Bateman ET 2

Tables of Integral Transforms CALIFORNIA INSTITUTE OF TECHNOLOGY BATEMAN MANUSCRIPT PROJECT A. ERDEL YI, Editor W. MAGNUS, F. 0BERHETTINGER, F. G. TRICOMI, Research Associates Higher Transcendental Functions, 3 volumes. Tables of Integral Transforms, 2 volumes. TABLES OF INTEGRAL TRANSFORMS Volume II Based, in part, on notes left by Harry Bateman St7.7S"" BA V.2 Ceto-=L A.rt. Late Professor of Mathematics, Theoretical Physics, and Aeronautics at the California Institute of Technology and compiled by the Staff of the Bateman Manuscript Project Prepared at the California Institute of Technology under Contract No. N 6onr-244 Task Order XIV with the Office of Naval Research Project Designation Number: NR 043-045 NEW YORK TORONTO LONDON McGRAW-HILL BOOK COMPANY, INC. 1954 TABLES OF lNTEGRAL TRANSFORMS, vol.IJ COPYRIGHT, 195,1, BY THE McGRAW-HILL ROOK COMPANY, INC. PRINTED IN THE UNITED STATES OF AMERICA All rights reserved except those granted to the United States GovPrnment. Otherwise, this book, or parts thereof, may not be rcprodueed in any form without pennission of the publishers . Library of Congress Catalog Card Number: 54-6214- 19550 9 10111213 141516-MAMM-7 6 54 This wNk is dedicated to the memory of HARRY B-ATEMAN as a tribute to the imagination which led him to undertake a project of this magnitude, and the scholarly dedication which inspired him to carry it so far toward completion. STAFF OF THE BATEMAN MANUSCRIPT PROJECT Director Arthur Erdelyi Research Associates Wilhelm Magnus (1948-50) Fritz Oberhettinger (1948-51) Francesco G. Tricomi (1948-51) Research Assistants David Bertin ( 1951-52) W.B. Fulks (1949-50) A.R. Harvey (1948-49) D.L. Thomsen, Jr. (1950-51) Maria A. Weber (1949-51) E.L. Whitney (1948-49) V ari-typis t Rosemarie Stampfel PREFACE The aims, the history, and the organization of these Tables of Integral Transforms were described in the Introduction to vol. I. A little more than one half of the present second, and last, volume consists of tables of further integral transforms, the remaining_ part of this volume contains integriJ}S of higher transcendental functions. Under tl1e generic name Bessel trans forms we list not only the famil­ iar Hankel transforms but also other transformations whose kernels are Bessel functions in the widest sense of the word. In addition to these we list fractional integrals, and also Stieltjes and Hilbert transforms. As far as we know, no extensive tables exist for any of the transformations included in this volume, in fact, for some of them there are comparatively few known transforrr . pairs. A list of all transforms included in this work is given on p. xi If. The second part of the volume contains miscellaneous integrals involving higher transcendental functions. Some of these integrals cannot be written as transforms, others Y\ere not included in the transform tables and are given here. Generally speaking, an integral which can be written as a transform is more likely to be found in the transform tables than among integrals of higher transcendental functions. The latter are ar­ ranged according to their integrands. The "hierarchy" of functions given on p.xii of vol. I has been followed and, as in vol. I, composite functions are classified according to the "highest" function occurring in them. A list of definitions of higher transcendental functions is given in the Appendix. Acknowledgments and thanks are due to the same persons and organ­ izations as in connection with vol. I. Acknowledgments are also due to Mr. John Q• Johnston who read the proofs and rendered other valuable technical assistance. Corrections of errors, additions, and suggestions for improvement will be received gratefully by the Editor. A, ERDELYI STANDARD FORMS Fourier cosine transform (£5 , Chapter I) c Joo f(x) cos (xy) dx 0 Fourier sine transform (£5 s, Chapter II) Joo f(x) sin (xy) dx 0 Exponential Fourier transform (u •, Chapter III) Joo f(x) e -U:y dx -oo Laplace transform (~, Chapter IV) Inverse Laplace transform (Chapter V) l fc+ioo --. g(p) ePtdp 2 7T L c-ioo Mellin transform em, Chapter VI) Joo f(x) X s-1 dx 0 Inverse Mellin transform (Chapter VII) l 1 c+ioo --. g(s)x-5ds 2 7T L c-ioo Hankel transform ($2 v' Chapter VIII) Joo f(x) J (xy) (xy)y, dx 0 v Y-transform (~ v, Chapter IX) Joo f (x) Y (xy) (xy) y, dx 0 v xi xii INTEGRAL THANSFOHI\15 K-transform (~ v' Chapter X) Joo f (x) K (xy) (xy) ){ dx 0 v H-transform (Chapter XI) f 0 00 f(x) H )xy) (xy )){ dx Kontoro.vich- Lebedev transform (Chapter XII) Joo f(x) K. (y) dx 0 1X Riemann-Liouville fractional integral (lH J.L' Chapter XIII) -- f(x) (y-x)f.L-1 dx l ly r(f.L) o Wey I fractional integral (~ , Chapter XIII) J.L l /00 -- f(x) (x-y)J.L-1 dx r(f.L) y Stieltjes transform (6, Chapter XIV) 100 f(x) --dx 0 X+ y Generalized Stieltjes transform (6P, Chapter XIV) f(x) (x + y )P dx Hilbert transform (Chapter XV) f(x) --dx x-y CONTENTS PREFACE STANDARD FORMS 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.11. 8.12. 8.13. 8.14. 8.15. 8.16. 8.17. 8.18. 8.19. BESSEL TRANSFORMS CHAPTER VIII HANKEL TRANSFORMS General fonnulas . . • . • • . . • • • Hankel transforms of order zero; Elementary functions • Hankel transfonns of order zero; Higher transcendental functions . • . . . . • . . . Hankel transfonns of order unity • • . • . . • . • • HANKEL TRANSFORMS OF ORDER 11 Algebraic functions and powers with arbitrary index Exponential and logarithmic functions . . . . • • Trigonometric and inverse trigonometric functions Hyperbolic and inverse hyperbolic functions Orthogonal polynomials .•.. Legendre functions • . • . • . . • Dessel functions of argument kx • • Bessel functions of other arguments ~1odified Bessel functions of argument kx ~lodified I3essel functions of other arguments Functions related to Bessel functions . Parabolic cylinder functions . . . Gauss' hypergeometric function Confluent hypergeometric functions Generalized hypergeometric series and miscellaneous functions • . . . . • • . . . • • . . • . . xiii ix xi 5 7 13 18 21 28 32 41 42 44 47 56 63 67 72 76 80 82 87 xiv 9.1. 9.2. 9.3. 9.4. 10.1. 10.2. 10.3. 11.1. 11.2. 11.3. INTEGRAL TRANSFORMS CHAPTER IX Y-TRANSFORMS General formulas. Algebraic functions and powers with an arbitrary index . Other elementary functions · . . Higher transcendental functions CHAPTER X K-TRANSFORMS General formulas. . Elementary functions . . . . •. Higher transcendental functions . CHAPTER XI H-TRANSFORMS General formulas. Elementary functions . . . . Higher transcendental functions CHAPTER XII KONTOROVICH- LEBEDEV TRANSFORMS 12.1. Formulas . . . • . . . . . . . . . . . . . . 13.1. 13.2. MISCELLANEOUS TRANSFORMS CHAPTER XIII FRACTIONAL INTEGRALS Riemann-Liouville fractional integrals Weyl fractional integrals . . . . . . 95 96 105 108 121 127 134 157 158 162 175 185 201 14.1. 14.2. 14.3. 14.4. 15.1. 15.2. 15.3. CONTENTS CHAPTER XIV STIEL T JES TRANSFORMS General formulas . . . . . . . Elementary functions . . . . . Higher transcendental functions Generalized Stieltjes transforms CHAPTER XV HILBERT TRANSFORMS General formulas. . . . . . . Elementary functions . . . . . Higher transcendental functions XV 215 216 224 233 243 243 253 INTEGRALS OF HIGHER TRANSCENDENTAL FUNCTIONS 16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 17.1. 17.2. CHAPTER XVI ORTHOGONAL POLYNOMIALS Tchebichef polynomials Legendre polynomials. . Gegenbauer polynomials Jacobi polynon1ials. . Hermite polynomials . Laguerre polynomials. CHAPTER XVII GAMMA FUNCTION, INCOMPLETE GAMMA FUNCTIONS AND RELATED FUNCTIONS The gamma function . . . . . . . . . The 'I'-function . . . . . . . . . . . 17.3. Incomplete gamma functions and related functions 271 276 280 284 288 292 297 305 306 xvi 18.1. 18.2. 18.3. 19.1. 19.2. 19.3. 19.4. 19.5. 19.6. 19.7. 19.8. 20.1. 20.2. 20.3. 20.4. 20.5. INTEGRAL TRANSFORMS CHAPTER XVIII LEGENDRE FUNCTIONS Legendre functions of variable ax + f3: finite intervals. Legendre functions of variable ax+ f3: infinite intervals Legendre functions of other variables • . • . . • . • . CHAPTER XIX BESSEL FUNCTIONS Bessel functions of argument x. Finite intervals. Bessel functions of argument x. Infinite intervals Bessel functions of arguments ax+ {3, x2, x-1• Bessel functions of other arguments . . . • l\1odified Bessel functions of argument x . . . Modified Bessel functions of other arguments • Bessel functions and modified Bessel functions of variable order Functions related to Bessel functions CHAPTER XX HYPERGEOMETRIC FUNCTIONS Parabolic cylinder functions · · • Gauss' hypergeometric series · Confluent hypergeometric functions MacRobert's E -function Meijer's G-function APPENDIX Notations and definitions of higher transcendental functions . • . INDEX OF NOTATIONS 313 320 326 333 339 349 358 364 372 379 383 395 398 401 414 417 423 449 BESSEL TRANSFORMS or integral transforms whose kernels are Bessel functions or functions related to Bessel functions. CHAPTER Vill HANKEL TRANSFORMS We call g(y; v) = ~v!f(x); y} = J~ f(x) Jv(xy) (xy)~ dx the Hankel transform of order v of f(x) and take y to he a positive real variable. For the sake of brevity we often write g (y) instead of g (y; v). This form of the Hankel transform has the advantage of reducing to the Fourier sine o~ cosine transform when v = ± ~. Many authors regard J"" f(x) J (xy) x dx 0 v or J"" [(x) Jv [2(xy)~] dx 0 as the Hankel transform of order v of f (x ). The Hankel transform is self­ reciprocal [se'1 8.1 (1)] and n·o table of inverse transforms is required. Hankel's inversion theorem is proved in detail, and many Hankel transforms are evaluated in Watson's (1922) hook on Bessel functions. The theory and application of Hankel transforms is described in several hooks on Fourier integrals, among which we mention Sneddon (1951) and Titchmarsh ( 1937). From the transform pairs given in this chapter further transform pairs may he derived by means of the methods indicated in ·the introduction to volume I of this work, and also by means of the general formulas given in sec. 8.1. Tricorni (1935) discovered the relation .13!t~v-!O g[(2t)~; v]; s} = s-v-1 .l3!t~v-!O f[(2t)~]; s-1} between Hankel transforms and Laplace transforms, and this relation may he used to evaluated Hankel transforms by means of the tables of Laplace transforms and inverse Laplace transforms given in chapters IV and V of volume I. 3 4 INTEGRAL TRANSFORMS REFERENCES Sneddon, I.N., 1951: Fourier transforms, McGraw-Hill, New York. Titchmarsh, E.C., 1937: Introduction to the theory of Fourier integrals, Oxford. Tricomi, Francesco, 1935: Rend. dei Lincei (6) 22, 564-571. Watson, G. N ., 1922: A treatise on the theory of Bessel functions, Cambridge. HANKEL TRANSFORMS 8.1. General formulas f(x) Joo f(x) J (xy) (xy)~ dx 0 v = g(y; v) y>O (l) Joo g(y) J (xy)(xy)~ dy 0 v g (y) Re v >-~~ (2) ((ax) a>O a-1 g(a-1 y; v) (3) x • f(x) m = 0, l, 2, ••• y~-v( ~) • [yv-~+a g(y; v+m)] ydy (4) x • f(x) m = 0, l, 2, ,,, (-1)• y~+v( ~ )• ydy x [y-~+a-v g(y; v-m)] (5) 2vx-1 f(x) y g (y; v-1) + y g (y; v+ 1) (6) x-1 f(x) y X -v J Y 77 v-X g (77; v-1) d71 0 (7) x -1 f(x) yX+v Joo77-v-~ g('r/, v+l)d77 y (8) x -J.L f(x) 21-J.L[f'(IL)J-1 yX-v Re v + 1 > Re 1.1. > 0 X f y l1v-J.L+X (y2-772)J.J.-1g(77;v-IL) d77 0 5 6 INTEGRAL TRANSFORMS 8.1 General fonnulas (cont'd) f(x) Joo f(x) J (xy) (xy)~ dx 0 11 = g (y; v) y>O (9) x -J.L f(x) 21 -J.L [f' (tL)] -1 Y ~ +v Re v-3/2 > Re tL > 0 X Joo f)l{-J.L-v(,l-yl)J.L-1 g (f); v+J.L)dYJ y .. (10) 2vf'(x) (v-Yz) y g (y; v+ 1) -(v+Yz) yg(y; v-1) (ll) x ~ -v ( ~ )" [x v+a -X f(x)] xdx y• g(y; v+m) m = 0, 1, 2, ... (12) x y, +v( ~)• [x • -v-Y, f(x)] xdx (-y)"' g (y; v-m) m = 0, 1, 2, ••• (13) x'A-v J 0% ~v-J.L+'A(x2-S:2)J.L-1 21-L-1 r' (tL) y-J.L g (y; v-tL) x r<e> de Re v + Yz > Re tL > 0 (14) xX+v Joo ~Y,-J.L-ll(~2-x2),u-1 21-L-1 r' (tL) y-J.L g (y; v+ tL) % X{(~) d~ Re v + 1 > Re tL > 0 (15) 2"-[' (,\) x y, -v 2J.4 [' (tL) Y y, -v X J% eY,-A.-J.L+ll(r- e)J.L-1 0 Jy Y,-A.-J.L+ll( 2 2)A.-1 Xof) y-YJ X f(t) dt x g(YJ; v->..-tL)dYJ ReA> 0, Re tL > 0 Re v > Re (>.. + tL) -Yz 8.2 HANKEL TRANSFORMS 7 General fonnulas (cont'd) f(x) Joo [(x) J (xy)(xy)~ dx 0 ll = g(y; v) r > o (16) 2/c r (.\) X~ +v 21-l r (ll) r ~ +v X Joo g~ -/c-J.l-v(g 2_x 2)J.l-l % X Joo r(t-tc-J.l-ll (TJ 2 _ y 2)tc-I y X {(g) dg Re .\ > 0 x g(TJ; v+A+Il) dTJ Re ll>O, Rev> JRe (.\-ll)J-1 8.2. Hankel transforms of order zero; Elementary functions f(x) f 000 f(x) J0 (xy) (xy) ~ dx y>O (l) X-~ y-~ (2) x-~ 0 <x < 1 Y~ Jo(y) + ~71Y~ [J,(y) Ho(y) 0 1<x<oo -Jo(y) H, (y)] (3) 0 0<x<1 y-~-y~ Jo(y)+~7Ty~[Jo(y)H,(r) X-~ 1<x<oo -J 1 (y) H o (y)] (4) x X (a2+x2)-X Rea> 0 y-~ e-ay (5) x~(a2-x2)-~ O<x<a r-~ sin (ay) 0 a<x<oo (6) 0 0 <x <a y-X cos (ay) x~(x2-a2)-~ a<x<oo (7) x~(x2+ a2)-3/2 Rea> 0 a-' yX e-ay (8) x ~ (x 4 + a 4) -' JargaJ < ~" -a-2 yX kei 0(ay) - 8 INTEGRAL TRANSFORMS 8.2 Elementary functions; Order zero (cont'd) f(x) J''" f(x) J (xy)(xy)~ dx 0 0 y>O (9) x 5/2 (x 4 + a 4) -1 iargal<~rr yV. ker 0(ay) (10) x 5/2 (x 4-a 4) -1 a>O ~y V. [K 0 (ay)-~ rr Y 0 (ay)] The integral is a Cauchy Principal Value (x 2+a2) ~ -x y-1/2+2a-2 y-5/2(aye-aY+e-ay_l) (ll) X-V, Rea> 0 (x 2+a2) ~ +x (12) x~ (x4+2a2x2+b4)-~ y~ l 0[2-V.(a2-b2)V.y] a>b>O xK [2-V.(a2+b2)V. y] 0 (13) x~ (x4+2a2x2+b4)-~ yV. Jo[2-~(b2-a2)~ y] b>a>O x K 0[2-V.(a2+b2)V.y] (14) x 1/2 (b 2-x 2)(x"' ±2a2x2+ b"')-312 2-112 (b2+ a2)-1/2 y312 0 <a< b x J1 [(~b2 + ~a2)1/2 y] xKo[(~b2 ±~a2)112y] (15) x -~(a 2-x 2) -~l[x + (x2-a2)V.]2" (-1)" rra2n+V.yV. [Jn(~ay)]2 + [x-(x2-a2)~]2nj 0 <x <a 0 a<x<oo (16) x -V. (a2 +x 2) -v, [(a 2+x 2) ~ +x f/.L y~ a21-LK/.L(~ay) 1_1-L(~ay) Rea> 0, Re 11. < ~ (17) xV. (x2+a2)-~ (x2+{32)-V. yV.(a2-{32)1-LK [~y(a+/3)] /.L x [(x 2 +a 2) ~ + (x 2 + {3 2) ~] x I_/.L[~y(a-{3)] Rea> 0, Re {3 > 0 Rep.<~ 8.2 HANKEL TRANSFORMS 9 Elementary functions; Order zero (cont'd) f (x) Joo f(x) J (xy)(xy)~dx 0 0 y>O (18) X-~ e -ax Rea> 0 r l{ (y 2+a2)-~ ( 19) X -312 (1-e -ax) Rea> 0 y X sinh -1 (a/y) (20) xn-X e-ax Rea> 0 n!y~ p n [(a2:y2)~J (a2+y2)~n+~ (21) 2 x2J1.-3/2 e-x /2 Re IL > 0 211--1 r (IL) r ~ 1 F; (IL; 1; -)·h 2) (22) x-1 exp(-2,Bx~) Re ,B > 0 "-1 .Br-~ K'4 (~e ~·7T,B2y-1) x K'4(~e-~i1T ,B2y-1) (23) x112 exp[-a(x2+,B2)112] ay 112 (y 2+ a2)-3/2 exp [-.B (y 2 +a 2)112] Rea> 0, Re ,B > 0 x [l+,B(y2+a2)112] (24) x~ (x2+,B2)-~ Y ~ (y 2 +a 2) -~ exp [-.8 (y 2 +a 2) ~] x exp [-a (x 2 + ,B 2) ~] Rea> 0, Re ,B > 0 (25) x~(x2+,B2)-~ ± iy~ (a2-y2)-~ x exp [ ± ia (x 2 + ,B 2) X] x exp [ ± i ,B (a 2-y 2) ~] 0 <y <a a> 0, Re {3 > 0 yX(y2-a2)- l{ x exp [-,B(y2-a2)l{] a<y<oo (26) ±ixl{(b 2-x2)-l{ yl{(y2+a2)-l{ exp[±ib(y 2+a2)l{] xexp[±ia(b2-x2)l{] 0 <y < b x ~ (x2-b 2)-l{ x exp [-a (x 2-b 2) l{] b<y<oo Rea> 0 10 INTEGRAL TRANSFORMS 8.2 Elementary functions; Order zero (cont•d) f(x) f 000 [(x) J 0 (xy)(xy)~ dx y>O (27) X-~ log X -y-~ log(2yy) (28) x-~(x2+a2)""'~ y~ [Y2K~(Y2ay) x log [x + (x 2 +a 2) ~] a>O + log a I 0 (Y2 ay) K 0 (Y2 ay)] (29) x-~(x2+a2)-~ y~ K~ (Y2ay) (x 2+a2)~ +x x log (x 2+a 2)~ -x Rea> 0 (30) x~ log(l+a 2x-2) Rea> 0 2y-~ [y-1-a K 1 (ay)] (31) x~ log[ax-1+(l+a2x-2)~] y-3/2 (1-e -ay) Rea·> 0 (32) x-~ sin (ax) a>O y~ (a2-y2)-~ 0 < y <a 0 a<y<oo (33) x-312 sin (ax) a>O Y2rry ~ 0 <y <a y~ sin -1 (a/y) a<y<oo (34) x-~ (l+x)-1 sin(l+x) Y2rr.Y ~ J0 (y) l.:::;y <oo (35) x-~(f32+x2)-1 sin(ax) y~ {r1 sinh (t3a) K0 (f3y) a<y<oo a> 0, Re f3 > 0 (36) x~ (/32+x2)-1 sin(ax) Y2rrr ~ e -a/3 I 0 (y {3) 0 <y <a a> 0, Re t3 > 0 (37) x-3/2e-bx sin (ax) y~sin-1(2 a ~ r 1 + r 2 r~ = b 2 + (a+y)2, r 1 > 0 r; = b 2 + (a -y) 2, r 2 > 0 8.2 HANKEL TRANSFORMS ll Elementary functions; order zero (cont•d) f(x) J oo ( (x) J (xy) (xy) ~ dx 0 0 y>O (38) x ~ sin (~ a 2 x 2) a>O a-2 r~ cos (~a-2 y2) (39) x-312sin(~a2 x2) ~)~ si (%a-2 y2) (40) --ax){ · 1 %y-~ al~ (~ a2/y) K !4 (~ a2/r) x 1 e sm(ax~) jargaj < ~rr (41) xl{(.82+x2)-~ yl{ (a2-y2)-X cos [f3(a2-y2)~] x sin [a (/3 2+x 2) ~] 0 <r <a a> 0, Re /3 > 0 0 a<y<oo (42) x-X cos (ax) a>O 0 O<y<a y ){ (y 2 _a 2) -X a<y<oo (43) x-312 [1-cos (ax)] a>O y X cosh -1 (a/y) O<y<a 0 a<y<oo (44) x-~(x2+/32)-1 cos(ax) ~/3-1rry~ e-a/3 Io(f3y) a> 0, Re /3 > 0 0 <y <a (45) x~ (f32+x2)-1 cos(ax) y~ cosh(f3a) K0 ({3y) a> 0, Re {3 > 0 a<y<oo (46) x-J{ e-bx cos (ax) y~ 2-~[(b2+y2-a2)2+4a2b2]-'h xI[ (b 2+ y 2_ a 2) 2 + 4 a 2b 2] X +b2+y2-a2!~ (47) xX cos(%a2x2) a>O a-2yX sin(%a- 2y2) (48) X -3/2 (1-COS (%a 2 X 2)] -~yX Ci (%a-2 y2) 12 INTEGRAL TRANSFORMS 8.2 Elementary fWtctions; Order zero (cont'd) f(x) Joo r (x) J (xy) (xy) ~ dx 0 0 y > 0 ~ X ay-~ I (~ a2y-1) (49) x-1 e-ax cos(ax ~) 2 -~ " Jarga\ <~77 x K ~ (~ a 2 y _, ) (50) x~ (,82+x2)- ~ -y ~(a 2-y 2) -~sin [,8 (a 2-y 2) ~] x cos[a(,82+x2)~] 0 <y <a a> 0, Re ,B > 0 ~ ( 2 2)-~ -,B(y2-a2 )~ y y -a e a<y<oo (51) x -312 e -<u sinh (ax) Rea> 0 XyX log[2ay-1+(1+4a2y-2)X] (52) x -X e -a..r sinh (,Bx) (a,By)Xr-1 r-1(r -r )X(r +r )-~ Re a> \Re 13\ 1 2 2 1 2 1 r , = [y2 + (,8 -a) 2] X r 2 = [y 2 + (,8 + a) 2] X (53) xX e-a.;.r sinh(ax-1) Rea> 0 2ay-~ J (2a~yX)K (2aXyX) 1 1 (54) x-X e -a..r cosh (,Bx) (a/3y) ~ r-1 r-1(r+r )X(r-r)- X Rea> \Re 13\ 1 2 2 1 2 1 r 1 = [y 2 + (,8 -a) 2] ~ r 2 = [y 2 + (/3 + a) 2] X (55) x~ sinh-1 (ax-1) Rea> 0 y -3/2 ( 1-e -a. Y) (56) x-X (l+x2)-X 11 _, sin (771-L) y ~ [K (X y)J2 x sinh(21L sinh-1 x) /L \Re ILl< X (57) x-~(l+x2)-y, Xry,KIL(Xy) x cosh(21L sinh-1 x) x[IIL(Xy) + 1_/L(Xy)] \ReJ.LJ <X 8.3 HANKF:L TRANSF'OHMS 13 8.3. Hankel tcansfonns of order zero; Higher transcendental functions f(x) Jo"" f(x) J0 (xy)(xy) ~dx y>O (l) x~ P (1-2x2) 0 <x < 1 Y-~ J2n+1 (y) n 0 1<x<oo (2) X S/2 p ( 1-2 X 2) 0 <x < 1 y-~(2n+1)-1 n 0 1<x<oo x [(n+ 1) J;n+2 (y)-n J;n (y)] x~ e-Y,x2 L 2 (3) n (x2) (-1)n e -Y,y y~ L (y 2) n x~ exp(- ~ax2) Ln '(~f3x2) (a-f3) n (4) y~ exp(- ~a-1x2) Rea> 0 an+1 x L [ ~y2 j n 2 a(~-a) (5) x~ e-x2 L n (x 2) [n 1]-1 2-2n-1 2n +~ -!4y 2 y e (6) x-~ si(ax) a>O -y-~ sin-1 (y/a) 0 <r <a 0 a<y<oo (7) x ~ si (a 2 x 2) a>O -2y-312 sin (~x2ja2) (8) xX Ci (a2x2) a>O 2y -312 [1-cos(~ x 2 /a 2)] (9) x-~ Ci(a2x2} a>O y-~ [Ci (~x2/a2)+log(~ y2x2ja2)] (10) x ~ (l+x 2)-11-1 (211 r<v+ l)] -2y 211 +~ K (y) xP [(1-x2)(1+x2)-1] 0 11 Re v > 0 (ll) xX 1Pt-_-~[(1+a2x2)~]J2 277-:zCOs (.).77) a-1 y-~ [Kt-_(~y/a)]2 Rea> 0, IRe>.l <~ 14 INTEGRAL TRANSFOHMS 8.3 Higher functions; Order zero (cont'd) f(x) J"" f(x) J (xy)(xy)Xdx 0 0 y>O (12) x X P [( 1 +a 2 x 2) X] o--X a-1y-X Io-(~2yla) Ko-(~~yla) X Q o--Y, ((1+a2X 2)Y,] He a> 0, Rea>-!4 (13) xy,lPJL [(l+a2x2)Y.Jl2 -i 17-1 y -312 W (y I a) o--y, Jl•O" He a> 0 x [W (e77iyla)-W (e-17iyla)] JReaJ < ~. Rep. <1 ~.u ~.a (14) x X p Jl [ ( 1 + a 2 x 2) y,] o--Y, 217-1 Y-312 cos (a 17) X. po--~Y,[(1+a2x2)Y,] x W (y I a) W (y I a) f.1.,CT -J.L,O" Rea> 0, JReaJ < !4 (15) xXPJL [(1+a2x2)y,] JL1Ti f'(~+a-p.) -3/2 e y a--x f'(1+2a) x QJL [(1+a2x2) X] o--X X wJl,o-(yl a) M -J,L,o-(yl a) Rea> 0 Rea>- ~4, Rep.< 1 (16) x -312 [1-J (ax)] 0 a>O 0 y>a y y, log (a/y) y<a (17) -Y, J ( ) -/3z x 0 ax e 2"-1yy,K(2aXyY, Ik) le-y, Re (3 > JlmaJ k == [(a + y) 2 + (3 2] y, (18) x-X J (ax) 1 a>O a-1 yX 0 <y <a 0 a<y<oo (19) x -312 J (ax) 1 a>O 277-1yX E(y/a) 0 <y <a 2::12 [K0~-~-;:) EG) J a<y<oo 8.3 HANKEL TRANSFORMS 15 Higher functions; Order zero (cont'd) f(x) Joo f(x) J (xy) (xy) ~ dx 0 0 y>O (20) .x~ J:(~ax) Re 11 > -~ 2rr-'r-~ (a2-y2)-~ x cos [2 11 sin_, (y ja)] O<y<a 0 a<y<oo (21) x~ J 0(ax) Y 0(ax) a>O 0 0 < r < 2a -2rr-1 r-y, (y2-4a2)-~ 2a < y < oo (22) x-~ [H 0(ax)-Y 0(ax)] a> 0 417'_, (a+y)-1 y~ x K [la-rl (a+ y)-'] (23) x-~ cosh (,Bx) K0 (ax) yY. (u +v)-Y, K(k) Rea> IRe /31 u= ~l[(a2+ /3 2+y 2) 2_4 a 213 2] ~ +a2-f32-y2} v = ~l[(a2+/32+y2)2_ 4a2/32]~ -a2 + 132 + y2} k2=v(u+v)-' (24) x-~ sinh($x) K, (ax) a-1 y~ [uE (k)-K(k) E (u) Rea> IRe/31 + K(k) snu dnuj(cnu)] cn2u = 2y2l[(a2 + /32 + y2)2 _ 4 a 2/3 2] ~_a 2 + /3 2 + y2}-, k2=~1l-(a2-$2-y2) X ((a2+ {3 2 +y 2)2-4 a2 f3 2r~} (25) x ~ J0 (ax) K 0 ({3x) r~ (,8<4+a4+r<4-2 a2y2 Ref3>1Imal + 2 a 2 f3 2 + 2/'F r 2) -~ (26) x312 J1 (ax) K0(f3x) 2ay~ (a2+ f32-y2) Re f3 > IImal, Rea> 0 X ((a2+ $ 2+y 2) 2_ 4y 2 a2r3/2 16 INTEGRAL TRANSFORMS 8.3 Higher functions; Order zero ccont•d) f(x) Joo [(x) J (xy)(xy)~dx 0 0 y>O (27) x~ I0(ax) K0({3x) y~ (a4+{3"'+y4-2a2{32 Re {3 >Rea + 2 a 2 y 2 + 2 $ 2 y 2) -~ (28) x312 I0(ax) K1 ({3x) 2y~ {3({32+y2-a2) Re {3 >\Rea\ X ((y2+a2+ $2.)2-4 a2. {32.r3/2. (29) x~ I1 (ax) K1($x) 2-,y~ a-1 13-11(a2+/32.+y2.) Re {3 >Rea> 0 x [(a2 + {32 +y 2) 2-4 a2{3 2] -X-11 (30) xX K (ax) I ({3x) y~r-1 r-1 (r -r )~ (r +r )-~ ~ iJ- 1 2. 2. 1 z 1 Re p. > -1, Rea> \Re/3\ r1 = [y2 + (f3-a)2]~ r 2 = [y 2 + ({3 +a) 2.] X (31) x-X I (~ax) K (~ax) ~ ~ a-1 yX p JJ--~ [(l+y2./a2)~] Rea> 0, Rep.>-~ x Q ~-~ [(l+y2/a2)~] (32) x~ K2(ax) 0 Rea> 0 -~ ( 2+4a2)-X lo (y2.+4a2)~+y y y g (y 2+4 a2~-y (33) x~ K2 (ax) rr2-I-2.JJ- a -2.~ (sin p.rr) _, y -X ~ Rea> 0, \Rep.\ < 1 x(y2+4a2)-~ l[(y2+4a2)X+y]~ -[(y2+4a2)X -y] 21-LI (34) x-X J (a2x-1) J (a2x-1) ~ -~ -i csc(2p.rr) y-~ a> 0, \Re11-\ < ~ x[e27ri~J (2ayXe-~77i) ~ x J (2ay~ e~77i) -2.J.L -e-2.1Tii.LJ (2ayX el(77i) 2.j.l. x J_ 2~(2ayX e-~77i)] 8.3 HANKEL TRANSFORMS 17 Higher functions; Onler zero (cont•d) f(x) Joo f(x) J (xy)(xy) ~ dx 0 0 y>O (35) x-~ [J2(a2x-1)-J2 (a2x-1)] J.l -J.l sec (~.LIT)y -~ a> 0, \ReiL\<~ x[J (2ay~e1171i)J (2ay~e~71i) 2J.l ~ -J (2ay~e~71i) J (2ay~e~71i)] -zj.l -'ljJ, (36) x -~ H 01(a2 x -t) H 121(a 2 x-1) J.l J.l 16 1T -2 COS (ILTT) y -~ \arg a\ < ~ 11, \ReiL\ < ~ xK (2ay~e~71i)K (2ay~e-'A'7Ti) 2J.l 2J.l (37) x-~ I (a2x-1) K (a2x-1) J.l J.l 2y-~ J (2ay~)K (2ay~) 2J.l 2J.l \arga\ < ~"• Re 1L >-~ (38) x -~ J (ax~) K (ax~) J.l J.l ~y-~ I (~a2y-1)K (~a2y-1) 2 ~ J.l ~ ~ J.l 'l \arg a\ < ~ 11, Re IL > -1 (39) x -~ Y (ax~) K (ax~) 0 0 -~TT-t y-~ [Ko(~a2/y)]2 \arg a\ <~IT (40) x-~ Y (ax~) K (ax~) J.l J.l -Yzy-~ sec(YziLTT)K~J.l(~a2y-1) \arg a\ < ~ 11, \Re ILl < 1 x[TT-t K~J.l(~a2y-1 +sin (Yz ILTT) I~)~ a2y-')] (41) x-~K (ae!471ix~) 2-4112 [cos {Yz ILTT)] -t x K J.l(ae-~ 71ixX) xH ~~(~ a2/y) H ~~ (~ a2/y) J.l \arga\ < ~11, \Re ILl < 1 (42) x-~ Dn (ax) Dn+t (ax) (-1)n Y-~ D n (y/ a) Dn+t (y/a) \arga\ < ~ 11 (43) x-1 D (a~x~) D (a~x~) 2-31211a -t/2 y t/2 11 -v-t xP_!',f+t/4 [(1+4y 2; a2) t/2] Rea> 0 x p:_v;z-t/4 [(1 + 4y 2/ a2) t/2] 18 INTEGRAL TRANSFORMS 8.3 Higher functions; Order zero (cont•d) f(x) J: f(x) J 0 (xy)(xy) ~ dx y>O (44) x-s/2 wK.~/ax) M -K ... (ax) e-iKrr rC1+2~) ~ y Rea> 0 r<~+ ~+K) Re~>-~, Re K< ~ xP~-~ [(1+y2/a2)~] x Q~-~ [(1+y2/a2)~] (45) x-312 w. (ax) w_. (ax) #J.L ,JJ. . ~ rr cos (wr)y ~ P !-~ [(l+y 2/ a2) ~] Rea> 0, -~ < Re ~ < ~ x p-• [(1+y2/a2)~] ~-~ (46) X~ F (A: 1· -x2) ' 1 ' ' Re ,\ > 0 [2 z.\-1 r (..\)]-1 y zA-3/2 exp (-~ y z) 8.4. Hankel tzansfonns or order unity f(x) f 0"" f(x) J1 (xy)(xy) ~ dx y>O (l) X-~ 0 < x< a y-~ [1-J 0 (ay)] 0 a<x<oo (2) 0 0 <x <a y-~ J (ay) 0 X-~ a<x<oo (3) x -~ (x 2+ a2)-~ Rea> 0 a -1 r -~ (1-e -aY) (4) x-~ (a2-x2)-~ 0 <x <a a-1 r-~ [1-<:os(ay)] 0 a<x<oo (5) 0 0 <x <a a-1 r-~ sin(ay) x-~ (x 2-a 2)-~ a<x<oo 8.4 HANKEL TRANSFORMS 19 Order unity (cont'd) f(x) J;' f(x) J1 (xy) (xy) l4 dx y>O (6) x-l4 e -az Rea> 0 y-l4 [l-a(a 2+y2)-l4] (7) x-~ e-I.4ax2 Rea> 0 y-~ (l-e"1'2/a) (8) x3/2 e-ax2/4 Rea> 0 4a-2 y3/2 e"1'21a (9) x -l4 exp [-a (x 2+ fF) l4] y-X [e-,8a_ a(a2+yz)-X Rea> 0, Re (3 > 0 x e -JX.a2 +y 2)~] (10) x-X(f32 +x2)-X y-X (3-1 [e -,8a_ e -,B(a2 +y 2)X] x exp [-a ((3 2 + x 2) X] Rea> 0, Re {3 > 0 (ll) x-X log x -y-l4 log(Xyy) (12) x -x log (a 2 + x 2) 2y-~ [K0(ay) +log a] (13) x-~ log(l +x4)X 2y-Xker 0y (14) x-X sin (ax) a>O 0 O<y<a ay-~ (y2- a2)-~ a<y<oo (15) x -312 e -ax sin (bx) by-X (1-r) az y2 b2==----l-r2 r2 (16) X -X sin (~ax 2) a>O y-~ sin{y2/a) (17) x -X sin 2 (~ax 2) a>O Xy~X cos (Xy2ja) 20 INTEGRAL TRANSFORMS 8.4 Order unity (cont'd) f(x) Joo f(x) J1 (xy) (xy) ~ dx 0 y>O (18) x-~(x2+a2)-% a-1 y-% lsin(ab)- sin[a(b2-y2)~]1 x sin[b(x 2 + a2)~] o <r < b Re a> 0, b>O a-1 y-% sin (ab) b<y<oo (19) x -~ cos (ax) a>O y -% [1 -a (a 2 -y 2) -~] 0 <r <a y-~ a<y<oo (20) x -~ cos(~ ax 2) a>O 2y-~ sin 2 (~y 2/a) (21) x-~ (x 2 + a2)-~ a-1 y-%1-cos[a(b2-y2)~] x cos [b (x 2 + a 2) ~] + cos(ab)l O<y<b Rea> 0, b>O a-1 y-~ Ieos (a b) -exp[-a(y2-b2)~]1 b<y<oo (22) x -~ tan_, (x 2) -2y ~ kei0 y (23) x312p (l-2x2) 0 <x < 1 (2n + 1)-1 y-~ [(n+1) J 211+2(y) n 0 1<x<oo -nJ2n(y)] (24) x-~ [D (ax)] 2 n largal <~IT (-1) n-1 Y -% [D n (y/ a)J2 (25) x-% si(a2x2) a>O y-~ [-si (~x2ja2)-~ IT] (26) x-~J0(ax) a>O 0 0 <y <a y-~ a<y<oo (27) x -3/2 J (ax) 0 2a IT_, y-% [E (y /a)-(l-y2;02) K(yfo}] 0 <r <a 21T-ly%E(a/y) a<y<oo 8.5 HANKEL TRANSFORMS 21 Order unity (cont'd) ((x) Joe ((x) J (xy)(xy)~dx 0 1 y>O (28) x-512[J0(ax)-1] a>O -~ y 312 [1 + 2log (a/y)] 0 <y <a -~y 1/2a2 a<y<oe (29) x-~ J 0 (ax) J 0 (bx) a, b > 0 0 O<y<\a-b\ 77-1 y-~ cos-1 [(a2+ b2-y2)/(2 ab)] \a-b\<y<a+b y-~ a+b<y<oe (30) x -512 J (ax) a>O y y, (y + a) [ E ( 2 iy ~ a j 1 rr \r-a\ _ K (2iy~a~ )] \r-a\ . (31) x-V. Y 0 (ax) a>O -rr-1 y-V. log(l- y2ja2) 0 <y <a (32) x y, kei ox -~ y-~ tan_, (y 2) (33) x-~ ker 0 x ~y-~ log(1 + y4)~ HANKEL TRANSFORMS OF ORDER ll 8.5. Algebraic functions and powers with arbitrary index f(x) Joe ((x) J (xy)(xy) ~ dx 0 y y>O (l) l 0 <x < 1 2~ -1 r (~ + ~ v) l J y r(~ + ~ v) +(v-~) )r) 0 1<x<oe Rev> -3/2 xS_~. v-I (y)-Jv_1(y) Sy,, )r) 22 INTEGRAL TRANSFORMS 8,5 Algebraic functions (cont'd) f(x) Joo {(x) J (xy)(xy) X dx 0 Jl y>O (2) 0 O<x<l Jv-1 (y) S X, v (y) 1 1<x<oo + (~-11) Jv (y) S -lL v-1 (y) (3) x-X Re 11 > -1 y-X (4) x X -v 0 <x < 1 21 -vyv-312 -y-1/2 Jv-1 (y) r(11) 0 l<x<oo (5) xv-X 0 <x < 1 2v-l y X -v rr~ r (11+ ~) 0 1<x<oo x[Jv(y) 8v-l (y)-HJy)Jv_l(y)] Re 11 > -~ (6) xv+X 0 <x < 1 Y-x Jv+1 (y) 0 l<x<oo Re 11> -1 (7) xJJ. -Re 11-3/2 < Re ~t<-1/2 2J.£+X y-J.£-1 r(~IL+~11+~) r<~ ~~-~ ~t+~) . y-J.£-1 [ (v+~t-~)y Jv(y) (8) xJ.£ 0 <x < 1 0 1<x<oo xS J.£-X.v-1 (y)-y Jv-!(y)SJJ.+X•a,(y) Re (IL + v) ::> -3/2 + 2J.£+X r(~IL+~v+~) J rnh-~~t+~) (9) x v-1( (x +a) -I largal < rr ~rra11sec(vrr)yX [H_)ay) -1/2 < Re v < 3/2, y f. 1/2 -Y _ 11(ay)] H.S HAN.k:EL TRANSFORMS 23 Algebraic functions (cont'd) f(x) Joo {(x) J (xy)(xy) ~ dx 0 11 y>O (10) x p-.3/2 (x +a) -J.L-' largal<rr y ~ rraP-J.L-1 sin (p+ v-IJ.) TT r (IJ. + 1) Re (p + v) > 0 Re(p- IJ.) < 5/2 {I (-1)· (~., r ... r(p +v+2m) x •= 0 m !r(v+m +1) r(p +v-11 +2m) I (~ay)J.L+I-phr<t.t+m+1) ··== 0 m! n~<t.t+v-p+m+3)] sin l;(p+v-p-m)"} X r [~ <tt-v-p+m+3)] (ll) x-~(x2+a2)-~ y~ I~)~ay) K~)~ay) Rea> 0, Re v > -1 (12) xv+~(x2+a2)-' a11 y~ K)ay) Rea> 0, -1 < Re v < 3/2 (13) xv-~ (x2+a2)-' ~rra11-1 sedvrr) y~ [111(ay)-L_)ay)] Rea> 0, -1/2 < Re v < 5/~ (14) x -v-~ (x 2+ a a) -t ~rra-11-1 y~ [I)ay)-L)ay)] Rea> 0, Re l-'>-~ (15) xv+~ (x2 + a2)-~ 2 ~ -~ v+~ K ( ) Rea> 0, -1 <Rev<~ TT a v+~ ay (16) x ~ -v(x2+ a2) -~ rr~2-~a~-11(1 (ay)-L (ay)] Rea> 0, Rev>-~ v-~ ' v-)!; (17) x-v-~ (x2+a2)-v-~ 211 -zv ~ +v r(v+ 1) a y Rea> 0, Rev>-~ r(2v+ 1) x I)~ ay) K 11 (~ ay) 24 INTEGRAL TRANSFORMS 8.5 Algebraic flBlctions (cont'd) f(x) J'"' f(x) J (xy)(xy)X dx 0 1.1 y>O (18) xv+X(x2+a2)-v-X 77X yv-X Rea> 0, Re v > -~ 21.1 eay f'(v + ~) (19) xv+1/2 (x2+a2)-v-3/2 y v+X 77X Rea> 0, Rev> -1 2v+1 aeay f'(v+3/2) (20) xv+X (x2+a2)-J.L-1 av-1-Lyi-L+Y. K (ay) Rea> 0 v-IJ -1 < Re v < 2 Re 11 + 3/2 21-L [' (11 + 1) (21) xA.-3/2 (x 2 + a2) -1-L-1 Rea> 0 ! v+1/2 ['(~,\ +~v) ['(11-~A -~v+1) -Re v < Re ,\ < 2Re 11 + 7/2 21.1+1 a2,~.L-A.-v+2 ['(11 +1) n v+1) X 1F/~A+~v; ~A+~v-IJ., v+1; ~y2~ y 2~-t-"N-512 ['(~,\ +~v-11-1) + 22J.L-43['(~v-~A+I1+2) x 1F2(11+1; 11+2+~v-~A, 11 +2-~,\ -~v; ~y 2 a 2) (22) x-X(a2-x2)-X 0 < x <a ~rryX [JXv(~ay)]2 0 a<x<oo Rev> -1 (23) 0 O<x<a -~rryX JXv(~ay) Yy,)~ay) x-X (x2-a2)- X a<x<oo (24) x X -v(a 2_x 2) -X 0 <x <a 2-y, "X aX-vu v-X (ay) 0 a<x<oo (25) x v-X (a 2_x 2) v-X 0 <x <a 2v-1 77Xf'(v+~)a2vyX-v 0 a<x<oo x [J (~ay)]2 v Rev >-~ 8.5 HANKEL TRANSFORMS 25 Algebraic functions (cont•d) f(x) F" f(x) J (xy)(xy)~dx 0 II y>O (26) 0 0 <x <a -2-v-t a-2v ['(~- v)yv+~ 77~ x-v-~ (x 2_a 2)-v-~ a<x<oo x Jv(~ay) Y v(~ay) IRe vi < ~ (27) xv+~ (a2-x2)-v-~ 0 < x <a 77 -~ 2 -vr(~- v)y v-~ sin (ay) 0 a<x<oo IRe vi < ~ (28) 0 0 <x <a 77 -~ 2-vr (Yz + v)y -v-~ cos (ay) x -v+~ (x 2 -a 2) v-~ a<x<oo IRe vi<~ (29) xv+ll2(a2-x 2)-v-3/2 0 < x <a 2-1 -v 77-~ r (-~-v)a-1 cos(ay) 0 a<x<oo X Y v+~ -1 <Rev<-~ (30) 0 O<x<a 2-v-t 77-~ a-1 f'(v-~)y~-vsin(ay) X~ -v (x 2_ a 2) v-3/2 a<x<oo 1/2 < Re v < 5/2 x~-v(a2-x2)JL 2' -vaJL-v+l s v+t!:·t!:-v+l (ay) (31) 0 <x <a yJL+\{ f'(v) 0 a<x<oo Re ll > -1 (32) 0 0 <x <a 211-f'(!J.+1)a'+JL-vy-.u.-~J (ay) v-JL-1 x~-v(x2-a2)JL a<x<oo Re ll > -1, Re (v-21J.) > ~ 26 INTEGRAL TRANSFORMS 8.5 Algebraic runctions (cont•d) f(x) J 00 f(x) J (xy)(xy) ~ dx 0 v y>O (33) xv+~ (a2-x2)t.t 0 <x <a 2/.L r (11+ 1) y-J..L-~ a v+J.,L+1 0 a<x<oo X JV+J.,L+1 (ay) Rev> -1, Re 11 > -1 (34) xJ..L-~ (a2-x2)"- 0 <x <a a2"-+J1.+v+1yv+~ B(A.+1, ~11+~v+~) 2v+ 1 r (v+ 1) 0 a<x<oo ReA.> -1, Re <11 + l-!) >-1 ~+JL+II 3+JL+ll a2yj x 1F ---;v+1,--+A.;-2 2 2 4 (35) x -v-~ (a 2 +2x)-~ 2vr (v+ ~) rr -~ D (ae !>( 77 iy ~) x[(a2+2x)~-apv -v-~ x D (ae-1(77iy~) Rev>-~ -v-~ (36) x-~ (x2+a2)-~ 2rr-112av-3/2 sinh (~ay)Kv-~ (~ay) x [(x 2+ a2) ~ +x]v-1 Rea> 0, -1 < Re v < 5/2 (37) x-~ (x2+a2)-~ rr~ a~-v e -~ ay I (~ ay) x [(x 2+a2)~ +x] 1-v v-~ 2 Re a > 0, -Re v > -~ (38) x-~ (x2+a2)-~ r~ aJ..LI (~ay)K (~ay) x[(x2+a2)~ ±x]J..L Rea> 0 ~ (vf:J..L) ~(v ±J..L) Re v>-1, Re 11 < 3/2 (39) x-v+~ (x2+a2)-~ r-~ e-ay • x [(x2+a2)~-a]11 Rea> 0, Re v > -1 x-J..L-~ (x 2+a2)-~ r<~ +~v-~11) (40) ar~ r2(v+z1) W~J..L.~)ar)M~J.,L.~v(ay) x [(x2+ a2)~ +a]J..L Rea> 0, Re(v-11)>-1 8.5 HANKEL TRANSFORI\15 27 Aliebraic functions (cont'd) f(x) Joo f(x) J (xy)(xy)~ dx 0 11 y>O (41) 0 0 <x <a -17 112 a11-312 [sin(~ ay) Jv+!.{ (~ ay) x -It; (x 2-a 2) -~I [x +(x2-a2)~] v-+1 +cos(~ay) Y v+~(~ay)] + [x-(x2-a2)~]v+q a<x<oo Rev<~ (42) 0 0 <x <a 1Tt12av-3/2 [cos (~ay)Jv-~ (~ay) x -!.{(x 2-a 2) -!.{1 [x +(x 2_ a 2)!.{] v-t -sin (~ay) Y 11_!.{ (~ay)] +[x-(x2-a2)l{]v-tl a<x<oo Rev< 5/2 (43) x -~(a 2-x 2) -~I [x + i(a 2-x 2)!-{]P. 17ati-y~ J!{(v+p.)(~ay) J~(v-J.J~ay) +[x-i(a2-x2)~]1·tj 0 <x<a 0 a<x<oo Re(ll+v)>-1 (44) 0 0 <x <a -~ 1TY It; atL[J ~Ct.t.+v )(~ ay) Y l{(v-,uJ~ay) x -~(x 2-a 2) -~1 [x +(x 2-a 2) l{]p. + J!.{(v-p.)( ~ay)Y l{(v+p.f~ay)] + [x-(x 2-a2) l{]tJ.! a<x<oo .Re ll < 3/2 (45) x -2p.-X (a 2_x 2)-l{ a11 B (~ +~ v+ ILt ~+~ v-ll) yv+lt; xl[a+ (a 2-x 2) l{Fp. [' (1 + v) +[a-(a2-x2)!.{]2tLI 0 <x <a x 1~(~+ ~v- ll; v+ 1;-iay) 0 a<x<oo Re(21l) <Rev+ 1 x 1F 1(~+~v-ll;v+1;iay) (46) xtL-lt;(1-2ax+a2)-~ 0 <x <1 see Bose, S. K., 1946: Bull. Calcuta 0 1<x<oo Math. $oc., 38,177-180. Re (v + ll + ~) > 0 28 INTEGHAL THA!'.SFOHMS 8.5 Algebraic functions (cont'd) f(x) J"" f (x) J (xy )(xy) ~ dx 0 II y>O (47) x11+5/2 (x 4+4 a4) -11-112 rr~Y~ +11J 11_ 1 (ay) K 11_ 1 (ay) largal <rr/4, Re 11 > 1/6 2311-1 a2 11-2 [' (11+ ~) (48) x11+~ (x4+4a4)-~~- ~ y11+~ rr~ J (ay) K (ay) II II Jarg al < ~ rr, Re 11 >-~ a 2 112 311 [' ( 11 + ~) (49) x11+~ (x4 ±2a2x 2+b 4)-~ (b2+a2)-11 211 y~ K [(~b2 ±~a2)~y] x [b 2+x 2+ (x4 ±2a2x2 + b4)~]-211 x J [(~b2+~a2)~y] O<a<b II Re 11>-~ (50) 0 0 <x <a 2y -~ cos (ay -X 1111) x ~ -~~ (x 2 -a 2)-~ xt[a+(a2-x2)~]11 +[a-(a2-x2)~]111 a<x<oo Re 11 > -1 8.6. Exponential and logarithmic functions (l) X-~ e -ax y~-~~(a2+y2)-~ [(a2+y2)~-a]ll Rea> 0, Re 11 > -1 (2) x-3/2 e -ax 11-1 y ~-~~[(a 2 + y 2) ~-a] II Rea> 0, Re 11 > 0 da+1 (3) x •+~ e-<1% Re 11 >-m-2 (-1)•+1y~-~~-- l(a2+y2)-~ da•+1 x [(a 2+y 2)~ -a]111 8.6 HANKEL TRANSFORMS 29 Exponential and logarithmic functions (cont'd) f(x) Joo f(x) J (xy)(xy) ~ dx 0 11 y>O (4) x 11+~ e -a>: 1T-1 /2 2 11 + 1 r ( tl + 3/2) a y 11 + 1 /2 Rea> 0, Re v > -1 x (a 2 + y 2) -11-3/2 (5) x 11-~ e -a>: 2111T- ~ r (v+ ~)y11+~ (a2+y2)-11-~ Rea> 0, Re v >-~ (6) xJ.L-3/2 e-ar y~ (a2+y2)- ~J.L['(J.L+ v) a> 0, Re(J.L+ v) > 0 xP-11 [a(a2+y2)- ~] J.L-1 (7) xJ.L-31'2 e -a>: y11+~ ['(11 + v) Rea> 0, Re(J.L + v) > 0 211 a).J.+1Inv+1) ~+v 11+v+1 y2j x F -- ·v+1·--2 1 2 ' 2 ' , a2 y11+~ f'(J.L+v) = 211 (a 2 + y 2) ~ (.u+11) r ( y+ 1) X F---· +1· ~+y 1-"+Y r' j 21 2' 2 ,v '(a2+y2) (8) x -~ e -ax 2 .... ('') (8 --exp --I Rea> 0, Rev> -1 2a~ Ba ~11 8a (9) X~ e -ax 2 TT 1/2 y 3/2 exp~ ~:) Rea> 0, Rev> -2 8a3/2 x [~,--~:)-I,,..(~: j] (10) x 11+~ e -ax 2 y11+~ exp (-:~) Rea> 0, Rev >-1 (2a)11+1 30 INTEGRAL TRANSFORMS 8.6 Exponential and lo&arithmic functions (cont'd) f(x) Joo f(x) J (xy)(xy) ~ dx 0 II y>O ( ll) x v-312 e -ax 2 211-1 ~-v ( y2 ) y y v,- Rea> 0, Rev> 0 4a (12) xv+~e±iax2 yY+~ [ (v+l y2 ~ J a> 0, -1 <Rev<% (2a)Y+I exp ± i -2-11 -4a 2 2 (13) x2n+v+~ e-~x 22n+v+1 n!yv+~ e-1' LY(y2) n Rev>-l-2n (14) xJ.L-~ e -ax 2 r v+~r<%v+ Xtt + %) Rea> 0, Re <tt + v) > - 1 2v+l a~C/.L+v+llr(v+1) ~+tt+ 1 y2 ) x F ---· v+l· ---I I 2 ' ' 4a = r(%v+%~t+%) exp(-y21 y~ a~J.Lr(v+ 1) Ba ~2) xM - ~J.L• ~~~ 4a (15) x-312 e-aj:c Rea> 0 2y ~ J [(2 ay) ~] K [(2 ay) ~] II II (16) x-3/2 e-aj:c-j3:c 2y~ J)(2a) ~[(f32+y2)~-13]~1 Rea> 0, Re /3 > 0 x K 111 (2a) X [(/3 2 + r 2) ~ + /3] ~I {17) -I -ax~ 11-~ 2Xr{v+%)D (2-~ae~71;y~) x e Rea> 0, Rev>-% 2 -v-~ D (2-~ -l(7Ti -X) X -v-X ae y 8.6 HANKEL TRANSFOHMS 31 Exponential and logarithmic functions (cont'd) f(x) F" f(x) J (xy)(xy)~ dx 0 IJ y>O (18) xv+~ ea(1-x2) 0 <x < 1 (2 · ) -v-1 ~ +v [U (2 · ) ta y 1.1+1 ta,y 0 1<x<oo -i U v+2 (2 i a, y)] Rev >-~ (19) x v+~ exp [-a (x 2+ /3 2) ~] (~ 17)-~ af3v+3!2yv+1!2 (y2 +a2)-v-3 /4 Rea>O, Ref3>0, Re v>-1 x K [f3 (y 2 + a 2) 1 /2] 1.1+3/2 (20) X-~ ({32+X 2)-~ y~ I~) ~f3[(a2+y2)~-a]l x exp[-a(f32+x2)~] x K~)~f3[(a2+y2)~+a]l Rea> 0, Re /3 > 0 Rev> -1 (21) xY+~(f32+x2)-~ · 2-~ ~ f3 ~ +vc 2 2)-l{ -~ v ~ +v t 1T a -y r x exp[ia(f32+x2)~] a>O x H<0 [f3(a2-y2)~] Re f3 > 0, -1 <Rev<~ -v-~ O<y<a 2~ 17-~ f3~ tvy~ +v(y2-a2tl{ -~ 1.1 x K [.B(y2-a2)~] v+~ a<y<oo (22) xv+~(f32+x2)-~ (~ rr)-~ /3v+~yv+~ (a2+y2)-~v-l{ x exp [-a (f3 2+x 2) ~] x K [f3 (a 2+y 2) ~] Rea>O, Ref3>0, Rev>-1 v+~ (23) x-v+~ (x2+ f32)-~ y v+~ [a+(y 2+ a2)~rvcr 2+ a2)-~ x [(x 2+ f3 2) ~-/3)1.1 x exp[-f3 (y2+a2)~] x exp[-a(x2+f32)~] Rea>O, Ref3>0, Rev>-1 (24) xu-~(x2+f32)-~ [(x2+ff)~+f3F r(~v+~a+~) M lf3[(y2+~)~-a]l x exp [-a (x 2 + f3 2) ~] {3r(v+1)y~ ~.~~.~ Rea> 0, Re f3 > 0 X W_~,..~)f3((y2+a2)~+aJl Re (v +a)> -1 32 INTEGRAL TRANSFORMS 8.6 Exponential and logarithmic ·functions (cont•d) f(x) J"" f(x) J (xy)(xy)~ dx 0 )I y>O For other Hankel transforms cont~ining exponential functions see Laplace transforms. (25) xlllog x 21l-~ f'(~ IL+ ~ v+%) f'(~ v-~IL+~)yJ<+1 X (1/J (~ IL+~ v+%) -Re v-3/2 < Re 1L < 0 + ljJ (~v- ~IL +~)-log(~ y 2)] 8. 7. Trigonometric and inverse trigonometric functions (l) x-~ sin (ax) cos(~ITV) y11+~ (a2-y2)-~ a> 0, Rev> -2 x [a +(a 2 -y 2)lq-v 0 <y <a y~ (y2-a2)-~ sin[vsin- 1 (a/y)] a<y<oo (2) x-312 sin(ax) 11-1 sin (~w)y11+~ a> 0, Rev> -1 x [a+(a 2_y 2)~rv 0 <y ~a 11-1 y ~sin [vsin -1 (a/y)] a<y<oo (3) x v+~ sin (ax) a> 0, -3/2 < Re v<-l/2 -21 +v "-112 sin (vn-) r (v+ 3/2) a xyv+ 112 (a 2 -y 2) -v-3/2 0 <y <a -21 +v"-112f'(v+ 3/2) x ay 11+112 (y 2_ a 2)-v-312 a<y<oo (4) x v-)( sin (ax) [f'(~-v)r1")( 2vyv+)( a> 0, -l<Rev<~ x (a 2_y 2)-v-~ 0 <r <a 0 a<y<oo 8.7 HANKEL TRANSFORMS 33 Trigonometric functions (cont'd) f(x) Joo f(x) J (xy)(xy) ~dx 0 v y>O (5) x Y, -v sin (ax) 0 O<y<a a> 0, He v > J-i 2t-v 17112 a[[' (v-J-i)r' y t/2+v x (y 2 _a 2) v-3/2 a<y<oo (6) x -v+2n+Y, sin (ax) 0 0 < y <a a> 0, He v > 2n + J,i (-l)n 2v-2n-2Y 2n-v+3/2 (2 n + 1)! xr (v-2n -1) [f'(2v-2n -1)] -t x (y 2_ a 2) v-2n-3/2 C v~,...., (ay-') 2n+t a<y<oo (7) xJ.J.-312 sin (ax) y v+~ r (v + Jl) sin[~ ~ dv+ J1)] a> 0, -He v <He J1 < 3/2 2"a"+J.J.f'(v+1) (1 +v+ J1 v+Jl y 2 j x F -----·v+1·--2 t 2 ' 2 ' 'a2 0 < y <a 2J.J. a ['(Yz+ Yz v+ X J1) yJ.J.+Y,['(Yz+Xv-Yz/1) x ~+v+" 1+•-v . .". a'j 2Ft 2 ' 2 , 2 , y 2 a<y<oo (8) xv-Y, (x 2+ {3 2)-1 si~ (ax) a>O {3 v-t sinh (a {3)y ~ K ({3y) y;::a v Re {3 > 0, -1 <He v < 3/2 (9) x ~ -v (x 2 + {3 2) -t sin (ax) a>O J-i1T{3-ve-a f3y'!. I)f3y) He {3 > 0, He v > -~2 O<y~a (10) x-312 e -xacos <f> cosljlsin(xasin t/1) a'!. v-1 (sin¢) '/, (tan X ¢)"sin(vt/J) Hev>-1, a>O, () < 0, L'; < 1 c TT y =a sin¢ 34 INTF:GRAL TRANSFORMS 8.7 'Il'igonometric functions (cont•d) {(x) J"" f(x) J (xy)(xy) l{ dx 0 v y>O (ll) XV+\{ e-ax cos¢ cos ljJ sin(ax sin r#) 2v+l17-l/2r(v+3 /2)a-v-3/2 a> 0, 0 < ¢, t/1 < ~17 x (sin ¢)""+112 (cos2 t{l Rev> -3/2 +sin2 t{l cos2cf>)-v-312 xsin[(v+3/2)a] y =a sin¢ tan (~a) = tan t{l cos ¢ (12) xv-l{ e -xacOIJ ¢ cosljlsin(xa sin 1/J) 2vrr-l{ a-v-X r(v+ ~)(sin¢)v+ X a> 0, 0<¢,1/J<~ rr x(cos2t{l+sin21/J cos2¢)-v- l{ Rev> -1 x sin[(v+ 3/2)a] y =a sin¢ tan (~a)= tan 1/J cos¢ (13) x -X sin (ax 2) -17Xy'l{ sin~2- 11+1 ~ a> 0, Re 11 > -3 2aY. 8a 4 ~2) xJ -l{v Sa (14) x l{ sin (ax 2) •'nyM [ G' ~) ~'j cos--- J -- a> 0, Re 11 > -4 Sa312 Sa 4 l{v-X Sa I _ s in(j 2 _ 11 ") J (f 2 ) J 8a 4 y, v+X Ba 05) x v+Y, sin (ax 2) yv+x «2 1117) a> 0, -2 < Re 11 < }2 2v+1 av+l cos 4a-2 (16) xv+X sin (ax2) 0 <X< b (2a)-v-l yv+X [sin (ab2)Uv+1(2ab~by) 0 li<x<oo -cos(ab2)Uv+2 (2ab,2by)] Re v>-2 ( 17) x-1 e-axX sin (ax X) ·2-x -Xr·< ~) n < -v, £ 17 11+ 2 -v-Y. ay ·) a> 0, Re 11 > -1 x [D -v-Y, (iay-X)-D_v- Y, (-iay-X)] 8.7 HANKEL TRANSFORMS 35 Trigonometric functions (cont•d) f(x) Joo f(x) J (xy)(xy) U dx 0 v y>O (18) xv+U sin[a(x2+/32)U] a>O (~ rr) 1/2af3 V+3/2yV+1 /2(a2-y2rV/~ /4 Re /3 > 0, -1<Rev <-~ x l sin (vrr) J [13 (a 2 -y 2) 112] v+3!2 +cos (vrr) Y v+312 [/3 (a 2-y 2) 1/2]1 0 <y <a _ (~ 7T r 1/2 af3v+3/2 y V +1/2(y2-a2j ll/2-3/4 xK [f3(y2-a2)112] v+3/2 a<y<oo (19) x-u (x 2 + 13 2) -u ~ rry U JU )~ 13 [a-(a 2_y2)U]1 x sin [a (x 2 + /3 2) U ] a>O x J -U) ~13 [a+ (a 2 -y 2) U]i Re /3 > 0, Rev> -1 0 <y <a (20) xv+U (f32+x2)- U 2-!-( 71U 13 U +vy'U +v(a 2 -y 2)-!4 -U v x sin [a (f32+x 2)U] a>O x J [f3(a2-y2) U] 0 <y <a Re /3 > 0, -1 <Rev<~ -v-U 0 a<y<oo (21) xv+U (b 2+x2)-2 ay!-( bV K (yb) y>a x sin [a (x 2 + b 2) U] a>O v b > 0, -1 <Rev< 7/2 (22) X -){ (a 2 _X 2) -){ ~rryU Ju)~a[(b2+y2) U-b]l xsin [b(a2-x2)U] O<x<a x Yu) ~a[(b2+y2)~+b]! -x-u (x 2_a 2)-U x exp [-b (x 2-a 2)U] a<x<oo b > 0, Rev> -1 (23) xv+Y,(a2-x2)-U 2-Y, 71U av+U (b2+y2)-Uv-l( yv+U x sin[b(a 2-x2)y,] 0 <x <a x Y [a(b2+y2)Y.] v+Y, - -xv+Y, (x2-a 2)-Y. x exp[-b(x 2-a2)Y.] a<x<oo b > 0, Rev> -1 36 INTEGRAL TRANSF0!1!\IS 8.7 Trigonometric functions (cont'd) f(x) Joo f(x) J (xy)(xy) Y. dx 0 )I y>O (24) 0 0 <x <a 0 0 <y < b x y, -11sin [b (x 2-a 2) Y.] 2-112 17112 a 3/2-v by 112 -v a<x<oo X (y 2_ b 2) V/2-3/4 b > 0, Rev>~ x J [a (y 2 _ b 2) 1 /2] v-3/2 b<y<oo (25) x Y. -v(x 2+ f3 2) -Y, [(x2+f3 z)Y,-{3] v y v+Y. [a+ (a 2_y 2) Y.rv (a z_y 2) -y, x sin[a(x2+{:32)Y.] a>O x cos [{3 (a 2-y 2)Y. + ~ rrv] Re f3 > 0, Rev> -1 0 <y <a y -Y, (y2-a2)- Y. exp[-{:3(y2-a2)Y.] x sin [v sin -1 (a/y)] a<y<oo (26) 0 0 <x < c 2 2 y, ~rryY.b-ve-a(c +b) Iv(by) x Y, -v (x 2 + b 2) -1 0 < y <a x sin [a (x 2-c 2) y,] c<x<oo .. Rev> -3/2 For other Hankel transforms containing sines see the table of Fourier sine transforms. (27) x -312 cos (ax) a>O v-1 cos(~vrr}y11+Y, [a+(a2-y2)Y.]-v Rev> 0 O<y_sa v-1 yY. cos[vsin-1(a/y)] a<y<oo (28) xv+Y. cos (ax) 21 +v17 112a [r(-~-v)r1 yv+112 a> 0, -1 <Rev <-~ x (a 2_y 2) -v-3/2 0 <y <a 0 a<y<oo (29) x v-Y, cos (ax) -2vrr-Y. sin(vrr)r( ~+v) a> 0, jRe vj < ~ xyv+Y, (a2-y 2)-v-Y, 0 < y <a 2V 17-Y. r(~+ v)yv+ Y,(y2-a2)-v- Y, a<y< oo 8.7 HANKEL TRANSFORMS 37 Trigonometric functions (cont•d) f(x) Joo f(x) J (xy)(xy) ~ dx 0 v y>O (30) x-v-Y, cos (ax) 0 O<y<a a> 0, Rev >-~ 17 Y, (y 2 _a 2) v-Y, 2vyv-Y.r(v+ ~) a<y<oo (31) x-v+2n-Y, cos(ax) 0 0 < y <a a> 0, Re v > 2n- ~~ (-l)"y-v+2n+Y, 2v-2n-1 r (v-2n) x[r(2v-2n)]-1 (2n)! x(y2-a 2)v-2n- Y, cv-2n(ay-1) 2n a<y<oo (32) xJL-312 cos (ax) y v+~ r (v+ 11) cos[% 7T (v+ 11)] a> 0, -Rev< Re Jl. < 3/2 2V aV+JL r(v+ l) x F ~+" v+"+l r'j 2 1 -----·v+l·-2 ' 2 ' 'a2 0 <y <a 21L-1 yY.-JLr(~v+~11) r (l + ~ v-% 11) x F (:+1111-v.L a2) 21 2'2'2'y2 a<y<oo (33) xv+Y, (x 2+ (32)-1 cos (ax) {3v cosh (a {3) y y, K ({3y) y ~a v a> 0, Re (3>0, -l<Rell<~ (34) x-v-Y, (x 2+ {3 2) -1 ~os (ax) ~rr{3-v-1e-a.ByY. I ({3y) a>O, Re (3>0, Rev>-3/2 v 0 < y <a 38 INTEGRAL THANSFORMS 8.7 'D'igonometric functions (cont'd) f(x) Joo f(x) J (xy)(xy)Y. dx 0 lJ y>O (35) X-3/2 e-xacos¢cost/J ay, v -1 (sin ¢)Y. (tan ~¢) v cos ( vt/J) x cos (xa sin t/J) a>O y =a sin ¢ 0 < cp, tP < Y~ TT, Rev> 0 (36) X v+Y, e -ax cos¢ cost/J 2v+1 rr-112r(v+3/2)a-v-3/2 x cos (ax sin t/J) a>O x (sin¢) v+112 (cos 2 t/J 0 < cp, t/J < ~ TT, . Rev> -1 +sin 2 t/J cos 2 ¢)-11-312 xcos [(v+3/2)aJ y =a sin¢ tan (~ a) = tan t/J cos ¢ (37) X v-Y, e -xacos¢ cost/J 211 TT-Y, a -v-Y, r ( V+ ~)(sin c/J)V+Y, x cos (xa sin t/J) a>O x (cos 2 t/J +sin 2t/J cos 2¢) -v-l<; 0 < cp, t/J < ~ TT, Rev >-~ x cos [(v+ Y2)a] y =a sin¢ tan (~ a) = tan t/J cos¢ (38) x-Y, cos (ax 2) rry,Yy, cos(Y2 _ v+1 ~ a> 0, Rev> -1 2ay, 8a 4 (£2) xJ - Y,v 8a (39) x v. cos (ax 2) TT 1/2 y J/2 [ ~2 VTT) a> 0, Re v > -2 8 a 3/2 cos 8a -4 X JY,v+Y,E:: j . ~2 VTT) Gy2)] +sm --- J 8a 4 Y,v-Y, 8;; (40) x v+Y, cos (ax 2) -1 <Rev<~ yv+Y, sin« 2 _ VTT) 2v+1 av+1 4a 2 8.7 IIANI\.EL TRANSFORMS 39 Trigonometric functions (coot •d) f(x) Joo f(x) J (xy)(xy) ~ dx 0 'II y>O (41) x v+~ cos (ax 2) O<x<b (2a)-v-1 Yv+~ 0 b<x<oo x[sin(ab2) uv+~(2ab,2 by) Rev> -1 + cos(ab2) uv+1 (2ab,2by)] (42) ~ ~ x-1 e-ax cos(ax ) 2-~ -~1( +X)D (a -y,) TT v 2 -v-Y, y a> 0, Rev> -X x [D -v-Y, (iay -~) + D -v-~ (-iay -~)] (43) xv+Y, cos[a(x2+.f:32)~] a>O (X TT) 1/2a .Bv+312yv+1/2 Re .B > 0, -l <Rev< -X X (a 2_y 2)-v/2-3/4 xI cos (TTv) J v+312 [/:3 (a 2 -y 2) 112] -sin (TTv) Y v+3/2 [/:3 (a 2 -y 2) 1/2]1 0 <y <a 0 ll<y.