ + 1) (n+ + 1)
((2n+ + + 1) (n+ + + 1)n!). For further implications of the parameter
constraints see the Note inx18.5(iii).
Name pn(x) (a;b) w(x) hn kn~kn.
kn Constraints
Jacobi P( ; )
n(x) ( 1;1) (1 x) (1 +x) A(n+ + + 1)n
2nn!n( )
2n+ + ; > 1
Ultraspherical
(Gegenbauer)C()
n(x) ( 1;1) (1 x2) 1
221 2 (n+ 2)
(n+) ( ())2n!2n()n
n!0> 1
2;6= 0
Chebyshev
of rst kindTn(x) ( 1;1) (1 x2) 1
2(
1
2; n> 0
; n = 0(
2n 1; n> 0
1; n = 00
Chebyshev
of second kindUn(x) ( 1;1) (1 x2)1
21
2 2n0
Chebyshev
of third kindVn(x) ( 1;1) (1 x)1
2(1 +x) 1
2 2n 1
2
Chebyshev
of fourth kindWn(x) ( 1;1) (1 x) 1
2(1 +x)1
2 2n 1
2
Shifted Chebyshev
of rst kindT
n(x) (0;1) ( x x2) 1
2(
1
2; n> 0
; n = 0(
22n 1; n> 0
1; n = 0 1
2n
Shifted Chebyshev
of second kindU
n(x) (0;1) ( x x2)1
21
8 22n 1
2n
Legendre Pn(x) ( 1;1) 1 2/(2 n+ 1) 2n 1
2
n
n! 0
Shifted Legendre P
n(x) (0;1) 1 1/(2 n+ 1) 22n 1
2
n
n! 1
2n
Laguerre L( )
n(x) (0;1)e xx (n+ + 1)/n! ( 1)n/n! n(n+ ) > 1
Hermite Hn(x) ( 1;1)e x21
22nn! 2n0
Hermite Hen(x) ( 1;1)e 1
2x2(2)1
2n! 1 0
440 Orthogonal Polynomials
For exact values of the coecients of the Jacobi
polynomials P( ; )
n(x), the ultraspherical polynomials
C()
n(x), the Chebyshev polynomials Tn(x) andUn(x),
the Legendre polynomials Pn(x), the Laguerre poly-
nomialsLn(x), and the Hermite polynomials Hn(x),
see Abramowitz and Stegun (1964, pp. 793{801). The
Jacobi polynomials are in powers of x 1 forn=
0;1;:::; 6. The ultraspherical polynomials are in pow-
ers ofxforn= 0;1;:::; 6. The other polynomials are
in powers of xforn= 0;1;:::; 12. See alsox18.5(iv).
Chebyshev
In this chapter, formulas for the Chebyshev polynomials
of the second, third, and fourth kinds will not be given
as extensively as those of the rst kind. However, most
of these formulas can be obtained by specialization of
formulas for Jacobi polynomials, via (18.7.4){(18.7.6).
In addition to the orthogonal property given by
Table 18.3.1, the Chebyshev polynomials Tn(x),n=
0;1;:::;N , are orthogonal on the discrete point set com-
prising the zeros xN+1;n;n= 1;2;:::;N +1, ofTN+1(x):
18.3.1N+1X
n=1Tj(xN+1;n)Tk(xN+1;n) = 0,
0jN, 0kN,j6=k,
where
18.3.2xN+1;n= cos
(n 1
2)=(N+ 1)
:
Whenj=k6= 0 the sum in (18.3.1) is1
2(N+ 1). When
j=k= 0 the sum in (18.3.1) is N+ 1.For proofs of these results and for similar properties
of the Chebyshev polynomials of the second, third, and
fourth kinds see Mason and Handscomb (2003, x4.6).
For another version of the discrete orthogonality
property of the polynomials Tn(x) see (3.11.9).
Legendre
Legendre polynomials are special cases of Legendre
functions, Ferrers functions, and associated Legendre
functions (x14.7(i)). In consequence, additional proper-
ties are included in Chapter 14.
18.4 Graphics
18.4(i) Graphs
Figure 18.4.1 : Jacobi polynomials P(1:5; 0:5)
n (x),n=
1;2;3;4;5.
Figure 18.4.2 : Jacobi polynomials P(1:25;0:75)
n (x),n=
7;8. This illustrates inequalities for extrema of a Jacobi
polynomial; see (18.14.16). See also Askey (1990).
Figure 18.4.3 : Chebyshev polynomials Tn(x),n=
1;2;3;4;5.
18.4 Graphics 441
Figure 18.4.4 : Legendre polynomials Pn(x),n=
1;2;3;4;5.
Figure 18.4.5 : Laguerre polynomials Ln(x),n=
1;2;3;4;5.
Figure 18.4.6 : Laguerre polynomials L( )
3(x), =
0;1;2;3;4.
Figure 18.4.7 : Monic Hermite polynomials hn(x) =
2 nHn(x),n= 1;2;3;4;5.
18.4(ii) Surfaces
Figure 18.4.8 : Laguerre polynomials L( )
3(x), 0 3,
0x10.
Figure 18.4.9 : Laguerre polynomials L( )
4(x), 0 3,
0x10.
442 Orthogonal Polynomials
18.5 Explicit Representations
18.5(i) Trigonometric Functions
Chebyshev
Withx= cos,
18.5.1 Tn(x) = cos(n);
18.5.2 Un(x) = (sin (n+ 1))/sin;
18.5.3 Vn(x) = (sin (n+1
2))
sin 1
2
;
18.5.4Wn(x) = (cos (n+1
2))
cos 1
2
:
18.5(ii) Rodrigues Formulas
18.5.5pn(x) =1
nw(x)dn
dxn(w(x)(F(x))n):
In this equation w(x) is as in Table 18.3.1, and F(x),
nare as in Table 18.5.1.
Table 18.5.1 : Classical OP's: Rodrigues formulas
(18.5.5).
pn(x)F(x) n
P( ; )
n(x) 1 x2( 2)nn!
C()
n(x) 1 x2( 2)n
+1
2
nn!
(2)n
Tn(x) 1 x2( 2)n 1
2
n
Un(x) 1 x2( 2)n 3
2
n
n+ 1
Vn(x) 1 x2( 2)n 3
2
n
2n+ 1
Wn(x) 1 x2( 2)n 1
2
n
Pn(x) 1 x2( 2)nn!
L( )
n(x)x n !
Hn(x) 1 ( 1)n
Hen(x) 1 ( 1)nRelated formula:
18.5.6
L( )
n1
x
=( 1)n
n!xn+ +1e1/xdn
dxn
x 1e 1/x
:
18.5(iii) Finite Power Series, the
Hypergeometric Function, and
Generalized Hypergeometric Functions
For the de nitions of 2F1,1F1, and 2F0seex16.2.
Jacobi
18.5.7
P( ; )
n(x)
=nX
`=0(n+ + + 1)`( +`+ 1)n `
`! (n `)!x 1
2`
=( + 1)n
n!2F1 n;n+ + + 1
+ 1;1 x
2
;
18.5.8
P( ; )
n(x)
= 2 nnX
`=0n+
`n+
n `
(x 1)n `(x+ 1)`
=( + 1)n
n!x+ 1
2n
2F1 n; n
+ 1;x 1
x+ 1
;
and two similar formulas by symmetry; compare the
second row in Table 18.6.1.
Ultraspherical
18.5.9C()
n(x) =(2)n
n!2F1 n;n+ 2
+1
2;1 x
2
;
18.5.10
C()
n(x) =bn=2cX
`=0( 1)`()n `
`! (n 2`)!(2x)n 2`
= (2x)n()n
n!2F1 1
2n; 1
2n+1
2
1 n;1
x2
;
18.5.11C()
n(cos) =nX
`=0()`()n `
`! (n `)!cos((n 2`))
=ein()n
n!2F1 n;
1 n;e 2i
:
18.6 Symmetry, Special Values, and Limits to Monomials 443
Laguerre
18.5.12L( )
n(x) =nX
`=0( +`+ 1)n `
(n `)!`!( x)`
=( + 1)n
n!1F1 n
+ 1;x
:
Hermite
18.5.13Hn(x) =n!bn=2cX
`=0( 1)`(2x)n 2`
`! (n 2`)!
= (2x)n
2F0 1
2n; 1
2n+1
2
; 1
x2
:
For corresponding formulas for Chebyshev, Legen-
dre, and the Hermite Henpolynomials apply (18.7.3){
(18.7.6), (18.7.9), and (18.7.11).
Note. The rst of each of equations (18.5.7) and
(18.5.8) can be regarded as de nitions of P( ; )
n(x) when
the conditions > 1 and > 1 are not satis ed.
However, in these circumstances the orthogonality prop-
erty (18.2.1) disappears. For this reason, and also in the
interest of simplicity, in the case of the Jacobi polyno-
mialsP( ; )
n(x) we assume throughout this chapter that
> 1 and > 1,unless stated otherwise . Similarly
in the cases of the ultraspherical polynomials C()
n(x)
and the Laguerre polynomials L( )
n(x) we assume that
> 1
2;6= 0, and > 1,unless stated otherwise .
18.5(iv) Numerical Coecients
Chebyshev
18.5.14T0(x) = 1; T 1(x) =x; T 2(x) = 2x2 1;
T3(x) = 4x3 3x; T 4(x) = 8x4 8x2+ 1;
T5(x) = 16x5 20x3+ 5x;
T6(x) = 32x6 48x4+ 18x2 1:
18.5.15U0(x) = 1; U 1(x) = 2x; U 2(x) = 4x2 1;
U3(x) = 8x3 4x; U 4(x) = 16x4 12x2+ 1;
U5(x) = 32x5 32x3+ 6x;
U6(x) = 64x6 80x4+ 24x2 1:Legendre
18.5.16P0(x) = 1; P 1(x) =x; P 2(x) =3
2x2 1
2;
P3(x) =5
2x3 3
2x; P 4(x) =35
8x4 15
4x2+3
8;
P5(x) =63
8x5 35
4x3+15
8x;
P6(x) =231
16x6 315
16x4+105
16x2 5
16:
Laguerre
18.5.17
L0(x) = 1; L 1(x) = x+ 1; L 2(x) =1
2x2 2x+ 1;
L3(x) = 1
6x3+3
2x2 3x+ 1;
L4(x) =1
24x4 2
3x3+ 3x2 4x+ 1;
L5(x) = 1
120x5+5
24x4 5
3x3+ 5x2 5x+ 1;
L6(x) =1
720x6 1
20x5+5
8x4 10
3x3+15
2x2 6x+ 1:
Hermite
18.5.18
H0(x) = 1; H 1(x) = 2x; H 2(x) = 4x2 2;
H3(x) = 8x3 12x; H 4(x) = 16x4 48x2+ 12;
H5(x) = 32x5 160x3+ 120x;
H6(x) = 64x6 480x4+ 720x2 120:
18.5.19He0(x) = 1;He1(x) =x; He2(x) =x2 1;
He3(x) =x3 3x; He4(x) =x4 6x2+ 3;
He5(x) =x5 10x3+ 15x;
He6(x) =x6 15x4+ 45x2 15:
For the corresponding polynomials of degrees 7
through 12 see Abramowitz and Stegun (1964, Ta-
bles 22.3, 22.5, 22.9, 22.10, 22.12).
18.6 Symmetry, Special Values, and Limits
to Monomials
18.6(i) Symmetry and Special Values
For Jacobi, ultraspherical, Chebyshev, Legendre, and
Hermite polynomials, see Table 18.6.1.
Laguerre
18.6.1 L( )
n(0) =( + 1)n
n!:
444 Orthogonal Polynomials
Table 18.6.1 : Classical OP's: symmetry and special values.
pn(x)pn( x) pn(1) p2n(0) p0
2n+1(0)
P( ; )
n(x) ( 1)nP( ; )
n(x) ( + 1)n/n!
P( ; )
n (x) ( 1)nP( ; )
n (x) ( + 1)n/n! ( 1
4)n(n+ + 1)n
n! ( 1
4)n(n+ + 1)n+1
n!
C()
n(x) ( 1)nC()
n(x) (2)n/n! ( 1)n()n/n! 2( 1)n()n+1
n!
Tn(x) ( 1)nTn(x) 1 ( 1)n( 1)n(2n+ 1)
Un(x) ( 1)nUn(x)n+ 1 ( 1)n( 1)n(2n+ 2)
Vn(x) ( 1)nWn(x) 2n+ 1 ( 1)n( 1)n(2n+ 2)
Wn(x) ( 1)nVn(x) 1 ( 1)n( 1)n(2n+ 2)
Pn(x) ( 1)nPn(x) 1 ( 1)n 1
2
n
n! 2( 1)n 1
2
n+1.
n!
Hn(x) ( 1)nHn(x) ( 1)n(n+ 1)n2( 1)n(n+ 1)n+1
Hen(x) ( 1)nHen(x) ( 1
2)n(n+ 1)n ( 1
2)n(n+ 1)n+1
18.6(ii) Limits to Monomials
18.6.2 lim
!1P( ; )
n(x)
P( ; )
n(1)=1 +x
2n
;
18.6.3 lim
!1P( ; )
n(x)
P( ; )
n( 1)=1 x
2n
;
18.6.4 lim
!1C()
n(x)
C()
n(1)=xn;
18.6.5 lim
!1L( )
n( x)
L( )
n(0)= (1 x)n:
18.7 Interrelations and Limit Relations
18.7(i) Linear Transformations
Ultraspherical and Jacobi
18.7.1C()
n(x) =(2)n
+1
2
nP( 1
2; 1
2)
n (x);
18.7.2P( ; )
n (x) =( + 1)n
(2 + 1)nC( +1
2)
n (x):Chebyshev, Ultraspherical, and Jacobi
18.7.3Tn(x) =P( 1
2; 1
2)
n (x).
P( 1
2; 1
2)
n (1);
18.7.4
Un(x) =C(1)
n(x) = (n+ 1)P(1
2;1
2)
n (x).
P(1
2;1
2)
n (1);
18.7.5Vn(x) = (2n+ 1)P(1
2; 1
2)
n (x).
P(1
2; 1
2)
n (1);
18.7.6Wn(x) =P( 1
2;1
2)
n (x).
P( 1
2;1
2)
n (1):
18.7.7 T
n(x) =Tn(2x 1);
18.7.8 U
n(x) =Un(2x 1):
See also (18.9.9){(18.9.12).
Legendre, Ultraspherical, and Jacobi
18.7.9 Pn(x) =C(1
2)
n(x) =P(0;0)
n(x):
18.7.10 P
n(x) =Pn(2x 1):
Hermite
18.7.11 Hen(x) = 2 1
2nHn
2 1
2x
;
18.7.12 Hn(x) = 21
2nHen
21
2x
:
18.8 Differential Equations 445
18.7(ii) Quadratic Transformations
18.7.13P( ; )
2n(x)
P( ; )
2n(1)=P( ; 1
2)
n
2x2 1
P( ; 1
2)
n (1);
18.7.14P( ; )
2n+1(x)
P( ; )
2n+1(1)=xP( ;1
2)
n
2x2 1
P( ;1
2)
n (1):
18.7.15C()
2n(x) =()n 1
2
nP( 1
2; 1
2)
n
2x2 1
;
18.7.16C()
2n+1(x) =()n+1 1
2
n+1xP( 1
2;1
2)
n
2x2 1
:
18.7.17 U2n(x) =Vn
2x2 1
;
18.7.18 T2n+1(x) =xWn
2x2 1
:
18.7.19H2n(x) = ( 1)n22nn!L( 1
2)
n
x2
;
18.7.20H2n+1(x) = ( 1)n22n+1n!xL(1
2)
n
x2
:
18.7(iii) Limit Relations
Jacobi!Laguerre
18.7.21 lim
!1P( ; )
n(1 (2x/ )) =L( )
n(x):18.7.22 lim
!1P( ; )
n((2x= ) 1) = ( 1)nL( )
n(x):
Jacobi!Hermite
18.7.23 lim
!1 1
2nP( ; )
n
1
2x
=Hn(x)
2nn!:
Ultraspherical!Hermite
18.7.24 lim
!1 1
2nC()
n
1
2x
=Hn(x)
n!:
18.7.25 lim
!01
C()
n(x) =2
nTn(x),n1.
Laguerre!Hermite
18.7.26
lim
!12
1
2n
L( )
n
(2 )1
2x+
=( 1)n
n!Hn(x):
See Figure 18.21.1 for the Askey schematic represen-
tation of most of these limits.
18.8 Di erential Equations
See Table 18.8.1 and also Table 22.6 of Abramowitz and
Stegun (1964).
Table 18.8.1 : Classical OP's: di erential equations A(x)f00(x) +B(x)f0(x) +C(x)f(x) +nf(x) = 0.
f(x) A(x) B(x) C(x) n
P( ; )
n(x) 1 x2 ( + + 2)x 0 n(n+ + + 1)
sin1
2x +1
2
cos1
2x +1
2
P( ; )
n(cosx)1 01
4 2
4 sin21
2x+1
4 2
4 cos21
2x
n+1
2( + + 1)2
(sinx) +1
2P( ; )
n (cosx) 1 0 (1
4 2)=sin2x (n+ +1
2)2
C()
n(x) 1 x2 (2+ 1)x 0 n(n+ 2)
Tn(x) 1 x2 x 0 n2
Un(x) 1 x2 3x 0 n(n+ 2)
Pn(x) 1 x2 2x 0 n(n+ 1)
L( )
n(x) x + 1 x 0 n
e 1
2x2x +1
2L( )
n
x2
1 0 x2+ (1
4 2)x 24n+ 2 + 2
Hn(x) 1 2x 0 2 n
e 1
2x2Hn(x) 1 0 x22n+ 1
Hen(x) 1 x 0 n
446 Orthogonal Polynomials
18.9 Recurrence Relations and Derivatives
18.9(i) Recurrence Relations
18.9.1pn+1(x) = (Anx+Bn)pn(x) Cnpn 1(x):
Forpn(x) =P( ; )
n(x),
18.9.2An=(2n+ + + 1)(2n+ + + 2)
2(n+ 1)(n+ + + 1);
Bn=( 2 2)(2n+ + + 1)
2(n+ 1)(n+ + + 1)(2n+ + );
Cn=(n+ )(n+ )(2n+ + + 2)
(n+ 1)(n+ + + 1)(2n+ + ):
For the other classical OP's see Table 18.9.1; compare
alsox18.2(iv).
Table 18.9.1 : Classical OP's: recurrence relations
(18.9.1).
pn(x)AnBnCn
C()
n(x)2n+
n+10n+2 1
n+1
Tn(x) 2 n;0 0 1
Un(x) 2 0 1
T
n(x) 4 2n;0 2 +n;0 1
U
n(x) 4 2 1
Pn(x)2n+1
n+10n
n+1
P
n(x)4n+2
n+1 2n+1
n+1n
n+1
L( )
n(x) 1
n+12n+ +1
n+1n+
n+1
Hn(x) 2 0 2 n
Hen(x) 1 0 n
18.9(ii) Contiguous Relations in the Parameters
and the Degree
Jacobi
18.9.3P( ; 1)
n (x) P( 1; )
n (x) =P( ; )
n 1(x);
18.9.4
(1 x)P( +1; )
n (x) + (1 +x)P( ; +1)
n (x) = 2P( ; )
n(x):
18.9.5
(2n+ + + 1)P( ; )
n(x)
= (n+ + + 1)P( ; +1)
n (x) + (n+ )P( ; +1)
n 1(x);
18.9.6(n+1
2 +1
2 + 1)(1 +x)P( ; +1)
n (x)
= (n+ 1)P( ; )
n+1(x) + (n+ + 1)P( ; )
n(x);
and a similar pair to (18.9.5) and (18.9.6) by symmetry;
compare the second row in Table 18.6.1.Ultraspherical
18.9.7 (n+)C()
n(x) =
C(+1)
n (x) C(+1)
n 2(x)
;
18.9.84(n++ 1)(1 x2)C(+1)
n (x)
= (n+ 1)(n+ 2)C()
n+2(x)
+ (n+ 2)(n+ 2+ 1)C()
n(x):
Chebyshev
18.9.9 Tn(x) =1
2(Un(x) Un 2(x));
18.9.10 (1 x2)Un(x) = 1
2(Tn+2(x) Tn(x)):
18.9.11Wn(x) +Wn 1(x) = 2Tn(x);
18.9.12 Tn+1(x) +Tn(x) = (1 +x)Wn(x):
Laguerre
18.9.13 L( )
n(x) =L( +1)
n (x) L( +1)
n 1(x);
18.9.14xL( +1)
n (x) = (n+ 1)L( )
n+1(x)
+ (n+ + 1)L( )
n(x):
18.9(iii) Derivatives
Jacobi
18.9.15d
dxP( ; )
n(x) =1
2(n+ + + 1)P( +1; +1)
n 1 (x);
18.9.16
d
dx
(1 x) (1 +x) P( ; )
n(x)
= 2(n+ 1)(1 x) 1(1 +x) 1P( 1; 1)
n+1 (x):
18.9.17(2n+ + )(1 x2)d
dxP( ; )
n(x)
=n( (2n+ + )x)P( ; )
n(x)
+ 2(n+ )(n+ )P( ; )
n 1(x);
18.9.18
(2n+ + + 2)(1 x2)d
dxP( ; )
n(x)
= (n+ + + 1) ( + (2n+ + + 2)x)P( ; )
n(x)
2(n+ 1)(n+ + + 1)P( ; )
n+1(x):
Ultraspherical
18.9.19d
dxC()
n(x) = 2C(+1)
n 1(x);
18.9.20
d
dx
(1 x2) 1
2C()
n(x)
= (n+ 1)(n+ 2 1)
2( 1)(1 x2) 3
2C( 1)
n+1(x):
18.10 Integral Representations 447
Chebyshev
18.9.21d
dxTn(x) =nUn 1(x);
18.9.22
d
dx
(1 x2)1
2Un(x)
= (n+ 1)(1 x2) 1
2Tn+1(x):
Laguerre
18.9.23d
dxL( )
n(x) = L( +1)
n 1(x);
18.9.24
d
dx
e xx L( )
n(x)
= (n+ 1)e xx 1L( 1)
n+1(x):Hermite
18.9.25d
dxHn(x) = 2nHn 1(x);
18.9.26d
dx
e x2Hn(x)
= e x2Hn+1(x):
18.9.27d
dxHen(x) =nHen 1(x);
18.9.28d
dx
e 1
2x2Hen(x)
= e 1
2x2Hen+1(x):
18.10 Integral Representations
18.10(i) Dirichlet-Mehler-Type Integral Representations
Ultraspherical
18.10.1P( ; )
n (cos)
P( ; )
n (1)=C( +1
2)
n (cos)
C( +1
2)
n (1)=2 +1
2 ( + 1)
1
2
+1
2(sin) 2 Z
0cos
(n+ +1
2)
(cos cos) +1
2d , 0<< , > 1
2.
Legendre
18.10.2 Pn(cos) =21
2
Z
0cos
(n+1
2)
(cos cos)1
2d , 0 << .
Generalizations of (18.10.1) are given in Gasper (1975, (6),(8)) and Koornwinder (1975b, (5.7),(5.8)).
18.10(ii) Laplace-Type Integral Representations
Jacobi
18.10.3P( ; )
n(cos)
P( ; )
n(1)=2 ( + 1)
1
2 ( )
+1
2
Z1
0Z
0
(cos1
2)2 r2(sin1
2)2+irsincos n(1 r2) 1r2 +1(sin )2 d dr ,
> > 1
2.
Ultraspherical
18.10.4P( ; )
n (cos)
P( ; )
n (1)=C( +1
2)
n (cos)
C( +1
2)
n (1)= ( + 1)
1
2 ( +1
2)Z
0(cos+isincos )n(sin )2 d , > 1
2.
Legendre
18.10.5 Pn(cos) =1
Z
0(cos+isincos )nd :
Laguerre
18.10.6 L( )
n
x2
=2( 1)n
1
2
+1
2
n!Z1
0Z
0(x2 r2+ 2ixrcos )ne r2r2 +1(sin )2 d dr , > 1
2.
Hermite
18.10.7 Hn(x) =2n
1
2Z1
1(x+it)ne t2dt:
448 Orthogonal Polynomials
18.10(iii) Contour Integral Representations
Table 18.10.1 gives contour integral representations of the form
18.10.8 pn(x) =g0(x)
2iZ
C(g1(z;x))ng2(z;x)(z c) 1dz
for the Jacobi, Laguerre, and Hermite polynomials. Here Cis a simple closed contour encircling z=conce in the
positive sense.
Table 18.10.1 : Classical OP's: contour integral representations (18.10.8).
pn(x) g0(x) g1(z;x)g2(z;x)c Conditions
P( ; )
n(x) (1 x) (1 +x) z2 1
2(z x)(1 z) (1 +z) x1 outsideC.
C()
n(x) 1 z 1(1 2xz+z2) 0
eioutsideC
(wherex= cos).Tn(x) 1 z 11 xz
1 2xz+z20
Un(x) 1 z 1(1 2xz+z2) 10
Pn(x) 1 z 1(1 2xz+z2) 1
20
Pn(x) 1z2 1
2(z x)1 x
L( )
n(x) exx z(z x) 1z e zx 0 outsideC.
Hn(x)=n! 1 z 1e2xz z20
Hen(x)=n! 1 z 1exz 1
2z20
18.10(iv) Other Integral Representations
Laguerre
18.10.9L( )
n(x) =exx 1
2
n!Z1
0e ttn+1
2 J
2p
xt
dt,
> 1.
For the Bessel function J(z) seex10.2(ii).
Hermite
18.10.10
Hn(x) =( 2i)nex2
1
2Z1
1e t2tne2ixtdt
=2n+1
1
2ex2Z1
0e t2tncos
2xt 1
2n
dt:
See alsox18.17.
18.11 Relations to Other Functions
18.11(i) Explicit Formulas
Seexx18.5(i) and 18.5(iii) for relations to trigonometric
functions, the hypergeometric function, and generalized
hypergeometric functions.Ultraspherical
18.11.1
Pm
n(x) = 1
2
m( 2)m(1 x2)1
2mC(m+1
2)
n m(x)
= (n+ 1)m( 2) m(1 x2)1
2mP(m;m)
n m(x),
0mn.
For the Ferrers function Pm
n(x), seex14.3(i).
Compare also (14.3.21) and (14.3.22).
Laguerre
18.11.2
L( )
n(x) =( + 1)n
n!M( n; + 1;x)
=( 1)n
n!U( n; + 1;x)
=( + 1)n
n!z 1
2( +1)e1
2zMn+1
2( +1);1
2 (z)
=( 1)n
n!z 1
2( +1)e1
2zWn+1
2( +1);1
2 (z):
For the con
uent hypergeometric functions M(a;b;z )
andU(a;b;z ), seex13.2(i), and for the Whittaker func-
tionsM;(z) andW;(z) seex13.14(i).
18.12 Generating Functions 449
Hermite
18.11.3Hn(x) = 2nU
1
2n;1
2;x2
= 2nxU
1
2n+1
2;3
2;x2
= 21
2ne1
2x2U
n 1
2;21
2x
:
18.11.4Hen(x) = 21
2nU
1
2n;1
2;1
2x2
= 21
2(n 1)xU
1
2n+1
2;3
2;1
2x2
=e1
4x2U
n 1
2;x
:
For the parabolic cylinder function U(a;z), seex12.2.
18.11(ii) Formulas of Mehler{Heine Type
Jacobi
18.11.5
lim
n!11
n P( ; )
n
1 z2
2n2
= lim
n!11
n P( ; )
n
cosz
n
=2
z J (z):
Laguerre
18.11.6 lim
n!11
n L( )
nz
n
=1
z1
2 J
2z1
2
:
Hermite
18.11.7 lim
n!1( 1)nn1
2
22nn!H2nz
2n1
2
=1
1
2cosz;
18.11.8 lim
n!1( 1)n
22nn!H2n+1z
2n1
2
=2
1
2sinz:
For the Bessel function J(z), seex10.2(ii). The
limits (18.11.5){(18.11.8) hold uniformly for zin any
bounded subset of C.
18.12 Generating Functions
With the notation of xx10.2(ii), 10.25(ii), and 15.2,
Jacobi
18.12.1
2 +
R(1 +R z) (1 +R+z)
=1X
n=0P( ; )
n(x)zn,R=p
1 2xz+z2,jzj<1.
18.12.2 1
2(1 x)z 1
2 J p
2(1 x)z
1
2(1 +x)z 1
2 I p
2(1 +x)z
=1X
n=0P( ; )
n(x)
(n+ + 1) (n+ + 1)zn:18.12.3
(1 +z) 1
2F11
2( + + 1);1
2( + + 2)
+ 1;2(x+ 1)z
(1 +z)2
=1X
n=0( + + 1)n
( + 1)nP( ; )
n(x)zn,jzj<1,
and a similar formula by symmetry; compare the second
row in Table 18.6.1. For the hypergeometric function
2F1seexx15.1, 15.2(i).
Ultraspherical
18.12.4
(1 2xz+z2) =1X
n=0C()
n(x)zn
=1X
n=0(2)n
+1
2
nP( 1
2; 1
2)
n (x)zn,
jzj<1.
18.12.5
1 xz
(1 2xz+z2)+1=1X
n=0n+ 2
2C()
n(x)zn,jzj<1.
18.12.6
+1
2
ezcos(1
2zsin)1
2 J 1
2(zsin)
=1X
n=0C()
n(cos)
(2)nzn, 0.
Chebyshev
18.12.71 z2
1 2xz+z2= 1 + 21X
n=1Tn(x)zn,jzj<1.
18.12.81 xz
1 2xz+z2=1X
n=0Tn(x)zn,jzj<1.
18.12.9 ln
1 2xz+z2
= 21X
n=1Tn(x)
nzn,jzj<1.
18.12.101
1 2xz+z2=1X
n=0Un(x)zn,jzj<1.
Legendre
18.12.111p
1 2xz+z2=1X
n=0Pn(x)zn,jzj<1.
18.12.12exzJ0
zp
1 x2
=1X
n=0Pn(x)
n!zn:
Laguerre
18.12.13
(1 z) 1expxz
z 1
=1X
n=0L( )
n(x)zn,jzj<1.
18.12.14
( + 1)(xz) 1
2 ezJ
2pxz
=1X
n=0L( )
n(x)
( + 1)nzn:
450 Orthogonal Polynomials
Hermite
18.12.15 e2xz z2=1X
n=0Hn(x)
n!zn;
18.12.16 exz 1
2z2=1X
n=0Hen(x)
n!zn:
18.13 Continued Fractions
We use the terminology of x1.12(ii).
Chebyshev
Tn(x) is the denominator of the nth approximant to:
18.13.1 1
x+ 1
2x+ 1
2x+;
andUn(x) is the denominator of the nth approximant
to:
18.13.2 1
2x+ 1
2x+ 1
2x+:
Legendre
Pn(x) is the denominator of the nth approximant to:
18.13.3a1
x+ 1
2
3
2x+ 2
3
5
3x+ 3
4
7
4x+;
wherea1is an arbitrary nonzero constant.
Laguerre
Ln(x) is the denominator of the nth approximant to:
18.13.4
a1
1 x+ 1
2
1
2(3 x) + 2
3
1
3(5 x) + 3
4
1
4(7 x)+;
wherea1is again an arbitrary nonzero constant.
Hermite
Hn(x) is the denominator of the nth approximant to:
18.13.51
2x+ 2
2x+ 4
2x+ 6
2x+:
See also Cuyt et al. (2008, pp. 91{99).
18.14 Inequalities
18.14(i) Upper Bounds
Jacobi
18.14.1jP( ; )
n(x)jP( ; )
n(1) =( + 1)n
n!,
1x1, > 1, 1
2;
18.14.2jP( ; )
n(x)jjP( ; )
n( 1)j=( + 1)n
n!,
1x1, > 1, 1
2.
18.14.3 1
2(1 x)1
2 +1
4 1
2(1 +x)1
2 +1
4jP( ; )
n(x)j
(max( ; ) +n+ 1)
1
2n!
n+1
2( + + 1)max( ; )+1
2,
1x1, 1
2 1
2, 1
2 1
2.Ultraspherical
18.14.4
jC()
n(x)jC()
n(1) =(2)n
n!, 1x1,>0.
18.14.5jC()
2m(x)jjC()
2m(0)j= ()m
m! ,
1x1, 1
2<< 0,
18.14.6jC()
2m+1(x)j< 2()m+1
((2m+ 1)(2+ 2m+ 1))1
2m!,
1x1, 1
2<< 0.
18.14.7(n+)1 (1 x2)1
2jC()
n(x)j<21
(),
1x1, 0<< 1.
Laguerre
18.14.8e 1
2x L( )
n(x) L( )
n(0) =( + 1)n
n!,
0x<1, 0.
Hermite
18.14.91
(2nn!)1
2e 1
2x2jHn(x)j1, 1<x<1.
For further inequalities see Abramowitz and Stegun
(1964,x22.14).
18.14(ii) Turan-Type Inequalities
Legendre
18.14.10 (Pn(x))2Pn 1(x)Pn+1(x), 1x1.
Jacobi
LetRn(x) =P( ; )
n(x)=P( ; )
n(1). Then
18.14.11
(Rn(x))2Rn 1(x)Rn+1(x), 1x1, > 1.
Laguerre
18.14.12
(L( )
n(x))2L( )
n 1(x)L( )
n+1(x), 0x<1, 0.
Hermite
18.14.13 (Hn(x))2Hn 1(x)Hn+1(x), 1<x<1.
18.14(iii) Local Maxima and Minima
Jacobi
Let the maxima xn;m,m= 0;1;:::;n , ofjP( ; )
n(x)jin
[ 1;1] be arranged so that
18.14.14 1 =xn;0<xn;1<<xn;n 1<xn;n= 1:
When ( +1
2)( +1
2)>0 choosemso that
18.14.15xn;m( )=( + + 1)xn;m+1:
Then
18.15 Asymptotic Approximations 451
18.14.16jP( ; )
n(xn;0)j>jP( ; )
n(xn;1)j>>jP( ; )
n(xn;m)j;
jP( ; )
n(xn;n)j>jP( ; )
n(xn;n 1)j>>jP( ; )
n(xn;m+1)j, > 1
2; > 1
2:
18.14.17jP( ; )
n(xn;0)j<jP( ; )
n(xn;1)j<<jP( ; )
n(xn;m)j;
jP( ; )
n(xn;n)j<jP( ; )
n(xn;n 1)j<<jP( ; )
n(xn;m+1)j, 1< < 1
2; 1< < 1
2:
Also,
18.14.18
jP( ; )
n(xn;0)j<jP( ; )
n(xn;1)j<<jP( ; )
n(xn;n)j,
1
2, 1< 1
2,
18.14.19
jP( ; )
n(xn;0)j>jP( ; )
n(xn;1)j>>jP( ; )
n(xn;n)j,
1
2, 1< 1
2,
except that when = = 1
2(Chebyshev case)
jP( ; )
n(xn;m)jis constant.
Szeg o{Sz asz Inequality
18.14.20 P( ; )
n(xn;n m)
P( ; )
n(1) > P( ; )
n+1(xn+1;n m+1)
P( ; )
n+1(1) ,
= > 1
2,m= 1;2;:::;n .
For extensions of (18.14.20) see Askey (1990) and Wong
and Zhang (1994a,b).
Laguerre
Let the maxima xn;m,m= 0;1;:::;n 1, ofjL( )
n(x)j
in [0;1) be arranged so that
18.14.21 0 =xn;0<xn;1<<xn;n 1<xn;n=1:
When > 1
2choosemso that
18.14.22 xn;m +1
2xn;m+1:
Then
18.14.23
jL( )
n(xn;0)j>jL( )
n(xn;1)j>>jL( )
n(xn;m)j;
jL( )
n(xn;n 1)j>jL( )
n(xn;n 2)j>>jL( )
n(xn;m+1)j:
Also, when 1
2
18.14.24
jL( )
n(xn;0)j<jL( )
n(xn;1)j<<jL( )
n(xn;n 1)j:
Hermite
The successive maxima of jHn(x)jform a decreasing se-
quence forx0, and an increasing sequence for x0.
18.15 Asymptotic Approximations
18.15(i) Jacobi
With the exception of the penultimate paragraph, we
assume throughout this subsection that , , andM(= 0;1;2;:::) are all xed.
18.15.1
sin1
2 +1
2
cos1
2 +1
2P( ; )
n(cos)
= 122n+ + +1B(n+ + 1;n+ + 1)
M 1X
m=0fm()
2m(2n+ + + 2)m+O
n M!
;
asn!1 , uniformly with respect to 2[; ]. Here,
and elsewhere inx18.15,is an arbitrary small positive
constant. Also, B( a;b) is the beta function ( x5.12) and
18.15.2fm() =mX
`=0Cm;`( ; )
`!(m `)!cosn;m;`
sin1
2`
cos1
2m `;
where
18.15.3
Cm;`( ; ) = 1
2+
` 1
2
` 1
2+
m ` 1
2
m `;
and
18.15.4n;m;` =1
2(2n+ + +m+1) 1
2( +`+1
2):
When ; 2( 1
2;1
2), the error term in (18.15.1) is less
than twice the rst neglected term in absolute value.
See Hahn (1980), where corresponding results are given
whenxis replaced by a complex variable zthat is
bounded away from the orthogonality interval [ 1;1].
Next, let
18.15.5 =n+1
2( + + 1):
Then asn!1 ,
18.15.6(sin1
2) +1
2(cos1
2) +1
2P( ; )
n(cos)
= (n+ + 1)
21
2 n!
1
2J ()MX
m=0Am()
2m
+3
2J +1()M 1X
m=0Bm()
2m+1+"M(;)!
;
whereJ(z) is the Bessel function ( x10.2(ii)), and
18.15.7
"M(;) =(
O
2M (3=2)
; c 1 ;
+(5=2)O
2M+
;0c 1;
withcdenoting an arbitrary positive constant. Also,
18.15.8A0() = 1; B 0() =1
4g();
A1() =1
8g0() 1 + 2
8g()
1
32(g())2;
452 Orthogonal Polynomials
where
18.15.9
g() = 1
4 2
cot 1
2
1
2 1
1
4 2
tan 1
2
:
For higher coecients see Baratella and Gatteschi
(1988), and for another estimate of the error term see
Wong and Zhao (2003).
For large , xed , and 0n= c, Dunster
(1999) gives asymptotic expansions of P( ; )
n(z) that
are uniform in unbounded complex z-domains contain-
ingz=1. These expansions are in terms of Whittaker
functions (x13.14). This reference also supplies asymp-
totic expansions of P( ; )
n(z) for large n, xed , and
0 =nc. The latter expansions are in terms of
Bessel functions, and are uniform in complex z-domains
not containing neighborhoods of 1. For a complemen-
tary result, see Wong and Zhao (2004). By using the
symmetry property given in the second row of Table
18.6.1, the roles of and can be interchanged.
For an asymptotic expansion of P( ; )
n(z) asn!1
that holds uniformly for complex zbounded away from
[ 1;1], see Elliott (1971). The rst term of this expan-
sion also appears in Szeg o (1975, Theorem 8.21.7).
18.15(ii) Ultraspherical
For xed2(0;1) and xed M= 0;1;2;:::;
18.15.10
C()
n(cos) =22
+1
2
1
2 (+ 1)(2)n
(+ 1)n
M 1X
m=0()m(1 )m
m! (n++ 1)mcosn;m
(2 sin)m+
+O1
nM!
;
asn!1 uniformly with respect to 2[; ], where
18.15.11n;m= (n+m+) 1
2(m+):For a bound on the error term in (18.15.10) see Szeg o
(1975, Theorem 8.21.11).
Asymptotic expansions for C()
n(cos) can be ob-
tained from the results given in x18.15(i) by setting
= = 1
2and referring to (18.7.1). See also Szeg o
(1933) and Szeg o (1975, Eq. (8.21.14)).
18.15(iii) Legendre
For xedM= 0;1;2;:::,
18.15.12
Pn(cos) =2
sin1
2M 1X
m=0 1
2
mm 1
2
ncos n;m
(2 sin)m
+O1
nM+1
2
;
asn!1 , uniformly with respect to 2[; ],
where
18.15.13 n;m= (n m+1
2)+ (n 1
2m 1
4):
Also, when1
6 < <5
6, the right-hand side of
(18.15.12) with M=1converges; paradoxically, how-
ever, the sum is 2 Pn(cos) and notPn(cos) as stated
erroneously in Szeg o (1975, x8.4(3)).
For these results and further information see Olver
(1997b, pp. 311{313). For another form of the asymp-
totic expansion, complete with error bound, see Szeg o
(1975, Theorem 8.21.5).
For asymptotic expansions of Pn(cos) and
Pn(cosh) that are uniformly valid when 0
and 0 <1seex14.15(iii) with = 0 and=n.
These expansions are in terms of Bessel functions and
modi ed Bessel functions, respectively.
18.15(iv) Laguerre
In Terms of Elementary Functions
For xedM= 0;1;2;:::, and xed ,
18.15.14L( )
n(x) =n1
2 1
4e1
2x
1
2x1
2 +1
4
cos( )
n(x) M 1X
m=0am(x)
n1
2m+O1
n1
2M!
+ sin( )
n(x) M 1X
m=1bm(x)
n1
2m+O1
n1
2M!!
;
asn!1 , uniformly on compact x-intervals in (0 ;1), where
18.15.15 ( )
n(x) = 2(nx)1
2 1
2 +1
4
:
The leading coecients are given by
18.15.16 a0(x) = 1; a 1(x) = 0; b 1(x) =1
48x1
2
4x2 12 2 24 x 24x+ 3
:
In Terms of Bessel Functions
De ne
18.15.17 = 4n+ 2 + 2;
18.15 Asymptotic Approximations 453
18.15.18 =1
2p
x x2+ arcsin (px)
, 0 x1.
Then for xed M= 0;1;2;:::, and xed ,
18.15.19
L( )
n(x) =e1
2x
2 x1
2 +1
4(1 x)1
4
1
2J ()M 1X
m=0Am()
2m+ 1
2J +1()M 1X
m=0Bm()
2m+1+1
2envJ ()O1
2M 1!
;
asn!1 uniformly for 0x1 . HereJ(z) denotes the Bessel function ( x10.2(ii)), env J(z) denotes
its envelope (x2.8(iv)), and is again an arbitrary small positive constant. The leading coecients are given by
A0() = 1 and
18.15.20 B0() = 1
2
1 4 2
8+1 x
x1
2
4 2 1
8+1
4x
1 x+5
24x
1 x2!!
:
In Terms of Airy Functions
Again de ne as in (18.15.17); also,
18.15.21=
3
4
arccos px
p
x x22
3, 0 x1,
=
3
4p
x2 x arccosh px2
3, x1.
Then for xed M= 0;1;2;:::, and xed ,
18.15.22L( )
n(x)( 1)ne1
2x
2 1
2x1
2 +1
4
x 11
40
@Ai
2
3
1
3M 1X
m=0Em()
2m+Ai0
2
3
5
3M 1X
m=0Fm()
2m+ envAi
2
3
O1
2M 2
31
A;
asn!1 uniformly for x<1. Here Ai denotes the Airy function ( x9.2), Ai0denotes its derivative, and envAi
denotes its envelope ( x2.8(iii)). The leading coecients are given by E0() = 1 and
18.15.23 F0() = 5
482+x 1
x1
2
1
2 2 1
8 1
4x
x 1+5
24x
x 12!
, 0x<1.
18.15(v) Hermite
De ne
18.15.24 = 2n+ 1;
18.15.25n=(
(n+ 1)
1
2n+ 1
; n even;
(n+ 2).
1
2 1
2n+3
2
; n odd;
and
18.15.26 !n;m(x) =1
2x 1
2(m+n):
Then for xed M= 0;1;2;:::,
18.15.27Hn(x) =ne1
2x2 M 1X
m=0um(x) cos!n;m(x)
1
2m
+O1
1
2M!
;
asn!1 , uniformly on compact x-intervals on R. The
coecients um(x) are polynomials in x, andu0(x) = 1,u1(x) =1
6x3.
For more powerful asymptotic expansions as n!1
in terms of elementary functions that apply uniformly
when 1 +t <1, 1 +t1 , or
1<t 1 , wheret=xp2n+ 1 andis again
an arbitrary small positive constant, see xx12.10(i){
12.10(iv) and 12.10(vi). And for asymptotic expansions
asn!1 in terms of Airy functions that apply uni-
formly when 1 +t <1or 1< t1 ,
seexx12.10(vii) and 12.10(viii). With =p2n+ 1 the
expansions in Chapter 12 are for the parabolic cylinder
functionU
1
22;tp
2
, which is related to the Her-
mite polynomials via
18.15.28Hn(x) = 21
4(2 1)e1
22t2U
1
22;tp
2
;
compare (18.11.3).
For an error bound for the rst term in the Airy-
function expansions see Olver (1997b, p. 403).
454 Orthogonal Polynomials
18.15(vi) Other Approximations
The asymptotic behavior of the classical OP's as x!
1 with the degree and parameters xed is evident
from their explicit polynomial forms; see, for example,
(18.2.7) and the last two columns of Table 18.3.1.
For asymptotic approximations of Jacobi, ultras-
pherical, and Laguerre polynomials in terms of Hermite
polynomials, see L opez and Temme (1999a). These
approximations apply when the parameters are large,
namely and (subject to restrictions) in the case
of Jacobi polynomials, in the case of ultraspherical
polynomials, and j j+jxjin the case of Laguerre poly-
nomials. See also Dunster (1999).
18.16 Zeros
18.16(i) Distribution
Seex18.2(vi).
18.16(ii) Jacobi
Letn;m,m= 1;2;:::;n , denote the zeros of
P( ; )
n(cos) with
18.16.1 0<n;1<n;2<<n;n<:
Thenn;mis strictly increasing in and strictly de-
creasing in ; furthermore, if = , thenn;mis strictly
increasing in .
Inequalities
18.16.2(m 1
2)
n+1
2n;mm
n+1
2, ; 2[ 1
2;1
2],
18.16.3
(m 1
2)
nn;mm
n+ 1,
= , 2[ 1
2;1
2],m= 1;2;:::;1
2n
.
Also, withde ned as in (18.15.5)
18.16.4
m+1
2( + 1)
<n;m<m
, ; 2[ 1
2;1
2],
except when 2= 2=1
4.
18.16.5n;m>
m+1
2 1
4
n+ +1
2,
= , 2( 1
2;1
2),m= 1;2;:::;1
2n
.
Letj ;mbe themth positive zero of the Bessel func-
tionJ (x) (x10.21(i)). Then
18.16.6
n;mj ;m
2+1
12(1 2 3 2)1
2, ; 2[ 1
2;1
2],
18.16.7
n;mj ;m
2+1
4 1
2( 2+ 2) 2(1 4 2)1
2,
; 2[ 1
2;1
2],m= 1;2;:::;1
2n
.Asymptotic Behavior
Let m=j ;m/. Then asn!1 , with (> 1
2) and
( 1 ) xed,
18.16.8
n;m= m+
2 1
41 mcot m
2 m
1
4( 2 2) tan 1
2 m1
2+ 2
mO1
3
;
uniformly for m= 1;2;:::;bcnc, wherecis an arbitrary
constant such that 0 <c< 1.
18.16(iii) Ultraspherical and Legendre
For ultraspherical and Legendre polynomials, set =
and = = 0, respectively, in the results given in
x18.16(ii).
18.16(iv) Laguerre
The zeros of L( )
n(x) are denoted by xn;m,m=
1;2;:::;n , with
18.16.9 0<xn;1<xn;2<<xn;n:
Also,is again de ned by (18.15.17).
Inequalities
Forn= 1;2;:::;m , and withj ;mas inx18.16(ii),
18.16.10 xn;m> j2
;m
;
18.16.11xn;m<(4m+ 2 + 2)
2m+ + 1
+
(2m+ + 1)2+1
4 21
2.
:
The constant j2
;min (18.16.10) is the best possible since
the ratio of the two sides of this inequality tends to 1 as
n!1 .
For the smallest and largest zeros we have
18.16.12xn;1>2n+ 2 (1 + 4(n 1)(n+ 1))1
2;
18.16.13xn;n<2n+ 2 + (1 + 4(n 1)(n+ 1))1
2:
Asymptotic Behavior
Asn!1 , with andm xed,
18.16.14
xn;n m+1=+ 22
3am1
3+1
524
3a2
m 1
3+O
n 1
;
whereamis themth negative zero of Ai( x) (x9.9(i)). For
three additional terms in this expansion see Gatteschi
(2002). Also,
18.16.15xn;m<+ 22
3am1
3+ 2 2
3a2
m 1
3;
when =2( 1
2;1
2).
18.17 Integrals 455
18.16(v) Hermite
All zeros of Hn(x) lie in the open interval
( p2n+ 1;p2n+ 1). In view of the re
ection for-
mula, given in Table 18.6.1, we may consider just the
positive zeros xn;m,m= 1;2;:::;1
2n
. Arrange them
in decreasing order:
18.16.16 (2n+ 1)1
2>xn;1>xn;2>>xn;bn=2c>0:
Then
18.16.17xn;m= (2n+ 1)1
2+ 2 1
3(2n+ 1) 1
6am+n;m;
whereamis themth negative zero of Ai( x) (x9.9(i)),
n;m<0, and asn!1 withm xed
18.16.18 n;m=O
n 5
6
:
For an asymptotic expansion of xn;masn! 1
that applies uniformly for m= 1;2;:::;1
2n
, see Olver
(1959,x14(i)). In the notation of this reference xn;m=
ua;m,=p2n+ 1, and = 4
3am. For an error
bound for the rst approximation yielded by this ex-
pansion see Olver (1997b, p. 408).
Lastly, in view of (18.7.19) and (18.7.20), results for
the zeros of L(1
2)
n(x) lead immediately to results for the
zeros ofHn(x).
18.16(vi) Additional References
For further information on the zeros of the classical or-
thogonal polynomials, see Szeg o (1975, Chapter VI),Erd elyi et al. (1953b,xx10.16 and 10.17), Gatteschi
(1987, 2002), L opez and Temme (1999a), and Temme
(1990a).
18.17 Integrals
18.17(i) Inde nite Integrals
Jacobi
18.17.1
2nZx
0(1 y) (1 +y) P( ; )
n(y)dy
=P( +1; +1)
n 1 (0) (1 x) +1(1+x) +1P( +1; +1)
n 1 (x):
Laguerre
18.17.2Zx
0Lm(y)Ln(x y)dy=Zx
0Lm+n(y)dy
=Lm+n(x) Lm+n+1(x):
Hermite
18.17.3Zx
0Hn(y)dy=1
2(n+ 1)(Hn+1(x) Hn+1(0));
18.17.4Zx
0e y2Hn(y)dy=Hn 1(0) e x2Hn 1(x):
18.17(ii) Integral Representations for Products
Ultraspherical
18.17.5C()
n(cos1)
C()
n(1)C()
n(cos2)
C()
n(1)=
+1
2
1
2 ()Z
0C()
n(cos1cos2+ sin1sin2cos )
C()
n(1)(sin )2 1d ,>0.
Legendre
18.17.6 Pn(cos1)Pn(cos2) =1
Z
0Pn(cos1cos2+ sin1sin2cos )d :
For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974,
1977).
18.17(iii) Nicholson-Type Integrals
Legendre
18.17.7 (Pn(x))2+ 4 2(Qn(x))2= 4 2Z1
1Qn
x2+ (1 x2)t
(t2 1) 1
2dt, 1<x< 1.
For the Ferrers function Qn(x) and Legendre function Qn(x) seexx14.3(i) and 14.3(ii), with = 0 and=n.
Hermite
18.17.8 (Hn(x))2+ 2n(n!)2ex2
V
n 1
2;21
2x2
=2n+3
2n!ex2
Z1
0e (2n+1)t+x2tanht
(sinh 2t)1
2dt:
For the parabolic cylinder function V(a;z) seex12.2. For similar formulas for ultraspherical polynomials see Durand
(1975), and for Jacobi and Laguerre polynomials see Durand (1978).
456 Orthogonal Polynomials
18.17(iv) Fractional Integrals
Jacobi
18.17.9(1 x) +P( +; )
n (x)
( ++n+ 1)=Z1
x(1 y) P( ; )
n(y)
( +n+ 1)(y x) 1
()dy,>0, 1<x< 1,
18.17.10x +(x+ 1)n
( ++n+ 1)P( ; +)
nx 1
x+ 1
=Zx
0y (y+ 1)n
( +n+ 1)P( ; )
ny 1
y+ 1(x y) 1
()dy,>0,x>0,
18.17.11 (n+ + + 1)
xn+ + +1P( ; )
n
1 2x 1
=Z1
x (n+ + + 1)
yn+ + +1P( ; )
n
1 2y 1(y x) 1
()dy,
+ + 1>> 0,x>1,
and three formulas similar to (18.17.9){(18.17.11) by symmetry; compare the second row in Table 18.6.1.
Ultraspherical
18.17.12 ( )C( )
n
x 1
2
x +1
2n=Z1
x ()C()
n
y 1
2
y+1
2n(y x) 1
()dy, >> 0,x>0,
18.17.13x1
2n(x 1)+ 1
2
++1
2C(+)
n
x 1
2
C(+)
n (1)=Zx
1y1
2n(y 1) 1
2
+1
2C()
n
y 1
2
C()
n(1)(x y) 1
()dy, >0,x>1.
Laguerre
18.17.14x +L( +)
n (x)
( ++n+ 1)=Zx
0y L( )
n(y)
( +n+ 1)(x y) 1
()dy, >0,x>0.
18.17.15 e xL( )
n(x) =Z1
xe yL( +)
n (y)(y x) 1
()dy, >0.
18.17(v) Fourier Transforms
Throughout this subsection we assume y>0; sometimes however, this restriction can be eased by analytic continu-
ation.
Jacobi
18.17.16Z1
1(1 x) (1 +x) P( ; )
n(x)eixydx
=(iy)neiy
n!2n+ + +1B(n+ + 1;n+ + 1) 1F1(n+ + 1; 2n+ + + 2; 2iy):
For the beta function B( a;b) seex5.12, and for the con
uent hypergeometric function 1F1see (16.2.1) and Chapter
13.
Ultraspherical
18.17.17Z1
0(1 x2) 1
2C()
2n(x) cos(xy)dx=( 1)n (2n+ 2)J+2n(y)
(2n)! ()(2y);
18.17.18Z1
0(1 x2) 1
2C()
2n+1(x) sin(xy)dx=( 1)n (2n+ 2+ 1)J2n++1(y)
(2n+ 1)! ()(2y):
For the Bessel function Jseex10.2(ii).
Legendre
18.17.19Z1
1Pn(x)eixydx=inr2
yJn+1
2(y);
18.17.20Z1
0Pn
1 2x2
cos(xy)dx= ( 1)n1
2Jn+1
2 1
2y
J n 1
2 1
2y
;
18.17.21Z1
0Pn
1 2x2
sin(xy)dx=1
2
Jn+1
2 1
2y2
:
18.17 Integrals 457
Hermite
18.17.221
2pZ1
1e 1
4x2Hen(x)e1
2ixydx=ine 1
4y2Hen(y);
18.17.23Z1
0e 1
2x2He2n(x) cos(xy)dx= ( 1)nq
1
2y2ne 1
2y2;
18.17.24Z1
0e x2He2n(2x) cos(xy)dx= ( 1)n1
2pe 1
4y2He2n(y):
18.17.25Z1
0e 1
2x2Hen(x)Hen+2m(x) cos(xy)dx= ( 1)mq
1
2n!y2me 1
2y2L(2m)
n
y2
;
18.17.26Z1
0e 1
2x2Hen(x)Hen+2m+1(x) sin(xy)dx= ( 1)mq
1
2n!y2m+1e 1
2y2L(2m+1)
n
y2
:
18.17.27Z1
0e 1
2x2He2n+1(x) sin(xy)dx= ( 1)nq
1
2y2n+1e 1
2y2;
18.17.28Z1
0e x2He2n+1(2x) sin(xy)dx= ( 1)n1
2pe 1
4y2He2n+1(y):
Laguerre
18.17.29Z1
0x2me 1
2x2L(2m)
n
x2
cos(xy)dx= ( 1)mq
1
21
n!e 1
2y2Hen(y)Hen+2m(y):
18.17.30Z1
0x2ne 1
2x2L(n 1
2)
n 1
2x2
cos(xy)dx=q
1
2y2ne 1
2y2L(n 1
2)
n 1
2y2
:
18.17.31Z1
0e axx 2nL( 2n)
2n 1(ax) cos(xy)dx=i( 1)n ()
2(2n 1)!y2n 1
(a+iy) (a iy)
, >2n 1,a>0,
18.17.32Z1
0e axx 1 2nL( 1 2n)
2n (ax) cos(xy)dx=( 1)n ()
2(2n)!y2n
(a+iy) + (a iy)
, >2n,a>0.
18.17(vi) Laplace Transforms
Jacobi
18.17.33Z1
1e (x+1)zP( ; )
n(x)(1 x) (1 +x) dx
=( 1)n2 + +n+1 ( +n+ 1) ( +n+ 1)
( + + 2n+ 2)n!zn
1F1 +n+ 1
+ + 2n+ 2; 2z
, z2C.
For the con
uent hypergeometric function 1F1see (16.2.1) and Chapter 13.
Laguerre
18.17.34Z1
0e xzL( )
n(x)e xx dx= ( +n+ 1)zn
n!(z+ 1) +n+1, <z> 1.
Hermite
18.17.35Z1
1e xzHn(x)e x2dx=1
2( z)ne1
4z2, z2C.
18.17(vii) Mellin Transforms
Jacobi
18.17.36Z1
1(1 x)z 1(1 +x) P( ; )
n(x)dx=2 +z (z) (1 + +n)(1 + z)n
n! (1 + +z+n),<z>0.
458 Orthogonal Polynomials
Ultraspherical
18.17.37Z1
0(1 x2) 1
2C()
n(x)xz 1dx=21 2 z (n+ 2) (z)
n! () 1
2+1
2n++1
2z
1
2+1
2z 1
2n,<z>0.
Legendre
18.17.38Z1
0P2n(x)xz 1dx=( 1)n 1
2 1
2z
n
2 1
2z
n+1, <z>0,
18.17.39Z1
0P2n+1(x)xz 1dx=( 1)n
1 1
2z
n
2 1
2+1
2z
n+1, <z> 1.
Laguerre
18.17.40Z1
0e axL( )
n(bx)xz 1dx= (z+n)
n!(a b)na n z
2F1 n;1 + z
1 n z;a
a b
,<a>0,<z>0.
For the hypergeometric function 2F1seexx15.1 and 15.2(i).
Hermite
18.17.41Z1
0e axHen(x)xz 1dx= (z+n)a n 2
2F2
1
2n; 1
2n+1
2
1
2z 1
2n; 1
2z 1
2n+1
2; 1
2a2!
,
<a>0. Also,<z>0,neven;<z> 1,nodd.
For the generalized hypergeometric function 2F2see (16.2.1).
18.17(viii) Other Integrals
Chebyshev
18.17.42Z1
1Tn(y)(1 y2) 1
2
y xdy=Un 1(x);
18.17.43Z1
1Un 1(y)(1 y2)1
2
y xdy= Tn(x):
These integrals are Cauchy principal values ( x1.4(v)).
Legendre
18.17.44Z1
1Pn(x) Pn(t)
jx tjdt= 2
1 +1
2++1
n
Pn(x), 1x1.
The casex= 1 is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972).
18.17.45 (n+1
2)(1 +x)1
2Zx
1(x t) 1
2Pn(t)dt=Tn(x) +Tn+1(x);
18.17.46 (n+1
2)(1 x)1
2Z1
x(t x) 1
2Pn(t)dt=Tn(x) Tn+1(x):
Laguerre
18.17.47Zx
0t L( )
m(t)
L( )
m(0)(x t) L( )
n(x t)
L( )
n(0)dt= ( + 1) ( + 1)
( + + 2)x + +1L( + +1)
m+n(x)
L( + +1)
m+n(0):
Hermite
18.17.48Z1
1Hm(y)e y2Hn(x y)e (x y)2dy=1
22 1
2(m+n+1)Hm+n
2 1
2x
e 1
2x2:
18.17.49Z1
1H`(x)Hm(x)Hn(x)e x2dx=21
2(`+m+n)`!m!n!p
(1
2`+1
2m 1
2n) ! (1
2m+1
2n 1
2`) ! (1
2n+1
2` 1
2m) !;
provided that `+m+nis even and the sum of any two of `;m;n is not less than the third; otherwise the integral is
zero.
18.18 Sums 459
18.17(ix) Compendia
For further integrals, see Apelblat (1983, pp. 189{
204), Erd elyi et al. (1954a, pp. 38{39, 94{95, 170{
176, 259{261, 324), Erd elyi et al. (1954b, pp. 42{44,
271{294), Gradshteyn and Ryzhik (2000, pp. 788{806),
Gr obner and Hofreiter (1950, pp. 23{30), Marichev
(1983, pp. 216{247), Oberhettinger (1972, pp. 64{67),
Oberhettinger (1974, pp. 83{92), Oberhettinger (1990,
pp. 44{47 and 152{154), Oberhettinger and Badii (1973,
pp. 103{112), Prudnikov et al. (1986b, pp. 420{617),
Prudnikov et al. (1992a, pp. 419{476), and Prudnikov
et al. (1992b, pp. 280{308).
18.18 Sums
18.18(i) Series Expansions of Arbitrary
Functions
Jacobi
Letf(z) be analytic within an ellipse Ewith foci
z=1, and
18.18.1an=n!(2n+ + + 1) (n+ + + 1)
2 + +1 (n+ + 1) (n+ + 1)
Z1
1f(x)P( ; )
n(x)(1 x) (1 +x) dx:
Then
18.18.2 f(z) =1X
n=0anP( ; )
n(z);
whenzlies in the interior of E. Moreover, the series
(18.18.2) converges uniformly on any compact domain
withinE.
Alternatively, assume f(x) is real and continuous
andf0(x) is piecewise continuous on ( 1;1). Assume
also the integralsR1
1(f(x))2(1 x) (1 +x) dxandR1
1(f0(x))2(1 x) +1(1 +x) +1dxconverge. Then
(18.18.2), with zreplaced by x, applies when 1<x<
1; moreover, the convergence is uniform on any compact
interval within ( 1;1).Chebyshev
Seex3.11(ii), or set = =1
2in the above results for
Jacobi and refer to (18.7.3){(18.7.6).
Legendre
This is the case = = 0 of Jacobi. Equation (18.18.1)
becomes
18.18.3 an=
n+1
2Z1
1f(x)Pn(x)dx:
Laguerre
Assumef(x) is real and continuous and f0(x)
is piecewise continuous on (0 ;1). Assume alsoR1
0(f(x))2e xx dxconverges. Then
18.18.4 f(x) =1X
n=0bnL( )
n(x), 0<x<1,
where
18.18.5bn=n!
(n+ + 1)Z1
0f(x)L( )
n(x)e xx dx:
The convergence of the series (18.18.4) is uniform on
any compact interval in (0 ;1).
Hermite
Assumef(x) is real and continuous and f0(x) is
piecewise continuous on ( 1;1). Assume alsoR1
1(f(x))2e x2dxconverges. Then
18.18.6 f(x) =1X
n=0dnHn(x), 1<x<1,
where
18.18.7dn=1p2nn!Z1
1f(x)Hn(x)e x2dx:
The convergence of the series (18.18.6) is uniform on
any compact interval in ( 1;1).
18.18(ii) Addition Theorems
Ultraspherical
18.18.8C()
n(cos1cos2+ sin1sin2cos )
=nX
`=022`(n `)!2+ 2` 1
2 1(()`)2
(2)n+`(sin1)`C(+`)
n `(cos1)(sin2)`C(+`)
n `(cos2)C( 1
2)
`(cos ),
>0,6=1
2.
For the case =1
2use (18.18.9); compare (18.7.9).
Legendre
18.18.9Pn(cos1cos2+ sin1sin2cos )
=Pn(cos1)Pn(cos2) + 2nX
`=1(n `)! (n+`)!
22`(n!)2(sin1)`P(`;`)
n `(cos1)(sin2)`P(`;`)
n `(cos2) cos(` ):
460 Orthogonal Polynomials
For (18.18.8), (18.18.9), and the corresponding formula for Jacobi polynomials see Koornwinder (1975a). See also
(14.30.9).
Laguerre
18.18.10 L( 1++ r+r 1)
n (x1++xr) =X
m1++mr=nL( 1)
m1(x1)L( r)
mr(xr):
Hermite
18.18.11(a2
1++a2
r)1
2n
n!Hn
a1x1++arxr
(a2
1++a2r)1
2!
=X
m1++mr=nam1
1amrr
m1!mr!Hm1(x1)Hmr(xr):
18.18(iii) Multiplication Theorems
Laguerre
18.18.12L( )
n(x)
L( )
n(0)=nX
`=0n
`
`(1 )n `L( )
`(x)
L( )
`(0):
Hermite
18.18.13 Hn(x) =nbn=2cX
`=0( n)2`
`!(1 2)`Hn 2`(x):
18.18(iv) Connection Formulas
Jacobi
18.18.14P(
; )
n(x) =( + 1)n
( + + 2)nnX
`=0 + + 2`+ 1
+ + 1( + + 1)`(n+ +
+ 1)`
( + 1)`(n+ + + 2)`(
)n `
(n `)!P( ; )
`(x);
18.18.151 +x
2n
=( + 1)n
( + + 2)nnX
`=0 + + 2`+ 1
+ + 1( + + 1)`(n `+ 1)`
( + 1)`(n+ + + 2)`P( ; )
`(x);
and a similar pair of equations by symmetry; compare
the second row in Table 18.6.1.
Ultraspherical
18.18.16
C()
n(x)
=bn=2cX
`=0+n 2`
()n `
(+ 1)n `( )`
`!C()
n 2`(x);
18.18.17
(2x)n=n!bn=2cX
`=0+n 2`
1
(+ 1)n ``!C()
n 2`(x):
Laguerre
18.18.18L( )
n(x) =nX
`=0( )n `
(n `)!L( )
`(x);
18.18.19 xn= ( + 1)nnX
`=0( n)`
( + 1)`L( )
`(x):Hermite
18.18.20 (2x)n=bn=2cX
`=0( n)2`
`!Hn 2`(x):
18.18(v) Linearization Formulas
Chebyshev
18.18.21Tm(x)Tn(x) =1
2(Tm+n(x) +Tm n(x)):
Ultraspherical
18.18.22
C()
m(x)C()
n(x)
=min(m;n)X
`=0(m+n+ 2`)(m+n 2`)!
(m+n+ `)`! (m `)! (n `)!
()`()m `()n `(2)m+n `
()m+n `(2)m+n 2`C()
m+n 2`(x):
18.18 Sums 461
Hermite
18.18.23
Hm(x)Hn(x) =min(m;n)X
`=0m
`n
`
2``!Hm+n 2`(x):
The coecients in the expansions (18.18.22) and
(18.18.23) are positive, provided that in the former case
>0.
18.18(vi) Bateman-Type Sums
Jacobi
With
18.18.24bn;`=n
`(n+ + + 1)`( n)n `
2`( + 1)n;
18.18.25
P( ; )
n(x)
P( ; )
n(1)P( ; )
n(y)
P( ; )
n(1)=nX
`=0bn;`(x+y)`
P( ; )
`((1 +xy)/(x+y))
P( ; )
`(1);
18.18.26P( ; )
n(x)
P( ; )
n(1)=nX
`=0bn;`(x+ 1)`:
18.18(vii) Poisson Kernels
Laguerre
18.18.271X
n=0n!L( )
n(x)L( )
n(y)
( + 1)nzn
= ( + 1)(xyz) 1
2
1 z
exp (x+y)z
1 z
I
2(xyz)1
2
1 z!
,
jzj<1.
For the modi ed Bessel function I(z) seex10.25(ii).
Hermite
18.18.281X
n=0Hn(x)Hn(y)
2nn!zn
= (1 z2) 1
2exp2xyz (x2+y2)z2
1 z2
,
jzj<1.
These Poisson kernels are positive, provided that
x;yare real, 0z <1, and in the case of (18.18.27)
x;y0.18.18(viii) Other Sums
In this subsection the variables xandyare not con ned
to the closures of the intervals of orthogonality; compare
x18.2(i).
Ultraspherical
18.18.29nX
`=0C()
`(x)C()
n `(x) =C(+)
n (x):
18.18.30nX
`=0`+ 2
2C()
`(x)xn `=C(+1)
n (x):
Chebyshev
18.18.31nX
`=0T`(x)xn `=Un(x):
18.18.32 2nX
`=0T2`(x) = 1 +U2n(x);
18.18.33 2nX
`=0T2`+1(x) =U2n+1(x):
18.18.34 2(1 x2)nX
`=0U2`(x) = 1 T2n+2(x);
18.18.35 2(1 x2)nX
`=0U2`+1(x) =x T2n+3(x):
Legendre and Chebyshev
18.18.36nX
`=0P`(x)Pn `(x) =Un(x):
Laguerre
18.18.37nX
`=0L( )
`(x) =L( +1)
n (x);
18.18.38nX
`=0L( )
`(x)L( )
n `(y) =L( + +1)
n (x+y):
Hermite and Laguerre
18.18.39
nX
`=0n
`
H`
21
2x
Hn `
21
2y
= 21
2nHn(x+y);
18.18.40
nX
`=0n
`
H2`(x)H2n 2`(y) = ( 1)n22nn!Ln
x2+y2
:
18.18(ix) Compendia
For further sums see Hansen (1975, pp. 292-330), Grad-
shteyn and Ryzhik (2000, pp. 978{993), and Prudnikov
et al. (1986b, pp. 637-644 and 700-718).
462 Orthogonal Polynomials
Askey Scheme
18.19 Hahn Class: De nitions
Hahn, Krawtchouk, Meixner, and Charlier
Tables 18.19.1 and 18.19.2 provide de nitions via orthogonality and normalization ( xx18.2(i), 18.2(iii)) for the Hahn
polynomials Qn(x; ; ;N ), Krawtchouk polynomials Kn(x;p;N), Meixner polynomials Mn(x; ;c), and Charlier
polynomials Cn(x;a).
Table 18.19.1 : Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP's: discrete sets, weight
functions, normalizations, and parameter constraints.
pn(x) X w x hn
Qn(x; ; ;N ),
n= 0;1;:::;Nf0;1;:::;Ng( + 1)x( + 1)N x
x!(N x)!,
; > 1 or ; < N( 1)n(n+ + + 1)N+1( + 1)nn!
(2n+ + + 1)( + 1)n( N)nN!
If ; < N, then ( 1)Nwx>0 and
( 1)Nhn>0.
Kn(x;p;N),
n= 0;1;:::;Nf0;1;:::;NgN
x
px(1 p)N x,
0<p< 11 p
pn,N
n
Mn(x; ;c)f0;1;2;:::g( )xcx/x!,
>0, 0<c< 1c nn!
( )n(1 c)
Cn(x;a)f0;1;2;:::gax/x!,a>0 a nean!
Table 18.19.2 : Hahn, Krawtchouk, Meixner, and Char-
lier OP's: leading coecients.
pn(x) kn
Qn(x; ; ;N )(n+ + + 1)n
( + 1)n( N)n
Kn(x;p;N)p n/( N)n
Mn(x; ;c) (1 c 1)n
( )n
Cn(x;a) ( a) n
Continuous Hahn
These polynomials are orthogonal on ( 1;1), and
with<a>0,<b>0 are de ned as follows.
18.19.1 pn(x) =pn
x;a;b;a;b
;
18.19.2
w(z;a;b;a;b) = (a+iz) (b+iz) (a iz)
b iz
;
18.19.3w(x) =w(x;a;b;a;b) =j (a+ix) (b+ix)j2;
18.19.4
hn=2 (n+a+a)
n+b+b
j
n+a+b
j2
(2n+ 2<(a+b) 1) (n+ 2<(a+b) 1)n!;18.19.5 kn=(n+ 2<(a+b) 1)n
n!:
Meixner{Pollaczek
These polynomials are orthogonal on ( 1;1), and are
de ned as follows.
18.19.6 pn(x) =P()
n(x; );
18.19.7w()(z; ) = (+iz) ( iz)e(2 )z;
18.19.8w(x) =w()(x; ) =j (+ix)j2e(2 )x,
>0, 0< < ,
18.19.9hn=2 (n+ 2)
(2 sin )2n!; kn=(2 sin )n
n!:
18.20 Hahn Class: Explicit Representations
18.20(i) Rodrigues Formulas
For comments on the use of the forward-di erence op-
erator x, the backward-di erence operator rx, and
the central-di erence operator x, seex18.2(ii).
18.21 Hahn Class: Interrelations 463
Hahn, Krawtchouk, Meixner, and Charlier
18.20.1
pn(x) =1
nwxrn
x
wxn 1Y
`=0F(x+`)!
,x2X.
In (18.20.1) Xandwxare as in Table 18.19.1. For the
Hahn polynomials pn(x) =Qn(x; ; ;N ) and
18.20.2
F(x) = (x+ + 1)(x N); n= ( N)n( + 1)n:
For the Krawtchouk, Meixner, and Charlier polynomi-
als,F(x) andnare as in Table 18.20.1.
Table 18.20.1 : Krawtchouk, Meixner, and Charlier
OP's: Rodrigues formulas (18.20.1).
pn(x)F(x)n
Kn(x;p;N)x N ( N)n
Mn(x; ;c)x+ ( )n
Cn(x;a) 1 1
Continuous Hahn
18.20.3w(x;a;b;a;b)pn
x;a;b;a;b
=1
n!n
x
w(x;a+1
2n;b+1
2n;a+1
2n;b+1
2n)
:
Meixner{Pollaczek
18.20.4w()(x; )P()
n(x; ) =1
n!n
x
w(+1
2n)(x; )
:
18.20(ii) Hypergeometric Function and
Generalized Hypergeometric Functions
For the de nition of hypergeometric and generalized hy-
pergeometric functions see x16.2.
18.20.5
Qn(x; ; ;N ) = 3F2 n;n+ + + 1; x
+ 1; N; 1
,
n= 0;1;:::;N .
18.20.6Kn(x;p;N) = 2F1 n; x
N;p 1
,
n= 0;1;:::;N .
18.20.7Mn(x; ;c) = 2F1 n; x
; 1 c 1
:
18.20.8 Cn(x;a) = 2F0 n; x
; a 1
:
18.20.9pn
x;a;b;a;b
=in(a+a)n
a+b
n
n!
3F2 n;n+ 2<(a+b) 1;a+ix
a+a;a+b; 1
:(For symmetry properties of pn
x;a;b;a;b
with respect
toa,b,a,bsee Andrews et al. (1999, Corollary 3.3.4).)
18.20.10
P()
n(x; ) =(2)n
n!ein
2F1 n;+ix
2; 1 e 2i
:
18.21 Hahn Class: Interrelations
18.21(i) Dualities
Duality of Hahn and Dual Hahn
18.21.1Qn(x; ; ;N ) =Rx(n(n+ + + 1); ; ;N ),
n;x= 0;1;:::;N .
For the dual Hahn polynomial Rn(x;
;;N ) seex18.25.
Self-Dualities
18.21.2Kn(x;p;N) =Kx(n;p;N),n;x= 0;1;:::;N .
Mn(x; ;c) =Mx(n; ;c),n;x= 0;1;2;:::.
Cn(x;a) =Cx(n;a), n;x= 0;1;2;:::.
18.21(ii) Limit Relations and Special Cases
Hahn!Krawtchouk
18.21.3 lim
t!1Qn(x;pt;(1 p)t;N) =Kn(x;p;N):
Hahn!Meixner
18.21.4
lim
N!1Qn
x;b 1;N(c 1 1);N
=Mn(x;b;c):
Hahn!Jacobi
18.21.5 lim
N!1Qn(Nx; ; ;N ) =P( ; )
n(1 2x)
P( ; )
n(1):
Krawtchouk!Charlier
18.21.6 lim
N!1Kn
x;N 1a;N
=Cn(x;a):
Meixner!Charlier
18.21.7 lim
!1Mn
x; ;a(a+ ) 1
=Cn(x;a):
Meixner!Laguerre
18.21.8 lim
c!1Mn
(1 c) 1x; + 1;c
=L( )
n(x)
L( )
n(0):
Charlier!Hermite
18.21.9 lim
a!1(2a)1
2nCn
(2a)1
2x+a;a
= ( 1)nHn(x):
Continuous Hahn!Meixner{Pollaczek
18.21.10
lim
t!1t npn(x t;+it; ttan ; it; ttan )
=( 1)n
(cos )nP()
n(x; ):
18.21.11
pn
x;a;a+1
2;a;a +1
2
= 2 2n(4a+n)nP(2a)
n
2x;1
2
:
464 Orthogonal Polynomials
Meixner{Pollaczek!Laguerre
18.21.12 lim
!0P(1
2 +1
2)
n
(2 ) 1x;
=L( )
n(x):A graphical representation of limits in xx18.7(iii),
18.21(ii), and 18.26(ii) is provided by the Askey scheme
depicted in Figure 18.21.1.
Figure 18.21.1 : Askey scheme. The number of free real parameters is zero for Hermite polynomials. It increases
by one for each row ascended in the scheme, culminating with four free real parameters for the Wilson and Racah
polynomials. (This is with the convention that the real and imaginary parts of the parameters are counted separately
in the case of the continuous Hahn polynomials.)
18.22 Hahn Class: Recurrence Relations
and Di erences
18.22(i) Recurrence Relations in n
Hahn
With
18.22.1 pn(x) =Qn(x; ; ;N );
18.22.2
xpn(x) =Anpn+1(x) (An+Cn)pn(x) +Cnpn 1(x);
where
18.22.3An=(n+ + + 1)(n+ + 1)(N n)
(2n+ + + 1)(2n+ + + 2);
Cn=n(n+ + +N+ 1)(n+ )
(2n+ + )(2n+ + + 1):Krawtchouk, Meixner, and Charlier
These polynomials satisfy (18.22.2) with pn(x),An, and
Cnas in Table 18.22.1.
Table 18.22.1 : Recurrence relations (18.22.2) for
Krawtchouk, Meixner, and Charlier polynomials.
pn(x)AnCn
Kn(x;p;N)p(N n)n(1 p)
Mn(x; ;c)c(n+ )
1 cn
1 c
Cn(x;a)a n
18.22 Hahn Class: Recurrence Relations and Differences 465
Continuous Hahn
With
18.22.4qn(x) =pn
x;a;b;a;b
pn
ia;a;b;a;b
;
18.22.5(a+ix)qn(x)
=~Anqn+1(x) (~An+~Cn)qn(x) +~Cnqn 1(x);
where
18.22.6
~An= (n+ 2<(a+b) 1)(n+a+a)(n+a+b)
(2n+ 2<(a+b) 1)(2n+ 2<(a+b));
~Cn=n(n+b+a 1)(n+b+b 1)
(2n+ 2<(a+b) 2)(2n+ 2<(a+b) 1):
Meixner{Pollaczek
With
18.22.7 pn(x) =P()
n(x; );
18.22.8(n+ 1)pn+1(x) = 2 (xsin +(n+) cos )pn(x)
(n+ 2 1)pn 1(x):
18.22(ii) Di erence Equations in x
Hahn
With
18.22.9 pn(x) =Qn(x; ; ;N );
18.22.10A(x)pn(x+ 1) (A(x) +C(x))pn(x)
+C(x)pn(x 1)
n(n+ + + 1)pn(x) = 0;
where
18.22.11A(x) = (x+ + 1)(x N);
C(x) =x(x N 1):
Krawtchouk, Meixner, and Charlier
18.22.12A(x)pn(x+ 1) (A(x) +C(x))pn(x)
+C(x)pn(x 1) +npn(x) = 0:
ForA(x),C(x), andnin (18.22.12) see Table 18.22.2.
Table 18.22.2 : Di erence equations (18.22.12) for
Krawtchouk, Meixner, and Charlier polynomials.
pn(x)A(x)C(x)n
Kn(x;p;N)p(x N) (p 1)x n
Mn(x; ;c)c(x+ )x n (1 c)
Cn(x;a)a x nContinuous Hahn
With
18.22.13 pn(x) =pn
x;a;b;a;b
;
18.22.14A(x)pn(x+i) (A(x) +C(x))pn(x)
+C(x)pn(x i)
+n(n+ 2<(a+b) 1)pn(x) = 0;
where
18.22.15
A(x) = (x+ia)(x+ib); C (x) = (x ia)(x ib):
Meixner{Pollaczek
With
18.22.16 pn(x) =P()
n(x; );
18.22.17A(x)pn(x+i) (A(x) +C(x))pn(x)
+C(x)pn(x i) + 2nsin pn(x) = 0;
where
18.22.18A(x) =ei (x+i); C (x) =e i (x i):
18.22(iii)x-Di erences
Hahn
18.22.19
xQn(x; ; ;N )
= n(n+ + + 1)
( + 1)NQn 1(x; + 1; + 1;N 1);
18.22.20
rx( + 1)x( + 1)N x
x! (N x)!Qn(x; ; ;N )
=N+ 1
( )x( )N+1 x
x! (N+ 1 x)!
Qn+1(x; 1; 1;N+ 1):
Krawtchouk
18.22.21 xKn(x;p;N) = n
pNKn 1(x;p;N 1);
18.22.22rxN
x
px(1 p)N xKn(x;p;N)
=N+ 1
x
px(1 p)N xKn+1(x;p;N+ 1):
Meixner
18.22.23 xMn(x; ;c) = n(1 c)
cMn 1(x; + 1;c);
18.22.24rx( )xcx
x!Mn(x; ;c)
=( 1)xcx
x!Mn+1(x; 1;c):
Charlier
18.22.25 xCn(x;a) = n
aCn 1(x;a);
18.22.26rxax
x!Cn(x;a)
=ax
x!Cn+1(x;a):
466 Orthogonal Polynomials
Continuous Hahn
18.22.27 x
pn
x;a;b;a;b
= (n+ 2<(a+b) 1)pn 1
x;a+1
2;b+1
2;a+1
2;b+1
2
;
18.22.28x
w(x;a+1
2;b+1
2;a+1
2;b+1
2)pn(x;a+1
2;b+1
2;a+1
2;b+1
2)
= (n+ 1)w(x;a;b;a;b)pn+1(x;a;b;a;b):
Meixner{Pollaczek
18.22.29x
P()
n(x; )
= 2 sin P(+1
2)
n 1(x; );
18.22.30x
w(+1
2)(x; )P(+1
2)
n (x; )
= (n+ 1)w()(x; )P()
n+1(x; ):
18.23 Hahn Class: Generating Functions
For the de nition of generalized hypergeometric func-
tions seex16.2.
Hahn
18.23.1
1F1 x
+ 1; z
1F1x N
+ 1;z
=NX
n=0( N)n
( + 1)nn!Qn(x; ; ;N )zn,x= 0;1;:::;N .
18.23.22F0 x; x+ +N+ 1
; z
2F0x N;x + + 1
;z
=NX
n=0( N)n( + 1)n
n!Qn(x; ; ;N )zn,
x= 0;1;:::;N .
Krawtchouk
18.23.3
1 1 p
pzx
(1 +z)N x=NX
n=0N
n
Kn(x;p;N)zn,
x= 0;1;:::;N .
Meixner
18.23.4
1 z
cx
(1 z) x =1X
n=0( )n
n!Mn(x; ;c)zn,
x= 0;1;2;:::,jzj<1.
Charlier
18.23.5ez
1 z
ax
=1X
n=0Cn(x;a)
n!zn,x= 0;1;2;:::.Continuous Hahn
18.23.61F1a+ix
2<a; iz
1F1b ix
2<b;iz
=1X
n=0pn
x;a;b;a;b
(2<a)n(2<b)nzn:
Meixner{Pollaczek
18.23.7(1 ei z) +ix(1 e i z) ix
=1X
n=0P()
n(x; )zn,jzj<1.
18.24 Hahn Class: Asymptotic
Approximations
Krawtchouk
Withx=Nand=n=N, Li and Wong (2000) gives
an asymptotic expansion for Kn(x;p;N) asn! 1 ,
that holds uniformly for andin compact subinter-
vals of (0;1). This expansion is in terms of the parabolic
cylinder function and its derivative.
With=N=n andx xed, Qiu and Wong (2004)
gives an asymptotic expansion for Kn(x;p;N) asn!
1, that holds uniformly for 2[1;1). This expan-
sion is in terms of con
uent hypergeometric functions.
Asymptotic approximations are also provided for the
zeros ofKn(x;p;N) in various cases depending on the
values ofpand.
Meixner
For two asymptotic expansions of Mn(nx; ;c) asn!
1, with andc xed, see Jin and Wong (1998). The
rst expansion holds uniformly for x1 +, and
the second for 1 x1 + 1,being an arbitrary
small positive constant. Both expansions are in terms
of parabolic cylinder functions.
For asymptotic approximations for the zeros of
Mn(nx; ;c) in terms of zeros of Ai( x) (x9.9(i)), see Jin
and Wong (1999).
Charlier
Dunster (2001b) provides various asymptotic expan-
sions forCn(x;a) asn!1 , in terms of elementary
functions or in terms of Bessel functions. Taken to-
gether, these expansions are uniformly valid for 1<
x<1and forain unbounded intervals|each of which
contains [0;(1 )n], whereagain denotes an arbitrary
18.25 Wilson Class: Definitions 467
small positive constant. See also Bo and Wong (1994)
and Goh (1998).
Meixner{Pollaczek
For an asymptotic expansion of P()
n(nx; ) asn!1 ,
with xed, see Li and Wong (2001). This expansion
is uniformly valid in any compact x-interval on the real
line and is in terms of parabolic cylinder functions. Cor-
responding approximations are included for the zeros of
P()
n(nx; ).
Approximations in Terms of Laguerre Polynomials
For asymptotic approximations to P()
n(x; ) asjx+
ij!1 , withn xed, see Temme and L opez (2001).
These approximations are in terms of Laguerre polyno-
mials and hold uniformly for ph( x+i)2[0;]. Com-
pare also (18.21.12). Similar approximations are in-
cluded for Jacobi, Krawtchouk, and Meixner polyno-
mials.18.25 Wilson Class: De nitions
18.25(i) Preliminaries
For the Wilson class OP's pn(x) withx=(y): if the
y-orthogonality set is f0;1;:::;Ng, then the role of the
di erentiation operator d/dxin the Jacobi, Laguerre,
and Hermite cases is played by the operator yfol-
lowed by division by y((y)), or by the operator ry
followed by division by ry((y)). Alternatively if the
y-orthogonality interval is (0 ;1), then the role of d/dx
is played by the operator yfollowed by division by
y((y)).
Table 18.25.1 lists the transformations of vari-
able, orthogonality ranges, and parameter constraints
that are needed in x18.2(i) for the Wilson polynomi-
alsWn(x;a;b;c;d ), continuous dual Hahn polynomials
Sn(x;a;b;c ), Racah polynomials Rn(x; ; ;
; ), and
dual Hahn polynomials Rn(x;
;;N ).
Table 18.25.1 : Wilson class OP's: transformations of variable, orthogonality ranges, and parameter constraints.
pn(x) x=(y)Orthogonality
range foryConstraints
Wn(x;a;b;c;d ) y2(0;1)<(a;b;c;d )>0;
nonreal parameters in conjugate pairs
Sn(x;a;b;c ) y2(0;1)<(a;b;c )>0;
nonreal parameters in conjugate pairs
Rn(x; ; ;
; )y(y+
++ 1)f0;1;:::;Ng + 1 or ++ 1 or
+ 1 = N;
for further constraints see (18.25.1)
Rn(x;
;;N )y(y+
++ 1)f0;1;:::;Ng
;> 1 or< N
Further Constraints for Racah Polynomials
If + 1 = N, then the weights will be positive i one
of the following eight sets of inequalities holds:
18.25.1 1< <
+ 1< N+ 1:
N 1< 1< <
+ 1:
;> 1; >N +
:
;> 1; < N :
N 1<N +
< < N :
N+
< < N < N 1:
;< N; > 1 :
;< N; <
+ 1:
The rst four sets imply
+ > 2, and the last four
imply
+< 2N.18.25(ii) Weights and Normalizations:
Continuous Cases
18.25.2Z1
0pn(x)pm(x)w(x)dx=hnn;m:
Wilson
18.25.3 pn(x) =Wn(x;a1;a2;a3;a4);
18.25.4 w(y2) =1
2y Q
j (aj+iy)
(2iy) 2
;
18.25.5hn=n! 2Q
j<` (n+aj+a`)
(2n 1 +P
jaj)
n 1 +P
jaj:
468 Orthogonal Polynomials
Continuous Dual Hahn
18.25.6 pn(x) =Sn(x;a1;a2;a3);
18.25.7 w(y2) =1
2y Q
j (aj+iy)
(2iy) 2
;
18.25.8 hn=n! 2Y
j<` (n+aj+a`):
18.25(iii) Weights and Normalizations: Discrete
Cases
18.25.9NX
y=0pn(y(y+
++1))pm(y(y+
++1))
++ 1 + 2y
++ 1 +y!y=hnn;m:
Racah
18.25.10 pn(x) =Rn(x; ; ;
; ), + 1 = N,
18.25.11
!y=( + 1)y( ++ 1)y(
+ 1)y(
++ 2)y
( +
++ 1)y( +
+ 1)y(+ 1)yy!;
18.25.12
hn=( )N(
++ 2)N
( +
+ 1)N(+ 1)N(n+ + + 1)nn!
( + + 2)2n
( +
+ 1)n( + 1)n( + 1)n
( + 1)n( ++ 1)n(
+ 1)n:Dual Hahn
18.25.13 pn(x) =Rn(x;
;;N );
18.25.14!y=( 1)y( N)y(
+ 1)y(
++ 1)2
(N+
++ 2)y(+ 1)yy!;
18.25.15 hn=n! (N n)! (
++ 2)N
N! (
+ 1)n(+ 1)N n:
18.25(iv) Leading Coecients
Table 18.25.2 provides the leading coecients kn
(x18.2(iii)) for the Wilson, continuous dual Hahn,
Racah, and dual Hahn polynomials.
Table 18.25.2 : Wilson class OP's: leading coecients.
pn(x) kn
Wn(x;a;b;c;d ) ( 1)n(n+a+b+c+d 1)n
Sn(x;a;b;c ) ( 1)n
Rn(x; ; ;
; )(n+ + + 1)n
( + 1)n( ++ 1)n(
+ 1)n
Rn(x;
;;N )1
(
+ 1)n( N)n
18.26 Wilson Class: Continued
18.26(i) Representations as Generalized
Hypergeometric Functions
For the de nition of generalized hypergeometric func-
tions seex16.2.
18.26.1Wn
y2;a;b;c;d
= (a+b)n(a+c)n(a+d)n4F3 n;n+a+b+c+d 1;a+iy;a iy
a+b;a+c;a+d; 1
:
18.26.2Sn
y2;a;b;c
(a+b)n(a+c)n=3F2 n;a+iy;a iy
a+b;a+c; 1
:
18.26.3Rn(y(y+
++ 1); ; ;
; ) = 4F3 n;n+ + + 1; y;y+
++ 1
+ 1; ++ 1;
+ 1; 1
,
+ 1 or ++ 1 or
+ 1 = N;n= 0;1;:::;N .
18.26.4 Rn(y(y+
++ 1);
;;N ) = 3F2 n; y;y+
++ 1
+ 1; N; 1
, n= 0;1;:::;N .
18.26(ii) Limit Relations
Wilson!Continuous Dual Hahn
18.26.5 lim
d!1Wn(x;a;b;c;d )
(a+d)n=Sn(x;a;b;c ):
18.26 Wilson Class: Continued 469
Wilson!Continuous Hahn
18.26.6 lim
t!1Wn
(x+t)2;a it;b it;a+it;b+it
( 2t)nn!=pn
x;a;b;a;b
:
Wilson!Jacobi
18.26.7 lim
t!1Wn 1
2(1 x)t2;1
2 +1
2;1
2 +1
2;1
2 +1
2+it;1
2 +1
2 it
t2nn!=P( ; )
n(x):
Continuous Dual Hahn !Meixner{Pollaczek
18.26.8 lim
t!1Sn
(x t)2;+it; it;tcot
tn=n!(csc )nP()
n(x; ):
Racah!Dual Hahn
18.26.9 lim
!1Rn(x; N 1; ;
; ) =Rn(x;
;;N ):
Racah!Hahn
18.26.10 lim
!1Rn(x(x+
++ 1); ; ; N 1;) =Qn(x; ; ;N ):
Dual Hahn!Krawtchouk
18.26.11 lim
t!1Rn(x(x+t+ 1);pt;(1 p)t;N) =Kn(x;p;N):
Dual Hahn!Meixner
With
18.26.12 r(x; ;c;N ) =x(x+ +c 1(1 c)N);
18.26.13 lim
N!1Rn
r(x; ;c;N ); 1;c 1(1 c)N;N
=Mn(x; ;c):
See also Figure 18.21.1.
18.26(iii) Di erence Relations
For comments on the use of the forward-di erence operator x, the backward-di erence operator rx, and the
central-di erence operator x, seex18.2(ii).
For each family only the y-di erence that lowers nis given. See Koekoek and Swarttouw (1998, Chapter 1) for
further formulas.
18.26.14y
Wn
y2;a;b;c;d
y(y2) = n(n+a+b+c+d 1)Wn 1
y2;a+1
2;b+1
2;c+1
2;d+1
2
:
18.26.15 y
Sn
y2;a;b;c
y(y2) = nSn 1
y2;a+1
2;b+1
2;c+1
2
:
18.26.16
y(Rn(y(y+
++ 1); ; ;
; ))
y(y(y+
++ 1))=n(n+ + + 1)
( + 1)( ++ 1)(
+ 1)Rn 1(y(y+
++ 2); + 1; + 1;
+ 1;):
18.26.17y(Rn(y(y+
++ 1);
;;N ))
y(y(y+
++ 1))= n
(
+ 1)NRn 1(y(y+
++ 2);
+ 1;;N 1):
18.26(iv) Generating Functions
For the hypergeometric function 2F1seexx15.1 and 15.2(i).
Wilson
18.26.18 2F1a+iy;d+iy
a+d;z
2F1b iy;c iy
b+c;z
=1X
n=0Wn
y2;a;b;c;d
(a+d)n(b+c)nn!zn,jzj<1.
470 Orthogonal Polynomials
Continuous Dual Hahn
18.26.19 (1 z) c+iy
2F1a+iy;b+iy
a+b;z
=1X
n=0Sn
y2;a;b;c
(a+b)nn!zn, jzj<1.
Racah
18.26.20
2F1 y; y+
++ 1;z
2F1y N;y+
+ 1
N;z
=NX
n=0( N)n(
+ 1)n
( N)nn!Rn(y(y+
++ 1); N 1; ;
; )zn:
Dual Hahn
18.26.21 (1 z)y
2F1y N;y+
+ 1
N;z
=NX
n=0(
+ 1)n( N)n
( N)nn!Rn(y(y+
++ 1);
;;N )zn:
18.26(v) Asymptotic Approximations
For asymptotic expansions of Wilson polynomials of
large degree see Wilson (1991), and for asymptotic ap-
proximations to their largest zeros see Chen and Ismail
(1998).
Other Orthogonal Polynomials
18.27q-Hahn Class
18.27(i) Introduction
Theq-hypergeometric OP's comprise the q-Hahn class
OP's and the Askey{Wilson class OP's ( x18.28). For
the notation of q-hypergeometric functions see xx17.2
and 17.4(i).
Theq-Hahn class OP's comprise systems of OP's
fpn(x)g,n= 0;1;:::;N , orn= 0;1;2;:::, that are
eigenfunctions of a second-order q-di erence operator.
Thus
18.27.1
A(x)pn(qx) +B(x)pn(x) +C(x)pn(q 1x) =npn(x);
whereA(x),B(x), andC(x) are independent of n, and
where thenare the eigenvalues. In the q-Hahn class
OP's the role of the operator d/dxin the Jacobi, La-
guerre, and Hermite cases is played by the q-derivative
Dq, as de ned in (17.2.41). A (nonexhaustive) classi ca-
tion of such systems of OP's was made by Hahn (1949).
There are 18 families of OP's of q-Hahn class. These
families depend on further parameters, in addition to
q. The generic (top level) cases are the q-Hahn polyno-
mials and the big q-Jacobi polynomials, each of which
depends on three further parameters.All these systems of OP's have orthogonality prop-
erties of the form
18.27.2X
x2Xpn(x)pm(x)jxjvx=hnn;m;
whereXis given byX=faqygy2I+orX=faqygy2I+[
f bqygy2I . Herea;bare xed positive real numbers,
andI+andI are sequences of successive integers, nite
or unbounded in one direction, or unbounded in both
directions. If I+andI are both nonempty, then they
are both unbounded to the right. Some of the systems
of OP's that occur in the classi cation do not have a
unique orthogonality property. Thus in addition to a
relation of the form (18.27.2), such systems may also
satisfy orthogonality relations with respect to a contin-
uous weight function on some interval.
Here only a few families are mentioned. They are
de ned by their q-hypergeometric representations, fol-
lowed by their orthogonality properties. For other for-
mulas, including q-di erence equations, recurrence re-
lations, duality formulas, special cases, and limit rela-
tions, see Koekoek and Swarttouw (1998, Chapter 3).
See also Gasper and Rahman (2004, pp. 195{199, 228{
230) and Ismail (2005, Chapters 13, 18, 21).
18.27(ii)q-Hahn Polynomials
18.27.3
Qn(x) =Qn(x; ; ;N ;q) =3 2q n; qn+1;x
q;q N;q;q
,
n= 0;1;:::;N .
18.27.4NX
y=0Qn(q y)Qm(q y)( q;q N;q)y( q) y
(q; 1q N;q)y
=hnn;m, n;m = 0;1;:::;N:
Forhnsee Koekoek and Swarttouw (1998, Eq. (3.6.2)).
18.27q-Hahn Class 471
18.27(iii) Big q-Jacobi Polynomials
18.27.5Pn(x;a;b;c ;q) = 3 2q n;abqn+1;x
aq;cq;q;q
;
and
18.27.6P( ; )
n(x;c;d;q)
=cnq ( +1)n
q +1; q +1c 1d;q
n
(q; q;q)n
Pn
q +1c 1dx;q ;q ; q c 1d;q
:
The orthogonality relations are given by (18.27.2), with
18.27.7 pn(x) =Pn(x;a;b;c ;q);
18.27.8X=faq`+1g`=0;1;2;:::[fcq`+1g`=0;1;2;:::;
18.27.9vx=(a 1x;c 1x;q)1
(x;bc 1x;q)1,
0<a<q 1, 0<b<q 1,c<0;
and
18.27.10 pn(x) =P( ; )
n(x;c;d;q)
18.27.11X=fcq`g`=0;1;2;:::[f dq`g`=0;1;2;:::;
18.27.12
vx=(qx=c; qx=d;q)1
(q +1x=c; q +1x=d;q)1, ; > 1,c;d> 0:
Forhnsee Koekoek and Swarttouw (1998, Eq. (3.5.2)).
18.27(iv) Little q-Jacobi Polynomials
18.27.13
pn(x) =pn(x;a;b;q) = 2 1q n;abqn+1
aq;q;qx
:
18.27.141X
y=0pn(qy)pm(qy)(bq;q)y(aq)y
(q;q)y
=hnn;m, 0 <a<q 1;b<q 1.
Forhnsee Koekoek and Swarttouw (1998, Eq. (3.12.2)).
18.27(v)q-Laguerre Polynomials
18.27.15
L( )
n(x;q) =
q +1;q
n
(q;q)n1 1q n
q +1;q; xqn+ +1
:
The measure is not uniquely determined:
18.27.16Z1
0L( )
n(x;q)L( )
m(x;q)x
( x;q)1dx
=
q +1;q
n
(q;q)nqnh(1)
0n;m, > 1,whereh(1)
0is given in Koekoek and Swarttouw (1998,
Eq. (3.21.2), and
18.27.171X
y= 1L( )
n(cqy;q)L( )
m(cqy;q)qy( +1)
( cqy;q)1
=
q +1;q
n
(q;q)nqnh(2)
0n;m, > 1,c>0,
whereh(2)
0is given in Koekoek and Swarttouw (1998,
Eq. (3.21.3).
18.27(vi) Stieltjes{Wigert Polynomials
18.27.18Sn(x;q) =nX
`=0q`2( x)`
(q;q)`(q;q)n `
=1
(q;q)n1 1q n
0;q; qn+1x
:
(Sometimes in the literature xis replaced by q1
2x.)
The measure is not uniquely determined:
18.27.19Z1
0Sn(x;q)Sm(x;q)
( x; qx 1;q)1dx=ln
q 1
qn(q;q)1
(q;q)nn;m;
and
18.27.20Z1
0Sn
q1
2x;q
Sm
q1
2x;q
exp
(lnx)2
2 ln(q 1)
dx
=p
2q 1ln(q 1)
qn(q;q)nn;m:
18.27(vii) Discrete q-Hermite I and II
Polynomials
Discreteq-Hermite I
18.27.21hn(x;q) = (q;q)nbn=2cX
`=0( 1)`q`(` 1)xn 2`
(q2;q2)`(q;q)n 2`
=xn
2 0q n;q n+1
;q2;x 2q2n 1
:
18.27.22
1X
`=0
hn
q`;q
hm
q`;q
+hn
q`;q
hm
q`;q
q`+1; q`+1;q
1q`
= (q;q)n(q; 1; q;q)1qn(n 1)=2n;m:
472 Orthogonal Polynomials
Discreteq-Hermite II
18.27.23
~hn(x;q) = (q;q)nbn=2cX
`=0( 1)`q 2n`q`(2`+1)xn 2`
(q2;q2)`(q;q)n 2`
=xn
2 1q n;q n+1
0;q2; x 2q2
:
18.27.24
1X
`= 1
~hn
cq`;q~hm
cq`;q
+~hn
cq`;q~hm
cq`;qq`
( c2q2`;q2)1
= 2
q2; c2q; c 2q;q2
1
(q; c2; c 2q2;q2)1(q;q)n
qn2n;m,
c>0.
(For discrete q-Hermite II polynomials the measure is
not uniquely determined.)
18.28 Askey{Wilson Class
18.28(i) Introduction
The Askey{Wilson class OP's comprise the four-
parameter families of Askey{Wilson polynomials and of
q-Racah polynomials, and cases of these families ob-
tained by specialization of parameters. The Askey{
Wilson polynomials form a system of OP's fpn(x)g,
n= 0;1;2;:::, that are orthogonal with respect to a
weight function on a bounded interval, possibly sup-
plemented with discrete weights on a nite set. The
q-Racah polynomials form a system of OP's fpn(x)g,
n= 0;1;2;:::;N , that are orthogonal with respect
to a weight function on a sequence fq y+cqy+1g,
y= 0;1;:::;N , withca constant. Both the Askey{
Wilson polynomials and the q-Racah polynomials can
best be described as functions of z(resp.y) such that
Pn(z) =pn(1
2(z+z 1)) in the Askey{Wilson case, and
Pn(y) =pn(q y+cqy+1) in theq-Racah case, and both
are eigenfunctions of a second-order q-di erence opera-
tor similar to (18.27.1).
In the remainder of this section the Askey{Wilson
class OP's are de ned by their q-hypergeometric repre-
sentations, followed by their orthogonal properties. For
further properties see Koekoek and Swarttouw (1998,
Chapter 3). See also Gasper and Rahman (2004,pp. 180{199) and Ismail (2005, Chapter 15). For the
notation of q-hypergeometric functions see xx17.2 and
17.4(i).
18.28(ii) Askey{Wilson Polynomials
18.28.1
pn(cos)
=pn(cos;a;b;c;djq)
=a nnX
`=0q`
abq`;acq`;adq`;q
n `
q n;abcdqn 1;q
`
(q;q)`` 1Y
j=0(1 2aqjcos+a2q2j)
=a n(ab;ac;ad ;q)n
4 3q n;abcdqn 1;aei;ae i
ab;ac;ad;q;q
:
Assumea;b;c;d are all real, or two of them are real
and two form a conjugate pair, or none of them are real
but they form two conjugate pairs. Furthermore, jabj,
jacj,jadj,jbcj,jbdj,jcdj<1. Then
18.28.2Z1
1pn(x)pm(x)w(x)dx=hnn;m,jaj;jbj;jcj;jdj1,
where
18.28.3 2sinw(cos) =
e2i;q
1
(aei;bei;cei;dei;q)1 2
;
18.28.4 h0=(abcd;q)1
(q;ab;ac;ad;bc;bd;cd ;q)1;
18.28.5
hn=h0(1 abcdqn 1) (q;ab;ac;ad;bc;bd;cd ;q)n
(1 abcdq2n 1) (abcd;q)n,
n= 1;2;:::.
More generally, without the constraints in (18.28.2),
18.28.6Z1
1pn(x)pm(x)w(x)dx+X
`pn(x`)pm(x`)!`=hnn;m;
withw(x) andhnas above. Also, x`are the points
1
2( q`+ 1q `) with any of the a;b;c;d whose ab-
solute value exceeds 1, and the sum is over the `=
0;1;2;::: withj q`j>1. See Koekoek and Swarttouw
(1998, Eq. (3.1.3)) for the value of !`when =a.
18.28 Askey{Wilson Class 473
18.28(iii) Al-Salam{Chihara Polynomials
18.28.7
Qn(cos;a;bjq)
=pn(cos;a;b;0;0jq)
=a nnX
`=0q`
abq`;q
n `(q n;q)`
(q;q)`
` 1Y
j=0(1 2aqjcos+a2q2j)
=(ab;q)n
an3 2q n;aei;ae i
ab;0;q;q
=
be i;q
nein
2 1q n;aei
b 1q1 nei;q;b 1qe i
:
18.28.81
2Z
0Qn(cos;a;bjq)Qm(cos;a;bjq)
e2i;q
1
(aei;bei;q)1 2
d=n;m
(qn+1;abqn;q)1,
a;b2Rora=b;jabj<1;jaj;jbj1.
More generally, without the constraints jaj;jbj 1
discrete terms need to be added to the right-hand
side of (18.28.8); see Koekoek and Swarttouw (1998,
Eq. (3.8.3)).
18.28(iv)q 1-Al-Salam{Chihara Polynomials
18.28.9
Qn 1
2(aq y+a 1qy);a;bjq 1
= ( 1)nbnq 1
2n(n 1)
(ab) 1;q
n3 1q n;q y;a 2qy
(ab) 1;q;qnab 1
:
18.28.10
1X
y=0(1 q2ya 2)
a 2;(ab) 1;q
y
(1 a 2) (q;bqa 1;q)y(ba 1)yqy2
Qn 1
2(aq y+a 1qy);a;bjq 1
Qm 1
2(aq y+a 1qy);a;bjq 1
=
qa 2;q
1
(ba 1q;q)1
q;(ab) 1;q
n(ab)nq n2n;m:
Eq. (18.28.10) is valid when either
18.28.11 0<q< 1;a;b2R;ab> 1;a 1b<q 1;
or
18.28.12
0<q< 1;a/i;b/i2R;(=a)(=b)>0;a 1b<q 1:
If, in addition to (18.28.11) or (18.28.12), we have
a 1bq, then the measure in (18.28.10) is uniquely
determined. Also, if q < a 1b < q 1, then (18.28.10)holds witha;binterchanged. For further nondegenerate
cases see Chihara and Ismail (1993) and Christiansen
and Ismail (2006).
18.28(v) Continuous q-Ultraspherical
Polynomials
18.28.13
Cn(cos; jq)
=nX
`=0( ;q)`( ;q)n `
(q;q)`(q;q)n `ei(n 2`)
=( ;q)n
(q;q)nein
2 1q n;
1q1 n;q; 1qe 2i
:
18.28.14
Cn(cos; jq)
=
2;q
n
(q;q)n 1
2n4 3
q n; 2qn; 1
2ei; 1
2e i
q1
2; ; q1
2;q;q!
:
18.28.15
1
2Z
0Cn(cos; jq)Cm(cos; jq)
e2i;q
1
( e2i;q)1 2
d
=( ; q ;q)1
( 2;q;q)1(1 )
2;q
n
(1 qn) (q;q)nn;m, 1< < 1.
These polynomials are also called Rogers polynomi-
als.
18.28(vi) Continuous q-Hermite Polynomials
18.28.16Hn(cosjq) =nX
`=0(q;q)nei(n 2`)
(q;q)`(q;q)n `
=ein
2 0q n;0
;q;qne 2i
:
18.28.17
1
2Z
0Hn(cosjq)Hm(cosjq)
e2i;q
1 2d
=n;m
(qn+1;q)1:
18.28(vii) Continuous q 1-Hermite Polynomials
18.28.18hn(sinhtjq) =nX
`=0q1
2`(`+1)(q n;q)`
(q;q)`e(n 2`)t
=ent
1 1q n
0;q; qe 2t
=i nHn
isinhtjq 1
:
For continuous q 1-Hermite polynomials the orthog-
onality measure is not unique. See Askey (1989) and
Ismail and Masson (1994) for examples.
474 Orthogonal Polynomials
18.28(viii)q-Racah Polynomials
Withx=q y+
qy+1,
18.28.19
Rn(x) =Rn(x; ; ;
;jq)
=nX
`=0q`
q n; qn+1;q
`
( q; q;
q;q ;q)`` 1Y
j=0(1 qjx+
q2j+1)
=4 3q n; qn+1;q y;
qy+1
q; q;
q;q;q
,
q, q, or
q=q N;n= 0;1;:::;N:
18.28.20NX
y=0Rn(q y+
qy+1)Rm(q y+
qy+1)!y
=hnn;m, n;m = 0;1;:::;N .
For!yandhnsee Koekoek and Swarttouw (1998,
Eq. (3.2.2)).
18.29 Asymptotic Approximations for
q-Hahn and Askey{Wilson Classes
Ismail (1986) gives asymptotic expansions as n!1 ,
withxand other parameters xed, for continuous
q-ultraspherical, big and little q-Jacobi, and Askey{
Wilson polynomials. These asymptotic expansions
are in fact convergent expansions. For Askey{Wilson
pn(cos;a;b;c;djq) the leading term is given by
18.29.1(bc;bd;cd ;q)n
Qn(ei;a;b;c;djq)
+Qn(e i;a;b;c;djq)
;
where with z=ei,
18.29.2
Qn(z;a;b;c;djq)zn
az 1;bz 1;cz 1;dz 1;q
1
(z 2;bc;bd;cd ;q)1,
n!1 ;z;a;b;c;d;q xed.
For a uniform asymptotic expansion of the Stieltjes{
Wigert polynomials, see Wang and Wong (2006).
For asymptotic approximations to the largest zeros
of theq-Laguerre and continuous q 1-Hermite polyno-
mials see Chen and Ismail (1998).
18.30 Associated OP's
In the recurrence relation (18.2.8) assume that the co-
ecientsAn,Bn, andCn+1are de ned when nis a
continuous nonnegative real variable, and let cbe an
arbitrary positive constant. Assume also
18.30.1 AnAn+1Cn+1>0, n0.
Then the associated orthogonal polynomials pn(x;c) are
de ned by
18.30.2 p 1(x;c) = 0; p 0(x;c) = 1;and
18.30.3pn+1(x;c) = (An+cx+Bn+c)pn(x;c)
Cn+cpn 1(x;c),n= 0;1;:::.
Assume also that Eq. (18.30.3) continues to hold,
except that when n= 0,Bcis replaced by an arbitrary
real constant. Then the polynomials pn(x;c) generated
in this manner are called corecursive associated OP's .
Associated Jacobi Polynomials
These are de ned by
18.30.4 P( ; )
n(x;c) =pn(x;c),n= 0;1;:::,
wherepn(x;c) is given by (18.30.2) and (18.30.3), with
An,Bn, andCnas in (18.9.2). Explicitly,
18.30.5
( 1)n( + +c+ 1)nn!P( ; )
n(x;c)
( + + 2c+ 1)n( +c+ 1)n
=nX
`=0( n)`(n+ + + 2c+ 1)`
(c+ 1)`( +c+ 1)` 1
2x+1
2`
4F3` n;n+`+ + + 2c+ 1; +c;c
+`+c+ 1;`+c+ 1; + + 2c; 1
;
where the generalized hypergeometric function 4F3is
de ned by (16.2.1).
For corresponding corecursive associated Jacobi
polynomials see Letessier (1995).
Associated Legendre Polynomials
These are de ned by
18.30.6 Pn(x;c) =P(0;0)
n(x;c),n= 0;1;:::.
Explicitly,
18.30.7Pn(x;c) =nX
`=0c
`+cP`(x)Pn `(x):
(These polynomials are not to be confused with associ-
ated Legendre functions x14.3(ii).)
For further results on associated Legendre polyno-
mials see Chihara (1978, Chapter VI, x12); on associ-
ated Jacobi polynomials, see Wimp (1987) and Ismail
and Masson (1991). For associated Pollaczek polynomi-
als (comparex18.35) see Erd elyi et al. (1953b,x10.21).
For associated Askey{Wilson polynomials see Rahman
(2001).
18.31 Bernstein{Szeg o Polynomials
Let(x) be a polynomial of degree `and positive when
1x1. The Bernstein{Szeg o polynomials fpn(x)g,
n= 0;1;:::, are orthogonal on ( 1;1) with respect
to three types of weight function: (1 x2) 1
2((x)) 1,
(1 x2)1
2((x)) 1, (1 x)1
2(1 +x) 1
2((x)) 1. In con-
sequence,pn(cos) can be given explicitly in terms of
(cos) and sines and cosines, provided that ` < 2n
in the rst case, ` < 2n+ 2 in the second case, and
`<2n+ 1 in the third case. See Szeg o (1975, x2.6).
18.32 OP's with Respect to Freud Weights 475
18.32 OP's with Respect to Freud Weights
AFreud weight is a weight function of the form
18.32.1 w(x) = exp( Q(x)), 1<x<1,
whereQ(x) is real, even, nonnegative, and continu-
ously di erentiable. Of special interest are the cases
Q(x) =x2m,m= 1;2;:::. No explicit expressions
for the corresponding OP's are available. However, for
asymptotic approximations in terms of elementary func-
tions for the OP's, and also for their largest zeros, see
Levin and Lubinsky (2001) and Nevai (1986). For a
uniform asymptotic expansion in terms of Airy func-
tions (x9.2) for the OP's in the case Q(x) =x4see Bo
and Wong (1999).
18.33 Polynomials Orthogonal on the Unit
Circle
18.33(i) De nition
A system of polynomials f n(z)g,n= 0;1;:::, where
n(z) is of proper degree n, isorthonormal on the unit
circle with respect to the weight function w(z) (0) if
18.33.11
2iZ
jzj=1 n(z) m(z)w(z)dz
z=n;m;
where the bar signi es complex conjugate. See Simon
(2005a,b) for general theory.
18.33(ii) Recurrence Relations
Denote
18.33.2 n(z) =nzn+nX
`=1n;n `zn `;
wheren(>0), andn;n `(2C) are constants. Also
denote
18.33.3
n(z) =nzn+nX
`=1n;n `zn `;
where the bar again signi es compex conjugate. Then
18.33.4nz n(z) =n+1 n+1(z) n+1(0)
n+1(z);
18.33.5n n+1(z) =n+1z n(z) + n+1(0)
n(z);
18.33.6n n(0) n+1(z) +n 1 n+1(0)z n 1(z)
= (n n+1(0) +n+1 n(0)z) n(z):
18.33(iii) Connection with OP's on the Line
Assume that w(ei ) =w(e i ). Set
18.33.7w1(x) = (1 x2) 1
2w
x+i(1 x2)1
2
;
w2(x) = (1 x2)1
2w
x+i(1 x2)1
2
:Letfpn(x)gandfqn(x)g,n= 0;1;:::, be OP's with
weight functions w1(x) andw2(x), respectively, on
( 1;1). Then
18.33.8
pn 1
2(z+z 1)
= (const.)
z n 2n(z) +zn 2n(z 1)
= (const.)
z n+1 2n 1(z) +zn 1 2n 1(z 1)
;
18.33.9
qn 1
2(z+z 1)
= (const.)z n 1 2n+2(z) zn+1 2n+2(z 1)
z z 1
= (const.)z n 2n+1(z) zn 2n+1(z 1)
z z 1:
Conversely,
18.33.10
z n 2n(z)
=Anpn 1
2(z+z 1)
+Bn(z z 1)qn 1 1
2(z+z 1)
;
18.33.11
z n+1 2n 1(z)
=Cnpn 1
2(z+z 1)
+Dn(z z 1)qn 1 1
2(z+z 1)
;
whereAn,Bn,Cn, andDnare independent of z.
18.33(iv) Special Cases
Trivial
18.33.12 n(z) =zn; w (z) = 1:
Szeg o{Askey
18.33.13
n(z)
=nX
`=0(+ 1)`()n `
`! (n `)!z`=()n
n!2F1 n;+ 1
n+ 1;z
;
with
18.33.14w(z) =
1 1
2(z+z 1);
w1(x) = (1 x) 1
2(1 +x) 1
2;
w2(x) = (1 x)+1
2(1 +x)1
2,> 1
2.
For the hypergeometric function 2F1seexx15.1 and
15.2(i).
Askey
18.33.15 n(z) =nX
`=0
aq2;q2
`
a;q2
n `
(q2;q2)`(q2;q2)n `(q 1z)`
=
a;q2
n
(q2;q2)n2 1aq2;q 2n
a 1q2 2n;q2;qz
a
;
with
18.33.16w(z) =
qz;q2
1.
aqz;q2
1 2
,a2q2<1.
For the notation, including the basic hypergeometric
function 2 1, seexx17.2 and 17.4(i).
Whena= 0 the Askey case is also known as the
Rogers{Szeg o case .
476 Orthogonal Polynomials
18.33(v) Biorthogonal Polynomials on the Unit
Circle
See Baxter (1961) for general theory. See Askey
(1982) and Pastro (1985) for special cases extending
(18.33.13){(18.33.14) and (18.33.15){(18.33.16), respec-
tively. See Gasper (1981) and Hendriksen and van
Rossum (1986) for relations with Laurent polynomials
orthogonal on the unit circle. See Al-Salam and Ismail
(1994) for special biorthogonal rational functions on the
unit circle.
18.34 Bessel Polynomials
18.34(i) De nitions and Recurrence Relation
For the con
uent hypergeometric function 1F1and the
generalized hypergeometric function 2F0seex16.2(ii)
andx16.2(iv).
18.34.1
yn(x;a) = 2F0 n;n+a 1
; x
2
= (n+a 1)nx
2n
1F1 n
2n a+ 2;2
x
:
Other notations in use are given by
18.34.2yn(x) =yn(x; 2); n(x) =xnyn(x 1);
and
18.34.3
yn(x;a;b) =yn(2x=b;a); n(x;a;b) =xnyn(x 1;a;b):
Often only the polynomials (18.34.2) are called Bessel
polynomials , while the polynomials (18.34.1) and
(18.34.3) are called generalized Bessel polynomials . See
alsox10.49(ii).
18.34.4
yn+1(x;a) = (Anx+Bn)yn(x;a) Cnyn 1(x;a);
where
18.34.5An=(2n+a)(2n+a 1)
2(n+a 1);
Bn=(a 2)(2n+a 1)
(n+a 1)(2n+a 2);
Cn= n(2n+a)
(n+a 1)(2n+a 2):
18.34(ii) Orthogonality
Because the coecients Cnin (18.34.4) are not all pos-
itive, the polynomials yn(x;a) cannot be orthogonal
on the line with respect to a positive weight function.
There is orthogonality on the unit circle, however:
18.34.6
1
2iZ
jzj=1za 2yn(z;a)ym(z;a)e 2=zdz
=( 1)n+a 1n! 2a 1
(n+a 2)!(2n+a 1)n;m,a= 1;2;:::,the integration path being taken in the positive rota-
tional sense.
Orthogonality can also be expressed in terms of mo-
ment functionals ; see Dur an (1993), Evans et al. (1993),
and Maroni (1995).
18.34(iii) Other Properties
18.34.7
x2y00
n(x;a)+(ax+2)y0
n(x;a) n(n+a 1)yn(x;a) = 0;
where primes denote derivatives with respect to x.
18.34.8 lim
!1P( ;a 2)
n (1 + x)
P( ;a 2)
n (1)=yn(x;a):
For uniform asymptotic expansions of yn(x;a) as
n! 1 in terms of Airy functions ( x9.2) see Wong
and Zhang (1997) and Dunster (2001c). For uniform
asymptotic expansions in terms of Hermite polynomials
see L opez and Temme (1999b).
For further information on Bessel polynomials see
x10.49(ii).
18.35 Pollaczek Polynomials
18.35(i) De nition and Hypergeometric
Representation
18.35.1P()
1(x;a;b) = 0; P()
0(x;a;b) = 1;
and
18.35.2
(n+ 1)P()
n+1(x;a;b) = 2((n++a)x+b)P()
n(x;a;b)
(n+ 2 1)P()
n 1(x;a;b),
n= 0;1;:::.
Next, let
18.35.3 a;b() =acos+b
sin, 0<< .
Then
18.35.4
P()
n(cos;a;b)
=( i a;b())n
n!ein
2F1 n;+i a;b()
n + 1 +i a;b();e 2i
=nX
`=0(+i a;b())`
`!( i a;b())n `
(n `)!ei(n 2`):
For the hypergeometric function 2F1seexx15.1, 15.2(i).
18.36 Miscellaneous Polynomials 477
18.35(ii) Orthogonality
18.35.5Z1
1P()
n(x;a;b)P()
m(x;a;b)w()(x;a;b)dx= 0,
n6=m,
where
18.35.6
w()(cos;a;b) = 122 1e(2 ) a;b()
(sin)2 1j (+i a;b())j2,
ab a,> 1
2, 0<< .
18.35(iii) Other Properties
18.35.7(1 zei) +i a;b()(1 ze i) i a;b()
=1X
n=0P()
n(cos;a;b)zn,jzj<1, 0<< .
18.35.8 P()
n(x; 0;0) =C()
n(x);
18.35.9P()
n(cos ; 0;xsin ) =P()
n(x; ):
For the polynomials C()
n(x) andP()
n(x; ) seexx18.3
and 18.19, respectively.
See Bo and Wong (1996) for an asymptotic ex-
pansion of P(1
2)
n
cos (n 1
2);a;b
asn! 1 , witha
andb xed. This expansion is in terms of the Airy
function Ai( x) and its derivative ( x9.2), and is uni-
form in any compact -interval in (0 ;1). Also in-
cluded is an asymptotic approximation for the zeros of
P(1
2)
n
cos (n 1
2);a;b
.
18.36 Miscellaneous Polynomials
18.36(i) Jacobi-Type Polynomials
These are OP's on the interval ( 1;1) with respect to
an orthogonality measure obtained by adding constant
multiples of \Dirac delta weights" at 1 and 1 to the
weight function for the Jacobi polynomials. For further
information see Koornwinder (1984a) and Kwon et al.
(2006).
Similar OP's can also be constructed for the La-
guerre polynomials; see Koornwinder (1984b, (4.8)).
18.36(ii) Sobolev OP's
Sobolev OP's are orthogonal with respect to an in-
ner product involving derivatives. For an introduc-
tory survey to this subject, see Marcell an et al. (1993).
Other relevant references include Iserles et al. (1991)
and Koekoek et al. (1998).18.36(iii) Multiple OP's
These are polynomials in one variable that are orthog-
onal with respect to a number of di erent measures.
They are related to Hermite-Pad e approximation and
can be used for proofs of irrationality or transcendence
of interesting numbers. For further information see Is-
mail (2005, Chapter 23).
18.36(iv) Orthogonal Matrix Polynomials
These are matrix-valued polynomials that are orthogo-
nal with respect to a square matrix of measures on the
real line. Classes of such polynomials have been found
that generalize the classical OP's in the sense that they
satisfy second-order matrix di erential equations with
coecients independent of the degree. For further in-
formation see Dur an and Gr unbaum (2005).
18.37 Classical OP's in Two or More
Variables
18.37(i) Disk Polynomials
De nition in Terms of Jacobi Polynomials
18.37.1
R( )
m;n
rei
=ei(m n)rjm njP( ;jm nj)
min(m;n)
2r2 1
P( ;jm nj)
min(m;n)(1),
r0,2R, > 1.
Orthogonality
18.37.2ZZ
x2+y2<1R( )
m;n(x+iy)R( )
j;`(x iy) (1 x2 y2) dxdy
= 0, m6=jand/orn6=`.
Equivalent De nition
The following three conditions, taken together, deter-
mineR( )
m;n(z) uniquely:
18.37.3 R( )
m;n(z) =min(m;n)X
j=0cjzm jzn j;
wherecjare real or complex constants, with c06= 0;
18.37.4ZZ
x2+y2<1R( )
m;n(x+iy)(x iy)m j(x+iy)n j
(1 x2 y2) dxdy = 0,
j= 1;2;:::; min(m;n);
18.37.5 R( )
m;n(1) = 1:
478 Orthogonal Polynomials
Explicit Representation
18.37.6
R( )
m;n(z) =min(m;n)X
j=0( 1)j( + 1)m+n j( m)j( n)j
( + 1)m( + 1)nj!
zm jzn j:
18.37(ii) OP's on the Triangle
De nition in Terms of Jacobi Polynomials
18.37.7P ; ;
m;n (x;y) =P( ; +
+2n+1)
m n (2x 1)
xnP( ;
)
n
2x 1y 1
,
mn0, ; ;
> 1.
Orthogonality
18.37.8ZZ
0<y<x< 1P ; ;
m;n (x;y)P ; ;
j;`(x;y)
(1 x) (x y) y
dxdy = 0,
m6=jand/orn6=`.
See Dunkl and Xu (2001, x2.3.3) for analogs of
(18.37.1) and (18.37.7) on a d-dimensional simplex.
18.37(iii) OP's Associated with Root Systems
Orthogonal polynomials associated with root systems
are certain systems of trigonometric polynomials in sev-
eral variables, symmetric under a certain nite group
(Weyl group), and orthogonal on a torus. In one vari-
able they are essentially ultraspherical, Jacobi, continu-
ousq-ultraspherical, or Askey{Wilson polynomials. In
several variables they occur, for q= 1, as Jack polyno-
mials and also as Jacobi polynomials associated with root
systems ; see Macdonald (1995, Chapter VI, x10), Stan-
ley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman
(1991). For general qthey occur as Macdonald polyno-
mials for root system An, asMacdonald polynomials for
general root systems , and as Macdonald-Koornwinder
polynomials ; see Macdonald (1995, Chapter VI), Mac-
donald (2000, 2003), Koornwinder (1992).
Applications
18.38 Mathematical Applications
18.38(i) Classical OP's: Numerical Analysis
Approximation Theory
The scaled Chebyshev polynomial 21 nTn(x),n1,
enjoys the \minimax" property on the interval [ 1;1],
that is,j21 nTn(x)jhas the least maximum valueamong all monic polynomials of degree n. In conse-
quence, expansions of functions that are in nitely dif-
ferentiable on [ 1;1] in series of Chebyshev polynomi-
als usually converge extremely rapidly. For these results
and applications in approximation theory see x3.11(ii)
and Mason and Handscomb (2003, Chapter 3), Cheney
(1982, p. 108), and Rivlin (1969, p. 31).
Quadrature
Classical OP's play a fundamental role in Gaussian
quadrature. If the nodes in a quadrature formula with
a positive weight function are chosen to be the zeros of
thenth degree OP with the same weight function, and
the interval of orthogonality is the same as the integra-
tion range, then the weights in the quadrature formula
can be chosen in such a way that the formula is exact
for all polynomials of degree not exceeding 2 n 1. See
x3.5(v).
Di erential Equations
Linear ordinary di erential equations can be solved di-
rectly in series of Chebyshev polynomials (or other
OP's) by a method originated by Clenshaw (1957). This
process has been generalized to spectral methods for
solving partial di erential equations. For further infor-
mation see Mason and Handscomb (2003, Chapters 10
and 11), Gottlieb and Orszag (1977, pp. 7{19), and Guo
(1998, pp. 120{151).
18.38(ii) Classical OP's: Other Applications
Integrable Systems
The Toda equation provides an important model of a
completely integrable system. It has elegant structures,
includingN-soliton solutions, Lax pairs, and B acklund
transformations. While the Toda equation is an impor-
tant model of nonlinear systems, the special functions
of mathematical physics are usually regarded as solu-
tions to linear equations. However, by using Hirota's
technique of bilinear formalism of soliton theory, Naka-
mura (1996) shows that a wide class of exact solutions
of the Toda equation can be expressed in terms of var-
ious special functions, and in particular classical OP's.
For instance,
18.38.1Vn(x) = 2nHn+1(x)Hn 1(x)
(Hn(x))2;
withHn(x) as inx18.3, satis es the Toda equation
18.38.2(d2
dx2) lnVn(x) =Vn+1(x) +Vn 1(x)
2Vn(x),n= 1;2;:::.
Complex Function Theory
The Askey{Gasper inequality
18.38.3
nX
m=0P( ;0)
m(x)0, 1x1, > 1,n= 0;1;:::,
18.39 Physical Applications 479
was used in de Branges' proof of the long-standing
Bieberbach conjecture concerning univalent functions
on the unit disk in the complex plane. See de Branges
(1985).
Zonal Spherical Harmonics
Ultraspherical polynomials are zonal spherical harmon-
ics. As such they have many applications. See, for
example, Andrews et al. (1999, Chapter 9). See also
x14.30.
Random Matrix Theory
Hermite polynomials (and their Freud-weight analogs
(x18.32)) play an important role in random matrix the-
ory. See Fyodorov (2005) and Deift (1998, Chapter 5).
Riemann{Hilbert Problems
See Deift (1998, Chapter 7) and Ismail (2005, Chap-
ter 22).
Radon Transform
See Deans (1983, Chapters 4, 7).
18.38(iii) Other OP's
Group Representations
For group-theoretic interpretations of OP's see Vilenkin
and Klimyk (1991, 1992, 1993).
Coding Theory
For applications of Krawtchouk polynomi-
alsKn(x;p;N) and q-Racah polynomials
Rn(x; ; ;
;jq) to coding theory see Bannai (1990,
pp. 38{43), Leonard (1982), and Chihara (1987).
18.39 Physical Applications
18.39(i) Quantum Mechanics
Classical OP's appear when the time-dependent
Schr odinger equation is solved by separation of vari-
ables. Consider, for example, the one-dimensional form
of this equation for a particle of mass mwith potential
energyV(x):
18.39.1 h2
2m@2
@x2+V(x)
(x;t) =ih@
@t (x;t);
where his the reduced Planck's constant. On substitut-
ing (x;t) =(x)(t), we obtain two ordinary di eren-
tial equations, each of which involve the same constant
E. The equation for (x) is
18.39.2d2
dx2+2m
h2(E V(x))= 0:
For a harmonic oscillator, the potential energy is given
by
18.39.3 V(x) =1
2m!2x2;where!is the angular frequency. For (18.39.2) to have a
nontrivial bounded solution in the interval 1< x <
1, the constant E(the total energy of the particle)
must satisfy
18.39.4 E=En=
n+1
2
h!,n= 0;1;2;:::.
The corresponding eigenfunctions are
18.39.5n(x) = 1
42 1
2n(n!b) 1
2Hn(x=b)e x2=2b2;
whereb= (h=m! )1=2, andHnis the Hermite polyno-
mial. For further details, see Seaborn (1991, p. 224) or
Nikiforov and Uvarov (1988, pp. 71-72).
A second example is provided by the three-
dimensional time-independent Schr odinger equation
18.39.6r2 +2m
h2(E V(x)) = 0;
when this is solved by separation of variables in spher-
ical coordinates ( x1.5(ii)). The eigenfunctions of one of
the separated ordinary di erential equations are Legen-
dre polynomials. See Seaborn (1991, pp. 69-75).
For a third example, one in which the eigenfunctions
are Laguerre polynomials, see Seaborn (1991, pp. 87-93)
and Nikiforov and Uvarov (1988, pp. 76-80 and 320-
323).
18.39(ii) Other Applications
For applications of Legendre polynomials in
uid dy-
namics to study the
ow around the outside of a pu of
hot gas rising through the air, see Paterson (1983).
For applications and an extension of the Szeg o{
Sz asz inequality (18.14.20) for Legendre polynomials
( = = 0) to obtain global bounds on the varia-
tion of the phase of an elastic scattering amplitude, see
Cornille and Martin (1972, 1974).
For physical applications of q-Laguerre polynomials
seex17.17.
For interpretations of zeros of classical OP's as
equilibrium positions of charges in electrostatic prob-
lems (assuming logarithmic interaction), see Ismail
(2000a,b).
Computation
18.40 Methods of Computation
Orthogonal polynomials can be computed from their ex-
plicit polynomial form by Horner's scheme ( x1.11(i)).
Usually, however, other methods are more ecient, es-
pecially the numerical solution of di erence equations
(x3.6) and the application of uniform asymptotic ex-
pansions (when available) for OP's of large degree.
480 Orthogonal Polynomials
However, for applications in which the OP's appear
only as terms in series expansions (compare x18.18(i))
the need to compute them can be avoided altogether
by use instead of Clenshaw's algorithm ( x3.11(ii)) and
its straightforward generalization to OP's other than
Chebyshev. For further information see Clenshaw
(1955), Gautschi (2004, xx2.1, 8.1), and Mason and
Handscomb (2003, x2.4).
18.41 Tables
18.41(i) Polynomials
ForPn(x) (= Pn(x)) seex14.33.
Abramowitz and Stegun (1964, Tables 22.4, 22.6,
22.11, and 22.13) tabulates Tn(x),Un(x),Ln(x), and
Hn(x) forn= 0(1)12. The ranges of xare 0:2(:2)1
forTn(x) andUn(x), and 0:5;1;3;5;10 forLn(x) and
Hn(x). The precision is 10D, except for Hn(x) which is
6-11S.
18.41(ii) Zeros
ForPn(x),Ln(x), andHn(x) seex3.5(v). See also
Abramowitz and Stegun (1964, Tables 25.4, 25.9, and
25.10).
18.41(iii) Other Tables
For tables prior to 1961 see Fletcher et al. (1962) and
Lebedev and Fedorova (1960).
18.42 Software
Seehttp://dlmf.nist.gov/18.42 .
References
General References
The main references for writing this chapter are An-
drews et al. (1999), Askey and Wilson (1985), Chi-
hara (1978), Koekoek and Swarttouw (1998), and Szeg o
(1975).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.x18.2 Andrews et al. (1999, Chapter 5), Szeg o (1975,
xx2.2(i), 3.2), Chihara (1978, Chapter 1, Theo-
rem 3.3 and p. 21).
x18.3 In Table 18.3.1, for Row 2 see Szeg o (1975, x2.4,
Item 1, (4.3.3), and (4.21.6)): the entry in the
last column follows from (18.5.7); for Row 3 see
Szeg o (1975, (4.7.1), (4.7.14), and (4.7.9)): the
entry in the last column follows from the symme-
try in the fourth row of Table 18.6.1; for Rows
4{7 see Andrews et al. (1999,x5.1 and Remark
2.5.3); for Row 10 specialize Row 2 to = = 0;
for Row 12 see Szeg o (1975, x2.4, Item 2, (5.1.1),
and (5.1.8)): the entry in the last column follows
from (18.5.12); for Row 13 see Szeg o (1975, x2.4,
Item 3, (5.5.1), and (5.5.6)): the entry in the last
column follows from the symmetry in the tenth
row of Table 18.6.1.
x18.4 These graphics were produced at NIST.
x18.5 To verify (18.5.1){(18.5.4) substitute them in the
fourth through seventh rows, respectively, of Ta-
ble 18.3.1. Alternatively, combine Szeg o (1975,
(4.1.7), (4.1.8)) with (18.7.5), (18.7.6). In Table
18.5.1, for Rows 2, 3, 9, 10 see Szeg o (1975, (4.3.1),
(4.7.12), (5.1.5), (5.5.3)); Rows 4{7 follow from
Row 2 combined with (18.5.1){(18.5.4) and Table
18.6.1, second row; Row 8 is the case = = 0
of Row 2. For (18.5.6) see Truesdell (1948, x18,
(5)). For (18.5.7) see Szeg o (1975, (4.21.2)). For
(18.5.8) see Szeg o (1975, (4.3.2)). For (18.5.9)
see Szeg o (1975, (4.7.6)). For (18.5.10) see Szeg o
(1975, (4.7.31)). For (18.5.11) see Andrews et al.
(1999, (6.4.11)). For (18.5.12) see Szeg o (1975,
(5.3.3)). For (18.5.13) see Szeg o (1975, (5.5.4)).
For (18.5.14){(18.5.19) apply the recurrence rela-
tions (x18.9(i)), with initial values obtained from
the values of knand~kn=kngiven in Table 18.3.1
withn= 0;1.
x18.6 For (18.6.1) see Szeg o (1975, (5.1.7)). For Ta-
ble 18.6.1, Rows 2, 4, 10, see Szeg o (1975, (4.1.3),
(4.1.1), (4.7.4), (2.3.3), (4.7.3), (5.5.5)); the en-
tries in the fourth and fth columns of Rows 3
and 4 of Table 18.6.1 follow from (18.5.9) com-
bined (for Row 3) with (18.7.2); the other entries
of Row 3 of Table 18.6.1 are the case = of
Row 2; Rows 5{8 follow from (18.5.1){(18.5.4);
Row 9 is the case = 0 of Row 3. (18.6.2) follows
from (18.6.3) by the second row of Table 18.6.1.
(18.6.3) follows from (18.5.7) together with the
second row of Table 18.6.1. (18.6.4) follows from
(18.5.10) together with the fourth row of Table
18.6.1. (18.6.5) follows from (18.5.12) together
with (18.6.1).
References 481
x18.7 For (18.7.1){(18.7.6) and (18.7.13), (18.7.14) see
Szeg o (1975, (4.1.5), (4.1.7), (4.1.8), (4.7.1)).
(18.7.7){(18.7.12) follow from the de nitions given
by Table 18.3.1. (18.7.15) and (18.7.16) fol-
low from (18.7.13) and (18.7.14), combined with
(18.7.1). (18.7.17) follows from (18.7.4), (18.7.5),
and (18.7.13). (18.7.18) follows from (18.7.3),
(18.7.6), and (18.7.14). For (18.7.19) and
(18.7.20) see Szeg o (1975, (5.6.1)). For (18.7.21)
see Szeg o (1975, (5.3.4)). (18.7.22) follows from
(18.7.21) and the symmetry in Row 2 of Ta-
ble 18.6.1. (18.7.23) follows from (18.7.24) and
(18.7.1). For (18.7.24) see Szeg o (1975, (5.6.3)).
For (18.7.25) see Szeg o (1975, (4.7.8)). For
(18.7.26) see Calogero (1978).
x18.8 For Table 18.8.1, Rows 2, 3, 5, 9{10, 11{12,
see Szeg o (1975, (4.2.4), (4.24.2), (4.7.5), (5.1.2),
(5.5.2)), respectively; Row 4 is the special case
= of Row 2; Rows 6, 7, 8 are the special cases
= = 1
2, = =1
2, = = 0, respectively,
of Row 2.
x18.9 For (18.9.1), (18.9.2) see Szeg o (1975, (4.5.1)).
For Table 18.9.1, Rows 2, 9, 10, see Szeg o (1975,
(4.7.17), (5.1.10), (5.5.8)), respectively; Rows 3
and 4 are rewritings of elementary trigonometric
identities in view of (18.5.1), (18.5.2); Row 7 is the
special case = = 0 of (18.9.2). For (18.9.3){
(18.9.5) see Rainville (1960, x138, (17), (16), (14)).
For (18.9.6) see Szeg o (1975, (4.5.4)). For (18.9.7)
see Szeg o (1975, (4.7.29)). For (18.9.8) sub-
stitute (18.7.15) or (18.7.16); the resulting for-
mula is a special case of Rainville (1960, x138,
(11)). (18.9.9){(18.9.12) are rewritings of elemen-
tary trigonometric identities in view of (18.5.1){
(18.5.4). For (18.9.13), (18.9.14) see Szeg o (1975,
(5.1.13), (5.1.14)). For (18.9.15) see Szeg o (1975,
(4.21.7)). (18.9.16) is an immediate corollary of
(18.5.5) and Table 18.5.1, Row 2. For (18.9.17)
and (18.9.18) see Koornwinder (2006, x4). For
(18.9.19) see Szeg o (1975, (4.7.14)). (18.9.20) is an
immediate corollary of (18.5.5) and Table 18.5.1,
Row 3. (18.9.21) and (18.9.22) are rewritings of
elementary trigonometric di erentiation formulas.
For (18.9.23) see Szeg o (1975, (5.1.14)). (18.9.24)
is an immediate corollary of (18.5.5) and Table
18.5.1, Row 9. For (18.9.25) see Szeg o (1975,
(5.5.10)). (18.9.26) is an immediate corollary of
(18.5.5) and Table 18.5.1, Row 10.
x18.10 For (18.10.1) combine (14.12.1) and (14.3.21).
For (18.10.2) see Szeg o (1975, (4.8.6)). For
(18.10.3) see Askey (1975, (4.20)). For (18.10.4)
see Andrews et al. (1999, Theorem 6.7.4). For(18.10.5) see Szeg o (1975, (4.8.10)). (18.10.6) can
be obtained as a limit case of (18.10.3) in view of
(18.7.21). (18.10.7) can be obtained as a limit case
of (18.10.4) in view of (18.7.23). Table 18.10.1
follows from the corresponding Rodrigues formu-
las (x18.5(ii)) or generating functions ( x18.12); see
Szeg o (1975, (4.4.6), (4.82.1), (4.8.16), (4.8.1),
(5.4.8)). For (18.10.9), (18.10.10) see Andrews
et al. (1999, (6.2.15), (6.1.4)).
x18.11 For (18.11.1) see Andrews et al. (1999, (9.6.7)).
(18.11.2) is a rewriting of (18.5.12). For (18.11.3)
see Temme (1990a, (3.1)). For (18.11.5) see
Szeg o (1975, Theorem 8.1.1). For (18.11.6) see
Szeg o (1975, Theorem 8.22.4). For (18.11.7) and
(18.11.8) see Szeg o (1975, Theorem 8.22.8).
x18.12 For (18.12.1) see Andrews et al. (1999, (6.4.3)).
For (18.12.2) see Bateman (1905, pp. 113{114)
and Koornwinder (1974, p. 128). For (18.12.3)
see Andrews et al. (1999, (6.4.7)). For (18.12.4)
see Szeg o (1975, (4.7.23)). (18.12.5) follows
by combining (18.12.4) and its z-di erentiated
form. For (18.12.6) see Rainville (1960, x144,
(7)). For (18.12.7) see Andrews et al. (1999,
(5.1.16)). (18.12.8) is an immediate consequence
of (18.12.7). For (18.12.9) see Szeg o (1975,
(4.7.25)). (18.12.10) is the special case = 1
of (18.12.4) in view of (18.7.4). (18.12.11) is the
special case =1
2of (18.12.4). (18.12.12) is the
special case =1
2of (18.12.6). For (18.12.13) see
Andrews et al. (1999, (6.2.4)). For (18.12.14) see
Szeg o (1975, (5.1.16)). For (18.12.15) see Andrews
et al. (1999, (6.1.7)).
x18.13 Lorentzen and Waadeland (1992, pp. 446{448).
x18.14 For (18.14.1) and (18.14.2) see Szeg o (1975,
Theorem 7.32.1). For (18.14.3) see Chow et al.
(1994). For (18.14.4), (18.14.5), and (18.14.6)
see Szeg o (1975, Theorem 7.33.1). For (18.14.7)
see Lorch (1984). For (18.14.8) see Koornwinder
(1977, Remark 4.1). For (18.14.9) see Sz asz
(1951). For (18.14.10) see Szeg o (1948). For
(18.14.11) see Gasper (1972). For (18.14.12),
(18.14.13) see Skovgaard (1954). For (18.14.14){
(18.14.19) see Szeg o (1975, discussion following
Theorem 7.32.1). For (18.14.20) see Sz asz (1950).
For (18.14.21){(18.14.24) see Szeg o (1975, Theo-
rem 7.6.1). For the last statement about the suc-
cessive maxima of jHn(x)jsee Szeg o (1975, The-
orem 7.6.3).
x18.15 Frenzen and Wong (1985, 1988), Szeg o (1975,
Theorems 8.21.11, 8.22.2, 8.22.6).
482 Orthogonal Polynomials
x18.16 Szeg o (1975, Theorems 6.21.1, 6.21.2, 6.21.3,
6.3.2). For (18.16.6), (18.16.7) see Gatteschi
(1987). For (18.16.8) see Frenzen and Wong
(1985). For (18.16.10) and (18.16.11) see Szeg o
(1975, Theorem 6.31.3). For (18.16.12) and
(18.16.13) see Ismail and Li (1992). For (18.16.14)
see Tricomi (1949). For (18.16.15) see Szeg o
(1975, Theorem 6.32). See also Gatteschi (2002).
For (18.16.16){(18.16.18) see Szeg o (1975, Theo-
rem 6.32) and Sun (1996).
x18.17 For (18.17.1) see Erd elyi et al. (1953b,
x10.8(38)). For the rst equation in (18.17.2) ap-
ply the convolution property of the Laplace trans-
form (x1.14(iii)) to (18.17.34) with = 0. For
the second equation combine (18.9.23), (18.9.13),
and (18.6.1). For (18.17.3) and (18.17.4) use
(18.9.25) and (18.9.26). For (18.17.5) see Ismail
(2005, (9.6.2)). (18.17.6) is the case = 0 of
(18.17.5). For (18.17.7), (18.17.8) see Durand
(1975). For (18.17.9) and (18.17.10) see Andrews
et al. (1999, Theorem 6.7.2). For (18.17.11){
(18.17.15) see Askey and Fitch (1969). For
(18.17.16) use (18.5.5), integrate by parts ntimes,
expande iy(1 x)in a Maclaurin series, and inte-
grate term by term. For (18.17.17) use (18.5.5),
integrate repeatedly by parts, expand cos( xy) in
a Maclaurin series, and integrate term by term;
the proofs of (18.17.18) and (18.17.19) are simi-
lar. For (18.17.20) expand cos( xy) in a Maclau-
rin series, make the change of integration vari-
able 1 2x2=t, apply (18.5.5), integrate by
partsntimes, and use (10.8.3); the proof of
(18.17.21) is similar. For (18.17.22) see Strichartz
(1994,x7.6). For (18.17.23) use (18.12.16) and
the fact that the Fourier transform of e 1
2x2is
e 1
2y2; the proofs of (18.17.24), (18.17.27), and
(18.17.28) are similar, except that in the case
of (18.17.24), (18.12.15) replaces (18.12.16). For
(18.17.25) use (18.18.23); similarly for (18.17.26).
For (18.17.29) take the inverse Fourier transform
and apply (18.17.25). For (18.17.30) consider the
Fourier transform of this function instead of the
cosine transform, and replace L(n 1
2)
n 1
2x2
by its
explicit form (18.5.12); then integrate term by
term and rearrange the the consequential nite
double sum into a single sum. For (18.17.31)
use (18.5.5) and then integrate by parts; simi-
larly for (18.17.32). For (18.17.33) expand the
exponential in the integral as a power series in z
and interchange integration and summation. The
resulting integral can be evaluated by consider-
ing the term `= 0 in(18.18.14). (18.17.33) may
also be veri ed by applying Kummer's transfor-mation (13.2.39) to (18.17.16). (18.17.34) follows
by substituting (18.5.12) into the integrand and
performing termwise integration. (18.17.35) fol-
lows by use of (18.5.5) and integration by parts.
For (18.17.36) use (18.5.7) and apply (16.4.3). For
(18.17.37) use (18.5.5) and integrate by parts n
times. For (18.17.38) use the rst equality in
(18.5.10), with =1
2andnreplaced by 2 n, in-
tegrate term by term, then apply (16.4.3); the
proof of (18.17.39) is similar. For (18.17.40) use
(18.5.12), integrate term by term, then apply
(15.8.6). For (18.17.41) use (18.5.13) and inte-
grate term by term. (18.17.42) and (18.17.43)
follow from the case = =1
2of Szeg o
(1975, Theorem 4.61.2), where the hypergeomet-
ric function on the right-hand side is rewritten as
in Erd elyi et al. (1953b, 19.8(19)). For (18.17.44)
see Tuck (1964). (18.17.46) is obtained from
(18.17.9) for = = 0,=1
2, together with
(18.7.5), (18.7.3), and the second row of Table
18.6.1. (18.17.45) is obtained from (18.17.46)
by symmetry; compare Rows 5 and 9 of Table
18.6.1. (18.17.47) and (18.17.48) follow by use
of (18.17.34) and (18.17.35). For (18.17.49) see
Andrews et al. (1999, p. 328).
x18.18 For (18.18.2) see Szeg o (1975, Theorem 9.1.2
and the Remarks on p. 248). For (18.18.3){
(18.18.7) see Lebedev (1965, pp. 68{71 and 88{
89) and Nikiforov and Uvarov (1988, pp. 21 and
59). For (18.18.8) see Carlson (1971). (18.18.9)
is the case = 0 of (18.18.8). (18.18.10) fol-
lows from (18.12.13). (18.18.11) follows from
(18.12.15). (18.18.12) follows by computingR1
0L( )
n(x)L( )
`(x)e xx dxwith use of the Ro-
drigues formula (Table 18.5.1), integration by
parts, and (18.5.12). (18.18.13) follows from
(18.18.12) for =1
2by (18.7.19), (18.7.20).
For (18.18.14) see Andrews et al. (1999, The-
orem 7.1.3). For (18.18.15) see Askey (1974).
For (18.18.16) see Andrews et al. (1999, Theo-
rem 7.1.40). (18.18.17) follows from (18.18.16),
(18.6.4), and the fourth row of Table 18.6.1.
(18.18.18) follows from (18.12.13). (18.18.19) fol-
lows from (18.18.12) by dividing both sides by n
and letting !1 . (18.18.20) is the case =1
2
of (18.18.19) in view of (18.7.19), (18.7.20). For
(18.18.21){(18.18.23) see Andrews et al. (1999,
(5.1.6), Theorems 6.8.2 and 6.8.1 and Remarks
6.8.2 and 6.8.1). For (18.18.25), (18.18.26) see
Koornwinder (1974). For (18.18.27), (18.18.28)
see Andrews et al. (1999, (6.2.25), (6.1.13)). For
the positivity of the Poisson kernels see Askey
(1975, p. 16). (18.18.29) follows from (18.12.4).
References 483
(18.18.30) follows from (18.12.4) and (18.12.5).
(18.18.31) is the limiting case of (18.18.30) as
!0. Each of the formulas (18.18.32){(18.18.35)
is equivalent to a di erence formula together with
a trivialn= 0 case, and each di erence for-
mula can be rewritten via (18.5.1), (18.5.2) as a
well-known trigonometric identity. (18.18.36) is
the special case ==1
2of (18.18.29). For
(18.18.37) see Szeg o (1975, (5.1.13)). (18.18.38)
follows from (18.12.13), and is the special case
r= 2 of (18.18.10). For (18.18.39) see Szeg o
(1975, (5.5.11)). (18.18.40) is the special case
= 1
2of (18.18.38) in view of (18.7.19).
x18.19 For Table 18.19.1 see Ismail (2005, (6.2.4),
(6.2.35), (6.1.4), (6.1.21)). For Table 18.19.2,
Rows 2, 4, see Ismail (2005, (6.2.7), (6.1.7));
Row 3 follows from (18.20.6); Row 5 follows
from (18.20.8). For (18.19.1){(18.19.4) see Askey
(1985, (4), (5)). (18.19.5) follows from (18.20.9).
For (18.19.6){(18.19.9) see Ismail (2005, (5.9.8),
(5.9.9)). The formula for knin (18.19.9) follows
from (18.20.10).
x18.20 For (18.20.2) see Karlin and McGregor (1961,
(1.8)). For Table 18.20.1, Rows 2, 3, see Is-
mail (2005, (6.2.42), (6.1.17)); for Row 4 see Chi-
hara (1978, Chapter V, (3.2)). (18.20.3) follows
by iteration of (18.22.28). (18.20.4) follows by
iteration of (18.22.30). For (18.20.5){(18.20.8)
and (18.20.10) see Ismail (2005, (6.2.3), (6.2.34),
(6.1.3), (6.1.20), (5.9.5)). For (18.20.9) see Askey
(1985).
x18.21 For (18.21.3), (18.21.5), (18.21.7), (18.21.8)
see Ismail (2005, x6.2, unnumbered formula af-
ter (6.2.34), also (6.2.17), (6.1.19), (6.1.18)). For
(18.21.1) see Karlin and McGregor (1961, (1.19)).
The three identities in (18.21.2) follow from
(18.20.6), (18.20.7), (18.20.8). (18.21.4) follows
from (18.20.5) and (18.20.7). (18.21.6) follows
from (18.20.6) and (18.20.8). (18.21.9) follows
from (18.22.2), Row 4 in Table 18.22.1, (18.9.1),
and Row 10 in Table 18.9.1. (18.21.10) follows
from (18.20.9) and (18.20.10). For (18.21.11)
see Koornwinder (1989, (2.6)). (18.21.12) follows
from (18.20.10) and (18.5.12). For Figure 18.21.1
see Askey and Wilson (1985, p. 46), together with
correction in Askey (1985).
x18.22 For (18.22.1){(18.22.3) see Ismail (2005, (6.2.8)
and (6.2.9)). For Table 18.22.1 see Ismail
(2005, (6.2.36), (6.1.5), and (6.1.25)). (18.22.4){
(18.22.6) is a limiting case of Andrews et al.
(1999, (3.8.2)), in view of (18.26.6). For (18.22.7){
(18.22.8) see Ismail (2005, (5.9.1)). For (18.22.9){(18.22.11) see Ismail (2005, (6.2.16)). For Ta-
ble 18.22.2, Rows 2 and 3, see Ismail (2005,
(6.2.38), (6.1.15)); Row 4 follows from Ta-
ble 18.22.1, Row 4, and the third identity in
(18.21.2). (18.22.13){(18.22.15) is a limiting case
of Koekoek and Swarttouw (1998, (1.1.6)) in view
of (18.26.6). Koekoek and Swarttouw (1998,
(1.1.6)) is a limiting case of Askey and Wil-
son (1985, (5.7)) in view of Koekoek and Swart-
touw (1998, (5.1.1)). (18.22.16){(18.22.17) fol-
low from (18.22.14) in view of (18.21.10). For
(18.22.19){(18.22.25) see Ismail (2005, (6.2.5),
(6.2.13), (6.2.39), (6.2.40), (6.1.13), x6.1, unnum-
bered formula following (6.1.15), also (6.1.23)).
(18.22.26) follows from (18.22.24) and (18.21.7).
(18.22.27) follows from (18.20.9). (18.22.28) fol-
lows from (18.22.14) and (18.22.27). (18.22.29)
follows from (18.20.10). (18.22.30) follows from
(18.22.28) in view of (18.21.10).
x18.23 For (18.23.3){(18.23.5), (18.23.7) see Is-
mail (2005, (6.2.43), (6.1.8), (6.1.22), (5.9.3)).
(18.23.1) and (18.23.2) follow by expanding the
factors on the left as power series in z, and sub-
stituting (18.20.5) on the right. (18.23.6) is a lim-
iting case of (18.26.18) via (18.26.6).
x18.25 For Table 18.25.1, Rows 2, 3, 4, see Wilson
(1980); for Row 5 see Ismail (2005, (6.2.20)).
(18.25.1) follows from (18.25.11). For (18.25.3){
(18.25.5) see Wilson (1980) and Andrews et al.
(1999, (3.8.3)). For (18.25.6){(18.25.8) see Wil-
son (1980). For (18.25.10){(18.25.12) see Wilson
(1980). For (18.25.13){(18.25.15) see Ismail (2005,
(6.2.20)). Table 18.25.2 follows from x18.26(i).
x18.26 For (18.26.1) see Andrews et al. (1999, Def-
inition 3.8.1). For (18.26.2) and (18.26.3) see
Wilson (1980). For (18.26.4) see Ismail (2005,
(6.2.19)). For (18.26.5) and (18.26.7) see Wil-
son (1980). (18.26.6) follows from (18.26.1) and
(18.20.9). (18.26.8) follows from (18.26.2) and
(18.20.10). (18.26.9) follows from (18.26.3) and
(18.26.4). (18.26.10) follows from (18.26.3) and
(18.20.5). For (18.26.11) see Karlin and McGregor
(1961, (1.21)). (18.26.12), (18.26.13) follow from
(18.26.4) and (18.20.7). (18.26.14){(18.26.17) fol-
low fromx18.26(i). For (18.26.18) see Ismail et al.
(1990, (6.1)). (18.26.19) follows by expanding
both factors on the left as power series in z, and
substituting (18.26.2) on the right. (18.26.20) fol-
lows from (18.26.18), (18.26.1), (18.26.3). For
(18.26.21) see Ismail (2005, (6.2.31)).
x18.27 For (18.27.3), (18.27.4) see Gasper and Rah-
man (2004, (7.2.21), (7.2.22)) and Ismail (2005,
484 Orthogonal Polynomials
(18.5.1), (18.5.2)). For (18.27.8){(18.27.11) see
Gasper and Rahman (2004, (7.3.10), (7.3.12))
and Ismail (2005, (18.4.7), (18.4.14)). For
(18.27.14) see Gasper and Rahman (2004, (7.3.1),
(7.3.3)). For (18.27.16), (18.27.17) see Ismail
(2005, (21.8.2), (21.8.4)) and Moak (1981, Theo-
rem 2). For (18.27.18){(18.27.20) see Ismail (2005,
(21.8.3), (21.8.46)). For (18.27.21){(18.27.24) see
Al-Salam and Carlitz (1965).
x18.28 For (18.28.1){(18.28.6) see Askey and Wil-
son (1985), Gasper and Rahman (2004, (7.5.2),
(7.5.15), (7.5.21)), Ismail (2005, (15.2.4),
(15.2.5)). For (18.28.7), (18.28.8) see Ismail
(2005, (15.1.5), (15.1.6), (15.1.11)). For (18.28.9){
(18.28.12) see Askey and Ismail (1984, Chapter 3).
For (18.28.13){(18.28.15) see Gasper and Rahman
(2004, (7.4.2), (7.4.14){(7.4.16)) and Ismail (2005,
(13.2.3){(13.2.5), (13.2.11)). For (18.28.16){
(18.28.18) see Ismail (2005, (13.1.7), (13.1.11),(21.2.1), (21.2.5)). For (18.28.19), (18.28.20) see
Gasper and Rahman (2004, (7.2.11)) and Ismail
(2005, (15.6.1), (15.6.7)).
x18.30 For (18.30.5) see Wimp (1987, Theorem 1).
(18.30.7) is mentioned in Chihara (1978, Chap-
ter VI, (12.6)), and proved in Barrucand and Dick-
inson (1968).
x18.33 For (18.33.1), (18.33.4), (18.33.8), and (18.33.9)
see Szeg o (1975, (11.1.8), (11.4.6), (11.4.7),
(11.5.2)). (18.33.6) follows from (18.33.4),
(18.33.5). (18.33.10), (18.33.11) follow from
(18.33.8), (18.33.9). For (18.33.13){(18.33.16) see
Askey (1982) and Pastro (1985).
x18.34 Ismail (2005, Chapter 4).
x18.35 Ismail (2005, Chapter 5).
x18.37 Dunkl and Xu (2001, x2.4.3), Koornwinder
(1975c).
Chapter 19
Elliptic Integrals
B. C. Carlson1
Notation 486
19.1 Special Notation . . . . . . . . . . . . . 486
Legendre's Integrals 486
19.2 De nitions . . . . . . . . . . . . . . . . . 486
19.3 Graphics . . . . . . . . . . . . . . . . . . 488
19.4 Derivatives and Di erential Equations . . 490
19.5 Maclaurin and Related Expansions . . . . 490
19.6 Special Cases . . . . . . . . . . . . . . . 491
19.7 Connection Formulas . . . . . . . . . . . 491
19.8 Quadratic Transformations . . . . . . . . 492
19.9 Inequalities . . . . . . . . . . . . . . . . 494
19.10 Relations to Other Functions . . . . . . . 494
19.11 Addition Theorems . . . . . . . . . . . . 495
19.12 Asymptotic Approximations . . . . . . . . 495
19.13 Integrals of Elliptic Integrals . . . . . . . 496
19.14 Reduction of General Elliptic Integrals . . 496
Symmetric Integrals 497
19.15 Advantages of Symmetry . . . . . . . . . 497
19.16 De nitions . . . . . . . . . . . . . . . . . 497
19.17 Graphics . . . . . . . . . . . . . . . . . . 499
19.18 Derivatives and Di erential Equations . . 500
19.19 Taylor and Related Series . . . . . . . . . 501
19.20 Special Cases . . . . . . . . . . . . . . . 50219.21 Connection Formulas . . . . . . . . . . . 503
19.22 Quadratic Transformations . . . . . . . . 504
19.23 Integral Representations . . . . . . . . . 506
19.24 Inequalities . . . . . . . . . . . . . . . . 506
19.25 Relations to Other Functions . . . . . . . 507
19.26 Addition Theorems . . . . . . . . . . . . 509
19.27 Asymptotic Approximations and Expansions 510
19.28 Integrals of Elliptic Integrals . . . . . . . 511
19.29 Reduction of General Elliptic Integrals . . 512
Applications 514
19.30 Lengths of Plane Curves . . . . . . . . . 514
19.31 Probability Distributions . . . . . . . . . 515
19.32 Conformal Map onto a Rectangle . . . . 515
19.33 Triaxial Ellipsoids . . . . . . . . . . . . . 515
19.34 Mutual Inductance of Coaxial Circles . . . 516
19.35 Other Applications . . . . . . . . . . . . 516
Computation 517
19.36 Methods of Computation . . . . . . . . . 517
19.37 Tables . . . . . . . . . . . . . . . . . . . 518
19.38 Approximations . . . . . . . . . . . . . . 519
19.39 Software . . . . . . . . . . . . . . . . . . 519
References 519
1Mathematics Department and Ames Laboratory (U.S. Department of Energy), Iowa State University, Ames, Iowa.
Acknowledgments : The parts of this chapter that deal with Legendre's integrals are based in part on Abramowitz and Stegun (1964,
Chapter 17) by L. M. Milne-Thomson. I am greatly indebted to R. C. Winther for indispensable technical support and to F. W. J. Olver
for long-sustained encouragement of a new approach to elliptic integrals. I thank E. Neuman for improvements to xx19.9 and 19.24(i).
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
485
486 Elliptic Integrals
Notation
19.1 Special Notation
(For other notation see pp. xiv and 873.)
l;m;n nonnegative integers.
real or complex argument (or amplitude).
k real or complex modulus.
k0complementary real or complex modulus,
k2+k02= 1.
2real or complex parameter.
B(a;b) beta function ( x5.12).
All square roots have their principal values. All
derivatives are denoted by di erentials, not by primes.
The rst set of main functions treated in this chapter
are Legendre's complete integrals
19.1.1 K(k); E (k);
2;k
;
of the rst, second, and third kinds, respectively, and
Legendre's incomplete integrals
19.1.2F( ;k); E ( ;k);
; 2;k
;
of the rst, second, and third kinds, respectively. This
notation follows Byrd and Friedman (1971, 110). We
use also the function D( ;k), introduced by Jahnke
et al. (1966, p. 43). The functions (19.1.1) and (19.1.2)
are used in Erd elyi et al. (1953b, Chapter 13), except
that
2;k
and
; 2;k
are denoted by 1(;k)
and ( ;;k ), respectively, where = 2.
In Abramowitz and Stegun (1964, Chapter 17) the
functions (19.1.1) and (19.1.2) are denoted, in order, by
K( ),E( ), (nn ),F( n ),E( n ), and (n; n ),
where = arcsinkandnis the 2(not related to k) in
(19.1.1) and (19.1.2). Also, frequently in this reference
is replaced by mandn byjm, wherem=k2. However,
it should be noted that in Chapter 8 of Abramowitz and
Stegun (1964) the notation used for elliptic integrals dif-
fers from Chapter 17 and is consistent with that used in
the present chapter and the rest of the NIST Handbook
and DLMF.
The second set of main functions treated in this
chapter is
19.1.3RC(x;y) ,RF(x;y;z ) ,RG(x;y;z ) ,
RJ(x;y;z;p ) ,RD(x;y;z ) ,
R a(b1;b2;:::;bn;z1;z2;:::;zn):
RF(x;y;z ),RG(x;y;z ), andRJ(x;y;z;p ) are the sym-
metric (inx,y, andz) integrals of the rst, second, and
third kinds; they are complete if exactly one of x,y, and
zis identically 0.
R a(b1;b2;:::;bn;z1;z2;:::;zn) is a multivariate
hypergeometric function that includes all the functions
in (19.1.3).A third set of functions, introduced by Bulirsch
(1965a,b, 1969a), is
19.1.4el1(x;kc);el2(x;kc;a;b);
el3(x;kc;p);cel(kc;p;a;b ):
The rst three functions are incomplete integrals of the
rst, second, and third kinds, and the cel function in-
cludes complete integrals of all three kinds.
Legendre's Integrals
19.2 De nitions
19.2(i) General Elliptic Integrals
Lets2(t) be a cubic or quartic polynomial in twith sim-
ple zeros, and let r(s;t) be a rational function of sand
tcontaining at least one odd power of s. Then
19.2.1Z
r(s;t)dt
is called an elliptic integral . Becauses2is a polynomial,
we have
19.2.2r(s;t) =(p1+p2s)(p3 p4s)s
(p3+p4s)(p3 p4s)s=
s+;
wherepjis a polynomial in twhileandare rational
functions of t. Thus the elliptic part of (19.2.1) is
19.2.3Z(t)
s(t)dt:
19.2(ii) Legendre's Integrals
Assume 1 sin2 2Cn( 1;0] and 1 k2sin2 2
Cn( 1;0], except that one of them may be 0, and
1 2sin2 2Cnf0g. Then
19.2.4F( ;k) =Z
0dp
1 k2sin2
=Zsin
0dtp
1 t2p
1 k2t2;
19.2.5E( ;k) =Z
0p
1 k2sin2d
=Zsin
0p
1 k2t2
p
1 t2dt:
19.2.6D( ;k) =Z
0sin2dp
1 k2sin2
=Zsin
0t2dtp
1 t2p
1 k2t2
= (F( ;k) E( ;k))=k2:
19.2 Definitions 487
19.2.7
; 2;k
=Z
0dp
1 k2sin2(1 2sin2)
=Zsin
0dtp
1 t2p
1 k2t2(1 2t2):
The paths of integration are the line segments connect-
ing the limits of integration. The integral for E( ;k) is
well de ned if k2= sin2 = 1, and the Cauchy principal
value (x1.4(v)) of
; 2;k
is taken if 1 2sin2 van-
ishes at an interior point of the integration path. Also,
ifk2and 2are real, then
; 2;k
is called a circular
orhyperbolic case according as 2( 2 k2)( 2 1) is
negative or positive. The circular and hyperbolic cases
alternate in the four intervals of the real line separated
by the points 2= 0;k2;1.
