[Jerome_Spanier,_Keith_B._Oldham]_An_Atlas_Of_Func(Bookos.org)

AN ATLASOF FUNCTIONS Jerome SpanierThe Claremont Graduate SchoolClaremont. California. U.S.A. Keith B. OldhamTrent UniversityPeterborough. Ontario. Canada O HE1VHSPHERE PUBLISHING CORPORATIONA subsidiary of Harper & Row. Publishers, Inc.WashingtonNew YorkLondon Distribution Outside North AmericaSPRINGER-VERLAGBerlinHeidelbergNew YorkLondonParisTokyo AN ATLAS OF FUNCTIONSCopyright G 198' bs Hemisphere Publishing Corporation. All rights reservedPrinted in the Urated States of America. Except as permitted under theUnited StatesCopyright Act of 1976.no pan of this publication may bereproduced or distributed in any form or by any means. or stored jr. a data base or retneNai system. without the prior written permission of the publisher.1234567890 EBEB 89876 This book was set in Times Roman by Edwards Brothers. Inc.The editor was Sandra Tamburrino. Edwards Brothers. Inc.. was printer and binder. Libran of Cow Cataloging in Publication Data Spanter. Jerome. dateAn atlas of functions.Bibliography: p.Includes indexes1. Oldham. Keith B. D. Title.QA331.S695 1997,51586.18294ISBN 0.89116-573-8Hemisphere Publishing CorporationDISTRIBUTION O( WE NORTH AMERICA:ISBN 3-540-17393-1 Berlin CONTENTS Preface lx0General Considerations I 1The Constant Function c 112The Factorial Function n! and Its Reciprocal 193The Zeta Numbers and Related Functions 254 The Bernoulli Numbers, B, 355The Euler Numbers, En 396The Binomial Coefficients (m ) 437The Linear Function bx - c and Its Reciprocal 538The Unit-Step u(x - a) and Related Functions 639The Integer-Value Int(x) and Fractional-Value frac(x) Functions 7110The Dirac Delta Function S(x - a) 79 11The Integer Powers (bx + c)" and x" 8312The Square-Root Function -.'bx + c and Its Reciprocal 91 13The Noninteger Powers x' 9914The b.ia= - x: Function and Its Reciprocal 107 15The bv'.t= + a Function and Its Reciprocal 115 CONTENTS vi16The Quadratic Function ax' + bx + c and Its Reciprocal 123 17The Cubic Function x' + ar= + bx + c and Higher Polynomials 13118The Pochhammer Polynomials (x). 149 19The Bernoulli Polynomials B,(x) 16720The Euler Polynomials E,(x) 17521The Legendre Polynomials P,(x) 18322The Chebyshev Polynomials T,(x) and U,(x) 19323The Laguerrre Polynomials L,(x) 20924The Hermite Polynomials H,(x) 21725The Logarithmic Function ln(x) 22526 The Exponential Function exp(bx + c) 23327Exponentials of Powers exp(- arxD 25328 The Hyperbolic Sine sinh(x) and Cosine cosh(x) Functions 26329 The Hyperbolic Secant sech(x) and Cosecant csch(x) Functions 27330 The Hyperbolic Tangent tanh(x) and Cotangent coth(x) Functions 27931The Inverse Hyperbolic Functions 28532The Sine sin(x) and Cosine cos(x) Functions 29533The Secant sec(x) and Cosecant csc(x) Functions 31134The Tangent tan(x) and Cotangent cot(x) Functions 31935The Inverse Trigonometric Functions 33136Periodic Functions 34337 The Exponential Integral Ei(x) and Related Functions 35138Sine and Cosine Integrals 36139 The Fresnel Integrals S(x) and C(x) 37340 The Error Function erf(x) and Its Complement erfc(x) 38541The exp(x) erfc(f) and Related Functions 39542Dawson's Integral 405 vU CONTENTS43The Gamma Function r(x) 41144 The Digamma Function $(x) 42345The Incomplete Gamma y(v;x) and Related Functions 43546The Parabolic Cylinder Function D,(r) 44547 The Kummer Function M(a;c;x) 45948 The Tricomi Function U(a;c;x) 47149 The Hyperbolic Bessel Functions L(x) and I#) 47950 The General Hyperbolic Bessel Function 1,(x) 48951The Basset Function K,(x) 49952The Bessel Coefficients J,(x) and J,(x) 50953The Bessel Function J,(x) 52154The Neumann Function Y,(x) 53355The Kelvin Functions 54356The Airy Functions Ai(x) and Bi(x) 55557The Struve Function 56358The Incomplete Beta Function B(v;µ;x) 57359 The Legendre Functions P,(x) and Q(x) 58160 The Gauss Function F(a,b;c;x) 59961The Complete Elliptic Integrals K(p) and E(p) 60962The Incomplete Elliptic Integrals F(p;b) and E(p;4) 62163The Jacobian Elliptic Functions 63564The Hurwitz Function ((v;u) 653Appendix A Utility Algorithms 665Appendix B Some Useful Data 673 References and Bibliography 679Subject Index 681Symbol Index 691 PREFACE The majority of engineers and physical scientists must consult reference books containing information on a variety ofmathematical functions. This reflects the fact that all but the most trivial quantitative work involves relationships that arebest described by functions of various complexities. Of course, the need will depend on the user, but all will requireinformation about the general behavior of the function in question and its values at a number of arguments.Historically. this latter need has been met primarily by tables of function values. However, the ubiquity of computersand programmable calculators presents an opportunity to provide reliable, fast and accurate function values without theneed to interpolate. Computer technology also enables graphical presentations of information to be made with digitalaccuracy. An Atlas of Functions exploits these opportunities by presenting algorithms for the calculation of most functionsto more than seven-digit precision and computer-generated maps that may be read to two or three figures. Of course, the need continues for ready access to the many formulas and properties that characterize a specific function. This need ismet in the Atlas through the display of the most important definitions, relationships, expansions and other properties ofthe 400 functions covered in this book.The Atlas is organized into 64 chapters, each of which is devoted to one function or to a number of closely relatedfunctions; these appear roughly in order of increasing complexity. A standardized format has been adopted for each chapter to minimize the time required to locate a sought item of information. A description of how the chapters aresectioned is included in Chapter 0. Two appendixes, a references/bibliography section and two indexes complete thevolume.It is a pleasure to acknowledge our gratitude to Diane Kaiser and Charlotte Oldham who prepared the typescript andto Jan Myland who created the figures. Their skill and unfailing good humor in these exacting tasks have exceeded allexpectations. We would also like to record our indebtedness to our families for their support and patience during themany years that the preparation of the Atlas took time that was rightfully theirs. To them, and especially to Bunny andCharlotte, we dedicate this book.Jerome SpanierKeith B. Oldham ix CHAPTERTGENERAL CONSIDERATIONS In this chapter are collected some considerations that relate to all, or most, functions. The general organization ofthe Atlas is also explained here. Thus, this could be a good starting point for the reader. However, the intent ofthe authors is that the information in the Atlas be immediately available to an unprepared reader. There are nospecial codes that must be mastered in order to use the book, and the only conventions that we adopt are thosethat are customary in scientific writing.Each chapter in the Atlas is devoted to a single function or to a small number of intimately related functions.The preamble to the chapter exposes any such relationships and introduces special features of the subject function. 0:1 NOTATIONThe nomenclature and symbolism of mathematical functions are bedevilled by ambiguities and inconsistencies.Several names may be attached to a single function, and one symbol may be used to denote several functions. In the first section of each chapter the reader is alerted to such sources of confusion.For the sake of standardization, we have imposed certain conventions relating to symbols. Thus, we haveeschewed boldface and similar typographical niceties on the grounds that they are difficult to reproduce by pencil on paper, or with office machines. We reserve the use of italics to represent numbers (such as constants. c: coef- ficients, a a,, a3, ...and function arguments. x) and avoid their use in symbolizing functions. Another instanceof a convention intended to add clarity is our distinction in using commas and semicolons as separators, as inF(a.b;c;x). Elements in a string may be interchanged when they are separated by commas but not where a semicolonserves as separator.The symbol z(=x + iy) is reserved to denote a complex variable. All other variables are implicitly real. 0:2 BEHAVIORSome functions are defined for all values of their variable(s). For other functions there are restrictions, such as - I< x s I or n = 1, 2, 3...., on these values that specify the range of each variable and thereby the domain ofthe function. Likewise, the function itself may be restricted in range and may be real valued, complex valued oreach of these in different domains. Such considerations are discussed in the second section of each chapter.This section also reveals how the function changes in value as its variables change throughout their ranges.thereby exposing the general "shape" of the function. This information is conveyed by a verbal description, sup- 1 0:3 GENERAL CONSIDERATIONS 2plemented by a graphical "map." All diagrams have been computer generated and are precisely drawn. Each graphhas been scaled so that the graticule spacing is either ten or fifteen millimetres (2.00 or 3.00 mm between adjacent dots), which permits interpolation to an accuracy of two or three significant digits. Much greater precision isavailable from the algorithms of Section 8.A univariate function (i.e., a function depending only on a single variable, its argument) is generally illustratedby a simple graph of that function versus its argument. Bivariate functions of two continuous variables are oftendiagrammed by contour maps whose curves correspond to specified values of the function and whose axes are theargument and the second variable.In discussing the behavior of some functions it is convenient to adopt the terminology quadrant. There arefour quadrants defined as illustrated in Figure 0-1. FIG 0-1x <0f (x) >0x>0f (x) >0 secondfirstquadrantquadrant ........................................thirdfourthquadrantquadrant f(x)<0f(x)<0X <0x>0 0:3 DEFINITIONS0 Often there arc several formulas relating a function to its variable(s), although they may not all apply over theentire domain of the function. These various interrelationships are listed in the third section of each chapter underthe heading Definitions even though, from a strictly logical viewpoint, some might prefer to select one as theunique definition and cite the others as "equivalences."Several types of definition are encountered in Section 3. For example, a function may be defined:(a) by an equation that explicitly defines the function in terms of "simpler" functions and algebraic operations;(b) by a formula relating the function to its variable(s) through a finite or an infinite number of arithmetic oralgebraic operations;(c) as the derivative or indefinite integral of a simpler function;(d) as an integral transform of the form 0:3:1 f(x) = Jg(xi)dtNwhere g is a function having one more variable than f. to and t, being specified limits of integration; 3 GENERAL CONSIDERATIONS(e) through a generating function, G(xt), that defines a family of functions f;(x) via the expansion 0:3:2 G(x;t) _f,(x)g,(t) where g,(t) is a simpler set of functions such as t';(f) as the inverse of another function F(z) so that the implicit equation 0:3:3 F(f(x)) = x0:5 is used to define f(x);(g) as a special case or a limiting case of a more general function;(h) parametrically through a pair of equations that separately relate the function f(x) and its argument x to a thirdvariable;(i) implicitly via a differential equation, the solution (or one of the solutions) of which is the subject function;and(j) through concepts borrowed from geometry or trigonometry.Some definitions generate two or more values of a function from a single value of the argument. The Atlasminimizes this complication by concentrating, wherever possible, on single-valued functions. 0:4 SPECIAL CASESIf the function reduces to a simpler function under special circumstances, this is noted in the fourth section of eachchapter. 0:5 INTRARELATIONSIIIPSAn equation linking the two functions f(x) and g(x) is an interrelationship between them. In contrast, one speaksof an interrelationship in case there exists a formula that provides a link between values of a single function attwo or more values of one of its variables, for example, between f(x,) and f(x:). In this Atlas intrarelationshipswill be found in Section 5 of each chapter, interrelationships mainly in Sections 3 and 13.An equation expressing the relationship between f(-x) and f(x) is called a reflection formula. Less commonlythere exist reflection formulas relating f(a - x) to f(a + x) for nonzero values of a.A second class of intrarelationships are translation formulas: these relate f(x + a) to f(x). The most generaltranslation formula, in which a is free to vary continuously, becomes an argument-addition formula that relatesf(x + y) to f(x) and f(y). However, many translation formulas are restricted to special values of a such as a = Ior a = nw; the relationships are then known as recurrence relations or recursion formulas. Such relationships arecommon in bivariate functions; a recursion formula then normally relates f(v;x) to f(v - I;x) or to both f(v - I;x)and f(v - 2;x).A very general argument-addition formula is provided by the Taylor expansion:_+ x _&f(y)x' d'(y) + .. . 0:5:1f(y- x) = f(y),- xaj(y)dx2! dx=3!x'Expressions for the remainder after the series is truncated to a finite number of terms are provided by Abramowitzand Stegun (Section 3.61.A third class of intrarelationships are argument-multiplication formulas that relate finx) to f(x). More rarelythere exist function-multiplication formulas orfunction-addition formulas that provide expressions for f(x) f(y) andf(x) + f(y), respectively.Yet other intrarelationships are those provided by finite and infinite series. With bivariate and multivariatefunctions there may be a great number of such formulas, and functions other than f may be involved. 0:6 GENERAL CONSIDERATIONS0:6 EXPANSIONS4 The sixth section of each chapter is devoted to ways in which the function(s) may be expressed as a finite or infinitearray of terms. Such arrays are normally series, products or continued fractions.Notation such as 0:6:1 fi(x) _ ', g(j;x)J=Ois used to represent a convergent infinite series, where g is a function of j and x. Unless otherwise qualified, 0:6:1implies that, for values of x in a specified range, the numerical value of the finite sum0:6:2g(0;x) +g(l;x) + g(2;x) ++ g(j:x) ++ g(J;x)can be brought indefinitely close to f(x) by choosing J to be a large enough integer.Frequently encountered are convergent series whose successive terms, for sufficiently large j, decrease in mag-nitude and alternate in sign. We shall loosely call such series alternating series. A valuable property of suchalternating series that is often used in this Atlas enables the remainder after a finite number of terms are summedto be estimated in terms of the first omitted term. When l (- I)'g(j;x) is used to represent an alternating series,this result: 0:6:3G(-I)'g(j;x)-(-l)'g(j:x)i-0/=0< 1g(J + l;x)t plays a particularly important role in the design of many algorithms of Sections 8.In contrast to 0:6:1. the symbolism 0:6:4f(x) -g(j;x)j = 0, 1, 2....which is reserved for asymptotic series, implies that for every J the numerical value of 0:6:2 can be broughtindefinitely close to f(x) by making x, not J, sufficiently large. It is this restriction on the magnitude of x thatmakes an asymptotic expansion, though of great utility in many applications, rather treacherous for the incautioususer.If the function g(j:x) in 0:6:1 or 0:6:4 can be written as the product c x°'s', where c, is independent of .% whilea and 0 are constants, then the expansions 0:6:1 and 0:6:4 are called power series. The evaluation of the finiteanalog of such series is aided by rewriting 0:6:2 in the form 0:6:5 (((...((ctxa + ct-,)xa + cJ-,)xa + ... + c:ks + c1)xa + ca)x"which is termed a concatenation or nested sum. Such nested sums will also frequently be found in algorithmsappearing in Section 8.The infinite product notation 0:6:6 f(x) = 1 1 g(j;x)=oimplies that the numerical value of the finite product 0:6:7 g(O:x)g(l:x)g(2:x) g(J;x) approaches f(x) indefinitely closely as J takes larger and larger integer values.The notation 0:0.8 aIa,a,ao+---...Rt+ 132t P3+is a standard abbreviation for the continued fraction GENERAL CONSIDERATIONS 0:7 0:6:9Q1 P2+ a3P3 + a.P.+...in which each as and P3 may denote constants or variables. Continued fractions may be infinite, as denoted in 0:6:8,or finite (or "terminated"): 0:6:10ao +P, + a, as+a,---...--P,+ P2+ P3+P,_,+ P,Of great utility in working with continued fractions is the equivalence 0:6:11(1,a,a3aYia, YiY2a2 Y2Y3a3P,+ P:+ 03+...P. -ao7101+ Y2P2+ Y,P,+Y.P.Demonstrating the interchangeability of continued fractions and power series is the identity Ixx2 X.1aoxa,xa2xa,-,x0:6:12- + - + - + +aoaoa,a0a,a2a0a,a2a.a0- a, + x- a2 + x- a3 + xa + x 0:7 PARTICULAR VALUESIf certain values of the variable(s) of a function generate noteworthy function values, these are cited in the seventhsection of each chapter, often as a table.In Section 7 of many chapters we include information about those arguments that lead to inflections, minima,maxima and zeros of the subject function f(x). The term extremum is used to mean either a maximum or a minimum.An inflection of a function occurs at a value of its argument at which the second derivative of the function iszero. that is: 0:7:1 d'fdx'(.r;) = 0f(x,) = inflection of f(x)A minimum and a maximum of a function are characterized respectively by 0:7:2dfd'f-0, - (x,,) >0f(x,) = minimum of f(x) dxd rand 0:7:3dfd:f) <) = 0(x(s0r) = maximum of f(x)(, dx2MMdxM.A zero of a function is a value of its argument at which the function vanishes; that is, if0:7:4 f(r) = 0 then r = a zero of f(x)Equivalent to the phrase "a zero of f(x)" is "a root of the equation f(x) = 0." A double zero or a repeated rootoccurs at a value r of the argument such that 0:7:5 dff(r)(r) = 0r = a double zero of f(x)dxThe concept extends to multiple zeros; thus, if 0:7:6dfd'ff(r)=-(r)-(r)=0dxdx"r = a zem of f(x) of multiplicity n + IA value r of x that satisfies 0:7:4 but not 0:7:5 corresponds to a simple zero or a simple root. 0:8 GENERAL CONSIDERATIONS0:8 NUMERICAL VALUES6 The prime goal of this section is to enable the reader to calculate precise values of the function in question formost argument values. Usually this is accomplished with the aid of an algorithm, it being assumed that the readerhas at his disposal a programmable calculator or computer of modest power.Because such devices invariably incorporate means of evaluating the functions \, In(x), exp(x), sin(x), cos(x).tan(s), and arctan(x), no algorithms are presented for these seven functions. By the same token, free use is madeof these seven ubiquitous functions in the algorithms of other functions. Several other functions (x`, Int(x), frac(x) and (xj) that are usually available are also assumed in the algorithms, but short subalgorithms for these are included to cover cases in which they may be absent from a particular device.No programs, as such, are presented because this would restrict the applicability of the algorithms to a particularprogramming language. Instead we report the algorithms as a list of straightforward commands that can easily beconverted to statements in any programming language or into a sequence of calculator keystrokes in either reversed Polish or algebraic logic.Traditionally, the goal in algorithm design has been to minimize the computation time required to generate afunction value to some specified precision. The algorithms of this Atlas are designed to meet quite a differentcriterion: to minimize the programmer's time.The algorithms of Section 8 are mostly based on the mathematical properties of the function in question.Usually, a series is summed. the number of summands being determined by an empirical formula or by makingsome test(s) after each addition. There are two advantages, other than sacrificing computer time in favor of "peopletime," to this style of algorithm. First, most of our algorithms may be easily modified to provide more, or less,precision. Second, since there are few multidigit numbers to be keyed, there is less chance of a mistake in pro-gramming.Many algorithms in this Atlas claim a precision of "24-bits." By this is meant that the algorithm gives anapproximation fix) that is related to the true value f(x) of the function by 0:8:1 -2_245f(x)-f(x)52_24f(x)That is. the relative error in the algorithm never exceeds about 6 X 10-b. It should be emphasized that the precisionclaimed is that of the algorithm and may not necessarily be reflected in a calculation based on the algorithm.Degraded precision could result, for example, from inadequate accuracy in the built-in functions of the computingdevice being used, or from variations in rounding error from computer to computer. A related way in which theprecision of implementation of an algorithm may be degraded is through the subtraction of two nearly equal num-bers. Where possible we have designed algorithms to minimize this loss of significance.A typical algorithm has the following features:(a) provision for loading the argument and other input parameters, if any:(b) indication of the restrictions, if any, on the input values:(c) a list of the constants and variables used by the algorithm;(d) a list of easily understood sequential commands, such as "Setf = 0" or "Replace g by N g/(1 + x)";(e) labels in the form of parenthesized numbers, such as "(2)," preceding certain commands;(f) commands in the form "if . .. go to (#)" where (#) is a label and . .. is some condition, it being understoodthat if the condition is not met, one proceeds to the next command in sequence;(g) provision for output; and(h) test values to be used in confirming that a program derived from the algorithm is functioning correctly.The test values supplied with each algorithm have been carefully chosen to exercise the algorithm maximally.Test outputs are usually reported with two or more terminal digits italicized. A correctly programmed algorithmshould always generate a test output that reproduces at [east those digits that are not italicized. On the other hand,italicized digits may fail to be reproduced by a correctly programmed algorithm. This is because changes in thesedigits correspond to errors that lie within our permitted range of ±6 x 10-8. For example, our algorithm fromSection 43:8 lists the test value f(0.6) = 1.48919225; however, an output of 1.48919228, having a relative errorof only 2 x 10-r, is no cause for concern. 7 GENERAL CONSIDERATIONS 0:10The techniques employed in the design of the algorithms for this Atlas are so diverse that it is inappropriateto attempt to summarize them here. However, each algorithm is accompanied by at least a brief synopsis of itsmode of operation.Throughout the Atlas there are also a number of general-purpose algorithms. These arc recapitulated in Ap-pendix A, which also contains several additional utility algorithms of wide applicability. 0:9 APPROXIMATIONSEntries in this section enable function values to be calculated for stated ranges of the argument. Often, the precisionis specified to be "8-bits," which implies(x) - f(x ) 0:9:1-2_Bcf (x)"1(x)(i.e., a relative error of no more than about ±4 x 10-'). where i(x) is the value given by the approximation andf(.r) is the value of the function. 0:10 OPERATIONS OF THE CALCULUSSome of the most important properties of functions are associated with their behavior when subjected to the variousoperations of the calculus. Accordingly, the tenth is often one of the largest sections of a chapter.For most functions defined for a continuous range of argument x, the derivativedf 0:10:1-dxexists and is reported in Section 10. The derivatives of function combinations such as f(.r)g(x), f(x)/g(x) or f(g(.r))are seldom reported because they may be readily evaluated via the product rule:0:10:2 d- f(x)g(x) = 1(x)dg- + g(x)df-dxdxd.r the quotient rule:0:10:3 d f(x)1dff(x) dg--_----d.r g(.0g(x) dxg--(x)dxor the chain rule: 0:10:4 ddxf(g(x)) =dfd ddgxg = g(x)1?If f(x) is differentiated n times, the nth derivative 0:10:5 d"fd.t"is generated. and expressions for this result are sometimes reported. The nth derivative of a product is given bythe Leibni: theorem:d"d"gn df d"-'gn(n - 1) d2f d='-gn d"-'f dgd"f 0:10:6- f(x)g(x) = f(x)+ - - - +-+ - + -- + - dx"dx1! dx dx"2!dx' dx"-'1! dx"-' dxdx"g(.r)_nd"-'f d'g-J/ dx'-' dx' 0:11 GENERAL CONSIDERATIONS Swhere (,) denotes a binomial coefficient (Chapter 61 and n! the factorial function [Chapter 2]. Another usefuloperation bearing the name of Leibniz gives a rule for differentiating an integral:d(I U) `Wddtd 0:10:7dxf,f(x;t)dt =J'- f(x;t)dt + dxdxf(x;t,(x)) - f(x;to(x))Ns,Reference books often express the results of indefinite integration in a form such as 0:10:8 J f(x)dx = c + e(x)where 4b(x) is a function that gives f(x) on differentiation (i.e., d4s/dx = f(x)), and c is an arbitrary constant. Toachieve closer unity with the representationf(t)di 0:10:9 I,JJ.oof a definite integral, the Atlas adopts the formulation 0:10:101+f(t)dt = th(x) - 4 (xo)for indefinite integration. In 0:10:9 and 0:10:10..t and x, are specific lower and upper limits. xo in 0:10:10 oftenbeing chosen to make db(xo) vanish. Of course, the information contained in 0:10:8 is also present in 0:10:10. Aninvaluable tool for integration is the formula for integration by parts:f.,0:10:11JAf(t)g(t)dtf(t)dG(t) = f(x,)G(x,) - f(xo)G(xo) - df(t)G(t)dt where g is the derivative of G.Frequently, integrals cannot be expressed in terms of known functions. In such cases it is convenient to evaluatethe integral numerically, for example by utilizing the algorithm presented in Appendix A [see Section A:71.Perhaps the most general operation of the calculus is that named diuerintegration and denoted by the symbolism 0:10:12 d`f(d(x - xo)10for the with differintegral of f(x) with respect to x, using x0 as the lower limit. The value of v is unrestricted: it maybe positive or negative, integer or noninteger. Differintegration, as its name suggests, generalizes the operationsof differentiation and integration so that 0:10:1. 0:10:5 and 0:10:10 are the v = 1. v = n and v = -1 instancesof 0:10:12. Note that the diffcrintegral 0:10:12 is independent of the lower limit xo only when v = 0, I. 2.....For further information on differintegration, the reader is referred to a monograph on the subject (Oldham andSpanier]. The symbol 0:10:12 collapses to 0:10:13 d"fdwhen xo is chosen to be zero.The most important noninteger order instances of differintegration are generated when v or v = -; andare termed semidifferentiation and semiintegration, respectively. Some examples of semiderivatives and semiin- tegrals are reported in Sections 10, and most often with a lower limit of zero or minus infinity.An algorithm that generates approximate values of the differintegral 0:10:12 may be found in Section A:7. 0:11 COMPLEX ARGUMENTOther than in the eleventh section of each chapter, the argument of a function is treated as real. Moreover, weusually restrict the domain of the function to ensure function values that are real. GENERAL CONSIDERATIONS 0:14In Sections 11, however, the effect of replacing the real argument x by the imaginary iy or the complex x +iy is often reported. The real and imaginary components of f(x) or f(x + iy) are sometimes exposed as well. 0:12 GENERALIZATIONSFunctions can be arranged in a hierarchy in which lower members are subsumed by more general higher members.In the chapter dealing with the f(x) function, special cases of f(x)-that is, functions lower in the hierarchy-arereported in Section 4. Conversely, functions that occur higher than f(x) in some hierarchy are reported in Section12; that is, the functions reported here are ones of which f(x) is a special case. 0:13 COGNATE FUNCTIONSThe thirteenth section of each chapter is devoted to citing functions-other than those reported in Sections 4 and12-that are related to the subject function, and to exploring the properties of some of them. 0:14 SPECIAL TOPICSAn application, or some other special feature of the function, is elaborated in the final section of many chapters.Such applications often amount to brief discourses on important topics in science, engineering and applied math-ematics. For example, certain topics in approximation and curve-fitting. Laplace and Fourier transforms, facts about permutations, combinations and distributions, and geometric properties of various sorts are to be found among the many exposed in Section 14. CHAPTER1THE CONSTANT FUNCTION c The constant function is the simplest, and an almost trivial, function. 1:1 NOTATIONConstants are also known as invarianrs and are represented by a wide variety of symbols, mostly letters drawnfrom the early members of the English and Greek alphabets. 1:2 BEHAVIORThe graphical representation (see Figure 1-1) of the constant function fix) = c is a horizontal line, extendingindefinitely in the x - =x directions. C ............................. 0 1:3 DEFINITIONSThe constant function is defined for all values of its argument x and has the same value. c, irrespective of x. 11 1:4 THE CONSTANT FUNCTION c 121:4 SPECIAL CASESWhen c is zero, the constant function is sometimes termed the zero function. Likewise, the function f(x) = c = 1is known as the unity function. 1:5 INTRARELATIONSHIPSBeing relations between function values at different values of the argument, intrarelationships are of no consequencefor the constant function. 1:6 EXPANSIONSA constant may be represented as a finite sum by utilizing the formulas for an arithmetic series:cb1 : 6 : 1 a=- a geometric series:J-ojI.62 c=a+aR+aR2+...+aR+='nRJ01=or an arithmetic-geometric series: 1:6:3c(R - 1) c(R - 1 ) - J 6 0 - 1+ &R(R' - 1)/(0 - 1) R'"' - IIn these formulas R and b may take any values and J may be any positive integer.Any constant greater than ! may be expanded as the infinite geometric sum 1:6:4c= 1+ I c cI I+ I cc1I-+ (cc'Y + ... _rccl lJc>2 or as the infinite product 1:6:5 [1+(CcJSimilarly, a constant is expansible as the infinite continued fraction 1:6:6 aaaC = - - - ...R+ R+ R+[see Section 0:6] in 8 variety of ways, some of which are indicated in Table 1.6.1. 1:7 PARTICULAR VALUESFour irrational constants that are frequently encountered are Archimedes' number. Catalan's constant, the base ofnatural logarithms and Euler's constant. Archimedes' number IT, also known as Rudolf's number or simply as pi,may be defined by the infinite sum 1:7:1 (-1)3.141592654357,_0 2j + I 13 THE CONSTANT FUNCTION cTable 1.6.1 09Constraint c1 - c - I<- c< I c=0<cs<IC+cc2:r-ccc: or the infinite product4163664 - 1:7:2a=2x-x ISx35x63x...=2H1=3.141592654J=1l4The definition of Catalan's constant G is similar to 1:7:1: 1:7:3G=1-9+2549+ 1)'0.9159655942-o (2JThe base of natural logarithms e and Euler's constant y are defined by the limit operations/I\ 1:7:4 e = limI + - I= 2.718281828\ohand1:10 1:7:5 i -In(n))=0.5772156649The latter is also known as Mascheroni's constant and is often denoted by the symbol C. There are many alternativeformulations of these four constants [see Gradshteyn and Ryzhik. Chapter 0. for some of these]. Also of widespreadoccurrence throughout this Atlas is the ubiquitous constant1:7:6U = common mean [Section 61:81 of I and 1/f = 0.8472130848The most important family of rational constants is the family of natural numbers, n = I. 2, 3. ... [see Section1:141. Other families that occur principally in coefficients of series expansions are the factorials n! [Chapter 2],Bernoulli numbers B. [Chapter 41 and Euler numbers E, [Chapter 5]. 1:8 NUMERICAL VALUESThis simplest of all functions has the same numerical value at all arguments. 1:9 APPROXIMATIONSNone are needed for the constant function. 1:10 OPERATIONS OF THE CALCULUSDifferentiation gives 1:10:1 d-c=0dx 1:11 THE CONSTANT FUNCTION cwhile indefinite and definite integration produce 1:10:2 and 1:10:3cdt = cx f cdt=c(x,-xo)respectively.The results of semidifferentiation and semiintegration [Section 0:101 with a lower limit of zero ared1/'c1:10:4 anddx'l:CVIIX d_'/2x 1:10:5 dx-ii: c =2c-Differintegration [see Section 0:101 with a lower limit of zero yields14 d'cx-'.1:10:6 dx'c_f(1 - v)where r is the gamma function [see Chapter 431. In fact, equations 1:10:1, 1:10:2, 1:10:4 and 1:10:5 are the v =1, -1, 1 and -i instances of 1:10:6. 1:11 COMPLEX ARGUMENTA complex constant can be expressed as1:11:1 c=a+i13where a and 0 are real constants and i = V 1. The rules for performing arithmetic operations upon the pair ofcomplex constants a + i13 and A + tB are1:11:2(a+i13)±(A+iB)=(a-A)+i(13±B)1:11:3 (a + i13)(A + iB) _ (aA - 13B) + i(aB + A13) 1114a+i13_aA+138iA13-aB ::A+iB+A+B'/A'+B'Powers may be handled making use of the identity1:11:5 (a + i13)"'B = exp[(A + iB)Ln(a + i13)]where exp and Ln are the exponential and logarithmic functions treated in Chapters 26 and 25, respectively. Ap-plying equations 25:11:1, 25:11:2 and 25:11:3 produces the multiple-valued function1:11:6(a + i13)4= (a2 + 132)A'2[cos(d) + i sin(4)]exp[-B(8 + 2kar)]where k is any integer and 1:11:7sgn(13)arccot(a/I13I)R*013=0 1:11:8 m=-In(a'+13) +A(8+2k-r)2 is THE CONSTANT FUNCTION c 1:13Setting k = 0 in equations 1:11:6 and 1:11:8 defines a single-valued function. When this is done and whenA + iB = v, a real number, the result is1:11:9 (a + ip)' = (a' + p'),12[cos(v9) + i sin(ve)[ an expression of de Moivre's theorem [see Section 32:111. 1:12 GENERALIZATIONSThe constant function is the special b = 0 case of the linear function discussed in Chapter 7. 1:13 COGNATE FUNCTIONSWhereas the constant function has the same value for all r, the related pulse function is zero at values of theargument outside a "window" of width h. and is a nonzero constant, c, within this window (see Figure 1-2): y0A1t01100.%FIG 1-2 : C p (ct he x-a 0 it0x<a--2hh 1:13:1p(c;h:x-a)=ca-;<x<a+2hx > a + -2The value of a establishes the "location" of the pulse. while c and it are termed the pulse height and pulse width,respectively. The pulse function can be expressed as1//h 1:13:2p(c;h;.r-a)=c ul.r -a+hhl-ux-a-h) in terms of the unit-step function of Chapter 8.The concept of a general 'window function' is discussed in Section 8:12.The addition of a number of pulse functions, having various locations, heights and widths, produces a functionwhose map consists of horizontal straight line segments. Such a function, known as a piecewise-constant functionand illustrated in Figure 1-3. may be used to approximate a more complicated or incompletely known function. It is the approximation usually adopted, for example, whenever a varying quantity is measured by a digital instrument. 1:14 THE CONSTANT FUNCTION c 16i`+, iy+ry+1b C2 FIG 1-3 :C1 C9 ............. Co 1:14 RELATED TOPICSThe natural numbers 1, 2. 3. ... are ubiquitous in mathematics and science. We record here several results forfinite sums of their powers. 1:14:1 n(n + 1)2n=1,2,3....j- 1:14:2I''+2'+3'+ +n'j'= n(n + 1)(2n + 1)n=1.2,3.6... 1:14:31' +2'+ 3' + ''+nn'(n + l)n= I,2.3.J=j...4Similarly, the sums of fourth and fifth powers of the rust n natural numbers are n(n + 1)(2n + 1)(3n' + 3n -1)/4 and n''(n + 1)2(2n2 + 2n - 1)/12, respectively. The general case is+ 1:14:4 B"+1(n1)B"+1n.m=1,2.3.m...where B. denotes a Bernoulli number [see Chapter 41 and B"(s) denotes a Bernoulli polynomial [Chapter 19]. Thesum of the reciprocals of the first n natural numbers islI1I11:14:5 n=I,2,3....123nj-1 Jwhere y is Euler's constant [Section 1:7] and *(x) denotes the digamma function [Chapter 44]. When continuedindefinitely, the sum 1:14:5 defines the divergent harmonic series. If m is not an integer, summation 1:14:4 maybe evaluated, for large n, by use of equation 13:5:5.The corresponding expressions when the signs alternate are 1:14:6 (n1)/2n=1.3.5,...j-1-n12n = 2, 4, 6, ... x:= -,z-_ n(n+I)/2n=1,3,5,... 1:14:7I'-2' +3' - 4 1)j-n(n+1)/2n=2.4.6.... J-11:14:81'-2'+33-4'+ ...±n'= (2n'+3n'- 1)/4n= 1,3,5....-n2(2n+3)/4n=2,4,6,...)-(-1)"E(n+1)n,m=1.2,3,... 1:14:91"-2"+3"-4"+...±n"(-1)jj"=-E2 2 17 THE CONSTANT FUNCTION cwhere E,"(x) denotes an Euler polynomial [Chapter 20], and1:14 n + I 11I(_1)is(n+l)-2n1,3,5,...1:14:101234n.,jVn+I)-*1Z+1n=2.4.6....The numbers 2, 4, 6, ... are called the even numbers. Sums of their powers are easily found by using theidentity(n 1:14:11 2'+4"'+6'+...+n'=2'1 1'+2'+3'+...+\2l/]n=2.4,6....in conjunction with equations 1:14:1-1:14:5. Likewise, use of these equations together with the identity 1:14:121"+3'+5'+ +n"=[I'+2' \n2n= 1.3.5,...permits sums of powers of the odd numbers, 1, 3, 5..... to be evaluated.For the infinite sums Ej-" where j runs from I to x, see Chapter 3. The same chapter also addresses the relatedinfinite sums E(-I)'j-", E(2j - I) and I(- 1)1(2j - 1)-". CHAPTER2THE FACTORIAL FUNCTION n! AND ITS RECIPROCAL Factorials occur widely in function theory: for example. in the power series expansions of many algebraic andtranscendental functions. They also play an important role in combinatorics [see Section 2:141. 2:1 NOTATIONThe factorial function of n is symbolized Lit in some older literature. The symbol fl(n) is occasionally encountered. 2:2 BEHAVIORThe factorial function is defined only for nonnegative integer arguments and is itself a positive integer. Figure2-I (with a logarithmic vertical scale) and the table of rounded values (Table 2.2.1) demonstrate the explosiveincrease of n! as n increases beyond n = 2.f' o oayh'os^00 40cc(I<(I<'010FIG 2-1:..............:....:..............:....:....: 105 Table 2.2.1 .:....:....:....:....:....:....*... ...:...: 109 :....:....:... :.......nl+t .........:....:....:.100 :....:......... :....:.... ....:....:.......100I104 x 10'202 x 10"303 x l0`408x I0'303 x I0°'608 x 10"70I x loll807x 10"'90IxIO'°1009 x 10,! FE 2:3 THE FACTORIAL FUNCTION n! AND ITS RECIPROCAL2:3 DEFINITIONSThe factorial of the positive integer n equals the product of all positive integers up to and including n: 2:3:1 n=1,2,3,...J.,20 This definition is supplemented by the value2:3:2 0! = 1conventionally accorded to factorial zero.The exponential function [Chapter 261 is a generating function [see Section 0:3] for the reciprocal of the factorialt" 2:3:3 exp(r) = i --0n! 2:4 SPECIAL CASES There are none. 2:5 INTRARELATIONSHIPSThe most important property of the factorial function is its recurrence2:5:1(n+ l)!=n!(n+1)n=0,1,2,...which may be iterated to produce the argument-addition formula2:5:2(n+m)!=n!(n+ 1)Tn,m=0. 1,2....where (n + I) is a Pochhammer polynomial [see Chapter 18]. Formula 2:5:2 leads to an expression for the ratioof two factorials. An alternative expression is 2:5:3 n.(n - m)! i-0whereis a Stirling number of the first kind, of which a short table is included in Chapter 18.Because reciprocals of factorials occur frequently as coefficients of power series, many expressions exist forinfinite sums involving these reciprocals. For example:I11 1 2:5:4-+-+-+i-=exp(1)=2.7182818280!1!2!J-0 j! 2rZ 2:5:5(1)/l11+I-) +II+==1(2)=2.279585302!\111/21i-o (j!)' 2:5:6 +++_= cosh(l) = 1.5430806350!2!4!ra (2# 111 12:5:7 sinh(l) = 1.1752011941!3!5!;-0 (2j+1)!where exp is the exponential function of Chapter 26, l0 is a hyperbolic Bessel function [see Chapter 49] and coshand sinh arc the hyperbolic cosine and sine functions of Chapter 28.The corresponding summations with alternating plus and minus signs give exp(- 1), J0(2), cos(l) and sin(]),where the Bessel coefficient J0 is discussed in Chapter 52, and the cosine and sine functions, cos and sin, appear in Chapter 32. 21THE FACTORIAL FUNCTION n! AND ITS RECIPROCAL2:6 EXPANSIONSStirling's formula provides an asymptotic expansion [see Section 0:61 1 1139 2:6:1n!-2anexp(-n)n" I+-+--Jn-- 12n288n'51840n'for the factorial n! as n tends to infinity. This formula and a similar one for the logarithm of n!: 1 1 2:6:2ln(n!)-nln(n)-n+-ln(2nn)+I--I-n--. c212n360n'1260n'produce remarkable accuracy, even for small n values.2:7 PARTICULAR VALUES 2:7:1 0!=1!=1 2:8 NUMERICAL VALUES I Storage needed: n, fMany calculators compute n! from n by a single keystroke, and some computer languages have similar facilities.The simple algorithm Input n >> f=n!<<Setf = IIf n = 0 go to (2)(1) Replace f by fnReplace n by n - IIfn*0goto(1)(2) Output f2:9 IInput restrictions: n must be a nom, ative integer. Test values:0! = 1(12)! = 479001600(50)! = 3.04140932 x IOs' based on definition 2:3:1, is precise, but may be tediously slow. Alternatively, one may use a second algorithm Input n >>b»»Setf = [I - (I/30n')]/12nReplace f byexp(f + nll n(n)Replace f by 1 + Int(fy'2an)Output ff=n!<<Storage needed: n,f 1Input restrictions: n must be a positive integer.Test value:12! = 479001600 which is based on exponentiating 2:6:2, truncation after the n-' term and rounding to the next higher integer. Thisalgorithm has a precision [see Section 0:8] better than 6 x 10-". Because of identities 2:12:1 and 2:12:2, the algorithms of Sections 18:8 and 43:8 may also be used to calculate factorials. 2:9 APPROXIMATIONSBased on the asymptotic expansion 2:6: I is the approximation 1 2:9:1n! =2nn exp(-n)n'I + -12n8-bit precisionit a 2 2:10 THE FACTORIAL FUNCTION n! AND ITS RECIPROCAL2:10 OPERATIONS OF THE CALCULUS22 No operations of the calculus are possible on a function such as n! that is defined only for discrete arguments. 2:11 COMPLEX ARGUMENTIn view of relation 2:12:2, the gamma function formulas given in Section 43:11 may be used to generate expressionsfor (im)! and (n + im)!. 2:12 GENERALIZATIONSThe factorial is a special case of Pochhammer's polynomial W. [Chapter 18]:2:12:1 n!=(I)"n=0,1.2....and of the gamma function [Chapter 43]:2:12:2 n! = r(n + I)n=0,1.2.... 2:13 COGNATE FUNCTIONSThe factorial function and the binomial coefficient [Chapter 61 are closely related.The double factorial or semifaclorial function is defined by 2:13:11n-1,0 n ! ! =n(n - 2) x (n - 4) x x 5 x 3 x 1 n = 1, 3. 5....n2,4.6....For even argument it reduces to 2:13:2n!!=2"'2( )n=0.2,4.... while for odd n it may be expressed in terms of factorials, or as a gamma function (Chapter 43J or as a Pochhammerpolynomial [Chapter 18] 2:13:3n!!=n.=2"/'-r1I+-J=2n(21"n=1.3,5....22Of frequent occurrence [for example in Sections 6:4, 32:5, 61:6 and 62:121 is the ratio (n - 1)!!/n!! of the doublefactorials of consecutive integers. For odd n the ratio is expressible by the integral 2:13:4=-II)!]z !=sin"(t)dtn = 1,3.5,... n!!n!LLL \2/owhile for even it it is given by Wallis' formula- 1)!!n!2 2:13:5(n==sin"(t)dtit = 0, 2, 4, ... n!!2"[(n/2)!]2ITf 23THE FACTORIAL FUNCTION n! AND ITS RECIPROCAL 2:14This important ratio has the asymptotic expansion 2:13:62I(n-I)!!_.V rn4n32.22]n!!evenn -rx n111--+--2n4.32nJFinite sums of such ratios obey the simple rule 1315(2n - I)!! (2j - 1)!!(2n + I)!!2:13:71 + - + - + - + +_'= n = 0, 1, 2,...2848(2n)!!j=o(2j)!!(2n)!!and there is the related infinite summation due to Ross: 1315105(2j - I )!!2:13:8 ++++' " == In(4)2161441536j(2j)!!The triple factorial is defined analogouslyIn=-2.-1,0 X ! !n(n - 3) x (n - 6) x x 7 x 4 x I n = 1, 4, 7,...2:13:9 n...n(n - 3) x (n - 6) x . x 8 x 5 x 2 n = 2, 5, 8,... and finds application, for example, in connection with Airy functions [Chapter 56J. The extension to a quadruplefactorial n!!!! is obvious; it is useful in Sections 43:4 and 59:7. 2:14 RELATED TOPICSThe factorial function appears very often in applications involving combinatorics. For example, the number ofpermutations (arrangements) of is objects, all different, is W. If not all of the is objects are different, the numberof permutations is reduced ton! 2:14:1 (ni)!(n2)! ... (nj)!where there are n, samples of object I. n, samples of object 2..... nj samples of object J.If from a group of is objects, all different, one withdraws m objects, one at a time, the number of variations(possible withdrawal sequences) is n!2:14:2 m <- n(n - m)!If one ignores the order of withdrawal, 2:14:2 is reduced ton! 2:14:3 m c n(n - m)!m!which then represents the number of ways in which m objects can be chosen from among is, all different, and isknown as the number of combinations.The number of partitions (different ways in which n distinct objects may be placed in m identical boxes sothat each box contains at least one object) is given by a Stirling number of the second kind: 2:14:4 v." i = -;_a (m - Yj! 2:14Table 2.14.1THE FACTORIAL FUNCTION n! AND ITS RECIPROCAL 24 I00000000000000 m= Io111II1I1I1I1I 1m = 200 137IS31631272555111023204740958191 m=3000 I62590301966302593302850186526261625788970 m40000 110653501701777034105145750611501253253010391745m = 500000I1514010506951425252467301379400750850140075035 m60000001212662624228271794871323652932131263436373 m70000000 128462588063987627396571542449329280 Some examples of Stirling numbers of the second kind are given in Table 2.14.1. Others may be calculated by theexact (but rather slow) algorithm Input m >>Set a0 = I Storage needed: m + 5 registers are required for a0,at. a2, .... a". m. k. n and j.Input n >> a"=o."'<D»»Seta,=a:= a,==a"=0If n = 0 go to (3)Set a0 = 0Seta,=k=IIf n = I go to (3)(1) Setj = the lesser of m and k + 1(2) Replace a, by ja, [or by -A-alReplace a, by a, + a, - 1Replace j by j - Ilfj(m+k-n-j) *0goto(2)Replace k by k + I Ifk*ngoto(1)(3) Output a"4:««(or S"') 1Test values:Qa. = Q.',=000'o - QiUA= Io0'7' = 301o'1;' = 9528822303S;" = -13132Si°o = 63273 which utilizes the recursion formula2:14:5v;"'=c;'%"+mcr,"',m=1.2.3,...n=1.2,3....When the alternative command shown in green is selected, the algorithm will generate S;,"', a Stirling number ofthe first kind [see Section 18:6]. CHAPTER3THE ZETA NUMBERS AND RELATED FUNCTIONS As detailed in Section 3:14, the four number families !;(n). X(n). 11(n) and [3(n) occur as coefficients in many powerseries expansions. In this context it is only positive integer orders. n = 1. 2. 3, .... that are encountered, and thischapter therefore emphasizes these cases. However, one is able to extend the definitions of all four functions toaccept noninteger and negative orders, and these possibilities are also addressed here.The first three functions are interrelated by the simple proportionalities 3:0:1 f(v)->,(v)-r)(v)2"2' - 12'-2and by the consequential identity3:0:2 i;(v) + q(v) = 2X(v)but there are no corresponding relationships involving 13(v). Because they are so easily related to 7,(v) via 3:0:1.few formulas for M v) and r1(v) are exhibited in this chapter. 3:1 NOTATIONThe numbers I,(n). Mn). lI(n) and p(n) do not appear to have acquired definitive names: we shall call them :etanumbers, lambda numbers, eta numbers and beta numbers. When the order is unrestricted, we adopt the symbolismf,(v), A(v), q(v) and (3(v) and the names :eta function, lambda function, eta function and beta function. The lastshould not be confused with the complete beta function [Section 43:13] or the incomplete beta function [Chapter62) to which it is totally unrelated.The symbols o and L have been used, respectively, in place of 11 and 0. Riemann's name is associated withthe zeta function, which is often known as "Riemann's function" or 'Riemann's zeta function." We avoid thesenames to prevent confusion with the bivariate function of Chapter 64, with which Riemann's name is also commonly associated. 3:2 BEHAVIORThe zeta and lambda functions are infinite only at v = I, whereas rl(v) and 13(v) are always finite. Figure 3-I mapsthe behaviors of the four functions. All approach unity rapidly as the order v increases. For v c 0. the four functionsu 3:3 THE ZETA NUMBERS AND RELATED FUNCTIONS 26Oy4i'1-,9i~.......................... ..................3 are oscillatory, with ever-increasing amplitudes as v -. -x. The zeros of Z(v), x(v) and q(v) occur when v is aneven negative integer, whereas 13(v) = 0 at v = -1. -3. -5..... 3:3 DEFINITIONSThe four functions may be defined by the definite integrals 3:3:1 3:3:2eti, aj 1t"-'dt;(v) =r(v) Jo exp(r) - I X(v) _t""' csch(t) dr2r(v)Jv>1 v>I Iv > 01-1d, 3:3:3 q(v) =r(v)fexp(r) + I 27THE ZETA NUMBERS AND RELATED FUNCTIONS and3:5 3:3:4 30) =2r(v) 1[ot"-' sech(r) dtv > 0 involving functions discussed in Chapters 13, 26,29 and 43.The most commonly encountered definitions of zeta, lambda, eta and beta numbers are via the infinite series3:6:1-3:6:4. These series may be reformulated as limits, for example: IIIll' 3:3:5Y(v)=lim=limYj"J-=2"3 J -=and apply generally for v > I in the cases of i(v) and A(v), or for v > 0 in the q(v) and p(v) cases. However,when sufficient additional terms are included in series 3:3:5, to give 3:3:6;(v) = limI +I+l++1 11+vv(v + 1)(v + 2)J-=2"3`(J - 1)"+(v - 1)J"'+2J"121"''720J""the limit provides a definition of the zeta function for any order v: positive, negative or zero. The general expressionfor the kth appended term in 3:3:6 involves functions from Chapters 2, 4 and 18 and is (v)j_1B5/(k!Jr*'-'). It isnecessary to include only those appended terms for which v + k - I is negative or zero, but a few extra will speedthe approach to the limit. The corresponding definition of the beta function is 1I 1 lvv(v + I)(v + 2) 3:3:7p(v)=lim I--+--...-+ - +-+...1J- =3"5"(J - 2)'2J"2J'-'6J"3the kth appended term being (v)t_,2'(2' - I)BR/(2k!J'-"-') in this case, J being a number equal to 4n + I wheren is a natural number. 3:4 SPECIAL CASESThe zeta, lambda, eta and beta numbers may be regarded as special cases of the corresponding functions. No furtherspecialization is fruitful. 3:5 INTRARELATIONSHIPSThe zeta and beta functions satisfy the reflection formulas 3:5:1V'Tr(1 - v) =2r(v)4(v)cost -(2a)"2 3:5:2 p(I - v) = I -J r(v)p(v) sineinvolving the gamma function [Chapter 43) and the functions of Chapter 32.With f(n) representing any one of the four numbers I,(n), Mn). 11(n) or (3(n). one may sum the infiniteseries E(-1)"f(n)/n, as well as the series of complements E[I - f(n)], E(-1)"(1 - f(n)]. E[I - f(n)]/n and F(-1)"[1 - f(n)]/n. With the lower summation limit taken as n = 2, these sums are as shown in Table 3.5.1.These sums involve the logarithmic function [Chapter 25] of various constants. Archimedes number it (Section1:7), Euler's constant y (Section 1:7) and the ubiquitous constant U = rr(.'-)/V ar = 0.8472130848 [see Section 1:7). Also listed in Table 3.5.1 are the sums E(- I)"f(n). Strictly, these particular series do not converge, thetabulated entries being the limits 1 3:5:3 limff(2)- f(3) + f(4) -= f(J - I) ± - f(J)2which do converge and whose values may be associated with E(-1)"f(n). 3:6 THE ZETA NUMBERS AND RELATED FUNCTIONS 28Table 3.5.1 f=; f=A f - df(n)(1), fin)(-1)' n11 - f(n)])-1)'[I - f(n))1 - f(n) CCI - fin)n 1Y-1-Y- 11- IM2) - Y 22 _ In(2)7+ In (-)In(2) - I-- In(2)Y- In(\n) - II - I n 4 ) - Y 2\'n. 22\\n28 In(4) - IIn (4In(4) - 13z - In(4)In(n) -II - In(-) r,nV_`IT Innnn -In(V2)-- In(-I-+In(\2)-I-+In(\2)---+In(-)-I1-In(V2U)-- 44U4 244.\'8U 4 3:6 EXPANSIONSThe series 3:6:1 3:6:2 3:6.31C(v)+21+3v>I 11v>13"5" 11TI(v)=1_' _v>0 ]1 3:6:4 v>03''5'are the most useful representations of the four functions and serve as definitions of the zeta and lambda numbers,l(n) and X(n), for n = 2, 3, 4, ... as well as for the eta and beta numbers, q(n) and [3(n), for n = 1. 2. 3, ....Zeta and lambda functions are expansible as the infinite products 3:6:5 3:6:61 1 1 I6(v)11 -21 -3 "I -5-" 1 - 7-"ri 17r III1 1I -3-''1 -5-"1 -7-"l - 11-where a; is the jth prime number. 3:7 PARTICULAR VALUES In Table 3.7.1 Z has the valuev>I v>1 3:7:1 Z = 4(3) = 1.202056903and G is Catalan's constant [see Section 1:7].For n = 2, 4, 6, ... all zeta, lambda and eta numbers equal e' multiplied by a proper fraction [the fractionis related by equation 3:13:1 to the Bernoulli number B. of Chapter 4]. Similarly, for n = 1, 3, 5, ..., P(n) isproportional to n", the proportionality constant being a proper fraction related to the Euler number E,_, [see Chapter 29THE ZETA NUMBERS AND RELATED FUNCTIONSTable 3.7.1 WY)3:8 v=-5v=-4v=-3'=-2v=-1v= 0v= Iv-2v-3v=4v=s -I0I 1 1 a' a'252 1200122 6Z90310-7010a'72u'252 120 12 88% I-1 1 1n'327a' 0 0 In(2) 4 8 42 1247205-1 1rrrr' 0-0-02dG32A(4) 51 by equation 3:13:2. For negative integer orders, the zeta and beta functions are related much more simply tothe Bernoulli and Euler numbers; thus: 3:7:2 1,(-n) = and 3:7:3 3:8 NUMERICAL VALUES-B-"+ In=1,2,3,... n=0, 1,2.... We present two algorithms. The first is designed to calculate zeta, lambda and eta numbers for n >- 5. so that, inconjunction with the table in Section 3:7. one may evaluate 4(n), x(n) or rl(n) for all positive integer orders. Thisalgorithm is based on expansion 3:6:6, using primes as large as 43, and on relation 3:0:1. The precision exceeds24 bits. Input n >-, =Storage needed: n, k and fSet k = 7 + 2 Int(9/n)Input c >Set f=(c-2")/(1-2)Input restrictions: n >- 5; code c = 0. 1 or 2code, cfunctionk_(1) Iffrac(k )fracO= 0 go to (3)(2) Replace f by f/(t - k-")0;(n) 1x(n)2r((n)(3) Replace k by k - 2Ifk>8goto(1)fo = ;(n)If k > 2 go to (2)Output fTest values:;(6) = 1.01734306f.Mn)K<<<K A(6) = 1.00144708 Jrl(n) YO) = 0.985551091 The second algorithm is much more sophisticated, being designed to calculate P(v), 4(v), a(v) or q(v) accordingto the value given to a code c that is input prior in v. Because C(l) = a(1) = ±-, the algorithm returns the value10" when c = 0 or I and v = 1. Otherwise, the absolute algorithmic accuracy is better than t6 x 10-1 for allfour functions and for all orders. For v ? i, the algorithm utilizes equation 3:3:7 (or the X-analog of 3:3:6, together 3:8 THE ZETA NUMBERS AND RELATED FUNCTIONS 30with 3:0:1) with J set equal to 13 and with appended terms up to k = 9. For v < ;, P(1 - v) or M(1 - v) iscomputed and this value is transformed into 0(v) or a(v) (and hence to 4(v) or n(v) via 3:0:11 by use of equation3:5:2 or the k-analog of 3:5:1. The gamma function r(I - v) needed for this transformation is computed by aroutine described in Section 43:8. To avoid "divide by zero" errors, special measures are adopted if v = 0 or 1. Input c >> Input v >> f, = P(v)fo = s(v)f = k(v)Set f = 1cc ln(2)LI+ (+ c)10"3J+2J [61 Replace k by k - 2If v = I go to (7)Set f = (3HcH-2 - c)/4Ifv=0goto(7)Set f = (1cl - 2")/(I - 2") 1If v- go to (3)Replace v by I - vSet w = vifc<0goto(I)Replace f byf[(2 - 2")/(2" - 1)] cos(90v)Go to (2)(1) Set f = 2 sin(90v)(2) Replace f by f/wReplace w by w + IIf w<3 oto(2)I2fl/Setg= 1+- (3%- 30w27w'Replace g byg+ wlIn(w) - I]12wReplace f by [f e%p(g)\ 2*w]/n"(3) If c < 0 go to (4)Set c = 1(4) Set J = 13Set k = I1Set g = 0(5) Replace g by 8g/(297 - 92.6k + 13k2 - 0.6k') Replace g by (v + k - 1)I 2`(1 - c) + c - g(v; k)]/JIf k * I go to (5)+ C JReplace g by3+ I +2(v -)2J'(6) Replace J by J - 4Replace g by g + (c/(J + 2)"] + [I /J"] If J * I go to (6) Replacef by fg(7) Output fStorage needed: c, f, v,g, k and J Use degree mode or change90 to rr/2. Input restriction: c is a codethat must equal - I , 0, 1 or2, as follows:code, cfunction-I(3(v)0?;(v)Ik(v)2q(v) Test values:(3(1) = 0.7853981640.500000000P(-3)=0 (-3) = 0.00833333333 k(-3)-0.0583333333-9(-3) _ -0.125000000(3(5) = 0.996157828;(5) = 1.03692776k(S) = 1.00452376ri(S) = 0.972119770 P(2) = 0.667691457C(;) = -1.460354511t() _ -0.4277279330.604898644 31THE ZETA NUMBERS AND RELATED FUNCTIONS3:9 APPROXIMATIONSThe formulas3:9:1i(v) = 1+ 2-'+ 3-8-bit precisionv? 53:9:2X(v) - I + 38-bit precisionv? 43:9:3 TI(v) = 1 - 28-bit precisionv? 53:9:40(v) - I - 3-"8-bit precisionv a 43:12 are based upon the truncation of expansions 3:6:1 through 3:6:4 and apply for large positive orders. For largenegative ordersv/va`3:9:5{(-v) _ -2sins 2 I v)(V)'141\/l 3:9:6a(-v)rl(2_V-eXp(-v) sin(Z V-. 3:9:70(-v) - 2 1vl". exp(-v)(Vir 2)v -. xIn the vicinity of v = I. the zeta function is well approximated by 1 13:9:8 ;(v) = y +v11 = 0.577 + v 18-bit precision 0.79 S v s 1.24where -y is Euler's constant (Section 1:71. 3:10 OPERATIONS OF THE CALCULUSThe derivatives and indefinite integrals of the four functions may be expressed as infinite series, for example:dIn(j')3:10:1 - (v) = - , - v > Idvivand 3:10:2((-I)'10(t)dr = v +In(2j + 1)[1 - (2j + 1)-']v -> 00but not as established functions. At v = 0 the derivative of the zeta function equals -InV 22rr. 3:11 COMPLEX ARGUMENTThe representation of 4(v + iµ) in terms of its real and imaginary parts is given bycos{µ In(k)}sin{µ ln(k)}3:11:1 l;(v + iµ) = > 1,_,k' k" 3:12 GENERALIZATIONSThe four functions of this chapter are special cases of the Hurwitz function of Chapter 64. Thus:3:12:1 {(v) = (v:I) 3:13 THE ZETA NUMBERS AND RELATED FUNCTIONS 32 3:12:2 3:12:3 3:12:4M(v) = 2-%v,1)2 1*v) = 2 " v: 2) - ;(v:I) = J(v:1) S(v) = 4 "LL1v:4)- C v:4)]= 2v:2)n The last pair of equations shows that the bivariate eta fu ction [see Section 64:13] is also a generalization of theeta and beta functions. 3:13 COGNATE FUNCTIONSWhen n is even t(n) is related to the Bernoulli number B. [Chapter 4] by(2a)"IB"I 3:13:1 I(n)=n=2,4,6,...2n!For odd n the zeta number is related to the Bernoulli polynomial [Chapter 191 via the integral)" 3:13:2{(n) _(2aB"(r)cot('nt)drn = 1. 3, 5, ... 2n!The beta number of odd argument is related to the Euler number E"_1 [Chapter 5) byn 3:13:3 [3(n)=IE.-,Iln=1,3,5....2/ 2(n - 1)!while for even n the relationship is to the integral of an Euler polynomial [Chapter 201 3:13:4 [i(n)= 4(n- 1)!E,(t)scc(at)dr0 3:14 RELATED TOPICSn=2.4,6,.. The four number families occur as coefficients in power series expansions of trigonometric and hyperbolic functionsof argument ax or arx/2: 3:14:1 3:14:2 3:14:3 3:14:4 3:14:5 3:14:612cot(ax)_---jlQn)x--1<x<17rx7rx "_,csc(.rx) =1+211(2n)x2n-1 <x< I1rXTX/arxtanl-)=4-X(2n)xv"-1 <x<2ax "_, sec()- I coth(trx) =-?1E (-1)"{(2n).rzn- l <x<1rxaz "_, csch(ax) _I- +2-(-1)"i(2n)x2w-1 <x< Iarx7rx "_, 33THE ZETA NUMBERS AND RELATED FUNCTIONS(ax)=-4 3:14:7 tank-(-1)"a(2n).iy'-1 <x< I 2Trx "_,and 3:14:8sechI-I=4 i(-I)"(3(2n-1)x'"-'-1<x<1 \\\ 2 /nxSimilar series represent the logarithms of trigonometric functions1-3:14:9In{csc(nx)} _-In{sin(ax)} _-ln(irx) +x'"-1 < x < 1 3:14:10In{tan()J-Inicot( 2)1=ln1 211(2-) x,-1<x<I 3:14:11Inisee(2 I}-lnicost-l<x<I3:14 Notice that itis invariably the zeta, eta and lambda numbers of even argument, and the beta numbers of oddargument, that appear in such expansions. CHAPTER4THE BERNOULLI NUMBERS, B,t Bernoulli numbers constitute a family of rational numbers that occur principally as coefficients in power series. 4:1 NOTATIONThere are two systems for indexing and assigning signs to Bernoulli numbers. Unfortunately, both systems are inwidespread use, although the one we adopt is more generally used than its rival. Table 4.1.1 compares the twosystems. Some authors employ both systems, using B. for one set of Bernoulli numbers and some modified sym-bolism such as B: or B" for the other. These latter are sometimes called auxiliary Bernoulli numbers. 4:2 BEHAVIORWe define B. for all nonnegative integers n. The accompanying map, Figure 4-1, plots the first sixteen Bernoullinumbers.In our notation, all Bernoulli numbers with odd index, except B,, are zero. All B. for which the index n is amultiple of 4, except Bo, are negative rational numbers. All other even-indexed Bernoulli numbers are positiverational numbers. The absolute values of the Bernoulli numbers of even index. IB,,,I, acquire a minimum value of,A when 2n = 6. Table 4.2.1 shows that the magnitude of even-indexed Bernoulli numbers increases rapidly withincreasing even n.Ba is the only Bernoulli number that is a nonzero integer but 4:2:12(n + 10.= odd integern = 2, 4, 6...(n/2)!2"1' 4:3 DEFINITIONSThe Bernoulli numbers are defined through the generating functionx_e 4:3:1 exp(x) - l -B"n!m 4:4Table 4.1.1THE BERNOULLI NUMBERS, B. 36 ValueOursystemRivalsystem -1Bo B,C0C* CTable 4.2.1B, B,.*....:....:......1.0 60B,FIG 4-1':BernoullinumberRoundedvalue-IB.-B: OSB,"8l0-:30 .B-5 x 100B, B6 x 10' *B.g-2 x 10'B. B, B.8x10"42 B.-2 x 10"- 0B; B.3 x 10B.,-2 x 10"B.,,4 x 10'°30 .............. -0.5 B,,,,-3 x 10" The integral representations of the even-indexed Bernoulli numbers include 4:3:2B. _ (-1)'"''' -srJt"csch'(t)drn = 2, 4, 6, ...vwhere csch denotes the hyperbolic cosecant function [Chapter 291. For others, the reader is directed to Erdelyi.Magnus Oberhettinger and Tricomi [Higher Transcendental Functions. Volume 1, pages 38-391. 4:4 SPECIAL CASESWhen n is an even integer, the Bernoulli number B. is expressible as a zeta number (Chapter 3] 4:4:1 B. _z,n n C(n)n = 2, 4, 6,(2ar)" 4:5 LNTRARELATIONSHIPSBernoulli numbers satisfy the recursion formula -1B, 4:5:1B"=-n!Zn=2,3.4....i-a j!(n -I - j)!with Ba = 1. Recursion 4:5:1 is easily derived from the more compact expression 4:5:2 (n"B,=On=2.3.4....Jwhere (,) is the binomial coefficient [Chapter 6]. 37 THE BERNOULLI NUMBERS, B. 4:94:6 EXPANSIONSEven-indexed Bernoulli numbers are expansible as r4:6:1B.2n= (-1)a+z12 (22)"1 +-+3"+(-1)1;212 2n! i (2!'27)n = 2, 4, 6, ...12' J=1If the multiplier (-1)1"+2)2 is replaced by - cos(nw/2), the expansion is valid for all values of n except 0 and 1. 4:7 PARTICULAR VALUES BaBIBIBIB.B,B.B,B.B.B.B,,BizB B,. BB.B,,B,.B.B,,B,,-11-II-i5-6917-561743867-17"611 000-000000026xr42m6627706510 798 330 4:8 NUMERICAL VALUESThe algorithm below produces Bernoulli numbers with a precision [see Section 0:8) better than 6 x 10-8. Forn = 2, 4, 6, ... the algorithm uses relationship 4:4:1 with the very crude approximation ?;(n) _ (1 + 3-")/ (1 - 2-"). The crude B, value that results is then corrected by a rounding procedure based on 4:2:1. Input n >>If frac(n/2) * 0 go to (1)Setf= IIfn=0goto(2)Set g = (n + 1)!/[(n/2)!2"21Replace f by (f + 3-")n!/[rr"(2" - 1)]1Replace f by 2 + Int(2fg)Replacef by f[4frac(n/4) - l]/gGo to (2)(1) Replace n by n - 2Set f = [(n/mni) - 11/4(2) Output fStorage needed: n, f and g1 f=B.<<a«« 4:9 APPROXIMATIONSBased on expansion 4:6:1 the approximation 11 +21214:9:1B. =((227)"227.I + 2"Input restriction: n must be a nonnegative integer. Test values:B0 = IB, = -0.5B9=0B14 = 1.16666667B. _ -4.03380719 x 1019 8-bit precisionn = 6, 8, 10, ...is useful for all even n exceeding 4. It may be supplemented by values drawn from the table in Section 4:7. Theapproximate recursion formula 4:9:2B,-k k-12B2A_2/1T28-bit precisionk = 5, 6, 7, ...becomes increasingly accurate as k -* -. 4:10 THE BERNOULLI NUMBERS. B.4:10 OPERATIONS OF THE CALCULUSNeither differentiation nor integration may be applied to discretely defined functions such as B,,. 4:11 COMPLEX ARGUMENTThe Bernoulli number B. has been defined only for real integer n. 4:12 GENERALIZATIONSBernoulli numbers are the special x = 0 cases of the Bernoulli polynomials B.(x) (Chapter 1914:12:1 B.=B,(0)n=0,1,2,...Apart possibly from sign, the Bernoulli numbers are also the x = I values of the Bernoulli polynomials4:12:2 B,=(-1)"B,(I)n=0.1.2.... 4:13 COGNATE FUNCTIONS38 Bernoulli numbers are closely related to Euler numbers [Chapter 51 and to the number families discussed in Chap-ter 3. 4:14 RELATED TOPICSIn addition to their roles as coefficients in power series expansions of trigonometric and hyperbolic functions (seeChapters 28-34] Bernoulli numbers occur as coefficients in the F_uler-Maclaurin formula, which provides a valuablelink between a definite integral and a sum. To exhibit this formula, let x°, x x:...., x, be evenly spaced arguments withx,_1 -x,=handj=0. 1,2,...,J- 1. ThenJ-11,hh2dfdf h f(x;) -J af(1) dr- (f(xj) -f(xo)] +12[dx (x,) - d ( X 0 ) 4:14:120(x,) -dz(xo)J +30240[dxs(z')dx' (x°)J + .. . h"B.d" jd" 'fn!dx"_'dx"_'(zo)Jprovided that the function f(x) is sufficiently differentiable. CHAPTER5THE EULER NUMBERS, En Euler numbers occur as coefficients in the power series expansions of the secant [Chapter 331 and hyperbolic secant[Chapter 291 functions. 5:1 NOTATIONAs with Bernoulli numbers (Section 4:1 J, there are (at least) two notational systems in use for Euler numbers.Table 5.1.1 explains the differences between the two systems. Some authors use both systems and introduce sup-plementary notation such as E, or E. to distinguish from E. The former symbols are sometimes called aueiliaryEuler numbers. 5:2 BEHAVIORThe Euler number E, is defined for all nonnegative integer index n. All Euler numbers are themselves integers.Euler numbers of odd index are invariably zero. Even-indexed Euler numbers are positive integers, or negativeintegers, according as n is, or is not, a multiple of 4. In the accompanying map, Figure 5-I. either E. (red points) <1(110.........1030Table 5.1.1FIG 5-1:** ValueOursystemRivalsystem ............ * ..1020 1E. E. *0E, . -E *..........*........1010-1oE;E,-E, 5E. E;0E, * -61F,-E,0E. ****............1 1385F,E. 39 5:3 THE EULER NUMBERS. E. 40 or -E" (green points) is plotted logarithmically versus n for even n values. Note the very rapid increase in theabsolute value with increasing even n.The least significant digit of each negative Euler number is a "1"; that is: 1 5:2:1 frac(-E"/10) = 10n = 2, 6, 10, ...where frac denotes the fractional value function [Chapter 9]. The least significant digit of each positive Eulernumber, except E0, is a "5"; that is: 5:2:2 5:3 DEFINITIONS The generating function 5:3:11frac(E,/10) = Z n = 4, 8, 12, sech(t)E. r may be used to define the Euler numbers. Here sech is the hyperbolic secant discussed in Chapter 29.An integral definition of even-indexed Euler numbers is 5:3:2E"_(-1)"''(2)Jesech(t)dtn=0,2,4,...0 5:4 SPECIAL CASESThe Euler numbers of even index are related by the formula/2-' 5:4:1E"=(-1)"1221-In!P(n+1)n=0.2,4,... w to the beta numbers discussed in Chapter 3. 5:5 INTRARELATIONSHIPS The compact expression 5:5:1 =0n= 1,2,3....(2n2J/where (;) is the binomial coefficient treated in Chapter 6, gives rise to the recursion formula for even-indexedEuler numbersE,, n 5:5:2E"=-n!n=2.4.6,J=--1 with Ev = 1.t-o (n - 2j)!(2j)! 2 41 THE EULER NUMBERS, E. 5:115:6 EXPANSIONSIt follows from equations 5:4:1 and 3:6:4 that Euler numbers of even index may be expanded as/} rr t5:6:1E"=(-1)"/22n!(l"L1-3, i+51)"/2_nii(-I)1R+ 5:7 PARTICULAR VALUES E. 1E, 00F. 50Eb -61 5:8 NUMERICAL VALUESE, 0F. 1385E. 0E. -50521E 0E 2702765E13 0n=0,2,4,... EuIE,, -199360981 I0 For n = 2, 4, 6, ... the accompanying algorithm begins by using equation 5:4:1 with S(n + 1) replaced by unity.This produces a very crude estimate of E", which is then refined by making use of rules 5:2:1 and 5:2:2. The finaloutput has a precision better than 6 X 10-s. Input n >>Setf=0p»»If frac(n/2) + 0 go to (1)Set f = (2/7r)"+' n!/5 Replace f by -1 - 10 Int(f)If frac(n/4) + 0 go to (1)Replace f by 4 - fIfn+0goto(1)Setf= 1(1) Output ff=E"<< 5:9 APROXIMATIONS Based on 5:6:1, one is led to5:9:1E. =(-1)"n 2n! (2/a)"+i 5:10 OPERATIONS OF THE CALCULUSStorage needed: n, fE,e 19391512145 1 Input restriction: n must be a nonnegative integer Test values:Ea = IES=0E10 = -50521Em = 3.70371188 X 10'" 8-bit precisionn = 4, 6, 8, ... Neither differentiation nor integration may be applied to discretely defined functions such as E. 5:11 COMPLEX ARGUMENTEuler numbers have been defined only for real integer argument. 5:12 THE EULER NUMBERS. L, 425:12 GENERALIZATIONSEuler polynomials E"(x) [Chapter 20] represent a generalization of Euler numbers, to which they are related by5:12:1 E" = 2E.(1/2)n = 0. 1.2... . 5:13 COGNATE FUNCTIONSEuler numbers have much in common with beta numbers [Chapter 3J and Bernoulli numbers [Chapter 4]. CHAPTER6THE BINOMIAL COEFFICIENTS (m) Binomial coefficients occur widely throughout mathematics: for example, in the expansions discussed in Section6:14 and in the Leibniz theorem, equation 0:10:7. 6:1 NOTATIONWhen v is the positive integer n, the binomial coefficient (;) is sometimes denoted C. or C". These symbols havetheir origins in the role played by the binomial coefficient (:) in expressing the number of combinations of m ob-jects selected from a group of n different objects [see Section 2:14]. 6:2 BEHAVIORThe lower index m of the binomial coefficient (;) is invariably a nonnegativc integer, whereas the upper index vmay take any value. Positive integer values of the upper index are common, however, and we shall write (a) asthe general expression for these cases.The binomial coefficient (.) is always a positive integer when m = 0, 1, 2, ..., n and zero when m =n r 1,n + 2, n + 3, .... Values of (;) for n values up to 16 and m up to 8 are tabulated in Section 6:8 (Table6.8.1). For a given n. (e) assumes its maximum value when m = n/2 if n is even. or when m = (n ± 1)/2 if nis odd; this is illustrated in Figure 6-1.When v is not a positive integer or zero, the binomial coefficient (;) may be positive or negative, integer ornoninteger. Its values are zero when v = 0, 1, 2, ..., m - I and, as illustrated in Figure 6-2, arc close to zerofor all other v values in the range -k < v < m - i The binomial coefficient (;,) increases without limit (exceptwhen m = 0) asv - x. 6:3 DEFINITIONSThe binomial coefficient is defined by the finite product 1am -I 6:3:1I1 43 6:4 THE BINOMIAL COEFFICIENTS (;) 44 supplemented by the definition)= 1 6:3:2 (011of a binomial coefficient with zero lower index. Binomial coefficients may also be defined by the generatingfunction [Section 0:3]/6:3:3 -1<I<ImFor positive integer upper index the expression 6:3:4 \A/ =nzmm/m!(n - m)!in terms of factorials (see Chapter 2], also serves as a definition. 6:4 SPECIAL CASESReduction to an expression involving the double factorial [Section 2:13] occurs when the upper index is equal totwice the lower index 6:4:1 2m_2"(2m - 1)!!4"(2,n - I)"Mm!(2m)!!or differs by unity from twice the lower index 45 THE BINOMIAL COEFFICIENTS (.) 6:5 6:4:22m - 1)_2m - 1_2"'-'(2m - I)!'_22i'-'(2m - 1)"m - I-(m )m!(2m)!! 6:5 INTRARELATIONS H I PSThere exist reflection formulas for the upper 6:5:1and lower 6:5:2indices, as well as recursion formulas6:5:3 6:5:4for each index. The addition formula6:5:5(n+ ) = (-1)^(rnn - 1) (nn m)-(m) (vm1/- \m/ + V+(v - m)ymI)(m+1)(m) (v mW)-,-0(J)(--j)known as Vandermonde's convolution, applies to the upper index.m= 1,2,3,... 6:6 THE BINOMIAL COEFFICIENTS (;,)There are a number of intrarelationships involving sums of binomial coefficients and including+ +m _ n+ 1 6:5:6GIm1n>m 6:5:7 and46 i,,(-I)i=(-l)I(v-1 i-o(j)m J+ I +v(J+vH 6:5:8`J+mJ+m J Formula 6:5:5 provides an expression for a sum of products of binomial coefficients. a second such formula is 6:5:9\m/\J!v/-\mm v/Jtmv/JJ?m When the upper index is an integer, finite or infinite series of binomial coefficients frequently have simpleexpressions; examples include 6:5:10 (n)=YJanj 6:5:111(-1),(n)=0J?n/111(,)6:5:12 nsJ=2.4.6. 6:5:13(7) +131+nsJ=1,3,5,.. and\\\ 6:5:14Vin) =2"-'nJznroSimilarly, the sum of squares of binomial coefficients gives6:5:15(n)(2n)JanJn 6:6 EXPANSIONSA binomial coefficient may be expanded as a power series in its upper index by the formula 6:6:1(v/I iS''Dmm! '.ain which S'';' is a Stirling number of the first kind [see Section 18:6). As well, definition 6:3:1 constitutes theexpansion of (;,) as a finite product.When v, but not in. is large in magnitude, the asymptotic expansionvv rrm(m - 1)m(m - 1)(m - 2)(3m - 1)6:6:2(rn) = - Ii -2v+24v2+ ...v-- 'x M! I 47 THE BINOMIAL COEFFICIENTS (.')applies. Conversely, when m, but not v, is large, we have 6:6:3\m)(v(-I)"v(v+ 1)v(v+ 1Hv +2)(3v+ 1)+'m"' (1(-v)[ 1 +2m+24m'... where f is the gamma function [Chapter 43].m-s 706:8 As noted in Section 6:2, the maximum value of ( ), for a given even value of n. occurs when m = n/2. Anasymptotic series expansion for this maximum is 6:6:4 n\n/2/2111-+-+even an4n32n'Similarly, the asymptotic expansion for the maximum value of Q) when n is odd is 6:6:5 n2"V--2 1-3+ 2, oddn (n/2±1/2)urn[4n32n2J 6:7 PARTICULAR VALUESThe rules 6:7:1 and 6:7:2 =v are valid for all values of the upper index. For integer values of this index, as well as for v = ±12.there areadditional particular values of the binomial coefficient as given in Table 6.7.1. 6:8 NUMERICAL VALUES For integer upper index, values of the binomial coefficient (.) are given in Table 6.8.1 for 0 s n < 16 and Table 6.7.1 m-0 I I 1I 1 1 M - I 1-102 Iisl-I)"(2m - 1)!!0--(-I) (2m - 3)!!0n! (-I)(2m)!! (2m)^m!(n - m)!-1-1)"(2n - 3)!!(-Il"(2n - 5)!! m - n - I--(-1) 0 0n (2n - 2)!! (2n - 2)!!- I)!!-(-I)'(2n - 3)^m - n(- I)' o 0 Iz(2n)!!(2n)m- n+I.n+2,... (- 0'(- IY(2m - 1)!!0-(-Ir(2m - 3)!!00 (2m)!! (2m)!I 6:9 THE BINOMIAL COEFFICIENTS (;,) 48Table 6.8.1 (,)(,)()(,)(,)(.,I(:,)(,j)(,)l,)('"1f!,=)(;,') 11IIII1IIIIIIIIII01234567891011121314151600 I3610152128364555667891IOS120000 1410203556841201652202863644555600000 15IS357012621033049571510011365182000000 I621561262524627921287200230034368000000 I7288421046292417163003500580080000000 I8361203307p21716343264351144000000000 I9451654 512873003643512870m - 0m1m2m3 m4m5m6m7m8 Recursion 6:5:3 provides a useful way of evaluating binomial coefficients. In a construction known as Pascal'striangle. binomial coefficients are listed in such a way that each entry is the sum of the two above.010011001210013310014641001510105I0016152015610The simple algorithm. Input v >>Input m >:f =IIf m = 0 go to (2)Setj=m Replace m by m - v(1) Replace fbyf/(I-ml\j/Replace j by j - 1Ifj*0goto(1)(2) Output f which is based on definitions 6:3:1 and 6:3:2, is exact. 6:9 APPROXIMATIONSI Storage needed:v, f. mandjInput restrictions: rn must be a nonnegative integer;v is unrestricted Test values:(u) = I(9')=-715(';') = 0.0234375 When both indices are small, the binomial coefficient is so easily calculated that no approximations are needed.When only one of v and m is large in magnitude, the asymptotic expansion 6:6:2 or 6:6:3 may be truncated toprovide an approximation to (;).When v is large. (;) is very large for values in the vicinity of v/2. Values of such binomial coefficients areapproximated by the Laplace-de Moivre formulaI 6:9:1I v l = 2`expi -2 I m - - l}v large mV vir111 v\2J M THE BINOMIAL COEFFICIENTS (;) 6:12which finds statistical applications [see Sokolnikoff and Redheffer, pages 623-626]. Even for v as small as 10,the approximation leads to small absolute errors, as illustrated in Figure 6-3. 6:10 OPERATIONS OF THE CALCULUSDifferentiation with respect to the upper index gives a derivative involving a difference of two digamma functions[Chapter 441:(f1(1 6:10:1a\v)-\v)L_+ l+ 1 +...+1 av minvv-1v-2v - m+l111 m/ 6:11 COMPLEX ARGUMENTThe equivalence presented in equation 6:13:2 permits the formulas of Section 43:11 to be used to evaluate a binomialcoefficient when one or both of the indices are imaginary or complex.6:12 GENERALIZATIONSThe reciprocal of the complete beta function B (Chapter 43] generalizes the binomial coefficient to noninteger lowerindex because of the relationship6:12:1 (m)mB(m,vim+1)m#0between the two functions. If the upper index is an integer, this generalization simplifies to 6:12:2I-n!sin(irµ)µB(µ.n - µ + 1)a(-µ).-1wheredenotes a Pochhammer polynomial (Chapter 18). This formulation was used to draw the curveslinking the binomial coefficient points in Figure 6 1.A generalization in a different direction is provided by mulrinomial coefficients which are the integer coefficientsthat arise in such expansions as6:12:3(x+y+Z)5=[x5+y5+z5]+5[x4y+x'z+y'.r+z'y+y'z+z4y]+ lO[x3y= + x322 + v3x' + Z3.r2 + y3Z2 + z5y2]+ 20[x3vz + y3xz + z3xy] + 30[x'y2z + x2:2V + y2Z2z] 6:13 THE BINOMIAL COEFFICIENTSThe general expression may be written6:12:4(xi + x_ + x3 + ... + x.)° = E M(N;m,,m,.m3,....m.)[x'7,XT,:... x 7n x'; ";':.... +where the summation embraces all combinations of nonnegative integer m's that satisfy6:12:5 =NThe multinomial coefficient is given by6:12:6 N!(m,)!(mz)!(m3)! ... (m,)!and the number of terms multiplied by that coefficient is6:12:7 # of terms =n!.1EE (W -d)!(W(mz))!(W(m3W ... 44-0)!where It(m) is the number of occurrences of a specific value of m;. (Thus, the denominator in 6:12:7 is unity ifall the in values are distinct.) Tables 6.12.1 and 6.12.2 give a small number of trinomial coefficients (as exemplifiedin 6:12:3) and quadrinomial coefficients: see Abramowitz and Stegun [pages 831-8321 for a more comprehensivetabulation. 6:13 COGNATE FUNCTIONSThe formulas 6:13:1 and(v)(v-m+1),,1mm! 6:13:2 v_r(v + I) r(v + 1)mm!r(v-m+ l)F.1l)r(v-m+1)relate the binomial coefficient to the Pochhammer polynomial [Chapter 18] and to the gamma function [Chap-ter 43J. 6:14 RELATED TOPICSThe binomial theorem, also known as New+wton's formula. permits a sum or difference to be raised to any power: 6:14:1(xx°r1 i (m)x-x<y<x The series terminates if v is a positive integer but is infinite otherwise. Some commonly encountered binomialexpansions are6:14:2(1 =x)°= 1 ±4x+6x2±4x3+x`6:14:3 (1 tx)3= 12: 3x+3x1±x36:14:4 (1 ±x)2= I ±2x+x233133 6:14:5( l ±x)312= 1 =2X16x3+ 128x4_-x5+'-I x I256 51 THE BINOMIAL COEFFICIENTS (;) 6:14 Table 6.12.1 No. ofNm m m, terms 00,0,0 1 1 1110.0 1 3 22.0,0 1 321. 1. 0 23 33,0.0 1 332. 1. 0 3631. 1. 1 6 I 44,0,0 1343. 1, 0 4 642, 2. 0 6 342. 1, 1 123 55.0,0 I354, 1. 0 5653.2.0 10 653. 1.1 20 352. 2. 1303 66.0,0 I365. 1. 0 6664, 2. 0 15664. 1.1 30 363. 3. 0 20 363, 2, 160662. 2. 290I 77.0.0 1376. 1. 0 7675. 2, 0 21675.1.142374, 3. 035674, 2, 1 105 673, 3.1140373. 2. 22103 ITable 6.12.2 No. ofNm,. MI. MI. M, terms 00.0.0,0 I 1 11.0,0,0 14 22,0.0.0 1 421. 1. 0. 0 2 4 33. 0, 0. 0 1432, 1. 0. 0 3 1231. 1. 1. 0 64 44.0.0.0 1443.1,0.0 41242, 2, 0, 0 6 642, 1. 1, 0 12 1241. 1. 1. 1 24 1 55.0.0.0 1454. 1.0.0 5 1253, 2.0.0 101253,1,1.0 201252. 2. 1, 0 30 1252.1,1,1604 66.0.0.0 I465, 1, 0, 0 6 1264.2.0,0 151264, 1, 1, 0 301263, 3. 0, 0 20 663, 2. 1. 0 60 2463. 1. 1.1 120 462. 2. 2. 0 90462, 2, 1,11806 77.0,0.0 I476, 1.0.0 71275. 2, 0, 0 21 1275. 1. 1, 0 421274, 3. 0. 0 351274, 2. 1. 0 1052474,1,1.1210473, 3. 1, 0 140 1273. 2, 2, 02101273. 2. 1.14201272. 2, 2,16304 II,I57 6:14:6(1 ±x)'12 = I ±-x--x-r5+... -1 K r 1 6:14:7 6:14:8 6:14:92816128256 13_53563x'x+-(l+x)t:=1+mo: x5+...-1<±x<1 x3+x4 -1628128256(1 ±x)-' = I Z x+x' -Px3+x'+x5 +31535315693-1<x<I(1 =x)-112= 1 w 2x+ 8x'+16x3+128x+256x5+...-1 <x< I 6:14 THE BINOMIAL COEFFICIENTS (;,) S2 6:14:10 (I ± x)-' = 12x + 3x' 4x' + 5x46x5 +_-1<x<16:14:11(l=x)-3=13x+6x'IOx'+15x°21x'+-1<x<1 6:14:12 (1 ± x)-' = 1W. 4x + 10x'20x3 +35x'56x5 +-1 <x< 1Equation 6:14:1 may be regarded as a special case of the Taylor expansion, cquation 0:5:1. The functioi(I'- x)", or sometimes (v + x)', is known as the binomial junction. CHAPTER THE LINEAR FUNCTION bx + c AND ITS RECIPROCAL Many relationships in science and engineering take the form f(x) = bx + c. and many others can be manipulatedinto this form by a redefinition of the variables. Data analysis problems can therefore often be reduced to thedetermination of the coefficients b and c from pairs (x.f(x)) of experimental values. The least squares method forperforming this analysis is presented in Section 7:14.7:1 NOTATIONThe constant c of the linear function bx + c is termed the intercept, whereas b is known as the slope or gradient.These terms derive, of course, from the observation that, if the linear function bx + c is plotted, its graph is astraight line that cuts the vertical axis at altitude c and whose inclination from the horizontal is characterized bythe number b [see Figure 7- I ]. Occasionally the word slope is used to mean arctan(b) [see Chapter 35]. The symbolm frequently replaces b. 33 7:2 THE LINEAR FUNCTION bx + c AND ITS RECIPROCAL 54The name inverse linear function is sometimes given to the function 1 /(bx + c). Throughout this Atlas. how-ever, we reserve the phrase inverse function for the relationship described in Section 0:3. The name reciprocallinear function unambiguously denotes 1/f(x) where f(x) = bx + c. 7:2 BEHAVIORThe linear function bx + c is defined for all values of x and (unless b is zero) itself assumes all values. The sameis true of the reciprocal linear function I /(bx + c) which, unlike the linear function, is discontinuous at x =-c/b. The accompanying Figure 7-1 is a map showing typical behavior of the linear function and its reciprocalwhen both b and c are positive. Figure 7-2 illustrates how the linear function is positioned with respect to the axeswhen b and/or c changes sign. i+*O 7:3 DEFINITIONSThe arithmetic operations of multiplication by b and addition of c define f(x) = bx + c for all values of x. Thefunction f(x) = 1l(bx + c) is defined by the same operations followed by division into unity. The linear functionis completely characterized when its values, f, and f2. are known at two (distinct) arguments, x, and x2. The slopeand intercept may be inferred from the formula 7:3:1bx+c=(f:-f.)x+xf, - x,fix2-xlx2-x,7:4 SPECIAL CASESWhen b = 0. both the linear function and its reciprocal reduce to a constant function [Chapter 1]. 7:5 INTRARELATIONSHIPSBoth the linear function and its reciprocal obey the reflection formula 7:5:1 f(-x)=-flx- b 1 55THE LINEAR FUNCTION bx + c AND CPS RECIPROCAL 7:6The sum or difference of two linear functions is a third linear function7:5:2 (b,x + c,) ± (b2x + c2) = (b, t b2)x + (c, ± c2)The product of two linear functions is a quadratic function (Chapter 16]7:5:3 (b,x + c,)(b2x + c2) = b1b2x2 + (b1c2 + b,c1)x + c1c2and (unless b2c, = b1c2) the quotient of two linear functions is an infinite series c,b2c, - b,c2 b2x ' II b1x + c,,c2b2c2,_, \c2/1XI <b7:5:4 b,x + c2 lI b2b2c2b2x b2The inverse function [Section 0:3] of the linear function f(x) = bx + c is another linear function 7:5:5 xcF(x)=--b+0-bwhile the inverse of the f(x) = 1 /(bx + c) function is 7:5:6IF(x) =-cbb * 0As explained in Section 16:3, the difference of two reciprocal linear functions that share a common b parameterproduces a reciprocal quadratic function. Finite and infinite sums of certain reciprocal linear functions may beevaluated in terms of the digamma or Bateman's G functions [Chapter 441 as follows:`s14j 7:5:71++I+...+I=L,1 7:5:8cx+c2x +cJx+c;_oJx+cxxx cx + c2x+cJx+c;G0Jxx+c2xIlL`(2+(2x120x(2+2+2x)J 2xG(J + 1 + C)Jwhere the upper signs are taken if J is even, the lower if J is odd, and 111(-1)' 11cc_I - 1 7:5:9- - - + +G cx + c2x + c;_o .x + c2x22x)2x)2x(c)X 7:6 EXPANSIONSThe reciprocal linear function is expansible as a so-called geometric series for either small 7:6:1 or large 7:6:2IIbx62x263x3Ibx 'bx + ccc'c3c'c\ c1 i<cb 1c 1_1cc'cbx + cbxb2x2b'x363x3bxbxJ.r1 >cb 7:7 THE LINEAR FUNCTION be + c AND ITS RECIPROCALvalues of the argument. The linear function may be expanded 7:6:3 bx + c = c + 2J,(bx) + 6J3(bx) + 10J3(bx) + = c + 2(2j - 1)Jz,_,(bx)56 =1as an infinite series of Bessel coefficients [Chapter 521 of odd order, although this representation is rarely employed.The reciprocal linear function may also be expanded as an infinite product cc-bx 11II + \bx)2i]bx ILLI\/IJ- I < - < IJ-7:6:4 =rfi,lbx+cbx-,c111+lbx> 1 b2x-L\bxICFor example. if } < I. (1 ± x)-' = (I z x)(1 + x2)(1 + x)(1 + x8)(1 + x1 b) ... . 7:7 PARTICULAR VALUES 7:7:1bx+c=0x=-c/bb * 07:7:2 bx+c=cx=0 7:8 NUMERICAL VALUESThese are easily calculated by direct substitution. 7:9 APPROXIMATIONSFor small absolute values of x, the reciprocal linear function may be approximated by a linear function 7:9:1I= -I-bx8-bit precisionIcl4x1 < - bx + ccc 16[b1Similarly, the approximation 7:9:21-bx - c8-bit precisionxJ >161c-bx + cb''x' JbIis useful for large arguments of either sign. These two results follow directly from expansions 7:6:1 and 7:6:2,respectively. 7:10 OPERATIONS OF THE CALCULUSDifferentiation of the linear function producesd 7:10:1-(bx+c)=bwhile the derivative of the reciprocal linear function isd1_-b 7:10:2 dx bx + c(bx + c)2 57THE LINEAR FUNCTION bx + c AND ITS RECIPROCALIntegration of the same pair results in 7:10:3 and dt_1o (Bt +C)(bt + c)bC - BcIn7:10 7:10:4IIdtI Inl I1 +bxllu bt+cb\cJIrespectively. If 0 < (-c/b) < x. the integrand in 7:10:4 (and in some of the integrals below) encounters an infinity;in this case the integral is to be interpreted as a Cauchy limit, as explained later in this section. Other importantintegrals include 7:10:5 7:10:6 and 7:10:7bx2(bt + c)dt = 2 + cx bx1 +-cBC *bC1 + Cj/Bt+Cdt =Bx+bC - Bc(1 +bx--}U bt+cbb2In\c ufedt-(-c)"br + cb"'InbxI+ -c+lr bxlj\/] n=0,1,2....,-IfcSemidifferentiation and semiintegration [see Section 0:10] with lower limit zero gived1f22bx + c 7:10:8 dx'/2(bx+c)Vnxand/\\7:10:9d;7Fi(bx+c)= \43x+2c1when applied to the linear function, anddV2It - Vx 7:10:10 =dx"2 bx + c\(br + c)d-1/2126 7:10:11 =dx '2bx+cyzrbwhen applied to the reciprocal linear function, where the form of the function d depends on the magnitude ofc/b, as follows:aresin(bx/c)c-x < - < -XU-x - (c/b) barcosh(-bx/c)c 7:10:126_i-x<-<0V.r + (c/b)barsinh(Vbx/c)cbVIn formulas 7:10:8-7:10:12 it is understood that x > 0. 7:11 THE LINEAR FUNCTION bx + c AND ITS RECIPROCALThe semiderivative, with lower limit -x, of 1/(bx + c) is 7:10:13 while the semiintegral is 7:10:14din11-rb[d(x+x)]1/2bx+c 2V(bx+c)' d-1/2I[d(x + x)]-1/2 bx + c01-Vn-b(bx + c)0 Definite integrals of the formf(t) 7:10:15 dtf-bt + ccan be evaluated by a Hilbert transform defined as 7:10:16 fH(s) =I J = f(t)dtitI-sTherefore, integral 7:10:12 can be evaluated by means of the identity 7 f(t)dt=abfH(-c/b) 7:10:1T.bt+CFor most functions f the integraldt 7:10:18 J Yf(t) -r - s-cx<b-cx>-b -CX < -b-cx>-b58 encounters an infinity at x = s and therefore the definition of the Hilbert transform requires a so-called Cauchylimit interpretation of the integral in 7:10:16;t( s,that is:rl 7:10:19Jif(t)-= limaJ'f(t)-+If(t)dt}e > 0t-s'-°(1t-st - s)Occasionally in this Atlas [e.g.. in Section 22:10] a Hilbert transform will be encountered with integration limitsother than ±x. The shorthand notation 7:10:20fH(s) _If(t)dt=JiF(t) - whererrt - snt-s0t<toF(t) =f(t)tott10t > t,is then intended. The Hilbert transforms of many functions are tabulated in Erdelyi, Magnus, Oberhettinger andTricomi [Tables of Integral Transforms, Volume 2, Chapter 15]. For example, the Hilbert transform of a pulsefunction [Section 1:13] is a logarithm [Chapter 25] 721 (dtIIs1 :10:1 7:11 COMPLEX ARGUMENTaJp( 1;2;r)t - sn(as+I The formulas of Section 1:11 can easily be applied to bx + c or 1/(bx + c) if x (or, indeed, if b or c) is complex. S9THE LINEAR FUNCTION bx + c AND iTS RECIPROCAL7:12 GENERALIZATIONS7:14 The linear function is the special a = 0 case of the quadratic function discussed in Chapter 16.A linear function is a member of most of the families of functions discussed in Chapters 17 through 24; thesepolynomials may therefore be regarded as generalizations of the linear function. 7:13 COGNATE FUNCTIONSA frequent need in science and engineering is to approximate a function whose values f(.to), f(x,), f(x,)..... f(x")are known only at a limited number of arguments xo, x, x2, ..., x.. One of the simplest ways of making such an approximation, but one that is nevertheless adequate in many applications, is by using the piecewise-linear function. For arguments between two adjacent points, x, and x,.,, at which the function has known values, the interpolationformula(xJ., - x)f(xJ) + (x - x,)f(x)_,)7:13:1 f(x) = x, < x < xJ,x,., - x,defines the piecewise-linear approximation. In graphical terms, this piecewise-linear function is constructed by"connecting the dots," as in Figure 7-3. FIG 7-3 ............ ......:.............. ........... .......... f(xJ,t) ...... ............f................. _.......... f(xf-1) .......... \........................................ f(xJ) Usually a piecewise-linear function has discontinuous derivatives at all the interior values x,. x,, x,...., .t"_The alternative interpolation formulas discussed in Section 17:14 do not suffer from this defect.The reciprocal linear function is related to the function discussed in Section 15:4 because the shape of each isa rectangular hyperbola. Thus, counterclockwise rotation through an angle tr/4 of the curve 1/(bt - c) about thepoint x = -c/b on the x axis produces a new function7:13:2,Jlx+bl'3that is a rectangular hyperbola of the class discussed in Section 15:4. 7:14 RELATED TOPICSFrequently. experimenters collect data that are known, or believed, to obey the equation f = f(x) = bx + c butthat incorporate errors. From the data, which consist of n pairs (.t,, fi) (x,, f-), (x3, f3), ..., (x". f) of numbers, thescientist needs to determine the "best" values of b and c. If the errors obey a Gaussian distribution (Section 27:141 7:14 THE LINEAR FUNCTION bx + c AND ITS RECIPROCALand are entirely associated with the measurement off (that is. the x values are exact), then the formulas 7:14:1 andnIxf - ExIfb =n E xz - (Et)z _fxzIf - ExExfEf- b.x 7:14:2 cnExz-(Fx)zitgive these best values. An abbreviated notation, exemplified by 7:14:360 is used in the formulas of this section. The term "best' is employed here in the sense of producing the minimumsum of squared deviations E(bx + c - f)z, and the procedure is known as least squares fitting or linear regression[see also Appendix A. Section A:4J. Figure 7-4 shows an example of five data pairs and the "best straight line"through them. byA measure of how well the data obey the linear relationship is provided by the correlation coefficient, givennExf - ExIfbnExz-(Ez)z7:14:4r=z-z=nIfz - (Ef)z [nEx(Ez) J(nEf2(Ef) 21Values of r close to ± I imply a good fit of data to the linear function, whereas r will be close to zero if there islittle or no correlation between f and x.Frequently, there is a need to know not only what the best values are of the slope b and intercept c but alsowhat uncertainties are associated with these best values. Such uncertainties are often expressed in terms of thestandard error [see Section 40:14] in b and c, these being reported as 61THE LINEAR FUNCTION bx + c AND ITS RECIPROCAL 7:14 7:14:5and7:14:6slope = b ± dwhered = standard error in b intercept = c ± e wheree = standard error in cThe significance to be attached to these statements is that the probability is approximately 68% that the true slopelies between b - d and b + d. Similarly, there is approximately a 68% probability that the true intercept liesbetween c - e and c + e. The formulas 7:14:7 and 7:14:8d-nXf'-(Ef)'nTxf-ExTf=bII Fx Ef22Ex2I'.xZ.xfZf+ Ex2(fxf)2_Fx= enF.x2 - (Ex)'[nlx2 - (Ex)22- d permit these standard errors to be estimated.Commonly, data are gathered with equal spacing, that is. x2 - x, = x3 - x2 = x, - x3 x, - x,_,.The formulas of this section simplify considerably under these circumstances, and these simplified formulas arethe basis of the algorithm presented here for the calculation of the best values of b and c. If the portions printedin green are included, the algorithm also generates the correlation coefficient, as well as values of the standarderrors of b and c. j (as cue) <Input f, >>Setb=c=r=j=0Set d>>>>Set e = (x. + d)/2 Replace d by (e - d)/3(1) Replace j by j + IOutput jG<GGReplace c by c + fReplace r by r + J';Replace b by b + j f,1fj<ngoto(1)2bReplace b byn + 1c /ndSet j=dV3(n-1)/(n-I)Replacer by nbj/V nr - c'Output rcorrelationr=coefficientb = slope 5Output bK««Set d = bV (I /r) - IOutput dd = slope ««<errorStorage needed: n, b, c, r, j. d and e.Several of these registers are also used for temporary storage. Test values:Lrr11-0.11.02 1.63 4.44 5.5n = 50.38.0 cReplace c by - - ben 7:14 THE LINEAR FUNCTION be + c AND ITS RECIPROCAL IOutput cc = intercept <<<<Replace e b% d\ eOutput eintercept<<<<errorutput:r=0.9134b - d = 17.9 - 3.3c-e=2.31=0.5762 A related, but simpler, problem that occurs frequently is the construction of the best straight line through thepoints (x f, ), (x2, f2). (x3. f3), .... (x,,. f) with the added constraint that the line must pass through the point (xo, fo)In practical problems this obligatory point is often the coordinate origin. The slope and intercept in this case aregiven by 7:14:9 andbE(x - Xo)(f - A)=E(x - xo)2 7:14:10 c = fo - bxoEquations 7:14:1-7:14:8 are based on the assumption that all data points are known with equal reliability, acircumstance that is not always valid. Variable reliability can be treated by assigning different weights to the points;thus, if the value off, is more reliable than f2, one assigns a larger weight w3 to the data pair (xr, f,) than the weightw2 assigned to pair (x2, f_). The weights then appear as multipliers in all summations, so that, for example, equations7:14:1 and 7:14:2 become replaced by 7:14.11 b =lwlwXf - lwxIwf1WYWX - (I14'x)2and 7:14:12 Ewx=Ewf-Ewxfwxf Ewf - blwxc =_Ew!wx` - (Ewx)21wOnly the relative weights are of importance; absolute magnitudes of w,, w2, w3, ..., w have no significance beyondthis. In practice, one attempts to attach a weight w, to point (x f,) such that w, is inversely proportional to thesquare of the error or uncertainty associated with f. Notice that 7:14:1 and 7:14:2 are the special cases of equations7:14:11 and 7:14:12 in which all weights equal unity, and that 7:14:9 and 7:14:10 are similarly special cases inwhich the weight of one point, (xo, fo), is overwhelming in comparison with the other n weights, which are uniform. CHAPTER0THE UNIT-STEP u(x - a) AND RELATED FUNCTIONS This chapter concerns the unit-step functions u(x - a) and u(x), the signum function sgn(x) and the absolute valuefunction lxl. These functions are closely interrelated through the simple formulas 8:0:1 sgn(x) = 2u(x) - I and8:0:2 Ixl = x sgn(x) = 2x u(x) - xand it is therefore appropriate to treat them together.The main utility of the unit-step and related functions is in modifying other functions, and they are usuallyencountered as in the following expressions: 8:0:3 8:0:4 8:0:5 and8:0:6 If(x)I 8:1. NOTATION The unit-step function is also known as the Heaviside function, or Heaviside's step function, and is sometimesdenoted by H(x - a) or S,(x). It should not be confused with the unity function [Section 1:4]. The notation u(x)replaces u(x - a) when a is zero.The signum function is also called the sign function and symbolized sign(x) or sg(x).Alternative names for the absolute value function are the magnitude of f(x) or, especially when f(x) is complex,its modulus. The term "modulus" should not be confused with "modulo" [see Section 9:13]. The symbolism ABS(x)sometimes replaces lxl, especially in computer applications. 63 8:2 THE UNIT-STEP u(x - a) AND RELATED FUNCTIONS 648:2 BEHAVIORFigures 8-1 through 8-4 illustrate the effect of modifying f(x) to each of the expressions 8:0:3-8:0:6. The modi-fication is shown in red and the original (arbitrary) function f(x) in green. In Figures 8-3 and 8-4, r is a zero of the function f. Figure 8-4 relates to a function f that is real because, as explained in Section 8:11, a specialsignificance attaches to jf(x)j when f is complex.A6 FIG 8-1 u (x-o) f (x) .............p 8:3 DEFINITIONSThe unit-step function is defined by(0x<a8:3:1u(x-a)= 1/2Ix = a 1x>aand therefore the effect of multiplying a function by u(x - a) is to nullify the function for arguments less than aand leave it unchanged for arguments greater than a:0x<a8:3:2 u(x - a)f(x) = f(x)/2 x = af(x)x > aThe signum function extracts the sign of its argument so thatf-If(x)<08:3:3 sgn(f(x))0f(x) = 0l1f(x) > 0 65THE UNIT-STEP u(x - a) AND RELATED FUNCTIONS 8:3 if-,*1CFIG 8-4f(x) / An integral may also be used to define the signum function8:3;4 sgn(x) = nr sin(xt) dz 0The absolute value function equals its argument when the latter is positive and equals the negative of its ar-gument when the latter is negative. Hence 8:4 THE UNIT-STEP u(x - a) AND RELATED FUNCTIONS ,,,,(f(-x)x50 andand 8:3:6 f(x)f(x) c 0If(x) =f(x)f(x) z 0with the requirement that, for 8:3:5 to be valid, f(x) must be real. 8:4 SPECIAL CASES66 The unit-step function may be used to construct many special discontinuous functions. For example, the pulsefunction (Section 1:13] can be represented by a difference of two unit-step functions:// 118:4:1p(c;h;x-a)=c uIx-a+2f -ulx - a - Z)J Similarly, the functions 8:4:2u(x - j)and(x - j)(u(x - j) + u(j + I - x)] are examples of the staircase and sawtooth functions, respectively [see Section 9:13].Acting on a sinusoidal function (Chapter 32], the signum function generates a `square wave" such as sgn(sin(x)).while the absolute value function effects "full rectification" as in Icos(x)I. See Section 36:14 for an explanation ofboth of these concepts. 8:5 INTRARELATIONSHIPS The definitions lead to the following reflection formulas:8:5:1 u(-x) = I - u(x)8:5:2 sgn(-x) _ -sgn(x)and8:5:3 -x1 = IxiFrom 8:5:1 it follows that u(x - a)f(x) and u(a - x)f(x) are complementary functions in the sense that8:5:4 u(x - a)f(x) + u(a - x)f(x) = f(x) 8:6 EXPANSIONS If the function f(x) is expansible as a power series (Section 11:14], the unit-step function may be distributed throughthe series 9:6:1 u(x - a)f(x) = u(x - a) i a,x' _ i a, u(x - a)x'-0,-0but the same is not true of the signum or absolute-value function. 67THE UNIT-STEP u(x - a) AND RELATED FUNCTIONS8:7 PARTICULAR VALUES x< 0x- 0x> 0 1 Ulx)011 s$n(x)-10 xl-x0X 8:8 NUMERICAL VALUES8:10 Many programmable calculators have a key that directly extracts the absolute value of the register's contents.Similarly, computer languages usually permit the finding of an absolute value via an operation such as ABS(x).When this is not so, the operations of squaring followed by taking the square root will give the absolute value,because 8:8:1 \ _ IxI 8:9 APPROXIMATIONSThe unit-step and signum functions may be approximated by a variety of continuous functions. For example, as btakes ever larger positive values, the approximations 18:9:1 u(x - a)-(1 + tanh(bx - ba)) and8:9:2 sgn(x) - erf(bx)become increasingly valid. The functions in 8:9:1 and 8:9:2 are the hyperbolic tangent [Chapter 30) and the errorfunction [Chapter 40). respectively. 8:10 OPERATIONS OF THE CALCULUSDifferentiation producesd 8:10:1 -u(x-a)=S(x-a) 8:10:2 8:10:3 8:10:40x<ad-[u(x - a)f(x)j = df(x)x > adxd-sgn(f(x))=0x*rn= 1.2.3,... dx Idfd-(x)x > 0dxfgxl) =df- - (-x)x < 0 8:11 THE UNIT-STEP u(x - a) AND RELATED FUNC71ONS and68 ddf 8:10:5-If(x)I =sgn(f(x)) - (x)x * r" dx dxwhere r,. r2. r ... are the zeros of f(x), that is, f(r") = 0, n = 1, 2. 3, .... The delta function, 8. occurring informula 8:10:1 is the Dirac delta function [see Chapter 10].Integration is also easily accomplished: 8:10:6fu(t-a)f(t)dt=Jtf(t)dtxo<asx Jsgn(f(r))1a8:10:7dt=2sgn(f(xo))Z+r,-r,+r3--(-1)"(r"-f(xo)*0 rr JJ'J8:10:8ff(I:I)dt=Jxf(t)dt-sgn(xo)f(t)dt (o8:10:9f. If(t)I di =ff(r) dtIJf(t) dt+Jf(t) dtI++J^f(r) drwhere r r,, r3, ..., r" are those zeros of f(t) that lie between x0 and x.Differintegration with a lower limit x0 gives d" [x - a] 8:10:10u(x-a)=xo<a<x [d(xr(1 - v)andd' d" 8:10:11[u(x - a) f(x)] = u(x - a) f(x)xo < a < x [d(x - xo)]' (d(x - a)]'This last result is a generalization of 8:10:1 and 8:10:5.8:11 COMPLEX ARGUMENTA special significance is accorded to the absolute value function of complex argument: thus8:11:1 1x+ryi=x2+.1'8:12 GENERALIZATIONSThe function defined by8:12:1 [u(x - x0) - u(x - x,)] f(x) xo < x,is equal to f(x) inside the "window" xo < x < x, and to zero outside. Such a window function is exhibited in Figure8-5.40444% .0 69THE UNIT-STEP u(x - a) AND RELATED FUNCTIONS 8:13The pulse function (Section/1:131 is the simplest example of such a window function: 8:12:2plc;xi - x0;x -xo2x,I = ((x - -to) - u(x -The concept may be generalized to 8:12:3(u(x-u(x-x,.1)lf;(x)which constitutes a piecewise-defined function, as illustrated in Figure 8-6. The properties of such a function maybe deduced in a straightforward fashion from those of u(x - a)f(x) as discussed throughout this chapter.;0y44%if 4' 8:13 COGNATE FUNCTIONSThe value of the unit-step function at the point where the step rises has little effect on the properties of the function.Hence, the function defined by 8:13:1u(x-a)=r0x < a 1xi?ais very similar to u(x - a): it may be called the alternative unit-step function and occurs in the expansions of thefunctions discussed in Chapter 9. CHAPTER9THE INTEGER-VALUE Int(x) ANDFRACTIONAL-VALUE frac(x) FUNCTIONS The two discontinuous functions of this chapter satisfy the relationship9:0:1 Int(x) + frac(x) = xand have their main utility as modifiers of other functions. Therefore, in addition to the simple integer-value andfractional-value functions themselves, this chapter will also discuss f(Int(x)), f(frae(x)), lnt(f(x)) and frae(f(x)), fbeing some unspecified function. 9:1 NOTATIONThe integer-value function is also symbolized [x], E(x) or int(x). The fractional-value function is sometimes sym-bolized (x). See Section 9:13 for the very similar integer-pan and fractional-part functions; these are sometimesalso denoted Int(x) and frac(x). 9:2 BEHAVIORFigures 9-1, 9-2, 9-3 and 9-4 depict maps of f(Int(x)), f(frac(x)), Int(f(x)) and frac(f(x)). They are sketched fora function f that is monotonic and continuous in the region graphed: a monotonic function has no maximum orminimum. Several of the formulas in this chapter are restricted to values of x for which f is monotonic.The functions f(lnt(x)) and Int(f(x)) are piecewise constant [Section 1:13]. Because its segments are of equalwidth, the f(Int(x)) function has a graph that resembles a histogram.The f(frac(x)) and frac(f(x)) functions assume values in restricted ranges. Thus, provided f is monotonic inthe 0 s x s 1 range, f(frac(x)) takes values between f(O) and f(l). Similarly, 0 s frac(f(x)) < 1. 9:3 DEFINITIONSThe integer-value function extracts from its argument the largest integer that does not exceed the argument; thatis:9:3:1Int(x)=nn_x<n+1n=0,±1,±2,... 71 9:3THE INTEGER-VALUE lnt(x) AND FRACTIONAL-VALUE frac(x) FUNCTIONS 72 4*4i 14404* FIG 9-1 44x44x14404414? FIG 9-2 ff (froc (x)) f(x)so that 9:3:2andf(Int(x)) = f(n)nsx<n+ In=0, ±1, ±2,... 9:3:3 Int(f(x)) = n f monotonic n s f(x) < n + I n = 0, ± 1, ± 2,...The fractional-value function is defined by9:3:4and therefore9:3:5frac(x) - x - nn sx<n+1n=0,±1,-t 2... f(frac(x)) = f(x - n) 0 5 x - n < I n = 0, ± 1, t2, . ,. 73THE INTEGER-VALUE Int(x) AND FRACTIONAL-VALUE frac(x) FUNCTIONS and9:3:6frac(f(x)) = f(x) - nf(x) - nf monotonic0:5 f(x) - n < 1n = 0, =1, t2, ... 9:4 SPECIAL CASES9:4 When f(x) is the linear function, bx + c [Chapter 71, f(Int(x)) and Int(f(x)) are similar in shape and each is knownby the descriptive name staircase function. Equally descriptive is the name sawtooth function given to bfrac(x)+ c or frac(bx + c). Figure 9-5 depicts this pair of functions. .....................-3 FIG 9-4::f (x) .............................. frac(f(x).).:........0 9:5THE INTEGER-VALUE Int(x) AND FRACTIONAL-VALUE frac(x) FUNCTIONS 74kwFIG 9-5 9:5 INTRARELATIONSHIPSs't'dsawtooth function It follows from equation 9:0:1 that Int and frac are complementary functions in the same sense that 9:5:1and 9:5:2f(Int(x) + frac(x)) = f(x) Int(f(x)) - frac(f(x)) = f(x) The following recurrence formulas apply:... 9:5:3f(Int(x + n)) = f(Int(x) + n) it = 0, =1. ±2, 9:5:4 f(frac(x + n)) = f(frac(x)) it = 0, ± 1, - 2, ... 9:6 EXPANSIONSWhen f is a monotonic function, lnt(f(x)) may be expanded in terms of the alternative unit-step function [Section8:13] 9:6:1Int(f(x)) = i v(x - F(j)) + i [v(x + F(-j)) - I ] f monotonicdx > 0 df 9:6:2Int(f(x)) =(I - v(x - F(j))] -v(x + F(-j))f monotonicdx< 0).ui-where F is the inverse function of f, that is, F(f(x)) = x.Because f(frac(x)) is a periodic function of x, it may be expanded by the methods described in Chapter 36. 9:7 PARTICULAR VALUESFor integer arguments one has the simplification 9:7:1and 9:7:2f(Int(n)) = f(n)n = 0, 21, :t2,. f(frac(n)) = f(0)it = 0, tl, t2, ...If, as before, F is the inverse function of f, then9:7:3Int(f(x)) = it = 0, t 1, t2, ... x = F(n)f monotonicand 9:7:4 frac(f(x)) = 0 x = F(n)f monotonic n = 0, t 1, t 2, ,,. 75THE INTEGER-VALUE Int(x) AND FRACTIONAL-VALUE frac(x) FUNCTIONS9:8 NUMERICAL VALUES9:10 Many calculators and computers are able to implement the Int(x) and frac(x) functions directly. Beware, however,that such operations are not actually implementing the "integer-part' or "fractional-part" functions discussed in Section 9:13, rather than those defined in Section 9:3. In view of relationship 9:13:1, the rounding facilities ofother calculators can be adapted to give Int(x). In yet other devices "shifting" to the right or left may be used togenerate Int(x) or frac(x), respectively. If all else fails the short algorithm Input x >> f = frac(x)Setf=0.5A»»If cos( 180x) = 0 go to (I )Setf = arctan(tan(180x))/180Iffa0goto(1)Replace f by I + f(1) Output fK««Storage needed: f, x1Use degree mode or change 180 to rr.Test values:frac(3.5) = 0.5frac(1.7) = 0.7frac(- 1. 7) = 0. 3 may be used to find frac(x). The difference x - frac(x) then gives Int(x). This algorithm uses functions fromChapters 32, 34 and 35 with the identity 9:8:1 arctan(tan(a[n + x])) _axp<x<-2n=0,±I, t2,...a(x-1)2<x51The commands shown in green guard against overflow; they may be omitted with many computing devices. 9:9 APPROXIMATIONSThere are none that are sufficiently useful to be included here. 9:10 OPERATIONS OF THE CALCULUSWith F designating the inverse function to f, as before, the following formulas indicate how the functions Int andfrac affect differentiation: 9:10:1 9:10:2 9:10:3 9:10:4d-f(Int(x))=0x*0.±1,±2,... ddf-f(frac(x))=-(x-n)n5x<n+ln=0,±1,±2, dxdxddxInt(f(x)) = 0x t- F(0), F(= 1), F(!'2), ... dfrac(f(x) - d (x)x * F(0), F(±1), F(±2), .. ,Integration leads to formulas that are more complicated:r X-19:10:5Jf(Int(r))dr = (n + I - a)f(n) + (x - N)f(N) + f(j)n = Int(a)N = Int(x) 9:11THE INTEGER-VALUE Int(x) AND FRACTIONAL-VALUE frac(x) FUNCTIONS 76 9:10:6J1 f(frac(t))dr=JQpf(t)df + (N - n - 2)Jf(t)dr + Jf(r)dtn = Int(a)N = lnt(x) J.o 9:10:7X-A({Xr-Aa-E F(A + j)f monotonicdf> 0JInt(f(t))d1 - dxA-x d.Xx - Aa +F(X + j)f monotonicdx< 0i_ where A = Int(f(a)). X = Int(f(x)) and F is the inverse function of f. Finally, by virtue of 9:5:2 9:10:8 f frac(f(t))dr = Jtf(t)dr - JFInt(f(t))dt 9:11 COMPLEX ARGUMENTThe integer-value and fractional-value functions are not defined for imaginary or complex arguments- 9:12 GENERALIZATIONSThe function f(frac(x)) is an example of a periodic function (see Chapter 36]. Its period is unity. Likewise f(Int(x))and Int(f(x)) are examples of piecewise-constant functions [see Section 1:131. 9:13 COGNATE FUNCTIONSThe rounding function (or nearest-integer function) (x) is defined as the integer nearest to x, with rounding upwardto n + I in case x = n + }. It is related by9:13:1 (x) = Int(x + 1/2)to the integer-value function.The integer-par: function Ip and fractional-part function Fp are very similar to the integer-value and fractional-.value functions. In fact, when x is positive, Ip(x) = Int(x) and Fp(x) = frac(x). The distinction for negativeargument is evident from the following example:9:13:2 lnt(-2.34) = -39:13:3 Ip(-2.34) _ -2 = I + Int(-2.34)9:13:4 frac(-2.34) = 0.669:13:5 Fp(-2.34) = -0.34 = -1 + frac(-2.34)This Atlas makes no use of the integer-part or fractional-part functions.The modulo function (or remainder function) is a bivariate function of two integer variables. This function.denoted n(mod m). is the remainder upon subtraction of m from n as many times as possible without leaving anegative difference. The modulo function is related to the fractional-value function by 9:13:6-1 11'x-N ln(mod m) = m frac-!mn/ Occasionally the symbol n(m) is encountered and the name modulus is used to describe m. 77THE INTEGER-VALUE Int(x) AND FRACTIONAL-VALUE frac(x) FUNCTIONS9:14 RELATED TOPICS9:14 The integer-value and fractional-value functions occur widely in number theory. In addition they lie at the heartof such devices as analog-to-digital convertors that digitize measured signals of various kinds. They also play rolesin the conversion of a number x from one number system to another; from the decimal system to the binary, forexample, or from the hexadecimal to the decimal.Any positive number can be represented as 9:14:1 X 0"where n is an integer and each N. takes one of the integer values 0, 1, 2, ..., (p - 1), where P is the base of thenumber system and where N. * 0. Such a representation is termed floating point or, if 0 = 10, scientific notation. An alternative is the fixed point representation9:14:2 where it is a sufficiently large integer and, as before, each N is an integer drawn from the set 0, 1. 2. , (R- 1).The algorithm below digitizes any positive number x by generating a sequence of integers. The first integer isn, a decimal integer of either sign. The other integers are digits of the base 02 number system. These may beinterpreted as the No, N_ N_2, ... digits of the floating point representation 9:14:1, or as the N. N,_,,digits of the fixed point representation 9:14:2. Input (32 >Input x >> n <<<<< digit <<<Set n = 0(3) If Int(x) * 0 go to (4)Replace x by 32r Replace it by it - 1 Go to (3)(4) Replace x by x/32If Int(x) = 0 go to (5)Replace n by n + 1Go to (4)(5) Output n (6) Replace x by 32xOutput Int(x) Replace x by frac(x)Ifx=0stopGo to (6)Storage needed: 02, x and n Intervention to halt output is needed if x is recurrentin base 32. Test values:x= a,02=2Output: n = 1followed by the digits1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0 The second algorithm accepts a number in base R, representation and converts it to a decimal number x. The inputis in the form of the signed decimal number n followed by a sequence of digits in base 31. All these digits must,of course, be positive. Following the entry of the last true digit a negative digit is input as a cue that the data entryis complete. This negative digit is ignored in the computation of x.The two algorithms are designed so that they can be combined, with the second preceding the first. The con-joined routine will then convert any positive number from base B, to base S2.I 9:14THE INTEGER-VALUE Int(x) AND FRACTIONAL-VALUE frac(x) FUNCTIONSInput R, > »»>Set x = 0Input n >> »»Set g = (3,"(I) Wait for inputInput digit N »»IfN<Ogoto(2)Replace x by x + NgReplace g by g/R,Go to (1)(2) Output xx <<<<< <<<<Storage needed:P1, x and g Test values:P,= 16,n=2followed by the digits 7.12,0,8,-1Output: x = 1984.578 CHAPTER10THE DIRAC DELTA FUNCTION S(x - a) This function, strictly speaking, is not a function at all because it violates rules that would be valid if it were afunction. For example, while it takes the value zero at all arguments but one, its integral has the value unity.Nevertheless, the Dirac delta 'function" is a useful concept that has found widespread application, especially inclassical and quantum mechanics, where it occurs mainly as a multiplier of an integrand. as in 10:0:1 10:1 NOTATIONf. S(x - a)f(x)dx The impulse function or unit impulse function is an alternative name for S(x - a).Dirac's name is associated with this function to avoid confusion with the Kronecker delta function [Section10:13]. 10:2 BEHAVIORThe Dirac delta function cannot be graphed: S(x - a) is zero for all values of x except x = a, where it is infinite. 10:3 DEFINITIONSThe Dirac delta function may be defined in terms of a limit in many ways, for example: 1 10:3:1 S(x - a) = limp -;h;x - ae-ohwhere p is the pulse function [Section 1:13], or 10:3:2 S(x - a) - him[-a)2)]J-9 79 10:4 [see Chapter 271, or 10:3:3THE DIRAC DELTA FUNCTION 8(x - a) -Lh=\xca)]8(x - a) = li m[L2cBe [see Chapter 29). All of these definitions-and many others-describe a function peaked at x - a that, as the limitis approached, becomes infinitely high and infinitesimally wide but whose area remains constant and equal to unity. Figure 10-1 illustrates the progress toward the limit in the case of definition 10:3:3. It follows that 10:3:4 J1x8(r - a)dt = I xo < a < xi Another interpretation of the Dirac delta function is as the derivative 10:3:5d8(x-a)-u(x-a) of the unit-step function [Chapter 8]. Yet another representation is as the definite integral 10:3:6 8(x - a) = Jycos[2irr(x - a)t]dt which is important in the evaluation of certain Fourier transforms [see Section 32:14]. 10:4 SPECIAL CASESWhen a = 0, the 8(x - a) symbol is replaced by 8(x). 81 THE DIRAC DELTA FUNCTION 8(x - a)10:5 INTRARELATIONSHIPSOne may derive the reflection formula10:5:1and the multiplication formula8(a - x) = 8(x - a) 10:5:2 8(vx) =8(x)- .10Vfrom the definition of the Dirac delta function. Another result is 10:5:3 10:6 EXPANSIONS There arc none.8(x2 - a2) =2a[8(x - a) + 8(x + a)) a > 0 10:7 PARTICULAR VALUESIf the Dirac delta function is interpreted conventionally, then10:7:18(x-a)=0xa10:7:28(x-a)=xxa 10:8 NUMERICAL VALUESNo finite nonzero value attaches to S(x - a) for any value of x. 10:9 APPROXIMATIONS10:10 No approximation of 8(x - a) itself is of any use. However, expressions 10:3:1-10:3:3 can sometimes be of valuein approximating integrals in which 8(x - a) appears as a factor of the integrand. 10:10 OPERATIONS OF THE CALCULUSDifferentiation of the Dirac delta function produces the unit-moment function discussed in Section 10:12. Integrationgives the unit-step function (Chapter( 81 10:10:1f+&(t-a)dt=u(x-a)-x5xo<aand, with f any function, 10:10:2 of which a special case is 10:10:3 S(t - a)f(t)dt = f(a)r.8(t - a)f(t)dt = u(x - a)f(a) 10:11 THE DIRAC DELTA FUNCTION S(x - a)It is this last relationship, known as its sifting properh', that renders the Dirac delta function so useful.The notation f(x) * g(x) and the definition 10:10:4f(x) * g(x) = J=f(t)g(x - r)dr = J=f(x - t)g(t)dt82 relate to the so-called convolution of the functions f and g. The convolution of two Dirac delta functions obeys therule 10:10:5 6(x- a) * S(x - b) = S(x -a - b) 10:11 COMPLEX ARGUMENTThe Dirac delta function of complex argument S(x + yi - a - bi) is nonzero only when x = a and y = b. Itobeys 10:11:1 JJ&(x+yi-a-bi)dxdy= Ias well as other relations that parallel its behavior as a function of a real argument. 10:12 GENERALIZATIONSBy utilizing the concept of differintegration [Section 0:10) it is possible to define a continuum of functions of whichthe unit-step function and the Dirac delta function are respectively the v = 0 and v = 1 instances. The generaldefinition is 10:12:1which evaluates tod"[d(x + x)]°u(x - a) 10:12:2u(x-a)(x-a).v * 1,2,3....r(1 - v)except when v is a positive integer. The v = 2 case, symbolized S'(x - a), may be regarded as the limith\\/hllS x-a--l-8(x-a+2/210:12:3 S'(x - a) = lim //\/ *-ahand is named the unit-moment function. It satisfies the integral identityr10:12:4 J aS'(r - a)f(t)dt = -F(a)df_ - dx(a)f The Dirac delta function may be regarded as a bivariate function of the variables x and a, each of which can adoptany real value: it is zero except when these two variables are equal. The Kronecker delta function b(n,m) or 5,,,,,is an analogous bivariate function but its two variables are restricted to integer values. It is defined by0n*m 10:13:1 S(n,m) =1n = m CHAPTER11THE INTEGER POWERS (bx + c)" AND x" With n = 0, ±1, ±2, ... this chapter concerns the function (bx + c)" and its special b = 1, c = 0 case. Thepowers 1, x, x2, ... and the reciprocal powers 1, x-1,x-2,... are the units from which most expansions are built;such expansions are the subjects of Section 11:14. 11:1 NOTATIONThe two symbolisms (bx + c)-" and 1/(bx + c)" are equivalent in all respects. The powers x2 and x' are termedthe square and the cube of x, respectively.In the general notation P°, [i is known as the base and a as the power or exponent. In this chapter [and inChapter 13] the family of functions in which the base is the primary variable is treated: In contrast, Chapter 19 isconcerned with functions in which the exponent varies and the base is held constant. 11:2 BEHAVIORThe power function is defined for all values of x and for all integer n except that x" is undefined when both x andn are zero. Figures 11-1 and 11-2 illustrate the behavior of x" for n = 0, ± 1, ±2, ±3, ±4, ±7 and ±12. Noticethe contrasting behavior of the positive and negative powers. Note also how the reflection properties depend onwhether it is even or odd. 11:3 DEFINITIONSThe function (bx + c)" for n = 1, 2, 3, ... is the product of n factors, each equal to bx 11:3:1 n=1,2,3,...(bx+c)"= 11(bx+c)i_IWhen n is a negative integer, the definition- is the quotient 11:3:2 (bx + c)" = 1 1+ c: 83 11:4 THE INTEGER POWERS (bx + ci' AND f 84'LO0COtiA, A e?,ea .,.:. :..jlx0.2X12 12: :....:......;/ ..:.. /...:....:... .......:....:.-0.2 :....:....:....:....:....:.........:.-0. 4FIG 11-1 These two definitions are supplemented by11:3:3 (bx+c)"= 1n=0Alternatively, the recurrences 11:5:3 and 11:5:4, coupled with 11:3:3, may be regarded as the definitions. 11:4 SPECIAL CASESWhen n=0:11:4:1 (bx + c)° = Ix # -c/band when it - ± 1, reduction occurs to the functions treated in Chapter 7.When b = 0, (bx + c)" reduces to a constant [Chapter 1] for all values of c and it. 11:5 INTRARELATIONSHIPSThe function x" obeys the simple reflection formulaxn = 0, ±2, ±4, ...1-x"n = ±1, ±3, ±5.... as THE INTEGER POWERS (bx + cy' AND x' 11:5For the (bx + c)" functions, reflection occurs about x = -c/b:/\1 11:5:2+c+xJ+c]The recurrences11:5:3and 11:5:4(bx + c)" = (bx + c)(bx + c)"-' (bx + 0-' =(bx + c)-'bx + capply, as do the laws of exponents (x")(x') = x"", x"/x' = x"-", (x')' = x"" and their extensions 11:5:5 11:5:6and 11:5:7(bx + c)'(bx + c)' = (bx + c)"" (bx + c)"/(bx + c)' = (bx + c)"-' [(bx = c)"]' = (bx + c)"" 11:6 THE INTEGER POWERS (bx + c)" AND x"The following finite products:12jn 11:5:8x"ty(x+y)711IX2 :t 2xy cos(+y=rin 11:5:9z +})+}xIn2z 11:5:10x"-y"=(x + y)(x- y)fl[x-2xycos()+,n/n= 1,3,5,...=2J+ 1n=2,4.6,...=2Jn=2,4,6,... =2J+2186 constitute function-addition and function-subtraction formulas for integer powers, cos being the cosine function ofChapter 32. Simple instances are x2 - y2 = (x + y)(x - y), x3 t y3 = (x t y)(x + xy + y2), x` + y` _ (x +f2ry+y2)(x- V2xy+y2)andx4-y` = (x + y)(x - y)(x2 +y2).Finite sums of positive or negative powers may be evaluated as the geometric sums 1 1 : 5 : 1 11 j x + x2 ++ x"-' + X. =n = 1, 2, 3,...1-x or 11:5:12 1 + x-' + x-2 + ... + x'-" + x n = 1, 2. 3,...x-1 11:6 EXPANSIONSIf n is positive, (bx + c)" may be expanded binomially as the finite sum 11:6:1(bx+c)"=c"+nc"'bx+ cj n(n2.1)(J)(bx)i n=0.1,2,...for all x, where (')is the binomial coefficient of Chapter 6. If n is negative, the sum is infinite and takes the form 11:6:2 (bx + c)" = c" + nc"-'bx +n(n - I) 2!e-262x-+... c"j-;-1)(-bx)n=-I,-2,-3....1< or 11:6:3 j-n-11c2! bxn=-1,-2,-3....[rj> depending on the magnitude of x. Section 6:14 presents some specific examples.Positive integer powers may be expanded in terms of Pochhammer polynomials [Chapter 18] 11:6:4x"=io;i'(x-j+l),=i(-I)"Q'(-x)j n=0,1,2,.../-o J-oclb cb where o4 is a Stirling number of the second kind [Section 2:14], or in terms of Chebyshev polynomials of thefirst kind (Chapter 22]: 87 11:6:5THE'INTEGER POWERS (bx + c)" AND x" 11:9 "rl'" x +U' T _ x +++n=0:2 4:_1 = Y :Y.-z ")-Y.'-'.T.-4(X)Y i 1t(x)n =13',5..._ 2 y",T,(x)J=0The coefficients y"' are zero whenever j and n have unlike parities. The coefficient yo' equals unity, while y." iszero for m > n. The numerical values of all other coefficients may be calculated from the recursions Yo' _y('-0/2, y(') = (,y('-')/2) + yo-" and for m ? 2 y." = (y."_-." + Expansions similar to 11:6:5, butinvolving the Chebyshev polynomials of the second kind U"(x), also hold. In fact, such expansions exist with anyset of orthogonal polynomials [Chapters 21-241. 11:7 PARTICULAR VALUES For b * 0 one has Ibx + cf. n = -2. -4 -6....(bx+c)".a=-1,-3.-5..-.(bx+c)'(bx+c)".n- 1.3.5....Ibx + c)", n = 2. 4. 6, ... 11:8 NUMERICAL VALUESc - 1-c_ -I - c X --=x Xbbx - -bx . x 0l 100-I= 10 I 1 1x-I0 1x= 10 1= These arc readily calculated using the "y'" key present on most programmable calculators or using the correspondingcomputer instruction. Where such a facility is absent, the simple algorithm below may be employed. Input n >>Input x >> f=X,<<SetfIfn=0goto(2)Ifn>0goto(I)Replace x by I /xReplace n by -n(1) Replace fby xfReplace n by n - 1Ifn*0goto(I)(2) Output fStorage needed: n, x and f Input restrictions: n must be a nonnegative integer butx is unrestricted. Test values:(1.5)' = 17.0859375(1.5)"' = 0.0585276635 This algorithm. which is based on recurrences 11:5:3 and 11:5:4, is exact. 11:9 APPROXIMATIONSIfn*0:/11:9:1(bx + c)" = c"I l +.rl8_bit precisionxI <c.Icl12n(n - 1)IbI1 11:10 THE INTEGER POWERS (bx + c)" AND j'v n(Icy11:9:2(bx + c)" = b"" I\ I +ncbx)8-bit precisionxt >12ibi 11:10 OPERATIONS OF THE CALCULUSDifferentiation and integration of the integer powers are easily accomplished:88 11:10:1 d(bx+c"= (bx + c)"-'n=0.1.2....11:10:2(bt + c)"dt =xb(n + 1)n=-1.