<oo (44) x -~ (x 2 + .B 2)-~ -Xny ~J ~ J X .B [a-(a 2-y 2) ~]I x cos [a (x 2+ {32)~] a>O xY -~viX,B[a+(a2-y2)~]1 Re {3 > 0, Rev> -1 0 <y <a (45) xv+~(x2+ .82)-~ _ 2 -~ 17~ .B¥, +vy Y, +v (a 2_y 2)-~- ~ v xcos [a(x2+/:32)y,] a>O x y [{3(a2-y2) Y,] 0 <y <a Re {3 > 0, -1 <Rev< X -v-~ 2~ 17-~ .BY, +vy ~ .+v(y 2_a 2)-~- Y, 11 x Kv+~ [{3(y2-a2) ~] a<y<oo (46) xv+112(x 2 + b 2)-3/2 y~ b v K (by) y>a x cos [a(x 2+b 2)~] 'II a>O b > 0, -1 <Rev< 5/2 (47) x -~(a 2-x 2) -~ cos[b(a 2-x 2) y,] X TTy~ J~) X a [(b2+y 2)~ -b]l 0 <x <a x J 1Xa[(b2+y2)~+b]l 0 Y,v a<x<oo Rev> -1 40 INTEGRAL TRANSFORMS 8.7 Trigonometric functions (cont•d) f(x) J"" f(x) J (xy)(xy) X dx 0 11 y > 0 (48) x11+~(a2-x2)-X TX rrX a 11+X y11+X (b 2 +y 2) -X 11-!4 x cos [b (a 2-x 2) X] 0 <x <a X J (a(b2+y2)X] 11+% 0 a<x<oo Rev> -1 (49) 0 O<x<a 0 0 <y < b x X -11(x 2 _a 2) -~ 2-X TTX ay, -11yX -11(y2-b 2)X 11-!4 xcos[b(x 2-a2)Y.] a<x<oo xJ [a(y2-b2)Y.] 11-X b<y<oo b > 0, Rev >-~ (50) x Y, -11 (x 2 + f3 2) -Y, -y11+Y. [a+(a2-y2) Xr11(a2-r2r Y. x [(x 2 + f3 2) ~ _ {3]11 x sin [{3 (a 2-y 2) y, + ~ rrv] xcos [a (x 2+ {3 2) ~] a>O 0 < y <a Re {3 > 0, Rev> -1 y -y, (y 2 _a 2)-X exp [-f3(y 2_a 2) Y.] xcos[vsin-1 (ajy)] a<y< oo (51) 0 0 <x < c ~rryX b-11(c 2+b2)- Y. x X-11(x2+b 2)-1 (x 2_c 2)-Y. xe-a(c2+b2) Y. I (by) 0< y <a x cos [a (x 2-c 2)Y.] 11 c<x<oo Rev> -5/2 For other Hankel transforms containin g cosines see the table of Fourier cosine transforms. (52) x-Y. (x2-1)-Y. rry, sin (~y) J11_X (~y) xcos [(v-1) cos -1 x] 0 <x < 1 0 1<x<oo Rev> 0 (53) x-y, (l-x2)- X 0 <x < 1 TTY, cos (Yzy) J11+Y, (y2y) xcos[(v+ 1) cos-1 x] 0 1<x<oo Rev> -1 8.8 HANKEL TRANSFORM~ 41 Trigonometric functions (cont'd) f(x) Joo f(x) J (xy)(xy) X dx 0 11 y>O (54) x -X (1-x 2) -x cos <11 cos_, x) ~tryX JXVJ.+vl(~y) JX(v-t.L>(~y) 0 <x < 1 0 1<x<oo Re(ll + v) > -1 (55) 0 0 <x < 1 y-X cos(y- ~vtr) x X (x 2-1)-X cos (vcos-1 x-1) 1<x<oo Rev> -1 8.8. Hyperbolic and inverse hyperbolic functions (l) x11-X e-X1Tx csch( ~trx) 71-X zv+l r(v+ ~)yv+X Rev >-~ 00 (n 2 712 + y 2) -v-X X ~ n= 1 xX x coshx + sinhx For this and similar integrals (2) sinh (2x) + 2x see Boit, M. A., 1935: f. Appl. Phys., 6, 367-375. (3) x v+X sinh (ax) csch (11x) 211-1 yX 00 ~ (-l)n-! n v+l sin (n a) IRe al < 11, Rev> -l n= 1 x K 11(ny) For other similar integrals see Weber, H., 1873 : ]. of Math. 75, 75-105. (4) x -x (1 +x 2)-X sinh (2/lsinh-1 x) ~y X [ Iy, v-t.L (~y) Kx v;t.L (~y) Rev> -1, IReJ.LI < ~ -IXv+t.L(~y) Kxv-t.L(~y)] 42 INTEGRAL TRANSFORMS 8.8 HYperbolic fUnctions (cont'd) f(x) J"" f(x) J (xy)(xy)X dx 0 . II y > 0 (5) x -X (1 +x 2) -X cosh (21£ sinh4 x) ~yx [I~~~-~}~r)Kxv+.u(~y) Re 11 > -1, IRe ILl <%: +IX v+J}~y) K~v-,u.(~y)] (6) 0 0 <x < 1 ~"X [cos (~y) J11_~ (~y) x-X (x2-1)-~ -sin (~y) Y v-~ (~y)] x cosh [(11-1) cosh_, x] 1<x<oo -1/2 < Re 11 < 5/2 (7) 0 0<x<1 -~"X [sin (~y) J11+~ (~y) x-X (x2-1)-X +cos (~y) Y v+X (~y)] x cosh [(II+ 1) cosh_, x] 1<x<oo -5/2 < Re 11 < 1/2 (8) 0 0 <x < 1 -~ 1TY ~ [J ~ (,u+v )(~ Y) Y X<v-,.,)~ y) x -X (x 2 -1) -X cosh(l£ cosh_, x) + J~(v-,u.)(~y)Y ~(v+,u.)(~y)] 1<x<oo IRe Ill < 3/2 8.9. Orthogonal polynomials (1) x-X(1-x2)-~T (x) 0 <x < 1 ~"YX J~(v+n)(~y) JX<v-n)(~y) n 0 1<x<oo Re 11 >-n-1 2 2-2n-v-1 (n !)-1 y 2n+v+X exp(- ~y 2) (2) xv+X e-x Lll(x2) Re 11>-1 n (3) x v+X e -~x 2 L ll(x 2) Rev> -1 (-1) ne -~y 2 y v+X L ll(y 2) n n 8.9 HANKEL TRANSFORMS 43 Orthogonal polynomials (cont•d) f(x) Joo f(x) J (xy)(xy)~dx 0 II y>O 4) 2 x2n+11+X e-Y,x L~+n(~x2) Re v > -1 2 y2n+11+~ e-y,y L ,:'+n(~y2) (5) x 11+~ e -{3x 2 L 11(ax 2) n 2-11-1 {3-~~-n-1 ({3-a)nyii+X Re {3 > 0, Re v > 0 ~ r') [ •r' j xex -- L11 P 4{3 n 4(3 (a-{3) (6) xii+X e-a.x2[LXII(ax2))2 n (2 a) -Il-l r II+~ Rea> 0, Re v > -1 x .x{ ::J [L~·«:) ]' X +11 (7) x II+~ e -{3JC 2 [LX II (ax 2)]2 _r_ r (n+ 1+ ~ vH2f3)_ 11_1 n rr n ! Re {3 > 0, Re v > -1 x exp (-;;) n (-1) l r (n-l+ ~) r (l+ ~) X I r (l + 1 + ~ v)(n -l)! t $)" [ ' j a- ar x -{3- L ~~ 2{3 (2a-{3) 2 (-1) "+n(2a)-11-l y 11+X exp (-::) (8) x II+X e -a.x L ~-a (ax 2) L~ (ax 2) Rea> 0, Re v > -l xLa-•+n~2 )tv-a+--n(~) n 4a • 4a (9) 2 x11+X e-x La(x2)L11-a(x2) n n 2-~~-1 y II+X e ..1_4y 2 L ~(~ y 2) Re v > -1 xL~-a(~y2) 44 INTEGRAL TRANSFORMS 8.9 Orthogonal polynomials (cont'd) f(x) J 00 f(x) J (xy)(xy) ~ dx 0 II y>O (lO) 0 0 <x <a (-1) n 2 2n-11+1 r (2 v-2 n) x 2n+~ -~~(x 2_a 2)11-2n- ~ X ((2n)! r(v-2n)r1y-ll+2n- ~ x cos (ay) X C11-2n (ajx) a<x<oo 2n 2n-~ < Re v < 2n + ~ (ll} 0 0 < x <a (-1) n 2 2n-ll+2 r (2 v-2n -1) x 2n-11+3/2 (x 2_ a 2) ~~-2n-3/2 x[(2n+ 1)!r (v-2n-1)]- 1 X Cll-2n-1 (ajx) a<x<oo X y -II +2n +~ sin (ay) 2n+1 2n + 1/2 < Re v < 2n + 3/2 (12) x 11+~ (1-x 2) -~sin [a(1-x 2) ~] (-1)n2- ~ rr~y~~+~ (a2+y2)- ~1J-!( XC~'+~ ((l-x2)~] 0 <X< l C 11+~ [ ( 2 2)-~] 2n+1 X 2n+1 a y +a 0 l<x<oo X JIJ+3 /2+2n((a2+y2) ~] Rev >-~ (13) x11+~(1-x2)-~cos[a(1- x2)~] · (-l)n2- ~ rr~y~'+~ (a2+ y2)-~11-l( X C11+~ ((l-x2)~] 0 <x < 1 X ell+~ (a(y2+a2)- ~] 2n 2n 0 1<x<oo x Jll+~ +2J(a2+y2)~] Rev >-~ 8.10. Legendre functions 2y, -11 -~ [K (2-~ )] 2 (1) (x 2+ 2) -Y, ~~-l( p -~~-~ (x 2 + l) " t.t.+X r tJ. Rev> -1 r (v:+-11 + 3/2) r (v-11+ l/2) -3/2 -Re v < Re 11 < Rev+ l/2 (2) 0 0 <x <a 2~ rr -~ y -t.L-~ cos [ay + ~ (v-11) rr] ( 2_ 2)~t.t.-!4p~-t.L( -1) x a -~+11 ax a<x<oo IRe Ill < ~. Rev> -1 8.10 HANKEL TRANSFORMS 45 Legendre functions (cont'd) f(x) F" f(x) J (xy)(xy) X dx 0 II y>O (3) x 11-X (l-x 2) X 11+!4 r (3/2 + IJ. + 11) r (~ + 11-ll)(2y) II+X X P -~~-x (2x -2-l) 0 <x < l (277) X [r (3/2+ 11))2 I.J. 0 l<x<oo x 1F1 (11+1J.+3/2;211+2;iy) -3/2-Re 11 < Re 11 < Re 11+ l/2 X 1F', (v+ll+3/2; 211+2;-iy) (4) x X (a2+x2)-XJ.L yJ.l-3/2 e -ay xP~~ 1 [a(a2+x2)-X] Rea>O r (JJ.+ 11) Re 11 > -1, Re IJ. > ~ (5) x 11+X (x 2 +a 2) X 11 (2 af+1 y-11-X [K 11+x (X arW [ x2+2a2 j 1rr(-11) xPII 2a(x2+a2)X Rea> 0, -l < Re 11 < 0 (6) x X -~~(x 2+ a2) -X 11 (2 a) 1 -~~ 11-x I (1 ) [ x2+2a2 r Cv) y II-X X ay xP J X K11_X (~ ay) 11-1 2 a (x 2 + a 2) X Rea> 0, 0 <He 11 < l (7) x X lP-~ 11((l+a2x 2)l4]J2 2 [Kg+~(~ a -1 y)]2 I.J. TTar<l+JJ.+X11) r(~11-IJ.) y~ Rea> 0 -~ < Re IJ. <-~. Re 11 > -1 (8) x X ( 1 +a 2 x 2) -X yx Kg+1AXa-1y) Kg+312(Xa-1y) x p-X 11((1+ a2x2)X] TTa2 r (2+ ~ 11+ JJ.) r (~ 11-IJ.) J.1- x p-'1.11 [U+a2x2)X] J,L+1 Re 11 > -1, Re a> 0 -7/4<Re ll<-1/4 46 INTEGRAL TRANSF'OH~IS R.lO Legendre functions · (cont'd) f(x) Joo f(x) J (xy)(xy) ~ d~ 0 )I y>O (9) xX(1+a2x2)-~ Y~ [KJ,L+~ <xa-1 y)]2 xP-~-Xv[(1+a2x2)~] rra2 r (v/2+ ll + 3/ 2) r (v/2-ll+ l/2) j..L X p~-~11((1+a2x2)X] J.L Re v > -1, Rea> 0 -5/4 < Re ll < 1/4 (10) Q [(a2+x2)x-1] T~ rry-~ exp[-(a2-~)Xy] v-x Re v>-X X J)I(Xy) (ll) xX-J..L(l+a2x2)-XJ..L-~ '(2 ) X i11(J,L+X v+!O -1 J..L-X t rr e a y x Q :~:j (± i ax) ne a> 0 xI <xa-1y) K <xa-1y) 11 J.L -~4-X Re v < Re J.J. < 1 +Re v (12) (x2+2)- Xv-!4 Qv+X (x2+1) j..L 2-v-X 77X e (v+l07Tiyv+X Re v > -1 X KJ.L+X (2-Xy) IJ..L+X (2-Xy) Re(2J.L+ v) >-5/2 03) x-v-~ Q~~X (1+2 a2/x 2) -ie i7TV rrX 2-v(y/ a)v-~ Rea> 0, 0 <Rev< 3/2 x 111_~ (Xay) K 11_x (Xay) (14) xv-lf(a2+ x2)~+~v -ie i7Tvrr-1/2 2 v[r (3/2+ J.L+ v)JZ X ov+X (1+ 2a2/x2) Rea> 0 X r (1/2+ v-J.L) a v-1/2 y-v-3/2 J.L Re(J.L+ v)>-3/2 X If/ -J.L-X,v+X (ay) Re(v-ll)>-1/2 x [ cos (wr) r (2+ 2 v) M J.L+l4,v+% (ay) sin (rrv) ~ +r(v+J.L+3 /2) WJ..L+l{,v+X(ay) 8.11 HANKEL TRANSFOHMS 47 Legendre functions (cont'd) f(x) J00[(x) J (xy)(xy) %dx 0 II y>O (15) x -~~-% (x 2+ a 2) !4-X 11 ie-illrr rr'/2r(3/2+J.L-v) y -~~-3/2 xQ%-11(1+2a2/x2) Re a>O 211 a11+112r(2v) J..L 0 < Re v < Re J.L + 3/2 X M J..L+%,11-% (ay) w_J..L-Y,,v-Y,(ay) (16) xy, p -Xv[(l+a2x2) X] e-y, vrrir (1+ J.L+ Y,: v) IJ..L+Y,~;a J J..L ar(1+ 11-Y,: v) yX X Q-Xv[(l+a2x2) %] Rea>O J..L x KJ..L+Y,(;a 1 Re 11 > -~, Rev> -1 8.11. Bessel functions of argument kx (1) x-X J (ax) 0 0 < y <a v-1 a> 0, Rev> -1 av-1 y-v+Y, a<y<oo (2) x-312 J (ax) y,;11-1 a-vyv+ Y, O<y.=:;a v a> 0, Rev> 0 Y.:v-1 avy-v+ Y, a.=:;y<oo (3) x-X Jv+1(ax) a-v-1 y11+X 0 <y <a a > 0, Rev> -3/2 0 a<y<oo (4) x-2A.-x J (ax) a>O a11y11+Y, r(v-A.+ Y.:) v 2 2A(a + y) 2 v-2A.+ 1 r<v+1) rex.+ y,;) Re v + Y,: > Re A >-Y,: x F [ 1 1 4ay ~ 2 1 v-A +i v+-; 2v+1; ---2 (a+y)2 (5) x-X Jv+2n+1 (ax) y11+% a-v-1 p(11,0)(1-2y2 /a2) n a> 0, Rev>-1-n 0 < y <a 0 a<y<oo 48 INTEGRAL TIIANSFO H~lS 8.11 Bessel functions of kx (cont•d) f(x) J"" ((x) J (xy)(xy) ~dx 0 v y>O y"+~ a-v-1 1 n~t-L+ Xv+ X) (6) x-~ J (ax) f'(v+1) nXt-L-Xv+ X) 1-L a> 0, Re (1-L + v) > -1 ~+v+1 v-tL+1 1 Y2j x F ---,---;v+ ;-2-2 1 2 2 a 0 <y <a For y >a interchange ll and V• 2v-J-L+1 yv+~ (a2-y2)J-L-v-1 (7) xv-J.L+% J (ax) f' (1-L-v) a/.L 1-L a> 0, -1 < Re v < Re tL 0 <y <a 0 a<y<oo (8) xJ-L-v+ ~ J (ax) j.L 0 0 <y <a a> 0, Re v > Re tL > -1 2iJ.-v+1 J.L a ( 2_a2)V-J-L-1 ( ) v-~ Y f' v-IL Y a<y<oo x-f..,..x J (ax) a>O f'[X(t-L+v-,\+1)] Yv+% (9) 2A.av-A.+1 f' (v+1) f' [X (,\+t-L-v+1)] 1-L Re(t-L+ v) + 1 > Re ,\ >-1 ~+tr-.\+1 tr-.\-1J.+1. 1·~ x. F , ,v+ • 2 2 1 2 2 a 0 < y <a f' [X (~+v-,\+1)] al-L 2A.y1-L-A.+~n11 +1) f'[ ~2 (.\ +v-1J.+1)] ~+tr-,\+1 ~.\-v+l 1 a 2 j x2F1 2 ' 2 ; IJ.+ ;yz a<y<oo 8.11 HANKEL TRANSF'OH\1S 49 Bessel functions of kx (cont•d) f(x) Joo f(x) J (xy)(xy) ~ dx y > 0 0 II (10) x~(x2+/32)-1 J (ax) a>O y ~ I (y /3) K (a /3) 0 < y <a II ll II Re(3>0, Re v > -1 yl4 I (a/3) K (y /3) II II a<y<oo (ll) x~-2n (/32 + x2)-1 J (ax) (-1)" /3-2n y~ I (y /3) K (a/3) II II II a> 0, Re /3 > 0 0 < y <a Re v > n-1, n = 0, 1, 2, ••• (-1)" /3-2nyl4 I (a/3) K (y/3) V II a<y<oo (12) xii-J.L+~ (/32+x2)-1 J (ax) 13v-J.Ly ~ I (a(3) K (y/3) J.L J.L II a> 0, Re /3 > 0 a<y<oo 1 + Re /1 > Re v > -1 (13) xv-J.L+2n+~(/32+x2)-1 J (ax) (-1) n (311-J.L+2n Y ~ I (a /3) K (y /3) J.L J.L II a> 0, Re /3 > 0 a<y<oo Re 11-2n + 1 > Re v >-n -1 n integer (14) xll-v+~(/32+x2)-1 J (ax) y~ 13wv I (y /3) K (a (3) J.L v J.L a> 0, Re /3 > 0 0 <y <a 1 + Re v > Re /1 > -1 (15) xJ.L-v+2n+~ (/32+x 2)-1 J (ax) J.L (-1)" (3J.L-v+2n Y ~ I (y (3) K (a /3) II J.L a> 0, Re /3 > 0 0 <y <a Re v-2n + 1 > Re /1 >-n -1 n integer (16) x~(x2+(32)-1J (ax) 11-2n (-1) n Y l{ Ill (y /3) Kll-2n (a /3) a> 0, Re /3 > 0 0 <y <a Rev> n-1 (-1)"y ~ Iv-2n(a(3)Kv(y{J) a<y<oo 50 IN 'lEG HAL TRANSFOHMS 8.11 Bessel functions of kx (cont'd) f(x) J: f(x) J)xy)(xy)~ dx y>O X -~ e -ax J 1,(f3x) (a2+f32+y2j (17) rr-1 rry, Q Re a > Im {3 > 0, Re 11 > -~ ~~-~ 2 {3y (18) xJ.J..-312 e -ax J ({3x) W'r v+Y, 1 (ll + 2 11) v 1TaJ.1..+2V[' (211 + 1) Re a> \Im (3\, Re(ll+211) > 0 j" ~ "+1 u'~ x 2F, -+11, ----+11; 11+1;--2 2 a2 0 x (sin rf>)211 drf> u 2 = {3 2 + y 2 - 2 f3 y cos ¢ (19) x-1 e-xacosrf>cost/;J (axsinrf>) 1 (ll + 11+ ~)(sin 1/1) y, J.l.. x P~~~ (cos rf>) P:~y, (cos 1/f) a> 0, 0 < rf>, "' < ~ I'T Re (ll + 11) >-~ y =a sin 1/J (20) x-~ e -{3x J (ax) 2rr-1 aJ.l.. {3 yv+Y, f'/,1T(2f3sec O).U+v .u Re {3 > \Ima\ 0 Re (tL + 11 + 1) > 0 x ({3 2 sec 2.0+y 2- a2+ u)-.u x ({32sec 2 0+ a2-y2+u)-v xsec2 e cos [(ll-11) O]u -1 dO u2 = (sec2 Ob 2+a2+y2)2-4a2y2 (21) x jJ.-v-Y, e -ax J ({3 x) f3J.l..yv+~r<tr+~) 11T J.l.. 2v-J.J..rrr(~~+~) o (sinrf>)2v Rea> \lm{3\, Re ll >-~ x[(a+iycosrf>)2+f32r.u-Y. drf> (22) x .\-3/2 e -ax J ({3 x) y'l. I 1<A+tr+ll+2m) (-{32) • .u Rea> lm {3 > 0 •=o m !1(1J.+m+1) 4a2 Re (,.\ + ll + 11) > 0 x 2F, (-m,-ll-m;ll+1;y2 /3-2) 8.11 HANKEL THAN SFOHMS 51 Bessel fWlcUons of kx (cont•d) f(x) Joo f (x) J (xy )(xy) ~ dx 0 I.J y>O 2 r' ( a'+r') Gar j (23) x ~ e -jft J (ax) --exp ---- I --11 2{3 4{3 11 2{3 Re f3 > 0, Re 11>-1 (24) xA+~ e -ax 2 J (f3x) Rea> 0 yX f r(m+Y711+Y7J.L+Y7..\) (-(32)• !J. • = 0 m ! r (m + /.L + 1) 4a Re (J.L + 11 + ,\) >-2 x 2F1(-m,-11-m;11+1;y2(3-2) (25) xA J (ax) cos (bx) see under Fourier transfonns. !J. sin (bx) (26) x X sin (ax 2) J (bx) \ yX cos~ 2+b 2-li7T) (~ I.J 2a 4a 2 J11 2a a> 0, b > 0, Re 11 > -2 (27) x ~ cos (ax 2) J (bx) yX . (b2+y2 11") Eby 1 -s1n -----J --I.J 2a 4a 2 11 2a a> 0, b > 0, Re 11 > -1 (28) x -~ [J (Y7 ax)]Z r-~ IP ~ 11_~ [(1-a 2 /r 2) ~]}2 0 a> 0, Re 11 > -1 a<y<oo (29) xX [Jli)Y7ax)]2 2 "-1 y-~ (a 2_y2) -x 0 < r <a a> 0, Re 11 > -1 0 a<y<oo (30) x ~-11 [J11(Y7ax)]2 21-11 y11-~ (a2-y2)11-~ 0 <r <a a> 0, Re 11 >-Y7 "X a 211r (11+ Y7) 0 a<y<oo 52 INTEGRAL TRANSFORMS 8.11 Bessel functions of kx (cont•d) f(x) Joo f(x) J (xy)(xy)~ dx 0 II y>O (31) x~-~~ J (ax) J (bx) [y 2 -(a-b )2] ~~-~[(a+ b)2 -y 2] ~~-~ II II yll-~ 2311-1 71~ (ab)ll r(v + ~) a, b > 0, Rev >-~ \a -b \ < y < a + b 0 O<y<\a-b\ or a+ b < y <oo (32) x ~ J~(ll+n)(~ax) Jy, 'li-n)(~ ax) 2rr-1 y-~ (a2-y2)-~ Tn(a-1y) a> 0, Re v > -1 O<y<a Q a<y<oo (33) x -~ J2 (~ax} a > 0 (~a)2J.Ly-2J.L- ~ r<~+~v+J.L) J.L W<J.L+l)J 2 r~ +~~rJ.L) Rev+ Re 2J.L>-l x12F1 [~ -Y7v+ J.L, ~ + ~v+ J.L; J.L +1; ~-~ (1-a2 jy2)~)12 a<y<oo (34) x ~ -J.L J (ax) J (bx) a, b > 0 yJ.L-~ (sinhu)J.L- ~ e (JJ.-Y,mi J.L II (~rr3)~ aJ.Lb1-J.L Rev> -1, Re 11>-~ X sin [(v-J.L) rr) Q~:~ (cosh u) O<y<a-b bJ..L-1 yJ..L-~ (. )J..L-~p~- (2rr)~ aJ..L smv 11_t'(cosv} \a -b \ < y < a + b 0 O<y<b-a or a+ b < y < oo 2 by cosh u = a 2 -b 2 _ y 2 2 by cos v = b 2 + y 2-a 2 8.11 HANKEL TRANSFORMS 53 Bessel fUnctions of kx (cont'd) f(x) J 00 f(x) J (xy)(xy)X dx 0 II y>O (35) x X-v J (ax) J (bx) J.L /.1. a, b > 0 0 0 < y < Ia-bl He /.1. > -1, He v > -% (ab) v-1 (sinu)v-!1 pl1-v(cosu) (2rr)X yv-X J.L-X la-bl<y<a+b (ab)11-1 (sinhv)11-X e (v-Y.l7T i (X 173) X yv-X x sin [(/.1.-v) rr] Q ~:::Y, (cosh v) a+b<y<oo 2 ab cos u = a 2 + b 2 -y 2 2 ab cosh v = y 2 -a 2 -b 2 (36) xP-J.L-v+Y, J (ax) J (bx) J.L P b>a>O 0 O<y<b-a Rep> -1, Re(p-jJ.-v) <% (37) xP-J.L-v- 312 J (ax) J (bx) J.L Pb>a>O 2P-J.L-v-1 yv+X aJ.L [' (p) bPr(/.1.+1) r(v+1) Rep> 0, He(p-1J.-v) < 5/2 O<y<b-a (38) x-X J (xa sin¢ cost/;)J (ax) a -X r [% (1 +a+ p)] J.L p r(/.1.+ l) r<v+ l) r[~20-a+ p)] a> 0, 0 < ¢, t/J <% 7T He (/.1. + v + p) > - 1 x (sin¢ cos tj;)f.l.(sin t/J cos¢) v+X (l+a-p l+a+p . ~ X 2F, 2 ' 2 ; jJ.+l; Sin 2q) 0+a-p l+a+p ) x 2F', -2-, - 2-; v+1; sin 2tf; a= IJ.+ v, y = a cos ¢ sin t/J 54 INTEGRAL TRANSFORMS 8.11 Bessel functions of kx (cont'd) f(x) Joo {(x) J (xy)(xy) ~ dx 0 !I y > 0 (39) x ~ J (xa sin <f> cos 1/J) J (ax) 2rr-1 a-312 sin(t.t7T)(sin <f>)J..L(sin 1/J )11+112 J..L V-J..L x (cos¢)112-11 (cosl/J)-J..L a > 0, 0 < <f>, 1/J < ~~ 11 Re 11 > -1 x[cos(ct>+ 1/J) cos(<f>-1/J)]-1 y = a cos <f> sin 1/J (40) x"-J (ax) J (bx) see Bailey, W. N., 1936: Proc. J..L p London Math. Soc. (2), 40, 37-48. (41) X 2n-J..L-3/2 (x 2 + c 2) -1 (-1)n+1c2n-J..L-2y~ I (be) x J (ax) J (bx) a > b > 0 J..L !I J..L I xI (yc) K (ac) O<y<a-b Re 11 >Xi-n !I II Re 11 > 2n -9/2 X v-M+~ II II (42) n J (a.x) 0 ~ a.<y<oo i= 1 J..Li t i= t l II ai > 0, ~ lli= M i= 1 -1 < Re 11 <ReM+ Yzk-Xi II J..L· X v-M-3/2 II 2v-M-1 y X -vr (11) n at (43) I1 J (a.x) t i= 1 J.Li t r (1 + t-t i) i= 1 II II ai > 0, M= ~ lli ~a .<y<oo i= 1 i= 1 1 0 <He 11 < Re M + k/2 + 3/2 (44) x-"--~ Y (ax) J..L see under Mellin transforms (45) x~(x2+132)-1 Y (ax) (-1)n yX Iv(y 13) Kv-2n-1 (at3) v-2n -1 a> 0, fie 13 > 0 0 <y <a 1-Re 11 > n-Xi 8.11 HANKEL TRANSFORMS 55 Bessel functions of kx (cont'd) f(x) f""f(x) J (xy)(xy)l{dx 0 II y>O (46) X l{ (x 2 + 13 2) -I y l{ I (13y) K (a {3) 0 <y:::; a xI cos[~ (v-IL) 11] J (ax) II ).L ).L + sin[~(v-IL)77] Y (ax)l f-L a> 0, Re 13 > 0 Re(v±IL)>-2 (47) xp+l{ (132+x 2)-1 13Py'!. I (13y) K (a/3) 0 <y <a xlcos[~(p-IL+vh] J (ax) II ).L 1-L +sin[~ (p-IL+ v) 77] Y (ax)l 1-L a> 0, Re 13 > 0 Re(v±IL+p)>-2, Rep<l (48) xy, Jl{ 11(~ax) Yl{ 11(~ax) 0 0 < y <a a> 0, Rev> -1 -277 -t y-'1. (y 2_a2)-Y. a<y<oo (49) x11+Y, J (~ax) Y (~ax) 0 0 < y <a II II a> 0, iRe vi<~ 211+1 a211y-11-Y, (y2-a2)-11-'!. - 77l{ r (~-v) a<y<oo (50) xP+'!.(x 2+y2)-1 J (bx) f-L yPy'!. I (by) I (yy) K (ay) ).L V II x Ieos [~ (p+ IL) 77] J11 (ax) O<y<a-b + sin[~(p+IL)rr] Y (ax)! II a> b > 0, Rep< 3/2 Re (IL + p + 2 v) >-2 Re (IL + p) > - 2 56 INTEGRAL TRANSFORMS 8.11 Bessel fwctions of kx (cont•d) f(x) F" f(x) J (xy)(xy)~ dx 0 v y>O xP+~ (x 2+ {32)-1 k k (51) II [J (c .x)] {Yy~ I){3r) KJ.L(a{3) i~ 1 IJ.Li({3ci) i= 1 J.Li l xl cos[~ (p+ M -ll) rr] JJ.L (ax) O<y<a-k ~c. +sin[~(p+M-Il)rr]Y (ax)l i= 1 1 J.L a> 0, ci> 0 k v+ "' lli= M, Re p<(k+ 3)/2 .. i= 1 Re (p + M) > IRe Ill - 2 8.12. Bessel functions of other arguments (l) x~ J (~ax2) 2a-1 Y~ J~)y2ja) ~v 'l a> 0, Rev> -1 (2) x~ exp (-~ ax2)J~ 11(~ {3x2) (~ 2(a2+{32)-~y~ exp - Rea> 1Im{31, Rev> -1 a2+{32 (~ xJ~v a2+{32 (3) x (~-v )/3 exp (~ax 2 i) I t+1 y2 ) a<v-2)/3y(~-v)/3exp 6"i-4ai x J(v-~ )/3 (~ax 2) lm a> 0, Rev> -1 (y2) xJ -(v-)0.'3 4a (4) x <~-v l/3 sin(~ ax 2) (v-2)/3 (~-v )/3 · ~ +1 Y 2 ) a y sm --rr-- x J(v-~)/3 e_~ax2) 6 4a a> 0, Re v>-5/2 x J(v-~ )/3(;~) R.l2 HANKEL THANSFORMS 57 Bessel functions of other argUments (cont'd) ((x) J''" f(x) J (xy)(xy) Y. dx 0 II y>O x (Y, -~~ )/3 cos 04 ax 2) 1 (11+1 y2j (5) a (v-2 )/3 y (Y, -v )/3 cos --rr--- x J (11-'1. )/3 (~~ax 2) 6 4a a> 0, Rev> -1 (y2) xJ --(11-'1.)/J 4 a (6) xy, [J~)~ax2)]2 y'l. (y2) (y2~ --J -y ·- a> 0, Rev> -1 a ~~~ 4a ~~~ 4a (7) x'h J~11(~ax2)J_~)~ax2 ) y '1. ( y 2 ) ~ Gy 2 ) . tTTV) -J -J -sm --a !4t-4a ~~-4a 4 a > 0, Rev> -2 -Y!411(;~) cos(:V) J x l~ J (ax 2) J (ax 2) 2 I . (8) ~[e ~11 7T'W ~ (u)W_ !4 (u) !.( 11-J.L !( II+J.L TTY j.L. II j.L. II a> 0, Rev >-~ +e-!.(117TiW (v)W (v)] J.L•!411 -j.L,!.(II y2 I . 2 y -Y. 7Ti u=-e Y.7T• v=-e 8a ' 8a (9) x-Y, J (ax-1) y-'1. J (2ay, y y,) II 21J a> 0, He v > -~ ( lO) x -512 J (ax -1) a -1 y y, J (2 a y, y y,) II 211 a> 0, Re v > -~ (ll) -3/2 J ( _,, a-Y. J (2aY.yY.) x 11_1 ax , 211-1 a> 0, Re v >-~ 58 11'\TEGHAL THANSFO n~1S 8.12 Bessel functions of other arguments (cont•d) f(x) Joo f(x) J (xy)(xy) l{ dx 0 v y>O (12) x-2VJ (ax-1) l{ -v -Xi csc(2vrr)(y ja)v-l{ a> 0, -X<Rt>v<3 x[e 2v7Ti J, -2)u) J2v-1 (v) -e-2v7TiJ (u)J (v)] 2v-1 1-2v u = (~ay) l{ e~7Ti v = (Xay)l{ e-~7Ti (13) xP-312 J (x-1) X rr esc[ X (IL-v-p) 1T ]y v +Y. IJ. -3/2 -Rev< Rep< Re11+ 3/2 [ 0 ~j.L+I/ j.L+v+p y 2 j x A F l+v,l+--,1+--;-0 3 2 2 16 0 I"~P ~~P r ~ -y11-8 OF3 1+j.L, 1+--,1+--;- 2 2 16 A -1 = 2v+pr (l+ v) r [1 + ~ (p-IL+ v)] X r[1+X (p+ j.L+ v)] B -1 = 2211--Pr (1+ ~ r [l+ X (IL+ v-p)] xr [l+~(j.L-v-p)] (14) xy,(/32+x2)- Y.exp(-a2/3) y-Y.e-lh J (2ayl{) /32+xz 2v ~a2 x ) xJ --v 2+x~ Re /3 > 0, Rev >-~ (15) J2v-1 (ax Y.) Re v > -~ Xay-312Jv-1 ()<{ a2y-1) (16) x-l{ J (ax l{) 2v Rev>-% y-l{ Jv()<{ a2y-1) (17) x-l{ e-f3x J (2axl{) r' (r'+/3')_ , oxp 0 ~ 2v /32+y2 Re /3 > 0, Re v > --% E a2 y ) X Jv /32+y2 8.12 HANKEL TRANSFOml S 59 Bessel functions of other arguments (cont'd) f(x) r f(x) J (xy)(xy) l1 dx 0 v y>O (18) x v+~ (x 2 + {3 2) -~ JL a-!Lyv+ ~ {3-IJ.+v+l (a2-y2) Y.JL-~v-~ x J [a(x2+{32)Y.] a>O xJ [{3(a 2_y 2)~] 0 <y <a IJ. JJ.-v-1 Re {3 > 0, Re 11 > Re v > -1 0 a<y<oo (19) x v+ X (x 2 + {3 2)-Y. JL -1 (~a)JJ.-• f3vW(Il)J-• yy, K ({3y) x J JL-1 [a (x 2 + {3 2) X] a>O v a<y<oo Re {3>0, Re(IJ.+2) > Rev>-1 (20) x v-312 (x 2 + {3 2) -JL/2 {3-IJ.2v-lf'(v)y ~-vJ (a{3) x J [a (x 2 + {3 2) 1 /2] a>O JL IJ. a<y<oo Re {3 > 0, Re(IJ.+2) > Rev>O (21) xv+Y, (x2+a2)-l (x2+{32)- XJJ. avy ~ ({3 2_ a 2)-X JL J [c ({3 2_ a2)'h] x J [c (x 2 + {3 2) X] JL JL x Kv(ay) c,Sy<oo Rea> 0, Re {3 > 0, c>O -1 < Re v < 2 + Re 11. (22) x v+2n-312 (x 2+ a2)-•(x2+ {32)-IJ./2 (-l)n+ly 'h av+2n-2 ({32-a2)-l1JL x J [c (x 2 + {3 2) 112) )1. x J [c ({3 2- a2) X]K (ay) JL v Rea> 0, Re {3 > 0, c > 0 c<y<oo -n <Rev< 4-2n + Re 11. (23) xv+X(x2+{32)- Xv-~ (-l)n2X TT-Y, aX -vyv+Y, (a2-y2)- ~ x C v+Y, [{3 (x 2 + {3 2) -X] x sin [{3 (a 2-y 2) ~] 2n+l xcv+'h[(1-y2ja2) X] x Jv+312 +2n[a(x2+{32) X] 0 <y <a 2n+l a> 0, Re {3 > 0, Rev> -1 0 a<y<oo (24) xv+Y, (x2+ {3 2)-X v-~ (-l)n2 Y, TT-Y, aX-vyv+ Y, (a2-y 2)-y, xc;:,+x [{3 (x2+ {32)-11] X cos [{3 (a2-y2)X] x C v+Y, [(l-y2ja2) X] 0 < y <a x Jv+Y,+2n[a(x2+{32) X] 2n 0 a<y<oo a >0, Re {3 > 0, Re v > -1 60 INTEGRAL THANSFOR~1S 8.12 Bessel fwtctions of other arguments (cont•d) f(x) (2S) xv-312(x2+t32)-n_u/2 (26) x ll J [a .(x2+ t32)~] i= 1 )1. 1 ai > 0, He t3 > 0 Re (n Jl. + ~n + ~) > Re v > 0 Xv+~ n -.u;J ( ) II z. a.z. i= 1 1 j.J.i l 1 n ai > 0, Re t3; > 0 z . = (x 2 + a 2) ~ 1 fJ 1 ~n+}; Jl.;-~>Rev>-1 i= 1 n -_u. (27) x v-312 II z . 1 J (a. z .) i= 1 1 .U; 1 1 a;>O, Ret3;>0 z.= (x2 + a2)~ 1 fJ 1 n ~n+ }; Jl.;+3/2>Rev>0 i=1 (28) xv+~(l-x2)~!1-J [a(1-x2)~] !1-0<x<1 0 1<x<oo Re Jl. > -1, Re v > -1 (29) 0 0 <X< C xX-v(x2-c2)~!1-J [a(x2-c2)~] .u c<x<oo a > 0, Re v > Re Jl. > -1 fo"" f(x) J)xy)(xy) ~ dx 2v-1 t3-n.u r (v) y ~ -v ii J (a .m i = 1 .u 1 n 0 0 }; a.<y<oo i= 1 1 n }; a.<y<oo i= 1 1 n -_u. x II [t3 . 1 J (a. t3 . )] i = 1 1 .U; 1 1 n }; a.<y<oo i= 1 1 0 <r <a a<y<oo 8.12 HANKEL TRANSFOBMS 61 Bessel functions of other arguments (cont'd) f(x) J"" f (x) J (xy )(xy) ~ dx 0 II y>O (30) 0 0 <X< C rrv(c2+f32)~1-Ly ~K [a(c2+f32) ~] /.1. x ~ -v (x z + f3 2) - 1 (x 2 _ c 2) ~ 1-L X Ill(yf3) 0 <y <a x J [a(x2-c2)~] c<x<oo 1-L a> 0, Re {3 > 0 -1 < Re /.1. < 2 + Re v (31) 0 0 <x < c (-l)n+l /3-v({32+c2) ~iJ.+n-l Y~ x ~ -v(x 2+ /3 2)-1 (x2-c 2)~ J..L+n-1 X K)a(f32+c2)~] I)by) x J [a (x 2 -c 2) ~] c<x<oo 0 <y <a 1-L Re {3 > 0 a> 0, -n < Re /.1. < 4-2n +Rev (32) xv+zn+ ~ (1-x2) ~A.+a a-A.y-v+~(~)•( ~n x J1Ja(1-x2)~] 0 <x < 1 ada ydy 0 1<x<oo xla2A.+2• y2v+2n a>O, ReA. >-1, Re v>-l x(a2+y2)-~t\+v+a +n+tl X JA.+v+a +n+l [(a2+y2) ~JI (33) xP(l-x 2)1-L J,\[a (l-x 2) ~] see Bailey, W. N., 1938: Quart. ]. 0 <x < l Math. Oxford Series 9, 141-147• 0 l<x<oo (34) x~ J l%a[(x2+{32)~-{3]1 2"-1 y-~ (a2-y2)- ~ ~~~ xcos[{3(a2-y2)~] 0 < y <a x J ~)%a [(x 2 + {3 2) ~ + {3]1 a> 0, Rev> -l 0 a<y<oo (35) x~ Y~)~ax2) -2a-1 y~ "~v(yz/a) a> 0, Rev> -l (36) x~ J (~ax2) Y (~ax2) -2a-1y~ [J~~~~:~) r ~y " ~~~ " a> 0, Rev> -1 62 INTEGRAL TRANSFORMS 8.12 Bessel fmctions of other argm~ents (cont•d) f(x) Joof(x)J (xyXxy)~dx 0 II y>O (37) x -~ Y (ax-1) -2rr-1 y-~ [K (2a~ y~) II 211 . a > 0, JRevJ < ~ -~rr Y (2a~ y~)] 2J.L (3S) x-!1/2 Y (ax-1) 2y~ a-1rr-1 [K (2a~y~) II 211 a > 0, JRevJ < ~ +~11Y (2a~y~)] 211 (39) x-~ Y (2axl{) 2sec(vrr)y-~~cos(vrr) Y (a2/y) 211 II a> 0, Rev>-~ -Y _)a2/y) + H_)a2/y)] (40) 0 0 <x < c (-l)n+1 {3-lly l{ ({3 2 + C 2) ~ J.L+n-~ x ~ -~~(x 2+ {3 2)-1 (x 2_c 2)~J.L+n-~ x K [a(f32+c2)~] I ({3y) J.L II x Y [a(x2-c2)~] c<x<oo 0 <y <a J.L a> 0, Re {3 > 0 -~-n < Re IJ. < 3-2n + Re v (41) x~ J l~a[(x2+/F)l{-{3]1 2rr-1 y-~ (a 2 -y 2) -~ ~~~ x Y~ 1)~a[(x2+{32)~+{3]1 xsin [{3 (a2-y2)~] 0 <y <a a> 0, Re v> -1 -2rr-1y-~ (y2-a2)-~ x exp [-f3 (y 2 -a 2) ~] a<y<oo (42) x ~ [H (I) (ax 2) H (1) (ax 2) Sf'(~-!J.+ ~ v) f' (~+ !J.+ ~ v) ~ 11+J.L ~ 11-J.L i rr [f' (~ v+ 1)]2 y 312 -H 121 (ax 2) H <2 1 (ax 2)] ~II+J.L ~~~-J.L x M (y2 e~71) Re v > -~ Re (~ ± IJ. + ~ v) > 0 J.L• ~~~ Sa xM c~e-~11) J.L• ~J.. Sa 8.B HANKEL THANSFORl\1S 63 8.13. Modified nessel functions of argument kx f(x) J ""f(x) J (xy )(xy) l{ dx 0 11 y>O 2 yl{ (a2-y) (ay) ( l) xX e-f3x I (ax) 2{3 exp ~ J11 2{3 11 He {3 > 0, Rev> -1 (2) x~ K (ax) y11+~ 11 a11(y2+a2) He a> 0, Rev> -1 (3) x J.L+1I+l{ K (ax) 211+1-Lf'(t-t+v+l) y11+l{ J.L a -J.L(y 2+ a 2)J.L+11+1 He a> 0, Re(v+ 1) > \l1et-t\ (4) x-A-1{ K (ax) Rea> 0 1 [~ (v-.\+ t-t+1)] r [~~ (v-.\-t-t+1)] J.L 2A+1 a11-A+1 [' (v+ 1)y-11-~ Re(v-.\+1)> \Ret-t\ cv-A+t-t+l v->--t-t+l y2) X F ·v+1·--2 1 2 ' 2 ' ' a2 (5) xA K (ax) cos ({3x) see under Fourier transforms J.L sin ({3 x) (6) x ~ K (ax) J ({3x) 0 11 y y, r -1 r-1 (r -r ) 11 (r + r ) -11 1 2 2 1 2 1 Re v > -1, Rea> \Im{3\ r 1 = [a 2 + ({3 -y) 2] ~ r 2 = [a 2 + ({3 + y) 2] X (7) x 11+Y, J (~ax) K (~ax) a211211f'(v+ ~) y11+Y, 11 11 77 X (y .-+ a 4) 11 + y, \arga\ <rr/4, Rev >-~ (8) x11+Y, J (ax) K ({3x) 2 311 (a{3) 11 y 11+Y. [' (v+ ~) 11 11 77X [(a2+ /32+y2) z_4azy2]11+ X Re {3 > \1m a\, Rev>-~ 64 INTEGTIAL THANSFORMS 8.13 Modified Bessel functions of kx (cont'd) f(x) Joo ((x) J (xy)(xy)~ dx 0 'II y>O (9) x11+~ J 11_1 (ax) K 11_1 (ax) 23v-1 a2v-2r(v+Yz) yv+s/2 \arg a\ < 77/4, 0 < Re v < Yz 77~ (y4+ a-c)v+Y, (10) x ~ J (xa sin¢) (sin ¢)tL(sin l/f)11+Y,(cos ¢)11-J.L(cos 1/f)J.L-11 J.L a 312 (1-sin 2¢ sin 2 1/f) xK v-J.L (xa cos¢ cos 1/f) y = a sin 1/f a> 0, 0<¢,1/f<Yz77 Re 11-> -1, Rev> -1 ( ll) xv+Y, J (xa sin ~/f) 2'~~r(J.L+v+1) [sin¢ cos 2(Yz a)] v+~ J.L av+3!2 (cos 1/f) 2vt2 xK 1-L (xa cos¢ cos 1/f) a> 0, 0 < ¢, "'< Yz 77 x P -J.L(cos a) y =a sin¢ ll Rev> -1, Re(J.L + v) > -1 tan (Yz a) = tan 1/f cos ¢ (12) xJ.L+~ J 11({3x) KJ.L(ax) (277)-~ atL[3-u-1 y-J.L-Y, e -(J.L+~ }n-i Rea> \Im {3\ x(u2-1)-~IJ.-\( QJ.L+Y.(u) Rev> -1, Re (11-+ v) > -1 v-Y, 2{3 yu = a 2 + {3 2 + y 2 (13) x -Y, J (xa sin¢) (sin ¢)tL(sin 1/f) v+Y, J.L 2ay, (cos¢ cos 1/f)P x K)xa cos¢ cos 1/f) a> 0, 0 < ¢, "'< Yz 7T X r (Yz (l+ iJ.+ v-p)]r (Yz (1+ J.L+ v+ p)] Re (iJ. + v + 1) > Re p r(1+J.L) r(1+v) F0+1-£+zrp 1+J.rzrp 0 x 2 1 2 ' 2 ; 1-£+1; sin 2 ¢ FC+I-£+-zrp 1+zrJ.rp ) x2 1 2 , 2 ;v+1;sin21/J y a sin 1/f 8.13 HANKEL THANSFORMS 65 Modified Bessel functions of kx (cont'd) f(x) Joo f(x) J (xy)(xy) !1 dx y > 0 0 11 (14) xP+1I-J.L+% J (ax) K (f3x) 2P+11-J.L-1 [f'(J.L+ 1)]-1 r (p+ II+ 1) J.L p xf'(p+ 1)f'(v+ l)aJ.L-p-11-2y~ Re {3 > Jim aJ, Rev> -1 Rep> -1, Re J.L > -1 x (cosh a-cos(}) P -p (cos(}) p+11-J.L Re(p + v) > -1 x PP:~-J.L (cosh a) y + i {3 = i a ctn [~ ((}+ i a)] (15) x"-J)ax) Kp(f3x) see Bailey, W. N., 1936: Proc. London Math. Soc. (2), 40, 37-48. (16) ~ x I%)ax) K%)ax) y-~ (y2+4a2)-!1 Rea> 0, Rev> -1 (17) x11+% I (~ax) K (~ax) 211 a211f'(v+~) 11 11 TT% (y3+a2y)11+!1 Rea> 0, JRevJ<~ (18) x11+% I (ax)K ({3x) 2311(af3)11y11+% f'(v+ ~) 11 11 TT% [({3 2_a 2 +y 2)2 + 4 a2y 2] 11+){ Re f3 >Rea, Rev>-~ (19) x11-Y, Iv-Y,(~ ax) K11_y, (~ax) r ( v) (2 a) 11-1 p [ 2a 2 + y 2 j Rea> 0, 0 <Rev< 3/2 yv-Y, -11 2a(a2+y2)Y, (20) x-y, I (~ax) K (~ax) eJL7Ti['(~v+J.L+X) J.L J.L ['(~ v-J.L+ X) y!1 Rea> 0, Rev> -1 xP -J.L [(1+ a2/y2)X] Re (v + 2J.L) > -1 Y, v-X x <r,I.L [(l+a2/y2)~J y, v-Y, (21) xJ.L+Y, I (X ax) K (~ax) (~ 71fX ~ -1y -J.L-Xe- Ctt-X 11+){ m i 11 J.L x(l+y2/ a2) -v, J.L-!4' QJ.L+X (iy/ a) Rea> 0, Rev> -1 v-V, -Re v-1 < Re J.L < ~ 66 INTEGRAL TRANSFORMS 8.13 Modified Bessel functions of kx (cont'd) f(x) Joo f(x) J (xy)(xy)~ dx 0 IJ y>O (22) x.u+~ I (ax) K ({3x) (217)-~ a -,u.-t f3.Uy-.u-'l. e-(,u.-'1. v+lOrr i IJ .u x(v2+ 1)-'l..u-~ o.u+'l. (iv) Re f3 > IRe al Re 11 > -1, Re (ll + 11) > -1 v-~ 2a yv = f3 2 -a 2 + y 2 (23) X 'I. I'l.(v-.u )(%_ax) K '1. (v+,u.)(%. ax) a-J.Ly-'1. (y2+a2)-'l. [y+(y2+a2)'1.].u Rea> 0, Re 11 > -1 Re (11-ll) >-2 (24) xv+'l. I (f3x) K (ax) (277) -'I, (a{3) -v-I y v+'l. e-(v+)Orr i J.L J.L x(u2-1)-'l.v-~ ov+'l. (u) Rea> IRe /31, Re 11 > -1 J.L-'1. Re (ll + 11) > -1 2a{3 u = a2 + f3 2 + y 2 (25) xA. I (ax) K (f3x) see Bailey, W. N., 1936: Proc. J.L p London Math. Soc. (2) 40, 37-48. (26) x-v-'.1, [Kv+'l. (%_ ax))2 "'I. (2a)-v-l r (-v)yl.l+'l. (a2+y2) 'I. v Rea> 0, -1 < Re 11 < 0 [ Z..'+r' l xP v 2a(a2+y2)'1.j (27) x'I.[K (%_ax)]Z Rea>O e 2!J.7Tiy '1. r (1+%. 11+ ll) J.L (y 2 +a 2) '1. r (%. 11-ll) Re (%_ 11 ± ll) > -1 X Q-!J.[(1+a2/y2)'1.] 'l.v x Q~~-1 [(1+a2/y2)~] (28) x-X [K (%.ax)] 2 Re a> 0 e 2,u.7Ti r(%.+ %.11+ ll) .u r(%.+%.11-/l) y'l. Re(%.11 ±ll) >-%. x IQ~~-'1. [(l+a2/y2)'1.]12 8.14 HANKEL TRANSFOHMS 67 Modified Bessel functions of kx (cont'd) f(x) J"" f(x) J (xy)(xy)~ dx 0 ll y>O (29) xy, K J.L-~ (~ax} KJ.L+'h (~ax) e 2 J.L71 i r (~ v+ J1. + 1) y X r(~v-J.L) (y2 + a2)'h Rea> 0, He v > -1 x Q-J.L+~ [(l+a2/y2)Y.] )ReJ.L) < 1 +~He v y, v-~ x Q~~-=-~ [(1+ a2/y2)Y.] (30) xv+'h K (ax) K ({3x) TTY, y v+Y, r (v+ J.L+ 1) r (v-J.L+ 1) J.L J.L 2 3/2 (a/3) v + 1 (u 2 _ 1)'~~ /2+ 1/-4 He a> 0, He {3 > 0 p-v-'h( ) Re (v ± J.L) > -1, Re v > -1 X J.L-Y, U 2a{3u = y2 + /32 + a2 (31) x"-K (ax} K ({3x} see J3ailey, W. N., 1936: Proc. J.L p London Math. $oc. (2) 41, 215-220. 8.14. Modified Bessel functions of other argwnents ( l) x y, -v exp (-~a 2 x 2) (~ )-X -1 v-Y, ( Y2 j 2 rr a y exp --- x I)~ a2x 2) 4a2 xD_211t) )arg a) < ~ rr, Re v > -~ (2) x-v-3/2 exp(-~a2x2) (Y. )-'h v+Y, ( y2 ) 2 rr y · exp --- x I (~ a2x 2) 4a2 v+1 " (y) )arg a) < ~ rr, Hev>-1 X 0-2v-3--;;- (3) x y, exp (-~ax 2) Iy, 11(~ ax 2) {Y.rray)-X exp (-;: ) Rea> 0, Re v >-l 68 INTF:G llA L THAN SFOH~IS 8.14 Modified Bessel functions (cont'd) [(x) J"" [(x) J (xy)(xy)~dx 0 ll y>O (4) x1113+116 exp (-~ax 2) -1 -v/3-2/3 v/3+1/6 tr2 j rr a y exp -- x Iv/3+1/6 (~ax 2) 4a X K11;3+116(:: ) H.e a> 0, -1 < Re v < 512 (5) x 116-v/3 exp (-~ax 2) av/3...>.213y 1/6-v/3 exp ~ :: ) x Iv/3-1/e(~ ax2) Rea> 0, Rev> -1 X Ill/3-1/6(::) (6) x ~ +2J.1.-v exp (-~ax 2) 22JJ.-v+~ (rra)-l{ f'(~+IL) x I (~ax 2) Re a > 0 X (f' (~-11 + v)r1 Y v-2J.1.-~ J.1. X 1F 0+/L;~-/L+ll;- y2 j Re v>2Re IL+~>-~ 1 2a (7) x~+v-2JJ. exp(- ~ a2x2) TT-~ 2li(3+2v-6J.1.) a-~-v+JJ.yJJ.-1 xi(~a2x2 ) !arga!<~rr xexpGL) w •• a(~~2J J.1. -1 < Re 11 < 2H.e IL + ~ 4a2 2k=~+v-3J.1., 2711=-~+/L-11 x A. exp (-~ a 2 x 2) I J.1. (~ a 2 x 2) (20A.+1 ~211-/L,l+, (8) (2rr)-~ -G21 -23 2 2 !arga! < ~rr Y a h ~ k ' 2, -312-Re (2/L + v) < H.e ,\ < 0 h=%'+~,\+~v k=%'+~,\-~v (9) x ~ K y, ,.,(~ax 2) rra-1 Y ~ [ I~ 11 (y 2 I a) -L ~ )Y 2 I a)] Rea> 0, Rev> -1 (lO) 3/2 K (~ 2) x ~ v+Y, "ax 2rra -2y 3/2 [I~ v-~ (y 21 a) Rea> 0, Rev> -1 -L~11-~(y2la)] 8.14 69 Modified Bessel functions (cont'd) f(x) Joo f(x) J (xy)(xy) ~ dx 0 ll y>O (ll) xll/3+1/6 exp(-~ax2) rra -v/3-2/3 y v/3 + 1/6 exp t Y 2 j x Kv/3+1/6 (~ax 2) 2 4a x Iv/3+1/S~:a) Rea> 0, Rev> -1 (12) xv/3+1/6exp (~ax 2) a-v/3-213ylll3+1!6 expC:) x Kv/3+1/6 (~ax 2) t2) -1 <Rev< 5/2 xK -v/3 + 1/6 4a (13) x 2J.L+v+~ exp (-~a 2 x 2) 17~ 21-La-2W2ll-2yv+~ f'(l+21J.+ v) xK J.L(~ a2x2) largal < ~1T [f' (ll + v+ 3/2)] -I Rev> -1, Re(21J.+v)>-1 ( 3 y2) x1F1 1+21J.+V;IJ.+v+-;--- 2 2a2 (14) x 2J.L+v+~ exp (~a 2 x 2) rr 1/2 f'(1+ 21J.+v) [f'(~-IJ.)rl 23/2-• xK (~ a2x2) xa-2• y-J.L-1 expEy2) w •.• c::1 J.L Rev> -1 Rea> 0, · 4a2 -1 < Re(21J. + v) < -~ 2k=-~-31J.-V, 2m=~+IJ.+V (15) x>--exp(-~a2x2)K (~a2x2) ct G''l'-•. l+J (~rr)~ -G 12 - J.L y 23 2a2 h, ~. k larg al < ~ 1T Re(,\+ v ±21J.) >-3/2 h = ~ + ~ ,\ + ~ v, k=%' +~ ,\-%1/ (16) x>--exp (~ a2x 2) K J.L (~ a2x2) (2rr) -~cos (1J.77)(2/y)A.+I Rea> 0 X c::e: ~1-., 1+") -3/2-Re (v ± 21J.) < Re ,\ < 0 2a h % k ' 2, h=%'+~,\+%v k=~+~,\-~v 70 IN 1EGRAL TRANSFOHMS 8.14 Modified Bessel functions (cont•d) f(x) Joo f(x) J (xy)(xy)~ dx 0 ll y>O x~I!l)~ ax2)K \.{)~ ax2 ) y~ (y2) ~y2 j (17) -I -K -- a !4 11 4a !4 "' 4a Rea> 0, Rev>-1 x ~ I!l (li-J.L )(~ax 2) 2r(~+~~~-~J.L) ~r2) (18) r(1+~v)y312 w~J.L·~~~ ~ x K~ (li+;.J~ ax 2 ) Rea> 0 ~2) Re v>-1, Re(v-J.L) >-2 xM -- -l{J.L.~ll 4a (19) x-512 K (ax-1) ll · _, ~[ y,11.,;K (2 y, !4.,; y,) ta y e 211 a e y Rea> 0, IRe vi < 5/2 -e -y, 11"'; K (2ay, e -!4 .,;y '/,)) 211 (20) -211-2 K ( -1} x y, _11 ax (2rr)y, a-11-y, y11+Y, K 211(2y, ay,yy,) Rea> 0, -~<Re v<2 xJ (2y,ay,yy,) 2ll (21) K_211_ 1 (2axy,) -~ rra sec (vrr) y -312 [H _11_1 (a 2/y) Rea> 0, Rev> -1 -Y_ll_t (a2/y)] (22) x-y, K (2axy,) 2ll ~ rr sec (vrr) y -y, [H -v(a2/y) Rea> 0, Rev>-~ -y _)a2/y)] (23) xy, J 11(2ay,xy,)K 11(2ay,xy,) Re a> 0, Re v > -1 ~y-3/2 e-2a.fy (24) x11+Y, J 211(2a~ xy,)K 211(2ay,xy,) 17 -~ 211 all+'/, y-2v-2 K~ _)2a/y) Rea> 0, Rev>- ~ (25) -y-'.1, J (2 '/, ~) ~ 2-11-2 -v-'.1, 2li(J (2a/ ) X 211+1 a X rr a y y+Y, y ( '/, ~) -Lv+Y, (2a/y)) xK211+1 2a x Rea> 0, Rev> -1 8.14 HANKEL TRANSFORMS 7l Modified Bessel functions (cont•d) f(x) Joo f(x) J (xy)(xy)~ dx 0 Jj y>O (26) x -~ J (2a ~ x l-1) K (2a l-1 x ~) [' (~ + ~ /1 + ~ JJ) y ~ JTI_~J.I,~j.i.(2ya) J.i. J.i. 4 a[' (1 + /1) Rea> 0, Rev> -1 R e ( v + 11) > -1 (2a) xM -Xv,XJ.i. y (27) x-X [K2)2al-lx~) -~17y-~ Yv(a/y) -~17 Y2)2a~x~)] a> 0, Rev>-~ (28) x-~ K (2alfx~) Y (2a~xl-l) Jj Jj -~a-t y ~ W ~ v.~ v (2a/y) Rea> 0, Rev>-~ x W -y, y, (2a/y) 2 lit v (29) x-Y, K (2aXxY.) -~a-1yXWX y, (2a/y) 1-L v, 2j.L x !sin[~ (11-v) 17] JJ.i. (2a~ x ~) X W_l\v,l\J.i.(2a/y) +cos[~(/1-v)rr]Y (2allx~)l J.i. Rea> 0, Re(v ± 11) > -1 (30) x-~K [(2ax)~e~7Ti] ~a-T y~ r(1+;+ V) r(1-;+v~ J.i. xK [(2ax)~e-~7Ti] J.i. x W (ay-t e Y,7Ti) Rea> 0, Re(v ± Jl) > -1 -~ v,Y,J.i. xW ( -te-~7Ti) -~ v,Y, 1-L ay (31) -v-~ K [(2 )~ ~7Ti] x w+t ax e -2-s/2 1T 3/2 esc (v17) a -v-112 y 2 v x K [(2ax) ~ e -~ 1ri] w+t x[Hv+~ (a/y)-Yv+~ [a/y)] Re a > 0, -1 < Re v < 0 (32) x v+~ (x 2 + ,13 2) -y, v-~ 1T~ 2-l-1 a -v-~ y v+~ (a 2 +y 2) -l-1 x K [a(x2+,132)l-l] X e-f3(a2 +y2)l-\ v+~ Re a > 0, Re ,13 > 0 Rev> -1 72 INTEGRAL TRANSFORMS 8.14 Modified Bessel functions (cont•d) f(x) J;,oo f(x) J )xy )(xy) X dx y>O (33) x v+~ (x 2 + ,8 2) -~ v-~ 17112 2-112 a-v-3/2 ,8-1 yv+1/2 X Kv+312[a(x2+,82)~] X e -j3(a2 +y 2)% Rea> 0, Re ,8 > 0 Rev> -1 (34) x v +~ (x 2 +a 2) -~ (v+1 J yv+~ (y2+a2)-~<v+1) x K~(v+1)[a(x2+a2)~] x K~(v+1)[a(y2+a2)~] Rea> 0, Rev> -1 (35) xv+~ (x2+,82)-~f..L a-f..L,8v+1-J..LyV+~ (a2+y2)~J..L-% v-% x K [a(x2+,82)X] Rea >0 x Kf..L-v-1 (/3(a2+y2)X] f..L ... Re /3 > 0, Rev> -1 (36) x v+% (b 2_x 2) X f..L y [a(b2-x 2)X] af..LbJ..L+v+1 yv+X (a2+y 2)-X(u+v+1) f..L x y [b (a2+y2) X] 0 <x < b J..L+v+1 -2rr-1xv+X (x2-b2)X.U x K [a(x2-b2)X] .u b<x<oo Re a>O, Re v>-1, ReJL>-1 (37) xX lx11l~/3[(a2+x2)X -a]l y-X (/32+y2)-% exp [-a(/32+y2)~] x Kx)~/3[(a2+x2)X +all Rea> 0, Re /3 > 0 Rev> -1 8.15. FWlctions related to Bessel functions (l) "v-X(ax) 2 ~ 17 -~ a v-% y X -v (a 2 -y 2)-Y, a> 0, -1 <Rev< 3/2 0 < y <a 0 a<y<oo 8.15 HANKEL TRANSFORMS 73 Functions related to Bessel functions (cont'd) f(x) J00[(x)J (xy)(xy)~dx 0 lJ y>O (2) xAH (ax) j.J. a>O 2A.+~ y-A-t -5/2-Re v < Re (,\ + IL) < 0 ~I-, " "J --1--1+-2 ' 2' 2 c21 x 33 ~+ ,\+v 1-IL ~+ ,\-v 4 2' 2 '4 2 (3) x ~ H ~ )J4 ax 2) -2a-'y~ Yx~.~<r2/a) a> 0, -2 <Rev< 3/2 (4) xA H (a/x) 2A.H y-A-t j.J. Re(,\ + v) > -2 a> 0, I+, J -Re v-5/2 < Re (,\-IL) < 1 ~' 2 xG2' -- 15 1+/L IL 1L h----k ' 2 ' 2' 2' 3 .\+ v 3 ,\-v h =-+-- k=-+-4 2 ' 4 2 (5) x X [H (ax)-Y (ax)] -~.~ -~.~ 2a -~.~ rr _, cos (vrr) y ~.~-~ (y+ a)_, \arga\ < rr, -X! < Re v (6) xA[u (ax) -Y (ax)] j.J. j.J. 2A+X "-2 cos (wr) y-A-t \arga\ < TT, Re (,\ + IL) < 1 ~~1-· " -~ Re (,\ + v) + 3/2 > \Re ILl --1--1+-2 ' 2' 2 xG 23 33 3 + A+v 1-IL ~+A-v 4 .2 ' 2 '4 2 (7) x-X [H (ax-1)-Y (ax-')] -~.~ -~.~ 4rr-1 cos(vrr) y-X K2~.~(2aXyX) \arg a\ < rr, \Rev\< X! 74 INTEGRAL TRANSFORMS 8.15 Functions related to Bessel functions (cont'd) f(x) J"" f(x) J (xy)(xy)~ dx 0 II y>O (8) x-312[8_11_1 (ax-1) -477-1 a-~ cos(V7T) K (2a~y~) -211-1 -y_ll-1 (ax-1)] Jarg a\ < 77, \Rev\<~ (9) x 211[811+~ (ax -1)- y~~+~(ax-1)] -2 s/277-3/2 a11+112y-11-112 sin (v77) \arga\ <77, -l<Re v< -l/6 xK (2~a~e!471iy~) 211+1 xK (2~ a~ e-!471iy~) 211+1 (10) xA. [H (ax-1)-Y (ax-1)] j.J. j.J. 2A.+~ 17-2 cos (~tiT) y -A.-1 \argaj <77, Re ,\ < -\Rell\ ~ 1+• J Re(v-ll + ,\) > -5/2 2 2 2 c·" r X 15 -6- l+IL ll ll h----k ' 2' 2'2' 3 ,\+v 3 ,\-v h =-+-- k =-+--4 2 ' 4 2 (ll) I11_~ (ax)- L11_~ (ax) 2~ 77-~ a~~-~y~-~~(y2+a2)-~ Rea> 0, \Rev\<~ (12) x ~[I (ax)-L (ax)] II II 277-1 a11+1 y-~~-~ (y2+a2)-1 Rea> 0, -1 <Rev<-~ (13) xJ.l.-ll+~ [I (ax) -L (ax)] 2J.1.-11+1aJ.1.-1yii-2J.l.-~ j.J. j.J. 17~ r(v-ll+ ~) Rea> 0 - 1 < 2 Re 11 + 1 < Re v + ~ x 2F1 (1, ~; v-IJ.+ X; -y 2/ a2) (14) x11-J.1.+~ [IJ.l.(ax)- LJ.l.(ax)] 2II-J.1.+1 r (3/2+ V) aj.J..+1 Rea> 0, -1<Rev<-~ 17 rC3/2+ IL) r IIH/2 x 2F, (1, 3/2+ v; 3/2+ p.;-a2 /y 2) 8.15 HANKEL TRANSFORMS 75 Functions related to Bessel functions (cont•d) rex) Joo f(x) J (xy)(xy)~ dx 0 II y>O x v-J.L-~ [I )ax) -L ~(ax)] 211-J.Lr (~ + v) a~ (15) ,.~ r (1+ ~)y v+~ Rea> 0, iRe vi < ~ x2F, (~+v,~;1+~;-a2/y2 ) (16) xA.[I (ax)-L (ax)] Rea>O 2A.+~ -1 -A.-1 J.L ~ " r -Rev- 3/2 < Re(A. + ~) < 0 ~ "1-· "0 1---1+-2' 2 ' 2 G22 x 2 ~+ A.+v 1-~ ~+ A-v 4 2. 2 '4 2 (17) x~ [I)ax)- L_)ax)] 2 ,.-1 a 1-vy v-K cos (vrr)(y 2 + a2)-1 Rea> 0, Re 11 > ~ (18) x~-v+~ [I (ax) -L (ax)] 2p.-v+1 a-p.-1yv-~ ~ -~ rc~- ~) r (~ + v) Re a>O, Rev>-~. Re~>-1 x 2F; (1, ~+IL; ~+v;-y2fa2) (19) x11-~+~ [I (ax)-L (ax)] J.L -p. 2 2 +!1-J.Lrr -3/2 cos (~rr) r (3/2 + v-~) Rea> 0, Rev>-~ x a 1-~y -s/2 +2~-v Re(v-~-t)>-1, Re (v-2~-t) <~ x 2F,(3/2+ v-p.,1; 3/2;-a2/y 2) (20) xJ.L+v-~[I (ax)-L (ax)] 2J.L+vr(~+~+v) ~ · a~y--2p.-2v ~ -p. r u + ~) r c~-~) Rea> 0, -1 < Re v < 3/2 Re (~ + v) > -1/2 x 2F1(~+~+v,~+~; 1+~;a2/y2 ) 76 INTEGRAL TRANSFORMS 8.15 Functions related to Bessel functions (cont'd) f(x) Joo { (x) J (xy )(xy) ~ dx 0 11 y>O (21) x"-[IJi.(ax)- L_Ji.(ax)] 2"-+X 11-1 cos (Jl.rr) y -A.-1 Rea>O, Re(Jl.+v+..\) > -3/2 ~ l+" " " ) -Rev- 5/2 < Re(..\-Jl.) < 1 --1--1+-2 ' 2' 2 xG22-33 ~+ ..\+v 1+u ~+ ..\-v 4 2' 2 '4 2 (22) x211[I (ax-1)-L (ax-1)] 2 3/217-112 y -v-1/2 av+l/2 v+~ v+~ x J2v+1 [(2ay) 1/2]K2v+1[(2ay)'l2] Rea> 0, -1 <Rev<~ (23) x11+~u (2a2{3 ax} v+l ' (2tn v+l y v+~ cos [{3 (a 2 -y 2)] a> 0, Rev> -1 O<y<a 0 a<y<oo (24) xv+~u (2a2{3 ax) v+z ' (2{3) v+l y v+~ sin [{3 (a 2-y 2)] a> 0, Rev> -1 0 <y <a 0 a<y<oo 8.16. Parabolic cylinder functions (l) xv-X exp(-~x2)D2v-1 (x} -~sec(v77')yv-Y, exp(-~y2) Rev>-~ x[D 2v-1 (y)-D 2v-1 (-y)] (2) xv-X exp(-~x2) yv-X exp(-~yz) xl [l-2cos(VTT)] Dzv-l (x) xl [1-2cosCvrr)] D2v-l (y) -D2v-1 (-x)l Rev>-~ -D2v-1 (-y)l (3) xv-X exp(-~x2) -yv-Y, exp(-~y2) xI [l + 2cos(vrr)] D211_1 (x) xI [l + 2cos(VTT)] D2v-l (y) -D 2v-1 (-x)l Rev>-~ -D2v-1 (-y)l 8.]6 HANKEL THANSI•OHMS 77 Parabolic cylinder functions (cont'd) f(x) J 00 f (x) J (xy Xxy) ~ dx 0 v y>O (4) x v-~ exp (~ x 2) D2v-l (x) 2Y,-v77 sin(vrr) f'(2v) yY,-v -~~ < Re v < Y~ x exp(~y2 ) Kv(~y2 ) (5) xv-~ exp(-~ x 2) D 2 v+l (x) Xsec(vrr) exp(-~ y2) yv+Y, Rev> -X X (D2)y)+D2)-y)] (6) xv-Y, exp(-~x2) yv+Y, exp(-~y2) x I [l + 2 cos ( vrr)] D 2 v + 1 (x) xl[1 + 2cos(vrr)]D2v(x) -D zv+l (-x )! Rev> -X + D 2)-x)l (7) xv-Y, exp(-~x2) -yv+Y, exp(-~y2) xI [1-2cos(vrr)] Dzv+l (x) xl [1-2cos(vrr)] D 2)y) -D 2v+1 (-x)! Rev> -X + D 2)-y)! (8) xv-Y, exp(- ~x2)D_2v(x) 2-Y, 77y, YY,-v exp(-~y2) Jv(~y2) Rev> -X (9) x v-Y, exp (~ x 2) D _2v (x) Yv-Y, exp(~y2)D_2v(y) Rev> -X (lO) xv-Y, exp(~x2)D_2v_2(x) (2v+1)-l Yv+Y, exp(~y2)D_2v-l(y) Rev>-X (ll) xv-Y, exp(-~a2x2)D2)ax) 2J.L-Y, f' (v + X) y v+Y, [' ( v-11 + 1 ) a I + 2 v largal < ~ rr, Rev> -X ( 1 y2 j X F v+-· v-11+1·--- I I 2' ' 2a2 78 INTEGRAL TRANSFOHMS 8.16 Parabolic cylinder functions (cont'd) f(x) Joo f(x) J (xy)(xy ·)~dx 0 11 y>O x11-~ exp(~a2x2)D2,/ax) r(%+v) a2k 2"+).L exp c::2~ (12) r (%-11) r).L+1 Jargaj < ~77 (y2) -%<Rev< Re(%-211) xW --k,• 2a2 2 k = Yz + 11-v, 2m=Yz+I1+V (13) x 11+~ exp (-~ x 2) D 211 (x) %sec(vrr)y11-~ exp(-~y2) Rev> -1 xW211+1 (y)-0211+1 (-y)] (14) x11+Y,exp(-~x2) y11-Y, exp (-~y2) xI [1 + 2cos(vrr)] D211(x) xl [1 + 2cos (vrr)] D 211+1 (y) +0211(-x)l Rev> -1 -D 211+1 (-y)l (15) x11+~ exp(-~x2) -y 11-~ exp (-~ y 2) xI [1-2 cos (vrr)] D 2)x) xI [1-2cos (vrr)] D 211+1 (y) + D2)-x)l Rev> -1 -D 211+1 (-y)l (16) x11+~ exp (-~ x 2) D 211+2 (x) -Yzsec (vrr) y11+Y, ex-p(-~ y2) Rev> -1 X [D 211+2 (y) + D 211+2 (-y.)] (17) xv+Y, exp(-~x2) y11+~ exp(-~y2) x 1[1-2 cos (VTT)] D211+2 (x) xI [1-2cos(vrr)] D211+2 (y) + D211+2(-x)l Re v > -1 + D 211+2 (-y)! (18) x"+~ exp(-~x2) -y11+Y, exp(-~y2) xI [1 + 2cos (v77)] D 211+2 (x) X I [1 + 2 cos (v17 )] D 2 11+2 (y) + D 211+2 (-x)! Rev> -1 + D 211+2 (-y)! (19) x11+~exp(~x2)D211+2(x) rr -1 sin (v17 )r (2 v+ 3) y -11-312 -1 < Re v < -5/6 xexp(~y2)K11+ 1 (~y2) 8.16 HANKEL TRANSFORMS 79 Parabolic cylinder functions (cont'd) f(x) Joo f(x.) J (xy) (xy )X dx 0 v y>O (20) xv+Y, exp(~x2)D_2v_ 1(x) (2 v+1) yv-V, exp (~y2) D _2v_2 (y) Rev>-%; (21) xv+Xexp(- ~x2 ) D -2v-3 (x) 2-112 rr112 y-v-3/2exp(-~y2) Rev> -1 X Iv+1 (~ y2) (22) xv+X exp(~xz)D_zv-3(x) Yv+X exp (~y2) D -zv-3 (y) Rev> -1 (23) x v+X exp (-~ a2 x2) D 2,/ax) 2''T(v+3 /2) yv+X [' (v-f1+3/2) a2v+2 largal < ~rr, Rev> -1 x F (v+~· V-f1+~·- y2 :1 1 1 2' 2' 2 a2 xv+X exp(~a2x2)D2Jl.(ax) f'(3/2+v) 2X+m+J1. a2k+1 (24) [' (-/1). y J1. +3/2 larg al <% rr x exp ( -s) W k ( L_) -1 < Re v <-Y2-2 Re f1 4a •" 2a2 2k = !1-v-1, 2m=f1+v+1 (25) x>--exp(-~a2x2)D (ax) 2-c-3v/2 7T1/2 f'(2b) Yv+1/2 J1. f' (v+ 1) f' (c + %;) a2b larg al < ~ 7T c 1 1 y2 ) Re(..\ + v) > -3/2 x F b b+-· v+1 c+-· ----2 2 ' 2' ' 2' 2a2 2b=A+v+3 /2, 2c=A-f1+3 /2 80 INTEGRAL TRANSFORMS 8.16 Parabolic cylinder functions (cont'd) {(x) (26) xA_exp(~a2x2) D)ax) largal<~rr Re fl. <-Re >..<Rev+ 312 (28) -X D ( ~7Ti -X) x -X -v ae x D ( -~7Ti -~) x -~-vae x Rea>O, Rev>-~ y>O 2-v"l{ [r(v+~·m-1 Y-v-l{ x(a2+2y)-X [(a2+2y)X-a]211 2~ "x [1(v+~)r1 Y-1 x exp[-a(2y)X] 8.17. Gauss' hypergeometric function (l) X 2a+v-X ii(~+a) i(~+a+v) y -v-3/2 x 2F1 (a-v-~.a; 2 a;->.. 2x2) 17 21-v-2a >.. 2a-1 Rev<-~, Re >..> 0 X W X-a, -~ -v (y 1>..) Re(a+v)>-~ X [W X -a • -X -v (e -i 7T y I>..) W. (e i7T I>..)] -X-a, -X -v Y x2a-v-X 22a-vr(~+a) Yv-3/2 (2) >..2a 1 1 (2 v) x 2F1(v+a-~, a; 2a;-A.2x2) Rea>-~, Rev>~ xAh-x,v-~(yi>..)W%-a,v-X (yl>..) Re >.. > 0 8.17 HANKEL TRANSFORMS 81 Gauss' hypergeometric function (cont'd) f(x) J"" f(x) J (xy Hxy))l; dx 0 11 y>O x 11+)1; 2F1 (a, {3; v+ 1; -A 2x 2) 211-a-{3+2 f'(v+1) Ya+/3-11-3/2 (3) A a+/3 f' (a) f' ({3) -1 <Rev < 2 max (Re a, Re {3)-3/2 x Ka-ir!A) ReA> 0 (4) x v+X 2F,[a, {3; %({3 + v)+1; -A2x 2] f'(%{3+%v+1) yf3-l<; rr)l; ['(a) 1({3) 2/::H A11+f3+1 -1 <Rev x [K)I;w-{3+1) (:A) r < 2max(Re a, Re{3)-3/2 (5) 11+X F ( {3 A 2 2) 211+1 f'(y) Y-11-3/2 X 2 1 a, ; y;- X ['(a) [' ({3) -1 <Rev G' I y J < 2 max (Rea, Re {3)-3/2 c3o Y ReA> 0 x 13 4A2v+1,a,f3 (6) xs-x F (a f3·y--A2x2) 28 f'(y) -8-X 2 1 ' ' ' ['(a) [' ({3) y -Rev- 1 <Reo < 2max (Rea, Re {3)-% ~' l,y ~ 31 y ReA> 0 xGu -2 1+o+v 1+8-v A 2 'a, {3, 2 (7) X -2a-3/2 A-2ayX I)l;11+a(Ay) K)/;11-a(Ay) (1 4A2) x F -+a, 1+a; 1+2a;--2 1 2 x2 Rev> -1, ReA> 0 Rea>-% (8) X 11-4a+X f'(v) 211A 1-20. 