The cases with ==2 are the complete integrals :
19.2.8K(k) =F(=2;k); E (k) =E(=2;k);
D(k) =D(=2;k) = (K(k) E(k))=k2;
2;k
=
=2; 2;k
;
19.2.9K0(k) =K(k0); E0(k) =E(k0); k0=p
1 k2:
Ifmis an integer, then
19.2.10F(m ;k) = 2mK(k)F( ;k);
E(m ;k) = 2mE(k)E( ;k);
D(m ;k) = 2mD(k)D( ;k):
19.2(iii) Bulirsch's Integrals
Bulirsch's integrals are linear combinations of Legen-
dre's integrals that are chosen to facilitate computa-
tional application of Bartky's transformation (Bartky
(1938)). Two are de ned by
19.2.11cel(kc;p;a;b )
=Z=2
0acos2+bsin2
cos2+psin2dq
cos2+k2csin2;
19.2.12el2(x;kc;a;b)
=Zarctanx
0a+btan2p
(1 + tan2)(1 +k2ctan2)d:
Herea;b;p are real parameters, and kcandxare real or
complex variables, with p6= 0,kc6= 0. If 1<p< 0,
then the integral in (19.2.11) is a Cauchy principal value.
With
19.2.13kc=k0; p = 1 2; x = tan ;
special cases include
19.2.14
K(k) = cel(kc;1;1;1);
E(k) = cel
kc;1;1;k2
c
; D (k) = cel(kc;1;0;1);
(E(k) k02K(k))=k2= cel(kc;1;1;0);
2;k
= cel(kc;p;1;1);and
19.2.15F( ;k) = el1(x;kc) = el2(x;kc;1;1);
E( ;k) = el2
x;kc;1;k2
c
;
D( ;k) = el2(x;kc;0;1):
The integrals are complete ifx=1. If 1<k1=sin ,
thenkcis pure imaginary.
Lastly, corresponding to Legendre's incomplete inte-
gral of the third kind we have
19.2.16
el3(x;kc;p)
=Zarctanx
0d
(cos2+psin2)q
cos2+k2csin2
= (arctan x;1 p;k), x26= 1=p:
19.2(iv)RC(x;y)
Letx2Cn( 1;0) andy2Cnf0g. We de ne
19.2.17RC(x;y) =1
2Z1
0dtpt+x(t+y);
where the Cauchy principal value is taken if y<0. For-
mulas involving
; 2;k
that are customarily di er-
ent for circular cases, ordinary hyperbolic cases, and
(hyperbolic) Cauchy principal values, are united in a
single formula by using RC(x;y).
In (19.2.18){(19.2.22) the inverse trigonometric and
hyperbolic functions assume their principal values
(xx4.23(ii) and 4.37(ii)). When xandyare positive,
RC(x;y) is an inverse circular function if x<y and an
inverse hyperbolic function (or logarithm) if x>y :
19.2.18RC(x;y) =1py xarctanr
y x
x
=1py xarccosp
x=y, 0x<y;
19.2.19
RC(x;y) =1px yarctanhr
x y
x
=1px ylnpx+px ypy, 0<y<x:
The Cauchy principal value is hyperbolic:
19.2.20
RC(x;y) =rx
x yRC(x y; y)
=1px yarctanhrx
x y
=1px ylnpx+px yp y,y<0x:
488 Elliptic Integrals
If the line segment with endpoints xandylies in
Cn( 1;0], then
19.2.21RC(x;y) =Z1
0(v2x+ (1 v2)y) 1=2dv;
19.2.22
RC(x;y) =2
Z=2
0RC
y;xcos2+ysin2
d:
19.3 Graphics
19.3(i) Real Variables
See Figures 19.3.1{19.3.6 for complete and incomplete
Legendre's elliptic integrals.
Figure 19.3.1 :K(k) andE(k) as functions of k2for
2k21. Graphs of K0(k) andE0(k) are the mirror
images in the vertical line k2=1
2.
Figure 19.3.2 :RC(x;1) and the Cauchy principal value
ofRC(x; 1) for 0x5. Both functions are
asymptotic to ln(4 x)=p
4xasx!1 ; see (19.2.19) and
(19.2.20). Note that RC(x;y) =y 1=2RC(x=y;1),
y>0.
Figure 19.3.3 :F( ;k) as a function of k2and sin2 for
1k22, 0sin2 1. If sin2 = 1 (k2), then
the function reduces to K(k), becoming in nite when
k2= 1. If sin2 = 1=k2(<1), then it has the value
K(1=k)=k: putc=k2in (19.25.5) and use (19.25.1).
Figure 19.3.4 :E( ;k) as a function of k2and sin2
for 1k22, 0sin2 1. If sin2 = 1
(k2), then the function reduces to E(k), with value 1
atk2= 1. If sin2 = 1=k2(<1), then it has the value
kE(1=k) + (k02=k)K(1=k), with limit 1 as k2!1+:
putc=k2in (19.25.7) and use (19.25.1).
19.4 Derivatives and Differential Equations 489
Figure 19.3.5 :
2;k
as a function of k2and 2for
2k2<1, 2 22. Cauchy principal val-
ues are shown when 2>1. The function is un-
bounded as 2!1 , and also (with the same sign
as 1 2) ask2!1 . As 2!1+ it has the limit
K(k) (E(k)=k02). If 2= 0, then it reduces to K(k). If
k2= 0, then it has the value1
2=p
1 2when 2<1,
and 0 when 2>1. Seex19.6(i).
Figure 19.3.6 : ( ;2;k) as a function of k2and sin2
for 1k23, 0sin2 < 1. Cauchy princi-
pal values are shown when sin2 >1
2. The function
tends to +1as sin2 !1
2, except in the last case be-
low. If sin2 = 1 (> k2), then the function reduces to
(2;k) with Cauchy principal value K(k) 1
2k2;k
,
which tends to 1 ask2!1 . See (19.6.5) and
(19.6.6). If sin2 = 1=k2(<1), then by (19.7.4) it
reduces to
2=k2;1=k
=k,k26= 2, with Cauchy prin-
cipal value ( K(1=k) 1
2;1=k
)=k, 1< k2<2, by
(19.6.5). Its value tends to 1ask2!1+ by (19.6.6),
and to the negative of the second lemniscate constant
(see (19.20.22)) as k2(= csc2 )!2 .19.3(ii) Complex Variables
In Figures 19.3.7 and 19.3.8 for complete Legendre's el-
liptic integrals with complex arguments, height corre-
sponds to the absolute value of the function and color
to the phase. See also p. xiv.
Figure 19.3.7 :K(k) as a function of complex k2for
2<(k2)2, 2=(k2)2. There is a branch
cut where 1 <k2<1.
Figure 19.3.8 :E(k) as a function of complex k2for
2<(k2)2, 2=(k2)2. There is a branch
cut where 1 <k2<1.
For further graphics, see http://dlmf.nist.gov/
19.3.ii .
490 Elliptic Integrals
19.4 Derivatives and Di erential Equations
19.4(i) Derivatives
19.4.1dK(k)
dk=E(k) k02K(k)
kk02;
d(E(k) k02K(k))
dk=kK(k);
19.4.2
dE(k)
dk=E(k) K(k)
k;d(E(k) K(k))
dk= kE(k)
k02;
19.4.3d2E(k)
dk2= 1
kdK(k)
dk=k02K(k) E(k)
k2k02;
19.4.4@
2;k
@k=k
k02(k2 2)(E(k) k02
2;k
):
19.4.5
@F( ;k)
@k=E( ;k) k02F( ;k)
kk02 ksin cos
k02p
1 k2sin2 ;
19.4.6@E( ;k)
@k=E( ;k) F( ;k)
k;
19.4.7
@
; 2;k
@k=k
k02(k2 2)
E( ;k) k02
; 2;k
k2sin cos p
1 k2sin2 !
:
19.4(ii) Di erential Equations
LetDk=@/@k. Then
19.4.8
(kk02D2
k+(1 3k2)Dk k)F( ;k) = ksin cos
(1 k2sin2 )3=2;
19.4.9 (kk02D2
k+k02Dk+k)E( ;k) =ksin cos p
1 k2sin2 :
If ==2, then these two equations become hyper-
geometric di erential equations (15.10.1) for K(k) and
E(k). An analogous di erential equation of third order
for
; 2;k
is given in Byrd and Friedman (1971,
118.03).
19.5 Maclaurin and Related Expansions
Ifjkj<1 andj j<1, then
19.5.1
K(k) =
21X
m=0 1
2
m 1
2
m
m!m!k2m=
22F1 1
2;1
2; 1;k2
;
where 2F1is the Gauss hypergeometric function ( xx15.1
and 15.2(i)).
19.5.2
E(k) =
21X
m=0
1
2
m 1
2
m
m!m!k2m=
22F1
1
2;1
2; 1;k2
;19.5.3
D(k) =
41X
m=0 3
2
m 1
2
m
(m+ 1)!m!k2m=
42F1 3
2;1
2; 2;k2
;
19.5.4
2;k
=
21X
n=0 1
2
n
n!nX
m=0 1
2
m
m!k2m 2n 2m
=
2F1 1
2;1
2;1; 1;k2; 2
;
whereF1( ; ; 0;
;x;y) is an Appell function ( x16.13).
ForJacobi's nome q:
19.5.5
q= exp( K0(k)=K(k)) =r+8r2+84r3+992r4+,
r=1
16k2, 0k1.
Also,
19.5.6
q=+25+159+15013+170717+, 0k1,
where
19.5.7 = (1 p
k0)=(2(1 +p
k0)):
Coecients of terms up to 49are given in Lee (1990),
along with tables of fractional errors in K(k) andE(k),
0:1k20:9999, obtained by using 12 di erent trun-
cations of (19.5.6) in (19.5.8) and (19.5.9).
19.5.8 K(k) =
2
1 + 21X
n=1qn2!2
,jqj<1,
19.5.9
E(k) =K(k) +22
K(k)P1
n=1( 1)nn2qn2
1 + 2P1
n=1( 1)nqn2,jqj<1.
An in nite series for ln K(k) is equivalent to the in-
nite product
19.5.10 K(k) =
21Y
m=1(1 +km);
wherek0=kand
19.5.11 km+1=1 p
1 k2m
1 +p
1 k2m,m= 0;1;:::.
Series expansions of F( ;k) andE( ;k) are sur-
veyed and improved in Van de Vel (1969), and the case
ofF( ;k) is summarized in Gautschi (1975, x1.3.2).
For series expansions of
; 2;k
whenj 2j<1 see
Erd elyi et al. (1953b,x13.6(9)). See also Karp et al.
(2007).
19.6 Special Cases 491
19.6 Special Cases
19.6(i) Complete Elliptic Integrals
19.6.1K(0) =E(0) =K0(1) =E0(1) =1
2;
K(1) =K0(0) =1; E (1) =E0(0) = 1:
19.6.2
k2;k
=E(k)=k02, k2<1,
( k;k) =1
4(1 +k) 1+1
2K(k), 0k2<1.
19.6.3
2;0
==(2p
1 2);(0;k) =K(k),
1< 2<1:
19.6.4
2;k
!+1, 2!1 ,
2;k
!1 sign
1 2
,k2!1 .
If 1< 2<1, then the Cauchy principal value
satis es
19.6.5
2;k
=K(k)
k2= 2;k
;
and
19.6.6
2;0
= 0;
2;k
!K(k)
E(k)=k02
, 2!1+,
2;k
! 1 , k2!1 .
Exact values of K(k) andE(k) for various special
values ofkare given in Byrd and Friedman (1971, 111.10
and 111.11) and Cooper et al. (2006).
19.6(ii)F( ;k)
19.6.7F(0;k) = 0; F ( ;0) = ; F 1
2;1
=1;
F 1
2;k
=K(k);lim
!0F( ;k)/ = 1:
19.6.8F( ;1) = (sin )RC
1;cos2
= gd 1( ):
For the inverse Gudermannian function gd 1( ) see
x4.23(viii). Compare also (19.10.2).
19.6(iii)E( ;k)
19.6.9E(0;k) = 0; E ( ;0) = ; E 1
2;1
= 1;
E( ;1) = sin ; E 1
2;k
=E(k):
19.6.10 lim
!0E( ;k)/ = 1:19.6(iv)
; 2;k
Circular and hyperbolic cases, including Cauchy prin-
cipal values, are uni ed by using RC(x;y). Letc=
csc2 6= 2and =p
1 k2sin2 . Then
19.6.11
0; 2;k
= 0;( ;0;0) = ;( ;1;0) = tan :
19.6.12
; 2;0
=RC
c 1;c 2
;
; 2;1
=1
1 2
RC(c;c 1) 2RC
c;c 2
;
( ;1;1) =1
2(RC(c;c 1) +pc(c 1) 1):
19.6.13
( ;0;k) =F( ;k);
;k2;k
=1
k02
E( ;k) k2
sin cos
;
( ;1;k) =F( ;k) 1
k02(E( ;k) tan ):
19.6.14
1
2; 2;k
=
2;k
;lim
!0
; 2;k
= 1:
For the Cauchy principal value of
; 2;k
when
2>c, seex19.7(iii).
19.6(v)RC(x;y)
19.6.15
RC(x;x) =x 1=2; RC(x;y ) = 1=2RC(x;y);
RC(x;y)!+1,y!0+ ory!0 ,x>0,
RC(0;y) =1
2y 1=2, jphyj<,
RC(0;y) = 0, y<0.
19.7 Connection Formulas
19.7(i) Complete Integrals of the First and
Second Kinds
Legendre's Relation
19.7.1E(k)K0(k) +E0(k)K(k) K(k)K0(k) =1
2:
Also,
19.7.2
K(ik=k0) =k0K(k); K (k0=ik) =kK(k0);
E(ik=k0) = (1=k0)E(k); E (k0=ik) = (1=k)E(k0):
19.7.3
K(1=k) =k(K(k)iK(k0));
K(1=k0) =k0(K(k0)iK(k));
E(1=k) = (1=k)
E(k)iE(k0) k02K(k)ik2K(k0)
;
E(1=k0) = (1=k0)
E(k0)iE(k) k2K(k0)
ik02K(k)
;
492 Elliptic Integrals
where upper signs apply if =k2>0 and lower signs if
=k2<0. This dichotomy of signs (missing in several
references) is due to Fettis (1970).
19.7(ii) Change of Modulus and Amplitude
Reciprocal-Modulus Transformation
19.7.4F( ;k1) =kF( ;k);
E( ;k1) = (E( ;k) k02F( ;k))=k;
; 2;k1
=k
;k2 2;k
,
k1= 1=k, sin =k1sin 1.
Imaginary-Modulus Transformation
19.7.5
F( ;ik) =0F(;);
E( ;ik) = (1=0)
E(;) 2(sincos)
(1 2sin2) 1/2
;
; 2;ik
= (0= 2
1)
2F(;) +02 2
; 2
1;
;
where
19.7.6=kp
1 +k2; 0=1p
1 +k2;
sin=p
1 +k2sin p
1 +k2sin2 ; 2
1= 2+k2
1 +k2:
Imaginary-Argument Transformation
With sinh = tan ,
19.7.7F(i ;k) =iF( ;k0);
E(i ;k) =i
F( ;k0) E( ;k0)
+ (tan )q
1 k02sin2
;
i ; 2;k
=i
F( ;k0)
2
;1 2;k0
=(1 2):
For two further transformations of this type see
Erd elyi et al. (1953b, p. 316).
19.7(iii) Change of Parameter of
; 2;k
There are three relations connecting
; 2;k
and
;!2;k
, where!2is a rational function of 2. Ifk2
and 2are real, then both integrals are circular cases
or both are hyperbolic cases (see x19.2(ii)).
The rst of the three relations maps each circular
region onto itself and each hyperbolic region onto the
other; in particular, it gives the Cauchy principal valueof
; 2;k
when 2>csc2 (see (19.6.5) for the
complete case). Let c= csc2 6= 2. Then
19.7.8
; 2;k
+
;!2;k
=F( ;k) +pcRC
(c 1)(c k2);(c 2)(c !2)
,
2!2=k2.
Sincek2cwe have 2!2c; hence 2> cimplies
!2<1c.
The second relation maps each hyperbolic region
onto itself and each circular region onto the other:
19.7.9
(k2 2)
; 2;k
+ (k2 !2)
;!2;k
=k2F( ;k)
2!2p
c 1RC
c(c k2);(c 2)(c !2)
,
(1 2)(1 !2) = 1 k2.
The third relation (missing from the literature of
Legendre's integrals) maps each circular region onto the
other and each hyperbolic region onto the other:
19.7.10(1 2)
; 2;k
+ (1 !2)
;!2;k
=F( ;k) + (1 2 !2)p
c k2
RC
c(c 1);(c 2)(c !2)
,
(k2 2)(k2 !2) =k2(k2 1).
19.8 Quadratic Transformations
19.8(i) Gauss's Arithmetic-Geometric Mean
(AGM)
Whena0andg0are positive numbers, de ne
19.8.1
an+1=an+gn
2; gn+1=pangn,n= 0;1;2;:::.
Asn! 1 ,anandgnconverge to a common
limitM(a0;g0) called the AGM ( Arithmetic-Geometric
Mean ) ofa0andg0. By symmetry in a0andg0we may
assumea0g0and de ne
19.8.2 cn=p
a2n g2n:
Then
19.8.3 cn+1=an gn
2=c2
n
4an+1;
showing that the convergence of cnto 0 and of anand
gntoM(a0;g0) is quadratic in each case.
The AGM has the integral representations
19.8.41
M(a0;g0)=2
Z=2
0dq
a2
0cos2+g2
0sin2
=1
Z1
0dtp
t(t+a2
0)(t+g2
0):
The rst of these shows that
19.8.5 K(k) =
2M(1;k0), 1<k2<1.
19.8 Quadratic Transformations 493
The AGM appears in
19.8.6E(k) =
2M(1;k0)
a2
0 1X
n=02n 1c2
n!
=K(k)
a2
1 1X
n=22n 1c2
n!
,
1<k2<1,a0= 1,g0=k0,
and in
19.8.7
2;k
=
4M(1;k0)
2 + 2
1 21X
n=0Qn!
,
1<k2<1, 1< 2<1,
wherea0= 1,g0=k0,p2
0= 1 2,Q0= 1, and
19.8.8pn+1=p2
n+angn
2pn; "n=p2
n angn
p2n+angn;
Qn+1=1
2Qn"n, n= 0;1;:::.
Again,pnand"nconverge quadratically to M(a0;g0)
and 0, respectively, and Qnconverges to 0 faster than
quadratically. If 2>1, then the Cauchy principal
value is
19.8.9
2;k
=
4M(1;k0)k2
k2 21X
n=0Qn,
1<k2<1, 1< 2<1,
where (19.8.8) still applies, but with
19.8.10 p2
0= 1 (k2= 2):
19.8(ii) Landen Transformations
Descending Landen Transformation
Let
19.8.11k1=1 k0
1 +k0;
1= + arctan(k0tan )
= arcsin
(1 +k0)sin cos p
1 k2sin2 !
:
(Note that 0 <k< 1 and 0< <= 2 implyk1<kand
< 1<2 , and also that ==2 implies 1=.)
Then
19.8.12K(k) = (1 +k1)K(k1);
E(k) = (1 +k0)E(k1) k0K(k):
19.8.13F( ;k) =1
2(1 +k1)F( 1;k1);
E( ;k) =1
2(1 +k0)E( 1;k1) k0F( ;k)
+1
2(1 k0) sin 1:19.8.14
2(k2 2)
; 2;k
=!2 2
1 +k0
1; 2
1;k1
+k2F( ;k)
(1 +k0) 2
1RC
c1;c1 2
1
;
where
19.8.15!2=k2 2
1 2; 2
1= 2!2
(1 +k0)2; c 1= csc2 1:
Ascending Landen Transformation
Let
19.8.16k2= 2p
k=(1 +k);
2 2= + arcsin(ksin ):
(Note that 0 <k< 1 and 0< =2 implyk<k 2<1
and 2< .) Then
19.8.17
F( ;k) =2
1 +kF( 2;k2);
E( ;k) = (1 +k)E( 2;k2) + (1 k)F( 2;k2) ksin :
19.8(iii) Gauss Transformation
We consider only the descending Gauss transformation
because its (ascending) inverse moves F( ;k) closer to
the singularity at k= sin = 1. Let
19.8.18k1= (1 k0)=(1 +k0);
sin 1=(1 +k0) sin
1 + ; =q
1 k2sin2 :
(Note that 0 < k < 1 and 0< < = 2 implyk1< k
and 1< , and also that ==2 implies 1==2,
thus preserving completeness.) Then
19.8.19
F( ;k) = (1 +k1)F( 1;k1);
E( ;k) = (1 +k0)E( 1;k1) k0F( ;k) + (1 ) cot ;
19.8.20
; 2;k
=4
1 +k0
1; 2
1;k1
+ ( 1)F( ;k) RC
c 1;c 2
;
where
19.8.21=p
1 (k2= 2);
2
1= 2(1 +)2=(1 +k0)2; c = csc2 :
If 0< 2<k2, thenis pure imaginary.
494 Elliptic Integrals
19.9 Inequalities
19.9(i) Complete Integrals
Throughout this subsection 0 < k < 1, except in
(19.9.4).
19.9.1ln 4K(k) + lnk0=2;1E(k)=2:
1(2=)p
1 2
2;k
1=k0, 2<1.
19.9.2 1 +k02
8<K(k)
ln(4=k0)<1 +k02
4;
19.9.3 9 +k2k02
8<(8 +k2)K(k)
ln(4=k0)<9:096:
The left-hand inequalities in (19.9.2) and (19.9.3) are
equivalent, but the right-hand inequality of (19.9.3) is
sharper than that of (19.9.2) when 0 <k20:922.
19.9.4
1 +k03=2
2!2=3
2
E(k)
1 +k02
2!1=2
for 0k1. The lower bound in (19.9.4) is sharper
than 2=when 0k20:9960.
19.9.5 ln(1 +p
k0)2
k<K0(k)
2K(k)<ln2(1 +k0)
k:
For a sharper, but more complicated, version of (19.9.5)
see Anderson et al. (1990).
Other inequalities are:
19.9.6 (1 3
4k2) 1=2<4
k2(K(k) E(k))<(k0) 3=4;
19.9.7(1 1
4k2) 1=2<4
k2(E(k) k02K(k))
<min((k0) 1=4;4=);
19.9.8 k0<E(k)
K(k)<1 +k0
22
:
Further inequalities for K(k) andE(k) can be found in
Alzer and Qiu (2004), Anderson et al. (1992a,b, 1997),
and Qiu and Vamanamurthy (1996).
The perimeter L(a;b) of an ellipse with semiaxes a;b
is given by
19.9.9 L(a;b) = 4aE(k),k2= 1 (b2=a2),a>b .
Almkvist and Berndt (1988) list thirteen approxima-
tions toL(a;b) that have been proposed by various au-
thors. The earliest is due to Kepler and the most ac-
curate to Ramanujan. Ramanujan's approximation and
its leading error term yield the following approximation
toL(a;b)=((a+b)):
19.9.10 1 +32
10 +p
4 32+310
217,=a b
a+b.
Even for the extremely eccentric ellipse with a= 99
andb= 1, this is correct within 0.023%. Barnard et al.
(2000) shows that nine of the thirteen approximations,
including Ramanujan's, are from below and four are
from above. See also Barnard et al. (2001).19.9(ii) Incomplete Integrals
Throughout this subsection we assume that 0 <k < 1,
0 =2, and =p
1 k2sin2 >0.
Simple inequalities for incomplete integrals follow
directly from the de ning integrals ( x19.2(ii)) together
with (19.6.12):
19.9.11 F( ;k)min( =;gd 1( ));
where gd 1( ) is given by (4.23.41) and (4.23.42). Also,
19.9.12 max(sin ; )E( ;k) ;
19.9.13
; 2;0
; 2;k
min(
; 2;0
=;
; 2;1
):
Sharper inequalities for F( ;k) are:
19.9.143
1 + + cos <F( ;k)
sin <1
( cos )1=3;
19.9.151<F( ;k)
(sin ) ln4
+ cos
<4
2 + (1 +k2) sin2 :
19.9.16F( ;k) =2
K(k0) ln4
+ cos
2,
(sin )=8<< (ln 2)=(k2sin ).
(19.9.15) is useful when k2and sin2 are both close
to 1, since the bounds are then nearly equal; otherwise
(19.9.14) is preferable.
Inequalities for both F( ;k) andE( ;k) involving
inverse circular or inverse hyperbolic functions are given
in Carlson (1961b, x4). For example,
19.9.17LF( ;k)p
UL1
2(U+L)U;
where
19.9.18L= (1=) arctanh(sin ),=p
(1 +k2)=2,
U=1
2arctanh(sin ) +1
2k 1arctanh(ksin ):
Other inequalities for F( ;k) can be obtained from
inequalities for RF(x;y;z ) given in Carlson (1966,
(2.15)) and Carlson (1970) via (19.25.5).
19.10 Relations to Other Functions
19.10(i) Theta and Elliptic Functions
For relations of Legendre's integrals to theta func-
tions, Jacobian functions, and Weierstrass functions, see
xx20.9(i), 22.15(ii), and 23.6(iv), respectively. See also
Erd elyi et al. (1953b, Chapter 13).
19.11 Addition Theorems 495
19.10(ii) Elementary Functions
Ify>0 is assumed (without loss of generality), then
19.10.1ln(x=y) = (x y)RC 1
4(x+y)2;xy
;
arctan(x=y) =xRC
y2;y2+x2
;
arctanh(x=y) =xRC
y2;y2 x2
;
arcsin(x=y) =xRC
y2 x2;y2
;
arcsinh(x=y) =xRC
y2+x2;y2
;
arccos(x=y) = (y2 x2)1=2RC
x2;y2
;
arccosh(x=y) = (x2 y2)1=2RC
x2;y2
:
In each case when y= 1, the quantity multiplying RC
supplies the asymptotic behavior of the left-hand side
as the left-hand side tends to 0.
For relations to the Gudermannian function gd( x)
and its inverse gd 1(x) (x4.23(viii)), see (19.6.8) and
19.10.2 (sinh )RC
1;cosh2
= gd( ):
19.11 Addition Theorems
19.11(i) General Formulas
19.11.1 F(;k) +F( ;k) =F( ;k);
19.11.2E(;k) +E( ;k) =E( ;k) +k2sinsin sin :
Here
19.11.3sin =(sincos )( ) + (sin cos)()
1 k2sin2sin2 ;
() =p
1 k2sin2:
Also,
19.11.4cos =coscos (sinsin )()( )
1 k2sin2sin2 ;
tan 1
2
=(sin)( ) + (sin )()
cos+ cos :
Lastly,
19.11.5
; 2;k
+
; 2;k
=
; 2;k
2RC(
;
);
where
19.11.6
= ((csc2) 2)((csc2 ) 2)((csc2 ) 2);
= 2(1 2)( 2 k2):19.11(ii) Case ==2
19.11.7 F( ;k) =K(k) F(;k);
19.11.8E( ;k) =E(k) E(;k) +k2sinsin ;
where
19.11.9 tan= 1=(k0tan ):
19.11.10
; 2;k
=
2;k
; 2;k
2RC(
;
);
where
19.11.11
= (1 2)((csc2) 2)((csc2 ) 2);
= 2(1 2)( 2 k2):
19.11(iii) Duplication Formulas
If =inx19.11(i) and ( ) is again de ned by
(19.11.3), then
19.11.12 F( ;k) = 2F(;k);
19.11.13E( ;k) = 2E(;k) k2sin2sin ;
19.11.14 sin = (sin 2)()=(1 k2sin4);
19.11.15cos = (cos (2) +k2sin4)=(1 k2sin4);
tan 1
2
= (tan)();
sin= (sin )=p
(1 + cos )(1 + ( ));
cos=s
(cos ) + ( )
1 + ( );
tan= tan 1
2 s
1 + cos
(cos ) + ( );
19.11.16
; 2;k
= 2
; 2;k
+ 2RC(
;
);
19.11.17
= ((csc2) 2)2((csc2 ) 2);
= 2(1 2)( 2 k2):
19.12 Asymptotic Approximations
With (x) denoting the digamma function ( x5.2(i)) in
this subsection, the asymptotic behavior of K(k) and
E(k) near the singularity at k= 1 is given by the fol-
lowing convergent series:
19.12.1
K(k) =1X
m=0 1
2
m 1
2
m
m!m!k02m
ln1
k0
+d(m)
,
0<jk0j<1,
496 Elliptic Integrals
19.12.2
E(k) = 1 +1
21X
m=0 1
2
m 3
2
m
(2)mm!k02m+2
ln1
k0
+d(m) 1
(2m+ 1)(2m+ 2)
,
jk0j<1,
where
19.12.3
d(m) = (1 +m) 1
2+m
;
d(m+ 1) =d(m) 2
(2m+ 1)(2m+ 2),m= 0;1;:::.
For the asymptotic behavior of F( ;k) andE( ;k)
as !1
2 andk!1 see Kaplan (1948, x2), Van de
Vel (1969), and Karp and Sitnik (2007).
Ask2!1
19.12.4
(1 2)
2;k
=
ln4
k0
1 +O
k02
2RC
1;1 2
,
1< 2<1,
19.12.5
(1 2)
2;k
=
ln4
k0
RC
1;1 2
1 +O
k02
,
1< 2<1.
Asymptotic approximations for
; 2;k
, with
di erent variables, are given in Karp et al. (2007). They
are useful primarily when (1 k)/(1 sin ) is either
small or large compared with 1.
Ifx0 andy>0, then
19.12.6
RC(x;y) =
2py px
y
1 +Orx
y
,x=y!0;
19.12.7RC(x;y) =1
2px
1 +y
2x
ln4x
y
y
2x
(1 +O
y2=x2
),y=x!0.
19.13 Integrals of Elliptic Integrals
19.13(i) Integration with Respect to the
Modulus
For de nite and inde nite integrals of complete ellip-
tic integrals see Byrd and Friedman (1971, pp. 610{
612, 615), Prudnikov et al. (1990,xx1.11, 2.16), Glasser
(1976), Bushell (1987), and Cvijovi c and Klinowski
(1999).
For de nite and inde nite integrals of incomplete el-
liptic integrals see Byrd and Friedman (1971, pp. 613,
616), Prudnikov et al. (1990,xx1.10.2, 2.15.2), and Cvi-
jovi c and Klinowski (1994).19.13(ii) Integration with Respect to the
Amplitude
Various integrals are listed by Byrd and Friedman (1971,
p. 630) and Prudnikov et al. (1990,xx1.10.1, 2.15.1).
Cvijovi c and Klinowski (1994) contains fractional in-
tegrals (with free parameters) for F( ;k) andE( ;k),
together with special cases.
19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the com-
plete elliptic integrals K(k),E(k), andD(k) see Prud-
nikov et al. (1992a,x3.31) and Prudnikov et al. (1992b,
xx3.29 and 4.3.33), respectively.
19.14 Reduction of General Elliptic
Integrals
19.14(i) Examples
In (19.14.1){(19.14.3) both the integrand and cos are
assumed to be nonnegative. Cases in which cos < 0
can be included by application of (19.2.10).
19.14.1Zx
1dtp
t3 1= 3 1=4F( ;k),
cos =p
3 + 1 xp
3 1 +x,k2=2 p
3
4:
19.14.2Z1
xdtp
1 t3= 3 1=4F( ;k),
cos =p
3 1 +xp
3 + 1 x,k2=2 +p
3
4:
19.14.3Zx
0dtp
1 +t4=sign(x)
2F( ;k),
cos =1 x2
1 +x2,k2=1
2:
19.14.4Zx
ydtp
(a1+b1t2)(a2+b2t2)=1p
F( ;k),
k2= (
)/(
).
In (19.14.4) 0y < x , each quadratic polynomial is
positive on the interval ( y;x), and ; ;
is a permuta-
tion of 0;a1b2;a2b1(not all 0 by assumption) such that
. More generally in (19.14.4),
19.14.5 sin2 =
U2+
;
where
19.14.6(x2 y2)U=xp
(a1+b1y2)(a2+b2y2)
+yp
(a1+b1x2)(a2+b2x2):
Symmetric Integrals 497
There are four important special cases of (19.14.4){
(19.14.6), as follows. If y= 0, then
19.14.7 sin2 =(
)x2
a1a2+
x2:
Ifx=1, then
19.14.8 sin2 =
b1b2y2+
:
Ifa1+b1y2= 0, then
19.14.9 sin2 =(
)(x2 y2)
(x2 y2) a1(a2+b2x2):
Ifa1+b1x2= 0, then
19.14.10 sin2 =(
)(y2 x2)
(y2 x2) a1(a2+b2y2):
(These four cases include 12 integrals in Abramowitz
and Stegun (1964, p. 596).)
19.14(ii) General Case
Legendre (1825{1832) showed that every elliptic in-
tegral can be expressed in terms of the three in-
tegrals in (19.1.2) supplemented by algebraic, loga-
rithmic, and trigonometric functions. The classical
method of reducing (19.2.3) to Legendre's integrals
is described in many places, especially Erd elyi et al.
(1953b,x13.5), Abramowitz and Stegun (1964, Chapter
17), and Labahn and Mutrie (1997, x3). The last refer-
ence gives a clear summary of the various steps involving
linear fractional transformations, partial-fraction de-
composition, and recurrence relations. It then improves
the classical method by rst applying Hermite reduction
to (19.2.3) to arrive at integrands without multiple poles
and uses implicit full partial-fraction decomposition and
implicit root nding to minimize computing with alge-
braic extensions. The choice among 21 transformations
for nal reduction to Legendre's normal form depends
on inequalities involving the limits of integration and
the zeros of the cubic or quartic polynomial. A similar
remark applies to the transformations given in Erd elyi
et al. (1953b,x13.5) and to the choice among explicit
reductions in the extensive table of Byrd and Friedman
(1971), in which one limit of integration is assumed to
be a branch point of the integrand at which the integral
converges. If no such branch point is accessible from the
interval of integration (for example, if the integrand is
(t(3 t)(4 t)) 3=2and the interval is [1,2]), then no
method using this assumption succeeds.Symmetric Integrals
19.15 Advantages of Symmetry
Elliptic integrals are special cases of a particu-
lar multivariate hypergeometric function called Lau-
ricella'sFD(Carlson (1961b)). The function
R a(b1;b2;:::;bn;z1;z2;:::;zn) (Carlson (1963)) re-
veals the full permutation symmetry that is partially
hidden inFD, and leads to symmetric standard inte-
grals that simplify many aspects of theory, applications,
and numerical computation.
Symmetry in x;y;z ofRF(x;y;z ),RG(x;y;z ),
andRJ(x;y;z;p ) replaces the ve transformations
(19.7.2), (19.7.4){(19.7.7) of Legendre's integrals; com-
pare (19.25.17). Symmetry uni es the Landen trans-
formations ofx19.8(ii) with the Gauss transforma-
tions ofx19.8(iii), as indicated following (19.22.22) and
(19.36.9). (19.21.12) uni es the three transformations
inx19.7(iii) that change the parameter of Legendre's
third integral.
Symmetry allows the expansion (19.19.7) in a series
of elementary symmetric functions that gives high pre-
cision with relatively few terms and provides the most
ecient method of computing the incomplete integral
of the third kind ( x19.36(i)).
Symmetry makes possible the reduction theorems of
x19.29(i), permitting remarkable compression of tables
of integrals while generalizing the interval of integration.
(Compare (19.14.4){(19.14.10) with (19.29.19), and see
the last paragraph of x19.29(i) and the text following
(19.29.15).) These reduction theorems, unknown in the
Legendre theory, allow symbolic integration without im-
posing conditions on the parameters and the limits of
integration (seex19.29(ii)).
For the many properties of ellipses and triaxial el-
lipsoids that can be represented by elliptic integrals,
any symmetry in the semiaxes remains obvious when
symmetric integrals are used (see (19.30.5) and x19.33).
For example, the computation of depolarization factors
for solid ellipsoids is simpli ed considerably; compare
(19.33.7) with Cronemeyer (1991).
19.16 De nitions
19.16(i) Symmetric Integrals
19.16.1 RF(x;y;z ) =1
2Z1
0dt
s(t);
19.16.2RJ(x;y;z;p ) =3
2Z1
0dt
s(t)(t+p);
498 Elliptic Integrals
19.16.3
RG(x;y;z ) =1
4Z2
0Z
0
xsin2cos2 +ysin2sin2
+zcos21
2sindd ;
wherep(6= 0) is a real or complex constant, and
19.16.4 s(t) =p
t+xpt+yp
t+z:
In (19.16.1) and (19.16.2), x;y;z2Cn( 1;0] except
that one or more of x;y;z may be 0 when the corre-
sponding integral converges. In (19.16.2) the Cauchy
principal value is taken when pis real and negative.
In (19.16.3)<x,<y,<z0. It should be noted that
the integrals (19.16.1){(19.16.3) have been normalized
so thatRF(1;1;1) =RJ(1;1;1;1) =RG(1;1;1) = 1.
A fourth integral that is symmetric in only two vari-
ables is de ned by
19.16.5RD(x;y;z ) =RJ(x;y;z;z ) =3
2Z1
0dt
s(t)(t+z);
with the same conditions on x,y,zas for (19.16.1), but
nowz6= 0.
Just as the elementary function RC(x;y) (x19.2(iv))
is the degenerate case
19.16.6 RC(x;y) =RF(x;y;y );
andRDis a degenerate case of RJ, so isRJa degenerate
case of the hyperelliptic integral ,
19.16.73
2Z1
0dtQ5
j=1pt+xj:
19.16(ii)R a(b; z)
All elliptic integrals of the form (19.2.3) and many mul-
tiple integrals, including (19.16.3), are special cases of
a multivariate hypergeometric function
19.16.8R a(b;z) =R a(b1;:::;bn;z1;:::;zn);
which is homogeneous and of degree ain thez's, and
symmetric when the same permutation is applied to
both sets of subscripts 1 ;:::;n . ThusR a(b;z) is sym-
metric in the variables zjandz`if the parameters bj
andb`are equal. The R-function is often used to make
a uni ed statement of a property of several elliptic inte-
grals. Before 1969 R a(b;z) was denoted by R(a;b;z).
19.16.9
R a(b;z) =1
B(a;a0)Z1
0ta0 1nY
j=1(t+zj) bjdt
=1
B(a;a0)Z1
0ta 1nY
j=1(1 +tzj) bjdt,
a;a0>0,zj2Cn( 1;0],where B(x;y) is the beta function ( x5.12) and
19.16.10 a0= a+nX
j=1bj:
19.16.11R a(b;z) = aR a(b;z);
R a(b;x1) =x a, 1= (1;:::; 1).
Whenn= 4 a useful version of (19.16.9) is given by
19.16.12
R a
b1;:::;b 4;c 1;c k2;c;c 2
=2(sin2 )1 a0
B(a;a0)Z
0(sin)2a 1(sin2 sin2)a0 1
(cos)1 2b1(1 k2sin2) b2(1 2sin2) b4d;
where
19.16.13
c= csc2 ;a;a0>0;b3=a+a0 b1 b2 b4:
For further information, especially representation
of theR-function as a Dirichlet average, see Carlson
(1977b).
19.16(iii) Elliptic Cases of R a(b; z)
R a(b;z) is an elliptic integral if the z's are distinct
and exactly four of the parameters a;a0;b1;:::;bnare
half-odd-integers, the rest are integers, and none of a,
a0,a+a0is zero or a negative integer. The only cases
that are integrals of the rst kind are the four in which
each ofaanda0is either1
2or 1 and each bjis1
2. The
only cases that are integrals of the third kind are those
in which at least one bjis a positive integer. All other
elliptic cases are integrals of the second kind.
19.16.14RF(x;y;z ) =R 1
2 1
2;1
2;1
2;x;y;z
;
19.16.15RD(x;y;z ) =R 3
2 1
2;1
2;3
2;x;y;z
;
19.16.16RJ(x;y;z;p ) =R 3
2 1
2;1
2;1
2;1;x;y;z;p
;
19.16.17RG(x;y;z ) =R1
2 1
2;1
2;1
2;x;y;z
;
19.16.18 RC(x;y) =R 1
2 1
2;1;x;y
:
When one variable is 0 without destroying convergence,
any one of (19.16.14){(19.16.17) is said to be complete
and can be written as an R-function with one less vari-
able:
19.16.19R a(b1;:::;bn; 0;z2;:::;zn)
=B(a;a0 b1)
B(a;a0)R a(b2;:::;bn;z2;:::;zn),
a+a0>0,a0>b1.
19.17 Graphics 499
Thus
19.16.20RF(0;y;z) =1
2R 1
2 1
2;1
2;y;z
;
19.16.21RD(0;y;z) =3
4R 3
2 1
2;3
2;y;z
;
19.16.22RJ(0;y;z;p ) =3
4R 3
2 1
2;1
2;1;y;z;p
;
19.16.23RG(0;y;z) =1
4R 1
2 1
2;1
2;y;z
=1
4zR 1
2
1
2;3
2;y;z
:
The lastR-function has a=a0=1
2.
Each of the four complete integrals (19.16.20){
(19.16.23) can be integrated to recover the incomplete
integral:
19.16.24
R a(b;z) =za0 b1
1
B(b1;a0 b1)Z1
0tb1 1(t+z1) a0
R a(b; 0;t+z2;:::;t +zn)dt,
a0>b1,a+a0>b1>0.19.17 Graphics
See Figures 19.17.1{19.17.8 for symmetric elliptic inte-
grals with real arguments.
Because the R-function is homogeneous, there is no
loss of generality in giving one variable the value 1 or
1 (as in Figure 19.3.2). For RF,RG, andRJ, which
are symmetric in x;y;z , we may further assume that zis
the largest of x;y;z if the variables are real, then choose
z= 1, and consider only 0 x1 and 0y1.
The casesx= 0 ory= 0 correspond to the complete
integrals. The case y= 1 corresponds to elementary
functions.
To viewRF(0;y;1) and 2RG(0;y;1) for complex y,
puty= 1 k2, use (19.25.1), and see Figures 19.3.7{
19.3.8.
Figure 19.17.1 :RF(x;y;1) for 0x1,y=
0;0:1;0:5;1.y= 1 corresponds to RC(x;1).
Figure 19.17.2 :RG(x;y;1) for 0x1,y=
0;0:1;0:5;1.y= 1 corresponds to1
2(RC(x;1) +px).
Figure 19.17.3 :RD(x;y;1) for 0x2,y=
0;0:1;1;5;25.y= 1 corresponds to3
2(RC(x;1) px)=(1 x),x6= 1.
Figure 19.17.4 :RJ(x;y;1;2) for 0x1,y=
0;0:1;0:5;1.y= 1 corresponds to 3( RC(x;1)
RC(x;2)).
500 Elliptic Integrals
Figure 19.17.5 :RJ(x;y;1;0:5) for 0x1,y=
0;0:1;0:5;1.y= 1 corresponds to 6( RC(x;0:5)
RC(x;1)).
x
/Minus4
/Minus8
/Minus120 .1.510 1 .5Figure 19.17.6 : Cauchy principal value of
RJ(x;y;1; 0:5) for 0x1,y= 0;0:1;0:5;1.
y= 1 corresponds to 2( RC(x; 0:5) RC(x;1)).
Figure 19.17.7 : Cauchy principal value of RJ(0:5;y;1;p)
fory= 0;0:01;0:05;0:2;1, 1p <0.y= 1 corre-
sponds to 3( RC(0:5;p) (=p
8))=(1 p). Asp!0
the curve for y= 0 has the nite limit 8:10386:::; see
(19.20.10).
Figure 19.17.8 :RJ(0;y;1;p), 0y1, 1p2.
Cauchy principal values are shown when p < 0. The
function is asymptotic to3
2=pypasp!0+, and
to (3
2=p) ln(16=y) asy!0+. Asp!0 it has the
limit ( 6=y)RG(0;y;1). When p= 1, it reduces to
RD(0;y;1). Ify= 1, then it has the value3
2=(p+pp)
whenp>0, and3
2=(p 1) whenp<0. See (19.20.10),
(19.20.11), and (19.20.8) for the cases p!0,y!0+,
andy= 1, respectively.
19.18 Derivatives and Di erential
Equations
19.18(i) Derivatives
19.18.1@RF(x;y;z )
@z= 1
6RD(x;y;z );
19.18.2
d
dxRG(x+a;x+b;x+c) =1
2RF(x+a;x+b;x+c):Let@j=@/@zj, and ejbe ann-tuple with 1 in the jth
place and 0's elsewhere. Also de ne
19.18.3wj=bj nX
j=1bj; a0= a+nX
j=1bj:
The next two equations apply to (19.16.14){(19.16.18)
and (19.16.20){(19.16.23).
19.18.4@jR a(b;z) = awjR a 1(b+ej;z);
19.19 Taylor and Related Series 501
19.18.5 (zj@j+bj)R a(b;z) =wja0R a(b+ej;z):
19.18(ii) Di erential Equations
19.18.6@
@x+@
@y+@
@z
RF(x;y;z ) = 1
2pxyz;
19.18.7@
@x+@
@y+@
@z
RG(x;y;z ) =1
2RF(x;y;z ):
19.18.8nX
j=1@jR a(b;z) = aR a 1(b;z):
19.18.9
x@
@x+y@
@y+z@
@z
RF(x;y;z ) = 1
2RF(x;y;z );
19.18.10
(x y)@2
@x@y+1
2@
@y @
@x
RF(x;y;z ) = 0;
and two similar equations obtained by permuting x;y;z
in (19.18.10).
More concisely, if v=R a(b;z), then each of
(19.16.14){(19.16.18) and (19.16.20){(19.16.23) satis es
Euler's homogeneity relation :
19.18.11nX
j=1zj@jv= av;
and also a system of n(n 1)=2Euler{Poisson di eren-
tial equations (of which only n 1 are independent):
19.18.12 (zj@j+bj)@lv= (zl@l+bl)@jv;
or equivalently,
19.18.13 ((zj zl)@j@l+bj@l bl@j)v= 0:
Herej;l= 1;2;:::;n andj6=l. For group-theoretical
aspects of this system see Carlson (1963, xVI). If
n= 2, then elimination of @2vbetween (19.18.11) and
(19.18.12), followed by the substitution ( b1;b2;z1;z2) =
(b;c b;1 z;1), produces the Gauss hypergeometric
equation (15.10.1).The next four di erential equations apply to
the complete case of RFandRGin the form
R a 1
2;1
2;z1;z2
(see (19.16.20) and (19.16.23)).
The function w=R a 1
2;1
2;x+y;x y
satis es
anEuler{Poisson{Darboux equation :
19.18.14@2w
@x2=@2w
@y2+1
y@w
@y:
AlsoW=R a 1
2;1
2;t+r;t r
, withr=p
x2+y2,
satis es a wave equation :
19.18.15@2W
@t2=@2W
@x2+@2W
@y2:
Similarly, the function u=R a 1
2;1
2;x+iy;x iy
satis es an equation of axially symmetric potential the-
ory:
19.18.16@2u
@x2+@2u
@y2+1
y@u
@y= 0;
andU=R a 1
2;1
2;z+i;z i
, with=p
x2+y2,
satis es Laplace's equation :
19.18.17@2U
@x2+@2U
@y2+@2U
@z2= 0:
19.19 Taylor and Related Series
ForN= 0;1;2;::: de ne the homogeneous hypergeo-
metric polynomial
19.19.1TN(b;z) =X(b1)m1(bn)mn
m1!mn!zm1
1zmn
n;
where the summation extends over all nonnegative in-
tegersm1;:::;mnwhose sum is N. The following
two multivariate hypergeometric series apply to each
of the integrals (19.16.14){(19.16.18) and (19.16.20){
(19.16.23):
19.19.2R a(b;z) =1X
N=0(a)N
(c)NTN(b;1 z),
c=Pn
j=1bj,j1 zjj<1,
19.19.3R a(b;z) =z a
n1X
N=0(a)N
(c)NTN(b1;:::;bn 1; 1 (z1=zn);:::; 1 (zn 1=zn)),c=Pn
j=1bj,j1 (zj=zn)j<1.
Ifn= 2, then (19.19.3) is a Gauss hypergeometric series
(see (19.25.43) and (15.2.1)).
De ne the elementary symmetric function Es(z) by
19.19.4nY
j=1(1 +tzj) =nX
s=0tsEs(z);and de ne the n-tuple1
2= (1
2;:::;1
2). Then
19.19.5
TN(1
2;z) =X
( 1)M+N 1
2
MEm1
1(z)Emnn(z)
m1!mn!;
whereM=Pn
j=1mjand the summation extends
over all nonnegative integers m1;:::;mnsuch thatPn
j=1jmj=N.
502 Elliptic Integrals
This form of TNcan be applied to (19.16.14){
(19.16.18) and (19.16.20){(19.16.23) if we use
19.19.6RJ(x;y;z;p ) =R 3
2 1
2;1
2;1
2;1
2;1
2;x;y;z;p;p
as well as (19.16.5) and (19.16.6). The number of
terms inTNcan be greatly reduced by using variables
Z=1 (z=A) withAchosen to make E1(Z) = 0. Then
TNhas at most one term if N5 in the series for RF.
ForRJandRD,TNhas at most one term if N3, and
two terms if N= 4 or 5.
19.19.7R a 1
2;z
=A a1X
N=0(a)N 1
2n
NTN(1
2;Z);
where
19.19.8A=1
nnX
j=1zj; Zj= 1 (zj=A);
E1(Z) = 0, jZjj<1.
A special case is given in (19.36.1).
19.20 Special Cases
19.20(i)RF(x;y;z )
In this subsection, and also xx19.20(ii){19.20(v), the
variables of all R-functions satisfy the constraints spec-
i ed inx19.16(i) unless other conditions are stated.
19.20.1RF(x;x;x ) =x 1=2;
RF(x;y;z ) = 1=2RF(x;y;z );
RF(x;y;y ) =RC(x;y);
RF(0;y;y) =1
2y 1=2;
RF(0;0;z) =1:
The rst lemniscate constant is given by
19.20.2Z1
0dtp
1 t4=RF(0;1;2) =
1
42
4(2)1=2
= 1:31102 87771 46059 90523 ::::
Todd (1975) refers to a proof by T. Schneider that this is
a transcendental number. The general lemniscatic case
is
19.20.3RF(x;a;y ) =R 1
4 3
4;1
2;a2;xy
,a=1
2(x+y).
19.20(ii)RG(x;y;z )
19.20.4RG(x;x;x ) =x1=2;
RG(x;y;z ) =1=2RG(x;y;z );
RG(0;y;y) =1
4y1=2;
RG(0;0;z) =1
2z1=2;
19.20.5 2RG(x;y;y ) =yRC(x;y) +px:19.20(iii)RJ(x;y;z;p )
19.20.6
RJ(x;x;x;x ) =x 3=2;
RJ(x;y;z;p ) = 3=2RJ(x;y;z;p );
RJ(x;y;z;z ) =RD(x;y;z );
RJ(0;0;z;p) =1;
RJ(x;x;x;p ) =RD(p;p;x ) =3
x p
RC(x;p) 1px
,
x6=p,xp6= 0.
19.20.7
RJ(x;y;z;p )!+1,p!0+ or 0 ;x;y;z> 0.
19.20.8
RJ(0;y;y;p ) =3
2(ypp+ppy), p>0,
RJ(0;y;y; q) = 3
2py(y+q), q>0,
RJ(x;y;y;p ) =3
p y(RC(x;y) RC(x;p)),p6=y,
RJ(x;y;y;y ) =RD(x;y;y ):
19.20.9RJ(0;y;z;pyz) =3
2pyzRF(0;y;z):
19.20.10
lim
p!0+ppRJ(0;y;z;p ) =3
2pyz;
lim
p!0 RJ(0;y;z;p ) = RD(0;y;z) RD(0;z;y)
= 6
yzRG(0;y;z):
19.20.11
RJ(0;y;z;p )3
2ppzln16z
y
,y!0+;p(6= 0) real.
19.20.12 lim
p!1pRJ(x;y;z;p ) = 3RF(x;y;z ):
19.20.13
2(p x)RJ(x;y;z;p ) = 3RF(x;y;z ) 3pxRC
yz;p2
,
p=xp
(y x)(z x),
wherex;y;z may be permuted.
When the variables are real and distinct, the vari-
ous cases of RJ(x;y;z;p ) are called circular (hyperbolic)
cases if (p x)(p y)(p z) is positive (negative), because
they typically occur in conjunction with inverse circular
(hyperbolic) functions. Cases encountered in dynamical
problems are usually circular; hyperbolic cases include
Cauchy principal values. If x;y;z are permuted so that
19.21 Connection Formulas 503
0x < y < z , then the Cauchy principal value of RJ
is given by
19.20.14(q+z)RJ(x;y;z; q)
= (p z)RJ(x;y;z;p ) 3RF(x;y;z )
+ 3xyz
xy+pq1=2
RC(xy+pq;pq );
valid when
19.20.15 q>0; p =z(x+y+q) xy
z+q;
or
19.20.16p=wy+ (1 w)z; w =z x
z+q;0<w< 1:
Sincex<y<p<z ,pis in a hyperbolic region. In the
complete case ( x= 0) (19.20.14) reduces to
19.20.17(q+z)RJ(0;y;z; q)
= (p z)RJ(0;y;z;p ) 3RF(0;y;z),
p=z(y+q)=(z+q),w=z=(z+q).
19.20(iv)RD(x;y;z )
19.20.18RD(x;x;x ) =x 3=2;
RD(x;y;z ) = 3=2RD(x;y;z );
RD(0;y;y) =3
4y 3=2;
RD(0;0;z) =1:
19.20.19 RD(x;y;z )3(xyz) 1=2,z=pxy!0.
19.20.20RD(x;y;y ) =3
2(y x)
RC(x;y) px
y
,
x6=y,y6= 0,
19.20.21
RD(x;x;z ) =3
z x
RC(z;x) 1pz
,x6=z,xz6= 0.
The second lemniscate constant is given by
19.20.22Z1
0t2dtp
1 t4=1
3RD(0;2;1) =
3
42
(2)1=2
= 0:59907 01173 67796 10371 ::::
Todd (1975) refers to a proof by T. Schneider that this
is a transcendental number. Compare (19.20.2). The
general lemniscatic case is
19.20.23
RD(x;y;a ) =R 3
4 5
4;1
2;a2;xy
,a=1
2x+1
2y.19.20(v)R a(b; z)
De nec=Pn
j=1bj. Then
19.20.24R0(b;z) = 1; RN(b;z) =N!
(c)NTN(b;z),
N= 0;1;2;:::,
whereTNis de ned by (19.19.1). Also,
19.20.25 R c(b;z) =nY
j=1z bj
j;
19.20.26R a(b;z) =nY
j=1z bj
jR a0
b;z 1
,
a+a0=c,z 1= (z 1
1;:::;z 1
n).
See also (19.16.11) and (19.16.19).
19.21 Connection Formulas
19.21(i) Complete Integrals
Legendre's relation (19.7.1) can be written
19.21.1RF(0;z+ 1;z)RD(0;z+ 1;1)
+RD(0;z+ 1;z)RF(0;z+ 1;1) = 3=(2z),
z2Cn( 1;0].
The casez= 1 shows that the product of the two lem-
niscate constants, (19.20.2) and (19.20.22), is =4.
19.21.2 3RF(0;y;z) =zRD(0;y;z) +yRD(0;z;y):
19.21.3
6RG(0;y;z) =yz(RD(0;y;z) +RD(0;z;y))
= 3zRF(0;y;z) +z(y z)RD(0;y;z):
The complete cases of RFandRGhave connection
formulas resulting from those for the Gauss hyperge-
ometric function (Erd elyi et al. (1953a,x2.9)). Up-
per signs apply if 0 <phz < , and lower signs if
<phz<0:
19.21.4RF(0;z 1;z) =RF(0;1 z;1)iRF(0;z;1);
19.21.5
2RG(0;z 1;z) = 2RG(0;1 z;1)i2RG(0;z;1)
+ (z 1)RF(0;1 z;1)
izRF(0;z;1):
Lety,z, andpbe positive and distinct, and permute
yandzto ensure that ydoes not lie between zandp.
The complete case of RJcan be expressed in terms of
RFandRD:
19.21.6
(prp=z)RJ(0;y;z;p ) = (r 1)RF(0;y;z)RD(p;rz;z )
+RD(0;y;z)RF(p;rz;z ),
r= (y p)=(y z)>0.
If 0< p < z andy=z+ 1, then as p!0 (19.21.6)
reduces to Legendre's relation (19.21.1).
504 Elliptic Integrals
19.21(ii) Incomplete Integrals
RD(x;y;z ) is symmetric only in xandy, but either
(nonzero)xor (nonzero) ycan be moved to the third
position by using
19.21.7(x y)RD(y;z;x ) + (z y)RD(x;y;z )
= 3RF(x;y;z ) 3p
y=(xz);
or the corresponding equation with xandyinter-
changed.
19.21.8RD(y;z;x ) +RD(z;x;y ) +RD(x;y;z )
= 3(xyz) 1=2;
19.21.9xRD(y;z;x ) +yRD(z;x;y ) +zRD(x;y;z )
= 3RF(x;y;z ):
19.21.10
2RG(x;y;z ) =zRF(x;y;z )
1
3(x z)(y z)RD(x;y;z ) +p
xy=z ,
z6= 0.
BecauseRGis completely symmetric, x;y;z can be
permuted on the right-hand side of (19.21.10) so that
(x z)(y z)0 if the variables are real, thereby
avoiding cancellations when RGis calculated from RF
andRD(seex19.36(i)).
19.21.116RG(x;y;z ) = 3(x+y+z)RF(x;y;z )
X
x2RD(y;z;x )
=X
x(y+z)RD(y;z;x );
where both summations extend over the three cyclic
permutations of x;y;z .
Connection formulas for R a(b;z) are given in Carl-
son (1977b, pp. 99, 101, and 123{124).
19.21(iii) Change of Parameter of RJ
Letx;y;z be real and nonnegative, with at most one of
them 0. Change-of-parameter relations can be used to
shift the parameter pofRJfrom either circular region
to the other, or from either hyperbolic region to the
other (x19.20(iii)). The latter case allows evaluation of
Cauchy principal values (see (19.20.14)).
19.21.12(p x)RJ(x;y;z;p ) + (q x)RJ(x;y;z;q )
= 3RF(x;y;z ) 3RC(;);
where
19.21.13
(p x)(q x) = (y x)(z x); =yz=x; =pq=x;
andx;y;z may be permuted. Also,
19.21.14
=p+q y z=(p y)(p z)
p x=(q y)(q z)
q x
=(p y)(q y)
x y=(p z)(q z)
x z:For each value of p, permutation of x;y;z produces
three values of q, one of which lies in the same region as
pand two lie in the other region of the same type. In
(19.21.12), if xis the largest (smallest) of x;y, andz,
thenpandqlie in the same region if it is circular (hy-
perbolic); otherwise pandqlie in di erent regions, both
circular or both hyperbolic. If x= 0, then==1
andRC(;) = 0; hence
19.21.15
pRJ(0;y;z;p ) +qRJ(0;y;z;q ) = 3RF(0;y;z),pq=yz.
19.22 Quadratic Transformations
19.22(i) Complete Integrals
Let<x>0,<y>0,a= (x+y)=2, andp6= 0. Then
19.22.1
RF
0;x2;y2
=RF
0;xy;a2
;
19.22.2
2RG
0;x2;y2
= 4RG
0;xy;a2
xyRF
0;xy;a2
;
19.22.3
2y2RD
0;x2;y2
=1
4(y2 x2)RD
0;xy;a2
+ 3RF
0;xy;a2
:
19.22.4(p2
p2
)RJ
0;x2;y2;p2
= 2(p2
a2)RJ
0;xy;a2;p2
3RF
0;xy;a2
+ 3=(2p);
where
19.22.5 2p=p
(p+x)(p+y)p
(p x)(p y);
and hence
19.22.6p+p =pa; p2
++p2
=p2+xy;
p2
+ p2
=p
(p2 x2)(p2 y2);
4(p2
a2) = (p
p2 x2p
p2 y2)2:
Bartky's Transformation
19.22.72p2RJ
0;x2;y2;p2
=v+v RJ
0;xy;a2;v2
+
+ 3RF
0;xy;a2
,
v= (p2xy)=(2p).
Ifp=y, then (19.22.7) reduces to (19.22.3), but if p=x
orp=y, then both sides of (19.22.4) are 0 by (19.20.9).
Ifx<p<y ory<p<x , thenp+andp are complex
conjugates.
19.22 Quadratic Transformations 505
19.22(ii) Gauss's Arithmetic-Geometric Mean
(AGM)
The AGM, M(a0;g0), of two positive numbers a0and
g0is de ned inx19.8(i). Again, we assume that a0g0
(except in (19.22.10)), and de ne cn=p
a2n g2n. Then
19.22.82
RF
0;a2
0;g2
0
=1
M(a0;g0);
19.22.9
4
RG
0;a2
0;g2
0
=1
M(a0;g0)
a2
0 1X
n=02n 1c2
n!
=1
M(a0;g0)
a2
1 1X
n=22n 1c2
n!
;
and
19.22.10RD
0;g2
0;a2
0
=3
4M(a0;g0)a2
01X
n=0Qn;
where
19.22.11 Q0= 1; Qn+1=1
2Qnan gn
an+gn:
Qnhas the same sign as a0 g0forn1.
19.22.12RJ
0;g2
0;a2
0;p2
0
=3
4M(a0;g0)p2
01X
n=0Qn;
wherep0>0 and
19.22.13pn+1=p2
n+angn
2pn; "n=p2
n angn
p2n+angn;
Q0= 1; Qn+1=1
2Qn"n:
(Ifp0=a0, thenpn=anand (19.22.13) reduces to
(19.22.11).) As n!1 ,pnand"nconverge quadrati-
cally toM(a0;g0) and 0, respectively, and Qnconverges
to 0 faster than quadratically. If the last variable of RJ
is negative, then the Cauchy principal value is
19.22.14RJ
0;g2
0;a2
0; q2
0
= 3
4M(a0;g0)(q2
0+a2
0)
2 +a2
0 g2
0
q2
0+g2
01X
n=0Qn!
;
and (19.22.13) still applies, provided that
19.22.15 p2
0=a2
0(q2
0+g2
0)=(q2
0+a2
0):
19.22(iii) Incomplete Integrals
Letx,y, andzhave positive real parts, assume p6= 0,
and retain (19.22.5) and (19.22.6). De ne
19.22.16a= (x+y)=2;
2z=p
(z+x)(z+y)p
(z x)(z y);
so that
19.22.17z+z =za; z2
++z2
=z2+xy;
z2
+ z2
=p
(z2 x2)(z2 y2);
4(z2
a2) = (p
z2 x2p
z2 y2)2:Then
19.22.18RF
x2;y2;z2
=RF
a2;z2
;z2
+
;
19.22.19
(z2
z2
)RD
x2;y2;z2
= 2(z2
a2)RD
a2;z2
;z2
3RF
x2;y2;z2
+ (3=z);
19.22.20(p2
p2
)RJ
x2;y2;z2;p2
= 2(p2
a2)RJ
a2;z2
+;z2
;p2
3RF
x2;y2;z2
+ 3RC
z2;p2
;
19.22.212RG
x2;y2;z2
= 4RG
a2;z2
+;z2
xyRF
x2;y2;z2
z;
19.22.22RC
x2;y2
=RC
a2;ay
:
Ifx;y;z are real and positive, then (19.22.18){
(19.22.21) are ascending Landen transformations when
x;y<z (implyinga<z <z+), and descending Gauss
transformations when z<x;y (implyingz+<z <a).
Ascent and descent correspond respectively to increase
and decrease of kin Legendre's notation. Descending
Gauss transformations include, as special cases, trans-
formations of complete integrals into complete integrals;
ascending Landen transformations do not.
Ifp=xorp=y, then (19.22.20) reduces to 0 = 0
by (19.20.13), and if z=xorz=ythen (19.22.19)
reduces to 0 = 0 by (19.20.20) and (19.22.22). If
x < z < y ory < z < x , thenz+andz are com-
plex conjugates. However, if xandyare complex con-
jugates and zandpare real, then the right-hand sides of
all transformations in xx19.22(i) and 19.22(iii)|except
(19.22.3) and (19.22.22)|are free of complex numbers
andp2
p2
=jp2 x2j6= 0.
The transformations inverse to the ones just de-
scribed are the descending Landen transformations and
the ascending Gauss transformations. The equations
inverse to (19.22.5) and (19.22.16) are given by
19.22.23
x+y= 2a; x y= (2/a)q
(a2 z2
+)(a2 z2
);
z=z+z /a;
and the corresponding equations with z,z+, andz re-
placed byp,p+, andp , respectively. These relations
need to be used with caution because yis negative when
0<a<z +z
z2
++z2
1=2.
506 Elliptic Integrals
19.23 Integral Representations
In (19.23.1){(19.23.3) we assume <y>0 and<z>0.
19.23.1RF(0;y;z) =Z=2
0(ycos2+zsin2) 1=2d;
19.23.2RG(0;y;z) =1
2Z=2
0(ycos2+zsin2)1=2d;
19.23.3
RD(0;y;z) = 3Z=2
0(ycos2+zsin2) 3=2sin2d:
19.23.4RF(0;y;z) =2
Z=2
0RC
y;zcos2
d
=2
Z1
0RC
ycosh2t;z
dt:19.23.5
RF(x;y;z ) =2
Z=2
0RC
x;ycos2+zsin2
d,
<y>0,<z>0,
19.23.6
4RF(x;y;z )
=Z2
0Z
0sindd
(xsin2cos2 +ysin2sin2 +zcos2)1=2;
wherex,y, andzhave positive real parts|except that
at most one of them may be 0.
In (19.23.7){(19.23.10) one or more of the variables
may be 0 if the integral converges. In (19.23.8) n= 2,
and in (19.23.9) n= 3. Also, in (19.23.8) and (19.23.10)
B denotes the beta function ( x5.12).
19.23.7 RG(x;y;z ) =1
4Z1
01pt+xpt+ypt+zx
t+x+y
t+y+z
t+z
tdt,x;y;z2Cn( 1;0].
19.23.8 R a(b;z) =2
B(b1;b2)Z=2
0(z1cos2+z2sin2) a(cos)2b1 1(sin)2b2 1d,b1;b2>0;<z1;<z2>0.
Withl1;l2;l3denoting any permutation of sin cos , sinsin , cos,
19.23.9 R a(b;z) =4 (b1+b2+b3)
(b1) (b2) (b3)Z=2
0Z=2
00
@3X
j=1zjl2
j1
A a3Y
j=1l2bj 1
j sindd ,bj>0,<zj>0.
19.23.10
R a(b;z) =1
B(a;a0)Z1
0ua 1(1 u)a0 1nY
j=1(1 u+uzj) bjdu,a;a0>0;a+a0=Pn
j=1bj;zj2Cn( 1;0].
For generalizations of (19.16.3) and (19.23.8) see
Carlson (1964, (6.2), (6.12), and (6.1)).
19.24 Inequalities
19.24(i) Complete Integrals
The condition yzfor (19.24.1) and (19.24.2) serves
only to identify yas the smaller of the two nonzero
variables of a symmetric function; it does not restrict
validity.
19.24.1
ln 4pzRF(0;y;z) + lnp
y=z1
2, 0<yz,
19.24.21
2z 1=2RG(0;y;z)1
4, 0yz,
19.24.3y3=2+z3=2
22=3
4
RG
0;y2;z2
y2+z2
21=2
,
y>0,z>0.Ify,z, andpare positive, then
19.24.4
2pp(2yz+yp+zp) 1=24
3RJ(0;y;z;p )(yzp2) 3=8:
Inequalities for RD(0;y;z) are included as the case
p=z.
A series of successively sharper inequalities is ob-
tained from the AGM process ( x19.8(i)) with a0g0>
0:
19.24.51
an2
RF
0;a2
0;g2
0
1
gn,n= 0;1;2;:::,
where
19.24.6an+1= (an+gn)=2; gn+1=pangn:
Other inequalities can be obtained by applying
Carlson (1966, Theorems 2 and 3) to (19.16.20){
(19.16.23). Approximations and one-sided inequalities
forRG(0;y;z) follow from those given in x19.9(i) for the
lengthL(a;b) of an ellipse with semiaxes aandb, since
19.24.7 L(a;b) = 8RG
0;a2;b2
:
19.25 Relations to Other Functions 507
Forx >0,y >0, andx6=y, the complete cases of
RFandRGsatisfy
19.24.8RF(x;y;0)RG(x;y;0)>1
82;
RF(x;y;0) + 2RG(x;y;0)>:
Also, with the notation of (19.24.6),
19.24.91
2g2
1RG
a2
0;g2
0;0
RF(a2
0;g2
0;0)1
2a2
1;
with equality i a0=g0.
19.24(ii) Incomplete Integrals
Inequalities for R a(b;z) in Carlson (1966, Theorems
2 and 3) can be applied to (19.16.14){(19.16.17). All
variables are positive, and equality occurs i all vari-
ables are equal.
Examples
19.24.103px+py+pzRF(x;y;z )1
(xyz)1=6;
19.24.115px+py+pz+ 2pp3
RJ(x;y;z;p )
(xyzp2) 3=10;
19.24.12
1
3(px+py+pz)
RG(x;y;z )min r
x+y+z
3;x2+y2+z2
3pxyz!
:
Inequalities for RC(x;y) andRD(x;y;z ) are included as
special cases (see (19.16.6) and (19.16.5)).
Other inequalities for RF(x;y;z ) are given in Carl-
son (1970).
Ifa(6= 0) is real, all components of bandzare pos-
itive, and the components of zare not all equal, then
19.24.13
Ra(b;z)R a(b;z)>1; Ra(b;z) +R a(b;z)>2;
see Neuman (2003, (2.13)). Special cases with a=1
2
are (19.24.8) (because of (19.16.20), (19.16.23)), and
19.24.14RF(x;y;z )RG(x;y;z )>1;
RF(x;y;z ) +RG(x;y;z )>2:
The same reference also gives upper and lower bounds
for symmetric integrals in terms of their elementary de-
generate cases. These bounds include a sharper but
more complicated lower bound than that supplied in
the next result:
19.24.15
RC
x;1
2(y+z)
RF(x;y;z )RC(x;pyz),x0,
with equality i y=z.19.25 Relations to Other Functions
19.25(i) Legendre's Integrals as Symmetric
Integrals
Letk02= 1 k2andc= csc2 . Then
19.25.1
K(k) =RF
0;k02;1
; E (k) = 2RG
0;k02;1
;
E(k) =1
3k02
RD
0;k02;1
+RD
0;1;k02
;
K(k) E(k) =k2D(k) =1
3k2RD
0;k02;1
;
E(k) k02K(k) =1
3k2k02RD
0;1;k02
:
19.25.2
2;k
K(k) =1
3 2RJ
0;k02;1;1 2
:
19.25.3
2;k
=1
2R 1
2
1
2; 1
2;1;k02;1;1 2
;
with Cauchy principal value
19.25.4
2;k
= 1
3(k2= 2)RJ
0;1 k2;1;1 (k2= 2)
,
1<k2<1< 2.
19.25.5 F( ;k) =RF
c 1;c k2;c
;
19.25.6@F( ;k)
@k=1
3kRD
c 1;c;c k2
:
19.25.7E( ;k) = 2RG
c 1;c k2;c
(c 1)RF
c 1;c k2;c
p
(c 1)(c k2)=c;
19.25.8E( ;k) =R 1
2 1
2; 1
2;3
2;c 1;c k2;c
;
19.25.9
E( ;k)
=RF
c 1;c k2;c
1
3k2RD
c 1;c k2;c
;
19.25.10E( ;k) =k02RF
c 1;c k2;c
+1
3k2k02RD
c 1;c;c k2
+k2p
(c 1)=(c(c k2)),c>k2,
19.25.11E( ;k) = 1
3k02RD
c k2;c;c 1
+p
(c k2)=(c(c 1)), 6=1
2.
Equations (19.25.9){(19.25.11) correspond to three
(nonzero) choices for the last variable of RD; see
(19.21.7). All terms on the right-hand sides are non-
negative when k20, 0k21, or 1k2c,
respectively.
19.25.12@E( ;k)
@k= 1
3kRD
c 1;c k2;c
:
19.25.13 D( ;k) =1
3RD
c 1;c k2;c
:
19.25.14
; 2;k
F( ;k) =1
3 2RJ
c 1;c k2;c;c 2
;
19.25.15
; 2;k
=R 1
2 1
2;1
2; 1
2;1;c 1;c k2;c;c 2
:
508 Elliptic Integrals
If 2>c, then the Cauchy principal value is
19.25.16
; 2;k
= 1
3!2RJ
c 1;c k2;c;c !2
+s
(c 1)(c k2)
( 2 1)(1 !2)
RC
c( 2 1)(1 !2);( 2 c)(c !2)
,
!2=k2= 2.
The transformations in x19.7(ii) result from the sym-
metry and homogeneity of functions on the right-hand
sides of (19.25.5), (19.25.7), and (19.25.14). For exam-
ple, if we write (19.25.5) as
19.25.17 F( ;k) =RF(x;y;z );
with
19.25.18 (x;y;z ) = (c 1;c k2;c);
then the ve nontrivial permutations of x;y;z that leave
RFinvariant change k2(= (z y)=(z x)) into 1=k2,k02,
1=k02, k2=k02, k02=k2, and sin (=p
(z x)=z) into
ksin , itan , ik0tan , (k0sin )=p
1 k2sin2 ,
iksin =p
1 k2sin2 . Thus the ve permutations
induce ve transformations of Legendre's integrals (and
also of the Jacobian elliptic functions).
The three changes of parameter of
; 2;k
in
x19.7(iii) are uni ed in (19.21.12) by way of (19.25.14).
19.25(ii) Bulirsch's Integrals as Symmetric
Integrals
Letr= 1=x2. Then
19.25.19
cel(kc;p;a;b ) =aRF
0;k2
c;1
+1
3(b pa)RJ
0;k2
c;1;p
;
19.25.20
el1(x;kc) =RF
r;r+k2
c;r+ 1
;
19.25.21
el2(x;kc;a;b) =ael1(x;kc)
+1
3(b a)RD
r;r+k2
c;r+ 1
;
19.25.22
el3(x;kc;p) = el1(x;kc)
+1
3(1 p)RJ
r;r+k2
c;r+ 1;r+p
:
19.25(iii) Symmetric Integrals as Legendre's
Integrals
Assume 0xyz,x<z , andp>0. Let
19.25.23 = arccosp
x/z= arcsinp
(z x)/z;
k=r
z y
z x; 2=z p
z x;with 6= 0. Then
19.25.24 (z x)1=2RF(x;y;z ) =F( ;k);
19.25.25
(z x)3=2RD(x;y;z ) = (3=k2)(F( ;k) E( ;k));
19.25.26
(z x)3=2RJ(x;y;z;p ) = (3= 2)(
; 2;k
F( ;k));
19.25.27
2(z x) 1=2RG(x;y;z ) =E( ;k) + (cot )2F( ;k)
+ (cot )q
1 k2sin2 :
19.25(iv) Theta Functions
For relations of symmetric integrals to theta functions,
seex20.9(i).
19.25(v) Jacobian Elliptic Functions
For the notation see xx22.2, 22.15, and 22.16(i).
With 0k21 and p;q;r any permutation of the
letters c;d;n, de ne
19.25.28 (p;q) = ps2(u;k) qs2(u;k) = (q;p);
which implies
19.25.29 (n;d) =k2;(d;c) =k02;(n;c) = 1:
If cs2(u;k)0, then
19.25.30 am (u;k) =RC
cs2(u;k);ns2(u;k)
;
19.25.31u=RF
ps2(u;k);qs2(u;k);rs2(u;k)
;
compare (19.25.35) and (20.9.3).
19.25.32
arcps (x;k) =RF
x2;x2+ (q;p);x2+ (r;p)
;
19.25.33
arcsp (x;k) =xRF
1;1 + (q;p)x2;1 + (r;p)x2
;
19.25.34arcpq (x;k) =pwRF
x2;1;1 + (r;q)w
,
w= (1 x2)
(q;p),
where we assume 0 x21 ifx= sn, cn, or cd; x21
ifx= ns, nc, or dc; xreal ifx= cs or sc; k0x1
ifx= dn; 1x1=k0ifx= nd;x2k02ifx= ds;
0x21=k02ifx= sd.
For the use of R-functions with (p ;q) in unifying
other properties of Jacobian elliptic functions, see Carl-
son (2004, 2006a,b, 2008).
Inversions of 12 elliptic integrals of the rst
kind, producing the 12 Jacobian elliptic functions,
are combined and simpli ed by using the properties
ofRF(x;y;z ). See (19.29.19), Carlson (2005), and
(22.15.11), and compare with Abramowitz and Ste-
gun (1964, Eqs. (17.4.41){(17.4.52)). For analogous
integrals of the second kind, which are not invertible
in terms of single-valued functions, see (19.29.20) and
(19.29.21) and compare with Gradshteyn and Ryzhik
(2000,x3.153,1{10 andx3.156,1{9).
19.26 Addition Theorems 509
19.25(vi) Weierstrass Elliptic Functions
For the notation see x23.2.
19.25.35z=RF(}(z) e1;}(z) e2;}(z) e3);
provided that
19.25.36 }(z) ej2Cn( 1;0],j= 1;2;3,
and the left-hand side does not vanish for more than
one value of j. Also,
19.25.37
(z) +z}(z) = 2RG(}(z) e1;}(z) e2;}(z) e3):
In (19.25.38) and (19.25.39) j;k;` is any permuta-
tion of the numbers 1 ;2;3.
19.25.38 !j=RF(0;ej ek;ej e`);
19.25.39j+!jej= 2RG(0;ej ek;ej e`):
Lastly,
19.25.40z=(z)RF
2
1(z);2
2(z);2
3(z)
;
where
19.25.41
j(z) = exp( jz)(z+!j)=(!j),j= 1;2;3.
19.25(vii) Hypergeometric Function
19.25.42 2F1(a;b;c;z) =R a(b;c b; 1 z;1);
19.25.43
R a(b1;b2;z1;z2) =z a
22F1(a;b1;b1+b2; 1 (z1=z2)):
For these results and extensions to the Appell function
F1(x16.13) and Lauricella's function FDsee Carlson
(1963). (F1andFDare equivalent to the R-function of
3 andnvariables, respectively, but lack full symmetry.)
19.26 Addition Theorems
19.26(i) General Formulas
In this subsection, and also xx19.26(ii) and 19.26(iii), we
assume that ;x;y;z are positive, except that at most
one ofx;y;z can be 0.
19.26.1RF(x+;y+;z+)
+RF(x+;y+;z+) =RF(x;y;z );
where>0 and
19.26.2
x+= 2p
(x+)yz+p
x(y+)(z+)2
;
with corresponding equations for y+andz+ob-
tained by permuting x;y;z . Also,
19.26.3pz=0+0 0
p0+p00;where
19.26.4(;; ) = (x+;y+;z+);
(0;0;0) = (x+;y+;z+);
withpxandpyobtained by permuting x,y, andz.
(Note that 0+0 0=0+0 0.) Equivalent
forms of (19.26.2) are given by
19.26.5= 2pxyz+p
(x+)(y+)(z+)2
x y z;
and
19.26.6 ( xy xz yz)2= 4xyz(++x+y+z):
Also,
19.26.7
RD(x+;y+;z+) +RD(x+;y+;z+)
=RD(x;y;z ) 3p
z(z+)(z+);
19.26.8
2RG(x+;y+;z+) + 2RG(x+;y+;z+)
= 2RG(x;y;z ) +RF(x+;y+;z+)
+RF(x+;y+;z+) +p
++x+y+z:
19.26.9RJ(x+;y+;z+;p+)
+RJ(x+;y+;z+;p+)
=RJ(x;y;z;p ) 3RC(
;
);
where
19.26.10
=p(p+)(p+); = (p x)(p y)(p z):
Lastly,
19.26.11
RC(x+;y+) +RC(x+;y+) =RC(x;y);
where>0,y>0,x0, and
19.26.12x+= 2(p
x+y+px(y+))2;
y+= (y(y+)=2)(px+p
x+)2:
Equivalent forms of (19.26.11) are given by
19.26.13RC
2; 2
+RC
2; 2
=RC
2;2
,= ( +)=( + ),
where 0<
2 <
2for
= ; ; , except that 2
can be 0, and
19.26.14
(p y)RC(x;p) + (q y)RC(x;q)
= ( )RC(;),x0,y0;p;q2Rnf0g,
where
19.26.15(p x)(q x) = (y x)2; =y2=x;
=pq=x; =p+q 2y:
510 Elliptic Integrals
19.26(ii) Case x= 0
Ifx= 0, then=yz. For example,
19.26.16
RF(;y+;z+)
=RF(0;y;z) RF(;y+;z+),=yz.
An equivalent version for RCis
19.26.17p RC( ; + ) +p
RC( ; + )
==2, ; 2Cn( 1;0), + >0:
19.26(iii) Duplication Formulas
19.26.18RF(x;y;z ) = 2RF(x+;y+;z+)
=RFx+
4;y+
4;z+
4
;
where
19.26.19 =pxpy+pypz+pzpx:
19.26.20
RD(x;y;z ) = 2RD(x+;y+;z+) +3pz(z+):
19.26.212RG(x;y;z ) = 4RG(x+;y+;z+)
RF(x;y;z ) px py pz:
19.26.22RJ(x;y;z;p ) = 2RJ(x+;y+;z+;p+)
+ 3RC
2; 2
;
where
19.26.23
=p(px+py+pz) +pxpypz; =pp(p+);
= (pppx)(pppy)(pppz);
2 2= (p x)(p y)(p z);
either upper or lower signs being taken throughout.
The equations inverse to z+= (pz+px)(pz+py)
and the two other equations obtained by permuting
x;y;z (see (19.26.19)) are
19.26.24z= (+ )2=(4),
(;; ) = (x+;y+;z+),
and two similar equations obtained by exchanging z
withx(andwith), orzwithy(andwith).
Next,
19.26.25
RC(x;y) = 2RC(x+;y+),=y+ 2pxpy.
Equivalent forms are given by (19.22.22). Also,
19.26.26RC
x2;y2
=RC
a2;ay
,
a= (x+y)=2,<x0,<y>0,
and
19.26.27
RC
x2;x2
= 2RC
s2;s2
,
s=x+p
x2 ,6=x2ors2.19.27 Asymptotic Approximations and
Expansions
19.27(i) Notation
Throughout this section
19.27.1
a=1
2(x+y); b =1
2(y+z); c =1
3(x+y+z);
f= (xyz)1=3; g = (xy)1=2; h = (yz)1=2:
19.27(ii)RF(x;y;z )
Assumex,y, andzare real and nonnegative and at
most one of them is 0. Then
19.27.2
RF(x;y;z ) =1
2pz
ln8z
a+g
1 +Oa
z
,a=z!0.
19.27.3
RF(x;y;z ) =RF(0;y;z) 1p
hrx
h+Ox
h
,
x=h!0.
19.27(iii)RG(x;y;z )
Assumex,y, andzare real and nonnegative and at
most one of them is 0. Then
19.27.4RG(x;y;z ) =pz
2
1 +Oa
zlnz
a
,a=z!0.
19.27.5
RG(x;y;z ) =RG(0;y;z) +pxOp
x=h
,x=h!0.
19.27.6RG(0;y;z)
=pz
2+y
8pz
ln16z
y
1
1 +Oy
z
,
y=z!0.
19.27(iv)RD(x;y;z )
Assumexandyare real and nonnegative, at most one
of them is 0, and z>0. Then
19.27.7
RD(x;y;z ) =3
2z3=2
ln8z
a+g
2
1 +Oa
z
,
a=z!0.
19.27.8
RD(x;y;z ) =3pxyz 6
xyRG(x;y;0)
1 +Oz
g
,
z=g!0.
19.27.9
RD(x;y;z ) =3pxz(py+pz)
1 +Ob
xlnx
b
,
b=x!0.
19.28 Integrals of Elliptic Integrals 511
19.27.10
RD(x;y;z ) =RD(0;y;z) 3px
hz
1 +Orx
h
,
x=h!0.
19.27(v)RJ(x;y;z;p )
Assumex,y, andzare real and nonnegative, at most
one of them is 0, and p>0. Then
19.27.11
RJ(x;y;z;p ) =3
pRF(x;y;z ) 3
2p3=2
1 +Orc
p
,
c=p!0.
19.27.12
RJ(x;y;z;p ) =3
2pxyz
ln4f
p
2
1 +Op
f
,
p=f!0.
19.27.13
RJ(x;y;z;p ) =3
2pzp
ln8z
a+g
2RC
1;p
z
+Oa
z+a
p
lnp
a
,
max(x;y)=min(z;p)!0.
19.27.14
RJ(x;y;z;p ) =3pyzRC(x;p) 6
yzRG(0;y;z)
+Opx+ 2p
yz
,
max(x;p)=min(y;z)!0.
19.27.15
RJ(x;y;z;p ) =RJ(0;y;z;p )
3px
hp
1 +Ob
h+h
prx
h
,
x=min(y;z;p )!0.
19.27.16
RJ(x;y;z;p ) = (3=px)RC
(h+p)2;2(b+h)p
+O1
x3=2lnx
b+h
,
max(y;z;p )=x!0.
19.27(vi) Asymptotic Expansions
The approximations in xx19.27(i){19.27(v) are furnished
with upper and lower bounds by Carlson and Gustafson
(1994), sometimes with two or three approximations of
di ering accuracies. Although they are obtained (with
some exceptions) by approximating uniformly the in-
tegrand of each elliptic integral, some occur also as
the leading terms of known asymptotic series with er-
ror bounds (Wong (1983, x4), Carlson and Gustafson
(1985), L opez (2000, 2001)). These series converge butnot fast enough, given the complicated nature of their
terms, to be very useful in practice.
A similar (but more general) situation prevails for
R a(b;z) when some of the variables z1;:::;znare
smaller in magnitude than the rest; see Carlson (1985,
(4.16){(4.19) and (2.26){(2.29)).
19.28 Integrals of Elliptic Integrals
In (19.28.1){(19.28.3) we assume <>0. Also, B again
denotes the beta function ( x5.12).
19.28.1Z1
0t 1RF(0;t;1)dt=1
2
B
;1
22;
19.28.2Z1
0t 1RG(0;t;1)dt=
4+ 2
B
;1
22;
19.28.3Z1
0t 1(1 t)RD(0;t;1)dt=3
4+ 2
B
;1
22:
19.28.4Z1
0t 1(1 t)c 1R a(b1;b2;t;1)dt
= (c) () (+b2 a)
(+c a) (+b2),
c=b1+b2>0,<>max(0;a b2).
In (19.28.5){(19.28.9) we assume x;y;z , andpare
real and positive.
19.28.5Z1
zRD(x;y;t )dt= 6RF(x;y;z );
19.28.6Z1
0RD
x;y;v2z+ (1 v2)p
dv=RJ(x;y;z;p ):
19.28.7Z1
0RJ
x;y;z;r2
dr=3
2RF(xy;xz;yz );
19.28.8Z1
0RJ(tx;y;z;tp )dt=6ppRC(p;x)RF(0;y;z):
19.28.9Z=2
0RF
sin2cos2(x+y);sin2cos2(x y);1
d
=RF
0;cos2x;1
RF
0;cos2y;1
;
19.28.10Z1
0RF
(ac+bd)2;(ad+bc)2;4abcdcosh2z
dz
=1
2RF
0;a2;b2
RF
0;c2;d2
,a;b;c;d> 0.
See also (19.16.24). To replace a single component
ofzinR a(b;z) by several di erent variables (as in
(19.28.6)), see Carlson (1963, (7.9)).
512 Elliptic Integrals
19.29 Reduction of General Elliptic
Integrals
19.29(i) Reduction Theorems
These theorems reduce integrals over a real interval
(y;x) of certain integrands containing the square root
of a quartic or cubic polynomial to symmetric integrals
over (0;1) containing the square root of a cubic poly-
nomial (compare x19.16(i)). Let
19.29.1X =p
a +b x; Y =p
a +b y,
x>y , 1 5,
19.29.2 d =a b a b ,d 6= 0 if 6= ,
and assume that the line segment with endpoints a +
b xanda +b ylies in Cn( 1;0) for 1 4. If
19.29.3 s(t) =4Y
=1p
a +b t
and ; ;
; is any permutation of the numbers
1;2;3;4, then
19.29.4Zx
ydt
s(t)= 2RF
U2
12;U2
13;U2
23
;
where
19.29.5
U = (X X Y
Y+Y Y X
X)=(x y);
U =U =U
=U
; U2
U2
=d d
:
There are only three distinct U's with subscripts 4,
and at most one of them can be 0 because the d's are
nonzero. Then
19.29.6
U =p
b p
b Y
Y+Y Y p
b
p
b,x=1,
U =X X p
b
p
b+p
b p
b X
X,
y= 1.
19.29.7Zx
ya +b t
a+btdt
s(t)=2
3d d
RD
U2
;U2
;U2
+2X Y
XYU , U 6= 0.
19.29.8Zx
ya +b t
a5+b5tdt
s(t)
=2
3d d
d
d 5RJ
U2
12;U2
13;U2
23;U2
5
+ 2RC
S2
5;Q2
5
,S2
52Cn( 1;0),
where
19.29.9
U2
5=U2
d
d d 5
d 5=U2
d d
d5
d 56= 0;
S 5=1
x yX X
X
X Y2
5+Y Y
Y
Y X2
5
;
Q 5=X5Y5
X Y U 56= 0; S2
5 Q2
5=d 5d
5d5
d 5:The Cauchy principal value is taken when U2
5orQ2
5
is real and negative. Cubic cases of these formulas are
obtained by setting one of the factors in (19.29.3) equal
to 1.
The advantages of symmetric integrals for tables
of integrals and symbolic integration are illustrated by
(19.29.4) and its cubic case, which replace the 8 + 8 +
12 = 28 formulas in Gradshteyn and Ryzhik (2000,
3.147, 3.131, 3.152) after taking x2as the variable of in-
tegration in 3.152. Moreover, the requirement that one
limit of integration be a branch point of the integrand
is eliminated without doubling the number of standard
integrals in the result. (19.29.7) subsumes all 72 for-
mulas in Gradshteyn and Ryzhik (2000, 3.168), and its
cubic cases similarly replace the 18 + 36 + 18 = 72 for-
mulas in Gradshteyn and Ryzhik (2000, 3.133, 3.142,
and 3.141(1-18)). For example, 3.142(2) is included as
19.29.10Zb
us
a t
(b t)(t c)3dt= 2
3(a b)(b u)3=2RD
+2
b cr
(a u)(b u)
u c,
a>b>u>c ,
where the arguments of the RDfunction are, in order,
(a b)(u c), (b c)(a u), (a b)(b c).
19.29(ii) Reduction to Basic Integrals
(19.2.3) can be written
19.29.11
I(m) =Zx
yhY
=1(a +b t) 1=2nY
j=1(aj+bjt)mjdt;
wherex > y ,h= 3 or 4,nh, andmjis an integer.
De ne
19.29.12 m= (m1;:::;mn) =nX
j=1mjej;
where ejis ann-tuple with 1 in the jth position and
0's elsewhere. De ne also 0= (0;:::; 0) and retain
the notation and conditions associated with (19.29.1)
and (19.29.2). The integrals in (19.29.4), (19.29.7), and
(19.29.8) are I(0),I(e e), andI(e e5), respec-
tively.
The only cases of I(m) that are integrals of the rst
kind are the two ( h= 3 or 4) with m=0. The only
cases that are integrals of the third kind are those in
which at least one mjwithj > h is a negative integer
and those in which h= 4 andPn
j=1mjis a positive
integer. All other cases are integrals of the second kind .
I(m) can be reduced to a linear combination of ba-
sic integrals and algebraic functions. In the cubic case
(h= 3) the basic integrals are
19.29.13 I(0);I( ej), 1jn.
19.29 Reduction of General Elliptic Integrals 513
In the quartic case ( h= 4) the basic integrals are
19.29.14I(0);I( ej), 1 jn;
I(e ), 1 4.
Basic integrals of type I( ej), 1jh, are not lin-
early independent, nor are those of type I(ej), 1j
4.
The reduction of I(m) is carried out by a relation de-
rived from partial fractions and by use of two recurrence
relations. These are given in Carlson (1999, (2.19),
(3.5), (3.11)) and simpli ed in Carlson (2002, (1.10),
(1.7), (1.8)) by means of modi ed de nitions. Partial
fractions provide a reduction to integrals in which m
has at most one nonzero component, and these are then
reduced to basic integrals by the recurrence relations.
A special case of Carlson (1999, (2.19)) is given by
19.29.15
bjI(el ej) =dljI( ej) +blI(0),j;l= 1;2;:::;n ,
which shows how to express the basic integral I( ej)
in terms of symmetric integrals by using (19.29.4) and
either (19.29.7) or (19.29.8). The rst choice gives a
formula that includes the 18+9+18 = 45 formulas in
Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and
the second choice includes the 8+8+8+12 = 36 formu-
las in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137,
3.157) (after setting x2=tin some cases).
Ifh= 3, then the recurrence relation (Carlson (1999,
(3.5))) has the special case
19.29.16b b
I(e ) =d d
I( e )
+ 2b s(x)
a +b x s(y)
a +b y
;
where ; ;
is any permutation of the numbers 1 ;2;3,
and
19.29.17 s(t) =3Y
=1p
a +b t:
(This shows why I(e ) is not needed as a basic integral
in the cubic case.) In the quartic case this recurrence
relation has an extra term in I(2e ), and hence I(e ),
1 4, is a basic integral. It can be expressed
in terms of symmetric integrals by setting a5= 1 and
b5= 0 in (19.29.8).
The other recurrence relation is
19.29.18
bq
jI(qel) =qX
r=0q
r
br
ldq r
ljI(rej),j;l= 1;2;:::;n ;
see Carlson (1999, (3.11)). An example that uses
(19.29.15){(19.29.18) is given in x19.34.
For an implementation by James FitzSimons of the
method for reducing I(m) to basic integrals and exten-
sive tables of such reductions, see Carlson (1999) and
Carlson and FitzSimons (2000).Another method of reduction is given in Gray
(2002). It depends primarily on multivariate recurrence
relations that replace one integral by two or more.
19.29(iii) Examples
The rst formula replaces (19.14.4){(19.14.10). De ne
Qj(t) =aj+bjt2,j= 1;2;and assume both Q's are
positive for 0y<t<x . Then
19.29.19Zx
ydtp
Q1(t)Q2(t)=RF
U2+a1b2;U2+a2b1;U2
;
19.29.20Zx
yt2dtp
Q1(t)Q2(t)
=1
3a1a2RD
U2+a1b2;U2+a2b1;U2
+ (xy=U );
and
19.29.21Zx
ydt
t2p
Q1(t)Q2(t)
=1
3b1b2RD
U2+a1b2;U2+a2b1;U2
+ (xyU) 1;
where
19.29.22
(x2 y2)U=xp
Q1(y)Q2(y) +yp
Q1(x)Q2(x):
If both square roots in (19.29.22) are 0, then the inde-
terminacy in the two preceding equations can be re-
moved by using (19.27.8) to evaluate the integral as
RG(a1b2;a2b1;0) multiplied either by 2=(b1b2) or by
2=(a1a2) in the cases of (19.29.20) or (19.29.21), re-
spectively. If x=1, thenUis found by taking the
limit. For example,
19.29.23Z1
ydtp
(t2+a2)(t2 b2)=RF
y2+a2;y2 b2;y2
:
Next, forj= 1;2, de neQj(t) =fj+gjt+hjt2,
and assume both Q's are positive for y<t<x . If each
has real zeros, then (19.29.4) may be simpler than
19.29.24Zx
ydtp
Q1(t)Q2(t)
= 4RF(U;U +D12+V;U +D12 V);
where
19.29.25
(x y)2U=S1S2;
Sj=q
Qj(x) +q
Qj(y)2
hj(x y)2;
Djl= 2fjhl+ 2hjfl gjgl; V =q
D2
12 D11D22:
(The variables of RFare real and nonnegative unless
bothQ's have real zeros and those of Q1interlace
those ofQ2.) IfQ1(t) = (a1+b1t)(a2+b2t), where
both linear factors are positive for y < t < x , and
514 Elliptic Integrals
Q2(t) =f2+g2t+h2t2, then (19.29.25) is modi ed
so that
19.29.26S1= (X1Y2+Y1X2)2;
Xj=p
aj+bjx; Yj=p
aj+bjy;
D12= 2a1a2h2+ 2b1b2f2 (a1b2+a2b1)g2;
D11= (a1b2 a2b1)2= d2
12;
with other quantities remaining as in (19.29.25). In the
cubic case, in which a2= 1,b2= 0, (19.29.26) reduces
further to
19.29.27
S1= (X1+Y1)2; D 12= 2a1h2 b1g2; D 11= b2
1:
For example, because t3 a3= (t a)(t2+at+a2),
we nd that when 0 ay<x
19.29.28Zx
ydtp
t3 a3
= 4RF
U;U 3a+ 2p
3a;U 3a 2p
3a
;
where
19.29.29
(x y)2U= (p
x a+py a)2
(+)2 (x y)2
;
=p
x2+ax+a2; =p
y2+ay+a2:
Lastly, de ne Q(t2) =f+gt2+ht4and assume Q(t2)
is positive and monotonic for y<t<x . Then
19.29.30Zx
ydtp
Q(t2)
= 2RF
U;U g+ 2p
fh;U g 2p
fh
;
where
19.29.31
(x y)2U=p
Q(x2) +p
Q(y2)2
h(x2 y2)2:
For example, if 0 yxanda40, then
19.29.32Zx
ydtp
t4+a4= 2RF
U;U + 2a2;U 2a2
;
where
19.29.33
(x y)2U=p
x4+a4+p
y4+a42
(x2 y2)2:
Applications
19.30 Lengths of Plane Curves
19.30(i) Ellipse
The arclength sof the ellipse
19.30.1 x=asin ; y =bcos , 0 2,witha>b , is given by
19.30.2 s=aZ
0p
1 k2sin2d:
When 0 1
2,
19.30.3
s=a=E( ;k)
=RF
c 1;c k2;c
1
3k2RD
c 1;c k2;c
;
where
19.30.4 k2= 1 (b2=a2); c = csc2 :
Cancellation on the second right-hand side of (19.30.3)
can be avoided by use of (19.25.10).
The length of the ellipse is
19.30.5L(a;b) = 4aE(k) = 8aRG
0;b2=a2;1
= 8RG
0;a2;b2
= 8abRG
0;a 2;b 2
;
showing the symmetry in aandb. Approximations and
inequalities for L(a;b) are given inx19.9(i).
Leta2andb2be replaced respectively by a2+and
b2+, where2( b2;1), to produce a family of confo-
cal ellipses. As increases, the eccentricity kdecreases
and the rate of change of arclength for a xed value of
is given by
19.30.6@s
@(1=k)=p
a2 b2F( ;k)
=p
a2 b2RF
c 1;c k2;c
,
k2= (a2 b2)=(a2+),c= csc2 .
19.30(ii) Hyperbola
The arclength sof the hyperbola
19.30.7 x=ap
t+ 1; y =bp
t, 0t<1,
is given by
19.30.8s=1
2Zy2=b2
0s
(a2+b2)t+b2
t(t+ 1)dt:
From (19.29.7), with a= 1 andb= 0,
19.30.9s=1
2I(e1) = 1
3a2b2RD
r;r+b2+a2;r+b2
+yr
r+b2+a2
r+b2,r=b4=y2.
Forsin terms of E( ;k),F( ;k), and an algebraic
term, see Byrd and Friedman (1971, p. 3). See Carlson
(1977b, Ex. 9.4-1 and (9.4-4)) for arclengths of hyper-
bolas and ellipses in terms of R athat di er only in the
sign ofb2.
19.31 Probability Distributions 515
19.30(iii) Bernoulli's Lemniscate
For 01
4, the arclength sof Bernoulli's lemnis-
cate
19.30.10 r2= 2a2cos (2), 02,
is given by
19.30.11
s= 2a2Zr
0dtp
4a4 t4=p
2a2RF(q 1;q;q+ 1),
q= 2a2=r2= sec(2),
or equivalently,
19.30.12s=aF
;1=p
2
,
= arcsinp
2=(q+ 1) = arccos(tan ).
The perimeter length Pof the lemniscate is given by
19.30.13P= 4p
2a2RF(0;1;2) =p
2a25:24411 51:::
= 4aK
1=p
2
=a7:41629 87::::
For other plane curves with arclength representable
by an elliptic integral see Greenhill (1892, p. 190) and
Bowman (1953, pp. 32{33).
19.31 Probability Distributions
RG(x;y;z ) andRF(x;y;z ) occur as the expectation val-
ues, relative to a normal probability distribution in R2
orR3, of the square root or reciprocal square root of
a quadratic form. More generally, let A(= [ar;s]) and
B(= [br;s]) be real positive-de nite matrices with n
rows andncolumns, and let 1;:::;nbe the eigen-
values of AB 1. Ifxis a column vector with elements
x1;x2;:::;xnand transpose xT, then
19.31.1 xTAx=nX
r=1nX
s=1ar;sxrxs;
and
19.31.2Z
Rn(xTAx)exp
xTBx
dx1dxn
=n=2
+1
2n
p
detB 1
2nR 1
2;:::;1
2;1;:::;n
,
> 1
2n.
x19.16(iii) shows that for n= 3 the incomplete cases of
RFandRGoccur when = 1=2 and= 1=2, respec-
tively, while their complete cases occur when n= 2.
For (19.31.2) and generalizations see Carlson
(1972b).19.32 Conformal Map onto a Rectangle
The function
19.32.1z(p) =RF(p x1;p x2;p x3);
withx1;x2;x3real constants, has di erential
19.32.2dz= 1
20
@3Y
j=1(p xj) 1=21
Adp,
=p>0; 0<ph(p xj)<,j= 1;2;3.
If
19.32.3 x1>x2>x3;
thenz(p) is a Schwartz{Christo el mapping of the open
upper-halfp-plane onto the interior of the rectangle in
thez-plane with vertices
19.32.4z(1) = 0;
z(x1) =RF(0;x1 x2;x1 x3) (>0);
z(x2) =z(x1) +z(x3);
z(x3) =RF(x3 x1;x3 x2;0)
= iRF(0;x1 x3;x2 x3):
Aspproceeds along the entire real axis with the upper
half-plane on the right, zdescribes the rectangle in the
clockwise direction; hence z(x3) is negative imaginary.
For further connections between elliptic integrals
and conformal maps, see Bowman (1953, pp. 44{85).
19.33 Triaxial Ellipsoids
19.33(i) Surface Area
The surface area of an ellipsoid with semiaxes a;b;c ,
and volume V= 4abc= 3 is given by
19.33.1 S= 3V RG
a 2;b 2;c 2
;
or equivalently,
19.33.2
S
2=c2+ab
sin
E( ;k) sin2 +F( ;k) cos2
,
abc,
where
19.33.3 cos =c
a; k2=a2(b2 c2)
b2(a2 c2):
Application of (19.16.23) transforms the last quan-
tity in (19.30.5) into a two-dimensional analog of
(19.33.1).
For additional geometrical properties of ellipsoids
(and ellipses), see Carlson (1964, p. 417).
516 Elliptic Integrals
19.33(ii) Potential of a Charged Conducting
Ellipsoid
If a conducting ellipsoid with semiaxes a;b;c bears an
electric charge Q, then the equipotential surfaces in the
exterior region are confocal ellipsoids:
19.33.4x2
a2++y2
b2++z2
c2+= 1,0.
The potential is
19.33.5V() =QRF
a2+;b2+;c2+
;
and the electric capacity C=Q=V (0) is given by
19.33.6 1=C=RF
a2;b2;c2
:
A conducting elliptic disk is included as the case
c= 0.
19.33(iii) Depolarization Factors
Let a homogeneous magnetic ellipsoid with semiaxes
a;b;c , volumeV= 4abc= 3, and susceptibility be
placed in a previously uniform magnetic eld Hparallel
to the principal axis with semiaxis c. The external eld
and the induced magnetization together produce a uni-
form eld inside the ellipsoid with strength H=(1+Lc ),
whereLcis the demagnetizing factor, given in cgs units
by
19.33.7Lc= 2abcZ1
0dp
(a2+)(b2+)(c2+)3
=V RD
a2;b2;c2
:
The same result holds for a homogeneous dielectric
ellipsoid in an electric eld. By (19.21.8),
19.33.8 La+Lb+Lc= 4;
whereLaandLbare obtained from Lcby permutation
ofa,b, andc. Expressions in terms of Legendre's inte-
grals, numerical tables, and further references are given
by Cronemeyer (1991).
19.33(iv) Self-Energy of an Ellipsoidal
Distribution
Ellipsoidal distributions of charge or mass are used to
model certain atomic nuclei and some elliptical galaxies.
Let the density of charge or mass be
19.33.9
(x;y;z ) =fp
(x2= 2) + (y2= 2) + (z2=
2)
;
where ; ;
are dimensionless positive constants. The
contours of constant density are a family of similar,
rather than confocal, ellipsoids. In suitable units the
self-energy of the distribution is given by
19.33.10
U=1
2Z
R6(x;y;z )(x0;y0;z0)dxdydzdx0dy0dz0
p
(x x0)2+ (y y0)2+ (z z0)2:Subject to mild conditions on fthis becomes
19.33.11U=1
2(
)2RF
2; 2;
2Z1
0(g(r))2dr;
where
19.33.12 g(r) = 4Z1
rf(t)tdt:
19.34 Mutual Inductance of Coaxial Circles
The mutual inductance Mof two coaxial circles of ra-
diusaandbwith centers at a distance hapart is given
in cgs units by
19.34.1
c2M
2=abZ2
0(h2+a2+b2 2abcos) 1=2cosd
= 2abZ1
1tdtp
(1 +t)(1 t)(a3 2abt)= 2abI(e5);
wherecis the speed of light, and in (19.29.11),
19.34.2a3=h2+a2+b2; a 5= 0; b 5= 1:
The method ofx19.29(ii) uses (19.29.18), (19.29.16),
and (19.29.15) to produce
19.34.3
2abI(e5) =a3I(0) I(e3) =a3I(0) r2
+r2
I( e3)
= 2ab(I(0) r2
I(e1 e3));
wherea1+b1t= 1 +tand
19.34.4 r2
=a32ab=h2+ (ab)2
is the square of the maximum (upper signs) or minimum
(lower signs) distance between the circles. Application
of (19.29.4) and (19.29.7) with = 1,a +b t= 1 t,
= 3, anda
+b
t= 1 yields
19.34.53c2
8abM= 3RF
0;r2
+;r2
2r2
RD
0;r2
+;r2
;
or, by (19.21.3),
19.34.6
c2
2M= (r2
++r2
)RF
0;r2
+;r2
4RG
0;r2
+;r2
:
A simpler form of the result is
19.34.7M= (2=c2)(a2)(b2)R 3
2 3
2;3
2;r2
+;r2
:
References for other inductance problems solvable
in terms of elliptic integrals are given in Grover (1946,
pp. 8 and 283).
19.35 Other Applications
19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis
of modular theorems of Ramanujan (Anderson et al.
(2000)); analysis of Selberg integrals (Van Diejen and
Spiridonov (2001)); use of Legendre's relation (19.7.1)
to compute to high precision (Borwein and Borwein
(1987, p. 26)).
Computation 517
19.35(ii) Physical
Elliptic integrals appear in lattice models of critical
phenomena (Guttmann and Prellberg (1993)); theories
of layered materials (Parkinson (1969));
uid dynam-
ics (Kida (1981)); string theory (Arutyunov and Stau-
dacher (2004)); astrophysics (Dexter and Agol (2009)).
Computation
19.36 Methods of Computation
19.36(i) Duplication Method
Numerical di erences between the variables of a sym-
metric integral can be reduced in magnitude by succes-
sive factors of 4 by repeated applications of the duplica-
tion theorem, as shown by (19.26.18). When the di er-
ences are moderately small, the iteration is stopped, the
elementary symmetric functions of certain di erences
are calculated, and a polynomial consisting of a xed
number of terms of the sum in (19.19.7) is evaluated.
ForRFthe polynomial of degree 7, for example, is
19.36.11 1
10E2+1
14E3+1
24E2
2 3
44E2E3
5
208E3
2+3
104E2
3+1
16E2
2E3;
where the elementary symmetric functions Esare de-
ned by (19.19.4). If (19.36.1) is used instead of its rst
ve terms, then the factor (3 r) 1=6in Carlson (1995,
(2.2)) is changed to (3 r) 1=8.
For a polynomial for both RDandRJseehttp:
//dlmf.nist.gov/19.36.i .
Example
Three applications of (19.26.18) yield
19.36.3 RF(1;2;4) =RF(z1;z2;z3);
where, in the notation of (19.19.7) with a= 1
2and
n= 3,
19.36.4
z1= 2:10985 99098 8 ;
z3= 2:15673 49098 8 ;
Z1= 0:00977 77253 5 ;z2= 2:12548 49098 8 ;
A= 2:13069 32432 1 ;
Z2= 0:00244 44313 4 ;
Z3= Z1 Z2= 0:01222 21566 9 ;
E2= 1:25480 1410 4; E 3= 2:921210 7:
The rst ve terms of (19.36.1) suce for
19.36.5 RF(1;2;4) = 0:68508 58166 ::::
All cases of RF,RC,RJ, andRDare computed
by essentially the same procedure (after transforming
Cauchy principal values by means of (19.20.14) and
(19.2.20)). Complex values of the variables are allowed,with some restrictions in the case of RJthat are suf-
cient but not always necessary. The computation is
slowest for complete cases. For details see Carlson
(1995, 2002) and Carlson and FitzSimons (2000). In
the Appendix of the last reference it is shown how to
computeRJwithout computing RCmore than once.
Because of cancellations in (19.26.21) it is advisable to
computeRGfromRFandRDby (19.21.10) or else to
usex19.36(ii).
Legendre's integrals can be computed from symmet-
ric integrals by using the relations in x19.25(i). Note the
remark following (19.25.11). If (19.25.9) is used when
0k21, cancellations may lead to loss of signi cant
gures when k2is close to 1 and > = 4, as shown
by Reinsch and Raab (2000). The cancellations can be
eliminated, however, by using (19.25.10).
Accurate values of F( ;k) E( ;k) fork2near 0
can be obtained from RDby (19.2.6) and (19.25.13).
19.36(ii) Quadratic Transformations
Complete cases of Legendre's integrals and symmet-
ric integrals can be computed with quadratic conver-
gence by the AGM method (including Bartky transfor-
mations), using the equations in x19.8(i) andx19.22(ii),
respectively.
The incomplete integrals RF(x;y;z ) andRG(x;y;z )
can be computed by successive transformations in which
two of the three variables converge quadratically to a
common value and the integrals reduce to RC, accom-
panied by two quadratically convergent series in the case
ofRG; compare Carlson (1965, xx5,6). (In Legendre's
notation the modulus kapproaches 0 or 1.) Let
19.36.62an+1=an+p
a2n c2n;
2cn+1=an p
a2n c2n=c2
n=(2an+1);
2tn+1=tn+p
t2n+c2n;
wheren= 0;1;2;:::, and
19.36.7 0<c0<a0; t 00; t2
0+a2
00; =1:
Then (19.22.18) implies that
19.36.8 RF
t2
n;t2
n+c2
n;t2
n+a2
n
is independent of n. Asn!1 ,cn,an, andtnconverge
quadratically to limits 0, M, andT, respectively; hence
19.36.9
RF
t2
0;t2
0+c2
0;t2
0+a2
0
=RF
T2;T2;T2+M2
=RC
T2+M2;T2
:
Ift0=a0and= 1, so thattn=an, then this pro-
cedure reduces to the AGM method for the complete
integral.
The step from nton+ 1 is an ascending Landen
transformation if = 1 (leading ultimately to a hyper-
bolic case of RC) or a descending Gauss transformation
518 Elliptic Integrals
if= 1 (leading to a circular case of RC). Ifx,y, and
zare permuted so that 0 x < y < z , then the com-
putation of RF(x;y;z ) is fastest if we make c2
0a2
0=2
by choosing = 1 wheny <(x+z)=2 or= 1 when
y(x+z)=2.
Example
We compute RF(1;2;4) by setting = 1,t0=c0= 1,
anda0=p
3. Then
19.36.10
c2
3= 6:6510 12; a2
3= 2:46209 30206 0 = M2;
t2
3= 1:46971 53173 1 = T2:
Hence
19.36.11
RF(1;2;4) =RC
T2+M2;T2
= 0:68508 58166 ;
in agreement with (19.36.5). Here RCis computed ei-
ther by the duplication algorithm in Carlson (1995) or
via (19.2.19).
For an error estimate and the corresponding pro-
cedure forRG(x;y;z ), see http://dlmf.nist.gov/19.
36.ii .
F( ;k) can be evaluated by using (19.25.5). E( ;k)
can be evaluated by using (19.25.7), and RDby using
(19.21.10), but cancellations may become signi cant.
Thompson (1997, pp. 499, 504) uses descending Lan-
den transformations for both F( ;k) andE( ;k). A
summary for F( ;k) is given in Gautschi (1975, x3).
For computation of K(k) andE(k) with complex ksee
Fettis and Caslin (1969) and Morita (1978).
(19.22.20) reduces to 0 = 0 if p=xorp=y, and
(19.22.19) reduces to 0 = 0 if z=xorz=y. Near
these points there will be loss of signi cant gures in
the computation of RJorRD.
Descending Gauss transformations of
; 2;k
(see (19.8.20)) are used in Fettis (1965) to compute a
large table (seex19.37(iii)). This method loses signif-
icant gures in if 2andk2are nearly equal unless
they are given exact values|as they can be for tables.
If 2=k2, then the method fails, but the function can
be expressed by (19.6.13) in terms of E( ;k), for which
Neuman (1969) uses ascending Landen transformations.
Computation of Legendre's integrals of all three
kinds by quadratic transformation is described by
Cazenave (1969, pp. 128{159, 208{230).
Quadratic transformations can be applied to com-
pute Bulirsch's integrals ( x19.2(iii)). The func-
tion cel(kc;p;a;b ) is computed by successive Bartky
transformations (Bulirsch and Stoer (1968), Bulirsch
(1969b)). The function el2( x;kc;a;b) is computed by
descending Landen transformations if xis real, or by
descending Gauss transformations if xis complex (Bu-
lirsch (1965a)). Remedies for cancellation when xis
real and near 0 are supplied in Midy (1975). See also
Bulirsch (1969a) and Reinsch and Raab (2000).Bulirsch (1969a,b) extend Bartky's transformation
to el3(x;kc;p) by expressing it in terms of the rst
incomplete integral, a complete integral of the third
kind, and a more complicated integral to which Bartky's
method can be applied. The cases k2
c=2p <1and
1<p<k2
c=2 require di erent treatment for numeri-
cal purposes, and again precautions are needed to avoid
cancellations.
19.36(iii) Via Theta Functions
Lee (1990) compares the use of theta functions for com-
putation of K(k),E(k), andK(k) E(k), 0k21,
with four other methods. Also, see Todd (1975) for a
special case of K(k). For computation of Legendre's
integral of the third kind, see Abramowitz and Stegun
(1964,xx17.7 and 17.8, Examples 15, 17, 19, and 20).
For integrals of the second and third kinds see Lawden
(1989,xx3.4{3.7).
19.36(iv) Other Methods
Numerical quadrature is slower than most methods for
the standard integrals but can be useful for elliptic in-
tegrals that have complicated representations in terms
of standard integrals. See x3.5.
For series expansions of Legendre's integrals see
x19.5. Faster convergence of power series for K(k) and
E(k) can be achieved by using (19.5.1) and (19.5.2) in
the right-hand sides of (19.8.12). A three-part computa-
tional procedure for
; 2;k
is described by Franke
(1965) for 2<1.
When the values of complete integrals are known,
addition theorems with ==2 (x19.11(ii)) ease the
computation of functions such as F( ;k) when1
2
is small and positive. Similarly, x19.26(ii) eases the com-
putation of functions such as RF(x;y;z ) whenx(>0) is
small compared with min( y;z). These special theorems
are also useful for checking computer codes.
19.37 Tables
19.37(i) Introduction
Only tables published since 1960 are included. For ear-
lier tables see Fletcher (1948), Lebedev and Fedorova
(1960), and Fletcher et al. (1962).
19.37(ii) Legendre's Complete Integrals
FunctionsK(k)andE(k)
Tabulated for k2= 0(:01)1 to 6D by Byrd and Friedman
(1971), to 15D for K(k) and 9D for E(k) by Abramowitz
and Stegun (1964, Chapter 17), and to 10D by Fettis
and Caslin (1964).
19.38 Approximations 519
Tabulated for k= 0(:01)1 to 10D by Fettis and
Caslin (1964), and for k= 0(:02)1 to 7D by Zhang and
Jin (1996, p. 673).
Tabulated for arcsin k= 0(1)90to 6D by Byrd
and Friedman (1971) and to 15D by Abramowitz and
Stegun (1964, Chapter 17).
FunctionsK(k),K0(k), andiK0(k)=K(k)
Tabulated with k=ReiforR= 0(:01)1 and=
0(1)90to 11D by Fettis and Caslin (1969).
Function exp( K0(k)=K(k))(=q(k))
Tabulated for k2= 0(:01)1 to 6D by Byrd and Friedman
(1971) and to 15D by Abramowitz and Stegun (1964,
Chapter 17).
Tabulated for arcsin k= 0(1)90to 6D by Byrd
and Friedman (1971) and to 15D by Abramowitz and
Stegun (1964, Chapter 17).
Tabulated for k2= 0(:001)1 to 8D by Bel akov et al.
(1962).
19.37(iii) Legendre's Incomplete Integrals
FunctionsF( ;k)andE( ;k)
Tabulated for = 0(5)90,k2= 0(:01)1 to 10D by
Fettis and Caslin (1964).
Tabulated for = 0(1)90,k2= 0(:01)1 to 7S
by Belakov et al. (1962). (F( ;k) is presented as
( ;0;k).)
Tabulated for = 0(5)90,k= 0(:01)1 to 10D by
Fettis and Caslin (1964).
Tabulated for = 0(5)90, arcsink= 0(1)90
to 6D by Byrd and Friedman (1971), for =
0(5)90, arcsink= 0(2)90and 5(10)85to 8D by
Abramowitz and Stegun (1964, Chapter 17), and for
= 0(10)90, arcsink= 0(5)90to 9D by Zhang and
Jin (1996, pp. 674{675).
Function
; 2;k
Tabulated (with di erent notation) for = 0(15)90,
2= 0(:1)1, arcsink= 0(15)90to 5D by Abramowitz
and Stegun (1964, Chapter 17), and for = 0(15)90,
2= 0(:1)1, arcsink= 0(15)90to 7D by Zhang and
Jin (1996, pp. 676{677).
Tabulated for = 5(5)80(2:5)90, 2=
1(:1) 0:1;0:1(:1)1,k2= 0(:05)0:9(:02)1 to 10D by
Fettis and Caslin (1964) (and warns of inaccuracies in
Selfridge and Max eld (1958) and Paxton and Rollin
(1959)).
Tabulated for = 0(1)90, 2=
0(:05)0:85;0:88(:02)0:94(:01)0:98(:005)1,k2= 0(:01)1
to 7S by Bel akov et al. (1962).
19.37(iv) Symmetric Integrals
FunctionsRF
x2;1;y2
andRG
x2;1;y2
Tabulated for x= 0(:1)1,y= 1(:2)6 to 3D by Nellis
and Carlson (1966).FunctionRF
a2;b2;c2
withabc= 1
Tabulated for = 0(:05)0:5(:1)1(:2)2(:5)5, cos(3
) =
1(:2)1 to 5D by Carlson (1961a). Here
2=2
3((lna)2+ (lnb)2+ (lnc)2), cos(3
) =
(4=3)(lna)(lnb)(lnc), anda;b;c are semiaxes of an
ellipsoid with the same volume as the unit sphere.
Check Values
For check values of symmetric integrals with real or com-
plex variables to 14S see Carlson (1995).
19.38 Approximations
Minimax polynomial approximations ( x3.11(i)) forK(k)
andE(k) in terms of m=k2with 0m < 1
can be found in Abramowitz and Stegun (1964, x17.3)
with maximum absolute errors ranging from 4 10 5to
210 8. Approximations of the same type for K(k)
andE(k) for 0< k1 are given in Cody (1965a)
with maximum absolute errors ranging from 4 10 5to
410 18. Cody (1965b) gives Chebyshev-series expan-
sions (x3.11(ii)) with maximum precision 25D.
Approximations for Legendre's complete or incom-
plete integrals of all three kinds, derived by Pad e ap-
proximation of the square root in the integrand, are
given in Luke (1968, 1970). They are valid over parts
of the complex kand planes. The accuracy is con-
trolled by the number of terms retained in the approxi-
mation; for real variables the number of signi cant g-
ures appears to be roughly twice the number of terms
retained, perhaps even for near=2 with the improve-
ments made in the 1970 reference.
19.39 Software
Seehttp://dlmf.nist.gov/19.39 .
References
General References
The main references used for writing this chapter are
Erd elyi et al. (1953b), Byerly (1888), Cazenave (1969),
and Byrd and Friedman (1971) for Legendre's integrals,
and Carlson (1977b) for symmetric integrals. For addi-
tional bibliographic reading see Cayley (1895), Green-
hill (1892), Legendre (1825{1832), Tricomi (1951), and
Whittaker and Watson (1927).
520 Elliptic Integrals
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x19.2 Bulirsch (1965a, 1969a,b), Bulirsch and Stoer
(1968). To prove (19.2.20) evaluate the two parts
of the Cauchy principal value (intervals (0 ; y )
and ( y+;1)) using Carlson (1977b, (8.2-2)),
and reduce the rst part to RCby Carlson (1977b,
(9.8-4)) with B=C. Apply (19.12.7) to both
parts as!0 and combine the two logarithms.
For (19.2.21) see (19.16.18) and put cos =vin
(19.23.8). For (19.2.22) put z=xin (19.23.5) and
interchange xandy.
x19.3 The graphics were produced at NIST.
x19.4 Cazenave (1969, p. 175). (19.4.1){(19.4.7) follow
by di erentiation of the de nitions in x19.2(ii).
(19.4.8) agrees also with Edwards (1954, vol. 1,
p. 402) and with expansion to rst order in k.
The term on the right side in Byrd and Friedman
(1971, 118.01) has the wrong sign.
x19.5 For (19.5.1){(19.5.4) put sin = 1 andt=pxin (19.2.4){(19.2.7). Then compare with
Erd elyi et al. (1953a, 2.1.3(10) and 2.1.1(2)) in the
rst three cases, and with Erd elyi et al. (1953a,
5.8.2(5) and 5.7.1(6)) in the fourth case. For
(19.5.5) and (19.5.6) see Kneser (1927, (12) and
p. 218); Byrd and Friedman (1971, 901.00) is in-
correct. (19.5.8) and (19.5.9) follow from Borwein
and Borwein (1987, (2.1.13) and (2.3.17), respec-
tively). For (19.5.10) iterate (19.8.12).
x19.6 For the rst line of (19.6.2) put =kin the
rst line of (19.25.2) and use the last line of
(19.25.1). For the second line of (19.6.2), and also
for (19.6.5), use (19.7.8) and (19.6.15). For the
rst line of (19.6.6) use (19.6.5) and (19.6.2). For
more detail as k2!1 seex19.12. For (19.6.7),
(19.6.8) use (19.2.4), (19.16.6), and (19.25.5). For
(19.6.9), (19.6.10) use (19.2.5). For (19.6.11){
(19.6.14) Byrd and Friedman (1971, 111.01 and
111.04, p. 10) also needs sin < 1. Start with
(19.25.14). For the second equation of (19.6.12)
use (19.20.8). For (19.6.13) use (19.16.5) with
(19.25.10) and (19.25.11).
x19.7 Three proofs of (19.7.1) are given in Duren
(1991). To prove it from (19.21.1) put z+
1 = 1=k2, use homogeneity, and apply the
penultimate equation in (19.25.1) twice. For(19.7.4){(19.7.7) see the penultimate paragraph
inx19.25(i). (19.7.8){(19.7.10) follow from the
change of parameter for the symmetric integral
of the third kind; see x19.21(iii) and (19.25.14).
x19.8 Cox (1984, 1985), Borwein and Borwein (1987,
Chapter 1), Cazenave (1969, pp. 114{127). To
prove the second equality in (19.8.4), put tan =p
t=g0. (19.8.7) is derived from (19.22.12) and
(19.25.14), and (19.8.9) is derived from (19.6.5)
and (19.8.7); see also Carlson (2002). For
(19.8.16) and (19.8.17) replace ( ;k) by ( 2;k2),
and then ( 1;k1) by ( ;k) in (19.8.11) and
(19.8.13). See also Hancock (1958, pp. 74{77) for
proof of (19.8.13) and (19.8.17).
x19.9 For (19.9.1) see Erd elyi et al. (1953b,
x13.8(9),(11)), (19.9.13), (19.6.12), and (19.6.15).
For (19.9.2) and (19.9.3) see Qiu and Vamana-
murthy (1996). For (19.9.4) see Barnard et al.
(2000, (6)); the rst inequality was given earlier
by Qiu and Shen (1997, Theorem 2). For (19.9.5)
see Lehto and Virtanen (1973, p. 62). For (19.9.6)
and (19.9.7) see (19.25.1) and (19.16.21) and then
apply Carlson (1966, (2.15)), in which H < H0
for 0< k1 in both cases. In (19.9.7) the up-
per bound 4 =, which is the smaller of the two
whenk20:855:::, is given by Anderson and
Vamanamurthy (1985). For (19.9.8) see (19.25.1),
Neuman (2003, (4.2)), and (19.24.9). For (19.9.9)
see (19.30.5). For (19.9.14) see (19.24.10) and
(19.25.5). For (19.9.15) and (19.9.16) see Carlson
and Gustafson (1985, (1.2), (1.22)).
x19.10 For (19.10.1) see (19.2.17). For (19.10.2) use
(19.6.8).
x19.11 Byerly (1888, pp. 243{245, 256{258), Edwards
(1954, v. 2, pp. 511{513), Cazenave (1969, pp. 83{
85). (19.11.5) can be derived from (19.26.9),
(19.25.26), and (19.11.1).
x19.12 For (19.12.1) and (19.12.2) see Cayley (1895,
p. 54) and Cazenave (1969, pp. 165{169). For
(19.12.4) and (19.12.5) use (19.25.2), (19.27.13),
and (19.6.5). For (19.12.6) and (19.12.7) see Carl-
son and Gustafson (1994, (22),(24)).
x19.14 For (19.14.1){(19.14.3) see Cazenave (1969,
pp. 286,276). For (19.14.4) use (19.29.19) and
(19.25.24).
x19.16 See Carlson (1977b, (6.8-6), Ex. 6.8-8, and (5.9-
1)). To prove (19.16.12) put t= csc2 csc2 in
the rst integral in (19.16.9). For (19.16.19) and
(19.16.23) see Carlson (1977b, (5.9-19) and (8.3-
4)). To derive (19.16.24) exchange subscripts 1
References 521
andnin Carlson (1963, (7.4)), put t=s=z1, and
use (19.16.19).
x19.17 The graphics were produced at NIST.
x19.18 (19.18.1) is derived from (19.16.1), (19.16.5),
and (19.18.4). (19.18.2) follows from (19.18.8).
For (19.18.4) and (19.18.5) put t= aand
c=a+a0in Carlson (1977b, (5.9-9),(5.9-10)).
(19.18.6) comes from (19.18.8) and (19.20.25). For
(19.18.8) and (19.18.11) see Carlson (1977b, (5.9-
2)). For (19.18.12){(19.18.17) see Carlson (1977b,
x5.4).
x19.19 To prove (19.19.2) expand the product in
(19.23.10) in powers of u. (19.19.3) is derived from
(19.16.11) and (19.19.2). For (19.19.5) see Carlson
(1979, (A.12)). For (19.19.6) compare (19.16.2)
and (19.16.9).
x19.20 In (19.20.2) put t= 1=ps+ 1; alternatively
use (19.29.19). For the second equality replace
t4bytand apply (5.12.1). For (19.20.3) use Carl-
son (1977b, Ex. 6.9-5 and p. 309) and (19.25.42).
For (19.20.4) use (19.20.5) and (19.16.3). For
(19.20.5) put z=yin (19.21.10). For (19.20.6)
substitute in (19.16.2) and (19.16.5). In (19.20.7)
see (19.27.12) for p!0+; forp!0
use (19.20.17) and (19.6.15). In (19.20.8) the
third equation is proved by partial fractions, and
also implies the rst two equations by (19.6.15).
For (19.20.9) put x= 0 in (19.20.13). For
(19.20.10) interchange xandzin (19.27.14) and
use (19.6.15). For (19.20.11) use (19.27.13),
(19.20.17), and (19.27.2). For (19.20.12) see
(19.27.11) and (19.21.12). For (19.20.13) let q=p
in (19.21.12). For (19.20.14) exchange xandzin
(19.21.12) and use (19.2.20). For the third equa-
tion in (19.20.18) put t=ytan2in (19.16.5); for
the fourth equation see (19.27.7). For (19.20.19)
see (19.27.8). For (19.20.20) and (19.20.21) use
(19.16.15), (19.16.9), and Carlson (1977b, Table
8.5-1). In (19.20.22) put t= 1=ps+ 1; alterna-
tively use (19.29.20). For the second equality re-
placet4bytand apply (5.12.1). For (19.20.23)
use Carlson (1977b, Ex. 6.9-5 and p. 309) and
(19.25.42). For (19.20.24){(19.20.26) see Carlson
(1977b, (6.2-1),(6.8-15)).
x19.21 To prove (19.21.1) see the text following
(19.21.6), use (19.20.10), and analytic continua-
tion. For (19.21.2) put x= 0 in (19.21.9). For
(19.21.3) put x= 0 in (19.21.11) and (19.21.10).
(19.21.6) is equivalent to Zill and Carlson (1970,
(7.15)). For (19.21.8) and (19.21.9) see Carlson
(1977b, (5.9-5),(5.9-6)) and (19.20.25). To obtain(19.21.7) eliminate RD(z;x;y ) between (19.21.8)
and (19.21.9), which follow from Carlson (1977b,
(5.9-5, (6.6-5), and (5.9-6)). For (19.21.10) see
Carlson (1977b, Table 9.3-1). To prove (19.21.11)
writext=(t+x) =x (x2=(t+x)) in (19.23.7)
and similarly for yandz. Then use (19.21.9). For
(19.21.12){(19.21.15) see Zill and Carlson (1970,
(4.6)).
x19.22 In (19.22.18), (19.22.21), and (19.22.20), put
z= 0 to obtain (19.22.1), (19.22.2), and (19.22.4),
respectively. (19.22.3) is derivable from (19.22.2)
and (19.21.3), or more directly by putting p=
yin (19.22.7). For (19.22.7) see Carlson
(1976, (4.14),(4.13)), where ( =4)RL(y;z;p ) =
RF(0;y;z) (p=3)RJ(0;y;z;p ). For (19.22.8){
(19.22.15) iterate the results given in x19.22(i);
see also (19.16.20), (19.16.23), and Carlson (2002,
Section 2). For (19.22.18) see Carlson (1964,
(5.13)). For (19.22.19) put p=zin (19.22.20).
For (19.22.20) see Zill and Carlson (1970, (5.7))
and Carlson (1990, (8.5)). For (19.22.21) see Carl-
son (1964, (5.16)). For (19.22.22) put z=yin
(19.22.18). In the ascending Landen case let k2=
(z2
+ z2
)=(z2
+ a2) andk2
1= (z2 y2)=(z2 x2)
to get the second equation in (19.8.11). In the de-
scending Gauss case let k2
1= (a2 z2
)=(a2 z2
+)
andk2= (z2 y2)=(z2 x2) to get the rst equa-
tion in (19.8.11).
x19.23 For (19.23.8) and (19.23.9) see Carlson (1977b,
Exercises 5.9-19, 5.9-20, and p. 306). By
x19.16(iii), (19.23.8) implies (19.23.1){(19.23.3),
and (19.23.9) implies (19.23.6). Use (19.23.8) to
integrate over in (19.23.6) and then permute
variables to prove (19.23.5). To prove (19.23.4)
putz= 0 in (19.23.5), relabel variables, and sub-
stitute cos= secht. For (19.23.7) and (19.23.10)
see Carlson (1977b, (9.1-9) and (6.8-2), respec-
tively).
x19.24 For (19.24.1){(19.24.3) use (19.9.1) and
(19.9.4). For (19.24.4) see (19.16.22) and Carl-
son (1966, (2.15)). For (19.24.3) see (19.30.5).
(19.24.8) is a special case of (19.24.13). For
(19.24.9) see Neuman (2003, (4.2)).
x19.25 (19.25.1), (19.25.2), and (19.25.3) are derived
from the incomplete cases. For (19.25.4) put c= 1
in (19.25.16). (19.25.5) and (19.25.7) come from
Carlson (1977b, (9.3-2) and (9.3-3)). For (19.25.6)
and (19.25.12) apply (19.18.4) to (19.25.5) and
(19.25.8), respectively. (19.25.8) and (19.25.15)
are special cases of (19.16.12). To get (19.25.9),
(19.25.10), and (19.25.11), let ( c 1;c k2;c) =
(x;y;z ) and eliminate RGbetween (19.25.7) and
522 Elliptic Integrals
each of the three forms of (19.25.10) obtained
by permuting x;yandz. For (19.25.13) com-
bine (19.2.6) and (19.25.9). For (19.25.14) see Zill
and Carlson (1970, (2.5)). For (19.25.16) sub-
stitute (19.25.14) in (19.7.8) and use (19.2.20).
For (19.25.19){(19.25.22) rewrite Bulirsch's inte-
grals (x19.2(iii)) in terms of Legendre's integrals,
then usex19.25(i) to convert them to R-functions.
For (19.25.24){(19.25.27) de ne c= csc2 , write
(x;y;z;p )/(z x) = (c 1;c k2;c;c 2), then
use (19.25.5), (19.25.9), (19.25.14), and (19.25.7)
to prove (19.25.24), (19.25.25), (19.25.26), and
(19.25.27), respectively. To prove (19.25.29) use
(cs;ds;ns) = (cn ;dn;1)=sn (suppressing vari-
ables (u,k)). For (19.25.30) see Carlson (2006a,
Comments following proof of Proposition 4.1).
For (19.25.31) see Carlson (2004, (1.8)). In
(19.25.32), (19.25.33), and (19.25.34), substitute
x= ps (u;k), sp (u;k), and pq ( u;k), respec-
tively, to recover (19.25.31). To prove (19.25.35)
use (23.6.36), with z=}(w) as prescribed in
the text that follows (23.6.36), substitute u=
t+}(w) and compare with (19.16.1). Then put
z=!jto obtain (19.25.38). For (19.25.37) and
(19.25.39) see Carlson (1964, (3.10) and (3.2)).
For (19.25.40) combine Erd elyi et al. (1953b,
xx13.12(22), 13.13(22)) and (19.25.35).
x19.26 Addition theorems (and therefore duplication
theorems) for the symmetric integrals are proved
by Zill and Carlson (1970, x8). For other proofs
of (19.26.1) see Carlson (1977b, x9.7) and Carl-
son (1978, Theorem 3). To prove (19.26.13)
use (19.2.9) to show that 2p
RC
2;2
=
ln
(+p
)=( p
)
, then apply this to all
three terms. To prove (19.26.14) put z=yin
(19.21.12) and use (19.20.8). For (19.26.17) put
= in (19.26.13) and use homogeneity. For
(19.26.18){(19.26.27) put =in the formulas
ofx19.26(i). For proofs of (19.26.18) not invoking
the addition theorem, see Carlson (1977b, x9.6)
and Carlson (1998, x2). Equations (19.26.25) and
(19.26.20) are degenerate cases of (19.26.18) and
(19.26.22), respectively.
x19.27 Carlson and Gustafson (1994). For (19.27.2)
see Carlson and Gustafson (1985).
x19.28 To prove (19.28.1){(19.28.3) from (19.28.4) use
x19.16(iii). To prove (19.28.4) expand the R-
function in powers of 1 tby (19.19.3), inte-
grate term by term, and use Erd elyi et al. (1953a,
2.8(46)). (19.28.5) is equivalent to (19.18.1). In
(19.28.6) let v=puand use Carlson (1963,(7.9)). In (19.28.7) substitute (19.16.2), change
the order of integration, and use (19.29.4). Use
Carlson (1963, (7.11)) and (19.16.20) to prove
(19.28.8) and (19.28.10). In the rst case Carl-
son (1977b, (5.9-21)) is needed; in the sec-
ond case put ( z1;z2;1;2) = (a2;b2;c2;d2),
use Carlson (1977b, (9.8-4)), and substitute
t= (ab=cd ) exp(2z). To prove (19.28.9) from
(19.28.10), put a= exp (ix) = 1=b,c= exp (iy) =
1=d, coshz= 1=sin, and on the right-hand side
use (19.22.1).
x19.29 For (19.29.4) see Carlson (1998, (3.6)). For
(19.29.7), a special case of (19.29.8), see also
Carlson (1987, (4.14)). For (19.29.8) see Carl-
son (1999, (4.10)) and Carlson (1988, (5.6)). For
(19.29.10) see Byrd and Friedman (1971, p. 76,
Eq. (234.13), and p. 74) for notation. Then use
Carlson (2006b, (3.2)) with ( p;q;r ) = (n;d;c ) for
reduction to RD. For (19.29.19){(19.29.33) take
t2as a new variable where appropriate. Then fac-
tor quadratic polynomials, use (19.29.4), and ap-
ply (19.22.18) to remove any complex quantities.
For (19.29.20) use (19.29.7) with a +b t=tand
a+bt= 1. For (19.29.21) use (19.29.7) with
a +b t= 1 anda+bt=t. With regard to
(19.29.28) see Carlson (1977a, p. 238).
x19.30 Carlson (1977b,x9.4 and Ex. 8.3-7, with solu-
tion on p. 312). For (19.30.5) see (19.25.1). For
(19.30.6) use (19.4.6).
x19.32 Carlson (1977b, pp. 234{235). For (19.32.2) use
(19.18.6).
x19.33 Carlson (1977b, pp. 271, 313, (9.4-10), and
Ex. 9.4-3) and Carlson (1961a). For other proofs
of (19.33.1) and (19.33.2) see Watson (1935b),
Bowman (1953, pp. 31{32), and Carlson (1964,
p. 417). For the rst equality in (19.33.7) see
Becker and Sauter (1964, p. 106).
x19.34 For (19.34.1) see Becker and Sauter (1964,
p. 194). For (19.34.7) see Carlson (1977b, Ex. 9.3-
2 and p. 313); alternatively, substitute Carlson
(1977b, (9.2-3) and (9.2-2)) in (19.34.6) and use
Carlson (1977b, Table 9.3-2).
x19.36 For the quadratic transformations see Carl-
son (1965, (3.1), (3.2), Sections 5, 6). To
obtain (19.36.6) and (19.36.8) from (19.22.18),
let (x2;y2;z2) = (t2
n;t2
n+c2
n;t2
n+a2
n) and
(a2;z2
;z2
+) = (t2
n+1;t2
n+1+c2
n+1;t2
n+1+a2
n+1).
Then use the expression for z2
a2from (19.22.17)
and the de nition of afrom (19.22.16).
Chapter 20
Theta Functions
W. P. Reinhardt1and P. L. Walker2
Notation 524
20.1 Special Notation . . . . . . . . . . . . . 524
Properties 524
20.2 De nitions and Periodic Properties . . . . 524
20.3 Graphics . . . . . . . . . . . . . . . . . . 525
20.4 Values at z= 0 . . . . . . . . . . . . . . 529
20.5 In nite Products and Related Results . . 529
20.6 Power Series . . . . . . . . . . . . . . . . 530
20.7 Identities . . . . . . . . . . . . . . . . . 530
20.8 Watson's Expansions . . . . . . . . . . . 531
20.9 Relations to Other Functions . . . . . . . 53220.10 Integrals . . . . . . . . . . . . . . . . . . 532
20.11 Generalizations and Analogs . . . . . . . 532
Applications 533
20.12 Mathematical Applications . . . . . . . . 533
20.13 Physical Applications . . . . . . . . . . . 533
Computation 534
20.14 Methods of Computation . . . . . . . . . 534
20.15 Tables . . . . . . . . . . . . . . . . . . . 534
20.16 Software . . . . . . . . . . . . . . . . . . 534
References 534
1University of Washington, Seattle, Washington.
2American University of Sharjah, Sharjah, United Arab Emirates.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 16), by L. M. Milne-Thomson.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
523
524 Theta Functions
Notation
20.1 Special Notation
(For other notation see pp. xiv and 873.)
m,n integers.
z(2C) the argument.
(2C) the lattice parameter, = >0.
q(2C) the nome, q=ei , 0<jqj<1. Since is
not a single-valued function of q, it is
assumed that is known, even when qis
speci ed. Most applications concern the
rectangular case< = 0,= >0, so that
0<q< 1 and andqare uniquely related.
q ei for 2R(resolving issues of choice of
branch).
S1=S2 set of all elements of S1, modulo elements of
S2. Thus two elements of S1=S2are
equivalent if they are both in S1and their
di erence is in S2. (For an example see
x20.12(ii).)
The main functions treated in this chapter are the
theta functions j(zj ) =j(z;q) wherej= 1;2;3;4
andq=ei . When is xed the notation is often ab-
breviated in the literature as j(z), or even as simply j,
it being then understood that the argument is the pri-
mary variable. Sometimes the theta functions are called
the Jacobian or classical theta functions to distinguish
them from generalizations; compare Chapter 21.
Primes on the symbols indicate derivatives with
respect to the argument of the function.
Other Notations
Jacobi's original notation: ( zj ), 1(zj ), H(zj ),
H1(zj ), respectively, for 4(uj ),3(uj ),1(uj ),
2(uj ), whereu=z=2
3(0j ). Here the symbol H de-
notes capital eta. See, for example, Whittaker and Wat-
son (1927, p. 479) and Copson (1935, pp. 405, 411).
Neville's notation: s(zj ),c(zj ),d(zj ),
n(zj ), respectively, for 2
3(0j )1(uj )/0
1(0j ),
2(uj )/2(0j ),3(uj )/3(0j ),4(uj )/4(0j ),
where again u=z=2
3(0j ). This notation simpli es
the relationship of the theta functions to Jacobian el-
liptic functions (x22.2); see Neville (1951).
McKean and Moll's notation: #j(zj ) =j(zj ),
j= 1;2;3;4. See McKean and Moll (1999, p. 125).
Additional notations that have been used in the
literature are summarized in Whittaker and Watson
(1927, p. 487).Properties
20.2 De nitions and Periodic Properties
20.2(i) Fourier Series
20.2.11(zj ) =1(z;q)
= 21X
n=0( 1)nq(n+1
2)2sin((2n+ 1)z);
20.2.22(zj ) =2(z;q) = 21X
n=0q(n+1
2)2cos((2n+ 1)z);
20.2.33(zj ) =3(z;q) = 1 + 21X
n=1qn2cos(2nz);
20.2.44(zj ) =4(z;q) = 1 + 21X
n=1( 1)nqn2cos(2nz):
Corresponding expansions for 0
j(zj ),j= 1;2;3;4,
can be found by di erentiating (20.2.1){(20.2.4) with
respect toz.
20.2(ii) Periodicity and Quasi-Periodicity
For xed , eachj(zj ) is an entire function of zwith
period 2;1(zj ) is odd in zand the others are even.
For xedz, each of1(zj )/sinz,2(zj )/cosz,3(zj ),
and4(zj ) is an analytic function of for= >0, with
a natural boundary = = 0, and correspondingly, an an-
alytic function of qforjqj<1 with a natural boundary
jqj= 1.
The four points (0 ;; + ; ) are the vertices of
thefundamental parallelogram in thez-plane; see Figure
20.2.1. The points
20.2.5 zm;n= (m+n ),m;n2Z,
are the lattice points . The theta functions are quasi-
periodic on the lattice:
20.2.6
1(z+ (m+n )j ) = ( 1)m+nq n2e 2inz1(zj );
20.2.7
2(z+ (m+n )j ) = ( 1)mq n2e 2inz2(zj );
20.2.8
3(z+ (m+n )j ) =q n2e 2inz3(zj );
20.2.9
4(z+ (m+n )j ) = ( 1)nq n2e 2inz4(zj ):
20.3 Graphics 525
Figure 20.2.1 :z-plane. Fundamental parallelogram. Left-hand diagram is the rectangular case ( purely imaginary);
right-hand diagram is the general case. zeros of1(zj ),zeros of2(zj ),Nzeros of3(zj ),zeros of4(zj ).
20.2(iii) Translation of the Argument by
Half-Periods
With
20.2.10 MM(zj ) =eiz+(i =4);
20.2.111(zj ) = 2
z+1
2
= iM 4
z+1
2
= iM 3
z+1
2+1
2
;
20.2.122(zj ) =1
z+1
2
=M 3
z+1
2
=M 4
z+1
2+1
2
;
20.2.133(zj ) =4
z+1
2
=M 2
z+1
2
=M 1
z+1
2+1
2
;
20.2.144(zj ) =3
z+1
2
= iM 1
z+1
2
=iM 2
z+1
2+1
2
:
20.2(iv)z-Zeros
Form;n2Z, thez-zeros ofj(zj ),j= 1;2;3;4,
are (m+n ), (m+1
2+n ), (m+1
2+ (n+1
2) ),
(m+ (n+1
2) )respectively.20.3 Graphics
20.3(i)-Functions: Real Variable and Real
Nome
See Figures 20.3.1{20.3.13.
Figure 20.3.1 :j(x;0:15), 0x2,j= 1;2;3;4.
Figure 20.3.2 :1(x;q ), 0x2,q= 0.05, 0.5, 0.7,
0.9. ForqqDedekind,1(x;q ) is convex in xfor
0< x < 1. HereqDedekind=e y0= 0:19 approx-
imately, where y=y0corresponds to the maximum
value of Dedekind's eta function (iy) as depicted in
Figure 23.16.1.
Figure 20.3.3 :2(x;q ), 0x2,q= 0.05, 0.5, 0.7,
0.9.
526 Theta Functions
Figure 20.3.4 :3(x;q ), 0x2,q= 0.05, 0.5, 0.7,
0.9.
Figure 20.3.5 :4(x;q ), 0x2,q= 0.05, 0.5, 0.7,
0.9.
Figure 20.3.6 :1(x;q), 0q1,x= 0, 0.4, 5, 10, 40.
Figure 20.3.7 :2(x;q), 0q1,x= 0, 0.4, 5, 10, 40.
Figure 20.3.8 :3(x;q), 0q1,x= 0, 0.4, 5, 10, 40.
Figure 20.3.9 :4(x;q), 0q1,x= 0, 0.4, 5, 10, 40.
20.3 Graphics 527
Figure 20.3.10 :1(x;q ), 0x2, 0q0:99.
Figure 20.3.11 :2(x;q ), 0x2, 0q0:99.
Figure 20.3.12 :3(x;q ), 0x2, 0q0:99.
Figure 20.3.13 :4(x;q ), 0x2, 0q0:99.
20.3(ii)-Functions: Complex Variable and Real Nome
See Figures 20.3.14{20.3.17. In these graphics, height corresponds to the absolute value of the function and color to
the phase. See also p. xiv.
Figure 20.3.14 :1(x+iy;0:12), 1x1, 1
y2:3.
Figure 20.3.15 :2(x+iy;0:12), 1x1, 1
y2:3.
528 Theta Functions
Figure 20.3.16 :3(x+iy;0:12), 1x1, 1
y1:5.
Figure 20.3.17 :4(x+iy;0:12), 1x1, 1
y1:5.
20.3(iii)-Functions: Real Variable and Complex Lattice Parameter
See Figures 20.3.18{20.3.21. In these graphics this subsection, height corresponds to the absolute value of the function
and color to the phase. See also p. xiv.
Figure 20.3.18 :1(0:1ju+iv), 1u1, 0:005v
0:5. The value 0.1 of zis chosen arbitrarily since 1
vanishes identically when z= 0.
Figure 20.3.19 :2(0ju+iv), 1u1, 0:005v
0:1.
Figure 20.3.20 :3(0ju+iv), 1u1, 0:005v
0:1.
Figure 20.3.21 :4(0ju+iv), 1u1, 0:005v
0:1.
20.4 Values at z= 0 529
20.4 Values at z= 0
20.4(i) Functions and First Derivatives
20.4.11(0;q) =0
2(0;q) =0
3(0;q) =0
4(0;q) = 0;
20.4.20
1(0;q) = 2q1=41Y
n=1
1 q2n3;
20.4.32(0;q) = 2q1=41Y
n=1
1 q2n
1 +q2n2;
20.4.43(0;q) =1Y
n=1
1 q2n
1 +q2n 12;
20.4.54(0;q) =1Y
n=1
1 q2n
1 q2n 12:
Jacobi's Identity
20.4.60
1(0;q) =2(0;q)3(0;q)4(0;q):
20.4(ii) Higher Derivatives
20.4.700
1(0;q) =000
2(0;q) =000
3(0;q) =000
4(0;q) = 0:
20.4.8000
1(0;q)
0
1(0;q)= 1 + 241X
n=1q2n
(1 q2n)2:
20.4.900
2(0;q)
2(0;q)= 1 81X
n=1q2n
(1 +q2n)2;
20.4.1000
3(0;q)
3(0;q)= 81X
n=1q2n 1
(1 +q2n 1)2;
20.4.1100
4(0;q)
4(0;q)= 81X
n=1q2n 1
(1 q2n 1)2:
20.4.12000
1(0;q)
0
1(0;q)=00
2(0;q)
2(0;q)+00
3(0;q)
3(0;q)+00
4(0;q)
4(0;q):
20.5 In nite Products and Related Results
20.5(i) Single Products
20.5.1
1(z;q)
= 2q1=4sinz1Y
n=1
1 q2n
1 2q2ncos(2z) +q4n
;
20.5.2
2(z;q)
= 2q1=4cosz1Y
n=1
1 q2n
1 + 2q2ncos(2z) +q4n
;
20.5.3
3(z;q) =1Y
n=1
1 q2n
1 + 2q2n 1cos(2z) +q4n 2
;20.5.4
4(z;q) =1Y
n=1
1 q2n
1 2q2n 1cos(2z) +q4n 2
:
20.5.5
1(zj ) =0
1(0j ) sinz1Y
n=1sin(n +z) sin(n z)
sin2(n );
20.5.6
2(zj ) =2(0j ) cosz1Y
n=1cos(n +z) cos(n z)
cos2(n );
20.5.7
3(zj )
=3(0j )1Y
n=1cos
(n 1
2) +z
cos
(n 1
2) z
cos2
(n 1
2) ;
20.5.8
4(zj )
=4(0j )1Y
n=1sin
(n 1
2) +z
sin
(n 1
2) z
sin2
(n 1
2) :
Jacobi's Triple Product
20.5.9
3(zj ) =1X
n= 1p2nqn2
=1Y
n=1
1 q2n
1 +q2n 1p2
1 +q2n 1p 2
;
wherep=eiz,q=ei .
20.5(ii) Logarithmic Derivatives
Whenj=zj<= ,
20.5.10
0
1(z;q)
1(z;q) cotz= 4 sin(2z)1X
n=1q2n
1 2q2ncos(2z) +q4n
= 41X
n=1q2n
1 q2nsin(2nz);
20.5.11
0
2(z;q)
2(z;q)+ tanz= 4 sin(2z)1X
n=1q2n
1 + 2q2ncos(2z) +q4n
= 41X
n=1( 1)nq2n
1 q2nsin(2nz):
The left-hand sides of (20.5.10) and (20.5.11) are re-
placed by their limiting values when cot zor tanzare
unde ned.
Whenj=zj<1
2= ,
20.5.12
0
3(z;q)
3(z;q)= 4 sin(2z)1X
n=1q2n 1
1 + 2q2n 1cos(2z) +q4n 2
= 41X
n=1( 1)nqn
1 q2nsin(2nz);
530 Theta Functions
20.5.13
0
4(z;q)
4(z;q)= 4 sin(2z)1X
n=1q2n 1
1 2q2n 1cos(2z) +q4n 2
= 41X
n=1qn
1 q2nsin(2nz):With the given conditions the in nite series in
(20.5.10){(20.5.13) converge absolutely and uniformly
in compact sets in the z-plane.
20.5(iii) Double Products
20.5.14 1(zj ) =z0
1(0j ) lim
N!1NY
n= Nlim
M!1MY
m= M
jmj+jnj6=0
1 +z
(m+n )
;
20.5.15 2(zj ) =2(0j ) lim
N!1NY
n= Nlim
M!1MY
m=1 M
1 +z
(m 1
2+n )
;
20.5.16 3(zj ) =3(0j ) lim
N!1NY
n=1 Nlim
M!1MY
m=1 M
1 +z
(m 1
2+ (n 1
2) )
;
20.5.17 4(zj ) =4(0j ) lim
N!1NY
n=1 Nlim
M!1MY
m= M
1 +z
(m+ (n 1
2) )
:
These double products are not absolutely convergent;
hence the order of the limits is important. The order
shown is in accordance with the Eisenstein convention
(Walker (1996,x0.3)).
20.6 Power Series
Assume
20.6.1 jzj<minjzm;nj;
wherezm;nis given by (20.2.5) and the minimum is for
m;n2Z, exceptm=n= 0. Then
20.6.21(zj ) =z0
1(0j ) exp0
@ 1X
j=11
2j2j( )z2j1
A;
20.6.32(zj ) =2(0j ) exp0
@ 1X
j=11
2j 2j( )z2j1
A;
20.6.43(zj ) =3(0j ) exp0
@ 1X
j=11
2j 2j( )z2j1
A;
20.6.54(zj ) =4(0j ) exp0
@ 1X
j=11
2j
2j( )z2j1
A:
Here the coecients are given by
20.6.62j( ) =1X
n= 11X
m= 1
jmj+jnj6=0(m+n ) 2j;
20.6.7 2j( ) =1X
n= 11X
m= 1(m 1
2+n ) 2j;20.6.8 2j( ) =1X
n= 11X
m= 1(m 1
2+ (n 1
2) ) 2j;
20.6.9
2j( ) =1X
n= 11X
m= 1(m+ (n 1
2) ) 2j;
and satisfy
20.6.10 2j( ) = 22j2j(2 ) 2j( );
2j( ) = 22j
2j(2 )
2j( ):
In the double series the order of summation is impor-
tant only when j= 1. For further information on 2j
seex23.9: since the double sums in (20.6.6) and (23.9.1)
are the same, we have 2n=cn=(2n 1) whenn2.
20.7 Identities
20.7(i) Sums of Squares
20.7.1
2
3(0;q)2
3(z;q) =2
4(0;q)2
4(z;q) +2
2(0;q)2
2(z;q);
20.7.2
2
3(0;q)2
4(z;q) =2
2(0;q)2
1(z;q) +2
4(0;q)2
3(z;q);
20.7.3
2
2(0;q)2
4(z;q) =2
3(0;q)2
1(z;q) +2
4(0;q)2
2(z;q);
20.7.4
2
2(0;q)2
3(z;q) =2
4(0;q)2
1(z;q) +2
3(0;q)2
2(z;q):
Also
20.7.5 4
3(0;q) =4
2(0;q) +4
4(0;q):
20.8 Watson's Expansions 531
20.7(ii) Addition Formulas
20.7.62
4(0;q)1(w+z;q)1(w z;q)
=2
3(w;q)2
2(z;q) 2
2(w;q)2
3(z;q);
20.7.72
4(0;q)2(w+z;q)2(w z;q)
=2
4(w;q)2
2(z;q) 2
1(w;q)2
3(z;q);
20.7.82
4(0;q)3(w+z;q)3(w z;q)
=2
4(w;q)2
3(z;q) 2
1(w;q)2
2(z;q);
20.7.92
4(0;q)4(w+z;q)4(w z;q)
=2
3(w;q)2
3(z;q) 2
2(w;q)2
2(z;q):
For these and similar formulas see Lawden (1989, x1.4)
and Whittaker and Watson (1927, pp. 487{488).
20.7(iii) Duplication Formula
20.7.101(2z;q) = 21(z;q)2(z;q)3(z;q)4(z;q)
2(0;q)3(0;q)4(0;q):
20.7(iv) Transformations of Nome
20.7.111(z;q)2(z;q)
1(2z;q2)=3(z;q)4(z;q)
4(2z;q2)=4
0;q2
;
20.7.12
1
z;q2
4
z;q2
1(z;q)=2
z;q2
3
z;q2
2(z;q)=1
22(0;q):20.7(v) Watson's Identities
20.7.13
1(z;q)1(w;q) =3
z+w;q2
2
z w;q2
2
z+w;q2
3
z w;q2
;
20.7.14
3(z;q)3(w;q) =3
z+w;q2
3
z w;q2
+2
z+w;q2
2
z w;q2
:
20.7(vi) Landen Transformations
With
20.7.15 AA( ) = 1/4(0j2 );
20.7.161(2zj2 ) =A1(zj )2(zj );
20.7.172(2zj2 ) =A1 1
4 z
1 1
4+z
;
20.7.183(2zj2 ) =A3 1
4 z
3 1
4+z
;
20.7.194(2zj2 ) =A3(zj )4(zj ):
Next, with
20.7.20BB( ) = 1
3(0j )4(0j )3 1
4
;
20.7.21 1(4zj4 ) =B1(zj )1 1
4 z
1 1
4+z
2(zj );
20.7.22 2(4zj4 ) =B2 1
8 z
2 1
8+z
2 3
8 z
2 3
8+z
;
20.7.23 3(4zj4 ) =B3 1
8 z
3 1
8+z
3 3
8 z
3 3
8+z
;
20.7.24 4(4zj4 ) =B4(zj )4 1
4 z
4 1
4+z
3(zj ):
20.7(vii) Derivatives of Ratios of Theta
Functions
20.7.25d
dz2(zj )
4(zj )
= 2
3(0j )1(zj )3(zj )
2
4(zj ):
See Lawden (1989, pp. 19{20). This reference also gives
ten additional identities involving permutations of the
four theta functions.
20.7(viii) Transformations of Lattice Parameter
20.7.26 1(zj + 1) =ei=41(zj );
20.7.27 2(zj + 1) =ei=42(zj );
20.7.28 3(zj + 1) =4(zj );
20.7.29 4(zj + 1) =3(zj ):In the following equations 0= 1= , and all square
roots assume their principal values.
20.7.30 ( i )1=21(zj ) = iexp
i 0z2=
1(z 0j 0);
20.7.31 ( i )1=22(zj ) = exp
i 0z2=
4(z 0j 0);
20.7.32 ( i )1=23(zj ) = exp
i 0z2=
3(z 0j 0);
20.7.33 ( i )1=24(zj ) = exp
i 0z2=
2(z 0j 0):
These are examples of modular transformations; see
x23.15.
20.8 Watson's Expansions
20.8.1
2(0;q)3(z;q)4(z;q)
2(z;q)= 21X
n= 1( 1)nqn2ei2nz
q ne iz+qneiz:
See Watson (1935a). This reference and Bellman (1961,
pp. 46{47) include other expansions of this type.
532 Theta Functions
20.9 Relations to Other Functions
20.9(i) Elliptic Integrals
Withkde ned by
20.9.1 k=2
2(0j )=2
3(0j )
and the notation of x19.2(ii), the complete Legendre in-
tegrals of the rst kind may be expressed as theta func-
tions:
20.9.2K(k) =1
22
3(0j ); K0(k) = i K(k);
together with (22.2.1).
In the case of the symetric integrals, with the nota-
tion ofx19.16(i) we have
20.9.3RF2
2(z;q)
2
2(0;q);2
3(z;q)
2
3(0;q);2
4(z;q)
2
4(0;q)
=0
1(0;q)
1(z;q)z;
20.9.4 RF
0;4
3(0;q);4
4(0;q)
=1
2;
20.9.5 exp
RF
0;k2;1
RF
0;k02;1!
=q:
20.9(ii) Elliptic Functions and Modular
Functions
Seexx22.2 and 23.6(i) for the relations of Jacobian and
Weierstrass elliptic functions to theta functions.
The relations (20.9.1) and (20.9.2) between kand
(orq) are solutions of Jacobi's inversion problem ;
see Baker (1995) and Whittaker and Watson (1927,
pp. 480{485).
As a function of ,k2is the elliptic modular func-
tion; see Walker (1996, Chapter 7) and (23.15.2),
(23.15.6).
20.9(iii) Riemann Zeta Function
See Koblitz (1993, Ch. 2, x4) and Titchmarsh (1986b,
pp. 21{22). See also xx20.10(i) and 25.2.
20.10 Integrals
20.10(i) Mellin Transforms with respect to the
Lattice Parameter
Letsbe a constant such that <s>2. Then
20.10.1Z1
0xs 12
0 ix2
dx= 2s(1 2 s) s=2 1
2s
(s);
20.10.2Z1
0xs 1(3
0 ix2
1)dx= s=2 1
2s
(s);
20.10.3Z1
0xs 1(1 4
0 ix2
)dx
= (1 21 s) s=2 1
2s
(s):
Here(s) again denotes the Riemann zeta function
(x25.2).
For further results see Oberhettinger (1974, pp. 157{
159).20.10(ii) Laplace Transforms with respect to
the Lattice Parameter
Lets,`, and be constants such that <s >0,` >0,
and sinhj j`. Then
20.10.4Z1
0e st1
2` it
`2
dt
=Z1
0e st2(1 + )
2` it
`2
dt
= `pssinh
ps
sech
`ps
;
20.10.5Z1
0e st3(1 + )
2` it
`2
dt
=Z1
0e st4
2` it
`2
dt
=`pscosh
ps
csch
`ps
:
For corresponding results for argument derivatives of
the theta functions see Erd elyi et al. (1954a, pp. 224{
225) or Oberhettinger and Badii (1973, p. 193).
20.10(iii) Compendia
For further integrals of theta functions see Erd elyi et al.
(1954a, pp. 61{62 and 339), Prudnikov et al. (1990,
pp. 356{358), Prudnikov et al. (1992a,x3.41), and Grad-
shteyn and Ryzhik (2000, pp. 627{628).
20.11 Generalizations and Analogs
20.11(i) Gauss Sum
For relatively prime integers m;n withn > 0 andmn
even, the Gauss sum G(m;n) is de ned by
20.11.1 G(m;n) =n 1X
k=0e ik2m=n;
see Lerch (1903). It is a discrete analog of theta func-
tions. If both m;n are positive, then G(m;n) allows
inversion of its arguments as a modular transformation
(compare (23.15.3) and (23.15.4)):
20.11.2
1pnG(m;n) =1pnn 1X
k=0e ik2m=n
=e i=4
pmm 1X
j=0eij2n=m=e i=4
pmG( n;m):
This is the discrete analog of the Poisson identity
(x1.8(iv)).
Applications 533
20.11(ii) Ramanujan's Theta Function and
q-Series
Ramanujan's theta function f(a;b) is de ned by
20.11.3f(a;b) =1X
n= 1an(n+1)=2bn(n 1)=2;
wherea;b2Candjabj<1. With the substitutions
a=qe2iz,b=qe 2iz, withq=ei , we have
20.11.4 f(a;b) =3(zj ):
In the case z= 0 identities for theta functions be-
come identities in the complex variable q, withjqj<1,
that involve rational functions, power series, and contin-
ued fractions; see Adiga et al. (1985), McKean and Moll
(1999, pp. 156{158), and Andrews et al. (1988,x10.7).
20.11(iii) Ramanujan's Change of Base
As inx20.11(ii), the modulus kof elliptic integrals
(x19.2(ii)), Jacobian elliptic functions ( x22.2), and
Weierstrass elliptic functions ( x23.6(ii)) can be ex-
panded inq-series via (20.9.1). However, in this case
qis no longer regarded as an independent complex vari-
able within the unit circle, because kis related to the
variable = (k) of the theta functions via (20.9.2).
This is Jacobi's inversion problem of x20.9(ii).
The rst of equations (20.9.2) can also be written
20.11.52F1 1
2;1
2; 1;k2
=2
3(0j );
seex19.5. Similar identities can be constructed for
2F1 1
3;2
3; 1;k2
,2F1 1
4;3
4; 1;k2
, and 2F1 1
6;5
6; 1;k2
.
These results are called Ramanujan's changes of base .
Each provides an extension of Jacobi's inversion prob-
lem. See Berndt et al. (1995) and Shen (1998). For
applications to rapidly convergent expansions for see
Chudnovsky and Chudnovsky (1988), and for applica-
tions in the construction of elliptic-hypergeometric se-
riessee Rosengren (2004).
20.11(iv) Theta Functions with Characteristics
Multidimensional theta functions with characteristics
are de ned inx21.2(ii) and their properties are described
inxx21.3(ii), 21.5(ii), and 21.6. For specialization to the
one-dimensional theta functions treated in the present
chapter, see Rauch and Lebowitz (1973) and x21.7(iii).Applications
20.12 Mathematical Applications
20.12(i) Number Theory
For applications of 3(0;q) to problems involving sums
of squares of integers see x27.13(iv), and for exten-
sions see Estermann (1959), Serre (1973, pp. 106{109),
Koblitz (1993, pp. 176{177), and McKean and Moll
(1999, pp. 142{143).
For applications of Jacobi's triple product (20.5.9)
to Ramanujan's (n) function and Euler's pentagonal
numbers see Hardy and Wright (1979, pp. 132{160) and
McKean and Moll (1999, pp. 143{145). For an applica-
tion of a generalization in ane root systems see Mac-
donald (1972).
20.12(ii) Uniformization and Embedding of
Complex Tori
For the terminology and notation see McKean and Moll
(1999, pp. 48{53).