-2.-3. rxn = -1, 0. 1. 2.11:10:3JOr + c)'dt = (bx + c)n-2.-3.-4,b(-n - 1) "(bx, + c)"'' - (bxo + c)"n = 0, 1, ±2,±3....b(n + 1)(bt11:10:4J+ c )'dt =1bx, + cbxo+c)bn= -1 Differintegrals of the simple powers x' are given by the formulanom-' 11:10:5 dx'f(n-v+l)n=0,1,2....where r is the gamma function of Chapter 43. 11:11 COMPLEX ARGUMENTForn= 1,2,3,... (x + iy) = x" -n(n 2!- 1) x"-'y2+n(n - 1)(n5!- 2)(n - 3) rn(n - 1)(n - 2)n(n - I)(n - 2)(n - 3)(n - 4)+ i nx"''y - X. 'y' + x" `y' -3! 4!(n)()2)k(n )(-?) where each upper limit in the summation is an integer chosen to make the final binomial coefficient (:) orSome examples are 11:11:2 (x + iy)2 = (x2 - y2) + i(2xy)11:11:3 (x + iy)' = (x' - 3x 2) + i(3x2y - y')11:11:4 (x + iy)4 _ (x' - 6x2y2 + y') + i(4x'y - 4xy3) 89 THE INTEGER POWERS (bas + c)' AND xThe corresponding formulas for negative n involve infinite sums 11:11:5i[?]* (x+iy)"=x'(2k-- 1J ) r ? 1_-,y j(1k2k k-0+1) LJ xAn example is 13y'5v'/2- ... I 11:11:6(x + iy)-x = (X-2 - x +xbxr-ss +jr,11:12 GENERALIZATIONS11:14 In the present chapter the power is restricted to be an integer. This restriction is removed in Chapter 13. 11:13 COGNATE FUNCTIONSAny polynomial, including all the functions discussed in Chapters 16 through 24. consists of a finite sum of thefunctions of this chapter. 11:14 RELATED TOPICSMany functions of x can be expanded as an infinite series: 1 1 : 1 4 : 1 f(x) = ao + a,x + axxx + _ i a,x'I-oof terms of which ax' is typical, j being a nonnegative integer and a, a constant. Such sums are known as powerseries and examples will be found in Section 6 of most chapters of this Atlas. In the notation of Chapter 17, powerseries are polynomial functions of infinite degree, and could be symbolized p.(x). The somewhat more complicatedseries 11:14:2 aox° + a,x°'B + a,X'-' + a1.t-omay also be treated as a power series because a redefinition of the argument and isolation of the factor x' relates11:14:2 to the 11:14:1 function 11:14:3 Za,x°la=xf(xe)i-oDepending on the values of the coefficients a,, power series may converge for all x, only over a certain rangeof argument values. or may represent an asymptotic expansion near some x = xo value [see Section 0:6]. Providedthat a power series converges for argument x, it may be differentiated term by term:d11:14:4 1)a;.,x, dx,-oor integrated: 11:14:5Jdt=o i-ai-o ji-ijto yield another power series.Power series may also be raised to a power 11:14:6 whereco=ao,c,=(jn-kn-k)a,_,c,j= 1,2,3,.,.oo jao t-o 11:14 THE INTEGER POWERS (bx + c)' AND x"Similarly, two power series may be multiplied: 11:14:7 wherec,Y, akbj_2 or divided: 11:14:8a1x a1'bx,'-a"X'whereci =bobok-0bi-icA.i1-090 If f is given by the power series Ea,x', then the inverse function (see Section 0:3] is another power series ofargument If - ao)/a namely 11:14:9 c1(fsoa2where c, = a,, c2 = -a,a,, c, = 2ala22 - alla,, c. = Sa a2a, - Sa,a; - a3la., c, = 14a,a2 + 3aja3 +21 a;a';a, - a;a, and ce = 84a;a, + 7o;a,a. + 7a;a a, - 42a,a2 - 28a3,a - 28a3a2'a, - a;ab. The procedureof converting a power series for f(x) into a power series for x is known as reversion of series.Any function that can be repeatedly differentiated may be expanded as a power series by utilizing the Maclaurinseries (the special y = 0 case of the Taylor series 0:5:1]:' dJf11:14:10 f(x) = f(0) + xdf(0) + -d2f(0) +=x(0)dx2T21_o j! dx'Whether or not such a series is convergent, it provides an asymptotic representation of f(x) as x -a 0. Functionsexpansible as power series may also be represented as series of Bessel functions [see Section 53:14] or as continuedfractions [Section 0:61. CHAPTER12THE SQUARE-ROOT FUNCTIONbx + c AND ITS RECIPROCAL Functions involving noninteger powers are known as algebraic functions. The square-root function V and thereciprocal square root 1/Vx are the simplest algebraic functions. In this chapter, as in the previous one, we gen-eralize the argument of these simplest algebraic functions and consider mainly the bx + c and 1/vi"" + c func-tions.A graph of 'V bx + c versus x generates a curve known as a parabola and the adjective parabolic is thereforeappropriately applied to the (bx + c)'r function. Some geometric properties of the parabola are noted in Section12:14. 12:1 NOTATION Especially in computer applications V' is sometimes denoted SQRT(x) or SQR(x). The notation lx is also oc-casionally encountered.The symbols x"' and V x- often interpreted as defining equivalent functions but in this Atlas we make adistinction between the two. If x is positive x 12 has two values, one positive and one negative. The square-rootfunction f, however, is single valued and equal to the positive of the two x"= values. Hence the relation betweenthe two functions is12:1:1 bx+c=I(bx+c)"'1or12:1:2 (bx + c)"2 = *_ bx +Siillmary 12:1:3 I= Ox + c)-'"I andbx+c 12:1:4 (bx +c)-1"2= 91 12:2THE SQUARE-ROOT FUNCTION bx --c AND ITS RECIPROCAL12:2 BEHAVIOR92 Figure 12-I is a map of the functions V box + c and I /V bT under standard conditions, that is, when b and care both positive. The orientation of bx + cchanges to those shown in Figure 12-2 when b and/or care negative.The Vbx + c function is not defined for values of x more negative than -c/b. The function itself takes allnonnegative values. As Figure 12-I shows, V b x+ is zero at x - -c/b, at which point it has an infinite slope.Likewise, 1/\'bx + c is defined only for bx > -c and takes all positive values. Figure 12-1 reveals that thereciprocal square-root function steadily declines from an infinite value at -c/b toward zero as x - x. FIG 12-1 12:3 DEFINITIONS The square-root function is defined as the inverse of the square function (Chapter I11. Thus, V'bx + c is thenonnegative number whose square equals bx + c; that is: 12:3:1 bx + c = Illwheref2 = bx + cA parabola is defined geometrically as constituting all points P whose distance PF from a fixed point F (calledthe focus of the parabola) equals the shortest distance from P to a straight line DD (the directrix of the parabola).If the x-axis of a Cartesian coordinate system is placed along DF. the shortest line joining the directrix to the focus(see Figure 12-3), and if d is the length of that shortest line, then the equation of the parabola is f = bx + cwhere b = 2k and c = k2 + lay, y being the distance from the focus to the coordinate origin. 93THE SQUARE-ROOT FUNCTION bx T -c AND ITS RECIPROCAL 12:4 SPECIAL CASES When b = 0 bothbx + c and I/V bx c reduce to constants. 12:5 INTRARELATIONSHIPS The multiplication and division of square-root functions generate root-quadratic functions [see Section 15:121: 12:5:1b,x + c,byr + r. _(b,.,, - c,)(b,x + c,) =b,b x1- + (bic: + b.ci)x + c,c,and%+ bx ' + (bbc)x + c lb cc 12:5:2bix+ci/b.r+c,_(b,x+c,)/(b,x+c,)_,, ,.., ,, while the rulebx+c,12:5 12:5:36.r+c)j(bx+c)""n=0,2.4....J(br+c)":In=1.3.5.-..governs the raising ofV bx + cto an integer power.If f(x) = V'bx + c, then the reflected function f(-x) coexists with f(x) only if c is positive and then only inthe range -c/jbI <.r <_ c/Ib{. Within this range the product f(.r)f(-x) is a semiellipse [see Chapter 141: 5:4Vbbx+c=JbI-x2 12: 12:6THE SQUARE-ROOT FUNCTION bx -+c AND ITS RECIPROCAL12:6 EXPANSIONSIf c is positive the expansions/bx62x2b3x 31/ 2) bx' 12:6:1 l12c8c'16x-c ,io \ j` e_\1cc(2j-3)!'bx)'-c5bxsc,-i(2j)!!c I[see Chapter 6 for (;,) and Section 2:13 for n!!] and I1bx3b'x'Sb3x 3 11/2)bx 12:6:2--+ z - z +_bx+c Vc2c8c16rjcJ 1j(2j - 1)!! (-bx)'- csbxsc-o(2j)!!c are valid for small values of (bx/cI, whereas 12:6:3bx + c = V rbs//+cc=-vrbx(I/2(cll2bx862x'+j\ j / \bx/-bxscsbx and 12:6:4I/c3c'l1/ c'(-1/2)c\cVb,2bx bx)94 apply when jbx/cj is large, bx being positive.The square-root function can also be expanded as a ratio of two exponential series [see Section 27:13):-+expl z I+expl2+exp(x12:6:5_\/\/\///-+ exp(-ax) + exp(-41rx) + 2+exp(-j2lrx)both of which converge rapidly. 12:7 PARTICULAR VALUESFor b * 0 12:7:1 12:7:2 andbx+c=0x= bbx+c=I= 1bx+cx= 1 12:7:3 =0bx+c 95THE SQUARE-ROOT FUNCTION bx + c AND ITS RECIPROCAL 12:1012:8 NUMERICAL VALUES Computer languages provide instructions for implementing the square-root operation. Programmable calculators,indeed, even nonprogrammable ones, usually have keys that generate N /X. 12:9 APPROXIMATIONSIf p is an approximate value of Vx, then (p2 + x)/2p is a better approximation. This is the basis of Newton'smethod for estimating square roots.The first four equations of Section 12:6 lead directly to the approximations b1291 x++8 b=V-biii<c1b :: cx -t precsonc x12c 1292-rI-bx)8bitiic1b1 < ::''--precsonV1\bx+cV cx10 1293+Vbb+8biiib> 71 1 ::x -t precson. xc 12492Vb; 1_l(cbiii 1l- c8> 101bi :: ont precs-xc bx+Vbx 12:10 OPERATIONS OF THE CALCULUSDifferentiation of the square-root and reciprocal square-root functions givesdb12:10:1 - bx + c =dx2br+candd1-b 12:10:2 dxbr + c2(bx + c)3i:while the corresponding indefinite integrals may be written bt + c dt =236(bx +C))/, 12:10:3f,band 12:10:4 The related indefinite integralsf-"I-'Vbtdr+c2bx+cb 2(3b12:10:5 tbt + c dt =1x56-`2c)(bx+ c)31 12:11THE SQUARE-ROOT FUNCTION bx + c AND ITS RECIPROCAL % and 12:10:62arcoshc > 0d t7bxc/bfbl+C2farccos C < 0CVbxare two examples drawn from a large class of integrals of the general function (bx + c)" 2(Bx + C)"/2 in which atleast one of the integers n and N is odd. A list of such integrals may be found in Bronshtein and Semendyayev (pages 423-4241 and a longer list in Gradshteyn and Ryzhik [Sections 2.21-2.24].Semidifferentiation using a lower limit of zero yields d'/=cbI 12:10:7bx+c=+d,>0 dx''2axTr and 208dl/'I 10 1:1:12dxbx+cbx+c ,ac> The function d' in 12:10:7 is given by 12:10:9d _b > 0x > -c/b b < 0x < -c/b and appears also in the expressions for the semiintegrals of the two functions, again with zero lower limit: d- '1,2b+ 200 b+cxxc+>0 1:1:1 xds 7Fc d-'/212 12:10:11 c > 0 dx-'/2bx + csIbl 12:11 COMPLEX ARGUMENT When the real argument x of the square-root function is replaced by a complex argument x + iy. the functionbecomes complex valued:+X)r+)'* -X11121+ i=+i ::xy2sgn(y)2 Similarly: 1x +y +s1777 12:11:22- sgn(y)i22 x + iy2(x+?)2(x+ y) 97THE SQUARE-ROOT FUNCTION bx + c AND ITS RECIPROCAL12:12 GENERALIZATIONS12:14 The functions discussed in Chapter 13, being fractional powers of x, generalize the functions x-rn, which are thesimplest fractional powers. 12:13 COGNATE FUNCTIONSThe functions (bx + c)'3/2, (bx + c)"5/2, (bx + c)-`, .. have properties similar to those of (bx + c)r"2. 12:14 RELATED TOPICSA very useful property of the parabola may be illustrated by reference to Figure 12-3. If, as shown, the horizontalline DP is extrapolated to E, then the lines FP and EP make equal angles with the parabolic curve at P. Thus, ifthe parabola represents a mirror and EP a ray of light, the ray will be reflected towards the focus F, irrespectiveof the position of P. This "focusing" principle lies behind the parabolic design, for example, of telescopes, search-lights and radar antennas.The area bounded by the parabola ±\,i c and an arbitrary x ordinate, as delineated by the shading in Figure12-4, may be found with the aid of integral 12:10:3; thus: 12:14:1 rshaded area = 2Jbt + c dt =4(bx + c)'/'=b 3b ,044FIG 12-4 The curved portion of the perimeter of the shaded area is found from(d12:14:2curved perimeter = 2 1 +`-bt + c Idtf,.,b /2\/c b_\/bb b)This length is to be increased by 2V'bx + c, the length of the x ordinate, to give the total perimeter of the shadedregion in Figure 12-4. CHAPTER13 THE NONINTEGER POWERS x" Relationships in science and engineering frequently involve fractional exponents. In this chapter we deal with thepower function x', where v is unrestricted. 13:1 NOTATION When v is the reciprocal 1/n of a positive integer other than one, the symbol is sometimes used to mean x'or k'i, but we employ only the latter notations in this Atlas. The names cube-root, fourth-root, fifth-root, etc., aregiven to x1/3, x1/4, xit3, etc. The symbols x-' and 1/x' represent identical functions. 13:2 BEHAVIORThe behavior of x' is affected by whether or not the number v is rational. If v is irrational (that is, incapable ofbeing expressed as the ratio m/n of two integers), x" is defined only for x ? 0 and takes positive values only. Inother words, a graph of x" versus x lies entirely within the first quadrant [see Section 0:2 and Figure 0-1 for adefinition of the four quadrants). Two irrational examples, x° and x°t4, are plotted in Figure 13-1. Notice bycomparing this figure with Figure 11-2 that the x° curve ties roughly where one would expect: between the x3and X-4 curves, but closer to the former.If v is a rational nonzero number it may, by definition, be equated to m/n where n is a positive integer andm is a nonzero integer of either sign. When, by cancellation of any common factors, the m/n fraction has been reduced to its lowest terms, it can be classified into three cases according to the parities of m and n: even/odd,odd/odd and odd/even (any even/even fraction must be reducible by dividing the numerator and denominator by 2 a sufficient number of times). We use 1, i and j as typical examples of these three classes of v values and displaythe corresponding x" graphs in Figure 13-2.When m is even but n is odd, x'"l" is defined for all arguments x but takes positive values only since it is thesquare of the real function x'1'". Hence the graph of x" versus x occupies the first and second quadrants.The function x" = x'1" with m and n both odd is defined for all values of x and takes the same sign as x.Hence, this type of x" graph ties wholly within the first and third quadrants.When m is odd and n even, the x''l' function is restricted to nonnegative arguments but adopts both positiveand negative values; that is, it is a two-valued function. Accordingly, its graph occupies the first and fourth quad-rants. 99 13:2 THE NONINTEGER POWERS x" 100 101 THE NONINTEGER POWERS x'13:3 DEFINITIONS13:5 When v is the reciprocal of an integer n greater than unity, the definition of x" relies on the concept of an inversefunction [Section 0:3]. Thus, for each x, P. equals any real number that generates x when raised to the power n;that is:13:3:1x11" =fwheref" - xn =2,3,4,...This condition is satisfied by a unique real value off when n is odd. (In addition it is satisfied by other n - Icomplex values off, as explained in Section 13:14.) When n is even. 13:3:1 is satisfied by exactly two real numbersf (equal in magnitude but opposite in sign) when x is positive, but by no real number at all when x is negative.(Once again a supplementary set of complex values off exists, giving a total of n solutions to 13:3:1.)Rational powers other than x'/" are all expressible as x"/" and are defined as the m" power [as in Chapter I I ]of x'1"; thus:13:3:2 X.I.= (.x'/")"m = ±2, ±3, ±4, ...This definition applies irrespective of the parities of m and n.When v is irrational, the definition of x' presents greater difficulties because it must be expressed in terms ofa limiting operation. Let m,/n m2/n,, m3/n7, ..., m//n/, ... be fractions that constitute ever more accurate ap-proximations to the irrational number v. Then x' is defined as13:3:3 .r' = lim X"""For example, x' could be defined as the limit of a sequence of positive numbers of which the initial members arex3 x31110 x157/50, x157/500' X3927/130. 311159/100000.-- 13:4 SPECIAL CASESThe values v = s1 and ±1 correspond to x' functions that are discussed in Chapters 12 and 6, respectively. Otherinteger powers are treated in Chapter I I. 13:5 LNTRARELATIONSHIPSIf v is irrational, the function a' is defined only for x ? 0 [as is explained in Section 13:21 and no reflectionprinciple is possible. The same is true when v = m/n and n is even. For odd n, however, one has13:5:1(-x)"/"=(-1)"x"/"m=±1.±2. ±3. ...n= 1.3.5,...The following laws of exponents hold for all values of v and µ: 13:5:2 X'T' = x''"13:5:3 r'/r° = r'-'and13:5:4 (x')' = r"'For x > 0 these formulas apply irrespective of v and µ. but when x < 0, application requires that all the functionsinvolved by well defined.The asymptotic expansion!J'"J"vJ'-'Iv' - 3v2 - 2v)J'-3(-v)# 13:5:5 j b(-v)+lyv+ *12-720+.k!B;J'''+v*-I rvalid for large J, permits finite (or infinite if v < -1) sums of arbitrary powers of the natural numbers [see Section1:14] to be expressed in terms of the zeta function [Chapter 3]. Pochhammer polynomials [Chapter 181, factorials 13:6 THE NONINTEGER POWERS x"[Chapter 21 and Bernoulli numbers (Chapter 4). The related summationsiif.,J.vi. -I-(-v), 13:5:6Z(j +u)C(-v;l+u)+I+v+2+ 12k1BtJ+...v*-I i-Iand 13:5:7(j + u)"x' = $(x;-v;u) - x'(P(x;-v;J + U) v * 0, -1, -2, .. ./=0involve the Hurwitz and Lerch functions [Chapter 64 and Section 64:12). 13:6 EXPANSIONSThe series102 13:6:1x'=[I+(x-l)1'=I+v(x-l)+v(v21) (x-1)`*0<x<2 where (,) is the binomial coefficient [Chapter 61 terminates only if v is a nonnegative integer.Unless v is a negative integer jr' may be expanded as an infinite series of Bessel functions via 53:3:1 and53:14:5. 13:7 PARTICULAR VALUESThe following special values obtain: vnofln=1,3,5,...Jn-l, -2. -3,0(-1)" I0 v- undefundef=_ 10aln=2.4.6....(rational v < 0 undefundef 10 -00 1 Iundef IIrrational v > 0 undefundef0 1nfm =1,2.3_]vJ Aln2,4,6....underundef0±1m n123...v=-.,.(-1)"s(-1)"0 1nn-1,3,5,... 13:8 NUMERICAL VALUESComputer language instructions such as x ' v or permit values of x" to be found quickly and easily, as dokeys on most programmable calculators. Such operations often accept only positive x values and yield only first-quadrant [see Section 0:21 values of x". Care is therefore required to interpret the result correctly when v is rational.Of course, the reflection formula 13:5:1 may be used to calculate x' when x is negative and v is a suitable rationalnumber. 103 THE NONINTEGER POWERS x' 13:10Alternatively, the following simple algorithm, based on the formula13:8:1 x' = exp(v In(x))[see Chapters 25 and 26 for the In and exp functions], may be used.Input x »» Input v »»f=x'««<Setf = ln(x) Replace f by exp(vf)Output f 13:9 APPROXIMATIONSStorage needed: v, x and fInput restrictions: The argument x must be positivebut v is unrestricted. Output values are positive.Test value:(3.7)-"' = 0.456119842 When v is small and x is close to unity: 13:9:1x" - I +2v(x - 1)8-bit precisionIvI s 0.20.7 L x s 1.4 .x + IThe right-hand side results from approximating the In and cxp functions in equation 13:8:1. 13:10 OPERATIONS OF THE CALCULUS Differentiation gives 13:10:1 X. = vx'-dxwhile indefinite integration produces rx" 13:10:2It'dt=-v> - lx >0v+land 13:10:3 1(-t)'dt =i-( -.t)"v+1v< - Ix <0 Equation 13:10:2 is restricted to v > -1; otherwise we have 13:10:4 J`t'dt=1n(x)v=-I and 13:10:5 It'dt = - v < - Iv+1I A general expression for the indefinite integral of a power function multiplied by a linear function raised to anypower involves the incomplete beta function [see Chapter 58]: 13:10 THE NONINTEGER POWERS x" 104 13:10:6c-'.' bxB v+ l;-µ-v+1;b""bx+c t"(bt + c)"dt ='cBv+l;µ+):-bxb<0 (-b)"" cSemidifferentiation with lower limit of zero or minus infinity results in 13:10:7 or 13:10:8d'"'_r(v + 1)5T'n-r(v+'lx,._v>-1x>0z) d"2 2_[d(x + x)]r(-v) V < -1/20<bx .<cc > 0v>-<cz x<0 respectively. Here r denotes the gamma function [Chapter 43). Similarly, semiintegration producesdr(v + 1)dx-'/-x' =r(v + 3h)x""1'v>-1x>0 d-i'2f(-v - '/,)(-x)Id(x + ;77n,')V < -'/2x<0for the two lower limits. Equations 13:10:2, 13:10:7 and 13:10:9, in which x must generally be positive, are specialcases of the differintegration result 13:10:11f('/-v) d"f(v+ l)dx"xr(v-µ+I)xv>-1while 13:10:3, 13:10:8 and 13:10:10. in which x is generally negative, are the special µ = 1, -1, and i casesof 13:10:12d")1"(-x)" = r r(-v)) (-x)"V < µ[d(xxFor a variety of functions f, definite integration between the limits x = 0 and x = x converts the productf(x)x'-' to a function fM of s, known as the Mellin transform of the function f: 13:10:13Jzf(x)x'-'dx = fM(s)0For example. provided s > I, the Mellin transform of I /[exp(x) - I) is r(s)C(s) [see Chapters 26, 43 and 64 forthe exp, r and { functions]. A tabulation of over 250 Mellin transforms is given by Erdelyi, Magnus, Oberhettingerand Tricomi [Tables of Integral Transforms, Volume 1, Chapter 6], together with some general transformationrules. Via the substitution jin(x)I = t, one can convert definition 13:10:13 into 13:10:14Us) = Jf(exp(-t)) exp(-sodr +J¢f(exp(t)) exp(sr)dr0 ob>00<-b11 and thereby use tables of Laplace transforms [Section 26:14] to calculate many Mellin transforms. For example,the Mellin transform of the function f(x) = In'(x)u(I - x) [where the In and u functions are those treated in Chapters25 and 8) may be obtained by noting that, for 0 < t < -. f(exp(t)) = 0 while f(exp(-t)) = (-t)". Equation 13:10:14then shows that fM(s) is simply the Laplace transform of (-t)', namely -n!/(-s)"". 105 THE NONINTEGER POWERS x'13:11 COMPLEX ARGUMENT13:14 The complex equivalent of equation 13:8: L: 13:11:1 (x + iy)' = exp[v Ln(x + iY)]where Ln is the multivalued logarithmic function discussed in Section 25:11, expresses an arbitrary real power ofa complex number. When v is a nonnegative integer, 13:11:1 reduces to 11:11:1. If v = m/n where in is an integer and n = 1, 2, 3, ..., formula 13:11:1 defines n complex numbers. The m = 1. y = 0 case is elaborated in Section13:14.When v is irrational, the power function (x + iy)" adopts infinitely many complex values. Making use ofrelationship 13:11:1 and equations 25:11:1. 25:11:2 and 25:11:3 dealing with the logarithmic function of complexargument, one finds13:11:2 (x + iy)" = (x' + y-)'''-[cos(6) + isin(4)]where13:11:3 46 =v(0+2ka)k=0,±1,±2,...andsgn(Y)arccot(x/IIyI)Y # 013:11:4 it2-[1 - sgn(x)] y = 0 The choice k = 0 produces13:11:5 (x + iv)' = (x2 + y2)"1[cos(v0) + isin(vO)]an expression of de Moivre's theorem [see Section 32:11]. 13:12 GENERALIZATIONSThe functions (bx + c)' and (ax' + bx + c)' generalize x", and their properties can sometimes be deduced byappropriate substitutions. 13:13 COGNATE FUNCTIONSThere is a multitude of functions of the form f(x) involving an arbitrary power of a function f(x). The v = ±instances are normally the most important noninteger cases. The examples (a2 t x)"1Z are the subjects of the nexttwo chapters. 13:14 RELATED TOPICSIn this section we briefly discuss the complex values of x'1'. When x is real and n = 2, 3, 4, .....r'l" takes two,one or no real values, as discussed in Sections 13:1 and 13:3, according to the sign of x and the parity of n.However x'l" always has n complex values. The formulau" =i;"r(7l I - sgn(x)1)tr[ 1 - sgn(x)1)13:14:1x- (xIcosn+ 2k+ i sinn2+ 2kk = 0, 1. 2, ..., n - Ifrom which these values may be calculated, is derived by setting y = 0, v = 1/n in equation 13:11:1, utilizingthe expression Ln(x) = ln(IxI) + i[2kir + ar(I - sgn(x))/2] (which derives from 25:11:1) and finally replacing the exponential function via Euler's formula, 26:11:1. 13:14 13:14:3(-8)'"' =41/4 =THE NONINTEGER POWERS x' 106 141"4Ilcos(0) + i sin(0)1 = f + Oi = V2-14114I[cos(zr/2) + i sin(ir/2)) = 0 + V 2-i = VZi [41'4IIcos(Tr) + i sin(ce)] = -V + Oi = -f I41J4ilcos(3a/2) + i sin(3a/2)] = 0 -2i = V2i (1V/3\1 18'"R1cos(-a/3) + i sin(-a/3)] = 212 + rl 2 IJ - 1 -V3-i(1V1/181/31[cos(a/3) + i sin(a/3)] = 212 + it 2 I = 1 + V-3il8113I1cos(a) + i sin(ce)] = 21-1 + Oil = -2/JJJ CHAPTER14THE b\/a2 - x2 FUNCTION AND ITS RECIPROCAL Together with the parabola [Chapter 12] and the hyperbola [Chapter 15]. the ellipse discussed in this chaptercompletes the family of curves of the second degree. These are also known as conic sections because they aregenerated by intersecting a cone with various planes. 14:1 NOTATIONAs explained in Section 12:1, the notations b,(a2 - x')1721 and 1(a2 - x2)-i' I/bare equivalent to ba - x andl /(ba - x ), respectively. The parameters Jal and lab] are known as the semiaxes of the by a s function,the larger being the semima'or axis and the smaller the semiminor axis.A graph of tb a - x- versus x is termed an ellipse, and the adjective semielliptical is therefore appropriatelyapplied to the b a - x function. Some geometric properties of the ellipse are discussed in Sections 14:3 and14:4. 14:2 BEHAVIORThe functions W-7 - x- and 1 /(b a - x) are defined (i.e., take real values) only for -la] K x < Jal. Theyadopt the sign of b. Map 14-1 shows typical shapes of the two functions when b is positive and less than unity. Typical shapes of b\/a2 - x- when b = 1 and b > I are depicted in Map 14-2. 14:3 DEFINITIONSThe algebraic operations of squaring [Chapter I l] and takin the square root [Chapter 12], together with arithmeticoperations, fully define the semielliptic function b a2 - x' and its reciprocal.A parametric definition [Section 0:3] of the f = ±b a' - x function in terms of trigonometric functions[Chapter 32] is provided by14:3:1 f = ab sin(w) x = a cos(w)An ellipse may he defined geometrically in two distinct ways. The first defines an ellipse as the locus of allpoints P whose distance PF from a fixed point F (called a focus of the ellipse) and whose distance PD from a 107 14:3 THE bV a - s' FUNCTION AND ITS RECIPROCAL 1blalFIG 14-1b<1b alb>1108 straight line DD (called a directrix of the ellipse) obey the relationshipPEa constant less __ the eccentricity 14:3:2p= than unityof the ellipse0 < e < ILet the x-axis of a cartesian coordinate system be chosen to lie along the shortest straight line joining the focus tothe directrix and let X be this shortest distance between F and DD. Create the origin of the coordinate system atpoint 0 (see Figure 14-3] such that the length of the line OF = >\e2/(1 - e'). Then the equation of the ellipse isf=tb a- x' where b-e- anda=>,e/ l-EThe second geometric definition of an ellipse is as the locus of points P such that the sum PF + PF' of thedistances from P to two points F and F' (the foci of the ellipse) obeys the relationship14:3:3 PF + PF'= a constant = oOf course, a must exceed the interfocal distance FF' = 4. If a cartesian coordinate system is erected with its 109THE 6V.' -72 FUNCTION AND ITS RECIPROCAL 14:7origin 0 at the midpoint of the line FF', which is chosen as the x-axis, then the equation of the ellipse isagain ±ba- - x- where a = a/2 and b = I - ((b/a)'. 14:4 SPECIAL CASESThe semielliptic function bV a -i becomes the semicircular function V-. when b = I (see Figure 14-21.The parameter jal is then known as the radius.The geometrical definition of a circle is as the locus of all points P lying at a constant distance (equal to theradius p) from a fixed point called the center of the circle. If the origin of a cartesian coordinate system is placedat the center, then the equation of the circle is f = x a- - r- where a = ±p. Other geometrical properties ofthe circle are discussed in Section 14:14. 14:5 INTRARELATIONSHIPSThe function f(x) = W a - x is an even function, that is, the reflection relationship14:5:1is satisfied.The multiplication formula14:5:2f(-x) = f(x) f(vx) = by(a/v)' - x'shows that multiplication of the argument of a semielliptic function by a constant generates another semiellipse,one semiaxis being unchanged. Choosing v = 1/b generates a semicircle of radius jabl. 14:6 EXPANSIONSTwo useful infinite expansions are 14:6:1 andX2x'16xa'-x==la1b 1 ------2a'8a'16a°128x°\x'1 IIx'3x'5x°35x° 1-1/2x 14:6:2=1+-+-+-+-+ = ll-a<x<a ba' - xZjajb2a'8a'16a°128x"Ialb J=u \JI a 1 14:7 PARTICULAR VALUES b a -xb a'- x's=- a'--x-+b- Px=-aablat lx-0;ablaI x = a 0 sgn(b) alb sgn(b) 0 ( m sgn(b) son(b)lalbsgn(b) sgn(b) 14:8 THE bVa' - x' FUNCTION AND ITS RECIPROCAL 11014:8 NUMERICAL VALUESThese are readily calculated with a programmable calculator or any other computing device. 14:9 APPROXIMATIONSBased upon truncation after the first two terms, expansions 14:6:1 and 14:6:2 yield- x= 14:9:1W;'x2 =b(2a=)21al8-bit precision Itar+x: 14:9:2ba' - x22bla'8-bit precision 14:10 OPERATIONS OF THE CALCULUSDifferentiation and integration give14:10:1bVa'- 14:10:2 14:10:3 and 14:10:4 The integrals 14:10:5 and-bxax2xa< 0.10 dlx Jdx btiab V(.2 -!x2)'\b't'dt?a-+ laresinlIQ.I-lalsxal\\\ rdt I=x(am)Jbaresino b1 a //////-lal s x s lal t%/;7-7-3(aZ-x')'-jai sx<lal 14:10:6Jr=- larsechl x l-'al s x s lalb! ta' - t'lal\a/are typical examples of a general class of indefinite integrals represented by jf (-a- t )T dt where n is an integerand m is an odd integer. Such integrals evaluate to simple algebraic terms, as 14:10:5, or may contain either anarcsin(x/lal) term [Chapter 35] or an arsech(x/a) term [Chapter 31]: Gradshteyn and Ryzhik [Section 2.27] listsmore than fifty such integrals, including some general formulas. Note that if x > 0. the integral in 14:10:6 is tobe interpreted as the Cauchy limit: 14:10:7dt dt-0f f--.,ltar-lr+J.ta'-t'0r<x<la Among other important integrals are 14:10:8drF(p;aresin(x/a))oa 2 - tZa' - p'r'a IllTHE ba=x2 FUNCTION AND ITS RECIPROCAL 14:11 anda' - p2at14:10:9 p2P= a E(p;arcsin(x/a))Jav7=1which serve as definitions of the incomplete elliptic integrals of the first and second kinds [see Chapter 621.Semidiffcrentiation produces 1/2{2E(P;O)dxb2=b 0<x<a dl/21=1 I10<x<a dx'n ba' - x'b(a + x)2aa - zaaxThe functions F(p;8) and E(p;O) that occur in the above semiderivatives are incomplete elliptic integrals [Chapter621 of argument 8 = aresin(2x/(a + x)) and parameter p = (a + x)/2a. The same functions occur in theexpressionsu2rb 14:10:12ax-1J1b3 YI (a - x)F(p;8) + 2xE(p;8) - (2x - a) ?a]0 < x < a andU214:10:13 d 1F(p;8)0< x< a dxbfor the semiintegrals with zero lower limit. 14:11 COMPLEX ARGUMENTReplacement of the real argument x, in the semielliptic function or its reciprocal, by the complex argumentx + iv yields 14:11:1b a2-(x+iy)2=b_(a+x) +y+y +a2_x2+y2 ib sgn(xy)VV5 +y 2v1or 14:11:21_lVV (a+x)-+y'(a - x)+ y+a2_x2+y2W;=( x+ iy) -b7[(a+x)2+Y'][(a-x)2+y2] isgn(xy)V (a + + y2 - a22 + x'- - y2 b v 2 [(a + x)2 + Y21[(a - x)2 + Y2)where the sgn(.ry) function [Chapter 81 equals the sign of the xy product, or is zero if either x or y is zero.When the argument is purely imaginary, equation 14:11:1 reduces to14:11:3 ba''-(iy)2=b1/'+y2which corresponds to a semihyperbola [see Chapter 15]. 14:12 THE b\/ a= - x' FUNCTION AND ITS RECIPROCAL14:12 GENERALIZATIONS112 The ellipse, b(a2 - x2)' 2, is the special n = 2 case of the more general function b(a" - x")'"". When n is evenand greater than 2, the curves obtained by plotting b(a" - x")"" versus x have been called superefiipses: the casesn = 2. 4 and 8 are depicted in Figure 14-4. As n approaches infinity the superellipse shape becomes more andmore rectangular.FIG 14-4 n=8The reader is referred to Section 15:12 for a discussion of the circumstances under which the functionax + bx + c is a generalization of the semiellipse W a- - x-- 14:13 COGNATE FUNCTIONSThe other conic sections, discussed in Chapters 12 and 15, resemble the semielliptic functions in many respects. 14:14 RELATED TOPICSIn this section we discuss geometric properties of the semicircle and the ellipse.The area of the shaded segment of a semicircle of radius a [see Figure 14-5] is easily found from 14:10:3 tobe/ \ l114:14:1area=2of2[2+ arcsinl - ) I + 2a2 - x2 shaded segmentLLL\ I f zo-zathe total area of the semicircle being rra2/2.The curved perimeter of the shaded region has a length 14:14:2 curved= al Z +IJ perimeterL 113THE W; =x' FUNCTION AND ITS RECIPROCAL 14:14while the entire semicircumference has a length ira. An important property of a semicircle is that the angle APBin Figure 14-5 is a right angle for any point P on the semicircle. Thus the triangles APB, AQP and PQB are similar.As explained in Section 14:3. the ellipse ±bV a - r- for b < I has foci F and F' located on the x-axisat x = waV I b . The sum of the distances to each focus from any point P on the ellipse is a constant. 2a.Moreover, the lines PF and PF' make equal angles with the ellipse itself at point P, as is illustrated in Figure 14-6. This means that any wave motion that originates at one focus F' and is reflected at the surface of the ellipsewill be "focused" at the other focus F and all the reflected waves will arrive at F at the same time, though froma variety of directions [Figure 14-61. This property of an ellipse is exploited in furnace design and in several acousticand optical devices.FIG 14-6 A segment of an ellipse, such as that shaded in Figure 14-7, has an area that can be determined with the helpof integral 14:10:3 asx1xV a' -- x area of=rrrIf = ba'-r= 14:14:3shaded segment2J Qfdi = a'1bj[2 + aresinlaI +a1 O0 FIG 14-7 The entire ellipse thus has an area ira`Ibf. The curved perimeter of the shaded area in Figure 14-7 may be evaluatedin terms of an incomplete elliptic integral by utilizing equations 14:10:1 and 14:10:13: 14:14:4curved= 2J a1 +y dr =2aLE1 - b2;aresin perimeter (- fI - EI1 - br; 2JJ/ /\//Because [see Chapter 621 E(p;7r/2) = -E(p;-a/2) = E(p) where E(p) denotes the com lp ete elliptic integral[Chapter 611 of p. the entire perimeter of an ellipse with semiaxes ial and Jab equals 4aE( I -- bZ). CHAPTER15THE b\/x2 + a FUNCTION AND ITS RECIPROCAL Al hou h the properties of theb x + a function depend on the sign of the constant a, a graph ofb x + a is a curve called a hyperbola whether a is positive or negative, the sign of a determining only theorientation of the hyperbola. 15:1 NOTATIONThe notations bl(x2 + a)'I2I and I(xz + a)-"21/b are equivalent to b x -+a and I /(b x + a), respectively. Be-cause the curve ±b\+x a is a hyperbola, the adjective semihyperbolic is appropriately applied to theb x + a function. 15:2 BEHAVIORWhen a is positive, bVx- a and its reciprocal are defined for all values of the argument x and both functionsshare the sign of b. Figure 15-1 shows typical behaviors of these functions when b is positive. 115 15:3 THE bV x- + a FUNCTION AND ITS RECIPROCAL 116When a is negative, neither bx -+a nor its reciprocal is defined in the interval -V a < x < V - a andtherefore each of these functions has two branches, as illustrated in Figure 15-2 for positive b.As x -+ x, the function b x` + a acquires values ever closer to bx. The linear function bx is said to be anasymptote of the bV +xfunction. Similarly, -bx is an asymptote of b x' + a as x -> -x. These asymptotesare shown in Figures 15-I and 15-2. 15:3 DEFINITIONSAlgebraic and arithmetic operations define b x + a and its reciprocal in a straightforward fashion.A hyperbola may be defined geometrically as the locus of all points P such that the length of the line PFconnecting P to a fixed point F (a focus of the hyperbola) obeys the relationshipPFa constant_the eccentricity_ 15:3: IPD = greater than unityof the hyperbola -e1 < E < xwith respect to the length PD of the shortest line from P to the straight line DD (called a directrix of the hyperbola).The eccentricity of the hyperbola ±bV +xis1 + b-. An illustration of relationship 15:3:1 is shown in Figure15-3. Note that the hyperbola of Figure 15-3 has two branches and that there exists a second directrix D'D' anda second focus F' symmetrically positioned with respect to a line parallel to both directrices and midway betweenthem.An alternative geometric definition of a hyperbola is as the locus of all points P such that15:3:2 IPF - PF'I = constantwhere F and F' are the foci. 15:4 SPECIAL CASESWhen a - 0, the bV7x 'T-aand I /(bx' -+a) functions reduce to the linear bx and reciprocal linear 1/(bx)functions, respectively. When b = 1, the function tbV +xbecomes tV+xand is termed a rectangularhyperbola or an equilateral hyperbola and has the interesting rotational properties now to be described. 117THE bx2 + a FUNCTION AND ITS RECIPROCAL 15:4 If a function g(x), interpreted as a cartesian graph of g(x) versus x, is rotated through a positive angle 0 in acounterclockwise direction about the point (X,G) then a new function f = f(x) is created that is related to g(x) bythe general rotation formula15:4:1fcos(O) _ [x - Xjsin(B) - [1 - cos(e)]G + g(y) y = X + [x - X]cos(h) + If - Gjsin(O)When rotation occurs about the origin, so that X = G = 0, the simpler relationship15:4:2 f cos(h) _ x sin(O) + g(x cos(O) + f sin(g))holds. This formula will be used to rotate the function g(x) = ±Vx +a about the origin through angles of-ir/4 and -n/2.The equation f/V 2 = -x/V2- _ [(x/V - f/V)[' + a, obtained by setting g(x) = ± x + a and 0 =-n/4 in 15:4:2, simplifies to 15:4:3 af=-2.xSimilarly, the equation 0 = -x ± V(f)2 a, which arises from setting g(x) to ±V +xa and 0 to -e/2 inequation 15:4:2. reduces to15:4:4 f= ±x'"-aThus, the three functions x V+x a, a/2r and ±\/x a all have exactly the same shape and differ only in theirorientations, as illustrated in Figure 154. The function -a/2r is also a member of the quartet of identically shaped rectangular hyperbolas.It mabe apparent from Figure 15-4 that the curve f(x) = ±Vlx' - a can be obtained from g(x) _± x' + a not only by rotation but also by reflection in their common asymptote. A very general result states thatthe reflection of the function g(x) in the straight line bx + c generates the function f = f(x) where 15.4.511-b2[f=2bx+2c-II+b2[g(v)y=2bf + x(1 - b2] - 2bc1+62When c = 0 and b = 1. this general reflection formula collapses to15:4:6 .x = g(f(x))showing that the reflection of a function g(x) in the straight line x generates the inverse function [see Section 0:31of g(x). Hence, the rectangular hyperbola ±V a is the inverse function of the rectangular hyperbolaand conversely. 15:5 THE W x* * a FUNCTION AND ITS RECIPROCAL 15:5 INTRARELATI ONSIQPSThe f(x) = b x ++ a function is even and therefore satisfies the reflection formula15:5:1The multiplication formula15:5:2f(-x) = f(x) f(vx) = bpvl+ a/(v'')118 shows that multiplication of the argument x of the semih perbolic function by a constant generates another semi-hyperbolic function. The inverse function of ±b x + a is also another hyperbola 15:5:3±1f=bx2-ab'wherex= tbVj'+aThe two hyperbolas tb -x + a and ±bV? - a are said to be conjugate to each other. As shown in Section15:4, they share the same asymptotes. 15:6 EXPANSIONSThe semihyperbolic function may be expanded binomially asx`x65x8(112) X'15:6:1b x=+a=bVa I +8a'+16a'28a`Usx'sa when x is small, and as/Iaaa3(2) /bgx'+l1+..)-bXjlx')1r!'when x is large. It is this latter equation that describes the approach of the semihyperbola to its linear asymptote. 119THE bx=a FUNCTION AND ITS RECIPROCAL 15:9Similarly, the reciprocal function is expansible in the two following ways: 1x'3x'5x635x" 11/2x' 15:6:3(1+_'+'_))0<_x=<abx +aba8a'l6a128abaJa or I(a3a5a'35a',15:6:4 --+) bx= + aiix8x'16x°128x'jX 15:7 PARTICULAR VALUES x - -ab'Ix =ab'-X=ab-, s Ix=- V iajs= 0ab' s I .c bx' + aa < 0. sgn(b)sgn(b)0undcf0sgn(b)- w(b)W;+ aa>0 s sgn(b) sgn(b)bV2abVbV s sgn(b) - sgn(b) 1W7 x a 0 sgn(b) = sgn(b)undef sgn(b) sgn(b) 0a<0 1b + a0sgn(b)IbV 2aIbVnIbsgn(b)0 a>0 15:8 NUMERICAL VALUESThese are readily calculated with a programmable calculator or other computing device. 15:9 APPROXIMATE VALUESThe following approximations are obtained by truncating expansions 15:6:1, 15:6:3, 15:6:2 and 15:6:4, respectively.x+15:9:1ba24=+ xz8-bit precisionICI s 0.4N/-aa> 0 2Va 12a - x= 15:9:2=8-bit precisionxj s 0.3 Vaa > 0 b x2+2bVa 15:9:3 15:9:4bx2 +bx +8-bit precisionW a 2.5Vj 12x=-a8-bit precisionx a 3.3 V ladbx= + a2bWI 15:10 THE bV x a FUNCTION AND ITS RECIPROCAL 12015:10 OPERATIONS OF THE CALCULUSDifferentiation givesdbx 15:10:1-b x'+adxx'+-anddI-x 15:10:2-_dxW;'+.b(x'+a)'The simplest formulas for indefinite integration must employ differing limits according to the sign of a. Thus: 15:10:31.bxabx bt2+adt= 2 z2+a+ 2 arsinh a>0V; 15:10:4 15:10:5 15:10:6 15:10:7Jedr=I arcosh(' x 1a< O.- br'+abV-a///d t 1//`ft'+a'arcsch(a>0x>0 15:10:8 Iadt-a arcsecla < 0-,V,' +aV-a/where the functions arsinh. arcosh. arcsch and arcsec are discussed in Chapters 31 and 35. These six indefiniteintegrals are simple members of the class ft"(ti7 -+a)" dr where n = 0. ± I. ±2. ... and m = t 1, ±3, ±5, ...:a long list of such integrals will be found in Gradshteyn and Ryzhik [Section 2.27] 15:11 COMPLEX ARGUMENTReplacement of the real argument x in b x' + a by the complex argument x + iy yields 15:11:1b(x+iy)'+a=72x'-y ib sgn(x), )+y2 - x= - a + Vy +2aa)y- + (x' + .7-V'2where sgn(x),) equals the sign of the ay product, or is zero if either x or Y is zero. Similarly: 15:11:2J_bt'+adt=bx+a+abarcoshlxIa<0a22-adtI/x\_ b arsinhl I->O Jo bt -+aba 1x'-y'+a+N,yW.(x + r)' + abN/2i sgn(xy)JbVv'+2(x2-a)y'+(x22+a)2-x2+a+Vy +2(x.'+2(x2- a)y' + (x' +a)' 121THE b\/ s= + a FUNCTION AND ITS RECIPROCALIf the argument in 15:11:1 is purely imaginary, the function reduces to a semiellipse (Chapter 14]15:11:3 b(ry)' + a = b(V)' - y'provided that a is positive.15:12 GENERALIZATIONS15:14 The function b(x - a) + a differs from b x- + a only by being translated a distance a alon the x-axis; it is,therefore, a semihyperbola centered at x = a rather than at x = 0. The function ax + bx + c may similarlyrepresent a translated semihyperbola; however, this so-called root-quadratic function may represent any conic sec-tion (or degenerate instances of the conic sections) according to the magnitudes of the coefficients a, b and c. Thesituation is summarized in Table 15.12.1. Table 15.12.1 b' > 4ac b' - 4ac b' < 4aca>04ac11b'ffbVbVa Ix+ -I (Vax+Vc)-4ac - b'V.alx+bl + 2a./ 11( \2a/4a=a * 1Semihyperbola of two branches Two straight lines Isee Chapter 7] Semihyperbola of one branchcentered at x = -b/2a [see this centered at x = -b/2a (see thischapter] chapterla=]//``IIIIzIx+21-d -e)--(x+V)//+IcIx+ l 42/``Rectangular semihyperbola (see Two straight lines of slope unity Rectangular semihyperbo[a of oneSection 15:41 of two branches branch [see Section 15:41 centeredcentered at x = -b/2 at x = -b/2a=0'VV_-C "ireSemiperabola [see Chapter 12] Constant [see Chapter I] Not possible ba=-1+-c- x--0Semicircle of radius V (b/ 2)- + c Not possible Not possiblecentered at x - b/2 [see Section14:4]V 'b'a < 0 1-I x+\)2a a * -1Semiellipse with semiaxes of Not possible Not possibleV b- - dac (-2a) andVc -centered at .r =-b/2a [see Chapter 141 15:13 COGNATE FUNCTIONSThe functions b(x" + a)'I" for it = 3, 4, 5. ... arc hyperbola-like, especially if n is even. The straight line bx isan asymptote in all cases, as is -bx when n is even.15:14 SPECIAL TOPICSIn this sectionwe treat geometrical properties of the hyperbola.The area enclosed by the rightmost branch of the function ±bV t3 + a and an x ordinate, for 0 < NI-a < .x, 15:14 THE bN+xFUNCTION AND ITS RECIPROCALis shown shaded in Figure 15-5 and may be evaluated directly from the integral 15:10:4 asshaded 1 15:14:1area= 2 L Ibt+ a Idt = bx+ a + aIbIarcoshJr xa<0 The curved perimeter of the shaded area has a length given by 15:14:2perimeter= 2f'1 +(b/')iJ2-a - ab2F(.40) rimeter ITZI + b122 /I xZ+a1 +62 - EIIb,0 1 + X sin(0)14o = arctanl b)a < 0 where F and E denote incomplete elliptic integrals [Chapter 62]. b t2+a FIG 15-5 Also shown in Figure 15-5 are the asymptotes OS and OT of the hyperbola and the tangent MN to the hyperbolicbranch at an arbitrary point P (M and N being the points at which the tangent meets the asymptotes, as depictedin Figure 15-5). Two remarkable properties of the hyperbola are that P bisects the line MN, so that MP PN,and that the area of the triangle MNO equals JabI and is independent of the position of P along the hyperbola. CHAPTER16THE QUADRATIC FUNCTIONaxe + bx + c AND ITS RECIPROCAL Next to the constant function [Chapter 11 and the linear function [Chapter 71. the quadratic function is the simplestof the large class of polynomial functions; others are discussed in the next eight chapters. 16:1 NOTATION The parameters a, b and c are called the coefficients of the quadratic function ax' + bx + c. The value of thequantity16:1:1 A=4ac-b'called the discriminant of the quadratic function, strongly influences the properties of the function and its reciprocal.Some authors define the discriminant to have the opposite sign to that in 16:1:1. 16:2 BEHAVIOR Graphs of ax' + bx + c are shown in Figure 16-1 for a > 0 and a < 0. The function is defined for all x and, forpositive a, assumes values that are not less than c - (b=/4a). Whether the value zero is part of the range of thequadratic function is determined by the sign of its discriminant.The behavior of the reciprocal quadratic function is profoundly affected by the sign of J. Figure I6-2 shows,for a > 0. typical graphs of 1/(ax' + bx + c) when the discriminant is positive, zero or negative. 16:3 DEFINITIONS The quadratic function is defined by the arithmetic and algebraic operations signified in the expression ax' + bx+ c. The reciprocal quadratic function is defined as the quotient of unity by the quadratic function.A reciprocal quadratic function may also be defined as the difference of two reciprocal linear functions [see 123 16:3THE QUADRATIC FUNCTION ax2 + bx - c AND ITS RECIPROCAL FIG 16-2 $ 21ox +bx+c :4oc<0 Chapter 7] that share the same b coefficient: I16:3:1 =Ibx+C bx+c /bc-C\Ccc -C1x''+b(1x+1c - Cc- C404ob124 Not every reciprocal quadratic function may be defined in this way, however, since the decomposition 125THE QUADRATIC FUNCTION ax' + bx + c AND ITS RECIPROCAL 16:3:2ax'+bx+c bb'-4acx+2c+2a( b2-4ac-b) Ibb2-4acx-2c+-( b'-4ac+b)is possible (using real arithmetic) only if the discriminant 4ac - b'- is negative. 16:4 SPECIAL CASES16:5 The linear function [Chapter 71 is the special a = 0 case of the quadratic function; further specialization to theconstant function [Chapter 11 occurs when a = b = 0. When c = b2/4a, the quadratic function reduces to {Y'ax(b/2V a)}-, a square function [Chapter II ] .Similarly, the reciprocal quadratic function reduces to a reciprocal linear function when a = 0. to a constantwhen a = b = 0 and to a power function of exponent -2 when c = b2/4a. The decomposition 16:4:11_Iaax'+bxbrabx+b'is always possible when c = 0. Equation 16:4:1 is a special case of a more general partial fraction decompositionthat provides a powerful tool for integration [see Section 17:13). 16:5 INTRARELATIONSHIPSUnless A = +a. the sum or difference of two quadratic functions16:5:1(ax2+bx+c)±(Ax'+B.x+C)=(a±A)x'+(b-B).r+c_Cis another quadratic function, while their product 16:5:2(ax2+bx+c)(Ax2 +Bx+C)=aAx'+(aB+Ab)x3+(aC+bB+Ac)x'+(bC+Bc)x+cCis a quartic function [Chapter 171. The quotient of two quadratic functions can be simplified. provided the dis-criminant of the denominator is negative, via the following formula into a constant and two reciprocal linear func-tions: 16:5:3Ax'+B.r+CAAr,-Br,+CAr=+Br2+Cax2+bx+ca6=-4ac(x-r,)b'-4ac(x-r2)b' > 4ac where r, = (-b + Vb` - 4ac)/2a and r2 = (-b - b- - 4ac)/2a.The inverse [Section 0:31 of the quadratic function is a constant combined with the square-root function [Chapter121 16:5:4 16:6THE QUADRATIC FUNCTION axe + bx + c AND ITS RECIPROCAL 12616:6 EXPANSIONSFor small x we have 16:6:1 Ij[-x(ar + b)1'ax'cbxC'=0c11bx(b2 - ac)x2(b3 - 2abc)x3(b4 - 3ab2c + a2c2)x°ax + bx + ccc2c3c'cs<Iwhile for large x 16:6:2 holds.1Ibb2 - acb3-2abcb'-3ab2c+a2c2ax 2 + bx + Caxa2x32x°a'x3ay1/-bx-c)'lbx+cl<I ax'1`ax'axe 16:7 PARTICULAR VALUES ax' + bx + c ax'+bx - cx = -x - r; 2awhen b' > 4ac x-0x - r, --b+Vbb-4ac2awhen b' > 4ac x . x (4ac - b')/4ax sin(a)0min if a> 0 058n(a) $ max if a < 04a/(4ac - b')0x Slrmax if. > 01min if a < 01ic 0 The values of x that make the quadratic function equal to zero are known as the zeros of the quadratic function oras the roots of the quadratic equation axe + bx + c = 0. Note that "root" is used here in a sense different fromthat of Chapter 13. There are two zeros-b+b2-4ac-b-b'-4ac16:7:1r1 _andr; _2a 2aif 4ac < b2, one zero-b16:7:2 r = -2aif 4ac = b2 and no (real) zeros if 4ac > b2. The reciprocal quadratic function becomes infinite at the zeros of thequadratic function, the so-called poles of the reciprocal quadratic function. 16:8 NUMERICAL VALUESThese are easily calculated from the definitions. 127THE QUADRATIC FUNCTION axz + bx + c AND ITS RECIPROCAL16:9 APPROXIMATE VALUES Ic-bx16:9:1 z8-bit precision Ixl < ax- b8-bit precisionjxj > axz+bx+ca:x3 16:10 OPERATIONS OF THE CALCULUS Differentiation produces d16:10:1 d (axz+ bx - c) = tax + b andIcf16ibz - ac + abxi bxe16b' -ac+-II lal 16:10:2 dI-tax - bdx ax' + bx + c(ax'' + bx + c)zwhile integration results in 16:10:3 (at' + bt + c)dt = and 16:10:4dtJ.,atz+bt+c The 16:10:4 integral is infinite if t 16:10:5An important integral is7nr3 + 1hr2 + frr u 6 2(2ax+b)= 4ac - b' > 0-2r2ax+biartanh`I< 0x <-b = 0 but dt2J at'+bt+c2ax + bb' = 4ac Iax'+bx+cbxInlarctanl9=4ac-b'>0 nIr2aC/a V abx + 2c116:10:6 ,_uat-+bt+cIh&+bx+c\b/xv-1InlI -artanhl I1 < 0 2a`c/a\bx + _c/16:10 and others of the general form f t"(aY + bt + c)'dt, where n and m are integers, will be found in Gradshteyn andRyzhik [Section 2.171. 16:11THE QUADRATIC FUNCTION ax' + bx + c AND ITS RECIPROCAL 12816:11 COMPLEX ARGUMENTReplacement of the real argument x of the quadratic and reciprocal quadratic functions by the complex argumentx + iy leads to16:11:1 16:11:2a(x + iv)' + b(x + iv) + c = (ax- - ay' + bx + c) + i(2axy + by) 1_(ax- - ay' + bx + c) - i(2axy + by)a(x + iy)' + b(x + iy) + c (ax' + av' + bx + c)' + v'(b' - 4ac) 16:12 GENERALIZATIONSThe quadratic function is a special can of the functions treated in Chapter 17. A quadratic function is a memberof all the function families addressed in Chapters 18-24. 16:13 COGNATE FUNCTIONSThe root-quadratic function V axx + bx + c and its reciprocal are functions of some importance. It is demonstratedin Section 15:12 that the root-quadratic function is equivalent to various conic sections depending on the magnitudes of the a. b and c coefficients. Some integrals involving the reciprocal root-quadratic function are 16:13:1(1tax + b\diIarsinha>0A=4ac-b'>0Lv- I Vat' +bt+cl-laresintax + b\IcaV 16:13:2r'dr='1=arcosh l2ax + blV_A JaI+br+cva\) 16:13:3 andJzelbtatbt+cs 1 1b x + 2cb x + 2c arcsiny-c(x)xV-O 16:13:4_-1arcoshrbx - 2cc, > 0p < 0x0 =2cgotat+bt+c \0 bOthers are given in Gradshteyn and Ryzhik [Section 2.26]. 16:14 RELATED TOPICSa<01<0 a>0A<0tax + b1 aV-b2a lbx + 2cvcarsinhx Va>00=4ac-b'>0 A quadratic function describes the trajectory of a body that travels at constant speed in one direction, while ex-periencing a constant acceleration (or force) at right angles to that direction. Thus, ignoring air resistance, the pathof a projectile launched at x = x0 with initial velocity v at an angle 8 to the horizontal travels along the are illustrated 1«9VHF Q17AG'RATIC Fu\Cr 100 ai' 1 0 + r AND ITS RECIPROCAL I504v2sin(2e) to Fieure I1-3 and is airhian for a rimeThe equation of the tra}ecturv is CHAPTER17THE CUBIC FUNCTION x3 + axe + bx + cAND HIGHER POLYNOMIALS The function a"x" + a"_,x'-1 + - - + a,x + ao, in which the a multipliers arc specified constants, is known asa polynomial function of degree n. This chapter treats such functions generally and pays particular attention to theit = 3 case with, for added simplicity, a3 taken as unity. 17:1 NOTATIONA polynomial function is often simply called a polynomial; sometimes the term integral function is used to describethe same family.We shall use the general notation 17:1:1 a._,x"-' ++ a,x + ao = aixia * 0to denote a polynomial of argument x and of degree n. The constants a,,, a"_,, - - ., a,, ao are called thecoefficientsof the polynomial function; there are it -1 such coefficients, some of which may equal zero.Special names are given to polynomials of degrees 5, 4, 3, 2, t or 0; they are known as quintic functions,quartic functions, cubic functions, quadratic functions, linear functions or constant functions.The quantitya2babca` 17:1:2D = Qt - P3whereP =9-andQ=-----9 227is known as the discriminani of the cubic function r3 + ax= + bx + c. Its value affects the number of (real) zerosof the function as explained in Section 17:7. 17:2 BEHAVIORPolynomial functions are defined for all values -- < x < x of their arguments. Those of odd degree acquire allreal values, whereas polynomials of even degree have a restricted range.Figure 17-1 shows the three possible shapes that the cubic function x3 + are + br + c may exhibit accordingto the sign of a2 - W. Notice that an inflection [see Section 0:71 occurs at x = -a/3 in all three cases. A localmaximum and minimum occur on either side of this inflection if, but only if, a' > 3b. (i.e, if P, given in 17:1:2, 131 17:3THE CUBIC FUNCTION x' - ax= + bx - c AND HIGHER POLYNOMIALS 132 is positive). The cubic function always acquires the value zero at least once: that is, there is at least one real rootto the equation p3(x) = 0. Section 17:7 discusses the circumstances under which a second or third mat root exists.The behavior of a polynomial of higher degree is determined in a complicated fashion by its coefficients andby its degree. Generally, a graph of a polynomial p (x) exhibits a number of inflections (at the zeros of the poly-nomial of degree n - 2 that is the second derivative of p (x)-see Section 0:7), and local maxima and minima (atthe zeros of the polynomial dp,(x)/dx of degree n - 1). Some general rules concerning the number of zeros ofp,(x) (i.e., the number of real roots of the equation 0) are given in Table 17.2.1. 17:3 DEFINITIONSThe operations of addition, multiplication and raising to a power fully define a polynomial. Written as a conca-tenation 133THE CUBIC FUNCTION x' + ax, - bx + c AND HIGHER POLYNOMIALSTable 17.2.1 Number N, of inflectionsNumber N. of minima Number N. of maxima Number N, of zerosrt=3,5.7....n=2,4,6....n-2,a, > 0a, < 0I S.N,sn-2OSN,SA-205N,sn-2n - InOSN,S 2 IsN,a - N,NM-l nNMN.N.-N.-11<_Nr<_-21aN,snOaN,snOsN,sn17:5 17:3:1 p,(x) = a,-1)x + a,_2)x + ... + a2)x + a,)x + aothe operations of addition and multiplication suffice.The product of it linear functions [Chapter 7] with real coefficients 17:3:2 1 1 (bx+ c)defines a polynomial of degree n, but not every polynomial function of degree it may be defined in this way.However, every polynomial of degree n can be defined as a product of 1 linear functions and (n - 1)/2 quadraticfunctions [Chapter 16]:rK17:3:3nf P,(x)=Fj (b,.r+c) F1 (AkxZ+Bkx+Ck) K=Z -Z.lk=1where I = N the number of (real) zeros of p,(x). The identification of a suitable set of linear and quadraticcoefficients {b,,c,,Ak,B,,C5} from a given set of polynomial coefficients a, will generally require iteration (or in- spiration!). 17:4 SPECIAL CASESThe functions p,(x) for n = 0, 1 and 2 are treated in Chapters 1, 7 and 16. If the polynomial coefficients obey therelationship 17:4:1 a^,= ()()i/.forj = 1,2,3..... n- 1then the polynomial Y-a,x' reduces to the power function (bx + c)" of Chapter l1 with b = (a,)"" and c =(ao)'/"Each of Chapters 18-24 deals with a family of polynomials with special coefficients.17:5 INTRARELATIONSHIPSThe cubic function p3(x) = x3 + axt + bx + c obeys the reflection formula/a4a32aba17:5:1p31 3x= 27 - 3 + 2c - p, 3 + xbut no comparable formula exists for higher polynomials.The polynomial function obeys the simple argument-multiplication formula 17:5:2 p,(vx) _(v1a,)x'wherep,(x) _a,x'-aro 17:6THE CUBIC FUNCTION x' * axe + bx + c AND HIGHER POLYNOMIALS 134showing that multiplying (or, of course, dividing) the argument of any polynomial by a constant generates anotherpolynomial function of the same degree.The addition or subtraction of two polynomials of degrees n and m yields a polynomial of degree equal to thelarger of n and m (unless m = n and the highest degree coefficients cancel); thus: 17:5:3 while their multiplication 17:5:4Jn?m',=L(a) ±AikiSJ_m Gajxl) (L. Ax')= G' a,x,roro,-on5m where2a,A,-,k-oyields a polynomial function of degree n + in. The quotient of two polynomials is termed a rational function andis discussed in Section 7:13.The converse of 17:5:4-the factoring of a polynomial into two polynomials of strictly lower degrees-is always possible for n a 3 because of 17:3:3. Thus, a cubic function may be factored into the product of a linearfunction and a quadratic function17:5:5x3+ax2+bx+c=(x-r)[x2+(a+r)x-c/r]where r is a (real) zero of the cubic function. (Further factorization of the quadratic function into a pair of linearfunctions is possible if its discriminant [see Chapter 16] is nonpositive, i.e., if it has two real zeros). This factoriza-tion of higher polynomials into ones of lower degree is the key to the decomposition of reciprocal polynomials[see Section 17:13]. 17:6 EXPANSIONSA polynomial function may be expanded as a continued fraction 17:6:1 01x1 =aoa,xaoa,xa,ax,_o1- ao + aix- a, + a,x- a2 + a3x- an_I + a,xA polynomial function may be written as the product of n linear functions with complex coefficients 17:6:2 p,(x) = a. fl (x - p;),=1Each p, is known as a complex zero of pn(x); those that are not real occur as complex conjugate pairs [see Section17:11]. If p, and pk are such a conjugate pair then the product (x - p,)(x - pk) is the quadratic function x2 -(p, + pk)x + P,P5 with real coefficients inasmuch as pi + P+ = (k + ip.) + (k - iµ) = 2k and P,Pk = (k + iµ)(k - iµ) = k2 + µ2. Real and/or complex zeros may be "repeated" in expansion 17:6:2. that is. p, and p, maybe identical for j * k. The zeros p, are related to the a, coefficients by the formulas 17:6:3 17:6:4 17:6:5ipi-PI+P2+...+P.-I+P.=-a,-,aa,_2iPiPk'PIP2+PIPS+PIP4+...+P.-2P.-i+P.--P-k-1, lnnn -an-)P,PsPs = P,P2P3 + PIP2P. + PIP2P3 + + P.-A.-IN-I + P.-2P.-&._=1t.k-I a. 135THE CUBIC FUNCTION x' + axe + bx + c AND HIGHER POLYNOMIALS 17:6:6 PIP2P.-I + PIP2P.-2P. + PIP2 P,-3P.-1P. + " 'I-(-1)"a2 +PIP3P4P.+P2P3P.=FT PJI-=plk=1 PAa. 17:6:7 P,P2PI " ' P. _ I1 P, _(-1) a017:7 -Ia.The functions occurring on the left-hand sides of equations 17:6:3-17:6:7 arc the so-called elementary symmetricfunctions of p1, p2, p3, ..., p. [see Kom and Korn, page 11]. An advantage in expressing the polynomial as in17:6:2 is that its reciprocal can be decomposed thereby into a sum of n terms, known as partial fractions: 17:6:8C=iC'whereC,=1p.(x)a. J-,x - p,e=1Pi-Ptt.,Alternatively, C, may be equated to a./p;,(p,) where p.(x) denotes the derivative of p.(x) [see 17:10:1 or 17:10:41.This decomposition must be modified if any of the zeros of p.(x) are repeated. 17:7 PARTICULAR VALUESFor the general polynomial function of degree n, as defined in equation 17:1:I, we have-.c° -1.r=0 x= 1x=x p,(.r1I -It' - sgn(a,)W a,anE a,x sgn(a.) The polynomial displays an inflection at any argument x, that satisfies the equation 17:7:1 (j + l)(j + 2)a;+2x; = 0p,(x,) = inflectionJ-0Minima and maxima occur at arguments x and xM that satisfy the conditions.-I17:7:2E (j + 1)a,+,x;. = 0andF, (j + 1)(j + 2)a;.2x;. > 0 p.(x.,) = minimumJ-0 J-0-I -17:7:3(j + 1)a). I.r!,r = 0and(j + 1)(j + 2)aJ,:xM < 0 p,(XM) =maximumi-0 J-0Equation 17:7:1 is equivalent to the condition dzp.(x,)/dx2 = 0, while equations 17:7:2 and 17:7:3 state that dp.(x,,,)/dx = 0, d2p.(x.)/dx2 > 0 and dp.(xM)/dx = 0, d'p.(x. )/d.c2 < 0, respectively [see Section 0:71. Zeros of p,(x)correspond to the roots of the equation p.(x) = 0, that is. r is a zero if 17:7:4 7- a,r'=0p.(r)=0J-0The numbers of inflections, minima, maxima and zeros depend not only on the degree of the polynomial but onthe values of the coefficients; see Table 17.2.1.For the cubic function x3 + axe + bx + c, an inflection occurs at x = -a/3 [see Figure 17-I]. Provided a2exceeds 3b. a maximum and a minimum occur at arguments (- a - 3b - a)/3 and (a- - 3b - a)/3, respec-tively. This cubic function has the single zero 17:7:5r=(Q+1r)"'+(Q-Vt1)1J3-3D>0 17:7THE CUBIC FUNCTION x' + ar' + bx - c AND HIGHER POLYNOMIALSif the discriminant D [defined in equation 17:1:2] is positive, and three distinct zeros r, = 2VPI cos(4/) -317:7:6r_ = -2V IPI cosl d' +3)3where¢ = 3 autos I - ID < 0136 rr = 2V cos0-3)3Jif the discriminant is negative. The constants P and Q are defined in equation 17:1:2. When the discriminant equalszero, r_ and r3 of 17:7:6 become identical and there are then two zeros:_ar,-2Q3 17:7:7 D=O_a-Q but these coalesce to a single value if a' = 3b.Formulas exist [see Abramowitz and Stegun, pages 17-18, for example] that permit the zeros of a quarticfunction to be determined algebraically, but no such formulas are available for quintics or higher polynomials. Numerical methods must therefore be used in these cases and, in practice, iterative numerical methods are appliedfor these higher polynomials and, in fact, for quartics and even cubits as well.A popular method for determining the zeros of a polynomial function p (x) = Ea .r! requires that each zerofirst be located approximately (often from a crudely drawn graph of the polynomial versus x). The approximation is then improved iteratively. The accompanying algorithm may be used to carry out the iteration. it requires values of the n + 1 coefficients of the polynomial as well as a value of the degree n and ro, the approximate zero. Improvement ceases when two successive approximants to the sought zero differ by less than one part in 10s. Thealgorithm may fail to locate a multiple zero [see Section 0:21 and will fail to locate a zero at x = 0. It is basedon Newton's formula that gives 17:7:8 ro -f(ro)/-(ro) as an improved approximation to a zero of the f(x) function (which actually need not be restricted to a polynomial).For more information on Newton's method (as this iterative technique is called) and on alternative techniques forfinding zeros, see Bronshtein and Semendyayev (pages 164-171). Section A:5 presents a program for finding zerosof an arbitrary function f that also avoids the need to evaluate the derivative of f. If the input parameters n, a,,,a,_,, ..., ao of the algorithm are replaced by it - 1, no., In - a then this algorithm will yield azero of dp,(x)/dx, that is, a minimum or a maximum of p.(x). Input n >> j (as cue) <Input a, >>Input ro >:Setj=n+ I(1) Replace j by j - IOutput j»»>Ifj*0goto(I)Set rr0(2) Set j = nSetf=a,Storageneeded:n + 6 registers are requiredfor n. r.j f, g and the coefficients a,,, a,, ao. 137THE CUBIC FUNCTION x' + axe + bx + c AND HIGHER POLYNOMIALS 17:9(3) Replace j by j - 1Replace f by fr + aiIfj * 0 go to (3)Setj=nSetg=jq(4) Replace j by j - IReplace g by gr + ja,Ifj*0goto(4)Replace r by r[1 - (f/g))if If I a 10-1 go to (1)Output rr <<<<< <<<< 17:8 NUMERICAL VALUESTest values: Input n = 4. a4 = 2, al = 5. a2 = 2, a,= -9 and ao = -18. There are two zeros: r, = -2and r2 = i. Any input ro < 0.5 will give r,; any inputro > 0.6 will give r2. These are easily calculated for the polynomial function, especially with the aid of the concatenation formula 17:3:1.The simple algorithm Input n >>Set)=n+1(1) Replace j by j - IOutput jIfj*0goto(1)Storage needed: n + 5 registers are required for n, j, j (as cue) <Input aJ > >Input x >> I = P.(x) <p»»Set) = nSetf=a;(2) Replace j by j - 1Replace f by fx + asIf j*0go to (2)Output f uses this formula. 17:9 APPROXIMATIONSI x. f and a,,. a,_,..... a,, ao. Testvalues: Input n= 4, a, = 2, a, = 5, a, = 2, a,=-9,ao=-18,x=2or-l.Output: p,(2) = 44, pP(-1) = -10 One may construct a polynomial p (x) to approximate a polynomial p,(x) of higher degree over a specified intervalxo s x s x, by a process known as economizarion. The procedure is fully described in Section 22:14 and a versatilealgorithm is presented there for calculating the economized coefficients. The degree m of the economized poly-nomial is determined by the acceptable error e of the approximation17:9:1 Ip.(x) - P.,(x)I < axo <- x <- x,The economization procedure has five steps, each of which would usually be carried out numerically.First, one replaces the argument of p,(x) by a new argument y = (2x - x, - xo)/(x, - xo) 17:9:2P.()a1x' _b,y'wherek =(x, - xo/'\k/(x'x")ak )-0 x, + Xok=Jf2so that the region of interest comes to lie in the standardized range - I < y <- 1. 17:10THE CUBIC FUNCTION x' - ax' + br + c AND HIGHER POLYNOMIALS 138Second, the given polynomial of degree n is replaced by a sum of Chebyshev polynomials T, (y), where j takesvalues up to n. The replacement uses equation 11:6:5 and leads to 17:9:3p,(x) _by' _c,T,(y)whereb,-I,"'1=o/.0 !-oand where y,°' is the coefficient of T, (y) in the expansion of ?. About half of these coefficients equal zero.Now, because for -I s y :s 1 the absolute value of T,(y) never exceeds unity, it follows that17:9:4 Ec,T,(y) s 7Ic,IIT,(y)I < Y1c,J-I s y s Ifor any set of j values. Therefore. if m is the smallest integer that satisfies 17:9:5 14 <eit follows that, to within the acceptable error specified by 17:9:1: 17:9:6 i c,T,(y) = p,,(x),-oThe third step in the economization procedure consists of finding the smallest integer m that satisfies 17:9:5.The fourth and fifth steps achieve the reverse of the second and first steps, using the economized polynomial.Thus, in the fourth step, one enumerates the coefficients B, defined by 17:9:7p (x) = i c,T,(v) _B,r'whereB, =,p.trL-0wherep)l"' denotes the coefficient of v' in the polynomial expansion of TR(y) [see Section 22:51. Finally, in thefifth step, one determines 17:9:8p,,(x) _B,y' =i A,x'where-2xr+xol'A,=()(I)`()B+xi+xe-xxoand evaluates the coefficients A, of the economized polynomial. 17:10 OPERATIONS OF THE CALCULUSThe derivative, indefinite integral and differintegral of the polynomial function: 17:10:1 17:10:2 andax' =ax'-'dx7fF7fi=oi-)('dr -a,x-',-01-0i+ I r17:10:3d'jax,_ j7a,x' `dx' _o.,r(j-v+l)follow directly from the results of Section 11:10. The cubic function provides an exemplary subject for the threeoperations, yielding 3x2 + 2az + b on differentiation, (3x' + 4ax3 + 6bx2 + 12cx)/12 on integration and (((3x/(3 - v) + a)2x/(2 - v) + b)x/(l - v) + c)/x"f(l - v) on differintegration.Operations of the calculus may also be carried out on the polynomial function and on its reciprocal via theexpansion 17:6:2. Thus: = 17:10:4 dxP.(x) = p (x)x'p; 139THE CUBIC FUNCTION x' + ax- + bx + c AND HIGHER POLYNOMIALS 17:10:5 where p,;(x) is thereby defined, and 17:10:6dtC ,o p,(t)a, i=idIIIdx p,(x)p.(x)1x - ptp' .(x) P, p,xP;)l+InIP,P)-P'Or')whereC, _I4-i P, - Pi4.j17:12 as in 17:6:8. Similarly, we have the Laplace inversion formula [see Section 26:14]: 17:10:71= Iexp(p,t)exp(-st)dtP.(s)Jo,-iP.(p,)sometimes known as the Heaviside expansion theorem. Recall that. in general, the p, need not be real. The fourequations above are useful only if each p, is a simple zero. This limitation does not affect the alternative methodsof handling reciprocal polynomial functions that are discussed in Section 17:13.Historically, it was the need to integrate the root-cubic v p3(x) and root-quartic functions as well astheir reciprocals. that led to the discovery of elliptic integrals [Chapters 61 and 62). Section 62:14 may be consultedfor further details and for explicit formulas for fdt/\1±p3(t). 17:11 COMPLEX ARGUMENTIf the argument of the polynomial function is complex, the coefficients remaining real, then 17:11:1p.(x + iv)()ajx1_1y2(4la'x-4v4 - .. . + (i)ax'-'y' +(,...J35 In the case of the cubic:17:11:2(x+iy)'+a(x+iy)2+b(x+iy)+c=x'+ax2+bx+c-y2(3x+a)+iy(3x2-y'+2ax+b)A zero p, of the polynomial Ea,x' may have real and imaginary components even when the argumentx and the coefficients a, are real. If p, _ X + iµ where k and p. are real, then, invariably, there will also be a zeropr = h - iµ. The complex numbers (X iµ) and ()L - iµ) are said to conjugate to each other. 17:12 GENERALIZATIONSThe power series discussed in Section 11:14 may in many respects be regarded as polynomials of infinite degree.Sums of negative or fractional powers can often be expressed as polynomial functions of altered argument,possibly with a power multiplier. For example: 17:12:1 a, x-" + a.-,x-"" + ao =P.(x) and17:12:2-Uµ + aox 1/4 = x-"4P.(Vx-) 17:13THE CUBIC FUNCTION x3 + ax' + bx + c AND HIGHER POLYNOMIALS 14017:13 COGNATE FUNCTIONSThe ratio of two polynomial functions is termed a rational function. We use the special notation 17:13:1 R.(x) = A,x'/ax';.oi-oIf m > n the rational function is said to be improper and may be resolved into a polynomial and a proper rationalfunction (i.e., one whose numeratorial degree is less than its denominatorial degree) by the procedure of algebraicdivision (see Bronshtein and Semendyayev, pages 150-151].To simplify a proper rational function, one first factorizes its denominatorial polynomial into linear and qua-dratic functions, thus: 17:13:2 p,(x) = I a,x' = a, f j (x - r,) fl (x2 - p,x + q,) J22 i-0 i-,;-,where each r; is a zero of p,(x) and the discriminant 4q, - pp of each quadratic factor is positive. Next, the r,values are inspected to see if there are duplicates. Similarly, the p;, q, values are searched to see if there are anyrepeated complex zeros. If all zeros of p,(x) are simple (i.e., of multiplicity one) then the proper rational functionmay be decomposed into (n + 1)/2 so-called partial fractions as follows: 17:13:3,.,x-r,-,x - pie+q,_nI22m<n The new constants in this relationship (the a,, P,, y of which there are n all told) are then determined, completingthe simplification- Most often the method of undetermined coefficients [see Bronshtein and Semendyayev, pages151-153 for details and examples] is used to identify the constants, although other methods can be employed.Equation 17:13:3 is appropriate only if there are no repeated zeros of p,(x). U two r values, say r, and r,. areidentical, then in 17:13:3 the termsa,a, a,a2 17:13:4 +should be replaced by + x - r,x - r2x - r,(x - r,)2If the three r values r, r, and r, are identical, that is, if p,(x) has a triple zero at r, [corresponding to d2p,(r,)/dx'= dp,(r,)/dx = p,(r,) = 0, see Section 0:2]. then the termsa,a,a, a,a,a, 17:13:5+ - +should be replaced by ++ x-r,x-r,x-r, x - r,(x-r,)2(x-r,)SSimilarly, if p, = p2 and q, = q2, 17:13:3 is to be modified so that _ 17:13:6,x+ y,+x+ y,x2 - p,x + q,x' - p,x + q2should be replaced by R,x' + -1,+02x + 72x2-Pox+q,(x2-p,x+q,)'Otherwise the partial fraction decomposition is unchanged.With R (x) defined as in 17:13:1, the infinite series R.'(0) + R.'(1) + R.(2) + can be summed providedn ? m + 2. The procedure is described in Section 44:14.A rational function frequently serves as a convenient approximation to a transcendental function. For specifiednumeratorial and denominatorial degrees, m and n, the Pade approximant is the best such approximation for smallvalues of the argument x. If the transcendental function f(x) can be represented by the power series EA,x', thenthe Ra Pade approximant is simply the sum of the first m + I terms in this series17:13:7 g=A,x'Below is shown an assemblage of Pade approximants to the exponential function [exp(x), see Chapter 26].Ax'i=E a, +E "-t+ 141THE CUBIC FUNCTION x' + ax' + bx + c AND HIGHER POLYNOMIALS 17:13 Ra(z)=I+x 12+xR°(x) -l-x 2-x26+2x2-2.x-x'6-4x+x' R°(x)'66-6x+3x'-x'R,(z) =24 + 6xRl(x)I+x+-26+4x+x'6-2x12-6xx'R(x) _12-6x+r60 - 24x + 3.rRl(x) _24 - 18x + 6x' - x' 60 - 36x + 9x' - x'X2X,R'a(x)=I+x+2624 + ISx - Fix' +24-6x60+36x+9x'+x'R,'.(x) _60 - 24x + 3x°120 + 60x + 12x' + x'R (x) _120 - 60c + 12r2 - x'Such an assemblage, extended indefinitely, is known as a Pade table [see Wall, Chapter XX]. Though they maynot always be optimal, the diagonal Pad' approtimants Ra(x). R:(x), RZ(x), ..., Rk(x), ... will be our especialconcern [see Section A:6]: we will report a general procedure for determining successive members of this family. For the exponential function example. the diagonal Padd approximants are given by the formula 17:13:8 RU(x) =P`(x)wherept(x) = Y(2 k- j)'xi;-I (k - j)!j!For the present purpose itis necessary to augment the Pad6 table by including entries corresponding to;(x), where m and n are odd multiples of i, These new entries are not approximations to f(t), though they arerational functions. Continuing with our f(x) = exp(x) example, we list below part of the augmented Pad' table:R111;(x) = 0It".'a(x) = 0 R "121AX) = 0Rx)=lR,(x)=I+xRa(x)'1+x+-Ro(x)=I+x+x'-+ i-2 26 1 2 6 24 R;!'(x)R;'(x) =x'R; ,`.'(x) =s'Ri;'(x) = z RI(x) _2-xRi(x) =6+4x+ x'24-I8x+6r+x'2-x6-2x24-6x- 12 + 12x - 2x'-72 + 48x - 6r-480 4- 240x - 24x' 'R3',t(x) -x'Rii(x) -R (x)x' Note that the identity 17:13:9R'.(x) =12+bx-.x'60+36x+9x'+x'12-6x+.r R"_'I1z(x) = 0R'.(X)60 - 24x + 3.rX' is true for all functions. Use of this rule and rule 17:13:7 enables the top two rows of the augmented Padi tableto be completed for any function f(x) whose power series is known. Every other entry in the augmented table maythen be found from those immediately above it by use of the rule [see Macdonald for details] 17:13:10R"(x) = R;_1(x) +1RR-i12(x) - R, ; (x)13 13m=-, 1,-....n= -. 1,-,...,m2We will use the phrase "Padd operation on R;(x)" to describe the procedure, indicated by equation 17:13:10, bywhich R; is constructed from R;_,. R,'-','/122 andBy carefully sequencing Pad6 operations, it is possible to generate an unlimited number of diagonal Padeapproximants corresponding to any power series. If we assign the verb "papoo" to mean "perform a Padi operation on" then our goal may be achieved economically by the following sequence: load R. load K. null R' ; JZ, papooRii, load R;,, null R-,/z, papoo R3,;;, papoo R load K, null 1111/_, papoo Ri, , papoo R;, papoo R3j , load R.nullR papoopapoopapoo R. papoopapoo R,"2!,papoo R', papoo R55/;, load R,, null R-"/2, Papoo R; z, papoo R;, papoo Riii, papoo R, papoo R'/2, papoo k',1/22etc. 17:13THE CUBIC FUNCTION x' + ax' + bx + c AND HIGHER POLYNOMIALS 142Because the number of papoo steps increases (0,3,7,11,15,...) for each successive diagonal Pade approximant,the algebra soon becomes daunting. imposing a limit to the usefulness of this particular procedure for the evaluationof expressions for R}(x). On the other hand, the procedure is extremely useful for calculating numerical values ofthe diagonal PadE approximants, for some argument x of interest, because, unlike the algebra, the arithmetic maybe performed efficiently by a computing device. The algorithm below calculates and outputs numerical values ofK. R;, R22, ....R; from input values of to, t,, t2, t2j, ... 12J- Set k = s = 0(1) Set j = 2kOutput jj (as cue)1fj=0goto(2)Sets=r,-,(2) Set r,=s+1fj=0goto(6) Sett=0(3) Replace j by j - 1Sets=r,Ifs * r,., go to (4)Set r, = lOwIfs * 1090 go to (5)Set r,=tGo to (5)(4) Set r, = I + (1/(r,_, - s))(5) Set t=sIfj*0goto(3)If frac(k) * 0 go to (7)(6) Output roInput t, >> ro=R:<'(7) Replace k by k + 1 /2Go to (1)Storage needed:k, j, t(stores all t, and serves as a temporary store),s and ro, r1, ..., r2,-,, r2 j Input restriction:j <- J where 2J + 1 is the size of the allocated storagearray. Test values (correspond to exp(x) with x = 1):input to = Ioutput Ro = 1.00000000inputs t, = 1. t2 = 1 /2output R; = 3.00000000inputst, = 1/6, t, = 1/24output R2 = 2.71428571inputst, = 1/120, rb = 1/720output R3 = 2.71830986inputs t, = 1/5040, is = 1/40320outputR; = 2.71828172inputs t9= 1/362880, to = 1/3628800output R5 = 2.71828183 The algorithm makes use of two running indices, k and j, that have simple interpretations in terms of the augmentedPadE table. It may be noticed that the entries in the augmented table define two sets of diagonals. Each diagonal in one set is composed of R; members such that m + n is a constant. this constant equals 2k. The other set comprisesdiagonals that are infinitely long, each diagonal being characterized by a constant difference m - n between thedegrees of the table members. this difference equals the index j. Thus, the identities 17:13:11 m = k + 2andn = k - 2k = 0.-, 1,... j = 0, 1,2,...2 relate the indices to the degrees.Though the algorithm is slow, it is so economical in storage requirements that it may be implemented by mostprogrammable calculators. The commands shown in green guard against a 'divide by zero' error, should the termsin the denominator of the Pade operation 17:13:12 Set r,= t+1 be equal (either coincidentally or by loss of arithmetic significance); often these five commands may be omitted.The diagonal Pad approximants Ro(x), R:(x), R?(x), ... converge toward f(x) much more rapidly than do fo(x),fi(x), f2(x), ..., where f,(x) is the partial sum defined by 17:13:7. This is illustrated by the test values that accompany 143THE CUBIC FUNCTION x' + axe + bx + c AND HIGHER POLYNOMIALS 17:14 Table 17.13.1 J f'(x) k 03.602598514 x 10-' 0.0360259851 0.0360259851 0-1.501082714 x 1O"' 0.034524902422.814530089 x 10"' 0.0348063554 0.0347619155 13-9.772673920 x 10'' 0.034708628744.988135647 x 10"' 0.0347585101 0.0347405098 25-3.366991560 x 10" 0.034724840262.829208187 x 10' 0.0347531322 0.0347395777 37-2.846048712 x 10' 0.034724671783.335213335 x 10'' 0.0347580239 0.0347395117 49-4.462391917 x 10-' 0.0347133999106.712181177 x 10-' 0.0347805218 0.0347395053 511-1.121239355 x 10" 0.0346683978122.059498676 x 10"' 0.0348743477 0.0347395045 613-4.125598309 x 10" 0.0344617878148.951074902 x 10" 0,0353568953 0.0347395044 715-2.091070553 x 10' 0.0332658248165.233121879 x 10"' 0.0384989467 0.0347395044 8 the algorithm. Here. t, = 1/j! and so Rt is an approximant to exp(1), equal to 2.718281828. The output data showthat Rk has converged to nine significant figures already by k = 5. It requires a partial sum of twelve terms, to +t, + t2 + ... + ill to achieve comparable accuracy. This advantage of the PadE approach is largely illusory,however, because terms up to r1o were needed to calculate the Rs approximant! In fact, for the summation ofconvergent series [Section 0:61 such as this, the Padd procedure will seldom be worth the effort.The great virtue of the Pade procedure comes from its ability to handle asymptotic series [see Section 0:6],and the Atlas takes advantage of this in a few of the algorithms in Sections 8 [for instance in Section 51:8]. Ademonstration of the efficacy of this procedure is shown in Table 17. 13. 1, which lists approximations to the zero- order Basset function [Chapter 51 ] of argument 3:17:13:13 Ko(3) = 0.03473950439calculated as partial sums of the asymptotic series 51:6:8 and as the corresponding diagonal Pad@ approximants.Evidently, whereas the former procedure can never give a precision better than about three significant figures, thelatter has achieved nine significant figures of convergence by k = 7.Even more remarkable is the effect of the Pad6 procedure on series that are not even asymptotic. Consider theGauss function [Chapter 60J F(;,=;1;-2) expanded via 60:6:1:(2j- 1)!!12 I922511025 17:13:14F(11;1;-2)2,2[(2j(2j)!!I(-2) = 1 -2 + 16288 + 9216 - =o LThis series is so wildly divergent that 17:13:14 would not usually be considered a valid representation of F(2,21;1;-2).Nevertheless, the diagonal Pade approximants converge to 0.745749189, a value that can be identified with(1/f)F(J,1;1 3) [or with (2/trV3)K( 2/3), where K denotes the complete elliptic integral of Chapter 61], towhich F(1,1;1;-2) is equivalent via transformation 60:5:3. 17:14 RELATED TOPICSTechnologists and engineers commonly collect extensive lists of values of a function f(x) without knowing the formof the relationship between f(x) and its argument x. Table 17.14.1 shows such a list, with f(x) abbreviated to f. A frequent need is to use these data so that a value of the function may be estimated at an argument where nomeasurement was made. Two situations arise in this setting. In the first, the tabular data are regarded as exact andthe problem is one of interpolation. This means the selection of a relationship between f(x) and x that is satisfiedby all (or by a subset of) the data. The relationship is then taken to apply equally well between the given data 17:14THE CUBIC FUNCTION x' + axe + br + c AND HIGHER POLYNOMIALS 144Table 17.14.1 X.AX,f points. In the second situation, error is assumed to contaminate the f(xj) data and a (usually rather simple) rela-tionship f(x) is sought that does not exactly reproduce f(x) at the x = xo, x,. X2, .... Xj points but is close. Such aprocedure is known as regression. Polynomials are commonly employed for both interpolation and regression.There exist many interpolative schemes, but one that provides a smooth interpolation without undue complexityis the sliding cubic, or Lagrange four-point interpolate. In this technique a cubic expression is fitted to the dataat x,- x,, x, and x;.2 but is used to describe the relationship only between the middle two points, x; and x;.,.The cubic is 17:14:1f(x)(x-x,)(x-x,")(x-x")f.p3(x)_ xsxi ie(xR - xr)(X& - x.)(xx - x")where, in turn, k takes the values j - 1, j, j + I and j + 2. while 1, in and n are the three of those four integersother than k. Observe that p,(xk) = f, for xi = x,_ x;, x,-,, x,.2.Figure 17-2 illustrates this interpolation scheme in the frequently encountered case in which the data are evenlyspaced so that x,.2 - xl., = x,., - x, = x; - x;_, = h. The sliding cubic then takes the formj- 2JJ-N xl/J/-If +f f(x) -X, - .f 17:14:2f, +_Xl/623J22+rx=x:1 Lf rf +jJh/ L 6226 Jx;szsz.,which is convenient for computation. When the argument passes from the x, s x s x;., range into the x;., s x sx;.2 range, the cubic "slides" with it and the interpolation now uses the f,, f,-,, f,,.2, f,, 3 data.An alternative, and popular, interpolation scheme is provided by the cubic spline. This is also a piecewisecubic polynomial that is, in general, even "smoother" than the sliding cubic. For this reason it is preferred fortreating data that display discontinuities. For further information on the cubic spline, reference may be made toHamming [Section 20.9].The remainder of this section will deal with the technique of polynomial fitting in which a K-degree polynomial17:14:3 f(x) = aKx` + aK_ix"' +-+ a,x + auis fitted to the evenly spaced data (xo,fo),(xi,ji)..... (x,_ ,,f_,), (xj,fj). If K is chosen to equal J. the fit is exactand the procedure is one of interpolation. If K is chosen to be less than J, the procedure minimizes the sum ofsquares 17:14:4 (f - f(xj)12j-oand is said to produce a least-squares regression. The integer K is restricted to values not exceeding J since, with 145THE CUBIC FUNCTION x' + ax' - bx + c AND HIGHER POLYNOMIALS 17:14 K = J, the fit is already exact. Figure 17-3 shows an example of the least-squares cubic function fitted to a set ofnine data pairs.An algorithm follows that permits the coefficients a5, aK_,. ..., a,. an to be found. There is no limit toJ + 1, the number of data that may be input, The degree K of the fitted polynomial is limited only by the registersavailable and by the need for K not to exceed J. There is no lower limit to K: K = I will return the coefficientsa, = b, ao = c of the best straight line fit (see Section 7:14; K = 0 will return the constant that is the average ofall the f, values.The algorithm is lengthy because it has five phases.The first phase simply allows the input of J, xo, .r, and K. The x,) and x, parameters are not stored as such butas (x0 + x,)/2 = u and (x, - xo)/2 = h. The independent variable x is subsequently regarded as having beenreplaced by y = (u - x)/h so that all the data lie in the range l ? y a - I and are characterized by y, = I - (2j/J)In the second phase x, is output to prompt the operator to load f,. These input values are each multiplied byvalues that the algorithm calculates for tt(y) for k = 0. 1. 2...., K. and are accumulated as the sums2j 17:14:5eA=Fft,,(y,)v,=1-k =0,1, 2, .... Ki-o 17:14THE CUBIC FUNCTION x' - axe - bx - c AND HIGHER POLYNOMIALS 146in registers bo, b,, b2, .... bK. Here t,(y) denotes the discrete Chebyshev polynomial of degree k and argument y;its definition and properties are given in Section 22:13. The algorithm uses recursion formula 22:13:8 to calculatet5(yj).As explained in Section 22:13, the 'best" Kdegree polynomial through the points (fo,yo). (ff,)y), ..., (f,-i,yj-J(f,,y,) is given by K(2k + 1)(J - k + 1), 17:14:6PK(y) _d,t,(y)wheredA =(J +and where (J - k + 1), and (J + 1),., are Pochhammer polynomials [see Chapter 18). The third phase of thealgorithm is devoted to replacing each e, value by the corresponding d,.The fourth phase effects the replacement of the stored d, values by b, values where 17:14:7KKIb,y4=Zdt,(y)k_oA second set of registers ao, a,, a2, ..., aK is employed for this purpose. For j = 0, 1, 2, ..., K the register isset equal to c' k", the coefficient of y' in the polynomial expansion of t,(y). These coefficients are zero for j < kand ifj and k have unlike parities. They are constructed using the recursion formula 17:14:8 Setr=sSet s = rGo to (3)with coo' = cnI = 0.oIn the fifth phase the coefficients ak that satisfy 17:14:9 Input J >Input X ,"Input xf Input K -, x, (as cue)Input f >v. _(2k - 1)JCV a(k - 1)(J + k)nk(J-k+l)k(J-k+l) KKKZ akx` =` bkVk= I(x - u)`,-ok==.ok-0 (-»»»Set h=xoSet u = (xj + h)/2Replace h by u - hSet k=K(1) Set bk = 0Replace k by k - 1If k a 0 go to (1)----------------------Setj=-1(2) Replace j by j + IIfj>Jgoto(4)Set k = r = 0Set s = 1Output [(2j/J) - 1)h + u«««Set f=fReplace b, by b, + f(3) Replace k by k + IIf k > K go to (2)Sett= r+ (sJ- rJ - 2sj)(2k - l)/[k(J- k + 1))Replacebk by bk + ifi IA ata ----------------------------Storage needed: As written,the algorithm requires 2K +12 registers: for the param-eters J, h, u, K, k, bo, b b2,..,bK. j, r, s. f r, ao, a,,a2, ..., ax. However, the min-imum number of registers re- quired for K z 3 is 2K + 8because r and ao may share thesame register, as may s anda,, f and a2 and r and a3. 147THE CUBIC FUNCTION x3 + ax' + bx + c AND HIGHER POLYNOMIALS (4) Set k = 0--------------------------Replacebk by bk/(J + I)(5) Replace k by k + 1If k > K go to (7)Replace bk by bk(2k + 1) s Setj=O °(6) Replace j by j + IReplace bk by bk/(J + j)Ifk<jgoto(5)Replace bk by bk(J - k + j)Goto(6) ----------------------------(7) Set ao - k = 1(8) Set at = 0Replace k by k + 1If k:Kgo to (8)Setj=k=0Go to (10)(9) Replace j by j + 1 s Ifj>Kgoto(ll)Setk=jSet ale = at-,(2k - l)(J/k)/(J - k + 1) u0Replace bk by bleak(10) Replace k by k + 2If k > K go to (9)Set at = ak_. - (at-2 - ak_,)(2k - 1)(J/k)/(J - k + 1)Replaceb, by b; +bleakGo to (10) ----------------------------(11) Setk=K+ 1(12) Replace k by k - 1Set at = k!bk Set) = k(13) Replace j by j + 1If j > K go to (14)Replace at by at + bj!(u/h)'-k/(j - k)!Go to (13)(14) Replace at. by ale/[k!(-h)']Output atax,a5-1, ..., as < < <If k It 0 go to (12)------------------------Test values:J= 8,xa=0,x1=80,K = 3, andx01020304050607080I 142026222632325217:14 Output: a3 = 0.0003754, a2 =-0.04464, a, = 1.803 and as= -0. 1010 are calculated via a binomial expansion of the (x - u)k term. The expression used is 17:14:10 _IKb1j!u' katkr .(-h)kk, .bk +(j - k)! \hThe at coefficients are output in the sequence aK, air-,, ..., a,, ao and also remain in the at registers. They maybe used in conjunction with the Section 17:8 algorithm to compute values of f(x). CHAPTER18 THE POCHHAMMER POLYNOMIALS (x)n This family of polynomials has very simple recursion properties [Section 18:5] and plays an important role ascoefficients in the series that define hypergeometric functions, discussed in Section 18:14. 18:1 NOTATIONThese polynomial functions were studied in 1730 by Stirling and later by Appell, who used the symbolism (x,n).The name Pochhammer polynomial recognizes the inventor of the now conventional (x), notation. An alternativename is shifted factorial function. 18:2 BEHAVIORThe Pochhammer polynomial W. is defined for all arguments x and for all nonnegative integer values of n [butsee Section 18:12 for a generalization]. In common with other polyl nial functions of degree n, the Pochhammerpolynomial adopts all real values when n is odd. but has a restricted range for even n.The Pochhammer polynomial (x) has exactly Intl(n - 1)/21 maxima, lnt(n/2) minima and n zeros, these latteroccurring at x = 0, -1, -2, ..., -n + I as illustrated in Figure 18-1. Outside the range -n < x < 1, (x), generallyincreases rapidly in magnitude. 18:3 DEFINITIONSThe Pochhammer polynomial is defined as the n-fold product 18:3:1 n=1,2,3,...-osupplemented by the definition18:3:2of the Pochhammer polynomial of zero degree. 149 18:4 THE POCHHAMMER POLYNOMIALS W. ISO An alternative definition18:3:3 n=0.1.2....relates Pochhammer's polynomial to a factorial function [Chapter 2] and a binomial coefficient [Chapter 61.The exponential function [Chapter 26] is used to define a generating function [see Section 0:31 for (x),: 18:3:4 18:4 SPECIAL CASES(1 - t)-` = exp[-x ln(I - t)] _ (x)n. The first eight Pochhammer polynomials are18:4:1 (x)o = 151 THE POCHHAMMER POLYNOMIALS W. 18:4:2 (x), = x 18:4:3 (x)2 = x2 + x 18:4:4 (x), = x' + 3x2 + 2x18:4:5 (x)4 =x4 +6x'+I1x2+6x 18:4:6 (x)3 =.r' + 10x' + 35x3 + 50x2 + 24x 18:4:7 (x)6 = x6 + l5x' + 85x' + 225x' + 274 + 120x 18:4:8 (x), = x' + 21x6 + 175x° + 735x' + 1624x' + 1764x2 + 720x 18:5 LNTRARELATIONSHIPSPochhammer polynomials obey the reflection formula18:5:1 (-x), _ (-1)"(x - n + 1).and the duplication formula 18:5:2(2x). =2"(x).,z(x+ 1.r. 12"(x),.-na(x+ - I2/n=0,2,4, n = 1. 3. 5....18:5 Formulas similar to 18:5:2 may be derived for (3x)., (4x). and generally for (mx), where m is an integer.Equation 18:5:2 may be reformulated as a duplication formula for the degree of the Pochhammer polynomial llI + x18:5:3x2))and similarlyll(l 18:5:4(xh.-i =4"x( 2 .r).(I+ 2/.2~.'(2)\I2x)JLikewise, formulas for (x)3., (x),,.1, (x)3.,2, (x),., etc., may be derived.Simple recursion formulas exist for both the argument 18:5:5 (x + 1). _(1+n)W.Xand the degree18:5:6 (x).. 1 = (n + x)(x). = Ax + 1).of a Pochhammer polynomial. The last formula generalizes to the expression 18:5:7 (x).=J(x+lm)._,.n > mW.(x + n).,_.n < mfor the quotient of two Pochhammer polynomials of common argument. Likewise. 18:5:5 may be generalized tothe formulas 18:5:8(x+m)._(x+n)m=0.1.2....W.W. 18:6 THE POCHHAMMER POLYNOMIALS W. and152 (Xm).(Xm).(I -x)18:5:9 ==m=0,1,2,... W.(x-m+n)(1 -n-x)for the quotient of two Pochhammer polynomials of common degree whose arguments differ by an integer. Theselast formulas are useful when m is small.Addition formulas exist for both the argument and degree of a Pochhammer polynomial. The expression 18:5:10 (x+y).= (n)(x),(y),-,Jis known as Vandermonde's theorem, while the rule 18:5:11 (x),.. = (x).(x + n)is a simple consequence of the definition 18:3:1 (or of the recursion formula 18:5:6). 18:6 EXPANSIONSThe polynomial expansion of a Pochhammer function 18:6:1W. _ (-1)" i, S,"'(-x)"m-1-0involves the numbers S,", known as the Stirling numbers of the first kind. A small display of these integers appearsin Table 18.6. 1. It may be extended by use of the recursion formula for these Stirling numbers18:6:2S"',=S;," -nS,'"'n=0,1,2,...m=1,2,3,...The same formula is utilized in our algorithm for S,", which may be found in Section 2:14 where it is incorporatedinto the algorithm for Stirling numbers of the second kind.Stirling numbers of the first kind satisfy the summations 18:6:3S-'-0n=2.3.4,,.. 18:6:4 1S.-I=n!n= 1,2,3,... For further information and an extended table see Abramowitz and Stegun [Chapter 24]. Table 18.6.1 Sa'S; 's 'Sr 'S':S1.1,$.-IS',-SrSa ,S',o' m010000000000M = 101-12-624-120720-504040320-362880200 1-311-50274-176413068-1095841026576m = 30001-635-2251624-13132118124-1172700m40000 1-1085-7356769-67284723680 M- 5000001-15175-196022449-269325 M6000000I-21322-453663273m700000001-28546-9450 15318:7 PARTICULAR VALUESTHE POCHHAMMER POLYNOMIALS W. 1 _ n(-m).(zl.(-n).2m = 1, 2. 3.18:9 1(0).2)(1).(2).(n + 1).(z). n=0 1I 1IIIIIIII-(2n - 3)!!(2n)!(2n)! n- 1.3.5.._z_n!000n!(n + I I!Y4'n! 2n!-(2n - 3)!!(2n)!(2n)! zni 00n!(n + 1)!2' 44"n! 2n! 18:8 NUMERICAL VALUESThe following algorithm, based on definition 18:3:1, is exact:Input x >>Input n >> f - W. <'Setf= 1Ifn=0goto(2)(1) Replace f by fxReplace x by x + 1Replace it by n - 1Ifn*0goto(1)(2) Output f 18:9 APPROXIMATIONSStorsee needed: x, f and itInput restriction: The degree it must be a nonnegativeinteger. Test values:(17)0 = I[ (-0.1)7 = -55.9122291 When x or it or both are of large magnitude, relationship 18:12:1, together with results from Section 40:9, may beused to derive asymptotic expressions for Pochhammer polynomials. Thus, for large it and modest x, the asymptoticapproximationn`-'n!I I +x(x - 1)+x(x - 1)(x - 2)(3x - 1)+ ... in -- x 18:9:1W. _f (x)2n24n2Jis useful and shows, for example. that 18:9:2A second asymptotic approximation for (1)" 18:9:3n!~ Vnnn-.x follows from 18:7:7, providing a link between what are probably the three most important irrational numbers: a,f and e.For large it and x close to -n/2, the Pochhammer polynomial approximates a sine function [see Chapter 321: 18:9:4(x)n -n' e2p(`n) sin('nx)+ 2 <<I2 -. x1 The development of this sinusoidal behavior is evident in Figure 18-1, even for n as small as 4. 18:10 THE POCHHAMMER POLYNOMIALS W.18:10 OPERATIONS OF THE CALCULUS154 The rules given in Section 17:10 apply since (x)" is, after all, a polynomial function. In particular, the effects ofdifferentiation and indefinite integration are 18:10:1dW. = W.1= (x)"[4(n + x) - t[r(x))dxj-0 x + jwhere 4 is the digamma function [Chapter 44] and 18:10:2f(:)dt= (-1)ESi=i-1+ li-2jwhere S," is a Stirling number of the first kind, as discussed in Section 18:6.18:11 COMPLEX ARGUMENTBecause of 18:6:1, an expression for (x + iy)" may be derived from equation 17.11:1 on replacement of ai by theStirling number (-1 8:12 GENERALIZATIONSPochhammer polynomials can be expressed as the quotient of two gamma functions [Chapter 43) 18:12:1 W.C(n + x)=1700whose arguments differ by the nonnegative integer n. Hence, the quotient fly + x)/f(x), with y unrestricted, isa generalization of the Pochhammer polynomial. A more restricted generalization, permitting n to be a negativeinteger, may be developed from recursion formula 18:5:4.18:13 COGNATE FUNCTIONSFactorial functions [Chapter 2), binomial coefficients [Chapter 61, gamma functions [Chapter 43], complete betafunctions [Chapter 43) and Pochhammer polynomials are all closely interrelated.A factorial polynomial, as defined by Tuma [Section 1.03):18:13:1 n=1.2.3,...is a Pochhammer polynomial of changed argument. One has (x)" = X'";(x) and X'(x) = (-h)"(-x/h)". The symbolx<"' is also used for the factorial polynomial defined as18:13:2xl"1=x(x-Yet another confusing symbolism, due to Kramp, is18:13:3None of these notations is employed in this Atlas. 18:14 RELATED TOPICSPochhammer polynomials occur in the coefficients of so-called hypergeometric functions. The most general rep-resentation of such a function is as the sum 1 R:14: I±(a,), (a2), (a3)i ... (ax),xi7 (ct);(c2);(c3)t ...(CL), 155 THE POCHHAMMER POLYNOMIALS (x), 18:14Table 18.14.1 K. number ofL, number ofValues ofValues ofnumeratorialdenominatorialnumeratorialdenominatorialparametersparametersparametersparameters 000 10 a I 2Va aIc 1. C2 1a. b I 22a. b1, c33a,. a,, a,I. C" c,KLa,. a a,,1. C1. C'.a,..., c,Chapter orName of function and usual notation section no.infinite geometric sum I/( I - x) 1:6exponential function explxl 26incomplete gamma function 45r(v)cxp(x)y"rn - 1:x)binomial function (I - x)" 6:14incomplete beta function 58 (c - Ilx'-'tI - x)'-"-'B(c - I:I +a-c;x)Kummer function M(a:c:x) 47Tricotni function 48(-.rrU(a;I+ a - b:-1/x)Gauss function F(a.b:c:x) 60Clausen functionsgeneralized hypergeometric 60:13function F,-,(a,..... a,:c....ca:x1 where x is the argument. a,, a, a3, ..., ax are prescribed numeratorial parameters and c c2, c3, .... cL areprescribed denominatorial parameters. If one of the numeratorial parameters is a negative integer, the 18:14:1series may terminate, but generally hypergeometric functions are represented by infinite series.If the number of denominatorial parameters exceeds the number of numeratorial parameters, that is. if L > K,the hypergeometric series necessarily converges for all finite values of x. If L = K, convergence is generally limitedto the argument range (xl < 1. If L < K the series diverges (unless it terminates) for all nonzero arguments, butit may nevertheless usefully represent a function asymptotically [see equation 37:6:5 for an example].The name "hypergeometric" arises because the 18:14:1 series can be regarded as an extension of the geometricseries [see equation 1:6:5J to which it reduces when K = L = 0. Other small values of K and L give rise to certainwell-known families of functions, discussed elsewhere in this Atlas, as well as a number of generic classes offunctions. Table 18. 14.1 summarizes these types of functions. Notice that the so-called generalized hypergeomerricfunction is not, in fact, fully general because one of its denominatorial parameters is constrained to be unity.Hypergeometric functions are important because most of the so-called special functions of mathematical physics(i.e., functions that arise as solutions to differential equations of practical importance) are instances of this class.The cylinder functions of Chapters 49-56 provide examples, all being K at 0, L = 2 hypergeometric functions.Cases with L = 4 and K = 0, 2 or 4 occur in the theory of elasticity and are encountered in Sections 47:11, 53:6and 55:6 of this Atlas.Functions that are expressible as hypergeometric series may be evaluated by exploiting convenient features ofexpansion 18:14:1. If the abbreviation 18:14:2 G =(a, + j)(a2 +j)(a3 + j) ... (aK + j)(Cl + j)(cr + j)(cs + j) ... (cr. + j)is adopted, then a hypergeometric function of argument X may be represented as18:14:3where R, is the remainder when the hypergeometric series 18:14:1 is truncated after the j = J - I term. If J ischosen to be larger in magnitude than any of the a or c parameters, and also to exceed IXIIIIL-10. then the dominant term in G, is JK-L. Unless L < K, the R, remainder will become steadily smaller in magnitude as J increases. For large enough J, the R, term may be well approximated by a geometric series18:14:4 +G,X+GIG, IX2+G,G,.,G,.2X'+Go = GoG,GI ... fJ,-,X'[1 +jK-LX + j2,1-21X2 + JM-XX) + ...] .I - JK-LX 18:14 THE POCHHAMMER POLYNOMIALS (x), 156With approximation 18:14:4 incorporated when appropriate, expression 18:14:3 forms the basis of a universalhypergeometric algorithm that may be used to calculate numerical values of a wide variety of functions. Thealgorithm, which is detailed below, requires the following inputs:(a) K, the number of numeratorial parameters;(b) L, the number of denominatorial parameters;(c) a a2, a3, ..., aK, C1, c2, c3, ..., cL, the values of the parameters (these are stored serially in the algorithm asb1, b,, b3... , bK.L); and(d) X, either the value of the argument x or some simple variant of it, such as I - x or x2/4. Input K >>Input L >>Seti=0la»»(1) Replace i by i + 1If i > K + L go to (2)Storage needed: 7 + K + L registers are required tostore the following quantities: K, L, i, br, b2, b3, ...,bK+L+ X, t, f and j. Input b,>parameter Input X >>Go to (1)(2) Set j = 0I Intervention to halt output is necessary.1 Y»»Setr=f= IGo to (4)(3) Replace t by t/(b; + J)(4) Replace i by i - IIfi>Kgoto(3)Ifi=0goto(5)Replace t by t(b; + J)Go to (4)(5) Replace t by tXReplace f by f + t Replace jbyj+lIf frac(j/5) * 0 go to (7)If I >- j L-K/X go to (6)Set i = r/(1 - (jL_x/X)1(6) Output f - iApproximatef(x) value(7) Seti=K+L+IGo to (4)Test values:input: K = 1, L = 1;parameters: 1, Z; X = 4output: 1.57229437, 1.57081694,1.57079671,1.57079633,1.57079633(=Tr/2), ...input: K = 0, L = 2;parameters: 21,1; X = 1output: 3.76219871, 3.76219569.3.76219569(=cosh(2)), ...input: K= 1,L=0;parameter: 1; X = 0.04output: 1.021340580, 1.0213407431.021340745,1.0213407441.021340744(=10 daw(5)), ... Because the performance of the algorithm depends on many factors, it is not possible generally to predict thevalue of J required to achieve a sought precision in f(X). Accordingly. the universal hypergeometric algorithm setsJ = 5. 10. 15, ... and generates an output for each of these J values. leaving it to the operator to observe whenconvergence has been achieved, or to judge when the output value has sufficient accuracy for his purpose. In somecases, convergence may occur but be tediously slow. Note that when L < K, the algorithm may fail to producean acceptable value unless X is sufficiently small; in this case the output values may initially tend toward the correctanswer but later diverge.Table 18.14.2 lists some of the functions that may be evaluated using the universal hypergeometric algorithm.For many of these functions alternative algorithms will be found in the appropriate chapter of this Atlas. Such"custom" algorithms are generally more satisfactory than the universal algorithm, but the latter has the supremeadvantage of versatility. 1S7 THE POCHHAMMER POLYNOMIALS W.Table 18.14.218:14 No. of ChapterPuamnen ortiKLParameter valuesb.. b,. b,.... b,,.,. Xsought run"" ftxi Restrictionsonsec 31: 1:72 2 1 I < 1:7 22 113 3IG 1:7 22 2 2I .... .I : 2.... 2- 1'g.)n - 1. 2. 3.... 3I133n - 1fi(n)n - I. 2. 3.... 322 22r.I %x-bx(I m x)°r-1<x<1 6:14 00-cb4+ckl < I.-Al 7 00 1 !<t > 1061 76xbx In;I-bxr(b4 + of/' 141 < k-/b{ I I IIn;I-rbx(bx + cP/b'.1 141 > k/bl I I 1: I-V'bx + </V cki < r/bl 12 1; 1-rV bx + r/V lu :ri > Ic/bl 12z Ibx- 1 141 < ION 12 1cVbicNc2'1blVb4+1>k/M 12 IV:1x-1x5I<z 13 II:I±-Va'=x'/a-a<x<a14.15I_x22+I: 1.1 -1 <x<3 21 222-4. -n; -2n, I n!P.(x)/12n - wu - w x < -1 or x > 3 21 1-xn1 +nI222 .2: Z .1(-[)*"n!!P,Ix)/(n - U!! IW l < 1; n = 0. 2. 4.... 211_n 2+13n-1I: 2222:I .2r`!!p tx)/<n = I. 3. 5.... 21!!II221.!P.4.)/(,- - I i!!.1 (x1 > 1 21222 22-n, 8;211ZxT,(x)-1 < r< 3 22 III_1/2222x+1 1+.x 22-n.n + 2: 1.i1x+I-1<x<3 222 18:14 THE POCHHAMMER POLYNOMIALS W. 158Table 18.14.2 (Continued) No. ofparxmetrnParameter valueKLb,. b:, b,,b,.,XSough function fW 22-n.n+v+p+I;1,I+v 22-n - v. - n. 1, l+v1.3F-1222x-1I+xZ xp..w/lnnY 1 22-n.n+211: I.-1-+2A-12Z 2-n: 1.1xL.(x)2 1n: 1.I-xexp(-x)L.(x) 2-n: I. I+ p xL;'fr1/(nnp) 2-v: 1,1 LJx)1 - n -a FI,(r)/f2rr_r 2nI-. Ir1Glrl/f-2146fn - w!221 - n3 un!. 2 02 Ia -n -22 21 Hedx)/f2x)' 1:21-x 1; 2L-2x 1.2.2 Infx) 11I;I v 22 1, 1; 2. 2 331. 1, 1.2.2.2 0 1 I 0 -a -n, n + 1;I 02I. 2 02-. 12 102Infx)x_1In(x)r - I Itxx)r-1Ils) - -f - InOlnfxN) In(s)In(x)Iilx) -(- In,(rlxs-1RestrictionsQnplerttf.ectionno. 22 22:12 22:12 22:12 2323 23:12 23:14 24n=0.2.4.... 24N. 1.3.5..-. 24 24:130<xa2 25 250<x+125.37 positive x, large or small 25. 37 25:12 thin(s)1-x 0<s225:13 s-1trim(s)1-x 0<s225:13 :-1xexpel 26nIe.(x)/t 26:13 u * expfx) K.. "AX) n0. 1. 2...26:132xrsslnMxlx284YcorMF1 28AV;-/2 (?n - 11!!4- rfxl/f>O :0 .1 .2 , 28:134 159 THE POCHHAMMER POLYNOMIALS W.Table 18.14.2 (Continued)18:14 No. 0( Chapterpvatnetm oeParanveter values sectionKLb,. b, b,..-. b,_,XSaug1 function f(x) Rcstncsioos I! rsm6lx) 22I.--.r ;2 22 -I<xsI 31 32 221.; 2. 2-4r[arsinhtxl - IM2x11 x > 1 31 22I.32: 2. 214.r(Inl2x1 - atshfx)I x > 1 31 221. 2: 2. 2x4[l12/x) - arseclex)1/x' 0 < x < I 31 I13 -1 l2 22 x atcscKx) I!cl > I 31 13 ananbulr-I<<1 31 2 2 1 I I I---2. x-t 3x arcothlx) vial.,) IA > 1 31 02I.- 32.3324xocsc(x) I-r 102 -,1 tos(s) - - 32. 332 4 sects)3_.' 02 I. - sinclxl 32:132 4 IY 021.2+ n4Vi/2 (2n - x > 0: n = 0. I. 2.... 32:13 1-r 02I .24 U!!x > 0: n - 0. 1. 2.... 32;13 112 0<x<43314 Irwcx)1--2 2 - 1 5 x u 1 35 I3 mint.,)221.2.r -I <x<I 35 2 2 13 1222. 2:I.2 x atccsctxl > I35 I12--x arccot(xl ld a l 352 2 22; 1. 3I-xo)ccos(x)/V 2- 2a-1 <.s 1 35 222 2 1132x_I22 -. I. - atcsec(x) I 35 2 22Zr2x - 2 Einl:)121: 2. 2-x 37 0 1 x exp-x) Hi(x) large 1.rI 373 3-x°SO)322 2 4 x383 3iShifxl1322 2 4 x 18:14 THE POCHHAMMER POLYNOMIALS W.Table 18.14.2 (Continued) So ofp4rametersParameter valuesKLb1. bj, b), ..., bx-ix 331;2.2,2-Al4 3r3 I:22. 24 0. 1-42r 3-420 1. 2 I33.1.3 .7-x-- -42 4 4 I15I3-, 1.-4 2441+V3 3-vI3-:1.-,-2224 V1+ VSI3-.1.-2 2241-v 2-v-4r 22v 3-v-420225 023 -_i4 44 0257- 4 4 4 13-420-4 4i3 S -420447 I3I221.2-r 0 13- 2 2 122 1 -I2160 or Sought function Rx) Reunctions.sawnno. 4 Cin(x)/r 38 4 Chio(x)/r 38 x fi(x) large x38:13 r ghx) large x38:13 Vrt/'_ Ax)/x? 39 \`n/2 Clxl/x 391+vIslO:v) - S(svll IC(O:Y) - Ctrv)) sirt)Cfx:v)] - (aualxlSlx:v) + cc41x)C(+o))1-v41ring + 4)- \r Cmet(x)J/ 2.v<1.x>039:12v<I.x>039:12 v < 1. large x39:12 Y < 1. large x39:12 39:13 //3v'. \2 Fres(x) - crnlr+ n-)/4x' 39:13V x Fralx) large x39:13 \'8a Si Ores(x) large x39:13 V'; aft,)/2x 40 \a exp0r`wrffx)/2x 40 \e x exp(rxrktix) large x40 II4:2.1-rV i I: + terfelx)] 40:13 I3\'4 rI 11_-2'2-r Ix `4 + 2iterfdx)J 40:13 3!rx+ i'6f+x11103 .I:2cedx)V a434:1 3 -12\a r exp(.r)berfc(x) large x40:13 r I \e, exp(.WrfcIV;) large x41 I + Vex exp(xlerf(\ x) e > 041 161 THE POCHHAMMER POLYNOMIALS W.Table 18.14.2 (Continued) No. ofparametersParameter values18:14 KLb,, b. b ., be.,xSuu1N function f(x1 Restnctioru 3 d..(.)0 -r 42 I0 2x daw(x) large x422 13 explx) 12-; I. -x_ daw(V x) 4222 V.v22I- v. x: I. 1+ x Ir B(x.yl v> 043:13 21.v:1+v.2 IIirtv)+7)v-I44 22I.I+r-y;l+r.2 - (Mx) - a(>9t v>oa4 22I- x. I+ s 1x'b'(sl 44:12nx... - . x; I + r..I + x 1 11'n = 2. 3. 4... -44:12 IIv, v+ l- I4G(.)/2 44.13 IIv;I + v-vt('i(vl/2' 44:13 0 11+ r x .e 'exp(xMv;xl 45 I2r. I. 1v-xrx-'7(v;x) 43011 + v r11 + vtexptx)y(wx) -1. -2. -3....45 12I. t + u-rr(I +-1. -2. -3....45 I01- v-Ix-'expxlrfv:x) large x45 12v: I.I+ u-xvIrvx-'7(vsl-dr1`large v45 12;I 2'a)exp(ID,(-x) + D,(x)1/2Vr( v * I. 3, s. 46 ,2:=rv3I4 '_I.- ,r1D.I-x) -v*0.2.4.... 4622 IvI42 1v_2:2. 12r(.exp(D,Ix)I/2V2`z2 * 1. 3.5....46 3-r e.''f(-}expl 4 a,}1D,(-xl- W.)J xV2+0.2.4....46 21v, 12 v:I2 large x46 12 a; c.I A Mta:c:x) 47 I2c - a; c,l-xexp(-x)M(ar;x) 47_Ixr"Uta:r:xl large x 48 2+I+2µ.I II-l46:13 rs"'exDl 2)W.,,(x) large x48:13 I,I Ivfx) 494 02 I, 24I,Lv/ 49 18:14 THE POCHHAMMER POLYNOMIALS (x),Table 18.14.2 (Continued) No. ofParametersParameter valuesb1. bl. b,, ..., br.t XSough function f(xl162 Resu.ctions KL 02 1,1+ R R421x 1"1,1x1 n0. 1. 2....49 4 2 V 2 = exp-x)1 jx) Imp x49_I I213-:I V'2trx exp(-x')1i(x) latgt x492'_2x 02 1,1 + r(t + vN2/ 19,(x)r = -l. -2. -3, ... 50 a I2- + v: 1. 1+ 2.2vAlIK2/x)*tap(x)ldx1v + -I. -2. -3.... 50 12 1- -- + v,I-V2wxexp(-x)1,(x) Urge302v 2 2s 3 x7 021.-+a (2R+I)!!x-"itxlm1.0.1.2.... 50:42 4 11 -1 +1- - v. - - v: 1-txplx)ILIx) large x 5122 21-R. 1+ R: 1 sexp(a IkJxlw = 0. 1. 2.... 51:4-x'02 1.1 lyfx) 52 4-x:02 1. 2 1,(x1 52 4-r 02I.I + n n!(2/x)"1,(x)n - 0, 1. 2, ... 52 4-x'021.1 + r(1 + vK2/x)9,Ix) v-1. -2, -3.... 53 132+ v; 1. 1 + v.I + 2.-JrrYl + vg4/x°YY;1xIv + -I. -2. -3.... 53 13-: 1 - r. 1, 1 + v-x:avcvcfav)1,(xv-Wv * s1, x2. x3, ... 53-x 02 1.3- 4 R (2n +I )!!x""J,(x)R - -1. 0. 1. 2.... 53:4 02 1.1 + v-xr0 + vIC,(xlv a -1. -2. -3.... 53:6 I31-1 3v42aI - a.- - a; -.IPtn:x1 with- - 2large x53:6 4 2 133-12xQ(v:x)3vI42a.a+-.--a.2-a:1.--wither= ---va: -. larger 53:6 2 2 42 I-x'04I2,=. 1.1236bes(xl 553 3-i4 04I. I. -.2256rbei(xl 55 04!, 1.,1-\'e [bee(x) + bei:fx)] 552 161+v-x04t + --r(I + vl(Fe,(xl 55:6222256x 163 THE POCHHAMMER POLYNOMIALS (x). 18:14Table 18.14.2 (Continued) No. o(parametersParameter valuesKLbi, b;. b,..., bx.t X 04Jv3-v222Sough (union r(s) r12 + v)t.-)OeJslx3"'r(*)Btu)(airs) - - 1 - + Ailx)2LLL V3gal(s)310[1;) rBils) 11I- Aifx)2rLV3Chapter/lrsectionRestrictionstto 55:6 02 13 94021.394 5 ?hai(s>2 0_- --- = - (Hi(s) - faifx) - gai(s1I 339s'e 2 II5.I-36 6 4s". 15 31 :I6 6 4x1 4215711II-912 12' 12 12 24x'56 36 56:13 2><Vi expl 3 x':2 IAi(xl large positive x56 211VrV r explBilsllarge positive x56 Ql2a1 IJAI(-xl + cBif -x11.: alarge positive x56 7I I13 173-912 4-I. --*Vx (sBN-s) - cAif-x11. c - cm4s'12' 12 12 122 115-9316.2.6 I4x' 0213-rZ 2 4 33-r02- 2 2 4J S-.r 02 2 43 3-xt02-v-large positive x56 large positive s56 57 57 57 3-r(-+,11-) ,. >057 2 24 22x13 40- - - (Y .4x1 - A.,(x)l Iarge x57 2 II-420--2 2 5--11-4 02,2 20I-I-4xl3 3x'0*2 2v4(hi(s) - Y{.rll A- (hdx) -' 4x11 Y+r r(2 *(1i.1x1 - YJx)IVa3/114r(2+oil-IIJ.r)large x 57 large x 57 large x $7 s > 057.13 12'202-2V; r(I * o)(x)ILJ.r) - 13x)( larger57:13 v'r 2lv; 1+ vxvr-'B)v:osl 0 s s< I58:4 Iu * µ: 1 + v xvs .(t - s) Bl u;µ; s1 0 <- s < I58Ir+ µ; I+ rI- xµs-'(BI v.µ1 - Ble4..xp0 < x s 158I-x22 P.lx)-I<s<359 1 *3I11(5- e)22-.1+-: I. - +r _ x>159222 VvrU + v) 18:14 THE POCHHAMMER POLYNOMIALS (x) 164Table 18.14.2 (Continued) No. ofParameters Sought function fix) RestrictionsChapterot sectionno.Parameter valuesKLb,, b, b,..... b- XrI+ Y122- -: -. 1Xlefts) -1< 159:6 3 1221_V.1 +v;I. -- kW.)- 1< '<159:6 222 2x 221.I.rlef,"Y:1 - I <x < 139:13 22 2.leg'."Vx)-1 <x< I59:13tc - IIBic - I;) - a - r:xlIIa;rFI,ar;xl- -o < x<160:4 X'41 -22a, b. I. r Ra.b:r:x1- < x < 1 60 rr(I1a a. b:l+a-b,I-I\`n r(l+o-h)/Irrl+--b)r( ]a-bs-I.-_.-3.. 60:7I+b)/[r'llllllaba 22a. b:.IZvsr(2l-)r-)Ja + b s -1. -3, -5....W7 Ia*rl+r-a.2.607 20. 1rlr)/ 2' r()r()0.1111d +Ia.....,1. qcr 60:13 221,1.1 Kw -1 61 2 2222-1 1:1.1r'-E1x1-1 cx< 1 6122 n 1x'2VI22 K(x) < x < 1 61 227-1 2 3221.1 Elx)-1 <x<1 61 2 2 aft - x-1 1I13 233 -. 1. I. - x'-Klrldr 0 x 161:10 ._2 nr 11333- - -: 1, 1. -x'-Ewdr 0 s x s 161:10 2'2'22 nxnaw..u; I +u. _I +ur,P4'IXA; I-1< I:n= 1.2.3,...64;I2rtln:uln - 1. 2. 3... 64:13 Operations of the calculus are easily applied to functions that are expressible as hypergeometric functions. Forexample, values of the derivative of any function f(x) that appears in Table 18.14.2 may be calculated by takingadvantage of the ruled(a,),(a,),---(aa,a:...a, dX(2),(a,+ I),(a,+ I),..(a.+ l)18:14:5-X = X' dx dxIc,- 1),(c.+ I),...(cc+ 1),(1),which shows the derivative to be another hypergeometric function with an altered and augmented list of numeratorialand denominatorial parameters. The first two rows in Table 18.14.3 show explicitly how the universal hypergeo-metric algorithm may be adapted to calculate values of df(x)/d.r. To economize on space, the table makes use ofabbreviation 18:14:2.Similarly, but more usefully, the universal hypergcometric algorithm may often be adapted to generate values 165 THE POCHHAMMER POLYNOMIALS (x)" 18:14of the indefinite integral of R.O. The situation is more complicated than for differentiation because the form adoptedby the hypergeometric series for the integral depends on the relationship between .x, the argument of the function.and X, the argument of the algorithm. The most commonly encountered relationships are X = ±x, X = bx" (b is a constant such as I or -2; is is a positive integer greater than unity), X = b/.t^. X = b/x and X = b( I - x). Thesefive possibilities are all addressed in Table 18.14.3. which itemizes the changes needed to enable the universal hypergeometric algorithm to generate values of an indefinite integral of fix). For example, using the Jj(x) entry from Table 18.14.2 and the fourth row of Table 18.14.3, one sees that 18:14:6 IJ' t)dtX oOther operations of the calculus may also be applied to functions expressible as hypergeometric functions.Table 18.14.3 reveals the effects of semidifferentiation and semiintegration; see Oldham and Spanier for other casesof generalized differintegration. Moreover, the Laplace transform [Section 26:141 of the hypergeometric function18:14:1 is yet another hypergeometric functioni(I),(al),(a2),(ar),... (aK),IIS:14:7S j=0(01(C_)j(C1)j ... (CL).S)that has an extra numeratorial parameter.The table suggests that operations of the calculus invariably increase the number of parameters. This is notnecessarily so because, for example, a "new" numeratorial parameter may equal an "old" denominatorial parameter, permitting the two to be "cancelled." Table 18.14.3 No. ofparuneurs Parameter values xAlgonthm evaluates KLa,. a...... ar, c n..... ci Xfix)fdX K+IL+I2.a,+La;+L....c;+l.. .. rxLIX±1Ex)G"dXd.rdx KIL + I1. al. a....., -.r-fu)ds I K + IL + I, u u;...., ur: AMU e n An_I 2n-1b(n - IU'' K +IL+1 ,a,I. a; + I.aK + r, r, + I, r.+I.-... rt + 1. nSbGI[AnIIdsn "K + IL + II. a, + 2. a. o 2..... ax + 2; c, + 222, c. + 2..... ct + 22.2 -f(On- I - - dt K + IL+1I. a a',.... aK; C. C'.C'. 2MI - x)-IJ fields-, 1_ d" K+1t. *II. a,. a2. .... aK; c,. r_..... s,- bx 71x)d-2 a....., a,, c'. n....-1nXIL + 11, u. re. 2 2x ds-,:.f7x1 CHAPTER19THE BERNOULLI POLYNOMIALS Included among the useful properties of the Bernoulli polynomials is their utility for expressing sums of powersof the integers [see Section 19:141. 19:1 NOTATIONB,(x) is the usual symbol for a Bernoulli polynomial of degree n and argument x, but 13,(x) is sometimes used byauthors who adopt the "rival notation" [see Section 4:11 for Bernoulli numbers. The name "Bernoulli polynomial" and the symbol 4A,(x) have been used for the quantity that we represent by B,(x) - B., where B. is the n"Bernoullinumber [Chapter 4]. 19:2 BEHAVIORBernoulli polynomials are defined for all nonnegative degrees it and all real values of the argument x. However,0 s x s 1is the most important range of the argument for B,(x).For even it, B,(x) never exceeds the Bernoulli number B, in magnitude in the range 0 x s 1:19:2:1 IB,(x)1s1B,10<xsln=0.2.4....and equals B. at both ends of this range. Apart from Bo(.x), all Bernoulli polynomials of even degree display alocal extremum [see Section 0:7] at x = z, as exemplified in Figure 19.1. Bernoulli polynomials of odd degree (except B1(x)) vanish at x = 0. x = I and x = 1.The magnitude of B,(x) for 2 is n <_ 11 and 0 s x s 1 is small, never exceeding }. Figure 19-2 shows thetypical behavior of Bernoulli polynomials outside these ranges. 19:3 DEFINITIONSThe Bateman manuscript [Erd6lyi, Magnus, Oberhettinger and Tricomi, Higher Transcendental Functions, Volume1, pages 38 and 391 gives several integral representations of the Bernoulli polynomials, but their usual definitionis via the generating function 19:3:1 t exp(xt)t"T, B,(x) -exp(t) - I=un! 167 19:3 THE BERNOULLI POLYNOMIALS !4s(x} ....:.-0.02:....:..... 0 :....:. ..:....:'1C ':-0.04//...........:....:1 ....................... -006..FIG 19-1 :..:.... :..... ........ .............. :.............................:....:....:.-(L 10Bq(x):Bz(x): : .:.... :.... :.... :NI- 20FIG 19-2.............. .15 iJB.(x)...:....:....1/.,10(x)............I... 5 B7 (x)..................-10 Another definition 19:3:2i ( n)B,x"-'n = 0. 1. 2, ..J-0)168 relates the polynomial coefficients to binomial coefficients Chapter 61 and Bernoulli numbers [Chapter 41. 169 THE BERNOULLI POLYNOMIALS B^(x) 19:519:4 SPECIAL CASESThe first eight Bernoulli polynomials are19:4:1 19:4:2 19:4:3 19:4:4 19:4:5 19:4:6 19:4:7 and 19:4:8Bo(x) = 1 1B1(x)=x--2B2(x)=x2-x+6 3x2xB3(x)=X3-+22B4(X)=x4 -2x3+x2-1305x'5x3xBs(x) = x3 -+ 3625x4x21B6(x)=x6-3x3++2242 B7(x) = x7 -7x6+7x3-7x3+x----2266Coefficients of all terms in B^(x) for n 5 15 are listed in Abramowitz and Stegun (Table 23. 1J. 19:5 INTRARELATIONSHIPSThe Bernoulli polynomials have even or `odd symmetry about x = 1 /2 19:5:1B^(!-x)=(-1)^B^(2+x)n=0,1,2,...and obey the reflection formula19:5:2B.(-x) _ (-1)^[B^(x) + nx^-'Jn = 0, 1, 2'...The general argument-addition and argument-multiplication formulas 19:5:3 and 19:5:4B^(x + y)(n.)B;(x)y i)B^(mx)=m m=1,2,3,...-o(mhave the important special cases19:5:5 B^(x + I) _('7)B1(x)= B^(x) + nx^' andr19:5:6 B^(2x) = 2^-'IB*(x+ 2), 19:6 THE BERNOULLI POLYNOMIALS B.(x)Equation 19:5:5 leads to the summation formula 19:5:7 ± (n + I )BJ(x)= (n + 1)x"J.ajJwhen n is replaced by n + 1. 19:6 EXPANSIONSBernoulli polynomials may be expanded as finite power series in x:(.-s 19:6:1B(x) =-2x' +n(nl2 1)x30(4)+ 426-n+n(n - 1)B.-,x2 + nB.-Ix + B.2!in (x - 1):/1n(n - 1) r 17 (n1l19:6:2B"(x) = 1 x -224\x2/"+ 2404)(x2J"-170 r1n(n - 1)l(IlI 12!B"\x-2/ -r L(1-2;__)nB"-i lj(x21) -(I2"')B"or in (x- 1): 19:6:3B"(x)=(x- I)"+-(x-1)"_'+n(n12 1) (x- w-2-30(4)(x-1)"-0+...rn(n1)B"__J(x - 1)' + [nB"_,)(x - I) + B.IL2!In these series many terms vanish because the Bernoulli number B. [see Chapter 4] is zero when n = 3, 5, 7, .. .Expansion 19:6:2 can be written concisely in terms of the eta numbers (Chapter 31 if q(0) is interpreted as 1:I 19:6:4B"(x)=2n!(-1)Jn(n-!)l(2ar)'j =0,2.4,...,n--±-22A Fourier expansion [see Section 36:6] of a Bernoulli polynomial leads to the general formula 19:6:5B"(x) = -2n!(2ja)cos(2jwrx-2)n = 2, 3, 4, ...0:5 Xs Ii-which simplifies to 19:6:6B"(x) = (-1)'"'31''2n! i (2ja)-" cos(2jarx) n = 2, 4, 6, ...0 S X :S Ii.,or to 19:6:7B"(x) _ (-1)'""'n2n! i (2jir) sin(2j rrx)n = 3. 5. 7....0 s x s IJ='according as n is even or odd. The last formula may be extended to include n = I if the points x = 0 and x =are excluded. 171 THE BERNOULLI POLYNOMIALS B"(x) 19:9Table 19.7.1 n-81W-.0B.(-11Ba 1)3.181048.4)B.(!)8,11)e.)4)8,12)B,Ini "B.B.F-- a.B. 19:7 PARTICULAR VALUES0"E-U (Y -')B.t B.Y' 1 B.B,-8.n ' B. . The particular values of B,(x) depend on the parity of n and are mostly related to the corresponding Bernoullinumber B" [see Chapter 4) or the Euler number En_1 [see Chapter 5), as in Table 19.7.1.19:8 NUMERICAL VALUESFor small values of n, it is easiest to calculate Bn(x) using the exact formulas of Section 19:4. For n 5, thefollowing algorithm may be used. The green segments may be omitted if the argument lies in the 0 5 x 5 1 range. Input n >>Input x >> f = BI(x) <I>>>>>Setf=g=0(1) If x<- 1 go to (3)Replace .r by x - 1Replaceg by g +nx"-'Go to (I)(2) Replace g by g - nx"Replace x by x + I(3) Ifx<0goto(2)Set j = 3 + lnt[30/(3n - 13))(4) Replace f by f + cos[90(4jx - n)]/j"Replace j by j + IIfj *0goto(4)Replace f by [-2n!/(2,rr)'IfReplace f by f + gOutput fFs-t,,age needed: n. x. f, g and jInput restrictions: The degree n must be an in-teger not less than 5. The argument x is unre-stricted if the green commands are included.Use degree mode or change 90 to rr/2. Test values:B,1(0.25) = 0.13249660B,/(1/6) _ -264.561616B, , -J7= 0B-or) = 690.849557 The algorithm uses expansion 19:6:5 with the infinite summation limit replaced by an empirical function of n thatensures an absolute algorithmic accuracy of 6 x 10-B for n ? 5. The relative error may exceed 6 x 10-1, especiallynear the zeros of the Bernoulli polynomials. The colored portions of the algorithm make use of the recursion 19:5:5,or the equivalent19:8:1 B"(x - 1) = B,(x) - n(x - 1)"-'to bring an arbitrary argument into the range of applicability of the expansion.19:9 APPROXIMATIONSI I Approximate values for many Bernoulli polynomials in the range -0.1 5 Bn(x) 5 0.1 can be read directly fromFigure 19-1. For large n and arguments between zero and unity, the leading term of expansion 19:6:5 provides theapproximation 19:10 THE BERNOULLI POLYNOMIALS B,(x) 172 19:9:1 -2n!naB,() = - cos 21Tx - - n -°0:5X5 1 (2,rr)"2For x values close to 0, i or 1, approximation formulas are readily derived from expansions 19:6:1-19:6:4.For x values of magnitude large in comparison with n, the approximation 19:9:2 follows from 19:6:2.B,(x) - (X -28-bit precision 19:10 OPERATIONS OF THE CALCULUSxZ>_ 2.3(n - 1) Additional to the general formulas of Section 17:10 are the following special results for the operations of differ-entiation and integration of Bernoulli polynomials 19:10:1d"dx"B"() = n' d 19:10:2 dxB,(x) = nB"_,(x) 19:10:3 B"(t)dt =B,_1(x) - B,_1 ron + l 19:10:4 nx*0 1 19:10:5 I'B"(t)dr = fo0n=0n= 1,2,3,... 19:10:6Jf B"(t)B.(t)dt=m!W !B.."m,n= 1,2,3,...u (m + n)! 19:11 COMPLEX ARGUMENTBernoulli polynomials are defined for real arguments only. 19:12 GENERALIZATIONS A class of functions defined by the generating functiont' exp(xt)19:12:1 2,... is known as Bernoulli pohnomials of order m [see Korn and Kom, page 824]. They represent a generalization ofthe Bernoulli polynomials discussed in this chapter, which are of first order. A still more general set of higher order Bernoulli polynomials is discussed by Erdelyi, Magnus, Oberhettinger and Tricomi (Higher TranscendentalFunctions, Volume 1. pages 39 and 40]. 173 THE BERNOULLI POLYNOMIALS B.(x)19:13 COGNATE FUNCTIONS19:14 Bernoulli functions are closely related to the Euler polynomials that are the subject of Chapter 20. Equations 20:3:3and 20:3:4 make this connection explicit. 19:14 RELATED TOPICSFrom integrals 19:10:4 and 19:10:3 it follows that 19:14:1 B". (m+x)-B"+,(x) " '+...+(m-1+x)=x+(1+x)+(2+x) n+1The most important applications of this general formula arise by setting x equal to unity;++ (-19:14:2 1" + 2" + 3" + + m" =B"+ (m1)1)'B"+1+ 1n = 0, 1, 2, p... or a moiety: 19:14:31" + 3" t 5" + - + (Zm - 1)" 1)2"B"+,(m)(2"1)B"., = n + 1n=0.1,2,...Further simplification occurs if is = 2. 4. 6. ... because B"_ r is then zero. Some examples of these finite summationsare presented in Section 1:14. See Section 20:14 for cases of alternating signs. CHAPTER20THE EULER POLYNOMIALS En(x) Certain series involving the natural numbers can be conveniently expressed using Euler polynomials, as explainedin Section 20:14. 20:1 NOTATIONEuler polynomials of degree n and argument x are generally denoted E (x), although the symbol E,(x) is occasionallyencountered. The E (x) symbolism is also used to denote Schldmilch functions, which are quite unrelated to Euler polynomials. The Schldmilch functions are briefly discussed in Section 37:14, but they are not used elsewhere inthis Atlas. 