2a-11-)l; I(% A ) x 2F, (a, a+%; v+ 1;-A 2x -2) f'(2a) y 11 2 y Re a-1<Re v<4Re a-3/2 xK2a_11_1l}2Ay) ReA> 0 82 IN 1EGRAL TRANSFORMS 8.17 Gauss' hypergeometric function (cont'd) f(x) foe f (x) J (xy Hxy )~ dx 0 v y>O (9) 8 -x F ( (3 A 2 -2) 2 8 r (y) y-8-~ X 2 I a, ; y;- X r (a) r ((3) -1-Rev-2min(Re a, Re,B) uC'r' 1-a,l-~ ~ <Reo<-~ xG24 --ReA> 0 4 1+o+v 1+8-v ---,0,1-y, 2 2 (10) xv+J{ (1+x)-2a r (v+ 1) r (v-a+ 1) 2 2v-2a+1 r (a) [ 1 4x J x 2F, a,v+2;2v+l;(l+x) 2 xy2a-2v-3/2 J (x) v -1 < Re v < 2Re a-3/2 8 .18. Confluent hypergeometric functions (l) x-1 exp(e. ~x '-.) (2v+ l) Tv y v-X Erfc(~y) xM~ v-\L ~ v+~ (x 2) Rev>-~ (2) x -312 exp (-~ x 2·) r (v+ 2) y v+~ Erfc (~y) xM X v+X,X v+X (x 2) r (v+ 3/2) 2v Rev> -l (3) x 2JL-v-~ exp (-~ x 2) y 2JL-v-X exp (-~ y 2) xM3JL-v+~ ,JL (~ x 2) xM 3 JL-v+~, JL (~ y 2) Re ll >-~. Re(41l-v)>-~ (4) x"'-2JL-~ exp(-~x2) r<211+ 1) yv-~ exp (-~y2) x M (~x 2) 2v-JLr (v+ ~) v-JL, JL 2 Rev>-~ X D 2 v-4 JL (y) 8,18 HANKEL TRANSFORMS 83 Confluent hypergeometric functions (cont'd) (5) (6) (7) (8) (9) f(x) x v-2!-L-X exp (-~ x 2) x M v-ll+1o!l ()i:;x2) x-A-1 exp(-~x2) Re v > -1 x MA+ (~~x2 ) !loll ..\=2f1-V-)i:; -1<Rev<4Ref1 x-Y, exp(-~x2) x MK I/ ()i:;x2) '12 v Rev> -1, x v-21-L-y, exp (-~4 x 2) X MK e~x2 ) oil Re K < ~ -1 <He v < 2Re(K+f1)- ~~ (10) x2p-Y,exp(-~ {ax2)MKo!l(ax2) -1-Re (~~ v+ f1)<Re p<R e K-~4 Rea>O J ""f(x) J (xy )(xy )X dx 0 v y>O 77-X 2-5(u+v/3) f'(2f1+ 1) y"-+zll f'(4f1-V) x exp (-~ y2) KA(~ y 2) 2-Kf'(v+1) 2K-l/ y /2 l (K + ~ V + ~) . x exp(- ~y2 ) [' (2 f1+ 1) 2 Y, (X -K+3!l-v) ['(~~+K-f1+ v) y 1-K+!l x exp (-~i y 2) M Cto,B(~ y 2) 2a=~+K+3f1-V 2f3=-Y 2+K-f1+V ')Y,(Y,-K-3!-L+V) f'(2f1 + 1) YK+Il-1 ~ [' (f1+ K+ )!:;) xexp(-~y2 ) wa..,B(~y2 ) 2a=K-3f1+V+~ 2{3=K+f1-V-~ 84 INTEGRAL TRANSFORMS 8.18 Confluent hypergeometric functions (cont'd) f(x) J000 f(x) J )xy)(xy )~ dx y>O (ll) x v-2.u-% exp (-J4' x 2) 77!12v-a,uy4,u-v-!1 exp(-J4'y2) X W3,u-v-!-\,±)~x2) X I ()4' 2) v-2,u+Y, 'l Y Re.v>-1, Re(v- 2/J.) > -1 (12) x v-2.u-% exp ()4' x 2) 77-!-1 2v-a.u f'(1+v-2!J.) y4,u-v-Y, X W (~ 2) f'(4!J.-V) v-a,u+~. ±.u 2X x exp(J4'y2) Kv-2,u+~ (J4'y2) Rev> -1, Re (v-2!J.) > -1 Re (3 J)-8!J.) <-3/2 (13) x v-2!-'--Y, exp ()4' x 2) y v-2,u-~ exp ()4' y 2) xW (~x2 ) 3,u-v-X,±.u 2 X W (~ 2) a,u-v-X,±.u 2Y Rev>-1, Re (v-2/J.) > -1 Re(v- 4!J.) >-~ (14) x v-2,u-Y, exp (-J4' x 2) f'(1+v-2/J.) 2[3-.u YK+,u-1 [' (1 + 2 {3) X W K ± (~X 2) • .u xexp(-J4'y2) Ma,f3(~y2 ) Rev> -1, Re(v- 2!J.) > -1 2a=~+K+V-3!J. 2{3=~-K+V-!J. (15) x v-2.u-% exp ()4' x 2) f'(1+v-2!J.) 2 Y, (Y, +K-3 ,u+v) y,u-K-1 ['(~+IL-K) x WK ± (~x2 ) • .u x exp ()4' y 2) W a,f3 (~y 2) Rev>-1, Re(v-2/J.).>-1 Re (K -IL + ~ v) <-J4' 2a=K+3~J.-v-~ 2{3=K-!J.+V+~ 8.18 HANKEL TRANSFORMS 85 Confluent hypergeometl'ic functions (cont'd) f(x) J''" f (x·) J (xy )(xy) ~ dx 0 v y > 0 16) x2p-~ exp(-~ax2) r(l + J.L+v/2+p)r (1-J.L+v/2+p) xWK (ax2) Rea>O r(v+1) r(3/2-K+v/2+p) ,J.L Re (p ± 11 + ~ v) > -1 x2-v-1 a-~ v-p-~ Yv+~ xF~ 1 y2 ) 2 2 A+/1 >..-w v+1 --K+A·--' ' '2 ' 4a A=1+~v+p' (17) x 2p-~ exp (~ax 2) 22py-2p-~ r<~+11-K) r<~-11-K) xWK,J.L(ax2) iarg ai < TT G' %-•• %+" ) -1 -Re (~ v ± 11) xG22 - <Rep <-~-ReK 23 a ~+p+~v,-K,~+p- ~v (18) x -~ M K, )\v (-iax) M_K .~v(-iax) ae-~<v+lmi [r(1+v)J2 y-~-2K r(~+K+~v) r(~-K+~v) a> 0, Rev>-1, IReKi <~ x(a2 _ y2)-~l[a+(a2-y2)){]2 K + [a-(a 2-r2)J{J2Kl 0 < y <a 0 a<y<oo (19) x-~ M (ax) r(v+1) -J.L-~ -~J.L . ~v aro~-~ll+ ~v) y X W~ Y, (ax•) Rev> -1 J.1.1 2V x[a +(a2+y2)J{]J.i.(a2+y2)-~ Re 11 < ~. Rea> 0 (20) 2J.L-v- ~ W ( )M ( ) x K,J.L ax -K,J.L ax 22J.i.-v+2 K a2K Yv-2J.L-2 K-){ Re 11 > -~, Rea> 0 r<211+1) Re(2J.L+2K-v) <~ X r(v-K-!J. + ~) X 3F2 (~-K, 1-K, ~-K+ 11; 1 - 2 K, ~ -K-11 + v; -y 2/ a 2) INTEGRAL TRANSFORMS 8.18 Confluent hypergeometric functions (coot 'd) f(x) J: f(x) J)xy)(xy)l{ dx y>O (21) x2p-v-s/2 W (ax) 22p-v-2 f'(2fL+1) Yv-2p+3/2 K,JJ-1Tl{ ['(~-K+ fL) x M-K,)ax) Rea> 0 0' ~.o.~-,.1<:+, ) Rep> 0, Re (p + fL) > 0 xG23 - 44 2 Re(2p + 2K-v) < 5/2 a p-~. -K, K, p-v-~~ (22) x 2p-v-sl2 W (ax) f'Cp+ fL) [' (p-fL) ['(2p) K,!J-f'(~+ K+ p)f'(%-K+p)f' (l+ v) x W -K,)ax) X 2-v-1 a 1-2p Yv+l{ Re p > IRe fLI, Rea> 0 X 4F3 (p, p + ~. p+ fL, p-fL; ~+ K+ p, ~-K+ p, 1 +v; -y2/ a2·) (23) 2p~v-5/2 W ( · ) 22p-v-2 Yv-2p+3/2 x K,JJ-tax 1T112f'(%-K+fL) ['(~-K-fL) xWK (-iax) Rea> 0 ,j.k ~' 14,0,~-•• ~+. J Rep> IRefLI Re(2p + 2K-v) < 5/2 xG24 - 44 2 I ~ p-~,-K,K,p-v- 2 (24) X-312M -JJ-, ~)~x 2) f'(l+~v)yl{ I (1 2 K (1 2) (~ I ) ',4v-j.k ~y ) ',4V+j.k ~y xW li (~x2 ) Rev> -1 l 2+~ V-fL j.k, v (25) -312 M (~ 2) 1(1+~ v-2y) -312M J~ 2) X a-{3,'4 v-y 2X ['( l+ ~ v-2 ,B) Y a-y.~v- 2Y X Wa+f3,\\ v+y (~x 2) X Wa+y, ~'Y+f3(~y2) Re {3 < 1/8, Re v > -1 Re(v-4y)>-2 (26) xl{ M~V,!J-(2/x)W_~v.)2/x) 4f'(l+2fL) y-l{ J (2y~)K (2y~) -[' (~ + ~ v+ p) 21J- 21J- Re v > -1, Re fL > -~ 8.19 HANKEL TRANSFORMS 87 Confluent hypergeometric functions (cont'd) f(x) fo"" f(x) J)xy)(xy)y, dx y>O (27) xy, WY,v,IL(2/x)W_Y,v,)2/x) -4y-X I sin [(Jl- ~2 v)rr] c'z)2yX) R e (v ± 2 11) > -1 +cos [(Jl-~ v) rr] Y 2fL (2y y, )} K2)2yX) (28) x Y, W _ Y, . (i a/ X) v,,.._ 4ay-x [f' (~+ Jl+ ~ v)!(~- J1+ ~v)r1 X JI'I-Y,v,i./-i a/x) Rea> 0 xK [(2i ay)X] K [(-2i ay)Yz] J.L IL IRe ILl<~. Rev> -1 (29) x-y,M -J.L, Y,vl a [({3 2+x 2)Y,-{3]! a 1(1+v) [(a2+y2)y, +aJ21L [' (~+ ~ v-IL} y Y, +211-(a 2+y 2)X xW ~ la[({32+x2)X+{3]l j.L, v x exp[-,B(a2+y2)X] Re v>-1, Re IL < )4 Rea> 0, Re {3 > 0 8.19. Generalized hypergeometric series and miscellaneous functions xv+Y, 1F, (2 a-v; a+1;- ~x 2) 2v-a+Y, 1(a+1) Y 2a-v-Y, (l) rry, 1(2a-v) Rev> -1, Re(4a- 3v)>~ xexp(-J4y2 ) Ka-v-Y, ()4y2) (2) a-Y, ( 1+a+v x2) x F a·---·--1 1 , 2 , 2 a-Y, (. l+a+v._ y2 ) y , F1 a, 2 , -2- Rea >-~. Re(a + v) > -1 xv+X-2a F (a·l+v-a ·-~x2) rrX!(l+v-a) Y 2a-v-Y, (3) 1 1 ' ' 2 2a-v-Y, ['(a) Re a-1 < Re v < 4 Re a-~ xexp(-~y2 ) Ia-Y, (~y2 ) 88 INTEGRAL TRANSFORMS 8.19 Miscellaneous functions (cont'd) f(x) J oo f(x) J (xy )(xy )~ dx 0 v y>O xv+~ 1F, (a;(3; -Ax2) 2 1 -a [' ({3) ~ Y 2 j (4) a-3/2 ex f' (a) A~ a+~ v Y P -8A -1 < Re v < 2 Re a-~ ReA> 0 ~2j xW -- K,j.L 4A 2K=a-2{3+v+2 2/l=a-v-1 (5) x2f3-v-3/2 F( ·(3·-A 2) 22{3-2a-v-1 [' ({3) Y2a-2f3+v+~ 1 t a, ' x ['(a -{3 + v + 1) A a 0 < Re (3 < ~ + Re(a + ~v) ReA >0 X F ( •. l+a-/l+v ·-r') 1 1 ' ' 4A (6) x2p-~ F(a·{3·-Ax2) 2 2p [' ((3) 1 1 ' , ['(a) y 2p+~ -1-Rev < 2Rep < ~ + 2Rea ReA> 0 G' I '· ll ) c21 x 23 A ~+p+~v,a,~+p-~v (7) xv+~ 1F2(a;(3,v+1;-~x2) f'(v+ 1) f'((3) 2v+1 Y2a-v-3/2 Rev> -1, Re (3 >Rea> 0 f' (a) f' ((3 -a) X (1-y2) f3-a-1 0<y<1 0 1<y<oo (8) xv+~ (1-x2)11--v-1 2 211--v-1 a -1-L[' (1 + fL) [' (ll-v )y v+~ x 1F2[fL+~; 2fL+1, ll-v;-a(l-x2)] xJ [~(y2+4a2)~+~y] 1-L 0 <x < 1 xJ [~(y2+4a2)~-~y] 0 1<x<oo 1-L Re fL > Re v > -1 8.19 HANKEL TRANSFORMS 89 Miscellaneous fWICtions (cont'd) f(x) Jo"" f(x) J11(xy)(xy)~ dx y >0 (9) x v+!/2 (l-X 2)J.L-3/2 2-J.L-2v-1 r ( ) -11--v v+~ u (4 ) 11 a Y v+11-a,y ~ 11 11+1 2( 21 x F 1·---·-a 1-x 1 2 '2' 2 ' 0<x<1 0 1<x<oo Re v > -1, Re 11 > 0 (10) xP-~ (-l)m Yp-~ G p+m+1 x2F2 p, 2 ; c p+m+1 X 2F2 p, 2 ; p -m +1 p+ v+1 . .:..~.-) 2 ' 2 ' 2 p-m+1 p+v+1. _y2) 2 ' 2 ' 2 Re(p+ v+ 1) > 0, Rep>~ Re(p-m+ 1) > 0 (ll) X v+~ 211-2a+2 r (a+.~) r (v+ 1) x 3F2(a,a+f3,a-{3; 17~ r (a) r (a+ {3) r (a-{3) a+~, v+ 1;-x 2) X Y -2a-v-3/2 [K ,a<~ Y )F Re a > ~ Re v + ~ >-~ JRe(3J <Rea- ~Rev-~ (12) xv+~ 2 v+3-2.B r (2{3) yz.B-v-3/2 x 3F2 ({3 + p, {3-p, 2 {3-v-1; r({3+p) r({3-p) r(2{3-v-1) (3,(3+ ~;-~ xz) xKv+!-.B+p (y)K11+1 -.B-p (y) IRepJ <Re{3-~Rev-~ Rev>-1, Re(4{3-3 v)>5/2 90 INTEGRAL TRANSFORMS 8.19 Miscellaneous functions (cont'd) f(x) J 000 f(x) J )xy )(xy) ~ dx y>O (13) xv+~ f'(a+ ~+K) f' (a+ ~-K) 2v+1 x 3F2(a, a+~' 2a-v-1; r (2 a) r (2 a -v -l) a+ ~+K, a+ ~-K;-x 2) xy2a-v-s /2 W ( ) -1<Rev<2Rea-l/2 K,a-v-1 Y Re (4 a-3 v) > 5/2 xW_K,a-v-1 (y) (14) xv+~ f'(v+K+l) f'(v-K+1) 771/221-v X 3F2(v+~,v+ ~+/l,V+ ~-/l; f'(v+ ~) f'(v+ ~+/l) f'(v+ ~-/l) V+1+K, v+1-K;- X 2) X Y v-3/2 W ( ) W ( ) Rev >-~ K,!J. y -K,!J. y \Re/.L\ <~Rev+~ (15) X v+~ c 3 f'(v+K+3/2) f'(v-K+3 /2)rr~ 2-v x 3F2 v+ 2"' v+1+/l, v+1-/l; r (v+3/2) r (v + 1 + /.L)f'(v+ 1-/.L) V+~ + K, v+~ -K'-X~ xyv-~ WK,!J.(y) W-K,!J.(y) 2 2 ' Rev> -1 \Re/.L\ <~Rev+ %' (16) xv+X 2v-4a+3 f'(2a) f'(v+ 1) Y2a-v-3/2 X 4F3 (a+ f3, a-f3, a+ y, a-y; r (a+ f3) r (a-f3) r (a-y) r (a+ y) a,a+~,v+1;-x2) X K f3+y (~y) K f3-y (~y) -1 < Re v < 2Re a-~ \Ref3\ <Rea- ~Rev-~ \Rey\ <Rea- ~Rev-~ 8.19 HANKEL TRANSFORMS 91 Miscellaneous fuoctions (cont'd) f(x) Joo f(x) J (xy)(xy)~ dx 0 v y>O (17) xv+~ r(~+a+K) r(~+a-K) r(v+1) x 4F3 (a, a+~. a+ p., a-11; r (2 a) r (a+ p.) r (a-p.) ~+a+K,~+a-K;-x2 ) x2v+1 Y2a-v-s/2 W (y)W_ (y) -1 < Re v < 2 Re a -~ K,J1. K,J1. \Rep.\< Rea- ~Rev-~ (18) x2p-~ r <13 1) .•• r <13) 2 2p r -2p-~ x PFP(a1, ••• , ap; r (a ) ••• r (a ) 1 p f3 1' • • • ' f3 p; -,\X 2) ~' ,,.~, ..... ~ ) -1-Rev <2 Rep< ~+2 Rear xcp+1o1 - p p+1, p +2 4A ReA>O, r = 1, ••• , p h, a 1, ••. , a p' k h = ~ + p + ~ v, k=~+p-~v (19) x2p-~ 2 2Pr <13 1) ••• r<f3.;) r(a ) ••• r(a +) y2p+~ x 11 + 1 F,. (a 1' ••• , a • + 1 ; 1 • 1 f3 1' ••• ' f3. ; -A 2 X 2) 1.~ , ..... ~.) Re(2p+ v)>-1, (f' ReA> 0 xc•+2,1 - Re(p-a)<~, r= 1, ••• , m+ 1 • +1,a +3 4,\2 h,a1, ••• ,a.+1,k h = ~ + p + ~ v, k=~+p-~v x'P-• C;; (Ax'l a,' ... a,) 22P (20) y2p+~ 13, ... '/3q p+q<2(m+n) (4,\ lh,a1, ••• ,a ,k) \argA\ < (m + n-~p- ~q)rr xc•,n+1 - p p+2,q 2 Re(f3 1+ p.+ ~v) > -~ y f31, ••• ,f3q 1 = 1, ••• , m Re(a +p)< %, } j = 1, •.• , n h = ~-p-~ v, k=~-p+~ll 92 INTEGRAL TRANSFORMS 8.19 Mise ell aoeous functions ( cont 'd) f(x) J"" f (x) J (xy) (xy) ){ dx 0 v y>O (21) xv+zn-1 [B(a+x,a-x)]-1 0 rr:Sy<oo -1<Rev<2a-2n-7/2 (22) x11+~ Erfc(ax) -v [' (v+ 3/2) -3/2 ( y2 ) largal < ~ rr, Rev> -1 a r(v+2l r( r,') -8,;' xM -- ~ v+~.~v+l{ 4az (23) x v-~ Erfc (ax) -~ l{-vf'(v+l/2) -1 ( yz) 2 a y exp --- I arg al< ~ rr, Rev>-~ f'(v+3/2) 8a2 ( y2j xM -- ~v-!C~v+J( 4az (24) x -Ji.-~ s (x) v+JL.-v+JL+1 2v-1 [' (v) Y ){ -v (1-y z.)Ji. Rep. >-1, -1 <Rev< 3/2 0 <y < 1 0 1<y<oo CHAPTER IX Y-TRANSFORMS We call [' )f(x); yl = J: f(x) Y ,)xy) (xy )y, dx the ¥-transform of order v of f(x) and regard y as a positive real variable. The inversion formula 9.1 (l) has been given by Titchmarsh (1937, P• 215). The reciprocal transform is the H-transform (see chapter XI). From the transform pairs given in this chapter further transform pairs may be derived by means of the methods indicated in the introduction to volume I, and also by the general formulas of sec. 9.1. Moreover, ¥­ transforms and Hankel transforms are connected by the relation ~ 1,\[(x ); yl = ctn (vrr) S;? vlf(x ); yl -esc (vrr) S;? -vlf(x); y I which is an immediate consequence of the relation between Bessel functions of the first and second kind and may be used to evaluate Y­ transfonns by means of the table of Hankel transforms given in chapter VIII. REFERENCE Titchmarsh, E.C., 1937: Introduction to the theory of Fourier integrals, Oxford. 93 Y-TRANSFORMS 9.1. General formulas f(x) Joo f(x) Y (xy) (xy)y, dx 0 v = g (y; v) y>O 00 g(y) Hv(xy)(xy)y, dy ( l) Jo g (y) -~2 < Re v < ~2 (2) [(ax) a>O a-1g(a-1y;v) (3) X m [(x) m = 0, l, 2, ••• YY,-v -- [yv-Y.+m g(y; v+ m)J (d)'" y dy ( t) x'" f(x) m = O, l, 2, ••• (-l)'" YY,+v (-d-)'" y dy x [y -Y.:+m -v g (y; v-m)J (5) 2 V X-! [ (x) yg(y; v-l)+yg(y; v+ l) (6) 2 v ['(x) (v-X)yg(y; v+ l) -(v+ ~2)yg (y; v-l) 95 96 JNTF:GRAL TfiANSFO!lMS 9.1 General formulas (cont'd) f(x) I: [(x) Yv(xy)(xy)~ dx = g (y; v) y>O (7) x~-v ( ~ )m x dx y" g (y; v + m) x[xv+.-~ f(x)] m = O, l, 2, ... (R) x~+v (-d)" x rl-r. (-y)'" g {y; v-m) X lx" -v-~ f(x )] m = 0, l, 2, ,,, (9) x~ -v Jx .;v-J..L+~ (x2 _ .;2)1-L-1 2~-'_, n11> r -J..L g <r; v-11> 0 x r<.;> r1.; fie v + 3/2 > Re 11 > 0 (lO) x -J..L f(x) 2 , -J..L [I , (11)]-, y v + ~ He11>0, He v > -3/2 j"" X-J..L-v( 2 2)J..L-1 X y TJ TJ -y X g(Tj; V + 11) dTJ 9, 2, Algebraic functions and powers with an arbitrary index ((x) I: ((x) Yv(xy)(xy) y,dx y>O ( 1) x-y, -l<Hev<l -tan n~ vrr) y -~ 9.2 Y-TRANSFORMS 97 Algebraic functions (cont'd) f(x) J: f (x) Y )xy ) (xy) ){ dx y>O (2) 0 0 <x <a v+ 1 -){ y ( ) -a y v+l ay xl{ +v a<x<oo Rev<-~ (3) xi-L IRevl-3/2<11<0 2J.L+Y, ctn [~(v + ~ -11) rr] y-1-L-I ['(% + ~j) + ~11) X r e~ + ~ v -~ 11) (4) 0 O<x<a ay -1-L[Y v-I (ay) SJ.L+Y,,)ay) xi-L a<x<oo -(11+V- ~) Y)ay) Re 11 < 0 x S 1-L-Y,, 11_1 (ay)] (5) x-l{ (x + a)-1 rr y ){ csc(vrr) [E)ay)+ Y 11(ay) J larg al < rr, -l<Hev<l + 2 ctn (vrr) [J)ay)-J)ay)] j) I= 0, ± ~ (6) xv-l{ (x + a)-1 I arg al < 77 2v+1 -1 v Y.r( l) - 77 a y v+ -l/2<Rev<3/2 X S -v-1, v (ay) (7) x -v-Y, (x + a)-1 larg al < 77 a -v y ){I~ 77 tan (JJ7T)[Y 11 (ay) -3/2 <Rev< l/2 -Hv(ay)]- 21-v 77 -1 cos(vrr) X r (l-v) Sv-l,v(ay)j 98 INTEGRAL TRANSFORMS 9.2 Algebraic functions (cont'd) f(x) J"" f(x) Y (xy)(xy) X dx 0 v y>O (8) x~-'--X(x+a)-1 largal<rr (2a)J.Lrr-1yX (9) Re (fL ± v) > -1, Re fL < 3/2 xlsin[Jr2rr(jl-v)]1( ~+ ~fL+ ~ v) X l (~+ ~:!jl-~ v) S_J.L, )ay) -1 <Rev< 1 -2cos [~rr(fL-v)] 1(1+ ~fL+ ~v) x 1 ( 1 + ~fL -~ v) S _ J.L _1 , v (a y) l rryX lctn(vrr) [Y v(ay) + E)ay)] + Jv(ay) + 2[ctn(vrr)f x [Jv(ay)- J)ay)J! The integral is a Cauchy Principal Value. (10) xv-X (x-a)-1 avyX[rrJv(ay)-2v+1 rr-1 f'(v+ 1) x S -v-1 'v (ay) J a > 0, - 1/2 < Re v < 3/2 The integral is a Cauchy Principal Value. ( ll) x -v-X (x -a) -1 a -v y X I ~ 7T tan ( vrr) [H v (ay) a> 0, -3/2 <Rev< 1/2 -Y)ay)] + rr J)ay) -21-vrr-1 cos(vrr)1(1-v) X Sv-1,V(ay)} The integral is a Cauchy Principal Value. 9.2 Y-TRANSF'OflMS 99 Algebraic functions (cont'd) f(x) J: f(x) Y)xy)(xy)X dx y>O (12) xJ.L-X (x-a)-1 a>O rraJ.LyX J)ay)-(2a)J.Lrr-1 yX Re(l1 ±v)>-1, Re 11 > 3/2 X I sin [~17(11-v)J r(~+ ~11+ ~iv) X r(~+ ~11-~v) 5 -J.L, v(ay) + 2 COS [ ~ 1T <11-V) J r ( 1 + ~11 + ~V) X r (l+ ~11-~v) 5 _J.L_1, v(ay)} The integral is a Cauchy Principal Value. X [-2"a tan(; )lv(ay) (13) x-X (x2 + a2)-1 y I' Rea> 0, -1 < Re v < 1 COS 12V1T 1 y sin (~vrr) --K (ay)+ a v 1-v 2 ( 3-v X 1F2 1;-2-, ~-~] 2 ' 1 (14) xv-X (x2 + a2)-1 -av-1 yy, K)ay) Rea> 0, -l/2 < Re v < 5/2 (15) xv+3/2(x2 + a2)-1 v+1 X ( a y Kv ay) Rea> 0, -3/2 <Rev< 1/2 (16) x-v-Y,(x2 + a2)-1 Rea> 0 a-v-1 yy, ~~~TT tan(vrr) [L)ay) -5/2 < Re v < l/2 -I)ay)]- sec (vrr) K )ay)! 100 INTEGRAL TRANSFORMS 9.2 Algebraic functions (cont'd) f(x) (17) x.u-312(x2 + a2)-1 He a> 0 IHe vi< He J1 < 7/2 (18) x-x(x2 + a2)-x Re a > 0, -1 < Re v < 1 (19) xl{+v(x2 + a2).U Rea> 0 -1 < He v <- 2 Re J1 (20) xl{-v(x2 + a2).U Rea> 0 ~ + 2 Re J1 < He v < 1 J~ f(x) Y)xy)(xy) l{ dx y>O -~~rra.u-2 y 112 csc[~rr(Jl+v)] x ctn [~~ rr(Jl-v)] I)ay) -a.u-2 y 112 esc [~27T(J1-v)] Kv(ay) -rr -1 y l{ sec ( ~ vrr) K X v ( ~ a y) x [Kx)~ay) + rrsin( ~vrr) x Ixv<~ay)] 2v-1 77-1 a2,u+2(1 + J1)-1['(v) YX-v x 1F2(1; 1-v, 2+J1; ~4a2y2 ) -2.u a.u+v+1 (sin vrr)-1 [' (Jl+ 1) x y -x -.u [I.u+v+1 (ay) - 2 cos (Jlrr) K ,u+v+1 (ay)J 2.U ,u-v+1 -1{-,ul -1 ( ) a y TT cos vrr xf'(Jl+ 1) f'(v) "Z-J.L-1 (ay) -2csc (vrr)[['(-Jl)]-1 ~-.u- 1(ay) l a2.u+2 ctn (vrr) yX +v -2v+l (Jl+ 1) [' (v+ 1) 9.2 Y-TRANSFORMS 101 Algebraic functions (cont'd) f(x) (21) x -~ (x 2 -a 2) _, a > 0, -1 < Re v < 1 (22) xv-X(x2-a2}-t a > 0, -1/2 < Re v < 5/2 (23) xv+~(x2-a2)-' a> 0, -1 <Rev< 3/2 (24) x-v-~(x2-a2)-1 a> 0, -5/2 < Re v < l/2 (25) xJ.L-3/2 (x2-a2}-' a > 0, IRe vi < Re fl < 7/2 y>O ~1Ta-1 y~ [Jv(ay} + tan(~v77) I tan(~V7T)[J)ay} -J)ay)]- E)ay)- Y)ay)}J The integral is a Cauchy Principal Value. The integral is a Cauchy Principal Value. ~17 av y~ Jv(ay} -2v+t 17 -t f'(v + 1) av y~ X S -v-t, v (ay) The integral is a Cauchy Principal Value. ~1T a-v-t y~ sec (V7T} [J_v(ay} + sin (V7T) H)ay)] The integral is a Cauchy Principal Value ~17 ai-L-2 y~ J)ay} + 2J.L-t "_, ai-L-2 y~ cos[~7T(fL-v)] ( fl-1/) ( fl + 1/ ) xf' --r --s, (ay} 2 2 -J.L,V The integral is a Cauchy Principal Value 102 INTEGRAL TRANSFORMS 9.2 Algebraic functions (cont'd) f(x) J 00 f (x) Y (xy) (xy) ~ dx 0 II y>O (26) x-~(a2 -x2)-~ O<x<a ~77Y-~ Jo(~ay) Yo(~ay) 0 a<x<oo v=O (27) 0 0 <x <a ~ 11y~ I [J~)~ay)f x-~(x2 -a2)-~ a<x<oo -[Y~11(~ay)fl (28) xv+~ (a2- x2)-~ O<x<a (~77)~ a11+~ csc(v77) 0 a<x<oo x [cos (v77) J11+~ (ay) Rev> -1 -"-~~-~ (y)J (29) 0 0 < x <a (~ ) ~ v+~ J ( ) 211 a v+~ ay x v+~ (x2-a 2)-~ a<x<oo Rev<~ (30) x~ -v(a 2-x2)-~ O<x<a (~ 11)~ a~ -vi ctn (V77)[H 11_~(ay) 0 a<x<oo -Yv-~ (ay)]- Jv-~ (y)! Rev< l (31) X~~-~ (a 2 -X 2)11-~ O<x<a 211-177 ~ a211y~-~~r(v+ ~) 0 a<x<oo x J)~ay) Y11(~ay) Rev >-~ 9.2 Y-TRANSFORMS 103 Algebraic functions (cont'd) f(x) J000 f(x) Y )xy) (xy )y, dx y>O (32) 0 0 <x <a 2v-2 17Y, a 211 Yy, -v f'(v + ~) x v-Y, (x 2 _ a 2)v-Y, a<x<oo x [J 11 ( ~ ay) J _) ~ ay) -~<Rev<~ -Y)~ay) f_11(~ay)] (33) 0 0 <x <a 211 17-Y,y-Y,-vf'(v+~) sin(ay) x Y, -v(x2 -a 2)v-Y, a<x<oo -~<Rev<~ (34) 0 0 < x <a 2-v-2 Y, -2v v+Y, ['(~ ) 11 a y 2-v x -v-Y, (x 2 _a 2)-v-Y, x I[J11 (~ayW- [Y )~ay)]Z! a<x<oo -~<Rev<~ (35) x v+Y, (a 2-x 2)JL O<x<a a1L+v+1 Y -JL-Y, [21L [' (11+ l) 0 a<x<oo x YJL+v+1(ay)+ 211+177-1f'(v+l) Re 11 > -1, Rev> -l X SJL-v,JL+v+1 (ay)] (36) 0 O<x<a -21L aJL+v+ 1 y -JL-Y, [' (11 + l) x v+Y, (x 2 -a 2)JL a<x<oo x [sin(/117) JJL+v+1 (ay) -2 < 2 Re 11 <-~-Re v +cos (/117) YJL+v+1 (ay)] 104 INTEGRAL TRANSFORMS 9.2 Algebnic functions (cont'd) f(x) J: f(x) Y)xy)(xy)~ dx y>O (37) x:.>-v(a2- x2)J.L O<x<a J.L-v+1 -J.L-~ [21-v -1 ( ) a y 7T CQS 117T 0 a<x<oo x [' ( 1-11) s J.L +v , J.L -v + 1 ( ay) Re Jl > -1, Re 11 < 1 -2J.1. esc (11rr) f'(Jl+ 1) X J J.L-v+ 1 (ay )] (38) 0 O<x<a 2J.L aJ.L-v+1 y -J.L-:0 l~ (J1 + 1) x:.>-v(x2- a2)J.L a<x<oo x Y v-J.L-1 (ay) -1 < Re Jl < ~2Re 11-~ (39) x2n+v+4J.L-:O (x4 + a4)-J.L-1 see Watson, G.N., 1922: Bessel Functions, P• 432, Cambridge. (40) x -:.> (x 2 +a 2)-~ a J.L y ~ [ c tn ( 11rr) I~ J.L + y, v ( ~ a y) x [(x2 + a2)~-x]J.L x K~J.L-~)~ay) Rea> 0, Re Jl > -3/2 -csc(11rr) Iy,J.L-~V(~ay) -1 < Re 11 < 1 x K~ J.L+:O)~ay)J (41) x-~(x2 + a2)-~ -2rr-1 a!Ly~ cos(~Jlrr) x I [(x 2 + a 2) ~ + x ]!L x [K~J.L(~ay )f + [(x 2 + a2)y, -xJ!L! v=O -3/2 < Re Jl < 3/2 9.3 Y-TRANSFORMS 105 Algebraic fimctions (cont'd) f(x) J'; f(x) Y11(xy)(xy)~ dx y>O (42) x-~(x2 + a2)-~ -1 -~ w ( ) -a y -k.~vay x [(x2 + a2)~-a]2k {rc~ + ~v + k) Rea> 0, IRe vi< X+Re k x tan [(~v -k )rr] r (v + l) x Mk, ~)ay) +sec [(~v-k) rr] xwk.~)ay)} (43) 0 0 < x <a ~ 1T af.l. y~ [J~ v+~)~ay) x-~(x2-a2)-~ x J~ v-~ )~ ay)- Y~ v+~)~ ay) x l[x + (x 2 -a 2 )~ ]f.l. x Y~~~-~f.l.(~ay)] + [x-(x2-a 2)~]f.l.} a<x<oo -3/2 < Re p. < 3/2 9.3. Other elementary functions (l) -~ -ax x e y ~ (y 2 + a2) -~ esc (vrr) Rea> 0, -l < Re v < l xly11[(y2 + a2)~ + ar11cos(vrr) -y-v[(y2 + a2)~ + a]vl (2) xf.l.-3/2 e -ax -2rr-1 r(p.+v)y~ (y2+a2)-~f.l. a> 0, Re p. > IRe vi o-v [ ( 2 2)-~] x f.l._1 a y +a 106 (3) (4) INTEGRAL TRANSFORMS 9.3 Elementary ftmctions (cont'd) -~ -ax 2 x e f(x) Re a> 0, -1 < Re v < l !1--~ -ax 2 x e Rea>O, Re11>IRev1-1 Joo f(x) Y (xy)(xy)~ dx y > 0 0 v +rr-1sec VTT K~ (~)] 2 v Sa x exp (-;: ) X {[' (7f + Y2/l + 7fv) ------sin [7f(v-11)rr] f'(1+v) ( y2 ) xM, , -- ~!1-. Y,v 4a (5) x -312 e -a/ x (6) x -~ (x 2 + {3 2) -~ x exp [-a (x 2 + {3 2) -~ J Rea>O, Ref3>0 -1 <Rev < 1 -y~ sec(7fvrr) xKY,v l7f{3[(y2 + a2)y, + a]J x (rr-1 K~)7f{3[(y2 + a2)~-a]J + sin(7fvrr) X Iy,vl7f{3[(y2 + a2)~-am 9.3 Y-TRANSFORMS 107 Elementary functions (cont'd) f(x) J~ f(x) Yv(xy)(xy)X dx y>O (7) x -~ sin ax2 a> 0, -3 <Rev< 3 (ll; ) ( y z ) (y z ll -1 ) x Jy, ---sin --+ --rr v Ba Ba 4 (8) -~ z x cos ax a> 0, -1 < Re v < 1 (9) x -x (a z -x z) -X ~rry~ Y~v1Xa[(y2 + b2)~ + bJ! X sin [b (a 2 ~X 2) X) O<x<a -~rryX J~vlll:;a[(y2+b2)~ -bJ! -x-X (xz _ az)-X x exp[-b2(x2 -a2)~J a<x<oo b > 0, -1 < Re v < 1 For other transfonns containing trigonometric functions see the tables of Fourier transforms, 108 INTEGHAL TRANSFORMS 9.4 9.4. Higher transcendental functions f(x) J 0"" f (x) Y )xy )(xy) X dx y>O (l) x l{ P ( 1 -2x 2) n 0 <X< 1 77-1 Y-:4 [S2n+1 (y)+TTY2n+1 (y)] 0 1<x<oo n = 0, 1, 2, ••• , v=O (2) v-2J1.+2n+3/2 ( 2)r( 2) r (3/2 -,1+ v+n) r (3/2-J.L+ n) X exp X J.L, X (-1)" l n integer r x r n-J.L) Re (v-11 + n) >-3/2 X exp ( :2) WJL-Xv-n-1,Xv (Y42 ) Re(-J.L+n) >-3/2 Rev<X-2n X (3) 0 O<x<a ( ~) [cos (~2ay) J (~2ay) 2y v p v-X (a -1 x) a<x<oo -sin (X ay) Y)Xay)] Rev< X (4) 0 0 < x <a 2-3/2 7T 1/2 a 1-Jl. YJl. [Jv(X ay) -Jl.( 2 2)-x J1. pJL ( 1 ) x X -a v-X X a xJJL_X(Xay)-Y)Xay) a<x<oo xYJL_X(Y:!ay)] -~<ReJ.L <1 Re(2J.L- v) >-X (5) 0 0 < x <a X v-2 X -v[ (1 7T 2 ay JJL+X Xay) (x2-a~ Xv-)c( pX-v(2a-2x2-1) J1. x J_Jl._'f. (Xay)-YJL+X (Xay) a<x<oo x Y -Jl.-:4 (Xay)] Rev >-~~ Rev+ \2Re 11 + 1\ < 3/2 9.4 Y-TRANSFORMS 109 Higher transcendental functions (cont'd) f(x) Joo f(x) Y (xy) (xy )~ dx 0 v y>O (6) xf....J (ax) see under Mellin transforms !.L (7) sin ax J v+~ (ax) (rr cos e)-~ (2a sine)-1 a > 0, Rev> -3/2 X COS ((v + 1) e) y = 2a cos e, O<e<~rr 0 2a < y < oo (8) xv+~ [Jv(ax)]2 0 0 < y < 2a a> 0, -~<Rev <~ 23v+1 a 2v -v-~( 2 4 2)-v-~ 7T ~ [' (~ -v) y y -a 2a < y < oo ~ [r~v(;~) (9) ~ 2 y x J~v(ax ) -4a a > 0, Rev> -1 -tan ( : ) J ~ v (; ~ ) +sec(:) "-~v (;: )] ( 2) (10) 5/2 J ( 2) -2 ~ y x ~v-~ ax a y J~v+~ ~ a > 0, Rev> -3/2 110 INTEGRAL TRANSFORMS 9.4 Higher transcendental functions (cont'd) f(x) J "" f (x) Y (xy) (xy) X dx 0 v y>O X 2 ( 2) yX sec(~ vrr) ~ (ll) x J !4 v (ax ) J -~ v ax [l + 2 cos(XVTT)] l6a a> 0, -2 < Re v < 2 X [J!4V c~:a) r +2sin(~V7T) x J~v (~2 a) Y!4v (~2 a ) -[ y ~ v ~~62 a ) T } (12) x -x J (a 2 x-, ) v Y-x [Y 2)2ayx) a> 0, -l/2 < Re v < 3/2 + 2rr-1 K2)2ayx)] (13) X-5/2 J (a 2 X-1) v a-2 YX [Y2)2ayx) a > 0, -Yz < Re v < Yz - 2 TT-1 K 2) 2 ay X ) ] (14) x-x Y (a2 x-1) v -Y -x J2)2ayx) a> 0, -X< Re v < Yz (15) x-s/2 y (a2 x_,) v -a-2 Yx J2v(2ayx) a> 0, -~2 < Re v < X (16) x-3/2 y (a2 x_,) v+l _, J ( X -a 2v+1 2ay ) a > 0, -3/2 < Re v < l/2 9.4 Y-TRANSFORMS Higher tcanscendental functions (cont'd) f(x) a> 0, Rev >-~ (18) x-l{ J2v(axl{) a> 0, Rev >-~ (19) x-l{ Y2)axl{) (20) a > 0, -~ < Re v < ~ xv+2n-l{ (x2 +A 2)-1 x(x2+a2)-l{J.L xJJ.L[b(x2+a2)l{] b>O Re A > 0, n = 0, l, 2, ... -~-n < Re11 < 3-2n+Rev (21) 0 xv+l{ (x2-az)l{J.L xJJ.L[b(x2-a2)l{] a<x<oo b>O -1 < Re 11 <-Rev Joo f(x) Y (xy)(xy)l{ dx y > 0 0 v a ( a2 ~ ---H --2y 3/2 v-1 4Y l{ a ( 2 ) -y-B -­ v 4y ~y-l{ Gec(vrr) J_v c::) + csc(vrr) "-v ( :: ) -2 ctn(2vrr) Hv ( :: J J (-l)n+1 Av+Zn-1 Yl{ Kv (Ay) x (A 2-a2)-l{J.L I [b(A 2-a2)l{] J.L y>b X (b 2 _ y2)-l{(J.I.+v+1) x KJ.L+v+1 [a(bz-yz)l{] O<y<b 111 112 INTEGRAL TRANSFORMS 9.4 • Higher transcendental functions (cont'd) f(x) J 00 r (x) y (xy) (xy) l{ dx 0 v y>O (22) 0 O<x<a J.L+v+l bi-L v+l{ ( 2 b 2)-l{(J.L+v+t) -a y y - xv+l{ (x2 _ a2)XJ.L [ 2 2 X xlsin(fL77)JJ.L+v+t a(y -b)] x J [b (x 2 -a 2)X] a < x < oo ).L + cos(fL77) YJ.L+v+t [a(y2~b2) X]l b>O b<y<oo -1 < Re fL <-Rev (23) 0 O<x<a _277-t aJ.L-v+l bi-LyX-v xY,-v(x2- a2)l{J.L X (b2 _ y2)-X(J.L-v+t) xJ [b(x2-a2)l{ ).L a<x<oo x K J.L-v+ 1 [a (b 2 -y 2) l{] O<y<b b > 0, - 1 < Re Jl < Re v (24) 0 0 <x <a a!-L-v+J bi-Lyl{-v(y2 _ b2)X(v-J.L-I) xl{-v(x2- a2)l{J.L X Y v-J.L-1 [a(y2- b2)] b<y<oo x J)b (x2-a2)l{] a<x<oo b > 0, - 1 < Re Jl < Re v (25) l{ 2 77Yy, [ ( 2 ) x Kl{)ax ) ~ csc(v77) L-l{v :a Rea> 0, -1 < Re v < 1 -ctn(v77) Ll{v( :: ) -tan ( 12 1/77) I l{ ( ~) v 4a -77-1 sec ( 12 1/77) K y, v ( :: ) J 9.4 Y-TRAN SFORMS 113 Higher transcendental functions (cont'd) f(x) ] 000 f(x) Y 1,(xy) (xy )~ dx y>O ( 2) ( 2) X-~ exp(~ax2 ) K0(~ax2 ) I -~ ~ y y (26) -~77 a y exp ~ K0 a-;;- v=O x -~ -2J.1. exp (~ax 2) aj.J. 17~ [r (~-2J.LW (27) - r~r(l-2J.L) x K (~ax 2) J.1. x exp ( ~) W 2J.1. 0 ( _c_) 1/ = 0, -%<ReJ.L <~ 8a ' 4a (28) x-~ K (ax-1) ll -2y- ~ [sin (3 V7T/2) ker 211(2 a~ y~) Rea> 0, -~<Rev < ~ + cos(3V7T /2) kei211(2a~ y~)] (29) x-512 K (ax-1) ll 2a-1 y~[sin(377v /2)kei211(2a~y ~) Rea> 0, -5/2 <Rev< 5/2 -cos(37TV /2) ker211(2a~ y~)] (30) -2vK ( -1) x v-~ ax (277)~ a~-v y11-~ Y 211_ 1[(2ay)~] Rea> 0, Rev> l/6 x K 211_1 [(2 ay )~] (31) -2v-2 K ( -1) x v-~ ax (277)~ a-~-11y~+11Y 2)(2ay)~J Rea> 0, Rev >-~ x K2)(2ay)~] (32) 2v-2 K ( -1) x v+~ ax ( ~ 77) ~ c s c ( v 77) a v-~ y ~ -v Rea> 0, Rev<~ x K 2)(2ay)~] IJ2)(2ay)~] -J_211[(2ay)~]l 114 INTEGRAL TRANSFORMS 9.4 Higher transcendental functions (cont'd) f(x) Joo f(x) Y (xy)(xy)~ dx 0 v y>O (33) x-~ [K (a2 x-1)f jJ. 2rry -~ [cos (llrr) J 21} 2 ay l{) largal < ~4 1T -sin(llrr) Y2JJ.(2ay~)] -~ < Re !L < ~. v=O (34) -~ K ( ~) -~ rry -~[sec (vrr) J -v ( :: J x 2v ax Rea> 0, -~<Rev<~ -esc (vrr) "-v ( :: ) + 2 csc(2vrr) Hv c:: ) J (35) 11-~ ( ~ ( ~) x J211-1 ax ) K 211-1 ax 2-11-1 1T~ a211-1 y-211 csc(vrr) largal<~rr, Rev> 0 x [Ly,-11(;;)-111_ ~(8] (36) x-y, H v-1 (ax) · 11-1 Y, -v -a y, 0 < y <a a> 0, -h <He v < !~ 0 a<y<oo (37) x11-JJ.+Y, H (ax) 211+1-JJ. Y 11+~ (a 2 _ y 2 )IJ. -11-1 jJ. aJJ. r(IL-v) a> 0, Re11>Hev -3/ 2 < n e v < l/ 2 0 <y <a 0 a<y<oo (38) -s/2 S ( 2 -1) 2-v-2 -2 Y, K ( ~) x _11_3 11 a x Re a>O rr a y · 211 2 ay '-3/2 <Rev< l/2 x [i(v + 2)r1 9.4 Y-TRANSFORMS 115 Higher transcendental functions (cont'd) f(x) j000 f(x) Y)xy) (xy)~ dx y>O (39) xv-~ exp(!4a2 x2) -rr-1 2~v+~ a-v y-~ r(v + l) X D ~v-~ (ax) X exp ( ::2 ) W-~ v-~ ~ v ( Y 2 2 ) larg al <% rr ' 2a -l/2 < Re v < 2/3 (40) ( -~) ( -~) D v-~ ax D -v-~ ax y-1 exp(-ay ~) largal < ~ 7T x sin [ay~-~(v-~) rrJ (41) xv-.. exp(- ~x2 ) (-1)111 2~v-~111 r(3/2-m) (~y2) /\ X MK' )( -~m (~X 2) m integer r(3/4 + K-m/2) R e (2 K -v) > -m ~-l x exp (-~y2) W a,,B(~y2) Rev> m-3/2 a= K/2 + m/4 + v/2 + 5/8 f3 = K/2 + m/4-v/2-3/8 ,\ = K/2 + m/ 4 -5/8 (42) x-m exp(-~x2) (-l)m r(v-m + 3/2) 2-~m (~ y 2)/\ xMK,~v-~ .. +)( (~x2) r (K + v/2-m/2 + 3/4) m integer, 2ReK>-m2-l X exp(-~y2 ) Wa,J3(~y2 ) Rev> m-3/2 a= K/2-3m/4 + v/4 + 5/8 f3 = K/2 + m/4 + v/4-3/8 ,\ = K/2 + m/4-v/4-5/8 116 INTEGRAL TRANSFORMS 9.4 Higher transcendental functions (cont'd) f(x) f" f(x) Y (xy)(xy)y,dx 0 v y>O (43) x21-L+v-Y, exp(-)ix2) 17-l 21-L+f3yK-!-L -I t(2j.t + l) x MK (~x2 ) ./-L X l (~-jL-K) exp (-)i y 2) - l < 2 Re IL < R e (2 K-v) + ~ { t(2j.t+V+ l) Re (21-l + v) > -1 X COS ( 2j.t 77) l(j.t+V-K+3/2) x M a,j3(~y2 ) +sin [(j.t-K)77] X Wa,f3 (~y2)} 2a=3jL+V+K+~ 2{3=j.t+V-K+~ (44) 2/-L-v-Y, ( )i 2) x exp -~.x -1 21-L+/3 K-/-L-1 ( )i 2) 77 y exp -~ y x MK (~x2 ) ./-L x r (21-l + l) r <~ -K -fL) -l < 2 Re IL < Re (2 K + v) + ~ { r (21-l -v -1) Re(2j.t-v)>-l x cos[(v-2j.t)77] r (2 f3 + l) x M 13(~y2)-sin [(v + K-j.t)17J X W ::(3(~y21 2a=3j.t-V+K+~ 2{3=j.t-V-K+~ (45) x2,\ exp(-)ix2)MK (~x2) 2,\1(2j.t+ l) ./-L l'(~+K+j.t) Re(K-,\)> 0 Re(2,\+ 2j.t ±v)>-5/2 X G ~~ ( ~ I -IL -,\, IL -,\, l ) 2 h, k, K-A-~' { h = )i + ~~ v, k=)i-~v l=-)i-~v 9.4 Y-TRANSFORMS 117 Higher transcendental functions (cont'd) f(x) (46) x2A. exp(-~~x2 ) WK,p_e ~x2 ) Re (2A ± 211 ± v) > -5/2 (47) x2A. exp(~x2 ) WK (~x2) ,J-L Tie (K +A)< 0 Re(2A±211 ±v)>-5/2 (48) X)i W )iv,)2/x) W_)i II,J-Lt2/x) -~ < Re 11 < ~ (50) XV+3/2 Rea> 0 -3/2 < Re v < -l/2 x 2F1(1,2v+3/2;v+2; -a2x2) He a > 0, -!-2 < R e v < J!2 J 00 f (x) Y (xy) (xy) ~ dx 0 v y>O C22 ,--,,_.' (y2~-u-A u-A l) X 34 -2-h, k, K -A-~:;. l h = ~ + ~ v, k = ~4 -~ v l=-~- Y:!v h = ~ + Y:! v, k = ~ -Y:! v l=-~-~v 4y-)i K 2)2y)i) x Ieos [(11- ~~ v) rrJ J2)2y ~) -sin [(11-%v) rrJ Y2J-L(2y ~)l -Y, 2-v -zv-3 r (v + 2) 77 ' a r (2v + 3/2) xyv+Y.[K,.(:a )T 118 INTEGRAL TRANSFORMS 9.4 Higher transcendental functions (cont'd) f(x) (51) X v+3/2 X 2F, (l,/1+v+3 /2;3/2;-a2x2) Rea> 0, -3/2 <Rev< 1/2 Re(211+ v)>-3/2 (52) xa 2F,(a, {3;y;-/..2x2) Re.\>0, Rea> lite vi-3/2 Rea< 2Rea, Rea< 2Hef3 x PFP_1 (al' •.• , ap; f3 " ••• 'f3 p-1; -Ax 2) larg "-I < rr, Rea> IRe vi Rea.>~Rea- ~ J 1(11 + v + 3/2) x y~+v+X K P (: ) ,\-a-1 f'(y) 2x r (a)[' ({3) y>O C41 (y21l-p, y-p, l ) X 35 -2- 1,,\ h,k,a-p,f3-p,l h = ~ + Y:!v, k = ~ -~ v l =-~ -~ v, p = ~ + ~a j = l, ... 'p a f3j = f3 j-2' j = l, ... ' p -l v h =-, 2 v k=--2 ' l+ v l=---2 9.4 Y-TRANSFORMS 119 Higher transcendental functions (cont'd) f(x) (54) :xa-3/2 x PF/a 1, ••• ,a P; f3" ... ,(3P; -,\x 2) Re,\>0, Rea>IRevl Re a.> Y2Re a-% } j=l, ••• ,p (55) XU-3/2 (56) x F(a., ... ,a; (31, ••• ,(3 ; p q • p q -,\x2) p _$ q -l, Re a> IRe vi p + q < 2(m + n) larg,\1 < (m + n-).fp-~q)rr Rea. <l /·-1 n } -' ... ' Re((3 ±Y~v )>-% } j = l, ... , m J"" f(x) Y (xy)(xy)X dx 0 v y>O [' ({3 ) ••• :[' ({3 ) ' p [' (a ) • • • ['(a ) ' p a a (3* = l --' a*= a.-- 0 2 J } 2 v h =-, 2 v k=--2 ' j = l, ... 'p l+v l = -----2 ( a+v a-v x +2F a" ... ,a , --,--· p q p 2 2 ' 4,\ \ f3 " ••• ,(3 q; -TJ h = ~ + ~ v, k = ~ -~ v l=-~-~v ¢ CHAPTER X K-TRANSFORMS We call the K-transform of order v of f(x) and regard y as a con.plex variable. This transformation was introduced by C.S. Meijer (1940) who gave the inversion formula 10.1 (1) and representation theorems: the transformation was further investigated by Boas (l942a, 1942b) and Erdelyi (1950-51). By virtue of the connection between Bessel functions of the first and second kinds, and the modified Bessel function of the third kind K 11, the ~11 transfonn may be expressed as a linear combination of any two of the transforms S;> 11, Sj _11, t'11, g' _11• However, the variable y in the Hankel and Y -transforms occurring in these expressions is negative, and very few J!ankel or Y-transforms converge for negative (or complex) values of y. Conversely, 77 &;?11l{(x); yl = eY,i (1I+Y,)7T S'i: 11l[(x); iyl + e -Y,i (11+Y,m ~111f(x); -iyl 77 [\11l{(x); yl =-e Y,i (1I+Y,)7T ~ 11l{(x); iyl -e -Y,i (1I-Y,)7T ft 11lf(x);-iyl, and these relations enable us to evaluate Hankel and Y-transforms by means ofatable ofK-transforms, although in many cases the K-transforms involved are to be taken on the boundary of the half-plane of convergence, and additional restrictions on the parameters must be introduced to 121 122 INTEGRAL Til.ANSFOHMS secure convergence. If v = ± ~. the K-transfonn reduces to the Laplace transform, sr±!{ lf(x); yl = (~rr) y, 1_5!f(x); yl and the above relations become the expressions of Fourier's sine and cosine transforms in terms of Laplace integrals. From the transform pairs given in this chapter further transform pairs may be derived by means of the methods indicated in the introduction to volume I, and also by the general formulas of sec. 10.1. The connection with the Laplace transformation in either of the two forms H.e v >-~ Rev >-~ may be used to evaluate K-transforms by means of the tables of Laplace transforms given in chapter IV. K-TRANSFORMS REFERENCES Boas, R.P., 1942a: Proc. Nat. Acad. Sci. U.S.A. 28, 21-24. Boas, R.P., 1942b: Bull. Amer. Math. Soc. 48, 286-294. Erd~lyi, Arthur, 1950-51: Rend. Sem. Mat. Univ. Torino 10, 217-234. Meijer, C.S., 1940: Proc. Amsterdam Akad. Wet. 43, 599-608 and 702-711. 123 K-TRANSFORMS 10.1. General formulas f(x) J 00 f (x) K (xy) (xy) l<2 dx 0 v = g(y; v) l f ~ :00 g (y) ~(xy )(xy) v, dy g (y) (l) - rn (2) f(ax) a> 0 -I g (y/a; v) a (3) x" f(x) m = 0, l, 2, ..• l<;-v ~ _.:__) • [y"'"-'g (y; v+m )] y ydy (4) x" f(x) m = 0, l, 2, •.. ( d). ~ +v m-v-1 y --- [y l<>g(y;v-m)] ydy (5) 2vx-1 f(x) yg(y; v+ l)-yg(y; v-l) (6) x _, f(x) Y v+V. Joo TJ -v-V, g (7]; v + l) d7J y 125 126 INTEGRAL TRANSFORMS 10.1 General formulas (cont'd) f(x) f 000 [(x) K)xy)(xy)~ dx =g(y; v) (7) x-1-Lf(x) Re /1-> 0 21-J-L[['(JL)r1 r"+~ Joo ~-1-L-v( 2 2))-L-1 X y Tf Tf -y X g(ry; V + JL) dry (8) 2 v ['(x) (v-~)yg(y;v+l) + (v + ~) y g(y; v-l) (9) ·'-"( ~)" x dx y'" g (y; v + m) X [xm+v-~ f(x)J m = 0, l, 2, ••• (10) •''{-~-)" x dx y" g (y; v-m) x [x m-v-~ f(x )] m = 0, l, 2, ••• (ll) x~-v Jx .;v-J.L+~ (x2 _ .;2)!-L-1 0 211--1 f'(JL)y-1-Lg(y; v-JL) x[(.;)d.; Re /1-> 0 10.2 K-TRANSFORMS 127 10.2. Elementary functions f(x) f : f (x) K v (xy) (xy) ~ dx (1) X p-1 He p >!Rev!- )'2 2p-3/2 -pr(P v l) y -+-+-2 2 4 xI ( £_-~+2_) 2 2 4 Re y > 0 (2) 0 0 <x <a v+1 -Y. K ( ) a y v+1 ay Re y > 0 X v+Y. a<x<oo (3) 0 O<x<a -~ -o--Y. 710" i[ ( ay e Kv_1 ay)So-+1,v(iay) x" +Y. a<x<oo + i (v +a) K)ay)S o-, v-1 (iay)J Re y > 0 (4) xJ.L-Y. (x + a)-1 largal < rr 2M-2 I c~ + ~) 1 (~ -~ j y ~ -J.L Re /1-> I nevi -l ( 11 v 11 v ~2Y2) X F 1·1----1--+-·-- 12, 22' 22'4 J.L-3 (/1-v l) ~ v l j -2 I 2-2-2 l~ 2+2-2 ay312-J.L x F 0· ~-~-_:: ~-~+ v. a2y2) 1 2 , 2 2 2' 2 2 2'_4_ u ~ -rra y esc [rr(p.-v)] lK v(ay) + TT cos (p.rr) esc [rr(v + p.)J Jv(ay)} Re y > 0 128 INTEGRAL TRANSFORMS 10.2 Elementary functions (cont'd) f(x) {"' f(x) K (xy)(xy)X dx 0 v (5) x-X(x+a) -1 ~rr2 [esc (v77)f yX [Iv(ay) \arg a\ < 77, -1 <Rev< 1 + I_)ay)- e-Xiv7T Jv(iay) -eX iv7T J -v (iay) J Re y > 0 (6) x-x (a2 + x2)-x ~ rr2 sec(~vrr)y x l[Jx)~ay)f Rea> 0, -1<Rev<1 + [Yxv<~ay)]2! Re y > 0 (7) x -X -v(x 2 + a2)-l ~rr2sec(v77) a-v-I yX [H)ay) Rea> 0, Rev<~ -Yv(ay)J Re y > 0 (8) x X + v (x 2 + a 2 )J.I. 2vf'(v+ 1) av+J.L+I Y-X-J.L Rea> 0, Rev> -1 xS ).1.-V,j.I.+V+I (ay) Re y > 0 p+2J.L X (9) xp-312(x2 + a2)J.I. a y [f(v) + f(-v)J+ 22J.L+p-2 Re a> 0, Rep> \Hev\ 41(-Jl) (p v) ~p v) Y,-p-2J.L X[' 2+J1- 2 [' 2 +J1+2 y ( p v p v X IF2 -J1; 1-J1,2-2' 1-jl--+-. 2 2' 2 2) a y --- 4 Continued on the following page. 10.2 K-TRANSFORMS 129 Elementary functions (cont'd) f(x) {'" f(x) K (xy)(xy)~ dx 0 v (9) Continued from the pre- f(v)= (~a)vr(-v)r(; + ;) ceding page. ( v p ) v x r _2_2_ fl r (p v p v X ,F2 -+-;--+ fl + 1 +-, 2 v 2 2 1 + v; 2 2 ) a y Re y > 0 ---- 4 (10) [x (a2- x2)]v-~ 0 < x <a ~ 2v-t 2v. ~-vr( ~) TT a y V+ 2 0 a<x<oo x Iv(~ay) K)~ay) Rev >-~ (ll) 0 0 < x <a -~2v-t 2v ~-vr( ~) TT a y V+ 2 [x(x2- a2)]v-~ a<x<oo x lK)~ayW Re y > 0 Rev >-~ (12) x~-v(a2-x2)f.1. 0 < x <a 2-v-2 a2f.1.+2 yv+~ (11 + l)-1 r(-v) 0 a<x<oo x 1F2(1; v+ 1, fl+ 2; ~a2y2 ) fle fl > -1, Rev< 1 + TT 2f.i.-l af.i.-v+ I y -f.l.-~ esc (vrr) X r (fl + 1) If.l._V+I (ay) ( 13) 0 0 <x <a 2f.l. f.l.-v+ I -f.l.-~ r ( ) a y fl+1 , xY,-v(x2- a2)f.1. a <x <"" xKf.l.-v+t(ay) fle y > 0 Re fl > -1 13{) INTEGRAL TRANSFORMS 10.2 Elementary functions (cont'd) f(x) J~ f(x) Kv(xy)(xy)!-S dx (14) x-!-S (x 2 + a 2)-!-S -Jirr a-21-Ly!-S { J (~ay)_a_ X [(x2 + a2)Y, + xr2J.L J.L all a Re a> 0, v=O x [Y 1_/~ay)]- Y J.L(~ay) - x [J)~ay)J} all Rey>O (15) x-y, (x2 + a2)-!-S Jirr2 a-21-Ly!-S csc(vrr) X ((x2 + a2)Y, + xr2J.L x [JJ.L+ !-Sv(~ay ) Y J.L-!-Sv(~ay) Rea> 0 -Y J.L+Y,)~ay) JJ.L_!-S)7~ay)J Re y > 0 (16) x -!-S (x2 + a2)-y, ~rr2 a2J.Ly!-S I[JJ.L(7~ay)]2 xll(x2+ a2)!-S +xfi-L + [Y )~ay)]2} Re y > 0 + L(x2 + a2)!-S-xfi-LJ Re a> 0, )/ = 0 ( 17) x -!-S (x 2 + a 2)-!-S Jirr2 a2J.Ly !-S [JY,v+J.L(~ ay)J!-Sv-j~ay) x l[(x2 + a2)X + xJ21-L + YXv+J.L(~ay) Yxv-)~ay)] +cos [(~v-ll)rr] + [(x 2 + a2)X -xfi-L Re y > 0 X COS((~ 1/ + ll) 7T )} Rea> 0 (18) x -y, -2,u(x 2 + a2)-X y, -1 -x ( l+ v ) c-)/ ) 2a y l ---ll [' -2--ll x ~(x2 + a2)!-S + af!L Re a>O 2 2 Re ll + fRe vf < l X If~, y, )i a y) W J.L • Y,)-i a y) Re y > 0 10.2 K-THANSF'OflMS 131 Elementary functions (cont'd) f(x) ( 19) 0 O<x<a x-y,(x2-a2)-y, x l [x + (x 2 -a 2) y, J2 J.1. + [x-(x 2-a 2)y, fJ.Lj a<x<oo J'XJ f (x) K (xy )(xy) y, dx 0 v Re y > 0 1--+------------+------ -·----------j (20) 0 0 <x <a _x-Y.-2J.L(x2-a2)-Y, x I [a + i (x 2 -a 2 )y, f J.1. + [a -i (x 2 -a 2)y, fJ.Lj a<x<oo rr a-1 y-Y, rf!J.L,Y,v(ay) W_J.L,Y,v(ay) Re y > 0 v=O yy,(y2-a2)-y, cos-1(a/y) (22) X -Y, e -ax -1 < Re v < 1 Re f1 > IRe vi -~ Re(a+y) >O ( 2 2)-x _, ( I ) v y y -a cos a y -+ 12 rr as y -+ oo. rry -X sin (ve) sin (vrr) sin e Re(a + y) > 0 TTY, 2vyv+Y, (a+y)f.1.+v+Y, r<fl+ v+ ~)[' (f1-v+ ~) [' (f1 + 1) ~ a-yJ x 2F, fl+ v+ ~. v+ ~; f1+ 1; --- a+y Re(a + y) > 0 132 INTEGRAL TRANSFORMS 10.2 Elementary functions (cont'd) f(x) (24) x -~ exp (-ax 2) Re a > 0, -l < Re v < l (25) x-~-2 J.Lexp(-ax2) Re a>O 2Re /.1. < l-\Rev\ (26) x-~(x2+ a2)-:~ 2 2 ~] x exp [-{:3 (x + a ) Re a> 0, Re {:3 > 0 -l < Re v < l -~<Rev<~ fry) ~ ~ sec(~ vrr) \---;- x exp(~) K~ fL) 8a v\Ba x exp(~) W l/ fL) 8a J1.,/2v\4a ~y~ sec(~vrr) x K~vl~a[{:3 + ({:32 -y2) ~]} X K~)~a[{:3-({:32-y2)~]1 Re (y + {:3) > 0 ~77 sec(vrr) [ Dv-~ (2~ay~) x D -v-~ (-2 ~ ay ~ ~ + Dv-~ (-2~ay~) D-v-~(2~ay~) J Re y > 0 10.2 K-TRANSFORMS 133 Elementary functions (cont'd) f(x) f oo f (x) K (xy) (xy) !4 dx 0 v (28) x-1 exp(-axy,) (~rr)!4 1(~-v) Dv-Y, (ay-!4el<71i) x cos(ax!4 + ~rr-~VIT) D ( -y, -~7T i) Re y > 0 x v-Y, ay e -~<Rev<~ (29) 0 0 <x <a (~rr)1/2 a3!2-v {3y 1/2-v(y2+ {32)vl2-3/4 x !4 -v sin [{3 (x 2 -a 2 )!4] x K [a(y2 + {32)112] v-3/2 a<x<oo Re y > lim /31 (30) x-y,(a2-x2)-!4 2 y, -~ 7T y esc (VIT) [J!4 v(u) J!4 )v) [ 2 2 -Y,] -J-Y,v(u) J_!4)v)] X COS {3 (a -X ) 0 <x <a u =~a [{3 + ({32-y2)!4] 0 a<x<oo v = ~a [{3 -({3 2 -y 2)!4] -1 < Re v < l (31) 0 O<x<a (1/ )!4 Y,-v !4-v( 2 {32) !4v-~ 12 7T a y y + xY,-v(x2- a2)-y, x Kv-!4 [a(y2 + {32)Y,] [ 2 2 !4 x cos {3 (x -a ) J Re y > I lm /31 a<x<oo (32) x -!4 sinh (ax) -2 <Rev< 2 !4 ( 2 2 -!4 ~rry y -a) csc(~vrr) x sin lv sin -1 (a/y)] Re y > IRe al 134 INTEGRAL TRANSFORMS 10.2 Elementary fWictions (cont'd) f(x) J: f(x) K)xy)(xy)X dx (33) x -X cosh (ax) 77YX cos[v sin-1(a/y)} 2(y2-a2)X cosO ~v77) -1 < Re v < 1 Rey>IReal x-312 sinh(ax) 77 sec (X v77) sin [v sin -I (a/y)] (34) 2vyx -1 < Re v < 1 Re y ~ IReal (35) x-X(a2-x2)-x ~ 772 y X esc (X v77) LI_x)u) I-xv<v) [ ( 2 2 X] x cosh f3 a -x ) -I x)u) I ~v (v)} 0 < x <a u =X a [({32 + y2)X + {3] 0 a<x<oo v = X a [ ({3 2 + y 2) X -f3] -1 < Re v < 1 10.3. Higher transcendental functions ( 1) XX p (1-2x2) 0 <X< 1 y-x [(-1)"+, K 2n+1 (y)+Xi S2n+,(iy)] n 0 1<x<oo I/= 0, n = 0, 1, 2, ••• (2) 0 0 < x <a , _, -x ( ) ~ 77 a y W Xn, ~ v ay x-X (x2-a2)-X T (a/x) n X W_Xn,X)ay) Re y > 0 a<x<oo n = 0, 1, 2, ••• 10.3 K-TRANSFORMS 135 Higher transcendental functions ( cont 'd) 00 I f(x) J f(x) K (xy)(xy) X dx 0 v (3) 0 0 <x <a (~rr)Xy-1e-Xayw (ay) J.L,V x!-L(x2-a2)-XJ.L Pj:.x(x/a) Re y > 0 a<x<oo Re f.1 < l (4) 0 0 <x <a (~rr)X a-1 e-l<lay WJ.L_1,)ay) xJ.L-2(x2-a2)- lc!iJ.L Pj:.X(x/a) Re y > 0 a<x<oo Re f.1 < l (5) 0 O<x<a (2rr)-X a I-J.LYJ.L Kv(~ay) x -J.L(x 2-a 2)-XJ.L Pj:.X (x/a) x KJ.L_X (~ay) Re y > 0 a<x<oo Re f.1 < l (6) 0 O<x<a (2:y )X -'hay W ( ) e J.L-X,v-X Y xJ.L-1(x2-a2)- XJ.LpJ.L (x/a) v-3/2 a<x<oo Re f.1 < l (7) xx(x2 + a2)l<;v pv(l+2x2a-2) J.L -v -v-l<! ( ) 2 ay S 2v, 2J.L+1 ay Rea> 0, Rev< l Re y > 0 136 INTEGRAL TRANSFORMS 10.3 Higher transcendental functions (cont'd) f(x) (B) XX (x2 + a2)Xv X ( (f.L-V) p V ( 1 + 2 X 2 a-2) j.J. + (f.L + v) P_~ (1 + 2 x 2 a-2)] Re a > 0, Re v < 1 (g) XX (x2 + a2)Xv-1 (10) 0 (ll) 0 X [pv(1+ 2x2 a-2) j.J. + pv (1+2x2a-2)] -j.J. Re a > 0, Re v < 1 a<x<oo Rev< 1 O<x<a a<x<oo Rev>-~ (l2) x-v-X(x2+a2)~-Xv QX-v(1 2 2 -2) x -x + a x Rea>O, Rev<1 J"" f(x) K)xy )(xy)X dx 0 1 -v -v-3/2 S ( ) 2 fLY 2v+1, 2J.L ay Re y > 0 2 1 -v X -v S (a ) Y 2v-1, 2J.L Y Re y > 0 Re y > 0 -X 2v-1 X-v [K (1/ )]2 rr ay J.L+X /2 ay Re y > 0 xa112-vyv-1/2 [1(1-v)f x l[Jv-X (~ay)f+[Yv_X( ~ay)]2l Re y > 0 10.3 K-TRANSFORMS 137 Higher transcendental functions (cont'd) f(x) J 00[(x) K (xy)(xy)!c; dx 0 v (13) x-v-X(x2+ a2)l4'-Xv ie-i'TTV TT112 2-v-1 a-v-1/2 Y,-v( 2 -2) xQ; l+2a x x yv-312 [f'(3/2 + f1.-11)]2 Re a> 0, Re f.l>-3/2 X W_J.L-X,v-X(i ay) W_J.L-Y,,v- X(-i ay) Re(f.l-11) >-3/2 Re y > 0 (14) XV, pv[(l + x2)X] Re 11 < l -1 Re y > 0 J.L Y Sv+X,J.L+X (y) (15) xv, (l + x2)-X pv[(l+x 2)v,] J.L s v-X,J.L+Y, (y) Re y > 0 Re 11 < l ( 16) xJ.L+v+X J (ax) J.L 2J.L+v aJ.Lyv+X f'(f.l + 11+ l) Re fl.> \Re11\-l X (y2 + a2)-J.L-v-1 Re y > \Ima\ xa+X J (ax) 2a aJ.L Y -a -J.L-3/2 (17) ['(fl.+ l) J.L Re(f1.+a) > \Re11["-2 (f1.+11+a ) cf1.-11+a ) xr +l r +l 2 2 (fl.+ 11+ a fl. -11 + a X 2F1 + l, + l; 2 2 fl.+ l; -;:) Re y > \Im a\ 138 INTEGRAL TRANSFORMS 10.3 Higher transcendental functions (cont'd) f(x) (18) x-l{ [J (ax)f JL 2 Rep.> IRe vi-1 (19) xl{ [J (ax)F JL 2Rep.>IRevl-2 (20) x~ JJL (ax) JJJ.+1 (ax) 2 Re f1. > IRe vi -3 ~r (p.+ ~ v+ ~0 r (p.-~ v+ 72) y -x 1 -JL l( 2 -")X]j2 x <Pxv-X 1+4a y · Re y > 2/Imal r (p. + ~ v + 1) r (p.-~ v + 1) x r -3;2 <1 + 4a2 r -2r 112 x p~~[(l + 4a2 y-2)112] x px-i:-1 [(1 + 4 az Y -2)1/2] Re y > 2/Imaj ( 3+v) ( 3-v\ r p.+-2- r p.+-2-; x y -3;2 (1 + 4 a2 y -2)112 xP-IL [(1+4a2y-2)112] X v-X x px-;:_:~ [(1 + 4 a2 Y -2) t/2 J Re y > 2Jlm aj (21) x-l{ JJL(ax)J_)ax) ~rry-X sec(~vrr) -1 <Rev< 1 x P~v- x [(1 + 4a2 y-2)X] x P;~-x [(l + 4a2 y -2)l{] Re y > 2/Im a/ 10.3 K-TRANSFORMS 139 Higher transcendental functions ( cont 'd) f(x) J: f(x) K)xy)(xy) lS dx (22) xX J 1./ax) J_)ax) -~21T y-312 z-1 esc (~2Im) -2 <Rev< 2 x[(J.L-r2v) P.f)z) ?:;:~_, (z) -( ~ v + J.L) P,: v-1 (z) P ~';:, (z )] z =(1+4a2y-2)X, Rey> 21Imal (23) xX JJ.l.(ax) J,_)ax) -3 <Rev< 3 a['(~) r(y) r !;/2 r (2-11) r < 1 + 11) (3+v 3-v 3 X 4F3 -2-,-2-'1,2; Re y > 21Imal (24) xX JJ.l.~ax) J_J.l._, (ax) ~" y-312 z-' sec (~v77) -1<Rev<1 x[P:;:~- X(z)P~:~lS(z) + (~v-~-J.L) (~v+~+p) x pX-;:_:~ (z) P~v-X (z)] ~ ~ 7T _, y '12 sin (J.L77) sec(~ V7T) (1 2 -2 X z = + 4a y ) , R e y > 2 I Im a I 140 INTEGRAL 1RANSFORMS 10.3 Higher transcendental functions (cont'd) f(x) (25) x.o-+Y, J.,_(ax) JA_(ax) Re(a+!l+A) > \Hev\-2 (27) xy, Jy,)ax2) a > 0, Re v > -1 J"" [(x) K (xy)(xy) ~ dx 0 11 r(1+11) r(l+A) (11+ A+ v+ a ) x r + 1 2 (11+ A+ a-v 0 xr +1 2 ~!l+A+ 1 11+A 11+A+v+a X ,_F3 --,-+1, +1, 2 2 2 /l+A-v+a ---+ 1; 1+11, 1+>.., 1+wA; 2 -4;:) Re y > \Im a\ see Bailey, W .N ., 1936: Proc. London Math. Soc. 40, 37-48; f. London Math. Soc. 11, 16-20. 