The space of complex tori C=(Z+ Z) (that is,
the set of complex numbers zin which two of these
numbersz1andz2are regarded as equivalent if there
exist integers m;n such thatz1 z2=m+ n) is
mapped into the projective space P3via the identi ca-
tionz!(1(2zj );2(2zj );3(2zj );4(2zj )). Thus
theta functions \uniformize" the complex torus. This
ability to uniformize multiply-connected spaces (mani-
folds), or multi-sheeted functions of a complex variable
(Riemann (1899), Rauch and Lebowitz (1973), Siegel
(1988)) has led to applications in string theory (Green
et al. (1988a,b), Krichever and Novikov (1989)), and
also in statistical mechanics (Baxter (1982)).
20.13 Physical Applications
The functions j(zj ),j= 1;2;3;4, provide periodic
solutions of the partial di erential equation
20.13.1 @(zj )/@ =@2(zj )
@z2;
with= i=4.
For =it, with ;t;z real, (20.13.1) takes the form
of a real-time tdi usion equation
20.13.2 @/@t= @2
@z2;
with di usion constant ==4. Letz; ;t2R. Then
the nonperiodic Gaussian
20.13.3 g(z;t) =r
4 texp
z2
4 t
is also a solution of (20.13.2), and it approaches a Dirac
delta (x1.17) att= 0. These two apparently di erent
534 Theta Functions
solutions di er only in their normalization and bound-
ary conditions. From (20.2.3), (20.2.4), (20.7.32), and
(20.7.33),
20.13.4r
4 t1X
n= 1e (n+z)2=(4 t)=3(zji4 t= );
and
20.13.5r
4 t1X
n= 1( 1)ne (n+z)2=(4 t)=4(zji4 t= ):
Thus the classical theta functions are \periodized", or
\anti-periodized", Gaussians; see Bellman (1961, pp. 18,
19). Theta-function solutions to the heat di usion equa-
tion with simple boundary conditions are discussed in
Lawden (1989, pp. 1{3), and with more general bound-
ary conditions in K orner (1989, pp. 274{281).
In the singular limit = !0+, the functions j(zj ),
j= 1;2;3;4, become integral kernels of Feynman path
integrals (distribution-valued Green's functions); see
Schulman (1981, pp. 194{195). This allows analytic
time propagation of quantum wave-packets in a box, or
on a ring, as closed-form solutions of the time-dependent
Schr odinger equation.
Computation
20.14 Methods of Computation
The Fourier series of x20.2(i) usually converge rapidly
because of the factors q(n+1
2)2orqn2, and provide a
convenient way of calculating values of j(zj ). Simi-
larly, their z-di erentiated forms provide a convenient
way of calculating the corresponding derivatives. For
instance, the rst three terms of (20.2.1) give the value
of1(2 iji) (=1(2 i;e )) to 12 decimal places.
For values ofjqjnear 1 the transformations of
x20.7(viii) can be used to replace with a value that
has a larger imaginary part and hence a smaller value of
jqj. For instance, to nd 3(z;0:9) we use (20.7.32) with
q= 0:9 =ei , = iln(0:9)=. Then 0= 1= =
i=ln(0:9) andq0=ei 0= exp
2=ln(0:9)
=
(2:07:::)10 41. Hence the rst term of the series
(20.2.3) for 3(z 0j 0) suces for most purposes. In
theory, starting from any value of , a nite number
of applications of the transformations ! + 1 and
! 1= will result in a value of with= p
3=2;
seex23.18. In practice a value with, say, = 1=2,
jqj0:2, is found quickly and is satisfactory for numer-
ical evaluation.20.15 Tables
Theta functions are tabulated in Jahnke and Emde
(1945, p. 45). This reference gives j(x;q),j= 1;2;3;4,
and their logarithmic x-derivatives to 4D for x= =
0(:1)1, = 0(9)90, where is the modular angle
given by
20.15.1 sin =2
2(0;q)=2
3(0;q) =k:
Spenceley and Spenceley (1947) tabulates
1(x;q)=2(0;q),2(x;q)=2(0;q),3(x;q)=4(0;q),
4(x;q)=4(0;q) to 12D for u= 0(1)90, = 0(1)89,
whereu= 2x=(2
3(0;q)) and is de ned by (20.15.1),
together with the corresponding values of 2(0;q) and
4(0;q).
Lawden (1989, pp. 270{279) tabulates j(x;q),j=
1;2;3;4, to 5D for x= 0(1)90,q= 0:1(:1)0:9, and
alsoqto 5D fork2= 0(:01)1.
Tables of Neville's theta functions s(x;q),c(x;q),
d(x;q),n(x;q) (seex20.1) and their logarithmic x-
derivatives are given in Abramowitz and Stegun (1964,
pp. 582{585) to 9D for "; = 0(5)90, where (in ra-
dian measure) "=x=2
3(0;q) =x=(2K(k)), and is
de ned by (20.15.1).
For other tables prior to 1961 see Fletcher et al.
(1962, pp. 508{514) and Lebedev and Fedorova (1960,
pp. 227{230).
20.16 Software
Seehttp://dlmf.nist.gov/20.16 .
References
General References
The main references used in writing this chapter are
Whittaker and Watson (1927), Lawden (1989), and
Walker (1996). For further bibliographic reading see
McKean and Moll (1999).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x20.2 Whittaker and Watson (1927, pp. 463{465) and
Lawden (1989, Chapter 1).
x20.3 These graphics were produced at NIST.
x20.4 Lawden (1989, pp. 12{23), Walker (1996, pp. 90{
92), and Whittaker and Watson (1927, pp. 470{
473). (20.4.1){(20.4.5) are special cases of
(20.5.1){(20.5.4).
References 535
x20.5 Lawden (1989, pp. 12{23), Walker (1996, pp. 86{
98), Whittaker and Watson (1927, pp. 469{473),
and Bellman (1961, p. 44). Equations (20.5.14){
(20.5.17) follow from (20.5.5){(20.5.8) by use of
the in nite products for the sine and cosine
(x4.22).
x20.6 Walker (1996,x3.2) andx23.9. (20.6.2){
(20.6.5) may be derived by termwise expansion
in (20.5.14){(20.5.17). (20.6.10) may be de-
rived from (20.6.6) and (20.6.8) by subtraction
of terms with even j, in a similar manner toP1
n=1( 1)n 1n j= (1 21 j)P1
n=1n j.
x20.7 Lawden (1989, pp. 5{23), Whittaker and Wat-
son (1927, pp. 466{477), McKean and Moll (1999,
pp. 129{130), Watson (1935a), Bellman (1961,
p. 61), and Serre (1973, p. 109). The rst equal-
ities in (20.7.11) and (20.7.12) follow by transla-
tion ofzby1
2as in (20.2.11){(20.2.14). The
second equalities follow from (20.5.5){(20.5.8) and
the identityQ1
n=1(1 +qn)(1 q2n 1) = 1 (Walker
(1996, p. 90)).
x20.9 Walker (1996, p. 156), Whittaker and Wat-
son (1927, pp. 480{485), Serre (1973, p. 109),
and McKean and Moll (1999, xx3.3, 3.9). For
(20.9.3) combination of (20.4.6) and (23.6.5) {
(23.6.7) yields }(z) ej=
v0
1(0;q)j+1(v;q)
z1(v;q)j+1(0;q)2
,j= 1;2;3, wherev=z=(2!1). Then by ap-
plication of (19.25.35) and use of the proper-
ties thatRFis homogenous and of degree 1
2
in its three variables ( xx19.16(ii), 19.16(iii)), we
derivez=z1(v;q)
v0
1(0;q)RF
2
2(v;q)
2
2(0;q);2
3(v;q)
2
3(0;q);2
4(v;q)
2
4(0;q)
.
This equation becomes (20.9.3) when the z's
are cancelled and vis renamed z. For
(20.9.4), from (19.25.1) and Erd elyi et al. (1953b,
13.20(11)) we have K(k) =RF
0;4
4(0;q)
4
3(0;q);1
=
2
3(0;q)RF
0;4
3(0;q);4
4(0;q)
, where the sec-
ond equality uses the homogeneity and symme-
try ofRF. Comparison with (20.9.2) proves
(20.9.4). For (20.9.5), by (19.25.1) the left side is
exp( K(k0)=K(k)), which equals qby Erd elyi
et al. (1953b, 13.19(4)).
x20.10 Bellman (1961, pp. 20{24). For (20.10.1) and
(20.10.3) usex20.7(viii) with appropriate changes
of integration variable. For (20.10.2) use (20.2.3)
withz= 0, =it, Bellman (1961, pp. 28{
32), Koblitz (1993, pp. 70{75), and/or Titchmarsh
(1986b,x2.6).
x20.11 Bellman (1961, pp. 38{39), Walker (1996,
pp. 181{182), and McKean and Moll (1999,
pp. 140{147 and 151{152).
x20.13 Whittaker and Watson (1927, p. 470).
Chapter 21
Multidimensional Theta Functions
B. Deconinck1
Notation 538
21.1 Special Notation . . . . . . . . . . . . . 538
Properties 538
21.2 De nitions . . . . . . . . . . . . . . . . . 538
21.3 Symmetry and Quasi-Periodicity . . . . . 539
21.4 Graphics . . . . . . . . . . . . . . . . . . 539
21.5 Modular Transformations . . . . . . . . . 541
21.6 Products . . . . . . . . . . . . . . . . . . 542Applications 543
21.7 Riemann Surfaces . . . . . . . . . . . . . 543
21.8 Abelian Functions . . . . . . . . . . . . . 545
21.9 Integrable Equations . . . . . . . . . . . 545
Computation 546
21.10 Methods of Computation . . . . . . . . . 546
21.11 Software . . . . . . . . . . . . . . . . . . 546
References 546
1Department of Applied Mathematics, University of Washington, Seattle, Washington.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
537
538 Multidimensional Theta Functions
Notation
21.1 Special Notation
(For other notation see pp. xiv and 873.)
g;h positive integers.
ZgZZ Z(gtimes).
RgRR R(gtimes).
Zghset of allghmatrices with integer
elements.
ggcomplex, symmetric matrix with =
strictly positive de nite, i.e., a Riemann
matrix.
; g-dimensional vectors, with all elements in
[0;1), unless stated otherwise.
ajjth element of vector a.
Ajk (j;k)th element of matrix A.
ab scalar product of the vectors aandb.
a
b[
a]b= [
b]a.
0gggzero matrix.
Igggidentity matrix.
J2g0gIg
Ig0g
.
Sgset ofg-dimensional vectors with elements
inS.
jSj number of elements of the set S.
S1S2 set of all elements of the form
\element of S1element ofS2".
S1=S2 set of all elements of S1, modulo elements of
S2. Thus two elements of S1=S2are
equivalent if they are both in S1and their
di erence is in S2. (For an example see
x20.12(ii).)
ab intersection index of aandb, two cycles
lying on a closed surface. ab= 0 ifaandb
do not intersect. Otherwise abgets an
additive contribution from every intersection
point. This contribution is 1 if the basis of
the tangent vectors of the aandbcycles
(x21.7(i)) at the point of intersection is
positively oriented; otherwise it is 1.H
a! line integral of the di erential !over the
cyclea.
Lowercase boldface letters or numbers are g-
dimensional real or complex vectors, either row or col-
umn depending on the context. Uppercase boldface let-ters areggreal or complex matrices.
The main functions treated in this chapter are the
Riemann theta functions (zj
), and the Riemann theta
functions with characteristics
(zj
).
The function ( jB) =( =(2i)jB=(2i)) is also
commonly used; see, for example, Belokolos et al. (1994,
x2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
Properties
21.2 De nitions
21.2(i) Riemann Theta Functions
21.2.1 (zj
) =X
n2Zge2i(1
2n
n+nz):
Thisg-tuple Fourier series converges absolutely and uni-
formly on compact sets of the zand
spaces; hence
(zj
) is an analytic function of (each element of) z
and (each element of)
.(zj
) is also referred to as
a theta function with gcomponents, a g-dimensional
theta function or as a genus gtheta function.
For numerical purposes we use the scaled Riemann
theta function ^(zj
), de ned by (Deconinck et al.
(2004)),
21.2.2 ^(zj
) =e [=z][=
] 1[=z](zj
):
^(zj
) is a bounded nonanalytic function of z. Many
applications involve quotients of Riemann theta func-
tions: the exponential factor then disappears.
Example
21.2.3
z1;z2 i 1
2
1
2i
=1X
n1= 11X
n2= 1e (n2
1+n2
2)e in1n2e2i(n1z1+n2z2):
Withz1=x1+iy1,z2=x2+iy2,
21.2.4^
x1+iy1;x2+iy2 i 1
2
1
2i
=1X
n1= 11X
n2= 1e (n1+y1)2 (n2+y2)2
ei(2n1x1+2n2x2 n1n2):
21.3 Symmetry and Quasi-Periodicity 539
21.2(ii) Riemann Theta Functions with
Characteristics
Let ; 2Rg. De ne
21.2.5
(zj
) =X
n2Zge2i(1
2[n+ ]
[n+ ]+[n+ ][z+ ]):
This function is referred to as a Riemann theta func-
tion with characteristics
. It is a translation of the
Riemann theta function (21.2.1), multiplied by an ex-
ponential factor:
21.2.6
(zj
) =e2i(1
2
+ [z+ ])(z+
+ j
);
and
21.2.7 0
0
(zj
) =(zj
):
Characteristics whose elements are either 0 or1
2are
called half-period characteristics . For given
, there are
22gg-dimensional Riemann theta functions with half-
period characteristics.
21.2(iii) Relation to Classical Theta Functions
Forg= 1, and with the notation of x20.2(i),
21.2.8 (zj
) =3(zj
);
21.2.9 1(zj
) = 1
2
1
2
(zj
);
21.2.10 2(zj
) =1
2
0
(zj
);
21.2.11 3(zj
) =0
0
(zj
);
21.2.12 4(zj
) =0
1
2
(zj
):21.3 Symmetry and Quasi-Periodicity
21.3(i) Riemann Theta Functions
21.3.1 ( zj
) =(zj
);
21.3.2 (z+m1j
) =(zj
);
when m12Zg:Thus(zj
) is periodic, with period 1,
in each element of z. More generally,
21.3.3
(z+m1+
m 2j
) =e 2i(1
2m2
m2+m2z)(zj
);
withm1,m22Zg. This is the quasi-periodicity prop-
erty of the Riemann theta function. It determines the
Riemann theta function up to a constant factor. The
set of points m1+
m 2form ag-dimensional lattice,
theperiod lattice of the Riemann theta function.
21.3(ii) Riemann Theta Functions with
Characteristics
Again, with m1,m22Zg
21.3.4 +m1
+m2
(zj
) =e2i m1
(zj
):
Because of this property, the elements of and are
usually restricted to [0 ;1), without loss of generality.
21.3.5
(z+m1+
m 2j
)
=e2i( m1 m2 1
2m2
m2 m2z)
(zj
):
For Riemann theta functions with half-period charac-
teristics,
21.3.6
( zj
) = ( 1)4
(zj
):
See alsox20.2(iii) for the case g= 1 and classical theta
functions.
21.4 Graphics
Figure 21.4.1 provides surfaces of the scaled Riemann theta function ^(zj
), with
21.4.1
=1:69098 3006 + 0 :95105 6516 i 1:5 + 0:36327 1264 i
1:5 + 0:36327 1264 i 1:30901 6994 + 0 :95105 6516 i
:
This Riemann matrix originates from the Riemann surface represented by the algebraic curve 3 7+ 23= 0;
comparex21.7(i).
540 Multidimensional Theta Functions
(a1) (b 1) (c 1)
(a2) (b 2) (c 2)
(a3) (b 3) (c 3)
Figure 21.4.1 :^(zj
) parametrized by (21.4.1). The surface plots are of ^(x+iy;0j
), 0x1, 0y5 (sux
1);^(x;yj
), 0x1, 0y1 (sux 2); ^(ix;iyj
), 0x5, 0y5 (sux 3). Shown are the real part
(a), the imaginary part (b), and the modulus (c).
For the scaled Riemann theta functions depicted in Figures 21.4.2{21.4.5
21.4.2
1=
i 1
2
1
2i
;
and
21.4.3
2=2
4 1
2+i1
2 1
2i 1
2 1
2i
1
2 1
2i i 0
1
2 1
2i 0i3
5:
21.5 Modular Transformations 541
Figure 21.4.2 :<^(x+iy;0j
1), 0x1, 0y5.
(The imaginary part looks very similar.)
Figure 21.4.3 : ^(x+iy;0j
1) , 0x1, 0y2.
Figure 21.4.4 : A real-valued scaled Riemann theta func-
tion: ^(ix;iyj
1), 0x4, 0y4. In this
case, the quasi-periods are commensurable, resulting in
a doubly-periodic con guration.
Figure 21.4.5 : The real part of a genus 3 scaled Riemann
theta function:<^(x+iy;0;0j
2), 0x1, 0
y3. This Riemann matrix originates from the genus
3 Riemann surface represented by the algebraic curve
3+ 2 4= 0; comparex21.7(i).
21.5 Modular Transformations
21.5(i) Riemann Theta Functions
LetA,B,C, and Dbeggmatrices with integer
elements such that
21.5.1 =A B
C D
is asymplectic matrix , that is,
21.5.2 J2g T=J2g:Then
21.5.3 det = 1;
and
21.5.4
[C
+D] 1Tz [A
+B][C
+D] 1
=( )p
det[C
+D]eiz[[C
+D] 1C]z(zj
):
Here( ) is an eighth root of unity, that is, ( ( ))8= 1.
For general , it is dicult to decide which root needs to
be used. The choice depends on , but is independent
542 Multidimensional Theta Functions
ofzand
. Equation (21.5.4) is the modular transfor-
mation property for Riemann theta functions.
The modular transformations form a group under
the composition of such transformations, the modular
group , which is generated by simpler transformations,
for which( ) is determinate:
21.5.5 =A 0g
0g[A 1]T
)
Az A
AT
=(zj
):
(Ainvertible with integer elements.)
21.5.6 =
IgB
0gIg
)(zj
+B) =(zj
):(Bsymmetric with integer elements and even diagonal
elements.)
21.5.7
=IgB
0gIg
)(zj
+B) =
z+1
2diagB
:
(Bsymmetric with integer elements.) See Heil (1995,
p. 24).
21.5.8
=0g Ig
Ig0g
)
1z
1
=p
det [ i
]eiz
1z(zj
);
where the square root assumes its principal value.
21.5(ii) Riemann Theta Functions with Characteristics
21.5.9D C +1
2diag[CDT]
B +A +1
2diag[ABT]
[C
+D] 1Tz [A
+B][C
+D] 1
=( ; ; )p
det[C
+D]eiz[[C
+D] 1C]z
(zj
);
where( ; ; ) is a complex number that depends on
, , and . However, ( ; ; ) is independent of
zand
. For explicit results in the case g= 1, see
x20.7(viii).
21.6 Products
21.6(i) Riemann Identity
LetT= [Tjk] be an arbitrary hhorthogonal matrix
(that is, TTT=I) with rational elements. Also, let Z
be an arbitrary ghmatrix. De ne
21.6.1K=ZghT=(ZghT\Zgh);
that is,Kis the set of all ghmatrices that are obtained
by premultiplying Tby anyghmatrix with integer
elements; two such matrices in Kare considered equiva-
lentif their di erence is a matrix with integer elements.
Also, let
21.6.2D=jTTZh=(TTZh\Zh)j;
that is,Dis the number of elements in the set contain-
ing allh-dimensional vectors obtained by multiplying
TTon the right by a vector with integer elements. Two
such vectors are considered equivalent if their di erenceis a vector with integer elements. Then
21.6.3hY
j=1 hX
k=1Tjkzk
!
=1
DgX
A2KX
B2Ke2itr[1
2AT
A+AT[Z+B]]
hY
j=1(zj+
aj+bjj
);
where zj,aj,bjdenote respectively the jth columns of
Z,A,B. This is the Riemann identity . On using theta
functions with characteristics, it becomes
21.6.4
hY
j=1Ph
k=1TjkckPh
k=1Tjkdk hX
k=1Tjkzk
!
=1
DgX
A2KX
B2Ke 2iPh
j=1bjcjhY
j=1aj+cj
bj+dj
(zjj
);
where cjanddjare arbitrary h-dimensional vectors.
Many identities involving products of theta functions
can be established using these formulas.
Example
Leth= 4 and
21.6.5 T=2
6641 1 1 1
1 1 1 1
1 1 1 1
1 1 1 13
775:
Then
Applications 543
21.6.6x+y+u+v
2
x+y u v
2
x y+u v
2
x y u+v
2
=1
2gX
21
2Zg=ZgX
21
2Zg=Zge2i(2
+ [x+y+u+v])
(x+
+ j
)(y+
+ j
)(u+
+ j
)(v+
+ j
);
and
21.6.7
1
2[c1+c2+c3+c4]
1
2[d1+d2+d3+d4]x+y+u+v
2
1
2[c1+c2 c3 c4]
1
2[d1+d2 d3 d4]x+y u v
2
1
2[c1 c2+c3 c4]
1
2[d1 d2+d3 d4]x y+u v
2
1
2[c1 c2 c3+c4]
1
2[d1 d2 d3+d4]x y u+v
2
=1
2gX
21
2Zg=ZgX
21
2Zg=Zge 2i [c1+c2+c3+c4]c1+
d1+
(xj
)c2+
d2+
(yj
)c3+
d3+
(uj
)c4+
d4+
(vj
):
21.6(ii) Addition Formulas
Let , ,
,2Rg. Then
21.6.8
(z1j
)
(z2j
)
=X
2Zg=(2Zg)1
2[ + +]
+
(z1+z2j2
)
1
2[ +]
(z1 z2j2
):
Thusis ag-dimensional vector whose entries are ei-
ther 0 or 1. For this result and a generalization see
Koizumi (1976) and Belokolos et al. (1994, pp. 38{41).
For addition formulas for classical theta functions see
x20.7(ii).
Applications
21.7 Riemann Surfaces
21.7(i) Connection of Riemann Theta
Functions to Riemann Surfaces
In almost all applications, a Riemann theta function is
associated with a compact Riemann surface. Although
there are other ways to represent Riemann surfaces (see
e.g. Belokolos et al. (1994,x2.1)), they are obtainable
from plane algebraic curves (Springer (1957), or Rie-
mann (1851)). Consider the set of points in C2that
satisfy the equation
21.7.1 P(;) = 0;
whereP(;) is a polynomial in andthat does
not factor over C2. Equation (21.7.1) determines aplane algebraic curve in C2, which is made compact by
adding its points at in nity. To accomplish this we write
(21.7.1) in terms of homogeneous coordinates:
21.7.2 ~P(~;~;~) = 0;
by setting=~=~,= ~=~, and then clearing frac-
tions. This compact curve may have singular points,
that is, points at which the gradient of ~Pvanishes.
Removing the singularities of this curve gives rise to
a two-dimensional connected manifold with a complex-
analytic structure, that is, a Riemann surface. All com-
pact Riemann surfaces can be obtained this way.
Since a Riemann surface is a two-dimensional man-
ifold that is orientable (owing to its analytic structure),
its only topological invariant is its genusg(the number
ofhandles in the surface). On this surface, we choose 2 g
cycles (that is, closed oriented curves, each with at most
a nite number of singular points) aj,bj,j= 1;2;:::;g ,
such that their intersection indices satisfy
21.7.3ajak= 0; bjbk= 0; ajbk=j;k:
For example, Figure 21.7.1 depicts a genus 2 surface.
1 a2
2b1ba
Figure 21.7.1 : A basis of cycles for a genus 2 surface.
On a Riemann surface of genus g, there are glin-
early independent holomorphic di erentials !j,j=
544 Multidimensional Theta Functions
1;2;:::;g . If a local coordinate zis chosen on the Rie-
mann surface, then the local coordinate representation
of these holomorphic di erentials is given by
21.7.4 !j=fj(z)dz,j= 1;2;:::;g ,
wherefj(z),j= 1;2;:::;g are analytic functions. Thus
the di erentials !j,j= 1;2;:::;g have no singularities
on . Note that for the purposes of integrating these
holomorphic di erentials, all cycles on the surface are a
linear combination of the cycles aj,bj,j= 1;2;:::;g .
The!jare normalized so that
21.7.5I
ak!j=j;k,j;k= 1;2;:::;g .
Then the matrix de ned by
21.7.6
jk=I
bk!j,j;k= 1;2;:::;g ,
is a Riemann matrix and it is used to de ne the cor-
responding Riemann theta function. In this way, we
associate a Riemann theta function with every compact
Riemann surface .
Riemann theta functions originating from Riemann
surfaces are special in the sense that a general g-
dimensional Riemann theta function depends on g(g+
1)=2 complex parameters. In contrast, a g-dimensional
Riemann theta function arising from a compact Rie-
mann surface of genus g(>1) depends on at most
3g 3 complex parameters (one complex parameter for
the caseg= 1). These special Riemann theta functions
satisfy many special identities, two of which appear inthe following subsections. For more information, see
Dubrovin (1981), Brieskorn and Kn orrer (1986, x9.3),
Belokolos et al. (1994, Chapter 2), and Mumford (1984,
x2.2{2.3).
21.7(ii) Fay's Trisecant Identity
Let , be such that
21.7.7@
@z1
(zj
)
z=0;:::;@
@zg
(zj
)
z=0
6=0:
De ne the holomorphic di erential
21.7.8 =gX
j=1!j@
@zj
(zj
)
z=0:
Then the prime form on the corresponding compact
Riemann surface is de ned by
21.7.9
E(P1;P2) =
ZP2
P1!
!,p
(P1)p
(P2)
;
whereP1andP2are points on , != (!1;!2;:::;!g),
and the path of integration on from P1toP2is
identical for all components. Herep
(P) is such thatp
(P)2=(P),P2 . Either branch of the square
roots may be chosen, as long as the branch is consistent
across . For all z2Cg, and allP1,P2,P3,P4on ,
Fay's identity is given by
21.7.10
z+ZP3
P1!
!
z+ZP4
P2!
!
E(P3;P2)E(P1;P4) +
z+ZP3
P2!
!
z+ZP4
P1!
!
E(P3;P1)E(P4;P2)
=(zj
)
z+ZP3
P1!+ZP4
P2!
!
E(P1;P2)E(P3;P4);
where again all integration paths are identical for all
components. Generalizations of this identity are given
in Fay (1973, Chapter 2). Fay derives (21.7.10) as a spe-
cial case of a more general class of addition theorems for
Riemann theta functions on Riemann surfaces.
21.7(iii) Frobenius' Identity
Let be a hyperelliptic Riemann surface . These are
Riemann surfaces that may be obtained from algebraic
curves of the form
21.7.11 2=Q();
whereQ() is a polynomial in of odd degree 2 g+ 1
(5). The genus of this surface is g. The zeros j,j= 1;2;:::; 2g+ 1 ofQ() specify the nite branch
pointsPj, that is, points at which j= 0, on the Rie-
mann surface. Denote the set of all branch points by
B=fP1;P2;:::;P 2g+1;P1g. Consider a xed subset
UofB, such that the number of elements jUjin the set
Uisg+1, andP1=2U. Next, de ne an isomorphism
which maps every subset TofBwith an even number of
elements to a 2 g-dimensional vector (T) with elements
either 0 or1
2. De ne the operation
21.7.12T1 T2= (T1[T2)n(T1\T2):
Also,Tc=BnT,1(T) = (1(T);2(T);:::;g(T)),
and2(T) = (g+1(T);g+2(T);:::; 2g(T)). Then the
21.8 Abelian Functions 545
isomorphism is determined completely by:
21.7.13 (T) =(Tc);
21.7.14 (T1 T2) =(T1) +(T2);
21.7.15 41(T)2(T) =1
2(jT Uj g 1) (mod 2) ;
21.7.16
4(1(T1)2(T2) 2(T1)1(T2)) =jT1\T2j(mod 2):
Furthermore, let (P1) = 0and(Pj) =
(fPj;P1g). Then for all zj2Cg,j= 1;2;3;4, such
thatz1+z2+z3+z4= 0, and for all j, j2Rg, such
that 1+ 2+ 3+ 4= 0 and 1+ 2+ 3+ 4= 0,
we have Frobenius' identity :
21.7.17X
Pj2U4Y
k=1 k+1(Pj)
k+2(Pj)
(zkj
)
=X
Pj2Uc4Y
k=1 k+1(Pj)
k+2(Pj)
(zkj
):
21.8 Abelian Functions
An Abelian function is a 2 g-fold periodic, meromor-
phic function of gcomplex variables. In consequence,
Abelian functions are generalizations of elliptic func-
tions (x23.2(iii)) to more than one complex variable.
For every Abelian function, there is a positive integer n,
such that the Abelian function can be expressed as a ra-
tio of linear combinations of products with nfactors of
Riemann theta functions with characteristics that share
a common period lattice. For further information see
Igusa (1972, pp. 132{135) and Markushevich (1992).
21.9 Integrable Equations
Riemann theta functions arise in the study of integrable
di erential equations that have applications in many
areas, including
uid mechanics (Ablowitz and Segur
(1981, Chapter 4)), magnetic monopoles (Ercolani and
Sinha (1989)), and string theory (Deligne et al. (1999,
Part 3)). Typical examples of such equations are the
Korteweg{de Vries equation
21.9.1 4ut= 6uux+uxxx;
and the nonlinear Schr odinger equations
21.9.2 iut= 1
2uxxjuj2u:
Here, and in what follows, x;y, andtsuxes indicate
partial derivatives.
Particularly important for the use of Riemann
theta functions is the Kadomtsev{Petviashvili (KP)
equation, which describes the propagation of two-
dimensional, long-wave length surface waves in shallow
water (Ablowitz and Segur (1981, Chapter 4)):
21.9.3 ( 4ut+ 6uux+uxxx)x+ 3uyy= 0:Herexandyare spatial variables, tis time, and
u(x;y;t ) is the elevation of the surface wave. All quanti-
ties are made dimensionless by a suitable scaling trans-
formation. The KP equation has a class of quasi-
periodic solutions described by Riemann theta func-
tions, given by
21.9.4u(x;y;t ) =c+ 2@2
@x2ln((kx+ly+!t+ j
));
wherecis a complex constant and k,l,!, and are
g-dimensional complex vectors; see Krichever (1976).
These parameters, including
, are not free: they are
determined by a compact, connected Riemann surface
(Krichever (1976)), or alternatively by an appropri-
ate initial condition u(x;y;0) (Deconinck and Segur
(1998)). These solutions have been compared success-
fully with physical experiments for g= 1;2 (Wiegel
(1960), Hammack et al. (1989), and Hammack et al.
(1995)). See Figures 21.9.1 and 21.9.2.
Figure 21.9.1 : Two-dimensional periodic waves in a shal-
low water wave tank, taken from Hammack et al. (1995,
p. 97) by permission of Cambridge University Press.
The original caption reads \Mosaic of two overhead pho-
tographs, showing surface patterns of waves in shallow
water."
Figure 21.9.2 : Contour plot of a two-phase solution of
Equation (21.9.3). Such a solution is given in terms of a
Riemann theta function with two phases; see Krichever
(1976), Dubrovin (1981), and Hammack et al. (1995).
546 Multidimensional Theta Functions
Furthermore, the solutions of the KP equation solve
theSchottky problem : this is the question concerning
conditions that a Riemann matrix needs to satisfy in
order to be associated with a Riemann surface (Schot-
tky (1903)). Following the work of Krichever (1976),
Novikov conjectured that the Riemann theta function
in (21.9.4) gives rise to a solution of the KP equation
(21.9.3) if, and only if, the theta function originates from
a Riemann surface; see Dubrovin (1981, xIV.4). The
rst part of this conjecture was established in Krichever
(1976); the second part was proved in Shiota (1986).
Computation
21.10 Methods of Computation
21.10(i) General Riemann Theta Functions
Although the de ning Fourier series (21.2.1) is uni-
formly convergent on compact sets, its evaluation is
cumbersome when one or more of the eigenvalues of
=(
) is near zero. Furthermore, for xed
di erent
terms of the Fourier series dominate for di erent values
ofz.
To overcome these obstacles, we compute instead the
scaled function ^(zj
) (x21.2(i)) from the expansion
21.10.1
^(zj
) =X
n2S()ei[n [Y 1y]]X[n [Y 1y]]
e2i[n [Y 1y]]xe [n+[Y 1y]]Y[n+[Y 1y]];
whereis the tolerated maximum absolute error for
^(zj
). Here X=<(
),Y==(
),x=<(z),
y==(z), and
21.10.2S() =n
m2Zg
m+ [Y 1y]
Y
m+ [Y 1y]
R()o
:
ThusS() is the set of all integer vectors that are con-
tained in an ellipsoid centered at the fractional part of
Y 1y, and whose size is determined by the allowed ab-
solute error. The value of R() is determined as follows.
Letrbe the length of the shortest vector of the lattice
=fpTmjm2Zgg, and TTT=Ybe the Cholesky
decomposition of Y(Atkinson (1989, p. 254)). Then
R() is the greater ofp
g=2 +rand the smallest posi-
tive root of the equation
21.10.3 1
2g;R2
=(2grg) =:
For the incomplete gamma function ( a;z), seex8.2(i).
The construction (21.10.2) amounts to determining
all integer vectors in a g-dimensional ellipsoid. For thispurpose it is convenient to have the ellipsoid as spherical
as possible (Siegel (1973, pp. 144{159), Heil (1995)).
Usually, (21.10.1) can also be used for the ecient
evaluation of ^(zj
) for xed
and varying z, by ad-
dition of a few vectors to the set S().
21.10(ii) Riemann Theta Functions Associated
with a Riemann Surface
In addition to evaluating the Fourier series, the main
problem here is to compute a Riemann matrix originat-
ing from a Riemann surface. Various approaches are
considered in the following references:
Belokolos et al. (1994, Chapter 5) and references
therein. Here the Riemann surface is represented
by the action of a Schottky group on a region of
the complex plane. The same representation is
used in Gianni et al. (1998).
Tretko and Tretko (1984). Here a Hurwitz sys-
tem is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane al-
gebraic curve representation of the Riemann sur-
face is used.
21.11 Software
Seehttp://dlmf.nist.gov/21.11 .
References
General References
The main references used in writing this chapter are
Mumford (1983, 1984), Igusa (1972), and Belokolos
et al. (1994). For additional bibliographic reading see
Dubrovin (1981), Siegel (1971, 1973), and Fay (1973).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x21.3 Mumford (1983, pp. 120{122).
x21.4 These graphics were computed by the author,
using the algorithms described in Deconinck et al.
(2004).
x21.5 Arnol0d (1997, p. 222), Mumford (1983, pp. 189{
210), Igusa (1972, pp. 78{85).
References 547
x21.6 Mumford (1983, pp. 211{216), Dubrovin (1981,
pp. 22{23).
x21.7 Mumford (1984, pp. 106{120 and 207{260).x21.9 Dubrovin (1981), Belokolos et al. (1994).
x21.10 Deconinck et al. (2004).
Chapter 22
Jacobian Elliptic Functions
W. P. Reinhardt1and P. L. Walker2
Notation 550
22.1 Special Notation . . . . . . . . . . . . . 550
Properties 550
22.2 De nitions . . . . . . . . . . . . . . . . . 550
22.3 Graphics . . . . . . . . . . . . . . . . . . 550
22.4 Periods, Poles, and Zeros . . . . . . . . . 553
22.5 Special Values . . . . . . . . . . . . . . . 554
22.6 Elementary Identities . . . . . . . . . . . 556
22.7 Landen Transformations . . . . . . . . . 556
22.8 Addition Theorems . . . . . . . . . . . . 557
22.9 Cyclic Identities . . . . . . . . . . . . . . 558
22.10 Maclaurin Series . . . . . . . . . . . . . . 558
22.11 Fourier and Hyperbolic Series . . . . . . . 559
22.12 Expansions in Other Trigonometric Se-
ries and Doubly-In nite Partial Fractions:
Eisenstein Series . . . . . . . . . . . . . . 55922.13 Derivatives and Di erential Equations . . 560
22.14 Integrals . . . . . . . . . . . . . . . . . . 560
22.15 Inverse Functions . . . . . . . . . . . . . 561
22.16 Related Functions . . . . . . . . . . . . . 561
22.17 Moduli Outside the Interval [0,1] . . . . . 563
Applications 563
22.18 Mathematical Applications . . . . . . . . 563
22.19 Physical Applications . . . . . . . . . . . 564
Computation 566
22.20 Methods of Computation . . . . . . . . . 566
22.21 Tables . . . . . . . . . . . . . . . . . . . 567
22.22 Software . . . . . . . . . . . . . . . . . . 567
References 567
1University of Washington, Seattle, Washington.
2American University of Sharjah, Sharjah, United Arab Emirates.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapters 16,18) by L. M. Milne-Thomson and
T. H. Southard respectively.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
549
550 Jacobian Elliptic Functions
Notation
22.1 Special Notation
(For other notation see pp. xiv and 873.)
x;y real variables.
z complex variable.
k modulus. Except in xx22.3(iv), 22.17, and
22.19, 0k1.
k0complementary modulus, k2+k02= 1. If
k2[0;1], thenk02[0;1].
K,K0K(k),K0(k) =K(k0) (complete elliptic
integrals of the rst kind ( x19.2(ii))).
q nome. 0q<1 except inx22.17; see also
x20.1.
iK0=K.
All derivatives are denoted by di erentials, not
primes.
The functions treated in this chapter are the three
principal Jacobian elliptic functions sn ( z;k), cn (z;k),
dn (z;k); the nine subsidiary Jacobian elliptic functions
cd (z;k), sd (z;k), nd (z;k), dc (z;k), nc (z;k), sc (z;k),
ns (z;k), ds (z;k), cs (z;k); the amplitude function
am (x;k); Jacobi's epsilon and zeta functions E(x;k)
and Z(xjk).
The notation sn ( z;k), cn (z;k), dn (z;k) is due to
Gudermann (1838), following Jacobi (1827); that for the
subsidiary functions is due to Glaisher (1882). Other
notations for sn ( z;k) are sn(zjm) and sn(z;m) with
m=k2; see Abramowitz and Stegun (1964) and Walker
(1996). Similarly for the other functions.
Properties
22.2 De nitions
The nomeqis given in terms of the moduluskby
22.2.1 q= exp( K0(k)=K(k));
whereK(k),K0(k) are de ned inx19.2(ii). Inversely,
22.2.2k=2
2(0;q)
2
3(0;q); k0=2
4(0;q)
2
3(0;q); K (k) =
22
3(0;q);
wherek0=p
1 k2and the theta functions are de ned
inx20.2(i).With
22.2.3 =z
2K(k);
22.2.4 sn (z;k) =3(0;q)
2(0;q)1(;q)
4(;q)=1
ns (z;k);
22.2.5 cn (z;k) =4(0;q)
2(0;q)2(;q)
4(;q)=1
nc (z;k);
22.2.6 dn (z;k) =4(0;q)
3(0;q)3(;q)
4(;q)=1
nd (z;k);
22.2.7 sd (z;k) =2
3(0;q)
2(0;q)4(0;q)1(;q)
3(;q)=1
ds (z;k);
22.2.8 cd (z;k) =3(0;q)
2(0;q)2(;q)
3(;q)=1
dc (z;k);
22.2.9 sc (z;k) =3(0;q)
4(0;q)1(;q)
2(;q)=1
cs (z;k):
As a function of z, with xed k, each of the 12 Ja-
cobian elliptic functions is doubly periodic, having two
periods whose ratio is not real. Each is meromorphic in
zfor xedk, with simple poles and simple zeros, and
each is meromorphic in kfor xedz. Fork2[0;1], all
functions are real for z2R.
Glaisher's Notation
The Jacobian functions are related in the following way.
Let p, q, r be any three of the letters s, c, d, n. Then
22.2.10 pq (z;k) =pr (z;k)
qr (z;k)=1
qp (z;k);
with the convention that functions with the same two
letters are replaced by unity; e.g. ss ( z;k) = 1.
The six functions containing the letter s in their two-
letter name are odd in z; the other six are even in z.
In terms of Neville's theta functions ( x20.1)
22.2.11 pq (z;k) =p(zj )/q(zj );
where
22.2.12 =iK0(k)/K(k);
and p, q are any pair of the letters s, c, d, n.
22.3 Graphics
22.3(i) Real Variables: Line Graphs
See Figures 22.3.1{22.3.4 for line graphs of the functions
sn (x;k), cn (x;k), dn (x;k), and nd (x;k) for represen-
tative values of real xand realkillustrating the near
trigonometric ( k= 0), and near hyperbolic ( k= 1) lim-
its. For corresponding graphs for the other 8 Jacobian
elliptic functions see http://dlmf.nist.gov/22.3.i .
22.3 Graphics 551
Figure 22.3.1 :k= 0:4, 3Kx3K,K= 1:6399:::.
Figure 22.3.2 :k= 0:7, 3Kx3K,K= 1:8456:::.
For cn (x;k) the curve for k= 1=p
2 = 0:70710:::is a
boundary between the curves that have an in
ection
point in the interval 0 x2K(k), and its translates,
and those that do not; see Walker (1996, p. 146).
Figure 22.3.3 :k= 0:99, 3Kx3K,K=
3:3566:::.
Figure 22.3.4 :k= 0:999999, 3Kx3K,K=
7:9474:::.
22.3(ii) Real Variables: Surfaces
See Figure 22.3.13 for sn ( x;k) as a function of real arguments xandk. The period diverges logarithmically as
k!1 ; seex19.12. For the corresponding surfaces for cn ( x;k) and dn (x;k) see http://dlmf.nist.gov/22.3.ii .
Figure 22.3.13 : sn (x;k) fork= 1 e n,n= 0 to 20, 5x5.
552 Jacobian Elliptic Functions
22.3(iii) Complex z; Realk
In Figure 22.3.16 height corresponds to the absolute
value of the function and color to the phase. See p. xiv.
Figure 22.3.16 : sn (x+iy;k) fork= 0:99, 3Kx
3K, 0y4K0.K= 3:3566:::,K0= 1:5786:::.
For the corresponding surfaces for the copolar func-
tions cn (z;k) and dn (z;k) and the coperiodic functions
cd (z;k), dc (z;k), and ns (z;k) withz=x+iysee
http://dlmf.nist.gov/22.3.iii .22.3(iv) Complex k
Figure 22.3.22 :<sn (x;k),x= 120, as a function of
k2=i, 04.
Figure 22.3.23 :=sn (x;k),x= 120, as a function of
k2=i, 04.
In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase.
See p. xiv.
Figure 22.3.24 : sn (x+iy;k) for 4x4, 0y
8,k= 1 +1
2i.K= 1:5149:::+i0:5235:::,K0=
1:4620::: i0:3552:::.
Figure 22.3.25 : sn (5;k) as a function of complex k2,
1 < (k2)3:5, 1 = (k2)1. Compare
x22.17(ii).
22.4 Periods, Poles, and Zeros 553
Figure 22.3.26 : Density plot of jsn (5;k)jas a function
of complex k2, 10 <(k2)20, 10 =(k2)
10. Grayscale, running from 0 (black) to 10 (white),
withj(sn (5;k))j>10 truncated to 10. White spots
correspond to poles.
Figure 22.3.27 : Density plot of jsn (10;k)jas a function
of complex k2, 10 <(k2)20, 10 =(k2)
10. Grayscale, running from 0 (black) to 10 (white),
withjsn (10;k)j>10 truncated to 10. White spots
correspond to poles.
For corresponding density plots with arguments 20 and 30 see http://dlmf.nist.gov/22.3.iv .
22.4 Periods, Poles, and Zeros
22.4(i) Distribution
For each Jacobian function, Table 22.4.1 gives its peri-
ods in thez-plane in the left column, and the position of
one of its poles in the second row. The other poles are at
congruent points , which is the set of points obtained by
making translations by 2 mK+ 2niK0, wherem;n2Z.
For example, the poles of sn ( z;k), abbreviated as sn in
the following tables, are at z= 2mK+ (2n+ 1)iK0.
Table 22.4.1 : Periods and poles of Jacobian elliptic func-
tions.
Periodsz-Poles
iK0K+iK0K 0
4K, 2iK0sn cd dc ns
4K, 2K+ 2iK0cn sd nc ds
2K, 4iK0dn nd sc cs
Three functions in the same column of Table 22.4.1
arecopolar , and four functions in the same row are cope-riodic .
Table 22.4.2 displays the periods and zeros of the
functions in the z-plane in a similar manner to Table
22.4.1. Again, one member of each congruent set of
zeros appears in the second row; all others are gener-
ated by translations of the form 2 mK+ 2niK0, where
m;n2Z.
Table 22.4.2 : Periods and zeros of Jacobian elliptic func-
tions.
Periodsz-Zeros
0K K +iK0iK0
4K, 2iK0sn cd dc ns
4K, 2K+ 2iK0sd cn ds nc
2K, 4iK0sc cs dn nd
Figure 22.4.1 illustrates the locations in the z-plane
of the poles and zeros of the three principal Jaco-
bian functions in the rectangle with vertices 0, 2 K,
2K+ 2iK0, 2iK0. The other poles and zeros are at
the congruent points.
554 Jacobian Elliptic Functions
(a) sn (z;k) (b) cn ( z;k) (c) dn ( z;k)
Figure 22.4.1 :z-plane. Poles and zeros of the principal Jacobian elliptic functions.
22.4(ii) Graphical Interpretation via Glaisher's
Notation
Figure 22.4.2 depicts the fundamental unit cell in the
z-plane, with vertices s = 0, c = K, d =K+iK0,
n =iK0. The set of points z=mK+niK0,m;n2Z,
comprise the lattice for the 12 Jacobian functions; all
other lattice unit cells are generated by translation of
the fundamental unit cell by mK+niK0, where again
m;n2Z.
Figure 22.4.2 :z-plane. Fundamental unit cell.Using the p,q notation of (22.2.10), Figure 22.4.2
serves as a mnemonic for the poles, zeros, periods, and
half-periods of the 12 Jacobian elliptic functions as fol-
lows. Let p,q be any two distinct letters from the set
s,c,d,n which appear in counterclockwise orientation at
the corners of all lattice unit cells. Then: (a) In any
lattice unit cell pq ( z;k) has a simple zero at z= p
and a simple pole at z= q. (b) The di erence between
p and the nearest q is a half-period of pq ( z;k). This
half-period will be plus or minus a member of the triple
K;iK0;K+iK0; the other two members of this triple
are quarter periods of pq ( z;k).
22.4(iii) Translation by Half or Quarter Periods
See Table 22.4.3.
For example, sn ( z+K;k) = cd (z;k). (The modu-
luskis suppressed throughout the table.)
For the other nine functions see http://dlmf.nist.
gov/22.4.iii .
Table 22.4.3 : Half- or quarter-period shifts of variable for the Jacobian elliptic functions.
u
z+K z +K+iK0z+iK0z+ 2K z + 2K+ 2iK0z+ 2iK0
snu cdz k 1dcz k 1nsz snz snz snz
cnu k0sdz ik0k 1ncz ik 1dsz cnz cnz cnz
dnuk0ndz ik0scz icsz dnz dnz dnz
22.5 Special Values
22.5(i) Special Values of z
Table 22.5.1 gives the value of each of the functions sn ( z;k), cn (z;k), dn (z;k), together with its z-derivative (or at a
pole, the residue), for values of zthat are integer multiples of K,iK0. For example, at z=K+iK0, sn (z;k) = 1=k,
dsn (z;k)/dz= 0. (The modulus kis suppressed throughout the table.)
For the other nine functions see http://dlmf.nist.gov/22.5.i .
22.5 Special Values 555
Table 22.5.1 : Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.
z
0K K +iK0iK02K 2K+ 2iK02iK0
snz0;1 1;0 1=k;01, 1=k 0; 1 0; 1 0;1
cnz1;0 0; k0 ik0=k;01, i=k 1;0 1;0 1;0
dnz1;0k0;0 0;ik01, i 1;0 1;0 1;0
Table 22.5.2 gives sn ( z;k), cn (z;k), dn (z;k) for other special values of z. For example, sn 1
2K;k
= (1+k0) 1=2.
For the other nine functions ratios can be taken; compare (22.2.10).
Table 22.5.2 : Other special values of Jacobian elliptic functions.
z
1
2K1
2(K+iK0)1
2iK0
snz (1 +k0) 1=2
(1 +k)1=2+i(1 k)1=2
=(2k)1=2ik 1=2
cnz (k0=(1 +k0))1=2(1 i)k01=2=(2k)1=2(1 +k)1=2k 1=2
dnzk01=2k01=2((1 +k0)1=2 i(1 k0)1=2)=21=2(1 +k)1=2
z
3
2K3
2(K+iK0)3
2iK0
snz (1 +k0) 1=2(1 +i)((1 +k)1=2 i(1 k)1=2)=(2k1=2) ik 1=2
cnz (k0=(1 +k0))1=2(1 i)k01=2=(2k)1=2 (1 +k)1=2k 1=2
dnzk01=2( 1 +i)k01=2((1 +k0)1=2+i(1 k0)1=2)=2 (1 +k)1=2
22.5(ii) Limiting Values of k
Ifk!0+, thenK!=2 andK0!1 ; ifk!1 , thenK!1 andK0!=2. In these cases the elliptic functions
degenerate into elementary trigonometric and hyperbolic functions, respectively. See Tables 22.5.3 and 22.5.4.
Table 22.5.3 : Limiting forms of Jacobian elliptic functions as k!0.
sn (z;k)!sinzcd (z;k)!coszdc (z;k)!seczns (z;k)!cscz
cn (z;k)!coszsd (z;k)!sinznc (z;k)!seczds (z;k)!cscz
dn (z;k)!1 nd (z;k)!1 sc (z;k)!tanzcs (z;k)!cotz
Table 22.5.4 : Limiting forms of Jacobian elliptic functions as k!1.
sn (z;k)!tanhzcd (z;k)!1 dc (z;k)!1 ns (z;k)!cothz
cn (z;k)!sechz sd (z;k)!sinhznc (z;k)!coshzds (z;k)!cschz
dn (z;k)!sechznd (z;k)!coshzsc (z;k)!coshzcs (z;k)!cschz
Expansions for K;K0ask!0 or 1 are given in xx19.5, 19.12.
For values of K;K0whenk2=1
2(lemniscatic case) see x23.5(iii), and for k2=ei=3(equianharmonic case) see
x23.5(v).
556 Jacobian Elliptic Functions
22.6 Elementary Identities
22.6(i) Sums of Squares
22.6.1
sn2(z;k) + cn2(z;k) =k2sn2(z;k) + dn2(z;k) = 1;
22.6.2 1 + cs2(z;k) =k2+ ds2(z;k) = ns2(z;k);
22.6.3k02sc2(z;k) + 1 = dc2(z;k) =k02nc2(z;k) +k2;
22.6.4
k2k02sd2(z;k) =k2(cd2(z;k) 1) =k02(1 nd2(z;k)):
22.6(ii) Double Argument
22.6.5 sn (2z;k) =2 sn (z;k) cn (z;k) dn (z;k)
1 k2sn4(z;k);
22.6.6cn (2z;k) =cn2(z;k) sn2(z;k) dn2(z;k)
1 k2sn4(z;k)
=cn4(z;k) k02sn4(z;k)
1 k2sn4(z;k);
22.6.7dn (2z;k) =dn2(z;k) k2sn2(z;k) dn2(z;k)
1 k2sn4(z;k)
=dn4(z;k) +k2k02sn4(z;k)
1 k2sn4(z;k):
For corresponding results for the other nine func-
tions see http://dlmf.nist.gov/22.6.ii . See also
Carlson (2004).
22.6.171 cn (2z;k)
1 + cn (2z;k)=sn2(z;k) dn2(z;k)
cn2(z;k);
22.6.181 dn (2z;k)
1 + dn (2z;k)=k2sn2(z;k) cn2(z;k)
dn2(z;k):
22.6(iii) Half Argument
22.6.19
sn2 1
2z;k
=1 cn (z;k)
1 + dn (z;k)=1 dn (z;k)
k2(1 + cn (z;k))
=dn (z;k) k2cn (z;k) k02
k2(dn (z;k) cn (z;k));
22.6.20
cn2 1
2z;k
= k02+ dn (z;k) +k2cn (z;k)
k2(1 + cn (z;k))
=k02(1 dn (z;k))
k2(dn (z;k) cn (z;k))
=k02(1 + cn (z;k))
k02+ dn (z;k) k2cn (z;k);22.6.21
dn2 1
2z;k
=k2cn (z;k) + dn (z;k) +k02
1 + dn (z;k)
=k02(1 cn (z;k))
dn (z;k) cn (z;k)
=k02(1 + dn (z;k))
k02+ dn (z;k) k2cn (z;k):
Iffp,q,rgis any permutation of fc,d,ng, then
22.6.22
pq2 1
2z;k
=ps (z;k) + rs (z;k)
qs (z;k) + rs (z;k)
=pq (z;k) + rq (z;k)
1 + rq (z;k)=pr (z;k) + 1
qr (z;k) + 1:
For (22.6.22) and similar results, see Carlson (2004).
22.6(iv) Rotation of Argument (Jacobi's
Imaginary Transformation)
Table 22.6.1 : Jacobi's imaginary transformation of Ja-
cobian elliptic functions.
sn (iz;k) =isc (z;k0)dc (iz;k) = dn (z;k0)
cn (iz;k) = nc (z;k0) nc (iz;k) = cn (z;k0)
dn (iz;k) = dc (z;k0) sc (iz;k) =isn (z;k0)
cd (iz;k) = nd (z;k0)ns (iz;k) = ics (z;k0)
sd (iz;k) =isd (z;k0)ds (iz;k) = ids (z;k0)
nd (iz;k) = cd (z;k0) cs (iz;k) = ins (z;k0)
22.6(v) Change of Modulus
Seex22.17.
22.7 Landen Transformations
22.7(i) Descending Landen Transformation
With
22.7.1 k1=1 k0
1 +k0;
22.7.2 sn (z;k) =(1 +k1) sn (z=(1 +k1);k1)
1 +k1sn2(z=(1 +k1);k1);
22.7.3
cn (z;k) =cn (z=(1 +k1);k1) dn (z=(1 +k1);k1)
1 +k1sn2(z=(1 +k1);k1);
22.7.4 dn (z;k) =dn2(z=(1 +k1);k1) (1 k1)
1 +k1 dn2(z=(1 +k1);k1):
22.8 Addition Theorems 557
22.7(ii) Ascending Landen Transformation
With
22.7.5 k2=2p
k
1 +k; k0
2=1 k
1 +k;
22.7.6
sn (z;k)
=(1 +k0
2) sn (z=(1 +k0
2);k2) cn (z=(1 +k0
2);k2)
dn (z=(1 +k0
2);k2);
22.7.7 cn (z;k) =(1 +k0
2)(dn2(z=(1 +k0
2);k2) k0
2)
k2
2dn (z=(1 +k0
2);k2);
22.7.8 dn (z;k) =(1 k0
2)(dn2(z=(1 +k0
2);k2) +k0
2)
k2
2dn (z=(1 +k0
2);k2):
22.7(iii) Generalized Landen Transformations
See Khare and Sukhatme (2004).
22.8 Addition Theorems
22.8(i) Sum of Two Arguments
Foru;v2C, and with the common modulus ksup-
pressed:
22.8.1 sn (u+v) =snucnvdnv+ snvcnudnu
1 k2sn2usn2v;
22.8.2 cn (u+v) =cnucnv snudnusnvdnv
1 k2sn2usn2v;
22.8.3 dn (u+v) =dnudnv k2snucnusnvcnv
1 k2sn2usn2v:
See also Carlson (2004).
For the other nine functions see http://dlmf.nist.
gov/22.8.i .
22.8(ii) Alternative Forms for Sum of Two
Arguments
Foru;v2C, and with the common modulus ksup-
pressed:
22.8.13 sn (u+v) =sn2u sn2v
snucnvdnv snvcnudnu;
22.8.14 sn (u+v) =snucnudnv+ snvcnvdnu
cnucnv+ snudnusnvdnv;
22.8.15 cn (u+v) =snucnudnv snvcnvdnu
snucnvdnv snvcnudnu;
22.8.16 cn (u+v) =1 sn2u sn2v+k2sn2usn2v
cnucnv+ snudnusnvdnv;
22.8.17 dn (u+v) =snucnvdnu snvcnudnv
snucnvdnv snvcnudnu;
22.8.18 dn (u+v) =cnudnucnvdnv+k02snusnv
cnucnv+ snudnusnvdnv:
See also Carlson (2004).22.8(iii) Special Relations Between Arguments
In the following equations the common modulus kis
again suppressed.
Let
22.8.19 z1+z2+z3+z4= 0:
Then
22.8.20 snz1cnz1dnz11
snz2cnz2dnz21
snz3cnz3dnz31
snz4cnz4dnz41 = 0;
and
22.8.21k02 k02k2snz1snz2snz3snz4
+k2cnz1cnz2cnz3cnz4
dnz1dnz2dnz3dnz4= 0:
A geometric interpretation of (22.8.20) analogous
to that of (23.10.5) is given in Whittaker and Watson
(1927, p. 530).
Next, let
22.8.22 z1+z2+z3+z4= 2K(k):
Then
22.8.23 snz1cnz1cnz1dnz1cnz1dnz1
snz2cnz2cnz2dnz2cnz2dnz2
snz3cnz3cnz3dnz3cnz3dnz3
snz4cnz4cnz4dnz4cnz4dnz4 = 0:
For these and related identities see Copson (1935,
pp. 415{416).
If sums/di erences of the zj's are rational multiples
ofK(k), then further relations follow. For instance, if
22.8.24 z1 z2=z2 z3=2
3K(k);
then
22.8.25(dnz2+ dnz3)(dnz3+ dnz1)(dnz1+ dnz2)
dnz1+ dnz2+ dnz3
is independent of z1,z2,z3. Similarly, if
22.8.26z1 z2=z2 z3=z3 z4=1
2K(k);
then
22.8.27 dnz1dnz3= dnz2dnz4=k0:
Greenhill (1959, pp. 121{130) reviews these results in
terms of the geometric poristic polygon constructions of
Poncelet. Generalizations are given in x22.9.
558 Jacobian Elliptic Functions
22.9 Cyclic Identities
22.9(i) Notation
The following notation is a generalization of that of
Khare and Sukhatme (2002).
Throughout this subsection mandpare positive in-
tegers with 1mp.
22.9.1s(2)
m;p= sn
z+ 2p 1(m 1)K(k);k
;
22.9.2c(2)
m;p= cn
z+ 2p 1(m 1)K(k);k
;
22.9.3d(2)
m;p= dn
z+ 2p 1(m 1)K(k);k
;
22.9.4s(4)
m;p= sn
z+ 4p 1(m 1)K(k);k
;
22.9.5c(4)
m;p= cn
z+ 4p 1(m 1)K(k);k
;
22.9.6d(4)
m;p= dn
z+ 4p 1(m 1)K(k);k
:
In the remainder of this section the rank of an iden-
tity is the maximum number of elliptic function factors
in each term in the identity. The value of pdetermines
the number of points in the identity. The argument zis
suppressed in the above notation, as all cyclic identities
are independent of z.
22.9(ii) Typical Identities of Rank 2
In this subsection 1 mpand 1np.
Three Points
With
22.9.7 = dn (2K(k)=3;k);
22.9.8s(4)
1;2s(4)
2;2+s(4)
2;2s(4)
3;2+s(4)
3;2s(4)
1;2=2 1
k2;
22.9.9c(4)
1;2c(4)
2;2+c(4)
2;2c(4)
3;2+c(4)
3;2c(4)
1;2= (+ 2)
(1 +)2;
22.9.10d(2)
1;2d(2)
2;2+d(2)
2;2d(2)
3;2+d(2)
3;2d(2)
1;2
=d(4)
1;2d(4)
2;2+d(4)
2;2d(4)
3;2+d(4)
3;2d(4)
1;2=(+ 2):
These identities are cyclic in the sense that each of
the indices m;n in the rst product of, for example, the
forms(4)
m;2s(4)
n;2aresimultaneously permuted in the cyclic
order:m!m+1!m+2!p!1!2!m 1;
n!n+ 1!n+ 2!p!1!2!n 1. Many
of the identities that follow also have this property.
22.9(iii) Typical Identities of Rank 3
Two Points
22.9.11
d(2)
1;22
d(2)
2;2
d(2)
2;22
d(2)
1;2=k0
d(2)
1;2d(2)
2;2
;
22.9.12 c(2)
1;2s(2)
1;2d(2)
2;2+c(2)
2;2s(2)
2;2d(2)
1;2= 0:Three Points
Withde ned as in (22.9.7),
22.9.13s(4)
1;3s(4)
2;3s(4)
3;3= 1
1 2
s(4)
1;3+s(4)
2;3+s(4)
3;3
;
22.9.14c(4)
1;3c(4)
2;3c(4)
3;3=2
1 2
c(4)
1;3+c(4)
2;3+c(4)
3;3
;
22.9.15d(2)
1;3d(2)
2;3d(2)
3;3
=2+k2 1
1 2
d(2)
1;3+d(2)
2;3+d(2)
3;3
;
22.9.16s(4)
1;3c(4)
2;3c(4)
3;3+s(4)
2;3c(4)
3;3c(4)
1;3+s(4)
3;3c(4)
1;3c(4)
2;3
=(+ 2)
1 2
s(4)
1;3+s(4)
2;3+s(4)
3;3
:
Four Points
22.9.17
d(2)
1;4d(2)
2;4d(2)
3;4d(2)
2;4d(2)
3;4d(2)
4;4+d(2)
3;4d(2)
4;4d(2)
1;4d(2)
4;4d(2)
1;4d(2)
2;4
=k0
d(2)
1;4+d(2)
2;4d(2)
3;4+d(2)
4;4
;
22.9.18
d(2)
1;42
d(2)
3;4
d(2)
2;42
d(2)
4;4+
d(2)
3;42
d(2)
1;4
d(2)
4;42
d(2)
2;4=k0
d(2)
1;4d(2)
2;4+d(2)
3;4d(2)
4;4
;
22.9.19c(2)
1;4s(2)
1;4d(2)
3;4+c(2)
3;4s(2)
3;4d(2)
1;4
=c(2)
2;4s(2)
2;4d(2)
4;4+c(2)
4;4s(2)
4;4d(2)
2;4= 0:
For identities of rank 4 and higher see http://dlmf.
nist.gov/22.9.iv .
22.10 Maclaurin Series
22.10(i) Maclaurin Series in z
Initial terms are given by
22.10.1
sn (z;k) =z
1 +k2z3
3!+
1 + 14k2+k4z5
5!
1 + 135k2+ 135k4+k6z7
7!+O
z9
;
22.10.2cn (z;k) = 1 z2
2!+
1 + 4k2z4
4!
1 + 44k2+ 16k4z6
6!+O
z8
;
22.10.3dn (z;k) = 1 k2z2
2!+k2
4 +k2z4
4!
k2
16 + 44k2+k4z6
6!+O
z8
:
Further terms may be derived by substituting
in the di erential equations (22.13.13), (22.13.14),
(22.13.15). The full expansions converge when jzj<
min (K(k);K0(k)).
22.11 Fourier and Hyperbolic Series 559
22.10(ii) Maclaurin Series in kandk0
Initial terms are given by
22.10.4
sn (z;k) = sinz k2
4(z sinzcosz) cosz+O
k4
;
22.10.5
cn (z;k) = cosz+k2
4(z sinzcosz) sinz+O
k4
;
22.10.6 dn (z;k) = 1 k2
2sin2z+O
k4
;
22.10.7sn (z;k) = tanhz k02
4(z sinhzcoshz) sech2z
+O
k04
;
22.10.8
cn (z;k) = sechz+k02
4(z sinhzcoshz) tanhzsechz
+O
k04
;
22.10.9
dn (z;k) = sechz+k02
4(z+ sinhzcoshz) tanhzsechz
+O
k04
:
Further terms may be derived from the di eren-
tial equations (22.13.13), (22.13.14), (22.13.15), or from
the integral representations of the inverse functions in
x22.15(ii). The radius of convergence is the distance to
the origin from the nearest pole in the complex k-plane
in the case of (22.10.4){(22.10.6), or complex k0-plane
in the case of (22.10.7){(22.10.9); see x22.17.
22.11 Fourier and Hyperbolic Series
Throughout this section qandare de ned as inx22.2.
Ifqexp(2j=j)<1, then
22.11.1 sn (z;k) =2
Kk1X
n=0qn+1
2sin((2n+ 1))
1 q2n+1;
22.11.2 cn (z;k) =2
Kk1X
n=0qn+1
2cos((2n+ 1))
1 +q2n+1;
22.11.3 dn (z;k) =
2K+2
K1X
n=1qncos(2n)
1 +q2n:
For the other nine functions see http://dlmf.nist.
gov/22.11 .
Next, with E=E(k) denoting the complete
elliptic integral of the second kind ( x19.2(ii)) and
qexp(2j=j)<1,
22.11.13
sn2(z;k) =1
k2
1 E
K
22
k2K21X
n=1nqn
1 q2ncos(2n):Similar expansions for cn2(z;k) and dn2(z;k) follow im-
mediately from (22.6.1).
For further Fourier series see Oberhettinger (1973,
pp. 23{27).
A related hyperbolic series is
22.11.14
k2sn2(z;k)
=E0
K0
2K021X
n= 1
sech2
2K0(z 2nK)
;
whereE0=E0(k) is de ned byx19.2.9. Again, similar
expansions for cn2(z;k) and dn2(z;k) may be derived
via (22.6.1). See Dunne and Rao (2000).
22.12 Expansions in Other Trigonometric
Series and Doubly-In nite Partial
Fractions: Eisenstein Series
Witht2Cand
22.12.1 =iK0(k)=K(k);
22.12.2
2Kksn (2Kt;k ) =1X
n= 1
sin
(t (n+1
2) )
=1X
n= 1 1X
m= 1( 1)m
t m (n+1
2) !
;
22.12.3
2iKkcn (2Kt;k ) =1X
n= 1( 1)n
sin
(t (n+1
2) )
=1X
n= 1 1X
m= 1( 1)m+n
t m (n+1
2) !
;
22.12.4
2iKdn (2Kt;k )
= lim
N!1NX
n= N( 1)n
tan
(t (n+1
2) )
= lim
N!1NX
n= N( 1)n
lim
M!1MX
m= M1
t m (n+1
2) !
:
The double sums in (22.12.2){(22.12.4) are convergent
but not absolutely convergent, hence the order of the
summations is important. Compare x20.5(iii).
For corresponding expansions for the subsidiary
functions see http://dlmf.nist.gov/22.12 .
560 Jacobian Elliptic Functions
22.13 Derivatives and Di erential
Equations
22.13(i) Derivatives
Table 22.13.1 : Derivatives of Jacobian elliptic functions
with respect to variable.
d
dz(snz) = cnzdnzd
dz(dcz) =k02sczncz
d
dz(cnz) = snzdnzd
dz(ncz) = sczdcz
d
dz(dnz) = k2snzcnzd
dz(scz) = dczncz
d
dz(cdz) = k02sdzndzd
dz(nsz) = dszcsz
d
dz(sdz) = cdzndzd
dz(dsz) = csznsz
d
dz(ndz) =k2sdzcdzd
dz(csz) = nszdsz
Note that each derivative in Table 22.13.1 is a constant
multiple of the product of the corresponding copolar
functions. (The modulus kis suppressed throughout
the table.)
For alternative, and symmetric, formulations of
these results see Carlson (2004, 2006a).
22.13(ii) First-Order Di erential Equations
22.13.1d
dzsn (z;k)2
=
1 sn2(z;k)
1 k2sn2(z;k)
;
22.13.2d
dzcn (z;k)2
=
1 cn2(z;k)
k02+k2cn2(z;k)
;
22.13.3d
dzdn (z;k)2
=
1 dn2(z;k)
dn2(z;k) k02
:
For corresponding equations for the subsidiary functions
seehttp://dlmf.nist.gov/22.13.ii .
For alternative, and symmetric, formulations of
these results see Carlson (2006a).
22.13(iii) Second-Order Di erential Equations
22.13.13
d2
dz2sn (z;k) = (1 +k2) sn (z;k) + 2k2sn3(z;k);
22.13.14
d2
dz2cn (z;k) = (k02 k2) cn (z;k) 2k2cn3(z;k);
22.13.15
d2
dz2dn (z;k) = (1 +k02) dn (z;k) 2 dn3(z;k):
For corresponding equations for the subsidiary func-
tions see http://dlmf.nist.gov/22.13.iii .
For alternative, and symmetric, formulations of
these results see Carlson (2006a).22.14 Integrals
22.14(i) Inde nite Integrals of Jacobian Elliptic
Functions
Withx2R,
22.14.1Z
sn (x;k)dx=k 1ln(dn (x;k) kcn (x;k));
22.14.2Z
cn (x;k)dx=k 1Arccos(dn ( x;k));
22.14.3Z
dn (x;k)dx= Arcsin(sn ( x;k)) = am (x;k):
The branches of the inverse trigonometric functions are
chosen so that they are continuous. See x22.16(i) for
am (z;k).
For alternative, and symmetric, formulations of
these results see Carlson (2006a).
For the corresponding results for the subsidiary func-
tions see http://dlmf.nist.gov/22.14.i .
22.14(ii) Inde nite Integrals of Powers of
Jacobian Elliptic Functions
Seex22.16(ii). The inde nite integral of the 3rd power
of a Jacobian function can be expressed as an elemen-
tary function of Jacobian functions and a product of
Jacobian functions. The inde nite integral of a 4th
power can be expressed as a complete elliptic integral,
a polynomial in Jacobian functions, and the integra-
tion variable. See Lawden (1989, pp. 87{88). See also
Gradshteyn and Ryzhik (2000, pp. 618{619) and Carl-
son (2006a).
For inde nite integrals of squares and products of
even powers of Jacobian functions in terms of symmet-
ric elliptic integrals, see Carlson (2006b).
22.14(iii) Other Inde nite Integrals
In (22.14.13){(22.14.15), 0 <x< 2K.
22.14.13Zdx
sn (x;k)= lnsn (x;k)
cn (x;k) + dn (x;k)
;
22.14.14Zcn (x;k)dx
sn (x;k)=1
2ln1 dn (x;k)
1 + dn (x;k)
;
22.14.15Zcn (x;k)dx
sn2(x;k)= dn (x;k)
sn (x;k):
For additional results see Gradshteyn and Ryzhik (2000,
pp. 619{622) and Lawden (1989, Chapter 3).
22.15 Inverse Functions 561
22.14(iv) De nite Integrals
22.14.16ZK(k)
0ln(sn (t;k))dt= 1
4K0(k) 1
2K(k) lnk;
22.14.17ZK(k)
0ln(cn (t;k))dt= 1
4K0(k) +1
2K(k) ln(k0=k);
22.14.18ZK(k)
0ln(dn (t;k))dt=1
2K(k) lnk0:
Corresponding results for the subsidiary functions
follow by subtraction; compare (22.2.10).
22.15 Inverse Functions
22.15(i) De nitions
The inverse Jacobian elliptic functions can be de ned in
an analogous manner to the inverse trigonometric func-
tions (x4.23). With real variables, the solutions of the
equations
22.15.1 sn (;k) =x, 1x1,
22.15.2 cn (;k) =x, 1x1,
22.15.3 dn (;k) =x,k0x1,
are denoted respectively by
22.15.4
= arcsn(x;k); = arccn(x;k); = arcdn(x;k):
Each of these inverse functions is multivalued. The prin-
cipal values satisfy
22.15.5 Karcsn(x;k)K;
22.15.6 0arccn(x;k)2K;
22.15.7 0arcdn(x;k)K;
and unless stated otherwise it is assumed that the in-
verse functions assume their principal values. The gen-
eral solutions of (22.15.1), (22.15.2), (22.15.3) are, re-
spectively,
22.15.8 = ( 1)marcsn(x;k) + 2mK;
22.15.9 =arccn(x;k) + 4mK;
22.15.10 =arcdn(x;k) + 2mK;
wherem2Z.
Equations (22.15.1) and (22.15.4), for arcsn( x;k),
are equivalent to (22.15.12) and also to
22.15.11x=Zsn (x;k)
0dtp
(1 t2)(1 k2t2),
1x1, 0k1.
Similarly with (22.15.13){(22.15.14) and also the other
nine Jacobian elliptic functions.22.15(ii) Representations as Elliptic Integrals
22.15.12
arcsn(x;k) =Zx
0dtp
(1 t2)(1 k2t2), 1x1,
22.15.13
arccn(x;k) =Z1
xdtq
(1 t2)(k02+k2t2), 1x1,
22.15.14
arcdn(x;k) =Z1
xdtq
(1 t2)(t2 k02),k0x1 .
For the corresponding results for the subsidiary func-
tions see http://dlmf.nist.gov/22.15.ii .
The integrals (22.15.12){(22.15.14) can be regarded
asnormal forms for representing the inverse functions.
Other integrals, for example,Zb
xdtp
(a2+t2)(b2 t2)
can be transformed into normal form by elementary
change of variables. Comprehensive treatments are
given by Carlson (2005), Lawden (1989, pp. 52{55),
Bowman (1953, Chapter IX), and Erd elyi et al. (1953b,
pp. 296{301). See also Abramowitz and Stegun (1964,
p. 596).
For representations of the inverse functions as sym-
metric elliptic integrals see x19.25(v). For power-series
expansions see Carlson (2008).
22.16 Related Functions
22.16(i) Jacobi's Amplitude ( am) Function
De nition
22.16.1 am (x;k) = Arcsin(sn ( x;k)),x2R,
where the inverse sine has its principal value when
KxKand is de ned by continuity elsewhere.
See Figure 22.16.1. am ( x;k) is an in nitely di eren-
tiable function of x.
Quasi-Periodicity
22.16.2 am (x+ 2K;k) = am (x;k) +:
Integral Representation
22.16.3 am (x;k) =Zx
0dn (t;k)dt:
Special Values
22.16.4 am (x;0) =x;
22.16.5 am (x;1) = gd(x):
For the Gudermannian function gd( x) seex4.23(viii).
Approximation for Small x
22.16.6 am (x;k) =x k2x3
3!+k2
4 +k2x5
5!+O
x7
:
562 Jacobian Elliptic Functions
Approximations for Small k,k0
22.16.7 am (x;k) =x 1
4k2(x sinxcosx) +O
k4
;
22.16.8am (x;k) = gdx 1
4k02(x sinhxcoshx) sechx
+O
k04
:
Fourier Series
Withqas in (22.2.1) and =x=(2K),
22.16.9 am (x;k) =
2Kx+ 21X
n=1qnsin(2n)
n(1 +q2n):
Relation to Elliptic Integrals
If KxK, then the following four equations are
equivalent:
22.16.10 x=F( ;k);
22.16.11 am (x;k) = ;
22.16.12 sn (x;k) = sin = sin(am (x;k));
22.16.13 cn (x;k) = cos = cos(am (x;k)):
ForF( ;k) seex19.2(ii).
22.16(ii) Jacobi's Epsilon Function
De nition
Forx2R
22.16.14E(x;k) =Zx
0r
1 k2t2
1 t2dt;
compare (19.2.5). See Figure 22.16.2.
Other Integral Representations
22.16.15E(x;k) =x k2Zx
0sn2(t;k)dt;
22.16.16E(x;k) =k02x+k2Zx
0cn2(t;k)dt;
22.16.17E(x;k) =Zx
0dn2(t;k)dt:
For corresponding formulas for the subsidiary func-
tions see http://dlmf.nist.gov/22.16.ii .
Quasi-Addition and Quasi-Periodic Formulas
22.16.27
E(x1+x2;k) =E(x1;k) +E(x2;k)
k2sn (x1;k) sn (x2;k) sn (x1+x2;k);
22.16.28
E(x+K;k) =E(x;k) +E(k) k2sn (x;k) cd (x;k);
22.16.29E(x+ 2K;k) =E(x;k) + 2E(k):
ForE(k) seex19.2(ii).Relation to Theta Functions
22.16.30E(x;k) =1
2
3(0;q)4(;q)d
d4(;q) +E(k)
K(k)x;
where=x=2
3(0;q). Forjseex20.2(i). For E(k) see
x19.2(ii).
Relation to the Elliptic Integral E( ;k)
22.16.31 E(am (x;k);k) =E(x;k), KxK.
ForE( ;k) seex19.2(ii). See also (22.16.14).
22.16(iii) Jacobi's Zeta Function
De nition
WithE(k) andK(k) as inx19.2(ii) and x2R,
22.16.32 Z(xjk) =E(x;k) (E(k)=K(k))x:
See Figure 22.16.3. (Sometimes in the literature Z( xjk)
is denoted by Z(am ( x;k);k2).)
Properties
Z(xjk) satis es the same quasi-addition formula as the
functionE(x;k), given by (22.16.27). Also,
22.16.33 Z(x+Kjk) = Z(xjk) k2sn (x;k) cd (x;k);
22.16.34 Z(x+ 2Kjk) = Z(xjk):
22.16(iv) Graphs
Figure 22.16.1 : Jacobi's amplitude function am ( x;k) for
0x10andk= 0:4;0:7;0:99;0:999999. Values of
kgreater than 1 are illustrated in Figure 22.19.1.
22.17 Moduli Outside the Interval [0,1] 563
Figure 22.16.2 : Jacobi's epsilon function E(x;k) for
0x10andk= 0:4;0:7;0:99;0:999999. (These
graphs are similar to those in Figure 22.16.1; com-
pare (22.16.3), (22.16.17), and the graphs of dn ( x;k)
inx22.3(i).)
Figure 22.16.3 : Jacobi's zeta function Z( xjk) for 0x
10andk= 0:4;0:7;0:99;0:999999.
22.17 Moduli Outside the Interval [0,1]
22.17(i) Real or Purely Imaginary Moduli
Jacobian elliptic functions with real moduli in the in-
tervals ( 1;0) and (1;1), or with purely imaginary
moduli are related to functions with moduli in the in-
terval [0;1] by the following formulas.
First
22.17.1 pq (z;k) = pq (z; k);
for all twelve functions.
Secondly,
22.17.2 sn (z;1=k) =ksn (z=k;k );
22.17.3 cn (z;1=k) = dn (z=k;k );
22.17.4 dn (z;1=k) = cn (z=k;k ):
Thirdly, with
22.17.5 k1=kp
1 +k2; k 1k0
1=k
1 +k2;22.17.6 sn (z;ik) =k0
1sd (z=k0
1;k1);
22.17.7 cn (z;ik) = cd (z=k0
1;k1);
22.17.8 dn (z;ik) = nd (z=k0
1;k1):
In terms of the coecients of the power series of
x22.10(i), the above equations are polynomial identities
ink. In (22.17.5) either value of the square root can be
chosen.
22.17(ii) Complex Moduli
Whenzis xed each of the twelve Jacobian elliptic
functions is a meromorphic function of k2. For illus-
trations see Figures 22.3.25{22.3.27. In consequence,
the formulas in this chapter remain valid when kis
complex. In particular, the Landen transformations
inxx22.7(i) and 22.7(ii) are valid for all complex val-
ues ofk, irrespective of which values ofp
kandk0=p
1 k2are chosen|as long as they are used consis-
tently. For proofs of these results and further informa-
tion see Walker (2003).
Applications
22.18 Mathematical Applications
22.18(i) Lengths and Parametrization of Plane
Curves
Ellipse
22.18.1
x2=a2
+
y2=b2
= 1;
withab>0, is parametrized by
22.18.2 x=asn (u;k); y =bcn (u;k);
wherek=p
1 (b2=a2) is the eccentricity, and 0
u4K(k). The arc length l(u) in the rst quadrant,
measured from u= 0, is
22.18.3 l(u) =aE(u;k);
whereE(u;k) is Jacobi's epsilon function ( x22.16(ii)).
Lemniscate
In polar coordinates, x=rcos ,y=rsin , the lem-
niscate is given by r2= cos(2 ), 0 2. The arc
lengthl(r), measured from = 0, is
22.18.4 l(r) = (1=p
2) arccn
r;1=p
2
:
Inversely:
22.18.5 r= cnp
2l;1=p
2
;
and
22.18.6x= cnp
2l;1=p
2
dnp
2l;1=p
2
;
y= cnp
2l;1=p
2
snp
2l;1=p
2.p
2:
564 Jacobian Elliptic Functions
For these and other examples see Lawden (1989,
Chapter 4), Whittaker and Watson (1927, x22.8), and
Siegel (1988, pp. 1{7).
22.18(ii) Conformal Mapping
Withk2[0;1] the mapping z!w= sn (z;k) gives a
conformal map of the closed rectangle [ K;K ][0;K0]
onto the half-plane =w0, with 0;K;K+iK0;iK0
mapping to 0 ;1;k 2;1respectively. The half-open
rectangle ( K;K )[ K0;K0] maps onto Ccut along
the intervals ( 1; 1] and [1;1). See Akhiezer (1990,
Chapter 8) and McKean and Moll (1999, Chapter 2)
for discussions of the inverse mapping. Bowman (1953,
Chapters V{VI) gives an overview of the use of Jacobian
elliptic functions in conformal maps for engineering ap-
plications.
22.18(iii) Uniformization and Other
Parametrizations
By use of the functions sn and cn, parametrizations of
algebraic equations, such as
22.18.7ax2y2+b(x2y+xy2) +c(x2+y2)
+ 2dxy+e(x+y) +f= 0;
in whicha;b;c;d;e;f are real constants, can be achieved
in terms of single-valued functions. This circumvents
the cumbersome branch structure of the multivalued
functionsx(y) ory(x), and constitutes the process of
uniformization ; see Siegel (1988, Chapter II). See Bax-
ter (1982, p. 471) for an example from statistical me-
chanics. Discussion of parametrization of the angles
of spherical trigonometry in terms of Jacobian elliptic
functions is given in Greenhill (1959, p. 131) and Law-
den (1989,x4.4).
22.18(iv) Elliptic Curves and the Jacobi{Abel
Addition Theorem
Algebraic curves of the form y2=P(x), wherePis
a nonsingular polynomial of degree 3 or 4 (see McK-
ean and Moll (1999, x1.10)), are elliptic curves , which
are also considered in x23.20(ii). The special case
y2= (1 x2)(1 k2x2) is in Jacobian normal form .
For any two points ( x1;y1) and (x2;y2) on this curve,
their sum (x3;y3), always a third point on the curve, is
de ned by the Jacobi{Abel addition law
22.18.8
x3=x1y2+x2y1
1 k2x2
1x2
2;
y3=y1y2+x2( (1 +k2)x1+ 2k2x3
1)
1 k2x2
1x2
2+x32k2x1y1x2
2
1 k2x2
1x2
2;
a construction due to Abel; see Whittaker and Wat-
son (1927, pp. 442, 496{497). This provides an abeliangroup structure, and leads to important results in num-
ber theory, discussed in an elementary manner by Sil-
verman and Tate (1992), and more fully by Koblitz
(1993, Chapter 1, especially x1.7) and McKean and Moll
(1999, Chapter 3). The existence of this group struc-
ture is connected to the Jacobian elliptic functions via
the di erential equation (22.13.1). With the identi -
cationx= sn (z;k),y=d(sn (z;k))/dz, the addition
law (22.18.8) is transformed into the addition theorem
(22.8.1); see Akhiezer (1990, pp. 42, 45, 73{74) and
McKean and Moll (1999, xx2.14, 2.16). The theory
of elliptic functions brings together complex analysis,
algebraic curves, number theory, and geometry: Lang
(1987), Siegel (1988), and Serre (1973).
22.19 Physical Applications
22.19(i) Classical Dynamics: The Pendulum
With appropriate scalings, Newton's equation of motion
for a pendulum with a mass in a gravitational eld con-
strained to move in a vertical plane at a xed distance
from a fulcrum is
22.19.1d2(t)
dt2= sin(t);
being the angular displacement from the point of sta-
ble equilibrium, = 0. The bounded ( )
oscillatory solution of (22.19.1) is traditionally written
22.19.2 sin 1
2(t)
= sin 1
2
sn
t;sin 1
2
;
for an initial angular displacement , withd/dt= 0 at
time 0; see Lawden (1989, pp. 114{117). The period is
4K
sin 1
2
. The angle =is aseparatrix , separat-
ing oscillatory and unbounded motion. With the same
initial conditions, if the sign of gravity is reversed then
the new period is 4 K0
sin 1
2
; see Whittaker (1964,
x44).
Alternatively, Sala (1989) writes:
22.19.3 (t) = 2 am
t;p
2=E
;
for the initial conditions (0) = 0, the point of stable
equilibrium for E= 0, andd(t)/dt=p
2E. Here
E=1
2(d(t)/dt)2+1 cos(t) is the energy , which is a
rst integral of the motion. This formulation gives the
bounded and unbounded solutions from the same for-
mula (22.19.3), for k1 andk1, respectively. Also,
(t) is not restricted to the principal range .
Figure 22.19.1 shows the nature of the solutions (t) of
(22.19.3) by graphing am ( x;k) for both 0k1, as
in Figure 22.16.1, and k1, where it is periodic.
22.19 Physical Applications 565
Figure 22.19.1 : Jacobi's amplitude function am ( x;k) for
0x10andk= 0:5;0:9999;1:0001;2. When
k<1, am (x;k) increases monotonically indicating that
the motion of the pendulum is unbounded in , cor-
responding to free rotation about the fulcrum; com-
pare Figure 22.16.1. As k!1 , plateaus are seen
as the motion approaches the separatrix where =n,
n=1;2;:::, at which points the motion is time inde-
pendent for k= 1. This corresponds to the pendulum
being \upside down" at a point of unstable equilibrium.
Fork>1, the motion is periodic in x, corresponding to
bounded oscillatory motion.
22.19(ii) Classical Dynamics: The Quartic
Oscillator
Classical motion in one dimension is described by New-
ton's equation
22.19.4d2x(t)
dt2= dV(x)
dx;
whereV(x) is the potential energy, and x(t) is the co-
ordinate as a function of time t. The potential
22.19.5 V(x) =1
2x21
4 x4
plays a prototypal role in classical mechanics (Law-
den (1989,x5.2)), quantum mechanics (Schulman (1981,
Chapter 29)), and quantum eld theory (Pokorski
(1987, p. 203), Parisi (1988, x14.6)). Its dynamics for
purely imaginary time is connected to the theory of in-
stantons (Itzykson and Zuber (1980, p. 572), Sch afer
and Shuryak (1998)), to WKB theory, and to large-
order perturbation theory (Bender and Wu (1973), Si-
mon (1982)).
For real and positive, three of the four possible
combinations of signs give rise to bounded oscillatory
motions. We consider the case of a particle of mass 1,
initially held at rest at displacement afrom the origin
and then released at time t= 0. The subsequent po-
sition as a function of time, x(t), for the three cases
is given with results expressed in terms of aand the
dimensionless parameter =1
2 a2.Case I:V(x) =1
2x2+1
4 x4
This is an example of Dung's equation ; see Ablowitz
and Clarkson (1991, pp. 150{152) and Lawden (1989,
pp. 117{119). The subsequent time evolution is always
oscillatory with period 4 K(k)=p1 +:
22.19.6x(t) =acnp
1 +t;1=p
2 + 1
:
Case II:V(x) =1
2x2 1
4 x4
There is bounded oscillatory motion near x= 0, with
period 4K(k)=p1 , for initial displacements with
jajp
1= :
22.19.7x(t) =asnp
1 t;1=p
1 1
:
Asa!p
1= from below the period diverges since
a=p
1= are points of unstable equilibrium.
Case III:V(x) = 1
2x2+1
4 x4
Two types of oscillatory motion are possible. For an ini-
tial displacement withp
1= jajp
2= , bounded
oscillations take place near one of the two points of sta-
ble equilibrium x=p
1= . Such oscillations, of pe-
riod 4K(k)=p, are given by:
22.19.8 x(t) =adnpt;p
2 1
:
Asa!p
2= from below the period diverges since
x= 0 is a point of unstable equlilibrium. For initial
displacement with jajp
2= the motion extends over
the full range axa:
22.19.9x(t) =acnp
2 1t;1=p
2 1
;
with period 4 K(k)=p2 1. Asjaj !p
2= from
above the period again diverges. Both the dn and cn
solutions approach asechtasa!p
2= from the ap-
propriate directions.
22.19(iii) Nonlinear ODEs and PDEs
Many nonlinear ordinary and partial di erential equa-
tions have solutions that may be expressed in terms of
Jacobian elliptic functions. These include the time de-
pendent, and time independent, nonlinear Schr odinger
equations (NLSE) (Drazin and Johnson (1993, Chap-
ter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the
Korteweg{de Vries (KdV) equation (Kruskal (1974), Li
and Olver (2000)), the sine-Gordon equation, and oth-
ers; see Drazin and Johnson (1993, Chapter 2) for an
overview. Such solutions include standing or station-
ary waves, periodic cnoidal waves, and single and multi-
solitons occurring in diverse physical situations such as
water waves, optical pulses, quantum
uids, and elec-
trical impulses (Hasegawa (1989), Carr et al. (2000),
Kivshar and Luther-Davies (1998), and Boyd (1998, Ap-
pendix D2.2)).
566 Jacobian Elliptic Functions
22.19(iv) Tops
The classical rotation of rigid bodies in free space or
about a xed point may be described in terms of ellip-
tic, or hyperelliptic , functions if the motion is integrable
(Audin (1999, Chapter 1)). Hyperelliptic functions u(z)
are solutions of the equation z=Ru
0(f(x)) 1=2dx,
wheref(x) is a polynomial of degree higher than 4.
Elementary discussions of this topic appear in Lawden
(1989,x5.7), Greenhill (1959, pp. 101{103), and Whit-
taker (1964, Chapter VI). A more abstract overview is
Audin (1999, Chapters III and IV), and a complete dis-
cussion of analytical solutions in the elliptic and hyper-
elliptic cases appears in Golubev (1960, Chapters V and
VII), the original hyperelliptic investigation being due
to Kowalevski (1889).
22.19(v) Other Applications
Numerous other physical or engineering applications in-
volving Jacobian elliptic functions, and their inverses, to
problems of classical dynamics, electrostatics, and hy-
drodynamics appear in Bowman (1953, Chapters VII
and VIII) and Lawden (1989, Chapter 5). Whittaker
(1964, Chapter IV) enumerates the complete class of
one-body classical mechanical problems that are solv-
able this way.
Computation
22.20 Methods of Computation
22.20(i) Via Theta Functions
A powerful way of computing the twelve Jacobian el-
liptic functions for real or complex values of both the
argumentzand the modulus kis to use the de nitions
in terms of theta functions given in x22.2, obtaining the
theta functions via methods described in x20.14.
22.20(ii) Arithmetic-Geometric Mean
Given real or complex numbers a0;b0, withb0=a0not
real and negative, de ne
22.20.1an=1
2(an 1+bn 1); bn= (an 1bn 1)1=2;
cn=1
2(an 1 bn 1);
forn1, where the square root is chosen so that
phbn=1
2(phan 1+ phbn 1), where ph an 1and
phbn 1are chosen so that their di erence is numeri-
cally less than . Then as n! 1 sequencesfang,
fbngconverge to a common limit M=M(a0;b0), the
arithmetic-geometric mean ofa0;b0. And since
22.20.2 max (jan Mj;jbn Mj;jcnj)(const.)2 2n;convergence is very rapid.
Forxreal andk2(0;1), use (22.20.1) with a0=
1,b0=k02(0;1),c0=k, and continue until
cNis zero to the required accuracy. Next, compute
N; N 1;:::; 0, where
22.20.3 N= 2NaNx;
22.20.4 n 1=1
2
n+ arcsincn
ansin n
;
and the inverse sine has its principal value ( x4.23(ii)).
Then
22.20.5sn (x;k) = sin 0;cn (x;k) = cos 0;
dn (x;k) =cos 0
cos( 1 0);
and the subsidiary functions can be found using
(22.2.10).
See also Wachspress (2000).
Example
To compute sn, cn, dn to 10D when x= 0:8,k= 0:65.
Four iterations of (22.20.1) lead to c4= 6:5
10 12. From (22.20.3) and (22.20.4) we obtain
1= 1:40213 91827 and 0= 0:76850 92170.
Then from (22.20.5), sn (0 :8;0:65) = 0:69506 42165,
cn (0:8;0:65) = 0 :71894 76580, dn (0 :8;0:65) =
0:89212 34349.
22.20(iii) Landen Transformations
By application of the transformations given in xx22.7(i)
and 22.7(ii), kork0can always be made sucently small
to enable the approximations given in x22.10(ii) to be
applied. The rate of convergence is similar to that for
the arithmetic-geometric mean.
Example
To compute dn ( x;k) to 6D for x= 0:2,k2= 0:19,
k0= 0:9.
From (22.7.1), k1=1
19andx=(1 +k1) =
0:19. From the rst two terms in (22.10.6) we nd
dn
0:19;1
19
= 0:999951. Then by using (22.7.4) we
have dn
0:2;p
0:19
= 0:996253.
If needed, the corresponding values of sn and cn
can be found subsequently by applying (22.10.4) and
(22.7.2), followed by (22.10.5) and (22.7.3).
22.20(iv) Lattice Calculations
If either orq=ei is given, then we use
k=2
2(0;q)=2
3(0;q),k0=2
4(0;q)=2
3(0;q),K=
1
22
3(0;q), andK0= i K, obtaining the values of
the theta functions as in x20.14.
Ifk;k0are given with k2+k02= 1 and=k0==k<0,
thenK;K0can be found from
22.20.6K=
2M(1;k0); K0=
2M(1;k);
using the arithmetic-geometric mean.
22.21 Tables 567
Example 1
Ifk=k0= 1=p
2, then three iterations of (22.20.1)
giveM= 0:84721 30848, and from (22.20.6) K=
=(2M) = 1:85407 46773 |in agreement with the value
of
1
42=(4p); compare (23.17.3) and (23.22.2).
Example 2
Ifk0= 1 i, then four iterations of (22.20.1) give
K= 1:23969 74481 + i0:56499 30988.
22.20(v) Inverse Functions
See Wachspress (2000).
22.20(vi) Related Functions
am (x;k) can be computed from its de nition (22.16.1)
or from its Fourier series (22.16.9). Alternatively, Sala
(1989) shows how to apply the arithmetic-geometric
mean to compute am ( x;k).
Jacobi's epsilon function can be computed from its
representation (22.16.30) in terms of theta functions and
complete elliptic integrals; compare x20.14. Jacobi's
zeta function can then be found by use of (22.16.32).
22.20(vii) Further References
For additional information on methods of computation
for the Jacobi and related functions, see the introduc-
tory sections in the following books: Lawden (1989),
Curtis (1964b), Milne-Thomson (1950), and Spenceley
and Spenceley (1947).
22.21 Tables
Spenceley and Spenceley (1947) tabulates sn ( Kx;k ),
cn (Kx;k ), dn (Kx;k ), am (Kx;k ),E(Kx;k ) for
arcsink= 1(1)89andx= 0 1
90
1 to 12D, or 12
decimals of a radian in the case of am ( Kx;k ).
Curtis (1964b) tabulates sn ( mK=n;k ),
cn (mK=n;k ), dn (mK=n;k ) forn= 2(1)15, m=
1(1)n 1, andq(notk) = 0(:005)0:35 to 20D.
Lawden (1989, pp. 280{284 and 293{297) tabulates
sn (x;k), cn (x;k), dn (x;k),E(x;k), Z(xjk) to 5D for
k= 0:1(:1)0:9,x= 0(:1)X, whereXranges from 1.5 to
2.2.
Zhang and Jin (1996, p. 678) tabulates sn ( Kx;k ),
cn (Kx;k ), dn (Kx;k ) fork=1
4;1
2andx= 0(:1)4 to
7D.
For other tables prior to 1961 see Fletcher et al.
(1962, pp. 500{503) and Lebedev and Fedorova (1960,
pp. 221{223).
Tables of theta functions ( x20.15) can also be used
to compute the twelve Jacobian elliptic functions by ap-
plication of the quotient formulas given in x22.2.22.22 Software
Seehttp://dlmf.nist.gov/22.22 .
References
General References
The main references used for the mathematical proper-
ties in this chapter are Bowman (1953), Copson (1935),
Lawden (1989), McKean and Moll (1999), Walker
(1996), Whittaker and Watson (1927), and for phys-
ical applications Drazin and Johnson (1993), Lawden
(1989), Walker (1996).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections in this chapter. These sources supplement the
references that are quoted in the text.
x22.2 Lawden (1989,x2.1), Whittaker and Watson
(1927,x22.1), Walker (1996, xx5.1, 6.2), Walker
(2003).
x22.3 These graphics were produced at NIST and by
the authors.
x22.4 Lawden (1989,xx2.1, 2.2), Whittaker and Wat-
son (1927,xx22.1{22.3), Walker (1996, x6.2).
x22.5 Lawden (1989,xx2.1{2.2, 2.6), Whittaker and
Watson (1927,x22.3).
x22.6 Lawden (1989,xx2.4{2.6), Whittaker and Wat-
son (1927,xx22.1, 22.4), Walker (1996, x6.2). For
(22.6.6) and (22.6.7) set u=v=zin (22.8.2) and
use (22.6.1) repeatedly.
x22.7 Lawden (1989,x3.9), Whittaker and Watson
(1927,x22.42), Walker (1996, p. 148).
x22.8 Lawden (1989,x2.4 and p. 43), Whittaker and
Watson (1927,x22.2 and p. 530), Walker (1996,
x6.2).
x22.9 Khare and Sukhatme (2002), Khare et al. (2003).
x22.10 Lawden (1989,xx2.5, 3.1). The expansions in
powers ofk0follow from those in powers of kby
use of Table 22.6.1 and (4.28.8){(4.28.10).
568 Jacobian Elliptic Functions
x22.11 Walker (1996,x5.4), Whittaker and Watson
(1927,x22.6). For (22.11.13) see Deconinck and
Kutz (2006, Eq. (48)). The version of this formula
in Byrd and Friedman (1971, p. 307, Eq. (911.01))
is incorrect; see Tang (1969).
x22.12 For the rst right-hand sides of (22.12.2){
(22.12.4) see Lawden (1989, x8.8). The sec-
ond right-hand sides of (22.12.2){(22.12.4) can
be obtained from the corresponding rst right-
hand sides by substituting, as appropriate, the
expansions csc() =P1
m= 1( 1)m=( m)
orcot() = limM!1PM
m= M1=( m); com-
pare (4.22.5) and (4.22.3).
x22.13 Lawden (1989, x2.5), Walker (1996, x6.2),
Whittaker and Watson (1927, xx22.1{22.2).(22.13.13){(22.13.15) are obtained by di erentia-
tion of (22.13.1){(22.13.3).
x22.14 Lawden (1989,x2.7), Whittaker and Watson
(1927,xx22.5, 22.72).
x22.15 Bowman (1953, Chapter 1), Lawden (1989,
xx3.1, 3.2), Whittaker and Watson (1927, x22.72).
x22.16 Lawden (1989,xx3.4{3.6), Walker (1996, x6.5),
Whittaker and Watson (1927, xx22.72{22.73).
The gures were produced at NIST.
x22.17 Lawden (1989,x3.9).
x22.19 Lawden (1989,xx5.1{5.2). The gure was pro-
duced at NIST.
x22.20 Walker (1996, pp. 141{143).
Chapter 23
Weierstrass Elliptic and Modular Functions
W. P. Reinhardt1and P. L. Walker2
Notation 570
23.1 Special Notation . . . . . . . . . . . . . 570
Weierstrass Elliptic Functions 570
23.2 De nitions and Periodic Properties . . . . 570
23.3 Di erential Equations . . . . . . . . . . . 571
23.4 Graphics . . . . . . . . . . . . . . . . . . 571
23.5 Special Lattices . . . . . . . . . . . . . . 574
23.6 Relations to Other Functions . . . . . . . 574
23.7 Quarter Periods . . . . . . . . . . . . . . 576
23.8 Trigonometric Series and Products . . . . 576
23.9 Laurent and Other Power Series . . . . . 577
23.10 Addition Theorems and Other Identities . 577
23.11 Integral Representations . . . . . . . . . 578
23.12 Asymptotic Approximations . . . . . . . . 578
23.13 Zeros . . . . . . . . . . . . . . . . . . . 579
23.14 Integrals . . . . . . . . . . . . . . . . . . 579Modular Functions 579
23.15 De nitions . . . . . . . . . . . . . . . . . 579
23.16 Graphics . . . . . . . . . . . . . . . . . . 579
23.17 Elementary Properties . . . . . . . . . . 580
23.18 Modular Transformations . . . . . . . . . 580
23.19 Interrelations . . . . . . . . . . . . . . . 581
Applications 581
23.20 Mathematical Applications . . . . . . . . 581
23.21 Physical Applications . . . . . . . . . . . 582
Computation 583
23.22 Methods of Computation . . . . . . . . . 583
23.23 Tables . . . . . . . . . . . . . . . . . . . 584
23.24 Software . . . . . . . . . . . . . . . . . . 584
References 584
1University of Washington, Seattle, Washington.
2American University of Sharjah, Sharjah, United Arab Emirates.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 18) by T. H. Southard.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
569
570 Weierstrass Elliptic and Modular Functions
Notation
23.1 Special Notation
(For other notation see pp. xiv and 873.)
L lattice in C.
`;n integers.
m integer, except in x23.20(ii).
z=x+iy complex variable, except in xx23.20(ii),
23.21(iii).
[a;b] or (a;b) closed, or open, straight-line segment
joiningaandb, whether or not aandb
are real.
primes derivatives with respect to the variable,
except where indicated otherwise.
K(k),K0(k) complete elliptic integrals ( x19.2(i)).
2!1;2!3 lattice generators ( =(!3=!1)>0).
!2 !1 !3.
=!3=!1 lattice parameter ( = >0).
q=ei!3=!1
=ei nome.
g2;g3 lattice invariants.
e1;e2;e3 zeros of Weierstrass normal cubic
4z3 g2z g3.
discriminant g3
2 27g2
3.
nZ set of all integer multiples of n.
S1=S2 set of all elements of S1, modulo
elements of S2. Thus two elements of
S1=S2are equivalent if they are both
inS1and their di erence is in S2. (For
an example seex20.12(ii).)
GH Cartesian product of groups GandH,
that is, the set of all pairs of elements
(g;h) with group operation
(g1;h1) + (g2;h2) = (g1+g2;h1+h2).
The main functions treated in this chapter are the
Weierstrass }-function}(z) =}(zjL) =}(z;g2;g3); the
Weierstrass zeta function (z) =(zjL) =(z;g2;g3);
the Weierstrass sigma function (z) =(zjL) =
(z;g2;g3); the elliptic modular function ( ); Klein's
complete invariant J( ); Dedekind's eta function ( ).
Other Notations
Whittaker and Watson (1927) requires only =(!3=!1)6=
0, instead of=(!3=!1)>0. Abramowitz and Ste-
gun (1964, Chapter 18) considers only rectangular and
rhombic lattices ( x23.5);!1,!3are replaced by !,!0for
the former and by !2,!0for the latter. Silverman and
Tate (1992) and Koblitz (1993) replace 2 !1and 2!3
by!1and!3, respectively. Walker (1996) normalizes
2!1= 1, 2!3= , and uses homogeneity ( x23.10(iv)).
McKean and Moll (1999) replaces 2 !1and 2!3by!1
and!2, respectively.Weierstrass Elliptic Functions
23.2 De nitions and Periodic Properties
23.2(i) Lattices
If!1and!3are nonzero real or complex numbers such
that=(!3=!1)>0, then the set of points 2 m!1+2n!3,
withm;n2Z, constitutes a lattice Lwith 2!1and 2!3
lattice generators .
The generators of a given lattice Lare not unique.
For example, if
23.2.1 !1+!2+!3= 0;
then 2!2, 2!3are generators, as are 2 !2, 2!1. In gen-
eral, if
23.2.2 1=a!1+b!3; 3=c!1+d!3;
wherea;b;c;d are integers, then 2 1, 2 3are generators
ofLi
23.2.3 ad bc= 1:
23.2(ii) Weierstrass Elliptic Functions
23.2.4}(z) =1
z2+X
w2Lnf0g1
(z w2) 1
w2
;
23.2.5(z) =1
z+X
w2Lnf0g1
z w+1
w+z
w2
;
23.2.6(z) =zY
w2Lnf0g
1 z
w
expz
w+z2
2w2
:
The double series and double product are absolutely and
uniformly convergent in compact sets in Cthat do not
include lattice points. Hence the order of the terms or
factors is immaterial.
Whenz =2Lthe functions are related by
23.2.7 }(z) = 0(z);
23.2.8 (z) =0(z)/(z):
}(z) and(z) are meromorphic functions with poles
at the lattice points. }(z) is even and (z) is odd. The
poles of}(z) are double with residue 0; the poles of (z)
are simple with residue 1. The function (z) is entire
and odd, with simple zeros at the lattice points. When
it is important to display the lattice with the functions
they are denoted by }(zjL),(zjL), and(zjL), respec-
tively.
23.3 Differential Equations 571
23.2(iii) Periodicity
If 2!1, 2!3is any pair of generators of L, and!2is
de ned by (23.2.1), then
23.2.9 }(z+ 2!j) =}(z),j= 1;2;3.
Hence}(z) is an elliptic function , that is,}(z) is mero-
morphic and periodic on a lattice; equivalently, }(z) is
meromorphic and has two periods whose ratio is not
real. We also have
23.2.10 }0(!j) = 0, j= 1;2;3.
The function (z) is quasi-periodic: for j= 1;2;3,
23.2.11 (z+ 2!j) =(z) + 2j;
where
23.2.12 j=(!j):
Also,
23.2.13 1+2+3= 0;
23.2.143!2 2!3=2!1 1!2=1!3 3!1=1
2i:
Forj= 1;2;3, the function (z) satis es
23.2.15 (z+ 2!j) = e2j(z+!j)(z);
23.2.16 0(2!j) = e2j!j:
More generally, if j= 1;2;3,k= 1;2;3,j6=k, and
m;n2Z, then
23.2.17
(z+ 2m!j+ 2n!k)/(z)
= ( 1)m+n+mnexp((2mj+ 2nk)(m!j+n!k+z)):
For further quasi-periodic properties of the -
function see Lawden (1989, x6.2).
23.3 Di erential Equations
23.3(i) Invariants, Roots, and Discriminant
The lattice invariants are de ned by
23.3.1g2= 60X
w2Lnf0gw 4;
23.3.2g3= 140X
w2Lnf0gw 6:
The lattice roots satisfy the cubic equation
23.3.3 4z3 g2z g3= 0;
and are denoted by e1;e2;e3. The discriminant
(x1.11(ii)) is given by
23.3.4 =g3
2 27g2
3= 16(e2 e3)2(e3 e1)2(e1 e2)2:
In consequence,
23.3.5 e1+e2+e3= 0;
23.3.6g2= 2(e2
1+e2
2+e2
3) = 4(e2e3+e3e1+e1e2);23.3.7 g3= 4e1e2e3=4
3(e3
1+e3
2+e3
3):
Letg3
26= 27g2
3, or equivalently be nonzero, or
e1;e2;e3be distinct. Given g2andg3there is a unique
lattice Lsuch that (23.3.1) and (23.3.2) are satis ed.
We may therefore de ne
23.3.8 }(z;g2;g3) =}(zjL):
Similarly for (z;g2;g3) and(z;g2;g3). As functions of
g2andg3,}(z;g2;g3) and(z;g2;g3) are meromorphic
and(z;g2;g3) is entire.
Conversely, g2,g3, and the setfe1;e2;e3gare deter-
mined uniquely by the lattice Lindependently of the
choice of generators. However, given any pair of gener-
ators 2!1, 2!3ofL, and with!2de ned by (23.2.1), we
can identify the ejindividually, via
23.3.9 ej=}(!jjL), j= 1;2;3.
In what follows, it will be assumed that (23.3.9) al-
ways applies.
23.3(ii) Di erential Equations and Derivatives
23.3.10 }02(z) = 4}3(z) g2}(z) g3;
23.3.11}02(z) = 4(}(z) e1)(}(z) e2)(}(z) e3);
23.3.12 }00(z) = 6}2(z) 1
2g2;
23.3.13 }000(z) = 12}(z)}0(z):
See also (23.2.7) and (23.2.8).
23.4 Graphics
23.4(i) Real Variables
See Figures 23.4.1{23.4.7 for line graphs of the Weier-
strass functions }(x),(x), and(x), illustrating the
lemniscatic and equianharmonic cases. (The gures in
this subsection may be compared with the gures in
x22.3(i).)
Figure 23.4.1 :}(x;g2;0) for 0x9,g2= 0.1, 0.2,
0.5, 0.8. (Lemniscatic case.)
572 Weierstrass Elliptic and Modular Functions
Figure 23.4.2 :}(x; 0;g3) for 0x9,g3= 0.1, 0.2,
0.5, 0.8. (Equianharmonic case.)
Figure 23.4.3 :(x;g2;0) for 0x8,g2= 0.1, 0.2,
0.5, 0.8. (Lemniscatic case.)
Figure 23.4.4 :(x; 0;g3) for 0x8,g3= 0.1, 0.2,
0.5, 0.8. (Equianharmonic case.)
Figure 23.4.5 :(x;g2;0) for 5x5,g2= 0.1, 0.2,
0.5, 0.8. (Lemniscatic case.)
Figure 23.4.6 :(x; 0;g3) for 5x5,g3= 0.1, 0.2,
0.5, 0.8. (Equianharmonic case.)
Figure 23.4.7 :}(x) with!1=K(k),!3=iK0(k) for
0x9,k2= 0.2, 0.8, 0.95, 0.99. (Lemniscatic case.)
23.4 Graphics 573
23.4(ii) Complex Variables
See Figures 23.4.8{23.4.12 for surfaces for the Weierstrass functions }(z),(z), and(z). Height corresponds to
the absolute value of the function and color to the phase. See also p. xiv. (The gures in this subsection may be
compared with the gures in x22.3(iii).)
Figure 23.4.8 :}(x+iy) with!1=K(k),!3=iK0(k)
for 2K(k)x2K(k), 0y6K0(k),k2= 0:9.
(The scaling makes the lattice appear to be square.)
Figure 23.4.9 :}(x+iy; 1;4i) for 3:8x3:8,
3:8y3:8:(The variables are unscaled and the
lattice is skew.)
Figure 23.4.10 :(x+iy; 1;0) for 5x5, 5y
5:
Figure 23.4.11 :(x+iy; 1;i) for 2:5x2:5,
2:5y2:5:
574 Weierstrass Elliptic and Modular Functions
Figure 23.4.12 :}(3:7;a+ib;0) for 5a3, 4
b4. There is a double zero at a=b= 0 and double
poles on the real axis.
23.5 Special Lattices
23.5(i) Real-Valued Functions
The Weierstrass functions take real values on the real
axis i the lattice is xed under complex conjugation:
L=L; equivalently, when g2;g32R. This happens in
the cases treated in the following four subsections.
23.5(ii) Rectangular Lattice
This occurs when both !1and!3=iare real and posi-
tive. Then >0 and the parallelogram with vertices
at 0, 2!1, 2!1+ 2!3, 2!3is a rectangle.
In this case the lattice roots e1,e2, ande3are real
and distinct. When they are identi ed as in (23.3.9)
23.5.1 e1>e2>e3; e 1>0>e3:
Also,e2andg3have opposite signs unless !3=i!1, in
which event both are zero.
As functions of=!3,e1ande2are decreasing and
e3is increasing.
23.5(iii) Lemniscatic Lattice
This occurs when !1is real and positive and !3=i!1.
The parallelogram 0, 2 !1, 2!1+ 2!3, 2!3is a square,
and
23.5.2 1=i3==(4!1);
23.5.3e1= e3=
1
44=(32!2
1); e 2= 0;
23.5.4g2=
1
48=(2562!4
1); g 3= 0:
Note also that in this case =i. In consequence,
23.5.5k2=1
2; K (k) =K0(k) =
1
42.
4p
:23.5(iv) Rhombic Lattice
This occurs when !1is real and positive, =!3>0,
<!3=1
2!1, and <0. The parallelogram 0, 2 !1 2!3,
2!1, 2!3, is a rhombus: see Figure 23.5.1.
The lattice root e1is real, and e3= e2, with=e2>0.
e1andg3have the same sign unless 2 !3= (1 +i)!1
when both are zero: the pseudo-lemniscatic case. As a
function of=e3the roote1is increasing. For the case
!3=ei=3!1seex23.5(v).
23.5(v) Equianharmonic Lattice
This occurs when !1is real and positive and !3=
ei=3!1. The rhombus 0, 2 !1 2!3, 2!1, 2!3can be
regarded as the union of two equilateral triangles: see
Figure 23.5.2.
23.5.6 1=ei=33=
2p
3!1;
and the lattice roots and invariants are given by
23.5.7e1=e2i=3e3=e 2i=3e2=
1
36
214=32!2
1;
23.5.8 g2= 0; g 3=
1
318
(4!1)6:
Note also that in this case =ei=3. In consequence,
23.5.9
k2=ei=3; K (k) =ei=6K0(k) =ei=1231=4
1
33
27=3:
23.6 Relations to Other Functions
23.6(i) Theta Functions
In this subsection 2 !1, 2!3are any pair of generators of
the lattice L, and the lattice roots e1,e2,e3are given
by (23.3.9).
23.6.1 q=ei ; =!3=!1:
23.6.2e1=2
12!2
1
4
2(0;q) + 24
4(0;q)
;
23.6.3e2=2
12!2
1
4
2(0;q) 4
4(0;q)
;
23.6.4e3= 2
12!2
1
24
2(0;q) +4
4(0;q)
:
23.6.5
}(z) e1=3(0;q)4(0;q)2(z=(2!1);q)
2!11(z=(2!1);q)2
;
23.6.6
}(z) e2=2(0;q)4(0;q)3(z=(2!1);q)
2!11(z=(2!1);q)2
;
23.6.7
}(z) e3=2(0;q)3(0;q)4(z=(2!1);q)
2!11(z=(2!1);q)2
:
23.6 Relations to Other Functions 575
Figure 23.5.1 : Rhombic lattice. <(2!3) =!1.
Figure 23.5.2 : Equianharmonic lattice. 2 !3=ei=32!1,
2!1 2!3=e i=32!1.
23.6.8 1= 2
12!1000
1(0;q)
0
1(0;q):
23.6.9(z) = 2!1exp1z2
2!11(z=(2!1);q)
0
1(0;q);
23.6.10(!1) = 2!1exp 1
21!1
2(0;q)
0
1(0;q);
23.6.11(!2) = 2!1iexp 1
21!1 2
3(0;q)
q1=40
1(0;q);
23.6.12(!3) = 2!1exp 1
21!1
4(0;q)
q1=40
1(0;q):
Withz=u/(2!1),
23.6.13 (u) =1
!1u+
2!1d
dzln1(z;q);
23.6.14}(u) =
2!12000
1(0;q)
30
1(0;q) d2
dz2ln1(z;q)
;
23.6.15
(u+!j)
(!j)= exp
ju+ju2
2!1j+1(z;q)
j+1(0;q),j= 1;2;3.
For further results for the -function see Lawden
(1989,x6.2).
23.6(ii) Jacobian Elliptic Functions
Again, in Equations (23.6.16){(23.6.26), 2 !1;2!3are
any pair of generators of the lattice Lande1;e2;e3are
given by (23.3.9).
23.6.16 k2=e2 e3
e1 e3; k02=e1 e2
e1 e3;23.6.17K2= (K(k))2=!2
1(e1 e3);
K02= (K(k0))2=!2
3(e3 e1):
23.6.18 e1=K2
3!2
1(1 +k02);
23.6.19 e2=K2
3!2
1(k2 k02);
23.6.20 e3= K2
3!2
1(1 +k2):
23.6.21 }(z) e1=K2
!2
1cs2Kz
!1;k
;
23.6.22 }(z) e2=K2
!2
1ds2Kz
!1;k
;
23.6.23 }(z) e3=K2
!2
1ns2Kz
!1;k
:
23.6.24}(z+!1) e1=Kk0
!12
sc2Kz
!1;k
;
23.6.25}(z+!2) e2= Kkk0
!12
sd2Kz
!1;k
;
23.6.26}(z+!3) e3=Kk
!12
sn2Kz
!1;k
:
In (23.6.27){(23.6.29) the modulus kis given and
K=K(k),K0=K(k0) are the corresponding complete
elliptic integrals ( x19.2(ii)). Also, L1,L2,L3are the lat-
tices with generators (4 K;2iK0), (2K 2iK0;2K+
2iK0), (2K;4iK0), respectively.
23.6.27(zjL1) (z+ 2KjL1)+(2KjL1) = ns (z;k);
576 Weierstrass Elliptic and Modular Functions
23.6.28(zjL2) (z+ 2KjL2)+(2KjL2) = ds (z;k);
23.6.29
(zjL3) (z+ 2iK0jL3) (2iK0jL3) = cs (z;k):
Similar results for some of the other nine Jacobi
functions can be constructed with the aid of the trans-
formations given by Table 22.4.3, or for all nine by
referring to the augmented version of Table 22.4.3 at
http://dlmf.nist.gov/22.4.t3 .
For representations of the Jacobi functions sn, cn,
and dn as quotients of -functions see Lawden (1989,
xx6.2, 6.3).
23.6(iii) General Elliptic Functions
For representations of general elliptic functions
(x23.2(iii)) in terms of (z) and}(z) see Lawden (1989,
xx8.9, 8.10), and for expansions in terms of (z) see
Lawden (1989,x8.11).
23.6(iv) Elliptic Integrals
Rectangular Lattice
Letzbe on the perimeter of the rectangle with vertices
0;2!1;2!1+ 2!3;2!3. Thent=}(z) is real (xx23.5(i){
23.5(ii)), and
23.6.30z=1
2Z1
tdup
(u e1)(u e2)(u e3),
te1,z2(0;!1],
23.6.31z !1=i
2Ze1
tdup
(e1 u)(u e2)(u e3),
e2te1,z2[!1;!1+!3],
23.6.32z !3=1
2Zt
e3dup
(e1 u)(e2 u)(u e3),
e3te2,z2[!3;!1+!3],
23.6.33z=i
2Zt
1dup
(e1 u)(e2 u)(e3 u),
te3,z2(0;!3].
23.6.342!1=Z1
e1dup
(u e1)(u e2)(u e3)
=Ze2
e3dup
(e1 u)(e2 u)(u e3);
23.6.352!3=iZe1
e2dup
(e1 u)(u e2)(u e3)
=iZe3
1dup
(e1 u)(e2 u)(e3 u):
For (23.6.30){(23.6.35) and further identities see Law-
den (1989,x6.12).See alsoxx19.2(i), 19.14, and Erd elyi et al. (1953b,
x13.14).
For relations to symmetric elliptic integrals see
x19.25(vi).
General Lattice
Letzbe a point of Cdi erent from e1;e2;e3, and de ne
wby
23.6.36w=Z1
zdup
4u3 g2u g3
=1
2Z1
zdup
(u e1)(u e2)(u e3);
where the integral is taken along any path from zto
1that does not pass through any of e1;e2;e3. Then
z=}(w), where the value of wdepends on the choice
of path and determination of the square root; see McK-
ean and Moll (1999, pp. 87{88 and x2.5).
23.7 Quarter Periods
23.7.1} 1
2!1
=e1+p
(e1 e3)(e1 e2)
=e1+! 2
1(K(k))2k0;
23.7.2} 1
2!2
=e2 ip
(e1 e2)(e2 e3)
=e2 i! 2
1(K(k))2kk0;
23.7.3} 1
2!3
=e3 p
(e1 e3)(e2 e3)
=e3 ! 2
1(K(k))2k;
wherek;k0and the square roots are real and positive
when the lattice is rectangular; otherwise they are de-
termined by continuity from the rectangular case.
23.8 Trigonometric Series and Products
23.8(i) Fourier Series
Ifq=ei!3=!1,=(z=!1)<2=(!3=!1), andz =2L, then
23.8.1}(z) +1
!1 2
4!2
1csc2z
2!1
= 22
!2
11X
n=1nq2n
1 q2ncosnz
!1
;
23.8.2(z) 1z
!1
2!1cotz
2!1
=2
!11X
n=1q2n
1 q2nsinnz
!1
:
23.9 Laurent and Other Power Series 577
23.8(ii) Series of Cosecants and Cotangents
Whenz =2L,
23.8.3}(z) = 1
!1+2
4!2
11X
n= 1csc2(z+ 2n!3)
2!1
;
23.8.4(z) =1z
!1+
2!11X
n= 1cot(z+ 2n!3)
2!1
;where in (23.8.4) the terms in nand nare to be brack-
eted together (the Eisenstein convention orprincipal
value : see Weil (1999, p. 6) or Walker (1996, p. 3)).
23.8.51=2
2!1
1
6+1X
n=1csc2n! 3
!1!
;
with similar results for 2and3obtainable by use of
(23.2.14).
23.8(iii) In nite Products
23.8.6 (z) =2!1
exp1z2
2!1
sinz
2!11Y
n=11 2q2ncos(z=! 1) +q4n
(1 q2n)2;
23.8.7 (z) =2!1
exp1z2
2!1
sinz
2!11Y
n=1sin((2n!3+z)=(2!1)) sin((2n!3 z)=(2!1))
sin2(n! 3=!1):
23.9 Laurent and Other Power Series
Letz0(6= 0) be the nearest lattice point to the origin,
and de ne
23.9.1cn= (2n 1)X
w2Lnf0gw 2n,n= 2;3;4;:::.
Then
23.9.2 }(z) =1
z2+1X
n=2cnz2n 2, 0<jzj<jz0j,
23.9.3(z) =1
z 1X
n=2cn
2n 1z2n 1, 0<jzj<jz0j.
Here
23.9.4 c2=1
20g2; c 3=1
28g3;
23.9.5cn=3
(2n+ 1)(n 3)n 2X
m=2cmcn m,n4.
Explicit coecients cnin terms of c2andc3are given
up toc19in Abramowitz and Stegun (1964, p. 636).
Forj= 1;2;3, and with ejas inx23.3(i),
23.9.6}(!j+t) =ej+ (3e2
j 5c2)t2+ (10c2ej+ 21c3)t4
+ (7c2e2
j+ 21c3ej+ 5c2
2)t6+O
t8
;
ast!0. For the next four terms see Abramowitz and
Stegun (1964, (18.5.56)). Also, Abramowitz and Ste-
gun (1964, (18.5.25)) supplies the rst 22 terms in the
reverted form of (23.9.2) as 1 =}(z)!0.
Forz2C
23.9.7
(z) =1X
m;n=0am;n(10c2)m(56c3)nz4m+6n+1
(4m+ 6n+ 1)!;wherea0;0= 1,am;n= 0 if either morn<0, and
23.9.8am;n= 3(m+ 1)am+1;n 1+16
3(n+ 1)am 2;n+1
1
3(2m+ 3n 1)(4m+ 6n 1)am 1;n:
Foram;nwithm= 0;1;:::; 12 andn= 0;1;:::; 8, see
Abramowitz and Stegun (1964, p. 637).
23.10 Addition Theorems and Other
Identities
23.10(i) Addition Theorems
23.10.1}(u+v) =1
4}0(u) }0(v)
}(u) }(v)2
}(u) }(v);
23.10.2(u+v) =(u) +(v) +1
200(u) 00(v)
0(u) 0(v);
23.10.3(u+v)(u v)
2(u)2(v)=}(v) }(u);
23.10.4(u+v)(u v)(x+y)(x y)
+(v+x)(v x)(u+y)(u y)
+(x+u)(x u)(v+y)(v y) = 0:
For further addition-type identities for the -function
see Lawden (1989, x6.4).
Ifu+v+w= 0, then
23.10.5 1}(u)}0(u)
1}(v)}0(v)
1}(w)}0(w) = 0;
and
23.10.6 ((u)+(v)+(w))2+0(u)+0(v)+0(w) = 0:
578 Weierstrass Elliptic and Modular Functions
23.10(ii) Duplication Formulas
23.10.7 }(2z) = 2}(z) +1
4}00(z)
}0(z)2
;
23.10.8
(}(2z) e1)}02(z) =
(}(z) e1)2 (e1 e2)(e1 e3)2:
(23.10.8) continues to hold when e1,e2,e3are permuted
cyclically.
23.10.9 (2z) = 2(z) +1
2000(z)
00(z);
23.10.10 (2z) = }0(z)4(z):
23.10(iii)n-Tuple Formulas
Forn= 2;3;:::,
23.10.11n2}(nz) =n 1X
j=0n 1X
`=0}
z+2j
n!1+2`
n!3
;23.10.12n(nz) = n(n 1)(1+3)
+n 1X
j=0n 1X
`=0
z+2j
n!1+2`
n!3
;
23.10.13
(nz)
=Ane n(n 1)(1+3)zn 1Y
j=0n 1Y
`=0
z+2j
n!1+2`
n!3
;
where
23.10.14An=nn 1Y
j=0n 1Y
`=0
`6=j1
((2j!1+ 2`!3)=n):
Equivalently,
23.10.15 An=2G2
!1n2 1qn(n 1)=2
in 1exp
(n 1)1
3!1
(2n 1)(!2
1+!2
3) + 3(n 1)!1!3
;
where
23.10.16q=ei!3=!1; G =1Y
n=1(1 q2n):
23.10(iv) Homogeneity
For any nonzero real or complex constant c
23.10.17 }(czjcL) =c 2}(zjL);
23.10.18 (czjcL) =c 1(zjL);
23.10.19 (czjcL) =c(zjL):
Also, when Lis replaced by cLthe lattice invariants g2
andg3are divided by c4andc6, respectively.
For these results and further identities see Lawden
(1989,x6.6) and Apostol (1990, p. 14).
23.11 Integral Representations
Let =!3/!1and
23.11.1f1(s; ) =cosh2 1
2 s
1 2e scosh( s) +e 2s;
f2(s; ) =cos2 1
2s
1 2ei scoss+e2i s:
Then
23.11.2}(z) =1
z2+ 8Z1
0s
e ssinh2 1
2zs
f1(s; )
+ei ssin2 1
2zs
f2(s; )
ds;and
23.11.3(z) =1
z+Z1
0
e s(zs sinh(zs))f1(s; )
ei s(zs sin(zs))f2(s; )
ds;
provided that 1<<(z+ )<1 andj=zj<= .
23.12 Asymptotic Approximations
Ifq(=ei!3=!1)!0 with!1andz xed, then
23.12.1}(z) =2
4!2
1
1
3+ csc2z
2!1
+ 8
1 cosz
!1
q2+O
q4
;
23.12.2(z) =2
4!2
1z
3+2!1
cotz
!1
8
z !1
sinz
!1
q2+O
q4
;
23.12.3
(z) =2!1
exp2z2
24!2
1
sinz
2!1
1 2z2
!2
1 4 sin2z
2!1
q2+O
q4
;
provided that z =2Lin the case of (23.12.1) and
(23.12.2). Also,
23.12.4 1=2
4!11
3 8q2+O
q4
;
with similar results for 2and3obtainable by use of
(23.2.14).
23.13 Zeros 579
23.13 Zeros
For information on the zeros of }(z) see Eichler and
Zagier (1982).
23.14 Integrals
23.14.1Z
}(z)dz= (z);
23.14.2Z
}2(z)dz=1
6}0(z) +1
12g2z;
23.14.3Z
}3(z)dz=1
120}000(z) 3
20g2(z) +1
10g3z:
For further integrals see Gr obner and Hofreiter
(1949, Vol. 1, pp. 161{162), Gradshteyn and Ryzhik
(2000, p. 622), and Prudnikov et al. (1990, pp. 51{52).
Modular Functions
23.15 De nitions
23.15(i) General Modular Functions
Inxx23.15{23.19, kandk0(2C) denote the Jacobi
modulus and complementary modulus, respectively, and
q=ei (= > 0) denotes the nome; compare xx20.1
and 22.1. Thus
23.15.1 q= exp
K0(k)
K(k)
;
23.15.2 k=2
2(0;q)
2
3(0;q); k0=2
4(0;q)
2
3(0;q):
AlsoAdenotes a bilinear transformation on , given
by
23.15.3 A =a +b
c +d;
in whicha;b;c;d are integers, with
23.15.4 ad bc= 1:
The set of all bilinear transformations of this form is
denoted by SL(2 ;Z) (Serre (1973, p. 77)).
Amodular function f( ) is a function of that is
meromorphic in the half-plane = > 0, and has the
property that for all A2 SL(2;Z), or for allAbelong-
ing to a subgroup of SL(2 ;Z),
23.15.5 f(A ) =cA(c +d)`f( ),= >0;wherecAis a constant depending only on A, and`(the
level) is an integer or half an odd integer. (Some ref-
erences refer to 2 `as the level). If, as a function of q,
f( ) is analytic at q= 0, thenf( ) is called a modular
form. If, in addition, f( )!0 asq!0, thenf( ) is
called a cusp form .
23.15(ii) Functions ( ),J( ),( )
Elliptic Modular Function
23.15.6 ( ) =4
2(0;q)
4
3(0;q);
compare also (23.15.2).
Klein's Complete Invariant
23.15.7J( ) =
8
2(0;q) +8
3(0;q) +8
4(0;q)3
54 (0
1(0;q))8;
where (as inx20.2(i))
23.15.8 0
1(0;q) =@1(z;q)/@zjz=0:
Dedekind's Eta Function (or Dedekind Modular Function)
23.15.9
( ) = 1
20
1(0;q)1=3=ei =123 1
2(1 + ) 3
:
In (23.15.9) the branch of the cube root is chosen to
agree with the second equality; in particular, when
lies on the positive imaginary axis the cube root is real
and positive.
23.16 Graphics
See Figures 23.16.1{23.16.3 for the modular functions
,J, and. In Figures 23.16.2 and 23.16.3, height cor-
responds to the absolute value of the function and color
to the phase. See also p. xiv.
Figure 23.16.1 : Modular functions (iy),J(iy),(iy) for
0y3. See also Figure 20.3.2.
580 Weierstrass Elliptic and Modular Functions
Figure 23.16.2 : Elliptic modular function (x+iy) for
0:25x0:25, 0:005y0:1.
Figure 23.16.3 : Dedekind's eta function (x+iy) for
0:0625x0:0625, 0:0001y0:07.
23.17 Elementary Properties
23.17(i) Special Values
23.17.1 (i) =1
2;
ei=3
=ei=3;
23.17.2 J(i) = 1; J
ei=3
= 0;
23.17.3
(i) = 1
4
23=4;
ei=3
=31=8
1
33=2
2ei=24:
For further results for J( ) see Cohen (1993, p. 376).
23.17(ii) Power and Laurent Series
Whenjqj<1
23.17.4 ( ) = 16q(1 8q+ 44q2+);
23.17.5
1728J( ) =q 2+ 744 + 1 96884 q2+ 214 93760 q4+;
23.17.6 ( ) =1X
n= 1( 1)nq(6n+1)2=12:
In (23.17.5) for terms up to q48see Zuckerman
(1939), and for terms up to q100see van Wijngaarden
(1953). See also Apostol (1990, p. 22).
23.17(iii) In nite Products
23.17.7 ( ) = 16q1Y
n=11 +q2n
1 +q2n 18
;
23.17.8 ( ) =q1=121Y
n=1(1 q2n);
withq1=12=ei =12.23.18 Modular Transformations
Elliptic Modular Function
(A ) equals
23.18.1( );1 ( );1
( );
1
1 ( );( )
( ) 1;1 1
( );
according as the elementsa b
c d
ofAin (23.15.3) have
the respective forms
23.18.2o e
e o
;e o
o e
;o e
o o
;
e o
o o
;o o
e o
;o o
o e
:
Here e and o are generic symbols for even and odd inte-
gers, respectively. In particular, if a 1;b;c, andd 1
are all even, then
23.18.3 (A ) =( );
and( ) is a cusp form of level zero for the correspond-
ing subgroup of SL(2 ;Z).
Klein's Complete Invariant
23.18.4 J(A ) =J( ):
J( ) is a modular form of level zero for SL(2 ;Z).
23.19 Interrelations 581
Dedekind's Eta Function
23.18.5(A ) ="(A) ( i(c +d))1=2( );
where the square root has its principal value and
23.18.6"(A) = exp
ia+d
12c+s( d;c)
;
23.18.7s(d;c) =c 1X
r=1
(r;c)=1r
cdr
c dr
c
1
2
,c>0.
Here the notation ( r;c) = 1 means that the sum is con-
ned to those values of rthat are relatively prime to c.
Seex27.14(iii) and Apostol (1990, pp. 48 and 51{53).
Note that( ) is of level1
2.
23.19 Interrelations
23.19.1 ( ) = 16
2(2 ) 1
2
3( )!8
;
23.19.2 J( ) =4
27
1 ( ) +2( )3
(( ) (1 ( )))2;
23.19.3 J( ) =g3
2
g3
2 27g2
3;
whereg2;g3are the invariants of the lattice Lwith gen-
erators 1 and ; seex23.3(i).
Also, with de ned as in (23.3.4),
23.19.4 = (2)1224( ):
Applications
23.20 Mathematical Applications
23.20(i) Conformal Mappings
Rectangular Lattice
The boundary of the rectangle R, with vertices 0, !1,
!1+!3,!3, is mapped strictly monotonically by }onto
the real line with 0 !1 ,!1!e1,!1+!3!e2,
!3!e3, 0! 1 . There is a unique point z02
[!1;!1+!3][[!1+!3;!3] such that }(z0) = 0. The
interior ofRis mapped one-to-one onto the lower half-
plane.Rhombic Lattice
The two pairs of edges [0 ;!1][[!1;2!3] and [2!3;2!3
!1][[2!3 !1;0] ofRare each mapped strictly mono-
tonically by }onto the real line, with 0 !1 ,!1!e1,
2!3! 1 ; similarly for the other pair of edges. For
each pair of edges there is a unique point z0such that
}(z0) = 0.
The interior of the rectangle with vertices 0, !1, 2!3,
2!3 !1is mapped two-to-one onto the lower half-
plane. The interior of the rectangle with vertices 0,
!1,1
2!1+!3,1
2!1 !3is mapped one-to-one onto the
lower half-plane with a cut from e3to} 1
2!1+!3
(=
} 1
2!1 !3
). The cut is the image of the edge from
1
2!1+!3to1
2!1 !3and is not a line segment.
For examples of conformal mappings of the function
}(z), see Abramowitz and Stegun (1964, pp. 642{648,
654{655, and 659{60).
For conformal mappings via modular functions see
Apostol (1990,x2.7).
23.20(ii) Elliptic Curves
An algebraic curve that can be put either into the form
23.20.1 C:y2=x3+ax+b;
or equivalently, on replacing xbyx=zandybyy=z
(projective coordinates), into the form
23.20.2 C:y2z=x3+axz2+bz3;
is an example of an elliptic curve (x22.18(iv)). Here a
andbare real or complex constants.
PointsP= (x;y) on the curve can be parametrized
byx=}(z;g2;g3), 2y=}0(z;g2;g3), whereg2= 4a
andg3= 4b: in this case we write P=P(z).
The curve Cis made into an abelian group (Mac-
donald (1968, Chapter 5)) by de ning the zero ele-
mento= (0;1;0) as the point at in nity, the nega-
tive ofP= (x;y) by P= (x; y), and generally
P1+P2+P3= 0 on the curve i the points P1,P2,
P3are collinear. It follows from the addition formula
(23.10.1) that the points Pj=P(zj),j= 1;2;3, have
zero sum i z1+z2+z32L, so that addition of points
on the curve Ccorresponds to addition of parameters zj
on the torus C=L; see McKean and Moll (1999, xx2.11,
2.14).
In terms of ( x;y) the addition law can be expressed
(x;y) +o= (x;y), (x;y) + (x; y) =o; otherwise
(x1;y1) + (x2;y2) = (x3;y3), where
23.20.3x3=m2 x1 x2; y 3= m(x3 x1) y1;
and
23.20.4m=(
(3x2
1+a)=(2y1); P 1=P2;
(y2 y1)=(x2 x1); P 16=P2:
Ifa;b2R, thenCintersects the plane R2in a curve
that is connected if 4a3+ 27b2>0; if <0,
582 Weierstrass Elliptic and Modular Functions
then the intersection has two components, one of which
is a closed loop. These cases correspond to rhombic
and rectangular lattices, respectively. The addition law
states that to nd the sum of two points, take the third
intersection with Cof the chord joining them (or the
tangent if they coincide); then its re
ection in the x-axis
gives the required sum. The geometric nature of this
construction is illustrated in McKean and Moll (1999,
x2.14), Koblitz (1993, xx6, 7), and Silverman and Tate
(1992, Chapter 1, xx3, 4): each of these references makes
a connection with the addition theorem (23.10.1).
Ifa;b2Q, then by rescaling we may assume a;b2
Z. LetTdenote the set of points on Cthat are of nite
order (that is, those points Pfor which there exists a
positive integer nwithnP=o), and letI;K be the
sets of points with integer and rational coordinates, re-
spectively. Then ;TIKC. BothT;K are
subgroups of C, thoughImay not be. Kalways has the
formTZr(Mordell's Theorem : Silverman and Tate
(1992, Chapter 3, x5)); the determination of r, the rank
ofK, raises questions of great diculty, many of which
are still open. Both TandIare nite sets. Tmust
have one of the forms Z=(nZ), 1n10 orn= 12, or
(Z=(2Z))(Z=(2nZ)), 1n4. To determine T, we
make use of the fact that if ( x;y)2Ttheny2must be
a divisor of ; hence there are only a nite number of
possibilities for y. Values of xare then found as integer
solutions of x3+ax+b y2= 0 (in particular xmust be
a divisor of b y2). The resulting points are then tested
for nite order as follows. Given P, calculate 2 P, 4P,
8Pby doubling as above. If any of these quantities is
zero, then the point has nite order. If any of 2 P, 4P,
8Pis not an integer, then the point has in nite order.
Otherwise observe any equalities between P, 2P, 4P,
8P, and their negatives. The order of a point (if nite
and not already determined) can have only the values 3,
5, 6, 7, 9, 10, or 12, and so can be found from 2 P= P,
4P= P, 4P= 2P, 8P=P, 8P= P, 8P= 2P,
or 8P= 4P. If none of these equalities hold, then P
has in nite order.
For extensive tables of elliptic curves see Cremona
(1997, pp. 84{340).
23.20(iii) Factorization
x27.16 describes the use of primality testing and factor-
ization in cryptography. For applications of the Weier-
strass function and the elliptic curve method to these
problems see Bressoud (1989) and Koblitz (1999).
23.20(iv) Modular and Quintic Equations
The modular equation of degreep,pprime, is an al-
gebraic equation in =(p ) and =( ). Forp= 2;3;5;7 and with u= 1=4,v= 1=4, the mod-
ular equation is as follows:
23.20.5 v8(1 +u8) = 4u4, p= 2,
23.20.6 u4 v4+ 2uv(1 u2v2) = 0, p= 3,
23.20.7
u6 v6+ 5u2v2(u2 v2) + 4uv(1 u4v4) = 0,p= 5,
23.20.8 (1 u8)(1 v8) = (1 uv)8,p= 7.
For further information, including the application of
(23.20.7) to the solution of the general quintic equation,
see Borwein and Borwein (1987, Chapter 4).
23.20(v) Modular Functions and Number
Theory
For applications of modular functions to number theory
seex27.14(iv) and Apostol (1990). See also Silverman
and Tate (1992), Serre (1973, Part 2, Chapters 6, 7),
Koblitz (1993), and Cornell et al. (1997).
23.21 Physical Applications
23.21(i) Classical Dynamics
Inx22.19(ii) it is noted that Jacobian elliptic functions
provide a natural basis of solutions for problems in
Newtonian classical dynamics with quartic potentials
in canonical form (1 x2)(1 k2x2). The Weierstrass
function}plays a similar role for cubic potentials in
canonical form g3+g2x 4x3. See, for example, Law-
den (1989, Chapter 7) and Whittaker (1964, Chapters
4{6).
23.21(ii) Nonlinear Evolution Equations
Airault et al. (1977) applies the function }to an in-
tegrable classical many-body problem, and relates the
solutions to nonlinear partial di erential equations. For
applications to soliton solutions of the Korteweg{de
Vries (KdV) equation see McKean and Moll (1999,
p. 91), Deconinck and Segur (2000), and Walker (1996,
x8.1).
23.21(iii) Ellipsoidal Coordinates
Ellipsoidal coordinates ( ;; ) may be de ned as the
three roots of the equation
23.21.1x2
e1+y2
e2+z2
e3= 1;
Computation 583
wherex;y;z are the corresponding Cartesian coordi-
nates ande1,e2,e3are constants. The Laplacian oper-
atorr2(x1.5(ii)) is given by
23.21.2
( )( )( )r2= ( )f()f0()@
@
+ ( )f()f0()@
@
+ ( )f()f0()@
@;
where
23.21.3f() = 2 (( e1)( e2)( e3))1=2:
Another form is obtained by identifying e1,e2,e3as
lattice roots (x23.3(i)), and setting
23.21.4=}(u); =}(v); =}(w):
Then
23.21.5(}(v) }(w)) (}(w) }(u)) (}(u) }(v))r2
= (}(w) }(v))@2
@u2+ (}(u) }(w))@2
@v2
+ (}(v) }(u))@2
@w2:
See alsox29.18(ii).
23.21(iv) Modular Functions
Physical applications of modular functions include:
Quantum eld theory. See Witten (1987).
Statistical mechanics. See Baxter (1982, p. 434)
and Itzykson and Drou e (1989, x9.3).
String theory. See Green et al. (1988a,x8.2) and
Polchinski (1998, x7.2).
Computation
23.22 Methods of Computation
23.22(i) Function Values
Given!1and!3, with=(!3=!1)>0, the nome qis
computed from q=ei!3=!1. For}(z) we apply (23.6.2)
and (23.6.5), generating all needed values of the theta
functions by the methods described in x20.14.
The functions (z) and(z) are computed in a simi-
lar manner: the former by replacing uandzin (23.6.13)
byzandz=(2!1), respectively, and also referring to
(23.6.8); the latter by applying (23.6.9).
The modular functions ( ),J( ), and( ) are also
obtainable in a similar manner from their de nitions in
x23.15(ii).23.22(ii) Lattice Calculations
Starting from Lattice
Suppose that the lattice Lis given. Then a pair of gen-
erators 2!1and 2!3can be chosen in an almost canon-
ical way as follows. For 2 !1choose a nonzero point of
Lof smallest absolute value. (There will be 2, 4, or 6
possible choices.) For 2 !3choose a nonzero point that
is not a multiple of 2 !1and is such that = > 0 and
j jis as small as possible, where =!3=!1. (There
will be either 1 or 2 possible choices.) This yields a pair
of generators that satisfy = > 0,j< j1
2,j j>1.
In consequence, q=ei!3=!1satis esjqje p
3=2=
0:0658:::. The corresponding values of e1,e2,e3are
calculated from (23.6.2){(23.6.4), then g2andg3are ob-
tained from (23.3.6) and (23.3.7).
Starting from Invariants
Suppose that the invariants g2=c,g3=d, are given,
for example in the di erential equation (23.3.10) or
via coecients of an elliptic curve ( x23.20(ii)). The
determination of suitable generators 2 !1and 2!3is
the classical inversion problem (Whittaker and Wat-
son (1927,x21.73), McKean and Moll (1999, x2.12); see
alsox20.9(i) and McKean and Moll (1999, x2.16)). This
problem is solvable as follows:
(a) In the general case, given by cd6= 0, we com-
pute the roots , ,
, say, of the cubic equation
4t3 ct d= 0; seex1.11(iii). These roots are nec-
essarily distinct and represent e1,e2,e3in some
order.
Ifcanddare real, then e1,e2,e3can be identi ed
via (23.5.1), and k2,k02obtained from (23.6.16).
Ifcanddare not both real, then we label , ,
so
that the triangle with vertices , ,
is positively
oriented and [ ;
] is its longest side (chosen arbi-
trarily if there is more than one). In particular, if
, ,
are collinear, then we label them so that
is on the line segment ( ;
). In consequence,
k2= (
)=(
),k02= ( )=(
) satisfy
=k20=k02(with strict inequality unless ,
,
are collinear); also jk2j,jk02j1.
Finally, on taking the principal square roots of k2
andk02we obtain values for kandk0that lie in
the 1st and 4th quadrants, respectively, and 2 !1,
2!3are given by
23.22.12!1M(1;k0) = 2i!3M(1;k)
=
3s
c(2 +k2k02)(k02 k2)
d(1 k2k02);
whereMdenotes the arithmetic-geometric mean
(seexx19.8(i) and 22.20(ii)). This process yields 2
584 Weierstrass Elliptic and Modular Functions
possible pairs (2 !1, 2!3), corresponding to the 2
possible choices of the square root.
(b) Ifd= 0, then
23.22.22!1= 2i!3=
1
42
2pc1=4:
There are 4 possible pairs (2 !1, 2!3), correspond-
ing to the 4 rotations of a square lattice. The
lemniscatic case occurs when c>0 and!1>0.
(c) Ifc= 0, then
23.22.32!1= 2e i=3!3=
1
33
2d1=6:
There are 6 possible pairs (2 !1, 2!3), correspond-
ing to the 6 rotations of a lattice of equilateral
triangles. The equianharmonic case occurs when
d>0 and!1>0.
Example
Assumec=g2= 4(3 2i) andd=g3= 4(4 2i).
Then = 1 2i, = 1,
= 2i;k2= (7 + 6i)/17,
andk02= (10 6i)/17. Working to 6 decimal places
we obtain
23.22.42!1= 0:867568 +i1:466607;
2!3= 1:223741 +i1:328694;
= 0:305480 +i1:015109:
23.23 Tables
Table 18.2 in Abramowitz and Stegun (1964) gives val-
ues of}(z),}0(z), and(z) to 7 or 8D in the rectangu-
lar and rhombic cases, normalized so that !1= 1 and
!3=ia(rectangular case), or !1= 1 and!3=1
2+ia
(rhombic case), for a= 1.00, 1.05, 1.1, 1.2, 1.4, 2, 4.
The values are tabulated on the real and imaginary z-
axes, mostly ranging from 0 to 1 or iin steps of length
0.05, and in the case of }(z) the user may deduce val-
ues for complex zby application of the addition theorem
(23.10.1).
Abramowitz and Stegun (1964) also includes other
tables to assist the computation of the Weierstrass func-
tions, for example, the generators as functions of the
lattice invariants g2andg3.
For earlier tables related to Weierstrass functions see
Fletcher et al. (1962, pp. 503{505) and Lebedev and Fe-
dorova (1960, pp. 223{226).
23.24 Software
Seehttp://dlmf.nist.gov/23.24 .References
General References
The main references used in writing this chapter are
Lawden (1989, Chapters 6, 7, 9), McKean and Moll
(1999, Chapters 1{5), Walker (1996, Chapter 7 and
xx3.4, 8.4), and Whittaker and Watson (1927, Chap-
ter 20 andx21.7). For additional bibliographic reading
see Apostol (1990, Chapters 1{6), Copson (1935, Chap-
ters 13 and 15), Erd elyi et al. (1953b,xx13.12{13.15 and
13.24), and Koblitz (1993, Chapters 1{4).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x23.2 Whittaker and Watson (1927, xx20.2{20.21,
20.4{20.421), Walker (1996, x3.1), Lawden (1989,
Chapter 6). For (23.2.16) di erentiate (23.2.15)
and use (23.2.6). For (23.2.17) use (23.2.15) and
induction.
x23.3 Whittaker and Watson (1927, xx20.22, 20.32,
21.73), Lawden (1989, x6.7), Walker (1996, x3.4).
(23.3.11) follows from (23.3.3), (23.3.10). For
(23.3.12), (23.3.13) di erentiate (23.3.10).
x23.4 These graphics were produced at NIST.
x23.5 Walker (1996,xx7.5, 8.4.2). (Some errors in x7.5
are corrected here.)
x23.6 For (23.6.2){(23.6.7) see Walker (1996, pp. 94
and 103). For (23.6.8) and (23.6.9) see Whit-
taker and Watson (1927, x21.43). For (23.6.10){
(23.6.12) use (23.6.9). For (23.6.13) and (23.6.14)
see Lawden (1989, x6.6). For (23.6.15) com-
bine (20.2.6) and (23.6.9). For (23.6.16) and
(23.6.17) combine (23.6.2){(23.6.4) with (22.2.2)
and (20.7.5). For (23.6.18){(23.6.20) combine
(20.9.1) and (20.9.2) with (23.6.2){(23.6.4). For
(23.6.21){(23.6.23) combine (23.6.5){(23.6.7) with
(22.2.4){(22.2.9). For (23.6.24){(23.6.26) com-
bine (23.6.21){(23.6.23) with x22.4(iii). (23.6.27){
(23.6.29) can be veri ed by matching periods,
poles, and residues as in Lawden (1989, x8.11).
x23.7 Lawden (1989, p. 182).
x23.8 Lawden (1989,x6.5, pp. 183{184, x8.6).
References 585
x23.9 For (23.9.2){(23.9.5) equate coecients in
(23.2.4), (23.2.5), and also apply (23.10.1),
(23.10.2). The rst two coecients in the Maclau-
rin expansion (23.9.6) are given by (23.3.9),
(23.2.10); the others are obtained from x23.3(ii)
combined with (23.9.4). (23.9.7) follows from
(23.2.8) and (23.9.3).
x23.10 Whittaker and Watson (1927, xx20.3{20.311,
20.41), Lawden (1989, pp. 152{158, 161{162).
For (23.10.7), (23.10.9), (23.10.10) let v!u
in (23.10.1){(23.10.3). For (23.10.8) see Walker
(1996, p. 83). For (23.10.11) and (23.10.12) com-
pare the poles and residues of the two sides.
(23.10.13) follows by integration. For (23.10.15)
combine (23.10.14), (23.8.7), and (4.21.35).
x23.11 Dienstfrey and Huang (2006).
x23.12 These approximations follow from the expan-
sions given inx23.8(ii). For (23.12.4) use Lawden
(1989, Eq. 6.2.7) and (20.4.8).
x23.14 To verify these results di erentiate and use
(23.2.7),x23.3(ii).x23.15 Apostol (1990, Chapters 1, 2), Walker (1996,
Chapter 7), McKean and Moll (1999, Chapters 4,
6). For (23.15.9) use (20.5.3) and (23.17.8).
x23.16 These graphics were produced at NIST.
x23.17 Walker (1996,x7.5). For (23.17.4){(23.17.6)
combinex23.15(ii) with the q-expansions of the
theta functions obtained by setting z= 0
inx20.2(i). For (23.17.7), (23.17.8) combine
(23.15.6), (23.15.9), and (20.5.1){(20.5.3).
x23.18 See Walker (1996, Chapter 7), Ahlfors (1966,
pp. 271{274), and Serre (1973, Chapter 7). For
(23.18.4){(23.18.7) see Apostol (1990, pp. 17, 52).
x23.19 Apostol (1990, Chapters 2, 3), Serre (1973,
Chapter 7). (23.19.4) follows from (20.5.3) and
(23.17.8).
x23.20 McKean and Moll (1999, x2.8).
x23.21 Jones (1964, pp. 31{33).
x23.22 For (23.22.1) combine (23.6.2){(23.6.4) and
x23.10(iv). (23.22.2) and (23.22.3) follow from
(23.5.3) and (23.5.7), respectively.
Chapter 24
Bernoulli and Euler Polynomials
K. Dilcher1
Notation 588
24.1 Special Notation . . . . . . . . . . . . . 588
Properties 588
24.2 De nitions and Generating Functions . . 588
24.3 Graphs . . . . . . . . . . . . . . . . . . . 589
24.4 Basic Properties . . . . . . . . . . . . . . 589
24.5 Recurrence Relations . . . . . . . . . . . 591
24.6 Explicit Formulas . . . . . . . . . . . . . 591
24.7 Integral Representations . . . . . . . . . 592
24.8 Series Expansions . . . . . . . . . . . . . 592
24.9 Inequalities . . . . . . . . . . . . . . . . 593
24.10 Arithmetic Properties . . . . . . . . . . . 593
24.11 Asymptotic Approximations . . . . . . . . 593
24.12 Zeros . . . . . . . . . . . . . . . . . . . 59424.13 Integrals . . . . . . . . . . . . . . . . . . 594
24.14 Sums . . . . . . . . . . . . . . . . . . . 595
24.15 Related Sequences of Numbers . . . . . . 595
24.16 Generalizations . . . . . . . . . . . . . . 596
Applications 597
24.17 Mathematical Applications . . . . . . . . 597
24.18 Physical Applications . . . . . . . . . . . 598
Computation 598
24.19 Methods of Computation . . . . . . . . . 598
24.20 Tables . . . . . . . . . . . . . . . . . . . 598
24.21 Software . . . . . . . . . . . . . . . . . . 598
References 598
1Dalhousie University, Halifax, Nova Scotia, Canada.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 23) by E. V. Haynsworth and K.
Goldberg.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
587
588 Bernoulli and Euler Polynomials
Notation
24.1 Special Notation
(For other notation see pp. xiv and 873.)
j;k;`;m;n integers, nonnegative unless stated
otherwise.
t;x real or complex variables.
p prime.
pjm p dividesm.
(k;m) greatest common divisor of m;n.
(k;m) = 1kandmrelatively prime.
Unless otherwise noted, the formulas in this chapter
hold for all values of the variables xandt, and for all
nonnegative integers n.
Bernoulli Numbers and Polynomials
The origin of the notation Bn,Bn(x), is not clear. The
present notation, as de ned in x24.2(i), was used in Lu-
cas (1891) and N orlund (1924), and has become the
prevailing notation; see Table 24.2.1. Among various
older notations, the most common one is
B1=1
6,B2=1
30,B3=1
42,B4=1
30;:::.
It was used in Saalsch utz (1893), Nielsen (1923),
Schwatt (1962), and Whittaker and Watson (1927).
Euler Numbers and Polynomials
The secant series ((4.19.5)) rst occurs in the work of
Gregory in 1671. Its coecients were rst studied in Eu-
ler (1755); they were called Euler numbers by Raabe in
1851. The notations En,En(x), as de ned inx24.2(ii),
were used in Lucas (1891) and N orlund (1924).
Other historical remarks on notations can be found
in Cajori (1929, pp. 42{44). Various systems of notation
are summarized in Adrian (1959) and D'Ocagne (1904).Properties
24.2 De nitions and Generating Functions
24.2(i) Bernoulli Numbers and Polynomials
24.2.1t
et 1=1X
n=0Bntn
n!, jtj<2.
24.2.2B2n+1= 0 , ( 1)n+1B2n>0,n= 1;2;:::.
24.2.3text
et 1=1X
n=0Bn(x)tn
n!, jtj<2.
24.2.4Bn=Bn(0);
24.2.5Bn(x) =nX
k=0n
k
Bkxn k:
See alsoxx4.19 and 4.33.
24.2(ii) Euler Numbers and Polynomials
24.2.62et
e2t+ 1=1X
n=0Entn
n!,jtj<1
2,
24.2.7 E2n+1= 0 , ( 1)nE2n>0 .
24.2.82ext
et+ 1=1X
n=0En(x)tn
n!,jtj<,
24.2.9 En= 2nEn 1
2
= integer;
24.2.10 En(x) =nX
k=0n
kEk
2k(x 1
2)n k:
See also (4.19.5).
24.2(iii) Periodic Bernoulli and Euler Functions
24.2.11eBn(x) =Bn(x) ,eEn(x) =En(x), 0x<1,
24.2.12eBn(x+ 1) =eBn(x);eEn(x+ 1) = eEn(x),
x2R.
24.3 Graphs 589
24.2(iv) Tables
Table 24.2.1 : Bernoulli and Euler numbers.
n Bn En
0 1 1
1 1
20
21
6 1
4 1
305
61
42 61
8 1
301385
105
66 50521
12 691
273027 02765
147
6 1993 60981
16 3617
5101 93915 12145Table 24.2.2 : Bernoulli and Euler polynomials.
n B n(x) En(x)
0 1 1
1 x 1
2x 1
2
2x2 x+1
6x2 x
3x3 3
2x2+1
2x x3 3
2x2+1
4
4x4 2x3+x2 1
30x4 2x3+x
5x5 5
2x4+5
3x3 1
6x x5 5
2x4+5
2x2 1
2
For extensions of Tables 24.2.1 and 24.2.2 see http://dlmf.nist.gov/24.2.iv .
24.3 Graphs
Figure 24.3.1 : Bernoulli polynomials Bn(x),n=
2;3;:::; 6.
Figure 24.3.2 : Euler polynomials En(x),n= 2;3;:::; 6.
24.4 Basic Properties
24.4(i) Di erence Equations
24.4.1 Bn(x+ 1) Bn(x) =nxn 1;
24.4.2 En(x+ 1) +En(x) = 2xn:
24.4(ii) Symmetry
24.4.3 Bn(1 x) = ( 1)nBn(x);
24.4.4 En(1 x) = ( 1)nEn(x):
24.4.5 ( 1)nBn( x) =Bn(x) +nxn 1;24.4.6 ( 1)n+1En( x) =En(x) 2xn:
24.4(iii) Sums of Powers
24.4.7mX
k=1kn=Bn+1(m+ 1) Bn+1
n+ 1;
24.4.8mX
k=1( 1)m kkn=En(m+ 1) + ( 1)mEn(0)
2:
590 Bernoulli and Euler Polynomials
24.4.9
m 1X
k=0(a+dk)n=dn
n+ 1
Bn+1
m+a
d
Bn+1a
d
;
24.4.10
m 1X
k=0( 1)k(a+dk)n
=dn
2
( 1)m 1En
m+a
d
+Ena
d
:
24.4.11
mX
k=1
(k;m)=1kn=1
n+ 1n+1X
j=1n+ 1
j
0
@Y
pjm(1 pn j)Bn+1 j1
Amj:
24.4(iv) Finite Expansions
24.4.12Bn(x+h) =nX
k=0n
k
Bk(x)hn k;
24.4.13En(x+h) =nX
k=0n
k
Ek(x)hn k;
24.4.14En 1(x) =2
nnX
k=0n
k
(1 2k)Bkxn k;
24.4.15
B2n=2n
22n(22n 1)n 1X
k=02n 1
2k
E2k;
24.4.16
E2n=1
2n+ 1 nX
k=12n
2k 122k(22k 1 1)B2k
k;
24.4.17
E2n= 1 nX
k=12n
2k 122k(22k 1)B2k
2k:
24.4(v) Multiplication Formulas
Raabe's Theorem
24.4.18Bn(mx) =mn 1m 1X
k=0Bn
x+k
m
:
Next,
24.4.19En(mx) = 2mn
n+ 1m 1X
k=0( 1)kBn+1
x+k
m
,
m= 2;4;6;:::,
24.4.20
En(mx) =mnm 1X
k=0( 1)kEn
x+k
m
,m= 1;3;5;:::.24.4.21Bn(x) = 2n 1
Bn 1
2x
+Bn 1
2x+1
2
;
24.4.22En 1(x) =2
n
Bn(x) 2nBn 1
2x
;
24.4.23En 1(x) =2n
n
Bn 1
2x+1
2
Bn 1
2x
;
24.4.24
Bn(mx) =mnBn(x) +nnX
k=1k 1X
j=0( 1)jn
k
m 1X
r=1e2i(k j)r=m
(1 e2ir=m )n!
(j+mx)n 1,
n= 1;2;:::,m= 2;3;:::.
24.4(vi) Special Values
24.4.25Bn(0) = ( 1)nBn(1) =Bn;
24.4.26En(0) = En(1) = 2
n+ 1(2n+1 1)Bn+1:
24.4.27 Bn 1
2
= (1 21 n)Bn;
24.4.28 En 1
2
= 2 nEn:
24.4.29B2n 1
3
=B2n 2
3
= 1
2(1 31 2n)B2n:
24.4.30
E2n 1 1
3
= E2n 1 2
3
= (1 31 2n)(22n 1)
2nB2n,
n= 1;2;:::.
24.4.31
Bn 1
4
= ( 1)nBn 3
4
= 1 21 n
2nBn n
4nEn 1,
n= 1;2;:::.
24.4.32
B2n 1
6
=B2n 5
6
=1
2(1 21 2n)(1 31 2n)B2n;
24.4.33E2n 1
6
=E2n 5
6
=1 + 3 2n
22n+1E2n:
24.4(vii) Derivatives
24.4.34d
dxBn(x) =nBn 1(x), n= 1;2;:::,
24.4.35d
dxEn(x) =nEn 1(x), n= 1;2;:::.
24.4(viii) Symbolic Operations
LetP(x) denote any polynomial in x, and after expand-
ing set (B(x))n=Bn(x) and (E(x))n=En(x). Then
24.4.36P(B(x) + 1) P(B(x)) =P0(x);
24.4.37 Bn(x+h) = (B(x) +h)n;
24.4.38P(E(x) + 1) +P(E(x)) = 2P(x);
24.4.39 En(x+h) = (E(x) +h)n:
For these results and also connections with the umbral
calculus see Gessel (2003).
24.5 Recurrence Relations 591
24.4(ix) Relations to Other Functions
For the relation of Bernoulli numbers to the Riemann
zeta function seex25.6, and to the Eulerian numbers see
(26.14.11).
24.5 Recurrence Relations
24.5(i) Basic Relations
24.5.1n 1X
k=0n
k
Bk(x) =nxn 1,n= 2;3;:::,
24.5.2nX
k=0n
k
Ek(x) +En(x) = 2xn,n= 1;2;:::.
24.5.3n 1X
k=0n
k
Bk= 0,n= 2;3;:::,
24.5.4nX
k=02n
2k
E2k= 0,n= 1;2;:::,
24.5.5nX
k=0n
k
2kEn k+En= 2:
24.5(ii) Other Identities
24.5.6
nX
k=2n
k 2Bk
k=1
(n+ 1)(n+ 2) Bn+1,n= 2;3;:::;
24.5.7nX
k=0n
kBk
n+ 2 k=Bn+1
n+ 1,n= 1;2;:::;
24.5.8nX
k=022kB2k
(2k)!(2n+ 1 2k)!=1
(2n)!,n= 1;2;::::
24.5(iii) Inversion Formulas
In each of (24.5.9) and (24.5.10) the rst identity im-
plies the second one and vice-versa.
24.5.9an=nX
k=0n
kbn k
k+ 1; bn=nX
k=0n
k
Bkan k:
24.5.10an=bn/2cX
k=0n
2k
bn 2k;
bn=bn/2cX
k=0n
2k
E2kan 2k:24.6 Explicit Formulas
The identities in this section hold for n= 1;2;:::.
(24.6.7), (24.6.8), (24.6.10), and (24.6.12) are valid also
forn= 0.
24.6.1B2n=2n+1X
k=2( 1)k 1
k2n+ 1
kk 1X
j=1j2n;
24.6.2Bn=1
n+ 1nX
k=1kX
j=1( 1)jjnn+ 1
k j,n
k
;
24.6.3B2n=nX
k=1(k 1)!k!
(2k+ 1)!kX
j=1( 1)j 12k
k+j
j2n:
24.6.4E2n=nX
k=11
2k 1kX
j=1( 1)j2k
k j
j2n;
24.6.5
E2n=1
2n 1n 1X
k=0( 1)n k(n k)2nkX
j=02n 2j
k j
2j;
24.6.6
E2n=2nX
k=1( 1)k
2k 12n+ 1
k+ 1j1
2k 1
2k
X
j=0k
j
(k 2j)2n:
24.6.7Bn(x) =nX
k=01
k+ 1kX
j=0( 1)jk
j
(x+j)n;
24.6.8En(x) =1
2nn+1X
k=1k 1X
j=0( 1)jn+ 1
k
(x+j)n:
24.6.9Bn=nX
k=01
k+ 1kX
j=0( 1)jk
j
jn;
24.6.10En=1
2nn+1X
k=1n+ 1
kk 1X
j=0( 1)j(2j+ 1)n:
24.6.11Bn=n
2n(2n 1)nX
k=1k 1X
j=0( 1)j+1n
k
jn 1;
24.6.12E2n=2nX
k=01
2kkX
j=0( 1)jk
j
(1 + 2j)2n:
592 Bernoulli and Euler Polynomials
24.7 Integral Representations
24.7(i) Bernoulli and Euler Numbers
The identities in this subsection hold for n= 1;2;:::.
(24.7.6) also holds for n= 0.
24.7.1
B2n= ( 1)n+1 4n
1 21 2nZ1
0t2n 1
e2t+ 1dt
= ( 1)n+1 2n
1 21 2nZ1
0t2n 1e tsech(t)dt;
24.7.2
B2n= ( 1)n+14nZ1
0t2n 1
e2t 1dt
= ( 1)n+12nZ1
0t2n 1e tcsch(t)dt;
24.7.3
B2n= ( 1)n+1
1 21 2nZ1
0t2nsech2(t)dt;
24.7.4
B2n= ( 1)n+1Z1
0t2ncsch2(t)dt;
24.7.5
B2n= ( 1)n2n(2n 1)
Z1
0t2n 2ln
1 e 2t
dt:
24.7.6
E2n= ( 1)n22n+1Z1
0t2nsech(t)dt:
24.7(ii) Bernoulli and Euler Polynomials
The following four equations hold for 0 <<x<1.
24.7.7
B2n(x) = ( 1)n+12n
Z1
0cos(2x) e 2t
cosh(2t) cos(2x)t2n 1dt,
n= 1;2;:::,
24.7.8
B2n+1(x) = ( 1)n+1(2n+ 1)
Z1
0sin(2x)
cosh(2t) cos(2x)t2ndt:
24.7.9
E2n(x) = ( 1)n4Z1
0sin(x) cosh(t)
cosh(2t) cos(2x)t2ndt;
24.7.10
E2n+1(x) = ( 1)n+14
Z1
0cos(x) sinh(t)
cosh(2t) cos(2x)t2n+1dt:
Mellin{Barnes Integral
24.7.11
Bn(x) =1
2iZ c+i1
c i1(x+t)n
sin(t)2
dt, 0<c< 1.24.7(iii) Compendia
For further integral representations see Prudnikov et al.
(1986a,xx2.3{2.6) and Gradshteyn and Ryzhik (2000,
Chapters 3 and 4).
24.8 Series Expansions
24.8(i) Fourier Series
Ifn= 1;2;::: and 0x1, then
24.8.1B2n(x) = ( 1)n+12(2n)!
(2)2n1X
k=1cos(2kx)
k2n;
24.8.2B2n+1(x) = ( 1)n+12(2n+ 1)!
(2)2n+11X
k=1sin(2kx)
k2n+1:
The second expansion holds also for n= 0 and 0<x<
1.
Ifn= 1 with 0 < x < 1, orn= 2;3;::: with
0x1, then
24.8.3Bn(x) = n!
(2i)n1X
k= 1
k6=0e2ikx
kn:
Ifn= 1;2;::: and 0x1, then
24.8.4
E2n(x) = ( 1)n4(2n)!