20:2 BEHAVIORThe Euler polynomialsare defined for all nonnegative integer n and for all real values of x. Figure 20-1shows the behavior of the first few Euler polynomials in the range 0 < x t 1, which is the most important rangefor these functions. Over this range the inequality 20:2:1 0E , (Enx )20 5 x s 1n= 0, 2, 4, ...is valid for even n, E. being the nth Euler number [Chapter 51.As is evident from the diagram, all Euler polynomials of even degree (except Eo(.r)) are zero at x = 0 andx = I and exhibit a local extremum [see Section 0:7) at x = 1. In complementary fashion, all Euler polynomialsof odd degree are zero at x = 1 and (except for E,(x)) display a local maximum or minimum at x = 0 and x = I.Figure 20-2 demonstrates the behavior of typical Euler polynomials outside the 0 rt x -S I range. 20:3 DEFINITIONS Euler polynomials are defined by the generating function [Section 0:31 20:3:1 l2+xexp(t)E.(x)n!-o 175 p,01 a O* O O O* O a 00444444444r44444444444444444444 ... . .1. 1 .. . . .'. . . . . .. . . .. . . . .1.0:....:....:............................. .................:. 0.9 FIG 20-1 : :::....:....:... ........:... :.............. .......0.8.......... :.... ................. ........................ .0.7 : '................ .................... ... .... . ....:... :....:....:....:....:.............: ..;....: ... .... 0.5: SCx: j ............. 0. 3 177 THE EULER POLYNOMIALS E.(x) 20:4 An alternative definition in terms of Euler numbers E; [Chapter 51 is and2"''E.(x) _ -n+1E.(.x)=2(n)(2x-1)"-'E, (x'I)B.,1 (2- B.,,(-x)n = 1. 2. 3, ... 20:3:4 n + 11B,-,(x) - 2`1313.,,(2)Jn = 1, 2. 3, ...relating Euler polynomials to Bernoulli polynomials (Chapter 19] also serve as definitions. 20:4 SPECIAL CASESThe first eight Euler polynomials are20:4:1 Eo(x) = 1 20:4:2 E1(x)=x-2 20:4:3 E2(x) = xz - x3x2120:4:4 E3(x)-x3-+42 20:4:5 E4(x) = x4 - 2x3 + x5x45x2120:4:6 E,(x)=x5-+ 222 20:4:7 E6(x)=x6-3x5+5x3-3x 20:5 THE EULER POLYNOMIALS E"(x) and7x635x'21x217 20:4:8 ET(x) = x' -++2248Coefficients of all terms in E"(x) for n 5 15 are listed by Abramowitz and Stegun [Table 23.11. 20:5 INTRARELATIONSHIPSThe Euler polynomials possess even or odd symmetry about x = Z: 20:5:1and obey the reflection formula20:5:2The argument-addition formula(-1)"[2x" - E"(x)] 20:5:3 E"(x + y) =I " JE,(x)ti "Itohas the special cases 20:5:4E"(1+x)_\nlE,(x)=2x"-E"(x)_(-I)"E"(-x). andno 1n 20:5:5 E"(I2+xl -x"!)E,(2x)'Because all Euler numbers of odd index are zero, about half of the terms in the last summation [which is equivalentto 20:3:21 vanish.The argument-multiplication formula takes different forms: 20:5:6E"(mx)=Z(-1)'B".,Ix+- l m=2,4,6,... n1 ;_(1`m//,(20:5:7 lm= 1,3,5,...maccording to whether the multiplier is even or odd. Definition 20:3:3 is a special case of formula 20:5:6. 20:6 EXPANSIONSDefinition 20:3:2 leads immediately to the polynomial(1n(n - 1) 15n(n - 1)(n - 2)(n - 3)20:6:1E"(x) = 1 x -8x-2!"384(x- 2 +n(n - 1) E_2 InE"_,I+E.lx --+-lx--l-22"-' \22"-'\2/2"in x - Z; all terms containing odd-indexed Euler numbers are zero. Polynomials in x: 20:6:2 E"(x)=x._2x.,+4(3)x" ,_2(5)x"`+8171x" z+ 179 THE EULER POLYNOMIALS E,.(x) 20:8or x - I : 20:6:3E"(x)=(x- I)"+Z(x- 1)'-'-4(3)(x- 1)"'+2(5)(x- I)" s- 1717)(x- I)"-'+... g \7may also be written; these contain (n + 2)/2 terms if n is even, or (n + 3)/2 terms if n is odd; see Section 20:4for the coefficients in expansion 20:6:2 when n adopts specific small values.The Fourier expansion (see Section 36:614n! nir 20:6:4E.(x) = -(2j +sin (2j + 1)ax -2]n = 1 , 2, 3 , ...0x1reduces tox(- I)"1-4n!sin[(2j + 1)inx] 20:6:5E"O=r.-Ii2j+I".In=2,4,6,...05xsl when the degree is even, and to(-1)4a`11n4n! . cos[(2j + I)irx]20:6:6 E"(x) = I "1n = 1, 3, 5, ... 0 <- X :S I'=o(2j + t)when n is odd. 20:7 PARTICULAR VALUESAs is the case with the Bernoulli polynomials [Chapter 19], particular values of the Euler polynomials depend onthe parity of n and are most easily related to the Bernoulli numbers B. [see Chapter 4] and the Euler numbers (seeChapter 51. Some values of E"(x) may be found in Table 20.7.1. Table 20.7.1 "=0r--' - 2".IE.4-1)E.I JIE.401E01t1E.411E.Ij1E.121E.l=1 a." -2-I2-r-I 2'-"+12^ - 2B..I 0E. 20:8 NUMERICAL VALUESr-'-22 When n is small, exact values of E"(x) are best found by using the polynomial coefficients [Section 20:4] in con-junction with the general algorithm of Section 17:8. The algorithm presented in this section may be more convenient if n is large.Input n >>Input x > >Sets=f= 1Ifn=0goto(5)Setf=g=0(I) If xI go to (3)Replace x by x - 1Replace g by g + 2sr"Replace s by -sGoto(1)Storage needed: n, x, s, f, g and jInput restrictions: n must be an integer not lessthan 5. x is unrestricted if the green commandsare included. Use degree mode or change 90 to n/2.1 1 20:9 THE EULER POLYNOMIALS E"(x) 180 f=E"(x)<(2) Replace g by g + 2sx"Replace s by -s Replace x by x - 1(3) Ifx<0goto(2)Set j = 1.5 + lnt(30/n)(4) Replace f by f + sin[90(4jx - n)]/jn+1Replace j by j - IIfj>0goto(4) Replace f by [4sn!/(2wrr)"'1]fReplace f by f - g(5) Output fTest values:E6(0.4) = -0.906624E13(1) = 2730.5E1(-1.5) = 56.66015625 E9(rr) = 1902.63483 For arguments in the range 0 s x s 1, the portions of the algorithm shown in green may be omitted. Thealgorithm is based on the Fourier expansion 20:6:4. truncated at j = 1 + Int(30/n). This choice ensures that theabsolute error in E"(x) does not exceed 6 x 10' for n > 5, although the relative error may be greater than this,particularly near a zerb of E"(x). When the green portions of the algorithm are included, arguments of any magnitudecan be accommodated. Recursion 20:5:4, or its equivalent:20:8:1 E"(x - 1) = 2(x - 1)" - E"(x)is employed to shift the argument into the range covered by the Fourier expansion. 20:9 APPROXIMATIONSFor small n and arguments between zero and unity, Figure 20-1 can provide approximate values of E"(x). For largen and arguments between zero and unity, the leading term of expansion 20:6:4 provides the approximation r20:9:1 E"(x) =4n.sin i r x -nnn-. x0 5 x s 12Approximation formulas may be derived from 20:6:2, 20:6:1 or 20:6:3 for x values close to 0. i or 1, respec-tively. For x values of large magnitude, the approximation 20:9:2E"(x) _ I x - Z I8-bit precisionIx-12> 4(n - 1)(1 follows from 20:6:1. 20:10 OPERATIONS OF THE CALCULUSIn addition to results that follow from the general formulas of Section 17:10, there are several special results fromapplying the differentiation and integration operators to the Euler polynomial. These include 20:10:1d"dx"E"(x)n! d20:10:2 dxE"(x) = nE"-1(x)=0 r E"-3(x)inn1,3,5,xa20:10:3n+1n+10,2,4,...x0 =-2 181 THE EULER POLYNOMIALS E"(x) and 20.10:4fJ, E"(t)E",(t)dt = 1)0 20:11 COMPLEX ARGUMENTEuler polynomials are defined for real argument only. 20:12 GENERALIZATIONS4m!n!(m+n+2)!B° .*_20:14 Euler polynomials may be generalized in the manner described in Erdelyi, Magnus, Oberhettinger and Tricomi[Higher Transcendental Functions, Volume 1, page 43J. 20:13 COGNATE FUNCTIONSBernoulli polynomials [Chapter 19] are closely related to Euler polynomials. Equations 20:3:3 and 20:3:4 establisha direct link. 20:14 RELATED TOPICS Iteration of recurrence formula 20:5:4 leads toE,(.r) ± E"(m + 20:14:1 x)=where the upper/lower signs correspond to odd/even m. Setting x = I in this relationship yields the useful result 20:14:2 1while setting x = ¢ generates 20:14:3 1)'=E. +2The first of these results is presented in Chapter 1 as equation 1:14:9. and equations 1:14:6-1:14:8 are specialcases. CHAPTER21THE LEGENDRE POLYNOMIALS Pn(x) The Legendre polynomials arise in several branches of applied mathematics (as, for example, in describing elec-trostatic or other fields), especially in problems involving spherical symmetry. The Legendre polynomials are amongthe simplest of the families of orthogonal polynomials [see Section 21:14]. 21:1 NOTATIONThe symbol P,(x) is standard for the Legendre polynomial of degree n and argument x, although the P is oftenitalicized. The name spherical polynomial is also encountered. Sometimes normalized Legendre polynomials arespecified: the polynomial of this chapter is normalized through multiplication by (2n + 1)/2 [see Section21:141. Yet another name is zonal surface harmonic [see Section 59:141.The symbol P,(x), where v is not necessarily an integer, denotes a member of the function family known asLegendre functions of the first kind, which are a subject of Chapter 59. When v takes any integer value, positive,negative or zero, P,(x) reduces to a Legendre polynomial21:1:1 Po(x) = P,(x)v = n = 0, 1. 2...21:1:2 P,(x) =v = -n = -1, -2, -3, ...The polynomials of the present chapter are thus special instances of Legendre functions of the first kind and some-times are so identified. The reader is referred to Sections 21:12 and 21:13 for explanations of the related notationsP:(x), P,'(x). P.*,"(x) and P"(x).For -1 s x s I the argument x is often replaced by the cosine of a subsidiary variable 0:21:1:3 0 = arccos(x)as a reflection of the fact that Legendre polynomials often arise naturally in scientific and engineering applicationswith arguments that are cosines of salient angles. 21:2 BEHAVIORIn this chapter we restrict the degree n to be a nonnegative integer; n = 0, 1, 2..... Although the argument x ofthe Legendre polynomial P (x) may take any real value, the most important range is -1 s x 5 1; this is the onlyrange accessible when x is replaced by cos(9). Within this range the polynomials never exceed unity in magnitude 183 21:2 THE LEGENDRE POLYNOMIALS P,(x) 21:2:1 s l-1 5 x 5 1184 as illustrated in Figure 21-1. The Legendre polynomial has exactly n real zeros, all of which lie in the interval -1<x<I.Outside the - I s x s I range, the Legendre polynomial P (x) increases in magnitude without limit (exceptorAj'Ltititi'h010_%Oti , ,o,ooo oo o oo-oo o 4 4 4 4 4G 4 4 4 4 49 4y46 444..41.2 185 THE LEGENDRE POLYNOMIALS P"(x) 21:3 for Po(x)) as x - --. This behavior is exemplified in Figure 21-2 and conforms to the rules21:2:21 =Po(x)<PA)<P2(x)<x>121:2:31=Po(x)<-P1(x)<P2(x)<-PA(x)< x<-1 21:3 DEFINITIONSLegendre polynomials may be defined by means of the generating function [see Section 0:31 21:3:1 "=oThe Rodrigues' formula for Legendre polynomials 21.3:2also serves as a definition.The explicit representation as a polynomiall-2xt-t=1P"(x)t"-1<t<1-I sx - I=o P"(x) _d"(x 2 - 1),2"n! dx" rr21:3:3 -1)..x"_Jj=0.2.4,...,n--±-j!!(n - j)! 22shows that Legendre polynomials contain (n + 2)/2 terms if n is even and (n + 1)/2 terms if n is odd. An equivalent 21:4 THE LEGENDRE POLYNOMIALS P,(x) 186expression for Ix: < IisI(2j- 1)!!(2n-2j- 1)!'21:3:4 P,(cos(9)) = - cos[(n - 2j)B]2"1-oj!(n-j)!Two representations of Legendre polynomials by integrals are V2RsinRn+2)tdt21:3:5 P,(cos(8)) =itIacos(8) - cos(t)and 21:3:6 P,(x) _!J*(x +x' - 1 cos(t))"dtx > 10the latter being known as Laplace's integral representation.The Legendre polynomial function f = P,(x) satisfies Legendre's differential equation 21:3:7 (1 - x2)--- 2x+ n(n + l)f = 0The most general solution of this equation is c,P,(x) + c2Q,(x). where c, and c2 are arbitrary constants and Q,(x)is the function discussed in Section 21:13.Figure 21-3 illustrates a geometric definition that explains how Legendre polynomials arise in certain appli-cations. The triangle shown has two of its sides, one of unity length and one of length r, enclosing the angle e.By the cosine law [see Section 34:14], the length z of the third side equals 1 - 2r cos(9) + r-. In some physicalproblems it is necessary to expand z-' as a power series in r-1. Formula 21:3:1 shows this series to be 1Po(cos(8))P,(cos(8))P2(cos(6)) 21:3:8 +++zrr=r'if r > 1, or a similar series if r < 1. 1 21:4 SPECIAL CASESThe first eight polynomials are 21:4:1 Po(x) = I = Pa(cos(9)) 21:4:2 P,(x) = x = cos(8)3x' - I1 * 3 cos(29)21:4:3 P2(x) =2=4 187 THE LEGENDRE POLYNOMIALS P.(x) 21:4:421:55x3 - 3x3 cos(8) + 5 cos(36)P3(x) =2=835x4 - 30x' + 3 9 + 20 cos(29) + 35 cos(49)21:4:5 P.(x) = =86463x5 - 70x3 + 15x_30 cos(O) + 35 cos(30) + 63 cos(50) 21:4:6P5(x) =8 128231x6 - 315x' + 1051' - 5 50 + 105 cos(20) + 126 cos(49) + 231 cos(60) 21:4:7P6(x) =16=512429x' - 693x' + 315x' - 35x 175 cos(8) + 189 cos(30) + 231 cos(58) + 429 cos(78) 214:8PT(x) =16=1024The formulas given above for P.(x) apply for all values of x, whereas those for P"(cos(h)) are restricted to - I Scos(h) S 1. 21:5 INTRARELATIONSIIPSLegendre polynomials are even or odd21:5:1 P.(-.r) _ (-1)"P.(x)according to the parity of n. The recurrence formula2n-1n-1 21:5:2P.(x)=XP._1(x)--P._2(x)n=2.3,4,...isnrelates three polynomials of consecutive degrees.Positive integer powers can be written as finite series of Legendre polynomials. For example: 21:5:3 x2 =13 Po() W +23P2(x) 21:5:4 x' = 5 Po(x) + 5 P3(x) 21:5:5 X4=lPo(x) + 4 P2(x) +8P.(-V)5735Other well-bchavcd functions of x can be expressed as infinite series of Legendre polynomials P.(x), and a collectionof such formulas is presented by Gradshteyn and Ryzhik [Section 8.92J. Many such series expansions may beobtained by the procedure explained in Section 21:14.Among summation formulas for Legendre polynomials arc21:5:6(2n - 1)P._1(x) + (2n - 3)P._2(x) + (Zn - 5)P"_3(x) + - + 3P,(x) + Po(x)n(P.-i(x) - P.(x)]=x*II - xand21:5:7(2n - 1)P._;(x) - (2n - 3)P._2(x) + (2n - 5)PP_3(x) - ± 3P,(x) + Po(x)_n[P._,(x) + P.(x)]1 + xx + -1 21:6 THE LEGENDRE POLYNOMIALS P"(x) 18821:6 EXPANSIONSDefinition 21:3:3 leads to the finite series(n - 1)!! rn(n2!+ 1)(n - 2)n(n +1)(n + 3)x2+4!X4-n=0,2,4,... (n/2)!1 _U12 21:6:1P"(x)n!!(n - 1)(n + 2)zt+(n - 3)(n - 1)(n + 2)(n + 4) xa...]- n - 13!5!2 n= 1,3,5.that are valid for all arguments x. The infinite Fourier series 21:6:2 P"(cos(8)) =?G(+1)"sin[(n + I - 2j)01it j_0 (j+ Y2)".,involves coefficients that are ratios of Pochhammer polynomials (Chapter 18]. A third expansion is 21:6:3P"(x)=1E(n+j)!2211Other expansions can be found in Gradshteyn and Ryzhik [Section 8.9111 and yet others may be derived from theformulas in Section 59:6 with v = n, or by combining equations 21:12:1-21:12:4 with the expansions given inChapter 60. 21:7 PARTICULAR VALUES n - 0Pa-z)P.(-1)P,(0)P,(I)P.(.) I 1 1 II-x 10 1(- 1)*"(" - 1)^ 21:8 NUMERICAL VALUESFor small n, values of P"(x) are easily found from the expressions in Section 21:4. The algorithm Input n >>Input x >> f = P"(x) <>>>>Setf=g=xIf n = I go to (2)Setf=h=j= 1If n = 0 go to (2)(1) Replace j by j + IReplace f by [(2j - 1)xg - (j - 1)h]/jIfj=ngoto(2)Set h = gSetg=fGo to (1)(2) Output fStorage needed: n. x, f. g, h and j Test values:Po(any x) = 1P,(x) = xP,2(=3) = 251595969P,I +/35632800 (but see text) 189 THE LEGENDRE POLYNOMIALS P.(x) 21:10 utilizes recurrence 21:5:2 and is exact although, of course, rounding may cause large relative errors close to thezeros of P.(x). 21:9 APPROXIMATE VALUESFor 0 S n 5 6 and 0 5 x 5 1, approximate values of P.(x) may be read from Figure 21-1, and extension to therange -1 w x w 0 is accomplished using 21:5:1. For arguments of large magnitude Legendre polynomials areapproximated by 21:9:1 (2n - 1)!!xP.(x) s8-bit precisionIxI > 8V n.which follows from expansion 21:3:3. 21:10 OPERATIONS OF THE CALCULUSDifferentiation of a Legendre polynomial givesdn 21:10:1 dxP"(x =1 - x,[P.-c(x) - xP.(x)ln = 1, 2. 3, ...while integration produces- 21:10:2 JPjr)dt =P.-1(X)+P1.-i(x)n = 1. 2. 3.... iFrom this last result it follows that 21:10:3 JP.(t)dr-0in view of 21:7:4. Other definite integrals involving Legendre polynomials include 21:10:4 21:10:5TTIP.(t)dt _N/"8- ,2n+I 'P.(r)dr[(n - 1)!J-nn=0.2.4....-Itzn!!=0I, 3. 5, ... 21:10:6JIt'P.(t)dt = andm!m=n - 1,n-2,n-3,... 2n!(m + n +m=n,n+2,n+4,...1)!!m=n+ I,n+3,n+5,...f(µ-n+2)(µ-n+4)(µ-n+6).. tµ-12±12) 21:10:71t"P.(t)dt = /\µ> -2 ±2 (µ+n+1)(µ+n-1)(µ+n-3)Iµ+2+ZI 21:11 THE LEGENDRE POLYNOMIALS P"(x) 190where the upper/lower signs apply according as n is even/odd. Gradshteyn and Ryzhik devote several pages [Sec-tions 7.22-7.251 to a listing of yet other definite integrals involving Legendre polynomials.The integral of a product of P,(x) with some other function f(x) is frequently evaluable by utilizing Rodrigues'formula 21:3:2 and parts integration ([Section 0:10]. Whenf the integration limits are ±1. the simple formula 21:10:8 J'f(t)P,(t)dt =1d"f(t)(1- tz)"dt2'n!dt'emerges.The integral 21:10:9f0mnP,(t)P"(t)dr =2inIt2n + 1establishes the orthogonalirv of Legendre polynomials over the interval - I <- x s 1, a valuable property that isdiscussed in Section 21:14.The Hilbert transform [Section 7:10] of a Legendre polynomial 21:10:10I'P"(t)dt=-2 OJs)WT,r-snproduces a Legendre function of the second kind [see Section 21:13]. a result known as Neumann's formula. 21:11 COMPLEX ARGUMENTThe argument of Legendre polynomials is generally taken to be real. See Section 59:11. however, for a discussionof P,(x + iy). 21:12 GENERALIZATIONSAs explained in Section 21:1, Legendre polynomials are the special v = integer cases of the Legendre functionsof the first kind, dealt with in Chapter 59. However, Legendre polynomials may be generalized in a number ofother directions.Legendre polynomials generalize to give associated Legendre functions of the first kind, P,, '(x) (see Section59:13], of which they are the v = integer. µ = 0 instances. Likewise. Legendre polynomials result as the v = µ= 0 instances of the Jacobi polynomials P,` "'(x) that are discussed in Section 22:12.Legendre polynomials are also special cases of the Gauss function [Chapter 60]. the simple identity 21:12:1 /1-xP,(x)=FI-n.n+1:I; 2/being known as Murphy's formula. Other relationships between Legendre polynomials and Gauss functions are 21:12:2 P,(x) _(2)7- 1)!! (x - 1)"F( -n, -n;-2n:2n!1 -X , 21:12:3 P,(x) _(2n - 1)!! x"F1n 1n I - 2n 1n!222x.(n-1)!!F((/-n n + I1x'1\n=0.2.4.... n!!``222 21:12:4P,(x) _ n!!l-n n+2 3..r')n = I. 3. 5.... (n - I)!!xF(-2' 22'and still other formulas may be developed using the relationships of Chapter 60. 191 THE LEGENDRE POLYNOMIALS21:13 COGNATE FUNCTIONSThe polynomials sometimes denoted P,*(x) and P"(x) are minor variants of P.W. defined as follows:21:13:1 Po(x) =I)known as the shifted I egendre polynomials [they are orthogonal, see Section 21:14, on 0 5 x s 11, and 21:13:2 n!(2n - I)"21:13 Neither variant is used in this Atlas.The functionshave been cited as solutions to differential equation 21:3:7 and in connection with Neu-mann's formula 21:10:8. Their behavior is illustrated in Figure 21-4; note that Q,(x) is infinite at x = I (and at x= -1). They are the special v = 0. 1. 2.... cases of the functions Legendre's functions of the second kind[see Chapter 591. Even for integer v, the functions are not polynomials, the first few cases being21:13:3 Q0(x) = artanh(x)-1 <x< 121:13:4 Q1(x) = x artanh(x) - 1-1 <x< I3x2 - l3x21:13:5 Q2(x) =2artanh(x) - 2-1 <X < 1 and5x'-3x15x2-4 21:13:6Q3(x) =2artanh(x) -6-1 <x< 1where artanh is the inverse hyperbolic tangent function [see Chapter 311. The general case is 21:13:7Q(x) = P . ( - r ) artanh(x) - 2 12n - 2j + Ij = 1, 3, 5, .... n - # }-- i°A" y®O0.83x) C): : D0 p (x) :.........-0.2 ..:....1.....X:.7..:.r .:.... :.... :.... :.............. :........... -0.4: FIG 21-4 : ............. :A.: .... A...:1.4..:....:.... .. ...:.............. . ....:.... ..-0.6JJ(2nl)'L1,a600'La0001.U6 ...:.... :.... :.... : ............. 0.4Dt <x):..0. 2 l 3 (x): 21:14 THE LEGENDRE POLYNOMIALS P,(x) 192for - I < x < 1. For j > 1, the artanh function is to be replaced by arcoth. Many of the properties of P,(x) aremimicked by Q,(x); for example, the recurrence formula 21:13:82n-In-1xQ,-i(x) - -Q,-2(x)n = 2, 3, 4, ...n nis obeyed. However, the symmetry properties of .(x) differ from those in 21:5:1 because Q,(x) is an even functionwhen n is odd, and vice versa. 21:14 RELATED TOPICSA family of functions 'Po(x),'P3(x), 4 2(x), ... is said to be orthogonal on an interval x0 s x s x, when its familymembers satisfy the relationship 21:14:1f j0m * n but 21:14:2ff2 s 0The family is said to be orthonormal if 21:14:1 and 21:14:2 hold and if f2, = 1, n = 0. 1. 2. .... The functionw(x) is known as the weight function; it is required to be nonnegative on the interval xo s x s x,. For example,integral 21:10:7 shows that the Legendre polynomials constitute an orthogonal family on the interval - I x :s Iwith a weight function of unity.The orthogonality property permits the functions 'Po(x), 'P,(x), 'P2(x), ... to be used as a basis set for theexpansion21:14:3 f(x) = c0'P0(x) + c,W,(x) + c=Yr2(x) +of a wide range of functions f(x). The constants c0, c,. C2. ... in this so-called orthogonal series or generalizedFourier expansion may be evaluated by the integral 21:14:4 C. = z, JFor example, the function l/ 1 - x2 may be expanded as a series 1 21:14:5 = c0P0()+ CAW+ c2P2(x) +1-z2of Legendre polynomials, the constants being evaluable, via 21:10:5, aslit <1 21:14:6 c, =(n+1)Tr[(n - 1)!!J 12n!!n=0,2,4,... withci=c,=cs==0. CHAPTER22THE CHEBYSHEV POLYNOMIALS T,t(x) AND U,t(x) Chebyshev polynomials of the first kind, T (x), constitute an orthogonal family [Section 21:141 on the interval -1s x 5 1 with a weight function I / 1 -x2. Similarly, the second kind of Chebyshev polynomials, U,(x), areorthogonal on the same interval with weight function 1 - x2. Chebyshev polynomials are extensively used in"fitting" techniques, such as those discussed in Section 22:14. 22:1 NOTATIONAlternative transliteration from the Cyrillic alphabet leads to spellings ranging from Chebyshev to Tschebischefffor the names of these polynomials.Unfortunately, mathematicians differ in their definitions of Chebyshev polynomials, and there is not even una-nimity on which family constitutes the first and which the second kind. The notations L(x) and Ujx) may beencountered with meanings equivalent to the T,(x)/2'-' and nU,_,(x) of this Atlas. Confusingly, the symbol U,(x)and the name Chebyshev polynomials of the second kind are commonly applied to functions defined by ourV'1 -x i U,_i(x) for n = 1, 2, 3.... and by arcsin(x) for n = 0, despite these not being polynomial functions atall.Several supplementary notations are encountered. T; (x) and U! (x) are so-called shifted Chebyshev polynomialsequal to our T,(2x - t) and U,(2r - 1). The symbols C,(x) and S,(x) have been used for 2T,(x/2) and U,(x/2),respectively. Chebyshev functions. T.00 and .(x) are defined identically to the corresponding polynomials, but nis not restricted to be an integer. None of these supplementary notations is employed in this Atlas. 22:2 BEHAVIORThe degree n of a Chebyshev polynomial may take any nonnegative integer value, although U0(x) is not a memberof the orthogonal family [see Section 22:101. Interest in Chebyshev polynomials is confined to the - I s x S I range, as portrayed in Figures 22-1 and 22-2, although several of the definitions remain valid outside this argumentrange.Chebyshev polynomials of each kind, T,(x) and U,(x). have exactly n zeros, Int(n/2) minima and lnt[(n - 1)/2] maxima, all of which lie within the -I < .r < I range. The T,(x) polynomial takes the value -I at eachminimum and + I at each maximum, but no comparable rule applies to the Chebyshev polynomials of the second kind. The polynomials are confined to the ranges22:2:1 -1T,(x)S1-1SxSI 193 22:3 THE CHEBYSHEV POLYNOMIALS AND0OMti44'14'44r44' and22:2:2 22:3 DEFINITIONS1.O0440'440'440'440' T4(x) \ ,TS(z) FIG 22-1: -n- 1 1-1 sx I Chebyshev polynomials are given by the trigonometric definitions194 22:3:1T,(x) = cos(nO) 22:3:2 csc(0) sin[(n + 1)010 = arccos(x) - 1 5 x s 1 22:4THE CHEBYSHEV POLYNOMIALS T"(x) AND U"(x)The purely algebraic definition 22:3:3T"(x)=2 (x+z'-1)"+Z(x-holds for Chebyshev polynomials of the first kind.These polynomials may be defined by the generating functions1% I -rx < 2234=T"- :: 1<r"(x)t 1I -2ix+r 13225::or by the Rodrigues formu1-21x+r2"=olas 22:3:6 T"(x)d"(1 - x')"(2n - 1)!!"dx 22:3:7 (z)=(-1)"(n+ 1)U d"(I -x2)"-ir,(2n +ifFor n = 1. 2, 3.the most general solution of Chebpshev's differential equationdf 22:3:8(1-x2)--xdf-+nf 0dx2dxis f = c,T"(x) + c21 - z2 U"_,(x). where c, and c2 are arbitrary constants. For n = 0 the solution is f = c, +C2 arcsin(x). 22:4 SPECIAL CASESThe first eight polynomials of each kind are22:4:1 T6(x) = U6(x) = 122:4:2 TI(x) = x22:4:3 U,(x) = 2x22:4:4 T2(x) = 2x2 - 122:4:5 U2(x) = 4x' - I22:4:6 T3(x) = 4x 3 - 3x22:4:7 U3(x) = 8x3 - 4x22:4:8 T4(x) = 8x4 + 8x2 + 122:4:9 U4(z) = 16x4 - 12x2 + 122:4:10 T5(x) = 16x5 - 20x3 + 5x22:4:11 Us(x) = 32x' - 32x3 + 6x 22:4:12 T6(x)=32x`-48x4- 18x2 - 122:4:13 U6(x) = 64x6 - 80x4 + 24x2 - 122:4:14 T7(x) = 64x' - 112x5 + 56x' - 7x22:4:15 U7(x) = 128x' - 192x' + 80x 3 - 8x 197THE CHEBYSHEV POLYNOMIALS T.(x) AND22:5 INTRARELATIONSHIPSChebyshev polynomials are even or odd22:5:1 f,(-x) _ (- I)"f.(x)f = T" or U.according to the parity of the degree n. The recurrence formula22:5:2 f.(x) = 2xf _,(x) - f._2(x)n = 2, 3, 4....also applies equally to both kinds of Chebyshev polynomials.The formulas22:5:3and 22:5:4T.(x) = U.(x) - xU._,(x)n = 1, 2, 3, ... U (z) =xT._,(x) - T.(x)1 -x2n=1,2.3....22:5 permit one kind of Chebyshev polynomial to be expressed in terms of the other.Positive integer powers can be expressed as finite series of Chebyshev polynomials of either kind. For example: 22:5:51IIx= = 2 T0(x) + 2 T2(x) =4Uo(xl+4U2(x) 3 1 1I3 22:5:6 U1(x)+T,(x)=T,(x)+U;(x)x'=I444and generally22:5:7x" = Y:'T,(x) + y."22f"_2(x) + y'.2,T"_.(x) + ... + {'vFo(x)Yi'TI(x)n0,2.4....n = 1.3. 5....- L,('r)i-0where0 if j > n or if j and n have unlike parities. The simplest coefficient yo' equals unity, and others canbe calculated by means of the recursions y;"' _ 2 y;=;" + 2 y;"-i" except -y"' = Yo 1) + 2 yz and Yoyi"-11There are formulas relating the product of two Chebyshev polynomials to (usually the sum of two) other Che-byshcv polynomials: 1 1 22:5:8T.(x)T,.(x) = 2 T".,.() + - T._ (x) n L, m 22:5:9 U,(z)U,.(x) =T_(x) -n s m2(1 - x2) 1 I- U..,,(x) + - U_(x)22n:5 m 1 22:5:10T,(x)U,.(x) =2 U_() n = m + I 1 12 U..,.()2 U._._2(x)n >- m+ 2as well as the so-called formulas for sums of products of Chebyshev polynomials of differentarguments: 22:5:111T; (x)T; (y) _T..j(x)T,(y) - T,(x)T,.dy)2x - yU,.,(x)U,(y) - U.(x)U,.,(y)UU+7- 22:5:12 I,(x),(y)-- 22:6 THE CHEBYSHEV POLYNOMIALST.0) AND U.(x)Series of Chebyshev polynomials of even degree have the sums 11 22:5:13To(x) + T2(x) + T.(x) + ... + T.(x) =-+-U.(x)n = 0, 2, 4, -.22 22:5:14U(x) + U2(x + U4(x) +. + U, x1 - T,.2(x)2(1 - x2)while the corresponding sums for odd degree are 122:5:15T,(x) + T)(x) + T3(x) ++ T,(x) _ - U,(x)n = 1, 3, 5, .. .2x - T..2(x) 22:5:16U1(x) + U3(x) + U5(x) ++ U.(x) =n = 1. 3. 5....2(1 - x2) 22:6 EXPANSIONSExplicit expressions for the two Chebyshev polynomials are 22:6:1T.(x) = 2 Y n ' . (njj)(2x)'-2'n = 1, 2, 3, .. . and 22:6:2U.(x)=I(-I)' nj)(2x)w2Jn=0.1.2....J=o`where J = Int(n/2). but these expansions may also be written/ 22:6:3T,(x) = X. - 1 2 1 x"-2(I - x2) + 14 1 x"-4(I - x2)2 - and\J111////`22:6:4U"(x) =rn + I)X.- (n3I)x"''-(I - x 2) + 1n + II x"-4(l - x')= - .. .If t;," represents the coefficient of x1 in the expansion of T.(x) so that22:6:5 T.(x) = i t."x,J=othen tw1' = 0 when j and n are of unlike parity. All nonzero values of t," are integers given by((n+-J 22:6:6(-1)4.-))/2n(2j-12J-1/\n)!J!j = 1, 2. 3..... nexcept that t`,° equals unity. 22:7 PARTICULAR VALUES198 T,(-1)T.(0)Tj1)U,(-1)U,(0)U,(1)n=0 1I 1 1II1 ,3 . 5 ,-I0 1-n- 10n-I n=2.4.6.... i1f-1Y'' 1n+l(-I)n + I 199THE CHEBYSHEV POLYNOMIALS T,(x) AND U,(x)The zeros of the Chebyshev polynomials occur at arguments given by 22:7:1 and- 1)a(2ji T,(r1) = 0r1 = cost2nJj= 1, 2, 3,...,n a 22:7:2U,(r,)=0rj =cosn+1j=1,2,3,...,n 22:8 NUMERICAL VALUES22:10 For small degree, values of the Chebyshev polynomials are readily calculated from the expressions in Section 22:4.The following algorithm employs recurrence formula 22:5:2 and provides exact values of the Chebyshev poly-nomials. T,(x) is generated if the command in black is executed; U,(x) is produced by the green alternative (the two algorithms differ only by a factor of 2 in the command that sets up the valueFor - I < x < 3, values of the Chebyshev polynomials may also be found using the universal hypergeometricalgorithm [Section 18:14). Input it >>Input x >> llf=U,(x))31»»»»Setf=g={2xIf n = l go to (2)Setf=h=j= 1If n = 0 go to (2)(1) Replace j by j + IReplace f by 2xg - hIf j = it go to (2)Replace h by gReplace g by fGo to (1)(2) Output f 22:9 APPROXIMATIONSStorage needed: n, x, f, g, h and j Input restrictions: n must be a nonnegative integer. xmay adopt any value. Test values:To(any x) = 1U1(x)=2xT12(0.2) = -0.748302037U,(0.5) = 01.239131013 Figures 22-1 and 22-2 can provide approximate values of T,(x) and U,(x) for degrees up to 5. For large values ofthe argument, the following approximations are valid: 22:9:1T,(x) - 2(2x-I) n8-bit precisionx > (18n)'14it -- 2 22:9:2lU,(x) = 2x1 2x - 2x) 8-bit precisionx > 2(n - 1)'I4n ? 2 22:10 OPERATIONS OF THE CALCULUSDifferentiation givesd_it22:10:1 dxT,(x)1 - xr[',(x) -T,+i(x)) = nU,-,(X)1 22:11 THE CHEBYSHEV POLYNOMIALS AND U.(x) 22:10:2 d 1dxU.(x) =7- i[(n + 1)U.-i(x) - nxU.(x)]The Chebyshev polynomial of the second kind may be integrated indefinitely r'1 -T.- I (x) 22:10:3 U(r)dt=,Jn+1but we know of no corresponding integral for T.W.The definite integralf0m*nf' Tj:)T,(t) 22:10:4 dr= J irm=n=0ITm=n=1,2,3,...2establishes that the polynomials T0(x), T,(x), T,(x), ... are orthogonal (see Section 21:14] on the interval -1 S xS I with respect to the weight function I / 1 - x2. Likewise, the integrals 22:10:5 1 - t2 dt0or,2,m3,=..n.=0= a 1 2mm #=nn=1a2establish the orthogonality of the polynomials U,(x). U2(x), U3(x).... on the same interval with respect to theweight function V 1- x2.Many other definite integrals are listed by Gradshteyn and Ryzhik [Sections 7.34 to 7.36]. includingT4n2-22:10:6T"(r)dr =4n2 - 1f't T.(t)trr(v + 1) 22:10:7 dr =\v > - 101-r22,rlv2n+llr(vn}I(-1)./2a 22:10:8J.2b>0n=0,2,4,...01 - rrandfsin(bt)T (22:10:9dr=b>0n=1,3,5, 1-t222where r denotes the gamma function [Chapter 43] and J. the Bessel coefficient [Chapter 52] of order it.On Hilbert transformation (Section 7:10] the product of one kind of Chebyshev polynomial and its weightfunction gives the other kind: 1T,(t)dr 22:10:10 J= U.-I(s)-1 <s< I a112 (tS) r d 22:10:11r+__ds)-T rJ77it- s).- 22:11 COMPLEX ARGUMENTSApplications of Chebyshev polynomials are usually restricted to real arguments. 201THE CHEBYSHEV POLYNOMIALS T"(x) AND U"(x)22:12 GENERALIZATIONS22:12 Chebyshev polynomials may be generalized to Gegenbauer polynomials, which are themselves special cases ofJacobi polynomials. This section briefly discusses both of these generalizations.The Jacobi polynomial or hypergeometric polynomial is a quadrivariate function, its value being de-pendent on: the argument x, which may take any real value with interest being concentrated on the -1 s x <- 1range; the degree n, which takes nonnegative integer values; and two distinct parameters, v and µ, both of whichexceed -1. These polynomials may be defined by the Rodrigues expression 1d" 22:12:1 PA":w(X) = [ (1 - X)"' "( I + x)"(-2)"n!(1 - x)"(1 + x)" dx"They obey the reflection and recurrence formulas22:12:2 P;,"(-x) _ (-1)"P;; `'(x) 22:12:3 pA.1:w(x)=(2n+v+µ- 1)((2n+v+µ-2)(2n+v+µ)x+v2-µ(X)2n(n+v+µ)(2n+v+IL -2)(n+v-1)(n+µ-I)(2n+v+ µ)n(n+v+µ)(2n+v+µ-2)P"i(x) with the first members being22:12:4andP101 "(x)= 1 + 2)x+µ 22:12:5 P;" "'(x) = (v + Z2A binomial coefficient (Chapter 6] expresses the value of the Jacobi polynomial at unity argument 22:12:6 P'":w(1) =!n + v)ItJacobi polynomials are orthogonal [Section 21:14] on the interval - I s x s 1 with weight function (I - .r)' (I + x)"f0 m+n22.12:7(I - t)"(I + 1)"P" w(t)P(t)dt =2"''l'(n + v + I)f(n + µ + 1)(2n+v+µ+l)n'f(n+v+µ+ 1) m=nFor more information about Jacobi polynomials, the reader is referred to more advanced sources. particularly Szego.Jacobi polynomials can be expressed in several ways as Gauss functions [Chapter 60]. For example:22:12:8P;,""'(x)=1nn)F(-n,n+v+µ+1;v+1;12xn//I\ /Jand numerical values are therefore calculable using the algorithm of Section 18:14.Gegenbauer polynomials, or ultraspherical polynomials, are related to those Jacobi polynomials in which thetwo parameters are equal: v = A. They may be defined by 22:12:9(2),)"P'a-,rza- 1120(x)-!< 1, + 0 (X + '/2)" 2supplemented by the definition 22:12:10 C01(x) = lim=A-ok(2n - 1)!'Most of the properties of Gegenbauer polynomials may be deduced as special cases of those of Jacobi polynomials. 22:13 THE CHEBYSHEV POLYNOMIALS AND 202Legendre polynomials [Chapter 21 ] and each kind of Chebyshev polynomial are special cases of Gegenbauerpolynomials22:12:11 P (x) = C°'21(x) = V0.91(X) 22:12:12 no(2n)!!-112-v2) T.(x) = - C(x) = P'(x) 2(2n - 1).(2n + 2)!'Pn/2:1/2)(x) 22:12:13 U (x) = C;'(x) =2(2n + I)'!while Hermite polynomials [Chapter 24) are generated from Gegenbauer polynomials by the limiting process 22:12:14 22:13 COGNATE FUNCTIONSOrthogonality defined in terms of integral 21:14:1 is generally satisfied by families of polynomials, each orthogonalfamily containing an infinite number of members. However, another variety of orthogonality, defined by may be satisfied by a finite set of polynomials `l'o(y), 4'1(Y), 'Y,,(y), known as discrete orthogonalpolynomials.The simplest of these are the discrete Chebyshev polynomials to(y), t (y), t2(y), ..., tt(y), which correspondto weights w(y) in 22:13:1 equal to unity and should not be confused with the coefficients discussed in Section22:6. With the variable y confined to the range -1 :5 y s 1, the orthogonality condition is 22:13:2 0mn j.0where 22:13:3G^=(J+n+l)!(J-n)!- 1(J+I)..1(2n+1)(J!)22n+1(J-n+1)Values of the discrete Chebyshev polynomials generally depend on J as well as on the argument y. The firstfew air:22:13:4 to(y) = 122:13:5 tt(y) = y 22136 =3Jy2-(J+2):: t2(y)2(J - 1) 22137 '5J2y3-(3J2+6J-4)y:: )tr02(J-1)(J-2)and others may be calculated via the recursion formula 22:13:8t.0') -(2n - 1)n(J - n + I)[yt,.-i(y) - t.-2(y)1 + t.-2(y)The practical importance of discrete Chebyshev polynomials depends on the role that they play in the fittingof low-degree polynomials to equally spaced data points [see Section 17:14]. Let a K-degree polynomial pk(y) besought that passes close to the points (yo,fo), (yi.fi). (y2.f2), , (yj,fj), as shown in Figure 22-3, so as to satisfy 203THE CHEBYSHEV POLYNOMIALS T"(x) AND U"(x) 22:141O1< .o!9.....A ..:..:..:..:...:...... . ..:....:.........:....+!....:....:....:.. the least squares condition, that is, to minimize the quantity J 2% 22:13:9 [f,-pK(Y1)12Y,-1--1=0Then this "best" polynomial is given by 22:13:10wherePK(Y) _ak4(Y)k-0J 1J22:13:11as=-j4(Y1)f1k J-0These relationships are exploited in the algorithm presented in Section 17:14.Despite their name, discrete Chebyshev polynomials have less in common with Chebyshev polynomials thanthey do with Legendre polynomials, to which they reduce as J approaches infinity:22:13:12 1im t"(y) = P"(Y)J-- "22:14 RELATED TOPICSAs inspection of Section 6 of most of the chapters of this Atlas will attest, very many functions f(x) can be rep-resented by infinite (or less frequently by finite) power series of the form E a,x'. If only the summands ao + a,x+ a2x2 ++ a"x" are used, and n is large, an extremely accurate approximation to the true value of the functionf(x) may be obtained: 22:14:1 f(x) -a,x'I1-0where S is very small. For computational purposes, however. it may be undesirable to require as many as n + Iterms when n is large. Indeed, it may be unnecessary to use more than a few terms, especially if interest in thefunction f(x) is restricted to a small range xo s x s x, of argument. Economization of series is a procedure thatreplaces a very accurate (or even exact) polynomial approximation I a,x' of degree n by an "economized" poly-nomial I e1x' of a smaller degree m such that, in the range of interest, the absolute error introduced by the re-placement is less than some acceptable value, e: 22:14:2 Ea x' - Eejx'I< ex0x < x,1°01.0 22:14 THE CHEBYSHEV POLYNOMIALS T,(x) AND U,(x) 204The procedure of economization, or telescoping as it is sometimes called, is accomplished by utilizing theproperties of Chebyshev polynomials of the first kind. A general algorithm follows by which any polynomial maybe economized to any extent. The algorithm is lengthy because it has seven distinct phases.The first phase merely allows the input parameters x0, x n, so, a,, a2, ..., a, to be loaded. Note that x0 andx, are not stored as such but as (x0 + x,)/2 = u and (x, - xo)/2 = h.The second phase calculates the coefficients bo, b,, b2, ..., b of a second polynomial of degree n 22:14:3ib,y'=ia,x'wherex=u+hyi-oj-owhose argument has been defined so that the region of interest is -1 S y S 1. The second phase uses the binomialrelationship' 22:14:4b,= h'k)akuk-'_ h,[aiJI+akk!u-'/(k-j)!ij[k-j- IIto calculate the new coefficients and stores them in place of the original set.The third phase computes the coefficients co. c,, c2, .... c, such that 22:14:5 1""}1i-oi-oi=0k-0by first exploiting the fact that, because only the T,(y) polynomial contains a term in y", c" = b,12"'(if n * 0). Using expression 22:6:6, c,t'1 is now subtracted from each stored b, other than b" so that the degreeof the I b,v polynomial is reduced to n - 1. The procedure is then repeated until all c, coefficients have beencalculated and their values used to replace the now-redundant b, coefficients.To start the fourth phase of the algorithm, the maximum permissible error a is input. The smallest integer mthat satisfies 22:14:6 i 10 sei-.-1is then calculated by the algorithm, output, and used to replace n in storage. Thereafter, no further use is made ofcoefficients c,.,, c,,.2, c,.The fifth phase reverses the process of the third phase, replacing I c,Tj(y) by the polynomial E d,y' of degreem. The properties of tk' (see Section 22:61 are used in the identity 22:14:7 c,tik,vk=t` tiic, ttil +,) ckt1j1LrLJ J.01=0k-0 a 1I'0kwhere in the final summation k takes the values j + 2, j + 4, j + 6, ..., (m or m- 1) to establish that 22:14:8dj = 2'-'c, +(-1)'k-iU2kck(k + j- 1!2i-'2 2lL\k,l/!j !except that, when j = 0, the 2'-'c1 term is to be doubled. The algorithm replaces co, c1, c2, ..., c in storage bydo, d1, d2, .., d,,.The sixth phase of the algorithm reverses the effect of the second phase in that it converts I d, y' to E e,x' andreplaces the m + I stored values of d, by the corresponding ej coefficients. The binomial identity 22:14:9e1xidj(x-u)j(-u)'-idkk!i.0j.0hhllJ-0 j ! k-j(k - j )!hkshows that 22:14:10 e1,j (kdkk(_u)1li!dk-ihi`ik-i,1j)!h1and this is the relationship used by the algorithm.The seventh phase simply outputs the economized coefficients in the order e0, e e2, ..., e,,. Input xo »»»Setu=xaInput x, »»»Set At = (x, - u)/2Replace u by u + hInput n >>>>>>Setj = 0I(1) Output j ,4 j(as cue) <<<<<<Input a;>>>>>>Replace j by j + IIfj :5ngoto(I)Seth=0(2) Replace a; by a,j!Ifj = n go to (4)Setk=j+ I(3) Replace a, by aj + akk!uk-'/(k - j)! °Replace k by k + I c Ifk:5 n go to (3)(4) Replace aj by a,h'/j!Replace j by j + I Ifj:5 n go to (2)Set kn(5) Set j = kReplace a, by a,/21-'(6) Replace j by j - 2Ifj<0goto(7)bete=Vc-j)/e 2Replacea,bya, -(-I)'kak(E + j - 1)!2'-'/(e!j!)Go to (6)(7) Replace k by k - 1If k * l go to (5)Setj=nInput a »»»(8) Replace a by e - Ia,IReplace j by j - 1Ife>0goto(8)Setm=j+1 0Output mM <<<<<<<<<<-----------------------------------------Replace ao by 2aoSetj= -I(9) Replace j by j + IIfj>mgoto(11)Replace a, by a, 2'-' Setk=j(10) Replace k by k + 2Ifk>mgoto(9)Set s = (k - j)/2II Replace a, by a, + (-1)'kak(e + j - l)!2'-'/(e!j!)Go to (10)----------------------------------------Storage needed: n + 7 registers arerequired for u, h, n, j, k, e, m andf o r the coefficients ao, a a2, .a,,. Note that the n register may beused to store m and that the a reg-ister is used for temporary storagetwice by the algorithm. The coef-ficients denoted bo, b,, b2, ..., b,:co, ca, C2, - , c,; do,d1,d2, , d.,and e0, e,, e2, ..., e,, in the text arestored in the same registers as ao,a a2, .... (a or a,,) and are de-noted a, in the algorithm. 205 22:14 THE CHEBYSHEV POLYNOMIALS T,(x) AND U,(x)(II) Setj=0(12) Replace a, by aj!If j = m go to (14)Set k = j + 1(13) Replace a, by a, + q,k!(-u/h)'-'/(k - j)!Replace k by k + IIf k :5 m go to (13)(14) Replace a, by a,/(hJ!)Replace j by j + 1if jsmgo to (12)- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ---Set j = 0(15) Output a,.5Replace j by j + 1Test values: xo = 0, x, = 1, n =9.ao= 1, a, _ -1,a2= 1/2,a,= -1/6, a, = 1/24, a, = 1/120,a6 = 1/720, a, _ -1 /5040, as =1/40320, a9 =1/362880 [i.e..f(x) = exp(-x)], e = 1/256Output: m = 2, eo = 0.99658, e,= -0.93532, e2 = 0.30963 Because of the very large number of arithmetic operations executed by the algorithm, rounding errors mayaccumulate and the values of the economized coefficients e0, e,, e2, ..., e generated by a program may not bequite the "best." To evaluate this effect, the magnitude of which will depend on the characteristics of the particularcomputing device employed, one may repeat the calculation with e = 0. Then m will equal n and the coefficientseo, e,, e2, ..., e should, but will probably not exactly, equal ab, at, a2, ..., a,.Apart from these rounding errors, the economized series E e,x' is guaranteed to differ from the original F a,x'series by no more than a over the xp s x s x, range. However, the actual absolute error 1E a,xJ - E e,x'I will often be less than e. This actual economization error E is found in the following algorithm by calculating themaximum value of 22:14:11i Axj1.0 1at one hundred points in the xo < x s x, range. The input coefficients A, are given by 22:14:12 Input xa > >>>>>Setx=x0Input x, > >>>>>Set h = (x, - x)/99Setj=k=E=0Input n >> »»(1) Output jj(as cue) < ««Input A, > >>>>>Replace j by j + I If j5ngo to (I)(2) Set f = 0Replace j by n + 1(3) Replace j by j - 1Replaccfbyfx+A, Ifj*0goto(3)If E> If go to (4)A'a, - e,j =0, 1,2,...,Ina,j = m + 1,m+2,m+3,...,n Storage needed: n + 8 registers are required for Ao,A,. A2, ..., A,,, x, h, j. k, n, E and f.1 207THE CHEBYSHEV POLYNOMIALS T,(x) AND U,(x) 22:14 E <<<<<Replace E by Ill(4) Replace x by x + hReplace k by k + I Ifk* IOOgoto(2)Output ETest values (taken from the preceding algorithm): xo= 0, x, = 1, n = 9, A0 = 0.00342, A, = -0.06468,A2=0.19037,A3=-0.16667, A,= 0.04167, As=- 0.00833, A9 = 0.00139, A7 = -0.00020, A, _0.00002, A9 = -0.00000Output: E = 0.00342 CHAPTER23THE LAGUERRE POLYNOMIALS Ln(x) The Lagucrre polynomials are orthogonal [see Section 21:14) on the interval 0 s x s x with a weight functionexp(-x) [see Chapter 261. They arise in a variety of scientific applications, for example, in solutions to the waveequation of quantum physics.23:1 NOTATIONThough the L (x) symbolism is used for both, Laguerre polynomials are defined in two different ways. Someauthorities use definitions that differ by a multiplicative factor of n! from those adopted in this Atlas. The termi-nology Laguerre polynomial is sometimes applied to the more general class of polynomials that we discuss inSection 23:12 under the name associated Laguerre polynomials.Although this Atlas avoids the ambiguity, the symbols L (x) and L,.(x) are sometimes used to denote the hy-perbolic Strove function [see Section 57:131.23:2 BEHAVIORWhile interest is usually confined to positive arguments, the Laguerre polynomial L (.r) is defined for all valuesof the argument x and all nonnegative integer degrees n. Laguerre polynomials of odd degree adopt all values,whereas those of even degree have a restricted range. Figure 23- I shows the behavior of the first few membersof this family.The Laguerre polynomial L (x) has exactly n zeros. lnt(n/2) minima and Int[(n - 1)/21 maxima, all of whichare located in the range 0 < x < 2n + I + 4n2 + 4n + (5/4). For x > 2n + I + V57+ 4. L (x)increases or decreases monotonically according to whether n is even or odd, remaining bounded by 23:2:1 -exp (--2)s L,(x) K exp(;)x >_ 0 23:3 DEFINITIONSLaguerre polynomials are defined by the generating function 23:3:1 11rexp1xr!)-L,(x)t"- I < t < I 209 it"tt4vb??' L4'I .5 .4 FIG•0 —1 —2 —3 •. —523—1 211 THE LAGUERRE POLYNOMIALS L (x)or by Rodrigues' formulad" 23:3:2L,(x) _exp(x)x" exp(-x)n = 0, I, 2, ...n !dx"11 A third definition is provided by expansion 23:6:1. The limiting operation 23:3:3 L,(x)=limP;°'" I-links the Laguerre polynomials to Jacobi polynomials [Section 22:121, while the integral representation23:3:4 L,(x) =exp(x)J;t' exp(-t)Jo(2\ tx)dtn!uestablishes a connection with the zero-order Bessel coefficient of Chapter 52.One solution of Laguerre's differential equation 23:3:5xaf+(I -x)af +nf=0dx2dxis f = cL,(x), c being an arbitrary constant. 23:4 SPECIAL CASESThe first eight members of the family of Laguerre polynomials are 23:4:123:4:2 23:4:3 23:4:4 23:4:5 23:4:6 23:4:7 23:4:8_x'2x'L.(-_)243+ 3x2 - 4x + l x55x'5x'L5(x)= - 120 + 24 - 3 +5x - 5x + l x°x`5x'lox'15x22Lb(x)72020+ 8 - 3+2 -6x+1 x'7x°7x535x'35x'21x'L,(x)5040 + 72040 + 24 -6+2- 7x + 1 23:5 INTRARELATIONSHIPS23:5 We know of no reflection or recursion formulas that express L,(-x) or L,(x + 1) in terms of L,(x). but there existsthe recurrence relation 23:5:1 2n-i-xn-IL,(x) =L,_,(x) - - L,_2(x)n = 2. 3, 4... nnrelating a Laguerre polynomial to its congenitors of lower degree but identical argument. 23:6 THE LAGUERRE POLYNOMIALS L,(x) 212Integer powers of x can be expressed as sums of Laguerre polynomials. The identity23:5:2x" = 720[Lo(x) + LF(x)] - 4320[L,(x) + LS(x)) + 10800[L,(x) + L4(x)] - 14400 L3(x)provides an example. The reader is referred to Abramowitz and Stegun [Table 22.101 for a tabulation of the coef-ficients that occur in such expressions. Other series of Laguerre polynomials include 23:5:3 L1(x) = x-n L"(x) + - L' "(x)/-0xxwhere L°' is an associated Laguerre polynomial [Section 23:12]. A series of Laguerre polynomials also occurs inthe multiplication formula 23:5:4L"(bx)=(1-b)"1while the addition formula(ll(n) 23:5:5L(x + )) = H,,(vc)H"-t,(\).aiinvolves a sum of products of Hermite polynomials [Chapter 241. The formula for Laguerrepolynomials isn+l 23:5:6, L (x)L (v) _ - IL"(x)L".,(y)-x - y 23:6 EXPANSIONSAn explicit expression for the Laguerre polynomial isn(n - 1)x"-2-1 23:6:1L,(x)=(-I)" ++(-1)"n!1!(n - 1)1.2!(n - 2)! i_o 23:7 PARTICULAR VALUES n=0L,1--)L,(0)L,1=) 1 1 1_l We know of no exact expressions for the arguments of the zeros, minima or maxima of the Laguerre poly-nomials, beyond the first few. 23:8 NUMERICAL VALUESWhen the portions of the algorithm that are shown in green are omitted, exact values of L"(x) are generated, viaformula 23:5:1, when n and x are input. Laguerre polynomials, and their associated counterparts [see Section 23:12],may also be evaluated through the universal hypergeometric algorithm that is presented in Section 18:14. Set h = f = j = I Storage needed: h, f, j, n, m, x Input n >>»» and g 213 THE LAGUERRE POLYNOMIALS L.(x) Input m >>Input x >> f = L`-(x)If n = 0 go to (2)t»»Replace f by I + m - xIf n = I go to (2)(1) Replace jby j + ISet g = fReplace f by [(2j + m - I - x)g - (j + m - I)hl/jReplace h by gIfj<ngoto(I)(2) Output f 23:9 APPROXIMATIONSFor large x one can approximate23:10 Input restrictions: n must be anonnegative integer. x and m areunrestricted. Test values:l4any x) = 1. L,(9) = -8L6(8.5) = 12.6229384L'(1) = 2.33333333 23:9:1L.(x) =8-bit precisionx ? (6n - 2)3'2while for small x and large nxx"'123:9:2L.(x) =exp(\x2/l[Jo(V)+2N JWNwhere N = 4n + 2. For further details of the asymptotic expansion 23:9:2 and for approximations to L,(x) as n- z for other ranges of x, see Erddlyi. Magnus, Oberhettinger and Tricomi [Higher Transcendental Functions.Section 10.151. 23:10 OPERATIONS OF THE CALCULUSDerivatives and indefinite integrals of L.(x) can be expressed in terms of other Laguerre polynomials or. moresuccinctly, as associated Laguerre polynomials [Section 23:12) of unity orderdn 23:10:1 drL,(x) = x [L.(x) - L.-,(x)] = -L'.u dx) 23:10:21L.(t)dt = L (x) - L..,(x) = + 1L," WnOther important integrals are 23:10:3 andf exp(-t)L.(t)dt = exp(-x)[L.(.r) - L.-,(x)1 = -n exp(-x)Q (x)X 23:10:4I`L.(t)L.,(x - t)dt = IL...,(t)dt =xL;,°. jx)0 on+ m+ Iand many more are listed by Gradshteyn and Ryzhik [Sections 7.41. 7.42). The orthogonality of the Laguerrepolynomials is established by the definite integral23:10:5jexp(-t)LrU)L.(z:={0m * n 1mn 23:11 THE LAGUERRE POLYNOMIALS L,(x)23:11 COMPLEX ARGUMENT Laguerre polynomials are usually encountered with real arguments only. 23:12 GENERALIZATIONS214 In previous sections of this chapter we refer to the family of associated Laguerre polynomials L; "(x) of which theLaguerre polynomial is the m = 0 case23:12:1 L'.Q1(.x) = L"(x)Each of the definitions 23:3:1-23:3:5 may be generalized as follows: 221xtL;""-1 << I 23:1: exp ,(x)tt 23:12:3 exp(x) d"n!x" dx 23:12:4 L;,'`(x) = limI -2-rIµ-=µl11 23:12:5exp(x) n 23:12:6x- +(m+I -x)-+nf=0f=cL'"'(x)+The associated Laguerre polynomials may be expanded via 23:12:7 ()n-J1!and the early members are listed in Table 23.12.1. Numerical values of L'. '(x) are given by the algorithm in Section23:8 if the portions printed in green are included. Analogs exist of most of the formulas of Sections 23:5, 23:9and 23:10, including the following recursion, differentiation and orthogonality formulas:In +m- I -xn+m - 123:12:8L;'(x) =L;;_',(x) -nTable 23.12.1 m-0m=1 m = 2m=3m=4L. (x)L°'(x)L:.-(x)L'," (x Ix-nn+,n- IL'.7- "(x) +L' 1 "(x)xx x'3x'-x+ l2 - 2 x ' 1- 6 + 2 - 3x+ 1 X2x'-x-2- -3.x+3- 6-2x'-6x+4 -x3x- 4x + 6-x'S.r+- lox + 10262X2-x+4-5x+I0--+3x=-15x+20267x'-x+5-6x+I5-a+--21x+35262n LT'(x)ix'.c-+3x'-4x+ I243X,+Sx'-lox+5246x'15x=-x'+--20x+152427x'21x'- + - - 35x + 352462x'4x'+ 14x' - 56x + 70246m=0 m = 1 m = 2m=3 m = 4 215 THE LAGUERRE POLYNOMIALS L (x) 23:14 d.,mL-n_ _L_,, 23:12:9 L 23:12:10q(x) =1(xfr" exp(-t) L`"'(t)L,"(t)dt =x) 0(x);n # NJU lnNCaution is necessary because the associated Laguerre polynomials are sometimes defined asd'23:12:11 n!-L,(x)dx'which definition leads to properties radically different from those of the polynomials defined by 23:12:2-23:12:6.According to Sneddon [Section 44], the definitions adopted in this Atlas are generally favored by pure mathema-ticians, whereas 23:12:11 is preferred by applied mathematicians. Although interest concentrates on nonnegativeinteger values of the order m. definitions 23:12:2-23:12:7 may remain valid for negative integer m. and even fornoninteger values provided m > - 1. Polynomials so defined are often named generalized Laguerre polynomials.the simplest instances are 23:12:12 and24n!x) 1H2,(Vx)23:12:13 4)"nwhere H is the Hermite polynomial [Chapter 241.The Kummer function [Chap(er 47) is a broader generalization of the Laguerre polynomial and of its associatedcounterparts. The relationship is23:12:14 L'`(x) =(nymlM(-n;m + l;x)J 23:13 COGNATE FUNCTIONSSeveral formulas are presented in previous sections that relate Laguerre polynomials to Jacobi polynomials [Section22:121, to Hermite polynomials [Chapter 241 and to Bessel coefficients [Chapter 52]. To the last category thegenerating function 23:13:1 should be added. 23:14 RELATED TOPICSexp(t) t" The term Laguerre function is encountered with two distinct meanings. It is used to signify the L,(x) function inwhich n is not restricted to integer values, being then an example of the Kummer function (Chapter 471(-v)jxi23:14:1 M(-v; I;x) _The generalized Laguerre function is addressed in Section 47:1. Alternatively, the expression/xldam"23:14:2 (n + 1)! expl2 /x'- --jL"'(x) 23:14 THE LAGUERRE POLYNOMIALS 216is defined as the Laguerre function. It is a solution to the radial portion of the Schrlidinger equation for a hydrogen-like atom 23:14:3dZj+2dj_In+f(I+ 1)dx2x dx4xxJ f = 0n?1+ Iwhere it and I are the appropriate quantum numbers [see Sneddon. Section 44]. CHAPTER24THE HERMITE POLYNOMIALS H,t(x) The Hermite polynomials are orthogonal [see Section 21:14] on the interval -x < x < x with a weight functionexp(-.x2) [see Chapter 271. They occur in several applications to physical problems, for example, in the solutionof the Schrodinger equation [see Section 24:141. 24:1 NOTATIONThe notation H.(x) is in general use for the Hetmite polynomial of degree n and argument x. An alternative Hermitepolynomial, symbolized is discussed in Section 24:13; unfortunately, it also is sometimes denotedespecially in texts on mathematical statistics.Occasionally, Hermite polynomials of odd degree are defined with the opposite sign to that adopted in thisAtlas.Although this Atlas avoids the ambiguity, the symbol H (x) is sometimes used for the Struve function (seeChapter 57). 24:2 BEHAVIORThe Hermite polynomial has exactly n zeros, Int(n/2) minima and Int((n - 1)/2) maxima. These featuresare symmetrically disposed about x = 0. and all occur in the region -V2n < x < V -2n. Outside this range IH,(x)lincreases monotonically. except for Ho(x), remaining bounded globally by24:2:1 IH,(x)I < I .092" n! exp(x=)The range spanned by the oscillations of H.(x) increases so dramatically with increasing n that Figure 24-1portrays the behavior of H.(x)/n!. rather than that of the Hermite polynomials themselves. 24:3 DEFINITIONSHermite polynomials are defined by the generating function 24:3:1 exp(2tx - t=) = ir_on. , 217 ....:....: ...: ...: H4(x)/41: 219or by theRodrigues formula 24:3:2 As well, the integral representation 24:3:3THE HERMITE POLYNOMIALS H"(x) _ (- 1)" exp(x')"exp(-x2)24:5 2r nw H"(x) _exp(x')Je exp(-tt) cos( 2x1 - 2 Idt may be regarded as defining H,(x), as may expansion 24:6:1 or the limiting operation24:3:4 H,(x) = IimC 'r applied to the Gegenbauer polynomials [Section 22:13].One solution of Hermite's differential equation 24:3:5--2x-+2nf=0is f = cH,(x), where c is an arbitrary constant; a second solution is a closely related infinite series [see Murphy.page 321, but note that his definition of H,(x) differs from ours]. Another important differential equationdf 24:3:6 '= (x' - 2n - 1)fdxis solved by f = c exp(-x2/2)H"(x) [see Section 24:141. 24:4 SPECIAL CASESThe first eight members of the family of Hermite polynomials are24:4:1 Ho(x) = 124:4:2 H1(x) = 2x24:4:3 H(x) = 4x2 - 224:4:4 H3(x) = 8x' - 12x24:4:5 H,(x) = 16x' - 48x= + 1224:4:6 H5(x) = 32x5 - 160x' + 120x24:4:7 H6(x) = 64x° - 480x° + 720x= - 12024:4:8 H;(x) = 128x' - 1344x5 + 3360x' - 1680x 24:5 INTRARELATIONSHIPSThe reflection formula24:5:1 H,(-x) _ (-I)"H,(x)shows that a Hermitc polynomial is an even or odd function according to whether its degree is even or odd. Theaddition formula24:5:2H,(x + y) = 2-"'(n)H,(xV2)H,.. (yV2) 24:6 THE HERMITE POLYNOMIALS H.(x)and the Chrisroffel-Darboux formulaH"-i(x)H,(y) - H"(x)H,.1(y) 24:5:32j!2°-'n!(x - v) -oJare obeyed by Hermite polynomials. Other summation formulas include the infinite series24:5:4 }lH2(x)+H4( x)Ho(x) _ Z 4 _ ... = e cos(2x) and 24:5:5 H.(x)HS(x)He(x) - = + - - ... = e sin(2x)3!5!220 where e is the base of natural logarithms [Chapter 11, Series akin to 24:5:4 and 24:5:5 in which the signs do notalternate sum to (cosh(2x)]/e, and [sinh(2x)]/e, respectively.The recursion formula24:5:6H,(x) = 2xH,_,(x) - 2(n - 1)H,_2(x) n = 2, 3, 4, ...together with H0(x) = 1 and H1(x) = 2x, permits the polynomial form of any Hermite polynomial function to beevaluated. 24:6 EXPANSIONSThe explicit expansion of the Hermite polynomial may be written in several equivalent ways; for example:n(n - 1) n(n - 1)(n - 2)(n - 3) "_, 24:6:1H,(x) = (2x)° -1!(2x).-2 +2!(2x) (_2)"I2(n- 1)!!n=0.2,4,...-(-2)'^. ,r2n!!xn = 1, 3, 5, ...zrn!. (-I)J(2x)"-J = Int(n/2)1-0 j!(n - 2j)! 24:7 PARTICULAR VALUES H.(--)H.(O)H,l=)n-0 rrI0(-2Y" (n - 1)' Because of relationship 24:10:1, certain characteristic values of the argument correspond to features in a numberof consecutive Hermite polynomials. For example, the four values ±V(3 ± V)/2 are the zeros of H4(x), generatemaxima or minima of H5(x) and correspond to inflection points of H6(x).The asterisks in Figure 24-2 show the exact locations of the zeros for H1(x) through H,2(x). The colored linesin this diagram are explained in Section 24:9.24:8 NUMERICAL VALUESThe algorithm ISeth=f=j= 1 Storage needed: h, f, j, n, x and g1 221 THE HERMITE POLYNOMIALS H.(x)Input n >> »»Ifn=0goto(2)Input x >> »»Set f = 2x lfn=Igoto(2)(1) Replace j by j + 1Set g = fSet f = 2[xg + (1 - j)hJSet It = gIfj<ngoto(I)(2) Output ff=H"(x)< ««24:9 Input restrictions: n must be a nonnegative integer; xis unrestricted. Test values:Ho(any x) = 1H,(-12)-24H7(4.2) = 1436292.99 utilizes equation 24:5:6 to generate exact values of the Hermite polynomial. Hermite polynomials may also beenumerated via the universal hypergeometric algorithm of Section 18:14. 24:9 APPROXIMATE VALUESFor large (positive or negative) values of x. the approximation 24:9:1H"(x) =[ZrL-n218-bit precisionjx{ > 2.3(n - 1)"'holds. For large it, the Hermite polynomial is approximated by the oscillatory function 24:9:2 H"(x) =(-2)`" "/'- n!!exp(x2/2) sin(x2n + 1)odd nxV2n(-2r'2(n - 1)!! cxp('-/2) cos(x 2n + 1)even n -, x This approximation is best for small I and fails hopelessly for lxI > N/:);;, that is, when the Hermite polynomialleaves its oscillatory region [see discussion in Section 24:2J. 44+i4i444444444C4Iroej%.440'441440 44ti441: +40\n14.. a ..,t x\. x:.. a 12*n=10.....:....:. *:*\:*. *......nab FIG 24-2 n-2:.........:.... :....:....:....:.... .... ..' n=0 ..... 24:10 THE HERMITE POLYNOMIALS H"(x) 222From 24:9:2 it follows that the n zeros of H"(x) are approximated byj7r1=±1,t3,±5,...,±(n-1) neven 24:9:3r=22n+1 J=0,±2,±4,...,±(n-1) noddThe lines drawn on Figure 24-2 connect the zeros predicted by this approximation, blue for even n and green forodd n. Evidently, the approximation is good for the inner zeros but worsens the further the zero is from x = 0. 24:10 OPERATIONS OF THE CALCULUSWe have the following simple results for differentiation and indefinite integration: 24:10:1 24:10:2 24:10:3 andH It'-it =( H".,(x) rrJ1dH"(x) = 2nH"_,(x)n = 1, 2, 3... . ddx[exp(-x')H"(x)] = -exp(-x')H"-,(x) 2(n + 1)n=0.2,4,...H" ,(x) - (-2)". "n2(n+ 1)n= 1.3,5, exp(-x2)H".1(x) n = 2, 4, 6, .. . 24:10:4J exp(-t')H"(tI = io(-2)'"-'11'(n - 2)!! - exp(-s-)H"_,(x) n = I, 3. 5, .. .Three important definite integrals are 24:10:5 24:10:6 and 24:10:7exp(-t')H"(bt)dt =V; n!(b' - 1)"2/(n/2)!n = 0, 2, 4, ...J=0n= 1,3,5,...r0n=0,2,4,...J=t exp(-t2 )H"(bt)dt =\cn"b(2b'-2)"-" I'n=1,3,5,...f.r" exp(-t')H"(bt)dt = \n!P"(b) where P. is the Legendre polynomial of degree n (Chapter 21], and many others are listed by Gradshteyn andRyzhik [Sections 7.37 and 7.38].The orthogonality of Hermite polynomials is established by 24:10:8r0m * nJ =exp(-t2)H"(t)H,"(t)dt =jV'n 2"n!m = R 24:11 COMPLEX ARGUMENTAs for other orthogonal polynomials, the argument of the Hermite polynomials is generally encountered as a realvariable. 223 THE HERMITE POLYNOMIALS H"(x)24:12 GENERALIZATIONS24:14 Hermite polynomials are special cases of the parabolic cylinder function [Chapter 46J in which the order is anonnegative integer: 24:12:1 2"12 expl2In = 0, 1, 2, ...11Because the parabolic cylinder function itself generalizes to the Tricomi function [Chapter 481 or the Kummerfunction [Chapter 471 these latter functions may also be regarded as generalizations of Hermite polynomials I-n3 24:12:2 H"(x) = 2"U22x`/-n1.\24:12:3 /1-n 3-(-2)`" 1)n!! M 2 ; 2; x'n = 1. 3. 5.... 24:13 COGNATE FUNCTIONSThe alternative Hermite polynomial He"(x) is related to H"(x) by a simple variable change24:13:1 He"(x) = 2-"1=H"(x/\) 24:14 RELATED TOPICSEquation 24:3:6 arises in the application of Schradinger's equation to simple harmonic oscillators. The solution 24:14:1f=expl2'IH"(.t)=f(x)i is known as a Hermire function. In addition to sat sfying a typical orthogonality relationship 24:14:2 1 "f(t)f"(t)dt=IV'0m * n'nn!2"m=nHermite functions also satisfy such related expressions as0m* n t 1 24:14:3f (tt -f(t)m=n-! CHAPTER25THE LOGARITHMIC FUNCTION ln(x) The logarithmic function may be regarded as the simplest transcendental function, that is, ln(x) is the simplestfunction of x that cannot be expressed as a finite combination of algebraic terms. Prior to the advent of electroniccalculators, logarithms [especially decadic logarithms. see Section 25:14] were extensively used as aids to arithmeticcomputation, both in the form of tables and in the design of devices such as slide rules.In addition to its treatment of the logarithmic function, this chapter includes brief discussions of the logarithmicintegral li(x) and the dilogarithm diln(x).25:1 NOTATIONThe name logarithm of x is used synonymously with logarithmic function of x to describe ln(x). Alternative no-tations are log(x) and log,(x). although the former is also used to describe a decadic logarithm- The initial letteris sometimes written in script type to avoid possible confusion with the numeral "one." See Section 25:11 for thesignificance of Ln(:).To emphasize the distinction from logarithms to other bases [Section 25:141, In(s) is variously referred to asthe natural logarithm, the Naperian logarithm, the hyperbolic logarithm or the logarithm to base e.25:2 BEHAVIORThe logarithm In(x) is not defined as a real-valued function for negative argument [but see Section 25:11] and takesthe value -x when x is zero. As shown in Figure 25-1, In(x) is negative for 0 s x < I and positive for x > 1.The slope dln(x)/dx continuously decreases as x increases so that, even though ln(x) increases without limitas x tends to infinity, its rate of increase becomes ever more leisurely. In fact, In(s) increases more slowly thanany positive power of x: 25:2:1In(x)-.0x-,xv>0x''25:3 DEFINITIONSAs illustrated in Figure 25-2, the logarithmic function is defined through the integral 1 25:3:1 In(s)-dtx > 0 us 25:4 THE LOGARITHMIC FUNCTION ln(x) 226 4444444444444444 :....:....:....:....:....:....:....:....:....:....:....:.... ....:..-5.:....:....:........:........:....:....:....:1...:....:....:....:.. 4 It may also be defined as the limit25:3:2 ln(x) = lim[v(xU' - I)]x > 0v>0or by the rule25:3:3 ln(x)=fifx=elwhere e is the number 2.1718281828 defined in Chapter 1. 25:4 SPECIAL CASESThere are none. 25:5 INTRARELATIONSHIPSThe logarithmic function of any argument in the range zero to unity may be related to ln(x) where I < x < x bythe reciprocation formula 227 THE LOGARITHMIC FUNCTION In(x) 25:5:1 In(-'= -ln(x)X>0XThe logarithmic functions of products, quotients and powers may be expressed by the formulas25:5:2 ln(xv) = ln(x) + ln(y)X>0y > 025:5:3ln(x/y) = ln(x) - In(y)x > 0y > 025:5:4 In(x) = vin(x)X>025:8 If a function f(x) is expressible as the finite (or infinite) product IIf,(x), its logarithm is the finite (or infinite)sum25:5:5 ln(f(x)) _In(f1(x))all f;(x) > 0provided, in the infinite case, that the series in question converges. 25:6 EXPANSIONSThe logarithmic function may be expanded as a power series in a variety of ways, of which the following arerepresentative:x2x3(-x)' 25:6:1 Ix -1(x - 1)'(x - 1)'(x - 1)' 1 25:6:2ln(x) _ - +++_r a - x273.r'jx'2X2- 1(x' - W(x2 - 1)S Ir' - 1)7j" 25:6:3ln()=++ x>0 x + I3(x + 1)'5(x' + 1)5i=o2j + IlAs well, the logarithm is expansible through the continued fractions Ixx4x9x16x25:6:4 In(1 + x) = 1 - -I+ I-x+ 2-x+ 3-2x+ 4-3x+ 5-4x+and 25:6:5In(l +x)_Ixx 4x 4x 9x 9x 16xx1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+ 25:7 PARTICULAR VALUES In(O) 25:8 NUMERICAL VALUESIn(I) 0In(e) 1We') n Computer languages invariably permit logarithms to be calculated by a single command. Likewise, scientific cal-culators mostly incorporate a key that generates the value of In(x) from an x value in the calculator's register.Sometimes a "log" key must be struck and the result multiplied by 2.302585093 (see Section 25:14]. Because of the widespread availability of such devices, we include no algorithm for the logarithmic function, although theuniversal hypergeometric algorithm [Section 18:14] does permit logarithms to be evaluated to any sought precision. 25:9 THE LOGARITHMIC FUNCTION In(s)25:9 APPROXIMATIONS A number of approximations, including 25:9:1 and 25:9:2x - I 34 ln(x) =8-bit precision- s x :S -x 43(6)3f5ln(x)(x - I)8-bit precision- s x 5 2 1 + 5x 2are available when x is close to unity.Based on the limiting expression 25:9:3In(n) -. -y + E 7,n-' xy = 0.5772156649/=1 1where y is Euler's constant [Chapter I), we have the approximationfrac(x)`n"'125:9:4In(x) _- 0.58 +-8-bit precisionx z 31Xj.1 Jvalid for large arguments. 25:10 OPERATIONS OF THE CALCULUSSingle and multiple differentiation give 25:10:1 25:10:2dbrIn(bx+c)=bx+c/ -b 1- ln(bx + c) = -(n - I)!dx" bx + cn= 1,2,3,... while indefinite integration yields the following results: 25:10:3 25:10:4 25:10:5 25:10:6 25:10:7I:/In(bt + cdr = x +b)[ln(bx + c) - 1)C )/b[-In(x)1'In"(t)dt = (- I)"n!xn = 1.2.3....fl " J=r)j ! +fvl1IV.In(x) -+v * -1r' ln(r)dr =In'(x)2V = -I228 dt_li(bx + c)J-/bin(bt+c)bt''dr,0In(t) -{In(ln(x))v = - 1The Ii function in the last two formulas is the logarithmic integral function [see Section 25:131. In addition todefinite integrals that can be evaluated as special cases of the last five formulas, the following are of interest: 229 THE LOGARITHMIC FUNCTION ln(x) 25:12 25:10:8JIn(-In(t))dr = -y0ln(r)dt(-mj/12 25:10:9 '- r(In(t)drl -a'/6(-G25:10:10 _Jot ± t-1 25:10:11 In(t)ln(1 t r)dt =2 - (ir'/12) - In(4)f2- (a'/6)Others are given by Gradshteyn and Ryzhik [Section 4.2-4.41. The constants appearing in these expressions aredefined in Section 1:7.Semidifferentiation and semiintegration, with a lower limit of zero, yieldd'rln(4bx)25:10:12 inIn(bx)=- andd1/`x 25:10:13 T7, ln(bx) = 2(ln(4bx) - 21 25:11 COMPLEX ARGUMENTWhen the argument of the logarithmic function is replaced by the complex number x + iy, the result is a functionthat adopts infinitely many complex values for each pair of x.y values. To emphasize this many-valued property.the first letter of the function's symbol is capitalized in writing:25:11:1Ln(x+iy)=ln(x2+)')+i[2kar+9)-tr<gsirx2+y2*0k=0,±1,±2,...where B is the polar angle corresponding to the Cartesian coordinates x, v. In selecting a single value, the so-calledprincipal value, to represent the logarithm of a complex argument, it is usual to take k = 0 in 25:11:1. Thedetermination of 6 uniquely in terms of x and y leads to25:11:2ln(x + iy) =12 ln(x2 + y2) + i sgn(y) arccot(x/Lvf) y * 0in the general case of a complex argument and to 25:11:3 ln(x) = ln(Ixi) +2ra[I - sgn(x))x * 0when v = 0 (for example. In(-1) = ia).Setting x = 0 in result 25:11:2 yields the formula25:11:4 In(iy) = ln((yj) + t* sgn(y)for the logarithm of an imaginary argument (for example, ln(±i) _ :tin/2). 25:12 GENERALIZATIONSWe defer to Section 25:14 a discussion of the generalization of ln(x) "to other bases." Polylogarithms are addressedin Section 25:13. 25:13 THE LOGARITHMIC FUNCTION ln(x)The logarithmic function is the special v = I case of the function defined by 25:12:11x-1x}1j+vxJ2230 which has been termed the generalized logarithmic function and is denoted ln,.(x) [see Oldham and Spanier, Section10.51. This generalized function is also representable asIx-1 25:12:2 ln,(x) = B v; 0;xin terms of an incomplete beta function [Chapter 58). 25:13 COGNATE FUNCTIONSThe logarithmic function is closely related to its inverse, the exponential function [Chapter 261. and to the inversehyperbolic functions [Chapter 311.The logarithmic integral, defined by 25:13:1 li(x) = f AIn(t) = xIn(x)atIn(t)is a related function with importance in number theory. Figure 25-3 illustrates its definition, and its behavior isportrayed in Figure 37-1. Because of the identity25:13:2 li(x) = Ei(ln(x))x > 0the properties of the logarithmic integral follow from those of the exponential integral of Chapter 37. Numericalvalues of li(xjmay be generated via the algorithm of that chapter. 231 THE LOGARITHMIC FUNCTION In(s) 25:13Yet another related function is the dilogarithm (or Spence's integral)ln(t)ft-In(t)wr'25:13:3 diln(x) dt =dt - -ftl 16which occurs in radiation theory, among other applications. Its definition is illustrated in Figure 25-4 and its be-havior in Figure 25-I. The dilogarithm is also expressible as the sum .(I -x)' 25:13:4diln(x) = - ,0:S x:5 2from which the particular values diln(O) = -4(2) _ -a'/6 and diln(2) = q(2) = w2/12 follow. Here 4 and qsignify zeta and eta numbers [Chapter 31. Confusingly, there are a number of alternative definitions and notationsfor the dilogarithm. Series 25:13:4 is rapidly convergent for 1/2 S x < 1 and advantage of this fact is taken inthe algorithm ti Input x >> f = diln(x)4'#4 FIG 25-4 Setf=g=0Set s = IIf x<Igo to (1)Set g = -[ln2(x)]/2Set s = -1Replace x by l/x(1) If x ? 1/2 go to (2)Replace g by g + s[(a=/6) - ln(x)In(1 - x)JReplaces by -sReplace x by I - x(2) Set j = 2 + Int(4/x2)(3) Replace f by [f+(1/j'-)J(1 - x)Replace j by j - 11fj*0goto(3)Replace! by -g - sfOutput fStorage needed: f, g, s. x and j Input restriction: x must be pos-itive. Test values:diln(0.7) = -0.326129511diln(0.2) _ -1.07479460diln(10) = +3.95066378 by replacing the infinite upper limit in 25:13:4 by 2 + Int(4/x2). The algorithm calculates dilogarithms of argumentsin the 0 < x < ; range by exploiting the reflection formulaa= 25:13:5diln(1 - x) = ln(x)ln(1 - x) - diln(x) - 60 S x S 1 25:14 THE LOGARITHMIC FUNCTION In(s) 232 and those in the range x > I are accessed via the reciprocation formula(x)25:13:6 diln(I/x) =In=2- diln(x)x > 0The algorithm generates values with 24-bit precision (relative error s6 x 10'8) for all positive arguments. Theuniversal hypergeoretrc algorithm [Section 18:141 may also be used to evaluate the dilogarithm in the range 0 sxs2.One may similarly define a trilogarithmr diln(t)- 25:13:7 triln(x)=Jdt(I=-e- IJ'x)' it occurs in distribution theory [Section 27:14] and is also evaluable. for 0 <_ x 2. by the universal hypergeometricalgorithm of Section 18:14. The logarithm, dilogarithm and trilogarithm generalize to the polylogarithm 25:13:8 polnjx)(1 - s)',-J ,where v is not necessarily an integer. The polylogarithm is encountered also in Section 64:12 and, in the alternativenotation -F(I - x,v). some of its properties were reported by Bateman [see Erdilyi, Magnus, Obcrhettinger andTricomi, Higher Transcendental Functions, Volume 1, pages 30 and 311. 25:14 RELATED TOPICSIf (3 is a positive number other than unity and x = [31, then f is termed the logarithm of x to the base B and isdenoted loga(x)25:14:1 logs(x) = logo([3') = f0 < 0 * 1A logarithm to any base is proportional to the logarithmic function In(s) 25:14:2 logg(x) = log5(e)ln(x) =ln(x)In(p)0<0* IOther than e, the most commonly encountered logarithmic bases are 2 and 10.Logarithms to base 2 are called binary logarithms 25:14:3(s)1og2(x) = (1.44269504)In(x) =In0.6931471806Logarithms to base 10 are called common logarithms. Briggsian logarithms or decadic logarithms. The sub-script 10 is often omitted so that log,o(x) is written log(s) or even lg(x). The identities 25:14:4 log,o(x) = (0.4342944819) In(s) = In(s)2.302585093obtain. The decadic logarithm of a number in scientific notation [Section 9:141 is easily found because25:14:51og10(no . n_,n_2n_3 ... X I0'") = N + log,o(no - n_,n..n_3 ...)The final term in 25:14:5, which necessarily lies in the range .000... through .999..., is called the mantissa ofthe logarithm, while N is its characteristic. Chemists use the p notation25:14:6 px = log1o(1/s) = -1og1o(x)For example, pH = -log,o(H). where H is the activity (or concentration) of hydrogen ions. The name cologarithmof x is sometimes applied to px. CHAPTER26THE EXPONENTIAL FUNCTION exp(bx + c) Many natural and manmade assemblies (populations of bacteria, bank balances, collections of radionuclei, arraysof lightbulbs) increase or decrease at a rate that is proportional to their size. This characteristic, described byequation 26:3:4. leads to exponential growth or exponential decay, whose functional dependences are expressedby exp(bx + c) with b positive for growth and negative for decay.The function exp(bx + c), or sometimes the special cases exp(±x), are considered in this chapter. The ex-ponential functions of more complicated arguments are deferred to the next chapter. 26:1 NOTATIONThe symbolism e', where e is the base of natural logarithms [Section 1:41, often replaces exp(x). The name "naturalantilogarithm" and the symbol In-'(x) are occasionally encountered. Colloquially, exponential functions are oftencalled "exponentials." 26:2 BEHAVIORThe exponential function accepts arguments of either sign and any magnitude, but itself adopts only positive values.As illustrated in Figure 26-1, exp(x) rapidly increases while exp(-x) decreases towards zero, as the magnitude ofx increases.The exponential function shares with the unity function the remarkable property f(x) f(-x) = 1.Figure 26-1 also displays a map of the self-exponential function x`. This exhibits a minimum value ofexp(- I /e) = 0.69220at x = exp(-1) = 0.36787 -. 26:3 DEFINITIONSThe exponential function is defined as the inverse of the logarithmic function; that is:26:3:1 f = exp(x)wherex = In(f )Alternatively, the definition as a power of the number e [see Section 1:7126:3:2 exp(x) = e' = (2.7182818284.)' 233 26:4 THE EXPONENTIAL FUNCTION cxp(bx + c) may be used, or expansion 26:6:1 may serve instead. The limiting operation/x 26:3:3 exp(x) = liml I +-nis yet another definition of the exponential function, albeit not a very useful one in practice.The differential equation 26:3:4 of= bfdx234 is solved by f = exp(bx + c) where c is an arbitrary constant. Thus, exp(x + c) is the only function (other thanthe zero function) that remains unchanged on differentiation. 26:4 SPECIAL CASES There are none. 26:5 INTRARELATIONSHIPSThe remarkably simple reflection, addition, subtraction and involution formulas 235 THE EXPONENTIAL FUNCTION cxp(bx + c) 26:6 26:5:1 exp(-x) _exp(x) 26:5:2 exp(x + y) = exp(x)exp(y)exp(x) 26:5:3 exp(x - y) =exp(y)26:5:4 exp(x) = exp(vx)partly explain the widespread utility of the exponential function.Infinite series of exponential functions of negative argument that may be summed geometrically [see equation1.6:51 or as hyperbolic functions [Chapters 29 and 30] include 6++-2-3+hlI-2> 0 2:5:5exp(-x)exp(x)exp(x) 2exP(x)-12cotx +---2-3-=hl=1-> 0 25:5:6exp(x)exp(x)exp(x) 2tanezpO122/x 26:5:7 (-5x) +. (-x) + ex(-3x) + exex =exp()= 1csch(x)x > 0 pppexp(2x) - 12-=exp(x)_I +-3 26:5:8exp(-x) - exp(x)exp(-5x) sech(x)exp(2x) + 12X>0 Similar sums in which the exponential arguments involve the squares of the natural (or the odd) numbers lead totheta functions [see Section 27:131 26:5:9exp(-x) + exp(-4x) + exp(-9x) + _exp(-j 2x) =2931 0; a2 I -Zx > 0 Ix > 0 (-4x) + ex26:5:10ex(-x) - ex(-9x) -(-1)1 exp(-j2x) = 2 - 2 0(0! ppp 42 26:5:11ex-x) + ex (-9x) + ex25x) +=ex21x P(Pp(--P[-(2j - 1)2x] = - B2 0; Z x > 0 26:5:12exp(-x) - exp(-9x) + exp(-25x) -_ -(- I)' exp[-(2j - 1)2x] = 2 021 2; 7 I x > 0 26:6 EXPANSIONSThe expansion bx + c(bx + c)2 (bx + c)'26:6:1 exp(bx + c) = I + ++=21 1-oj1is valid for all arguments. Another expansion is in terms of hyperbolic Bessel functions [Chapter 491: 26:6:2exp(=x) = I5(x) -x 211(x) + 212(x) t 213(x) + (±1)'11(x) Among continued fraction expansions of the exponential function are 26:7 THE EXPONENTIAL FUNCTION exp(bx + c) 236 26:6:3 and 26:6:4 26:7 PARTICULAR VALUES 26:8 NUMERICAL VALUES1xxxx xexp(x) = - - - - - ---1- 1+ 2- 3+ 2- 5+ Xxxxxexp(x)= I +-----...1- 2+ 3- 2+ 5- exp(-z) 0exp(-I) t/eexp(o) 1exp(I) eexp(-) Most calculators and computers incorporate means by which exp(x) can be calculated directly. Therefore, no al-gorithm for the exponential function is included in this chapter although the universal hypergeometric algorithm(Section 18:141 may be employed to calculate exp(x) for any finite value of the argument x. 26:9 APPROXIMATIONS Relaxing the n -p x condition in definition 26:3:3 leads to13026:9:1exp(x) = 1+8-bit precision-I 5 X:5 1 130See the test values for an 'economized polynomial" approximation eo + e,x + e2x2 to exp(-x) in the range0 5 x 5 1. This approximation has a precision superior to 8 bit, and is presented in Section 22:14. Some rationalfunction approximations to exp(x) are given in Section 17:13. 26:10 OPERATIONS OF THE CALCULUSSingle differentiation, multiple differentiation and indefmite integration of the function exp(bx + c) give 26:10:1 26:10:2 and 26:10:3ddxexp(bx + c) = b exp(bx + c) d-a;exp(bx + c) = b" exp(bx + c) f exp(bt + c)dt =exp(bx + c)bb>0These three results may be generalized to the ruled"26:10:4 exp(bx + c) = b"exp(bx + c) b > 0[d(x + °C)]"for differintegration (Section 0:101 with a lower limit of -x. The generalized differintegral with a lower limit of 237 exp(bx + c) - exp(c)zero is a function containing an incomplete gamma function [Chapter 45]d" Y(-v bx) 26:10:5dr"exp(bx + c) = b" exp(bx + c)T(-v)= x 'exp(bx + c)y*(-v;bx)Differentiation of the self-exponential function givesd26:10:6 dxx' = x`[1 + ln(x)JFor positive b, indefinite integrals of t"exp(br + c) are generally evaluable as Kummer functions [Chapter 471 26:10:7Jr` exp(bt + c)dt = xvep(c)M(v + l;v + 2;bx)v > - l and the following special cases apply: 26:10:8 26:10:9 26:10:10 26:10:11 26:10:12Jexp(bt + c)dt =o br" exp(bt + c)dt =n! exp(bx + c)[exp(-bx)-a"(-bz)Jn=0.1.2....exp(btc)dt=-exp(bx+c)dawVb>0 t + c)exp(bTHE EXPONENTIAL FUNCTION exp(bx + c) 26:10 J=dt = exp(c) Ei(bx)b > 0 exp(br + cdz)_(nb"I) exp(bx + c)Lexp(-bx) Ei(bx) 0!1!(n - 2)!bx(bx)r(bx)"-'n=2,3,4,.b>0 the a", daw and Ei functions being explained in Section 26:13, Chapter 42, and Chapter 37. See Table 37.14.1for a compilation of indefinite integrals of .r"exp(x) for commonly encountered values of v.Equations 26:10:7-26:10:9 apply also when b is negative, but the general expressions 26:10:13 26:10:14Lexp(c)I ;v > -1b < 0 I exp(c)Jt"exp(bt+c)dtr(v+ I;-bx)v<---Ib<0in terms of incomplete gamma functions [Chapter 42] are then more useful. Special cases include 26:10:15 26:10:16exp(bt + c) ndt =V -b exp(c) erf V -bxb < 01fexp(bt + cIdt = -exp(c) Ei(bx)b < 0 )=bexp(bx + c) 0!1!- fexP(b:26:10:17+ ct"(n - 1)!bx+(bx)Z+l+exp(-bx)Ei(bx)ln - 2. 3, 4, ...b < 0 (bx)" 26:11 THE EXPONENTIAL FUNCTION exp(bx + c)and specific expressions for C r" exp(-t)dt will be found in Table 37.14.1 for a variety of v values.Some other important indefinite integrals includedtx1a + exp(bx + c)f-lI 26:10:18 anda + exp(br + c)aabna + exp(c) 1/ezp(bx)arctana > 0jrdtb''Va26:10:19a exp(bt) + a exp(-b:)1 Ih(!FPartana)a< 0bVZ/238 Some thirty pages are devoted by Gradshteyn and Ryzhik [Sections 3.3 and 3.4] to definite integrals of ex-ponential functions. Here we cite only four general examples:n!26:10:201t" exp(-xt)dr =exp(-x) e,(x)n = 0. 1, 2....X > 0 n!26:10:21Jt"exp(-xt)dt = -, [exp(x)e"(-x) - exp(-x)e"(x)] n = 0. 1, 2....X>0r exp(r"-xt)((n -x-)""'1)!0!1!Ei(-x) + exp(-4 - - Z 26:10:22Jdt = -Ixx+...+(-1)"(n-2)!Jjn=2,3.4,...x>0 and 26:10:23rr"dtdi-Jn!T(n+1)1n=0,1,2,...oexp(r) ± In! l;(n + 1) The 71 and { functions are those discussed in Chapter 3.Definite integrals of the form 26:10:24 f(t) exp(bt + c)dt1which generally exist only for a restricted range of b values, can be evaluated via the Laplace transforms that arelisted in Section 14 of this chapter. A Laplace transform is defined by26:10:25 fL(s) = Jf(t) exp(-st)dt and integral 26:10:24 can therefore be evaluated via the identity 26:10:26 26:11 COMPLEX ARGUMENTf(t) exp(bt + c)dr = exp(c) fi,(-b) When the argument of the exponential function is the complex number x + iy, the relationship26:11:1 exp(x + iv) = exp(x)[cos(y) + i sin(y)]known as Euler's formula, is obeyed. The special cases 239 THE EXPONENTIAL FUNCTION exp(bx + c) 26:1326:11:2 exp(ti1r) _ - I 26:11:3 exp =i 2 1tt are noteworthy. 26:12 GENERALIZATIONSThe function B. where 0 is a positive constant other than unity or e, is known as the general exponential functionor antilogarithm to base P. It is equivalent to an exponential function with changed argument26:12:1 W = exp(bx) b = ln((3)This formula enables the properties of B` to be deduced simply from those of exp(bx + c). The function 10' isfrequently encountered and is the common antilogarithm of x.Generalization in a different direction is provided by the concept of hypergeometric functions introduced inSection 18:14. In the nomenclature of that section, exp(x) is the K = 0. L = I hypergeometric function in whichthe sole parameter equals unity. Generalizing this parameter to v gives the function C(v) exp(x)y*(v - 1;x) (see Chapter 45 for the 'y* function], which could therefore claim to be a generalized exponential. 26:13 COGNATE FUNCTIONSThe exponential polynomial e,(x) may be derived from the 26:6:1 expansion by truncation after the .r"/n! term:xx'x^x' 26:13:1 I!2!n!;-o J!Thus, exp(x) = e.W. The concatenation 26:13:2e,(x)=11 II-+l1 x +11 x``\\n//n-I/n-2 1provides a convenient method of evaluating the exponential polynomial if its degree n is not too large. Section18:14 shows how the universal hypergeometric algorithm may be employed to calculate n!x-e,(x). For large n theasymptotic approximation 26:13:3 e,_2(x) - exp(x) -n:z -'n-xn!(n - x) is useful.The self-exponential function x` was mentioned in Section 26:2 and is graphed in Figure 26-1. The relatedfunction x"' displays a maximum value at x - e. The inverse function (Section 0:3] of the latter function is thelimit of the sequence26:13:4 x, x' x"`' x""",x""I"I",corresponding to infinitely repeated exponentiation. Figure 26-2 maps these interesting functions.Those Basset functions [see Chapter 51] K,(x), in which v is an odd multiple of i, are related in a verystraightforward way to the exponential function exp(-x). Thus, we have 26:13:5 KIr(x) _ .jexp(-x) 26:13:6 K3r(x) _- exp(-x)I 1 +IJ2x`z 26:13 THE EXPONENTIAL FUNCTION exp(bx + c) 240 26:13:7 Ks/2(x) _exp(-x)I I +3+xi3J YYY 2x LLand further members of the sequence may be constructed via the recurrence formula 26:13:8 2v - 2K,(x) = K. -2(x) +K,._,xMaps of the functions K,(x) for v = ,andare displayed in Figure 26.3. The symbol and the namemodified spherical Bessel function of the third kind are sometimes applied to the function a/2x K.. ,,2(x). 241 THE EXPONENTIAL FUNCTION exp(bx + c)26:14 RELATED TOPICS26:14 Laplace transformation is one of the most valuable techniques available for the solving of practical problems inengineering and the physical sciences. Particularly susceptible to this approach are situations characterized by a setof simultaneous ordinary differential equations, or by a partial differential equation with attendant boundary con-ditions. The overall strategy and motive for the use of Laplace transformation are illustrated in the following scheme: Original probleminvolving thevariable rfficuhFinal solutioninvolving thevariable rLaplacetransformation LaplaceinversionTransformed probleminvolving thevariable sfkileSolution to the transformedproblem involvingthe variable s In this section we present some general rules for Laplace transformation and inversion; the classic book by Churchillmay be consulted for further information. A short table of transform pairs is also included.In the following we use If and g to represent functions of t whose Laplace transforms ft, and IL are functionsof the variable s. which is sometimes called the dummy variable. Equation 26:10:25 shows how fL(s) is related tof(t), but it is often more economical to use the operator notation 26:14:1 L{f(t)}=ff(:)exp(-st)dtfn(s)to represent this relationship. In most applications t is a real variable whereas s is often complex. Here, however.we shall not pursue the complex nature of the dummy variable.It should be recognized that not all functions have Laplace transforms. To be transformable, a function f(t)must at least be defined as a real function over the entire range 0 s t s -. Any function that is finite throughoutthis semiinfinite range has a Laplace transform, although fL(s) is not necessarily a named function. Moreover, manyfunctions that encounter infinities in the 0 c t s - range may nevertheless be transformed. For example, an infinityat t = 0 will not impede Laplace transformation provided that Of(f) approaches zero as r tends to zero from positivevalues.Later in this section a table [Table 26.14. 11 of Laplace transforms is presented. Its utility may be greatlyenhanced by making use of certain general properties that obtain under Laplace transformation. Some of thesegeneral rules are explored in the next few paragraphs.The weighted sum or difference of two (or more) functions may be Laplace transformed by using the linearityproperty26:14:2 L{ctf(t) ± c;g(:)} = c,fL(s) ± c.gL(s)Provided that both series converge, this rule may be extended to the infinite summation Ec,f,(t), which therebygives Eci f; .(s) on transformation.The scaling property 26:14:3 L{f(bt)} _fr(bJb > 0of Laplace transformation is easily implemented but the linear shift property 26:14:4L{f(bi + c)} = b expl I[()J f(t)explbIdt]b > 0is generally less useful than the corresponding rule [equation 26:14:2\3]for Laplace inversion. Of course. if f isa function (such as sin or exp) for which an argument-addition formula [Section 0:51 exists, transformation off(bt + c) may be accomplished by that route. 26:14 THE EXPONENTIAL FUNCTION exp(bx + c) 242With a > 0 and u representing the unit-step function [Chapter 81. the function u(t - a)f(t - a) may be obtainedby nullifying the t < 0 portion of f(t) and then translating the residue along the t-axis as portrayed in Figure 26-4. Such a function may be Laplace transformed by the delay property26:14:5L{u(t - a)f(t - a)} = exp(-as)fL(s) a > 0 For certain simple functions g(t), the composite function f{g(t)} may be Laplace transformed as the definiteintegral 26:14:6 L{f{g(t)}}G(s.t)far)dt0where G(v) is a bivariate function, some examples of which are shown in Table 26.14.1. The most general chainrule for Laplace transformation is 26:14:7 L{f{g(t)}} = JG(s,t)fL(1)dttIowhere G(u;t) is the function resulting from Laplace inversion of exp{-uG(s)}dG/ds, G being the inverse functionof g [that is, G{g(x)} = x: see Section 0:3].There is no simple product rule for Laplace transformation: that is. the transform of f(t)g(t) cannot be simplywritten in terms of 1L(s) and gas). For certain instances of f(t), however, simple relationships do exist. Thus:26:14:8 L{exp(bt + c)g(t)} = exp(c)gL(s - b) Table 26.14.1 g(r)G(sx) exp(r)- I sinh(r)1 (2\ sr)exP(_4r /V exp(-r)I'(I+s)J,(r) 243 THE EXPONENTIAL FUNCTION exp(bx + c) 26:14 26:14:9 L{sinh(bt)g(t)} = I (gL(s - b) - gi(s + b)]and similarly the Laplace transform of cosh(bt)g(t) is [gL(s - b) + gL(s + b)]/2.When f(t) is a power of r, Laplace transformation of the product f(t)g(t) corresponds to performing operationsof the calculus on gi(s). The simplest cases26:14:10and 26:14:11 iterate to 26:14:12dL{tg(t)} = -dsgc(s) g(r)rL71) =Jgc(s)ds L{t"g(t)} _ (- 1)"gr.(s)n = 0, 1, 2, .. .ds'and (lr .. . 26:14:13LS gr")} =JJL(s)(dsrn = 2,3,4,and all these rules are subsumed in the general power multiplication propertyd' 26:14:14 L{t'g(t)} =[d(-s + x)]' AL(S)which corresponds to differintegration [Section 0:10] with respect to -s. the lower limit being -x.If f(t) is the derivative of a function F(t), then L{f(t)} = sFL(s) - F(0) and if f(t) is an indefinite integral ofti(t), then L{f(t)} = [IL(s) + f(0)J/s, Both these rules of Laplace transformation are special cases of the generaldifferintegral transformation property [see Oldham and Spanier, Section 8.11 (l i26:14:15Lif(t) } = s fc(s) - Z s' 'tit,If(0)J = -Int(-v)This rule applies for all is, but the summation is empty if v s 0.The periodicity property permits the Laplace transform of a periodic function [Chapter 361 to be replaced byan integral over a single periodI/ `26:14:16 L{per(t)}per(t) exp(-st)dt =I-Icothl Ps 1per(t) exp(-st)dt I-exp(-Ps) 222 ///Many important periodic functions are symmetrical in the sense that translation by a half-period changes the sign:that is, if ger(t) is such a symmetrical periodic function of period Q, qer[t + (Q/2)J = - Ber(t). In such casesLaplace transformation may be accomplished by the formula 26:14:17 L{per(t)} = gerL(s) =Qsger(t) exp(-st)dto1 + expI2=[.!.+tanhlQsJJ Jqer(t) exp(-st)dt\/oin which integration is over a half-period. The half- and fully rectified analogs [Section 36:131 of such a symmetricalperiodic function have transforms equal to [Z - coth(Qs/4)] gerL(s) and coth(Qs/4) gerL(s). respectively, wheregerL(s) is given by 26:14:17.The rules above, in conjunction with Table 26.14.2, permit a very large number of Laplace transforms to be 26:14 Table 26.14.2 ChapterorsectionnoTHE EXPONENTIAL FUNCTION exp(bx - c) 244 Chapterorsectionfm 10 r8( t) 1:13p(c:h:t - a) 8£u4t - Wit)j - 0. 1. 2.... 7bt + c2chsjexp(-as)sinh1 `1isteepeningstaircase - {(slscs+b s' 7 10:12 26 8u(t - a)a >0-1cscsIbcxp b) Ei .b/cs(brs,Euler'sfunctionbs +cbbs,c- exp(-asl 82£(-1)'uU - b - 2jb1j = 0, I. 2, ...pulses - sech(bs) 82£u(r -(j +)'7r-)j = 0. 1. 2....Staircases 840:s) with 81 + 2!u)r - j:w')j = 1, 2. 3....lengthening g,(Os) treads2b-ccs - bno 7 28 I II 37 7 7 26 29 27:13 27:13 8c - btb > 0exp(-) +- 26 sbs' 9Int(bt)staircaseb Mhr s)-1J 30 2s12b.r 1 1 9fretP/sawtooth + I - coth Ps 30 2Pss(2)J 9Int(bt - 1)staircase ?t csch(-s) 29tri 9+ (-1)1-,, frac(2i) - J angular 2ta,nh\l P-// J 30 2wave Ps'4 2 LLLP /106(r - a)a > 0 exp(-as) 2610:12611 - a)a > 0 s exp(-as) 2611r"n = 0. I. 2.. _n'/r"' I I 11(br+cY^=2.3.4. -. (n!)'sj=0.1.2.....nII An -j)'(bCS,)' 11bc'-'(br + c)-n - 2. 3.4. ..exp(rs\E.(cs\Schl6milch 37:13 bb/1functions 26 16t"' exp( b1)at'+bi+c(^ - 1)!b'(bs + rY'n = 2. 3. 4, ...cb2a SS'71 I 245 THE EXPONENTIAL FUNCTION exp(bx + c) 26:14Table 26.14.2 (Continued) Chapterorsection 23 12 12 12 12 12 12 12 26 41.42 41 41 40 13 13 13 13 13 13 13n = 0. 1. 2. Vr b1+ 1°C1 b1+c7 (1 - Wt - b)jVr exp(-bf) erf(V -br)ordaw(V br)VR a cxp(drl erfc(a V 1)V Rt exp(a-r) erfc(aVr) erf(Vbf) Ib1+c)"v>-1lt-aY-'u(t-a)v>0b""f'v>-1bt\\+ c(rbbl`ulr-b)v>-1r(I-u(f-b)jv>-1b'P-'01 +cr-'v>0<It- + -ex -216b) bsexp(b) erfclbJ exp(Cs) erfclcEb. `.b,J-erf(VbJ)YY J R erfclV bsl Rbbs+cbg0 V,I VJ +a)1 bss - brl o /s,exp(b) r( I +v: b rl vlexp(-as)/r(I +b.\br(1 + v)r(-vbs)r(v)b"-y-(1 + v;bs)r(v)c""-'u(v;v+cs\ µ;Chapterorsection 11 12 13 41 41 41 40 40 12 12 12 12 12 13 45 26 45 45 45 48 26:14Table 26.14.2 (Continued)THE EXPONENTIAL FUNCTION exp(bx + c) 246 Chapterorsectionno. lr) 26r'-' expibr - c)v > 0 45exp(hr)y(vbr)v > 0 47s "-' exp(-a»M(v;v + war)v + it > 0 46Ibr4211-u(r- a))14WNa' -r>0 15 a IS 15Vr - aa>0u(r - b)V :-b49.52WIN' -0)orJ.(,%; 49. 52I1UV -alor],(IV o) 50. 53I,uVa)or3,AVa) 39S(br)orC(bt) 55N", ber(brlorV2 bei(hr) 512b'K.(br)v - 0. I. 2.... l 26exp((V -A - b)rexpl -(V -A I blr1} ll2a2aN/A 322 expl-bi)sinl -\\` 2a\,2a 28. 32sinhuV-aiorsin(rVa) 28. 32cosh(rV -a)orcos(tV o) 16:13u0 - b) - u(r - 3b)V-r=+4br-3b=DO ex"",Is - b)' I//Irhrly)1bslbs - crlv+µ) V2itn(V 2s + \/ yWas) - (das) ho(sVal -VaI1c-+1!5 exp(as)erfc(Vas)2v Vs+aVla`s-11a§0a 111111Vs2 - a IrVal(Vs=as) LJs'+aK.(bs ( a§o VbVs= + b' + s)/(s 4 b') +1sVvs b +S +b'a§0 Vas - +bs+cv -as'-bs+cV4ac-6A<0Sac-h'=A>0a r 0J"+a a S 0 W bs)Chapterorsectionno. 13 13 13 13 57 57 41 51 IS 15 15 15 IS 15 16 16 16 16 49 247 THE EXPONENTIAL FUNCTION exp(bx + c) 26:14Table 26.14.2 (Continued) Chapterorsectionno. 49. 52 17-24 52 25 25 37 38 27 27 27 50. 53 40:13 41 28. 32 28. 32 46 27:13 27:13btv __1exPor\ a2aJot2aTat'a, = b/%'t'hd)v> -1 + In(t) = In(ar)a = 1.78107. In(bt + c) ctIn(c) - E -bCi(bt) - 1Mb) eexp(-,:'It) (20"exp-af)a>0I _ba.`b"t" 'expl2Iv> 0 `[I,(2VboorJ.(2V_-W)1v> -1t""i erfc(V)n = -1. 0, 1. 2... exp(br + c) erfc(V bt +2Vb-t Icosh(2V ) orcos(2)i/ y t sinh(2\)orsin12\ b,) El.f xP(Q 0v:t)vk s2fa4ac - b-'A 3: 0as'+bs+c s2b`b, - Jta,r(1 + v)b1bs...exp4s /1'(4s1 In(11 =-IMsIrsss 1I s I In(c)-a%P(\Jb/Et( bs)J In(bs + c) S2(K,.(2af1 R2 + 20o'aexp 8s-D.;,_;Jv > -1r"'\ 3s r(v)exp(4b) D "\b)(blexp\s/b 3: 0exp[-2Vr(Z expc J-' exp(-b 2 0SS) E1,11 exp(b)b 1E 0rtS. (2s)", CXP(- V'2-b;)s sech(V S) stnh(2v y s) Isech(\/-S) sinh(Vs - 2vVs)VsChapterorsectionno. 16:13 17-24 23:14 25 37 25 25 51 46 46 27 27 27 27 27 27 28. 29 28. 29 27:13 O,(v:r)0 K v s 1 csch(V) cosh(Vs - 2vVst 28. 29 Vs 26:14 THE EXPONENTIAL FUNCTION exp(bx + c) 248Table 26.14.2 (Continued) Chapterorsectionno. 27:13 32 29 29 29 29, 30 29. 30 30 32 30:13 31 31 51 54. 57 32 32 32 55 32:13 35f(r) B.(v;r)IvI s 2 sgnjsin(V ;)) lengtheningsquare wave/bt21x-br)"exPl 2) csch(b)22 sech(br) (2hr1' csch(br)v > 0 b'?I I + csch(br) - coth(br)) b"r"-'II- csch(br) - cothlbr)) tanh(br)arsgn sinsquarewave(26coth(bi) - br 2 arstnh(br) arcosh(l+ br) K (bi) iY,(br)orrh(br)-sinrlory'rl2rl2r)r" sin(bi)0 * v > -I Sin(V'h) bc02V br)orbei(2Vbr1 I)-J1b sincw= I sin(bi) 2 arctan(V)csch(V s) cosh1206)Vs e.(o;s)y..(b)n0.I.2... 1 G(s+ bl2bl 2bfll+v) Is+h)bt1 + v:2b l G12 I_12bbs 1tanh(bs)1n(!-ly(s -Ih(-2b)bZh.s[h.(!)-Y,(s)Jb.``lb. J Sexp(b/K ( )Il-arcoshll\\Vs'-h=b arsinhor or arsinh)Vt+bbs,, expl-sr(sin varccot(-J)(s + b')"rb in erf(V b%4s ) I/6 I\i-cos - or-sinIgSsSSsbarccatb- I = arcten-s.r l 5exp\6/ e1fc1b1lChapterorsectionno. 28. 29 27:13 44:12 44:13 64 64 64:13 44:13 30 44 57 SI 31 31 32 35 40 32 35 41 249 THE EXPONENTIAL FUNCTION exp(bx + c)Table 26.14.2 (Continued)26:14 Chapter Chapteror orsection sectionno. f(r) US) no.MsCirbl 38 35 arctan(br) cos(SI- Si(-)Iu/b) 57:13 srC,lbt)IlL 2s)l1 /J 35V7 \b's/ 38 Si(br)sarccot(/b)sarctanlsI 35br1\\\s' 422 daw(2 bEi(b' ) 37 40 erf(bt) I exp(s _) erfcl s/41s4b'2b/b=r'1 f(2v)s- `s 452"yl v:Iv > 0 expl - I D_,.l - I 46 2 / 2s4bbb 20s)' 45f 2v;-+///) K_.(2Vbs) 51 }}lrl 56 2 r(3/ (2)"'Jexpr \b/ r(3:45 47r MIa;2c br)c > -1f(I+ c) FI a, I + c:2c: b1 I 60r(c)'vbr"-"'r \ 50 1._,(2V' br) f(a)- M ax;b- 47 00' s"I$ 48t'"'exp(-Br)U(c - ba + a - 6:811 B''" F a,b:c:sB 602b sinc(vtb - s)'-'tb + s)-"48:4 k B (I - v;I + v:2b 49 s K(-) 61 49 bin(br)l,lbr) s lI-I E(26) 6II 50LLL r251 KdV 'b,) ab 1f 51 exp(8s/ K.(±) 51K,;.."(br)-I S v t 0 acsc(va)P.(b) $9YYYar 59\ P,11 + b,) exp(b) K. :.,(bl 51 26:14 THE EXPONENTIAL FUNCTION exp(bx + c) 250determined. The table lists pairs f(t), fL(s) of functions and their Laplace transforms. To use this table for trans-formation, first locate the chapter or section of the Atlas in which the f function appears- Then scan the numericalindex that constitutes the first column of the table to find entries containing the f function. Of course, there existmany Laplace transform pairs that are not listed in Table 26.14.2 and that cannot be conveniently deduced viarules 26:14:2-26:14:18. We recommend the comprehensive tabulation by Roberts and Kaufman as a source ofseveral thousand varied transform pairs.A useful check on the accuracy of an f(t), fL(s) pair is provided by their limiting forms as the argumentsapproach zero and infinity. These limiting properties are26:14:18L {lim{f(t)}} = lim{sfL(s)} andL f lim{f(r)}} = lim{sir(s)} ,-.o ,-u An even simpler check is provided by the requirement that the inverse Laplace transform of any continuous functionf(t) must tend to zero as s -. X.Section 18:14 demonstrates that the majority of the functions in this Atlas may be expressed as hypergeometricfunctions. Let f(t) be such a function and let us adopt (a,_x), as an abbreviation for the product of K Pochhammerpolynomials 26:14:19f(t) -(a )j(w),(a3); ... (ax), (a;-.k);P=o (c1);(c.) (c3); ... (CL),,_o (ci-L),Then fL(s) is also a hypergeometric function, namely that given by expression 18:14:7. Moreover, many functionsclosely related to f(t) may also be Laplace transformed to hypergeometric functions with one or more extra nu- meratorial parameters. Thus:r(1 + v)(1+ v) (ai_x) 26:14:20L{t"f(br)} =i''(b)'-v > -1S,=u(cl.L)jsand 112o fII+2/(ai-.x),+4b'' 26:14:21L{t`f(sb r)} =r(l + v)(-sj=0(ci-.L);`s'while the transform of ?f(±\/) is the sum 26:14:22r(1 + v)(I + v)j (;a _x) (. + ;ai .x)\4x-ib r(# + V)a,.K. o ±(4 + v)3 (} + }ai_K)j(l + ial-x), 4x_Lbsin"cI-L;=o(i +;ci_L),(1 + 1c'-L), sJof two hypergeometric functions, each with 2K + 1 numeratorial and 2L denominatorial parameters.The table of transform pairs 26:14:2 is also valuable for Laplace inversion. To use it for that purpose, firstscan the right-most column to locate the chapter or section number of the Atlas where the function fL is discussed.Then the second column lists the inverse transform of the function in the third column. As for transformation, thereare a number of general properties of Laplace inversion that permit the table to be used to invert many more thanthe 120 entries that are displayed. Some of these general rules are presented in the following paragraphs.The linearity property of Laplace inversion follows immediately from equation 26:14:2. The linear shift prop-erty(// \ l26:14:23 fL(bs + c) = LS b expl b tt)fl L I }of Laplace inversion incorporates the scaling property,, which``is the c=1110instance of 26:14:23.The most general chain rule for Laplace inversion gives the inverse transform of the composite function fL{g(s)}as the definite integral of the product F(ru)f(u), where f(t) and F(ru) are respectively the inverse transforms offL(s) and exp{-ug(s)}. That is: 251 (THE EXPONENTIAL FUNCTION exp(bx + c) 26:14 26:14:24fc{g(s)} = LiJmF(t;u)f(u)du}whereexp(-ug(s)} = L{F(ru)}0Unfortunately, there exist rather few g(s), F(t:u) pairs that can be used to exploit relationship 26:14:24. Table26.14.3 lists three.Whereas there is no such simple rule for Laplace transformation, there does exist a general product rule forLaplace inversion. This is the so-called convolution property 26:14:25 fL(s)gc(s) = L{ J,f(t-u)g(u)du} = L{f(t)*g(t)}a 111Two straightforward applications of this property lead to the inversion rules26:14:26 exp(-bs)gL(s) = L{u(t - b)g(t - b)} b > 0and//bsr Il/t 26:14:27cothl -2) gi(s) = Li g(t) + 21 u(t - jb)g(r - jb) } J = intl-)`111 J.1 1\bTo Laplace invert the product s"gL(s) involves operations of the calculus applied to the inverse transform. Thesimplest cases 26:14:28 and 26:14:29 iterate to 26:14:30 andl(rlk(s) = L jJg(t)dts 0 dsgc(s) = Lddtt(t) +9(0)8(1) AL(s)(l= L{r... J0g(t)(dt)" }n = 2. 3.4....lJJJJ-'; 26:14:31s"gt(s) = L2{dtg (t) +dg(0)8)(t)}n = 1, 2, 3, ...-oHere 8'`(t) is the n" derivative of the Dirac delta function (Section 10:131 at t = 0. away from the immediatevicinity of t = 0 all such functions are zero. Ignoring the contributions from these delta derivatives, the inverseLaplace transform of S'L(s) is the differintegral d'g/dt' for all values of v.A number of useful rules exist for the Laplace inversion of the quotients ft{g(s)}/g(s) and fL{g(s)}/h(s) whereg(s) and h(s) are such simple functions as f, I/s, s- . o-. etc. Inversion usually leads to definite or indefinite Table 26.14.3 g(s)F(ru) In(s)rim)expQ AiIIIIs(3r'')"'\(3:t' 26:14 THE EXPONENTIAL FUNCTION cxp(bx + c)integrals of the product of f(t), the inverse transform of 0s), with other functions. Examples include 26:14:32 fL\s/(l= Lic jcos(2tu)f(u)du }252 fL(s'+a)_( 126:14:33s + a=LS i Jo(ar - aue)f(u)du ia ? 0l 1with Io(au2 - ate) replacing Jo(at - au2) if a is negative; Roberts and Kaufman [pages 171-1741 may beconsulted for others.If fL(s) is the hypergeometric function 26:14:34k(s) _I j (a)/(a2), ... (aK)/I/1(a .r );s ,-o (e1,(c., ... (CL),ss(c,_L)/(S')then inverse transformation of fL(s), and a number of related functions, give hypergeometric functions of t. Theresults 26:14:35 1." fL=Lv>s -1 bT'(1 + v) ,_o (1 + v), (c1-L),/t et`(a1~x),(±b't /4)'26:14:36 S.fLl bs I = LC(1+ v)J-0 (I+ V),(V\'v > -1 1s_t"(lair)/(i + 'a,-x), (br14X"L)'26:14:37 f(±LL\b/f(1 + v),_o(1 +. + ;c 1-L), LS1112 "ai-x(# + #a1-x)/(1 + lahl )/(bt/4K LI'( + v) l bci_c(j + v),(! +ICI-J,are seen to be strict counterparts of the transforms 26:14:20-26.14:22.We conclude this section by drawing to the attention of the reader the Heaviside expansion theorem that isdiscussed in Section 17:10. This theorem provides a procedure for the Laplace inversion of whereisa polynomial function of degree n. A similar procedure may be employed to invert the rational function p,,(s)/provided m < n, via the decomposition detailed in Section 17:13. The Heaviside theorem may even beapplied for infinite values of m and n, provided that the denominatorial degree exceeds that of the numerator, andhence may be employed to invert the quotient of two transcendental functions that may be written, for example. as (Aa + A1s + A2s2 +- )/(a1s + a2s2 + as' +- ). CHAPTER27EXPONENTIALS OF POWERS exp(- ax" ) Many of the results of this chapter follow by combining the properties of the functions discussed in Chapters 12,13 and 26. It is appropriate to devote a chapter to the functions exp(-ax"), however, because they are of suchwidespread importance, particularly when a is positive. For example, the temperature dependence of many physicalproperties obeys this functionality with v = - 1. Random events often involve the v = 2 instance and lead to important distributions, as discussed in Section 27:14. 27:1 NOTATIONNo special notation or symbolism need be considered beyond that addressed in Sections 12:1 and 26:1. 27:2 BEHAVIORAs with the functions discussed in Chapter 13, the range of exp(-ax") depends in a rather detailed way on thecharacteristics of the number v. In this chapter, however, attention will be confined to the range x >- 0. exceptwhen v is an integer.Figures 27-1 and 27-2 are maps of the functions exp(x') and exp(-.r") for assorted values of v. The importantgraph of exp(-x2) versus x, known as a Gauss curve, displays a maximum value of unity at x = 0 and points ofinflection [see Section 0:71 at x = ±1/V2. Similarly, exp(-1/x) has an inflection at x = #. 27:3 DEFINITIONSWith x replaced by -ax', any of the definitions in Section 26:3 can serve to define exp(-ax'). 27:4 SPECIAL CASESWhen v = 0 or 1, the exp(-ax') function reduces respectively to a constant [Chapter I] or a simple exponential[Chapter 26]. 27:5 EXPONENTIALS OF POWERS exp(-ax") 254O00Rti 11.'r0' p4ti+pN +p 00+NOv4~ACOX10i44i.44, 4b44444i4 exp U /x) :exp(-1/x7 O0Coatia06 O.0tiCjtitia4bv ififi44fifif4fifif.....1.0 27:5 INTRARELATIONSHIPSThe relationships of Section 26:5 hold when x is replaced by -ax".Summation formulas for series of exponentials of I/x includeaz/4 )rtx - 127:5:1exp(X1I + expl x + exp( z9)+= 2 93(0;x) - 2 = 2+ \ exp(_ r2x)x >0 27:5:2explX1I - expls I+ expl9J -=Z- zz 03(0:x)Z- N/Trx expl4 x Ix > 0 and\///\/ 255 EXPONENTIALS OF POWERS exp(-ax') 27:9 27:5:3ex - + ex - +e.p(-- +_ - B. 0;-- exX>0 P( xl)p\ x /p(x5)4(4/42p4=x/in addition to variants of equations 26:5:5-26:5:12. The B functions are explained in Section 27:13. The finalapproximations improve as x increases but are valid to better than I part in 109 for x z 1. 27:6 EXPANSIONSThe series expansion 27:6:1exp(-ax')ax"+a'x''-= i(-ax')'=1---I!2!i-sj,holds for all a, for all v and for all x for which x' is defined.The continued fraction expansion 27:6:2 is rapidly convergent.2Vx/3 x/15 x/35 x/(4j' - 1)exp(V)=1+--2 - + 2+2+2+2+ 27:7 PARTICULAR VALUESChoosing a = I or a = - I and taking care that x' is defined we find X = 0x = tx = x exp(x')JxIv < 0`v>0 v<0{x1v>00 v<0v<Ol1 exp(-x'){1v>0 {0v>0 27:8 NUMERICAL VALUESValues of exp(-ax') are calculable by first evaluating -ax', followed by exponentiation. 27:9 APPROXIMATIONSThough crude, the approximation 27:9:1exp(-x')-I---0IrI;!: V-1ris never in error by more than 0.09. This triangular approximation to the Gauss curve has an area of V, equalto that under the curve. 27:10 EXPONENTIALS OF POWERS exp(-(xx')27:10 OPERATIONS OF THE CALCULUSDifferentiation gives256 27:10:1dd exp(-ax') =-avx"_iexp(-ax")Indefinite integration of the general expressions cxp(-at') and t" exp(-at') can be accomplished by making thesubstitution -at' _y and utilizing the general formulas contained in equations 26:10:7. 26:10:13 and 26:10:14.Below are listed some important indefinite integrals: 27:I0:2Jexp(t')dt = exp(x2) daw(x)o 27:10:3J exp(V)dt = 2(f - 1) exp( ) 27:10:4Jj exp(Idt = (x + f)expQ10-Ei(VIx)x > 0of27:10:5exp(-dt= xexp(1Ei11fx > 0 txx 27:10:6 J exp(-r)dt = 2 erf(x)f27:10:7exp(-V)dt= 2(f + 1) exp(-Vx) 27:10:8 27:10:9fexp(:'dt=(x-V)exp(-:1-Ei1Vzlx>0 71 N/_XI111exp(-) dt=xexpl-)+Ei-Ix>0t x///x/J!/B2- 4ay27:10:10J6 exp(-at' - At - y)dt =12expl4aerfctax +a l2Va> 0 Chapters 42, 37 and 40 are devoted to the daw, Ei and eerfe functions.Some important definite integrals include vJ-(D+ll 27:10:11j:exP(_w)dt = 1> 0a> 0 anddt=111exp(-2V)a>0< 27:10:12ot'2re 1 is the gamma function [see Chapter 43].whe 27:11 COMPLEX ARGUMENTWhen the arguments of the two most important functions of this chapter are replaced by x + iy, we find27:11:1exp[-a(x + iy)2] = exp(ay2 - ax2)[cos(2axy) - i sin(2axy)] 257 EXPONENTIALS OF POWERS exp(-ax") and/-a-ax \/ ayy1a27:11:2exp(I = explI)+ i sincost(F)x2+ Yiy/\x+\J =+y2x+? 27:12 GENERALIZATIONS27:13 The function exp(-ax") may be generalized "to other bases" as described in Section 26:12 so that evaluation of0'-°"' is possible. 27:13 COGNATE FUNCTIONSThe four theta functions are bivariate, depending on the parameter v as well as on the argument x. We define 1=27:13:1 (-1)y exp[-(v - '/2 + j)22/xjaxe=_. 271329_+2+ '/- IJ-= 0 ::2(v;x)arxJ___j)(v/xj,;x) ()exp[1(v 7 9_+2- 2:13:3 3(v;x) Iexp[(vj)/xJ 27:13:4 94(v;x)exp[-(v +'/2 +j)2/xl = 93(vbut the reader should be alert to the wide variety of symbolisms in use in the literature. Thus, the quantity definedin equation 27:13:1 would be denoted 9,(nv,exp(-n2x)) by Abramowitz and Stegun [Sections 16.27-16.30J.It is evident that these four functions are dependent periodically on v so that 9(v + 2;x) _ -9(v + I:x) _0(v:x) for 9, and 92, whereas 9(v + ';X) = 9(v;x) for 93 and 94. This periodicity is brought out more clearly inthe alternative representations 27:13:591(v:x) = 2(- I)' exp(-(j + '/2)2n2.rj sin(2(j + '/2)vaJ=o27:13:6 9.,(v-.x) = 2exp[-(j + '/2)2n2xJ cos[2(j + '/,)vnj-o27:13:7 93(v;x) = I + 2exp[-j2n2xj cos[2jmjj-I27:13:894(v;x) = l + 2(-1)' exp[-j'a'xj cos[2jvwl'_of the theta functions.The theta functions satisfy the intriguing quadruplication formulas 7 v(v I 1}( 14= :13:92 - 9r(v;x)-4 x)-2 e424x22 e3 27:13:10 92(v;4x) = Z 0 I Z; x I-2 94(2;x /\ 27:13:1112' x 1 93(v;4x) =203 -:xI +2941 27:14 EXPONENTIALS OF POWERS cxp(-ax') 258 1I-- 1 11-1 27:13:12 9.(v;4x) =2 e3 24x) +24x)that may be rephrased in several alternative ways.The series in 27:13:1-27:13:4 converge so rapidly if x K 1/ir that it is rarely necessary to use Iii > 3 tocompute very accurate values of the theta functions. On the other hand, if x ? I /-rr. the series in 27:13:5-27:13:8converge even more rapidly. This means that accurate computation of theta functions is always a simple matter, making these functions useful in a variety of applications. Their utility is aided by the widespread occurrence oftheta functions in Laplace transformation [see Section 26:141. As examplesr27:13:1393(v;q exp(-sr)dt =csch(Vs)cosh[(2v - 10 127:13:1493(0;s) = JII + 2u(t - j'a2)J exp(-st)drs > 0so L=ibut for a more comprehensive listing see Roberts and Kaufman, in which the notation for theta functions is almostidentical with that adopted in this Atlas.The identity of expressions 27:13:5-27:13:8 with definitions 27:13:1-27:13:4 permits a reformulation of seriesof exponential functions, as follows:(l l 27:13:15I (2jI1 + 21 (-1)' cxp\x / J= 92(0;x) = 27-exp-41)'n'xL J=o( 27:13:161 2I + 2i explxIJ = 93(0;x) = I + 21 exp(-j'a'x)x>0x>0 27:13:17eXP((24x1)')=94(0:x) = I + 27_ (-1)' exp(-j-ar2x) x > 0=o ,-ITheta functions with v = 0 are important in their own right. Several of the representations coalesce when xleading to the particular values 27:13:182'/'e2(0; !)=93(0; !)=2'"9.(0;i)= 1.086434811It IT zr where U is the ubiquitous constant discussed in Section 1:7.Theta functions play an important role in the theory of elliptic functions [Chapter 63]. As well as the thetafunctions discussed above, there are other versions known as Neville's theta functions and Jacobi's theta functions[see Sections 63:8 and 63:131. 27:14 RELATED TOPICSWhen a measurement or observation is repeated a large number N of times, it frequently happens that the valuesfound are not identical. We say that the measured quantity x has a distribution. Sometimes (as in rolling dice) onlya finite set of values is available for x. Here, however, we consider the continuous case (exemplified by sizes ofraindrops) in which possible x values are limited in proximity only by the discrimination of the measuring device.Let the measurements be arranged in order of size along the line -x < x < X. Then, as N -' x. it oftenbecomes possible to delineate a density function or frequency function f(x) with the property that 2f(x)dx gives theapproximate probability that any single measurement of x will lie in the range x - dx to x + dx. Associated witheach distribution is a mean defined by 27:14:1 µ = Jjtf(t)dtand a variance defined by 259 EXPONENTIALS OF POWERS exp(-ax') 27:14 27:14:2 v= =Jfj(I -p.)'f(t)dt = -12 +ft2f(t)dt ? 0The limits, xo to x,, on the integrals demarcate the range that is accessible to x, often -x to x or 0 to x. Themean of a distribution provides a measure of the average value of the property x, while the variance describes thedispersion of the distribution about the mean. If the distribution has too great a dispersion, no finite variance exists.The Lorentz distribution [see Table 27.14.11 is a case in point: for this particular distribution, depicted in Figure27-3 for a = 8, the integral in 27:14:2 diverges. Even integral 27:14:1 diverges for the Lorentz distribution, butthe so-called Cauchy principal value of the integral, namelyc27:14:3 lim f rf(r)di= LOh4)Og .gxa (z3:....: 0.8....: X. ....:........... 0.8:....:....: 0.4 FIG 27-3 is as tabled.Accompanying each distribution is a cumulative function or distribution function, given by 27:14:4 Kr) = f x f(t)dtwhere, by definition of f(x), F(x,) = 1. The value of F(x) necessarily lies in the range 0 s F(x) s 1 and expressesthe fraction of the measurements that (for N -. x) will lie in the interval between ro and x. Statisticians speak ofpercentiles or percentage points: the p'" percentile is the value x,1100 of x such that 27:14:5 F(x,i,oo) = 100The fiftieth percentile .r1j2 is also called the median of the distribution and satisfies the condition 27:14:6 f(t)dt =ff(t)dt = 2Though with symmetrical distributions they are the same, the mean and the median are not, in general, identical,nor does either necessarily correspond to the peak that often exists in a graph of f(x) versus x. Where such a peakexists, its s-coordinate is known as the mode or most probable value of the distribution.Distributions occur widely, especially in statistics and physics. Examples from both of these fields have beenassembled into Table 27.14.1, and a selection is displayed graphically. The archetypal distribution is the normaldistribution shown in Figure 27-4 for a = 3. The Gauss distribution is the p. = 0 instance of this. Notwithstanding 27:14 EXPONENTIALS OF POWERS exp(-axe) 260 Table 27.14.1 Llmm Dense/ luncuon MphYarulwc Cunwlwn< fuacu0nD01nMron,n In al n+r 0 Fn U. l*mnnanpularlan c,l Gws. dl\¢al2v - .1.,110 Gems! tap-normal Raylclcircular nonnel Bcllxmann 1¢e%p0ntnltall Fenru Dlrac Burt-Eimmn01o= 010 0 1,, 0 n= 010=`!scap{- I.r-411l`'! 0 exp -0 ln'(I i1v aµ!160 tar cap - an' Inp-a l weh'Ivu - r11 tapl-0p - µll o tap-0r! n0IM1 - BXB - enpvnldan,J -Br01a1 01 -v0MI I -B)dlln! l -BIn lafl - B1ae w010 =,rln/210.-tw(-vµ r101, B01a0.-Blv.B! Sntdaa s+ow.w>11 IfeQelasa1rU.all1.0agh- C -C .. by, rc µawkpnwipalvalvefkll.la CwMll 4, 66.-a10%-0)In-II'f1..2\rn-2l"`µ SIum,-r la - II!!na i°-1.:n=3.5.7...-sa:I11r\iln - 21!' 0µ tap(- 11 r0a. v tMl - Blo lnl a - B, nhwI -0.Inl1 -Blr v0.(.Cl17\ain-µ1I 7 Lµa\eI -cap -.5 1 rr, \` eap -o r 1 I - ecpl Iwhlvl, - wl II - c.M-=1 - rlll1spa.. - yl 1 - eap-nn - IMB' cap,,')In. I' B l - Inkapvnl - 01In! i - BI *0 LMt\ ul, - on2A S[ 12//=11!l!' I1_-aacuw,/\ nl]121 f' 121W 261 EXPONENTIALS OF POWERS exp(-ax") 27:14,, 6,'L6,6x6xti6x44ig 4q 44ig.tiqt :....:....:...0.4 0. 2 rL6,x6x10Itq'}4:....:....:....:..............:....:....:.1.0 .:....:....:....:.0.6 :....;. /.:.....:....:. ...:....: 0.4 0. 2: FIG 27-5 ::.:....:.............. :....:....:-.. 0 ,66it4:igit...............:....:....:.0.e .:....\ .. Z ....: ....................0.6FIG 27-6 :....:....: ...................0.2r.f Cx) . 27:14 EXPONENTIALS OF POWERS exp(-ar") 262its name, the normal distribution is encountered in practice rather seldom. The asymmetrical log-normal distri-bution, in contrast, finds more frequent application. An example of this latter distribution in which µ = 2v isillustrated in Figure 27-5. Even less symmetrical is the Boltzmann distribution depicted in Figure 27-6: such adistribution describes, for example, the variation of the numbers of molecules with height in the earth's atmosphere.The shape of the Weibull distribution is a very strong function of a (which must be positive) and this distributiontherefore finds widespread empirical application in failure analysis. CHAPTER28THE HYPERBOLIC SINE sinh(x)AND COSINE cosh(x) FUNCTIONS This chapter and the next two chapters address the six so-called hyperbolic functions. The present chapter dealswith the two most important of the six: the hyperbolic sine and the hyperbolic cosine. These two functions are interrelated by 28:0:1 cosh''(x) - sinh'(x) = Iand by each being the derivative of the other [see equations 28: 10:1 and 28:10:2]. 28:1 NOTATIONThe names of these functions arise because of their complex algebraic relationship [see Section 32:111 to the sineand cosine functions. Their association with the hyperbola is explained in Section 28:3.The notations sh(x) and ch(x) sometimes replace sinh(x) and cosh(x). Although they cause confusion, the sym-bolisms Sin(x) and Cos(x) are occasionally encountered. 28:2 BEHAVIORBoth functions are defined for all arguments but, whereas the hyperbolic sine adopts all values, the hyperboliccosine is restricted in range to cosh(x) ? 1. Figure 28-1 shows the behavior of the functions for rather smallarguments. For arguments of large absolute magnitude, both functions tend exponentially towards infinite values. 28:3 DEFINITIONSThe hyperbolic sine and cosine functions are defined in terms of the exponential function of Chapter 26 by 28:3:1 sinh(x) =exp(x) - exp(-x)2 28:3:2 cosh(x) =exp(x) + exp(-x)2 263 28:4THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS 264 ti44044 To provide a geometric definition of the hyperbolic functions, consider the positive branch of the rectangularhyperbola [Section 15:41x - 1 depicted in Figure 28-2. The green area is bounded by the hyperbola and bya pair of straight lines OP and OP' through the origin with slopes that are equal in magnitude but opposite in sign.Let a denote this shaded area: it can take values between zero (corresponding to points P and P' coinciding withA) and infinity (corresponding to lines OP and OP' having slopes of +I and -1 and constituting the asymptotesof the hyperbola). The lengths PQ and OQ may then be regarded as functions of a and are, in fact, the hyperbolicsine and cosine of a28:3:3 PQ = sinh(a) = sinh (green area)28:3:4 OQ = cosh(a) = cosh (green area)The second-order differential equation 28:3:5 d2f= b2xdx2has the general solution f = cl sinh(bx) + c2 cosh(bx), where b, cl and c2 are constants. 28:4 SPECIAL CASESThere are none. 28:5 INTRARELATIONSHIPSThe hyperbolic cosine function is even 28:5:1 cosh(-x) = cosh(x) 265THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONSwhereas the hyperbolic sine is an odd function28:5:2 sinh(-x)-sinh(x)The duplication and triplication formulas28:5:3 cosh(2x) = cosh 2(X) + sinh2(x) = 2 cosh2(x) - 1 = 1 + 2 sinh((x)28:5:4 sinh(2x) = 2 sinh(x) cosh(x) = 2 sinh(x) 1 + sinh2(x) 28:5:5 cosh(3x) = 4 cosh((x) - 3 cosh(x)28:5:6 sinh(3x) = 4 sinh((x) + 3 sinh(x) = sinh(x)[4 cosh2(x) - 1] generalize to 28:5:7 cosh(nx) = TA(cosh(x)) = i t' cosh(nx)!-aand28:5 28:5:8 sinh(nx) = sinh(x)Un_,(cosh(x)) = i(tp'cosh(x) - tt_',] cosh((x)j_asinh(x)where the T. and U. Chebyshev polynomials are discussed in Chapter 22, as are the Chebyshev coefficientsDe Moivre's theorem28:5:9cosh(nx) ± sinh(nx) _ [cosh(x) ± sinh(x)]" = exp(±nx)is also useful.Equations 28:5:3 and 28:5:4 may be regarded as special cases of the argument-addition formulas28:5:10 cosh(x ± y) = cosh(x) cosh(y) ± sinh(x) sinh(y)28:5:11 sinh(x ± y) = sinh(x) cosh(y) ± cosh(x) sinh(y)From 28:5:3 one may derive the expressions cosh)x)=/cosh(x) + 128:5:1222 x2-1 28:5THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS and 28:5:13 sinhl 2 I = sgn(x)cosh(2) - I for the hyperbolic functions of half argument`,as well as the formulas28:5:14 coshz(x) =cosh(2x) + 12 and 28:5:15cosh(2x) - 1sinh'(x) =2266 for the squares. These latter may be generalized to the expressions 28:5:16cosh"(x) =2111 (J) cosh[(n and 28:5:17a-nr (\ 1}nlcoshl(n-2j)x]n = 1,3,5,... 2"1,=oj n=2,4,6,... 1(-1NI n J sinh[(n - 2&1 =2"--1 1 (-I)'I n I sinh[(n - 2j)x]2",=o _on = 1,3.5,...sinh'(x) =I(n)(-l)"/=(n -2"cosh[(n - 2j)x1 =n!!a/A-1+1)I n I cosh[(n - 2j)r1 n = 2, 4, 6, ... 2"-1,-oj)for any positive integer power of the hyperbolic cosine or sine.The function-addition formulas28:5:18 cosh(x) i- sinh(x) = exp(±x)//(xZ,J 28:5:19 cosh(x) + cosh(y) = 2 cosh( x 2 ,_ I cosh 28:5:20 cosh(x) - cosh(y) = 2sink( Z) sinh \\(x2 y) 28:5:21sinh(x) ± sinh(y) = 2 sinhl Z Y)cosh lx2)and the function-multiplication formulas \\ 28:5:22 28:5:23 28:5:24sinh(x)sinh(y)1=2cosh(x + y) -2cosh(x - y) 1sinh(x) cosh(y) =2sinh(x + y) + 2 sinh(x - y) 1 1cosh(x) cosh(y) = - cosh(x + y) + - cosh(x - y)22complete our listing of intrarelationships between these most maleable functions. 267THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS28:6 EXPANSIONSThe hyperbolic sine and cosine functions may be expanded as infinite seriesx3x5x2/*r28:6:1sinh(x)=x+-+-+3!5!,=o (2j + 1)!x2x4x2j28:6:2 cosh(x) = 1 + - +-+_ i -2!4!/-o(2j)!or as infinite productsx2//x2 x2 28:6:3sinh(x) = xI +_)I1 +Z)= x n I + 22zr\\4n;_1j n 28:6:4cosh(x) = 11 +4x2)(I+4x)\I +4x22)... _I +x2zz\\\\\\'+71\9 rr\25.rj.1(j + 0) rr 28:7 PARTICULAR VALUES sinh(x) coah(x) 28:8 NUMERICAL VALUESx= -mx =-1x=0x=Ix=m Ie2e-102e 2eI+ e'2e12e28:9 These are easily calculated via equations 28:3:1 and 28:3:2. There is an algorithm in Section 29:8 that enables anyone of the six hyperbolic functions, including sinh(x) or cosh(x), to be evaluated. As well, the universal hyper- geometric algorithm [Section 18:14] permits values of sinh(x)/x and of cosh(x) to be found. 28:9 APPROXIMATIONSThe hyperbolic sine and cosine may be approximated by polynomials; for example,x328:9:1 sinh(x) = x +68-bit precision jxj < 0.84 /z28:9:2 cosh(x) = 1 I+4)8-bit precision jxl < 0.70at small arguments and by exponential functions 28:9:3sinh(x) = sgn(x)28-bit precisionjxj > 2.78 2849=exp2lxi)h8biiili > 278 ::cos(x)-t precsonx.when the argument is large. 28:10THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS28:10 OPERATIONS OF THE CALCULUSDifferentiation and indefinite integration of sinh(bx) and cosh(bx) gived 28:10:1 dxsinh(bx) = b cosh(bx) 28:10:2 28:10:3 28:10:4The general formulasdcosh(bx) = b sinh(bx)268 Icosh(bx) - Isinh(bt)dt =b0 J. cosh(bt)dr =sinh(bx) ob-id:cosh2'(xln = 135.., ... " n!!_o(2j)!!28:10:5cosh (t)dt = w12 _ ro(n - I)u rr(2j)!!Lx + sinh(x)cosh'-'"(x)n = 2, 4, 6, .. . n,.o(2j + 1)!! (-1)u-1)/2(n- 1)!! cosh(x)r"-1W2(2j - 1)!r .sinh2'(x)n = 1, 3, 5, ... n!!E (- Iy.r28:10:6sink"(t)dt = nn'_o(2')r(-1)"'2(n - 1)!![x+cosh(x)r"(2j)!! 1(-1)'sinh2'+'(x)n!!;_o(2j + 1)!!n = 2, 4, 6, ...permit the indefinite integration of integer powers of the hyperbolic sine and cosine functions. Alternative expres-sions may be derived by integration of equations 28:5:16 and 28:5:17. Noninteger powers are treated in Section58:14. Other important classes of indefinite integral includen!sinh(bx)(bx)"(b.)" -2 1b"'-++...+1JM!(n - 2)!)-n! cosh(bx) r (bx)"-' (bx)"-3IL 1+n=2.4.6,...n! sinh(bx) r(bx)"(bx)"'2 28:10:7 t° cosh(br)dt = I - +++ bxb1L n!(n - 2)!l-n!cosh(bx) [(bxr+ (bx)sJ(n --I)!(n --3)!n!+-n=1.3,5,...a!cosh(bx) 1(bx)"+(bx)" 2+ . + 1 b0+1n!(n - 2)!]-n! cosh(bx) r (bx)"-'(bx)"-,j;77- L(n - I)!+ (n- 3)!+ ... i.bxn!28:10:8Jt" sinh(bt)dr =- b-+,n = 2, 4, 6. ...0n! cosh(bx)(bx)°(bx)r'2++ bxL n!(n - 2)! ]-n!sinh(bx)(bx)"''+(bx)°''n=1,3,5,I(n... - 1)!(n - 3)! 269THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS 28:13The bracketed series in the above integrals may be expressed as [e (bx) ± e.