17yy, [ ( 2 \ 8acos(~vrr) "-Y,v :a J Re y > 0 10.3 K-TRANSFORMS 141 Higher transcendental functions (cont'd) f(x) X 2 (28) x Y X )ax ) a > 0, -1 < Re v < 1 (29) xy, J~)ax2)J_~)ax2) a>O, -2<Hev<2 (30) xX JJ1.~~v(ax2 )JJ1.-~V(ax2 ) a> 0, 4Re 11> IRe vi-2 (31) x-X J)a/x) a> 0 -5/2 < Re l/ < 5/2 J''" f(x) K (xy )(xy )X dx 0 v Re y > 0 y-X eXi(v+T)7T K2)2(ay)X e!<i7TJ +y-X e-Xi(v+I)7T K2vl2(ayfe-!<i7TJ Hey> 0 142 INTEGRAL TRANSFORMS 10.3 Higher transcendental functions (cont'd) f(x) .(" [(x) K11(xy)(xy) ~ dx (32) x -yz J (a/x) v a-1 y~ e~iv7T K 2J2(ay)~ e!l:i7T] a> 0, -~<Rev<~ -1 ~ -Y,iv7TK [2( )~ -!4i7T] +a y e zv ay e Re y > 0 (33) zv-z J (a/x) x v+Y, (2rr)~ (y/a)-v+Y, J2)(2ay)~] a> 0, Re v > -l/3 x K 211[(2ay) 'la] Re y > 0 (34) -zv / ) x J11_~ (a x (2rr)~ (y/a)v-~ K 211_ 1 [(2ay)y,] a> 0, Rev< l x lsin(vrr) J211_1 [(2ay)~] +cos (vrr) Y 211_1 [(2ay)~]l Re y > 0 (35) x211 Jy, +)a/x) (2rr)~ (y/a)-v-Y, J 1+2)(2ay)~] a> 0, Rev> -1 ~ xK1+zvl(2ay) ] Re y > 0 (36) xcr-1 J (a/x) 1.1. 2-cr -3/z a cr a> 0, Rea> IRevl- 2 00~'r'J •-a 1 v xG ----,-+-04 4 2 4 2' l V Jl+O) 4-2,--2- Re y > 0 10.3 K-THANSFORMS 143 Higher transcendental functions (cont'd) f(x) Joo [(x) K11(xy)(xy)~ dx 0 (37) x-~ Y)a/x) -y-~ e~ivrrK2v[2(ay) ~ e!4irr] a> 0, -5/2 < Re v < 5/2 -~ -~iV7T K [2( )~ -!4irr] -y e 211 ay e Re y > 0 (38) x-512 Y)a/x) - t ~ ~ i (v+ t ) 1T a y e a> 0, -X< Rev< X xK [2(ay) ~e!4irr] 2V -t ~ -~i (v+t)rr +a y e x K [2(ay) ~ e-!4irr] 2V Re y > 0 (39) x 2 v-2 y v+~ (a/x) (2rr)~ (y/a)~-v Y 2)(2ay) y,] a> 0, Rev> -l/3 x K2)(2ay) ~] Re y > 0 (40) x -2v Y v-~ (a/x) -(Xrr)y, (y/a)v-~ sec(vrr) a> 0, Re v < l x K 211_1 l(2ay) y,] IJ211_ 1 [(2ay)y,] -J, -2)(2ay)X]! (41) x2v Yv+~ (a/x) (2rr)y, (y/a)-v- ~ Y 2v+t [(2ay) y,] a> 0, Re v < -1 [ y, xK2v+t (2ay)'] Re y > 0 (42) x-~ J (a/x) Y (a/x) 1-L 1-L -2y -~ J21-L (2a~ y~) a> 0, v=O x K (2a~ yy,) Re y > 0 2J.L 144 INTEGRAL TRANSFORMS 10.3 Higher transcem1ental functions (cont'd) f(x) (43) x-~I[J (a/xW-lY (a/xW! JJ. JJ. a> 0, v = 0 Re v > -~ Re v >-~ (47) 0 0 < x <a x~-v(x2 -a2)~JJ. x J)fHx 2-a 2)~} a<x<oo Re p. > -1 f oo f (x) K (xy )(xy) \{ dx 0 v -~ 1\.y rra 4y3/2 TT ~ Re y > 0 [ Iv-• (:;)-L._, (:: j J Re y > 0 [I.~::)- L,~::) J Re y > 0 Re y > 0 aJJ.-v+t {3JJ.y~-v(y2+{32)~(v-JJ.-I) x K v-JJ.-1 [a (y 2 + {3 2)~] Re y > lm {3 10.3 K-TRANSFORMS 145 Higher transcendental functions (cont'd) f(x) Joo f(x) K (xy)(xy) ~dx 0 v -v ~ v 2v 2v x~ Kv(ax) rra y a -y (48) -1 <Rev< 1 2 2 2sin(vrr) a -y Re (y +a)> 0 xa-v2 K (ax) 2a-3 a -v-a ~~+;+v) 1 ~-;+v) (49) J.1. !(a) Rea> IRe pi+ IRe vi ~a+p-v) t-p-v) +~ x 1 · 1 yv 2 2 0+e+v a-e+v. . ~ x 2F, --,-- ,a,1-2 2 2 a Re(y +a)> 0 (50) x~ l2rr-1 K0(ax)-Y0(ax)] 2 77-1 y ~ [ (y 2 +a 2) -1 + (y 2 _a 2) -1 ] v=O x log (y/ a) Rey>IIrnal, Re (y +a)> 0 (51) xa+~ J)ax) KA.(f3x) see Bailey, W.N., 1936: !. London +~ Math. Soc. ll, 16-20; Proc. London xa K )ax) K A_({3x) Math. Soc. 40, 37-48. 146 INTEGRAL TRANSFORMS 10.3 Higher transcendental functions (cont'd) f(x) Joo f(x) K (xy)(xy)~ dx 0 v ' { c) x~ K~v(ax2 ) TTy ' y (52) -- sec(~vrr) K~v --8a 4a Rea> 0, -1 < Re v < 1 + TT csc(vrr) [ L-~v (::) -L (~)]} ~v 4a (53) x ZJ.L+v+~ exp (-~ax 2) -~ J.L-~ -~ J.L-~ v-~ -J.L-1 rr 2 a y x l)~ax2 ) Rea> 0 x r<2J.L+v+1)r(J.L+~) Re J.L>-~. Re (2ft+ v )>-1 X exp~2 )Wk c~) 8a ·• 4a 2k=-3JL-V-~ 2m=JL+v+~ (54) x-~ Kv(a/x) Rea> 0 rry-~ K2)2a~ y~) Re y > 0 (55) x -!Vz K (a/x) v Rea> 0 -1 ~K (2 ~ ~) TT a y zv a y Re y > 0 (56) x2v Kv+~ (a/x) Rea> 0 (2rr)~ (y/a)-v-~ K [(2ay)~e ~i71] 2v+1 x K [(2ay)~ e-~i71] 2v+1 Re y > 0 (57) zv-z ( / ) x Kv+~ ax Rea >0 (2rr)~(y/a)~-vK2vl(2ay)~ e~i71] x K [(2ay )~ e -~ i71] 2V Re y > 0 10.3 K-TRANSFORMS 147 Higher transcendental functions (cont'd) f(x) (" f(x) K (xy)(xy)X dx 0 v (58) xu-l K (a/x) 1-' Rea> 0 2-u -s/2 au x C 40 E a 2 y 21 fl-a ~ + ~ ~-~ -fl+aj 04 4 2 '4 2'1 2' 2 (59) x -y, [K .u (a/xW 2TTy-x K2)2ax Yx e!4i7T) Rea> 0, v=O x K (2ax yx e-!4i77) Re y > 0 2,U x-X I (axx) 1T [ Iv(4~:~ + Lv(:; ) J (60) Rev>-~ 2yx 2V Re y > 0 x-X [J2)axX) + I 2 . ..,(axX)] 1T I c~) (61) - Re y > 0 Rev>-?i yy, v 4y x-X[I (axy,)-J (axy,)] 1T Lv(;;) (62) -- Re y > 0 2 V 2V y, Rev >-~;, y (63) x-Y, K (axy,) 1T y -y, { KVC::) 2v 4 cos (vTT) -~~<He v < ~,;, + 2 si:(vrr) [ L-v~4~; ) -Lv ~:)]} Re y > 0 • r 148 INTEGRAL TRANSFORMS 10.3 Higher transcendental functions (cont'd) f(x) f000 f(x) K v (xy) (xy )~ dx (64) x v+~ I2)axY.) J2v(ax~) ~ -v-1 2v+1 -zv-2 rr 2 a y Rev>-~2 X Jv-Y, (;:) Re y > 0 (65) x v-Y, y, y, Izv-1 (ax ) Jzv-1(ax ) TT~ 2-v a2v-1 y -zv Rev> 0 X Jv-~ ~:: ) Re y > 0 (66) v-~ ( ~) ~ x I2v_1 ax Y2v_1tax ) ~ 2-v-1 zv-1 -2v ( ) rr a y esc vrr Rev> 0 x [u,_.(;:) + cos (vrr) Jv-~ ~::) + sin(vrr) Yv-Y,(::) J Re y > 0 (67) v-~ ( ~ ~ x J2v_1 ax ) K2v_1 tax ) TT3/2 2-v-z azv-1 y-zv csc(vrr) Rev> 0 x [u,_v(;: )-Y._,(f) J Re y > 0 10.3 K-TRANSFORMS 149 Higher transcendental functions (coot 'd) f(x) J"'' f(x) K (xy)(xy)X dx 0 v (69) -v-x 1 ( y,) x -2v-1 ax X x J211+1 (ax ) Re v < ~ (70) xX-v[I;)ax X) J_2v(axy,) -77X 2va1-2vsin(V7l')vzv-2 -J2v(axx) I_2)axX)] x Jv+x~;:) Rev< 3/2 Re y > 0 2 -x 1110 ~a2 ) (71) x-X K (axx) 77 y -f-L 16 cos 02 f177) y,f-L 4y x[sin(~flrr)J (axy,) f-L + cos(~f177) Y (axy,)] xi/,!Zl ( ~) f-L Re y > 0 -1 < Re 11 < l, v=O Y,J-L 4y 150 INTEGRAL TRANSFORMS 10.3 Higher transcendental functions ( cont 'd) f(x) Joo f(x) K (xy)(xy) y, dx 0 v x-X K (axy,) c+~-v) (~ (72) -Xa-2yy, I~ -- 2-r 2 J.L x lsin[~ 2(~-v) rr] J (axy,) J.L (' ) + cos[X(fl""""v)rr] YJ.L(ax y,)! X If/ !!____ Xi7T \Re~\ +\Rev\ < l Y,v, XJ.'. 2y e w c~ -"·) x y, v, ~J..<. 2y e Re y > 0 (73) x!<l H (ax) v Rev> -3/2 v+1 -v-Y, ( 2 2)-1 a y y +a Re y > \Ima\ (74) xJ.L+v+X H (ax) J.L He~ > -3/2 77 -Y, 2J.L+v+ 1 aJ.L+ 1 Re(~+ v)>-3/2 x y -2J.L-v-512 1( J1 + V + 3/2) X 2F;(H v+ ~, 1-~--~) ' 2 ' y 2 Re y > \Ima\ (75) xy, H (ax2) yy, 1(1 + Xv) (y2 j Y,v 21-Y,varr 5-Xv-1,!<!t-~ a> 0, Re v > -2 Re y > 0 (76) 3/2 ( 2) t' I 2+ :/2 y 3/2 1 ( 3 + v) x HY,v+X ax a> 0, Re v > -3 a rr 2 ~2) xS - Re y > 0 _ v+S ~ -1 2 • 2 _.a 10.3 K-TRANSFORMS 151 Higher lranscendental functions (cont'd) [(x) (77) x512 U (ax2) Xv a> 0, Rev> -3 (78) x ~ s v (ax 2) a > 0 f..L,nv Re11>Y2!Rev1-2 (79) 312 ( 2) a > 0 x s f..L,Y,v+~ ax 2Re 11> IRevl-5 (80) x 512 s (ax 2) a > 0 f..L.~ v Re 11 > Y21Re vi -3 (81) D v-~ (ax -~)D -v-Y, (ax-~) largal < ~ 7T Joo f(x) K (xy) (xy )~ dx 0 v Re y > 0 (r2 ) xS -- -f..L-1,~v 4a Re y > 0 y3/2 ( 3-v\ ~ 11+-2--; ( 1-v) ( 3+v) X[''(+ -2- [' \:+-2-- Re y > 0 Re y > 0 7T 2y Re y > 0 152 INTEGRAL TRANSFORMS 10.3 Higher transcendental functions (cont'd) f(x) (83) x-312 MK.0(iax2)MK,0(-iax2) a> 0, v = 0 a> 0, Re J1 >-Y:;, v = 0 2k=-3j1-V-K -Y:; 2m=J1+V-K+Y:; Re y > 0 (85) X~ W ~v.J.L(a/x) IT' -~v.)a/x) 2ay- ~ K2)(2ay)~ e!{i7T] Rea> 0 X K2J.L[(2ay) ~ e-!{i7T] Re y > 0 (86) x v+~ F ( (3 l ' 2 2) 21a,·;v+ ;-"x 2v+1 , -a-j3Ya+,B-v-3/2 ( " r v + l) Re A.> 0, Re v > -1 X S 1 -a-j3. a-r}Y/A) Re y > 0 10.3 K-TRANSFORMS 153 Higher transcendental functions (cont'd) f(x) (87) xv+2y -312 x 3F2(l,a,,B;y,y+v;-A2x2) Re ,\ > 0, Re y > 0 Re(y+v) >O (88) (89) XJ.L-3/2 x E (a 10 ... ,a P: p 1, ... ,p q: ax -2) a> 0, Re J1 >!He vi (90) cmn (,\x21 al'"''ap) pq .B 1''"',8 q p + q < 2(m + n) largAI < (m + n-~p -. ~q)rr Re ,B . > ~IRe vi -% J j = 1, ... , m fooo f(x) K v (xy) (xy )X dx 2v+2y -2 ,\ -a-(3 Ya+j3-2y -v+Y, X r(y)r(y+ v)St-a-j3,a-ir!A) Re y > 0 4,\~ {31' ... ,{3 ;-2-q y Re y > 0 2J.I.-2 a-J.LYY, xE(al'"'ap+z:pi'"''Pq: ~ay2 ) J1 + v ap+t --2-, jl-V r:p+2 =-2- Re y > 0 xcn+2,m (y21 ~-{31' .... ~-{3q ) q,p+z 4,\ h k il ll , ,;2-a1, ... ,;2-ap h = ~ + ~ v, k = ~ -~ v Re y > 0 CHAPTER XI H-TRANSFORMS We call the H-transform of order 1.1 of f(x) and regard y as a positive real variable. The inversion formula ll.l(l) was given by Titchmarsh (1937, p. 215). The H-transform is the reciprocal of the Y-transform (see chapter IX). From the transform pairs given in this chapter further transform pairs may be derived by means of the methods indicated in the introduction to vol. I, and also by means of the general formulas of sec. ll.l. Moreover, U-transforms being reciprocal to Y-transforms when-X < Re 1.1 < X, many further formulas may be obtained from the tables in chapter IX: the extension of such formulas by means of analytic continuation to a wider range of Re 1.1 (the range of absolute convergence of the integral) is immediate. REFERENCE Titchmarsh, E .C., 1937: Introduction to the theory of Fourier integrals. Oxford. 155 H-TRANSFORMS 11.1. General formulas f(x) fo 00 f(x) H., (xy )(xy )y, dx = g (y; v) y>O (1) Joo g(y; v) Y (xy)(xy)y,dy 0 v g (y; v) -X< Rev< X (2) f(ax) a>O a-1 g(a-1 y; v) (3) x" f(x) m = 0, 1, 2, ••• Y, -v ( d ) m [ v-Y,+., ( )] y -- y u y·v+m y dy b ' (4) xY,+v ( ~ )" [x"-v-Y, f(x)] x dx (-y)" g (y; v-m) m = O, 1, 2, ••• (5) Xi.-+y, Joo ,;Y,-v-Ji-(,;2 -x2)J.L-1 2J.L-l 1 (p.) y -J.L g (y; v + p.) % X f(,;) d,; Rep.> 0, Rev> -3/2 (6) x -J.L f(x) 21-J.L[f'(p.)r, Yy,-., Re v + 3/2 > Re p. > 0 JY Y,-J.L+V( 2 2)J.L-1 X TJ y-TJ 0 X g ( TJ; V-p.) dTJ 157 158 INTEGRAL THANSFORMS 11.2 11.2. Elementary functions f(x) J 00 f(x) H (xy )(xy )l? dx 0 v y>O (l) x-l? -2 <He v < 0 -ctn (~vrr) y-X (2) xv+Y, 0 < x <a v+ 1 -Y, H ( ) a y v+1 ay 0 a<x<oo Rev> -3/2 (3) xl?-v 0 < x <a ayv-Y, 2v-l77 l? i(v+ ~) 0 a<x<oo -a1-vy- Y, Hv_,(ay} (4) XA-Y, ReA<~ 2"-y-A.-l? tan OHA + v + l) rr] -2 < H e (A + v) < 0 1(~+ ~A+ ~v) X 1(~- ~A+ ~~v) A+v+2 v+3/2 (5} x"--l? O<x<a a y 2v rry, i(v + 3/2}(A + v + 2) 0 a<x<oo Re(A+v) >-2 0 A.+ v 3 X 2F3 l, -2-+ l; 2' 3 A+v a2y2) v+---+2·---2' 2 , 4 (6) x -l? (x 2 + a 2)-, rry y, --[ 11 (ay)-L 1 (ay)] 2a Rea> 0, v=l 11.2 H-TRANSFORMS Elementary functions (cont'd) f(x) (7) x-y,(x2+a2)-1 Rea>O, -2<Rev<2 (B) xv+Y,(x2+a2}u-1 Rea> 0, Re v >-3/2 Re(fL + v) < l/2 Re (2/L + v) < 3/2 (9) xY,-v(x2 + a2)J.L-1 Rea>O, Rejl<l/2 lle(2jL- v) < 3/2 (10) x,\-~ (x2 + a2)J.L-l Rea>O, lle(.\+v) >-2 Re (.\ + 2fL) < 5/2 ll e (.\ + 2/L + v) < 2 J'><> f(x) H)xy)(xy)~ dx 0 rry 112 ------- L)ay) + 2J.L-l TTUJ.L+V YY,-J.L r(l-jl) cos [(jL + v) rr] xG~ l v h=-+-4 2' 3 v l=-+-4 2' 159 y>O l v k=--- 4 2 3 .\ m=--- 4 2 160 INTEGRAL TRANSFORMS 11.2 Elementary functions (cont'd) f(x) Joo f(x) H (xy)(xy) X dx 0 J) y > 0 (x2 + a2)-X 77 X a,_. f X [sinh 0':2 ay) I v+X (r2 ay) (ll) X x[x + (x2 + a2)Y,]v+1 y sin (vrr) Rea> 0, -2<Rev<O -coshe~ay) I-v-x( r2ay)] (12) xv+X (a2 _ x2)p.-1 0 < x <a 21-L-1 a p.+v y X -p. r (11) H p.+v (ay) 0 a<x<oo Re 11 > 0, Rev>-3/2 (13) x,\_-Y,(a2-x2)p.-1 O<x<a a 2p.+v+A. y v+3!2 f' (11) r (A.;v + 1) 0 a<x<oo 2v+1 rr112 r(v + 3/2)r(A.;v +11+ 1) Re 11 > 0, Re (.\ + v) >-2 ~ .\ + v 3 3 x2F3 l,--+1;-,v+-, 2 2 2 A+v a2y2) --+ 11 + 1; ---- 2 ·1 (14) 0 O<x<a 2-v-1 X -2v v+X 77 a y x -v-Y, (x 2 _a 2)-v-X X f' (r2-V) ( J )r2 ay W a<x<oo -r2 <.Rev< 1 ( 15) 0 0 <x <a (-1)'"+1 2'" a" +v+1 y-,.-X m! X v+X (x2 _ a2)m a<x<oo X 8v+m+1 (ay) m = 0, 1, 2, ... , Rev<-2m-r2 11.2 H-TRANSFORMS 161 Elementary functions (cont'd) f(x) (16) 0 0 < x <a xv+Y, (x2-a2),u-l a< x < oo Re11>0, Re(fL+v) <~ Re (2/1 + v) < 3/2 Joo f(x) H (xy)(xy) y, dx 0 v y>O 2,u-l a,u+vyY,-,u['(fL) sec[(fL+v)rr] x [sin (fLrr) J -,u-)ay) + cos(V7T) H,u+v(ay)] (17) 0 O<x<a -rr22,u-vyv-2,u-Y, x -v-Y, (x 2 -a 2).U a <x <oo x[f'(~-fL)f'(~+v-fL) sin(fLrr)r1 -l<Re/1<0 Rev> 2Re 11-~ (18) 0 0 < x <a x,\-y, (x2-a 2),u-l a < x < oo Re11>0, Re(A+2fL) <5/2 R e (A + 2 11 + v) < 2 (19) x,\-Y, e -ax Re a> 0, Re (A+ v) >-2 2-112[' (fL) a 2,u+A-3/2 Q2 21l ) 21 a y 'm x c24 ---4 l, m-fL, h, k l v h=-+-4 2' 3 v l =-+-4 2' Y v+3/2 ['(A + v + 2) l v k =---4 2 3 A m=--- 4 2 162 INTEGRAL TRANSFORMS 11.2 Elementary functions (cont'd) f(x) Joo f(x) H (xy )(xy )\{ dx 0 v y > 0 (20) ,\ +Y, 2) x ' exp(-ax 2-v-1 -112 -""+v+3)/2 v+3/2 rr a y Rea> 0, Re(,.\ + v) > -3 [' cf',+v+3) X 2 [' (v + 3/2) ~ A+v+3 3 3 r'j x 2F2 ' 2 ; 2' v+2; -4a (21) x -v-\{ sin (ax) 0 O<y<a a> 0, Rev >-~ Try, 2-vy\{-v[f'(v+ ~)r, ( 2 2)v-Y, x y -a a<y<oo (22) xy, cos [(v + l) e]! sin e 77y, a y, sin (~ay) Jv+Y, (~ay) 0 <x <a 0 a<x<oo o < e < ~ "• X= a COS 8 Rev> -2 11.3. Higher transcendental functions (l) Jv+\{ (ax) 0 O<y<a a> 0, -3/2 <Rev< l (~) \{ ( )+\{ ~ (y2 _ a2)-y, a<y<oo 11.3 H-TRANSFORMS 163 Higher transcendental functions (cont'd) r (x) f 00 f (x) H (xy) (xy) lS dx 0 v y>O (2) x -y, Y (ax) v+1 0 O<y<a a> 0, -3/2 <He v < 3/~ -v-1 lS+v -a y a<y<oo (3) xJ.L-v+Y, Y (ax) 0 O<y<a J.L a> 0, He(v-11) >0 21 +J.L-v aJ.L -3/2 <He 11 < l/2 Y,-v( 2 2)v-J.L-1 f'(v-11) y' y -a a<y<oo (4) x y, -J.L[sin (1177) J J.L+)ax) 0 O<y<a +cos(I177)Y +(ax)] Y, +v( 2 2)J.L-1 J.L v a> 0, l<Rel1<3/2 y y -a 2J.L-1 aJ.L+v [' (11) a<y<oo Rev>-3/2, Re (v-11) < l/2 (5) xv+Y, J (ax) Y (ax) r~ (2v + 3/2) yv+312 v v 77 3/2 2v+2 a 2 v+3 [' (v + 2) a> 0, -% <He v < 0 ( 3 y2 j x 2F1 1, 2v +-; v + 2;--2- 2 tla 0 < y < 2a (6) xv+Y, l[Jv(ax)P- [Y)ax)]2l 0 0 < y < 2a a> 0, -~4 <Rev< 0 23v+2 2v -v-Y, 'a y (y2-4a2)-v- Y, 77 y, r e1-v) 2a < y < oo 164 INTEGRAL TRANSFORMS 11.3 Higber transcendental functions (cont'd) f(x) J"" f(x) H (xy)(xy)~ dx 0 v y>O (7) x~ I [J~ (ax)Jl-[Y~ (axW! 0 0 < y < 2a v ,v 4 -1 -~ ( 2 4 2)-~ a> 0, -3/2 < Re v < 0 rry y-a' 2a<y<oo (8) ~ . x [J~v+~)ax) J~v-~)ax) 0 0 < y < 2a -Y~v+~JJ.(ax) Y~v-~JJ.(ax)] 4 -1 -~ ( 2 4 2)-~ ·TT y y -a cosh(f.1u) a> 0, -3/2 < Re v < 0 y = 2a cosh u, u>O (9) ~ a Yv+1 ~::) J2v+1 (ax ) -2y 3/2 a> 0, -3/2 <Rev< l/4 (10) -~ J ( ~) -y-~ y (~ x 211 ax a> 0, -l<Re v<5/4 v 4y ( ll) x~ [IJ~ [b(z-a)]!2 4 sin[a(4b2 -y2)~] 0 < y < 2b - rry~(4b2-y2)~ ,v -IY~11[b (z + a)W] 4exp[-a(y2-4b2)~] Re a> 0, b > 0 2b<y<oo -3/2 < Re v < l ~ 2 2 ~ z=(x2+a2)~ TTy 2 (y - 4 b ) (12) x~ K (ax) v a-v-1 Yv+3/2(y2 + a2)-1 Rea> 0, Re v > -3/2 11.3 H-TRANSFORMS 165 Higher transcendental functions (cont'd) f(x) J 00 f (x) H (xy) (xy) y. dx 0 v y>O ( 13) xJl.+v+Y, K (ax) J1. 2Jl.+v+1rr-Y, a-J1.-2v-3 He a> 0, He v >-3/2 X Yv+J/2 ['(J-L + v + 3/2) lle (J-L + v) >-3/2 ( 3 3 ') X 2f~ 1 J-L+V+- '-'-~ ' 2' 2' a2 ( 14) xJl.-v+Y, K (ax) 2Jl.-v Yv+3/2 [' (J-L + 3/2) J1. aJ1.+3 l'.(v + 3/2) Re a> 0, ReJ-L>-3/2 0 3 3 y2 ) x2F1 l,J-L+-;v+ -;--2- 2 2 a (15) xa-Y, K (ax) Rea> 0 2o-rr-112 a-v-a-2 Yv+3!2 J1. Re(a+ v)> IRe J-LI-2 l'(l + v+;+i!:) f'(l + v+~-1£) X f'(v+3/2) X 3f~ 0• l+ v+a+J-L ,1+ v+a-J-L 2 2 ; 3 V+~--~) 2' 2' a2 (16) a-Y, -ax K ( ) x e v ax see Mohan, Brij, 1942: Bull. Calcutta Math. Soc. 34, 55-59. (17) xv+Y, [K)ax)f rr 1/2 2-v-3 d-2v-3 Y v+3!2 He a> 0, Rev>-% x f'(2v + 3/2) [l'(v + 2)r1 ~ 3 r') x 2F, l, 2v+2; v+ 2;- 4a2 166 INTEGRAL TRANSFORMS 11.3 Higher transcendental functions (cont'd) f(x) (18) x~ [K (ax)f v = 0 1-' Rea>O, -3/2<Ref1<3/2 (19) x-v-~ K 0(ax)K 1(ax) Rea> 0 (20) x-v-~ K)ax) Kv+1 (ax) Re a > 0, Re v < ~~ (21) xa--s;2 KA_(ax) KJJ-(ax) Rea> 0 Re(a+ v) > \Re ..\\ + \Re 11\ fo""' f(x) H)xy )(xy)~ dx y>O -rr 2-~-'-1 a-2~-'y-~ z-1 sec(f.lrr) x [(z + y)2 ~-' + (z-y)2 ~-'] z=(y2+4a2)y, TT 112 aa-+v r (v+ 3/2) r (a+ v) a+v-A+fl a+v-..\-11 2 '--2--; 2_ v+~ a+v a+v+l._ y2 \ 2' 2' 2 ' 2 ' 4a27 11.3 H-TRANSFORMS 167 Higher transcendental functions (cont'd) f(x) (22) xv+312 K)ax) Kv+l (ax) Rea>O, Rev>-5/4 (23) xu-512 exp (-Yz a 2 x 2) x K)~a2 x2 ) largal < ~ 7T Re(a+v) >21Refll (24) x X exp (a: X ) K X v (a: X ) larg al <% 7T -3/2 <Rev< 0 Joo f(x) H (xy)(xy)X dx 0 v y > 0 77112 2-v-3 a-2v-4 Yv+3/2 x r(2v + 5/2) [f'(v + 2)r1 ~ 5 y20 X 2F, l, 2 V + - ; V + 2; ---2- 2 4a x----------- v + 2._ v+ a . _ _i___. \ 2' 2 ' 4a2 J 2 -v -Xv-1 vv-v 77 n a 2 yn /2 x cos (Yz vrr) f'(-Yz v) k = ~ v, m = Yz + ~ v 168 (25) INTEGRAL TRANSFORMS 11.3 Higher transcendental functions (cont'd) f(x) xo-exp (ax 2) K (ax 2) JL Jarga\<377/2, Rea<l jRe11\-5/2 < Re(a+ v) < l/2 Joo f (x) H (xy)(xy)y, dx 0 v y>O xG23(r2 34 ~a l v h=-+-4 2' l 1-o- 1-o-) ' -2-+ p., -2--11 l,-%' h, k l v k =--- 4 2' 3 v l=-+- 4 2 (26) K2v_1 (2axy,) 2v+1 77-1 a y-312 r(v + l) Rea>O, Rev>-l xS_v-2,v_1(a2/y) (27) x-y, K2)2axy,) 2v77-1 y-Y. r(v+ l)S_v- 1,)a2/y) Re a> 0, Rev> -l (28) xy, K2)2axy,) 2v 77-1 a2 y-sr2 r(v + 2) Rea>O, Rev>-2 xS_v-3,)a2/y) (29) xo-K)2axy,) Rea>O 2Re(a + v)> \Re11J- 5 x G1s c~ 1l, (31, •·· • (34) 53 a4 l,h,k l v h=-+-4 2' l v k =---4 2' 3 v l=-+-4 2 2 (3 1 = l-a+ Yz fl, 2(3 2 = l-a-Yz 11 2 (3 3 = -a+ Yz fl, 2(3 4 = -a-Yz11 11.3 H-TIIANSFORMS 169 Higher transcendental functions (cont'd) f(x) 1a 00 f(x) H)xy )(xy )X dx y > 0 (30) x-X [2 rr-1 K 2 v (2 ax X ) Y-x J)a2/y) + Y 2)2axx)] a> 0, -~ < Re v < Y~ (31) xX-v[J (axY:) 2v 2v77-Y, a1-2vy2v-2 X y, x sin (vrr) K +X (a2 ) -J_2)ax )] K2)ax ') largal < ~4 TT v 2y -3/2 < Re v < 3/2 xy, Y (axy,) K (axy,) 1 exp (-::) (32) v v 2y 3/2 I arg a I < ~~ rr, Re v >-3/2 xv-Y, Y (axy,) K (axy,) a 2v-1 (::) (33) 2v-1 2v-1 Y, 2v 2v K v-Y, largal < ~4 rr, Ilev >-~4 TT y 2v+1 Kv-x(:: 7 xv+Y, Y (axy,) K (axy,) a (34) 2V 2V Y, .-,v+1 2v+2 largal <}.; rr, Rev>-% TT ..:.. y ( 2) (35) x-Y, lcosD2(/l- v)rr] J (axY.) -2 Y: a a y Wy, y, -- J.L v. 11 2y -sin [~2 (ll-v)rr] Y)axy,)! x K (axy,) largal < )4rr Ga2 ) J.L X IT' Rev> IHelll- 2 -Y, v, Y,J.L 2y 170 INTEGRAL TRANSFORMS 11.3 Higher transcendental functions (cont'd) f(x) J"" f(x) H (xy)(xy) X dx 0 v y>O (36) v-x K ( y, !4i7T) x 2v_1 ax e -X 3v-1 -v-Y, TT 2 y x K (ax y, e -!4i7T) xr<v+ l)l (2v+ ~) 2v-1 Rea> 0, Rev >-~ X S_3v-X,v- Y, (2Y,ayy, ~ (37) x-y, H )a 2/x) -y-Y, J 2)2ayy,) a> 0, Rev>-3/2 (38) x-312 ll (a 2 I x) v-1 -1 J ( y,) -a 2v-1 2ay a> 0, Re v >-~2 (39) x-X[J (a2/x) -v y-y, [2rr-1 K 2)2ay y,) + sin (vrr) H)a 2/x)] -Y 2)2ayy,)] a> 0, -3/2 <He v < 0 2A.+a-Y, a-a-1 (40) xa SA. (ax) rO ':!-~>.-~fl) r<~-!,H+~2fl) ,IJ. largal <rr, Re(,\ + cr) < l Re(cr+ v)> IRe Ill-5/2 xc~~ l -"-'" = •-o-") ' 2 ' 2 ' 2 -3/2 < Re (,\ + P + cr) < l/2 44 az l, -'A.~a, h, k l v l v 3 v h =-+-, k =---l=-+-4 2 4 2' 4 2 11.3 H-THANSFORM S 171 Higher transcendental functions (cont'd) f(x) (41) x-v-~exp(- 7,i'x2) x[D/.L(x)-D)-x)] (42) n e (Jl + j)) > -3/2 Re 11 > -1 x2.\exp(-~x2)MK (x;x2) ,/.L Re ( 2 A+ 211 + v) > -7/2 Re (K-A)> 0 Re(2A-2K +v)<-~2 (44) x-y, exp (x;x2) X W-Y,v-Y,,Y,)x2 ) Re v > -1 2-11. r<211 + 1) r(x;+K+11) y>O c 22 (r 21z, -!1-A, 11-A) X 34 \.2 l, K-A-x;, h, k 1 1/ h =-+-4 2' 1 j) k =---4 2' 21/4 -A-v/2 -1/2 v+ 3/2 TT y 3 j) l=-+- 4 2 X r<-!r+ ~+A+ 11) rc!r +~+A-11) r(v+ t) r(~ +A-K-~) 172 INTEGRAL TRANSFORMS 11.3 Higher transcendental functions ( cont'd) f(x) (45) -~ (11 2) w (11 2) X exp ;4 X K .~ V !2X -3/2 <Rev<-2Re K Re K < ~ (46) X 21\_ exp (~ x2) WK,A_(~ x2) Re(2A. + v) > 2\Rell\- 7/2 He(2K + 2,\ + v) < -l/2 Re(K+A) <O (47) GP"; (A.x21 al' ···' aP) (31' ••• ' {3 q p + q < 2(m + n) \ arg A.\ < (m + n -~ p -~ q) 77 Re a. <min (1, %-~~ v) J j = l, ... , n · R e (2 {3. + v) > -5/2 J j = l, ... , m J 00 f(x) H (xy Hxy )~ dx 0 v y>O X exp0~y2 ) lT/k,m (~y2 ) 2k=K+~ :lv, 2m=K+~v+l [2A_ 1U·2-K + /l) ['(~-K-/l)]-1 c~ ' r 'r Qy 2 ll -u-,\ ,,_ ,\ ) X 34 2 [, -K-A- ~, h, k l v h=-+-4 2' (2,\)- ~ l v k =---4 2' 3 v l=-+- 4 2 x G n+1,m+1 Qr 21[, ~-f31' ... • ~'2-f3q) q + 1, p +3 4\ l 11 11 h k 1\ , 12-a1' ... ,12-a , , p l v l v 3 v h = -+-, k =---'--, l = -+- 4 2 4 2 4 2 CHAPTER XII KONTOROVICH-LEBEDEV TRANSFORMS The pair of reciprocal formulas g (y) = Joo f(x) 1\ . (y) dx 0 tx f(x) = 2 77 -z x sinh (rr x) J 00 g (y) f( . (y) y -1 dy 0 u was given by Kontorovich and Lebedev (1938, 1939) who used these formulas in the solution of certain boundary value problems, Further applications to boundary value problems were given by Lebedev and Kontorovich, and the mathematical theory was developed by Lebedev (1946, 1949). It should be noted that K ix (y) is real when x is real andy is positive. Alternative forms of this inversion were stated in the papers referred to above. See also Erdelyi, eta), (1953, p. 75). In this chapter we give a short list of integrals corresponding to the first of the above formulas; integrals corresponding to the second formula may be evaluated by means of the tables given in chapter X. We take y to be a positive real variable, although some of the integrals given below are valid for complex y. 173 174 INTEGRAL TRANSFORMS REFERENCES Erd~lyi, Arthur, Wilhelm Magnus, Fritz Oberhettinger, and FoGo Tricomi, 1953: Higher transcendental functions, vol. II, McGraw-Hill. Kontorovich, Mo] o and NoN o Lebedev, 1938: 1 o Exper. Theor. Phys. USSR, 8, 1192-12060 Kontorovich, MoJo and NoN o Lebedev, 1939: A cad. Sci. USSR, 10 Phys. 1, 229-241. Lebedev, NoNo and MoJo Kontorovich, 1939: 1. Exper. Theor. Phys. USSR, 9, 729-741. Lebedev, NoN o' 1946: A cad. Sci. USSR, Doklady 52, 655-658. Lebedev, NoN o' 1949: A cad. Sci. USSR, Doklady 65, 621-6240 KONTOROVICH-LEBEDEV TRANSFORMS 12.1. Formulas f(x) J"" f(x) K. (y) dx 0 u y>O ( 1) x sin (ax) · lim al < ~17 ~11y sinh a exp (-y cosh a) (2) cos ax lima!< ~17 ~17 exp(-y cosh a) (3) x tanh (17x) P -!,H ix (z) (?':l11y)~ e-zy (4) x tanh (77x) K. ((3) u: }277((3y) ~ ((3 + y)-1 exp(-(3- y) larg (31 < 17 11312 a exp ~ y _ 8a; ) (5) x sinh ( 11 x) K 2 ix (a) 2 5/2 y 1/2 largal < ~ 11 (6) x sin (~77x) K~ix(a) 173/2 y exp (-a-:: ) 2112 a112 largal < ~17 (7) cosh(ax) K ix((3) ~17K0[(y2+ (32+ 2(3y cosa) ~] IRe al + larg (31 < 11 175 176 INTEGflAL THANSFORMS 12.1 Formulas (cont'd) f(x) Joo f(x) K. (y) dx 0 u y>O (8) x (x 2 + n 2)-1 sinh (rrx) Xrr2 I (y) K (a) n n O<y<a x K . (a) Xrr2 I (a) K (y) a<y<oo tx n n a> 0, n = 0, 1, 2, 3, ... (9) x sinh (rr x) K ix (a) K ix ({3) ,_'_ exp [ _2:_( ~+_!'_+a~) J I arg al + larg{31 < ~'21T 4 2 {3 a y 2 ex [-(a+f'h J (lO) x sinh (X rrx) K X )a) K Xix(,B) 1T y -- 2z P 2(a{3)x largal + largfl < 77 2 y, z=(y +4a{3)' (ll) x sinh(rrx) KXix+A. (a) 0 0 < y < 2a x K Xix-r._(a) a>O 2 1T 2:: [(y + z)2A.+ (y-z)2A.] 2 2A.+1 a 2A. z 2a < y < oo z = (y 2-4a 2)y, (12) x sinh(rrx)i(A+ ix) 27.>-1 rr3/2(ay) A. (y +a)-A_ X r (A.-ix) K ix (a) X r (A. + X) K r._<r + a) larg al < rr, ReA.> 0 X sinh(2rrx) rCA.+ ix) 2(.,_ 1T 5/2 (-lr~ai) A. Kr._Cir-al) (13) r < x-A.) x1(A.-ix)Kix(a) a> 0, 0 <ReA.< X 12.1 KONTOROVICH-LEBEDEV TRANSFORMS 177 Fonnulas (cont'd) f(x) Joo f(x) K. (y) dx 0 t% y>O (14) x sinh (rrx) ['(,\ + ~ix) 2772 (;: )2 ~ K 2~ (z) x f'(A-Yzix)Kix(a) z=(y2+a2)X \arg a\ < Yz rr, ReA> 0 x tanh (rrx) K ix (a) ( )X (15) 1 rr ay 2 2 , - 2 2 exp [-(y + a ) X] ['(% + ~ ix) [' (%- ~ix) 2 y +a \arg a\ < ~" (16) x sinh (rrx) f'(A + ix) ['(A-ix) T112 17312(ay /z)~ (B2 _ l)X~-~ X p :X-~ix ({3) K ix (a) x K~(z) ~ =(y2 + a2 + 2a{3y)X \arg a\ < Yz rr \arg({3- 1)\ < 7T ReA> 0 MISCELLANEOUS TRANSFORMS 179 CHAPTER XIII FRACTIONAL INTEGRALS We call l Jy g(y; fl.)= lR)fCx); rl =m f(x) (y-x).U-1 dx J1 0 the Riemann-Liouville (fractional) integral of order fl., and l !00 1 h (y; Jl) = ~ lf(x); y} =-- f(x) (x-y).u-dx .u ['(fl.) y the Weyl (fractional) integral of order fl., of f(x). In general, J1 andy are envisaged as complex numbers, the path of integration being the segment x = yt, 0 < t < l in g, and one of the rays x = yt, t > l or x = y + t, t > 0 in h. Many authors denote g (y; Jl) by I.U[ or I~[, and h (y; Jl) by K .U[ or K ~f. The integral 1 fa -- f(x) (x-y),u-1 dx, [' (p) y sometimes denoted by I!:[, may be expressed as lR )f(a -x); a-y} by a change of variables; alternatively it may be written as ~)f(x); y} by adopting the convention that f(x) = 0 when x >a. Fractional derivatives of order a may be defined by the formulas d" D0~ f(x) =--lR lf(t); x} dx n n-a n-l <Rea< n d" Da·f(x) =- ~ alf(t); x} oo dx" n-n-l <Rea< n so that tables of fractional integrals may be used to evaluate fractional derivatives. 181 182 INTEGHAL THANSFORJVIS On Jl• WI!. we give a brief selection of books and papers which contain information on the theory of fractional integrals and derivatives, Hardy and Littlewood (1928) give further references, As far as we know, there is no extensive table of fractional integrals although numerous integrals of this kind occur in aln:ost any table of integrals. An extension of the operators ~~ and lffi was introduced by Kober J.L J.L 0940) and Frdcl yi ( 1940). 1-:ober (l941 b) also discussed fractional in- tegrals of imaginary order, Fractional integration by parts over a finite interval is expressed by the formula fag I (x; p) {2(a-x) dx =fa {1 (a-x) g2 (x; Jl) dx, 0 0 and ~·as discussed by Young and Love (1938), For the infinite interval the forn1ula is foo {1 (x) g 2 (x; p) dx = foo lz I (x; J.i.) {2 (x) dx, 0 0 and was discussed by Kober (1940). In these forn1ulas g 1 2 = 3t[1 2, and /z I= 'l£[1' ' ' The operators ~J.L' ThJ.L are connected with differentiation and integra­ tion, and with each other, by a nurr.ber of relations, We list a few here, others being given in the list of general formulas in sections 13.1 and 13.2. g (x; 1) = r f(t) dt, h (x; 1) = foo f(t) dt 0 X d d dx g (x; p) = g (x; 11-1), -· -lz (x · 11) = h (x · 11 -l) dx 'r 'r The functions g (x; 11) and h (x; p) may be regarded as J.l. times repeated indefinite integrals of f(x), the fixed limit being 0 in the case of g, and oo in the case of h. The connection of fractional integrals with other integral transforms may be seen from the following formulas, D lg(t; p); pl = p -J.L !J lf(t); pl ~· lh (x-11)· yl = e 'lzJ.Lniy-IL ').; lf(x)· yl u e ' ' (, e ' FRACTIONAL INTEGRALS 183 [' (l-s -fl.) ;llllg(x· u)· sl = 9Jilf(x)·, s + u1 ,,..., 1(1-s) ,..., [' (s) 9Il!h(x; f1.); sl = 9Il!f(x); s + fl.l [' (s + fl.) which may be used in conjunction with the tables of vo], I to evaluate fractional integrals by means of tables of Fourier, Laplace, Mellin trans­ forms and their inversions, These formulas may also be used to derive from a known pair of, say, Fourier transforms a new pair by means of integration of fractional order. The connection of fractional integrals and Laplace transforms is dis­ cussed in Doetsch (1937,p. 293-305) and Widder(1941, p. 70-75), Doetsch also discusses Abel's integral equation, g = 3-i JLf, For the connection of fractional integrals and Fourier transforms see Kober (1941 a, Lemma 3), For the connection of fractional integrals and Mellin transforms see Kober (1940), For the connection of fractional integrals and llankel trans­ forms see Erdelyi and Kober (1940) and Erdelyi (1940); see also 8.1 (13) to 8.1 (16), For fractional integrals in the theory of Fourier series see Zygmund (1935, P• 222 ff,), Fractional integrals occur in the solution by definite integrals of linear differential equations, In this context fractional integrals are often called Euler transforms (see, for instance, lnce, 1927, p. 191 fT.), '5?J1-f being the Euler transform of the first kind, and Sj]Ji-f the Euler transform of the second kind, off, ~I. n iesz ( 1949) has developed a theory of fractional integrals of functions of several variables: this theory has been applied by Riesz and others to the solution of partial differential equations , (See, for instance, ~ 13aker and Copson, 1950, Chap. I, ~ 7.) From the fractional integrals given in the tables, further fractional integrals may be derived by the general methods enumerated in the intro­ duction to vol, I, Ly means of the general formulas stated above and in sections 13.1 and 13.2, and by means of the connection, also stated above, between fractional integrals and other integral transforms. 184 INTEGRAL TRANSFORMS REFERENCES Baker, B.B. and E.T. Copson, 1950: The mathematical theory of Huygens' prin­ ciple. Oxford, Clarendon Press. Doetsch, Gustav, ·1937: Theorie und Anwendung der Laplace-Transformation. J:!erlin, Springer. Erde'lyi, Arthur, 1940: Quart. ] • Math., Oxford Ser. ll, 293-303. Erd~lyi, Arthur and Hermann Kober, 1940: Quart]. Math., Oxford Ser. ll, 212- 221. Hardy, G.H., 1918: Messenger of Math. 47, 145-150. Hardy, G.H. and J .E. Littlewood, 1925: Proc. London Math. Soc. (2) 24, xxxvii­ xli. Hardy, G.H. and J.E. Littlewood, 1928: Math. Z. 27, 565-606. Hardy, G.H; and J .E. Littlewood, 1932: Math. z. 34, 403-439. Hardy, G.H., Littlewood, J.E. and G. Polya, 1934: Inequalities, Cambridge. Ince, E.L., 1927: Ordinary differential equations. Longmans, Green & Co., London. Kober, Hermann, 1940: Quart.]. Math., Oxford Ser. ll, 193-211. Kober, Hermann, 1941a: Quart.]. Math., Oxford Ser., 12, 78-85. Kober, Hermann, 1941b: Trans. Amer. Math. Soc. 50, 160-174. Kuttner, Brian, 1953: ?roc. London Math. Soc. (3) 3, 480-497. Love, E.R ., 1938: Pro c. London Math. Soc. (2) 44, 363-397. Riesz, Marcel, 1949: Acta Math. 81, 1·223. Weyl, Hermann, 1917: Vierteljschr. Nat~rforsch. Ges. Ziirich, 62, 296-302. Widder, O.V ., 1941: The Laplace transform. Princeton University Press, Prince- ton, New Jersey. Young, L.C. and E.R. Love, 1938: Proc. London Math. Soc. (2) 44, 1-28. Zygmund, Antoni, 1935: Trigonometrical series, Warszawa-Lwow. FRACTIONAL INTEGRALS 13 .1. Riemann-Liouville fractional integrals f(x) [1(11)]-1 JY f(x) (y-x)i-L-1 dx 0 = g(y; 11) ( l) f(ax) a -f.L g (ay; 11) (2) f(a/x) ayf-L-1 Th)t_f.L_1 f(t); a/y! For tables see sec 13.2. (3) f'(x) g (y; 11-1) -f(O) yi-L-1 /1 (11) (4) J% f(t) dt g (y; 11 + 1) 0 (5) g (x; v) g(y;l1+v) (6) l Re 11 > 0 yf-L ['(11 + 1) (7) xv-1 Re 11 > 0, Re v > 0 i(v) f.L+v-1 1(11+v) y (8) (x + a)11 Re 11 > 0 av yf-L 2F1 (1,-v; 1+11; -y/a) [' (11 +"1) larg(y/a)l < rr 185 186 INTEGRAL TRANSFORMS 13.1 Riemann-Liouville integrals (cont'd) f(x) (9) xv-1 (x + a)A. Re 11 > 0, Rev> 0 (10) xv-T(x2+az)A. Re 11 > 0, Re v > 0 k = 1, 2, ••• ' Re 11 > 0, Re v > 0 (12) x -Y, (x + 2)-y, l[(x + 2)y, + xY,]zv + [(x+ 2)y, -xY,Fv! fie 11 > 0 a A. y.f.L+v-1 1 (v) 1(/1 + v) x 2F, (-.\, v; 11+ v; -y/ a) a zA. y.f.L +v-1 1 (v) 1(/1 + v) I arg y /a I < rr ( v v+ 1 11+ v 11+ v+ 1 X F -A,---·- . 3 2 2' 2 , 2 ' 2 , -:: ) Re (y/ a)> 0 akA. y.f.L+v+1 1 (v) 1 (/1 + v) ( v v+ 1 v+k-1 X k+1 Fk -.\, k '-k-, ... '-k--; 11+ v+k-1 -lc -, --k-' ... ' ---,-k--; larg(y/a)l <rr/k 2.f.L+X TTY, [y (y + 2)]XJ.L-l( X p Y, -J.L(y + 1) v-x largyl < rr 13.1 FHACTIONAL INTEGH ALS 187 Riemann-Liouville integrals (cont'd) f(x) [I'(Jl)r1 J Y f(x) (y-x)J..L-1 dx 0 (13) xJ..L-1 ea"' He 11 > 0 Y, Y, I rr '(y/a)11--' exp(~ay) xiJ..L_y,(~ 2ay) Xv-1 eax l~(v) yJ..L+v-1 1F1(v; Jl+ v; ay) (14) Re /l > 0, He v > 0 1(/l + v) x"'-1 exp(axk) l~ (v) J..L+v-1 (15) r (/l + v) y He/l>O, Re v > 0 k = 2, 3, 4, ..• (v v+l v + k -1 xkFk k' -k-, ••• ' k ; Jl+V /l + v+ 1 Jl+v+k -1 ~--k-, ... , k ; ark) (16) x-11--1 exp(-a/x) Re /l > 0 a-11-yJ..L-1 exp(-a/y) larg rl < rr (17) x-211-exp(-a/x) Re/l>O (rry)-y, ay,-11-exp (- 2ay) xK J..L-Y, (;y) Re(a/y)>O (18) xv-1 exp(-a/x) Re/l>O Y,v-Y, -K ( a ) a 2 2 y exp -~ x WK,Y.v (;) K = ~-Jl-~ V, Re(a/y)>O 188 INTEGRAL TRANSFORMS 13.1 Riemann-Liouville 'integrals (cont'd) f(x) [1(/1)]-1 JY f(x) (y-x)J.L-I dx 0 J.L (19) exp(ax ~) Re 11 > 0 y ~(~)~-J.L~J.L +~ +17 2a y • l(fl + 1) [ ~ ~ x IJ.L+~(ay ) + LJ.L+~ (ay )] (20) x-~ exp(ax ~) ~(~ )~-J.L ~J..L-~ 17 2a y ' [ ~ X x IJ..L-~ (ay ) + LJ..L-~ (ay )] (21) xv-l exp(ax ~) YJ..L+v-1 I (v) I' (fl + v) Re 11 > 0, Rev> 0 ( 1 a2 y) x IFz v;2, fl+v;-4- ayJ..L+v-~ r (v + ~) + l(fl+V+~) X ( l 3 1 a2y) IFz v+2;2, fl+V+2;4 (22) x-312 exp(-a x-~) f1=1 2a-1 exp(-ay- ~) (23) x-J..L-Y, exp(-a x-Y,) 2J..1.+~ 17-~ a~-J..Ly~J..L -~ He 11 > 0 x KJ..L_~(ay-~) Re (ay- ~) > 0 (24) ~v-I log x yJ.L+v-1 r(v) Re 11 > 0, Rev> 0 l' (fl + v) x [logy + 1U(v)-t/; (fl + v)] 13.1 FRACTIONAL INTEGRALS 189 Riemann- Liouville integrals (coot 'd) f(x) [r(IL)]-1 J:r f(x)(y- x)JJ.-1 dx 0 (25) xiL-1 sin (ax) Re IL > 0 1T~(y/a)IL-~ sin(~ay)JJL_ ~(~ay) x11-1 sin(ax) YJL+v-1 r(v) . (26) [ 1 F, (v; IL + v; t a y) Re IL > 0, Rev> -1 2 i r (IL + v) -1F, (v; IL + v;-iay)] (27) sin(ax ~) Re IL > 0 2Ji.-~ ~ ~ -JL ~JL+~i J ( ~) 11 a y JL+Y, ay (28) x-~ sin(ax~) Re IL > 0 y, 2Ji.-~ ~ -JL Y,JL-~ ( ~) 1T a y HJL-~ ay (29) x11-1 sin(ax ~) ayJL+v- ~ r<v+ ~) r <~I+ v + ~) Re IL > 0, Rev >-~ ( l 3 l a2y) x F v+-·- IL+v+-·---12 2'2' 2' 4 (30) x-JL-~ sin(ax-~) 2Ji.-~ ~ ~ -JL XJL-~ J. ( -~) 1T a y ~-JL ay 0 < Re IL < l, a>O jargy\ <11 (31) xiL-1 cos(ax) Re 1L > 0 ;,~(y/a)JL-~ cos(~ay) JJL_~(~ay) (32) x11-1 cos(ax) YJL+v-1 r(v) [1F1(v; IL+v; iay) Re IL > 0, Rev> 0 2r(IL+v) + 1F, (v; IL+v;-iay)] 190 INTEGRAL TRANSFORMS 13.1 Riemann-Liouville integrals (cont'd) f(x) [r(/l)r1 J Y f(x) (y-x ).U_, dx 0 .u (33) cos(ax~) Re 11 > 0 y 2.u-~ ~ -.u ~ - a TT r<11+ 1) ~ ,u+~ ~ xy H,u+~(ay ) (34) x-~ cos(ay ~) Re /1 > 0 2.u-~ ~ ~-.u ~.u-~ J ( ~) 1T a y .u-~ ay (35) xv-1 cos(ax ~) yj..l.+v-1 r(v) r(/1 + v) Re 11 > 0, Rev> 0 ( 1 a2 y ) x I Fz v; 2' /1 + v; --4-- (36) x-.u-~ cos (ax-~) 2J..I.-~ ~ ~ -.u ~.u-?ri y ( -~) - TT a y ~-.u ay 0 < Re Jl < 1, a>O larg y I < 1T (37) Pn(l-yx) Re p. > 0 n!y.U p<.u. -P->n-rr) r (/1 + n + 1) n (38) xv-1 pn (1-yx) r(v) y.u+v-1 r (p. + v) Re 11 > 0, Rev> 0 x 3F/-n,n +1,v;1,/l+v; ~yy) (39) X A_-~ c"-(l -y X) (2A\ r<A + ~) n r(A+/l+n+~) ReA> -1, A I= 0, -~ Re 11>0 xy"-+.u -~ p(a,,B)(l-yy) n a= A+ 11-~. {3=A-/1-~ 13.1 FRACTIONAL INTEGRALS 191 Riemann-Liouville integrals (cont'd) f(x) [f'(ll)r1 JY f(x) (y -x).u-t dx 0 (40) xv-t C,\(1_ yx) (2..\)n f'(v) .u+v-1 y n n! [' (11 + v) 2..\ I= 0, -1, -2, ••• Re 11 > 0, He v > 0 x 3F2(-n,n+2..\,v;..\+~;11+v;~yy) (41) v-tC,\( X) ( )" (..\)" f'(v) .u+v-1 x 2n yx -1 y He 11 > 0, He v> 0 n! [' (11 + v) x 3F2(-n,n+..\,v; ~. 11+v; y2y) (42) v-t cA. ( y,) X 2n+l yx 2(-1)" yy.u+v-Y, He 11 > 0, He v >-~ (..\) +t f'(v+ ~) X n n! ['(11 + v + ~) x 3F2 (-n, n + ..\ + 1, v + ~ ; 3 1 2 ) 2' /l+V+Z; y y X a P (a' ,L3 ) ( 1 -y X) ['(a + n + 1) Ya+.u (43) f'(a+/1+n+1) n Rea> -1, Re p. > 0 x p<a+.u, J3-.u>u-yy) n (44) x,L3 p(a,J3)(yx-1) [' ({3 + n + 1) J3+.u y n [' ({3 + 11 + n + 1) Re {3 > -1, fle 11 > 0 x p<a-.u, ,L3+.u)(yy- 1) n 192 INTEGRAL TRANSFORMS 13.1 Riemann-Liouville integrals (cont'd) f(x) [1(J.L)r1 Jy f(x) (y-x).IL-1 dx 0 (45) -/3-rn-1 (1 ~ )/3 1(-/3-J.L-n) -/3-n-1(1 ~ )/3+_1.1. X -2yx r <-/3 -J.L) Y -2YY x p~a,f3)(1-yx) O<ReJ.L<-Re/3-n X pn(a,f3+.1L)(1-yy) (46) xt\-1 p<a,f3)(l_ yx) 1 (a + n + 1) 1 (,\) y"'-+_~.~.-1 n n! 1 (a + 1) 1 (,\ + J.L) Re ,\ > 0, Re J1 > 0 x 3F2 (-n, n + a+ /3 + 1, ,\; a+ 1, ,\ + J.L; ~ y y) (47) x"'--1 P (a,f3)(yx-1) (-1)" 1(/3 + n + 1) 1(,\) A.+_~.~.-1 n n! 1 (/3 + 1) 1 (,\ + J.L) y Re ,\ > 0, Re J1 > 0 x 3F2 (-n, n + a+ /3 + 1, ,\; /3 + 1, ,\ + J1; ~ yy) (48) x"'--1 (1-~yx)f3 pn(a,/3\1-yx) ['(n +a+ 1) 1(,\) A.+_~.~.- 1 n! 1~ (a+ 1) 1 (,\ + p.) y Re ,\ > 0, Re J1 > 0 x 3F2 (a+ n +1, -/3-n, ,\; a+1, ,\ + p.; ~yy) (49) xa L a(f3x) 1(a + n + 1) + + Y a .1-L L a .IL(/3 Y) n 1(a+p.+n+1) n Re a> -1, Re J1 > 0 (50) x"'--1 La (f3x) 1(a+n +1)1(,\) A.+_~.~.-1 y n n! 1 (a+ 1) 1 (,\ + p.) Re ,\ > 0, Re J1 > 0 x 2F2(-n, A; a+1, A.+p.; f3y) 13.1 FRACTIONAL INTEGRALS 193 Riemann-Liouville integrals (cont'd) f(x) [r(/'L)r1 Jy f(x) (y-x)f.L-1 dx 0 (51) x,\-1 e -f3x L a({3x) r(a+n+1)r(> .. ) .\+J.L-1 y n n! ['(a+ 1) r(A + 11) ReA> 0, Re 11 > 0 x 2F2(a+n+1,A; a+1,A+/'L; -f3y) (52) [x (1 + ~~ yx)r~" P~(l + yx) (2/y)~f.L[y(l + 72yy)]~J.L-~,\. ReA< 1, Re 11 > 0 X p~-f.L(l + yy) larg YYI < TT (53) xK+~,\-1 (1 + 72 yx)-~A (~y)-~Ar(K) YK+j.L-1 x P~(l + yx) ro-A) r(K + 11) Re K > 0, Re 11 > 0 X 3F2(-v, 1+v,K; 1-.\,K+/'L;-72yy) IYYI < 1 (54) [x(1-x)r~ " p.\(1_ 2x) ll [y U-y)]~J.L- ~.\ p~-f.L(1- 2y) ReA< 1, Re 11 > 0 0 < y < 1 (55) xK+~.\-1 (1-x)-~ApA(l-2x) r(K) rK:!-J.L-1 ll nK + 11) r (1 -A) Re K > 0, Re 11 > 0 X 3F/-v, 1+ v, K; 1-.\, K+/'1; y) 0<y<1 (56) x,\-1 J (ax) r(A+v) (~a)"yA+J.L+v-1 ll r(v+1) r(A+/'L+v) Re 11 > 0, Re(A+v) >O (A+v A+v+1 x2F3 --, ;v+1, 2 2 A+/'L+V A+/'L+v+1._a2y2 ) 2 ' 2 ' 4 194 INTEGRAL TRANSFORMS 13.1 Riemann-Liouville inteiU'als (cont'd) f(x) [f'(ll)r1 JY f(x) (y -x)J.L-1 dx 0 (57.) xv e±iax J (ax) (2a)v YJ.L+2v f'(v + ~) v 7T y, [' <!l + 2 v +1) Re /l > 0, Rev >-~ x 1F1 (v + ~; /-(+ 2v+1; ± 2i ay) (58) kv-1 ±iax J ( ) 2-v av YA.+J.L-1 ['(A) x e v ax ['(A + /l) [' (v + 1) ReA> 0, Re /l > 0 x 2F;_ (A, v+ ~;A+/-(, 2v+ 1; ± 2i ay) (59) -y, J ( y,) x 2v ax 77X [J)~ ayX)].2 /l = ~. Rev >-~ (60) xXv-Y, J (axy,) 7Ty, (2~ X [Jv(~ayy,)f v /l = v + ~. Rev >-~ (61) xY,v-y, J (axy,) v (~a)1-v17X Yv-Y, J)~ayy,) /-(=V-~, Rev>~ x J (~ayy,) v-1 2 (62) x-Xv-X J (axy,) "X (2ay y Jv(~ayy,) J_v(~ayy,) v /l = ~-v, Rev<~ (63) xXv J (axy,) v 21-L -J.L XJ.L+XvJ ( X) a y J.L+v ay Re /l > 0, Rev> -l (64) x-Xv J (axy,) Re /l > 0 22-v a-J.LyX J.L-Y,v s (ayy,) v [' (/-() [' (v) J.L+v-1, J.L-v .13.1 FRACTIONAL INTEGRALS 195 Riemann-Liouville integrals((cont'd) f(x) [l(p)r1 JY f(x) (;y-x)J.L-1 dx 0 (65) x"--~v-1 J (ax~) av y"-+J.L-1 ['(,\) v 2v ['(,\ + p) ['(v + 1) Re ,\ > 0, Re p > 0 x 1F2 (,\; v+1, A+ p; -~a 2 y) (66) x"-.-v-1 [J (ax ~w (~a)2v y"-+J.L-1 ['(,\) v [[' (v + 1)]2 ['(A.+ p) Re ,\ > 0, Re p > 0 For several particular cases x 2F;(A, v+~ ; A.+p, v+1, 2v+ 1; 2 ) . see I3ailey, W.N., 1938: -a y Quart.!. Math. Oxford Ser., 9, 141-147. (67) x"--1 J (ax~) J (ax~) [' (,\) sin ( vrr) "-.+J.L-1 v -v J.I7T [' (,\ + p) y Re ,\ > 0, Rep> 0 x 2F3(~, A.; 1+v, 1-v, A+p; -a\) (68) x-~ Y (ax~) v rr~ ctn(vrr) [J~)~ay~)]2 ll = ~. -1<Rev<1 -rr~ csc(vrr) [J_~v(~ay~W (69) x~ v Y (ax X) v 2~ a-X y~v+!4 csc(vrr) ll = ~. Rev> -1 x [cos(vrr) Jv+~ (ay~) -"-v-~ (ay~)] (70) x-~v Y (ax~) v 2~ -~ ~ -~ v ( ) a y esc vrr ll = ~. He v < 1 x [cos(ll7T) "v-~ (ay~) ~ ] -J~-v(ay ) 196 INTEGRAL TRANSF ORMS 13.1 Riemann-Liouville integrals ( cont'd) f(x) [f'(/1)]-1 JY f(x) (y-x)IL-1 dx 0 (71) x~v-~ Y (ax~) 77~ c:y)v J)Xay~) Y )Xay~) v 11 = v + X, Rev>-X (72) x-~v-~ Y (ax~) 77~ ( ~ )v J (~ay~) Y (~ay~) v 2y -v 2 v 2 11 = ~-v, Rev<~ (73) x~v Y (axy,) (2_ J yY,I-L+Y,v ctn(V77) J + (ayy,) v a 1-L v Re 11 > 0, Rev> -1 2v+2y~I-L+~v['(v+ 1) + 77 al-L [' (11) x s (ay~) !-L-v-1, 1-L+v (74) x~v Y (ax~) (2_ J Y.!-L+~v Y ( ~) v Y 1-L+v ay a Re11>0, Re v > -1 2v+2 y~I-L+~vf'(v+ 1) + 77 al-L [' (11) s ( ~) x 1-L-v-1,!-L+v ay (75) x-~v Y (ax~) y~I-L-~v ctn(v77) y, v 2v-2 ai-LI'(/1) f'(v) s!-L+v-1,).4-v(ay ') Re 11 > 0, Rev< 1 -(2/ a)!-L y y, g-~ v esc (v77) x J (ayy,) 1-L-v 13.1 FRACTIONAL INTEGRALS 197 Riemann-Liouville integrals (cont'd) f(x) [f'(/l)r1 J Y f(x) (y-x)J.L-1 dx 0 (76) x,\-1 Y (axl-:) v 2-v v A.+J.L+l-:v-1 t ( ) ay cnv77 Re A> ~IRe vi, Re /l > 0 f'(A+ ~v) X f'(l + v) ['(A+ /l + ~ v) x 1J<;(A+ ~v; l + v, A+ /l+ ~v; -~a2y) 2v -v A.+J.L-l{v-1 ( ) -a y esc V77 f'(A- ~v) X r n -v) r (A + ll -~ v) x 1F2(A-~v; 1-v, A+/l-~v; -~ a2y) (77) xv e ±ax I (ax) (2 a)v YJ.L+2v [' (v + ~) v 17 l{ [' (/l + 2 v + l) Re /l > 0, Rev >-~ x 1F1(v+~; /l+2v+l; ±2ay) (78) ,\-1 ±ax I ( } (~a)vy A.+J.L+v-1 f'(A+ v) x e v ax [' (v + l) f'(A + /l + v) Re /l > 0, Re (A+ v) > 0 x 2F2 (v+ ~. A+ v; 2 v+ l, /l+ A+ v; ± 2ay) (79) x-~ I (ax~) 2V 77~ [I)~ ay~}F /l = ~. Rev >-~ (80) x ~ v-~ I (ax~) 17~ ( 2: )v [Iv(~ayl-:)]2 v /l = v + ~. Rev >-~ 198 INTEGRAL TRANSFORMS 13.1 Riemann-Liouville integrals (cont'd) f(x) [r(Jl)r1 JY f(x) (y-x)J.L-1 dx 0 (81) x~v-~ I (axl{) v (~ )1-v l{ v-l{I (~ l{) 2 a TT y · v 2 ay Jl = v-~. Rev>~ x Iv_1 (~ ayl{) (82) x-l{v-~ I (axl{) rrl{ ( ;y) v Iv(~ ay l{) I_J~ ay l{) v Jl = ~-v, Rev<~ (83) xXv I (axx) v 2J.L -J.L XJ.L+Xv I ( X) a y J.L+v ay Re Jl > 0, Rev> -1 (84) xA.-Xv-1 I (axX) avyA.+J.L-1 r(.\) v 2v r(v + l) r(..\ + Jl) Re ,.\ > 0, Re Jl > 0 x 1F2(.\; v+1, A+Jl; ~a2y) (85) x-~ K 2)axx) ~ sec(vrr) [I)~ayX) Jl = ~. -~<Rev<~ + I_v(~ayx)] Kv(~ ayX) (86) xXv-X K (axX) v rr~(2y/a)v I)~ayx) K)~ayx) Jl = v + ~. Rev >-~ (87) x"--1 K (axx) 2v-1 -v \+J.L-~v-1 ['(v)r(,.\-~v) v Re ,.\>~\Rev\, Re Jl > 0 a y r(A.+ JL-~v) ( v v a2y) x F .\--·1-v A+JL--· --1 2 2' ' 2' 4 1-v v A.+J.L+Xv-1 r<-v)r(.\+ ~v) +2 a y r(A.+Jl+ ~v) ~ v v a2 y~ x 1F2 A+-; 1+v, A+JL+-; -- 2 2 4 13.1 FRACTIONAL INTEGRALS 199 Riemann-Liouville integrals (cont'd) f(x) [1(1l)r1 JY [(x) (y -x'•~-'-1 dx 0 (88) x~11 H)yx~) (%y)-1-Ly~J.L+ ~ v "J.L+v(yy~) Re v >- 3/2, Re ll > 0 (89) i\-V/2-3/2 H ( ~) ['(A) Yv+ 1 Yi\. +J.L-1 x v yx 211 11~ ['(v + 3/2) t(A + ll) ReA> 0, Re ll > 0 ~ 3 3 y'y) x 2F3 1, A;-, v+-, A+ ~I;--- 2 2 4 (90) x~v L)yx~) (% )-1-L ~J.L+~ v ( ~) 2Y y LJ.L+v yy Re ll > 0, Rev> -3/2 (91) xi\.-v/2-3/2 L)yx~) Yv+1 y"-+J.L-1 ['(A) 211 11~ [' (v + 3/2) ['(A+ ll) ReA> 0, Re ll > 0 ~ 3 3 y2 y) x 2F3 1, .\;-, v+-, A+ll; --2 2 4 (92) i\-~K-~ ( ~) aK yi\+J.L['(A+ 1) x sK,v ax (K-V + 1) (K + V + 1) t(A+/l+1) ReA> -1, Re ll > 0 ( K-V-1-3 K+V+3 X 2F3 1,A+1;~--,A +JL+1; 2 2 -a:y) 200 INTEGRAL TRANSFORMS 13.1 Riemann-Liouville integrals (cont'd) (94) f(x) v-1 F ( x a1, ••• ,a; p q p v, b2, •••, bq; ax) psq+1 Re 11 > 0, Re v > 0 "9 ) v-1 F ( ,5 x pqa1, ••• ,ap; (96) (97) b 1 , ••• , b q·; ax) p,Sq+1 Re 11 > 0, Re v > 0 G •n (ax la1' •••' a P\ pq b1, ••• ,bq) p s q, Re 11 > 0 Re lJ. > -1 j = l, ... , m 1 c•n ~axial' ••• 'a p) pq . b b ' ' ••• ' q p + q < 2 (m + n ), R e 11 > 0 Reb.> -1 j = 1, ... , m 1 x M K-J.L ,A. (ay) + cos[(K-A)i7] WK-J.L,A.(ay)} YJ.L+v-1 r(v) r (Jl + v) YJ.L+v-1 r(v) r(Jl + v) larl < 1 if p = q + 1 x p+!Fq+l (v, a1, ••• , ap; Jl+V, b1, •••, bq; ay) I a r 1 < 1 if P = q + 1 I-LG•·n+1 (ay ,0, a1' •••' ap) y p+1, q+1 b, ••• ,bq,-11 larl < 1 if p = q I-LG••n+1 (a IO,a1, ... ,ap) y p + 1, q + 1 y. b b ,, ••• , q'-Jl largayl < (m + n-~p-~q)rr 13.2 FRACTIONAL INTEGRALS 201 13 .2. Weyl fractional integrals f(x) [r(f.L)r1 f" f(x) (x-y)J.L-I dx y =h(y; f.L) (l) {(ax) a -J.L h (ay; f.L) (2) f(a/x) ayJ.I.-I !R)t_J.I._I f(t); a/yl For tables see sec. 13.1. (3) f'(x) -h (y; fL-l) (4) J"" f(t) dt h (y; !l + l) X (5) h (x; v) h(y;f.L+V) (6) X-A_ 0 < Re fL <ReA r<A-11) J.I.-A. r(A) y (7) (x + a)-A. 0 < Re 11 <ReA r (A -fL) ( + a)J.I.