2n+11X
k=0sin((2k+ 1)x)
(2k+ 1)2n+1;
24.8.5
E2n 1(x) = ( 1)n4(2n 1)!
2n1X
k=0cos((2k+ 1)x)
(2k+ 1)2n:
24.8(ii) Other Series
24.8.6
B4n+2= (8n+ 4)1X
k=1k4n+1
e2k 1, n= 1;2;:::,
24.8.7
B2n=( 1)n+14n
22n 11X
k=1k2n 1
ek+ ( 1)k+n,n= 2;3;:::.
Let =2. Then
24.8.8B2n
4n( n ( )n) = n1X
k=1k2n 1
e2 k 1
( )n1X
k=1k2n 1
e2 k 1,
n= 2;3;:::.
24.8.9E2n= ( 1)n1X
k=1k2n
cosh 1
2k
41X
k=0( 1)k(2k+ 1)2n
e2(2k+1) 1,n= 1;2;:::.
24.9 Inequalities 593
24.9 Inequalities
Except where otherwise noted, the inequalities in this
section hold for n= 1;2;:::.
24.9.1jB2nj>jB2n(x)j, 1 >x> 0,
24.9.2
(2 21 2n)jB2njjB2n(x) B2nj, 1x0.
(24.9.3){(24.9.5) hold for1
2>x> 0.
24.9.3 4 njE2nj>( 1)nE2n(x)>0;
24.9.4
2(2n+ 1)!
(2)2n+1>( 1)n+1B2n+1(x)>0,n= 2;3;:::,
24.9.5
4(2n 1)!
2n22n 1
22n 2>( 1)nE2n 1(x)>0:
(24.9.6){(24.9.7) hold for n= 2;3;:::.
24.9.6 5pnn
e2n
>( 1)n+1B2n>4pnn
e2n
;
24.9.7
8rn
4n
e2n
1+1
12n
>( 1)nE2n>8rn
4n
e2n
:
Lastly,
24.9.8
2(2n)!
(2)2n1
1 2 2n( 1)n+1B2n2(2n)!
(2)2n1
1 2 2n
with
24.9.9 = 2 +ln
1 6 2
ln 2= 0:6491::::
24.9.10
4n+1(2n)!
2n+1>( 1)nE2n>4n+1(2n)!
2n+11
1 + 3 1 2n:
24.10 Arithmetic Properties
24.10(i) Von Staudt{Clausen Theorem
Here and elsewhere in x24.10 the symbol pdenotes a
prime number.
24.10.1 B2n+X
(p 1)j2n1
p= integer;
where the summation is over all psuch thatp 1 divides
2n. The denominator of B2nis the product of all these
primesp.
24.10.2 pB2np 1 (modp`+1);
wheren2, and`(1) is an arbitrary integer such that
(p 1)p`j2n. Here and elsewhere two rational numbers
arecongruent if the modulus divides the numerator of
their di erence.24.10(ii) Kummer Congruences
24.10.3Bm
mBn
n(modp);
wheremn60 (modp 1).
24.10.4 (1 pm 1)Bm
m(1 pn 1)Bn
n(modp`+1);
valid when mn(mod (p 1)p`) andn60 (modp
1), where`(0) is a xed integer.
24.10.5 EnEn+p 1(modp);
wherep(>2) is a prime and n2.
24.10.6 E2nE2n+w(mod 2`);
valid for xed integers `(0), and for all n(0) and
w(0) such that 2`jw.
24.10(iii) Voronoi's Congruence
LetB2n=N2n/D2n, withN2nandD2nrelatively
prime andD2n>0. Then
24.10.7(b2n 1)N2n
2nb2n 1D2nM 1X
k=1k2n 1kb
M
(modM);
whereM(2) andbare integers, with brelatively prime
toM.
For historical notes, generalizations, and applica-
tions, see Porubsk y (1998).
24.10(iv) Factors
WithN2nas inx24.10(iii)
24.10.8 N2n0 (modp`);
valid for xed integers `(1), and for all n(1) such
that 2n60 (modp 1) andp`j2n.
24.10.9E2n(
0 (modp`) ifp1 (mod 4) ;
2 (modp`) ifp3 (mod 4) ;
valid for xed integers `(1) and for all n(1) such
that (p 1)p` 1j2n.
24.11 Asymptotic Approximations
Asn!1
24.11.1 ( 1)n+1B2n2(2n)!
(2)2n;
24.11.2 ( 1)n+1B2n4pnn
e2n
;
24.11.3 ( 1)nE2n22n+2(2n)!
2n+1;
24.11.4 ( 1)nE2n8rn
4n
e2n
:
594 Bernoulli and Euler Polynomials
Also,
24.11.5
( 1)bn=2c 1(2)n
2(n!)Bn(x)!(
cos(2x); n even;
sin(2x); n odd;
24.11.6
( 1)b(n+1)=2cn+1
4(n!)En(x)!(
sin(x); n even;
cos(x); n odd;
uniformly for xon compact subsets of C.
For further results see Temme (1995b) and L opez
and Temme (1999b).
24.12 Zeros
24.12(i) Bernoulli Polynomials: Real Zeros
In the interval 0x1 the only zeros of B2n+1(x),n=
1;2;:::, are 0;1
2;1, and the only zeros of B2n(x) B2n,
n= 1;2;:::, are 0;1.
For the interval1
2x <1denote the zeros of
Bn(x) byx(n)
j,j= 1;2;:::, with
24.12.11
2x(n)
1x(n)
2:
Then the zeros in the interval 1<x1
2are 1 x(n)
j.
Whenn(2) is even
24.12.23
4+1
2n+2<x(n)
1<3
4+1
2n+1;
24.12.3 x(n)
1 3
41
2n+1, n!1 ,
and asn!1 withm(1) xed,
24.12.4x(n)
2m 1!m 1
4; x(n)
2m!m+1
4:
Whennis oddx(n)
1=1
2,x(n)
2= 1 (n3), and as
n!1 withm(1) xed,
24.12.5 x(n)
2m 1!m 1
2; x(n)
2m!m:
LetR(n) be the total number of real zeros of Bn(x).
ThenR(n) =nwhen 1n5, and
24.12.6 R(n)2n=(e), n!1 .
24.12(ii) Euler Polynomials: Real Zeros
For the interval1
2x <1denote the zeros of En(x)
byy(n)
j,j= 1;2;:::, with
24.12.71
2y(n)
1y(n)
2:
Then the zeros in the interval 1<x1
2are 1 y(n)
j.
Whenn(2) is eveny(n)
1= 1, and as n!1 with
m(1) xed,
24.12.8 y(n)
m!m:Whennis oddy(n)
1=1
2,
24.12.93
2 n+1
3(n!)<y(n)
2<3
2,n= 3;7;11;:::,
24.12.103
2<y(n)
2<3
2+n+1
3(n!),n= 5;9;13;:::,
and asn!1 withm(1) xed,
24.12.11 y(n)
2m!m 1
2:
24.12(iii) Complex Zeros
For complex zeros of Bernoulli and Euler polynomials,
see Delange (1987) and Dilcher (1988). A related topic
is the irreducibility of Bernoulli and Euler polynomials.
For details and references, see Dilcher (1987b), Kimura
(1988), or Adelberg (1992).
24.12(iv) Multiple Zeros
Bn(x),n= 1;2;:::, has no multiple zeros. The only
polynomial En(x) with multiple zeros is E5(x) = (x
1
2)(x2 x 1)2.
24.13 Integrals
24.13(i) Bernoulli Polynomials
24.13.1Z
Bn(t)dt=Bn+1(t)
n+ 1+ const.;
24.13.2Zx+1
xBn(t)dt=xn,n= 1;2;:::,
24.13.3Zx+(1=2)
xBn(t)dt=En(2x)
2n+1;
24.13.4Z1=2
0Bn(t)dt=1 2n+1
2nBn+1
n+ 1;
24.13.5Z3=4
1=4Bn(t)dt=En
22n+1:
Form;n = 1;2;:::,
24.13.6Z1
0Bn(t)Bm(t)dt=( 1)n 1m!n!
(m+n)!Bm+n:
24.13(ii) Euler Polynomials
24.13.7Z
En(t)dt=En+1(t)
n+ 1+ const.;
24.13.8Z1
0En(t)dt= 2En+1(0)
n+ 1=4(2n+2 1)
(n+ 1)(n+ 2)Bn+2;
24.13.9Z1=2
0E2n(t)dt= E2n+1(0)
2n+ 1=2(22n+2 1)B2n+2
(2n+ 1)(2n+ 2);
24.14 Sums 595
24.13.10Z1=2
0E2n 1(t)dt=E2n
n22n+1,n= 1;2;:::.
Form;n = 1;2;:::,
24.13.11Z1
0En(t)Em(t)dt
= ( 1)n4(2m+n+2 1)m!n!
(m+n+ 2)!Bm+n+2:
24.13(iii) Compendia
For Laplace and inverse Laplace transforms see Prud-
nikov et al. (1992a,xx3.28.1{3.28.2) and Prudnikov
et al. (1992b,xx3.26.1{3.26.2). For other integrals see
Prudnikov et al. (1990, pp. 55{57).
24.14 Sums
24.14(i) Quadratic Recurrence Relations
24.14.1
nX
k=0n
k
Bk(x)Bn k(y) =n(x+y 1)Bn 1(x+y)
(n 1)Bn(x+y);
24.14.2
nX
k=0n
k
BkBn k= (1 n)Bn nBn 1:
24.14.3
nX
k=0n
k
Ek(h)En k(x) = 2(En+1(x+h)
(x+h 1)En(x+h));
24.14.4
nX
k=0n
k
EkEn k= 2n+1En+1(0)
= 2n+2(1 2n+2)Bn+2
n+ 2:
24.14.5
nX
k=0n
k
Ek(h)Bn k(x) = 2nBn 1
2(x+h)
;
24.14.6
nX
k=0n
k
2kBkEn k= 2(1 2n 1)Bn nEn 1:
Letm+nbe even with mandnnonzero. Then
24.14.7mX
j=0nX
k=0m
jn
kBjBk
m+n j k+ 1
= ( 1)m 1m!n!
(m+n)!Bm+n:24.14(ii) Higher-Order Recurrence Relations
In the following two identities, valid for n2, the
sums are taken over all nonnegative integers j;k;` with
j+k+`=n.
24.14.8
X (2n)!
(2j)!(2k)!(2`)!B2jB2kB2`
= (n 1)(2n 1)B2n+n(n 1
2)B2n 2;
24.14.9
X (2n)!
(2j)!(2k)!(2`)!E2jE2kE2`=1
2(E2n E2n+2):
In the next identity, valid for n4, the sum is taken
over all positive integers j;k;`;m withj+k+`+m=n.
24.14.10X (2n)!
(2j)!(2k)!(2`)!(2m)!B2jB2kB2`B2m
= 2n+ 3
3
B2n 4
3n2(2n 1)B2n 2:
For (24.14.11) and (24.14.12), see Al-Salam and Car-
litz (1959). These identities can be regarded as higher-
order recurrences. Let det[ ar+s] denote a Hankel (or
persymmetric )determinant , that is, an ( n+ 1)(n+ 1)
determinant with element ar+sin rowrand column s
forr;s= 0;1;:::;n . Then
24.14.11
det[Br+s] = ( 1)n(n+1)=2 nY
k=1k!!6, 2n+1Y
k=1k!!
;
24.14.12
det[Er+s] = ( 1)n(n+1)=2 nY
k=1k!!2
:
See also Sachse (1882).
24.14(iii) Compendia
For other sums involving Bernoulli and Euler numbers
and polynomials see Hansen (1975, pp. 331{347) and
Prudnikov et al. (1990, pp. 383{386).
24.15 Related Sequences of Numbers
24.15(i) Genocchi Numbers
24.15.12t
et+ 1=1X
n=1Gntn
n!;
24.15.2 Gn= 2(1 2n)Bn:
See Table 24.15.1.
596 Bernoulli and Euler Polynomials
24.15(ii) Tangent Numbers
24.15.3 tant=1X
n=0Tntn
n!;
24.15.4
T2n 1= ( 1)n 122n(22n 1)
2nB2n,n= 1;2;:::,
24.15.5 T2n= 0, n= 0;1;:::.
Table 24.15.1 : Genocchi and Tangent numbers.
n 0 1 2 3 4 5 6 7 8
Gn0 1 1 0 1 0 3 0 17
Tn0 1 0 2 0 16 0 272 0
24.15(iii) Stirling Numbers
The Stirling numbers of the rst kind s(n;m), and the
second kind S(n;m), are as de ned in x26.8(i).
24.15.6Bn=nX
k=0( 1)kk!S(n;k)
k+ 1;
24.15.7Bn=nX
k=0( 1)kn+ 1
k+ 1
S(n+k;k) n+k
k
;
24.15.8nX
k=0( 1)n+ks(n+ 1;k+ 1)Bk=n!
n+ 1:
In (24.15.9) and (24.15.10) pdenotes a prime. See
Horata (1991).
24.15.9
pBn
nS(p 1 +n;p 1) (modp2), 1np 2,
24.15.102n 1
4np2B2nS(p+ 2n;p 1) (modp3),
22np 3.
24.15(iv) Fibonacci and Lucas Numbers
The Fibonacci numbers are de ned by u0= 0,u1= 1,
andun+1=un+un 1,n1. The Lucas numbers are
de ned byv0= 2,v1= 1, andvn+1=vn+vn 1,n1.
24.15.11
bn/2cX
k=0n
2k5
9k
B2kun 2k=n
6vn 1+n
3nv2n 2;
24.15.12
bn/2cX
k=0n
2k5
4k
E2kvn 2k=1
2n 1:
For further information on the Fibonacci numbers
seex26.11.24.16 Generalizations
24.16(i) Higher-Order Analogs
Polynomials and Numbers of Integer Order
For`= 0;1;2;:::,Bernoulli and Euler polynomials of
order`are de ned respectively by
24.16.1t
et 1`
ext=1X
n=0B(`)
n(x)tn
n!,jtj<2,
24.16.22
et+ 1`
ext=1X
n=0E(`)
n(x)tn
n!,jtj<.
Whenx= 0 they reduce to the Bernoulli and Euler
numbers of order `:
24.16.3 B(`)
n=B(`)
n(0); E(`)
n=E(`)
n(0):
Also for`= 1;2;3;:::,
24.16.4ln(1 +t)
t`
=`1X
n=0B(`+n)
n
`+ntn
n!,jtj<1.
For this and other properties see Milne-Thomson (1933,
pp. 126{153) or N orlund (1924, pp. 144{162).
For extensions of B(`)
n(x) to complex values of x,n,
and`, and also for uniform asymptotic expansions for
largexand largen, see Temme (1995b).
Bernoulli Numbers of the Second Kind
24.16.5t
ln(1 +t)=1X
n=0bntn,jtj<1,
24.16.6 n!bn= 1
n 1B(n 1)
n ,n= 2;3;:::.
Degenerate Bernoulli Numbers
For suciently small jtj,
24.16.7t
(1 +t)1/ 1=1X
n=0 n()tn
n!;
24.16.8
n() =n!bnn+bn/2cX
k=1n
2kB2ks(n 1;2k 1)n 2k,
n= 2;3;:::.
Heres(n;m) again denotes the Stirling number of the
rst kind.
N orlund Polynomials
24.16.9t
et 1x
=1X
n=0B(x)
ntn
n!,jtj<2.
B(x)
nis a polynomial in xof degreen. (This notation is
consistent with (24.16.3) when x=`.)
Applications 597
24.16(ii) Character Analogs
Let be a primitive Dirichlet character mod f(see
x27.8). Then fis called the conductor of . General-
ized Bernoulli numbers and polynomials belonging to
are de ned by
24.16.10fX
a=1 (a)teat
eft 1=1X
n=0Bn; tn
n!;
24.16.11 Bn; (x) =nX
k=0n
k
Bk; xn k:
Let 0be the trivial character and 4the unique (non-
trivial) character with f= 4; that is, 4(1) = 1,
4(3) = 1, 4(2) = 4(4) = 0. Then
24.16.12 Bn(x) =Bn; 0(x 1);
24.16.13En(x) = 21 n
n+ 1Bn+1; 4(2x 1):
For further properties see Berndt (1975a).
24.16(iii) Other Generalizations
In no particular order, other generalizations include:
Bernoulli numbers and polynomials with arbitrary com-
plex index (Butzer et al. (1992)); Euler numbers and
polynomials with arbitrary complex index (Butzer et al.
(1994)); q-analogs (Carlitz (1954b), Andrews and Foata
(1980)); conjugate Bernoulli and Euler polynomials
(Hauss (1997, 1998)); Bernoulli{Hurwitz numbers (Katz
(1975)); poly-Bernoulli numbers (Kaneko (1997)); Uni-
versal Bernoulli numbers (Clarke (1989)); p-adic in-
teger order Bernoulli numbers (Adelberg (1996)); p-
adicq-Bernoulli numbers (Kim and Kim (1999)); pe-
riodic Bernoulli numbers (Berndt (1975b)); cotangent
numbers (Girstmair (1990a)); Bernoulli{Carlitz num-
bers (Goss (1978)); Bernoulli-Pad e numbers (Dilcher
(2002)); Bernoulli numbers belonging to periodic func-
tions (Urbanowicz (1988)); cyclotomic Bernoulli num-
bers (Girstmair (1990b)); modi ed Bernoulli numbers
(Zagier (1998)); higher-order Bernoulli and Euler poly-
nomials with multiple parameters (Erd elyi et al. (1953a,
xx1.13.1, 1.14.1)).
Applications
24.17 Mathematical Applications
24.17(i) Summation
Euler{Maclaurin Summation Formula
Seex2.10(i). For a generalization see Olver (1997b,
p. 284).Boole Summation Formula
Let 0h1 anda;m, andnbe integers such that
n>a ,m> 0, andf(m)(x) is absolutely integrable over
[a;n]. Then with the notation of x24.2(iii)
24.17.1
n 1X
j=a( 1)jf(j+h) =1
2m 1X
k=0Ek(h)
k!
( 1)n 1f(k)(n)
+ ( 1)af(k)(a)
+Rm(n);
where
24.17.2
Rm(n) =1
2(m 1)!Zn
af(m)(x)eEm 1(h x)dx:
Calculus of Finite Di erences
See Milne-Thomson (1933), N orlund (1924), or Jordan
(1965). For a more modern perspective see Graham
et al. (1994).
24.17(ii) Spline Functions
Euler Splines
LetSndenote the class of functions that have n 1
continuous derivatives on Rand are polynomials of de-
gree at most nin each interval ( k;k+ 1),k2Z. The
members ofSnare called cardinal spline functions . The
functions
24.17.3 Sn(x) =eEn
x+1
2n+1
2
eEn 1
2n+1
2,n= 0;1;:::,
are called Euler splines of degree n. For each n,Sn(x)
is the unique bounded function such that Sn(x)2Sn
and
24.17.4 Sn(k) = ( 1)k, k2Z.
The function Sn(x) is also optimal in a certain sense;
see Schoenberg (1971).
Bernoulli Monosplines
A function of the form xn S(x), withS(x)2Sn 1
is called a cardinal monospline of degree n. Again with
the notation ofx24.2(iii) de ne
24.17.5Mn(x) =(eBn(x) Bn; n even;
eBn
x+1
2
; n odd:
Mn(x) is a monospline of degree n, and it follows from
(24.4.25) and (24.4.27) that
24.17.6 Mn(k) = 0, k2Z.
For eachn= 1;2;::: the function Mn(x) is also
the unique cardinal monospline of degree nsatisfying
(24.17.6), provided that
24.17.7 Mn(x) =O(jxj
),x!1 ,
for some positive constant
.
598 Bernoulli and Euler Polynomials
For anyn2 the function
24.17.8 F(x) =eBn(x) 2 nBn
is the unique cardinal monospline of degree nhaving
the least supremum norm kFk1onR(minimality prop-
erty).
24.17(iii) Number Theory
Bernoulli and Euler numbers and polynomials occur in:
number theory via (24.4.7), (24.4.8), and other identi-
ties involving sums of powers; the Riemann zeta func-
tion andL-series (x25.15, Apostol (1976), and Ireland
and Rosen (1990)); arithmetic of cyclotomic elds and
the classical theory of Fermat's last theorem (Riben-
boim (1979) and Washington (1997)); p-adic analysis
(Koblitz (1984, Chapter 2)).
24.18 Physical Applications
Bernoulli polynomials appear in statistical physics
(Ord o~ nez and Driebe (1996)), in discussions of Casimir
forces (Li et al. (1991)), and in a study of quark-gluon
plasma (Meisinger et al. (2002)).
Euler polynomials also appear in statistical physics
as well as in semi-classical approximations to quan-
tum probability distributions (Ballentine and McRae
(1998)).
Computation
24.19 Methods of Computation
24.19(i) Bernoulli and Euler Numbers and
Polynomials
Equations (24.5.3) and (24.5.4) enable BnandEnto be
computed by recurrence. For higher values of nmore ef-
cient methods are available. For example, the tangent
numbersTncan be generated by simple recurrence rela-
tions obtained from (24.15.3), then (24.15.4) is applied.
A similar method can be used for the Euler numbers
based on (4.19.5). For details see Knuth and Buckholtz
(1967).
Another method is based on the identities
24.19.1N2n=2(2n)!
(2)2n0
@Y
p 1j2np1
A Y
pp2n
p2n 1!
;
24.19.2 D2n=Y
p 1j2np; B 2n=N2n
D2n:
IfeN2ndenotes the right-hand side of (24.19.1) but with
the second product taken only for p
(e) 12n
+ 1,thenN2n=l
eN2nm
forn2. For proofs and further
information see Fillebrown (1992).
For other information see Chellali (1988) and Zhang
and Jin (1996, pp. 1{11). For algorithms for comput-
ingBn,En,Bn(x), andEn(x) see Spanier and Oldham
(1987, pp. 37, 41, 171, and 179{180).
24.19(ii) Values of BnModulop
For number-theoretic applications it is important to
computeB2n(modp) for 2np 3; in particular
to nd the irregular pairs (2n;p) for which B2n0
(modp). We list here three methods, arranged in in-
creasing order of eciency.
Tanner and Wagsta (1987) derives a congruence
(modp) for Bernoulli numbers in terms of sums
of powers. See also x24.10(iii).
Buhler et al. (1992) uses the expansion
24.19.3t2
cosht 1= 21X
n=0(2n 1)B2nt2n
(2n)!;
and computes inverses modulo pof the left-hand
side. Multisectioning techniques are applied in im-
plementations. See also Crandall (1996, pp. 116{
120).
A method related to \Stickelberger codes" is ap-
plied in Buhler et al. (2001); in particular, it
allows for an ecient search for the irregular
pairs (2n;p). Discrete Fourier transforms are used
in the computations. See also Crandall (1996,
pp. 120{124).
24.20 Tables
Abramowitz and Stegun (1964, Chapter 23) includes
exact values ofPm
k=1kn,m= 1(1)100, n= 1(1)10;P1
k=1k n,P1
k=1( 1)k 1k n,P1
k=0(2k+ 1) n,n=
1;2;:::, 20D;P1
k=0( 1)k(2k+1) n,n= 1;2;:::, 18D.
Wagsta (1978) gives complete prime factorizations
ofNnandEnforn= 20(2)60 and n= 8(2)42, respec-
tively. In Wagsta (2002) these results are extended
ton= 60(2)152 and n= 40(2)88, respectively, with
further complete and partial factorizations listed up to
n= 300 andn= 200, respectively.
For information on tables published before 1961 see
Fletcher et al. (1962, v. 1,x4) and Lebedev and Fedorova
(1960, Chapters 11 and 14).
24.21 Software
Seehttp://dlmf.nist.gov/24.21 .
References 599
References
General References
The main references used in writing this chapter are
Erd elyi et al. (1953a, Chapter 1), N orlund (1924, Chap-
ter 2), and N orlund (1922). Introductions to the sub-
ject are contained in Dence and Dence (1999) and
Rademacher (1973); see also Apostol (2008). A com-
prehensive bibliography on the topics of this chapter
can be found in Dilcher et al. (1991).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x24.2 N orlund (1924, Chapter 2), Milne-Thomson
(1933, Chapter 6). Tables are from Abramowitz
and Stegun (1964, pp. 809{810).
x24.3 These graphs were produced at NIST.
x24.4 N orlund (1924, Chapter 2), Milne-Thomson
(1933, Chapter 6), Howard (1996b), Slavutski
(2000), Apostol (2006), Todorov (1991). For
(24.4.12){(24.4.17) use (24.2.3) and (24.2.8). For
(24.4.25){(24.4.33) use xx24.4(ii) and 24.4(v). For
(24.4.34) and (24.4.35) use (24.2.3) and (24.2.8).
x24.5 Erd elyi et al. (1953a, Chapter 1), N orlund (1924,
pp. 19 and 24), Apostol (2008), Apostol (1976,
p. 275), Riordan (1979, p. 114).
x24.6 Gould (1972, pp. 45{46), Horata (1989), Todorov
(1978), Schwatt (1962, p. 270), Carlitz (1961b,
p. 134), Todorov (1991, pp. 176{177), Jordan
(1965, p. 236).
x24.7 Erd elyi et al. (1953a, Chapter 1), Ramanujan
(1927, p. 7), Paris and Kaminski (2001, p. 173).x24.8 Apostol (1976, p. 267), Erd elyi et al. (1953a,
p. 42), Berndt (1975b, pp. 176{178).
x24.9 Olver (1997b, p. 283), Temme (1996a, p. 16),
Lehmer (1940, p. 538), Leeming (1989), Alzer
(2000). For (24.9.5) use (24.9.8), (24.4.35),
(24.2.7), and (24.4.26). For (24.9.10) use (24.4.28)
and (24.8.4) with x=1
2; see also Lehmer (1940,
p. 538).
x24.10 Ireland and Rosen (1990, Chapter 15), Wash-
ington (1997, Chapter 5), Carlitz (1953, p. 167),
Ernvall (1979, pp. 36 and 24), Slavutski (1995,
1999), Uspensky and Heaslet (1939, p. 261),
Ribenboim (1979, p. 105), Girstmair (1990b),
Carlitz (1954a).
x24.11 Leeming (1977), Dilcher (1987a).
x24.12 Olver (1997b, p. 283), Inkeri (1959), Leeming
(1989), Delange (1991), Lehmer (1940), Dilcher
(1988, p. 77), Howard (1976), Delange (1988),
Dilcher (2008), Brillhart (1969).
x24.13 N orlund (1922, p. 143), Apostol (1976, p. 276),
N orlund (1924, pp. 31 and 36). For (24.13.1) and
(24.13.2) use (24.4.34) and (24.4.1).
x24.14 N orlund (1922, pp. 135{142), Carlitz (1961a,
p. 992), Dilcher (1996), Sitaramachandrarao and
Davis (1986), Huang and Huang (1999).
x24.15 Dumont and Viennot (1980), Graham et al.
(1994, Chapter 6), Knuth and Buckholtz (1967),
Todorov (1984, pp. 310 and 343), Gould (1972,
pp. 44 and 48), Kelisky (1957, pp. 32 and 34).
x24.16 Howard (1996a), Washington (1997, pp. 31{
34), Dilcher (1988, pp. 8 and 9).
x24.17 Temme (1996a, pp. 17 and 18), N orlund
(1924, pp. 29{36), Schumaker (1981, pp. 152{153),
Schoenberg (1973, pp. 40{41 and 101).
x24.19 For (24.19.3) use (24.2.1).
Chapter 25
Zeta and Related Functions
T. M. Apostol1
Notation 602
25.1 Special Notation . . . . . . . . . . . . . 602
Riemann Zeta Function 602
25.2 De nition and Expansions . . . . . . . . 602
25.3 Graphics . . . . . . . . . . . . . . . . . . 603
25.4 Re
ection Formulas . . . . . . . . . . . . 603
25.5 Integral Representations . . . . . . . . . 604
25.6 Integer Arguments . . . . . . . . . . . . 605
25.7 Integrals . . . . . . . . . . . . . . . . . . 606
25.8 Sums . . . . . . . . . . . . . . . . . . . 606
25.9 Asymptotic Approximations . . . . . . . . 606
25.10 Zeros . . . . . . . . . . . . . . . . . . . 606
Related Functions 607
25.11 Hurwitz Zeta Function . . . . . . . . . . 60725.12 Polylogarithms . . . . . . . . . . . . . . 610
25.13 Periodic Zeta Function . . . . . . . . . . 612
25.14 Lerch's Transcendent . . . . . . . . . . . 612
25.15 Dirichlet L-functions . . . . . . . . . . . 612
Applications 613
25.16 Mathematical Applications . . . . . . . . 613
25.17 Physical Applications . . . . . . . . . . . 614
Computation 614
25.18 Methods of Computation . . . . . . . . . 614
25.19 Tables . . . . . . . . . . . . . . . . . . . 614
25.20 Approximations . . . . . . . . . . . . . . 615
25.21 Software . . . . . . . . . . . . . . . . . . 615
References 615
1California Institute of Technology, Pasadena, California.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
601
602 Zeta and Related Functions
Notation
25.1 Special Notation
(For other notation see pp. xiv and 873.)
k;m;n nonnegative integers.
p prime number.
x real variable.
a real or complex parameter.
s=+itcomplex variable.
z=x+iycomplex variable.
Euler's constant ( x5.2(ii)).
(x) digamma function 0(x)= (x) except in
x25.16. Seex5.2(i).
Bn;Bn(x) Bernoulli number and polynomial
(x24.2(i)).
eBn(x) periodic Bernoulli function Bn(x bxc).
mjn m dividesn.
primes on function symbols: derivatives with
respect to argument.
The main function treated in this chapter is the Rie-
mann zeta function (s). This notation was introduced
in Riemann (1859).
The main related functions are the Hurwitz zeta
function(s;a), the dilogarithm Li 2(z), the polylog-
arithm Li s(z) (also known as Jonqui ere's function
(z;s)), Lerch's transcendent ( z;s;a ), and the Dirich-
letL-functionsL(s; ).
Riemann Zeta Function
25.2 De nition and Expansions
25.2(i) De nition
When<s>1,
25.2.1 (s) =1X
n=11
ns:
Elsewhere(s) is de ned by analytic continuation. It is
a meromorphic function whose only singularity in Cis
a simple pole at s= 1, with residue 1.
25.2(ii) Other In nite Series
25.2.2 (s) =1
1 2 s1X
n=01
(2n+ 1)s,<s>1.
25.2.3 (s) =1
1 21 s1X
n=1( 1)n 1
ns,<s>0.25.2.4(s) =1
s 1+1X
n=0( 1)n
n!
n(s 1)n,<s>0,
where
25.2.5
n= lim
m!1 mX
k=1(lnk)n
k (lnm)n+1
n+ 1!
:
25.2.6 0(s) = 1X
n=2(lnn)n s,<s>1.
25.2.7
(k)(s) = ( 1)k1X
n=2(lnn)kn s,<s>1,k= 1;2;3;:::.
For further expansions of functions similar to
(25.2.1) (Dirichlet series) see x27.4. This includes, for
example, 1=(s).
25.2(iii) Representations by the
Euler{Maclaurin Formula
25.2.8(s) =NX
k=11
ks+N1 s
s 1 sZ1
Nx bxc
xs+1dx,
<s>0,N= 1;2;3;:::.
25.2.9(s) =NX
k=11
ks+N1 s
s 1 1
2N s
+nX
k=1s+ 2k 2
2k 1B2k
2kN1 s 2k
s+ 2n
2n+ 1Z1
NeB2n+1(x)
xs+2n+1dx,
<s> 2n;n;N = 1;2;3;:::.
25.2.10(s) =1
s 1+1
2+nX
k=1s+ 2k 2
2k 1B2k
2k
s+ 2n
2n+ 1Z1
1eB2n+1(x)
xs+2n+1dx,
<s> 2n,n= 1;2;3;:::.
ForB2kseex24.2(i), and for eBn(x) seex24.2(iii).
25.2(iv) In nite Products
25.2.11 (s) =Y
p(1 p s) 1,<s>1,
product over all primes p.
25.2.12(s) =(2)se s (
s=2)
2(s 1) 1
2s+ 1Y
1 s
es=;
product over zeros ofwith<>0 (seex25.10(i));
is Euler's constant ( x5.2(ii)).
25.3 Graphics 603
25.3 Graphics
Figure 25.3.1 : Riemann zeta function (x) and its
derivative0(x), 20x10.
Figure 25.3.2 : Riemann zeta function (x) and its
derivative0(x), 12x 2.
Figure 25.3.3 : Modulus of the Riemann zeta function
j(x+iy)j, 4x4, 10y40.
Figure 25.3.4 :Z(t), 0t50.Z(t) and 1
2+it
have
the same zeros. See x25.10(i).
Figure 25.3.5 :Z(t), 1000t1050.
Figure 25.3.6 :Z(t), 10000t10050.
25.4 Re
ection Formulas
Fors6= 0;1,
25.4.1(1 s) = 2(2) scos 1
2s
(s)(s);
25.4.2(s) = 2(2)s 1sin 1
2s
(1 s)(1 s):
Equivalently,
25.4.3 (s) =(1 s);
604 Zeta and Related Functions
where(s) isRiemann's -function , de ned by:
25.4.4(s) =1
2s(s 1) 1
2s
s=2(s):
Fors6= 0;1 andk= 1;2;3;:::,
25.4.5
( 1)k(k)(1 s)
=2
(2)skX
m=0mX
r=0k
mm
r
<(ck m) cos 1
2s
+=(ck m) sin 1
2s
(r)(s)(m r)(s);
where
25.4.6 c= ln(2) 1
2i:
25.5 Integral Representations
25.5(i) In Terms of Elementary Functions
Throughout this subsection s6= 1.25.5.1(s) =1
(s)Z1
0xs 1
ex 1dx,<s>1.
25.5.2(s) =1
(s+ 1)Z1
0exxs
(ex 1)2dx,<s>1.
25.5.3(s) =1
(1 21 s) (s)Z1
0xs 1
ex+ 1dx,<s>0.
25.5.4(s) =1
(1 21 s) (s+ 1)Z1
0exxs
(ex+ 1)2dx,
<s>0.
25.5.5(s) = sZ1
0x bxc 1
2
xs+1dx, 1<<s<0.
25.5.6
(s) =1
2+1
s 1+1
(s)Z1
01
ex 1 1
x+1
2xs 1
exdx,
<s> 1.
25.5.7(s) =1
2+1
s 1+nX
m=1B2m
(2m)! (s+ 2m 1)
(s)+1
(s)Z1
0
1
ex 1 1
x+1
2 nX
m=1B2m
(2m)!x2m 1!
xs 1
exdx,
<s> (2n+ 1),n= 1;2;3;:::.
25.5.8(s) =1
2(1 2 s) (s)Z1
0xs 1
sinhxdx,<s>1.
25.5.9(s) =2s 1
(s+ 1)Z1
0xs
(sinhx)2dx,<s>1.
25.5.10(s) =2s 1
1 21 sZ1
0cos(sarctanx)
(1 +x2)s=2cosh 1
2xdx:
25.5.11
(s) =1
2+1
s 1+ 2Z1
0sin(sarctanx)
(1 +x2)s=2(e2x 1)dx:
25.5.12(s) =2s 1
s 1 2sZ1
0sin(sarctanx)
(1 +x2)s=2(ex+ 1)dx:
25.5(ii) In Terms of Other Functions
25.5.13
(s) =s=2
s(s 1) 1
2s
+s=2
1
2sZ1
1
xs=2+x(1 s)=2!(x)
xdx,
s6= 1,
where
25.5.14!(x) =1X
n=1e n2x=1
2(3(0jix) 1):For3seex20.2(i). For similar representations involving
other theta functions see Erd elyi et al. (1954a, p. 339).
In (25.5.15){(25.5.19), 0 <<s < 1, (x) is the
digamma function, and
is Euler's constant ( x5.2).
(25.5.16) is also valid for 0 <<s<2,s6= 1.
25.5.15(s) =1
s 1+sin(s)
Z1
0(ln(1 +x) (1 +x))x sdx;
25.5.16(s) =1
s 1+sin(s)
(s 1)
Z1
01
1 +x 0(1 +x)
x1 sdx;
25.5.17(1 +s) =sin(s)
Z1
0(
+ (1 +x))x s 1dx;
25.5.18(1 +s) =sin(s)
sZ1
0 0(1 +x)x sdx;
25.5.19(m+s) = ( 1)m 1 (s) sin(s)
(m+s)
Z1
0 (m)(1 +x)x sdx,
m= 1;2;3;:::.
25.6 Integer Arguments 605
25.5(iii) Contour Integrals
25.5.20
(s) = (1 s)
2iZ(0+)
1zs 1
e z 1dz,s6= 1;2;:::,
where the integration contour is a loop around the neg-
ative real axis; it starts at 1, encircles the origin once
in the positive direction without enclosing any of the
pointsz=2i,4i, . . . , and returns to 1. Equiv-
alently,
25.5.21
(s) = (1 s)
2i(1 21 s)Z(0+)
1zs 1
e z+ 1dz,s6= 1;2;:::.
The contour here is any loop that encircles the origin
in the positive direction not enclosing any of the points
i,3i, . . . .
25.6 Integer Arguments
25.6(i) Function Values
25.6.1
(0) = 1
2; (2) =2
6; (4) =4
90; (6) =6
945:
25.6.2(2n) =(2)2n
2(2n)!jB2nj,n= 1;2;3;:::.
25.6.3( n) = Bn+1
n+ 1, n= 1;2;3;:::.
25.6.4( 2n) = 0, n= 1;2;3;:::.25.6.5
(k+ 1) =1
k!1X
n1=1:::1X
nk=11
n1nk(n1++nk),
k= 1;2;3;:::.
25.6.6
(2k+ 1) =( 1)k+1(2)2k+1
2(2k+ 1)!Z1
0B2k+1(t) cot(t)dt,
k= 1;2;3;:::.
25.6.7 (2) =Z1
0Z1
01
1 xydxdy:
25.6.8 (2) = 31X
k=11
k2 2k
k:
25.6.9 (3) =5
21X
k=1( 1)k 1
k3 2k
k:
25.6.10 (4) =36
171X
k=11
k4 2k
k:
25.6(ii) Derivative Values
25.6.11 0(0) = 1
2ln(2):
25.6.1200(0) = 1
2(ln(2))2+1
2
2 1
242+
1;
where
1is given by (25.2.5).
Withcde ned by (25.4.6) and n= 1;2;3;:::;
25.6.13 ( 1)k(k)( 2n) =2( 1)n
(2)2n+1kX
m=0mX
r=0k
mm
r
=(ck m) (r)(2n+ 1)(m r)(2n+ 1);
25.6.14 ( 1)k(k)(1 2n) =2( 1)n
(2)2nkX
m=0mX
r=0k
mm
r
<(ck m) (r)(2n)(m r)(2n);
25.6.15 0(2n) =( 1)n+1(2)2n
2(2n)!(2n0(1 2n) ( (2n) ln(2))B2n):
25.6(iii) Recursion Formulas
25.6.16
n+1
2
(2n) =n 1X
k=1(2k)(2n 2k),n2.
25.6.17
n+3
4
(4n+ 2) =nX
k=1(2k)(4n+ 2 2k),n1.25.6.18
n+1
4
(4n) +1
2((2n))2=nX
k=1(2k)(4n 2k),
n1.
25.6.19
m+n+3
2
(2m+ 2n+ 2)
= mX
k=1+nX
k=1!
(2k)(2m+ 2n+ 2 2k),
m0,n0,m+n1.
606 Zeta and Related Functions
25.6.20
1
2(22n 1)(2n) =n 1X
k=1(22n 2k 1)(2n 2k)(2k),
n2.
For related results see Basu and Apostol (2000).
25.7 Integrals
For de nite integrals of the Riemann zeta function see
Prudnikov et al. (1986b,x2.4), Prudnikov et al. (1992a,
x3.2), and Prudnikov et al. (1992b,x3.2).
25.8 Sums
25.8.11X
k=2((k) 1) = 1:
25.8.21X
k=0 (s+k)
(k+ 1)!((s+k) 1)
= (s 1), s6= 1;0; 1; 2;:::.
25.8.31X
k=0 (s+k)(s+k)
k! (s)2s+k= (1 2 s)(s),s6= 1.
25.8.4
1X
k=1( 1)k
k((nk) 1) = ln0
@n 1Y
j=0
2 e(2j+1)i=n1
A,
n= 2;3;4;:::.
25.8.51X
k=2(k)zk=
z z (1 z),jzj<1.
25.8.61X
k=0(2k)z2k= 1
2zcot(z),jzj<1.
25.8.71X
k=2(k)
kzk=
z+ ln (1 z),jzj<1.
25.8.81X
k=1(2k)
kz2k= lnz
sin(z)
,jzj<1.
25.8.91X
k=1(2k)
(2k+ 1)22k=1
2 1
2ln 2:
25.8.101X
k=1(2k)
(2k+ 1)(2k+ 2)22k=1
4 7
42(3):
For other sums see Prudnikov et al. (1986b, pp. 648{
649), Hansen (1975, pp. 355{357), Ogreid and Osland
(1998), and Srivastava and Choi (2001, Chapter 3).25.9 Asymptotic Approximations
Ifx1,y1, 2xy =t, and 01, then as
t!1 with xed,
25.9.1(+it) =X
1nx1
ns+ (s)X
1ny1
n1 s
+O
x
+O
y 1t1
2
;
wheres=+itand
25.9.2 (s) =s 1
2 1
2 1
2s
= 1
2s
:
If=1
2,x=y=p
t=(2), andm=bxc, then
(25.9.1) becomes
25.9.3 1
2+it
=mX
n=11
n1
2+it
+ 1
2+itmX
n=11
n1
2 it+O
t 1=4
:
For other asymptotic approximations see Berry and
Keating (1992), Paris and Cang (1997); see also Paris
and Kaminski (2001, pp. 380{389).
25.10 Zeros
25.10(i) Distribution
The product representation (25.2.11) implies (s)6= 0
for<s > 1. Also,(s)6= 0 for<s= 1, a prop-
erty rst established in Hadamard (1896) and de la
Vall ee Poussin (1896a,b) in the proof of the prime num-
ber theorem (25.16.3). The functional equation (25.4.1)
implies( 2n) = 0 forn= 1;2;3;:::. These are called
thetrivial zeros . Except for the trivial zeros, (s)6= 0
for<s0. In the region 0 <<s < 1, called the
critical strip ,(s) has in nitely many zeros, distributed
symmetrically about the real axis and about the critical
line<s=1
2. The Riemann hypothesis states that all
nontrivial zeros lie on this line.
Calculations relating to the zeros on the critical line
make use of the real-valued function
25.10.1 Z(t) = exp(i#(t)) 1
2+it
;
where
25.10.2 #(t)ph 1
4+1
2it
1
2tln
is chosen to make Z(t) real, and ph 1
4+1
2it
assumes
its principal value. Because jZ(t)j=j 1
2+it
j,Z(t)
vanishes at the zeros of 1
2+it
, which can be sepa-
rated by observing sign changes of Z(t). Because Z(t)
changes sign in nitely often, 1
2+it
has in nitely
many zeros with treal.
Related Functions 607
25.10(ii) Riemann{Siegel Formula
Riemann developed a method for counting the total
numberN(T) of zeros of (s) in that portion of the
critical strip with 0 < t < T . By comparing N(T)
with the number of sign changes of Z(t) we can de-
cide whether (s) has any zeros o the line in this re-
gion. Sign changes of Z(t) are determined by multiply-
ing (25.9.3) by exp( i#(t)) to obtain the Riemann{Siegel
formula :
25.10.3Z(t) = 2mX
n=1cos(#(t) tlnn)
n1=2+R(t);
whereR(t) =O
t 1=4
ast!1 .
The error term R(t) can be expressed as an asymp-
totic series that begins
25.10.4
R(t) = ( 1)m 12
t1=4cos
t (2m+ 1)p
2t 1
8
cos p
2t
+O
t 3=4
:
Riemann also developed a technique for determin-
ing further terms. Calculations based on the Riemann{
Siegel formula reveal that the rst ten billion zeros of
(s) in the critical strip are on the critical line (van de
Lune et al. (1986)). More than one-third of all the ze-
ros in the critical strip lie on the critical line (Levinson
(1974)).
For further information on the Riemann{Siegel ex-
pansion see Berry (1995).Related Functions
25.11 Hurwitz Zeta Function
25.11(i) De nition
The function (s;a) was introduced in Hurwitz (1882)
and de ned by the series expansion
25.11.1
(s;a) =1X
n=01
(n+a)s,<s>1,a6= 0; 1; 2;:::.
(s;a) has a meromorphic continuation in the s-
plane, its only singularity in Cbeing a simple pole at
s= 1 with residue 1. As a function of a, withs(6= 1)
xed,(s;a) is analytic in the half-plane <a >0. The
Riemann zeta function is a special case:
25.11.2 (s;1) =(s):
For most purposes it suces to restrict 0 <<a1
because of the following straightforward consequences
of (25.11.1):
25.11.3 (s;a) =(s;a+ 1) +a s;
25.11.4
(s;a) =(s;a+m) +m 1X
n=01
(n+a)s,m= 1;2;3;:::.
Most references treat real awith 0<a1.
25.11(ii) Graphics
Figure 25.11.1 : Hurwitz zeta function (x;a),a= 0.3,
0.5, 0.8, 1, 20x10. The curves are almost indis-
tinguishable for 14<x< 1, approximately.
Figure 25.11.2 : Hurwitz zeta function (x;a), 19:5
x10, 0:02a1.
608 Zeta and Related Functions
25.11(iii) Representations by the Euler{Maclaurin Formula
25.11.5 (s;a) =NX
n=01
(n+a)s+(N+a)1 s
s 1 sZ1
Nx bxc
(x+a)s+1dx,s6= 1,<s>0,a>0,N= 0;1;2;3;:::.
25.11.6 (s;a) =1
as1
2+a
s 1
s(s+ 1)Z1
0eB2(x)
(x+a)s+2dx,s6= 1,<s> 1,a>0.
25.11.7
(s;a) =1
as+1
(1 +a)s1
2+1 +a
s 1
+nX
k=1s+ 2k 2
2k 1B2k
2k1
(1 +a)s+2k 1 s+ 2n
2n+ 1Z1
1eB2n+1(x)
(x+a)s+2n+1dx,
s6= 1,a>0,n= 1;2;3;:::,<s> 2n.
ForeBn(x) seex24.2(iii).
25.11(iv) Series Representations
25.11.8
s;1
2a
=
s;1
2a+1
2
+ 2s1X
n=0( 1)n
(n+a)s,
<s>0,s6= 1, 0<a1.
25.11.9(1 s;a) =2 (s)
(2)s1X
n=11
nscos 1
2s 2na
,
<s>1, 0<a1.
25.11.10(s;a) =1X
n=0 (n+s)
n! (s)(n+s)(1 a)n,
s6= 1,ja 1j<1.
Whena=1
2, (25.11.10) reduces to (25.8.3); compare
(25.11.11).
25.11(v) Special Values
Throughout this subsection <a>0.
25.11.11
s;1
2
= (2s 1)(s), s6= 1.
25.11.12
(n+ 1;a) =( 1)n+1 (n)(a)
n!,n= 1;2;3;:::.25.11.13 (0;a) =1
2 a:
25.11.14 ( n;a) = Bn+1(a)
n+ 1,n= 0;1;2;:::.
25.11.15
(s;ka) =k sk 1X
n=0
s;a+n
k
,s6= 1,k= 1;2;3;:::.
25.11.16
1 s;h
k
=2 (s)
(2k)skX
r=1coss
2 2rh
k
s;r
k
,
s6= 0;1;h;kintegers, 1hk.
25.11(vi) Derivatives
a-Derivative
25.11.17@
@a(s;a) = s(s+ 1;a),s6= 0;1;<a>0.
s-Derivatives
In (25.11.18){(25.11.24) primes on denote deriva-
tives with respect to s. Similarly inxx25.11(viii) and
25.11(xii).
25.11.18 0(0;a) = ln (a) 1
2ln(2),a>0.
25.11.190(s;a) = lna
as1
2+a
s 1
a1 s
(s 1)2+s(s+ 1)Z1
0eB2(x) ln(x+a)
(x+a)s+2dx (2s+ 1)Z1
0eB2(x)
(x+a)s+2dx,
<s> 1,s6= 1,a>0.
25.11.20( 1)k(k)(s;a) =(lna)k
as1
2+a
s 1
+k!a1 sk 1X
r=0(lna)r
r!(s 1)k r+1 s(s+ 1)Z1
0eB2(x)(ln(x+a))k
(x+a)s+2dx
+k(2s+ 1)Z1
0eB2(x)(ln(x+a))k 1
(x+a)s+2dx k(k 1)Z1
0eB2(x)(ln(x+a))k 2
(x+a)s+2dx,
<s> 1,s6= 1,a>0.
25.11 Hurwitz Zeta Function 609
25.11.21
0
1 2n;h
k
=( (2n) ln(2k))B2n(h=k)
2n ( (2n) ln(2))B2n
2nk2n+( 1)n+1
(2k)2nk 1X
r=1sin2rh
k
(2n 1)r
k
+( 1)n+12(2n 1)!
(2k)2nk 1X
r=1cos2rh
k
0
2n;r
k
+0(1 2n)
k2n;
whereh;kare integers with 1 hkandn= 1;2;3;:::.
25.11.22 0
1 2n;1
2
= B2nln 2
n4n (22n 1 1)0(1 2n)
22n 1, n= 1;2;3;:::.
25.11.23
0
1 2n;1
3
= (9n 1)B2n
8np
3(32n 1 1) B2nln 3
4n32n 1 ( 1)n (2n 1) 1
3
2p
3(6)2n 1
32n 1 1
0(1 2n)
232n 1,n= 1;2;3;:::.
25.11.24k 1X
r=10
s;r
k
= (ks 1)0(s) +ks(s) lnk, s6= 1,k= 1;2;3;:::.
25.11(vii) Integral Representations
25.11.25 (s;a) =1
(s)Z1
0xs 1e ax
1 e xdx, <s>1,<a>0.
25.11.26 (s;a) = sZ1
ax bxc 1
2
(x+a)s+1dx, 1<<s<0, 0<a1.
25.11.27 (s;a) =1
2a s+a1 s
s 1+1
(s)Z1
01
ex 1 1
x+1
2xs 1
eaxdx,<s> 1,s6= 1,<a>0.
25.11.28(s;a) =1
2a s+a1 s
s 1+nX
k=1 (s+ 2k 1)
(s)B2k
(2k)!a 2k s+1
+1
(s)Z1
0
1
ex 1 1
x+1
2 nX
k=1B2k
(2k)!x2k 1!
xs 1e axdx,<s> (2n+ 1),s6= 1,<a>0.
25.11.29 (s;a) =1
2a s+a1 s
s 1+ 2Z1
0sin(sarctan(x=a))
(a2+x2)s=2(e2x 1)dx, s6= 1,<a>0.
25.11.30 (s;a) = (1 s)
2iZ(0+)
1eazzs 1
1 ezdz, s6= 1,<a>0,
where the integration contour is a loop around the negative real axis as described for (25.5.20).
25.11(viii) Further Integral Representations
25.11.311
(s)Z1
0xs 1e ax
2 coshxdx= 4 s
s;1
4+1
4a
s;3
4+1
4a
,<s>0,<a> 1.
25.11.32Za
0xn (x)dx= ( 1)n 10( n) + ( 1)nh(n)Bn+1
n+ 1 nX
k=0( 1)kn
k
h(k)Bk+1(a)
k+ 1an k
+nX
k=0( 1)kn
k
0( k;a)an k, n= 1;2;:::,<a>0,
where
25.11.33 h(n) =nX
k=1k 1:
25.11.34 nZa
00(1 n;x)dx=0( n;a) 0( n) +Bn+1 Bn+1(a)
n(n+ 1),n= 1;2;:::,<a>0.
610 Zeta and Related Functions
25.11(ix) Integrals
See Prudnikov et al. (1990,x2.3), Prudnikov et al.
(1992a,x3.2), and Prudnikov et al. (1992b,x3.2).
25.11(x) Further Series Representations
25.11.35
1X
n=0( 1)n
(n+a)s
=1
(s)Z1
0xs 1e ax
1 +e xdx
= 2 s
s;1
2a
s;1
2(1 +a)
,
<a>0,<s>0; or<a= 0,=a6= 0, 0<<s<1.
Whena= 1, (25.11.35) reduces to (25.2.3).
25.11.361X
n=1 (n)
ns=k skX
r=1 (r)
s;r
k
,<s>1,
where (n) is a Dirichlet character (mod k) (x27.8).
See also Srivastava and Choi (2001).
25.11(xi) Sums
25.11.37
1X
k=1( 1)k
k(nk;a ) = nln (a)
+ ln0
@n 1Y
j=0
a e(2j+1)i=n1
A,
n= 2;3;4;:::,<a1.
25.11.381X
k=1n+k
k
(n+k+ 1;a)zk
=( 1)n
n!
(n)(a) (n)(a z)
,
n= 1;2;3;:::,<a>0,jzj<jaj.
25.11.391X
k=2k
2k
k+ 1;3
4
= 8G;
whereGisCatalan's constant :
25.11.40G=1X
n=0( 1)n
(2n+ 1)2= 0:91596 55941 772 ::::
For further sums see Prudnikov et al. (1990, pp. 396{
397) and Hansen (1975, pp. 358{360).
25.11(xii)a-Asymptotic Behavior
Asa!0 withs(6= 1) xed,
25.11.41(s;a+ 1) =(s) s(s+ 1)a+O
a2
:
As !1 withs xed,<s>1,
25.11.42 (s; +i )!0;uniformly with respect to bounded nonnegative values
of .
Asa!1 in the sectorjphaj (< ), with
s(6= 1) and xed, we have the asymptotic expansion
25.11.43
(s;a) a1 s
s 1 1
2a s1X
k=1B2k
(2k)! (s+ 2k 1)
(s)a1 s 2k:
Similarly, as a!1 in the sectorjphaj1
2 (<
1
2),
25.11.440( 1;a) 1
12+1
4a2 1
12 1
2a+1
2a2
lna
1X
k=1B2k+2
(2k+ 2)(2k+ 1)2ka 2k;
and
25.11.450( 2;a) 1
12a+1
9a3 1
6a 1
2a2+1
3a3
lna
1X
k=12B2k+2
(2k+ 2)(2k+ 1)2k(2k 1)a (2k 1):
For the more general case 0( m;a),m= 1;2;:::, see
Elizalde (1986).
For an exponentially-improved form of (25.11.43) see
Paris (2005b).
25.12 Polylogarithms
25.12(i) Dilogarithms
The notation Li 2(z) was introduced in Lewin (1981) for
a function discussed in Euler (1768) and called the dilog-
arithm in Hill (1828):
25.12.1 Li2(z) =1X
n=1zn
n2,jzj1.
25.12.2 Li2(z) = Zz
0t 1ln(1 t)dt,z2Cn(1;1):
Other notations and names for Li 2(z) includeS2(z)
(K olbig et al. (1970)), Spence function Sp( z) ('t Hooft
and Veltman (1979)), and L 2(z) (Maximon (2003)).
In the complex plane Li 2(z) has a branch point at
z= 1. The principal branch has a cut along the in-
terval [1;1) and agrees with (25.12.1) when jzj1;
see alsox4.2(i). The remainder of the equations in this
subsection apply to principal branches.
25.12.3
Li2(z) + Li 2z
z 1
= 1
2(ln(1 z))2,z2Cn[1;1).
25.12.4
Li2(z) + Li 21
z
= 1
62 1
2(ln( z))2,z2Cn[0;1).
25.12.5Li2(zm) =mm 1X
k=0Li2
ze2ik=m
,
m= 1;2;3;:::,jzj<1.
25.12 Polylogarithms 611
25.12.6
Li2(x) + Li 2(1 x) =1
62 (lnx) ln(1 x), 0<x< 1.
Whenz=ei, 02, (25.12.1) becomes
25.12.7 Li2
ei
=1X
n=1cos(n)
n2+i1X
n=1sin(n)
n2:
The cosine series in (25.12.7) has the elementary sum
25.12.81X
n=1cos(n)
n2=2
6
2+2
4:By (25.12.2)
25.12.91X
n=1sin(n)
n2= Z
0ln
2 sin 1
2x
dx:
The right-hand side is called Clausen's integral .
For graphics see Figures 25.12.1 and 25.12.2, and for
further properties see Maximon (2003), Kirillov (1995),
Lewin (1981), Nielsen (1909), and Zagier (1989).
Figure 25.12.1 : Dilogarithm function Li 2(x), 20x<
1:
Figure 25.12.2 : Absolute value of the dilogarithm func-
tionjLi2(x+iy)j, 20x20, 20y20. Prin-
cipal value. There is a cut along the real axis from 1 to
1.
25.12(ii) Polylogarithms
For real or complex sandzthepolylogarithm Lis(z) is
de ned by
25.12.10 Lis(z) =1X
n=1zn
ns:
For each xed complex sthe series de nes an ana-
lytic function of zforjzj<1. The series also converges
whenjzj= 1, provided that <s>1. For other values of
z, Lis(z) is de ned by analytic continuation.
The notation (z;s) was used for Li s(z) in Truesdell
(1945) for a series treated in Jonqui ere (1889), hence
the alternative name Jonqui ere's function . The special
casez= 1 is the Riemann zeta function: (s) = Lis(1).
Integral Representation
25.12.11 Lis(z) =z
(s)Z1
0xs 1
ex zdx;
valid when<s>0 andjph(1 z)j<, or<s>1 and
z= 1. (In the latter case (25.12.11) becomes (25.5.1)).Further properties include
25.12.12
Lis(z) = (1 s)
ln1
zs 1
+1X
n=0(s n)(lnz)n
n!,
s6= 1;2;3;:::,jlnzj<2,
and
25.12.13
Lis
e2ia
+eisLis
e 2ia
=(2)seis=2
(s)(1 s;a);
valid when<s >0,=a >0 or<s >1,=a= 0. When
s= 2 ande2ia=z, (25.12.13) becomes (25.12.4).
See also Lewin (1981), K olbig (1986), Maximon
(2003), Prudnikov et al. (1990,xx1.2 and 2.5), Prud-
nikov et al. (1992a,x3.3), and Prudnikov et al. (1992b,
x3.3).
25.12(iii) Fermi{Dirac and Bose{Einstein
Integrals
The Fermi{Dirac and Bose{Einstein integrals are de-
ned by
612 Zeta and Related Functions
25.12.14Fs(x) =1
(s+ 1)Z1
0ts
et x+ 1dt,s> 1,
25.12.15Gs(x) =1
(s+ 1)Z1
0ts
et x 1dt,
s> 1,x<0; ors>0,x0,
respectively. Sometimes the factor 1 = (s+ 1) is omit-
ted. See Cloutman (1989) and Gautschi (1993).
In terms of polylogarithms
25.12.16Fs(x) = Lis+1( ex); Gs(x) = Lis+1(ex):
For a uniform asymptotic approximation for Fs(x)
see Temme and Olde Daalhuis (1990).
25.13 Periodic Zeta Function
The notation F(x;s) is used for the polylogarithm
Lis
e2ix
withxreal:
25.13.1 F(x;s) =1X
n=1e2inx
ns;
where<s>1 ifxis an integer,<s>0 otherwise.
F(x;s) is periodic in xwith period 1, and equals
(s) whenxis an integer. Also,
25.13.2F(x;s) = (1 s)
(2)1 s
ei(1 s)=2(1 s;x)
+ei(s 1)=2(1 s;1 x)
,
0<x< 1,<s>1,
25.13.3
(1 s;x) = (s)
(2)s
e is=2F(x;s) +eis=2F( x;s)
,
0<x< 1,<s>0.
25.14 Lerch's Transcendent
25.14(i) De nition
25.14.1(z;s;a ) =1X
n=0zn
(a+n)s,
a6= 0; 1; 2;:::;jzj<1;<s>1;jzj= 1.
For other values of z, (z;s;a ) is de ned by analytic
continuation. This is the notation used in Erd elyi
et al. (1953a, p. 27). Lerch (1887) used K(a;x;s ) =
e2ix;s;a
.
The Hurwitz zeta function (s;a) (x25.11) and the
polylogarithm Li s(z) (x25.12(ii)) are special cases:
25.14.2(s;a) = (1;s;a),<s>1,a6= 0; 1; 2;:::,
25.14.3 Lis(z) =z(z;s;1),<s>1,jzj1.25.14(ii) Properties
With the conditions of (25.14.1) and m= 1;2;3;:::,
25.14.4 (z;s;a ) =zm(z;s;a +m) +m 1X
n=0zn
(a+n)s:
25.14.5(z;s;a ) =1
(s)Z1
0xs 1e ax
1 ze xdx,
<s>0,<a>0,z2Cn[1;1).
25.14.6
(z;s;a ) =1
2a s+Z1
0zx
(a+x)sdx
2Z1
0sin(xlnz sarctan(x=a))
(a2+x2)s=2(e2x 1)dx,
<s>0 ifjzj<1;<s>1 ifjzj= 1;<a>0.
For these and further properties see Erd elyi et al.
(1953a, pp. 27{31).
25.15 Dirichlet L-functions
25.15(i) De nitions and Basic Properties
The notation L(s; ) was introduced by Dirichlet (1837)
for the meromorphic continuation of the function de-
ned by the series
25.15.1 L(s; ) =1X
n=1 (n)
ns,<s>1,
where (n) is a Dirichlet character (mod k) (x27.8).
For the principal character 1(modk),L(s; 1) is an-
alytic everywhere except for a simple pole at s= 1 with
residue (k)=k, where (k) is Euler's totient function
(x27.2). If 6= 1, thenL(s; ) is an entire function of
s.
25.15.2 L(s; ) =Y
p
1 (p)
ps 1
,<s>1,
with the product taken over all primes p, beginning with
p= 2. This implies that L(s; )6= 0 if<s>1.
Equations (25.15.3) and (25.15.4) hold for all sif
6= 1, and for all s(6= 1) if = 1:
25.15.3L(s; ) =k sk 1X
r=1 (r)
s;r
k
;
25.15.4L(s; ) =L(s; 0)Y
pjk
1 0(p)
ps
;
where 0is a primitive character (mod d) for some pos-
itive divisor dofk(x27.8).
When is a primitive character (mod k) theL-
functions satisfy the functional equation:
25.15.5
L(1 s; ) =ks 1 (s)
(2)s
e is=2+ ( 1)eis=2
G( )L(s; );
Applications 613
where is the complex conjugate of , and
25.15.6 G( ) =kX
r=1 (r)e2ir=k:
25.15(ii) Zeros
SinceL(s; )6= 0 if<s >1, (25.15.5) shows that for a
primitive character the only zeros of L(s; ) for<s<0
(the so-called trivial zeros) are as follows:
25.15.7L( 2n; ) = 0 if ( 1) = 1,n= 0;1;2;:::,
25.15.8
L( 2n 1; ) = 0 if ( 1) = 1,n= 0;1;2;:::.
There are also in nitely many zeros in the critical
strip 0<s1, located symmetrically about the criti-
cal line<s=1
2, but not necessarily symmetrically about
the real axis.
25.15.9 L(1; )6= 0 if 6= 1;
where 1is the principal character (mod k). This re-
sult plays an important role in the proof of Dirichlet's
theorem on primes in arithmetic progressions ( x27.11).
Related results are:
25.15.10L(0; ) =8
><
>: 1
kkX
r=1r (r); 6= 1;
0; = 1:
Applications
25.16 Mathematical Applications
25.16(i) Distribution of Primes
In studying the distribution of primes px, Chebyshev
(1851) introduced a function (x) (not to be confusedwith the digamma function used elsewhere in this chap-
ter), given by
25.16.1 (x) =1X
m=1X
pmxlnp;
which is related to the Riemann zeta function by
25.16.2 (x) =x 0(0)
(0) X
x
+o(1),x!1 ,
where the sum is taken over the nontrivial zeros of
(s).
The prime number theorem (27.2.3) is equivalent to
the statement
25.16.3 (x) =x+o(x), x!1 .
The Riemann hypothesis is equivalent to the state-
ment
25.16.4 (x) =x+O
x1
2+
,x!1 ,
for every>0.
25.16(ii) Euler Sums
Euler sums have the form
25.16.5 H(s) =1X
n=1h(n)
ns;
whereh(n) is given by (25.11.33).
H(s) is analytic for<s > 1, and can be extended
meromorphically into the half-plane <s > 2kfor ev-
ery positive integer kby use of the relations
25.16.6 H(s) = 0(s) +
(s) +1
2(s+ 1) +kX
r=1(1 2r)(s+ 2r) +1X
n=11
nsZ1
neB2k+1(x)
x2k+2dx;
25.16.7H(s) =1
2(s+ 1) +(s)
s 1 kX
r=1s+ 2r 2
2r 1
(1 2r)(s+ 2r) s+ 2k
2k+ 11X
n=11
nZ1
neB2k+1(x)
xs+2k+1dx:
For integer s(2),H(s) can be evaluated in terms
of the zeta function:
25.16.8 H(2) = 2(3); H (3) =5
4(4);
25.16.9H(a) =a+ 2
2(a+ 1) 1
2a 2X
r=1(r+ 1)(a r),
a= 2;3;4;:::.Also,
25.16.10
H( 2a) =1
2(1 2a) = B2a
4a,a= 1;2;3;:::.
H(s) has a simple pole with residue (1 2r) (=
B2r=(2r)) at each odd negative integer s= 1 2r,
r= 1;2;3;:::.
614 Zeta and Related Functions
H(s) is the special case H(s;1) of the function
25.16.11 H(s;z) =1X
n=11
nsnX
m=11
mz,<(s+z)>1,
which satis es the reciprocity law
25.16.12H(s;z) +H(z;s) =(s)(z) +(s+z);
when both H(s;z) andH(z;s) are nite.
For further properties of H(s;z) see Apostol and Vu
(1984). Related results are:
25.16.131X
n=1h(n)
n2
=17
4(4);
25.16.141X
r=1rX
k=11
rk(r+k)=5
4(3);
25.16.151X
r=1rX
k=11
r2(r+k)=3
4(3):
For further generalizations, see Flajolet and Salvy
(1998).
25.17 Physical Applications
Analogies exist between the distribution of the zeros of
(s) on the critical line and of semiclassical quantum
eigenvalues. This relates to a suggestion of Hilbert and
P olya that the zeros are eigenvalues of some operator,
and the Riemann hypothesis is true if that operator is
Hermitian. See Armitage (1989), Berry and Keating
(1998, 1999), Keating (1993, 1999), and Sarnak (1999).
The zeta function arises in the calculation of the
partition function of ideal quantum gases (both Bose{
Einstein and Fermi{Dirac cases), and it determines
the critical gas temperature and density for the Bose{
Einstein condensation phase transition in a dilute gas
(Lifshitz and Pitaevski (1980)). Quantum eld theory
often encounters formally divergent sums that need to
be evaluated by a process of regularization: for example,
the energy of the electromagnetic vacuum in a con ned
space ( Casimir{Polder e ect ). It has been found pos-
sible to perform such regularizations by equating the
divergent sums to zeta functions and associated func-
tions (Elizalde (1995)).
Computation
25.18 Methods of Computation
25.18(i) Function Values and Derivatives
The principal tools for computing (s) are the expan-
sion (25.2.9) for general values of s, and the Riemann{
Siegel formula (25.10.3) (extended to higher terms) for 1
2+it
. Details are provided in Haselgrove and Miller
(1960). See also Allasia and Besenghi (1989), Butzer
and Hauss (1992), Kerimov (1980), and Yeremin et al.
(1985). Calculations relating to derivatives of (s)
and/or(s;a) can be found in Apostol (1985a), Choud-
hury (1995), Miller and Adamchik (1998), and Yeremin
et al. (1988).
For the Hurwitz zeta function (s;a) see Spanier and
Oldham (1987, p. 653).
For dilogarithms and polylogarithms see Jacobs and
Lambert (1972), Os acar et al. (1995), and Spanier and
Oldham (1987, pp. 231{232).
For Fermi{Dirac and Bose{Einstein integrals see
Cloutman (1989), Gautschi (1993), Mohankumar and
Natarajan (1997), Natarajan and Mohankumar (1993),
Paszkowski (1988, 1991), Pichon (1989), and Sagar
(1991a,b).
25.18(ii) Zeros
Most numerical calculations of the Riemann zeta func-
tion are concerned with locating zeros of 1
2+it
in
an e ort to prove or disprove the Riemann hypothesis,
which states that all nontrivial zeros of (s) lie on the
critical line<s=1
2. Calculations to date (2008) have
found no nontrivial zeros o the critical line. For re-
cent investigations see, for example, van de Lune et al.
(1986) and Odlyzko (1987). For earlier work see Hasel-
grove and Miller (1960).
25.19 Tables
Abramowitz and Stegun (1964) tabulates: (n),
n= 2;3;4;:::, 20D (p. 811); Li 2(1 x),
x= 0(:01)0:5, 9D (p. 1005); f(),=
15(1)30(2)90(5)180,f() +ln,=
0(1)15, 6D (p. 1006). Here f() denotes
Clausen's integral, given by the right-hand side
of (25.12.9).
Morris (1979) tabulates Li 2(x) (x25.12(i)) for
x= 0:02(:02)1(:1)6 to 30D.
Cloutman (1989) tabulates ( s+ 1)Fs(x), where
Fs(x) is the Fermi{Dirac integral (25.12.14), for
s= 1
2;1
2;3
2;5
2,x= 5(:05)25, to 12S.
Fletcher et al. (1962,x22.1) lists many sources for
earlier tables of (s) for both real and complex s.
x22.133 gives sources for numerical values of coef-
cients in the Riemann{Siegel formula, x22.15 de-
scribes tables of values of (s;a), andx22.17 lists
tables for some Dirichlet L-functions for real char-
acters. For tables of dilogarithms, polylogarithms,
and Clausen's integral see xx22.84{22.858.
25.20 Approximations 615
25.20 Approximations
Cody et al. (1971) gives rational approximations
for(s) in the form of quotients of polynomials
or quotients of Chebyshev series. The ranges cov-
ered are 0:5s5, 5s11, 11s25,
25s55. Precision is varied, with a maximum
of 20S.
Piessens and Branders (1972) gives the coecients
of the Chebyshev-series expansions of s(s+ 1)
and(s+k),k= 2;3;4;5;8, for 0s1 (23D).
Luke (1969b, p. 306) gives coecients in
Chebyshev-series expansions that cover (s) for
0s1 (15D),(s+ 1) for 0s1 (20D),
and ln 1
2+ix
(x25.4) for 1x1 (20D).
For errata see Piessens and Branders (1972).
Morris (1979) gives rational approximations for
Li2(x) (x25.12(i)) for 0 :5x1. Precision is
varied with a maximum of 24S.
Antia (1993) gives minimax rational approxima-
tions for (s+ 1)Fs(x), whereFs(x) is the Fermi{
Dirac integral (25.12.14), for the intervals 1<
x2 and 2x <1, withs= 1
2;1
2;3
2;5
2. For
eachsthere are three sets of approximations, with
relative maximum errors 10 4;10 8;10 12.
25.21 Software
Seehttp://dlmf.nist.gov/25.21 .
References
General References
The main references used in writing this chapter are
Apostol (1976), Erd elyi et al. (1953a), and Titch-
marsh (1986b). For additional bibliographic reading see
Edwards (1974), Ivi c (1985), Karatsuba and Voronin
(1992).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.x25.2 Apostol (1976, Chapter 12). For (25.2.2){
(25.2.7) see also Hardy (1912). For (25.2.8){
(25.2.10) see also Knopp (1948, p. 533). (25.2.9)
follows from (25.2.8) by repeated integration by
parts. For (25.2.11), (25.2.12) see also Titchmarsh
(1986b, p. 30).
x25.3 These graphics were constructed at NIST.
x25.4 Apostol (1976, Chapter 12).
x25.5 Apostol (1976, Chapter 12), Erd elyi et al.
(1953a, Chapter I). For (25.5.2) and (25.5.4)
integrate (25.5.1) and (25.5.3) by parts. For
(25.5.5) see Titchmarsh (1986b, p. 15). (25.5.6)
comes from (25.5.1) by using the identity e x=
(1 e x)=(ex 1) in the integral ( s) =R1
0e xxs 1dxtogether with (5.5.1). (25.5.7)
follows from (25.5.6) because ( s+ 2m 1) =R1
0e xxs+2m 2dx. For (25.5.10) and (25.5.11)
see Lindel of (1905, p. 103). For (25.5.12) see Sri-
vastava and Choi (2001, p. 12). For (25.5.13)
see Titchmarsh (1986b, p. 22). For (25.5.14){
(25.5.19) see de Bruijn (1937). For (25.5.21) see
Erd elyi et al. (1953a, p. 32).
x25.6 For (25.6.1){(25.6.4) see Apostol (1976, pp. 266{
268). For (25.6.5) see Mordell (1958). For (25.6.6)
see N orlund (1924, p. 66). For (25.6.7) see Apostol
(1983). For (25.6.8){(25.6.10) see van der Poorten
(1980, pp. 271, 274). For (25.6.11){(25.6.14) see
Apostol (1985a). For (25.6.15) see Miller and
Adamchik (1998). For (25.6.16){(25.6.20) see
Basu and Apostol (2000).
x25.8 Titchmarsh (1986b, Chapter IV), Adamchik and
Srivastava (1998), Erd elyi et al. (1953a, pp. 45
and 51). For (25.8.2) see Landau (1953, p. 274).
For (25.8.3) see Srivastava (1988). For (25.8.7),
(25.8.8) divide by xin (25.8.5), (25.8.6) and inte-
grate. For (25.8.9) see Srivastava and Choi (2001,
p. 212). For (25.8.10) see Ewell (1990).
x25.9 Titchmarsh (1986b, Chapter XV), Berry (1995).
x25.10 Apostol (1976, Chapter 12), Titchmarsh
(1986b, pp. 89 and 263).
x25.11 Apostol (1976, Chapter 12). Analytic prop-
erties of(s;a) with respect to afollow from
(25.11.30). For (25.11.5){(25.11.6) see Apostol
(1985a). For (25.11.7) take N= 1 in (25.11.5)
and integrate by parts. For (25.11.8){(25.11.9)
see Srivastava and Choi (2001, p. 89). For
(25.11.10) use Taylor's theorem ( xx1.4(vi), 1.10(i))
and (25.11.17). For (25.11.11) apply (25.2.2) and
(25.11.1). For (25.11.12) see Erd elyi et al. (1953a,
616 Zeta and Related Functions
p. 45). For (25.11.13) and (25.11.14) see Apostol
(1976, pp. 268, 264). For (25.11.15) use (25.11.1)
and analytic continuation. For (25.11.16) see
Apostol (1976, p. 263). For (25.11.17) di eren-
tiate (25.11.1). For (25.11.18) see Erd elyi et al.
(1953a, p. 26). For (25.11.19){(25.11.23) see
Apostol (1985a, p. 231) and Miller and Adam-
chik (1998). For (25.11.24) use (25.11.15) with
a= 1=k, multiply by ksand di erentiate.
For (25.11.25) see Srivastava and Choi (2001,
p. 89) For (25.11.26) see Berndt (1972). For
(25.11.27) and (25.11.28) argue as indicated above
for (25.5.6) and (25.5.7). For (25.11.29) see Lin-
del of (1905, p. 106). For (25.11.30) assume <s>
1, collapse the integration path onto the real
axis, apply (25.11.25) and (5.5.3) followed by an-
alytic continuation. For (25.11.31) use (25.11.25).
For (25.11.32){(25.11.34) see Adamchik (1998).
For (25.11.35) use (25.11.25) and (25.11.8). For(25.11.36) see Apostol (1976). For (25.11.37){
(25.11.40) see Adamchik and Srivastava (1998).
For (25.11.41) and (25.11.42) see Apostol (1952).
For (25.11.43) see Paris (2005b). For (25.11.44)
and (25.11.45) see Elizalde (1986). The graphics
were constructed at NIST.
x25.12 Erd elyi et al. (1953a, pp. 27, 29), Maximon
(2003). For (25.12.13) see Erd elyi et al. (1953a,
p. 31) with change of notation. The graphics were
constructed at NIST.
x25.13 Apostol (1976, Chapter 13).
x25.15 Apostol (1976, Chapter 12), Apostol (1985b).
For (25.15.9) see Apostol (1976, pp. 142, 149).
x25.16 Apostol (1976, Chapter 13). For (25.16.2) see
Apostol (2000). For (25.16.4) see Ingham (1932,
p. 84). For (25.16.5){(25.16.15) see Apostol and
Vu (1984) and Basu and Apostol (2000).
Chapter 26
Combinatorial Analysis
D. M. Bressoud1
Notation 618
26.1 Special Notation . . . . . . . . . . . . . 618
Properties 618
26.2 Basic De nitions . . . . . . . . . . . . . 618
26.3 Lattice Paths: Binomial Coecients . . . 619
26.4 Lattice Paths: Multinomial Coecients
and Set Partitions . . . . . . . . . . . . . 620
26.5 Lattice Paths: Catalan Numbers . . . . . 620
26.6 Other Lattice Path Numbers . . . . . . . 621
26.7 Set Partitions: Bell Numbers . . . . . . . 623
26.8 Set Partitions: Stirling Numbers . . . . . 624
26.9 Integer Partitions: Restricted Number and
Part Size . . . . . . . . . . . . . . . . . . 626
26.10 Integer Partitions: Other Restrictions . . 627
26.11 Integer Partitions: Compositions . . . . . 62826.12 Plane Partitions . . . . . . . . . . . . . . 629
26.13 Permutations: Cycle Notation . . . . . . 631
26.14 Permutations: Order Notation . . . . . . 632
26.15 Permutations: Matrix Notation . . . . . . 633
26.16 Multiset Permutations . . . . . . . . . . 634
26.17 The Twelvefold Way . . . . . . . . . . . 634
26.18 Counting Techniques . . . . . . . . . . . 634
Applications 635
26.19 Mathematical Applications . . . . . . . . 635
26.20 Physical Applications . . . . . . . . . . . 635
Computation 635
26.21 Tables . . . . . . . . . . . . . . . . . . . 635
26.22 Software . . . . . . . . . . . . . . . . . . 635
References 635
1Macalester College, Saint Paul, Minnesota.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
617
618 Combinatorial Analysis
Notation
26.1 Special Notation
(For other notation see pp. xiv and 873.)
x real variable.
h;j;k;`;m;n nonnegative integers.
integer partition.
plane partition.
jAj number of elements of a nite set A.
jjk j dividesk.
(h;k) greatest common divisor of positive
integershandk.
The main functions treated in this chapter are:
m
n
binomial coecient. m
n1;n2;:::;nk
multinomial coecient.
m
n
Eulerian number.
m
n
qGaussian polynomial.
B(n) Bell number.
C(n) Catalan number.
p(n) number of partitions of n.
pk(n) number of partitions of ninto at most k
parts.
pp(n) number of plane partitions of n.
s(n;k) Stirling numbers of the rst kind.
S(n;k) Stirling numbers of the second kind.
Alternative Notations
Many combinatorics references use the rising and falling
factorials:
26.1.1xn=x(x+ 1)(x+ 2)(x+n 1);
xn=x(x 1)(x 2)(x n+ 1):
Other notations for s(n;k), the Stirling numbers
of the rst kind, include S(k)
n(Abramowitz and Ste-
gun (1964, Chapter 24), Fort (1948)), Sk
n(Jordan
(1939), Moser and Wyman (1958a)), n 1
k 1
B(n)
n k(Milne-
Thomson (1933)), ( 1)n kS1(n 1;n k) (Carlitz
(1960), Gould (1960)), ( 1)n kn
k
(Knuth (1992),
Graham et al. (1994), Rosen et al. (2000)).
Other notations for S(n;k), the Stirling numbers of
the second kind, include S(k)
n(Fort (1948)), Sk
n(Jor-
dan (1939)), k
n(Moser and Wyman (1958b)), n
k
B( k)
n k
(Milne-Thomson (1933)), S2(k;n k) (Carlitz (1960),
Gould (1960)),n
k
(Knuth (1992), Graham et al.
(1994), Rosen et al. (2000)), and also an unconventional
symbol in Abramowitz and Stegun (1964, Chapter 24).Properties
26.2 Basic De nitions
Permutation
Apermutation is a one-to-one and onto function from
a non-empty set to itself. If the set consists of the in-
tegers 1 through n, a permutation can be thought
of as a rearrangement of these integers where the inte-
ger in position jis(j). Thus 231 is the permutation
(1) = 2,(2) = 3,(3) = 1.
Cycle
Given a nite set Swith permutation , acycle is an
ordered equivalence class of elements of Swherejis
equivalent to kif there exists an `=`(j;k) such that
j=`(k), where1=and`is the composition of
with` 1. It is ordered so that (j) followsj. If, for
example, a permutation of the integers 1 through 6 is
denoted by 256413, then the cycles are (1 ;2;5), (3;6),
and (4). Here (1) = 2;(2) = 5, and (5) = 1. The
functionalso interchanges 3 and 6, and sends 4 to
itself.
Lattice Path
Alattice path is a directed path on the plane integer lat-
ticef0;1;2;:::gf 0;1;2::::g. Unless otherwise speci-
ed, it consists of horizontal segments corresponding to
the vector (1 ;0) and vertical segments corresponding to
the vector (0 ;1). For an example see Figure 26.9.2.
Ak-dimensional lattice path is a directed path com-
posed of segments that connect vertices in f0;1;2;:::gk
so that each segment increases one coordinate by exactly
one unit.
Partition
Apartition of a set Sis an unordered collection of
pairwise disjoint nonempty sets whose union is S. As
an example,f1;3;4g,f2;6g,f5gis a partition of
f1;2;3;4;5;6g.
Apartition of a nonnegative integer nis an un-
ordered collection of positive integers whose sum is n.
As an example,f1;1;1;2;4;4gis a partition of 13. The
total number of partitions of nis denoted by p(n). See
Table 26.2.1 for n= 0(1)50. For the actual partitions
() forn= 1(1)5 see Table 26.4.1.
The integers whose sum is nare referred to as the
parts in the partition. The example f1;1;1;2;4;4ghas
six parts, three of which equal 1.
26.3 Lattice Paths: Binomial Coefficients 619
Table 26.2.1 : Partitions p(n).
n p (n)n p (n)n p (n)
0 1 17 297 34 12310
1 1 18 385 35 14883
2 2 19 490 36 17977
3 3 20 627 37 21637
4 5 21 792 38 26015
5 7 22 1002 39 31185
6 11 23 1255 40 37338
7 15 24 1575 41 44583
8 22 25 1958 42 53174
9 30 26 2436 43 63261
10 42 27 3010 44 75175
11 56 28 3718 45 89134
12 77 29 4565 46 1 05558
13 101 30 5604 47 1 24754
14 135 31 6842 48 1 47273
15 176 32 8349 49 1 73525
16 231 33 10143 50 2 04226
26.3 Lattice Paths: Binomial Coecients
26.3(i) De nitions
m
n
is the number of ways of choosing nobjects from
a collection of mdistinct objects without regard to or-
der. m+n
n
is the number of lattice paths from (0 ;0)
to (m;n). The number of lattice paths from (0 ;0) to
(m;n),mn, that stay on or above the line y=xis m+n
m
m+n
m 1
:
26.3.1m
n
=m
m n
=m!
(m n)!n!,mn,
26.3.2m
n
= 0, n>m:
For numerical values of m
n
and m+n
n
see Tables
26.3.1 and 26.3.2.
Table 26.3.1 : Binomial coecients m
n
.
mn
0 1 2 3 4 5 6 7 8 9 10
01
11 1
21 2 1
31 3 3 1
41 4 6 4 1
51 5 10 10 5 1
61 6 15 20 15 6 1
71 7 21 35 35 21 7 1
81 8 28 56 70 56 28 8 1
91 9 36 84 126 126 84 36 9 1
101 10 45 120 210 252 210 120 45 10 1Table 26.3.2 : Binomial coecients m+n
m
for lattice
paths.
mn
0 1 2 3 4 5 6 7 8
01 1 1 1 1 1 1 1 1
11 2 3 4 5 6 7 8 9
21 3 6 10 15 21 28 36 45
31 4 10 20 35 56 84 120 165
41 5 15 35 70 126 210 330 495
51 6 21 56 126 252 462 792 1287
61 7 28 84 210 462 924 1716 3003
71 8 36 120 330 792 1716 3432 6435
81 9 45 165 495 1287 3003 6435 12870
26.3(ii) Generating Functions
26.3.3mX
n=0m
n
xn= (1 +x)m,m= 0;1;:::,
26.3.41X
m=0m+n
m
xm=1
(1 x)n+1,jxj<1.
26.3(iii) Recurrence Relations
26.3.5m
n
=m 1
n
+m 1
n 1
,mn1,
26.3.6m
n
=m
nm 1
n 1
=m n+ 1
nm
n 1
,
mn1,
26.3.7m+ 1
n+ 1
=mX
k=nk
n
,mn0,
26.3.8m
n
=nX
k=0m n 1 +k
k
,mn0.
26.3(iv) Identities
26.3.9n
0
=n
n
= 1;
26.3.10m
n
=nX
k=0( 1)n km+ 1
k
,mn0,
26.3.112n
n
=2n(2n 1)(2n 3)31
n!:
See alsox1.2(i).
26.3(v) Limiting Form
26.3.122n
n
4n
pn, n!1 .
620 Combinatorial Analysis
26.4 Lattice Paths: Multinomial
Coecients and Set Partitions
26.4(i) De nitions
n
n1;n2;:::;nk
is the number of ways of placing n=
n1+n2++nkdistinct objects into klabeled boxes
so that there are njobjects in the jth box. It is also the
number ofk-dimensional lattice paths from (0 ;0;:::; 0)
to (n1;n2;:::;nk). Fork= 0;1, the multinomial coe-
cient is de ned to be 1. For k= 2
26.4.1n1+n2
n1;n2
=n1+n2
n1
=n1+n2
n2
;
and in general,
26.4.2n1+n2++nk
n1;n2;:::;nk
=(n1+n2++nk)!
n1!n2!nk!
=k 1Y
j=1nj+nj+1++nk
nj
:
Table 26.4.1 gives numerical values of multinomi-
als and partitions ;M 1;M2;M3for 1mn5.
These are given by the following equations in which
a1;a2;:::;anare nonnegative integers such that
26.4.3 n=a1+ 2a2++nan;
26.4.4 m=a1+a2++an:
is a partition of n:
26.4.5 = 1a1;2a2;:::;nan:
M1is the multinominal coecient (26.4.2):
26.4.6M1=0
@n
a1z}|{
1;:::; 1;:::;anz}|{n;:::;n1
A
=n!
(1!)a1(2!)a2(n!)an:
M2is the number of permutations of f1;2;:::;ngwith
a1cycles of length 1, a2cycles of length 2, :::, andan
cycles of length n:
26.4.7M2=n!
1a1(a1!) 2a2(a2!)nan(an!):
(The empty set is considered to have one permutation
consisting of no cycles.) M3is the number of set parti-
tions off1;2;:::;ngwitha1subsets of size 1, a2subsets
of size 2,:::, andansubsets of size n:
26.4.8M3=n!
(1!)a1(a1!) (2!)a2(a2!)(n!)an(an!):
For eachnall possible values of a1;a2;:::;anare cov-
ered.Table 26.4.1 : Multinomials and partitions.
nmM1M2M3
1 1111 1 1
2 1211 1 1
2 2122 1 1
3 1311 2 1
3 211;213 3 3
3 3136 1 1
4 1411 6 1
4 211;314 8 4
4 2226 3 3
4 312;2112 6 6
4 41424 1 1
5 1511 24 1
5 211;415 30 5
5 221;3110 20 10
5 312;3120 20 10
5 311;2230 15 15
5 413;2160 10 10
5 515120 1 1
26.4(ii) Generating Function
26.4.9(x1+x2++xk)n
=Xn
n1;n2;:::;nk
xn1
1xn2
2xnk
k;
where the summation is over all nonnegative integers
n1;n2;:::;nksuch thatn1+n2++nk=n.
26.4(iii) Recurrence Relation
26.4.10n1+n2++nm
n1;n2;:::;nm
=mX
k=1n1+n2++nm 1
n1;n2;:::;nk 1;nk 1;nk+1;:::;nm
,
n1;n2;:::;nm1.
26.5 Lattice Paths: Catalan Numbers
26.5(i) De nitions
C(n) is the Catalan number. It counts the number of
lattice paths from (0 ;0) to (n;n) that stay on or above
the liney=x.
26.5.1
C(n) =1
n+ 12n
n
=1
2n+ 12n+ 1
n
=2n
n
2n
n 1
=2n 1
n
2n 1
n+ 1
:
(Sixty-six equivalent de nitions of C(n) are given in
Stanley (1999, pp. 219{229).)
See Table 26.5.1.
26.6 Other Lattice Path Numbers 621
Table 26.5.1 : Catalan numbers.
n C (n)n C (n)n C (n)
0 1 7 429 14 26 74440
1 1 8 1430 15 96 94845
2 2 9 4862 16 353 57670
3 5 10 16796 17 1296 44790
4 14 11 58786 18 4776 38700
5 42 12 2 08012 19 17672 63190
6 132 13 7 42900 20 65641 20420
26.5(ii) Generating Function
26.5.21X
n=0C(n)xn=1 p1 4x
2x,jxj<1
4:
26.5(iii) Recurrence Relations
26.5.3C(n+ 1) =nX
k=0C(k)C(n k);
26.5.4C(n+ 1) =2(2n+ 1)
n+ 2C(n);26.5.5C(n+ 1) =bn=2cX
k=0n
2k
2n 2kC(k):
26.5(iv) Limiting Forms
26.5.6 C(n)4n
p
n3, n!1 ,
26.5.7 lim
n!1C(n+ 1)
C(n)= 4:
26.6 Other Lattice Path Numbers
26.6(i) De nitions
Dellanoy Number D(m;n)
D(m;n) is the number of paths from (0 ;0) to (m;n)
that are composed of directed line segments of the form
(1;0), (0;1), or (1;1).
26.6.1
D(m;n) =nX
k=0n
km+n k
n
=nX
k=02km
kn
k
:
See Table 26.6.1.
Table 26.6.1 : Dellanoy numbers D(m;n).
mn
0 1 2 3 4 5 6 7 8 9 10
01 1 1 1 1 1 1 1 1 1 1
11 3 5 7 9 11 13 15 17 19 21
21 5 13 25 41 61 85 113 145 181 221
31 7 25 63 129 231 377 575 833 1159 1561
41 9 41 129 321 681 1289 2241 3649 5641 8361
51 11 61 231 681 1683 3653 7183 13073 22363 36365
61 13 85 377 1289 3653 8989 19825 40081 75517 1 34245
71 15 113 575 2241 7183 19825 48639 1 08545 2 24143 4 33905
81 17 145 833 3649 13073 40081 1 08545 2 65729 5 98417 12 56465
91 19 181 1159 5641 22363 75517 2 24143 5 98417 14 62563 33 17445
10 1 21 221 1561 8361 36365 1 34245 4 33905 12 56465 33 17445 80 97453
Motzkin Number M(n)
M(n) is the number of lattice paths from (0 ;0) to (n;n) that stay on or above the line y=xand are composed of
directed line segments of the form (2 ;0), (0;2), or (1;1).
26.6.2 M(n) =nX
k=0( 1)k
n+ 2 kn
k2n+ 2 2k
n+ 1 k
:
See Table 26.6.2.
622 Combinatorial Analysis
Table 26.6.2 : Motzkin numbers M(n).
n M (n)n M (n)n M (n)n M (n)n M (n)
0 1 4 9 8 323 12 15511 16 8 53467
1 1 5 21 9 835 13 41835 17 23 56779
2 2 6 51 10 2188 14 1 13634 18 65 36382
3 4 7 127 11 5798 15 3 10572 19 181 99284
Narayana Number N(n;k)
N(n;k) is the number of lattice paths from (0 ;0) to (n;n) that stay on or above the line y=x, are composed of
directed line segments of the form (1 ;0) or (0;1), and for which there are exactly koccurrences at which a segment
of the form (0 ;1) is followed by a segment of the form (1 ;0).
26.6.3 N(n;k) =1
nn
kn
k 1
:
See Table 26.6.3.
Table 26.6.3 : Narayana numbers N(n;k).
nk
0 1 2 3 4 5 6 7 8 9 10
01
10 1
20 1 1
30 1 3 1
40 1 6 6 1
50 1 10 20 10 1
60 1 15 50 50 15 1
70 1 21 105 175 105 21 1
80 1 28 196 490 490 196 28 1
90 1 36 336 1176 1764 1176 336 36 1
10 0 1 45 540 2520 5292 5292 2520 540 45 1
Schr oder Number r(n)
r(n) is the number of paths from (0 ;0) to (n;n) that stay on or above the diagonal y=xand are composed of
directed line segments of the form (1 ;0), (0;1), or (1;1).
26.6.4 r(n) =D(n;n) D(n+ 1;n 1), n1.
See Table 26.6.4.
Table 26.6.4 : Schr oder numbers r(n).
n r (n)n r (n)n r (n)n r (n)n r (n)
0 1 4 90 8 41586 12 272 97738 16 2 09271 56706
1 2 5 394 9 2 06098 13 1420 78746 17 11 18180 26018
2 6 6 1806 10 10 37718 14 7453 87038 18 60 03188 53926
3 22 7 8558 11 52 93446 15 39376 03038 19 323 67243 17174
26.7 Set Partitions: Bell Numbers 623
26.6(ii) Generating Functions
For suciently small jxjandjyj,
26.6.51X
m;n=0D(m;n)xmyn=1
1 x y xy;
26.6.61X
n=0D(n;n)xn=1p
1 6x+x2;
26.6.71X
n=0M(n)xn=1 x p
1 2x 3x2
2x2;
26.6.81X
n;k=1N(n;k)xnyk
=1 x xy p
(1 x xy)2 4x2y
2x;
26.6.91X
n=0r(n)xn=1 x p
1 6x+x2
2x:
26.6(iii) Recurrence Relations
26.6.10D(m;n) =D(m;n 1) +D(m 1;n)
+D(m 1;n 1),m;n1,
26.6.11M(n) =M(n 1) +nX
k=2M(k 2)M(n k),
n2.
26.6(iv) Identities
26.6.12 C(n) =nX
k=1N(n;k);
26.6.13M(n) =nX
k=0( 1)kn
k
C(n+ 1 k);
26.6.14C(n) =2nX
k=0( 1)k2n
k
M(2n k):
26.7 Set Partitions: Bell Numbers
26.7(i) De nitions
B(n) is the number of partitions of f1;2;:::;ng. For
S(n;k) seex26.8(i).
26.7.1 B(0) = 1;
26.7.2 B(n) =nX
k=0S(n;k);
26.7.3 B(n) =mX
k=1kn
k!m kX
j=0( 1)j
j!,mn,26.7.4B(n) =e 11X
k=1kn
k!= 1 +$
e 12nX
k=1kn
k!%
:
See Table 26.7.1.
Table 26.7.1 : Bell numbers.
n B (n)n B (n)
0 1 10 1 15975
1 1 11 6 78570
2 2 12 42 13597
3 5 13 276 44437
4 15 14 1908 99322
5 52 15 13829 58545
6 203 16 1 04801 42147
7 877 17 8 28648 69804
8 4140 18 68 20768 06159
9 21147 19 583 27422 05057
26.7(ii) Generating Function
26.7.51X
n=0B(n)xn
n!= exp(ex 1):
26.7(iii) Recurrence Relation
26.7.6 B(n+ 1) =nX
k=0n
k
B(n):
26.7(iv) Asymptotic Approximation
26.7.7
B(n) =NneN n 1
(1 + lnN)1=2
1 +O(lnn)1=2
n1=2
,n!1 ,
where
26.7.8 NlnN=n;
or, equivalently, N=eWm(n), with properties of the
Lambert function Wm( n) given inx4.13. For higher
approximations to B(n) asn!1 see de Bruijn (1961,
pp. 104{108).
624 Combinatorial Analysis
26.8 Set Partitions: Stirling Numbers
26.8(i) De nitions
s(n;k) denotes the Stirling number of the rst
kind: ( 1)n ktimes the number of permutations of
f1;2;:::;ngwith exactly kcycles. See Table 26.8.1.
26.8.1 s(n;n) = 1, n0,
26.8.2 s(1;k) =1;k;
26.8.3
( 1)n ks(n;k) =X
1b1<<bn kn 1b1b2bn k,
n>k1.S(n;k) denotes the Stirling number of the second
kind: the number of partitions of f1;2;:::;nginto ex-
actlyknonempty subsets. See Table 26.8.2.
26.8.4 S(n;n) = 1, n0,
26.8.5 S(n;k) =X
1c12c2kck;
where the summation is over all nonnegative integers
c1;c2;:::;cksuch thatc1+c2++ck=n k:
26.8.6 S(n;k) =1
k!kX
j=0( 1)k jk
j
jn:
Table 26.8.1 : Stirling numbers of the rst kind s(n;k).
nk
0 1 2 3 4 5 6 7 8 9 10
01
10 1
20 1 1
30 2 3 1
40 6 11 6 1
50 24 50 35 10 1
60 120 274 225 85 15 1
70 720 1764 1624 735 175 21 1
80 5040 13068 13132 6769 1960 322 28 1
90 40320 1 09584 1 18124 67284 22449 4536 546 36 1
10 0 3 62880 10 26576 11 72700 7 23680 2 69325 6327 9450 870 45 1
Table 26.8.2 : Stirling numbers of the second kind S(n;k).
nk
0 1 2 3 4 5 6 7 8 9 10
01
10 1
20 1 1
30 1 3 1
40 1 7 6 1
50 1 15 25 10 1
60 1 31 90 65 15 1
70 1 63 301 350 140 21 1
80 1 127 966 1701 1050 266 28 1
90 1 255 3025 7770 6951 2646 462 36 1
10 0 1 511 9330 34105 42525 22827 5880 750 45 1
26.8(ii) Generating Functions
26.8.7nX
k=0s(n;k)xk= (x n+ 1)n;where (x)nis the Pochhammer symbol: x(x+1)(x+
n 1).
26.8.81X
n=0s(n;k)xn
n!=(ln(1 +x))k
k!,jxj<1;
26.8 Set Partitions: Stirling Numbers 625
26.8.91X
n;k=0s(n;k)xn
n!yk= (1 +x)y,jxj<1:
26.8.10nX
k=1S(n;k)(x k+ 1)k=xn;
26.8.11
1X
n=0S(n;k)xn=xk
(1 x)(1 2x)(1 kx),jxj<1=k;
26.8.121X
n=0S(n;k)xn
n!=(ex 1)k
k!;
26.8.131X
n;k=0S(n;k)xn
n!yk= exp (y(ex 1)):
26.8(iii) Special Values
Forn1,
26.8.14s(n;0) = 0; s(n;1) = ( 1)n 1(n 1)!;
26.8.15s(n;2) = ( 1)n(n 1)!
1 +1
2++1
n 1
;
26.8.16 s(n;n 1) =S(n;n 1) =n
2
;
26.8.17S(n;0) = 0; S(n;1) = 1; S(n;2) = 2n 1 1:
26.8(iv) Recurrence Relations
26.8.18s(n;k) =s(n 1;k 1) (n 1)s(n 1;k);
26.8.19k
h
s(n;k) =n hX
j=k hn
j
s(n j;h)s(j;k h),
nkh,
26.8.20s(n+ 1;k+ 1) =n!nX
j=k( 1)n j
j!s(j;k);
26.8.21s(n+k+ 1;k) = kX
j=0(n+j)s(n+j;j):
26.8.22S(n;k) =kS(n 1;k) +S(n 1;k 1);
26.8.23k
h
S(n;k) =n hX
j=k hn
j
S(n j;h)S(j;k h),
nkh,
26.8.24S(n;k) =nX
j=kS(j 1;k 1)kn j;
26.8.25S(n+ 1;k+ 1) =nX
j=kn
j
S(j;k);
26.8.26S(n+k+ 1;k) =kX
j=0jS(n+j;j):26.8(v) Identities
26.8.27s(n;n k)
=kX
j=0( 1)jn 1 +j
k+jn+k
k j
S(k+j;j);
26.8.28nX
k=1s(n;k) = 0, n>1,
26.8.29nX
k=1( 1)n ks(n;k) =n!;
26.8.30nX
j=ks(n+ 1;j+ 1)nj k=s(n;k):
26.8.311
k!dk
dxkf(x) =1X
n=ks(n;k)
n!nf(x);
whenf(x) is analytic for all x, and the series converges,
where
26.8.32 f(x) =f(x+ 1) f(x);
comparex3.6(i).
26.8.33S(n;n k)
=kX
j=0( 1)jn 1 +j
k+jn+k
k j
s(k+j;j);
26.8.34nX
j=0jkxj=kX
j=0S(k;j)xjdj
dxj1 xn+1
1 x
;
26.8.35nX
j=0jk=kX
j=0j!S(k;j)n+ 1
j+ 1
;
26.8.36nX
k=0( 1)n kk!S(n;k) = 1:
26.8.371
k!kf(x) =1X
n=kS(n;k)
n!dn
dxnf(x);
whenf(x) is analytic for all x, and the series converges.
LetAandBbe thennmatrices with ( j;k)th
elementss(j;k), andS(j;k), respectively. Then
26.8.38 A 1=B:
26.8.39nX
j=ks(j;k)S(n;j) =nX
j=ks(n;j)S(j;k) =n;k:
26.8(vi) Relations to Bernoulli Numbers
Seex24.15(iii).
626 Combinatorial Analysis
26.8(vii) Asymptotic Approximations
26.8.40
s(n+ 1;k+ 1)( 1)n kn!
k!(
+ lnn)k,n!1 ,
uniformly for k=o(lnn), where
is Euler's constant
(x5.2(ii)).
26.8.41 s(n+k;k)( 1)n
2nn!k2n,k!1;
n xed.
26.8.42 S(n;k)kn
k!, n!1 ,
k xed.
26.8.43 S(n+k;k)k2n
2nn!,k!1 ,
uniformly for n=o
k1=2
.
For asymptotic approximations for s(n+ 1;k+ 1)
andS(n;k) that apply uniformly for 1 knas
n!1 see Temme (1993).
For other asymptotic approximations and also ex-
pansions see Moser and Wyman (1958a) for Stirling
numbers of the rst kind, and Moser and Wyman
(1958b), Bleick and Wang (1974) for Stirling numbers
of the second kind.
For asymptotic estimates for generalized Stirling
numbers see Chelluri et al. (2000).
26.9 Integer Partitions: Restricted Number
and Part Size
26.9(i) De nitions
pk(n) denotes the number of partitions of ninto at most
kparts. See Table 26.9.1.
26.9.1 pk(n) =p(n), kn:
Unrestricted partitions are covered in x27.14.
Table 26.9.1 : Partitions pk(n).
nk
0 1 2 3 4 5 6 7 8 9 10
01 1 1 1 1 1 1 1 1 1 1
10 1 1 1 1 1 1 1 1 1 1
20 1 2 2 2 2 2 2 2 2 2
30 1 2 3 3 3 3 3 3 3 3
40 1 3 4 5 5 5 5 5 5 5
50 1 3 5 6 7 7 7 7 7 7
60 1 4 7 9 10 11 11 11 11 11
70 1 4 8 11 13 14 15 15 15 15
80 1 5 10 15 18 20 21 22 22 22
90 1 5 12 18 23 26 28 29 30 30
10 0 1 6 14 23 30 35 38 40 41 42
A useful representation for a partition is the Ferrers
graph in which the integers in the partition are eachrepresented by a row of dots. An example is provided
in Figure 26.9.1.
Figure 26.9.1 : Ferrers graph of the partition 7 + 4 + 3 +
3 + 2 + 1.
The conjugate partition is obtained by re
ecting
the Ferrers graph across the main diagonal or, equiv-
alently, by representing each integer by a column of
dots. The conjugate to the example in Figure 26.9.1 is
6 + 5 + 4 + 2 + 1 + 1 + 1. Conjugation establishes a one-
to-one correspondence between partitions of ninto at
mostkparts and partitions of ninto parts with largest
part less than or equal to k. It follows that pk(n) also
equals the number of partitions of ninto parts that are
less than or equal to k.
pk(m;n) is the number of partitions of ninto
at mostkparts, each less than or equal to m. It is
also equal to the number of lattice paths from (0 ;0) to
(m;k) that have exactly nvertices (h;j), 1hm,
1jk, above and to the left of the lattice path. See
Figure 26.9.2.
(0;0)(6;5)
Figure 26.9.2 : The partition 5 + 5 + 3 + 2 represented as
a lattice path.
Equations (26.9.2){(26.9.3) are examples of closed
forms that can be computed explicitly for any positive
integerk. See Andrews (1976, p. 81).
26.9.2 p0(n) = 0, n>0;
26.9.3p1(n) = 1; p 2(n) = 1 +bn=2c;
p3(n) = 1 +n2+ 6n
12
:
26.10 Integer Partitions: Other Restrictions 627
26.9(ii) Generating Functions
In what follows
26.9.4m
n
q=nY
j=11 qm n+j
1 qj,n0,
is the Gaussian polynomial (orq-binomial coecient );
comparexx17.2(i){17.2(ii). In the present chapter m
n0 in all cases. It is also assumed everywhere that
jqj<1.
26.9.5
1X
n=0pk(n)qn=kY
j=11
1 qj= 1 +1X
m=1k+m 1
m
qqm;
26.9.61X
n=0pk(m;n)qn=m+k
k
q:
Also, whenjxqj<1
26.9.71X
m;n=0pk(m;n)xkqn= 1 +1X
k=1m+k
k
qxk
=mY
j=01
1 xqj:
26.9(iii) Recurrence Relations
26.9.8 pk(n) =pk(n k) +pk 1(n);
equivalently, partitions into at most kparts either have
exactlykparts, in which case we can subtract one from
each part, or they have strictly fewer than kparts.
26.9.9pk(n) =1
nnX
t=1pk(n t)X
jjt
jkj;
where the inner sum is taken over all positive divisors
oftthat are less than or equal to k.
26.9(iv) Limiting Form
Asn!1 withk xed,
26.9.10 pk(n)nk 1
k!(k 1)!:
26.10 Integer Partitions: Other Restrictions
26.10(i) De nitions
p(D;n) denotes the number of partitions of ninto dis-
tinct parts. pm(D;n) denotes the number of partitions
ofninto at most mdistinct parts. p(Dk;n) denotes
the number of partitions of ninto parts with di erence
at leastk.p(D03;n) denotes the number of partitions
ofninto parts with di erence at least 3, except that
multiples of 3 must di er by at least 6. p(O;n) denotesthe number of partitions of ninto odd parts. p(2S;n)
denotes the number of partitions of ninto parts taken
from the set S. The setfn1jnj(modk)gis de-
noted byAj;k. The setf2;3;4;:::gis denoted by T. If
more than one restriction applies, then the restrictions
are separated by commas, for example, p(D2;2T;n).
See Table 26.10.1.
26.10.1p(D;0) =p(Dk;0) =p(2S;0) = 1:
Table 26.10.1 : Partitions restricted by di erence condi-
tions, or equivalently with parts from Aj;k.
p(D;n)p(D2;n)p(D2;2T;n)p(D03;n)
n and and and and
p(O;n)p(2A1;5;n)p(2A2;5;n)p(2A1;6;n)
0 1 1 1 1
1 1 1 0 1
2 1 1 1 1
3 2 1 1 1
4 2 2 1 1
5 3 2 1 2
6 4 3 2 2
7 5 3 2 3
8 6 4 3 3
9 8 5 3 3
10 10 6 4 4
11 12 7 4 5
12 15 9 6 6
13 18 10 6 7
14 22 12 8 8
15 27 14 9 9
16 32 17 11 10
17 38 19 12 12
18 46 23 15 14
19 54 26 16 16
20 64 31 20 18
26.10(ii) Generating Functions
Throughout this subsection it is assumed that jqj<1.
26.10.21X
n=0p(D;n)qn
=1Y
j=1(1 +qj) =1Y
j=11
1 q2j 1
= 1 +1X
m=1qm(m+1)=2
(1 q)(1 q2)(1 qm)
= 1 +1X
m=1qm(1 +q)(1 +q2)(1 +qm 1);
where the last right-hand side is the sum over m0
of the generating functions for partitions into distinct
parts with largest part equal to m.
628 Combinatorial Analysis
26.10.3(1 x)1X
m;n=0pm(k;D;n)xmqn
=kX
m=0k
m
qqm(m+1)=2xm=kY
j=1(1 +xqj),
jxj<1;
26.10.4
1X
n=0p(Dk;n)qn= 1 +1X
m=1q(km2+(2 k)m)=2
(1 q)(1 q2)(1 qm);
26.10.51X
n=0p(2S;n)qn=Y
j2S1
1 qj:
26.10(iii) Recurrence Relations
26.10.6p(D;n) =1
nnX
t=1p(D;n t)X
jjt
joddj;
where the inner sum is the sum of all positive odd divi-
sors oft.
26.10.7X
( 1)kp
D;n 1
2(3k2k)
=(
( 1)r; n = 3r2r;
0; otherwise;
where the sum is over nonnegative integer values of k
for whichn 1
2(3k2k)0.
26.10.8
X
( 1)kp
D;n (3k2k)
=(
1; n =1
2(r2r);
0;otherwise;
where the sum is over nonnegative integer values of k
for whichn (3k2k)0.
In exact analogy with (26.9.8), we have
26.10.9pm(D;n) =pm(D;n m) +pm 1(D;n);
26.10.10p(Dk;n) =X
pm
n 1
2km2 m+1
2km
;
where the sum is over nonnegative integer values of m
for whichn 1
2km2 m+1
2km0.
26.10.11p(2S;n) =1
nnX
t=1p(2S;n t)X
jjt
j2Sj;
where the inner sum is the sum of all positive divisors
oftthat are in S.
26.10(iv) Identities
Equations (26.10.13) and (26.10.14) are the Rogers{
Ramanujan identities . See alsox17.2(vi).
26.10.12 p(D;n) =p(O;n);26.10.13 p(D2;n) =p(2A1;5;n);
26.10.14p(D2;2T;n) =p(2A2;5;n),T=f2;3;4;:::g,
26.10.15 p(D03;n) =p(2A1;6;n):
Note thatp(D03;n)p(D3;n), with strict inequal-
ity forn9. It is known that for k > 3,p(Dk;n)
p(2A1;k+3;n), with strict inequality for nsuciently
large, provided that k= 2m 1;m= 3;4;5, ork32;
see Yee (2004).
26.10(v) Limiting Form
26.10.16 p(D;n)ep
n=3
(768n3)1=4,n!1 .
26.10(vi) Bessel-Function Expansion
26.10.17
p(D;n)
=1X
k=1A2k 1(n)
(2k 1)p24n+ 1I1
2k 1r
24n+ 1
72!
;
whereI1(x) is the modi ed Bessel function ( x10.25(ii)),
and
26.10.18Ak(n) =X
1<hk
(h;k)=1eif(h;k) (2inh=k );
with
26.10.19f(h;k) =kX
j=12j 1
2kh(2j 1)
k
;
and
26.10.20 [ [x] ] =(
x bxc 1
2; x =2Z;
0; x2Z:
The quantity Ak(n) is real-valued.
26.11 Integer Partitions: Compositions
Acomposition is an integer partition in which order is
taken into account. For example, there are eight compo-
sitions of 4: 4 ;3+1;1+3;2+2;2+1+1;1+2+1;1+1+2,
and 1 + 1 + 1 + 1. c(n) denotes the number of composi-
tions ofn, andcm(n) is the number of compositions into
exactlymparts.c(2T;n) is the number of compositions
ofnwith no 1's, where again T=f2;3;4;:::g. The in-
teger 0 is considered to have one composition consisting
of no parts:
26.11.1 c(0) =c(2T;0) = 1:
Also,
26.11.2 cm(0) =0;m;
26.11.3 cm(n) =n 1
m 1
;
26.11.41X
n=0cm(n)qn=qm
(1 q)m:
26.12 Plane Partitions 629
The Fibonacci numbers are determined recursively
by
26.11.5F0= 0 ,F1= 1 ,Fn=Fn 1+Fn 2,
n2.
26.11.6 c(2T;n) =Fn 1, n1.
Explicitly,
26.11.7 Fn=(1 +p
5)n (1 p
5)n
2np
5:
Additional information on Fibonacci numbers can
be found in Rosen et al. (2000, pp. 140{145).
26.12 Plane Partitions
26.12(i) De nitions
Aplane partition ,, of a positive integer n, is a par-
tition ofnin which the parts have been arranged in
a 2-dimensional array that is weakly decreasing (non-
increasing) across rows and down columns. Di erent
con gurations are counted as di erent plane partitions.
As an example, there are six plane partitions of 3:
26.12.1
3 , 2 1 ,2
1,1 1 1 ,1 1
1,1
1
1.
An equivalent de nition is that a plane partition is
a nite subset of NNNwith the property that
if (r;s;t )2and (1;1;1)(h;j;k )(r;s;t ), then
(h;j;k ) must be an element of . Here (h;j;k )(r;s;t )
meanshr,js, andkt. It is useful to be able
to visualize a plane partition as a pile of blocks, one
block at each lattice point ( h;j;k )2. For example,
Figure 26.12.1 depicts the pile of blocks that represents
the plane partition of 75 given by (26.12.2).
Figure 26.12.1 : A plane partition of 75.26.12.26 5 5 4 3 3
6 4 3 3 1
6 4 3 1 1
4 2 2 1
3 1 1
1 1 1
The number of plane partitions of nis denoted by
pp(n), with pp(0) = 1. See Table 26.12.1.
Table 26.12.1 : Plane partitions.
npp(n)n pp(n)n pp(n)
0 1 17 18334 34 281 75955
1 1 18 29601 35 416 91046
2 3 19 47330 36 614 84961
3 6 20 75278 37 903 79784
4 13 21 1 18794 38 1324 41995
5 24 22 1 86475 39 1934 87501
6 48 23 2 90783 40 2818 46923
7 86 24 4 51194 41 4093 83981
8 160 25 6 96033 42 5930 01267
9 282 26 10 68745 43 8566 67495
10 500 27 16 32658 44 12343 63833
11 859 28 24 83234 45 17740 79109
12 1479 29 37 59612 46 25435 35902
13 2485 30 56 68963 47 36379 93036
14 4167 31 85 12309 48 51913 04973
15 6879 32 127 33429 49 73910 26522
16 11297 33 189 74973 50 1 04996 40707
We de ne the rstboxB(r;s;t ) as
26.12.3
B(r;s;t ) =f(h;j;k )j1hr;1js;1ktg:
Then the number of plane partitions in B(r;s;t ) is
26.12.4
Y
(h;j;k )2B(r;s;t)h+j+k 1
h+j+k 2=rY
h=1sY
j=1h+j+t 1
h+j 1:
A plane partition is symmetric if (h;j;k )2im-
plies that (j;h;k )2. The number of symmetric plane
partitions in B(r;r;t ) is
26.12.5rY
h=12h+t 1
2h 1Y
1h<jrh+j+t 1
h+j 1:
A plane partition is cyclically symmetric if (h;j;k )2
implies (j;k;h )2. The plane partition in Fig-
ure 26.12.1 is an example of a cyclically symmetric plane
partition. The number of cyclically symmetric plane
partitions in B(r;r;r ) is
26.12.6rY
h=13h 1
3h 2Y
1h<jrh+ 2j 1
h+j 1;
630 Combinatorial Analysis
or equivalently,
26.12.7rY
h=10
@3h 1
3h 2rY
j=hr+h+j 1
2h+j 11
A:
A plane partition is totally symmetric if it is both
symmetric and cyclically symmetric. The number of
totally symmetric plane partitions in B(r;r;r ) is
26.12.8Y
1hjrh+j+r 1
h+ 2j 2:
The complement ofB(r;s;t ) isc=
f(h;j;k )j(r h+ 1;s j+ 1;t k+ 1)=2g. A
plane partition is self-complementary if it is equal to its
complement. The number of self-complementary plane
partitions in B(2r;2s;2t) is
26.12.90
@rY
h=1sY
j=1h+j+t 1
h+j 11
A2
;
inB(2r+ 1;2s;2t) it is
26.12.100
@rY
h=1sY
j=1h+j+t 1
h+j 11
A0
@r+1Y
h=1sY
j=1h+j+t 1
h+j 11
A;
inB(2r+ 1;2s+ 1;2t) it is
26.12.110
@r+1Y
h=1sY
j=1h+j+t 1
h+j 11
A0
@rY
h=1s+1Y
j=1h+j+t 1
h+j 11
A:
A plane partition is transpose complement if it is
equal to the re
ection through the ( x;y)-plane of its
complement. The number of transpose complement
plane partitions in B(r;r;2t) is
26.12.12t+r 1
r 1Y
1hjr 2h+j+ 2t+ 1
h+j+ 1:
The number of symmetric self-complementary plane
partitions in B(2r;2r;2t) is
26.12.13rY
h=1rY
j=1h+j+t 1
h+j 1;
inB(2r+ 1;2r+ 1;2t) it is
26.12.14rY
h=1r+1Y
j=1h+j+t 1
h+j 1:
The number of cyclically symmetric transpose com-
plement plane partitions in B(2r;2r;2r) is
26.12.15r 1Y
h=0(3h+ 1) (6h)! (2h)!
(4h+ 1)! (4h)!:The number of cyclically symmetric self-
complementary plane partitions in B(2r;2r;2r) is
26.12.16 r 1Y
h=0(3h+ 1)!
(r+h)!!2
:
The number of totally symmetric self-
complementary plane partitions in B(2r;2r;2r) is
26.12.17r 1Y
h=0(3h+ 1)!
(r+h)!:
Astrict shifted plane partition is an arrangement of
the parts in a partition so that each row is indented one
space from the previous row and there is weak decrease
across rows and strict decrease down columns. An ex-
ample is given by:
26.12.186 6 6 4 3
3 3
2
Adescending plane partition is a strict shifted plane
partition in which the number of parts in each row is
strictly less than the largest part in that row and is
greater than or equal to the largest part in the next
row. The example of a strict shifted plane partition
also satis es the conditions of a descending plane par-
tition. The number of descending plane partitions in
B(r;r;r ) is
26.12.19r 1Y
h=0(3h+ 1)!
(r+h)!:
26.12(ii) Generating Functions
The notationP
B(r;s;t)denotes the sum over all plane
partitions contained in B(r;s;t ), andjjdenotes the
number of elements in .
26.12.20X
NNNqjj=1Y
k=11
(1 qk)k;
26.12.21X
B(r;s;t)qjj=Y
(h;j;k )2B(r;s;t)1 qh+j+k 1
1 qh+j+k 2
=rY
h=1sY
j=11 qh+j+t 1
1 qh+j 1;
26.12.22X
B(r;r;t)
symmetricqjj
=rY
h=11 q2h+t 1
1 q2h 1Y
1h<jr1 q2(h+j+t 1)
1 q2(h+j 1):
26.13 Permutations: Cycle Notation 631
26.12.23X
B(r;r;r)
cyclically symmetricqjj
=rY
h=11 q3h 1
1 q3h 2Y
1h<jr1 q3(h+2j 1)
1 q3(h+j 1)
=rY
h=10
@1 q3h 1
1 q3h 2rY
j=h1 q3(r+h+j 1)
1 q3(2h+j 1)1
A:
26.12.24
X
B(r;r;r)
descending plane partitionqjj=Y
1h<jr1 qr+h+j 1
1 q2h+j 1:
26.12(iii) Recurrence Relation
26.12.25 pp(n) =1
nnX
j=1pp(n j)2(j);
where2(j) is the sum of the squares of the divisors of
j.
26.12(iv) Limiting Form
Asn!1
26.12.26
pp(n)(3)
211n251=36
exp
3(3)n2
41=3
+0( 1)!
;
whereis the Riemann -function (x25.2(i)).
26.13 Permutations: Cycle Notation
Sndenotes the set of permutations of f1;2;:::;ng.2
Snis a one-to-one and onto mapping from f1;2;:::;ng
to itself. An explicit representation of can be given
by the 2nmatrix:
26.13.11 2 3 n
(1)(2)(3)(n)
:
In cycle notation, the elements in each cycle are put
inside parentheses, ordered so that (j) immediately fol-
lowsjor, ifjis the last listed element of the cycle, then
(j) is the rst element of the cycle. The permutation
26.13.21 2 3 4 5 6 7 8
3 5 2 4 7 8 1 6
is (1;3;2;5;7)(4)(6;8) in cycle notation. Cycles of
length one are xed points . They are often dropped
from the cycle notation. In consequence, (26.13.2) can
also be written as (1 ;3;2;5;7)(6;8).
An element of Snwitha1 xed points, a2cy-
cles of length 2 ;:::;ancycles of length n, where
n=a1+ 2a2++nan, is said to have cycle type(a1;a2;:::;an). The number of elements of Snwith
cycle type ( a1;a2;:::;an) is given by (26.4.7).
The Stirling cycle numbers of the rst kind, de-
noted byn
k
, count the number of permutations of
f1;2;:::;ngwith exactly kcycles. They are related
to Stirling numbers of the rst kind by
26.13.3hn
ki
=js(n;k)j:
Seex26.8 for generating functions, recurrence relations,
identities, and asymptotic approximations.
Aderangement is a permutation with no xed
points. The derangement number ,d(n), is the number
of elements of Snwith no xed points:
26.13.4d(n) =n!nX
j=0( 1)j1
j!=n! +e 2
e
:
Atransposition is a permutation that consists of a
single cycle of length two. An adjacent transposition
is a transposition of two consecutive integers. A per-
mutation that consists of a single cycle of length kcan
be written as the composition of k 1 two-cycles (read
from right to left):
26.13.5
(j1;j2;:::;jk) = (j1;j2)(j2;j3)(jk 2;jk 1)(jk 1;jk):
Every permutation is a product of transpositions. A
permutation with cycle type ( a1;a2;:::;an) can be writ-
ten as a product of a2+ 2a3++ (n 1)an=
n (a1+a2++an) transpositions, and no fewer.
For the example (26.13.2), this decomposition is given
by (1;3;2;5;7)(6;8) = (1;3)(2;3)(2;5)(5;7)(6;8):
A permutation is even oroddaccording to the parity
of the number of transpositions. The sign of a permu-
tation is + if the permutation is even, if it is odd.
Every transposition is the product of adjacent trans-
positions. If j <k , then (j;k) is a product of 2 k 2j 1
adjacent transpositions:
26.13.6(j;k) = (k 1;k)(k 2;k 1)(j+ 1;j+ 2)
(j;j+ 1)(j+ 1;j+ 2)(k 1;k):
Every permutation is a product of adjacent transpo-
sitions. Given a permutation 2Sn, the inversion
number of, denoted inv( ), is the least number of
adjacent transpositions required to represent . Again,
for the example (26.13.2) a minimal decomposition into
adjacent transpositions is given by (1 ;3;2;5;7)(6;8) =
(2;3)(1;2)(4;5)(3;4)(2;3)(3;4)(4;5)(6;7)(5;6)(7;8)
(6;7): inv((1;3;2;5;7)(6;8)) = 11.
632 Combinatorial Analysis
26.14 Permutations: Order Notation
26.14(i) De nitions
The set Sn(x26.13) can be viewed as the collec-
tion of all ordered lists of elements of f1;2;:::;ng:
f(1)(2)(n)g. As an example, 35247816 is an el-
ement of S8:The inversion number is the number of
pairs of elements for which the larger element precedes
the smaller:
26.14.1inv() =X
1j<kn
(j)>(k)1:
Equivalently, this is the sum over 1 j < n of
the number of integers less than (j) that lie in po-
sitions to the right of the jth position: inv(35247816) =
2 + 3 + 1 + 1 + 2 + 2 + 0 = 11 :
Adescent of a permutation is a pair of adjacent ele-
ments for which the rst is larger than the second. Thepermutation 35247816 has two descents: 52 and 81. The
major index is the sum of all positions that mark the
rst element of a descent:
26.14.2maj() =X
1j<n
(j)>(j+1)j:
For example, maj(35247816) = 2 + 6 = 8. The major
index is also called the greater index of the permutation.
The Eulerian number , denoted
n
k
, is the number
of permutations in Snwith exactly kdescents. An ex-
cedance in2Snis a position jfor which(j)>j. A
weak excedance is a position jfor which(j)j. The
Eulerian number
n
k
is equal to the number of permu-
tations in Snwith exactly kexcedances. It is also equal
to the number of permutations in Snwith exactly k+1
weak excedances. See Table 26.14.1.
Table 26.14.1 : Eulerian numbers
n
k
.
nk
0 1 2 3 4 5 6 7 8 9
01
11
21 1
31 4 1
41 11 11 1
51 26 66 26 1
61 57 302 302 57 1
71 120 1191 2416 1191 120 1
81 247 4293 15619 15619 4293 247 1
91 502 14608 88234 1 56190 88234 14608 502 1
10 1 1013 47840 4 55192 13 10354 13 10354 4 55192 47840 1013 1
26.14(ii) Generating Functions
26.14.3X
2Snqinv()=X
2Snqmaj()=nY
j=11 qj
1 q:
26.14.4
1X
n;k=0 n
k
xktn
n!=1 x
exp((x 1)t) x,jxj<1;jtj<1:
26.14.5n 1X
k=0 n
k x+k
n
=xn:
26.14(iii) Identities
In this subsection S(n;k) is again the Stirling number
of the second kind ( x26.8), andBmis themth Bernoullinumber (x24.2(i)).
26.14.6 n
k
=kX
j=0( 1)jn+ 1
j
(k+ 1 j)n,n1,
26.14.7 n
k
=n kX
j=0( 1)n k jj!n j
k
S(n;j);
26.14.8 n
k
= (k+ 1) n 1
k
+(n k) n 1
k 1
,n2,
26.14.9 n
k
= n
n 1 k
,n1,
26.14.10n 1X
k=0 n
k
=n!, n1.
26.15 Permutations: Matrix Notation 633
26.14.11
Bm=m
2m(2m 1)m 2X
k=0( 1)k m 1
k
,m2.
26.14.12
S(n;m) =1
m!n 1X
k=0 n
k k
n m
,nm,n1.
26.14(iv) Special Values
26.14.13 0
k
=0;k;
26.14.14 n
0
= 1;
26.14.15 n
1
= 2n n 1, n1,
26.14.16 n
2
= 3n (n+ 1)2n+n+ 1
2
,n1.
26.15 Permutations: Matrix Notation
The set Sn(x26.13) can be identi ed with the set of
nnmatrices of 0's and 1's with exactly one 1 in each
row and column. The permutation corresponds to the
matrix in which there is a 1 at the intersection of row
jwith column (j), and 0's in all other positions. The
permutation 35247816 corresponds to the matrix
26.15.12
666666666640 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 03
77777777775
The sign of the permutation is the sign of the de-
terminant of its matrix representation. The inversion
number ofis a sum of products of pairs of entries in
the matrix representation of :
26.15.2 inv() =X
aghak`;
where the sum is over 1 g<knandnh>`1.
The matrix represents the placement of nnonat-
tacking rooks on an nnchessboard, that is, rooks
that share neither a row nor a column with any other
rook. A permutation with restricted position speci es a
subsetBf1;2;:::;ngf 1;2;:::;ng. If (j;k)2B,
then(j)6=k. The number of derangements ofnis
the number of permutations with forbidden positions
B=f(1;1);(2;2);:::; (n;n)g.
Letrj(B) be the number of ways of placing jnonat-
tacking rooks on the squares of B. De ner0(B) = 1.For the problem of derangements, rj(B) = n
j
. The
rook polynomial is the generating function for rj(B):
26.15.3 R(x;B) =nX
j=0rj(B)xj:
IfB=B1[B2, where no element of B1is in the
same row or column as any element of B2, then
26.15.4 R(x;B) =R(x;B 1)R(x;B 2):
For (j;k)2B,Bn[j;k] denotesBafter removal of
all elements of the form ( j;t) or (t;k),t= 1;2;:::;n .
Bn(j;k) denotesBwith the element ( j;k) removed.
26.15.5R(x;B) =xR(x;Bn[j;k]) +R(x;Bn(j;k)):
Nk(B) is the number of permutations in Snfor
which exactly kof the pairs ( j;(j)) are elements of
B.N(x;B) is the generating function:
26.15.6 N(x;B) =nX
k=0Nk(B)xk;
and
26.15.7N(x;B) =nX
k=0rk(B)(n k)!(x 1)k:
The number of permutations that avoid Bis
26.15.8N0(B)N(0;B) =nX
k=0( 1)krk(B)(n k)!:
Example 1
The probl eme des m enages asks for the number of ways
of seatingnmarried couples around a circular table
with labeled seats so that no men are adjacent, no
women are adjacent, and no husband and wife are ad-
jacent. There are 2( n!) ways to place the wives. Let
B=f(j;j);(j;j+1)j1j <ng[f(n;n);(n;1)g. Then
26.15.9 rk(B) =2n
2n k2n k
k
:
The solution is
26.15.10
2(n!)N0(B) = 2(n!)nX
k=0( 1)k2n
2n k2n k
k
(n k)!:
Example 2
The Ferrers board of shape (b1;b2;:::;bn), 0b1
b2bn, is the setB=f(j;k)j1jn;1k
bjg. For this set,
26.15.11nX
k=0rn k(B)(x k+ 1)k=nY
j=1(x+bj j+ 1):
IfBis the Ferrers board of shape (0 ;1;2;:::;n 1),
then
26.15.12nX
k=0rn k(B)(x k+ 1)k=xn;
and therefore by (26.8.10),
26.15.13 rn k(B) =S(n;k):
634 Combinatorial Analysis
26.16 Multiset Permutations
LetS=f1a1;2a2;:::;nangbe the multiset that has aj
copies ofj, 1jn.SSdenotes the set of permuta-
tions ofSfor all distinct orderings of the a1+a2++an
integers. The number of elements in SSis the multi-
nomial coecient ( x26.4) a1+a2++an
a1;a2;:::;an
. Additional in-
formation can be found in Andrews (1976, pp. 39{45).
The de nitions of inversion number and major in-
dex can be extended to permutations of a multi-
set such as 351322453154 2Sf12;22;33;42;53g. Thus
inv(351322453154) = 4+8+0+3+1+1+2+3+1+0+1 =
24, and maj(351322453154) = 2 + 4 + 8 + 9 + 11 = 34 :
Theq-multinomial coecient is de ned in terms of
Gaussian polynomials ( x26.9(ii)) by
26.16.1a1+a2++an
a1;a2;:::;an
q=n 1Y
k=1ak+ak+1++an
ak
q;and again with S=f1a1;2a2;:::;nangwe have
26.16.2X
2SSqinv()=a1+a2++an
a1;a2;:::;an
q;
26.16.3X
2SSqmaj()=a1+a2++an
a1;a2;:::;an
q:
26.17 The Twelvefold Way
The twelvefold way gives the number of mappings f
from setNofnobjects to set Kofkobjects (putting
balls from set Ninto boxes in set K). See Table 26.17.1.
In this table ( k)nis Pochhammer's symbol, and S(n;k)
andpk(n) are de ned inxx26.8(i) and 26.9(i).
Table 26.17.1 is reproduced (in modi ed form) from
Stanley (1997, p. 33). See also Example 3 in x26.18.
Table 26.17.1 : The twelvefold way.
elements of N elements of Kfunrestricted fone-to-one fonto
labeled labeled kn(k n+ 1)nk!S(n;k)
unlabeled labeledk+n 1
nk
nn 1
n k
labeled unlabeledS(n;1) +S(n;2)
++S(n;k)(
1nk
0n>kS(n;k)
unlabeled unlabeled pk(n)(
1nk
0n>kpk(n) pk 1(n)
26.18 Counting Techniques
LetA1;A2;:::;Anbe subsets of a set Sthat are not necessarily disjoint. Then the number of elements in the set
Sn(A1[A2[[An) is
26.18.1 jSn(A1[A2[[An)j=jSj+nX
t=1( 1)tX
1j1<j2<<jtnjAj1\Aj2\\Ajtj:
Example 1
The number of positive integers Nthat are not divisible by any of the primes p1;p2;:::;pn(x27.2(i)) is
26.18.2 N+nX
t=1( 1)tX
1j1<j2<<jtnN
pj1pj2pjt
:
Applications 635
Example 2
With the notation of x26.15, the number of placements
ofnnonattacking rooks on an nnchessboard that
avoid the squares in a speci ed subset Bis
26.18.3 n! +nX
t=1( 1)trt(B)(n t)!:
Example 3
The number of ways of placing nlabeled objects into k
labeled boxes so that at least one object is in each box
is
26.18.4 kn+nX
t=1( 1)tk
t
(k t)n:
Note that this is also one of the counting problems for
which a formula is given in Table 26.17.1. Elements of
Nare labeled, elements of Kare labeled, and fis onto.
For further examples in the use of generating func-
tions, see Stanley (1997, 1999) and Wilf (1994). See also
P olya et al. (1983).
Applications
26.19 Mathematical Applications
Combinatorics has applications to analysis, algebra,
and geometry. Examples can be found in Beckenbach
(1981), Billera et al. (1996), and Lov asz et al. (1995).
Partitions and plane partitions have applications to rep-
resentation theory (Bressoud (1999), Macdonald (1995),
and Sagan (2001)) and to special functions (Andrews
et al. (1999) and Gasper and Rahman (2004)).
Other areas of combinatorial analysis include graph
theory, coding theory, and combinatorial designs. These
have applications in operations research, probability
theory, and statistics. See Graham et al. (1995) and
Rosen et al. (2000).
26.20 Physical Applications
An English translation of P olya (1937) on applications
of combinatorics to chemistry has been published as
P olya and Read (1987). Other articles on this subject
are de Bruijn (1981) and Rouvray (1995). The latter
reference also describes chemical applications of other
combinatorial techniques.
Applications of combinatorics, especially integer and
plane partitions, to counting lattice structures and other
problems of statistical mechanics, of which the Ising
model is the principal example, can be found in Mon-
troll (1964), Godsil et al. (1995), Baxter (1982), andKorepin et al. (1993). For an application of statistical
mechanics to combinatorics, see Bressoud (1999).
Other applications to problems in engineering, crys-
tallography, biology, and computer science can be found
in Beckenbach (1981) and Graham et al. (1995).
Computation
26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates
binomial coecients m
n
formup to 50 and nup to 25;
extends Table 26.4.1 to n= 10; tabulates Stirling num-
bers of the rst and second kinds, s(n;k) andS(n;k),
fornup to 25 and kup ton; tabulates partitions p(n)
and partitions into distinct parts p(D;n) fornup to
500.
Andrews (1976) contains tables of the number of
unrestricted partitions, partitions into odd parts, par-
titions into parts 62 (mod 5), partitions into parts
61 (mod 5), and unrestricted plane partitions up to
100. It also contains a table of Gaussian polynomials
up to12
6
q.
Goldberg et al. (1976) contains tables of binomial
coecients to n= 100 and Stirling numbers to n= 40.
26.22 Software
Seehttp://dlmf.nist.gov/26.22 .
References
General References
Comprehensive references include Graham et al. (1995)
and Rosen et al. (2000). Most of this chapter is treated
in detail in Comtet (1974), Riordan (1958), and Stanley
(1997, 1999).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x26.2 Table 26.2.1 is from Abramowitz and Stegun
(1964, Table 24.5).
x26.3 Comtet (1974, pp. 8{10, 22{23, 292), Riordan
(1958, pp. 4{11), and (5.11.7). Tables 26.3.1 and
26.3.2 are from Abramowitz and Stegun (1964, Ta-
ble 24.1).
636 Combinatorial Analysis
x26.4 Comtet (1974, pp. 28{29). Table 26.4.1 is from
Abramowitz and Stegun (1964, Table 24.2).
x26.5 Comtet (1974, pp. 52{54), Riordan (1979,
p. 157). For (26.5.6) and (26.5.7) use (26.3.12)
and (26.5.1). Table 26.5.1 was computed by the
author.
x26.6 Comtet (1974, pp. 80{81), Stanley (1999,
pp. 237{241). (26.6.4) is a consequence of Andr e's
re
ection principle; see Comtet (1974, pp. 22{23).
For (26.6.11) use (26.6.7). Tables 26.6.1{26.6.4
were computed by the author.
x26.7 Comtet (1974, pp. 210{211). For (26.7.3) see
Wilf (1994, p. 22). For (26.7.7) see de Bruijn
(1961, pp. 104{108) and Olver (1997b, pp. 329{
331). Table 26.7.1 was computed by the author.
x26.8 Comtet (1974, pp. 206{216), Riordan (1979,
pp. 195 and 203{227), Graham et al. (1994,
pp. 264{265). For (26.8.31) use (26.8.37) and
(26.8.39). For (26.8.40){(26.8.43) see Jordan
(1939, pp. 161{174) (as corrected here). Tables
26.8.1 and 26.8.2 are from Abramowitz and Ste-
gun (1964, Tables 24.3 and 24.4).
x26.9 Andrews (1976, Chapter 6 and pp. 1{13, 36,
47, 81). For (26.9.9) see Bressoud (1999, p. 60,Eq. (2.23)). Table 26.9.1 was computed by the
author.
x26.10 Andrews (1976, pp. 5, 11{12, 16{17, 19, 36, 82,
97, 104, 116), Bressoud (1999, pp. 60, 78{79). Ta-
ble 26.10.1 was computed by the author.
x26.11 Andrews (1976, Chapter 4).
x26.12 Bressoud (1999, pp. 11, 13{18, 22, 57, 197{199
(with corrections)), Andrews (1976, p. 199), An-
drews (1979, p. 195). Table 26.12.1 was computed
by the author.
x26.13 Cameron (1994, pp. 77, 80{84), Stanley (1997,
pp. 20{21, 67).
x26.14 Andrews (1976, pp. 39{42), Graham et al.
(1994, pp. 267{272), Riordan (1958, pp. 38{
39), Stanley (1997, pp. 20{23). For (26.14.10)
and (26.14.11) use (26.14.4) and (24.2.1). Table
26.14.1 was computed by the author.
x26.15 Stanley (1997, pp. 71{76), Tucker (2006,
pp. 335{345).
x26.16 Andrews (1976, pp. 39{45).
x26.18 Riordan (1958, pp. 50{65).
Chapter 27
Functions of Number Theory
T. M. Apostol1
Notation 638
27.1 Special Notation . . . . . . . . . . . . . 638
Multiplicative Number Theory 638
27.2 Functions . . . . . . . . . . . . . . . . . 638
27.3 Multiplicative Properties . . . . . . . . . 640
27.4 Euler Products and Dirichlet Series . . . 640
27.5 Inversion Formulas . . . . . . . . . . . . 641
27.6 Divisor Sums . . . . . . . . . . . . . . . 641
27.7 Lambert Series as Generating Functions . 641
27.8 Dirichlet Characters . . . . . . . . . . . . 642
27.9 Quadratic Characters . . . . . . . . . . . 642
27.10 Periodic Number-Theoretic Functions . . 642
27.11 Asymptotic Formulas: Partial Sums . . . 643
27.12 Asymptotic Formulas: Primes . . . . . . 644Additive Number Theory 644
27.13 Functions . . . . . . . . . . . . . . . . . 644
27.14 Unrestricted Partitions . . . . . . . . . . 645
Applications 647
27.15 Chinese Remainder Theorem . . . . . . . 647
27.16 Cryptography . . . . . . . . . . . . . . . 647
27.17 Other Applications . . . . . . . . . . . . 647
Computation 648
27.18 Methods of Computation: Primes . . . . 648
27.19 Methods of Computation: Factorization . 648
27.20 Methods of Computation: Other Number-
Theoretic Functions . . . . . . . . . . . . 649
27.21 Tables . . . . . . . . . . . . . . . . . . . 649
27.22 Software . . . . . . . . . . . . . . . . . . 649
References 649
1California Institute of Technology, Pasadena, California.
Acknowledgments : The author thanks Basil Gordon for comments on an earlier draft, and David Bressoud for providing xx27.12,
27.18, 27.19, and 27.22.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
637
638 Functions of Number Theory
Notation
27.1 Special Notation
(For other notation see pp. xiv and 873.)
d;k;m;n positive integers (unless otherwise
indicated).
djn d dividesn.
(m;n) greatest common divisor of m;n. If
(m;n) = 1,mandnare called relatively
prime, or coprime.
(d1;:::;dn) greatest common divisor of d1;:::;dn.P
djn,Q
djnsum, product taken over divisors of n.P
(m;n)=1sum taken over m, 1mnandm
relatively prime to n.
p;p1;p2;::: prime numbers (or primes): integers
(>1) with only two positive integer
divisors, 1 and the number itself.P
p,Q
psum, product extended over all primes.
x;y real numbers.P
nxPbxc
n=1.
logx natural logarithm of x, written as ln xin
other chapters.
(s) Riemann zeta function; see x25.2(i).
(njP) Jacobi symbol; see x27.9.
(njp) Legendre symbol; see x27.9.
Multiplicative Number Theory
27.2 Functions
27.2(i) De nitions
Functions in this section derive their properties from the
fundamental theorem of arithmetic , which states that
every integer n > 1 can be represented uniquely as a
product of prime powers,
27.2.1 n=(n)Y
r=1par
r;
wherep1;p2;:::;p(n)are the distinct prime factors of
n, each exponent aris positive, and (n) is the number
of distinct primes dividing n. ((1) is de ned to be 0.)
Euclid's Elements (Euclid (1908, Book IX, Proposition
20)) gives an elegant proof that there are in nitely many
primes. Tables of primes ( x27.21) reveal great irregular-
ity in their distribution. They tend to thin out among
the large integers, but this thinning out is not com-
pletely regular. There is great interest in the function(x) that counts the number of primes not exceeding x.
It can be expressed as a sum over all primes px:
27.2.2 (x) =X
px1:
Gauss and Legendre conjectured that (x) is asymp-
totic tox=logxasx!1 :
27.2.3 (x)x
logx:
(See Gauss (1863, Band II, pp. 437{477) and Legendre
(1808, p. 394).)
This result, rst proved in Hadamard (1896) and
de la Vall ee Poussin (1896a,b), is known as the prime
number theorem . An equivalent form states that the
nth primepn(when the primes are listed in increasing
order) is asymptotic to nlognasn!1 :
27.2.4 pnnlogn:
(See alsox27.12.) Other examples of number-theoretic
functions treated in this chapter are as follows.
27.2.51
n
=(
1; n = 1;
0; n> 1:
27.2.6 k(n) =X
(m;n)=1mk;
the sum of the kth powers of the positive integers mn
that are relatively prime to n.
27.2.7 (n) = 0(n):
This is the number of positive integers nthat are
relatively prime to n; (n) isEuler's totient .
If (a;n) = 1, then the Euler{Fermat theorem states
that
27.2.8 a (n)1 (modn);
and if (n) is the smallest positive integer fsuch that
af1 (modn), thenais aprimitive root modn. The
(n) numbersa;a2;:::;a (n)are relatively prime to n
and distinct (mod n). Such a set is a reduced residue
system modulon.
27.2.9d(n) =X
djn1
is the number of divisors of nand is the divisor func-
tion. It is the special case k= 2 of the function dk(n)
that counts the number of ways of expressing nas the
product ofkfactors, with the order of factors taken into
account.
27.2.10 (n) =X
djnd ;
is the sum of the th powers of the divisors of n, where
the exponent can be real or complex. Note that
0(n) =d(n).
27.2.11Jk(n) =X
((d1;:::;dk);n)=11;
27.2 Functions 639
is the number of k-tuples of integers nwhose greatest
common divisor is relatively prime to n. This is Jordan's
function . Note that J1(n) = (n).
In the following examples, a1;:::;a(n)are the ex-
ponents in the factorization of nin (27.2.1).
27.2.12(n) =8
><
>:1; n = 1;
( 1)(n); a 1=a2==a(n)= 1;
0; otherwise:
This is the M obius function .
27.2.13(n) =(
1; n = 1;
( 1)a1++a(n); n> 1:This is Liouville's function .
27.2.14 (n) = logp, n=pa,
wherepais a prime power with a1; otherwise
(n) = 0. This is Mangoldt's function .
27.2(ii) Tables
Table 27.2.1 lists the rst 100 prime numbers pn. Ta-
ble 27.2.2 tabulates the Euler totient function (n), the
divisor function d(n) (=0(n)), and the sum of the
divisors(n) (=1(n)), forn= 1(1)52.
Table 27.2.1 : Primes.
n pnpn+10pn+20pn+30pn+40pn+50pn+60pn+70pn+80pn+90
1 2 31 73 127 179 233 283 353 419 467
2 3 37 79 131 181 239 293 359 421 479
3 5 41 83 137 191 241 307 367 431 487
4 7 43 89 139 193 251 311 373 433 491
5 11 47 97 149 197 257 313 379 439 499
6 13 53 101 151 199 263 317 383 443 503
7 17 59 103 157 211 269 331 389 449 509
8 19 61 107 163 223 271 337 397 457 521
9 23 67 109 167 227 277 347 401 461 523
10 29 71 113 173 229 281 349 409 463 541
Table 27.2.2 : Functions related to division.
n (n)d(n)(n)n (n)d(n)(n)n (n)d(n)(n)n (n)d(n)(n)
1 1 1 1 14 6 4 24 27 18 4 40 40 16 8 90
2 1 2 3 15 8 4 24 28 12 6 56 41 40 2 42
3 2 2 4 16 8 5 31 29 28 2 30 42 12 8 96
4 2 3 7 17 16 2 18 30 8 8 72 43 42 2 44
5 4 2 6 18 6 6 39 31 30 2 32 44 20 6 84
6 2 4 12 19 18 2 20 32 16 6 63 45 24 6 78
7 6 2 8 20 8 6 42 33 20 4 48 46 22 4 72
8 4 4 15 21 12 4 32 34 16 4 54 47 46 2 48
9 6 3 13 22 10 4 36 35 24 4 48 48 16 10 124
10 4 4 18 23 22 2 24 36 12 9 91 49 42 3 57
11 10 2 12 24 8 8 60 37 36 2 38 50 20 6 93
12 4 6 28 25 20 3 31 38 18 4 60 51 32 4 72
13 12 2 14 26 12 4 42 39 24 4 56 52 24 6 98
640 Functions of Number Theory
27.3 Multiplicative Properties
Except for (n), (n),pn, and(x), the functions in
x27.2 are multiplicative , which means f(1) = 1 and
27.3.1 f(mn) =f(m)f(n), (m;n) = 1.
Iffis multiplicative, then the values f(n) forn>1
are determined by the values at the prime powers.
Speci cally, if nis factored as in (27.2.1), then
27.3.2 f(n) =(n)Y
r=1f(par
r):
In particular,
27.3.3 (n) =nY
pjn(1 p 1);
27.3.4Jk(n) =nkY
pjn(1 p k);
27.3.5 d(n) =(n)Y
r=1(1 +ar);
27.3.6 (n) =(n)Y
r=1p (1+ar)
r 1
p r 1, 6= 0.
Related multiplicative properties are
27.3.7 (m) (n) =X
dj(m;n)d mn
d2
;
27.3.8 (m) (n) = (mn) ((m;n))=(m;n):
A function fiscompletely multiplicative iff(1) = 1
and
27.3.9 f(mn) =f(m)f(n),m;n = 1;2;:::.
Examples areb1=ncand(n), and the Dirichlet char-
acters, de ned in x27.8.
Iffis completely multiplicative, then (27.3.2) be-
comes
27.3.10 f(n) =(n)Y
r=1(f(pr))ar:
27.4 Euler Products and Dirichlet Series
The fundamental theorem of arithmetic is linked to
analysis through the concept of the Euler product. Ev-
ery multiplicative fsatis es the identity
27.4.11X
n=1f(n) =Y
p
1 +1X
r=1f(pr)!
;
if the series on the left is absolutely convergent. In this
case the in nite product on the right (extended over all
primesp) is also absolutely convergent and is called theEuler product of the series. If f(n) is completely multi-
plicative, then each factor in the product is a geometric
series and the Euler product becomes
27.4.21X
n=1f(n) =Y
p(1 f(p)) 1:
Euler products are used to nd series that generate
many functions of multiplicative number theory. The
completely multiplicative function f(n) =n sgives the
Euler product representation of the Riemann zeta func-
tion(s) (x25.2(i)):
27.4.3(s) =1X
n=1n s=Y
p(1 p s) 1,<s>1.
The Riemann zeta function is the prototype of series
of the form
27.4.4 F(s) =1X
n=1f(n)n s;
called Dirichlet series with coecients f(n). The func-
tionF(s) is a generating function , or more precisely, a
Dirichlet generating function , for the coecients. The
following examples have generating functions related to
the zeta function:
27.4.51X
n=1(n)n s=1
(s),<s>1,
27.4.61X
n=1 (n)n s=(s 1)
(s),<s>2,
27.4.71X
n=1(n)n s=(2s)
(s),<s>1,
27.4.81X
n=1j(n)jn s=(s)
(2s),<s>1,
27.4.91X
n=12(n)n s=((s))2
(2s),<s>1,
27.4.101X
n=1dk(n)n s= ((s))k,<s>1,
27.4.11
1X
n=1 (n)n s=(s)(s ),<s>max(1;1 +< ),
27.4.121X
n=1(n)n s= 0(s)
(s),<s>1,
27.4.131X
n=2(logn)n s= 0(s),<s>1.
In (27.4.12) and (27.4.13) 0(s) is the derivative of (s).
27.5 Inversion Formulas 641
27.5 Inversion Formulas
If a Dirichlet series F(s) generates f(n), andG(s) gen-
eratesg(n), then the product F(s)G(s) generates
27.5.1 h(n) =X
djnf(d)gn
d
;
called the Dirichlet product (orconvolution ) offand
g. The set of all number-theoretic functions fwith
f(1)6= 0 forms an abelian group under Dirichlet multi-
plication, with the function b1=ncin (27.2.5) as identity
element; see Apostol (1976, p. 129). The multiplicative
functions are a subgroup of this group. Generating func-
tions yield many relations connecting number-theoretic
functions. For example, the equation (s)(1/(s)) = 1
is equivalent to the identity
27.5.2X
djn(d) =1
n
;
which, in turn, is the basis for the M obius inversion
formula relating sums over divisors:
27.5.3g(n) =X
djnf(d)()f(n) =X
djng(d)n
d
:
Special cases of M obius inversion pairs are:
27.5.4n=X
djn (d)() (n) =X
djndn
d
;
27.5.5 logn=X
djn(d)() (n) =X
djn(logd)n
d
:
Other types of M obius inversion formulas include:
27.5.6G(x) =X
nxFx
n
()F(x) =X
nx(n)Gx
n
;
27.5.7
G(x) =1X
m=1F(mx)
ms()F(x) =1X
m=1(m)G(mx)
ms;
27.5.8g(n) =Y
djnf(d)()f(n) =Y
djn
gn
d(d)
:
For a general theory of M obius inversion with appli-
cations to combinatorial theory see Rota (1964).
27.6 Divisor Sums
Sums of number-theoretic functions extended over divi-
sors are of special interest. For example,
27.6.1X
djn(d) =(
1; n is a square;
0;otherwise:
Iffis multiplicative, then
27.6.2X
djn(d)f(d) =Y
pjn(1 f(p)),n>1.Generating functions, Euler products, and M obius
inversion are used to evaluate many sums extended over
divisors. Examples include:
27.6.3X
djnj(d)j= 2(n);
27.6.4X
d2jn(d) =j(n)j;
27.6.5X
djnj(d)j
(d)=n
(n);
27.6.6X
djn k(d)n
dk
= 1k+ 2k++nk;
27.6.7X
djn(d)n
dk
=Jk(n);
27.6.8X
djnJk(d) =nk:
27.7 Lambert Series as Generating
Functions
Lambert series have the form
27.7.11X
n=1f(n)xn
1 xn:
Ifjxj<1, then the quotient xn=(1 xn) is the sum of a
geometric series, and when the series (27.7.1) converges
absolutely it can be rearranged as a power series:
27.7.21X
n=1f(n)xn
1 xn=1X
n=1X
djnf(d)xn:
Again withjxj<1, special cases of (27.7.2) include:
27.7.31X
n=1(n)xn
1 xn=x;
27.7.41X
n=1 (n)xn
1 xn=x
(1 x)2;
27.7.51X
n=1n xn
1 xn=1X
n=1 (n)xn;
27.7.61X
n=1(n)xn
1 xn=1X
n=1xn2:
642 Functions of Number Theory
27.8 Dirichlet Characters
Ifk(>1) is a given integer, then a function (n) is
called a Dirichlet character (modk) if it is completely
multiplicative, periodic with period k, and vanishes
when (n;k)>1. In other words, Dirichlet characters
(modk) satisfy the four conditions:
27.8.1 (1) = 1;
27.8.2 (mn) = (m) (n),m;n = 1;2;:::,
27.8.3 (n+k) = (n), n= 1;2;:::,
27.8.4 (n) = 0, ( n;k)>1.
An example is the principal character (modk):
27.8.5 1(n) =(
1;(n;k) = 1;
0;(n;k)>1:
For any character (modk), (n)6= 0 if and only
if (n;k) = 1, in which case the Euler{Fermat theorem
(27.2.8) implies ( (n)) (k)= 1. There are exactly (k)
di erent characters (mod k), which can be labeled as
1;:::; (k). If is a character (mod k), so is its com-
plex conjugate . If (n;k) = 1, then the characters
satisfy the orthogonality relation
27.8.6 (k)X
r=1 r(m) r(n) =(
(k); mn(modk);
0; otherwise:
A Dirichlet character (modk) is called primitive
(modk) if for every proper divisor dofk(that is, a
divisord < k ), there exists an integer a1 (modd),
with (a;k) = 1 and (a)6= 1. Ifkis prime, then ev-
ery nonprincipal character (modk) is primitive. A
divisordofkis called an induced modulus for if
27.8.7 (a) = 1 for all a1 (modd), (a;k) = 1.
Every Dirichlet character (modk) is a product
27.8.8 (n) = 0(n) 1(n);
where 0is a character (mod d) for some induced mod-
ulusdfor , and 1is the principal character (mod k).
A character is realif all its values are real. If kis odd,
then the real characters (mod k) are the principal char-
acter and the quadratic characters described in the next
section.
27.9 Quadratic Characters
For an odd prime p, the Legendre symbol (njp) is de-
ned as follows. If pdividesn, then the value of ( njp)
is 0. Ifpdoes not divide n, then (njp) has the value
1 when the quadratic congruence x2n(modp) has
a solution, and the value 1 when this congruence has
no solution. The Legendre symbol ( njp), as a functionofn, is a Dirichlet character (mod p). It is sometimes
written as (n
p). Special values include:
27.9.1 ( 1jp) = ( 1)(p 1)=2;
27.9.2 (2jp) = ( 1)(p2 1)=8:
Ifp;qare distinct odd primes, then the quadratic
reciprocity law states that
27.9.3 (pjq) (qjp) = ( 1)(p 1)(q 1)=4:
If an odd integer Phas prime factorization P=Q(n)
r=1parr, then the Jacobi symbol (njP) is de ned by
(njP) =Q(n)
r=1(njpr)ar, with (nj1) = 1. The Jacobi
symbol (njP) is a Dirichlet character (mod P). Both
(27.9.1) and (27.9.2) are valid with preplaced by P; the
reciprocity law (27.9.3) holds if p;qare replaced by any
two relatively prime odd integers P;Q.
27.10 Periodic Number-Theoretic
Functions
Ifkis a xed positive integer, then a number-theoretic
functionfisperiodic (modk) if
27.10.1 f(n+k) =f(n),n= 1;2;:::.
Examples are the Dirichlet characters (mod k) and the
greatest common divisor ( n;k) regarded as a function
ofn.
Every function periodic (mod k) can be expressed as
a nite Fourier series of the form
27.10.2 f(n) =kX
m=1g(m)e2imn=k;
whereg(m) is also periodic (mod k), and is given by
27.10.3 g(m) =1
kkX
n=1f(n)e 2imn=k:
An example is Ramanujan's sum :
27.10.4 ck(n) =kX
m=1 1(m)e2imn=k;
where 1is the principal character (mod k). This is
the sum of the nth powers of the primitive kth roots of
unity. It can also be expressed in terms of the M obius
function as a divisor sum:
27.10.5 ck(n) =X
dj(n;k)dk
d
:
More generally, if fandgare arbitrary, then the
sum
27.10.6 sk(n) =X
dj(n;k)f(d)gk
d
27.11 Asymptotic Formulas: Partial Sums 643
is a periodic function of n(modk) and has the nite
Fourier-series expansion
27.10.7 sk(n) =kX
m=1ak(m)e2imn=k;
where
27.10.8 ak(m) =X
dj(m;k)g(d)fk
dd
k:
Another generalization of Ramanujan's sum is the
Gauss sum G(n; ) associated with a Dirichlet charac-
ter (modk). It is de ned by the relation
27.10.9 G(n; ) =kX
m=1 (m)e2imn=k:
In particular, G(n; 1) =ck(n).
G(n; ) isseparable for somenif
27.10.10 G(n; ) = (n)G(1; ):
For any Dirichlet character (modk),G(n; ) is
separable for nif (n;k) = 1, and is separable for every
nif and only if G(n; ) = 0 whenever ( n;k)>1. For a
primitive character (modk),G(n; ) is separable for
everyn, and
27.10.11 jG(1; )j2=k:
Conversely, if G(n; ) is separable for every n, then
is primitive (mod k).
The nite Fourier expansion of a primitive Dirichlet
character (modk) has the form
27.10.12 (n) =G(1; )
kkX
m=1 (m)e 2imn=k:
27.11 Asymptotic Formulas: Partial Sums
The behavior of a number-theoretic function f(n) for
largenis often dicult to determine because the func-
tion values can
uctuate considerably as nincreases. It
is more fruitful to study partial sums and seek asymp-
totic formulas of the form
27.11.1X
nxf(n) =F(x) +O(g(x));
whereF(x) is a known function of x, andO(g(x)) repre-
sents the error, a function of smaller order than F(x) for
allxin some prescribed range. For example, Dirichlet
(1849) proves that for all x1,
27.11.2X
nxd(n) =xlogx+ (2
1)x+O px
;
where
is Euler's constant ( x5.2(ii)). Dirichlet's divi-
sor problem (unsolved in 2009) is to determine the least
number0such that the error term in (27.11.2) is O
x
for all> 0. Kolesnik (1969) proves that 012
37.Equations (27.11.3){(27.11.11) list further asymp-
totic formulas related to some of the functions listed
inx27.2. They are valid for all x2. The error terms
given here are not necessarily the best known.
27.11.3X
nxd(n)
n=1
2(logx)2+ 2
logx+O(1);
where
again is Euler's constant.
27.11.4X
nx1(n) =2
12x2+O(xlogx):
27.11.5X
nx (n) =( + 1)
+ 1x +1+O
x
,
>0, 6= 1, = max(1; ).
27.11.6X
nx (n) =3
2x2+O(xlogx):
27.11.7X
nx (n)
n=6
2x+O(logx):
27.11.8X
px1
p= log logx+A+O1
logx
;
whereAis a constant.
27.11.9X
px
ph(modk)1
p=1
(k)log logx+B+O1
logx
;
where (h;k) = 1,k>0, andBis a constant depending
onhandk.
27.11.10X
pxlogp
p= logx+O(1):
27.11.11X
px
ph(modk)logp
p=1
(k)logx+O(1);
where (h;k) = 1,k>0.
Lettingx!1 in (27.11.9) or in (27.11.11) we see
that there are in nitely many primes ph(modk) if
h;kare coprime; this is Dirichlet's theorem on primes
in arithmetic progressions .
27.11.12X
nx(n) =O
xe Cplogx
,x!1;
for some positive constant C,
27.11.13 lim
x!11
xX
nx(n) = 0;
27.11.14 lim
x!1X
nx(n)
n= 0;
27.11.15 lim
x!1X
nx(n) logn
n= 1:
644 Functions of Number Theory
Each of (27.11.13){(27.11.15) is equivalent to the
prime number theorem (27.2.3). The prime num-
ber theorem for arithmetic progressions |an extension
of (27.2.3) and rst proved in de la Vall ee Poussin
(1896a,b)|states that if ( h;k) = 1, then the number
of primespxwithph(modk) is asymptotic to
x=( (k) logx) asx!1 .
27.12 Asymptotic Formulas: Primes
pnis thenth prime, beginning with p1= 2.(x) is the
number of primes less than or equal to x.
27.12.1 lim
n!1pn
nlogn= 1;
27.12.2 pn>nlogn,n= 1;2;:::.
27.12.3
(x) =bxc 1 X
pjpxx
pj
+X
r2( 1)rX
pj1<pj2<<pjrpxx
pj1pj2pjr
,
x1,
where the series terminates when the product of the rst
rprimes exceeds x.
Asx!1
27.12.4 (x)1X
k=1(k 1)!x
(logx)k:
Prime Number Theorem
There exists a positive constant csuch that
27.12.5
j(x) li(x)j=O
xexp
cp
logx
,x!1 .
For the logarithmic integral li( x) see (6.2.8). The best
available asymptotic error estimate (2009) appears in
Korobov (1958) and Vinogradov (1958): there exists a
positive constant dsuch that
27.12.6j(x) li(x)j
=O
xexp
d(logx)3=5(log logx) 1=5
:
(x) li(x) changes sign in nitely often as x!1 ;
see Littlewood (1914), Bays and Hudson (2000).
The Riemann hypothesis (x25.10(i)) is equivalent to
the statement that for every x2657,
27.12.7j(x) li(x)j<1
8pxlogx:
Ifais relatively prime to the modulus m, then there
are in nitely many primes congruent to a(modm).
The number of such primes not exceeding xis
27.12.8x
(m)+O
xexp
( )(logx)1=2
,
m(logx) , >0,where( ) depends only on , and (m) is the Euler
totient function ( x27.2).
AMersenne prime is a prime of the form 2p 1.
The largest known prime (2009) is the Mersenne prime
243;112;609 1. For current records online, see http:
//dlmf.nist.gov/27.12 .
Apseudoprime test is a test that correctly identi-
es most composite numbers. For example, if 2n62
(modn), thennis composite. Descriptions and com-
parisons of pseudoprime tests are given in Bressoud and
Wagon (2000,xx2.4, 4.2, and 8.2) and Crandall and
Pomerance (2005, xx3.4{3.6).
ACarmichael number is a composite number nfor
whichbnb(modn) for allb2N. There are in nitely
many Carmichael numbers.
Additive Number Theory
27.13 Functions
27.13(i) Introduction
Whereas multiplicative number theory is concerned
with functions arising from prime factorization, addi-
tive number theory treats functions related to addition
of integers. The basic problem is that of expressing a
given positive integer nas a sum of integers from some
prescribed set Swhose members are primes, squares,
cubes, or other special integers. Each representation of
nas a sum of elements of Sis called a partition ofn,
and the number S(n) of such partitions is often of great
interest. The subsections that follow describe problems
from additive number theory. See also Apostol (1976,
Chapter 14) and Apostol and Niven (1994, pp. 33{34).
27.13(ii) Goldbach Conjecture
Every even integer n>4is the sum of two odd primes.
In this case, S(n) is the number of solutions of the equa-
tionn=p+q, wherepandqare odd primes. Gold-
bach's assertion is that S(n)1 for all even n > 4.
This conjecture dates back to 1742 and was undecided
in 2009, although it has been con rmed numerically up
to very large numbers. Vinogradov (1937) proves that
every suciently large odd integer is the sum of three
odd primes, and Chen (1966) shows that every su-
ciently large even integer is the sum of a prime and a
number with no more than two prime factors.
For an online account of the current status of Gold-
bach's conjecture see http://dlmf.nist.gov/27.13.
ii.
27.14 Unrestricted Partitions 645
27.13(iii) Waring's Problem
This problem is named after Edward Waring who, in
1770, stated without proof and with limited numerical
evidence, that every positive integer nis the sum of four
squares, of nine cubes, of nineteen fourth powers, and
so on. Waring's problem is to nd, for each positive in-
tegerk, whether there is an integer m(depending only
onk) such that the equation
27.13.1 n=xk
1+xk
2++xk
m
has nonnegative integer solutions for all n1. The
smallestmthat exists for a given kis denoted by
g(k). Similarly, G(k) denotes the smallest mfor which
(27.13.1) has nonnegative integer solutions for all su-
ciently large n.
Lagrange (1770) proves that g(2) = 4, and during
the next 139 years the existence of g(k) was shown for
k= 3;4;5;6;7;8;10. Hilbert (1909) proves the exis-
tence ofg(k) for every kbut does not determine its
corresponding numerical value. The exact value of g(k)
is now known for every k200;000. For example,
g(3) = 9,g(4) = 19,g(5) = 37,g(6) = 73,g(7) = 143,
andg(8) = 279. A general formula states that
27.13.2 g(k)2k+3k
2k
2;
for allk2, with equality if 4 k200;000. If
3k=q2k+rwith 0< r < 2k, then equality holds
in (27.13.2) provided r+q2k, a condition that is
satis ed with at most a nite number of exceptions.
The existence of G(k) follows from that of g(k) be-
causeG(k)g(k), but only the values G(2) = 4 and
G(4) = 16 are known exactly. Some upper bounds
smaller than g(k) are known. For example, G(3)7,
G(5)23,G(6)36,G(7)53, andG(8)73.
Hardy and Littlewood (1925) conjectures that G(k)<
2k+ 1 whenkis not a power of 2, and that G(k)4k
whenkis a power of 2, but the most that is known (in
2009) isG(k)<ck logkfor some constant c. A survey
is given in Ellison (1971).
27.13(iv) Representation by Squares
For a given integer k2 the function rk(n) is de ned
as the number of solutions of the equation
27.13.3 n=x2
1+x2
2++x2
k;
where thexjare integers, positive, negative, or zero,
and the order of the summands is taken into account.
Jacobi (1829) notes that r2(n) is the coecient of
xnin the square of the theta function #(x):
27.13.4 #(x) = 1 + 21X
m=1xm2,jxj<1.(Inx20.2(i),#(x) is denoted by 3(0;x).) Thus,
27.13.5 (#(x))2= 1 +1X
n=1r2(n)xn:
One of Jacobi's identities implies that
27.13.6 (#(x))2= 1 + 41X
n=1(1(n) 3(n))xn;
where1(n) and3(n) are the number of divisors of
ncongruent respectively to 1 and 3 (mod 4), and by
equating coecients in (27.13.5) and (27.13.6) Jacobi
deduced that
27.13.7 r2(n) = 4 (1(n) 3(n)):
Hencer2(5) = 8 because both divisors, 1 and 5, are
congruent to 1 (mod 4). In fact, there are four rep-
resentations, given by 5 = 22+ 12= 22+ ( 1)2=
( 2)2+ 12= ( 2)2+ ( 1)2, and four more with the
order of summands reversed.