(-bx)J/2, where the e function isdiscussed in Section 26:13. Similar indefinite integrals for is = -1, -2, -3, ... are listed by Gradshteyn andRyzhik [Section 2.4751; they involve the chi and shi functions defined in Chapter 38.A large number of indefinite and definite integrals involving the hyperbolic sine and cosine functions exist.The reader is referred to Chapters 2.4 and 3.5 of Gradshteyn and Ryzhik. 28:11 COMPLEX ARGUMENTWhen the argument x of sinh(x) or cosh(x) is replaced by x + iy, we have28:11:1 sinh(x + iy) = sinh(x) cos(y) + i cosh(x) sin(y)28:11:2 cosh(x + y) = cosh(x) cos(y) + i sinh(x) sin(y)For a purely imaginary argument28:11:3 sinh(iy) = i sin(y)28:11:4 cosh(iy) = cos(y) 28:12 GENERALIZATIONSThe Jacobian elliptic functions nc(x;p) and nd(x;p) may be regarded as generalizations of cosh(x), to which theyreduce when p = 1. Likewise, sc(x;p) and sd(x;p) reduce to sinh(x) when p = I and therefore generalize thehyperbolic sine. See Chapter 63 for all these Jacobian elliptic functions. 28:13 COGNATE FUNCTIONS The expressionssgn(x)1 - sech2(x) 1tanh(x) sgn(x)28:13:1sinh(x) = sgn(x) I - cosh2(x) _sech(x) cosh(x)) + tanh2(x) coth2(x) - 1 )1 + csch(x) Icoth(x)I 28:13:2cosh(x) ='1+ sinh2(s) =sech(x)Icsch(x)l1 - tanh2(x)Vcoth2(x)- 1relate the hyperbolic sine and cosine to the other hyperbolic functions (see Chapters 29 and 301.The hyperbolic sine and cosine functions are closely related to those hyperbolic Bessel functions I.(x) in whichv is an odd multiple of ± . Examples are 28:13:3 lt 2(x) =Ysinh(x)YYYY Irx228:13:4 1_112(x) = cosh(x)Trx 28:13:5 I,,,(x) =VI cosh(x) -sinh(x)1 VVVVTrxLXand others may be constructed by use of the recursion formula2v 28:13:6 L. A) + - L(x) - I.-AX) = 0XThese functions, some of which are graphed in Figure 28-3. share the properties of all hyperbolic Bessel functionsas discussed in Chapter 50. The name modified spherical Besse! function and the symbol i,(x) is sometimes givento the function V/Tr/2r I,,,i,:(x). 28:13THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS 270 28:14 RELATED TOPICSIf a heavy rope or flexible chain of length 2L is freely suspended from two points, separated by a horizontal distance2h, but at the same level, then the rope adopts a characteristic shape known as acatenaryiscosh(bx) - cosh(bh)28:14:1 f(x) =band its shape is illustrated in Figure 28-4. The coefficient b is related to the lengths L and h by the implicit definition 28:14:2 bL = sinh(bh) 271THE HYPERBOLIC SINE sinh(x) AND COSINE cosh(x) FUNCTIONS 28:14 titer..............:..............1.pFIG 28-4 CHAPTER29THE HYPERBOLIC SECANT sech(x)AND COSECANT csch(x) FUNCTIONS Of the six hyperbolic functions, the two treated in this chapter are perhaps the least frequently encountered. Theproperty 29:0:1 (1 - sech'(x))(t + csch2(x)) =interrelates the two. Two features of this chapter-sec Sections 29:3 and 29:8-deal with all six hyperbolic func-tions. 29:1 NOTATIONThe notation cosech(x) is sometimes used for the hyperbolic cosecant. Some authors admit only four hyperbolicfunctions, using 1/cosh(x) and 1/sinh(x) to represent the secant and cosecant. 29:2 BEHAVIORFigure 29-1 shows the behavior of the two functions. Both approach zero as their arguments tend to ±x. Bothfunctions accept any argument but, whereas csch(.r) adopts all values, the hyperbolic secant is restricted in rangeto 0 : sech(x) s 1. 29:3 DEFINITIONSThe relationships 29:3:1 sech(x) _2exp(x) - exp(-.x) cosh(x)and229:3:2 csch(x) = _exp(x) - exp(-x)sinh(x)are the usual definitions of the hyperbolic secant and cosecant. 273 29:3THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS 274Oh 41ififififi? cech (x>....:.. 1.5 sech(x) .......................0 :.........:..............:....:.-0.5cacti (x): FIG 29-1:....:.......:.............. :....:.-1.0 X//each (x) of FIG 29-2 Figure 29-2 depicts three similar right-angled triangles. If the sides that are drawn as dashed lines are of unitlength, then each of the other six sides equals one of the hyperbolic functions as labeled. Pythagorean and similarityproperties thus enable the interrelationship between any two hyperbolic functions to be deduced and may serve as the definition of any one function in terms of any other. Note that the argument x is shown in the diagram, but it is not related to the triangles by any simple construction. It is, in fact, related to the angles in the triangles via thegudcrmannian function (see Section 33:14).The differential equations 29:3:3 d+oz-_j'-0dxare satisfied respectively by f = a csch(ax) and f = a sech(ax). 275THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS29:4 SPECIAL CASES There are none. 29:5 INTRARELATIONSHIPSThe hyperbolic secant and cosecant obey the reflection formulas29:5:1 sech(-x) = sech(x)29:5:2 csch(-x) _ -csch(x)and the duplication formulas 29:5:3 29:5:4sech(2x) =sech2(x)2 - scch2(x) csch(2x) =sech(x) csch(x)229:6 Other relationships may be derived via the equations of Section 28:5. but these are generally more complicatedand less useful than are the intrarclationships of the hyperbolic sine or cosine. 29:6 EXPANSIONSThe functions sech(x) and x csch(x) may be expanded as power seriesx'Sx`6lx°/4x2 \iE,x2jitis 29:6:1 --<x<-224720,=o` a /=o (2j)!22andx7x231x5I2 =x='(2-4)29:6:2csch(x)=I 1-+ --++TI(2j)2Byx2'-a<x<a x636015120xxa'(2j).The 0 and t coefficients are defined in Chapter 3 and the Euler E and Bernoulli B numbers in Chapters 5 and 4.respectively.Expansions as exponentials take the forms 29:6:3sech(.x) = 2 exp(-!rI) - 2 exp(-3*xl) + 2 exp(-grl) - = 2 i (-1)'expt-(2j + I)1xll x s 0 and 29:6:4csch(x) = 2 exp(-3(xI) + exp(-51x1) +1 = 2 sgn(x)exp((I - 2j)(.rl]As well, the hyperbolic secant and cosecant can be expanded as partial fractions 29:6:5 sech(x) _41r20tr(-I)'(2j+ 1)nrr'+4x225ir+4x' =oU+#)tn2+x2x*0 29:6:6 csch(x) _I2x2x2x- -, + ,, +- ra2 + s-4tr' + x-x'j-n- +x,' 29:7THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS29:7 PARTICULAR VALUES csch(xl sech(x( 29:8 NUMERICAL VALUESx= -xx = -1x= 0x- lx= z - 2e 2e 0 0e' - 12e 2e 0 0r' + Ie.+1276 Calculation via definitions 29:3:1 and 29:3:2 is usually a simple matter.If many values of the hyperbolic functions are needed, the following algorithm may be useful. It generatesexact values of any one of the six hyperbolic functions according to the value (0. 1. 2, 3. 4 or 5) of a code thatis input in addition to the argument x.Input x >>»»Set f = exp(x)Set g = 1 /fReplace f by If - g)/2Input code c>>>If cs1go to (2)Replace g by f + gIf c > 3 go to (1)Replace f by g Go to (2)(1) Replace f by f/gfo = sinh(x)(2) If frac(c/2) = 0 go to (3)f, = csch(x) Replace f by I/ff, = cosh(x)(3) Output ff3 = sech(x) <<<<fy = tanh(x)fs = cosh(x) 29:9 APPROXIMATIONSFor large values of x, one can approximateStorage needed: x, f, g and cInput restriction: x0code, cfunction0sinhcschcoshsechtanhcothTest valucx=afo = 11.54873936f. = 11.59195328f, = 1.003741873 29:9:1sech(x) = 2 exp(-Ixl)8-bit precisionz 2.829:9:2csch(x) = 2 sgn(x) exp(-kI) 8-bit precisionjxl z 2.8 29:10 OPERATIONS OF THE CALCULUSDifferentiation of the hyperbolic secant or cosecant givesd29: 1 0:I dxsech(x) = -sech(x) tanh(x) = -sech(x) 1 - sechz(x) 277THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS 29:10:2dY csch(x) _ -csch(x) coth(x) = -csch(x) 1 + csch2(x)The indefinite integration of these functions yields rather complicated results: 29:10:3 sech(t)dt = arctan(sinh(x)) = gd(x)where gd is the gudermannian functionn discussed in Section (33::1\4, and29:10:4 Jcsch(t)dt = lnl cothl 2 I I--0but their squares integrate more simply 29:10:5 Jtsech'(t)dt = tanh(x)029:10:6 csch2(t)dt = -coth(x)x > 029:12 Indefinite integration of the square roots of the hyperbolic secant and cosecant functions generates special cases ofthe incomplete elliptic integral [see Chapter 62] in which the parameter equals I /V2.r /1 29:10:7Jsech(1) dt = V2 FI_:O Isin(O) = Vril - sech(x)a 29:10:8Icsch(t) dt = F V2: d 1cos((b) =csch(x) - Icsch(x) + IFor a general ization to arbitrary power, see Section 58:14.Useful definite integrals include 29:10:9 Jt" sech(t)dt = 2n! 0(n + 1)n = 0. 1. 2.ro29:10:10 Jt"csch(t)dt = 2n! J1 (n + 1)n = 1, 2, 3, ...awhere Chapter 3 describes the beta and lambda numbers. 29:11 COMPLEX ARGUMENTEquations 28:11:1 and 28:11:2 may be employed to evaluate csch(x + iv) and sech(x + iv), respectively. For purelyimaginary argument29:11:1 sech(iv) = sec(v)29:11:2 csch(iy) = -icsc(y)where sec and esc are the functions to which Chapter 33 is devoted. 29:12 GENERALIZATIONSThe Jacobian elliptic functions cn(x;p) and dn(.x:p). discussed in Chapter 63. are generalizations of sech(x). whilecs(xp) and ds(x;p) similarly generalize csch(x). 29:13THE HYPERBOLIC SECANT sech(x) AND COSECANT csch(x) FUNCTIONS 27829:13 COGNATE FUNCTIONS The expressions 1 I)csch(x)( coth'(x) - 129:13:1sech(x) _ __= VI - tanh'(x) _till - sinh'(r cosh(x)I + csch'(x) Icoth(x)I 1sgn(x)sgn(x) sech(x) 1 - tanh2(x)29:13:2csch(x) =sinh(xl tanh(x)= sgn(x)coth2(x) - Ieosh'(x) - I N /'l - sech (x)relate the hyperbolic secant and cosecant to the other hyperbolic functions (Chapters 28 and 30). CHAPTER30THE HYPERBOLIC TANGENT tanh(x)AND COTANGENT coth(x) FUNCTIONS The two functions of this chapter are the reciprocals of each other30:0:1 tanh(.r) coth(x) = Iand are closely related to the other hyperbolic functions [Chapters 28 and 291. 30:1 NOTATIONThe symbolism th(x) sometimes replaces tanh(x); cth(x) or ctnh(x) sometimes replaces coth(x). The misleadingnotations Tan(x) and Cot(x) are occasionally encountered. 30:2 BEHAVIORThese functions are defined for all values of the argument x, but both are restricted in range. The hyperbolic tangenttakes values only in the range - l to + I, whereas coth(x) assumes all values except those between - I and + 1.As shown in Figure 30-1, both functions lie exclusively in the first and third quadrants [see Section 0:21, andboth approach + I as x -. x and - I as x -i -x. The diagram also shows the coth(x) - I /x function, which alsoapproaches ± I at its limits, and which is cited in Section 30:13. 30:3 DEFINITIONSThe hyperbolic tangent and cotangent may be defined in terms of the exponential functionexp(2x) - 1_1 - exp(-2r)_exp(:r) - exp(-.r)30:3:1tanh(x) =exp(2.0 + 1l + exp(-2r)exp(.x) + exp(-.r)exp(2r) + II + exp(-20exp(x) + exp(-x)30:3:2 co(h(x)exp(2r) - I-I - exp(-20exp(x) - exp(-x)or in terms of the other hyperbolic functions 279 30:4THE HYPERBOLIC TANGENT tanh(x) AND COTANGENT coth(x) FUNCTIONS 280 30:3:3 30:3:4sinh(x)sech(x) I tanh(x)cosh(x)csch(x)coth(x) coth(x)cosh(x)csch(x) Isinh(x)sech(x)tanh(x)Figure 28-2 may be used to provide a geometric definition of the hyperbolic tangent, namely30:3:5 RA = tanh(a) = tanh (shaded area)The hyperbolic tangent and cotangent each satisfy the differential equation 30:3:6 30:4 SPECIAL CASES There are none. 30:5 INTRARELATIONSHIPSBoth functions are odd:30:5:1The duplication formulasdf-__1 -fdx f(-x) _ -f(x)f = tanh or cosh 2 tanh(x)_230:5:2 tanh(2x) =T+ tanh2(x)tanh(x) + coth(x) 281THE HYPERBOLIC TANGENT tanh(x) AND COTANGENT coth(x) FUNCTIONS and 30:5:3 coth(2x) =coth2(x) + I-coth(x) + tanh(x)2 coth(x) 2are special cases of the argument-addition expressions 30:5:4 and 30:5:5tanh(x ± y) _tanh(x) ± tanh(y)I ± tanh(x) tanh(y) coth(x ± y) =1 ± coth(x) coth(y)coth(x) ± coth(y)The equations in this paragraph may be used to build formulas for tanh(3x), coth(4x), etc.The half-argument formulasI -1 - tanh2(x)30:5:6tanh(x- )=tanh(x)= coth(x) - sgn(x) coth2(x) - 1 = coth(x) - csch(x) 2and30:6 (x I +1 - tanh2(x)30:5:7cothl - I = coth(x) + sgn(x) coth2(x) - I =tanh(x)= coth(x) + csch(x) are useful.The sums and differences of the functions of this chapter may be expressed in terms of hyperbolic sines andcosines: 30:5:8 30:5:9 30:5:10 30:6 EXPANSIONSsinh(x ± y)tanh(x) ± tanh(y) =cosh(x) cosh(y)cosh(x ± y)coth(x) ± tanh(y) =sinh(x) cosh(y)sinh(x -!- y)coth(x) ± coth(y) =sinh(x) sinh(y) The functions tanh(x)/x and x coth(x) may be expanded as power series in x2:x'2x517x'-2-4x2 Y_Im(41 - l)Bu(4x2)'an 30:6:1 X(2!')--<x<-315315x2x1(2j)!22 1xx32x512( x21By(4x2)' 30:6:2I =--ar<x<ar x345945xx J.1\ ar/x ;_o(2j)!where the lambda and zeta functions are discussed in Chapter 3 and the Bernoulli numbers in Chapter 4.The hyperbolic tangent and cotangent may alternatively be expanded as series of exponentials. The tanh(x)expression 30:6:3tanh(x) = sgn(x)[l - 2 exp(-2kl) + 2 exp(-41x1) - 2 exp(-6IxxI) + J = sgn(x)(- I)'exp(-2jkI)f--ghas alternating signs; the expression for coth(x) is similar but has uniformly positive signs: 30:7THE HYPERBOLIC TANGENT tanh(x) AND COTANGENT coth(x) FUNCTIONS 28230:6:4coth(x) = sgn(x)[I + 2 exp(-2Ix1) + 2 exp(-4x1) + 2 exp(-6Ixl) + - -j= sgn(x)exp(-2j)xj)x * 0Partial fraction expansions exist for both functions8x8x8x =8x30:6:5 tanh(x) = ,+ ,--- = I it- + 4x'9ar' + 4x'25ir + 4x-(2j + l)'ar + 4x'Zr2x2xx 30:6:6coth(x) _ - ++++_ xar +x'4ir+x'9ar=+x'and the continued fraction expansion 30:6:7holds for the hyperbolic tangent. 30:7 PARTICULAR VALUESx x' x' x'tanh(x) =1+3+5+7+ tanh(x)I_ 11- e'1- e0 coth(x) - I 30:8 NUMERICAL VALUESI+e'l+eI+e'l+e_1-e'I-e0x= 9x - IX -e- 1e'- 11a+1e'+1a+Ie'+Ia-1e'-I The hyperbolic tangent and cotangent functions may be evaluated via formulas 30:3: I and 30:3:2 or by the algorithmin Section 29:8. 30:9 APPROXIMATIONS For small arguments, the approximations 30:9:1x\tanh(x) = xI - 3 I8-bit precision4x] S 0.41 1 30:9:2coth(x) = X1 +x'38-bit precisionIxj 5 0.65 are valid. For large arguments, one may use30:9:3tanh(x) = sgn(x)[I - 2 exp(-2Ixj)j 8-bit precision1.630:9:4coth(x) = sgn(x)[1 + 2 exp(-2kl)j 8-bit precision[xI z 1.6 283THE HYPERBOLIC TANGENT tanh(x) AND COTANGENT coth(x) FUNCTIONS 30:1130:10 OPERATIONS OF THE CALCULUSDifferentiation and indefinite integration give 30:10:1-tanh(bx) = b sech'(bx) = b[I - tanh2(bx)] 30:10:2 ddrcoth(bx)-b csch2(bx) = b( I - coth2(bx)J 30:10:3 30:10:4 30:10:5Ltanh(bt)dt = b In(cosh(bx)) coth(r)dt = ln(sinh(.r))x > 0xo = ln(1 + V2) = 0.88137 ... 10Icoth(t) - -t1`Jdt =m/sinh(x))x/J1 30:10:6J[coth(t) - l]dt = In[I - exp(-2r)] x > 0 30:10:7 30:10:8 30:10:9r.[I - tanh(i)Jdt = In[Icxp(-2x)]rtanh(bx)b 0 coth2(t)dt = x - coth(x) x0 = 1.19967864f30:10 10tanh(t)dtfxIn(cosh(x)) - on = I. 3. 5...crtank"'(x)- v n-2j+1The indefinite integrals of tanh"(x) and coth"(x), where X is an arbitrary power, are discussed in Section 58:14.The semiderivative, with lower limit -x, of [I + tanh(x/2)1/2 is an important function, known as the Randles-Sevcik function in electrochemistryI+tanh(-Id''//;n 30:10:11rsf(x) =(-1) VJexp(jx) 1/22= -xd)](x +(X- AX + 2r)VrX where X =(21 - 1)2a- + x2. It is related to Lerch's function [Section 64:121 by the identity rsf{In(.r)} = r(D(-.r:-1/2;1).30:11 COMPLEX ARGUMENTWhen the argument of the hyperbolic tangent and cotangent functions becomes x + iy, we have 30:11:1 tanh(x + iy) =sinh(2x)cosh(2r)++i sin(2y)cos(2v) 30:11:2 coth(x + iv) =sinh(2r)cosh(2x)--i sin(2v)cos(2v)112 30:12THE HYPERBOLIC TANGENT tanh(x) AND COTANGENT coth(x) FUNCTIONS 284For purely imaginary argument30:11:3 tanh(ii') = i tan(v)30:11:4 coth(iy) _ -i cot(v) 30:12 GENERALIZATIONSThe Jacobian elliptic functions sn(xp) and ns(xp) ]see Chapter 63] may be regarded as generalizations of tanh(x)and coth(x), respectively. Asp - 1, sn(x;p) -> tanh(x) and ns(x;p) -> coth(x). 30:13 COGNATE FUNCTIONSThe expressions 30:13:1 and 30:13:2sinh(x)cosh(x) - I sgn(x)tanh(x) sgn(x)1 - sech2(x)717 sinh22(x) sgn(x)cosh(x) 1 + csch2(x)coth(x) coth(x) =I + sinh2(x)-sgn(x)cosh(x)-sgn(x)= g (x)1 + csc 2(s nhx)=sinh(x) cosh2(x) - I 1 - sech2(x) tanh(x)relate the tangent and cotangent to other members of the hyperbolic family. Figure 29-2 is useful in expressingthese relationships.The function coth(x) - (1/x), which occurs in the theory of dielectrics, is known as the Langevin function. Itis mapped in Figure 30-1 and its integral is given in 30:10:5. The Langevin function can be expanded via 30:6:6and its reciprocal as 13-1 30:13:3 =-+2x 2coth(x) - (I/x) xx' + r,'(1)where r,(1) denotes the j ° positive root of the equation tan(y) _ y 1see Section 34:7]. CHAPTER31THE INVERSE HYPERBOLIC FUNCTIONS The six functions of this chapter are interrelated by the following permutations of argument: 1 1z 31:0:1arsinh(x) = sgn(x) arcosh(1 + x2) = sgn(x) arsech = aresc i - = artanhl -+X1x+ x2 = arcothll + x2x 31:0:2arcosh(x) = arsinh( x2 -(1=(I 1/x2 - 1) = arsechl xJ - areschl xZ) =artanh)x x= arcothx >_ 1x2- 1 Xx2I = artanh(1 - x')IW I=arcoth7,-05xs Ix2 31:0:4aresch(x) = arsinh - 1 = sgn(x) arcosh( 1 + 1/x2) = sgn(x) arscch( x2/(1 + x2))x1/ = sgn(x) artanh= sgn(x) arcoth(V1+ x2)x * 0 II+ x2 31:0:5/x1 1 artanh(x) = arsinhl1 - x2I = sgn(x) arcosh1 - x2= sgn(x) arsech( 1--X 2)1-x2 1 = areschlxI = arcothl - I-1 s x < 1 285 31:1 THE INVERSE HYPERBOLIC FUNCTIONS 286 31:0:6arcoth(x) = sgn(x) arsinh +.)= sgn(x) arcosh(x2/(x2 - 1)) = sgn(x) arsech( 1 - 1/x2) x-- 1` = sgn(x) arcssh(x=1) = artanhIIGxi > 1x/ 31:1 NOTATIONThe prefix `ar" means "area," and its pertinence can be appreciated by reference to Figure 28-2. The inversehyperbolic sine of x, for example, denotes that area which. in the Figure 28-2 construct, has a hyperbolic sine ofa. That is:31:1:1 arsinh(x) = a = shaded area if x = sinh(a) = PQThe symbolisms arcsinh(x), agsinh(x), arsh(x) and sinh-'(x) are all used in place of arsinh(x). Correspondingvariants are encountered for the other inverse hyperbolic functions. The same symbolisms with a capitalized initialletter-Arcsinh(x), Arsh(x), Sinh-'(x), etc.-are sometimes used synonymously with arsinh(x). but more oftenthey denote the multiple-valued functions that we discuss in Section 31:12.31:2 BEHAVIORThe behaviors of the six inverse hyperbolic functions are shown in Figure 31-1. Notice that all functions exist inthe first and third quadrants (see Section 0:2) only.The inverse hyperbolic sine has a behavior that is simpler than the other five. It accepts all arguments andadopts the sign of its argument.The inverse hyperbolic cosine is normally defined only for arguments x ? I although some authors extend itsdomain of definition to W a I via arcosh(-x) = arcosh(x). It adopts only positive values.The inverse hyperbolic secant is likewise normally defined only for 0 s x s 1 although its definition may beextended to 0 s i s I by means of arsech(-x) = arsech(x). The values of arsech(x) are always positive.The inverse hyperbolic cosecant has two branches, as mapped in Figure 31-1. It adopts the sign of its argumentand exhibits a discontinuity at x = 0.The inverse hyperbolic tangent is defined only for -1 5 x 5 1 and approaches ±x as x -+ ± 1.The inverse hyperbolic cotangent has two branches. For I s x <_ x, arcoth(x) is positive; for -x s x s -1,arcoth(x) is negative. The function is not defined in the -I < x < 1 gap.31:3 DEFINITIONSIndefinite integrals define each of the six inverse hyperbolic functions. Small diagrams, Figures 31-2 through 31-7, illustrate these definitions. 31:3:1dtf1 -+rarsinh(x) = dr 31:3:2 arcosh(x) =Jt2 - 1x >- I ridr31:3:3 arsech(x) = 10 s x 5 1 drfIN/l/l+ t2 '141%.Aamsdi(x =dt-=t1+t2 287 THE INVERSE HYPERBOLIC FUNCTIONS 31:3OhOh`,A'o 4*iyyhO0O44f4f+ l,PI, 31:3 THE INVERSE HYPERBOLIC FUNCTIONS 288 FIG 31-3cosh (x) aresch(z)FIG 31-4 -I s x s 1 31:3:5 artanh(x) =JI 31:3:6 arcoth(x) _< x c xr=- 1dtx5X5-t'i Another approach to inverse hyperbolic functions is to use definitions as the logarithms of algebraic functionsof the argument x. Thus: 289 THE INVERSE HYPERBOLIC FUNCTIONS 31:5_ 31:3:7 arsinh(x) = ln(x + x2+l)31:3:8 arcosh(x) = hi(x +x2 - 1)x > I 3139 = Ih0 s5 I :: arsec(x)n I\xX: 31:3:10 /arcoch(x) = Inl 1 + 1 +lZx\\\x 31:311(h(= ItrI+ x)-1 -- 1 : aranx)n1-VXx ./x+1\31:3:12 arcoth(x) = lnIVX-- )W ? IThe definition from which the names of these functions arise must be treated with caution. The arsinh, arcsch,artanh, and arcoth functions are straightforward inverses of the corresponding hyperbolic functions31:3:13 f = arsinh(x)wherex = sinh(f)31:3:14 f = aresch(x)wherex = csch(f)31:3:15 f = artanh(x)wherex = tanh(f)31:3:16 f = arcoth(x)wherex = coth(f)For the arcosh and arsech functions, however, a proviso is needed:31:3:17 f = arcosh(x)wherex = cosh(f) and f >- 031:3:18 f = arsech(x)wherex = sech(f) and f ? 0 31:4 SPECIAL CASES There are none. 31:5 INTRARELATIONSHIPSNo reflection formulas exist for the arcosh or arsech functions [although see Section 31:2], but the other four inversehyperbolic functions are odd:31:5:1f(-x) = -f(x)f = arsinh, arcsch, artanh, arcothA number of function-addition formulas exist for the inverse hyperbolic functions. These include31:5:2arsinh(x) ± arsinh(y) = arsinh(x 1 + y2 ± y vi -+",)31:5:3 arsinh(x) ± arcosh(y) = arsinh(xy ± (x2 + 1)(y2 - 1)) 31:5:4 arcosh(x) ± arcosh(y) = arcosh(xy t (x2 - 1)(y2 - 1)) 31:5:5 artanh(x) ± artanh(y) = artanh x± yI±xy/Ixy 31:5:6 artanh(x) ± arcoth(y) = artanh 1(.X±y 31:5:7 arcoth(x) ± arcoth(y) = arcoth(1 ± q)x±y 31:6 THE INVERSE HYPERBOLIC FUNCTIONS 290and many others may be constructed using the equivalences listed in formulas 31:0:1-31:0:7. Use of these formulasrequires, of course, that each argument lie within the domain of its function.Special cases of the relationships of the last paragraph may be converted into such formulas as(VI+x2-131:5:8 arsinh(x) = 2 sgn(x) arsinhl\2 31:6 EXPANSIONSSeries expansions exist for the inverse hyperbolic sine and tangent 311X'3x'--+--Sx'+..(2j-I)"(-x2Y-1< = <1 ih:6: xx(x)arsn640112 ,_o(2j)!!x2j + I 31:6:2X3xsx'xrr-1<x<l357'zL' 2j + IReplacing x by 1/x and restricting I > 1, these same series depict arcsch(x) and arcoth(x), respectively. We alsohave/2l x23x45x6 (2j - I)" x2j 31:6:3 0<x<l 41696 ,_.(2j)!!2jReplacing x by 1/x and restricting x to exceed unity converts this series into an expression for ln(2x) - arcosh(x).The similar series with alternating signs is 31:6:4sgn(x) arsinh(x) - 1n(2x)3+5(2j - I)!'4x216x'96x6., 2j(2j)!!(-x2)'The inverse hyperbolic sine 'anted tangent may each be expressed as continued fractionsarsinh(V x) II x 2x I x 2x 3 x 4x 3 x 4x 5 x 6x 31:6:5I +xI+3+5+7+9+11+ and 31:6:6 31:7 PARTICULAR VALUESartanh(x)_Ix2 (2x)2 (3x)2 (4x)2x1-3- 5- 7- 9- In Table 31.7.1 sundcf" means that the function is not defined for the argument in question. Table 31.7.1 arsinh(x)atcosh(x)anech(X)aresch(x)artanh(x)arcoth(x)W L, I x= -xX = -Ix = 0 x = 1 x =-:In(V - 1)0ln(ti 2 + 1)xundef undef undef 0undef undef+x0undefo In(\ + 1) oundef-a0+xundef0_xundefS0 291 THE INVERSE HYPERBOLIC FUNCTIONS 31:1031:8 NUMERICAL VALUESThe algorithm below, which is exact, uses definitions 31:3:7-31:3:12 to calculate values of the inverse hyperbolicfunctions. The universal hypergeometric algorithm [Section 18:14) may also be used to evaluate these functionsfor suitable arguments. Input x >>>>Input code >>; fo = arsinh(x)f, = arcsch(x)f- = arcosh(x)f, = arsech(x)f4 = artanh(x)Is = arcoth(x)If frac(c/2) = 0 go to (1)Replacex b1 x(1) Setf=.r+ IIf c s I go to (3)If cs3go to (2)Setf = V(1 +X)/(1 - x)Go to (4)(2)Setf=x'-1(3) Replace f by f + x(4) Replace f by In(j)Output fK««Storage needed:x, c and fInput restrictions: Argument must lie in the range:code, cfunctionargument0arsinhany 1arcochx*02arcoshX ? 13arsech0<xsI4artanh-1 <x< l5arcothW> I Test values:arsinh(-w) _ -1.862295743arcosh(ir) = 1.811526272arcoth(ir) = 0.3297653150 31:9 APPROXIMATIONS For large x we can use equations 31:6:4 and 31:6:3 to approximate 31:9:1arsinh(x) = sgn(x)I ln(2k1) +1J8-bit precisionz 2L4,2 31.9:2arcosh(x) = In(2x) - 4s=8-bit precision2.2For small x the approximation 31:9:3artanh(x) is useful.x 31:10 OPERATIONS OF THE CALCULUSII 8-bit precision-2< x S ` Differentiation of the six inverse hyperbolic functions givesd 131:10:1 arsinh(x) =&1+xd 131:10:2 a rcosh(x) = x >- Idxx ' - 1 X1x''0Sx<-I d-1- arcsch(x)_dxxI V l + x'd- artanh(x) _- 1x1 dx1 -xTHE INVERSE HYPERBOLIC FUNCTIONS d_-arsech(x) 31:10:6darcoth(x) =ILrI a Idx1 - x'The corresponding indefinite integrals arearsinh(t)dt = x arsinh(x) - + I 31:10:7f31:10:8Jaosh(tz.x arcoch(x) -xI 31:10:9J'arsech(r)dt = x arsech(x) + arcsin(x) 0 s x <- 1031:10:10 31:10:11 31:10:12 The semiderivativecarcsch(r)dt = x arcsch(x) + Iarsinh(x)I 31:10:13,artanh(V ) =2I - xlinks the inverse hyperbolic tangent to a simple algebraic function. 31:11 COMPLEX ARGUMENT- I<x<I I <x<x292 Either the definitions 31:3:1-31:3:6 or 31:3:7-31:3:12 may be used to extend the inverse hyperbolic functions tocomplex arguments. For example, using 31:3:7, 12:11:1 and 25:11:1, one obtains the multivalucd function 31:11:1 Arsinh(x + iv) = 2 In(X' + Y') + i(2kir + 0)where k is any integer:sgn(Y) arccot(X/IYI)Y * 031:11:2 9T<2[1-sgn(X)]Y=0x'-y2+1+(1++y+y-)--4-1:11:33X = x + and2Joartanh(t)dt = x artanh(x) + ln(V I '-X2) J`arcoth(t)dt = x arcoth(x) + In\\ll r27 293 THE INVERSE HYPERBOLIC FUNCTIONS 31:11:4 Y = y + sgn(xy)2-x2- I(1+x2+y2)2-4y2231:13 The resulting explicit expressions in terms of x and y are rather complicated. However, several useful special casesof such formulas arise upon selecting k = 0 and by restricting the argument to be purely imaginary. This produces31:11:5 arsinh(iy) = i aresin(y)31:11:6 arcsch(iy) _ -i arccsc(v)31:11:7 artanh(iy) = i arctan(v)31:11:8 arcoth(iv) = - i arccot(y)where the functions appearing on the right-hand sides are inverse trigonometric functions [see Chapter 35]. 31:12 GENERALIZATIONSThe inverse hyperbolic tangent is a special case of the generalized logarithmic function [Section 25:12] 31:12:1 artanh(x) =sgX)111x-and of the incomplete beta function [Chapter 581gn(x)B(ii;;Oxi-1 < x < 1 31:12:2artanh(x) =2while the inverse hyperbolic sine is an instance of the Gauss function [Chapter 601 31:12:3 arsinh(x) = xF(2 , 2;2;-x-1 <x< IThe generalization of arsinh to the multivalued Arsinh function is discussed in Section 31:11. The multivaluedArcosh, Arsech, Arcsch, Artanh and Arcoth functions may be constructed analogously. 31:13 COGNATE FUNCTIONSThe six inverse hyperbolic functions are closely related to the logarithmic function and to the six inverse trigo-nometric functions that are discussed in Chapter 34. CHAPTER32THE SINE sin(x) AND COSINE cos(x) FUNCTIONS The functions of this chapter are of paramount importance: they are the units from which all periodic functions[Chapter 361 can be built. The sine and cosine functions are interrelated by 32:0:1 sin2(x) + COs-I(x) = 1and by 32:0:2 32:1 NOTATIONsin x +2= cos(x) The symbolism sin(x) and cos(x) is universal. The notations Sin(x) and Cos(x) refer to the hyperbolic functions ofChapter 28, not to those presently under consideration. Sometimes the names circular sine and circular cosine areemployed to emphasize the distinction from the corresponding hyperbolic functions.In this Atlas we generally use x to represent the argument of a function, and this convention is retained in thepresent chapter. As explained in Section 32:2, however, the arguments of the sine and cosine functions [and ofthe functions addressed in Chapters 33 and 34] are often regarded as angles, rather than simply as numbers towhich no special geometric significance is to be attached. We shall write sin(O) and cos(O) [as well as sec(O), tan(d),etc. in Chapters 33 and 34] when we particularly want to emphasize the angular interpretation of the argument.The sine and cosine functions, and any "mixture" of them c, sin(x) + c2 cos(x), are known as sinusoidalfunctions or sinusoids.. Via equatjon 32:5:26, such a "mixture" may be expressed as c; -+c:, sin(x + (b) wherethe angle 4, is known as the pharb of the sinusoid and VT r + Z as its amplitude.Collectively, the functions of Chapters 32, 33 and 34 are often known as the trigonometric functions [seeSection 34:14]. 32:2 BEHAVIORThe sinusoids are periodic functions with period equal to 2a, that is, their values at x ± 2w, x ± 4n, x ± 6w, .. .exactly equal their values at argument x. This is evident from Figure 32-1. 295 32:2 THE SINE sin(x) AND COSINE cos(x) FUNCTIONS 296 In each period the sinusoidal functions display two zeros, one maximum and one minimum. The sine functionhas maxima of sin(x) = I at x = a/2. -3w/2, 5,r/2. -7tr/2, ... and minima of sin(x) = -1 at x = -a/2,3,r/2, -5w/2, 7w/2, ..., a zero being located midway between each adjacent maximum-minimum pair (i.e., atx = 0. ±w, ±2ar, ...). The behavior of the cosine function is similar. cos(x) = I maxima occur at x = 0, ±2w,±4n, ...; cos(x) = -I minima at x = ±w, ±3w, ±5w, ...; and zeros at x = ±ir/2, ±3tr/2, ±5w/2.....The periodicity of the sine and cosine functions can be appreciated perhaps more easily when the argument isregarded as an angle. Since the angle 0 is coincident with the }nglcs 0 ± 2a, 0 ± 4w, 0 ± trrr, ..., it comes asno surprise that 32:2:1 sin(o ± 2zr) = sin(0 ± 4ir) = sin(6 ± 6w) _ = sin(0)and similarly for the cosine [and, in fact, for the other four circular functions discussed in Chapters 33 and 341.Angles in the ranges 32:2:2 It2ka<0<-+2kwk=0, ±1, w2, ...2are said to be in the first quadrant. Similarly the second, third and fourth quadrants encompass the angles32:2:3(ir/2) + 2ktr < 0 < w + 2k second quadrantk = 0, ±1, ±2, .. .32:2:4it + 2kw < 0 < (37r/2) + 2kw third quadrantk = 0, ± 1, ±2, .. .and32:2:5(3w/2) + 2ku < 0 < 2(k + ow fourth quadrantk = 0, ±1, ±2, ...This usage of the team `quadrant' is related to, but differs from, that described in Section 0:2. Some simple rulesrespecting the signs of the trigonometric functions in the four quadrants are assembled in Section 33:2. 297 THE SINE sintx) AND COSINE cos(x) FUNCTIONS32:3 DEFINITIONS32:3 The trigonometric definitions of the sine and cosine function are illustrated in Figure 32-2. Consider the point Pto have arrived at its present location by having moved from A along the circular path AP such that the distance from point 0 has remained constant and equal to unity. Then the sine and cosine functions are defined as the lengths32:3:1 sin(x) = PQ[negative if P lies below line 0A132:3:2 cos(x) = OQ[negative if Q lies to the left of 01PQO being a right angle. The argument x may be interpreted in three distinct ways: first, as the length of the arcAP, this being assigned a negative value if the rotation of OP was clockwise; second, as the angle POA. measuredin radians, this again being considered negative if P rotated clockwise to arrive at its present location; and third,as the area (shaded in Figure 32-21 enclosed by the arc AP, the line OP and a mirror-symmetrical construct below the OA line. It is this third interpretation of x that parallels the one used in Section 28:2 as a geometric definitionof the hyperbolic sine and cosine. Expansions 32:6:1 and 32:6:2 may serve as definitions of the cosine and sine functions. Alternatively, thesefunctions may be defined via the formulas 32:3:3 sin(x) _exp(ix) - exp(-ix)2i 32:3:4 cos(x)exp(ix) -2exp(-ix)=which utilize exponential functions of imaginary arguments [see Section 26:111.Several polynomial functions exhibit sinusoidal behavior in the limit of large order: see equation 18:9:4 for theexample of Pochhammer polynomials. Similarly, for Euler polynomials [Chapter 201it232:3:5 sin(x) = - lima (-)E2( x)4 !-=1 (2n)! ITand 32:3:6 cos(x)lims n(-E..-,( ')IFor negative b, the differential equation 32:3:7dfdx '=bf+cb<0 32:4 THE SINE sin(x) AND COSINE coa(x) FUNCTIONS 298is solved by the general sinusoid f = -(c/b) + c, sin(xV b) + c2 cos(xV b), where c, and c2 are arbitraryconstants. Similarly, for suitable values of the coefficients, the differential equation 32:3:8 =aft+bf+ca<0b2>4acis solved by f = -(b/2a) + [ b2 - 4ac/2a][VA sin(xV a) - N /I cos(x\)] where A is an arbitraryconstant in the range 0 <- A <- 1. Hence, equation 32:3:7 or 32:3:8 can be considered to define the sinusoidalfunction. 32:4 SPECIAL CASES There are none. 32:5 INTRARELATIONSIIIPSThe cosine function is even32:5:1 cos(-x) = cos(x)whereas the sine is an odd function32:5:2 sin(-x) _ -sin(x)The duplication and triplication formulas 32:5:3 cos(2x) = cos2(x) - sin2(x) = 2 cos2(x) - I = 1 - 2 sin 2(X) 32:5:4 sin(2x) = 2 sin(x) cos(x) = 2 sin(x) 1 - sin2(x) 32:5:5 cos(3x) = 4 cos3(x) - 3 cos(x) 32:5:6 generalize tosin(3x) = 3 sin(x) - 4 sin'(x) = sin(x)[4 cos2(x) - I) 32:5:7 andcos(nx) =t.`r' cos'(x)J-o.-I32:5:8sin(nx) = sin(x)U.-,(coa(x)) = csc(x) 7, It,'_'2 coa(x) - !'1 co0x)1J-owhere T. and U. are Chebyshev polynomials discussed in Chapter 22, as are the Chebyshcv coefficients t '.Equations 32:5:3 and 32:5:4 are special cases of the argument-addition formulas 32:5:9 cos(x ± y) = cos(x) cos(y) + sin(x) sin(y) 32:5:10 sin(x ± y) = sin(x) cos(y) ± cos(x) sin(y)which, in turn, have the important special casescos(x)n = 0, 4, 8, .. .FIT:;sin(x)n = 1, 5, 9, ... 32:5:11Cos x "-' 2) _-oos(x)is = 2, 6. 10, .. .±sin(x)n= 3, 7, 1l, ... 299 THE SINE sinLt AND COSINE cos4x) FUNCTIONS 32:5:12sin(s)n = 0.4. 8....sin(.rmrt`_-cos(.%)n = 1, 5. 9....-121-cm(x)n=2,6.10.-..=costs)n = 3. 7. 11....constituting recursion formulas.From equation 32:5:3 one may derive the expressionsrV+ cos(y)/a + xl1 32:5:13cos_)(y(-1)'"1m = Intl2,1 - cos(x)\! 32:5:14 sinr(2=1-1)TC2m = VVVfor the cosine and sine of half argument. as well as the formulas 32:5:15 cos'(.r) = 32:5:16 sin'(x) =I - cost 2x)21 - cos(2rl32:5 2for the squares. These latter may be generalized to the expressions I(n) 32:5:17cos"(x) ="Y/cos((n - 21,x1 =01 andIT_ 71ij(n)cos((n - 2j)xJn=2.N+I=1.3.5.In - 1)!:_In!!2"V-1cos((n - 2j)xln= 2N=2_,4.6. 1)') sinf(n- 2j)xJ = -, Z(-]" (2j>xl_o ,2 1.n=2N- I = 1.3.5.32:5:18sin"(.r) = { (- 1)'.v nin - I)!'-(-1)'(Icos[(n-2j41=-0j n 1)-v(n)- 212"--ufor positive integer powers of the cosine and sine functions.The cosine and sine functions satisfy the function-addition formulas 32:5:19 32:5:20 32:5:21 32:5:22ITcos(.r) = sin(x) = tip' sin(x = V 2 costTr`ryCOs(y) -cos(y) = 2 cosyh) cos(2 )y\kr) cos(x) - cos(y) _ -2 sin (X2I sin(xv r±vriysin(x) t sin(y) = 2 sin;} cos(-) 32:6 THE SINE sin(x) AND COSINE cos(x) FUNCTIONS 300as well as the function-multiplication expressions 32:5:23 32:5:24I Icos(x) cos(y) = - cos(x + y) + - cos(x - y)22 1 1coa(x) sin(y) = 2 sin(x + y) - -sin(x - y) 32:5:25 sin(x) sin(y) _ cos(x - y) -Icos(x + y)22Equation 32:5:19 is a special case of the important formula32:5:26c, sin(x) + c; cos(x) = c; + r; sin(x + arctan(c2/c,)) = Vc; + c cos(x - arccot(c2/c,))by which any sinusoid may be expressed as a sine or as a cosine. The arctan and arccot functions occurring in32:5:26 are discussed in Chapter 35.Infinite series of the forms 32:5:272+ c, cos(x) + c2 cos(jx) + c3 cos(3x) + =2+c; cos(jx) or32:5:28 s, sin(x) + s2 sin(2r) + s3 sin(3x) + _ I s; sin(jx)i-jor sometimes of the combination zc0 + I c, cos(jx) + s, sin(jx), with the c, and s, being specified coefficients,are termed Fourier series and are discussed in Chapter 36. Here we merely quote two examples with relevance tothe present chapter. If v is any number other than an integer, then 32:5:29Cos(vx) _2vsin(wrt)f I+cos(x)-cos(jx)+cos(3x)-1- J2v2I-v29- =r I(-1),cos(jx)1it2vr,j 2 - v2232:5:30 sm(vx)sm(vtr)sin(x)2 sin(2x)+3sin(3x)IT1-v24-i?9-v2_2sin(wn)(-1)2j sin(jx)-,n < x < nrThe infinite series 32:5:31s(Fr)n=1,3,5,...and-)n=2,4,6.... i-rJ* i=Ifmay be summed in terms of the Hurwitz function (1 - n;x/21r) [see equation 64:6:21. The sum E cos(jx)/j equalsln{csc(x/2)/2}, while T sin(jx)/j2 is the integral of this function, known as Clausen's integral [see Abramowitz and Stegun, Section 27.8]. 32:6 EXPANSIONSTaylor series [see equation 0:5:1] exist for the cosine and sine functions and for their logarithms. Those for thefunctionsX2x`x6(-x2))32:6:1 cas(t) = 1 - - + - - - + _2!4!6!1-0 (2j)! 301 THE SINE sin)x) AND COSINE cosCxt FUNCTIONS 32:8 32:6:2sin(x)=.r--+---3!5!7!,=0 (2j+1)'converge for all .x, but more rapidly for small arguments. The logarithmic series have limited domains of conver-gence:X-X'to32:6:3 ln(cos(r))=2-1245 21(21)! 32:6:4ITIT-<x<-22sin(x)x`x'To4(2j)x Inix/61802835- -j\e/ (1r)'',-Ir<.r<Ir-1 2j(2j)!and the general terms involve eta, zeta or Bernoulli numbers [see Chapters 3 and 4].The sine and cosine functions are also expansible as infinite products:/x'\I=x I1 -- 32:6:5sin(x)x11 --JI l -r TX-..2/4x'\ /4x'/4x21// 32:6:6cos(x)I - - III - - 11 - -J = 11 1I -}Ir- // 1`91r'.`25-'. \\\\\\(2j - 021r2,both of which are encompassed by the general formula\32:6:7f(x) _ fi (l -x)f(x) =sin(x), cos(x) ,r,.rwhere the r values are the zeros of f(x). Yet another expansion as an infinite product issin(x)rr11r) 32:6:8 =0.4-!) cos( 4 cos(8... = IJcos(r x ///The cosine and sine functions may be expressed as infinite sums of Bessel functions [Chapter 52] 32:6:9cos(x) = J,,(.r) - 2.1.,(11 + 2J,(x) - = J4x) + 2f -1))J(_c) 32:6:10sin(s) = 211(x) - 2J,(-r) - Mix) - _ 2(-1)7;,-,(x) 32:7 PARTICULAR VALUESTable 32.7.1 evaluates the cosine and sine functions for special values of the argument x in the range --/2 s xIr/2 (angles 0 in the range -90° to 90°).For particular values outside this range, use the recursion formulas 32:5:11 and 32:5:12, 32:8 NUMERICAL VALUESAlmost all calculators have keys by which sin(s) and cos(.r) may be computed. and almost all computers incorporatesine and cosine functions; therefore, no algorithms are presented here. Sometimes the computer generates sin(0)or cos(0) where 8 is an angle in degrees: in this case sin(s) can be calculated as sin(0) = sin( 180x/-). Computing 32:9 Table 32.7.1THE SINE sin(x) AND COSINE cos(x) FUNCTIONS eo:IPIrX22':0*3e:is`:Sr+6o'-67t'n''73°t9o 12to 8654to3B5 122l+lS+VS2+_f+\5t\r\r-V2IIVSVS-I6 fa22B\r9i2eVM 213. \15+1mt\rxta Vi.viztvi±t6,f.\.tB56\+82 BS1 devices often offer the user a choice of "radian mode" or `degree mode" when evaluating trigonometric or inversetrigonometric functions; be aware that, because of rounding errors inherent in the internal routines of the device.the "degree thode" is usually the more precise of the two modes. (For example. sin(1440°) will be given as zero.whereas sin(8ar) will often fail to be calculated as zero.)The universal hypergeometric algorithm [Section 18:141 permits cos(x) and sin(x)/x to be evaluated for any x. 32:9 APPROXIMATIONSFor small values of the argument, the approximations' 32:9:1sin(x) - x I - -8-bit precisionx S 1.12X2andI/z``32:9:2coa(x) c1 I- 3 18-bit precision-0.9:%:S 0.9 are useful. \//// 32:10 OPERATIONS OF THE CALCULUSThe differentiation formulas 32:10:1 & sio(bx)= b cos(bx) = b sin(bx + 2 ) 32:10:2 & cos(bx)-b sin(bx) = b cost b., + - I can be generalized to /// 32:10:3- f(bx) =bmbx+ Z In = 0. 1. 2....f = sin, cosIndefinite integration gives \\ 32:10:4 32:10.5I - cos(bx)sin(bt)dt =b0 cos(bt)dt -sinItx) 303 THE SINE sin(.r) AND COSINE cos(-v) FUNCTIONSwhich are special cases of the more general results 32:10:6sin"tt)dt =(n-I)!!r12j-W!1 - cos(s)( 2J Wsin='(.r1Jn = 2V + I = 1. 3. 5, nnL in - l)" t2j}n-cos(."in"-'(.V)n = 2N = 2. 4, 6.n !!'=n (2j + 1 )!!(n - l)!!(2j - U"sin(x)cos21)x)n = 2N - 1 =I. 3. 5...i=o2j)! !2j 32:10:7f"' cos'Wdt =vIn -n!!1rr (2j)...r + sin(x) E ,cos_`"(.r)Jn = 2N = 2, 4, 6. n..1=0 (2j + I)..Other important indefinite integrals include dtIbxa 32:10:8I+ a sin(bt) -b[tan(2+4111Jfarcosh(a)-aILLL \\\V a' - 1 2/atan(bx/2)`arctanl\) -arcsin(a) bV la' 2 (a ' lbxartanh-cot(-))bV'a'-Iva - I2 32:10:9 ran- I b2lal a = 1;1 -abr-,arctan ltan- Ia < I bVI -a' )t'I +a232:10 Indefinite integrals of the sine or cosine of the quadratic function ar' + br + c may be expressed in terms ofFresnel integrals (Chapter 391r += ! - I costb' - 4a,J S(tarb 32:10:10Iysin(ar - br + c)dtIT2a4a 1112V ab'-4ucear-b-sinICI)a>0a 32:10:11 as may 32:10:12fl dtb n r& - 4acf,cu + b costar -bt + c)dt =V1 3acost4aJC1, b/ n 2 V a' b' - 4acb+sinSIa>0_V ar.- -Isin(btldtj2ax / b S(Vbr)nV'tV 32:10 THE SINE sin(s) AND COSINE cos(t) FUNCTIONS0r cos(bt)dt2n 32:10:13J,=C() With Si and Ci denoting the sine and cosine integrals [Chapter 381, we also havesin(bt) 32:10:14 dr = Si(bx)ot304 and(¢ oos(bt) 32:10:15 Jdt = -Ci(bx)frThe technique exemplified in 34:10:19 is useful for evaluating many integrals of sine and cosine functions.Definite integrals involving integrands of the forms f(x) cos(sx) and f(x) sin(sx) are called Fourier transformsand are disctssed in Section 32:14. One definition of so-called cosine transformation is 32:10:16fds) =1 Jf(t) exp(-ist)dr +IJ¢f(t) exp(ist)dto o owhile the corresponding sine transformation is defined by1;232:10:174(s) =Jf(t) sin(st)dt = i'Jf(t) exp(-ist)dt - l Jf(t) exp(ist)dtoV1, o;V1;oErd6lyi, Magnus, Oberhettinger and Tricomi [Tables of lnte ral Transforms, Volume 1. Chapters I and 11) listmany such transforms, although their definitions omit the 2/ar multiplier. The final equalities in 32:10:16 and32:10:17 show how cosine transforms and sine transforms are related to the Laplace transforms of Section 26:14.Thereby the tabulation in that section may be used to evaluate many cosine and sine transforms. Consider, forexample, the function1/V for which the Laplace transform is fr(s) It follows then from 32:10:16that f<s) _/VZar +/= I/f.The general formula 32:10:18r.(l+vf(bt )dt = Ivb-it.f2vb> ov> 1f =sin, cos also holds.'With a lower limit of -, differintegration to order v generally causes the addition of va/2 to the argument 32:10:19d'f(bx) =b'f(lbx +mv > -1f = sin, cos ld(x + 00)1'\2 ///and multiplication by the va power of the argument multiplier. Equations 32:10:1-32:10:3 are instances of thisrule; another example is 32:10:20d-'n 1arlsin(bx) - oos(bx)[d(x +sin(bx) _sin( bx -4/b>0 With a lower limit of zero, semidifferentiation and semiintegration generate auxiliary Fresnel integrals (see Chapter391 32:10:21 32:10:22d1/2sta(bs) = V b sin( bx + 4 I - V Tb tms(V bx) d 1/2dTllcos(bx) _+cos bx + 4/ - V '2b Fres(\)art 305 THE SINE sin(.o) AND COSINE cos(x) FUNCTIONS 32:12 32:10:23 sin(bx)sins bx--l+Fres(Ni' br) dx\/h4!bd Ia(232:10:24 cos(bx) =cos(bx-yb4)- Grestibx)\ bThat the functionsI = cos(0.x). sin(x). cos(x). sin(2r), cos(2x). sin(3x).... form an orthogonal family [seeSection 21:14] on the interval 0 :s x :s 27r (or alternatively on -a <- x s n) is established by the integrals32:10:251,icos(nt) sin(ntt>dt = 0n = 0, 1, 2.m = 1, 2, 3, .. .n 32:10:26(0m*n1*cos(nt) cos(mt)dt =2am = n = 0n, m = 0, 1, 2, .. .onm=n+032:10:27 1"1sin(nt) sin(mt)dt =J Omnln, m = I, 2, 3,...Inm=nOn the interval 0 s x i- it. the functions cos(Ox), cos(.x), cos(2r), - or the functions sin(x). sin(2x), sin(3x).are orthogonal. but the conjoined families are not. 32:11 COMPLEX ARGUMENTFor argument (x + iv) we have32:11:1 sin(x - iv) = sintx) cosh) v) - i cos(x) sinh( v)32:11:2 cos(xiv) = coslx) cosh( v) - i sin(x) sinhi v)and therefore for purely imaginary argument32:11:332:11:4sin(iv) = i sinh( v)cos(iv) = cosh(y)Useful relationships are embodied in De Moivre's theorem.32:11:5[cost:) + i sin):)]' = exp(iv:) = cos(v:) i sin(v=)where : itself may be complex. Unless v is an integer. the real part of = must lie between -rr and it. 32:12 GENERALIZATIONSAs explained in Chapter 36, any periodic function can be regarded as a generalized sinusoid.The Jacobian elliptic functions sntp;.r) and sd(p:x) generalize the sine function, to which they reduce when pis zero:32:12:1 sn(O:x) = sd(O:x) = sin(x)Likewise. because32:12:2 cn(0:.r) = cd(0:x) = cos(.x)the functions cnl p;x) and cd(p;x) are generalizations of the cosine function. The Jacobian elliptic functions are thesubject of Chapter 63. Finally, the sine function is a special case of the incomplete elliptic integral of the second kind [Chapter 62]:32:12:3 sin(x) = E(x;l) 32:13 THE SINE sin(x) AND COSINE cos(x) FUNCTIONS 30632:13 COGNATE FUNCTIONSThe sine and cosine functions are related to the other circular functions those treated in Chapters 32, 33 and 34]bya,sec2(x) - 1_1_tan(x)a232:13:1sin(x) = a21 -.2'x) 04 =sec(x)csc(x)_W101)ti cot2(x) + I Ia3csc'(x) - I a4a_ cot(x)32:13:2cos(x) =94VI - sin (x)sec(x)csc(x)V I + tan2(x) cot2(x) + Iwhere the multipliers a2, a, and a, take the values - I or - I according to the quadrant (sec Section 33:2] in whichx (interpreted as an angle) lies. All multipliers are positive in the first quadrant. or, also in the second, a3 in thefirst and third, (r. in the first and fourth. The explicit formulas32:13:3 a2 = (-1)'° '/"'32:13:4 a3 =1)'"'12r/.r32:13:5 a4 =(-1)W42,+II)/2.I =ata.permit the multipliers to be calculated for any x value. Figure 33-2 provides a geometric interpretation of theinterrelationships among the circular function.The versine function vers(x), coversine function covers(x), and haversine function hav(x), defined by32:13:6 vers(x) = I - cos(x)32:13:7 covers(x) = I - sin(x)and 32:13:8 hav(x) =2[1 - cos(x)] =sin2(2/are rather archaic functions seldom encountered nowadays. The function sinc(x), sometimes known as the samplingfunction, is important in spectral theory. It is usually defined by 32:13:9 sinc(x) =sin(rrx)axalthough the definition sin(x)/x is also encountered. Figure 32-1 includes a graph of sinc(x), and equations 32:6:2,32:6:4. 32:6:5, 32:6:7 and 32:6:8 are readily adapted to provide information on this function.Bessel functions [Chapter 53] and Neumann functions [Chapter 54] of orders equal to one-half of a (positiveor negative) odd integer are related in a very simple way to sines and cosines. The simplest cases are 32:13:10 32:13:11J,12(x) = Y_,/2(x) =Ysin(x)arx Yia(x) _ -J-112(x) = cos(x)F1TXand the general expressions for n = 1. 2. 3. ... are 32:13:12J,,1 (x) _ (- Ii sin x - 2 W,() +cos x - 2 V (x)Y\nom/ 12/a 1mrl2na.132:13:13Y,.,/2(x) sin] x - 2/V,(x) -Zrxcosy-2 2IWr(x)\ \// 307 THE SINE sin(x) AND COSINE cos(x) FUNCTIONSwhere W"(x) andare the finite sums 32:13:14 W"(x) _(n+ 2 ).r-I lI = Int;-a(2j)!(n -2j)'4x2r (121) 32:13:15V(x);(n+2j+1)! 1`n - I-(2lt3 =Int(l-oJ +2j - 1).4x')/\2For example, W1(x) = 1, V1(x) = 2, W2(x) = 1 - 3x-2 and V2(x) = 6. Of course, all the general properties ofthe Bessel and Neumann functions [Chapter 53 and 541 apply to these spherical functions, the first few of whichare mapped in Figure 32-3. The symbols and y"(x) and the names spherical Bessel function of the first kindand spherical Bessel function of the second kind, respectively, are sometimes applied to the composite functionsVir/2xandThe significance of the adjective "spherical" will be evident from Section59:14. . ... ... . .........0.7FIG32-3 <x> :....:f._:....:.\ ,n..................... .:....... 1` 0.6 :....:....:.......0.4Y. ,(x).:....... 0.3 32:14 RELATED TOPICSIn this section we are concerned with the transform 32:14:1J=f(t) exp(-2ilrst)dt =J 9f(t)[cos(21rst) - i sin(2irst)1dt = F(s)which converts a function f of t into a function F of s and is known as the exponential Fourier transform. Ingeneral, f and F may be complex, but t and s are usually real. Properties of the transform and examples for specific 32:14 THE SINE sin(s) AND COSINE tos(s) FUNCTIONS 308f(t) functions are tabulated [for example, by Beyer, Handbook of Mathematical Sciences, page 597, who, however,uses a definition slightly different from 32:14:1]. the function f(t) may be regenerated by the inversion formula32:14:2f F(s) exp(2iirst)ds = JF(s)Icos(2 rst) + i sin(21rst)]ds = f(t)which differs from 32:14:1 only in signs.The operations known as Fourier transformation and Fourier inversion that are of such practical importancein science and engineering are discrete-and-finite analogs of the continuous-and-infinite formulas 32:14:1 and 32:14:2.Most applications are in the fields of spectroscopy and acoustics where f and F describe the intensity of a wavemotion: F in the space of frequency. s: and fin the space of time, t. Usually, measurements of f consist of recordingf o , fl, f2, ..., f, .., fN-I atequally spaced values of i, namely t = 0, T/N, 2T/N.... , jT/N, ..., (N - 1)T/N,each value possibly having a real component r, and an imaginary component i, The equations1 N--I' r/2njk/2srjk32:14:3R,=Lr,costN+ysin N 32:14:41, = -[i,r sinl2k we used to transform these data into a set of real and imaginary components of the variable F. The subscript k isintimately related to the frequency variable s: in fact, R. + i!, = Ft is the value of F sampled at s = k/T. Thehighest k value that is accessible corresponds to the Nyquist frequency N/2T; see Hamming for further discussions on this topic.Equations 32:14:3 and 32:14:4 are the fundamental formulas used for numerical Fourier transformation. WhenN is large, these equations call for vast computational resources, and to alleviate this demand the fast fouriertransform has been developed. This invention Ifor full details, Section 2.3.2 of Stoer and Bulirsch. or the originalwork of Cooley and Tukey, may be consulted) takes advantage of the symmetry properties of the sine and cosinefunctions to abbreviate the volume of arithmetic. Such abbreviation is especially pronounced when N, the numberof data pairs, is a power of 2 (e.g., N = 27 = 128 or N = 210 = 1024) and this is the condition incorporated intothe algorithm below, which performs fast fourier transformation.The algorithm first requires the input of the integer p. equal to log2(N). followed by the sampled data ro. io,r,, is, r2. ---. iN-2, rN-,, iN- ,. The ultimate output is in the order Ro, 10, R,, !,, R2, !N-2, RN- 1N-1.The two portions of the algorithm shown in green are virtually identical and could therefore be programmedas a subroutine. They are used to set register M equal to the bit-reversed complement of the integer K. This means that the bit pattern of the number K written in binary notation (e.g.. 0101001 if K = 41 and p = 7) is reversed inM (e.g., 1001010 = 74 = M).Even though the algorithm makes provision for imaginary components of the sampled input function, manyapplications of Fourier transformation utilize real data only. Then, of course, one enters the values io = is = i2 == iu-, = 0 into the fast fourier transform algorithm. In such applications, only the first half of the output hasuseful information content because the second half merely duplicates the first in magnitude: RN-k = Rk,1N-k = -1k. Input p > j(as cue)IInput ri >Input it >»»>Setk=N=210Setj=0(1) Output j»»>»»>Replace j by j + IIfj*kgoto(1)(2) Replace k by k/2Set m = -2kStorage needed: 2N + 9 registers are required to store:p, k. N, j. to, io, r,. is. r2...., iN-2, rx-1, iN-i, m, K,M, ! and R. FUse degree mode or replace 360 by 2w.1 309 THE SINE sin(x) AND COSINE cos(r) FUNCTIONS 32:14 k««R=R,<(3) Replace m by m + 2kSetK=m/kSet M = 0Set j = P1(4) Replace j by j - IReplace .M by M + K2'Replace K by Int(K/2)ReplaceM by M - 2K211fj*0goto(4) Set ! = 360M/NSet R = cos(t)Replace I by (p/Ipl) sin(1)Setj=k+m - 1(5) Replace j by j + ISetK=r,R+i,! SetM=i,R-r,lReplace r,_,, by r,_, + KSet r,=r,-4-2KReplace i,-4 by i,-4 + MSet i,=y-t-2MIf)+ I <2k+mgoto(5)If m + 2k < N go to (3)If k > I go to (2) Set m = -1(6) Replace m by m + ISet KmSet,M = 0Set) = pli7) Replace j by j - lReplace M by ,M - K2'Replace K by Int(K/2)Replace M by M - 2K21Ifj*0goto(7)JIM Sm go to(8)SetR=r,Setr,,,rMSet ',RSet ! = ImSet i = i,w Set i,w = I(8) If m < N - I go to (6)Set k = -1Ifp>0goto(9)Set p = 0Set N= 1(9) Replace k by k + IOutput kSet R = rk/NOutput R«G«Set I = i4/NOutput IInput restriction: The parameter p must be an integer. Test values for transformation:p=4ro=rs=2.io=is=3r,n,V2.i,=iy3r,=r,o=0.i.=iio 3r) =r_ -V2.isill =3 r4r1:-2,i4=ic=3rs = ris = -V2, is = iii = 3r6ri4 =0.i6=iu= 3r. = ris = V 2. i. = i,s = 3Output:Io=Po=3R, = P, = IR14=P14= 1All other outputs are zero. 32:14 THE SINE sin(x) AND COSINE cos(x) FUNCTIONSI = It « ««<Ifp=0goto 10Set M =R2 + !2Output MM=Pk< ««<(10) If k < m go to (9)310 To test inversion, input p = -4 followed by the Ro.lo, R,, ..., I,s values listed above. Output values: thoselisted above as to, io, r , ,. .., il,. The physically significant result of the transformation is usually the real quantity ti R which, as a functionof the frequency k/t, is known as the power spectrum of the input data. The algorithm also outputs this quantity,Pk.Many applications of Fourier transformation require that the transformed data be processed in some way in thefrequency domain and then inverted back to the time domain. The equations describing the inversion areN-I r(1kj2irkj\1 32:14:5 r, =I Rkcos)- /t sing N rrL/2arkj/2nkj/32:14:6ij_ EL/k coslN+ Rk sinlNand are so similar to equations 32:14:3 and32:14:44that the same algorithm may be used to effect inversion astransformation. Thus, the algorithm presented above is actually a fast fourier transformation/inversion algorithm.To select inversion, one merely inputs -p instead of p, followed by Ro, Io, R1. I,. R,...., IN-2- RN_ IN_,. Theoutput is to, b, r1, il, r2, ... iv-2, rN-1, iv-1- CHAPTER33THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS These functions are the reciprocals of those discussed in Chapter 32. The secant and cosecant functions are inter-related by 33:0:1 [sec2(x) - II[csc2(x) - 11 = Iand by 33:0:2 esc(.r + -)= sec(x)IT 33:1 NOTATIONThe symbol cosec(x) sometimes replaces csc(x). To avoid possible confusion with the functions of Chapter 29. thenames circular secant and circular cosecant may be used. Because of their applicability to triangles [see Section34:141. the functions of Chapters 32-34 are known collectively as trigonometric functions. 33:2 BEHAVIORFigure 33-1 shows that sec(x) and csc(x) adopt all values except those between - I and + 1. The two functionsmay receive any argument, but sec(x) is ill defined when x = ±tr/2. ±31r/2. ±5w/2.... and csc(x) is similarlyindefinite at x = 0, '-n, ±21r, .... Both functions are periodic with period 2n. With the argument interpreted asan angle, the signs acquired by the sec and csc functions depend on the quadrant [Section 32:21 in which theargument lies, as illustrated in Table 33.2.1. which includes the other four trigonometric functions. 33:3 DEFINITIONSThe secant and cosecant functions may be defined as the reciprocals of the functions of Chapter 32: 33:3:1 sec(x) _cos(x) 311 33:3 THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS 312 FIG 33-1 : : : 33:3:21csc(x) =sin(x)Three similar right-angled triangles are shown in Figure 33-2. If the sides represented by the dashed lines areall of unity length, then the other six sides have lengths that define the six trigonometric functions as depicted in the diagram. The argument x of these functions may be interpreted either as the marked angle or as the length of the arc of a unity-radius circle subtended by the angle, as diagrammed. Besides serving to define the six functions,many interrelations between them may be derived by applying similarity and Pythagorean relationships to the tri-angles. For example: 33:3:3Isec(x)csc(x)cos(x)Icot(s)follows by equating the ratio of the red to the blue sides in the three triangles, while 33:3:4 csc2(x) = I + cot2(x)is a consequence of applying Pythagoras' theorem to the third triangle. Table 33.2.1 FustSecondThirdFourthquadrantquadrantquadrantquadrantcog and sec I+--sin and csc++-m and cot+-+ 313THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS 33:5As well, the secant and cosecant functions may be defined in terms of exponential functions of imaginaryargument 353 2 exp(ix)::3 sec(x) _exp(2ix) + I2i exp(ix) 33:3:6 vithitl tfoscsc(x) =exp(2ix) - Ioraenegraransrm2r'`I °dt-aa33:3:7 sec(x)_ < x<2 tZ + 12n 0 338 0< 3:: 33:4 SPECIAL CASESThere are none.csc(x)x< aoJ+r 33:5 INTRARELATIONSHIPSWhereas the secant function is even:33:5:1the cosecant is an odd function:sec(-x) = sec(x) 33:5:2 csc(-x) = -csc(x) 33:6 THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS 314The recursion formulassec(x)n = 0, 4, 8, .. .,cc(,:,nir\=+csc(x)n = 1, 5, 9, .. .2-sec(x)n = 2, 6, 10, ...tcsc(x)n = 3. 7, 11, .., csc(x)it = 0. 4. 8, .. .nn_±se(x)n=1,5,9....csc(x+ -2)-csc(x)it = 2, 6, 10.-;: sec(x)n=3,7,11,...parallel those of the cosine and sine functions, but them are no simple formulas, akin to 32:5:9 or 32:5:10, toexpress sec(x ± y) or csc(x ± v). Such expressions-as well as those for sec"(x), csc(x) ± esc(v), etc.-are bestevaluated via the formulas of Section 32:5, making use of identities 33:3:1 and 33:3:2.The secants and cosecants of double argument and half argument may be evaluated through the formulas 33.55 sect) x) =scC2(x)2 - sec2(x)sec(x) -(x) esc2(x)33:5:6) = 22csc (x) - I(x12 sec(x)/ n + \ 33:5:7 sectI = (-1)"1 + c(x)m = lntl21 andxlsec(x)(w,( J 33:5:8 csc= (-1)'"m = bit 2sec(x) - I 2zr 33:6 EXPANSIONSThe powerseries expansion of the secantx2Sx'61x6 33:6:1soc(x) = I + - + - + - +_ - X2'224720(2j)!-an 2Z <x<2=2ft(kx2x/tr)tk=1,3,5,... can be written in terms of Euler numbers [Chapter 5] or beta numbers (Chapter 3], whereas that of the cosecant 33:6:2I+x+7x531x5p(4' - 2)B22{,r2j-l csc(x) _ -- -_x6360+15120+(2j)! 12 `+ - E *I(2j)(x/ir)2iXx r1-n<x<9r involves the Bernoulli numbers [Chapter 4) or the eta numbers [Chapter 3]. Series for the logarithms of sec(x) andof x csc(x) invoke the lambda and zeta numbers [Chapter 31k(21)(7)2x\'-.aIr33:6:3 1n(sec(x)) 2 <X<-,-IJ 315THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS 33:6:4 ln(x cec(x))/xx; I - 1-W < x < n;-11\The secant and cosecant functions may be expanded as partial fractions 33:6:5sec(x) _4ir-l21r+20w-= a(-1)'(2j + 1)rr2 - 4x297r2 - 4x225.r - 4x2(i + )zirx - s2I2x2x2xI(-1)' 33:6:6CSC(x)+xx-z+- 2xx;°1 )-2'R2- xz X'R -x4a -X9'tr2-x-as may their squares 33:6:7sec-,(x) =4+4+4+4 1+_ (w - 2x)2(n + 2x)2(3Tr - 2x)22x)2._ [x + (j + 2)a]2 1II1 1'133:6:8 csc-(x)+++++_x)2Fir + x)2(2Trx)2(2a + x)2[x + jwf33:7 PARTICULAR VALUES33:9 Table 33.7.1 lists values of the secant and cosecant functions for special values of the argument x in the range-n/2 s x s a/2 (angles 0 in the range -90° s 0 5 90°). For particular values outside this range. use the recursion formulas 33:5:3 and 33:5:4. 33:8 NUMERICAL VALUESThese are easily found by taking the reciprocals of the values of the cosine or sine [see Section 32:8]. 33:9 APPROXIMATIONSFor small values of the argument, the approximations-3/2/x-\\ 33:9:1sec(x) - I i- 38-bit precision-0.9 5 X:5 0.9 Ix 2233:9:2cse(x) = - + -8-bit precision- -:S X:5 -are valid. Table 33.7.1 6r-1r-1r22I'-xr-J6°-ar-5a'-°V-67'X72''-7r-6t? 12 10 865 1038 5 12 2 21V6-V32-5Va-VS-3V6-V20V22--vs V V =V;V1-V6-V20-V4- V5-22-- '-V'2.V6-VIO -'1-Va-VS,5sV0=V2'5\\ 33:10 THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS33:10 OPERATIONS OF THE CALCULUSDifferentiation of the secant and cosecant functions gives 33:10:1dsec(x) = sec(x) tan(x) _csc(x)and 33:10:2 whereas integration leads to 33:10:3 anddcsc(x) = -csc(x) cot(x) =-csc2(x)dx sec(x) fjcsc(t)di =ln[tanl2l]= ln[csc(x) - cot(x)l r . \I316 33:10:4J,sec(t)dt =In{tanl 2+ 4)] = ln[sec(x) + tan(x)) = invgd(x) where invgd is the inverse gudermannian function discussed in Section 33:14. Indefinite integrals of the squaresof the secant and cosecant functions have simpler formulations: 33:10:5 and 33:10:6 33:11 COMPLEX ARGUMENT The formulasJAsec2(t)dt = tan(x)0F xcsc2(t)dt = cot(x) coth(y) + i tan(x) 33:11:1 sec(x + ry) =sec(x) sinh(y) + coc(x) cech(y) 33:11:2 csc(x + iy) =coth(y) - i cot(x)sin(x) csch(y) + csc(x) sinh(y)reduce to33:11:3 sec(iy) = sech(y)33:11:4 csc(iy) _ -i cscb(y)when the argument is purely imaginary. 33:12 GENERALIZATIONS The Jacobian elliptic functions [see Chapter 631 nc(p;x) and dc(p:x) are generalizations of sec(x). to which theyreduce as p - 0. Similarly, csc(x) is the p = 0 limit of the ns(p;x) and ds(p;x) functions- 317THE SECANT sec(xt AND COSECANT csc(x) FUNCTIONS33:13 COGNATE FUNCTIONSThe secant and cosecant functions are related to the other circular functions (Chapters 32, 33 and 341 by33:14 Q4 1v3csc(x) ,92V cc33:13:1sec(x) _ _ = v.V I + tan-(x) _ V 1 - sin-(x)cos(x)V csc (^ x7) - I cot(s) 1(720r3 sec(x)-.V -I 33:13:2csc(x) _ - =-J--c0-S7(-=r_== cr,rn (x) + 1 sin(x)Vx)V sec_-(x)- Itan(x)The multipliers ore, Q3 and rr. equal w 1 according to the magnitude of x. as explained in Section 32:13; for 0 <- x<- 1T/2 (i.e.. in the first quadrant), they are all plus unity.The secant and cosecant functions are related to hyperbolic functions in two distinct ways: through the ima-ginary-argument formulas 33:11:3 and 33:11:4, and via the gudermannian function of Section 33:14.Of course, sec and csc are closely related to their inverses, the arcsec and arccsc functions that are the subjectof Chapter 35. Occasionally encountered is the exsecant function 33:13:3 exsec(x) = sec(x) - I = tan(x)tan(g) 2 <x<2 33:14 RELATED TOPICSComparison of Figures 29-2 and 33-2 suggests that the six hyperbolic functions and the six circular functions areclosely interrelated. In fact, one has33:14:1 sin(s) = tanh(y)33:14:2 coc(x) = sech(y)33:14:3 sectx) = cosh(y) trii--<x<-33:14:4 csclx) = coth(y) 2233:14:5 tantx) = sinh(y)33:14:6 cot(x) = csch(y)provided that the arguments x and y are suitably related. The relationship involves the gudermannian function33:14:7or the inverse ,qudermannian functionx = gd(y) 33:14:8 y = invgd(x)also denoted gd-'(x).The gudermannian function may be defined in a variety of ways, including 1 33:14:9 gd(y) = 2 arctan(exp(y)) - 2 = 2 arctan(tanh( /I2. /and33:14:10 gd(y) =sech(t)dt0Similarly, the inverse gudermannian function may be defined by 33:14:11invgd(x) =In(tin+In(sec(x) -- tan(x))- - < x < 4 33:14 THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONSor by the integral318 33:14:12 invgd(x) sec(r)dt-2 <x< 20The same function may be realized as a special case of the incomplete elliptic integral of the first kind [Chapter62]:33:14:13 invgd(x) = F(l;x)As Figure 33-3 demonstrates, both the gudermannian and its inverse are odd functions. The gudermannianfunction approaches -7r/2 as its argument approaches ±¢. Conversely, invgd(x) approaches =x as x -* w tr/2.The derivatives of gd(x) and invgd(x) are sech(x) and sec(x), respectively. The power series of gd(x) and invgd(x) involve Euler numbers [Chapter 51 and are remarkably similar:xsx'61x1Etjxx'"33:14:14gd(x) = x - - ++_71(2+ 1-1 <x< 1 -6245040l)!xrx}61x'JEjWjaIT 33:14:15invgd(x) = x ++++_--<x<-6245040,.o (2j + 1)!22Several other expansions exist for the gudermannian function and its inverse [see Beyer, Handbook of MathematicalSciences, pages 323-325]. CHAPTER34THE TANGENT tan(x) AND COTANGENTcot(x) FUNCTIONS The functions of this chapter are the reciprocals of each other.34:0:1 tan(x) cot(s) = ITogether with the sine, cosine, secant and cosecant functions [Chapters 32 and 331 they constitute the six trigo-nometric functions that play important roles in the mensuration of triangles [see Section 34:141. 34:1 NOTATIONThe alternative notation tg(x) is occasionally encountered for the tangent function, while cotan(x) or ctg(x) some-times replaces cot(x). To emphasize the distinction from the functions addressed in Chapter 30. the names circulartangent and circular cotangent are often applied to tan(x) and cot(.t). As in Chapters 32 and 33, we use tan(N) andcot(H) as alternatives to tan(g) and cot(x) whenever we wish to stress the angular interpretation of the argument [A.B and C are used as angular arguments in Section 34:141. 34:2 BEHAVIORLike the other four trigonometric functions, the tangent and cotangent are periodic functions, but unlike the otherfour the period is it, not 21r. This is evident in Figure 341. The signs acquired by tan(O) and cot(9) in the fourquadrants [as defined in Section 32:21 are tabulated in Section 33:2.The tangent function has zeros at .r = 0. t1r, =21T, ... but is undefined at x = z7r/2, =3a/2, t5n/2, ....Conversely, cot(s) has zeros at x = tir/2, t3a/2, t5w/2.... but is undefined at x = 0. =ir, t2tt 34:3 DEFINITIONSGeometric definitions of the tangent and cotangent functions are possible with the aid of Figure 32-2. namely 34:3:1 tan(x) _ 319 34:3 THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS 320 34:3:2 cot(x) =-The functions may also be defined in terms of the sine and cosine functions: 34:3:3sin(x)tan(x) =cos(x) 34:3:4 cot(x) = or via exponentials of imaginary argument: 34:3:5 34:3:6cos(x)sin(x) 2itan(x) =exp(2ix) + I - i2icot(x)+ iexp(2ix) - Ior trigonometrically as in Figure 33-2. 321THE TANGENT tantx) AND COTANGENT cot(s) FUNCTIONS 34:5The indefinite integrals34:3:70i234:3:8 cot(x)csc2(t)dt also define the tangent and cotangent functions, as do the definite integrals2it 34:3:9 tan(x) =-tZ - 1dtx < 2034:3:10For appropriate values of a, 34:3:11 is solved bycot(s) =nt - tx<7t6 and c, the differential equationdafZ+bf+eb2 < 4ac 4ac - b/x 4ac - b \ bVac - b/z 4ac --b-' b34:3:12f =2atool I -2aor-cot(2- -When b2 > 4ac,a solutionis f = -(b/2a) - 4ac/2a) tanh(xVb c4ac4a/2) or-(b/2a) -(Vb- - 4ac/2a) [see equation 30:3:6 for a special case]. When b2 = 4ac. a solution is f =-(1 + bx)/4ax. 34:4 SPECIAL CASESThere are none. 34:5 INTRARELATIONSHIPSThe tangent and cotangent functions are both odd:34:5:1The argument-addition formulas 34:5:2 and 34:5:3tan(x) = J sec2(Odt f(-x) = -f(x)f = tan or cot tan(x ± v)tan(x) ± tan(y)I = tan(x) tan(y) cot(x ± y) =cot(x) cot(y) 7- 1cot(y) ± cot(x)have the special cases 34:5:4 tan(2x) _2 tan(x)2 cot(x)IT- tan2(x)cot-(x) - Icott2x)and2t-1-dtJ 3 tan(x) - tan'(x)3 cot'(x) - 1_1 34:5:5tan(3x) =I - 3 tan-(x)coe(x) - 3 cot(x)cot(3.r) .44:5 THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS 322 and also generate the recursion formulas 34:5:6 andtan(x)tan(x) ± 1tan(x +/ =I r tan(x)``4-cot(x)tan(x) = 11 ± tan(x) cot(x)cot(s) w 1/I = cot(x) 34:5:7 cotl x t 4 _-tan(x)cot(x) ± I I-+ cot(x)The tangent of half argument, which we represent here by 34:5:8 ((x1cos(x)T -tan2,= (-1)T1 + cos(x)n=0.4,8....n= 1.5.9,...n=2,6,10,...n=3,7,11.... n=0.4,8,...n= 1.5.9,...n=2,6,10....n=3,7.11.... is a useful quantity because it is related in a very simple fashion to the following trigonometric functions of Jr 2.734:5:9 sin(x)1+ r1 - T -=--3 4 : 5 : 1 0 cos(x): T 'T+ 34:5:11 I +T2sec(x) =7---Ti 34512 I+ r2= :: cec(x)2T 1134:5:13 tan(x) 34:5:141-TI+TT'cot(x) _2T234:5:15 I + cos(x) =2l + T 1134:5:16sec(x) + 1 = - + -1 - T1 + T 34:5:17 I+Tsec(x) + tan(x) = 1835I -TI- T-= 4:: sec(x)tan(x)I + T 323THE TANGENT tan(x) AND COTANGENT coUx) FUNCTIONS 34:5:191csc(x) + cot(s) = -T34:6 34:5:20 CSC(x) - COt(x) = TThese equations apply irrespective of the magnitude of x. that is, in all four quadrants, as defined in Section 32:2.The series T3r'r1 ' 1invgd(x) 34:5:21T+-+-+352j + l2sums to the inverse gudermannian function [Section 33:141.The function-addition formulas34:5:22 tan(x) ± tan(y) = sin(x ± y) sec(x) sec(y)34:5:23 cot(x) t cot(y) = sin(y ± x) csc(x) csc(y)34:5:24 cot(x) + tan(x) = 2 csc(2x)34:5:25 cot(x) - tan(x) = 2 cot(2x)and function-multiplication formulas 34:5:26 34:5:27 34:5:28tan(s) tan(y) =cos(x - y) - cos(x + y)cos(x - y) + cos(x + y) cot(x) cot(y) =cos(x - y) + cos(x _ y)cos(x - y) - cos(x + y) tan(x) cot(y) =sin(x+y) + sin(x - y)sin(x + Y) - sin(x - y)complete our list of intrarelationships. 34:6 EXPANSIONSPower series corresponding to the tangent and cotangent functions and their logarithms are 34:6:1tan(s) = z +x3+2.r'+17x'+27(4x')`315315-xtr-,4'(4'-I)IB:i-aITx'3(2j)! 22 Ix.r32xs 12r'34:6:2 cot(s) = - - - - - - - - ... = - - - Z 4(2j)x345945xx =Iif 1(2))!and-ar<x<IT [can(s)]--r7x62x64'(4) - 2)IB_j 34:6:3In-l-ln[xcot(x)] x x 39028352j(2j)!=±rl(21)(1-:n-<x<-irr 12T2where the X. C and Tj numbers are considered in Chapter 3 and the Bernoulli numbers in Chapter 4. 34:7 THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONSThe partial fraction expansions of the tangent and cotangent functions8x8x8x34:6:4 tan(x) =zz +iz ++ 34:6:5tr - 4x'9w- - 4x25tr - 4x- 1_2x_ZrZr cot(x) =x;7774a2 - x29tr2 - x324 can be concisely written 34:6:6 f(x) _ f(x) = cot(x) or -tan(x), x-x,where x, are the values of x that make f(x) infinite: for example, for cot(x): x0 = 0, x, = it, x2 = -IT.x3= 2a,....The continued fraction expansion 34:6:7Xxz x X2 IT37r5ir tanx =----'.,x 4 ±-. ±-. - -, ...1-3-5-7- 222holds for the tangent function. 34:7 PARTICULAR VALUESValues of the tangent and cotangent functions for particular values of the argument x in the range -7r/2 <- x <tr/2 (angles 0 between -90° and 90°) are listed in Table 34.7.1, and particular values outside this range arecalculable via the recursion formulas 34:5:6 and 34:5:7.The values of x that satisfy the equation34:7:1 tan(x)=bx-x<b<xarise in the solutions to certain problems in applied mathematics. These values, the so-called roots of equation34:7:1, depend on It and will be denoted r,(b). They form an infinite set, symmetrically disposed about x = 0. viz 34:7:2x = 0, ±ro(b), ±rr(b), ±r2(b), ..., ±rr(b), ... trtr wherejrr--<r,<jrr+-22except that the roots ±ro(b) exist only if b a 1. The positive members of this set may be found by utilizing thefollowing simple algorithm, which is based on inversion of equation 34:7:1 to 34:7:3 x = Arctan(bx) = jtr + arctan(bx) x = r,(b)where the Arctan and am-tan functions are discussed in Chapter 35. Starting with a crude estimate of r,(b). namelyfir (or A/4 for j = 0), repeated application of 34:7:3 converges to the exact value of the j' positive root. Thealgorithm halts when two successive estimates of r,(b) differ by less than 10-9. Table 34.7.1 60.:13'=10'=221'230'x36'x44234'560'=67":72'x73':9P ttxtasxtx:3ttx:3taltx3n01210t654103a5 12- 0=n-\5I1-7: s-v20si; : =(N,5 1)==c:3tQ+f,2a,rl--0ievs -7207(T42-V-30 r5 325THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS 34:9Input b >>Input j > > r-r,(b)<Setr=firIf r*0go to (1)If bsIgo to (1)Set r = A/4(1) Set q = rReplace r by arctan(br) + jwIfIr - qI> 10-9 go to (1)Output rK««Storage needed: b. j, r acid q Use radian mode or change arctan(br) to(w/ 180) arctan(br). Test values:r9(-1) = 26.7409160r0(2) = L 16556118I The case b = I is especially important and these roots are given by 121314678134:7:4r,") = J -J3J'15J'05!'15!9whereJ = (j± '/=)arWith j taking values from 1 through x, the sums Er; "(1) occur in certain problems and have the values ,a, .4 ands for n = 2, 4 and 6. The values of r,(I) occur in the expansion of the Langevin function [Section 30:13] andcorrespond to the zeros of the spherical Bessel functions of the first kind [see Section 53:7]:Similarly, the roots of the equation34:7:5which we denotecot(x) = bx-x<b<x 34:7:6x = ±pi(b), ±p2(b), ±p3(b), ..., ±pib), ... wheref r - it < p,(b) <jirare determinable via the recursive algorithm Input b >=input j> >D»»Set p = (j - '.!firIf b = 0 go to (2)Replace p by p - rrjbj/2b(1) Set q = pReplace p by p + arctan(I /bp) If lp-gj?10-9 go to (1)(2) Output pP"P,(b)q 34:8 NUMERICAL VALUESStorage needed: b. j, p and q Use radian mode or change arctan(l /bp) to (-n/ 180)arctan(1 /bp). Test values:p5(l) = 12.6452872P,(-t) = 6.12125047 Computer languages and programmable calculators invariably permit the tangent function to be evaluated eitherstraightforwardly or via definition 34:3:3. Accordingly, we present no algorithms here. 34:9 APPROXIMATIONSFor small values of the argument, the tangent and cotangent functions can be approximated byI X 34:9:1 tan(x)8-bit precision-0.6 ts x < 0.6 3 34:10 THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS .and326 1x2l' (6/34:9:2cot(x) =8-bit precision-0.5x0.5 xFrom expansion 34:6:6 it is evident that I /(x - x,) is an approximation to cot(x) or -tan(x) whenever x isclose to one of the arguments, x,, at which the function is infinite. In fact: 34:9:3tan(x) =1- x, = ±a/2, ±3a/2, t5tr/2, ... 8-bit precision-0.1 s x, - x s 0.1 xJ - x 134:9:4cot(x) _x, = 0, ±n. =2Tr, ...8-bit precision x - x, 34:10 OPERATIONS OF THE CALCULUSDifferentiation and indefinite integration give 34:10:1 ddxtan(x) = sec'(x) = I + tan-'(x) 34:10:2 cot(x) _ -csc2(x) _ - I - cot2(x)ddr 134:10:3Jtan(t)dr = ln(sec(x)) In(1 + tan2(x))034:10:4rcot(t)dt = ln(csc(x)) = 2 ln(1 + cot-'(x)) integer powers of the tangent function integrate to give-0.1 sx-x;50.1 - 2 tan'(x) + ln(sec(x))n = 1, 5, 9, ... 34:10:5tan"(t)dt =tan"-'(x)tan"-'(x)+tan"-'(x)+tan(x) - xn = 2, 6, 10, ... --on-ln- 3n- 5 1 tan2+2(x) - In(sec(x))n = 3, 7, 11, .. .-tan(x) +.rit = 4, 8, 12, ...The integral of eot"(t) is similar, but without alternating signs. Section 58:14 discusses the indefinite integrals ofan arbitrary power of the tangent and cotangent functions.Some important definite integrals include 34:10:6(tenntan'(t)dt =Jcot'(r)d = 2sec(2)- I < v < 10 0 73410 v1 t'(twa'(t)=(If:: ancr2 0Z/Jand 34:10:8pqW4ln(Ntan(t))dtln(cot(t))dr = G0 0 327THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS 34:13and involve the secant function [Chapter 33], the beta function [Chapter 3] and Catalan's constant [Section 1:51.The use of formulas 34:5:9-34:5:20, together with the differential identity2dT 34:10:9 dt=T=tan(2) can often aid the integration of expressions involving trigonometric functions by converting the integrand to a purelyalgebraic function. As a simple example (x2/\/l 34:10:10f x[ 1 + (r)]z=- r/n(1 + Tr)di= 2 tanI + 6 tan3l2/ 34:11 COMPLEX ARGUMENTWith argument x + y the tangent and cotangent functions becomesin(2x) + i sinh(2y)34:11:1 tan(x +ry) =cos(2x) + cosh(2y) sin(2x) - i sinh(2y)34:11:2 cot(x + iy) =cosh(2y) - cos(2x)For purely imaginary arguments, these results reduce to34:11:3and34:11:4 34:12 GENERALIZATIONStan(iy) = itanh(y) cot(iy) = -i coth(y) As periodic functions, the tangent and cotangent are special cases of the functions of Chapter 36.Respectively. tan(x) and cot(x) generalize to the Jacobian elliptic functions sc(p:x) and cs(p;x) that are dis-cussed in Chapter 63. When p = 0, we have34:12:1 sc(O;x) = tan(x)34:12:2 cs(O:x) = cot(x) 34:13 COGNATE FUNCTIONSThe tangent and cotangent functions are related to the other circular functions [those addressed in Chapters 32 and33] bya4 sin(x)az1 - Cos (z) a3I34:13:1tan(x) = a3sec (x) - l =_ - I - sin (x)cos(x) csc (.r) - 1cot(x) 34:13:2a,V 1 -sin(x)o`2 cos() 0`3 1 cot(x) ___= a3CSCix) - I = -sin(x) 1 - cos'(x) sec (x) - 1 tan(x)where a2, a3 and a4 take the values + I or -1 according to the magnitude of x. as discussed in Section 32:13.The inverse functions of the tangent and cotangent are among the functions that are the subject of the nextchapter. 34:14 THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS34:14 RELATED TOPICS326 Trigonometric functions and their inverses [Chapter 35] play an indispensable role in-the mensuration of triangles.Historically, it was the need to determine the angles and sides of triangles that led to the invention of the trigo-nometric functions.Figure 34-2 shows a triangle in which angle C is a right angle. Note that angle A is the angle opposite sidea, and similarly for angles B and C. The side c, opposite the right angle. is known as the hypotenuse. There arefive undefined parameters: the sides a, b and c, and the angles A and B. If any two of these five parameters arespecified, then the other three are calculable via Table 34.14.1, provided that at least one of the two known pa- rameters is the length of a side. Chockmarks indicate the given parameters in the table, which also includes acolumn giving the area of the triangle. Of course, the most important relationship applicable to a right-angled triangle is the theorem of Pythagoras, a'+ b- = c2. Sets of integers that satisfy this relationship are known as Pythagorean trios; examples are (3,4;5). (5,12;13), (8,15;17), (7,24;25), (20,21:29). (12,35.37). (9.40:41), (28.45;53),(11.60:61). (16.63;65), (33,56;65), (48,55;73), (13,84;85), (36,77;85), (39,80;89), (65.72;97), (20.99;101).(60,91;109), (15.112;113), (44.117:125). (88.105:137) and (119,120:169). aTable 34.14.1 a V - P b tantA) b cor(B)a b CAB Area b d+b'arctan(a/b)arctan(b/a) a2arcsin(a/c)arcnn(c/a)a V'- --"22 r. a cot(A)a cw(A) 2 - A-2 ca(A)a 42 a tan(g)a sec(B)--B tan(g)22atccos(b/c)aresin(b/c)-2b' Jb sec(A) v2--Ab'2ton(A)n b' b cee(8)__H,-cot(B)A f c sin(A)c coa(A) 2A4sin(2A)c' C ca(B)c ara(B)Z- B 4sin(2B) 329THE TANGENT tan(s) AND COTANGENT cot(s) FUNCTIONS 34:14Table 34.14.2 Known parameters Formula for unknown parametem Area The three sides a, b andc:-c<a-b<cA - 2rtaF-1-(b lb --a<a+barn B = 2b'- V (b+cY'-a' arctan(c+a)-b' 4 C2 arctan=c'-(a-b)'(a+b) -c'Two sides and the angleopposite the longer ofthose two sides. e.g..a, b, A where a a b Two sides and the angleopposite the shorter ofthose two sides. e.g..a, b, B where aab>_ a sin(B) Two sides and the anglebetween them, e.g..a, b. c Two angles and the sideopposite one of them.e.g.. a. A, b Twles and the sidego anbetween them, e.g.,a, B, Cc = V a + b'cos(24) - 2b cos(A) a -sin-(A)[ b sin (A)1B = aresinaJ esin(A) a'-Arrbb'11 C = arccasl - sin'(A) + cos(A)1 --sin'(A)J c =a'cos(2B) + b' ± 2a cos(B)-sin B)ra sin(B)1Aa± arccoalJ26aa' C =atccos{sin' (B) + cos(B)1- ! sin'(B)cas(A)stn ()J ---[ooe(B)+- sin'(B)J sia(C)I A =aruia(Va+9,J- 2abcos(C)-sia(C)(vi'b sin(C)1 B = atcsinIV.'+ b' - 2abcos(C)b - a csc(A) sin(B)c - a(cot(A) sin(B) + cost8)]C=+r - A - Busin'(B)[cot(A) + cot(B)J 6 =a cse(C) Ccag8) + cogC)a cac(B)l 2cot(B) + cot(C) cot(B) + cot(C) Ca - A -B For a triangle that is not necessarily right angled, there are six parameters, and three of these (including atleast one side length) must be known in order for the triangle to be determined. For example, if the side a is ofknown length and the angles B and C are also known [see Figure 34-3J, sides b and c are calculable by the formulasgiven in Table 34.14.2, as is the angle A. Notice that prescribing the magnitudes of a, b and B does not fully delineate the triangle if a exceeds b. The line of length b has two alternative positions, in that case, as illustratedin Figure 34-3 by the full green line and the dashed green line, and consequently the magnitudes of c, A and Ceach have two alternative values. The alternative magnitudes of these quantities are given in Table 34.14.2, as arethe two possible areas. The formulas in this table are based on the cosine law34:14:1 2bc cos(A) = b' + c' - a2thesine law 34:14:2 sin(A)sin(B)sin(C)abc 34:14THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS 330and the area identitiesbe -- -34:14:3 area =2sin(A) = Vs(s - a)(s - b)(s - c)where s is the semiperimeter, s = (a + b + c)/2. Another useful rule in the mensuration of triangles is the tangentrule 34:14:4 tanf(A - B)/2]a - btanl(A + B)/2]a + bThree constructions that are important in trigonometry are shown in Figure 34-4. The altitude AO. that is theline through A perpendicular to BC, has a length34:14:5 AO = b sin(C) = c sin(B)(angle BOA) = a/2A line through a vertex of the triangle that bisects the angle there is known as a bisector: the bisector of angle Ain Figure 34-4 has a length 34:14:6 A AN =2bcb s(Ac /2)(angle BAN) = (angle NAC) =2Line AM. which bisects the line BC, is termed a median and has the lengtht a_ 34:14:7AM =2b`--c'+2bccos(A)BM=MC=2 CHAPTER35THE INVERSE TRIGONOMETRIC FUNCTIONS The fundamental interrelationships between the six subjects of this chapter are 35:0:1 arctan(x) = arcsinmix.,) = arccsc and 35:0:2 arctan(x) + arccot(x) = arcsin(x) + arccos(x) = arccsc(x) + arcsec(x) = 2 but some of the many alternative links are presented below: JX\xz+ t\35:0:3arctan(x) = aresinlI =arccsc arccotQ 1=arccos(I)= aresec(x2 -+I)z > 0 I arccot1-- n = -arccos= -arcsec(r'' + 1)x < 0 35:0:4arcsin(x) = arctan x\(1)sgn(x)I = atccsc=arccos(l - 2x')1-x/x21 -z' I arccotl1 = arccos(I - x') = aresecl0 < x < Il -x' 1 arccot(I - a = -arccos(1 - x') = -aresecl I- l:5x s 0 331 35:1 THE INVERSE TRIGONOMETRIC FUNCTIONS35:0:5arccsc(x) =(sn(x))=(I)>x aresec fx2 - 1Ix > 1 x arccot(x= - 1) = arccos\ arccot(x2 - 1) - a = arccoslsx1I - a = arcsec 1zx VT:1z- I332 - itx < -1/x\/ I + xx35:0:6arccot(x) = arccosl )= aresecl\I+xz\X arstanlJ= aresinl 1I = arccsc(1 x'I+)z > 0 Q\1+xzr + arctanl ') = n - aresinl 11I = n - arccsc( 1 x + x=)x < 0 35:0:7arccos(x) = aroootl_X/777I =arcsec( !)= 2 arcsinl l 2 I-1 s x s 1 arctanlxI = aresin(I - x2) =arcesc(1 __X2J0 < x < I __X t 1 e + arctan= a - aresin(I - x2) = IT - arccsc )-1 < x < 0-x 35:0:8aresec(x) = arccotlsi(x))= arccoslxIW a I\21Q arctan(xZ - 1) = arcsinlx) =arccsclx' - 1Ix > Ix - 1x n - atctan) = rr + aresin 1 = a + arccscx x'x < -1 Many more interrelationships may be constructed by combining equations 35:0:3-35:0:8 with those in 35:0:2. Forexample:/`/x'35:0:9arctan(x) = 2 - arccoslx+I =- 2- areseclxx /r\I/l\rl = sgn(x)I 2 - aresinl 11_+X1)sgn(x)I 2 - arccsc(x' + 1)J follows from combining 35:0:2 with Ltheright-hand members of equation 35:0:3.35:1 NOTATIONAn alternative collective title for the functions of this chapter-the inverse tangent, the inverse sine, the inversecosecant, the inverse cotangent, the inverse cosine, and the inverse secant-is the inverse circular functions. The 333 THE INVERSE TRIGONOMETRIC FUNCTIONS 35:2symbolism tan-'(x), sin-'(x), etc., often replaces arctan(x), arcsin(x), etc. Variants such as arctg(x). arcctg(x) andarccosec(x) are occasionally encountered. The origin of the "arc" prefix is made evident in Section 35:3.The notation arccot(x) is sometimes used to denote a function defined, for negative x, somewhat differentlythan here, being equal to our arccot(x) - n. Similar discrepancies may be encountered for other inverse trigono-metric functions.The multivalued functions Arctan(x), Arcsin(x), etc., are discussed in Section 35:12, but, confusingly, theseare often also denoted arctan(x), aresin(x), etc.We use arctrig(x) to represent any one of the six inverse trigonometric functions. OhOggOgOto ieii........................................... 3.0arecot(x) FIG 35-1 aretan (x)...................... .1.0wcoecarccsc(x)......................... ........ ......0.5arc=t(x) arccec (x) - -.................. ..................... -0.5 arcein(x)-.. :.......:....:....: ..............-1.0 ...... ................................... Figure 35-1 depicts the behavior of the six functions. Note that both the inverse cosecant and inverse secant func-tions have two branches. The domains and ranges of all the functions are summarized in Table 35.2.1. 35:3 THE INVERSE TRIGONOMETRIC FUNCTIONS 334 Table 35.2.1 f(x)Domain of f(x) Range of f(x) arctan(x)--<X<--A12 < arctan(x) < n12aresm(x)-1 s X !CI-a/2 s aresm(x) s ar/2 arccsc(x){< x 5- I-'r/2 5 arccsc(x) s 0 l 1X<z0<arccsc(x)5r/2artaa(x)-z < X <0 < arvcottx) < 9rarccas(X)-I s x s l0 5 arccos(x) 5 'r arcseclX)rz < x 5 -1a/2<aresec(x) = al I± x<0aresec(x) <a/2} 35:3 DEFINITIONSIndefinite integrals define each of the six inverse trigonometric functionsdt 35:3:1 arctan(x) ="1 + t1dt35:3:2 aresin(x) _-1x 5 101 -t=dt=tt-- 135:3:3 arccsc(x) = dt1sx<xtt'1dt35:3:4 arccot(x) =1 + 35:3:5 arccos(x) _-1 sxE 1 35:3:6 atrsec(x) =dtx a 1ttt2 - 1Diagrams illustrating these definitions, for x > 0, are included as Figures 35-2, 35-3 and 35-4.Geometric definitions of the inverse trigonometric functions may be illustrated by reference to Figure 35-5 forarctan(x). A line of length x is made part of a right-angled triangle, the side shown dashed being of unity length.Thus, x is the tangent of the marked angle. The arc of a unity-radius circle subtended by this marked angle, shownin black on the diagram, is then defined as the inverse tangent of x. Alternatively, the angle itself may be identifiedwith arctan(x).The names "inverse trigonometric functions" imply that these functions are the inverses of the functions ofChapters 32, 33 and 34. This is true, however, only if appropriate restrictions, namely35:3:7f = arctan(x)wherex = tan(f )ar 35:3:8f= arcsin(x)wherex= sin(f)- Z s f s 235:3:9f = arccsc(x)wherex = csc(f ) 335 THE INVERSE TRIGONOMETRIC FUNCTIONS 35:3:10f = arccot(x)wherex = cot(f)35:3:11f= arccos(x)when:x = cos(f)0 5 f s a35:3:12f =arcsec(x)wherex= sec(f )are placed upon the magnitudes of the arguments of the trigonometric functions. .................... 35:4 THE INVERSE TRIGONOMETRIC FUNCTIONS 33635:4 SPECIAL CASESThere are none. 35:5 INTRARELATI ONSHIPSThe inverse tangent, sine and cosecant are odd functions:35:5:1f(-x) _ -f(x)f = arctan. arcsin or arccecwhereas the other three inverse trigonometric functions obey the reflection formula35:5:2f(-x) = v - f(x)f = arccot, arccos or arcsecThe inverse tangent and cotangent functions obey the simple reciprocation formulasg 35:5:3 arctanzI=-arctan(x) +s(x)x * 02 35:5:4arccot(s)_ -arccot(x) + .. -w sgn(x)2x * 0For the other inverse trigonometric functions, formulas for arctrig( I /x) will be found among the equations in Section35:0. Other formulas relating two values of a single inverse trigonometric function include 1 2-1 1 35:5:5 aresin(x) = 2 arcsin(2xI - x)72s x :72and/l +x\35:5:6 arocos(x) = 2 arccosl 2 I - 1 s x S IIf f is an inverse trigonometric function, then f(x) ± f(y) may lie within the range of f, or above it, or belowit. These three possibilities are incorporated into the following function-addition formulas by allowing the integerk to adopt one of three values: 35:5:7 35:5:8 35:5:9arctan(x) ± arctan(y) = krr + arctanl x Y I1x y- a r e s i n ( x ) ± arcsin(y) = kit + aresin(x V " t - y- ± y I - x)k = 0, ±1 arccsc(x) ± arccsc(y) = ka + arccsc x9(y--r±xt-IThe appropriate k value may be found from the formula/ 35:5:10k = Int12 +f(x) ± f(y)1Jf = arctan, arcsin or arccscthat utilizes the integer-value function (Chapter 9). Similarly:11x±y35:5:11arccot(x) ± arccot(y) =2- k W + arccotG_k)u+ 35:5:12cox) ± arccoy) = arccssxy +1 - x'1 - y)k = 0, ±1 XY 35:5:13aresec(x) w atcsec(y)I- k)w+ arcsec2(1,;:x'-1 yz-1 337 THE INVERSE TRIGONOMETRIC FUNCTIONSwherei-35:5:14 k =IntI f(x)f(Y)1Jf = arecot, arccos or arccscaApplication of the formulas in Section 35:0 enables many more intrarelationships to be constructed. 35:6 EXPANSIONS35:8 There are three basic power series expansions for inverse trigonometric functions: 35:6:1arctan(x) = x - 3+5-7 +.- x2--)-1< x<1-oJ1 35:6:2 arctan(x) _sgn(x)-1+1-1+ ..=rrsgn(x)-I(---'> l2x3i5x52x,so 2j+ I1 x3I X 3 x°I x 3 x 5.e(2j - l)!! x" 35:6:3aresin(x)=x+--++232x45 2x4x67 (2J)!!2j+Ibut the multifarious interrelationships between the six functions allow easy adaptation of these series to other inversetrigonometric functions. For example, in tight of equation 35:0:7, replacement of x on the right side of 35:6:3 by(I --x)/2 provides an expansion for 2 arccos(x).Continued fraction expansions of the inverse trigonometric functions includexjr' 4x' 9xr 16x'35:6:4 arctan(x) = - - - - - .--1+ 3+5+ 7+ 9+and, for -1 < x < 1 35:6:5 arcsin(x)_x V ` - x= (I x 2)x (1 x 2)x (3 x 4)x (3 X 4)x (5 x 6)x1-3-5-7-9-II- 35:7 PARTICULAR VALUESThe entry "undef" in Table 35.7.1 indicates that the function is undefined at the argument in question. 35:8 NUMERICAL VALUESCalculators usually incorporate keys for evaluating the arctan, arcsin and arccos functions, but computer languagesrarely incorporate any inverse trigonometric function other than arctan. Accordingly, we present an algorithm that Table 35.7.1 amtan(x)aresin(.)amcsc(x)arccot(x)arwos(x)aresec(x)x=-xx=-V2-IV2x=01x=-V2x=1x=V2x=s -1/2-0.955...-n/4-0.615... 00.615._.n/40.955...n/2undefundef-n/2-n/40n/4n/2undcfundef0-n/4-n/2undefundefundefn/2n/40 it2.526...3n/42.186...n/20.955...n/40.615...0undefundefn3n/4n/2n/40undefundefn/23n/4nundefundefundef0n/4n/2 35:9 THE INVERSE TRIGONOMETRIC FUNCTIONS 338computes any one of the six inverse trigonometric functions by utilizing a built-in arctan function. The algorithm,which is exact, uses relationships 35:0:1 and 35:0:2.-ts-well, the universal hypergeometric algorithm (Section 18:141 may be used to calculate inverse trigonometricfunctions.Input x »4»»Setf = IInput code c »» (1) (2)(3) 1(4)fo = arctan(x) f, = arccot(x)If c <-1 go to (2)Setf= 1 -x: Iff*0goto(1) Setf = irx/2 Go to (3)Ifcs3goto(2)Replace f by -x2fReplace f by arctan(x/\If )if frac(c/2) = 0 go to (4)Replace f by (ir/2) - fOutput f f: = arcsin(x)A = arccos(x)f. = arccsc(x) A = aresec(x) 35:9 APPROXIMATIONSStorage needed:x, c (the code) and fUse radian modeInput restrictions: Argument must lie in the range:1 codefunctionargument0arctan anyIarccot any2arcsin-1 is x{ I3arccos-1<<x<I4arccsc1rI > I5aresec4xi ? 1Test values:arccot(ar) = 0.30816907arcsin(-0.7) _ -0.77439750arccsc(4) = 0.25268026 arcsec(-1) = 3.14159265 Among the approximations to inverse trigonometric functions are 35:9:1 35:9:2 35:9:33xarctan(x) =+ x28-bit precision3-0.45<x50.45 3xarccot(x) =3X2 + 18-bit precisionx Z 1.8Fx3in(x) = x8-bit precision- 0.5 5 xs 0.5 ares 35:10 OPERATIONS OF THE CALCULUSDerivatives of the six inverse trigonometric functions are 35:10:1 35:10:2dd- arctan(bx + c)arrrnt(bx + c) _bdxdx1+(bx+c)2ddb arcsin(bx + c)arocos(bx + c)dx_1 - (bx + c)2dd-b35:10:3arccsc(bx + c)aresec(bx + c)dsdx_Ibx+ci(bx+cY- I 339 THE INVERSE TRIGONOMETRIC FUNCTIONSwhile the corresponding indefinite integrals are 35:10:4Jtarctan(bt + c)dtx + b ) arctan(bx + c) -!In v,(bx + c)235:10 35:10:5J `arccot(bt + c)dt = (x +bI arccot(bx + c) + b In VI + (bx + c)ZcI1c\ 35:10:6Jcaresin(bt+c)dt=Ix+-)aresin(bx+c)--+ b-(X+bI-c-lsbxsI-cr\//i 35 :10:7Jarccos(bt + c)dt =(x +barccos(bx + c) +-tiZ- t x +b c I-c - 1 s bx < 1 - c/b b 35:10:8Jfjarccoc(br + c)dt = (x + b) arccsc(bx + c) + b arccos(bx + c) bx > I - c\35:10:9Jjarzsec(bt + c)dt = I x +61 aresec(bx + c) - - arccos(bx + c) bur > 1 - c0 -c)/b bIndefinite integrals of the form ftarctrig(bt + c)dt includer::I+z-arctaa(bx + c) -bc +l01 + (bx + c)22b,t /b35:10:10J tt arctan(bt + Odt = Jtarccot(b:+c)dJ=1 +62x2-C22bx+2c-irc 235:10:11262arccot(bx + c) +462- bileI + (bx + c)BY _2c'c 35:10:12t arcsin(bt + c)dt =2b xt4b- 1 arcsin(bx + c) +bt r/bbx - 3c+462I - (bx + c)2-c - I s bx s I - cr22_xarccos(bx + c) +- 35:10:13Ict arccos(bt + c)dt =x4b-I 111/bbx - 3c-462I-(bx+c)2-c-Isbx51-c but none of ft-'arctrig(bt + c)dt has been evaluated as a finite number of terms. See Spiegel [pages 82-841 for along list of indefinite integrals of the form ft="arctrig(t)dt for integer n. Gradshteyn and Ryzhik [Section 2.81 listsimilar integrals and include additional entries such as farctrig"(t)dt and f(bt + c)-'arctrig(t)dt.Gradshteyn and Ryzhik [Section 4.5] also present over 100 definite integrals involving inverse trigonometricfunctions; 35:10:14 and 35:10:15Jacctan(t)dt -f arccot(t)dt = G r' aresin(t)a Irdt = 2 ln(2)0are typical examples. G is Catalan's constant (see Section 1:7).The inverse trigonometric functions play a role in the fractional calculus [see Oldham and Spanierl. For ex-ample: 35:11 THE INVERSE TRIGONOMETRIC FUNCTIONS 340 d1/2 1aretan(1x)351016::dx'nV V xatrcot(V x) 1 + x) dr 'n 35:10:18 'd'R aresin(V x-)V;dx'/22(1 - x) 35:11 COMPLEX ARGUMENTA variety of ways may be used to extend the inverse trigonometric functions to complex argument. One mightcouple the definitions 35:3:1-35:3:6 with integration in the complex plane or, alternatively, permit power seriessuch as 35:6:1 and 35:6:3 to accept a complex argument. Another route makes use of the relationships 35:13:1-35:13:6, which are valid for complex argument as well. The results, however one proceeds, are rather formidable looking. The most useful of the resulting multivalued functions are35:11:1Arcsin(x + iy) = kir + (-1)4arcsin(Y) + (-1)ri In[X + X' --I]35:11:2Arccos(x + iy) = 2k7r ± {arccos(Y) - i In[X + XZ - ]}235:11:3Arctan(x + ry) = kir +1arctan2x+ t Inx2 + (y + 1)2x2 + (V - 1)2 * 0 21-x2-y-4x2+(y-1)where k is any integer and 35:11:4X=1(x+l)2+y2+(x-1)2+yry=! (x+1)2+y^-(x-1)Z+y22222The logarithmic function In occurring in the above equation is the single-valued logarithm defined in Chapter 25. 35:12 GENERALIZATIONSBecause the trigonometric functions we periodic, their unrestricted inverses have infinitely many values for eachacceptable argument. These multivalued inverse functions are distinguished from the single-valued functions treatedelsewhere in this chapter by having their initial letter capitalized. The definitions are encompassed by35:12:1 Arctrig(x) = ywhentrig(y) = xWith k = 0, ±1, ±2, ..., the relationships35:12:2Aretrig(x) = arctrig(x) + 2kir arctrig = aresin, arccos, aresec, arcsec35:12:3Arctrig(x) = arctrig(x) + for arctrig = arctan, arccothold.The inverse tangent and inverse sine functions are each special cases of the Gauss F function [see Chapter 60): 35:12:4 arctan(x) =xF(2,1; 2;-x2I-1 < x < 1 35:12:5 arcsin(x) = F1 2, 2; 2;x2)-1 <x < 1 Also, the inverse sine and cosine are special cases`of the incomplete beta function of Chapter 58: 341 35:12:6 35:12:7THE INVERSE TRIGONOMETRIC FUNCTIONSraresin(x)= 1 2Bl2;12;x' !11arccos(x) =2B 2, 2;1 - xr 35:13 COGNATE FUNCTIONSThe inverse trigonometric functions are closely related to the inverseChapter 30. One has 35:13:1 arctan(x) _ -i artanh(ix)35:13:2 arccot(x) = i arcoth(ix)35:13:3 arcsin(x)-i arsinh(ix)35:13:4 arccos(x)±i arcosh(x)35:13:5 arccsc(x) = i arcsch(ix)35:13:6 aresec(x) _ ±i arsech(x)35:13 hyperbolic functions that are considered in CHAPTER36PERIODIC FUNCTIONS Periodic functions play an important role in the solutions of many difficult problems in applied mathematics. Also,periodic functions, or nearly periodic functions, form the medium by which a good many information transfers take place. For example, beams of light, sound waves, and a variety of telecommunications signals are instancesof periodic functions, often "modulated" in some way. 36:1 NOTATIONWe shall use per(x) to represent any periodic function, and occasionally qer(x) to represent a second periodicfunction. Throughout this chapter P will denote the period of per(.r). The quantity 27r/P is sometimes known asthe frequency of the periodic function and is often denoted by w. 36:2 BEHAVIORApart from their characteristic of indefinitely repeating, periodic functions share no common behavior. Periodicfunctions may be continuous or discontinuous, simple or complicated. Examples of periodic functions are shownin Figures 36-1 through 36-7, exhibited later in this chapter. 36:3 DEFINITIONSA function of argument x that satisfies the condition36:3:1 f(x) = f(x + kP)k=0 .±1 .± 2 ,for all x is a periodic function of period equal to the smallest positive value of P that satisfies equality 36:3:1.A periodic function may be 'naturally" periodic, as are each of the functions cited in Section 36:4, or it maybe "synthesized" from an aperiodic function. An example of a "synthetic" periodic function, in this case createdf r o m the square function [Chapter I I ], is36:3:2per(x)=(x-2k)22ksx<2+2kk=0, ±1,±2,. 343 36:4 PERIODIC FUNCTIONS 344 ................1.........r..4 This particular periodic function, which is mapped in Figure 36-1, has a period of 2. As here, synthetic periodicfunctions often exhibit a discontinuity at one point, at least, in the 0 rt x < P range. 36:4 SPECIAL CASESA number of periodic functions are considered in previous chapters. The fractional value function frac(x) is treatedin Chapter 9, as is f(frac(x)); these have periods of unity. The functions sin(x), cos(x), csc(x) and sec(x), discussedin Chapters 32 and 33, have periods of 2w. In Chapter 34 the tan(x) and cot(x) functions, which are periodicfunctions of period a, are considered. The theta functions [Section 27:131 are periodic, as are the functions ofChapter 63,The constant function [Chapter 1] is a degenerate case of a periodic function for which no smallest nonzerovalue of P exists. 36:5 INTRARELATIONSHIPSThe fundamental intrarelationship of periodic functions is their periodicity, which constitutes a recurrence formula36:5:1 per(x + P) = per(x)If two periodic functions per(x) and qer(x), with periods P and Q, are added, subtracted, multiplied or divided,the resultant function will be periodic only if the quotient P/Q is rational. If P/Q is rational, the period of per(x)± qer(x), per(x)ger(x) and per(x)/qer(x) will normally be the least common multiple of P and Q. There are, how-ever, frequent exceptions to this rule; for example, sin(x) and cos(x), each of period 2w, have products and quotients of period it, not 2w as predicted by the rule.If the argument x of a periodic function per(x) is replaced by a linear function of x, to give per(bx + c),periodicity is maintained but the period is altered from P to P/b. Replacement of the argument by any otheraperiodic function of x destroys the periodicity; the functions per(ax2 + bx + c) and per(] /x), for example. are not periodic. Multiplication or division by, or addition or subtraction of, any aperiodic function other than a constant likewise destroys the periodicity.On the other hand, composites of periodic functions, such as a per(bx) + b per(x) + c, exp(per(x)) or 1/[c +per(x)] are themselves periodic. Such functions have the same period P as per(x) or, occasionally, a submultipleof it such as P/2. 345 PERIODIC FUNCTIONS 36:636:6 EXPANSIONSIn a process known as harmonic analysis, any periodic function per(x) of period P may be represented, exactly orapproximately, as the Fourier series(2jlrx)_ per()(2jirx)+ s s 36:6:1F(x) _ - +Ccosinxi-The c and s coefficients in this series, known as Fourier coefficients, are given by the so-called Euler formulas 36:6:2c =?fkos(2dtPJPer(P)j = 0. 1.2... . 2-ryarper(r)sin2 pj = 1, 2, 3, .. . 36:6:3 si = where xo is arbitrary. The Fourier coefficients satisfy Parseval's relation2°.Jpee(r)dt 36:6:4 2 +c; + s; =Po provided that per(x) is everywhere finite. A brief tabulation of Fourier coefficients for a variety of periodic functionsis included in Section 36:14.The Fourier series 36:6:1 may be written in the alternative formco 36:6:5F(x) = z + (2J1rxP+ arctan(si/ci) I or. in terms of exponential functions of imaginary argument, asl 36:6:6 F(x)ai exp/O 36:6:7v2ijarlc111-sgn(j)is!iqPIper(t) exp1/2 oWe shall not discuss the convergence of Fourier series in detail [see Hamming, Chapter 321. Suffice it to statethat the Fourier series of a finite continuous periodic function converges to the function, whereas the Fourier seriesof a finite discontinuous periodic function converges to the function except at the points of discontinuity. If the periodic function per(x) has a discontinuity at x = d (as well as at x = d t P, x = d ± 2P, etc., of course), suchthat36:6:8andlim per(d - e) = per(d-)a-oe > 0 36:6:9 lim per(d + e) = per(d+)oe > 0then the Fourier series converges to [per(d-)+per(d+)[/2 at x = d. For example, the periodic function defined in36:3:2 and mapped in Figure 36-1 has the Fourier series44436:6:10 - + - sin(jwx)3j'tr'.PRThis evaluates, for x = 0, to s + 4;(2)/7rr = 2, which is the mean of the values, 4 and 0. on either side of thediscontinuity at per(0). 36:7 PERIODIC FUNCTIONS 34636:7 PARTICULAR VALUESThe particular values of per(x) depend on the identity of the periodic function. In terms of the Fourier coefficients,however, it is evident that 36:7:1 per(0) = per(P) =2+ ci + c, + cy + c4 + c. + ce + - 36:7:2 Pper4c o=2+si-c.-sy+c.+ss-ce-... 36:7:3 Pper 2co=2-cl+c_-ey+c.-cy+c6-... 36:7:4per(3P)=co-si-cr+sy+c.-sy-ce+...42 36:8 NUMERICAL VALUESThe calculation of values of per(x) generally presents no special problems. 36:9 APPROXIMATIONSA truncated Fourier seriesco'2jrrxl(2jirY\36:9:1Fi(x) = Z +c, costP /+s, sin(P1 = per(x) produces an approximation to per(x) that is best in the least-squares sense [see Section 7:141. 36:10 OPERATIONS OF THE CALCULUSThe derivative of a periodic function per(x) is another periodic function of identical period. Discontinuities in per(x)will generate Dirac delta functions [Chapter 10] in d per(x)/dx.The co/2 Fourier coefficient represents the average value of per(x) over the period, that is: Ico 36:10:1 Pfoper(t)dt =Zx arbitraryOnly if this coefficient is zero will indefinite integration of per(x) give rise to another periodic function.After subtraction of co/2, differintegration with lower limit minus infinity converts per(x) into another periodicfunction of identical period. In terms of its Fourier series, the new function is 36:10:2d"co]2jrrl/2vu 1[d(x + -)]'[per(x)2 J(P/[c, cos(P+ 2l+ s sinlp+ 2)2!)2jirx2jax`- If(c,C +s,S)cos(I + j"(s,C -c;S)sin( =(2!)'P /,.,P /where C = cos(wr/2) and S = sin(va/2). This formulation is valid for v > - l . 347 PERIODIC FUNCTIONS36:11 COMPLEX ARGUMENT36:14 For constant y, the periodic function per(x + iy) remains periodic in x. The Fourier coefficients of such functionsare then complex numbers that can be evaluated with the help of equations 32:11:1 and 32:11:2.Certain functions, such as In and sinh, which are aperiodic for real argument, become periodic when theargument is imaginary or complex.The Jacobian elliptic functions [Chapter 63] are doubly periodic when their argument is complex. That is, theysatisfy the recurrence relations36:11:1per(x + iy) - per(x + P + iy) = per(x + iv + IQ) = per(x + P + iy + iQ)where P and Q are the real and imaginary periods. 36:12 GENERALIZATIONSApart from the double periodicity cited in Section 36:11. no generalization of the concept of a periodic functionhas been made. 36:13 COGNATE FUNCTIONSThe terms full-rectification and half-rectification, which have their origin in the technology of alternating electricalcurrents, are sometimes encountered in connection with periodic functions. Using the notation of the absolute-value function [see Chapter 8), full- and half-rectification converts the function per(x) into36:13:1and 36:13:2lper(x)Ifull-rectification 1Z {Iper(x)I + per(x)}half-rectificationFigure 36.2 shows an example of each. Full- or half-rectification of a periodic function maintains its periodicityand, generally, the period remains unchanged, Sometimes, as in the full-rectification example shown in Figure 36-2, rectification decreases the period (increases the frequency) by a factor of 2 (or more). sin(2nx/P) 36:14 RELATED TOPICSFIG 36-27half rectification Table 36.14.1 lists some frequently encountered periodic functions, together with their Fourier coefficients (seeequations 36:6:1-36:6:31. For further examples of Fourier coefficients, see equations 32:5:29, 32:5:30, 20:6:4-20:6:6 and 19:6:5-19:6;7. 36:14 PERIODIC FUNCTIONS 348Table 36.14.1 C StName of(unctionFigurepert.)(O uare waveS 4 q(odd)3631-1)b"iSie0JnwaveS /4x 4 P quare(even)36-4(-I)' n = Intl2P0iw0 Sawtoot6 Ix _Iwave36-5frac 2 P 01rI _4 Triangularwave36.6t - , )rs'' 2 frac(2+ PP_l]2 ch:r-kj4crlrrh0 0 Pulse train367-puha--- JP) Pt -- sin -hP- sinll-IsP.0 0 Full-rectified / xx--4sine waveHalf-rectified36-2sinl` pIxsr.!!2arl10 0(j: _ I M0ut2-J)0 0sine wave zizin l36.22P) +2ng/Pr :....;....:FIG 36-3:....:..p :....:....:FIG 36-4;......p ;Q{s,44Q4+Q4;... 0. 5FIG365;......0 349 PERIODIC FUNCTIONS 36:14 y4Q;4tiA? ti44,Q,y4p+4ey440+4q? :FIG 36-7 : CHAPTER37THE EXPONENTIAL INTEGRAL Ei(x)AND RELATED FUNCTIONS Indefinite integrals of the form ft-'exp(±t)dt cannot be expressed in terms of elementary functions for n = I. 2,3..... The exponential integral function Ei(x) fills this deficiency. A closely related function, the entire exponentialintegral, is also discussed in this chapter, it is related to Ei(x) by an identity37:0:1Ein(x) = y + ln(kl) - Ei(-x) y = 0.5772156649involving Euler's constant y [Section 1:71 and the logarithm of the absolute value of the argument x.The logarithmic integral 11(x) is discussed in Chapter 25 but, because of its simple relationship37:0:2 li(x) = Ei(ln(x)) to the exponential integral, many of its properties are given in this chapter. 37:1 NOTATIONNotations abound for the exponential integral function: Ei*(x), Es(x), Ei(x), i(x), E`(x) and E_(x) have all beenused to denote Ei(x) or some very similar function. When one of these symbols is encountered, care is needed toascertain whether its definition differs from that which is employed for Ei(x) in this Atlas.When x is negative, the expression -E,(-x) is often used to replace Ei(x) in view of relationship 37:13:7. TheE,(x) symbol here represents a Schl6milch function [see Section 37:13] and should not be confused with the identicalsymbol used for Euler polynomials [Chapter 20]. To add to the confusion, the notation ei(x) is sometimes en-countered for the Schli milch E,(x) function. 37:2 BEHAVIORFigure 37-1 maps the behaviors of the exponential integral Ei, entire exponential integral Ein and logarithmicintegral li functions. All three functions increase without limit as x - x but, following an inflection [Section 0:7]at x = 1, the increase is dramatic for the exponential integral function (e.g., Ei(10) - 2 x 10' and Ei(100) - 3X 10"). Note also that Ei(x) rapidly approaches zero as x -. - x and exhibits a discontinuity at x = 0. Observealso that li(x) is discontinuous at x = 1 and is defined only for x ? 0. 351 FIG 37-1 Black37:3THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS 352 37:3 DEFINITIONSThe exponential integral function is defined by the indefinite integral 37:3:11exp(t)Ei(x) = JadtIfor all x, as illustrated in Figure 37-2. At zero argument the integrand in 37:3:1 encounters an infinity so that forx > 0 the integral is to be interpreted as the Cauchv limitl37:3:2limSEi(-e)+ezp(r)jrdt}x>0a>0The exponential integral may also be eexpressed as a definite integral in a number of ways, including 37:3:3 -IdtEi(±x) = exp(±x)Jo ln(t) ± xx>0Gradshteyn and Ryzhik (Section 8.212) list many other integral representations of Ei(x).The definition of the entire exponential integral1 -37:3:4 Em()exp(-t)dtot 353THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS 37:4 is illustrated in Figure 37-3. Its relationship, equation 37:0:1, to the exponential integral Ei permits several alter-native definitions of the Ein function.Definitions of the logarithmic integral function were presented in Section 25:13. Some authorities regard thelogarithmic integral as defined only for arguments exceeding unity, but in this Atlas li(x) exists for 0 s x < 1 witha Cauchy limit interpretation, similar to 37:3:2. serving to extend the definition to .t > 1. 37:4 SPECIAL CASESThere are none. 1A 37:5THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS37:5 INTRARELATIONSHIPSRelating the exponential integral and its entire analog is the identity37:5:1 Ei(x) + Ein(-x) = Ei(-x) + Ein(x)that follows from equation 37:0:1. or through definitions 37:3:1 and 37:3:4.354 We know of no reflection, recurrence, addition or multiplication formulas for the exponential integral functionsthat involve a finite number of elementary functions. However the reflection formulas37:5:2and37:5:3Ei(-x) = Ei(x) - 2 Shi(x) Ein(-x) = Ein(x) - 2 Shi(x) = -Ein(x) - 2 Chin(x)may be written in terms of the functions of Chapter 38. Also, the argument-addition formulaj! [exp(y)e/-y)-1 ] 37:5:4Ei(x + y) = Ei(x) + exp(x) X ',I1x1 > Lvi-0for the exponential integral exists and utilizes the exponential polynomial [see Section 26:13]. 37:6 EXPANSIONSTwo alternative power series: 37:6:1 andxrx3X(-X)'Ein(x)=x-+18 -+964i-j!j 3x2Ilx'25x'II1x'37:6:2exp43628823j/ j!['y+yU+1)]X!are available to express the entire exponential integral function. In 37:6:2, 0 is the digamma function described inChapter 44.The exponential integral function is expansible in terms of Lagucrre polynomials [Chapter 23J:(-x) 37:6:3 Ei(x) = -exp(x)L; (-x)< 0i-a j+1as the continued fraction:exp(x)1122337:6:4 Ei(x) _- - - - -X-1- x- 1- x- I-or as the asymptotic series 37:6:5Ei(x) - exp(x) II+I+?+6++j!+Ix - xIXx=x3xx'"Jvalid for large argument x. For small arguments, combination of expansion 37:6:1 with relation 37:0:1 leads to X-2x3 37:6:6 418which describes the behavior of the exponential integral close to x = 0.Expansions for li(x) arise by replacing x in 37:6:3-37:6:6 by ln(x). 355THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONSTable 37.7.1 Ei(s)Ein(x)(((x)s=-x x = 0 x = 1x=x 0-=1.895117816 =00.7965995993undef0-sx 37:7 PARTICULAR VALUESAs well as the values given in Table 37.7.1, note that Ei(0.3725074) = li(1.4513692) = 0. 37:8 NUMERICAL VALUES37:8 The algorithm below is designed to generate values of Ei(x). for x values of any magnitude and either sign, with24-bit precision. However, less precise values may be encountered in the immediate vicinity of the zero of theEi(x) function, that is, between x = 0.370 and 0.375. For -2 < x < 22, the algorithm uses series 37:6:1 in thetruncated and concatenated form + 1-x+(-x) 37:8:1Ein(-x)J.r+ 1(J - I)x+ 1(J - 2)x++ 12r =-1 - (J + 1)'J2)(J - 1)=941 Input x >> f=JEi(x)1li(x) JSet f = -10wD»»Replace x by 1n(x)If x- 101 2:12 go to (2)Ifx=0goto(4)Set j = Int(IO + 2Gx1)Set fI/(j+ 1)2(I) Replace f by (fjx + I)/j1Replace j by j - IIfj*0goto(1)Replace f by fx + In( I.78107241 Slxl )Go to (4)(2) Set j = Int(5 + 20/Ixi)Setf = x(3) Replace f by [ 1 /(1 If - I /J)] + xReplace j by j - IIfj+0goto(3)Replace f by [exp(x)]/f(4) Output fStorage needed: x, j and f Input restrictions: For the calculation ofmust be positive. Test values:Ei(25) = 3.00595092 x 109Ei(-1.5) = -0.100019585li(7) = 4.75705176 with J assigned the empirical value int(10 + 21x1), and then employs interrelation 37:0:1 to generate Ei(x). For xs -2 or x >_ 22, the algorithm employs the continued fraction 37:6:4 terminated at (J - 2)/((1 - (J - 1)/(x -J/(1 - J/x))) where J = Int(5 + The algorithm will generate li(x) when the additional instruction shownin green is included. Because Ei(0) = Ii(l) = -x. the algorithm returns -1099 at these arguments.The universal hypcrgcomctric algorithm [Section 18:141 also permits Ei(x), Ein(x) and li(x) to be determinedfor suitable values of the argument x. 37:9THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS 35637:9 APPROXIMATIONBased on the identity 37:0:1 and expansion 37:6:1, the approximation 36x + x= 1.05 s x < 037:9:1Ei(x) ^+ ln(1.78[xl)8-bit precision0 < x s 0.36 x36 - 80.38 5 x s 1.25is valid for most small values of x, while the approximationexp(x) X:5-9 37:9:2Ei(x) =8-bit precisionX? 15based on expansion 37:6:5, is useful for arguments of large magnitude. The very simple approximation to the entireexponential integral 37:9:336x - xrEin(x) =8-bit precision-1.1 :5x:5 1.4 36 + 8xmay be used near the origin. 37:10 OPERATIONS OF THE CALCULUSThe following rules apply for differentiation and indefinite integration:dexp(hx + c) 37:10:1 d Ei(bx + c) =x + c/b() 37:10:2 37:10:3 37:10:4 37:10:5 37:10:6 37:10:7Jexp(Br + C) Ei(br + c)dt =/bbIexp(Bx+b)Ei(bx+c)-Ei1 (B+b)Ix+b)B+bb B+b*0ex C+c/c)l\l 1l)lnlx++y-exp(-bx-c)Ei(bx+c)b J \\\/ /B+b=0When n = -1, the indefinite integral fI"Ei(bl + c)dl cannot be evaluated in a finite number of terms, but for n= 0, 1, 2, 3, ... we haved- Ein(bxdxc)I - exp(-bx - c)x + (c/b)dx"-'-li(x")=-vf0dxIn(x)Ei(br + Odr = (X +b1 Ei(bx + c) +I - cxpbbx + c) Ein(br+c)dt=Ix+b)[Ein(bx+c)- 11+1 - exb bx-c) Jli(bi + c)dtx + b l li(bx + c) -I li(bY+ 2bcx + c')h\/BCexp1`C -Bc c 357THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS 37:11 i37:10:8 Ei(bx + c)x"+'(l c+n!jc"-'[exp(bx + c)e,(-bx - c)- l)JtEi(bt + e)dt -l-pien + lI\ b /J(-b)"+'_o(j + l)(n - j)!while the n = -2, -3. -4, ... cases can be evaluated via the lengthy expressionEi(bt + c)Ei(bx + c)cl l 37:10:9J`is1-1dt =n(-c/b)"[I- (bx/J1 [e,(-c) 11 r[exp(bx)-exp(-c)I +exp(c)e"_,(-cEi(-c) - Ei(bx) +j!J-J f n(-c/b)" )_o(bx)j*'(-c)'*'in which e,(x) denotes the exponential polynomial [Section 26:13].Included among the definite integrals tabulated by Gradshteyn and Ryzhik [Section 6.21-6.231 are37:10:10 JEi(bt)dt =b1b > 0 )6+s`37:10: l l =Ei(bt) Et(B:)dt = - In(bb"Ba)b > 0B>0fbB( 37:10:12 37:10:13 37:10:14 37:10:15 37:10:16I ln(Bt)Ei(-bt)dtI + y + ln(b/B)=bb > 0B > 00 It' Ei(-bt)dt =-fY1+v)b z 0v > - I,10(1 + v)b'-" Izt"-'exp(-Bt) Ei(-bt)dt = BI 0.v; bBBIb + B > 0v> 0o \/ '-Jt" li(t)dtIn(2 + v)=oI+vv> -2ff li(t)dt =-ln(v - 2)Jv>2t"v- IThe r and B functions occurring in integrals 37:10:13 and 37:10:14 are, respectively, the gamma function [Chapter43] and the incomplete beta function (Chapter 58]. 37:11 COMPLEX ARGUMENTEven with a complex argument, the Ein function is single valued and continuous. For small complex arguments,the series 37:11:1[x-- y x3-3xy3x'-6x'y'+v Ein(x +iv) = x -+-+2!23!34!42xy+3xy- - y'4.x'y - 4xy'+i[y2'23'3-4!4+ .. .l converges rapidly and can be used to find values of Ein(x + iy). For purely imaginary argument, one has37:11:2 Ein(iv) = Cin(v) + iSi(y)where the Cin and Si functions are discussed in Chapter 38.The exponential integral Ei is a many-valued function, when its argument is complex, because equation 37:0:1generalizes to37:11:3Ein(x + iy) + Ei(-x - iv) = y + Ln(x + iv) 37:12THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONSwhere Ln is the multivalued logarithmic function discussed in Section 25:11. However, as explained in that section,one usually selects a principal value of the Ln function, which leads to the single value 1 37:11:4Ei(x+iy)=y+2ln(x'+y)-Ein(-x-iy)+iOwhere 0 is the angle defined in Section 25:11. Notice that, according to 37:11:4, the exponential integral may havea complex value even when its argument is real. In fact:37:11:5Ei(x + Oi) = y + In(x) - Ein(-x) + in x > 0The imaginary term in this expression is ignored in the rest of the chapter. Abramowitz and Stegun [Tables 5.6and 5.7] provide tables from which the real and imaginary parts of Ei(x + iy) may be determined. For purelyimaginary argument, one has 37:11:6 Ei(iy) = Ci(y) + iSi(y) -n2sgn(y)in terms of the functions of Chapter 38. 37:12 GENERALIZATIONSThe exponential integral function is a special case:37:12:1 Ei(x) = -f(0;-x)of the complementary incomplete gamma function [Chapter 45J and of the Tricomi function [Chapter 48137:12:2 Ei(x) _ -exp(x) U(l;l;-x)The entire exponential integral function is a hypergeometric function [Section 18:14]:' 37:12:3 Ein(x) = x(I)i(-x)-o(2),(2), 37:13 COGNATE FUNCTIONSIn addition to the Schl6milch functions (discussed later in this section], two other function families, related toexponential integrals, are sometimes encountered.The alpha exponential integral function of order n is defined by 37:13:1 Jt"exp(-xt)dtn = 0, 1, 2, ...The first member ao(x) equals [exp(-x)]/x and because of the recurrence relationn37:13:2 a"(x) _ - a"-,(x) + ao(x)n = 1, 2, 3, .. .xall of these functions can be reduced to the elementary functions 37:13:3 a"(x) =n! exp(-x)x'.ejx)n = 0, 1, 2, .. . where e, is the exponential polynomial (Section 26:13].The beta exponential integral function family is defined by a similar integral, but with changed limits: 37:13:4Nx) = J t" exp(-xt)dtn = 0, 1, 2, .. . 359THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONSThe first member po(x) = 2[sinh(x)]/x and all members are reducible, via the expressionn! 37:13:5 [3"(x)[exp(x)e"(-x) - exp(-x)e"(x)] n = 0, 1, 2. ...to more elementary functions.A family of functions defined by37:14 37:13:6 E"(x) = Iexp(--rt)dtn = 0, 1, 2, ...t.was introduced by Schldmilch. The first member E0(x) = [exp(-x)]/x is elementary, while the second37:13:7 E,(x) = -Ei(-x)is related in a simple way to the exponential integral function. These identities, coupled with the recurrence re-lationship.r 37:13:8 E"(x) _[Eo(x) - E,-,(x)]n = 2, 3, 4, .. .n - Ienable the properties of E"(x) to be deduced from those of Ei(-x). The general relationship is 37:13:9-(-x)" ' rEi(exp(-x) III2!(n - 2)!1 ) 1 E"(x)=-x) +--+ + In=2,3,4,... (n - 1). XxX2(-x)Some familial properties of Schldmilch functions are 1 37:13:10E"(0)=-n=2,3,4,...n-1 37:13:11ddxE"(x)E.(x)n = 1. 2, 3.... 37:13:12 E"(x) = x"-'1'(1 - n;x)n = 0, 1, 2....and the asymptotic expansionInn(n + 1)(n), 37:13:13E"(.r)exp(-x) Is -z'+Jr,x)I,iThe function 37:13:14xl exp(/r/lEi( Xll - 1 - x + 2x= - 6r' + ... + j!(-.r)1 + ...has been named Euler's function and symbolized E(x). It should not be confused with the complete elliptic integralof the second kind [Chapter 61 ] for which the same notation is adopted.The avoid an unnecessary proliferation of functions, none of the functions discussed in this section is usedelsewhere in the Atlas. 37:14 RELATED TOPICSVery general expressions for indefinite integrals of the form f t`exp(bt)dt were presented in Section 26:10. Never-theless. because of the very widespread occurrence of the integrals 1337:14:1Jt"exp(tt)dt v = 0, t 2, t I, t 2, ... in applied mathematics, Table 37.14.1, containing such indefinite integrals, may be useful. Integrals of the forms 37:14THE EXPONENTIAL INTEGRAL Ei(x) AND RELATED FUNCTIONS 360 +I 1337:14:2 andu"cxp- duuv=0,±-2,+I,a- - 2, 37:14:3 J v" exp(±v2)dvn = 0, ±1, 22, ±3, ...are similarly ubiquitous. The substitutions u = 1/1 or v = vt convert these into the form of 37:14:1 so that Table37.14.1 may also be used to evaluate integrals of the 37:14:2 and 37:14:3 families.The erfc and daw functions occurring in Table 37.14.1 are the error function complement [Chapter 401 andDawson's integral [Chapter 421. The lower limits xM and x; are, respectively, the argument values correspondingto the maximum and the point of inflection of a graph of daw(V') (see Section 42:71. Table 37.14.1 V -3xuJr' exp(odrtEi(x)(x + 1) exp(x)2 Ir2x'lIcxp(x) d.-(N/;)3L-xEi(x) --2 X.-x 0 0 0exp(x I X2 exp(x) 2 daw(V) -VxB(x) )2 exp(x) daw(Vx) exp(x)exp(x)UV -daw(Vx)l(x - I) exp(x) rr3Vr3\r22(x2 - 2x + 2) exp(x)Je exp(-10 (1 - x)exp(-x) Ei(-x)2x22 exp(-x) rr 1 114V,Lx-21 3 erfNl`l3VxLEi(-x) +Aexp(-x) 2 exp(-x)- 2 V a erfc(Vl)Vx v'- erfc(V/X) + V. exp(-x)2(x 1 1) exp(-x)- erfc(v;) + VT x + - cxp(-x)34A(2)(x= + 2x + 2) exp(-x) CHAPTER38SINE AND COSINE INTEGRALS This chapter concerns functions defined in terms of indefinite integrals of sin(s)/x. cos(x)/x and their hyperbolicanalogs. Whereas the definitions of the hyperbolic sine integral Shi(x) and the sine integral Si(x) present no dif-ficulty, the corresponding integrals of cosines diverge at zero argument. Accordingly, in addition to the hyperboliccosine integral Chi(x) and the cosine integral Ci(x), it is useful also to define an entire hyperbolic cosine integralChin(x) and an entire cosine integral Cin(x). These entire integrals are related by38:0:1Chin(x) = Chi(x) - 1n(lxl) - y = Chi(x) - In(ix;) - 0.5772156649and38:0:2Cin(x) = y + ln(Ixl) - Ci(x) = ln(I.781072418kxi) - Ci(x)to the Chi and Ci functions.For large arguments. certain auxiliary functions discussed in Section 38:13 are more convenient than Ci andSi.Because of their wider applicability, this chapter places more emphasis on the Si and Ci functions than on theirhyperbolic counterparts. 38:1 NOTATIONThe initial letter of Shi and Chi is not always capitalized. The "h" that is used to identify the hyperbolic integralsmay occur elsewhere in the function's symbol. For example, Sih(x) sometimes replaces Shi(x). and the anagramCinh(x) is often used instead of Chin(x).Some authors use ci(x) synonymously with Ci(x), but others employ it to denote -Ci(x). The notation si(x) isusually encountered with the meaning 38:1:1 nsi(x) = Si(.r) - - Neither ci nor si is utilized in this Atlas. 361 38:2 SINE AND COSINE INTEGRALS 36238:2 BEHAVIORFigure 38-1 maps the Shi, Chi and Chin functions. Note that for large positive arguments Shi(x) and Chi(x) convergeand approach the value ; Ei(x), where Ei is the exponential integral [Chapter 37].The damped oscillatory behavior of the Si and Ci functions is evident in Figure 38-2. As x -+ ± x. Si(x)approaches ±0r/2) and Ci(x) approaches zero, whereas Cin(x) approaches infinity logarithmically via a series ofplateaus. FIG 38-1 :..:.... :.... :.... :.... :.... :.... :.... : J.:.. 3 2 38:3 DEFINITIONS In addition to their definitions as integrals, the hyperbolic sine and cosine integrals may be defined in terms of thefunctions of Chapter 37: 363 SINE AND COSINE INTEGRALS 38:3 -.1 A60'L-ti01'L4ti4474 38:3:1 Shi(x) =Ei(x)ZEi(-x)=` sinh(t)dt0tEi(.r) + Ei(-x)(cosh(:) 38:3:2Chi(x) == Jdtxa = 0.523822572Ein(x) + Ein(-x)f cosh(:) - 1 38:3:3Chin(x) _ - dt 2=otThe sine integral, cosine integral and entire cosine integral are defined byf0.sin(t)1sin(:) 38:3:4Si(x) = - dt = -J+sinc(t)dt =It- - J - dt1m o2t 38:4 SINE AND COSINE INTEGRALS 38:3:5 Ci(x) _ - andcos(t) r1 - 38:3:6 Cin(x)COW)= IdtotSome other definitions may be constructed by employing relationships 38:0:1 and 38:0:2. 38:4 SPECIAL CASES There are none. 38:5 INTRARELATIONSHIPS All cosine integrals, like the cosine itself, are even functions:38:5:1whereas the sine integrals are odd:38:5:2as is the sine function.Expressions for the Ci and38:5:3 38:5:4 38:6 EXPANSIONSf(-x) _ -f(x)f = Shi. Sif(-x) = f(x)f = Chi, Chin, Ci, Cin Si functions in terms of the auxiliary functions of Section 38:13 arcCi(x) = sin(x) fi(x) - cos(x) gi(x)Si(x) =2- cos(x) fi(x) - sin(x) gi(x)364 The sine integral may be expanded as the power series 38:6:1 x3x3x7=(-x ZSi(x)=x--+--1860035280 ,_o (2j + 1)(2j + 1)!The similar series without alternating signs, namely E.r2J"/(2j + 1)(2j + I)!. represents Shi(x). Likewise, re-placement of all the - signs by + in the expansion X2x4x6x6 (-x2)' 38:6:2Cin(x) = - - - + - -4964320322560!_0 23(2j)!of the entire cosine integral produces the series F.xz'/(2j)(2j)!, which represents Chin(x).The sine integral is also expansible in terms of spherical Bessel functions of the first kind [Section 32:13J:( l4(l16[I l/llz38:6:3Si(x) = itJ .1/22/= -sins\2! + -sin12/-Z cost_)]+5s6L\1-121sin12J-2cos(2) Jz+... 365 SINE AND COSINE INTEGRALSMore rapidly convergent even than 38:6:1 and 38:6:2 are the composite power series2 - cos(x)sin(x)xx3x5(-xY 38:6:4SO) -x+x2=3-180 + 12600 -a (2j + 1)(2j + 3)!and 38:6:5Cin(x) ++Asymptotic expansions of Ci(x) and of Si(x) - (a/2) may be constructed by combining expressions 38:5:3and 38:5:4 with the expansions 38:13:5 and 38:13:6. 