-A. r (A) r larg(y/a)l < TT (8) x-A.(x + a)v A. r (A-11-v) YJ.L+v- 0 < R e 11 < Re (A-v) r (A-v) x 2F; (-v, A-!l-v; ,.\-v; -a/y) larg(a/y)l < TT or la/yl < 1 202 INTEGRAL TRANSFORMS 13.2 Weyl integrals (cont'd) f(x) [r(f!))-1 J00f(x)(x-y)J.L-1 dx y (9) x -A.(x 2 + a 2) v r(.\-fL-2v) J.L-A.+2v r<A-f!) y 0 < Re fL < Re(,\-2·v) . ( A-ft l+A-f! x3F2 -v,- 2--v, 2 -v; ~-v l+,\ -v--a2 ) 2 ' 2 ' y2 \r\ >\a\ or Re(a/y) > 0 ( 10) (x 2 -l)-~ [ (x + l) ~ -(x -l) ~ f v 2v+~ 17 -~ e (J.L-~ l7r i 0 < Re fL < l + Re v x(y2-l)~J.L-l( Q::~(y) \arg (y -l)\ < TT (ll) e -ax Re fL > 0 a-:-J.L e -ay Re(ay)>O (12) xJ.L-1 e -a% Re fL > 0 TT-~ (y/a)J.L-~ exp(-X ay) x KJ.L-~ (Xay) Re(ay)>O (13) -A. _;,.% Re fL > 0 v-~ -v-~ -~ayw ( ) x e a y e K.v ay 2 K = l -,\-fL, 2v=A-f! Re(ay) >O (14) x -2J.L exp (a/x) Re fL > 0 (rr/y)~ a~-J.Lexp(;y) x IJ.L-~ (;y) 13.2 FRACTIONAL INTEGHALS 203 Weyl integrals (cont'd) f(x) [1(/1)r1 Joo f(x) (x-y)Jl.-1 dx y (15) x-A exp(a/x) [' (,\ -/1) )1.-A F ( · · I ) [' (,\) r 1 1 ,\-11· ,\, a r 0 < Re 11 < Re ,\ (16) exp (-ax~) Re 11 > 0 2)1.+~ -~ ~-)1. ~J1.+l( K ( ~) Tr a r )1.+~ ay Re(ay~)>O (17) x-~ exp(-ax~) Re 11 > 0 2J1.+~ -~ ~-)1. ~)1.-l(K ( ~) Tr a r )1.-~ ay Re(ay ~)>O (18) x-A log x 0 < Re 11 < Re ,\ [' (,\-/1) YJ1.-A(logy+ rf;(,\)-rf;(,\-11)] ['(,\) (19) sin (ax) a> 0, 0 < Re 11 < l a -J1. sin (ay + ~ 11") (20) xJl.-1 sin (ax) ~ rr~ (y/a)Jl.-~ [cos(~ay) J~ _J1.(~ ay) a> 0, 0 < Re 11 < ~ -sin(~ay) Y~_Jl.(~ay)] (21) x-2J1. sin(a/x) Re 11 > 0 (;)~ a~-J1.sin(2:) X JJ1.-~ (;y) (22) sin (ax~) 2)1.-~ ~ ~ -)1. ~)1.+!< y ( ~) a> 0, 0 < Re 11 < ~ rr a y -y,-Jl.ay (23) x-~sin(ax~) 2)1.-~ ~ ~-)1. ~)1.-l( J. ( ~) a> 0, 0 < Re 11 < l rr a y ~ _J1. ay 204 INTEGRAL TRANSFORMS 13.2 Weyl integrals (cont'd) f(x) [f' (11)r 1 J"" f(x) (x -y )J.L-1 dx y (24) cos(ax) a> 0, 0 < Re 1-< < 1 a -J.L cos (ay + ~ 11") (25) J.L-1 cos (ay) -~ 77 ~ (y /a )J.L-X [sin (~ay ) JX _J.L(~ay) X a> 0, 0 < Re 11 < ~ +cos (~ay) Yx _J.L (~ay)] (26) x-2J.i.cos(a/x) Re 11 > 0 (77/y)X ax -J.L cos(a/2y) JJ.L-X (a/2y) (27) cos (ax X) 2J.L-X X X-J.L XJ.L+!O J ( ~) a> 0, 0 < Re 11 < ~ 77 a y -~ _J.L ay (28) x-X cos (ax~) 2J.L-X ~ X-J.L XJ.L-!0 y ( ~) - 77 a y X _J.L ay a> 0, 0 < Re 11 < 1 (29) Qv(x) 0 < Re 11 < 1 + Re v eJ.L7Ti(y2 _ 1)xJ.LQ~J.L(y) larg(y- 1)1 < 77 (30) (x2-1)~,\ Q~"-(x) eJ.L7Ti(y2 -l)X.\+XJ.LQ:"--J.i.(y) 0 < Re 11 < 1 + Re (v-A.) larg (y -1)1 < 77 (31) x-v e ia.x J (ax) eXiJ.L7T (2a)v-J.L['(~- 11 + v) v "X 1(1-11 + 2v) a> 0, 0 < Re 11 < ~2 + Re v x 1F1 (~-11+v; 1-11+2v; 2aiy) y > 0 (32) xXv-~ J (ax~) a>O -~· (':) [Jv~~'}-v~~") v 11 = ).1 + ~. -~<Rev<~ + J-v~;") Y v ~~")] y>O 13.2 FRACTIONAL INTEGRALS 205 Weyl integrals (cont'd) f(x) [r(/1)]-1 J oo f(x) (x-y)J..L-1 dx y (33) x~11-~ J (ax~) a>O -•'C:f J-v~;") -11 11 = v + }2, -}2 < Re v < h ~ay~) y>O xY -- -11 2 (34) x-~11 J (ax~) a>O 21-La-J.Ly ~J.L-~11 J (ay~) y>O 11 11-J.L 0 < Re J.L < }2 Re v + ~ (35) x-~11 J_ 11(ax~) a>O 2J.La-J.Ly ~J.L-~11[cos(vrr) J 11_jay ~) 0 < Re 11 < Y2Re v + ~ -sin (vrr) Y 11_J.L (ay ~ )] y>O (36) x!': J)ax ~) a>O 2 2/': -2/': J.L a y 0 < Re 11 < ~-Re A. {'yl 0 ) x cl3 --4 -jl,A.+}lv,>..-}lv y>O (37) x-11[J (ax~)JZ a>O -~ -11 -~11-~ (2 ~) 11 rr a y H 11 ay 11 = v-}2, Re v > }2 y>O (38) x~11-~ Y (ax~) a>O •: ~:' )" ~v~~}- v~~') 11 11 = v + }2, -}2 < Re v < }2 -y ~y") y ~y ") J 11 2 -,· 2 y>O 206 INTEGRAL TRANSFORMS 13.2 Weyl integrals (cont'd) f(x) [r(l1)]-1 J"" f(x)(x -y)f.L-1 dx y (39) xY,v Y (axy,) a>O 2J.L a-f.L YY,J.L+Y, v v 0 < Re 11 < % -~ Re v x [cos (vrr) Y_J.L_V(ayy,) -sin (vrr) J_J.L_)ay y,)] y>O (40) x -Y, v Y (ax y,) a>O 21-La-f.Ly Y,J.L-Y,v Yv_)ayy,) y>O v 0 < Re 11 < ~Re v +% (41) x-v J (axy,) Y (axy,) v v a>O -rr-Y, a-v y-Y,v-Y, J)2ay y,) 11=V-~, Rev>~ (42) x-"'[Y (axy,)F v a>O -Y, -v -Y,v-Y, [ ( Y,) rr a y H11 2ay /1 = v-~. Rev>~ -2Y 11(2ayy,)] y>O ' ( )" ( ') (43) xY,v-Y, H(1) (axy,) 1T 2 i :y H ~1) a: v 11 = v + ~. Rev >-~ xH<n(a ry,) lm (ayy,) > 0 -v 2 (44) xY,v-Y, H<0(axy,) 1Ty, i (;} [HC~~t) ]' -- -v 2 11 = v + ~. Rev >-~ lm(ay y,)>O (45) x-Y,v l/(11(axy,) Re J.L > 0 21-L a-f.LY.Y,J.L-Y, v H (1) (ayy,) v . v-J.L Im(ayy,) >O (46) xy, v-Y, H (21(axy,) rr:i ~:r )"' H~2~(~ ry,) v J.L = V+ ~' Rev >-~ x H~~ (a:y,) lm(ay y,)<O 13.2 FRACTIONAL INTEGRALS 207 Weyl integrals (cont'd) f(x) [r(p)r' Joo f(x) (x-y)~'--' dx y (47) x~ v-~ H<2>(ax~) rr:i ~~ )v [H~~ (~r~) J 2 -v /1 = v + x, Rev>-X Im(ay ~)<O (48) x-~v H<2>(ax~) v Re 11 > 0 2~'-a-~'- ~~.~--~v f/(2) (a ~) r v-~.~-r Im(ay ~) < 0 (49) x -v e -ax I) ax) (2a)v-~.~-r<X- 11 + v) rr~ r{l-p + 2v) 0 <He 11 < lt2 + Re v x 1F1(X-p+v; l-p+2v; -2ay) Re(ay) >O (50) X-A. e -ax I (a X) v rr -~ (2 a)A.y~'- 0 < Re 11 < X+ Re ..\ "(I ~-~0 ) x G 23 2ay -p,v-..\,-v-,\ Re(ay) >O (51) x -~.~--~ e -ax K (ax) Re 11 > 0 rr~ (2a)-~ y-1 e-ay w (2ay) v -J.L,ll Re(ay) >O ~ ~~.~--v-~ r('' ) (52) x-vearK (ax) rr y 12-p+v v (2a)~~.~-+ ~ r(X + v) 0 < Re 11 < ~2 + Re v X eay W~IJ.,v-~)2ay) larg(ay)l < 3rr/2 208 INTEGHAL THANSFORMS 13.2 Weyl integrals (cont'd) f(x) [f'(ll)]-1 J oo f(x) (x-y)J.L-1 dx y (53) x-v e-ax Kv(ax) Re 11 > 0 rrX (2 a)-y, J.L-X y y, J.L-v-Y, X e-ay lf'_XJ.L,v-XJ.L(2ay) He(ay) >O (54) x-A._ea"K)ax) rr-y, (2a)A..yJ.L cos(V7T) 0 < Re 11 < ~ + Re A. C I "'-~0 ) X G31 2 a y -1-4 v-A., -v -A. 23 \arg(ay)\ < 3rr/2 (55) x-A._e-ax K (ax) 1/ Re 11 > 0 rry, (2a)A..yJ.L ~ ( I o, Y.-A ) xc23 2ay -1-4 v-A, -v -A. Re(ay) >O (56) x-y, K (axX) 21/ ll = ~ -y, [K (1/ X)F 1T ·v12ay Re(ayy,)>O (57) XX v-X K (ax X) 1T-Yc(;)v [K.(:r') ]' 1/ ll = v + ~. Rev >-~ He(ayy,) >O (58) XX v-Y, K (ax y,) 17-Y, (2/a)v-1 Yv~X 1/ ll = v-~. He v > ~2 x K)~ayy,) Kv_1(Y:tay y,) He(ayx) >O 13.2 FRACTIONAL INTEGRALS 209 Weyl integrals (cont'd) f(x) [f'(/l)]-1 f" f(x) (x-y)Jl.-1 dx y (59) x-~v K (ax~) 1.1 Re /l > 0 2Jl.a-Jl.y~J1.- Y,v KV_Jl.(ay~) Re(ay ~)>O (60) x-A.K (ax~) Re /l > 0 2-2/-...-1 2/-... J1. 1.1 a y (a2y xG~ --13 4 --· ~v~>., -~v-J Re(ay ~)>O (61) x-v I (ax~) K (ax~) 1.1 1.1 X Y, -v -y, v-Y, 2 TT a y 11 = v-X, Rev> X x U)2 ayy,)-L)2 ayy,)] Re(ayy,) >O (62) x-v [K (axY.)f 1.1 ~ -v -~ v-~ K (2 y,) TT a. y v ay 11=v-X, Rev> X Re(ay~) >O (63) x~Jl.n -)ax~) (~r)Jl. [r_2Jl.(ay~) a> 0, 0 < Re 11 <X . 2 ~ J y>O +-So 2 (ay ) TT • J1. (64) x~v-Y, H (ax~) -v a>O TTY,~~ )1.1 ~-v (~~)] 2 11=v+X, -~<Rev< X y>O 210 INTEGRAL TRANSFORMS 13.2 Weyl integrals (coot'd) f(x) IJ' (/l)]-1 J"" f(x) (x -y )J.L-1 dx y x~ v H (ax~) (2/a)f.Ly~v+~J.L , (65) a>O [( ) ] [cos(vrr)HJ.L+)ay~) v Re ll > 0, Re (/l + v) < ~ cos !l+ v 1T Re(/l + ~v) <% +sin (!l7T) J_J.L_v(ay~)] y>O x~f.L[H (ax~)- Y (ax~)] 2 (2y )J.L (66) - - S (ay~) -J.L -J.L 17 a o,2J.L 0 < Re ll < ~ iarg(ay~)l < 1T (67) x~v-~ [H_)ax~)-Y_v(ax~)] ·: (2: )" {~. (·;') ]' ll = v + ~. -~<Rev< X + [y .(~' )]'} iarg(ay~)l < rr (68) x~v[H (ax~)- Y (ax~)] (2/a)f.Lcos(V7T) ~ +~ y v J.L v v cos [(/l + v) 1T] 0 < Re ll < ~-Re v x [BJ.L+)ay~)- Y J.L+)ay~)] larg(ay~)l < rr (69) x~v-~[I (ax~)-L (ax~)] ~(2r) I (•r') -v -v 1T~ a -v 2 ll = v + ~. -~<Rev<~ (ay~) Re (ay~) > 0 xK --v 2 .. :'. ·. 13.2 FRACTIONAL INTEGRALS 211 Weyl integrals (cont'd) f(x) [f' (11)]-1 J'>e f (x) (x -Y )~-1 dx y (70) x~v[I_v(ax~)- L)ax~)] cos (vrr) (:) Y~~+~v cos [(11 + v)rr] o < Re 11 < X -Re v x [I_~-v(ay ~)-L ~+)ay~)] Re(ay ~)>O (71) x~v S (ax~) f'(X-XA-11-Xv) a-~y ~~+~v f...,v f'(X-XA-Xv) 0 < 2 Re 11 < l -Re (,\ + v) ~ x SA.+ + (ay ) ~ .~ v larg(ay~)l < rr (72) x/...-~ e ~ax W (ax) f'(X-K-,\-11) -~~ ~~+/...-~ K,>\. ['(X-K-,\) a y 0 < Re 11 < X-Re (K + ,\) ~ ay W ( ) x e K+~ A ~ ay ~. + ~ larg(ay)l < 3n/2 (73) K-~-1 -~ax W ( ) K-1 -~ ay W ( ) x e K,A.ax y e K-~.A. ay Re 11 > 0 Re(ay)>O (74) /...-~ -~ax W ( ) -~~ ~~+/...-~ -~ay x e K,'Aax a y e Re 11 > 0 x WK-~~.r...-~~(ay) Re(ay)>O 212 INTEGRAL TRANSFORMS 13.2 Weyl integrals (cont'd) f(x) (75)x-peY,axw (ax) K,'fl. 0 < Re 11 < Re (p -K) Re 11 > 0 (77) x -A pFq (a,. ... , a P; b 1 , .. , , b q ; -a/ x) 0 < Re 11 < Re ,\, p ::; q + 1 [1(1l)r1 Joo f(x) (x-y)IL_, dx y YJL-p l (~ + A-K) l (~-A-K) xc:: Hp~: ~: .. llJ larg(ay)l < 377/2 JL-Pc30 , ~ I P 1-K ) y 23 a y p -ll• ~ + A., ~ -,\ Re(ay)>O x p+IFq+1(,\-llo a,, ... ,aP; ,\, b1, ... , bq; -a/y) lrl > lal or larg(a/y)l < 7T if p = q + 1 JLcm+1,n (a la,,,.,aP,O) p 2 q y p+1, q+1 y -b b p., 1 ' ••• ' q O<Re/1<1-Re a. J • 1 = l, ... , n (79) c;; ~· 1:·,: :::: :; ) p + q < 2(m + n) 0 < Re 11 < 1 -Re a. J j = 1, ... , n larl > 1 if p = q +1 ( Ia,, ... ,a ,0) JLGm ,n a P y p+1,q+1 y b b -J1, 1' ••• ' q larg(ay)l < (m +n-~ p-~q )TT CHAPTER XIV STIEL TJES TRANSFORMS We call g(y) =51 f(x); yl = Joo f(x) (x + y)-1 dx 0 the Stieltjes transform of f(x), Here integration is over the positive real x-axis, and y is a complex variable ranging over the complex y-plane cut along the negative real axis. Stieltjes transforms are iterated Laplace transforms, Glf(x); yl = .Qt.QI[(x); d; yl and accordingly , information about Stieltjes transforms is found in works on Laplace transforms, in particular in Widder (1941, Chapter VIII) and Titchmarsh (1937, sections 11.8, 1L9). Stieltjes transforms are also connected with the moment problem for the semi-infinite interval (Shohat and Tamarkin, 1943) and hence with certain continued fractions, We also give a brief list of generalized Stieltjes transforms of order p g (y; p) = 5 lf(x);yl = Joo f(x) (x + y)-p dx p 0 where x andy are as before, and p is a complex parameter, For the theory see Widder ( 1941, Chapter VIII). Generalized Stieltjes transforms of different orders are connected with each other, and with Stieltjes trans­ forms, by fractional integration accordin g to the formulas rCp) mp_1 Gp = 5 f'(p) SffiJ.L Gp = f'(p-fl.) GP_J.L. From the transform pairs given in the tables, further integrals may be derived by the methods mentioned in the introduction to vol. I, by the general formulas given in the tables, and by using the above formulas in connection with tables of Laplace transforms and fractional integrals. 213 214 INTEGRAL TRANSFORMS REFERENCES Shohat, J.A. and J.D. Tamarkin, 1943: 'f.he problem of moments. Amer. Math. Soc. New York. Titchmarsh, E.C., 1937: Introduction to the theory of Fourier integrals. Oxford. Widder, D.V., 1941: The Laplace transform. Princeton University Press, Prince­ ton, N.J. STIEL TJES TRANSFORMS 14.1. General fonnulas f(x) J'>O 1 f(x) (x + y)-dx 0 !argyl < TT (l) f(x) g (y) (2) x f(x) Joo f(x) dx-yg(y) 0 (3) (x + a)-1 f(x) largal < TT (y-a)-1 [g (a)-g (y)] (4) f(x)-f(a) a>O (y + a)-1 [~g(ae i7T) + ~g(ae-i7T) x-a -f(a) log(y/a)-g (y)] (5) g (xe i7T) -g (xe-i 7T) 2 TT i g (y) (6) f(ax) a>O g (ay) (7) x-1 f(a/x) a>O -1 g(a/y) y (8) f(x ~) g.(iyy,) + g(-iy y,) (9) f'(x) -y-1 f (O) -g '(y) 215 216 INTEGRAL TRANSFORMS 14.2 14.2. Elementary functions f(x) Joo 1 f(x) (x + y)-dx 0 \argy\ < rr ( l) -1 2n < x < 2n + 1 logl~y[rO~y)/r(~y + ~Wl 1 2n + 1 < x < 2n + 2 n = 0, l, 2, ,,, (2) (a+x)-1 \arg a\ < rr (a-y)-1 log(a/y) l 1 [ ~: -log~: ) J (3) 2 2 Rea> 0 a2 + y~ a + x X 2 1 2 [ ~ + Y log( 2_)] (4) 2 2 Rea> 0 a + x a + y 2 a (5) XV -1<Rev<0 -77 yv csc(rrv) XV rr(av-yv) (6) -- a+x (a-y) sin(VTT) \arg a\ < rr, -1 <Rev< 1 XV v-1 7T [ a y (7) a2 + x2 2 2 2 cos (~~vrr) a + y Rea> 0, -1 < Re v < 2 av yv ] + - 2sin(~~vrr) sin(vrr) xv-av 7T [ Yv -av ctn (VTT) (8) --- -1 <Rev< 1 x-a a+ y sin (vrr) + a: log(; ) J 14.2 STIEL T JES TRANSFORMS 217 Elementary fuoctions (cont'd) f(x) s= 1 ((x)(x+y)- dx 0 !argyl < TT r' < v) r <11 -v) y v-1 (9) xv-1(a+x)1-J.L l' (JL) aJ.L-1 largal < rr, O<Rev<ReJL x /'~ (JL-1, v; JL; 1-y/a) (10) x -p (a+ x) -a larg a! < rr TT csc(prr)y- P(a-y)-a 11_y/a(a, p) -Hea<Hep<1 (ll) e -ax He a> 0 -eaYf_:i(-ay) (12) e-ax 0 <X< b eay [Ei(-ab-ay)-l:.:i(-ay)] 0 b<x<oo (13) 0 0 <x < b -e ay Ei(-ab- ay) e -ax b<x<oo He a> 0 (14) xn e -ax He a> 0 (-l)n+1 y" eay F:i(-ay) + i (-1)"-r(r-1)!a-ryn-r r= 1 (15) X -X: e -ax Rea> 0 rry -y, eay l<:rfc (ay, y y,) (16) X -ax x e Rea> 0 y, -y, y, a ' ( y, y,) TT a -rry e Y Erfc a ' y ' (17) x-ve -ax f'(1-v)y-v eay·r(v, ay) Rea> 0, Rev< 1 ( 18) x-1 (1-e-ax) Rea> 0 y-1 [log(ayy)- eay Ei(-ay)] 218 INTEGRAL TRANSFORMS 14.2 Elementary functions (cont'd) f(x) {"' 1 r (x) (x + y)-dx 0 !argyl < rr ( 19) X v-1 e -a/x [' (l -v) y v-1 e a/y [' (v; a/y) l1ea>O, Hev<l (20) exp(-ax ~) Rea> 0 y, y, 2 cos(ay ') ci(ay ') -2 sin(ayy,) si(ay y,) (21) x -y, exp (-ax y,) He a> 0 -2y-y, [sin(ay y,) ci(ayy,) + cos(ay y,) si(ayX)] (22) lc y, x exp(-ax ) 1(2,\ + l)yA[exp(iay y, + Arri) fie a> 0, Re>..>-1 x l' (-2 >.., i a y ~) + exp (-i ay ~ -,\ rri) [' (-2>.., -i ay ~)] (23) [exp(ax y,)-l]-1 Rea> 0 log(ay y,)-(2ayy,)-1-tjJ(ay~) (24) (a+ x)-1 log x !argal < TT ~ (y-a)-1 [(logy)2-Oog a)2] (a+x)-1 log(x/a) l [log(:) J (25) 2 (y-a) !arga! <rr (26) (x-a)-1 log(x/a) a>O ~ (y + a)-1 lrr2 +[log (y/a}F l (27) x-Y, log(ax + {3) 2rry-~ log(a~ y~ + {3~) Rea> 0, Re {3 > 0 (28) XV Jog X -1 <Rev< 0 -TT yv esc (vrr) [logy- rr ctn (VTT)] 14.2 STIEL TJES TRANSFORMS 219 Elementary functions (cont'd) f(x) J 00 r (x) (x + y)-1 dx 0 !argyl < 77 (29) xv(a+x)-1 !ogx -77 esc (v77) (y-a)-1 [ av log a largal <77, -1 <Rev< 1 -yv logy- 77 ctn(V77) (av-yv)] (30) xv(a + x)-1 log(x/a) 77 csc(v77) (y-a)-1 [yv log(y/a) larg al < 77, -1 <Rev< 1 + 77 ctn (V77) (av- y V)] (31) sin (ax) a>O -sin (ay) ci (ay)-cos (ay) si (ay) (32) x~ sin (ax) a>O 77Y~ [sin (ay)-2 ~ sin (ay+~77) C(ay) + 2 ~ cos (ay + ~ 77) S (ay )] -2-~ 77~ a-~ (33) x -~ sin (ax) a>O 77Y-~ [2~ sin (ay :;-~ 77) C(ay) -2~ cos (ay+ ~ 77) S(ay)-sin (ay)] (34) x -v sin (ax) ~i 1(1-v)y-v[ e-iay ['(v,-iay) a> 0, -1 < Re v < 2 -e iay [' ( v, iay)] (35) 0 0 < x <a See Erd~lyi, Arthur, 1939: Proc. (x 2-a 2)-~ sin (bx) a<x<oo Edinburgh Math. Soc. (2) 6, 94-104. (36) sin (ax~) a>O 77 exp (-ay ~) (37) x-1 sin (ax~) a>O 77y-1 [1-exp(-ay ~)] (38) x-~ sin (ax~) a>O y -~ [exp (-ay ~) Ei (ay ~) -exp (ay ~) Ei (-ay ~)] 220 INTEGRAL TRANSFORMS 14.2 Elementary functions (cont'd) f(x) Joo 1 f(x) (x + y)-dx 0 \argy\ < TT (39) x'A. sin (ax~) -TTy'A.sec(ATT) sinh(ay~) a> 0, -3/2<ReA<l/2 -a -2'A. r (2 A) sin (A TT) x [1F1 (1; 1-2A; ay~) + 1F1 (1; 1-2A;- ay~)] (40) (x + {3)-1 sin (ax~) TT (y -{3)-1 [exp (-a {3 ~)-exp(-ay ~ )] a > 0, \arg{3\ <TT (41) x-j3sin(ax~ +f3TT) TTY -j3 exp (-ay ~) a > 0, -~ < Re {3 < 1 (42) sin (ax~- bx-~} a, b > 0 TT exp(-ay~- by-~) (43) x-~ [sin (ax~)]2 a>O ~TTY-~ U-exp(-2ay~)] (44) x-~ sin(ax~) sin(bx~) ~TTy-~lexp(-\a-b\ y~) a> 0, b>O -exp [-(a+ b) y~]l (45) x -~ sin (ax~) sin (bx y,) TTY-~ exp(-ay~) sinh(by~) a?_b>O (46) log({3x) sin (ax~) TT[Iog({3y) exp(-ayy,) a> 0, \arg{3\ < rr -exp (ay y,) Ei (-ay~) -exp(-ay~) Ei(ay~)] (47) x-y, log\sin(ax~)\ a>O TTY-y, log[~-~exp(-2 ay~)] 14.2 STIEL T JES TRANSFORMS 221 Elementary functions (coot'd) f(x) Joo 1 f(x) (x + y)-dx 0 Jargyj < TT (48) cos (ax) a>O cos (ay) ci (ay)-sin (ay) si (ay) (49) x -1 [cos (bx)-cos (ax)] y-1 [-ci (by) cos (by)+ si(by)sin(by) a, b > 0 +ci(ay) cos(ay)-si(ay) sin(ay) + log(ab-1)] (50) xX cos (ax) a>O 2-X TTX a-X-TTyYz[cos(ay) -2X cos(ay + ~ TT) C(ay) -2X sin (ay + ~ TT) S (ay)] (51) x -X cos (ax) a>O TTY -y, [cos (ay )-2 y, cos (ay + ~TT) C(ay) -2y, sin (ay + ~ TT) S (ay)] (52) x-11 cos(ax) r:;ro- v) y-11 [e iay r(v, iay) a> 0, -1<Rev<1 + e -iay r (v' -iay)] (53) 0 0 < x <a See Erd!dyi, Arthur, 1939: Proc. (x2-a2)-X cos(bx) a<x<oo Edinburgh Math. Soc. (2) 6, 94-104. (54) cos (ax y,) a>O -exp(-ayy,) Ei(ayy,) -exp (ayy,) Ei (-ayX) (55) x-X cos(axy,) a>O TTY-y, exp(-ayx) (56) x'A cos (ax y,) A_ ( ) X -TTy esc An cosh(ay ) a> 0, -1<Re-\<r:l -a-2/\_cos(ATT) r(2,\) x[1F, 0; 1-2,\; ayx) + 1F1 (1; 1-2,\;-ayYz)] 222 INTEGRAL TRANSFORMS 14.2 Elementary functions (cont'd) f(x) J"" f(x)(x +y)-1 dx I argyl < rr 0 (57) x-~ (x + {3)-1 cos(ax ~) rr(y-{3)-1 ur~ exp (-a{3 ~) a> 0, largf31 < rr -y-~ exp(-ay~)] (58) x-~ cos (ax~-bx-~) rry-~ exp(-ay~-by-~) a, b > 0 (59) x -312 [cos (ax 112)-cos (bx 112)] rry-:Y2[(b -a)y112 a> 0, b>O + exp (-by 112) -exp (-ay 1/2 )] (60) x-~ [cos (ax~)]Z a>O ~rry-~ [1-exp(-2ay ~)] (61) X-~ cos (ax~) cos (bx~) ~ rry -~I exp (-Ia-b I y ~) a> 0, b>O + exp [-(a+ b) y~]l (62) X-~ cos (ax~) cos (bx ~) rry-~ exp(-ay~) cosh(by~) a?_b>O X-~ rry -~[~({3/y-y/{3)sinh(2ay ~ )-1] (63) [{3 sin(ax ~W+ [ycos (ax~ )f [{3 sinh (ay ~ )]2-[y cosh (ay~ )]2 larg({3/y)l < rr (64) sin (2 ax~) rr[({3-y) /({3+y)- exp (-2ay~)] [{3 sin (ax~ )]2 + [y cos (ax~ )]2 [{3 sinh (ay~)]2-[y cosh(ay~)f larg({3/y)l < rr (65) X-~ log ({3 x) cos (ax~) rry-!-S [iog({3y) exp(-ay~) a> 0, larg f31 < rr + exp (ay~) Ei (-ay ~) -exp(-aylS) Ei(ay~)] 14.2 STIELTJES TRANSFORMS 223 Elementary fWlctions (cont'd) f(x) J 00 f (x) (x + y)-1 dx 0 !argyl <TT (66) x-~ log I cos (ax~)l a>O TTY-~ log[~+~ exp (-2ay ~)] (67) x-~ log[l+2f3 cos(ax ~) + {32] 2TTy- ~ log[l+f3 exp(-ay ~)] a> 0, 1131 < l (68) x -~ log I [b sin (ax ~)f 2 TTY-~ log [b sinh (ay ~) + [c cos (ax~ )]2} a, b, c > 0 + c cosh (ay~)]-2TTa (69) [cos(ax ~)]" sin (nax~) 2-n TTI[l + exp(-2ay ~)]" -ll a> 0, n = l, 2, ••• (70) x-% logltan(ax~)l a>O TTY-~ log[tanh(ay ~)] (71) x-~ logll + [b tan (axl{)JZ l 2 TTY-~ log [l + b tanh (ay ~ )] a, b > 0 (72) x -){ log 11 + [b ctn (ax~ WI 2TTy-% log[l + b ctnh(ay ~)] a, b > 0 (73) csch (TTx ~) -y-~ + t/J(~y ~+~)-tj;(~y~) (74) x-~ sech(TTx ~) y -~ [tf; (~ y ~ + %) -t/1 (X y ~ + ~)] (75) x-~ sin(ax~) See Ramanujan, Srinivasa, 1914: sinh (bx ~) Messenger of Math. 44, 75-85. x-~ cos(ax ~) cosh (bx~) x-~ cos (ax~) c +cosh (bx~) 224 INTEGRAL TRANSFORMS 14.3 14.3. Higher transcendental functions f(x) Joo 1 0 f(x) (x + y)-dx \argy\ < 77 (l) ci (ax) a>O ~ [ci (ay)F + ~ [si (ay)Jl (2) J)ax) a> 0, Rev> -1 77 csdvrr) [J)ay)- J)ay)] (3) xv Jv (ax) ~ rry v sec (vrr) [H _)ay) -Y -v (ay )] a> 0, -l/2 < Re v < 3/2 (4) xv+1 J (ax) v 2v 77-~ a-v-1 I'(v + ~) a> 0, -1<Rev<~ +~rrsec(vrr)yv+1 x [Y _)ay)- H_)ay)] (5) x-v J)ax) ~ 77 y-v[H)ay)- Y v(ay)] a> 0, Re v>-3/2 21-v Y-v - sv-1, )ay) r (v) (6) x'A J (ax) v a> 0, Re ,\ < 3/2 -rr/'-esc[(,\+ v)rr] J (ay) v R e (,\ + v) > - 1 2,\-1 a-,\ r (~ ,\ + ~ v) + r (1 -~2 .\ + ~ v) ( ,\ + v ,\-v a 2y ~ X F 1·1--- 1-·---· ---1 2 ' 2 ' 2 ' 4 2,\-2 a 1-,\y ['(~ ,\ + ~ v-~) -['(3/2-~,\+ ~v) 0 2 2) .3-.\-v 3-.\+v. a y X 1F2 1, ' '---2 2 4 14.3 STIEL TJES TRANSFORMS 225 Higher transcendental functions (cont'd) f(x) Joo t f (x) (x + y)-dx !argyl < TT 0 (7) xv sin (ax) J,,(ax) ll:; rr y v sec (vrr) [cos (ay-vrr) J )ay) a> 0, -1<Rev<li:; +sin (ay- 1.177) Y)ay)] (8) xv cos (ax) J (ax) v ll:;rryv sec(vrr) [sin(ay- vrr) J)ay) a> 0, -ll:; <Rev< ll:; -cos(ay- vrr) Y)ay)] (9) xv cos(ax + {3) J)ax) ll:; 77 y v sec( vrr) [sin (ay-vrr-{3) J)ay) a> 0, -~2<llev <li:; -cos (ay-vTT-{3) Y)ay)] (10) xy, v+k J (axy,) v 2(-l)k YY,v+k K)ay y,) a> 0, k = 0, 1, 2 -k-1<Rev<-2k+3 /2 (ll) xY,v+k- Y, J (axy,) v (-1)k TT sec (vrr) y y, v+k-Y, a> 0, k = 0, 1, 2 x [I)ay y,)-L_)ay y,)] -k-ll:; < R e v <-2k + 5/2 (12) xk-Y.v-Y, J (axy,) v rryk-Y,v-Y, [lv(ay y,)-L)ay y,)] a> 0, k = 0, 1, 2, ... Rev> 2k-5/2 (13) x'AJ (axy,) (: )2'A_ 1(,\ +X v) v 1(1-.\+ Xv) a> 0, R e (,\ + ll:; v) > -1 lle ,\ <% ( v v a 2y) X F. 1·1-.\-- 1-A+-·-- t 2 ' 2' 2' 4 -TT c s c [( ,\ + 12 v) TT] y 'A I )ay y,) .• 226 INTEGRAL TRANSFORMS 14.3 Higher transcendental functions ( cont'd) f(x) Joo f(x) (x + y)-1 dx \argy\ < TT 0 (14) sin (ax X) J0 (bxX) O<b<a rr exp (-ayX) ]0 (by X) (15) x -X sin (ax X) J 0 (bx X) 2y-x sinh(ayx) K 0(byx) 0 <a< b (16) cos (ax X) J 0 (bxx) 0 <a< b 2 cosh(ayX) K0(byX) (17) x-X cos (ax X) J 0 (bxx) rr y-x exp(-ayx) 1 0 (byx) 0 < b <a (18) xX v-X sin (axX) J (bxx) v 2yXv-X sinh(ay x) K )byx) 0 <a< b, -1 <Rev< 3/2 (19) x-Xvsin(axX)J (bxX) v rry-Xv exp(-ayX) Iv(byX) 0 < b <a, Rev >-~ (20) X X X 2yXv cosh(ayX)Kv(byX) x v cos (ax ) J (bx ) v 0 <a< b, -1 <Rev<~ (21) x -x v-X cos (ax X) J (bx X) v -X v-X ( X) I (b X) rry exp -ay v y 0 < b <a, Rev> -3/2 (22) [J)ax)]2 a>O 2 Iv(ayx) Kv(ayx) (23) Jv (ax X) J v(bx X) 2 lv(ayx) K)byx) b>a a, b > 0, Rev> -1 2 I)byx) K)ayx) b<a 14.3 STIELTJES TRANSFORMS 227 Higher transcendental functions (cont'd) f(x) J''" 1 f(x) (x + y)-dx jargy\ < TT 0 (24) x~ v-~J.L J (bx~) J (ax~) ).J J.L 2yY,v-~J.L IJ.L(ay ~) K)by~) 0 <a< b 2+Rep>Rev>-l (25) xf.._J (ax~) J (axy,) -2A_ -~ G23 ( 2 10, A, A+~ ) a TT 35 a y J.L . ).J O,p, q, r, s a> 0, ReA< l Re(2A+ p+ v)>-2 p = A+~p+~ v, q=A+~p-~v r=A- ~p+~v, s =A-~p-~v (26) Y,J.L+n( )-~11J ( Y,) x x + y J.L ax 2(-l)n Y Y, J.L+n (y _ y)-Y, 11 x J )b (x + y )y,] x K (ay~) I [b(y-y)y,] J.L J.L a> b > 0, n = 0, l, 2, ••• -1-n < Re p < 2-2 n + Re 11 (27) x-y, [sin (ax) J (ax) TT sec (vrr) y -~[-sin (ay) J (ay) 11 11 + cos (ax) Y 11(ax)] +cos (ay) Y11(ay)] a> 0, -~<Rev<~ (28) x -Y, [cos (ax) J )ax) rrsec(vrr)y-~[cos(ay)J (ay) 11 -sin (ax) Y)ax)] +sin(ay) Y11(ay)] a> 0, -~<Rev<~ 228 INTEGRAL TRANSFORMS 14.3 Hililier transcendental functions (cont'd) f(x) J"" f (x) (x + y)-1 dx 0 Jargyj < TT TT ctn (vrr) y'A. Jv (a 2y ) (29) x'A.Y (ax) a> 0 ----::--------:- v -l + jRevj < Re .\ < 3/2 sin [(v + .\)rr) -sin (vrr~ys~ [(v-.\) TT) J -v (a;) -2kt TT-t a-'A.cos[~(.\-v)rr) (.\ -v) (.\ + v) Xl -- l -- 2 2 ( .\-v .\+v a2y2 ) X 1F;_ \l; l--2-, l--2-; -- 4-- 2k2 -t t-'A. . (.\-v ) + TT a y sm --rr 2 (30) x~v-~ Yv(ax~) -2y~v-~ K)ay~) a>O, -l/2<Rev<5/2 (31) x~v+~ Y)ax~) 2y~v+~ K)ay~) a> 0, -3/2 <Rev< l/2 (32) x'A. Y (ax~) a> 0 v -l+~JRevj <Re.\<% -.\ -~ J ' 2 _ ,\, _ ..::_ ~ _ v+ l 2' 2' 2 • 14.3 STIEL T JES TRANSFORMS 229 Higher transcendental functions (cont'd) f(x) Joo 1 f(x) (x + y)-dx 0 I argyl < 7T (33) xA.-1Icos[(.\- ~v)TT] J)ax }!;) -2yA.-1 K)ay }!;) +sin[(.\- ~ v)TT] Yv (ax X)! a> 0, !Rev! <2Re.\<7/2 (34) xXJ1-+n-}!; (x + y)-}!;v Y (ax}!;) j1. 2(-1)"+1 yXJ1-+n-X (y _ y)-}!;v X J )b (x + y) X ] x K J1-(ayx) I)b (y-y)X] a> b > 0, n = 0, 1, 2, ... -~-n <Rep.< 3-2n +Rev (35) XA_-1(x+ y)-XJ1-J [b(x+ y)X] 2 A.-1c )-xj..l. -y y-y j1. xiJ1-[b(y-y)X] K)ay }!;) x lcus [(,\-~ v)TT] J)axX) +sin[(,\- ~v)TT] Y )ax}!;)l a>b>O IRe vi < 2 Re .\ < 4 +Rep. (36) xv e-ax Iv(ax) yv sec(v77) eay K)ay) Rea> 0, -~<Rev <~ xA. e -ax I)ax) -}!; A. ( I -A, ~ ) (37) 7T y G ~ 2ay Rea> 0, Re ,\ < ~ -A, v, -v Re(.\+v) >-1 (38) xA. e ax K)ax) 77-}!; cos(v77) yA. I arg al < 3 7T /2 ( I -A,~) Re.\-IRevl >-1 x G~~ 2ay -A, v, -j) 230 INTEGRAL TRANSFORMS 14.3 Higher transcendental functions (cont'd) f(x) J"" I { (x) (x + y)-dx 0 \argy\ < TT (39) x-l<; e-az K (ax) 11 TT sec(vrr) y-~ eay K,_,(ay) Rea> 0, -~ < Re v < ~~ (40) xA.e-az K (ax) rr~yA.c~(2ay~-A,~) 11 -A, v, -v Rea> 0, Re /...-\Rev\ > -1 (41) x-~11-~K (ax~) 11 ~rr2 y-Y,11-y, sec (vrr) Rea> 0, Rev<~ x [B)ay y,)-Y,_,(ay~)] (42) xA. K (ax~) 11 22A.+I yA.['(l +'A+ ~v)['(1+/...-~v) Rea> 0, Re /... > ~\Rev\ - 1 y, x5_2r..__,,)ay ) (43) x-~ [2rr-1 K0(ax~)-Y0(ax~)] 4y-~ ker(ay~) \arga\ <~ rr (44) xy,11 H (ax~) y, 11 y, y, 11 rrsec(vrr)y [I_ 11(ay )-L)ay)] a> 0, -~2 <He v < ~ (45) x-Y,11H (axy,) 11 TT y-y,11[I)ayy,)- L 11(ay~)] a> 0, Re v>-3/2 (46) X A. H (ax~) a > 0, Re /... <% TT [ (2/a)2A. 11 cos[(/..+ ~v)rr] ['(1-/...+~v)['(l:--/...-~v) -~ < Re('A+ ~v) < ~~ x 1F2(1;1-'A+ ~v,1-'A- ~v; ~a2y) -yr._ L,_,(ay~)] 14.3 STIEL T JES TRANSFORMS 231 Higher transcendental functions (cont'd) f(x) Joo f (x) (x + y)-1 dx 0 largyl < 77 (47) x-~ [cos(~vrr) J)ax y,) rry-~ [I)ay~)- L,}ay~)] + sin(~vrr) H,_.,(ax ~)] a> 0, -~<Rev < 2 (48) x-~ [I (ax~)- L (ax~)] -1 csc(~vrr)[H,_.,(ay~)+ E,}ay ~)] rra v v Rea> 0, -l <Rev< 2 (49) xA.[I (ax~)-L (ax~)] YA. 32(a2y,-A,~v+~ ) 1-- G24 -4--A, ~v, ~v+~,-~v v v 7T Rea> 0 -2<Re(2A+v)<l For other integrals with Dessel functions see Wa,tson, G.N., 1922: A treatise on the theo•ry of Bessel functions, Cambridge, in partie- ular sections 13.5 to 13.6. (50) x,u-Y, e-~ax M (ax) K,,u l (2/l + l) l (K-/l + ~2) y,u-Y, Rea> 0 x e y, ax IT' (ax) -~ < Re 11 < Re K + ~ -K,,u (51) xA.e-Y,ax M (ax) Rea> 0 [' (2 ll + l) y A. 22 ~ I -A, l-K ) K,/-L [' (K + 11 + ~) C 23 ax -A, 11+ ~. ~ -11 -3/2 -n e 11 < Re A < Re K (52) xA.eXax IV (ax) K,iJ-rA. w <~ + 11-K) r <~~-11-K)r 1 largal < 3rr/2 32 ~ I -A, K + l ) Re (K + A) < 0 xG23 ay 1' i' n e A > I Be Ill -3/ 2 ->., i'2 + 11· /2-11 232 INTEGRAL TRANSFORMS 14.3 Higher transcendental functions (coot 'd) f(x) Joo f(x) (x + y)-1 dx JargyJ < rr 0 (53) K-1 -~ax W ( ) x e K,J.L ax [' (K + fl + X) [' (K -fl + X) YK-T Rea> 0, Re K > IRe fli -X x e~ax W -K,J.L(ax) (54) x"-e -~ax W K (ax) Re a > 0 R~ A.> IRefli- 3/2 A ~ I -\ 1-< ) y G~ ay I I ->.., X+fl> X-fl (55) G an ~x /aT ' ••• ' ap ) G • + 1, n + 1 ~ ~' aT ' • • • ' a p) pq bl, ••• ,bq p+T,q+T y 0 b b ' 1 , ••• , q p + q < 2(m + n) JargaJ < (m+n-Xp-Xq)rr Rea.< l j = l, ..• , n Reb 1 > -1 j = l, •.• , m J . (56) -~'j"······") ~G"+z,n+z ~ z,O, X,aT" •• ,aP) G ax P pq bl, ••• ,bq 2 p+Z,q+Z y 0 X b b 1T ' 2, 1' ••• ' q p + q < 2(m + n) I arg a I < (m + n-X p-X q) rr Rea.< l j = l, ••• , n Reb\ -X j = l, ••. , m J 14.4 233 14.4. Generalized Stieltjes tl'ansforms f(x) J 00 f(x) (x + y)-Pdx 0 !argyl < 17 ( l) f(x) g (y; p) (2} x f(x) g(y; p-1)-yg(y; p) (3) f (ax) a>O aP-1g(ay ;p) (4) xp-2 f(a/x) a>O aP-1 y-p g(a/y) (5) ['(x) p g(y; p+l)-y-p f(O) (6) J "f(t) dt (p -1} -1 g (y; p -l) Rep> l 0 (7) [r<11>r1 f" f(t)(x-t)~.~--1 dt r (p-11> g (y; p -11) 0 r(p) 0 < Re 11 <Rep (8) v-1 Rev> 0 r(v) r (p-v) Yv-p Rep> Rev X r (p) (9) xv-1 (a+ x)-11-r(v)r(l1- V+p)yv-p r (11 + p) al-L largal. < "• Rev> 0 x 2F, (llt v; 11 + p; l -y I a) Re p > Re (v-11) (10) e -ax Rea> 0 ap-1 ea.y r(l-p, ay) (ll) x-pe-a." TT-~ r(l-p)(a /y)p-~ e~a.y Rea> 0, Rep< l x Kp-~ (Y:; ay) 234 INTEGRAL THANSFOHMS 14.4 Generalized Stieltjes transforms (cont'd) f(x) Joo f(x) (x + y)-p dx Jargyj <rr 0 (12) XA. e -ax 1(A+ 1) a~p-~A.-1 y~A.-~p Rea> 0, ReA> -1 x e~ay W (ay) k,m 2k =-A-p, 2m= A-p + 1 (13) xA. exp(-a/x) Red> 0 1(p-A-1) a~A.y-~A.-1 x exp ( ~) Wk ( ~) 2y •" y k =~A-p + 1, m=~A+~ Rep> ReA+ 1 (14) x-Y, exp(-ax ~) Rea> 0 TT~ (2y~/a)~ -p 1(1-p) [ y, y, x "~-p(ay )-Y~-p(ay )] xA. exp(-axy,) kp+1 ( 2 I A ) (15) Y G 31 !!:._.! - Rea> 0, ReA> -1 TTY,1(p) 13 tl. p-A-1,0, ~ (16) sin (axy,) a>O 2 TTY, y ~ (2:Y,)Y,-p K p-312 (ayy,) 1 (p) Rep>~ y, kp+1 ( 2 I ) (17) xA. sin (ax y,) TT y G 21 a y -A a> 0, ReA >-~ 1 (p) 13 4 p-A-1, ~,0 Rep> lle A+ 12 x-~ cos(ax y,) 2 y, ~2 ~ )Y,-p ( 18) a>O TT y Y, 1 (p) -a- K p-~ (ay ') Rep> 0 14.4 STIEL TJES TRANSFORMS 235 Generalized Stieltjes transforms (cont'd) {~) J 00 r (x) (x + y)-p dx 0 \argyl <rr X A._-p+ 1 ~a2 yl -A ) ( 19) xA._ cos (ax y,) 77 y c21 __ a> 0, Re A> -1 1 (p) 13 4 p-A-1, 0, ~ Rep> ReA+ ~; x X v J (axy,) ap-1 (20) YY,v+Y,-Xp K (ayy,) v 2P1(p) v-p+1 a> 0, Rev> -1 R e p > ~2 He v + ~ (21) xA._J (axX) 22A.y1-p 21 (a 2 y I 0 ) v a2AI'(p) G 13 4 p-1, A+ ~~v, A-~~, a> 0, Re (,.\ + ~'f v) > -1 He p >He A+ )c4 (22) xv e -ax I (ax) 1 (v+ ~2) 1 (p-v-~2) y, 1 (2 a) ,p-v rry, I'(p) Rea> 0, He v >-!2 x y v-Y,p e a Y Irk ( 2 a y) ,m k = ~-~:ip, m=~- 1~p+v He p >Rev+ !2 (23) XA._ e -ax I (ax) yA.+1-p ( I -,.\,~ ) , C22 2ay v 77y, f'(p) 23 p-A.-1, v, -v flea>O, Ile (A+ v) > -1 Rep> Re ,.\ + ~ (24) XA._ eax K )ax) \arga\ < 3rr/2 cos(V77) A.+1-p y, y Ile ,.\ > /Re vi -1 77 2 1 (p) , ( I -~ ~; ) x G 23 2 ay p-,.\-1, v, -v Hep >Re,.\+~2 236 INTEGRAL TRANSFORMS 14.4 Generalized Stieltjes transforms (cont'd) f(x) {" f(x) (x + y)-p dx 0 !argyl < TT (25) xp-3/2 e-ax K (ax) TTX f'(p+v- X) f'(p-v- X) v (2a)x y r (p) Rea> 0, Re p > IRe vi + X X eay W1_p,v(2ay) X (26) xA.e-ax Kv(ax) TT A_+1-p f'(p) y Rea> 0, Re >..> IRevl-1 31 ( I ->..,X ) x G23 2ay p->..-1, v, -v xA. K (ax X) kp+1 (~ >.. ) (27) Rea> 0 Y ___ G31 a y - v 21(p) 13 4 p->..-1, Xv, -Xv Re >.. > X IRe vi - 1 (28) J.L-X -X ax M ( ) x e K,J.L ax r<2fl+ 1)r <K+p- fl-~Hr (p)r1 He a> 0, He fl >-r2 Xp-X A.+X -Xp X ay W ( ) xa y e k,m ay k =X-Xp-K, m=X-Xp+fl Hep>He(f.L-K)+X (29) XA. e-X ax MK (ax) 1(2f.L+ 1) yA.+1-p ,J.L f'(p) i(K + fl +X) He a> 0, He(>..+ f.L) > -3/2 "( I _,_ 1-K ) x G23 ay p->..-1,X+fl,X-fl He p > He(>..- K) + 1 14.4 STIEL T JES TRANSFORMS 237 Generalized Stieltjes transforms (cont'd) f(x) J"" f(x) (x + y)-p dx 0 largyl < rr ;\+1-p (30) x,\e~axw (ax) y K,J.l. r(p) ro~-K+11) ro~-K-11) largal < 3rr/2 Re ..\ > IRe 111 -3/2 n ~ I -~ l+K ) xG23 ay 1 1 p -,.\ -1, ~ + 11· ~ -11 Re p > Re (..\ + K) + 1 (31) K+p-2 -~ax W ( ) x e K,J.l. ax r(K+ p+ 11-~)r (K+p-11-~) [f'(pW1 Rea> 0 X YK-1 e~ay w1-K-p,J.1.(ay) Re p > IRe 111 -Re K + ~ A.+1-p 0 I ) (32) x,\ e -~ax W (ax) Rea> 0 y -..\, 1-K K,J.l. r (p) G ~~ a y P -..\ -1, ~ + /1. ~ -11 Re ..\ > IRe 111 -3/::: (33) Gm"~x~a1, ... ,aP) y 1-p +1 +1 ( I 0, a 1' ... 'a ) --G\1'"+1 ay P pq b,. ... ,bq r(p) p ,q p-1,b1, ... ,b q p + q < 2(m + n) Rep> Rea. j = 1, ••. , n I arg al < (m + n-~ p ~ ~ q) TT J Re b.> -1 j = 1, ... , m J CHAPTER XV HILBERT TRANSFORMS We call g(y) = 77-1 foo f(x) (x-y)-1 dx -oo the Hilbert transform of [(x). Here x andy are real variables, and Joo = lim <J Y -E + J 00 ) -oo €->+0 -oo y+€ is the Cauchy Principal Value of Joo • -oo For the theory of Hilbert transforms see chapter V of Titchmarsh 's book (1937) and the references given there. Additional references to papers which appeared after the publication of Titchmarsh 's book are given below. The finite Hilbert transform, g (y) = ;:--1 J b f (x) (x -y)-1 dx a and its application to airfoil theory was discussed recently by Tricomi (1951 a, b) and Nickel (1951, 1953); the latter author gives references to earlier work on this subject. In the above relation, g (x) is said to be conjugate to f (x ): the relation­ ship is skew-reciprocal, i.e., -f(x) is conjugate tog (x). For the relation of Hilbert transforms to Fourier integrals see Titchmarsh (1937) and Kober (1942, 1943 a, b). The connection with Laplace transforms may be expressed by stating that, formally, the imaginary part of a Laplace trans­ form evaluated on a line parallel to the imaginary axis is conjugate to the real part of that Laplace transform evaluated on the same line. Hilbert transforms may be evaluated by means of tables of Stieltjes transforms (Chapter XIV) using the formulas 239 240 INTEGRAL TRANSFORMS g(y)=rr-1 61f(x);-yl-(2rr)-161[(-x); lrl eirrl -(2rr)-1 61[(-x); lrl e-i7TI g(y) = (2rr)-1 61f(x); ye i7TI + (2rr)-1 61[(x); ye-i7TI -rr -1 61[(-x); yl Related transforms are :f 7T f(x) ctn [If (x-y)] dx -7T r7T -1 J". f(x) (cosx- cosy) dx. 0 -oo<y<O O<y<oo These can Le reduced to Ililbert transforn :s by a change of the variables of integration. From the transform pairs given in the tables, further transform pairs may be derived by the methods mentioned in the introduction to vol. I, by the general formulas given in sec, 15. l, and by exploiting the con­ nection with other transforms (see above), HILBERT TRANSFORMS REFERENCES Cossar, James, 1939: Proc. London Math. Soc. (2) 45, 369-381. Kober, Hermann, 1942: Bull. Amer. Math. Soc. 48, 421-426 . Kober, Hermann, 1943 a: 1. London Math. Soc. 18, 66-71. Kober, Hermann, 1943 b: Quart. 1. Math. Oxford Ser. 14, 49-54. Nickel, 1\.arl, 1951: Math. Z. 54, 81-96. Nickel, Karl, 1953: Math. Z. 58, 49-62. 241 Titchmarsh, E.C., 1937: Introduction to the theory of Fourier integrals. Oxford. Tricomi, F.G., 1951a: Quart. 1. Math. Oxford Ser. (2) 2, 199-211. Tricomi, F .G., 1951 b: Z. Angew. Math. Physik 2, 402-406. fl HILBERT TRANSFORMS 15 .1. General formulas f(x) 77-1 :f_ 00 f (x) (x -y)-1 dx * -oo (l) f(x) g (y) (2) g (x) -[(y) (3) f(a + x) a real g (a+ y) (4) [(ax) a>O g(ay) (5) f(-ax) a>O -g(-ay) (6) x f(x) y g (y) + 77-I f"" f(x) dx -oo (7) (x +a) f(x) (y+a)g(y)+77-1 J"" f(x)dx -oo (8) ['(x) g '(y) 15.2. Elementary functions (l) l 0 (2) 0 1 I b-y I -oo<x<a -log -- l a< x < b 11 a-y 0 b<x<oo * y is real, and the integral is a Cauchy Principal Value. 243 244 INTEGRAL TRANSFORMS 15.2 Elementary functions (cont'd) f(x) 77-1 f oo f (x) (x -y)-1 dx -oo l logl-a I (3) 0 -oo<x<a -_, 77Y a-y X a<x<oo a>O y,..; 0, y ,.fa X-1 l I (y-a)b I (4) -oo<x<a -log 0 a <x < b 77Y a (b -y) X-1 b<x<oo y 1: 0, a, b a< 0 < b l I a I l (5) 0 -oc <x <a 77Y2 log ------ -2 a-y 77ay X a<x<oo a>O y 1: 0, y/:a (6) (x + a)-1 Im a> 0 i (y + a)-1 (7) (x + a)-1 lm a< 0 -i(y+a)-1 1 log I :y I (8) 0 -oo<x<O (ax+ b)-1 O<x<oo 77(ay +b) a, b > 0 y ,.f -b/a, y;iO l I b I l (9) 0 -oo<x<O 77(ay + b)2 log -- (ax+ b)-2 O<x<oo ay 77 b (ay + b) a, b > 0 y,..; 0, y/:-b/a y is real, and the integral is a Cauchy Principal Value, 15.2 HILBERT TRANSFORMS 245 Elementary functions (cont'd) f(x) 7T-1 { 00 f (x) (x -y) -1 dx -oo (lO) (x 2 + a2)-1 Rea> 0 y -a (y 2 + a2) X a (ll) x2 + a2 Rea> 0 2 2 y +a (12) ,\x + f1 a Rea> 0 ,\a-flY 2 2 y2 +a 2 x +a (13) 0 -oo<x<O ey + d log I a: I 77(ay + b )2 ex+ d O<x<oo (ax+ b)2 ad-be - a, b > 0 TTab (ay +b) y ~ 0, y ~ -b/a (14) (a -X)~ -(b -X)~ -oo<x<a 0 -oo <y <a -(b-x)~ a <x < b (y-a)~ a <y <b 0 b<x<oo ~ ~ (y -a) -(y -b ) b<y<oo ( 15) 0 -oo<x<a ~ ~ (b -y) ' -(a -y) -oo<y<a (x-a)~ a< x < b (b-y) ~ a< y < b (x -a) ~ -(x -b) ~ 0 b<y<oo b<x<oo (16) \a-x\ ~ -\b -x\~ ~ ~ (b -y) -(a -y) ' -oo<y<a a> 0, b>O (b-y) ~ + (y-a)~ a <y < b (y -a)~ -(y -b)~ b<y<oo y is real, and the integral is a Cauchy Principal Value, 246 f(x) (17) 0 (ax+ b)-~ 0 0 INTEGRAL TRANSFORMS 15.2 Elementary functions (cont'd) -oo<x<O O<x<oo a, b > 0 -oo<x<O O<x<a a<x<oo 2rr-1 (-ay- b)-~ x tan -1l [-(ay + b)/b]~ l -oo<y<-b/a 1 ~ log (ay +b) y = -b/a I b ~ + (ay + b)~ I b ~ -(ay + b)~ -b/a < y < oo -1 I/ -i( 2 2)~ -rr a -/2Y-77 y -a xcos-1(-a/y) -oo<y<-a -77-1 a -~ y + rr -1 (a 2 -y 2) ~ a+ (a -y ) I 2 2 ~ I xlog -y -a<y<a -rr-1a-~y+ rr-1 (y2-a2)~ x cos-1 (-a/y) a< y < oo 0 < cos -1 < 17 -oo<x<-a -y-(y2-a2) Y. -oo<y<-a -a< x <a -y -a< y <a a < x < oo -y + (y 2-a 2)~ a<y<oo y is real, and the integral is a Cauchy Principal Value, 15.2 HILBERT TRANSFORMS 247 Elementary functions (cont'd) f(x) 17-1 joo f(x)(x- y)-1 dx -00 (20) 0 -oo<x<O cos -1 (-a/y) rr(y2-a2)Y, -oo<y<-a (a2-x2)-Y, O<x<a l I a + (a 2 -y 2) y, I 0 a<x<oo 17 (a 2 _ y 2)Y, log -y -a< y <a cos-1 (-a/y) -rr(y2-a2) Y, a<y<oo 0 <cos -1 < 1T (21) 0 -oo<x<-a (y2-a2)-y, -oo<y<-a (a 2-x 2)-Y, -a< x <a 0 -a< y <a 0 a<x<oo ( 2 2)-Y, -y -a a<y<oo l ~-y + (y 2 -a 2) y, I (22) 0 -oo<x<a rr(y2-a2)X log (x2 _ a2)-x a a<x<oo a>O -oo<y<-a l -1 c-y) rr(a2-y2)Y, cos --;;-. -a< y <a l ~-y+(y2-a2) X I rr(y2-a2)Y, log a a<y<oo 0 <cos -1 < 1T y is real, and the integral is a Cauchy Principal Value. 248 INTEGRAL TRANSFOHMS 15.2 Elementary fwtctions (cont'd) f(x) 1T -1 :F"" -oo f(x) (x-y)-1 dx (23) -(x2-a2)-Yc -oo <x <-a 0 -oo<y<-a 0 -a<x<a ( 2 2)-~ a -y -a< y <a (x2-a2)- ~ a<x<oo 0 a<y<oo (24) 0 -oo<x<O -~+:_I~ I~ cos -1 (-;) (a-x)y, (a+x)- y, O<x<a 2 rr a+y 0 a<x<oo -oo<y<-a -~+!_~-y )y, I a+ (a 2-y 2) ~ I log 2 rr a+y -y -a<y<a --+--- cos-1 1 1~-a)' 2 n y+a (-;) a<y<oo 0 <cos -1 < 1T (25) 0 -oo<x<-a -l+(a-y)y, Jy + aJ-~ (a-x)~ (a+ x)-y, -a< x <a -oo<y<-a 0 a<x<oo -1 -a< y <a -l+(y-a) ~ (y + a)-y, a<y<oo y is real, and the integral is a Cauchy Principal Value. 15.2 HILBERT TRANSFORMS 249 Elementary functions (cont'd) f(x) 1T-1.Joo -oo f(x) (x-y)-1 dx (26) 0 -oo<x<O 2-~1~1 ~ cos-1 c-~) (a+ x)~ (a-x)-~ 0 <x <a 2 1ra-y y 0 a<x<oo -oo <y <-a 1 l G + r )" i•+(a'-r'l " I -+--- log 2 " a-y -y -a< y <a ~-~~+a )X cos -1 (-~) 2 " y-a y a<y<oo 0 <cos -1 < 1T (27) 0 I a y ~~ -oo<x<-a a-y+y -- -oo<y<-a x(a-x)~ (a+x)-~ a+y -a< x <a a-y -a< y <a 0 a<x<oo (y a)~ a-y+y -- a<y<oo y+a (28) 0 -oo<x<O esc (Jm) (-y)v-1 -oo<y<O xv-1 O<x<oo -ctn(v77) y v-1 O<y<oo O<Rev<l (29) jxjv-1 0 <Rev< l -ctn 0~ V7T) sgn y Jyjv-1 (30) sgn x jxj v-1 O<Rev<l tan(~~ v1r) Jyj v-1 y is real, and the integral is a Cauchy Principal Value. 250 INTEGRAL TRANSFORMS 15.2 Elementary ftmctions (cont'd) f(x) 77-1 foo f(x) (x-y)-1 dx -oo (31) 0 -oo<x<a osd~{l-c=:)"] (x-a)v(b -x)-v a <x < b 0 b<x<oo -oo<y<a IRevl < l osd~l [1-co•(~)G =:YJ a<y<b csc(vrr) [l-G=:)v] b<y<oo (32) 0 -oo<x<a esc (vrr) ~~I v-1 (x-a)v-1 (b-x)-v a< x < b b-y b-y 0 b<x<oo -oo<y<a or b<y<oo 0 <Rev< l -(y-a) v-1 (b-y)-vctn(vrr) a<y<b (33) 0 -oo<x<a r (p) r (a) (b -a )P +a--1 (x-a)P-1 (b -X )a--1 (b -y) "r (p + a) a <x < b ( b-a) x 2F, l, a; p +a; -- 0 b<x<oo b-y Rep> 0, Rea> 0 -oo<y<a or b<y<oo (y-a)P-1 (b -y)o--1 ctn(arr) r(p)r(a-l) ( p+o--2 - b-a) rr I' (p +a-l) ( b-y) x F 2-p-a l· 2-a· --2 1 ' , , b-a a<y<b Y JS real, and the integral is a Cauchy Principal Value, 15.2 HILBERT TRANSFORMS 251 Elementary functions (coot'd) f(x) 17-1 -{_: f(x) (x -y)-1 dx (34) 0 -oo<x<O r(fl-v) r (v) (-y)v-J x v-1 (x + a) I -11-O<x<oo 17 r (fL) all--I a> 0, O<Rev<Refl x 2F, (fL-l, v; fL; l + y/a) -oo<y<O yv-l (y + a)1-11-ctn [(fL-v)rr] r (fL-1/-l) r (v) a l-iJ.+v - (y + a) 17 r (fL-l) X 2FI ( 2 -fl, l; 2 -fl + 1/; _a_) y+a O<y<oo (35) exp (-alxl) a>O 17-1 sgn y[exp(alyl) Ei(-alyl) -exp (-alyl) Ei (alyi)] (36) sgn x exp(-alxl) a>O -17-1 [exp(alyl) Ei(-alyl) + exp(-alyl) Ei(alyi)J (37) 0 -oo<x<a -17-1 e.,_by Ei(by-ab) -oo<y<a e -bx a<x<oo -17-1 e-by Ei(by- ab) a< y <oo b > 0 (38) e ia.x a>O ie iay y is real, and the integral is a Cauchy Principal Value. 252 INTEGRAL TRANSFORMS 15.2 Elementary functions (cont'd) f(x) 77-1:foo -oo [ (x) (x -y) -1 dx (39) 0 -oo<x<O 277-1 cos(a\y\ l{) ci(a\y\ y,) exp(-ax l{) O<x<oo -277-1 sin(a\y\ l{) si(a\y\ y,) a>O -oo<y<O -77-l exp(ay l{) Ei(-ay y,) -77-1 exp(-ay y,)Ei(ay l{) O<y<oo lb -X I (40) log-- a <b 0 -oo<y<a x-a -77 a <y < b 0 b<y<oo l 11+axl -I (41) -log-- a> 0, b>O -77y -oo <y <-a-1 x 1-bx 0 -a-1 <y<b-1 -77y-l -b -I < y < 00 (42) log ~ 0 <a< b -77 -b<y<-a a b 77 a <y < b 0 elsewhere (43) sin (ax) a>O cos (ay) (44) sin (ax) a>O cos (ay)-1 X y (45) 0 -oo<x<O exp(-a\r\ l{) -oo<y<O sin (axy,) O<x<oo cos(ay l{) O<y<oo a>O y is real, and the integral is a Cauchy Principal Value. 15.3 HILBERT TRANSFORMS 253 Elementary functions (cont'd) f(x) 7T-1.t"" f(x) (x-y)-1 dx -oo (46) sgn x sin(a\x\~) a>O cos(a\y\ ~)+ exp(-a\y\ ~) (47) cos (ax) a>O -sin (ay) (48) 1-cos (ax) a>O sin (ay) X y 15.3. Higher transcendental functions (l) e -= Ei (ax) -oo<x<O 0 -oo<y<O e -ax Ei (ax) O<x<oo rre -ay O<y<oo (2) ci (a\x \) a>O sgnysi(a\y\) (3) sgn x si (a\x\) a>O Ci (a\y\) (4) cos(ax) ci(a\x\) sgn y cos (ay) si(a\y\) -sin(a\x\) si (a\x\) a>O +sin (ay) ci(a\y\) (5) sin(ax) ci(a\x\) sin(a\y\) si (a\y\)- cos (ay) ci (a\y\) + sgn x cos (ax) si(a\x\) a>O (6) 0 -oo <X< -1, 1<x<oo -2rr-1 Q (y) n p (x) -1<x<1 -oo < y < -1, 1<y<oo n n = 0, 1, 2, ... -2rr-1 Qn(y) -1<y<1 (7) 0 -oo <X < -1, 1<x<oo un-1 (y) -1<y<1 (1-x2)-~ T (x) n -1<x<1 n=1,2, ... Y is real, and the integral is a Cauchy Principal Value. 254 !NTEGHAL TRANSFORMS 15.3 Higher transcendental functions (cont'd) f(x) 77-1 {oo [(x) (x-y)-1 dx -oo (8) 0 -oo <X< -1, 1<x<oo -Tn+l (y) -1<y<1 ( 1 -x 2) y, U (x) n -1<x<1 n = 0, 1, 2, •.• (9) 0 -oo <X< -1, 1<x<oo -2 77-, (y-l)a(y + 1),8 Q ~a,,B)(y) (1 -x)a (1 + x),B P (a,,B) (x) n -00 < y < -1, 1<y<oo -1<x<1 -277-, (1-y)a(1 + y),B Q (a,,B)(y) n Re a> -1, Re f3 > -1 -1<y<1 ( 10) 0 -oo<x<O esc (v77) [J)-ay)- J)-ay)] J)ax) O<x<oo -oo<y<O a> 0, Re v > -1 esc (v77) [J)-ay)- cos(v77) Jv(ay)] O<y<oo (ll) -J_)-ax) -oo<x<O -Y -)-ay) -oo<y<O Jv(ax) O<x<oo -Y )ay) O<y<oo a> 0, -1<Rev<1 ( 12) 0 -oo<x<O Jt:!\y\v [tan(v77) sgn y Jv(a\y\) xv J)ax) O<x<oo -Y)a\y\)- sec(v77) sgn y H_v(a\y\)] a> 0, -1/2 < n e 1/ < 3/2 (13) \x\ v J)a\x\) sgn y \y\v [tan (v77) J)a\y\) a> 0, -l/2 < He v < 3/2 -sec (v77) H_v(a\y\)] ( 14) sgn x \x\v J)a\x\) -\y\ v Y)a\y\) a> 0, -1/ 2 < n e 1/ < 3/ 2 y is real, and the integral is a Cauchy Principal Value. 15.3 HILBERT TRANSFORMS 255 Higher transcendental functions (cont'd) f(x) 77-1 joo f (x) (x -y)-1 dx -00 (15) \x\-v J)a\x\) - sgn y \y\-v Hv(a\y\) a> 0, Rev> -3/2 (16) 2A.-1 r (Y:; >. + Y:; v) 0 -oo<x<O 77 a A_ r (l -Y:; ,\ + Y:; v) xA_J (ax) O<x<oo ( A+v ,\-v a2y2 ) J) a> 0 X F 1· 1---1---·--- -1-Rev< Re ,\ < 3/2 1 2 , 2 ' 2 , 4 2A.-2 y r (Y:; >. + Y:; v-Y:;) + 77 a/\-1 r (3/2- Y:; ,\ + Y:; v) ( 3 -,\-v 3 -,\ + v a 2y 2 ) X F 1·------·---1 2 , 2 ' 2 , 4 -h(y) \y\A_J)a\y\) {" [ (.\ +V hI -oo<y<O h(y) = ctn [(,\ + v) 77] O<y<oo ( 17) sin (ax) J1 (ax) a>O cos (ay) J1 (ay) (18) sin (ax) Jn (by) cos (ay) J n (by) 0 < b <a, n = 0, 1, 2, ... (19) cos (ax) J1 (ax) a>O -sin (ay) J1 (ay) (20) cos (ax) Jn (bx) -sin(ay)Jn(by) 0 < b <a, n = 0, 1, 2, ... y is real, and the integral is a Cauchy Principal Value. 256 INTEGRAL TRANSFORMS 15.3 Higher transcendental functions (cont'd) f(x) 77-1 :foo f (x) (x -y)-1 dx -oo (21) sgn x lxlv sin(alxl- 77v) lrlv cos(alrl- 77v) J)alyi) x J)alxl) a> 0, -~~ < Re v < ~ (22) lxlv cos(alxl- 77v) J)alxl) -sgn y lrl v sin (air I -TTV) J)alyi) a> 0, -~ < Re v < ~2 (23) lxl-v sin(ax) J)aixl) lrl-v cos (ay) J)alrl) a> 0, Rev >-~ (24) lxl-v cos (ax) J)alxi) -lrl-v sin (ay) J)alyi) a> 0, Rev >-~ (25) lxl~ Jv-~(alxi)J_ v-~(alxl) -sgn Y lrl ~ J"-+)alyi) J>t.-v(aiyi) a>O (26) sgn x lxl ~ Jp)aixi) lrl ~ Jv-~ (aiyi) J-v-'4 (air I) x J~ _)alxl) a>O (27) 0 -oo<x<O 277-1 (-y)~v K)a(-y) ~] x ~ v J (ax~) O<x<oo -oo<y<O v -y~v Y)ay ~) a> 0, -1 <Rev< 3/2 O<y<oo (28) lx I ~ v J v (a lx I ~) -sgn y IYI ~v [277-1 Kv(ajyl ~) a> 0, -1 < Re v < 3/2 + Yv(ajyi il)] (29) sgn x lxl ~v J)alxl ~) 277-1 lrl~v K)airl il) a> 0, · -1 <Rev< 3/2 -lrlilv Y)alrl il) y is real, and the integral is a Cauchy Principal Value. 15.3 HILBERT TRANSFORMS 257 Higher transcendental functions (cont'd) f(x) 1T-1 f. "" f (x) (x -y)-1 dx -oo (30) 0 -oo <X< 0 lrll{ ,,.