By similar methods Jacobi proved that r4(n) =
81(n) ifnis odd, whereas, if nis even,r4(n) = 24
times the sum of the odd divisors of n. Mordell (1917)
notes that rk(n) is the coecient of xnin the power-
series expansion of the kth power of the series for #(x).
Explicit formulas for rk(n) have been obtained by sim-
ilar methods for k= 6;8;10, and 12, but they are more
complicated. Exact formulas for rk(n) have also been
found fork= 3;5, and 7, and for all even k24.
For values of k > 24 the analysis of rk(n) is consider-
ably more complicated (see Hardy (1940)). Also, Milne
(1996, 2002) announce new in nite families of explicit
formulas extending Jacobi's identities. For more than
8 squares, Milne's identities are not the same as those
obtained earlier by Mordell and others.
27.14 Unrestricted Partitions
27.14(i) Partition Functions
A fundamental problem studies the number of ways n
can be written as a sum of positive integers n, that
is, the number of solutions of
27.14.1 n=a1+a2+,a1a2 1.
The number of summands is unrestricted, repetition is
allowed, and the order of the summands is not taken into
account. The corresponding unrestricted partition func-
tion is denoted by p(n), and the summands are called
parts ; seex26.9(i). For example, p(5) = 7 because there
are exactly seven partitions of 5: 5 = 4 + 1 = 3 + 2 =
3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1.
The number of partitions of ninto at most kparts
is denoted by pk(n); again seex26.9(i).
646 Functions of Number Theory
27.14(ii) Generating Functions and Recursions
Euler introduced the reciprocal of the in nite product
27.14.2 f(x) =1Y
m=1(1 xm),jxj<1;
as a generating function for the function p(n) de ned in
x27.14(i):
27.14.31
f(x)=1X
n=0p(n)xn;
withp(0) = 1. Euler's pentagonal number theorem
states that
27.14.4f(x) = 1 x x2+x5+x7 x12 x15+
= 1 +1X
k=1( 1)k
x!(k)+x!( k)
;
where the exponents 1, 2, 5, 7, 12, 15, :::are the pen-
tagonal numbers , de ned by
27.14.5 !(k) = (3k2k)=2,k= 1;2;3;:::.
Multiplying the power series for f(x) with that for
1=f(x) and equating coecients, we obtain the recur-
sion formula
27.14.6
p(n) =1X
k=1( 1)k+1(p(n !(k)) +p(n !( k)))
=p(n 1) +p(n 2) p(n 5) p(n 7) +;
wherep(k) is de ned to be 0 if k<0. Logarithmic dif-
ferentiation of the generating function 1 =f(x) leads to
another recursion:
27.14.7 np(n) =nX
k=11(n)p(n k);
where1(n) is de ned by (27.2.10) with = 1.
27.14(iii) Asymptotic Formulas
These recursions can be used to calculate p(n), which
grows very rapidly. For example, p(10) = 42;p(100) =
1905 69292, and p(200) = 397 29990 29388. For large n
27.14.8 p(n)eKpn=(4np
3);
whereK=p
2=3 (Hardy and Ramanujan (1918)).
Rademacher (1938) derives a convergent series that also
provides an asymptotic expansion for p(n):
27.14.9
p(n)
=1
p
21X
k=1p
kAk(n)"
d
dtsinh
Kp
t
k
p
t#
t=n (1=24);
where
27.14.10Ak(n) =kX
h=1
(h;k)=1exp
is(h;k) 2inh
k
;ands(h;k) is a Dedekind sum given by
27.14.11s(h;k) =k 1X
r=1r
khr
k hr
k
1
2
:
27.14(iv) Relation to Modular Functions
Dedekind sums occur in the transformation theory of
theDedekind modular function ( ), de ned by
27.14.12 ( ) =ei =121Y
n=1(1 e2in ),= >0.
This is related to the function f(x) in (27.14.2) by
27.14.13 ( ) =ei =12f
e2i
:
( ) satis es the following functional equation: if
a;b;c;d are integers with ad bc= 1 andc>0, then
27.14.14a +b
c +d
="( i(c +d))1
2( );
where"= exp(i(((a+d)=(12c)) s(d;c))) ands(d;c)
is given by (27.14.11).
For further properties of the function ( ) see
xx23.15{23.19.
27.14(v) Divisibility Properties
Ramanujan (1921) gives identities that imply divisibil-
ity properties of the partition function. For example,
the Ramanujan identity
27.14.15 5(f
x5
)5
(f(x))6=1X
n=0p(5n+ 4)xn
impliesp(5n+ 4)0 (mod 5). Ramanujan also found
thatp(7n+ 5)0 (mod 7) and p(11n+ 6)0
(mod 11) for all n. After decades of nearly fruitless
searching for further congruences of this type, it was
believed that no others existed, until it was shown in
Ono (2000) that there are in nitely many. Ono proved
that for every prime q > 3 there are integers aandb
such thatp(an+b)0 (modq) for alln. For example,
p(1575 25693 n+ 1 11247)0 (mod 13).
27.14(vi) Ramanujan's Tau Function
The discriminant function ( ) is de ned by
27.14.16 ( ) = (2)12(( ))24,= >0;
and satis es the functional equation
27.14.17 a +b
c +d
= (c +d)12( );
ifa;b;c;d are integers with ad bc= 1 andc>0.
The 24th power of ( ) in (27.14.12) with e2i =x
is an in nite product that generates a power series in
Applications 647
xwith integer coecients called Ramanujan's tau func-
tion (n):
27.14.18 x1Y
n=1(1 xn)24=1X
n=1 (n)xn,jxj<1.
The tau function is multiplicative and satis es the more
general relation:
27.14.19
(m) (n) =X
dj(m;n)d11 mn
d2
,m;n = 1;2;:::.
Lehmer (1947) conjectures that (n) is never 0 and ver-
i es this for all n<21 49286 39999 by studying various
congruences satis ed by (n), for example:
27.14.20 (n)11(n) (mod 691) :
For further information on partitions and generating
functions see Andrews (1976); also xx17.2{17.14, and
xx26.9{26.10.
Applications
27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of
congruences xa1(modm1);:::;xak(modmk),
always has a solution if the moduli are relatively prime
in pairs; the solution is unique (mod m), wheremis the
product of the moduli.
This theorem is employed to increase eciency in
calculating with large numbers by making use of smaller
numbers in most of the calculation. For example, sup-
pose a lengthy calculation involves many 10-digit inte-
gers. Most of the calculation can be done with ve-digit
integers as follows. Choose four relatively prime mod-
ulim1;m2;m3, andm4of ve digits each, for example
216 3, 216 1, 216+ 1, and 216+ 3. Their prod-
uctmhas 20 digits, twice the number of digits in the
data. By the Chinese remainder theorem each integer
in the data can be uniquely represented by its residues
(modm1), (modm2), (modm3), and (mod m4), re-
spectively. Because each residue has no more than ve
digits, the arithmetic can be performed eciently on
these residues with respect to each of the moduli, yield-
ing answers a1(modm1),a2(modm2),a3(modm3),
anda4(modm4), where each ajhas no more than ve
digits. These numbers, in turn, are combined by the
Chinese remainder theorem to obtain the nal result
(modm), which is correct to 20 digits.
Even though the lengthy calculation is repeated four
times, once for each modulus, most of it only uses ve-
digit integers and is accomplished quickly without over-
whelming the machine's memory. Details of a machineprogram describing the method together with typical
numerical results can be found in Newman (1967). See
also Apostol and Niven (1994, pp. 18{19).
27.16 Cryptography
Applications to cryptography rely on the disparity in
computer time required to nd large primes and to fac-
tor large integers.
For example, a code maker chooses two large primes
pandqof about 100 decimal digits each. Procedures
for nding such primes require very little computer time.
The primes are kept secret but their product n=pq, a
200-digit number, is made public. For this reason, these
are often called public key codes. Messages are coded
by a method (described below) that requires only the
knowledge of n. But to decode, both factors pandq
must be known. With the most ecient computer tech-
niques devised to date (2009), factoring a 200-digit num-
ber may require billions of years on a single computer.
For this reason, the codes are considered unbreakable,
at least with the current state of knowledge on factoring
large numbers.
To code a message by this method, we replace each
letter by two digits, say A= 01,B= 02,:::,Z= 26,
and divide the message into pieces of convenient length
smaller than the public value n=pq. Choose a prime
rthat does not divide either p 1 orq 1. Liken, the
primeris made public. To code a piece x, raisexto the
powerrand reduce xrmodulonto obtain an integer y
(the coded form of x) between 1 and n. Thus,yxr
(modn) and 1y<n .
To decode, we must recover xfromy. To do this,
letsdenote the reciprocal of rmodulo (n), so that
rs= 1 +t (n) for some integer t. (Here (n) is Euler's
totient (x27.2).) By the Euler{Fermat theorem (27.2.8),
x (n)1 (modn); hencext (n)1 (modn). But
ysxrsx1+t (n)x(modn), soysis the same
asxmodulon. In other words, to recover xfromywe
simply raise yto the power sand reduce modulo n. Ifp
andqare known, sandyscan be determined (mod n)
by straightforward calculations that require only a few
minutes of machine time. But if pandqare not known,
the problem of recovering xfromyseems insurmount-
able.
For further information see Apostol and Niven
(1994, p. 24), and for other applications to cryptography
see Menezes et al. (1997) and Schroeder (2006).
27.17 Other Applications
Reed et al. (1990, pp. 458{470) describes a number-
theoretic approach to Fourier analysis (called the arith-
metic Fourier transform ) that uses the M obius inversion
648 Functions of Number Theory
(27.5.7) to increase eciency in computing coecients
of Fourier series.
Congruences are used in constructing perpetual cal-
endars, splicing telephone cables, scheduling round-
robin tournaments, devising systematic methods for
storing computer les, and generating pseudorandom
numbers. Rosen (2004, Chapters 5 and 10) describes
many of these applications. Apostol and Zuckerman
(1951) uses congruences to construct magic squares.
There are also applications of number theory in
many diverse areas, including physics, biology, chem-
istry, communications, and art. Schroeder (2006) de-
scribes many of these applications, including the de-
sign of concert hall ceilings to scatter sound into broad
lateral patterns for improved acoustic quality, precise
measurements of delays of radar echoes from Venus and
Mercury to con rm one of the relativistic e ects pre-
dicted by Einstein's theory of general relativity, and the
use of primes in creating artistic graphical designs.
Computation
27.18 Methods of Computation: Primes
An overview of methods for precise counting of the
number of primes not exceeding an arbitrary integer
xis given in Crandall and Pomerance (2005, x3.7).
T. Oliveira e Silva has calculated (x) forx= 1023, us-
ing the combinatorial methods of Lagarias et al. (1985)
and Del eglise and Rivat (1996); see Oliveira e Silva
(2006). An analytic approach using a contour integral
of the Riemann zeta function ( x25.2(i)) is discussed in
Borwein et al. (2000).
The Sieve of Eratosthenes (Crandall and Pomerance
(2005,x3.2)) generates a list of all primes below a given
bound. An alternative procedure is the binary quadratic
sieve of Atkin and Bernstein (Crandall and Pomerance
(2005, p. 170)).
For small values of n, primality is proven by showing
thatnis not divisible by any prime not exceedingpn.
Two simple algorithms for proving primality re-
quire a knowledge of all or part of the factorization
ofn 1;n+ 1, or both; see Crandall and Pomerance
(2005,xx4.1{4.2). These algorithms are used for test-
ing primality of Mersenne numbers , 2n 1, and Fermat
numbers , 22n+ 1.
The APR (Adleman{Pomerance{Rumely) algorithm
for primality testing is based on Jacobi sums. It runs in
timeO
(logn)clog log logn
. Explanations are given in
Cohen (1993,x9.1) and Crandall and Pomerance (2005,
x4.4). A practical version is described in Bosma and
van der Hulst (1990).The AKS (Agrawal{Kayal{Saxena) algorithm is the
rst deterministic, polynomial-time, primality test.
That is to say, it runs in time O((logn)c) for some
constantc. An explanation is given in Crandall and
Pomerance (2005, x4.5).
The ECPP (Elliptic Curve Primality Proving) algo-
rithm handles primes with over 20,000 digits. Explana-
tions are given in Cohen (1993, x9.2) and Crandall and
Pomerance (2005, x7.6).
27.19 Methods of Computation:
Factorization
Techniques for factorization of integers fall into three
general classes: Deterministic algorithms ,Type I prob-
abilistic algorithms whose expected running time de-
pends on the size of the smallest prime factor, and
Type II probabilistic algorithms whose expected running
time depends on the size of the number to be factored.
Deterministic algorithms are slow but are guaran-
teed to nd the factorization within a known period of
time. Trial division is one example. Fermat's algorithm
is another; see Bressoud (1989, x5.1).
Type I probabilistic algorithms include the Brent{
Pollard rho algorithm (also called Monte Carlo method ),
the Pollardp 1algorithm , and the Elliptic Curve
Method (ecm). Descriptions of these algorithms are
given in Crandall and Pomerance (2005, xx5.2, 5.4, and
7.4). As of January 2009 the largest prime factors found
by these methods are a 19-digit prime for Brent{Pollard
rho, a 58-digit prime for Pollard p 1, and a 67-digit
prime for ecm.
Type II probabilistic algorithms for factoring nrely
on nding a pseudo-random pair of integers ( x;y) that
satisfyx2y2(modn). These algorithms include the
Continued Fraction Algorithm (cfrac ), the Multiple
Polynomial Quadratic Sieve (mpqs ), the General Num-
ber Field Sieve (gnfs ), and the Special Number Field
Sieve (snfs). A description of cfrac is given in Bres-
soud and Wagon (2000). Descriptions of mpqs ,gnfs ,
andsnfs are given in Crandall and Pomerance (2005,
xx6.1 and 6.2). As of January 2009 the snfs holds the
record for the largest integer that has been factored by
a Type II probabilistic algorithm, a 307-digit compos-
ite integer. The snfs can be applied only to numbers
that are very close to a power of a very small base.
The largest composite numbers that have been factored
by other Type II probabilistic algorithms are a 63-digit
integer by cfrac , a 135-digit integer by mpqs , and a
182-digit integer by gnfs .
For further information see Crandall and Pomerance
(2005) andx26.22.
For current records online, see http://dlmf.nist.
gov/27.19 .
27.20 Methods of Computation: Other Number-Theoretic Functions 649
27.20 Methods of Computation: Other
Number-Theoretic Functions
To calculate a multiplicative function it suces to de-
termine its values at the prime powers and then use
(27.3.2). For a completely multiplicative function we
use the values at the primes together with (27.3.10).
The recursion formulas (27.14.6) and (27.14.7) can be
used to calculate the partition function p(n). A similar
recursion formula obtained by di erentiating (27.14.18)
can be used to calculate Ramanujan's function (n), and
the values can be checked by the congruence (27.14.20).
For further information see Lehmer (1941, pp. 5{83)
and Lehmer (1943, pp. 483{492).
27.21 Tables
Lehmer (1914) lists all primes up to 100 06721. Bres-
soud and Wagon (2000, pp. 103{104) supplies tables and
graphs that compare (x);x/logx, and li(x). Glaisher
(1940) contains four tables: Table I tabulates, for all
n104: (a) the canonical factorization of ninto pow-
ers of primes; (b) the Euler totient (n); (c) the divisor
functiond(n); (d) the sum (n) of these divisors. Ta-
ble II lists all solutions nof the equation f(n) =m
for allm2500, where f(n) is de ned by (27.14.2).
Table III lists all solutions n104of the equation
d(n) =m, and Table IV lists all solutions nof the
equation(n) =mfor allm104. Table 24.7 of
Abramowitz and Stegun (1964) also lists the factoriza-
tions in Glaisher's Table I(a); Table 24.6 lists (n);d(n),
and(n) forn1000; Table 24.8 gives examples of
primitive roots of all primes 9973; Table 24.9 lists all
primes that are less than 1 00000.
The partition function p(n) is tabulated in Gupta
(1935, 1937), Watson (1937), and Gupta et al. (1958).
Tables of the Ramanujan function (n) are published
in Lehmer (1943) and Watson (1949). Lehmer (1941)
gives a comprehensive account of tables in the theory of
numbers, including virtually every table published from
1918 to 1941. Those published prior to 1918 are men-
tioned in Dickson (1919). The bibliography in Lehmer
(1941) gives references to the places in Dickson's His-
tory where the older tables are cited. Lehmer (1941)
also has a section that supplies errata and corrections
to all tables cited.
No sequel to Lehmer (1941) exists to date, but many
tables of functions of number theory are included in Un-
published Mathematical Tables (1944).
27.22 Software
Seehttp://dlmf.nist.gov/27.22 .References
General References
The main references used in writing this chapter are
Apostol (1976, 1990), and Apostol and Niven (1994).
Further information can be found in Andrews (1976),
Erd elyi et al. (1955, Chapter XVII), Hardy and Wright
(1979), and Niven et al. (1991).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references quoted in the text.
x27.2 Apostol (1976, Chapter 2). For (27.2.11) see
Erd elyi et al. (1955, p. 168). Tables 27.2.1 and
27.2.2 are from Abramowitz and Stegun (1964,
Tables 24.6 and 24.9).
x27.3 Apostol (1976, Chapter 2).
x27.4 Apostol (1976, Chapter 11). For (27.4.10) see
Titchmarsh (1986b, p. 4).
x27.5 Apostol (1976, Chapter 2 and p. 228). For
(27.5.7) use (27.5.2) and formal substitution.
x27.6 Apostol (1976, Chapter 2).
x27.7 Apostol (1990, Chapter 1).
x27.8 Apostol (1976, Chapter 6).
x27.9 Apostol (1976, Chapter 9).
x27.10 Apostol (1976, Chapter 8).
x27.11 Apostol (1976, Chapters 3, 4). For (27.11.12),
(27.11.14), and (27.11.15) see Prachar (1957,
pp. 71{74).
x27.12 Crandall and Pomerance (2005, pp. 131{152),
Davenport (2000), Narkiewicz (2000), Rosser
(1939). For (27.12.7) see Schoenfeld (1976) and
Crandall and Pomerance (2005, pp. 37, 60).
For the proof that there are in nitely many
Carmichael numbers see Alford et al. (1994).
x27.13 Apostol (1976, Chapter 14), Ellison (1971),
Grosswald (1985, pp. 8, 32). For (27.13.4) see
(20.2.3).
x27.14(ii) Apostol (1976, Chapter 14), Apostol (1990,
Chapters 3{5).
Chapter 28
Mathieu Functions and Hill's Equation
G. Wolf1
Notation 652
28.1 Special Notation . . . . . . . . . . . . . 652
Mathieu Functions of Integer Order 652
28.2 De nitions and Basic Properties . . . . . 652
28.3 Graphics . . . . . . . . . . . . . . . . . . 655
28.4 Fourier Series . . . . . . . . . . . . . . . 656
28.5 Second Solutions fen,gen. . . . . . . . 657
28.6 Expansions for Small q. . . . . . . . . . 659
28.7 Analytic Continuation of Eigenvalues . . . 661
28.8 Asymptotic Expansions for Large q. . . . 661
28.9 Zeros . . . . . . . . . . . . . . . . . . . 663
28.10 Integral Equations . . . . . . . . . . . . . 663
28.11 Expansions in Series of Mathieu Functions 664
Mathieu Functions of Noninteger Order 664
28.12 De nitions and Basic Properties . . . . . 664
28.13 Graphics . . . . . . . . . . . . . . . . . . 665
28.14 Fourier Series . . . . . . . . . . . . . . . 666
28.15 Expansions for Small q. . . . . . . . . . 666
28.16 Asymptotic Expansions for Large q. . . . 666
28.17 Stability as x!1 . . . . . . . . . . . 667
28.18 Integrals and Integral Equations . . . . . 667
28.19 Expansions in Series of me+2nFunctions 667
Modi ed Mathieu Functions 667
28.20 De nitions and Basic Properties . . . . . 66728.21 Graphics . . . . . . . . . . . . . . . . . . 669
28.22 Connection Formulas . . . . . . . . . . . 669
28.23 Expansions in Series of Bessel Functions . 670
28.24 Expansions in Series of Cross-Products of
Bessel Functions or Modi ed Bessel Func-
tions . . . . . . . . . . . . . . . . . . . . 671
28.25 Asymptotic Expansions for Large <z. . . 672
28.26 Asymptotic Approximations for Large q. 672
28.27 Addition Theorems . . . . . . . . . . . . 672
28.28 Integrals, Integral Representations, and In-
tegral Equations . . . . . . . . . . . . . . 672
Hill's Equation 674
28.29 De nitions and Basic Properties . . . . . 674
28.30 Expansions in Series of Eigenfunctions . . 676
28.31 Equations of Whittaker{Hill and Ince . . 676
Applications 677
28.32 Mathematical Applications . . . . . . . . 677
28.33 Physical Applications . . . . . . . . . . . 678
Computation 679
28.34 Methods of Computation . . . . . . . . . 679
28.35 Tables . . . . . . . . . . . . . . . . . . . 680
28.36 Software . . . . . . . . . . . . . . . . . . 681
References 681
1Fachbereich Mathematik, University Duisburg-Essen, Essen, Germany.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 20) by G. Blanch.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
651
652 Mathieu Functions and Hill's Equation
Notation
28.1 Special Notation
(For other notation see pp. xiv and 873.)
m;n integers.
x;y real variables.
z=x+iycomplex variable.
order of the Mathieu function or modi ed
Mathieu function. (When is an integer
it is often replaced by n.)
arbitrary small positive number.
a;q;h real or complex parameters of Mathieu's
equation with q=h2.
primes unless indicated otherwise, derivatives
with respect to the argument
The main functions treated in this chapter are the
Mathieu functions
ce(z;q), se(z;q), fen(z;q), gen(z;q), me(z;q),
and the modi ed Mathieu functions
Ce(z;q), Se(z;q), Fen(z;q), Gen(z;q),
Me(z;q), M(j)
(z;h), Mc(j)
n(z;h), Ms(j)
n(z;h),
Ien(z;h), Ion(z;h), Ken(z;h), Kon(z;h).
The functions Mc(j)
n(z;h) and Ms(j)
n(z;h) are also known
as the radial Mathieu functions.
The eigenvalues of Mathieu's equation are denoted
by
an(q); bn(q); (q):
The notation for the joining factors is
ge;n(h); g o;n(h); f e;n(h); f o;n(h):
Alternative notations for the parameters aandqare
shown in Table 28.1.1.
Table 28.1.1 : Notations for parameters in Mathieu's
equation.
Reference a q
Erd elyi et al. (1955) h
Meixner and Sch afke (1954) h2
Moon and Spencer (1971) q
Strutt (1932) h2
Whittaker and Watson (1927) a8q
Alternative notations for the functions are as follows.Arscott (1964b) and McLachlan (1947)
Feyn(z;q) =q
1
2ge;n(h) cen(0;q) Mc(2)
n(z;h);
Me(1;2)
n(z;q) =q
1
2ge;n(h) cen(0;q) Mc(3;4)
n(z;h);
Geyn(z;q) =q
1
2go;n(h) se0
n(0;q) Ms(2)
n(z;h);
Ne(1;2)
n(z;q) =q
1
2go;n(h) se0
n(0;q) Ms(3;4)
n(z;h):
Arscott (1964b) also uses ifor.
Campbell (1955)
inn= fen; cehn= Cen; inhn= Fen;
jnn= gen; sehn= Sen; jnhn= Gen:
Abramowitz and Stegun (1964, Chapter 20)
F(z) = Me(z;q):
NBS (1967)
Withs= 4q,
Sen(s;z) =cen(z;q)
cen(0;q);Son(s;z) =sen(z;q)
se0n(0;q):
Stratton et al. (1941)
Withc= 2pq,
Sen(c;z) =cen(z;q)
cen(0;q);Son(c;z) =sen(z;q)
se0n(0;q):
Zhang and Jin (1996)
The radial functions Mc(j)
n(z;h) and Ms(j)
n(z;h) are de-
noted by Mc(j)
n(z;q) and Ms(j)
n(z;q), respectively.
Mathieu Functions of Integer Order
28.2 De nitions and Basic Properties
28.2(i) Mathieu's Equation
The standard form of Mathieu's equation with param-
eters (a;q) is
28.2.1 w00+ (a 2qcos(2z))w= 0:
With= sin2zwe obtain the algebraic form of Math-
ieu's equation
28.2.2
(1 )w00+1
2
1 2)w0+1
4(a 2q(1 2)
w= 0:
This equation has regular singularities at 0 and 1, both
with exponents 0 and1
2, and an irregular singular point
at1. With= coszwe obtain another algebraic form:
28.2.3 (1 2)w00 w0+
a+ 2q 4q2
w= 0:
28.2 Definitions and Basic Properties 653
28.2(ii) Basic Solutions wI,wII
Since (28.2.1) has no nite singularities its solutions are
entire functions of z. Furthermore, a solution wwith
given initial constant values of wandw0at a pointz0
is an entire function of the three variables z,a, andq.
The following three transformations
28.2.4z! z;z!z;z!z1
2;q! q;
each leave (28.2.1) unchanged. (28.2.1) possesses a fun-
damental pair of solutions wI(z;a;q);wII(z;a;q) called
basic solutions with
28.2.5wI(0;a;q)wII(0;a;q)
w0
I(0;a;q)w0
II(0;a;q)
=1 0
0 1
:
wI(z;a;q) is even and wII(z;a;q) is odd. Other proper-
ties are as follows.
28.2.6 WfwI;wIIg= 1;
28.2.7wI(z;a;q) =wI(;a;q)wI(z;a;q)
w0
I(;a;q)wII(z;a;q);
28.2.8wII(z;a;q) =wII(;a;q)wI(z;a;q)
+w0
II(;a;q)wII(z;a;q);
28.2.9 wI(;a;q) =w0
II(;a;q);
28.2.10wI(;a;q) 1 = 2w0
I(1
2;a;q)wII(1
2;a;q);
28.2.11wI(;a;q) + 1 = 2wI(1
2;a;q)w0
II(1
2;a;q);
28.2.12 w0
I(;a;q) = 2wI(1
2;a;q)w0
I(1
2;a;q);
28.2.13 wII(;a;q) = 2wII(1
2;a;q)w0
II(1
2;a;q):
28.2(iii) Floquet's Theorem and the
Characteristic Exponents
Letbe any real or complex constant. Then Mathieu's
equation (28.2.1) has a nontrivial solution w(z) such
that
28.2.14 w(z+) =eiw(z);
i eiis an eigenvalue of the matrix
28.2.15wI(;a;q)wII(;a;q)
w0
I(;a;q)w0
II(;a;q)
:
Equivalently,
28.2.16 cos() =wI(;a;q) =wI(;a; q):
This is the characteristic equation of Mathieu's equa-
tion (28.2.1). cos( ) is an entire function of
a;q2. The solutions of (28.2.16) are given by =
1arccos(wI(;a;q)). If the inverse cosine takes its
principal value (x4.23(ii)), then =b, where 0<b
1. The general solution of (28.2.16) is =b+ 2n,
wheren2Z. Eitherboris called a characteristic ex-
ponent of (28.2.1). If b= 0 or 1, or equivalently, =n,
thenis a double root of the characteristic equation,
otherwise it is a simple root.28.2(iv) Floquet Solutions
A solution with the pseudoperiodic property (28.2.14)
is called a Floquet solution with respect to . (28.2.9),
(28.2.16), and (28.2.7) give for each solution w(z) of
(28.2.1) the connection formula
28.2.17w(z+) +w(z ) = 2 cos()w(z):
Therefore a nontrivial solution w(z) is either a Floquet
solution with respect to , orw(z+) eiw(z) is a
Floquet solution with respect to .
Ifq6= 0, then for a given value of the correspond-
ing Floquet solution is unique, except for an arbitrary
constant factor (Theorem of Ince; see also 28.5(i)).
The Fourier series of a Floquet solution
28.2.18 w(z) =1X
n= 1c2nei(+2n)z
converges absolutely and uniformly in compact subsets
ofC. The coecients c2nsatisfy
28.2.19
qc2n+2
a (+ 2n)2
c2n+qc2n 2= 0,n2Z.
Conversely, a nontrivial solution c2nof (28.2.19) that
satis es
28.2.20 lim
n!1jc2nj1=jnj= 0
leads to a Floquet solution.
28.2(v) Eigenvalues an,bn
For givenandq, equation (28.2.16) determines an in-
nite discrete set of values of a, the eigenvalues orchar-
acteristic values , of Mathieu's equation. When b= 0
or 1, the notation for the two sets of eigenvalues corre-
sponding to each bis shown in Table 28.2.1, together
with the boundary conditions of the associated eigen-
value problem. In Table 28.2.1 n= 0;1;2;:::.
Table 28.2.1 : Eigenvalues of Mathieu's equation.
bBoundary Conditions Eigenvalues
0w0(0) =w0(1
2) = 0 a2n(q)
1w0(0) =w(1
2) = 0a2n+1(q)
1w(0) =w0(1
2) = 0b2n+1(q)
0w(0) =w(1
2) = 0b2n+2(q)
An equivalent formulation is given by
28.2.21w0
I(1
2;a;q) = 0; a =a2n(q);
wI(1
2;a;q) = 0; a =a2n+1(q);
and
28.2.22w0
II(1
2;a;q) = 0; a =b2n+1(q);
wII(1
2;a;q) = 0; a =b2n+2(q);
654 Mathieu Functions and Hill's Equation
wheren= 0;1;2;:::. Whenq= 0,
28.2.23 an(0) =n2, n= 0;1;2;:::,
28.2.24 bn(0) =n2, n= 1;2;3;:::.
Nearq= 0,an(q) andbn(q) can be expanded in power
series inq(seex28.6(i)); elsewhere they are determined
by analytic continuation (see x28.7). For nonnegative
real values of q, see Figure 28.2.1.
Figure 28.2.1 : Eigenvalues an(q),bn(q) of Mathieu's
equation as functions of qfor 0q10,n= 0;1;2;3;4
(a's),n= 1;2;3;4 (b's).
Distribution
28.2.25
forq>0:a0<b1<a1<b2<a2<b3<;
forq<0:a0<a1<b1<b2<a2<a3<:
Change of Sign of q
28.2.26 a2n( q) =a2n(q);
28.2.27 a2n+1( q) =b2n+1(q);
28.2.28 b2n+2( q) =b2n+2(q):
28.2(vi) Eigenfunctions
Table 28.2.2 gives the notation for the eigenfunctions
corresponding to the eigenvalues in Table 28.2.1. Pe-
riodmeans that the eigenfunction has the property
w(z+) =w(z), whereas antiperiod means that
w(z+) = w(z).Even parity meansw( z) =w(z),
and odd parity meansw( z) = w(z).Table 28.2.2 : Eigenfunctions of Mathieu's equation.
Eigenvalues Eigenfunctions Periodicity Parity
a2n(q) ce 2n(z;q) Period Even
a2n+1(q) ce 2n+1(z;q) Antiperiod Even
b2n+1(q) se 2n+1(z;q) Antiperiod Odd
b2n+2(q) se 2n+2(z;q) Period Odd
Whenq= 0,
28.2.29ce0(z;0) = 1=p
2;cen(z;0) = cos(nz);
sen(z;0) = sin(nz), n= 1;2;3;:::.
For simple roots qof the corresponding equations
(28.2.21) and (28.2.22), the functions are made unique
by the normalizations
28.2.30Z2
0(cen(x;q))2dx=;Z2
0(sen(x;q))2dx=;
the ambiguity of sign being resolved by (28.2.29) when
q= 0 and by continuity for the other values of q.
The functions are orthogonal, that is,
28.2.31Z2
0cem(x;q) cen(x;q)dx= 0, n6=m,
28.2.32Z2
0sem(x;q) sen(x;q)dx= 0, n6=m,
28.2.33Z2
0cem(x;q) sen(x;q)dx= 0:
For change of sign of q(compare (28.2.4))
28.2.34 ce2n(z; q) = ( 1)nce2n 1
2 z;q
;
28.2.35 ce2n+1(z; q) = ( 1)nse2n+1 1
2 z;q
;
28.2.36 se2n+1(z; q) = ( 1)nce2n+1 1
2 z;q
;
28.2.37 se2n+2(z; q) = ( 1)nse2n+2 1
2 z;q
:
For the connection with the basic solutions in
x28.2(ii),
28.2.38cen(z;q)
cen(0;q)=wI(z;an(q);q),n= 0;1;:::,
28.2.39sen(z;q)
se0n(0;q)=wII(z;bn(q);q),n= 1;2;:::.
28.3 Graphics 655
28.3 Graphics
28.3(i) Line Graphs: Mathieu Functions with Fixed qand Variable x
Even-Periodic Solutions
Figure 28.3.1 : ce2n(x;1) for 0x=2,n= 0;1;2;3.
Figure 28.3.2 : ce2n(x;10) for 0x=2,n= 0;1;2;3.
Even-Antiperiodic Solutions
Figure 28.3.3 : ce 2n+1(x;1) for 0x=2,n=
0;1;2;3.
Figure 28.3.4 : ce 2n+1(x;10) for 0x=2,n=
0;1;2;3.
Odd-Antiperiodic Solutions
Figure 28.3.5 : se 2n+1(x;1) for 0x=2,n=
0;1;2;3.
Figure 28.3.6 : se 2n+1(x;10) for 0x=2,n=
0;1;2;3.
656 Mathieu Functions and Hill's Equation
Odd-Periodic Solutions
Figure 28.3.7 : se2n(x;1) for 0x=2,n= 1;2;3;4.
Figure 28.3.8 : se2n(x;10) for 0x=2,n= 1;2;3;4.
For further graphs see Jahnke et al. (1966, pp. 264{265 and 268{275).
28.3(ii) Surfaces: Mathieu Functions with Variable xandq
Figure 28.3.9 : ce0(x;q) for 0x2, 0q10.
Figure 28.3.10 : se1(x;q) for 0x2, 0q10.
For further graphics see http://dlmf.nist.gov/28.3.ii .
28.4 Fourier Series
28.4(i) De nitions
The Fourier series of the periodic Mathieu functions
converge absolutely and uniformly on all compact sets
in thez-plane. For n= 0;1;2;3;:::,
28.4.1 ce2n(z;q) =1X
m=0A2n
2m(q) cos 2mz;
28.4.2 ce2n+1(z;q) =1X
m=0A2n+1
2m+1(q) cos (2m+ 1)z;
28.4.3 se2n+1(z;q) =1X
m=0B2n+1
2m+1(q) sin (2m+ 1)z;
28.4.4 se2n+2(z;q) =1X
m=0B2n+2
2m+2(q) sin (2m+ 2)z:28.4(ii) Recurrence Relations
28.4.5aA0 qA2= 0;(a 4)A2 q(2A0+A4) = 0;
(a 4m2)A2m q(A2m 2+A2m+2) = 0,
m= 2;3;4;:::,a=a2n(q),A2m=A2n
2m(q).
28.4.6
(a 1 q)A1 qA3= 0;
a (2m+ 1)2
A2m+1 q(A2m 1+A2m+3) = 0,
m= 1;2;3;:::,a=a2n+1(q),A2m+1=A2n+1
2m+1(q).
28.4.7
(a 1 +q)B1 qB3= 0;
a (2m+ 1)2
B2m+1 q(B2m 1+B2m+3) = 0,
m= 1;2;3;:::,a=b2n+1(q),B2m+1=B2n+1
2m+1(q).
28.5 Second Solutions fen,gen 657
28.4.8
(a 4)B2 qB4= 0;
(a 4m2)B2m q(B2m 2+B2m+2) = 0,
m= 2;3;4;:::,a=b2n+2(q),B2m+2=B2n+2
2m+2(q):
28.4(iii) Normalization
28.4.9 2
A2n
0(q)2+1X
m=1
A2n
2m(q)2= 1;
28.4.101X
m=0
A2n+1
2m+1(q)2= 1;
28.4.111X
m=0
B2n+1
2m+1(q)2= 1;
28.4.121X
m=0
B2n+2
2m+2(q)2= 1:
Ambiguities in sign are resolved by (28.4.13){(28.4.16)
whenq= 0, and by continuity for the other values of q.
28.4(iv) Case q= 0
28.4.13A0
0(0) = 1=p
2; A2n
2n(0) = 1, n>0,
A2n
2m(0) = 0, n6=m,
28.4.14 A2n+1
2n+1(0) = 1; A2n+1
2m+1(0) = 0, n6=m,
28.4.15 B2n+1
2n+1(0) = 1; B2n+1
2m+1(0) = 0, n6=m,
28.4.16 B2n+2
2n+2(0) = 1; B2n+2
2m+2(0) = 0, n6=m.
28.4(v) Change of Sign of q
28.4.17 A2n
2m( q) = ( 1)n mA2n
2m(q);
28.4.18B2n+2
2m+2( q) = ( 1)n mB2n+2
2m+2(q);
28.4.19A2n+1
2m+1( q) = ( 1)n mB2n+1
2m+1(q);
28.4.20B2n+1
2m+1( q) = ( 1)n mA2n+1
2m+1(q):
28.4(vi) Behavior for Small q
For xeds= 1;2;3;::: and xedm= 1;2;3;:::,
28.4.21A0
2s(q) =( 1)s2
(s!)2q
4s
+O
qs+2
A0
0(q);
28.4.22
Am
m+2s(q)
Bm
m+2s(q)
=( 1)sm!
s!(m+s)!q
4s
+O
qs+1Am
m(q);
Bm
m(q);
28.4.23
Am
m 2s(q)
Bm
m 2s(q)
=(m s 1)!
s!(m 1)!q
4s
+O
qs+1Am
m(q);
Bm
m(q):
For further terms and expansions see Meixner and
Sch afke (1954, p. 122) and McLachlan (1947, x3.33).28.4(vii) Asymptotic Forms for Large m
Asm!1 , with xed q(6= 0) and xed n,
28.4.24
A2n
2m(q)
A2n
0(q)=( 1)m
(m!)2q
4m
1 +O
m 1
wII(1
2;a2n(q);q);
28.4.25
A2n+1
2m+1(q)
A2n+1
1(q)=( 1)m+1
1
2
m+12q
4m+12
1 +O
m 1
w0
II(1
2;a2n+1(q);q);
28.4.26
B2n+1
2m+1(q)
B2n+1
1(q)=( 1)m
1
2
m+12q
4m+12
1 +O
m 1
wI(1
2;b2n+1(q);q);
28.4.27
B2n+2
2m(q)
B2n+2
2(q)=( 1)m
(m!)2q
4mq
1 +O
m 1
w0
I(1
2;b2n+2(q);q):
For the basic solutions wIandwIIseex28.2(ii).
28.5 Second Solutions fen,gen
28.5(i) De nitions
Theorem of Ince (1922)
If a nontrivial solution of Mathieu's equation with q6= 0
has periodor 2, then any linearly independent solu-
tion cannot have either period.
Second solutions of (28.2.1) are given by
28.5.1 fen(z;q) =Cn(q) (zcen(z;q) +fn(z;q));
whena=an(q),n= 0;1;2;:::, and by
28.5.2 gen(z;q) =Sn(q) (zsen(z;q) +gn(z;q));
whena=bn(q),n= 1;2;3;:::. Form= 0;1;2;:::, we
have
28.5.3f2m(z;q)-periodic, odd ;
f2m+1(z;q)-antiperiodic, odd ;
and
28.5.4g2m+1(z;q)-antiperiodic, even ;
g2m+2(z;q)-periodic, even;
comparex28.2(vi). The functions fn(z;q),gn(z;q) are
unique.
The factors Cn(q) andSn(q) in (28.5.1) and (28.5.2)
are normalized so that
28.5.5(Cn(q))2Z2
0(fn(x;q))2dx
= (Sn(q))2Z2
0(gn(x;q))2dx=:
Asq!0 withn6= 0,Cn(q)!0,Sn(q)!0,
Cn(q)fn(z;q)!sinnz, andSn(q)gn(z;q)!cosnz.
This determines the signs of Cn(q) andSn(q). (Other
normalizations for Cn(q) andSn(q) can be found in
658 Mathieu Functions and Hill's Equation
the literature, but most formulas|including connec-
tion formulas|are una ected since fe n(z;q)=Cn(q) and
gen(z;q)=Sn(q) are invariant.)
28.5.6C2m( q) =C2m(q);
C2m+1( q) =S2m+1(q);
S2m+2( q) =S2m+2(q):
Forq= 0,
28.5.7fe0(z;0) =z;fen(z;0) = sinnz;
gen(z;0) = cosnz, n= 1;2;3;:::;
compare (28.2.29).
As a consequence of the factor zon the right-hand
sides of (28.5.1), (28.5.2), all solutions of Mathieu'sequation that are linearly independent of the periodic
solutions are unbounded as z!1 onR.
Wronskians
28.5.8 Wfcen;feng= cen(0;q) fe0
n(0;q);
28.5.9 Wfsen;geng= se0
n(0;q) gen(0;q):
See (28.22.12) for fe0
n(0;q) and gen(0;q).
For further information on Cn(q),Sn(q), and expan-
sions offn(z;q),gn(z;q) in Fourier series or in series of
cen, senfunctions, see McLachlan (1947, Chapter VII)
or Meixner and Sch afke (1954, x2.72).
28.5(ii) Graphics: Line Graphs of Second Solutions of Mathieu's Equation
Odd Second Solutions
Figure 28.5.1 : fe0(x;0:5) for 0x2and (for com-
parison) ce 0(x;0:5).
Figure 28.5.2 : fe0(x;1) for 0x2and (for compar-
ison) ce 0(x;1).
Figure 28.5.3 : fe1(x;0:5) for 0x2and (for com-
parison) ce 1(x;0:5).
Figure 28.5.4 : fe1(x;1) for 0x2and (for compar-
ison) ce 1(x;1).
28.6 Expansions for Small q 659
Even Second Solutions
Figure 28.5.5 : ge1(x;0:5) for 0x2and (for com-
parison) se 1(x;0:5).
Figure 28.5.6 : ge1(x;1) for 0x2and (for compar-
ison) se 1(x;1).
28.6 Expansions for Small q
28.6(i) Eigenvalues
Leading terms of the power series for am(q) andbm(q) form6 are:
28.6.1 a0(q) = 1
2q2+7
128q4 29
2304q6+68687
188 74368q8+;
28.6.2 a1(q) = 1 +q 1
8q2 1
64q3 1
1536q4+11
36864q5+49
5 89824q6+55
94 37184q7 83
353 89440q8+;
28.6.3 b1(q) = 1 q 1
8q2+1
64q3 1
1536q4 11
36864q5+49
5 89824q6 55
94 37184q7 83
353 89440q8+;
28.6.4 a2(q) = 4 +5
12q2 763
13824q4+10 02401
796 26240q6 16690 68401
45 86471 42400q8+;
28.6.5 b2(q) = 4 1
12q2+5
13824q4 289
796 26240q6+21391
45 86471 42400q8+;
28.6.6 a3(q) = 9 +1
16q2+1
64q3+13
20480q4 5
16384q5 1961
235 92960q6 609
1048 57600q7+;
28.6.7 b3(q) = 9 +1
16q2 1
64q3+13
20480q4+5
16384q5 1961
235 92960q6+609
1048 57600q7+;
28.6.8 a4(q) = 16 +1
30q2+433
8 64000q4 5701
27216 00000q6+;
28.6.9 b4(q) = 16 +1
30q2 317
8 64000q4+10049
27216 00000q6+;
28.6.10 a5(q) = 25 +1
48q2+11
7 74144q4+1
1 47456q5+37
8918 13888q6+;
28.6.11 b5(q) = 25 +1
48q2+11
7 74144q4 1
1 47456q5+37
8918 13888q6+;
28.6.12 a6(q) = 36 +1
70q2+187
439 04000q4+67 43617
9293 59872 00000q6+;
28.6.13 b6(q) = 36 +1
70q2+187
439 04000q4 58 61633
9293 59872 00000q6+:
Leading terms of the of the power series for m= 7;8;9;::: are:
28.6.14am(q)
bm(q)
=m2+1
2(m2 1)q2+5m2+ 7
32(m2 1)3(m2 4)q4+9m4+ 58m2+ 29
64(m2 1)5(m2 4)(m2 9)q6+:
The coecients of the power series of a2n(q),b2n(q) and alsoa2n+1(q),b2n+1(q) are the same until the terms in
q2n 2andq2n, respectively. Then
28.6.15 am(q) bm(q) =2qm
(2m 1(m 1)!)2
1 +O
q2
:
Higher coecients in the foregoing series can be found by equating coecients in the following continued-fraction
equations:
28.6.16
a (2n)2 q2
a (2n 2)2 q2
a (2n 4)2 q2
a 22 2q2
a= q2
(2n+ 2)2 a q2
(2n+ 4)2 a ,a=a2n(q),
660 Mathieu Functions and Hill's Equation
28.6.17
a (2n+ 1)2 q2
a (2n 1)2 q2
a 32 q2
a 12 q= q2
(2n+ 3)2 a q2
(2n+ 5)2 a ,a=a2n+1(q),
28.6.18
a (2n+ 1)2 q2
a (2n 1)2 q2
a 32 q2
a 12+q= q2
(2n+ 3)2 a q2
(2n+ 5)2 a ,a=b2n+1(q),
28.6.19
a (2n+ 2)2 q2
a (2n)2 q2
a (2n 2)2 q2
a 22= q2
(2n+ 4)2 a q2
(2n+ 6)2 a ,a=b2n+2(q).
Numerical values of the radii of convergence (j)
nof the power series (28.6.1){(28.6.14) for n= 0;1;:::; 9 are given
in Table 28.6.1. Here j= 1 fora2n(q),j= 2 forb2n+2(q), andj= 3 fora2n+1(q) andb2n+1(q). (Table 28.6.1 is
reproduced from Meixner et al. (1980,x2.4).)
Table 28.6.1 : Radii of convergence for power-series expansions of eigenvalues of Mathieu's equation.
n (1)
n (2)
n (3)
n
0 or 1 1 :46876 86138 6 :92895 47588 3 :76995 74940
2 7:26814 68935 16 :80308 98254 11 :27098 52655
3 16:47116 58923 30 :09677 28376 22 :85524 71216
4 30:42738 20960 48 :13638 18593 38 :52292 50099
5 47:80596 57026 69 :59879 32769 58 :27413 84472
6 69:92930 51764 95 :80595 67052 82 :10894 36067
7 95:47527 27072 125 :43541 1314 110 :02736 9210
8 125:76627 89677 159 :81025 4642 142 :02943 1279
9 159:47921 26694 197 :60667 8692 178 :11513 940
It is conjectured that for large n, the radii increase in proportion to the square of the eigenvalue number n; see
Meixner et al. (1980,x2.4). It is known that
28.6.20 lim inf
n!1(j)
n
n2kk0(K(k))2= 2:04183 4:::;
wherekis the unique root of the equation 2 E(k) =K(k) in the interval (0 ;1), andk0=p
1 k2. ForE(k) and
K(k) seex19.2(ii).
28.6(ii) Functions cenandsen
Leading terms of the power series for the normalized functions are:
28.6.21 21/2ce0(z;q) = 1 1
2qcos 2z+1
32q2(cos 4z 2) 1
128q3 1
9cos 6z 11 cos 2z
+;
28.6.22ce1(z;q) = cosz 1
8qcos 3z
+1
128q2 2
3cos 5z 2 cos 3z cosz
1
1024q3 1
9cos 7z 8
9cos 5z 1
3cos 3z+ 2 cosz
+;
28.6.23se1(z;q) = sinz 1
8qsin 3z
+1
128q2 2
3sin 5z+ 2 sin 3z sinz
1
1024q3 1
9sin 7z+8
9sin 5z 1
3sin 3z 2 sinz
+;
28.6.24 ce2(z;q) = cos 2z 1
4q 1
3cos 4z 1
+1
128q2 1
3cos 6z 76
9cos 2z
+;
28.6.25 se2(z;q) = sin 2z 1
12qsin 4z+1
128q2 1
3sin 6z 4
9sin 2z
+:
Form= 3;4;5;:::,
28.6.26cem(z;q) = cosmz q
41
m+ 1cos (m+ 2)z 1
m 1cos (m 2)z
+q2
321
(m+ 1)(m+ 2)cos (m+ 4)z+1
(m 1)(m 2)cos (m 4)z 2(m2+ 1)
(m2 1)2cosmz
+:
28.7 Analytic Continuation of Eigenvalues 661
For the corresponding expansions of se m(z;q) for
m= 3;4;5;::: change cos to sin everywhere in
(28.6.26).
The radii of convergence of the series (28.6.21){
(28.6.26) are the same as the radii of the corresponding
series foran(q) andbn(q); compare Table 28.6.1 and
(28.6.20).
28.7 Analytic Continuation of Eigenvalues
As functions of q,an(q) andbn(q) can be continued
analytically in the complex q-plane. The only singu-
larities are algebraic branch points, with an(q) and
bn(q) nite at these points. The number of branch
points is in nite, but countable, and there are no -
nite limit points. In consequence, the functions can be
de ned uniquely by introducing suitable cuts in the q-
plane. See Meixner and Sch afke (1954, x2.22). The
branch points are called the exceptional values , and the
other points normal values . The normal values are sim-
ple roots of the corresponding equations (28.2.21) and
(28.2.22). All real values of qare normal values. To
4D the rst branch points between a0(q) anda2(q) are
atq0=i1:4688 witha0(q0) =a2(q0) = 2:0886, and
betweenb2(q) andb4(q) they are at q1=i6:9289 with
b2(q1) =b4(q1) = 11:1904. For real qwithjqj<jq0j,
a0(iq) anda2(iq) are real-valued, whereas for real qwith
jqj>jq0j,a0(iq) anda2(iq) are complex conjugates.
See also Mulholland and Goldstein (1929), Bouwkamp
(1948), Meixner et al. (1980), Hunter and Guerrieri
(1981), Hunter (1981), and Shivakumar and Xue (1999).
For a visualization of the rst branch point of a0(i^q)
anda2(i^q) see Figure 28.7.1.
Figure 28.7.1 : Branch point of the eigenvalues a0(i^q)
anda2(i^q): 0^q2:5.
All thea2n(q),n= 0;1;2;:::, can be regarded as be-
longing to a complete analytic function (in the large).
Thereforew0
I(1
2;a;q) is irreducible, in the sense that it
cannot be decomposed into a product of entire functionsthat contain its zeros; see Meixner et al. (1980, p. 88).
Analogous statements hold for a2n+1(q),b2n+1(q), and
b2n+2(q), also for n= 0;1;2;:::. Closely connected
with the preceding statements, we have
28.7.11X
n=0
a2n(q) (2n)2
= 0;
28.7.21X
n=0
a2n+1(q) (2n+ 1)2
=q;
28.7.31X
n=0
b2n+1(q) (2n+ 1)2
= q;
28.7.41X
n=0
b2n+2(q) (2n+ 2)2
= 0:
28.8 Asymptotic Expansions for Large q
28.8(i) Eigenvalues
Denoteh=pqands= 2m+1. Then as h!+1with
m= 0;1;2;:::,
28.8.1
am
h2
bm+1
h2
2h2+ 2sh 1
8(s2+ 1) 1
27h(s3+ 3s)
1
212h2(5s4+ 34s2+ 9)
1
217h3(33s5+ 410s3+ 405s)
1
220h4(63s6+ 1260s4+ 2943s2+ 486)
1
225h5(527s7+ 15617s5+ 69001s3
+ 41607s) +:
For error estimates see Kurz (1979), and for graphical
interpretation see Figure 28.2.1. Also,
28.8.2bm+1
h2
am
h2
=24m+5
m!2
1/2
hm+( 3/2 )e 4h
1 6m2+ 14m+ 7
32h+O1
h2
:
28.8(ii) Sips' Expansions
Letx=1
2+h 1/4, whereis a real constant
such thatjj<21/4. Also let = 2p
hcosxand
Dm() =e 2/4Hem() (x18.3). Then as h!+1
28.8.3cem
x;h2
=bCm(Um() +Vm());
sem+1
x;h2
sinx=bSm(Um() Vm());
where
662 Mathieu Functions and Hill's Equation
28.8.4Um()Dm() 1
26h
Dm+4() 4!m
4
Dm 4()
+1
213h2
Dm+8() 25(m+ 2)Dm+4() + 4! 25(m 1)m
4
Dm 4() + 8!m
8
Dm 8()
+;
28.8.5Vm()1
24h
Dm+2() m(m 1)Dm 2()
+1
210h2
Dm+6() + (m2 25m 36)Dm+2()
m(m 1)(m2+ 27m 10)Dm 2() + 6!m
6
Dm 6()
+;
and
28.8.6 bCmh
2(m!)21/4
1 +2m+ 1
8h+m4+ 2m3+ 263m2+ 262m+ 108
2048h2+ 1/2
;
28.8.7 bSmh
2(m!)21/4
1 2m+ 1
8h+m4+ 2m3 121m2 122m 84
2048h2+ 1/2
:
These results are derived formally in Sips (1949, 1959, 1965). See also Meixner and Sch afke (1954, x2.84).
28.8(iii) Goldstein's Expansions
Letx=1
2 h 1/4, whereis a constant such that 1, ands= 2m+ 1. Then as h!+1
28.8.8cem
x;h2
cem(0;h2)=2m ( 1/2 )
m
W+
m(x)(Pm(x) Qm(x)) +W
m(x)(Pm(x) +Qm(x))
;
sem+1
x;h2
se0
m+1(0;h2)=2m ( 1/2 )
m+1
W+
m(x)(Pm(x) Qm(x)) W
m(x)(Pm(x) +Qm(x))
;
where
28.8.9 W
m(x) =e2hsinx
(cosx)m+1(
cos 1
2x+1
42m+1;
sin 1
2x+1
42m+1;
and
28.8.10m1 +s
23h+4s2+ 3
27h2+19s3+ 59s
211h3+; m+12h 1
4s 2s2+ 3
26h 7s3+ 47s
210h2 ;
28.8.11 Pm(x)1 +s
23hcos2x+1
h2s4+ 86s2+ 105
211cos4x s4+ 22s2+ 57
211cos2x
+;
28.8.12 Qm(x)sinx
cos2x1
25h(s2+ 3) +1
29h2
s3+ 3s+4s3+ 44s
cos2x
+:
28.8(iv) Uniform Approximations
Barrett's Expansions
Barrett (1981) supplies asymptotic approximations for
numerically satisfactory pairs of solutions of both Math-
ieu's equation (28.2.1) and the modi ed Mathieu equa-
tion (28.20.1). The approximations apply when the pa-
rametersaandqare real and large, and are uniform
with respect to various regions in the z-plane. The
approximants are elementary functions, Airy functions,
Bessel functions, and parabolic cylinder functions; com-
parex2.8. It is stated that corresponding uniform ap-
proximations can be obtained for other solutions, in-
cluding the eigensolutions, of the di erential equationsby application of the results, but these approximations
are not included.
Dunster's Approximations
Dunster (1994a) supplies uniform asymptotic approxi-
mations for numerically satisfactory pairs of solutions
of Mathieu's equation (28.2.1). These approximations
apply when qandaare real and q!1 . They are uni-
form with respect to awhen 2qa(2 )q, where
is an arbitrary constant such that 0 < < 4, and
also with respect to zin the semi-in nite strip given by
0<zand=z0.
The approximations are expressed in terms of Whit-
taker functions W;(z) andM;(z) with=1
4; com-
28.9 Zeros 663
parex2.8(vi). They are derived by rigorous analysis
and accompanied by strict and realistic error bounds.
With additional restrictions on z, uniform asymptotic
approximations for solutions of (28.2.1) and (28.20.1)
are also obtained in terms of elementary functions by re-
expansions of the Whittaker functions; compare x2.8(ii).
Subsequently the asymptotic solutions involving ei-
ther elementary or Whittaker functions are identi ed in
terms of the Floquet solutions me (z;q) (x28.12(ii)) and
modi ed Mathieu functions M(j)
(z;h) (x28.20(iii)).
For related results see Langer (1934) and Sharples
(1967, 1971).
28.9 Zeros
For realqeach of the functions ce 2n(z;q), se 2n+1(z;q),
ce2n+1(z;q), and se 2n+2(z;q) has exactly nzeros in
0< z <1
2. They are continuous in q. Forq!1
the zeros of ce 2n(z;q) and se 2n+1(z;q) approach asymp-
totically the zeros of He2n
q1=4( 2z)
, and the zeros
of ce 2n+1(z;q) and se 2n+2(z;q) approach asymptotically
the zeros of He2n+1
q1=4( 2z)
. Here Hen(z) de-
notes the Hermite polynomial of degree n(x18.3). Fur-
thermore, for q > 0 cem(z;q) and sem(z;q) also have
purely imaginary zeros that correspond uniquely to the
purely imaginary z-zeros ofJm
2pqcosz
(x10.21(i)),
and they are asymptotically equal as q!0 andj=zj!
1. There are no zeros within the strip j<zj<1
2other
than those on the real and imaginary axes.
For further details see McLachlan (1947, pp. 234{
239) and Meixner and Sch afke (1954, xx2.331, 2.8, 2.81,
and 2.85).
28.10 Integral Equations
28.10(i) Equations with Elementary Kernels
With the notation of x28.4 for Fourier coecients,
28.10.12
Z/2
0cos(2hcoszcost) ce2n
t;h2
dt
=A2n
0(h2)
ce2n 1
2;h2ce2n
z;h2
;
28.10.22
Z/2
0cosh(2hsinzsint) ce2n
t;h2
dt
=A2n
0(h2)
ce2n(0;h2)ce2n
z;h2
;
28.10.32
Z/2
0sin(2hcoszcost) ce2n+1
t;h2
dt
= hA2n+1
1(h2)
ce0
2n+1 1
2;h2ce2n+1
z;h2
;28.10.4
2
Z/2
0coszcostcosh(2hsinzsint) ce2n+1
t;h2
dt
=A2n+1
1(h2)
2 ce2n+1(0;h2)ce2n+1
z;h2
;
28.10.5
2
Z/2
0sinh(2hsinzsint) se2n+1
t;h2
dt
=hB2n+1
1(h2)
se0
2n+1(0;h2)se2n+1
z;h2
;
28.10.6
2
Z/2
0sinzsintcos(2hcoszcost) se2n+1
t;h2
dt
=B2n+1
1(h2)
2 se2n+1 1
2;h2se2n+1
z;h2
;
28.10.7
2
Z/2
0sinzsintsin(2hcoszcost) se2n+2
t;h2
dt
= hB2n+2
2(h2)
2 se0
2n+2 1
2;h2se2n+2
z;h2
;
28.10.8
2
Z/2
0coszcostsinh(2hsinzsint) se2n+2
t;h2
dt
=hB2n+2
2(h2)
2 se0
2n+2(0;h2)se2n+2
z;h2
:
28.10(ii) Equations with Bessel-Function
Kernels
28.10.9Z/2
0J0
2q
q(cos2 sin2)
ce2n( ;q)d
=wII(1
2;a2n(q);q) ce2n(;q);
28.10.10Z
0J0(2pq(cos + cos)) cen( ;q)d
=wII(;an(q);q) cen(;q):
28.10(iii) Further Equations
Seex28.28. See also Prudnikov et al. (1990, pp. 359{
368), Erd elyi et al. (1955, p. 115), and Gradshteyn and
Ryzhik (2000, pp. 755{759). For relations with variable
boundaries see Volkmer (1983).
664 Mathieu Functions and Hill's Equation
28.11 Expansions in Series of Mathieu
Functions
Letf(z) be a 2-periodic function that is analytic in an
open doubly-in nite strip Sthat contains the real axis,
andqbe a normal value ( x28.7). Then
28.11.1
f(z) = 0ce0(z;q) +1X
n=1( ncen(z;q) + nsen(z;q));
where
28.11.2 n=1
Z2
0f(x) cen(x;q)dx;
n=1
Z2
0f(x) sen(x;q)dx:
The series (28.11.1) converges absolutely and uniformly
on any compact subset of the strip S. See Meixner and
Sch afke (1954,x2.28), and for expansions in the case
of the exceptional values of qsee Meixner et al. (1980,
p. 33).
Examples
With the notation of x28.4,
28.11.3 1 = 21X
n=0A2n
0(q) ce2n(z;q);
28.11.4 cos 2mz=1X
n=0A2n
2m(q) ce2n(z;q),m6= 0,
28.11.5 cos (2m+ 1)z=1X
n=0A2n+1
2m+1(q) ce2n+1(z;q);
28.11.6 sin (2m+ 1)z=1X
n=0B2n+1
2m+1(q) se2n+1(z;q);
28.11.7 sin (2m+ 2)z=1X
n=0B2n+2
2m+2(q) se2n+2(z;q):
Mathieu Functions of Noninteger
Order
28.12 De nitions and Basic Properties
28.12(i) Eigenvalues +2n(q)
The introduction to the eigenvalues and the functions
of general order proceeds as in xx28.2(i), 28.2(ii), and
28.2(iii), except that we now restrict b6= 0;1; equiva-
lently6=n. In consequence, for the Floquet solutions
w(z) the factor eiin (28.2.14) is no longer 1.For given(or cos()) andq, equation (28.2.16)
determines an in nite discrete set of values of a, de-
noted by+2n(q),n= 0;1;2;:::. Whenq= 0
Equation (28.2.16) has simple roots, given by
28.12.1 +2n(0) = (+ 2n)2:
For other values of q,+2n(q) is determined by ana-
lytic continuation. Without loss of generality, from now
on we replace + 2nby.
For change of signs of andq,
28.12.2 ( q) =(q) = (q):
As inx28.7 values of qfor which (28.2.16) has simple
rootsare called normal values with respect to . For
real values of andqall the(q) are real, and qis
normal. For graphical interpretation see Figure 28.13.1.
To complete the de nition we require
28.12.3m(q) =(
am(q); m = 0;1;:::;
b m(q); m = 1; 2;::::
As a function of with xedq(6= 0),(q) is discon-
tinuous at=1;2;:::. See Figure 28.13.2.
28.12(ii) Eigenfunctions me(z;q)
Two eigenfunctions correspond to each eigenvalue a=
(q). The Floquet solution with respect to is denoted
by me(z;q). Forq= 0,
28.12.4 me(z;0) =eiz:
The other eigenfunction is me ( z;q), a Floquet so-
lution with respect to witha=(q). Ifqis a
normal value of the corresponding equation (28.2.16),
then these functions are uniquely determined as ana-
lytic functions of zandqby the normalization
28.12.5Z
0me(x;q) me( x;q)dx=:
They have the following pseudoperiodic and orthogonal-
ity properties:
28.12.6 me(z+;q) =eime(z;q);
28.12.7Z
0me+2m(x;q) me+2n( x;q)dx= 0,m6=n.
For changes of sign of ,q, andz,
28.12.8 me (z;q) = me( z;q);
28.12.9 me(z; q) =ei=2me
z 1
2;q
;
28.12.10 me(z;q) = me ( z;q):
(28.12.10) is not valid for cuts on the real axis in the
q-plane for special complex values of ; but it remains
valid for small q; comparex28.7.
28.13 Graphics 665
To complete the de nitions of the me functions we
set
28.12.11men(z;q) =p
2 cen(z;q),n= 0;1;2;:::,
me n(z;q) = p
2isen(z;q),n= 1;2;:::;
compare (28.12.3). However, these functions are notthe
limiting values of me (z;q) as!n(6= 0).
28.12(iii) Functions ce(z;q),se(z;q), when
=2Z
28.12.12 ce(z;q) =1
2(me(z;q) + me( z;q));
28.12.13 se(z;q) = 1
2i(me(z;q) me( z;q)):These functions are real-valued for real , realq, and
z=x, whereas me (x;q) is complex. When =s=m is
a rational number, but not an integer, all solutions of
Mathieu's equation are periodic with period 2 m.
For change of signs of andz,
28.12.14 ce(z;q) = ce( z;q) = ce (z;q);
28.12.15 se(z;q) = se( z;q) = se (z;q):
Again, the limiting values of ce (z;q) and se(z;q) as
!n(6= 0) are notthe functions ce n(z;q) and sen(z;q)
de ned inx28.2(vi). Compare e.g. Figure 28.13.3.
28.13 Graphics
28.13(i) Eigenvalues (q)for General
Figure 28.13.1 :(q) as a function of qfor= 0:5(1)3:5
andan(q);bn(q) forn= 0;1;2;3;4 (a's),n= 1;2;3;4
(b's). (Compare Figure 28.2.1.)
Figure 28.13.2 :(q) for 2< < 2, 0q10.
28.13(ii) Solutions ce(x;q),se(x;q), and me(x;q)for General
Figure 28.13.3 : ce(x;1) for 1< < 1, 0x2.
Figure 28.13.4 : se(x;1) for 0< < 1, 0x2.
666 Mathieu Functions and Hill's Equation
Figure 28.13.5 : mei(x;1) for 0:10:4, x
:
28.14 Fourier Series
The Fourier series
28.14.1 me(z;q) =1X
m= 1c
2m(q)ei(+2m)z;
28.14.2 ce(z;q) =1X
m= 1c
2m(q) cos (+ 2m)z;
28.14.3 se(z;q) =1X
m= 1c
2m(q) sin (+ 2m)z;
converge absolutely and uniformly on all compact sets
in thez-plane. The coecients satisfy
28.14.4qc2m+2
a (+ 2m)2
c2m+qc2m 2= 0,
a=(q);c2m=c
2m(q);
and the normalization relation
28.14.51X
m= 1(c
2m(q))2= 1;
compare (28.12.5). Ambiguities in sign are resolved by
(28.14.9) when q= 0, and by continuity for other values
ofq.
The rate of convergence is indicated by
28.14.6c
2m(q)
c
2m2(q)= q
4m2
1 +O1
m
,m!1 .
For changes of sign of ,q, andm,
28.14.7 c
2m(q) =c
2m(q);
28.14.8 c
2m( q) = ( 1)mc
2m(q):
Whenq= 0,
28.14.9 c
0(0) = 1; c
2m(0) = 0, m6= 0.
Whenq!0 withm(1) and xed,
28.14.10
c
2m(q) =( 1)mqm (+ 1)
m! 22m (+m+ 1)+O
qm+2
c
0(q):28.15 Expansions for Small q
28.15(i) Eigenvalues (q)
28.15.1
(q) =2+1
2(2 1)q2+52+ 7
32(2 1)3(2 4)q4
+94+ 582+ 29
64(2 1)5(2 4)(2 9)q6+:
Higher coecients can be found by equating powers
ofqin the following continued-fraction equation, with
a=(q):
28.15.2a 2 q2
a (+ 2)2 q2
a (+ 4)2
=q2
a ( 2)2 q2
a ( 4)2 :
28.15(ii) Solutions me(z;q)
28.15.3
me(z;q)
=eiz q
41
+ 1ei(+2)z 1
1ei( 2)z
+q2
321
(+ 1)(+ 2)ei(+4)z
+1
( 1)( 2)ei( 4)z 2(2+ 1)
(2 1)2eiz
+;
comparex28.6(ii).
28.16 Asymptotic Expansions for Large q
Lets= 2m+ 1,m= 0;1;2;:::, andbe xed with
m< <m + 1. Then as h(=pq)!+1
28.16.1
h2
2h2+ 2sh 1
8(s2+ 1) 1
27h(s3+ 3s)
1
212h2(5s4+ 34s2+ 9)
1
217h3(33s5+ 410s3+ 405s)
1
220h4(63s6+ 1260s4+ 2943s2+ 486)
1
225h5(527s7+ 15617s5+ 69001s3+ 41607s)
+:
For graphical interpretation, see Figures 28.13.1 and
28.13.2.
See alsox28.8(iv).
28.17 Stability as x!1 667
28.17 Stability as x!1
If all solutions of (28.2.1) are bounded when x!1
along the real axis, then the corresponding pair of pa-
rameters (a;q) is called stable . All other pairs are un-
stable .
For example, positive real values of awithq= 0
comprise stable pairs, as do values of aandqthat cor-
respond to real, but noninteger, values of .
However, if=6= 0, then (a;q) always comprises an
unstable pair. For example, as x!+1one of the
solutions me (x;q) and me ( x;q) tends to 0 and the
other is unbounded (compare Figure 28.13.5). Also, all
nontrivial solutions of (28.2.1) are unbounded on R.
For realaandq(6= 0) the stable regions are the open
regions indicated in color in Figure 28.17.1. The bound-
ary of each region comprises the characteristic curves
a=an(q) anda=bn(q); compare Figure 28.2.1.
Figure 28.17.1 : Stability chart for eigenvalues of Math-
ieu's equation (28.2.1).
28.18 Integrals and Integral Equations
Seex28.28.
28.19 Expansions in Series of me+2n
Functions
Letqbe a normal value ( x28.12(i)) with respect to ,
andf(z) be a function that is analytic on a doubly-
in nite open strip Sthat contains the real axis. Assume
also
28.19.1 f(z+) =eif(z):
Then
28.19.2 f(z) =1X
n= 1fnme+2n(z;q);
where
28.19.3fn=1
Z
0f(z) me+2n( z;q)dz:
The series (28.19.2) converges absolutely and uniformly
on compact subsets within S.Example
28.19.4eiz=1X
n= 1c+2n
2n(q) me+2n(z;q);
where the coecients are as in x28.14.
Modi ed Mathieu Functions
28.20 De nitions and Basic Properties
28.20(i) Modi ed Mathieu's Equation
Whenzis replaced byiz, (28.2.1) becomes the modi-
ed Mathieu's equation :
28.20.1 w00 (a 2qcosh(2z))w= 0;
with its algebraic form
28.20.2
(2 1)w00+w0+
4q2 2q a
w= 0,= coshz.
28.20(ii) Solutions Ce,Se,Me,Fen,Gen
28.20.3 Ce(z;q) = ce(iz;q),6= 1; 2;:::,
28.20.4 Se(z;q) =ise(iz;q),6= 0; 1;:::,
28.20.5 Me(z;q) = me( iz;q);
28.20.6 Fen(z;q) =ifen(iz;q),n= 0;1;:::,
28.20.7 Gen(z;q) = gen(iz;q), n= 1;2;:::.
28.20(iii) Solutions M(j)
Assume rst that is real,qis positive, and a=(q);
seex28.12(i). Write
28.20.8 h=pq(>0):
Then fromx2.7(ii) it is seen that equation (28.20.2)
has independent and unique solutions that are asymp-
totic to1/2e2ihas!1 in the respective sectors
jph(i)j3
2 ,being an arbitrary small positive
constant. It follows that (28.20.1) has independent and
unique solutions M(3)
(z;h), M(4)
(z;h) such that
28.20.9 M(3)
(z;h) =H(1)
(2hcoshz) (1 +O(sechz));
as<z!+1with +=z2 , and
28.20.10 M(4)
(z;h) =H(2)
(2hcoshz) (1 +O(sechz));
as<z!+1with 2+=z . Seex10.2(ii)
for the notation. In addition, there are unique solutions
M(1)
(z;h), M(2)
(z;h) that are real when zis real and
have the properties
28.20.11
M(1)
(z;h) =J(2hcoshz)+ej=(2hcoshz)jO
(sechz)3=2
;
668 Mathieu Functions and Hill's Equation
28.20.12
M(2)
(z;h) =Y(2hcoshz)+ej=(2hcoshz)jO
(sechz)3=2
;
as<z!+1withj=zj .
For other values of z,h;andthe functions
M(j)
(z;h),j= 1;2;3;4;are determined by analytic con-
tinuation. Furthermore,
28.20.13 M(3)
(z;h) = M(1)
(z;h) +iM(2)
(z;h);
28.20.14 M(4)
(z;h) = M(1)
(z;h) iM(2)
(z;h):
28.20(iv) Radial Mathieu Functions Mc(j)
n,
Ms(j)
n
Forj= 1;2;3;4,
28.20.15 Mc(j)
n(z;h) = M(j)
n(z;h),n= 0;1;:::,
28.20.16 Ms(j)
n(z;h) = ( 1)nM(j)
n(z;h),n= 1;2;:::.
28.20(v) Solutions Ien,Ion,Ken,Kon
28.20.17 Ien(z;h) =i nMc(1)
n(z;ih);
28.20.18 Ion(z;h) =i nMs(1)
n(z;ih);
28.20.19
Ke2m(z;h) = ( 1)m1
2iMc(3)
2m(z;ih);
Ke2m+1(z;h) = ( 1)m+11
2Mc(3)
2m+1(z;ih);
28.20.20
Ko2m(z;h) = ( 1)m1
2iMs(3)
2m(z;ih);
Ko2m+1(z;h) = ( 1)m+11
2Ms(3)
2m+1(z;ih):
28.20(vi) Wronskians
28.20.21
Wn
M(1)
;M(2)
o
= Wn
M(2)
;M(3)
o
= Wn
M(2)
;M(4)
o
= 2/;
Wn
M(1)
;M(3)
o
= Wn
M(1)
;M(4)
o
= 1
2Wn
M(3)
;M(4)
o
= 2i/:28.20(vii) Shift of Variable
28.20.22 M(j)
z1
2i;h
= M(j)
(z;ih), =2Z.
Forn= 0;1;2;:::,
28.20.23 Mc(j)
2n
z1
2i;h
= Mc(j)
2n(z;ih);
Ms(j)
2n+1
z1
2i;h
= Mc(j)
2n+1(z;ih);
28.20.24 Mc(j)
2n+1
z1
2i;h
= Ms(j)
2n+1(z;ih);
Ms(j)
2n+2
z1
2i;h
= Ms(j)
2n+2(z;ih):
Fors2Z,
28.20.25
M(1)
(z+si;h ) =eisM(1)
(z;h);
M(2)
(z+si;h ) =e isM(2)
(z;h)
+ 2icot() sin(s) M(1)
(z;h);
M(3)
(z+si;h ) = sin((s 1))
sin()M(3)
(z;h)
e isin(s)
sin()M(4)
(z;h);
M(4)
(z+si;h ) =eisin(s)
sin()M(3)
(z;h)
+sin((s+ 1))
sin()M(4)
(z;h):
Whenis an integer the right-hand sides of (28.20.25)
are replaced by the their limiting values. And for
the corresponding identities for the radial functions use
(28.20.15) and (28.20.16).
28.21 Graphics 669
28.21 Graphics
Radial Mathieu Functions: Surfaces
Figure 28.21.1 : Mc(1)
0(x;h) for 0h3, 0x2.
Figure 28.21.2 : Mc(1)
1(x;h) for 0h3, 0x2.
For further graphics see http://dlmf.nist.gov/28.21 .
28.22 Connection Formulas
28.22(i) Integer
28.22.1
Mc(1)
m(z;h) =r
2
1
ge;m(h) cem(0;h2)Cem
z;h2
;
28.22.2
Ms(1)
m(z;h) =r
2
1
go;m(h) se0m(0;h2)Sem
z;h2
;
28.22.3
Mc(2)
m(z;h) =r
2
1
ge;m(h) cem(0;h2)
fe;m(h) Cem
z;h2
+2
Cm(h2)Fem
z;h2
;
28.22.4
Ms(2)
m(z;h) =r
2
1
go;m(h) se0m(0;h2)
fo;m(h) Sem
z;h2
2
Sm(h2)Gem
z;h2
:
The joining factors in the above formulas are given by
28.22.5 ge;2m(h) = ( 1)mr
2
ce2m 1
2;h2
A2m
0(h2);
28.22.6ge;2m+1(h) = ( 1)m+1r
2
ce0
2m+1 1
2;h2
hA2m+1
1 (h2);28.22.7go;2m+1(h) = ( 1)mr
2
se2m+1 1
2;h2
hB2m+1
1 (h2);
28.22.8go;2m+2(h) = ( 1)m+1r
2
se0
2m+2 1
2;h2
h2B2m+2
2 (h2);
28.22.9 fe;m(h) = p
/2ge;m(h) Mc(2)
m(0;h);
28.22.10 fo;m(h) = p
/2go;m(h) Ms(2)
m0(0;h);
whereAm
n(h2),Bm
n(h2) are as inx28.4(i), and Cm(h2),
Sm(h2) are as inx28.5(i). Furthermore,
28.22.11Mc(2)
m0(0;h) =p
2/ge;m(h);
Ms(2)
m(0;h) = p
2/go;m(h);
28.22.12
fe0
m
0;h2
=1
2Cm(h2) (ge;m(h))2cem
0;h2
;
gem
0;h2
=1
2Sm(h2) (go;m(h))2se0
m
0;h2
:
28.22(ii) Noninteger
28.22.13 M(1)
(z;h) =M(1)
(0;h)
me(0;h2)Me
z;h2
:
Here me
0;h2
(6= 0) is given by (28.14.1) with z= 0,
and M(1)
(0;h) is given by (28.24.1) with j= 1,z= 0,
andnchosen so thatjc
2n(h2)j= max(jc
2`(h2)j), where
the maximum is taken over all integers `.
28.22.14
M(2)
(z;h) = cot() M(1)
(z;h) 1
sin()M(1)
(z;h):
See also (28.20.13) and (28.20.14).
670 Mathieu Functions and Hill's Equation
28.23 Expansions in Series of Bessel Functions
We use the following notations:
28.23.1 C(1)
=J;C(2)
=Y;C(3)
=H(1)
;C(4)
=H(2)
;
comparex10.2(ii). For the coecients c
n(q) seex28.14. ForAm
n(q) andBm
n(q) seex28.4.
28.23.2 me
0;h2
M(j)
(z;h) =1X
n= 1( 1)nc
2n(h2)C(j)
+2n(2hcoshz);
28.23.3 me0
0;h2
M(j)
(z;h) =itanhz1X
n= 1( 1)n(+ 2n)c
2n(h2)C(j)
+2n(2hcoshz);
valid for all zwhenj= 1, and for<z>0 andjcoshzj>1 whenj= 2;3;4.
28.23.4 me 1
2;h2
M(j)
(z;h) =ei/21X
n= 1c
2n(h2)C(j)
+2n(2hsinhz);
28.23.5 me0
1
2;h2
M(j)
(z;h) =iei/2cothz1X
n= 1(+ 2n)c
2n(h2)C(j)
+2n(2hsinhz);
valid for all zwhenj= 1, and for<z>0 andjsinhzj>1 whenj= 2;3;4.
In the case when is an integer
28.23.6 Mc(j)
2m(z;h) = ( 1)m
ce2m
0;h2 11X
`=0( 1)`A2m
2`(h2)C(j)
2`(2hcoshz);
28.23.7 Mc(j)
2m(z;h) = ( 1)m
ce2m 1
2;h2 11X
`=0A2m
2`(h2)C(j)
2`(2hsinhz);
28.23.8 Mc(j)
2m+1(z;h) = ( 1)m
ce2m+1
0;h2 11X
`=0( 1)`A2m+1
2`+1(h2)C(j)
2`+1(2hcoshz);
28.23.9 Mc(j)
2m+1(z;h) = ( 1)m+1
ce0
2m+1 1
2;h2 1cothz1X
`=0(2`+ 1)A2m+1
2`+1(h2)C(j)
2`+1(2hsinhz);
28.23.10 Ms(j)
2m+1(z;h) = ( 1)m
se0
2m+1
0;h2 1tanhz1X
`=0( 1)`(2`+ 1)B2m+1
2`+1(h2)C(j)
2`+1(2hcoshz);
28.23.11 Ms(j)
2m+1(z;h) = ( 1)m
se2m+1 1
2;h2 11X
`=0B2m+1
2`+1(h2)C(j)
2`+1(2hsinhz);
28.23.12 Ms(j)
2m+2(z;h) = ( 1)m
se0
2m+2
0;h2 1tanhz1X
`=0( 1)`(2`+ 2)B2m+2
2`+2(h2)C(j)
2`+2(2hcoshz);
28.23.13 Ms(j)
2m+2(z;h) = ( 1)m+1
se0
2m+2 1
2;h2 1cothz1X
`=0(2`+ 2)B2m+2
2`+2(h2)C(j)
2`+2(2hsinhz):
Whenj= 1, each of the series (28.23.6){(28.23.13) converges for all z. Whenj= 2;3;4 the series in the even-
numbered equations converge for <z >0 andjcoshzj>1, and the series in the odd-numbered equations converge
for<z>0 andjsinhzj>1.
For proofs and generalizations, see Meixner and Sch afke (1954, xx2.62 and 2.64).
28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions 671
28.24 Expansions in Series of Cross-Products of Bessel Functions or Modi ed Bessel
Functions
Throughout this section "0= 2 and"s= 1,s= 1;2;3;:::.
WithC(j)
,c
n(q),Am
n(q), andBm
n(q) as inx28.23,
28.24.1 c
2n(h2) M(j)
(z;h) =1X
`= 1( 1)`c
2`(h2)J` n
he z
C(j)
+n+`(hez);
wherej= 1;2;3;4 andn2Z.
In the case when is an integer,
28.24.2"sMc(j)
2m(z;h) = ( 1)m1X
`=0( 1)`A2m
2`(h2)
A2m
2s(h2)
J` s
he z
C(j)
`+s(hez) +J`+s
he z
C(j)
` s(hez)
;
28.24.3 Mc(j)
2m+1(z;h) = ( 1)m1X
`=0( 1)`A2m+1
2`+1(h2)
A2m+1
2s+1(h2)
J` s
he z
C(j)
`+s+1(hez) +J`+s+1
he z
C(j)
` s(hez)
;
28.24.4 Ms(j)
2m+1(z;h) = ( 1)m1X
`=0( 1)`B2m+1
2`+1(h2)
B2m+1
2s+1(h2)
J` s
he z
C(j)
`+s+1(hez) J`+s+1
he z
C(j)
` s(hez)
;
28.24.5 Ms(j)
2m+2(z;h) = ( 1)m1X
`=0( 1)`B2m+2
2`+2(h2)
B2m+2
2s+2(h2)
J` s
he z
C(j)
`+s+2(hez) J`+s+2
he z
C(j)
` s(hez)
;
wherej= 1;2;3;4;ands= 0;1;2;:::.
Also, withInandKndenoting the modi ed Bessel functions ( x10.25(ii)), and again with s= 0;1;2;:::,
28.24.6"sIe2m(z;h) = ( 1)s1X
`=0( 1)`A2m
2`(h2)
A2m
2s(h2)
I` s
he z
I`+s(hez) +I`+s
he z
I` s(hez)
;
28.24.7 Io2m+2(z;h) = ( 1)s1X
`=0( 1)`B2m+2
2`+2(h2)
B2m+2
2s+2(h2)
I` s
he z
I`+s+2(hez) I`+s+2
he z
I` s(hez)
;
28.24.8 Ie2m+1(z;h) = ( 1)s1X
`=0( 1)`B2m+1
2`+1(h2)
B2m+1
2s+1(h2)
I` s
he z
I`+s+1(hez) +I`+s+1
he z
I` s(hez)
;
28.24.9 Io2m+1(z;h) = ( 1)s1X
`=0( 1)`A2m+1
2`+1(h2)
A2m+1
2s+1(h2)
I` s
he z
I`+s+1(hez) I`+s+1
he z
I` s(hez)
;
28.24.10"sKe2m(z;h) =1X
`=0A2m
2`(h2)
A2m
2s(h2)
I` s
he z
K`+s(hez) +I`+s
he z
K` s(hez)
;
28.24.11 Ko2m+2(z;h) =1X
`=0B2m+2
2`+2(h2)
B2m+2
2s+2(h2)
I` s
he z
K`+s+2(hez) I`+s+2
he z
K` s(hez)
;
28.24.12 Ke2m+1(z;h) =1X
`=0B2m+1
2`+1(h2)
B2m+1
2s+1(h2)
I` s
he z
K`+s+1(hez) I`+s+1
he z
K` s(hez)
;
28.24.13 Ko2m+1(z;h) =1X
`=0A2m+1
2`+1(h2)
A2m+1
2s+1(h2)
I` s
he z
K`+s+1(hez) +I`+s+1
he z
K` s(hez)
:
The expansions (28.24.1){(28.24.13) converge absolutely and uniformly on compact sets of the z-plane.
672 Mathieu Functions and Hill's Equation
28.25 Asymptotic Expansions for Large <z
For xedh(6= 0) and xed ,
28.25.1M(3;4)
(z;h)ei(2hcoshz (1
2+1
4))
(h(coshz+ 1))1
2
1X
m=0D
m
(4ih(coshz+ 1))m;
where the coecients are given by
28.25.2 D
1= 0; D
0= 1;
and
28.25.3
(m+ 1)D
m+1
+
(m+1
2)2(m+1
4)8ih+ 2h2 a
D
m
(m 1
2) (8ihm)D
m 1= 0,m0:
The upper signs correspond to M(3)
(z;h) and the lower
signs to M(4)
(z;h). The expansion (28.25.1) is valid for
M(3)
(z;h) when
28.25.4<z!+1, +phh+=z2 ,
and for M(4)
(z;h) when
28.25.5<z!+1, 2+phh+=z ,whereagain denotes an arbitrary small positive con-
stant.
For proofs and generalizations see Meixner and
Sch afke (1954,x2.63).
28.26 Asymptotic Approximations for Large
q
28.26(i) Goldstein's Expansions
Denote
28.26.1Mc(3)
m(z;h) =ei
(hcoshz)1/2
(Fcm(z;h) iGcm(z;h));
28.26.2iMs(3)
m+1(z;h) =ei
(hcoshz)1/2
(Fsm(z;h) iGsm(z;h));
where
28.26.3 = 2hsinhz
m+1
2
arctan(sinh z):
Then ash!+1with xed zin<z > 0 and xed
s= 2m+ 1,
28.26.4Fcm(z;h)1 +s
8hcosh2z+1
211h2s4+ 86s2+ 105
cosh4z s4+ 22s2+ 57
cosh2z
+1
214h3
s5+ 14s3+ 33s
cosh2z 2s5+ 124s3+ 1122s
cosh4z+3s5+ 290s3+ 1627s
cosh6z
+;
28.26.5Gcm(z;h)sinhz
cosh2zs2+ 3
25h+1
29h2
s3+ 3s+4s3+ 44s
cosh2z
+1
214h3
5s4+ 34s2+ 9 s6 47s4+ 667s2+ 2835
12 cosh2z+s6+ 505s4+ 12139s2+ 10395
12 cosh4z
+:
The asymptotic expansions of Fs m(z;h) and Gsm(z;h)
in the same circumstances are also given by the right-
hand sides of (28.26.4) and (28.26.5), respectively.
For additional terms see Goldstein (1927).
28.26(ii) Uniform Approximations
Seex28.8(iv). For asymptotic approximations for
M(3;4)
(z;h) see also Naylor (1984, 1987, 1989).
28.27 Addition Theorems
Addition theorems provide important connections be-
tween Mathieu functions with di erent parameters and
in di erent coordinate systems. They are analogous to
the addition theorems for Bessel functions ( x10.23(ii))
and modi ed Bessel functions ( x10.44(ii)). For a com-
prehensive treatment see Meixner et al. (1980,x2.2).28.28 Integrals, Integral Representations,
and Integral Equations
28.28(i) Equations with Elementary Kernels
Let
28.28.1w= coshzcostcos + sinhzsintsin :
Then
28.28.2
1
2Z2
0e2ihwcen
t;h2
dt=incen
;h2
Mc(1)
n(z;h);
28.28.3
1
2Z2
0e2ihwsen
t;h2
dt=insen
;h2
Ms(1)
n(z;h);
28.28 Integrals, Integral Representations, and Integral Equations 673
28.28.4ih
Z2
0@w
@ e2ihwcen
t;h2
dt
=ince0
n
;h2
Mc(1)
n(z;h);
28.28.5ih
Z2
0@w
@ e2ihwsen
t;h2
dt
=inse0
n
;h2
Ms(1)
n(z;h):
In (28.28.7){(28.28.9) the paths of integration Ljare
given by
28.28.6L1: from 1+i1to 2 1+i1;
L3: from 1+i1to2 i1;
L4: from2 i1to 2 1+i1;
where1and2are real constants.
28.28.71
Z
Lje2ihwme
t;h2
dt
=ei=2me
;h2
M(j)
(z;h),j= 3;4;
28.28.81
Z
Lj2ih@w
@ e2ihwme
t;h2
dt
=ei=2me0
;h2
M(j)
(z;h),j= 3;4;
28.28.91
2Z
L1e2ihwme
t;h2
dt
=ei=2me
;h2
M(1)
(z;h):In (28.28.11){(28.28.14)
28.28.10 0<ph(h(coshz1))<:
28.28.11Z1
0e2ihcoshzcoshtCe
t;h2
dt
=1
2ieice
0;h2
M(3)
(z;h);
28.28.12Z1
0e2ihcoshzcoshtsinhzsinhtSe
t;h2
dt
=
4hei=2se0
0;h2
M(3)
(z;h);
28.28.13Z1
0e2ihcoshzcoshtsinhzsinhtFem
t;h2
dt
=
4himfe0
m
0;h2
Mc(3)
m(z;h);
28.28.14Z1
0e2ihcoshzcoshtGem
t;h2
dt
=1
2im+1gem
0;h2
Ms(3)
m(z;h):
In particular, when h > 0 the integrals (28.28.11),
(28.28.14) converge absolutely and uniformly in the half
strip<z0, 0=z.
28.28.15Z1
0cos(2hcosycosht) Ce 2n
t;h2
dt= ( 1)n+11
2Mc(2)
2n(0;h) ce2n
y;h2
;
28.28.16Z1
0sin(2hcosycosht) Ce 2n
t;h2
dt= A2n
0(h2)
2 ce2n 1
2;h2
ce2n
y;h2
2
C2n(h2)fe2n
y;h2
;
where the upper or lower sign is taken according as
0yory2. ForA2n
0(q) andC2n(q)
seexx28.4 and 28.5(i).
For details and further equations see Meixner et al.
(1980,x2.1.1) and Sips (1970).
28.28(ii) Integrals of Products with Bessel
Functions
With the notations of x28.4 forAn
m(q) andBn
m(q),
x28.14 forc
n(q), and (28.23.1) for C(j)
,j= 1;2;3;4,
28.28.171
Z
0C(j)
+2s(2hR)e i(+2s) me
t;h2
dt
= ( 1)sc
2s(h2) M(j)
(z;h), s2Z,
whereR=R(z;t) and = (z;t) are analytic functions
for<z>0 and realtwith
28.28.18R(z;t) = 1
2(cosh(2z) + cos(2t))1/2;
R(z;0) = coshz;and
28.28.19e2i =cosh(z+it)
cosh(z it);
(z;0) = 0:
In particular, for integer and`= 0;1;2;:::,
28.28.202
Z
0C(j)
2`(2hR) cos(2` ) ce2m
t;h2
dt
="`( 1)`+mA2m
2`(h2) Mc(j)
2m(z;h);
where again "0= 2 and"`= 1,`= 1;2;3;:::.
28.28.21
2
Z
0C(j)
2`+1(2hR) cos((2`+ 1) ) ce2m+1
t;h2
dt
= ( 1)`+mA2m+1
2`+1(h2) Mc(j)
2m+1(z;h);
28.28.22
2
Z
0C(j)
2`+1(2hR) sin((2`+ 1) ) se2m+1
t;h2
dt
= ( 1)`+mB2m+1
2`+1(h2) Ms(j)
2m+1(z;h);
674 Mathieu Functions and Hill's Equation
28.28.23
2
Z
0C(j)
2`+2(2hR) sin((2`+ 2) ) se2m+2
t;h2
dt
= ( 1)`+mB2m+2
2`+2(h2) Ms(j)
2m+2(z;h):
28.28(iii) Integrals of Products of Mathieu
Functions of Noninteger Order
With the parameter hsuppressed we use the notation
28.28.24
D0(;;z ) = M(3)
(z) M(4)
(z) M(4)
(z) M(3)
(z);
D1(;;z ) = M(3)
0(z) M(4)
(z) M(4)
0(z) M(3)
(z);
and assume =2Zandm2Z. Then
28.28.25
sinhz
2Z2
0costme
t;h2
me 2m 1
t;h2
sinh2z+ sin2tdt
= ( 1)m+1ih (0)
;mD0(;+ 2m+ 1;z);
28.28.26
coshz
2Z2
0sintme
t;h2
me 2m 1
t;h2
sinh2z+ sin2tdt
= ( 1)m+1ih (1)
;mD0(;+ 2m+ 1;z);
where
28.28.27
(0)
;m=1
2Z2
0costme
t;h2
me 2m 1
t;h2
dt
= ( 1)m2i
me
0;h2
me 2m 1
0;h2
hD0(;+ 2m+ 1;0);
28.28.28
(1)
;m=1
2Z2
0sintme
t;h2
me 2m 1
t;h2
dt
= ( 1)m+12i
me0
0;h2
me 2m 1
0;h2
hD1(;+ 2m+ 1;0):
For further integrals see http://dlmf.nist.gov/
28.28.iii .
28.28(iv) Integrals of Products of Mathieu
Functions of Integer Order
Again with the parameter hsuppressed, let
28.28.35
Ds0(n;m;z ) = Ms(3)
n(z) Ms(4)
m(z) Ms(4)
n(z) Ms(3)
m(z);
Ds1(n;m;z ) = Ms(3)
n0(z) Ms(4)
m(z) Ms(4)
n0(z) Ms(3)
m(z);
Ds2(n;m;z ) = Ms(3)
n0(z) Ms(4)
m0(z) Ms(4)
n0(z) Ms(3)
m0(z):
Then
28.28.36sinhz
2Z2
0costsen
t;h2
sem
t;h2
sinh2z+ sin2tdt
= ( 1)p+1ihb (s)
n;mDs0(n;m;z );28.28.37coshz
2Z2
0sintse0
n
t;h2
sem
t;h2
sinh2z+ sin2tdt
= ( 1)p+1ihb (s)
n;mDs1(n;m;z );
wherem n= 2p+ 1,p2Z;m;n = 1;2;3;:::. Also,
28.28.38b (s)
n;m=1
2Z2
0costsen
t;h2
sem
t;h2
dt
= ( 1)p2
ise0
n
0;h2
se0
m
0;h2
hDs2(n;m; 0):
For further integrals see http://dlmf.nist.gov/
28.28.iv and Sch afke (1983).
28.28(v) Compendia
See Prudnikov et al. (1990, pp. 359{368), Gradshteyn
and Ryzhik (2000, pp. 755{759), Sips (1970), and
Meixner et al. (1980,x2.1.1).
Hill's Equation
28.29 De nitions and Basic Properties
28.29(i) Hill's Equation
A generalization of Mathieu's equation (28.2.1) is Hill's
equation
28.29.1 w00(z) + (+Q(z))w= 0;
with
28.29.2 Q(z+) =Q(z);
and
28.29.3Z
0Q(z)dz= 0:
Q(z) is either a continuous and real-valued function for
z2Ror an analytic function of zin a doubly-in nite
open strip that contains the real axis. is the minimum
period ofQ.
28.29(ii) Floquet's Theorem and the
Characteristic Exponent
The basic solutions wI(z;),wII(z;) are de ned in the
same way as inx28.2(ii) (compare (28.2.5), (28.2.6)).
Then
28.29.4
wI(z+;) =wI(;)wI(z;) +w0
I(;)wII(z;);
28.29.5
wII(z+;) =wII(;)wI(z;) +w0
II(;)wII(z;):
Letbe a real or complex constant satisfying (with-
out loss of generality)
28.29.6 1<<1
throughout this section. Then (28.29.1) has a nontrivial
solutionw(z) with the pseudoperiodic property
28.29.7 w(z+) =eiw(z);
28.29 Definitions and Basic Properties 675
i eiis an eigenvalue of the matrix
28.29.8wI(;)wII(;)
w0
I(;)w0
II(;)
:
Equivalently,
28.29.9 2 cos() =wI(;) +w0
II(;):
This is the characteristic equation of (28.29.1), and
cos() is an entire function of . Giventogether with
the condition (28.29.6), the solutions of (28.29.9)
are the characteristic exponents of (28.29.1). A solu-
tion satisfying (28.29.7) is called a Floquet solution with
respect to(orFloquet solution ). It has the form
28.29.10 F(z) =eizP(z);
where the function P(z) is-periodic.
If(6= 0;1) is a solution of (28.29.9), then F(z),
F (z) comprise a fundamental pair of solutions of Hill's
equation.
If= 0 or 1, then (28.29.1) has a nontrivial solu-
tionP(z) which is periodic with period (when= 0)
or 2(when= 1). Let w(z) be a solution linearly
independent of P(z). Then
28.29.11 w(z+) = ( 1)w(z) +cP(z);
wherecis a constant. The case c= 0 is equivalent to
28.29.12wI(;)wII(;)
w0
I(;)w0
II(;)
=( 1)0
0 ( 1)
:
The solutions of period or 2are exceptional in the
following sense. If (28.29.1) has a periodic solution with
minimum period n,n= 3;4;:::, then all solutions are
periodic with period n.
Furthermore, for each solution w(z) of (28.29.1)
28.29.13w(z+) +w(z ) = 2 cos()w(z):
A nontrivial solution w(z) is either a Floquet solution
with respect to , orw(z+) eiw(z) is a Floquet
solution with respect to .
In the symmetric case Q(z) =Q( z),wI(z;) is an
even solution and wII(z;) is an odd solution; compare
x28.2(ii). (28.29.9) reduces to
28.29.14 cos() =wI(;):
The cases= 0 and= 1 split into four subcases
as in (28.2.21) and (28.2.22). The -periodic or -
antiperiodic solutions are multiples of wI(z;);wII(z;),
respectively.
For details and proofs see Magnus and Winkler
(1966,x1.3).
28.29(iii) Discriminant and Eigenvalues in the
Real Case
Q(x) is assumed to be real-valued throughout this sub-
section.The function
28.29.154() =wI(;) +w0
II(;)
is called the discriminant of (28.29.1). It is an entire
function of . Its order of growth for jj!1 is exactly
1
2; see Magnus and Winkler (1966, Chapter II, pp. 19{
28).
For a given , the characteristic equation 4()
2 cos() = 0 has in nitely many roots . Conversely,
for a given , the value of4() is needed for the com-
putation of . For this purpose the discriminant can
be expressed as an in nite determinant involving the
Fourier coecients of Q(x); see Magnus and Winkler
(1966,x2.3, pp. 28{36).
To every equation (28.29.1), there belong two in-
creasing in nite sequences of real eigenvalues :
28.29.16n; n= 0;1;2;:::; with4(n) = 2;
28.29.17n; n= 1;2;3;:::; with4(n) = 2:
In consequence, (28.29.1) has a solution of period i
=n, and a solution of period 2 i =n. Both
nandn!1 asn!1 , and interlace according to
the inequalities
28.29.18
0<12<12<34<34<:
Assume that the second derivative of Q(x) in
(28.29.1) exists and is continuous. Then with
28.29.19 N=1
Z
0(Q(x))2dx;
we have for m!1
28.29.202m 1 (2m 1)2 N
(4m)2=o
m 2
;
2m (2m 1)2 N
(4m)2=o
m 2
;
28.29.212m 1 (2m)2 N
(4m)2=o
m 2
;
2m (2m)2 N
(4m)2=o
m 2
:
IfQ(x) haskcontinuous derivatives, then as m!1
28.29.222m 2m 1=o
1
mk
;
2m 2m 1=o
1
mk
;
see Hochstadt (1963).
For further results, especially when Q(z) is analytic
in a strip, see Weinstein and Keller (1987).
676 Mathieu Functions and Hill's Equation
28.30 Expansions in Series of
Eigenfunctions
28.30(i) Real Variable
Letbm,m= 0;1;2;:::, be the set of characteristic val-
ues (28.29.16) and (28.29.17), arranged in their natural
order (see (28.29.18)), and let wm(x),m= 0;1;2;:::,
be the eigenfunctions , that is, an orthonormal set of
2-periodic solutions; thus
28.30.1 w00
m+ (bm+Q(x))wm= 0;
28.30.21
2Z2
0wm(x)wn(x)dx=m;n:
Then every continuous 2 -periodic function f(x) whose
second derivative is square-integrable over the interval
[0;2] can be expanded in a uniformly and absolutely
convergent series
28.30.3 f(x) =1X
m=0fmwm(x);
where
28.30.4 fm=1
2Z2
0f(x)wm(x)dx:
28.30(ii) Complex Variable
For analogous results to those of x28.19, see Sch afke
(1960, 1961b), and Meixner et al. (1980,x1.1.11).28.31 Equations of Whittaker{Hill and Ince
28.31(i) Whittaker{Hill Equation
Hill's equation with three terms
28.31.1W00+
A+Bcos(2z) 1
2(kc)2cos(4z)
W= 0
and constant values of A;B;k , andc, is called the Equa-
tion of Whittaker{Hill . It has been discussed in detail
by Arscott (1967) for k2<0, and by Urwin and Arscott
(1970) fork2>0.
28.31(ii) Equation of Ince; Ince Polynomials
Whenk2<0, we substitute
28.31.22= 4k2c2; A = 1
82; B = (p+ 1);
W(z) =w(z) exp
1
4cos(2z)
;
in (28.31.1). The result is the Equation of Ince :
28.31.3w00+sin(2z)w0+ ( pcos(2z))w= 0:
Formal 2-periodic solutions can be constructed as
Fourier series; compare x28.4:
28.31.4we;s(z) =1X
`=0A2`+scos (2`+s)z,s= 0;1,
28.31.5wo;s(z) =1X
`=0B2`+ssin (2`+s)z,s= 1;2,
where the coecients satisfy
28.31.6 2A0+ (2 +p)A2= 0; pA 0+ (4 )A2+ 1
2p+ 2
A4= 0;
(1
2p `+ 1)A2` 2+
4`2
A2`+ (1
2p+`+ 1)A2`+2= 0, `2,
28.31.7
1 + 1
2p+1
2
A1+ 1
2p+3
2
A3= 0;
(1
2p `+1
2)A2` 1+
(2`+ 1)2
A2`+1+ (1
2p+`+3
2)A2`+3= 0, `1,
28.31.8
1 1
2p+1
2
B1+ 1
2p+3
2
B3= 0;
(1
2p `+1
2)B2` 1+
(2`+ 1)2
B2`+1+ (1
2p+`+3
2)B2`+3= 0, `1,
28.31.9(4 )B2+ 1
2p+ 2
B4= 0;
(1
2p `+ 1)B2` 2+ (4`2 )B2`+ (1
2p+`+ 1)B2`+2= 0, `2.
Whenpis a nonnegative integer, the parameter
can be chosen so that solutions of (28.31.3) are trigono-
metric polynomials, called Ince polynomials . They are
denoted by
28.31.10C2m
2n(z;) withp= 2n;
C2m+1
2n+1(z;) withp= 2n+ 1;
28.31.11S2m+1
2n+1(z;) withp= 2n+ 1;
S2m+2
2n+2(z;) withp= 2n+ 2;andm= 0;1;:::;n in all cases.
The values of corresponding to Cm
p(z;),Sm
p(z;)
are denoted by am
p(),bm
p(), respectively. They are real
and distinct, and can be ordered so that Cm
p(z;) and
Sm
p(z;) have precisely mzeros, all simple, in 0 z< .
The normalization is given by
28.31.12
1
Z2
0
Cm
p(x;)2dx=1
Z2
0
Sm
p(x;)2dx= 1;
Applications 677
ambiguities in sign being resolved by requiring Cm
p(x;)
andSm
p0(x;) to be continuous functions of xand posi-
tive whenx= 0.
For!0, withx xed,
28.31.13
C0
p(x;)!1=p
2; Cm
p(x;)!cos(mx);
Sm
p(x;)!sin(mx),m6= 0;am
p(); bm
p()!m2:
Ifp!1 and!0 in such a way that p!2q,
then in the notation of xx28.2(v) and 28.2(vi)
28.31.14Cm
p(x;)!cem(x;q); Sm
p(x;)!sem(x;q);
28.31.15am
p()!am(q); bm
p()!bm(q):
For proofs and further information, including con-
vergence of the series (28.31.4), (28.31.5), see Arscott
(1967).
28.31(iii) Paraboloidal Wave Functions
With (28.31.10) and (28.31.11),
28.31.16 hcm
p(z;) =e 1
4cos(2z)Cm
p(z;);
28.31.17 hsm
p(z;) =e 1
4cos(2z)Sm
p(z;);
are called paraboloidal wave functions . They satisfy the
di erential equation
28.31.18
w00+
1
82 (p+ 1)cos(2z) +1
82cos(4z)
w= 0;
with=am
p(),=bm
p(), respectively.
For change of sign of ,
28.31.19
hc2m
2n(z; ) = ( 1)mhc2m
2n(1
2 z;);
hc2m+1
2n+1(z; ) = ( 1)mhs2m+1
2n+1(1
2 z;);
and
28.31.20
hs2m+1
2n+1(z; ) = ( 1)mhc2m+1
2n+1(1
2 z;);
hs2m+2
2n+2(z; ) = ( 1)mhs2m+2
2n+2(1
2 z;):
Form16=m2,
28.31.21Z2
0hcm1
p(x;)hcm2
p(x;)dx
=Z2
0hsm1
p(x;)hsm2
p(x;)dx= 0:
More important are the double orthogonality relations
forp16=p2orm16=m2or both, given by
28.31.22Zu1
u0Z2
0hcm1
p1(u;)hcm1
p1(v;)hcm2
p2(u;)hcm2
p2(v;)
(cos(2u) cos(2v))dvdu = 0;and
28.31.23Zu1
u0Z2
0hsm1
p1(u;)hsm1
p1(v;)hsm2
p2(u;)hsm2
p2(v;)
(cos(2u) cos(2v))dvdu = 0;
and also for all p1;p2;m1;m2, given by
28.31.24Zu1
u0Z2
0hcm1
p1(u;)hcm1
p1(v;)hsm2
p2(u;)hsm2
p2(v;)
(cos(2u) cos(2v))dvdu = 0;
where (u0;u1) = (0;i1) when >0, and (u0;u1) =
(1
2;1
2+i1) when<0.
For proofs and further integral equations see Urwin
(1964, 1965).
Asymptotic Behavior
For > 0, the functions hcm
p(z;),hsm
p(z;) behave
asymptotically as multiples of exp
1
4cos(2z)
(cosz)p
asz!i1. All other periodic solutions behave as
multiples of exp 1
4cos(2z)
(cosz) p 2.
For > 0, the functions hcm
p(z; ),
hsm
p(z; ) behave asymptotically as multiples of
exp 1
4cos(2z)
(cosz) p 2asz!1
2i1. All
other periodic solutions behave as multiples of
exp
1
4cos(2z)
(cosz)p.
Applications
28.32 Mathematical Applications
28.32(i) Elliptical Coordinates and an Integral
Relationship
If the boundary conditions in a physical problem relate
to the perimeter of an ellipse, then elliptical coordinates
are convenient. These are given by
28.32.1x=ccoshcos; y =csinhsin:
The two-dimensional wave equation
28.32.2@2V
@x2+@2V
@y2+k2V= 0
then becomes
28.32.3@2V
@2+@2V
@2+1
2c2k2(cosh(2) cos(2))V= 0:
The separated solutions V(;) =v()w() can be ob-
tained from the modi ed Mathieu's equation (28.20.1)
forvand from Mathieu's equation (28.2.1) for w, where
ais the separation constant and q=1
4c2k2.
This leads to integral equations and an integral re-
lation between the solutions of Mathieu's equation (set-
ting=i,z=in (28.32.3)).
678 Mathieu Functions and Hill's Equation
Letu() be a solution of Mathieu's equation (28.2.1)
andK(z;) be a solution of
28.32.4@2K
@z2 @2K
@2= 2q(cos(2z) cos(2))K:
Also letLbe a curve (possibly improper) such that the
quantity
28.32.5 K(z;)du()
d u()@K(z;)
@
approaches the same value when tends to the end-
points ofL. Then
28.32.6 w(z) =Z
LK(z;)u()d
de nes a solution of Mathieu's equation, provided that
(in the case of an improper curve) the integral converges
with respect to zuniformly on compact subsets of C.
KernelsKcan be found, for example, by separat-
ing solutions of the wave equation in other systems of
orthogonal coordinates. See Schmidt and Wolf (1979).
28.32(ii) Paraboloidal Coordinates
The general paraboloidal coordinate system is linked
with Cartesian coordinates via
28.32.7
x1=1
2c(cosh(2 ) + cos(2 ) cosh(2
));
x2= 2ccosh cos sinh
; x 3= 2csinh sin cosh
;
wherecis a parameter, 0 <1, < , and
0
<1. When the Helmholtz equation
28.32.8 r2V+k2V= 0
is separated in this system, each of the separated equa-
tions can be reduced to the Whittaker{Hill equation
(28.31.1), in which A;B areseparation constants . Two
conditions are used to determine A;B. The rst is the
2-periodicity of the solutions; the second can be their
asymptotic form. For further information see Arscott
(1967) fork2<0, and Urwin and Arscott (1970) for
k2>0.
28.33 Physical Applications
28.33(i) Introduction
Mathieu functions occur in practical applications in two
main categories:
Boundary-values problems arising from solution
of the two-dimensional wave equation in elliptical
coordinates. This yields a pair of equations of the
form (28.2.1) and (28.20.1), and the appropriate
solution of (28.2.1) is usually a periodic solution
of integer order. See x28.33(ii).
Initial-value problems, in which only one equation
(28.2.1) or (28.20.1) is involved. See x28.33(iii).28.33(ii) Boundary-Value Problems
Physical problems involving Mathieu functions in-
clude vibrational problems in elliptical coordinates; see
(28.32.1). We shall derive solutions to the uniform, ho-
mogeneous, loss-free, and stretched elliptical ring mem-
brane with mass per unit area, and radial tension
per unit arc length. The wave equation
28.33.1@2W
@x2+@2W
@y2
@2W
@t2= 0;
withW(x;y;t ) =ei!tV(x;y), reduces to (28.32.2) with
k2=!2= . In elliptical coordinates (28.32.2) becomes
(28.32.3). The separated solutions Vn(;) must be 2 -
periodic in , and have the form
28.33.2
Vn(;) =
cnM(1)
n(;pq) +dnM(2)
n(;pq)
men(;q);
whereq=1
4c2k2andan(q) orbn(q) is the sepa-
ration constant; compare (28.12.11), (28.20.11), and
(28.20.12). Here cnanddnare constants. The boundary
conditions for =0(outer clamp) and =1(inner
clamp) yield the following equation for q:
28.33.3M(1)
n(0;pq) M(2)
n(1;pq)
M(1)
n(1;pq) M(2)
n(0;pq) = 0:
If we denote the positive solutions qof (28.33.3) by
qn;m, then the vibration of the membrane is given by
!2
n;m= 4qn;m
(c2). The general solution of the prob-
lem is a superposition of the separated solutions.
For a visualization see Guti errez-Vega et al. (2003),
and for references to other boundary-value problems see:
McLachlan (1947, Chapters XVI{XIX) for appli-
cations of the wave equation to vibrational sys-
tems, electrical and thermal di usion, electromag-
netic wave guides, elliptical cylinders in viscous
uids, and di raction of sound and electromag-
netic waves.
Meixner and Sch afke (1954, xx4.3, 4.4) for elliptic
membranes and electromagnetic waves.
Daymond (1955) for vibrating systems.
Troesch and Troesch (1973) for elliptic mem-
branes.
Alhargan and Judah (1995), Bhattacharyya and
Shafai (1988), and Shen (1981) for ring antennas.
Alhargan and Judah (1992), Germey (1964),
Ragheb et al. (1991), and Sips (1967) for electro-
magnetic waves.
More complete bibliographies will be found in
McLachlan (1947) and Meixner and Sch afke (1954).
Computation 679
28.33(iii) Stability and Initial-Value Problems
If the parameters of a physical system vary periodically
with time, then the question of stability arises, for ex-
ample, a mathematical pendulum whose length varies
as cos(2!t). The equation of motion is given by
28.33.4w00(t) + (b fcos(2!t))w(t) = 0;
withb,f, and!positive constants. Substituting z=!t,
a=b
!2, and 2q=f
!2, we obtain Mathieu's stan-
dard form (28.2.1).
As!runs from 0 to + 1, withbandf xed, the
point (q;a) moves from1to 0 along the ray Lgiven
by the part of the line a= (2b=f)qthat lies in the
rst quadrant of the ( q;a)-plane. Hence from x28.17
the corresponding Mathieu equation is stable or unsta-
ble according as ( q;a) is in the intersection of Lwith
the colored or the uncolored open regions depicted in
Figure 28.17.1. In particular, the equation is stable for
all suciently large values of !.
For points ( q;a) that are at intersections of Lwith
the characteristic curves a=an(q) ora=bn(q), a pe-
riodic solution is possible. However, in response to a
small perturbation at least one solution may become
unbounded.
References for other initial-value problems include:
McLachlan (1947, Chapter XV) for amplitude dis-
tortion in moving-coil loud-speakers, frequency
modulation, dynamical systems, and vibration of
stretched strings.
Vedeler (1950) for ships rolling among waves.
Meixner and Sch afke (1954, xx4.1, 4.2, and 4.7)
for quantum mechanical problems and rotation of
molecules.
Alyet al. (1975) for scattering theory.
Hunter and Kuriyan (1976) and Rushchitsky and
Rushchitska (2000) for wave mechanics.
Fukui and Horiguchi (1992) for quantum theory.
Jager (1997, 1998) for relativistic oscillators.
Torres-Vega et al. (1998) for Mathieu functions in
phase space.Computation
28.34 Methods of Computation
28.34(i) Characteristic Exponents
Methods available for computing the values of
wI(;a;q) needed in (28.2.16) include:
(a) Direct numerical integration of the di erential
equation (28.2.1), with initial values given by
(28.2.5) (xx3.7(ii), 3.7(v)).
(b) Representations for wI(;a;q) with limit formu-
las for special solutions of the recurrence relations
x28.4(ii) for xed aandq; see Sch afke (1961a).
28.34(ii) Eigenvalues
Methods for computing the eigenvalues an(q),bn(q),
and(q), de ned inxx28.2(v) and 28.12(i), include:
(a) Summation of the power series in xx28.6(i) and
28.15(i) whenjqjis small.
(b) Use of asymptotic expansions and approximations
for largeq(xx28.8(i), 28.16). See also Zhang and
Jin (1996, pp. 482{485).
(c) Methods described in x3.7(iv) applied to the dif-
ferential equation (28.2.1) with the conditions
(28.2.5) and (28.2.16).
(d) Solution of the matrix eigenvalue problem for each
of the ve in nite matrices that correspond to the
linear algebraic equations (28.4.5){(28.4.8) and
(28.14.4). See Zhang and Jin (1996, pp. 479{482)
andx3.2(iv).
(e) Solution of the continued-fraction equations
(28.6.16){(28.6.19) and (28.15.2) by successive ap-
proximation. See Blanch (1966), Shirts (1993),
and Meixner and Sch afke (1954, x2.87).
28.34(iii) Floquet Solutions
(a) Summation of the power series in xx28.6(ii) and
28.15(ii) whenjqjis small.
(b) Use of asymptotic expansions and approximations
for largeq(xx28.8(ii){28.8(iv)).
Also, once the eigenvalues an(q),bn(q), and(q)
have been computed the following methods are applica-
ble:
680 Mathieu Functions and Hill's Equation
(c) Solution of (28.2.1) by boundary-value meth-
ods; seex3.7(iii). This can be combined with
x28.34(ii)(c).
(d) Solution of the systems of linear algebraic equa-
tions (28.4.5){(28.4.8) and (28.14.4), with the
conditions (28.4.9){(28.4.12) and (28.14.5), by
boundary-value methods ( x3.6) to determine the
Fourier coecients. Subsequently, the Fourier se-
ries can be summed with the aid of Clenshaw's
algorithm (x3.11(ii)). See Meixner and Sch afke
(1954,x2.87). This procedure can be combined
withx28.34(ii)(d).
28.34(iv) Modi ed Mathieu Functions
For the modi ed functions we have:
(a) Numerical summation of the expansions in series
of Bessel functions (28.24.1){(28.24.13). These se-
ries converge quite rapidly for a wide range of val-
ues ofqandz.
(b) Direct numerical integration ( x3.7) of the di eren-
tial equation (28.20.1) for moderate values of the
parameters.
(c) Use of asymptotic expansions for large zor large
q. Seexx28.25 and 28.26.
28.35 Tables
28.35(i) Real Variables
Blanch and Clemm (1962) includes values of
Mc(1)
n
x;pq
and Mc(1)
n0
x;pq
forn= 0(1)15
withq= 0(:05)1,x= 0(:02)1. Also Ms(1)
n
x;pq
and Ms(1)
n0
x;pq
forn= 1(1)15 with q=
0(:05)1,x= 0(:02)1. Precision is generally 7D.
Blanch and Clemm (1965) includes values of
Mc(2)
n
x;pq
, Mc(2)
n0
x;pq
forn= 0(1)7,x=
0(:02)1;n= 8(1)15, x= 0(:01)1. Also
Ms(2)
n
x;pq
, Ms(2)
n0
x;pq
forn= 1(1)7,x=
0(:02)1;n= 8(1)15,x= 0(:01)1. In all cases
q= 0(:05)1. Precision is generally 7D. Approxi-
mate formulas and graphs are also included.
Blanch and Rhodes (1955) includes Ben(t),
Bon(t),t=1
2pq,n= 0(1)15; 8D. The range of t
is 0 to 0.1, with step sizes ranging from 0.002 down
to 0.00025. Notation: Ben(t) =an(q)+2q (4n+
2)pq,Bon(t) =bn(q) + 2q (4n 2)pq.Ince (1932) includes eigenvalues an,bn, and
Fourier coecients for n= 0 or 1(1)6, q=
0(1)10(2)20(4)40; 7D. Also ce n(x;q), sen(x;q) for
q= 0(1)10, x= 1(1)90, corresponding to the
eigenvalues in the tables; 5D. Notation: an=
ben 2q,bn=bon 2q.
Kirkpatrick (1960) contains tables of the modi-
ed functions Ce n(x;q), Sen+1(x;q) forn= 0(1)5,
q= 1(1)20,x= 0:1(:1)1; 4D or 5D.
NBS (1967) includes the eigenvalues an(q),bn(q)
forn= 0(1)3 with q= 0(:2)20(:5)37(1)100, and
n= 4(1)15 with q= 0(2)100; Fourier coe-
cients for ce n(x;q) and sen(x;q) forn= 0(1)15,
n= 1(1)15, respectively, and various values of q
in the interval [0 ;100]; joining factors ge;n(pq),
fe;n(pq) forn= 0(1)15 with q= 0(:5 to 10)100
(but in a di erent notation). Also, eigenvalues for
large values of q. Precision is generally 8D.
Stratton et al. (1941) includes bn,b0
n, and the cor-
responding Fourier coecients for Se n(c;x) and
Son(c;x) forn= 0 or 1(1)4, c= 0(:1 or:2)4:5.
Precision is mostly 5S. Notation: c= 2pq,bn=
an+ 2q,b0
n=bn+ 2q, and for Se n(c;x), Son(c;x)
seex28.1.
Zhang and Jin (1996, pp. 521{532) includes
the eigenvalues an(q),bn+1(q) forn= 0(1)4,
q= 0(1)50; n= 0(1)20 ( a's) or 19 ( b's),
q= 1;3;5;10;15;25;50(50)200. Fourier co-
ecients for ce n(x;10), sen+1(x;10),n=
0(1)7. Mathieu functions ce n(x;10), sen+1(x;10),
and their rst x-derivatives for n= 0(1)4,
x= 0(5)90. Modi ed Mathieu functions
Mc(j)
n
x;p
10
, Ms(j)
n+1
x;p
10
, and their rst x-
derivatives for n= 0(1)4,j= 1;2,x= 0(:2)4.
Precision is mostly 9S.
28.35(ii) Complex Variables
Blanch and Clemm (1969) includes eigenvalues
an(q),bn(q) forq=ei ,= 0(:5)25, =
5(5)90,n= 0(1)15; 4D. Also an(q) andbn(q)
forq=i,= 0(:5)100,n= 0(2)14 and
n= 2(2)16, respectively; 8D. Double points for
n= 0(1)15; 8D. Graphs are included.
28.35(iii) Zeros
Blanch and Clemm (1965) includes the rst and
second zeros of Mc(2)
n
x;pq
, Mc(2)
n0
x;pq
for
n= 0;1, and Ms(2)
n
x;pq
, Ms(2)
n0
x;pq
for
n= 1;2, withq= 0(:05)1; 7D.
28.36 Software 681
Ince (1932) includes the rst zero for ce n, senfor
n= 2(1)5 or 6, q= 0(1)10(2)40; 4D. This ref-
erence also gives zeros of the rst derivatives, to-
gether with expansions for small q.
Zhang and Jin (1996, pp. 533{535) includes the
zeros (in degrees) of ce n(x;10), sen(x;10) forn=
1(1)10, and the rst 5 zeros of Mc(j)
n
x;p
10
,
Ms(j)
n
x;p
10
forn= 0 or 1(1)8, j= 1;2. Preci-
sion is mostly 9S.
28.35(iv) Further Tables
For other tables prior to 1961 see Fletcher et al. (1962,
x2.2) and Lebedev and Fedorova (1960, Chapter 11).
28.36 Software
Seehttp://dlmf.nist.gov/28.36 .
References
General References
The main references used in writing this chapter are Ar-
scott (1964b), McLachlan (1947), Meixner and Sch afke
(1954), and Meixner et al. (1980). Forxx28.29{28.30
the main source is Magnus and Winkler (1966).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x28.2 Arscott (1964b, Chapter II), Erd elyi et al. (1955,
xx16.2, 16.4), McLachlan (1947, Chapter II),
Meixner and Sch afke (1954, x2.1), Meixner et al.
(1980, Chapter 2). Figure 28.2.1 was produced at
NIST.
x28.3 These graphics were produced at NIST.
x28.4 Arscott (1964b, Chapter III), McLachlan (1947,
Chapter III), Meixner and Sch afke (1954, xx2.25,
2.71), Wolf (2008).
x28.5 Arscott (1964b,x2.4), McLachlan (1947, Chap-
ter VII), Meixner and Sch afke (1954, x2.7). The
graphics were produced at NIST.
x28.6 McLachlan (1947, Chapter II), Meixner and
Sch afke (1954,x2.2), Meixner et al. (1980,x2.4),
Volkmer (1998).x28.7 Meixner and Sch afke (1954, xx2.22, 2.25). Figure
28.7.1 was provided by the author.
x28.8 Goldstein (1927), Meixner and Sch afke (1954,
xx2.33, 2.84).
x28.10 Arscott (1964b, Chapter IV), Meixner and
Sch afke (1954,x2.6), Meixner et al. (1980,x2.1.2).
x28.11 Arscott (1964b,x3.9.1).
x28.12 Arscott (1964b, Chapter VI), McLachlan (1947,
Chapter IV), Meixner and Sch afke (1954, x2.2).
x28.13 These graphics were produced at NIST.
x28.14 Arscott (1964b, Chapter VI), McLachlan (1947,
Chapter IV), Meixner and Sch afke (1954, x2.2).
x28.15 Meixner and Sch afke (1954, x2.2).
x28.16 Meixner and Sch afke (1954, x2.2).
x28.17 Arscott (1964b,x6.2), McLachlan (1947, Chap-
ter III), Meixner and Sch afke (1954, x2.3). Figure
28.17.1 was recomputed by the author.
x28.19 Meixner and Sch afke (1954, x2.28).
x28.20 Arscott (1964b, Chapter VI), Meixner and
Sch afke (1954,x2.4).
x28.21 These graphics were produced at NIST.
x28.22 Meixner and Sch afke (1954, xx2.29, 2.65, 2.73,
2.76).
x28.23 Meixner and Sch afke (1954, x2.6).
x28.24 Meixner and Sch afke (1954, xx2.6, 2.7).
x28.26 Goldstein (1927), Meixner and Sch afke (1954,
x2.84), NBS (1967, IV).
x28.28 Arscott (1964b, Chapters IV and VI), McLach-
lan (1947, Chapters IX and XIV), Meixner and
Sch afke (1954,x2.7), Meixner et al. (1980,x2.1),
Sch afke (1983). There is a sign error on p. 158 of
the last reference.
x28.29 Magnus and Winkler (1966, Part I, pp. 1{43),
Arscott (1964b, Chapter VII), McLachlan (1947,
x6.10).
x28.30 Magnus and Winkler (1966, Part I, x2.5).
x28.31 Arscott (1967), Urwin and Arscott (1970), Ur-
win (1964, 1965).
x28.32 Arscott (1967,xx1.3 and 2.6), Meixner and
Sch afke (1954,x1.135).
Chapter 29
Lam e Functions
H. Volkmer1
Notation 684
29.1 Special Notation . . . . . . . . . . . . . 684
Lam e Functions 684
29.2 Di erential Equations . . . . . . . . . . . 684
29.3 De nitions and Basic Properties . . . . . 685
29.4 Graphics . . . . . . . . . . . . . . . . . . 686
29.5 Special Cases and Limiting Forms . . . . 688
29.6 Fourier Series . . . . . . . . . . . . . . . 688
29.7 Asymptotic Expansions . . . . . . . . . . 689
29.8 Integral Equations . . . . . . . . . . . . . 689
29.9 Stability . . . . . . . . . . . . . . . . . . 690
29.10 Lam e Functions with Imaginary Periods . 690
29.11 Lam e Wave Equation . . . . . . . . . . . 690
Lam e Polynomials 690
29.12 De nitions . . . . . . . . . . . . . . . . . 69029.13 Graphics . . . . . . . . . . . . . . . . . . 691
29.14 Orthogonality . . . . . . . . . . . . . . . 692
29.15 Fourier Series and Chebyshev Series . . . 692
29.16 Asymptotic Expansions . . . . . . . . . . 693
29.17 Other Solutions . . . . . . . . . . . . . . 693
Applications 693
29.18 Mathematical Applications . . . . . . . . 693
29.19 Physical Applications . . . . . . . . . . . 694
Computation 694
29.20 Methods of Computation . . . . . . . . . 694
29.21 Tables . . . . . . . . . . . . . . . . . . . 694
29.22 Software . . . . . . . . . . . . . . . . . . 695
References 695
1Department of Mathematical Sciences, University of Wisconsin{Milwaukee, Milwaukee, Wisconsin.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
683
684 Lame Functions
Notation
29.1 Special Notation
(For other notation see pp. xiv and 873.)
m;n;p nonnegative integers.
x real variable.
z complex variable.
h;k; real parameters, 0 <k< 1, 1
2.
k0p
1 k2, 0<k0<1.
K,K0complete elliptic integrals of the rst kind
with moduli k;k0, respectively (see x19.2(ii)).
All derivatives are denoted by di erentials, not by
primes.
The main functions treated in this chapter are
the eigenvalues a2m
k2
,a2m+1
k2
,b2m+1
k2
,
b2m+2
k2
, the Lam e functions Ec2m
z;k2
,
Ec2m+1
z;k2
,Es2m+1
z;k2
,Es2m+2
z;k2
, and
the Lam e polynomials uEm
2n
z;k2
,sEm
2n+1
z;k2
,
cEm
2n+1
z;k2
, dEm
2n+1
z;k2
, scEm
2n+2
z;k2
,
sdEm
2n+2
z;k2
,cdEm
2n+2
z;k2
,scdEm
2n+3
z;k2
. The
notation for the eigenvalues and functions is due to
Erd elyi et al. (1955,x15.5.1) and that for the polynomi-
als is due to Arscott (1964b, x9.3.2). The normalization
is that of Jansen (1977, x3.1).
Other notations that have been used are as follows:
Ince (1940a) interchanges a2m+1
k2
withb2m+1
k2
.
The relation to the Lam e functions L(m)
c,L(m)
sof Jansen
(1977) is given by
Ec2m
z;k2
= ( 1)mL(2m)
c( ;k02);
Ec2m+1
z;k2
= ( 1)mL(2m+1)
s ( ;k02);
Es2m+1
z;k2
= ( 1)mL(2m+1)
c ( ;k02);
Es2m+2
z;k2
= ( 1)mL(2m+2)
s ( ;k02);
where = am (z;k); seex22.16(i). The relation to the
Lam e functions Ecm
, Esm
of Ince (1940b) is given by
Ec2m
z;k2
=c2m
(k2)Ec2m
(z;k2);
Ec2m+1
z;k2
=c2m+1
(k2)Es2m+1
(z;k2);
Es2m+1
z;k2
=s2m+1
(k2)Ec2m+1
(z;k2);
Es2m+2
z;k2
=s2m+2
(k2)Es2m+2
(z;k2);
where the positive factors cm
(k2) andsm
(k2) are deter-
mined by
(cm
(k2))2=4
ZK
0
Ecm
x;k22dx;
(sm
(k2))2=4
ZK
0
Esm
x;k22dx:Lam e Functions
29.2 Di erential Equations
29.2(i) Lam e's Equation
29.2.1d2w
dz2+ (h (+ 1)k2sn2(z;k))w= 0;
wherekandare real parameters such that 0 <k< 1
and 1
2. For sn (z;k) seex22.2. This equation has
regular singularities at the points 2 pK+ (2q+ 1)iK0,
wherep;q2Z, andK,K0are the complete elliptic inte-
grals of the rst kind with moduli k,k0(= (1 k2)1=2),
respectively; seex19.2(ii). In general, at each singular-
ity each solution of (29.2.1) has a branch point ( x2.7(i)).
See Figure 29.2.1.
0
3iK0 2iK0 iK0iK02iK03iK0
4K 2K 2K 4K
Figure 29.2.1 :z-plane: singularities of Lam e's
equation.
29.2(ii) Other Forms
29.2.2d2w
d2+1
21
+1
1+1
k 2dw
d
+hk 2 (+ 1)
4( 1)( k 2)w= 0;
where
29.2.3 = sn2(z;k):
29.2.4(1 k2cos2 )d2w
d 2+k2cos sin dw
d
+ (h (+ 1)k2cos2 )w= 0;
where
29.2.5 =1
2 am (z;k):
For am (z;k) seex22.16(i).
29.3 Definitions and Basic Properties 685
Next, lete1;e2;e3be any real constants that satisfy
e1>e2>e3and
29.2.6e1+e2+e3= 0;(e2 e3)/(e1 e3) =k2:
(These constants are not unique.) Then with
29.2.7 g= (e1 e3)h+(+ 1)e3;
29.2.8 = (e1 e3) 1=2(z iK0);
we have
29.2.9d2w
d2+ (g (+ 1)}())w= 0;
and
29.2.10d2w
d2+1
21
e1+1
e2+1
e3dw
d
+g (+ 1)
4( e1)( e2)( e3)w= 0;
where
29.2.11 =}(;g2;g3) =}();
with
29.2.12g2= 4(e2e3+e3e1+e1e2); g 3= 4e1e2e3:
For the Weierstrass function }seex23.2(ii).
Equation (29.2.10) is a special case of Heun's equa-
tion (31.2.1).
29.3 De nitions and Basic Properties
29.3(i) Eigenvalues
For each pair of values of andkthere are four in-
nite unbounded sets of real eigenvalues hfor which
equation (29.2.1) has even or odd solutions with periods
2Kor 4K. They are denoted by a2m
k2
,a2m+1
k2
,
b2m+1
k2
,b2m+2
k2
, wherem= 0;1;2;:::; see Ta-
ble 29.3.1.
Table 29.3.1 : Eigenvalues of Lam e's equation.
eigenvaluehparity period
a2m
k2
even 2K
a2m+1
k2
odd 4K
b2m+1
k2
even 4K
b2m+2
k2
odd 2K
29.3(ii) Distribution
The eigenvalues interlace according to
29.3.1 am
k2
<am+1
k2
;
29.3.2 am
k2
<bm+1
k2
;
29.3.3 bm
k2
<bm+1
k2
;
29.3.4 bm
k2
<am+1
k2
:The eigenvalues coalesce according to
29.3.5 am
k2
=bm
k2
,= 0;1;:::;m 1.
Ifis distinct from 0 ;1;:::;m 1, then
29.3.6
am
k2
bm
k2
( 1)( m+ 1)>0:
Ifis a nonnegative integer, then
29.3.7
am
k2
+a m
1 k2
=(+ 1),m= 0;1;:::;;
29.3.8
bm
k2
+b m+1
1 k2
=(+ 1),m= 1;2;:::; .
For the special case k=k0= 1p
2 see Erd elyi et al.
(1955,x15.5.2).
29.3(iii) Continued Fractions
The quantity
29.3.9 H= 2a2m
k2
(+ 1)k2
satis es the continued-fraction equation
29.3.10 p H p 1
p
p 1 H p 2
p 1
p 2 H
= p
p+1
p+1 H p+1
p+2
p+2 H ;
wherepis any nonnegative integer, and
29.3.11 p=(
( 1)(+ 2)k2; p = 0;
1
2( 2p 1)(+ 2p+ 2)k2; p1;
29.3.12 p= 4p2(2 k2);
p=1
2( 2p+ 2)(+ 2p 1)k2:
The continued fraction following the second negative
sign on the left-hand side of (29.3.10) is nite: it equals
0 ifp= 0, and if p > 0, then the last denominator
is 0 H. Ifis a nonnegative integer and 2 p,
then the continued fraction on the right-hand side of
(29.3.10) terminates, and (29.3.10) has only the solu-
tions (29.3.9) with 2 m. Ifis a nonnegative inte-
ger and 2p > , then (29.3.10) has only the solutions
(29.3.9) with 2 m> .
For the corresponding continued-fraction equations
fora2m+1
k2
,b2m+1
k2
, andb2m+2
k2
seehttp:
//dlmf.nist.gov/29.3.iii .
29.3(iv) Lam e Functions
The eigenfunctions corresponding to the eigenvalues of
x29.3(i) are denoted by Ec2m
z;k2
,Ec2m+1
z;k2
,
Es2m+1
z;k2
,Es2m+2
z;k2
. They are called Lam e
functions with real periods and of order , or more sim-
ply, Lam e functions . See Table 29.3.2. In this table
the nonnegative integer mcorresponds to the number
of zeros of each Lam e function in (0 ;K), whereas the
superscripts 2 m, 2m+ 1, or 2m+ 2 correspond to the
number of zeros in [0 ;2K).
686 Lame Functions
Table 29.3.2 : Lam e functions.
boundary conditionseigenvalue
heigenfunction
w(z)parity of
w(z)parity of
w(z K)period of
w(z)
dw/dzjz=0=dw/dzjz=K= 0a2m
k2
Ec2m
z;k2
even even 2 K
w(0) =dw/dzjz=K= 0a2m+1
k2
Ec2m+1
z;k2
odd even 4 K
dw/dzjz=0=w(K) = 0 b2m+1
k2
Es2m+1
z;k2
even odd 4 K
w(0) =w(K) = 0 b2m+2
k2
Es2m+2
z;k2
odd odd 2 K
29.3(v) Normalization
29.3.18ZK
0dn (x;k)
Ec2m
x;k22dx=1
4;
ZK
0dn (x;k)
Ec2m+1
x;k22dx=1
4;
ZK
0dn (x;k)
Es2m+1
x;k22dx=1
4;
ZK
0dn (x;k)
Es2m+2
x;k22dx=1
4:
For dn (z;k) seex22.2.
To complete the de nitions, Ecm
K;k2
is positive
anddEsm
z;k2
dz
z=Kis negative.
29.3(vi) Orthogonality
Form6=p,
29.3.19ZK
0Ec2m
x;k2
Ec2p
x;k2
dx= 0;
ZK
0Ec2m+1
x;k2
Ec2p+1
x;k2
dx= 0;
ZK
0Es2m+1
x;k2
Es2p+1
x;k2
dx= 0;
ZK
0Es2m+2
x;k2
Es2p+2
x;k2
dx= 0:For the values of these integrals when m=pseex29.6.
29.3(vii) Power Series
For power-series expansions of the eigenvalues see Volk-
mer (2004b).
29.4 Graphics
29.4(i) Eigenvalues of Lam e's Equation: Line
Graphs
Figure 29.4.1 :am
(0:5),bm+1
(0:5) as functions of for
m= 0;1;2;3.
29.4 Graphics 687
Figure 29.4.2 :a3
(0:5) b3
(0:5) as a function of .
Figure 29.4.3 :am
1:5
k2
,bm+1
1:5
k2
as functions of k2for
m= 0;1;2.
For additional graphs see http://dlmf.nist.gov/29.4.i .
29.4(ii) Eigenvalues of Lam e's Equation: Surfaces
Figure 29.4.9 :a0
k2
as a function of andk2.
Figure 29.4.10 :b1
k2
as a function of andk2.
For additional surfaces see http://dlmf.nist.gov/29.4.ii .
29.4(iii) Lam e Functions: Line Graphs
Figure 29.4.13 :Ecm
1:5(x;0:5) for 2Kx2K,m=
0;1;2.K= 1:85407:::.
Figure 29.4.14 :Esm
1:5(x;0:5) for 2Kx2K,m=
1;2;3.K= 1:85407:::.
For additional graphs see http://dlmf.nist.gov/29.4.iii .
688 Lame Functions
29.4(iv) Lam e Functions: Surfaces
Figure 29.4.25 :Ec0
1:5
x;k2
as a function of xandk2.
Figure 29.4.26 :Es1
1:5
x;k2
as a function of xandk2.
For additional surfaces see http://dlmf.nist.gov/29.4.iv .
29.5 Special Cases and Limiting Forms
29.5.1 am
(0) =bm
(0) =m2;
29.5.2 Ec0
(z;0) = 2 1
2;
29.5.3Ecm
(z;0) = cos
m(1
2 z)
, m1;
Esm
(z;0) = sin
m(1
2 z)
, m1:
Let= max ( m;0). Then
29.5.4 lim
k!1 am
k2
= lim
k!1 bm+1
k2
=(+ 1) 2;
29.5.5
lim
k!1 Ecm
z;k2
Ecm
(0;k2)= lim
k!1 Esm+1
z;k2
Esm+1
(0;k2)
=1
(coshz)F
1
2 1
2;1
2+1
2+1
2
1
2; tanh2z!
,
meven,
29.5.6
lim
k!1 Ecm
z;k2
dEcm
(z;k2)/dzjz=0
= lim
k!1 Esm+1
z;k2
dEsm+1
(z;k2)
dz
z=0
=tanhz
(coshz)F
1
2 1
2+1
2;1
2+1
2+ 1
3
2; tanh2z!
,
modd,
whereFis the hypergeometric function; see x15.2(i).Ifk!0+ and! 1 in such a way that
k2(+ 1) = 4(a positive constant), then
29.5.7limEcm
z;k2
= cem 1
2 z;
;
limEsm
z;k2
= sem 1
2 z;
;
where cem(z;) and sem(z;) are Mathieu functions; see
x28.2(vi).
29.6 Fourier Series
29.6(i) Function Ec2m
z;k2
With =1
2 am (z;k), as in (29.2.5), we have
29.6.1 Ec2m
z;k2
=1
2A0+1X
p=1A2pcos(2p ):
Here
29.6.2 H= 2a2m
k2
(+ 1)k2;
29.6.3 ( 0 H)A0+ 0A2= 0;
29.6.4
pA2p 2+ ( p H)A2p+ pA2p+2= 0,p1,
with p, p, and
pas in (29.3.11) and (29.3.12), and
29.6.51
2A2
0+1X
p=1A2
2p= 1;
29.6.61
2A0+1X
p=1A2p>0:
29.7 Asymptotic Expansions 689
When6= 2n, wherenis a nonnegative integer, it
follows fromx2.9(i) that for any value of Hthe sys-
tem (29.6.4){(29.6.6) has a unique recessive solution
A0;A2;A4;:::; furthermore
29.6.7
lim
p!1A2p+2
A2p=k2
(1 +k0)2,6= 2n, or= 2nandm>n .
In addition, if Hsatis es (29.6.2), then (29.6.3) applies.
In the special case = 2n,m= 0;1;:::;n , there is
a unique nontrivial solution with the property A2p= 0,
p=n+ 1;n+ 2;:::. This solution can be con-
structed from (29.6.4) by backward recursion, start-
ing withA2n+2= 0 and an arbitrary nonzero value of
A2n, followed by normalization via (29.6.5) and (29.6.6).
Consequently, Ec2m
z;k2
reduces to a Lam e polyno-
mial; comparexx29.12(i) and 29.15(i).
An alternative version of the Fourier series expan-
sion (29.6.1) is given by
29.6.8
Ec2m
z;k2
= dn (z;k)
1
2C0+1X
p=1C2pcos(2p )!
:
Here dn (z;k) is as inx22.2, and
29.6.9 ( 0 H)C0+ 0C2= 0;
29.6.10
pC2p 2+ ( p H)C2p+ pC2p+2= 0,p1,
with p; p, and
pnow de ned by
29.6.11 p=(
(+ 1)k2; p = 0;
1
2( 2p)(+ 2p+ 1)k2; p1;
p= 4p2(2 k2);
p=1
2( 2p+ 1)(+ 2p)k2;
and
29.6.12
1 1
2k2
1
2C2
0+1X
p=1C2
2p!
1
2k21X
p=0C2pC2p+2= 1;
29.6.131
2C0+1X
p=1C2p>0;
29.6.14
lim
p!1C2p+2
C2p=k2
(1 +k0)2,
6= 2n+ 1, or= 2n+ 1 andm>n ,
29.6.15
1
2A0C0+1X
p=1A2pC2p=4
ZK
0
Ec2m
x;k22dx:
For the corresponding expansions for Ec2m+1
z;k2
,
Es2m+1
z;k2
, and Es2m+2
z;k2
seehttp://dlmf.
nist.gov/29.6.ii .29.7 Asymptotic Expansions
29.7(i) Eigenvalues
As!1 ,
29.7.1am
k2
p 0 1 1 2 2 ;
where
29.7.2=k((+ 1))1=2; p = 2m+ 1;
29.7.3 0=1
23(1 +k2)(1 +p2);
29.7.4 1=p
26((1 +k2)2(p2+ 3) 4k2(p2+ 5)):
The same Poincar e expansion holds for bm+1
k2
, since
29.7.5
bm+1
k2
am
k2
=O
m+3
21 k
1 +k
,!1 .
See also Volkmer (2004b).
For higher terms in (29.7.1) see http://dlmf.nist.
gov/29.7.i .
29.7(ii) Lam e Functions
M uller (1966a,b) found three formal asymptotic expan-
sions for a fundamental system of solutions of (29.2.1)
(and (29.11.1)) as !1 , one in terms of Jacobian
elliptic functions and two in terms of Hermite polyno-
mials. In M uller (1966c) it is shown how these expan-
sions lead to asymptotic expansions for the Lam e func-
tions Ecm
z;k2
andEsm
z;k2
. Weinstein and Keller
(1985) give asymptotics for solutions of Hill's equation
(x28.29(i)) that are applicable to the Lam e equation.
29.8 Integral Equations
Letw(z) be any solution of (29.2.1) of period 4 K,w2(z)
be a linearly independent solution, and Wfw;w 2gde-
note their Wronskian. Also let xbe de ned by
29.8.1x=k2sn (z;k) sn (z1;k) sn (z2;k) sn (z3;k)
k2
k02cn (z;k) cn (z1;k) cn (z2;k) cn (z3;k)
+1
k02dn (z;k) dn (z1;k) dn (z2;k) dn (z3;k);
wherez;z1;z2;z3are real, and sn, cn, dn are the Jaco-
bian elliptic functions ( x22.2). Then
29.8.2w(z1)w(z2)w(z3) =Z2K
2KP(x)w(z)dz;
where P(x) is the Ferrers function of the rst kind
(x14.3(i)),
29.8.3 =2
Wfw;w 2g;
690 Lame Functions
and(=1) and are determined by
29.8.4w(z+ 2K) =w(z);
w2(z+ 2K) = w(z) +w2(z):
A special case of (29.8.2) is
29.8.5Ec2m
z1;k2w2(K) w2( K)
dw2(z)/dzjz=0
=ZK
KP(y)Ec2m
z;k2
dz;
where
29.8.6 y=1
k0dn (z;k) dn (z1;k):
For results corresponding to (29.8.5) for Ec2m+1
,
Es2m+1
,Es2m+2
seehttp://dlmf.nist.gov/29.8 .
For further integral equations see Arscott (1964a),
Erd elyi et al. (1955,x15.5.3), Shail (1980), Sleeman
(1968a), and Volkmer (1982, 1983, 1984).
29.9 Stability
The Lam e equation (29.2.1) with speci ed values of
k;h; is called stable if all of its solutions are bounded
onR; otherwise the equation is called unstable . Ifis
not an integer, then (29.2.1) is unstable i ha0
k2
orhlies in one of the closed intervals with endpoints
am
k2
andbm
k2
,m= 1;2;:::. Ifis a nonnega-
tive integer, then (29.2.1) is unstable i ha0
k2
or
h2[bm
k2
;am
k2
] for somem= 1;2;:::; .
29.10 Lam e Functions with Imaginary
Periods
The substitutions
29.10.1 h=(+ 1) h0;
29.10.2 z0=i(z K iK0);
transform (29.2.1) into
29.10.3d2w
dz02+ (h0 (+ 1)k02sn2(z0;k0))w= 0:
In consequence, the functions
29.10.4Ec2m
i(z K iK0);k02
;
Ec2m+1
i(z K iK0);k02
;
Es2m+1
i(z K iK0);k02
;
Es2m+2
i(z K iK0);k02
;
are solutions of (29.2.1). The rst and the fourth func-
tions have period 2 iK0; the second and the third have
period 4iK0.
For these results and further information see Erd elyi
et al. (1955,x15.5.2).29.11 Lam e Wave Equation
The Lam e (orellipsoidal )wave equation is given by
29.11.1
d2w
dz2+ (h (+ 1)k2sn2(z;k) +k2!2sn4(z;k))w= 0;
in which!is another parameter. In the case != 0,
(29.11.1) reduces to Lam e's equation (29.2.1).
For properties of the solutions of (29.11.1) see Ar-
scott (1956, 1959), Arscott (1964b, Chapter X), Erd elyi
et al. (1955,x16.14), Fedoryuk (1989), and M uller
(1966a,b,c).
Lam e Polynomials
29.12 De nitions
29.12(i) Elliptic-Function Form
Throughoutxx29.12{29.16 the orderin the di erential
equation (29.2.1) is assumed to be a nonnegative integer .
The Lam e functions Ecm
z;k2
,m= 0;1;:::; ,
and Esm
z;k2
,m= 1;2;:::; , are called the Lam e
polynomials . There are eight types of Lam e polynomi-
als, de ned as follows:
29.12.1 uEm
2n
z;k2
=Ec2m
2n
z;k2
;
29.12.2 sEm
2n+1
z;k2
=Ec2m+1
2n+1
z;k2
;
29.12.3 cEm
2n+1
z;k2
=Es2m+1
2n+1
z;k2
;
29.12.4 dEm
2n+1
z;k2
=Ec2m
2n+1
z;k2
;
29.12.5 scEm
2n+2
z;k2
=Es2m+2
2n+2
z;k2
;
29.12.6 sdEm
2n+2
z;k2
=Ec2m+1
2n+2
z;k2
;
29.12.7 cdEm
2n+2
z;k2
=Es2m+1
2n+2
z;k2
;
29.12.8 scdEm
2n+3
z;k2
=Es2m+2
2n+3
z;k2
;
wheren= 0;1;2;:::,m= 0;1;2;:::;n . These func-
tions are polynomials in sn ( z;k), cn (z;k), and dn (z;k).
In consequence they are doubly-periodic meromorphic
functions of z.
The superscript mon the left-hand sides of
(29.12.1){(29.12.8) agrees with the number of z-zeros
of each Lam e polynomial in the interval (0 ;K), while
n mis the number of z-zeros in the open line segment
fromKtoK+iK0.
The pre xes u,s,c,d,sc,sd,cd,scdindicate the
type of the polynomial form of the Lam e polynomial;
compare the 3rd and 4th columns in Table 29.12.1. In
the fourth column the variable zand modulus kof the
Jacobian elliptic functions have been suppressed, and
P(sn2) denotes a polynomial of degree nin sn2(z;k)
(di erent for each type). For the determination of the
coecients of the P's seex29.15(ii).
29.13 Graphics 691
Table 29.12.1 : Lam e polynomials.
eigenvalue
heigenfunction
w(z)polynomial
formreal
periodimag.
periodparity of
w(z)parity of
w(z K)parity of
w(z K iK0)
2n a2m
k2
uEm
z;k2
P(sn2) 2K 2iK0even even even
2n+ 1a2m+1
k2
sEm
z;k2
snP(sn2) 4K 2iK0odd even even
2n+ 1b2m+1
k2
cEm
z;k2
cnP(sn2) 4K 4iK0even odd even
2n+ 1a2m
k2
dEm
z;k2
dnP(sn2) 2K 4iK0even even odd
2n+ 2b2m+2
k2
scEm
z;k2
sn cnP(sn2) 2K 4iK0odd odd even
2n+ 2a2m+1
k2
sdEm
z;k2
sn dnP(sn2) 4K 4iK0odd even odd
2n+ 2b2m+1
k2
cdEm
z;k2
cn dnP(sn2) 4K 2iK0even odd odd
2n+ 3b2m+2
k2
scdEm
z;k2
sn cn dnP(sn2) 2K 2iK0odd odd odd
29.12(ii) Algebraic Form
With the substitution = sn2(z;k) every Lam e poly-
nomial in Table 29.12.1 can be written in the form
29.12.9 ( 1)( k 2) P();
where,, are either 0 or1
2. The polynomial P()
is of degree nand hasmzeros (all simple) in (0 ;1)
andn mzeros (all simple) in (1 ;k 2). The functions
(29.12.9) satisfy (29.2.2).
29.12(iii) Zeros
Let1;2;:::;ndenote the zeros of the polynomial P
in (29.12.9) arranged according to
29.12.10 0<1<<m<1<m+1<<n<k 2:
Then the function
29.12.11
g(t1;t2;:::;tn)
= nY
p=1t+1
4pjtp 1j+1
4(k 2 tp) +1
4!Y
q<r(tr tq);
de ned for ( t1;t2;:::;tn) with
29.12.12 0t1tm1tm+1tnk 2;
attains its absolute maximum i tj=j,j= 1;2;:::;n .
Moreover,
29.12.13
+1
4
p++1
4
p 1+ +1
4
p k 2+nX
q=1
q6=p1
p q= 0,
p= 1;2;:::;n .
This result admits the following electrostatic inter-
pretation: Given three point masses xed at t= 0,
t= 1, andt=k 2with positive charges +1
4,+1
4,and +1
4, respectively, and nmovable point masses at
t1;t2;:::;tnarranged according to (29.12.12) with unit
positive charges, the equilibrium position is attained
whentj=jforj= 1;2;:::;n .
29.13 Graphics
29.13(i) Eigenvalues for Lam e Polynomials
Figure 29.13.1 :am
2
k2
,bm
2
k2
as functions of k2for
m= 0;1;2 (a's),m= 1;2 (b's).
For additional graphs see http://dlmf.nist.gov/
29.13.i .
692 Lame Functions
29.13(ii) Lam e Polynomials: Real Variable
Figure 29.13.5 :uEm
4(x;0:1) for 2Kx2K,
m= 0;1;2.K= 1:61244:::.
For additional graphs see http://dlmf.nist.gov/
29.13.ii .
29.13(iii) Lam e Polynomials: Complex Variable
Figure 29.13.21 :juE1
4(x+iy;0:1)jfor 3Kx3K,
0y2K0.K= 1:61244:::,K0= 2:57809:::.
For additional graphics see http://dlmf.nist.
gov/29.13.iii .
29.14 Orthogonality
Lam e polynomials are orthogonal in two ways.
First, the orthogonality relations (29.3.19) apply; see
x29.12(i). Secondly, the system of functions
29.14.1fm
n(s;t) =uEm
2n
s;k2
uEm
2n
K+it;k2
,
n= 0;1;2;:::,m= 0;1;:::;n;
is orthogonal and complete with respect to the inner
product
29.14.2hg;hi=ZK
0ZK0
0w(s;t)g(s;t)h(s;t)dtds;
where
29.14.3w(s;t) = sn2(K+it;k) sn2(s;k):For the corresponding results for the other seven
types of Lam e polynomials see http://dlmf.nist.
gov/29.14 .
29.15 Fourier Series and Chebyshev Series
29.15(i) Fourier Coecients
Polynomial uEm
2n
z;k2
When= 2n,m= 0;1;:::;n , the Fourier series
(29.6.1) terminates:
29.15.1 uEm
2n
z;k2
=1
2A0+nX
p=1A2pcos(2p ):
A convenient way of constructing the coecients, to-
gether with the eigenvalues, is as follows. Equations
(29.6.4), with p= 1;2;:::;n , (29.6.3), and A2n+2= 0
can be cast as an algebraic eigenvalue problem in the
following way. Let
29.15.2 M=2
66666664 0 0 0 0
1 1 1......
0......... 0
......
n 1 n 1 n 1
0 0
n n3
77777775
be the tridiagonal matrix with p, p,
pas in (29.3.11),
(29.3.12). Let the eigenvalues of MbeHpwith
29.15.3 H0<H 1<<Hn;
and also let
29.15.4 [A0;A2;:::;A 2n]T
be the eigenvector corresponding to Hmand normalized
so that
29.15.5 1
2A2
0+nX
p=1A2
2p= 1
and
29.15.6 1
2A0+nX
p=1A2p>0:
Then
29.15.7 a2m
k2
=1
2(Hm+(+ 1)k2);
and (29.15.1) applies, with again de ned as in (29.2.5).
For the corresponding formulations for the other
seven types of Lam e polynomials see http://dlmf.
nist.gov/29.15.i .
29.16 Asymptotic Expansions 693
29.15(ii) Chebyshev Series
The Chebyshev polynomial Tof the rst kind ( x18.3)
satis es cos( p ) =Tp(cos ). Since (29.2.5) implies that
cos = sn (z;k), (29.15.1) can be rewritten in the form
29.15.43 uEm
2n
z;k2
=1
2A0+nX
p=1A2pT2p(sn (z;k)):
This determines the polynomial Pof degreenfor which
uEm
2n
z;k2
=P(sn2(z;k)); compare Table 29.12.1.
The set of coecients of this polynomial (without nor-
malization) can also be found directly as an eigenvector
of an (n+ 1)(n+ 1) tridiagonal matrix; see Arscott
and Khabaza (1962).
For the corresponding expansions of the other seven
types of Lam e polynomials see http://dlmf.nist.
gov/29.15.ii .
For explicit formulas for Lam e polynomials of low
degree, see Arscott (1964b, p. 205).
29.16 Asymptotic Expansions
Hargrave and Sleeman (1977) give asymptotic approx-
imations for Lam e polynomials and their eigenvalues,
including error bounds. The approximations for Lam e
polynomials hold uniformly on the rectangle 0 <z
K, 0=zK0, whennkandnk0assume large real
values. The approximating functions are exponential,
trigonometric, and parabolic cylinder functions.
29.17 Other Solutions
29.17(i) Second Solution
If (29.2.1) admits a Lam e polynomial solution E, then
a second linearly independent solution Fis given by
29.17.1 F(z) =E(z)Zz
iK0du
(E(u))2:
For properties of these solutions see Arscott (1964b,
x9.7), Erd elyi et al. (1955,x15.5.1), Shail (1980), and
Sleeman (1966a).
29.17(ii) Algebraic Lam e Functions
Algebraic Lam e functions are solutions of (29.2.1) when
is half an odd integer. They are algebraic functions of
sn (z;k), cn (z;k), and dn (z;k), and have primitive pe-
riod 8K. See Erd elyi (1941c), Ince (1940b), and Lambe
(1952).
29.17(iii) Lam e{Wangerin Functions
Lam e{Wangerin functions are solutions of (29.2.1) with
the property that (sn ( z;k))1=2w(z) is bounded on the
line segment from iK0to 2K+iK0. See Erd elyi et al.
(1955,x15.6).Applications
29.18 Mathematical Applications
29.18(i) Sphero-Conal Coordinates
The wave equation
29.18.1 r2u+!2u= 0;
when transformed to sphero-conal coordinates r; ;
:
29.18.2
x=krsn ( ;k) sn (
;k); y =ik
k0rcn ( ;k) cn (
;k);
z=1
k0rdn ( ;k) dn (
;k);
with
29.18.3
r0; =K+i 0;0 02K0;0
4K;
admits solutions
29.18.4 u(r; ;
) =u1(r)u2( )u3(
);
whereu1,u2,u3satisfy the di erential equations
29.18.5d
dr
r2du1
dr
+ (!2r2 (+ 1))u1= 0;
29.18.6d2u2
d 2+ (h (+ 1)k2sn2( ;k))u2= 0;
29.18.7d2u3
d
2+ (h (+ 1)k2sn2(
;k))u3= 0;
with separation constants hand. (29.18.5) is
the di erential equation of spherical Bessel functions
(x10.47(i)), and (29.18.6), (29.18.7) agree with the Lam e
equation (29.2.1).
29.18(ii) Ellipsoidal Coordinates
The wave equation (29.18.1), when transformed to el-
lipsoidal coordinates ; ;
:
29.18.8x=ksn ( ;k) sn ( ;k) sn (
;k);
y= k
k0cn ( ;k) cn ( ;k) cn (
;k);
z=i
kk0dn ( ;k) dn ( ;k) dn (
;k);
with
29.18.9
=K+iK0 0, 0 0<K,
=K+i 0, 0 02K0;0
4K,
admits solutions
29.18.10 u( ; ;
) =u1( )u2( )u3(
);
whereu1,u2,u3each satisfy the Lam e wave equation
(29.11.1).
694 Lame Functions
29.18(iii) Spherical and Ellipsoidal Harmonics
See Erd elyi et al. (1955,x15.7).
29.18(iv) Other Applications
Triebel (1965) gives applications of Lam e functions to
the theory of conformal mappings. Patera and Winter-
nitz (1973) nds bases for the rotation group.
29.19 Physical Applications
29.19(i) Lam e Functions
Simply-periodic Lam e functions ( noninteger) can be
used to solve boundary-value problems for Laplace's
equation in elliptical cones. For applications in antenna
research see Jansen (1977). Brack et al. (2001) shows
that Lam e functions occur at bifurcations in chaotic
Hamiltonian systems. Bronski et al. (2001) uses Lam e
functions in the theory of Bose{Einstein condensates.
29.19(ii) Lam e Polynomials
Ward (1987) computes nite-gap potentials associated
with the periodic Korteweg{de Vries equation. Shail
(1978) treats applications to solutions of elliptic crack
and punch problems. Hargrave (1978) studies high fre-
quency solutions of the delta wing equation. Macfadyen
and Winternitz (1971) nds expansions for the two-
body relativistic scattering amplitudes. Roper (1951)
solves the linearized supersonic
ow equations. Clark-
son (1991) solves nonlinear evolution equations. Strutt
(1932) describes various applications and provides an
extensive list of references.
See alsox29.12(iii).
Computation
29.20 Methods of Computation
29.20(i) Lam e Functions
The eigenvalues am
k2
,bm
k2
, and the Lam e func-
tions Ecm
z;k2
,Esm
z;k2
, can be calculated by di-
rect numerical methods applied to the di erential equa-
tion (29.2.1); seex3.7. The normalization of Lam e func-
tions given inx29.3(v) can be carried out by quadrature
(x3.5).
A second approach is to solve the continued-fraction
equations typi ed by (29.3.10) by Newton's rule or
other iterative methods; see x3.8. Initial approxima-
tions to the eigenvalues can be found, for example, fromthe asymptotic expansions supplied in x29.7(i). Sub-
sequently, formulas typi ed by (29.6.4) can be applied
to compute the coecients of the Fourier expansions
of the corresponding Lam e functions by backward re-
cursion followed by application of formulas typi ed by
(29.6.5) and (29.6.6) to achieve normalization; compare
x3.6. (Equation (29.6.3) serves as a check.) The Fourier
series may be summed using Clenshaw's algorithm; see
x3.11(ii). For further information see Jansen (1977).
A third method is to approximate eigenvalues and
Fourier coecients of Lam e functions by eigenvalues
and eigenvectors of nite matrices using the methods
ofxx3.2(vi) and 3.8(iv). These matrices are the same as
those provided in x29.15(i) for the computation of Lam e
polynomials with the di erence that nhas to be cho-
sen suciently large. The approximations converge ge-
ometrically (x3.8(i)) to the eigenvalues and coecients
of Lam e functions as n!1 . The numerical computa-
tions described in Jansen (1977) are based in part upon
this method.
29.20(ii) Lam e Polynomials
The eigenvalues corresponding to Lam e polynomials are
computed from eigenvalues of the nite tridiagonal ma-
trices Mgiven inx29.15(i), using methods described in
x3.2(vi) and Ritter (1998). The corresponding eigen-
vectors yield the coecients in the nite Fourier series
for Lam e polynomials. x29.15(i) includes formulas for
normalizing the eigenvectors.
29.20(iii) Zeros
Zeros of Lam e polynomials can be computed by solv-
ing the system of equations (29.12.13) by employing
Newton's method; see x3.8(ii). Alternatively, the ze-
ros can be found by locating the maximum of function
gin (29.12.11).
29.21 Tables
Ince (1940a) tabulates the eigenvalues am
k2
,
bm+1
k2
(witha2m+1
andb2m+1
interchanged)
fork2= 0:1;0:5;0:9,= 1
2;0(1)25, and m=
0;1;2;3. Precision is 4D.
Arscott and Khabaza (1962) tabulates the co-
ecients of the polynomials Pin Table 29.12.1
(normalized so that the numerically largest coef-
cient is unity, i.e. monic polynomials), and the
corresponding eigenvalues hfork2= 0:1(:1)0:9,
n= 1(1)30. Equations from x29.6 can be used
to transform to the normalization adopted in this
chapter. Precision is 6S.
29.22 Software 695
29.22 Software
Seehttp://dlmf.nist.gov/29.22 .
References
General References
The main references used in writing this chapter are Ar-
scott (1964b), Erd elyi et al. (1955), Ince (1940b), and
Jansen (1977). For additional bibliographic reading see
Hobson (1931), Strutt (1932), and Whittaker and Wat-
son (1927).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x29.2 Erd elyi et al. (1955,x15.2).
x29.3 Erd elyi (1941b), Erd elyi et al. (1955,x15.5.1),
Ince (1940b), Jansen (1977, x3.1), Magnus and
Winkler (1966, x2.1), Volkmer (2004b). For
(29.3.19) combine Hochstadt (1964, p. 148) and
Table 29.3.2.x29.4 The graphics were produced at NIST.
x29.5 Erd elyi et al. (1955,x15.5.4), Volkmer (2004b).
x29.6 Erd elyi et al. (1955,x15.5.1), Ince (1940b),
Jansen (1977).
x29.7 Ince (1940a), M uller (1966a). For (29.7.5) see
Volkmer (2004b).
x29.8 Erd elyi et al. (1955,x15.5.3), Volkmer (1982,
1983).
x29.9 Magnus and Winkler (1966, x2.1).
x29.12 Arscott (1964b, Chapter 9), Whittaker and
Watson (1927, Chapter 23).
x29.13 The graphics were produced at NIST.
x29.14 Arscott (1964b, x9.4), Erd elyi et al. (1955,
x15.7).
x29.15 The method for constructing the Fourier coe-
cients follows from x29.6. For the Chebyshev coef-
cients see Arscott (1964b, x9.6.2), Ince (1940a).
x29.18 Arscott (1964b, x9.8.1), Erd elyi et al. (1955,
xx15.1.2, 15.1.3, 15.7), Hobson (1931, Chapter
XI), Jansen (1977).
Chapter 30
Spheroidal Wave Functions
H. Volkmer1
Notation 698
30.1 Special Notation . . . . . . . . . . . . . 698
Properties 698
30.2 Di erential Equations . . . . . . . . . . . 698
30.3 Eigenvalues . . . . . . . . . . . . . . . . 698
30.4 Functions of the First Kind . . . . . . . . 699
30.5 Functions of the Second Kind . . . . . . 700
30.6 Functions of Complex Argument . . . . . 700
30.7 Graphics . . . . . . . . . . . . . . . . . . 700
30.8 Expansions in Series of Ferrers Functions 702
30.9 Asymptotic Approximations and Expansions 702
30.10 Series and Integrals . . . . . . . . . . . . 703
30.11 Radial Spheroidal Wave Functions . . . . 70330.12 Generalized and Coulomb Spheroidal
Functions . . . . . . . . . . . . . . . . . 704
Applications 704
30.13 Wave Equation in Prolate Spheroidal Co-
ordinates . . . . . . . . . . . . . . . . . . 704
30.14 Wave Equation in Oblate Spheroidal Co-
ordinates . . . . . . . . . . . . . . . . . . 705
30.15 Signal Analysis . . . . . . . . . . . . . . 706
Computation 707
30.16 Methods of Computation . . . . . . . . . 707
30.17 Tables . . . . . . . . . . . . . . . . . . . 708
30.18 Software . . . . . . . . . . . . . . . . . . 708
References 708
1Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 21) by A. N. Lowan.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
697
698 Spheroidal Wave Functions
Notation
30.1 Special Notation
(For other notation see pp. xiv and 873.)
x real variable. Except in xx30.7(iv), 30.11(ii),
30.13, and 30.14, 1<x< 1.
2real parameter (positive, zero, or negative).
m order, a nonnegative integer.
n degree, an integer n=m;m + 1;m+ 2;:::.
k integer.
arbitrary small positive constant.
The main functions treated in this chapter are the
eigenvalues m
n
2
and the spheroidal wave functions
Psm
n
x;
2
,Qsm
n
x;
2
,Psm
n
z;
2
,Qsm
n
z;
2
, and
Sm(j)
n(z;
),j= 1;2;3;4. These notations are similar to
those used in Arscott (1964b) and Erd elyi et al. (1955).
Meixner and Sch afke (1954) use ps, qs, Ps, Qs for Ps,
Qs,Ps,Qs, respectively.
Other Notations
Flammer (1957) and Abramowitz and Stegun (1964) use
mn(
) form
n
2
+
2,R(j)
mn(
;z) forSm(j)
n(z;
), and
30.1.1S(1)
mn(
;x) =dmn(
)Psm
n
x;
2
;
S(2)
mn(
;x) =dmn(
)Qsm
n
x;
2
;
wheredmn(
) is a normalization constant determined
by
30.1.2S(1)
mn(
;0) = ( 1)mPm
n(0), n meven;
d
dxS(1)
mn(
;x)
x=0= ( 1)md
dxPm
n(x)
x=0,
n modd:
For older notations see Abramowitz and Stegun (1964,
x21.11) and Flammer (1957, pp. 14,15).