38:7 PARTICULAR VALUES With n = 1, 2, 3...., Table 38.7.1 lists particular values and features of the six functions. Included in Table38.7.1 are the arguments at which Si(x) and Ci(x) acquire values that are locally maximal or minimal. These extremaalso correspond to special values of the auxiliary integrals discussed in Section 38:13:n 38:7:1Si(±2na) _2 + fi(2na) 38:7:2 38:7:3 38:7:4Si(±(2n - l)ir) _ ±it± fi((2n - 1)a) Cii±l2n - 1)- I = W . gil 1 2n - D-) 22CiI±(2n-2)a1= +gtl l,2»21rIminimummaximum maximumminimum minimummaximumof Ci(x)of Si(.t) maximumminimum.n= 1.2,3.... At x = :2a, ±4a, a horizontal inflection is displayed by Cin(x), that is, d Cin(x)/dx and d' Cin(x)/d 2 are both zero at these values of the argument [see Section 0:71.The zeros of the Chi(x) function occur at x = ±0.52382257. The first zeros of the Ci(x) function are foundat =0.61650549, and others occur close to the points of inflection of the function, which are given byd2 38:7:5ar Ci(x) = 0.r = ±p,(-l)j = 1, 2. 3, ...Similarly, the sine integral inflects at Table 38.7.1 SM.)CMtx)Chin(.) Si(x)20x - -2n.11 - 2n)438:7 sin(x)I - cos(x)3x'xxxb;'(-x')'-xx22241440120960 )-1j(2j + 2)! (1 - 2,,),- 2n)nx -0 0 0mu min 0 min max12n -)a2n - i). max2n - 1)n maxminhonz.innc 38:8 SINE AND COSINE INTEGRALS 366 38:7:6ad,Si(x)=0x=0.±r,(I)j= 1,2,3, x ..Here p,(-I) and r,(1) are, respectively, the roots of the equations cot(x) = -x and tan(x) = x (see Section 34:7]. 38:8 NUMERICAL VALUESThe hyperbolic integrals Shi(x) and Chi(x) arc conveniently evaluated by using the algorithm in Section 37:8 togenerate values of Ei(±x) and then employing the identities given in equations 38:3:1 and 38:3:2. Input x >>If 14<jxj go to (3)If(xj<0.14/goto(2)Set J = 2lnt 3 +S4I)Set f = -x '/(J + 2)'(1) Replace f by(f+J_)(J(;21))I Storageneeded:x, J. fand g Use radian mode.1Replace J by J - 2if J * 0 go to (I)sin(x)--cos(x)Replace f by 2(1 - f)XGo to (5)(2) Set f = x exp(-x2/18)Go to (6)/(3) Set J = 2 Intl 2 +6)Set f = g = [(J +2)//x]'(4) Replacegby 1- g(J +IV/x2Replacef by I - fJ(J - 1)/x2Replace J by J - 2IfJ*0goto(4)g sin(x)Test values: Replace f byf cos(x) - Si(n) = 1.85193705(5) Replace f by f/x Si(0.1) = 0.0999444641(6) Output f Si(-20)-1.54824170 f=Si(x)<K«« The algorithm above permits Si(x) to be calculated to 24-bit precision (i.e., the relative error never exceeds 6X 10-') for any argument. For values of x in the range -0.14 < x < 0.14. the approximation formula 38:9:1 isused. When j lies in the 0.14 S 14 range, the algorithm uses expansion 38:6:4 in the concatenated andtruncated form 38:8:1/(((((((-x2Il Si(x) +-x21l-x2 =\\\\ "\\(J+2)'J-1/J(J+1)+J-3/(J-2)(J-1) 1-x'1x2+\12 - cossin(x)1+. +3/4x5+1/2x31(x) - xxwith J = 2 Int(3 + 5x1/4). For ki > 14 the expression 367 SINE AND COSINE INTEGRALS 38:8 W((((((+1)J(J-1)+1(J - 2)(J - 3)38:8:2SO) =-s-s2- x4x3+1\2x Icos(x)1 -\\\...(((J+2)2+ 11 (J+ I)l+1) (J- 1)(J-2)-x-X-X2+...+115x4+IJ+llsin(x)l-x'J -xZ/IIx(which follows from equations 38:5:4, 38:13:5 and 38:13:6) is used with J = 2 Int(2 + 56/[x J).The algorithm below is designed to compute values of Cin(x) to 24-bit precision. It is very similar to thealgorithm for Si(x), being based on equations 38:9:2, 38:6:5, 38:13:5, 38:13:6, 38:5:3 and 38:0:2. Input x >>3»»If 16 < kI go to (3) If[tl<0.17goto(2)Set J = 2 [nt(3 + 4Ixi/,rr)Set f = -x2/(J + 3)' 1-x2(1) Replace f by1f+J/(J + 1)(J + 2)Replace Jby J - 2IfJ*0goto(1)3sin(x)I - cos(x) Replace f by2-f-xx'Go to (6 or 5)(2) Setf = 1x2 exp(-x'/24)1/4Go to (6 or 5)(3) Set J = 2 1ntc2 +Self = g = 1(J + 2)/-x12(4) Replace g by I - g(J + I )J/x2Replace j by l - fJ(J - I)/x2Replace J by J - 2If J * 0 go to (4)sin(x)cos()Replacej by fx- g X2Goto(5or6)(5) Replace f by ln(1.781072418k1) - f(6) Output ff = Cin(x)or Ci(.r)Storage needed: x, J, f and g Use radian mode. Test values:Cin(ir) = 1.64827764Cin(0.l) = 0.00249895856Cin(20) = 3.5285212Ci(2) = 0.422980829Ci(-0.15) = -1.3'-5524050(20) = 0.0444199210I When the alternate commands shown in green are substituted, the algorithm generates values of Ci(x) instead ofCin(x). However, the fractional error in the calculated values of Ci(x) may exceed 6 x 10-1, especially near thezeros of the cosine integral.Also, values of Shi(x), Chin(x), Si(x) and Cin(x) are calculable via the universal hypergeometric algorithm ofSection 18:14. 38:9 SINE AND COSINE INTEGRALS38:9 APPROXIMATIONSClose to x = 0, the formulas 38:9:1Si(x) - x exp(182)8-bit precisionIx s 2.2 and 38:9:2Cin(x) =4exp( 142)8-bit precisionIxj s 2.5are useful, while for large x the approximationsnx cos(x)sin(s)large x 38:9:3 Si(x) = Z -xx + 2x' +6and 38:9:4 may be used.x sin(x)cos(x)Ci(x) =-large xx-+2x'+6 38:10 OPERATIONS OF THE CALCULUS The differentiation and indefinite integration operators, applied to the six functions of this chapter, give 38:10:1dShi(bx + c=sinh(bx + c)dxx + c b 38:10:2 dChi(br + c) =cosh(bx + cdxx + c b)dcosh(bx + c) - 1 38:10:3 dxChin(bx + c) =x + (c/b) 38:10:4 dSi(bx + c=sin(bx + c_ bsinc(irbx + ac) dxx + (c/b) 38:10:5 dCi(bx + c) =cos(bx + cdx x + (c/b)d1 - cos(bx + c) 38:10:6rdxCin(bx + c)\=x + (c/b) 38:10:7Shi(bt + c)dt = I x + bc)Shi(bx + c) _cosh(bx6+ c) - 1 / 38:10:8 Chi(bt + c/c`sinh(bx + c) Jc)d<=1x+blChi(bx+c)- b 38:10:9J/bChin(bt + c)dt = I x +Chin(bx + c)) -sinh(bx + c)b\/ 38:10:10 Si(bt + c)dt = I x + b)Si(bx + c) -1 - co bbx + c) c/b368 369 SINE AND COSINE INTEGRALS 38:11 38:10:11 cCi(bt + c)dt = (x + b) Ci(bx + c) - sin(bb + c)/bCin(bt + c)dt = I x +b)[Cin(bx + c) - 1] +sin(bxb + c) 38:10:12Other indefinite integrals are listed by Gradshteyn and Ryzhik [Section 5.3]. The most useful of these arc probablythe integrals of products of sinusoidal functions with the sine and cosine integrals. We have the results 38:10:13Jx sin(Bt + C) Si(bt + c)dt = 8 cos(Bx + C)I Si(bx + c) - 2 sgn(b)J BcSi(+) - Si(-)BcCi(+) - Ci(-) - cos C - b2B- sink - b2Br/1\F 138:10:14Jcos(Bt + C) Si(bt + c)dt = Bsin(Bx + C)irI 2 sgn(b) - Si(bx + c)J + sin C -BcSi(+) - Si(-)- cos CBcCi(+) - Ci(-)-- b2B 2Br38:10:15Jsin(Bt+ C) Ci(bt +c)dt =Bcos(Bx + C) Ci(br + c) + sin(C - Bcb/l Si(+) + Si(-) Ci(+) + Ci(-)-lC-b2Br- Icos(``/38:10:16Jcos(Bt + C) Ci(bt - c)dt = 8 sin(Bx + C) Ci(bx + c) + cost C - Bc Si(+) + Si(-)b-sin C -BcCi(+) + Ci(-)b2Bwhere Ci(±) is an abbreviation for Ci[(B t b)(x + c/b)i and Si(±) similarly abbreviates Si[(B t b)(x + c/b)) -(w/2) sgn(B -t b).Definite integrals of the sine and cosine integral functions include38:10:17 f Ci(Bt) Ci(bt)dt =n/B 2: b{a/'b2b. b a Band many others are listed by Gradshteyn and Ryzhik [Sections 6.26 and 6.27]. 38:11 COMPLEX ARGUMENTSThe six functions of this chapter acquire complex values when their arguments are complex and even, in the caseof the Chi function, for example, for real values: 38:11:1 /n Chi(x + Or) = Y +Chin(x) + -With argument x + iy. values of the functions may be calculated from those of the Chapter 37 functions viathe identities 38:11:2Shi(x + iy) = I [Ei(x t iv) - Ei(-x - iv)) 38:12 SINE AND COSINE INTEGRALS 1 38:11:3 Chi(x + iv) _[Ei(x + iy) + Ei(-x - iy)] 38:11:4 Chin(x + iy) =2[Ein(x + iv) + Ein(-x - iv)] zr1 38:11:5 Si(x + iy) =Z+ - [Ei(y - ix) - Ei(-y + ix)] 1 38:11:6Ci(x+iy)=2[Ei(y-ix)+Ei(-y+ix)) 1 38:11:7 Cin(x + iv) = 2 [Ein(y - ix) - Ein(-r + ix)]For purely imaginary arguments, one hasn 38:11:8 Shi(iv) = i Si(y) -238:11:9 Chi(iv) = Ci(y)a 38:11:10 Si(iv) = i Shi(y) + - i-i 38:11:11 Ci(iv) = Chi(y) +2 38:12 GENERALIZATIONSThe sine integral is the special a = 0 instance of the more general function 38:12:1sin(a + t')dtsin(t)dta'+1'-If -a'370 (see Erddlyi, Magnus, Oberhettinger and Tricomi. Higher Transcendental Functions, Volume 2, page 147].The Shi, Chin, Si and Cin functions are all instances of hypergeometric functions, as discussed in Section18:14.Boehmer integrals [Section 39:12) also generalize the sine and cosine integrals. We have38:12:2 Ci(x) _ -C(x;0)and 38:12:3 aSi(x) = Z - S(x;0) 38:13 COGNATE FUNCTIONSThe so-called auxiliary sine integral and auxiliary cosine integral functions are defined by the definite integralsfsin(:)_ rr 38:13:1fi(x)J t+xdtexp(-xt)-JO r+1dtfand 371 SINE AND COSINE INTEGRALS 38:13Icos(t)r exp(-xt)38:13:2 gi(x)I(t + x)dtIt2+ IdtThere appears to be no standard notation. Abramowitz and Stegun [Section 5.21 use the f(x) and g(x) symbolism.Figure 38-3 maps the two functions, which are defined for positive real argument only. The functions are related byr11 38:13:3 ft(x) = sin(x) Ci(x) +cos(x)12- Si(.r) andrl38:13:4 gi(x) = sin(x)f - Si(z)J - cos(x) Ci(x)to the sine and cosine integrals. The simple asymptotic formulas, valid for large argument I224720(2j)! 38:13:5 r1(X)s_X+--x'+...++...x-x andI61205040(2j + I)!+ ... ++x - - ;x+ sx- ex'(-x')'rX-'permit convenient calculation of the auxiliary functions and are incorporated into the hypergeometric representation[Section 18:141.The differential identitiesd 38:13:7 dxr(.r) = -80) 38:13 SINE AND COSINE INTEGRALSand 38:13:8 dIdxgi(x) = fi(x) -xrelate the two auxiliary integrals, which therefore satisfy the differential equations 38:13:9df1+ f = -f = i(x) + c, sin(x) + c, cos(x)dx'xand 38:13:10ds + f =S2f = gi(x) + c, sin(x) + c, coa(x) respectively, where c, and c2 are arbitrary constants. One also hasa38:13:11 Jgi(t)dt =nbut the corresponding integral for fi is infinite.372 CHAPTER39THE FRESNEL INTEGRALS S(x) AND C(x) Among other applications, Fresnel integrals occur in theories of physical optics [see Section 39:141. The so-calledauxiliary Fresnel integrals, discussed in Section 39:13, are also of some importance. 39:1 NOTATION The symbols S(x) and C(.r) are almost universally adopted for the Fresnel sine-integral and Fresnel cosine-integral,respectively. However, there is a wide variation in the definitions chosen for these functions by various authors. Thus, in addition to our definition 39:3:1, each of the following integrals is cited as the "Fresnel sine-integral" byat least one authority: 9:1:1 sin(t')dt S(x) 3 !2Jo 39:1:2Isin(t)dt=S(V.r)x10V2aVtx 39:1:3 I; sin(2tIdt =S(V2 o 39:1:4Irr sin(t)dt =S(-V,)x } 0 2Vx oVtV YThe right-hand member of each of the above equations is the expression for the left-hand integral in the notationof this Atlas. In all cases the definitions adopted for the Fresnel cosine-integral involves straightforward replacementof sin in the integrand by cos. Some authors adopt multiple definitions of these functions and utilize subscripts (asin S,(x). C:(x), etc.) to distinguish the options. There is, however, no greater unanimity in this secondary notationthan in the primary.The reader will note that the definitions of S(x) and C(x) adopted in this Atlas agree with those used by Grad-shteyn and Ryzhik, and by Spiegel, but differ from those selected by Abramowitz and Stegun. and by a numberof other authors. 373 39:2 THE FRESNEL INTEGRALS S(x) AND C(x)39:2 BEHAVIOR374 Figure 39-1 maps the behavior of the two functions. Both S(x) and C(x) display damped oscillations that decrease in both period and amplitude as x - is approaching the values ±i in these limits. .....................................................0.8 -C (x) ............ ....... .. ................... . ...................... :Free (x).......................:....:....... ......0.1Zras (Z):........... .... : 0. . For x ? 0, the S(x) function lies entirely between the curves ; - Fres(x) - Gres(x) and ; + Fres(x) + Gres(x) where Fres and Gres are the functions discussed in Section 39:13. The same bounds apply for C(x). and a similar relationship holds for x s 0. For all arguments39:2:1sgn(x)- Fres(jxI) - Gres(kf) <_ f(x) <_ !9! !r2+Gres(jxJ)f= S or C 239:3 DEFINITIONSThe Fresnel sine-integral is defined by either of the equivalent representations 39:3:1 S(x) =ITsV2,roVt?Jsin(r)dl =sgn(x)Isin(r)dt 375 THE FRESNEL INTEGRALS S(x) AND C(x)and similarly for the Fresnel cosine-integrala cos()1r`sgn(x)I39:3:2 C(x) =VJcos(r=)dt = dt'noThe integrands may be regarded as spherical Bessel functions [Section 32:13] 39:3:3 S(x) =sg2(z)Jr J,1,(t)dt =sg2(x)JY_11,(t)dt0 0 39:3:4 39:4 SPECIAL CASESThere are none.C(x) =sgn(z)JYJ_,r(tk1t =-sgn(x) rY1r:(t)dr22Jo 39:5 INTRARELATIONSIIIPSBoth Fresnel integrals are odd functions39:5:1 f(-x) = -f(x)f= S or CThey may be expressed in terms of the auxiliary Fresncl integrals (Section 39:13] by 39:5:2S(x) = sgn(x)I2- cos(x')sin(x2) Gres(lxl) and 139:5:3C(x) = sgn(x)12+ sin(x') Fres(',.rj) - cos(x2) Gres(Ixl) 39:6 EXPANSIONSPower series expansions of the Fresnel integrals are42.rrs 39:6:1S(x)+ -ir3421320and,2xsx°(_x')39:6:2 C(x) =x - - + - - xj\10216/a(4j + 1)(2j)'More rapidly convergent than these are the composite expansionssin(.r2) + 2x' cos(_r') 3x3x'x'1-3339:6:3- S(x) ++ - - - +-_24.r34x241120739204.r(4j - 1)(2j + I)!and39:6 cos(x) - 2r= sin(x2) I3.rxsx"-3(-x')'_ - - - + - -_ 39:6:4- C(x) +24x34x3+816086404.r3,=o (4j - 3)(2j)!Via relationships 39:5:2 and 39:5:3, one can utilize the various expansions of the auxiliary Fresnel integrals[Section 39:13] to produce the summations 39:7 THE FRESNEL INTEGRALS Six) AND Clx) 376 39:6:5 and 39:6:6S(x) = xV[sln(X2)(-4x2')'- 2x2cos(x')(-4x2`)1-o (4j + 1)!!(4j + 3)!! C(x) = x[COS(X2)(-4x')'+''(-4x')' 1o (4j + I)!!!-o (4j +3)!!Jas well as the asymptotic representations 39:6:7S(x) -sgn(x)-cos(x') rl - 3 + 105+(4j)!!+...l2xV L4x216x2(-4x')'Jsin(x') r15945(4j + 1)!!1x3\4x`16x"(_4X4 )' and 39:6:8 sgn(x)sin(x2) r3105(4j - 1)!!1 C(xl-+ILI -Z;+--...+i+...2x16x"(-4x2 )1cos(x2) 115945(4j + 1)!!11 --+--...++...x xx2°The Fresnel integrals are expansible in terms of spherical Bessel functions [see Section 32:131 39:6:9S(x) =sgn(x)[hn_(x)+ 1712(x=) + J1112(x2') +1 = sgn(x)1,,.3;2(x)j-oand 39:6:10 C(x) = sgn(x)[1r12(x') + Jsr_(x-) + 1912(x2) + ...1 = sgn(x) J2r 112(x2)=o 39:7 PARTICULAR VALUES Included in the Table 39.7.1 are the values at which the Fresnel integrals exhibit maxima, minima and points ofinflection. The functions also inflect at the origin x = 0. In this table n represents any positive integer, n = 1, 2,3, .... Table 39.7.1 3 x-x-V-2na - 2-2na - n2mr-2022mr2mr - a2n+r - Z V x S(x)innmininn0innmaxinnmin 2 2C).t)Innmexinnmin0mnxinnmininn2 2 377 THE FRESNEL INTEGRALS S(x) AND C(x) 39:8As x - ±z, the values of the maxima and minima converge towards ±= according to the formulas 39:7:1S(±2na - a)-± Fres(V2nir - TO maximuni2 minimum 39:7:2S(± V 2na) _ `- 2 ± Fres(2na) 39:7:3 39:7:4C(±2na-Zal=t-2±Fres)2na-322/1/ C(2na -2J =± 2±FresV2na- 2minimummaximum maximumminimum minimummaximumof S(x) a= 1,2.3.... of C(x) where the Fres function is the auxiliary Fresnel cosine-integral discussed in Section 39:13. The values at the pointsof inflection converge more rapidly towards ±j' and arc easily calculated from 39:7:544ma-2)J 39:7:6C(±V) = ±12+(_lrGres(V rt) where Gres is the auxiliary Fresnel sine-integral [Section 39:13). 39:8 NUMERICAL VALUESm= 1,2.3,... In Section 18:14 we show how the universal hypergeometric algorithm may be utilized to generate values of theFresnel integrals (as well as those of the auxiliary Fresnel integrals) in several different ways.Because applications of Fresnel integrals usually require values of both S(x) and C(x), the algorithm presentedbelow generates numerical values of the Fresnel sine-integral and the Fresnel cosine-integral simultaneously. Thealgorithm has 24-bit precision (i.e., the relative error never exceeds 6 x 10"). For very small absolute values ofx. approximations 39:9:1 and 39:9:2 are used by the algorithm. For intermediate values of W, the composite ex-pansions 39:6:3 and 39:6:4 are used in the truncated and concatenated forms. 39:8:1 39:8:2SW ((((".(((2J-1)(J+1)J+2J1 5J(J-I)(J-2)+2J`9/(!-3)(J-4) 1-x4I-x4I-3sin(x-) 1cos(x-)17 5x43 3x2-122x-/x 2ar-x4 1_xeI-x' C(.c)((..(2J-3)J(J-1)+2J-7J (J-2)(J-3)+2J-11/(1-4)(J-5) aa_++1-x + I-x +I-3- 2x- sin(x-) - COs(t) I 5 4x312x 1-31x'V8n1) 39:8 THE FRESNEL INTEGRALS S(x) AND C(x)with J given the empirical value 2 Int(5 + &2/5). Similarly, the concatenated expansions 39:8:3 39:8:4x`((...r((J - z)(JJ)+11 (J - i)( - )y1\ (J - 2)(s - 4)\\\\"\2X\\1_JX)/I-xII-x1 1` 1\-x'+ Ix)-'+ 1 JI-xX 12X 2+...+I1-x,+1/X. +378 are used to evaluate the two asymptotic series in 39:6:7 and 39:6:8, and these latter are then employed to evaluateS(x) and C(x) for large values of jx4. Input x >>1f0.1 <x2goto(1)Set f = (2x'13) exp(-x'/14)Set g= 2x exp(-x4/10)Go to (5)(1) If x2 > 15 go to (3)Set J = 2 Int(5 + 1.2x2)Set f = 11(2J - 1)Setg - 1/(2J-3)(2) Replace f by1Xf2J - 5J(J + 1)IXgReplace g by2J - 7(J - 1)JReplace J by J - 2 1fJ*0goto(2)1 Replace f by [-st2)- cos(x2) -ZJ/xReplace g by (2x2 sin(x2) - cos(x2) - 3gJ/2x3Go to (5)(3) Set J = 2 Int(4 + 60/x2)Setf=g= 1 (4) Replace f by I - f I(i2JReplace g by I --4\)\/xsReplace J by J - 2If J * 0 go to (4)Set J = f sin(x2)Storage needed: x, f, g and J Use radian mode.1 g sin(x)I Replace f byff cos(x2) -2x2=+22J /x 379 THE FRESNEL INTEGRALS S(x) AND C(x) 39:10 f = S(x) < g - C(x) <Replace g by + J +g cos(x2) (5) Replace f by f/V 2Output fReplace g by g/V 2Output gTest values:S(0.2) = 0.00212744901C(0.2) = 0.159551382S(-n) = -0.616486872C(-,r) = -0.451358120S(5) = 0.421217048C(5) = 0.487879892 This algorithm is readily adapted to generate values of the auxiliary integrals Fres(x) and Gres(x) [see Section39:13].39:9 APPROXIMATIONSFor arguments of small magnitude one has 39:9:1 XS(x) = V-Tr 3exp( 14)8-bit precisionjxI1.3:5 39:9:2 I8-bit precision C(x)x expl10Ix s 1.3while for large arguments Icos(x')sin(x2) 39:9:3 S(x) = - -2xu2nIsin(X2)-x'V 8a cos(x2)8-bit precision 1rI > as 39:9:4C(x) =+2xV 2a-xT8-bit precisionjx ? n The approximation 39:9:5r1=[C(X)I21S(x) - 2] +- 2] =tax'8-bit precisionx 2:3es greater than 3, a cartesian graph of the Fresnel sine-integral versus the shows that, for a small range of x valuFresnel cosine-integral is roughly a circle, centered at the point (,) and with radius I/xV 2a. The precise shapeof such a graph is discussed in Section 39:14. 39:10 OPERATIONS OF THE CALCULUS The derivatives and indefinite integrals of the Fresnel integrals are 39:10:1 39:10:2 39:10:3 39:10:4dS(bx) = b- sin(b-x2)dC(bx) = b- cos(b-.,6drIT IS(br)dt = x S(bx) -I - cos(b'x2)Jo 62n IC(biKlt = x C(bx)0bV 2a 39:11 THE FRESNEL INTEGRALS S(x) AND C(x)Among the definite integrals listed by Gra((dshteynl and(Ryzhik [Section 6.32) are 1r139:10:5Iti-S(br)Jdt=/-I <v<2b>0 0 02 V 2w(1 + v)b'-'\239:10:6Jo tC(bt)-1Jdr=-1<v<2b>012\(l+ v)b"and 39:10:7r a0<BJ3 sin(Bt'') S(bt)dtcos(Br') C(bt)dt =32B0 0 0B > b: 39:11 COMPLEX ARGUMENTFresnel integrals of complex argument may be related via the formulas I + ix - }x +r1-ix+yx-c) 39:11:1Six + iy)erf(V:1+ i%/2+4erfN/_2 - i 4 L'2and 39:11:21-ix-Sx+yI+ix+yx-y C(x+iy)=4erf+i- +-e--i-lti4V2V2/to the error function of complex argument [Section 40:11). 39:12 GENERALIZATIONS380 Fresnel integrals may be generalized to Boehmer integrals, also known as generalized Fresnel integrals, and definedby 39:12:1 S(x;v) = Jt'-' sin(t)drx z 0v< 1 39:12:2 C(x:v) =Jt'-' cos(t)dtx > 0v < I When the parameter v is zero, these integrals reduce to sine and cosine integrals [Chapter 38) 39:12:3 itS(x:0) = Z - Si(x) 39:12:4 C(x;0) = -Ci(x)and reduction to the Fresnel integrals occurs when the parameter is a moiety: 1 39:12:5 S(x;i) = V 1-2 - S( x) 39:12:6 C(x;}) = V2 1-2C(Vx)Jf 1sin(-) <b' 381The recurrence relationships39:12:7 39:12:839:13 C(x;v)S(x;v + 1)x" cos(x)VVpermit any indefinite integral of t-"12sin(t) or t-"ncos(t) to be evaluated for n = 1, 2. 3. ... in terms of the specialcases 39:12:3-39:12:6. Boehmer integrals have the particular values 39:12:9 C(0;v) = F (V) cos 2 var 39:12:10 S(0;v) = ['(v)sin(2at zero argument.Convergent expansions in power series 39:12:11 39:12:12 C(x;v) = C(0;v) - x"S(x;v) = S(0;v) - xI *'Z ()'=u (2j + 1)!(2j + I + v) as well the asymptotic series-2 '(_x),j=o (2j)1(2j + v)(I -v)(2-v) 39:12:13S(x;v) - x"-' cos(z)I l -5-THE FRESNEL INTEGRALS S(x) AND C(x) S(xC(X,v + 1)x'sin(x)v) = --VV 'X +++ (I - v)x"(I - v). 1 2 sin(x)LrI -(2 - v)(3 - v) -X-1(2 - v)(3 - v)(4 - v)(5 - v)-+(2 - v)2,+1 X3(-X2)')(1 - v)(2 - v)(3 - v)(4 - v) 39:12:14C(x;v) ^ x"-' sin(x) 1(1 - v)(2 - v+-x x'(1 - v) 1 2r(2 - v)(3 - v)(-x2) x-(2-v)(3-v)(4-v)(5-v)(2-v);,x'(-x2)'valid as xx, hold for the Boehmer integrals. These expansions permit the numerical evaluation of S(x;v) andC(x;v): for example, via the hypergeometric algorithm of Section 18:14. 39:13 COGNATE FUNCTIONSThere appears to be no definitive symbolism for the auxiliary Fresnel cosine-integral or the auxiliary Fresnel sine-integral, for which this Atlas uses the notations Fres(x) and Gres(x), respectively. Abramowitz and Stegun (Chapter71 use f(V 2/irx) and g(2/ax) for the same functions. Maps of these functions, which we define only for positiveargument. are included in Figure 39-I.The auxiliary Fresnel integrals are defined by the transforms(1 -v)(2-v)(3-v)(4-v) 39:14 39:13:1 39:13:2THE FRESNEL INTEGRALS S(x) AND CU) Fres(x) =Jexp(-2x1) cos(t')dtx0 Gres(x) =Jexp(-2xt) sin(r')dtx >- 0o382 or as the nonperiodic component of the semiintegral and semiderivative, respectively, of the sine function 39:13:3 39:13:4d-v2sin(x) = sinx 4d+ \ Fres(f) v2&T5 sin(x) =sinfl\x + 4 - with zero lower limit. Their relationships to the ordinary Fresnel integrals 39:13:5Fres(x) = cos(x'-)I2- S(x)1 - sin(x-)I2-C(x)jx ? 0 111 39:13:6Gres(x) =sin(x2)12- S(W )J +cos(x2)12- C(x)x ? 0 may also serve as definitions.The auxiliary Fresnel integrals may be expanded as the convergent power series 2 39:13:7Fres(x)cos(x') - sin(x2)2 3105 +32x"103958_o((4j-+4x`)3)!!1 r1r r4.r'16x°(-4,e)' 39:13:8-sin(x2) + cos(x2)- Gres(x)V aILx+-V2 15945a,=o (4j + 1)!!or as asymptotic inverse-power series 39:13:9Fres(x) --13+105+(4j - 1)!!+x16x°(-4x')' 39:13:10Gres(x)-115-+945+(4j + I)!!4x`16x'(-4x`)1the latter being valid for large x. 39:14 RELATED TOPICSFresnel integrals play a paramount role in the theory of the diffraction of light. Indeed, it was his studies in physicaloptics that led Augustin Fresnel to construct the integrals that now bear his name.A very useful construct in diffraction theory is Cornu's spiral, also known as the clotoid curve. This doublespiral is defined parametrically by39:14:1 f = S(w)whereC(w) = xand is shown, in part, in Figure 39-2. The two points x = ±_, f = t (marked A and B on the figure) are approachedasymptotically by the ever-tightening spirals. Cornu's spiral has the interesting property that its curvature, that is,the quantity 383 THE FRESNEL INTEGRALS S(x) AND C(x) 39:14010,1 IO0ha'btia%ti'bahmA0 . ... . .. ....... .... ... ... . ...... .... . .. .. . ....... ... FIG 39-2: . ... . . . .... ... .... . . .. .. .... .... . ........................................................... ............ ...... 0.2 :........:....:... ...............:....:..... ....... 0.1S(r). x-C(w) ...... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :.... :...... -0.3 :.... :.... :.... :.... ..... ..... ..... .... ......-0.5 ....:...:....:....:... ...:....:....:....:....:....:.-0.7 39:14:2 39:14:3dzfdx2T/2 Il + (dx/ J01 +dt) dtbetween the origin 0 and point P. These two quantities are, in fact, w y tar and wv 2%a, respectively, where wis the paramctcr dcfincd in 39:14:1. CHAPTER40THE ERROR FUNCTION erf(x)AND ITS COMPLEMENT erfc(x) The functions of this chapter are interrelated by 40:0:1 erf(x) + erfc(x) = Iand occur widely in problems of heat conduction and similar instances of the diffusion of matter or energy. Thename of the function arises from its importance in probability theory [see Section 40:141. 40:1 NOTATIONThe function erf(x) is also known as the probability integral and is sometimes denoted H(x) or 4)(x). The relatednotations (b1(x), 4);(x), etc., then denote successive derivatives of the error function 40:1:1 4)"(x) =derf(x)Sometimes the initial letter of the erf and erfc notation is capitalized without change of meaning, but Erf(x)and Erfc(x) may also denote ('/2) erf(x) and ('c/2) erfc(x), respectively. Changed arguments are common,the name "probabilityintegral"or the Gauss probability integral often being givento erf(x/'v 2) or[erf(x/V 2)]/2.40:2 BEHAVIORFigure 40-1 includes maps of erf(x) and erfc(x). both of which are sigmoidal functions that rapidly approach limitsas x -, _w. These limits are ± I for erf(x) and 0 or 2 for crfc(x).40:3 DEFINITIONSThe most useful definitions of these functions are as the indefinite integralssgn(x) f ezp(-t) 40:3:1erf(.r) _ - Jexp(-t')dt =Jdta oVtroVr 395 40:4THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x) 386 ............................................... 1.2 and 40:3:2 erfc(x) =Jmexp(-r2)dt = I - erf(x)but definitions as the definite integrals2 rcxp(-r2) sin(2xt) 40:3:3 erf(x) = -Jdtu2x 40:3:4 erf(x) _ - J exp(-x-t2)dtoas well as others. also apply. 40:4 SPECIAL CASESThere are none. 387THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x)40:5 INTRARELATIONSIIIPSThe error function is an odd function40:5:1 erf(-x) = -erf(x)while its complement obeys the reflection formula40:5:2 erfc(-x) = 2 - erfc(x) 40:6 EXPANSIONSThe error function may be expanded in many ways, includingszj40:6:1erf(x)?x-x+ x -_ x(-x)310v"Goj!(j+i)22x34xs xzi+140:6:2elf(x)- exp(-xz) x + - + - + = exp(-x')315l'(j+ 40:6:3erf(x) = V-2[11 12(x') - l312(xz) - I...(xz) + 1712(x2) + ...) _ , (-1)'li,i/z(x2)1-040:8 2 +l=lntr4while its complement is expansible asymptotically aszexp(-x) 1315(2j - 1)!! 40:6:4erfc(x)- x- x X V'ir2xZ4X4ix-;(-2x2)The functions 1 and l,, ,z are discussed in Chapter 43 and Section 28:13, respectively. See Section 41:6 for otherrelated expansions. 40:7 PARTICULAR VALUESIncluded in Table 40.7.1 are values that have a relevance to probability theory [Section 40:14]. 40:8 NUMERICAL VALUESThe following algorithm calculates numerical values of the error function or its complement with 24-bit precision(i.e., the relative error in erf(x) or erfc(x) never exceeds 6 x 10-1) for any value of the argument. The erf(x) function is computed if the code c is set equal to 0; erfc(x) is evaluated when c = I is input. For -1.5 s x <1.5, the algorithm uses a concatenated form of expansion 40:6:1, truncated at j = 3 + lnt(91xl). Note that thistruncation actually generates more precision in erf(.r) than is necessary but that this gratuitous accuracy is neededto preserve sufficient precision on complementation to give erfc(x) when x is close to 1.5. For x > 1.5, the algorithmutilizes the continued fraction expression 41:6:3 in the truncated form Table 40.7.1 x e 0x a 0.476936276 erf(x)-10 0.682689492 1 erfc(x) 2 1 0.317310508 0 40:9THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x) 40:8:1erfc(x)2/a exp(-x) 1232 + Int(32/x) _.v-2+x2xV2+ xa%2+x\'2388 The same continued fraction is employed for x < -1.5, with -x replacing x. and the reflection formula 40:5:2 isthen utilized.Some care is needed in programming this algorithm. For instance. the final parentheses in the penultimatecommand should not be omitted. If they are, the altered order of operation may cause insufficient precision in anoutput value of the error function complement. For example, erfc(3.5) = 7.43098372 x 10-7 but, on a computingdevice that retains only 10 significant digits, the result of the calculation erfc(3.5) + I - 1 is 7.43000000 x 10-'.Numerical values of [n erf(x)1/2x are also available via the universal hypergeotnetric algorithm of Section18:14. Input code cInput x »:Storage needed: c (the code),x..iandf1f1.5<1x;goto(2)Set j = 3 + lnt(94xj)Set f = I(1) Replace f by 1 + fx2(_: - j)/[ j(1 + ill codeoutputReplace j by j - 1lfj*0goto(I)0erf(x)Ierfc(x)Replace f by c + fx(2 - 4c)/V rGo to (4)(2) Replacec bySet j = 3 + lnt(32/[xl) Setf=OTest values:erf(1.45) = 0.959695026(3) Replace f by 1 /[ j + 2x21 erfc( 1.45) = 0.0403049740Replacejbyj - 1Ifj*0goto(3)Replace f by f(c2 + c - 1)(V L/'R) exp(-x2) + (I - c)(4) Output ffo - erf(x) 1A - erfc(x) 40:9 APPROXIMATIONSFor small arguments, one can use 40:9:1erf(-0.5)-0.520499879crfc(-0.5) = 1.52049988erfc(5) = 1.53745979 x 10_12 3.372xerf(x)+8-bit precision-0.96 < x s 0.963xwhereas for large values of x40:92 1. 132x exp(-x2)erfc(x) = 8-bit precisionx z 2.72x22 + 1 40:10 OPERATIONS OF THE CALCULUSDifferentiation gives 40:10:1dx erf(bx + c)-dx erfc(bx +c) _-exp(-b2x'' - 2bcx - c'') while successive differentiation gives rise to Hermits polynomials [Chapter 241 389THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT crfc(x) d"d"2 40:10:2-axe erf(bx) =dx"erfc(bx)(-b)"n = 1, 2, 3, ...y'aIndefinite integration gives40:10:3 and1 - exp(-b'xz)rerf(btkis = x erf(bx) -obV40:12 ierfc(bx)exp(-bzxz)40:10:4 jerfc(bt)dt =b=b- x erfc(bx) where the icrfe function is discussed in Section 40:13. The same section discusses repeated integrals of the errorfunction complement.Of widespread occurrence are indefinite and definite integrals of the product of an exponential function withan error function or complementary error function. Such integrals will be found in Section 41:10 or may be evaluated via the table of Laplace transforms in Section 26:14. Other definite integrals are listed by Gradshteyn and Ryzhik[Sections 6.28-6.311 and include-rl 1 +)40:10:5 Jt` erfc(bt)dt =\bb > 0v > - Io( l + V )zz40:10:6 sin(Bt) erfc(bt)dt =1 - expp ( B 14b)B > 0b > 00 40:11 COMPLEX ARGUMENTIf the argument in definition 40:3:1 is replaced by the complex variable x + iv. one obtains a complex functiongiven by(((l40:11:1erf(x + iv) =2I exp(y') J exp(-rcos(2vt)dt + exp(-x') Jo exp(t') sin(2xt)dzJ V a Iexp(yr) J exp(-r=) sin(2vt)dt - exp(-x') J Vexp(C) cos(2xt)dtJin terms of four integrals that are not evaluable in simpler terms. When the real part of the argument is either zeroor infinity, however, reduction occurs to the simple results 40:11:2 2ierf(iv) _ - exp(v') daw(y)Vii40:11:3 erf(- + iy) = I - i erf(y)where daw is Dawson's integral, discussed in Chapter 42.Confusingly, the function generally known as the error function of complex argument, and denoted by thesymbolism W(x + iy), is not that given by 40:11:1 but is instead defined by equation 41:11:1 of the next chapter. 40:12 GENERALIZATIONSThe error function and its complement generalize to the incomplete gamma functions of Chapter 45, of which theyare the v = = cases 40:13THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x) 39040:12:1 V v erf(x) = sgn(x) Y(2;x2)40:12:2 \ erfc(x) = r(2;x2)x > 0The error function complement may also be considered as a special case of the parabolic cylinder functiondiscussed in Chapter 46:2x2 40:12:3 erfc(x) =aexpl 2Because the incomplete gamma functions and the parabolic cylinder function themselves generalize to theKummer function [Chapter 47] and Tricomi function [Chapter 48], the error function and its complement may beregarded as special cases of those functions. The relationships areVa 40:12:4 erf(x) = M(2;-2;-x2) = exp(-x2) M(1;-,;x)it 40:12:51/a erfc(x) = exp(-x2) U(2;;;x2)=exp(/'x')U(1;x2) 40:13 COGNATE FUNCTIONSx>0 A set of functions is defined by the integral2 40:13:1i'erfc(x) _Ja (tx)n exp(-t2)dtn = 2. 3. 4... . n!The notation is sometimes extended to embrace 40:13:2i'erfc(x) = icrfc(x) _ 2J=(t - x) exp(-t2)dt = Jerfc(Odt 40:13:3 i°erfc(x)2exp(-t2)dt = erfc(x)Vuand even2 40:13:4 i'erfc(x) _exp(-x2)The ierfc and i"erfc functions are known as the complementary error function integral and the repeated integralsof the error function complement, respectively. The behavior of some of these functions is shown in Figure 40-1.Because values for x < 0 are seldom of importance in practical applications, these are excluded from the diagram.The power series expansion 40:13:51(-24'i'erfc(x) _2 1-0;!r(l+12,)shows that the functions have the particular values 40:13:6 ierfc(0) =12T(I+ 2/of which the first few are shown in Table 40.13.1. For n a 1 i'erf/c(x)approaches - as x -o -a and approacheszero asymptotically as x -+ « according to the asymptotic expansion 391THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x) 40:13Table 40.13.1 Mi'erfoo) 40:13:7i"erfc(x) -2 exp(-x2) r(n + IXn + 2)(n + 1)(n + 2)(n + 3)(n + 4)IL 1 -V n(2x)'4x+32x4(n + 2j)!- ... ++ ...X- xn! j!(-4x2)JThe functions obey the recurrence relationship 40:13:8i"erfc(x) =ni""'erfc(x) + 2rt i"-''erfc(x)n = 1, 2, 3, ...and sufficient applications of this formula permit any of the i"erfc(x) functions to be expressed in terms of erfc(x)and exp(-x2), and hence evaluated. Early examples are 40:13:9zierfc(x) = ex )- x erfc(x)n1+2x2x 40:13:10 i2erfc(x) =4erfe() - i,;,= exp(-x') andl+x'3x +2x32-'- 40:13:11 x)12 ierfc(x) =6Virexp(erfc(x) Alternatively, these functions are evaluable via the universal hypergeometric algorithm of Section 18:14.The repeated integrals of the error function complement satisfy the following operations of the calculus:d 40:13:12 - i"erfc(x) = -i"-'erfc(x)dx 40:13:13Ji"erfc(t)dt = i'-'crfc(x)it being the latter property that explains the name of the function family. It is possible to generalize the family toi'erfc(x), where v is not necessarily an integer. by using procedures of the fractional calculus [see Oldham andSpanier. Section 10.5]. The differential equation2fdf 40:13:14 ad-+x--nf=0a>0dx'dxis solved by f = c, i'erfe(x/V 2a) + c2 i"erfc(-x/V 2a), where c, and c, are arbitrary constants.A useful definite integral is 40:13:15 exp(Bt)i"erfe(bt + c)di = Bi"erfc(c) +-uexp(Bt)i"-'erfe(bt + c)dt8fb 40:14THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x)40:14 RELATED TOPICS As discussed in Section 27:14. the so-called normal distribution function is given by_z40:14:1 f(x) = 1e(x - 10ay'2ar\2a'392 in terms of the mean µ and variance o= of thedistribution, the corresponding cumulative function being40:14:2 F(x) = J=f(t)dr = + l erfl x - ILairRandom events often obey. or are assumed to obey, distribution 40:14:1, and the quantity Ix - µ1 is termed theerror of a single measurement or observation of x. Accordingly, a is known as the standard error: about 68% of all normally distributed samples lie in the range µ - or 5 x s µ + a. A similar concept is the probable error.equal to 0.6745u. Because erf(0.6745/\ 2) = erf(0.4769) it follows that the probability of a normally dis- tributed measurcmcnt x lying in the range µ - 0.6745a S x s µ + 0.6745a is exactly one-half so that as manyevents lie outside this interval as within it.The functions 40:14:31f(x)exti 2iP(2and 40:14:4F(x) = J"f(t)dt =l jerf1-I =1 -erfc(,X'2)are normalized versions of 40:14:1 and 40:14:2 known as the standard normal densin function and the cumulativestandard normal probability function, respectively.A common need in simulation studies is to create a set of numbers that mimic random errors. having a specifiedmean µ and variance az. Such a set may be constructed by the formula//N\//2w40:14:5 x,+2az InInfcoalN n,"I``where N is a large integer and n, and ni. , are each random integers in the range 0 < n s N. Knuth recommendsN = 199017 and the formula40:14:6n,., = 1 + (24298n, + 99991) mod(199017) j = 0, 1, 2, ...as a means of creating successive pseudorandom integers. The "seed" no is selected arbitrarily in the range 0 <no s 199017, and the integers n, are confined to the same range.An algorithm follows for generating a sequence of J successive numbers that are normally distributed about amean s with a variance ox. It uses formulas 40:14:5 and 40:14:6, taking 2J as the seed. Input µ >Input a >Input J >Set N = 199017Set S = 2azSet j=n=2J(1) Replace n by 24298n + 99991Replace n by I + n - N Int(n/N)Storage needed: µ, N, S, j, n and x Use radian mode.1 393THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x) 40:14 x=xi«Replace j by j - IIf frac(j/2) = 0 o to (2)Set x =S In(N/n)Goto(1)(2) Replace x by µ + x cos(2un/N)Output xIfj*0goto(1)Test values:µ= 5.rr=2,J=6x, = 5.46780187x, = 2.27940439x3 = 4.56323336x, = 3.85012900xs = 5.10798656xb = 6.18719458 CHAPTER41THE exp(x) erfc(V) AND RELATED FUNCTIONS Although the functions considered in this chapter are composites of those discussed in Chapters 12, 26 and 40,their practical importance warrants a separate treatment. Moreover, many of the properties of these functions tran-scend those of their component factors.Primarily, this chapter is devoted to the following functions, each of which involves the product of an expo-nential function and an error function complement:41:0:1 exp(x) erfc(V)41:0:2exp(x) erfc(-V) = 2 exp(x) - exp(x) erfc(\)41:0:3 \ rrxexp(x)erfe(f) 41:0:4 1+ exp(x) erfc(-V) = 2 exp(x) +1- exp(x) erfc(vx) and41:0:5 exp(x-) erfc(x)All these functions arise in physical applications, such as problems in diffusion. The similar set of functions withthe error function replacing its complement are also important; these are addressed more briefly in Section 41:13. 41:1 NOTATIONSymbol variants such as (2/V)e Erfc(P'') and [I - 4r(V )] cxp(x) occur, as explained in Sections 12:1, 26:1and 40:1. In addition, a unitary notation is sometimes adopted; for example, erc(x) and eerfc(x) have both beenused as abbreviations for exp(xr) erfc(x). 41:2 BEHAVIORMost of the functions of this chapter are defined as real functions only for arguments x ? 0, and this is the onlyrange shown in Figure 41-1.The function exp(x) increases rapidly in magnitude as x increases, whereas erfc(Vx) suffers a dramatic decrease.The competition between these two effects causes exp(x) erfc(Vx) to diminish in a very leisurely fashion toward 395 41:3 THE exp(x) erfc(f) AND RELATED FUNCTIONS 396 O'LORCOOO0IV11Y'll...................:........................:....... 1.0exp (x) erfc xp (x2) erfc (x) 0.8 :...... :....:....:....:................. 0.6:../: ....:....:....:......-ex (x) erfc (rx) ....:....:. .04. ' ....................................... :....:....... 0.2FIG 41-1 : zero as x -. x, whereas \x exp(x) erfc() slowly approaches a unity asymptote in the same limit. These trendsare mapped in Figure 41-1.No such compensation influences the functions exp(x) erfc(- V) or (I / V) + ex,Lx) erfc(- V x-), both ofwhich increase exponentially as x - x as depicted in Figure 41-2. The function (1/Vex) + exp(x) erfc(-V)displays a minimum value of about 3.1 close to x = 0.20. The same argument corresponds to a point of inflection(Section 0:71 in exp(x) erfc(-f), as reported in Section 41:7. 41:3 DEFINITIONS The exp(x) erfc(V) function may be defined as the indefinite integral 1- t)dt2cxp(x - ts)dt 41:3:1exp(x) erfc(V)Jz exp(x_"7 eor via the definite integrals I 41:3:2exp(x) erfc(Vx)exp(-t)dt2\/Xexp(-tt)dt,fat+xifotr+x2I 41:3:3exp(x) erfc(\) -Jexp(-,2 - 2tf )dtoandxexp(-xt)dtIexp(-xt)41:3:4exp(x) erfc(\)Jor + 1A Jo (f + 1)The integrals in 41:3:4 may be regarded as Laplace transforms [Section 26:14], in applications of which the func-tions of this chapter play a distinguished role. 397THE exp(x) erfc(V) AND RELATED FUNCTIONS 41:3Functions 41:0:1-41:0:5 all satisfy rather simple first order differential equations, namely 41:3:5 41:3:6df+1f = cxp(x)[crfc(--V) + cldx- f=:Trx dx-\1 + 2x/f =-1f = V exp(x)[erfc(Vx) + c)iPtttyOy FIG 41-2!:! 1: 41:4 THE exp(x) erfc(f) AND RELATED FUNCTIONS 398 41:3:7 41:3:8dff --1f =+ exp(x) [erfc(- V) + c]dx2ax' - 2xf = f = exp(x2)[erfc(x) + c]where c is an arbitrary constant. Thus, the various functions may be defined as the special c = 0 cases of thesolutions to these differential equations.The fractional calculus may be employed to define some of the functions of this chapter. Thus, with lowerlimit zero, semiintegration of the exponential function gives d- 1/241:3:9dx_j/, exp(x) = exp(x)erf(V) = exp(x) - exp(x) erfc(f)Function 41:0:4 is preserved on semidifferentiation; that is: 41:3:10d"'fdxFr= ff==+exp(x)erfc(-f)'faxso that this function plays a role in the semicalculus analogous to that played by the exponential function in theclassical calculus. 41:4 SPECIAL CASESThere are none.41:5 INTRARELATIONSHIPSThe reflection formula41:5:1exp(x) erfc(- V) = 2 exp(x) - exp(x) erfc(f)applies. The addition formula41:5:2ex (x + y) erfc(x +exp(y) , (2j - 1)!! y(j + I;y) involves a series of incomplete gamma functions [Chapter 45]. 41:6 EXPANSIONSA power series for exp(x) erfc(±V) X4'x=VXY 41:6:1 +x+-V3a2'-0r 1+?involves the gamma function [Chapter 43]. Alternate terms may be summed separately:xzr2r4x2 41:6:2exp(x) erfc(±Vx) = 1 + x + Z +r 2al +3+ 15 +Jx'+T(2x)i-j' +2 V/n(2j + 1)! i= exp(x) + exp(x)erf(1rx) Of course, replacement of f by x in 41:6:1 gives the power series expansion of exp(x2) erfe(±x).The exp(x) erfc(f) function may also be expanded as a continued fraction: 399THE cxp(x) erfc(Vx) AND RELATED FUNCTIONS 41:6:3 2/n1234 exp(x)V2x+ V2-x+ V -2x+ V2r+41:8 Truncated versions of the asymptotic series 41:6:413(2j - 1)!! Irx exp(x)1 -2x+4xZ++ ...X --O >o(-2xYare useful for large x. The truncation error is less in absolute value than the first neglected term and of the samesign.It follows f r o m 41:6:2 that the function (l / V nx) + exp(x) erfc(- Vx) is expansible as the series I-IfzrV-ulr = exp() +1(2x)V rcx =u (2j-1)!!with uniformly positive terms.41:7 PARTICULAR VALUES ezp(x) erfc(V x)exp(x) erfc(- Vx)V i exp(x) erfc V )+ exp(x) erfc(-f)V nxezp(xr) erfc(r)11Fj+Il2 x--xx - 0 x - 0.204053939x=x undefundef1 I0.6413044461.81142415 = inflection0 undef0 0.513465985 I undef 3.06039578 = minimum x I 0.805767647 0 41:S NUMERICAL VALUESThe algorithm presented below generates values of exp(x) erfc(Vx) to 24-bit precision (i.e., the relative error inthe output does not exceed 6 x 10-') for any input x ? 0. Because exp(x) erfc(V) does not exist as a real numberfor x < 0, the input of a negative number -x is treated by the algorithm as an instruction to calculate the alternativefunction exp(x) erfc(- Vx), using identity 41:0:2. If the facility to calculate this alternative function is not needed,the portion of the algorithm shown in green may be omitted.The algorithm resembles that in Section 40:8. For x < 2.25, it uses the concatenation 41:8:1erf(V)=((( r/(1-J)+Il x(?- J)+1}x(i - J) .(J - 1)(J - 1)(J - 2)(J - j)+...+II-l.r+II -dx+1122xIxVVVVrtwhich is based on expansion 40:6:1, truncated with J = 3 + 1nt(9V). For x > 2.25 a continued fraction expansionanalogous to 40:8:1 is utilized. Self= g = 0 Storage needed: j', g, x and jInput ± .s >>>I1f .s 0goto(l) 41:9 THE exp(x) erfc(U'x) AND RELATED FUNCTIONSReplace x by -xSet g = 2 ex[(-v)(I) If 1.5<Vxgo to (3)Set i = 3+ Int(9V )Setf= 1(2) Replace f by I + fx('12 - j)/[i('/2 + j)]Replace j by j - IIfj * 0 go to (2)Replace f by exp(x)[ I - 2f V x/n]Go to (5)(3) Set j = 3 + Int(32/f)(4) Replace f by I /(fj + 2xReplace j by j - 1 Ifj*0goto(4)Replace f by j(5) If g = 0 go to (6)Replace f by g - f(6) Output ff = exp(x) erfc(±ti x) 41:9 APPROXIMATIONS400 Input restrictions: When x 2: 0 is input, exp(x)erfc(Vx)is output. When -x 5 0 is input. exp(x)erfc(-V_x)is output. Test values:exp(l) erfc(l) = 0.427583576exp(rr)erfe(Vc) = 0.282059176exp(l) erfc(- I)5.00898008exp(rr) erfc(- V a) = 45.9993261 The approximation 40:9:2 is easily adapted to provide an estimate of the exp(x) erfc(V) function for large x. Amore accurate approximation, valid for all arguments, is 41:9:1V,-rx exp(x) erfc(V)2Cl+xexp(5x/7)/This approximation exploits the inequality 419.2 2< \ ex-(x) erfc(f) s 2F2+ ++arx10-bit precision 41:10 OPERATIONS OF THE CALCULUSDifferentiation yields 41:10:1dexp(bx) erfc(-- Vbx) = b exp(bx) erfc(±V )dx Vax 41:10:2exp(bx) erfc(V') =it(bx + '/2) exp(bx) erfc(V ) - Vb= 1 + b exp(bx) erfc(- Nlrb-x) - 41:10:3d\I+ exp(bx)erfc(-))Jax dcirbx 2bx 41:10:4d&exp(axs) erfc(bx) = 2 exp(ax')[ax erfc(bx) - bb exp(-b-x')n 401THE exp(x) erfc(f) AND RELATED FUNCTIONS 41:10Repeated differentiation of exp(x) erfc(±f) regenerates the original function together with a finite series of al-gebraic termsd" II3 41:10:5dx-" exp(x)erfc(±f)=exp(x)erfc(-f)+ V ax2V ax4-(2j -= exp(x) erfc(±f)1I)!!=o(-2x)iThe series is identical with the leading members of the asymptotic expansion 41:6:4.Indefinite integrals includer+41:10:6J,exp(-b212) erfc(br)dt = 4b [1 - erfc'-(±bx)]041:10:7f exp(Bt) erfc(bt + c)dt =exp(Bx)[erfe(bx + c) - erfe(c)]0-- cxp(46:-bcc -f - JJ2b2b(\-exp(2V aB))exp(-2aB)/41:10:8J exp(-Br) erfc,VYJ t/Idr =2BerfcY x+ x +erfc,\x -/Bx0ex Bx)erfclx)B > 0B J exp(Bt) erfc(bt + c)dt = B [exp(Bx) erfc(bx + c) - erfc(f) + 4,(x)[0 41:10:10dt(x) _ B>b Representing the error function by (P, Gradshteyn and Ryzhik (Sections 6.28-6.311 list over thirty definiteintegrals relevant to this chapter. Some of the more useful areBx+b)-erfc(c - b)]B<b 2exp(-c)[Vbbx+c- Ve]B=btr46 ctc1ir(B - b)exp(-c)I exp(Bz - br) dawl V B x + b I - daw( V-c)Ja>0x>0 a>0artanh(f/b) 41:10:11 Joexp(at) erfc(bt)dt =b1fa = 0arctan(V a/b)Vaaa<0 41:11 THE exp(x) erfc(f) AND RELATED FUNCTIONS 402 V/k(t =Frv+ I-v+2 v+3'-2lb2 > av > -1 41:10:12r" exp(at2) erfc(bt2 oV ar(1 + v)b""\22'ab2 41:10:13Jexp(Br) erfc(\)dr =0I /(b-B+b-bB)B <0 {b/(b - B) - 1J/B 0 < B < b exp(br) erfc\ sec(vi) r(v) 41:10:14Jdt=0<v<'1=ot"b` I,With zero lower limit, the operations of semidifferentiation and semiintegration gived'/2 1'ffX 41:10:15 exp(bx)Vbx) _z V exp(bx) erfc(±) andd-1/2 +141:10:16 dx_inexp(bx) erfc(±\, s) _7(1 - exp(bx) erfc(±V)] 41:11 COMPLEX ARGUMENTThe so-called error function of complex argument is. in fact, the function exp(x) erfe(-\) with x replaced by-(x + iv)2. It is defined by rr41:11:1W(z) = exp(-22)1 1 +2i- J exp(r')dtI= exp(-:) erfc(-:z). = x + iyoThis is an important complex-valued function, the real and imaginary portions of which arise in practical problems.Accordingly, its values have been tabulated (for example by Abramowitz and Stegun, Table 7.9). or, alternatively,numerical values may be calculated via expansions 41:6:2 and 41:6:5. which remain valid when x is replaced by-(x + w2-The real and imaginary parts of W(x + iy), defined by41:11:2 W(x + iv) = Re{W(x + iv)} + i lm{W(x + iy)}are expressible as the integralsexp(-t2)dt 41:11:3 Re{W(x T ry)} _7F(x - t)2 + Y2 and 41:11:4 Im{W(x + ivI(x - r) exp(-t2)dt)} _ -tr(x-t)2+y1When the argument of the W function is real (i.e., y = 0), the function has real and imaginary parts given by 41:11:5 W(x) = exp(-x2) +2idaw(x)but when the argument is purely imaginary (i.e., x = 0), the function is real41:11:6 W(iy) = exp(y2) erfc(v) 403THE exp(x) erfc(V) AND RELATED FUNCTIONS41:12 GENERALIZATIONSThe exp(x) erf(V) function is a special case of the Kummer function [Chapter 471a 41:12:1M(1 ;1;x) _ -jX exp(x) erf(f) =I2 Ya[exp(x) - exp(x) erfc(\)141:13 The asymptotic representation 41:6:4 of the \ exp(x) erfc(Vx) function generalizes to the simple hyper-geometric function [Section 18:14] 41:12:2of which it is the a =special case. 41:13 COGNATE FUNCTIONS(a),(-x)' Functions formed by replacement of erfc in formulas 41:0:1-41:0:5 by erf are almost as important as the originals.We do not explicitly treat such cognate functions in this Adas because their properties arc so easily deduced fromthose of the primary functions of this chapter via the identities41:13:1and 41:13:2exp(x) erf(V) = exp(x) - exp(x)erfc(V) exp(x) erf(V) _exp(x) erfc(-f) - exp(x) erfc(\)2In addition, explicit reference to the cxp(x) erf(f) function is made in equations 41:3:9.41:6:2 and 41:13:1. Figure17-2 includes a graph of exp(x2) erf(x).An expansion in terms of hyperbolic Bessel functions [Section 28:13] and particular values of the exponentialpolynomial [Section 26:131 is41:13:3exp(x) erf(f) _=owhile a more conventional expansion ise,(-1)(V)'1,.x,2(2\ r) expx cx -Jx [)x4x2-2J-I+-+-+.-x(2z)'- V315LJw,o (2j + 1)!rThe latter is useful for calculating values of cxp(x) erf(f ); for example via the universal hypergeometric algorithmof Section 18:14. CHAPTER42DAWSON'S INTEGRAL The functions of this chapter have much in common with those of the previous chapter. Dawson's integral daw(x)is related to (V'/2) exp(-x2) erf(x) via the identity-iV42:0:1 daw(x) =2exp(-x') erf(ix)in much the same way that sin(x) is related to sinh(x) [see 28:11:31. 42:1 NOTATIONBecause no standard notation exists, this Atlas uses the symbol daw(x) for Dawson's integral of argument x. Abra-mowitz and Stegun [Chapter 71 use F(s) and, in the light of relationship 42:0:1, Gradshteyn and Ryzhik adopt asymbolism equivalent to -i(V /2) exp(-x') erf(ir).Sometimes the name "Dawson's integral" is given to exp(x2)daw(x) and Erfi(x) or erfi(x) is emplod to denotethis function. A rescaling of the argument often leads to simpler formulas; the function Vi daw(Vx/2) has been symbolized D(x). 42:2 BEHAVIORThe functions daw(Vx), daw(x) and x daw(x) are all important and all three are mapped in Figure 42-1. Each ofthese functions is zero at x = 0, displays a maximum in the vicinity of x = 1 and then slowly declines, approachings asymptotically in the case of x daw(x) and zero in the other two cases. 42:3 DEFINITIONSThe usual definition of Dawson's integral is through the indefinite integral 42:3:1 daw(x) =rrexp(rxj)dt =sgn(x)`exp(: - xI)dt 02JoVJ-'t 405 42:4 DAWSON'S INTEGRAL 406hOhOhOhids44'44'1.4'1.4'.,P:i*:....:.... .................: 0.6xdaw (x) : :..... ..:......:..............:.... :....:.............. 0.3.. ......dart.....'0.2 FIG 42-1:......:. ..:....:....:..............:....:....:....:....:....:....:....:....:.0.1 0 but it may also be defined as an inverse Laplace transform [see integral 42:10:7], in terms of the error function ofcomplex argument [Section 41:11] or as the semiintegral 42:3:2with lower limit zero.Dawson's integral may be considered to arise as the complementary function [see Murphy, page 82J in thesolution of the very general first-order differential equation 42:3:3 of+bxf=cdxwhere b and c are constants. The general solution isIbs\bx'42:3:4f=2cb dawl 2 I+ (constant) exp 2 42:4 SPECIAL CASES There are none. 42:5 INTRARELATIONSHIPSDawson's integral is an odd function:42:5:1 daw(-x) = -daw(x) 42:6 EXPANSIONSThe power series expansion 42:6:1idaw(x) = x -+15- ... = x E(-2x')-u(2j+ 1)" 407 DAWSON'S INTEGRAL 42:7may be written in the alternative forms 42:6:2or as the continued fraction 42:6:3The expansiondaw(V x)nxi (-x)=V'_ i (-x)'2_o r(j + 1)J=O(1)f daw(12)=x/2x3xSx1+3-x+ 5-x+ 7-x+ 42:6:4daw(V x) = exp(x)+ - + - +_ V exp(-x)'x3xsJ310j! (2j + 1)may be developed into42:6:5daw(V x) ='I-, (2j - I)!!+(2n - 1)!!V exp(-x) x+2 V x i-o(2x)'(2x)";_o j!(2j - 2n + 1)for n = I. 2. 3, .... As n -y m, this becomes the asymptotic series 13(2j - 1)!! I 42:6:6daw(x) '2x + 4x'+ 8x5 + ...+ (2r-)Zx+ ...X-* xvalid for large x. 42:7 PARTICULAR VALUESAll three functions acquire the value daw(I) = 0.538079507 at unity argument: daw(V x)daw(x)x daw(x)x = -xx = -1x = 0.r = Ix = = un def undef0dawn)00-daw(l) 0daw(l)0 3daw(1)0daw(l) _ The values of the maxima and the inflection of each of the functions are related in a simple way to the cor-responding arguments: Maximum Inflection(s)Value of XValue offValue of xValue off2x, + I0.854032657 = xM 1.84365320 = .r; 0.924138873 = Vru 1.50197527 - x,4Vc 2VxMX 5956772781102.r - Ix' - I1.50197327 = x,2r'-I.11.95562141 J2xt - 3 The particular arguments xN and x which correspond respectively to the maximum and inflection point of thedaw(f) function, are cited in Section 37:14 as the lower limits of important indefinite integrals. 42:842:8 NUMERICAL VALUESDAWSON'S INTEGRAL 408 The algorithm below evaluates daw(x) to 24-bit precision (i.e., the relative error never exceeds 6 x 10-2) for anyinput value of x.For arguments in the range 0 < jxj < 5, the algorithm uses a double concatenation formula, based on expansion42:6:5. With J set to 1 + lnt(I2V) and n to Int(x2), the expression 42:8:1f=(( 1x'1++ - + - x2 exp(-x2)i - nI" - nIfx2+1x2Ix2+; -n) J-1-n/J-1 J-J-n J-2 is first evaluated. This value then initiates the second concatenationO - n)fnz-n-;-z142:8:2daw(x) =(I (...((-x=+ t/l-x'+ 1/l-X=+ ... +1)-x'+ 1)-x2+ I)2VFor GxI ? 5, the asymptotic expansion 42:6:6 is employed, being suitably terminated and evaluated by the sameset of commands that are used by 42:8:2. Input x >> f = daw(x)I,»»If 5 5 IxI go to (3) Set J = I + Int(I2-%L )Set P = J +Int(x2)Setf = x2/pIff=0goto(5)(1) Replace p by p - IReplace f by(1 /p)Jx2Replace J by J - 1IfJ*0goto(I) Replace f by f exp(-x2)(2) If p > O go to (4)Replace f by I - (pf/x)Replace p by p + IGo to (2)(3) Set p = -; - Int(30/IxI)Setf= IGo to (2)(4) Replace f by f/2x(5) Output fK««Storageneeded:x, J. pand f Test values:daw(O) = 0daw(0.7) = 0.510504058daw(-2) = -0.301340389daw(10) = 0.0502538471 Alternatively, the universal hypergeometric algorithm of Section 18:14 may be utilized to evaluate Dawson'sintegral. 42:9 APPROXIMATIONSFor small and large arguments, one may use the approximations 42:9:1daw(x) = 3 (3 - 2x2)8-bit precision -0.34 <_ x:5 0.34 409 DAWSON'S INTEGRAL 42:10 and 42:9:2 respectively.xdaw(x) -2x+ I8-bit precision ki z 3.94x3 42:10 OPERATIONS OF THE CALCULUSDifferentiation gives 42:10:1 42:10:2ddxdaw(bx) = b - 2bx daw(bx) d 1bdaw(V) = 2X - b daw(\)Repeated derivatives involve Hermite polynomials [Chapter 24112j2J+1=1,3,5,d^ ,_o(-2)1(2j)! 42:10:3- daw(x) _ (- l)" H^(x) daw(x) -dx'2n=2J=2,4,6,...-Q (-2)'(2j + 1)! Indefinite integrals include 42:10:4 and 42:10:5jxdaw(y nx)aw()dt=-d Jr daw(bt)dt =bx - daw(bx) o 2b2The indefinite integral jdaw(t)dt is not expressible in terms of any named function. It is, however, quite a simplehypergeometric function [see Section 18:141 42:10:6J;daw(t)dt = xL(1)J(-x=)'02The first three of the definite integrals listed below may be regarded as Laplace transforms [Section 26:141 butare included here because of the importance of these particular integrals: 42:10:7J daw(\)exp(-st)dt =Jo2Vs(s + b) 242:10:8daw(bt) exp(-st)dt =46 exp(\4b'1Ei(_.)b > 0s > 0jJt daw(bt) exp(-st)dt =Z1s +Eil -S Ib > 0s > 0o42:10:10J1 exp(-at')daw(bt)dt4(b'ba)a > 0b > 0042:10:11f sin(Bt) daw(bt)dt = b expl 4bB > 0 42:11 DAWSON'S INTEGRALWith lower limit zero, the semiderivative and semiintegral of daw(Vx) are 42:10:12dueV--jr ,daw(\) = 2 exp(-x) andd-'I2Vx=1 -- d 42:10:13 aw()exp(x)12 1 ue/2 42:11 COMPLEX ARGUMENT410 The relationship of Dawson's integral to the error function of complex argument W(x + iv) is discussed in Section41:11. With a purely imaginary argument one findsi V n42:11:1 daw(iv) =2exp(y) erf(y) 42:12 GENERALIZATIONSDawson's integral is a special case of the incomplete gamma function [Chapter 45] 42:12:1 daw(x) =2x exp(-x2)and of the Kummer function (Chapter 47)42:12:2 daw(x) = x M(1;;;-x2) 42:13 COGNATE FUNCTIONSDawson's integral is the n = 2 instance of functions defined by 42:13:1 rexp(r" - x")drn = 2, 3, 4, ...The other members also have some importance. All display a peak in the vicinity of x = 0.7. CHAPTER43THE GAMMA FUNCTION r(x) The gamma function is unusual in the simplicity of its recurrence properties. It is because of this that the gammafunction (and its special case, the factorial) plays such an important role in the theory of other functions. Thereciprocal 1/r(x) and the logarithm ht(r(x)) are also important and are discussed in this chapter, as is the relatedcomplete beta function B(x.y), which is addressed in Section 43:13.Formulas involving the gamma function often become simpler when written for argument 1 + x rather than x,and we have sometimes taken advantage of this fact. Because r(x) = r o + x)/x, a change of argument is readilyachieved. 43:1 NOTATION The gamma function is also known as Euler's integral of the second kind. r(1 + x) is sometimes symbolized x!or n(x) and termed the factorial function or pi function, respectively. To avoid possible confusion with the functionsof Chapter 45, r is distinguished as the complete gamma function. 43:2 BEHAVIORThe behavior of r(x) for -5 < x < 6 is shown on the accompanying map, Figure 43-1; it is complicated. Forpositive argument, the gamma function passes through a shallow minimum between x = I and x = 2 and increasessteeply as x = 0 or x = x is approached. On the negative side, r(x) is segmented: it has positive values for -2<x<-1.-4<x<-3,-6<x<-5,...butnegative values for-1<x<0.-3<x<-2,-5<.x<-4, ..., with discontinuities at x = 0, -1, -2, .... The gamma function never takes the value zero, but it comes very close between consecutive large negative integers.The reciprocal 1/r(x) has no discontinuities. It rapidly approaches zero as x -b x and equals zero at x = 0,-1. -2, .... As shown on the map, its oscillations become increasingly violent as the argument becomes moreand more negative.The logarithm In(r(x)) is usually only considered for x > 0. This is the convention adopted in drawing themap, which shows that ln(r(x)) is a positive function except between x = l and x = 2, where it is briefly andslightly negative. 411 43:3 THE GAMMA FUNCTION r (x) 4t4g0,'L412 FIG 43-1 43:3 DEFINITIONSAlthough it is restricted to positive arguments, the most useful definition of the gamma function is the Euler integral43:3:1 r(x) =Jr`-' exp(-t)dtx > 00More comprehensive are the Gauss limit definition 43:3:2r (x) = lim/Il rT(l+x)I+2(I...+n1/\l+n/ and the infinite product definition of Wcierstrass43:3:3 x exp(W) !l l+ 7 expl -x IC(x);-1 \!` Jwhere y is Euler's constant [Chapter 1]. 413 THE GAMMA FUNCTION r(x) 43:4The gamma function may be expressed as a definite integral in many ways apart from the one given above.Gradshteyn and Ryzhik [Section 8.311 give a long list of which the following are representative:(!1 43:3:4r(1 +x)= J In'I-Idtx>-1ot 43:3:5r(x) = s`Jt'-' exp(-st)dtx> Os> 0a43:3:6r(x) = s' sec(2IJt'"'cos(st)dt0 < x < 1s > 0Alternatively, in 43:3:6 sec and cos may be/replacedby csc and sin, respectively.Likewise, there are several ways of representing the logarithm of the gamma function by means of integrals.One is( ` - 43:3:7In(r(l+x))=II-II-xJIn(t)x>-10tand others may be found in Gradshteyn and Ryzhik [Section 8.34]. 43:4 SPECIAL CASESThe gamma function reduces to the factorial function [Chapter 2] when its argument is a positive integer:43:4:1 r(n) _ (n - 1)!n = 1, 2, 3....The gamma function of = is V n, and the gamma function of an odd multiple of i involves a double factorialfunction [Section 2:131. Comprehensive formulas are43:4:2r(n + 1) _ (2n - I)!! r(,)/2" = (2n - 1)!!\/2" 0, 1, 2, ...= 1.772453851and43:4:3r(, - n) _ (-2)" r(j)/(2n - 1)!! _ (-2)"Vr/(2n - 1)!! it = 0, 1, 2, ...Similar to 43:4:2 and 43:4:3 are the formulas43:4:4r(n + }) = (3n - 2)!!! r(1)/3" n = 0, 1, 2, ...43:4:5r(; - n) _ (-3)" r(J)1(3n - 1)!!! n = 0, 1, 2, ...43:4:6r(n + 3) = On - I)!!! r(32)/3" = 2zr(3n - T(3)it = 0, 1. 2,...and43:4:7r(, - n) _ (-3)" r(5)/(3n - 2)!!! = 2a(-3)"/Vi(3n - 2)!!! r(#) n = 0, 1, 2....involving the triple factorial function [Section 2:131 and where 43:4:8r(1) = 2.678938535andr(;) =2a73= 1.354117939r(ti)Likewise, for arguments that are odd multiples of ,, the gamma function takes the values43:4:9r(; + n) _ (4n - 3)!!!! r(14)/4"it = 0, 1, 2....43:4:10r(; + n) _ (4n - 1)!!!!r(i)/4" = V 2-.(4n - 1)!!v/4" r(,'-!) it = 0. 1. 2.43:4:11r(} - n) _ (-4)" r(1)1(4n - 1)!!!! n = 0, 1, 2, ...or 43:4:12r(j - n) = (-4)"(})/(4n - 3)!!!! = \rr(-4)"/(4n - 3)!!!! r(,) it = 0, 1, 2. 43:5 THE GAMMA FUNCTION r(x)involving the quadruple factorial function [Section 2:131 and where 43:4:13 r(}) = a'/' , 1- = 3.625609908 or 43:4:14r(:) = - = a'I V U = 1.225416702r(:)U being the ubiquitous constant [see Section 1:7]. 43:5 INTRARELATIONSHIPS The gamma function obeys the reflection formulas-iresc(ax)_lresc(-rrx) 43:5:1 C(-x) =xr(x)r(1 + x)and43:5:2The recurrence formulas43:5:3 and 43:5:4C(1 + x) = xr(x)r(x - I) =-x- Igeneralize to43:5:5r(n+x)=x(11 +x)r(x)=(x)"r(x)n=0. 1,2,..and(- 43:5:6r(x - n)r(x)I)" r(x)(x- (1 -x)"in terms of the Pochhammer polynomial [Chapter 18].The duplication and triplication formulas 43:5:7 and 43:5:82rm V3n=0.1.2.... are the n = 2 and 3 cases of the general Gauss-Legendre formula2an° 43:5:9r(nx) _H r(n +z In = 2. 3. 4... .ni.nasec(trx)r(1-x)=1(})+x r(2x) =4r(x) r(# + x)2V r(3x) =(27).r(x) r(# + x)r(i + x)r(x)414 which applies for any positive integer multiplier n.From 43:5:5 and 43:5:6, one may derive the expressions 415 THE GAMMA FUNCTION r(x) r(n + x) 43:5:10 r(x)=(x)n=1,2,3,... and43:6 r(x - n)_(-1)"43:5:11r(x)(1 - x)n = 1, 2, 3, ...for the ratio of the gamma functions of two arguments that differ by an integer. These formulas may be used evenwhen the individual gamma functions are infinite (e.g., r(x - 3)/r(x) -. -6 as x -. 0).Because of the frequent occurrence of the reciprocal gamma function in power series expansions of transcen-dental functions, particular values of the latter functions often serve as sums of infinite series of reciprocal gammafunctions. For example, on account of expansion 41:6:5, we have 111 11 43:5:12+-++ +eerfc(-l)=5.573169664 r(s)r(1)r(3)r( J/2)17 while, from the expansions in Section 45:6 43:5:1311_y(x- 1:-I)Fix)r(x + I) + r(x +T) ,zo r(x + j) e 43:6 EXPANSIONSThe power series expansions for the gamma function and its reciprocal are(Y2Tr2\ 43:6:1[r(x)J" = x'' j lYx + (2 - 12/x7 ..a_1x'='x-1, -2. -3, .where ao = 1 andya.;_, + (=1), k1, 2, 3, ...0.5772156649I 43:6:2a_, )..kj =Y=1Jk-oHere y denotes Euler's constant [Chapter I I and ;(n) is the n'" zeta number [Chapter 3). Numerical values of ao,a-,, a_2, .... a_2s are listed by Abramowitz and Stegun [page 256). The power series expansion of the logarithmof the gamma function is less complicated: 43:6:3 -1<xr= l12,.2JMore rapidly convergent is the similar series 43:6:4!n(r(1 + x)) = ( 1 - y)x +IInarx(l - x)'(J) - l,j = 3, 5, 7....-I <.r< 12(1 + .r) sin(irx),jThe gamma function may also be expanded as the infinite product 43:6:5//1\II + -1 \\Ir(1 + x) _ fl,_11+XJAn asymptotic expansion of the gamma function is provided by Stirling's formula: 43:6:6r(x)2,rrexp(-x)x,(1+I+1-1391-X\12x288x'51840x 43:7 THE GAMMA FUNCTION F(x)The corresponding asymptotic expansion for the gamma function's logarithm is416 (1II 43:6:7In(F(x)) - ln(\) - x + I x - - 2) ln(x) + 12x - 360x'+1_...++...x-*x1260x`2j(2j - l)x"-'where B_, denotes a Bernoulli number [Chapter 4 1. Though exact only in the x limit, these expansions areremarkably accurate for modest values of the argument [see 43:9:1, for example]. Relationship 43:6:7 is a special case of Barnes' asymptotic expansion(-+6c' - 6c + 1 43:6:8ln(F(x + c)) - (x + c) ln(x) - x +I- In2ir2\x12x2c3 - 3c2 + cBj_,(c)12x`j(j + 1)(-x)'where B,. 1(c) denotes a Bernoulli polynomial [Chapter 19]_ From this expansion one may also derive the usefulexpansionC(z + c)c(c - 1)c(c - l)(c - 2)(3c - I)c2(c - 1)2(c - 2)(c -3) 43:6:9-x` 1++++ x-+x F(x)2x24x'48x3for the ratio of two gamma functions of large, but not very different, arguments. For example: 43:6:10F(x + 1)X25X321X'399X` 1-1-X+-+---- + X=--+OV;F(x)22888x 43:7 PARTICULAR VALUESIn addition to those reported in 43:4:14 and 43:4:15. there are the following: X-3s-2-1 0 1234ft:-8Vaws4V'2x-2Vrz-_VTIV'nI3V+r215Vn6 (x)IS 3 24-8 The arguments yielding local maxima or minima of r(x) correspond to the zeros of the digamma function [seeSection 44:7]. 43:8 NUMERICAL VALUESFor x z 3. the following algorithm uses a truncated version of expansion 43:6:6 to calculate a sufficiently precisevalue of F(x), For smaller arguments, the recurrence 43:5:4 is used repeatedly until x is brought within the 3 < x< 4 range. Since the gamma functions of 0, -1, -2.... are infinite, the algorithm returns a value of 1079 at thesearguments. Apart from these cases, the algorithm has a precision of 24 bits (i.e., the relative error in r(x) neverexceeds 6 x 10-e). Set f = 1099 Storage needed: f, g and xSet gl 417 THE GAMMA FUNCTION r(x) 43:10Input x >> f = r(x) <(1) Ifx=0goto(2)Replace g by gxReplace x by x + I Ifx<3rrgoto(I)\1Setf//=ll-?II-230x LLLLL7x' \3x'1/1-fReplace f by?x+ x[ln(x) - 1] Replace f byexp(f)8ILX(2) Output f 43:9 APPROXIMATIONSTest values:1'(0.6) = 1.48919225r(7.4) = 1541.33619 R-4.2) = -0.164061051 From 43:6:5. the Stirling approximation 11+ - I8-bit precisionx ? 0.4 .r\12xmay be dcrivcd by truncation. By splitting the argument x into its integer-value I = lnt(x) and fractional-value f= frac(.r) components [Chapter 91. the gamma function may be approximated by 43:9:2r(x) = 11L1-f (121f )J8-bit precisionx z 2 an expression derived from 43:6:9 that permits the gamma function to be expressed in terms of the factorial function[Chapter 2].Close to its zeros, the reciprocal gamma function is approximated by 143:9:3= (-1)"n!(x + n)[ 1 - (x + n)ti(n + I )] x = -n = 0, - 1. -2... .r(x)where 6 denotes the digamma function [Chapter 44]. 43:10 OPERATIONS OF THE CALCULUSDifferentiation givesd43:10:1 -r(x) = 6(x) r(x) 43:10:2 43:10:3d-ln(r(x)) = 0(x) d 1Mr(x)) = Jr'"-n()n= 1,2,3,...where 0 and 0" are the digamma and polygamma functions [Chapter 441.Few simple integrals involving the gamma function have been established. although there are numerous integralsof products and quotients of gamma functions [see Gradshteyn and Ryzhik. Sections 6.41 and 6.42]. A number ofintegrals involving the logarithm of the gamma function, and including 43:11 THE GAMMA FUNCTION r(x) 43:10:4JrIln(r(t))dt = In(\) + x 1n(x) - x x> 0are also listed by Gradsheyn and Ryzhik [Section 6.44]. The latter formula leads ton- 1 43:10:5f In(r(r))dt =(ln(2ir) - n] + E j ln(j)n = 2, 3, 4, .2 43:11 COMPLEX ARGUMENT418 The gamma function with complex argument has real and imaginary components:1j + x1 43:11:1r(x + iy) = [cos(O) + i sin(0)1Ir(x)Ii-oyr+(j+x)2 where0 = V40) +Y- arctanl(,,-oj+x\j+xHere dj is the digamma function [Chapter 44]. Tables from which r(x + iv) may be evaluated are given by Abra-mowitz and Stegun (pages 277-2871. It follows from 43:11:1 that the product r(x + iv)r(x - iv) is real; specialcases are43:11:2and43:11:3For purely imaginary argumentr(I + iy)r(1 - iv) = pry csch(arv) r(12 + iy)r( - iv) = ir IT Y 43:11:4r(iv) = 1 /[sin(6) - i cos(9)]e = -yy +- arctanlI where y is Euler's constant [Section 1:7] 43:12 GENERALIZATIONSThe gamma function is a special case of the incomplete gamma function -y(v;x)43:12:1and of its complement r(v;x)43:12:2r(v) = y(v;x) r(v) = r(v;o)both of which are discussed in Chapter 45. Moreover, these two functions sum to give the gamma function43:12:3 y(v;x) + r(v;x) = r(v)for all values of X. 43:13 COGNATE FUNCTIONSThe functions discussed in Chapters 2, 6. 18. 44 and 45 are all closely related to the gamma function.Another important function that is intimately related to the gamma function is the complete beta function B(x,y).Also known as Eider's integral of the first kind or simply as the beta function, it is defined by the Euler integral 419 THE GAMMA FUNCTION r(x)r,t'-'(1 - t)'-'d:x > 0y > 0 43:13:1 B(x,Y)= J0and is related to the gamma function through 43:13:243:11 B(x,y) =r(x)r(Y)= B(y,x)r(x + y)As with the gamma function, the complete beta function may be expressed as a definite integral in many waysother than 43:13:1. These includert(1'-'++ r -' 43:13:3B(x,y) = Idtx< 0v> 0at)'andf43:13:4B(x,y) = 2r"-'(I - t2)'`'dt = 2 cosr*-'(t)dtx> 0> 00and extensive lists may be found in Gradshteyn and Ryzhik [Section 8.38J and in Magnus, Oberhettinger and Soni[page 71. The integral representations of B(.r.y) apply only when both arguments are positive, but relationship43:13:2 extends the definition to any pair of real arguments, as does the infinite-product expression 43:13:5 B(xY) _(jl)(x + Y + j);_a(x+j)(Y+j)If the two arguments of the complete beta function are equal in magnitude, or sum to a moiety, we have thespecial casesr l 4' I(2j _ 1)!! 43:13:6 and_(2B(.rx) = cos(t<x)B x, - -x)Bx) J=22'-o (2j)!!(j + x) 43:13:7 B(x.-x) =(-1)`/xx = 0, t 1, t2, .. .0otherwiseImportant special cases of 43:13:6 are B(;,;) = 21r/U, B(].]) = it and B(y,y) = 2U, where U is the ubiquitousconstant (Section 1:7J. When one (or both) of the arguments is a positive integer, the reciprocal of the completebeta function reduces to a binomial coefficient (Chapter 61 43:13:8 n+x-1=xB(x.n)n - I Intrarelationships of complete beta functions, such asxx43:13:9 B(.t + 1,y) = - B(x.y) _ - B(x,v + I)x+v vas well as expansions, such asI - v(1 - y)(2 - y)I43:13:10 B(x,y) =++2(2 + x)+ XI + x I_oj! (x + j)and infinite sums, such as 43:13:11B(x.y) + B(x + I,y) + B(x + 2,y) + B(x + j.y) = B(r.y - 1)'-omay be established via the identity 43:13:2. The same expression may be used to calculate numerical values ofB(x,y). The argument values that lead to B(x,y) = 0 and B(x.y) = tx are indicated by green and black lines, 43:14 THE GAMMA FUNCTION f(x) 420 respectively, in Figure 43-2. The areas shaded in red on this map show where B(x,y) is positive: blue shadingindicates that the complete beta function is negative. 43:14 RELATED TOPICSTogether with very few others, the functionsx,X > 0_43:14:1 }(1)itJo(2V-[)X250XI43:14:2 exp(X)I_o (1),I 43:14:3 -1<X<If-oI - Xand the asymptotic representation 43:14:4 1 X') Ei(')IXI smallcan claim a distinguished role in the theory of transcendental functions. The Bessel functions to and J. occurringin 43:14:1 are treated in Chapters 49 and 52, respectively, while the exponential integral Ei is the subject of Chapter37. These four have been termed basis hypergeomerric functions because they can be employed to generate almostany hypergeometric function (i.e., any of the functions tabulated in Section 18:14), as will now be demonstrated.Functions 43:14:1-43:14:4 have been expressed in terms of (I) the Pochhammer polynomial [Chapter 18] of 421 THE GAMMA FUNCTION r(x) 43:14unity argument, but since (1), = j! = C(1 + j), each of these functions can be written alternatively as EX'11(1+ j) with n = 2, 1, 0 or -1. Here X represents the argument x of interest, or some simple variant of it such as-2x or x2/4. Now, let µ represent the following sequence of operations: (a) division by X1'(1 - µ), (b) differ- integration [see Section 0:101 to order v - p. with respect to X using lower limit zero and (c) multiplication byX"1'(1 - v). Then, making use of equations 13:10:11 and 43:5:10, we find for the v = 0 case =X' 0(1 - µ),X'43:14:5 (1), I;-a(I)j--while for the µ = 0 caseX'vX'43:14:6 In the general case when neither v nor µ is necessarily zero 43:14:7=(1), µ ,-a (1),(1 - V),In the terminology of Section 18:14, the operation is seen to preserve the difference L - K between thenumbers of denominatorial and numeratorial parameters but to alter the values of some of those parameters. Hence.starting with the appropriate basis hypergeometric function, one may "synthesize" a sought hypergeometric functionby one or more v operations. Examples are 43:14:8 la(x)(x2/4)' :-= cosh(x)x = 2NX 01Y_a (1),(i),-2', (2r1'243:14:9exp()x)_()Oexp(x) 11(x)(1),(1),(3),x 43:14:10 and 43:14:I1Xx=- 1O(x'Yp 2- xr(X2), 1(I)i(1)= - E(x)x = NX,=o ; x exp(-x) Ei(r) - I +-+Z,+ ... +(1),+ ...I +-+xx'x' 2x4x'+...++.. -2 sdaw( s)x=-x' Xwhere cosh, I E and daw are the functions of Chapters 28, 49. 61 and 42, respectively. Further information onthis topic will be found in Oldham and Spanier [Chapter 9J. CHAPTER44 THE DIGAMMA FUNCTION 4j(x) The digamma function is derived by differentiation of the gamma function of Chapter 43. Multiple differentiationyields the polygamma functions that are addressed in Section 44:12. Digamma and polygamma functions are usefulin summing certain algebraic series, a subject discussed in Section 44:14. 44:1 NOTATIONThe digamma function is also known as the psi function. Some authors employ a translated argument and denoteour 4r(x) by ir(x - 1). 44:2 BEHAVIORFigure 44-1 includes a map showing the global behavior of the digamma function. The function 4o(x) has an infinitediscontinuity at x = 0 and at each negative integer argument. Otherwise, dir(x)/dx is always positive, this gradientdecreasing steadily as x increases through positive values. As x -+ x, y(x) approaches infinity logarithmically.Figure 44-2 shows the detailed behavior of *(x), and some other functions, for 0.5 s x s 5.5. 44:3 DEFINITIONSThe digamma function is the derivative of the logarithm of the gamma function (Chapter 431 44:3:1 A second definition is as the limit 44:3:2dId4Y(x) =ln(r(x)) =r(x) dxr(x)dx y(x) = limIn(n) - i'_oj+xRepresentations as definite integrals are numerous and include 423 44:3 THE DIGAMMA FUNCTION $(x) 424 425 THE DIGAMMA FUNCTION Xx) 44:3:3x) _ -y+l0I -t`fdt = -yl0+exp(-t) - exp(-x:)dtx > 0 1 - t1 - exp(-t)where y is Euler's constant [Section 1:71. The integralsexp(-t)exp(-xt)exp(-t) - (I + t)-` 44:3:4fix) = j[t1 -exp(-t)Jdt-ftdtx > 0 and44:5 44:3:54.(x) = ln(x) - I - 2 tdtX>0to(t'' + x2)[exp(21rt) - 1]are attributed respectively to Gauss, Dirichlet and Binet. For other definitions of 41(x) as a definite integral, seeErddlyi, Magnus. Oberhettinger and Tricomi [Higher Transcendental Functions, Volume 1, Section 1.7.2.). 44:4 SPECIAL CASESWhen its argument is a positive integer, the digamma function is given by 44:4:146(n)=-y+ 1+Z+I +...+ - =-y+E7 n=2,3,4,...y=0.5772156649n-1J-4Jan expression involving Euler's constant y and the first n - I terms of the harmonic series. The function 4i(n +1) + y is sometimes denoted 0(n).If x is a rational number (i.e.. capable of being expressed as a ratio m/n of two integers) then 41r(x) is expressibleas a finite sum of terms involving simpler functions (logarithmic and trigonometric) and constants (Archimedes'number it and Euler's constant y). For 0 < x = m/n < 1, where at least one of m and n is odd, the theorem ofGauss states`/\//44:4:2n-y-2cot/ (Icosl2Ilnl2sinlnJ\l)IJln(2n)n=2J+2=2,4,6,...In(n)n = 2J + l =3,5,7-.If m > n, recursion 44:5:5 should be used before 44:4:2. 44:5 I NTRA R ELATIONSHIPS The reflection formulas44:5:1and41(1 - x) = 414(x) + if cot(er) I\\I 44:5:2 y 2 - s l = Jr - + x - it tan(irrx) apply for any value of x, whereas the formula// 44:5:3r2-J/ 1J=0,1,2,...is restricted to nonnegative integer J.The recurrence relationship 44:6 THE DIGAMMA FUNCTION 4i(x) 1 44:5:4 01+x)=4.(x)+-xmay be generalized to 44:5:5 n= 1,2.3,...;=o+xSimilarly, the duplication formula 44:5:6 ilr(2x) = In(2) + 2 y(x) + 24(x+2)generalizes to426 44:5:7*(nx) = ln(n) +rt4r(x+ rt/n = 2, 3, 4,poThe difference between two digamma functions may be expressed as 1 4458 b-W_-5' :: (x)tv)(xY);-o (j + x)(j + Y)or as a hypergeometric function [Section 18:141 of unit argument:44:5:9fi(x)-4'(y)=x_y(1);(x-y+l);Y>0x(2);(x+ l),Infinite sums of quotients of digamma functions by gamma functions include4r(3)01)y(2) Ei(1)I)*j)44:5:10 _+ r(1) + r(2)r(3)==F(j)e0.697174883 dan(1)0(3)(-010(j)ffYo(2) 44:5:11r2(1) + r2(2)r2(3)+ =-r2()20.801696703where Ei is the exponential integral function [Chapter 37) and Yo is the zero-order Neumann function [Chapter54]. The corresponding series without alternating signs Eilr(j)/r"(j) sum to -eEi(-1) = 0.596347362 and KO(2)= 0.113893873 for n = I and 2, respectively, Ko being the zero-order Basset function [Chapter 51]. 44:6 EXPANSIONSThe Taylor expansion 44:6:1*r(x + l)+ 1)(-x)'-1 <x< 1 involves the zeta numbers [Chapter 3] and is valid only for arguments between 0 and 2. Of wider validity are theexpansions 44:6:2y(x)=-Y+=f1+1]+ z-1=-Y-l+xl o Lj + Ij + xi-o U + I)(j +x)xi-i j(j + x) 44:6:34r(x)=ln( x)-iI 1_]-In1I+x>0 andjL+jx\+ _J 427 THE DIGAMMA FUNCTION 4x)x2aIl1 44:6:4I[2(x)=-y--cot(az) -- +t)x6J-2<x<2k=3,5,7,... 1 -x22x12An asymptotic expansion for the digamma function, valid for large argument, is II I IB, 44:6:5ln(x) -2x12x'' +120x4252x6 + ... - Js' +x-- xwhere B, is a Bernoulli number [Chapter 41. 44:7 PARTICULAR VALUES44:8 Euler's constant, y = 0.5772156649, is a component of most particular values of the digamma function. Table44.7.1 includes particular values of the trigamma function (Section 44:121 and of Bateman's G function [Section44:131, as well as those of the digamma function fi(x).By exploiting formulas 44:4:1 and 44:5:4, it is possible to evaluate the digamma function of any rationalargument x = m/n. For small values of n, rather simple expressions result. In addition to those incorporated intoequation 44:5:3, we have the following examples: 44:7:1y(3+J) +2V3iT 0(2-J)-2V=-y-lnV27+3j32J=0.1,2....33- 44:7:2yl?+JI-=4#1-JI+ -= -y -lnr27+Y 3J=0,1,2,... l\3//2V3\\32V3;-1 3j - I 44:7:344'+J +-=3 Jl-=]n8+42-y()4j - 3(3)n_-1'4 44:7:4+J -40(lJI+=-y-In(8)+2l=0,1,2, 424Z ...r-14J _ 1The digamma function encounters zeros at argument values of + 1.461632145, -0.504083008, -1.57349847,-2.61072087, .... Denoting these zeros by r0, r1, r2, r3,/... then the following approximation holds: 44:7:5 r, = -j + - arctanl In(j) J large J 44:8 NUMERICAL VALUESThe algorithm below has an absolute accuracy of better than I X 10-6, except when x is a nonpositive integer, inwhich case the algorithm returns 1099 instead of ±x. For x z 5, the algorithm uses the expansion 44:6:5 in a Table 44.7.1 _-2.- '1'_-I q,l VU) 001_=3-Y-M4l2--In14) s wJ20 2S-J - r3-0..4 .-2 ,-3 63 -T-mi4)-1--7-.411-13-y-I.041--y`Y 40 SJI 026262964 IM414-a2-1NJ)n-t`c41 -10 44:9 THE DIGAMMA FUNCTION 4(x) 428truncated and compensated version. For smaller arguments, the algorithm uses the recursion 4o(x) = y(x + 1) -(I/x) a sufficient number of times until the argument is brought to 5 or above. Input x >>Set f = 10°°Set g = 0)a»»(1) If x = 0 go to (2)Replace g by g + -xReplace x by x + Ilfx<5goto(1)0.46Set f-4-[(k -- -I )/ lox-' Replace f by -LQx+1) /2x]+ In(x) - g(2) Outputff = 4L(x) <.44:9 APPROXIMATIONSFor large x the approximations 44:9:1 and 44:9:2 are useful.Storage needed:f, gand x Test values:dr(7) = 1.872784344(1.461632145) = 0ir(-1.05) = 20.2897713 4(-x) = In x + 2 I + IT cot(arx) large x s() -in x-128-bit precisionx z 4 44:10 OPERATIONS OF THE CALCULUSOne or more differentiations of the digamma function give the polygamma functions that are discussed in Section44:12: 44:10:1 d"*(x) =yiRi(x)n = 1.2.3, .. .Formulas for indefinite integration includefl(:)clz 44:10:2 = In(r(x))x > 0 f(r&=1n(x) 44:10:3 F'x > 0Many definite integrals, including 44:10:4r Yy(I + r>dr = -a csc(vvn);(v)I < v < 2 and 429 THE DIGAMMA FUNCTION $(x) 44:10:5r0n1J,(t) sin(nzrt)dt =n 02n=2,4,6,.. are listed by Gradshtcyn and Ryzhik [Sections 6.46 and 6.471. 44:11 COMPLEX ARGUMENTThe digamma function of argument x + iy is given by 44:11:1 x y+G2 +i L.x +Y ;=.J[(i+x)z +Y1,-.(i+x)z +y'Tables permitting the evaluation of 4,(x + iy) are presented by Abramowitz and Stegun [Table 6.81.For purely imaginary argument 44:11:2For the related special case 44:11:3+ Fy-+ it-+ - coth(ay)J,..1(1=+Yz)2y2 2It yitz+ -1 tanh(ay) + :+ - tanh(ay) q'(z + iy) = -y -T+ 4y4y [(zj + I) + Y21 2the imaginary part of the digamma function is seen to contain a hyperbolic function [see Chapter 301. 44:12 GENERALIZATIONS44:12 One differentiation of In([ (x)) gives the digamma function; multiple differentiations generate the family of functionsknown as the polygamma functions and denoted %W"(x). One has 44:12:1Another definition is 44:12:2d"u,(x) =dr(x)n = 1, 2, 3,...a t" exp(-xt)t-!dtn=1,2,3,...0The names trigamma function and tetragamma function, respectively, are given to 4, and +'2':-': these particularpolygamma functions are more commonly denoted y' and V. The functions 111th and 4,101 are known as the pen-tagamma function and hexagamma function. .4,14),For odd n the polygamma functions 4, are restricted to positive values, but 1,", 4+161,... take valuesof both signs. As illustrated in Figures 44-3 through 44-5. the polygamma functions are infinite at x = 0, -1, -2..... By combining the recurrence formula 44:12:3ili (x+I)=d, (x)- jn=1,2,3,... with the particular values44:12:40",(1)=(-1)"'In!(,(n+ I)n = 1,2,3....andV)12l 44:12:5/=(-2)"'In!A(n+ 1)n= 1,2,3,... 44:12 THE DIGAMMA FUNCTION $r(x) 430 -if14114V 431 THE DIGAMMA FUNCTION 4(x) 44:12involving zeta C and lambda A numbers (Chapter 3], one may derive the expressionsr/_1 1144:12:6yr"1(n=(-l)"'n!IOn +1)-n=1,2,3,...Lplj-! \ 31 !-1 144:12:7,yl.l(1-(-2)"*'n!IA(n+ l) -(2j - 1)-" J = n!GI, 2. 3,2 - lJlJ for the polygamma functions of positive integers and half-integers. Some such values are included in Table 44.7.1for the n = I case.For large positive values of their arguments, the polygamma functions are well approximated by44:12:8 4'(x) =large x(2Yet more refined approximations are provided by 44:12:12 for large x. As is evident in Figures 44-3-44-5, fornegative x values of any magnitude, the values of qs``(x) are largely determined by the fractional value [Chapter9] of x and hardly at all by its integer value. This is a consequence of the fact that, in the reflection formula44:12:901"1(-x) _ (-1)"a dt° cot(-) + I)the final term for x > 0 is generally negligible compared with the periodic derivative term.Among the relationships for the polygamma functions that can be derived by differentiation of the correspondingformulas for the digamma function are the following expansions: n! (n + j)! 44:12:10 t;(n+j+l)x'n=1.2.3....-1<x<I (-x)"-1j_0J 44:12:11 n=1,2,3,...j-0 !+x 44:12:12"(x)-(n-I)!+n+n(n+1)-n(n+l)(n+2)(n+3)(-x)"2x12x'720x4+...+(i+n_ ,Il Bj+...l the latter asymptotic expansion being valid for large x.The algorithm below is designed to calculate values of Ilr'"'(x) for small values of n (e.g., n = 1, 2, 3). Sincethe polygamma functions take infmite values for x = 0, -1, -2, ..., the algorithm returns 10" under theseconditions. Otherwise, the algorithm has high accuracy, although it becomes increasingly imprecise as n increases.Note that n must be a positive integer. Input n >> Input x >>Setf10"Set g = 0(I) Ifx=0goto(4)Seth = -l/x Setj = n(2) Replace h by -hj/xReplace j by j - 11fj*0go to (2) Replace g by g + hReplace x by x + IStorage needed: n, J. g, x. h and j Input restriction: n must be a positive in-teger. 44:13 THE DIGAMMA FUNCTION 4(x) 432 f = '"'(x)nSet j =Ifx<5goto(1)Set f = [I - (n + 4)(n + 5)/45x2]/60x'Replace f by (n + 1)[1 -f(n + 2)(n + 3)1/6xReplacefby-I1+f+ lJn2x(3) Replace f by fj/xReplace j by j - 1Ifj*0goto(3)Replace f by f + g(4) Output fTest values:x'(5.5) = 0.199342388%r(-0.5) = -0.828796644 403)(9) = 0.00323439609 The algorithm does not generate iIr(x) when n = 0 is input. The algorithm utilizes equation 44:12:3 together witha truncated, compensated and concatenated version of 44:12:10; it parallels the algorithm of Section 44:8 in itsmanner of operation. Some approximate values of 4,'"'(x) may also be read from maps in this chapter. 44:13 COGNATE FUNCTIONS In addition to its definition as the integral 44:13:1 G(x) = 2I'It +'t dt= 2 J%1+p( xr)t)P(-Bateman's G function is defined in terms of the digamma function asx+I 44:13:2 G(x) = 0x) -The notation 2B(x) is also used for this function, but confusion must be avoided with the dissimilar function denotedby B in Chapter 3. In fact it is to the bivariate eta function of Section 64:13 that G(x) is related: 44:13:3G(x)=2(=2-n(l;x)I-0 x +jSome properties of Bateman's G function are 44:13:4 and44:13:52G(1+x)=--G(x)x G(1 - x) = 2tt csc(irx) - G(x)and, in addition to 44:13:3, we have the expansions244:13:6 G(x) = 2 7,T1(j + 1)(1 - x)ij-0 (2j+ x)(2j + x + 1)!-0 2'(x)j+i-0and 44:13:7G(x) =1- 12x1+ csc(,rrx) + 211 - -n(2f + 1)]x22'i-owhere -n(j) is the eta number of Chapter 3. The behavior of G(x) is mapped in Figures 44-1 and 44-2; Section18:14 shows how the universal hypergeometric algorithm may be used to calculate its values. Some particularvalues of G(x) will be found in Section 44:7. 433 THE DIGAMMA FUNCTION y(x) 44:14Many properties of the successive derivatives G'"(x) of Bateman's G function are given by Erddlyi, Magnus,Oberhettinger and Tricomi [Higher Transcendental Functions, Volume 1, Section 1.16]. We mention only44:13:8G'`W = 2n!2(-1)"n!i(n + l;x),-o (j + x)"'where 71 is the bivariate eta function of Section 64:13. 44:14 RELATED TOPICSThe digamma function and Bateman's G function are useful for summing certain algebraic series involving recip-rocal linear functions. Thus: 44:14:1 and 44:14:21lI1I_'-1I_1/Jb + cccb+c 2b+c 3b+cJb-b+c-Jjb+cb'`b)- IIII 1=Z(-1)'cb+cf 2lb+c\3b+c/Jb-b+c -obj+cIIGI--bI -(-1)'G`Jbb c/Jp Because the G function approaches zero as its argument a proaches infinity, the sum to j = x of (-I)'/(bj + c)is simply G[(b + c)/b]/2b. Special cases of these results are presented as equations 1:14:5 and 1:14:10.Similarly, infinite series of reciprocal powers have the sums44:14:3 -+++ n=2,3,4, c"(b + c)"(2b + c)"(3b + c)" ,Y-_o ... (jb + c)"(-b)"(n - 1)! and- - V1-(b + c)c )2b2b44:14:4--+-+ C.(b + c)"(2b + c)"(3b + c)"_0 (jb + c)"z(-2b)"(n - I)!n= 1,2,3,...in terms of polygamma functions. A corresponding finite series may be expressed as the difference between twoinfinite series and hence may be summed via equation 44:14:3 or 44:14:4.Consider the summation of a series of rational functions [Section 17:131 of the summation index k 44:14:5S =pT(k)A k' + A.,_,k-' + - -- + Aik + Ao k.o p.(k)k=o a.k" + o._,k" ' + ... + ask + aowhere p," and p" are polynomial functions of degrees m and n. If n a m + 2. the series will generally converge,and, in these circumstances, S may be expressed in terms of digamma and/or polygamma functions, as we nowdemonstrate.In Section 17:6 it is shown that 1 /p.(k) may be expressed as the sum of n partial fractions, each of form C;/[a.(k - pi)] where pi is the ja zero of p. and Ci is a constant. A similar fractionation of p.,(k)/p.(k) is possible sothat44:14:6 S=C'=-1CJd-P,)a. ,..1 k=o k - pia. j-, 44:14 THE DIGA.MMA FUNCTION y(x) 434Because m s n - 2, it follows that IC, = 0, and this fact is used, together with the first of the 44:6:2 expansions,to establish 44:14:6. Some (or all or none) of the zeros p, may be complex, in which case the corresponding C,constants will also be complex.The foregoing presupposes that the zeros P. p,, ... p. are all distinct. When this is not the case, one followsthe procedures outlined in Section 17:13. For example, if the zeros p Pr and p, (but no others) were coincident,one would replace 44:14:6 by 44:14:7S =I- j rCi+G+JC.1+IC,- k - Pi(k - pi)-(k-P,)'a.,-44-0k-p,C1.Cid(-PI) +22I"(-Pi) - 33 d"(-Pi) - i C,,b(-p,)i-4equation 44:12:9 having been used. CHAPTER45 THE INCOMPLETE GAMMA 'y(v;x)AND RELATED FUNCTIONS Many of the functions discussed in preceding chapters are special cases of the functions considered in this chapter.Besides the incomplete gamma function y(v;x) itself, we shall make frequent use in this chapter of two closelyrelated functions: the complementary incomplete gamma function r(v;x) and the entire incomplete gamma function y'(v;x). The interrelationshipsy(v;x)f(v;x)45:0:1 fy(v;x) =r(v)= I -r(v)link the three, where f(v) denotes the (complete) gamma function (Chapter 43]. The entire incomplete gammafunction remains useful for negative x, whereas y(v;x) and r(v;x) are then often ill defined. 45:1 NOTATIONThe variables v and x are termed, respectively, the parameter and the argument of the functions. The adjective"incomplete" reflects the restricted ranges of integration in definitions 45:3:1 and 45:3:2 compared with that inEulei s integral of the second kind, 43:3:1.Alternative symbolisms abound. fAv) has been used for y(v:x); P(b,x) and Q(v,x) have been used for y(vx)/r(v) and r(v;x)/r(v). respectively. The notation EV(x) or is sometimes adopted for x"-'r(1 - v;x). 45:2 BEHAVIORBecause, for negative arguments, the functions y(v;x) and r(v;r) are often complex and/or multivalued, we restrictour discussion of these functions to x ? 0. Figure 45-1 maps typical behaviors of these two functions for a varietyof values of the parameter.For x > 0, the y(v;x) function is well defined except for v = 0, - 1, -2, ... when, like the complete gammafunction [Chapter 431, it encounters infinite discontinuities viewed as a function of its parameter. The y(v;x) func- tion is always a monotonically increasing function of x. For v > 0, -y(v;x) is zero at x = 0 and positive elsewhere. For -1 < v < 0, -3 < v < -2, -5 < v < -4, etc., the function is invariably negative, infinitely so at x = 0.For -2 < v < -1, -4 < v < -3, etc., y(v;x) equals -- at zero argument, then increases monotonically, crossingzero to reach the positive value r(v) at x = x. 435 45:3THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONS 436 Y(ZIx)...........FIG 45-1 .....................-2 The complementary incomplete gamma function is invariably positive and decreases monotonically with x,approaching zero asymptotically as x -+ x.The entire incomplete gamma function y*(v;x) is single valued, real and finite at all values of v and x that arereal and finite. Its behavior is comprehensively described by the contour map Figure 45-2. Notice that y* has nozeros for v > 0. has one zero (at a negative value of x) for -1 < v < 0, -3 < v < -2, etc., and two zeros (oneat positive and one at negative arguments) for -2 < v < -1, -4 < v < -3, etc. 45:3 DEFINITIONSThe y(v;x) and r(v;x) incomplete gamma functions are usually defined via the integrals 45:3:1 y(v;x) =J'r"-' exp(-r)drx a- 0v> 00 437THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONS 45:3 45:4THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONS 438 45:3:2 r(v;x) = Jt"-' exp(-t)dtx > 0Definition 45:3:1 may be extended to negative v by utilizing the recursion formula 45:5:2. The definite integralsx" exp(-x)t-' exp(-t) 45:3:3r(v;x) =r(1 - v)ft + xdtx > 0v < I0and 45:3:4 y*(v;x) =17(t'-' exp(-xt)dtv > 0v)oopen up other ranges of the parameter and argument. The more complicated integral representations 45:3:5y(v;x) = x'J}t"-''r exp(-t) J,(2V)dtv > 00453:6rvx) =2x'/' exp(-x)J('t-.12'exp(-t) K,(2V)dtv < 1 r(1 - v)ulink the incomplete gamma functions to Bessel [Chapter 531 and Basset [Chapter 51 ] functions.Often the series 45:6:1 is taken as the definition of the incomplete gamma function. As well, the entire in-complete gamma function may be defined succinctly by the differintegration [Section 0: 10] formulationd' 45:3:7-exp(±x) = x-' exp(±x) y*(-v;±x)dx' 45:4 SPECIAL CASESCertain values of the parameter convert the incomplete gamma function into simpler functions treated elsewherein this Atlas.When the parameter is a positive integer.45:4:1r(n;x) _ (n - 1)! exp(-x) a _1(x) n = 1, 2, 3, .where e .(x) denotes the exponential polynomial function [Section 26:131. For example, r(l;x) = exp(-x), so thaty(l;x) = I - exp(-x) and y*(l;x) = [1 - exp(-x)]/x. When v = 0, y(v;x) is undefined but r(O;x) _ -Ei(-x)and y*(0;x) = 1, where Ei is the exponential integral of Chapter 37.For a parameter equal to a negative integer:45:4:2 y*(-n,x) = x"n = 0, 1, 2, ...a simple result that provides a link to Chapter 11. The complementary incomplete gamma function for a negativeinteger parameter may be related to a Schlomilch function [Section 37::13], but an equivalent expression is 45:4:3r(-n;x)rexp(-x)I!+ Ei(-x)In = 0, 1, 2.... =In!x;_o (-x)in terms of the exponential integral function [Chapter 37).Incomplete gamma functions of moiety parameter reduce to the functions of Chapters 40 and 42:45:4:4 y(12;x)erf(V)7r erfc(y T) 45:4:5 r(;;x) = Vv-2 45:4:6 y*(,;-x) =7exp(x) daw(f ) 439THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONSNotice also that by 45:3:2 45:4:7 r(v + I ;x) = Jt° exp(-t)dtx > 045:6 Special cases of such integrals are tabulated in Table 37.14.1 for v = -3, -j, -2, ..., 2. Thus, this table servesto provide a listing of the special cases r(v;x) for v = -2. 9, -I, ..., 3. 45:5 INTRARELATIONSHIPSThe incomplete gamma function and its complementary cohort sum to the (complete) gamma function [Chapter43]45:5:1 Y(v;x) + r(v;x) = r(v)Recurrence formulas for the three functions are45:5:2 Y(v + l;x) = vY(v;x) - x'exp(-x)45:5:3 r(v + l;x) = vr(v;x) + x"exp(-x)and 45:5:4 Y*(v;x)exp(-x)Y*(v +Ix) xxxr(1 + v)The first of these may be generalized to I. -'x' 145:5:5Y(v + n;x) = (v).LY(v;x) - x" exp(-x) 2 -J = x"p._i(x) exp(-x)where p.-I(x) is the polynomial defined in 45:8:2.Nielsen's expansion provides an addition formula for the argument: 45:5:6Y(v;x + Y) = Y(v;x) +exp(Vx)i(1 - v);[I - exp(-Y) e,(Y)]IYI < xj.0(-x) 45:6 EXPANSIONSA large number of expansions exist for the incomplete gamma functions. The most important of these arex"x"'x"`'(-x)'45:6:1 Y(v;x) = - - - + -+= x"vI + v2(2 + v)6(3 + v)J.Oj!(j + v)andrxx",x'-"_exp(-x)Y45:6:2Y(v;x) = exp(-x)Iv+ v(v + I) + v(v + I)(v + 2) + = exp(-x) j (v)j+1 v(v + 1)j 1-0which lend themselves to expression as hypergeometric series [Section 18:14]. Involving the associated Laguerrepolynomials (Section 23:12] and hyperbolic Bessel functions [Chapter 501, respectively, are the expansionsLJ(x) 45:6:3 r(v;x) = x" exp(-x)-uj + Iand 45:6:4 y (v;x) =expx)x1izej(-I)1j+"(2Vx)12X,f.p 45:7THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONSThe complementary incomplete gamma function is expansible as the continued fraction 45:6:5r(v;x) _.s exp(-x) 1- v 12- v2 3- vx+1+x+1+x+l+or as the important asymptotic expansionexp(-x) r1 - v(I - v)(2 - v)(I - v)(2 - v)(3 - v) 45:6:6r(v;x) --I - - +-x1xX. x3 (-x)' valid for large x.Expansion 45:6:6 applies when x, but not v, is large. Conversely, there is an asymptotic expansion 1xx'x'(-x)'45:6:7y(v;x)--x--+-+-++v-'xIVv + I2(v + 2)6(v + 3)j!(v + j)that is valid when the parameter, but not the argument, is large. 45:7 PARTICULAR VALUES440 The values of the three functions for arguments of zero and infinity sometimes depend on the sign of the parameter.In Table 45.7.1 "undef" means undefined. Table 45.7.1x=ov<ov>ov<ov>o -0;x)-x0r(v) 174 V)r(v:x)undefr( Y)00y9v:x) 45:8 NUMERICAL VALUESf(l+v)r(I+v)0 By repeated applications of recursion formula 45:5:4, and using expansion 45:6:2, one can write145:8:1y'(v;x) = x"y"(v + n;x) + P"-,(X) exp(-x)=exp(-x)P"-,(X) + x"x'J Fin + v)r(n + v)r_, (n + v);.,where p, -,(x) is a polynomial function of x that may be written 45:8:2 P"-,(x) = v) + x)(2 + v) + x2)(3 + v) + + x"-')(n - 2 + v)+ x"-')(n - I + v) + x"-'These are the formulas used in the algorithm for y'(v;x) that is presented below. The integer n is chosen to be atleast I and such that n + v > 2. The infinite series in 45:8:1 is evaluated via the truncated concatenation 45:8:3i'r(rr+1l+1 0(n+v),,(((\ n+v+J-xIn+v+J - ln +v+J-2 441THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONS 45:10xxl+1 -n+v+2n+v+ln+vBy use of the empirical assignment J = Int(5 + (3 + kxl)/2) one ensures that approximation 45:8:3 has 24-bitprecision (the relative error is less than 6 X l0-"). The final segment of the algorithm is concerned with calculatingthe multiplier exp(-x)/r(n + v), using a procedure similar to that employed in Section 43:8.Input v >>Input x >> f = y*(v;x)Setg=p= I(1) Replace v by I + vIfv>2goto(2)Replace g by gxReplace p by pv + gGo to (1)(2) Set j = lnt(5(3 + !x1)/2)Set f = 1 /(j + v - x)(3) Replace j by j - IReplacef by (fx + 1)/(j + v)Ifj * 0 go to (3)Replace p by p + fgx\ 1Setg= 1-2(1_2 I] /30v27v'3v'Replace g by g- v[In(v) - I]12vSet f = p exp(g - V)v/2aOutput fStorage needed: v, g. p. x, j and f Test values:y*(#;-0.49) = 1.34326134y*(-3.9;0) _ -0.521258841y*(I;a) = 0.304554469 Of course, numerical values of y(v;x) and r(v:x) may be computed from y*(v;x) by using equations 45:0:1.Alternatively, the universal hypergeometric algorithm of Section 18:14 may be used in several versions. 45:9 APPROXIMATIONSWhereas it is exact only if the parameter is a nonpositive integer, the expression45:9:1 y*(v;x) - Xis a useful approximation for all v if x is positive and sufficiently large. 45:10 OPERATIONS OF THE CALCULUSDifferentiation of the incomplete gamma function and its complementary analog give 45:10:1 dddxy(v;x)dxF(v;x) = x°'exp(-x) = y(v;x) - (v - 1)-y(v - I;x)The differintegration [Section 0:10] formula for the entire incomplete gamma function 45:10:2 d"dx"If exp(x)y*(v;x)] = x"-" exp(x)y*(v - µ;x)holds for any value of p., positive or negative, integer or noninteger. 45:11THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONSTwo definite integrals are1/45:10:3 Jy(v;bt) exp(-st)dt = ysv) 1bb s\v> 0aand('r(v + µ)45:10:4 Jr"-'r(v;bt)di=v+ IL >0p.>00µbµand many others are listed by Gradshteyn and Ryzhik [Section 6.45). 45:11 COMPLEX ARGUMENT442 Here we shall cite the form acquired by the complementary incomplete gamma function when its argument isimaginary and omit the more general case of /complex argument- The result:cos(!_.) 5:11:1r(v;iy) = sinl 2 I S(y;v) + cost2/C(v;v) + it sing2C(y;v) -t)S(v;v)]involves Boehmer integrals (seeeSection39:12`].L\\\ 45:12 GENERALIZATIONSThe three incomplete gamma functions are special cases either of the Kummer function [Chapter 471x" x" 45:12:1y(v;x) _ - exp(-x) M(1;l + v:x) _ - M(v;l + v;-x)V Vexp(-x)M(v;I + v;-x) 45:12:2 y*(v'x)r(1 + v)M(1;1 + v;x) =r(1 + v)or of the Tricomi function (Chapter 48145:12:3r(v;x) = x"exp(-x) U(1;1 + v;x) = exp(-x) U(I - v;l - v;x) 45:13 COGNATE FUNCTIONSJust as the incomplete gamma function derives from Euler's integral of the second kind [the complete gammafunction, Chapter 431 by allowing the upper limit to become indefinite, so the incomplete beta function (Chapter58] derives from Euler's integral of the first kind (the complete beta function, Section 43:131 by similarly makingthe upper limit indefinite. 45:14 RELATED TOPICSSeveral functions are arranged below in a hierarchical chart. The higher placed functions are the more general:arrows represent the effect of restricting one variable to a specific value. Numbers in brackets refer to chapters orsections of this Atlas. This chart is not, of course, exhaustive; thus, the Basset function [K,(x), Chapter 51] isanother special case of the Tricomi function, while both y*(v,x) and K,(x) include the exponential function (Chapter26] among their special cases. 443THE INCOMPLETE GAMMA y(v;x) AND RELATED FUNCTIONSGeneral hypergeometric function [18:14] with K + L = 3, IK - LI = l.\Kummer function [47J Tricomi function 1481 IIncomplete gamma Parabolic cylinderfunction, etc. [451 function (46]I Dawson's Error function, Exponentialintegral [421 etc. [40, 411 integral (39145:14 Constants [11 CHAPTER46THE PARABOLIC CYLINDER FUNCTION This function arises in the solution to many practical problems that are conveniently expressed in parabolic cylin-drical coordinates. This coordinate system, and others. are discussed in Section 46:14. 46:1 NOTATIONThe parabolic cylinder function is also known as the Weber function or the Weber-Hermite function. The name"Whittaker's function' is also encountered, but confusion should be avoided with the identically named functionsof Section 48:13. An alternative to the usual D,(.r) notation is U(-v - k,x).The variables v and x are known, respectively, as the order of the parabolic cylinder function and its argument.The origin of the name "parabolic cylinder" function is made evident in Section 46:14. 46:2 BEHAVIORThe D,(x) function is defined for all real values of v and x. Figure 46-1 is a contour map showing some values ofthe parabolic cylinder function.For v less than about -0.20494, D,(x) is a monotonically decreasing positive function of x. For larger orders,the parabolic cylinder function displays a number of zeros, maxima and minima in the -x < x < x range, asshown in Table 46.2.1.Table 46.2.1 -0.20494 < v s 00<vs1I<vs22<vs3etc.n-1 < V s nNumberNumberNumberof zerosof maximaof minima 0 10 1 102I32I nn - Intl--IIntl-22 447 THE PARABOLIC CYLINDER FUNCTION D,(.r)When v is an even/odd nonnegative integer, the function D.(x) is even/odd:46.2.1D,(-x)=(-I)"D,(x)n=0,1.2....but otherwise the parabolic cylinder function is neither even nor odd. 46:3 DEFINITIONSThe parabolic cylinder function is defined by a number of integrals, including 46:3:1D,(x) _ , f?exp4 ///o\\\lx'IIt" expl -I l cost xt - v Id:v> -1w+ a`2 /`2 /and 46:3:246:4 D.(x)=I(1exp42/Jexpl-xt)-v<02e, 0Many other integral representations are listed by Erdelyi, Magnus, Oberhettinger and Tricomi [Higher Transcen-dental Functions, Volume 2, Section 8.31.The differential equation known as Weber's equation:d'f1x-1 46:3:3 dx3 =4 -v2fis satisfied by D,(x) and, if v is not an integer, independently by D,(-.x).The equations of Section 46:12 sometimes serve as the definition of the parabolic cylinder function. 46:4 SPECIAL CASESWhen v is a nonnegative integer, the parabolic cylinder function involves a Hermite polynomial [Chapter 241 oran alternative Hermite polynomial [Section 24:13]:1z\Th expx46:4:1D"(x) = He4H"xI = exp 4J"(x)n = 0, 1, 2, .. .and hence Do(x) = exp(-x2/4), DI(x) = x exp(-x2//4),etc.With v = -1 we have reduction to the product of an exponential function and an error function complement[but note that the product is not an instance of the functions of Chapter 411: 46:4:2 D .,(x) _expl xJerfclX;72)24This is the n = 0 case of a general expression for a parabolic cylinder function of negative integer order, whichmay be written as either(lr/\/1 l46:4:3D_ (x)n!2expl4ds"Lexpl 2,/ erfcl c IJn = 0. 1, 2.... or 46:4:4D_,-,(x)=22"1=cxp/4)i"erfclxIn=0,1,2,..1\V/in terms of either successive derivatives or repeated integrals [see Section 40:131 of the error function complement.When the argument of a parabolic cylinder function is positive and its order is an odd multiple of 2, it may 46:5 THE PARABOLIC CYLINDER FUNCTION D.(x) 448be expressed in terms of one or more Basset functions (Chapter 511 of orders that are odd multiples of #. Thesimplest cases are 46:4:5 and 46:4:6D_,12(x) =K,/.I41x > 0 a[KI/4(X42)Dv2(x )- +K3/44x>0 and others may be evaluated via recurrence 46:5:1.46:5 INTRARELATIONSHIPSThe parabolic cylinder function satisfies the recursion formula46:5:1 D,.,(x) = xD,(x) - YD,-,(x)and the argument-addition formulas 46:5:2D,(x + y) = exp4+y2ID,.j(x) = expl2xy11v.,y;D._,(x) /2xy=Qj!4- y2\)f=o \The sum or difference D,(x) t Dr(-x) can be expressed in terms of Kummer functions [Chapter 47] 2° 2 7-'nx'\-v1 x'\2`+2 R/xz\/1 + V; 1x \ 46:5:3D,(x) + D,(-x) =exp(-I Ml - - - I =expl - I MI-_r(12v\4 /22 2/r112v\\4/\222l;/'eJ/x2/1-v 3x2_-2yJ3nx2v 3_x2 46:5:4D,(x) - D,.(-x) =r(_vxexpl 4)MI 2 .2. 2r(-vxezp\4/ M\1+ -; 2;2 2)\`2) 46:6 EXPANSIONSA number of power series expansions exist for exp(±x2/4) D.(x). Thus, one hasr- zv f (-xV2)'46:6:1 2aP('2r(/v)'p(x)(-x V [)J46:6:2 D,(x) = 2\2.46:6:3D,(x) =Vexp(xI i cos(+7 1/4 / ro2/j!Alternatively, one may add equations 46:5:3 and 46:5:4 and then use expansion 47:6:1 so as to express the paraboliccylinder function as the sum of two power series.Expansion in terms of Hetmite polynomials [Chapter 24] is possible in two distinct ways: 449 THE PARABOLIC CYLINDER FUNCTION D,(x) 2i2'/()H2,(x/ V 2) (I '46:6:4D,(x) =r(/2)exp\ 44}x > 0j,(j and 46:6:5D,(x) =expl-(__!)x > 0 \\\2j!!2The asymptotic expansion of the parabolic cylinder functionD,(x) - ? expp(-x2)[I -(-v)(1 - v)+(-v)(1 - v)(2 - v)(3 - v)46:6:6 42x22!(2x)2(- v),,j!(-2x')Jvalid for large x, is a consequence of relationship 46:12:2 or 46:12:3, together with 48:6:1. 46:7 PARTICULAR VALUES D.(0)D,(-) 0<v<1,2<v<3.4<v<S.v=0.1,2....v<0 46:8 NUMERICAL VALUES046:8 From equation 46:6:1 one can develop the equation 46:8:1D,(.)=Vr+Go)e%p\4}+exp( 4')±(F,-f)+(G,-8;)}riwhere F0 =fov)/21, Go = So = -xv/r(-v/2), F/F,-, = [j - I - (v/2)1x2/j(2j - 1). G;/G,-, = If - 4 - (v/2)J.x2/j(2j + 1) and f/f-, = g;/gf-, = x1/2j. This expression is more convenient than anyof the expansions in Section 46:6 and is the basis of the algorithm presented below. In the algorithm the upperinfinite limit in equation 46:8:1 is replaced by an empirically determined integer dependent upon x and v. Generally,the precision of the algorithm exceeds 24 bits (i.e., the relative accuracy of D,(x) is 6 x 10-8 or better) but thismay be degraded near the zeros of D,(x) or when D,(x) has a very small magnitude (e.g., when x and -v are bothlarge and positive).The first part of the algorithm calculates V -r/r[(1 - v)/2J and V a/I'(-v/2) using essentially the same pro-cedure as in Section 43:8. Because these values are stored permanently (in registers a and b, respectively), thereis no need to access the first part of the algorithm if one wants to calculate another parabolic cylinder functionD,(x) without altering the order v.Set a= I Storage needed: Input v » »» a, v, x, f, g. b, w, d(1) Set x = (a - v)/2 G. J,F. hand jSet g = 1 46:9 THE PARABOLIC CYLINDER FUNCTION D,.(x)(2) If x = 0 go to (3)Replace g by gxReplace x by x + 1Ifx:S 3 go to (2)Setb=2r/x2/\1+7b ReplacefLbyf-l\-x[l/n(x)- 1112xReplace f by g V-./2 exp(f)(3) If a = 0 go to (4)Seta=0Set w = fGo to (I)(4) Set a = wSet b = fInput x >>»»Set G = -b.\/Set J = 3 + 61x1 + (I(x + 2)(2 - v)I/5) + (x2/2)Replace xby x 2/2//xSetw= explvln\2-x-Replace Gby``Gw//SetfF=awSet g=GSet d = (F + G) exp(x) Set w = (v + 1)/2Setj=0(5) Replace j by j + I// 1 Replace F by Ij-1! - N.Fx/jReplace f byfx``/jReplace g by gx/jReplace G by (j - Nw) Gx/j j +Replace dbyd+(F-f)+(G-g)If)SJgoto(5)Output dd = D,(x) ««<450 Test values:D,(0) = Dr(±l) = 013_,(1.7) = 0.0944123874 D_34(-2.8) = 85.6541111 For large arguments, the parabolic cylinder function is best determined via expansion 46:6:6 (see also Section48:8]. 46:9 APPROXIMATIONSIf x is small and of either sign 46:9:1 D,.(x) =2". 451 THE PARABOLIC CYLINDER FUNCTION D,(x)whereas for large positive arguments46:12 46:9:2 D.(x) _ .r" expl-x'JII +v - v' 14L2x'The latter approximation is valid even for moderate positive x if v is close to one of the integers 0, I. 2 or 3. 46:10 OPERATIONS OF THE CALCULUSRules for single and double differentiation are 46:10:1 dxD.(x) = - D..(x) - D,,,() = vD.-,(x) - - D,(x)2 2d 2(X1 46:10:2 ax, D.(x) =4 - 2 -v)D.(x) and we also have the elegant relationship46:10:3 4D.(x) _ (-IYcxpl4I D,.,()n = 0. 1, 2....The very general definite integral(aY2"ar(p+l)p+r(1-vp - va-l 46:10:4tp exp - D.(t)dt = F- l + -;l\(222a+I -4 )(a+p2v+ 11 yields a Gauss function (Chapter 601.46:11 COMPLEX ARGUMENTp> -1a>0 The parabolic cylinder function is complex when its argument is complex. Here we present only the formular(I + v)\ 1 46:11:1D.(iy) _cosrrvZI[D_._i(y) + D_._,(-c)1 - isin]Trv2I[D-,-i(y) - D-.- 1(-A a//for the case of imaginary argument.46:12 GENERALIZATIONSThe Kummer function [Chapter 47] and the Tricomi function [Chapter 48] may be regarded as generalizations ofthe parabolic cylinder function. The relationships arevI; x/1- v 3r'MIM2222?2-r'/'46:12:1D.(x) =Vr n explJ V tr4 /r(l 2 v)r(' 1+ v1x/\I/v3r/=2. 2 22 222"a expx l41l-l 1rr2vlr((2v/x?0 46:13 46:12:2 46:12:3THE PARABOLIC CYLINDER FUNCTION D,(x) D (x) = 2'n expl4 )UI - 2: z:2)x > 0 D,(x) = 2`00/2eXp(4)U(I2v; 3; 2)x ? 02 46:13 COGNATE FUNCTIONSThe function 46:13:1 C(-v)[D,.(-x) - cos(Tr0D,(x)1ITis sometimes encountered and is symbolized V(-v - ?,x). 46:14 RELATED TOPICS452 In many contexts there arises a need to map space using an origin and three coordinates. In this section we discussa number of orthogonal coordinate systems, which constitute the most useful ways of performing such mappings.If we use r, q and p to denote three general coordinates, specifying the triplet (r,q.p) of numbers locates aunique point in space. Specifying two of the coordinates, say q and p, but allowing the third to adopt any permissiblevalue, defines a line (generally a space curve) that we can denote (q,p). Specifying only one coordinate, say r,defines a surface (r). It is a characteristic of an orthogonal coordinate system that, at any point (r,q,p). the threesurfaces (r), (q) and (p) are mutually perpendicular. Likewise, the three lines (q,p), (r,p) and (r,q) are mutuallyperpendicular at (r,q,p).In simple physical applications, each (r) surface, defined by allocating a specific value to the r coordinate,may correspond to a particular value of some scalar quantity F (temperature, energy, concentration, electric po-tential, etc.). Such surfaces are sometimes called equipotentials. Some physical body, known as the generator ofthe coordinate system, may occupy the r = 0 surface. The line (p,q) corresponding to specified values of the pand q coordinates is known by a variety of names such as "line of force," "flux line," "field vector," "line ofsteepest descent," "streamline," etc., depending on the field of application. Here we use the name streamline.Of course, the most familiar set of orthogonal coordinates is the cartesian coordinate system, (x,y.z). We maythink of this as arising from a generator corresponding to the infinite plane y = 0 with uniformly spaced equipo-tentials y = ±1, y = t2, y = t3, .... The streamlines are straight lines. A cartesian coordinate system is depicted in Figure 46-2. As in the other diagrams of this section, streamlines arc shown in red, equipotentials in green andthe generator in blue; the z-coordinatc is perpendicular to the plane of the paper. The rectangular coordinate systemmay equally well be represented by Figure 46-2; it is the two-dimensional equivalent of the cartesian system.The cylindrical coordinate system is illustrated in Figure 46-3. The coordinates consist of two lengths, r andz, and one angle, 0. The relationship to cartesian coordinates is46:14:1x=rcos(8)y=rsin(8)05r<-Tr<85aThe generator is the line (r = 0, 0 arbitrary); streamlines are straight lines radiating from the generator; equipo-tentials are circular cylinders having the generator as their common axis. The polar coordinate system is the two-dimensional analog of the cylindrical coordinate system: equations 46:14:1 and Figure 46-3 apply equally to both.The generator of the parabolic cylindrical coordinate system is the half-plane (y = 0, x < 0) as depicted inFigure 46-4. The coordinates are r, q and z, the first two of these being related to cartesian coordinates by 46:14:2x=I(r2-q2)y=rq0sr<x-x<q<xThe equipotentials are parabolic cylinders with a common focal axis (.r = 0. y = 0) [or (r = 0, q = 0)], whilethe streamlines are semiparabolas [see Chapter 12] whose foci lie on the same axis.In the elliptic cylindrical coordinate system the generator is the strip (y = 0, -1 < x < 1). The r and 8 453 THE PARABOLIC CYLINDER FUNCTION 46:14 y=4 Y-2FIG 46-2y=2 y=1 y=o y=-ya- FIG 46-31 2 y-U r.. y=-1rot.......... a-2y rsA.:y=-3 coordinates are related by46:14:3x = cosh(r) cos(h)y = sinh(r) sin(O)0 5 r < x-if < $ !5 itto cartesian coordinates, while the third coordinate is again z. As shown in Figure 46-5. the equipotentials areelliptic cylinders and the streamlines are semihyperbolas [see Chapter 15].All the coordinate systems we have considered up to this point have had a single generator at r = 0. althoughone might postulate a second generator at r = x. The bipolar coordinate system has two generators, correspondingto r = +- and r - --, located on the lines Is = 0. y = l) and (x = 0, y = -1), respectively. Representing thethree coordinates by r, 8 and z, we have the relationships 46:14 THE PARABOLIC CYLINDER FUNCTION D,(x) 454sinh(r)sin(8) 46:14:4x = -x < r < x-V < 8 <_ V cosh(r) - cos(8) cosh(r) - cos(O)to cartesian coordinates. As Figure 46-6 shows, the equipotentials are mostly circular cylinders; their axes are thestraight lines (y = 0, x = coth(r)) and their radii are Icsch(r)I. Streamlines are arcs of circles whose centers lie onthe line (x = 0, z) at y = cot(O) with radii equal to Icsc(e)I.The spherical coordinate system employs one length r and two angles 8 and 4s as its three coordinates. Becauseit is inherently three dimensional, we do not include a diagrammatic representation of this system, which is relatedto cartesian coordinates by the equations 46:14:5x= rsin(8) cos(d,) y= r sin(8) sin(4) z= r cos(8)0 sr<-- <- 8<IT--n < 4 K a22The generator is a single point at the origin; equipotentials are spheres centered on the origin, and streamlines arestraight lines radiating in all directions from the origin.There are a number of other orthogonal coordinate systems that we shall not discuss in detail. The generatorof the paraboloidal coordinate system is the half-line (x = 0, y = 0, z > 0). The generator of the prolate spheroidalcoordinate system is the line segment (x = 0, y = 0, - I < z < 1). The generator of the oblate spheroidal coordinatesystem is the disc (x2 + y2 < 1, z = 0). The generator of the toroidal coordinate system is the hoop (x2 + v= _1, z = 0). For details of these systems, and three others, see Spiegel (pages 126-I30J.With k sometimes positive, sometimes negative and sometimes zero, the equation46:14:6 V2F=kFoften known as the Helmholtz equation, is ubiquitous throughout all of science. Special instances are the waveequation for the propagation of vibrations, Laplace's equation describing electric fields, the Fourier equation of i$,igbi4 1Ai'iO 44 4gp 44i4,O2 455 THE PARABOLIC CYLINDER FUNCTION D,.(x) 46:14 heat conduction, the Schrodinger equation of quantum physics and Fick's second law of diffusion. In all thesecases F is some scalar property whose value depends on the spatial coordinates (and often on time also, althoughhere we shall ignore the temporal variation of F). The so-called Laplacian operator V performs double differen-tiation with respect to the spatial coordinates, the precise form of the operator depending on the coordinate systemadopted. Table 46.14.1 lists the forms of 6''F for a variety of orthogonal coordinate systems. Table 46.14.1 Coordinate system Cartesian Cylindrical Parabolic cylindrical Elliptic cylindrical Bipolar SphericalVF a'Fd'Fd'F ax=ay'd:a'F1 aFI a-Fa'F-+--+--+-ar'r drr' aB' Ia'F+a'F1I+a"F--r + q'ay;aq'/a-Ia'Fd'Fa'Fsinh'(r) + sin'(6)a .,a9'aY(a'Fd-Fd'F lcosh(r) - cos(O)1'\ dr +d9' +d2F2 aFcot(B) dFI a'Fcsc'(B) a-F-+--+--+-+--ar'r arr'aer'-ae'r'ab' 46:14 THE PARABOLIC CYLINDER FUNCTION D,(x) 456it.i4,ittip,i4'Lt Because the cartesian coordinate system is undoubtedly the simplest, both conceptually and in the form of itsLaplacian, one might wonder why other systems are ever used. One answer is that, for a physical body that matchesthe corresponding generator, the Helmholtz equation is most easily solved in the appropriate orthogonal coordinatesystem. We shall illustrate the method of solution by reference to the parabolic cylindrical coordinate system.The function F that is a solution of the Helmholtz equation46:14:7VF=I(d=F+J=F\+dF=kFr2 + q' I\dr'dq'/IBz'will generally be a function of all three coordinates: r, q and z. Assume, however, that F = R(r) Q(q) Z(z) whereR is a function of r, but not of q or z, and similarly for Q and Z. Then equation 46:14:7 is easily transformed to46:14:81(1 d'R+I d2Q_ k-I d'Z----r' + q' R dr2 Q dq'Z dz'A consequence of our assumption is that the left-hand side of this equation is a function of r and q, but not of z.whereas the right-hand side is a function of z, but of neither r nor q. It follows that each side must equal the sameconstant, a so-called separation constant, say c4, so that 46:14:9 d_, = (k - C4) Z and2Q46:14:10R- -cr2=cq'-I dQ dq 457 THE PARABOLIC CYLINDER FUNCTION D,(x) 46:14Once again, we can argue that, because the left-hand side of 46:14:10 does not depend on q and the right-handside does not depend on r. each side must equal another separation constant. It is convenient to represent this newconstant by c2(2v + 1), whence 46:14:11 1 d'QI)Qc dq'and1 d'R 46:14:12 ( C 2ORc dr'Thus, the Helmholtz equation has been decomposed into three ordinary differential equations. It requires only aredefinition of the independent variable to convert equation 46:14:11 to 46:3:3. showing that a parabolic cylinder function can be a solution of the Helmholtz equation expressed in parabolic cylindrical coordinates.Section 59:14 presents a second example of the solution of the Helmholtz equation. CHAPTER47THE KUMMER FUNCTION M(a;c;x) The Kummer function is one of the most important instances of the hypergeometric function [Section 18:14] andis closely related to the Tricomi function of Chapter 48. 47:1 NOTATIONThe symbol $(a;c:x) often replaces M(a;c;x). The variables a and c are termed the numeratorial parameter anddenominatorial parameter, respectively; x is the argument.Alternative notations for the Kummer function make recourse to a generalization of the Laguerre function[Section 23:14]M(-v:g + l;x) 47:1:1 L"'(x) =r(v + I)or to a simple instance of the generalized hypergeometric function [Section 60:12147:1:2 1Ft(a;c;x) = M(a;c;x)Collectively, the Kummer function and the Tricomi function (Chapter 48] are known as confluent hypergeo-metric functions or as degenerate hypergeometric functions. These puzzling names have their origins in definitions47:3:2 and 48:3:5. 47:2 BEHAVIORThe Kummer function is defined for all real values of a and x and for all values of c except c = 0, -1, -2, ...Being trivariate. M(a;c;x) is incapable of having its behavior comprehensively described in a two-dimensionalmap. Some examples of the wide variety of behaviors are depicted in Figure 47-1. In its dependence upon x, theKummer function may, or may not, encounter zeros, maxima or minima, depending on the values of the twoparameters. Figure 47-2 may be used to find, for specific values of a and c, the number of negative zeros of theKummer function, that is, the number of times that M(a;c;x) acquires the value zero in the argument range -x <x < 0. Similarly, Figure 47-3 can be used to find the number of positive zeros of M(a;c;x). In the regions in whichthe Kummer function lacks zeros, it is invariably positive.For the behavior of the Kummer function as its argument approaches infinity in magnitude, see Section 47:7. lID 47:2 THE KUMMER FUNCTION M(a;c;x) 460 FIG 47-1<.: .... : .............. : .... :.1 M (-2. 6s 1. 21 x) ......-1H(1.6s-0.41x) 040'%'10ec-2 c=1 C-0 c--2 c--3 461 THE KUMMER FUNCTION M(a:c;x) 47:3 DEFINITIONS47:3 The Kummer function may be defined as a hypergeometric function (Section 18:14) with one numcratorial parameterand two denominatorial parameters, one of the latter being equal to unity: 47:3:1 M(a;c;x) _ x,J-o (c)/(l );It may also be defined in terms of the Gauss function [Chapter 601 by means of the limiting operationx47:3:2 M(a;c;x) = llim F a,b;c;bThe indefinite integralf(c)x't°"' exp(t)47:3:3M(a;c;x) = di0 < a < cr(c - a) r(a)u (x - tis a representation of the Kummer function, as are the definite integralsr(c)fr°'' exp(xt) 0 < a < c 47:3:4 M(a;c;z) =f(c - a) r(a)Joandand2'-`exp(x/2) f(c) (1 + t)°-'exp(xt/2)47:3:5M(a;c;x) = Jdt0 < a < c r(c - a) r(a)(1 -The so-called confluent h)pergeometric equation 47:4 THE KUMMER FUNCTION M(a;c;x) 462 47:3:6 d''f+ (c - x)-dfz -- of= 0dxdxhas the solution f = c,M(a;c;x) + c2x'-`M(a - c + 1;2 - c;x) provided that c is not an integer; c, and c_ are ar-bitrary constants.The Kummer function is obtained by differintegrating the function xa-' exp(x) with respect to x, using a lowerlimit of zero:r( 47:3:7 d° 47:3:7 M(a;c;x) =f (a)xl `dx°-`[X,-, exp(x)]Thus, in the notation of Section 43:14, the synthesisI-c47:3:8 exp(x) -- M(a;rx)1-agives rise to the Kummer function. 47:4 SPECIAL CASESThe exponential function [Chapter 26] is the special case of the Kummer function in which the parameters areidentical: 47:4:1 M(a;a;x) = exp(x)If the denominatorial parameter exceeds the numeratorial parameter by unity, the Kummer function reduces to anincomplete gamma function [Chapter 45]:47:4:2M(a; I + a;x) = a y(a; -x) = r(I + a)y*(a; -x)(-x)oThe Kummer function M(a;a + n;x) in which the denominatorial parameter exceeds the numeratorial parameterby a positive integer may also, by sufficient applications of recurrence formula 47:5:3, be expressed as an entireincomplete gamma function; for example: 47:4:3 M(a;2 + a;x) = as±xx r(2 + a)y*(a;-x) - all+ a) exp(x)x xWhen the denominatorial parameter is twice the numeratorial parameter, c = 2a, the Kurnmer functionspecializes to the product of the exponential function and the hyperbolic Bessel function (Chapter 50) of ordera-4:r(z +4)exp\2/47:4:4 M(a;2a;x) =(x/o 1n1-1/2With the numeratorial parameter equal to a negative integer, the Kummer function becomes a generalizedLaguerre polynomial [Section 23:12): 47:4:5M(-n;c;r) = L;`-(x)/\nc1/n = 1, 2, 3, ,..c > 0 nIf a = 0, the Kummer function equals unity, and if a = 1, specialization to the incomplete gamma function occursagain:47:4:6M(1;c;x) = I + x 'exp(x) y(c;x) = I + C(c)x exp(x) y*(c:x)Coupled with recurrence relationship 47:5:2, this last result also enables M(2;c;x), M(3;c;x), etc., to be expressed 463 THE KUMMER FUNCTION M(a:c;x) 47:5in terms of the entire incomplete gamma function or, alternatively, if x is positive, in terms of the incompletegamma function.Except perhaps when a is also a negative integer, setting c = 0. - 1, -2, ... causes M(a;c;x) to be undefined[but see Section 47:12). When c = 1, reduction occurs to a Laguerre function [Section 23:141.47:4:7 M(a;l;x) = L_o (x)Use of recurrence formula 47:5:3 then permits the special cases M(a:2;x), M(a;3;x), etc., to be expressed as La-guerre functions also.With a denominatorial parameter equal to ;, the Kummer function becomes the sum of two parabolic cylinderfunctions [Chapter 4612°-'r(a +z) 1exPI2 1[D_z(V tar) +D-.(-VT,))x 2 047:4:8M a;-;xa2r(1 - a)expl 2 IDm-,(-))x:5 02°The difference of two other parabolic cylinder functions is the special c = ; case of the Kummer function:-2"r(a - 1)/x3--expl 2 J1D,-<(\) - x > 047:4:9Ma;;zV s2ax\ /2r(1 - a)/x\expl 2 I[Dm_z(V -lx) - D2.