-){ sec(VIT) [l)alrll{) xl{v-l{ J (ax){) . v O<x<oo -L_)alrll{)] -C>O < y < 0 a> 0, -l/2 < Re v < 5/2 y l{ v-l{ [tan (vrr) J (ay l{) v -sec(vrr) H_v(ayl{)] 0 <y <"" (31) lxll{v-l{ J)alxll{) sgn y lrll{v-X ltan(vrr) J)alrll{) a> 0, -l/2<Re v<5/2 + sec (vrr) [L_)alrll{)- H_Jalrll{) -l)alrll{)]! (32) sgn x lxll{ v-X J)alxl X) lrl Xv-x I tan (vrr) Jv(airl X) a> 0, -l/2<Rev<5/2 + sec(vrr) [l)alrlx) -L_v(alrl X)-H_)alrl X)]! (33) 0 -oo <X< 0 lrl-l{v-X [lv(alrll{)-L)alriX)] x-Xv-XJ(axX) v O<x<oo -C>O < y < 0 a> 0, Rev> -5/2 _ y-Xv-X Hv(ayX) O<y<oo (34) lxl-x v-X J )alxl X) sgn Y lri-Xv-X[Lv(alrlx) a> 0, Re v>-5/2 -H)alrll{)-l)alrll{)] (35) sgn x lxl-x v-X J)alxl X) lrl-l{ v-l{ (Iv(alrll{) a> 0, Re v > -5/2 -L)alrll{)-H)alrll{)] y is real,aand the integral is a Cauchy Principal Value. 258 INTEGHAL THANSFOmlS 15.3 Higher transcendental functions (cont'd) f(x) (36) 0 xtc J (ax y.) v O<x<oo (37) 0 a>O -1- ~2 fie v <He A< Jl ( 2 2) y, v sgn x x -a ' x J )b (x 2-a 2)y,] -oo < x < -a or a < x < oo a> 0, b > 0 -1 <He v < 3/2 11 a 2A.. I· (l -A + 1• v) ~ v v a2y0 X F 1· ]-A-- 1-A+-·-- -h(y) I 2 ' 2' 2' 1). h (y)= JyJA..csc [('A+ ~2V)TT] I)ajyj Y.) -oo<y<O h (y)=yA..ctn [(A+ !2v)TT] J)ajyj y,) O<y<oo -a<y<a -(y2-a2) Y,v Y)b(y2 _ a2)Y,] -oo < y <-a or a < y < oo (38) r_, (-ax) Yv(ax) -oo < x < 0 -J _)-ay) 0 < x < oo J v (ay) -oo<y<O O<y<oo a>O, -1<Rev<1 (39) sin (}:I V7T) J )a jx J) + cos(}:lvTT) Y)ajxj) a> 0, -1 < Re v < 1 (40) sgnx[sin(}:lvTT)Y)ajxj) -cos(}:lvTT) J)ajxj)] a > 0, -1 <Rev< 1 sgn y (cos e-~ VTT) J)ajyj) -sin(}:; vTT) Y)ajyj)] cosnlvTT) Y)ajyj) +sin n12 vTT) cJ)ajyj) y is real, and the integral is a Cauchy Principal Value. 15.3 HILBERT TRANSFORMS .._ 259 Higher transcendental functions (cont'd) f(x) 17-1 f"" f (x) (x -y)-I dx -oo (41) lxl v Y)alxl) sgn Y lrlv Jv(alrl) a> 0, -l/2 < Re v < 3/2 (42) sgn x lxi-J.Lisin [~(IL+v)77] lri-1-L Ieos [~2(/L+ v)1T] Y)alrl) x Y )alxl) +sin [~(IL+v)77] J)alyl)l -cos[~(IL+vh]J)alxl)l a>O -~~ < Re 11 < l-IRe vi (43) lxi-1-L I sin[~(/!+ v)77] J)alxl) sgn y lri-1-Llcos[~(/L+vh] J)alrl) +cos [~(IL+vh] Yv(alxl)l -sin[~(/L+v)1T] Y)alrl)l a>O -3/2 < Re 11 <l-IRe vi (44) sgn x lxi-1-Licos[alxl lri-1-L I sin [alrl- ~ (IL+v)7T] J)b lrl) -~(IL+vh] J)blxl) -cos[alri-~(IL+v)77] Y)blrl)l + sin[alxi-~(/L+v)77] YJblxl)l a<b -3/2 < Re IL < l-IRe vi (45) lxi-J.Licos[alxl sgn y lri-I-Lisin[alri-~(IL+v)77] -~(IL+v)7T] Yv(b JxJ) xYv(bJyJ)+ cos[aJyJ-~(IL+v)77} -sin [alxi-Yz(/L+ v)1T]J)blxl)l x J)blrl)l a< b -3/2 < Re 11 <l-IRe vi y is real, and the integral is a Cauchy Principal Value, 260 INTEGRAL TRANSFORMS 15.3 Higher transcendental functions (cont'd) f(x) 7T-1:roo -oo f(x) (x-y)-1 dx (46) 0 -oo <X< 0 2"-1 lriJ.L K)alrl y,) -oo<y<O xJ.LI cos[(fL- ~ v) TT] J )ax y,) y, y, yJ.Lisin[(jl-~v)TT]Jv(ay) + sin[(fL- ~ v)TT] Y)ax )] y, -cos [(fL- ~v)TT] Y)ay )} O<x<oo O<y<oo a> 0, IRe vi-l < lle fL < ~ (47) 0 -oo<x<O 77-1 e-ay Ko(alyl) e -ax 10 (ax) O<x<oo a>O (48) exp(-alxl) 10(ax) a>O -277-1 sinh(ay) K0(alrl) (49) sgn x exp (-alxi) 10 (ax) 277-1 cosh (ay) K0 (alyl) a>O (50) 0 -oo<x<-a 277-1 (y2-a 2)Y,v e -by K )b (y2-a 2)y,] (a 2_x2)Y,v e-b:~: J [b(a 2_x2)Y,] v -oo<y<-a -a< x <a -(a2-y2)Y,ve-by Y)b(a2-y2)Y,] 2 (x 2 -a 2)Y,v cos (vTT) e -b:~: x 1)b(x2-a2)y,] a<x<oo -a <y <a a> 0; b>O 2(y 2 -a 2)Y,v e -by ITT -1 ~[b(y 2_a 2)!1,] -1 <Rev <~ + sin(VIT) 1)b(y2-a2)y,]! a<y<oo (51) eax K0(alxl) a>O 7T e ay 10 (ay) -oo<y<O 0 O<y<oo y is real, and the integral is a Cauchy Principal Value. 15.3 HILBERT TRANSFORMS 261 Higher transcendental functions ( cont 'd) f(x) 77-1 :f oo f (x) (x -y)-1 dx -oo (52) sinh(ax) K0(ajxj) a>O )1:2 77 exp(-ajyj) 10 (ay) (53) cosh (ax) K 0 (ajxj) a>O -?:277 sgn y exp(-ajyj) I0(ay) (54) jxj-11 e ax K)a jxj) )1:277 sec(v77) jyj-11 eay a> 0, -?:2 < Re v < 7'2 x [I)a\yj)+ J_11(ajyj)] -oo<y<O -77 tan (v 77) y -v e ay K )ay) O<y<oo (55) jxj-11 sinh(ax) K11(ajxj) \y\-v [?:277sec (vrr) exp(-ajyj) 1 11(ajyj) a> 0, -?:2 <Rev< l -tan(vrr) sinh(ajyj) K11(ajyj)] (56) jxj-11 cosh (ax) K)ajxj) -jyj-v sgn y [?:277 sec(v77) exp(-ajyj) a> 0, -?:2 <Rev< )1:2 x I11(ajyj) +tan (v77) cosh (ay) K )ajy j)] (57) jxj211 exp(-ax 2) [K)ax2) -77 cos(vrr) sgn y Jyj211 + 77 sin(vrr) I (ax2)] II x exp(-ay2 ) I)ay2) a> 0, -~<Rev<~ (58) sgn x Jx\211 exp(-ax2) I)ax2) Jy\211 exp (-ay 2 ) a> 0, -~<Rev< )1:2 x [77 -1 sec (v77) K)ay2) + tan(v77) I)ay2)] (59) sgnx \x\-11H)a\x\) \y\-v J)a\y\) a> 0, Rev> -3/2 y is real, and the integral is a Cauchy Principal Value. 262 INTEGRAL TRANSFORMS 15.3 Higher transcendental functions (cont'd) f(x) 11-1 f"" f (x) (x -y)-1 dx -oo (60) 0 -oo <X< 0 .-•c·"··" ~I !\"·"•""'",) p+l,q+l r, o b b ' ,, ••• ' q ~ 'a1' ... ,a) G "n ax P -C>O < y < 0 pq bl, ••• ,bq O<x<oo (-'/' m+l,n+l ~ I O,al, ... ,ap, ~+k) 1 G +2 +2 ay p + q < 'lf...m + n) P ,q O,b1, ... ,bq,~+k \ arg a\ < (m + n -~ p -~ q) 11 O<y<oo Rea < 1 j = 1, ••• , n J k integer Reb.>-1 j = 1, ••• , m J (61) G'"n ~x2,al' ···' aP) " , c·"··" ~ , 1 Y. ......... ,. 1) pq bl' ••• ,bq g y p+2,q+2 y ~ b b 1 2, 1 , ••• , q' p + q < 2(m + n) \arga\ < (m + n-~p-~q)rr Rea.< 1 j = 1, ••• , n Re b1> -~ j = 1, ••• , m J (62) ~ G ;~ ~x 21 a I' ••• ' a P) \ \Ga+l,n+l ~ 2~-~,al' ... ,ap,O) bl' ••• ,bq y p+2,q+2 y -~b b 0 2, t' ••• ' q' p + q < 2(m + n) \arga\ < (m + n-~p- ~q)rr Rea. <~ j = 1, ••• , n Reb1>-1 j = 1, ••• , m J y is real, and the integral is a Cauchy Principal Value. INTEGRALS OF HIGHER TRANSCENDENTAL FUNCTIONS This part contains mostly integrals which have not been listed in the tables of Chapters I to XV. 263 CHAPTER XVI ORTHOGONAL POLYNOMIALS In this chapter we list integrals involving the classical orthogonal polynomials. For the theory of these polynomials see 1-J.T.F. vol, II, Chapter X and the literature quoted there, especially Szego's book, The notation used in the present compilation for Hermite polynomials differs from that used in H .T .F, Further integrals may be evaluated by the methods mentioned in the introduction to vol, I, by the use of Rodrigues' formula and its analogues (given below) followed by repeated integrations by parts, by using gener­ ating functions (see below), and also by utilizing the relations (see below) between the various systems of orthogonal polynomials and between these polynomials and Legendre functions, hypergeometric series, confluent hypergeometric functions in conjunction with tables given in other chapters of this hook. Tchebichef polynomials Tn (x) = (-l)n Tn (-x) =cos (n e) = (1-x2) ~ ~ _!_)n [(1-x2)n-~] 2n(~)n dx = 2F, (-n, n; ~; ~-~x) nl = ~n C0(x) =-·- p<-~.-Xl(x) 2 n (~) n n sin [(n + 1)e] U (x) = (-1) n U (-X) = ----- n n sin e (n + 1) (1-x 2)-~ 2n+1 (~)n+l 265 x =cos e x =cos e 266 INTEGRAL TRANSFORMS ( 3 1-x\ U" (x) = (n + 1) 2F1 -n, n + 1; Z; -2-) (n + 1)! C 1 ( ) p('h., '/,){x) = n X = (11) n 2 /2 n+ 1 1-z 2 T (x) z n = 2 n 1-2xz + z 00 ~ n= 0 For other generating functions see H.T.F. vol II, P• 186. Legendre polynomials p (x) = (-1)" p (-x) =-1-(ddx)" [(x2-1)"] n n 2" n! = 2F1 (-n, n + 1; 1; ~-~x) ')"(I/,) ~ '2 n n ( II II II II -2) X 2F1 -;2n,;2-12n;12;x n! 00 2, n= 0 For tl1e connection with Legendre functions see 1-l.T.F. vol. I, P• 150 A; for additional hypergeom etric series representing Legendre polyno­ n>ials see H.T.F. vol. I, p. 124-131 (f1 = 0, v = n), and vol. II, P• 180; and for other gene):atin g functions see I-l.T.F. vol II, P• 182. For the definition of associated Legendre polynomials, and their properties see ll.T.F. vol. I, p. l481L and below. Gegenbauer polynomials These polynomials are also called ultraspherical polynomials and are denoted by pn(v)(x). ORTHOGONAL POLYNOMIALS C11(x) = (-1)" C11(-x) n n 211-~ f'(2v+ n)f'(v+ ~) (x2- 1)~-~v p~-v (x) n!f'(2v) n+v-~ (2v) =--" 2F1(-n, n + 2v; v+ ~; ~-~x) n! 267 2" (v) ( 1 2 ) = " (x -1)" F -n --n -v · 1 - 2 n -2 v · --n! 2 I '2 ' ' 1-X (2v) n = (v)n p (v-~. -~) (2x2-1) (~) n n Cv + (x) = (-1)" (v) n+ 1 2x F ( 1 / 2) 2n I 2 I -n, n + V + ; 3 2; X n! = (v)n+l xP (v-~. ~) (2x2-1) (~)n+l n ~ n= 0 For the connection with Legendre functions, and for additional hyper­ geometric expansions see H.T.F. vol. I, p. 175ff, P• 124-131, Vol. II, P• 176; for other generating functions see H.T.F. vol. II, p. 177. 268 INTEGRAL TRANSFORMS Jacobi polynomials P (a,,B)(x) = (-l)" P (,8, a)(-x) n n ~ n= 0 (n+a) ( l-x) n 2F1 -n, n + a+ {3 + l; a + l; -2- en+ {3) ( · l + x) = (-l) n n 2F, -n, n + a + {3 + l ; {3 + l ; -2- ( n : a) (l ~ x) n c : {3) (X~ l) n ( X -l) 2F1 -n, -n-{3; a+ 1;--x+l ( x+l) 2F, -n,-n-a;{3+l;--x-l Other expansions may he obtained from those given above by means of the transformations given in H .T .F, vol, I, sec, 2. 9. Hermite polynomials He (x) = (-l)" Jle (-x) = 2-Y.n H (2-Y. x) · n n n 2 ( d)" 2 = e Y.x -dx [e -Y,x ] =x" F c-~ ~-_2_\ 2 o 2' 2 ' x2) -2Y.n+~ -Y. l4x2 W Qx2) - x e v +'' v 12n 14 • -;.. 2 l/ 2 = e /4X D (x) n ORTHOGONAL POLYNOMIALS He2n (x) = (-2)" (Y:;)n 1F1 (-n; Y:;; Y:;x2) = (-2)" n! L-X (Y:; x 2) n He 2n+l (x) = (-2)" (3/2)" x 1F1 (-n; 3/2; Y:;x2) =(-2)"n!xLX(Y:;x2) n 00 L n=· 0 z" He (x)-= exp(-Y:;z2 + xz) n n! For other generating functions see H.T.F. vol. II, p. 194. Laguerre polynomials = (n +n a) 1F1(-n; a+ 1;x) (-1)" =--x" F(-n-a-n·-1 /x) ' 2 0 ' ' n .. (-1)" -Xa-X Xx 'I ( ) = --- X e If v +'/ + v X f n n a n. n a n. ~ n= 0 XZ L "\x) z" = (1-z)-a-l exp --- n Z-1 For other generating functions see B.T.F. vol. II, P• 189. 269 ORTHOGONAL POLYNOMIALS 16.1. Tchebichef polynomials The integrals in this section may also be expressed as integrals of trigonometric functions. In this section m and n are non-negative integers. ( l) J' y, 2a+2n+Y, rry, (n!)2 f'(a+ 1) f'(a+3/2) (1-x)- , (1+x)aT (x)dx= -1 " (2n)! f'(a+n+3 /2) l~(a-n+ 3/2) Rea> -1 (2) / (1-x)a(1 + x)f3 T (x) dx = 2a+j3+2n +1 (n!) 2 ['(a+ 1) l ((3 + l) -1 n (2n)! f'(a+f3+2) X 3F2(-n, n, a+1; ~. a+f3 + 2; l) Rea> -1, Re {3 > -1 (3) -f 1 (x-y)-1 U-x2)-y, T" (x) dx = rr U"_1 (y) -l<y<1 -1 (4) t 2 y, 2 y, ( _1sin(xyz) cos[(1-x) (1-y) z] T2n+1 x)dx = (-1)n 1T T2n+1 (y) J2n+1 (z) (5) f cos(xyz) cos[(1-x2)y, (1-y2)y, z] T2 (x)dx -1 n = (-1)" 1T T2n (y) J2n(z) (6) f [T (x)f dx = l-(4n 2-l)-1 -1 n 271 272 INTEGRAL TRANSFORMS 16.1 Tchebichef polynomials (cont'd) m, n = 0, 1, 2, ••• (7) f {l-x2)-l{ [T0(x)fdx=TT -1 (8) { (1-x2}-X [T (x)Fdx = 1:;rr n~O -1 n (9) J1 (1-x2}-x T (x) T (x)dx=O m~n _1 m n (10) J 1 ( 1 -x} -1 12( 1 + x) "-n-312 T (x) T (x) dx = 0 m > n _1 m n (ll) t (1-x)-112 (1 + x)"+n-312 T (x) T (x) dx _1 m n rr(2m + 2n-2)! m+n~O = 2"+n(2m- 1)! (2n-1)! (12) f1 (1 + x)-X (1-x)a-1 T" (x) T n(x) dx TTX 2a-l{ r(a) l,(n-a+ 1:;) = r (7:; -a) l, (a+ n + 7:;) x 4F3 (-m, m, a, a+ 1:;; 7:;, a+ n + 7:;, a-n+ 7:;; 1) Rea> 0 (13) t 0 x-Y, (1-x2)-X e -2a/x Tn (x) dx = TTX Dn-X (2!ly,) D-n-X (2aX) Rea> 0 (14) J' (l-x2)-X T (1-x2y)dx=7;;rr[P (1-y)+P (1-y)] -1 n n n-1 (15) ~00 (l + x2)-n sech(7:; rrx) T 2n [(1 + x2)-X] dx 2rr2n = (-l}n+1 __ (2n-1 _ 1) B (2n)! 2n 16.1 ORTHOGONAL POLYNOMIALS 273 Tchebichef polynomials (cont'd) m, n = 0, 1, 2, ... (16) f000 (1 + x2)-~n sech (~ rrx) TJ(l + x2)-~] dx = 21-n(1-21-n)((n) (17) Jo"" (1 + x2 )~-n [cosh(~rrx)r2 r2n-1 [(1 + x2)-~] dx 2n-1 :: 2(-l)n+1 7T2n-1 __ 8 (2n)! 2n (18) J"" (1 + x2)-~n [cosh(~rrx)r2 T [(1 + x2)-~] dx 0 n = 1T -1 n 2 1 -n ((n + 1) (19) J"" (a2 + x2)-~n sech(~rrx) T [a(a2 + x2)-~] dx o n 12n[(a+1) ~ a+3)] 1 ( a+1) =2- ( n,- 4--( n,- 4-=2 -n<l> -1,n,- 2- Rea> 0 (20) J"" (a2 + x2)-~n [cosh( ~rrx)r2 T [a(a2 + x2)-~ :kix 0 n 1 ( a+ 1) =rr-1n2 -n( n+1,- 2- Rea> 0 r ( 1 -X)~ ( 1 + X) a lj (x) dx = 7T1/2 2a+2n+3/2[(n+l)JY1(a +~)l(a+1) (21) (2n+ 2)!1(a+n+5/2)1(a-n+ 1/2) -1 n Rea> -1 r (1-x)a(1 + x)13 un (x) dx = 2a+/3+2n +2 [(n + 1) !] 2 r· (a+ 1) r ({3 + 1) (22) (2n + 2)! r (a+ {3 + 2) -1 x 3F2 (-n, n + ~ a+ 1; 3/2, a+ {3 + 2; 1) Rea> -1, Re{3>- 1 274 INTEGRAL TRANSFORMS 16.1 Tchebichef polynomials (cont'd) m, n = 0, 1, 2, ... (23) / (x-y)-1 (1-x2)-y, U (x) dx = -rr T +1 (y) _ 1 n n -1<y<1 (24) t 2 y, 2 y, ] _1cos(xyz)sin[(1- x )' (1-y) z U2n(x)dx =(-1)"rr(1-y2)Yz u2n(y)J2n+1(z) (25) J 1 sin (xyz) sin [(1-x2)y, (1-y2)y, z] u2 +1 (x) dx -1 n = (-1)" TT (1-y2)Y, lj2n+1 (y) J2n+2 (z) (26) r (l -X)-Y, ( 1 + X) Y, [U (x )F dx = (n + } ) TT -1 n (27) p (l-x 2)y, [U (x)f dx = ~ TT -1 n (28) f (l-x2)y, U (x) U (x) dx = 0 -t a n m -Fn (29) f 1 ( 1 -x )( 1 + x) y, U • (x) U n (x) dx 2 512 (m + 1 )(n + 1) = (m + n + 3/2)(m + n + 5/2)[1- 4 (m-n)2] (30) f (1-x)y, (1 + x)•-n-Yz U (x) U (x) dx = 0 m>n -1 • n (31) I\ <1-x) 112 <1 + x)•+n+312 u .. <x) u" <x) dx rr(2m + 2n + 2)! X 2•+n+2 (2m+ 1)! (2n + 1)! 16.1 ORTHOGONAL POLYNOMIALS 275 Tchebichef polynomials (cont'd) m, n = 0, 1, 2, ... (32) f (1+x)~(1-x)a-l U (x) U (x) dx -1 • n rr~ 2a-~ (m + 1)(n + 1) 1(a) 1(n-a+ 3/2) = 1(3/2- a) 1(3/2 +a+ n) x 4F3(-m,m+2,a,a-r:;;3/2, a+n+3/2, a-n-1/2;1) Rea> 0 (33) J I 2 -~ ( 2 ) (1-x) U2 (xz)dx=rrP 2z -1 -1 n n (34) ~ ~ 2 Un(y) Un(z) r u [x(1-y2) (1-z2) +yz]dx=---1 n n+1 (35) iooxU2 1[(1+x2)- ~] l (-1)"rr2n 0 (1 +n~2)"+~ (e71"' + 1) dx = 2(2n -1) + 2(2n)! B2n (36) IooxU[(l+x2)- ~] 1 -n-1(( ) n dx =-- 2 n + 1 (1 + x2)~n+l (e71"' + 1) 2n 0 ioo X lj2n-l ((l + x2)-~] (-1)"+1 (2rr)2n 1 1 (37) (1 + x2)n+~ (e 27Tx-1) dx = B ------ 4 (2n)! 2n 4 4n-2 loo xUn[(1+x2 )-~] 1 1 1 (38) (1 + x2)~n+l (e27Tx _1) dx =-((n + 1)---- 2 4 2n ioo 2 2 -~] a-n 1 ( a+ 1) X Un [a (a +X ) (39) (a2 + x2)~n+l (e7Tx + 1) dx =---z-n-' n + 1,--2n 2 Rea> 0 276 INTEGHAL TRANSFOHMS 16.1 Tchebichef polynomials (coot 'd) m, n = 0, 1, 2, ... (40) ,11 dx =-((n + 1 a)------- [ xU [a(a2+x2)-Y,] 1 a-"-1 a-n 0 (a2 + x2)Y,"+1 (e271x-1) 2 ' 4 2n Rea> 0 16.2. Legendre polynomials See also under Gegenbauer polynomials, Legendre functions, hyper­ geometric series. In this section m and n are non-negative integers. ( l) J1 A. ( (-1)" (-X A) ReA> -1 x P2 x) dx = " 0 • 2(X+XA),.+1 (2) J' A_ (-1)"' (X -X A). ReA> -2 x P 2 .. + 1 (x) dx = 0 2(1 +X A),.+1 2 3/2 (3) J' (1-x)-y, P (x) dx = -1 n 2n + 1 2 (4) J.' 2 -Y, [(X) m J _ 1 ( 1 -x ) P 2,. (x) dx = rr ~ (5) j1 x(1-x 2)-Y, P (x)dx= 77 (X),. (X)m+1 -1 2• + 1 m ! (m + 1) ! (6) J' (1-x)a-1 (1 + x)/3-1 P (x) dx = za+/3-1 ['(a) f'(f3) -1 n f' (a + (3) x 3F2 (-n, 1 + n, a; l, a + (3; l) Rea> 0, Re (3 > 0 16.2 ORTHOGONAL POLYNOMIALS 277 Legendre polynomials (cont'd) m, n = 0, l, 2, ... (7) f (z-x)-1 P (x) dx = 2Q (z) -1 n n z in the cut plane (8) f x (z -x)-1 P (x) dx = 2 Q (z) -1 0 1 z in the cut plane P (z-x)-1 x"+1 P (x) dx = 2z"+1 Q (z) 2n+1 (n!) 2 (9) (2n + l)! -1 n n z in the cut plane ( 10) r (z -X) -1 X • p (x) dx = 2 Z " Q (z) -1 n m m ::;n, z in the cut plane ( ll) t (a2+ b2-2abx)- l1 sin[.\(a2+ b2-2abx)l1]p (x)dx -1 n = rr(ab)-)1 Jn+l1 (a,\) Jn+l1 (b,\) a, b > 0 (12) r (a2+b2-2abx)- l1 cos[,\(a2+b2-2abx) Y.]p (x)dx -1 n = rr(ab)-)1 Jn+Y, (a,\) Yn+Y, (b.\) o::;a::;b (13) f [P (x)f dx = (n + X)-1 -1 n (14) f1 P" (x) P n (x) dx = 0 m ,J n + 2"+n+1 [(m-+ n)!]4 (15) p ( l + x)" n P (x) P (x) dx = 2 -1 " n (m!n!) (2m+2n+l)! The complex z-plane is cut along the real axis from -1 to 1. 278 INTEGRAL TRANSFORMS 16.2 Legendre polynomials (cont'd) m, n = 0, 1, 2, ... (16) J1 (1 + x)"'_"_1 P (x) P (x) dx = 0 m>n -1 m n 1 2a r (a) r (n -a + 1) ( 17) J (1-x)a-1 P (x) P (x) dx = -1 m n r (l-a) r (n + a + 1) x 4F3 (-m, m + 1, a, a; 1, a+ n + 1, a-n; 1) Rea> 0 (18) J' (z-x)-1 P (x) P (x) dx = 2P (x) Q (z) _1 m n m n m :S n, z in the cut plane ( 19) _[' 1 (z-x)-1 Pn (x) Pn+1 (x) dx = 2Pn+1 (z) Q n(z)-2(n + 1)-1 z in the cut plane (20) J'x(z-x)-1 [P (x)Fdx:=2zP (z)Q (z)-2(2n+l)-1 -1 n n n z in the cut plane (21) L:X<z -x)-1 Pm(x) Pn(x)dx = 2z Pm (z) Qn(z) m < n, z in the cut plane For other similar integrals see MacRo bert, T.M., 1948: Proc. Glasgow Math. Assoc. 1, 1Q-12. (22) J' x 2J.L-1 p (1-2x 2) dx = (-1)" [r (ll)]2 Re !l > 0 0 n 2r(!l+n)r(!l-n) [ ( 2 1)~r2n-1 (23) J' ( 2 2)-~ p ( 2) d . a+ a + Rea> 0 X a +X 2 1-2x X= 0 n 2n + 1 The complex z-plane is cut along the real axis from -1 to 1. 16.2 ORTHOGONAL POLYNOMIALS 279 Legendre polynomials (cont'd) k, l,m, n = 0, 1, 2, ... (24) J 1 (1-x)a(l + x).B P '"(x) dx -1 n See Shabde, N.G., 1940: Bull. Calcutta Math. Soc. 32, 121-128. (25) J.1 (z -x)-1 (1-x2)Y,m P'"(x)dx= (-2)" (z2 -1)y," Q" (z) -1 n n m ::;n, z in the cut plane (26) r xk (z-x)-1 (1-x2)y," P'" (x) dx = (-2)" zk (z 2-1)y," Q"(z) -1 n n m ::;n, k = 0, 1, ... , n -m, z in the cut plane r 2 2 (n + m)! (27) [P" (x)] dx = -- (n -m)! m ::;n -1." 2n+1 (28) f, P "n (x) P ~ (x) dx = 0 kf,n 1-(-1)k+n (k + m)! (29) J1 P" (x) Q~ (x) dx = (-1)" (k -m)! - 1 n (n -k) (n + k + 1) (30) 11 (1-x2)-1 P"'(x)Pk(x)dx=O 1 n n kf,m 1 (n + m)! (31) J (1-x2)-1 [P., (x)f dx = -1 n m (n-m)! The complex z-plane is cut along the real axis from -1 to l. 16.3 ORTHOGONAL POLYNOMIALS 281 Gegenbauer polynomials (cont'd) m, n = 0, 1, 2, .•• (4) f_1 1 (1-x)a(1 + x)/3 C~(x) dx = 2a+f3+1 r(a + 1) r({3 + 1) r(n + 2v) n ! r (2 1/) r (a + f3 + 2) x 3F2 (-n, n + 2v, a+ 1; v + ~. a+ f3 + 2; 1) Rea>-1, Re/3>-1 (5) J 1 x• (z-x)-1 (1-x2)v-~ cv(x) dx -1 n TT 1/2 2 3/2 -v -(v-~ )71 i " ( 2 _ 1)~ v-~ Qv-~ ( ) = r (v) e z z n+v-~ Z m ~n , Rev>-X, z in the cut plane (6) J 1 Xn+1 (z-x)-1 (1-X 2)v-~ cv(x) dx -1 n TT 1/2 2 3/2 -v -(v-~)71 i n+1 ( 2 _ 1)~ v-!4 Qv-~ ( ) = r (v) e z z n +v-~ Z 21-2v-n 1 TT n. - Rev> -X, z in. the cut plane r (v)r (v+n +1) Jh-x2r-~e iax cv(x) dx = TT 21-v in r (2 1/ + n) a -v J + (a) (7) n!r(v) -1 n v n Rev>-X TT21-2vr(2v+n) (8) l (l-x2)v-~ [Cv(xW dx = Rev >-~~ -1 n n! (n + v) [r (v )JZ r U-x)v-312 (1 + x)v-112 [CV(x)f dx = TT~ r(v-X)r(2v+n) (9) n ! r (v) r (2v) -1 n Rev> X The complex z-plane is cut along the real axis from -1 to l. 282 INTEGRAL TRANSFORMS 16.3 Gegenbauer polynomials (cont'd) m, n = 0, 1, 2, ••• (10) _t (l-x)v-X (1 + x)2v-1 [Cv(x)}2 dx -1 n 2311-x [1 (2v + n)Jl 1 (2n + v + ~) Rev> 0 = (n!)21(2v)1(3v +2n+~) (11) J1 (1-x)3v+2n-3/2 (1 + x)v-1/2 [Cv(x)f dx -1 n 7Tx [1 (v + ~)] 2 1 ( v + 2n + ~) 1 (2 v + 2 n) 1 (3 v + 2 n -~) = 2 211+2n [n! 1(v +' n + ~) 1(2v))2 1 (2v+2n+ ~) Rev>l/6 (12) f (l-x2)v-X C11(x)C11(x)dx=0 -1 ~ n m I= n, Rev>-~ (13) r {l-x)v-1/ 2 (l + x)v+m-n-3/2 c ll(x) cv(x) dx -1 m n = (-1)• 22-211-• +n rr312 1(2v+n)1(v- ~+m-n)1(~-v+m-n) m! (n-m)! [1 (v))21 (~ + v+ m)r(~-v--n )1 (~+ m -n) Rev>~ (14) f 1 (1-x)2v-1 (l + x)v-X C v(x) Cv(x) dx -1 a n 2311-X r(v+ ~)r(2v+ m)1(2v+n )I'(v+ ~+m+n )l{~-v+n-m) = m !n !r(2v)1(~-v)r(v+ ~+n-m) r(3v+ ~+m + n) Rev>~ (15) J' (1-x)v-1/2 (1 + x)3v-l,n +n-3/2 cv(x) cv(x) dx -1 m n 2411-t, +n-1 [r(v+ ~)l(2v+m +n)f 1(v+m +n +~)I '(3v+m +n-~) = l(v+m + ~~)l(v+n+ ~)r{2v+ m )1 (2v+n )f'(4v+ 2m+ 2n) He v > 1/6 16.3 ORTHOGONAL POLYNOMIALS 283 Gegenbauer polynomials (coot 'd) m, n = 0, 1, 2, ... (16) t (1-x)a(1 + x)v-~ c Jl(x) cv(x) dx -1 m. n 2a+v+Y, ['(a+ 1) f'(v + ~) [' (v-a+ n-~) ['(2JL+ m) f' (2v + n) = m! n! [' (v-a-1/2) f'(v-a+ n + 3/2) ['(2p) [' (2v) ( 3 1 3 3 0 x F -m m+2p a+1 a-v+- ·p+-v+a+n+- a-v-n+-·1 4 3 ' ' ' 2' 2' 2' 2' Rea>-1, Rev >-~ ( 17) r (z-x)-1 (1-x2)v-Y, c v(x) cv(x) dx -1 m n 77~ 2Y,-v e -(v-Y,)TT i(z 2 _ l)y, v-)4' C v(z) Qv-Y, (z) = [' (v) m n+v-Y, m .::;n, Rev >-~. z in the cut plane (18) p (1-x2)v-~ c V(x) cv(x) C~(x) dx _ 1 m n See Hsii, Hsien-Yii, 1938: Duke Math. f. 4, 374-383. 1 rry, f'(~ v) cV.v(2a2 -l) (19) J: (1-x2)Y,v-1C~(ax)dx= 2 Rev> 0 -1 n ['(~ v+ ~) n (20) J:1 (1-x2)v-l cv(cosacosf3+xsinasin{3)dx -1 n 22v-t n! [f'(v)f cv(cosa) cv(cos{3) Rev> 0 = [' (2v + n) n n (21) J lxzv(l-x z)o--1 0 2 (2v) f'(v+~)f'(a) (a .Bl( ) C v ( 1 -X y) dx = n P ' 1 -y n 2f'(n+v+a+~) n Rev>-~, Rea> 0, a= v +a-~. f3=v-a-~ The complex z-plane is cut along the real axis from -1 to 1. 284 INTEGRAL TRANSFORMS 16.4 16.4. Jacobi polynomials See also under hypergeometric series. In this section m and n are non-negative integers. ( l) J1 ( 1 -x) a ( l+ x )u P (a ,,8 >(x) dx -1 n 2a+u+l r(a + 1) r(a + n + 1) r(a-(3 + 1) = r (a -(3 -n + 1) r (a+ a+ n + 2) Rea> -1, Rea> -1 (2) f (1-z)P(l+x) ,Bp(a,,B)(x)dx -1 n 2,8+p+l r(p + 1) r({3 + n + 1) r(a-p + n) = n! r (a-p) r ((3 + p + n + 2) Rep> -1, Re (3 > -1 (3) r (1-x)P(l+x)up(a, ,B)(x)dx"' 2p+u +I r (p + l) r (a+ 1) -1 n r (p + a+ 2) x 3F2(-n, a+(3+n+1, p+1; a+1, p+a+2; 1) Rep>-1, Rea> -1 (4) J~ 1 (z-x)-1 (l-x)a(1 + x).B P ~a,,B)(x) dx 2a+,B+n+l r(a + n + 1) r({3 + n + 1) = r (a+ {3 + 2n + 2) (z -1)n +I X 2F, (n+l,a+n+1;a+f3+2n+2;--2--) z -1 Rea>-1, Re (3 > -1, z in the cut plane The complex z-plane is cut along the real aX:is from -1 to 1. 16.4 ORTHOGONAL POLYNOMIALS 285 Jacobi polynomials (cont'd) m, n = 0, 1, 2, ... (5) I' (l-X )a(1 + x)f3 [p<a./3 ><xW dx = 2a+f3+1 r (a+n + 1) r (/3 +n + 1) -1 n n!(a+J3+2n+1)r(a+J3+n+1) Rea> -1, Re J3 > -1 (6) f1 (1-x)a-1 (1 + x)f3 [P (a,f3)(x)]2 dx = 2a+f3r(a+n+ 1) r(J3+n+ 1) -1 n n! a r (a+J3+n+ 1) Rea> 0, Re J3 :> -1 (7) f (1-x)2a (l + x)f3 [P <a,f3>(x)f dx -1 n 24a+f3+1 rCa+~) [r(a+n+l)F r<J3+2n+1) = 1T l{ (n !) 2 r (a:+ 1) r ( 2a + J3 + 2n + 2) Rea>-~. Re J3 > -1 (8) p (1 -x )2a+f3+Zn ( 1 + x )f3 [P (a ,{3 )(x )JZ dx -1 n 2 za+zf3+2n +, r<J3 + 2ra1Hr<a+ 13+ 2n + 1W r (2a+ J3 + 2n+ 1) = [n! r (a+ J3+n + 1W r C2a+ 213 +4n + 2) Re J3 >-1, Re(2a+ J3) >- 1 (9) r (1-x)a(l + x)f3 p (a,f3>(x) p (a,f3)(x) dx = 0 -1 n 11. m /: n, Rea> -1, Re J3 > -1 (lO) f ( 1 -x) P ( 1 + x )f3 P <a· f3 >ex) P <p ./3 >ex) dx -1 n n 2p+f3+1 r (p+n + 1) r (J3 +n + 1) r (a+ J3 + 2n + 1) = n! r (p + (3 + 2 n + 2) r (a+ J3 + n + 1) Rep> -1, Re/3>-1 286 INTEGRAL THANSFORMS 16.4 Jacobi polynomials ( cont 'd) m, n = 0, 1, 2, ... ( ll) J' 0-x)P-1 (1 + x)f3 P (a,f3)(x) P (p,f3)(x) dx -1 n n 2p+f3r(a+n+1) r(,l3+n+1) r(p) Re ,13 > -1, Rep> 0 = n! r(a+ 1) r(p+,B+n+ 1) (ll) {1 (l-x)a(l + x)cr P ~a,f3)(x) P .,<a,cr)(x) dx 2a+cr +1 r (a+n + l)r (a+ (3 +m+n+ l) r (a+m+ 1)1_' (a-,13+ 1) = m!(n-m)!r(a+f3+n+l)r(a+a+m+n+2)r(a -,B+m + 1) Re a> -1, Rea> -1 (13) P (1-x)a(l + x)f3+cr P (a,f3Jcx) P (a,cr)(x) dx _1 n 1n 2a+f3+cr +1 r (a+m+n + 1) r(,B+n+ 1) r(,B +a+ 1) r (a+ m + 1) = m!n! r(a+f3+a+m+n+2) r(,l3-m+n+1) r(a+m -n + 1) Rea> -1, Re(,B+a)>-1 (14) J 1 (1-x)a(l + x)a+f3+cr +,.+n p (a,J3 l(x) p (a ,cr l(x) dx -1 n • 22a+J3+cr +m+n+1 r (a+ ,13 +a +m + n + 1) r (a+ a+ m +n + 1) = m! n! r(a+ ,13 +n + 1) r (a+a+n+ 1) r(a+m+n+ 1) r(a+.B+m+n+ 1) X r(2a+,l3+a+2m+2n + 2) Rea> -1, Re(a+,B+a) >-1 (15) J1 (1-x)a(l + x)cr +..-n-1 p (a,f3)(x) p (a,cr)(x) dx _ 1 n m 2a+cr+.t-n r(a+n+1) r(,B+n+ 1) r(a + m-n) r(a-,B+m-n) = n !(n -m) !r (a+. a+ m + 1)r (,13-m+ n + 1) r (a-,13 +2m -2n) He a> 0, Rea> n-m 16.4 ORTHOGONAL POLYNOMIALS 287 Jacobi polynomials (cont'd) m, n = 0, 1, 2, ... (16) r (1-x)P (l + x),B P (a,,B)(x) P (p, ,BJ(x) dx -1 n m 2,B+p+ll~ (a+ /3+m +n+l)i (/3 +n +1)[' (p+m +1)['(p-a-m+n) = n! (n-m)!i(a+/3 +n+1)['({3 +p+m+n+ 2)['(p- a) Re /3 > -1, Rep> -1 (17) f (1-x)a+p (1 + x),B P (a,,B)(x) P (p,,B)(x) dx -1 n m. (-1)"' +n 2a+,B+p+l [' (a+n +1)1 (a+ p + 1) ['({3 + m + n +1)['(p+m +1) = m! n !1 (a-m +n +1 )['(a+ /3 + p+ m + n +2)1(p + m -n +1) Re (a + p) > -1, Re /3 > -1 (18) f (l-x )a+,B+p+,.+n (1 + x ),8 P (a ,,8 )(x) P (p,,B l(x) dx _ 1 n m (-l),.+n 2a+,B+p+m+n+1 ['(a+{3+m+n + 1) = m!n!1(a+/3+n+ 1) !(a+ /3+ p+m+n+ 1)['({:3+m +n+ 1) 1(/3+ p+m+n + 1) X !(a+ 2{3+p+ 2m+ 2n+ 2) ['({3+p+m + 1) Re /3 > -1, Re(a+ /3 + p) > -1 (10) j_1 1 (1 -x )p+m-n-l (1 +x ),8 P~a ,,8 )(x) P ~p,,B l(x) dx 2,B+p+m-nr (a+ n +1)1 ({3 +n + 1) [' (p + m-n) na-p- 2m+ 2n+1) = n! (n-m)![' (a-m +n + 1) ['(a-p-m +n + 1) ['({3 +p +m +1) Re /3 > -1, Rep> n-m 16.5 OHTHOCONAL POL YNmliALS 289 Hermite polYnomials (cont'd) m, n = 0, 1, 2, ... (6) Joo 2 }' exp(-~r ) cos((:3:r:) }lc (x) dx = (-l)n e~rr)' (J2nexp(-12()2) 0 ~ (7) Iaoo exfl(-1~x2 ) sinh ((h) He2n+t (x) dx = 0'2rr)Yc j32n+t exp (/2{32) (~') Joo I 2 1/ X 2 2 exp(-;;z::) cosh(f;x) lle2 (x)dx = (nrr) /3 n exp(~2/3) 0 n (9) Joo exp(-!_x2) [l!cn(x)Jl dx = (2rr)X n! -oo Joo e-x 2 lie (x) He (x) dx = (-1)X"' -Xn r( m + 2n + 1) 00) -oo m n m + rz even (: l) J"" 2 m l=n exp (-~'~ ; ) Hem (x) He n(x) dx = 0 -oo ( 12) Joo 2 2 ( exp(-a x) lle,.(x) Hen x) dx -oo X +Y, (m+n+1 ~ = a-m-n -1 ( 1-2a 2) m m [' 2 ~ 1-m-n a2 ) x 2F1 -m, -n; ; 2 2 2a - 1 fie a2 > 0, m + n even (13) Joo exp [-1 ~ (x -y) 2] 1-1 e (x) He (x) dx = (2rr) y, m ! y n-" L n-'" (-y 2) I -oo m n n m<n 290 INTEGRAL TRANSFORMS 16.5 Hennite polynomials (cont'd) k, m, n = 0, 1, 2, ... (14) J:'oo exp (-x 2) II e k (x) II e m (x) II e n (x) dx = 1T -I [' (s -k) r (s -m) [' (s -n) k + m + n even, 2s=k+m+n+1 (2 )~ k I I I (15) Joo 2 1T • m . n . exp (-X.'\: ) He k(x) He (x) lie (x) dx = -oo m " (s-k)!(s-m)!(s-n)! k + m + n = 2 s even (1G) Joo exp(-a2x2) lie (x) lie (x) Bek(x) ••• dx -oo Wl n See Busbridge, I.W., 1948:]. London Math. Soc. 23, 135-141. (17) J00exp[-X(x-y)2]He"(ax)dx=(2rr) ~ (1-a2)y,"He{ uy j -oo (1-a2) ~ (18) fooo exp(-Xx2) sin({3x) He2n+l (ax) dx 1 ~ 2 +X 2 [ a f3 J = (-1)"(Xrr) (a -1)" exp(-X/3) lle2 +I 2 ~ n (a -l)' (19) fooo exp(-Xx2) cos ({3x) He 2" (ax) dx =(X rr)~ (1-a2)" exp(-Yz/32) lle 2" [(a:~ 1)~ ] (20) Joo exp(-Xx2) H (ax) H (x) = 0 m<n -oo 111 n 16.5 ORTHOGONAL POLYNOMIALS 291 Hermite polynomials (cont'd) m, n = 0, l, 2, ... oo 1 2 1 (2m+n)! (21) J exp(- ~x ) H2 + (ax) H (x) dx = rrX -X (a2-l)" a" -oo ,. " " m! 2" Re(a2 + {32) > 0, m + n even (23) Joo exp(-)·h2) He (ax) He ({3x)dx -oo • n (24) Joo exp(-A2x2) He,.(ax) He"({3x) dx = 0 -oo m + n odd See Buchholz, Herbert, 1953: Die konfluente hypergeometrische Funktion. Springer-Verlag, Berlin, Gottinger, Heidelberg, Sec. 13. (26) f_:exp[- ~(x-y)2 ) He.,(ax) He"(ax) dx min( .. , n) =(2rr)X k~o k!(:)(~)(l-az) Xm+Xn-k X He m+n-Zk [ (l a: az)x] 292 INTEGRAL TRANSFORMS 16.5 Hennite polynomials (cont'd) k, m, n = 0, l, 2, ... (27) Joo 2 2 _ 00exp(-A x ) He,.(ax) He/f3x) Ilek(yx) dx = 0 m + n + k odd (23) Joo 2 2 _ 00exp(-A x ) llek(ax) He.,(f3x) llen(yx) ••• dx See I3ailey, v;.N., 1948: J. London Math. Soc. 23, 291-297. Lord, R.D., 1949: ]. London Math. Soc. 24, 101-112. (2<l) Joo p-1 2 2 x exp (-A x ) l!e k (ax) He ({3x) lle (yx) •.. dx 0 m n See Appell, Paul and ~l.j. Kan•pe deFeriet, 1926: Fonctions hyper- g~om~t riques et hypersph~riques. Polynomes d' 1/ermite. Gautltier- Vi liars, P• 3 '1.3. F:rdelyi, Arthur, 1936: 11ath. Z. 40, 693-702. ("')) Joo exp(-~2 '· 2) lle (x + y) He (x + z) dx 111 n -oo = (2rr)y, m! zn-m L:-"(-yz) m::; n (31) J 77 y; (-l)n 77 (2n)! 2 (cos x)n l!e2 [u(l-secx) '] dx = 2 [I.e (u)] 0 n 2 n (n!) n 16.6. Laguerre polynomials See also confluent hypergeometric functions In this section m and n are non-negative integers. ( l) Joo {3-1 -x LG.f.. ) d 1(a-f3 + n + l) 1({3)_ Re f3 > 0 X e X X= 0 n n ! 1 (a-(3 + l) (2) J"" xae-" [La(x)]2 dx = l'(a+n+l) He a> 0 0 n n! 16.6 OHTHOGONAL POLYNOMIALS 293 Laguerre polynomials ( cont 'd) m, n = 0, 1, 2, ... {3) Joo xae-x La(x) La(x) dx = 0 0 m n m ,J n, Rea> -1 (4) fo";a+j3 e-x L ~(x) L~(x) dx = (-1)"'+" (a: m) c~:n) Re (a+~) > -1 (5) t a( ).B-a-1 a( r(a+ n + 1)r(~- a) j3 x 1 -x L xy) dx = L (y) 0 n r (~ + n + 1) n Re ~>Rea > -1 (o) _l' x11 e-x L a(Ax) L a(/lx) dx 0 n n See Buchholz, Herbert, 1953: Die konfluente hypergeometrische Funktion, Springer Verlag. Berlin, Gottingen, Heidelberg. Sec. 12. (7) f xa(1-x)f3 L a(xy) Lf3[(1- x) y] dx 0 • n (m + n)! r (a+ m + 1) r (~ + n + 1) L a+j3+1 ( ) = m ! n ! r (a + ~ + m + n + 2) m+n Y Rea> -1, Re ~ > -1 (8) loo X"-" exp [-Y:; (x-y)2] L ,.-n (x 2) dx oo n (2rr)x = --i"-"' He (iy) He (iy) n! n • (9) Joo exp (-Y:;x2) [L -)4' (Y:;x 2)]2 cos (xy) dx 0 n =(Y:;rr)X exp(-Y:;y2) [L:l4'(Y:;y2)F 294 INTEGRAL TRANSFORMS 16.6 Laguerre polynomials (cont'd)) m, n = 0, l, 2, ... (10) Joo 2 ~ 2 2 x exp(-~x )[L (~x )] sin(xy)dx 0 n =(~rr)~ y exp(-~y2)[L~(~y2)J2 (ll) Joo x exp(-~x2) L a(~x 2) L ~ -a(~x2) sin (xy) dx 0 n n = (~rr)l{ y exp(-~y2) L~(~y2) L~-a(~y2) (12) J"' exp(-~x2 ) La(~x2)L-~-a(~x2 ) cos(xy)dx 0 n n = (~~ rr)~ cxp (-~ y 2) L ~(~ y.2) L :a-~ (~y 2) (13) f 000 exp(-~x2)L n (~x2 ) He2n+t (~x) sin (xy) dx = (~ rr)~ exp(-~y2 ) L n (~y2 ) He 2n+l (~y) (14) Joo exp(-~x2 ) L (~x2 ) He 2n (~x) cos (xy) dx 0 n = (~rr)~ exp(-~y2 ) Ln(~y2 ) He2n(~y) (15) Joo 1 a1 a x p-e -x L (A x) • • • L n (A. x) dx o m1 1 mn n See Erd;lyi, Arthur, 1936: Math. Z. 40, 693-702. CHAPTER XVII GAMMA FUNCTION, INCOMPLETE GAMMA FUNCTIONS, AND RELATED FUNCTIONS For these functions see J-l.T .F. vol. I, Chapter I and vo I. II, Chapter IX. The expressions, given below, of incomplete gamma functions and related functionE' in terms of confluent hypergeometric functions will assist in the eva!uation of integrals involving these functions. For this reason, only a s•nall selection of integrals involving incomplete gamma fnnctions and their particular cases is given here. Error functions and Fresnel integrals Erf(x) = 11-x y(~, x2) = 211 -x x ,F; (l/2; 3/2; -x 2) 2 = 211-x xe-x ,F, (l; 3/2; x2) _ 2 -x -x -y, x 2 M · ( 2) -77 X e -~.~ X = 1-Erfc (x) -X -1 -x2 r. (1 .1 • -2) = 77 x e 2r0 , ;;z, -x 2 -x -x -y, x If ( 2) = TT X e -~, !~ X = 1-Erf(x) 295 296 INTEGRAL TRANSFORMS C (x) ± i S (x) = e ± !4 77 i Erf (e + ~ 77 i x ~) = 277-~ x~ 1F1 (1/2; 3/2; ± ix) Exponential integral and related functions -Ei(-x)=£1(x)=r(O, x) -1 -x F(l l -1) = x e 2 0 , ; -x -~ -~x W ( ) = x e -~.ox Ei(x) = ~[Ei(x + iO) + Ei(x-iO)] -1 x F (1 1 -1) =x e 20 , ;x Ci(x) ±i si(x)=- ci(x) ±i si(x) = Ei(± ix) =-r(O,::;: ix) -. -1 ± u F (1 1 -. -1) = + tX e 2 O , ; + tX Incomplete gamma: functions y(a, x) = a-1 xa 1F1 (a; a+l; -x) -1 ~cr-Y, -~,. M ( ) = a x e Y,a-Y,, Y,a X = r(a)-r(a, x) _ Y;a-Y, -Y,x W -x e Ua-U,Ua (x) = r(a)-y(a, x) (l) (2) (3) (4) (5) (6) GAMMA FUNCTION, INCOMPLETE GAMMA FUNCTIONS, AND RELATED FUNCTIONS 17 .1. The gamma function F" ['(a+ X) r ({3 -X) dx = 0 -oo Re (a+ {3) < 1 and either Im a< 0 < Im f3 or lm f3 < 0 < lm a J 00 ['(a + x) [' ({3 -x) dx = iTT 21 -a-,13 ['(a + {3) --oo He (a+ {3) < 1, lm a, lm f3 < 0 Joo !(a+ x) 1({3-x) dx =-iTT 21-a-,13 i(a + {3) ..,., Re (a+ {3) < 1, Im a, Imf3>0 foo ['(a+ x) lm a~ 0, Re (a-{3) < -1 dx = 0 1({3 + x) 00 [ dx 2a+,i3-z = Re (a+ {3) > 1 [' (a + x) I' ({3 -x) ['(a + f3 -1) J: l'(a + x) Re ({3-a)> 0 exp [(2rrn + TT-261) xi] dx = 0 [' ({3 + X) . -~ 1T < {) < ~ TT, n integer, (n + ~) lm (a) > 0 297 298 INTEGRAL TRANSFORMS 17 .l The gamma function (cont'd) (7) J''"1(a + x) 1(,8-x) exp[2(77n +e) xi] dx -oo = 2 71 i 1 (a+ ,8)(2 cos e)-a-,8 exp [(,8-a) 8i) x [ryn(/3) exr:-A2n77,Bi)-ryn(-a) exp(-2n77ai)) Re(a + ,8) < l, -~ 11 < e < ~ 11, n integer 7] n (() = 0 if (~-n)lm( >O 7] n (() = sgn (~ -n) if (~-n)lm (<0 (8) I00 1(a+x) 1(,8 + x) -oo exp [ ( 2 71 n + 71 -2 e) xi) dx (2 cos e),B-a-l = 277i sgn(n + ~) exp[-(277n+77-8)ai+e(,B-l)i) 1 (,8-a) Re(,B- a)> 0, -~2 11 < e < ~ 11, n integer, (n+ ~)lm a<O (9) foo sin(cx)dx Re(a + ,8) > l, -0 c > 71 ['(a+ x) ['(,8-x) -oo (10) [ sin (ex) dx [ 2 cos 0'2 c na+,B-z sin[~ c (,8-a)) 001(a+x)1(,8-x) = i(a+ ,8-l) fie (a+ ,8) > l, O<c<11 1~ sin(2n77x) dx ( ll) =0 sin (77x) 1(a + x) 1(,8-x) Re(a+ ,8) > l, n integer f: sin [(2n + l)77x) dx 2a+,B-z ( 12) = sin (77x) 1(a + x) 1(,8-x) 1(a+,8-l) Re(a + f3) > l, n integer 17 .l GAMMA AND RELATED FUNCTIONS 299 The gamma function (coot 'd) f'"" cos (ex) dx Re(a + {3) > 1, (13) -0 e > TT 00 r(a + x) ['({3-x) ( 14) [ cos (ex) dx [2cos( ~e)]a+B-z cos [~2 e ({3 -a)] = r(a+x)r({3-x) r(a+/3-1) Re(a + {3) > 1, O<e<rr (15) I~ P (x) e icx dx , P (x) polynomial r(a + x) r({3-x) See Ramanujan, Srinivasa, 1920: Quart.]. Math. 48, 294-310. (16) f~ <P(x) exp[(2rrn + e)xi] dx r (a + X) r ({3 -X) [2 cos(~e)]a+ ,B-z exp [~ e({3-a) i] f ¢ (t) exp ( 2 TT nti) dt = r(a+/3-1) 0 Re(a + {3) > 1, -TT < e < TT, n integer, ¢ (x + 1) = Cl>(x) (17) J~ ¢ (x) e ixc dx Cl> (x) periodic, period real ' ['(a + X) r ({3 -X) See Ramanujan, Srinivasa, 1920: Quart.]. Math. 48, 294-310. (18) foo r(y+x)r(8 +x) dx = 0 00 r(a+x)r({3+x) Re (a+ {3 -y -8) > 1, lm y, lm 8 > 0 300 INTEGRAL TRANSFORMS 17 .l The gamma function (cont'd) (19) l ""['(y + x) f'(o + x) dx f'(a+x)f'({:3+x) 00 ± 2 rr 2 i ['(a + {:3 -y -o -l) = sin [rr (y-o)] ['(a-y) ['(a-o) [' ({:3-y) [' ({:3 -o) Re (a+ {:3 -y -o) > l, lm y, lm o<O ±according as lm y ~ lm o (20) 1: f' (a-{:3 -y + x + l) dx f'(a + x) 1({:3-x) f'(y + x) rr exp [± ~ rr(o-y)i] = [' ({:3 + y-l) ['(a.:&) ['(Y -zs +1) Re({:3 + y) > l, o =a-{:3-y + l, Im of 0 ± according as lm o~O J: dx (21) f'(a + x) 1({:3-x) f'(y + x) f'(o-x) f'(a+ {:3+ y+ o-3) = f'(a + {:3-l) 1({:3 + y-l) f'(y + o-l) f'(o +a-l) Re (a + {:3 + y + o) > 3 (22) f~ sin (rrx) dx l (a + X) l ({:3 -X) l (y + X) l (o -X) sin[~ rr({:3 -a)] = 2f'(a.;ll) f'~ f'(a + o-l) a+o=f:3+y, Re(a + {:3 + y + o) > 2 7 .I GAMMA AND RELATED FUNCTIONS 301 The gamma function (cont'd) :23) J_~ cos (rrx) dx r (a+ x) r (/3 -x) r (y + x) r (8 -x) cos [X rr ({3 -a)] = 2r(a;B} r(Y;8) r(a + 8 -1) a+ 8 = f3 + y, Re(a + f3 + y + 8) > 2 (24) 1: <I> (x) dx r (a+ x) r (/3 -x) r (y + x) r (8-x) r (a+ f3 + y + 8-3) t <I> (t) dt ·= r (a+ {3-1) r ({3+ y-l) r(y+ 8-1) r(8+ a-1) 0 Re (a+ f3 + y + 8) > 3, <I> (x + 1) = <I> (x) (25) J: <I> (x) dx r<a + x) r ({3 -x) r (y + x) r (8 -x) J' 0 <l>(t) cos[Xrr(2t +a-{3)] dt = r~r~r(a+8-1) a+ 8 = f3 + y, Re (a + f3 + y + 8) > 2, <I> (x + 1) = -<I> (x) (26) J"" [r(a+x)r({3-x)r(y+kx)r(8-kx)]-1 exp(rrcxi) dx = 0 -oo Re (a+ f3 + y + 8) > 2, c, k real, Jcj > Jkj + 1 Fur further similar integrals see Ramanujan, Srinivasa, '1920: Quart. ]. Math. 48, 294-310. 302 INTEGRAL TRANSFORMS 17.1 The gamma function (cont'd) (27) ["' jr(a + ix) ['(b + ixW dx = ~rrX ['(a) ['(a+~) f'(b) f'(b + ~) 0 . . X l (a + b)/[' (a + b + ~) a> 0, b>O (28) ioo dx = I f'(a + ix) 12 ~rrX f'(a) f'(a+~) ['(b-a-~) f'(b + ix) f'(b) f'(b-~) ['(b-a) O<a<b-~ (29) (2 rri)-1 f i~ [' (s-K-A) ['(A+ J.l-S + ~) [' (,\-J.l-S + ~) z s ds -100 = 1(~-K-J.l) 1(~-K+J.l) zA.eXz WK (z) ,J.L Re (K + ,\) < 0, Re ,\ > IRe J.ll -~. largzl < 3rr/2 (30) l iioo f'(A+J.l-S+~)f'(A-J.l-S+~) A. -X -- z • ds = z e z W (z) 2rri f'(A-K-S +l) K,J.L -ioo Re ,\ > IRe J.ll -~. largzl<~rr lioo (31) _l_ f'(K-A+s)f'(A+J.l-S+~) (2rri) ioo f'(J.L-A+s+~) z • ds f'(K+J.l+~) z A. e-X • M (z) = r (2J.L+ l) K,J.L Re (K -,\) > O, Re(,\ + J.l) > -~. largzl < ~ 7T l 1:: f'(a+s) 1(f3+s) f'(y-s) ['(8-s) ds (32) - 2 rri f'(a+y) f'(a+O) f'(/3+y) 1(/3+ 8) f'(a+/3+y+8) Rea, Re /3, Re y,_ Re 8 > 0 17.1 GAMMA AND RELATED FUNCTIONS 303 The gamma function (cont'd) 1 1 ioo[f'(~-s)]2 (33)-2 TT i _ ioo f' (s) z>O n II f' (1-a . + s) } . IT r (b.-s) 1 1'00 j=l } j=l (34) 2rri -ioo __ q ________ P _____ _ II f'(1-b+s) II f'(a-s) z • ds (35) j= a+l 1 j=n+l 1 ( la1, ••• ,a) = G an Z p pq b,, ••• ,bq p + q < 2 (m + n ), I arg z I < (m + n -~ p -~ q) TT Rea.< 1 } j = 1, •.. ,n, Reb.> 0 } IT r <b . -s) ft r n -a . + s) 1 fioo __ i_=_l __ J ___ J_·=_l ____ J __ _ 2 TT i _ ioo q p n r (l -b . + s) II r (a . -s) j=,.+l 1 j=n+l 1 z • ds = c•n pq (z I a 1 , ••• , a P) \.' b, ••• ,bq j=1, ••• ,m p + q::; 2(m + n), j = l, ... , n, largzl ~ (m + n-~p-~q)rr Rea.< 1 } Reb > 0 j = l, ... , m } p q Re( I. a.-I. b.)>~p-~q+l j=l } j=l } For further integrals of this type see sec. 7.3. An empty product is interpreted as l. 304 INTEGRAL 1RANSFORMS 17.1 Tbe gamma function (cont'd) (36) J.' sin (2rrnx) log rr (a+ x)] dx 0 =-(2n rr) -I [log a+ cos (2nrra) ci(2nrra)- sin(2nrra) si(2nrra )] a> 0, n=1,2,3, .•• (37) fo' cos(2rrnx) log[i(a+x)]dx \= -(2nrr)-1 [sin (2nrra) ci (2nrra) +cos (2n rra) si (2nrra)] a> 0, n = 1, 2, 3, ••• (38) J' exp(2rrnxi) log [1 (a + x )] dx 0 = (2nrri)-1 Dog a-exp(-2rrnai) Ei(2nrria)]- a> 0, n = ±1, ±2, .•. (39) f' log[I'(x)] dx = ~ log(2rr) 0 (40) J' log[f'(a + x)] =a log a-a+~ log(2rr) a>O 0 (41) n n-1 J log[i(a+x)]dx=!. (a+k)log(a+k)-na 0 k=O + ~n log(2rr)- ~n(n-1) a;:: 0, n = 1, 2, 3, ..• (42) J.' sin(2rrnx) log[i(x)] dx = (2rrn)-1 log(2rryn) 0 n = 1, 2, ••• 17.2 GAMMA AND RELATED FUNCTIONS 30!:> The gamma function (cont'd) (43) f sin[(2n + 1)rrx] log[r(x)]dx 0 1 [log~17J+2 (1+2_+···+-1 ) 1 J = +--(2n + 1)rr 2 3 2n-1 2n + 1 n = 0, 1, 2, ••• { 1 n = 1, 2, 3, ••• (44) cos(2rrnx) log[r(x)] dx =- 0 4n 17 .2. The t/J function ( l) / 0 1/J(a+x)dx=loga a>O (2) F e 2"71zi t/J (a+ x) dx = e -2nrrai Ei (2n rrai) 0 a> 0, n = ±1, ±2, ••• (3) p 0 sin ( 2 n TT x) 1/J (x) dx = -~ rr n = 1, 2, ••• (4) I J sin(2nnx) 1/J(a +x)dx = sin(2nrra) ci(2nrra) 0 + cos (2nrra) si (2nrra) a~ 0, n=1,2, ••• (5) / 0 cos(2nrrx) 1/J(a + x) dx = sin(2nrra) si(2nrra) -cos(2nrra) ci(2nrra) a> 0, n = 1, .2, ••• (6) J"" x -a [C + 1/J (1 + x)] =-TT csdrra) ((a) • 1 <Rea< 2 0 306 INTEGRAL TRANSFORMS 17.2 Tbe ..p function (cont'd) (7) Joo x -a [log x -..p (l + x )] dx = TT esc (rra) ((a) 0 <Rea< l 0 (8) Joo x-a[log(l + x)-1/J(l + x)]dx = TT csc(rra) [((a) -{a -1)-'] 0 O<Rea<l (9) Joo x-a[(l + x)-1-1/J'(l + x)] dx = -rra csc(rra) [((l +a)-a-1] 0 -1 <Rea< l (10) Joo x-a[x-1-1/J'(l + x)] dx = -rra csc(arr) ((l +a) 0 -2<Rea<0 (ll) () rri(a+n) Joo x-a..p n (l + x) dx = (-l)"-1 ((a+ n) o ['(a) sin rra d"t/1 n = l, 2, ... , ..p(n)(z) =- 0 <Rea< l dz n' (12) Joo [1/J (x + l)-log x] cos (2rrxy) dx = ~ [..p (y + l)-logy] 0 17.3. Incomplete gamma functions and related functions ( l) Joo p-'Ef < )d _r(~p+~n x r c ax x- ~ 0 TT paP Rea> 0, Rep> 0 ~~ (v v+l v ·~ 2 (2) Joo xv-t exp({32 x2) Erfc(ax) dx = ~ 2F, 2'-2-;Z+l; a2 o TT ~v Rev> 0, 0, Re (3 2 < Re a2 17.3 GAMMA AND RELATED FUNCTIONS 307 Incomplete gamma functions etc. (cont'd) (3) f" 1 xv-sin({3x) Erfc(ax) dx 0 I r(l+~v)b = rr~ (v + 1) av+1 (v+l v 3 v+3 ~'j F. ---+1·---·---2 2 2 ' 2 '2' 2 ' 4a2 Rea> 0, Rev> -1 (4) Joo v-1 ( ) x cos (3 x Erfc (ax) dx 0 r (~ + ~ v) = rrX vav (VV+llv ~'j 2F2 2' -2-; 2' 2 + 1; -~ Rev> 0, Rea> 0 (5) fooe,s%Erfc(axxx)dx=_:_ [ ax x-1] o f3 (a-{3) 0, Re (3 <Rea (6) 00 1 1 1 ~~a )X 1 1 J sin({3x)Erfc(a xxx)dx=--2 2 [(a2+f32)x-arx o (3 a +13 Rea> IIm/31 ~ ~ )X (7) oo 1 1 2a · 2 2 1 1 fo cos({3x) Erfc(ax xx)dx = 2 2 [(a + (3 )X+ arx a + (3 Rea> IIm/31 (8) _(' sin (bx) Erfc (ax x -X) dx = b -1 exp [-(2ab )X] cos [(2 ab )X] 0 Rea> 0, b > 0 (9) Joo cos(bx) Erfc(aX x-X)dx =-b-1 exp[-(2ab) X] sin[(2ab) X] 0 Rea> 0, b>O 308 INTEGRAL TRANSFORMS 17.3 Incomplete gamma functions etc. (cont'd) (lO) Joo 2 cosh (2 v t) exp [(a cosh t) ] Erfc (a cosh t) dt 0 =~sec (v77) exp (~a2 ) K)a2) Re a> 0, -~<Rev <~ For integrals involving products of error functions and other con- fluent hypergeometric functions see Bock, Philipp, 1939: Compositio Math. 7, 123-134. Note that the function denoted by Erfc by Bock is ~77~ Erfc in the notation used in the present work. (ll) fa ex Ei(-x) dx = -log(ay) + ea Ei(-a) 0 (12) r e -f3x Ei (-ax) dx = -{3-1le -f3c Ei(-ac) + log(l + {3/a) 0 -Ei [-(a+ {3) c ]! (13) Joo x11 ex Ei (-x) dx = 77 esc (77v) r(v+ 1) -1 <Rev< 0 0 (14) Joo v-1 -j3x '( r(v) x e Et-ax)dx=- o v(a+f3)v 2F', (1, v; v+1; ~) a+ f3 larg al < 77, Re(a + {3) > 0, Rev> 0 For further integrals involving- Ei(-x) = E 1 (x) see LeCaine, j., 1948: National Research Council of Canada, Division of Atomic Energy, Document No. MT -131(NRC 1553), 45 PP•· and Busbridge, I.W ., 1950: Quart. J. Math. Oxford Ser. (2) 1, 176-184. ( 15) Joo xil--1 e -j3x y(v, ax) dx = a11r(,.av) 2F1 (1,J.L+v ; v+1; _a_) 0 v(a + (3)11-+v a+ {3 Re (a + (3) > 0, Re (3 > 0, Re(J.L+v) >O 17.3 GAMMA AND RELATED FUNCTIONS 309 Incomplete gamma functions etc. (cont'd) oo 1 f3 av['(f1+v) ( {3 ~ (16) J xiL-e-xf'(v, ax)dx= 2F, l, f1+v; f1+l; -- o f1 (a + (3)11-+v a + {3 Re(a+/3) >0, Re11>0, Re(f1+v) >O (17) Joo e-f3x y(v, ax2)dx = 21-v {3-1 f'(2v) exp( {32 )D 2 [ {3 ~ J 0 \a a -v (2a) Re a> 0, Re {3 > 0, v f. 0, Re v >-~1 (18) Joo x1-2v exp(ax2) sin(bx) f'(v, ax2) dx 0 = rr~ 2-v av-1 r(~- v) exp(!:_)D2 _2 [~] 2 \sa v (2a) ' largal < 3rr/2, 0 <Rev< l For other integrals involving E (x) = x"-1 [' (1-n, x) n see LeCaine, J,, 1948: National Research Council of Canada, Division of Atomic Energy, Document No. l\H -131 (NRC 1553), 45 pp., and 13usbridge, I.W., 1950: Quart.]. Math. Oxford Ser. (2) 1, 176-184. (19) ~00 e-f3x y(v, axy,) dx = 2-Y,v av {3-Y,v-1 f'(v) exp(:~) x D -v [ (2a{3)'/. J Re{3>0, Rev>O CHAPTER XVIll LEGENDRE FUNCTIONS For the theory of these functions see ll.T .F. vol. I, Chapter III and the literature quoted there, especially the hooks by Hobson, ~1acRohert, Whittaker and Wntson. Numerous expansions of Legendre functions in hypergeometric series are listed in H.T.F. vol. J, P• 124-139, and these may he used to reduce integrals involving Legendre functions to integrals involving hypergeometric series. 311 LEGENDRE FUNCTIONS 18.1. Legendre functions of variable ax+ {3: finite intervals ReA.> 0 \. 1 7Tx 2-\.1(>..) (l) J~ x -P (x) dx =------------ v 1 <~ + ~ >.. -~ v) 1 n + ~ >.. + ~ v) (2) J1 \.-1 (-l)" 7TX 2-z.. -1 1(~A.) 1(1 + m + v) x P" (x) dx = o v 10~+~m)['(1+~A+m)1(1-m+v) (m+v+1 m-v m A.+m ) x 3F2 , --, -+ 1; m + 1, --+ 1 ; 1 2 2 2 2 Re A. > 0, m = 0, 1, 2, ... (3) X F ---1--·1-11 --+1·1 (v -11 + 1 11 + v 11 A. -11 ) 3 2 2 ' 2 ' 2' ,...., 2 , Re A.> 0, Re 11 < 2 (4) ]1 x\.-1 (l-x2)X" P"(x)dx 0 v (-1)" 7TX 2-\.-,. ['(A.) 1(1 + m + v) =----------------------------------------- 1(~ +~A+ ~m-~ v) 1(1 +~A+ ~m + ~v) 1(1-m + v) R e A. > 0, m = 0, 1, 2, .•. 313 314 INTEGRAL TRANSFORMS 18.1 Variable ax+ f3: finite intervals (cont'd) (5) t xA.-1 (1-x2)-~11-p11-(x) dx 0 v rr ~ 211--A. 1 (,\) = [' (~ + ~ ,\-~ 11-~ v) [' (1 + ~ ,\-~ 11 + ~ v) Re ,\ > 0, Re 11 < 1 (6) r A-1 2 K 2J.L-1['(1+K-~11)['(~,\) x (1-x) p11-(x)dx= o v ['(1-11)1(1+K+ ~A-~11) (v-11+ 1 11+ v 11 ,\-11 ) X 3F2 , ---, 1+ K--; 1-11• 1 + --+ K; 1 2 2 2 2 Re (K-~ 11) > -1, Re ,\ > 0 (7) J1 xA.