Properties
30.2 Di erential Equations
30.2(i) Spheroidal Di erential Equation
30.2.1
d
dz
(1 z2)dw
dz
+
+
2(1 z2) 2
1 z2
w= 0:
This equation has regular singularities at z=1 with
exponents1
2and an irregular singularity of rank 1 at
z=1(if
6= 0). The equation contains three real pa-
rameters,
2, and. In applications involving prolatespheroidal coordinates
2is positive, in applications in-
volving oblate spheroidal coordinates
2is negative; see
xx30.13, 30.14.
30.2(ii) Other Forms
The Liouville normal form of equation (30.2.1) is
30.2.2d2g
dt2+
+1
4+
2sin2t 2 1
4
sin2t
g= 0;
30.2.3z= cost; w (z) = (1 z2) 1
4g(t):
With=
zEquation (30.2.1) changes to
30.2.4
(2
2)d2w
d2+ 2dw
d+
2
2
22
2
2
w= 0:
30.2(iii) Special Cases
If
= 0, Equation (30.2.1) is the associated Legendre
di erential equation; see (14.2.2). If 2=1
4, Equation
(30.2.2) reduces to the Mathieu equation; see (28.2.1). If
= 0, Equation (30.2.4) is satis ed by spherical Bessel
functions; see (10.47.1).
30.3 Eigenvalues
30.3(i) De nition
With=m= 0;1;2;:::, the spheroidal wave functions
Psm
n
x;
2
are solutions of Equation (30.2.1) which are
bounded on ( 1;1), or equivalently, which are of the
form (1 x2)1
2mg(x) whereg(z) is an entire function
ofz. These solutions exist only for eigenvalues m
n
2
,
n=m;m + 1;m+ 2;:::, of the parameter .
30.3(ii) Properties
The eigenvalues m
n
2
are analytic functions of the
real variable
2and satisfy
30.3.1m
m
2
<m
m+1
2
<m
m+2
2
<;
30.3.2m
n
2
=n(n+ 1) 1
2
2+O
n 2
,n!1 ,
30.3.3 m
n(0) =n(n+ 1);
30.3.4 1<dm
n
2
d(
2)<0:
30.4 Functions of the First Kind 699
30.3(iii) Transcendental Equation
Ifpis an even nonnegative integer, then the continued-
fraction equation
30.3.5 p p 2
p
p 2 p 4
p 2
p 4
= p
p+2
p+2 p+2
p+4
p+4 ;
where k, k,
kare de ned by
30.3.6 k= (k+ 1)(k+ 2);
k= (m+k)(m+k+ 1)
2;
k=
2;
has the solutions =m
m+2j
2
,j= 0;1;2;:::. Ifp
is an odd positive integer, then Equation (30.3.5) hasthe solutions =m
m+2j+1
2
,j= 0;1;2;:::. Ifp= 0
orp= 1, the nite continued-fraction on the left-hand
side of (30.3.5) equals 0; if p>1 its last denominator is
0 or 1 .
For a di erent choice of p, p,
pin (30.3.5) see
http://dlmf.nist.gov/30.3.iii .
30.3(iv) Power-Series Expansion
30.3.8 m
n
2
=1X
k=0`2k
2k,j
2j<rm
n.
For values of rm
nsee Meixner et al. (1980, p. 109).
30.3.9`0=n(n+ 1);2`2= 1 (2m 1)(2m+ 1)
(2n 1)(2n+ 3);
2`4=(n m 1)(n m)(n+m 1)(n+m)
(2n 3)(2n 1)3(2n+ 1) (n m+ 1)(n m+ 2)(n+m+ 1)(n+m+ 2)
(2n+ 1)(2n+ 3)3(2n+ 5):
For additional coecients see http://dlmf.nist.gov/30.3.iv .
30.4 Functions of the First Kind
30.4(i) De nitions
The eigenfunctions of (30.2.1) that correspond to the
eigenvalues m
n
2
are denoted by Psm
n
x;
2
,n=
m;m + 1;m+ 2;:::. They are normalized by the con-
dition
30.4.1Z1
1
Psm
n
x;
22dx=2
2n+ 1(n+m)!
(n m)!;
the sign of Psm
n
0;
2
being ( 1)(n+m)=2whenn m
is even, and the sign of dPsm
n
x;
2
dxjx=0being
( 1)(n+m 1)=2whenn mis odd.
When
2>0Psm
n
x;
2
is the prolate angular
spheroidal wave function , and when
2<0Psm
n
x;
2
is the oblate angular spheroidal wave function . If
= 0,
Psm
n(x;0) reduces to the Ferrers function Pm
n(x):
30.4.2 Psm
n(x;0) = Pm
n(x);
comparex14.3(i).
30.4(ii) Elementary Properties
30.4.3 Psm
n
x;
2
= ( 1)n mPsm
n
x;
2
:
Psm
n
x;
2
has exactly n mzeros in the interval
1<x< 1.
30.4(iii) Power-Series Expansion
30.4.4
Psm
n
x;
2
= (1 x2)1
2m1X
k=0gkxk, 1x1,where
30.4.5 kgk+2+ ( k m
n
2
)gk+
kgk 2= 0
with k, k,
kfrom (30.3.6), and g 1=g 2= 0,
gk= 0 for even kifn mis odd and gk= 0 for odd k
ifn mis even. Normalization of the coecients gkis
e ected by application of (30.4.1).
30.4(iv) Orthogonality
30.4.6Z1
1Psm
k
x;
2
Psm
n
x;
2
dx=2
2n+ 1(n+m)!
(n m)!k;n:
Iff(x) is mean-square integrable on [ 1;1], then for-
mally
30.4.7 f(x) =1X
n=mcnPsm
n
x;
2
;
where
30.4.8cn= (n+1
2)(n m)!
(n+m)!Z1
1f(t)Psm
n
t;
2
dt:
The expansion (30.4.7) converges in the norm of
L2( 1;1), that is,
30.4.9 lim
N!1Z1
1 f(x) NX
n=mcnPsm
n
x;
2 2
dx= 0:
It is also equiconvergent with its expansion in Ferrers
functions (as in (30.4.2)), that is, the di erence of cor-
responding partial sums converges to 0 uniformly for
1x1.
700 Spheroidal Wave Functions
30.5 Functions of the Second Kind
Other solutions of (30.2.1) with =m,=m
n
2
,
andz=xare
30.5.1 Qsm
n
x;
2
,n=m;m + 1;m+ 2;:::.
They satisfy
30.5.2 Qsm
n
x;
2
= ( 1)n m+1Qsm
n
x;
2
;
and
30.5.3 Qsm
n(x;0) = Qm
n(x);
comparex14.3(i). Also,
30.5.4W
Psm
n
x;
2
;Qsm
n
x;
2
=(n+m)!
(1 x2)(n m)!Am
n(
2)A m
n(
2) (6= 0);
withAm
n(
2) as in (30.11.4).
For further properties see Meixner and Sch afke
(1954) andx30.8(ii).
30.6 Functions of Complex Argument
The solutions
30.6.1 Psm
n
z;
2
;Qsm
n
z;
2
;
of (30.2.1) with =mand=m
n
2
are real when
z2(1;1), and their principal values ( x4.2(i)) are ob-
tained by analytic continuation to Cn( 1;1].
Relations to Associated Legendre Functions
30.6.2 Psm
n(z;0) =Pm
n(z);Qsm
n(z;0) =Qm
n(z);
comparex14.3(ii).
Wronskian
30.6.3W
Psm
n
z;
2
;Qsm
n
z;
2
=( 1)m(n+m)!
(1 z2)(n m)!Am
n(
2)A m
n(
2);
withAm
n(
2) as in (30.11.4).
Values on ( 1;1)
30.6.4 Psm
n
xi0;
2
= (i)mPsm
n
x;
2
;
30.6.5Qsm
n
xi0;
2
= (i)m
Qsm
n
x;
2
1
2iPsm
n
x;
2
:
For further properties see Arscott (1964b).
For results for Equation (30.2.1) with complex pa-
rameters see Meixner and Sch afke (1954).30.7 Graphics
30.7(i) Eigenvalues
Figure 30.7.1 : Eigenvalues 0
n
2
,n= 0;1;2;3, 10
210.
For additional graphs see http://dlmf.nist.gov/30.
7.i.
30.7(ii) Functions of the First Kind
Figure 30.7.5 :Ps0
n(x;4),n= 0;1;2;3, 1x1.
For additional graphs see http://dlmf.nist.gov/30.
7.ii .
Figure 30.7.9 :Ps0
2
x;
2
, 1x1, 50
250.
30.7 Graphics 701
For an additional surface see http://dlmf.nist.
gov/30.7.ii .
30.7(iii) Functions of the Second Kind
Figure 30.7.11 :Qs0
n(x;4),n= 0;1;2;3, 1<x< 1.
For additional graphs see http://dlmf.nist.gov/30.
7.iii .
Figure 30.7.15 :Qs0
1
x;
2
; 1<x< 1; 10
210.30.7(iv) Functions of Complex Argument
Figure 30.7.16 :jPs0
0(x+iy;4)j, 2x2, 2y
2.
For additional surfaces see http://dlmf.nist.gov/30.
7.iv .
Figure 30.7.20 :jQs0
0(x+iy;4)j, 2x2, 2y
2.
For an additional surface see http://dlmf.nist.
gov/30.7.iv .
702 Spheroidal Wave Functions
30.8 Expansions in Series of Ferrers
Functions
30.8(i) Functions of the First Kind
30.8.1 Psm
n
x;
2
=1X
k= R( 1)kam
n;k(
2)Pm
n+2k(x);
where Pm
n+2k(x) is the Ferrers function of the rst
kind (x14.3(i)),R=1
2(n m)
, and the coecients
am
n;k(
2) are given by
30.8.2am
n;k(
2) = ( 1)k
n+ 2k+1
2(n m+ 2k)!
(n+m+ 2k)!
Z1
1Psm
n
x;
2
Pm
n+2k(x)dx:
Let
30.8.3Ak=
2(n m+ 2k 1)(n m+ 2k)
(2n+ 4k 3)(2n+ 4k 1);
Bk= (n+ 2k)(n+ 2k+ 1)
2
2(n+ 2k)(n+ 2k+ 1) 1 +m2
(2n+ 4k 1)(2n+ 4k+ 3);
Ck=
2(n+m+ 2k+ 1)(n+m+ 2k+ 2)
(2n+ 4k+ 3)(2n+ 4k+ 5):
Then the set of coecients am
n;k(
2),k= R; R+
1; R+ 2;::: is the solution of the di erence equation
30.8.4Akfk 1+
Bk m
n
2
fk+Ckfk+1= 0;
(note thatA R= 0) that satis es the normalizing con-
dition
30.8.51X
k= Ram
n;k(
2)a m
n;k(
2)1
2n+ 4k+ 1=1
2n+ 1;
with
30.8.6a m
n;k(
2) =(n m)!(n+m+ 2k)!
(n+m)!(n m+ 2k)!am
n;k(
2):
Also, ask!1 ,
30.8.7k2am
n;k(
2)
am
n;k 1(
2)=
2
16+O1
k
;
and
30.8.8m
n
2
Bk
Akam
n;k(
2)
am
n;k 1(
2)= 1 +O1
k4
:
30.8(ii) Functions of the Second Kind
30.8.9Qsm
n
x;
2
= N 1X
k= 1( 1)ka0m
n;k(
2)Pm
n+2k(x)
+1X
k= N( 1)kam
n;k(
2)Qm
n+2k(x);where Pm
nand Qm
nare again the Ferrers functions and
N=1
2(n+m)
. The coecients am
n;k(
2) satisfy
(30.8.4) for all kwhen we set am
n;k(
2) = 0 fork< N.
Fork Rthey agree with the coecients de ned
inx30.8(i). For k= N; N+ 1;:::; R 1 they
are determined from (30.8.4) by forward recursion us-
ingam
n; N 1(
2) = 0. The set of coecients a0m
n;k(
2),
k= N 1; N 2;:::, is the recessive solution of
(30.8.4) as k! 1 that is normalized by
30.8.10A N 1a0m
n; N 2(
2)
+
B N 1 m
n
2
a0m
n; N 1(
2)
+C0am
n; N(
2) = 0;
with
30.8.11C0=8
>><
>>:
2
4m2 1; n meven;
2
(2m 1)(2m 3); n modd:
It should be noted that if the forward recursion (30.8.4)
beginning with f N 1= 0,f N= 1 leads to f R= 0,
thenam
n;k(
2) is unde ned for n< Rand Qsm
n
x;
2
does not exist.
30.9 Asymptotic Approximations and
Expansions
30.9(i) Prolate Spheroidal Wave Functions
As
2!+1, withq= 2(n m) + 1,
30.9.1m
n
2
2+
q+ 0+ 1
1+ 2
2+;
where
30.9.28 0= 8m2 q2 5;26 1= q3 11q+ 32m2q;
210 2= 5(q4+ 26q2+ 21) + 384 m2(q2+ 1);
214 3= 33q5 1594q3 5621q
+ 128m2(37q3+ 167q) 2048m4q:
For additional coecients see http://dlmf.nist.
gov/30.9.i .
For the eigenfunctions see Meixner and Sch afke
(1954,x3.251) and M uller (1963).
For uniform asymptotic expansions in terms of Airy
or Bessel functions for real values of the parameters,
complex values of the variable, and with explicit error
bounds see Dunster (1986). See also Miles (1975).
30.10 Series and Integrals 703
30.9(ii) Oblate Spheroidal Wave Functions
As
2! 1 , withq=n+ 1 ifn mis even, orq=n
ifn mis odd, we have
30.9.4m
n
2
2qj
j+c0+c1j
j 1+c2j
j 2+;
where
30.9.5
2c0= q2 1 +m2;8c1= q3 q+m2q;
26c2= 5q4 10q2 1 + 2m2(3q2+ 1) m4;
29c3= 33q5 114q3 37q+ 2m2(23q3+ 25q) 13m4q:
For additional coecients see http://dlmf.nist.
gov/30.9.ii .
For the eigenfunctions see Meixner and Sch afke
(1954,x3.252) and M uller (1962).
For uniform asymptotic expansions in terms of ele-
mentary, Airy, or Bessel functions for real values of the
parameters, complex values of the variable, and with
explicit error bounds see Dunster (1992, 1995). See also
Jorna and Springer (1971).
30.9(iii) Other Approximations and Expansions
The asymptotic behavior of m
n
2
andam
n;k(
2) as
n!1 in descending powers of 2 n+ 1 is derived in
Meixner (1944). The cases of large m, and of large m
and largej
j, are studied in Abramowitz (1949). The
asymptotic behavior of Psm
n
x;
2
and Qsm
n
x;
2
as
x! 1 is given in Erd elyi et al. (1955, p. 151). The
behavior of m
n
2
for complex
2and largejm
n
2
j
is investigated in Hunter and Guerrieri (1982).
30.10 Series and Integrals
Integrals and integral equations for Psm
n
x;
2
are given
in Arscott (1964b, x8.6), Erd elyi et al. (1955,x16.13),
Flammer (1957, Chapter 5), and Meixner (1951). For
product formulas and convolutions see Connett et al.
(1993). For an addition theorem, see Meixner and
Sch afke (1954, p. 300) and King and Van Buren (1973).
For expansions in products of spherical Bessel functions,
see Flammer (1957, Chapter 6).
30.11 Radial Spheroidal Wave Functions
30.11(i) De nitions
Denote
30.11.1 (j)
k(z) =
2z1
2C(j)
k+1
2(z),j= 1;2;3;4,
where
30.11.2
C(1)
=J;C(2)
=Y;C(3)
=H(1)
;C(4)
=H(2)
;withJ,Y,H(1)
, andH(2)
as inx10.2(ii). Then solu-
tions of (30.2.1) with =mand=m
n
2
are given
by
30.11.3
Sm(j)
n(z;
) =(1 z 2)1
2m
A mn(
2)X
2km na m
n;k(
2) (j)
n+2k(
z):
Herea m
n;k(
2) is de ned by (30.8.2) and (30.8.6), and
30.11.4Am
n(
2) =X
2km n( 1)kam
n;k(
2) (6= 0):
In (30.11.3) z6= 0 when j= 1, andjzj>1 when
j= 2;3;4.
Connection Formulas
30.11.5Sm(3)
n(z;
) =Sm(1)
n(z;
) +iSm(2)
n(z;
);
Sm(4)
n(z;
) =Sm(1)
n(z;
) iSm(2)
n(z;
):
30.11(ii) Graphics
Figure 30.11.1 :S0(1)
n(x;2),n= 0;1, 1x10.
For additional graphs see http://dlmf.nist.gov/30.
11.ii .
30.11(iii) Asymptotic Behavior
For xed
, asz!1 in the sectorjphzj (<),
30.11.6
Sm(j)
n(z;
) =(
(j)
n(
z) +O
z 2ej=zj
; j= 1;2;
(j)
n(
z)
1 +O
z 1
; j = 3;4:
For asymptotic expansions in negative powers of z
see Meixner and Sch afke (1954, p. 293).
30.11(iv) Wronskian
30.11.7 Wn
Sm(1)
n(z;
);Sm(2)
n(z;
)o
=1
(z2 1):
704 Spheroidal Wave Functions
30.11(v) Connection with the PsandQs
Functions
30.11.8Sm(1)
n(z;
) =Km
n(
)Psm
n
z;
2
;
30.11.9
Sm(2)
n(z;
) =(n m)!
(n+m)!( 1)m+1Qsm
n
z;
2
Kmn(
)Amn(
2)A mn(
2);
where
30.11.10
Km
n(
) =p
2
2m ( 1)ma m
n;1
2(m n)(
2)
3
2+m
A mn(
2)Psm
n(0;
2),
n meven,
or
30.11.11
Km
n(
) =p
2
2m+1
( 1)ma m
n;1
2(m n+1)(
2)
5
2+m
A mn(
2)(dPsm
n(z;
2)/dzjz=0),
n modd.
30.11(vi) Integral Representations
Whenz2Cn( 1;1]
30.11.12A m
n(
2)Sm(1)
n(z;
)
=1
2im+n
m(n m)!
(n+m)!zm(1 z 2)1
2m
Z1
1e i
zt(1 t2)1
2mPsm
n
t;
2
dt:
For further relations see Arscott (1964b, x8.6), Con-
nett et al. (1993), Erd elyi et al. (1955,x16.13), Meixner
and Sch afke (1954), and Meixner et al. (1980,x3.1).
30.12 Generalized and Coulomb Spheroidal
Functions
Generalized spheroidal wave functions and Coulomb
spheroidal functions are solutions of the di erential
equation
30.12.1d
dz
(1 z2)dw
dz
+
+ z+
2(1 z2) 2
1 z2
w= 0;
which reduces to (30.2.1) if = 0. Equation (30.12.1)
appears in astrophysics and molecular physics. For the
theory and computation of solutions of (30.12.1) see Fal-
loon (2001), Judd (1975), Leaver (1986), and Komarov
et al. (1976).Another generalization is provided by the di erential
equation
30.12.2d
dz
(1 z2)dw
dz
+
+
2(1 z2)
( + 1)
z2 2
1 z2
w= 0;
which also reduces to (30.2.1) when = 0. See Leitner
and Meixner (1960), Slepian (1964) with = 0, and
Meixner et al. (1980).
Applications
30.13 Wave Equation in Prolate Spheroidal
Coordinates
30.13(i) Prolate Spheroidal Coordinates
Prolate spheroidal coordinates ;; are related to
Cartesian coordinates x;y;z by
30.13.1x=cp
(2 1)(1 2) cos ;
y=cp
(2 1)(1 2) sin ; z =c;
wherecis a positive constant. The ( x;y;z )-space with-
out thez-axis corresponds to
30.13.2 1<<1; 1<< 1;0 <2:
The coordinate surfaces = const:are prolate ellipsoids
of revolution with foci at x=y= 0,z=c. The co-
ordinate surfaces = const:are sheets of two-sheeted
hyperboloids of revolution with the same foci. The focal
line is given by = 1, 11, and the rayszc,
x=y= 0 are given by =1,1.
30.13(ii) Metric Coecients
30.13.3h2
=@x
@2
+@y
@2
+@z
@2
=c2(2 2)
2 1;
30.13.4h2
=@x
@2
+@y
@2
+@z
@2
=c2(2 2)
1 2;
30.13.5h2
=@x
@ 2
+@y
@ 2
+@z
@ 2
=c2(2 1)(1 2):
30.14 Wave Equation in Oblate Spheroidal Coordinates 705
30.13(iii) Laplacian
30.13.6
r2=1
hhh @
@hh
h@
@
+@
@hh
h@
@
+@
@ hh
h @
@
=1
c2(2 2)@
@
(2 1)@
@
+@
@
(1 2)@
@
+2 2
(2 1)(1 2)@2
@ 2
:
30.13(iv) Separation of Variables
The wave equation
30.13.7 r2w+2w= 0;
transformed to prolate spheroidal coordinates ( ;; ),
admits solutions
30.13.8 w(;; ) =w1()w2()w3( );
wherew1,w2,w3satisfy the di erential equations
30.13.9
d
d
(1 2)dw1
d
+
+
2(1 2) 2
1 2
w1= 0;
30.13.10
d
d
(1 2)dw2
d
+
+
2(1 2) 2
1 2
w2= 0;
30.13.11d2w3
d 2+2w3= 0;
with
2=2c20 and separation constants and2.
Equations (30.13.9) and (30.13.10) agree with (30.2.1).
In most applications the solution whas to be a
single-valued function of ( x;y;z ), which requires =m
(a nonnegative integer) and
30.13.12w3( ) =a3cos(m ) +b3sin(m ):
Moreover,whas to be bounded along the z-axis away
from the focal line: this requires w2() to be bounded
when 1< < 1. Then=m
n
2
for some
n=m;m + 1;m+ 2;:::, and the general solution of
(30.13.10) is
30.13.13w2() =a2Psm
n
;
2
+b2Qsm
n
;
2
:
The solution of (30.13.9) with =mis
30.13.14w1() =a1Sm(1)
n(;
) +b1Sm(2)
n(;
):
Ifb1=b2= 0, then the function (30.13.8) is a twice-
continuously di erentiable solution of (30.13.7) in the
entire (x;y;z )-space. Ifb2= 0, then this property holds
outside the focal line.30.13(v) The Interior Dirichlet Problem for
Prolate Ellipsoids
Equation (30.13.7) for 0, and subject to the bound-
ary condition w= 0 on the ellipsoid given by =0,
poses an eigenvalue problem with 2as spectral param-
eter. The eigenvalues are given by c22=
2, where
is determined from the condition
30.13.15 Sm(1)
n(0;
) = 0:
The corresponding eigenfunctions are given by
(30.13.8), (30.13.14), (30.13.13), (30.13.12), with b1=
b2= 0. For the Dirichlet boundary-value problem of
the region 12between two ellipsoids, the
eigenvalues are determined from
30.13.16 w1(1) =w1(2) = 0;
withw1as in (30.13.14). The corresponding eigenfunc-
tions are given as before with b2= 0.
For further applications see Meixner and Sch afke
(1954), Meixner et al. (1980) and the references cited
therein; also Ong (1986), M uller et al. (1994), and Xiao
et al. (2001).
30.14 Wave Equation in Oblate Spheroidal
Coordinates
30.14(i) Oblate Spheroidal Coordinates
Oblate spheroidal coordinates ;; are related to
Cartesian coordinates x;y;z by
30.14.1x=cp
(2+ 1)(1 2) cos ;
y=cp
(2+ 1)(1 2) sin ; z =c;
wherecis a positive constant. The ( x;y;z )-space with-
out thez-axis and the disk z= 0,x2+y2c2corre-
sponds to
30.14.2 0<<1; 1<< 1;0 <2:
The coordinate surfaces = const:are oblate ellipsoids
of revolution with focal circle z= 0,x2+y2=c2.
The coordinate surfaces = const:are halves of one-
sheeted hyperboloids of revolution with the same focal
circle. The disk z= 0,x2+y2c2is given by = 0,
11, and the raysz0,x=y= 0 are given
by=1,0.
30.14(ii) Metric Coecients
30.14.3 h2
=c2(2+2)
1 +2;
30.14.4 h2
=c2(2+2)
1 2;
30.14.5 h2
=c2(2+ 1)(1 2):
706 Spheroidal Wave Functions
30.14(iii) Laplacian
30.14.6
r2=1
c2(2+2)@
@
(2+ 1)@
@
+@
@
(1 2)@
@
+2+2
(2+ 1)(1 2)@2
@ 2
:
30.14(iv) Separation of Variables
The wave equation (30.13.7), transformed to oblate
spheroidal coordinates ( ;; ), admits solutions of the
form (30.13.8), where w1satis es the di erential equa-
tion
30.14.7
d
d
(1 +2)dw1
d
+
2(1 +2) 2
1 +2
w1= 0;
andw2,w3satisfy (30.13.10) and (30.13.11), respec-
tively, with
2= 2c20 and separation constants
and2. Equation (30.14.7) can be transformed to
equation (30.2.1) by the substitution z=i.
In most applications the solution whas to be a
single-valued function of ( x;y;z ), which requires =m
(a nonnegative integer). Moreover, the solution whas to
be bounded along the z-axis: this requires w2() to be
bounded when 1<< 1. Then=m
n
2
for some
n=m;m + 1;m+ 2;:::, and the solution of (30.13.10)
is given by (30.13.13). The solution of (30.14.7) is given
by
30.14.8w1() =a1Sm(1)
n(i;
) +b1Sm(2)
n(i;
):
Ifb1=b2= 0, then the function (30.13.8) is a twice-
continuously di erentiable solution of (30.13.7) in the
entire (x;y;z )-space. Ifb2= 0, then this property holds
outside the focal disk.
30.14(v) The Interior Dirichlet Problem for
Oblate Ellipsoids
Equation (30.13.7) for 0together with the bound-
ary condition w= 0 on the ellipsoid given by =0,
poses an eigenvalue problem with 2as spectral param-
eter. The eigenvalues are given by c22=
2, where
2is determined from the condition
30.14.9 Sm(1)
n(i0;
) = 0:
The corresponding eigenfunctions are then given by
(30.13.8), (30.14.8), (30.13.13), (30.13.12), with b1=
b2= 0.
For further applications see Meixner and Sch afke
(1954), Meixner et al. (1980) and the references cited
therein; also Kokkorakis and Roumeliotis (1998) and Li
et al. (1998).30.15 Signal Analysis
30.15(i) Scaled Spheroidal Wave Functions
Let (>0) and(>0) be given. Set
= and de ne
30.15.1
n(t) =r
2n+ 1
2 p
nPs0
nt
;
2
,n= 0;1;2;:::,
30.15.2 n=2
K0
n(
)A0
n(
2)2;
seex30.11(v).
30.15(ii) Integral Equation
30.15.3Z
sin(t s)
(t s) n(s)ds= n n(t):
30.15(iii) Fourier Transform
30.15.4Z1
1e it! n(t)dt= ( i)nr
2
n n
!
(!);
30.15.5Z
e it! n(t)dt= ( i)nr
2 n
n
!
;
where
30.15.6 (!) =(
1;j!j;
0;j!j>:
Equations (30.15.4) and (30.15.6) show that the func-
tions nare-bandlimited , that is, their Fourier trans-
form vanishes outside the interval [ ;].
30.15(iv) Orthogonality
30.15.7Z
k(t) n(t)dt= nk;n;
30.15.8Z1
1 k(t) n(t)dt=k;n:
The sequence n,n= 0;1;2;::: forms an orthonormal
basis in the space of -bandlimited functions, and, after
normalization, an orthonormal basis in L2( ; ).
Computation 707
30.15(v) Extremal Properties
The maximum (or least upper bound) B of all numbers
30.15.9 =1
2Z
Z1
1e it!f(t)dt 2
d!
taken over all f2L2( 1;1) subject to
30.15.10Z1
1jf(t)j2dt= 1;Z
jf(t)j2dt= ;
for ( xed) 0< 1, is given by
30.15.11 arccosp
B + arccosp = arccosp
0;
or equivalently,
30.15.12 B =p
0 +p
1 0p
1 2
:
The corresponding function fis given by
30.15.13f(t) =a 0(t) (t) +b 0(t)(1 (t));
a=r
0; b =r
1
1 0:
If 0< 0, then B = 1.
For further information see Frieden (1971), Lyman
and Edmonson (2001), Papoulis (1977, Chapter 6),
Slepian (1983), and Slepian and Pollak (1961).
Computation
30.16 Methods of Computation
30.16(i) Eigenvalues
For smallj
2jwe can use the power-series expansion
(30.3.8). Sch afke and Groh (1962) gives corresponding
error bounds. Ifj
2jis large we can use the asymptotic
expansions inx30.9. Approximations to eigenvalues can
be improved by using the continued-fraction equations
fromx30.3(iii) andx30.8; see Bouwkamp (1947) and
Meixner and Sch afke (1954, x3.93).
Another method is as follows. Let n mbe even.
Fordsuciently large, construct the ddtridiagonal
matrix A= [Aj;k] with nonzero elements
30.16.1
Aj;j= (m+ 2j 2)(m+ 2j 1)
2
2(m+ 2j 2)(m+ 2j 1) 1 +m2
(2m+ 4j 5)(2m+ 4j 1);
Aj;j+1=
2(2m+ 2j 1)(2m+ 2j)
(2m+ 4j 1)(2m+ 4j+ 1);
Aj;j 1=
2 (2j 3)(2j 2)
(2m+ 4j 7)(2m+ 4j 5);and real eigenvalues 1;d, 2;d,:::, d;d, arranged in
ascending order of magnitude. Then
30.16.2 j;d+1 j;d;
and
30.16.3 m
n
2
= lim
d!1 p;d,p=1
2(n m)
+ 1.
The eigenvalues of Acan be computed by methods in-
dicated inxx3.2(vi), 3.2(vii). The error satis es
30.16.4 p;d m
n
2
=O
4d
42d+1((m+ 2d 1)!(m+ 2d+ 1)!)2
,
d!1 .
Example
Form= 2,n= 4,
2= 10,
30.16.5 2;2= 14:18833 246; 2;3= 13:98002 013;
2;4= 13:97907 459; 2;5= 13:97907 345;
2;6= 13:97907 345;
which yields 2
4(10) = 13:97907 345. If n mis odd,
then (30.16.1) is replaced by
30.16.6Aj;j= (m+ 2j 1)(m+ 2j)
2
2(m+ 2j 1)(m+ 2j) 1 +m2
(2m+ 4j 3)(2m+ 4j+ 1);
Aj;j+1=
2(2m+ 2j)(2m+ 2j+ 1)
(2m+ 4j+ 1)(2m+ 4j+ 3);
Aj;j 1=
2 (2j 2)(2j 1)
(2m+ 4j 5)(2m+ 4j 3):
30.16(ii) Spheroidal Wave Functions of the
First Kind
Ifj
2jis large, then we can use the asymptotic expan-
sions referred to in x30.9 to approximate Psm
n
x;
2
.
Ifm
n
2
is known, then we can compute Psm
n
x;
2
(not normalized) by solving the di erential equation
(30.2.1) numerically with initial conditions w(0) = 1,
w0(0) = 0 ifn mis even, or w(0) = 0,w0(0) = 1 if
n mis odd.
Ifm
n
2
is known, then Psm
n
x;
2
can be found
by summing (30.8.1). The coecients am
n;r(
2) are com-
puted as the recessive solution of (30.8.4) ( x3.6), and
normalized via (30.8.5).
A fourth method, based on the expansion (30.8.1), is
as follows. Let Abe theddmatrix given by (30.16.1)
ifn mis even, or by (30.16.6) if n mis odd. Form the
eigenvector [ e1;d;e2;d;:::;ed;d]TofAassociated with
the eigenvalue p;d,p=1
2(n m)
+ 1, normalized
according to
30.16.7dX
j=1e2
j;d(n+m+ 2j 2p)!
(n m+ 2j 2p)!1
2n+ 4j 4p+ 1
=(n+m)!
(n m)!1
2n+ 1:
708 Spheroidal Wave Functions
Then
30.16.8 am
n;k(
2) = lim
d!1ek+p;d;
30.16.9 Psm
n
x;
2
= lim
d!1dX
j=1( 1)j pej;dPm
n+2(j p)(x):
For error estimates see Volkmer (2004a).
30.16(iii) Radial Spheroidal Wave Functions
The coecients am
n;k(
2) calculated inx30.16(ii) can
be used to compute Sm(j)
n(z;
),j= 1;2;3;4 from
(30.11.3) as well as the connection coecients Km
n(
)
from (30.11.10) and (30.11.11).
For another method see Van Buren and Boisvert
(2002).
30.17 Tables
Stratton et al. (1956) tabulates quantities closely
related tom
n
2
andam
n;k(
2) for 0m8,
mn8, 64
264. Precision is 7S.
Flammer (1957) includes 18 tables of eigenvalues,
expansion coecients, spheroidal wave functions,
and other related quantities. Precision varies be-
tween 4S and 10S.
Hanish et al. (1970) gives m
n
2
andSm(j)
n(z;
),
j= 1;2, and their rst derivatives, for 0 m2,
mnm+ 49, 1600
21600. The range
ofzis given by 1z10 if
2>0, orz= i,
02 if
2<0. Precision is 18S.
EraSevskaja et al. (1973, 1976) gives
Sm(j)(iy; ic),Sm(j)(z;
) and their rst deriva-
tives forj= 1;2, 0:5c8,y= 0;0:5;1;1:5,
0:5
8,z= 1:01;1:1;1:4;1:8. Precision is
15S.
Van Buren et al. (1975) gives 0
n
2
,Ps0
n
x;
2
for 0n49, 1600
21600, 1x1.
Precision is 8S.
Zhang and Jin (1996) includes 24 tables of eigen-
values, spheroidal wave functions and their deriva-
tives. Precision varies between 6S and 8S.
Fletcher et al. (1962,x22.28) provides additional infor-
mation on tables prior to 1961.
30.18 Software
Seehttp://dlmf.nist.gov/30.18 .References
General References
The main references used in writing this chapter are
Arscott (1964b), Erd elyi et al. (1955), Meixner and
Sch afke (1954), and Meixner et al. (1980). For addi-
tional bibliographic reading see Flammer (1957), Ko-
marov et al. (1976), and Stratton et al. (1956).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x30.2 Meixner and Sch afke (1954, x3.1), Arscott
(1964b,x8.1).
x30.3 Meixner and Sch afke (1954, xx3.2, 3.531).
x30.4 Meixner and Sch afke (1954, x3.2), Arscott
(1964b,x8.2).
xx30.5, 30.6 Meixner and Sch afke (1954, x3.6).
x30.7 These graphics were produced at NIST with the
aid of Maple procedures provided by the author.
x30.8 Arscott (1964b,xx8.2, 8.5), Meixner and Sch afke
(1954,xx3.542, 3.62).
x30.9 Meixner and Sch afke (1954, xx3.251, 3.252),
M uller (1962, 1963).
x30.11 Arscott (1964b, x8.5), Meixner and Sch afke
(1954,xx3.64{3.66, 3.84), Erd elyi et al. (1955,
x16.11). Figure 30.11.1 was produced at NIST
with the aid of Maple procedures provided by the
author.
x30.13 Erd elyi et al. (1955,x16.1.2), Meixner and
Sch afke (1954,xx1.123, 1.133, Chapter 4).
x30.14 Erd elyi et al. (1955,x16.1.3), Meixner and
Sch afke (1954,xx1.124, 1.134, Chapter 4).
x30.15 Frieden (1971, pp. 321{324, x2.10), Meixner
et al. (1980, p. 114), Papoulis (1977, pp. 205{210),
Slepian (1983).
x30.16 Meixner and Sch afke (1954, x3.93), Volkmer
(2004a), Van Buren et al. (1972).
Chapter 31
Heun Functions
B. D. Sleeman1and V. B. Kuznetsov2
Notation 710
31.1 Special Notation . . . . . . . . . . . . . 710
Properties 710
31.2 Di erential Equations . . . . . . . . . . . 710
31.3 Basic Solutions . . . . . . . . . . . . . . 711
31.4 Solutions Analytic at Two Singularities:
Heun Functions . . . . . . . . . . . . . . 712
31.5 Solutions Analytic at Three Singularities:
Heun Polynomials . . . . . . . . . . . . . 712
31.6 Path-Multiplicative Solutions . . . . . . . 712
31.7 Relations to Other Functions . . . . . . . 713
31.8 Solutions via Quadratures . . . . . . . . 713
31.9 Orthogonality . . . . . . . . . . . . . . . 714
31.10 Integral Equations and Representations . 71431.11 Expansions in Series of Hypergeometric
Functions . . . . . . . . . . . . . . . . . 716
31.12 Con
uent Forms of Heun's Equation . . . 717
31.13 Asymptotic Approximations . . . . . . . . 718
31.14 General Fuchsian Equation . . . . . . . . 718
31.15 Stieltjes Polynomials . . . . . . . . . . . 718
Applications 719
31.16 Mathematical Applications . . . . . . . . 719
31.17 Physical Applications . . . . . . . . . . . 720
Computation 720
31.18 Methods of Computation . . . . . . . . . 720
References 721
1Department of Applied Mathematics, University of Leeds, Leeds, United Kingdom.
2Department of Applied Mathematics, University of Leeds, Leeds, United Kingdom.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
709
710 Heun Functions
Notation
31.1 Special Notation
(For other notation see pp. xiv and 873.)
x,y real variables.
z,,w,W complex variables.
j,k,`,m,n nonnegative integers.
a complex parameter, jaj1;a6= 1.
q; ; ;
;;; complex parameters.
The main functions treated in this chapter
areH`(a;q; ; ;
; ;z), (s1;s2)Hfm(a;qm; ; ;
; ;z),
(s1;s2)Hf
m(a;qm; ; ;
; ;z), and the polynomial
Hpn;m(a;qn;m; n; ;
; ;z). These notations were in-
troduced by Arscott in Ronveaux (1995, pp. 34{44).
Sometimes the parameters are suppressed.
Properties
31.2 Di erential Equations
31.2(i) Heun's Equation
31.2.1
d2w
dz2+
z+
z 1+
z adw
dz+ z q
z(z 1)(z a)w
= 0, + + 1 =
++.
This equation has regular singularities at 0 ;1;a;1, with
corresponding exponents f0;1
g,f0;1 g,f0;1 g,
f ; g, respectively (x2.7(i)). All other homogeneous
linear di erential equations of the second order having
four regular singularities in the extended complex plane,
C[f1g , can be transformed into (31.2.1).
The parameters play di erent roles: ais the singu-
larity parameter ; ; ;
;; areexponent parameters ;q
is the accessory parameter . The total number of free
parameters is six.
31.2(ii) Normal Form of Heun's Equation
31.2.2w(z) =z
=2(z 1) =2(z a) =2W(z);
31.2.3d2W
dz2=A
z+B
z 1+C
z a+D
z2+E
(z 1)2
+F
(z a)2
W,
A+B+C= 0,31.2.4
A=
2
2a+q
a; B =
2
2(a 1) q
a 1;
C=
2a+
2(a 1) a q
a(a 1); D =1
2
1
2
1
;
E=1
2 1
2 1
; F =1
2 1
2 1
:
31.2(iii) Trigonometric Form
31.2.5 z= sin2;
31.2.6d2w
d2+
(2
1) cot (2 1) tan
sin(2)
a sin2dw
d+ 4 sin2 q
a sin2w= 0:
31.2(iv) Doubly-Periodic Forms
Jacobi's Elliptic Form
With the notation of x22.2 let
31.2.7 a=k 2; z = sn2(;k):
Then (suppressing the parameter k)
31.2.8d2w
d2+
(2
1)cndn
sn (2 1)sndn
cn
(2 1)k2sncn
dndw
d
+ 4k2( sn2 q)w= 0:
Weierstrass's Form
With the notation of xx19.2(ii) and 23.2 let
31.2.9k2= (e2 e3)=(e1 e3);
=iK0+(e1 e3)1=2; e 1=}(!1);
e2=}(!2); e 3=}(!3); e 1+e2+e3= 0;
where 2!1and 2!3with=(!3=!1)>0 are generators
of the lattice Lfor}(zjL). Then
31.2.10w() = (}() e3)(1 2
)=4(}() e2)(1 2)=4
(}() e1)(1 2)=4W();
whereW() satis es
31.2.11d2W
d2+ (H+b0}() +b1}(+!1)
+b2}(+!2) +b3}(+!3))W= 0;
with
31.2.12b0= 4 (
++ 1
2)(
++ 3
2);
b1= ( 1
2)( 3
2); b 2= ( 1
2)( 3
2);
b3= (
1
2)(
3
2);
H=e1(
+ 1)2+e2(
+ 1)2
+e3(+ 1)2 4 e 3 4q(e2 e3):
31.3 Basic Solutions 711
31.2(v) Heun's Equation Automorphisms
F-Homotopic Transformations
w(z) =z1
w1(z) satis es (31.2.1) if w1is a solution
of (31.2.1) with transformed parameters q1=q+ (a+
)(1
); 1= + 1
, 1= + 1
,
1= 2
.
Next,w(z) = (z 1)1 w2(z) satis es (31.2.1) if w2
is a solution of (31.2.1) with transformed parameters
q2=q+a
(1 ); 2= + 1 , 2= + 1 ,
2= 2 . Lastly,w(z) = (z a)1 w3(z) satis es
(31.2.1) ifw3is a solution of (31.2.1) with transformed
parameters q3=q+
(1 ); 3= +1 , 3= +1 ,
3= 2 . By composing these three steps, there result
23= 8 possible transformations of the dependent vari-
able (including the identity transformation) that pre-
serve the form of (31.2.1).
Homographic Transformations
There are 4! = 24 homographies ~ z(z) = (Az+
B)=(Cz+D) that take 0 ;1;a;1to some permutation
of 0;1;a0;1, wherea0may di er from a. If ~z= ~z(z)
is one of the 3! = 6 homographies that map 1to1,
thenw(z) = ~w(~z) satis es (31.2.1) if ~ w(~z) is a solu-
tion of (31.2.1) with zreplaced by ~ zand appropriately
transformed parameters. For example, if ~ z=z=a, then
the parameters are ~ a= 1=a, ~q=q=a;~=, ~=.
If ~z= ~z(z) is one of the 4! 3! = 18 homographies
that do not map 1to1, then an appropriate prefac-
tor must be included on the right-hand side. For exam-
ple,w(z) = (1 z) ~w(z=(z 1)), which arises from
~z=z=(z 1), satis es (31.2.1) if ~ w(~z) is a solution of
(31.2.1) with zreplaced by ~ zand transformed parame-
ters ~a=a=(a 1), ~q= (q a
)=(a 1);~ = +1 ,~= + 1 .
Composite Transformations
There are 824 = 192 automorphisms of equation
(31.2.1) by compositions of F-homotopic and homo-
graphic transformations. Each is a substitution of de-
pendent and/or independent variables that preserves
the form of (31.2.1). Except for the identity automor-
phism, each alters the parameters.
31.3 Basic Solutions
31.3(i) Fuchs{Frobenius Solutions at z= 0
H`(a;q; ; ;
; ;z) denotes the solution of (31.2.1) that
corresponds to the exponent 0 at z= 0 and assumes the
value 1 there. If the other exponent is not a positive in-
teger, that is, if
6= 0; 1; 2;:::, then fromx2.7(i)
it follows that H`(a;q; ; ;
; ;z) exists, is analytic in
the diskjzj<1, and has the Maclaurin expansion
31.3.1 H`(a;q; ; ;
; ;z) =1X
j=0cjzj,jzj<1,
wherec0= 1,
31.3.2 a
c1 qc0= 0;
31.3.3Rjcj+1 (Qj+q)cj+Pjcj 1= 0,j1,
with
31.3.4Pj= (j 1 + )(j 1 + );
Qj=j((j 1 +
)(1 +a) +a+);
Rj=a(j+ 1)(j+
):
Similarly, if
6= 1;2;3;:::, then the solution of (31.2.1) that corresponds to the exponent 1
atz= 0 is
31.3.5 z1
H`(a;(a+)(1
) +q; + 1
; + 1
;2
;;z):
When
2Z, linearly independent solutions can be constructed as in x2.7(i). In general, one of them has a
logarithmic singularity at z= 0.
31.3(ii) Fuchs{Frobenius Solutions at Other Singularities
With similar restrictions to those given in x31.3(i), the following results apply. Solutions of (31.2.1) corresponding
to the exponents 0 and 1 atz= 1 are respectively,
31.3.6 H`(1 a; q; ; ;;
; 1 z);
31.3.7 (1 z)1 H`(1 a;((1 a)
+)(1 ) + q; + 1 ; + 1 ;2 ;
; 1 z):
Solutions of (31.2.1) corresponding to the exponents 0 and 1 atz=aare respectively,
31.3.8 H`a
a 1; a q
a 1; ; ;; ;a z
a 1
;
31.3.9a z
a 11
H`a
a 1;(a(+
)
)(1 )
a 1+ a q
a 1; + 1 ; + 1 ;2 ;;a z
a 1
:
712 Heun Functions
Solutions of (31.2.1) corresponding to the exponents and atz=1are respectively,
31.3.10 z H`1
a; ( ) +
a( ) q
a; ;
+ 1; + 1;;1
z
;
31.3.11 z H`1
a; ( ) +
a( ) q
a; ;
+ 1; + 1;;1
z
:
31.3(iii) Equivalent Expressions
Solutions (31.3.1) and (31.3.5){(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. Each
is related to the solution (31.3.1) by one of the automorphisms of x31.2(v). There are 192 automorphisms in all, so
there are 192 =8 = 24 equivalent expressions for each of the 8. For example, H`(a;q; ; ;
; ;z) is equal to
31.3.12 H`(1=a;q=a ; ; ;
; + + 1
;z=a);
which arises from the homography ~ z=z=a, and to
31.3.13 (1 z) H`a
a 1; q a
a 1; ; + 1 ;
; + 1 ;z
z 1
;
which arises from ~ z=z=(z 1), and also to 21 fur-
ther expressions. The full set of 192 local solutions of
(31.2.1), equivalent in 8 sets of 24, resembles Kummer's
set of 24 local solutions of the hypergeometric equation,
which are equivalent in 4 sets of 6 solutions ( x15.10(ii));
see Maier (2007).
31.4 Solutions Analytic at Two
Singularities: Heun Functions
For an in nite set of discrete values qm,m=
0;1;2;:::, of the accessory parameter q, the function
H`(a;q; ; ;
; ;z) is analytic at z= 1, and hence also
throughout the disk jzj<a. To emphasize this property
this set of functions is denoted by
31.4.1 (0;1)Hfm(a;qm; ; ;
; ;z),m= 0;1;2;::::
The eigenvalues qmsatisfy the continued-fraction
equation
31.4.2q=a
P 1
Q1+q R1P2
Q2+q R2P3
Q3+q ;
in whichPj;Qj;Rjare as inx31.3(i).
More generally,
31.4.3 (s1;s2)Hfm(a;qm; ; ;
; ;z),m= 0;1;2;:::;
with (s1;s2)2f0;1;a;1g, denotes a set of solutions of
(31.2.1), each of which is analytic at s1ands2. The set
qmdepends on the choice of s1ands2.
The solutions (31.4.3) are called the Heun functions .
See Ronveaux (1995, pp. 39{41).31.5 Solutions Analytic at Three
Singularities: Heun Polynomials
Let = n,n= 0;1;2;:::, andqn;m,m= 0;1;:::;n ,
be the eigenvalues of the tridiagonal matrix
31.5.12
66666640a
0::: 0
P1 Q1R1::: 0
0P2 Q2...
.........Rn 1
0 0 ::: Pn Qn3
7777775;
wherePj;Qj;Rjare again de ned as in x31.3(i). Then
31.5.2
Hpn;m(a;qn;m; n; ;
; ;z) =H`(a;qn;m; n; ;
; ;z)
is a polynomial of degree n, and hence a solution of
(31.2.1) that is analytic at all three nite singularities
0;1;a. These solutions are the Heun polynomials . Some
properties are included as special cases of properties
given inx31.15 below.
31.6 Path-Multiplicative Solutions
A further extension of the notation (31.4.1) and (31.4.3)
is given by
31.6.1 (s1;s2)Hf
m(a;qm; ; ;
; ;z),m= 0;1;2;:::;
with (s1;s2)2f0;1;ag, but with another set of fqmg.
This denotes a set of solutions of (31.2.1) with the prop-
erty that if we pass around a simple closed contour in
thez-plane that encircles s1ands2once in the positive
sense, but not the remaining nite singularity, then the
solution is multiplied by a constant factor e2i. These
solutions are called path-multiplicative . See Schmidt
(1979).
31.7 Relations to Other Functions 713
31.7 Relations to Other Functions
31.7(i) Reductions to the Gauss
Hypergeometric Function
31.7.1
2F1( ; ;
;z) =H`(1; ; ; ;
; ;z)
=H`(0;0; ; ;
; + + 1
;z)
=H`(a;a ; ; ;
; + + 1
;z):
Other reductions of H`to a 2F1, with at least one free
parameter, exist i the pair ( a;p) takes one of a nitenumber of values, where q= p. Below are three
such reductions with three and two parameters. They
are analogous to quadratic and cubic hypergeometric
transformations ( xx15.8(iii){15.8(v)).
31.7.2H`(2; ; ; ;
; + 2
+ 1;z)
=2F1 1
2 ;1
2 ;
; 1 (1 z)2
;
31.7.3H`
4; ; ; ;1
2;2
3( + );z
=2F1 1
3 ;1
3 ;1
2; 1 (1 z)2(1 1
4z)
;
31.7.4H`
1
2+ip
3
2; (1
2+ip
3
6); ; ;1
3( + + 1);1
3( + + 1);z
=2F1
1
3 ;1
3 ;1
3( + + 1); 1
1
3
2 ip
3
2
z3
:
For additional reductions, see Maier (2005). Joyce
(1994) gives a reduction in which the independent vari-
able is transformed not polynomially or rationally, but
algebraically.
31.7(ii) Relations to Lam e Functions
Withz= sn2(;k) and
31.7.5a=k 2; q = 1
4ah; = 1
2;
=1
2(+ 1);
===1
2;
equation (31.2.1) becomes Lam e's equation with inde-
pendent variable ; compare (29.2.1) and (31.2.8). The
solutions (31.3.1) and (31.3.5) transform into even and
odd solutions of Lam e's equation, respectively. Sim-
ilar specializations of formulas in x31.3(ii) yield solu-
tions in the neighborhoods of the singularities =K,
K+iK0, andiK0, whereKandK0are related to k
as inx19.2(ii).
31.8 Solutions via Quadratures
For half-odd-integer values of the exponent parameters:
31.8.1
=m0+1
2;
= m1+1
2; = m2+1
2;
= m3+1
2,m0;m1;m2;m3= 0;1;2;:::,
the Hermite{Darboux method (see Whittaker and Wat-
son (1927, pp. 570{572)) can be applied to construct so-
lutions of (31.2.1) expressed in quadratures, as follows.
Denote m= (m0;m1;m2;m3) and= 4q. Then
31.8.2
w(m;;z)
=q
g;N(;z)
exp
i()
2Zz
z0tm1(t 1)m2(t a)m3dt
g;N(;t)p
t(t 1)(t a)!are two independent solutions of (31.2.1). Here
g;N(;z) is a polynomial of degree ginand of degree
N=m0+m1+m2+m3inz, that is a solution of the
third-order di erential equation satis ed by a product
of any two solutions of Heun's equation. The degree g
is given by
31.8.3g=1
2max
2 max
0k3mk;1 +N
(1 + ( 1)N)
1
2+ min
0k3mk
:
The variables andare two coordinates of the associ-
ated hyperelliptic (spectral) curve : 2=Q2g+1
j=1(
j). (Thisis unrelated to the inx31.6.) Lastly, j,
j= 1;2;:::; 2g+ 1, are the zeros of the Wronskian of
w+(m;;z) andw (m;;z).
By automorphisms from x31.2(v), similar solutions
also exist for m0;m1;m2;m32Z, and g;N(;z) may
become a rational function in z. For instance,
31.8.4
1;2=z2+z+a; 2= (+a+ 1)(2 4a),
m= (1;1;0;0),
and
31.8.5 1; 1=
z3+ (+ 3a+ 3)z+a
=z3;
2= (+ 4a+ 4)
(+ 3a+ 3)2 4a
,
m= (1; 2;0;0).
Form= (m0;0;0;0), these solutions reduce to Her-
mite's solutions (Whittaker and Watson (1927, x23.7))
of the Lam e equation in its algebraic form. The curve
re
ects the nite-gap property of Equation (31.2.1)
when the exponent parameters satisfy (31.8.1) for mj2
Z. When= 4qapproaches the ends of the gaps,
the solution (31.8.2) becomes the corresponding Heun
polynomial. For more details see Smirnov (2002).
714 Heun Functions
The solutions in this section are nite-term Liouvil-
lean solutions which can be constructed via Kovacic's
algorithm; seex31.14(ii).
31.9 Orthogonality
31.9(i) Single Orthogonality
With
31.9.1wm(z) = (0;1)Hfm(a;qm; ; ;
; ;z);
we have
31.9.2Z(1+;0+;1 ;0 )
t
1(1 t) 1(t a) 1
wm(t)wk(t)dt=m;km:
Hereis an arbitrary point in the interval (0 ;1). The
integration path begins at z=, encirclesz= 1 once
in the positive sense, followed by z= 0 once in the
positive sense, and so on, returning nally to z=.
The integration path is called a Pochhammer double-
loop contour (compare Figure 5.12.3). The branches of
the many-valued functions are continuous on the path,
and assume their principal values at the beginning.
The normalization constant mis given by
31.9.3m= (1 e2i
)(1 e2i)
(1 )( a)
f0(q;)
f1(q;)@
@qWff0(q;);f1(q;)g
q=qm;
where
31.9.4
f0(qm;z) =H`(a;qm; ; ;
; ;z);
f1(qm;z) =H`(1 a; qm; ; ;;
; 1 z);
andWdenotes the Wronskian ( x1.13(i)). The right-
hand side may be evaluated at any convenient value, or
limiting value, of in (0;1) since it is independent of .
For corresponding orthogonality relations for Heun
functions (x31.4) and Heun polynomials ( x31.5), see
Lambe and Ward (1934), Erd elyi (1944), Sleeman
(1966b), and Ronveaux (1995, Part A, pp. 59{64).
31.9(ii) Double Orthogonality
Heun polynomials wj=Hpnj;mj,j= 1;2, satisfy
31.9.5Z
L1Z
L2(s;t)w1(s)w1(t)w2(s)w2(t)dsdt
= 0, jn1 n2j+jm1 m2j6= 0,where
31.9.6(s;t) = (s t)(st)
1((s 1)(t 1)) 1
((s a)(t a)) 1;
and the integration paths L1,L2are Pochhammer
double-loop contours encircling distinct pairs of singu-
laritiesf0;1g,f0;ag,f1;ag.
For further information, including normalization
constants, see Sleeman (1966b). For bi-orthogonal re-
lations for path-multiplicative solutions see Schmidt
(1979,x2.2). For other generalizations see Arscott
(1964b, pp. 206{207 and 241).
31.10 Integral Equations and
Representations
31.10(i) Type I
Ifw(z) is a solution of Heun's equation, then another
solutionW(z) (possibly a multiple of w(z)) can be rep-
resented as
31.10.1 W(z) =Z
CK(z;t)w(t)(t)dt
for a suitable contour C. The weight function is given
by
31.10.2 (t) =t
1(t 1) 1(t a) 1;
and the kernelK(z;t) is a solution of the partial di er-
ential equation
31.10.3 (Dz Dt)K= 0;
whereDzisHeun's operator in the variable z:
31.10.4Dz=z(z 1)(z a)(@2
@z2) + (
(z 1)(z a)
+z(z a) +z(z 1)) (@/@z) + z:
The contour Cmust be such that
31.10.5p(t)@K
@tw(t) Kdw(t)
dt
C= 0;
where
31.10.6 p(t) =t
(t 1)(t a):
Kernel Functions
Set
31.10.7 cos=zt
a1=2
;sincos =i(z a)(t a)
a(1 a)1=2
;sinsin =(z 1)(t 1)
1 a1=2
:
31.10 Integral Equations and Representations 715
The kernelKmust satisfy
31.10.8 sin2@2K
@2+
(1 2
) tan+2(+ 1
2) cot@K
@ 4 K
+@2K
@ 2+((1 2) cot (1 2) tan )@K
@ = 0:
The solutions of (31.10.8) are given in terms of the Riemann P-symbol (seex15.11(i)) as
31.10.9K(; ) =P8
<
:0 1 1
01
2 cos2
1
1
2 + 9
=
;P8
<
:0 11
0 0 1
2++cos2
1 1 1
2+ 9
=
;;
whereis a separation constant . For integral equations satis ed by the Heun polynomial Hpn;m(z) we have=
1
2 j,j= 0;1;:::;n .
For suitable choices of the branches of the P-symbols in (31.10.9) and the contour C, we can obtain both integral
equations satis ed by Heun functions, as well as the integral representations of a distinct solution of Heun's equation
in terms of a Heun function (polynomial, path-multiplicative solution).
Example 1
Let
31.10.10
K(z;t)
= (zt a)1
2
2F11
2 + ;1
2 +
;zt
a
2F1 1
2++; 1
2+
;a(z 1)(t 1)
(a 1)(zt a)
;
where<
>0,<>0, andCbe the Pochhammer double-loop contour about 0 and 1 (as in x31.9(i)). Then the inte-
gral equation (31.10.1) is satis ed by w(z) =wm(z) andW(z) =mwm(z), wherewm(z) = (0;1)Hfm(a;qm; ; ;
; ;z)
andmis the corresponding eigenvalue.
Example 2
Fuchs{Frobenius solutions Wm(z) = ~mz H`(1=a;qm; ;
+ 1; + 1;; 1=z) are represented in terms of
Heun functions wm(z) = (0;1)Hfm(a;qm; ; ;
; ;z) by (31.10.1) with W(z) =Wm(z),w(z) =wm(z), and with
kernel chosen from
31.10.11K(z;t) = (zt a)1
2 (zt/a) 1
2++
2F11
2 + ;3
2 +
+ 1;a
zt
P8
>><
>>:0 11
0 0 1
2++(z a)(t a)
(1 a)(zt a)
1 1 1
2+ 9
>>=
>>;:
Here ~mis a normalization constant and Cis the contour of Example 1.
31.10(ii) Type II
Ifw(z) is a solution of Heun's equation, then another
solutionW(z) (possibly a multiple of w(z)) can be rep-
resented as
31.10.12W(z) =Z
C1Z
C2K(z;s;t)w(s)w(t)(s;t)dsdt
for suitable contours C1,C2. The weight function is
31.10.13(s;t) = (s t)(st)
1((1 s)(1 t)) 1
((1 (s=a))(1 (t=a))) 1;
and the kernelK(z;s;t) is a solution of the partial dif-
ferential equation
31.10.14 ((t z)Ds+ (z s)Dt+ (s t)Dz)K= 0;whereDzis given by (31.10.4). The contours C1,C2
must be chosen so that
31.10.15p(t)@K
@tw(t) Kdw(t)
dt
C1= 0;
and
31.10.16p(s)@K
@sw(s) Kdw(s)
ds
C2= 0;
wherep(t) is given by (31.10.6).
716 Heun Functions
Kernel Functions
Set
31.10.17
u=(stz)1=2
a; v =(s 1)(t 1)(z 1)
1 a1=2
;
w=i(s a)(t a)(z a)
a(1 a)1=2
:
The kernelKmust satisfy
31.10.18@2K
@u2+@2K
@v2+@2K
@w2+2
1
u@K
@u
+2 1
v@K
@v+2 1
w@K
@w= 0:
This equation can be solved in terms of cylinder func-
tionsC(z) (x10.2(ii)):
31.10.19
K(u;v;w ) =u1
v1 w1 C1
(up1)
C1 (vp2)C1
iwp1+2
;
where1and2are separation constants.
Transformation of Independent Variable
A further change of variables, to spherical coordinates,
31.10.20
u=rcos; v =rsinsin ; w =rsincos ;
leads to the kernel equation
31.10.21
@2K
@r2+2(
++) 1
r@K
@r+1
r2@2K
@2
+(2(+) 1) cot (2
1) tan
r2@K
@
+1
r2sin2@2K
@ 2+(2 1) cot (2 1) tan
r2sin2@K
@ = 0:
This equation can be solved in terms of hypergeometric
functions (x15.11(i)):
31.10.22
K(r;; ) =rmsin2pP8
<
:0 11
0 0acos2
1
2(3
)c b9
=
;
P8
<
:0 11
0 0 a0cos2
1 1 b09
=
;;with
31.10.23m2+ 2( + )m 1= 0;
p2+ ( +
1
2)p 1
42= 0;
a+b= 2( + +p) 1;
ab=p2 p(1 ) 1
41;
c=
1
2 2( + +p);
a0+b0=+ 1; a0b0= 1
42;
and1and2are separation constants.
For integral equations for special con
uent Heun
functions (x31.12) see Kazakov and Slavyanov (1996).
31.11 Expansions in Series of
Hypergeometric Functions
31.11(i) Introduction
The formulas in this section are given in Svartholm
(1939) and Erd elyi (1942a, 1944).
The series of Type I ( x31.11(iii)) are useful since
they represent the functions in large domains. Series of
Type II (x31.11(iv)) are expansions in orthogonal poly-
nomials, which are useful in calculations of normaliza-
tion integrals for Heun functions; see Erd elyi (1944) and
x31.9(i).
For other expansions see x31.16(ii).
31.11(ii) General Form
Letw(z) be any Fuchs{Frobenius solution of Heun's
equation. Expand
31.11.1 w(z) =1X
j=0cjPj;
where (x15.11(i))
31.11.2Pj=P8
<
:0 11
0 0 +j z
1
1 j9
=
;;
with
31.11.3 +=
+ 1 = + :
The coecients cjsatisfy the equations
31.11.4 L0c0+M0c1= 0;
31.11.5 Kjcj 1+Ljcj+Mjcj+1= 0,j= 1;2;:::;
where
31.11.6 Kj= (j+ 1)(j+ 1)(j+
1)(j+ 1)
(2j+ 1)(2j+ 2);
31.12 Confluent Forms of Heun's Equation 717
31.11.7Lj=a(+j)( j) q+(j+ )(j+ )(j+
)(j+)
(2j+ )(2j+ + 1)
+(j +)(j +)(j
+)(j )
(2j+ )(2j+ 1);
31.11.8 Mj= (j ++ 1)(j ++ 1)(j
++ 1)(j + 1)
(2j+ + 1)(2j+ + 2):
,must also satisfy the condition
31.11.9 M 1P 1= 0:
31.11(iii) Type I
Here
31.11.10 = ; = ;
or
31.11.11 = ; = :
Then condition (31.11.9) is satis ed.
Every Fuchs{Frobenius solution of Heun's equation
(31.2.1) can be represented by a series of Type I. For
instance, choose (31.11.10). Then the Fuchs{Frobenius
solution at1belonging to the exponent has the ex-
pansion (31.11.1) with
31.11.12Pj= ( +j) (1
+ +j)
(1 + ++ 2j)z j
2F1 +j;1
+ +j
1 + ++ 2j;1
z
;
and (31.11.1) converges outside the ellipse Ein thez-
plane with foci at 0, 1, and passing through the third
nite singularity at z=a.
Every Heun function ( x31.4) can be represented by a
series of Type I convergent in the whole plane cut along
a line joining the two singularities of the Heun function.
For example, consider the Heun function which is
analytic at z=aand has exponent at1. The ex-
pansion (31.11.1) with (31.11.12) is convergent in the
plane cut along the line joining the two singularities
z= 0 andz= 1. In this case the accessory parameter
qis a root of the continued-fraction equation
31.11.13 (L0=M0) K1=M1
L1=M1 K2=M2
L2=M2 = 0:
The case = nfor nonnegative integer ncorresponds
to the Heun polynomial Hpn;m(z).
The expansion (31.11.1) for a Heun function that is
associated with any branch of (31.11.2)|other than a
multiple of the right-hand side of (31.11.12)|is conver-
gent inside the ellipse E.31.11(iv) Type II
Here one of the following four pairs of conditions is sat-
is ed:
31.11.14 =
+ 1; = 0;
31.11.15 =
; = 1;
31.11.16 =; =
1;
31.11.17 = 1; =
+ 2:
In each case Pjcan be expressed in terms of a Ja-
cobi polynomial ( x18.3). Such series diverge for Fuchs{
Frobenius solutions. For Heun functions they are con-
vergent inside the ellipse E. Every Heun function can
be represented by a series of Type II.
31.11(v) Doubly-In nite Series
Schmidt (1979) gives expansions of path-multiplicative
solutions (x31.6) in terms of doubly-in nite series of hy-
pergeometric functions.
31.12 Con
uent Forms of Heun's Equation
Con
uent forms of Heun's di erential equation (31.2.1)
arise when two or more of the regular singularities merge
to form an irregular singularity. This is analogous to
the derivation of the con
uent hypergeometric equation
from the hypergeometric equation in x13.2(i). There are
four standard forms, as follows:
Con
uent Heun Equation
31.12.1d2w
dz2+
z+
z 1+dw
dz+ z q
z(z 1)w= 0:
This has regular singularities at z= 0 and 1, and an
irregular singularity of rank 1 at z=1.
Mathieu functions (Chapter 28), spheroidal wave
functions (Chapter 30), and Coulomb spheroidal func-
tions (x30.12) are special cases of solutions of the con-
uent Heun equation.
Doubly-Con
uent Heun Equation
31.12.2d2w
dz2+
z2+
z+ 1dw
dz+ z q
z2w= 0:
This has irregular singularities at z= 0 and1, each of
rank 1.
718 Heun Functions
Bicon
uent Heun Equation
31.12.3d2w
dz2+
z++zdw
dz+ z q
zw= 0:
This has a regular singularity at z= 0, and an irregular
singularity at1of rank 2.
Tricon
uent Heun Equation
31.12.4d2w
dz2+ (
+z)zdw
dz+ ( z q)w= 0:
This has one singularity, an irregular singularity of rank
3 atz=1.
For properties of the solutions of (31.12.1){(31.12.4),
including connection formulas, see B uhring (1994), Ron-
veaux (1995, Parts B,C,D,E), Wolf (1998), Lay and
Slavyanov (1998), and Slavyanov and Lay (2000).
31.13 Asymptotic Approximations
For asymptotic approximations for the accessory param-
eter eigenvalues qm, see Fedoryuk (1991) and Slavyanov
(1996).
For asymptotic approximations of the solutions of
Heun's equation (31.2.1) when two singularities are close
together, see Lay and Slavyanov (1999).
For asymptotic approximations of the solutions of
con
uent forms of Heun's equation in the neighborhood
of irregular singularities, see Komarov et al. (1976),
Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik
(1997), Slavyanov and Veshev (1997), and Lay et al.
(1998).
31.14 General Fuchsian Equation
31.14(i) De nitions
The general second-order Fuchsian equation withN+1
regular singularities at z=aj,j= 1;2;:::;N , and at
1, is given by
31.14.1
d2w
dz2+0
@NX
j=1
j
z aj1
Adw
dz+0
@NX
j=1qj
z aj1
Aw= 0,
PN
j=1qj= 0.
The exponents at the nite singularities ajare
f0;1
jgand those at1aref ; g, where
31.14.2 + + 1 =NX
j=1
j; =NX
j=1ajqj:
The three sets of parameters comprise the singularity
parameters aj, the exponent parameters ; ;
j, and
theN 2 free accessory parameters qj. Witha1= 0
anda2= 1 the total number of free parameters is 3 N 3.
Heun's equation (31.2.1) corresponds to N= 3.Normal Form
31.14.3w(z) =0
@NY
j=1(z aj)
j=21
AW(z);
31.14.4
d2W
dz2=NX
j=1~
j
(z aj)2+~qj
z aj
W,PN
j=1~qj= 0,
31.14.5 ~qj=1
2NX
k=1
k6=j
j
k
aj ak qj;~
j=
j
2
j
2 1
:
31.14(ii) Kovacic's Algorithm
An algorithm given in Kovacic (1986) determines if a
given (not necessarily Fuchsian) second-order homoge-
neous linear di erential equation with rational coe-
cients has solutions expressible in nite terms (Liouvil-
lean solutions). The algorithm returns a list of solutions
if they exist.
For applications of Kovacic's algorithm in spatio-
temporal dynamics see Rod and Sleeman (1995).
31.15 Stieltjes Polynomials
31.15(i) De nitions
Stieltjes polynomials are polynomial solutions of the
Fuchsian equation (31.14.1). Rewrite (31.14.1) in the
form
31.15.1
d2w
dz2+0
@NX
j=1
j
z aj1
Adw
dz+(z)QN
j=1(z aj)w= 0;
where (z) is a polynomial of degree not exceeding
N 2. There exist at most n+N 2
N 2
polynomials V(z)
of degree not exceeding N 2 such that for ( z) =V(z),
(31.15.1) has a polynomial solution w=S(z) of degree
n. TheV(z) are called Van Vleck polynomials and the
corresponding S(z)Stieltjes polynomials .
31.15(ii) Zeros
Ifz1;z2;:::;znare the zeros of an nth degree Stieltjes
polynomial S(z), then every zero zkis either one of the
parameters ajor a solution of the system of equations
31.15.2NX
j=1
j=2
zk aj+nX
j=1
j6=k1
zk zj= 0,k= 1;2;:::;n:
Iftkis a zero of the Van Vleck polynomial V(z), cor-
responding to an nth degree Stieltjes polynomial S(z),
andz0
1;z0
2;:::;z0
n 1are the zeros of S0(z) (the derivative
Applications 719
ofS(z)), thentkis either a zero of S0(z) or a solution
of the equation
31.15.3NX
j=1
j
tk aj+n 1X
j=11
tk z0
j= 0:
The system (31.15.2) determines the zkas the points
of equilibrium of nmovable (interacting) particles with
unit charges in a eld of Nparticles with the charges
j=2 xed ataj. This is the Stieltjes electrostatic inter-
pretation .
The zeroszk,k= 1;2;:::;n; of the Stieltjes polyno-
mialS(z) are the critical points of the function G, that
is, points at which @G/@k= 0,k= 1;2;:::;n , where
31.15.4G(1;2;:::;n)
=nY
k=1NY
`=1(k a`)
`/2nY
j=k+1(k j):
If the following conditions are satis ed:
31.15.5
j>0; aj2R,j= 1;2;:::;N ,
and
31.15.6 aj<aj+1,j= 1;2;:::;N 1,
then there are exactly n+N 2
N 2
polynomials S(z), each
of which corresponds to each of the n+N 2
N 2
ways of dis-
tributing its nzeros among N 1 intervals ( aj;aj+1),
j= 1;2;:::;N 1. In this case the accessory parame-
tersqjare given by
31.15.7 qj=
jnX
k=11
zk aj,j= 1;2;:::;N .
See Marden (1966), Alam (1979), and Al-Rashed
and Zaheer (1985) for further results on the location
of the zeros of Stieltjes and Van Vleck polynomials.
31.15(iii) Products of Stieltjes Polynomials
If the exponent and singularity parameters satisfy
(31.15.5){(31.15.6), then for every multi-index m=
(m1;m2;:::;mN 1), where each mjis a nonnegative
integer, there is a unique Stieltjes polynomial with
mjzeros in the open interval ( aj;aj+1) for eachj=
1;2;:::;N 1. We denote this Stieltjes polynomial by
Sm(z).
LetSm(z) andSl(z) be Stieltjes polynomials
corresponding to two distinct multi-indices m=
(m1;m2;:::;mN 1) and l= (`1;`2;:::;`N 1). The
products
31.15.8Sm(z1)Sm(z2)Sm(zN 1),zj2(aj;aj+1);
31.15.9 Sl(z1)Sl(z2)Sl(zN 1),zj2(aj;aj+1);
are mutually orthogonal over the set Q:
31.15.10Q= (a1;a2)(a2;a3) (aN 1;aN);with respect to the inner product
31.15.11 (f;g)=Z
Qf(z)g(z)(z)dz;
with weight function
31.15.12
(z) =0
@N 1Y
j=1NY
k=1jzj akj
k 11
A0
@N 1Y
j<k(zk zj)1
A:
The normalized system of products (31.15.8) forms an
orthonormal basis in the Hilbert space L2
(Q). For fur-
ther details and for the expansions of analytic functions
in this basis see Volkmer (1999).
Applications
31.16 Mathematical Applications
31.16(i) Uniformization Problem for Heun's
Equation
The main part of Smirnov (1996) consists of
V. I. Smirnov's 1918 M. Sc. thesis \Inversion problem
for a second-order linear di erential equation with four
singular points". It describes the monodromy group of
Heun's equation for speci c values of the accessory pa-
rameter.
31.16(ii) Heun Polynomial Products
Expansions of Heun polynomial products in terms of Ja-
cobi polynomial ( x18.3) products are derived in Kalnins
and Miller (1991a,b, 1993) from the viewpoint of inter-
relation between two bases in a Hilbert space:
31.16.1
Hpn;m(x)Hpn;m(y)
=nX
j=0Ajsin2j
P(
++2j 1; 1)
n j (cos 2)P( 1;
1)
j (cos 2 );
wheren= 0;1;:::,m= 0;1;:::;n , and
31.16.2x= sin2cos2 ; y = sin2sin2 :
The coecients Ajsatisfy the relations:
31.16.3 Q0A0+R0A1= 0;
31.16.4PjAj 1+QjAj+RjAj+1= 0,j= 1;2;:::;n ,
where
720 Heun Functions
31.16.5 Pj=( j+n)j( +j 1)(
++j 2)
(
++ 2j 3)(
++ 2j 2);
31.16.6Qj= aj(j+
+ 1) q+(j n)(j+ )(j+
)(j+
+ 1)
(2j+
+)(2j+
+ 1)
+(j+n+
+ 1)j(j+ 1)(j +
+ 1)
(2j+
+ 1)(2j+
+ 2);
31.16.7 Rj=(n j)(j+n+
+)(j+
)(j+)
(
++ 2j)(
++ 2j+ 1):
By specifying either or in (31.16.1) and (31.16.2)
we obtain expansions in terms of one variable.
31.17 Physical Applications
31.17(i) Addition of Three Quantum Spins
The problem of adding three quantum spins s,t, and
ucan be solved by the method of separation of vari-
ables , and the solution is given in terms of a product
of two Heun functions. We use vector notation [ s;t;u]
(respective scalar ( s;t;u )) for any one of the three spin
operators (respective spin values).
Consider the following spectral problem on the
sphereS2:x2=x2
s+x2
t+x2
u=R2.
31.17.1
J2 (x)(s+t+u)2 (x) =j(j+ 1) ( x);
Hs (x)( 2st (2/a)su) (x) =hs (x);
for the common eigenfunction ( x) = (xs;xt;xu),
whereais the coupling parameter of interacting spins.
Introduce elliptic coordinates z1andz2onS2. Then
31.17.2x2
s
zk+x2
t
zk 1+x2
u
zk a= 0,k= 1;2,
with
31.17.3x2
s=R2z1z2
a; x2
t=R2(z1 1)(z2 1)
1 a;
x2
u=R2(z1 a)(z2 a)
a(a 1):
The operators J2andHsadmit separation of variables
inz1;z2, leading to the following factorization of the
eigenfunction ( x):
31.17.4 (x) = (z1z2) s 1
4((z1 1)(z2 1)) t 1
4
((z1 a)(z2 a)) u 1
4w(z1)w(z2);
wherew(z) satis es Heun's equation (31.2.1) with aas
in (31.17.1) and the other parameters given by
31.17.5
= s t u j 1; =j s t u;
= 2s;
= 2t; = 2u;q=ahs+ 2s(at+u):For more details about the method of separation of
variables and relation to special functions see Olevski
(1950), Kalnins et al. (1976), Miller (1977), and Kalnins
(1986).
31.17(ii) Other Applications
Heun functions appear in the theory of black holes
(Kerr (1963), Teukolsky (1972), Chandrasekhar (1984),
Suzuki et al. (1998), Kalnins et al. (2000)), lattice sys-
tems in statistical mechanics (Joyce (1973, 1994)), dislo-
cation theory (Lay and Slavyanov (1999)), and quantum
systems (Bay et al. (1997), Tolstikhin and Matsuzawa
(2001)).
For applications of Heun's equation and functions
in astrophysics see Debosscher (1998) where di erent
spectral problems for Heun's equation are also consid-
ered. More applications|including those of generalized
spheroidal wave functions and con
uent Heun functions
in mathematical physics, astrophysics, and the two-
center problem in molecular quantum mechanics|can
be found in Leaver (1986) and Slavyanov and Lay (2000,
Chapter 4). For application of bicon
uent Heun func-
tions in a model of an equatorially trapped Rossby wave
in a shear
ow in the ocean or atmosphere see Boyd and
Natarov (1998).
Computation
31.18 Methods of Computation
Independent solutions of (31.2.1) can be computed in
the neighborhoods of singularities from their Fuchs{
Frobenius expansions ( x31.3), and elsewhere by numer-
ical integration of (31.2.1). Subsequently, the coe-
cients in the necessary connection formulas can be cal-
culated numerically by matching the values of solutions
and their derivatives at suitably chosen values of z; see
La (1994) and Lay et al. (1998). Care needs to be
taken to choose integration paths in such a way that the
wanted solution is growing in magnitude along the path
at least as rapidly as all other solutions ( x3.7(ii)). The
References 721
computation of the accessory parameter for the Heun
functions is carried out via the continued-fraction equa-
tions (31.4.2) and (31.11.13) in the same way as for the
Mathieu, Lam e, and spheroidal wave functions in Chap-
ters 28{30.
References
General References
The main references used in writing this chapter are
Sleeman (1966b) and Ronveaux (1995). For additional
bibliographic reading see Erd elyi et al. (1955).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x31.2 Erd elyi et al. (1955, Chapter XV), Ronveaux
(1995, Part A, Chapters 1 and 2).x31.3 Snow (1952), Ronveaux (1995, Part A, Chapters
2 and 3).
x31.4 Erd elyi et al. (1955, Chapter XV), Arscott
(1964b, Chapter IX).
x31.5 Erd elyi et al. (1955, Chapter XV), Arscott
(1964b, Chapter IX).
x31.7 Ronveaux (1995, Part A, Chapter 1).
x31.9 Becker (1997).
x31.10 Lambe and Ward (1934) Erd elyi (1942b), Va-
lent (1986), Sleeman (1969), An error in the last
reference is corrected here.
x31.12 The process of con
uence is discussed in
Ince (1926, Chapter XX). See Decarreau et al.
(1978a,b) for the classi cation of con
uent forms.
x31.14 Ince (1926, Chapter XV).
x31.15 Marden (1966).
x31.17 Gaudin (1983), Kuznetsov (1992).
Chapter 32
Painlev e Transcendents
P. A. Clarkson1
Notation 724
32.1 Special Notation . . . . . . . . . . . . . 724
Properties 724
32.2 Di erential Equations . . . . . . . . . . . 724
32.3 Graphics . . . . . . . . . . . . . . . . . . 726
32.4 Isomonodromy Problems . . . . . . . . . 728
32.5 Integral Equations . . . . . . . . . . . . . 729
32.6 Hamiltonian Structure . . . . . . . . . . 729
32.7 B acklund Transformations . . . . . . . . 730
32.8 Rational Solutions . . . . . . . . . . . . . 732
32.9 Other Elementary Solutions . . . . . . . . 734
32.10 Special Function Solutions . . . . . . . . 73532.11 Asymptotic Approximations for Real Vari-
ables . . . . . . . . . . . . . . . . . . . . 736
32.12 Asymptotic Approximations for Complex
Variables . . . . . . . . . . . . . . . . . . 738
Applications 738
32.13 Reductions of Partial Di erential Equations 738
32.14 Combinatorics . . . . . . . . . . . . . . . 739
32.15 Orthogonal Polynomials . . . . . . . . . . 739
32.16 Physical . . . . . . . . . . . . . . . . . . 739
Computation 739
32.17 Methods of Computation . . . . . . . . . 740
References 740
1School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, United Kingdom.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
723
724 Painlev e Transcendents
Notation
32.1 Special Notation
(For other notation see pp. xiv and 873.)
m;n integers.
x real variable.
z complex variable.
k real parameter.
Unless otherwise noted, primes indicate derivatives
with respect to the argument.
The functions treated in this chapter are the solu-
tions of the Painlev e equations P I{PVI.Properties
32.2 Di erential Equations
32.2(i) Introduction
The six Painlev e equations P I{PVIare as follows:
32.2.1d2w
dz2= 6w2+z;
32.2.2d2w
dz2= 2w3+zw+ ;
32.2.3d2w
dz2=1
wdw
dz2
1
zdw
dz+ w2+
z+
w3+
w;
32.2.4d2w
dz2=1
2wdw
dz2
+3
2w3+ 4zw2+ 2(z2 )w+
w;
32.2.5d2w
dz2=1
2w+1
w 1dw
dz2
1
zdw
dz+(w 1)2
z2
w+
w
+
w
z+w(w+ 1)
w 1;
32.2.6d2w
dz2=1
21
w+1
w 1+1
w zdw
dz2
1
z+1
z 1+1
w zdw
dz
+w(w 1)(w z)
z2(z 1)2
+ z
w2+
(z 1)
(w 1)2+z(z 1)
(w z)2
;
with , ,
, andarbitrary constants. The solutions
of P I{PVIare called the Painlev e transcendents . The
six equations are sometimes referred to as the Painlev e
transcendents, but in this chapter this term will be used
only for their solutions.
Let
32.2.7d2w
dz2=F
z;w;dw
dz
;
be a nonlinear second-order di erential equation in
whichFis a rational function of wanddw/dz, and
islocally analytic inz, that is, analytic except for iso-
lated singularities in C. In general the singularities of
the solutions are movable in the sense that their loca-
tion depends on the constants of integration associated
with the initial or boundary conditions. An equation is
said to have the Painlev e property if all its solutions are
free from movable branch points ; the solutions may have
movable poles or movable isolated essential singularities
(x1.10(iii)), however.
There are fty equations with the Painlev e property.
They are distinct modulo M obius (bilinear) transforma-
tions
32.2.8W() =a(z)w+b(z)
c(z)w+d(z); = (z);
in whicha(z),b(z),c(z),d(z), and (z) are locally an-alytic functions. The fty equations can be reduced
to linear equations, solved in terms of elliptic functions
(Chapters 22 and 23), or reduced to one of P I{PVI.
For arbitrary values of the parameters , ,
, and
, the general solutions of P I{PVIaretranscendental ,
that is, they cannot be expressed in closed-form ele-
mentary functions. However, for special values of the
parameters, equations P II{PVIhave special solutions in
terms of elementary functions, or special functions de-
ned elsewhere in this Handbook.
32.2(ii) Renormalizations
If
6= 0 in P III, then set
= 1 and= 1, without
loss of generality, by rescaling wandzif necessary. If
= 0 and 6= 0 in P III, then set = 1 and= 1,
without loss of generality. Lastly, if = 0 and
6= 0,
then set = 1 and
= 1, without loss of generality.
If6= 0 in P V, then set= 1
2, without loss of
generality.
32.2(iii) Alternative Forms
In P III, ifw(z) = 1=2u() with=z2, then
32.2.9d2u
d2=1
udu
d2
1
du
d+u2( +
u)
42+
4+
4u;
32.2 Differential Equations 725
which is known as P0
III.
In P III, ifw(z) = exp( iu(z)), = , and=
,
then
32.2.10d2u
dz2+1
zdu
dz=2
zsinu+ 2
sin(2u):
In P IV, ifw(z) = 2p
2(u())2with=p
2zand
= 2+ 1, then
32.2.11d2u
d2= 3u5+ 2u3+ 1
42 1
2
u+
32u3:
When = 0 this is a nonlinear harmonic oscillator.
In P V, ifw(z) = (cothu())2with= lnz, then
32.2.12d2u
d2= coshu
2(sinhu)3 sinhu
2(coshu)3
1
4
esinh(2u) 1
8e2sinh(4u):
See also Okamoto (1987c), McCoy et al. (1977), Bas-
som et al. (1992), Bassom et al. (1995), and Takasaki
(2001).
32.2(iv) Elliptic Form
PVIcan be written in the form
32.2.13
z(1 z)I Zw
1dtp
t(t 1)(t z)!
=p
w(w 1)(w z)
+ z
w2+
(z 1)
(w 1)2+ ( 1
2)z(z 1)
(w z)2
;
where
32.2.14I=z(1 z)d2
dz2+ (1 2z)d
dz 1
4:
See Fuchs (1907), Painlev e (1906), Gromak et al. (2002,
x42); also Manin (1998).
32.2(v) Symmetric Forms
Let
32.2.15df1
dz+f1(f2 f3) + 21= 0;
df2
dz+f2(f3 f1) + 22= 0;
df3
dz+f3(f1 f2) + 23= 0;
where1,2,3are constants, f1,f2,f3are functions
ofz, with
32.2.16 1+2+3= 1;
32.2.17 f1(z) +f2(z) +f3(z) + 2z= 0:
Thenw(z) =f1(z) satis es P IVwith
32.2.18 ( ; ) = (3 2; 22
1):
See Noumi and Yamada (1998).Next, let
32.2.19zdf1
dz=f1f3(f2 f4) + (1
2 3)f1+1f3;
zdf2
dz=f2f4(f3 f1) + (1
2 4)f2+2f4;
zdf3
dz=f3f1(f4 f2) + (1
2 1)f3+3f1;
zdf4
dz=f4f2(f1 f3) + (1
2 2)f4+4f2;
where1,2,3,4are constants, f1,f2,f3,f4are
functions of z, with
32.2.20 1+2+3+4= 1;
32.2.21 f1(z) +f3(z) =pz;
32.2.22 f2(z) +f4(z) =pz:
Thenw(z) = 1 (pz=f1(z)) satis es P Vwith
32.2.23 ( ; ;
; ) = (1
22
1; 1
22
3;4 2; 1
2):
32.2(vi) Coalescence Cascade
PI{PVare obtained from P VIby a coalescence cascade:
32.2.24PVI ! PV ! PIV
# #
PIII ! PII ! PI
For example, if in P II
32.2.25 w(z; ) =W() +1
5;
32.2.26 z=2 6
10; =4
15;
then
32.2.27d2W
d2= 6W2++6(2W3+W);
thus in the limit as !0,W() satis es P Iwithz=.
If in P III
32.2.28 w(z; ; ;
; ) = 1 + 2W(;a);
32.2.29z= 1 +2; = 1
2 6;
=1
2 6+ 2a 3;
= =1
4 6;
then as!0,W(;a) satis es P IIwithz=, =a.
If in P IV
32.2.30w(z; ; ) = 22=3 1W(;a) + 3;
32.2.31
z= 2 2=3 3; = 2a 1
2 6; = 1
2 12;
then as!0,W(;a) satis es P IIwithz=, =a.
If in P V
32.2.32w(z; ; ;
; ) = 1 +W(;a;b;c;d );
32.2.33z=2; =1
4a 1+1
8c 2;
= 1
8c 2;
=1
4b; =1
82d;
then as!0,W(;a;b;c;d ) satis es P IIIwithz=,
=a, =b,
=c,=d.
If in P V
32.2.34w(z; ; ;
; ) =1
2p
2W(;a;b);
726 Painlev e Transcendents
32.2.35z= 1 +p
2; =1
2 4; =1
4b;
= 4; =a 2 1
2 4;
then as!0,W(;a;b) satis es P IVwithz=, =a,
=b.Lastly, if in P VI
32.2.36 w(z; ; ;
; ) =W(;a;b;c;d );
32.2.37z= 1 +;
=c 1 d 2; =d 2;
then as!0,W(;a;b;c;d ) satis es P Vwithz=,
=a, =b,
=c,=d.
32.3 Graphics
32.3(i) First Painlev e Equation
Plots of solutions wk(x) of P Iwithwk(0) = 0 and w0
k(0) =kfor various values of k, and the parabola 6 w2+x= 0.
For analytical explanation see x32.11(i).
Figure 32.3.1 :wk(x) for 12x1:33 andk= 0:5,
0:75, 1, 1:25, and the parabola 6 w2+x= 0, shown in
black.
Figure 32.3.2 :wk(x) for 12x2:43 andk= 0:5,
0:25, 0, 1, 2, and the parabola 6 w2+x= 0, shown in
black.
Figure 32.3.3 :wk(x) for 12x0:73 andk=
1:85185 3, 1:85185 5. The two graphs are indistinguish-
able whenxexceeds 5:2, approximately. The parabola
6w2+x= 0 is shown in black.
Figure 32.3.4 :wk(x) for 12x2:3 andk=
0:45142 7, 0:45142 8. The two graphs are indis-
tinguishable when xexceeds 4:8, approximately. The
parabola 6w2+x= 0 is shown in black.
32.3 Graphics 727
32.3(ii) Second Painlev e Equation with = 0
Herewk(x) is the solution of P IIwith = 0 and such that
32.3.1 wk(x)kAi(x), x!+1;
comparex32.11(ii).
Figure 32.3.5 :wk(x) andkAi(x) for 10x4 with
k= 0:5. The two graphs are indistinguishable when x
exceeds 0:4, approximately.
Figure 32.3.6 :wk(x) for 10x4 withk= 0:999,
1:001. The two graphs are indistinguishable when x
exceeds 2:8, approximately. The parabola 2 w2+x= 0
is shown in black.
32.3(iii) Fourth Painlev e Equation with = 0
Hereu=uk(x;) is the solution of
32.3.2d2u
dx2= 3u5+ 2xu3+ 1
4x2 1
2
u;
such that
32.3.3 ukU
1
2;x
, x!+1.
The corresponding solution of P IVis given by
32.3.4 w(x) = 2p
2u2
k(p
2x;);
with = 0, = 2+ 1, and
32.3.5 w(x)2p
2k2U2
1
2;p
2x
, x!+1;
compare (32.2.11) and x32.11(v). If we set d2u
dx2= 0 in (32.3.2) and solve for u, then
32.3.6 u2= 1
3x1
6p
x2+ 12+ 6:
Figure 32.3.7 :uk(x; 1
2) for 12x4 withk=
0:33554 691, 0 :33554 692. The two graphs are indis-
tinguishable when xexceeds 5:0, approximately. The
parabolasu2+1
2x= 0,u2+1
6x= 0 are shown in black
and green, respectively.
Figure 32.3.8 :uk(x;1
2) for 12x4 withk=
0:47442, 0:47443. The two graphs are indistinguish-
able whenxexceeds 2:2, approximately. The curves
u2+1
3x1
6p
x2+ 12 = 0 are shown in green and black,
respectively.
728 Painlev e Transcendents
Figure 32.3.9 :uk(x;3
2) for 12x4 withk=
0:38736, 0:38737. The two graphs are indistinguish-
able whenxexceeds 1:0, approximately. The curves
u2+1
3x1
6p
x2+ 24 = 0 are shown in green and black,
respectively.
Figure 32.3.10 :uk(x;5
2) for 12x4 withk=
0:24499 2, 0:24499 3. The two graphs are indistinguish-
able whenxexceeds 0:6, approximately. The curves
u2+1
3x1
6p
x2+ 36 = 0 are shown in green and black,
respectively.
32.4 Isomonodromy Problems
32.4(i) De nition
PI{PVIcan be expressed as the compatibility condition
of a linear system, called an isomonodromy problem or
Lax pair . Suppose
32.4.1@
@=A(z;) ;@
@z=B(z;) ;
is a linear system in which AandBare matrices and
is independent of z. Then the equation
32.4.2@2
@z@=@2
@@z;
is satis ed provided that
32.4.3@A
@z @B
@+AB BA= 0:
(32.4.3) is the compatibility condition of (32.4.1).
Isomonodromy problems for Painlev e equations are not
unique.
32.4(ii) First Painlev e Equation
PIis the compatibility condition of (32.4.1) with
32.4.4A(z;) = (44+ 2w2+z)1 0
0 1
i(42w+ 2w2+z)0 i
i0
2w0+1
20 1
1 0
;
32.4.5 B(z;) =
+w
1 0
0 1
iw
0 i
i0
:32.4(iii) Second Painlev e Equation
PIIis the compatibility condition of (32.4.1) with
32.4.6A(z;) = i(42+ 2w2+z)1 0
0 1
2w00 i
i0
+
4w
0 1
1 0
;
32.4.7 B(z;) = i w
w i
:
See Flaschka and Newell (1980).
32.4(iv) Third Painlev e Equation
The compatibility condition of (32.4.1) with
32.4.8A(z;) =1
4z 0
0 1
4z
+ 1
21u0
u11
211
+v0 1
4z v1v0
(v0 1
2z)
v11
4z v01
2;
32.4.9B(z;) =1
40
0 1
4
+0u0
u101
z
v0 1
4z v1v0
(v0 1
2z)
v11
4z v01
z;
where1is an arbitrary constant, is
32.4.10 zu0
0=1u0 zv0v1;
32.4.11zu0
1= 1u1 (z(2v0 z)/(2v1));
32.4.12zv0
0= 2v0u1v1+v0+ (u0(2v0 z)=v1);
32.4.13 zv0
1= 2u0 2u1v2
1 1v1:
Ifw= u0=(v0v1), then
32.4.14zw0= (4v0 z)w2+ (21 1)w+z;
32.5 Integral Equations 729
andwsatis es P IIIwith
32.4.15 ( ; ;
; ) = (20;2(1 1);1; 1);
where
32.4.160=4v0
z
1
1 z
4v0
+z 2v0
2v0v1u0+u1v1
:
Note that the right-hand side of the last equation is a
rst integral of the system (32.4.10){(32.4.13).
32.4(v) Other Painlev e Equations
For isomonodromy problems for P IV, PV, and P VIsee
Jimbo and Miwa (1981).
32.5 Integral Equations
LetK(z;) be the solution of
32.5.1
K(z;)
=kAiz+
2
+k2
4Z1
zZ1
zK(z;s) Ais+t
2
Ait+
2
dsdt;
wherekis a real constant, and Ai( z) is de ned inx9.2.
Then
32.5.2 w(z) =K(z;z);
satis es P IIwith = 0 and the boundary condition
32.5.3 w(z)kAi(z), z!+1.
32.6 Hamiltonian Structure
32.6(i) Introduction
PI{PVIcan be written as a Hamiltonian system
32.6.1dq
dz=@H
@p;dp
dz= @H
@q;
for suitable (non-autonomous) Hamiltonian functions
H(q;p;z ).
32.6(ii) First Painlev e Equation
The Hamiltonian for P Iis
32.6.2 HI(q;p;z ) =1
2p2 2q3 zq;
and so
32.6.3 q0=p;
32.6.4 p0= 6q2+z:
Thenq=wsatis es P I. The function
32.6.5 = H I(q;p;z );
de ned by (32.6.2) satis es
32.6.6 (00)2+ 4 (0)3+ 2z0 2= 0:
Conversely, if is a solution of (32.6.6), then
32.6.7 q= 0;
32.6.8 p= 00;
are solutions of (32.6.3) and (32.6.4).32.6(iii) Second Painlev e Equation
The Hamiltonian for P IIis
32.6.9 HII(q;p;z ) =1
2p2 (q2+1
2z)p ( +1
2)q;
and so
32.6.10 q0=p q2 1
2z;
32.6.11 p0= 2qp+ +1
2:
Thenq=wsatis es P IIandpsatis es
32.6.12pp00=1
2(p0)2+ 2p3 zp2 1
2( +1
2)2:
The function (z) = H II(q;p;z ) de ned by (32.6.9) sat-
is es
32.6.13 (00)2+ 4 (0)3+ 20(z0 ) =1
4( +1
2)2:
Conversely, if (z) is a solution of (32.6.13), then
32.6.14 q= (400+ 2 + 1)/(80);
32.6.15 p= 20;
are solutions of (32.6.10) and (32.6.11).
32.6(iv) Third Painlev e Equation
The Hamiltonian for P IIIis
32.6.16
zHIII(q;p;z ) =q2p2
1zq2+ (20+ 1)q 0z
p
+1(0+1)zq;
and so
32.6.17zq0= 2q2p 1zq2 (20+ 1)q+0z;
32.6.18zp0= 2qp2+ 21zqp
+ (20+ 1)p 1(0+1)z:
Thenq=wsatis es P IIIwith
32.6.19 ( ; ;
; ) =
211;20(0+ 1);2
1; 2
0
:
The function
32.6.20=zHIII(q;p;z ) +pq+2
0 1
201z2
de ned by (32.6.16) satis es
32.6.21(z00 0)2+ 2
(0)2 2
02
1z2
(z0 2)
+ 80101z0= 42
02
1(2
0+2
1)z2:
Conversely, if is a solution of (32.6.21), then
32.6.22q=0(z00 (20+ 1)0+ 2011z)
2
021z2 (0)2;
32.6.23 p= (0+01z)/(20);
are solutions of (32.6.17) and (32.6.18).
730 Painlev e Transcendents
The Hamiltonian for P0
III(x32.2(iii)) is
32.6.24HIII(q;p; ) =q2p2
1q2+0q 0
p
+1
21(0+1)q;
and so
32.6.25 q0= 2q2p 1q2 0q+0;
32.6.26p0= 2qp2+ 21qp+0p 1
21(0+1):
Thenq=usatis es P0
IIIwith
32.6.27
( ; ;
; ) =
411;40(0+ 1);42
1; 42
0
:
The function
32.6.28=HIII(q;p; ) +1
42
0 1
201
de ned by (32.6.24) satis es
32.6.292(00)2+
4(0)2 2
02
1
(0 )
+01010=1
42
02
1(2
0+2
1):
Conversely, if is a solution of (32.6.29), then
32.6.30 q=0(00 200+011)
2
021 4(0)2;
32.6.31 p= (20+01)/(20);
are solutions of (32.6.25) and (32.6.26).
The Hamiltonian for P IIIwith
= 0 is
32.6.32zHIII(q;p;z ) =q2p2+ (q 0z)p 1zq;
and so
32.6.33 zq0= 2q2p+q 0z;
32.6.34 zp0= 2qp2 p+1z:
Thenq=wsatis es P IIIwith
32.6.35 ( ; ;
; ) =
21;0( 1);0; 2
0
:
The function
32.6.36=zHIII(q;p;z ) +pq+1
4(+ 1)2
de ned by (32.6.32) satis es
32.6.37(z00 0)2+ 2(0)2(z0 2)
401(+ 1)1z0= 42
02
1z2:
Conversely, if is a solution of (32.6.37), then
32.6.38q=0(z00 0+ 201z)
(0)2;
32.6.39 p=0/(20);
are solutions of (32.6.33) and (32.6.34).32.6(v) Other Painlev e Equations
For Hamiltonian structure for P IVsee Jimbo and Miwa
(1981), Okamoto (1986); also Forrester and Witte
(2001).
For Hamiltonian structure for P Vsee Jimbo and
Miwa (1981), Okamoto (1987b); also Forrester and
Witte (2002).
For Hamiltonian structure for P VIsee Jimbo and
Miwa (1981) and Okamoto (1987a); also Forrester and
Witte (2004).
32.7 B acklund Transformations
32.7(i) De nition
With the exception of P I, aB acklund transformation
relates a Painlev e transcendent of one type either to
another of the same type but with di erent values of
the parameters, or to another type.
32.7(ii) Second Painlev e Equation
Letw=w(z; ) be a solution of P II. Then the trans-
formations
32.7.1 S:w(z; ) = w;
and
32.7.2T:w(z; 1) = w 2 1
2w22w0+z;
furnish solutions of P II, provided that 6=1
2. PIIalso
has the special transformation
32.7.3 W(;1
2") =2 1=3"
w(z; 0)d
dzw(z; 0);
or equivalently,
32.7.4
w2(z; 0) = 2 1=3
W2(;1
2") "d
dW(;1
2") +1
2
;
with= 21=3zand"=1, whereW(;1
2") satis es
PIIwithz=, =1
2", andw(z; 0) satis es P IIwith
= 0.
The solutions w =w(z; ),w 1=w(z; 1),
satisfy the nonlinear recurrence relation
32.7.5 +1
2
w +1+w + 1
2
w +w 1+ 2w2
+z= 0:
See Fokas et al. (1993).
32.7 B acklund Transformations 731
32.7(iii) Third Painlev e Equation
Letwj=w(z; j; j;
j;j),j= 0;1;2, be solutions of
PIIIwith
32.7.6 ( 1; 1;
1;1) = ( 0; 0;
0;0);
32.7.7 ( 2; 2;
2;2) = ( 0; 0; 0;
0):
Then
32.7.8 S1:w1= w0;
32.7.9 S2:w2= 1/w0:
Next, letWj=W(z; j; j;1; 1),j= 0;1;2;3;4,
be solutions of P IIIwith
32.7.10 1= 3= 0+ 2; 2= 4= 0 2;
1= 2= 0+ 2; 3= 4= 0 2:
Then
32.7.11
T1:W1=zW0
0+zW2
0 W0 W0+z
W0(zW0
0+zW2
0+ W0+W0+z);
32.7.12
T2:W2= zW0
0 zW2
0 W0 W0+z
W0(zW0
0 zW2
0 W0+W0+z);
32.7.13
T3:W3= zW0
0+zW2
0+ W0 W0 z
W0(zW0
0+zW2
0+ W0+W0 z);
32.7.14
T4:W4=zW0
0 zW2
0+ W0 W0 z
W0(zW0
0 zW2
0 W0+W0 z):
See Milne et al. (1997).
If
= 0 and 6= 0, then set = 1 and= 1,
without loss of generality. Let uj=w(z; 1; j;0; 1),
j= 0;5;6, be solutions of P IIIwith
32.7.15 5= 0+ 2; 6= 0 2:
Then
32.7.16T5:u5= (zu0
0+z ( 0+ 1)u0)
u2
0;
32.7.17T6:u6= (zu0
0 z+ ( 0 1)u0)
u2
0:
Similar results hold for P IIIwith= 0 and
6= 0.
Furthermore,
32.7.18w(z;a;b;0;0) =W2(; 0;0;a;b); z =1
22:32.7(iv) Fourth Painlev e Equation
Letw0=w(z; 0; 0) andw
j=w(z;
j;
j),j=
1;2;3;4, be solutions of P IVwith
32.7.19
1=1
4
2 2 03p
2 0
;
1= 1
2
1 + 01
2p
2 02
;
2= 1
4
2 + 2 03p
2 0
;
2= 1
2
1 01
2p
2 02
;
3=3
2 1
2 03
4p
2 0;
3= 1
2
1 01
2p
2 02
;
4= 3
2 1
2 03
4p
2 0;
4= 1
2
1 01
2p
2 02
:
Then
32.7.20T
1:w
1=w0
0 w2
0 2zw0p 2 0
2w0;
32.7.21T
2:w
2= w0
0+w2
0+ 2zw0p 2 0
2w0;
32.7.22T
3:w
3=w0+2
1 01
2p 2 0
w0
w0
0p 2 0+ 2zw0+w2
0;
32.7.23T
4:w
4=w0+2
1 + 01
2p 2 0
w0
w0
0p 2 0 2zw0 w2
0;
valid when the denominators are nonzero, and where
the upper signs or the lower signs are taken throughout
each transformation. See Bassom et al. (1995).
32.7(v) Fifth Painlev e Equation
Letwj(zj) =w(zj; j; j;
j;j),j= 0;1;2, be solu-
tions of P Vwith
32.7.24
z1= z0; z 2=z0;( 1; 1;
1;1) = ( 0; 0;
0;0);
( 2; 2;
2;2) = ( 0; 0;
0;0):
Then
32.7.25S1:w1(z1) =w(z0);
32.7.26S2:w2(z2) = 1/w(z0):
LetW0=W(z; 0; 0;
0; 1
2) andW1=
W(z; 1; 1;
1; 1
2) be solutions of P V, where
32.7.27
1=1
8
0+"1
1 "3p
2 0 "2p
2 02
;
1= 1
8
0 "1
1 "3p
2 0 "2p
2 02
;
1="1
"3p
2 0 "2p
2 0
;
732 Painlev e Transcendents
and"j=1,j= 1;2;3, independently. Also let
32.7.28 =zW0
0 "2p
2 0W2
0+"3p
2 0
+
"2p
2 0 "3p
2 0+"1z
W0;
and assume 6= 0. Then
32.7.29T"1;"2;"3:W1= ( 2"1zW0)/;
provided that the numerator on the right-hand side does
not vanish. Again, since "j=1,j= 1;2;3, indepen-
dently, there are eight distinct transformations of type
T"1;"2;"3.
32.7(vi) Relationship Between the Third and
Fifth Painlev e Equations
Letw=w(z; ; ; 1; 1) be a solution of P IIIand
32.7.30 v=w0 "w2+ ((1 " )w/z);
with"=1. Then
32.7.31W(; 0; 0;
0;0) =v 1
v+ 1; z =p
2;
satis es P Vwith
32.7.32
( 0; 0;
0;0)
=
( " + 2)2=32; ( +" 2)2=32; ";0
:
32.7(vii) Sixth Painlev e Equation
Letwj(zj) =wj(zj; j; j;
j;j),j= 0;1;2;3, be solu-
tions of P VIwith
32.7.33 z1= 1=z0;
32.7.34 z2= 1 z0;
32.7.35 z3= 1=z0;
32.7.36 ( 1; 1;
1;1) = ( 0; 0; 0+1
2;
0+1
2);
32.7.37 ( 2; 2;
2;2) = ( 0;
0; 0;0);
32.7.38 ( 3; 3;
3;3) = ( 0; 0;
0;0):
Then
32.7.39S1:w1(z1) =w0(z0)=z0;
32.7.40S2:w2(z2) = 1 w0(z0);
32.7.41S3:w3(z3) = 1=w0(z0):
The transformations Sj, forj= 1;2;3, generate a group
of order 24. See Iwasaki et al. (1991, p. 127).
Letw(z; ; ;
; ) andW(z;A;B;C;D ) be solutions
of P VIwith
32.7.42 ( ; ;
; ) = 1
2(1 1)2; 1
22
0;1
22
1;1
2(1 2
2)
;32.7.43
(A;B;C;D ) = 1
2(1 1)2; 1
22
0;1
22
1;1
2(1 2
2)
;
and
32.7.44 j= j+1
2;
forj= 0;1;2;1, where
32.7.45
=0+1+2+1 1 = 1 (0+ 1+ 2+ 1):
Then
32.7.46
w W=z(z 1)W0
W(W 1)(W z)+0
W+1
W 1+2 1
W z
=z(z 1)w0
w(w 1)(w z)+0
w+1
w 1+2 1
w z:
PVIalso has quadratic and quartic transformations.
Letw=w(z; ; ;
; ) be a solution of P VI. The
quadratic transformation
32.7.47u1(1) =(1 w)(w z)
(1 +pz)2w; 1=1 pz
1 +pz2
;
transforms P VIwith = and
=1
2 to P VIwith
( 1; 1;
1;1) = (4 ; 4
;0;1
2). The quartic transfor-
mation
32.7.48u2(2) =(w2 z)2
4w(w 1)(w z); 2=z;
transforms P VIwith = =
=1
2 to P VIwith
( 2; 2;
2;2) = (16 ;0;0;1
2). Also,
32.7.49u3(3) =1 z1=4
1 +z1=42pw+z1=4
pw z1=42
;
32.7.50 3=1 z1=4
1 +z1=44
;
transforms P VIwith = = 0 and
=1
2 to P VI
with 3= 3and
3=1
2 3.
32.7(viii) Ane Weyl Groups
See Okamoto (1986, 1987a,b,c), Sakai (2001), Umemura
(2000).
32.8 Rational Solutions
32.8(i) Introduction
PII{PVIpossess hierarchies of rational solutions for spe-
cial values of the parameters which are generated from
\seed solutions" using the B acklund transformations
and often can be expressed in the form of determinants.
See Airault (1979).
32.8 Rational Solutions 733
32.8(ii) Second Painlev e Equation
Rational solutions of P IIexist for =n(2Z) and are
generated using the seed solution w(z; 0) = 0 and the
B acklund transformations (32.7.1) and (32.7.2). The
rst four are
32.8.1 w(z; 1) = 1/z ;
32.8.2 w(z; 2) =1
z 3z2
z3+ 4;
32.8.3w(z; 3) =3z2
z3+ 4 6z2(z3+ 10)
z6+ 20z3 80;
32.8.4
w(z; 4) = 1
z+6z2(z3+ 10)
z6+ 20z3 80 9z5(z3+ 40)
z9+ 60z6+ 11200:
More generally,
32.8.5 w(z;n) =d
dz
lnQn 1(z)
Qn(z)
;
where theQn(z) are monic polynomials (coecient of
highest power of zis 1) satisfying
32.8.6Qn+1(z)Qn 1(z)
=zQ2
n(z) + 4 (Q0
n(z))2 4Qn(z)Q00
n(z);
withQ0(z) = 1,Q1(z) =z. Thus
32.8.7Q2(z) =z3+ 4;
Q3(z) =z6+ 20z3 80;
Q4(z) =z10+ 60z7+ 11200z;
Q5(z) =z15+ 140z12+ 2800z9+ 78400z6
3 13600z3 62 72000;
Q6(z) =z21+ 280z18+ 18480z15+ 6 27200z12
172 48000z9+ 14488 32000 z6
+ 1 93177 60000 z3 3 86355 20000 :
Next, let pm(z) be the polynomials de ned by
pm(z) = 0 form< 0, and
32.8.81X
m=0pm(z)m= exp
z 4
33
:
Then forn2
32.8.9 w(z;n) =d
dz
ln n 1(z)
n(z)
;
where n(z) is thenndeterminant
32.8.10
n(z) = p1(z)p3(z)p2n 1(z)
p0
1(z)p0
3(z)p0
2n 1(z)
............
p(n 1)
1 (z)p(n 1)
3 (z)p(n 1)
2n 1(z) :
For plots of the zeros of Qn(z) see Clarkson and
Mans eld (2003).32.8(iii) Third Painlev e Equation
Special rational solutions of P IIIare
32.8.11 w(z;; 2;; 4) =;
32.8.12 w(z; 0; ;0;) =z;
32.8.13w(z; 2+ 3; 2+ 1;1; 1) =z+
z++ 1;
with,, andarbitrary constants.
In the general case assume
6= 0, so that as in
x32.2(ii) we may set
= 1 and= 1. Then P IIIhas
rational solutions i
32.8.14 = 4n;
withn2Z. These solutions have the form
32.8.15 w(z) =Pm(z)/Qm(z);
wherePm(z) andQm(z) are polynomials of degree m,
with no common zeros.
For examples and plots see Milne et al. (1997); also
Clarkson (2003a). For determinantal representations
see Kajiwara and Masuda (1999).
32.8(iv) Fourth Painlev e Equation
Special rational solutions of P IVare
32.8.16 w1(z;2; 2) =1/z ;
32.8.17 w2(z; 0; 2) = 2z;
32.8.18 w3(z; 0; 2
9) = 2
3z:
There are also three families of solutions of P IVof
the form
32.8.19w1(z; 1; 1) =P1;n 1(z)/Q1;n(z);
32.8.20w2(z; 2; 2) = 2z+ (P2;n 1(z)/Q2;n(z));
32.8.21w3(z; 3; 3) = 2
3z+ (P3;n 1(z)/Q3;n(z));
wherePj;n 1(z) andQj;n(z) are polynomials of degrees
n 1 andn, respectively, with no common zeros.
In general, P IVhas rational solutions i either
32.8.22 =m; = 2(1 + 2n m)2;
or
32.8.23 =m; = 2(1
3+ 2n m)2;
withm;n2Z. The rational solutions when the param-
eters satisfy (32.8.22) are special cases of x32.10(iv).
For examples and plots see Bassom et al. (1995);
also Clarkson (2003b). For determinantal representa-
tions see Kajiwara and Ohta (1998) and Noumi and
Yamada (1999).
734 Painlev e Transcendents
32.8(v) Fifth Painlev e Equation
Special rational solutions of P Vare
32.8.24w(z;1
2; 1
22;(2 ); 1
22) =z+;
32.8.25 w(z;1
2;2;2; ) ==(z+);
32.8.26w(z;1
8; 1
8; ; ) = (+z)=( z);
withandarbitrary constants.
In the general case assume 6= 0, so that as in
x32.2(ii) we may set = 1
2. Then P Vhas a ratio-
nal solution i one of the following holds with m;n2Z
and"=1:
(a) =1
2(m+"
)2and = 1
2n2, wheren > 0,
m+nis odd, and 6= 0 whenjmj<n.
(b) =1
2n2and = 1
2(m+"
)2, wheren > 0,
m+nis odd, and 6= 0 whenjmj<n.
(c) =1
2a2, = 1
2(a+n)2, and
=m, withm+n
even.
(d) =1
2(b+n)2, = 1
2b2, and
=m, withm+n
even.
(e) =1
8(2m+ 1)2, = 1
8(2n+ 1)2, and
=2Z.
These rational solutions have the form
32.8.27w(z) =z++ (Pn 1(z)/Qn(z));
where,are constants, and Pn 1(z),Qn(z) are poly-
nomials of degrees n 1 andn, respectively, with no
common zeros. Cases (a) and (b) are special cases of
x32.10(v).
For examples and plots see Clarkson (2005). For de-
terminantal representations see Masuda et al. (2002).
For the case = 0 see Airault (1979) and Luka sevi c
(1968).
32.8(vi) Sixth Painlev e Equation
Special rational solutions of P VIare
32.8.28w(z;; 2;1
2;1
2 ( 1)2) =z;
32.8.29 w(z; 0;0;2;0) =z2;
32.8.30 w(z; 0;0;1
2; 3
2) =/z ;
32.8.31 w(z; 0;0;2; 4) =
z2;
32.8.32
w(z;1
2(+)2; 1
2;1
2( 1)2;1
2(2 )) =z
+z;
withandarbitrary constants.
In the general case, P VIhas rational solutions if
32.8.33 a+b+c+d= 2n+ 1;
wheren2Z,a="1p
2 ,b="2p 2 ,c="3p2
,
andd="4p
1 2, with"j=1,j= 1;2;3;4, inde-
pendently, and at least one of a,b,cordis an integer.
These are special cases of x32.10(vi).32.9 Other Elementary Solutions
32.9(i) Third Painlev e Equation
Elementary nonrational solutions of P IIIare
32.9.1 w(z;;0;0; 3) =z1=3;
32.9.2
w(z; 0; 2;0;4 2) =z((lnz)2+lnz+);
32.9.3w(z; 2;0;2(2 4);0) =z 1
z2+z+;
with,,, andarbitrary constants.
In the case
= 0 and 6= 0 we assume, as in
x32.2(ii), = 1 and= 1. Then P IIIhas algebraic
solutions i
32.9.4 = 2n;
withn2Z. These are rational solutions in =z1=3of
the form
32.9.5 w(z) =Pn2+1()/Qn2();
wherePn2+1() andQn2() are polynomials of degrees
n2+ 1 andn2, respectively, with no common zeros.
For examples and plots see Clarkson (2003a) and Milne
et al. (1997). Similar results hold when = 0 and
6= 0.
PIIIwith == 0 has a rst integral
32.9.6z2(w0)2+ 2zww0= (C+ 2 zw +
z2w2)w2;
withCan arbitrary constant, which is solvable by
quadrature. A similar result holds when =
= 0.
PIIIwith = =
== 0, has the general solution
w(z) =Cz, withCandarbitrary constants.
32.9(ii) Fifth Painlev e Equation
Elementary nonrational solutions of P Vare
32.9.7w(z;; 1
8; 2;0) = 1 +z1=2;
32.9.8 w(z; 0;0;; 1
22) =exp(z);
withandarbitrary constants.
PV, with= 0, has algebraic solutions if either
32.9.9 ( ; ;
) = (1
22; 1
8(2n 1)2; 1);
or
32.9.10 ( ; ;
) = (1
8(2n 1)2; 1
22;1);
withn2Zandarbitrary. These are rational solutions
in=z1=2of the form
32.9.11 w(z) =Pn2 n+1()/Qn2 n();
wherePn2 n+1() andQn2 n() are polynomials of de-
greesn2 n+1 andn2 n, respectively, with no common
zeros.
PV, with
== 0, has a rst integral
32.9.12z2(w0)2= (w 1)2(2 w2+Cw 2 );
32.10 Special Function Solutions 735
withCan arbitrary constant, which is solvable by
quadrature. For examples and plots see Clarkson
(2005). P V, with = = 0 and
2+ 2= 0, has
solutionsw(z) =Cexp
p
2z
, withCan arbitrary
constant.
32.9(iii) Sixth Painlev e Equation
An elementary algebraic solution of P VIis
32.9.13w(z;1
22; 1
22;1
22;1
2(1 2)) =z1=2;
withandarbitrary constants.
Dubrovin and Mazzocco (2000) classi es all alge-
braic solutions for the special case of P VIwith =
=
0,=1
2. For further examples of algebraic solutions see
Andreev and Kitaev (2002), Boalch (2005, 2006), Gro-
mak et al. (2002,x48), Hitchin (2003), Masuda (2003),
and Mazzocco (2001b).
32.10 Special Function Solutions
32.10(i) Introduction
For certain combinations of the parameters, P II{PVI
have particular solutions expressible in terms of the so-
lution of a Riccati di erential equation, which can be
solved in terms of special functions de ned in other
chapters. All solutions of P II{PVIthat are expressible in
terms of special functions satisfy a rst-order equation
of the form
32.10.1 (w0)n+n 1X
j=0Fj(w;z)(w0)j= 0;
whereFj(w;z) is polynomial in wwith coecients that
are rational functions of z.
32.10(ii) Second Painlev e Equation
PIIhas solutions expressible in terms of Airy functions
(x9.2) i
32.10.2 =n+1
2;
withn2Z. For example, if =1
2", with"=1, then
the Riccati equation is
32.10.3 "w0=w2+1
2z;
with solution
32.10.4 w(z;1
2") = " 0(z)= (z);
where
32.10.5 (z) =C1Ai
2 1=3z
+C2Bi
2 1=3z
;
withC1,C2arbitrary constants.Solutions for other values of are derived from
w(z;1
2) by application of the B acklund transforma-
tions (32.7.1) and (32.7.2). For example,
32.10.6 w(z;3
2) = 1
22+z;
32.10.7w(z;5
2) =1
22+z+2z2+ +z2
43+ 2z 1;
where = 0(z)= (z), with (z) given by (32.10.5).
More generally, if n= 1;2;3;:::, then
32.10.8w(z;n+1
2) =d
dz
ln n(z)
n+1(z)
;
where n(z) is thenndeterminant
32.10.9 n(z) = (z) 0(z) (n 1)(z)
0(z) 00(z) (n)(z)
............
(n 1)(z) (n)(z) (2n 2)(z) ;
and
32.10.10 w(z; n 1
2) = w(z;n+1
2):
32.10(iii) Third Painlev e Equation
If
6= 0, then as inx32.2(ii) we may set
= 1 and
= 1. P IIIthen has solutions expressible in terms of
Bessel functions ( x10.2) i
32.10.11 "1 +"2 = 4n+ 2;
withn2Z, and"1=1,"2=1, independently. In
the case"1 +"2 = 2, the Riccati equation is
32.10.12zw0="1zw2+ ( "1 1)w+"2z:
If 6="1, then (32.10.12) has the solution
32.10.13 w(z) = "1 0(z)= (z);
where
32.10.14 (z) =z(C1J() +C2Y());
with=p"1"2z,=1
2 "1, andC1,C2arbitrary con-
stants.
For examples and plots see Milne et al. (1997). For
determinantal representations see Forrester and Witte
(2002) and Okamoto (1987c).
32.10(iv) Fourth Painlev e Equation
PIVhas solutions expressible in terms of parabolic cylin-
der functions (x12.2) i either
32.10.15 = 2(2n+ 1 +" )2;
or
32.10.16 = 2n2;
withn2Zand"=1. In the case when n= 0 in
(32.10.15), the Riccati equation is
32.10.17 w0="(w2+ 2zw) 2(1 +" );
which has the solution
32.10.18 w(z) = " 0(z)= (z);
736 Painlev e Transcendents
where
32.10.19
(z) =
C1U
a;p
2z
+C2V
a;p
2z
exp 1
2"z2
;
witha= +1
2", andC1,C2arbitrary constants. When
a+1
2is zero or a negative integer the Uparabolic cylin-
der functions reduce to Hermite polynomials ( x18.3)
times an exponential function; thus
32.10.20
w(z; m; 2(m 1)2) = H0
m 1(z)
Hm 1(z),m= 1;2;3;:::,
and
32.10.21
w(z; m; 2(m+1)2) = 2z+H0
m(z)
Hm(z),m= 0;1;2;:::.
If 1 +" = 0, then (32.10.17) has solutions
32.10.22w(z) =8
>><
>>:2 exp
z2
p(C ierfc(iz)); "= 1;
2 exp
z2
p(C erfc(z)); " = 1;
whereCis an arbitrary constant and erfc is the com-
plementary error function ( x7.2(i)).
For examples and plots see Bassom et al. (1995). For
determinantal representations see Forrester and Witte
(2001) and Okamoto (1986).
32.10(v) Fifth Painlev e Equation
If6= 0, then as inx32.2(ii) we may set = 1
2. PV
then has solutions expressible in terms of Whittaker
functions (x13.14(i)), i
32.10.23 a+b+"3
= 2n+ 1;
or
32.10.24 (a n)(b n) = 0;
wheren2Z,a="1p
2 , andb="2p 2 , with
"j=1,j= 1;2;3, independently. In the case when
n= 0 in (32.10.23), the Riccati equation is
32.10.25 zw0=aw2+ (b a+"3z)w b:
Ifa6= 0, then (32.10.25) has the solution
32.10.26 w(z) = z 0(z)=(a (z));
where
32.10.27 (z) =C1M;() +C2W;()
(a b+1)=2exp 1
2
;
with="3z,=1
2(a b+ 1),=1
2(a+b), andC1,
C2arbitrary constants.
For determinantal representations see Forrester and
Witte (2002), Masuda (2004), and Okamoto (1987b).32.10(vi) Sixth Painlev e Equation
PVIhas solutions expressible in terms of hypergeometric
functions (x15.2(i)) i
32.10.28 a+b+c+d= 2n+ 1;
wheren2Z,a="1p
2 ,b="2p 2 ,c="3p2
, and
d="4p
1 2, with"j=1,j= 1;2;3;4, indepen-
dently. Ifn= 1, then the Riccati equation is
32.10.29w0=aw2
z(z 1)+(b+c)z a c
z(z 1)w b
z 1:
Ifa6= 0, then (32.10.29) has the solution
32.10.30 w(z) = 1
a ()d
d; =1
1 z;
where
32.10.31
() =C1F(b; a;b+c;) +C2 b+1 c
F( a b c+ 1; c+ 1; 2 b c;);
withC1,C2arbitrary constants.
Next, let = ( u;z) be the elliptic function
(xx22.15(ii), 23.2(iii)) de ned by
32.10.32 u=Z
0dtp
t(t 1)(t z);
where the fundamental periods 2 1and 2 2are lin-
early independent functions satisfying the hypergeomet-
ric equation
32.10.33z(1 z)d2
dz2+ (1 2z)d
dz 1
4 = 0:
Then P VI, with = =
= 0 and=1
2, has the
general solution
32.10.34w(z; 0;0;0;1
2) = (C1 1+C2 2;z);
withC1,C2arbitrary constants. The solution
(32.10.34) is an essentially transcendental function of
both constants of integration since P VIwith = =
= 0 and=1
2does not admit an algebraic rst inte-
gral of the form P(z;w;w0;C) = 0, with Ca constant.
For determinantal representations see Forrester and
Witte (2004) and Masuda (2004).
32.11 Asymptotic Approximations for Real
Variables
32.11(i) First Painlev e Equation
There are solutions of (32.2.1) such that
32.11.1w(x) = q
1
6jxj+djxj 1=8sin( (x) 0)
+o
jxj 1=8
, x! 1 ,
where
32.11.2 (x) = (24)1=4
4
5jxj5=4 5
8d2lnjxj
;
anddand0are constants.
32.11 Asymptotic Approximations for Real Variables 737
There are also solutions of (32.2.1) such that
32.11.3 w(x)q
1
6jxj, x! 1 .
Next, for given initial conditions w(0) = 0 and
w0(0) =k, withkreal,w(x) has at least one pole on
the real axis. There are two special values of k,k1and
k2, with the properties 0:45142 8<k1< 0:45142 7,
1:85185 3<k2<1:85185 5, and such that:
(a) Ifk<k 1, thenw(x)>0 forx0<x< 0, wherex0
is the rst pole on the negative real axis.
(b) Ifk1<k<k 2, thenw(x) oscillates about, and is
asymptotic to, q
1
6jxjasx! 1 .
(c) Ifk2<k, thenw(x) changes sign once, from pos-
itive to negative, as xpasses from x0to 0.
For illustration see Figures 32.3.1 to 32.3.4, and for
further information see Joshi and Kitaev (2005), Joshi
and Kruskal (1992), Kapaev (1988), Kapaev and Kitaev
(1993), and Kitaev (1994).
32.11(ii) Second Painlev e Equation
Consider the special case of P IIwith = 0:
32.11.4 w00= 2w3+xw;
with boundary condition
32.11.5 w(x)!0, x!+1.
Any nontrivial real solution of (32.11.4) that satis es
(32.11.5) is asymptotic to kAi(x), for some nonzero
realk, where Ai denotes the Airy function ( x9.2). Con-
versely, for any nonzero real k, there is a unique solu-
tionwk(x) of (32.11.4) that is asymptotic to kAi(x) as
x!+1.
Ifjkj<1, thenwk(x) exists for all suciently large
jxjasx! 1 , and
32.11.6wk(x) =djxj 1=4sin( (x) 0) +o
jxj 1=4
;
where
32.11.7 (x) =2
3jxj3=2 3
4d2lnjxj;
andd(6= 0),0are real constants. Connection formulas
fordand0are given by
32.11.8 d2= 1ln
1 k2
;
32.11.9
0=3
2d2ln 2 + ph
1 1
2id2
+1
4(1 2 sign(k));
where is the gamma function ( x5.2(i)), and the branch
of the ph function is immaterial.
Ifjkj= 1, then
32.11.10 wk(x)sign(k)q
1
2jxj,x! 1 .Ifjkj>1, thenwk(x) has a pole at a nite point
x=c0, dependent on k, and
32.11.11 wk(x)sign(k)(x c0) 1,x!c0+.
For illustration see Figures 32.3.5 and 32.3.6, and for
further information see Ablowitz and Clarkson (1991),
Bassom et al. (1998), Clarkson and McLeod (1988), De-
ift and Zhou (1995), Segur and Ablowitz (1981), and
Sule manov (1987). For numerical studies see Miles
(1978, 1980) and Rosales (1978).
32.11(iii) Modi ed Second Painlev e Equation
Replacement of wbyiwin (32.11.4) gives
32.11.12 w00= 2w3+xw:
Any nontrivial real solution of (32.11.12) satis es
32.11.13w(x) =djxj 1=4sin( (x) )
+O
jxj 5=4lnjxj
,x! 1 ,
where
32.11.14 (x) =2
3jxj3=2+3
4d2lnjxj;
withd(6= 0) and arbitrary real constants.
In the case when
32.11.15 +3
2d2ln 2 1
4 ph 1
2id2
=n;
withn2Z, we have
32.11.16 w(x)kAi(x), x!+1,
wherekis a nonzero real constant. The connection for-
mulas forkare
32.11.17 d2= 1ln
1 +k2
, sign(k) = ( 1)n.
In the generic case
32.11.18 +3
2d2ln 2 1
4 ph 1
2id2
6=n;
we have
32.11.19w(x) =q
1
2x+(2x) 1=4cos( (x) +)
+O
x 1
, x!+1,
where,(>0), andare real constants, and
32.11.20 (x) =2
3p
2x3=2 3
22lnx:
The connection formulas for ,, andare
32.11.21 = sign(=s);
32.11.22 2= 1ln
(1 +jsj2)=j2=sj
;
32.11.23= 3
4 7
22ln 2 + ph
1 +s2
+ ph
i2
;
where
32.11.24
s=
exp
d2
11=2
exp
i 3
2d2ln 2 1
4+ ph 1
2id2
:
738 Painlev e Transcendents
32.11(iv) Third Painlev e Equation
For P III, with = = 2(2R) and
= = 1,
32.11.25
w(x) 1
+1
2
2 2x (1=2)e 2x,x!+1,
whereis an arbitrary constant such that 1=<<
1=, and
32.11.26 w(x)Bx, x!0,
whereBandare arbitrary constants such that B6=
0 andj<j<1. The connection formulas relating
(32.11.25) and (32.11.26) are
32.11.27 = (2=) arcsin();
32.11.28B= 2 2 2 1
2(1 )
1
2(1 +) +
2 1
2(1 +)
1
2(1 ) +:
See also Abdullaev (1985), Novoksh enov (1985),
Its and Novoksh enov (1986), Kitaev (1987), Bobenko
(1991), Bobenko and Its (1995), Tracy and Widom
(1997), and Kitaev and Vartanian (2004).
32.11(v) Fourth Painlev e Equation
Consider P IVwith = 2+ 1 (2R) and = 0, that is,
32.11.29w00=(w0)2
2w+3
2w3+ 4xw2+ 2(x2 2 1)w;
and with boundary condition
32.11.30 w(x)!0, x!+1.
Any nontrivial solution of (32.11.29) that satis es
(32.11.30) is asymptotic to hU2
1
2;p
2x
asx!
+1, whereh(6= 0) is a constant. Conversely, for any
h(6= 0) there is a unique solution wh(x) of (32.11.29)
that is asymptotic to hU2
1
2;p
2x
asx!+1.
HereUdenotes the parabolic cylinder function ( x12.2).
Now suppose x! 1 . If 0h<h, where
32.11.31 h= 1.
1=2 (+ 1)
;
thenwh(x) has no poles on the real axis. Furthermore,
if=n= 0;1;2;:::, then
32.11.32 wh(x)h2nx2nexp
x2
,x! 1 .
Alternatively, if is not zero or a positive integer, then
32.11.33wh(x) = 2
3x+4
3dp
3 sin( (x) 0)+O
x 1
,
x! 1 ,
where
32.11.34 (x) =1
3p
3x2 4
3d2p
3 lnp
2jxj
;
andd(>0) and0are real constants. Connection for-
mulas fordand0are given by
32.11.35 d2= 1
4p
3 1ln
1 jj2
;
32.11.360=1
3d2p
3 ln 3 +2
3+7
12
+ ph+ ph
2
3ip
3d2
;where
32.11.37= 1 +
2ih3=2exp( i).
( )
;
and the branch of the ph function is immaterial.
Next ifh=h, then
32.11.38 wh(x) 2x, x! 1 ,
andwh(x) has no poles on the real axis.
Lastly ifh > h, thenwh(x) has a simple pole on
the real axis, whose location is dependent on h.
For illustration see Figures 32.3.7{32.3.10. In terms
of the parameter kthat is used in these gures h=
23=2k2.
32.12 Asymptotic Approximations for
Complex Variables
32.12(i) First Painlev e Equation
See Boutroux (1913), Kapaev and Kitaev (1993), Takei
(1995), Costin (1999), Joshi and Kitaev (2001), Kapaev
(2004), and Olde Daalhuis (2005b).
32.12(ii) Second Painlev e Equation
See Boutroux (1913), Novoksh enov (1990), Kapaev
(1991), Joshi and Kruskal (1992), Kitaev (1994), Its and
Kapaev (2003), and Fokas et al. (2006, Chapter 7).
32.12(iii) Third Painlev e Equation
See Fokas et al. (2006, Chapter 16).
Applications
32.13 Reductions of Partial Di erential
Equations
32.13(i) Korteweg{de Vries and Modi ed
Korteweg{de Vries Equations
The modi ed Korteweg{de Vries (mKdV) equation
32.13.1 vt 6v2vx+vxxx= 0;
has the scaling reduction
32.13.2z=x(3t) 1=3; v(x;t) = (3t) 1=3w(z);
wherew(z) satis es P IIwith a constant of integration.
The Korteweg{de Vries (KdV) equation
32.13.3 ut+ 6uux+uxxx= 0;
has the scaling reduction
32.13.4z=x(3t) 1=3; u(x;t) = (3t) 2=3(w0+w2);
wherew(z) satis es P II.
32.14 Combinatorics 739
Equation (32.13.3) also has the similarity reduction
32.13.5z=x+ 3t2; u(x;t) =W(z) t;
whereis an arbitrary constant and W(z) is express-
ible in terms of solutions of P I. See Fokas and Ablowitz
(1982) and P. J. Olver (1993b, p. 194).
32.13(ii) Sine-Gordon Equation
The sine-Gordon equation
32.13.6 uxt= sinu;
has the scaling reduction
32.13.7 z=xt; u (x;t) =v(z);
wherev(z) satis es (32.2.10) with =1
2and
= 0. In
consequence if w= exp( iv), thenw(z) satis es P III
with = =1
2and
== 0.
32.13(iii) Boussinesq Equation
The Boussinesq equation
32.13.8 utt=uxx 6(u2)xx+uxxxx;
has the traveling wave solution
32.13.9 z=x ct; u (x;t) =v(z);
wherecis an arbitrary constant and v(z) satis es
32.13.10 v00= 6v2+ (c2 1)v+Az+B;
withAandBconstants of integration. Depending
whetherA= 0 orA6= 0,v(z) is expressible in terms of
the Weierstrass elliptic function ( x23.2) or solutions of
PI, respectively.
32.14 Combinatorics
LetSNbe the group of permutations of the numbers
1;2;:::;N (x26.2). With 1m1<< mnN,
(m1);(m2);:::;(mn) is said to be an increasing
subsequence ofoflengthnwhen(m1)<(m2)<
<(mn). Let`N() be the length of the longest
increasing subsequence of . Then
32.14.1 lim
N!1Prob
`N() 2p
N
N1=6s!
=F(s);
where the distribution function F(s) is de ned here by
32.14.2F(s) = exp
Z1
s(x s)w2(x)dx
;
andw(x) satis es P IIwith = 0 and boundary condi-
tions
32.14.3 w(x)Ai(x), x!+1,
32.14.4 w(x)q
1
2x, x! 1 ,
where Ai denotes the Airy function ( x9.2).The distribution function F(s) given by (32.14.2)
arises in random matrix theory where it gives the lim-
iting distribution for the normalized largest eigenvalue
in the Gaussian Unitary Ensemble of nnHermitian
matrices; see Tracy and Widom (1994).
See Forrester and Witte (2001, 2002) for other in-
stances of Painlev e equations in random matrix theory.
32.15 Orthogonal Polynomials
Letpn(),n= 0;1;:::, be the orthonormal set of poly-
nomials de ned by
32.15.1Z1
1exp
1
44 z2
pm()pn()d=m;n;
with recurrence relation
32.15.2an+1(z)pn+1() =pn() an(z)pn 1();
forn= 1;2;:::; comparex18.2. Then un(z) = (an(z))2
satis es the nonlinear recurrence relation
32.15.3 (un+1+un+un 1)un=n 2zun;
forn= 1;2;:::, and also P IVwith = 1
2nand
= 1
2n2.
For this result and applications see Fokas et al.
(1991): in this reference, on the right-hand side of
Eq. (1.10), ( n+
)2should be replaced by n+
at
its rst appearance. See also Freud (1976), Br ezin et al.
(1978), Fokas et al. (1992), and Magnus (1995).
32.16 Physical
Statistical Physics
Statistical physics, especially classical and quantum
spin models, has proved to be a major area for research
problems in the modern theory of Painlev e transcen-
dents. For a survey see McCoy (1992). See also McCoy
et al. (1977), Jimbo et al. (1980), Essler et al. (1996),
and Kanzieper (2002).
Integrable Continuous Dynamical Systems
See Bountis et al. (1982) and Grammaticos et al. (1991).
Other Applications
For the Ising model see Barouch et al. (1973).
For applications in 2D quantum gravity and related
aspects of the enumerative topology see Di Francesco
et al. (1995). For applications in string theory see
Seiberg and Shih (2005).
740 Painlev e Transcendents
Computation
32.17 Methods of Computation
The Painlev e equations can be integrated by Runge{
Kutta methods for ordinary di erential equations; see
x3.7(v), Butcher (2003), and Hairer et al. (2000). For
numerical studies of P Isee Holmes and Spence (1984)
and Noonburg (1995). For numerical studies of P IIsee
Kashevarov (1998, 2004), Miles (1978, 1980), and Ros-
ales (1978). For numerical studies of P IVsee Bassom
et al. (1993).
References
General Reference
The survey article Clarkson (2006) covers all topics
treated in this chapter.
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x32.2 Adler (1994), Hille (1976, pp. 439{444), Ince
(1926, Chapter XIV), Kruskal and Clarkson
(1992), Iwasaki et al. (1991, pp. 119{126), Noumi
(2004, pp. 13{23).
x32.3 The graphs were produced at NIST. See also Bas-
som et al. (1993).
x32.4 Jimbo and Miwa (1981), Fokas et al. (2006,
Chapter 5), Its and Novoksh enov (1986).x32.5 Ablowitz and Clarkson (1991), Ablowitz and Se-
gur (1977, 1981).
x32.6 Forrester and Witte (2002), Jimbo and Miwa
(1981), Okamoto (1981, 1986, 1987c).
x32.7 Cosgrove (2006), Fokas and Ablowitz (1982),
Gambier (1910), Gromak (1975, 1976, 1978,
1987), Gromak et al. (2002,xx25,34,39,42,47),
Luka sevi c (1967a, 1971), Okamoto (1987a).
x32.8 Flaschka and Newell (1980), Gromak (1987),
Gromak et al. (2002,xx20,26,35,40), Gromak and
Luka sevi c (1982), Kajiwara and Ohta (1996), Ki-
taev et al. (1994), Luka sevi c (1967a,b), Mazzocco
(2001a), Murata (1985, 1995), Vorob'ev (1965),
Yablonski (1959).
x32.9 Gromak et al. (2002,xx33,38), Gromak and
Luka sevi c (1982), Hitchin (1995), Luka sevi c
(1965, 1967b).
x32.10 Airault (1979), Albrecht et al. (1996), Flaschka
and Newell (1980), Fokas and Yortsos (1981),
Gambier (1910), Gromak (1978, 1987), Gro-
mak et al. (2002, Chapter 6, xx35,40,44), Gro-
mak and Luka sevi c (1982), Luka sevi c (1965,
1967a,b, 1968), Luka sevi c and Yablonski (1967),
Mans eld and Webster (1998), Okamoto (1986,
1987a), Umemura and Watanabe (1998), Watan-
abe (1995).
x32.11 Ablowitz and Segur (1977), Bassom et al.
(1992), Bender and Orszag (1978, pp. 158{166),
Deift and Zhou (1995), Fokas et al. (2006, Chap-
ters 9, 10, 14), Hastings and McLeod (1980),
Holmes and Spence (1984), Its et al. (1994), Its
and Kapaev (1987, 1998), McCoy et al. (1977).
For (32.11.2) see Qin and Lu (2008).
x32.13 Ablowitz and Segur (1977).
x32.14 Baik et al. (1999).
Chapter 33
Coulomb Functions
I. J. Thompson1
Notation 742
33.1 Special Notation . . . . . . . . . . . . . 742
Variables; 742
33.2 De nitions and Basic Properties . . . . . 742
33.3 Graphics . . . . . . . . . . . . . . . . . . 743
33.4 Recurrence Relations and Derivatives . . 744
33.5 Limiting Forms for Small , Smalljj, or
Large`. . . . . . . . . . . . . . . . . . 744
33.6 Power-Series Expansions in . . . . . . . 745
33.7 Integral Representations . . . . . . . . . 745
33.8 Continued Fractions . . . . . . . . . . . . 745
33.9 Expansions in Series of Bessel Functions . 745
33.10 Limiting Forms for Large or Largejj. 746
33.11 Asymptotic Expansions for Large . . . . 747
33.12 Asymptotic Expansions for Large . . . . 747
33.13 Complex Variable and Parameters . . . . 748
Variablesr; 748
33.14 De nitions and Basic Properties . . . . . 74833.15 Graphics . . . . . . . . . . . . . . . . . . 749
33.16 Connection Formulas . . . . . . . . . . . 751
33.17 Recurrence Relations and Derivatives . . 752
33.18 Limiting Forms for Large `. . . . . . . . 752
33.19 Power-Series Expansions in r. . . . . . . 752
33.20 Expansions for Small jj. . . . . . . . . . 752
33.21 Asymptotic Approximations for Large jrj. 753
Physical Applications 753
33.22 Particle Scattering and Atomic and Molec-
ular Spectra . . . . . . . . . . . . . . . . 753
Computation 755
33.23 Methods of Computation . . . . . . . . . 755
33.24 Tables . . . . . . . . . . . . . . . . . . . 755
33.25 Approximations . . . . . . . . . . . . . . 756
33.26 Software . . . . . . . . . . . . . . . . . . 756
References 756
1Lawrence Livermore National Laboratory, Livermore, California.
Acknowledgments : This chapter is based in part on Abramowitz and Stegun (1964, Chapter 14) by M. Abramowitz.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
741
742 Coulomb Functions
Notation
33.1 Special Notation
(For other notation see pp. xiv and 873.)
k;` nonnegative integers.
r;x real variables.
nonnegative real variable.
; real parameters.
(x) logarithmic derivative of ( x); seex5.2(i).
(x) Dirac delta; see x1.17.
primes derivatives with respect to the variable.
The main functions treated in this chapter are
rst the Coulomb radial functions F`(;),G`(;),
H
`(;) (Sommerfeld (1928)), which are used in the
case of repulsive Coulomb interactions, and secondly the
functionsf(;`;r),h(;`;r),s(;`;r),c(;`;r) (Seaton
(1982, 2002)), which are used in the case of attractive
Coulomb interactions.
Alternative Notations
Curtis (1964a): P`(;r) = (2`+ 1)!f(;`;r)=2`+1,
Q`(;r) = (2`+ 1)!h(;`;r)=(2`+1A(;`)).
Greene et al. (1979):f(0)(;`;r) = f(;`;r),
f(;`;r) =s(;`;r),g(;`;r) =c(;`;r).
Variables;
33.2 De nitions and Basic Properties
33.2(i) Coulomb Wave Equation
33.2.1
d2w
d2+
1 2
`(`+ 1)
2
w= 0,`= 0;1;2;:::.
This di erential equation has a regular singularity at
= 0 with indices `+ 1 and `, and an irregu-
lar singularity of rank 1 at =1(xx2.7(i), 2.7(ii)).
There are two turning points, that is, points at which
d2w
d2= 0 (x2.8(i)). The outer one is given by
33.2.2tp(;`) =+ (2+`(`+ 1))1=2:
33.2(ii) Regular Solution F`(;)
The function F`(;) is recessive (x2.7(iii)) at = 0,
and is de ned by
33.2.3F`(;) =C`()2 ` 1(i)`+1Mi;`+1
2(2i);
or equivalently
33.2.4
F`(;) =C`()`+1eiM(`+ 1i;2`+ 2;2i);whereM;(z) andM(a;b;z ) are de ned inxx13.14(i)
and 13.2(i), and
33.2.5C`() =2`e =2j (`+ 1 +i)j
(2`+ 1)!:
The choice of ambiguous signs in (33.2.3) and (33.2.4)
is immaterial, provided that either all upper signs are
taken, or all lower signs are taken. This is a consequence
of Kummer's transformation ( x13.2(vii)).
F`(;) is a real and analytic function of on the
open interval 0 <<1, and also an analytic function
ofwhen 1<<1.
The normalizing constant C`() is always positive,
and has the alternative form
33.2.6
C`() =2`
(2=(e2 1))Q`
k=1(2+k2)1/2
(2`+ 1)!:
33.2(iii) Irregular Solutions G`(;);H
`(;)
The functions H
`(;) are de ned by
33.2.7H
`(;) = (i)`e(=2)i`()Wi;`+1
2(2i);
or equivalently
33.2.8
H
`(;)
=ei`(;)(2i)`+1iU(`+ 1i;2`+ 2;2i);
whereW;(z),U(a;b;z ) are de ned inxx13.14(i) and
13.2(i),
33.2.9`(;) = ln(2) 1
2`+`();
and
33.2.10 `() = ph (`+ 1 +i);
the branch of the phase in (33.2.10) being zero when
= 0 and continuous elsewhere. `() is the Coulomb
phase shift .
H+
`(;) andH
`(;) are complex conjugates, and
their real and imaginary parts are given by
33.2.11H+
`(;) =G`(;) +iF`(;);
H
`(;) =G`(;) iF`(;):
As in the case of F`(;), the solutions H
`(;) and
G`(;) are analytic functions of when 0< <1.
Also,ei`()H
`(;) are analytic functions of when
1<<1.
33.2(iv) Wronskians and Cross-Product
With arguments ;suppressed,
33.2.12 WfG`;F`g=W
H
`;F`
= 1:
33.2.13F` 1G` F`G` 1=`=(`2+2)1=2,`1.
33.3 Graphics 743
33.3 Graphics
33.3(i) Line Graphs of the Coulomb Radial Functions F`(;)andG`(;)
Figure 33.3.1 :F`(;),G`(;) with`= 0,= 2.
Figure 33.3.2 :F`(;),G`(;) with`= 0,= 0.
Figure 33.3.3 :F`(;),G`(;) with`= 0,= 2. The
turning point is at tp(2;0) = 4.
Figure 33.3.4 :F`(;),G`(;) with`= 0,= 10. The
turning point is at tp(10;0) = 20.
In Figures 33.3.5 and 33.3.6
33.3.1 M`(;) = (F2
`(;) +G2
`(;))1=2= H
`(;) :
Figure 33.3.5 :F`(;),G`(;), andM`(;) with`=
0,=p
15=2. The turning point is at tpp
15=2;0
=
p
30 = 5:47:::.
Figure 33.3.6 :F`(;),G`(;), andM`(;) with`=
5,= 0. The turning point is at tp(0;5) =p
30 (as in
Figure 33.3.5).
744 Coulomb Functions
33.3(ii) Surfaces of the Coulomb Radial Functions F0(;)andG0(;)
Figure 33.3.7 :F0(;), 22, 05.
Figure 33.3.8 :G0(;), 22, 0<5.
33.4 Recurrence Relations and Derivatives
For`= 1;2;3;:::, let
33.4.1R`=r
1 +2
`2; S`=`
+
`; T`=S`+S`+1:
Then, with X`denoting any of F`(;),G`(;), or
H
`(;),
33.4.2R`X` 1 T`X`+R`+1X`+1= 0,`1,
33.4.3 X0
`=R`X` 1 S`X`, `1,
33.4.4 X0
`=S`+1X` R`+1X`+1,`0.
33.5 Limiting Forms for Small , Smalljj,
or Large`
33.5(i) Small
As!0 with xed,
33.5.1
F`(;)C`()`+1; F0
`(;)(`+ 1)C`()`:
33.5.2G`(;) `
(2`+ 1)C`(),`= 0;1;2;:::,
G0
`(;) ` ` 1
(2`+ 1)C`(),`= 1;2;3;:::.
33.5(ii)= 0
33.5.3F`(0;) =j`(); G`(0;) = y`():Equivalently,
33.5.4F`(0;) = (=2)1=2J`+1
2();
G`(0;) = (=2)1=2Y`+1
2():
For the functions j,y,J,Yseexx10.47(ii), 10.2(ii).
33.5.5
F0(0;) = sin; G 0(0;) = cos; H
0(0;) =ei:
33.5.6 C`(0) =2``!
(2`+ 1)!=1
(2`+ 1)!!:
33.5(iii) Smalljj
33.5.7 0()
, !0,
where
is Euler's constant ( x5.2(ii)).
33.5(iv) Large `
As`!1 withand(6= 0) xed,
33.5.8
F`(;)C`()`+1; G`(;) `
(2`+ 1)C`();
33.5.9C`()e =2
(2`+ 1)!!e =2e`
p
2(2`)`+1:
33.6 Power-Series Expansions in 745
33.6 Power-Series Expansions in
33.6.1F`(;) =C`()1X
k=`+1A`
k()k;
33.6.2F0
`(;) =C`()1X
k=`+1kA`
k()k 1;whereA`
`+1= 1,A`
`+2==(`+ 1), and
33.6.3(k+`)(k ` 1)A`
k= 2A`
k 1 A`
k 2,
k=`+ 3;`+ 4;:::,
or in terms of the hypergeometric function ( xx15.1,
15.2(i)),
33.6.4
A`
k()
=( i)k ` 1
(k ` 1)!2F1(`+ 1 k;`+ 1 i; 2`+ 2; 2):
33.6.5H
`(;) =ei`(;)
(2`+ 1)! ( `+i) 1X
k=0(a)k
(2`+ 2)kk!(2i)a+k(ln(2i) + (a+k) (1 +k) (2`+ 2 +k))
2`+1X
k=1(2`+ 1)!(k 1)!
(2`+ 1 k)!(1 a)k(2i)a k!
;
wherea= 1 +`iand (x) = 0(x)= (x) (x5.2(i)).
The series (33.6.1), (33.6.2), and (33.6.5) converge
for all nite values of . Corresponding expansions for
H
`0(;) can be obtained by combining (33.6.5) with
(33.4.3) or (33.4.4).
33.7 Integral Representations
33.7.1
F`(;) =`+12`ei (=2)
j (`+ 1 +i)jZ1
0e 2itt`+i(1 t)` idt;
33.7.2
H
`(;) =e i `
(2`+ 1)!C`()Z1
0e tt` i(t+ 2i)`+idt;
33.7.3
H
`(;)
= ie `+1
(2`+ 1)!C`()Z1
0exp( i(tanht 2t))
(cosht)2`+2
+i(1 +t2)`exp( t+ 2arctant)
dt;
33.7.4
H+
`(;)
=ie `+1
(2`+ 1)!C`()Z i1
1e it(1 t)` i(1 +t)`+idt:
Noninteger powers in (33.7.1){(33.7.4) and the arc-
tangent assume their principal values ( xx4.2(i), 4.2(iv),
4.23(ii)).33.8 Continued Fractions
With arguments ;suppressed,
33.8.1F0
`
F`=S`+1 R2
`+1
T`+1 R2
`+2
T`+2 :
ForR,S, andTsee (33.4.1).
33.8.2
H
`0
H
`=ci
ab
2( i) +(a+ 1)(b+ 1)
2( 2i) +;
where
33.8.3a= 1 +`i; b = `i; c =i(1 (=)):
The continued fraction (33.8.1) converges for all nite
values of, and (33.8.2) converges for all 6= 0.
If we denote u=F0
`/F`andp+iq=H+
`0.
H+
`,
then
33.8.4F`=(q 1(u p)2+q) 1=2; F0
`=uF`;
33.8.5G`=q 1(u p)F`; G0
`=q 1(up p2 q2)F`:
The ambiguous sign in (33.8.4) has to agree with that
of the nal denominator in (33.8.1) when the continued
fraction has converged to the required precision. For
proofs and further information see Barnett et al. (1974)
and Barnett (1996).
33.9 Expansions in Series of Bessel
Functions
33.9(i) Spherical Bessel Functions
33.9.1 F`(;) =1X
k=0akj`+k();
746 Coulomb Functions
where the function jis as inx10.47(ii),a 1= 0,
a0= (2`+ 1)!!C`(), and
33.9.2k(k+ 2`+ 1)
2k+ 2`+ 1ak 2ak 1
+(k 2)(k+ 2` 1)
2k+ 2` 3ak 2= 0,k= 1;2;:::.
The series (33.9.1) converges for all nite values of
and.
33.9(ii) Bessel Functions and Modi ed Bessel
Functions
In this subsection the functions J,I, andKare as in
xx10.2(ii) and 10.25(ii).
Witht= 2jj,
33.9.3
F`(;) =C`()(2`+ 1)!
(2)2`+1 `1X
k=2`+1bktk=2Ik
2p
t
,
>0;
33.9.4
F`(;) =C`()(2`+ 1)!
(2jj)2`+1 `1X
k=2`+1bktk=2Jk
2p
t
,
<0:
Hereb2`=b2`+2= 0,b2`+1= 1, and
33.9.542(k 2`)bk+1+kbk 1+bk 2= 0,
k= 2`+ 2;2`+ 3;:::.
The series (33.9.3) and (33.9.4) converge for all nite
positive values of jjand.
Next, as!+1with(>0) xed,
33.9.6
G`(;)
`
(`+1
2)`()C`()1X
k=2`+1( 1)kbktk=2Kk
2p
t
;
where
33.9.7 `()1X
k=2`+1( 1)k(k 1)!bk:
For other asymptotic expansions of G`(;) see
Fr oberg (1955,x8) and Humblet (1985).
33.10 Limiting Forms for Large or Large
jj
33.10(i) Large
As!1 with xed,
33.10.1F`(;) = sin(`(;)) +o(1);
G`(;) = cos(`(;)) +o(1);
33.10.2 H
`(;)exp(i`(;));
where`(;) is de ned by (33.2.9).33.10(ii) Large Positive
As!1 with xed,
33.10.3
F`(;)(2`+ 1)!C`()
(2)`+1(2)1/2I2`+1
(8)1/2
;
G`(;)2(2)`
(2`+ 1)!C`()(2)1/2K2`+1
(8)1/2
:
In particular, for `= 0,
33.10.4F0(;)e ()1/2I1
(8)1/2
;
G0(;)2e(/)1/2K1
(8)1/2
;
33.10.5F0
0(;)e (2)1/2I0
(8)1/2
;
G0
0(;) 2e(2/)1/2K0
(8)1/2
:
Also,
33.10.60() =(ln 1) +1
4+o(1);
C0()(2)1=2e :
33.10(iii) Large Negative
As! 1 with xed,
33.10.7F`(;) =(2`+ 1)!C`()
( 2)`+1
( 2)1/2
J2`+1
( 8)1/2
+o
jj1/4
;
G`(;) = ( 2)`
(2`+ 1)!C`()
( 2)1/2
Y2`+1
( 8)1/2
+o
jj1/4
:
In particular, for `= 0,
33.10.8
F0(;) = ()1/2J1
( 8)1/2
+o
jj 1/4
;
G0(;) = ()1/2Y1
( 8)1/2
+o
jj 1/4
:
33.10.9
F0
0(;) = ( 2)1/2J0
( 8)1/2
+o
jj1/4
;
G0
0(;) = ( 2)1/2Y0
( 8)1/2
+o
jj1/4
:
Also,
33.10.10
0() =(ln( ) 1) 1
4+o(1); C 0()( 2)1=2:
33.11 Asymptotic Expansions for Large 747
33.11 Asymptotic Expansions for Large
For large, with`and xed,
33.11.1H
`(;) =ei`(;)1X
k=0(a)k(b)k
k!(2i)k;
where`(;) is de ned by (33.2.9), and aandbare
de ned by (33.8.3).
With arguments ( ;) suppressed, an equivalent for-
mulation is given by
33.11.2F`=gcos`+fsin`; G`=fcos` gsin`;
33.11.3F0
`=bgcos`+bfsin`; G0
`=bfcos` bgsin`;
33.11.4 H
`=ei`(fig);
where
33.11.5 f1X
k=0fk; g1X
k=0gk;
33.11.6 bf1X
k=0bfk;bg1X
k=0bgk;
33.11.7 gbf fbg= 1:Heref0= 1,g0= 0,bf0= 0,bg0= 1 (=), and for
k= 0;1;2;:::,
33.11.8fk+1=kfk kgk;
gk+1=kgk+kfk;
bfk+1=kbfk kbgk (fk+1=);
bgk+1=kbgk+kbfk (gk+1=);
where
33.11.9
k=(2k+ 1)
(2k+ 2); k=`(`+ 1) k(k+ 1) +2
(2k+ 2):
33.12 Asymptotic Expansions for Large
33.12(i) Transition Region
When`= 0 and>0, the outer turning point is given
bytp(;0) = 2; compare (33.2.2). De ne
33.12.1x= (2 )=(2)1=3; = (2)2=3:
Then as!1 ,
33.12.2F0(;)
G0(;)1=2(2)1=6Ai(x)
Bi(x)
1 +B1
+B2
2+
+Ai0(x)
Bi0(x)A1
+A2
2+
;
33.12.3F0
0(;)
G0
0(;) 1=2(2) 1=6Ai(x)
Bi(x)B0
1+xA1
+B0
2+xA2
2+
+Ai0(x)
Bi0(x)B1+A0
1
+B2+A0
2
2+
;
uniformly for bounded values of ( 2)=1=3 . Here Ai and Bi are the Airy functions ( x9.2), and
33.12.4 A1=1
5x2; A 2=1
35(2x3+ 6); A 3=1
15750(21x7+ 370x4+ 580x);
33.12.5 B1= 1
5x; B 2=1
350(7x5 30x2); B 3=1
15750(264x6 290x3 560):
In particular,
33.12.6F0(;2)
3 1/2G0(;2) 1
3
!1=2
2p
12
35 2
3
1
31
!4 8
20251
!65792
46 06875 2
3
1
31
!10 !
;
33.12.7F0
0(;2)
3 1/2G0
0(;2) 2
3
2p!1=2
1 +1
15 1
3
2
31
!22
141751
!6+1436
23 38875 1
3
2
31
!8!
;
where!= (2
3)1=3.
For derivations and additional terms in the expan-
sions in this subsection see Abramowitz and Rabinowitz
(1954) and Fr oberg (1955).
33.12(ii) Uniform Expansions
With the substitution = 2z, Equation (33.2.1) be-
comes
33.12.8d2w
dz2=
421 z
z
+`(`+ 1)
z2
w:Then, by application of the results given in xx2.8(iii)
and 2.8(iv), two sets of asymptotic expansions can be
constructed for F`(;) andG`(;) when!1 .
The rst set is in terms of Airy functions and the ex-
pansions are uniform for xed `andz <1, where
is an arbitrary small positive constant. They would
include the results of x33.12(i) as a special case.
The second set is in terms of Bessel functions of or-
ders 2`+ 1 and 2`+ 2, and they are uniform for xed `
748 Coulomb Functions
and 0z1 , whereagain denotes an arbitrary
small positive constant.
Compare alsox33.20(iv).
33.13 Complex Variable and Parameters
The functions F`(;),G`(;), andH
`(;) may
be extended to noninteger values of `by generalizing
(2`+ 1)! = (2 `+ 2), and supplementing (33.6.5) by a
formula derived from (33.2.8) with U(a;b;z ) expanded
via (13.2.42).
These functions may also be continued analytically
to complex values of ,, and`. The quantities C`(),
`(), andR`, given by (33.2.6), (33.2.10), and (33.4.1),
respectively, must be de ned consistently so that
33.13.1
C`() = 2`ei`() (=2) (`+ 1 i)= (2`+ 2);
and
33.13.2 R`= (2`+ 1)C`()=C` 1():
For further information see Dzieciol et al. (1999),
Thompson and Barnett (1986), and Humblet (1984).
Variablesr;
33.14 De nitions and Basic Properties
33.14(i) Coulomb Wave Equation
Another parametrization of (33.2.1) is given by
33.14.1d2w
dr2+
+2
r `(`+ 1)
r2
w= 0;
where
33.14.2 r= ; = 1=2:
Again, there is a regular singularity at r= 0 with
indices`+1 and `, and an irregular singularity of rank
1 atr=1. When>0 the outer turning point is given
by
33.14.3rtp(;`) =p
1 +`(`+ 1) 1.
;
compare (33.2.2).
33.14(ii) Regular Solution f(;`;r)
The function f(;`;r) is recessive (x2.7(iii)) at r= 0,
and is de ned by
33.14.4f(;`;r) =`+1M;`+1
2(2r=)=(2`+ 1)!;
or equivalently
33.14.5
f(;`;r)
= (2r)`+1e r=M(`+ 1 ;2`+ 2;2r=)=(2`+ 1)!;whereM;(z) andM(a;b;z ) are de ned inxx13.14(i)
and 13.2(i), and
33.14.6 =8
><
>:( ) 1=2; < 0;r> 0;
( ) 1=2; < 0;r< 0;
i 1=2; > 0:
The choice of sign in the last line of (33.14.6) is imma-
terial: the same function f(;`;r) is obtained. This is a
consequence of Kummer's transformation ( x13.2(vii)).
f(;`;r) is real and an analytic function of rin the
interval 1< r <1, and it is also an analytic func-
tion ofwhen 1<<1. This includes = 0, hence
f(;`;r) can be expanded in a convergent power series
inin a neighborhood of = 0 (x33.20(ii)).
33.14(iii) Irregular Solution h(;`;r)
For nonzero values of andrthe function h(;`;r) is
de ned by
33.14.7
h(;`;r) = (`+ 1 )
`
W;`+1
2(2r=)
+( 1)`S(;r) (`+ 1 +)
2(2`+ 1)!M;`+1
2(2r=)
;
whereis given by (33.14.6) and
33.14.8S(;r) =8
>>><
>>>:2 cos
jj 1=2
; < 0;r> 0;
0; < 0;r< 0;
e 1=2; > 0;r> 0;
e 1=2; > 0;r< 0:
(Again, the choice of the ambiguous sign in the last line
of (33.14.6) is immaterial.)
h(;`;r) is real and an analytic function of each of r
andin the intervals 1<r<1and 1<<1,
except when r= 0 or= 0.
33.14(iv) Solutions s(;`;r)andc(;`;r)
The functions s(;`;r) andc(;`;r) are de ned by
33.14.9s(;`;r) = (B(;`)=2)1=2f(;`;r);
c(;`;r) = (2B(;`)) 1=2h(;`;r);
provided that `<( ) 1=2when<0, where
33.14.10
B(;`) =(
A(;`)
1 exp
2=1=2 1; > 0;
A(;`); 0;
and
33.14.11 A(;`) =`Y
k=0(1 +k2):
An alternative formula for A(;`) is
33.14.12 A(;`) = (1 +`+)
( `) 2` 1;
33.15 Graphics 749
the choice of sign in the last line of (33.14.6) again being
immaterial.
When <0 and` >( ) 1=2the quantity A(;`)
may be negative, causing s(;`;r) andc(;`;r) to be-
come imaginary.
The function s(;`;r) has the following properties:
33.14.13Z1
0s(1;`;r)s(2;`;r)dr=(1 2);
where the right-hand side is the Dirac delta ( x1.17).
When= 1=n2,n=`+ 1;`+ 2;:::,s(;`;r) is
exp( r=n) times a polynomial in r, and
33.14.14 n;`(r) = ( 1)`+1+n(2=n3)1=2s
1=n2;`;r
satis es
33.14.15Z1
0 2
n;`(r)dr= 1:
33.14(v) Wronskians
With arguments ;`;r suppressed,
33.14.16 Wfh;fg= 2=;Wfc;sg= 1=:33.15 Graphics
33.15(i) Line Graphs of the Coulomb Functions
f(;`;r)andh(;`;r)
Figure 33.15.1 :f(;`;r);h(;`;r) with`= 0;= 4.
Figure 33.15.2 :f(;`;r);h(;`;r) with`= 1;= 4.
Figure 33.15.3 :f(;`;r);h(;`;r) with`= 0;=
1=2;= 1:5.
Figure 33.15.4 :f(;`;r);h(;`;r) with`= 0;=
1=2;= 2.
Figure 33.15.5 :f(;`;r);h(;`;r) with`= 0;=
1=2;= 2:5.
750 Coulomb Functions
33.15(ii) Surfaces of the Coulomb Functions f(;`;r),h(;`;r),s(;`;r), andc(;`;r)
Figure 33.15.6 :f(;`;r) with`= 0; 2<< 2; 15<
r<15.
Figure 33.15.7 :h(;`;r) with`= 0; 2<< 2; 15<
r<15.
Figure 33.15.8 :f(;`;r) with`= 1; 2<< 2; 15<
r<15.
Figure 33.15.9 :h(;`;r) with`= 1; 2<< 2; 15<
r<15.
Figure 33.15.10 :s(;`;r) with`= 0; 0:15< <
0:10;0<r< 65.
Figure 33.15.11 :c(;`;r) with`= 0; 0:15< <
0:10;0<r< 65.
33.16 Connection Formulas 751
33.16 Connection Formulas
33.16(i)F`andG`in Terms offandh
33.16.1F`(;) =(2`+ 1)!C`()
( 2)`+1f
1=2;`;
;
33.16.2G`(;) =( 2)`
(2`+ 1)!C`()h
1=2;`;
;
whereC`() is given by (33.2.5) or (33.2.6).
33.16(ii)fandhin Terms ofF`andG`when
>0
When>0 denote
33.16.3 =1=2(>0);
and again de ne A(;`) by (33.14.11) or (33.14.12).
Then forr>0
33.16.4f(;`;r) =2
1 e 2=
A(;`)1/2
F`( 1= ; r );
33.16.5h(;`;r) =2
A(;`)
1 e 2= 1/2
G`( 1= ; r ):
Alternatively, for r<0
33.16.6
f(;`;r) = ( 1)`+12
e2= 1
A(;`)1/2
F`(1= ; r);
33.16.7
h(;`;r) = ( 1)`2
A(;`)
e2= 11/2
G`(1= ; r):
33.16(iii)fandhin Terms ofW;(z)when
<0
When<0 denote
33.16.8 = 1=( )1=2(>0);
33.16.9`(;r) =W;`+1
2(2r=);
`(;r) =<
eiW ;`+1
2
ei2r=
;
and again de ne A(;`) by (33.14.11) or (33.14.12).
Then forr>0
33.16.10f(;`;r) = ( 1)``+1
cos()`(;r)
(`+ 1 +)
+sin() ( `)`(;r)
;
33.16.11h(;`;r) = ( 1)``+1A(;`)sin()`(;r)
(`+ 1 +)
+cos() ( `)`(;r)
:Alternatively, for r<0
33.16.12
f(;`;r) =( 1)``+1
`( ;r)
(`+ 1 +)
+ sin() cos() ( `)`( ;r)
;
33.16.13
h(;`;r) = ( 1)``+1A(;`) ( `)`( ;r)=:
33.16(iv)sandcin Terms ofF`andG`when
>0
When > 0, again denote by (33.16.3). Then for
r>0
33.16.14s(;`;r) = ( ) 1=2F`( 1= ; r );
c(;`;r) = ( ) 1=2G`( 1= ; r ):
Alternatively, for r<0
33.16.15s(;`;r) = ( ) 1=2F`(1= ; r);
c(;`;r) = ( ) 1=2G`(1= ; r):
33.16(v)sandcin Terms ofW;(z)when
<0
When<0 denote,`(;r), and`(;r) by (33.16.8)
and (33.16.9). Also denote
33.16.16K(;`) =
2 (+`+ 1) ( `) 1=2:
Then forr>0
33.16.17s(;`;r) =( 1)`
21=2sin()
K(;`)`(;r)
cos()2K(;`)`(;r)
;
c(;`;r) =( 1)`
21=2cos()
K(;`)`(;r)
+ sin()2K(;`)`(;r)
:
Alternatively, for r<0
33.16.18
s(;`;r) =( 1)`+1
21=23=2
K(;`)`( ;r)
sin() cos()
1=2K(;`)`( ;r)
;
c(;`;r) =( 1)`
(2)1=2K(;`)`( ;r):
752 Coulomb Functions
33.17 Recurrence Relations and Derivatives
33.17.1 (`+ 1)rf(;` 1;r) (2`+ 1) (`(`+ 1) r)f(;`;r) +`
1 + (`+ 1)2
rf(;`+ 1;r) = 0;
33.17.2 (`+ 1)
1 +`2
rh(;` 1;r) (2`+ 1) (`(`+ 1) r)h(;`;r) +`rh(;`+ 1;r) = 0;
33.17.3 (`+ 1)rf0(;`;r) =
(`+ 1)2 r
f(;`;r)
1 + (`+ 1)2
rf(;`+ 1;r);
33.17.4 (`+ 1)rh0(;`;r) =
(`+ 1)2 r
h(;`;r) rh(;`+ 1;r):
33.18 Limiting Forms for Large `
As`!1 withandr(6= 0) xed,
33.18.1f(;`;r)(2r)`+1
(2`+ 1)!; h(;`;r)(2`)!