-,(- V -2x)) x < 0\ /Bessel functions [Chapter 53) and their hyperbolic counterparts (Chapter 51) arise from the limiting operationsymr-xlr(1 + e) 47410lim M a;c+ I; - = Jz z 0ax`/'- 47:4:11limM`ax+ l; -J - rJZI(2Vx)xa0 o-=axFurther specialization of incomplete gamma functions [Section 45:4), parabolic cylinder functions [Section46:4), hyperbolic Bessel functions (Section 51:41 and Bessel functions [Section 53:4) may also occur to yield still more elementary functions. 47:5 INTRARELATIONSHIPSKnown as Kummer's transformation, the important identity47:5:1 M(a;c;-x) = exp(-x) M(c - a;c;x)constitutes a reflection formula for the Kummer function.Recurrence formulas may be written interrelating three Kummer functions whose parameters differ by unity,as follows:2a - C +X c - a47:5:2M(a + l;c;x) =aa - c+lM(a;c;x) + - M(a - l;c;x)ac-IaM(a;c;x) + - M(a;c - l;x)a=c(c - I + x)M-c(c - 1)M=c- M-c-M + l- l- l 47:5:3;x)(a;c;x)(a;c;x)(a;c;x)(a;c;x) M(a;c(c - a)x(c - a)xxx 47:6 THE KUMMER FUNCTION M(a;c;x) 464 47:5:4 M(a;c;x) = a M(a + l;c;x) + (c - ajarM(a;c + l;x)a+xc(a+x)c-Ic-a=M(a;c - l;x) -M(a - l;c;x)a-I+xa - I+xThese relationships may be developed into formulas expressing, for example, M(a + n;c;x) where n is an integer.The Kummer function obeys a number of argument-addition relationships. One is 47:5:5 M(a;c;x +(a)')'i-o ciirand others are given by Erd6lyi, Magnus, Oberhettinger and Tricomi [Higher Transcendental Functions, VolumeI, Section 6:14]. By setting y = (v - 1)x, 47:5:5 becomes an argument-multiplication formula.A special linear combination of Kummer functions: r(l - c) r(c - I) 47:5:6r(a - c + 1)M(a;c;x) + r(ai M(a - c + 1;2 - c:x) = U(a;c;x) generates the Tricomi function [Chapter 481. Among the summablc infinite series of Kummer functions is 47:5:7M(a;c;x) + M(a - 1;c;x) + M(a - j;c;x) = c s 1 M(a;c - l;x) c >5-o 2and several others are reported by Erdelyi, Magnus, Oberhettinger and Tricomi [Higher Transcendental Functions.Volume 1, Section 6.15. ]. 47:6 EXPANSIONSThe power series expansionaxa(a + 1)x2a(a + 1)(a + 2)x' (a)x' 47:6:1M(a;c;x) = I +c+2!c(c + 1) + 3!c(c + 1)(c + 2) +-i=oknown as Kummer's series, is convergent for all arguments and for all parameter values except c = 0. -I.2, .... Using v as an abbreviation for c - a - _, a series expansion in hyperbolic Bessel functions [Chapter 50]is/4x1(-l)j(2v) (c - 2a)(x\ 47:6:2M(a;c;x) =r(v)1-)expl - I'' 1,;IX2i-ojl(c),2Expansions may also be made in Bessel functions of order c - I + j: see Abramowitz and Stegun [Section 13.3]for the coefficients involved. Another expansion is in terms of generalized Laguetre polynomials (Section 23:12]: 47:6:3 M(a;c;-x)(a),L"-" (x)x > 02'(c)1The Kummer function may be written as the sum of two terms involving Tricomi functions [Chapter 481(-1)° r(c)(- I)°-` r(c)47:6:4 M(a;c;x) = r(c - a)U(a;c;x) +r(a)exp(x) U(c - ac;-x)where powers of -1 are to be interpreted, as explained in Section 13:11, as the (generally complex) number47:6:5 (-1)` = cos(tar) + i sin(g)Because the Tricomi function has a simple asymptotic expansion [see 48:6:1], an expansion of the Kummer functionvalid for large x may be constructed using 47:6:4. 465 THE KUMMER FUNCTION M(a;c;x) 47:847:7 PARTICULAR VALUESThe Kummer function equals unity at zero argument, irrespective of the parameter values47:7:1 M(a;c;0) = IAs x - -x, the Kummer function acquires one of four values: -x, 0, 1 or + x depending on the values ofthe a and c parameters. From Figure 47-4 one can identify the limiting value acquired in any particular case. Note that M(O;c;-x) = I is illustrated in black on the map and that we have M(a;c;-x) = 0 if a - c is a nonnegativeinteger, as illustrated in green. Figure 47-5 similarly depicts the values of M(a;c;+x). Notice the exclusion from both diagrams of c = 0,-1, -2, ... where M(a;c;x) is undefined. 47:8 NUMERICAL VALUESBecause the Kummer series, equation 47:6:1, converges rapidly, it constitutes a convenient method for calculatingvalues of M(a;c;x). If x is negative, the terms in the expansion(a),x147:8:1 M(a;c;x) T,ultimately constitute an alternating series (see Section 0:61. One can show that, for all values of j greater thansome integer J, the series T, + T,. I + T,.2 + + T) + -is alternating provided that IT,I < rr,_,j and thatJ > 21al + lei + I. These considerations form the basis of the algorithm presented below, which exploits the uniqueadvantage, namely 47:8 THE KUMMER FUNCi7ON M(a;c;x) 466 IJ11 47:8:2 M(a;c;x) =2T, + E T,lerrorlZIT,Ii-othat alternating series possess for the placing of bounds on the error introduced by truncation. The algorithm testseach T, until a value T, is reached that satisfies IT,I < 10-' 1 To + T, + T2 + T,-. 1. thus ensuring a precisionof better than 6 x 10-8. The algorithm also checks to ascertain that the necessary conditions J > 2Ial + Icd + Iand IT,I < IT,_,I are met. If not, j is incremented until all three conditions are satisfied. Setf=tISetj=g=0Ifx<0goto(I)Set f = exp(x)Replace x by -xReplace a by c - a(1) Replace g by g + tSet p = (a + j)x/(c + j)(j + 1)Replace jby j + IReplace t by tp If 111 - 10-'IgI go to (1)If IpI > I go to (1)If )s2Ial+Icl+2go to (I)Replacef by f g +r2Storage needed: a, f, t. c, j, g, x and p Input restrictions: The c parameter cannot take values Test values:M(0.7;0.6;2) = 8.94061153M(-4;2;-1) = 4.175M(0.5:l;-2) = 0.465759608 467 THE KUMMER FUNCTION M(a.c:x)Output ff - M(a;c;x) < <<47:10 If x is positive, the Kummer transformation, equation 47:5:1, is invoked by the algorithm.For large argument, numerical values of M(a;c;.r) may be calculated via expansion 47:6:4 and the algorithmin Section 48:8. 47:9 APPROXIMATIONSIf either a or x, but not c, is small: 47:9:1 axM(a;c;x) = I + -cIf one of a. c or x is large, the other two remaining modest in magnitude, the following approximations hold:47:9:2M(a;c;x) =N (12- ax)expl 2 1 costIT-2 t2cx -4-)a large and negative 47:9:3M(a;c;x)r(c) axex-2)expxl,._,(4ox - 2cx)a large and positive/xl 47:9:4 M(a;rx)I - -cIc large, either sign\\\r(c) 47:9:5M(a;c;x)(-x)'x large and negativec - a * 0, -1, -2, .. . r(c - a) 47:9:6M(a:c;x) =r(c )r(o)xe-'exp(x)x large and positivea * 0, -1. -2, .. . 47:10 OPERATIONS OF THE CALCULUSSingle and multiple differentiation of the Kummer function satisfy the rules 47:10:1 - M(a;c;x) =a-M(1 + a;I + c;x)dxcand47:10:2 dx"M(a;c;x) _(c),M(n + am + c;x)Formulas for indefinite integration include 47:10:3[M(a;c;r)dr = c1 (M(a- 1:c - I;.r) - I)a # Ic * 1 a - 1 ot`-'M(a:c;r)r(c) r(c - c) 47:10:4 I(x - t)I, -cdr =r(C,)M(a:C:.r)c > 0C> 0and we also cite the important definite integral 47:10:5Jrt'-'M(a;c;-r)dt -r(v) r(c) r(a - v)0 < v < ar(a) r(c - v) 47:11 THE KUMMER FUNCTION M(a;c;x) 468With a lower limit of zero, differintegration of the Kummer function gives a hypergeometric function [Section18:14]; the formula 47:10:6d" M(a.c.x) =x-(a) xdx"r(1 - V) J.0 (c);(I - v);represents a generalization of 47:10:2. The differintegration formulad° J 'Xr(c) 47:10:7dx'x`-'M(a;c:x) =r(c - v)M(a:c - v;x) is equivalent to the integral transform 47:10:4. Hence, in the notation of Section 43:14, one Kummer function canbe synthesized from another by 47:10:8I-CM(a:c;x) -+ M(a;C;x) Similarly:I - C 1 -a 47:10:9 M(a;c;x) - M(A;c;x)1-A 47:11 COMPLEX ARGUMENT The replacement of x in M(a:c;x) by x + iy produces a complex-valued quadrivariate function, the properties ofwhich are not pursued in this Atlas.When its argument is purely imaginary, the Kummer function has real and imaginary parts that are higher-order (K = 2. L = 4; see Section 18:141 hypergeometric functions ,\ 4/, \2!N+2 1j(24 ' it.),\a21/;\a2')47:11:1M(a;c;iy)((((l+(l(f=u\c2I),(c22/(I),\2) 47:12 GENERALIZATIONS The entire Kummer function, defined by 47:12:1 M(a:c:x)r(c)is a modest generalization of the Kummer function designed to embrace those values of the denominatorial param-eter (c = 0,-1,-2,...) that are excluded from the definition of the Kummer function itself. Unlike M(a;c;x), theentire Kummer function encounters no discontinuities as c -. 1 - n = 0, -1, -2, ..., having the finite valuesM(a;l - n:x)(a),x"47:12:2r(1 - n)=nrM(a + n;l + n;x) n = 1, 2. 3,... 469 THE KUMMER FUNCTION M(a;c;x) 47:13at these points. It is unfortunate that the entire Kummer function is not widely used because its properties aresignificantly simpler than those of M(a:c;x).As well, all hypergeometric functions with L = K + 1. K >_ I [see Section 18:14] may be regarded as gen-eralizations of the Kummer function. 47:13 COGNATE FUNCTIONSClosely related to the Kummer function are the Tricomi function (Chapter 481 and the Whittaker function M,(x),which is discussed in Section 48:13. CHAPTER48THE TRICOMI FUNCTION U(a;c;x) Together with the Kummer function, the Tricomi function is a solution of an important differential equation. Thesetwo functions constitute the so-called confluent hvpergeomerric functions. The closely related Whittaker functionsare discussed in Section 48:13. 48:1 NOTATIONThe symbol ib(a;c;x) often replaces U(a;c:x), and Tricomi used the G symbol. On account of asymptotic expansion48:6:1, the notation x-°,F0(a.l + a - c:-1/x) is sometimes used for the Tricomi function.The variable x is the argument of the Tricomi function. The variables a and c are the parameters but, in contrastto their roles in the Kummer function, they cannot be separately assigned as numeratorial and denominatorialparameters. The composite variable I+ a - c is important in determining the properties of U(a;c:x) and willsometimes be denoted by b. 48:2 BEHAVIORThe Tricomi function is defined for all values of a, c and x. However, when x is negative. U(a;c;x) is generallycomplex and we therefore exclude the x < 0 range from most of the discussion of the Tricomi function.As a function of a and c, Figure 48-I shows the number of positive zeros of U(a;c;x), that is, the number oftimes the Tricomi function acquires the value zero in the range 0 < x < x. Where zeros are absent, U(a;c;.r) isinvariably positive, being a monotonically increasing function if a < 0 and monotonically decreasing for a > 0.The behavior of the Tricomi function when its positive argument is close to zero or infinity can be deducedfrom the approximations given in Section 48:9. 48:3 DEFINITIONSIn terms of the Kummer function [Chapter 471, the Tricomi function is defined as 48:3:1U(a;c;x) =r(1 - c)r(1 + a - c)M(a;c;.t) 4r(c - I)r(a).XM(I + a - c;2 - c;x) 471 48:3 THE TRICOMI FUNCTION U(a;c;x) This definition may be invalid if c is an integer. The complicated equation(n-1)!"(a-n)x'(-1)" 48:3:2U(a;n + l;x) =-{ In(s) M(a;n + 1;x)Foe .U (1 - n)jj!n!f(a - n) l472 (a);x' 1lld(a+j)-i1(I+j)-VI+n+j)1}n=0.1.2.... )vo (I + n)J!provides a supplementary definition to cover the cases c = 1. 2, 3... , while the c = 0, -1. -2, ... instancesmay be accessed via transformation 48:5:1. Here y is the digamma function (Chapter 44). When the argument isnegative, definition 48:3:1 is to be replaced by the complex expressionf(1 - c)48:3:3 U(a;c;-x) =f(b)M(a;c;-x) -r(a)x'_'M(b;2 - c;-x) x > 0where b = I + a - c and (-1)` = cos(4rc) + i sin(ac). This formula may be rewritten in a large number ofequivalent forms. Thus, either or both of the Kummer functions may be transformed via 47:5:1. Again, any of thegamma functions may be subjected to the reflection formula 43:5:1, and the resulting cosecant function may becombined with the trigonometric terms in (-1)`.The Tricomi function is also defined by the limiting operation 48:3:4 U(a:c;x) = x-° lim FI a,l + a - c;-y;l - y1Y- \ xperformed on the Gauss function [Chapter 601 or as the Laplace transform (Section 26:141 I 48:3:5 U(a;c;x) =f(a), t(I+ r`°t) drc > a > 0 The confluent hypergcomctric differential equation, 47:3:6, is solved by f = c,U(a;c;x) + cz exp(x) U(c - a;c;-x) for all values of the variables, where c, and c., are arbitrary constants. 473 THE TRICOMI FUNCTION U(a;c;x)48:4 SPECIAL CASES48:4 When the two parameters of the Tricomi function are identical, reduction occurs to the complementary incompletegamma function [Chapter 45148:4:1 U(a;a;x) = exp(x) r(1 - a;x)If the c parameter exceeds a by unity, the result is a simple power function (Chapter 13):48:4:2 U(a;a + l;x) = x-"Using these two formulas, and the recursion formulas in Section 48:5, one may deduce expressions for U(a;a ±n;x) with n = 1, 2, 3, ....When the c parameter is twice the a parameter, the Tricomi function specializes to a function involving theexponential and a Basset function [Chapter 5 I ]: 48:4:3 U(a;2a;x) = a-1/21K,_,n (2)axFurther specialization may occur [see Section 51:4).With the a parameter equal to a negative integer, the Tricomi function becomes a generalized Laguerre poly-nomial [Section 23:12]48:4:4 U(-n;c:x) _ (-1)"n!L;`-"(x)Simplifications of the Tricomi function for a = 0 or I are 48:4:5 U(O.c;x) = Iand48:4:6 U(l ;c;x) = x'-` exp(x) r(c - l;x)The latter function has an asymptotic expansion48:4:7U(l;c;x)--1(2-c),=Il-c)JxyxX ,.O(-x)jc - 1 J=i(-x)ithat represents the simplest of the K = L + I family of hypergeometric functions [Section 18:14). With the aid ofequations 48:4:5 and 48:4:6, recursion 48:5:2 permits expressions for U(2;c;x), U(3,c;x), etc., to be derived.Definition 48:3:2 can be simplified somewhat when c = 2 and, employing 48:5:1, the result can be transformedtoi48:4:8U(a;O;x) =f(1 + a)((j! x 11 + j 1n(x) + j 4)(a + j) - 2j 4'(j )]This special case of the Tricomi function is related to Bateman's k function: 48:4:9 U(a;0;x) = xU(l + a;2;x) = r(I - a)exp(2) k_,.().r>0which is defined by the integral 48:4:10 k,(x) =2J*cos(x tan(s) - vt)dt0With the c parameter equal to 1 or ?, the Tricomi function is a parabolic cylinder function (Chapter 46]: 48:4: 11 U(a !;x) = 2a exp 2 D_,(2V)(X)48412 U(a;];x) = 2a exp 2Di_y,(2) 48:5 THE TRICOMI FUNCTION U(a;c;x) 474Clearly, these last two expressions can be used in conjunction with recursion 48:5:3 to express U(a;n + 2;x) wheren is any integer.The limit operation 48:4:13lint{r(1+ a - c)UI a;c;I } = 2x!2K,. (2 V x)`a/11produces a Basset function ]Chapter 51] of order c - 1.In this section we have addressed the effect of specializing one of the two parameters. As will be clear fromSections 4 of Chapters 13, 23, 45, 46 and 51, still simpler functions are generated when both a and c are specialized. 48:5 INTRARELATIONSHIPSThe important transformation48:5:1 U(a;c;x) = x'-` U(I + a - c;2 - c;x)relates two Tricomi functions of common argument.Recurrence formulas may be written interrelating three Tricomi functions whose parameters differ by unity.Examples are2a-c+x I 1I48:5:2U(a + I;c;x) = U(a;c;x) - U(a - I;c;x) _ - U(a;c;x) - - U(a;c - 1;x)a(l+a-c)a(I+a-c) aac-l+xl+a-cc-a 148:5:3U(a;c + l;x) = U(a;c;x) + U(a;c - 1;x) _ U(a:c;x) + - U(a - 1; c;x)x x xXand 48:5:4 U(a:c;x) =all + a - c)U(a + I:c;x) + -xU(a;c + l;x) a+xa+x II+a-cU(a - l;c;x) -U(a;c - l;x) a - I+xa-I+xAnalogous to equation 47:5:5 is the argument-addition formula 48:5:5U(a;c;x + Y) = Z(a)'(y)U(a + j;c + j;x) LvI<IxI J=ujand it may be converted to an argument-multiplication formula in a similar manner. 48:6 EXPANSIONSThe function U(a;c;x) may be written as a power series in x by combining definition 48:3:1 with expansion 47:6:1.The Tricomi function may be expanded asymptotically as 1Iaba(a + 1)b(b + 1)-(a), (b)48:6:1 U(a;c;x) = x1 - - + .. +f+x -. x x2x'j!(-x)jif x is large, where b = I + a - c. 48:7 PARTICULAR VALUESWhen its argument equals zero or +x, the Tricomi function takes the valuesc<I 48:7:1 U(a;c;0)r(1 - c)/r(i + a - c)c?1 475 THE TRICOMI FUNCTIONU(a;c:x) 48:9ma< 048:7:2 U(a;c x) = 1a = 00a> 0 48:8 NUMERICAL VALUESFor small values of x, U(a;c;x) is best evaluated via definition 48:3:1 and the algorithm of Section 47:8.For large positive x the following algorithm may be employed.>a»»Set t = I3-»» Set f = .rSet)=g=0(1) Replace g by g + tSet p = (a + j)(b + j)/x(1 + j)31»»Setb=a+I - c Replace j by j + IReplace t by -tpIf 1pI < I go to (1)rr/2Replace f by Lg -S1- a - b - x -j)t/4pxIf Output f\Storage needed: a, t, b, x, f,j, g, t and p Test values:U(,;1: 3n) = 0.310662341U(4;3;5) = 0.101573807U(1;1;20) = 0.0477185455 f - U(a;c;x) The algorithm utilizes expansion 48:6:1 with an appropriate convergence factor: 48:8:1fU(a;c;x) = 1 - Ti + T7 - T3 + = Tj_, :t Tf34 +3 - 2a - 2b - 2J`Ib = I + a - cwhere T, = (a),(b);/j!x' and J is the positive integer such that T, is smaller in magnitude than any other T;. Noparticular accuracy is claimed for this algorithm, but the precision will increase with x, rarely being acceptable for x<5.48:9 APPROXIMATIONSFor small positive values of its argument, the behavior of the Tricomi function depends dramatically on the cparameter. This leads to a multiplicity of approximation formulas:r(1 - c)ar(-c)x 48:9:1U(a;x)-c --0r(I+a-c)r(1+a-c) 48:9:2 U(a;0;x) =-c = 0r(-a)r(I+a)r(1 - c)_r(c)x'0 48:9:3U(a:c:x) =r(l + a - c)(I - or(a)G c < I 48:9:4 -In(x)U(a;l;x) =2y - kP(a)c 4895r(a)r(c - l)U-r(a)r(2 - c)I << 2 ::(a;c;x) r(a)x`_'(c - 1) r(1 + a - c) c 48:10 THE TRICOMI FUNCTION U(a;c;x) 476 48:9:6 48:9:7I IU(a;2,x) =F(a)x + r(a - I)In(x)c = 2r(c - 1)(c - I - a) r(c - 2)U(a;c:x) =r(a).r`_I+r(a)x'-2c > 2all of which are valid for small x.The crude approximation x-" may be refined tor(x - v)"- 48:9:8 U(a;c;x)x,"-I_,v = 2 + 2a - cfor large positive argument. Abramowitz and Stegun [Section 13.5] give formulas valid when a (but not c or x) isof large magnitude, or when all three variables are large. 48:10 OPERATIONS OF THE CALCULUS Single and multiple differentiation of the Tricomi function lead tod 48:10:1 dxU(a;c;x) _ -aU(a + l;c + l;x) and 48:10:2 d"ix; U(a;c;x) _ (- I )"(a) U(a + n;c + n;x)Formulas for indefinite and definite integration includef 48:10:3 JU(a;c;t)dt =U(a - l;c - l;x)a < Ia-1 48:10:4 U(a;c;t)dr =r(2 - c)a > 1c < 2f(a-I)r(l+a-c)and (('r(1 +v)r(a- l-v)r(2+v-c) 48:10:5r"U(a;cbr)dr=b""r(a)r(l+a-c)b>00<1+v<ac<2tv0 48:11 COMPLEX ARGUMENT The Tricomi function generally is complex valued when its argument is complex or negative. The latter case wasdiscussed in connection with definition 48:3:3. The case of complex argument is not treated in this Atlas. 48:12 GENERALIZATIONSVia its asymptotic representation, equation 48:6:1, the Tricomi function generalizes to the unconstrained K = 2,L = I hypergeometric function [see Section 18:14)(a), (b), (x)' 48:12:1 6= I +a-cJ-o(Y)i 477 THE TRICOMI FUNCTION U(a;c;x) 48:1348:13 COGNATE FUNCTIONSIn this section we discuss the Whittaker functions M,.,,(x) and These are closely related, respectively, tothe Kummer and Tricomi functionssl 48:13:1 M,µ(x) =x'nw exp/ l 2 ) M 2 + µ - Y-.1 + 2µ;x 48:13:2W,:,,(x) = x'"r'" expl2JUI2+ µ - v;l + 2µ;xThus, the Whittaker functions have parameters related to those of their equivalent confluent hypergeometric func-tions by v = (c/2) - a, µ = (c/2) - 1. This choice of parameter definitions makes the Whittaker functions moresymmetrical under transformations equivalent to 47:5:1 and 48:5:1:48:13:3M,,,(-x) _ (-l)"z*µ M_,,M(x) _ [-sin(µa) + i cos(µtr)J M_,.w(x)48:13:4 W,,,(x) =(x)Most of the formulas of this, and the preceding, chapter may be redrafted in terms of Whittaker functions;sometimes the resulting expressions are simpler. For example, when v = 0: 48:13:5 Mo..(x) = 4"r(1 + µ) \ I. 12 I\ //x(x 48:13:6 Wo:r(x)K. 2l/ CHAPTER49THE HYPERBOLIC BESSEL FUNCTIONS I0(x) AND II(x) Although they are simply the v = 0 and v = 1 cases of the general function addressed in Chapter 50, the :ero-order hyperbolic Bessel function la(x) and the first-order hyperbolic Bessel function l (x) are sufficiently importantin their own right to warrant separate consideration. The present chapter also contains some results that relate to["(x), the hyperbolic Bessel function of arbitrary integer order. 49:1 NOTATIONThe names modified Bessel function of the first kind of order zero and modified Bessel function of the first kind oforder one are also applied to la(x) and I1(x). respectively.The adjective "hyperbolic" indicates that 10(x) and 1,(x) are related to the Bessel functions J0(x) and J,(x) [seeChapter 521 in the same way that the functions of Chapters 28. 29 and 30 are related to those of Chapters 32, 33and 34. 49:2 BEHAVIORFigure 49-1 shows that both functions increase rapidly in magnitude as the argument x increases in magnitude.That the rate of increase is somewhat less than exponential is evident from Figure 49-2. The latter map also includes a graph of I/2TCr, the common asymptote to which the product of exp(-x) with any hyperbolic Bessel function(including those of noninteger order (see Chapter 501) tends as x -. Z. 49:3 DEFINITIONSHyperbolic Bessel functions of integer order are defined by the generating functionsf1 49:3:1Z t'1i(x) = lo(x) + ( t + ) 10) + r' +I>(t) + 49:3:2exp(x cos(:)) = 1a(r) + 2 cos(t) I,(x) + 2 cos(2t) 1.(x) + =la(x) + 2cos( jt)h (x)i- 379 49:3 THE HYPERBOLIC BESSEL FUNCTIONS lo(x) AND I,(x) 49:3:3exp(x sin(r)) = lo(x) + 2 sin(t) II(x) + 2 (- I)' {cos(2jt) I.,, (x) + sin(r + 2jr) I.,_,(x)} A number of indefinite integrals, such as 49:3:4 13(x)I(x - t) exp(x - t)dt_- I'nxJO21-1'and definite integrals, exemplified by 49:3:5 lo(x)Jxexp(±x cos(t))dt =1 Icosh(x cos(t))dt ,Ro ofZff= 2exp(tx sin(t)Wt =ITJcosh(x sin(t))dt049:3:6 lo(x) =2cosh(xt)dt- JIT'--roV 1 49:3:72x ('II(X)_ - J \cosh(xt)dtifo480 481THE HYPERBOLIC BESSEL FUNCTIONS 10(x) AND 1,(x) 49:3may be employed to define the hyperbolic Bessel functions of orders zero and unity. Some of these latter definitionsmay be generalized: 49:3:81.(x) =-Jf cos(nt) exp(x cos(t))dtn = 0. -t 1, ±2, .. .0 49:3:91.(x) =xJ1 sin'"(r) exp(±x cos(t))dtn = 0, 1, 2, ... (2n - 1)!!ar0to arbitrary integer order.The hyperbolic Bessel functions of orders zero and unity satisfy some simple second order differential equa-tions, as follows: 49:3:10 49:3:11 49:3:12&fdfx-+--xf=0f=c,10(x)+c_Ko(x)dx2dx d2fdfx-+--f=0f=c,Io(2\[x)+c2Ko(2>/)dx2dxd'2fdjx - - - - xf = 0f= CIA(x) + c2xK,(x)dx2dx 0. 4:.:....:....:....:....:....:. 0.5nxp(-z)Io(z) 0.20.3 49:4 THE HYPERBOLIC BESSEL FUNCTIONS lo(x) AND 1,(x) 49:3:131('-fx-f= 0f= c, Vx1,(2V)+c2\K,(2V)1(x2482 where c, and c, are arbitrary constants and Ke and K, denote the Basset functions (Chapter 521 of orders zero andunity.Hyperbolic Bessel functions of orders zero and unity may be generated by the operations of semiintegrationor semidifferentiation acting upon functions involving exponentials [see Chapter 261: 49:3:14 2d'"'d-'/2 coshO10(V x) _ - - sinh(V x) = -1/249:3:15 11(x) =2dxi12'n'3error functions [Chapter 40]: 49:3:16d-"' exp(2x)Irxxexp(x) d'/2 exp(-2x) exp(x) d"2lo(x) __dxrerf(V 2x) 'exp(x) 1(d='I%%{erf(\/i).-pl49:3:17I (x) = ex (-2x) Jxdx-nor hyperbolic functions [Chapter 28): 49:3:18 49:3:19 49:4 SPECIAL CASESV, 1(x1(xV Wxd'/' coshO -1/2x1,(V/ dx-u2sinh(1T)= 2Yr TM There are no special cases, but the important role played by l0(2Vx) as a basis hypergeometric function [see Section43:14] should be noted. 49:5 INTRARELATIONSHIPSThe reflection formula49:5:1I,(-x) =n =0.'1. ±2,...shows that 10(x) is an even function, whereas 11(x) is odd. Hyperbolic Besselare identical with their counterparts of positive order.49:5:2 1_,(x) = 1,(x)n= 1.2.3.... but this is not true for noninteger orders [see Chapter 50).The multiplication formulas49:5:3and 49:5:4IbIx-x10(bx) _ I (x)r0 j.2 1b2xx),11(bx) = b i1'.1(x)!-0l.2functions of negative integer order 483THE HYPERBOLIC BESSEL FUNCTIONS lo(x) AND 1,(x)generate series of hyperbolic Bessel functions of integer order. Other series of such functions have the sums 49:5:5 49:5:6 49:5:7 49:5:8I10(x) + I,(x) + l.(x) + I3(x) + ... = l 1j(x) =exx2 22 1 l2 210(x)-lA)+1:(x)-130x)+...=2,(x)=exp(-x) 2 10(x) + I,(x) + 13(x) +.. =co (x) sinh(x)10) + 1,(x) + 16(x) +=2 1 1 49:5:9 49:5:102f2+++_ - {[y +In(l10(x) +K0(x)}122)144)166)4\2)49:6 and yet other similar expressions are derivable via the generating functions 49:3:1-49:3:3. Here K. is the zero-order Basset function of Chapter 51.By sufficient applications of the recursion formula 49:5:11 2n1.-1(x) = I.,(x) - - 1(x)xany hyperbolic Bessel function of integer order may be expressed in terms of 10(x) and 1,(x). The first fewexamples are2 49:5:12 12(x) = l0(x) - -1,(x)/x 49:5:13 IS(x)-410(x) + II +S)1i(x)/z\111x-``49:5:14 14(x) = (1 +24 110(x)-/\g +4J((x) \\\\\\x'/\xX3 49:5:1515(x)(12+192110(x) +(I+72+384) _ - - - - 10r)xxx--x1441920;r 1876840149:5:16 16(x) = l I + - + y J10(x) - l + 3 +38sJ I,(x)x'xxxx 49:6 EXPANSIONSThe hyperbolic Bessel functions of orders zero and one may be expanded as46/!49:6:110(x)=1+ +++...=4-2304(x'4)=o(j!)2and 49:6:2_,y7 (x2/4)Jxli(x)-+s + x-+ "+...--216384184322 ,_0 j!(j + 1)! 49:7 THE HYPERBOLIC BESSEL FUNCTIONS 10(x) AND I,(x) each of which is a special case of 49:6:3 l ,(X)(x2/4)'\x2/)_a j!(j + n)!Asymptotic expansions of lo(x) and 1,(x), valid for large argument. are 49:6:4 10(x) _exp(x)I +1+9+225+ ... +[(2j).]'+ ...Jx 8x128x''3072x'(j!)'(32x)' and484 49:6:5I ,(X) -exp(x)1 -3-15315(2j - 3)!!(2j + 1)!! 11x-+x----- 1 tax8x128x23072x' j!(8x)'The asymptotic expansion of the general hyperbolic Bessel function l,(x) of integer order is given by equation50:6:4 on replacement of v by it. 49:7 PARTICULAR VALUES x- -xx=0x=x Idx)x 1s1.(x), 6(x). Idx)._. 0t¢.d. L(x). tAx).... =0 49:8 NUMERICAL VALUESIf. for a given x, R, denotes the ratio 1,.,(x)/1,1x) of two successive hyperbolic Bessel functions of integer order.then from equation 49:5:5exp(x) 49:8:1 lo(x) =I + 2R0(l + R,(1 + R,(1 + R,(I + ))))A truncated version of this relationship is employed by the following algorithm to calculate 10(x). The approximationis made that R, = 0 for j } J = 11 + which is sufficiently accurate to ensure 24-bit precision in thegenerated valucs of 10(x) and I,(x). The recurrence 49:8:2 R,_, _ input x >>1R, +2ixp»»Set s = 1Ifx-0goto(1) Sets= -IReplace x by -x(1)Setr=f=0Set j = 10 + lnt(x)(2) Replace f by I + frReplace r by x/(xr + 2,pj=J,J-1,...,3,2,1 Storage needed: x. s. r. f and j 485THE HYPERBOLIC BESSEL FUNC71ONS lo(x) AND 1,(x) 49:10 f-10(x)<f='I,(x)<Replace j by j - 1Ifj * 0 go to (2)Replace f by exp(x)/[ I + 2rf )Output fReplace f by frsOutput fTest values:10(6.9) = 153.6989961,(6.9) = 142.079028lo(-3) = 4.880792591,(-3) = -3.95337022 [which follows from 49:5:101 is used to calculate R, values for successfully smaller values of j.The algorithm is designed to produce values of 10(x) and 1,(x) with a relative error of less than 6 x 10-" forany nonncgative argument x. When those portions of the algorithm shown in green are included, negative argumentsare also admissible.There are algorithms in Sections 50:8 and 51:8 that may also be used to generate values of 10(x) and 1,(x). Aswell, the universal hypergeometric algorithm of Section 18:14 can perform these tasks. 49:9 APPROXIMATIONSFor small arguments, the algebraic approximations 49:9:1 10(x)I +.c18-bit precision-1.5 s x s 1.5 49:9:21,(x) - 21 l + 2)8-bit precision-2.3 s x s 2.3are valid. For sufficiently large positive argument.the approximation 49:9:3 1"(x) =cxp(x)x - -taxis valid irrespective of the order of the hyperbolic Bessel function. 49:10 OPERATIONS OF THE CALCULUSThe differentiation formulasd 49:10:1 dx10(x) = 1,(x) d 1 49:10:2 dx10) = 10(x) - x 10) are special cases of the general expressionsd1"_,(x) + I".,(x) n1 (nx) = 1"-A) +U-0) 49:10:31"(x) = z.r dx2The formulas49:10:4 dx{x'" I"(x)} = x`" 1":,xare attractively symmetrical. 49:11 THE HYPERBOLIC BESSEL FUNCTIONS lo(x) AND 1,(x)Formulas for indefinite integration include 49:10:5 49:10:6raxlo(r)d[ = 2 [lo(x) a-,(x) - I,(x) eo(x)J0 Jo'1,(t)dt = lo(x) 49:10:7Jt[lo(r)d[ = x1,(x)049:10:8 tl,(t)dt =n(x)eo(x) - lo(x) ((x)1fHere a-,, Co and e, denote hyperbolic Struve functions [Section 57:13]. The indefinite integrals 49:10:9Jj exp(-t) l0(t)dt = x exp(-x)[l0(x) + 1,(x)]0and 49:10:10r.exp(-t) 1,(t)dr = (x + 1) exp(-x) b(x) + x cxp(-x) II(x) -1486 of products of the exponential function and the hyperbolic Bessel functions lo(x) and 1,(x) generalize to49:10:11J,exp(-t)exp(-x) [(x + n) lo(x) + xl,(x) + 2 In - j) 1, (x)Jn0 J.1Formulas for definite integrals involving the 10 and 11 functions may be obtained by specialization of those ofSection 50:10.With lower limit zero, the following are included among differintegration formulas:d1/2cosh(\ x)49:10:12dsxlo(V) _'Vmx 49:10:13For others, see Oldham and Spanier. 49:11 COMPLEX ARGUMENTd-u210(v x)_sinh(V x) When their arguments are complex, the hyperbolic Bessel functions of orders zero and unity are expressible as theseries (x-y2ixv)I421,_x' - y-x - 6x'y2 + y' 49:11:1I0(x+iv)=I 21++i-o(j!)464(xvx3v_xv)) x:v2tixvll'a2149:11:211(x+iv)=x+iy=(X+ x3-3xy +xs-IOx3y2+5xy'+2-oj!U + 1)!216384 487THE HYPERBOLIC BESSEL FUNCTIONS lo(x) AND 11(x)+i(y+3x'y-y'+5xy- 1Qx'y'+y'+...\216384For purely imaginary arguments, these formulas reduce to49:11:3 1o(iy) =1o(Y)49:11:4 1,(iY) = iJ,(y)which generalize to49:11:5 "(iv) = i"J"(y)49:13 and show that the hyperbolic Bessel functions of imaginary argument and integer order n are real or imaginaryaccording as n is even or odd. The functions J. are those discussed in Chapter 52.Equations 49:11:1 and 49:11:2 simplify when the magnitudes of the real and imaginary components of thearguments are equal. Then one has 49:11:6 rttx' /2)V2) lo(x ±: ix) = 2 = bero(x y 1) ± i1.0(j!)'x t ix(iix'/2)'49:1 1:71,(x ± ix) _i= bei,(x V 2) + i ber,(xV2)2_o j!(j + 1)!where ber" and bei" denote Kelvin functions, as discussed in Chapter 57. 49:12 GENERALIZATIONSThe next chapter discusses hyperbolic Bessel functions of arbitrary order. 49:13 COGNATE FUNCTIONSThe functions to and 11 are closely related to the Bessel coefficients Jo and I. (Chapter 52). to the Basset functionsKo and K, [Chapter 51] and to the hyperbolic Struve functions e_,, to and e, (Section 57:131. CHAPTER50THE GENERAL HYPERBOLIC BESSEL FUNCTION 1,(x) The function I,(x) is defined for all values of its order v. Many physically important functions arise when v acquiresspecial values. 50:1 NOTATION The names modified Bessel function of the first kind and Bessel function of imaginary argument are commonlyapplied to l,(x), but these titles are less informative than the name hyperbolic Bessel function that we adopt. The adjective "hyperbolic" indicates that the relationship [see 50:11:21 between I,(x) and the Bessel function itself [J,(x), see Chapter 531 resembles that between hyperbolic functions [Chapters 28-30] and circular functions (Chapters32-34). See Section 50:4 for the i,(x) symbol.The parameters v and x are named the order and argument, respectively, of the function. 50:2 BEHAVIOR Unless v is an integer, the hyperbolic Bcsscl function I,(x) is complex when its argument is negative. Accordingly,this chapter concentrates on positive values of x. and this is the only range depicted in the contour map, Figure50-1.When the order v is an integer, the functions I,(x) and 1_,(x) are identical, but these two functions are alsoclose to each other in value whenever x exceeds jvl. Moreover, for large enough argument, the hyperbolic Besselfunction I,(x) becomes almost ind ndent, not only of the sign of its order, but also of the magnitude of v, beingwell approximated by exp(x)/V 2ax.Whereas 1,(x) is generally complex for negative x. the composite function x-"I,(x) remains real for all realarguments and is, in fact, even so that 50:2:1 (-x)-, l,(-x) = x""1,(x)The evenness is evident on writing 50:2:1 as 2"C,(-x2/4), in terms of Clifford's notation [see Section 53:2]. 489 59:2 THE GENERAL HYPERBOLIC BESSEL FUNCTION I,(x) 490 491THE GENERAL HYPERBOLIC BESSEL FUNCTION I,(x)50:3 DEFINITIONS50:4 The hyperbolic Bessel function is defined by a number of definite integrals. including 50:3:11,(x) _(x/2)"I(I - r')"-'t' exp(±xt)dtv > - Var(#+v) iSubstituting t = cos(h) in 50:3:1 gives a second definition and some others are listed by Gradshteyn and Ryzhik[Section 8.4311.A Kummer function [Chapter 471 whose denominatorial and numeratorial parameters are in a two-to-one ratiois related to a hyperbolic Bessel function by 50:3:2 1,(x) =(x/2)" exp(-x)M( + v;l + 2v;2x)r(l + v)The operations discussed in Section 43:14 can generate any hyperbolic Bessel function from any other hyper-bolic Bessel function and hence from any of the functions discussed in Chapter 49. For example, in the notationof Section 43:14 50:3:3 lox)-v r(+ v)0(x/2), Mx)x =21`A solution of the modified Bessel equationd'fdf 50:3:4.r-+x--(x'+ir)f=0dx'd.ris c,I,(x) + c2unless v is an integer [see Section 51:3 for that ease]. The terms c, and c2 arc arbitrary constants.Other differential equations that are satisfied by hyperbolic Bessel functions are 50:3:5xa+ (2v + 1)of- xf = 0f = c,x `I,(x) + and 50:3:6xdZf+ (v + 1)df-f = 0f = c,(2\)-" 1.(2V) + c.,(2V)'I_.(2V )dx-dxunless v is an integer. 50:4 SPECIAL CASESThe hyperbolic Bessel functions of integer order are the subject of Chapter 49.When the order of the hyperbolic Bessel function is an odd multiple of =;, reduction occurs to the simplerfunctions discussed in Section 28:13. The simplest cases are 2cosh(z)1/x - 1exp(x)(x + 1exp(-x)50:4:1 1_312(x) =axsinh(x) -x1 s - 12az`tax!2!Lexp(-x)50:4:2 1_112(x) _- cosh(x)+ax2x2azP(z) 50:4:3 11,(x) _?sinh(x) =cxp(x)axtax2axsinh(x)x - !exp(x)exp(-x) 50:4:4I2 J1(x) =axI cosh(x) x +(x+l)s [ tax 2ax 50:5 THE GENERAL HYPERBOLIC BESSEL FUNCTION 11(x) 492and others may be constructed via recursion formula 50:5:1. Some of these functions are mapped in Figure 28-3.The s} mbol i (x) and the name modified spherical Bessel function of the first kind are sometimes applied to theNi it/2x I., 112W function.Hyperbolic Bessel functions of order ± are closely related to Airy functions [Chapter 561. One has 50:4:5n4X [Bi(X) = V /Ai(X)IX=(2x Hyperbolic Bessel functions of order ±14 are expressible as parabolic cylinder functions [Chapter 46] of order-j. The relationships are13,14-2N6) + D_112(2\)50:4:6 I-114(x) _ x1/4Similarly. 1:314(x) may be expressed in terms of parabolic cylinder functions by-VxD_,,2(2Vx) - VxD_,R(-2Vx) D,,2(2vx) - D12(-2V)50:4:71,1/4(x) 2.x7/4and, hence, with the help of recursion formula 50:5:1, so may 45/4(x), etc. 50:5 INTRARELATIONSHIPSHyperbolic Bessel functions satisfy the recursion relationship 50:5:1 2vI,. (x) = I,_,(x) - -11(x)Xand the argument-multiplication formula 50:5:2 11.(bx) =b''r-- )11.;(x)2j!Setting b = I + (y/x) converts 50:5:2 into an argument-addition formula, while setting b = i = generatesthe summation formula _ =l50:5:3I,() - x1,.1(x) +211..2(x) - ... _ `(-x)I,.1(x) = i "1,(ix) = J,(x)j=o%The hyperbolic Bessel functions 1,(x) and I_,(x) are identical if v is an integer; otherwise: 50:5:42I_,(x) = 1,(x) + - sin(vir) K,(x)itwhere K, is the Basset function [Chapter 51]. This equation may be regarded as an order-reflection formula, in-terrelating I_,(x) and I,(x). There are similar order-reflection formulas for the product of two hyperbolic Besselfunctions whose orders sum to ± 1. namely250:5:5 and1_,(x) 1,_1(x) = I,(x) 1_,1,(x) + - sin(vtr) ax 2 50:5:6 1-,-112(x)1,-112(x) = 1112..(x) 112_,(x) + - cos(va) 'nx 493THE GENERAL HYPERBOLIC BESSEL FUNCTION I,(x)50:6 EXPANSIONS50:8 The hyperbolic Bessel functions may be expanded in the convergent series(x/2Y(x/2)4..50:6:1I,(x) =+++= i r(1 + v)l!r(2 + v)2!r(3 + v);_,j!r(j+v+1)That this expansion involves a hypergeometric series [Section 18:141 becomes more evident when it is rewrittenin the form4 650:6:2I,(x)+x2+x+x+ r(1 + v)4(1 + v)32(1 + v)(2 + v)384(l +v)(2 + v)(3 + v)(x/2)" .(x2/4)'r(1 + o, (l),11 + v)iwhere (I + v); denotes a Pochhammer polynomial [Chapter 181. The expansionxi 50:6:3I,(x) = 1,(x) + xi,.1(x) +2J,+2(x) +_Jj.,(x)j=0j, in terms of Bessel functions [Chapter 531 is also convergent.An asymptotic expansion is provided by the seriesexp(z) (- v=(2 - y2)(1 - v)(114_ v2)(/- v2)(i - v2)50:6:4I,(z) -271 +2x+8x2+48x3(;-v))(;+v);1+j!(2x)'which is valid if x, but not jvi, is large. This series terminates when v = ±; ±? ±i .. but the expansion isnot exact even under those circumstances. 50:7 PARTICULAR VALUES 44c)v > 0b(x)I,(x)-I <v<0.-3<v<-2, -5<v<-4, ...1,(x)v=-1,-2.-3,.Ux)-2<v<-1.-4<v<-3.-6<v<-5.... As is evident from the contour map, Figure 50-I, the hyperbolic Bessel function 1,(x) encounters a zero, for.r > 0. only if the order is negative and in one of the ranges -2 < v < - I, -4 < v < -3, -6 < v < -5, etc.For other negative values of v, I,(x) generally displays a minimum. 50:8 NUMERICAL VALUESx c The following algorithm for I,(x) utilizes expansion 50:6:2 in the truncated and concatenated versionX2 X2 .\4J(J+0)4(J-l)(J-1+v)/4(J-2XJ-2+v) 50:8 THE GENERAL HYPERBOLIC BESSEL FUNCTION I,(x) 494 X2z'(x/2)''+...+I/8(2+v)4(1+v)+1i(1+v)when x is less than 8 + (v2/12). The I /17(1 + v) multiplier is calculated by a routine that is essentially the oneused in Section 43:8, but special measures are taken if v = 0 or if v is a negative integer. The integer J in 50:8:1 is calculated by an empirical expression to ensure 24-bit precision (the fractional error in the hyperbolic Besselfunction does not exceed 6 x 10-"). If x exceeds 8 + (v2/12), the hyperbolic Bessel function is calculated via theformula2)' - v2l (J -)2 - V2(J - 1)2 - v250:8:2l,(z) = ll2A+ I /2(J - 1)x+ I /2(J - 2)x v221 (2)2 - v2exp(x)4x2x2arxwhich is a concatenated version of the asymptotic expansion 50:6:4, truncated by an empirical choice of J, againto ensure 24-bit precision.V >>>>>X »»>Setf= IIf x a 8 + (v2/12) go to (4)If v + Intflvj) * 0 go to (1)Replace v by -v(1) Set j = Int(x + 3)3x\If(v-5)v+10+/2 1>0goto(2) Replace j by j + lntl 4/- 3vl(2) Replace f by I +fx2`/I4j(j+/JIv)]Replace j by j -Ifj*0goto(2)Ifv=0goto(6)Replace f by f/v(3) Replace f by 2fv/xReplace v by v + IIfvs3goto(3)Setj=L1+2 1--1 /30v'12L7y'3v2Replace j byL:11121+ v[ I - ln(2v/x)]Replace f byfexp(j)v/2aGo to (6)(4) Set j =lnt(5+--+ ll IIf x < 100 go to (5)r////\1 Replace j by) - IntLl 2 -Vx 4 IvIJrLL\V1/(5) Replace f by I + f(i2v2J /2jxReplace j by j -Storage needed: Y. f, x and j Input restrictions: Any value of v maybe input, but x must exceed zero. Test values:Id(5) = 27.23987181112(n) = 5.19875924L,(l) = 0.02216842481 495THE GENERAL HYPERBOLIC BESSEL FUNCTION 1,(x)If j*0go to(5)(6) Replace f by f exp(x)/ 2srxOutput ff - 1,(x) <C<<<<1-r,2(2) = -0.62800904912(10) = 2281.518971,00000) = 4.64153494 x 10211_,(125) = 6.88584377 x I0s250:10 A simpler alternative is available in Section 49:8 for calculating hyperbolic Bessel functions of integer order.Several versions of the universal hypergeometric algorithm Isce Section 18:141 are also useful for determiningnumerical values of l,(x). 50:9 APPROXIMATIONSIf v is not a negative integer, the hyperbolic Bessel function of small positive argument is approximated crudelyby 50:9:1 1,(x) =r(1 + v)X-0v # -I. -2, -3, ... and more accurately (especially if v is positive) by 50:9:2l,(x) =(x/2)`1 +X2small xr(1 + v)[4(1 + v)(2 +v),For large argument, the approximationr1r21 50:9:3 1,(x) =exp(x)I1 - - Jx » µ =2--2ax\ 8is valid and shows that increasing its argument makes the hyperbolic Bessel function increasingly independent ofits order and increasingly closer to exp(x)/ti x.Making use of the reflection property [Section 43:5] of the gamma function, approximation 50:9:1 shows that 50:9:4I,(x) 1_,(x) -Isin(vrr)x-0r(1 + v) r(1 - v)vaa result that remains valid for all orders. It may be combined with equation 50:5:4 to produce the approximation 11 50:9:5----2vK,(x)x-.0v*0.±1, ±2, ... I,(x)I_,(x) 50:10 OPERATIONS OF THE CALCULUSDifferentiation of a hyperbolic Bessel function gives the alternative formulations 50:10:1d1,_1(x) + vvdxI,(x) =2= I.-i(x) - -1.(x) = 1..1(x) +xI,(x)but the indefinite integral of the general l,(x) function is expressible only as the summation 50:10:2 fL(z)ch= 2(-I,(x)i=oand not in closed form. On the other hand, the indefinite integrals of the following products:r( +,u50:10:3jt`,(t)dt = [I,(x) C,_1(x) - 1,_i(x)l.(x)] v > -0 2 50:11 THE GENERAL HYPERBOLIC BESSEL FUNCTION 1.(x) 50:10:4 Jx`l.(x)v > 00 50:10:5 r.t-'I,. 1(t)dt = x-`1.(x) -2-" r(1 + v)4% v *1 50:10:6C' exp(I 1) I.(t)dt =x,- "exp(+z)±I._I(x) - l.(x)+21-'J0 2v - I- I(2v - I) r(v) 2are straightforward. The f functions in 50:10:3 are instances of the hyperbolic Struve function discussed in Section57:13, while r denotes the gamma function [Chapter 43).Among formulas for definite integration isrl/ z\50:10:7Jexp(-B'tz) !"(bt)dt =n2B exP\8B2J!"/zIii2Iv > -1oReplacing the argument x by 2N/-x in the first of the two Rayleigh formulas1 d 50:10:8 {x dx} [x"I.(x)) = x"-"I.-.(x) 50:10:9fxdxf [x_"I"(x)J = x-"-"1...(x)for multiple differentiation leads to a relationship that generalizes to the very simple expressiond" 50:10:10 dx"[xvn-I.(2f)J = xr"-" r_4 Here the operator d"/dx" signifies differintegration with lower limit zero to an arbitrary (positive or negative, integeror noninteger) order [see Section 0:10). 50:11 COMPLEX ARGUMENT Replacing x in expansion 50:6:1 by x + iy leads to.cos((b) + isin(do) (z' + yzl'""t2) 50:11:11.(x + iy) _ Ztj,=aj!r(j + I + v) 4where 4 = (2j + v) (8 + 2ka), 0 and k having retained their significances from Section 13:11.When the argument is purely imaginary, the hyperbolic Bessel function becomes a Bessel function [Chapter531 of real argument:r/\/l50:11:2 I.(iy) - i"J"(y) = I cosy 2I + i( 2)]2 )j My)Unless its order is an integer, the hyperbolic Bessel function is complex when its argument is negative. Wehave50:11:3 I.(-x) = (- 1)" I"(x) _ [cos(vw) + i sin(var)J 1.(x) 50:12 GENERALIZATIONSThe function 50:12:1(x "," . (x2/4)' \2)0 r(1 + µ + j)r(1 + v + j) 497THE GENERAL HYPERBOLIC BESSEL FUNCTION l,tr) 50:13closely related to the Kummer function [Chapter 47], is a generalization of the hyperbolic Bessel function, whichis the µ = 0 instance of 50:12:1. The hyperbolic Struve C function [Section 57:131 is also a particular instance ofthe general function 50:12:1. 50:13 COGNATE FUNCTIONSIn the next chapter the cognate Basset function is discussed. Also closely related to hyperbolic Bessel functionsarc the Kelvin functions (Chapter 55] and, of course, the Bessel functions [Chapter 531 themselves. CHAPTER51THE BASSET FUNCTION Because the Basset function of noninteger order is related so simply [see 51:3:51 to the hyperbolic Bessel functionof the previous chapter, this chapter concentrates on the Basset functions Ko(x), K,(x). K2(x), ... of integer order. 51:1 NOTATIONAlternative names for the Basset function are the modified Bessel function of the third kind, Bessel's function ofthe second kind of imaginary argument, Macdonald's function and the modified Hankel function.We use v generally to represent the order of a Basset function but replace this symbol by n to specify integerorder. 51:2 BEHAVIORThe Basset function K,(x) is infinite for x = 0 and complex for x < 0 [see equation 51:11:31. Accordingly, werestrict attention to x > 0 here and generally throughout this chapter.For all v, K,(x) is a positive and monotonically decreasing function of its argument x, approaching zero as x- %, in accordance with expression 51:9:6, in a manner increasingly independent of the order v. However, theapproach to infinity as x -. 0 is a strong function of v as detailed in Section 51:9.For a constant positive argument, K,(x) is an even function of its order v, as evidenced by equation 51:5:1.Moreover, for constant positive argument, K,(x) invariably increases as wj increases.Figure 51-1 shows maps of K,(x) for n = 0. 1, 2, 3. 4. 5 and 6. The curves for noninteger order Bassetfunctions smoothly interpolate between these mapped curves: for example, Ka12(x) lies intermediate between K4(x)and K5(x). 51:3 DEFINITIONSBasset functions may be defined via those Tricomi functions [Chapter 481 in which the a parameter is a moiety ofthe c parameter.51:3:1K,(.,) _ \(2x)" exp(-x) U(12 + v; I + 2v; 2x) v >- 0 499 51:3 THE BASSET FUNCTION K.(x) e1,UO0O$Ut0Otibr000'L ,y040Rr40 y s 4'*'.N fV,Vy'1.L4.,y.rtj.ififiifI.:...............:.........:.................... :.... ................. .2.42.22.0 0.2 or approaches infinity: 51:3:2K,.(x) = Z(21,1L4a{r(a- v) Ul a; 1 + v;/Iv0 Equivalent to 51:3:1, but somewhat simpler, is the definitionof a Basset function as a special case of Whittaker'sW function [see 48:13:61.Among definite integrals that define the Basset function are 51:3:3K,(x) _(2x)' fexp(-x/)dtv) -1F( , + v)2(r' - 1)2andr(3 + v)cos(t)dt I 51:3:4 K,(x) _c()" f.& +v2and a large number of alternatives were assembled in the Bateman manuscript (see Magnus, Oberhettingerand Tricomi, Higher Transcendental Functions, Volume 2. pages 82 and 831. Because of relationship 51:5:1. thesign of v may be changed in definitions 51:3:3 and 51:3:4.Basset functions of noninteger order may be defined in terms of hyperbolic Bessel functions by 501 THE BASSET FUNCTION K,(x) 51:5 51:3:5 Tr[l_,(x) - l,(x))2 sin(vrt)but this expression must be written as a limit:v*0,±1,-'2,... IT1.(x)1 51:3:6K(x)2limSlsin(m)Jrwhen the order is an integer.With c, and c, representing arbitrary constants, the differential equation 51:3:7x. d/+xdf-(.Y+n')f=0n=0,±1,±2....dx'dxis solved by f = c,l (x) + c;K (.r). Other differential equations whose solutions involve the Ko or K, functions arelisted as 49:3:10-49:3:13.Semiintegration with lower limit zero is a powerful method of generating Basset functions. Examples include 51:3:8uzI _, (1d-1 K.2xv'rrexp\2x d.r-"2 xeap r(lxvrt'1(l 51:3:9 K,2exp7vx_ I' /- 1d-I 51:3:10Kv< 2x> = x V to expx=)d.r ''=eXp .r 51:4 SPECIAL CASESWhen its order is an odd multiple of , the Basset function reduces to a simple function involving an exponential.The simplest cases arerr 51:4:1 K,,_(.r) = K-, _(s) _exp(-x) 51:4:2 K3 .(x) = K_};.(x) _x Ll +1xJ exp(-.r)[see Section 26:13 and Figure 26-2] and others may be constructed via recurrence formula 51:5:2. The symbolk,(.r) and the name modified spherical Bessel function of the third kind are sometimes applied to the V a//1"function.A Basset function of order 3 is related to an Airy function [Chapter 561:51:4:3 K1/3(x) = K-113(x) _'rX Ai(X)X =(32while that of order ; is related to a parabolic cylinder function (Chapter 461: 51:5 INTRARELATIONSHIPSWith respect to its order, the Basset function is even51:5:1 K_,(x) = K,.(x) 51:6 THE BASSET FUNCTION K,(x)so that v may be replaced by jv= in most of the formulas of this chapter.The Basset function obeys the recurrence relations2v51:5:2 andK,. ,(x) = K,_ ,(x) + - K,(x)x 1 51:5:3 K,.,r(x)1,(x) _ - - K,(x) l,.r(x)XThe zero-order Basset function satisfies the argument-addition and -subtraction formulas502 51:5:4Ka(x±y)=lo(y)K0(y)+2(rIYI,(y)K,(x) x > y > 0By sufficient applications of formula 51:5:2, any Basset function K"(x) of integer order may be expressed interms ofand K,(x). The first two examples are2 51:5:5 K,(x) = K((x) + - K,(x)x 51:5:6 /\K3(x) =4Kd(x) + 1I +8, IK,(x)x\x'/and expressions for K,(x). K5(x) and Ke(x) are analogous to 49:5:13-49:5:15. but with uniformly positive signs. 51:6 EXPANSIONSThe Basset function of noninteger order is expansible as the sum of two convergent series 51:6:1K,(x) =I(v) (xr(x'/4)1+r(-v, Ix\,(x=/4)'--I Iv*0,'_1,±2....22_O j!(1 - v),22=p j!(l + v),that coalesce only if v is an odd multiple of For integer order the series arer/x\x\\ 11 (x- 4)[-,yIx11 (x' 4)-' 51:6:2K0(x)y - In{ 2 I++ l -ln(21J(1 , ++ I + 2In(2)J(2 ). \ ///+-y + 1 +l+I-ln(x(x /4)+ ...23\2/(3!)2y1x(x-'14)312 1-y 51:6:3K1(x)+I-ln(x(x2/4)r12-+ I +In=----l- ---x22\2/0!I!242l!2! HII\ 1(x'/4)3''+I+2+6-1n\2.,J2!3!and generally 1(n -j- 11.xVb(l+j)dr(I+n+j)x(x/2)''" 51:6:4K"(x)=-(l r+I+-ln(Jrr 2 \x/ ;=oj!\ 4/L222)j!(n + j)!In these formulas r denotes the gamma function [Chapter 43], s the digamma function [Chapter 441, (1 - v), isa Pochhammer polynomial [Chapter 181 and y is Euler's constant (Section 1:71. Alternatively. Basset functions ofinteger order may be expanded as Neumann series in terms of hyperbolic Bessel functions: 51:6:5K0(x) - {--InJ10(x) + 2 iL"_(L)y = 0.5772156649;-rJ 503 THE BASSET FUNCTION K,(x) 51:6:6and generallyIK,(x) _ - -y + I -10) + - lo(x) -' (1 + 2j)It+2,(x)2lxJ=i J(1 +J)51:8 rr(x)] 1 (n + 2j) 51:6:7 K(x)I+(1+n)-In-I(x)+-j xI(.r)+(-I)LLLLLL22 \xk!(n,k)2J(n+J)The seriesfv'(1-y')(1v')(11 -v')(i-y')(1-v') 51:6:8K,(x) -- exp(-x)I - 42x+8xz-48x'0-v),(1+v)ilj!(-2x)' 1is generally asymptotic but it does terminate when of is an odd multiple of 1, and under these circumstances theexpansion is exact. If vj is not an odd multiple of 1 and lies between the numbers n - ; and n + 1, where n isan integer, then the terms in series 51:6:8 corresponding to j = O. j = 1, j = 2..... j = n are uniformly positive(we are treating only positive x), whereas the j = (n + I)m term is negative; thereafter, the terms alternate in sign. Accordingly, it follows from the properties of alternating series [see Section 0:5] that if the series 51:6:8 is truncated, terms after j = J being ignored, then the partial sum is related to the true value of K,(x) by one of the inequalities51:6:9(series of J terms) 5: K,(x) 5 (series of J + 1 terms) J ? n = lntl+2for large enough J. 51:7 PARTICULAR VALUES K, x) 51:8 NUMERICAL VALUESx - 0x * x 0 Two algorithms are presented in this section. The first generates values of K0(x) and K1(x) for sufficiently smallarguments, say x 5 3. From these, recurrence 51:5:2 shows how it is possible to compute the K,(x) function for any positive integer n. The second algorithm produces values of the Basset function for arguments exceeding 2.5 and for orders (integer or noninteger) less than about (24x/ln(x)J'". Neither algorithm is suitable for evaluatingK,(x) for small x and noninteger v, but definition 51:3:5 provides easy access to these values via the algorithm ofSection 50:8.With R, having the significance accorded to it in Section 49:8. the Neumann series 51:6:5 may be rewrittenas the concatenation 51:8:1K0(x) = lo(x)4RoRi(2+ RR, 4 + R4R56+)1 1- y - In (2)JThis is the formula used by the first algorithm to calculateK0(x).. the required value of 15(x) being determined viathe similar concatenation51:8:2cosh(.r) = 10(x)[1 + 2RoR,(l + R:R}(l + R4R5(l + )))1which follows from series 49:5:7. In practice, both concatenations are terminated by setting R, = 0 for j greater 51:8 THE BASSET FUNCTION K,(x) 504than an empirically chosen integer J. Recurrence 49:8:2 then serves to calculate all other R, values. The hyperbolicBessel function 1,(x) is calculated by the algorithm as R010(x) and thence the Basset function K,(x) as R0{[I/xl,(x)]- Ko(x)}, an identity that follows from 51:5:3. Many problems in applied mathematics require values of the func-tions 10(x) and 1,(x) as well as those of K0(x) and K,(x); the algorithm below delivers all four functions if theportions shown in green are included. Input x >> >>>>Set j = 82 lnt(x)SetI=K=r=R=0(1) ReplaceI by I + IrRReplace K by (I 1j) + KrR/Setr= 1R+2j)IAXReplace j by j -2jSetR= 1Ir+Replace j by j -Ifj*0goto(I) Replace I by [exp(x) + exp(-x)1/12 + 4IrR]Output I10(x)< ««Replace K by I14KrR - In(.890536209x)]Output KK = K0(x) ««Replace I by IROutput I<<<<<Replace K by R[(1/xl) - K]Output KK - K,(x) ««Storage needed: x, j, 1, K, r and R Input restrictions: x must exceed zero.If x exceeds about 3 (the precise limitdepends on the computing device),accuracy may be impaired. Test values:10(1) = 1.26606588K0(l) = 0.4210244381,(1) = 0.565 1 5 9 /04K1(1) = 0.601907230K0(3) = 0.0347395044 K,(3) = 0.0401564311 K((5) = 0.00369109833 K1(5) = 0.00404461344 Although the accuracy of this algorithm is inherently high, it may not deliver precise values of K0(x) and K,(x)unless x is small. The difficulty arises because equation 51:8:1 calls for the differencing of two terms whose valuesbecome increasingly close as x increases; therefore, significance is lost in their subtraction. The seriousness of thiseffect depends on the number of significant bits utilized by the computing device. Changes to the algorithm cannot ameliorate the problem.The second algorithm utilizes expansion 51:6:8 and overcomes its asymptoticity by invoking the Pad6 techniqueas described in Section 17:13. An empirical formula is used to determine the order J, up to a maximum of 30, of the diagonal Padd approximant Rjj that is generated. Within the permitted ranges of x and v, K,.(x) is generallyreturned with a precision better than 24 bits. .r >>>>>Sctj=t =h0=Set J = 2 lnt(6 + ]Ivj"ln(x)/x]}//11 (I) Replace t by rv2I- I j- 2 1]/(2Jx)Set h, = r + h,_,\//////Setq=0Set k=jStorage needed: v. j. r. x. J. k, p. q. ho, h,.h,..... h,,,. ha, SOS THE BASSET FUNCTION K.(x) 51:10 p - K,(x)(2) Set p = h4_1If p # hk go to (3)Set hk_, = l099If p * 1099 go to (4)Set hk_, = qGo to (4)(3) Set he-i = q + [1/(h5 - p))(4) Set q = pReplace k by k - 1Ifk*0goto(2)Replace j by j + IIf jsJgo to (1)Set p = h0V a/2x exp(-x)Output PInput restrictions:x z 2.5 !vi < [24x/ln(x)J213 Test values:K,(5) = 0.0053089437/K,,,2(a) = 0.0305568546K_,(10) = 1.86487735 x 10"5K,0 (100) = 9.27452265 X 10-iOK16(5) = 186233.583 51:9 APPROXIMATIONSFrom expansions 51:6:1-51:6:4, the following two-term approximations, valid as x -' 0, may be derived: 51:9:1K,(x) -I - y = 0.11593 - ln(x)v = 0small xC/ 51:9:2 r(v)x-"r(-v)x'K,(x) =+0 < v < Ismall x 2'2" 51:9:3 K,(x) =I+x InrV= Ismall .rX2(x2)/ 2l 51:9:4K,(x) = r(wl)lx/[2-81vl-8,w, > IsmallxMore accurate than 51:9:4, for ivl > 2, is the approximation 51:9:5K,(.t) = r(2 I)\x/[1401 - 1)(ivj -2)Jlvi > 2small x For large x, the approximation 51:9:6 K,.( x)exp(-.r)(1+~x >> µ =2-8J follows from the asymptotic expansionYY51:6:8. It is exact only for v = ± i51:10 OPERATIONS OF THE CALCULUSThe general differentiation formulasd I 11v 51:10:1dxK,(x) = 2[K,-I(x)+ K,_,(x)J = - K,.(x) - K". ,(.t) _ -K,._,() W -LvK,(.t)have the special casesd 51:10:2 dxKo(x) _K. (.0 and 51:11 51:10:3 and lead to the derivatives 51:10:4 The formulas 51.10:5 and 51:10:6 J=K,(t)dr = Ko(x)506 for the indefinite integrals of the zero-order and unity-order Basset functions may be obtained as special cases ofthe general expressions 51:10:7 and 51:10:9 below. In equations 51:10:5 and 51:10:7 f is the hyperbolic Struvefunction discussed in Section 57:13. Closed-form expressions are available for the following indefinite integrals ofthe Basset function multiplied by a power. 51:10:7it'K,(t)dt = 2'-' \ r (v + )xIK.(x) K,-,(x) f,.(x)]1v>-2 51:10:8 Jr"K"-,(t)dt = 2"-'T(v) - x"K,(x) v > 0051:10:9Jr 'K,,, (t)dt = z K,(x) as well as for the following expressions involving exponential functions:),(x)21'(v + 1)v > -1Jrt` exp(ct) K,U)dt =x'_' exp(±xIK± K,.,(x)]02v + I 2v + 12 x` exp(xJt-` cxp(t) K,(t)dt =IK,(x) * K,_,(x)]v > -2v- 1 2The importantKing's integralis the v = 0 instance of 51:10:10 with upper signs selected.For vi > Ithe definite integral fK,(t)dt between limits of r = 0 and t = x diverges. The convergent casesare the µ = 0 instances of the general formula 51:10:12 namelyf",t" K jolt = 2"-' [(µ+y+l\r(- v+II\/11\2 51:10:13JiK,(t)dt =ITsec(\-fIvI < Io/ 51:11 COMPLEX ARGUMENTTHE BASSET FUNCTION K,(x)dK,(x)K,(x) = -K0(x) -dxx ddxx_'K"(x) = -x" K,,,(x) rxKo(t)dr =2[Ko(x) f-,(x) + K,(x) fo(x)]0 Definition 51:3:5 may be combined with equation 50:11:1 to produce an expression for the Basset function ofcomplex argument when v is noninteger. 507 THE BASSET FUNCTION K,(x) 51:13For purely imaginary argument, the Basset function is related to the functions of Chapters 53 and 54 by 51:11:1K,(iy) =2IcosILT2Y,(Y)+ sin (7-)1.(Y)1 +2 Lsin(-2)Y,(Y) -cos( 2)J (Y)When the real and imaginary components of the argument' of a Basset function are equal in magnitude, we have51:11:2 K,(x ± ix) = i'` [ker,(x y 2) t i kei,(x V G)]where ker and kei are Kelvin functions [Chapter 55] and i'" is to be interpreted as cos(vw/2) ± i sin(vir/2).The Basset function is complex for negative real argument. The real and imaginary parts are given byis 51:11:3K,(-x) = cos(vir)K,(x) - Z [I,(x) + 1 _,(x)] v * 0, ± 1, ± 2, .. . 51:12 GENERALIZATIONSThe Basset function may be generalized to Whittaker's function [Section 48:131 51:12:1 K. \2/ =x Wa,(x)and thence to the Tricomi function [see Chapter 48]. 51:13 COGNATE FUNCTIONSThe Basset function is related to all the functions discussed in Chapters 49-57. CHAPTER52THE BESSEL COEFFICIENTS Jo(x) AND JI(x) Those Bessel functions [Chapter 53) in which the order is a nonncgative integer are known as "Bessel coefficients"and are the subject of this chapter. Emphasis is placed on the most important of the functions: those with n = 0 and 1. 52:1 NOTATIONBessel coefficients are also known as Bessel functions of the first kind of nonnegative integer order. 52:2 BEHAVIORAll Bessel coefficients are oscillatory functions whose oscillations become increasingly damped as their argumentsapproach large values of either sign. As Figure 52-1 shows, all Bessel coefficients except J0(x) are zero at x = 0.Moreover, as n increases, J,(x) retains near-zero values over an increasing range in the vicinity of x = 0; forexample, J9(x) does not exceed 0.01 in magnitude in the range -5.4 5 x 5 5.4.Some regularities are apparent in Figure 52-1. Notice that each local maximum or minimum of J0 correspondsto a zero of J, but this rule does not extend to larger orders. On the other hand, for n z 1, each local maximumor minimum of J. corresponds to a point of intersection of the and J,,,, curves. Moreover, at each zero of J.1,_, and 1,., have equal magnitudes but opposite signs.For x s 0, each Bessel coefficient encounters its first (and largest) maximum at an argument that, for smalln, is close to nir/2. This rule fails for larger n, however, and for very large order, the Bessel coefficient attains its first maximum close to x = n. Subsequent, and ever-smaller, maxima are encountered with a spacingof approximately 2a. Local minima occur approximately midway between consecutive maxima. For x > 0 the firstminimum is the largest and subsequent minima decrease progressively in size. A zero is encountered between eachminimum and its adjacent maxima so that the spacing of the zeros is approximately IT. 52:3 DEFINITIONS Bessel coefficients are defined by a number of generating functions. Those of even order arise from 52:3:1cos(x cos(t)) = Jo(x) - 2J,(x) cos(2t) + 2J,(x) cos(4t) - = Jo(x) + 2(- 1), J,t(x) cos(2jt) so 52:3 THE BESSEL COEFFICIENTS Jo(x) AND 11(x) 510 . . . .............1.0. ............. :Jo(x ----- ........... .......................................................................... . . ... ............ . . . . .............. .. ..........FIG 52-1 : .......... Jt(x}:;J ....:....:....:....:....:....:... ..a B :J.(x):J (): :....:.. /.....:. \..:. !.:.. :... ....Jz (x): - .Tw (z>' 0.4 and if the signs in this series are made uniformly positive, the terms are generated by cos(x sin(s)). Bessel coef-ficients of odd order arise from52:3:2sin[x cos(t)] = 2J,(x) cos(t) - 213(x) cos(3t) + 215(x) cos(5t) - -= 2 E (-I)'J,,,,(x) cos(t+2jt)i-oor from the corresponding series for sin[x sin(r)], which differs from 52:3:2 in that sines replace cosines, and allterms are positive. The\ generating /function 52:3:3explxt=2r xI = 10(x) + I r -J1(x) +r=+r.lz(x) +t'-Ily33(x) +NJ; (z) gives rise to Bcsscl coefficients of both parities.The Bessel coefficient Jo may be represented as a definite integral in a number of ways, including 52:3:4Jo(x) = -I*cos[x cos(t))dt = 21sin[x cosh(t)]dt0 o S11 THE BESSEL COEFFICIENTS Jo(x) AND J,(x) 52:5The following are among the integral definitions of the general Bessel coefficient 52:3:5 J"(x)Jcos[x sin(t) - nt]dtn = 0, 1. 2, .. .n o2n=0,2,4,...f=cos 52:3:6J"(x) = Jx lin(t)] f(nr)dt j'Rn1,3,5,...f=sinand others may be obtained by specializing the formulas of Section 53:3.The Bessel coefficients JO and 1, arise in a large number of physical contexts because they satisfy certain verysimple second-order differential equations, as follows: 52:3:7 d2fdfx- + - + xf = 0 f = c,J0(x) + c2Yo(x) dx'dx 52:3:8xdzf+df+f=0f=c,Jo(2V)+c2Yo(2Vx)dxdx 52:3:9 xdzf-df+ xf = 0f= c,4J,(x) + c2xY,(x)dx2dx 52:3:10x-dz-+f=0dX2''f=c, VxJo(2Vx)+c2VxYo(2Vx)Here c, and c2 are arbitrary constants and Yo and Y, are Neumann functions [Chapter 541 of zero and unity orders.The Bessel coefficients JO and J, are generated by applying the operations of semiintegration or semidiffer-entiation, with lower limit zero, upon certain functions involving sinusoids. Examples include sec(x)d-cos(2x)csc(x) d-'/'sin(2r) 52:3:11Jo(x) =dx'ir( )=dx-"2( ) 2d"1'52:3:12 Jo(V x)V. dx'2(sin(V x)) 1d-"'rrX 52:3:13 1,(1f) _dx uz(sin( 52:3:14 52:4 SPECIAL CASESThere are none. 52:5 INTRARELATIONSHIPSJo\l1-d12 (sin(l/\)/VrxJxdxI1\x Bessel coefficients are even or odd52:5:1 J"(-x)(-1)' J"(x)n = 0, 1, 2, .. .according to the parity of n. A similar reflection formula52:5:2 J -.(x) _ (-1)" J"(x)n = 0, 1 , 2, .. .exists for the order of Bessel functions of integer order. 52:5 THE BESSEL COEFFICIENTS Jo(x) AND J,(x)Many formulas exist for infinite sums of Bessel coefficients. Some of these, such as 52:5:31J,(x) - J3(x) + J5(x) - Jr(x) + = 2 sin(s) 1 1 52:5:4 2 J0(x) - J,(x) + J,(x) - J6(x) + = 2 cos(x) and 1 1-1 52:5:52J0(x) + J2(x) + l1(x) + J6(.r) + =2J,(x) =2may be derived as special cases of formulas 52:3:1-52:3:3. Others, for example: 52:5:6 52:5:7 52:5:8and, for n = 1, 2, 3. ...: 52:5:9) = 2'+1) J2,- A)x/=ox21,(x) 4J2(x) + 161,(x) + 36J6(x) + = 4j2J.,(x) = 2 (n- I) !nJ (x) + On + 2) J,,.,(x) + (n + 1)!(n + 4)2!x1,.2,(x)_-i-oj!2follow from the properties of Neumann series (see Section 53:14]. Yet others, such as 52:5:10 2 AX) + J-1(X) + 121(x) += 2 Jo(x) + i J,(x) = 2 and 52:5:11J0(x) J,(x) - Jdx) J2(x) + J2(x) J3(x) - ... _ (- I )` J,{x) J;.i(x)J,(2x);=omay be derived from Neumann's addition formula 52:5:12 n=0,±1,±2,...Included among still other types of expansions are 52:5:13 and 52:5:141,(x) + J2(2x) + J3(3x) + _J(ix) _x ,_,'2(1 - x) 121x)+1;(2x)+1;(3x)+...=iJ;(jx)=2[ 1- 1J512 that are examples of Kapteyn series [see Erd6lyi, Magnus, Oberhcttinger and Tricomi, Higher TranscendentalFunctions, Volume 2, pages 66-68). 513 THE BESSEL COEFFICIENTS J6(x) AND J,(x)By applying the recursion formula52:5:15 2n- J,(x) - J,-,(X)x52:6 for a sufficient number of times, any Bessel coefficient may be expressed in terms of J0 and J1. Examples are2 52:5:16 J2(x)- l,(x) - J0(x)x/8-14 52:5:17 JAx)xz1 J Ji(x) - - Jdx)x 52:5:18 Mx)488_-XIxX-),J,(x) -24_(xIJO() I394-721( x192-12152:5:19 JS(x) = Zx3+I)J,(x) - l-J Jo(x)xx 52:5:20 16(x)3840-768+181J,(x)!1920-144+11Jo(x)= s i -- `2 xxxxxThe ratio R. of two consecutive Bessel coefficients of common argument obeys the simple recursion formulaJ,+,(x)2n1 52:5:21J"(x)X -R--n= 1,2,3....which makes these ratios computationally useful [see Section 52:8].Of course, all formulas in Section 53:5 can be applied to Bessel coefficients by setting v = n, a nonnegativeinteger. For example, the argument-multiplication formula yieldsI /x - bzxlV 52:5:22 b" F _ (!(x)'-o J.252:6 EXPANSIONSAny Bessel coefficient may be expanded as the power series 52:6:1J(xllIxz/4(xz/4)z(x'/4)'(x^(-xz/4)' (.r) _2J[n!(n + 1)!+ 2!(n+ 2)!3!(n + 3)!+]- \2/-o j '(n + J)!of which the first two instances areX2 B(-xz/4); 52:6:2lo(x)=I-+.r- x6+x-...=4642304147456andxxX5xtx(-x/4); 52:6:3JI(x)=-+-+...=216384184322j!(j + 1)!Expansions as infinite products take the form 52:6:4 JO()I -/\1z\Izxz) ... _1 _ _LJo,JojR=,Jos/zzz52:6:5J,(x)=zll-fIl-Z=J'. :zJ,ar-,JI.*where j,x denotes the k'° zero of the Bessel coefficient J,(x) [see Section 52:7]. 52:7 THE BESSEL COEFFICIENTS Jo(x) AND 11(x) 514By making use of formula 52:5:22, it is possible to expand a Bessel coefficient J,(x) in terms of the Besselcoefficients J J,,,,, J,-,, ... at any fixed nonzero argument. For example, choosing 2 for this fixed argumentgives(z\`1 / 52:6:6J,(x) I1-x4 J,",(2)n = 0, 1, 2, 2J ...-oldSuch an expansion converges extremely rapidly.The Bessel coefficient ratio defined in equation 52:5:21 may be expanded in partial fractions2x2x2x 1 52:6:7R, = ,,+-,+l+.1-x-j:.2x-ja-x.-Ija-x' or as the infinite continued fraction 52:6:8 _R.x/2x'/4x2/4x2/4J,(x)l+n- 2+n- 3+n- 4+n-The effect of curtailing this continued fraction at any point may be represented as another Bessel coefficient ratio.For example: 52:6:9 R.x12x'/4xz/4xR,.3, _I +n- 2+n- 3+n- 2For large arguments, the asymptotic expansionscos[192251si11922552:6:10J((x) -I - - - - + ++I +--nx8x128x"3072x3 arx8x128x'3072x3 sin(x)315315cos(x)315315 52:6:11J()-I +-+--1 -+-+ 1 8x128x=3072x3 8x128x3072x3 hold. See equations 53:6:6-53:6:8 for the general formulation. 52:7 PARTICULAR VALUES 1,(x)...1i(x), J,(x), J ,(X),x= -xx= 0x= s0I0000X -' x Those positive argument values that cause 1,(x) to acquire the value zero are denoted j ,,is called the km zero of the na Bessel coefficient. The value of the derivative d1,(x)/dx at the zero is a quantitywhose value is needed in certain practical problems [see Section 52:141. Such a derivative is known as the associatedvalue of the zero, and the usual notation is 1' .0.,.). although two alternative notations follow from the identity52:7:1 J,(J,:A) = J,-1(jn:k) _ - 4*1(1.:A) 515 THE BESSEL COEFFICIENTS J0(x) AND 1,(x) 52:7Table 52.7.1i.Jdjo,)illVi,.) JiLJdja)jikJi(ju) 2.4048-0.5191 3.8317-0.4028k-I0.0000+1.0000 1.8412+0.58195.5201+0.3403 7.0156+0.3001k=23.8317-0.4028 5.3314-0.34618.6537-0.271510.1735-0.2497k37.0156+0.3001 8.5363+0.273311.7915+0.232513.3237+0.2184k=410.1735-0.249711.7060-0.233314.9309-0.206516.4706-0.1965k513.3237+0.218414.8636+0.207018.0711+0.187719.6159+0.1801k616.4706-0.196518.0155-0.188021.2116-0.173322.7601-0.1672k=719.6159+0.180121.1644+0.1735 which is a consequence of 52:10:1. Approximate values of the first seven zeros, and their associated values, forn = 0 and 1. constitute the left-hand side of Table 52.7.1. Other approximate values of may be read from themap in Section 53:7 (Figure 53-2). More precise values of j,.k and J,( j,:k) are available from the algorithm thatfollows.The algorithm operates in two modes according to the relative values of n and k. If k >: n - 3, the algorithmadopts x, = a(2k + n - 1)/2 as an approximate value of and then uses Newton's method [Section 17:7] togenerate the improved approximation 52:7:2J,(x1) x,J,(x,) x,_JR(xi) -x,nJ,(xi) - x,J,,.1(x,)r'n - x,R.and this procedure is repeated until two successive approximants differ by less than l0-'. The method used tocompute R,,, the Bessel coefficient ratio, is described in Section 52:8. When k < n - 3, the algorithm first calculates n + 2n'"' [see 53:7] as a crude approximation to j,,, and refines this approximation by a single application of 52:7:2.A crude approximation to j.4, is then calculated by adding 4 to the j,,., approximant and this, in turn, is refined bya single application of Newton's method. The procedure of adding 4 and refining is repeated until a crude valueof j,;k is reached. This is then improved by repeated applications of formula 52:7:2. The second half of the algorithm,which calculates -J,.,(j,;k), is essentially the same as the algorithm in Section 52:8.Input n »> »»>Set J = 0Input k »> »»> (I) Replace x by x + 4Set I = Intn2 + 2(x + 6)2Replace J by J + 1Go to (6)121 If J * I go to (3)Replace.r by.r - (n114)(3) IfJ<kgoto(1)Set J = 0(4) Set L = 2+ 2 Int[n'/4] + [(.r + 7)2/2](5) Set ! = L(6) Set R = 0 (7) If2121 - R go to (8)X/nSetx=a k+n-1Z `[or =alk-I+nIf .r=0goto(8) 11JIf k > n - 4 go to (4)Set x = n - 4 + 2n'13 [orn-4-0.8n"'[Storage needed: n, k, x, J, 1, R.L and D Input restrictions:n = 0. 1, 2....k1.2.3.... 52:7 THE BESSEL COEFFICIENTS Jo(x) AND J,(x) 516 ,,,21Replace R by I-AX - R)Replace I by I - IIf I > n go to (7)nIf - =R go to (8)XSetD=I/(n-R)xReplace D by x'/[D(n' - n - .r' + _rR)[Replace x by x - DIf J * 0 go to (3) (or to (2)) If IDS > 10-7 go to (5)(8) Output xxor } <SetD=J= IIf x=0go to (II)(9) Replace R by I/(-- R + 10-'SxIfL - n> 1 [or>0)goto(10) Test values:Relace J bJR j,-, = 8.41724414 py J:021) = 0.271382590(L) (10) Replace D by DR + 2 frac .ho-s = 28.88737512J;0(j10s) = 0.143812670Replace L by L - I jig, = 0.000000000 If L *0goto(9) Joljv.,) = 1.000000000 Replace J by J/(1 - 2D) [or by J/(2D - 1)) j., = 6.70613319((J'(j,;d(11) Output J 1,(j__) = -0.3 1 3 5 304 45Jor} «« jiu5 = 27.18202/5J,0.Al J,n(jios) = 0.158631021 Also important in a number of physical applications are those nonnegativc values of the argument that causeJ (x) to attain an extremum (a local maximum or minimum): we call these values extrema of the n' Bessel coefficient and denote the k' such value by j;,,. Some approximate values of the extrema of J0 and J, occupy the right-handside of the table above, together with the corresponding associated values of the extrema of the n a Bessel coefficientThese associated values obey the relationships 52:7:3xXJ.(x)J..,(x) = -x =nnNote also that ja;A.' = j and that Jo(jo,,-,) = J;(j,;k). Figure 53-2 is useful for finding approximate values of theextrema, and accurate values of both j,;,A and are given by the algorithm above if the additions and modi-fications shown in green are instituted. The procedures adopted by the modified algorithm generally resemble its unmodified counterpart. but the Newtonian improvement formula is 52:7:4 x;J,,.,(x,)_x,(n -n' - n - x2 + x,R in this case.Whether used for finding zeros or extrema, the algorithm of this section is slow, particularly for large n and/or k. 517 THE BESSEL COEFFICIENTS Jo(x) AND J,(x)52:8 NUMERICAL VALUES Recursion 52:5:19 may be rewritten as 152:8:1R,-,=-1=1,2,3,...--R,x52:9 and this formula has the useful property that an error in R, generally induces a smaller error in R,_,. This meansthat the concatenation52:8:21 = Jo(x)(1 + 2R0R,(1 + R,R3(1 + R.RS(1 + R6R7(1 + )))))that follows from 52:5:5, may be curtailed by setting R,,_, = I and RL = 0 with little error, provided that L iseven and large enough. Because J (x) = R._, R,ROJ0(x), the formula 52:8:3RT_iR._2p._3... R,R01)RL_3Rs_4 + 1)RL_3RL_6 + + 1)R3R2 + 1)2R,Ro + Ican therefore be used to calculate numerical values of Bessel coefficients.The algorithm that follows utilizes expressions 52:8:1 and 52:8:3. An empirical formula selects a value of L.depending on n and x, that is large enough to ensure that the error in J (x) does not exceed 6 x 10. X »»> J - J (x) <3-»»SetJ=D=R= I»»Set 1 =2+2 lnt[(kI+ 7)-/2) + [n2/4)(1) Replace R by x/(2! - Rx)If!>ngoto(I)Replace J by JR(2) Replace D by DR + 2 frac(1/2)Replace I by I - 1If ! * 0 go to (2)Replace J by J/(2D - 1)Output JK<<<<Storage needed: n, J, D, R, x and I Test values:Jo(8) = 0.171650807J,0o(-10) = 6.59731606 x 10-e9 The algorithm of Section 53:8 may also be used to calculate J.(x) for x > 0 and the universal hypergeometricalgorithm (Section 18:14) can perform this calculation as well. 52:9 APPROXIMATIONSFor small arguments, the first two Bessel coefficients are well approximated by ((:252:9:1J0(x) = 1- 8/8-bit precision-1.3 s x :s 1.3 \\/52:9:21,(x) = 2 1 l- 2418-bit precision-2 s x s 2 52:10 THE BESSEL COEFFICIENTS J0(x) AND 11(x) S18while the corresponding approximations for large x are 52:9:3 J0(x)+"/4)cos(x +'/4)(x) =-ax/24V2ax3sin(x - '/4)3 cos(x - "14)52:9:4 J,(x)+Tx/242;rx'For large k, the zeros and extrema of the zeroth and first Bessel coefficients may be conveniently computedfrom the asymptotic formulas 1313779 152:9:5loiK +8K384K3 + 15360K5K = k -4) 52:9:6 j. A K=(k+)w331179+- -8K128K35120K5{ 52:9:7 )U//3 i 743129893KIk-4 it1\52:9:8j;A-K-8K384K'15360K5_K= k-4/ 52:10 OPERATIONS OF THE CALCULUSThe derivative of a Bessel coefficient may be expressed as three equivalent formulationsdJ-0) - J,+,(x) nn52:10:1dx11(x) =2=(x) - X 11(x) = x M-0 - J., (x)of which d[J0(x))dx = -11(x) and d[J,(x))/dx = lo(x) - 111(x)1x] are special cases. The coefficients occurring inthe formulas 52:10:2 52:10:3d2 Idx=J.(x) = Zz [J.-_(x) - 2J,(x) + 1,,.,(x)1d' .J.(x) =233J,-,(x) + 3J.+,(.r) - 1.+3(x)) for multiple differentiation will be recognized as binomial coefficients [Chapter 61 of alternating sign.Indefinite integrals of Besse] coefficients include 52:10:4 52:10:5arxJo(i)dr = xJo(x) +2[J,(x) ho(.r) - Jo(x) h,(x))x > 0 CJ,(t)dt = I - Jo(x) and generally 52:10:6fJ(tt-2Jr(x) +Jt)dtn0, 2, 4....k = 1. 3, 5...., n -o1 -Jo(x)n= 1,3.5....k=2,4,6,..., n-I 519 THE BESSEL COEFFICIENTS Jo(x) AND J,(x) 52:12and some others may be obtained by specialization of formulas in Section 53:10. The ho and h, functions in formula52:10:4 are discussed in Chapter 57.Among important definite integrals areJ brI 52:10:7 n= 1,2,3,...r0B<b52:10:8J¢cos(Bt) J0(bt)dt = 1B > b 0 62_b2r' /-62 52:10:9Jexp(-Bt) Jo(b V r)dt =1Bexp(48oalong with many others that follow from formulas in Section 53:10.With n = 0 and 1, Erdblyi, Magnus, Oberheuinger and Tricomi [Tables of Integral Transforms, Volume 2,pages 9-21] list many definite integrals of the form 52:10:10 J.(.,,) f(t)dtthese being examples of Hankel transforms [see Section 53:10).The operations of semidiffercntiation and scmiintcgration frequently convert Bessel coefficients into functionsinvolving sinusoids [Chapter 32]. Examples are 'd1r2'cos(V x)52:10:11 dx'2Jo(Vx) andd"2 J0(V)d"2sin(V x) 52:10:12dx 'r2- dx' 2V; 52:11 COMPLEX ARGUMENT We discuss specific instances of iv), but not the general case.When the argument of the Jr Bessel coefficient is purely imaginary, one has a function that is real or imaginaryaccording to the parity of n:l(y)n0,4.8,...52:11:1 n1,5,9,..._ I (y)n = 2, 6, 10...-A,(y)n=3,7,11,-..Here I is the hyperbolic Bessel function of Chapter 49.When the real and imaginary components of its argument are equal in magnitude, the real and imaginarycomponents of a Bessel coefficient correspond to Kelvin_ (-functions [Chapter 55152:11:2 J .(x t ix)V2)iV l)) 52:12 GENERALIZATIONSThe generalization of Bessel coefficients to arbitrary order is the subject of the next chapter. 52:13 THE BESSEL COEFFICIENTS J0(x) AND 11(x)52:13 COGNATE FUNCTIONSSimilarities exist among all the functions discussed in Chapters 49-57. 52:14 RELATED TOPICSThe definite integral520 0J.(0) J.(a) - fJ.(a) J,(13) 52:14:1JJ.(Rt)dr =02 -a o[[aJ(a)] + (a2-n2)[J.(a)122a23 = ashows that the functions J.(j.;Ix), J.(j,.,x), ... are orthogonal (Section 21:14) on the interval 0 s x s Iwith weight function x. For, if we abbreviate J.(j.;kx) _ *k(x), thenf01k52:14:2 t V (t)`l',(t)dtIJ.(j.:k)l'/k2Hence, any function f(x) which exists in the 0 S x 1 range can be expanded in the orthogonal series 52:14:3f(x) = ci4'I(x) + c"Jr2(x) + c,'P (x) cki.(j.,x)k'- Iwhere21I52:14:4 Ca = tf(r) J.Ij.,r)drIJ.(J.:k)1"oNote that the orthogonal series may be based on a y Bessel coefficient: in practice. J. is often the most suitable.It is equally evident from 52:14:1 that r,01k52:14:5Jr Yt(t) *;(r)dr =(n l l IJ.(j.:.)I' where 'Y,(x) abbreviates J, (j;,,x). Hence, the functions 41;(x). W;(x). 41;(x), ... are also orthogonal. In fact, onemay generalize the orthogonality of Bessel coefficients more widely, as discussed by Spiegel (page 144). CHAPTER53THE BESSEL FUNCTION Jt,(x) Some authors classify transcendental functions (those that cannot be expressed by a finite number of algebraicoperations) as either "elementary" or 'higher." The Bessel function is generally regarded as the most important member of the latter category, the 'higher transcendental functions." This function was invented in the context ofplanetary astronomy but finds application in a wide variety of practical problems. 53:1 NOTATIONThe symbolism J,(x) is almost universal, v being termed the order and x the argument of the Bessel function. Thename cylinder function is sometimes applied to J,(x), but this name is also used to describe collectively the broadclass of functions that this Atlas treats in Chapters 49-57.A change of argument to 2V often simplifies formulas, and Clifford's notation53:1:1 Cx) = x-"/=J,(2 Vx) is sometimes useful in this context. 53:2 BEHAVIORExcept when v is an integer, the Bessel function J,(x) is complex for negative argument. Therefore, in this chapterwe deal exclusively with x a 0, and this is the only range depicted in the contour map, Figure 53-I.The map suggests that an infinite number of bands of alternating positive and negative values cross the r, vplane, and this is, in fact, the case. This means that the Bessel function, viewed either as a function of x or of v,displays an infinite number of zeros, separated by local maxima and minima.For noninteger v less than zero, J,(x) approaches ±x as x -. 0, while for positive v, J,(x) remains bounded.For v z 0, J,(x) never exceeds unity.At constant small v, J,(x) is oscillatory throughout the entire x > 0 domain, whereas if the order takes largevalues of either sign there is a range of the argument. approximately 0 < x < v, during which J,(x) increases (ifv > 0) or decreases (if v < 0) in magnitude prior to breaking into oscillations. Once they are established, theoscillations steadily attenuate, as x increases, remaining centered about zero and eventually conforming to theasymptotic expression $21 53:2 THE BESSEL FUNCTION ],(x) 522 523 THE BESSEL FUNCTION J,(x) 53:2:1 J,(x) --2cosx -m-x-+ marx\2453:3 If the argument x takes some constant value greater than zero, Figure 53-1 also illustrates how J,(x) varies withv. As v takes increasingly positive values, the oscillations of 1,(x) sooner or later cease and the Bessel functionbecomes a monotonically decreasing positive function that eventually obeys the asymptotic expression 53:2:21,(x) -IrexlV-. =x = constant > 02av2v/IIOn the other hand, as v acquires ever-more-negative values, 1,(x) remains oscillatory, and the oscillations becomeincreasingly amplified and eventually satisfy 53:2:3J,(x)V/2 /-ex'"sin(vir)v-- -xx = constant > 0avl\2vAn advantage of Clifford's notation [equation 53:1:1 ] is that whereas J(x) is generally defined as a real functiononly for x a 0, C,(x) is real for all values of its argument. In fact, for negative x. C,(x) becomes related to thehyperbolic Bessel function of Chapter 50:53:2:4C,(x) _ (-x)-"121.(2\) x !5 053:3 DEFINITIONSThe function (x/2)" is a generating function for the Bessel functions J,(x), J,,,(x), Thus: 53:3:1 (x/2)"r(1 + v)k_ok'Bessel functions are defined by a number of definite integrals,including 53:3:2 1,(x)V;(x/2),JAcos(x cos(t)) sin'"(t)dtr(v+I) u 53:3:3J,(x) ='2(x/2)"I(1- t=)'-''' cos(xt)dtv > -- Vrrr(v+0 0 2and2(2/x)" sin(xt)53:3:4 J.(x) - dir(I - V)f(t. - I)I 1-2Gv<2 The general solution to the differential equationddf1_d'-fdf(v' 53:3:5xdxxdx-x-dx=+xdx--x')fis f = c,J,(x) + cJ_,(x) if v is not an integer, c, and c_ being arbitrary constants. Equation 53:3:5 is known asBessel's equation but Bessel functions also satisfy the simpler Bessel-Clifford equation 53:3:6xdjax.+ (I + v)djdx+j= 0j= x-`2[c,J..(2N6) +as well as many other differential equations [see Murphy]. See Section 52:3 for the corresponding solutions whenv is an integer.The differintegration formula2d"' 53.3:7 J,(VA) _sin(Vx) 53:4 THE BESSEL FUNCTION J,(x) suis valid for all v and illustrates the close relationship of the Bessel function to the sine. Differintegration can alsobe employed [see 53:10:10] to convert a Bcsscl function of any order to a Bessel function of any other order.Hence, a Bessel function of arbitrary order may be regarded as derived from the Jo Bessel coefficient [Chapter 52]by the operation 53:3:8 d-'J,(2Vx) =x '12 -Ju(2\Ix)or, in Clifford's notation [see 53:1:11: 53:3:9dx' dC.(x) = x- - CO(x) 53:4 SPECIAL CASESdx' A Bessel function of nonnegative integer order is known as a Bessel coefficient [Chapter 521. When the order isa negative integer, we have53:4:1 n = 1, 2, 3....Those Bessel functions of order equal to one-half of an odd integer are termed spherical Bessel functions ofthe first kind, and the notation5342j,(x)=n=0,±1,±2,n is sometimes adopted. These functions may be expressed either as an n-fold derivative:d' sin(V x)0.:t 1. ±2..n =.. 53:4:3J.(Vx) = (-2Vx)'dx' Nx or as an n-fold integral:2d_"53:4:4 sin(Vx)n = 0, ±1. ±2....dx-'involving sin(\). Because of this intimate relation to the sine, spherical Bessel functions are addressed in Section32:13.Bessel functions of orders3 are related to Airy functions [Chapter 561/3x)21353:4:5J.In(x) °Ai(-X) -+ Bi(-X)1X- I`2 53:5 INTRARELATIONSHIPSBessel functions satisfy the recursion relationship 53:5:1 2vJ, 1(X) _ -J,(x) - J,_J(x)Xand the argument-multiplication formula/x - bZx\' l53:5:2 J,(bx) = b' II - J.+;(x),em\2/ J.Setting b = I + (y/x) converts 53:5:2 into an argument-addition formula; another is Neumann's addition theorem: 525 THE BESSEL FUNCTION J,(x) 53:6 53:5:3 J,(x ± Y)J,.,(x) ii(Y)1Y1 < 1x1Setting b = i =in relationship 53:5:2 generates an infinite seriesX2Xf 53:5:4J,(x) + x J.+,(x) +2l",z(x) +_- J,+,<x) = i-°l,(ix) = I,()j-o Jthat sums to give a hyperbolic Bessel function [Chapter 50]. Expression 53:5:4 is a special case of a Neumannseries [see Section 53:141; another similar example isX153:5:5J,(2x) + xJ",1(2x) +J,+2(2x) +_J,,,(2x) = 2!-or(1 + v)Unless v is an integer (in which case 53:4:1 applies), there is no straightforward order-reflection formula,although J_,(x) and J,(x) are related in terms of the Neumann function [Chapter 54)53:5:6 J_,(x) = cos(vtr) J,(x) - sin(rnr) Y,(x)and via the relationship 53:5:7 53:6 EXPANSIONS-J-,() J,+i(x)W+ J-1-,(x) Mx)2 sin(va)rx As a power series, the Bessel function is expressible by 53:6:1Mx) =(x/2)'(x/2)2+(_____,-(x)(_x2/4),-r(1 + v)l!r(2 + v)2!r(3 + v)- 2j!r(l + j + v)which is convergent for all values of v, although the leading terms vanish if v is a negative integer. That J,(x) maybe expressed in terms of a hypergeometric series [Section 18:14) is evident by rewriting 53:6:1 as [(x/2)/r(l + v)](-x'/4)'/((1),(l + v),] or, in Clifford's notation [see 53:1:11: 53:6:2C,(x) =AA l-= -+ r(l + v)l!r(2 + v)2!r(3 + v) r(1 + v)(1)(1 + v),The product of two Bessel functions of common argument is expressible as a hypergeometric series with twonumeratorial and four denominatorial parameters:/1+v+µL+±) =I1+(-x')'(x/2),+"22,53:6:3J x 44)°Or(1+v)r(l+u.)j(1),(l+v),(I+µ),(t+v+µ);From this it follows by setting µ = tv that 12.(x) or J,(x) J_,(x) may each be expressed as a K = I, L = 3 (seeSection I8:14) hypergeometric series.The Bessel function may be expanded as the infinite product 53:6:4J,(x)l1 -x- \f\l-x'`(x/2)`l -x2 =I--11--J(1..'=1I-r(1 + v) ` r:l/ \\\j;Ji illr(I + v),=Z.where j,;t denotes the k" zero of J, [see Section 53:71. A similar infinite product holds for the derivative J,(x) _dJ,(x)/dxJ,_,(x) - J,.,(x)(x/2)"''x 53:6:5 2= 1;.(x) =2r(v)n I -v > 0where j;A is the k's extremum of J. (also discussed in Section 53:7]. 53:7 THE BESSEL FUNCTION J,(x) 526Although they are of greatest utility when the order is an integer, equations 52:6:6-52:6:9 apply to Besselfunctions generally, on replacement of n by v. Also, see 50:5:3 for an expansion of 1,(x) in terms of hyperbolicBessel functions.When x, but not lvi, is large, the asymptotic expansion(v tral,. 53:6:6J,(x)2=axI P(v;x) cost x - 2 -4- Q(v;x) sin1 z -2 -4)]holds, where, as x -. x:(i - v2)(i - v2)(19- v2)(i - v2)(2 4 4 2- V2)(1 - v2)( - V)2,(1 + v)2i53:6:7P(v;x)--I-+-++ 2!(2x)2412x)'(2j)!(-4x2)'and(, - v)(- V2)(2 - v2)(i - v2)(iV - y2)(Y - y2)(- V2)(s - V2)(1 - v2) 44453:6:8Q(v;x)+ - + 1!(2x)3!(2x)' 5!(2x)'_(_ - 1):r16 + y)2j_,2(2j + l)!x(-4x2)'... Each of the functions P(v;x) and Q(v;x) is expressible as a K = 4, L = 2 hypergeometric function, as tabulatedin Section 18:14. Although they are generally valid only for sufficiently large x, the series 53:6:7 and 53:6:8terminate when v is an odd multiple of ± 12, and in this circumstance an exact expression for J,.(x) is provided forall x. 53:7 PARTICULAR VALUES x=0x=x J.(x)Jv > 0 00,(x)J,Jx)-I <v<0,-3<v<-2,-5<v<-4.,..I00l_.(x)n = 1. 2, 3,... 00J.(x)-2<v<-1,-4<v<-3,-6<v<-5. ... -=0 The values of x that cause J,(x) to acquire the value zero are denoted by j,,,, j,.j, j,..3,..., and the name k"'zero of the Bessel function J,(x) is given to j,x. This name is appropriate for is > -I but may be misleading foris < -1 because not all these zeros then exist (i.e., as real numbers). As an example, for -4 < is < -3 thereexists the zeros j,:,, js, j,b, ... but there is no real J,,.,, j,, or j,3. For nonnegative integer is. values of j,.,, arediscussed in Section 52:7, and an algorithm is presented there for their calculation. For order - I we have j-..! _j,:,. J-5! = ju, etc., and, generally:53:7:1J-.A =J,-k=n+l,n+2,n+3,...For orders ±-! we have the simple rules j,1u = ka and j_112., _ (k - '/2)a. The zeros of order ; are given byj3,2s = q(l) where the latter is discussed in Section 34:7. The same section discusses, and provides an algorithmfor. p,(b), which is related to the zeros of order by j_3na. = p,_,(-1). For small is of either sign j,,, is givenby the approximation formula 53:7:22 1smalljvl2as is evident from Figure 53-2, in which the red lines display values of is as a function of is for I S ks 9, permitting approximate values of the zeros to be found easily. Fur large positive orders, the first zero of J,(x)is given approximately by 527 THE BESSEL FUNCTION J,(x) 53:853:7:3 j,_, = v + 1.8558v't' +1.033v-'t3large vThe symbols j,_,,j;;3, ... are used to denote those values of the argument x that cause J,(x) to exhibit alocal maximum or minimum, and j;;, is known as the 0 °i extremum of the Bessel function J,(x). Again, this nameis not always appropriate because, for example, the "first" extremum of J_y(x) is j'_3;4. Values of j;,, were discussed in Section 52:7 for n = 0, 1, 2, ..., while for negative integer order we have the rule53:7:4j'_,,,=j;,,_k=n+1,n+2.n+3,...For moiety orders j;1,,, = r,,_,(2) and j'_12e = p,_,(-2), where r and p are the functions discussed in Section 34:7.For small orders of either sign, the approximation2 3 53:7:5-j;,k-v=2k--k=2,3,4,...small IVI 2holds except for k = 1. Figure 53-2 contains graphs of (2/n) j;;, - v versus v for the first nine extrema. Greenlines on this graph represent maxima, blue lines minima. The approximation53:7:6 j.;, - v + 0.8086v''3 + 0.072v-'13is valid for large v. 53:8 NUMERICAL VALUESFor arguments not exceeding 7.5 + 0.3 Ivj, the algorithm below utilizes the formula 11 53:8:1where to = I and t, = -x2:R_,/4k(k + v), which is a truncated version of expansion 53:6:1. Provided K > -v,series 53:8:1 eventually alternates and the properties of alternating series [see Section 0:61 then guarantee that the fractional error in approximation 53:8:1 is less than e/2 if ItK(2/x)" f(1 + v) J,(x)I < e. The algorithm increments K until it exceeds -v and until tK/(to + t, + + tx_, + 2! 1 ',)1 < 10', thus ensuring 24-bit precision in J,(x)except in the immediate vicinity of a zero of the Bessel function. The multiplier (x/2)`/t(1 + v) is calculated bya modified version of the algorithm in Section 43:8, but special measures are adopted if v is a negative integer orzero. Input v >: Input x >aSetf=h=2Set k = 0»»>If x > 7.5 + .31v1 go to (3)If v + Int(IvI) * 0 go to (1)Replace v by -vSet f = h = (- 1)'/2(1) Replace k by k + IReplace f by f + hReplace h by - hx2/4k(k + v)Replace f by f + hIf If/hI <2 x 10'goto(1)IfvS -k go to (1)Ifv=0 go to (5)(2) Replace v by v + IReplace f by 2fv/xIf v<-3go to (2)Storage needed: v, f, h, k. x. r and 0 aaaaaaaaaaaa ....:....:....:....:....:....:....................... ....Jy_y17 .............:... .2Jy e ;y........... .215 14 13 11 528 529 THE BESSEL FUNCTION J"(x) 53:10 f ` J"(.r) <Set h =I1+ 7v(3v=-1l] /30v3 Replace h byh---1+v[l/+ InI2v1]Replace f by f exp(h)/2TvGo to (5)(3) Set t =21rrxSet 0 = 90(v + h - 12x2)Setf = 0(4) Replace f by f + t cos(0)Replace t by t(h2 - v2)/[(2h + 1)x]Replace 0 by 0 + 90Replace h by h + IIf h < lvl go to (4)If Jfj > 2 X 10'Itj go to (5)If h2<(2h+ 1)x+ v2 go to (4)(5) Output fUse degree mode or replace 90 by a/2. Test values:J2(10) = 0.254630314Jo(7) = 0.300079271J_, (4) = 0.0660433280J5t2 (ir) = 0.429869376J_3.6(0.5) = -169.507814 For arguments greater than 7.5 + 0.3jvI, equations 53:6:6-53:6:8 are used in the form 53:8:2K/(}VirJ"(x)--EticoslkZ-x+2+4/where to = V2/ar and other tQ values obey the recurrence tk = [(k - 1)' - v2]t4_,/2kx. The number of termsincluded in this asymptotic series is determined by three criteria. First, to ensure that the terms in the P(v:x) andQ(v;x) series alternate in sign, we include terms in 53:8:2 up to at least k = Ivi - I. Second, we cease addingterms to the cumulative sum f when the magnitude of tt becomes less than (5 x l0 -")Ifs. Third, irrespective ofsize of the contribution of t, to f, we never allow k to reach the critical value x - ? + VT x + v2 after whicht4 increases in magnitude and the series diverges. 53:9 APPROXIMATIONSThe first three terms of expansion 53:6:1 are reproduced exactly by the approximation4XX31.53:9:1 J"(x)xT(1 + v)128(1 + v)(2 +v)]small .rand subsequent terms are close when v is large. An asymptotic approximation, valid for large x, is given by 53:2:1. 53:10 OPERATIONS OF THE CALCULUSThe derivative of the Bessel function is given by the equivalent formulasdI1vv 53:10:1dxJ"(x) =2J-0) -J,_,(x) - -J"(x) _ -1"(x) -1".,(x) The expressions 53:10:2 d d.t'"J'(x)_x'"J":,(x).r also hold. 53:11 THE BESSEL FUNCTION J.(x)Corresponding to 53:10:2 are the indefinite integrals 53:10:3 J,t'"J.(r)dt = x"7,..(x)v > -10and 53:10:4 t'-'7"(t)dr =- x' '7._,(x)02T(v)Important definite integrals include 53:10:5 JTJ,Ir)dt = Iv > -10r(VI +b'-b)"53:10:6 Jexp(-bt) J.(t)dt = v > - I0I + b' 53:10:7 53:10:8cos(v arcsin(b))/1-O r b< 11cos(bt)1.(t)dt =-sin(rv/2)b1(b+ b'-Ib>lIsin(v arcsin(b))/ 1b'0 < b < Isin(bt) J"(t)dt =cos(rv/2)bI[b+b'-II"b> Iv > -2 and many others are listed by Gradshteyn and Ryzhik (Sections 6.5-6.7). Definite integrals of the form 53:10:910f(t) J.(st)dt530 are known as Hankel transforms and tables off V st f(t)1,.(sr)d1 are given by ErdElyi, Magnus. Oberhettinger andTricomi [Tables of Integral Transforms, Volume 2. Chapter 8).The operations of differintegration are particularly simple when Clifford's notation [see 53:1:1) is adopted.Thus: d"53:10:10 - x"C,J.t) = x'""C._"(x)dxAand, in terms of the function discussed in Section 53:12:d"x'"(-x)'53:10:11 - C"(x) =dx"r(l+v)r(I- s)j_0(I+v),(I-µ), 53:11 COMPLEX ARGUMENTThe Bessel function adopts complex values when its argument is complex and, unless the order is an integer, evenwhen the argument is negative. We report the expressions53:11:1andJ.(-x) = (-I)"J,(x) = [cos(vir) + isin(vr)) J.(x) 53:11:2J,(iv) = i`1.(v) =[cos( 2)+ isin( 2JJl.(v) 531 THE BESSEL FUNCTION J,(x) 53:14for negative and imaginary argument, but exclude the case of general complex argument. I is the hyperbolic Besselfunction of Chapter 50. With n replaced by v, equation 52:11:2 applies for the special case of complex argumentsin which the real and imaginary parts have equal magnitudes. 53:12 GENERALIZATIONSAs a hypergeometric function [Section 18:14], the Bessel function may be written 53:12:1 J,(x) =(x/2)"G(-x2/4)'r(1) r(1 +v)_omill +v);so that K = 0, L = 2 and one of the denominatorial parameters is constrained to be unity. Removal of this constraintgenerates a function 53:12:2 (x/2)"(-x2/4)'r(I +µ)r(1+v-p.)j_o (1+µ);(1 +v-p.);that thus represents a generalization of the Bessel function. The µ = 12 instance of this generalized function is theStruve function h,-,(x) of Chapter 57. 53:13 COGNATE FUNCTIONSThe functions addressed in Chapters 49-57 all have common features. 53:14 RELATED TOPICSIf, for some restricted or unrestricted range of its arguments. a function f(x) is expansible as a power series [seeSection 11:14]: 53:14:1 f(x) _a xa =I d'f 0)ij! dx'then it may also be expanded as the Neumann series 53:14:2 f(x) = 12Ir j(v + k) b1J,.,(x)XA-0Here v is arbitrary except that it may not be a negative integer. The coefficient b; is given by4'r(J+j+v) 53:14:3bt=Ya.;2J=k=0,2,4,...J.o(J - j)!when k is even, and by' 4'r(J+j+I+v) 53:14:4b,=2Za..,2J + I = k = 1,3.5....;_o(J - j)!for odd k. Equations 52:5:3-52:5:7 are all examples of Neumann series.Replacing f(x) in 53:14:2 by the unity function leads to the expression 53:14:52'(2k + v)r(k + v)Js.,(.x)v * - I. -2. -3, .A_o k!as an expansion of an arbitrary power. Equations 52:5:6. 52:5:8 and 52:5:9 are special cases. 53:14 THE BESSEL FUNCTION J,.(x) 532As an alternative to the 53:14:2 expansion. the function f(x) may be expanded as the modified Neumann series 53:14:6f(x) _c°IIJ,V x)v * -1, -2. -3,2 where the coefficients are given by`r(I+j+v)53:14:7cc=4'a,(k-j)!Equations 53:5:2 and 53:5:4 are examples of modified Neumann series. Replacing f(x) in 53:14:6 by unity leadsto expression 53:3:1. CHAPTER54THE NEUMANN FUNCTION Y (x) In linear combination with a Bessel function [Chapter 531, the Neumann function satisfies Bessel's equation, and,accordingly, it appears in the solution to many physical problems, especially those involving cylindrical symmetry. 54:1 NOTATIONThe symbol N,(.r) is a common alternative to Y,(.r) to denote the Neumann function of order v and argument x.The names Bessel function of the second kind and Weber's function are also in use. 54:2 BEHAVIORBecause the Neumann function is complex for negative argument. we restrict attention to x >: 0. Behavior in thisrange can be appreciated best by consideration of the contour map, Figure 54-1.Viewed as a function either of x, or of v. the Y,(x) function displays an infinite number of oscillations. For aconstant value of v not very different from zero, the oscillations commence close to x = 0, and their amplitudessteadily diminish as the argument increases, eventually conforming to the asymptotic expression 54:2:1Y,(x)-? sin(x-irx\v-2n--4)lVAs v departs further from zero in either direction, the oscillatory behavior persists, but its onset becomes increasinglydelayed. This is very clearly illustrated in Figure 54-2. which depicts Neumann functions of nonnegative integerorders.At any constant positive value of .r, the Neumann function is an oscillatory function of v in the vicinity ofv = 0. and the oscillations become increasingly amplified as v becomes more negative until the asymptotic for-mulation2ex1,cos(v-rr)V_ -xX = constant > 0 54:2:2Y,(x) - -its \w` comes to he nheyed. On the other hand, as v/becomesincreasingly positive, the oscillations of Y,(x) eventuallycease, after which the Neumann function acquires negative values of increasing magnitude, given by 54:2 THE NEUMANN FUNCTION Y,(x) 534 THE NEUMANN FUNCTION Y,(x) 54:3 0.2 54:2:3 x = constant > 0Yy(x) y -2VIWVexV -0 JCComparison of Figures 53-I and 54-1 reveals considerable similarities. The major point of distinction occursfor positive v as x- 0. Whereas the Bessel function approaches zero in this limit, the Neumann function approaches--. Another difference concerns the phase of the oscillations, which generally differ by a/2 or thereabouts. Thisis evident on comparison of equation 53:2:1 with 54:2:1 and of equation 53:2:3 with 54:2:2. 54:3 DEFINITIONSFor noninteger order, the Neumann function is defined in terms of Bessel functionscos(VIT) J,(x) - J_,.(x) 54:3:1Y,(x)=vs0,±I,±2....sin(vrr)For integer order, this definition is to be replaced by its limit54:3:2Y,(x) = lim (cot(vir) J,(x) - csc(vrr) J_,(x)} n = 0, ±1, ±2, ...The combination c,J,(x) + provides a solution to Bessel's equation 53:3:5, for integer or nonintegerV. Similarly, c,J,(2Vx) + c,_Y,(2Vx) satisfies the Bessel-Clifford equation 53:3:6 for all v. The coefficients c, andc2 are arbitrary constants.Some integral representations of the Neumann function are given by Gradshteyn and Rhyzik (Section 8.4151.The close relationship of the Neumann function of zero order to the cosine function is evidenced by the semiintc-gration formula 54:4 THE NEUMANN FUNCTION Y,(x) d-'.1'1/1ar1 54:3:3a; is costXV xY'. 54:4 SPECIAL CASESKnown as spherical Bessel functions of the second kind, the functionsar 54:4:1 Y"(z)=Y,,,/,.(x)n=0. ±1.'2,-..are related to those of the first kind by the simple equivalence54:4:2Y"_0)=(-1)"j_"(x)n = 0. ±1. µ2. ... and the relationships given in Section 53:4 are therefore readily adapted to y"(x).Neumann functions of order t3 are related to Airy functions (Chapter 561 via the expressions 54:4:3 -3Bi(-X) + V3 Ai(-X) 3x Y-in(x) =2VXX =2)"Similarly, Neumann functions of order t3 are related to the derivatives of the Airy functions. 54:5 INTRARELATIONSHIPSS36 The order-reflection formula54:5:1 n=0.±1,±2....applies only for integer order, but the recurrence formula2v 54:5:2 Y- ,W-Y,(x) - Y,.-,(x)xapplies for all v. Because the form of 54:5:2 is identical to that of equation 52:5:15, expressions analogous to52:5:16-52:5:20 relate Y,(x), Y3(x), Y4(x), Y5(x) and Y6(x) to Yo(x) and Y,(x).A number of relationships interconnect Neumann and Bessel functions: 54:5:3 mY,(x) + Y-,(x) = cot 2 fJ,.(x) - J_,(x)1 54:5:4 Y,(x) - Y_,(x)a n Z IIJ (.r) + 1_,.(x)1\\\54:5:5 Y;() + J!() =Y=-(x)) + J2-,(X)254:5:6 J,,.1(x) Y,(x) - J,(x) Y,,,,(x) _ - ax 54:6 EXPANSIONSUsing definition 54:3:1 and expansion 53:6:1, Y,(x) may be expressed as the sum of two infinite series. Alterna-tively, by exploiting the properties of the gamma function [Section 43:51 and of the trigonometric functions, thetwo infinite series may be conjoined into a polynomial function and a single infinite series: 54:6:1Y,(x) = As(1vD Z(-x2/4)1=oil(, - IvI)j 537 THE NEUMANN FUNCTION Y,(x) 54:7Bs(-Ivl)Cs(Ivl-n)(x/2)"xz Vlv#0,±1,±2,... +'=O j!(1+(n + J)!(1 + it - jvj))(4 /where we use the abbreviation s(p) _ -(2/x)p r(p)/n. When v is positive, the coefficients A and C are unity andB is cos(va); when v < 0, A = C = cos(vn) and B = 1. Here n may take any positive integer value, but theexpansion is most useful when n is the integer closest to jvi, for then the B and C terms in 54:6:1 have similarmagnitudes, permitting the summation to be evaluated with little loss of precision.Both of the expansions discussed above become indeterminate when the order of the Neumann function is aninteger. For nonnegative integer orderx12 ;xz i54:6:2Y,(x) _2J (x) -(n - j - 1)!ITx)=i-vJ!41 (X) jy(j+l)+tp(j+n+l)xz'(ln=0, 1,2,...where J. and ts are the functions of Chapters 52 and 44. The polynomial term in 54:6:2 is absent when n = 0;under these conditions one has 7r 3x'11x6 1I1(-xz/4)' 54:6:3Y0(X)In(x)+-J x+-+ r+22Y ° O 4 12813824(23!/where y is Eulei s constant, 0.5772156649. Neumann functions of negative integer order can be expanded as in54:6:2 by taking advantage of relationship 54:5:1.Neumann functions of integer order are also expansible as an infinite series of Bessel coefficients [Chapter52]. The general case is2 54:6:4 (x/2)jJ,(x) +ITIln(-)J .(X) itx,_0(n - j)j! 22,,(-1)'(n+ 2j)ir ,=II(n + J)and the n = 0 case reduces to 54:6:5 2 Y0(x) =11n(2)+y]J0(x) - 21 )J J,,(x)When x. but not jvl, is large, the Neumann function obeys the asymptotic expansion2m av,IT 54:6:6Y,(x) -P(v;.r)sin(x- --l+ Q(v;x)cosrx- - - - Jx-' x az[.24)\24) where P(v;x) and Q(v;x) are the functions discussed in Section 53:6. 54:7 PARTICULAR VALUES A =0X = m Y,tc)v > -000-3-1 -7-5 -II-9 Y,(x)-<v<2 ,.<v<Z,2<v < 2....+.0 3-3 -9-7 -13-11 Y,(x)-<<-,-<v<-,-<v<-.... -0222222 54:8 THE NEUMANN FUNCTION Y,(x) 538The values y,..,, y,2, y,;3.... of the argument x that cause Y,(x) to acquire the value zero are known as thezeros of the Neumann function, whereas the values y:.,,, y,2, y;,;,.... of the argument that cause Y,(x) to exhibita local maximum or minimum are called extrema of the Neumann function. Qualitatively, y,., and y..A are very similar to j,,, and j;.j so that Section 53:7 contains useful information on these zeros and extrema. Numerically,the relationships 54:7:1v`j:.,<y..i<Y..1 <j<J:,_<Y,2<Y:.2<J..2<j 3<...hold, and for the more remote zeros and extrema: 54:7:2 54:7:3Y" - j.s= -+k -3itlargek2v4v1y;,,=j,k=- +k -- itlargek4)For large positive orders, the first zero of the Neumann function is approximated by54:7:4 y.;, - v + 0.9316v"' + 0.260v-"' large vwhile the first extremum, a maximum, obeys the approximation54:7:5y;., = v + 1.821 lv"' + 0.940v-"large v 54:8 NUMERICAL VALUESIn this section we present a lengthy algorithm for calculating Y,(x) for all orders v and for any nonnegative argumentx. The precision is generally 24 bits (i.e., the relative error does not exceed 6 x 10-8) except in the immediatevicinity of e zero of the Neumann function. Except when v = -;. 73, 2..... Y,. (0) equals -x or +x; the algorithmreturns 0, -109° or 1099, as appropriate, when x = 0 is input. The algorithm is long because it employs threedistinct procedures for calculating Y,(x), depending on the values of v and x.When v is the nonnegative integer n. the algorithm employs expansion 54:6:2. which may be rewritten as2 In(h) + 2-1 - rb(j) - 46(j + n) j!1' 54:8:1rY,(x) = h" i (h2)' -h"-2(l'_0J!(n + j)! ,-o (n - j - 1)! \h''/where h = x/2, 4i(0) = 0 and d,(k) = I + _ + ; + + Ilk. A satisfactory approximation to the right-hand sideof this equation is given by the finite sum S, = 2to + 2t, + 2t, + + 2t,_, + 21" ++ 21,_, + t, for somesuitably large J. The general term, t, in this summation may be expressed as A,,; B",, - C,.; where Ao,o = 2 In(h)+ 2Y> Bo:o = ;i, Co,o = 0 and C,;o =J /(2h); for m = 1 . 2, 3,- - - n, A,,;o = A_ 1:0 - (1 /m), B",;o =and C.,.:,, = hCo/m: and fork = 1. 2. 3..... j. A":x = A":k-, - (Ilk) - [11(n + k)], B,, = -h2B":.-1/k(n + k) and C.., = k(n - k)C",,_,/h2. For a sufficiently large J. the terms 2t 2t,.,, 2r,.2, ... will alternate in signand progressively diminish in magnitude. It then follows from the properties of convergent alternating series [Sec-tion 0:6] that the error in approximating irY,(x) by S, cannot exceed It, I in magnitude. The algorithm thereforetests, for j = 1, 2, 3, ..., until jyI/js,I -< 5 x l0-e. Additionally, three other tests must be passed before J can beidentified with certitude. First, j must be at least equal ton - 1, to ensure that all terms in the polynomial componentof 54:8:1 have been included. Second, alternation of sign will not occur consistently until d.(j) + e(n + j) exceeds2 ln(h) + 2: this is guaranteed if the sign of g is negative for even j. Third, it is necessary to confirm that the tterms are, indeed, alternating and convergent.Relationship 54:5:1 is invoked when the order is a negative integer.The asymptotic expansion 54:6:6 is employed whenever x exceeds both 8 and 5.6 + 0.71v1. or for all x whenv is an odd multiple of ± 1. Provided (j + !t)2 > v= the terms in the series for P(v;x) and Q(v;x) alternate after theja term, t;. Moreover, t, progressively diminishes in magnitude provided j < J = x + { + x= + -x+ v2. Incor-porating these two provisos, the algorithm appends terms to the series for P(v:x) 2/lrx and Q(v;x)2/rrx untilthe j° term satisfies the inequality ft,l < 5 x 10-s {P(v;x) sin[x - vrr/2 - it/4] + Q(v;x) cos[x - yr/2 - w/4]}2/ars. THE NEUMANN FUNCTION Y,(x) Set f = -109°Set h = x/2Iffracly+210 go to (16)Ifh+0goto(1)Ifv>0goto(19)Replace f by f cos[180 Int(2- v lGo to (19)\////(1) If h4 go to (2)If h > 0.35(8 + Jul) go to (17)(2) If frac(v) = 0 go to (11)Set B = cos(I80v)SetA=C=f=t=lIf v > 0 go to (3)Set A = C = B Set B = 1Replace v by -v(3) Set p = vSet n=Int (2 +V)(4) Set q = h-°(5) Replace q by q/pReplace p by p + IIfp<3goto(5)1lSets= [I+7p=\3p=-IIJ/30p2 LI-sReplaces byZp+p(In(p)-I]Replace s by -q exp(s)2/apIff<0goto(6)Iff=0goto(7)Replace A by AsSet f = -1Setp-vGo to (4)(6) Replace B by BsSetf=j=0Setp=v - nGo to (4)(7) Replace C by CsIf n = 0 go to (9)(8) Replace j by j + IReplace f by f + rReplace C by Ch/jReplace t by -the/j(j - v)Ifj+ngoto(8)(9) Replace A by AfSet j = f = 0(10) Replace f by f + (B + C)54:8Storage needed: v, f, h, A, B, C, t, p, n, q,s,jandJ Input restrictions: Order v may take any valuebut argument x may not be negative. Use degree mode or change 180 to it and 90to a/2. 54:8 THE NEUMANN FUNCTION Y,(x)Replace j by j + IReplace B by -Bh2/j(j + v)Replace C by -Ch2/(j + n)(j + n - v)If j < 4 + 4h go to (10)B+CReplace f by f + A +2Go to (19)(II)Setfq=1Ifv>0goto(12)Set f = cos(I 80v)Replace v by -v(12) Set A = 2 ln(I.78107242h)Set B = 1 /2Set C = 0Ifv=0 go to (14)Replace A by A - 1Replace B by Bh Set C = 1/(2h)If v = I go to (14)Set j1(13) Replace C by Ch/jReplace j by j + IReplace B by Bh/jReplace A by A - -JIf j * v go to (13)(14) Sets=r=AB - CSet j = 0(15) Replace j by j + 1Replace q by -q Replace s by s + r 1Jj + vReplace B by -h2B/j(v + j)Replace C by j(v - j)C/h2Setp=rSet:=AB - CReplace s by s + : If Isj <- 2 x 107ptj go to (15)Ifj < v go to (15)Ifgt>0goto(15)Ifgr<-gpgoto(15)Replace f by fs/IrGo to (19)(16) Ifh4Ogoto(17)If v > 0 go to (19)Sctf = 0Go to (19)IReplace A by A - - - - 1(17) Set A = v --Set B = - I - AISetC=2h+-2 541 f - Y,(x)Set t = 1/VhSet p = q = j = 0(18) Replace p by p + t cos(90j)Replace q by q - t sin(90j)Replace j by j + l Replace A by A + I Replace B by B + IReplace r by tAB/4hjIf AB<0goto(18)Set f = p sin(s) + q cos(s)If If]> 2 x 10'11] go to (19) If j < J go to (18)(19) Output fTest values:Y,13(4.93837580) = -0.163661435Y6(0) = -xY3(1) = -5.82151761Y-,(0) = -Yso(I) _ -2.19114281 x 10"Y_1(5) _ -0.147863143Y-9/t(0) = 0Y512(2.6) = 0.530168510Y,(50) = 0.095912027854:10 The third routine is used by the algorithm when x s 8 or when x :5 5.6 + 0.71v], except when v is an oddmultiple of ± Z. It employs expansion 54:6:1 with the infinite series terminated empirically at j = lnt(4 + 2x).The required gamma functions are calculated by a procedure that is essentially that utilized in Section 43:8. 54:9 APPROXIMATIONSFor a range of integer orders and moderate arguments, the polynomial term in expansion 54:6:2 is dominant. Thisleads to the approximation 54:9:1 I2- 18-bit precisionx < 0.73(n - 1)n = 5, 6, 7, ... roI 54:10 OPERATIONS OF THE CALCULUSThe derivative of the Neumann function may be expressed in the alternative formsdY,_,(x) - Y. i(x) vv 54:10:1dzY.(x) =2= Y.-i(x)-Y,(x) =-Y,(x) - Y,-i(x) and we have the simpler relationships 54:10:2 ddxIr'" Y,(x)1 =These results are identical to the corresponding equations for Bessel functions [Section 53:101.Indefinite integrals include2"r(v) 54:10:3 t"Y,_,(t)dt = x° Y,(.r) + v > 0o ITThe definite integral54:10:4THE NEUMANN FUNCTION Y,(x)SetI=C+C'1-ABSets=901E+4h1\a 1Y4:)dt = -tan(va/2)- I < v < 1shows that the infinite integral of Ya is zero. A more general definite integral is 54:11 THE NEUMANN FUNCTION Y,(x)r2"l+µ+v+µ-vaµ - arv\ 54:10:5Jt"Y(r)dt = f'I2Jr('21 sin(2Iµ< 2> -1and other definite integrals are listed by Gradshteyn and Ryzhik [Sections 6.5-6.7]. 54:11 COMPLEX ARGUMENT542 Because the Neumann function is complex even for real negative argument54:11:1 Y,(-x) = cos(va) Y,(x) + i[cos(va) J,(x) + J_,.(x)]we omit the general formula for Y\,.(x + iv) and present only that Ithe Neumann function of imaginary argument254:11:2Y,(iy) _cost 2 I Ke(y) - sing Z I Ii(i) +'I cos( Z I,(y) - sin(2I K,(y)JIn these equations J,. K, and 1, are/the functions addressed in Chapters 53, 51 and 50. 54:12 GENERALIZATIONSIn light of the limiting operation 54:12:1 -srlim {X"Q1j, (cos(kx))} = - Y_,(x) v> 0x> 0the Neumann function can be regarded as a special limiting case of the second Legendre function [Chapter 59]. 54:13 COGNATE FUNCTIONSThe two Hankel functions, also known as Bessel functions of the third kind, are generally complex functions definedas54:13:1 J,(x) ± iY,.(x)They satisfy Bessel's equation and are sometimes preferable to J. or Y. in such solutions. CHAPTER55THE KELVIN FUNCTIONS Of the four Kelvin functions, ber,(x), bei,(x), ker,(x) and kei,(x), those with zero order, v = 0, are most importantand are emphasized in this chapter. 55:1 NOTATIONThese functions are named after Lord Kelvin, the title adopted by William Thomson (1824-1907) on his elevationto the peerage. Accordingly, the name Thomson function is given to these functions by those with distaste for thearistocracy.The initial letter, b or k (after Bessel and Kelvin), is sometimes capitalized. The terminal letter, r or i, refersto "real" or "imaginary," as in definitions 55:3:3-55:3:6. The absence of a subscript, as in ber(x) and kei(x),indicates an order of zero:55:1:1 ker(x) = kero(x), bei(x) = beio(x). etc. 55:2 BEHAVIORFor unrestricted values of v and x, the functions ber,(x). bei,(x), ker,(x) and kei,(x) display rather complicatedbehaviors, and a comprehensive verbal description wil not be attempted. Moreover, we shall restrict attention toarguments x 2: 0, although her, and bei, are real for negative arguments if v is an integer.The four Kelvin functions are oscillatory functions of x when v is in the neighborhood of zero, but the onsetof oscillations becomes increasingly delayed as'vj increases. Once established, the periods of the oscillations ap-proach \ir for all four functions, but while the amplitudes increase rapidly for ber,(x) and bei,(x), those of ker,(x)and kei,(x) decline as x -, -.The magnitudes acquired by ber(x), bei(x), ker(x) and kci(x) arc so diverse that it is impractical to plot thefour zero-order Kelvin functions on the same map, except for arguments in the rather narrow range displayed inFigure 55-I. For larger arguments we make use of the asymptotic properties of the Kelvin functions by mapping,in Figure 55-2, the curves labeled A = 2ax exp(-x/V) ber(x), B = V 2ax exp(-x/V) bei(x), C =2x/a exp(x/'V-2) ker(x) and D = 2x7 exp(x/\/2) kei(x). As is evident from this diagram, or from equations55:9:15-55:9:18, these four products approach pure sinusoids (Chapter 32) at large arguments. 543 55:3 THE KELVIN FUNCTIONS 544 1. 4FIG 55-1 0. 8 0.8.:. 0.40.2 55:3 DEFINITIONSThe general her, and bei, functions are defined in terms of Bessel functions of comlex argument, as 55:3:1 ber(x) =lJ,-x-ix l+1J,-x + ix l2 \2 \ V/ 55:3:2 V2-N2)t that is wholly real for nonnegative Although it is not immediately apparent, each of hese formulas definesreal x.If we use the symbols Re{ } and Im{ } to signify "the real part or and "the imaginary part of," so that x =Re{x + iy} and y = Im{x + iiy},/we may define([cos(!!)/ 55:3:3ber,(x) = Re{+ i sinl 2> IJ I,1+f /\f l 111f\(!!)1\/ \ I55:3:4bei,(x)m{ Jl+= Im{ Iroos(2I+ i sinII,I x + I } l\7/\/\\\72V2in terms of the Bessel function [Chapter 531 or hyperbolic Bessel function (Chapter 50] of argument whose realand imaginary parts are of equal magnitude. Similar formulas involving the Basset function [Chapter 51] define 548 THE KELVIN FUNCTIONSthe ker, and kei, functions+ 'XI } 55:3:5ker,(x) = Re s Icos(2)- isin(- )J K,1 sV2 V1 } 55:3:6kei,(x) = Imi Icos( Z)- i einl2IJK,(V2= V1+)The ker and kei Kelvin functions of zero order may be defined as the following definite integrals:xZ r-xrt:310(x[)x 55:3:7ker(x)8 JaCil r I expl4)dto 11 + t°dt =4Jln(1 + t°)1,(xt)dtr/o\/1/\55:3:8kei(x) =8Jaw I Stlt1 - 1J expl 4'Idt = -J'+ 4dt = Z Jfarctan(12) 1,(xt)dtin terms of functions discussed\\in/Chapters 38\\\and 52. 55:4 SPECIAL CASES55:4 Kelvin functions of orders equal to odd multiples of ± 1 /2 may be expressed as elementary transcendental functions.For example: ... . . . .......... . .. . . . .... ... .. ... ...... ..1.2FIG 55-2 : 55:5 THE KELVIN FUNCTIONS 546 55:4:1 55:4:2 55:4:3 55:4:4expX ker112(x)2x\(sin(72_8 kei112(x)exp(cos(lit`V2\ J 81 1xa1 berln(x) =expcosx-+8+-kei1,2(x)taxx Ixxa1 bei1/2(x) =exp" /sin+8+-ker,,2(x)7(--) ()taxV2V2The Kelvin functions ber, and bei, become somewhat simpler when v is a multiple of 3. Only for these ordersis ber,(x) or bei,(x) expressible as a simple power series. When the order is an even multiple of 3 (i.e., a multipleof ;), the power series contains terms of alternating signs + - + - + -; when the order is an odd multipleof 3, signs occur in the sequence+ + - - + + - -. . Equations 55:6:1-55:6:4 provide details. 55:5 INTRARELATIONSHIPS The order-reflection formulas 55:5:1 55:5:22 ber_,(x) = cos(-) ber,(x) + sin(va) bei,(x) + - sin(vt) ker,(x) it2 bei_,r(x) = -sin(va) ber,(x) + cos(va) bei,(x) + - sin(va) kei,(x)IT55:5:3 ker_,(x) = cos(va) ker,(x) - sin(va) kei,(x)55:5:4 kei_,(x) = sin(va) ker,(x) + cos(va) kei,(x)all reduce to55:5:5f_"(x) _ (-1)" f;,(x)f = ber, bei, ker or kein = 0, ± 1. ±2,when the order is an integer.Negative arguments generally render the Kelvin functions complex. Exceptions are ber, and bei, for integerorder, to which the argument-reflection formula55:5:6f,(-x)_(-1)"f(x)f=berorbein=0,±1,±2,.applies.The recurrence relationships may be written 7 f-v V [I ffiff= bk 55:5: e ererer 5558 i_f-0/2+ f- fi fi fibikiI:: ,,1(x)e (,(x)er,(x)1e,_1(x) exe=eore 547 THE KELVIN FUNCTIONS55:6 EXPANSIONSAll four Kelvin functions are expansible in terms of two series for which we shall use the abbreviations(-x`/16)'Vjr(x/4)'(-x'/256)'55:6:1Fe,(x)_( l\2/_o (2j)!f(I + 2j + v) 1v;_o1vv2,(1);(2+2)'11+2)'f\((2+2f(1+2)(1)(-x4/16)'2 / (x/4)22'"(-x'/256)' 55:6:2Ge,(x)_(x)2*(1+2j)!f(2+2j+v) ((v3vr 2j-, (f`'+2)f(2+2)55:6 The second formulations above show that Fe, and Ge, are instances of K = 0, L = 4 hypergeometric functions[see Section 18:141. When v is a negative integer, formulas 55:6:1 and 55:6:2 are inconvenient and the identitiesFe_ ,(x) = (-1)'Fe,-2,(x) = (-1)' Ge2,_1(x), Ge_,,,(x) _may be used instead, where n = 1, 2, 3, .... The expansions/3wr/ `cost -+L- 55:6:3 / ber.(x) = I -IV\ 42lx2 I'= cos___)(3-Fe,(x)-sinm 1 _o j!r(1 +j+v) \4\ 42)'3wnjir 55:6:4bei,(x) =(X42)sC(I+14)=sinI34)2 Fe,(x)+ cost31 Ge,(x)-oj'.j+v) \\\Jare valid for all v, but a restriction to noninteger v is required to validate the expansionsn csc(vtr) r/3va\/3vir/vn\/va55:6:5ker,(x) =2Lcosl 4 I Fe-.(x) + sin(4)Ge_,(x) - cost 4 I Fe,(x) -sin(4)Ge,(x)v#0. +1,±2.... 55:6:6kei,(x) =sin - Fe_.(x) + cos - Ge_,(x) + sin - Fe,(x) -Ge,(.r)a c2(m) I134/\344)4cos__)Jv*0.±1.--2....For nonnegative integer order, ker,(x) and kei,(x) are given by the expansions/2I /2\n(n -J1)!/3nn l(x''55:6:7ker,(x) = InlXI ber,(x) + 42 I x I! cost4 +2)\ 4) 2(2)' =k4(1 + j) + y(I + It + j)cos(3naj7r\ fx2 +-(-+- I\n=0.1,2,...=oj!(n + j)!424 2nI(n3narax' 55:6:8kei(x)=In )bei.(.r)-4ber(x)-2IXIn1 J!sin(4+J2)(-)Irrr4,0+j)+y(I+n+ J') 3n7rnx'')^(n+j)!ll4)\4)n=0.1.2,-- +2 \2j!-o J 55:6 THE KELVIN FUNCTIONS 548where $ is the digamma function of Chapter 44.Asymptotic expansions of the Kelvin functions are provided by the formulas 55:6:9ber,(x)exp(x/ V [)(2 - v),Q+ v)'cosx +'7 -1n-lx-. xtax,=oj!(2x)'(248J 55:6:10bei,(x) -exp(x / V 1)(i - v), ( + v),sinr x+va2Trx,mooj!(2.)1\\2X_%c( -xv),+ v)jxwnj n) 55:6:11ker r)-exp lcos- +-+-+-x +x 2xV22xi-vj,(-)'V 2248 ITx)(1-v),(!+v),/ xma 55:6:12kei,(x) - --exp( sit)++ 4 + -1X _X with suitably chosen upper limits to the summation index j. Although the series in 55:6:9 does, indeed, asymp-totically represent ber,(x) for large x, it more closely represents ber,(x) + [sin(2va) ker,(x) + cos(2v,r) kei,(x)]/ I. Similarly, a better approximation to the series in 55:6:10 is bei,(x) - [cos(2var) ker,(x) - sin(2vir) kei,(x)]/a.When v = 0, the various expansions simplify to 55:6:13 55:6:14a/ 16Y64147456[(2j)!]2x'(-z'/16)'... beix2_s°+x1o++ (x) =42304147456004((2j + 1)!]2 55:6:15ker(x) =[ln()- yJ ber(x)+'r bei(x) + 11 +-+I++1 (-x`/16)'x4-, `232j11(2j)! I' 55:6:16kei(x) =-beix) - 7 ber(x) + sIII\ (-x'/16)' [In()x44;=u `232j + I J [(2j + 1)!]2 berexp(x/N/2) "[(2j)!]2cos!?r\J = lnt(2x)x-. x 55:6:17bei(.r)-G(_L_---2wxio(j!)2(32x)'sin48 55:6:18kc.(x) -V 2xexp(72)-xI(2J)!)2cos(-x+3Jn-? 1J = Int(2x)x-+ xo U!)'(32x)'sin,,248As well, ber(x) and bei(x) are expansible in terms of the functions of Chapters 49 and 52: 55:6:19ber(x) = 10()o(212(7) J,(--!z)+ (_I))17J(7)J2,(x) 549 THE KELVIN FUNCTIONS 55:6:20bei(x)=21,\x )J,Ix 1-2131 x111x)+2Is( x)Js(x)-55:7 )J2,,I() The sum of squares of the her and bei functions is the simple seriess r55:6:21ber=() + bei2(x) = Izx(x16)3224576,(j!)'I (j + :)which is rapidly convergent for small arguments. Asymptotic expansions exist for ber'(x) + bei2(x), for kerz(x) +kei2(x) and for a number of similar functions; see Abramowitz and Stegun [Section 9. 101 for these. 55:7 PARTICULAR VALUESKelvin functions display diverse behaviors as x -* 0, and this leads to very varied values for ber,(0), bei,(0), ker,(0)and kei,(0) as functions of v. For the most part these particular values are +x, 0 or -, but the following finitenonzero values are also realized: 55:7:1 ber(0) = I 55:7:2 -ker,,(0) =2 55:7:3 akei(0)4Figure 55-3 shows what value represents the limit of the four Kelvin functions of order v as x -. 0. The colorcoding used in this diagram is that green represents zero, red represents +x, blue represents -x and black representsone of the values given in equations 55:7:1-55:7:3. Notice the very elaborate behavior of ber,(0) and bei,(0) fornegative P. In this region these functions have a periodicity of 8 in their orders; that is: 55:7:4 f,(0) = G-00) = f,_16(0) = f = her or bei v < 0As x approaches infinity, ber,(x) and bei,(x) oscillate with ever-increasing amplitude, but ker,(x) and kci,(x)converge towards zero:55:7:5 ker,(x) = kei,(x) = 0All four Kelvin functions have an infinite number of zeros. For a sufficiently large x, these are given by3 55:7:6ber,(r) = 0r = V 2rr(k - 2 - 8k = large positive integer' 55:7:7bei,(r) - 0r - V to k -v2+18)k - large positive integer 55:7:8ker,(r) = 0r -5" Vk - 2 -g)k =large positive integer 55:7:9kei,(r) = 0r = V 2x k - 2 --8Ik = large positive integerThe first few zeros for v = 0 are included in Table 55.7.1 together with argument and function values of earlyextrema, as well as the function values at the x = 0 origin. The final line in Table 55.7.1 provides an approximationformula for calculatintthe k'" zero of each Kelvin function; it uses the abbreviations a = N/2(8k - 3)sr. b = N/2-(8k + 1)a, c = V2(8k - 5)ir and d = V (8k - Since the approximations are excellent, even for k assmall as 4. the table permits all the zeros of the zero-order Kelvin functions to be estimated. 1,4_ciobi .y=5 .. y.4 550 551 THE KELVIN FUNCTIONSTable 55.7.1 xbei(x)xbei(x)xker(x)x'kei(x)55:8 01.000000000 0-0.78539822.84891803.7726742.3461471.718543 03.914668 06.038711-8.8640365.02622402.665845-0.07102369 4.9318110.011216077.23882908.280989-36.165406.127279 08.344225 010.51364 153.78189.455406 07.1721200.001956681 9.4040510.000357423311.67396 012.74215670.160210.56294 012.78256 014.96845-2968.68113.89349011.63219-6.705969 x 10'' 13.858271.280427 x 10-'16.11356 017.19343-13305.5215.00269 017.22314 0aI bI CI.7.idIV '32 +-+-0+ - + -0+---0-+ ---0 8aa'8bb'8c 8dd2 55:8 NUMERICAL VALUESThe algorithm below calculates values of the zero order Kelvin function for all arguments x ? 0. Its precision is24 bits (relative error less than 6 X 10-'). but some loss of significance may be encountered on computing devices that carry insufficient digits. The algorithm is based on expansions 55:6:13-55:6:18. Input x >>SetB=K=b=kj=r=0Ifx<8goto(I)-\(1\-n)41Set r =90Sets=-r-45Set t = l /VsGo to (4)(1) SetB=r=1Set s = 1099Ifx=0goto(2)Set s = 0.115931516 - ln(x)(2) Replace j by j + IReplace t by t(x/2j)2Replace r by r + (I /j)Replace B by B + t cos(90j)Replace K by K + rt cos(90j)Replace b by b + t sin(90j)Replace k by k + rt sin(90j)If j < Zx + 4 go to (2)Replace K by K + sB + irb/4Replace k by k + sb - 7rB/4Go to (5)Storage needed: B, K, b, k, j, r, x, s and r1 (3) Replace r by 8x 14j +j+lReplace j by j+\\IReplace r by r - 45Replace s by s + 135(4) Replace B by B + t cos (r)Replace b by b + t sin(r)Use degree mode or alter 90 to ir/2, 45 to 7r/4 and135 to 37r/4. Input restriction: x > 01 55:9 B - bcr(x)b = bci(x)K - ker(x)k - kci(x)THE KELVIN FUNCTIONS 552Replace K by K + r cos(s)Replace k by k + t sin(s)70Ifj<3+- go to (3)xSet t = (1 //v) exp(x/ / 2)Replace K by K/tReplace k by kitkReplace B by Bt - -itKReplace b by br + -ITTest values:(5) Output B ber(2) = 0.751734183G«« bei(2) = 0.972291627Output b ker(2) = -0.0416645140kei(2) = -0.202400068Output K ber(10) = 138.840467bei(10) = 56.3704592Output k ker(10) = 0.000129466329kei(10) = -0.000307524569 55:9 APPROXIMATIONSFor small arguments, the following approximations generally hold: 55:9:1ber,(x) _cos(3vrr/4) xsin(3va/4) x2(lYv + -1, -2, -3, ... r(1 + v)2)Yr(2 + v)2)sin(3va/4)cos(3va/4) (x 55:9:2bei,(x) =+v * -1, -2, -3, ... r(1 + v)(2x),r(2 + v)2) 55:9:3ker,(x) -r(v) cos2(3va/4)\x\`-r(v - 1) 2n(3vsr/4) (x)`-2v > 2 55:9:4Iaxeker,(x) =2 - 16: t 2r(v) cos(3va/4)(x2)r(-v) cos(va/4x ` 55:9:5ker,(x) =2+2(2)-2 < v < 2v + 0, ±1 55:9:6ker,(x):x= in(!) +!!+,Y-I]+V= t l2 55:9:7 ker(x) = I n- yv = 0\xr(-v) cos(va/4)r(-v - I) sin(va/4) 55:9:8kerx)2(x)2/ +22)rv < -2 55:9:9kei,(x)-r(v) sin(3va/4) r2_r(v - 1) cos(3va/4) (2)'-' > 2 2\x)y2x 55:9:10kei,(x) = x +4[1.(X2)- y + 2]v s ±2 553 THE KELVIN FUNCTIONS 55:10-f(v) sin(3vir/4) (X2)'f(-v) sin(va/4) x55:9:11kei,(z) =2\-2(2)-2 < v < 2v * 0, ±1+I 1+IT4-Zv-_1 55:9:12kei x)x=-_[1,x 55:9:13kei(x) 4 +4LlnI-I -y+-Iv=0r(- Y) sin(va/4) (xlf(-v - 1) cos(vir/4) xv < -2 55:9:14kei,(x) -22)+2(2)r2 and these may be supplemented by using equation 55:5:5 for cases of negative integer order. Generally, equations55:9:1-55:9:14 give the first two terms in expansions of the Kelvin functions in ascending powers of the argumentx. For certain special values of v, however, some of these terms may vanish. The leading terms in equations 55:9:1.55:9:3 and 55:9:5, for example, become zero when the order adopts one of the values 3, 2, IF, i, 6, ....For large arguments, the approximations 1xxvaa+- 55:9:15ber,(x) =tax2expcoal8 %JV tI/x/ax+va-i) 55:9:16bei,(x) -tax2gnl -Lexpl - sV-2 55:9:17(x) =kerex( xcos(x+va+ap)"Zx 28 55:9:18kei,(x)expl)sinl++VVVV2x\V-2`,r228large x large x large x large x hold. From these, one may derive the order-independent relationships 55:9:19 [berr.