-1 (1-x2)-~11-sin(ax)P11-(x)dx 0 v rry, 211--A.-1 [' (,\ + 1) a = [' [1 + ~ (,\-11 -v )] [' [~ (3 + ,\-11 + v )] (1+,\ ~-~ ,\-11-v 3+,\-11+v. a2 ) X 2F3 ' 1 + ' , 1 + ' ., - -2 2 2 2 2 4 Re ,\ > -1, Re 11 < 1 (8) r XA.-1 (1-x2)-~11-cos(ax) p11-(x) dx 0 v rr~ 211--A. 1 (,\) = 1[1 + Yz(,\-11 + v)] 1[Y2(l + ,\-11- v)] (,\ A+1 1 X 2F3 2' -2-; 2' 1+A-11-I-' ,\-11+v a2 ) 1+ ·--2 ' 2 ' 1 Re ,\ > 0, He 11 < l 18.1 LEGENDnE FUNCTIONS 315 Variable ax + {3: finite intervals (coot' d) 1 n I (9) J (1-x2)-1 [pn-v(x)JZ dx =- · o v 2 (n -v) [' ( l -n + 2 v) n = 0, l, 2, ••• , Rev> n (10) { 1 Pr.._(x) P (x) dx = 2 [rr(A.-v) (,\ + v + l)r 0 v x[A sin(XM-)cos(XV7T)-A-1 cos(XA.rr)sin(Xvrr)] A= f'(X+ Xv)f'(l+ X>..) f'(X +X.\) f'(l + Xv) (ll) r p )x) Qr.._(x) dx = [(,\-v) (,\ + v + l)]-11 A -1 cos [X ( v-A.)rr ]-l I 0 A= f'(l +X>..) f'(X + Xv) f'(X +X>..) f'(l + Xv) (12) ]01 Qr.._(x) Qv(x) dx = [(,\-v)(,\ + v + l)]-11 tf;(v + l)-tf;(,\ + l) -X rr (A -A -1) sin [X(>.. + v) rr] + X rr(A + A -1) sin [X(>..-v )rr]l A f' (l + X>..) f' (X + X v) f'(X +X>..) f'(l + Xv) (13) J 1 (PIL(x)f dx, 0 v 1 ]0 P~(x) Q~(x) dx. See Barnes, E.W., 1908: Quart. f. Math. 39, 97-204. Note that Barnes' definition of Legendre functions of the second kind differs from the one used in this book. (14) r p .. (x) p \(x) dx 0 v See Shabde, N.G., 1937: Bull. Calcutta Math. Soc. 29, 33-40. 316 INTEGRAL TRANSFORMS 18.1 Variable ax+ (3: finite intervals (cont'd) (15) 1 A. 1 0-[r (.X.) F Re .X.> 0 J ( 1 + x) -P (x) dx = -1 v ['(.X.+v+1)1(.X.-v) (16) { (1-x2)A_-1 PJ.L(x)dx -1 v rr 2J.L1(.X. +X 11) !(.X.-X 11) = r (.X.+ X v + 1)1 (.X.-X v) 1(-X 11 +X v + 1) r (-X 11-X v + X) 2 Re (.X.)> \Re 11\ (17) f (1-x2)-XJ.L(z -x)-1 pJ.L (x)dx = 2e-¥L17(z2-1)-Y,J.LQJ.L (z) -1 J.L+n J.L+n n = 0, 1, 2, •.• , Re 11 + n > -1, z in the cut complex plane (18) J1 {l-x)-XJ.L(1+x)XJ.L- Y, (z +x)J.L-Y, pJ.L(x)dx -1 v 2e -zJ.L7Tir(X + Jl) = , (z-1)J.L TTX [' (Jl -v) [' (Jl + v + 1) xQ~[c;z)y,] Q~v-1 [C;z) x] -~~ < R e 11 < 1, z in the cut complex plane ( 19) J.1 {l-x)-YJ.L-1 (l + x)XJ.L-X (z + x)J.L-X [(v-Jl)P!1x)-(v +Jl)pu 1(x)]dx -1 v v- e-J.L7111 (Jl+ X)(z -1)J.L 1+ z 1+ z ' . t ~ "] ~ "] =(X rr)y, l(Jl+v) l(Jl-v)(z+ l)y, Q~ ( 2) Q'::_v ( 2) [('.,)"] [C'.,)"]} + Q~-1 -2-Q':::_v-1 -2- -X< Re 11 < o, z in the cut complex plane The complex z-plane is cut along the real axis from -1 to 1. 18.1 LF:G~:!'\Dllt: f'liNCTIONS 317 Variable ax+ {3: finite intervals (cont'd) 1 (20) J (1 -x )-!1 J.L (l + x )y, J.L-!1 (z + x)J.L-312 p1-L(x) dx -1 v • _ f'(p-H' -ll"-' (nl)- Y. { "[ C+<)'] "-' [ CH )'] 77 Y, e 2 J.L77' [' (Jl + v) }' (Jl-v _1) Q v 2 Q -v-1 2 + Q~-1 [ (1;z)y,] Q~v-t ~ ')']} -~2 < He 11 < 1, z in the cut complex plane (': l) r (1-X)-/ J.L (l + x)!1J.L+v-1 exp(-1-X r) -1 1+x P~ (x) dx -2v Y,Jt+v-Y, eY,y W ( ) -Y Y,J.L-v-!1, Y,J.L Y Re y > 0 (2:~) J_1 1 P )x) P,\(x) dx 4 sin ( vrr) sin (..\77) [If! ( v + l) -If!(..\ + l )] + 2 77 sin [ (..\ -v) 77] = 772 (..\-v) (..\ + v + 1) 2 -2[sin(vrr)J2 w'(v+ 1) .( 2 77 (23) _ 1 [P )x)] dx = 772(v+~2) (24) {1 P )x) P,\(xXl + x)A.+v dx = 2A.+v+1 [f'(..\+v+1)]4 [I~(..\+ 1)['(v+ 1)]2 f'(2..\+2v+2) He(..\+v) >-1 (25) J' P (x) Q (x) dx = -sin (277v) lfl'(v + 1) -1 1.1 v 77 (2 1/ + l) The complex z-plane is cut along the real axis from -1 to l. 318 INTEGRAL TRANSFORMS 18.1 Variable ax+ {3: finite intervals (cont'd) (26) / 1 _ 1 P )x) QA.(x) dx = [(v-A)(v+A+ l)r 11-cos [(A-v)rr] -2rr -1 sin(vrr) cos(Arr) [1/J(v+1)-1/J(A+ 1)]! (27) { 2 X rr2 -11-[cos(VTTW! 1/J '(v + 1) _ 1 [Qv(x)] dx = 2v + 1 (28) t Q (x)QA.(x)dx=[(A-v)(A+v+1)r11Xrrsin[(A-v)rr] -1 v + [1/J (v + 1)-1/J (A+ 1)] [1 + cos (.\rr) cos (vrr)]! (29) r(l-x2)Y.a-M-X --- pM(x)PM(x)dx (J x)Xv -1 1+x J.L A. See Shabde, N.G., 1940: Bull. Calcutta Math. Soc. 32, 121-128. (30) r P_\(x) pa-(x) P:(x) dx -1 J.L See Gaunt, ].A., 1929: Philos. Trans. Royal Soc. 228, 151-196. (31) f (1-x2/'--1 (1-a2x2)XJ.L P (ax) dx -1 v 1T 2J.L r(A) = 1(Y2+A)1(X-Xt-L-Xv)1(1- Xf.L+ Xv) ( f.L+V 1-f.L+V 1 2) x 2 F, --2-, 2 ; 2 + A; a fie A> 0, -1<a<1 18.1 LEGENDRE FUNCTIONS 319 Variable ax+ {3: finite intervals (cont'd) (32) Re J1 < ~. iargal < 77 Re J1 <-~. larg ai < TT = ;:y, r(Yz+Jl) a-XI-L ?~[(1 + a)y,] P:~-'[(1 + a)y,] Re J1 > -~, iargal < 77 (35) J 1 x~-'12-112 (1- x)~-'-312 (1 + ax)- ~-'12 P~(l + 2 ax) dx 0 = Yz77y, r(u- Yz)(1 + a)-y, ay,-y,~-'!?1-~-'[(l + a)X]p~-'[(1+a)X] r v v + (Ji.+ v)(1-Jl+ v) P -1-L((l+ a)y,] P~-'[( 1+ a)y, ]! v v R e J1 > ~. larg al < 77 + (l+V-Jl)p~-l (l+ 2ax)]dx= 277!1, r(Y:-(l)(1+a)- Y, aXJ.L+Y, x P~[(1+a) X] P~_1 [(1 + a)X] He J1 < ~. iargal < TT (37) J1 x-Xi-L-1 (1-x)-J.L- Y, (1 + ax)X!L-qp1-~-'(l + 2ax) 0 v-1 -P1-~-'(1+2ax)]dx= rrX r(Yz-(l)(1+a)- X aXJ.L+X v x1l (fl-v)P ~[(1+a)X] ?:~1 [(l+a)X] -(fl+v)P~_ 1 [(1+a)X] P:~-'[(l+a) X]! Re fl < ~2, larg al < TT 320 INTEGRAL TRANSFORMS 18.1 Variable ax+ {3: finite intervals (cont'd) 1 (38) J x-XJ.L-X (1-x)-J.L-Y, (1+ ax)Y,J.LQJ.L(1+ 2ax)dx 0 v = rry, I'(Yz-/l) ay,J.L PJ.L[(1 + a)X] QJ.L[(1 + a)X] v v Re !l < Yz, largal < 7T (39) J 01 xJ.L/2-112(1-x)-J.L-312 (1 + ax)J.L12 Q~(l + 2ax) dx = Yz rrx 1(-,.,.-Yz) (1 + a)-x aY,J.L+Y, x IPJ.L+I [(l + a)y,] QJ.L[(1 + a)y,] v v + PJ.L[(1 + a)y,] QJ.L+I [(1 + a)y,]! v v Re !l <-~~. iarg ai < rr 18.2. Legendre functions of variable ax + {3: infinite intervals (l) r>O 2 y, (x -1) •J.L sin (ax) PJ.L(x) dx v 0 2J.L rry, a -J.L-X = [' (Yz -Yz !l -Yz v) [' ( 1 -Yz !l + Yz v) S J.L+Y,, v +X (a) a> 0, Re !l < 3/2, Re (!l + v) < 1 (2) Joo 2 )A_-! 2J.L-I ['(.\- Yz,.,.)['(1-.\ +Yzv)['(Yz-.\- Yzv) (x -1 P J.L (x) dx = 1 v ['(1-Yz,.,.+Yzv)['(Yz-Yz,.,.-Yzv)['(1-.\- Yz,.,.) Re .\ > Re ,.,., Re(1-2.\-v) >O, Re(2-2.\+v) >O 2p+J.L-2 [' (~) [' ( p+g-v-1 ) (3). Joo x-P(x2-1)-Y,J.L PJ.L(x) dx = 2 2 I V 7Ty, [' (p) Re !l < 1, Re (p + !l + v) > 0, Re (p + !l-v) > 1 18.2 LEGENDRE FUNCTIONS 321 Variable ax+ {3: infinite intervals (cont'd) (4) Joo f-1 2 ~ 1 (x-1 -(x -1) 1-' P~(x) dx 2"-+~-'r(..\) r(-,.\-fl-v) r(l-,.\-f1 + v) = r (1-f1 + v) r (-fl-v) r (1-,.\-f1) Re ,.\ > 0, Re(..\+f1+v)<O, Re(..\+f1-v)<1 (5) Joo (x -l)A -1 (x 2 -l)-~ 1-' P 1-' (x) dx 1 v 2,\-1-' sin(vrr) r(..\-fl) r(-A+f1-v) r(l-..\+fl+V) =-rrr(l-,.\) ite(..\-f1)>0, Re(fl-,.\-v) > 0, Re (f1-,.\ + v) > -1 (6) Joo (x -l}-~1-' (x + 1)~1-'-~ (z + x)~-'-~ P~-'(x) dx 1 v = 77~ r<-11-v)r(1- 11+v) { [(1+z)~]}2 (z- 1)~-' pi-' -- r <~-11) v 2 Re(f1 + v) < 0, Re(f1-v) < 1, \arg(z + 1)\ < rr (7) Joo (x -l)-~1-' (x + 1)~1-'-~ (z + x)~-'-312 P~-'(x) dx 1 v ~C(1- 11-v)r(2-11+v) -~ -~ = rr (z- 1)~-' (z + 1) r (3/2-11) x p~ [ (1; z)X] p~-{ ( 1; z) X] Re f1 < 1, Re(fl + v) < 1, Re (f1-v) < 2, \arg(1 + z)\ < rr 322 INTEGRAL TRANSFOHI\IS 18.2 Variable ax+ {3: infinite intervals (cont'd) (8) J"" (x -1)-Y,JJ.-1 (x + l)Y,JJ.-Y, (z + x)JJ.-Y, [(v-fl) PfJ.(x) 1 v 1 rn-fl-V)r(l-fl+V) I -(v+fl)PfJ. 1(x)]dx=(2rr)y, (z-1)1-' (z+l)-y, v- r (7'2 -fl) Re fl < 0, Re fl <l-IRe vi, larg(z + l)l < 7T (9) J"" (x -l ),\ -1 (x 2 -l) y, fJ. (x + z)-P P JJ.(x) dx 1 v 2,\+JJ.-p r (,.\-p) r (p-,.\-fl-v) r (p-,.\-fl + v + l) =------~--------------------------------r (l -fl + v) r <-11 -v) 1~ n + P-A.-fl) HeA.>O Re(p-,.\-fl-v) > 0, He(p-,.\-fl+ v + 1) > 0, larg(z+l)l <rr sin (vrr)r (,.\-fl-p)r(p-A+ fl-V) r(p-A+ fl+v+l) =- 2p-M~-'rr ro + r-A.) X 3F2(p,p-A+fl-V,p-A+fl+V+l; l+p-A., l+p-A+fl; 7'2+~2Z ) x 3F2 (,.\-fl•-v, v+ l; l + ,.\...:. fl-p, l-fl; 7'2 + ~2 z) He(,.\-fl)> 0 He(p-A+fl-v) >O, Re(p-A.+fl+v+l) >O, larg(z+l)l <rr 18.2 LEGENDRE FUNCTIONS 323 Variable ax + {3: infinite intervals (cont 'd) Re a> 0, Re /1-< l (12) f 00 (X+ l) Y,JL __ e-ax PfL(x) dx = a-1 W +v(2a) X_ ~ V j..L, II /2 1 Re a> 0, Re /1-< l -A_-JL -a ( I ) a e 31 l + /1-• l = G 2a r(l-JL+v)r(-JL-V) 23 A+J.L,-v,l+v Re a> 0, Re A> 0 (14) J"" (x-l)A_-t (x 2-n-Y,JL e -ax Q~(x) dx , -v J,Lrri JL-A. -ac22 (2 I l-J.L, l ) -12 e a e 23 a A-J.L, v + l, -v Rea>O, ReA>O, Re(A-Ji-) >0 (15) J"" (x-l)A.-t (x2-1)-Y,JL e -ax PfL(x) dx 1 II -1 . ( ) JL-A_ -a G 31 (2 I l, l-/1-) = -rr Sin VTT a e 23 a A-Jl-, l + v, -v Re a> 0, Re A> Re /1- ( 16) J"" (a 2 + f3 2 + 2 af3 x) -Y, exp [-(a 2 + f3 2 + 2 af3 x )y,] P (x) dx 1 II = 2rr _, (af3)-y, K11+y, (a) Kv+Y, ({3) ·Re a> 0, Re f3 > 0 324 INTEGRAL TRANSFORMS 18.2 Variable ax+ {3: infinite intervals (cont'd) (17) Joo (x2-1)-~.U exp(a2 x2) Erfc(ax) p.U(x) dx 1 v = 17-1 2,u-1 r(1+~+v) rt;v) a,u-3/2 exp~a:) X W~-~,u.~+~v(a2 ) Rea> 0, Re J1. < 1, Re (f.£+ v) > -1, Re(Jl.-v) > 0 (18) J<>O 1 Qv(x) dx = [v(v + l)r 1 Rev> 0 (19) J<>O 1 P v(x) QA.(x) dx =[(A-v) (A+ v + l)r 1 Re(A-v) >O, Re(A+v) >-1 (20) Joo [Q)x)f dx = (2v + 1)-1 0 '(v + 1) Rev>-Yz 1 (21) J oo 0 (A+ 1)-0 (v + 1) Q)x) QA_(x) dx = 1 (A-v) (A+ v + 1) Re(A+v) >-1 (22) j1oo [Q~(x)f dx See Barnes, E.W., 1908: Quart. ]. Math. 39, 97-204. Note that Barnes' definition of Legendre functions of the second kind differs from the one used in this book. (23) J~ (x2-1)A.-1 Q~(x) dx . r(~+ ~ v+ ~ f.l)r(l-A+ ~ v)r (A+ ~f.£) r(A-~ Jl.) -elL'"' - 22/\ .ur(1+~v-~J1.)r(~+A+Yzv) IRe f-£1 < 2 Re A< Re v + 2 18.2 LEGENDRE FUNCTIONS 325 Variable ax + {3: infinite intervals (cont'd) -f'(v+p+1) J.mi -1\.-J-L -aC22(211+p,1 ) _ e a e 23 a 2 r (v -/l + 1) ,\ + p, v + 1, -v Re a> 0, Re ,\ > 0, Re (,\ + p) > 0 r (,\) r (1-.\-~ /l + ~ v) r (~-,\-~ /l-~ v) = ['( 1 -~ /l + ~ v) [' (~ -~ v -~ /l) [' (l -,\ -p) Rea>O, Re,\>0, Re(v-p-2.\)>-2, Re(2.\+p+v)<1 (27) J 100 (x2-l)A.-1 (a2 x2 -1)-Y,J-L Q~(ax) dx f'(J-L+V+1) ['(,\)['(1-A+~ 2 2 = ---------- 2J-L-2 e J-L7Ti a -J-L-v-1 f'(v + -}) (p.+v+ 1 p+v 3 _2) x 2F, , 1-,\ +~; v +-;a 2 2 2 larg(a-1)l<rr, Re.\>0, Re(2,\-p-v)<2 (28) Joo x -Y,J-L-Y, (x-1)-J-L-Y, (1 + ax)Y,J-L Q1-L(1 + 2ax) dx 1 v = 77-Y, e-J-L7Ti['(~ -p) ay,1-LIQ~[(1 + a)Yz]l2 largal <rr, Rep<~. Re(p+v)>-1 326 INTEGRAL TRANSFORMS 18.2 Variable ax+ {3: infinite intervals (cont'd) (29) J 00 x -J.L/2-112 (x -l)-J..<-312 (l + ax )1LI2 QJ.L(l + 2 ax) dx I V -~ -J.L7Tir< ~) ~J.L+~ (l 2)-~ =-rr e -!L-2a +a x QJ.Ltl [(l +a)~] QJ.L[(l +a)~] v v larg al < rr, Re !L <-~~. He (IL + v + 2) > 0 (30) J 100 (x-l)J.L-I P )ax) Q>c(ax) dx, J,oo (x-l)J.L-I Q)ax)Q~ax)dx See Shahde, N.G., 1937: Bull. Calcutta Math. Soc. 29,33-40. 18.3. Legendre functions of other variables (l) fa"" P v(2x2 a-2-l) sin (bx) dx rra { [Jv•<(~b)l- [J-v-< ~:)]'} =-4 cos (vrr) a, b > 0, -l <Rev< 0 (2) J 00 P (2x2 a -2-l) cos (bx) dx a V = -~ rra [Jv+~(~ab) J-v-~(~ ab)- Yv+~(~ ab) y_v-~(~ab )] a, b >.0, -l <Rev< 0 100 (a-x) ca-X) (3) (a+ x)-J.L-v-2 p --Pv -- dx J.L a+x a+x a-J.L-v-1 [r(IL + v + 1)]4 = [r(!L+ l)r(v+ l)JZr(2!L+ 2v+ 2) largal < rr, Re (IL + v) > -l 18.3 (4) (5) (6) (7) (8) (9) LEGENDRE FUNCTIONS 327 Other variables (cont'd) =-~ 77 esc (v77) 1F; (v + 1; 1; ai) 1F1 (v + 1; 1; -ai) a > 0, -1 < Re v < 0 2 00 ~ 77 f 0 x-1e-xQ_y,(1+2x-2)dx=BI[J0(~f3W+[Y0(~{3)fl Re f3 > 0 R e a> 0, Rev> -1 =----- 11 [' (-11-v) 1~~ 1-.\ ) 2' 2 0 ~-A+!l+V -A-/l+V ' 2' 2 ' 2 a > 0, Re f3 > 0, Re A> 0 Re a> 0, b > 0, -5/4 <Rev< l/4 f"' xy, sin (bx) p-'4 (y) Q-'4 (y) dx 0 v v (~77) X e-'.47Tif'(v + 5/4) aby,['(v+3/4) 328 IN1EGRAL TRANSFORMS 18.3 Other variables (cont'd) ( 10) Joo x~ y-1 sin (bx) ?:14 (y) P11-=_~ (y) dx 0 (2rr)-~ a-2 b~ (b) ( b J = r(5/4 + v) r(5/4-v) K11-~ 2a K11+'h ~ Rea> 0, b > 0, -5/4 < Re v < 5/4 (l l) j x~ y-1 sin(bx)P~(y)P:l((y)dx 0 [K"'' (:a)'] (2rr)-'h a-2 by, = r(7/4 + v) r(3/4-v) Rea> 0, b > 0, -7/4<Rev<3/4 (12) J"" x~ cos (bx) [P~(y)F dx a -1 (Yz rrb )-'/, r"'"c2~)J = 0 ro~ + v) r(-~- v) Rea> 0, b > 0, -%<Rev <-~ (13) J"" xy, cos (bx) ?14 (y) Q'A (y) dx = (Yz 1T) y, e 'A 7T i r (v + %) 0 11 11 a by, r(v + 5/4) x I11+x( ~ )K 11+y, ( ~ 2a 2 2a Re a> 0, b > 0, Rev>-% ( ) -Yo -2 X (14) J"" y, 1 14 l( 2rr a b · x 2 y-cos (bx) P-' (y) P ' (y) dx = 0 11 11 r(5/4+v)r(l/4-v) x [K 11+~i(:a) J He a> 0, b > 0, -5/4<Rev<l/4 18.3 (lS) (16) (17) ( 18) (19) (20) LEGENDRE FUNCTIONS 329 Other variables (cont'd) ~00 x~ y-1 cos (bx) P~ (y) P~_1 (y) dx (2rr)- ~a-2b~ b b = 1(% + v) 1(%-v) Kv-Y, (2";;) Kv+Y, (2a) Re a> 0, b > 0, -%<Rev<% 1 a [ si_n (n -x) J K _ _ dx P J..l.(cos x) P K [cos (a-x)] -v v . Sln X Sin X 0 2K 1 (Jl -K) 1 (K + ~) (sin a}K p-J..L (cos a} = 77'/, 1 (K + J1 + l) v Re J1 > Re K >-~ I a (sin x)P r sin (a -x)](7 pJ..L(cos x} p: [cos (a -x}] dx 0 v For this integral and several particular cases see Bailey, W.N., 1931: Proc. Cambridge Philos. Soc. 27, 184-189 and 381-386. Joo cos(ax} P (cosh x} dx 0 v sin(vrr) ,e+v+iaJ Q+v-iaJ ( v+ia) ·( v-iaJ =- ::-z I 1 1--- I·---4rr 2 2 2 2 fooo p -..,-y, (cos e) dx =~esc (~2e) foo p (cos e) dx = esc(% e) -oo .x , 2 2 '/, y=Cl+a x) a> 0, -l <Rev< 0 O<e<rr O<e<rr 330 INTEGRAL TRANSFORMS 18.3 Other variables (cont'd) (21) f000 cos (bx) P!:Y,+iz (cosh a) dx =0 0 <a< b (}27T)y, (sinh a)I-L O<b<a = r (~ -p.) (cosh a -cosh b )J.L+Y, (22) J""x-1 tanh(77x)P_Y.+iz(cosh a)dx= 2e-Y,a K(e-a) a>O 0 (23) f"" x tanh (7Tx) a2 + x2 P-Y.+ iz (cosh b) dx = Q a-Y, (cosh b) Rea> 0 0 (24) I; cos <bx) r <11 + ix) r <11-ix) P ~Y.-fiz <cosh a) dx (~7T) y, r (p.) (sinh a)J.L-Y, a, b > 0, Re J1 > 0 = (cosh a + cosh b )11- CHAPTER XIX BESSEL FUNCTIONS For Bessel functions and related functions see H .T .F., vol. II, Ct!apter VII and the literature quoted there, especially the books by Watson (the standard treatise on the subject), Gray and Mathew, McLachlan (r.ow in a second, revised, edition), and Weyrich. Integrals involving Bessel functions appear in almost every chapter of the present work, and the kernels of the integral transforms listed in Chapters VIII to XII arc Bessel functions. The present chapter contains mainly integrals which have not already appeared in the earlier chapters, although some of the i!ttegrals alreadylistedhave been included forthe sake of easy reference. Bessel functions are particular confluent hypergeometric functions, and the expressions which follow may be used to reduce integrals in­ volving Bessel functions to integrals involving hypergeometric functions. e-iz 1 F, ( v + Y.; 2 v + 1; 2 iz) f'(v + 1) z -Y, P. -Y,(v+Y,)rri = zv+Y, ( ) Mo v(2iz) 2 f'v+1 ' If ~ll (z) = (~ rr z )-y, e -Y,(v+Y, )rri W 0 ,v<-2iz) /-/~2l(z) = (~rrz)-y, eY,(v+XmiW 0,)2iz) 331 332 INTEGRAL TRANSFORMS O~z)v I (z)-0F 1(v+1;~z2 ) v r (v + 1) (Yzz)v -z ----e 1F1(v+ Yz; 2v+ 1; 2z) f'(v+1) z-Y. 2-2v-Y. -----~--M0,v(2z) r (v + 1) K (z)=(.:_)y,W0 (2z) v 2z ,v 2(z/2)v+l 2 B (z)= y, 1F2(l;v+3/2,3/2;-z /4) v rr f'(v+ 3/2) 2(z/2)v+l L (z)= y, 1F2(1;v+3/2,3/2;z2/4) v TT ' [' (v + 3/2) ZJ.L+I ~ 11 + JJ + 3 11-JJ + 3 z 2) s (z) = F l ; , ; -- J.L,V (11-jJ + 1) (11 + JJ + 1) 1 2 2 2 4 Expressions of various combinations of Bessel functions in terms of Meijer's G-function are given in the Appendix. 1\lanr. integrals involving Bessel functions may be obtained by specializing parameters in the small number of known integrals involving the G-function. Likewise, in the tables which follow, many fairly general integrals involving Bessel functions have been evaluated in terms of confluent hypergeometric functions or G-functions. For special value.s of the parameters these expressions simplify considerably. Frequently these particular cases are not given separately, and the user of these tables is expected to perform the necessary operations, the requisite formulas being given in the Appendix. BESSEL FUNCTIONS 19.1. Bessel functions of argument x. Finite intervals (1) Joa J)x) dx = 2 n~ o Jv+2n+t (a) Rev> -1 (2) Ja xv J (x) dx = 2v-t 0 v rr!-'2 f'(v +~)a [J)a) Hv_1(a) -Hv(a) Jv-1 (a)] Rev >-~ (3) Ja xv+t Jv(x) dx =a v+t Jv+t (a) Rev> -1 0 1 (4) Joa x t-v J)x) dx = 2v-t [' (v)-a t-v Jv-t (a) (5) J0a xiL Jv(x) dx = (/1 + v-1) a J)a) S JL-!, v-t (a) ['(1 +J.L+!:) 2 Re (/1 + v) > -1 -a Jv-t (a) S JL )a)+ 21L ' ['(1-JL+v) 2 (6) ~a (a-x)-l-? Jv(x) dx = rr(~a)X J!l>v+~ (~a) J!l> v-~ (~a) Rev> -1 (7) Ja x-v(a2- x2)-v-Y, J (x) dx = 7T!I; 2-v-t a-2v ['(~-v) 0 v x J)~a) J_v(~a) Rev<~ For other similar integrals see sections 8.5 and 13.1. 333 334 INTEGRAL TRANSFORMS 19.1 Bessel functions of x; finite intervals (cont'd) (8) fa xP(a2-2abx + b2)-~ J (x) dx 0 v See Bose, S.K., 1946: Bull. Calcutta Math. Soc. 38, 177-180. av+! (9) fa xv sin x J (x)dx =---sin a J (a)-cos a J +I (a) o v 2v + 1 v v Rev> -1 (10) f 0a sin (a-x) J2n (x) dx =a J2n+l (a)+ (-1)" 2n [cos a -J (a)-2 ± (-1)m J (a)] 0 m= 1 2m n = 0, 1, 2, .•. (ll) fa a sin (a -x) J 2 n + 1 (x) dx = a J 2" + z<a) + (-1)" (2n + 1) [sin a-2 n ~ (-l)m J2m+1(a)] n = 0, 1, 2, ... m= 0 fa sin(a- x) J)x) dx =a Jv+! (a)-2v 00 (12) ~ (-1)" Jv+2n+2 (a) 0 n=O Rev> -1 fa x-1 sin(a- x) J (x) dx = 2v-1 00 (13) L (-1)" J + 2 + 1 (a) Rev> 0 0 v n= 0 v n ( 14) fa x-312 sin(a- x) J (x) dx = (v2-)i)-1 a~ J (a) 0 v v Ile v > ~ v+1 (15) la v a Jv+l(a) Rev>-r 2 x sin (a -x) J (x) dx = o v 2v+1 19.1 BESSEL FUNCTIONS 335 Bessel functions of x; finite interval (cont'd) (16) r x.\sin(a-x)J (x)dx 0 v 00 = 2a.\+1 l (-1)" (v-,.\)2n (v + 2n + 1) Jv+2n+1 (a) n= 0 (v+..\+ 1)2n+2 Re(..\+v) >-1 Seealsoi3ailey, W.N., 1930:Proc. London Math. Soc. (2) 31, 20()-208. (17) fa (a2-x2)-X sin ({3x) J (x) dx 0 v 00 = TT L (-1)" J2n+1 (a{3) J X v+n+X (~a) JX v-n-X (a) n= 0 Rev> -2 (18) r +1 [I ( 2 2)] ( ) -v-1 uv+2(a2{3,a) x v sin ~ {3 a -x Jv x dx = {3 0 Rev> -1 (19) fa xv+1 sin[b(a2-x2)X] J (x) dx = (~rr)X av+312 b(l + b2)-Xv-~ 0 v x Jv+3/2 [a (1 + b 2)X] Re 1-1 > -1 v+1 a a (20) J xv cos x J (x) dx = [cos a J (a)+ sin a J +1 (a)] o v 2v+1 v v Re v >-Y2 (21) faa cos (a -X) J 2n (x) dx n-1 =aJ2 (a)-(-1)"2n[sina-2 L (-1)m J2 +1(a)) n m= 0 "'" n = 0, 1, 2, ... 336 INTEGRAL TRANSFORMS 19.1 Bessel functions of x; finite intervals (cont'd) (22) faa cos(a- x) J2n+1 (x) dx =a J2n+1 (a) n +(-l)n(2n +1Hcosa-J0(a)-2 ~ (-1)" J2.,(a)] n = 0, 1, 2, ... m=t f0a cos (a-x) J)x) dx =a J)a)-2v 00 (23) ~ (-1)nJv+2n+1(a) n= 0 Re v > -1 rx -1 cos(a-x)J (x.)dx=v-1 J (a)+2v-1 00 (24) ~ (-1)n Jv+2n (a) 0 v v n= 1 Re v > 0 v+1 (25) Ja v a Rev >-~ x cos(a-x) J (x) dx =---J (a) 0 v 2v+1 v (26) a A aA+1 J (a) J x cos (a-x) J (x) dx = v o v A+v+1 00 (-1)n (v-A)2n-1 + 2aA+1 l (v + 2n~ Jv+2n (a) n=1 (v +A+ 1)2n+1 Re(A+v) >-1 See also Bailey, W.N., 1930: Proc. London Math. Soc. (2) 31, 200- 2_08. (27) Ja 2 2 X (a -x )-cos(f3x) J (x)dx= ~rrJ0(af3) [JX (~aW 0 v v 00 + 7T n~1 (-1)n J2n(af3) JXv+n(~a) JXv-n(~a) Re v > -1 (28) Jaxv+1 cos[~f3(a2-x2)]J (x)dx=f3-v-1 U +1(a2{3,a) 0 v v Re v > -1 19.1 BESSEL FUNCTIONS 337 Bessel functions of x; finite intervals (cont'd) (29) fa { 2 2)-~ [ ( 2 2 ~ xa -x cos(3a -x) ]J(x)dx 0 0 = ((32 + l)-~ sin[a({32 + l)~] (30) fo a (a 2 -x 2) -~ cos [(3 (a 2 -x 2)~] J )x) dx = ~77 J~)~aeu)J~)~ae-u) {3 = sinh u, Rev> -l (31) r x'"-1 p (x/a) J (x) dx, o n V r x'"-1 p (x/a) J (x) J (x) dx o n JL v See Bose, S.K., 1946: Bull. Calcutta Math. Soc. 38, 177-180. (32) Ja xfL-1 p (2x2 a -2 -l) Jv(x) dx 0 n 2-v-1 a'"+v [r(~JL + ~v)]2 = r (v + l) r 0~ JL + ~ v + n + l) r(~+ ~ v-n) Q 2~ JL+V JL+V JL+V JL+V a x F ----· v+l --+n + l ---n· --23 2' 2' '2 '2 ' 4 fie (JL + ~) > 0 For particular cases see Bose, B.N ., 1944: Bull. Calcutta Math. Soc. 36, 125-132. (33) F ~-'"( 2 2)-~'"P'"( I ) J ( ) d O X a -X V X a v+~ X X =(~77)~ a1-ILJ~_JL(~a)Jv+~(~a) Re JL < l, fie (JL-v) < 2 (34) fa xX (a2-x2)-~v-~ pv+~ (2x2 a-2 -l) J (x) dx 0 JL v = 77~ 2-v-1 a JJL+~ (~a) J_JL-~ {'~a) -l < Re v < ~~ 338 INTEGRAL TRANSFORMS 19.1 Bessel functions of x; finite intervals (cont'd) (35) J a 2 0 J0(x)J 1(x)dx=~- ~[J0(a)] (36) ]0a Jn (x) Jn+1 (x) dx = Y2-~ [J0 (a)]2-"~ 1 [J,. (a)F n = 1, 2, 3, •.• (37) Ja oo 2 o J)x) Jv+1 (x) dx = n~ o [Jv+n+1 (a)] Rev> -1 (38) Jaxp-1 (a2-x2)o--1 J (x)J (x)dx 0 JL v See I3ailey, W .N ., 1938: Quart. ]. Math. Oxford Ser. 9, 141-147. (39) 2 Ja a x P (1-2x2 a-2) [J0(x)f dx = I[J (a)f+[J +1 (a}F} 0 n 2(2n + 1) n n n = 0, 1, 2, .•• (40) Ja x2v+1 p (2x2 a-2 -1) [J (x)]2 dx o n V See I3ose, B.N., 1944: Bull. Calcutta Math. Soc. 36, 125-132. ib dx rr [ Y)b) Y)a) J (41) x [J 11(x)F =------ 2 J 11(b) J 11(a) (42) fb a dx rr [ J_)a)J)b) J = log x J)x) J_)x) 2 sin(vrr) J 11(a) J_ 11(b) (43) r X11 y )x) dx = 211-1rr~ 1(v+Y2)a [Y)a) "v-1(a)- H,}a)Yv-1(a)] Re v >-~2 19.2 BESSEL FUNCTIONS 339 Bessel functions of x; finite intervals (cont'd) (44) foaxv+T Y)x)dx=av+T Yv+T(a)+2v+Tf'(v+ l) Rev> -l a T-v ctn(vrr) T-v (45) fo x Y )x) dx = 2v-T 1 (v)-a Y v-T (a) Rev< l b 00 (46) J Y)x)dx=2 L [Yv+2n+T(b)-Yv+2n+T(a)] a n= 0 = J4 rr [J)u) Y v (v) + J )v) Y v (u )] -J4rr tan(vrr )[Jv(u )J)v )+ Yv(u) Y)v )] U=~ 2a[({32+l) y,+f31 v=Xa[({32+l)y,-{3], -X<Hev<X a2 = ( )[J(a)Y (a)+J +T(a)Y +T(a)] 2 2n + l n n n n n = 0, l, 2, ... Jb dx rr [J (a) J (b) ] (49) x[Y(x)JZ=2 /(a)_Yv(b) a V 11 v 1------l--------------- -------------------l I b dx rr [J (a) Y (b)] (SO) 0 x J)x) Y)x) = 2log J>b) Y:(a) 19 .2. Bessel functions of argument x. Infinite intervals Re v > -l 340 INTEGRAL TRANSFORMS 19.2 Bessel functions of x; infinite intervals (cont'd) (2) Joo J-/x) 2 dx = IT[J)a)- J)a)] x +a asin(viT) Re a > O, Re v > -1 Re a> 0, Re {3 > 0, Re v >-% a>O (6) J xv(eax- 1)-1 J)x) dx = 2v IT-l{ ['(v + ~) ~ (1 + a2 n 2)-v-l{ 0 n= 1 Rea> 0, Rev> 0 Rea> 0 (8) 100 x :v {3 sin (x + {3) Jv(x) dx =~~IT sec (viT) {3v J_)f3) iarg f31 <IT, IRe vi < ~ (9) ~oo _L cos(x + {3) J (x) dx =-~IT sec(viT) {3v Y_ ({3) x+f3 v v iarg f31 <IT, IRe vi < ~ 19.2 BESSEL FUNCTIONS 341 Bessel functions of x; infinite intervals (cont'd) (10) j 000 x!4' sin (2ax~) J_!4' (x) dx = 77112 a 312 J 314 (a 2) a>O (ll) Joo x !4 sin (2 ax~) J 'A (x) dx = 77112 a 312 J (a 2) 0 -1/4 a>O (12) Joo x~ cos (2axy,) J (x) dx = 77112 a312 J (a2) 0 !4 -3/4 a>O (13) J'"' x!4' cos(2ax y,) J_ (x) dx = 77112 a312 J (a2) 0 'A 1~ a>O (14) Joo x-~ e-ax sin(2,8x y,)J_~(x)dx 0 • =77y, (-p )' ox{_o£) a2 + 1 a2 + 1 J_~ c~2 ~21) Rea> 0 (15) Joo x-~ e-ax sin(2,8x y,) J!4' (x) dx 0 ' ( P )' ( aP') E P' ) Rea> 0 = 77 -2- exp --2- J!4' 2 a+1 a+1 a+1 (16) Joo x-~ sinx sin(4ax~) J0(x)dx = (1277)~ cos(a2 + ~,;77) J0(a2) 0 a>O (17) J~ x-113 sinx sin(4ax112)J113(x)dx =-2-512 77112 a 113 [sin (a 2 + 77/ 12) J113 (a 2) + cos(a2 + 77/12) Y113(a2)] a>O (18) Joo x-Y, sinx cos (4ax~) J0(x) dx 0 =-2-312 77112 [cos (a 2-)i 77)J0(a2)-sin (a 2-~~ 77) Y0(a2)] a>O 342 INTEGRAL TRANSFOHMS 19.2 Bessel functions of x; infinite intervals (cont'd) (19) J 000 x-113 sinx cos (4ax 112) J_113 (x) dx =-2-312rr112 a113 sin(a2 -rr/12)J (a2) -1/3 a>O (20) Joo x-~ cosx sin (4ax~) J0(x) dx 0 = (~ rr)~cos (a 2 -~ rr) J0 (a 2) a>O (21) J 000x-113 cosx sin(4ax112)J113(x)dx = 2-5/2 rr 112 a 113 [cos (a 2 + rr/12) J (a 2) 1/3 -sin (a 2 + rr/12) Y113 (a 2)] a.>O (22) Joo x-~ cosx cos (4ax~) J 0(x) dx 0 =-2-312 TT 112 [sin (a 2 -~ rr) J0 (a 2)+ cos (a 2-~~ rr) Y0 (a 2)] a>O (23) Joo x-113 cosx cos(4ax112) J_ 113(x)dx 0 = 2-312 TT 112 a 113 cos (a 2-rr/12) J (a 2) a>O -1/3 (24) J 00 x -P J (x) J (x) dx 0 1-L v 2-Pf'(p)l~(~(fl+V-p+ 1)) = r [~ (p + fl+ v+ 1)] r [~ (p-fl+ v+ 1)] r [~ (p+ 11-v+ 1)] 0 <Rep< Re (fL + v) + 1 (25) Joo X 1-2v[J (x)]4 dx = l (v) f'(2v) o v 2rr[f'(v+~)ff'(3v) Rev> 0 (26) 1:2: a2 [Jv(x)f dx = I)a) K)a) Rea> 0, Rev> -1 0 19.2 BESSEL FUNCTIONS 343 Bessel functions of x; infinite intervals (cont'd) See Bouwkamp, C.S., 1950: Nederl. Akad. Wetensch., Proc. 53, 654-661. See Watson,G.N;, 1922:A treatise on the theory of Bessel functions, Cambridge, P• 436. (29) Joo XJ.L+V e -axJ (x) J (x) dx 0 J.L v See Watson, G.N., 1922:A treatise on the theory of Besselfunctions, Cambridge, P• 390. (30) Joo sin (2ax) [J (x)f dx 0 v = ~p v-l{ (1-2a2 ) =77-1 cos(V7T)Qv-Y,(2a2-1) (31) fcoo sin (2ax) [x v J )x )] 2 dx 0<a<1 a>1 fie v > -1 a-2vr(~+v) 2 ){ 2F., (~ + v, ~; 1 -v; a ) 0 < a < 1 '1.77 r(l-v) -~~<Rev<~ 344 INTEGRAL TRANSFORMS 19.2 Bessel functions of x; infinite intervals (cont'd) (32) {)0 2 cos (2ax) [J (x)] dx 0 v = rr-1 Q..,-~ (1-2a2) 0<a<1 = -rr -• sin (vrr) Q v-~ (2a 2-1) a>1 Rev >-~ (33) Joo 2 cos (2ax) [x11 J (x)] dx 0 v a-2vr(v) 2 = ~ ~ 2F1 (v + ~. ~; 1 -v; a ) 2rr r( 2-V) r(-v)r( ~+ 2v) + 2rrr( ~-v) 2F, (~ + v, ~ + 2 v; 1 + v; a 2) 0<a<1 sin(vrr) a_4V_1.r(~ + 2v) 2F 1(~+v , ~~+2v; 1+v; a-2) =- r(l+v)r( ~-v) a>1 -~<Rev<~ (34) J 00 (x 2 -a 2)-X J )x) dx = -~ rr J ~ )~2 a) Y Y. v (~a) Hev>-1 a . (35) Jaoox_1 (x2-a2)-XJ 0(x)dx=-si(a) 00 xX P v-~ (x/a) J)x) dx = -(Yta)-X [cos (Y:!a) Y)~~a) (36) J a +sin (~a) J)~a)] -~2 < Re v < ~ (37) J00 x~-f..l.(x2-a2)-Xf..l.pf..I._X(x/a)J (x)dx a v z 11 = -2-3/2"1/<a 1-f.i.[JJ..I._ Y,(~a)Yv( ~2a) + YJ..I.-Y, (~a) J)~a)] -~ < Re f.L < 1, jRevj <~~+2 llef.L 19.2 BESSEL FUNCTIONS 345 Bessel fWJctions of x; infinite intervals (cont'd) (38) J"" x11(x2-a2)~,\-~ P~-1 (x/a) J (x) dx a ll c}'-+v a11 rc~ + v) = rr~ r{l-A.) SA.-v,A.+)a) Re I/< 5/2, Re(2A. + v) < 3/2 (39) J 00 x ~ (x 2 -a 2) ~ v-l( P ~ -v (2 x 2 a-2 -l) J (x) dx a J.1. v = -211 -2 rr~ a sec(flrr) I[JJ.J.+ ~ (~a)]Z- [J_J.J._~(~a)]2 1 Rev> -l/2, Re v-3/2 < 2Re f1 < l/2-Re v 00 r(v- ~) H)2a) (40) J x1-2v(x2-a2)v-3/2 [J (x)]Zdx= Rev>~ a v 2rr~av+1 (41) Joo x2v+1 (a2-x2)-v-3/2 I[Jv(x)]2 + [J_)x)]21 dx a = 17-~ a11-1 r(-v- ~) sin(vrr) J)2a) Rev <-~ (42) 1 00 sin [a (x + {3)] x+f3 00 J0 (x) dx Ja 2 ~ =2 (1-u )-cos(f3u)du 0 O_::;a.::;l = rr Jo ({3) l.::;a<oo (43) loo !x! O<a<l ---sin [a (x + {3)] J0 (x) dx = 0 x+f3 00 (44) 1: sm [a (x + {3)] J (x) dx = rr/3 -v J ({3) x v (x + {3) v+2n v+2n l.::;a<oo n = 0, l, 2, ... , Re v > -3/2 346 INTEGRAL TRANSFORMS 19.2 Bessel functions of x; infinite intervals (cont'd) (45) I: sin [a (x + {3)] 2 2 2=:;a<oo [J +~ (x)] dx = 11 [J +X ({3)] x+f3 n n 2 n = 0, 1, 2, ••• (46) J: sin [a (x + ,13)] Jn+Y, (x) J_n-Y, (x) dx =77Jn+~(,B)J -n-~({3) x+,B n = 0, 1, 2, ••• , 2=:;a<oo (47) loo sin [a (x + ,13)] [J + (x)]2 dx = 11{3-2v [J + (,13)]2 2sa<oo X 2V(X + ,13) v n 1.1 n -oo n = 0, 1, 2, ••• , Rev> -1 (48) Joo Y)x)dx=-tan(~v77) 0 -1 <Rev< 1 (49) J;" xP(x' +a')-~(x) dx P v ) 1----ap-2J1. a2 2' 2 =-c3t _ r (11) 24 4 _£. ~ 1-v -~ /1 2' 2 ' 2 ' 2 IRe vi-1 <Rep< 2Re 11 + ~ (50) Joo _1 ( 2 a) y, y, 0 X exp ----;-Y)x)dx=2Yv(2a )K)2a') Rea> 0 (51) r X-1sin (~)(sin X J0 (x) + COSX f (x)] dx o 2x o = 11 J 0 (a~) Y 0 (a~) a>O 19.2 BESSEL FUNCTIONS 347 Bessel functions of x; infinite intervals (cont'd) (52) oo 1 (a) J x-cos- [sinxY0(x)-cosxJ0(x)]dx 0 2x = TT J 0 (a 'lz ) Y 0 (a 'lz) a>O (53) f 0 00 x!4 sin (2ax 'lz) Y :%'(x) dx = -TT 112 a312 H_)((a 2) a>O (54) f 000 x!4 cos (2ax'lz) Y!4 (x) dx = -rr112 a312 "-~ (a2) a>O (55) Joo x-'lz sin x cos (4ax'lz} Y 0 (x) dx 0 = 2-312 TT 112 [3 sin(a 2-Ji rr) J0 (a2)-cos (a2-Ji rr) Y0 (a2)] a>O (56) Joo x-'lz cosx cos(4ax'lz) Y0(x)dx 0 =-2-312 TT 112 [3 cos (a 2-Ji rr) J0 (a 2) +sin (a 2-Ji rr) Y0(a 2)] a>O (57) J. 00 J + (x) Y (x) dx = (-1)n+1 Re v >-~. n = 0, 1, 2, ••• 0 11 n v-n 2 (58) Joo e -2ax Jo (x) Yo (x) dx = 0 K[a(a2+ 1)-'lz] rr(a2 + l)~ Rea> 0 (59) Joo x211+1 exp(-ax2) J (x) Y (x) dx 0 11 11 =-~TT-y, a-311/2-1/2 exp (--1-) W ( ~) • 2a '/z11,'/z11 a Rea> 0, He v > -~ 348 INTEGRAL TRANSFORMS 19.2 Bessel functions of x; infinite intervals (cont'd) (60) f000 sin (2ax) J0 (x) Y 0 (x) dx =0 O<a<l K [(l-a -2)y,] =- a> l 77a (61) J 000 cos (2ax) J0 (x) Y0 (x) dx =-77-1 K (a) 0 <a< l = -(77a)-1 K(a-1) a>l (62) Joo cos (2 ax) [Y0 (x )f dx 0 =77-1 K[(l-a2)y,] 0 <a< l = 2(77a)-1 K[(l-a-2)y,] a> l (63) Joo x 1-2v sin (2ax) J (x) Y (x) dx 0 v v r(3/2-v) a (3 3 2) =-2 r(2 v :..._ ~) r (2-v) 2F1 2-v, 2-2 v; 2-v; a 0 <He v < 3/2, O<a<l (64) J0 00 x 1-2v sin (2ax) I[Jv(x)J2- [Y )x)f! dx sin (2 V77) r (3/2-v) r (3/2-2 v) a F (~-v ~-2v 2-v·a2) 77r(2-v) 2 1 2 ' 2 ' ' 0 < f!e v < ~h· O<a<l (65) Joo x2-2v sin (2ax) [J)x) Jv_1(x)-Y)x) Yv_1 (x)] dx 0 sin (2v77)r(3 /2-v)r(5/2-2v)a (3 5 . ) =- F --v --2v · 2-v · a 2 77r(2-v) 2 1 2 '2 ' ' l/2<Re v<5/4, O<a<l 19.3 BESSEL FUNCTIONS 349 Bessel functions of x; infinite intervals (cont'd) (66) J 000 x2-2v sin (2ax) [Jv(x) Yv_1 (x) + Y)x) Jv_1 (x)] dx 1(3/2-v)a ( 3 5 2) = - F --v --2v · 2-v ·a [' (2v-3/2) f' (2-v) 2 1 2 , 2 , , l/2 < Re v < 5/2, O<a<l (67) j xy,-1L(a2-x2)-Y,!Lp!L (x/a)Y (x)dx a v-~ v =2-3/217112a1-IL[J (~a)J y,C~a)-Y (~a)Y y,CY:;a)] v jJ.-2 v jJ.- -~ < Re J1 < l, Re(2J1-v) >-Y:; 19.3, Bessel functions of arguments ax + {3, x 2, x-I (l) Joo xp-1 J (ax) J (bx) dx = 2p-1 aiL b -IL-Pf'02 J1 + ~v+ Y:;p) 0 jJ. v 1(/Hl)f'(l-~ ~Jl+Xv- Y:;p) (j!+v+p Jl-V+p. . a2 0 X 2F1 2 ' 2 , J1 + ~ , ~ Re (Jl + v + p) > 0, Be p < 2, 0 <a< b (0\ <..) Joo xy, (x2 + >.. 2)-Y, J (ax) J (bx) dx 0 jJ. v See Couv.kamp , ·c.S., 1950: Nederl. Akad. Wetensch ., Pro c. 53, 654-661. (3) Joo x 1 +v[J (ax)f J (2bx) dx 0 v v ['(~+v ) sin (v77) a2v b -v-1 (b2-a2)-v-Y, O<a<b =-2773/2 r·e2 + v) a 2v b -v-1 (a 2 _ b2)-v-Y, 0 < b <a = 2 77312 -~2 < Re v < r2 350 INTEGRAL TRANSFORMS 19.3 Bessel functions of ax+ f3, x2, x-1(cont'd) (4) J 000 x 1 +v Jv(ax) J _)ax) J,(2bx) dx =0 0 <a< b a2v b -v-1 I (a2-b2)- v-Y, 0 < b <a 277y, rO.-f-v) -1<Hev <~2 (5) J00xJy, ~(ax)J y, +~(ax)J (2bx)dx o v-4 v 4 v =0 0 <a< b = 2-3/2 77-1 a-112 b-1(a- 6)-Y, O<b<a lle v > -1 (6) J~ x 1 +v JJJ.(ax) J _JJ.(ax) J)2bx) dx =0 O<a<b (a2 _ b2) Y,v-~ = 1 1 pv+Y,(262a-2-1) O<b<a 2 77y, ab y, JJ.-Y, -1 <He v < );2 (7) f000 xp-1 JA_(ax) JJJ.(ax) J)2bx) dx a/c+11-b -A_-JJ.-p [' ( ~'2 A.+~~ fl+ ~1 v+ ~ p) = 2:\+ JJ. f' (A.+ 1) I~ ( 11 + 1) f' ( 1-);2 A.-);2 11 + ~ v-);2 p) 0-+11+1 A.+11 A.+11+v+p A.+11-v+p X 4F3 '--+ 1, ' ; 2 2 2 2 a2 ) A.+ 1, 11+ 1, A.+ 11+ 1; ~ He(,\+ 11+ v+ p)> 0, O<a<b 19.3 (8) (9) BESSEL FllNCTWNS 351 Bessel functions of ax+ {3, x2, x-l,(cont'd) J""x1-11J 11(ax)J (bx)J (cx)dx 0 v v if a, b, c are sides of a triangle of area 6. if a, b, c are not sides of a triangle a, b, c > 0, Re v > ->2 J"" xP-1 J,(ax) J (bx) J (ex) dx 0 ~ ~ v Tie(A+fl+l '+p)>O, P.ep<5/2, a,b,c>O, c>a+b For particu!ar cases see \latson, G.!\'., 1922: A treatise on the theory of Bessel functions. Cambridge, Sec.l3.46; Dailey, \';'.N., 1936: Proc. London Math. Soc. (2) 40, 37-48. (lf)) J"" x 1-zv [J (ax) J (bx)f dx 0 v v a zv-1 b -1 f'(v) ( = . 2F1 v, 2 77 [' (v+ ~~) [' (2 v+ h) 1/ 2-1-', 2 v + 12. a 2 \ '62) Re v > 0, 0 <a< b (ll) J 0"" x 1-v [J)x) Y_ 11(x) + Y 11(x) J_)x)] J)2ax) dx =0 av-1(a2-l)v- Y, = rry,l'(v+~) CJ<a<l a > l -~2 < ne v < l 352 INTEGRAL TRANSFORI\JS 19.3 Bessel functions of ax+ {3, x 2, x -t (cont'd) (12) J"" +t xll-[J (x) Y (x) + J (x) Y (x)) J (2ax) dx 0 v 11- 11-v v =0 O<a<l -J.i--1 ( 2-1)- Y,J.L-~ a a +Y, a> l =- y, J.L+Y, P~-~ (a) 1T 2 -l < 11e J.L < ~2, He v > -1, He (J.L + v) > -l (13) J"" + x 1 11-Y (ax) J (bx) J (ex) dx = 0 0 11-v v 0 < b < c, O<a<c-b (14) J"" 2 x J (ax) J (bx) Y (bx) dx 0 11 v 11 =0 O<a<b =-(2TTab)-1 0 < b <a Rev>-~2 (15) J"" 2 +t 0 x v J)ax) Y)ax) J)bx) Yv(bx) dx a 21/ l (3 V + l) ~ l 3 a2 ) = 2TT b4v+2 ['(~-v) f'(2v + 3/2) 2F, v+ -, 3v+l; 2v+ -;-2- 2 2 b 0 <a < b, -l/3 < Re v < l/2 """ J)ax) Yv(bx)- Jv(bx) Y v(ax) dx = _ ~~ )v (1fi) i 0 < b <a xi[J)bxW+[Yv(bx)]2! 2 a For other sin:ilar integrals see sec. 6.8. 19.3 BESSEL FUNCTIONS 353 Bessel functions of ax+ (3, x2, x-1 (cont'd) !"" [J (ax) Y (bx) -J (bx) Y (ax)] (17) 11 11 11 11 [J 11(bx)F + [Y (bx)F 0 11 x [Jv+T (ex) Y)bx)-J)bx) Y11+1 (ex)] dx av b2v = ---;:;+! -11 11 + 1 e a e 18 [ [J)ax) Y )bx)..,. J)bx) Y11(ax)] ( ) [J (bx)f + [Y (bx)]Z 0 11 11 x [J11+1(ax) Y 11(bx)-J)bx) Y11+1 (ax)] dx l b 211 = 2a-a2v+1 77 J0(ae)J0(be)+Y0(ae)Y0(be) =2 [J 0(be)F + [Y 0(be)J2 (22) r J)x) J,_)a- x) dx = Jo(a)- cos a 0 x dx 2 2 e -X O<b<e<a O<b<a<e He..\>0, O<b<a 0 < b <a, e > 0 -1 <He v < 1 -1 <Rev< 2 354 INTEGRAL TRANSFORMS 19.3 Bessel functions of ax + {3, x 2, x -z( coot 'd) (23) JaJ (x)J (a-x)dx =2 Y (-1)" J + +z +1(a) o jJ. v m= 0 jJ. v m Re11>-1, Rev>-1 Re 11>0, He v>-1 (25) JaxA_- 1J (x)J (a-x)dx =2A_ I (-l)"['(A +Jl+m)(A)m o J-1. v m = 0 rn ! [' (Jl + rn + 1) Re (A+ J1) > 0, Re v > -1 __ 2A. ~ (-1)" i(A + 11 + rn) (A)" f.. (A+J1+v+2rn) va m=O m!t(J1+m+1) He 11 > -~~. R e v > -~~ He v> -1, Re 11> -~~ 19.3 BESSEL FUNCTIONS 355 Bessel functions of ax+ {3, x2, x-1 (cont'd) a _ 1 2Ml(fL+~)r(v-fL) (30) J xM(a-x) M-J (x)J (a-x)dx= X aMJ (a} o M v 7T r (fL+ v+ l) v Rev>RefL>-~ (31) JaxP-1(a-x}a--l J (x)J (a-x)dx o M v Ja xp-l (a-x)a--l J,(bx) J (ex) J (a-x) dx 0 A M v For these integrals and several particular cases see I3ail ey, W .N ., 1930: Proc. London Math. Soc. (2) 30, 422-424 and 31, 200-208; Rutgers, J.G., 1931: Nederl. Akad. Wetensch. Proc. 44, 75-85. (32) loo J)a(x + y)] (x + y )M 00 J [a (x + z )] v dx (x + z )v (2rr/a)X r<fL+ v) Ju+v-'6 [a(y-z)] (y _ z)M+v-'6 = r(fL+~) r(v+~) a > 0, Re (fL + v) > 0 roo 3 2 ) ( 2) (34) J0 x J2)2ax) Jv-'6 (x ) dx = ~ csc(vrr "-v-'6 a -~ctn(VTT)Jv+ '6(a2 )-~Yv+Y,(a2 ) a>O, Rev>-% 356 INTEGRAL TRANSFORMS 19.3 Bessel functions of ax + {3, x \ x-I (cont'd) a>O a>O a<v-l)/3 [ 2 ( 2 v-1) = 8 J(v+!t2)/3 (a ) sin a + -6--rr 2 ( 2 v-1 )] + Y(v+~)/3 (a ) cos a +-4-rr · -5/2 < Re v < -l/2 (40) Joo (v+2)/3 ( 2} J ( 2} T '2 } d 0 x cos x (v+~)/3 x uv~ ax x a (v-1 )/3 [ 2 ( 2 v-1 \ 8 J (v+~)/3 (a ) cos a ·+ -6--rr) 2 ( 2 v-1 )] + Y(v+~)/3 (a ) sin a +- 6-rr = Ji:!h sin(ac2h) Jv(bc2h) = J1! k cos (ac 2 k) J v (be 2 k) Joo 2 2 (42) 0 x cos (ax ) J)bx ) J2v(2cx) dx = Ji:!h cos (ac2h) Jv(bc2h) -1 < Re v < ll/2 0 <a< b O<b<a 0 <a< b = Ji:!k sin (ac2k) J)bc2k) Rev> -Ji:!, 0 < b <a h=(b2-a2)-X, k=(a2-b2)-~ 19.3 BESSEL FUNCTIONS 357 Bessel ftmctions of ax+ {3, x 2, x-I (cont'd) ia x2 (43) a2-x2 J'4 (x) J_)((x) J2v(2a2-2x2) dx a 2 2 = 4v Jv+)( (a ) Jv-)( (a ) Rev> 0 (44) 1-(') (" )"" ~a)' ('b') 0 Jv ~ Jv+l -;-----;= b J2v+1 ay, a, b > 0, Rev> -1 (45) Joo xp-l J (ax) J (bx-1) dx 0 J.L v p-1 -p 20 (a2b21 v P+/1 p-/1 v) =2 a G ---------04 16 2 ' 2 ' 2 ' 2 a, b > 0, Re (p-v) < 3/2, He (p + 11) > -3/2 (46) 1-J (-a-} (-=}xeb['l_K (2a}Y (2a')] v X z-· b TT 2v bY, 2v b X 0 a, b > 0, -~~ < Re v < ~ (4 7) 100 (a) (x) dx 1 [2 (2ay,) ~2a y,) J J- Y --=---K ---Y --0 V X V b X2 a 1T 2V b'/, 2V bY, a, b > 0, -)'2 <Rev< ~2 ,.oo (a) (x) E2a'hj (4f:l) Jo Y v ~ Yv b dx = -b J2 v ~ a, b > 0, -~ < Re v < ~~ 358 INTEGRAL TRANSFORMS 19.4 19.4. Bessel functions of other arguments (l) Joo xv+l y!L J (ax) J (by) dx 0 v !L = av b!l-([3/h)!L+v+l [sin(vrr) YJL+v+1({3h)-cos(vrr)J JL+v+1({3h)] 0 <a< b a,b>O, Re{3>0, Re(l1+v) <0, Rev>-1 (2) Joo xv+l y!L J (ax) Y (by) dx 0 v !L = -avb!l-([3 /h)!L+v+l [sin (vrr) JJL+v+ 1({3 h)+ cos (vrr) YJL+v+/{3h)] 0 <a< b =-2rr-1 avb!l-([3/k)!i-+v+l cos(l1rr)KJL+v+1({3k) 0 <b <a a,b>O, Re{3>0, Re(l1+v) <0, Rev>-1 (3) Joo xv+l y-!1-J (ax) Y (by)dx 0 v !L = av b -!1-([3/h)v-JL+l Y ({3h) JL-v-1 0 < b <a a, b > 0, Re {3 > 0, Re 11 > Re v > -1 (4) ~00 xp-ly-IL(x2+,\2)-1 [cos(p~v rr)Jv(ax) +sin (p ~ v rr) Y)ax)] J!L(by) dx J [b({32-,x_2)Y,] =-,\p-2 '({32-,\2)y,!L K)a-\) O<b<a IRe vi < Re p < Re 11 + 4, He ,\ > 0 19.4 (5) (6) (7) (8) (9) (lO) BESSEL FUNCTIONS 359 Bessel functions of other arguments (cont'd) 0 < b < a, Re v > -3/2 Joo sin [a(x-z)] x-z -00 =TT J)a(z2-2bz cos e+b2)y,] (z 2 -2 bz COS e + b 2)Y, v = -TT [J v+p(a) Yv-p(b) + Jv-p(b) Y v+p(a)] Rev >-~ a, b > 0, -% <Rep <% = rr[Jv+p(a) Jv-p(b)-Yv+p(a) Yv-p(b)] a, b > 0, -% < Re v <% f 00 (a+ bx)v 1 1 1 xp-! --- H<2l[x-y,(a+bx) y,(ax+b)y,]dx 0 ax+ b v =-irrf/(2) (a)H(2) (b) v+p v-p a,b>O, -%<Rev<% Joo coshx cos(2a sinhx)J (bex)J (be-x)dx 0 v v =~(b2-a2)- XJ [2(b2-a2) X] 2V =0 2 2 y, y = (x + (3 ) ' 0 <a< b O<b<a Re v>-l 360 INTEGRAL TRANSFORMS 19.4 Bessel functions of other arguments (cont'd) ( ll) J''" cosh x cos (2a sinh x) Y (bex) Y (be-x) dx 0 v v =-~(b2-a2)-x J [2(b2-a2) X] 2V 0 <a< b = 277-1 cos(v77) (a2-b2)-X K2J2(a2-b2)X] 0 < b <a -1 < Re v < 1 (12) ]000 cosh x sin (2 a sinh x) [J )be x) Y )be-x)-Y )be x)Jv(be -x)]dx =0 0 <a< b = -2 77-1 cos (v77 )(a 2 -b 2)-X K 2 v [2 (a 2 -b 2 )X] 0 < b <a -X<Rev<X (13) J7Tsin(21lx)J2 (2asinx)dx=77sin(ll77)J (a)J +(a) o v v-11-v 11- I He v>-1 ( 14) J7T cos (2/lx) J 2 (2 a sin x) dx = 77 cos (1177) J (a) J + (a) 0 v v-11- v 11- Re v >-X X7T ( 15) J cos~2nx) J0(2a sin x)dx = ~77[J (a)F n = 0, 1, 2, ••• o n (16) J X7T cos (2nx) Y0 (2 a sin x) dx = Yz 77 J (a) Y (a) o n n n = 0, 1, 2, ••• (17) fo7T [tan (Xx)]-2K e-{3cosx J2)a sin x) dx f' (X +K+ v)f' (X-K+ v) = M [{3 + (,8 2 _a 2) X] M [{3 _ ({3 2 _a 2)X] a [f' (2v + 1)]2 K ,v K ,v He v+~2> IRe Ki 19.4 BESSEL FUNCTIONS 361 Bessel functions of oUter arguments (cont'd) ~7T (18) f0 cos(2{3 cos x) J2)2a sin x) dx = ~ 77 J)(,B 2 + a2)~ + {:3) J)(,82 + a2)~ _ f?) He v > -~ (19) J~7T v+ (sin x) 1 cos (,8 cos x) J (a sin x) dx 0 v = 2-~ 77~ av(a2 + ,82)-~v-'4 Jv+~[(a2 + {32)Y.) He v > -1 (20) J~77 sin(2x) P (cos2x) J0(a sin x)dx = c:-1 J2 +1(a) 0 n . n n = 0, 1, 2, ••• (21) J~7T (sin x)v+l cos (a cos e cos x) c~+Y, (cos x) Jv(asin e sinx) dx 0 = (-l}n 2-y, 77y, (sin e)v a-~ C~n+Y, (cos 8) Jv+2n+Y, (a) n = 0, 1, 2, ••• , Re v > -1 (22) Jy,7T (sin x)v+l sin (a cos e cos x) c ~;;(cos x) Jv(asin 8sinx) dx 0 = (-l)n 2 -Y, 77y, (sin e)v a -Y, v+Y, ) ( ) c 2 n +I (COS e J V + 2n + 3/2 a n = 0, 1, 2, .•• , Re v > -1 Y,7T Re v >-~ (23) ~ ' cos (2/lx) J2v(2a cos x) dx = X77 Jv+)a) JV_J-L(a) (24) y,7T ~ cos (2/lx) Y 2v(2a cosx) dx = ~ 11 ctn (2v77) Jv+Jl.(a) JV_Jl.(a) -~77 esc (2v17') JJl._V(a) J_Jl._V(a) -~2 <He v < ~~ 362 INTEGRAL TRANSFORMS 19.4 Bessel functions of other arguments (cont'd) l->71 (25) J (sin x)J..L+l (cos x)v+l J (a sin x) J ((3 cos x) dx 0 J..L ).1 = aJ..L (3v(a2 + {32)-l>(J..L+v+l) JJ..L+v+l [(a2 + {32)}>) Re J1 > -1, Rev> -1 (26) l->71 J (sin x)P (cos x)o-J (a sin x) J ({3 sin x) dx 0 J..L ).1 l->71 J (sin x)P (cos x)o-J (a sin x) J ((3 cos x) dx 0 J..L ).1 See Bailey, W.N., 1938: Quart. J. Math. Oxford Ser. 9, 141-147. (27) J71 2 J (w) X J (a) J ).1 ({3) (sin x) v_v_ dx = 2v rr f'(v + ~) _v_ o wv av (3).1 w=(a2+{32-2a(3 cosx)y,, Rev>-~ (28) J71 2 Y (w) y, J (a) y )(3) (sin x) v_v __ dx = 2v rr' f'(v + ~) _v_ o wv av (3).1 ( 2 2 X li.J = a + (3 -2 a (3 cos x) , \a\ < \(3\, Rev>-~ (29) 11 2v v J (w) rr['(2v + n) Jv+n (a) Jv+}{3) J (sinx) C (cosx)-v-dx= 1 av (3).1 o n wv 2v n!f'(v) n = 0, 1, 2, ... , ( 2 2 y, w = a + (3 -2 a (3 cos x) , Rev >-~ (30) J11 2v v Y)w) rrf'(2v+n) Jv+n(a) Y v+n ({3) 0 (sinx) Cn(cosx) ---dx= wv 2v 1n!f'(v) av (3).1 n = 0, 1, 2, ••• , li.J = (a2 + {32-2a(3 cosx)y,, \a\ < \(-; \ Rev>-~ 19.4 BESSEL FUNCTIONS 363 Bessel functions of other arguments (cont'd) (31) J"" e-2J-Lz Y2 (2a sinh x) dx = ctn(2v77) I+ (a) K (a) 0 v J-L v J-L-v (33) -esc (2v77) IJ-L_)a) K J-L+)a) a>O, Rejl>-3/2, -1/2<Rev<l/2 + cos[(J.!- K)71] y2 (a sinh x)! dx =-a-1 WK (a) w_K (a) J-L ~ ~ a> 0, Re K > IRe 111-~ J"" sinhx [tanh (~x)]v e-,Bcosh z J)a sinhx) dx 0 2 2 -y, (a + fl ) ' + f3 [ 2 2 y, =(a + (3 ) (a2 + (32)Y,- (3 ]-Y,v 2 2 I exp [-(a + (3 )y,] Re (3 > IRe al, Re v > -1 (34) J"" [ctnh( ~ x )fK e -,B cosh z J 2 (a sinh x) dx 0 J-L = ['(~-K + J1) M_K [(a2+(32)Y,-(3] WK [(a2+(32) Y,+(3] a r (2 J1 + 1) .J-L ,J-L R e (3 > In e a I' R e ( J1 -K) > -~2 =-a-t sec((J.! + K)77) JTIK ,(h) Jl'I_K (k) ,,_ ,J.J.. tan[(J.!+K)77]r(~-K+J1) ( ) - If h) M (k a 1'(2J1 + 1) K,J-L -K,J-L h = (a 2 + (3 2)Y, + (3, k =(a 2 + (3 2)'1. -{3 Re (3 >IRe al, Re K < ~~-llle 111 364 INTEGRAL TRANSFORMS 19.4 Bessel fUD(:tions of other arguments (cont'd) (36) Joo (sinh x)J.L+1 (cosh x)';+1 J (a sinh x) II <2l(b cosh x) dx 0 J.L v (37) _ _ -J.L7Ti J.L b v h -,.-v-1 H (2) (h) -e a J.L+v+1 = 2irr-1 ev'TTiaJ.Lbvk-J.L-v -1 KJ.L+v+1(k) h=(b2-a2)y,, He f1 > -1, 0 <a< b O<b<a k=(a2-b2)y, Re (f1 + v) < 0 Joo (sinh x)J.L+1 (cosh x)1-v J (a sinh x) 11 (2) (b cosh x) dx 0 J.L v =af.Lb-vhv-J.L -!Jj( 2) (h) O<a<b v-J.L-1 = 2 i 1T-1 a J.L b -v k v-J.L-1 K (k ) 0 < b < a v-J.L-1 h = (b 2-a 2)y,, k =(a 2-b 2)y,, Re v > fie f1 > -l (38) Joo sech x e 2Kx-,Btanh x J 2 (a sech x) dx -oo J.L l (~ + K + f1) 1(~- K + f1) = [ ] 2 At K , (h ) M K Uc) a 1(2/1 + l) ,,.. ,J.L h + k = 2 (3, hk =a 2, He f1 > IRe Ki-!2 19.5. Modified Bessel functions of argument x For integrals involving ber v x, beiv x, ker v x, keiv x and similar functions see l\1cLachlan, N.K., 1954: Bessel functions for engineers. Cxford, Second edition. Ja I (x) dx = 2 00 (l) ~ (-l)n Jv+2n+1 (a) He v > -l 0 v n= 0 (2) J0a xv IJx)dx=2v-1rry,i(v+~)a[IJa)Lv_ 1(a)-Lv(a) Iv_;(a)] Re v >-~" 19.5 BESSEL FUNCTIONS 365 Modified functions of x ( cont 'd) (3) Joa xv+1 I)x) dx = av+1 Iv+1 (a) Rev> -1 (4) 21-v J a x 1 -v I (x) dx = a 1 -v I 1 (a) --- o v v- 1 (v) (5) Ja xv(a 2-x2}v-}i I (x) dx = 2~v-1 rr!-i a 2v 1(v+ ~)[I (~a)]2 0 v . v (6) Ja v+1 ( 2 2)'r-1 I)x) dx = 2o--1 av+o-1(a) Iv+o-(a) x a -x 0 Rev> -1, Rea> 0 (7) Ja xp-1 (a 2-x2)o--1 I (x) dx 0 ll 1(~) 1 (a) av+p+2o--2 ("+p v+p a') 2 F --; v+1,--+a;- = 2v+1 1(v + l) 1(11 ~p +a) 1 2 2 2 4 Re (p + v) > 0, Re(a)>O J.a n+1 -x 2 2 2 n (8) I (2ax)dx = )ian[ea -a }; I (2a 2)] 0 x e -e n r r= -n n = 0, 1, 2, ... }i 2v+1 (9) Ja +1 -1 1T a xv cos I x dx = o Y y 11( ) 211+1 1(v + 3/2) 2 2)!-i y =(a -x , Rev> -1 (10) Ja -1 0 y cosh (y sinh t) I211(x) dx = ~" I)ae ~ Iv (ae -t) y = (a2-x2)Y., Rev >-)~ 366 INTEGRAL TRANSFORMS 19.5 Modified functions of x (cont'd) -a (ll) J a 1 ae e-"p (l-2xa- )I0(x)dx=--[I (a)+ I +1(a)] o n 2n + 1 n n (12) Jaxf.i.e-"P (l-2x/a)I (x)dx 0 v J.i. See Bose, B.N., 1948: Bull. Calcutta Math. Soc. 40, 8-14. 