(2r)`:
33.19 Power-Series Expansions in r
33.19.1 f(;`;r) =r`+11X
k=0 krk;
where
33.19.2
0= 2`+1=(2`+ 1)!; 1= 0=(`+ 1);
k(k+ 2`+ 1) k+ 2 k 1+ k 2= 0,k= 2;3;:::.
33.19.3
2h(;`;r) =2`X
k=0(2` k)!
k
k!(2r)k ` 1X
k=0krk+`+1
A(;`) (2 lnj2r=j+< (`+ 1 +)
+< ( `+))f(;`;r),r6= 0.
Hereis de ned by (33.14.6), A(;`) is de ned by
(33.14.11) or (33.14.12),
0= 1,
1= 1, and
33.19.4
k
k 1+1
4(k 1)(k 2` 2)
k 2= 0,k= 2;3;:::.
Also,
33.19.50= ( 2`+1 2( (2`+ 2) + (1))A(;`)) 0;
1= ( 2`+2 2( (2`+ 3) + (2))A(;`)) 1;
33.19.6k(k+ 2`+ 1)k+ 2k 1+k 2
+ 2(2k+ 2`+ 1)A(;`) k= 0,k= 2;3;:::,
with 0= 1= 0, and
33.19.7 k k 1
+1
4(k 1)(k 2` 2) k 2+1
2(k 1)
k 2= 0,
k= 2;3;:::.
The expansions (33.19.1) and (33.19.3) converge for all
nite values of r, exceptr= 0 in the case of (33.19.3).33.20 Expansions for Small jj
33.20(i) Case = 0
33.20.1f(0;`;r) = (2r)1=2J2`+1p
8r
;
h(0;`;r) = (2r)1=2Y2`+1p
8r
,r>0,
33.20.2f(0;`;r) = ( 1)`+1(2jrj)1=2I2`+1p
8jrj
;
h(0;`;r) = ( 1)`(2=)(2jrj)1=2K2`+1p
8jrj
,
r<0.
For the functions J,Y,I, andKseexx10.2(ii), 10.25(ii).
33.20(ii) Power-Series in for the Regular
Solution
33.20.3 f(;`;r) =1X
k=0kFk(`;r);
where
33.20.4
Fk(`;r) =3kX
p=2k(2r)(p+1)=2Ck;pJ2`+1+pp
8r
,r>0,
33.20.5
Fk(`;r)
=3kX
p=2k( 1)`+1+p(2jrj)(p+1)=2Ck;pI2`+1+pp
8jrj
,
r<0.
The functions JandIare as inxx10.2(ii), 10.25(ii), and
the coecients Ck;pare given by C0;0= 1,C1;0= 0,
and
33.20.6Ck;p= 0, p<2korp>3k,
Ck;p= ( (2`+p)Ck 1;p 2+Ck 1;p 3)=(4p),
k>0, 2kp3k.
The series (33.20.3) converges for all rand.
33.21 Asymptotic Approximations for Large jrj 753
33.20(iii) Asymptotic Expansion for the
Irregular Solution
As!0 with`andr xed,
33.20.7h(;`;r) A(;`)1X
k=0kHk(`;r);
whereA(;`) is given by (33.14.11), (33.14.12), and
33.20.8
Hk(`;r) =3kX
p=2k(2r)(p+1)=2Ck;pY2`+1+pp
8r
,r>0,
33.20.9
Hk(`;r)
= ( 1)`+12
3kX
p=2k(2jrj)(p+1)=2Ck;pK2`+1+pp
8jrj
,
r<0.
The functions YandKare as inxx10.2(ii), 10.25(ii),
and the coecients Ck;pare given by (33.20.6).
33.20(iv) Uniform Asymptotic Expansions
For a comprehensive collection of asymptotic expansions
that cover f(;`;r) andh(;`;r) as!0and are
uniform in r, including unbounded values, see Curtis
(1964a,x7). These expansions are in terms of elemen-
tary functions, Airy functions, and Bessel functions of
orders 2`+ 1 and 2`+ 2.
33.21 Asymptotic Approximations for Large
jrj
33.21(i) Limiting Forms
We indicate here how to obtain the limiting forms of
f(;`;r),h(;`;r),s(;`;r), andc(;`;r) asr!1 ,
withand` xed, in the following cases:
(a) Whenr!1 with>0, Equations (33.16.4){
(33.16.7) are combined with (33.10.1).
(b) When r! 1 with < 0, Equations
(33.16.10){(33.16.13) are combined with
33.21.1`(;r)e r=(2r=);
`(;r)er=(2r=) , r!1 ,
33.21.2`( ;r)er=( 2r=) ;
`( ;r)e r=( 2r=),r! 1 .
Corresponding approximations for s(;`;r) andc(;`;r)
asr! 1 can be obtained via (33.16.17), and as
r! 1 via (33.16.18).
(c) Whenr!1 with= 0, combine (33.20.1),
(33.20.2) withxx10.7(ii), 10.30(ii).
33.21(ii) Asymptotic Expansions
For asymptotic expansions of f(;`;r) andh(;`;r) as
r!1 withand` xed, see Curtis (1964a, x6).Physical Applications
33.22 Particle Scattering and Atomic and
Molecular Spectra
33.22(i) Schr odinger Equation
Withedenoting here the elementary charge, the
Coulomb potential between two point particles with
chargesZ1e;Z2eand masses m1;m2separated by a
distancesisV(s) =Z1Z2e2=(40s) =Z1Z2 hc=s,
whereZjare atomic numbers, 0is the electric con-
stant, is the ne structure constant, and his the re-
duced Planck's constant. The reduced mass is m=
m1m2=(m1+m2), and at energy of relative motion
Ewith relative orbital angular momentum `h, the
Schr odinger equation for the radial wave function w(s)
is given by
33.22.1
h2
2md2
ds2 `(`+ 1)
s2
+Z1Z2 hc
s
w=Ew;
With the substitutions
33.22.2 k= (2mE= h2)1=2; Z =mZ1Z2 c=h; x =s;
(33.22.1) becomes
33.22.3d2w
dx2+
k2 2Z
x `(`+ 1)
x2
w= 0:
33.22(ii) De nitions of Variables
k Scaling
The k-scaled variables andofx33.2 are given by
33.22.4=s(2mE= h2)1=2; =Z1Z2 c(m=(2E))1=2:
At positive energies E > 0,0, and:
Attractive potentials :Z1Z2<0,<0.
Zero potential (V= 0):Z1Z2= 0,= 0.
Repulsive potentials :Z1Z2>0,>0.
Positive-energy functions correspond to processes
such as Rutherford scattering and Coulomb exci-
tation of nuclei (Alder et al. (1956)), and atomic
photo-ionization and electron-ion collisions (Bethe and
Salpeter (1977)).
At negative energies E < 0 and both andare
purely imaginary. The negative-energy functions are
widely used in the description of atomic and molecular
spectra; see Bethe and Salpeter (1977), Seaton (1983),
and Aymar et al. (1996). In these applications, the Z-
scaled variables randare more convenient.
754 Coulomb Functions
ZScaling
TheZ-scaled variables randofx33.14 are given by
33.22.5r= Z1Z2(mc = h)s; =E=(Z2
1Z2
2mc2 2=2):
ForZ1Z2= 1 andm=me, the electron mass,
the scaling factors in (33.22.5) reduce to the Bohr ra-
dius,a0= h=(mec ), and to a multiple of the Rydberg
constant,
R1=mec 2=(2h).
Attractive potentials :Z1Z2<0,r>0.
Zero potential (V= 0):Z1Z2= 0,r= 0.
Repulsive potentials :Z1Z2>0,r<0.
ik Scaling
Theik-scaled variables zandofx13.2 are given by
33.22.6
z= 2is(2mE= h2)1=2; =iZ1Z2 c(m=(2E))1=2:
Attractive potentials :Z1Z2<0,=<0.
Zero potential (V= 0):Z1Z2= 0,= 0.
Repulsive potentials :Z1Z2>0,=>0.
Customary variables are ( ;r) in atomic physics and
(;) in atomic and nuclear physics. Both variable sets
may be used for attractive and repulsive potentials: the
(;r) set cannot be used for a zero potential because this
would imply r= 0 for alls, and the (;) set cannot be
used for zero energy Ebecause this would imply = 0
always.
33.22(iii) Conversions Between Variables
33.22.7 r= ; = 1=2,Zfrom k.
33.22.8 z= 2i; =i,ikfrom k.
33.22.9 =z=(2i); ==i, kfromik:
33.22.10 r=z=2; = 1=2,Zfromik.
33.22.11 = 1=2; = r=, kfromZ.
33.22.12 =( ) 1=2; z = 2r=,ikfromZ.
Resolution of the ambiguous signs in (33.22.11),
(33.22.12) depends on the sign of Z=kin (33.22.3). See
alsoxx33.14(ii), 33.14(iii), 33.22(i), and 33.22(ii).
33.22(iv) Klein{Gordon and Dirac Equations
The relativistic motion of spinless particles in a
Coulomb eld, as encountered in pionic atoms and pion-
nucleon scattering (Backenstoss (1970)) is described by
a Klein{Gordon equation equivalent to (33.2.1); see
Barnett (1981a). The motion of a relativistic electron in
a Coulomb eld, which arises in the theory of the elec-
tronic structure of heavy elements (Johnson (2007)), is
described by a Dirac equation. The solutions to this
equation are closely related to the Coulomb functions;
see Greiner et al. (1985).33.22(v) Asymptotic Solutions
The Coulomb solutions of the Schr odinger and Klein{
Gordon equations are almost always used in the external
region, outside the range of any non-Coulomb forces or
couplings.
For scattering problems, the interior solution is
then matched to a linear combination of a pair of
Coulomb functions, F`(;) andG`(;), orf(;`;r)
andh(;`;r), to determine the scattering S-matrix and
also the correct normalization of the interior wave solu-
tions; see Bloch et al. (1951).
For bound-state problems only the exponentially de-
caying solution is required, usually taken to be the
Whittaker function W ;`+1
2(2). The functions n;`(r)
de ned by (33.14.14) are the hydrogenic bound states
in attractive Coulomb potentials; their polynomial com-
ponents are often called associated Laguerre functions ;
see Christy and Duck (1961) and Bethe and Salpeter
(1977).
33.22(vi) Solutions Inside the Turning Point
The penetrability of repulsive Coulomb potential bar-
riers is normally expressed in terms of the quantity
=(F2
`(;)+G2
`(;)) (Mott and Massey (1956, pp. 63{
65)). The WKBJ approximations of x33.23(vii) may
also be used to estimate the penetrability.
33.22(vii) Complex Variables and Parameters
The Coulomb functions given in this chapter are most
commonly evaluated for real values of ,r,,and
nonnegative integer values of `, but they may be con-
tinued analytically to complex arguments and order `
as indicated inx33.13.
Examples of applications to noninteger and/or com-
plex variables are as follows.
Scattering at complex energies. See for example
McDonald and Nuttall (1969).
Searches for resonances as poles of the S-matrix in
the complex half-plane =k<0. See for example
Cs ot o and Hale (1997).
Regge poles at complex values of `. See for exam-
ple Takemasa et al. (1979).
Eigenstates using complex-rotated coordinates
r!rei, so that resonances have square-
integrable eigenfunctions. See for example Halley
et al. (1993).
Solution of relativistic Coulomb equations. See for
example Cooper et al. (1979) and Barnett (1981b).
Computation 755
Gravitational radiation. See for example Berti
and Cardoso (2006).
For further examples see Humblet (1984).
Computation
33.23 Methods of Computation
33.23(i) Methods for the Con
uent
Hypergeometric Functions
The methods used for computing the Coulomb functions
described below are similar to those in x13.29.
33.23(ii) Series Solutions
The power-series expansions of xx33.6 and 33.19 con-
verge for all nite values of the radii andr, respec-
tively, and may be used to compute the regular and
irregular solutions. Cancellation errors increase with
increases in andjrj, and may be estimated by com-
paring the nal sum of the series with the largest par-
tial sum. Use of extended-precision arithmetic increases
the radial range that yields accurate results, but even-
tually other methods must be employed, for example,
the asymptotic expansions of xx33.11 and 33.21.
33.23(iii) Integration of De ning Di erential
Equations
When numerical values of the Coulomb functions are
available for some radii, their values for other radii may
be obtained by direct numerical integration of equations
(33.2.1) or (33.14.1), provided that the integration is
carried out in a stable direction ( x3.7). Thus the reg-
ular solutions can be computed from the power-series
expansions (xx33.6, 33.19) for small values of the radii
and then integrated in the direction of increasing val-
ues of the radii. On the other hand, the irregular solu-
tions ofxx33.2(iii) and 33.14(iii) need to be integrated
in the direction of decreasing radii beginning, for exam-
ple, with values obtained from asymptotic expansions
(xx33.11 and 33.21).
33.23(iv) Recurrence Relations
In a similar manner to x33.23(iii) the recurrence rela-
tions ofxx33.4 or 33.17 can be used for a range of values
of the integer `, provided that the recurrence is carried
out in a stable direction ( x3.6). This implies decreas-
ing`for the regular solutions and increasing `for the
irregular solutions of xx33.2(iii) and 33.14(iii).33.23(v) Continued Fractions
x33.8 supplies continued fractions for F0
`=F`and
H
`0=H
`. Combined with the Wronskians (33.2.12),
the values of F`,G`, and their derivatives can be ex-
tracted. Inside the turning points, that is, when <
tp(;`), there can be a loss of precision by a factor of
approximatelyjG`j2.
33.23(vi) Other Numerical Methods
Curtis (1964a,x10) describes the use of series, radial
integration, and other methods to generate the tables
listed inx33.24.
Bardin et al. (1972) describes ten di erent methods
for the calculation of F`andG`, valid in di erent re-
gions of the ( ;)-plane.
Thompson and Barnett (1985, 1986) and Thompson
(2004) use combinations of series, continued fractions,
and Pad e-accelerated asymptotic expansions ( x3.11(iv))
for the analytic continuations of Coulomb functions.
Noble (2004) obtains double-precision accuracy for
W ;(2) for a wide range of parameters using a com-
bination of recurrence techniques, power-series expan-
sions, and numerical quadrature; compare (33.2.7).
33.23(vii) WKBJ Approximations
WKBJ approximations ( x2.7(iii)) for > tp(;`) are
presented in Hull and Breit (1959) and Seaton and
Peach (1962: in Eq. (12) ( c)=cshould be ( c)=). A
set of consistent second-order WKBJ formulas is given
by Burgess (1963: in Eq. (16) 3 2+2 should be 3 2c+2).
Seaton (1984) estimates the accuracies of these approx-
imations.
Hull and Breit (1959) and Barnett (1981b) give
WKBJ approximations for F0andG0in the region in-
side the turning point: < tp(;`).
33.24 Tables
Abramowitz and Stegun (1964, Chapter 14) tab-
ulatesF0(;),G0(;),F0
0(;), andG0
0(;)
for= 0:5(:5)20 and= 1(1)20, 5S; C0() for
= 0(:05)3, 6S.
Curtis (1964a) tabulates P`(;r),Q`(;r) (x33.1),
and related functions for `= 0;1;2 and=
2(:2)2, withx= 0(:1)4 for<0 andx= 0(:1)10
for0; 6D.
For earlier tables see Hull and Breit (1959) and
Fletcher et al. (1962,x22.59).
756 Coulomb Functions
33.25 Approximations
Cody and Hillstrom (1970) provides rational approxi-
mations of the phase shift 0() = ph (1 + i) (see
(33.2.10)) for the ranges 0 2, 24, and
4 1 . Maximum relative errors range from
1:0910 20to 4:2410 19.
33.26 Software
Seehttp://dlmf.nist.gov/33.26 .
References
General References
The main references used in writing this chapter are
Hull and Breit (1959), Thompson and Barnett (1986),
and Seaton (2002). For additional bibliographic reading
see also the General References in Chapter 13.
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x33.2 Yost et al. (1936), Hull and Breit (1959, pp. 409{
410).
x33.3 These graphics were produced at NIST.
x33.4 Powell (1947).
x33.5 Yost et al. (1936), Hull and Breit (1959, pp. 435{
436), Wheeler (1937), Biedenharn et al. (1955).
For (33.5.9) combine the second formula in (5.4.2)
with (5.11.7).
x33.6 For (33.6.5) use the de nition (33.2.8) with
U(a;b;z ) expanded as in (13.2.9). For (33.6.4)
use (33.2.4) with Eq. (1.12) of Buchholz (1969).
x33.7 Hull and Breit (1959, pp. 413{416). For (33.7.1)
see also Lowan and Horenstein (1942), with
change of variable = 1 tin the integral that
follows Eq. (8). For (33.7.2) see also Hoisington
and Breit (1938). For (33.7.3) see also Bloch et al.
(1950). For (33.7.4) see also Newton (1952).x33.9 The convergence of (33.9.1) follows from the
asymptotic forms, for large k, ofak(obtained by
application ofx2.9(i)) and j`+k() (obtained from
(10.19.1) and (10.47.3)). For (33.9.3) see Yost
et al. (1936), Abramowitz (1954), and Humblet
(1985). For (33.9.4) see Curtis (1964a, x5.1). For
(33.9.6) see Yost et al. (1936) and Abramowitz
(1954).
x33.10 Yost et al. (1936), Fr oberg (1955), Hum-
blet (1984), Humblet (1985, Eqs. 2.10a,b and
4.7a,b). For (33.10.6) and (33.10.10) use (33.2.5),
(33.2.10), andx5.11(i).
x33.11 Fr oberg (1955).
x33.14 Curtis (1964a, pp. ix{xxv), Seaton (1983),
Seaton (2002, Eqs. 3, 4, 7, 9, 14, 22, 47, 49,
51, 109, 113{116, 122{125, 131, and x2.3). For
(33.14.11) and (33.14.12) see Humblet (1985,
Eqs. 1.4a,b), Seaton (1982, Eq. 2.4.4).
x33.15 These graphics were produced at NIST.
x33.16 Seaton (2002, Eqs. 104{109, 119{121, 130, 131).
(33.16.3){(33.16.7) are generalizations of Seaton
(2002, Eqs. 88, 90, 93, 95). For (33.16.14)
and (33.16.15) combine (33.14.9) with (33.16.4){
(33.16.7). For (33.16.17) and (33.16.18) com-
bine (33.14.6), (33.14.9){(33.14.12), (33.16.10){
(33.16.13), and (33.16.16).
x33.17 Seaton (2002, Eqs. 77, 78, 82).
x33.18 Combine (33.5.8) and (33.16.1), (33.16.2). For
f(;`;r) (33.19.1) can also be used.
x33.19 Seaton (2002, Eqs. 15{17, 31{48).
x33.20 Seaton (2002, Eqs. 58, 59, 64, 67{70, 96, 98,
100, 102 (corrected)).
x33.21 Seaton (2002, Eqs. 104, 107), or apply
(13.14.21) to (33.16.9).
x33.23 Stable integration directions for the di eren-
tial equations are determined by comparison of
the asymptotic behavior of the solutions as the
radii tend to in nity and also as the radii tend to
zero (xx33.11, 33.21;xx33.6, 33.19). Stable recur-
rence directions for x33.4 are determined by the
asymptotic form of F`(;)=G`(;) as`!1 ;
see (33.5.8) and (33.5.9). For x33.17 seex33.18.
Chapter 34
3j;6j;9jSymbols
L. C. Maximon1
Notation 758
34.1 Special Notation . . . . . . . . . . . . . 758
Properties 758
34.2 De nition: 3jSymbol . . . . . . . . . . . 758
34.3 Basic Properties: 3jSymbol . . . . . . . 759
34.4 De nition: 6jSymbol . . . . . . . . . . . 761
34.5 Basic Properties: 6jSymbol . . . . . . . 762
34.6 De nition: 9jSymbol . . . . . . . . . . . 763
34.7 Basic Properties: 9jSymbol . . . . . . . 764
34.8 Approximations for Large Parameters . . 764
34.9 Graphical Method . . . . . . . . . . . . . 76534.10 Zeros . . . . . . . . . . . . . . . . . . . 765
34.11 Higher-Order 3njSymbols . . . . . . . . 765
Applications 765
34.12 Physical Applications . . . . . . . . . . . 765
Computation 765
34.13 Methods of Computation . . . . . . . . . 765
34.14 Tables . . . . . . . . . . . . . . . . . . . 765
34.15 Software . . . . . . . . . . . . . . . . . . 766
References 766
1Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, D.C.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
757
758 3j;6j;9jSymbols
Notation
34.1 Special Notation
(For other notation see pp. xiv and 873.)
2j1;2j2;2j3;2l1;2l2;2l3nonnegative integers.
r;s;t nonnegative integers.
The main functions treated in this chapter are the
Wigner 3j;6j;9jsymbols, respectively,
j1j2j3
m1m2m3
;j1j2j3
l1l2l3
;8
<
:j11j12j13
j21j22j23
j31j32j339
=
;:
The most commonly used alternative notation for
the 3jsymbol is the Clebsch{Gordan coecient
(j1m1j2m2jj1j2j3 m3)
= ( 1)j1 j2 m3(2j3+ 1)1
2j1j2j3
m1m2m3
;
see Condon and Shortley (1935). For other notations see
Edmonds (1974, pp. 52, 97, 104{105) and Varshalovich
et al. (1988,xx8.11, 9.10, 10.10).
Properties
34.2 De nition: 3jSymbol
The quantities j1;j2;j3in the 3jsymbol are called an-
gular momenta . Either all of them are nonnegative in-
tegers, or one is a nonnegative integer and the othertwo are half-odd positive integers. They must form the
sides of a triangle (possibly degenerate). They therefore
satisfy the triangle conditions
34.2.1jjr jsjjtjr+js;
wherer;s;t is any permutation of 1 ;2;3. The corre-
sponding projective quantum numbers m1;m2;m3are
given by
34.2.2mr= jr; jr+ 1;:::;jr 1;jr,r= 1;2;3;
and satisfy
34.2.3 m1+m2+m3= 0:
See Figure 34.2.1 for a schematic representation.
Figure 34.2.1 : Angular momenta jrand projective
quantum numbers mr,r= 1;2;3.
If either of the conditions (34.2.1) or (34.2.3) is not satis ed, then the 3 jsymbol is zero. When both conditions
are satis ed the 3 jsymbol can be expressed as the nite sum
34.2.4j1j2j3
m1m2m3
= ( 1)j1 j2 m3(j1j2j3) ((j1+m1)!(j1 m1)!(j2+m2)!(j2 m2)!(j3+m3)!(j3 m3)!)1
2
X
s( 1)s
s!(j1+j2 j3 s)!(j1 m1 s)!(j2+m2 s)!(j3 j2+m1+s)!(j3 j1 m2+s)!;
where
34.2.5 (j1j2j3) =(j1+j2 j3)!(j1 j2+j3)!( j1+j2+j3)!
(j1+j2+j3+ 1)!1
2
;
and the summation is over all nonnegative integers ssuch that the arguments in the factorials are nonnegative.
Equivalently,
34.2.6j1j2j3
m1m2m3
= ( 1)j2 m1+m3(j1+j2+m3)!(j2+j3 m1)!
(j1j2j3)(j1+j2+j3+ 1)!(j1+m1)!(j3 m3)!
(j1 m1)!(j2+m2)!(j2 m2)!(j3+m3)!1
2
3F2( j1 j2 j3 1; j1+m1; j3 m3; j1 j2 m3; j2 j3+m1; 1);
where 3F2is de ned as inx16.2.
34.3 Basic Properties: 3jSymbol 759
For alternative expressions for the 3 jsymbol, written either as a nite sum or as other terminating generalized
hypergeometric series 3F2of unit argument, see Varshalovich et al. (1988,xx8.21, 8.24{8.26).
34.3 Basic Properties: 3jSymbol
34.3(i) Special Cases
When any one of j1;j2;j3is equal to 0 ;1
2, or 1, the 3 jsymbol has a simple algebraic form. Examples are provided
by
34.3.1j j 0
m m0
=( 1)j m
(2j+ 1)1
2;
34.3.2j j 1
m m0
= ( 1)j m 2m
(2j(2j+ 1)(2j+ 2))1
2, j1
2;
34.3.3j j 1
m m 1 1
= ( 1)j m2(j m)(j+m+ 1)
2j(2j+ 1)(2j+ 2)1
2
, j1
2:
For these and other results, and also cases in which any one of j1;j2;j3is3
2or 2, see Edmonds (1974, pp. 125{127).
Next de ne
34.3.4 J=j1+j2+j3:
Then assuming the triangle conditions are satis ed
34.3.5j1j2j3
0 0 0
=8
><
>:0; J odd;
( 1)1
2J(J 2j1)!(J 2j2)!(J 2j3)!
(J+ 1)!1
2 (1
2J)!
(1
2J j1)!(1
2J j2)!(1
2J j3)!; J even:
Lastly,
34.3.6j1j2j1+j2
m1m2 m1 m2
= ( 1)j1 j2+m1+m2(2j1)!(2j2)!(j1+j2+m1+m2)!(j1+j2 m1 m2)!
(2j1+ 2j2+ 1)!(j1+m1)!(j1 m1)!(j2+m2)!(j2 m2)!1
2
;
34.3.7j1j2j3
j1 j1 m3m3
= ( 1) j2+j3+m3(2j1)!( j1+j2+j3)!(j1+j2+m3)!(j3 m3)!
(j1+j2+j3+ 1)!(j1 j2+j3)!(j1+j2 j3)!( j1+j2 m3)!(j3+m3)!1
2
:
Again it is assumed that in (34.3.7) the triangle conditions are satis ed.
34.3(ii) Symmetry
Even permutations of columns of a 3 jsymbol leave it unchanged; odd permutations of columns produce a phase
factor ( 1)j1+j2+j3, for example,
34.3.8j1j2j3
m1m2m3
=j2j3j1
m2m3m1
=j3j1j2
m3m1m2
;
34.3.9j1j2j3
m1m2m3
= ( 1)j1+j2+j3j2j1j3
m2m1m3
:
Next,
34.3.10j1j2j3
m1m2m3
= ( 1)j1+j2+j3j1j2j3
m1 m2 m3
;
34.3.11j1j2j3
m1m2m3
=j11
2(j2+j3+m1)1
2(j2+j3 m1)
j2 j31
2(j3 j2+m1) +m21
2(j3 j2+m1) +m3
;
34.3.12j1j2j3
m1m2m3
=1
2(j1+j2 m3)1
2(j2+j3 m1)1
2(j1+j3 m2)
j3 1
2(j1+j2+m3)j1 1
2(j2+j3+m1)j2 1
2(j1+j3+m2)
:
Equations (34.3.11) and (34.3.12) are called Regge symmetries . Additional symmetries are obtained by applying
(34.3.8){(34.3.10) to (34.3.11)) and (34.3.12). See Srinivasa Rao and Rajeswari (1993, pp. 44{47) and references
given there.
760 3j;6j;9jSymbols
34.3(iii) Recursion Relations
In the following three equations it is assumed that the triangle conditions are satis ed by each 3 jsymbol.
34.3.13((j1+j2+j3+ 1)( j1+j2+j3))1
2j1j2j3
m1m2m3
= ((j2+m2)(j3 m3))1
2j1j2 1
2j3 1
2
m1m2 1
2m3+1
2
((j2 m2)(j3+m3))1
2j1j2 1
2j3 1
2
m1m2+1
2m3 1
2
;
34.3.14(j1(j1+ 1) j2(j2+ 1) j3(j3+ 1) 2m2m3)j1j2j3
m1m2m3
= ((j2 m2)(j2+m2+ 1)(j3 m3+ 1)(j3+m3))1
2
j1j2j3
m1m2+ 1m3 1
+ ((j2 m2+ 1)(j2+m2)(j3 m3)(j3+m3+ 1))1
2j1j2j3
m1m2 1m3+ 1
;
34.3.15(2j1+ 1) ((j2(j2+ 1) j3(j3+ 1))m1 j1(j1+ 1)(m3 m2))j1j2j3
m1m2m3
= (j1+ 1)
j2
1 (j2 j3)21
2
(j2+j3+ 1)2 j2
11
2
j2
1 m2
11
2j1 1j2j3
m1m2m3
+j1
(j1+ 1)2 (j2 j3)21
2
(j2+j3+ 1)2 (j1+ 1)21
2
(j1+ 1)2 m2
11
2j1+ 1j2j3
m1m2m3
:
For these and other recursion relations see Varshalovich et al. (1988,x8.6). See also Micu (1968), Louck (1958),
Schulten and Gordon (1975a), Srinivasa Rao and Rajeswari (1993, pp. 220{225), and Luscombe and Luban (1998).
34.3(iv) Orthogonality
34.3.16X
m1m2(2j3+ 1)j1j2j3
m1m2m3j1j2j0
3
m1m2m0
3
=j3;j0
3m3;m0
3;
34.3.17X
j3m3(2j3+ 1)j1j2j3
m1m2m3
j1j2j3
m0
1m0
2m3
=m1;m0
1m2;m0
2;
34.3.18X
m1m2m3j1j2j3
m1m2m3
j1j2j3
m1m2m3
= 1:
In the summations (34.3.16){(34.3.18) the summation variables range over all values that satisfy the conditions given
in (34.2.1){(34.2.3). Similar conventions apply to all subsequent summations in this chapter.
34.3(v) Generating Functions
For generating functions for the 3 jsymbol see Biedenharn and van Dam (1965, p. 245, Eq. (3.42) and p. 247,
Eq. (3.55)).
34.3(vi) Sums
For sums of products of 3 jsymbols, see Varshalovich et al. (1988, pp. 259{262).
34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
For the polynomials Plseex18.3, and for the functions Yl;mandY
l;mseex14.30.
34.3.19 Pl1(cos)Pl2(cos) =X
l(2l+ 1)
l1l2l
0 0 02
Pl(cos);
34.3.20Yl1;m1(; )Yl2;m2(; ) =X
l;m(2l1+ 1)(2l2+ 1)(2l+ 1)
41
2l1l2l
m1m2m
Y
l;m(; )l1l2l
0 0 0
;
34.4 Definition: 6jSymbol 761
34.3.21Z
0Pl1(cos)Pl2(cos)Pl3(cos) sind= 2l1l2l3
0 0 02
;
34.3.22Z2
0Z
0Yl1;m1(; )Yl2;m2(; )Yl3;m3(; ) sindd
=(2l1+ 1)(2l2+ 1)(2l3+ 1)
41
2l1l2l3
0 0 0l1l2l3
m1m2m3
:
Equations (34.3.19){(34.3.22) are particular cases of more general results that relate rotation matrices to 3 jsymbols,
for which see Edmonds (1974, Chapter 4). The left- and right-hand sides of (34.3.22) are known, respectively, as
Gaunt's integral and the Gaunt coecient (Gaunt (1929)).
34.4 De nition: 6jSymbol
The 6jsymbol is de ned by the following double sum of products of 3 jsymbols:
34.4.1j1j2j3
l1l2l3
=X
mrm0s( 1)l1+m0
1+l2+m0
2+l3+m0
3
j1j2j3
m1m2m3j1l2l3
m1m0
2 m0
3l1j2l3
m0
1m2m0
3l1l2j3
m0
1 m0
2m3
;
where the summation is taken over all admissible values of the m's andm0's for each of the four 3 jsymbols; compare
(34.2.2) and (34.2.3).
Except in degenerate cases the combination of the triangle inequalities for the four 3 jsymbols in (34.4.1) is equiv-
alent to the existence of a tetrahedron (possibly degenerate) with edges of lengths j1;j2;j3;l1;l2;l3; see Figure 34.4.1.
Figure 34.4.1 : Tetrahedron corresponding to 6 jsymbol.
The 6jsymbol can be expressed as the nite sum
34.4.2j1j2j3
l1l2l3
=X
s( 1)s(s+ 1)!
(s j1 j2 j3)!(s j1 l2 l3)!(s l1 j2 l3)!(s l1 l2 j3)!
1
(j1+j2+l1+l2 s)!(j2+j3+l2+l3 s)!(j3+j1+l3+l1 s)!;
where the summation is over all nonnegative integers ssuch that the arguments in the factorials are nonnegative.
Equivalently,
34.4.3j1j2j3
l1l2l3
= ( 1)j1+j3+l1+l3(j1j2j3)(j2l1l3)(j1 j2+l1+l2)!( j2+j3+l2+l3)!(j1+j3+l1+l3+ 1)!
(j1l2l3)(j3l1l2)(j1 j2+j3)!( j2+l1+l3)!(j1+l2+l3+ 1)!(j3+l1+l2+ 1)!
4F3 j1+j2 j3;j2 l1 l3; j1 l2 l3 1; j3 l1 l2 1
j1+j2 l1 l2;j2 j3 l2 l3; j1 j3 l1 l3 1; 1
;
where 4F3is de ned as inx16.2.
For alternative expressions for the 6 jsymbol, written either as a nite sum or as other terminating generalized
hypergeometric series 4F3of unit argument, see Varshalovich et al. (1988,xx9.2.1, 9.2.3).
762 3j;6j;9jSymbols
34.5 Basic Properties: 6jSymbol
34.5(i) Special Cases
In the following equations it is assumed that the triangle inequalities are satis ed and that Jis again de ned by
(34.3.4).
If any lower argument in a 6 jsymbol is 0,1
2, or 1, then the 6 jsymbol has a simple algebraic form. Examples are
provided by:
34.5.1j1j2j3
0j3j2
=( 1)J
((2j2+ 1)(2j3+ 1))1
2;
34.5.2j1j2j3
1
2j3 1
2j2+1
2
= ( 1)J(j1+j3 j2)(j1+j2 j3+ 1)
(2j2+ 1)(2j2+ 2)2j3(2j3+ 1)1
2
;
34.5.3j1j2j3
1
2j3 1
2j2 1
2
= ( 1)J(j2+j3 j1)(j1+j2+j3+ 1)
2j2(2j2+ 1)2j3(2j3+ 1)1
2
;
34.5.4j1j2j3
1j3 1j2 1
= ( 1)JJ(J+ 1)(J 2j1)(J 2j1 1)
(2j2 1)2j2(2j2+ 1)(2j3 1)2j3(2j3+ 1)1
2
;
34.5.5j1j2j3
1j3 1j2
= ( 1)J2(J+ 1)(J 2j1)(J 2j2)(J 2j3+ 1)
2j2(2j2+ 1)(2j2+ 2)(2j3 1)2j3(2j3+ 1)1
2
;
34.5.6j1j2j3
1j3 1j2+ 1
= ( 1)J(J 2j2 1)(J 2j2)(J 2j3+ 1)(J 2j3+ 2)
(2j2+ 1)(2j2+ 2)(2j2+ 3)(2j3 1)2j3(2j3+ 1)1
2
;
34.5.7j1j2j3
1j3j2
= ( 1)J+1 2(j2(j2+ 1) +j3(j3+ 1) j1(j1+ 1))
(2j2(2j2+ 1)(2j2+ 2)2j3(2j3+ 1)(2j3+ 2))1
2:
34.5(ii) Symmetry
The 6jsymbol is invariant under interchange of any two columns and also under interchange of the upper and lower
arguments in each of any two columns, for example,
34.5.8j1j2j3
l1l2l3
=j2j1j3
l2l1l3
=j1l2l3
l1j2j3
:
Next,
34.5.9
j1j2j3
l1l2l3
=
j11
2(j2+l2+j3 l3)1
2(j2 l2+j3+l3)
l11
2(j2+l2 j3+l3)1
2( j2+l2+j3+l3)
;
34.5.10j1j2j3
l1l2l3
=1
2(j2+l2+j3 l3)1
2(j1 l1+j3+l3)1
2(j1+l1+j2 l2)
1
2(j2+l2 j3+l3)1
2( j1+l1+j3+l3)1
2(j1+l1 j2+l2)
:
Equations (34.5.9) and (34.5.10) are called Regge symmetries . Additional symmetries are obtained by applying
(34.5.8) to (34.5.9) and (34.5.10). See Srinivasa Rao and Rajeswari (1993, pp. 102{103) and references given there.
34.5(iii) Recursion Relations
In the following equation it is assumed that the triangle conditions are satis ed.
34.5.11(2j1+ 1) ((J3+J2 J1)(L3+L2 J1) 2(J3L3+J2L2 J1L1))j1j2j3
l1l2l3
=j1E(j1+ 1)j1+ 1j2j3
l1l2l3
+ (j1+ 1)E(j1)j1 1j2j3
l1l2l3
;
where
34.5.12 Jr=jr(jr+ 1); Lr=lr(lr+ 1);
34.5.13 E(j) =
(j2 (j2 j3)2)((j2+j3+ 1)2 j2)(j2 (l2 l3)2)((l2+l3+ 1)2 j2)1
2:
For further recursion relations see Varshalovich et al. (1988,x9.6) and Edmonds (1974, pp. 98{99).
34.6 Definition: 9jSymbol 763
34.5(iv) Orthogonality
34.5.14X
j3(2j3+ 1)(2l3+ 1)j1j2j3
l1l2l3j1j2j3
l1l2l0
3
=l3;l0
3:
34.5(v) Generating Functions
For generating functions for the 6 jsymbol see Biedenharn and van Dam (1965, p. 255, eq. (4.18)).
34.5(vi) Sums
34.5.15X
j( 1)j+j0+j00(2j+ 1)j1j2j
j3j4j0j1j2j
j4j3j00
=j1j4j0
j2j3j00
;
34.5.16( 1)j1+j2+j3+j0
1+j0
2+l1+l2j1j2j3
l1l2l3j0
1j0
2j3
l1l2l0
3
=X
j( 1)l3+l0
3+j(2j+ 1)j1j0
1j
j0
2j2j3l3l0
3j
j0
1j1l2l3l0
3j
j0
2j2l1
:
Equations (34.5.15) and (34.5.16) are the sum rules . They constitute addition theorems for the 6 jsymbol.
34.5.17X
j(2j+ 1)j1j2j
j1j2j0
= ( 1)2(j1+j2);
34.5.18X
j( 1)j1+j2+j(2j+ 1)j1j2j
j2j1j0
=p
(2j1+ 1)(2j2+ 1)j0;0;
34.5.19X
l
j1j2l
j2j1j
= 0, 2 jodd,= min(j1;j2);
34.5.20X
l( 1)l+jj1j2l
j1j2j
=( 1)2
2j+ 1, = min(j1;j2);
34.5.21X
l( 1)l+j+j1+j2j1j2l
j2j1j
=1
2j+ 1(2j1 j)!(2j2+j+ 1)!
(2j2 j)!(2j1+j+ 1)!1
2
, j2j1;
34.5.22X
l( 1)l+j+j1+j21
l(l+ 1)j1j2l
j2j1j
=1
j1(j1+ 1) j2(j2+ 1)(2j1 j)!(2j2+j+ 1)!
(2j2 j)!(2j1+j+ 1)!1
2
,j2<j1:
34.5.23
j1j2j3
m1m2m3
j1j2j3
l1l2l3
=X
m0
1m0
2m0
3( 1)l1+l2+l3+m0
1+m0
2+m0
3j1l2l3
m1m0
2 m0
3l1j2l3
m0
1m2m0
3l1l2j3
m0
1 m0
2m3
:
Equation (34.5.23) can be regarded as an alternative de nition of the 6 jsymbol.
For other sums see Ginocchio (1991).
34.6 De nition: 9jSymbol
The 9jsymbol may be de ned either in terms of 3 jsymbols or equivalently in terms of 6 jsymbols:
34.6.18
<
:j11j12j13
j21j22j23
j31j32j339
=
;=X
allmrsj11j12j13
m11m12m13j21j22j23
m21m22m23j31j32j33
m31m32m33
j11j21j31
m11m21m31j12j22j32
m12m22m32j13j23j33
m13m23m33
;
34.6.28
<
:j11j12j13
j21j22j23
j31j32j339
=
;=X
j( 1)2j(2j+ 1)j11j21j31
j32j33jj12j22j32
j21j j 23j13j23j33
j j 11j12
:
764 3j;6j;9jSymbols
The 9jsymbol may also be written as a nite triple sum equivalent to a terminating generalized hypergeometric
series of three variables with unit arguments. See Srinivasa Rao and Rajeswari (1993, pp. 7 and 125{132) and
Rosengren (1999).
34.7 Basic Properties: 9jSymbol
34.7(i) Special Case
34.7.18
<
:j11j12j13
j21j22j13
j31j31 09
=
;=( 1)j12+j21+j13+j31
((2j13+ 1)(2j31+ 1))1
2j11j12j13
j22j21j31
:
34.7(ii) Symmetry
The 9jsymbol has symmetry properties with respect to permutation of columns, permutation of rows, and transpo-
sition of rows and columns; these relate 72 independent 9 jsymbols. Even (cyclic) permutations of either columns
or rows, as well as transpositions, leave the 9 jsymbol unchanged. Odd permutations of columns or rows introduce
a phase factor ( 1)R, whereRis the sum of all arguments of the 9 jsymbol.
For further symmetry properties of the 9 jsymbol see Edmonds (1974, pp. 102{103) and Varshalovich et al. (1988,
x10.4.1).
34.7(iii) Recursion Relations
For recursion relations see Varshalovich et al. (1988,x10.5).
34.7(iv) Orthogonality
34.7.2X
j12j34(2j12+ 1)(2j34+ 1)(2j13+ 1)(2j24+ 1)8
<
:j1j2j12
j3j4j34
j13j24j9
=
;8
<
:j1j2j12
j3j4j34
j0
13j0
24j9
=
;=j13;j0
13j24;j0
24:
34.7(v) Generating Functions
For generating functions for the 9 jsymbol see Biedenharn and van Dam (1965, p. 258, eq. (4.37)).
34.7(vi) Sums
34.7.3X
j13j24( 1)2j2+j24+j23 j34(2j13+ 1)(2j24+ 1)8
<
:j1j2j12
j3j4j34
j13j24j9
=
;8
<
:j1j3j13
j4j2j24
j14j23j9
=
;=8
<
:j1j2j12
j4j3j34
j14j23j9
=
;:
This equation is the sum rule . It constitutes an addition theorem for the 9 jsymbol.
34.7.4j13j23j33
m13m23m338
<
:j11j12j13
j21j22j23
j31j32j339
=
;=X
mr1;mr2;r=1;2;3j11j12j13
m11m12m13j21j22j23
m21m22m23
j31j32j33
m13m23m33j11j21j31
m11m21m31j12j22j32
m12m22m32
:
34.7.5X
j0(2j0+ 1)8
<
:j11j12j0
j21j22j23
j31j32j339
=
;
j11j12j0
j23j33j
= ( 1)2j
j21j22j23
j12j j 32
j31j32j33
j j 11j21
:
34.8 Approximations for Large Parameters
For large values of the parameters in the 3 j, 6j, and 9jsymbols, di erent asymptotic forms are obtained depending
on which parameters are large. For example,
34.8.1j1j2j3
j2j1l3
= ( 1)j1+j2+j3+l34
(2j1+ 1)(2j2+ 1)(2l3+ 1) sin1
2
cos
(l3+1
2) 1
4
+o(1)
,
j1;j2;j3 l3 1;
34.9 Graphical Method 765
where
34.8.2 cos=j1(j1+ 1) +j2(j2+ 1) j3(j3+ 1)
2p
j1(j1+ 1)j2(j2+ 1);
and the symbol o(1) denotes a quantity that tends to
zero as the parameters tend to in nity, as in x2.1(i).
Semiclassical (WKBJ) approximations in terms of
trigonometric or exponential functions are given in Var-
shalovich et al. (1988,xx8.9, 9.9, 10.7). Uniform ap-
proximations in terms of Airy functions for the 3 jand
6jsymbols are given in Schulten and Gordon (1975b).
For approximations for the 3 j, 6j, and 9jsymbols with
error bounds see Flude (1998), Chen et al. (1999), and
Watson (1999): these references also cite earlier work.
34.9 Graphical Method
The graphical method establishes a one-to-one corre-
spondence between an analytic expression and a dia-
gram by assigning a graphical symbol to each function
and operation of the analytic expression. Thus, any an-
alytic expression in the theory, for example equations
(34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be rep-
resented by a diagram; conversely, any diagram rep-
resents an analytic equation. For an account of this
method see Brink and Satchler (1993, Chapter VII).
For speci c examples of the graphical method of repre-
senting sums involving the 3 j;6j, and 9jsymbols, see
Varshalovich et al. (1988, Chapters 11, 12) and Lehman
and O'Connell (1973, x3.3).
34.10 Zeros
In a 3jsymbol, if the three angular momenta j1;j2;j3do
not satisfy the triangle conditions (34.2.1), or if the pro-
jective quantum numbers do not satisfy (34.2.3), then
the 3jsymbol is zero. Similarly the 6 jsymbol (34.4.1)
vanishes when the triangle conditions are not satis ed
by any of the four 3 jsymbols in the summation. Such
zeros are called trivial zeros . However, the 3 jand 6j
symbols may vanish for certain combinations of the an-
gular momenta and projective quantum numbers even
when the triangle conditions are ful lled. Such zeros
are called nontrivial zeros .
For further information, including examples of non-
trivial zeros and extensions to 9 jsymbols, see Srini-
vasa Rao and Rajeswari (1993, pp. 133{215, 294{295,
299{310).
34.11 Higher-Order 3njSymbols
For information on 12 j;15j,..., symbols, see Var-
shalovich et al. (1988,x10.12) and Yutsis et al. (1962,
pp. 62{65 and 122{153).Applications
34.12 Physical Applications
The angular momentum coupling coecients (3 j, 6j,
and 9jsymbols) are essential in the elds of nuclear,
atomic, and molecular physics. For applications in
nuclear structure, see de Shalit and Talmi (1963); in
atomic spectroscopy, see Biedenharn and van Dam
(1965, pp. 134{200), Judd (1998), Sobelman (1992,
Chapter 4), Shore and Menzel (1968, pp. 268{303), and
Wigner (1959); in molecular spectroscopy and chemi-
cal reactions, see Burshtein and Temkin (1994, Chap-
ter 5), and Judd (1975). 3 j;6j, and 9jsymbols are also
found in multipole expansions of solutions of the Laplace
and Helmholtz equations; see Carlson and Rushbrooke
(1950) and Judd (1976).
Computation
34.13 Methods of Computation
Methods of computation for 3 jand 6jsymbols include
recursion relations, see Schulten and Gordon (1975a),
Luscombe and Luban (1998), and Edmonds (1974, pp.
42{45, 48{51, 97{99); summation of single-sum expres-
sions for these symbols, see Varshalovich et al. (1988,
xx8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation
of the generalized hypergeometric functions of unit ar-
gument that represent these symbols, see Srinivasa Rao
and Venkatesh (1978) and Srinivasa Rao (1981).
For 9jsymbols, methods include evaluation of the
single-sum series (34.6.2), see Fang and Shriner (1992);
evaluation of triple-sum series, see Varshalovich et al.
(1988,x10.2.1) and Srinivasa Rao et al. (1989). A review
of methods of computation is given in Srinivasa Rao and
Rajeswari (1993, Chapter VII, pp. 235{265). See also
Roothaan and Lai (1997) and references given there.
34.14 Tables
Tables of exact values of the squares of the 3 jand 6j
symbols in which all parameters are 8 are given in
Rotenberg et al. (1959), together with a bibliography of
earlier tables of 3 j;6j, and 9jsymbols on pp. 33{36.
Tables of 3jand 6jsymbols in which all parameters
are17=2 are given in Appel (1968) to 6D. Some se-
lected 9jsymbols are also given. Other tabulations for
3jsymbols are listed on pp. 11-12; for 6 jsymbols on
pp. 16-17; for 9 jsymbols on p. 21.
766 3j;6j;9jSymbols
Biedenharn and Louck (1981) give tables of algebraic
expressions for Clebsch{Gordan coecients and 6 jsym-
bols, together with a bibliography of tables produced
prior to 1975. In Varshalovich et al. (1988) algebraic
expressions for the Clebsch{Gordan coecients with all
parameters5 and numerical values for all parameters
3 are given on pp. 270{289; similar tables for the 6 j
symbols are given on pp. 310{332, and for the 9 jsym-
bols on pp. 359, 360, 372{411. Earlier tables are listed
on p. 513.
34.15 Software
Seehttp://dlmf.nist.gov/34.15 .
References
General References
The main references used in writing this chapter are Ed-
monds (1974), Varshalovich et al. (1988), and de Shalit
and Talmi (1963).Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x34.2 Edmonds (1974, pp. 44{45).
x34.3 Edmonds (1974, pp. 46{50, 63), de Shalit and
Talmi (1963, pp. 515, 519), Thompson (1994,
p. 288).
x34.4 Varshalovich et al. (1988,x9.2.4), de Shalit and
Talmi (1963, p. 131).
x34.5 Edmonds (1974, pp. 94{98, 130{132), de Shalit
and Talmi (1963, pp. 517{518, 520), Varshalovich
et al. (1988,x9.8), Dunlap and Judd (1975).
x34.6 Edmonds (1974, p. 101), de Shalit and Talmi
(1963, p. 516).
x34.7 Edmonds (1974, pp. 103{106), de Shalit and
Talmi (1963, pp. 127, 517{518).
x34.8 Watson (1999), Chen et al. (1999).
Chapter 35
Functions of Matrix Argument
D. St. P. Richards1
Notation 768
35.1 Special Notation . . . . . . . . . . . . . 768
Properties 768
35.2 Laplace Transform . . . . . . . . . . . . 768
35.3 Multivariate Gamma and Beta Functions 768
35.4 Partitions and Zonal Polynomials . . . . . 769
35.5 Bessel Functions of Matrix Argument . . 769
35.6 Con
uent Hypergeometric Functions of
Matrix Argument . . . . . . . . . . . . . 770
35.7 Gaussian Hypergeometric Function of Ma-
trix Argument . . . . . . . . . . . . . . . 77135.8 Generalized Hypergeometric Functions of
Matrix Argument . . . . . . . . . . . . . 772
Applications 773
35.9 Applications . . . . . . . . . . . . . . . . 773
Computation 773
35.10 Methods of Computation . . . . . . . . . 773
35.11 Tables . . . . . . . . . . . . . . . . . . . 773
35.12 Software . . . . . . . . . . . . . . . . . . 773
References 773
1Department of Statistics, Pennsylvania State University.
Acknowledgments : With deep gratitude to Ingram Olkin for advice and support regarding the nal version of this material.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
767
768 Functions of Matrix Argument
Notation
35.1 Special Notation
(For other notation see pp. xiv and 873.)
All matrices are of order mm, unless speci ed oth-
erwise. All fractional or complex powers are principal
values.
a;b complex variables.
j;k nonnegative integers.
m positive integer.
[a] partitional shifted factorial ( x35.4(i)).
0 zero matrix.
I identity matrix.
S space of all real symmetric matrices.
S;T;X real symmetric matrices.
trX trace of X.
etr(X) exp(tr X).
jXj determinant of X(except when m= 1
where it means either determinant or
absolute value, depending on the context).
j(X)jjjth principal minor of X.
xj;k (j;k)th element of X.
dXQ
1jkmdxj;k.
space of positive-de nite real symmetric
matrices.
t1;:::;tmeigenvalues of T.
jjTjj spectral norm of T.
X>T X Tis positive de nite.
Z complex symmetric matrix.
U;V real and complex parts of Z.
f(X) complex-valued function with X2
.
O(m) space of orthogonal matrices.
H orthogonal matrix.
dH normalized Haar measure on O(m).
Z(T) zonal polynomials.
The main functions treated in this chapter are
the multivariate gamma and beta functions, respec-
tively m(a) and Bm(a;b), and the special functions
of matrix argument: Bessel (of the rst kind) A(T)
and (of the second kind) B(T); con
uent hyperge-
ometric (of the rst kind) 1F1(a;b;T) or 1F1a
b;T
and (of the second kind) ( a;b;T); Gaussian hyper-
geometric 2F1(a1;a2;b;T) or 2F1a1;a2
b;T
; gener-
alized hypergeometric pFq(a1;:::;ap;b1;:::;bq;T) or
pFqa1;:::;ap
b1;:::;bq;T
.
An alternative notation for the multivariate gamma
function is m(a) = m
a+1
2(m+ 1)
(Herz (1955,
p. 480)). Related notations for the Bessel func-
tions areJ+1
2(m+1)(T) =A(T)=A(0) (Faraut andKor anyi (1994, pp. 320{329)), Km(0;:::; 0;jS;T) =
jTjB(ST) (Terras (1988, pp. 49{64)), and K(T) =
jTjB(ST) (Faraut and Kor anyi (1994, pp. 357{358)).
Properties
35.2 Laplace Transform
De nition
For any complex symmetric matrix Z,
35.2.1 g(Z) =Z
etr( ZX)f(X)dX;
where the integration variable Xranges over the space
.
Suppose there exists a constant X02
such that
jf(X)j<etr( X0X) for all X2
. Then (35.2.1) con-
verges absolutely on the region <(Z)>X0, andg(Z) is
a complex analytic function of all elements zj;kofZ.
Inversion Formula
Assume thatR
Sjg(Z)jdVconverges, and also that
limU!1R
Sjg(Z)jdV= 0. Then
35.2.2f(X) =1
(2i)m(m+1)=2Z
etr(ZX)g(Z)dZ;
where the integral is taken over all Z=U+iVsuch
thatU>X0andVranges overS.
Convolution Theorem
Ifgjis the Laplace transform of fj,j= 1;2, theng1g2is
the Laplace transform of the convolution f1f2, where
35.2.3f1f2(T) =Z
0<X<Tf1(T X)f2(X)dX:
35.3 Multivariate Gamma and Beta
Functions
35.3(i) De nitions
35.3.1 m(a) =Z
etr( X)jXja 1
2(m+1)dX,
<(a)>1
2(m 1).
35.3.2
m(s1;:::;sm)
=Z
etr( X)jXjsm 1
2(m+1)m 1Y
j=1j(X)jjsj sj+1dX,
sj2C,<(sj)>1
2(j 1),j= 1;:::;m .
35.3.3
Bm(a;b) =Z
0<X<IjXja 1
2(m+1)jI Xjb 1
2(m+1)dX,
<(a);<(b)>1
2(m 1).
35.4 Partitions and Zonal Polynomials 769
35.3(ii) Properties
35.3.4 m(a) =m(m 1)=4mY
j=1
a 1
2(j 1)
:
35.3.5
m(s1;:::;sm) =m(m 1)=4mY
j=1
sj 1
2(j 1)
:
35.3.6 m(a;:::;a ) = m(a):
35.3.7 Bm(a;b) = m(a) m(b)
m(a+b):
35.3.8Bm(a;b) =Z
jXja 1
2(m+1)jI+Xj (a+b)dX,
<(a);<(b)>1
2(m 1).
35.4 Partitions and Zonal Polynomials
35.4(i) De nitions
Apartition= (k1;:::;km) is a vector of nonnegative
integers, listed in nonincreasing order. Also, jjdenotes
k1++km, the weight of;`() denotes the number of
nonzerokj;a+denotes the vector ( a+k1;:::;a +km).
The partitional shifted factorial is given by
35.4.1 [a]= m(a+)
m(a)=mY
j=1
a 1
2(j 1)
kj;
where (a)k=a(a+ 1)(a+k 1).
For any partition , the zonal polynomial Z:S!
Ris de ned by the properties
35.4.2
Z(I) =jj! 22jj[m=2]Q
1j<l`()(2kj 2kl j+l)
`()Q
j=1(2kj+`() j)!
and
35.4.3
Z(T) =Z(I)jTjkmZ
O(m)m 1Y
j=1j(HTH 1)jjkj kj+1dH,
T2S.
See Muirhead (1982, pp. 68{72) for the de nition
and properties of the Haar measure dH. See Hua
(1963, p. 30), Constantine (1963), James (1964), and
Macdonald (1995, pp. 425{431) for further information
on (35.4.2) and (35.4.3). Alternative notations for the
zonal polynomials are C(T) (Muirhead (1982, pp. 227{
239)),Y(T) (Takemura (1984, p. 22)), and (T) (Fa-
raut and Kor anyi (1994, pp. 228{236)).35.4(ii) Properties
Normalization
35.4.4 Z(0) =(
1; = (0;:::; 0);
0; 6= (0;:::; 0):
Orthogonal Invariance
35.4.5 Z
HTH 1
=Z(T), H2O(m).
ThereforeZ(T) is a symmetric polynomial in the eigen-
values of T.
Summation
Fork= 0;1;2;:::,
35.4.6X
jj=kZ(T) = (tr T)k:
Mean-Value
35.4.7Z
O(m)Z
SHTH 1
dH=Z(S)Z(T)
Z(I):
Laplace and Beta Integrals
ForT2
and<(a);<(b)>1
2(m 1),
35.4.8Z
etr( TX)jXja 1
2(m+1)Z(X)dX
= m(a+)jTj aZ
T 1
;
35.4.9Z
0<X<IjXja 1
2(m+1)jI Xjb 1
2(m+1)Z(TX)dX
=[a]
[a+b]Bm(a;b)Z(T):
35.5 Bessel Functions of Matrix Argument
35.5(i) De nitions
35.5.1 A(0) =1
m
+1
2(m+ 1),2C.
35.5.2
A(T) =A(0)1X
k=0( 1)k
k!X
jj=k1
+1
2(m+ 1)
Z(T),
2C,T2S.
35.5.3
B(T) =Z
etr
(TX+X 1)
jXj 1
2(m+1)dX,
2C,T2
.
35.5(ii) Properties
35.5.4Z
etr( TX)jXjA(SX)dX
= etr
ST 1
jTj 1
2(m+1),
S2S,T2
;<()> 1.
770 Functions of Matrix Argument
35.5.5Z
0<X<TA1(S1X)jXj1A2(S2(T X))jT Xj2dX=jTj1+2+1
2(m+1)A1+2+1
2(m+1)((S1+S2)T),
j2C,<(j)> 1,j= 1;2;S1;S22S;T2
.
35.5.6 B(T) =jTj B (T), 2C,T2
.
35.5.7Z
A1(TX)B 2(SX)jXj1dX=1
A1+2(0)jSj2jT+Sj (1+2+1
2(m+1)),<(1+2)> 1;S;T2
.
35.5.8Z
O(m)etr(SH)dH=A 1=2
1
4SST
A 1=2(0), Sarbitrary.
35.5(iii) Asymptotic Approximations
For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood
(2003).
35.6 Con
uent Hypergeometric Functions of Matrix Argument
35.6(i) De nitions
35.6.1 1F1a
b;T
=1X
k=01
k!X
jj=k[a]
[b]Z(T):
35.6.2 (a;b;T) =1
m(a)Z
etr( TX)jXja 1
2(m+1)jI+Xjb a 1
2(m+1)dX,<(a)>1
2(m 1),T2
.
Laguerre Form
35.6.3 L(
)
(T) = m
++1
2(m+ 1)
m
+1
2(m+ 1)1F1
+1
2(m+ 1);T
,<(
);<(
+)> 1.
35.6(ii) Properties
35.6.4 1F1a
b;T
=1
Bm(a;b a)Z
0<X<Ietr(TX)jXja 1
2(m+1)jI Xjb a 1
2(m+1)dX,<(a);<(b a)>1
2(m 1).
35.6.5Z
etr( TX)jXjb 1
2(m+1)
1F1a
b;SX
dX= m(b)jI ST 1j ajTj b,T>S,<(b)>1
2(m 1).
35.6.6Bm(b1;b2)jTjb1+b2 1
2(m+1)
1F1a1+a2
b1+b2;T
=Z
0<X<TjXjb1 1
2(m+1)
1F1a1
b1;X
jT Xjb2 1
2(m+1)
1F1a2
b2;T X
dX,<(b1);<(b2)>1
2(m 1).
35.6.7 1F1a
b;T
= etr( T)1F1b a
b; T
:
35.6.8Z
jTjc 1
2(m+1) (a;b;T)dT= m(c) m(a c) m
c b+1
2(m+ 1)
m(a) m
a b+1
2(m+ 1) ,
<(a)><(c) +1
2(m 1)>m 1,<(c b)> 1.
35.6(iii) Relations to Bessel Functions of Matrix Argument
35.6.9 lim
a!11F1a
+1
2(m+ 1); a 1T
=A(T)
A(0):
35.6.10 lim
a!1 m(a)
a+;+1
2(m+ 1);a 1T
=B(T):
35.7 Gaussian Hypergeometric Function of Matrix Argument 771
35.6(iv) Asymptotic Approximations
For asymptotic approximations for con
uent hypergeometric functions of matrix argument, see Herz (1955) and
Butler and Wood (2002).
35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7(i) De nition
35.7.1 2F1a;b
c;T
=1X
k=01
k!X
jj=k[a][b]
[c]Z(T), c+1
2(j+ 1)=2N, 1jm;jjTjj<1.
Jacobi Form
35.7.2P(
;)
(T) = m
++1
2(m+ 1)
m
+1
2(m+ 1)2F1
;
+++1
2(m+ 1)
+1
2(m+ 1);T!
,0<T<I;
;;2C;<(
)> 1.
35.7(ii) Basic Properties
Casem= 2
35.7.3 2F1a;b
c;t10
0t2
=1X
k=0(a)k(c a)k(b)k(c b)k
k! (c)2k
c 1
2
k(t1t2)k
2F1a+k;b+k
c+ 2k;t1+t2 t1t2
:
Con
uent Form
35.7.4 lim
c!12F1a;b
c;I cT 1
=jTjb
b;b a+1
2(m+ 1);T
:
Integral Representation
35.7.52F1a;b
c;T
=1
Bm(a;c a)Z
0<X<IjXja 1
2(m+1)jI Xjc a 1
2(m+1)jI TXj bdX,
<(a);<(c a)>1
2(m 1),0<T<I.
Transformations of Parameters
35.7.62F1a;b
c;T
=jI Tjc a b
2F1c a;c b
c;T
=jI Tj a
2F1a;c b
c; T(I T) 1
=jI Tj b
2F1c a;b
c; T(I T) 1
:
Gauss Formula
35.7.7 2F1a;b
c;I
= m(c) m(c a b)
m(c a) m(c b),<(c);<(c a b)>1
2(m 1).
Re
ection Formula
35.7.8 2F1a;b
c;T
= m(c) m(c a b)
m(c a) m(c b)2F1a;b
a+b c+1
2(m+ 1);I T
,<(c);<(c a b)>1
2(m 1).
35.7(iii) Partial Di erential Equations
Letf:
!C(a) be orthogonally invariant , so thatf(T) is a symmetric function of t1;:::;tm, the eigenvalues of the
matrix argument T2
; (b) be analytic in t1;:::;tmin a neighborhood of T=0; (c) satisfy f(0) = 1. Subject to
the conditions (a){(c), the function f(T) = 2F1(a;b;c;T) is the unique solution of each partial di erential equation
35.7.9
tj(1 tj)@2F
@tj2 1
2mX
k=1
k6=jtk(1 tk)
tj tk@F
@tk+0
B@c 1
2(m 1)
a+b 1
2(m 3)
tj+1
2mX
k=1
k6=jtj(1 tj)
tj tk1
CA@F
@tj=abF;
forj= 1;:::;m .
772 Functions of Matrix Argument
Systems of partial di erential equations for the 0F1
(de ned inx35.8) and 1F1functions of matrix argument
can be obtained by applying (35.8.9) and (35.8.10) to
(35.7.9).
35.7(iv) Asymptotic Approximations
Butler and Wood (2002) applies Laplace's method
(x2.3(iii)) to (35.7.5) to derive uniform asymptotic ap-
proximations for the functions
35.7.10 2F1 a; b
c;T
and
35.7.11 2F1a;b
c;I 1T
as !1 . These approximations are in terms of ele-
mentary functions.
For other asymptotic approximations for Gaussian
hypergeometric functions of matrix argument, see Herz
(1955), Muirhead (1982, pp. 264{281, 290, 472, 563),
and Butler and Wood (2002).
35.8 Generalized Hypergeometric Functions
of Matrix Argument
35.8(i) De nition
Letpandqbe nonnegative integers; a1;:::;ap2C;
b1;:::;bq2C; bj+1
2(k+1)=2N, 1jq, 1km.
The generalized hypergeometric function pFqwith ma-
trix argument T2S, numerator parameters a1;:::;ap,
and denominator parameters b1;:::;bqis
35.8.1
pFqa1;:::;ap
b1;:::;bq;T
=1X
k=01
k!X
jj=k[a1][ap]
[b1][bq]Z(T):
Convergence Properties
If aj+1
2(k+ 1)2Nfor somej;ksatisfying 1jp,
1km, then the series expansion (35.8.1) termi-
nates.
Ifpq, then (35.8.1) converges for all T.
Ifp=q+ 1, then (35.8.1) converges absolutely for
jjTjj<1 and diverges for jjTjj>1.
Ifp > q + 1, then (35.8.1) diverges unless it termi-
nates.
35.8(ii) Relations to Other Functions
35.8.2 0F0
;T
= etr( T), T2S.
35.8.3
2F1a;b
b;T
=1F0a
;T
=jI Tj a,0<T<I.35.8.4
A(T) =1
m
+1
2(m+ 1)0F1
+1
2(m+ 1); T
,
T2S.
35.8(iii) 3F2Case
Kummer Transformation
Letc=b1+b2 a1 a2 a3. Then
35.8.5
3F2a1;a2;a3
b1;b2;I
= m(b2) m(c)
m(b2 a3) m(c+a3)
3F2b1 a1;b1 a2;a3
b1;c+a3;I
,
<(b2);<(c)>1
2(m 1).
Pfa {Saalschutz Formula
Leta1+a2+a3+1
2(m+ 1) =b1+b2; one of the ajbe
a negative integer; <(b1 a1),<(b1 a2),<(b1 a3),
<(b1 a1 a2 a3)>1
2(m 1). Then
35.8.63F2a1;a2;a3
b1;b2;I
= m(b1 a1) m(b1 a2)
m(b1) m(b1 a1 a2)
m(b1 a3) m(b1 a1 a2 a3)
m(b1 a1 a3) m(b1 a2 a3):
Thomae Transformation
Again, letc=b1+b2 a1 a2 a3. Then
35.8.7
3F2a1;a2;a3
b1;b2;I
= m(b1) m(b2) (c)
m(a1) m(c+a2) (c+a3)
3F2b1 a1;b2 a2;c
c+a2;c+a3;I
,
<(b1),<(b2),<(c)>1
2(m 1).
35.8(iv) General Properties
Value at T = 0
35.8.8 pFqa1;:::;ap
b1;:::;bq;0
= 1:
Con
uence
35.8.9lim
!1p+1Fqa1;:::;ap;
b1;:::;bq;
1T
=pFqa1;:::;ap
b1;:::;bq;T
;
35.8.10
lim
!1pFq+1a1;:::;ap
b1;:::;bq;
;
T
=pFqa1;:::;ap
b1;:::;bq;T
:
Applications 773
Invariance
35.8.11
pFqa1;:::;ap
b1;:::;bq;HTH 1
=pFqa1;:::;ap
b1;:::;bq;T
,
H2O(m).
Laplace Transform
35.8.12Z
etr( TX)jXj
1
2(m+1)
pFqa1;:::;ap
b1;:::;bq; X
dX
= m(
)jTj
p+1Fqa1;:::;ap;
b1;:::;bq; T 1
,
<(
)>1
2(m 1).
Euler Integral
35.8.13Z
0<X<IjXja1 1
2(m+1)jI Xjb1 a1 1
2(m+1)
pFqa2;:::;ap+1
b2;:::;bq+1;TX
dX
=1
Bm(b1 a1;a1)p+1Fq+1a1;:::;ap+1
b1;:::;bq+1;T
,
<(b1 a1);<(a1)>1
2(m 1).
35.8(v) Mellin{Barnes Integrals
Multidimensional Mellin{Barnes integrals are estab-
lished in Ding et al. (1996) for the functions pFqand
p+1Fpof matrix argument. A similar result for the
0F1function of matrix argument is given in Faraut and
Kor anyi (1994, p. 346). These multidimensional in-
tegrals reduce to the classical Mellin{Barnes integrals
(x5.19(ii)) in the special case m= 1.
See also Faraut and Kor anyi (1994, pp. 318{340).
Applications
35.9 Applications
In multivariate statistical analysis based on the mul-
tivariate normal distribution, the probability density
functions of many random matrices are expressible in
terms of generalized hypergeometric functions of matrix
argument pFq, withp2 andq1. See James (1964),
Muirhead (1982), Takemura (1984), Farrell (1985), and
Chikuse (2003) for extensive treatments.
For other statistical applications of pFqfunctions
of matrix argument see Perlman and Olkin (1980),
Groeneboom and Truax (2000), Bhaumik and Sarkar
(2002), Richards (2004) (monotonicity of power func-
tions of multivariate statistical test criteria), Binghamet al. (1992) (Procrustes analysis), and Phillips (1986)
(exact distributions of statistical test criteria). These
references all use results related to the integral formu-
las (35.4.7) and (35.5.8).
For applications of the integral representation
(35.5.3) see McFarland and Richards (2001, 2002) (sta-
tistical estimation of misclassi cation probabilities for
discriminating between multivariate normal popula-
tions). The asymptotic approximations of x35.7(iv) are
applied in numerous statistical contexts in Butler and
Wood (2002).
In chemistry, Wei and Eichinger (1993) expresses
the probability density functions of macromolecules in
terms of generalized hypergeometric functions of matrix
argument, and develop asymptotic approximations for
these density functions.
In the nascent area of applications of zonal polyno-
mials to the limiting probability distributions of sym-
metric random matrices, one of the most comprehensive
accounts is Rains (1998).
Computation
35.10 Methods of Computation
For small values of jjTjjthe zonal polynomial expan-
sion given by (35.8.1) can be summed numerically. For
largejjTjjthe asymptotic approximations referred to in
x35.7(iv) are available.
Other methods include numerical quadrature ap-
plied to double and multiple integral representations.
See Yan (1992) for the 1F1and 2F1functions of ma-
trix argument in the case m= 2, and Bingham et al.
(1992) for Monte Carlo simulation on O(m) applied to
a generalization of the integral (35.5.8).
Koev and Edelman (2006) utilizes combinatorial
identities for the zonal polynomials to develop compu-
tational algorithms for approximating the series expan-
sion (35.8.1). These algorithms are extremely ecient,
converge rapidly even for large values of m, and have
complexity linear in m.
35.11 Tables
Tables of zonal polynomials are given in James (1964)
forjj6, Parkhurst and James (1974) for jj12,
and Muirhead (1982, p. 238) for jj5. Each table
expresses the zonal polynomials as linear combinations
of monomial symmetric functions.
35.12 Software
Seehttp://dlmf.nist.gov/35.12 .
774 Functions of Matrix Argument
References
General References
The main references used in writing this chapter are
Herz (1955), James (1964), Muirhead (1982), Gross and
Richards (1987), and Richards (1992). For additional
bibliographic reading see Vilenkin and Klimyk (1992)
and Faraut and Kor anyi (1994).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections in this chapter. These sources supplement the
references that are quoted in the text.
x35.2 G arding (1947), Herz (1955, p. 479), Muirhead
(1982, p. 252). See also Siegel (1935), Bochner
and Martin (1948, pp. 90{92, 113{132).
x35.3 Wishart (1928), Ingham (1933), Gindikin (1964),
G arding (1947), Herz (1955), Olkin (1959).x35.4 James (1964), Muirhead (1982, Chapter 7),
Macdonald (1995, p. 425). See also Constan-
tine (1963), Maass (1971, pp. 64{71), Macdonald
(1995, pp. 388{439).
x35.5 Herz (1955). See also Bochner (1952), Gross and
Kunze (1976), Terras (1988, pp. 49{63), Butler
and Wood (2003).
x35.6 Koecher (1954), Muirhead (1978), Muirhead
(1982, pp. 264{266, 472{473), Herz (1955). For
(35.6.6) apply (35.2.3) and (35.6.5). See also
Shimura (1982).
x35.7 Herz (1955), Muirhead (1982, pp. 264{281, 290,
472), Faraut and Kor anyi (1994, pp. 337{340).
For (35.7.8) see Zheng (1997). See also Macdon-
ald (1990), Ding et al. (1996), Koornwinder and
Sprinkhuizen-Kuyper (1978), Gross and Richards
(1991).
x35.8 Gross and Richards (1987, 1991), Faraut and
Kor anyi (1994, pp. 318{340), Herz (1955), Muir-
head (1982, pp. 259{262), Macdonald (1990),
James (1964), Ding et al. (1996).
Chapter 36
Integrals with Coalescing Saddles
M. V. Berry1and C. J. Howls2
Notation 776
36.1 Special Notation . . . . . . . . . . . . . 776
Properties 776
36.2 Catastrophes and Canonical Integrals . . 776
36.3 Visualizations of Canonical Integrals . . . 778
36.4 Bifurcation Sets . . . . . . . . . . . . . . 781
36.5 Stokes Sets . . . . . . . . . . . . . . . . 782
36.6 Scaling Relations . . . . . . . . . . . . . 785
36.7 Zeros . . . . . . . . . . . . . . . . . . . 785
36.8 Convergent Series Expansions . . . . . . 787
36.9 Integral Identities . . . . . . . . . . . . . 78736.10 Di erential Equations . . . . . . . . . . . 788
36.11 Leading-Order Asymptotics . . . . . . . . 789
Applications 789
36.12 Uniform Approximation of Integrals . . . 789
36.13 Kelvin's Ship-Wave Pattern . . . . . . . . 790
36.14 Other Physical Applications . . . . . . . 791
Computation 792
36.15 Methods of Computation . . . . . . . . . 792
References 792
1H H Wills Physics Laboratory, Bristol, United Kingdom.
2School of Mathematics, University of Southampton, Southampton, United Kingdom.
Copyright c
2009 National Institute of Standards and Technology. All rights reserved.
775
776 Integrals with Coalescing Saddles
Notation
36.1 Special Notation
(For other notation see pp. xiv and 873.)
l;m;n integers.
k;t;s real or complex variables.
K codimension.
xfx1;x2;:::;xKg, wherex1;x2;:::;xKare real
parameters; also x1=x,x2=y,x3=zwhen
K3.
Ai, Bi Airy functions ( x9.2).
complex conjugate.
The main functions covered in this chapter are cus-
poid catastrophes K(t;x); umbilic catastrophes with
codimension three (E)(s;t;x), (H)(s;t;x); canonical
integrals K(x), (E)(x), (H)(x); di raction catastro-
phes K(x;k), (E)(x;k), (H)(x;k) generated by the
catastrophes. (There is no standard nomenclature for
these functions.)Properties
36.2 Catastrophes and Canonical Integrals
36.2(i) De nitions
Normal Forms Associated with Canonical Integrals:
Cuspoid Catastrophe with Codimension K
36.2.1 K(t;x) =tK+2+KX
m=1xmtm:
Special cases: K= 1, fold catastrophe ;K= 2, cusp
catastrophe ;K= 3, swallowtail catastrophe .
Normal Forms for Umbilic Catastrophes with Codimension
K= 3
36.2.2(E)(s;t;x) =s3 3st2+z(s2+t2) +yt+xs,
x=fx;y;zg;
(elliptic umbilic).
36.2.3(H)(s;t;x) =s3+t3+zst+yt+xs,
x=fx;y;zg;
(hyperbolic umbilic).
Canonical Integrals
36.2.4 K(x) =Z1
1exp(iK(t;x))dt:
36.2.5 (U)(x) =Z1
1Z1
1exp
i(U)(s;t;x)
dsdt , U = E ;H.
36.2.6 (E)(x) = 2p
/3 exp
i 4
27z3+1
3xz 1
4Z1exp(i=12)
1exp( 7i=12)exp
i
u6+ 2zu4+ (z2+x)u2+y2
12u2
du;
with the contour passing to the lower right of u= 0.
36.2.7 (E)(x) =4
31=3exp
i 2
27z3 1
3xz
exp
i
6
F+(x) + exp
i
6
F (x)
;
F(x) =Z1
0cos
ryexp
i
6
exp
2ir2zexp
i
3
Ai
32=3r2+ 3 1=3exp
i
3 1
3z2 x
dr:
36.2.8 (H)(x) = 4p
/6 exp
i 1
27z3+1
6z(y+x) +1
4
Z1exp(i=12)
1exp(5i=12)exp
i
2u6+ 2zu4+ 1
2z2+x+y
u2 (y x)2
24u2
du;
with the contour passing to the upper right of u= 0.
36.2.9 (H)(x) =2
31=3Z1exp(i=6)
1exp(5i=6)exp
i(s3+xs)
Aizs+y
31=3
ds:
36.2 Catastrophes and Canonical Integrals 777
Di raction Catastrophes
36.2.10
K(x;k) =p
kZ1
1exp(ikK(t;x))dt,k>0.
36.2.11
(U)(x;k) =kZ1
1Z1
1exp
ik(U)(s;t;x)
dsdt ,
U = E;H;k>0.
For more extensive lists of normal forms of catas-
trophes (umbilic and beyond) involving two variables
(\corank two") see Arnol'd (1972, 1974, 1975).
36.2(ii) Special Cases
36.2.12 0=pexp
i
4
:
1is related to the Airy function ( x9.2):
36.2.13 1(x) =2
31=3Aix
31=3
: 2is the Pearcey integral (Pearcey (1946)):
36.2.14
2(x) =P(x2;x1) =Z1
1exp
i(t4+x2t2+x1t)
dt:
(Other notations also appear in the literature.)
36.2.15
K(0)
=2
K+ 2 1
K+ 28
>><
>>:exp
i
2(K+ 2)
; K even,
cos
2(K+ 2)
; K odd:
36.2.16
1(0) = 1:54669; 2(0) = 1:67481 +i0:69373
3(0) = 1:74646; 4(0) = 1:79222 +i0:48022:
36.2.17@p
@x1p K(0) =2
K+ 2 p+ 1
K+ 2
cos
2p+ 1
K+ 2+p
, Kodd,
@2q+1
@x12q+1 K(0) = 0, Keven,
@2q
@x12q K(0) =2
K+ 2 2q+ 1
K+ 2
exp
i
22q+ 1
K+ 2+ 2q
, Keven.
36.2.18 (E)(0) =1
3p 1
6
= 3:28868;
(H)(0) =1
3 2 1
3
= 2:39224:
36.2.19
2(0;y) =
2r
jyj
2exp
iy2
8
exp
i
8
J 1/4y2
8
sign(y) exp
i
8
J1/4y2
8
:
For the Bessel function Jseex10.2(ii).
36.2.20
(E)(x;y;0) = 22(2
3)2=3<
Aix+iy
121=3
Bix iy
121=3
;
36.2.21 (H)(x;y;0) =42
32=3Aix
31=3
Aiy
31=3
:36.2(iii) Symmetries
36.2.22
2K(x0) = 2K(x),x0
2m+1= x2m+1,x0
2m=x2m.
36.2.23
2K+1(x0) =
2K+1(x),x0
2m+1=x2m+1,x0
2m= x2m.
36.2.24 (U)(x;y;z ) = (U)(x;y; z), U = E;H.
36.2.25 (E)(x; y;z) = (E)(x;y;z ):
36.2.26
(E)
1
2xp
3
2y;p
3
2x 1
2y;z
= (E)(x;y;z );
(rotation by2
3inx;yplane).
36.2.27 (H)(x;y;z ) = (H)(y;x;z ):
778 Integrals with Coalescing Saddles
36.3 Visualizations of Canonical Integrals
36.3(i) Canonical Integrals: Modulus
(a) Density plot. (b) 3D plot.
Figure 36.3.1 : Modulus of Pearcey integral j 2(x;y)j.
For additional gures see http://dlmf.nist.gov/36.3.i .
(a) Density plot. (b) 3D plot.
Figure 36.3.5 : Modulus of swallowtail canonical integral function j 3(x;y; 7:5)j.
For additional gures see http://dlmf.nist.gov/36.3.i .
36.3 Visualizations of Canonical Integrals 779
(a) Density plot. (b) 3D plot.
Figure 36.3.8 : Modulus of elliptic umbilic canonical integral function j (E)(x;y;4)j.
For additional gures see http://dlmf.nist.gov/36.3.i .
(a) Density plot. (b) 3D plot.
Figure 36.3.12 : Modulus of hyperbolic umbilic canonical integral function j (H)(x;y;3)j.
780 Integrals with Coalescing Saddles
36.3(ii) Canonical Integrals: Phase
In Figure 36.3.13(a) points of con
uence of phase contours are zeros of 2(x;y); similarly for other contour plots in
this subsection. In Figure 36.3.13(b) points of con
uence of all colors are zeros of 2(x;y); similarly for other density
plots in this subsection.
(a) Contour plot, at intervals of =4. (b) Density plot.
Figure 36.3.13 : Phase of Pearcey integral ph 2(x;y).
For additional gures see http://dlmf.nist.gov/36.3.ii .
(a) Contour plot. (b) Density plot.
Figure 36.3.17 : Phase of elliptic umbilic canonical integral ph (E)(x;y;4).
For additional gures see http://dlmf.nist.gov/36.3.ii .
36.4 Bifurcation Sets 781
(a) Contour plot. (b) Density plot.
Figure 36.3.21 : Phase of hyperbolic umbilic canonical integral ph (H)(x;y;3).
36.4 Bifurcation Sets
36.4(i) Formulas
Critical Points for Cuspoids
These are real solutions tj(x), 1jjmax(x)K+1,
of
36.4.1@
@tK(tj(x);x) = 0:
Critical Points for Umbilics
These are real solutions fsj(x);tj(x)g, 1j
jmax(x)4, of
36.4.2@
@s(U)(sj(x);tj(x);x) = 0;
@
@t(U)(sj(x);tj(x);x) = 0:
Bifurcation (Catastrophe) Set for Cuspoids
This is the codimension-one surface in xspace where
critical points coalesce, satisfying (36.4.1) and
36.4.3@2
@t2K(t;x) = 0:
Bifurcation (Catastrophe) Set for Umbilics
This is the codimension-one surface in xspace where
critical points coalesce, satisfying (36.4.2) and
36.4.4@2
@s2(U)(s;t;x)@2
@t2(U)(s;t;x)
@2
@s@t(U)(s;t;x)2
= 0:
Special Cases
K= 1, fold bifurcation set:
36.4.5 x= 0:K= 2, cusp bifurcation set:
36.4.6 27x2= 8y3:
K= 3, swallowtail bifurcation set:
36.4.7
x= 3t2(z+ 5t2); y = t(3z+ 10t2), 1<t<1.
Swallowtail self-intersection line:
36.4.8 y= 0; z0; x =9
20z2:
Swallowtail cusp lines (ribs):
36.4.9z0; x = 3
20z2;10y2= 4z3:
Elliptic umbilic bifurcation set (codimension three):
for xedz, the section of the bifurcation set is a three-
cusped astroid
36.4.10x=1
3z2( cos(2 ) 2 cos );
y=1
3z2(sin(2 ) 2 sin ), 0 2.
Elliptic umbilic cusp lines (ribs):
36.4.11 x+iy= z2exp 2
3im
,m= 0;1;2.
Hyperbolic umbilic bifurcation set (codimension
three):
36.4.12
x= 1
12z2(exp(2 )2 exp( ));
y= 1
12z2(exp( 2 )2 exp( )), 1 <1:
The + sign labels the cusped sheet; the sign labels
the sheet that is smooth for z6= 0 (see Figure 36.4.4).
Hyperbolic umbilic cusp line (rib):
36.4.13 x=y= 1
4z2:
For derivations of the results in this subsection see
Poston and Stewart (1978, Chapter 9).
782 Integrals with Coalescing Saddles
36.4(ii) Visualizations
Figure 36.4.1 : Bifurcation set of cusp catastrophe.
Figure 36.4.2 : Bifurcation set of swallowtail catastrophe.
Figure 36.4.3 : Bifurcation set of elliptic umbilic catas-
trophe.
Figure 36.4.4 : Bifurcation set of hyperbolic umbilic
catastrophe.
36.5 Stokes Sets
36.5(i) De nitions
Stokes sets are surfaces (codimension one) in x
space, across which K(x;k) or (U)(x;k) acquires an
exponentially-small asymptotic contribution (in k), as-
sociated with a complex critical point of Kor (U).
The Stokes sets are de ned by the exponential domi-nance condition:
36.5.1
<(K(tj(x);x) K(t(x);x)) = 0;
<
(U)(sj(x);tj(x);x) (U)(s(x);t(x);x)
= 0;
wherejdenotes a real critical point (36.4.1) or (36.4.2),
anddenotes a critical point with complex tors;t,
connected with jby a steepest-descent path (that is, a
path where< = constant) in complex tor (s;t) space.
36.5 Stokes Sets 783
In the following subsections, only Stokes sets involv-
ing at least one real saddle are included unless stated
otherwise.
36.5(ii) Cuspoids
K= 1. Airy Function
The Stokes set consists of the rays ph x=2=3 in the
complexx-plane.
K= 2. Cusp
The Stokes set is itself a cusped curve, connected to the
cusp of the bifurcation set:
36.5.2y3=27
4p
27 5
x2= 1:32403x2:
K= 3. Swallowtail
The Stokes set takes di erent forms for z= 0,z <0,
andz>0.
Forz= 0, the set consists of the two curves
36.5.3x=Bjyj4=3; B= 10 1=3
2x4=3
1
2x 2=3
;
wherexare the two smallest positive roots of the equa-
tion
36.5.4 80x5 40x4 55x3+ 5x2+ 20x 1 = 0;
and
36.5.5B = 1:69916; B += 0:33912:
Forz6= 0, the Stokes set is expressed in terms of
scaled coordinates
36.5.6 X=x=z2; Y =y=jzj3=2;
by
36.5.7X=9
20+ 20u4 Y2
20u2+ 6u2sign(z);
whereusatis es the equation
36.5.816u5 Y2
10u+ 4u3sign(z) 3
10jYjsign(z)
+ 4t5+ 2t3sign(z) +jYjt2= 0;
in which
36.5.9t= u+jYj
10u u2 3
10sign(z)1=2
:
Forz <0, there are two solutions u, provided that
jYj>(2
5)1=2. They generate a pair of cusp-edged sheets
connected to the cusped sheets of the swallowtail bifur-
cation set (x36.4).
Forz >0 the Stokes set has two sheets. The rst
sheet corresponds to x <0 and is generated as a solu-
tion of Equations (36.5.6){(36.5.9). The second sheet
corresponds to x > 0 and it intersects the bifurca-
tion set (x36.4) smoothly along the line generated by
X=X1= 6:95643,jYj=jY1j= 6:81337. ForjYj>Y1
the second sheet is generated by a second solution of
(36.5.6){(36.5.9), and for jYj< Y 1it is generated by
the roots of the polynomial equation
36.5.10 160u6+ 40u4=Y2:36.5(iii) Umbilics
Elliptic Umbilic Stokes Set (Codimension three)
This consists of three separate cusp-edged sheets con-
nected to the cusp-edged sheets of the bifurcation set,
and related by rotation about the z-axis by 2=3. One
of the sheets is symmetrical under re
ection in the plane
y= 0, and is given by
36.5.11x
z2= 1 12u2+ 8u y
z2 1
3 u
u 2
3 u1=2:
Hereuis the root of the equation
36.5.12
8u3 4u2 y
3z2 u
2
3 u1=2
=y2
6wz4 2w3 2w2;
with
36.5.13w=u 2
3+
2
3 u2+ y
6z2 2
3 u
u1=2!1=2
;
and such that
36.5.14 0<u<1
6:
Hyperbolic Umbilic Stokes Set (Codimension three)
This consists of a cusp-edged sheet connected to the
cusp-edged sheet of the bifurcation set and intersecting
the smooth sheet of the bifurcation set. With coordi-
nates
36.5.15X= (x y)=z2; Y =1
2+
(x+y)=z2
;
the intersection lines with the bifurcation set are gen-
erated byjXj=X2= 0:45148,Y=Y2= 0:59693.
De ne
36.5.16Y(u;X) = 8u 24u2+Xu 1
6
u
u 1
31=2;
f(u;X) = 16u3 4u2 1
6jXju
u 1
31=2
:
WhenjXj>X 2the Stokes set YS(X) is given by
36.5.17 YS(X) =Y(u;jXj);
whereuis the root of the equation
36.5.18 f(u;X) =f( u+1
3;X);
such thatu >1
3. This part of the Stokes set connects
two complex saddles.
Alternatively, when jXj<X 2
36.5.19 YS(X) =Y( u; jXj);
whereuis the positive root of the equation
36.5.20 f( u;X) =X2
12w+ 4w3 2w2;
in which
36.5.21w= (1
3+u)
1
1 jXj
12u1=2(1
3+u)3=21=2!
:
784 Integrals with Coalescing Saddles
36.5(iv) Visualizations
In Figures 36.5.1{36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves
(bifurcation sets). Red and blue numbers in each region correspond, respectively, to the numbers of real and complex
critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. In Figure
36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. The distribution of
real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact
that there are four real saddles in the inner regions.
Figure 36.5.1 : Cusp catastrophe.
Figure 36.5.2 : Swallowtail catastrophe with z<0.
Figure 36.5.3 : Swallowtail catastrophe with z= 0.
Figure 36.5.4 : Swallowtail catastrophe with z>0.
36.6 Scaling Relations 785
Figure 36.5.5 : Elliptic umbilic catastrophe with z=
constant.
Figure 36.5.6 : Hyperbolic umbilic catastrophe with z=
constant.
For additional gures see http://dlmf.nist.gov/36.5.iv .
36.6 Scaling Relations
Di raction Catastrophe Scaling
36.6.1 K(x;k) =k K K(y(k));
(U)(x;k) =k (U) (U)
y(U)(k)
;
where
36.6.2
cuspoids: y(k) = (x1k
1K;x2k
2K;:::;xKk
KK);
umbilics: y(U)(k) =
xk2=3;yk2=3;zk1=3
:
Indices fork-Scaling of Magnitude of Kor (U)
(Singularity Index)
36.6.3 cuspoids: K=K
2(K+ 2);umbilics: (U)=1
3:
Indices fork-Scaling of Coordinates xm
36.6.4cuspoids:
mK= 1 m
K+ 2;
umbilics:
(U)
x=2
3;
(U)
y=2
3;
(U)
z=1
3:
Indices fork-Scaling of xHypervolume
36.6.5cuspoids:
K=KX
m=1
mK=K(K+ 3)
2(K+ 2);
umbilics:
(U)=3X
m=1
(U)
m=5
3:Table 36.6.1 : Special cases of scaling exponents for cus-
poids.
singularity K K
1K
2K
3K
K
fold 11
62
3 2
3
cusp 21
43
41
2 5
4
swallowtail 33
104
53
52
59
5
For the results in this section and more extensive
lists of exponents see Berry (1977) and Var cenko (1976).
36.7 Zeros
36.7(i) Fold Canonical Integral
This is the Airy function Ai ( x9.2).
36.7(ii) Cusp Canonical Integral
This is (36.2.4) and (36.2.1) with K= 2.
The zeros in Table 36.7.1 are points in the x= (x;y)
plane, where ph 2(x) is undetermined. All zeros have
y<0, and fall into two classes. Inside the cusp, that is,
forx2<8jyj3=27, the zeros form pairs lying in curved
rows. Close to the y-axis the approximate location of
these zeros is given by
36.7.1ym= p
2(2m+ 1), m= 1;2;3;:::;
x
m;n=r2
ym
2n+1
2+ ( 1)m1
21
4
,
m= 1;2;3;:::,n= 0;1;2;:::.
786 Integrals with Coalescing Saddles
Table 36.7.1 : Zeros of cusp di raction catastrophe to 5D.
Zerosx
y
inside, and zerosx
y
outside, the cusp x2=8
27jyj3.
0:52768
4:37804 2:35218
1:74360
1:41101
5:55470 2:36094
5:52321 4:42707
3:05791
0:43039
6:64285 3:06389
6:44624 3:95806
6:40312 6:16185
4:03551
1:21605
7:49906 2:02922
7:48629 4:56537
7:19629 5:42206
7:14718 7:72352
4:84817
0:38488
8:31916 2:71193
8:22315 3:49286
8:20326 5:96669
7:85723 6:79538
7:80456 9:17308
5:55831
More general asymptotic formulas are given in
Kaminski and Paris (1999). Just outside the cusp, that
is, forx2>8jyj3=27, there is a single row of zeros on
each side. With n= 0;1;2;:::, they are located ap-
proximately at
36.7.2xn=8
271=2
jynj3=2(1 +n);
yn= 3(8n+ 5)
9 + 8n1=2
;
wherenis the real solution of
36.7.3
3(8n+ 5)
9 + 8n3=2
n=27
163
21=2
ln1
n
+ 3 ln3
2
:
For a more extensive asymptotic analysis and further
tabulations, see Kaminski and Paris (1999).
36.7(iii) Elliptic Umbilic Canonical Integral
This is (36.2.5) with (36.2.2). The zeros are lines in
x= (x;y;z ) space where ph (E)(x) is undetermined.
Deep inside the bifurcation set, that is, inside the three-
cusped astroid (36.4.10) and close to the part of the z-
axis that is far from the origin, the zero contours form
an array of rings close to the planes
36.7.4zn=3(1
4(2n 1
2))1=3
= 3:48734(n 1
4)1=3,n= 1;2;3;:::.
Nearz=zn, and for small xandy, the modulus
j (E)(x)jhas the symmetry of a lattice with a rhombo-
hedral unit cell that has a mirror plane and an inversethreefold axis whose zandxrepeat distances are given
by
36.7.5 z=9
2z2n;x=6
zn:
The zeros are approximated by solutions of the equation
36.7.6exp
2iz zn
z+2x
x
2 exp 6ix
x
cos
2p
3y
x!
+ 1!
=p
3:
The rings are almost circular (radii close to ( x)=9
and varying by less than 1%), and almost
at (devi-
ating from the planes znby at most ( z)=36). Away
from thez-axis and approaching the cusp lines (ribs)
(36.4.11), the lattice becomes distorted and the rings are
deformed, eventually joining to form \hairpins" whose
arms become the pairs of zeros (36.7.1) of the cusp
canonical integral. In the symmetry planes (e.g., y= 0),
the number of rings in the mth row, measured from the
origin and before the transition to hairpins, is given by
36.7.7 nmax(m) =256
13m 269
52
:
Outside the bifurcation set (36.4.10), each rib is
anked
by a series of zero lines in the form of curly \antelope
horns" related to the \outside" zeros (36.7.2) of the cusp
canonical integral. There are also three sets of zero lines
in the plane z= 0 related by 2 =3 rotation; these are
zeros of (36.2.20), whose asymptotic form in polar co-
ordinates (x=rcos; y=rsin) is given by
36.7.8r= 3
(2n 1)
4jsin 3
2
j!2=3
(1 +O
n 1
),n!1 .
36.8 Convergent Series Expansions 787
36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals
The zeros of these functions are curves in x= (x;y;z ) space; see Nye (2007) for 3and Nye (2006) for (H).
36.8 Convergent Series Expansions
36.8.1 K(x) =2
K+ 21X
n=0exp
i(2n+ 1)
2(K+ 2)
2n+ 1
K+ 2
a2n(x), Keven,
K(x) =2
K+ 21X
n=0incos(n(K+ 1) 1)
2(K+ 2)
n+ 1
K+ 2
an(x), Kodd,
where
36.8.2 a0(x) = 1; an+1(x) =i
n+ 1min(n;K 1)X
p=0(p+ 1)xp+1an p(x), n= 0;1;2;:::.
For multinomial power series for K(x), see Connor and Curtis (1982).
36.8.332=3
42 (H)
31=3x
= Ai(x) Ai(y)1X
n=0( 3 1=3iz)ncn(x)cn(y)
n!+ Ai(x) Ai0(y)1X
n=2( 3 1=3iz)ncn(x)dn(y)
n!
+ Ai0(x) Ai(y)1X
n=2( 3 1=3iz)ndn(x)cn(y)
n!+ Ai0(x) Ai0(y)1X
n=1( 3 1=3iz)ndn(x)dn(y)
n!;
and
36.8.4 (E)(x) = 222
32=31X
n=0
i(2=3)2=3zn
n!<
fnx+iy
121=3;x iy
121=3
;
where
36.8.5
fn(;)
=cn()cn() Ai() Bi() +cn()dn() Ai() Bi0() +dn()cn() Ai0() Bi() +dn()dn() Ai0() Bi0();
with asterisks denoting complex conjugates, and
36.8.6 c0(t) = 1; d 0(t) = 0; cn+1(t) =c0
n(t) +tdn(t); dn+1(t) =cn(t) +d0
n(t):
36.9 Integral Identities
36.9.1 j 1(x)j2= 25=3Z1
0 1
22=3(3u2+x)
du;
equivalently,
36.9.2 (Ai(x))2=22=3
Z1
0Ai
22=3(u2+x)
du:
36.9.3 j 1(x)j2=r
8
3Z1
0u 1=2cos
2u(x+u2) +1
4
du:
36.9.4 j 2(x;y)j2=Z1
0
14u3+ 2uy+x
u1=3
+ 14u3+ 2uy x
u1=3du
u1=3:
36.9.5 j 2(x;y)j2= 2Z1
0cos(2xu) 1
2u2=3(y+ 2u2)du
u1=3:
36.9.6j 3(x;y;z )j2= 24=5Z1
1 3
24=5(x+ 2uy+ 3u2z+ 5u4);0;22=5(z+ 10u2)
du:
36.9.7j 3(x;y;z )j2=27=4
51=4Z1
0<
e2iu(u4+zu2+x) 2
27=4
51=4yu3=4;r
2u
5(3z+ 10u2)!!
du
u1=4:
788 Integrals with Coalescing Saddles
36.9.8 (H)(x;y;z ) 2
= 822
91=3Z1
1Z1
1Ai 4
31=3
(x+zv+ 3u2)!
Ai 4
31=3
(y+zu+ 3v2)!
dudv:
36.9.9 (E)(x;y;z ) 2
=82
32=3Z1
0Z2
0<
Ai1
31=3
x+iy+ 2zuexp(i) + 3u2exp( 2i)
Bi1
31=3
x iy+ 2zuexp( i) + 3u2exp(2i)
udud:
For these results and also integrals over doubly-in nite intervals see Berry and Wright (1980). This reference also
provides a physical interpretation in terms of Lagrangian manifolds and Wigner functions in phase space.
36.10 Di erential Equations
36.10(i) Equations for K(x)
In terms of the normal form (36.2.1) the K(x) satisfy
the operator equation
36.10.1 0
K
i@
@x1;x
K(x) = 0;
or explicitly,
36.10.2
@K+1 K(x)
@x1K+1+KX
m=1( i)m K 2mxm
K+ 2@m 1 K(x)
@x1m 1
= 0:
Special Cases
K= 1, fold: (36.10.1) becomes Airy's equation ( x9.2(i))
36.10.3@2 1
@x2 x
3 1= 0:
K= 2, cusp:
36.10.4@3 2
@x3 1
2y@ 2
@x i
4x 2= 0:
K= 3, swallowtail:
36.10.5@4 3
@x4 3
5z@2 3
@x2 2i
5y@ 3
@x+1
5x 3= 0:
36.10(ii) Partial Derivatives with Respect to
thexn
36.10.6
@ln K
@xmln=in(l m)@mn K
@xlmn, 1mK, 1lK.
Special Cases
K= 1, fold: (36.10.6) is an identity.
K= 2, cusp:
36.10.7@2n 2
@x2n=in@n 2
@yn:K= 3, swallowtail:
36.10.8@2n 3
@x2n=in@n 3
@yn;
36.10.9@3n 3
@x3n= ( 1)n@n 3
@zn;
36.10.10@3n 3
@y3n=in@2n 3
@z2n:
36.10(iii) Operator Equations
In terms of the normal forms (36.2.2) and (36.2.3), the
(U)(x) satisfy the following operator equations
36.10.11(U)
s
i@
@x; i@
@y;x
(U)(x) = 0;
(U)
t
i@
@x; i@
@y;x
(U)(x) = 0;
where
36.10.12(U)
s(s;t;x) =@
@s(U)(s;t;x);
(U)
t(s;t;x) =@
@t(U)(s;t;x):
Explicitly,
36.10.13 6@2 (E)
@x@y 2iz@ (E)
@y+y (E)= 0;
36.10.14
3@2 (E)
@x2 @2 (E)
@y2
+ 2iz@ (H)
@x x (E)= 0:
36.10.15 3@2 (H)
@x2+iz@ (H)
@y x (H)= 0;
36.10.16 3@2 (H)
@y2+iz@ (H)
@x y (H)= 0:
36.10(iv) Partial z-Derivatives
36.10.17 i@ (E)
@z=@2 (E)
@x2+@2 (E)
@y2;
36.10.18 i@ (H)
@z=@2 (H)
@x@y:
Equation (36.10.17) is the paraxial wave equation .
36.11 Leading-Order Asymptotics 789
36.11 Leading-Order Asymptotics
With real critical points (36.4.1) ordered so that
36.11.1 t1(x)<t2(x)<<tjmax(x);
and far from the bifurcation set, the cuspoid canonical integrals are approximated by
36.11.2 K(x) =p
2jmax(x)X
j=1exp
i
K(tj(x);x) +1
4( 1)j+K+1 @2K(tj(x);x)
@t2 1=2
(1 +o(1)):
Asymptotics along Symmetry Lines
36.11.3 2(0;y) =(p
/y
exp 1
4i
+o(1)
; y !+1;p
/jyjexp
1
4i
1 +ip
2 exp
1
4iy2
+o(1)
; y! 1:
36.11.4 3(x;0;0) =p
2
(5jxj3)1=8(
exp
2p
2(x/5)5=4
cos
2p
2(x/5)5=4 1
8
+o(1)
; x!+1;
cos
4(jxj/5)5=4 1
4
+o(1); x ! 1:
36.11.5 3(0;y;0) =
3(0; y;0) = exp 1
4ip
/y
1 (i=p
3) exp
3
2i(2y/5)5=3
+o(1)
, y!+1.
36.11.6 3(0;0;z) = 1
3
jzj1=3p
3+8
><
>:o(1); z !+1;
2p51=4
(3jzj)3=4
cos
2
33jzj
55=2
1
4!
+o(1)!
; z! 1:
36.11.7 (E)(0;0;z) =
z
i+p
3 exp4
27iz3
+o(1)
, z!1 ,
36.11.8 (H)(0;0;z) =2
z
1 ip
3exp1
27iz3
+o(1)
, z!1 .
Applications
36.12 Uniform Approximation of Integrals
36.12(i) General Theory for Cuspoids
The canonical integrals (36.2.4) provide a basis for uni-
form asymptotic approximations of oscillatory integrals.
In the cuspoid case (one integration variable)
36.12.1I(y;k) =Z1
1exp(ikf(u;y))g(u;y)du;
wherekis a large real parameter and y=fy1;y2;:::g
is a set of additional (nonasymptotic) parameters. As yvaries as many as K+1 (real or complex) critical points
of the smooth phase function fcan coalesce in clusters
of two or more. The function ghas a smooth ampli-
tude. Also, fis real analytic, and @K+2f.
@uK+2>0
for all ysuch that all K+ 1 critical points coincide. If
@K+2f.
@uK+2<0, then we may evaluate the complex
conjugate of Ifor real values of yandg, and obtain I
by conjugation and analytic continuation. The critical
pointsuj(y), 1jK+ 1, are de ned by
36.12.2@
@uf(uj(y);y) = 0:
The leading-order uniform asymptotic approxima-
tion is given by
36.12.3 I(y;k) =exp(ikA(y))
k1=(K+2)KX
m=0am(y)
km=(K+2)
m;0 (1 m;0)i@
@zm
K(z(y;k))
1 +O1
k
;
790 Integrals with Coalescing Saddles
whereA(y),z(y;k),am(y) are as follows. De ne a
mappingu(t;y) by relating f(u;y) to the normal form
(36.2.1) of K(t;x) in the following way:
36.12.4f(u(t;y);y) =A(y) + K(t;x(y));
with theK+ 1 functions A(y) and x(y) determined by
correspondence of the K+1 critical points of fand K.
Then
36.12.5f(uj(y);y) =A(y) + K(tj(x(y));x(y));
wheretj(x), 1jK+ 1, are the critical points of
K, that is, the solutions (real and complex) of (36.4.1).
Correspondence between the uj(y) and thetj(x) is es-
tablished by the order of critical points along the real
axis when yandxare such that these critical points
are all real, and by continuation when some or all of the
critical points are complex. The branch for x(y) is such
thatxis real when yis real. In consequence,
36.12.6 A(y) =f(u(0;y);y);
36.12.7z(y;k) =fz1(y;k);z2(y;k);:::;zK(y;k)g;
zm(y;k) =xm(y)k1 (m=(K+2));
36.12.8
am(y) =K+1X
n=1Pmn(y)Gn(y)
(tn(x(y)))m+1K+1Q
l=1
l6=n(tn(x(y)) tl(x(y)));
where
36.12.9Pmn(y) = (tn(x(y)))K+1
+KX
l=m+2l
K+ 2xl(y)(tn(x(y)))l 1;and
36.12.10
Gn(y) =g(tn(y);y)s
@2K(tn(x(y));x(y))
@t2
@2f(un(y))
@u2:
In (36.12.10), both second derivatives vanish when crit-
ical points coalesce, but their ratio remains nite. The
square roots are real and positive when yis such that all
the critical points are real, and are de ned by analytic
continuation elsewhere. The quantities am(y) are real
for real ywhengis real analytic.
This technique can be applied to generate a hierar-
chy of approximations for the di raction catastrophes
K(x;k) in (36.2.10) away from x= 0, in terms of
canonical integrals J((x;k)) forJ < K . For exam-
ple, the di raction catastrophe 2(x;y;k) de ned by
(36.2.10), and corresponding to the Pearcey integral
(36.2.14), can be approximated by the Airy function
1((x;y;k)) whenkis large, provided that xandy
are not small. For details of this example, see Paris
(1991).
For further information see Berry and Howls (1993).
36.12(ii) Special Case
ForK= 1, with a single parameter y, let the two criti-
cal points of f(u;y) be denoted by u(y), withu+>u
for those values of yfor which these critical points are
real. Then
36.12.11I(y;k) =1=4p
2
k1=3exp
ikef
g+p
f00
++g p
f00
!
Ai
k2=3
1 +O1
k
i
g+p
f00
+ g p
f00
!
Ai0
k2=3
k1=31=2
1 +O1
k!
;
where
36.12.12ef=1
2(f(u+(y);y) +f(u (y);y));
g=g(u(y);y); f00
=@2
@u2f(u(y);y);
= 3
4(f(u (y);y) f(u+(y);y))2=3:
For Ai and Ai0seex9.2. Branches are chosen so that
is real and positive if the critical points are real, or real
and negative if they are complex. The coecients of Ai
and Ai0are real ifyis real and gis real analytic. Also,
1=4=p
f00
+and 1=4=p
f00
are chosen to be positive
real whenyis such that both critical points are real,and by analytic continuation otherwise.
36.12(iii) Additional References
For further information concerning integrals with sev-
eral coalescing saddle points see Arnol'd et al. (1988),
Berry and Howls (1993, 1994), Bleistein (1967), Duis-
termaat (1974), Ludwig (1966), Olde Daalhuis (2000),
and Ursell (1972, 1980).
36.13 Kelvin's Ship-Wave Pattern
A ship moving with constant speed Von deep water
generates a surface gravity wave. In a reference frame
36.14 Other Physical Applications 791
where the ship is at rest we use polar coordinates rand
with = 0 in the direction of the velocity of the water
relative to the ship. Then with gdenoting the acceler-
ation due to gravity, the wave height is approximately
given by
36.13.1z( ;) =Z=2
=2cos
cos(+ )
cos2
d;
where
36.13.2 =gr
V2:
The integral is of the form of the real part of (36.12.1)
withy= ,u=,g= 1,k=, and
36.13.3 f(; ) = cos(+ )
cos2:
When > 1, that is, everywhere except close to
the ship, the integrand oscillates rapidly. There are two
stationary points, given by
36.13.4+( ) =1
2(arcsin(3 sin ) );
( ) =1
2( arcsin(3 sin )):
These coalesce when
36.13.5j j= c= arcsin 1
3
= 19:47122:
This is the angle of the familiar V-shaped wake. The
wake is a caustic of the \rays" de ned by the disper-
sion relation (\Hamiltonian") giving the frequency !as
a function of wavevector k:
36.13.6 !(k) =p
gk+Vk:
Herek=jkj, and Vis the ship velocity (so that
V =jVj).
The disturbance z(; ) can be approximated by the
method of uniform asymptotic approximation for the
case of two coalescing stationary points (36.12.11), us-
ing the fact that ( ) are real forj j< cand complex
forj j> c. (See alsox2.4(v).) Then with the de ni-
tions (36.12.12), and the real functions
36.13.7u( ) =s
1=2( )
2
1p
f00
+( )+1p
f00
( )!
;
v( ) =s
1
21=2( )
1p
f00
+( ) 1p
f00
( )!
;
the disturbance is
36.13.8
z(; ) = 2
1=3u( ) cos
ef( )
Ai
2=3( )
(1 +O(1=))
+ 2=3v( ) sin
ef( )
Ai0
2=3( )
(1 +O(1=))
, !1 .
See Figure 36.13.1.
Figure 36.13.1 : Kelvin's ship wave pattern, computed
from the uniform asymptotic approximation (36.13.8),
as a function of x=cos ,y=sin .
For further information see Lord Kelvin (1891, 1905)
and Ursell (1960, 1994).
36.14 Other Physical Applications
36.14(i) Caustics
The physical manifestations of bifurcation sets are caus-
tics. These are the structurally stable focal singularities
(envelopes) of families of rays, on which the intensi-
ties of the geometrical (ray) theory diverge. Di raction
catastrophes describe the (linear) wave amplitudes that
smooth the geometrical caustic singularities and deco-
rate them with interference patterns. See Berry (1969,
1976, 1980, 1981), Kravtsov (1964, 1988), and Ludwig
(1966).
36.14(ii) Optics
Di raction catastrophes describe the connection be-
tween ray optics and wave optics. Applications include
twinkling starlight, focusing of sunlight by rippling wa-
ter (e.g., swimming-pool patterns), and water-droplet
\lenses" (e.g., rainbows). See Adler et al. (1997), Berry
and Upstill (1980), Marston (1992, 1999), Nye (1999),
Walker (1983, 1988, 1989).
36.14(iii) Quantum Mechanics
Di raction catastrophes describe the \semiclassical"
connections between classical orbits and quantum wave-
functions, for integrable (non-chaotic) systems. Appli-
cations include scattering of elementary particles, atoms
792 Integrals with Coalescing Saddles
and molecules from particles and surfaces, and chemi-
cal reactions. See Berry (1966, 1975), Connor (1974,
1976), Connor and Farrelly (1981), Trinkaus and Drep-
per (1977), and Uzer et al. (1983).
36.14(iv) Acoustics
Applications include the re
ection of ultrasound pulses,
and acoustical waveguides. See Chapman (1999),
Frederickson and Marston (1992, 1994), and Kravtsov
(1968).
Computation
36.15 Methods of Computation
36.15(i) Convergent Series
Close to the origin x= 0 of parameter space, the series
inx36.8 can be used.
36.15(ii) Asymptotics
Far from the bifurcation set, the leading-order asymp-
totic formulas of x36.11 reproduce accurately the form
of the function, including the geometry of the zeros de-
scribed inx36.7. Close to the bifurcation set but far
fromx= 0, the uniform asymptotic approximations of
x36.12 can be used.
36.15(iii) Integration along Deformed Contour
Direct numerical evaluation can be carried out along
a contour that runs along the segment of the real t-
axis containing all real critical points of and is de-
formed outside this range so as to reach in nity along
the asymptotic valleys of exp( i). (For the umbilics,
representations as one-dimensional integrals ( x36.2) are
used.) For details, see Connor and Curtis (1982) and
Kirk et al. (2000). There is considerable freedom in the
choice of deformations.
36.15(iv) Integration along Finite Contour
This can be carried out by direct numerical evaluation
of canonical integrals along a nite segment of the real
axis including all real critical points of , with contribu-
tions from the contour outside this range approximated
by the rst terms of an asymptotic series associated with
the endpoints. See Berry et al. (1979).
36.15(v) Di erential Equations
For numerical solution of partial di erential equations
satis ed by the canonical integrals see Connor et al.
(1983).References
General References
There is no single source covering the material in this
chapter. An overview of some of the mathematical anal-
ysis is given in Arnol'd (1975, 1986). Many physical ap-
plications can be found in Poston and Stewart (1978).
For applications to wave physics, especially optics, see
Berry and Upstill (1980).
Sources
The following list gives the references or other indica-
tions of proofs that were used in constructing the various
sections of this chapter. These sources supplement the
references that are quoted in the text.
x36.2 The convergence of the oscillatory integrals
(36.2.4){(36.2.11) can be con rmed by rotating
the integration paths in the complex plane. For
(36.2.6) see Berry et al. (1979). For (36.2.7)
shift thesvariable in (36.2.5) (with (36.2.2)) to
remove the quadratic term, integrate, and then
deform the contour of the remaining tintegra-
tion. For (36.2.8) see Berry and Howls (1990).
For (36.2.9) integrate (36.2.5) (with (36.2.3)) with
respect to t. For (36.2.12) and (36.2.13) use
(4.10.11) and (9.5.4), respectively. For (36.2.15)
and (36.2.17) use (5.9.1). For (36.2.18) combine
(36.2.6), (36.2.8), and (5.9.1) For (36.2.19) use
(12.5.1) and (12.14.13). For (36.2.20) see Trinkaus
and Drepper (1977). For (36.2.21) use (36.2.9).
Eqs. (36.2.22){(36.2.27) follow from the de ni-
tions given inx36.2(i).
xx36.3, 36.4 The graphics were generated by the au-
thors.
x36.5 Wright (1980) and Berry and Howls (1990). The
common strategy employed in deriving the for-
mulas in this section involves using the critical-
point condition (36.4.1) to reduce the order of the
catastrophe polynomials in (36.2.1), then solving
(36.5.1) for the imaginary part of the complex crit-
ical point in terms of the value of the real criti-
cal point, which is itself determined by (36.4.1)
and then used to generate the Stokes sets para-
metrically. For (36.5.11){(36.5.21) we also use
the exponents in the representations (36.2.6) and
(36.2.8). The graphics were generated by the au-
thors. For Figures 36.5.2{36.5.6, Eqs. (36.5.11){
(36.5.21) were used in parametric form x=x(y),
and checked against the numerical computations
References 793
in Berry and Howls (1990) (which were based di-
rectly on the de nitions given in x36.5(i)).
x36.7 Berry et al. (1979). (36.7.2) and (36.7.3) may
be derived by setting to zero the stationary-phase
approximation (x2.3(iv)) of the Pearcey integral
2(x;y) just outside the caustic; this involves one
real saddle and one complex saddle. Table 36.7.1
was computed by the authors.
x36.8 Connor (1973) and Connor et al. (1983). For
(36.8.1), in the integral (36.2.4) retain the high-
est power of tin (36.2.1) in the exponent, expand
the rest of the exponential as a power series in
t, and evaluate the resulting integrals in terms
of gamma functions. For (36.8.3), in the integral
(36.2.5) with the polynomial (36.2.3) expand the
z-dependent part of the exponential in powers of z,
and then repeatedly use the di erential equation
(9.2.1) to express higher derivatives of the Airy
function in terms of Ai and Ai0. For (36.8.4), inthe integral (36.2.5) with the polynomial (36.2.2)
expand the z-dependent part of the exponential in
powers ofz, and then repeatedly use (9.2.1), and
(36.2.20).
x36.10 For (36.10.1) to (36.10.10) see Connor et al.
(1983). (36.10.11) to (36.10.18) are derived by
repeated di erentiations with respect to x,y, or
z, in combinations that generate exact derivatives
of the exponents in (36.2.5).
x36.11 The formulas in this section are derived by the
method of stationary phase, applied to the real
critical points of the integral representations in
x36.2. Seex2.3(iv) and also Berry and Howls
(1991). For (36.11.4) the integral is exponentially
small when x>0 and the dominant contribution
is from a critical point o the real axis.
x36.12 Berry and Howls (1993).
x36.13 The gure was generated by the authors.
Bibliography
A. S. Abdullaev (1985). Asymptotics of solutions of the gen-
eralized sine-Gordon equation, the third Painlev e equa-
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Notations
!
n!q:q-factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
ab: vector dot (or scalar) product . . . . . . . . . . . . . . . 9
fg: convolution for Fourier transforms . . . . . . . . . 27
fg: convolution for Laplace transforms . . . . . . . . . 28
fg: convolution for Mellin transforms . . . . . . . . . . 29
fg: convolution product . . . . . . . . . . . . . . . . . . . . . . . 53
GH: Cartesian product of groups GandH. . .570
ab: vector cross product. . . . . . . . . . . . . . . . . . . . . . . .9
=
S1=S2: set of all elements of S1modulo elements of
S2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
asymptotic equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
r
del operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
r2
Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Laplacian for cylindrical coordinates. . . . . . . . . . . . . . .7
Laplacian for polar coordinates . . . . . . . . . . . . . . . . . . . . 7
Laplacian for spherical coordinates . . . . . . . . . . . . . . . . 8
rf
gradient of di erentiable scalar function f. . . . . . . . 10
rF
curl of vector-valued function F. . . . . . . . . . . . . . . . . . 10
rF
divergence of vector-valued function F. . . . . . . . . . . .10
Rb
a
Cauchy principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
R(b+)
a
loop integral in C: path begins at a, encirclesbonce
in the positive sense, and returns to a. . . . . . . . . 139
R(1+;0+;1 ;0 )
P
Pochhammer's loop integral . . . . . . . . . . . . . . . . . . . . . 142R
dqx
q-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422z
complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
jzj
modulus (or absolute value) . . . . . . . . . . . . . . . . . . . . . . 15
kak
magnitude of vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
kAkp
p-norm of a matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
kxk2
Euclidean norm of a vector. . . . . . . . . . . . . . . . . . . . . . .74
kxkp
p-norm of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
kxk1
in nity (or maximum) norm of a vector . . . . . . . . . . 74
f(c+)
limit on right (or from above) . . . . . . . . . . . . . . . . . . . . . 4
f(c )
limit on left (or from below). . . . . . . . . . . . . . . . . . . . . . .4
f[n](z)
nthq-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
xn
falling factorial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .618
xn
rising factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
b0+a1
b1+a2
b2+
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
(njP)
Jacobi symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
(njp)
Legendre symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
(a;q)n
q-factorial (or q-shifted factorial) . . . . . . . . . . . 145, 420
(a;q)
q-shifted factorial (generalized). . . . . . . . . . . . . . . . . .420
(a;q)1
q-shifted factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
(a1;a2;:::;ar;q)n
multipleq-shifted factorial . . . . . . . . . . . . . . . . . . . . . . 420
(a1;a2;:::;ar;q)1
multipleq-shifted factorial . . . . . . . . . . . . . . . . . . . . . . 420
873
874 Notations
(j1m1j2m2jj1j2j3 m3)
Clebsch{Gordan coecient. . . . . . . . . . . . . . . . . . . . . .758 m
n
binomial coecient . . . . . . . . . . . . . . . . . . . . . . . . . . . 2, 619 n1+n2++nk
n1;n2;:::;nk
multinomial coecient . . . . . . . . . . . . . . . . . . . . . . . . . . 620j1j2j3
m1m2m3
3jsymbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .758
h; i
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
hf; i
tempered distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
h; i
Dirac delta distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 36
n
k
Eulerian number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
[z0;z1;:::;zn]
divided di erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
[a]
partitional shifted factorial . . . . . . . . . . . . . . . . . . . . . . 769
[p=q]f
Pad e approximant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98n
k
Stirling cycle number . . . . . . . . . . . . . . . . . . . . . . . . . . . 631n
m
q
q-binomial coecient (or Gaussian polynomial)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .421, 627a1+a2++an
a1;a2;:::;an
q
q-multinomial coecient . . . . . . . . . . . . . . . . . . . . . . . . 634
f:::g
sequence, asymptotic sequence (or scale), or enumer-
able set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
fz;g
Schwarzian derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27j1j2j3
l1l2l3
6jsymbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7618
<
:j11j12j13
j21j22j23
j31j32j339
=
;
9jsymbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .763
A
Glaisher's constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
A(z)
Anger{Weber function . . . . . . . . . . . . . . . . . . . . . . . . . . 295A(T)
Bessel function of matrix argument ( rst kind) . . 769
An(z)
generalized Airy function. . . . . . . . . . . . . . . . . . . . . . . .206
Ak(z;p)
generalized Airy function. . . . . . . . . . . . . . . . . . . . . . . .207
Am;s(q)
q-Euler number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
am;s(q)
q-Stirling number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
Ai(z)
Airy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
am (x;k)
Jacobi's amplitude function . . . . . . . . . . . . . . . . . . . . . 561
arccd(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
arccn(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
Arccosz
general arccosine function . . . . . . . . . . . . . . . . . . . . . . . 118
arccosz
arccosine function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Arccoshz
general inverse hyperbolic cosine function . . . . . . . 127
arccoshz
inverse hyperbolic cosine function . . . . . . . . . . . . . . . 127
Arccotz
general arccotangent function . . . . . . . . . . . . . . . . . . . 118
arccotz
arccotangent function . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Arccothz
general inverse hyperbolic cotangent function. . . .127
arccothz
inverse hyperbolic cotangent function . . . . . . . . . . . 127
arccs(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
Arccscz
general arccosecant function. . . . . . . . . . . . . . . . . . . . .118
arccscz
arccosecant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Arccschz
general inverse hyperbolic cosecant function . . . . . 127
arccschz
inverse hyperbolic cosecant function. . . . . . . . . . . . .127
arcdc(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
Notations 875
arcdn(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
arcds(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
arcnc(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
arcnd(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
arcns(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
arcsc(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
arcsd(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
Arcsecz
general arcsecant function. . . . . . . . . . . . . . . . . . . . . . .118
arcsecz
arcsecant function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Arcsechz
general inverse hyperbolic secant function . . . . . . . 127
arcsechz
inverse hyperbolic secant function . . . . . . . . . . . . . . . 127
Arcsinz
general arcsine function . . . . . . . . . . . . . . . . . . . . . . . . . 118
arcsinz
arcsine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Arcsinhz
general inverse hyperbolic sine function . . . . . . . . . 127
arcsinhz
inverse hyperbolic sine function . . . . . . . . . . . . . . . . . 127
arcsn(x;k)
inverse Jacobian elliptic function . . . . . . . . . . . . . . . . 561
Arctanz
general arctangent function . . . . . . . . . . . . . . . . . . . . . 118
arctanz
arctangent function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Arctanhz
general inverse hyperbolic tangent function. . . . . .127
arctanhz
inverse hyperbolic tangent function. . . . . . . . . . . . . .127
Bn
Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
B(`)
n
generalized Bernoulli numbers. . . . . . . . . . . . . . . . . . .596
B(x)
n
N orlund polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596B(n)
Bell number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
Bn(x)
Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
B(T)
Bessel function of matrix argument (second kind)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
Bn(z)
generalized Airy function. . . . . . . . . . . . . . . . . . . . . . . .206
eBn(x)
periodic Bernoulli functions . . . . . . . . . . . . . . . . . . . . . 588
Bk(z;p)
generalized Airy function. . . . . . . . . . . . . . . . . . . . . . . .207
B(`)
n(x)
generalized Bernoulli polynomials . . . . . . . . . . . . . . . 596
B(a;b)
beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Bm(a;b)
multivariate beta function. . . . . . . . . . . . . . . . . . . . . . .768
Bx(a;b)
incomplete beta function . . . . . . . . . . . . . . . . . . . . . . . . 183
Bq(a;b)
q-beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
bei(x)
Kelvin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
ber(x)
Kelvin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
n(x;q)
q-Bernoulli polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 422
Bi(z)
Airy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
C(n)
Catalan number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
C(I) orC(a;b)
continuous on an interval Ior (a;b) . . . . . . . . . . . . . . . 4
Cn(I) orCn(a;b)
continuously di erentiable ntimes on an interval Ior
(a;b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
C1(I) orC1(a;b)
in nitely di erentiable on an interval Ior (a;b) . . . 5
(n)
Dirichlet character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
C(z)
Fresnel integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
(n)
ratio of gamma functions. . . . . . . . . . . . . . . . . . . . . . . .198
876 Notations
c(n)
number of compositions of n. . . . . . . . . . . . . . . . . . . . 628
C(z)
cylinder function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
C`()
normalizing constant for Coulomb radial functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
cm(n)
number of compositions of ninto exactly mparts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
ck(n)
Ramanujan's sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
C()
(z)
Gegenbauer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
C()
n(x)
ultraspherical (or Gegenbauer) polynomial . . . . . . 439
c(condition;n)
restricted number of compositions of n. . . . . . . . . . 628
Cn(x;a)
Charlier polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . .462
Cm
n(z;)
Ince polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
c(;`;r)
irregular Coulomb function. . . . . . . . . . . . . . . . . . . . . .748
C(f;h)(x)
cardinal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Cn(x; jq)
continuous q-ultraspherical polynomial . . . . . . . . . . 473
cd (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
cdEm
2n+2
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
Ce(z;q)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 667
ce(z;q)
Mathieu function of noninteger order. . . . . . . . . . . .665
cen(z;q)
Mathieu function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
cEm
2n+1
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
cel(kc;p;a;b )
Bulirsch's complete elliptic integral. . . . . . . . . . . . . .487
Chi(z)
hyperbolic cosine integral . . . . . . . . . . . . . . . . . . . . . . . 150
Ci(z)
cosine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Ci(a;z)
generalized cosine integral. . . . . . . . . . . . . . . . . . . . . . .188
ci(a;z)
generalized cosine integral. . . . . . . . . . . . . . . . . . . . . . .188
Cin(z)
cosine integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
cn (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
cosz
cosine function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Cosq(x)
q-cosine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
cosq(x)
q-cosine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
coshz
hyperbolic cosine function. . . . . . . . . . . . . . . . . . . . . . .123
cotz
cotangent function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
cothz
hyperbolic cotangent function . . . . . . . . . . . . . . . . . . . 123
cs (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
cscz
cosecant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
cschz
hyperbolic cosecant function . . . . . . . . . . . . . . . . . . . . 123
curl
of vector-valued function . . . . . . . . . . . . . . . . . . . . . . . . . 10
D(I)
test function space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
D(k)
complete elliptic integral of Legendre's type . . . . . 487
d(n)
divisor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
d(n)
derangement number. . . . . . . . . . . . . . . . . . . . . . . . . . . .631
Dq
q-di erential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
D(z)
parabolic cylinder function . . . . . . . . . . . . . . . . . . . . . . 304
dk(n)
divisor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
dqx
q-di erential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
D
fractional derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Notations 877
D(m;n)
Dellanoy number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .621
D( ;k)
incomplete elliptic integral of Legendre's type . . . 486
Dj(;;z )
cross-products of modi ed Mathieu functions and
their derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
dc (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
@(f;g)
@(x;y)
Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
dEm
2n+1
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
( )
discriminant function . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
(x a)
Dirac delta (or Dirac delta function) . . . . . . . . . . . . . 37
div
divergence of vector-valued function . . . . . . . . . . . . . . 10
dn (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
ds (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
Dsj(n;m;z )
cross-products of radial Mathieu functions and their
derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
e
base of exponential function. . . . . . . . . . . . . . . . . . . . .105
En
Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
E(`)
n
generalized Euler numbers . . . . . . . . . . . . . . . . . . . . . . 596
E(k)
Legendre's complete elliptic integral of the second
kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
( )
Dedekind's eta function (or Dedekind modular func-
tion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .579, 646
Es(z)
elementary symmetric function. . . . . . . . . . . . . . . . . .501
En(x)
Euler polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .588
E1(z)
exponential integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Ep(z)
generalized exponential integral . . . . . . . . . . . . . . . . . 185Ea;b(z)
Mittag-Leer function . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Eq(x)
q-exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . 422
E(z)
Weber function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
eq(x)
q-exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . 422
eEn(x)
periodic Euler functions. . . . . . . . . . . . . . . . . . . . . . . . .588
E(`)
n(x)
generalized Euler polynomials . . . . . . . . . . . . . . . . . . . 596
E( ;k)
Legendre's incomplete elliptic integral of the second
kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
Ecm
z;k2
Lam e function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
Ei(x)
exponential integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Ein(z)
complementary exponential integral . . . . . . . . . . . . . 150
el1(x;kc)
Bulirsch's incomplete elliptic integral of the rst kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
el2(x;kc;a;b)
Bulirsch's incomplete elliptic integral of the second
kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
el3(x;kc;p)
Bulirsch's incomplete elliptic integral of the third kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
env Ai(x)
envelope of Airy function. . . . . . . . . . . . . . . . . . . . . . . . .59
env Bi(x)
envelope of Airy function. . . . . . . . . . . . . . . . . . . . . . . . .59
envJ(x)
envelope of Bessel function . . . . . . . . . . . . . . . . . . . . . . . 61
envY(x)
envelope of Bessel function . . . . . . . . . . . . . . . . . . . . . . . 61
envU( c;x)
envelope of parabolic cylinder function . . . . . . . . . . 367
envU( c;x)
envelope of parabolic cylinder function . . . . . . . . . . 367
jk`
Levi-Civita symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
E(x;k)
Jacobi's epsilon function . . . . . . . . . . . . . . . . . . . . . . . . 562
878 Notations
erfz
error function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
erfcz
complementary error function . . . . . . . . . . . . . . . . . . . 160
Esm
z;k2
Lam e function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
expz
exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Fn
Fibonacci number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .629
FD
Lauricella's multivariate hypergeometric function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
F(z)
Dawson's integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
F(z)
Fresnel integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Fs(x)
Fermi{Dirac integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
Fc(x)
Fourier cosine transform . . . . . . . . . . . . . . . . . . . . . . . . . 27
Fs(x)
Fourier sine transform. . . . . . . . . . . . . . . . . . . . . . . . . . . .27
Fp(z)
terminant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
fe;m(h)
joining factor for radial Mathieu functions. . . . . . .669
fo;m(h)
joining factor for radial Mathieu functions. . . . . . .669
F(x)
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
F( ;k)
Legendre's incomplete elliptic integral of the rst kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
F(x;s)
periodic zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
F`(;)
regular Coulomb radial function. . . . . . . . . . . . . . . . .742
F
a;b
c;z
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 384
F(a;b;c;z)
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 384
F
a;b
c;z
Olver's hypergeometric function. . . . . . . . . . . . . . . . .384
F(a;b;c;z)
Olver's hypergeometric function. . . . . . . . . . . . . . . . .384f(;`;r)
regular Coulomb function . . . . . . . . . . . . . . . . . . . . . . . 748
1F1 a
b;T
con
uent hypergeometric function of matrix argu-
ment ( rst kind). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .770
1F1(a;b;T)
con
uent hypergeometric function of matrix argu-
ment ( rst kind) . . . . . . . . . . . . . . . . . . . . . . . . . 768, 770
2F1
a;b
c;T
hypergeometric function of matrix argument . . . . 771
2F1(a;b;c;T)
hypergeometric function of matrix argument
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .768, 771
2F1(a;b;c;z)
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 384
pFq a
b;z
generalized hypergeometric function . . . . . . . . 404, 408
pFq
a1;:::;ap
b1;:::;bq;z
generalized hypergeometric function . . . . . . . . 404, 408
pFq
a1;a2;:::;ap
b1;b2;:::;bq;T
generalized hypergeometric function of matrix argu-
ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
pFq(a;b;z)
generalized hypergeometric function . . . . . . . . 404, 408
pFq(a1;:::;ap;b1;:::;bq;z)
generalized hypergeometric function . . . . . . . . 404, 408
pFq(a1;a2;:::;ap;b1;b2;:::;bq;T)
generalized hypergeometric function of matrix argu-
ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .768, 772
2F1(a;b;c;z)
Olver's hypergeometric function. . . . . . . . . . . . . . . . .384
pFq a
b;z
scaled (or Olver's) generalized hypergeometric func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
F1( ; ; 0;
;x;y)
Appell function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
F2( ; ; 0;
;
0;x;y)
Appell function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
F3( ; 0; ; 0;
;x;y)
Appell function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
F4( ; ;
;
0;x;y)
Appell function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
Fen(z;q)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 667
fen(z;q)
second solution, Mathieu's equation . . . . . . . . . . . . . 657
Notations 879
Gn
Genocchi numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
G(z)
Barnes'G-function (or double gamma function) . . 144
G(z)
Goodwin{Staton integral. . . . . . . . . . . . . . . . . . . . . . . .160
G(k)
Waring's function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
g(k)
Waring's function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Gs(x)
Bose{Einstein integral . . . . . . . . . . . . . . . . . . . . . . . . . . 612
Gp(z)
product of gamma and incomplete gamma functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199, 230
ge;m(h)
joining factor for radial Mathieu functions. . . . . . .669
go;m(h)
joining factor for radial Mathieu functions. . . . . . .669
G(n; )
Gauss sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
G`(;)
irregular Coulomb radial function . . . . . . . . . . . . . . . 742
Gm;n
p;q
z;a1;:::;ap
b1;:::;bq
MeijerG-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Gm;n
p;q(z;a;b)
MeijerG-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Euler's constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136
(z)
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
m(a)
multivariate gamma function. . . . . . . . . . . . . . . . . . . .768
q(z)
q-gamma function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145
(a;z)
incomplete gamma function . . . . . . . . . . . . . . . . . . . . . 174
(a;z)
incomplete gamma function . . . . . . . . . . . . . . . . . . . . . 174
(a;z)
incomplete gamma function . . . . . . . . . . . . . . . . . . . . . 174
gdx
Gudermannian function . . . . . . . . . . . . . . . . . . . . . . . . . 121
gd 1(x)
inverse Gudermannian function . . . . . . . . . . . . . . . . . 121Gen(z;q)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 667
gen(z;q)
second solution, Mathieu's equation . . . . . . . . . . . . . 657
Gi(z)
Scorer function (inhomogeneous Airy function) . . 204
grad
gradient of di erentiable scalar function . . . . . . . . . . 10
H(s)
Euler sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .613
H(x)
Heaviside function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Hn(x)
Hermite polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
H(z)
Struve function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
Hen(x)
Hermite polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
H(1)
(z)
Bessel function of the third kind (or Hankel function)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
H(2)
(z)
Bessel function of the third kind (or Hankel function)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
h(1)
n(z)
spherical Bessel function of the third kind . . . . . . . 262
h(2)
n(z)
spherical Bessel function of the third kind . . . . . . . 262
H(s;z)
generalized Euler sums . . . . . . . . . . . . . . . . . . . . . . . . . . 614
H(f;x)
Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
H(a;u)
line-broadening function . . . . . . . . . . . . . . . . . . . . . . . . 167
Hn(xjq)
continuous q-Hermite polynomial. . . . . . . . . . . . . . . .473
hn(xjq)
continuous q 1-Hermite polynomial . . . . . . . . . . . . . 473
hn(x;q)
discreteq-Hermite I polynomial . . . . . . . . . . . . . . . . . 471
~hn(x;q)
discreteq-Hermite II polynomial . . . . . . . . . . . . . . . . 472
H
`(;)
irregular Coulomb radial functions . . . . . . . . . . . . . . 742
h(;`;r)
irregular Coulomb function. . . . . . . . . . . . . . . . . . . . . .748
880 Notations
pHq
a1;:::;ap
b1;:::;bq;z
bilateral hypergeometric function . . . . . . . . . . . . . . . 408
hcm
p(z;)
paraboloidal wave function . . . . . . . . . . . . . . . . . . . . . . 677
(s1;s2)Hfm(a;qm; ; ;
; ;z)
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .712
(s1;s2)Hf
m(a;qm; ; ;
; ;z)
path-multiplicative solutions of Heun's equation . . 712
Hhn(z)
probability function . . . . . . . . . . . . . . . . . . . . . . . . 167, 308
Hi(z)
Scorer function (inhomogeneous Airy function) . . 204
H`(a;q; ; ;
; ;z)
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .711
Hpn;m(a;qn;m; n; ;
; ;z)
Heun polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .712
hsm
p(z;)
paraboloidal wave function . . . . . . . . . . . . . . . . . . . . . . 677
I
fractional integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35, 53
I(m)
general elliptic integral . . . . . . . . . . . . . . . . . . . . . . . . . . 512
I(z)
modi ed Bessel function . . . . . . . . . . . . . . . . . . . . . . . . 249
eI(x)
modi ed Bessel function of imaginary order . . . . . 261
i(1)
n(z)
modi ed spherical Bessel function . . . . . . . . . . . . . . . 262
i(2)
n(z)
modi ed spherical Bessel function . . . . . . . . . . . . . . . 262
Ix(a;b)
incomplete beta function . . . . . . . . . . . . . . . . . . . . . . . . 183
idem( 1; 2;:::; n)
idem function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Ien(z;h)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 668
inerfc(z)
repeated integrals of the complementary error func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
inv
inversion number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .632
inverfx
inverse error function . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
inverfcx
inverse complementary error function . . . . . . . . . . . 166Ion(z;h)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 668
j;m
zeros of the Bessel function J(x) . . . . . . . . . . . . . . . 235
j0
;m
zeros of the Bessel function derivative J0
(x) . . . . . 235
J( )
Klein's complete invariant. . . . . . . . . . . . . . . . . . . . . . .579
J(z)
Anger function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295
J(z)
Bessel function of the rst kind . . . . . . . . . . . . . . . . . 217
eJ(x)
Bessel function of imaginary order . . . . . . . . . . . . . . 248
Jk(n)
Jordan's function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
jn(z)
spherical Bessel function of the rst kind . . . . . . . . 262
K(k)
Legendre's complete elliptic integral of the rst kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
()
condition number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
K(z)
modi ed Bessel function . . . . . . . . . . . . . . . . . . . . . . . . 249
eK(x)
modi ed Bessel function of imaginary order . . . . . 261
K(z)
Struve function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
kn(z)
modi ed spherical Bessel function . . . . . . . . . . . . . . . 262
Kn(x;p;N)
Krawtchouk polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 462
Ken(z;h)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 668
kei(x)
Kelvin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
ker(x)
Kelvin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Ki (x)
Bickley function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Kon(z;h)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 668
L
lattice in C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
Notations 881
Ln
Lebesgue constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Ln(x)
Laguerre polynomial . . . . . . . . . . . . . . . . . . . . . . . 436, 439
L(z)
modi ed Struve function . . . . . . . . . . . . . . . . . . . . . . . . 288
L( )
n(x)
Laguerre (or generalized Laguerre) polynomial . . 439
L(s; )
DirichletL-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . .612
L(f;s)
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
L( )
n(x;q)
q-Laguerre polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 471
( )
elliptic modular function . . . . . . . . . . . . . . . . . . . . . . . . 579
(n)
Mangoldt's function. . . . . . . . . . . . . . . . . . . . . . . . . . . . .639
(n)
Liouville's function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .639
li(x)
logarithmic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Li2(z)
dilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
Lis(z)
polylogarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .611
Lnz
general logarithm function . . . . . . . . . . . . . . . . . . . . . . 104
lnz
principal branch of logarithm function . . . . . . . . . . 104
logx
logarithm to base e(Chapter 27 only). . . . . . . . . . .105
log10z
common logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
logaz
logarithm to general base a. . . . . . . . . . . . . . . . . . . . . 105
M(x)
Mills' ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163
M(n)
Motzkin number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
M(z)
modi ed Struve function . . . . . . . . . . . . . . . . . . . . . . . . 288
M;(z)
Whittaker con
uent hypergeometric function. . . .334
M(a;g)
arithmetic-geometric mean . . . . . . . . . . . . . . . . . . . . . . 492M(f;s)
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
M(j)
(z;h)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 667
M(a;b;z )
Kummer con
uent hypergeometric function . . . . . 322
M(a;b;z )
Olver's con
uent hypergeometric function . . . . . . . 322
Mn(x; ;c)
Meixner polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . .462
maj
major index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .632
Mc(j)
n(z;h)
radial Mathieu function . . . . . . . . . . . . . . . . . . . . . . . . . 668
Me(z;q)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 667
me(z;q)
Mathieu function of noninteger order. . . . . . . . . . . .664
men(z;q)
Mathieu function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
Ms(j)
n(z;h)
radial Mathieu function . . . . . . . . . . . . . . . . . . . . . . . . . 668
(n)
M obius function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
N
winding number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
N(n;k)
Narayana number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
nc (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
nd (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
ns (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
(n)
number of distinct primes dividing n. . . . . . . . . . . . 638
O(x)
order not exceeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
o(x)
order less than . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
On(x)
Neumann's polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 247
PI, PII, PIII, P0
III, PIV, PV, PVI
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 724
p(condition;n)
restricted number of partions of n. . . . . . . . . . . . . . . 627
882 Notations
P8
<
:
a1b1c1z
a2b2c29
=
;
Riemann's P-symbol for solutions of the generalized
hypergeometric di erential equation . . . . . . . . . . 396
}(z) (=}(zjL) =}(z;g2;g3))
Weierstrass }-function . . . . . . . . . . . . . . . . . . . . . . . . . . 570
p(n)
total number of partitions of n. . . . . . . . . . . . . . . . . . 618
P(x):P
(x) with= 0 . . . . . . . . . . . . . . . . . . . . 352, 353
Pn(x)
Legendre polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
P(z):P
(z) with= 0 . . . . . . . . . . . . . . . 352, 353, 375
pk(n)
number of partitions of ninto at most kparts. . .626
P
(x)
Ferrers function of the rst kind . . . . . . . . . . . . . . . . 353
P
(z)
associated Legendre function of the rst kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353, 375
P
n(x)
shifted Legendre polynomial . . . . . . . . . . . . . . . . . . . . 439
P( ; )
n(x)
Jacobi polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
P
1
2+i (x)
conical function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
P(a;z)
normalized incomplete gamma function . . . . . . . . . 174
Pn(x;c)
associated Legendre polynomial . . . . . . . . . . . . . . . . . 474
pk(m;n)
number of partitions of ninto at most kparts, each
less than or equal to m. . . . . . . . . . . . . . . . . . . . . . . 626
pk(D;n)
number of partitions of ninto at most kdistinct parts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
P( ; )
n(x;c)
associated Jacobi polynomial. . . . . . . . . . . . . . . . . . . .474
P()
n(x; )
Meixner{Pollaczek polynomial . . . . . . . . . . . . . . . . . . 462
P ; ;
m;n (x;y)
triangle polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
P()
n(x;a;b)
Pollaczek polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . .476
pn(x;a;b;q)
littleq-Jacobi polynomial . . . . . . . . . . . . . . . . . . . . . . . 471P( ; )
n(x;c;d;q)
bigq-Jacobi polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 471
pn
x;a;b;a;b
continuous Hahn polynomial . . . . . . . . . . . . . . . . . . . . 462
Pn(x;a;b;c ;q)
bigq-Jacobi polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 471
pn(x;a;b;c;djq)
Askey{Wilson polynomial . . . . . . . . . . . . . . . . . . . . . . . 472
ph
phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
(n)
Euler's totient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
1(t;x)
fold catastrophe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .776
2(t;x)
cusp catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
3(t;x)
swallowtail catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . 776
K(t;x)
cuspoid catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
k(n)
sum of powers of integers relatively prime to n. . 638
( ; )
(t)
Jacobi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
(E)(s;t;x)
elliptic umbilic catastrophe. . . . . . . . . . . . . . . . . . . . . .776
(H)(s;t;x)
hyperbolic umbilic catastrophe . . . . . . . . . . . . . . . . . . 776
(; ;z)
generalized Bessel function . . . . . . . . . . . . . . . . . . . . . . 261
(z;s;a )
Lerch's transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
(1)(a;b;b0;c;x;y)
rstq-Appell function. . . . . . . . . . . . . . . . . . . . . . . . . . .423
(2)(a;b;b0;c;c0;x;y)
secondq-Appell function . . . . . . . . . . . . . . . . . . . . . . . . 423
(3)(a;a0;b;b0;c;x;y)
thirdq-Appell function. . . . . . . . . . . . . . . . . . . . . . . . . .423
(4)(a;b;c;c0;x;y)
fourthq-Appell function . . . . . . . . . . . . . . . . . . . . . . . . 423
r+1 s
a0;a1;:::;ar
b1;b2;:::;bs;q;z
basic hypergeometric (or q-hypergeometric) function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
r+1 s(a0;a1;:::;ar;b1;b2;:::;bs;q;z)
basic hypergeometric (or q-hypergeometric) function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Notations 883
set of plane partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
(x)
number of primes not exceeding x. . . . . . . . . . . . . . .638
2;k
Legendre's complete elliptic integral of the third kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
; 2;k
Legendre's incomplete elliptic integral of the third
kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
pp(n)
number of plane partitions of n. . . . . . . . . . . . . . . . . 629
pq (z;k)
generic Jacobian elliptic function . . . . . . . . . . . . . . . . 550
Psm
n
z;
2
spheroidal wave function of complex argument . . 700
Psm
n
x;
2
spheroidal wave function of the rst kind. . . . . . . .699
(x)
Chebyshev -function. . . . . . . . . . . . . . . . . . . . . . . . . . .613
(z)
psi (or digamma) function . . . . . . . . . . . . . . . . . . . . . . 136
2(x)
Pearcey integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .777
K(x)
canonical integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
(n)(z)
polygamma functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
(E)(x)
canonical integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
(H)(x)
canonical integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
3(x;k)
swallowtail canonical integral function. . . . . .777, 778
K(x;k)
di raction catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . 777
(E)(x;k)
elliptic umbilic canonical integral function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .777, 779, 780
(H)(x;k)
hyperbolic umbilic canonical integral function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .777, 779, 781
(a;b;T)
con
uent hypergeometric function of matrix argu-
ment (second kind) . . . . . . . . . . . . . . . . . . . . . . 768, 770
r s
a1;a2;:::;ar
b1;b2;:::;bs;q;z
bilateral basic hypergeometric (or bilateral q-hyper-geometric) function . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
r s(a1;a2;:::;ar;b1;b2;:::;bs;q;z)
bilateral basic hypergeometric (or bilateral q-hyper-
geometric) function . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Q(x):Q
(x) with= 0 . . . . . . . . . . . . . . . . . . . . 352, 353
Qn(x; ; ;N )
Hahn polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Q(z):Q
(z) with= 0 . . . . . . . . . . . . . . . 352, 354, 375
Q
(x)
Ferrers function of the second kind . . . . . . . . . . . . . . 353
Q
(z)
associated Legendre function of the second kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .354, 375
Q
(z)
Olver's associated Legendre function . . . . . . . 354, 375
bQ
1
2+i (x)
conical function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
Q(a;z)
normalized incomplete gamma function . . . . . . . . . 174
Qn(x;a;bjq)
Al-Salam{Chihara polynomial. . . . . . . . . . . . . . . . . . .473
Qn
x;a;bjq 1
q 1-Al-Salam{Chihara polynomial . . . . . . . . . . . . . . 473
Qn(x; ; ;N ;q)
q-Hahn polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .470
Qsm
n
z;
2
spheroidal wave function of complex argument . . 700
Qsm
n
x;
2
spheroidal wave function of the second kind . . . . . 700
r(n)
Schr oder number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .622
rtp(;`)
outer turning point for Coulomb functions . . . . . . 748
R( )
m;n(z)
disk polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
R a(b1;b2;:::;bn;z1;z2;:::;zn)
multivariate hypergeometric function. . . . . . . . . . . .498
R a(b;z)
multivariate hypergeometric function. . . . . . . . . . . .498
Rn(x;
;;N )
dual Hahn polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 467
Rn(x; ; ;
; )
Racah polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .467
Rn(x; ; ;
;jq)
q-Racah polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
884 Notations
RC(x;y)
Carlson's elliptic integral with two variables . . . . . 487
RD(x;y;z )
elliptic integral symmetric in only two variables . . 498
RF(x;y;z )
symmetric elliptic integral of rst kind . . . . . . . . . . 497
RG(x;y;z )
symmetric elliptic integral of second kind. . . . . . . .498
tp(;`)
outer turning point for Coulomb radial functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
RJ(x;y;z;p )
symmetric elliptic integral of third kind . . . . . . . . . 497
Sn
set of permutations of f1;2;:::;ng. . . . . . . . . . . . . . 631
S(z)
Fresnel integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
S;(z)
Lommel function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295
s;(z)
Lommel function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294
Sm(j)
n(z;
)
radial spheroidal wave function. . . . . . . . . . . . . . . . . .703
S(f;s)
Stieltjes transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
S(n;k)
Stirling number of the second kind . . . . . . . . . . . . . . 624
s(n;k)
Stirling number of the rst kind. . . . . . . . . . . . . . . . .624
Sn(x;q)
Stieltjes{Wigert polynomial . . . . . . . . . . . . . . . . . . . . . 471
Sm
n(z;)
Ince polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
s(;`;r)
regular Coulomb function . . . . . . . . . . . . . . . . . . . . . . . 748
S(k;h)(x)
Sinc function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
Sn(x;a;b;c )
continuous dual Hahn polynomial . . . . . . . . . . . . . . . 467
sc (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
scdEm
2n+3
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
scEm
2n+2
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690sd (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
sdEm
2n+2
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
Se(z;q)
modi ed Mathieu function . . . . . . . . . . . . . . . . . . . . . . 667
se(z;q)
Mathieu function of noninteger order. . . . . . . . . . . .665
sen(z;q)
Mathieu function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
sEm
2n+1
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
secz
secant function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
sechz
hyperbolic secant function . . . . . . . . . . . . . . . . . . . . . . 123
Shi(z)
hyperbolic sine integral . . . . . . . . . . . . . . . . . . . . . . . . . 150
Si(z)
sine integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150
si(z)
sine integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150
Si(a;z)
generalized sine integral. . . . . . . . . . . . . . . . . . . . . . . . .188
si(a;z)
generalized sine integral. . . . . . . . . . . . . . . . . . . . . . . . .188
n()
Rayleigh function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
`()
Coulomb phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
(n)
sum of powers of divisors of n. . . . . . . . . . . . . . . . . . . 638
(z) (=(zjL) =(z;g2;g3))
Weierstrass sigma function . . . . . . . . . . . . . . . . . . . . . . 570
sinz
sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Sinq(x)
q-sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
sinq(x)
q-sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
sinhz
hyperbolic sine function. . . . . . . . . . . . . . . . . . . . . . . . .123
sn (z;k)
Jacobian elliptic function . . . . . . . . . . . . . . . . . . . . . . . 550
Tn
tangent numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
Notations 885
Tn(x)
Chebyshev polynomial of the rst kind . . . . . . . . . . 439
T
n(x)
shifted Chebyshev polynomial of the rst kind . . 439
tanz
tangent function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
tanhz
hyperbolic tangent function . . . . . . . . . . . . . . . . . . . . . 123
(n)
Ramanujan's tau function. . . . . . . . . . . . . . . . . . . . . . .647
j(zj )
theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
j(z;q)
theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
(zj
)
Riemann theta function . . . . . . . . . . . . . . . . . . . . . . . . . 538
^(zj
)
scaled Riemann theta function . . . . . . . . . . . . . . . . . . 538
(zj
)
Riemann theta function with characteristics . . . . . 539
Un(x)
Chebyshev polynomial of the second kind . . . . . . . 439
Um(t)
generalized Airy function. . . . . . . . . . . . . . . . . . . . . . . .207
U
n(x)
shifted Chebyshev polynomial of the second kind
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
U(a;z)
parabolic cylinder function . . . . . . . . . . . . . . . . . . . . . . 304
U(x;t)
Voigt function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
U(a;x)
parabolic cylinder function . . . . . . . . . . . . . . . . . . . . . . 305
U(a;b;z )
Kummer con
uent hypergeometric function . . . . . 322
uEm
2n
z;k2
Lam e polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
Va;b(f)
total variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
Vn(x)
Chebyshev polynomial of the third kind . . . . . . . . . 439
Vm(t)
generalized Airy function. . . . . . . . . . . . . . . . . . . . . . . .207
Vm(t)
generalized Airy function. . . . . . . . . . . . . . . . . . . . . . . .207V(a;z)
parabolic cylinder function . . . . . . . . . . . . . . . . . . . . . . 304
V(x;t)
Voigt function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
W
Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
W(x)
LambertW-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
w(z)
complementary error function . . . . . . . . . . . . . . . . . . . 160
Wn(x)
Chebyshev polynomial of the fourth kind. . . . . . . .439
Wp(x)
principal branch of Lambert W-function . . . . . . . . 111
Wm(x)
nonprincipal branch of Lambert W-function. . . . .111
W;(z)
Whittaker con
uent hypergeometric function. . . .334
W(a;x)
parabolic cylinder function . . . . . . . . . . . . . . . . . . . . . . 314
wI(z;)
basic solution, Hill's equation . . . . . . . . . . . . . . . . . . . 674
wII(z;)
basic solution, Hill's equation . . . . . . . . . . . . . . . . . . . 674
wI(z;a;q)
basic solution, Mathieu's equation. . . . . . . . . . . . . . .653
wII(z;a;q)
basic solution, Mathieu's equation. . . . . . . . . . . . . . .653
Wn(x;a;b;c;d )
Wilson polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
(s)
Riemann's -function . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
y;m
zeros of the Bessel function Y(x) . . . . . . . . . . . . . . . 235
y0
;m
zeros of the Bessel function derivative Y0
(x). . . . .235
Y(z)
Bessel function of the second kind. . . . . . . . . . . . . . .217
eY(x)
Bessel function of imaginary order . . . . . . . . . . . . . . 248
yn(z)
spherical Bessel function of the second kind . . . . . 262
yn(x;a)
Bessel polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
Yl;m(; )
spherical harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
886 Notations
Ym
l(; )
surface harmonic of the rst kind . . . . . . . . . . . . . . . 378
Z(z)
modi ed cylinder function . . . . . . . . . . . . . . . . . . . . . . 249
Z(T)
zonal polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .769
za
power function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Z(xjk)
Jacobi's zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
(s)
Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 602
x(s)
incomplete Riemann zeta function. . . . . . . . . . . . . . .189
(s;a)
Hurwitz zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
(z) (=(zjL) =(z;g2;g3))
Weierstrass zeta function. . . . . . . . . . . . . . . . . . . . . . . .570
Index
Abel means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Abel summability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33, 34
Abel{Plana formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Abelian functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
absolute error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
acceleration of convergence
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
for sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93{94
for series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93{94
limit-preserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
accumulation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
acoustics
canonical integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792
additive number theory . . . . . . . . . . . . . . . . . . . . . . . 644{647
Dedekind modular function . . . . . . . . . . . . . . . . . . . . . 646
Dedekind sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
discriminant function . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Euler's pentagonal number theorem . . . . . . . . . . . . . 646
Goldbach conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
Jacobi's identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
unrestricted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Ramanujan's identity . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Ramanujan's tau function. . . . . . . . . . . . . . . . . . . . . . .646
representation by squares . . . . . . . . . . . . . . . . . . . . . . . 645
Waring's problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
aerodynamics
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
ane Weyl groups
Painlev e equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
ship waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790
approximations
expansions in Chebyshev series . . . . . . . . . . . . . . . 211
in terms of elementary functions . . . . . . . . . . . . . . 211
in the complex plane. . . . . . . . . . . . . . . . . . . . . . . . . .212
asymptotic expansions. . . . . . . . . . . . . . . . . . . . . .198{199
error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 199
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .209{210
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .194
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .194for products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
initial values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .194
numerically satisfactory solutions . . . . . . . . . . . . . 194
Riccati form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
envelope functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
generalized. . . . . . . . . . . seegeneralized Airy functions.
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
integral identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
integral representations. . . . . . . . . . . . . . . . . . . . .196, 203
integrals
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 211, 212
asymptotic approximations . . . . . . . . . . . . . . . . . . . 202
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
of products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
repeated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Laplace transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
modulus and phase
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 200
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195
identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
relation to Bessel functions . . . . . . . . . . . . . . . . . . . 199
relation to zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
products
di erential equation . . . . . . . . . . . . . . . . . . . . . . . . . . 203
integral representations . . . . . . . . . . . . . . . . . . . . . . . 203
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Wronskian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203
relation to umbilics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
relations to other functions
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 196{197
con
uent hypergeometric functions . . 197, 328, 338
Hankel functions . . . . . . . . . . . . . . . . . . . . . . . . . 196{197
modi ed Bessel functions. . . . . . . . . . . . . . . . .196{197
Stieltjes transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
tables
complex variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .210
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
real variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201{202, 211
887
888 Index
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
zeros
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 201
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76, 210
di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
relation to modulus and phase . . . . . . . . . . . . . . . . 200
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201{202, 211
Airy transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Airy's equation
. . . . . . . . . . seeAiry functions, di erential equation.
Aitken's 2-process
for sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
iterated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Al-Salam{Chihara polynomials . . . . . . . . . . . . . . . . . . . . 473
algebraic curves
Riemann surface . . . . . . . . . . . . . . . . . . . . . . 543, 544, 546
algebraic equations
parametrization via Jacobian elliptic functions . . 563
spherical trigonometry . . . . . . . . . . . . . . . . . . . . . . . . 564
uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
algebraic Lam e functions . . . . . . . . . . . . . . . . . . . . . . . . . . 693
alternant
determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
amplitude (am) function . . . . . . . . . . . . . . . . . . . . . . . . . . 561
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .564
approximations
smallk;k0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
smallx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 561
quasi-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
relation to elliptic integrals. . . . . . . . . . . . . . . . . . . . . .562
relation to Gudermannian function . . . . . . . . . . . . . . 562
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .567
analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
by re
ection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
analytic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
at in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
in a domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Anger function. . . . . . . . . . . . seeAnger{Weber functions.
Anger{Weber functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
asymptotic expansions
large argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
large order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .298
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87, 299
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .295graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 295
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
relations to other functions
Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Lommel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .296
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
series expansions
power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
products of Bessel functions . . . . . . . . . . . . . . . . . . 297
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299
angle between arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
angular momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
angular momentum
generalized hypergeometric functions . . . . . . . . . . . . 418
angular momentum coupling coecients
. . . . . see3jsymbols, 6jsymbols, and9jsymbols.
angular momentum operator
spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
antenna research
Lam e functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
Appell functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .414
applications
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
de nition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .412{413
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 414
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
inverse Laplace transform. . . . . . . . . . . . . . . . . . . . .414
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
partial di erential equations . . . . . . . . . . . . . . . . . . . . 413
relation to Legendre's elliptic integrals . . . . . . . . . . 490
relation to symmetric elliptic integrals. . . . . . . . . . .509
relations to hypergeometric functions. . . . . . . . . . . .414
transformations of variables . . . . . . . . . . . . . . . . 414{415
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
reduction formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . .414
approximation techniques
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . . 97
least squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99{100
minimax polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
minimax rational functions . . . . . . . . . . . . . . . . . . . . . . . 97
Pad e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98{99
splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Index 889
arc length
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .563
arc(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
angle between. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
area of triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246
argument principle . . . . . . . . . . . . . . . . seephase principle.
arithmetic Fourier transform . . . . . . . . . . . . . . . . . . . . . . 647
arithmetic mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 13
arithmetic progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
arithmetic-geometric mean. . . . . . . . . . . . . . . . . . . . . . . . 492
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 492
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .566
Legendre's elliptic integrals . . . . . . . . . . . . . . . . . 492{493
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .505
arithmetics
complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
exact rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
oating-point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
level-index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Askey polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
Askey scheme for orthogonal polynomials. . . . . . . . . .464
Askey{Gasper inequality
Jacobi polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .478
Askey{Wilson class orthogonal polynomials . . . 472{474
as eigenfunctions of a q-di erence operator . . . . . . 472
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .474
interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .472
representation as q-hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472{474
Askey{Wilson polynomials . . . . . . . . . . . . . . . . . . . . . . . . 472
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .474
relation to q-hypergeometric functions . . . . . . 472{474
associated Anger{Weber function
. . . . . . . . . . . . . . . . . . . . . . seeAnger{Weber functions.
associated Laguerre functions . . . . . . . . . . . . . . . . . . . . . 754
associated Legendre equation . . . . . . . . . . . . . . . . . 352, 375
exponent pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
numerically satisfactory solutions. . . . . . . . . . .352, 375
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .352
standard solutions. . . . . . . . . . . . . . . . . . . . .352, 354, 375
associated Legendre functions. . . . . . . . . . . . . . . . . . . . . 352
. . . . . . . . . . . . . . . . . . . . . . . . . . see also Ferrers functions.
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 370, 377
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .376
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378{379
asymptotic approximations
. . . . . . . . . . seeuniform asymptotic approximations.
behavior at singularities . . . . . . . . . . . . . . . . . . . . 361, 375
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379connection formulas . . . . . . . . . . . . . . . . . . . . . . . . 362, 375
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .364
cross-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353{354, 375
degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
with respect to degree or order. . . . . . . . . . . . . . . .363
di erential equation
. . . . . . . . . . . . . . . . seeassociated Legendre equation.
expansions in series of. . . . . . . . . . . . . . . . . . . . . . . . . . .370
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .378
generating functions. . . . . . . . . . . . . . . . . . . . . . . .361, 375
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . 357{359, 375{376
Heine's formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
hypergeometric representations . . . . . . . . 353{354, 375
integer degree and order . . . . . . . . . . . . . . . 360{361, 375
integer order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360, 375
integral representations. . . . . . . . . . . . . . . . . . . . .363, 377
integrals
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
Mellin transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .370
products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .369
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
of the rst kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
Olver's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354, 375
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
principal values (or branches) . . . . . . . . . . . . . . . . . . . 375
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . 362, 375
relations to other functions
elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Gegenbauer function. . . . . . . . . . . . . . . . . . . . . . . . . .355
hypergeometric function. . . . . . . . . . . . .353, 354, 394
Jacobi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Legendre polynomials. . . . . . . . . . . . . . . . . . . . . . . . .360
Rodrigues-type formulas . . . . . . . . . . . . . . . . . . . . . . . . 360
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359, 360
sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370{371, 377
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .380
uniform asymptotic approximations
large degree . . . . . . . . . . . . . . . . . . . . . . . . . 366{368, 377
large order . . . . . . . . . . . . . . . . . . . . . . . . . . 365{366, 377
values on the cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .376
Whipple's formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352{353, 375
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368, 377
associated orthogonal polynomials . . . . . . . . . . . . . . . . . 474
corecursive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .474
Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
Legendre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .474
astrophysics
error functions and Voigt functions. . . . . . . . . . . . . .169
890 Index
Heun functions and Heun's equation . . . . . . . . . . . . 720
asymptotic and order symbols . . . . . . . . . . . . . . . . . . . . . . 42
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
asymptotic approximations and expansions. . . . see also
asymptotic approximations of integrals, asymptotic
approximations of sums and sequences, asymptotic
solutions of di erence equations, asymptotic solu-
tions of di erential equations, andasymptotic so-
lutions of transcendental equations.
algebraic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
cases of failure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52, 66
di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
double asymptotic properties
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Hankel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .258
Kelvin functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . 257
parabolic cylinder functions. . . . . . . . . . . . . . . . . . .311
exponentially-improved expansions. . . . . . . . . . . .67{69
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
hyperasymptotic expansions. . . . . . . . . . . . . . . . . . . . . .68
improved accuracy via numerical transformations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
logarithms of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42
null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
numerical use of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66, 69
Poincar e type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
powers of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
re-expansion of remainder terms . . . . . . . . . . . . . . 67{69
reversion of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Stokes phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
substitution of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
via connection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 66
asymptotic approximations of integrals . . . . . . . . . . 43{55
Bleistein's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chester{Friedman{Ursell method . . . . . . . . . . . . . . . . 48
coalescing critical points . . . . . . . . . . . . . . . . . . . . . . . . . 48
coalescing peak and endpoint. . . . . . . . . . . . . . . . . . . . .45
coalescing saddle points . . . . . . . . . . . . . . . . . . . . . . . . . . 48
distributional methods. . . . . . . . . . . . . . . . . . . . . . . .51{55
Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Haar's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
inverse Laplace transforms . . . . . . . . . . . . . . . . . . . . 46{47
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Laplace's method . . . . . . . . . . . . . . . . . . . . . . . . . 44{45, 47
Mellin transform methods . . . . . . . . . . . . . . . . . . . . . . . . 48
extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49{51
method of stationary phase . . . . . . . . . . . . . . . . . . . . . . 45extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
method of steepest descents . . . . . . . . . . . . . . . . . . . . . . 47
multidimensional integrals. . . . . . . . . . . . . . . . . . . . . . . .51
Stieltjes transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 52{53
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Watson's lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44, 46
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
asymptotic approximations of sums and sequences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63{66
Abel{Plana formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Darboux's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65{66
entire functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Euler{Maclaurin formula . . . . . . . . . . . . . . . . . . . . . . . . . 63
summation by parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
asymptotic scale or sequence . . . . . . . . . . . . . . . . . . . . . . . 43
asymptotic solutions of di erence equations . . . . . 61{63
characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
coincident characteristic values . . . . . . . . . . . . . . . . . . . 62
Liouville{Green (or WKBJ) type approximations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
transition points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
turning points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
with a parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62{63
asymptotic solutions of di erential equations . . . . 55{61
characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
coincident characteristic values . . . . . . . . . . . . . . . . . . . 57
error-control function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Fabry's transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
irregular singularities of rank 1 . . . . . . . . . . . . . . . . . . . 56
Liouville{Green approximation theorem . . . . . . . . . . 57
Liouville{Green (or WKBJ) approximations . . . . . . 57
numerically satisfactory solutions. . . . . . . . . . . . . . . . .58
resurgence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57, 68
with a parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58{61
classi cation of cases. . . . . . . . . . . . . . . . . . . . . . . . . . .58
coalescing transition points . . . . . . . . . . . . . . . . . . . . 61
connection formulas across transition points. . . .61
in terms of Airy functions. . . . . . . . . . . . . . . . . . . . . .59
in terms of Bessel functions of xed order. . .60{61
in terms of Bessel functions of variable order . . . 61
in terms of elementary functions . . . . . . . . . . . . . . . 59
Liouville transformation . . . . . . . . . . . . . . . . . . . . . . . 58
transition points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
turning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
asymptotic solutions of transcendental equations. . . .43
Lagrange's formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
atomic photo-ionization
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
atomic physics
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
atomic spectra
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
atomic spectroscopy
Index 891
3j;6j;9jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
attractive potentials
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . 753, 754
auxiliary functions for Fresnel integrals
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 164
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 163
Mellin{Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . 163
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162
auxiliary functions for sine and cosine integrals
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .151
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 153
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 154
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 157
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 152
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
relation to con
uent hypergeometric functions. . .153
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156
axially symmetric potential theory . . . . . . . . . . . . . . . . 501
B acklund transformations
classical orthogonal polynomials. . . . . . . . . . . . . . . . .478
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . 730{732
backward recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Bailey's 2F1( 1) sum
q-analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Bailey's 4F3(1) sum
q-analogs ( rst and second) . . . . . . . . . . . . . . . . . . . . . 427
Bailey's 2 2transformations
bilateralq-hypergeometric function. . . . . . . . . . . . . .429
Bailey's bilateral summations
bilateralq-hypergeometric function. . . . . . . . . . . . . .427
bandlimited functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
Barnes' beta integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Barnes'G-function
asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
in nite product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 144
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Barnes' integral
Ferrers functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Bartky's transformation
Bulirsch's elliptic integrals . . . . . . . . . . . . . . . . . . . . . . 487
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .504
basic elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
basic hypergeometric functions. . . seebilateralq-hyper-
geometric function andq-hypergeometric function.