(x) + bei,(x))[ker;(x) + kei(x)J = 4xl=large x and 55:9:20 ber,(x) ker"(x) + bci,(x) kci,(x) = ber,(x) kei,(x) - bei"(x) ker"(x) = large xx87 55:10 OPERATIONS OF THE CALCULUSThe differentiation formulas l[fer 55:10:1fer,(x)[fer.,(x) + fei,.,() - fer,,(x) fei"_,(x)] I= her or ker d 155:10:2- fei"(x) = - fer,.,(x) - fei"_,(x) + fer,_,(x)J fei = bei or kei dxVemay be combined with recursions 55:5:7 and 55:5:8 to produce a number of alternative formulations.Among indefinite integrals are-x'55:10:3 ti-" fer,(t)dt =;[fcr,_i(x) - fei,_,(x))0 55:11 THE KELVIN FUNCTIONSwhile the following definite integrals establish links with the functions of Chapters 32 and 38: 55:10:4Jaexp(-t/x) ber(2 V r)dt = x cos(x)0 55:10:5 exp(-t/x) bci(2V )dt = x sin(x)('/(zrr 55:10:6exp(-i/x) ker(2\t)dt ={cos(x) Ci(x) + sin(x)2lLSi(x) -2] j 111 55:10:7exp(-t/x) kei(2Vt)dr = Z { sin(z) Ci(x) - cos(x)1 Si(x) -7 -1}0 2L554 Other indefinite integrals are given by Abramowitz and Stegun [Section 9.9] and other definite integrals by Grad-shteyn and Ryzhik (Section 6.87). 55:11 COMPLEX ARGUMENTKelvin functions are seldom encountered with complex arguments, and this Arias does not address this circumstance. 55:12 GENERALIZATIONSIf the complex variable z = x + iy is represented by the polar equivalent z = r exp(i(l), one sees from definitions55:3:3 and 55:3:4 that ber,(r) and bei,(r) are the real and imaginary parts of J,.(z) with 0 = 3a/4. Thus, one maygeneralize these Kelvin functions by allowing 0 to adopt an arbitrary angular value. 55:13 COGNATE FUNCTIONSJust as the her. and bei. functions are defined in terms of the Bessel J, function to satisfy 55:13:1 ber,(x) + i bei,(x) = J,x + so Kelvin functions of the third kind are defined by a similar relationship but with the Hankel function [Section54:13] replacing the Bessel function. Thus: One has the identities her,(x) _ (2/a) kei,(x) and hei,(x) = (-2/n) ker,(x). CHAPTER56THE AIRY FUNCTIONS Ai(x) AND Bi(x) The Airy functions Ai(x) and Bi(x) are related to Bessel functions of order 3 and -3, with resealed arguments. Thetwo so-called auxiliary Airyfunctions are also important and are briefly discussed in this chapter. 56:1 NOTATIONIn formulas involving the Airy functions, the auxiliary argument 56:1:1 X = 3 (IxI)`/'is frequently more convenient than the usual argument x. Note that dX = tix I dx.There appears to be no standard notation for the auxiliary Airy functions. This Atlas employs fai(x) and gai(x)for the functions that Abramowitz and Stegun [Section 10.41 denote by f(x) and g(x). 56:2 BEHAVIORThe functions Ai(x) and Bi(x) exist for all real arguments, but their behaviors depend crucially on the sign of x,as illustrated in Figure 56-I.For x z 0, both Airy functions are positive, but whereas Bi(x) increases rapidly as x --> x, Ai(x) steadily decaystowards zero. For negative arguments, Ai(x) and Bi(x) are oscillatory functions with oscillations whose frequenciesgradually increase and whose amplitudes gradually decrease as x -+ -x.The auxiliary Airy functions are mapped in Figure 56-2. For positive arguments, both increase exponentiallyas x -> x, while for negative arguments each function exhibits oscillations similar to those of the Airy functionsthemselves. 56:3 DEFINITIONSThe Airy integral 56:3:1Ai(x)-J3)dto` 56:3 THE AIRY FUNCTIONS Ai(x) AND Bi(x) 556iAi0.I,i',.1.1444444444444IF44it44~ FIG 56-1 : : :............:..............:......... :....:.... .......0.6At (x)...................... ...:....: .....0.4Ell W : ..............:....:.. p ...............:.:....:......-0.2 ..:Bi (x):....:....:....:....:....:.. ...:....:......-0.4 defines the Ai function for all arguments, while the similar definite integral sum 56:3:2(' 3Bi(x) _ ! Jexpl xt - 3)d,+if ssin(xt + 3 Idr serves the same purpose for Bi.Using the 56:1:1 definition of an auxiliary argument, the two Airy functions may be defined in terms of hy-perbolic Bessel functions [Chapter 501, or the Basset function [Chapter 51], for positive arguments. The Besselfunctions [Chapter 531, or Neumann functions [Chapter 541, provide the corresponding definitions when the ar-gument is negative: 56:3:3Ai(x) _ 56:3:4Bi(x) =1.3[I-it3(X) - 11113M) =a3K,n(X)x> 0-xx3[1-1p(X) + 1.,3(x) = 3[Y-113(X) - Y.13(X)1 13 [I-1n(X) + 1113(X)1x > 0 P-1/3M - 1113(X )) = - 31Y_ '/)(X) + Yu3(X)1x < 0X < 0 A linear combination of Ai(x) and Bi(x) functions satisfies the Airy differential equation SS7 THE AIRY FUNCTIONS Ai(x) AND Bi(x) 56:3:5 drfdx' =xff = c,Ai(x) + c_Bi(x) where c, and cr are arbitrary constants.The auxiliary Airy functions are defined by 32/3r(23) [Bi(x)56:3:6fai(x) =+ Ai(x)2 3113r(i) [Bi(x)2- Ai(x)_I _,/3(X)x > 0 31/3J_,n(X)x < 0 x>0 3zpJ,/3(X)x < 0X = 1301)",36:3 where r denotes the gamma function [Chapter 431. In the notation of Section 43:14. these auxiliary functions may 56:4 THE AIRY FUNCTIONS Ai(x) AND Bi(x) 558 be synthesized from the zero-order hyperbolic Bessel function by the operations t56:3:8 I°I ?x312)fai(x)(9X)"3 = x -- 00 56:3:9 10(2rgai(x)(9X)'13 = x ? 0310xSyntheses similar to 56:3:8 and 56:3:9, but involving the Bessel J0 coefficient instead of its hyperbolic counterpart.provide definitions of fai(x) and gai(x) for negative arguments. 56:4 SPECIAL CASESThere are none. 56:5 INTRARELATIONSHIPSThe formulas 56:5:1 andAi(x) = Ai(0) fai(x) -gai(x)2irBi(0) 56:5:2 Bi(x) = Bi(0) fai(x) +gai(x)2iAi(0)relate Airy functions to their auxiliary counterparts. See Section 56:7 for values of Ai(0) and Bi(0). 56:6 EXPANSIONSThe auxiliary Airy functions are expansible as the convergent series 56:6:1x'x6x9fai(x)=1+-+(1x4)-+(1x4x7)-+ (3j + lf!x3i3!6! 9!i=o (3j + I)!X4x7x10(3j + 2)!!!3i+ 56:6:2 x 4!7!10!r0 (3j + 2)!in which the triple factorial [see Section 2:13] occurs. Notice that fai and gai are two members of a trio of functions,the third of which 56:6:3x2x!x6x11 (3J)1tlxrj-2(3j + 3)!!! x3i+22!5!8! 11!i_o (3j + 2)!i=o (3j + 3)! is encountered in Section 56:13.Expansions 56:6:1 and 56:6:2 may be used in conjunction with relationships 56:5:1 and 56:5:2 to generatepower series expansions of the Airy functions Ai and Bi. Terms involving x'', xs, xr, ... are absent from such expansions.Asymptotic expansions of the Airy functions take different forms according to whether the large argument ispositive or negative: 559 THE AIRY FUNCTIONS Ai(x) AND Bi(x) 56:6:4Ai(x) - 56:6:5Bi(x)exp(-X)aia2as2 aXX2X'XisinlX+ 41costX+- 'a2aq\f a,a,a X2X+-..LX XsXnla0+ +ss exp(X)a,a2a,a,VxXX''X'X'a a cosX +-sin X +'--.LX'X[XXX s - Vwapa + a`+7t V/a,a3 + a-56:8 X--* -x x-s -x where the auxiliary argument X is given by 56:1:1, a0 = 1, and other coefficients obey the definition and recursion 56:6:6a,=(6j-1)!!!1+5a-1j=1,2.3,... j!(23 - 1)!'(216)' 36j2An asymptotic expansion, namely 56:6:7Ai2(x) + BOW ) -11 +5} +1155W s++(6j - 1)!!+x -. -xa32x204gx j!(96x')'describes the sum of the squares of the Airy functions as the argument acquires very negative values. 56:7 PARTICULAR VALUES Ai(x) Bi(r) fai(s)gai(x)x=-z x=0 x =. 0T(i)= 0.3550280539 0 3''r(i)2a3,n r31n1'd)0 61492662760--0.T(i)2m3u 0 0x Table 56.7.1 records the first few zeros and extrema of Ai(x) and Bi(x), all of which occur at negative argu-ments. The final tabular entries, which use the abbreviations a = [31r(4k - I )J21' and b = (31r(4k - 3)1"', provideexcellent approximations for the k" zeros of the Ai and Bi functions, for k > 5. 56:8 NUMERICAL VALUESThe algorithm below calculates Ai(x) and Bi(x) with 24-bit precision, that is, the relative error does not exceed 6X 10', except when the value of the Airy function is close to zero. For arguments in the range -5 K x s 5, thealgorithm uses expansions 56:6:1 and 56:6:2, together with relationships 56:5:1 and 56:5:2. When the argumentexceeds 5 in absolute value, expansions 56:6:4 and 56:6:5 are utilized. When x < -5, these expansions are rewritten as 56:8 THE AIRY FUNCTIONS Ai(x) AND Bi(x)Table 56.7.1 Bi(x) -1.018793 0.5356567-1.173712 0-2.338107 0-2.294440-0.4549444-3.248198-0.4190155-3.2710930-4.087949 0-4.0731550.3965228-4.820099 0.3804065-4.830738 0-5.520560 0-5.512396-0.3679692-6.163307-0.3579079-6.169852 0-6.786708 0-6.7812940.3494992 56:8:1 Input x >> (4) Replace B by Bpl j - I +1 I /2X\\\7.2j/Replace A by-Ap(j- 1 +)/2X7.2jReplace t by t + AReplace a by u + BReplace p by pqReplace j by j + Ia543a0bS43b'0560 Ai(x)cos(X) a,a2a3a4Bi(x)2aMsoXX2±X3+X4 sin(X)aa2a3a4 +fmap + - - - - - +2;%I--xXX2X3X'Setp=q=j=If5<[xlgoto(2)Set t = .614926628Set u = x1vr3/2atSetA=t - uSetB=t+uSet X = x3/3(1) Replace t by tX/j(3j - 1)Replace u by uX/j(3j + 1)Replace A by A + t - uReplace B by B + t + uReplace j by j + IIfj<2+3W go to (I)Replace A by A/fGo to (6)(2) If x > O go to (3)Setq= -IReplace x by - x(3)SetA=t=B=u=1/ 2nfSet X = 2x3J2/3Storage needed: p, q, j, x. t. u, A. B and X Use radian mode. Input restrictions: There are no restrictionson the magnitude of x.I 561 THE AIRY FUNCTIONS Ai(x) AND Bi(x) 56:10 90If j < 3 + X go to (4)If q < 0 go to (5)Set B = of exp(X)Set A = t/[f exp(X))Test values:Ai(4) = 9.516074 x 10-'Go to (6) Bi(4) = 83.8470714(5) Set B = u cos(X) - t sin(X) Ai(-4) _ - 0.0702655329Set A = t cos(X) + u sin(X) Bi(-4) = 0.392234706(6) Output A Ai(6) = 9.94769437 x I0-6A = Ai(x) Bi(6) = 6536.44608Output B Ai(-6) = -0.3 29 1 45 1 74B - Bi(x) Bi(-6) = -0.146698376 The universal hypergeometric algorithm [Section 18:14) permits values of fai(x) and gai(x) to be determined. 56:9 APPROXIMATIONSFor small arguments, the approximations 56:9:1 Ai(x) - 0.355 - 0.259x + 0.059x3 - 0.022x' small56:9:2 Bi(x) - 0.615 + 0.448x + 0.102x3 + 0.037x' small 1XIhold, while for large positive argumentsexp(-2x312/3) 56:9:3 Ai(x) =large positive x2 n 56:9:4 Bi(x) _exp(2x"'/3)large positive x-V.N77so that the product Ai(x) Bi(x) = I/2a\/x. For large negative argument 56:9:5 56:9:6cos(X) + sin(K)2(-x)112 Ai(x) =X =large negative x3 cos(X) - sin(X)Bi(x) yX =3large negative x;V --xso that in this range Ai2(x) + Bi'(x) = 1/itV . a result that is also a consequence of expansion 56:6:7. 56:10 OPERATIONS OF THE CALCULUSDerivatives of the Airy functions involve Bessel and related functions of order 2: Xxd-[I-xn(X) - 1212(X)] _-c113 dxK213(X)x > 056:10:1Ai(x) =x3 [J-213(X) - J213(X))X < 0 56:11 THE AIRY FUNCTIONS Ai(x) AND Bi(x) 562x[1-2n(X) + 1213(X)]x>0dV3 56:10:2dxx[J-,n(X) +J2/3(X)]x < 0 The two derivatives are interrelated by 56:10:3 ddAi(x)dxBi(x) - Bi(x)dxAi(x) _itirrespective of sign of the argument. Second derivatives obey the simple relationshipd2 56:10:4f= xff = Ai(x) or Bi(x)dx'-Indefinite integrals of the Airy functions are expressible in terms of their derivatives and that of the functionHi(x), discussed in Section 56:13: 56:10:5J=Ai(t)dr =3+ JjAi(t)dt = I - JAi(r)dt =dx Hi(x)- Hi(x)dxAi(x)J 56:10:6J=Bi(t)dt = JBi(t)di ='rr1 Bi(x)dHi(x) - Hi(x) d Bi(x)JILdxdxThese relationships are valid for either sign of the argument x. Three useful definite integrals result from setting x= 0 in 56:10:5 and 56:10:6. 56:11 COMPLEX ARGUMENTBy using definitions 56:3:3 and 56:3:4, the information in Sections 50:11 and 53:11 may be applied to Airy func-tions. 56:12 GENERALIZATIONSInasmuch as Airy functions are Bessel functions of order ±3, they may be generalized to other fractional orders. 56:13 COGNATE FUNCTIONSThe two components of the integral definition, 56:3:2, of Bi(x) are sometimes regarded as distinct functions anddenoted as follows: 56:13:1Hi(x) =1 oexp (;)r6)(j - 2)!!!2Bi(0)ITxr -3dt =3201x1 =3[fai(.r) +gai(x) +hai(r)] 1(/t'\Bi(0) [fai(x)27\156:13:2Gi(x) =itf sinl xr + 3 Idt = - +2irBi2(0) -2 I gai(x) - 2haix)JOf course, Hi(x) + Gi(x) = Bi(x)./\ CHAPTER57THE STRUVE FUNCTION The Struve function h,(x) has many analogies with the Bessel function of Chapter 53. Likewise, the hyperbolicStruve function Q x), which is discussed briefly in Section 57:13. is analogous to the hyperbolic Bessel functionof Chapter 50. 57:1 NOTATIONThe usual notation for the Struve function is H,(x) with the capital H printed in bold type to avoid confusion withHermite polynomials (Chapter 241. However, because they are difficult to reproduce by hand or on an officemachine, this Atlas avoids abnormal fonts. Thus, we adopt h,(x) for the Struve function and e,(x), instead of themore usual boldface L,(x), for its hyperbolic counterpart.We use h,(x) to denote the Struve function of argument .r and unrestricted order v, but h (x) is employed whenthe order is constrained to be an integer. 57:2 BEHAVIOR Like the Bessel function, the Struve function is real for negative argument only when its order is an integer.Accordingly. the contour map Figure 57-1 is confined to x ? 0.Also like the Bessel function, h,(x) is an oscillatory function of its argument, the amplitude of the oscillationsdecreasing with increasing x. Unlike the Bessel function, however, the damped oscillations of the Struve functiondo not generally occur symmetrically about x = 0. For v > , this asymmetry is so pronounced that h,(x) is uniformlypositive when its argument is positive. 57:3 DEFINITIONS The definite integral 57:3:1h,(x) =2(-r/2)"(t -t1),- insin(.rr)dtv > -2 V;-r(-V+ -21)02 $63 57:3 THE STRUVE FUNCTION S64 545 THE STRUVE FUNCTION 57:5 or its equivalent on replacement of t by sin(O) or cos(6), defines the Struve function for orders exceeding - ;.Another integral 57:3:2h"(x) - Y"(x) (I + 12).- 1/2 exp(-xt)dtv > - 7. r(v+A)l0 2provides a representation of the difference between the Struve and Neumann functions of common order and ar-gument. This time tan(O) or sinh(u) may replace t, but the restriction to orders in excess of -; again applies.Using the notation of Section 43:14. the Struve function may be synthesized by 57:3:3-' -V - 14 F(12 + v)-x2J--o(-r) h"(.r)X 002(.r/2)' " 4from the zero order Bessel coefficient. This, then, constitutes yet another way of defining h"(x). 57:4 SPECIAL CASESWhen its order is half a negative odd integer, the Struve function becomes a spherical Bessel function [see Sections32:13, 53:4 and 54:41:Zc2x57:4:1 y_"(x) = (-1)"- j"(x)n = 0, 1, 2....TrrrReduction to elementary functions also occurs for orders equal to a moiety of a positive odd integer, thus: 57:4:2 h11,(x)(I - cos(x))TX2I - cos(x) 57:4:3 h)/,(x) =-- sin(x) +x-axx2and others may be derived from the recursion(x/2)" 57:4:4h"_,,,(x) =in - Ih"_,,,(x) - 1V1n = 0, 1, 2.....r Trxn!In some respects the Struve function is also "special" when its order is an integer. Because of the preeminentinterest in these cases, they are nevertheless treated in the main sections of this chapter. 57:5 INTRARELATIONSHIPSThe recurrence relationship 57:5:1 h" ,(X) _(.T/2)"+2vh.(x) - h"_,(x)Xis obeyed for all orders, but it takes the simpler forms 57:5:22t° h".,(x) + h,-,(x) _2n- h"(x)a(2n + 1)!'x2(-2n - 3)!n = -1, 0, 1. 2,n=-1,-2.-3,. when the order of the Struve function is an integer. The double factorial symbol, which occurs frequently in thischapter, is explained in Section 2:13. 57:6 THE STRUVE FUNCTION S66Reflection formulas exist only for integer orders. For argument-reflection, we have57:5:3 h"(-x) = -(-1)" h"(x)n = 0, t 1, t2, ...so that the Struve function of even order is odd, and vice versa. Order-reflection introduces an n-term polynomialfunction 57:5:41)" h_"(x) = h"(x) - -2x"-'+1!!x"-'+3!!x"-6++(2n - 3)!1xv((2n- 1)!!(2n - 3)!!(2n - 5)"1'!n= 1,2,3,.so that h_r(x) = (2/1r) - hi(x), h_2(x) = h2(x) - (2/ax) - (2x/3tr), etc. 57:6 EXPANSIONSFor all finite values of v, the expansionfx'(-x2/4Y_2(x/2)""(-x2/4)' 57:6:1h,(x)=I-i2 3 +( ', (2 +v) ,=ar(i+;)r(j+v+f r(2v) j-o,converges, provided x is real and positive; negative arguments are admissible if the order is an integer. The mostimportant instances correspond to v = - 1, 0 and 1, and are 57:6:2h_i(x) =2Ix2+.e-x6++(-x2),- ---- Tr3451575(2j - 1)!!(2j + I)!!2x'x`x'x(-x2)'57:6:3 x -+-+++l n922511025[(2j + 1)"]2-x"++(-x2)'" +X6 57:6:4hi(x) =2x2-X4- ---+345157599225(2j + 1)!! (2j + 3)10Integer order Struve functions are expansible in terms of Bessel coefficients (Chapter 52]; for example: 57:6:5(x)_4(Jx) +J=,(x)+ -Js(x)+) =4J,,, i(x)tr35rr ,_0 2j + 14 (1 - Jo(x)J,(-t)JI(x)J6(x)J'' (x)57:6:6 he(x) _+ = ++++ it2315354j2 - IAn asymptotic expansion exists for the difference between Struve and Neumann functions of common argumentand order:(x/2)"-'(1 - 2v3!!(l - 2v)(3 - 2v)(2j - 1)!! (2' - v),/57:6:7h,(x) -1 -'+x-+2V >r r( + v)x (-x /2))+x- xSee Section 54:6 for an asymptotic expansion of Y,(x) that may be used in conjunction with the 57:6:7 series. Theimportant cases of orders -1. 0 and I are 213451575(2j + 1)!!(2j - 1)" 57:6:8h_dx)-.(x)I _X,+s _X6+XB _...+ (-x2))-'+...r-,x2(1-19225[(2j - I)"]21 57:6:9h0(4-Y6(x)---+;;- Jx-,x rrx.exrx(-x2)l 567 THE STRUVE FUNCTION 57:8 and2 r 1345(2j - 1)!!(2j - 3)!! 57:6:10h,(x)-Y,(x)-- 1+-- Jx nx'x'.r°(-x2) 57:7 PARTICULAR VALUES x= -xx =0x=xh,(x)n - 3. 5, 7.... 0h,(x)n = 2, 4. 6,... 0h,(x)I <v*2,3,4,... undef0h,(x) 2/v02/vh.(r)I > v > - I: v0 undef00h.(x) 000h-.(x) 02/v0-3h,(x)-1 > v > - undefx0 2-3 -5 -7h.(x)222 .... undef00 -3 -5 -7 h.(x)->v>-2,-2>v> -, - >v>-4. undef-x0 h,(x) 0±x0-5 -7 -9 h,(x)->v>-3.-3>v--,->v>-5,.. undefx02 22h,(x) 0x0 Apart from its x = 0 value, the Struve function has no zeros for v > ;. For v ~= 1, however, there is an infiniteset of zeros that, for large arguments, correspond to the roots of the equation/\r'257:7:1sinlr - 2 +41=(I'(v)+Jh,(r)=0largerv<-2 and therefore ultimately occur close to r - ir(2k + v + #)/2 where k is a large positive integer. 57:8 NUMERICAL VALUESThe algorithm below is designed to calculate values of the Struve function in the approximate ranges - 15 5 v s15, 0 < x -- 30 to a 24-bit precision (i.e., the fractional error does not exceed 6 X 10-1) except in the immediatevicinity of the function's zeros. Input v >>Input x >>[or -x)Storage needed: v, a, x. J, t, p, f andj set J = 1xI1.5 - 40+ 5 - 3Replacex by(-x/2)(/Ix/21)Set t = 1/ax 57:9 THE STRUVE FUNCTION 568 f = h,(x) <[or f,(x)](1) Replacet by tx/aReplace a by a + 1Setp=a+vIf p +0 go to (1)(2) Replace t by tpReplace p by p + IIf p < 3 go to (2)/`\Set f= 1+13p,- 1I/30p' \\\\\\1/ /v + Replace f byf1-2p+ "p[I -- In(p)] +IZ ln(xj)Replace r by t exp(f) V pl2Setj=f=0(3) Replace f by f + IReplace t by tx/[(a + j)(a + j + v)]Replace j by j + IIfj<Jgoto(3)Output fInput restrictions: x > 0If -x, a negative number, is input,the algorithm computes f,(x) insteadof h,(x). Test values:h,(5) = 0.807811945h_5n(tr) =0.42986937805) = 23.7282158f_sj2(a) = 1.79596683 The algorithm computes h,(x) as the sum to + ti + t_ + - - - + t; +-+ p where to, equal to 2(vj/2)'+'/I'(2 + v). is evaluated by a modified version of the algorithm of Section 43:8, t, is calculated as (-x2/4)tj_,/(, + j)(_s + j + v) and J is given by an empirical function of x and v designed to ensure that the series hasconverged adequately. Special measures are taken if v has one of the values -. -;, -;..... This algorithm willalso compute the hyperbolic Struve function f,(x), discussed in Section 57:13, when -x is input in place of x. 57:9 APPROXIMATIONSExpansions 57:6:1-57:6:4 can readily provide approximations to the Struve function that are valid for small ar-guments. For large arguments, one may similarly use the asymptotic expressions 57:6:7 and 54:6:6. For the casesof orders equal to - 1. 0 or 1, theVWX leading asymptotic terms are/57:9:1h_,(x)=sin(.,,lcosx+2largex32rx'4///Irx h_2i-2- - + - -1- -(-I 57:9:2(x)u-snaxJx4arxJcos x32ax'4ar e xg 57:9:3h0) =2- +3sr)sin x - - +3-)cos z - -large x'Tiax432nx'4 57:10 OPERATIONS OF THE CALCULUSThe formula 57:10:1 - h,.(x) +dr2 V . 2for the derivative of the Struve function simplifies to dh.(x)/dx = h_ (x) for zero order. One also has 569 THE STRUVE FUNCTION 57:12 57:10:2 ddz(x'h.(x)) = xh.-.(x)The indefinite integral of t"h,(t) is expressible in closed form only when 57:10:3 57:10:4 57:10:5J t"`h.(t)dt = x"h,..(x)0µ = f t'+%(:t = -x"h. .(x)v < -12 J,tl-'h.(t)dt =2x 02" '+ v)± v:I but the corresponding definite integral-2"rvZ+µ2+2vaµn 1 57:10:6t"h,(t)dt=cot(-+-lµ<--2<v+µ<0o((vIl\22/2\222/holds for a small range of µ values. Selected indefinite and definite integrals are 57:10:7- Jjh1(t)dt = Jxh_.(t)dt = h0(x)ITo(a 57:10:8f1h0(t)di = x ? - h_i(x))= xh1(x)oa ho 57:10:9 (t)1dt=-0t2With a redefinition of the argument, equations 57:10:2 and 57:10:3 are the special µ = 1 and µof the general differintegration formula'/-l 57:10:10- {x` 'h.(2Vx)} = x'"'12h._"(2Vx)v > - - dx" 2in which .s may adopt any value. 57:11 COMPLEX ARGUMENT We present here only the case of the Struve function with a \purely imaginary argument: 57:11:1h.(iy) = i'-`e.( v) _ 1sinl 2I+ i cosl Z)JE.(v)Here (, denotes the hyperbolic Struve function discussed in /Section 577:13. 57:12 GENERALIZATIONS-1 cases The function introduced in Section 53:12 is a generalization of the Struve function. 57:13 THE STRUVE FUNCTION 57057:13 COGNATE FUNCTIONSThe hyperbolic Struve function is to the Struve function as the hyperbolic Bessel function I,(x) is to the Besselfunction J,(x). It may be defined by the definite integral 57:13:1e"(x) =2(x/2)"I(1 -t=)"-'l: sinh(xt)dtv > -2 V r(, + v)oor by its equivalent on replacement of t by sin(0) or cos(0).A similar definite integral represents the difference between the hyperbolic Struve and the hyperbolic Besselfunction of identical argument but order of opposite sign 57:13:2e"(x) - I_"(x) =-2(x/2)"r (I+sin(xt)dtv <IV it r(# + v)O 2Replacement of t by tan(0) or sinh(u) leads to equivalent definitions.The hyperbolic Struve function is defined (as a real function) for negative argument only if v is an integer: thefunctionis odd if n is even and vice versa. Figure 57-2 shows maps of (,(x) for n = 0, 1, 2, .... Thecorresponding hyperbolic Struve functions of negative integer order are related to those graphed by (- l)"(2n - 3)!!x'-" 57:13:3e_,(x) = f"(x) +2 rx°-'-1!!x"-'3!!x"-5- I\+it(2n - 1)!!(2n - 3)!!(2n - 5)!! 111n = 1,2,3,...for example, (_1(x) = e,(x) + (2/ir) and a-.(x) = fi(x) + (2x/31r) - (2/,rrx). 571 THE STRUVE FUNCTION 57:13For orders of -!, -;, ..., we have the identity where the latter functions are discussedin Sections 50:4 and 28:13.The expansion of e,(x) mirrors that given for h,(x) in 57:6:1 except that the negative signs should be expunged.Similarly, if the alternating signs on the right-hand side of 57:6:7 are replaced by uniformly positive signs, theseries is an asymptotic representation if I_,(x) - e,(x). Again, it is only by distinctions in sign that the recursionformula 57:13:4 e,+i(x) = e,_i(x) -2ve,(x) -(x/2)'xV++r(v+i)for the hyperbolic Struve function differs from 57:3:1 for h,-,(x).Because the Struve function and the hyperbolic Struve function are generally complex for negative argument,the algorithm in this chapter will accept only positive arguments. Accordingly, the input of a negative x into theSection 57:8 algorithm is interpreted as an instruction to compute t,(x), instead of the h,(x) that is calculated whenthe input x is a positive number.The differentiation formulade,-,(x) + f,.,(x)+(x/2)' 57:13:5-e,(x) =22Vr(v+ )produces d to(x)/dx = e_i(x) = ei(x) + (2/n) as a special case, and, apart from a sign, is analogous to the firstequation in Section 57:10. Similarly, allowing for changes of sign, all the other relationships in that section havetheir analogs for the hyperbolic Struve function. CHAPTER58THE INCOMPLETE BETA FUNCTION B(v;µ;x) Many simpler functions are special cases of the incomplete beta function. For example, as described in Section58:14, the indefinite integral of any trigonometric or hyperbolic function, raised to an arbitrary power, is expressibleas an incomplete beta function. 58:1 NOTATION Alternative symbolisms for B(v;µ;x) are B,(v,µ) and the product 1,(v,µ) B(v,p) where B(v. s) is the complete betafunction discussed in Section 43:13. The 1,(v,µ) function, which has statistical applications [see Section 27:141, isthe incomplete beta function ratio, equal to the quotientB(v;is,x)_r(v + µ) B(v;µ;x)58:1:1 I,(v.µ) =B(v,µ)r(v) r(µ)of the incomplete beta function by the complete beta function.The adjective incomplete reflects the fact that the upper limit of Eu(er's integral of the firs: kind [see definition58:3:1) is generally less than the value of unity that is required to "complete," that is, symmetrize, the integral.This incompleteness prevents the interchangeability of the v and µ parameters. 58:2 BEHAVIOR Being trivariate, the incomplete beta function has a behavior sufficiently complicated that it cannot be adequatelydescribed either in words or planar maps.While allowing µ to adopt any real value, we shall require v to be positive. For otherwise unrestricted valuesof the v and µ parameters, B(v;µ x) is defined only for arguments in the range 0 s x < 1. However, if v is apositive integer, the formula 58:2:1 B(n; xµ;-x)=(-1)"B n;l-µ-n:l+xs>0n=1,2,3,...permits extension of the argument to general negative values. Similarly, extension to arguments exceeding unity 573 58:3 THE INCOMPLETE BETA FUNCTION B(v; s;x)is possible if the µ parameter is a positive integer: 58:2:2B(v;m;x+1)=(m-1) -(-1),B(m;l-v-m;l+x x?0m=1,2,3...(v),,,Here (v),.. is a Pochhammer polynomial [Chapter 181. 58:3 DEFINITIONS The incomplete beta function is defined by the indefinite integrals 58:3:1 B(v;µ;x) = Jst"-'(1 - t)"-'dt0: x< 10t'dt 58:3:2B(v:µ;x) =0 s T = X < T58:3:3 B(v;µ;x) = 2 sin'-`-'(t) cost"-1(t)dt 0 s T = arcsin (V X) < 20574 andrr'58:3:4B(v;A;x) =2Jtanh2'(r) sech"(t)di0T =artanh(V x) <0As well, the incomplete beta function may be defined as a number of definite integrals, the simplest being 58:3:5 B(v;µ;x) = x"Jt"-'(l - xt)"-'dt0Using the notation of Section 43:14, the incomplete beta function may be synthesized from a reciprocal linearfunction:I-vvB(v;µ;x) 58:3:6 x=X0!5 x<l1-x 1-v-µx°(I-x)" 58:4 SPECIAL CASESWhen the v parameter equals unity: 58:4:1 B(1:µ;x) _(IFtBy sufficient applications of the recurrence formula 58:5:2, this case can also provide expressions for B(2;µ;x),B(3;µ;x). etc. Formula 58:4:1 becomes indeterminate if µ is zero, but, by considering the p. -* 0 limit, one con-cludes that B(1;0;x) = -ln(1 - x).When the µ parameter equals unity:58:4:2 B(v;l;x) = -XVThis formula may be used in conjunction with recursion 58:5:3 to produce the polynomial expression 58:4:3B(v;m;x)=x" m-1 O m= 1,2,3....Jj + vfor the incomplete beta function having a positive integer as its µ parameter. When µ is zero, one has 575 THE INCOMPLETE BETA FUNCTION B(v;µ;x)/1-1xf 58:4:4 B(v;0; x)In,1= x"$(x; I ;V),-0j+v58:5 where In, is the generalized logarithm discussed in Section 25:12 and denotes Lerch's function [see Section64:12J. Some important instances of this result arex2x'X.x'"58:4:5B(n + 1;0;x) = -In(1 - x) - x - - - - - - - = - n = 0, 1, 2,...+1 23 58:4:6 and58:4:7u2X%-ij+inJ=.xs rV.r_B(n + =;0;x) = 2l{artanh(V x) - V x - - - - -352n - In=0,1,2,... B(; -- 14;0;x) = 2 artanh(jx"j) + 2 arctan(Ix'''1)Negative integer µ cases can be deduced from 58:4:4 by one or more applications of the recurrence (µ - 1) B(v;µ- 1;.r) = (v - 1) B(v - 1;µ:x) - x'-'(1 - x)"-', which is a restatement of 58:5:2.Whenever v and µ are both positive multiples of ;, the incomplete beta function reduces to an elementaryfunction. There are four general cases. When both v and µ are even multiples of i (i.e., both are integers), thegeneral formula is` 58:4:8B(n + lm + Ix)1"(1 - t)dt =.r-'(-x)iIn, m = 0, 1, 2, ...oto\m1n+j+IWhen v is an odd but µ an even multiple of z, the general formula isVi58:4:9B(n + ;;m + Ix) = 2t2'(1 - t)"dt = x""/2(-x)/n, m = 0, 1, 2, ...0).o(-j)n+j+4When v is an even but p. an odd multiple of s, the general formula is 'n\ 1-(1-x)'" '112 58:4:10 In,m=0,1,2.... vi_.-0jm + j + 2fThe final case, when both v and µ are odd multiples of ;, is more complicated and invariably leads to anarcsin(V) term. The general formula can be obtained by combining('r/IT58:4:11B(n + l;m +Z;x) = 2 Jsine"(t) cos2T(t)dt = 2 i (-1), I -j) sin'"'=1(t)dtn, m = 0. 1. 2...oi-o\where T = aresin(\), with the indefinite integral 32:10:6. The simplest instance is B(# l;x) = 2 arcsin(V), andother simple cases can be obtained from this result and recursion formulas 58:5:2 and 58:5:3.When the two parameters differ only in sign, there is the simple result 58:4:12 B(v;- v;x) _ - I x,v \I -xbut there is no comparable general formula for B(v;v:,r).Many other special cases, while not evaluable as established functions, can be expressed as simple indefiniteintegrals. Some of these are cited in Section 58:14.58:5 INTRARELATIONSHIPSThe formular(µ)r(v)58:5:1 B(A;v;x) =r(v + µ) -B(v;µ;I - x) 58:6 THE INCOMPLETE BETA FUNCTION B(v;µ;x)shows the effect of interchanging the parameters. As well, it provides an argument-reflection formula.The recursion formulas 58:5:2576 rB(v + 1;µ;x) B(v;µ + );x) - x (I - x)µ 4tx`(1x)58:5:3 B(v;µ + l:x) = B(v + 1;µ:x) +Vvlink two incomplete beta functions. There is a large number of interrelationships connecting three incomplete betafunctions, of which58:5:4 B(v;µ;x) = B(v + I;µ:x) + B(v;µ + Ix)is the simplest. 58:6 EXPANSIONSThe important power series 58:6:1B(v;µ;x)=x(1-x)"1+ v+µ x+(v+1+)(v+µ+1)x2+._. =(1-x) (v+µxf+. Yv(v + 1)v(v + l)(v + 2) v(1 + v);converges for 0 s x < I. When x is close to unity. convergence may be slow and the alternative seriesr(µ) 1"(Y)x"(v + µ)58:6:2 B(v;µ;x) =f(µ + v)-µ(1 +may then be preferable. 58:7 PARTICULAR VALUESGenerally, we have B(v;µ;x)X = 0x = 1 0r(v) r(or(v+µ) but for certain interrelationships between the v and µ parameters, the incomplete beta function of moiety argumentalso acquires particular values. For example: 58:7:1 58:7:2 and58:7:3B(v;v;#)r2(v)V; no2r(2v)4°r(v + ;)B(v;-v;z) =1V B(v;l - v) _ ;G(v)where G is Bateman's function described in Section 44:13. The special cases B(;;;;,) = IT/U, B(!;..!) = a/2 andB(4'; ::) = U of equation 58:7:1 should be noted, U being the ubiquitous constant cited in Section 1:7. 577 THE INCOMPLETE BETA FUNCTION B(v;µ;x)58:8 NUMERICAL VALUES58:8 The algorithm below accepts any positive v value and any argument in the range 0 <- x < 1. The µ parameter isunrestricted except that if µ adopts any one of the values -I, -2, -3, ..., then v + p. must be positive.If x < 0.7, the algorithm uses expansion 58:6:1 in the form58:8:1where fo = 0, to = x(I - x)°/v and z = 1,x(v + µ + j)/(v + I + j). The RJ term is an approximation to theinfinite sum of the terms tt + tt,I + :1.2 + - which is given exactly by II + (v + µ)/J , ( 1 + (v + p. + 1)/J][I + (v + p.)/J1 58:8:2Rt=tt l+x+x`+1 + (v + 1)/J [1 + (v + 2)/J] 11 + (v + 1)/J]and approximately, for large J, by//xI58:8:3Rt=tt (l+x+2xJ2+3xJ')+smallerterms I(p. -1)xti++I -x Al-x)-...)If µ = I. this evaluation of RJ is exact, even for J = 1, and B(v;µ;x) is calculated as fo + to + ti/(1 - x). Oth-erwise, J is determined by adding terms into series 58:8:1 until t;(µ - 1)x/[jj(l - x)2J is less than 10-'. with f)= fo + to + tj ++ t;_1, whereupon J = j and the output value is B(v;µ;x) = fr + tt(l + (µ - 1)x/J(1 -X)1/0 - x).If µ is zero or a negative integer, the procedure of the last paragraph is adopted irrespective of the magnitudeof x. However, the recurrence B(v;p.;x) _ -x`-'(1 - x)"/µ + ((v - 1)/µ]B(v - 1;µ + l;x) is first applied asmany times as necessary until B(v + R, O;x) is reached. The sought incomplete beta function is then evaluated asfo + kB(v + µ;O;x) wherefo and k result from updating at each recursion stage.If x0.7 and µ is not a negative integer, the reflection formula 58:5:1 is utilized. The incomplete beta functionB(µ;v;l - x) is calculated as described above, and the three gamma functions are evaluated by a routine similarto that described in Section 43:8.Generally, the algorithm is of 24-bit precision (i.e., the fractional error does not exceed 6 x 10-8). Its imple-mentation may be slow, especially when the argument is close to unity and the µ parameter equals zero or a negativeinteger.Setf=j=0Set g=t=If µ +Int(lµI) * 0 go to (2)(1) IfA=0goto(7)Replace f by f - x"-'(I - .r)"/µReplace t by (v - 1) t/µReplace v by v - IReplace p. by µ + IGo to (1)(2) If x < 0.7 go to (7)Seta=v+µIf a + Int(jaI) * 0 go to (3)Set:=-1Go to (6)(3) Replace g by g/aReplace a by a + IStorage needed: v, f, j, µ, g, t. xand a Inputrestrictions: 0 <- x < 1, v >0. R unrestricted but if µ = -1,-2, -3, ... then (v + µ) > 0 58:9 THE INCOMPLETE BETA FUNCTION B(v;µ;x) S78 Replace j by j + 1(8)Replace fbyf+tReplace t by tx(v +If a < 3 o to (3)((Setf=I+7;3a:-I)Replace f by [(f - I)/12a] - a[ln(a) - 1]Replace g by g exp(-f)2./aIft<0goto(5)If r = 0 go to (4)Replace g by I/gSett=0Set a=vGo to (3)(4) Set r = -1Set a=ltGo to (3)(5) Set f = g(6) Set a = µSet p.=vSet y=aReplace x by I - x(7) Replacet by rx"(I - x)°/vSet a = 10a(1a - 1)x/(1 - x)2µ + j)/(v + I + j) Replace fbyf+t Q +11108jIxJIf j < jail go to (8) Output ff = B(v; µ;x) <<<< 58:9 APPROXIMATIONSFor small enough values of the argument, the approximation 58:9:111B(v;µ;x) = x"v +1 +")X1holds. The corresponding approximation when x is close to unity issmall xTest values:8(3;3;1) = 0.0166666667B(;:0:0.8) = 5.09468017B(2;-1;0.2) = 0.0268564487B(1;-1.5;0.9) = 24.0000000B(1:1;0.75) = 2.09439510 58:9:2B(v;µ;x) =II''(v)+µ)(I - x) I - + Il t µ )(I - x) Ismall (1 - x) 58:10 OPERATIONS OF THE CALCULUSDifferentiation of the incomplete beta function with respect to its argument gives 58:10:1 58:11 COMPLEX ARGUMENTddx The incomplete beta function is seldom encountered with an argument that is imaginary or complex. 579 THE INCOMPLETE BETA FUNCTION B(v:µ;x)58:12 GENERALIZATIONS58:14 The Gauss function of Chapter 60 represents a generalization of the incomplete beta function. The incomplete betafunction may be expressed as a Gauss function with a unity numeratorial parameter 158:12:1 B(v;µ;x) _ - x(1 - x)"F(l,v + µ;I + v;x)Vor with a numeratorial parameter less by unity than the dcnominatorial parameterl 58:12:2 58:13 COGNATE FUNCTIONSB(v;.t;x) _ - xF(v, I - µ; l + v;x)V The incomplete beta function has some similarities to the Legendre function of Chapter 59. Also, it is allied to theincomplete gamma function discussed in Chapter 45.58:14 RELATED TOPICSIf f(x) denotes any hyperbolic or trigonometric function (i.e., any one of the twelve functions addressed in Chapters28-30 and 32-34) and f5(x) represents such a function raised to an arbitrary power, then the indefinite integralf f5(t)dt is an incomplete beta function.Table 58.14.1fvµ sinh1+A-A22 cosh1-A2 2 sech1A2 2 csch1-A A2I + A2 tanh 0 2 I - Acosh 0 sin2I+A I cos2 I2I-S22 I1-A0 csc2 2I-A 12 2I-A1-A Inn cot2 2I - AI + A2 2 58:14The general formula is 58:14:1 for hyperbolic functions and 58:14:2THE INCOMPLETE BETA FUNCTION B(v;µ;x)If'(t)dt = I B(v; s;tanh2(x))x ? 0580 f f"(t)dt = 2 B(v;µ;sin'(x))0< x<IT 0for trigonometric functions. The parameters v and .s of the incomplete beta functions depend on a as shown inTable 58.14.1. For the integral to be finite, the v parameter of the corresponding incomplete beta function mustbe positive: this limits the range of X in certain cases; for example, it requires A < I for convergence of fcsch'(t)dtwith zero lower limit.Interchanging the roles of the v and µ parameters produces expressions for certain complementary indefiniteintegrals. Thus, for all hyperbolic functions, except tanh and coth, one has('I58:14:3 Jf"(f)dt = 2 B(µ;v;sech=(x)) while for any trigonometric function whatsoever 58:14:4 Jf'(t)dt = 2 B(µ;v;cos'(x))The same table still applies, but the restriction, if any, on A is now governed by the requirement that p. be positive. CHAPTER59THE LEGENDRE FUNCTIONS P (x) AND Qt,(x) The functions of this chapter arise in several physical contexts, notably to describe the properties of the surface ofa sphere [see Section 59:14). Most of the chapter is concerned with the bivariate P,(x) and Q,(x) functions, butattention is given in Section 59:13 to the associated Legendre functions P' "(x) and Q,which are trivariate. 59:1 NOTATIONP,(x) is known as the Legendre function of the first kind and Q,(-r) as the Legendre function of the second kind.The variables v and x are termed the degree and argument, respectively, of the Lcgcndre functions. Frequently, the argument is expressed as a cosine so that the symbols P,(cos(O)) and P,(cosh(X)) are often encountered.Sometimes the title "Legendre function" is taken to embrace the associated Legends: functions that we discussseparately in Section 59:13. The third variable, µ, in these functions is termed the order of P; '(x) or Q,"(x) andis zero for P,(x) and Q,(x).As explained in Section 59:11, slightly different definitions for the Legendre functions may be adopted ac-cording to whether the argument is regarded as an unrestricted complex variable or as a real number confined to- I < x < I. Slightly different notations sometimes reflect this distinction. Other notational conventions are sum-marized by Abramowitz and Stegun [Chapter 8). 59:2 BEHAVIORUnless the degree is an integer, the Legendre functions adopt complex values when the argument is real and lessthan - 1. Accordingly, we restrict consideration to the range .r a - 1. Perhaps it is better to consider two subranges,- l < x S 1 and 1 K x < x, inasmuch as the behaviors of the Lcgendre functions are substantially different inthe two. In practical applications - I < x < I is the more important subrange and accordingly it is emphasized inthis chapter.The contour maps Figures 59-1 and 59-2 depict the rather elaborate behaviors of the Legendre functions. Noticethat these diagrams segment naturally into rectangular domains. The boundaries of these domains are x = ± I andv = 0,1 for the P,(x) function; for the Q,(x) function they are x = ±1 and v = -1, -2, -3..... Section 59:7details the values adopted by the Legendre functions along these boundaries.The number of zeros of the Legendre function of the first kind is given by 581 59:2 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x) 582 o%.°`pro 0 vo C44b\,o'op Atiootiao 11,vIPis4g444a4?ya+o+y+o+o + ;a ' + ' +h ' + +:...:/7...:'tuipl--l\ .. \ .... \ .. .y-s 583 59:2:1THE LEGENDRE FUNCTIONS P"(x) AND Q"(x) onn - I<vsnI0lsvs0n=1,2,3,...n- n - Isv< - nand all these zeros lie in the range -1 < x < 1. For the Legendre function of the second kind, the formula 59:2:2N,(Q,) =3 1n--<vsn--22-1 <v5 -1-2 1_2n+I+ls-n<v< - n I-n- I<v:s -259:3 specifies the number of zeros as a'function of the degree v. Most of these zeros of Q. lie in the - l < x < I range,but the asterisk in 59:2:2 indicates that sometimes a single zero is located in the I < x < x range of real arguments.Note that for -'I < v < -1, -' < v < -2, ... one zero of Q,(x) is located so close to x = - I that it cannot beproperly portrayed on Figure 59-2.The values v = -1, -2, -3, ... are omitted from formula 59:2:2 [and also from the table of particular valuesof Q,(x) in Section 59:7] because Q,(x) is poorly defined for these degrees. Consider Q_r(x) and refer to Figure59-2. As v approaches - 3 from more positive values, Q,(x) -+ -x for -(1 f) < x < (i /V) whereas Q,(x) -yx for -1 < x < -(I /V) and (I /V) < x < 1. On the other hand, the approach of v to -3 from more negative values leads Q,(x) toward +x within the ±1/V range and toward -x outside. At the values x = ±l/V (theseare the zeros of P-3, the first kind of Legendre function), there is a flip in the signs of the infinities so that all thecontour lines crowd to these points and, as it were, penetrate the Q_3(x) = ±x barrier there.Behaviors for very large argument, and for x in the vicinities of ±1, may be deduced from the formulaspresented in Section 59:9.59:3 DEFINITIONSSeveral definite integrals, including 59:3:1 59:3:2P,(x) _J* [x +x' - i cos(t))'dtx ;'I (0Q,(x) = JY[x +it - 1 cosh(t)J-"-'drx? Iv> -10 59:3:3P,(x) _-sin(rnr)tdt-1 <x< 1- I <V<0ITf1 + 2xt + t' 59:3:4 Q.(x) =FT;r' (1 - r'-;-dtx > IV>-1 I( dt 59:3:6Q,(x) = ZJif[t+ itvI-x2x2]-'-'-[.r-it1-x'1-'-"}-l<xslv > -lt- - 1{[x+it1x1J1'+[x-ir1-x2]dt-l<xslv >-1t'- 1may be used to define Lcgcndrc functions. Despite appearances to the contrary, the integrands in the final twodefinitions above are wholly real: all imaginary terms cancel after binomial expansion. 59:3 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x) 584 O00U',tibco0O'Lco0O'LICJ*A*,yea 40 O O 90'O'4ti'y1yti 4411144444444444444!....x.../...:/...1.1..11111 .1 A.v ..\.:\..;.V-5 2 585 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x) 59:4The Legendre functions may also by the indefinite integralsV-2 rxcosh((v +s)r) 59:3:7P,(cosh(X)) = -Jdtcosh(X) = x > 1'aocosh([) - cosh(X) 1Iexp(-(v + )t)59:3:8 Q,(cosh(X)) _ - x > 1cosh(:) - cosh(X)and 59:3:9 P,(cos(e)) _Jocos((+ 1)r)dt0<6<-rrcos(t) - cos(9)the last being known as the Mehler-Dirichlet formula.The formulas of Section 59:12, and especially equations 59:6:7 and 59:6:8. often serve to define Legendrefunctions as special cases of the Gauss function.With c, and c2 as arbitrary constants, a linear combination of the two Legcndre functions generally satisfiesLegendre's differential equation: 59:3:10(I - x2) tixf -2x!Lfd+ v(v + 1) f = 0f = c,P,(x) + c:Q,(x)In the notation of Section 43:14, the Legendre function of the first kind may be synthesized from the binomialfunction (1 + x)` by the procedure 59:3:11rl +xl 0P,(x)2-vX= 59:4 SPECIAL CASESWhen v is the nonnegative integer n, the Legendre function P,(x) reduces to the Legendre polynomial discussedin Chapter 21. The same polynomials result when n is a negative integer, on account of the identity59:4:1 polynomial of degree nn = 0, 1, 2, ...The Legendre function of the second kind becomes an inverse hyperbolic function [Chapter 31 ] when its degreeis zero: 59:4:2 Qtr)artanh(x)-1 <x< 1arcoth(.x)[ri > IOther integer degrees lead to the following special cases:{P(x)Qox) - polynomial of degree (n - 1 ) n = 1, 2. 3... . 59:4:3 -x n=-1,-2, -3, ...See equation 59:5:9 for details and Section 21:13 for examples.When the degree v is an odd multiple of _4, Legendre functions of both kinds reduce to complete ellipticintegrals [see Chapter 61]. The simplest cases are 59:4:4 P-,12(x) =2K112r/-1 5 x 5 1 22Kx1xitx+1\x+l/> I 59:5 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x) 586 59:4:5 59:4:6Q_ 112(x) _ Pu2(x) = P-3/,(X) _K1+s2-1x sF2z?K( ?z+Ix+l> I 4P_,,2(x)-1 5 x 5 1nE(22 x' I 2x+Ex> Ix+x -/Iyx/l+x\Kl2-2E11 -lax _ 159:4:7 Q12(x) = Q_3;2(x) _ \ \ YY/22x + 2 EI Ix > Ix+land others may be deduced with the aid of recursion formula 59:5:5. 59:5 INTRARELATIONSI{IPSThe argument-reflection formulas59:5:1 (-1), P,(x)n = 0. 1. 2....59:5:2 -(-1)" Q.(x)n = 0, 1, 2. .. .are valid only for integer degree, whereas the degree-reflection formulas59:5:3 P-..-I(x) = P,(x)59:5:4 a cot(vvr) P,(x)have general validity for real arguments.The same recursion formula59:5:5 f=PorQis obeyed by Legendre functions of both kinds.The equationsn59:5:6Q,(±x) = Z [cot(vrr)P,(±x) - csc(va) P,(+x)] v * 0, ± 1, t2, ...-1 <x< 1-259:5:7P,(±x) _ - [cot(vir) Q,(tx) + csc(vu)nv*0, t1,±2,...-1<x<I 59:5:8 (-v);(1 t v)1 Qv(x) = P,(x)[Qo(x) - 4.(v + 1) - y] + (j!)2[y+ 4i (j+)1 (_2V-t 0, =I, -t 2, ...interrelate the two kinds of Legendre function when the degree is not an integer. See 59:4:2 for Q0(x), Chapter 44for J and y. When the degree is a positive integer: I(n+ 11- IJI ++++1x 59:5:9QO(x)-1--I----1+j)!23nIn - j)!(jp223j2n= 1,2,3,... S87 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x)59:6 EXPANSIONS59:6 As detailed in Section 59:12, Legendre functions may be expressed as Gauss functions in a multitude of ways.Because each of these latter functions may be expanded as the power series 60:6;1. the Legendre functions havevery many expansions. Here we present only those that are the most useful computationally.If. for -I < x < I, we define the auxiliary Legendre functionsrv 1+v 1l1_vl (I +v(2x)'' 59:6:1lef,(x) = FI2 ,22xZr1 -\ 22/j (2j)!and59:6:2 then1- vv 3I- vv(2x)2' leg,(x) = 2xF (2,I +2;2:x')_ 2)(1+ 2)(2j + I)! IIr(l2 vv)/varl I + ZJIvn\59:6:3P,(x)= 7 /cost - f lef,(x) +\sinl2I leg, (x) 221'(2,vlzr(I+-whereas 59:6:4Q,(x) =2rl I + 2 IrI 2-rl l 2 l)r\I1 +2//vrr\sin(7)lef,(x) ++ vcos( - IV/ For the same argument range, one may use the trigonometric series+ v)(1 + v)J ()f2C(1 59:6:5P,(cos(O)) = sin[(1 + v + 2j)O]0 < 0 < it/r(2 ++ v)1j!r(1 + v)x(1 + v),(4);59:6:6Q,(cos(O)) = cos[(1 + v + 2j)910 < 0 < nFor arguments in the range - I < x < 3, the series 59:6:7P,(x) = I _W + v1 - x)+(v4 + 2v' - s' 2v)(I - .r)'') (16(-v);(I + Vol (Ixl'1\2J-I <.r<3(j!)'is a convenient way of calculating values of the Legendre function of the first kind. Similarly, for x > 1. one mayutilize the seriesr ++ +j+r l+j+`1v222)C(I + v) +(2),(2 59:6:8QJX)_ - !!(+ + v)xai,,,. ir(3 + oar)-, x>Ito calculate values of the Legendre function of the second kind, except for negative integer degree (for which Q,(x)is infinite) or when v = Y, i ,i',.... In the latter event, the first (-v - j) terms of series 59:6:8 become zero,permitting a redefinition of the summation index and leading to 59:7 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x) .E2v))(1259:6:9Q.(x) _ = Q- -1(x)r(i - v)(2X),-oI!(i - v)jx2'in conformity with 59:5:4. 59:7 PARTICULAR VALUES i P,(xl2.4.6...2nSlv=-3.-5.-7....=-1-2n 1 <v<2, 3<v<4.5<v<6....P.(c)3<v<-2. 5<v<-4.-7<v<-6, v= 1.3.5....P,Ix I 10<v<1,2.<v<3.4<v<S.- -2<v<-I,-4<p<-3.-6<v P,[x) 5, Po(x). P.,(x) P.(x)-I <v<0 Q,(x) Q,(x)15 9 12n+2 2 2 2 I13< v << v < 5.7 c< 922 22 22 Q,(x)v = 0, 2, 4. Q.(x)-1 3711 I-2n--2'2'2' 22 Q.(x)- 1 <v<- I. I<3v<,5<v<72 22 22 Q,Ax)-3-7-11-<2v<-),-2<v<-3,-2<v<-5,-3 -5 -7v=222,.. x=-1x0 (2n - 1)" } }-x -x2)'n'I+2v(lVar(I _!) 0r(1cos2fr I+v)-2 (4n + 1)!!!!r(l +2 2r(Isin I -/2201-IT1?(4n-3)"!!22U(4n - 1)!!!'-V.-r(1v/ z _3 _7 _II I_nQ,4x)v -3= 2.2. 22- 2n2vn\sin2 V-. r2v) 2r (1-v)cos(z 2-4-1)'0(4n- I)!'!!(4n + 1)!^!x>I x = 1 x x588 x = z 0 0 0 00 0 -x 0 589 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x) 59:8 x= 0 x= Ix= x -3-7-11Q,(x)-2<v<2,-4<1, <2,-6<v<-,... 0.(x) QJx) Q,tx)-5-9-13222 -5 -9 -13 -I2V2. 22-2n2 5-9-13-3<v<2,-5<v<2.-7<v<2 ....r( 2v)roll'n 2r(1v)2V.r'yV)cosI/2r(2 W(-1)',r(4n-3)r'!! =022U(4n - WTar(2v!mxcos -2r(1-v)2 The quadruple factorial (4n - I)!!!!! that, with its analogs, appears in several of the tabular entries is defined inSection 2:13 and equals 3 x 7 x I I x x (4n - 5)(4n - I). The constant U [Section 1:71 equals 0.8472130848;note that P1,2(0) = 2U/a, P_,12(0) = 1/U, Q,,2(0) = U and Q_,,2(0) = 'rr/2U. 59:8 NUMERICAL VALUESEach auxiliary Legendre function, defined in equations 59:6:1 and 59:6:2, may be calculated as the sum of a seriesto + t, + t2 ++ ry + ..., where to = Ifor the lef, function while rr = 2x for the leg, function. The samerecursion formula, t)., = r;.r2[k2 - (k + .)2 - ;], applies to both series; however, k = 2j + for lef,,, whereas It= 2j + I for the leg, function. These relationships are used by the following algorithm, in conjunction with equa-tions 59:6:3 and 59:6:4, to calculate P,(x) and Q,(.x). The gamma function ratio, R = r[I + (v/2)]/r[(I + v)/2]is calculated with the aid of equation 43:6:10, if v > 6. If v 6. 43:6: 10 is not used until the identity {r[l + I + (v/2)]/r[l + (1 + v)/21} = {[I + (I + v)/2]/[l + I + (v/2)]}{r[/ + 2 + (v/2)]/r[l + (3 + v)/2]} has beenapplied L + 1 times with l = 0, 1, 2. .... L where 2 - (v/2) < L <_ 3 - (v/2). Should v be a negative integer,equation 59:5:3 is used to calculate P,(x) and a number of order 1041 is returned for Q,(x), in recognition of 59:4:3. Input v >>SetaR=1 Set K = 4 011v - v-1If 11 +vJ + Int(I +v)+Ogoto(1)Set a = 10"Replace v by -I - v(1) Set s = sin(90v)Set c = cos(90v)Setw=('),+ v)2(2) If v > 6 go to (3)Replace v by v + 2Replace R by R(v - 1)/vGo to (2)(3) Set X = 1/(4 + 4v)Set g = I + 5X(I - 3X(.35 + 6.IX))Replace R by R[I - X(1 - gX/2)]/VStorage needed: a, R, v. K. s, c, w, k, X,u, 1,x,fandg Use degree mode or change 90 to n/2. Input restrictions: - I < x < 1 To recalculate with unchanged v, simply in-put new x.I 59:9 THE LEGENDRE FUNCTIONS P,(x) AND 590Input x >> p - P,-W< q = Q,(x)Setg=u=2xSetf=t= ISetk - 'Set X = I + [10"/(1 - x2)](4) Replace t by tx2(k2 - w)/[(k + 1)2 -Replace k by k + IReplace f by f + t114 Replace u by u--, (k2 - w)/[(k + 1)2 - 411Replace k by k + IReplace g by g + uIf k < K go to (4) If (XtI > Ifi go to (4)Replace f by f + [x2t/(1 - x2)]Replace g by g + [xIu/(I l2)]Set p = I(sgR) + (cf/R)]/V aOutput PIK««Set q = aN/7rl(cgR) - (sf/R)1/2Output qTest values:P,(0.5) = 0.223144533Q7(0.5) = 0.343915293 P_2(-I/a) _ -0.318309886Q_ (-1/rr1 = 2xP3/2(0) = -0.393446867Q72(0) = -0.618024892P,(8/9) = 0.888888889Q1(8/9) = 0.259205931 For J > 2w2vl the infinite series t, + t,_, + t,_. + -is well approximated by the geometric series t,(I+ x2 + x4 + x° +) and the sum is therefore approximately t,/(1 - x''). Hence, the fractional error in truncatingthe infinite sum to + t, + t, + at the t,_, term is approximately [t,/(l - x2)1/Ito + It + + t,_, + t,(I -x)-']. The algorithm makes use of this principle in deciding when to truncate, thereby ensuring that lef, and leg,are computed to a precision of about 6 X 10-°. Close to the zeros of P,(x) and Q,,(x), this precision may not becarried over to the Legendre functions themselves.The algorithm is valid only for arguments in the range -1 < x < I and is extremely slow close to the boundariesof this range. 59:9 APPROXIMATIONSClose to x = I, the Legendre function of the first kind is approximated by the linear function 59:9:1P,(x) =[(1 - v)(2 + v) + v (I + v)x] I I - x{ smallwhile that of the second kind obeys 59:9:2Q,(x)-InII(1 +v)II -xlsmall The corresponding approximations close to x = -1 are 59:9:3P,.(x) = cos(vtr) +sin(ir)IInr t2+ xl+ y + 2d,(1 + v)I(I + x) smally = 0.57721566491 59:9:4Qdx) -coswa) r In(1Zx) + y + 2y(I +v)J -17 si 2(vw)(1 + x) small Here 'h is the digamma function [ Chapter 44] and y is Euler's constant [see Section 1:4]. 591 THE LEGENDRE FUNCTIONS P,(.r) AND Q,(x) 59:10As x acquires very large values, we have 59:9:5P,(x) = V.- r(1+v)r(21+ v)(2x)' 'v> -1 IT(-2v- 2)!!-3 -5 -7 59:9:6Q.(x)N/2v)!(4x)"v = 2 , 2 , 2 ,...V'nr(i -v)r(-v)cot(var)(2x) "'-1> v *3Z,57Z,2..x- z 59:10 OPERATIONS OF THE CALCULUS Differentiation or integration of a Legendre function gives an associated Legendre function [Section 59:13J: r-1fj'(x)-1 < x < 1 59:10:1 ; U X) = S 1 } f= P or Q I,'fl.(x)x > 1f59:10:2 P,.(r)dr =1 - x= P;-"(x)-1 <x< 1 59:10:3 JxP,(t)dt =x= - 1 P;-"(x)x > 59:10.4JxQ,(0(dl) (.r)x > IAlternatively, the derivatives may be expressed asdvx+xv+ lvvx 59:10:5- f(x)f(x) -f_I(W ) -f,(x) dx1 -xI -x'l -xI -x'a formulation that applies for either kind of Lcgcndre function and for any x exceeding - 1.Definite integrals of products of Legendre functions includei59:10:6P,(t)P (t)dt =4ti(1 + v) - 44o(l + w) + 2w cot(ve) - 27r cot(a)a) v + w * -1 T.I2(l + v + w)(w - v) csc(v7r) csc(w7r) ITcos(wlr - vw) - it + sin(2vir)[ijr(l + v) - ,(1 + w)J 59:10:7P,Q)Q (t)dt= v>0w>0_a 7r(1 + v + w)(w - v)2 + 2 cos(vrr) cos(wa)J[y(1 + v) - 4s(1 + w 59:10:8Q,(t) )1 - IT sin(m - wa)v + w * -1 T. all + v + w)((d - v) 59:11 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x) 592when the integration limits are - I < t < 1. These formulas are indeterminate when w = v; in that event the threeintegrals give [I - 2/1r2) sin2(vrr) ib'(I + v)]/(} + v), -sin(2vrr) qi'(1 + v)/[(I + 2v)tr] and {(w2/2) - (1 +cos2(v'rt)]y'(I + v)}/(l + 2v), respectively. Similarly, the integral between limits 1 < t < x: 59:10:94r(1 +- ir(1 + vJQ.(t) Q.(t)dt =w) f(I+v+w)(w-v)has y'(1 + v)/(1 + 2v) as its special w = v case. but 59:10:10 FP,(t)Q (t)dr =I i(I+v+w)(w-v)Jdiverges if the degrees are equal.w>V>0 Since Pa(t) = 1. P,(t) = t,{ P,(r) + } Po(t) = r', etc. [see Section 21:5], the above formulas may be adaptedto give many definite integrals of the form ft'f,.(t)dt for f, = P, or Q,. Very many other definite integrals are listed by Gradshteyn and Ryzhik [Sections 7.1 and 7.2]. 59:11 COMPLEX ARGUMENTWith the real argument x replaced by x + iv. equation 59:12:1 serves to define the complex-valued function P,(x+ iy) as F(-v,l + v;l;(l - x - iy)/2). The corresponding formulait1 - x - iy/I + x + iy15 9 : 1 1 : 1Q,(x + iy) = 2 [cot(vrr) z i] F -v,1 + v:l; 2-2csc(vrr) FI -v,1 + v;l; 2provides a valid definition of the Legendre function of the second kind for all combinationsof x and y except wheny = 0 and> 1. The alternative signs in 59:11:1 apply accordingly as y > 0 (upper sign) or y < 0 (lower sign).For y = 0 and - I < x < 1, definition 59:11:1 yields a function that is multiple valued and complex. To avoidthis difficulty, it is usual to redefine Q,(x) as the average value of the two limits Q,(x + iv) and Q,(x - iy) as y-+ 0. This removes the i from 59:11:1, which then reduces to 59:5:6. This convention has been followed in all equations of this chapter (except 59:11: 1), so that our Q,(x) is always real for real argument. 59:12 GENERALIZATIONSLegendre functions may be generalized to the Gauss function of the next chapter. Included among the ways inwhich Legendre functions may be expressed as a single Gauss function areI-x59:12:1 V ,+ v;1; _-1 <x < 3.2) /vI + v1-vv59:12:3P,(x)=Fl 2, 2;1;1-x=)=xF(,l+Z;l;l-x'0<x<V2 12- 159:12:41,14) = [x +x=I J'' Fv,-2:1;x+z'- 1(x -r' - 111/21. 12V7- I+ x- I`2x+xr- 1) 593 THE LEGENDRE FUNCTIONS P,(x) AND Q"(x) 2x'- 159:12:5P"(x) _ [x -x'11" F -v, 2;1;'-1-[x +V77-111/2-12x' - 13 FI 2,1 + v;l;_)x>x- x'-1\x'-l-xv I-v'/I+vv1 59:12:6P"(x) = x' F - -;1;1 - II = x-'-' FI -,1 + -:1:1 - - I2 '2-r-\22X-I>'r59:12 V; r(I + v) 259:12:7Q"(x)=F(I+v,1+v,2+2s';x>2+I r(i + v)(2x2)'-"\1 ± x/Var(1+v)/1+vv31 59:12:8Q"(x) =F(-,lx > I [(,' + v)(2x)''"22 2xr(I+v)fl+v 1+v3 I) 59:12:9Q"(x) _- F l- + v;1x > r(? + v)(2Vx' - t)i_ 2'21 - x-r(1 + v)IT(x -(113x' -.rl59:12:10Q"(x)=r(,+v)2V'-F\2'2'2+v'/ z12lV (r(I"F(j3 3 =VI+v,l+v,-+v;Vx>- 59:12:11Q"(x)F ,1 + V.+ v:x > I r(i+v)(x+x - I)2x+r - IThe restrictions on x attaching to the formulas above ensure that the Legendre functions are real and that the seriesexpansion of the Gauss functions [see 60:6:11 converges. As well, there are many formulas that express the Le-gendre functions as the sum of two Gauss functions; some arer(-v - !)r2159:12:12P"(x) =Fl I + v.I + v:2 + 2v: - J"" 1 V ir r(-,,)(2.2)t -rr(#+v)(2xt2)' / 2\+FI -v,-v;-2v;-Ix>2+ IVrrtr(1+ v)``1 ± x/r(-v - D/I+v I+v3 1 59:12:13P"(x) FI- + v;r(-v)(2 x1)'22'21r(; + v)(2z' - 1)"( -v-vII) +F -- v:Jx > \/2V;r(I+v)2' 221-x'v l+ vl 59:12:14P"(x)F -l - '2'2:2, x \\\222Vnx1-vv 3-FI,1 +-;-;z'-1 <x< 1 (l + v1r(-v}\22 2r2I2/ 59:13 THE LEGENDRE FUNCTIONS P,(x) AND Q,(x)r(-v-1)l+vv 31/59:12:15P,(x) =F'I+ -; - + v; ;1`r(-v)(2x)'"22 2x1 z + v)(2x)'/-v Iv II +FI`-- v; -/1x > 1 V'nr(l +v)22'2X.C(- v -(x -(11 31 Vc59:12:16P,(s) = F+ v; - -J r(-v)2a x- 12 2 222 x-594 r(+ v)(x-'(1 11Ix3 Fr(1+v)2n x'-l\-,2:222V77- -1Visr(-v-1. 359:12:17P,(x) =Fr-1+ v; - + v;fir(-v)(x+ 122+x=- 1r( + v)(x + 1 +F -,-v: - - v: x > 1r(I + v) 22x +x' - 1r( l2v) 59:12:18Q,(x) =sinl2JF( Zv, 12v:2zr(l+ z)r(i+ZI\\/va1 - vv 3-cos - xF -, I + -; -;x- -1 < x < 1r(1v\z2222/IMany of these formulas become invalid, or require modification, if v is an integer. For x > I the substitution x= cosh(X),x=I = sinh(X), s t V= exp(tX/2) often yields simplifications. 59:13 COGNATE FUNCTIONSThe associated Legendre functions P."(x) and Q,"'(s) represent a generalization of the Lcgcndre functions inasmuchas PY°'(x) = P.(x) and Q'.01(x) = Q.W. They are sometimes named "spherical harmonics" but we reserve that titlefor the extended functions discussed in Section 59:14. In linear combination, they satisfy the associated Legendreequation 59:13:1+[v(1+v)-1 µ'x2Jf=0f=c,P;"'(x)+c,Q;."'(x)[see 59:14:4 for a trigonometric equivalent of this equation]. Replacement of x by I - 2x and then f by (x= -x)"Rf leads to an example of the Gauss equation. 60:3:4, and accordingly associated Legendre functions are in-stances of the Gauss function of Chapter 60.Generally, these functions are complex even when their arguments are real. However, it is conventional toadopt redefinitions similar to that discussed in Section 59:11 to ensure that P;"'(x) and q."(x) are real for real xbetween - I and + 1. Here we discuss this range exclusively. Moreover, we shall emphasize cases in which v andµ are nonnegative integers, using n and m to represent the degree and order, respectively, in these cases.Calculation of values of the associated Legendre functions is possible via the definition 595 THE LEGENDRE FUNCTIONS P,(x) AND Q(x) 59:13 59:13:2(l+V+µ\II\I 2vtr + µa\ P;°'(x) = cosI Ief,'(x) V.-(1 -x'-)"r_r(I +vµ)22 r(i+v+µ\2/IsinIleg;,"'(x)r(l+v-2\2J 59:13:3 µa\ V n2"-'(vir +lef (1 - x')"/' sin2I,"'(r) 2rll+v2IL//+cos(m + µJ Ieg;1"(x)rrl+v-µl 1\22Jwhere the auxiliary associated Legendre functions are defined by(-V - µ 1+v-µ 1i yI+v- µI (2x)2J 59:13:4Ief;"r(x) =Fx-)-l-v-µv -µ 3I - v - µv- µ(2x)"=' 59:13:5Ieg;"'(x) = 2xF+ -; ;x_I +The algorithm of Section 59:8 may be modified to use these equations, although four distinct gamma functionevaluations are required instead of the single gamma function ratio that sufficed with equations 59:6:1-59:6:4.Some interrelationships between associated Legendre functions are59:13:6 ()Pt'-1W W x= 11."WW59:13:7(tan(") - tan(µa)] Q"0_,(x) _ [tan(var) + tan(An)1 Q."(.0 - TrPV`(x) 59138"r(1 + v - µ)",2 sin(µa)"P-P= :: (x) (x) itQ, (x)1 r(1 + v + µ)[cos(µa) 59139r(1 + v - µ):_" '+:"'=ir sin(µ-rr)P;"' ::(x)n) Q(x) Qr(1 + v + µ)Icos(µ2(x)1-2 sin(va + µa)P+,"-=` 59:13:10 µn);,cos(varP(x)WIT 115913 wv + nµ)-a sin(+''-_ -"'P' ::Q(x)cos(avaµ) Q;(x)2;(x) constituting degree-, order- and argument-reflection formulas.Some examples of associated Legendre functions are 59:14 THE LEGENDRE FUNCTIONS P.(x) AND Q.(.0 59:13:12P'i'(x) 1xQi"(x) =-1 - x' artanh(x)1 - x'2-3x' 59:13:13P;"(x) = -3x1 - x2Q;"(x) =- 3x1 - x' artanh(x) ` 59:13:14P':''(x) = 3(1 - x'-)Q," (x) =5x - 3x'+ 3(1 -xartanh(x)1 -x'59:13:15P;°'(x)=(-I)"(2n- 1)!!(1 -x22)"'n=0.1.2,...and others may be obtained via the differentiation 59:13:16G -(X) = (- I)"(I - .r2)"/'f,(x)f = P or Q or recursion 59:13:17(x) _(1 + 2rt)xf' '(x) +n + inf'; ',(x) 59:13:181 +n - m1 +n - m-f,"(x)2mx=f"'(x) - (I + n - m)(n + m) f'"-"(x)71=-=Tf=PorQ5% formulas. Formulas 59:13:12-59:13:15 demonstrate the inappropriateness of the name "associated Legendre poly-nomials" sometimes given to these functions. Be aware that the (- 1)" factor in 59:13:16 is often omitted so thatassociated Legendre functions of odd order may be encountered with signs opposite to those employed in this Atlas.The associated Legendre functions of the first kind satisfies the orthogonality relationships 59:13:19Jf P, '(t) P;,"' Will=2(n + m)!'2n + l(n - m)!but no such relationship holds for the second kind. 59:14 RELATED TOPICSN+nN=n In Section 46:14 we discuss the solution of the Helmholtz equation in various orthogonal coordinate systems. Inthe spherical system one findsr2 d'R2r dRI a'Ycot(o) aycsc2(o) d'Y59:14:1 - - + - - - kr- = h = - - - - - - - - R dr2R dr Y ao'YaoYa8'where R is a function only of the radial coordinate r, Y is a function of the two angular coordinates, 0 and +, andh is a separation constant. With m2 denoting a second separation constant. the second equality in 59:14:1 may befurther decomposed intosin2(o) d26sin(20) d6I d2I 59:14:2 Ado' +29do+ h sin'(o) = m =11.$12where 0 is a function only of o, and is a function only of 4).The second equality in 59:14:2 is satisfied by a sinusoidal function of the longitude (b59:14:3 4) = C. cos(me) + S. sin(m$) in = 0. 1. 2... .where C. and S. are constants and m is constrained to be an integer by the geometric constraint that (P(-a)NW).10 597 THE LEGENDRE FUNCTIONS P(x) AND Q,(x) 59:14The first equality in 59:14:2 may be rewritten as'8d8 59:14:4 dde. + cot(O)d8+ [h - m' csc=(6)]8 = 0which, by the substitution x = cos(8), may be converted into the associated Legendre equation 59:13:1. Accord-ingly, the general solution is59:14:50 = p, ,F,"(cos(0)) + q,,,,Q,"(cos(8)) v =; + h -where constant coefficients are denoted by p_ and The latter is usually constrained to be zero by physicalconsiderations: the prohibition on Y, and hence 0, being infinite at 0 = 0. Because the latitude of the sphere is(a/2) - 0, 0 = 0 represents its "north pole." At the "south pole," 0 = a, cos(8) = -1 and, since isinfinite unless v is an integer, we are led to conclude that v = n = 0, ± 1, ±2, .... Because the P;"' and P;'_,functions are identical, we can ignore negative n values, leading to59:14:60 = p,.,,P;"' (cos(8))n = 0, 1, 2....h = n(n + 1)as the only physically significant solutions of 59:14:4. Moreover, since the associated Legendre function P.,` iszero if Imi > n, we can discount all values of m except 0, ± 1. t2.... ±n_ Where only certain values of separationconstants are permitted, as n and m here, these values are named eigenvalues or quantum numbers.Because the eigenvalue in enters into the solutions for both and 0, it is often counterproductive to factorthe Y function into its longitudinal and latitudinal components. The general solution of the second equality in59:14:1 has now been shown to be 59:14:7Y = ±[S"," sin(m4,) + C",,, cos(md,)J P;,"'(cos(0)) where S",,, and C",,, are redefined constants. The components of this solution are known as spherical harmonics orsurface harmonics and, in light of orthogonality relationships 59:13:19, 32:10:25 and 32:10:26. they are usuallydefined in the normalized versions2n + 59:14:8 Y";"(8:m)21TII(n + m)!f(mm) P;,"'(cos(8))///f=sinorcosn=0,1,2,...However, normalization conventions sometimes differ from author to author.As their name suggests, spherical harmonics play a vital role in describing oscillatory behavior in systems ofspherical symmetry: for example, in the quantum mechanics of atomic electrons. One distinguishes tonal surfaceharmonics when m = 0, sectoral surface harmonics when in = n and tesseral surface harmonics when m = I, 2,Finally, let us return to the first equality in 59:14:1 with n(n + I) now substituted for the separation constanth. Replacement of the radial coordinate r by x/Vk (or by .r/' k for negative k) and the dependent variable Rbythen leads to 59:14:9d'fdfx--+x--[(n+?)-'-x'Jf=On=0,1,2.... dx-dxwhere the alternative upper/lower signs apply as k is positive/negative. This is the (hyperbolic Bessel)/(Bessel)equation 50:3:4/53:3:5. and the lower alternative is satisfied by 59:14:10 R =f=A.]+J.1/2_(x) = a"j"(x) + b"y"(x)where J, represents the Bessel function [Chapter 53J; A", B", a" and b, arc arbitrary constants; and j, and v" arespherical Bessel functions [see Sections 32:13, 53:4 and 54:4J. CHAPTER60THE GAUSS FUNCTION F(a,b;c;x) A surprisingly large number of simple intrarelationships make the quadrivariate Gauss function unusually flexible.It embraces many of the functions discussed in previous chapters and may be further generalized as explained inSection 18:14. 60:1 NOTATIONThe Gauss function is also known as the hypergeometric function or as the Gauss hypergeomerric function. Thesubscripted symbolism ;F1(a,b;c;x) is sometimes encountered; Section 60:13 contains an explanation of the "2"and "1" numerals.As usual, the x variable in the Gauss function is its argument. Because of their locations in expansion 60:6:1,the a and b variables are known as numeratorial parameters, while c is a denominatorial parameter. There is asecond denominatorial parameter, equal to unity, whose presence is not explicitly displayed in the F(a,b;c;x) no-tation but is brought out in the first of the following alternative symbolisms:C(c)f a- 1,b- 1a,b;x 60 115a)1'(b) Lx0, cJ = 2F11c= F(a,b;c;-r) 60:2 BEHAVIORUnless one of the four quantities a, b, c - a or c - b is a nonpositive integer (in which event, see 60:4:10, 60:4:11or their analogs), the Gauss function is defined only for real values of its argument in the range -- < x < 1. Thedomain may be extended to embrace x = I provided that c > a + b.When c is a nonpositive integer the Gauss function adopts infinite values [unless a or b is also an integer suchthat a - c or b - c equals 0, 1, 2, .... so that 60:4:11 applies[- Nevertheless, significance can always be attributedto the ratio F(a,b;c;x)f[(c), even when c = I - n = 0, -1, -2, ..., because of the limiting operationF(a,b;c;x)(a)"(b)"x"I ... 60:2:1limr(c)=(bF(n+a,n + b;n+l;x)n = 1, 2, 3,Because it is quadrivariate, the Gauss function displays such a wide variety of behaviors that it is impracticalto depict graphically. 60:3 THE GAUSS FUNCTION F(a,b;c;x)60:3 DEFINITIONS600 Expansion 60:6:1 provides the usual definition of the Gauss function for - I < x < 1. The transformation 60:5:3permits extension to -x < x < ;.The Euler hypergeometric integralr(c)f?-'di 60:3:1F(a,b;c;x) =r(b) r(c - b)(1 - t)1° `(1 - xt)°serves as a definition, as do the equivalent definite integralsc>b>0x<I 60:3:2 r(c)F(a,b;c;x) = (r1)-°-'r°-dtI + a > c > bx<l r(b) r(c - b)(r - x)°dan 60:3:3 t° 'dt F(a,b;c;x) =r(c) c> b> 0x< 1 r(b) r(c - b) JG (I + :)`-°(l + x - xt)'Because of identity 60:5: 1, the two numeratorial parameters may be interchanged in any of these definitions.A linear combination of two Gauss functions satisfies the h}pergeometric differential equation 60:3:4 x(1 - x) --- + [I - dfdxY-(1+a+0kldx-aRf=O f= c,F(a,P:I - y;x) + cx'F(a + y,3 + y;I + y;x)c, and c, being arbitrary constants.in the notation of Section 43:14. the Gauss function may be synthesized from the binomial function (Section6:141 by the operation 60:3:5 60:4 SPECIAL CASES1-c(1 - x)'°bF(a,b;c;x)-1 < x < I A substantial fraction of the functions treated in this Atlas are special cases of the Gauss function. Thus, any K =L = 1 hypergeometric function [see Section 18:141 is a Gauss function on account of the identity 60:4:1 (a)rx'= F(l,a;c;x);.o (c)1Similarly, any K = L = 2 hypergeometric function having unity as one of its denominatorial parameters is a Gaussfunction via 60:4:2 F(a,,a,;(,,;x)i-o (l)t(c2);A glance at Table 18.14.2 will reveal the large number of instances in which important functions conform to either60:4:1 or 60:4:2.If the two numeratorial parameters of the Gauss function are interrelated by any one of the three relationships60:4:36=a-;,a+;orl -aor if the denominatorial parameter is related by any one of12l+a+b 60:4:4c= 2a,2b,a+b-,a+b+2I +a - boil + b - a 601 THE GAUSS FUNCTION F(a,b;c;x) 60:4to the numeratorial parameters, then the Gauss function reduces to an associated Legendre function [Section 59:131or to one of the special cases (a Legendre function, a Legendre polynomial or a complete elliptic integral, amongothers) of the latter. Of these 10 possibilities, seven are detailed in Table 60.4.1 by listing the appropriate values of "factor," µ, v and X in the identity60:4:5 F(a,b;c;x) = (factor)Pr"'(X)The three omissions are easily derived by interchanging a and b.When the denominatorial parameter equals i (or ), reduction occurs to the sum (or difference) of two associatedLegendre functions; thus:60:4:6 F(a.b;l + ;;x) = (factor)[P!,`(X) :t PY1`(-X)1These cases arc also included in Table 60.4.1.Another trivariale function that is a common special case of the Gauss function is the incomplete beta function.If either of the numeratorial parameters of the Gauss function equals unity, or is less by unity than the denominatorialparameter, we have reduction as follows: 60:4:7 F(a. I :r- r) GO.4-8 Pa c -(c- I)B(c-l;l+a-c;x)Y '(I-.r)._. (c - I) B(c - l;1 - a;x)xr-1to instances of the function discussed in Chapter 58. Powers, polynomials, logarithms and inverse hyperbolic ortrigonometric functions may arise by further specialization of the incomplete beta function, as explained in Section58:4.When the denominatorial parameter c equals one of the numeratorial parameters, say b, we have Fbb ' == I(a 1 60:4:9 x (a,;:x)i-o (1),(I - x)°In effect, the two parameters have "cancelled." Table 60.4.1 Parameterrelation Factor µvXRestrictionb=a - ;2`-T(c)[r(1-x)'I-c2a-c-Ix<1b = I - aI'(c)(kl/(I -x)P'-" I- c-a,CI -21x < I2x c = 2a2"-'C(; + a).i-"(1 - x!-ab - u -0 < x < Icab -r-''r(a+b bb - a - i1 --1x <I c ab+C(;++b)(Vkj/2)`-'-"_-a-ba -b -=V 1-x<I Cl+a+b/)+a+blC I- zx 2 2 22c=I+a -b x)°b - a-bI+xx<II-xc a - ba- b - i& -\cam.-2"T(a-1)C(b-l)(I-xl'"'-°/128axa - ba- b -?v &-vc0<x<I 60:5 THE GAUSS FUNCTION F(a,b;c;x)if either of the numeratorial parameters is a nonpositivc integer, for example, if b - -n = 0, - 1,then the Gauss function reduces to a polynomial function [Chapter 17) of degree n:602 to 60:4:10F(a,-n;c;x) =n(a); (-x)'=(1 - 2x) ,! (c)1 (c).where P?"(x) is a Jacobi polynomial [Section 22:12). A similar polynomial combined with a binomial function[Section 6:14) is generated if a - c or b - c is a nonnegative integer; thus, for example:(c - a)I 60:4:11F(a,c + n;c;x) = (1 - x)-"-" n(-x),=oJ(c)J(1 -2x)n=0, 1, 2. ...(c).The cases in which the denominatorial parameter c is zero or a negative integer are addressed in Section 60:2;however, 60:4:11 may still be valid if one of the numcratorial parameters is a less negative integer. Equations22:12:11-22:12:14, combined with 60:4:11, show how a number of orthogonal polynomials are special cases of the Gauss function. 60:5 INTRARELATIONSHIPSThe Gauss function is symmetrical with respect to interchange of the two numeratorial parameters60:5:1 F(b,a;c;x) = F(a,b;c;x)The transformation 60:5:2 F(a,b;c;x) =F(c - a,c - b;c;x)(I -x)"ra-,which may be regarded as a reflection formula for both numeratorial parameters, provides a relationship betweentwo Gauss functions of common argument. On the other hand, the transformations 60:5:3X/xFI a,c - b;c; - Ix-1///(1 - x)"(I - x)brelate a Gauss function with argument in the range 0 t x s 1 to ones with arguments in the -x <_ x <_ 0 range.The six functions F(a ± I,b;c;x), F(a,b ± I:c;x) and F(a.b:e ± I:x) are said to be contiguous to the Gaussfunction F(a,b;c;x). The function F(a,b;c;x) is linearly related to any pair of its contiguous functions. Thus, thereare 15 contiguity relationships. Three of these are60:5:4(2a - c + bx - ax) F(a,b;c;x) _ (a - c) F(a - l,b;c;x) + (a - ax) F(a + l ,b;c;x)60:5:5(a - b) F(a.b;c;x) = aF(a + l,b;c;x) - bF(a,b + I ;c;x)and60:5:6[1 - c - (I + a + b - 2c)xl F(a.b;c;x) = (c - 1)(x - l) F(a,b;c - l;x)+(c - a)(c - b)xF(a,b;c + 1;x)cand the remaining 12 may be found in Erdilyi, Magnus, Oberhettinger and Tricomi (Higher Transcendental Func-tions, Volume 1, Section 2.8, formulas (33)-(44)1. These contiguity relationships constitute recursion formulas,enabling F(a + l,b;c;x), for example, to be expressed in terms of F(a,b:c;x) and F(a - I ,b;c;x). 603 THE GAUSS FUNCTION F(ab:c;x) 60:5The 15 contiguity relationships provide intrarelationships linking three Gauss functions of common argument,but there also exists a plethora of relationships between trios of Gauss functions with dissimilar arguments. Threetypical relationships of this class are/\F(X.l+A - c;I + 2X-a-b:1r(c) r(a + b - 2a) x60:5:7F(a,b;c;x) _ X<0 r(a+b-\)r(c-k) (-x)"+,c + A - a - b;l +2a-a-b;/r(c) r(a + b - 2x) 1 -xl60:5:8F(a,b;c;x) = s x < I,=.r(a + b - x) r(c - x) (1 - x)and br(c)r(a+b+c-2X)F(JX - a,X - b;l +2),-a-b-c;l -x)60:5:9F(a,b;c,x) _ Z r(a + c - K) r(b + c - k) (I - x)"-"x<I The E notation in these three equations indicates that their right-hand members consist of two terms that differ onlyby virtue of A adopting either of the two indicated values. These relationships may fail if the argument of one of the numeratorial gamma functions equals a nonpositive integer see Abramowitz and Stegun [equations 15.3.10-15.3.141 for expansions that apply in these exceptional cases.Because they interrelate Gauss functions whose arguments are linked by a linear relationship, formulas 60:5:2,60:5:3 and 60:5:7-60:5:9 are examples of linear transformations; the Bateman manuscript [see ErdElyi, Magnus, Oberhettinger and Tricomi, Higher Transcendental Equations. Volume I. Sections 2.9 and 2.11] contains a catalog of these transformations. Apart from the exceptional cases noted in the previous paragraph, linear transformationsof Gauss functions are valid for all values of the numeratorial and denominatorial parameters. In contrast, the set of so-called quadratic transformations, in which a quadratic relationship exists between the Gauss function argu- ments, holds only if one of the following restrictive conditions apply: 60:5:10IIa + b = 1,c+-,c--or 2c- 1 11a-b= -, --,c- lorc+ l2213c=2, 2, 2a or 2b These 12 conditions are precisely those detailed in Section 60:4 that cause reduction of the Gauss function to oneor two associated Legendre functions. Accordingly, the table in that section (Table 60.4.1) may be used to deducequadratic transformation formulas. For example, the first two tabular entries may be separately reformulated asl+vv - µ1 `2F21+2 :I-µ:l-1I 60:5:11P;" (X) = (I t-X2)r(1 - µ) xMX => 0 and ax260.5.12p"(X)-(I+X)wX=I-2x>-I1-XFv, 1 + v; I - µ;1-XrU - µ)The equality of the right-hand members of 60:5:11 and 60:5:12 constitutes a quadratic transformation, valid for X> 0. The Baseman manuscript [see reference above] includes a comprehensive listing of quadratic transformationsand information on cubic transformations. 60:660:6 EXPANSIONSThe Gauss seriesTHE GAUSS FUNCTION F(a,b;c;x) 604 abx(I + a)(1 + b)x (2 + a)(2 + b)x / 60:6:1F(a,b;c;x) = 1 + - I + I +(1 +c2(1 + c)3(2 + c)\+(/+a-1)(J+b-I)xl+to)'(b),Xj-l<x<I I(J + c - 1)..)/,); (c),is the fundamental expansion of the Gauss function, here expressed in concatenated form and in terms of thePochhammer polynomial )Chapter 18). The restriction on the magnitude of the argument x may be discarded if theseries terminates, a circumstance discussed in connection with equations 60:4:10 and 60:4:11.For large negative argument, the expansion'r(c)I'(a+b-2AA(I +A - c) 1 60:6:2F(a,b;c;x) = Y- 1 -+ -x --. -x ,,_j(a+b-A)r(c-A)(-x)"(1+2A-a-b)xJvalid unless a and b differ by an integer, is a consequence of linear transformation 60:5:7. Useful at the other endof the -x to I range of argument is the similar expansionr(c) r(a + b + c - 2X).r(A-a)(A-b)x 1 60:6:3 Ra,bxcI-x)=Ir(a+c-A)r(bI+1+2A-a-b-c+... that derives from 60:5:9. 60:7 PARTICULAR VALUES60:7:1 60:7:2 60:7:3F(a,b;c;0) =r(c) r(c - a - b)F(a,b;c;l) =r(c - a) r(c - b)c>a+b(0a<0,b<0F(a,b;c;-x) _ 1(a or b) = 0. (b or a) < 0{lx(aorb)>0These are the only particular values that can be formulated without placing severe restrictions on the parameters.Some particular values under such restrictive conditions includeV; r(1 +a-b)60:7:4F(a,b;l+a-b;-1)=/``a-b*-I,-2.-3, ...2'TII+Z-b\\ Ir(I2aJr/I+a+bl/1+a+b 1V I\2/J 60:7:5F a,b;l _a+b*-1,-3, -5, ... 22/r\(l + ar\}(l + bl2/2andr12' r(r)60:7:6F a,1 - a;c;1-I =c + 0, -1. -2... .22'rla+9(1+2C -a)2 605 THE GAUSS FUNCTION 60:960:8 NUMERICAL VALUESWhen 0 5 x < 1. the following algorithm uses the Gauss series 60:6:1 in the form60:8:1 t,+R,where to = I and t, = (j + a - I)(j + b - I) t;_,x/j(j + c - 1). An estimate of the remainder is provided by the geometric sum R, = t,x/(l - x). The number J + I of summed terms is incremented until J ? Jai + JbI +cl (ensuring that all t, have uniform signs for j > J) and until JR11 s 10-11to + t, + + t, I. A similar techniqueis employed for negative x, but transformation 60:5:3 is first invoked. Input a >>Input b >>Input c > >Input x >>D»»Self =t= ISetj = 0D»»Set d = at + JbI + 1clD»»Self =t=(I-x)If x ? 0 go to (1) Replace f by f + [xt/(I - x)]If IfI/l0° < x{tI/(I - x) go to (1)Ifj<dgoto(I)Replace f by f + tReplace j by j + I(I) Replace t by (a + j)(b + j) tx/(c + j)(IReplace x by x/(x - 1)Replace b by c - b )Output ff -j)I Storage needed: a. f, r, b, j, c, d and x Input restrictions: x < 1, c * 0, - 1, Test values:F(1,12;21;1) = 1.41421356F(-0.1,0.4;0.5;-1) = 1.05857815 No particular accuracy is claimed for this algorithm, but the precision will generally be better than 24 bits.The algorithm will often be intolerably slow if x > 0.9 or if x < -9. The linear transformations 60:5:9 or 60:5:8.respectively, are useful in these circumstances. 60:9 APPROXIMATIONSThe rational approximation 60:9:1F(a.b:c;x)+abx [I+2(c +II)-1(a + 1)(b + Oxsmall x is valid for sufficiently small argument. If one of the numeratorial parameters, say b, is large, then60:9:2 F(a.b;c;x) - M(a;c;bx)large bwhere M denotes the Kummer function [Chapter 47). Conversely, if the denominatorial parameter is large, wehave an approximation as a Tricomi function [Chapter 48): 60:9:3F(a,b;c;x) = i -toUI a;l +a - b; - I large c1-x/\1-x 60:10 THE GAUSS FUNCTION F(a,b:c;x)60:10 OPERATIONS OF THE CALCULUSThe formulas for differentiation: 60:10:1 multiple differentiation:d- F(a,b;c;x) _ab- F(a + 1,b + lc + l;x)dx c 60:10:2and integration:d"Ra.b;c;x) _(a(c)b)"F(a + n.b + n;c + n;x)606 c - l 60:10:3F(a,b;c;r)dr =1)(b1)[F(a - I.b - l;c - l:x) - 1]o(a --modify each of the three parameters equally. One may, however, selectively alter a single parameter of the Gaussfunction by such operations as 60:10:4 d"x'-b - {x"-1+h F(a,b;c;x)} _ (b)" F(a,b + n;c;x)dx"-1-bdox"-1+,-bX60:10:5LL-F(a,b;c;x)=(c - b)" F(a,b - n;c;x) Id"F(a.b;c:x) (c - a). (c - b)60:10:6-= F(a,b;c + n;x)(I - x)' +b-..-" dx" (1 - x)` (c)"d" 60:10:7x'-""- -{x-' F(a,b;c;x)} _ (c - n)" F(a,b;c - n;x)dx'Differintegration of the product of a Gauss function and a power obeys the rule 60:10:8 dX"x" F(a,b;c;x) =r(1 + µ -P)i(Iv),=o (1 + µ - v)j(c)j(1)j V+and generally creates a K = L = 3 hypergeometric function [Section 18:14]. By suitable choices of v and µ,however, the generated function may be of the K = L = 2 or even the K = L = I class. The notation of Section43:14 enables these conversions to be expressed succinctly; for example: 60:10:9 60:10:10I-aF(a,b;c;x) -' F(A,b;c;x) X = xI - A1 -bF(a,b;c;x) -+ (I - x)-" X = xI-c60:11 COMPLEX ARGUMENTGenerally, F(a,b;c;x + iy) is complex valued, but we shall not pursue this topic. 60:12 GENERALIZATIONSBecause 60:12:1 F(a,b;c;.r)(a),(b),X;izo (1)r(e), 607 THE GAUSS FUNCTION F(a,b;c;x) 60:13any of the following series:(a)i(b)i(al),(a2),(a,),(a),(b); 60:12:2 x',(1(c)r/(1) lc:) (c3)xi, etc.i-o)IJ!-oiirepresent functions that may be regarded as generalizations of the Gauss function. All of these are called hyper-geometric functions in this Atlas and are discussed in Section 18:14.There are several more profound generalizations of the Gauss function. These are the subject of a chapter byErdelyi, Magnus, Oberhettinger and Tricomi (Higher Transcendental Functions, Volume I, Chapter 4]. but theywill not be discussed here. 60:13 COGNATE FUNCTIONSThe misleading name "generalized" hypergeometric function is reserved for the class of hypergeometric functionsin which at least one of the dcnominatorial parameters equals unity. These are symbolized by F, in which n andd are integers equalling the numbers of numeratorial and denominatorial parameters. The unity denominatorial parameter is excluded from this count so that, for example: xJ60:13:1 oFo(x)- = exp(x)(1),and 60:13:2 2F1(a,b;c;x) _ x1 = F(a,b;c;x)=o (I);(c),In this Atlas the name "hypergeometric function" denotes any function that may be represented by a series ofthe form 60:13:3i(a1)i(a>1.. (axx),=u (c),(c:), ... (c,)iirrespective of whether one of the denominatorial parameters happens to be unity. Moreover, the integers K andL, which we use to characterize the numbers of numeratorial and denominatorial parameters, include any unityparameters. Thus, the Fd function generally has K = n and L = I + d. In the F. notation, function 60:13:3 wouldbe symbolized K.,F,(I,al,a_, ... aK;CI,C,, ... CL;x). CHAPTER61THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p) The complete elliptic integrals of the first and second kinds are simple. although important, univariate functions.More complicated. but less important, is the complete elliptic integral of the third kind, discussed in Section 61:12.All these functions arise in the canonical representation of integrals of certain rational functions that do not reduceto simpler expressions. 61:1 NOTATIONThroughout this Atlas we normally use the name "argument" and the symbol x to represent the single variable ofa univariate function. To achieve uniformity with Chapter 62. we here violate this rule and use the name modulus(module is also encountered) and the symbol p to represent the variable of K(p), the complete elliptic integral ofthe first kind, and of E(p), the complete elliptic integral of the second kind. Similarly, we denote the completeelliptic integral of the third kind (Section 61:121 by f1(v:p).An important auxiliary variable is the complementary modulus. defined by61:1:1 q =The symbols k and k' frequently replace p and q. As well, the role of the modulus is often taken over by themodular angle a or the parameter in, where 6 1 : 1 : 2 p = k = sin(s) ' V InThe complementary modulus then takes a symbol from the equivalences61:1:3yt-p =q=k'=coc(a)= yt -m= VmiEspecially when interest is primarily in the functions of Chapters 62 and 63, the naked symbols K and E arefrequently used to replace K(p) and E(p), a constant unspecified modulus p being understood. The notations K'and E' then mean K(q) and E(q), respectively. 61:2 BEHAVIORFigure 61-1 illustrates the simple behaviors of the complete elliptic integrals. They acquire values n/2 s K(p) 5= and n/2 ? E(p) ' 1 over their usual 0 - p - I domain. The inequality 61:3 THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p) 610 61:2:1IT1.57 = z a K(p) + In(q) >- In(4) - 1.39 0- 5P : 51shows that K(p) + ln(q) varies little. The behaviors of K(p) and E(p) asp -+ 0 or p -. I can be deduced fromthe expansions presented in Section 61:6.In many applications the complete elliptic integrals of complementary modulus are almost as important as K( p Iand E(p). For this reason. Figure 61-1 includes maps of K(q) and E(q).Normally, interest in complete elliptic integrals is confined to the zero to unity range of modulus. When pexceeds unity. K(p) and E(p) are complex valued; this circumstance is discussed in Section 61:11. The evenness of the functions, which is evident from equations 61:3:1 and 61:3:2, permits extension to negative moduli. Bothcomplete elliptic integrals acquire real values when their moduli are imaginary numbers of any magnitude: again,see Section 61:11. 61:3 DEFINITIONSThe complete elliptic integrals are defined byIdBdt61:3:1K(p)=1-p'sin'(0)o y'(! - t')(1 - p't) aanddt1` (1 + r)(1 + q=t') t 1 _ P:::1 _q 61:3:2E()1 -' sin2(0) dO =dr =dt p -Jop"-f1 - r"j1,(1 + r-the two integrals that involve the angle 0 being illustrated in Figures 62-2 and 62-3.Alternative definitions are as special cases of the Gauss function [Chapter 601:n11-n 61:3:3K(p)=2F( 2.2:l;p' F( 1 11 13 61:3:4 F-4 p) = 2F(2, 2:1;p') = 2FI2.2;l;p')or via Legendre functions of moiety order [see Section 59:4).The perimeter of the ellipse [see Section 14:14] shown in Figure 61-2, which has semiaxes of q and 1, equals4E(p). The common mean [Section 61:81 of the two semiaxes is 'rr/12K(p)].In the notation of Section 43:14, the complete elliptic integrals may be synthesized as follows:On" ,-061:3:5 K(p) i 2 V t- p 0 E(p) X= p'- 61:4 SPECIAL CASES There are none. 61:5 INTRARELATIONSIIIPSBoth complete elliptic functions are even:61:5:1 f(-p)=f(p)f=KorEThe four complete elliptic integrals K(p), E(p), K(q) and E(q) are linked by the remarkable Legendre relation 611THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p) 61:5 FL.alb%QuoQp f QpQp;~ ..... ................. 2.8 61:6 THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p) 612 IT 61:5:2 K(p) E(q) + K(q) E(p) _+ K(p) K(q)Complete elliptic integrals of modulus p are related to those of modulus -1Vp/(1 + p) by the transformationsp161:5:3KI1P)=(1+p)K(p)p) 61:5:4 EI2E(p) - 9'K(P)q= = 1- p= \\1+PI +Pand to those of modulus (I - q)/(1 - q) b 61:5:5 K(Iq)=1qK(p)1q.261:5:6 E(11- q) =E(p)+qK(p)1 +q.1 +q 61:6 EXPANSIONSAlternatives to the power seriesit1 5r- - 1), 61:6:1K(p)=-I +p +(1 X 3pa ry+p +pr l2\2 x4\12Xx34xx 6/= 2(2J(2j)!'((1 2p'1x3l2p'Ix3x51p5-1(2j-1)!161:6:2 ,--1[(=i1PP,J 2\,12 x. 4,32x4x652_o2jJmay be constructed by use of transformations 61:5:2-61:5:5. Whereas expansions 61:6:1 and 61:6:2 are most usefulfor moduli close to zero,/ the pair61:6:3K(P)=1n(q)+Ilnlq),][1q]1r.3q_1\L`Ix2291x23x4 2x4JIn4222(2j - 1)!! +- ----..._ q)+... qJI x 23 x 4(2j -1)(2j)(2j)!!Jandf( ll 261:6:4F(P) = I+ I Lln\q)Ix2][Iq]-+4[ln(4)23q1 x 23 x4][2x 4q=12'422 1J-1)! 2j -1[ln\q/_I X2(2j - 3)(2j --)(2j -1)(2j)][(2i(21)!!q+ .is more valuable as p 1.\ /A trigonometric expansion is provided by 61:6:5K(p) _ n sin(s) + - sin(5a) + -9sin(9a) += xrsin(a + 4 jot)a = arcsin(p) 464[2'j!]Forexpansions involving the 'nomc,' see Section 61:14. 61:7 PARTICULAR VALUESIn addition to listing values of the complete elliptic integrals at modular values of 0, and 1, Table 61.7.1includes p values at which the K(q)/K(p) ratio acquires particular values. These ratios correspond to special values 613THE COMPLETE ELLIPi7C INTEGRALS K(p) AND E(p)Table 61.7.161:8 I t2+V3 p=0 p-3-V8P=V6-V2P=Vi - IP 7 P=VVB-2 p= p=2VVI8-4-4 p - I Kip) Up) Klq)Kip)IT IF2 2UUA2 24U 2V3V2 11 1 1N2V3 20 of the "nome" [Section 61:14). The constant U that appears in this table is the ubiquitous constant [Section 1:7),equal to 0.8472130848. 61:8 NUMERICAL VALUESWe start this section by describing the procedure for determining the so-called common mean of two numbers.Let A) be the larger of the two positive numbers G and A,,. The geometric mean of these numbers is G, _V G0A0 while their arithmetic mean is A, = (Go + From the properties of means Isee, for example, Bron-shtein and Semendyayev, Section 11.14) it follows that Go < G, < A, < Aa. If the procedure of forming geometricand arithmetic means is repeated indefinitely so that61:8:1G,., =andA;., = (G, - A;)/2j = 0. 1, 2....one must have61:8:2and there is convergence towards a common mean. G. = Ate, of G and A,,.The common mean of the complementary modulus q and unity is related to the complete elliptic integral ofthe first kind byn 61:8:3K(p) = -G, = commonmeanof q and I 2G.This is the relationship that is exploited in the algorithm below. The complete elliptic integral of the second kindis evaluated via the expressionK(p) F_. _1w.7 Input p >>Set 11 = 0J77JSet g = L' 1 -P2(1) Seta=t=e=Replace e by e - g'(2) Set s = gaReplace a by (a + g)/2Set g = VsReplace t by 2:Replace e by e - r(a' - s)If t(a'-t)=6:< 10-0 go to (2)Set k = tr/2gsuccessive geometric (or arithmetic)means of q and I Storage needed:n. p, g. a. t. e. sand k Input restrictions:0 - p < I0 < p <.I 61:9 THE COMPLETE ELLIPTIC INTEGRALS Kip) AND E(p) 614 k = K(p)e-E(p)<Replace e by ek/2lfn=0goto(3)Output 4Output eSet it = kSet e=pGo toll)131 Replace r: he exp(-ak/nlOutput kOutput eOutput nTest values:K(1/2) = 1.68575035E(1/2) = 1.46746221Kt\ ? 2r = 2.156515r_._E(\ 3/2! - 1211056(1;of 1 "i = 0 01797_'3$7!, The algorithm ceases to compute successive values of the geometric and arithmetic means when sufficient havebeen calculated to ensure 24-bit precision in K(p) and E(p). If the green portions of the algorithm are included.values of K(q), E(q) are also evaluated, as well as that of the nome [Section 61:141 N(p).On account of the definitions 61:3:3 and 61:3:4, the universal hypergeometric algorithm of Section 18:14 mayalso be used to compute elliptic integrals of the first and second kinds. As well, the algorithm in Section 62:8generates values of K(p) and E(p). 61:9 APPROXIMATIONSFor rather wide ranges of small values of the modulus, the approximations/.1 61:9:1K(p) _sr16 - 5p-J8-bit precision0 s p fs 0.67216- 61:9:2it16 - 7p2E(p) --]6 -3p2)8-bit precision 0 s p :s 0.71 61:9:3K(q) = II + 4]In(4)-48-bit precision0 <p S 0.47 P61:9:4E(q) - I +2In(4)- 48-bit precision0 <p :5 0.35P/are valid, whereas for moduli close to unity we have z 61:9:5K(p)5-8 pz1016(1- z-1 -4p8-bit precision0.88 5 p < 1 P61:9:6E(p) _3 + pz1 -P21 _ pz44In168-bit precision0.945p<I T5pz 61:9:7K(q) -2\117++9pz)8-bit precision0.73 5 p 5 I 61:9:8E(q)2(19+3+73Pz)8-bit precision0.65 S p :s 61STHE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p) 61:10These results derive from expansions 61:6:1-61:6:4, from which more refined approximations we available. Be-cause of their rapid convergence, the expansions in Section 61:14 give accurate approximations even when severelytruncated. 61:10 OPERATIONS OF THE CALCULUSDifferentiation with respect to the modulus givesd 61:10:1PK(p) _E(P)p2) -KIP) d 61:10:2 - E(p) _PPdppIndefinite integration of the complete elliptic integrals yields K = L = 3 hypergeometric functions [Section 18:141 61:10:3 61:!0:6ap(Ot(i),(i)';K(t)dt =223p'o-'u (I),(I)j!2), E(1)d( = -o2-,I1),(1);(3),P fv 0tK(t)dt= E(p) -(1 - p'-)K(p)(1 +p:)Elp) - (1 -pK(p) 0tE(t)dt =3are simpler.Some definite integrals lead to Catalan's constant [Section 1:71 K(rldr = `+rE0dt = G 61:10:72f), aInteresting integrals involving the complementary modulus includera= 61:10:8 JKlti'I-t')dt=J Etv'Ir')dt=- > 61:10:9 61:10:101K(tIdt _K1a2V'1-t\Y'_I4U: V_/IraL' E(t)dlE(V'j)28U22 o' I - t-and more are listed by Gradshteyn and Ryzhik [Sections 6.14 and 6.15].The operations of semidifferentiation and scmiintegration with respect to p-, when applied to complete ellipticintegrals, produce elementary functions. Examples include 611011dir-E=('I-):: (p)(dpz)i.p 611012 Ki= V- :: (p)n(p),r arcs (dp_),_ 61:11 THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p)61:11 COMPLEX ARGUMENT616 The complete elliptic integrals of a complex modulus are themselves generally complex valued, but these functionshave real values when their modulus is purely imaginary. The transformations 1 61:11:1 K(ip) =KpN' pP V'YPand 61:11:2 E(ip) =V I + pr E(p1 +p'are valid for all real p. For example, K(i I = (l /\/2) K(l /\,2) and E(i) = V2 E(1 /V'2).The complete elliptic integrals acquire complex values when their real moduli exceed unity. The transformationformulas 61:11:3 pPp61:11:4F(P)Pfi\P,-CpPpermit the evaluation of the real and imaginary parts in this circumstance. They show, for example, that 61:11:5K(\/)( I+i )K( -=(l+i)V2-N/2)V8U 1i 11^ 1IISri 61:11:6E(VZ)=+[2E(=fl-i1-=V2 V2\ 2AV2V22K(\ 2)indicating that the real and imaginary parts are equal for a modular value of V2. 61:12 GENERALIZATIONSAllowing the upper limit to vary in the integral definitions 61:3:1 and 61:3:2 gives rise to the incomplete ellipticintegrals of the next chapter.In the remainder of this section we discuss the complete elliptic integral of the third kind f1(v;p) defined bydt-f tZdB61:12:1 II(v;p) =J J o (I + vt2)(l - t2)(1 - p2t2) o[I + v sin2(9)]l - p2 sin2(0)It is a generalization of the first kind of complete elliptic integral inasmuch as [1(O;p) = K(p). Be aware of thevariety of definitions and notation. Thus, the integral 61:12:1 is variously denoted 17,(v,k), f1(-iAaresin(p)), II(p,v, rr/2). It(a/2,-v,p) or even (-1/v)fl(-1/v,p,w/2) by different authors.The characteristic v may take real values in the range -- < v < x, although the integral is infinite for v =-1. Interest concentrates on the range 0 5 p 5 1 of the modulus. Figure 61-3 shows the behavior of 11(v;p) in its most important domain.One may evaluate the complete elliptic integral of the third kind via the incomplete elliptic integrals of thefirst and second kinds [Chapter 62]. The formulas permitting this are as follows:61:12:2I1(v;p) =K+pv+(1 + vxP + v){2+ [K(p) - E(p)]K(p) E(q;+) 1sin(41) =1+vv>0 aoa. ,a,moo TT(-0. 91p) : Tr(-(L 85op)Q....................... .rr<-20,p).5 1 -2 617 61:13 THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E4 p)61:12:3 l(O;p) = K(p)v = 0 61:12:4fI(v;p) = K(p) - 1(1 + v)(p' + v){E(p)F(p;4,) - K(p) E(p;d))}tivsin(6)PE( V---;)61:12:5fI(-p2;p) =p),=1 + vv = -PPvn 61:12:6fl(v;p) = K(p) +(1 + v)(P2 ' v){2[K(p) - E(p)) F(q:O - K(p) E(q.d,) 61:12:7 fl(- 1;P) = ± -V = -1Vt--vsin(k) _q618 -p2<s'<0 I <v<-p'` n 1 61:12:8II(v;p) _{E(p) F(p:d,) - K(p) E(p: )}sin(Or) =v < -1 (1 + v)(p + v)vwhere q =and F(p;d,) and E(p;d)) are incomplete elliptic integrals whose numerical values are calculableusing the algorithm in Section 62:8.For characteristics of magnitudes less than unity, the expansionI (p23vpgP411_ 5 (v2p23vp'Sp!) 61:12:9 II(v;p)=1-2\v2+8Iv2 + 8/16L2 +816J-(2j - Il!(2k -1)!!-p,'' (2j)!.' A_0(2k)!!vmay provide useful values of the complete elliptic integral of the third kind. 61:13 COGNATE FUNCTIONSIn addition to those of the first, second and third kinds, the following complete elliptic integrals may be encounteredin some works 61:13:1 61:13:2 61:13:3although not in this Atlas.D(p) = 1(`/'sin2(e)dOK(p) - E(p)=ZoI - p2 sin'(e) Pf.nto's2(9)deE(p) - g2K(P)1 - p2sin2(O)P2.nSing(e) CAS (e)de2 - P22 C(P) =J[1 - p2 sin2(6)]3n=iK(p) - P, E(p) 61:14 RELATED TOPICSAssociated with the functions of Chapter 61, 62 and 63 is the elliptic name. A common symbol is q, but to avoidconfusion with the complementary modulus, and to emphasize that the note is a function of the modulus, we shall use the N(p) notation. The nome is defined by 619THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p) 61:14 61:14:1N(p) =ezPaK(q)zZOsK(p))4 = V tpps land its utility arises from the large number of functions that can be expressed as infinite sums or products ofalgebraic functions of N(p). Except in the immediate vicinity of p = 1, the nome is a small positive number (0N(p) s ; for 0 5 p s 0.999995; N(p) s 1 for 0.999995 <- p <_ 11, so these infinite sums and productsgenerally converge very rapidly.The functions that can be expressed in terms of the note include the elliptic modulus:61:14:2p=4V (p)the complementary modulus: 61:14:3r l + N'(p)14r 1 + N'(p)lsr I + N6(p)14 .. . IL1+N(p)J L1+N'(p)J Ll+N'(p)J N(p)1411 - N'(p)14f 1 - Ns(p)144I + N(p)i + N'(p)I + N(P)the complete elliptic integrals of the first:K(p)(' 1 161:14:42=L-+N(p)+N4(p)+N9(p)+...1 and second:11SpI - N(p)i - N'(p)J L 1 - N°(p)]If,' ... (p) [ I) 61:14:5E(p) = Kip) -n LN(p) - 4N'(p) + 9N9(p) -K(p) :-N(p)=N'(p)-N°(p)+kinds, the incomplete elliptic integral [Chapter 621 expression:61.14:6K(p) E(p:b) - xE(p)N(p) sin(arx/K(p)J-N-(p) sin[2n.r/K(p)1+N'(p) sinj31rr/K(p)J2trI - N2(p)I - N4(p)1 - N°(p)where x = F(p:(b). the elliptic amplitude [Section 63:31:am(p;x)axN(p)N'(p) sin[2ax/K(p)JN3(p) sin[3.r/K(p)] 61:14:724K(p)I + ti=(p)+2[1 + N4(p)]+3[1 + N6(p)]+all four of Neville's theta functions [Section 63:81:2'R VN()az3ax5-rr6 1:1 4 : 8H,ip:.r)p9K(pIsin2K(p)I - N-(p)sin(2 ---- l =N"(p) sin(._'K(pl-...24\ N(p)arI, 31"Sax 61:14:96,P-..T) _- cos .+ N'( p ) costN'( P ) cos17K(p)2K1p))2K(p)2Krp)I1ax27ttSax6 1:1 4 :1 00,,(p: x)N(p) cosN4(p) cos(N9(p) cos(l Ktp)2-Kip),K(p). `K(p)/21r1axtaxI31rx 61:14:11H,( p;x) =- - N(p) cosl - l + N'(p) cos(IN9(p) cost -1 + ...J flgK(p)2\K(p)/1K(p)!K(p)!and all 12 Jacobian elliptic functions [Chapter 63J, as well as products. quotients and logarithms of such functions.Gradshteyn and Ryzhik (Section 8.1461 give a comprehensive listing of these, but the Atlas is content with thethree most important:61:14:12pK(p)snlp:z) =N"2(p) sin(-rrr/2K(p)j+N3, (p) ;in(3Trx/2K(p)lNs'(p) sin(5ax/2K(p)l2a1 - N(p)1 - N3(p)+l - N'(p)+ 61:14 THE COMPLETE ELLIPTIC INTEGRALS K(p) AND E(p) 620pK(p)N"(p) cos[rx/2K(p)]N'12(p) cos[3rx/2K(p)]N512(p) cos15rx/2K(p)J 61:14:13cn(p:x) = +++ 2r1 + N(p)I + N3(p)I - N5(p)K(p)I_N(p)cos[rx/K(p)]N2(p)cos[2rrx/K(p)]N3(p)cos[3rx/K(p)] 61:14:142idn(p:.r) -41 + N :(p)+I + N`(p)+I - N6(p)+A deterrent to the full exploitation of the formulas above is the lack of a simple method of evaluating N(p1from p. If p is sufficiently small, a truncated version of the series 61:14:15 AI(I-p21S0sp<12-r 211-p)will serve, but otherwise one must resort to iterative or numerical methods: the extended algorithm of Section 61:8provides one possibility. Some particular values of the nome are easily found by application of definition 61:14:1to the values of K(q)/K(p) listed in Table 61.7.1. CHAPTER62THE INCOMPLETE ELLIPTIC INTEGRALSAND The incomplete elliptic integrals of the first and second kinds are important bivariate functions. Together with moreelementary functions and the incomplete elliptic into teal of the third kind [see Section 62:12]. they may be usedto express any indefinite integral of the form fR(t,V p(t)dt (where p is a polynomial of degree 3 or 4 and R is a rational function) [see Section 62:14]. 62:1 NOTATIONThese functions are sometimes named Legendre's elliptic integrals. The adjective "incomplete" is often omitted:it refers to the fact that the upper limits in the defining integrals 62:3:1 and 62:3:2 are usually smaller than thosein the corresponding integrals 61:3:1 and 61:3:2 that define the 'complete" elliptic integrals. The symbols F and E are in general use for the incomplete elliptic integrals of the first and second kinds, but there is no unanimity on the ways in which the variables are specified.This Atlas uses the notation F(p;¢) for the incomplete elliptic integral of the first kind and we describe d asthe amplitude of the function and p as its modulus (k is often used). Other symbolisms are based on the parameterm = p` or the modular angle a = aresinl pl. so the notations F(4)lm), F(d)\a). as well as F(k..b), F(.b,k) and evenF(dt) are encountered. Since dt is often interpreted as an angle, it may be expressed in degrees, and occasionallysin((O) is treated as the variable instead of 4 itself.The situation is even more confused for the incomplete elliptic integral of the second kind for which the Atlasuses the E(p:4)) symbol. In addition to analogs of the variants discussed in the last paragraph, one also encountersE(x), where x represents the incomplete elliptic integral of the first kind, F(p:d,). As well, ambiguities can arisebecause the same E symbol is adopted for both the incomplete and complete (Chapter 611 elliptic integrals of thesecond kind. 62:2 BEHAVIORThe incomplete elliptic integrals are sometimes discussed only for 0 s p < I and 0 s d' n a/2 but they may bedefined for all moduli and amplitudes. They are real and finite, for all real p and 6, except that F(I :(.) is infinitewhen 41 a it/2. In fact. F(pab) and E(p:4b) may be real even when p is imaginary, as discussed in Section 62:11.Figure 62-1 shows graphs of F(p:e) and f o r p = 0. 0.5. 0.7, 0.9 and 1. 621 62:2THE INCOMPLETE ELLIPTIC INTEGRALS F(p:e) AND E(p;4)) 622 623THE INCOMPLETE ELLIPTIC INTEGRALS F(p;4') AND E(p;4) 62:3The incomplete elliptic integrals possess a high degree of symmetry. They are even with respect to p and oddwith respect to 46. The differences 62:2:1 F(p;(b) -240K(p)andE(P;4)) - - E(P)Itaare periodic (but not sinusoidal) with period it [see Chapter 36). Moreover, each of these difference functionsdisplays odd reflection symmetry about amplitude values of 0, ±ir/2, trr, -t31r/2, ..., as evident in equation62:5:2 and from the examples mapped in Figure 62-2. ......................................................0.2 01-0.2 62:3 DEFINITIONSThe indefinite integralsdefdt Jo62:3:1 F(p;dt) = = = 1 - p' sin2(O)=p"t21 + t21 + q=t' IMmI 6r62:3:2E(p:e) =1 - p' sin'(O)d0 = f0 0I V,V2qz = 1-P _261 =I-ptl+q -t Z, dt =, }dtq = 1 - p- rU1 +.define the incomplete elliptic integrals of the first and second kinds. The trigonometric integrals are illustrated byFigures 62-3 and 62-4. which also demonstrate the relationship to the complete elliptic integrals [Chapter 611.The incomplete elliptic integrals may be expressed as inverse functions [Section 0:31 of the Jacobian ellipticfunctions [Chapter 631. Thus, for example, if d, is the angle defined, in terms of variables p and x, by 62:3:3 (b = aresin{sn(p:x)} = arccos{cn(p:x)} = arcsinPthen F(p;4) = x andfr 62:3:4 E(p;4,) =dn2(p;r)dt = x - p'J`sn'(p;t)dtoIn discussions of elliptic functions, 4, is often denoted am(p:x).To define the incomplete elliptic integral of the first kind via a geometric construct, refer to Figure 62-5. Letpoint 0 be the r = 0 origin of a polar coordinate system [Section 46:14] (r,O). In this system let points I and P 62:3THE INCOMPLETE ELLIPTIC INTEGRALS F(p:(tI) AND E(p:d)) 624 411.4 04le...........................:...... 1/q I11-p2ein2(0) IIIIIIIIII111111113FIG 62-3 At .......................................D 111111111 unuuL11111u I MI"HILL11L1L1rn1-pzein2 (6) )E U .........................:...... pFIG 62-4.........................:........ q have coordinates (1,0) and (I/q,a/2), respectively, where q is the complementary modulus of F(p4). Now con-struct an ellipse [Section 14:141 with 01 and OP as its semiminor and semimajor axes. In polar coordinates theequation of this ellipse is r = I / l - p2 sin2(8). Let db be the amplitude and consider the average length f of theradius vector from the origin to the ellipse over the segment 0 s 8 s $. Then the product Pd; is defined as theincomplete elliptic integral F(p;$). That this definition is equivalent to 62:3:1 is evident by evaluating f as follows: 62:3:5E (pt 0) P=1Id8F(p;ib) `Y0rd8 =Jo1 - p2 sin2(6) It,A geometric definition of the second kind of incomplete elliptic integral is illustrated in Figure 61-2 of theprevious chapter. This shows a portion of the ellipse -q'V I --t 22 marked in green and delineated by the ordinatesi = 0 and r = sin(4). The length of this green arc defines the elliptic integral E(p:4)). The entire perimeter of theellipse is 4E(p). four times the complete elliptic integral of the second kind. 625THE INCOMPLETE ELLIPTIC INTEGRALS F(p;4) AND E(p:¢)62:4 SPECIAL CASES The incomplete elliptic integral becomes equal to its amplitude62:4:1 F(0;4o) = E(0;4,)62:4 when the modulus is zero, to the inverse gudermannian [Section 33:141 or sine function [Chapter 32] when themodulus is unity:62:4:2F(1;4) = invgd(4) = lnl tan\(; +2I4 s a/262:4:3 E(1;iO) = sin(4o)141 sir/2and to instances of the incomplete beta function [Chapter 58] when the modulus is V2 or I/V2: 11l1 62:4:4 F('V2;4O) =4 B/2'4'sin'(24b)/10:5 4 s ir/4 trl 31 62:4:5 4B``2 4' sin=(2-0) J0:5 4os a/4 I11I 62:4:6 FB Z; -: 1 - cos'(4) 0 <__< a/2 V2f 62:4:7EI 1cos'(-b)) +BI ! 3;I - cos'(d,) I0 5 4, < a/2V 24V22 442 4See Section 63:7 for Ftp:(b) and Elp;,b) when 4 takes particular values. When the amplitude and modulus areinterrelated such that cotta) = V q = i 1- p=)"', we have ICl 62:5THE INCOMPLETE ELLIPTIC INTEGRALS F(p:A) AND E(p:4) 1ITa 62:4:8F(p; arccot(V q)) = F(V I - cot4(d.):40 = 2K(p)4< d' <2 62:4:9E(p; arccot(Vq)) = E(V 62:5 INTRARELATIONSHIPS1Ar -cot'(1)4)=ZII-q+E(p)Jq -2626 The incomplete elliptic integrals are even with respect to their modulus but odd with respect to their amplitude:62:5:1fl-p:d,) = ftp:d,) = -f(p:-,b)f= ForEThe latter equality is a special case of the formula 171T/ nn' 62:5:2flp:-dt2nf(p;)-f(p:Td>}n=0,±1,±2, for reflection of theamplitudeacross any multiple of r/2. In the equation above. and that below. fi p,,-U12) is thecomplete elliptic integral K(p) or E(p). Both incomplete elliptic integrals obey the recurrence formula 62:5:3f(p;6 + nw) = 2nf P; 2) - f(P:d.) f = For En = ± 1, ±2, . .As a result, the difference functions 62:2:1 are periodic.Let p_ po and p, be three moduli interrelated by1-VI -po4625 =2Vpo :: P- andP, _1+VI-Pal+Poand dt- k andbe three amplitudes interrelated by 62:5:54,-, = 4ro + arctan{V I - p tan(4o)} andId+, _ - 14 + arcsin{pThen the corresponding incomplete elliptic integrals of the first kind are interrelated by62:5:6F(p-i;4-i)[IV I1 F(Po:d'o)andF(p,4,))PoF(po:(iio) and those of the second kind by 2 (I -1 po) E(P-A,-1)[E(po:dro) + \ P F(po:db)I - andE(Po:4J1 + po (I + pot62:5:7 1 -PoF(Id+Po:o2PO sin(40)IPo cos(4) +1 - pu sin2(db)1)I - Po + 2po cos(4MJ[Po cos(4,o) + VI - pp sin 2(4+o)1Going from elliptic integrals of variables po, dte to those with variables p-,, 4b_, is known as descending Landentransformation, while the conversion from variables po. dro to P. dr, is called ascending Landen transformation.The adjectives 'descending" and "ascending" recognize the fact that, for 0 < po < 1. the parameter P-, is smallerthan po, whereas p, is larger. Although it is not immediately apparent from 62:5:4, the numbers p_ : pr,: P, form 627THE INCOMPLETE E t tv'FIC INTEGRALS F(p*) AND E(p;4) 62:7a sequence, that is, the rule by which po is constructed from p_, is identical with the rule by which p, is formedfrom po. Similarly, though more obscurely, (b-1; din; (01 form another sequence, as do F(p-,;4o-,); F(Po;400); F(pl;4iJ and E(p-1;4_i); E(po;0o); E(p,;dij). It is evident that each of these sequences may be extended indefinitely in bothdirections. In one popular method of computing numerical values of elliptic integrals, one or other of the Landentransformations is implemented repeatedly, starting with po = p. 4o = di, and generating a sequence of moduli(either po; p-1; p-2; .. .; P-, or PO; Pi; p2; .p,) that satisfy the inequalities62:5:8 P < pi_, < ... < P-2 < P-1 < Pa < P1 < P2 < ... < P.-I < P.The limits as n - x of p-, and p are 0 and 1, respectively; hence, if the transformation is carried out a sufficientnumber of times, it becomes possible to approximate and E(p..pbt.) by use of equations 62:4:1-62:4:3.In this way F(p;(b) and E(p;di), as well as K(p) and E(p) are calculable. The "common mean' technique used inSections 61:8 and 62:8 is, in fact, an adaptation of the descending Landen transformation.Incomplete elliptic integrals with moduli exceeding unity may be related to ones in the standard 0 < p < Irange by the transformations62:5:9 FI 1:(b= PFI p;aresin{p1P\I.P/I(1I(sin(di)1 - psin(tb)62:5:10 E[P:40)= p EI p;aresin- P ) -pF p;aresinpp sYet another transformation is62:5:11 F(p;di) = K(p) - F(p;ii)62:5:12 E(p;di) = E(p) - E(padi) +p sin((i) sin(i(i)where sin(g) = cos(4)/l - p '' sin'O. 62:6 EXPANSIONSFor small values of the amplitude and modulus. there exist expansions of which 62:6:1F(P 4b) =2K(p)4) -[2KI p) -1J1sin($) cos((b) - ILr4K(P)_2P2 1- itIT 31T36Jsin3() cos(Qr) 2E(p) 11 4E(p)p2 62:6:2E(p; (b) =itdi +I -a2E(p)I''3 -6Jsin3(di) cos((b) +are the leading terms. Similarly, when p is close to unity: 62:6:3F(p; 4)) = !3K(qmvgd(di) -'K()- 1sinl<b)4K(q)-2-q2]9it ITcos'(rb)31T36cos'(di)1 - cos((i)N,'l - p' sin'ldi)2K(q) - 2E(q) [invgd(40 sin((i) 1q2 sin(di) 62:6:4E(p: di) = ++J - sinl(b)itcos'(di)2 cos2(di)Twhere invgd is the inverse gudermannian function [Section 33:14]. Gradshteyn and Ryzhik [Sections 8.117 and8.1181 may be consulted for the general terms in these expansions. 62:7 PARTICULAR VALUESTable 62.7.1 applies for n = I. 2. 3. ... and for any p in the range 0 < p < 1. See Section 62:4 for cases inwhich p acquires special values. 62:8THE INCOMPLETE ELLIPTIC INTEGRALS F(p:6) AND E(p:4))Table 62.7.1 h p: 6)FA p: 61 62:8 NUMERICAL VALUES-nnnn6= -= 6=- 6= 06'- 6= x9-_-nK(pi0nK(p)-_-nE(p)0n6( p)628 The algorithm of this section produces values of F(p:(b) and E(p:6) using an extension of the "common mean"procedure that is described in Section 61:8. Because applications of incomplete elliptic integrals often require valuesof the complete integrals as well, the algorithm also generates K(p) and E(p).Ut A, and G. retain their significances from Section 61:8 and let a set of their values, for a given p. be generatedby recursions 61:8:1. Further, let a set of angles d,,, 4;, d>,.... be defined by the recurrence formula 62:8:1 tan(21''6,_, - 2'4),) = G,tan(2'd)d j = 0, I. 2,...with db = 4), the amplitude of the sought elliptic integrals. Then. in addition to the results K(p) = a/(2G,) and 62:8:2E(P)=K(2)2-p'-Y2'(A;-C)- LLLLL1-1Jrowhich were given in the last chapter. we have62:8:3 F(p;2K(p)4,,/,rr = 4)x./GandE(p)F(p:d.) I 62:8:4E(p;dr1 =+VA - G,2 sin(2'd,;) (A, - G,) sin(2'-'d),. K(p)These equations are the results of applying Landen's descending transformation an indefinite number of times. Inpractice, the angles are calculated via the formula 1((A, - C,) tan(2'd,;l 62:8:54),,4 -are-A + G tan'(2'dt, l j'17'and converge toward d): as rapidly as A, and G, converge toward their common mean G. Input p > > Input 4 >>Sett=a= ISet E = 0»»Setg=Set e= 2(1+g2)(1) Set T = tan(4)Replace dr by d, - {arctanl(a - g)T/(a + gT')1}/(2t) Replace e by e - t(a - g)2Replace t by 2tReplace E by E + (a - g) sin(t4)Replace g byReplace a by l(g2/a) + al/210-1go to (I)Set It = 'R/(2g)Storage needed: t. a. E, g. e.d) and T Use radian mode. Input restrictions: -1 < p < I4) must be in radians, not de-grees 629k-K(p)< e = E(p) <- F-F(P;46)E-E(p;4)THE INCOMPLETE ELLIPTIC INTEGRALS F(p;4,) AND E(p;4,)Output kReplace e by re/(8g)Output eSet F = 4b/gOutput F«G<Replace E by (E/2) + 24e/a)Output E<GG<62:10 Test `values::KI//2\I = FI/2' 2`I = 1.68575036 EI/=EI`22/I= 1.467462212 I FI 2'4I=\0.80411366101 El-;4 I = 0.767 1 95 9862 The algorithm is designed to operate over the range IpI < 1 for all 4. The formulas of Section 62:5 may beemployed for moduli outside this range. For 1,01 s it/2, values of E(p;$) and F(p;4) generally are computed to24-bit precision. However, F(1;tr/2) = 00, and when both input variables are very close to these critical values,less accuracy may be produced. In this circumstance, transformations 62:5:11 and 62:5:12 may be applied withadvantage. 62:9 APPROXIMATIONSBased on expansions 62:6:1 and 62:6:2. the approximations 62:9:1 62:9:2PF(p:4,) _() 124,- sin(21b)1 + - sin(26) jpj0.59 ? :4,2Efp)E(p4.) =[24, - sin( 261] -Isin(26)n2pj = 0.72 > iiiare valid to 8-bit precision when both the modulus and amplitude fall within the specified range of small magnitudes.As p approaches unity, the approximations62:9:3F(p:o) _2K(q)[ln{tan(4,) + sect(b)} - tan((b)sec(,b)) + tan(6) secOd.) p-. IITandsec(6) -Ipp- sin'()2 62:9:4E(p:4,) -tang)+ - [K(q) - E(q)l[ln{tan((b) + sec(o)} + tan(6) sec(4,)1 P- Ibecome increasingly accurate, but the precision depends on the amplitude 4,. 62:10 OPERATIONS OF THE CALCULUSDifferentiation with respect to the amplitude gives 62:10:1d1E( p;6) == Vd(b- F(p:4s)ad)- p: sin'(() whereas, with respect to the modulus, we have 62:10:2 E(p:4,) - F(p:4,)ap E(p;.) =P 62:11THE INCOMPLETE ELLIPTIC INTEGRALS F(p;cb) AND E(p.4o) 630 and 62:10:3aE(p;d,)F(p;4)p sin(d)) cos(h) - F(P;cb) _ _--aPpqPqV I - p' sm'(dt)Indefinite integrals of incomplete elliptic integrals include 62:10:4 62:10:5 62:10:6arcsin(p sin(4))Jsin(g) F(p:B)dQ = - cos(y,) F(p:(b)nParcsin(p sin(d,))sin(do) sin(B)E(p:B)dO = +V I - p stn (d,) - cosld,) E(p:d,)f0F(p:B)d0_F'(p:d,)t( V1 - p` sin2(B) 2r. 0E'(p:4) 62:10:7 I -p sm'(B) E(p:B)dO =-'See Gradshteyn and Rvzhik [Sections 5.12 and 6.11-6.13] for a few other indefinite integrals and many definiteintegrals. 62:11 COMPLEX ARGUMENTThe incomplete elliptic integrals[K( of purely imaginary modulus are real:`l62:11:1Rip; (b)PF( d, JJ-F(b 1 *p1p%, IP2/V' 1 + p"V I + p62:11:2E(Ip:d,) = V 1[E(P1.'I+p".V1+p" 2Pp' sin([,) cos(y) 1 + p 2 EV l --P:717where tan(J.) =1 + p" tan(dr). However, for imaginary amplitude. the incomplete elliptic integrals are themselvesimaginary and involve the gudermannian function [Section 33:14) as well as hyperbolic functions (Chapters 30 and 28):62:11:3 F(p;id,) = IF(q;gd(d))q = V62:11:4E(p;id,) = i [F(q;gd(d,)) - E(q; gd(d,)) + tanh(d,) I + p2 sinh'(d,)]For a complete list of similar transformations, see Gradshteyn and Ryzhik [Section 8.1271. 62:12 GENERALIZATIONSThe incomplete elliptic integral of the third kind: 62:12:1dO o(I + v sin2(0)]1 - p '" sin2(0)is a generalization of the first kind because 17(0;p;4,) = F(p;d.) The notation is as confused as for the correspondingcomplete integral [Section 61:121. Some other special cases include 631THE INCOMPLETE ELLIPTIC INTEGRALS F(p;4) AND E(p;m) 1I 62:12:2HI(- I;p;(W = F(P4) - - E(p;40) + - tan(+) 1 - p2 sin=(+)q = V1 -p'qqIarctan{(162:12:3 f1(}p:p:4P) = 2 F(p:4) ± p) tan(+)/1 - p' sio=(.6)} +2 tE(P;))P= sin(+) cos(*) 62:12:4 q-(1 - p2)I - p2 sin2(4)( arctan{1 + v ttm(*)} 62:12:5-v 1-1V = -1 l(v;0;O) =1'tan(4,)artanh{-v - 1 tan(+)}-v- 1v< -1 invgd(tb) - V; artaah{ V v sin(.0)} v L. 0 I+v62:12:6II(v;1A )) _Iv = -1invgd(4) - V -v artanh{ V -v sin(4i)} 0 ? V * -1 andI +v 62:12:7 fIl v;p;= I1(v;p)262:13 Because itis trivariate, numerical values of this third kind of elliptic integral are not easy to calculate. Forsmall magnitudes of the characteristic, v, the expansion 62:12:811(v;p:iO)(2j - 1)!!(-v)11(2k - 1)!!()5e - cos(d)+ 1sine+i(4)[1-0(2 + 1)!!]may be useful. The double factorial ratio [see Section 2:131 occurs repeatedly in this expression. When jvt > 1,62:12:8 does not converge, and one must resort to rather complicated computational methods that are described byMime-Thomson [see Abramowitz and Stegun, Section 17.7, but note that his n equals our -vj. 62:13 COGNATE FUNCTIONSTwo functions that are closely related to the incomplete elliptic integrals are Jacobi's zeta function: 62:13:1 E(p:iO) -E(p)F(p;4b)K(p)and Heuman's lambda function:2K(p)2K(p) - 2E(p) 62:13:2 E(q;(b) -F(q;(b)q = VI - p- a 17Neither is used in this Atlas. 62:14THE INCOMPLETE ELLIPTIC INTEGRALS F(p;d') AND E(p:4)62:14 SPECIAL TOPICS632 Historically, incomplete elliptic integrals arose as an attempt to codify the indefinite integrals fR(t,\:j)dt wherep(t) represents either a cubic function or a quartic function:62:14:1p(t) = ±p3(t) = ±W + at' + bt + c) orp(t) = p4(t) = a.?a,? + a,t` + a,t + aeBy R(x,y) we mean a rational function [Section 17:13] of x and y, that is, a quotient of polynomials in x and Y.The name "elliptic integral" is sometimes given to this very general class of indefinite integrals.By straightforward algebra [see Korn and Kom, Section 21.6, for details] am- indefinite integral of the formfR(t,\")dt may be reduced to a weighted sum of integrals of the following forms: 62:14:2 tdt V (pt)dtf(t - constant)N p(t)Here R; (t) is a rational function oft as discussed in Section 17:13, and its indefinite integral is easily found viathe partial fraction decomposition discussed in that section.The first three integral types in 62:14:2 may be expressed as incomplete elliptic integrals of the first and/orsecond and/or third kinds. We shall illustrate how this occurs, using the first member of 62:14:2, with p(t)±p3(t) as our example.As discussed in Section 17:7. the cubic function p3(1) = r' + ar' + bt + c will have one, two or three distinctreal zeros according to whether the discriminant D is positive, zero or negative. The D = 0 case is easily treatedbecause the cubic function must then be of the form p,(t) = (t - r,)(t - r)2, where r, and r: are the zeros, leadingto-x-r, 62:14:3 dt-dt2=arcothf- -V'P)(t- r2)t- r,Vr;-r,if x > r2 > r,, or to some equally elementary result otherwise.If the cubic has a single (real) zero, r, then the sought integral involves the constants 1 2 - Or + a)h= 62:14:4h' =andP, = I - 4, =43rand takes one of the following forms: 62:14:5 dtJ= hF(p:2 arccot(h))P3(t)r - x62:14:6 hF(p;2V p3lrJ 62:14:7 62:14:8= hF(q;2 hF(q;2 arccot(h V r - x))dt-- V -p3(t)x5r according to the relative values of x and r.When the cubic function has three zeros, denote them r,, r2 and r3, where r, < r; < r3, and define the constants 62:14:9 h2 =4andr3-r,2-q2=h2(r2-r,)r2-r,4r3-r,R,(t)dt Then the following integrals are valid: 633THE INCOMPLETE ELLIPTIC INTEGRALS F(p;sb) AND E(p;+) 62:14 62:14:10 J='= hF1 p, arrcoslr3 sx 62:14:11 62:14:12 62:14:13 62:14:14 62:14:15 62:14:16 62:14:17r=hF(p;aresin,x-r Jry= hF q;arccosl- rZrjdtr(1z))= hF`q,aresin-J J Iq J ,2at= hFl p;arccoslp11\\/dt(rII=hFkp;aresinl IIVpslt)`r2-ri//// = hFlq;arccosrt - r)) dtJI-V-p(t)= hF`q;aresin`r,-xr, <xsr2 We refer the reader elsewhere [for example to Erddlyi. Oberhettinper. Magnus and Tricomi. Higher Tran-scendental Functions. Volume 2, Section 13.5] for a discussion of fdt/V p4(t) and for treatments of the other formsin 62:14:2. CHAPTER63THE JACOBIAN ELLIPTIC FUNCTIONS These 12 functions have several interesting properties. One is their ability to bridge the gap between circular func-tions [Chapters 32-341 and hyperbolic functions [Chapters 28-301. Another, discussed in Section 63:11, is theirdouble periodicity. The close interrelationship of the twelve elliptic functions is detailed in Table 63.0.1.The a terms in this table take the values + I or - 1 according to the value of the quantity 63:0:1 w =frac(4K(p))as shown in Table 63.0.2. Explicit expressions for the or terms are63:0:2a2 = (-I)mn2wia3 = (_I)",'a, = a,a3Notice the analogy to the concept [Section 32:131 of the sign of the circular functions being dependent on thequadrant" in which argument falls. This is one manifestation of the fact that K(p) plays the same role in elliptictrigonometry that tr/2 plays in circular trigonometry. 63:1 NOTATIONJacobi's name is not always attached to these functions. Alternatively, only the sn. en and do functions may beassociated with Jacobi, the other nine being attributed to Glaisher, who invented their notation. Individual ellipticfunctions do not usually carry a name, although sinus amplitudinis has been applied to sn. cosinus amplitudinis tocn and delta amplitudinis to dn.The Jacobian elliptic functions are bivariate, but one frequently encounters notations, such as sn(x), that suggestonly a single variable. The second variable is then implied and is regarded as a constant. We shall avoid thisoversimplification in the Atlas and write, for example, sn(p;x) where p is the modulus and x the argument of thefunction. Commonly k replaces p and a replaces x, as well, the order of citation is often reversed so that sn(u.k)may be encountered. Rarely in is used for sc.A number of supplementary univariate and bivariate functions arise in discussions of elliptic integrals. Theseinclude the complementary modulus q = V l - p2, the complete elliptic integrals [Chapter 611 K(p) and K(q), theelliptic amplitudel - dn ' (p,r) 63:1:1am(p;.r) = arcsin{sn(p;x)} = arccos{cn(p;x)} = aresin }_p 63:2 Table 63.0.1 7-7I-f-f-I-I-I-1- r-r- -a'lplldlp.,Isll f.+a?,dIO.1<af per:dY p,ianp:+!dm p.,rnst p:,,cIp,dIDsI aJV'I-c7'V 14 rr=a.:a,f a,y7'-I\II-&fa,7fV I - f %VI-df:Vq:-r.Vp'-p1 e,afa ID!VI-f:\- p:\ p: r OlrpepVf:_ 11VIIIVI.d1-VI-I°VI-p l' a1ratpf V'I - 9-f" V.I - v=flo,f77V I - f :\ l - / -f: - .alr... - aol p1I - mff.,i ,VI.r11 VI'dfVI-f°VI+f=v:`!:\Y:-oL(r!n,pl\/:-r\p"a')'a,p,VII-y-fV'I-II,ara,\/ -y%pIf\i.I\ 1V'I -a. N, q'7Pa,r1V 7--;-r--V'a: ":o,aV' y" ' Pr- o,yra,Df\ 77-77-7r777,THE JACOBIAN ELLIPTIC FUNCTIONS a,VI - l:a.fiVr+Fvl-vPTT \%oIyfIappa.f o,pl110 - r'a_\ r I fffVI=-rVI:+*IPV'rrVr=-f'fJUIf I- fIV'1-.,VV1.-nldP <Vl-a:r:VITaCIvl771 77- dlp,l -\'I. dfl 11117 fI7'Pnrpa,ra.p a,fV' I - p1(1 '7°- 1a,pr7-7V.If:Vfp%ff and the incomplete elliptic integrals [Chapter 621 F(p;cb) and E(p;4). The square root \'I - p2 sin2(b) _I - p2 sn2(p;x) occurs often in the theory of elliptic functions and is frequently abbreviated to A(p;cb) althoughwe use dn(p;x).The 12 Jacobian elliptic functions may be classified into four groups, each with three members, according tothe second letter of the function's name. Thus, the cs, ds and ns functions are said to be copolar: they all possessa pole of "type s."We shall use ef(p;x), fe(p;x), ge(p;x), hg(p;x), etc., to represent arbitrary Jacobian elliptic functions, it beingunderstood that e, f, g and h are letters selected from the quartet s, c, d and n. Often we shall reserve e to representthe pole type. 63:2 BEHAVIORThe Jacobian elliptic functions display interesting properties when the modulus and/or the argument is imaginaryor complex. In this section, however, we restrict p and x to real values. Moreover, as in Table 63.0.1, we shall Table 63.0.2 O s W Gt W Ct W GS W< I Qi+1+1+ 1a,*1I-1+1 637 THE JACOBIAN ELLIPTIC FUNCTIONS 63:3consider p to lie in the range 0 5 p 5 1. though equations 63:5:2 and 63:5:14-63:5:16 show that this restrictionis not mandatory.Except for some of the functions when p = 0 or 1, all Jacobian elliptic functions are periodic, their periodsbeing either 2K(p):63:2:1fe(p;2K(p) + x) = fe(p;x) fe = sc, cs, do or ndor 4K(p):63:2:2fe(p;4K(p) + x) = fe(p;x) fe = sd, sn, cd, cn, ds, dc, ns or ncThis is supported by Figures 63-1 and 63-2, which together map all the elliptic functions forp = 0.9. This particularvalue of the modulus was chosen to emphasize the nonsinusoidality of the sn, cn, ds, dc, do and nd functions. Forsmaller positive values of p, these six functions become increasingly sinusoidal [see Section 32:1 1 in shape, whilethe shapes of the other six Jacobian functions come to resemble those of the functions of Chapters 33 or 34. Asp - x, the periods increase indefinitely, and most of the functions degenerate into hyperbolic functions as explainedin Section 63:5.The ranges of the sc and es functions are unrestricted, but the other 10 elliptic functions are constrained asfollows, for jpj s 1:-1 163:2:3 - <_ sd(p;x) 5 - q =qq63:2:4 -1 s fe(p;.r) 5 1fe = sn. cd or en63:2:5 q s Ids(p;x)l s x63:2:6 q s dn(p:x)I63:2:7 1 s fe(p; r)I < xfe = dc. as or ncl 63:2:8 1 <- ndtp;x) 5 -qBecause the Jacobian elliptic functions have periods that are proportional to the elliptic integral K(p). it issometimes fruitful to treat the ratio v = x/K(p) as an independent variable in considering properties of ellipticfunctions. This has been done in Figures 63-3, 63-4 and 63-5, which, for a range of values of v and for the moduli in Table 63.2.1, display the copolar trio sn(p;x). cn(p;.r) and dn(p:x). These are, perhaps, the most important ofthe 12 functions; moreover, the other nine are easily calculable from them via equation 63:5:1. 63:3 DEFINITIONSThe Jacobian elliptic functions are defined via the inverse of the incomplete elliptic integral of the first kind [Chapter621. Thus, if 63:3:1 thenx = F(p:rb) _dOJ.,I p 63:3:2 sn(p;x) = sin((,) = sin(am(p;x))Jhe other I I elliptic functions can then be defined via the interrelationships reported in Table 63.0.1. For example:63:3:3cn(p;x) =1 - sn'(p;x) = cos(4)) = cos(am(p:.r))and63:3:4dn(p;x) = 1p' ,n2(p;.r) =I - p' sin'ob) _l - p' sin:i amt p:.r)) 63:3 THE JACOBIAN ELLIPTIC FUNCTIONS 638 639 THE JACOBEAN ELLIPTIC FUNCTIONS 63:3 63:3 THE JACOBIAN ELLIPTIC FUNCTIONS 640 ..:....:....:....:............Q.9 : F 1 6 6 3 - 3 :tx):....:....:..0.6-/t.:....:....:en(pv-x/K (p):.IFF/%//./ :....:....:....:....:....:....:....:....:....:..o.4 The remainder of this section is devoted to making a geometric construction that may be used to define all theJacobian elliptic functions. In Sections 29:3 and 33:3 the six hyperbolic and the six trigonometric functions aredefined in terms of the lengths of the sides of three similar right-angled triangles. In strict analogy- six of theJacobian functions may be defined as the nonunity lengths of the sides of the triangles OAC. OGI and OJL as depicted in Figure 63-6. Here the angle d, equals the elliptic amplitude am(p:x). and those sides of the trianglesthat are dashed have unity lengths.Further construction is needed to provide definitions of the other six Jacobian elliptic functions. First. super-impose the three triangles OAC, OGI and OJL, as shown in Figure 63-7. Then select the point K on line JL such that length LK equals the complementary modulus q. Join points 0 and K by a straight line and denote its points of intersection with lines AC and GI by B and H, respectively. Three more elliptic functions may now be definedin terms of this new line, namely63:3:5length OB = dn(p;x)length OH = dc(p;x)length OK = ds(p;x)Yet more construction is required to define the remaining three functions. Measure unity length from point 0toward K, and so create point E. Next draw line DEF through E perpendicular to OL to cut lines OJ and OL at Dand F.Then63:3:6length OD = nd(p;x)length DF = sd(p;x)length OF = cd(p;x)An alternative construction can locate line DEF without recourse to line OK. Superimpose Figure 63-4 onto Figure62-5, both diagrams being similarly scaled and sharing line 01 and angle d in common. Then the ellipse cuts lineOJ at D. as depicted.Finally, to emphasize the symmetry of the definitions and the logic of Glaisher's notation. disnren,ber Figurc63-7 into its four similar components, each consisting of a right-angled triangle with an additional line. These four 641 THE JACOBIAN ELLIPTIC FUNCTIONS 63:4components are illustrated in Figure 63-8. on which diagram all 16 lengths are identified. To better accentuate thesimilarities, we have adopted the notation63:3:7 nn(p;x) = dd(p;x) = cc(p;x) = ss(p;x) = 1for those lines in the diagram that have unity length. This notation is consistent with Glaisher's.Similarity and pythagorean rules applied to Figure 63-8 lead immediately to the relationships in Table 63.0.1 .The dismemberment has segregated the elliptic functions into copolar groups. Note that the triangle of pole typec is always larger than that of d, which, in turn, is necessarily larger than the triangle of pole n. The size of thes triangle, however, need not be larger than that of c or d, though it must exceed n. 63:4 SPECIAL CASESAll Jacobian elliptic functions reduce to a trigonometric function, or to unity, when p = 0. Similarly, all Jacobianelliptic functions reduce to a hyperbolic function, or to unity, when p = 1. Because of relationship 63:5:1, Table63.4.1 provides sufficient information to identify all these special cases. The examples 63:4:1 63:4:211dc(0;x) =_= sec(x)cd(O;x)coa(x)nd(l;x)cosh(x)ns(l;x) =_= coth(x)Sd(l:x)Slnh(.C)illustrate how the table may be extended.ti,a40 :b,o049O4Jga4*t1.0FIG 63-4 :cn(pIx);....;..0. 8v-x/K (p)...................... .............. :....:. 0. 6 ........................ ..:....:....:0.4 i:....:..0.2 63:5 THE JACOBIAN EWYrIC FUNCTIONS 642 "OtiOAsRe0.tp'0FIG 63-5 ::........ ;)....:....:.. 0.8i(;ntpt x :....4....X.\. 0.a09:....:...:...V V..\ .... :...:.... :.... .... ..:..0.4 63:5 INTRARELATIONSHIPSTable 63.0.1 gives explicitly the relationship between any two Jacobian elliptic functions. If e, f and g representletters drawn from the quartet s, c, d and n, then 63:5:1 fe(p;x)= fg(p;x)e, f, g = s, c, d, nge(p;x)As in 63:3:7, ff(p;x) is interpreted as unity. Table 63.2.1 pK(p) o 1.5710.8 1.9950.9 2.2810.95 2.5900.99 3.3570.999 4.4960.9999 5.6451 - 10' 11.401 643 THE JACOBIAN ELLIPTIC FUNCTIONS 63:5All elliptic functions are even with respect to their moduli:63:5:2 fe(-p;x) = fe(p:x)all feand either even or odd with respect to argument:63:5:3fe(p;-x) = fe(p;x)fe = cd, cn, dc,dn, nc or nd63:5:4fe(p;-x) _ -fe(p;x)fe = sc, sd, sn. cs, ds or nsThe periodicity of the Jacobian functions is expressed in equations 63:2:1 and 63:2:2. With n = 0, ±1, ±2....,we have63:5:5fe(p;nP + x) = fe(p;z) fPP = 2K(p)fe = sc, cs, dn, or nd= 4K(p)fe = sd, sn, cd, cn, ds, dc. ns or ncA great many reflection and recurrence formulas are summarized in Table 63.5.1; it may be extended by use of63:5:5.Addition/subtraction formulas for a copolar trio aresn(p;x) cn(p;y) dn(p;y) ± cn(p;x) dn(p;x) sn(p;y) 63:5:6sn(p;x ± y) =I - p2sn2(p;x) sn2(p;y) 9 63:5 THE JACOBIAN ELLIPTIC FUNCTIONS Ce (Pt X) 635:7cn( x t v) = 635-8dmt ' v) =cnr p:x) cn(p:v) ± sn(p:x) dn(p:x) sn(p:y) dn(p:v)1 - p sn ( p;x) sn (p: y)dn(p;x) dn(p: v) = p- sn(p:x) cn(p.x) sn( p: v) cn(p: v )p.-I - p' sn_( p;x) snI(p:y)The use of rule 63:5:1 enables all other fg(p.x ± v) to be evaluated as the reciprocal of one member of the trio.or as a quotient of two. Double-argument formulas may be obtained as fg(p;x + x). Thence one may derive 63:5:9JL_-cn(p-_'x)_I sn(p;x) dn(p;x)dn(p;x)_sn(p;x) I+ cn(p ?c) cn(p;x) Ics(p;x)andV1 - dn(p2x)63:5:10 =I + dn(p;2x)which lead to the half-argument formulas 63:5:11p sn(p;x) cn(p;x)dn(p;x)p cn(p;x)ds(p;x)(X _ 1 - cn(p;x)saZp' 21ddn(p;x) Table 63.4.1 sd( p:x)cd(P.xIa* p:x)P=0P - Isin(s)sinh(r)coslx)I 1cosh(x)p sn(p;x)dc(p;x) 645 THE JACOB LAN ELLU'llC FUNCTIONS 63:5 63512x\cn(p;x) + dn(p;x)I + cd(p:x)',I' :: cnp;21 + dn(p;x) 1 + nd(p;x) 6353/x 1_q' + p2 cn(p:x) + dn(p:x)' ::1 Idnp''1 + dn(z)Again, rule 63:5:1 may be used to extend these formulas to any fg2(p:x/2). Because of the interrelationships citedin Section 63:0. all such formulas may be expressed in a vast number of alternative ways.The so-called Jacobi real transformations: 63:5:14Ixsn p;x =psn p;p 63:5:15 cnll:x I= dnlxp/pp 63:5:16 dnl;x)= cnlIpp'Ppermit the evaluation of elliptic functions for moduli exceeding unity. Once again, rule 63:5:1 permits extensionfrom this copolar trio to all other fg(I/p.x) functions.The principles of Landen transformation are explained in Section 62:5. This transformation may be applied tothe delta amplitudinis in either the descending or ascending mode through the following equations. If1 - N I -p. 2Vpn63:5:17 PE = andp, =I - v I - p1 - p and I635181 + ` I -=dl+ P,,).,[,,_ :: r_ I(po)x0an.r, then1 p -63:5:19dn(p_,:.r-,) =(1 + v l -dn(po:.ro) Table 63.5.1 x(p:Xlsd1p.X) _sill p:X)csip:X) _cd(p:X) _cn(p:X) _d5(PA)dc(p:X) - dni p: X)ns(p:X) _nc(p.X) _ndl p:X)anddn(p,:x,) =dn(pn:-r0) - V dn-(p,,:-rn) - I + poI + Pn X=.r-_K(PiX=.r - Kip,X=KIp) - rX=Klpl -x.t'=]K(pl- rX=2K, pl - xs lP:xI-q cs(p:.r)q- cslp:.c)-q csl p:.ri-x(P:.r)scl P:.U- SJip'..r)-q cn(P:rlqcn(p:r)q 'cn(p:r)d(P:rl-sdip:x)-snlp:.r)-cd(p:rlcd(p:x)cdlp:-rlsn(p:.r)-snl p:.r)cu P:n-gsc(p:xlgsc(p:.rI-gsc(p.r)-cs(p.xlcs(p:.r)-cd(p:x)nip:.nsnip:.,)-sn(p:xl-cd(p..r)-cd(p:x)-cn(p:-qgdfP:xlqsd(p:x)-gsd(p:x)-cn(p..r)-cnlp:.r)ds4p:.r)- gnclp:.rlgnc(p:.t)gnclp:.tIdslpr)-ds(p:x)-dc(p:.c)ns(p:.r)ns(p:.r)-ns(p:r)-do p:.r)-do p:r)dnlp:.r)gnd(p:x)gnd(P:x)gndip:rldn(p:.rldnip:x)-ns(p:.r1-dclp:.r)dc(p:.r)Jc(p:-r!ns(p:.r)-ns(p:.r)-nc(p:x)q-,dlp:.rq_ ds(p:x)-q- ds(p:x)-nclp:x)-nc(p:.dn(fi p:xIq-dill P..rq 'dnip..t)q Jn(p..Und)p:.rind(p % 63:6 THE JACOBIAN ELLIPTIC FUNCTIONS 646A popular technique for evaluating elliptic integrals of modulus p and argument x is to set p = p(, and x = x,, in63:5:17 and 63:5:18 and then sequentially transform dn(p(,:x6) via 63:5:19 into dn(p_I:x_1), dn(p_2:s_2).._. (orinto dn(p,:x1), dn(p2:x2)....) until p" has become so close to zero (or p so close to unity i that the approximation 63:5:20 _a) = 12 (or 63:5:21dn(pa:xa) = sech(xa)l +1p" (sinh(x,) +x" tanh(x"))f 4Jin the case of the ascending transformation) becomes valid. 63:6 EXPANSIONSExpansions of the Jacobian elliptic functions exist under three circumstances: as a power series in x. valid for small.t: as a power series in p involving trigonometric coefficients, valid for small p: and as a power series in q involvinghypcrbolic coefficients, valid when p is close to unity. The leading terms in these three expansions are given inTable 63.6.1. In addition to the elliptic functions. the table lists expansions of the elliptic amplitude am p:x). SeeSection 61:14 for expansions in terms of the -norrte.- Table 63.6.1r-.0t2-p-1r6120n - 2p')xn + (4p - i 4f'ir'x - -61211 flis'rl - 14p- + p',6120I(2-p'Ir6360-(1-pal x'+(I-P). +..24I'0 + 4p't r'224 I0- 2p111 + ( + 8p' - Sp')1' +163601+(1-pyre+(5- 6p' 4p')i+..224x' (+ )224I(1+pi x+(7-22p'+lip)t6360 t2(5 - 4p')4"224+ P} +(2p' - p)t224p.6120P-0 12.4-am 2+17(_,(0414 -b uT-11 .041104 14,11 t2r - sinl2.o)p'114111--8 seal[21 - 104121)1 P-ea+ ..u)8 Sm'l1)CMI) ++-.4 .410x)121 - sm12o] ptcola)B 010(+1[canal - x) p'0101x1 ---4 u11(1) an(x) 1001x) -4 colt) COSIt)1 -4 -No[2. - mo(2o)] p'0401x) +8 tank) sing)]2x - 1(2t)] 080ota)0osa) 4 cocko[21 - 104)214] ptr8P- 1-ht,)snhl2, -?r1 q"B sechlxl .Inhl ,- ,Inh-1,. -.oIhl,)i _I -hl 21, - 21)184115 -.1.,)8 cosh'Ii[1046)21: - 211 g-cstlux)8 tanhu) sinhul + 2 csclri.[,1m1,2.11 - 2r] g'1(01$(1) . +..B cvN1$r1) 00116)x)[I - sinb'(x) - r 0(01$1x)1 q'cscha) -4 slnhl..1I+q+---2 00&mIsi41N2x0 + 2s) qt1001$)x) +B coth(I) 0046(.4 ))sinh(2r) - 2.1 q'coil(s) - 8 coshlt; 647 THE JACOBIAN ELLIPTIC FUNCTIONS63:7 PARTICULAR VALUES63:8 These occur when the argument of the Jacobian elliptic function is a multiple of the complete elliptic integral K(p).Let v = x/K(p), then for n = 0, t1, t2, ...: v-4nin+qv=4n * 1r-- 4n+4n+ 24n+v=4n+3v=4n+: sc(p:x) sd(p:x) ca P:-t) ds( p:x)dc(p:x)dn(p:x) ns( p:.c I no p:x) nd( p..)-I -I0 ° 1 -I-1-1°Vq(1 + q) 4q(I -+q)° q(Y -I- y) qq(I 1-1-107771+q01+q-I+q.nV90-VqZxVq0-Vq-1 -1 I1I+q0I+q-IVlq0V I + q1IIr_qq0-IQq-I 0CxVol +q)qVq(l -q)-Vq(I - q)-q-V ql q) 1VI+q2x-V1+q-I-=zVqVq qVq IVqqVq xVI -q IVI-q -VI-q -I-V1-ql_q 1q1- i_ `'tq I`q--`f4-i-\qxV9V f I IVqqV'q Vq qVq 63:8 NUMERICAL VALUESOur algorithm for calculating values of any one of the 12 Jacobian elliptic functions makes use of Veville's thetafunctions. There are four such functions and they are defined in terms of the theta functions of Section 27:18. Oneof the definitions:qId- 0,(tK(2q)) 63:3:10,1 p:.t) = a0,tKI( 2K(p)dx_Kip) ,TK(p)',_'Jis more complicated than the other three:xK(q) (KIq) 63:8:20,(p:x) = 0. II l0, 0:l= \/qO,(p:Kip) - x)\2K(p) rK(p)!-.,K(p)/63:8:3 0a(P:2r) =0,(` x-K1 q10, 0_K( q)K(p) irK(p)/\trK(p)(rK(q)/o(o.K(q) I 63:8:40,(p:x) =042K(P)IrK(P)trK(p)) =0a(P:K(p) - x) Here, as usual. K(p) and K(q) denote the complete elliptic integrals (Chapter 6l1 of the modulus and complementarymodulus, respectively. 63:8 THE JACOBLAN ELLIPTIC FUNCTIONSEvery one of the 12 Jacobian elliptic functions is expressible as the quotient of two distinct Neville thetafunctions, the literal subscripts serving as a mnemonic to identify the numerator and denominator of the quotient:thus:648 63:8:5 cs(p:x) =N,( p:x)nc(p:x) =p:x)etc.O,i. p:x)OJ p:x)By exploiting this principle, the algorithm becomes compact and rapid.A shortened version of the algorithm in Section 61:8 is first used to calculate r/[2K( p)] and the nome [seeSection 61:14] N(p). The user then supplies the argument x, followed by a two-digit code. The algorithm decodes the latter to identify the denominatorial Neville function then the numeratorial Neville function. If N, is called for.the algorithm computes \ qK(p)/2r 8,(p:x) via equation 61:14:8. Similarly. if 0 is needed. \ qK( p)/2r 0,(p:x)is evaluated by use of 61:14:11. The second equality in 63:8:2 or 63:8:4 is utilized whenever Ii, or 8,, is required.Finally. the two VqK(p)/2r/2r 0 values are divided to produce the sought Jacobian function.The series 61:14:8 and 61:14:11 are used truncated to only the terms actually listed in Section 61:14 so thatthe largest neglected term is of order N°7i p). Nevertheless, these series converge so rapidly that 24-bit precisionis virtually assured provided p does not exceed 0.99.Because intermediate numbers are stored. one need on]% input the new argument .t and the new code to calculatea second Jacobian function of unchanged modulus. Input p > >Set n = 0 Storage needed:n. p. g, a. s, q. x. e,r and Input x >>Jp»»Set g = p(1) Seta= I(2) Set s = gaReplace a by (g + a)/2Setg=\sIf a<l0'(a-g)goto(21 ifn40goto(3)Set n = gSet g =q= \ 1 -p'Go to (1)Replace n by exp(-rg/n)Sets=0 Set a = 10 frac(c/5) - 5(4) Set v = gx(I80/7.)Set f = IIf {al * I go to (5)Replacev by 90 - vSet f =(5) If a > 0 go to 6)Replace f byf(n/p{sin(v)+n2[n4sin(5v) - sin(3v)]}Go to (7)(6) Replace f by f {; + n4 cos(4v) - n[ne cos(6v) + cos(2v)]}(7) If s * 0 go to (8)Sets = ISet a = Int(c/5) - 5Go to (4)(8) Replace f by f/sOutput f«<<Input code c f '. fg(p;x)Input restrictions: 0 < p 5 0.99.c may only be one of the 12codes listed below ..v * 0 if c= 21, 31. or 41. To reuse thesame p. enter new x and c only. Use degree mode or change 90to r/2 and delete the (180/1T)factor. Test values:P=i andx1.311139165code, cou ut-fQ12Sc2.9552811113sd1.0755180614sn0.94724020221cs0.33837728623cd0.36393088024en0.32052456831ds0.92978448432do2.7477745234do0.88072924341as1.0556984442nc3.1198856543nd1.13542273 649 THE JACOBIAN ELLIPTIC FUNCTIONS63:9 APPROXIMATIONS63:10 The expansions in Section 63:6 provide raw material from which many useful approximations may be constructed.For example:.r' 63:9:1cn(p:x) = I - -8-bit precision 63:9:2ds(p;x) = csc(x)8-bit precision 63:9:3nd(p;x)= cosh(x)8-bit precisionL32 -3128p'J'/Ixi < ipi <81 - x cot(x) 1Iqi < 63:10 OPERATIONS OF THE CALCULUS8sinh2(x) + x tanh(x) The derivative of any elliptic function with respect to its argument is proportional to the product of the function'stwo copolar cohorts. Thus, if fe is a Jacobian function of pole type e, and e, f, g and h are all distinct:a 63:10:1- fe(p;x) = a ge(p:x) he(p x) e, f, g. h = s. c, d. naxwhere a is independent of x. Values of a depend on fe and are included in Table 63.10.1. The derivative withrespect to the modulus similarly involves a and the two copolar cohorts but has an additional factor equal to lxq'- 3 sn(p:x) cd(p:x) - E(p:(WJ/pq2. where 41 = am(p;x) and 0 is listed in Table 63.10.1. Thus, the derivativewith respect to p of a Jacobian function of pole type c is[xq- - 3 sn(p:.t) cd(p.x) - E(pxbiJ 63:10:2- fe(p:x) = a get p:-r) he( p:.raP pq-Integrals of the types 63:10:31, =Jrfe(p;t)dtand1._fc'(p:t)dtf = s, c. d or ne = c. d or nn oexist and are in Table 63.10.1. For functions having type s poles, however. these integrals diverge, and it is thecomplementary alternatives rKipi Kip,63:10:41, =Jfs(p: tkltandL = Ifsp: t)d(f= c. d or n that arc listed in the table. The tabulated integrals are valid for 0 < p < " . 1 and 0 < x < Ki pl but not necessarilcfor variables outside these ranges.Rule 63:10:1 is the key to the evaluation of a great many indefinite integrals. Thus. we have 1 63:10:5 gel p:1) he(p::l& =a[fe(p;.i) - fe(p:0)]e * snexcept that if e = s this should be replaced by the complementaryf63:10:6 gs(pr) hs(p:idt =- [fs(p:K(p))- fs(p:x)1aThe transformations63:10:7 Igh(pa) dr _ r hg(pa)dt -Igc(p:t) ltd p,t)drfeh'(p:t)J eg'(pa) 63:10 THE JACOBIAN ELLIPTIC FUNCTIONS 650Table 63.10.1 cd cn dodo DC nd fC sd Cs ds nsap -q'0- In(ndl P: x) + psd( p:x))P- artxas{dn(p.x)}P0In(sc(p:xl - nc(p:x))-p-do-(P;x)aresin{sn(p:x)} = d Ip'- in(dc(p:xl - q scl p;x) P. 1q& 1 P t l- arccas{cdf p;xl)qIIndclp:x) - q nop:xlp--qI - q p-dc=( p:x)- asin{ pq nd(p:x) - pq cdl p:.r) 1pq1 ln' dn(p:x) - p Cal p:x)pI-p -4Inns(p.x) - ds(p,x !IIn(ns(p;x)-cs(p:x))9In\ds(p:x)- cs(p:x)IE(pA6)- + an(p: x) cd(p. r l -p.q'xPP:P- x -sn(p;x, dc(p:tiElp.d)E(p;d,lsn(p:x) dc(p:xlx -+q- q-E(p:4)p'sn(p;x)cd(p:xl49- Isn(p: c) dc( p:.r) - EI p Am )1q"E(pAAI.tsn(p:xl dcl p:xlP'q'P-x - E(p:A) E(Ph) - cn(p:x) ds(p:x) - E(p) q'Klp) - E(p)Elp;d,l - qr + cn(p:.t) ds(p:x) Kip) - E(p( -- E(p,tt- cn(p:.n 64 p:.0 permit evaluation of these integrals via 63:10:5 or 63:10:6. Similarly, such transformations asfgh(p:t)drge(p:t)he(p:t)_Idfe(p;t) 63:10:8gf(p;r) he(p;r)dr =J lh(p;t) eh(p;t)Jfe(P;1)aJfe(p;t)lead to results involving the logarithm (1/a) In{fe(p;x)} and find many applications. In the formulas of this para-graph, the a constant is that appropriate to the fe(p;x) function.For the sake of completeness we include the following derivatives of functions commonly associated with theJacobian elliptic functions: 63:10:9 axam(P;x) = d = dn(p;x) aa+dn(p;x)22 63:10:10 x + psn(p;x) cd(p;x) - E(p;4t))()Pam(p;x) =aP=P42Ig al 63:10:11(aaxdn2(p;x) p cn2(p;x) '' 63:10:12( dp E(P;4) 1 =\/// xx sc 42lsc(p;x) dn(p;x) - E(p;d) - q (p;x)1 651 THE JACOBIAN ELLIPTIC FUNCTIONS63:11 COMPLEX ARGUMENTWith argument x + iy the trio of Jacobian elliptic functions of type n pole adopt the complex values 63:11:1 63:11:2 63:11:363:12 ns(p;x) cs(q;y) +P2 sn(p;x) sc(q;y)cs(p;x) ns(q;y) - i dn(p;x) dc(q;y)cn(p;x + iy) _ns(p;x) cs(q;y) + p2 sn(p;x)sc(q;y)dn(p;x + iy)ds(p;x) ds(q;y) - iP2 cn(p;x) nc(q;y)ns(p;x) cs(q;y) + p2 sn(p;x) sc(q;y)sn(p;x + iv) =ds(q;y) nc(q;y) + i ds(p;x) cn(p;x) The other nine functions may be derived by application of rule 63:5:1.The transformation of an elliptic function of imaginary argument to one of real argument is known as Jacobi'simaginary transformation. Those functions that involve an s become imaginary on transformation:63:11:4sc(p;iy) = i sn(q;v)sd(p;iv) = i sd(q;y)sn(p;iy) = i sc(q;y)63:11:5cs(p;iy) = -i ns(q;y)ds(p;iv) _ -i ds(q;y)ns(p;iv)-i cs(q;y)while those that do not, transform to real functions:63:11:6 cd(p:iy) = nd(q;y) cn(p;iy) = nc(q;y) nc(p:iy) = cn(q:y)63:11:7 dn(p:iv) = dc(q;v) dc(p;iy) = dn(q;y) nd(p;iy) = cd(q;y)The transformations 63:11:4-63:11:7 illustrate that elliptic functions have an imaginary period as well as thereal period discussed in Section 63:2. For example, one finds sc(p;x + iv + 4iK(q)) = sc(p;x + iy). Table 63.11.1summarizes the periods for each of the 12 Jacobian functions and explains why these functions are known as doublyperiodic functions.We have been discussing imaginary values of the argument. but one can also have an imaginary modulus.Jacobian elliptic functions of imaginary modulus are, in fact. real. The appropriate transformations are 63:11:8 63:11:9 63:11:10sn(ip:x) = 1sdlp;xV`+ p=Vl+pV+p-cn(ip;.r) =cd(P:.rV 1 + p'V`I +pdn(ip;x) =nd(Vp: xV l pl+p'for the three functions of type n pole. As usual, the other nine may be obtained via 63:5:1. 63:12 GENERALIZATIONS There are none. Table 63.11.1 RealImaginaryperiodperiodicl p:.r + iv). csi p: x - iv I. dn(p: c + ty 1. ndf p:x + iv)sd(p;x + ii, cnlp;x + W. ds(p;i - iv), nc(p:x + w)snl p:x + n:1). cd(p:.r + tv1. dc(p:.r rv). nsA p:xiv)2KI p l4i K(j)4Kip)4iKIq)4KipiIKigI 63:13 THE JACOBIAN ELLIP71C FUNCTIONS63:13 COGNATE FUNCTIONS652 The functions of Chapters 61. 62 and 63 arose. largely due to Legendre's work, in connection with the integralsdiscussed in Section 62:14, which themselves arise in a variety of geometric problems and in such physical contexts as the motion of a pendulum. However, there is a rival formalism, due to Weierstrass, that can accomplish thesame tasks as that of Legendre. Of course, the two systems are intimately related. We shall not describe theWeiersrrassian elliptic functions but simply point out some of the equivalences.There are three interrelated variables, usually symbolized e,, e_ and e3. that play a similar role in the Weierstrasssystem to that played by the modulus in Legendre's system. One has the equivalences 63:13:1p=qe,-e2+e;=0e, - e,e,e3Likewise, the role of determining the two (real and imaginary) periods is taken over by two new variables63:13:2 w, =2K(p)2iK(q)N /e, - e.V e, - e_The principal Weierstrdssian elliptic function. usually symbolized by P(:) with some fancy typographical renderingof the P, is then expressible as63:13:3e, + (e, - e3) cs2(p;:Ve, - e3) = e: + (e, - e3) ds'(p:zVe, - e,) = e3 + (e, - e3) sn(p;:Ve, - e3)in terms of Jacobian elliptic functions.Both the Weicrstrass and Jacobian elliptic functions are intimately related to the theta functions [Section 27:131but, beyond reporting the relationships given in Section 63:8, we do not explore these connections in the Atlas.Further to complicate a topic already overburdened by redundant symbolism, we should mention Jacobi's etafunctions and Jacobi's theta functions, which are linked to ordinary theta functions and to Neville's theta functions[Section 63:81 by the equivalences 63:13:4 63:13:5 63:13:6 63:13:7H(p:z) =e \2K(P) K p)/ IT[aH(l' x)]x-o9,(p:.r) XK*H1(p;x) =rtK(p) = H1(P:0) 8,(P:x) ei(P;x) = 63zK(q))ei(P;0)Od(p:x)2K(p) 1TK(p)/6(p;x) =K p) I =O(p;0)9(p;x) CHAPTER64THE HURWITZ FUNCTION (v;u) Closely related to the functions of Chapters 3 and 44, the Hurwitz function plays an invaluable role in the differ-integration of periodic functions, a topic discussed in Section 64:14. Along with the related bivariate eta function [Section 64:13] and Lerch's function [Section 64:121, the Hurwitz function provides a means of summing many series whose terms involve arbitrary powers. 64:1 NOTATION The symbol C followed by two parenthesized variables is standard, but the name generalised seta function is oftenencountered. Sometimes {(v;u) is called "Riemann's function" or "Riemann's zeta function" but we eschew thesenames to avoid confusion with the ;(v) function of Chapter 3.We shall refer to v and u as the order and parameter, respectively, of the Hurwitz function. It is to achieveunity with the notation for Lerch's function [Section 64:121 that we resist the temptation to replace the symbol uand name "parameter" by the symbol x and the name "argument." 64:2 BEHAVIORUnless the order is an integer, the Hurwitz function becomes complex when u < 0. Accordingly, this range ofparameter is excluded from our discussion, except in Section 64:11. The zero order function C(0,u) bccomcs illdefined as a approaches zero; similarly, ;(l .u) is ill defined as u approaches infinity.The Hurwitz function displays an infinite discontinuity at v = 1. and, as Figure 64-I shows, the behaviors forv > I and Y < I are quite distinct.For v > I, 4(v:u) is uniformly positive. When both the order and the parameter exceed unity, the Hurwitzfunction declines in value, asymptotically approaching zero, as either v or u increases. Conversely, l,(v;u) - x asv approaches unity from positive values or as u approaches zero from positive values.For the most part, the Hurwitz function assumes negative values for v < 1; however, there are "lobes" ofpositivity as mapped on the contour diagram. As a function of u. {(v;u) displays a small, finite number of zeros,mostly in the 0 < u < I range, for v < 1. 653 64:2 THE HURWTTZ FUNCTION {(v;w) 654 655 THE HURWITZ FUNCTION ;(v;u)64:3 DEFINITIONSThe Hurwitz function is defined by the integral transforms It'-' exp(-ut)dt_IIn'-'(t)dt 64:3:1Yv;u)=r(v) Jo1 - exp(-t)['(v)tit - 1)or by Hermite's integralv>lu>064:4 u-u'-sin(v arctan(t/u))dt 64:3:2(v;u)=-++2 u > 02v - 1o(u' + t2)'/2[exp(2irt) - 11Expansion 64:6:1 provides a definition of;(v;u) for v > 1, and Hurwitz's series 64:6:2, supplemented by recursion64:5:1, can fill a similar role for v < 0.Notice that none of these definitions is valid in the important domain 0 < v < 1. Contour integration [seeErdelyi, Magnus, Oberhettinger and Tricomi, Higher Transcendental Functions, Volume 1, pages 25 and 261 candefine the Hurwitz function in this region. More practically, the series 64:6:4 can be employed for all orders andparameters.The difference i(u) - t,(u - t) of two digamma functions [Chapter 441 provides a generating function:64:3:3 t[y(u) - 4.(u - t)] = 2 ;(n;u)t' u > 1-2for Hurwitz functions of integer orders 2, 3. 4, ... 64:4 SPECIAL CASESHere we discuss cases of 7(v;u) that correspond to special values of the order Y. See Section 64:7 for simplificationsof the Hurwitz function that arise when the parameter u acquires particular values.When v is a positive integer greater than unity, the Hurwitz function is equivalent to a polygamma function[Section 44:121: 64:4:1 (- I )" OR- u(u)Un;u) _(n - 1)!n = 2,3,4,...When v is a nonpositive integer, the Hurwitz function can be expressed in terms of a Bernoulli polynomial[Chapter 19):64:4:2 Y-n;u) =n+ 1n = 0, 1, 2, ... of which the first few instances are 1 64:4:3 I(0;u)=--u-1 +6u-6u264:4:4 >;(-l:u) =12-u+3u2-2u3 64:4:5 (-2;u) =6Although the Hurwitz function is infinite for v = I, the quotient;(v;u)/r(1 - v) and the product [v - 1]i;(v;u)both remain well behaved in the vicinity of v = 1. This is evident from the limiting expressions64:4:6 limVV*= _1-I r(1-v) 64:5 THE HURWITZ'FUNCTION Vv;u) 65664:4.7v1 I_Vu)Jwhere 4' is the digamma function (Chapter 44). 64:5 INTRARELATIONSHIPSThe recursion formula 64:5:1 1'(v;u + 1) = >;(v;u) - - u Z. 0U.may be iterated to 64:5:2 I,(v;u + J) + Z (j + u)-` u a 0J=Oand, provided v > I, becomes expansion 64:6:I when J = x. The duplication formula64:5:3 Uv,2u) = 2 "LZ(v;u) + rl v;2+u 1may be generalized to64:5:4{(v:mu)=m "m+um=2,3,4,...and used to demonstratefor examleth,p,at64:5:5C('; 34+u)=2"1'(v:2u) -V;I+ul\\\The following infinite series of Hurwitz functions of integer order 64:5:6 (2;u) + C(3u) + 4(4;u) + ... _ (n;u) = u > IU1 64:5:7i'(2;u) - 4(3;u) + Z(4;u)- ... _(- l)%(n;u) _u > 0.=2 uZ(2;u) ,(4;u) 64:5:8_ + - + 4(n:u)234+2=40)-In( u-1)u>I.-Inand 4(2;u)1'(3;u)C(4;u) 6459::23+ 4 _ ... = L,(-1)"In(u) - tr(u)u > 0,=2nmay be combined in several ways: for example, to sum the series I{(k;u) where k is restricted to a single party64:6 EXPANSIONSThe important series 64:6:1 IIIk`(I + U),(2 + u)`op 657 THE HURWITZ FUNCTION Ov;u) 64:7often serves as a definition of the Hurwitz function for orders exceeding unity. Hurwit_ formula:av1sin/( 2jau + 2/64:6:24(v;u)=2r(1-v)\V<005u51=i(2jar)'may also be written in the standard Fourier series [Section 36:61 form by replacing sin[2jau + (av/2)] by cos(av/2) sin(2jiru) + sin(7tv/2) cos(2jau). As written. 64:6:2 is valid only for a narrow range of parameters, but thismay be extended to larger u by sufficient applications of recurrence 64:5:1.An asymptotic expansion of the Hurwitz function may be written in the convenient form Ivv(v + I)v(v + I)(v + 2Xv + 3) kBk 64:6:3u v+l;u+l +-(v+u-.x ()u U.2u""l2u;;'720u`+4k!uk'"F.-where (v)k denotes a Pochhammer polynomial [Chapter 18] and B, is the k'" Bernoulli number [Chapter 41. Oddmembers of this latter family, except B i, are zero, and even members, except Bo, may be expressed as zeta numbers[Chapter 31. Thus also making use of 64:5:2, we may reformulate 64:4:3 as 1+l+(J + U)]-. II - 2 ZK(vI) A(2k) 64:6:4LJ i-02(J ;k_[ [-4az(u+ J)z]kwhere the first sum is "empty" if J - 0 (such sums are to be interpreted as zero). While this expansion technicallyremains asymptotic (i.e., valid only in the u + J -* x limit), it may be made arbitrarily exact by making J largeenough and choosing K at least to exceed (I - v)/2. Yet a further development leads to 64:6:512(2x)"-'4*+ui . r3- 2(v - l)-[C(2k) -I]zsk]J2u"V- Iv - 1where (* is related to the Boehmer integrals of Section 39:12 by64:6:6c* = cos(21ru) S(2au;2 - v) - sin(2au) C(21tu;2 - v)1kaIra\\(2au)k sin 2 - 2au//=r(2-v)sinl-+2au1-(2au)'"Gv*2.3,4,.,. \\ 21k=ok!(k + 2 - v) 64:7 PARTICULAR VALUESThe Hurwitz function of integer parameter is related to the zeta function of Chapter 3. The simplest cases areIxv>064:7:1 Y,(v;0) _(v).<O64:7:2 ;(V;l,(v)and64:7:3 4(v:2) = {(v) - 1while the general case involves a finite sum:64:7:4 t(v;m) = Y,(v) -m=2,3,4,..or. for orders exceeding unity, an alternative infinite sum: 64:7:5 ;(v;m) _ m = 1, 2, 3,... v > I!-m j 64:8 THE HURWITZ FUNCTION 2'(v;u) 658Similarly, the Hurwitz function of a parameter that is an odd multiple of a moiety is related to the lambdafunction of Chapter 3. The simplest case is 64:7:6 Ov;Z) = 2"a(v) = (2. -and others may be expressed either as a finite or an infinite sum 64:7:7{(v;m + Z) = 2`f a(v) - i(2j - I)-°J m = 1, 2, 3,L=2'm=0,1,2,...v>I ,_,, (2j +I)'The Hurwitz functions of parameters equal to or ; are related to the lambda and beta functions of Chapter3 by 64:7:8 Yv; i) =a(v)1 LL2 64:7:9 re(v) - (3(v)] Adding equations 64:7:8, 64:7:6. 64:7:9 and 64:7:2 leads to Yv;',) + Y OZ) + {(v;,) +the J = 4 case of the general result64:7:10v;j1J";(v) J=1.2,3,...As u increases, the following values are approached 64:7:11 {(v;x) = 64:8 NUMERICAL VALUES0v>l-xv<14(v;l) = 4`I(v), which is We present below an algorithm suitable for calculating {(v;u). It is based on the J = 0 version of asymptoticexpansion 64:6:4 and uses the Pade technique [see Section 17:13] to overcome the convergence problem. A fixednumber (thirty-one) of Pade approximants is calculated prior to output so that the computation is always rather leisurely. The algorithm may use equation 64:5:1 either to decrease or increase the parameter before the expansion is invoked. If v > 1, the parameter is increased, if necessary, until it exceeds the order v. If v < 1. the parameteris invariably adjusted to he in the range from I through 2. Input V >Input u >Setf = 16"If (v - 1)(u + wI - v) = 0 go to (10)Setf=k=0(I) Replace fbyf+uReplace u by u + IIf v < I go to (6) If u<vgo to (I)(2) Set ho = ISett=-2Set Z = 271/60Storage needed: v, f, u. k, r, Z, j,p, q and ho, h h2, ... hs9, hw input restriction: u > 0 1 659 THE HURWITZ FUNCTION 64:10 f = C(v;u)(3) Replace t by t(v + 2k)(1 - v - 2k)/(21ru)2Replace k by k + IIfk:55 go to (7)Set Z = I + (I + 4-*)(4-' + 9-') + 25-*(4) Set hf = hk_ i + tZSetq=0Setj = k(5) Set p = h1_,Ifh;*pgoto(8)Seth,-, = 1099If p * l099 go to (9)Set h2_, = q Go to (9)(6) IfuS2goto(2)Replace u by u - 1Replace f by f - uGo to (6)(7) Replace Z by 10 r2Z/(504 - 301k + 74k2 - 6k3)Go to (4)(8) Set h;_, = q + I 1 /(h; - p)](9) Set q = pReplace j by j - 1 Ifj+0goto(5) If k * 60 go to (3) Replace f by f + [ I /(2u")] + [h°u'-'/(v - 1)](10) OutputfTest values:402;0)1\0;ZI=0?;(10;10) = 1.69268613 x 101°2= 0.00384424603l6:4)0.306886545 The algorithm returns 1090, instead of infinity, if v = I or if v a 0 = u. Otherwise. the algorithm generallyhas a precision of 24 bits although this may be degraded near the zeros of the Hurwitz function or at extreme values (say Ivj > 30) of the order. 64:9 APPROXIMATIONSA useful approximation to the J = 0 version of expansion 64:6:4 is provided by its first three terms: I2uI (v2 - I)(v2 + 2v)v064:9:1f,(v;u)-u"[v- 1+ I +6u]8-bit precisionu >I3Bear in mind, however, that the expansion is asymptotic and is not itself always accurate. 64:10 OPERATIONS OF THE CALCULUSSingle and multiple differentiation with respect to the parameter obey the formulas 64:10:1 64:10:2aau0v;u) _ -0V + 1;u) a.au0Vv;u) _ (-1)"(v). {(v + n:u) 64:11 THE HURWITZ FUNCTION ;(v;u)Differentiation with respect to order gives 64:10:3d1-0l(J (+ u),v > land there is also the particular value 64:10:4 a W;u) = lnlr(u)Iv = 0 for zero order.Formulas for indefinite integration include 64:10:5 64:10:6r.;(v r)dt =v -1v > 2 IY;(v;r)dt=;(v- I;u)-;(v- 1)v<2 .11,I-v(64:10:7 JC(v;t)dr =;(v - I;u) - ;(v - I)V < I01 - vand lead to the interesting definite integrals 64:10:8 JC(v;t)dt = 0v < In 11 64:10:9J{(v:t)dr =all v . .v-IThe parts-integration procedure that produces the resultftrtu(v - l;u) -(v - 1)(v - 2u) -(v - 2) 64:10:10 =I - v(1 - v)(2 - v)v < 2 may be iterated to generate expressions for ft";(v:r)dt for n = 2, 3. 4, ....The Boehmer sine integral (Section 39:12) appears in the formula 64:10:11sin(27rt);(v:r)dr = (2a)"-' S(2trr;I - v) v > Ix > 0660 and a similar operation generates the Boehmer cosine integral from cos(2nt) ;(v;t). These results relate closely tothe discussion in Section 64:14. 64:11 COMPLEX ARGUMENTThe Hurwitz function is complex not only for complex parameter but even, unless the order is an integer, fornegative parameter. Here we address the latter circumstance only. If a is a positive number lying between theintegers m - I and in. then64:11:1;(v;-u) = ;(v:m - u) + (-1)"(;(v;u + I - m) - ;(v:u + I )) IM- I <u<m =;(v;m-u)+(cos(m)+isin(vw))Y`-0 (u -J)64:12 GENERALIZATIONSIn this section we briefly discuss the trivariate Lerch's function 4(.r;v;u), of which the Hurwitz function is thespecial case of unity argument, x = l: 661 THE HURWITZ FUNCTION ;(v;u)64:12:1 4)(I;v;u) = Y(v;u)Lerch's function may be defined by an integral or a series: 64:12:2 4)(.r;V;u) =Ir"exp(-ut) dt-x1'(v) joI - x exp(-r)-(j + u)°although these are generally real and convergent only for limited ranges of x. v and u. A third definition:d-"exp(ux) 64:12;3[d(t + x)] '° I - exp(x)= exp(ux) (Nexp(x):v:u}x < 0 utilizes the concept of differintegration [Section 0:101 with a lower limit of -X.The recurrence formula 64:12:4 4'(x:-2:u) =64:12 can clearly be iterated indefinitely often. When the order is a positive or negative integer, Lerch's function can beexpanded in terms of the logarithm of its argument: thus: 64:12:54){exp(y):n;u} = exp(-uy} [kO(n) - tL(u) - In(-v)] + i 4(1 + n - j; u)1(n- 1)!j_1 (j - 1)!n=2.3.4. nB,.1_"(n) l I !Y' 64:12:64){exp(y);-n;u) = exp(-uy)Sll-Jn = 1, 2. 3....Here B;,I."(u) is a Bernoulli polynomial [Chapter 191.In addition to 64:12:1. there are many special cases of Lerch's function. We conclude this section with anincomplete listing of such special casts. Following each identity is a reference to a chapter or section in which thespecial function is discussed: 64:12:7 64:12:8 64:12:9 64:12:10 64:12:11 64:12:12 64:12:13 64:12:14'(x;v:u)+ U) + 4 (x:0:u) _(Chapter 711-xxUVx:- l:u) _+ - (Chapter II ]1 -x 14'(0;v:u)(Chapter 131( m-0 2 x2-(2u2 -2u- 1)x+u2 u"( I + u)" 4)(x:l :1)In(1 - .r)xI1 -x)' I(u)" 4)(x;1:u) _-I.r B(u:0;x)-14)(x;2;1)diln(l - x)x[Section 17:131 [Section 18:141 [Chapter 251 [Sections 25:12 and 58:41 [Section 25:131 64:13 THE HURWITZ FUNCTION Uv;u) 662-164:12:1540m1) - x)[Section 25:13] x l1 2" 64:12:16(x;v;)2=x) --poln,(1 - V.)[Section 25:131 I1 64:12:17(x;l;)( -x;-;1=rsf(ln(x))[Section 30:10) 64:12:18=orarctan()[Chapters 31 and 35] 64:12:19 $(-l;v;u)= ii(v;u) [Section 64:13) 64:13 COGNATE FUNCTIONSThe bivariate era function is related to the eta function of Chapter 3 in the same way that the Hurwitz function isrelated to the zeta function. It is defined by 64:13:1q(v;u)=i (-1j1v>0u?0i-0 (J +u)°The recurrence formula 164:13:2 r(v;u + 1) = - - rl(v;u)Uresembles 64:5:1, apart from sign, but the duplication formula 64:13:3 ,(v;2u) = 2-"I Yv;u) - I v; 2 + u I Jprovides one of many links to the Hurwitz function. Another is 64:13:4c(-; 2U)+1(v;u) =2-1&;u)Special cases of the bivariate eta function include the eta function of Chapter 3: 64:13:564:13:664:13:7the beta function of the same chapter: 64:13:8 Bateman's G function [Section 44:13]: 64:13:9and successive derivatives of the last:I(v;0) ={-(v)00Y1(v;1) = 11(V)v<0v>0 T1(v;2) = I - TI(v) 71(v; 2)= 2°[i(v) 11(l;u) =2G(u) 663 THE HURWITZ FUNCTION {(v;u) 64:14 64:13:10The very simple algorithm Input v >>Input u >> f = q(v;u)rl(v;u) =2(n - I)! b»»r10u+2Set j = 2 Int`2 1Set f=-(u+j)"(1) Replace jbyj-2Replacef by f - (u + j + 1)+ (u + j)-"Ifj*0goto(I)Output f:««Storage needed:v. u, j and f Input restrictions:u > 0; v must be large and positive. Test values:1l(5;8) = 1.979 x10-5rl(8;0.2) = 390624.769 is based on definition 64:13:1 and uses the properties of alternating series [Section 0:61 to ensure an absoluteaccuracy of 6 x l0-8. Although the algorithm is valid for all positive v and u, it is tediously slow unless v is large.For orders that are not large and positive one may use identity 64:13:3 or 64:13:4 and the algorithm in Section64:8. 64:14 RELATED TOPICSThe differintegrals [Section 0:101 of periodic functions can be expressed as integrals involving the Hurwitz function,the integration range being a single period. The property that permits this simplification is illustrated by the trans-formationrJP/-I p/-164:14:1Ju" per(u)duf'.P-'pu"per(u)du = P.JI'per(t){(j +P) Idtr = u - jP /--p l j°0in which per(x) is a periodic function [Chapter 361 of period P.Differintegration with a lower limit of -x is especially facilitated by the Hurwitz transformation. Such dif-fcrintegrals are sometimes known as Weyl dijferintegrals. We report formulas for the two ranges, 0 < v < I and- I < v < 0, that embrace the important semidiffcrentiation and semiintegration casesd° per(x)I 64:14:2=J[per(x - Pt) - per(x)1 C(1 + v;t)dt 0 < v < I [d(x + x)]"P"1'(-v)0i64:14:3d" per(x)=1per(x - Pt) 4(1 + v:t)dt-1 < V < 0 [d(x + x)]"Pr(- v)oper(Pt)dt = 0Notice that the second result requires that the average value of the periodic function over its period be zero; oth-erwise, the differintegral is infinite. Weyl differintegrals of periodic functions are themselves periodic and the periodis unchanged on differintegration. We conclude this section (and the Atlas!) by reporting three examples of the application to simple periodic functions of equations 64:14:2 and 64:14:3 and their extensions. For the sine functionfor any sinusoid, see Chapter 321 differintegration with a lower limit of minus infinity corresponds simply to ascaling and phase shift:-(-1)"G")(u)n= 1,2,3,... 64:14:4 P)" sink 2Px+ 2/ ld(x 64:14 THE HURWITZ FUNCTION;(v;u) 664The other two examples report the Weyl semiderivative 64:14:5d'/:(_ IP[(IZ:fraclP/I --; frac(- +2/ / Jand Weyl semiintegral 64:14:6d-/x(-I)t4[C(-fracI-ll-;I- 1:fracI-+l [d(x +x)]-'/2VIr2\P2P2)of the square wave function that is mapped in Figure 36-3 and formulated in Section 36:14. The shapes of theselast two periodic functions are displayed in Figure 64-2, together with the square wave itself. yQ1tiyQb4ydQ APPENDIXAUTILITY ALGORITHMS Most of the algorithms in the chapters of this Atlas are devoted to generating numerical values of particular func-tions. However, a few of the algorithms scattered throughout the chapters perform operations that are more general than evaluating specific functions. This appendix provides a glossary of the latter kind of algorithm and includesa few additional instances of such general-purpose algorithms. A:1 BASE-CONVERSION ALGORITHMS Section 9:14 contains two algorithms designed to convert numbers from one base to another. The first converts adecimal number to any other base (to binary or hexadecimal, for example). The second algorithm performs thereverse conversion: it converts from any base into decimal.The two algorithms may be conjoined to permit conversion from any number base to any other. A:2 FRACTIONATION ALGORITHM Sometimes one needs to replace a decimal number v by the quotient n/d of two integers. Even if v is ostensiblya rational number, rounding errors in its decimal representation will usually prevent the equality v = n/d frombeing exact. Instead, there is a fractional error A:2:1 v - (n/d) Vassociated with the replacement of v by n/d.The algorithm below generates a sequence of ever-improving approximations n,/d, n./d:. n,/d..... to aninput number v. Each pair of output integers n. d is accompanied by an output of e. Although it is designed tohandle positive numbers only. these may be larger or smaller than unity. Output continues until, to the precisionof the computing device, (n/d) = v. "S A:2 UTILITY ALGORITHMS Input v >> d=d,«-5 V = £J <<Setd=D=1Set it = Int(v)SetN=n+ IGo to (3)(1) Ifr> I go to (2)Replace r by I/r(2) Replace N by N + it Int(r)Replace D by D + d Int(r)Replace n by n + NReplace d by d + D(3) Set r = 0If vd = n go to (4)Set r = (N - vD)/(vd - n)If r > I go to (4)Set :=NSet N=nSet n=: Set :=DSet D = dSet d=:(4) Output nOutput dSet e = 11 - (n/vd)!Ifr=0goto(6)Set in = 1(5) Replace m by tomIf me < I go to (5)/Intl2 +meReplace a byin(6) Output £Ifr*0goto(1)666Storage neeed: v, N, D, n, d, r, r, a and m input restriction: v > 0 Test values:input: v = zroutputs:n,3d,In,22e,=4x 10d,7355d3113n'_104348d.33215ea = 8 x10-e B. = I x 10-101 The commands shown in green perform the interchanges N .= n and D 1~ d; with many devices there are simplerways of achieving this.The algorithm first finds the two integers between which v lies so that this number is known to be bracketedby the fractions A:2:2with d = D = 1. The ratio A:2:3nN5v5dD N - vDr=vd - nis then calculated. If this exceeds unity, n/d is accepted as a valid approximant to v. and the numerator n anddenominator d of the (proper or improper) fraction n/d are output, along with a rounded value of the fractionalerror e. Moreover, if r lies between the integers m and m + 1, then it follows from A:2:3 that(m+1)n+Nmn+N A:2:4(m+1)d+D<v:5md+DI 667 UTILITY ALGORITHMS A:4On the other hand, if r 5 1, then N/D is accepted as the approximant and is output, followed by the correspondingfractional error in the approximation. From the definition of r, one can show that A:2:5 n+mNn + (m + I )N 1d+mDSV<d+(m+1)Dm-r<m+in the r <_I case. Thus, either A:2:4 or A:2:5 provides a narrower bracketing of v than did the original A:2:2.The algorithm continues this process of bracket narrowing until the computing device interprets r as either infinityor zero. A:3 ALGORITHMS THAT GENERATE NUMBERSAn algorithm in Section 40: 14 generates a succession of J numbers that are pseudorandomly distributed accordingto the "normal" distribution [see Section 27:14]. The mean and standard deviation of the distribution are selectedby the user.A subset of this same algorithm, namelyInput x0 > »»>Set N = 199017Input x, >>>>>>>Seth=x, - xaInput J >> »»Setj = n = J(I) Replace n by 24298n + 99991Replace n by n + I - N Int(n/N) Replace j by j - ISet .r = xo + (nh/N)Output xr <<<<< ««Ifj*0goto(1)Storage needed: xo, N, h, j and n Input restrictions:.ro < x 1 S J :5 199017 Test values: x _ -1, x, = 1. J = 5; outputs:x = -0.774240392x = -0.488174377x = 0.343844998x = 0.750624319 x = 0.674550415 generates a set of pseudorandom decimal numbers uniformly distributed on the interval r. < x s .r,. The arbitrary"seed" is taken as J, the number of sought outputs. A:4 DATA-HANDLING ALGORITHMSThe Atlas contains three algorithms that are designed to process large data sets.In Section 7:14 we present an algorithm for the linear regression analysis of equally spaced and equally weighteddata. This procedure gives the slope b and intercept c of the best straight line fit to data points (x1,f1), (x2,f2),(x,, f,)...., (x,,f) where x, - x,_, = x,_2 = .= x: - x, and 2 S n < x. Optionally, the algorithmalso generates the correlation coefficient of the fit and the standard errors of the slope and intercept. Although algorithms are not included, the same section discusses the linear regression analysis of data that are not equallyspaced and not equally weighted.A straight line may be considered a first degree polynomial function. In Section 17:14 we show how to fitpK(..r), a polynomial of any degree K, to a set of J + I data pairs, (x,,, fo), (x,, f,), ..., (x;, f)..... (x,, f,), providedthat J ? K. This polynomial fitting algorithm is also restricted to data that are equally spaced and equally weighted.Several popular methods of processing data involve Fourier transformation, Fourier inversion, or both. Al-though this Atlas does not discuss such data-processing methods, it does include, in Section 32:14, an algorithmfor fast fourier transformation/inversion. Once again, the algorithm assumes equally spaced and equally weighteddata, moreover, the number of data must be a power of two. A:5 UTILITY ALGORITHMSA:5 ALGORITHMS TO LOCATE FEATURES OF A FUNCTION668 In Section 17:7 an algorithm is presented for finding an accurate zero of a polynomial function from an initial crudeestimate. That algorithm uses Newton's method which utilizes the fact that/A:5:1 j(r)d-x(r,)is usually a better approximation to the local zero of f(x) than is r,. Newton's method may be used to find the zeros[but not double zeros-see Section 0:7] of any continuous function (or discontinuous function away from its dis-continuities) provided that the initial estimate r5 is close enough to the sought zero.However. Newton's method is sometimes inconvenient in that it requires numerical values of two functions:f(x) and its derivative. We present below an algorithm that needs values of f(x) only. It is based on approximatingthe derivative df(r,)/dx in equation A:5:1 by the difference quotient [f(r,) - f(r,_,)]/(r, - r;_,] and employs theresulting formula A:5:2 r_,f(r,) - rf(r,_,)f(r) - f(r,_,)The algorithm needs two (different) estimates rp and r, of the zero, but these can often be wild guesses and r, neednot be the closer of the two. In a kindred method [known as regula jalsi; see Bronshtein and Semendyayev. page1701, the values of r and r, must lie on either side of the zero, but this is not necessary here. Double zeros areaccessible by this method. though convergence is slower than for single zeros.The algorithm repeatedly outputs the improving values to. r, r., r, ... until two successive values of f(r) areidentical or until the user intervenes. Input r >> Input r, >> x=r;<<Set Fl0Set x=r0Go to (4)(1) If F1099 go to (2)Set r = r,Go to (3)(2) Replace r by (Rf - rF)/(j - F)(3) Set R = xSet x = r Set F = f Output x(4) Routine to calculate f = f(x) from xIfF*jgoto(1)Storage needed: F. x. r. R and f. in addition toany storage needed by the f routine. Test values:using f(x) = cxp(x) - x=. ro = -0.6,r, _ -0.7;outputs:-0.703613805-0.703467221r. _ -0.703467422ry = -0.703467423 The segment shown in green must be provided by the user; it could, for example, be one of the algorithms fromSection 8 of the chapter dealing with f(x).Our algorithm has the advantage of simplicity, but it will seldom be optimal and it may occasionally fail. Thereexist very many alternative procedures for locating zeros (or root finding as the process is often called); Ruckdeschel[Volume I]. Chapter 61 discusses some of these. The same reference [Chapter 7] gives methods for finding thecomplex :giros of functions.The algorithm is easily modified to provide information about features other than the values of the zeros. Thus.to find the value of x at which f(x) equals, say. it. one need only replace the green segment by a routine to calculatef(x) - IT. Again, to find the extrema (local maxima or minima) of the function, one simply changes the greensegment to "Routine to calculate j = df(x)/dx from x." 669 UTILITY ALGORITHMSA:6 ALGORITHMS TO MANIPULATE SERIESA very large class of functions can be represented as the power series or Maclaurin series [Section 11:14]A:7 A:6:1 f(x) = ao + a,x + a +_ I a,x,-oSuch a series may always be reformulated as a concatenation [see equation 0:6:5] or as a continued fraction [viaequation 0:6:12]. These reformulations frequently permit the numerical evaluation of f(x), for specific argumentsof interest, to be carried out more expeditiously, or more accurately, than by the original series A:6:1. As well,there are a couple of more profound ways in which power series may be manipulated as an aid to the numericalevaluation of f(x).If interest is confined to a small range xo <_ x s x, of arguments and if a restricted accuracy, If(x) - [(x)l te, is acceptable, then the economized polynamial A:6:2J-0-nay serve as a convenient means of calculating numerous approximate values of the f(x) function. Such economizediolynomials are the basis of many computer library subroutines for functional evaluation [see Ruckdeschel]. Be-:ause Chebyshev polynomials are utilized in the procedure, this Atlas's algorithm for calculating values of theJegree m and the coefficients eo, e,, e:, ..., e of the economized polynomial will be found in Section 22:14. Notehat coefficients generated by the Chebyshev procedure are "best" in the sense of minimizing the worst error)etween f(x) and f(x) rather than in the "least squares" sense.If the series A:6:1 is asymptotic [Section 0:6], accurate values of f(x) can be calculated by its use only forufficiently small values of the argument x. However, by using the procedure detailed in Section 17:13, it is possibleo convert the sequence (ao); (a0 + a,x): (a0 + a,x + a_x); ...; (a0 + a,x + + ajx1); ... of truncated powercries into a sequence R$(x); RI(x); R=(x); ...; Rj(x); ... of diagonal Pads' approximants. The latter sequence oftentas an advantage over the former in converging to f(.r) when A:6: I is asymptotic or even divergent. An algorithmn Section 17:13 exploits this advantage in evaluating numerical values of functions from their asymptotic seriesis the Padt operation. s:7 ALGORITHMS TO PERFORM OPERATIONS OF THE CALCULUShe derivative of any function f(x) discussed in this Atlas can always, and its integral can usually, be expressedterms of some other function(s) of the Atlas. Such expressions will be found in Section 10 of the relevant chapter.cases of difficulty, the properties of hypergeometric functions [Section 18:141 or Laplace transformation [Sectioni:14] can sometimes permit the operations of the calculus to be applied to otherwise intractable functions. Never-:eless, there will remain certain functions whose integrals are evaluable only by numerical methods. Such analuation is known as a numerical quadrature and we here present a simple algorithm for the numerical quadratureF a function f: 7:1 f(t)dtx0 * ±x * x, twcen two finite limits. We treat only the case in which f(x) is known (i.e., is calculable or infinite) at all pointsthe xo s x <- x, range. Not considered is the technically important situation in which the integral is sought offunction whose values are known at a finite set of (usually equally spaced) arguments only.The algorithm utilizes the trapezoidal approximation 7:2I f(t)dt ° 1, = x1xof1 xo + L (x, -.to))J = 1. 3. 5..... 2n - I h n given the successive values 9, 27, 81, .... The advantage of this threefold increase in the number of A:7 UTILITY ALGORITHMS 670summands in the approximation 1, to the integral is that all the summands that are needed as components of I. willbe reused in forming the 13,,. 19,,, 1.,,, ... sums. In the algorithm that follows, the green portion represents thecommands necessary to calculate values of the function f(x) from a value of x: Input xo >> >>>>Set r = 0Set n = IInput x, >>»»(1) Set d = 3(x, - xo)/nSets=0Set a = 1(2) Set x = x(, + (ad/6)(3) Calculate f = f(.r)Replace s by s + fReplace x by x + dIf x < x, go to (3)If n = I go to (6)If a = 5 go to (4) Set a = 5Go to (2)(4) If n = 3 go to (5)Set a = 3d(81s + 51! + 52r)/640Output afa = Jf(r)dr ««b 1(5) Replace r by r + 1(6) Set I = sReplace n by 3nGo to (1)Storage needed: x0, x,, r, n. d, s, a and 1. in addi-tion to any used by the f(x) routine Successive outputs become increasingly infrequent untilthe user intervenes. Input restriction: x, > it. Test values:with f(x) = I/x: inputs: x9 = 1, x, = 2;successive outputs:0.693144600, 0.693147174.0.693147181.... The algorithm does not output 1, values as such, but instead makes use of two levels of Richardson extrapolation[see Ruckdeschel. Volume Il, page 3171 and returns values of (7291 - 901,,,3 + 1,,,9)/640. For suitable functionsf. this double extrapolation compensates so effectively for the finiteness of n that an accurate approximation to theintegral is soon achieved. The "test values" illustrate such a case: the fourth output (corresponding to n = 243)already has converged to a value that reproduces the correct value of the integral A:7:321-JItdt = ln(2) = 0.693 1 47 1 806 to nine significant digits.Because the algorithm does not utilize f(xo) or f(x,), it is able to handle integrals that are indeterminate orinfinite at either end of the integration range, provided, of course, that the integral itself is finite. Nevertheless, itis better to dismember integrands in the latter class, as in the examplesin(tii)1" dtrdtVt - sin(V) A:7:4 rdt=JO tI=2a_ Irdtfinto analytically integrable functions and ones that are everywhere finite over the xo <- x <- x, interval. A similarstrategy often can aid the numerical quadrature of functions that encounter an infinity within the x9 < x < x, 671 UTILITY ALGORITHMS A:7range, as exemplified by tA:7:5rs=2Idt- I- Si=- In(2) - I(3r ?3r't'l1Ids tl3t1Jt- 1r13Our algorithm, as it stands. cannot be used if either (or both) of the integration limits is infinite. A suitableredefinition of the integration variable, however, will effect the conversion7:6 `f(t)dtg(u)dujA:wand often render the function amenable to the numerical quadrature algorithm. Table All gives examples ofsuitable transformations; there are many alternatives.The preceding algorithm is designed to evaluate definite integrals. If one wanted to use it to study the behaviorof the indefinite integral A:7:7 f(t)dt of f as a function of x, one would need to access the algorithm repeatedly, replacing the upper limit x, by different.r values until the range of interest had been scanned. Such a procedure would usually be impossibly tedious. Thealgorithm below, however, is designed for just such a survey: it generates rather crude values of the integral A:7:7at x values equal to xo + (x, - so)/n..r) + 2(x, - .ro)/n, ro + 3(x, - x.)/n, ..., xo + (n - 1)(.r1 - .xo)/n, xwhere x, and n are values provided by the user.Actually the algorithm is much more versatile than we have just suggested. Indefinite integration is just oneexample (the v = - I instance) of the generalized differintegration operator of the calculus [see Section 0:10], andour algorithm generates approximate values of A:7:8d"f(x) [d(x - .ro)]"atx=xo+J(x,-x0)forj=1.2.3,....nnfor any value of v [including v = -I when A:7:8 reduces to A:7:7]. The algorithm is based on the improvedGrumvald definition [Oldham and Spanier. pages 56-57] A:7:9 d"f(x)_limi lo/(((Jlt,v), f(.r(2j - v)(x - xo)-02Jof a differintegral, with the condition J - x relaxed and J equated to n. Table A.7.1 L,Ii -xxg(u) A:7Input x0 >>Input x, >>Input n >> Input v >> x <<<<<Set z = xID Set D = (x, - xo)/nXP»»(1) Sets=0Replace x by x + DSet d = (x - ro)/nSet j=nSeta=n - vSet y=xo+(vd/2)(2) Replace y by v + dReplace a by a - ICalculatc f = f(tiReplace s by f + (as/j)Replace j by j - IIfj*0goto(2)Output XSet F = s/d`Output Fd"f(F=(d(x -If .t < x, go to (1)UTILITY ALGORITHMS 672 Storage needed:x0, x,, x. n, D. v, s, d, j, and a, as well as that re-quired by the f routineI Input restrictions: x, > x0; n integerUse radian mode or replacesin(V) bysin(iry x/ 180). Test values:f(x) = sin(V ). xo = 0, x, = 10, 1it = 10, v = 2; outputs:x = 1. F = 0.681974314it = 2, F = 0.499347769x = 3, F = 0.340096522x = 4, F = 0.202248636x = 5, F = 0.0839564103x = 6, F = -0.0165098702x = 7. F = -0.1007679243x = 8. F = -0.170329209x = 9, F = -0.226604411x = 10. F = -0.2709087071 It should be emphasized that there exist many better algorithms for specific orders of differintegration but that theone given here has the advantage of versatility. Comparison of the listed test values with those of (V r/2)J.(x),the true semiderivative of sin(V) [see 52:3:13], reveals errors of up to 2%, which is typical for thisalgorithm using it = 10. Accuracy increases with increasing n, but with a concomitant speed penalty. APPENDIXWSOME USEFUL DATA Since 1960 Le Syst2me International d Unites has been the official system of units throughout most of the world.The SI system, as it is usually abbreviated, is used universally in science and increasingly in engineering andmedicine. This appendix contains information about the SI system and lists important physical constants and con-version factors. B:1 THE PRIMARY UNITSThe SI system recognizes seven base units and two supplementary units. These nine primary units are listed inTable B.1.1. All other quantities are measured in derived units that are combinations of the primary units. B:2 DERIVED UNITSSome quantities are expressed in units derived directly from primary SI units. See Table B.2.1 for a few examples.Eighteen derived units are given special names as listed in Table B.2.2. Notwithstanding their special names, theseunits are also derived directly from the nine primary units. Table B.1.1 Table B.2.1 Name ofSymbol of SymbolQuantityunitunitQuantityName of unitof unit LengthmetremAreasquare metrem'MasskilogramkgVolumecubic metrein,Time second sVelocitymetre per secondm s'Electric current ampereADensitykilogram per cubic metreits m-'Thermodynamic temperature KelvinKCurrent densityampere per square metreA m-'Amount of substance molemolTemperature gradient kelvin per metreK m'Luminous intensity candelacdConcentrationmole per cubic metremol m-'Plane angle radian radLuminancecandela per square metrecd in-'Solid angle steradiansrRotation rateradian per secondrad s-' 673 B3 SOME USEFUL DATA 674Table B.2.2 QuantityName of unitSymbol of unit and equivalent in primary units Frequency hertz Hz = s-'Forcenewton N= kg in s-'Pressure, stress Pascal PaN m-'kg in-' s-1Energy, work, heat jouleJ=Ninkg m's-'Power, radiant flux wattW = I s` = kg m' s-'Electric charge coulombC=JV-'=AsElectric potential voltV = WA-'=kgm2s'A"Electric resistance ohmf1=VA-'=Jig m's-'A-'Conductance siemcnsS=fl-'=s' A2 kg-' m-1Capacitance faradF=CV-' -s'A'kg'm-'Magnetic flux weberWb=Vskg m's-' A-'Magnetic flux density teslaT=Wbm-'= kg s'A-Inductance henryH - Wb A"kg m' s-' A-'Luminous flux lumen Im ° cd srIlluminance luxIx = Im m-' = cd sr m-'Radioactivity becquerelBq=Jkg-'= m's-'Absorbed dose grayGy = I kg-' - m' s-'Dose equivalent sievertSv = J kg-' = m' s-' Very many other SI units arise by combining specially named units with the primary units or with each other.A few examples are listed in Table B.2.3. B:3 PREFIXESThere is a set of 16 prefixes to the primary and special-name SI units that are used to create multiples and sub-multiples. These are listed in Table B.3.1.With one exception, these prefixes are attached to the primary or special-name Si units, as in the examplesB:3:1 l03 in s-' = kilometre per second = km s-'B:3:2 10-12 F m-2 = picofarad per square metre = pF m-2The exception is with mass units, for which the prefix is attached to the gram, rather than the kilogram, unit; forexample:B:3:3 10-9 kg = microgram = µg [not nkg]Note that gm-' means (µm)-t, not µ(m-'), and equals 106 m-', not 10-6 m'. Similarly, cm3 means (10-2m)3 = 10-6 m3, not 10-2 m3. Table B.2.3 Quantity Name of unitSymbol of unit and equivalent Energy densityHeat capacity, entropyThermal conductivityDynamic viscositySurface tensionPermittivitySpecific capacitancejoule per cubic metrejoule per kelvinwatt per metre kelvinPascal secondnewton per metrefarad per metrefarad per square metre1 m-' = kg m" s'J K-' - kg m' s-' K-'W m` K-' = kg m s-' K-'Pa s - kg m-' s-'N m-' - kg s-'F m-' = s' A' kg-' m-'F m-' a s' A' kg-' m-' 675 SOME USEFUL DATA B:4 NON-SI UNITSTable B.3.1 PrefixFactorSymbolPrefixFactorSymbol deka10dadeci0.1dhecto100hcent0.01ckiloto,kmini10-,mmega10'Mmicro10-9µgiga10'Gnano10-9ntera1012Tpico10-11ppets101'Pfemto10-"fexat0"Eano10'"a There are a few units, not part of the SI system, that are officially condoned. These includeB:4:1B:4:2 B:4:3 B:4:4 B:4:5 B:4:6B:4:7litre (L) = 10-3 m3 = dm3tonne (t) = 103 kg = Mgday (d) = 8.64 x 104 shectare (ha) = I0` m2 = hm2adegree (°) =rad180atomic mass unit (u) = 1.660566 x 10-27 kgclectronvolt (eV) = 1.602189 x 10-19 Jand the degree Celsius (°C), for which see B:7:11 below. B:5 UNIVERSAL CONSTANTSB:5 The choice of a fundamental set of universal constants is, to a certain extent, arbitrary. One selection, in SIunits, isB:5:1gravitational constant = G = 6.6720 X 10-" N m2 kg-'B:5:2 speed of light = c = 2.9979258 X 108 m s-'B:5:3 Planck constant = h = 6.62618 x 10-74 J Hz-'B:5:4 Boltzmann constant = k = 1.38066 x 10-231 K-1B:5:5Avogadro constant = L =10-3 kg mol-' u-' = 6.02245 x 102' mol-'B:5:6 elementary charge = q.-1.602189 x 10-19 CB:5:7electron rest mass = m, = 5.4858026 X 10-' u = 9.109534 X 10-" kgB:5:8proton rest mass = mp = 1.007276470 u = 1.672649 x 10-27 kgUncertain digits are italicized. Other constants, related to those above, or arbitrarily defined, includeB:5:9 permeability of free space = pv = 41r x 10-7 H m-'B:5:I0permittivity of free space = ea = 1/µac2 = 8.85418782 x 10-12 F m-' B:6 SOME USEFUL DATA B:5:11 fine structure constant = a = µacgq/2h = 7.297351 x I0-'B:5:12 Faraday constant = F = -Lq, = 9.648456 x 10' C mol-'B:5:13 gas constant = R = kL = 8.31441 J K-' mol-'B:5:14Stefan-Boltzmann constant = a = 2ir'k'/15h3c2 = 5.6703 x10-6W M-2 K-4B:5:15Rydberg constant R. = poc'q^/8h' = 1.09737318 x 10' m-' B:6 TERRESTRIAL CONSTANTS B:6:1 earth's mean radius = 6.370949 x 106 mB:6:2 earth's mass = 5.9732 x 102i kg B:6:3 earth's angular velocity = 7.292116 x 10-' rad s-'B:6:4 siderial year = 3.1558 x 10' sB:6:5gravitational acceleration (45° latitude) = g = 9.8062 m s-2B:6:6 standard atmospheric pressure = 1.01325 X 10' PaB:6:7standard laboratory temperature = 298.15 K = 25.00° C B:7 CONVERSION FACTORS676 To non-SI units other than those in Section B:4:B:7:1metre = 39.3701 inch = 3.28084 foot = 6.21371 X 10-' mileB:7:2104m2 = hectare = 0.247105 acre = 3.86102 x 10-' square mileB:7:310-'m3 = litre = 2.11338 U.S. pint = 0.264172 U.S. gallon = 0.219969 Imp. gallonB:7;4kilogram = 2.20462 pound = 0.0685218 slug = 1.10231 x 10-' short ton = 9.84207 x 10-' tonB:7:510-'kg m-' = kg L-' = gram per cubic centimetre = 62.4280 pound per cubic footB:7:6 metre per second = 2.23694 mile per hourB:7:7newton = 10' dyne = 7.23301 poundal = 0.224809 pound weightB:7:8joule = 10' erg = 23.7304 foot poundal = 0.239006 calorie = 9.86923 x 10-' litre atmosphere= 9.48451 X 10-' B.t.u. = 2.77778 X 10' kilowatt hourB:7:9watt = 3.41443 B.t.u. per hour = 1.34102 x 10-' horsepowerB:7:1010' Pa = kilopascal = 10-2 bar = 7.50062 torr = 0.750062 cm Hg = 0.334562 foot of water= 0.145038 pound per square inch = 9.86923 X 10-' atmosphereIf TK, Tc and TP are numbers expressing temperatures on the kelvin (absolute), Celsius and Fahrenheit scales 5B:7:11 (TK - 273.15) K = (Tc)°C =9(Tp - 32) °F 677 SOME USEFUL DATA B:8B:8 THE GREEK ALPHABETA, a alpha I. L iota P, p rhoB, p beta K, K kappa I, v sigmar, ry gamma A, X lambda T, T tauA, S delta M, µ mu Y, v upsilonE, e epsilon N, v nu <P, 0 phiZ,zeta E, !; xi X,chiH, , eta 0, o omicron Yr,, psie, 0 theta II, n pi d2, (a omega REFERENCES AND BIBLIOGRAPHY Abramowitz, M., and 1. A. Stegun (editors), Handbook of Mathematical Functions, National Bureau of Standards(Applied Mathematics Series #55), Washington, D.C. (1964 and subsequent revisions).Bartsch, H. S., Handbook of Mathematical Formulas, Academic Press, New York (1973).Bell, W. W., Special Functions for Scientists and Engineers, Van Nostrand, London (1967).Beyer, W. H., Handbook of Mathematical Sciences, CRC Press, Boca Raton, Fla. (1964 and subsequent editions).Beyer, W. H., Handbook of Tables for Probability and Statistics, CRC Press, Boca Raton, Fla. (1966 and sub-sequent editions).Bronshtein, 1. N., and K. A. Semendyayev, A Guidebook to Mathematics. Hatri Deutsch, Frankfurt (1971 andsubsequent editions).Carlson, B. C., Special Functions of Applied Mathematics, Academic Press, New York (1977). Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London and New York(1947).Churchill, R. V., Operational Mathematics, third edition, McGraw-Hill, New York (1972).Cooley, J. W., and J. W. Tukey, Mathematics of Computation. 19, 297 (1965).Erd6lyi, A., W. Magnus, F. Oberhettinger and F. G. Tricomi (editors), Higher Transcendental Functions (BatemanManuscript), three volumes, McGraw-Hill, New York (1955).Erd6lyi, A., W. Magnus, F. Oberhettinger and F. G. Tricomi (editors), Tables of Integral Transforms (BatemanManuscript), two volumes, McGraw-Hill, New York (1955).Fletcher, A., J. C. P. Miller and L. Rosenhead, An Index of Mathematical Tables, McGraw-Hill, New York (1946).Gradshteyn, I. S., and 1. M. Ryzhik, Table of Integrals Series and Products, second English edition, AcademicPress,York (1980).Hamming, R: W., Numerical Methods for Scientists and Engineers, second edition, McGraw-Hill, New York(1973).Hart, J. F., E. W. Cheney, C. L. Lawson, H. J. Maehly, C. K. Mesztenyi, J. R. Rice, H. G. Thacher, Jr. andC. Witzall, Computer Approximations, Wiley, New York (1968).Hastings, C., Jr., Approximations for Digital Computers, Princeton University Press, Princeton, N.J. (1955).Knuth, D. E., The Art of Computer Programming, Volume 2, Addison-Wesley, Reading, Mass. (1969).Kom, G. A., and T. M. Kom, Mathematical Handbook for Scientists and Engineers, second edition, McGraw-Hill, New York (1968).Luke, Y. L., Algorithms for the Computation of Mathematical Functions, Academic Press, New York (1977).Luke, Y. L., Mathematical Functions and their Approximations, Academic Press, New York (1975).Luke, Y. L., The Special Functions and their Approximations, two volumes, Academic Press, New York (1969).Macdonald, J. R., Journal of Applied Physics, 35, 3034 (1961). 679 REFERENCES AND BIBLIOGRAPHY 680Magnus, W., F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of MathematicalPhysics, third edition, Springer-Verlag, New York (1966).Murphy, G. M., Ordinary Differential Equations and their Solutions, Van Nostrand and Company, Princeton, N.J.(1960).Oldham, K. B., and 1. Spanier, The Fractional Calculus, Academic Press, New York (1974). Reichel, A., Special Functions, Science Press, Sydney (1968).Roberts, G. E., and H. Kaufman, Table of Laplace Transforms, Saunders, Philadelphia (1966).Ruckdeschel, F. R., Basic Scientific Routines, two volumes, Byte/McGraw-Hill, New York (1981).Sneddon, I. N., Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd, Edinburgh (1956).Sokolnikoff, I. S., and R. M. Redheffer, Mathematics of Physics and Modern Engineering, McGraw-Hill, NewYork (1966).Spiegel, M. R., Mathematical Handbook, McGraw-Hill, New York (1968). Stoer, J., and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York (1980).Szego, G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications #23, Providence,R.I. (1959).Tuma, J. 1., Engineering Mathematics Handbook, McGraw-Hill, New York (1979).Wall, H. S., Analytic Theory of Continued Fractions, Chelsea Publishing, Bronx, New York (1948).Weast, R. C. (editor), CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, Fla. (annual Ikditions). SUBJECT INDEX References are to chapters and sections, not pages. Where several references are cited, the first is the prime ref-erence; others are in numerical order. A citation such as "Sections 9" implies that the subject is encountered inthat section of most chapters. Where a chapter is cited, the subject occurs repeatedly throughout that chapter.Absolute value function, Chapter 8Airy functions, Chapter 56, 50:4, 51:4, 53:4, 54:4Airy integral, 56:3Algebraic functions, 12:0Algorithms, 0:8, Sections 8, Appendix AAlpha exponential integral, 37:13Alternating series, 0:6Alternative Hermite polynomials, 24:13Alternative unit-step function, 8:13Altitude of triangle, 34:14Amplitude. 32:1Antilogarithm, 26:12Appell's symbol, 18:1Approximations, Sections 9Arc length. 39:14 Archimedes' number, 1:7Area of ellipse, 14:14Area of hyperbolic segment, 15:14Area of parabolic segment, 12:14Area of semicircle, 14:14 Argument-addition formula, 0:5Argument-multiplication formula. 0:5Arithmetic mean, 61:8Arithmetic series, 1:6 Arithmetico-geometric series, 1:6Associated Laguerre polynomials, 23:12, 23:1Associated Legendre functions, 59:13, 21:12, 60:4Associated Legendre polynomials (see associated Le-gendre functions)Associated value of extremum of Bessel coefficient,52:7Associated value of zero of Bessel coefficient. 52:7Asymptote. 15:2Asymptote of hyperbola. 15:2. 15:14 Asymptotic series, 0:6Auxiliary Airy functions, Chapter 56Auxiliary associated Legendre functions, 59:13Auxiliary Bernoulli number, 4:1Auxiliary cosine integral. 38:13Auxiliary Euler number, 5:1Auxiliary Fresnel integrals, 39:13Auxiliary Legendre functions, 59:6Auxiliary sine integral, 38:13 Barnes' asymptotic expansion, 43:6Base-conversion algorithms. 9:14, A: I Base of natural logarithms, 1:7Base of number system, 9:14Basis hypergeometric function. 43:14Basset function. Chapter 51, 48:4. 48:13Basset functions of moiety orders, 26:13Basset functions of orders zero and one, Chapter 51,49:3, 49:5Basset functions of quarter orders. 46:4Bateman's G function, 44:13, 7:5, 58:7, 64:13Bateman's k function, 48:4Behavior, Sections 2 681 SUBJECT INDEX 682Bernoulli number. Chapters 4 and 19, 1:14, 3:3. 3:7.3:13, 13:5, 20:7Bernoulli polynomials. Chapter 19, 1:14, 64:4Bernoulli polynomials of order m. 19:12Bessel coefficients, Chapter 52, 43:14Bessel function. Chapter 53 (see also Chapter 52),47:4Besse] function of the second kind (see Neumannfunction)Bessel functions of imaginary argument (see hyper-bolic Bessel function or Basset function)Bessel functions of moiety orders, 32:13Bessel functions of the third kind, 54:13Bessel's equation. 53:3Bessel-Clifford equation, 53:3Best straight line, 7:14. A:4Beta distribution, 27:14Beta exponential integral. 37:13Beta function, Chapter 3, 64:7, 64:13 (see also com-plete beta or incomplete beta function)Beta number, Chapter 3. 5:4Binary logarithm, 25:14Binomial coefficient, Chapter 6, 0:10, 4:5, 5:5, 43:13 Binomial function, 6:14. 18:14Binomial theorem, 6:14Bipolar coordinate system, 46:14Bisector of angle in triangle. 34:14Bit-reversed complement, 32:14 Bivariate eta function, 64:13, 3:12, 64:12 Bivariate function, 0:2Boehmer integrals, 39:12, 64:6, 64:10Boltzmann distribution, 27:14Bose-Einstein distribution. 27:14Briggsian logarithm (see decadic logarithm)Calculus operations, 0:10, Sections 10Cartesian coordinate system, 46:14Catalan's constant, 1:7, 3:7, 61:10Catenary, 28:14Cauchy distribution, 27:14Cauchy limit, 7:10, 14:10, 37:3Cauchy principal value, 27:14Causal distribution. 27:14Center of circle, 14:4Chain rule for differentiation, 0:10Chain rule for Laplace transformation and inversion,26:14Characteristic of elliptic integrals of the third kind,61:12. 62:12Characteristic of logarithm, 25:14Chebyshev functions, 22:1 Chebyshev polynomials (of the first kind and secondkind), Chapter 22, 11:6, 17:9, 28:5, 32:5Chebyshev's differential equation, 22:3Chemists' p. 25:14Chi-square distribution, 27:14Christoffel-Darboux formula for Hermite polynomi-als. 24:5Christoffel-Darboux formula for Laguerre polynomi-als, 23:5Christoffcl-Darboux formulas for Chebyshevnomials. 22:5Circular functions. Chapters 32-34Circular normal distribution, 27:14Clausen functions, 18:14Clausen's integral, 32:5 Clifford's notation, 53:1Clotoid (see Comu's spiral)Coefficients of polynomial functions. 17:1Coefficients of quadratic function, 16:1Cognate functions, Sections 13Cologarithm, 25:14Combinations, 2:14, 6:1Combinatorics. 2:14Commands, 0:8Common antilogarithm, 26:12Common logarithm (see decadic logarithm) Common mean. 61:8. 1:7. 61:3. 62:5. 62:8Complementary error function integral, 40:13Complementary function. 42:3poly- Complementary incomplete gamma function, Chapter45Complementary modulus, 61:1, 63:1 Complete beta function, 43:13, 6:12Complete elliptic integral of the third kind, 61:12Complete elliptic integrals (of the first and secondkinds), Chapter 61, 59:4, 62:8Complete gamma function (see gamma function)Complex argument, Sections I1Complex arithmetic, 1:11Complex variable, 0:1Complex zero of polynomial function, 17:6Complex zeros, A:5Composite function. 26:14Concatenation, 0:6, A:6Confluent hypergeometric equation, 47:3Confluent hypergeometric functions (see Kummerfunction or Tricomi function)Conic section, 14:0Conjugate hyperbolas, 15:5Conjugate pair of complex zeros, 17:6Constant function, Chapter IConstant of integration, 0:10Contiguity relationships of Gauss functions. 60:5Continued fraction. 0:6, 1:6, 17:6, 25:6. 27:6, 34:6,35:6, 37:6, 41:6, 42:6, 45:6, A:6Conventions. figures, 0:2Conventions, symbolism, 0.1 '$3 SUBJECT INDEXConvergent series, 0:6Conversion factors, B:7Convolution of Dirac delta functions, 10:10Convolution property of Laplace inversion, 26:14Convolution, Vandermonde's, 6:5 Copolar elliptic functions, 63:1Comu's spiral, 39:14Correlation coefficient, 7:14Cosecant function, Chapter 33, 3:14. 43:5Cosine function, Chapter 32Cosine integral, Chapter 38Cosine law, 34:14, 21:3Cosine transformation, 32:10Cosinus amplitudinis, 63:1Cotangent function, Chapter 34, 3:14Coversine function, 32:13Cube function, 11:1Cube-root, 13:1 Cubic function, Chapter 17, 62:14 Cubic spline, 17:14Cumulative function, 27:14Cumulative standard normal probability function, 40:14Curvature, 39:14Curve of the second degree. 14:0Cylinder function (see Bessel function) Cylindrical coordinate system, 46:14 Dawson's integral, Chapter 42, 37:14, 40:11, 45:4,45:14de Moivre's theorem, 1:11, 28:5, 32:11 Decadic logarithm, 25:14 Definite integrals, 0:10, Sections 10, A:7Definitions, Sections 3 Degenerate hypergeometric functions (see Kummerfunction or Tricomi function)Delay property of Laplace transformation, 26:14Delta amplitudinis, 63:1 Delta distribution, 27:14 Delta function (see Dirac delta function or Kroneckerdelta function)Denominatorial parameter, 18:14Density function. 27:14Derivatives, 0:10, Sections 10Diagonal Pad6 approximant, 17:13, A:6Diagram conventions, 0:2Differentiation, 0:10, Sections 10Differentiation of integrals, 0:10Differintegral property of Laplace transformation, 26:14Differintegration, 0:10, 43:14Differintegration algorithm, A:7Differintegration of periodic functions, 64:14Differintegration of sinusoid. 64:14Differintegration of square wave function, 64:14Digamma function, Chapter 44, 1:14, 6:10, 7:5, 18:10,37:6Dilogarithm, 25:13, 64:12Dirac delta function, Chapter 10, 8:10, 26:14Directrix of ellipse, 14:3Directrix of hyperbola, 15:3 Directrix of parabola, 12:3Discontinuous function, 8:4Discrete Chebyshev polynomials, 22:13, 17:14Discrete orthogonal polynomials, 22:13Discriminant of cubic function, 17:1 Discriminant of quadratic function, 16:1Distribution function, 27:14Distributions, 27:14 Domain of function, 0:2Double factorial function, 2:13, 6:4, 43:4Double zero. 0:7Doubly periodic function, 63:11Dummy variable. 26:14Duplication formula for gamma function, 43:5 Eccentricity of ellipse, 14:3Eccentricity of hyperbola, 15:3Economization, 22:14, 17:9, A:6Eigenvalues, 59:14Elementary symmetric functions, 17:6Ellipse, 14:14, 61:3, 62:3Elliptic amplitude, 63:3, 61:14, 62:1, 63:1, 63:10Elliptic cylindrical coordinate system, 46:14Elliptic functions (see Jacobian elliptic functions orelliptic integrals)Elliptic integrals, 17:10 (see also complete elliptic in-tegrals or incomplete elliptic integrals)Elliptic modulus, 61:1. 63:1Elliptic parameter, 61:1. 62:1Entire cosine integral, Chapter 38Entire exponential integral, Chapter 37Entire hyperbolic cosine integral. Chapter 38Entire incomplete gamma function, Chapter 45Entire Kummer function. 47:12Equilateral hyperbola, 15:4Equipotential, 46:14Error, 40:14Error function, Chapter 40, 8:9, 45:4. 45:14Error function complement, Chapter 40, 37:14, 46:4Error function of complex argument. 41:11, 40:11 Eta function, Chapter 3. 64:13Eta number, Chapter 3. 44:13Euler function, 43:14Euler hypergeometric integral, 60:3Euler integral definition of gamma function, 43:3Euler number, Chapter 5. 3:7, 3:13. 19:7, 20:7, 33:14Euler polynomials, Chapter 20, 1:14 SUBJECT INDEX 684Euler's constant, 1:7, 1:14, 3:5Euler's formula for exponential function of complexargument, 26:11Euler's formula for Fourier coefficients, 36:6Euler's integral of the first kind, 58:1 (see also com-plete beta function)Euler's integral of the second kind (see gamma func-tion)Euler-Maclaurin formula. 4:14Even numbers, sum of powers, 1:14Expansions, 0:6, Sections 6Exponent, 11:1Exponential decay, 26:0 Exponential distribution, 27:14 Exponential Fourier transformation, 32:14Exponential function, Chapter 26, 18:14, 43:14Exponential growth, 26:0Exponential integral, Chapter 37, 45:4, 45:14Exponential polynomial. 26:13. 28:10. 37:5. 37:13.45:4Exponentials of powers, Chapter 27Exsecant function, 33:13Extrema of Airy functions. 56:7Extrema of Bessel coefficient, 52:7, 52:9Extrema of Bessel function. 53:7Extreme of Kelvin functions, 55:7Extrema of Neumann function. 54:7Extremum, 0:7, Sections 7Factorial function, Chapter 2. 0:5, 6:3. 43:1Factorial polynomial. 18:13Factoring of cubic function. 17:5Factoring of polynomial functions, 17:3Fast fourier transformation/inversion, 32:14, A:4 Fermi-Dirac distribution, 27:14Fick's second law, 46:14Field vector, 46:14Fifth-root, 13:1Figure conventions, 0:2First order hyperbolic Bessel function, Chapter 49Fixed point, 9:14Floating point, 9:14Flux line, 46:14Focus of ellipse, 14:3Focus of hyperbola, 15:3 Focus of parabola, 12:3Focusing property, 12:14. 14:14Fourier coefficients, 36:6, 36:14Fourier equation, 46:14Fourier inversion, 32:14Fourier series, 36:6, 19:6, 20:6, 32:5, 64:6Fourier transformation, 32:14Fourth-root. 13:1Fractional-part function, 9:13. 9:8Fractional-value function, Chapter 9, 5:2Fractional algorithm, A:2Frequency, 36:1Frequency function, 27:14Fresnel integrals, Chapter 39, 32:10Full rectification, 36:13, 8:4, 26:14Function-addition formula, 0:5Function index (see Symbol Index)Function-multiplication formula, 0:5Gamma distribution, 27:14Gamma function, Chapter 43, 2:12 Gauss curve, 27:2 Gauss distribution, 27:14 Gauss function, Chapter 60, 18:14, 31:12, 35:12, 59:12Gauss hypergeometric function (see Gauss function) Gauss limit definition of gamma function, 43:3Gauss probability integral, 40:1 Gauss series, 60:6Gauss' theorem, 44:4Gauss-Legendre multiplication formula for gammafunction, 43:5Gaussian distribution, 27:14, 7:14Gegenbauer polynomials, 22:12General exponential function, 26:12Generalizations, Sections 12 Generalized Fourier expansion, 21:14Generalized Fresnel integrals (see Boehmer integrals)Generalized hypergeometric function, 18:14, 60:13 Generalized Laguerre function, 47:1, 23:14Generalized Laguerre polynomials, 23:12, 47:4, 48:4Generalized logarithm, 64:12, 58:4 Generalized logarithmic function, 25:12, 31:12Generalized zeta function (see Hurwitz function)Generating function, 0:3, 18:3, 19:3, 20:3, 21:3, 22:3,23:3, 24:3, 49:3, 64:3Generator of coordinate system, 46:14Geometric mean, 61:8Geometric series, 1:6, 7:6, 18:14Glaisher's functions (see Jacobian elliptic functions)Gradient, 7:1Graticule, 0:2Greek alphabet, B:8Grunwald definition of differintegral. A:7 Gudermannian function, 33:14, 62:11Half rectification, 36:13, 26:14Hankel functions, 54:13Hankel transformation, 53:10, 52:10Harmonic analysis, 36:6 Harmonic oscillator, 24:14Harmonic series, 44:4Haversinefunction, 32:13Heavyside expansion theorem, 17:10, 26:14Heavyside function (see unit-step function)Heavyside's step function (see unit-step function) 685 SUBJE('1' INDEXHelmholtz equation, 46:14, 59:14Hermite function. 24:14Hermite polynomials, Chapter 24, 23:12, 40:10, 42:10,46:4, 46:6Hermite's differential equation, 24:3Heuman's lambda function, 62:13Hexagamma function, 44:12Hierarchy, 0:12Higher order Bernoulli polynomials, 19:12Hilbert transformation, 7:10, 21:10, 22:10Histogram, 9:2Hurwitz function, Chapter 64, 3:12, 13:5. 32:5Hurwitz' formula, 64:6Hyperbola, 15:14, 15:3Hyperbolic Bessel function, Chapter 50, 47:4, 48:13(see also Chapter 49)Hyperbolic Bessel function of zero order. Chapter 49,43:14Hyperbolic Bessel functions of moiety orders, 28:13Hyperbolic cosecant function, Chapter 29, 3:3, 3:14.4:3Hyperbolic cosine integral, Chapter 38Hyperbolic cotangent function, Chapter 30, 3:14Hyperbolic functions, Chapters 28-30Hyperbolic logarithm (see logarithmic function)Hyperbolic secant function. Chapter 29, 3:3, 3:14, 5:3Hyperbolic sine function. Chapter 28Hyperbolic sine integral, Chapter 38Hyperbolic Struve function, 57:13, 50:10, 50:12, 51:10Hyperbolic tangent function, Chapter 30, 3:14, 8:9Hypergeometric differential equation, 60:3 Hypergeometric function, 18:14, 45:14, 60:13 (see alsoGauss function)Hypergeometric polynomials, 22:12Hypotenuse, 34:14Imaginary argument, Sections 1 IImaginary period, 63:11Implicit definition, 0:3Improper rational function, 17:13 Impulse function (see Dirac delta function)Incomplete beta function, Chapter 58. 18:14, 31:12.35:12, 60:4, 62:4Incomplete elliptic integral of the third kind. 62:12 Incomplete elliptic integrals (of the first and secondkinds), Chapter 62, 14:10, 63:10Incomplete gamma function, Chapter 45, 18:14. 26:12,41:6, 42:12, 47:4, 48:4Indefinite integration, 0:10, Sections 10, A:7Infinite product, 0:6, 1:6, 3:6. 7:6. 32:6, 53:6Infinite series, 0:6Infinitely repeated exponentiation, 26:13Inflection, 0:7Integer powers, Chapter I IInteger-part function, 9:13, 9:8Integer-value function, Chapter 9Integral function (see polynomial function) Integral of error function complement, 40:13Integral transform, 0:3 Integrals, Sections 10Integration by parts, 0:10Integration constant. 0:10Intercept, 7:1Interpolation. 17:14Interrelationships, 0:5Interrelationships. 0:5Invariant (see constant function)Inverse circular functions, Chapter 35 Inverse cosecant function, Chapter 35Inverse cosine function, Chapter 35Inverse cotangent, Chapter 35 Inverse function, 0:3 Inverse gudermannian function. 33:14, 34:5, 62:4, 62:6Inverse hyperbolic cosecant function. Chapter 31. 15:10Inverse hyperbolic cosine function, Chapter 31, 15:10,16:13Inverse hyperbolic cotangent function, Chapter 31.21:13, 59:4Inverse hyperbolic functions, Chapter 31 Inverse hyperbolic secant function, Chapter 31, 14:10Inverse hyperbolic sine function. Chapter 31, 15:10,16:13Inverse hyperbolic tangent function, Chapter 31. 12:10,16:10, 21:13, 58:4. 59:4, 64:12Inverse linear function (see reciprocal linear function)Inverse secant function, Chapter 35, 15:10Inverse sine function, Chapter 35, 14:10. 16:13, 22:3.61:10Inverse tangent function. Chapter 35. 12:10, 16:10.58:4. 64:12Inverse trigonometric functions, Chapter 35 Jacobi polynomials, 22:12, 60:4Jacobi's cta functions, 63:13Jacobi's imaginary transformations, 63:11Jacobi's real transformations, 63:5Jacobi's theta functions, 63:13Jacobi's zeta function, 62:13Jacobian elliptic functions, Chapter 63, 61:14, 62:3Kapteyn series, 52:5Kelvin functions, Chapter 55, 49:11, 51:11, 52:11Kelvin functions of the third kind, 55:13 King's integral, 51:10Kramp's symbol, 18:13Kronecker delta function, 10:13Kummer function, Chapter 47, 18:14. 24:12, 40:12,41:12, 42:12, 45:14, 46:5, 46:12Kummer's series. 47:6Kummer's transformation. 47:5 SUBJECT INDEXLagrange four-point interpolation, 17:14Laguerre function, 23:14, 47:1, 47:4Laguerre polynomials, Chapter 23, 37:6Laguerre's differential equation, 23:3 Lambda function, Chapter 3, 64:7Lambda number, Chapter 3Landen transformations, 62:5, 63:5Langevin function, 30:13Laplace distribution, 27:14 Laplace inversion, 26:14 Laplace inversion of hypergeometric functions,26:14Laplace transformation, 26:14. 26:10Laplace transformation of hypergeometric functions,26:14, 18:14Laplace's equation, 46:14Laplace's integral representation of Legendre poly-nomials, 21:3Laplace-de Moivre formula, 6:9Laplacian operator, 46:14Latitude of sphere, 59:14Lawsofexponents, 11:5. 13:5Least squares, 7:14, 17:14, 22:13Legendre function of the first kind, Chapter 59, 21:1Legendre function of the second kind. Chapter 59.21:10, 21:13, 54:12Legendre polynomials, Chapter 21Legendre relation, 61:5Legendre's differential equation, 21:3, 59:3Legendre's elliptic integrals (see incomplete ellipticintegrals)Leibniz's rule for differentiating an integral, 0:10Leibniz's theorem for multiple differentiation of aproduct, 0:10Lerch's function, 64:12, 13:5, 30:10Library subroutines, A:6 Limiting properties of Laplace transformation and in-version. 26:14Limits of integration, 0:10Line of force, 46:14 Line of steepest descent, 46:14Linear function, Chapter 7Linear regression, 7:14, A:4 Linear shift property of Laplace transformation andinversion, 26:14Linear transformations of Gauss function, 60:5Linearity property of Laplace transformation and in-version, 26:14Log-normal distribution, 27:14Logarithm (see logarithmic function)Logarithm to base 2 (see binary logarithm)Logarithm to base e (see logarithmic function)Logarithmic function, Chapter 25Logarithmic integral, Chapter 37, 25:13Logarithm to base 10 (see decadic logarithm)Longitude of sphere, 59:14Lorentz distribution, 27:14686 Macdonald's function (see Basset function)Maclaurin series, 11:14, A:6Magnitude of function (see absolute value function)Mantissa, 25:14Many-valued logarithmic function, 25:11Map conventions, 0:2Mascheroni's constant (see Euler's constant)Maximum, 0:7, Sections 7Maxwell distribution, 27:14Mean. 27:14Median, 27:14Median of triangle, 34:14 Mehler-Dirichlet formula, 59:3Mellin transform, 13:10Mensuration of triangles, 34:14Method of undetermined coefficients, 17:13Minimum. 0:7, Sections 7Mode. 27:14Modified Bessel equation, 50:3Modified Bessel function of the first kind (see hy-perbolic Bessel functions)Modified Bessel function of the third kind (see Bassetfunction)Modified Hankel function (see Basset function)Modified Neumann series, 53:14Modified spherical Bessel function of the fast kind,28:13, 50:4Modified spherical Bessel function of the third kind,26:13, 51:4Modular angle. 61:1, 62:1Module (see modulus)Modulo function, 9:13Modulus (see absolute value function, modulo func-tion or elliptic modulus)Monotonic function, 9:2Most probable value, 27:14Multinomial coefficient, 6:12Multiple differentiation of a product, 0:10Multiple zero, 0:7Murphy's formula, 21:12Naperian logarithm (see logarithmic function)Natural logarithm (see logarithmic function)Natural numbers, 1:14 Nearest-integer function (see rounding function)Nested sum, 0:6Neumann function, Chapter 54, 52:3Neumann functions of moiety orders, 32:13Neumann series, 53:14, 51:6, 52:5Neumann's addition formula, 52:5, 53:5 Neumann's formula for Legendre functions, 21:10Nevillc's theta functions, 63:8, 61:14 697 SUBJECT INDEXNewton's formula, 6:14, 17:7Newton's method for improving a zero, 17:7, A:5Newton's method for square roots, 12:9Nielsen's expansion, 45:5Nome, 61:14, 61:7Nomenclature, Sections INoninteger powers. Chapter 13Normal distribution, 27:14, 40:14, A:3Normalized Legendre polynomial, 21:1Notation, Sections 1Number systems, 9:14Numeratorial parameter, 18:14Numerical quadrature, A:7Numerical values, Sections 8Nyquist frequency, 32:14Oblate spheroidal coordinate system, 46:14Odd numbers, sum of powers, 1:14Operations of calculus, 0:10, Sections 10Operations of calculus on hypergeometric functions,18:14Orthogonal coordinate systems, 46:14Orthogonal functions, 21:14Orthogonal series, 21:14Orthogonality of associated Legendre functions, 59:13Orthogonality of Bessel coefficients, 52:14Orthogonality of Chebyshev polynomials, 22:10Orthogonality of Hermite polynomials, 24:10Orthogonality of Jacobi polynomials, 22:12Orthogonality of Laguerre polynomials. 23:10Orthogonality of Legendre polynomials, 21:14, 21:10Orthonormality, 21:14Pade approximant, 17:13, 51:8, 64:8Pade operation, 17:13Pade table, 17:13Parabola. 12:0. 12:14Parabolic cylinder function, Chapter 46, 45:14. 47:4,48:4. 50:4. 51:4Parabolic cylindrical coordinate system, 46:14Paraboloidal coordinate system, 46:14Parameter (see elliptic parameter)Parametric definition. 0:3, 39:14Parseval's relation, 36:6Partial fractions, 17:13, 16:4, 17:6, 33:6, 34:6 Particular values, Sections 7Partitions, 2:14Pascal's triangle, 6:8Pentagamma function, 44:12Percentage point. 27:14Percentile, 27:14Perimeter of ellipse, 14:14, 61:3Perimeter of hyperbolic segment, 15:14Perimeter of parabolic segment, 12:14Perimeter of semicircle, 14:14Period, 36:1Periodic functions, Chapter 36Periodicity property of Laplace transformation. 26:14Permutations, 2:14Phase, 32:1Physical constants, B:5. B:6Pi (see Archimedes' number)Piecewise-constant function, 1:13Piecewise-defined function, 8:12Piecewise-linear function, 7:13Pochhammer polynomials, Chapter 18, 2:5, 2:12, 2:13,6:12, 6:13, 43:5Polar coordinate system, 46:14 Polygamma function, 44:12, 64:4 Polylogarithm, 25:13, 64:12 Polynomial fitting, 17:14, 22:13, A:4Polynomial function, Chapter 17Power, 11:1Power multiplication property of Laplace transfor-mation, 26:14Power series, 11:14, 0:6, A:6 Power spectrum, 32:14Precision, 0:8 Prime numbers, 3:6Probability integral (see error function)Probable error, 40:14Product rule for differentiation, 0:10Product rule for Laplace transformation and inver-sion, 26:14Projectile, 16:14Prolate spheroidal coordinate system, 46:14Proper rational function, 17:13Pseudorandom integer, 40:14Pseudorandom number, A:3Psi function (see digamma function)Pulse function. 1:13, 8:4Pulse height, 1:13Pulse width, 1:19Pythagoras' theorem, 34:14Pythagorean trios, 34:14Quadrant, 0:2, 13:2. 32:2Quadratic function, Chapter 16Quadratic transformations of Gauss function. 60:5Quadrature, A:7Quadrinomial coefficient. 6:12Quadruple factorial function, 2:13, 43:4. 59:7Quadruplication formulas of theta functions, 27:13Quantum numbers, 59:14Quartic function. 17:1. 16:5Quintic function, 17:1Quotient rule for differentiation, 0:10 Radius of circle. 14:4Randles-Sevcik function, 30:10, 64:12 SUBJECT INDEX 688Random number, 40:14Range of variable, 0:2Ratio of successive Bessel coefficients, 52:5, 52:6,52:8Rational function, 17:13, 44:14 Rayleigh distribution, 27:14Rayleigh formulas, 50:10Reciprocal linear function, Chapter 7Reciprocal square-root function, Chapter 12Rectangular coordinate system, 46:14Rectangular distribution, 27:14Rectangular hyperbola, 15:4, 7:13, 28:3Rectification (see full rectification or half rectifica-tion)Recurrence relation, 0:5Recursion formula, 0:5Reflection formula, 0:5, 15:4Regression, 17:14Regula falsi, A:5Relative error, 0:8Remainder function (see modulo function)Repeated integrals of error function complement,40:13, 46:4Repeated root, 0:7Richardson extrapolation, A:7Riemann function (see zeta function or Hurwitz func-tion)Riemann's zeta function (see zeta function or Hurwitzfunction)Rodrigues' formula for Chebyshev polynomials, 22:3Rodrigues' formula for Hermite polynomials, 24:3 Rodrigues' formula for Jacobi polynomials, 22:12Rodrigues' formula for Laguerre polynomials, 23:3Rodrigues' formula for Legendre polynomials, 21:3Root finding, A:5Root of equation, 0:7Root of quadratic equation, 16:7Root-quadratic function. 15:12Rotation formula, 15:4Rotation of a function, 7:13Rounding error, 0:8Rounding function. 9:13, 9:8Rudoirs number (see Archimedes' number)Sampling function, 32:13Sawtooth function, 9:13, 8:4Scaling property of Laplace transformation and in-version, 26:14Schlomilch functions, 37:13Schrodinger equation, 24:14, 23:14, 46:14Scientific notation, 9:14, 25:14Secant function, Chapter 33, 3:14, 43:5Sech-square distribution, 27:14Sectoral surface harmonics, 59:14Self-exponential function, 26:2, 26:13Semiaxis of ellipse, 14:1Semicircle, 14:14Semicircular function, 14:4Semielliptical function, Chapter 14Semifactorial function (see double factorial function)Semihypcrbolic function, Chapter 15Semiintegration, 0:10Semimajor axis of ellipse, 14:1Semimanor axis of ellipse, 14:1Semiperimeter of triangle, 34:14Separation constant, 46:14, 59:14Sequence, 62:5 Shifted Chebyshev polynomials, 22:1Shifted factorial function (see Pochhammer polyno-mial)Shifted Gauss distribution, 27:14Shifted Legendre polynomials, 21:13SI system of units, B:l, B:2, B:3Sifting property of Dirac delta function, 10:10Sign function (see signum function)Significance, 0:8Signum function, Chapter 8 Simple harmonic oscillator, 24:14Simple root, 0:7 Simple zero, 0:7, 17:13Sine function, Chapter 32 Sine integral, Chapter 38Sine law, 34:14 Sine transformation, 32:10Single-valued function, 0:3Sinus amplitudinis, 63:1Sinusoidal function, 32:1 Sliding cubic. 17:14Slope, 7:1Snedeoor's-F distribution, 27:14Special cases, Sections 4Special functions of mathematical physics, 18:14Special topics, Sections 14Spence's integral (see dilogarithm)Spherical Bessel functions, 32:13, 38:6, 39:3, 39:6,53:4, 54:4, 57:4, 59:14Spherical coordinate system, 46:14, 59:14Spherical harmonics, 59:14 (see also associated Le-gendre functions)Spherical polynomial, 21:1Spheroidal coordinate systems, 46:14Square function, 11:1Square wave function, 36:14, 8:4 Square-root function, Chapter 12Staircase function, 9:13, 8:4Standard error, 40:14, 7:14Standard normal density function, 40:14Stirling approximation, 43:9 689 SUBJECT INDEXStirling number of the first kind, 18:6, 2:5, 6:6Stirling number of the second kind, 2:14, 11:6Stirling's formula for factorials, 2:6Stirling's formula for gamma function, 43:6Streamline, 46:14Struve function, Chapter 57, 53:12Student's-t distribution, 27:14Subroutines, A:6Successive derivatives of Bateman's G function, 44:13,64:13Successive derivatives of error function, 40:1, 40: 10,46:4Sums of powers, 1:14, 19:14, 20:14Superellipse, 14:12 Surface harmonics, 59:14Symbolism, Sections 1Symbolism, conventions adopted, 0:1Symmetrical periodic function, 26:14Synthesis of functions, 43:14Tangent function, Chapter 34, 3:14Tangent rule, 34:14Taylor expansion, 0:5, 6:14Telescoping of series, 22:14Terrestrial constants, B:6 Tesseral surface harmonics, 59:14Test values, 0:8Tetragamma function, 44:12Theorem of Gauss, 44:4Theta functions, 27:13, 26:5Thomson functions (see Kelvin functions)Toroidal coordinate system, 46:14Trajectory, 16:14Transcendental function, 53:0Translation formula, 0:5Trapezoidal approximation. A:7Triangles, 34:14Tricomi function. Chapter 48, 18:14, 24:12. 40:12,45:14, 46:12Trigamma function, 44:12 Trigonometric functions. Chapters 32-34Trilogarithm. 25:13, 18:14Trinomial coefficient, 6:12Triple factorial function, 2:13, 43:4, 56:6Triplication formula for gamma function, 43:5 Ubiquitous constant, 1:7, 3:5, 27:13, 43:4, 59:7, 61:7,61:10Ultrespherical polynomials, 22:12Undetermined coefficients, 17:13Uniform distribution, 27:14 Unit-impulse function (see Dirac delta function)Unit-moment function, 10:12Unit-step function. Chapter 8Units, B:l, B:2, B:3Unity function, 1:4Universal constants. B:5 Universal hypergeometric algorithm, 18:14 Utility algorithms, Appendix AVandennonde's convolution, 6:5Vandermonde's theorem, 18:5Variance, 27:14Variations, 2:14Versine function, 32:13Wallis' formula. 2:13Wave equation. 46:14Weber function (see parabolic cylinder function orNeumann function)Weber-Hermite function (see parabolic cylinder func-tion)Weibull distribution, 27:14Weierstrass infinite product definition of gammafunction, 43:3Weierstrassian elliptic functions, 63:13Weight function of an orthogonal polynomial, 21:14Weight of a datum, 7:14Weighted least squares, 7:14Weyl differintegration, 64:14Whittaker functions, 48:13, 51:12Window function, 8:12Zero order hyperbolic Bessel function. Chapter 49Zeros of Airy functions, 56:7Zeros of Besse) coefficients, 52:7, 52:9Zeros of Bessel function, 53:7Zeros of functions, Sections 7, 8:10Zeros of Kelvin functions, 55:7Zeros of Legendre functions. 59:2Zeros of Neumann function, 54:7Zeros of quadratic function, 16:7Zeros of Struve function, 57:7Zeta function, Chapter 3, 64:7Zeta number, Chapter 3. 44:6Zonal surface harmonics, 59:14, 21:1 SYMBOL INDEX References are to the chapter and/or section in which the symbol is defined or introduced. It is intended that thisindex will provide a complete glossary of all the notations used in the Atlas. but temporary symbolism that isdefined locally is not indexed. Use of the word 'see" following a symbol implies that the notation is identified inthe referenced section but is not used elsewhere in the Atlas; such a symbol has a meaning identical (or related)to the notation that follows "see." This listing may also be used as a function index. the final section has beenadded for this purpose. CAPITAL LETTERS: ROMAN ALPHABET AArccos(=), Aresin(:), etc.Arctrig(x)Arcosh(x), Areoth(x), etc. Arsinh(:)A.B,CAi(x), Bi(x) B(x.y)B(v:µ:x)B.B,. B; 8,1 v:µ)B', '(.r)B(p). C(p). D(p)arithmetic mean. 61:8multivalued inverse trigonometric function of complex argument :. 35:11any multivalued inverse trigonometric function of argument x. 35:12multivalued inverse hyperbolic functions of argument x. 31:12multivalued inverse hyperbolic sine function of complex argument :. 31:11angles in a triangle: vertices of a triangle, 34:14Airy functions of argument x, Chapter 56complete beta function of arguments x and v, 43:13incomplete beta function of parameters v, µ and argument x. Chapter 58n" Bernoulli number, Chapter 4auxiliary Bernoulli number of index n. 4:1Bernoulli polynomial of degree n and argument x, Chapter 19see19:1see B(v:p.:x). 58:1Bernoulli polynomial of order in, degree n and argument x, 19:12complete elliptic integrals of modulus p. 61:13C see y, Euler's constant, 1:7C(x) Fresnel cosine integral of argument .x, Chapter 39C(x:v) Bochmcr cosine integral of degree v and argument x, 39:12Chi(x) hyperbolic cosine integral of argument x. Chapter 38Chin(x) entire hyperbolic cosine integral of argument x, Chapter 38Cih(x). Cinh(.r) see Chi(x). Chin(x), 38:1all SYMBOL INDEX 692Ci(x)Cin(x)Cos(x), Cot(s)Cu,C_,.. ,C,,...C.(x)Cdz)C,(x). C,(x).C_ C.C'''(x)DD. dD(x)D,.(x)E, E'E(p), E(q) E(x)E(x) Ei(x)E*(x). Ei(x), E'(x), etc.Ein(x)Erfi(x) FFF(x)Fp(x)F(a,b;c;x) .Fd(x) ,F,(a;c;x),Fo(a;c:x),F,(a,b;c;x)Fres(x), Gres(x) Fe,(x), Ge.(x)GGG G(x)G(x;t)Go, G,, ...,G,G..G`(x)Gi(x), Hi(x)H(x)H(x - a)cosine integral of argument x. Chapter 38entire cosine integral of argument x, Chapter 38see cosh(x), cosh(x), 30:1coefficients of partial fractions. 17:1see T.(x), 22:1Clifford's notation for Bessel functions of order v and argument 2\ z, 53: Isee C(x). 39:1number of combinations of m objects chosen from a group of n, 6:1Gegenbauer polynomial of parameter h. degree n and argument x, 22:12discriminant of cubic function. 17:1denominators of fractions, A:2see daw(x), 42:1parabolic cylinder function of order v and argument x, Chapter 46see E(p), E(q). 61:1complete elliptic integral (of the second kind) of modulus p.p, Chap-ter 61see lnt(x), 9:1Euler's function of argument x. 37:13exponential integral of argument x, Chapter 37see Ei(x), 37:1entire exponential integral, Chapter 37seedaw(x), 42:1n" Euler number, Chapter 5auxiliary Euler number of index n, 5:1Euler polynomial of degree n and argument x, Chapter 20n" Schlomilch function of argument x, 37:13see E.(x), Euler polynomial, 20:1see r(v;x), 45:1incomplete elliptic integrals (of the second, first kinds) of modulus p andamplitude d,, Chapter 62some scalar property, 46:14Faraday constant, B:5inverse function of f(x); see also f(x), f(t). 0:3fractional-part function of argument x. 9:13Gauss function of argument x with numeratorial parameters of a. b and de-nominatorial parameter c, Chapter 60"generalized" hypergeometric function of argument x having n unspecified numeratorial and d unspecified denominatorial parameters, 60:13seeM(a;c;x), 47:1see U(a;c;x), 48:1see F(a,b;c;x), 60:1auxiliary Frcsnel (cosine- and sine-) integrals of argument x, 39:13series associated with Kelvin functions of order v and argument x, 55:6geometric mean, 61:8gravitational constant. B:5Catalan's constant. 1:7Bateman's G function of argument x, 44:13generating function for f(x), 0:3terms in the j1° coefficient of the hypergeometric series, 18:14common mean, 61:8the na successive derivative of Bateman's G function of argument x, 44:13components of theAiry Bi(x) function, 56:13see erf(x), 40:1see u(x - a), 8:1 693 LL(x)L{ }Ln(z)L,(x)L,(x)L''(x)L;,-(x)L,"'(x)M(a;c:x)M(N:m......m")M,.&(x)NN, nN(p) N,N,(x)P(z)P"(x)P;(x), P"(x)P,"'(x)P, QP, QP(v;x), Q(v:x)P(v;x), Q(v:x)P,(x), Q,(.r)K`W, Q;.`(x)P(p). Q(q). R(r)SYMBOL INDEXJacobi's eta functions of modulus p and argument x, 63:13Hermite polynomial of degree n and argument x, Chapter 24see h,(x), 57:1alternative Hermite polynomial of degree n and argument x, 24:13integer value function of argument x. Chapter 9integer-part function of argument x, 9:13imaginary part operator, 55:3hyperbolic Bessel functions of integer orders and argument x. Chapter 49hyperbolic Bessel function of order v and argument x, Chapter 50incomplete beta function ratio of parameters v, p and argument x. 58:1largest value of the summation index j. 0:6Bessel coefficients of argument x, Chapter 52 Bessel function (of the first kind) of order v and argument x, Chapter 53associated value of the k" extremum of the n' Bessel coefficient. 52:7associated value of the k" zero of the n" Bessel coefficient, 52:7see K(p), K(q), 61:1complete elliptic integrals (of the first kind) of modulus p, 1 - . Chapter61Basset function of order v and argument x. Chapter 51see r(v;x), 45:1number of numeratorial, denominatorial parameters of hypergeometric func-tion, 18:14Avogadro constant, B:5 see (i(x), 3:1Laplace transform operator. 26:14multivalued logarithmic function of complex argument z, 25:11 Laguerre polynomial of degree n and argument x. Chapter 23Laguerre function of degree v and argument x, 23:14see e,(x), 57:1associated Laguerre polynomial of order m, degree n and argument x, 23:12generalized Laguerre polynomial of order v, degree n and argument x, 23:12generalized Laguerre function of order µ, degree v and argument x, 47:1 Kummer function of numeratorial parameter a. denominatorial parameter cand argument x, Chapter 47multinomial coefficient in the expansion of (-r, + x: + + x")", 6:12Whittaker function of parameters Y. µ and argument x. 48:13characteristic of a logarithm. 25:14numerators of fractions. A: Ielliptic nome of modulus p. 61:14j" integer drawn from the digit set 0. 1. 2, ..., 13 - 1, 9:14see Y,(x), 54:1Weierstrassian elliptic function of complex argument z, 63:13Legendre polynomial of degree n and argument x. Chapter 21see P"(x), 21:13Jacobi polynomial of parameters v. µ, degree n and argument x, 22:12components of the discriminant of a cubic function. 17:1periods of periodic functions. 36:1functions arising in the asymptotic expansion of Bcsscl functions, 53:6see y(v;x), r(v;x). 45:1Legendre functions (of the first and second kinds) of degree v and argumentx, Chapter 59associated Legendre functions of ordcr µ, degree v and argument x, 59:13functions that depend on a single coordinate. p, q or r. 46:14 SYMBOL INDEX 694 Q.(x) RRe{ }R(x,y ) R.R.R1RR,'(x) S(x)S(x;v) Shi(x)Si(x)Sin(x)S,(x). S.(x)Sa(x)STah(x)T,(x), ZLegendre function (of the second kind) of integer order n and argument x.21:13gas constant, B:5 real part of, 55:3rational function of arguments x and Y. 26:14 ratio of two consecutive hyperbolic Bessel functions, 49:8ratio of two consecutive Bessel functions of common argument, 52:5remainder after truncating power series to polynomial of order J - 1. 18:14Rydberg constant. 13:5the rational function Po(x)/P.(x). 17:13diagonal Pade approximants, 17:13Fresnel sine integral of argument x. Chapter 39Boehmer sine integral of degree v and argument x, 39:12hyperbolic sine integral of argument x. Chapter 38sine integral of argument x. Chapter 38see sinh(x). 28:1see S(x). 39:1see u(x - a). 8:1see U,(x), 22:1Stirling numbers (of the first kind). 18:6see tanh(x). 30:1Chebyshev polynomials (of the first and second kinds) of degree n and ar-gument x, Chapter 22see 22:1ubiquitous constant, 1:7Tricomi functions of parameters a, c and argument .r. Chapter 48function cognate to parabolic cylinder function, 46:13finite sums related to spherical Besse] functions, 32:13error function of complex argument :. 41:11Whittaker function of parameters v, µ and argument x, 48:13auxiliary argument of Airy function, 56:1auxiliary argument of hypergeometric function. 18:14factorial polynomial, 18:13Neumann function of order v and argument x. Chapter 54 spherical harmonics of two angular coordinates 0 and $, 59:14the zeta number 4(3). 3:7 LOWERCASE LETTERS: ROMAN ALPHABET abs(x)am(p;x)arccos(x), arccot(x), etc.arccosec(x). arcctg(x), etc.arccosh(x), arccoth(x), etc.arch(x), arch(x)arcosh(x), arcoth(x), etc.arctrig(x)argcosh(x), argcoth(x), etc.ao, a,, a2, .., a,, ...a., a,-,, a.-2, ..., aca,, a,, a,, ..., aAa, absee f 8:1elliptic amplitude of modulus p and argument x, 63:3 inverse trigonometric functions of argument x. Chapter 35see arccsc(x). arccot(x), etc.. 35:1see arcosh(x), arcoth(x), etc., 31:1see arcosh(x), arsinh(x). 31:1inverse hyperbolic functions of argument x. Chapter 31any inverse trigonometric function of argument x, 35:1 see arcosh(x), arcoth(x), etc.. 31:1coefficients of power series, 11:14coefficients of polynomial of degree n. 11:7numeratorial parameters of hypergcometric function. 18:14semiaxes of ellipse, 14:1 695a,b,ca,b,c bei(x), ber(x)bei,(x), ber,(x)b, ccCcd(p;x), cn(p;.r), cs(p;x)ch(x)ci(x) cot(x), cos(t))cosec(x), cosech(x)cosh(x)Cos -, (X), cot -, (X)cosh-'(x), cosh-'(x)cot(x), cot(e) cotan(x), ctg(x)cosh(x)covers(x)csc(x). csc(e)csch(x)csc-'(x), csch-'(x)cth(x), ctnh(x)co. ci, c2....c1, c2, c,, ..., CLc,, Sid, adaw(x)dc(pa), dn(p:x), ds(p,,x)d. ee ef(p:x). eg(p:x). gh(p:x). etc.eerfc(x). erc(x)ei(x)erf(x), erfc(x)crfi(x)_exp(x) erfc(V x), etc.exp(x). exp(br + c). exp(-ax")exsec(x)e e2. e,e (x)f(r). F(x)f(t). k1s)f(t), f (s)f(t), ft(s)f(0. 1Ml s)f(t). fsis)f(t). F(s)frac(x)ffei,F(x),, fer,(x)ff(x). g(x)SYMBOL INDEX coefficients of quadratic or cubic function, 16:1, 17:1lengths of the sides of a triangle, 34:14Kelvin functions of zero order and argument x, Chapter 55Kelvin functions of order v and argument x, Chapter 55slope, intercept of linear function. 7:1 constant function, Chapter Ispeed of light, B:5Jacobian elliptic functions of modulus p and argument x, Chapter 63see cosh(x), 28:Isee Ci(x), 38:1cosine function of argument x, angle 0, Chapter 32see csc(x), csch(x), 33:1, 29:1hyperbolic cosine function of argument x, Chapter 28see arccos(x), arccot(x), 35:1see arcoos(x), arcoth(x), 31:1cotangent function of argument x, angle 0, Chapter 34see cot(x), 34:1hyperbolic cotangent function of argument x, Chapter 30coversine function of argument x, 32:13cosecant function of argument x, angle 0. Chapter 33hyperbolic cosecant function of argument x, Chapter 29see arccsc(x), arcsch(x), 35:1, 31:1see coth(x), 30:1coefficients in orthogonal series, 21:14denominator parameters of hypergeometric series. 18:14Fourier coefficients. 36:6total, partial differential operators. 0:10Dawson's integral of argument x, Chapter 42Jacobian elliptic functions of modulus p and argument x, Chapter 63standard errors in slope, intercept, 7:14base of natural logarithms, 1:7 arbitrary Jacobian elliptic functions. 63:1 see exp(x) erfc(V/x), 41:1see E,(x) (Schl6milch function), 37:1error function, error function complement of argument x, Chapter 40see daw(x), 42:1products of exponentials and error function complements. Chapter 41exponential functions of various arguments, Chapter 26, 27exsecant function of argument x, 33:14variables of the Weierstrassian system, 63:13 exponential polynomial of degree n and argument x. 26:13density function of a distribution, the corresponding cumulative function.27:14a function of t. its cosine transform, 32:10a function of t, its Hilbert transform, 7:10a function of i, its Laplace transform. 26:14a function of t, its Mellin transform. 13:10a function of r, its sine transform. 32:10a function of t, its Fourier transform. 32:14fractional-value function of argument r, Chapter 9bei,(x) or kei,(x), ber,(x) or ker,(x). 55:5values of f(x) at .r - r .c2, x ..., x,, 17:14arbitrary functions of argument x, 0:2 SYMBOL INDEX 696 f(x;v), g(.x;r), etc.fai(x), gai(x) 6(x), gi(x)fe(p;x). ge(p:x), he(p:x)9g(j:x)gd(x)8d-'(x)hhhai(x)hav(x)he(x)hei,(x). her.,(x)iierfc(x) i'erfc(x)..... i"erfc(x)int(x)invgd(x)i"(x) J"(x)J":r, 1..kj,k,IItk, k'kei(x), ker(x)k"(x) In,(x)log,o(x), loge(s)log,(x)mm, mim., mom, nnc(p;x), nd(p;x), ns(p;x)Ppx, pHp(c;hrx - a)P"(x)poln,(x)P, qper(x), qMx)bivariate functions. 0:2auxiliary Airy functions of argument x. 56:3auxiliary sine, cosine integrals of argument x, 38:13arbitrary Jacobian elliptic functions of pole type "e," 63:1gravitational acceleration of the earth. B:6component of an infinite product, 0:6gudcrmannian function of argument x. 33:14see invgd(x), 33:14Planck constant, B:5width of an interval or pulse, 4:14, 1:13function of argument x. related to auxiliary Airy functions. 55:6haversine function of argument x, 32:13Struve function of order v and argument x. Chapter 57Kelvin functions (of the third kind) of order v and argument x, 55:13square-root of minus one; imaginary operator, 0:1 complementary error function integral of argument x, 40:13repeated integrals of error function complement, 40:13see Int(x), 9:1inverse gudermannian function of argument x, 33:14modified (or hyperbolic) spherical Bessel function of order n and argumentx, 28:13spherical Bessel function (of the first kind) of order n and argument x. 32:13k' zero, ka extremum of the n' Bessel coefficient, 52:7finite or infinite summation indices, 0:3Boltzmann constant, B:5see p, q (elliptic modulus, complementary elliptic modulus), 61:1Kelvin functions of zero order and argument x, Chapter 55modified spherical Bessel function (of the third kind) of order n and argu-ment x, 26:13Bateman's k function of order v and argument x, 48:4Kelvin functions of order v and argument x, Chapter 55logarithmic integral of argument x. 25:13 logarithmic function of argument x. Chapter 25see ln(x) or log,o(x) 25:1hyperbolic Struve function of order v and argument x, 57:13auxiliary Legendre functions of degree v and argument x. 59:6auxiliary associated Legendre functions of order µ, degree v and argumentx, 59:13 generalized logarithmic function of order v and argument x, 25:12decadic logarithm, logarithm to base 0 of argument x. 25:14see ln(x), 25:1see b (slope of linear function). 7:1see p, q (elliptic modulus, complementary elliptic modulus), 61:1integers appearing as powers in multinomial coefficients, 6:12rest mass of electron, proton, B:5 integers or natural numbers, 0:2, 1:14Jacobian elliptic functions of modulus p and argument x, Chapter 63percentage, 27:14chemists' p, 25:14pulse function of height c, width h and centered at x = a, 1:13polynomial function of degree n and argument x, Chapter 17polylogarithm of order v and argument x, 25:13elliptic modulus, complementary modulus (=gyp ). 61:1periodic functions of argument x, Chapter 36 697 4errrsf(x)r;ro(b), r1(b)....,rr(b)SSsc(p;x), sd(p;x), sn(p;x) sec(x), sec(O)sec-'(x), sech-'(x)sech(x)sg(x), sign(x)sgn(x)sh(x)si(x)sin(x), sin(O)sin"'(x), sinh-'(x)sinc(x)sinh(x)sqrt(x)s,, c;rtan(x), tan(O)tan-'(x), tanh-'(x)tanh(x)th(x)tn(p;x)triln(x)to, l r2,..., r;, ...t,(y)t;,iiu(x), u(x - a) w;xo, xixo, x1, X21 ..,x;,X1/2X, Y... Y..kSYMBOL INDEX elementary charge. B:5correlation coefficient, 7:14zero of f(x), root of f(x) = 0, 0:7Randles-Sevcik function of argument x. 30:10j1° zero of a polynomial (or other) function. 17:13roots of the equation tan(x) = bx, 34:7dummy variable of Laplace (or other) transformation, 26:14semiperimeter of triangle, 34:14Jacobian elliptic functions of modulus p and argument x, Chapter 63secant function of argument x, angle 0, Chapter 33see aresec(x), arsech(x), 35:1, 31:1hyperbolic secant function of argument x, Chapter 29see sgn(x), 8:1 signum function of argument x, Chapter 8see sinh(x), 28:1see Si(x), 38:1sine function of argument x, angle 9. Chapter 32see arcsin(x), arsinh(x), 35:1, 31:1sampling function of argument x, 32:13hyperbolic sine function of argument x, Chapter 28 see Xlx-, 12:1Fourier coefficients, 36:6variable of integration, any real argument, 0:10tangent function of argument x. angle 0. Chapter 34see arctan(x). artanh(x), 35:1. 31:1 hyperbolic tangent function of argument x, Chapter 30 see tanh(x), 30:1see sc(p;x), 63:1trilogarithm of argument x, 25:13terms in a power (or other) series expansion, 17:13 k' discrete Chebyshev polynomial of argument y, 22:13coefficient of x' in the expansion of T (x), 22:6unit-step functions at x = 0, x = a, Chapter 8x/K(p), auxiliary variable for Jacobian elliptic functions, 63:2versine function of argument x, 32:13frac{x/4K(p)}, auxiliary function for Jacobian elliptic functions, 63:0weight function of orthogonal polynomial. 21:14weight attaching to j1° datum, 7:14integration limits or limits of a distribution, 0:10specific values (often equally spaced) of the variable x, 17:14median (501° percentile) of the f(x) distribution, 27:14argument value yielding an inflection of a function, 0:7, 42:7argument values yielding a maximum, minimum of a function, 0:7, 42:7p" percentile of f(x) distribution, 27:14real variables. 0:1spherical Bessel function (of the second kind) of order n and argument x,32:13k" zero, k" extremum of the Neumann function Y,(x). 54:7x + iy, complex variable, 0:1GREEK LETTERS: CAPITAL AND LOWERCASE (see Section B:8 for names)a fine-structure constant. B:5a modular angle, 61:1, 62:1 SYMBOL INDEX 698 y"Ah (p;(b)8(x), &(x - a)A(n.m). 8-6U), S'(x - a) EoZ(x){(v:u)H(par), H,(" x)T((n). iq(x) TI(v;u)O(p;x). e,(P;x)e1(v;x), 6,(v;x), ...6.(p;x)e = s, c, d, nxMn). Mx)k, sP(m,). µ(m), ... Is. Vnn(n)n(x)n(v;p)n(v;P:4')alpha exponential function of order it and argument x, 37:13terms in continued fractions, 0:6complete beta function of arguments x and y. 43:13 incomplete beta function of parameters µ and v and argument x, Chapter 58see B(v;p.;x), 58:1base of a number system. 9:14base of a general logarithm, 25:14base of a general exponential function, 26:12 nih beta number, beta function of argument x, Chapter 3beta exponential function of order it and argument x. 37:13coefficient of x' in the polynomial expansion of T5(x). 17:9gamma function of argument x. Chapter 43complementary incomplete gamma function of parameter v and argument x.Chapter 45see y(v;x). 45:1Euler's constant. 1:7incomplete gamma function of parameter v and argument x, Chapter 45entire incomplete gamma function of parameter v and argument x, Chapter 45coefficient of Ta(x)in the expansionof x" as Chebyshev polynomials, 22:5discriminant of quadratic function, 16:1delta amplitudinis function of modulus p and amplitude 4. 63:1Dirac delta function at argument x = 0. x = a, 63:1Kronecker delta function of integer arguments n and m, 10:13unit-moment function at argument x = 0, x = a, 10:12ni5 derivative of the Dirac delta function at r = 0, 26:14eccentricity of ellipse or hyperbola, 14:3, 15:3error or fractional error, A:6permittivity of free space, B:5 n' zeta number, zeta function of argument x. Chapter 3Hurwitz function of order v and parameter u, Chapter 64Jacobi's eta functions of modulus p and argument x, 63:13n' eta number, eta function of argument x, Chapter 3bivariate eta function of order v and parameter u. 64:13Jacobi's theta functions of modulus p and argument x, 63:13theta functions of parameter v and argument x, 27:13Neville's theta functions of modulus p and argument x, 63:8shortest distance to directrix from focus of ellipse, 14:3n1h lambda number, lambda function of argument x. Chapter 3real and imaginary parts of a complex zero, 17:6, 17:11mean of a distribution, 27:14number of repetitions of m m2, ..., M. in 6:12permeability of free space, B:5numbers that are not (or not necessarily) integers, Chapter 13 product operator, 0:6see n!, 2:1see r(x), 43:1 complete elliptic integral (of the third kind) of characteristic v and modulusp, 61:12incomplete elliptic integral (of the third kind) of characteristic v, modulusp and amplitude d', 62:12Archimedes number, 1:7j°i prime number, 3:6radius of a circle, 14:4 699 P1Pi(b)a NONALPHABETIC SYMBOLSz, f(x) n!X!n!!, n!!!, n!!!!f(x)*g(x)x.NrxV2(r)(x) (:)(q,p)(x,n)(r,q,p)n(m)n(mod m)[x]xt"'W.(x) CI..L(a HK)/SYMBOL INDEXjih complex zero of a polynomial function, 17:6j'h root of the equation cot(x) = bx, 34:7summation operator, 0:6Stefan-Boltzmann constant, B:5standard error (0' = variance), 27:14see 11(x), 3:1equal to ±I depending on quadrant or magnitude of w, 33:13, 63:0Stirling numbers (of the second kind), 2:14tangent function of half argument, 34:5alternative unit-step function at argument x = a, 8:13see y(n), 44:4see erf(x), 40:1see M(a;c;x), 47:1Lerch's function of argument x, order v and parameter u, 64:12derivatives of the error function of argument x, 40:1see19:1elliptic amplitude, 62:1, 63:1indefinite integral of f(x), 0:10see U(a;c;x), 48:1orthogonal function family of argument x, 21:14 discrete orthogonal function family of 1 + I members, 22:13 digamma function of argument x, Chapter 44 trigamma, tetragamma functions of argument x, 44:12pentagamma, hexagamma functions of argument x, 44:12polygamma function of order n and argument x, 44:12normalizing factor of orthogonal polynomials, 21:14frequency of a periodic function, 36:1 approximate value of x, f(x), 0:9see n! 2:1factorial function of the number n, Chapter 2 see I'(x), 43:1double, triple, quadruple factorial function, 2:13the convolution of two functions, 10:10, 26:14square root of x, Chapter 12see x'i", 13:1Laplacian operator, 46:14a surface specified by the value of the r coordinate, 46:14 see frac(x), 9:1binomial coefficient of upper index v and lower index in, Chapter 6a line (space curve) specified by the values of the q and p coordinate, 46:14see (x),,, 18:1a point specified by the values of the three coordinates r, q and p, 46:14see n(mod rn), 9:13 number n modulo number m, 9:13see Int(x), 9:1factorial polynomial, 18:13Pochhammer polynomial of argument x and degree n, Chapter 18 rounding function, 9:13absolute-value function of argument x, Chapter 8an abbreviation for the product c,c2c3 cL_,cL, 26:14an abbreviation for the product (a,)J(a2)i(a3)/ . . 26:14 SYMBOL INDEX 700ALGEBRAICALLY DEFINED FUNCTIONS Some common functions are represented by straightforward algebraic notation, not requiring special symbols, thoughoften having special names. So that this index may serve as a comprehensive function index, we list such notations here.X2, x' 10,e'X'bx+c.1/(bx+C)axe + bx + c, 1/(ax' + bx + c)x'+ ax- +bx+c1 bx+c, 1/ bx+c1a xbVx +ab'va xxii+ bx + c.`a -I/ax' + bx + cVx-acax2+bx+c(1 + x)" (x + Y)"square, cube function of x, 1 1:1power function of argument x and power (exponent) n, v, Chapters 11. 13general exponential function of base P and argument (exponent) x, 26:12common antilogarithm of x, 26:12see exp(x), 26:1self-exponential function, 26:13linear function, reciprocal linear function. Chapter 7 quadratic function, reciprocal quadratic function, Chapter 16p,(x). cubic function of argument x, Chapter 17 square-root function, reciprocal square-root function of argument x, Chapter12semicircular function of argument x, 14:4sentihyperbolic function of argument x. Chapter 15semielliptic function of argument x, Chapter 14root-quadratic function, 15:12reciprocal root-quadratic function. 15:12root-cubic function of argument x, 17:10binomial functions of argument x, or x and v, 6:14Finally, and again with the motive of having this symbol index serve also as a comprehensive function index,we list functions that are described in the Atlas but for which it was not necessary to adopt any special notation.These are listed in alphabetical order, followed by a section reference: Clausen function (18:14), cumulative stan-dard normal probability function (40:14), elementary symmetric functions (17:6), Hankel functions (54:13). Hermitefunction (24:14), Heuman's lambda function (62:13), higher-order Bernoulli polynomials (19:12), hypergeometricfunction (18:14), Jacobi's zeta function (62:13), Langevin function (30:13), piecewise-constant function (1:13),piecewise-defined function (8:12), piecewise-linear function (7:13), standard normal density function (40:14), win-dow function (8:12).