Basset's integralmodi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 253
Bell numbers
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .623
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .623
Bernoulli monosplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
arithmetic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . .593
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .593
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
degenerate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
nite expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596, 597
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 592
inversion formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
irregular pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .598
Kummer congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
recurrence relations
linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
quadratic and higher order. . . . . . . . . . . . . . . . . . . .595
relations to
Eulerian numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
Genocchi numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
tangent numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .596
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589, 598
Bernoulli polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597{598
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .593
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
di erence equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
nite expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596, 597
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
in nite series expansions
Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
892 Index
other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 592
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
Laplace transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
multiplication formulas. . . . . . . . . . . . . . . . . . . . . . . . . .590
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
recurrence relations
linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
relation to Eulerian numbers . . . . . . . . . . . . . . . . . . . . 591
relation to Riemann zeta function . . . . . . . . . . . . . . . 591
representation as sums of powers . . . . . . . . . . . . . . . . 589
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
symbolic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589
zeros
complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
real. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .594
Bernoulli's lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
Bernstein{Szeg o polynomials . . . . . . . . . . . . . . . . . . . . . . 474
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see also cylinder func-
tions, Hankel functions, Kelvin functions, modi ed
Bessel functions, andspherical Bessel functions.
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .226
applications
asymptotic solutions of di erential equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274{275
electromagnetic scattering . . . . . . . . . . . . . . . . . . . . 275
Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Laplace's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
oscillation of chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
oscillation of plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
asymptotic expansions for large argument . . 228{230
error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229{230
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 230
asymptotic expansions for large order. . . . . . .231{235
asymptotic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Debye's expansions. . . . . . . . . . . . . . . . . . . . . . .231{232
double asymptotic properties. . . . . . . . . . . . .235, 258
resurgence properties of coecients . . . . . . . . . . . 233
transition region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232{235
branch conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276{277
computation by quadrature . . . . . . . . . . . . . . . . . . . . . . 83
computation by recursion . . . . . . . . . . . . . . . . . . . . . . . . 87connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .222
contiguous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226
cross-products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222, 223
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
de nite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217{218
derivatives
asymptotic expansions for large argument. . . . .229
asymptotic expansions for large order . . . . 231{232
explicit forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
uniform asymptotic expansions for large order
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
with respect to order . . . . . . . . . . . . . . . . . . . . . 227{228
zeros . . . . . . . . . . . . . . . . seezeros of Bessel functions.
di erential equations . . . . . . . . . . . . . . . . . . . . . . . 217, 226
. . . . . . . . . . . . . . . . . . . . . . . . see also Bessel's equation.
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
envelope functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
expansions in partial fractions. . . . . . . . . . . . . . . . . . .247
expansions in series of . . . . . . . . . . . . . . . . . . . . . . 247{248
Fourier{Bessel expansion. . . . . . . . . . . . . . . . . . . . . . . .248
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218{222
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
in nite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
integral representations
along the real line . . . . . . . . . . . . . . . . . . . . . . . . 223{224
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226
contour integrals . . . . . . . . . . . . . . . . . . . . . . . . . 224{225
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .225
integrals . . . . . see also integrals of Bessel and Hankel
functions andHankel transforms.
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279, 280
limiting forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223
minimax rational approximation . . . . . . . . . . . . . . . . . 98
modulus and phase functions
asymptotic expansions for large argument. . . . .231
basic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .230
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
relation to zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
monotonicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227
multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 246
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
of imaginary argument
. . . . . . . . . . . . . . . . . . . . seemodi ed Bessel functions.
of imaginary order
Index 893
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221{222
limiting forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
numerically satisfactory pairs . . . . . . . . . . . . . . . . . 248
uniform asymptotic expansions for large order
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
of matrix argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
asymptotic approximations . . . . . . . . . . . . . . . . . . . 770
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
of the rst and second kinds . . . . . . . . . . . . . . . . . . 768
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
relations to con
uent hypergeometric functions of
matrix argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
of the rst, second, and third kinds . . . . . . . . . 217{218
orthogonality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243, 244
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223
principal branches (or values). . . . . . . . . . . . . . .217{218
recurrence relations. . . . . . . . . . . . . . . . . . . . . . . . .222{223
relations to other functions
Airy functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .196{197
con
uent hypergeometric functions . . . . . . . . . . . 228
elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . .228
generalized Airy functions . . . . . . . . . . . . . . . . . . . . 206
generalized hypergeometric functions. . . . . . . . . .228
parabolic cylinder functions . . . . . . . . . . . . . . 228, 315
sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246{248
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . 246{247
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .248
expansions in series of Bessel functions . . . 247{248
multiplication theorem. . . . . . . . . . . . . . . . . . . . . . . .246
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278{279
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
zeros . . . . . . . . . . . . . . . . . . seezeros of Bessel functions.
Bessel polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 264, 476
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 476
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .476
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .476
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
relations to other functions
complex orthogonal polynomials . . . . . . . . . . . . . . . 83
con
uent hypergeometric functions . . . . . . . . . . . 476
generalized hypergeometric functions. . . . . . . . . .476
Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
Bessel transform . . . . . . . . . . . . . . . . seeHankel transform.
Bessel's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
inhomogeneous forms. . . . . . . . . . . . . . . . . .288, 294, 295
numerically satisfactory solutions . . . . . . . . . . . . . . . 218
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 217{218
Bessel's inequality
Fourier series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
Bessel's integral
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223
best uniform polynomial approximation . . . . . . . . . . . . 96
best uniform rational approximation. . . . . . . . . . . . . . . .97
beta distribution
incomplete beta functions . . . . . . . . . . . . . . . . . . . . . . . 189
beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
. . . . . . . . . . . . . . . . . see also incomplete beta functions.
applications
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145{146
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 142
multidimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
multivariate. . . . . . . . . . seemultivariate beta function.
beta integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142{143
Bickley function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .259
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
bicon
uent Heun equation. . . . . . . . . . . . . . . . . . . . . . . . .718
application to Rossby waves. . . . . . . . . . . . . . . . . . . . .720
Bieberbach conjecture . . . . . . . . . . . . . . . . . . . . . . . . 417, 479
Jacobi polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .479
bifurcation sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .781
visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782
bigq-Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 471
bilateral basic hypergeometric function
. . . . . . . . . . seebilateralq-hypergeometric function.
bilateral hypergeometric function. . . . . . . . . . . . . . . . . .408
bilateralq-hypergeometric function
Bailey's 2 2transformations . . . . . . . . . . . . . . . . . . . . 429
Bailey's bilateral summations . . . . . . . . . . . . . . . . . . . 427
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Ramanujan's 1 1summation. . . . . . . . . . . . . . . . . . . .427
special cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .427{428
transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
bilateral series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .408
bilinear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
cross ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
SL(2;Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
binary number system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
binary quadratic sieve
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
Binet's formula
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
binomial coecients
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .619
894 Index
limiting form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .619
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 619
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619, 635
binomial expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
binomial theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
binomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
black holes
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
Bohr radius
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
Bohr-Mollerup theorem
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
q-gamma function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145
Boole summation formula . . . . . . . . . . . . . . . . . . . . . . . . . 597
Borel summability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
Borel transform theory
applications to asymptotic expansions . . . . . . . . . . . . 68
Bose{Einstein condensates
Lam e functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
Bose{Einstein integrals
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
relation to polylogarithms. . . . . . . . . . . . . . . . . . . . . . .612
Bose{Einstein phase transition . . . . . . . . . . . . . . . . . . . . 614
bound-state problems
hydrogenic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
Whittaker functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .754
boundary points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 15
boundary-value methods or problems
di erence equations . . . . . . . . . . . . . . . . . . . . . . . . . . 86, 87
ordinary di erential equations. . . . . . . . . . . . . . . . . . . .88
parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . 317
bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Boussinesq equation
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
box
plane partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
branch
of multivalued function . . . . . . . . . . . . . . . . . . . . . . 20, 104
construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
branch cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
branch point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
movable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
Bromwich integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bulirsch's elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . 487
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
rst, second, and third kinds . . . . . . . . . . . . . . . . . . . . 486
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
relation to symmetric elliptic integrals. . . . . . . . . . .508
calculus
complex variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14{18
one variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4{7two or more variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 7{9
calculus of nite di erences. . . . . . . . . . . . . . . . . . . . . . . .597
canonical integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
applications
acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792
caustics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .791
integrals with coalescing critical points . . . 789{790
optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791
quantum mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . .791
asymptotic approximations . . . . . . . . . . . . . . . . . 789{790
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792
convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
integral identities. . . . . . . . . . . . . . . . . . . . . . . . . . .787{788
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
relations to other functions
Airy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
Pearcey integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
visualizations of modulus . . . . . . . . . . . . . . . . . . . 778{779
visualizations of phase . . . . . . . . . . . . . . . . . . . . . . 780{781
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785{787
cardinal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
cardinal monosplines. . . . . . . . . . . . . . . . . . . . . . . . . .597{598
cardinal spline functions. . . . . . . . . . . . . . . . . . . . . . . . . . .597
Carmichael numbers
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
Casimir forces
Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
Casimir{Polder e ect
Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 614
Catalan numbers
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .623
limiting forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .621
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 620
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .621
Catalan's constant
Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 610
Cauchy determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Cauchy principal values
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Cauchy's integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . 16
for derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
Cauchy's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Cauchy{Riemann equations . . . . . . . . . . . . . . . . . . . . . . . . 16
Cauchy{Schwarz inequalities for sums and integrals
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 13
caustics
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Index 895
canonical integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791
Cayley's identity for Schwarzian derivatives. . . . . . . . .27
central di erences in imaginary direction . . . . . . . . . . 436
Ces aro means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Ces aro summability. . . . . . . . . . . . . . . . . . . . . . . . . . . . .33, 34
chain rule
for derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 7
characteristic equation
di erence equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
characteristics
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 539
characters
number theory
Dirichlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642
induced modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
orthogonality relation. . . . . . . . . . . . . . . . . . . . . . . . .642
primitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
principal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642
quadratic Jacobi symbol . . . . . . . . . . . . . . . . . . . . . . 642
quadratic Legendre symbol . . . . . . . . . . . . . . . . . . . 642
real. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642
Charlier polynomials
. . . . . . . . . . . . . seeHahn class orthogonal polynomials.
Chebyshev -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
. .see also Chebyshev-series expansions andclassical
orthogonal polynomials.
applications
approximation theory . . . . . . . . . . . . . . . . . . . . . . . . . 478
solutions of di erential equations . . . . . . . . . . . . . 478
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
dilated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .437
expansions in series of . . . . . . . . . . . . . . . . . . 96, 459, 461
explicit representations . . . . . . . . . . . . . . . . . . . . . 442{443
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 448
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
interrelations with other classical orthogonal polyno-
mials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444{445
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
linearization formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
local maxima and minima. . . . . . . . . . . . . . . . . . . . . . .451
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
of the rst, second, third, and fourth kinds . . . . . . 439
orthogonality properties
with respect to integration . . . . . . . . . . . . . . . . 96, 439with respect to summation . . . . . . . . . . . . . . . . 97, 440
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . 96, 446
relations to other functions
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . 394
Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . 442
Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
scaled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .478
shifted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437, 439
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .480
of coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438, 440
Chebyshev-series expansions
complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
computation of coecients . . . . . . . . . . . . . . . . . . . . . . . 97
relation to minimax polynomials . . . . . . . . . . . . . . . . . 97
summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97
chemical reactions
3j;6j;9jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
chi-square distribution function
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
Chinese remainder theorem
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
Christo el coecients (or numbers) . . . . . . . . . seeGauss
quadrature, Christo el coecients (or numbers)
Christo el-Darboux formula
classical orthogonal polynomials. . . . . . . . . . . . . . . . .438
con
uent form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
Chu{Vandermonde identity
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 387
circular trigonometric functions
. . . . . . . . . . . . . . . . . . . . . . seetrigonometric functions.
classical dynamics
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .566
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
classical orthogonal polynomials . . . . . . . . . . . . . . . . . . 438
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .459
applications
approximation theory . . . . . . . . . . . . . . . . . . . . . . . . . 478
Bieberbach conjecture . . . . . . . . . . . . . . . . . . . . . . . . 479
integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
numerical solution of di erential equations . . . . 478
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
quantum mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . .479
Radon transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . 479
Riemann{Hilbert problems . . . . . . . . . . . . . . . . . . . 479
asymptotic approximations . . . . . . . . . . . . . . . . . 451{454
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
896 Index
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .460
contiguous relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438{439
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446{447
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
expansions in series of . . . . . . . . . . . . . . . . . . . . . . 459{461
explicit representations . . . . . . . . . . . . . . . . . . . . . 442{443
Fourier transforms. . . . . . . . . . . . . . . . . . . . . . . . . .456{457
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
in two or more variables . . . . . . . . . . . . . . . . . . . . . . . . 477
inequalities
local maxima and minima . . . . . . . . . . . . . . . . 450{451
Turan-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
upper bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450
integral representations. . . . . . . . . . . . . . . . . . . . .447{448
for products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455{459
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .459
interrelations
limiting forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
with other orthogonal polynomials . . . . . . . . . . . . 464
Laplace transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .457
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
limiting forms
Mehler{Heine type formulas . . . . . . . . . . . . . . . . . . 449
linearization formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
local maxima and minima . . . . . . . . . . . . . . . . . . 450{451
Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
multiplication theorems . . . . . . . . . . . . . . . . . . . . . . . . . 460
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
orthogonality properties . . . . . . . . . . . . . . . . . . . . 439, 443
parameter constraints . . . . . . . . . . . . . . . . . . . . . . 439, 443
Poisson kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
relations to other functions
con
uent hypergeometric functions . . . . . . . . . . . 442
generalized hypergeometric functions. . . . . . . . . .442
hypergeometric function . . . . . . . . . . . . . 393{394, 442
sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459{461
Bateman-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .480
of coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
zeros
asymptotic approximations . . . . . . . . . . . . . . . 454{455
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .438
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
classical theta functions . . . . . . . . . . . seetheta functions.Clausen's integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
Clebsch{Gordan coecients . . . . . . . . . . . see3jsymbols.
relation to generalized hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
Clenshaw's algorithm
Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
classical orthogonal polynomials. . . . . . . . . . . . . . . . .480
Clenshaw{Curtis quadrature formula. . . . . . . . . . . .79, 82
comparison with Gauss quadrature . . . . . . . . . . . . . . . 80
closed point set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 15
closure
of interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
of point sets in complex plane . . . . . . . . . . . . . . . . . . . . 15
coalescing saddle points. . . . . . . . . . . . . . . . . . . . . . .789{790
coaxial circles
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .516
coding theory
combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
Krawtchouk and q-Racah polynomials. . . . . . . . . . .479
cofactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seedeterminants.
coherent states
generalized
con
uent hypergeometric functions . . . . . . . . . . . 346
cols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seesaddle points.
combinatorial design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .635
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
generalized hypergeometric functions . . . . . . . . . . . . 417
hypergeometric identities. . . . . . . . . . . . . . . . . . . . . . . .400
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
compact set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
complementary error function. . . . . . seeerror functions.
complementary exponential integral
. . . . . . . . . . . . . . . . . . . . . . . . . seeexponential integrals.
completely multiplicative functions . . . . . . . . . . . . . . . . 640
complex numbers
arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
complex conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
DeMoivre's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
imaginary part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
polar representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
real part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
triangle inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
complex physical systems
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
complex tori
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533
computer arithmetic
Index 897
generalized exponentials and logarithms . . . . . . . . . 131
computer-aided design
Cornu's spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
conductor
generalized Bernoulli polynomials . . . . . . . . . . . . . . . 597
con
uent Heun equation. . . . . . . . . . . . . . . . . . . . . . . . . . .717
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
properties of solutions. . . . . . . . . . . . . . . . . . . . . . . . . . .718
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
con
uent hypergeometric functions . . see also Kummer
functions andWhittaker functions.
of matrix argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
asymptotic approximations . . . . . . . . . . . . . . . . . . . 771
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
rst kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
Laguerre form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
relations to Bessel functions of matrix argument
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
second kind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .768
relations to other functions
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Bessel and Hankel functions . . . . . . . . . . . . . . . . . . 228
classical orthogonal polynomials . . . . . . . . . . . . . . 442
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . 742, 748
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
exponential integrals. . . . . . . . . . . . . . . . . . . . . . . . . .153
generalized Bessel polynomials . . . . . . . . . . . . . . . . 476
generalized exponential integral . . . . . . . . . . . . . . . 186
Hahn class orthogonal polynomials. . . . . . . . . . . .466
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . 255
parabolic cylinder functions . . . . . . . . . . . . . . 308, 315
repeated integrals of error functions. . . . . . . . . . .167
sine and cosine integrals . . . . . . . . . . . . . . . . . . . . . . 153
conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16{17
generalized hypergeometric functions . . . . . . . . . . . . 417
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 399
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .564
modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .581
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .515
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 581
congruence of rational numbers. . . . . . . . . . . . . . . . . . . .593
conical functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379
asymptotic approximations
large degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
large order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .374
behavior at singularities. . . . . . . . . . . . . . . . . . . . . . . . .373
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .372
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .372generalized Mehler{Fock transformation. . . . . . . . .373
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 373
integrals with respect to degree . . . . . . . . . . . . . . . . . 375
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352, 372
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .380
trigonometric expansion. . . . . . . . . . . . . . . . . . . . . . . . .373
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375
connected point set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
constants
roots of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24{25
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
canonical denominator (or numerator). . . . . . . . . . . .24
contraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
convergents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
existence of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
determinant formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
equivalent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
even part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
fractional transformations . . . . . . . . . . . . . . . . . . . . . . . . 25
J-fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Jacobi fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
associated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
numerical evaluation
backward recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
forward recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
forward series recurrence. . . . . . . . . . . . . . . . . . . . . . .96
odd part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Pringsheim's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
quotient-di erence algorithm . . . . . . . . . . . . . . . . . . . . . 95
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
relation to power series . . . . . . . . . . . . . . . . . . . . . . . 94, 95
S-fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Stieltjes fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Van Vleck's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
continuous dual Hahn polynomials
. . . . . . . . . . . seeWilson class orthogonal polynomials.
continuous dynamical systems and mappings
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
continuous function
at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 7, 15
notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
of two variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7, 15
on a point set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
on a region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
on an interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
898 Index
on the left (or right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
piecewise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 7
removable discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
sectionally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
simple discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
continuous Hahn polynomials
. . . . . . . . . . . . . seeHahn class orthogonal polynomials.
continuous q-Hermite polynomials . . . . . . . . . . . . . . . . . 473
continuous q 1-Hermite polynomials. . . . . . . . . . . . . . .473
asymptotic approximations to zeros . . . . . . . . . . . . . 474
continuous q-ultraspherical polynomials . . . . . . . . . . . 473
contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
simple closed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
convergence
acceleration . . . . . . . . . seeacceleration of convergence.
cubic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90
geometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
of thepth order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
convex functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
coordinate systems
cylindrical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
ellipsoidal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582, 693
elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
elliptical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .677{678
oblate spheroidal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .705
parabolic cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
paraboloid of revolution. . . . . . . . . . . . . . . . . . . . . . . . .317
paraboloidal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346, 678
polar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
projective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
prolate spheroidal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .704
spherical (or spherical polar) . . . . . . . . . . . . . . . . . . . . . . 8
sphero-conal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
toroidal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371, 379
Cornu's spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
connection with Fresnel integrals . . . . . . . . . . . . . . . . 168
cosecant function . . . . . . . . . seetrigonometric functions.
cosine function. . . . . . . . . . . . seetrigonometric functions.
cosine integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .151
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 153
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 154
auxiliary functions . . . seeauxiliary functions for sine
and cosine integrals.
Chebyshev-series expansions . . . . . . . . . . . . . . . . 156{157
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150expansion in spherical Bessel functions . . . . . . . . . . 153
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188{189
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
hyperbolic analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . 151
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 152
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151
principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
relations to exponential integrals . . . . . . . . . . . . . . . . 151
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156
value at in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . 154
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
cosmology
con
uent hypergeometric functions . . . . . . . . . . . . . . 346
incomplete beta functions . . . . . . . . . . . . . . . . . . . . . . . 189
cotangent function . . . . . . . . seetrigonometric functions.
Coulomb excitation of nuclei . . . . . . . . . . . . . . . . . . . . . . 753
Coulomb eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
Coulomb functions
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
Coulomb functions: variables ;. . . . . . . . . . . . . . . . . 742
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753{755
asymptotic expansions
large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .747
large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .747
uniform expansions. . . . . . . . . . . . . . . . . . . . . . .747{748
case= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
complex variable and parameters . . . . . . . . . . . 748, 754
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .745
conversions between variables and parameters . . . 754
cross-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
expansions in Airy functions . . . . . . . . . . . . . . . . . . . . 747
expansions in Bessel functions. . . . . . . . . . . . . . . . . . .746
expansions in modi ed Bessel functions . . . . . . . . . 746
expansions in spherical Bessel functions . . . . . . . . . 745
functionsF`(;);G`(;);H
`(;). . . . . . . . . . . . .742
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743{744
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 745
limiting forms
large`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
largejj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746
large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746, 747
smalljj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
Index 899
normalizing constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
phase shift (or phase) . . . . . . . . . . . . . . . . . . . . . . 742, 756
power-series expansions in . . . . . . . . . . . . . . . . . . . . .745
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
relations to other functions
con
uent hypergeometric functions . . . . . . . . . . . 742
Coulomb functions with variables r;. . . . . . . . . 751
Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . 742
scaling of variables and parameters . . . . . . . . . 753, 754
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .755
transition region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
WKBJ approximations. . . . . . . . . . . . . . . . . . . . . . . . . .755
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742
Coulomb functions: variables r;. . . . . . . . . . . . . . . . . . 748
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753{755
asymptotic approximations and expansions for large
jrj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
asymptotic expansions as !0 . . . . . . . . . . . . . . . . . 753
uniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .753
case= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .752
complex variables and parameters . . . . . . . . . . . . . . . 754
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
conversions between variables and parameters . . . 754
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752
expansions in Airy functions . . . . . . . . . . . . . . . . . . . . 753
expansions in Bessel functions . . . . . . . . . . . . . . 752, 753
expansions in modi ed Bessel functions. . . . .752, 753
functionsf(;`;r);h(;`;r). . . . . . . . . . . . . . . . . . . . . .748
functionss(;`;r);c(;`;r) . . . . . . . . . . . . . . . . . . . . . . 748
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749{750
integral representations for Dirac delta . . . . . . . . . . 749
limiting forms for large `. . . . . . . . . . . . . . . . . . . . . . . . 752
power-series expansions in . . . . . . . . . . . . . . . . . . . . . 752
power-series expansions in r. . . . . . . . . . . . . . . . . . . . .752
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752
relations to other functions
con
uent hypergeometric functions . . . . . . . . . . . 748
Coulomb functions with variables ;. . . . . . . . . 751
Whittaker functions. . . . . . . . . . . . . . . . . . . . . .748, 751
scaling of variables and parameters . . . . . . . . . 753, 754
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .755
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
Coulomb phase shift . . . . . . . . . . . . . . . . 145, 742, 755, 756
Coulomb potential barriers . . . . . . . . . . . . . . . . . . . . . . . . 754
Coulomb potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 753{754
q-hypergeometric function. . . . . . . . . . . . . . . . . . . . . . .432
Coulomb radial functions
. . . . . . . . . . . . seeCoulomb functions: variables ;.
Coulomb spheroidal functions . . . . . . . . . . . . . . . . . . . . . 704
as con
uent Heun functions . . . . . . . . . . . . . . . . . . . . . 717
Coulomb wave equation
irregular solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 742, 748regular solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 742, 748
singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742, 748
turning points . . . . . . . . . . . . . . . . . . . . . . . . . 742, 748, 754
Coulomb wave functions . . seeCoulomb functions: vari-
ables;andCoulomb functions: variables r;.
counting techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
critical phenomena
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
critical points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
coalescing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789{790
cross ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
cryptography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .647
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
cubature
for disks and squares . . . . . . . . . . . . . . . . . . . . . . . . . 84{85
cubic equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
cubic equations
solutions as trigonometric and hyperbolic functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
curve
piecewise di erentiable . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
simple closed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
cusp bifurcation set
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .782
cusp canonical integral . . . . . . . . . . . . . . . . . . . . . . . 776, 785
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .785
table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
cusp catastophe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776, 784
cuspoids
normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
cyclic identities
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .558
cyclotomic elds
Bernoulli and Euler polynomials . . . . . . . . . . . . . . . . 598
cylinder functions
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
di erential equations . . . . . . . . . . . . . . . . . . . . . . . 217, 226
integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .240{241
multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 246
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
zeros. . . . . . . . . . . . . . . . . seezeros of cylinder functions.
cylindrical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
cylindrical polar coordinates
. . . . . . . . . . . . . . . . . . . . . . . seecylindrical coordinates.
900 Index
Darboux's method
asymptotic approximations of sums and sequences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65{66
Dawson's integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 162
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
relation to error functions . . . . . . . . . . . . . . . . . . . . . . . 162
relation to parabolic cylinder functions . . . . . . . . . . 308
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
de Branges{Wilson beta integral . . . . . . . . . . . . . . . . . . 143
De Moivre's theorem
trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 118
Dedekind modular function. . . . . . . . . . . . . . . . . . . . . . . .646
functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Dedekind sums
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Dedekind's eta function . . . . . . . . seemodular functions.
Dedekind's modular function . . . seemodular functions.
del operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Dellanoy numbers
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 621
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .621
delta sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
delta wing equation
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
derivatives
chain rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5, 7
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 7
distributional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Fa a di Bruno's formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
L'H^ opital's rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
left-hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Leibniz's formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 7
of distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
right-hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Descartes' rule of signs (for polynomials) . . . . . . . . . . . 22
determinants
alternants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
circulant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
cofactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3de nition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Hadamard's inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Hankel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93, 595
inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
in nite
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Hill's type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Krattenthaler's formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
persymmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Vandermonde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
diatomic molecules
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
di erence equations
asymptotic solutions
. .seeasymptotic solutions of di erence equations.
distinguished solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
minimal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85{88
backward recursion method . . . . . . . . . . . . . . . . 85, 86
boundary-value methods . . . . . . . . . . . . . . . . . . . 86, 87
homogeneous equations. . . . . . . . . . . . . . . . . . . . .85{86
inhomogeneous equations . . . . . . . . . . . . . . . . . . . . . . 86
normalizing factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
recessive solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
di erence operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
central in imaginary direction . . . . . . . . . . . . . . . . . . . 436
forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
di erentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 15
di erential equations
asymptotic solutions. . . . . seeasymptotic solutions of
di erential equations.
change of variables
elimination of rst derivative. . . . . . . . . . . . . . . . . . .26
Liouville transformation . . . . . . . . . . . . . . . . . . . . . . . 26
point at in nity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
classi cation of singularities . . . . . . . . . . . . . . . . . 56, 409
closed-form solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
dominant solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Fuchs{Frobenius theory . . . . . . . . . . . . . . . . . . . . . . . . . . 55
homogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 88
indices di ering by an integer . . . . . . . . . . . . . . . . . . . . 56
indicial equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
inhomogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 88
solution by variation of parameters. . . . . . . . . . . . .26
irregular singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
nonhomogeneous. . . . . . . . . . . . . . . . seeinhomogeneous.
numerical solution
boundary-value problems . . . . . . . . . . . . . . . . . . . . . . 88
eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Index 901
eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
initial-value problems . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Runge{Kutta method . . . . . . . . . . . . . . . . . . . . . . 89{90
stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Sturm{Liouville eigenvalue problems . . . . . . . . . . . 89
Taylor-series methods . . . . . . . . . . . . . . . . . . . . . . 88{89
numerically satisfactory solutions. . . . . . . . . . . . . . . . .58
of arbitrary order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
ordinary point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55, 409
rank of singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
recessive solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
regular singularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56
Schwarzian derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
solutions
existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
fundamental pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
in series of Chebyshev polynomials . . . . . . . 478, 480
in series of classical orthogonal polynomials . . . 479
linearly independent . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
subdominant solutions . . . . . . . seerecessive solutions.
with a parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
di erentiation
Cauchy{Riemann equations . . . . . . . . . . . . . . . . . . . . . . 16
numerical
analytic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
Lagrange's formula for equally-spaced nodes . . . 77
partial derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78
of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 21
partial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
di raction catastrophes . . . . . . . . . . . . . . . . . . . . . . 777, 789
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
di raction of light
Fresnel integrals and Cornu's spiral. . . . . . . . .161, 169
di raction problems
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .678
di usion equations
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533
di usion problems
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .678
digamma function . . . . . . . . . . . . . . . . . . . . seepsi function.
dilogarithms
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
principal branch (or value) . . . . . . . . . . . . . . . . . . . . . . 610
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
Dirac delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37{38
delta sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37{38
integral representationsAiry functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
Coulomb functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37{38
spherical Bessel functions . . . . . . . . . . . . . . . . . . . . . . 38
mathematical de nitions . . . . . . . . . . . . . . . . . . . . . . . . . 38
series representations
Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 38
spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Dirac delta distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
Dirac delta function. . . . . . . . . . . . . . . . . . . seeDirac delta.
Dirac equation
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
DirichletL-functions
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .613
Dirichlet characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
Gauss sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
Dirichlet problem
with toroidal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 379
Dirichlet product (or convolution) . . . . . . . . . . . . . . . . . 641
Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602, 640
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
Dirichlet's divisor problem
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
Dirichlet's theorem
prime numbers in arithmetic progression . . . . . . . . 643
discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
discrete Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . 99
discreteq-Hermite I and II polynomials. . . . . . . . . . . .471
discriminant
of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
discriminant function
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
disk
around in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
open . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
disk polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
dislocation theory
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
distribution function
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
distribution functions
connection with
incomplete beta functions. . . . . . . . . . . . . . . . . . . . .189
incomplete gamma functions . . . . . . . . . . . . . . . . . . 189
902 Index
distributional derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35{37
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35, 36
convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36
distributional derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Heaviside function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
linear functionals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
of derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
regular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36{37
singular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
tempered . . . . . . . . . . . . seetempered distributions., 52
test function space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
divergence theorem
. .seeGauss's theorem for vector-valued functions.
divergent integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51
regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
divided di erences
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
divisor function
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
Dixon's 3F2(1) sum
q-analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Dixon's sum
F. H. Jackson's q-analog . . . . . . . . . . . . . . . . . . . . . . . . 426
domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
simply-connected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
dominated convergence theorem
in nite series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
double gamma function . . . . . . . seeBarnes'G-function.
double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8{9
change of order of integration . . . . . . . . . . . . . . . . . . . . . 9
change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
in nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
double sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
double series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
doubly-con
uent Heun equation . . . . . . . . . . . . . . . . . . . 717
Dougall's 7F6(1) sum
F. H. Jackson's q-analog . . . . . . . . . . . . . . . . . . . . . . . . 427
Dougall's bilateral sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Dougall's expansion
associated Legendre functions . . . . . . . . . . . . . . . . . . . 371dual Hahn polynomials
. . . . . . . . . . . seeWilson class orthogonal polynomials.
Dung's equation
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .565
dynamical systems
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Dyson's integral
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
ecological systems
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
Einstein summation convention for vectors. . . . . . . . . .10
Eisenstein convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
Eisenstein series
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .559
electric particle eld
Stieltjes electrostatic interpretation . . . . . . . . . . . . . 719
electromagnetic scattering
Bessel functions and spherical Bessel functions . . 275
electromagnetic theory
sine and cosine integrals . . . . . . . . . . . . . . . . . . . . . . . . 155
electromagnetic waves
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .678
electron-ion collisions
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
electronic structure of heavy elements
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
electrostatics
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .566
zeros of classical orthogonal polynomials . . . . . . . . 479
elementary functions. . . . . . . . . . . . . seeexponential func-
tion, hyperbolic functions, inverse hyperbolic func-
tions, inverse trigonometric functions, Lambert W-
function, logarithm function, power function, and
trigonometric functions.
relation to RC-function . . . . . . . . . . . . . . . . . . . . . . . . . 495
elementary particle physics
conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
ellipse
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .514
ellipse arc length
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .563
ellipsoid
capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
depolarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .497, 516
potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .516
self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
surface area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515
triaxial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
ellipsoidal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
ellipsoidal harmonics
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
ellipsoidal wave equation . . . . . seeLam e wave equation.
Index 903
elliptic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
elliptic crack and punch problems
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
addition law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
Jacobi{Abel addition theorem. . . . . . . . . . . . . . . . . . .564
Jacobian normal form. . . . . . . . . . . . . . . . . . . . . . . . . . .564
Mordell's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 581
elliptic functions . . . see also Jacobian elliptic functions
andWeierstrass elliptic functions.
general. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .571
representation as Weierstrass elliptic functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
Weierstrass. . . . . . . . seeWeierstrass elliptic functions.
elliptic integrals . . seebasic elliptic integrals, Bulirsch's
elliptic integrals, general elliptic integrals, general-
izations of elliptic integrals, Legendre's elliptic in-
tegrals, andsymmetric elliptic integrals.
complete
quasiconformal mapping . . . . . . . . . . . . . . . . . . . . . . 399
relations to other functions
associated Legendre functions. . . . . . . . . . . . . . . . .360
Ferrers functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .360
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . 576
elliptic modular function . . . . . . . seemodular functions.
elliptic umbilic bifurcation set
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .782
elliptic umbilic canonical integral . . . . . . . . . . . . . . . . . 776
asymptotic approximations . . . . . . . . . . . . . . . . . 789{790
convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
formulas for Stokes set . . . . . . . . . . . . . . . . . . . . . . . . . . 783
integral identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .788
pictures of modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
pictures of phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780
scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .786
elliptic umbilic catastrophe. . . . . . . . . . . . . . . . . . . 776, 785
elliptical coordinates
Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . 677{678
entire functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Liouville's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
enumerative topology
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
epsilon function. . . . . . . . . . seeJacobi's epsilon function.
equation of Ince . . seeHill's equation, equation of Ince.
equiconvergent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
Erlang loss function
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189error-control function
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
applications
asymptotic approximation of integrals . . . . . . . . 168
physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
Stokes phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . .168
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 164
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 164
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83{84, 169
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
expansions in spherical Bessel functions . . . . . . . . . 162
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166
graphics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160, 161
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
integral representations. . . . . . . . . . . . . . . . . . . . .162{163
integrals
Fourier transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . .166
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 166
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . 166
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . . . 162
relations to other functions
con
uent hypergeometric functions . . 164, 328, 338
Dawson's integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . .162
Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
generalized exponential integrals . . . . . . . . . . . . . . 164
incomplete gamma functions . . . . . . . . . . . . . . . . . . 164
parabolic cylinder functions. . . . . . . . . . . . . . . . . . .308
probability functions. . . . . . . . . . . . . . . . . . . . . . . . . .160
Voigt functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
repeated integrals of. . . . seerepeated integrals of the
complementary error function.
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169, 170
values at in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 165
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165, 170
error measures
absolute error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
complex arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
molli ed error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
relative error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
relative precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
error term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43
904 Index
essential singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
. . . . . . . . . . . . . . . see also isolated essential singularity.
eta function . . . . . . . . . . . . . . seeDedekind's eta function.
Euler{Maclaurin formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
arithmetic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . .593
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .593
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
nite expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596, 597
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 592
inversion formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
Kummer congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
recurrence relations
linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
quadratic and higher order. . . . . . . . . . . . . . . . . . . .595
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589, 598
Euler polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597{598
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .593
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
di erence equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
nite expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596, 597
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
in nite series expansions
Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 592
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
Laplace transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
multiplication formulas. . . . . . . . . . . . . . . . . . . . . . . . . .590
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
recurrence relations
linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .591
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595representations as sums of powers . . . . . . . . . . . . . . . 589
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
symbolic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589
zeros
complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
real. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .594
Euler product
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
Euler splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Euler sums
Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 613
reciprocity law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
Euler's beta integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142
Euler's constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 140
Euler's homogeneity relation
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .501
Euler's integral
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Euler's pentagonal number theorem
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Euler's sums
q-hypergeometric function . . . . . . . . . . . . . . . . . . 423, 424
Euler's totient
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
Euler's transformation
applied to asymptotic expansions. . . . . . . . . . . . . . . . .69
of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Euler{Fermat theorem
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638, 647
Euler{Poisson di erential equations . . . . . . . . . . . . . . . 501
Euler{Poisson{Darboux equation
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .501
Euler{Tricomi equation
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Eulerian numbers
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .632
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
relation to Bernoulli numbers . . . . . . . . . . . . . . . . . . . 591
relation to permutations . . . . . . . . . . . . . . . . . . . . . . . . 632
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .632
evolution equations
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
exact rational arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Index 905
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 132
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
graphics
complex argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
real argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
exponential growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
exponential integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .151
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 153
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 153
re-expansion of remainder term . . . . . . . . . . . . . . . 153
Chebyshev-series expansions. . . . . . . . . . . . . . . .156, 157
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
expansion in inverse factorials . . . . . . . . . . . . . . . . . . . 153
expansions in modi ed spherical Bessel functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 152
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
interrelations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150, 151
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151
principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
relations to other functions
con
uent hypergeometric functions . . . . . . . . . . . 153
incomplete gamma function. . . . . . . . . . . . . . . . . . .153
logarithmic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
sine and cosine integrals . . . . . . . . . . . . . . . . . . . . . . 151
small argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
extended complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Fa a di Bruno's formula
for derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Fabry's transformation
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
factorials (rising or falling) . . . . . . . . . . . . . . . . . . . . . . . . 618
factorization
of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
via Weierstrass elliptic functions . . . . . . . . . . . . . . 582
Faddeeva function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
fast Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Fay's trisecant identity
Riemann theta functions with characteristics . . . . 544
generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .544
Fej er kernel
Fourier integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Fourier series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
Fermat numbers
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
Fermat's last theorem
Bernoulli and Euler numbers and polynomials . . . 598
Fermi{Dirac integrals
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
relation to polylogarithms. . . . . . . . . . . . . . . . . . . . . . .612
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
uniform asymptotic approximation . . . . . . . . . . . . . . 612
Ferrers board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
Ferrers function
of the rst kind
integral equation for Lam e functions . . . . . . . . . . 689
Ferrers functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .370
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .376
applications
spherical harmonics . . . . . . . . . . . . . . . . . . . . . . 378{379
spheroidal harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 378
asymptotic approximations
. . . . . . . . . . seeuniform asymptotic approximations.
behavior at singularities . . . . . . . . . . . . . . . . . . . . 361{362
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .362
cross-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353{354
degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
with respect to degree or order. . . . . . . . . . . . . . . .363
di erential equation
. . . . . . . . . . . . . . . . seeassociated Legendre equation.
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355{357
integer degree and order . . . . . . . . . . . . . . . . . . . . 360{361
906 Index
integer order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 363
integrals
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
Mellin transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .370
orthogonality properties . . . . . . . . . . . . . . . . . . . . . . 369
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
of the rst kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
re
ection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
relations to other functions
elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
hypergeometric function. . . . . . . . . . . . .353, 354, 394
Legendre polynomials. . . . . . . . . . . . . . . . . . . . . . . . .360
ultraspherical polynomials . . . . . . . . . . . . . . . . . . . . 448
Rodrigues-type formulas . . . . . . . . . . . . . . . . . . . . . . . . 360
special values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .359{360
sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370{371
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .380
trigonometric expansions. . . . . . . . . . . . . . . . . . . . . . . .364
uniform asymptotic approximations
large degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366{368
large order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365{366
Wronskians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .352{353
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .368
Ferrers graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .626
Feynman diagrams
Appell functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
Feynman path integrals
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .534
Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 596, 629
ne structure constant
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
nite Fourier series
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
xed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
oating-point arithmetic
bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
format width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
signi cant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
double precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
exponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
fractional part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
IEEE standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
machine epsilon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
machine number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
machine precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
over
ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
roundingby chopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
to nearest machine number . . . . . . . . . . . . . . . . . . . . 72
signi cand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
single precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
under
ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Floquet solutions
Hill's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
Mathieu's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
Floquet's theorem
Hill's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
Mathieu's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
uid dynamics
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517
Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 545
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
fold canonical integral. . . . . . . . . . . . . . . . . . . . . . . . 776, 785
bifurcation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .788
integral identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .787
relation to Airy function . . . . . . . . . . . . . . . . . . . . . . . . 777
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .785
fold catastrophe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776, 785
Fourier cosine and sine transforms
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .400
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Parseval's formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31, 32
Fourier integral
asymptotic expansions. . . . . . . . . . . . . . . . . . . . . . . .44, 45
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
Fej er kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Poisson kernel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13{14
Bessel's inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
compendia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
Fej er kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
nite
number theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .643
integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Parseval's formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
Poisson kernel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
Poisson's summation formula. . . . . . . . . . . . . . . . . . . . .14
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33{34
Index 907
uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27{28
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
discrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
fast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
group
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . 400
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Parseval's formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30, 32
tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Fourier{Bessel expansion
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .248
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Fourier-series expansions
nonuniformity of convergence . . . . . . . . . . . . . . . . . . . 155
piecewise continuous functions . . . . . . . . . . . . . . . . . . 155
fractals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92
fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
fractional integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 53{55
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
fractional linear transformation
. . . . . . . . . . . . . . . . . . . . . . seebilinear transformation.
Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
applications
Cornu's spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
interference patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 161
physics and astronomy. . . . . . . . . . . . . . . . . . . . . . . .169
probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 164
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 164
auxiliary functions
. . . . . . seeauxiliary functions for Fresnel integrals.
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
expansions in spherical Bessel functions . . . . . . . . . 162
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
integrals
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . . . 162
relations to other functions
Anger{Weber functions . . . . . . . . . . . . . . . . . . . . . . . 297
auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . 160, 162
con
uent hypergeometric functions . . . . . . . . . . . 164
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162generalized hypergeometric functions. . . . . . . . . .164
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
values at in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 165
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Freud weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
Frobenius' identity
Riemann theta functions with characteristics . . . . 544
Fuchsian equation
classi cation of parameters. . . . . . . . . . . . . . . . . . . . . .718
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
polynomial solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
relation to Heun's equation . . . . . . . . . . . . . . . . . . . . . 718
functions
analytic . . . . . . . . . . . . . . . . . . . . . . . seeanalytic function.
analytically continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
continuous . . . . . . . . . . . . . . . . . seecontinuous function.
continuously di erentiable . . . . . . . . . . . . . . . . . . . . . . 5, 7
convex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
decreasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
de ned by contour integrals . . . . . . . . . . . . . . . . . . . . . . 21
di erentiable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
entire . . . . . . . . . . . . . . . . . . . . . . . . . . seeentire functions.
harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
holomorphic . . . . . . . . . . . . . . . . . . seeanalytic function.
increasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
limits . . . . . . . . . . . . . . . . . . . . . . . . seelimits of functions.
many-valued . . . . . . . . . . . . . . seemultivalued function.
meromorphic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
monotonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
multivalued . . . . . . . . . . . . . . . seemultivalued function.
nondecreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
nonincreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
of a complex variable . . . . . . . . . . . . . . . . . . . . . . . . . 18{22
of bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
of compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
of matrix argument
. . . . . . . . . . . . . . . . seefunctions of matrix argument.
strictly decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
strictly increasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
strictly monotonic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
support of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vector-valued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10{11
functions of matrix argument . . . . . . . . . . . . . . . . . . . . . 768
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .768
orthogonal invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
fundamental theorem of arithmetic . . . . . . . . . . . . . . . . 638
fundamental theorem of calculus. . . . . . . . . . . . . . . . . . . . .6
gamma distribution
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
908 Index
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
. . . . . . . . . . . . . . see also incomplete gamma functions.
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145{146
approximations
Chebyshev series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147
complex variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .147
rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146, 147
asymptotic expansions. . . . . . . . . . . . . . . . . . . . . .140{142
error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 141
for ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Bohr-Mollerup theorem . . . . . . . . . . . . . . . . . . . . . . . . . 138
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
duplication formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Euler's integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
extrema
asymptotic approximation . . . . . . . . . . . . . . . . . . . . 138
table of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Gauss's multiplication formula . . . . . . . . . . . . . . . . . . 138
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136{137
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
integral representations. . . . . . .139{140, 143{144, 188
for derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
multidimensional . . . . . . . . . . . . . . . . . . . . . . . . . 143{144
logarithm
continued fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136, 138
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
integral representations . . . . . . . . . . . . . . . . . . . . . . . 140
Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
multiplication formulas. . . . . . . . . . . . . . . . . . . . . . . . . .138
multivariate. . . . . . . seemultivariate gamma function.
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
reciprocal
analytic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . .136
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136, 137
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
re
ection formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
relations to hypergeometric function . . . . . . . . . . . . 387
scaled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146
Gaunt coecient
3jsymbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .761
Gaunt's integral3jsymbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .761
Gauss quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80{83
Christo el coecients (or numbers) . . . . . . . . . . . . . . 80
comparison with Clenshaw{Curtis formula . . . . . . . 80
eigenvalue/eigenvector characterization . . . . . . . . . . . 82
for contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83
Gauss{Chebyshev formula. . . . . . . . . . . . . . . . . . . . . . . .80
Gauss{Hermite formula . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Gauss{Jacobi formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Gauss{Laguerre formula . . . . . . . . . . . . . . . . . . . . . . 80{81
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Gauss{Legendre formula . . . . . . . . . . . . . . . . . . . . . . . . . 80
logarithmic weight function . . . . . . . . . . . . . . . . . . . 81{82
nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80{83
remainder terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80{83
Gauss series
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 384
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Gauss sums
number theory
Dirichlet character. . . . . . . . . . . . . . . . . . . . . . . . . . . .643
separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .532
Gauss's 2F1( 1) sum
q-analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Gauss's theorem for vector-valued functions . . . . . . . . 12
Gauss{Christo el quadrature . . . seeGauss quadrature.
Gaussian
nonperiodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
Gaussian elimination. . . . . . . . . . . . . . . . . . . . . . . . . . . .73{74
back substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
forward elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
iterative re nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
partial pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
pivot (or pivot element) . . . . . . . . . . . . . . . . . . . . . . . . . . 73
residual vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
triangular decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 73
tridiagonal systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
Gaussian hypergeometric function
. . . . . . . . . . . . . . . . . see also hypergeometric function.
of matrix argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
asymptotic approximations . . . . . . . . . . . . . . . . . . . 772
basic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .771
casem= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
con
uent form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
Gauss formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
integral representation . . . . . . . . . . . . . . . . . . . . . . . . 771
Index 909
Jacobi form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .771
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
partial di erential equations. . . . . . . . . . . . . .771{772
re
ection formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
transformations of parameters . . . . . . . . . . . . . . . . 771
Gaussian noise
LambertW-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Gaussian polynomials
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .635
Gaussian probability functions . . . . . . . . . . . . . . . . . . . . 160
Gaussian unitary ensemble
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Gegenbauer function
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
relation to associated Legendre functions . . . . . . . . 355
relation to hypergeometric function . . . . . . . . . . . . . 394
Gegenbauer polynomials
. . . .seeultraspherical polynomials and also classical
orthogonal polynomials.
Gegenbauer's addition theorem
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 261
general elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . 486
reduction to basic elliptic integrals . . . . . . . . . . . . . . 512
reduction to Legendre's elliptic integrals . . . . 496{497
reduction to symmetric elliptic integrals . . . . 512{514
general orthogonal polynomials . . . . . . . . . . . . . . . . . . . 437
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
di erence operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
monic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .438
on nite point sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
on intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
orthonormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
sums of products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .438
weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
x-di erence operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .438
generalizations of elliptic integrals . . . . . . . . . . . . . . . . . 516
generalized Airy functions
from di erential equation . . . . . . . . . . . . . . . . . . . 206{207
asymptotic approximations . . . . . . . . . . . . . . . . . . . 206
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206, 207
relation to Bessel functions . . . . . . . . . . . . . . . . . . . 206
relation to modi ed Bessel functions . . . . . . . . . . 206
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
from integral representations . . . . . . . . . . . . . . . 207{208
connection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 208
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
di erence equation. . . . . . . . . . . . . . . . . . . . . . . . . . . .208
di erential equation . . . . . . . . . . . . . . . . . . . . . . . . . . 208
generalized exponential integral . . . . . . . . . . . . . . . . . . . 185
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .187applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
asymptotic expansions
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 187
large parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187
large variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 191
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
further generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 187
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185{186
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 185
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
of large argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
relations to other functions
con
uent hypergeometric functions . . . . . . . . . . . 186
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
incomplete gamma functions . . . . . . . . . . . . . . . . . . 185
series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190
generalized exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
generalized functions
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
generalized hypergeometric di erential equation . . . 409
con
uence of singularities . . . . . . . . . . . . . . . . . . . . . . . 410
connection formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .410
fundamental solutions. . . . . . . . . . . . . . . . . . . . . . . . . . .409
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .409
generalized hypergeometric function 0F2
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404, 408
of large argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
generalized hypergeometric functions. . . . . . . . . . . . . . 404
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .408
analytic properties . . . . . . . . . . . . . . . . . . . . 404, 405, 408
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
argument unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
as functions of parameters . . . . . . . . . . . . . . . . . . . . . . 405
asymptotic expansions
formal series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
large parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .412
large variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
910 Index
small variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .408
balanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
bilateral series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Dougall's bilateral sum . . . . . . . . . . . . . . . . . . . . . . . 408
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
contiguous balanced series . . . . . . . . . . . . . . . . . . . . . . 407
contiguous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
contiguous relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407
de nitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .404, 408
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
di erential equation . . . . . . . . . . . . . . . . . seegeneralized
hypergeometric di erential equation.
Dixon's well-poised sum. . . . . . . . . . . . . . . . . . . . . . . . .406
Dougall's bilateral sum. . . . . . . . . . . . . . . . . . . . . . . . . .408
Dougall's very well-poised sum . . . . . . . . . . . . . . . . . . 406
D zrbasjan's sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
expansions in series of. . . . . . . . . . . . . . . . . . . . . . . . . . .410
extensions of Kummer's relations. . . . . . . . . . . . . . . .407
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 408
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
inverse Laplace transform. . . . . . . . . . . . . . . . . . . . .408
Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
k-balanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
Kummer-type transformations. . . . . . . . . . . . . .407, 409
monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
of matrix argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
con
uence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
convergence properties . . . . . . . . . . . . . . . . . . . . . . . . 772
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
Euler integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .773
expansion in zonal polynomials . . . . . . . . . . . . . . . 772
general properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . .772
invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
Kummer transformation . . . . . . . . . . . . . . . . . . . . . . 772
Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
Mellin{Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . 773
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
Pfa {Saalschutz formula. . . . . . . . . . . . . . . . . . . . . .772
relations to other functions . . . . . . . . . . . . . . . . . . . 772
Thomae transformation. . . . . . . . . . . . . . . . . . . . . . .772
3F2case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
value at T=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .772
Pfa {Saalsch utz balanced sum . . . . . . . . . . . . . . . . . . 406
polynomial cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
principal branch (value). . . . . . . . . . . . . . . . . . . . . . . . .404
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
relations to other functions
associated Jacobi polynomials. . . . . . . . . . . . . . . . .474Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
classical orthogonal polynomials . . . . . . . . . . . . . . 442
Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
generalized Bessel polynomials . . . . . . . . . . . . . . . . 476
Hahn class orthogonal polynomials. . . . . . . . . . . .463
Kummer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
MeijerG-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . .416
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . 255
orthogonal polynomials andother functions. . .409
3j;6j;9jsymbols . . . . . . . . . . . . . . . . . . . . . . . . 407, 418
Wilson class orthogonal polynomials . . . . . . . . . . 468
Rogers{Dougall very well-poised sum. . . . . . . . . . . .406
Saalsch utzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
terminating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
transformation of variable. . . . . . . . . . . . . . . . . . . . . . .408
cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
very well-poised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .405
Watson's sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
well-poised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .405
Whipple's sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Whipple's transformation . . . . . . . . . . . . . . . . . . . . . . . 407
with two variables . . . . . . . . . . . . . . . . . . . . . . . . . . 412{415
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .410
generalized hypergeometric series . . . . . . . . . . . . . . . . . . 404
generalized integrals
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
generalized logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 73, 111
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131
generalized precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
generalized sine and cosine integrals . . . . . . . . . . . . . . 188
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
asymptotic expansions for large variable. . . . . . . . .189
auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
asymptotic expansions for large variable . . . . . . 189
integral representations . . . . . . . . . . . . . . . . . . . . . . . 189
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
de nitions
general values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
expansions in series of spherical Bessel functions . . 188
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 188
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . . . 188
relation to sine and cosine integrals . . . . . . . . . . . . . 188
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Genocchi numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .596
genus
Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
geometric mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 13
geometric progression (or series) . . . . . . . . . . . . . . . . . . . . . 2
geophysics
Index 911
spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Gibbs phenomenon
sine integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
Glaisher's constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63, 144
Glaisher's notation
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .550
Goldbach conjecture
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
Goodwin{Staton integral
asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
relations to Dawson's integral and exponential inte-
gral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162
Graf's addition theorem
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 261
Gram{Schmidt procedure
for least squares approximation . . . . . . . . . . . . . . . . . . 99
graph theory
combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
gravitational radiation
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
Green's theorem for vector-valued functions
three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
two dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
group representations
orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 479
group theory
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
Gudermannian function . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
relation to RC-function . . . . . . . . . . . . . . . . . . . . . . . . . 495
relation to amplitude (am) function . . . . . . . . . . . . . 561
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
Haar measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
Hadamard's inequality for determinants . . . . . . . . . . . . . 3
Hahn class orthogonal polynomials. . . . . . . . . . . .462{467
asymptotic approximations . . . . . . . . . . . . . . . . . 466{467
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
di erence equations on variable . . . . . . . . . . . . . . . . . 465
di erences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463, 464
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
limit relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463
normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .462
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464relations to con
uent hypergeometric functions and
generalized hypergeometric functions. . . . .328, 463
relations to hypergeometric function. . . . . . . .394, 463
Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Hahn polynomials
. . . . . . . . . . . seeHahn class orthogonal polynomials.
Hamiltonian systems
chaotic
Lam e functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
handle
Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
Hankel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 246{247
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .226
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
asymptotic expansions for large argument . . 229{230
error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229{230
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 230
asymptotic expansions for large order. . . . . . .231{235
asymptotic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
double asymptotic properties. . . . . . . . . . . . .235, 258
transition region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232{235
branch conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276{277
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .222
cross-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
asymptotic expansions for large argument. . . . .229
asymptotic expansions for large order . . . . 231{232
uniform asymptotic expansions for large order
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
. . . . . . . . . . . . . . . . . . . . . . . . see also Bessel's equation.
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220{221
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
integral representations
along real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224
integrals
. . . . . . seeintegrals of Bessel and Hankel functions.
limiting forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217, 223
multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 246
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223
principal branches (or values) . . . . . . . . . . . . . . . . . . . 217
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
relations to other functions
Airy functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .196{197
912 Index
con
uent hypergeometric functions . . . . . . . . . . . 228
elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . .228
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .238
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
with respect to order ( -zeros) . . . . . . . . . . . . . . . . 240
Hankel transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Hankel's expansions
for Bessel and Hankel functions. . . . . . . . . . . . .228{229
for modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . 255
Hankel's integrals
Bessel functions and Hankel functions . . . . . . . . . . . 226
Hankel's inversion theorem
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246
Hankel's loop integral
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
harmonic analysis
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 399
harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
maximum modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
mean value property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Poisson integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
harmonic mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3, 13
harmonic oscillators
Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
q-hypergeometric function. . . . . . . . . . . . . . . . . . . . . . .432
harmonic trapping potentials
parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . 317
heat conduction in liquids
Rayleigh function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
heat theory
conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Heaviside function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36, 54
derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Heine's formula
associated Legendre functions . . . . . . . . . . . . . . . . . . . 377
Heine's integral
Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Helmholtz equation
3j;6j;9jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
associated Legendre functions . . . . . . . . . . . . . . . . . . . 379
Bessel functions and modi ed Bessel functions . . 275
parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . 317
paraboloidal coordinates . . . . . . . . . . . . . . . . . . . . . . . . 678
Hermite polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
. . . . . . . . . . . see also classical orthogonal polynomials.
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .460
applications
integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . 479
Schr odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . 479
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .453computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
expansions in series of . . . . . . . . . . . . . . . . . . . . . . 459{461
explicit representations . . . . . . . . . . . . . . . . . . . . . 442{443
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450, 451
Turan-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
integral representations. . . . . . . . . . . . . . . . . . . . .447, 448
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455, 457{459
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
Nicholson-type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455
interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444{445, 463, 464
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .457
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
limiting forms as trigonometric functions . . . . . . . . 449
linearization formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
local maxima and minima. . . . . . . . . . . . . . . . . . . . . . .451
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
monic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81, 441
multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 460
normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .439
Poisson kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
relations to other functions
con
uent hypergeometric functions . . 328, 338, 449
derivatives of the error function . . . . . . . . . . . . . . . 163
generalized hypergeometric functions. . . . . . . . . .443
parabolic cylinder functions. . . . . . . . . . . . . . . . . . .308
repeated integrals of the complementary error func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .480
of coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438, 455
asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 455
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81
Hermite{Darboux method
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .713
Hermitian matrices
Gaussian unitary ensemble
limiting distribution of eigenvalues . . . . . . . . . . . . 739
Index 913
Heun equation . . . . . . . . . . . . . . . . . . . seeHeun's equation.
Heun functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719{720
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .718
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
di erential equation. . . . . . . . . . . . seeHeun's equation.
expansions in series of
hypergeometric functions. . . . . . . . . . . . . . . . .716{717
orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . 717
integral equations and representations . . . . . . 714{716
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
orthogonality
double. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .714
single. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .714
relations to hypergeometric function . . . . . . . . . . . . 713
relations to Lam e functions . . . . . . . . . . . . . . . . . . . . . 713
Heun polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .719
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
integral equations and representations. . . . . . . . . . .715
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
Heun's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
accessory parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . .710
asymptotic approximations . . . . . . . . . . . . . . . . . . . 718
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719{720
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
asymptotic approximations
eigenvalues of accessory parameters . . . . . . . . . . . 718
solutions near irregular singularities. . . . . . . . . . .718
solutions of con
uent forms . . . . . . . . . . . . . . . . . . . 718
solutions with coalescing singularities . . . . . . . . . 718
automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711{713
composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
F-homotopic transformations . . . . . . . . . . . . . . . . . 711
homographic transformations . . . . . . . . . . . . . . . . . 711
basic solutions
equivalent expressions . . . . . . . . . . . . . . . . . . . . . . . . 712
Fuchs{Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
bicon
uent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718
classi cation of parameters. . . . . . . . . . . . . . . . . . . . . .710
computation of solutions . . . . . . . . . . . . . . . . . . . . . . . . 720
con
uent forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .717{718
asymptotic approximations . . . . . . . . . . . . . . . . . . . 718
integral equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .716
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
doubly-con
uent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
doubly-periodic forms
Jacobi's elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710Weierstrass's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
eigenvalues of accessory parameter . . . . . . . . . . . . . . 712
expansions of solutions in series of
hypergeometric functions. . . . . . . . . . . . . . . . .716{717
orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . 717
exponent parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . 714{716
integral representation of solutions. . . . . . . . . .714{716
kernel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 714, 716
separation constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
inversion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
Jacobi's elliptic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
Liouvillean solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
monodromy group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
parameters
classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
path-multiplicative solutions . . . . . . . . . . . . . . . . . . . . 712
biorthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
expansions in series of hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
relation to Fuchsian equation . . . . . . . . . . . . . . . . . . . 718
relation to Lam e's equation . . . . . . . . . . . . . . . . . . . . . 685
separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 720
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .710
singularity parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
solutions analytic at three singularities
. . . . . . . . . . . . . . . . . . . . . . . . . . . seeHeun polynomials.
solutions analytic at two singularities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seeHeun functions.
solutions via quadratures. . . . . . . . . . . . . . . . . . . . . . . .713
tricon
uent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
trigonometric form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
uniformization problem . . . . . . . . . . . . . . . . . . . . . . . . . 719
Weierstrass's form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
Heun's operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
hexadecimal number system . . . . . . . . . . . . . . . . . . . . . . . . 72
high-frequency scattering
parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . 317
higher-order 3 njsymbols. . . . . . . . . . . . . . . . . . . . . . . . . .765
highway design
Cornu's spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
Hilbert space
interrelation between bases
Heun polynomial products . . . . . . . . . . . . . . . 719{720
L2
(Q) orthonormal basis . . . . . . . . . . . . . . . . . . . . . . . . 719
Hilbert transform
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Fourier transform of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Hill's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
. . . . . . . . . . . . . . . . . . . see also Whittaker{Hill equation.
914 Index
antiperiodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
basic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 675
characteristic exponents. . . . . . . . . . . . . . . . . . . . . . . . .675
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .675
equation of Ince. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .676
Fourier-series solutions. . . . . . . . . . . . . . . . . . . . . . . .676
polynomial solutions . . . . . . . . seeInce polynomials.
expansions in series of eigenfunctions. . . . . . . . . . . .676
Floquet solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
Floquet's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
pseudoperiodic solutions . . . . . . . . . . . . . . . . . . . . . . . . 674
real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
separation constants . . . . . . . . . . . . . . . . . . . . . . . 677, 678
symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
H older's inequalities for sums and integrals . . . . . 12, 13
holomorphic function . . . . . . . . . . . . seeanalytic function.
homogeneous harmonic polynomials . . . . . . . . . . . . . . . 379
homographic transformation
. . . . . . . . . . . . . . . . . . . . . . seebilinear transformation.
Horner's scheme for polynomials. . . . . . . . . . . . . . . . . . . .22
extended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Hurwitz criterion for stable polynomials . . . . . . . . . . . . 23
Hurwitz system
Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Hurwitz zeta function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
asymptotic expansions for large parameter . . . . . . 610
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
asymptotic expansions for large parameter . . . . 610
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607{608
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 609
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
relations to other functions
Lerch's transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . 612
periodic zeta function. . . . . . . . . . . . . . . . . . . . . . . . .612
polylogarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .611
Riemann zeta function. . . . . . . . . . . . . . . . . . . . . . . .607
representations by Euler{Maclaurin formula . . . . . 608
series representations. . . . . . . . . . . . . . . . . . . . . . .608, 610
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .610
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
hydrodynamics
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .566
hyperasymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 68
hyperbola
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .514hyperbolic cosecant function . . seehyperbolic functions.
hyperbolic cosine function. . . . seehyperbolic functions.
hyperbolic cotangent function
. . . . . . . . . . . . . . . . . . . . . . . . . seehyperbolic functions.
hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
addition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
graphics
complex argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
real argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123{124
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
integrals
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
inverse . . . . . . . . . . . . . seeinverse hyperbolic functions.
Laurent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
moduli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126
multiples of argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
partial fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126
periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
poles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
real and imaginary parts . . . . . . . . . . . . . . . . . . . . . . . . 126
relations to trigonometric functions . . . . . . . . . . . . . 123
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
squares and products . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .130
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
hyperbolic secant function . . . seehyperbolic functions.
hyperbolic sine function . . . . . . seehyperbolic functions.
hyperbolic tangent function . . seehyperbolic functions.
hyperbolic trigonometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . seehyperbolic functions.
hyperbolic umbilic bifurcation set
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .782
hyperbolic umbilic canonical integral. . . . . . . . . . . . . . 776
asymptotic approximations . . . . . . . . . . . . . . . . . 789{790
convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
formulas for Stokes set . . . . . . . . . . . . . . . . . . . . . . . . . . 783
Index 915
integral identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .787
pictures of modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
pictures of phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .787
hyperbolic umbilic catastrophe . . . . . . . . . . . . . . . 776, 785
hyperelliptic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .566
hyperelliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
hypergeometric di erential equation . . . . . . . . . . . . . . 394
equivalent equation for contiguous functions. . . . .388
fundamental solutions . . . . . . . . . . . . . . . . . . . . . . 394{395
Kummer's solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .395
hypergeometric equation
. . . . . . . . . seehypergeometric di erential equation.
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . . . 384
. . . . . . . . . see also Gaussian hypergeometric function.
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
asymptotic approximations
largea(orb) andc. . . . . . . . . . . . . . . . . . . . . . 397, 398
largeaandb. . . . . . . . . . . . . . . . . . . . . . . . . . . . .397, 398
largeaorb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
largea,b, andc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
largec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396{398
large variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
contiguous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .389
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387{388
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385{386
Hankel transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
integral representations. . . . . . . . . . . . . . . . . . . . .388{389
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . 388{389
integrals . . . . . . . . . . . . . . . . . . . . . 326, 327, 337, 398{399
compendia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398{399
Laplace transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .398
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
multivariate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .498
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Olver's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384
polynomial cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
principal value (or branch) . . . . . . . . . . . . . . . . . . . . . . 384
products
series expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
relations to other functions
associated Legendre functions . . . . . . . 353, 354, 394classical orthogonal polynomials . . . . . . . . . . . . . . 442
elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . .386
Ferrers functions . . . . . . . . . . . . . . . . . . . . 353, 354, 394
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Gegenbauer function. . . . . . . . . . . . . . . . . . . . . . . . . .394
Hahn class orthogonal polynomials. . . . . . . . . . . .463
Heun functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
incomplete beta functions. . . . . . . . . . . . . . . . . . . . .183
Jacobi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
orthogonal polynomials. . . . . . . . . . . . . . . . . . .393{394
Painlev e transcendents. . . . . . . . . . . . . . . . . . . . . . . .399
Pollaczek polynomials . . . . . . . . . . . . . . . . . . . . . . . . 476
psi function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .387
symmetric elliptic integrals . . . . . . . . . . . . . . . . . . . 509
Szeg o{Askey polynomials . . . . . . . . . . . . . . . . . . . . . 475
Wilson class orthogonal polynomials . . . . . . . . . . 469
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384
special cases
argument1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
argument a fraction. . . . . . . . . . . . . . . . . . . . . . . . . . .387
argumentsei=3. . . . . . . . . . . . . . . . . . . . . . . . 387, 400
elementary functions . . . . . . . . . . . . . . . . . . . . . 386{387
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399
transformation of variable
cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390{391, 400
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391{393
with two variables . . . . . . . . . . . . . seeAppell functions.
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .398
hypergeometric functions of matrix argument
. . . .seecon
uent hypergeometric functions of ma-
trix argument, Gaussian hypergeometric functions
of matrix argument, andgeneralized hypergeomet-
ric functions of matrix argument.
hypergeometric R-function. . . . . . . . . . . . . . . . . . . . . . . . 498
derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .501
elliptic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 498
implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Ince polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .676
Ince's equation . . . seeHill's equation, equation of Ince.
Ince's theorem . . . . . . . . . . . . . . . . . . . seeTheorem of Ince.
incomplete Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . 208
incomplete beta functions. . . . . . . . . . . . . . . . . . . . . . . . . 183
applications
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
statistical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189
asymptotic expansions for large parameters
general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
916 Index
inverse function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185
symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184
historical pro le. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 183
inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
relation to hypergeometric function . . . . . . . . . . . . . 183
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . . . 174
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .174
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
statistical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
asymptotic approximations and expansions
exponentially-improved . . . . . . . . . . . . . . . . . . 179, 181
for inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
large variable and/or large parameter
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179{180, 182
uniform for large parameter . . . . . . . . . . . . . . 181, 182
basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 191
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179
de nitions
general values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
expansions in series of
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 178
modi ed spherical Bessel functions. . . . . . . . . . . .178
generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
graphics
complex argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
real variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .175{176
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
integral representations
along real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .178
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
monotonicity properties . . . . . . . . . . . . . . . . . . . . . . . . . 176
normalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
of imaginary argument . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Pad e approximant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179power-series expansions . . . . . . . . . . . . . . . . . . . . . . . . . 178
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
relations to other functions
con
uent hypergeometric functions . . 177, 328, 338
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164
exponential integrals. . . . . . . . . . . . . . . . . . . . . . . . . .153
generalized exponential integral . . . . . . . . . . . . . . . 185
incomplete Riemann zeta function . . . . . . . . . . . . 189
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182
incomplete Riemann zeta function . . . . . . . . . . . . . . . . . 189
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .189
expansions in series of incomplete gamma functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189
inductance
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .516
inequalities
means. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
sums and integrals
Cauchy{Schwarz . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 13
H older's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12, 13
Jensen's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Minkowski's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12, 13
in nite partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Mittag-Leer's expansion . . . . . . . . . . . . . . . . . . . . . . . . 22
in nite products
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
absolute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
uniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
M-test for uniform convergence . . . . . . . . . . . . . . . . . . 21
relation to in nite partial fractions . . . . . . . . . . . . . . . 22
Weierstrass product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
in nite sequences
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
absolute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
pointwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
uniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
relation to in nite double series . . . . . . . . . . . . . . . . 18
in nite series
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see also power series.
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
absolute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
pointwise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
uniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
Weierstrass M-test. . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
divergent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
dominated convergence theorem . . . . . . . . . . . . . . . . . . 18
double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Index 917
doubly-in nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
summability methods . . . . . . . . . . . . . . . . . . . . . . . . . 33{34
term-by-term integration . . . . . . . . . . . . . . . . . . . . . . . . . 18
inhomogeneous Airy functions . . . . seeScorer functions.
initial-value problems
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679
integrable di erential equations
Riemann theta functions . . . . . . . . . . . . . . . . . . . 545{546
integrable equations
. . . . . . . . . . . . . . seeintegrable di erential equations.
integral equations
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 729
integral transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
. . . . . . . . . . . . . . . . . . . . . see also Fourier cosine and sine
transforms, Fourier transform, Jacobi transform,
Hankel (or Bessel) transform, Hilbert transform,
Kontorovich{Lebedev transform, Laplace trans-
form, Mellin transform, spherical Bessel transform,
andStieltjes transform.
compendia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
in terms of parabolic cylinder functions . . . . . . . . . 317
in terms of Whittaker functions . . . . . . . . . . . . . . . . . 344
integrals
asymptotic approximations
. . . . . . . seeasymptotic approximations of integrals.
Cauchy principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
absolute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 21
convolution product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
de nite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 8
double . . . . . . . . . . . . . . . . . . . . . . . . . seedouble integrals.
fundamental theorem of calculus . . . . . . . . . . . . . . . . . . 6
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
inde nite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
in nite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6, 9, 16
Jensen's inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
mean value theorems
rst. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
second . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
multiple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
over parametrized surface . . . . . . . . . . . . . . . . . . . . . . . . 12
path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
repeated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
square-integrable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
summability methods . . . . . . . . . . . . . . . . . . . . . . . . . 34{35
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
with coalescing saddle points . . . . . . . . . . . . . . . 789{790
integrals of Bessel and Hankel functions
compendia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242fractional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Hankel (or Bessel) transform . . . . . . . . . . . . . . . . . . . . 246
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240{241
orthogonal properties . . . . . . . . . . . . . . . . . . . . . . 243, 244
over nite intervals . . . . . . . . . . . . . . . . . . . . . . . . . 241{243
over in nite intervals . . . . . . . . . . . . . . . . . . 243{246, 326
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241{246
triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
trigonometric arguments . . . . . . . . . . . . . . . . . . . . . . . . 241
integrals of modi ed Bessel functions
compendia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
fractional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Kontorovich{Lebedev transform. . . . . . . . . . . . . . . . .260
over nite intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .258
over in nite intervals. . . . . . . . .205, 258{260, 326, 337
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279
integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5, 16
by parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
numerical . . . . . . . . . . seecubature, Gauss quadrature,
Monte-Carlo methods, andquadrature.
term by term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
interaction potentials
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
interior Dirichlet problem
for oblate spheroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
for prolate spheroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
interior points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75{77, 91
. . . . . . . . . . . . . . . . . . . . see also Lagrange interpolation.
based on Chebyshev points. . . . . . . . . . . . . . . . . . . . . . .77
based on Sinc functions . . . . . . . . . . . . . . . . . . . . . . . . . . 77
bivariate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
convergence properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Hermite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
inverse linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
spline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
trigonometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
interval
closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
interval arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Lagrange inversion theorem . . . . . . . . . . . . . . . . . . . . . . 21
extended. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
inverse Gudermannian function. . . . . . . . . . . . . . . . . . . .121
relation to Legendre's elliptic integrals . . . . . . . . . . 491
relation to RC-function . . . . . . . . . . . . . . . . . . . . . . . . . 491
inverse hyperbolic functions. . . . . . . . . . . . . . . . . . . . . . . 127
addition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
918 Index
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
branch cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 132
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
fundamental property . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
general values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127
graphics
complex argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
real argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123{124
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
logarithmic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
re
ection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
values on the cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128
inverse incomplete beta function. . . . . . . . . . . . . . . . . . .185
inverse incomplete gamma function. . . . . . . . . . . . . . . .182
inverse Jacobian elliptic functions . . . . . . . . . . . . . . . . . 561
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .563
as Legendre's elliptic integrals. . . . . . . . . . . . . . . . . . .561
as symmetric elliptic integrals . . . . . . . . . . . . . . . . . . . 561
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
equivalent forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . . . 561
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
inverse Laplace transforms
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 46{47
inverse trigonometric functions. . . . . . . . . . . . . . . . . . . . 118
addition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
branch cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 132
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
fundamental property . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
general values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
graphicscomplex argument . . . . . . . . . . . . . . . . . . . . . . . 113{115
real argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
logarithmic forms. . . . . . . . . . . . . . . . . . . . . . . . . . .119{120
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
real and imaginary parts . . . . . . . . . . . . . . . . . . . . . . . . 120
re
ection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
values on the cuts . . . . . . . . . . . . . . . . . . . . . . . . . . 119{120
Ising model
Appell functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
generalized hypergeometric functions . . . . . . . . . . . . 417
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
isolated essential singularity . . . . . . . . . . . . . . . . . . . . . . . . 19
. . . . . . . . . . . . . . . . . . . . . . . see also essential singularity.
movable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
isolated singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
iterative methods
Bairstow's method (for zeros of polynomials) . . . . . 91
bisection method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
convergence
cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
geometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
of thepth order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
eigenvalue methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xed-point methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Halley's rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Newton's rule (or method) . . . . . . . . . . . . . . . . . . . . . . . 90
regula falsi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
secant method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Ste ensen's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Jacobi fraction ( J-fraction) . . . . . . . . . . . . . . . . . . . . . . . . . 95
Jacobi function
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
relations to other functions
associated Legendre functions. . . . . . . . . . . . . . . . .355
conical functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . 394
Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
. . . . . . . . . . . see also classical orthogonal polynomials.
applications
Bieberbach conjecture . . . . . . . . . . . . . . . . . . . . . . . . 479
associated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .474
Index 919
asymptotic approximations . . . . . . . . . . . . . . . . . 451{452
Bateman-type sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
expansions in series of . . . . . . . . . . . . . . . . . . . . . . 459{461
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450, 451
Szeg o{Sz asz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Turan-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
integral representations. . . . . . . . . . . . . . . . . . . . .447, 448
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455{457, 459
fractional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444{445, 463, 464
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .457
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
limiting form
as Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .449
as Bessel polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 476
limits to monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
local maxima and minima . . . . . . . . . . . . . . . . . . 450{451
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
monic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .439
parameter constraint . . . . . . . . . . . . . . . . . . . . . . . 439, 443
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
relations to other functions
hypergeometric function . . . . . . . . . . . . . . . . . 393, 442
orthogonal polynomials on the triangle. . . . . . . .478
Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
shifted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444
tables of coecients. . . . . . . . . . . . . . . . . . . . . . . . . . . . .440
upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438, 454
asymptotic approximations . . . . . . . . . . . . . . . . . . . 454
Jacobi symbol
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
Jacobi transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379, 394
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
Jacobi's amplitude function
. . . . . . . . . . . . . . . . . . . . . seeamplitude (am) function.
Jacobi's epsilon function . . . . . . . . . . . . . . . . . . . . . . . . . . 562
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .563
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 562
quasi-addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . 562
quasi-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
relation to Legendre's elliptic integrals . . . . . . . . . . 562
relation to theta functions . . . . . . . . . . . . . . . . . . . . . . 562
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .567
Jacobi's identities
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Jacobi's imaginary transformation . . . . . . . . . . . . . . . . . 556
Jacobi's inversion problem for elliptic functions . . . . 532
Jacobi's nome
power-series expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 490
Jacobi's theta functions . . . . . . . . . . . seetheta functions.
Jacobi's triple product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
q-version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .427
Jacobi's zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
quasi-addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . 562
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .567
Jacobi{Abel addition theorem
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .564
Jacobi-type polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . .477
Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . 550
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .557
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . 550, 563
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563{564
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564{566
change of modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .566{567
congruent points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
coperiodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
copolar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
cyclic identities
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
simultaneously permuted . . . . . . . . . . . . . . . . . . . . . 558
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
di erential equations
rst-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
second-order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
double argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
elementary identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
equianharmonic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
expansions in doubly-in nite partial fractions . . . 559
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
920 Index
for squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .559
fundamental unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
Glaisher's notation . . . . . . . . . . . . . . . . . . . . . . . . . 550, 554
graphical interpretation via Glaisher's notation . . 554
graphics
complex modulus . . . . . . . . . . . . . . . . . . . . . . . . 552{553
complex variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .552
real variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .550{552
half argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .556
hyperbolic series for squares. . . . . . . . . . . . . . . . . . . . .559
integrals
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
inverse . . . . . . . seeinverse Jacobian elliptic functions.
Jacobi's imaginary transformation. . . . . . . . . . . . . . .556
Landen transformations
ascending . . . . . . . . . . . . . . . . . . . . . . . . . . . 557, 563, 566
descending . . . . . . . . . . . . . . . . . . . . . . . . . . 556, 563, 566
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
lemniscatic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
limiting forms as k!0 ork!1. . . . . . . . . . . . . . . .555
Maclaurin series
ink;k0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
inz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
change of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . .552, 553, 563
limiting values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
outside the interval [0 ;1] . . . . . . . . . . . . . . . . . . . . . . 563
purely imaginary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
real. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .563
nome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550, 553{554
poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553{554
poristic polygon constructions. . . . . . . . . . . . . . . . . . .557
principal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
relations to other functions
symmetric elliptic integrals . . . . . . . . . . . . . . . . . . . 508
theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . 575
rotation of argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
special values of the variable . . . . . . . . . . . 554{555, 557
subsidiary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .550
sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .567
translation of variable. . . . . . . . . . . . . . . . . . . . . . . . . . .554
trigonometric series expansions. . . . . . . . . . . . . . . . . .559
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .553Jensen's inequality for integrals. . . . . . . . . . . . . . . . . . . . .13
Jonqui ere's function . . . . . . . . . . . . . . . seepolylogarithms.
Jordan curve theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Jordan's function
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
Jordan's inequality
sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Julia sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92
Kadomtsev{Petviashvili equation
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 545
Kapteyn's inequality
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227
Kelvin functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
asymptotic expansions for large argument . . . . . . . 271
cross-products and sums of squares . . . . . . . . . . . 271
exponentially-small contributions . . . . . . . . . . . . . 271
asymptotic expansions for large order . . . seeuniform
asymptotic expansions for large order.
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276{277
cross-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
with respect to order . . . . . . . . . . . . . . . . . . . . . . . . . 269
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
expansions in series of Bessel functions . . . . . . . . . . 270
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 269
integrals
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .274
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
modulus and phase functions
asymptotic expansions for large argument. . . . .272
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
orders1
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .268
power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269{270
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270
cross-products and sums of squares . . . . . . . . . . . 270
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
re
ection formulas for arguments and orders . . . . 268
uniform asymptotic expansions for large order. . .273
double asymptotic property. . . . . . . . . . . . . . . . . . .273
exponentially-small contributions . . . . . . . . . . . . . 273
zeros
asymptotic approximations for large zeros. . . . .273
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Kelvin's ship-wave pattern . . . . . . . . . . . . . . . . . . . . 790{791
kernel equations
Index 921
Heun's equation. . . . . . . . . . . . . . . . . . . . . . . . . . . .715, 716
kernel functions
Heun's equation. . . . . . . . . . . . . . . . . . . . . . . . . . . .715, 716
Klein's complete invariant. . . . . . seemodular functions.
Klein{Gordon equation
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
Kontorovich{Lebedev transform
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 260
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Korteweg{de Vries equation
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .565
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 738
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 545
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
Kovacic's algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 713, 718
KP equation . . . . seeKadomtsev{Petviashvili equation.
Krattenthaler's formula for determinants . . . . . . . . . . . . 4
Krawtchouk polynomials
. . . . . . . . see also Hahn class orthogonal polynomials.
applications
coding theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .479
relation to hypergeometric function . . . . . . . . . . . . . 394
Kummer congruences
Bernoulli and Euler numbers . . . . . . . . . . . . . . . . . . . . 593
Kummer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
. . . . . . . . see also con
uent hypergeometric functions.
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .323
analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
applications
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
asymptotic approximations for large parameters
largea. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330{331
largeb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330{331
asymptotic expansions for large argument . . 328{329
error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 329
hyperasymptotic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 347
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .346{347
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .325
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325{326
di erential equation. . . . . . . . . seeKummer's equation
integer parameters. . . . . . . . . . . . . . . . . . . . . . . . . .322{323
integral representations
along the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
contour integrals . . . . . . . . . . . . . . . . . . . . . . . . . 326{327
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 327integrals
along the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Hankel transforms. . . . . . . . . . . . . . . . . . . . . . . .332{333
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Mellin transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .332
interrelations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .322, 325
Kummer's transformations . . . . . . . . . . . . . . . . . . . . . . 325
limiting forms
asz!0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323
asz!1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
multiplication theorems . . . . . . . . . . . . . . . . . . . . . . . . . 334
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
polynomial cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 322, 323
principal branches (or values) . . . . . . . . . . . . . . . . . . . 322
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
relations to other functions
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . .327
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .328
generalized hypergeometric functions. . . . . . . . . .328
incomplete gamma functions . . . . . . . . . . . . . . . . . . 328
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . 328
orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . 328
parabolic cylinder functions. . . . . . . . . . . . . . . . . . .328
Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . 334
series expansions
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
in modi ed Bessel functions. . . . . . . . . . . . . . . . . . .333
Maclaurin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
multiplication theorems. . . . . . . . . . . . . . . . . . . . . . .334
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .347
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
zeros
asymptotic approximations . . . . . . . . . . . . . . . . . . . 331
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
number of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Kummer's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
equivalent form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
fundamental solutions . . . . . . . . . . . . . . . . . . . . . . 323{324
numerically satisfactory solutions . . . . . . . . . . . 323{324
relation to hypergeometric di erential equation . . 322
relation to Whittaker's equation . . . . . . . . . . . . . . . . 334
standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Kummer's transformations
for3F2hypergeometric functions of matrix argument
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
for con
uent hypergeometric functions . . . . . . . . . . 325
L'H^ opital's rule for derivatives . . . . . . . . . . . . . . . . . . . . . . . 5
922 Index
Lagrange interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 75{76
abscissas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
equally-spaced nodes. . . . . . . . . . . . . . . . . . . . . . . . . .75{76
error term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Newton's interpolation formula. . . . . . . . . . . . . . . . . . .76
nodal polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
remainder terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75{76
via divided di erences. . . . . . . . . . . . . . . . . . . . . . . . . . . .76
Lagrange inversion theorem . . . . . . . . . . . . . . . . . . . . . . . . 21
extended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Lagrange's formula for reversion of series . . . . . . . . . . . 43
Laguerre functions
associated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .754
Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
. . . . . . . . . . . see also classical orthogonal polynomials.
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .460
applications
Schr odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . 479
asymptotic approximations . . . . . . . . . . . . . . . . . 452{453
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
expansions in series of. . . . . . . . . . . . . . . . . . . . . .459, 460
explicit representations . . . . . . . . . . . . . . . . . . . . . 442{443
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .436
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450, 451
Turan-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
integral representations. . . . . . . . . . . . . . . . . . . . .447, 448
integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455{457
fractional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .445, 463, 464
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .457
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
limiting form as a Bessel function . . . . . . . . . . . . . . . 449
limits to monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
local maxima and minima. . . . . . . . . . . . . . . . . . . . . . .451
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
monic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 460
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .439
parameter constraint . . . . . . . . . . . . . . . . . . . . . . . 439, 443Poisson kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
relation to con
uent hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328, 338, 443, 448
Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .480
of coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
tables of zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81
upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
value atz= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443
weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438, 454
asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 454
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81
Lam e functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
algebraic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .693
applications
conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
rotation group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
sphero-conal coordinates . . . . . . . . . . . . . . . . . . . . . . 693
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 689
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
di erential equation . . . . . . . . . . . seeLam e's equation.
eigenvalues
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 689
coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
continued-fraction equation . . . . . . . . . . . . . . . . . . . 685
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .685
graphics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .686{687
interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
limiting forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . 686
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688{689
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687{688
integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
limiting forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .688
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
parity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .686
period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
relations to Heun functions. . . . . . . . . . . . . . . . . . . . . .713
relations to Lam e polynomials. . . . . . . . . . . . . .689, 690
Index 923
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
with imaginary periods. . . . . . . . . . . . . . . . . . . . . . . . . .690
with real periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .685
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .685
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
algebraic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
applications
ellipsoidal harmonics. . . . . . . . . . . . . . . . . . . . . . . . . .694
physical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .691, 694
spherical harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . .694
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 693
Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
eigenvalues
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 693
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
elliptic-function form. . . . . . . . . . . . . . . . . . . . . . . . . . . .690
explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .684, 690, 691
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
relation to Lam e functions . . . . . . . . . . . . . . . . . 689, 690
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .694
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .690
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
electrostatic interpretation . . . . . . . . . . . . . . . . . . . . 691
Lam e wave equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .690
Lam e's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
algebraic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
eigenvalues . . . . . . . . . seeLam e functions, eigenvalues.
Jacobian elliptic-function form . . . . . . . . . . . . . . . . . . 684
other forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .684{685
relation to Heun's equation . . . . . . . . . . . . . . . . . . . . . 685
second solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .684
stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
trigonometric form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
Weierstrass elliptic-function form. . . . . . . . . . . . . . . .685
Lam e{Wangerin functions . . . . . . . . . . . . . . . . . . . . . . . . . 693
LambertW-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 111
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 111
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
principal branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111other branches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
Lambert series
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
Lanczos tridiagonalization of a symmetric matrix . . . 75
Lanczos vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Landen transformations
Jacobian elliptic functions . . . . . . . . 556, 557, 563, 566
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531
Laplace equation
3j;6j;9jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
Laplace transform
analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
asymptotic expansions for large parameters
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43, 44, 46
asymptotic expansions for small parameters . . . . . . 51
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
for functions of matrix argument . . . . . . . . . . . . . . . . 768
analytic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . .768
convolution theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 768
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
numerical inversion . . . . . . . . . . . . . . . . . . . . . . . 83{84, 99
of periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Laplace's equation
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275
for elliptical cones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .694
spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .501
toroidal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Laplace's method for asymptotic expansions of integrals
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 47
Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
ellipsoidal coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . .583
numerical approximations . . . . . . . . . . . . . . . . . . . . . . . . 78
oblate spheroidal coordinates. . . . . . . . . . . . . . . . . . . .706
parabolic cylinder coordinates. . . . . . . . . . . . . . . . . . .317
polar coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
prolate spheroidal coordinates. . . . . . . . . . . . . . . . . . .705
spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
lattice
for elliptic functions
. . . . . . . . . seeWeierstrass elliptic functions, lattice.
924 Index
lattice models of critical phenomena
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517
lattice parameter
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .524
lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618{623
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
k-dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
lattice walks
Appell functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
generalized hypergeometric functions . . . . . . . . . . . . 417
Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
asymptotic approximations for coecients. . . . . . . .65
Lauricella's function
relation to symmetric elliptic integrals. . . . . . . . . . .509
Lax pairs
classical orthogonal polynomials. . . . . . . . . . . . . . . . .478
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 728
layered materials
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517
least squares approximations . . . . . . . . . . . . . . . . . . . 99{100
conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
normal equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99
orthogonal functions with respect to weighted sum-
mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Lebesgue constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 97
asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
. .see also associated Legendre functions andFerrers
functions.
complex degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Legendre functions on the cut . . . seeFerrers functions.
Legendre polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
. . . . . . . . . . . see also classical orthogonal polynomials.
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .459
applications
Schr odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . 479
associated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .474
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .452
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .445
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
expansions in series of. . . . . . . . . . . . . . . . . . . . . .459, 461
explicit representations . . . . . . . . . . . . . . . . . . . . . 442{443
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
Turan-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
integral representations. . . . . . . . . . . . . . . . . . . . .447, 448
for products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455, 458
Nicholson-type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
large degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
monic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .439
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
relations to other functions
associated Legendre functions. . . . . . . . . . . . . . . . .360
Ferrers functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .360
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . 394
3jsymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760{761
Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
shifted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436, 439
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .480
of coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
value at argument zero. . . . . . . . . . . . . . . . . . . . . . . . . .285
weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438, 454
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
Legendre symbol
prime numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642
Legendre's elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . 486
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .495
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514{516
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
approximations (except asymptotic). . . . . . . . . . . . .519
arithmetic-geometric mean . . . . . . . . . . . . . . . . . . . . . . 492
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .495
change of amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . .492
change of modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
change of parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
circular cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .487, 492
complete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .487
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517{518
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .491
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
duplication formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
rst, second, and third kinds . . . . . . . . . . . . . . . . . . . . 486
Gauss transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488{489
hyperbolic cases. . . . . . . . . . . . . . . . . . . . . . . . . . . .487, 492
imaginary-argument transformations . . . . . . . . . . . . 492
imaginary-modulus transformations . . . . . . . . . . . . . 492
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
inequalities
Index 925
complete integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
incomplete integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 494
integration
with respect to amplitude. . . . . . . . . . . . . . . . . . . . .496
with respect to modulus . . . . . . . . . . . . . . . . . . . . . . 496
Landen transformations
ascending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .493
descending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .493
Laplace transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .496
limiting values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . . . 490
quadratic transformations . . . . . . . . . . . . . . . . . . . . . . . 492
reciprocal-modulus transformation . . . . . . . . . . . . . . 492
reduction of general elliptic integrals. . . . . . . .496{497
relations to other functions
am function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
Appell functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
inverse Gudermannian function . . . . . . . . . . . . . . . 491
inverse Jacobian elliptic functions . . . . . . . . . . . . . 561
Jacobi's epsilon function . . . . . . . . . . . . . . . . . . . . . . 562
Jacobi's zeta function. . . . . . . . . . . . . . . . . . . . . . . . .562
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . 494
symmetric elliptic integrals. . . . . . . . . . . . . . .507, 508
theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . 494
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518{519
Legendre's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
Legendre's relation
Legendre's elliptic integrals . . . . . . . . . . . . . . . . . . . . . 492
Legendre's relation for the hypergeometric function
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399
Leibniz's formula for derivatives . . . . . . . . . . . . . . . . . . . . . 5
lemniscate arc length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .563
lemniscate constants. . . . . . . . . . . . . . . . . . . . . . . . . .502, 503
lengths of plane curves
Bernoulli's lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .514
hyperbola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .514
Lerch's transcendent
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .612
relation to Hurwitz zeta function. . . . . . . . . . . . . . . .612
relation to polylogarithms. . . . . . . . . . . . . . . . . . . . . . .612
level-index arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Levi-Civita symbol for vectors . . . . . . . . . . . . . . . . . . . . . . 10
Levin's transformations
application to asymptotic expansions . . . . . . . . . . . . . 69
for sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94
Lie algebras
q-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
light absorptionVoigt functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
limit points (or limiting points). . . . . . . . . . . . . . . . . . . . .15
limits of functions
of a complex variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
of one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
of two complex variables . . . . . . . . . . . . . . . . . . . . . . . . . 15
of two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
line broadening function. . . . . . . . . . . . . . . . . . . . . . . . . . .167
linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73{75
. . . . . . . . . . . . . . . . . . . . . . see also Gaussian elimination.
condition numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . .74, 75
conditioning of linear systems . . . . . . . . . . . . . . . . . . . . 74
error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
a posteriori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
norms
Euclidean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
of arbitrary order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
of matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
linear functional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Liouville transformation for di erential equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26, 58
Liouville's function
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
Liouville's theorem for entire functions . . . . . . . . . . . . . 16
Liouville{Green (or WKBJ) approximation . . . . . . . . . 57
for di erence equations. . . . . . . . . . . . . . . . . . . . . . . . . . .62
littleq-Jacobi polynomials. . . . . . . . . . . . . . . . . . . . . . . . .471
local maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . 450
locally analytic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
locally integrable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
logarithm function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
branch cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Briggs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 132
common . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
general base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
general value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
graphics
complex argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
real argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
926 Index
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Napierian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
natural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
real and imaginary parts . . . . . . . . . . . . . . . . . . . . . . . . 104
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
values on the cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
logarithmic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
number-theoretic signi cance. . . . . . . . . . . . . . . . . . . .155
relation to exponential integrals . . . . . . . . . . . . . . . . . 150
Lommel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
asymptotic expansions for large argument . . . . . . . 295
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294{295
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .294
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 295
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294
re
ection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
relation to Anger{Weber functions . . . . . . . . . . . . . . 296
series expansions
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Lucas numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
M-test for uniform convergence
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
in nite series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
magic squares
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
magnetic monopoles
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 545
Mangoldt's function
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
many-body systems
con
uent hypergeometric functions . . . . . . . . . . . . . . 346
many-valued function . . . . . . . . seemultivalued function.
mathematical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 652,664
. . . . . . see also Mathieu's equation, modi ed Mathieu
functions, andradial Mathieu functions.
analytic properties . . . . . . . . . . . . . . . . . . . . 653, 661, 665
antiperiodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
applicationsmathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677{678
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678{679
asymptotic expansions for large q
. .see also uniform asymptotic approximations for
large parameters.
Goldstein's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
Sips'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .661
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679{680
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .665
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .652
expansions in series of. . . . . . . . . . . . . . . . . . . . . .664, 667
Fourier coecients
asymptotic forms for small q. . . . . . . . . . . . . 657, 666
asymptotic forms of higher coecients . . . . . . . . 657
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657, 666
recurrence relations . . . . . . . . . . . . . . . . . . . . . . 656, 666
re
ection properties in q. . . . . . . . . . . . . . . . . . . . . . 657
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
values atq= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . 653, 656, 666
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655{656, 665
integral equations
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .663
variable boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
with Bessel-function kernels. . . . . . . . . . . . . . . . . . .663
with elementary kernels . . . . . . . . . . . . . . . . . . 663, 672
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 672
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
integrals
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
of products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
of products with Bessel functions. . . . . . . . .673{674
irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
limiting forms as order tends to integers. . . . . . . . .665
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654, 664
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
of integer order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
of noninteger order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664
orthogonality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .654, 664
parity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .654
periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654, 664
power series in q. . . . . . . . . . . . . . . . . . . . . . . . . . . 660, 666
pseudoperiodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 653, 664
re
ection properties in . . . . . . . . . . . . . . . . . . . . . . . . 664
re
ection properties in q. . . . . . . . . . . . . . .654, 664, 665
re
ection properties in z. . . . . . . . . . . . . . . . . . . . . . . . 664
relations to other functions
basic solutions of Mathieu's equation . . . . . . . . . 654
con
uent Heun functions . . . . . . . . . . . . . . . . . . . . . 717
modi ed Mathieu functions . . . . . . . . . . . . . . . . . . . 667
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .680
uniform asymptotic approximations for large param-
eters
Index 927
Barrett's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
Dunster's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662{663
values atq= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .663
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
Mathieu's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
algebraic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
basic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
relation to eigenfunctions . . . . . . . . . . . . . . . . . . . . . 654
characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 653
characteristic exponents. . . . . . . . . . . . . . . . . . . . . . . . .653
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
eigenfunctions . . . . . . . . . . . . . . . seeMathieu functions.
eigenvalues (or characteristic values) . . . . . . . . . . . . 653
analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . 661
analytic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . .661
asymptotic expansions for large q. . . . . . . . 661, 666
branch points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .661
characteristic curves . . . . . . . . . . . . . . . . . . . . . . . . . . 667
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
continued-fraction equations . . . . . . . . . . . . . 659, 666
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654, 664
exceptional values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654, 665
normal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661, 664
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652, 653, 664
power-series expansions in q. . . . . . . . . 659{660, 666
re
ection properties in . . . . . . . . . . . . . . . . . . . . . . 664
re
ection properties in q. . . . . . . . . . . . . . . . . 654, 664
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
Floquet solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
Fourier-series expansions. . . . . . . . . . . . . . . . . . . . . .653
uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .653
Floquet's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .653
parameters
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
stability chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
stable pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
stable regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
unstable pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
second solutions
antiperiodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
expansions in Mathieu functions . . . . . . . . . . . . . . 658
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .657
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
periodicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .657
re
ection properties in q. . . . . . . . . . . . . . . . . . . . . . 658
values atq= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .652standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
Theorem of Ince . . . . . . . . . . . . . . . . . . . . . . . . . . . 653, 657
transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
matrix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . see also linear algebra.
augmented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
characteristic polynomial. . . . . . . . . . . . . . . . . . . . . . . . .74
condition number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74{75
characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . 74
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
condition numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
eigenvectors
left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
normalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
equivalent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .542
factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
nondefective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
Riemann. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .538
symmetric
tridiagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
symplectic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
triangular decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 73
tridiagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
local. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5, 8
maximum-modulus principle
analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
harmonic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
Schwarz's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
McKean and Moll's theta functions. . . . . . . . . . . . . . . .524
McMahon's asymptotic expansions
zeros of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . 236
error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
mean value property for harmonic functions . . . . . . . . 16
mean value theorems
di erentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
means. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seeAbel
means, arithmetic mean, Ces aro means, geometric
mean, harmonic mean, andweighted means.
measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .437
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Mehler functions . . . . . . . . . . . . . . . . seeconical functions.
Mehler{Dirichlet formula
Ferrers functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Mehler{Fock transformation . . . . . . . . . . . . . . . . . . 373, 379
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373, 379
Mehler{Sonine integrals
928 Index
Bessel and Hankel functions. . . . . . . . . . . . . . . . . . . . .224
MeijerG-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 417
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .417
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .416
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 415
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
relation to generalized hypergeometric function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415{416
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .416
Meixner polynomials
. . . . . . . . . . . . . seeHahn class orthogonal polynomials.
relation to hypergeometric function . . . . . . . . . . . . . 394
Meixner{Pollaczek polynomials
. . . . . . . . . . . . . seeHahn class orthogonal polynomials.
relation to hypergeometric function . . . . . . . . . . . . . 394
Mellin transform
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
convolution integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
de nition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29, 48
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 48
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Parseval-type formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Mellin{Barnes integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
meromorphic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Mersenne numbers
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
Mersenne prime
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
method of stationary phase
asymptotic approximations of integrals . . . . . . . . . . . 45
metric coecients
for oblate spheroidal coordinates . . . . . . . . . . . . . . . . 705
for prolate spheroidal coordinates . . . . . . . . . . . . . . . 704
Mill's ratio for complementary error function . . . . . . 163
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Miller's algorithm
di erence equations. . . . . . . . . . . . . . . . . . . . . . . . . . .85{87
minimax polynomial approximations. . . . . . . . . . . . . . . .96
computation of coecients . . . . . . . . . . . . . . . . . . . . . . . 96
minimax rational approximations . . . . . . . . . . . . . . . . . . . 97
computation of coecients . . . . . . . . . . . . . . . . . . . . . . . 98
type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
local. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5, 8
Minkowski's inequalities for sums and series. . . . .12, 13minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seedeterminants.
Mittag-Leer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Mittag-Leer's expansion
in nite partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 22
M obius transformation . . . . seebilinear transformation.
M obius function
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
M obius inversion formulas
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 248
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .253
applications
asymptotic solutions of di erential equations . . 274
wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
asymptotic expansions for large argument . . 255{256
error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255, 256
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 256
for derivatives with respect to order. . . . . . . . . . .255
for products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
asymptotic expansions for large order. . . . . . .256{258
asymptotic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
double asymptotic properties . . . . . . . . . . . . . 257{258
in inverse factorial series . . . . . . . . . . . . . . . . . . . . . . 257
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256{257
branch conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .249
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276{277
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .251
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253
cross-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
derivatives
asymptotic expansions for large argument. . . . .255
explicit forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
uniform asymptotic expansions for large order
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256{257
derivatives with respect to order . . . . . . . . . . . . . . . . 254
asymptotic expansion for large argument. . . . . .255
di erential equations . . . . . . . . . . . . . . . . . . . . . . . 248, 254
. . . . . . . . . . . . . . . see also modi ed Bessel's equation.
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
hyperasymptotic expansions. . . . . . . . . . . . . . . . . . . . .276
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
integral representations
along real line. . . . . . . . . . . . . . . . . . . . . . . . . . . .252{253
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253
integrals . . seeintegrals of modi ed Bessel functions.
limiting forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252
Index 929
monotonicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254
multiplication theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 260
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
of imaginary order
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250, 251
limiting forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
numerically satisfactory pairs . . . . . . . . . . . . . . . . . 261
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
uniform asymptotic expansions for large order
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252
principal branches (or values) . . . . . . . . . . . . . . . . . . . 249
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
relations to other functions
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
con
uent hypergeometric functions . . 255, 328, 338
elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . .254
generalized Airy functions . . . . . . . . . . . . . . . . . . . . 206
generalized hypergeometric functions. . . . . . . . . .255
parabolic cylinder functions . . . . . . . . . . . . . . 255, 308
sums
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261
expansions in series of . . . . . . . . . . . . . . . . . . . . . . . . 261
multiplication theorem. . . . . . . . . . . . . . . . . . . . . . . .260
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .258
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
modi ed Bessel's equation. . . . . . . . . . . . . . . . . . . . . . . . .248
inhomogeneous forms . . . . . . . . . . . . . . . . . . . . . . 288, 295
numerically satisfactory solutions . . . . . . . . . . . . . . . 249
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .248
standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
modi ed Korteweg{de Vries equation
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 738
modi ed Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . 667
. . . . . . . . . . . . . . . . . . see also radial Mathieu functions.
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .672
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .668
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678{679
asymptotic approximations
. .see also uniform asymptotic approximations for
large parameters.
for large<z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .667, 672
for largeq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680connection formulas . . . . . . . . . . . . . . . . . . . . . . . . 667, 669
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .667
expansions in series of
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670
cross-products of Bessel functions and modi ed
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
integral representations. . . . . . . . . . . . . . . . . . . . .672{674
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
of cross-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
joining factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .652, 669
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
relation to Mathieu functions . . . . . . . . . . . . . . . . . . . 667
shift of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .680
uniform asymptotic approximations for large param-
eters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662, 672
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
zeros
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
modi ed Mathieu's equation . . . . . . . . . . . . . . . . . . . . . . 667
algebraic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
modi ed spherical Bessel functions
. . . . . . . . . . . . . . . . . . . . seespherical Bessel functions.
modi ed Struve functions . . . . seeStruve functions and
modi ed Struve functions.
modi ed Struve's equation . . . seeStruve functions and
modi ed Struve functions, di erential equations.
modular equations
modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
modular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581{582
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582{583
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
cusp form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
general. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .579
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
Laurent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .580
level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
modular form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
modular transformations . . . . . . . . . . . . . . . . . . . . . . . . 580
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570, 579
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .580
relations to theta functions . . . . . . . . . . . . 525, 532, 579
930 Index
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
modular theorems
generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . . 516
molecular spectra
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
molecular spectroscopy
3j;6j;9jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
molli ed error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
moment functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
monic polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22, 80
monodromy groups
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .719
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
monosplines
Bernoulli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .597
cardinal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .597
monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Monte Carlo sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . .189
Monte-Carlo methods
for multidimensional integrals . . . . . . . . . . . . . . . . . . . . 84
Mordell's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
Motzkin numbers
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .623
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 621
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .622
multidimensional theta functions
. . . . . . . seeRiemann theta functions andRiemann
theta functions with characteristics.
multinomial coecients
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 620
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .620
multiple orthogonal polynomials. . . . . . . . . . . . . . . . . . .477
multiplicative functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
multiplicative number theory . . . . . . . . . . . . . . . . . 638{644
completely multiplicative functions . . . . . . . . . . . . . . 640
Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
Euler product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .640
fundamental theorem of arithmetic . . . . . . . . . . . . . . 638
multiplicative functions . . . . . . . . . . . . . . . . . . . . . . . . . 640
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
primitive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
multivalued function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 104
branch cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
closed de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
multivariate beta functionde nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .769
multivariate gamma function
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .769
multivariate hypergeometric function . . . . . . . . . . . . . . 498
mutual inductance of coaxial circles
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .516
n-dimensional sphere
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
N orlund polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
nanotubes
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Narayana numbers
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 622
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .622
negative de nite
Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 15
cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
of in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
punctured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
Neumann's addition theorem
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 260
Neumann's expansion
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247
Neumann's integral
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224
Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Neumann's polynomial
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247
Neumann-type expansions
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 261
Neville's theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
relations to Jacobian elliptic functions. . . . . . . . . . .550
Newton's interpolation formula . . . . . . . . . . . . . . . . . . . . . 76
Newton's rule (or method) . . . . . . . . . . . . . . . . . . . . . . . . . 90
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Nicholson's integral
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .225
Nicholson-type integral
parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . 313
9jsymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .764
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
approximations for large parameters . . . . . . . . . . . . 764
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
Index 931
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764
graphical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764
recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764
representation as
nite sum of 6 jsymbols . . . . . . . . . . . . . . . . . . . . . . 763
nite sum of 3 jsymbols . . . . . . . . . . . . . . . . . . . . . . 763
generalized hypergeometric functions. . . . . . . . . .764
special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764
sum rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764
summation convention . . . . . . . . . . . . . . . . . . . . . . . . . . 760
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .764
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .764
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
nodal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79, 80
nome
Jacobi's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .490
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .550
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .524
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 570
nonlinear equations
xed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
numerical solutions
iterative methods. . . . . . . . . . . . . . . . . . . . . . . . . . .90{92
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
nonlinear evolution equations
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
nonlinear harmonic oscillator
Painlev e equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
nonlinear ordinary di erential equations
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .565
nonlinear partial di erential equations
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .565
normal probability functions . . . . . . . . . . . . . . . . . . . . . . 160
Novikov's conjecture
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 546
nuclear physics
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
nuclear structure
3j;6j;9jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
number theory . . . . . . . see also additive number theory,
multiplicative number theory, andprime numbers.
Bernoulli and Euler numbers and polynomials . . . 598
generalized hypergeometric functions . . . . . . . . . . . . 417
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .564
modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
number-theoretic functions. . . . . . . . . . . . . . . . . . . .638{643
completely multiplicative. . . . . . . . . . . . . . . . . . . . . . . .640
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649Dirichlet character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
induced modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
Legendre symbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642
primitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
principal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642
Dirichlet divisor problem. . . . . . . . . . . . . . . . . . . . . . . .643
divisor function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
divisor sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
inversion formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .641
Lambert series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
M obius inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 641, 647
pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
multiplicative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
periodic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .642
Ramanujan's sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .649
numerical di erentiation
. . . . . . . . . . . . . . . . . . . . seedi erentiation, numerical.
oblate spheroidal coordinates . . . . . . . . . . . . . . . . . . . . . . 705
Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
metric coecients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .705
Olver's algorithm
di erence equations. . . . . . . . . . . . . . . . . . . . . . . . . . .86{87
Olver's associated Legendre function. . . . . . . . . .354, 375
Olver's con
uent hypergeometric function . . . . . . . . . 322
Olver's hypergeometric function . . . . . . . . . . . . . . 353, 384
OP's. . . . . . . . . . . . . . . . . . . . . . seeorthogonal polynomials.
open disks around in nity . . . . . . . . . . . . . . . . . . . . . . . . . . 16
open point set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 15
optical di raction
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
optics
canonical integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791
Orr{Sommerfeld equation
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
orthogonal matrix polynomials . . . . . . . . . . . . . . . . . . . . 477
orthogonal polynomials
complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
relations to con
uent hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .328, 338
relations to hypergeometric function . . . . . . . . 393{394
orthogonal polynomials associated with root systems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
orthogonal polynomials on the triangle . . . . . . . . . . . . 478
orthogonal polynomials on the unit circle
. . . . . . seepolynomials orthogonal on the unit circle.
orthogonal polynomials with Freud weights . . . . . . . . 475
oscillations of chains
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275
oscillations of plates
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276
}-function . . . . . . . . . . . seeWeierstrass elliptic functions.
932 Index
packing analysis
incomplete beta functions . . . . . . . . . . . . . . . . . . . . . . . 189
Pad e approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . .98{99
computation of coecients . . . . . . . . . . . . . . . . . . . . . . . 98
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Pad e table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Painlev e equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
. . . . . . . . . . . . . . . . . . . . see also Painlev e transcendents.
ane Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
alternative forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .724
B acklund transformations . . . . . . . . . . . . . . . . . . 730{732
coalescence cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
compatibility conditions . . . . . . . . . . . . . . . . . . . . 728{729
elementary solutions. . . . . . . . . . . . . . . . . . . . . . . .732{735
elliptic form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .725
graphs of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 726{728
Hamiltonian structure . . . . . . . . . . . . . . . . . . . . . . 729{730
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730{732
isomonodromy problems . . . . . . . . . . . . . . . . . . . . . . . . 728
compatibility condition . . . . . . . . . . . . . . . . . . . . . . . 728
Lax pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
rational solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 732{734
renormalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
special function solutions . . . . . . . . . . . . . . . . . . . 735{736
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735
Hermite polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .735
hypergeometric function . . . . . . . . . . . . . . . . . 399, 736
parabolic cylinder functions. . . . . . . . . . . . . . . . . . .735
Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . 736
symmetric forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
Painlev e property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .739
Painlev e transcendents. . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
. . . . . . . . . . . . . . . . . . . . . . . . see also Painlev e equations.
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .738
Boussinesq equation . . . . . . . . . . . . . . . . . . . . . . . . . . 739
combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .739
enumerative topology . . . . . . . . . . . . . . . . . . . . . . . . . 739
integrable continuous dynamical systems . . . . . . 739
integral equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .729
Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Korteweg{de Vries equation. . . . . . . . . . . . . . . . . . .738
modi ed Korteweg{de Vries equation . . . . . . . . . 738
orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . 739
partial di erential equations. . . . . . . . . . . . . .738{739
quantum gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .739
sine-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . 739
statistical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
asymptotic approximations . . . . . . . . . . . . . . . . . 736{738
complex variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .738
real variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .736{738
B acklund transformations . . . . . . . . . . . . . . . . . . 730{732computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740
di erential equations for . . . . . . . . . . . . . . . . . . . . . . . . 724
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726{728
Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729{730
Lax pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724, 730{732
parabolic cylinder functions . . . . . . . . . . . . . . . . . 304,314
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
asymptotic expansions for large parameter . . seeuni-
form asymptotic expansions for large parameter.
asymptotic expansions for large variable . . . . 309, 315
exponentially-improved . . . . . . . . . . . . . . . . . . 309, 317
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
connection formulas . . . . . . . . . . . . . . . . . . . . . . . . 304, 315
continued fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .308
de nitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .304, 305, 314
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
numerically satisfactory solutions . . . . . . . . 304, 314
standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
envelope functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
expansions in Chebyshev series. . . . . . . . . . . . . . . . . .318
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .317
graphics
complex variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .306
real variables . . . . . . . . . . . . . . . . . . . . . . . . 305{306, 314
Hermite polynomial case . . . . . . . . . . . . . . . . . . . 304, 308
integral representations
along the real line. . . . . . . . . . . . . . . . . . . . . . . .307, 315
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .308
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
integral transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .317
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
asymptotic methods . . . . . . . . . . . . . . . . . . . . . . . . . . 317
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313
Nicholson-type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313
modulus and phase functions . . . . . . . . . . . . . . . 305, 316
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
power-series expansions . . . . . . . . . . . . . . . . . . . . 307, 315
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
re
ection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
relations to other functions
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . .228, 315
con
uent hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308, 315, 328, 338
error and related functions. . . . . . . . . . . . . . . . . . . .308
Hermite polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .308
modi ed Bessel functions . . . . . . . . . . . . . . . . 255, 308
Index 933
probability functions. . . . . . . . . . . . . . . . . . . . . . . . . .308
repeated integrals of the complementary error func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .318
uniform asymptotic expansions for large parameter
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309{312, 315{316
double asymptotic property. . . . . . . . . . . . . . . . . . .311
in terms of Airy functions . . . . . . . . . . . 311{312, 316
in terms of elementary functions. . . . .310{311, 316
modi ed expansions in terms of Airy functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
modi ed expansions in terms of elementary func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
values atz= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304, 314
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304, 314
zeros
asymptotic expansions for large parameter . . . . 313
asymptotic expansions for large variable . . 312, 317
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .312
paraboloidal coordinates
wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
Whittaker{Hill equation . . . . . . . . . . . . . . . . . . . . . . . . 678
paraboloidal wave functions . . . . . . . . . . . . . . . . . . . . . . . 677
asymptotic behavior for large variable . . . . . . . . . . . 677
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .677
re
ection properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
parallelepiped
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
parallelogram
area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
parametrization of algebraic equations
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .563
parametrized surfaces
area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
integral over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
tangent vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
paraxial wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
Parseval's formula
Fourier cosine and sine transforms. . . . . . . . . . . . . . . .28
Fourier series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Parseval-type formulas
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 49
partial derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
partial di erential equations
nonlinear
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . 582
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 738
spectral methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .479partial di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
. . . . . . . . . . . . . . . . . . . see also in nite partial fractions.
particle scattering
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
partition . . . . . . . . . . . . . . . . . . . . . . . seepartition function.
partition function
asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
hadronic matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146
parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .645
Ramanujan congruences. . . . . . . . . . . . . . . . . . . . . . . . .646
unrestricted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .645
partitional shifted factorial . . . . . . . . . . . . . . . . . . . . . . . . 769
partitions. . . . . . . . . . . . . . . . . . . . . . .618{620, 624{631, 769
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .635
compositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .628
conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . 618{620, 624{626
of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618, 626{628
parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .618
plane. . . . . . . . . . . . . . . . . . . . . . . . . . . seeplane partitions.
restricted . . . . . . . . . . . seerestricted integer partitions.
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619, 629, 635
weight of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .769
path
integrals of vector-valued functions . . . . . . . . . . . . . . . 11
length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
PCFs. . . . . . . . . . . . . . . . . seeparabolic cylinder functions.
Pearcey integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
asymptotic approximations . . . . . . . . . . . . . . . . . 789{790
convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .788
formula for Stokes set . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
integral identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
picture of Stokes set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
pictures of modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
pictures of phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780
scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .785
table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
pendulum
amplitude (am) function . . . . . . . . . . . . . . . . . . . . . . . . 564
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .564
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679
pentagonal numbers
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
periodic Bernoulli functions . . . . . . . . . . . . . . . . . . . . . . . 588
periodic Euler functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
periodic zeta function
934 Index
relation to Hurwitz zeta function. . . . . . . . . . . . . . . .612
relation to polylogarithms. . . . . . . . . . . . . . . . . . . . . . .612
permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618, 631{634
adjacent transposition. . . . . . . . . . . . . . . . . . . . . . . . . . .631
cycle notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .631
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
derangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
derangement number. . . . . . . . . . . . . . . . . . . . . . . . . . . .631
descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
even or odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
excedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .632
weak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
xed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
greater index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .632
inversion numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 631{634
major index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632, 634
matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
multiset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .634
order notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
restricted position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .633
sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631, 633
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
transpositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .631
twelvefold way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
Pfa {Saalschutz formula
3F2functions of matrix argument . . . . . . . . . . . . . . . 772
phase principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 92
photon scattering
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
pi
computation to high precision via elliptic integrals
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Picard's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Picard{Fuchs equations
generalized hypergeometric functions . . . . . . . . . . . . 417
piecewise continuous. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
piecewise di erentiable curve . . . . . . . . . . . . . . . . . . . . . . . 11
pion-nucleon scattering
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
pionic atoms
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
plane algebraic curves . . . . . . . . . . . . seealgebraic curves.
plane curves
elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514{515
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .563
plane partitions
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .635
complementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
descending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
limiting form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .631recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
strict shifted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .629
plane polar coordinates . . . . . . . . . seepolar coordinates.
plasma dispersion function . . . . . . . . . . . . . . . . . . . . . . . . 169
plasma waves
error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
plasmas
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
Pochhammer double-loop contour. . . . . . . .326, 389, 714
Pochhammer's integral
beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .714
Pochhammer's symbol. . . . . . . . . . . . . . . . . . . . . . . . . . . . .136
point sets in complex plane
closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
compact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
exterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
open . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
points in complex plane
accumulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
at in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
limit (or limiting) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Poisson identity
discrete analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Gauss sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .532
Poisson integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16, 34
conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
harmonic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
Poisson kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Fourier integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Fourier series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
Poisson's equation
in channel-like geometries . . . . . . . . . . . . . . . . . . . . . . . 379
Poisson's integral
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224
Poisson's summation formula
Fourier series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
polar representation
complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
movable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
Pollaczek polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
Index 935
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
expansions in series of. . . . . . . . . . . . . . . . . . . . . . . . . . .477
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .477
relation to hypergeometric function . . . . . . . . . . . . . 476
relations to other orthogonal polynomials . . . . . . . 477
polygamma functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146
polylogarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 611
relations to other functions
Fermi{Dirac integrals . . . . . . . . . . . . . . . . . . . . . . . . . 612
Lerch's transcendent . . . . . . . . . . . . . . . . . . . . . . . . . . 612
periodic zeta function. . . . . . . . . . . . . . . . . . . . . . . . .612
Riemann zeta function. . . . . . . . . . . . . . . . . . . . . . . .611
series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
polynomials
characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
de
ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
monic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22, 80
nodal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
stable . . . . . . . . . . . . . . . . . . . . . . . seestable polynomials.
Wilkinson's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
zeros . . . . . . . . . . . . . . . . . . . . . . seezeros of polynomials.
zonal. . . . . . . . . . . . . . . . . . . . . . . . . seezonal polynomials.
polynomials orthogonal on the unit circle . . . . . 475{476
biorthogonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
connection with orthogonal polynomials on the line
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
population biology
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
poristic polygon constructions of Poncelet
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .557
positive de nite
Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
potential theory
conical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
symmetric elliptic integrals . . . . . . . . . . . . . . . . . 501, 516
power function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
branch cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
general bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
general value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
power series
addition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
circle of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
radius of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
di erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
of logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
of powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
of reciprocals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
primality testing
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
prime number theorem . . . . . . . . . . . . . . . . . . 638, 643, 644
equivalent statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
prime numbers
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .647
asymptotic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .648{649
counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613, 638
asymptotic estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 638
Euler{Fermat theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 638
in arithmetic progressions
Dirichlet's theorem. . . . . . . . . . . . . . . . . . . . . . .613, 643
Jacobi symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
largest known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
Legendre symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
Mersenne prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644, 648
prime number theorem . . . . . . . . . . . . . . . . 638, 643, 644
quadratic reciprocity law. . . . . . . . . . . . . . . . . . . . . . . .642
relation to logarithmic integral . . . . . . . . . . . . . . . . . . 155
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639, 649
primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . seeprime numbers.
primitive Dirichlet characters
relation to generalized Bernoulli polynomials . . . . 597
principal branches . . . . . . . . . . . . . . . . seeprincipal values.
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
. . . . . . . . . . . . . . . . . . . see also Cauchy principal values.
closed de nition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
principle of the argument. . . . . . . . . . seephase principle.
Pringsheim's theorem for continued fractions . . . . . . . 25
936 Index
probability distribution
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .515
probability functions. . . . . . . . . . . . . . . . . . . . .160, 167, 308
Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160
normal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160
relations to other functions
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160
parabolic cylinder functions. . . . . . . . . . . . . . . . . . .308
repeated integrals of the complementary error func-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167, 308
probl eme des m enages. . . . . . . . . . . . . . . . . . . . . . . . . . . . .633
projective coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
projective quantum numbers
3jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .758
prolate spheroidal coordinates . . . . . . . . . . . . . . . . . . . . . 704
Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
metric coecients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .704
Prym's functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
pseudoperiodic solutions
of Hill's equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
of Mathieu's equation . . . . . . . . . . . . . . . . . . . . . . 653, 664
pseudoprime test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
pseudorandom numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
psi function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
approximations
Chebyshev series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147
complex variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147
rational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146, 147
asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
expansions in partial fractions. . . . . . . . . . . . . . . . . . .139
graphics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136, 137
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 140
multiplication formula . . . . . . . . . . . . . . . . . . . . . . . . . . 138
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
re
ection formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
relation to hypergeometric function . . . . . . . . . . . . . 387
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146
Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136
asymptotic approximation . . . . . . . . . . . . . . . . . . . . 138
table of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
public key codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .647
punctured neighborhood. . . . . . . . . . . . . . . . . . . . . . . . . . . .19
q-beta function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145
q-factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145q-gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
q-Appell functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423
transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
q-Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
q-binomial coecient . . . . . . . . . . . . . . . . . . . . . . . . . 421, 627
q-binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
q-binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 421, 424
q-calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .420{422
q-cosine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
q-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
q-di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
q-Dyson conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
q-elementary functions. . . . . . . . . . . . . . . . . . . . . . . .422, 432
q-Euler numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
q-exponential function. . . . . . . . . . . . . . . . . . . . . . . . . . . . .422
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .432
q-hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 420
Andrews{Askey sum . . . . . . . . . . . . . . . . . . . . . . . 424, 426
Andrews'q-Dyson conjecture. . . . . . . . . . . . . . . . . . . .431
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Bailey chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Bailey lemma
strong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
weak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Bailey pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Bailey transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Bailey's 2F1( 1) sum
q-analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Bailey's 4F3(1) sum
q-analogs ( rst and second) . . . . . . . . . . . . . . . . . . . 427
Bailey's transformation of very-well-poised 8 7. . 429
Bailey{Daum q-Kummer sum . . . . . . . . . . . . . . . . . . . 424
balanced series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
bibasic sums and series. . . . . . . . . . . . . . . . . . . . . . . . . .429
bilateral . . . . seebilateralq-hypergeometric function.
Cauchy's sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Chu{Vandermonde sums ( rst and second)
q-analogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
constant term identities . . . . . . . . . . . . . . . . . . . . . . . . . 431
contiguous relations (Heine's) . . . . . . . . . . . . . . . . . . . 425
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .426
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
Dixon's 3F2(1) sum
q-analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Dixon's sum
F. H. Jackson's q-analog . . . . . . . . . . . . . . . . . . . . . . 426
Dougall's 7F6(1) sum
F. H. Jackson's q-analog . . . . . . . . . . . . . . . . . . . . . . 427
Euler's sums ( rst, second, third) . . . . . . . . . . 423, 424
F. H. Jackson's transformations . . . . . . . . . . . . . . . . . 428
Index 937
Fine's transformations ( rst, second, third). . . . . .424
Gauss's 2F1( 1) sum
q-analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Heine's transformations ( rst, second, third) . . . . 424
idem function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420, 429
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 426
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
k-balanced series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
mixed base Heine-type transformations. . . . . . . . . .429
nearly-poised . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
q-Pfa {Saalsch utz sum. . . . . . . . . . . . . . . . . . . . . . . . . .426
q-Saalsch utz sum
nonterminating form. . . . . . . . . . . . . . . . . . . . . . . . . .426
q-Sheppard identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
quintuple product identity . . . . . . . . . . . . . . . . . . . . . . 427
Ramanujan's integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 431
relations to other functions
Askey{Wilson class orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472{474
q-Hahn class orthogonal polynomials . . . . . 470{472
Rogers{Fine identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Saalsch utzian series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Sears' balanced 4 3transformation. . . . . . . . . . . . . .428
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
three-term 2 1transformation . . . . . . . . . . . . . . . . . . 425
transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
Vandermonde sum
nonterminating q-version. . . . . . . . . . . . . . . . . . . . . .425
very-well-poised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423
well-poised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423
Zeilberger{Bressoud theorem. . . . . . . . . . . . . . . . . . . .431
q-Hahn class orthogonal polynomials. . . . . . . . . .470{472
as eigenvalues of q-di erence operator . . . . . . . . . . . 470
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .474
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .470
relation to q-hypergeometric function . . . . . . . 470{472
q-Hahn polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
q-hypergeometric orthogonal polynomials . . . . . . . . . .470
q-integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .422
q-Laguerre polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . .471
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .432
asymptotic approximations to zeros . . . . . . . . . . . . . 474
q-Leibniz rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
q-multinomial coecient . . . . . . . . . . . . . . . . . . . . . . . . . . 634
q-Pochhammer symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
q-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
q-Racah polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
applications
coding theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .479
relation to q-hypergeometric function . . . . . . . . . . . 474
q-series
classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423q-sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
q-Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
q 1-Al-Salam{Chihara polynomials. . . . . . . . . . . . . . . .473
quadratic characters
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
quadratic reciprocity law
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
quadrature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78{84
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83{84
interpolatory rules (or formulas)
. . . . . . . . . . . . . . . . . . . . . . . see also Gauss quadrature.
Clenshaw{Curtis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
error term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Fej er's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Newton{Cotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
open. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
oscillatory integrals
Clenshaw{Curtis formula (extended) . . . . . . . . . . . 82
Filon's rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Longman's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
multidimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Romberg integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Simpson's rule
composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78, 79
elementary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79
steepest-descent paths . . . . . . . . . . . . . . . . . . . . . . . . 83{84
trapezoidal rule
composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78, 79, 84
elementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78, 79
improved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
via classical orthogonal polynomials . . . . . . . . . . . . . 478
quantum chemistry
generalized exponential integral . . . . . . . . . . . . . . . . . 190
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
quantum chromo-dynamics
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
quantum eld theory
modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 614
quantum gravity
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
quantum groups
q-series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
quantum mechanics
associated Legendre functions . . . . . . . . . . . . . . . . . . . 379
canonical integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791
classical orthogonal polynomials. . . . . . . . . . . . . . . . .479
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679
938 Index
nonrelativistic
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . 317
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Whittaker functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .346
quantum probability distributions
Euler polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .598
quantum scattering
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 400
quantum spin models
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
quantum spins
Heun's equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
quantum systems
Heun's equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
quantum wave-packets
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .534
quark-gluon plasma
Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
quartic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
quartic oscillator
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .565
quasiconformal mapping
complete elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . 399
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 399
queueing theory
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
quintic equations
modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
quotient-di erence algorithm . . . . . . . . . . . . . . . . . . . . . . . 95
rhombus rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95
stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95
quotient-di erence scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Raabe's theorem
Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
Racah polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
. . . . . . . . . . . seeWilson class orthogonal polynomials.
radial Mathieu functions . . . . . . . . . . . . . . . . . . . . . . . . . . 668
. . . . . . . . . . . . . . . . see also modi ed Mathieu functions.
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
expansions in series of Bessel functions . . . . . . . . . . 670
expansions in series of cross-products of Bessel func-
tions and modi ed Bessel functions. . . . . . .671{672
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
integral representations. . . . . . . . . . . . . . . . . . . . .672{674
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674
of cross-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
joining factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .652, 669
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
relation to modi ed Mathieu functions . . . . . . . . . . 668
shift of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
radial spheroidal wave functions . . . . . . . . . . . . . . . . . . 703
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .706
asymptotic behavior for large variable . . . . . . . . . . . 703computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .703
connection with spheroidal wave functions. . . . . . .704
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . 704
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .708
Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
radiative equilibrium
generalized exponential integral . . . . . . . . . . . . . . . . . 190
Radon transform
classical orthogonal polynomials. . . . . . . . . . . . . . . . .479
railroad track design
Cornu's spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
rainbow
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Ramanujan's 1 1summation
bilateralq-hypergeometric function. . . . . . . . . . . . . .427
Ramanujan's beta integral. . . . . . . . . . . . . . . . . . . . . . . . .143
Ramanujan's cubic transformation
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . . . 393
Ramanujan's partition identity
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Ramanujan's sum
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
Ramanujan's tau function
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646{647
random graphs
generalized hypergeometric functions . . . . . . . . . . . . 417
random matrix theory
Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
rational arithmetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
exact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
rational functions
summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145
Rayleigh function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276
RC-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .496
limiting values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
relation to elementary functions . . . . . . . . . . . . . . . . . 495
relation to Gudermannian function . . . . . . . . . . . . . . 495
relation to inverse Gudermannian function . . . . . . 491
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
reduced Planck's constant . . . . . . . . . . . . . . . 379, 479, 753
reduced residue system
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
reductions of partial di erential equations
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 738
Regge poles
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
Regge symmetries
Index 939
6jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .762
3jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .759
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
regularization
distributional methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
relative error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
relative precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
relativistic Coulomb equations. . . . . . . . . . . . . . . . . . . . .754
relaxation times for proteins
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
Remez's second algorithm
minimax rational approximations. . . . . . . . . . . . . . . . .98
removable singularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
repeated integrals of the complementary error function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 167
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .167
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
power-series expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 167
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
relations to other functions
con
uent hypergeometric functions . . . . . . . . . . . 167
Hermite polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .167
parabolic cylinder functions . . . . . . . . . . . . . . 167, 308
probability functions . . . . . . . . . . . . . . . . . . . . . 167, 308
scaled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
representation theory
partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
repulsive potentials
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . 753, 754
residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
resistive MHD instability theory
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
resolvent cubic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
resonances
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
restricted integer partitions
Bessel-function expansion . . . . . . . . . . . . . . . . . . . . . . . 628
conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .628
limiting form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .627, 628
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626, 627
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . 627, 628
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 626
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626, 627
resurgenceasymptotic solutions of di erential equations . . . . . 57
reversion of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Riccati{Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .240
Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
equivalent statements . . . . . . . . . . . . . . . . . 613, 614, 644
Riemann identity
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 542
Riemann theta functions with characteristics . . . . 542
Riemann matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .538
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
connection with Riemann theta functions. . .543, 546
cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
handle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
holomorphic di erentials . . . . . . . . . . . . . . . . . . . . . . . . 543
hyperelliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
intersection indices . . . . . . . . . . . . . . . . . . . . . . . . . 538, 543
prime form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .544
representation via Hurwitz system . . . . . . . . . . . . . . 546
representation via plane algebraic curve . . . . . . . . . 546
representation via Schottky group . . . . . . . . . . . . . . . 546
Riemann theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . 538
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543{546
components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .538
genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539{541
modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
modular transformations . . . . . . . . . . . . . . . . . . . 541{542
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
period lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
quasi-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
relation to classical theta functions. . . . . . . . . . . . . .539
Riemann identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
scaled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538, 546
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .539
Riemann theta functions with characteristics. . . . . . 539
addition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
applications
Abelian functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
half-period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .539
modular transformations . . . . . . . . . . . . . . . . . . . . . . . . 542
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
quasi-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Riemann identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .539
940 Index
Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
asymptotic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
connection with incomplete gamma functions. . . .189
critical line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
critical strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
integer arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
series expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .602
Euler-product representation . . . . . . . . . . . . . . . . . . . . 640
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
integer argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605
integral representations
along the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
recursion formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
re
ection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
relations to other functions
Bernoulli and Euler numbers and polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .598, 605
Hurwitz zeta function. . . . . . . . . . . . . . . . . . . . . . . . .607
polylogarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .611
representations by Euler{Maclaurin formula . . . . . 602
series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .606
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614
zeros
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607, 614
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .606
on critical line or strip . . . . . . . . . . . . . . . . . . . 606, 614
relation to quantum eigenvalues. . . . . . . . . . . . . . .614
Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 606
trivial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
Riemann's -function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .603
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
Riemann's di erential equation
general form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
reduction to hypergeometric di erential equation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .396
solutions
P-symbol notation. . . . . . . . . . . . . . . . . . . . . . . . . . . .396transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .396
Riemann{Hilbert problems
classical orthogonal polynomials. . . . . . . . . . . . . . . . .479
Riemann{Lebesgue lemma. . . . . . . . . . . . . . . . . . . . . . . . . .14
Riemann{Siegel formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
Riemann's P-symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
ring functions. . . . . . . . . . . . . . . . . . . seetoroidal functions.
Ritt's theorem
di erentiation of asymptotic approximations . . . . . 42
robot trajectory planning
Cornu's spiral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
Rodrigues formulas
classical orthogonal polynomials. . . . . . . . . . . . . . . . .442
Hahn class orthogonal polynomials . . . . . . . . . . . . . . 462
Rogers polynomials
. . . . . . . . seecontinuous q-ultraspherical polynomials.
Rogers{Ramanujan identities . . . . . . . . . . . . . . . . . 422, 430
constant term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .431
partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
Rogers{Szeg o polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 475
rolling of ships
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679
rook polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .633
roots
of equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90
Rossby waves
bicon
uent Heun functions . . . . . . . . . . . . . . . . . . . . . . 720
rotation matrices
relation to 3 jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . .761
Rouch e's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20, 92
round-robin tournaments . . . . . . . . . . . . . . . . . . . . . . . . . . 648
Runge{Kutta methods
ordinary di erential equations . . . . . . . . . . . . . . . . 89{90
Rutherford scattering
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Rydberg constant
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
S-matrix scattering
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754
saddle points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
coalescing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48, 789{790
sampling expansions
parabolic cylinder functions . . . . . . . . . . . . . . . . . . . . . 317
scaled gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
scaled Riemann theta functions
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
scaled spheroidal wave functions . . . . . . . . . . . . . . 706{707
bandlimited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .706
extremal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
Index 941
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
scaling laws
for di raction catastrophes . . . . . . . . . . . . . . . . . . . . . . 785
scattering problems
associated Legendre functions . . . . . . . . . . . . . . . . . . . 379
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . 753{755
scattering theory
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679
Schl a
i's integrals
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224, 225
Schl a
i{Sommerfeld integrals
Bessel and Hankel functions. . . . . . . . . . . . . . . . . . . . .224
Schl a
i-type integrals
Kelvin functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Schottky group
Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Schottky problem
Riemann surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
Schr oder numbers
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
relation to lattice paths . . . . . . . . . . . . . . . . . . . . . . . . . 622
table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .622
Schr odinger equation
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . 753{755
nonlinear
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . 565
Riemann theta functions. . . . . . . . . . . . . . . . . . . . . .545
q-deformed quantum mechanical . . . . . . . . . . . . . . . . 432
solutions in terms of classical orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .534
Schwarz re
ection principle. . . . . . . . . . . . . . . . . . . . . . . . .19
Schwarz's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Schwarzian derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
Scorer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .209
approximations
expansions in Chebyshev series . . . . . . . . . . . . . . . 212
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 205
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
computation by quadrature . . . . . . . . . . . . . . . . . . . . . . 84
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .205
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .204
initial values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204
numerically satisfactory solutions . . . . . . . . . . . . . 204
standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
integral representations. . . . . . . . . . . . . . . . . . . . .204{205
integrals
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 206tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
secant function. . . . . . . . . . . . seetrigonometric functions.
sectorial harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .378
Selberg integrals
generalized elliptic integrals . . . . . . . . . . . . . . . . . . . . . 516
Selberg-type integrals
gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
separable Gauss sum
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
Shanks' transformation
for sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
ship wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .790{791
sieve of Eratosthenes
prime numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .648
sigma function. . . . . . . seeWeierstrass elliptic functions.
signal analysis
spheroidal wave functions. . . . . . . . . . . . . . . . . . .706{707
simple closed contour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
simple closed curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
simple discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
simple zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
simply-connected domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Sinc function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
sine function . . . . . . . . . . . . . . seetrigonometric functions.
sine integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
applications
Gibbs phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 153
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 154
auxiliary functions . . . seeauxiliary functions for sine
and cosine integrals.
Chebyshev-series expansions . . . . . . . . . . . . . . . . 156{157
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
expansion in spherical Bessel functions . . . . . . . . . . 153
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188{189
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
hyperbolic analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 152
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Laplace transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151
relations to exponential integrals . . . . . . . . . . . . . . . . 151
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
942 Index
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156
value at in nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . 154
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
sine-Gordon equation
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .565
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
singularities
movable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
singularity
branch point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
essential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
isolated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
isolated essential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
pole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
removable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 19
6jsymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .763
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
approximations for large parameters . . . . . . . . . . . . 764
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
alternative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .763
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
graphical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
Regge symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .762
representation as
nite sum of algebraic quantities . . . . . . . . . . . . . .762
nite sum of 3 jsymbols . . . . . . . . . . . . . . . . . . . . . . 761
generalized hypergeometric functions. . . . . . . . . .761
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
sum rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
summation convention . . . . . . . . . . . . . . . . . . . . . . . . . . 760
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .763
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .762
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
SL(2;Z) bilinear transformation . . . . . . . . . . . . . . . . . . . 579
Sobolev polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .477
soliton theory
classical orthogonal polynomials. . . . . . . . . . . . . . . . .478
solitons
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .565
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . 582
spatio-temporal dynamics
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718
spectral problems
Heun's equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 720
spherical Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 262addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
applications
electromagnetic scattering . . . . . . . . . . . . . . . . . . . . 276
Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
asymptotic approximations for large order . . . . . . . see
uniform asymptotic expansions for large order
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276{277
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266
cross-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266, 280
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
numerically satisfactory solutions . . . . . . . . . . . . . 262
singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
duplication formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
explicit formulas
modi ed functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
sums or di erences of squares . . . . . . . . . . . . . . . . . 264
unmodi ed functions . . . . . . . . . . . . . . . . . . . . . . . . . 264
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 266
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
limiting forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265
modi ed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
of the rst, second, and third kinds . . . . . . . . . . . . . 262
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265
Rayleigh's formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
re
ection formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267
duplication formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 267
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .280
uniform asymptotic expansions for large order. . .266
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266
spherical Bessel transform . . . . . . . . . . . . . . . . . . . . . . . . . 278
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
spherical harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379
basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378{379
Index 943
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
Dirac delta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38
distributional completeness. . . . . . . . . . . . . . . . . . . . . .379
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
relation to 3 jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . .760
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379
zonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
spherical polar coordinates . . seespherical coordinates.
spherical triangles
solution of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
spherical trigonometry
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .564
sphero-conal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 693
spheroidal coordinates . . . . . . . . . . . seeoblate spheroidal
coordinates andprolate spheroidal coordinates.
spheroidal di erential equation. . . . . . . . . . . . . . . . . . . . 698
eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698{699
asymptotic behavior. . . . . . . . . . . . . . . . . . . . . .702{703
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
continued-fraction equation . . . . . . . . . . . . . . . . . . . 699
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700
power-series expansion . . . . . . . . . . . . . . . . . . . . . . . . 699
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
Liouville normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .698
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
with complex parameter . . . . . . . . . . . . . . . . . . . . . . . . 700
spheroidal harmonics
oblate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .378
prolate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .378
spheroidal wave functions . . . . . . . . . . . . . . . . . . . . . . . . . 698
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .703
applications
signal analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . .706{707
wave equation. . . . . . . . . . . . . . . . . . . . . . . . . . . .704{706
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
as con
uent Heun functions . . . . . . . . . . . . . . . . . . . . . 717
asymptotic behavior
asx!1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
for large
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702{703
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .707{708
convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
Coulomb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .704
de nitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .699, 700
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .698
eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .698
elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
expansions in series of Ferrers functions . . . . . . . . . 702
asymptotic behavior of coecients . . . . . . . . . . . . 702
tables of coecients . . . . . . . . . . . . . . . . . . . . . . . . . . 708
expansions in series of spherical Bessel functions . . 703
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .704
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700{701integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . 703, 706
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
oblate angular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
of complex argument. . . . . . . . . . . . . . . . . . . . . . . . . . . .700
of the rst kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
other notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
power-series expansions . . . . . . . . . . . . . . . . . . . . . . . . . 699
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
prolate angular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699
radial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
scaled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .706
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .708
with complex parameters . . . . . . . . . . . . . . . . . . . . . . . 700
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .699
spline functions
Bernoulli monosplines. . . . . . . . . . . . . . . . . . . . . . . . . . .597
cardinal monosplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
cardinal splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Euler splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
splines
B ezier curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
square-integrable function . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
stability problems
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .679
stable polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
Hurwitz criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
statistical analysis
multivariate
functions of matrix argument . . . . . . . . . . . . . . . . . 773
statistical applications
functions of matrix argument . . . . . . . . . . . . . . . . . . . 773
statistical mechanics
application to combinatorics . . . . . . . . . . . . . . . . . . . . 635
Heun functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .720
incomplete beta functions . . . . . . . . . . . . . . . . . . . . . . . 189
Jacobian elliptic functions. . . . . . . . . . . . . . . . . . . . . . .564
modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
q-hypergeometric function. . . . . . . . . . . . . . . . . . . . . . .432
solvable models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533
statistical physics
Bernoulli and Euler polynomials . . . . . . . . . . . . . . . . 598
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Steed's algorithm
for continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
steepest-descent paths
numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . 83{84
Stickelberger codes
Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
Stieltjes fraction ( S-fraction) . . . . . . . . . . . . . . . . . . . . . . . 95
944 Index
Stieltjes polynomials
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718
electrostatic interpretation . . . . . . . . . . . . . . . . . . . . 719
Stieltjes transform
analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 52{53
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
de nition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29, 52
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
representation as double Laplace transform. . . . . . .30
Stieltjes{Wigert polynomials . . . . . . . . . . . . . . . . . . . . . . 471
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .474
Stirling cycle numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . .631
Stirling numbers ( rst and second kinds)
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .626
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
generalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .626
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .625
notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
relations to Bernoulli numbers . . . . . . . . . . . . . . . . . . 596
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624, 635
Stirling's formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141
Stirling's series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Stokes line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Stokes multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Stokes phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
complementary error function . . . . . . . . . . . . . . . . . . . 164
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
smoothing of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Stokes sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .782{785
cuspoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782
umbilics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
visualizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .784{785
Stokes' theorem for vector-valued functions . . . . . . . . . 12
string theory
beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
elliptic integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517
modular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582
Painlev e transcendents . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 545
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533
Struve functions . . . . seeStruve functions and modi ed
Struve functions.
Struve functions and modi ed Struve functions . . . 288
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .291applications
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
argumentxe3i=4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
asymptotic expansions
generalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
large argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
large order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .293
remainder terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
with respect to order . . . . . . . . . . . . . . . . . . . . . . . . . 292
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
numerically satisfactory solutions . . . . . . . . . . . . . 288
particular solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289{291
half-integer orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
integral representations
along real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .293
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .292
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
integrals
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
inde nite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .293{294
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
with respect to order . . . . . . . . . . . . . . . . . . . . . . . . . 294
Kelvin-function analogs . . . . . . . . . . . . . . . . . . . . . . . . . 294
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .288
principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
relations to Anger{Weber functions . . . . . . . . . . . . . 297
series expansions
Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Chebyshev. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300
power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .292
Struve's equation . . . seeStruve functions and modi ed
Struve functions, di erential equations.
Sturm{Liouville eigenvalue problems
ordinary di erential equations. . . . . . . . . . . . . . . . . . . .89
summability methods for integrals
Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Ces aro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Index 945
Fourier integrals
conjugate Poisson integral . . . . . . . . . . . . . . . . . . . . . 34
Fej er kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Poisson integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
Poisson kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
fractional integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
summability methods for series
Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Ces aro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Fourier series
Abel means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Ces aro means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Fej er kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Poisson kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
regular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
Tauberian theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
summation by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
summation formulas
Boole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Euler{Maclaurin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
sums of powers
as Bernoulli or Euler polynomials . . . . . . . . . . . . . . . 589
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .598
supersonic
ow
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
support
of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
surface . . . . . . . . . . . . . . . . . . . . . seeparametrized surfaces.
surface harmonics of the rst kind . . . . . . . . . . . . . . . . . 378
surface-wave problems
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
swallowtail bifurcation set
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .782
swallowtail canonical integral . . . . . . . . . . . . . . . . . . . . . 776
asymptotic approximations . . . . . . . . . . . . . . . . . 789{790
convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788
formulas for Stokes set . . . . . . . . . . . . . . . . . . . . . . . . . . 783
integral identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
picture of Stokes set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
pictures of modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .787
swallowtail catastrophe . . . . . . . . . . . . . . . . . . . . . . 776, 784
symmetric elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . 497
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 509{510
advantages of symmetry . . . . . . . . . . . . . . . . . . . . . . . . 497
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514{516physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516{517
statistical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515
arithmetic-geometric mean . . . . . . . . . . . . . . . . . . . . . . 505
asymptotic approximations and expansions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53, 510{511
Bartky's transformation. . . . . . . . . . . . . . . . . . . . . . . . .504
change of parameter of RJ. . . . . . . . . . . . . . . . . . . . . . 504
circular cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502{504
complete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .486
computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517{519
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .503
degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
duplication formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
elliptic cases of R a(b;z) . . . . . . . . . . . . . . . . . . . . . . . 498
rst, second, and third kinds . . . . . . . . . . . . . . . . . . . . 486
Gauss transformations . . . . . . . . . . . . . . . . . . . . . 497, 505
general lemniscatic case . . . . . . . . . . . . . . . . . . . . 502, 503
graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499{500
hyperbolic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502{504
inequalities
complete integrals. . . . . . . . . . . . . . . . . . . . . . . .506{507
incomplete integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 507
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 506
integrals of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .511
Landen transformations . . . . . . . . . . . . . . . . . . . . 497, 505
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
permutation symmetry . . . . . . . . . . . . . . . . . . . . . 497, 498
power-series expansions. . . . . . . . . . . . . . . . . . . . .501{502
reduction of general elliptic integrals. . . . . . . .512{514
relations to other functions
Appell functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
Bulirsch's elliptic integrals . . . . . . . . . . . . . . . . . . . . 508
hypergeometric function . . . . . . . . . . . . . . . . . . . . . . 509
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . 508
Lauricella's function . . . . . . . . . . . . . . . . . . . . . . . . . . 509
Legendre's elliptic integrals . . . . . . . . . . . . . . 507, 508
theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . 509
special cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .502{503
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .519
transformations replaced by symmetry
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .497, 505, 508
symmetries
of canonical integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . .777
Szeg o{Askey polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .475
Szeg o{Sz asz inequality
Jacobi polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .451
tangent function . . . . . . . . . . seetrigonometric functions.
tangent numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .596
Taylor series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
asymptotic approximations for coecients. . . . . . . .65
946 Index
Taylor's theorem
one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 18
two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
tempered distributions. . . . . . . . . . . . . . . . . . . . . . . . . .36, 52
convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
term-by-term integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
terminant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
incomplete gamma functions . . . . . . . . . . . . . . . . . . . . 189
tesseral harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .378
test functions
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Theorem of Ince
Mathieu's equation. . . . . . . . . . . . . . . . . . . . . . . . .653, 657
theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
addition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529{530
of ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
discrete analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
double products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .530
duplication formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
fundamental parallelogram . . . . . . . . . . . . . . . . . . . . . . 524
generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
graphics
complex variables . . . . . . . . . . . . . . . . . . . . . . . . 527{529
real variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .525{527
in nite products. . . . . . . . . . . . . . . . . . . . . . . . . . . .529{530
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Jacobi's identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
Jacobi's inversion formula . . . . . . . . . . . . . . . . . . 532, 533
Jacobi's original notation . . . . . . . . . . . . . . . . . . . . . . . 524
Jacobi's triple product . . . . . . . . . . . . . . . . . . . . . . . . . . 529
Landen transformation. . . . . . . . . . . . . . . . . . . . . . . . . .531
Laplace transform with respect to lattice parameter
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
lattice parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .524
transformation of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
lattice points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .524
limit forms as= !0+. . . . . . . . . . . . . . . . . . . . . . . . .534
McKean and Moll's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
Mellin transform with respect to lattice parameter
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
modular transformations . . . . . . . . . . . . . . . . . . . . . . . . 531
multidimensional . . . . . . . . . . . . . . . . . . . seeChapter 21.
Neville's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524, 550
nome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
rectangular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
transformation of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .530
quasi-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
Ramanujan's. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533
Ramanujan's change of base . . . . . . . . . . . . . . . . . . . . 533
rectangular case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .524
relations to other functions
Dedekind's eta function. . . . . . . . . . . . . . . . . . . . . . .525
elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
elliptic modular function. . . . . . . . . . . . . . . . . . . . . .532
Jacobi's epsilon function . . . . . . . . . . . . . . . . . . . . . . 562
Jacobian elliptic functions. . . . . . . . . . . . . . . .532, 550
modular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Riemann zeta function. . . . . . . . . . . . . . . . . . . . . . . .532
symmetric elliptic integrals . . . . . . . . . . . . . . . . . . . 508
Weierstrass elliptic functions . . . . . . . . . . . . . 532, 574
Riemann. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .538
Riemann with characteristics. . . . . . . . . . . . . . . . . . . .539
sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .534
translation by half-periods . . . . . . . . . . . . . . . . . . . . . . 525
values atz= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .529
Watson's expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
Watson's identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
with characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .525
Thomae transformation
3F2functions of matrix argument . . . . . . . . . . . . . . . 772
3j;6j;9jsymbols
relation to generalized hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407, 418
3jsymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
angular momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
approximations for large parameters . . . . . . . . . . . . 764
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
Gaunt coecient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .761
Gaunt's integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .761
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
graphical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
projective quantum numbers . . . . . . . . . . . . . . . . . . . . 758
recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
Regge symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .759
relations to other functions
Legendre functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .760
rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
spherical harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . .760
representation as
nite sum of algebraic quantities . . . . . . . . . . . . . .758
generalized hypergeometric functions. . . . . . . . . .758
special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
Index 947
summation convention . . . . . . . . . . . . . . . . . . . . . . . . . . 760
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .760
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .759
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
triangle conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .765
Toda equation
Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
tomography
con
uent hypergeometric functions . . . . . . . . . . . . . . 346
tops
Jacobian elliptic, or hyperelliptic, integrals . . . . . . 566
toroidal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 371, 379
toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .379
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
hypergeometric representations. . . . . . . . . . . . . . . . . .371
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 371
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .372
Whipple's formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
torus
complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
transcendental equations
asymptotic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
transcendental functions. . . . . . . . . . . . . . . . . . . . . . . . . . .724
transition points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58, 63
transport equilibrium
generalized exponential integral . . . . . . . . . . . . . . . . . 190
triangle conditions
3jsymbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .758
triangle inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
triangles
solution of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
triangular matrices
con
uent hypergeometric functions . . . . . . . . . . . . . . 345
tricon
uent Heun equation . . . . . . . . . . . . . . . . . . . . . . . . 718
trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
addition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
applications
cubic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
solution of triangles and spherical triangles . . . 130
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Chebyshev-series expansions . . . . . . . . . . . . . . . . . . . . 132
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
conformal maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
elementary properties . . . . . . . . . . . . . . . . . . . . . . 115{116
graphics
complex argument . . . . . . . . . . . . . . . . . . . . . . . 113{115
real argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
integrals
de nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
inde nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
inverse . . . . . . . . . . seeinverse trigonometric functions.
Laurent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Maclaurin series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
moduli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
multiples of argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
partial fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
poles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
real and imaginary parts . . . . . . . . . . . . . . . . . . . . . . . . 118
relations to hyperbolic functions . . . . . . . . . . . . . . . . 123
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
squares and products . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
triple integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
truncated exponential series . . . . . . . . . . . . . . . . . . . . . . . 180
turning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58, 63
fractional or multiple. . . . . . . . . . . . . . . . . . . . . . . . . . . . .61
two-body relativistic scattering
Lam e polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
ultraspherical polynomials . . . . . . . . . . . . . . . . . . . . . . . . 438
. . . . . . . . . . . see also classical orthogonal polynomials.
addition theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .459
applications
zonal spherical harmonics . . . . . . . . . . . . . . . . . . . . . 479
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .452
case= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
di erential equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .445
expansions in series of. . . . . . . . . . . . . . . . . . . . . .460, 461
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
integral representations. . . . . . . . . . . . . . . . . . . . .447, 448
for products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444{445, 448
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
limits to monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
linearization formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
948 Index
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436, 437
orthogonality property . . . . . . . . . . . . . . . . . . . . . . . . . . 439
parameter constraint . . . . . . . . . . . . . . . . . . . . . . . 439, 443
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
relations to other functions
Ferrers functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .448
hypergeometric function . . . . . . . . . . . . . . . . . 393, 442
Rodrigues formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .444
tables of coecients. . . . . . . . . . . . . . . . . . . . . . . . . . . . .440
upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438, 454
umbilics
normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
umbral calculus
Bernoulli and Euler polynomials . . . . . . . . . . . . . . . . 590
uniformization
algebraic equations via Jacobian elliptic functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
unity
roots of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
vacuum magnetic elds
toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
validated computing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
Van Vleck polynomials
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718
Van Vleck's theorem for continued fractions . . . . . . . . 25
Vandermondian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
variation of parameters
inhomogeneous di erential equations . . . . . . . . . . . . . 26
variation of real or complex functions . . . . . . . . . . . . . . . . 6
bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
variational operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
vector
equivalent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .542
norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
vector-valued functions. . . . . . . . . . . . . . . . . . . . . . . . . . .9{12
. . . . . . . . . . . . . . . . . . . . . see also parametrized surfaces.
curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
del operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
divergence (or Gauss's) theorem. . . . . . . . . . . . . . . . . .12
gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
Green's theorem
three dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
line integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
reparametrization of integration pathsorientation-preserving. . . . . . . . . . . . . . . . . . . . . . . . . .11
orientation-reversing . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Stokes' theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
. . . . . . . . . . . . . . . . . . . . see also vector-valued functions.
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
right-hand rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Einstein summation convention . . . . . . . . . . . . . . . . . . 10
Levi-Civita symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 10
right-hand rule for cross products . . . . . . . . . . . . . . . . 10
scalar product . . . . . . . . . . . . . . . . . . . . . seedot product.
unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
vector product. . . . . . . . . . . . . . . . . . . . seecross product.
vibrational problems
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .678
Voigt functions
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168
relation to line broadening function . . . . . . . . . . . . . 167
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169
von Staudt{Clausen theorem
Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
Voronoi's congruence
Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
Waring's problem
number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
water waves
Kelvin's ship-wave pattern. . . . . . . . . . . . . . . . . .790{791
Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . . . 545
Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Watson integrals
Appell functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
generalized hypergeometric functions . . . . . . . . . . . . 417
Watson's 3F2sum
Andrews' terminating q-analog . . . . . . . . . . . . . . . . . . 427
Gasper{Rahman q-analog . . . . . . . . . . . . . . . . . . . . . . . 426
Watson's expansions
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531
Watson's identities
theta functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531
Watson's lemma
asymptotic expansions of integrals . . . . . . . . . . . . 44, 46
Watson's sum
generalized hypergeometric functions . . . . . . . . . . . . 406
wave acoustics
generalized exponential integral . . . . . . . . . . . . . . . . . 190
wave equation
Index 949
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . see also water waves.
Bessel functions and modi ed Bessel functions . . 276
con
uent hypergeometric functions . . . . . . . . . . . . . . 346
ellipsoidal coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . .693
Mathieu functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .678
oblate spheroidal coordinates . . . . . . . . . . . . . . . 705{706
paraboloidal coordinates . . . . . . . . . . . . . . . . . . . . . . . . 346
prolate spheroidal coordinates . . . . . . . . . . . . . . 704{705
separation constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
spherical Bessel functions . . . . . . . . . . . . . . . . . . . . . . . 276
sphero-conal coordinates . . . . . . . . . . . . . . . . . . . . . . . . 693
symmetric elliptic integrals. . . . . . . . . . . . . . . . . . . . . .501
wave functions
paraboloidal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Weber function . . . . . . . . . . . seeAnger{Weber functions.
Weber parabolic cylinder functions
. . . . . . . . . . . . . . . . . . seeparabolic cylinder functions.
Weber's function
. . . . . . . . . . seeBessel functions of the second kind.
Weber{Schafheitlin discontinuous integrals
Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244
Weierstrass M-test
. . . . . . . . . . . . . . seeM-test for uniform convergence.
Weierstrass elliptic functions . . . . . . . . . . . . . . . . . . . . . . 570
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .577
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
applications
mathematical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
physical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582{583
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .578
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
duplication formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
equianharmonic case. . . . . . . . . . . . . . . . . . .571{572, 574
Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
graphics
complex variables . . . . . . . . . . . . . . . . . . . . . . . . 573{574
real variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .571{572
homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
in nite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
integral representations . . . . . . . . . . . . . . . . . . . . . . . . . 578
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
equianharmonic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .574
generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .571
lemniscatic . . . . . . . . . . . . . . . . . . . . . . . . . . 571{572, 574
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570pseudo-lemniscatic. . . . . . . . . . . . . . . . . . . . . . . . . . . .574
rectangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
rhombic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
Laurent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .577
lemniscatic case . . . . . . . . . . . . . . . . . . . . . . . 571{572, 574
n-tuple formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
poles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .570
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .577
principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
pseudo-lemniscatic case . . . . . . . . . . . . . . . . . . . . . . . . . 574
quarter periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
quasi-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
relations to other functions
elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
general elliptic functions . . . . . . . . . . . . . . . . . . . . . . 576
Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . 575
symmetric elliptic integrals . . . . . . . . . . . . . . . . . . . 509
theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
rhombic case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .574
series of cosecants or cotangents. . . . . . . . . . . . . . . . .577
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .584
zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570, 579
Weierstrass }-function
. . . . . . . . . . . . . . . . . seeWeierstrass elliptic functions.
Weierstrass product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Weierstrass sigma function
. . . . . . . . . . . . . . . . . seeWeierstrass elliptic functions.
Weierstrass zeta function
. . . . . . . . . . . . . . . . . seeWeierstrass elliptic functions.
weight functions
cubature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84{85
de nition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79, 437
Freud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
least squares approximations . . . . . . . . . . . . . . . . . . . . . 99
logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81{82
minimax rational approximations. . . . . . . . . . . . . . . . .97
quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79{80
weighted means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Weniger's transformation
for sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94
Whipple's 3F2sum
Gasper{Rahman q-analog . . . . . . . . . . . . . . . . . . . . . . . 427
Whipple's formula
associated Legendre functions . . . . . . . . . . . . . . . . . . . 362
toroidal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
Whipple's sum
generalized hypergeometric functions . . . . . . . . . . . . 406
Whipple's theorem
Watson'sq-analog. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .429
Whipple's transformation
generalized hypergeometric functions . . . . . . . . . . . . 407
950 Index
Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
. . . . . . . . see also con
uent hypergeometric functions.
addition theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .345
analytic continuation. . . . . . . . . . . . . . . . . . . . . . . . . . . .334
analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
applications
Coulomb functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .346
groups of triangular matrices . . . . . . . . . . . . . . . . . 345
physical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .346, 754
uniform asymptotic solutions of di erential equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
asymptotic approximations for large parameters
imaginaryand/or. . . . . . . . . . . . . . . . . . . . . . . . .340
large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341{342
large. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339{341
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339{342
asymptotic expansions for large argument . . . . . . . 339
error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .339
exponentially-improved . . . . . . . . . . . . . . . . . . . . . . . 339
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
connection formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . .335
continued fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
di erential equation. . . . . . . seeWhittaker's equation.
expansions in series of. . . . . . . . . . . . . . . . . . . . . . . . . . .344
integral representations
along the real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .337
Mellin{Barnes type . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
integral transforms in terms of . . . . . . . . . . . . . . . . . . 344
integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
compendia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .344
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Hankel transforms. . . . . . . . . . . . . . . . . . . . . . . .343{344
Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Mellin transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .343
interrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
large argument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
limiting forms
asz!0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .335
asz!1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .335
multiplication theorems . . . . . . . . . . . . . . . . . . . . . . . . . 345
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
power series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .334
principal branches (or values) . . . . . . . . . . . . . . . . . . . 334
products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
relations to other functions
Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Coulomb functions . . . . . . . . . . . . . . . . . . 742, 748, 751
elementary functions. . . . . . . . . . . . . . . . . . . . . . . . . .338
error functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338
incomplete gamma functions . . . . . . . . . . . . . . . . . . 338Kummer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
modi ed Bessel functions . . . . . . . . . . . . . . . . . . . . . 338
orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . 338
parabolic cylinder functions. . . . . . . . . . . . . . . . . . .338
series expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 344{345
addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
in Bessel functions or modi ed Bessel functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
multiplication theorems. . . . . . . . . . . . . . . . . . . . . . .345
power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
zeros
asymptotic approximations . . . . . . . . . . . . . . . . . . . 343
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .342
inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
number of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Whittaker's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
fundamental solutions. . . . . . . . . . . . . . . . . . . . . . . . . . .335
numerically satisfactory solutions . . . . . . . . . . . . . . . 335
relation to Kummer's equation . . . . . . . . . . . . . . . . . . 334
standard solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
Whittaker{Hill equation. . . . . . . . . . . . . . . . . . . . . . . . . . .676
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .678
separation constants . . . . . . . . . . . . . . . . . . . . . . . . . . 678
Wigner 3j;6j;9jsymbols
. . . . . see3jsymbols, 6jsymbols, and 9 jsymbols.
Wilf{Zeilberger algorithm
applied to generalized hypergeometric functions . . 407
Wilkinson's polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . .92
Wilson class orthogonal polynomials . . . . . . . . . . 467{470
asymptotic approximations. . . . . . . . . . . . . . . . . . . . . .470
de nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
di erences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
interrelations with other orthogonal polynomials
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464, 468{469
leading coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467{468
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
orthogonality properties. . . . . . . . . . . . . . . . . . . . . . . . .467
relation to generalized hypergeometric functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468{469
transformations of variable . . . . . . . . . . . . . . . . . . . . . . 467
weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467{468
Wilson polynomials
. . . . . . . . . . . seeWilson class orthogonal polynomials.
winding number
of closed contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
WKB or WKBJ approximation
. .seeLiouville{Green (or WKBJ) approximation.
Wronskian
di erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Wynn's cross rule
Index 951
for Pad e approximations . . . . . . . . . . . . . . . . . . . . . . . . . 98
Wynn's epsilon algorithm
for sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
zero potential
Coulomb functions . . . . . . . . . . . . . . . . . . . . . . . . . 753, 754
zeros of analytic functions
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90{92
conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
multiplicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19, 90
simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
zeros of Bessel functions (including derivatives)
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
asymptotic expansions for large order
uniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237
asymptotic expansions for large zeros . . . . . . . . . . . 236
error bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
common . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235, 238
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . .235, 238{240
double . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
monotonicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
of cross-products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .238
asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . 238
purely imaginary . . . . . . . . . . . . . . . . . . . . . . . . . . . 235, 236
relation to inverse phase functions. . . . . . . . . . . . . . .235
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132, 278
with respect to order ( -zeros) . . . . . . . . . . . . . . . . . . 240
zeros of cylinder functions (including derivatives)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235{237
analytic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
asymptotic expansions for large order
uniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236asymptotic expansions for large zeros . . . . . . . . . . . 236
forward di erences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
monotonicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
relation to inverse phase functions. . . . . . . . . . . . . . .235
zeros of polynomials
. . . . . . . . . . . . . . . . . . . . . . . . see also stable polynomials.
computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91{92
conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
degrees two, three, four . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Descartes' rule of signs . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
division algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
elementary symmetric functions . . . . . . . . . . . . . . . . . . 22
explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Horner's scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
extended. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
resolvent cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
roots of constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
zeta function . . . . . . . . . . . . . . . seeHurwitz zeta function,
Jacobi's zeta function, periodic zeta function, Rie-
mann zeta function, andWeierstrass zeta function.
zonal polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .773
beta integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
Laplace integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .769
mean-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .769
tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .773
zonal spherical harmonics
ultraspherical polynomials . . . . . . . . . . . . . . . . . . . . . . 479