2 (13) J"" x -113 e -z sin (4ax~) I (x) dx = (277)-112a 113 e -a K (a 2) 0 1/3 1/3 a>O J"" x -ve-x sin (4ax~) I (x) dx = (2312 aY-1 e -a 2 (14) 0 v x W 112 -3v/2, 1/2-v/2 (2a 2) a> 0, Rev> 0 r X-~ e-" cos (4ax~) Io (x) dx = (277)-~ e -a 2 K o (a 2) (15) a ·> 0 0 (16) J"" x-v-~ e -z cos (4ax~) I (x) dx 0 v = 23v/2-1 av-1 e -a 2 W -3v/2, V/2 (2a 2) a> 0, Re v>-r:! (17) J"" 2 2)-~ _,) 17 (x -a T (ax K 2 (x) dx = -W~ (a ) a n f.i. 2a n,J.L w -~n,f.i. (a) n = 0, l, 2, .•• (18) J"" (l + x/a)f.i. e -z P-2f.i.(l + 2x/a) I (x) dx = 0 0 v J..L -Y:; < Re fl < 0, -~ + Re fl < Re 1.1 < -Y:;-Re fl 19.5 BESSEL FUNCTIONS 367 Modified functions of x (cont'd) (19) Joo (x + aP.Le-" P-2~-'(l + 2x/a) I (x)dx 0 v 1-' 2~-'-1 r (j.t + v + Yz) r <~L -v-Yz) e a W ~-I-'·~ +v (2 a) = 7Tx r(2j.t + v + l) r(2j.t- v) iarg ai < 771 Re jL > IRe v + ~21 00 2 2 2 n-1 (20) J 1-n -x I (2ax)dx =J4a-n[ea -a ~ I (2a 2)] x e -e a n r= 1-n r . n = l, 2, ••• (21) loa XV Kv(x) dx = 2v-1 77~ r(v + Yz) a [Kv(a) Lv-1 (a) +Lv(a)~_1(a)] Re v > -~2 (22) J 0axv+1 K)x)dx= 2vr(v+ l)-av+1 Kv+1(a) Rev> -l (23) Jax1-vK (x)dx=2-vr(l-v)-a1-vK (a) 0 v v-1 Rev< l (24) J a xll(a 2-x2)1l-l> K (x) dx = 2~-'-1 rry, a 21-Lr(j.t+ ~2) 0 ll xI (Yza) K (Yza) ll ll Re IL >-~~ (25) J ~ y -1 cosh (y sinh t) K 2v(x) dx = J4 772 esc (vrr)[I_)ae t) I_v(ae -r) 0 -I)ae t) I)ae-t)] 2 2 ~ y =(a -x ) , -~~ < Re v < Yz (26) J 0a x J)A.x) Kv(Kx) dx = (K2 +A 2)-1 [(.V K)v +A.aJv+1(A.a)K)Ka) -Ka J)>..a) Kv+1 (Ka)] He v > -l 368 INTEGRAL TRANSFORMS 19.5 Modified functions of x (cont'd) (27) J a x 2 v + 1 P ( l - 2 x 2 a -2) I (x) K (x) dx 0 J.L v v See Bose, B.N., 1948: Bull. Calcutta Math. Soc. 40, 8-14. (2R) Jax2v+1 p [(l-x2a-2 .)X]J (x)K (x.)dx o n 11 v See Bose, B.N., 1944: Bull. Calcutta Math. Soc. 36, 125-132. (29) a J oo -X -1 - 77 e x (x + a) e x K (x) dx = X K (a) o v a cos(v77) v largal <77, IRe vi<~ (30) 00 1 I (a77)12 2 f 0 x-!4 exp(-2ax X)K!4(x)dx= - 2-K!4(a) + ~773/2 a112[L_1/4(a2)-L1/4(a2)] oo 77312 a -112 sec ( VTT·) (31) J x-X y-1 e-y K (x) dx = K (a) 0 v r(%+~v)r(%-~v) v 2 2 X y=(x+a), Rea> 0, -~<Rev<~ (32) Joo x-X y-1 e-y K (x) dx =77a-X sec(v77) P x<-coscj>) K (a) o v v- v ( 2 2 X y = x + a -2ax cos¢) , larg al + IRe ¢1 < 77, IRe vi< X (33) Joo x-312 (l + a2/x)-112 exp [-(,8 + x)(l + a2/x) 112] K (x) dx 0 v = 4 a-1 K .,(,B) K2.,(2a,BX) Rea> 0, Re (a ,B)> 0 (34) foo x sin( 2ax)K0(x)dx = ~~77a J,(ay,) K1(ay,) a>O 0 19.5 BESSEL FUNCTIONS 369 Modified functions of x (cont'd) (35) J ""x cos( 2ax )J(0 (x) dx = -~rr a Y1 (a~) K 1 (a~) a>O 0 00 2 (36) J x-113 ex sin(4ax112) K113(x) dx = (~rr)112 a113 ea K (a 2) 0 1/3 a>O 2 (37) Joox-113e-xsin(4ax112)K (x)dx=2-112rr312a113e-a] (a2) 0 1~ 1~ (38) Joo x -vex sin (4ax ~) K (x) dx = (2312 a) v-1 rr f'(3/2-2v) . v [' (l/2 + v) 0 2 x ea lfl3v/2-112 , 1/2-v/2 (2a2) a> 0, 0 <Rev<%' (39) Joo xP-312 e-x sin (4 ax 112) K)x) dx = 77112 a [' (p + v) [' (p _ 1/) 0 2p-2 f'(p + ~) x 2F2(p + v, p-v; 3/2, p + l/2;-2a2) Rep> IRe vi ] 000 x -~ ex cos (4ax ~) J( 0 (x) dx = (~ 7T) y, ea 2 K o (a 2) (40) a>O 2 (41) f= x-y, e-x cos(4ax y,)K(x)dx=2- 112rr312 e-a J 0(a2) 0 0 (42) J=x-v-~excos(4ax~)K (x)dx 0 v [' (~-2 v) 2 = 23v12-177av-1 ea If/ (2a2) f'(~+V ) 3V/2,-V/2 a> 0, -~~ < Re v < )-4 370 INTEGRAL TRANSFORMS 19.5 Modified functions of x (cont'd) (43) Joo x p-l e -x cos ( 4 ax~ ) K (x) dx = 11y, ['(p + v) f'(p-v) 0 v 2Pf'(p + Y7) X 2F2 (p + v, p -v ; ~2. p + Y7 ; -2a 2) Rep> \Rev\ (44) J 00 x"+2v-Y, exp [-(1 + a)x] L 2v(ax) K (x) dx 0 n V 11 ~ [' (n + v + ~'2) [' (n + 3 v + Y:!) = 2n+2v+~ n! 1(2v+ 1) x 2F1 (n + v + Y7, n + 3 v + Y7; 2 v + 1; -~~a) Rea> -2, Re ( n + v) >-~. Re (n + 3 v) > -Y7 (45) Joo x -v (x 2 -a 2) )( -y, v P v-y, ( 2 a 2 x-2 -1) K (x) dx a J-1. v ~ 2-v -y, -vII" ( ) W ( ) =77 a 1-'+~,v-y,a -J-1.-~.v- Y.a ne v < 3/2 (46) r x -• e<p ( ~ -x) E<f{ a X J K)x) dx 2x (2x) 0 = ~ 11512 sec (V7T) l[J (aW + [Y (a)]2! v v He a> 0, -Y7<Hev<~ (47) Joo J (x) K (x) xJ-1.-v+l dx = ~12l(/l-v + 1) 0 J-1. v He /l > -1, Re (/l-v) > - l (48) J 00 e -2= I 0 (x) K 0 (x) dx 0 = Y7K[(l-a2)y,] 0 <a< l = (2a)-1 K [(1-a -2)y,] l<a<oo 19.5 BESSEL FUNCTIONS 371 Modified functions of x (cont'd) (49) fo 00 x exp ~ x2 ) [I (x) +I (x.)] K (x) dx = aeaK (a) 2a v -v v v Rea> 0, -1<Rev<1 (50) Joo 1 0 xp-sir (2ax) KI-L(x) Kv(x) dx 2p-1 a I~ (p + 11 + v + 1) r (p + 1) 2 x 1 (p+ 11~v+ 1) 1 ~-~+ v+ 1) r( p -112-v+ 1) X F . (p+J1+V+1 p+J1-V+1 p-J1+V+1 p-J1-V+1 43 2 ' 2 ' 2 ' 2 ' ~.p+1,!!._+1;-a2) 2 2 2 \Rea\< 1, Rep> jRe11\ + \Re v\-1 (51) 2P-3 (p + J1 + v) I~ xp-1 cos (2ax) KI-L(x) K)x) dx = i(p) r 2 x r (P +; -v) r (P -: + v) 1 ~ -; -v) (p + J1 + v p + Jl-v p -J1 + v p -Jl-~· 1 p p + 1 ~ x 4F3 2 ' 2 ' 2 '--2-;2'2'-2-; -a \Rea\ < 1, Rep> jRe11\ +\Rev\ (52) [~ x' oos (;~ )Y, (x) K ,(x) dx" -a' K0 (o) a>O 372 INTEGRAL TRANSFORMS 19.6 19.6. Modified Bessel functions of other arguments (l) 1= cos E;~)K 2)xe i7T/4) K2)xe -i7T14) dx r ()<t' + v) I' (J<t' -v) W 1 (ae i7T12) W 1 (ae- i7T12) = 8ay,rr-x !4,v ~,v a> 0, -J<t'<Rev<XI (2) J= x-X I (x) K (x) K (2x) dx 0 v v J.L rO~+~J.L) r(J<t'-~J.L) r(J<t'+v+~J.L) r(J<t'+v-YzJ.L) = 4r(%+v+~J.L) ro~ + v-~J.L) \Re J.L\ < ~. 2Re v > \ReJ.L\- ~2 (3) J= [J0(ax) Y1(bx)+ 2rr-1 I0(ax)K1(bx)]dx =0 0 <a< b 0 (4) J= xP[y (ax)± 2rr -I K (ax)] [Y (bx) ± 2rr -I K (bx)] dx 0 J.L J.L v J.L See Dixon, A.L. and W.L. Ferrar, 1930: Quart.}. Math. Oxford Ser. 1, 122-145. (5) J~ sinh (ex) K 1 (ax) J0 (bx) dx, ~ = cosh (ex) K 0(ax) J0 (bx) dx See Watson, G.N ., 1928: }. London Math. Soc. 3, 22-27. (6) J: x Jo(ax) Io(fx) Ko(yx)dx = L(a2 +/32 + y2)2 -1f,2y2r X l! e y > \ [m a\ + \He /3\ (7) J= x J0(ax) I1 ((:x) K 1 (yx) dx = -1-l(a2 + (32 + y2) 0 2{3y x [(a 2 + {3 2 + y2)2 _ 4,132 y2]'!, _ ll Re y > \lm a\+ \He(!\ 19.6 13ESSEL FUNCTIONS .'373 Modified functions of other arguments (cont'd) <)fc-2 !L 811 -f.-JL-11 (8) J''" xt\-l J (ax) J (l3x) A: p(yx) dx "" a ' y 0 JL II [' (f1+]) ~~ (v+ l) (A+f1+V-p) (A+f1+v+p ) x r I' 2 ~ tA+f1+v-p X F4 2 ' A+f1+v+p a2 N2) ; f1+ l, v+l; -2,--2- ~ y y He(>..+ f1 + v) > IHepl, Ite y > IImal ~ l!n: f11 J"" 2 1T l -I (b2-a2 ) (9) x [J0 (ax) K 0 (bx )] dx =----sin b 2 +a 2 o !3 ab 4ab a, b > 0 ( 10) J"" J (ax) J (bx) K (ax) K (bx) x211+1 dx 0 v v v v 211-Ja2111(11:1) l(v+1-)l(3v2+,) = b 411+2 TTY, !(v + l) c l 3v+ l X 2F1 V+?:'~; a4 ) 2v+ l · 1---' b 4 0 <a< b, He v>-l/3 For other similar integrals see sec. 6.8. (ll) J ""x -11-ex P 211-(l + 2x/ a) K (x + a) dx 0 II JL = 1T -Y, 211--1 cos(f1rr)l '(f1+v+Y2)I'(fL-v+Y:) lr'~-y,+11(2a) ~ J..L' 2 lnrg al < rr, He f1 > I Be v + ~-21 374 INTEGRAL TRANSFOR~1S 19.6 Modified functions of other arguments (cont'd) (12) Joo ~ ~ (a-X~ x-f.L(x+a)- e-xpJL_~ ---I< (a+x)dx o v, a+x v = (~rr)~ a-Xi-'r'(J.L, 2a) a> 0, Re J.L < 1 (13) Joo xJL-1 (x + (3)-JL I (x + (3) I< (x) dx 0 J1. v See 1\~acRober t, T .~!., 1950: Functions of a cor.tplex variable. ~1ac- millan, P• 379. (14) Joo xJl-1 Jx-bj-JL K (jx-bj) K (x) dx 0 J1. v = rr-~ ( 2 b ) -JL 1 ( ~ -J.L) 1~ (f.L + v) l ' (fl -v) K ) b ) b > 0, Re f.L < ~. Re11>JBevj (1S) oo _1 _ . 77~ l'(J.L+ v)f'(J.L-v) J xJ1. (x + (3) J1. K (x + (3) K (x) dx = K ((3) 0 J1. v 2J1.(3J1. l~(J.L+~) v Jarg(3J <rr, He fl > jBe vj (16) Jooo x 1+2 v J2v-1 (2ax) I< 2v-1 (2ax) Jv(x2) dx = 17-112 2v-2 a2v-1 I< ( r. 2) v-Y. "'a Jars aj < ~477, Rev> 0 (17) J~ x 1-2v J2v+1 (2ax) K 2v+1 (~ax) J)x 2) dx ~ 2-v-3 -2v-1 ( ) [I (0 2) (" 2)] = rr a esc vrr v+~ -a -Lv+~ .:.a Jarg aj < ~ rr, Rev> -1 (18) J00x1-2vY2 + (2ax)K2 + (2ax)J(x2)dx=rr:;,2-v-Ja-2v-1ctn(vrr) 0 v 1 v 1 v x [~+ y,(2a2)-Lv+ Y.(2a2)+ 2rr -1 sec (vrr)F< v+~(2a2 )) Jarg aJ < ~ rr, -1<Hev<0 19.6 BESSEL FUNCTIONS 375 Modified functions of other arguments (cont'd) _ -x 2-v-t t-2vK (2 2) -1T a ·v+X a largal < ~rr, -1 <Rev< 1 a> 0, Re v > -1 b > 0, Re a > 0, Rep > IRe Ill -3/2 (22) ;,~~~:~:~:'::~~:~: :~: dcx a2 b 2 1 ~ v-m J 15 16 v v p+v p-p 1-v ------- ---m 2' 2' 2 ' 2 ' 2 m integer, b > 0, Rea> 0, Rep> IRe Ill-3/2 Re a>O, Re{3>0 (24) J 000 x2v-X K X -v(ax) K)f3x -1) dx = (2rr)X a-v-X {3v K [(2a{3)X e~77i] K [(2a{3)X e-~77i] 2V 2V Re a·> 0, He {3 > 0 (25) Re /3 > v 376 INTEGRAL TRANSFORMS 19.6 Modified functions of other arguments (cont'd) (26) ] 000 x-2[K)ax)]2 J0(bx-1)dx=-2TTb-1 K2v(2ay, by,) x [sin(VTT) J2)2ay, b 'l:z) +cos (vTT) Y2)2ay, b 'l:z)] b > 0, Rea> 0, -~<Rev< l-4 (27) fa XJ-1.+1 y-J-1.-2 J (x) I (y)dx = f'(ll:; v-ll:;p.) (ll:;a)J-1. o J-1. v ['(ll:;v+ ll:;p.+ 1) Jv(a) y = (a 2-x 2)'/,' Re v > Re p. > -1 (28) Ja x2v y-2v-1 J2 1 (2x) K2 (2y) dx = -~ ['(ll:;+p.-v)['( !t2-p.-v) o v- J-1. x a 2v-1 I sin [(p.-v)TT] J·2J-1.(2a) +cos [(p.-v)TT] Y 2)2a)! 2 2 '/, y =(a -x ) , 0 < Re v < ll:;-\Rep.\ (29) Ja x 1-3v Y 2 v-1 J -3)2x) Iv (y) Iv-1 (y) dx = f'(v'/,+ ~)v Jv(a) J_v(a) o 2TT a y=(a2-x2)'1:z, 0 <Rev< l/3 (30) Jax1-2vy2v-3/2 I_ (x)K (x)J (2y)dx o v v 2v-3/2 = -~ [' (ll:; -v) a v-1 Y (2 a) 2 2 '/, ~ < Re v < ?2 v y =(a -x ) , a 2A+2J-1.+2 (31) JaxJ,\_(2x)I ,\_(2x)J (2y)I (2y)dx= 0 J-1. J-1. 2r(A+1)r(p.+1)1(.\+p.+2) ~A+ p+ l . A+p+ 3 ') x1F4 2 ;..\+1,!1+1,..\+!1+1, 2 ;-a y = (a 2 -x 2) y,' Re ..\, Re 11 > -1 19.6 BESSEL FUNCTIONS 377 Modified functions of other arguments (cont'd) 2 f' (Yz + K + f1) f' (Yz -K + f1) t = ['( )]2 MK,(Yzaet)M_K,O~ae-) a I 2 f1 + 1 ,,_ .,_ Rev>IRefll, Rea>O y = [ (x + a) 2 + ,8 2] y,' c = (a 2 + ,8 2) y, Re ,8 > lim al, Rev> IRe fll = a-1 f'(Yz-K+f1)['(Yz-K-f1)1T'K (Yzae i7712) WK (Yzae-i7712) ,f..J.. ,j.l. . larg al < rr, Re K + IRe fll < Yz (36) Joo x-y,(a+x)-',1, e-%coshtK [xy,(a+x) y,]dx 0 v = Yzsec (Yzvrr) e '/,a cosh t K Y,v(~ ae t) Ky,v<~ ae -t) -] <Rev< l (37) Joo x-K-)~ (a+x)K-',1, exp(-(3x)K 2 [xy,(a+x)y,]dx 0 J-L =a-1e',l,a,Bf'(Yz-K+f1)f'(Yz-K-f!)WK (z,)WK (z2) ,j..L ,j..L largal < rr, Re ,8 > -1, ReK + IRefll < Yz z " z 2 = Yz a [,8 ± (,8 2 -l) X] 378 INTEGRAL THANSFOili\IS 19.6 Modified functions of other arguments (cont'd) (38) ioo 1 (a+f3x) "' ~ ~ ~ xp- K2)x-(a+ {3x) (ax+ {3) ] dx ax+ (3 = 2/\ v+p(a) K v-p({3) Rea> 0, Re {3 > 0 (39) s:7T cos[(J--v)x]I + (2acosx)dx= Yzrri (a)Jv(a) j.J. tl j.J. R e (J1 + v) > -1 (-1-0) ~7T ~ cos[(J1-v)x]KJ.i.+v(2acosx)dx= Yzrr csc[(Jl+ v)rr] x [l_)a) I_v(a)- Ija) Iv(a)] -1<Re(J1+v) <1 (41) J~7T 7T ' secxcos(2Kx)K2 (asecx)dx=-W K (a)W_K (a) 0 j.J. 2a ,j.J. ,j.J. Rea> 0 (42) f 0"" cosh (2J1x) K 2v(2a coshx) dx = YzK J.i.+)a) KJ.i._V(a) Rea> 0 (13) Joo sech x cosh (2Kx) 12 (a sech x) 0 j.J. r0'2 + K + Jl) J'(l"-K + Jl) M K (a) 1H -K (a) IHeKI- He/1 < ~2 = 2a [r(2, + 1)]2 ,J.1. ,J.l. (44) J 00 (sinh x)J.i-+1 (cosh x)-2J.i.-312 p-fl[cosh (2x)] I ~(a sech x) dx 0 tl j.J.-l 2j.J.-~ l'(Jl-v) r(Jl+ v+ l) = 771/2 aJ.i-+3/2 [r(Jl+ 1)]2 11v+1/2,J.i.(a)M_v-1/2)a) He 11> !1e v, He11>-Bev-l 19.7 BESSEL FUNCTIONS 379 Modified functions of other arguments (cont'd) (45) (a + (3 ex) v 2 2 , epx x K 2 [(a + f3 + 2af3 cosh x)~] dx ae +/3 v flea, Hef3>0 19.7. Bessel functions and modified Bessel functions of variable order ( l) Joo Jv_x(a) JJ.L+)a) dx = JJ.L+,_/2a) He(fl + v) > 1 -oo (2) Joo a -J.L-x b -v+x e cxi J (a) J (b) dx ~ ~+x v-x [ 2 cosC2c) J ~1J.L+~v 1 • = 2 -~ · 2 y,. exp[Xc(v-f.L)L] a e c• + b e ,c, x J l[2c s(Xc) (a2 e-Y,ci+ b2 eY,ci)]y, I J.L+V 0 2 -rr<c<rr =0 C~TT or c ::; -11 He(fl + v) > 1 (3) J_: JK +x(a) J';,-)a) J J.L+}a) Jv-x (a) dx l'(K+A+f.L+ v + 1) = l'(K+A+1) l'(A+f1+1) I'(f1+v+1) I'(v+K+l) 0;+.\.+it+v+] K+A+f1+V+1 K+A+f.L+V K+A+fl+!l +1; X F ' ' + 1, 2 4 5 ~ 2 2 K+A+f1+V+1,K+A+1, A+!(+1, /L+ V+ 1, V+ K + l;-tJ.a2) l~e (K + A+ 11 + v) >-1 For similar integrals see Vol. I, P• 50fr. and r• 123fT. 380 INTEGRAL TRANSFORMS 19.7 Variable order (cont'd) (4) J~ Jx (xz) J_)xz) cos (rrx) dx = ~(l-z 2)-X \z \ < 1 (5) J00[J (xz)J x(xz)cos(rrx)-l]x-2dx=- Yzrr2 0 % - (6) loo I' a>O sech (12rrx) J. (a) dx = 2 sin a 00 u (7) foo cschn ~rrx) J. (a) dx = -2i cos a a>O -00 u: (8) f~ e Xnx cos (bx) J ix (a) dx = -i exp (ia cosh b) a, b > 0 sinh (rrx) (9) J_oooo e-cxi[Jv-ix(a) Yv+ix(b) + Yv-ix(a) Jv+;/b)] dx =-2(h/k)2v J2v(hk) a, b > 0, creal h = (aeXc+ be-'/,c)Y, ' k = (ae -Y,c + be Xc)Y: ( 10) l00e-cxi[Jv .(a)J +(b)-}' . (a)}'+ (b)]dx oo -u: v u v-tx v tx = '2(h/k)2V }' 2)h/r) a, b > 0, creal h = (ae Xc + be -Y,c), k = (ae -Y,c +be Y,c)Y, ( 11) Joo e -cxi H (2). (a) 1/ (2). (b) dx = 2i(h/k)2v H (2l(hk) -oo v-tx v+tx ... 2v a, b > 0, c real, h = (aeY,c + be-Y,c)X, k = (ae -Y,c + be Xc)X (12) Joo secii(rrx) l[J. (a)f + [Y. (a)fl dx = -}'0(2a)-E (2a) 0 1% l% 0 a>O 19.7 BESSEL F'UNCTIOl\S 381 Variable order (cont'd) (13) oo 71 2 2 2(ab) y, J xe xtanli(rrx)H<.>(a)f/<l(b)dx=- exp[-ik(a+b)] o a a rr (a+ b) a, b > 0 ( 1Lt) Joo xe71x sinh (17x) 1(11 + ix) 1(11-ix) H <21(a) H <21(b) dx 0 u \% = i 211 Try, f'(~ + 11) (ab)11 (a+ b)-11 K 11(a +b) a, b > 0, He 11 > 0 (15) J 00 xe 71x sinh (rr x) cosh (rr x) [' ( v + ix) [' (v -ix) 11 <21(a) H <21(b) dx 0 IX IX i T7 312 2 l/ = (6 -a)-11 f-!<21(b -a) 0 <a< 6, 0 < Re 11 < l~ I" (1·2-v) 11 (l6) (11+ix) (xe71xsinb(rrx)1 - 2-r t~ix) !I <21(a) 11 <21(6) dx lX IX = i 17 2 2-11(ab )11 (a 2 + 6 2)-Y, 11 H <21 [(a 2 + 6 2)y,] l/ a, b > 0, He 11 > 0 07) J:oo xe 71x tanh (rrx) P y, +. (-cos¢) H <21(a) If <21(6) dx 0 -. I% lX I.X 2(ab)y, -iR =---- e rrR a, b > 0, O<cp<rr, N = (a 2 + b 2 -'2 ab cos 9) Y. (lfl) ;;: xe71 % sinh (17x) r(ll + ix) r(ll-ix) !,:~~~%(-cos ¢) X 11 (21(a) i! (21(6) dx = i ( 2 17 )y, (sin 0) v-Y, (ab) l/ R -v // ~21 un l:C t:C ' a, b > 0, 0 < ¢ < rr, R = (a2 + b2 -2a6 cos~>)\ :'.e v > 0 382 INTEGRAL TRANSFORMS 19.7 Variable a-der (cont'd) ( 19) Joo cosh (Yz rrx) K . (a) dx = Yz 1T a>O 0 1.% (20) Joo x sinh(Yzrrx) K. (a) dx = ~!:! rra a>O 0 1X (21) Joo K . + . (a) K . + . ({3) dx = 1T K . . (a+ {J) -00 tx ty 1% 1% ty-1% \arga\ + \arg/3\ < rr (22) Joo e -rrx K . + . (a) K . + . (b) dx = rr e--rr z K . . (a -b.) a>b>O -00 tx ty u: tz ty-t% (23) oo . ~a+ {3 e P )v ..[ e tfJx K + . (a) K _ . ({3) dx = 1T K 2 (w) 00 v •x v u aeP + {') v \arga\ + \arg/3\ +\Imp\ < rr, 2 2 y, w =(a + {3 + 2a{3 coshp)' (24) J""exp[(rr-y)x]K. +· (a)K. +. (b)dx=rre-{3y-azK .. (c) -00 u ry u u ry-u where 0 < y < rr, a, b, c > 0, and a, {3, y are angles of the triangle with the sides a, b, c. (25) J_:(n + v + ix)-1 sin [(v + ix)rr] Kv+ix(a) Kv_ix(b) dx = rr2 I (a) K +2 (b) n n v 0 <a< b = rr2 K n+2)a) Jn(b) 0 < b <a n = 0, l, 2, •.• (26) Joo sin (bx) sinh (rrx) [K . (a)Jl dx = ~ 1T 2 Jo [2a sinh e'~ b)] 0 1.% a, b > 0 19.8 BESSEL FUJ\'CTIONS 383 Variable order (cont'd) (27) J: cos (bx} cosh (rrx) [K ix (a}]2 dx =-~ rr2 Y 0 [2a sinh (~2 b)] a, b > 0 (28) Joo cosh(px)K + (a)K . (a)dx= ~'~rrK2 [2a cos(~p)l 0 V tx v-tx v 2Jargaj + jHe pj < rr (29) oo X J (v-~ + ix )[' (~-ix )[' (2v-~ + ix) P11+7:-l (cos¢) -oo xi x+· (a)K x+· (b}dx=(2rr) X(sin¢)11-~(ab/w)11K (w) v-2 tx v-u v w = (a 2 + b 2 + 2ab cos¢) :.:; See also Chapter XII for similar integrals. 19.8. Functions related to Bessel functions oo sin (vrr) (l) J x J (ax) [J (x) -J (x)] dx = a> 0, He v > -1 o 11 11 11 1T a (a + l) ')-11-1 (2) Joo I ~ 1T Hev>-3/2 x -11-H (x) dx = o 11 ['(v+ l) J: sin [a(x +A.)] (3) x-11-1 H 11(x)dx= rr.\_11_1 H)A.) X+ A a?. l, Rev> 5/2 (4) (2v-l)2_J1._1I · Joo x I-Jl.-11 J (x} H (x) dx = o 11 J1. (f.l+v-l)f'(f.l+ Y2)f'(v+~ ) He v > ~. Re(f.l+ v)> l 384 INTEGRAL TRANSFORMS 19.8 Related fWJctions (cont'd) 00 ~ (5) J [cos(~V7T)J (x)+ sin(~v77)H (x)] 2 2 o v v x +a 17· [I)a)-L)a)] =- Rea> 0, -~2 < Re v < 2 2a (6) Joo XX (x2-a2)-X- Xv pv+X(2x2 a-2-l) ru (x)-y (x)] dx a J.J- 11 v = Tv-2 77X a csc(/.1.17) cos(v77) I[Y (72a)F-[J O~a)]2 ! . v v -1 < Re f1 < 0, l1e v < ~~ (7) Joo xp-l KJ.L(ax) K)ax) HA_(x) dx 0 See Mohan, 11., 19-12: Rull. Calcutta Math. Soc. 34, 55-59. For other integrals involving Cessel functions and Struve functions see r,Jc Lachlan, N .V: • and A.L. ~.!eyers , l~Xl6 : P hilos. ~lag . 21, 425-4·18. 2-rv 17Y, r (fl.+ v) C·l) jx_J.L_VH (x)H (x)dx= 0 . J1. v 1~ ( 11 + ~·2) l' ( 1/ + ~:) r. (fL + 1/ + ~ 2) l{c(/1+ v)>O (9) Jy,71 cos l(v + l)x] H (a cosx) dx = 77'1, 0 v -:;, . (I ) T (I ) a · s1n /2a vv+Y. :~a lte v>-2 (lO) f X11 cos(2vx) [H2)a secx)- Y 2)a sec x)J dx COS X 0 X _, rr a W (a e i77 /2) jf/ (a e-irr/2) = He v < ~ I'(2v + ~~) v,v v,v 19.8 BESSEL FUNCTIONS 385 Related functions (coot 'd) (ll) J"" exp[(v + 1)x] H (a sinhx) dx = 11y, a-Y, csc(v77) 0 v x [sinh( ~ a) Iv+Y, (~a)-cosh( ~ a) I_11_y, (~a)] Rea> 0, -2<Rev<0 (12) j 0"" x-1 cos (2ax-1) [I0(x)-L0(x)] dx = 2 J 0(2ay,) K 0(2ay,) a>O (13) J"" xv-Y, exp [-(1 + a)x] K0(ax) L 11(x) dx 0 = 11y, [f' (v + ~)F P _, (1 +a ) (2a)11+Xf'(v+1) v-Y, Rea> 0, Rev >-~ (14) j 0° x P n(l-2x2 a-2) [J0(x)-L0(x')]dx = (-l)n a [I2n+l (a)-L2n+l (a)] n = 0, 1, 2, ... (15) J""xY,(x2-a2)-~-Xvpv+ Y,(2x2a-2-1)[I (x)-L (x)]dx a If -v v = 2-11-1 11y, a csc(2J.m) cos(V7T) I[I)~a)]Z- [I_ 11(~a)Fl -1 < Re 11 < 0, Rev<~ i Y.1r cos(2vx) ( 16) · [I2)a secx)- L2)a secx)] dx COS X . 11·y, a_, = W v,v(a) M -v,v(a) Rev<~ ['(2v + l) ['(1 +.\.+f.L) ['( 1 -,\_-If) ['(I +f.L+V) ['(1 +u-11) (17) f"" x,\-1 s (x) dx 2 2 2 2 0 JL,V 22-A.-!1-['(1-A.+v) ['(1-f.--v) 2 2 -Re 11 < Re ,\ + 1 < 5/2 386 INTEGRAL TRANSFORMS 19.8 Related functions (cont'd) (18) Joo x-J.L-I cos (ax) s (x) dx 0 J.L,V -~ ~ (p.+ v+ 1) = 2J.L TT l 2 a> 1 [' ~-;+ 1~(1-az) ~J.L+~ P~~;,~(a) =0 =22J.L+2v-2 TT-~r(1+~+v) rC+~-v) X['~+ 3;+ l) r(/1 +2v + 1) S_J.L-2v- ~.~-v(a) 0 <a< 1 a > 1 0 <a< 1 Rev> -1 largal <)4rr, Re(p.-v) >-3, Re(p.+ 3v)>-1 (21) JO~ 7T COS [(p. + 1)x] S J.L,V (a COS x) dx = 2J.I.-Z TTl (p.+ ;+ 1) [' cp.-;+ 1) x J~(t.L+v+t) (~a) J~(t.L-v+l) (~a) Re p. > -2 (22) Joo exp [(p. + 1)x] s, v (a sinh x) dx = 2J.L-Z rr esc (p.rr) [' (p) ['(a) 0 ,.., x [IP(~a) Ia(~a)-I_P(~a) I_a(~a)] 2p=p.+v+l, 2a=p.-v+1, a>O, -2<Rep.<0 19.8 BESSEL FUNCTIO NS 387 Related functions (cont'd) (23) Joo x -1-L sin (ax) S (x) dx = 2-1-L-Y, 1Ty, 1 ( ~~II) [' ( ~-~~) 1-- 1---0 J.L,V 2 2 x (a 2-1)YzJ.L-l{ P/::y,y, (a) a> 1, Re ~ < 1-IRe11l (24) J.a xY,(v-J.L-1)(a2-x2)l{(v-J.L-2)p Y,(~-v+2)(x /a)S (x)dx o v- ~ J.1. ,v j (~ + v + 3) (11-3 11 + 3 j = 2/-L-3/2 1T 1/2 a (v-J.L) 2 [' 4 lo 1 cos[~,(~- 11)77] X [J11(~a) Y -Y,(JJ.-v+1tja)-l)~ ·2a) J-Y,(J.L-v+1)(~ 2a)] He(~- 11) < 0, -1 < ll e (~ + 11) < 1, He(~- 3 11) < 1 (25) Joox Yz (x 2-a 2)-y, /3 P !3(x/a) S , (x) dx a v J..L,!I:l ay, ['(~ + +n ·<.B-~-v _ -;i-) S J.L-/3+1, v+Y,(rr) = 77 112 2 3/2 -!3+J.L 1 Clz _ 11) He (-< < 1, fle(p +II-(n < -~:<, fle(/1- 11-(!) < ~2 (26) J 00 ( 2 -2 '-'1. v r v( ') 2 a -2 -1) 5 ( ' d a X X Q , A_ -·X 1-L ,v X, X a I . ( v -~ -1 -A) r ( v -~ + 1 + A) . = S + 2( + 1 (a) :2 l' ( 1 -f-"') l' ( 1 -r +II) J.L-V 1, c He v < 1, n c <1.l -11 + A) <-1, n e <tl -II -A) < (J (27) Joo -v( 2 2)l{-Y,v v-Yz ( 2 -2 1)'\ () i X X -a p Y, -Y, 2 a X -• X ~X a J.1. V J-L,V 77y, 2J.L-vr(3v~t!:-1) !r' (aei7T/2)ff (ae-i7T/2) av+Yzr(1+~- 1-L) p,a p,a p = ~ (~ + 1 -II), a=v-~ It e (~ -11) < 0, He 11 < 3/2, He(311-fJ) >] 388 INTEGRAL TRANSFORMS 19.8 Related functions (cont'd) (28) J 0"" x 1-JL-v J)ax) S JL,-JL_2)x) dx = 1Tx a v-1 r (l -/1 -v) 2JL+2v r(v + Yz) X (a 2 _ l)X(JL+v-1) pJL+v-1 (a) JL+V a> l, Re v >-~·2, Re (p. + v) < l X7T rr~2 1L-3a21Lcsc(2vrr) (29) J cos(2p.x) s2 1 2 (a cos x)dx= o IJ.-• v I'(l-11-v) rO-p.+v) x [J !J.+vO ~ a) Y IJ.-v<~2 a)-J IJ.-)~0. a) Y !J.+v(Yz a)] Rep.>-2, -l<Hev<l J y, 7T cos (2p.x) 1T 221J.-1 JTI (ae i7TI 2) (30) S 2 2 (a sec x) dx = COS X !J., v a !J.,V 0 x W (ae -i7T/2) !J.,V \arga\ < rr, Re 11 < l (31) J (sinh x)y, cosh(vx) S X (a cosh x) dx 0 !J., r, (__!_-IJ. + v) r (...!..-g-v) 4 2 4 2 S +Y, (a) = 21J.+3/2 a 1/2 r (Y:; -p.) J..L 2,V \arga\ < rr, Re 11 + \f!.e v\ < ~ (32) ~"" x2v-1 U)w, x) dx = 2v-1 r(v)wv cos( ~:!w) Rev> 0 (33) J"" x2v-3 U (w, x) dx = 2v-2 ['(v-l) wv-1 sin (Xw) 0 v Rev> l (34) J""x1-vsine 0.ax)U (x,z)dx 0 v =0 a>l =Yzrr(l-a) Y,v-1z2-vJ 2[z(l-a) y,] v-O<a<l 19.8 BESSEL FUNCTIONS 389 Related functions (con t'd) (35) Joo x -v cos (Y2ax) U (x, z) dx 0 v =0 a > l = Y.rr(l- a)Y,v-Y, z 1-v Jv_1 [z (l-a)y,] O<a<l (36) ( x2 ) wv (w) Jooo xv-1 JY,v-1 2w U v(w, x) dx = 2(v-l) Jy, v 2 Tie v > l I: sin[a(x+z)] U)w, x)dx = 77 Uv(w, z) a> l (37) X+Z CHAPTER XX HYPERGEOMETRIC FUNCTIONS ~lost of the higher transcendental functions of Chapters XVI to XIX are special hypergeometric functions" In the present chapter we list tl1ose special hypeq :;eor.~etric functions not included in the former chap­ ters, and also 'generalized hypergeometric functions" The integrals listed in sections 20.4 and 20.5 are key formulas from which an enormous number of integrals involvin g special hypergeometric functioqs may be derived. For particular c·1.ses of the £-function and of the G-function see the Appendix" 1\'e do not list in this chapter integrals involving the general­ ized hypergeometric series F : since p q pFq (a 1, ... , a P; b1, ••• , bq; x) r <b I) •.• nb ) --,----'-:-------"-:- E (a I' • • • , a P ; b I' na ) ... ['(a ) I p ... ' ] ' (b I ) • • • l ' (b ) ['(a ) • • • I' (a ) I p p, I ( l 11, b I' •• • , b ) G -- q q+l, p X a 1, ••• , a P such integrals may be derived from tl.ose given in sections 20.4 and 20.5" Cecause of the great importance of integrals involving £-functions and G-functions, \\·e have repeated integrals given in the earlier chapters, and in some ca~;es have given more elaborate conditions of validity. Parabolic cylinder functions For t!te theory of these functions see JT "T .F'" voL II, Chapter VIII and the literature quoted there, also Buchholz, Herbert, 1953: Die konfluente hypergeometrische Funktion. Springer Verlag. 391 392 INTEGRAL TRANSFORMS = 2-Y,v-1 ex c-~) c21 c~ll + ~~~ 11y, 1(-v) p 4 12 2 0, Yz } Other expressions in terms of the C-function, and expressions for pro­ ducts of parabolic cylinder functions may be derived by means of the formulas given in the Appendix. Gauss' hypergeometric series For the theory of these series see H.T.F. vol. I, Chapter II and the literature quoted there, especially the monographs by Gou~sat, Kampe de Feriet, Klein, Snow (no¥. available in a second edition), ·and Chapter XIV of Whittaker and Watson. I' (c) F (a b· c·x)=------- 2 1 ' ' , ['(a) ['(b) I' (c) I' (a) I' (b) G ~~ (-2__11, c) x a, b The evaluation cf integrals involving Gauss' series is often facilitated by the use of the transformation formulas ': for these see JLT .F •• vol. I, sections 2.9 and 2.11. Confluent hypergeometric functions For the theory of these functions see P.T .F. vol. I, Chapter VI and t!Je literature quoted there, especially Chapter XVI of \Yhittaker and V.atson, and also Tricomi, F .G., 1952: Lezioni s ulle funz ioni ipergeometriche confluenti. Torino, Gl:eroni and Duchholz, !lerbert, 1953: Die konfluente hypergeometrische F unktion. Springer Verlag. IIYPERGEOMETHIC FUNCTIONS M (z)=ziL+y,e-Xz F(~2-K+~~"2"+1·z) K,J.L 1 1 '' r ' +Y, y, = z JL • e ,z 1F, (~ + K + J1; 2 J1 + 1; -z) 1(2J1+1) z JL+X e Xz E (~'~ + .K + J1 : 2 J1 + 1 : z -1) r <~ + K + 11) 1' (2J1 + 1) ['(~ + K + f.l) I/. 11 ( I 1 -K ) e /2Z C z 12 ~ + f.l, ~~ -J1 . e-Xz 21(11+K ) = c z I' <12 -K + f.l) r <~ -K-11) 12 ~2 + p., ~·~ -f1 = e Xz C 20 (z I 1 -K ) 12 \ ~ + f.l, ~-f.l 393 For other expressions in terms of the C-function, and for expressions of products of confluent hypergeornetric functions see the Appendix. MacRobert's £-function A brief introduction to this function is given in Il.T .F. vol. T, sections 5..2-5..2.2, and a more detailed presentation of its theory may be found in ~lacHoLert, T .1\1., 1950: Functions of a complex variable. ~~acmillan, Appendix V and Miscellaneous Examples III. See also the papers by ~inc!lober t listed on P• 246 (f. of ILT .F. vol. I, and further papers by Professor ~iacflober t and his p·upils in Proc. Glasgow Math. Ass. vol. I, 1953. 1 (IJ,b, ... ,b) E(a,. •.• ,ap:b,. ••• ,bq:x) =C:·+1,p z q a 1, ••• , ap Nun1erous higher transcendental functions and son~e of th"eir conobinations are special instances of the £-function: a selection of these is given in the Appendix. 394 INTEGRAL TRANSFOHMS Meijer's G-function For the theory of this function see H.T.F. vol. I, sections 5,3-5,6 and the papers by Meijer listed on P• 24 7 of I! .T ,F. voL I, and also further papers by Professor ~Jeijer in recent voluntes of Proc. Nederl. Akad. Wetensch. It has already been mentioned that a very large number of integrals involving special functions may be reduced to integrals in­ volving the G-function. Examples of this process, and the necessary forrr ~ulas, are given in Professor ~1eijer's papers: a selection of reduction formulas is also given in the Appendix. HYPERGEOMETRIC FUNCTIONS 20.1. Parabolic cylinder functions See also under confluent hypergeornetric functions, £-function, G-function. (l) J~ 1 ( 3x2 ) o xv-exp -4 D )x) dx = 2-Y,vf'(v) cos (~~vrr) Rev> 0 (2) J~ ( 3x2 ) 0 x v exp - - 4-D v-I (x) dx = 2-Y,v-l l~(v) sin(~ V17') Rev> -l (3) .r X 2 v-1 (a 2 -X 2),\-1 exp (X:) D -2A-2V(x) dx = ['(,\) f'(2v) ,\-1 2A.+2v-2 C.a2)D ( ) [' (2 ,\ + 2 v) 2 a exp 4 _2v a Re ,\ > 0, Rev> 0 (4) J"" xv(x- a)Y,J.L-Y,v-1 exp [-~ (x-a)2] D (x) dx a J.L = 2J.i.-v-2 aJ.L-1 [' ~~ v) Dv(a) Re(f.L-v) >O (5) J~(x-ia)-1 exp ~ x2)D (x)dx=-(2rr)y, (-i)nn! ~ 4 n x exp (-a:)D-n_1 (a) n = 0, l, 2, ... , Rea> 0 395 396 INTEGRAL TRANSFORMS 20.1 Parabolic cylinder functions (cont'd) (6) (7) ( 2) 00 v-1 2 2 -1 x J x (x + a ) X exp --D (x) dx 0 4 v = av-1 f'(v) exp (a2)D (a) \4 -v Rea> 0, Rev> 0 (8) Joo x 2p-1 sin (ax) exp ~ x 2 ) D 2 (x) dx 0 4 v = 2v-p-X "X 1(2p + 1) ( 1 3 a2) a 2F2 p+-, p+1; -, p-v+1;--· [' (p -v ~ 1) 2 2 2 Rep> -X (9) ~00 x2p-1 sin (ax) exp (x 42 ) D 2)x) dx -2p-v-2 G22 (a21 X-p, 1-p) 1(-2v) 23\2 -p -v, X, 0 a> 0, Rep >-~2> Re (p + v) <X Rev> 0 (ll) ( 2) oo 2 -1 X f 0 x P cos (ax) exp 4 D 2)x) dx 2p-v-2 (az~X-p 1-p) czz 2 ' =1(-2v) 23 -2--p-v,O,X a> 0, Rep> 0, Re (p + v) < }j 20.1 HYPERGEOMETRIC FUNCTIONS 397 Parabolic cylinder functions ( cont 'd) (12) fooo xv-Y, exp [-(x + a)2] Iv-Y, (2ax) D )2x) dx = ~77-Y, 1(v) av-Y, D_v(2a) Rea> 0, Rev> 0 (13) I~ xv-312 exp[-(x + a)2] Iv_312(2ax) Dv(2x)dx = ~ 77-1/2 1 (v) a v-3/2 D (2 ) -v a Re a> 0, Rev> l 1/2 (14) 00 y, 77 I [D (x}fdx=( ~77)'1(v+l)+ 312 0 v 2 1 (-v) x [ ~ (v : l) -~ (; + l)] J 00 D (x) D (x) dx = 77 2Y,(IL+v+1) [ 1(-!::..)1 1(~)- 1(-l!:.)~~J (15) 0 v J.L fl-V 2 2 2 2 W;i) Ioo J0 (xy) D )x) D v-1 (-x.) dx = y -1 [D v(y) D v-1 (y) 0 + ~D)y)DV_1(-y)+ Y~D)-y)DV_1(y)] y>O (17) fa 00 J0 (xy) D )x) D v-1 (x) dx l = -W )y) D v-1 (-y)-D )-y) D v-1 (y)] 2y (18) Ioo J0 (xy) D v(-x) D v-1 (x) dx = y -1 [~ D )y) D v-1 (-y) 0 + ~Dv(-y)Dv-1(y)-D)y)Dv_1(y)] (19) Iy,'TT (sinx)·-v (cos x)_J.L_2 D (a sinx) D (acos x) dx 0 v J.L = -(~77)y, (f1 + l)-1 D J.L+v+1 (a) Rev< l, Re f1 < -l 398 INTEGRAL TRANSFORMS 20.1 Parabolic cylinder functions (cont'd) 00 (20) Jo cosh(2J.Lx) exp[-(a sinhx)2] D2K(2a cosh x) dx = 2K-3/Z 77112 a -1 lTI (2 az) K,J-L Re a2 > 0 (21) J"" cosh (2J.Lx) exp [(a sinh x)2] D ZK(2a coshx) dx 0 1(J.L-K) 1(-J.L- K) W K+Y, (2 a2 ) 2K+5/2 a 1(-2K) ,,J-L largal <3rr/4, ReK+IReJ.LI <O (22) J"" cos (ax) D x-Y, ((3) D -x-Y, ({3) dx o 1 G rr )' ( $' ) -~ TT < a < ~~ TT --· -- exp - 2 cos a 2 sec a =0 a<-~rr or a > ~2 rr 20.2. Gauss' hypergeometric series See also under Legendre functions, E-function, and G-function. (l) _( xa-y (l-x )'l -.B-1 2F, (a, (1; y; x) dx 1(1+ ~12a)1(y)I'(a-y+ l)I'(y- ~a-{3) = r(l+a) l'(l+~ ~a-(3) 1(y-~a) Re a+ l > Re y > Re (3, Re(y- ~~a-(3) > 0 (2) J 1 xp-1 (l-x)f3-y-n 2F1 (-n, (3; y; x) dx 0 1(y) 1(p) 1((3-y + l) 1(y-p + n) = 1(y+n)1(y-p)1((3-y+ p+ l) n = 0, l, 2, ... , Rep> 0, Re ((3 -y) > n -l 20.2 HYPERGEOMETRIC FUNCTIONS 399 Gauss' series (cont'd) (3) J 01 xp-l (1-x).B-p-l 2F1(a, (3; y; x)dx r(y) [' (p) ['((3-p) ['(y-a-p) = 1((3) f'(y-a) f'(y-p) Rep> 0, Re((3- p) > 0, Re (y -a -p) > 0 I (4) fa x'Y-1(1-x)p-l 2F1(a,(3;y;x)dx f'(y) f'(p) f'(y + P-a-(3) = f'(y + p-a)[' (y + p-(3) Re y > 0, Rep> 0, Re (y + p -a-(3) > 0 (5) J I xp-1 (l-x)a--1 2F1 (a, (3 ; y; x) dx 0 f'(p) f'(a) 3F2(a, (3, p; y, p +a; 1) f'(p+a) Rep> 0, Rea> 0, Re (y + a-a-fi) > 0 (6) J 1 x'Y -I (1-x)p-l (1-zx)-a-2f'. (a, (3; y; x) dx 0 f'(y) f'(p) f'(y + p-a-(3) (1 -z )0" = ['(y+p-a) f'(y + p-(3) x 3F;_ G• a, y+p-a-(3; y+p-a, y+p-(3; _z -) z -1 Re y > 0, Rep> 0, Re (y + p-a-(3) > 0, larg(1- z)l < TT (7) J1 xp-l (1-x)a--l 2f'.(a, {3; y; xz)dx 0 [' (p) ['(a) 3F2 (a, (3, p ; y, a; z) = f'(p+a) Rep> 0, Rea> 0, I arg ( 1 -z) I < TT 400 INTEGRAL TRANSFORMS 20.2 Gauss' series (cont'd) r ( y) r (p) r( y + p -a -f3) -z = e r (y + p -a) r (y + p-{3) x 2F2(p, y+p-a-{3; y+p-a, y+p-{3; z) Rey>O, Rep>O, Re(y+p-a-,A)>O (9) J00x'Y-'(1+x)-a-2F,(a,{3;y;-x)dx 0 r (y) r (a-y + a) r ({3 -y + a) r(a)r(a+/3-y+a) Hey>O, Ile(a-y+a)>O, Re(f3-y+a)>O (lO) J x 'Y -t (x + z) -a-2F1 (a, {3; y; -x) dx 0 r(a-y +a) r({3-y +a) r(y) r(a+f3-y+a)r(a) x 2F1 (a-y +a, {3 -y + a; a+ f3 -y + a; 1 -z) Rey>O, lle(a-y+a) >O, He(/3-y+a)>O, \argz\<rr (ll) r' y-1(1 )S-y-t F( r· .. ) Jo X -X 2 I a, I~, y, XZ x 2F,[o-a,o-{3;o-y;(1-x)(]dx r(y)r(o-y) 2 = (1-() a-S 2F1 (a, {3; o; z + s-z () r Co) 0 < Re y <Reo, \arg(1- z)\ < rr, \arg(l- ()\ < rr (12) J' xy-t (1-x)E-t (1-xz)-S 2F,(a,{3;y;xz) 0 . [ (1-x)z J r(y)r(f) x 2F1 8,{3-y;f; dx= 2F1(a+o,{3;y+f; z) 1-xz r(y + t) · Re y > 0, Ref> 0, \arg(z -1)\ < rr 20.3 HYPERGEOMETRIC FUNCTIONS 401 Gauss' series (cont'd) ( 13) f"" -Ax F ( {3· ~· 2) d -A. a+f3-t S (A.) 0 e 2 1 a, , 2, -x x- t-a-{3,a-{3 ReA.> 0 ( 14) Jo"" xe -Ax 2F; (a, {3; 3/2; -x 2) dx = A.a+{3-2 S '-a-{3, a-f3(A.) Ref..> 0 (15) j 000 x 'Y -' (x + y) -a (x + z)-f3 e-x 2 F; [ x(x+y+z) J dx a, {3; y; (x+y)(x+ z) =ny)(xy)- x-J..LeXy+XzwK (y)WA (z) ,j.J. ,j..L 2 K = l -a + {3 -y, 2 A = l + a -{3 -y, 2J.t=a+f3-y Re y > 0, largy I < rr, larg z I < rr (16) J""xa+{3-2v-t(x+ 1)-vexzK)(x+ l)z] 2F;(a,{3;a+{3-2v;-x)dx 0 = rr-X cos(vrr)1(~-a+v )1(~-f3+v)1(y) ( )-X-Xy W ( ) x 2z Xy, X(/3-a) 2z Re(cz+ {3-2v)>O, Re (~-a+ v) > 0, Re (~-{3 + v) > 0 largzl < 3rr/2, y=a+f3-2v 20.3. Confluent hypergeometric functions See also under F -function, G-function. For special confluent hypergeometric functions see also sections 16.5, 16.6, 17.3, 20.1, and Chapter XIX. ( 1) fa /3-t(- )'Y-1 F( ·{3· )d =['({3)['(y) f3+y-t x a x 1 1 a, ,x x a 0 ['(f3+y) x ,F; (a;{3 +y; a) Re {3 > 0, Re y > 0 402 INTEGRAL TRANSFORMS 20.3 Confluent hypergeometric functions (cont'd) (2) fa !3-• ( -)s-• F ( . {3· ) F ( . o· -) d O X a X I I a, 1 X I I y, , a X· X ~~ ({3) 1(0) af3+S-1 1F1(a+ y; {3 + o; a) Re {3 > 0, Reo> 0 = r<f3+o) (3) f~ xf3-1 (l-x)cr-{3-1 1F1 (a; {3; Ax) 1F1 [a-a; a-{3; ll(l-x)] dx r ({3) ['(a-{3) e A 1F1 (a; a; ll-,\) 0 < Re {3 <Rea = r(a) (4) f00cos(ax) 1F,(v+ l; l;ix) 1F1(v+l; 1;-ix)dx 0 =-a-1 sin(v77)P1)2a-2-l) O<a<l =0 l<a<oo -l < Re v < 0 (5) fa -1( )K-1 X(a-z)M ( )d r(K)l ~(21l+ l) "X aK-X I ()~a) x a-x e x x= o K,Jl. r (K + ll + X) J1. Re K > 0, Rell>-X (6) Ja XK-I(a-x) A-1 eX(a-z)M (x)dx 0 K+A,Jl. r (>..) r<K + ll + X) K+"A-• = a M (a) r (K + A+ ll + ~) K ,Jl. Re (K + ll) > -~~. ReA> 0 (7) fa xJl.-X (a-x)v-X MK (x) M)\ (a-x) dx 0 .J.L ,11 1(2/l+ l) l'(2v+ l) a J1.+v M K+A,Jl.+v+X (a) = r (21l + 2 v + 2) Re ll >-~~. Rev>-% 20.3 HYPERGEOI\IETRIC FUNCTIONS 403 Confluent hypergeometcic functions (cont'd) (8} J"" xp-1 [x~ +(a+ x)~fa e -~x M (x) dx 0 K~ af'(2f.L+l)aa 23(1 ~,1,1-K +p ) =-X c34 a I rr f'C!:l + K + f.L) ~ + f.L + p, -a, a, ~-f.L + p largal<rr, Re(f.L+p) >-~, Re(K-p-a) >O (9) largal < rr, Re(p + f.L) > -~, Re(K-p-a)>-%; ( 10) J"" x-J..L-~ e-~x sin(2ax ~)M (x)dx 0 K,J..L ~ K+,-1 f'(3-2f.L) (a2~W (2) = TT a ,... exp -- a [' (~ + K + f.L) 2 P ,a a>O, Re(K+f.L) >O, 2p=K-3f.L+1, 2a = K + f.L -1 x eY,a1F (a) p,a p=~-K+f.L, a=~+v larg~l < TT, Re f.L >-~. Re (K-f.L) > IRe v + hi ( 12) Joo x-y, (a+ x)-J..Le-Y,x P-2J..L(1 + 2x/a) Af (x) dx o v K,J..L f' (2 f.L + 1) f' (K + f.L + V + ~) ['(K + Jl-V -~) . = ['(K + f.L + ~) f'(2f.L + v + 1) f'(2f.L-v) largai<TT, Ref.L >-~, Re(K+f.L)>IRev+~ ~~ 404 INTEGRAL TRANSFORMS 20.3 Confluent hypergeometric functions (cont'd) (13) j''"x-~e-~xp -21-L[(l+x/a )~]MK (x)dx 0 v ,J.L f' (2fL + l) f' (K + ~ V) f' (K -~ V -~) x exaw~ -K.~+Y,v(a) \arga\<rr, ReK>~Rev-~, ReK >-~~Rev =----~~~----~------------------2 f' (~2 + K + fL) \arga\ < rr, Re fL >-~2, He (2fl-v) > -l 2 p = ~ -K -fL + 2 v, 2 a = K -3 fL -3/2 (1 -K+u-v )71 i u-K-v Y, '"+u-1) [' (~-V) [' (l+ 2 p.) [' (K + fL + V) = e ,... 2,... a "'' ,... · [' (K + fL + ~) X e Y, aw (a) p,O' Rep. >-~~. He(K+ p. + v) > 0 \arga\<rr, p=~-K-~2v, a=fL+~~v (16) j 0"" xv-Y, e-Y,xQ22/:_:22:_3[(l+x/a)y,]MK ,J.L(x)dx = e 2 {J.L-v )71 i 22J.L-2v- 1 aY,(K+J.L-1) e y, a f'(2p. + l) f'(v + l) f'(K + p.-2v-!':l) X lf'lp,O'(a) [' (K + fl + ~) \arga\<rr, Refl>-~, Rev>-1, Re(K+p.- 2v)>~ 2 p = l -K + fl -2 V, 2 a = K -fL -2 V -2 20.3 HYPERGEOMETRIC FUNCTIONS 405 Confluent hypergeometricfunctions (coot'd) (17) J00 xK-:v2exp[-%(a+ l)x]K (%ax)M K (x)dx 0 v ,v 7T~l(K)l(K+2v) -t = K+ 2F1 (K, K + 2v; 2v + l; -a ) a v i(K + v + %) Rea> 0, Re K > 0, Re(K+ 2v)>O (18) Joox-~ J (ax~)K~ (%x)MK (x)dx o v v-!J. ,!J. <C2e+ 1) (•') (') =ar(K+%v+l) w~V<-!J.),XK-~V 2 M~(K+jJ.),~K+~v 2 a> 0, Re K >-~, Re 11 > -%, Rev> -1 (19) J: x p-t e -~ x M 'Y + P, (3+ p+ ~ (x) 2 F, (a, (3; y ; ->J x) d x 1(a+f3+2p)1(2(3+2p)1(y) ~f3+p-X xA. = A e W (A) r ({3) r ((3 + y + 2 P) K ,jJ. larg AI < rr, Re ({3 + p) > 0, Re(a+f3+2p) >O, Re y > 0 K = % -a -~~ {3 -p, 11=%{3+p (20) J 000 x2A.-t (a +x)-J.l.-Y, e-Xx MK,!J.(a + x)dx a/\.-11--X 1(2A)r(211+ l)i(K+f.l- 2A+ ~0 M K-A., !J.-1\. (a) = r (K + 11 + %) r· < 1 -2 A + 2 11) He A> 0, He(K+f.l-2A) >-~ (21) J a X -K-k t (a -X )A.-t e Y.x W K (x) dx 0 ,J.L r (A) r (% -K -A + 11) r, <% -K -A-11) = JTI (a) aK+t f'(%-K+11)l '(%-K-f.l) . K+/c,!J. iteA>O, n e (K + A) < ~2 -IH e Ill 406 INTEGRAL TRANSFORMS 20.3 Confluent hypergeometric functions (cont'd) 100 dx rr312 2K sec (f.17T) (22) lf'K J.L(x)- = • X I'(%-YzK + Yzf.l) 1(%'-YzK-Yzf.l) 0 -Yz < Re f.1 < }2 (23) Joo K+ZJ.L-1 _3x/2 I(K+f.1+ Yz)1(~(2K+6f.1+5)] x e If' (x) dx = ° K,J.L (K+3f.1+h)1(~(2f.1-2K+3)] Re(K + f.l) > -~!::, Re (K + 3 f.1) > -Yz (2t1) Joo xP-1 [xX +(a+ x)X]Zcr e -Xx W (x) dx 0 K~ _, a ~~I \!:, 1, 1-K+p ) = -rr oa G a 34 Yz+f.l+p, Yz-f.l+p, -o, o \arga\ <rr, Rep> \Re11\- Yz (25) J xp-1 [xX +(a+ x)X]Zcr e Xx W (x) dx 0 K~ o rr -x acr 33 ( I Yz, ~ _l+K+p ) =-c34 a I I i(Yz-K+f.l) l(Yz-K-f.l) Yz+f.l+p, Yz-f.l+p,-o,o Jarga\ < rr, Rep> \Re11\- Yz, Re(K+p+o) <O (26) JooxP-1(a+x)- X[xX +(a+x)XJZcreXx WK (x)dx 0 ~ -X CT ( I Yz Yz ) TT a 0, z, z+K+p = C33 a 1(Yz-K+f.1) i(Yz-K-f.l) 34 -o, p+f.1, p-f-1, a \arg a\ < rr, Rep > \Re 11\ -~~. Re(K+p +o) < Yz (27) Joo xp-1 (a+ x)-Y, [xy, +(a+ X·)Y,]Zcr e -Xx W (x) dx 0 K,J.L _ -Y, CT 32 ( IQ, ~2, Yz-K+p ) \arga\ < rr, Rep> \Re 11\- ~:! -TT a C34 a -o, p+f.l, p-f.l, a 20.3 HYPERGEOMETRIC FUNCTIONS 407 Confluent hypergeometric functions (cont'd) (28) J,oo xp-1 sin (cxy,) e -Y,x W K (x) dx = f'(1 + p+ p) f'(l-p+ p) ' G '" 3 3 r(3/2-c: )p) xcF 1+p+p 1-p+p·- --K+p·--2 2 ' '2'2 ' 4 Rep> IHepl-1 (29) r Xp-1 sin (cxY,) e Y,x W K (x) dx 0 .~ TTY, 22 ( ;;._fX+p-p, X-p-p) = G 23 4 X o f'(X-K+p) f'(X-K-p) • 2,-K-p, c > 0, Rep> IRepl-1, Re (K + p) < X (30) J00xp-1 cos(cxy,)e- y,""WK (x)dx=. f'(X+p+p) f'(X-p+p) 0 ~ r (1-K+ p) (1 1 1 c2 j X F -+p+p --p+p·- 1-K+p· --2 2 2 ' 2 '2' ' 4 Re p > IRe 111 -X y, (31) 00 y, y, 1T fo xp-1cos(cx )e ""WK (x)dx= .~ f'(X-K+p) f'(X-K-p) X G22 ( ~, X+p-p, X-p-p) 23 4 0, -K-p, X c > 0, Rep> IRepl- X, Re (K + p) <X (32) Joo -Y,-Y,~-v( )Y,~ -Y,xp~ (1 2 -1)W ( )d 0 x a+x e K+v-312 + xa K,vx x f'(1-p-2v) a-~ +Y, K-Y, v e y, a W (a) = ['(3/2-K -p-v) p,a- 2p =X+ 2p + 1/-K, 2a = K + .'3 v-3/2 largal < "• Rep< 1, Re (p + 2v) < 1 408 INTEGRAL TRANSFORMS 20.3 Confluent bypergeomeU:ic functions (cont'd) r(l-JL-2v) = 1(3/2- K-JL-v) 2 p = Ji2 -K + V, 2 a= K + 2 JL + 3 V -3/2 \arga\ < rr, ReJL < l, Re(JL + 2v) < l 21'T(l-JL-2v) = r < 3/ 2 -K -JL -v) a-Y,+Y,K-Y,veY.aw (a) p,a- 2 p = l -K + JL + v, 2 a = K + JL + 3 V -2 \arga\<rr, ReJL<~ Re(JL+2v)<l (35) Joo x-Y,-Y,,u-v (a+ x)-y, e-Y," P~K+,u+2.v_2 [(l + x/a)y,] W K,)x) dx 0 =2.Uf'(l-JL-2v) a-Y,+Y,K-Y,veY.aw (a) l (3/2-K-JL-V) p,a- 2p=JL+V-K, \arga\ < rr, 2a=K+JL+3v-l ReJL>O, Rev>O (36) J:oo x-K-312 exp[-Ji2(a-l)x]K (Ji2ax)WK (x)dx 0 .u • .u 1T f'(-K)f'(2JL-K) f'(-2JL-K) = f'(Ji2-K) f'(}~+JL-K) f'(}:/-JL-K) 22K+1 K-v . a X 2F1 (-K, 2 JL-K ; -2 K ; l -a-1 ) Rea>O, ReK<2ReJL<-ReK 20.3 HYPERGEOMETRIC FUNCTIONS 409 Confluent hypergeometric functions (cont'd) (37) J""xp-1 e-~xJA.+ (ax~)JA_- (ax~)WK (x)dx 0 v v ,J.L (~a)2A.r(~+A+Jap) r(~+A-JL+p) = r(1+A+v) r(l+.\-v) r(1+A-K+p) X 4~(1+.\, ~+.\, ~+A+JL+p, ~+A-JL+p; 1+.\+v, 1+.\-v, 1+2.\, 1+.\-K+p; -a2) \ReJLI < Re(.\+ p) +~ (38) J"" xp-1 e-~.x IA_+ (ax~) KA_-(ax~) W K (x) dx 0 v v ,j.L 1 M ( 'I 0, X, l<i+ " -p, ~ -"-p) ---G a .\, -2rr~ 45 v, -.\, -v, K-p \ReJL\ < Re(.\+ p)+ ~. \ReJL\ < Re(v+ p)+ ~ (39) J"" 1 r (2 JL + 1) x -M (x) W A_ (x) dx = 0 K ,J.L ,J.L (K -A) r (~ + Jl -.\) ReJL > -~, Re (K-.\) > 0 (40) 00 1 1 [ r(~-K+Jl~ r(~-A-JL) J x-W (x) W A_ (x) dx = o K,J.L ,J.L (K-A) sin(2JLrr) -r<~-K-JL~ r<~-A+JL)] -~ < Re JL < ~ (41) J"" 1 r(p+1)r(~+JL)r(~-JL) x p-W (x) W (x) dx = 0 K,J.L -K,J.L 2r(l + ~p + K) r(l + ~p-K) Rep> 2\ReJL\-1 410 (42) INTEGRAL TRANSFORMS 20.3 Confluent hypergeometric functions (cont'd) Joo xp-1 ll"K (x) If, (x) dx O ,J..L 1\.,V l'(l+ !J.+ v+ p) 1'(1-JH v+ p) 1(-2v) x3F2(1+ !J.+ v+ p, 1-JH v+ p, l/2-A+ v; 1+ 2v, 3/2-K+ v+ p; l) f'(1+!J.-V+p) f'(1-!J.-V +p) 1'(2v) + 1(1/2-A+v) 1'(3/2 -K-v+p) X 3F/1+!J.-V+p, 1-!J.-V+p, l/2-A-v; I -2v, 3/2-K-v+p; l) !Hell!+ !Rev!< Rep+ 1 (43) Jooxp-1exp[-~ ·Ha+f3)x]M K tax)IT-\ ({3x)dx 0 ,j..L 1\.,V = f'(1+!J.+ V+ p) f'(l+!J.-V+p) aJ.L+Y: {TJ.L-p-Y, 1 (3/2-A+ !J.+ p) X 3F2(1/2+K+!J., 1+!J.+V+p, 1+!J.-V+p; 2!J.+1, 3/2-A+!J.+p;-a /{3) He a> 0, Re f3 > 0, He (p + !J.) > !He vi -1 (44) J''" xp-1 exp O·:l(a + rm !f'K (ax) f! \ v((-3x) dx 0 ~ A, !Hell!.+ !Rev!< Rep+ 1, Re(K +A+ p) <0 20.3 HYPERGEOMETRIC FUNCTIONS 411 Confluent hypergeometric functions (cont'd) (45) J00xp-1exp[-X(a-/3)x]WK (ax)W, ({3x)dx 0 'J..L f\.,11 (3 -p 23 ~(3~X + f1, X -fl• l + A.+ p) = I G33 -f'(X-A.+v)f'(X-A.-v) a X+v+p,~ 2-v+p,K Re a> 0, IRe fll + !He vi < Re p + l (46) Jooxp-1 exp[-J-Ha+f3)x]WK (ax)!fi, ({Jx)dx 0 ,j...L f\., 11 = -p 22 (£ I X+ fl, X-v, l -A+ p) (3 G 33 I/ I a /2+v+p,Y2-v+p ,K Re (a+ {3) > 0, IRe fll + IRe vi < Re p + l (47) Jooxzf\-1 (a+x)-J.l.-Y, e-Y,x!T'K (a+x)dx 0 ,j.l. largal < rr, ReA.> 0 (48) Joo xf\-1 (a+x)K-f\-1 e-Y,x WK (a+x)dx 0 ,j.l. = ['(A.) aK-1 W K-'A.J.l. (a) largal < rr, ReA.> 0 (49) Joo xp-1 (a+ x)-a e -Y,x WK (a+ x) dx 0 ,j.l. y, ( I 0, l-K-a ) = l' (p) aP e 'a G 30 a 23 _ p, X + f1 _a, ~l:i-f1-a largal < rr, Rep> 0 _ I'(2A.) f'(X-K + 11-2A.) a/1.-J.l.-Y, W (a) ['(X -K + f1) K+f\,j.l.-1\_ largal<rr, 0<2ReA.<J2-Re(K+f1) 412 INTEGRAL TRANSFORMS 20.3 Confluent hyper geometric functions (cont 'd) (52) J"" xr-1 (a+ x)-A. 2F1 (p, a; r; -x/a) e ±\Sx WK (a+ x) dx 0 . ~ (53) (54) (55) Express WK in terms of G and then see under ~]eijer's C-function. ,J-L Integrals involving products of e ±Y,x WK (a + x) with Legendre ,J-L functions. Express WK in tem1s of G, and the Legendre function ,J-L as a hypergeometric series, then see under r.1eijer's C-function. Rea>O, He,B>CJ = f3P G~~ (~I~+ ll• ~~--:.\~-+A: ~p, ~-v-P) Ilea>O, Re/3>0 J"" xp-1 0 exp[~(x-~)] W . (x) w, (13 )d 2 \a X K ,Jl ,-a f'c,V X X \arg a\ < 377/2, He {3 > 0, He (K + p) <-/Rev/ -~~ 20.3 HYPER GEO'IIETR IC FUNCTIONS 413 Confluent hypergeometric functions (cont'd) 1TY, l (-K-fl) l (-K + fl) (a/3)'.4 = 2Y,+2Kf'(l; )T"'(I; ) /2 -K + f1 1 /2 -K -f1 ~cfl)y,J . [~(f3 )y,J X exr L, ~ !(! 2K+Y,,z,u -" \--;:; larg a! < 3rr/2, larg f3l < 3rr/2, He K <-!Be 111 (57) exp [~ (:_ + f}_) J If (__:_) If' A ((3) dx 2 a X K,,u a ,v X f3P =----~------ -------------- ---------- [' (~-K+ f1) [' (~-K-f1) l~ (~1-A + v) [' (~2-A-v) ~c{31 l+K,l+A-p ) xG4 -24 I I I I/ a /~ + fl• ~ -fl, ~ + v-p, 12 -v -p iargal < 3rr/2, iarg/31 < 3rr/2 He(A-p) < ~z-!Hefli, He(K+ p)< ~2-IIlevl (58) l00 e-zpxi['( ~+v+ix)I'( ~+v-ix)M. (2a)dx 00 u,v = 2Y,-v rr{3v+Y, (coshp)-2v-l exp(-a tanhp) l'(2v + l) lim pI < ~~ rr, He v > -~' (59) jioo f'Ciz+v+f1+X)f'(1 ~+V+f1-x)r( ~+V-f1+X)f'( ~+V-f1-X) -ioc X M,u+ix, v(a) M ,u-ix ,)f3) dx 2rr(a{3)v+ Y, [l~(2v+ l)Jli' (2v+ 2J1+l) I'(2v-2f1 +l) (a+ (-3)2v+1l'(4v+ 2) x .112 +'1 (a+ {3) j..L, 2V 12 414 (60) (61) (62) ( 1) (2) INTEGRAL TRANSFORMS 20.3 Confluent hypergeometric functions (cont'd) J00e-2P"if'(X+v+ix)f'(X+v-ix)M. (a)M. ((3)dx -00 lX , 11 l% 1 11 2 rr(a{3) y, (? ay, (3y, ~ = exp[-(a+ (3) tanhp] J2., - cosh p cosh p JimpJ < Y:;rr, Rev >-~2 00 (a{3)y, J_oo sech (rrx) W ix, 0 (a) W _ ix, 0 ((3) dx = 2 --exp [-~ -~(a + {3)] a+ {3 _r_: f'(ix) f'(2K + ix) W K+ix,K-Y, (a) JTi_K-ix,K-)(.({3) dx = 2rry, 1(2K) (a{3)K (a+ {3)Y.-2K K2K-Y, (a: {3) For numerous integrals with respect to parameters see 13uchholz, Herbert, 1953: Die konfluente hypergeometrische Funktion. Springer Verlag. Chapter VI. 20.4. MacRobert's £-function See also under G -function. fo1 13-1 ( 1 yr 13-1 E ( ) X -X a I , , , , , a p ; p I , , , , , p q ; XZ dx See ~lacRobert, T.M., 1953: ?roc. Glasgow Math. Assoc. { j3-l ( )'Y -/3-1 £ ( -m ) d 0 x 1-x al' ... ,ap:p1' ... ,pq:x z x = 1 (y -(3) m 13-y E (a~' ... , a P +m: pI' ... , p q +m: z) (3+k-1 y+k-1 lc = 1, a p+k-' p p+k = ' m m l, 118. ... 'm Re y > Re (3 > 0, m = 1, 2, ... , 20.4 HYPERGEOMETRIC FUNCTIONS 415 MacRobert's £-function (cont'd) = r (p) E (a,. ... ,a P, a-p: pI' ••• , p 9, a: z) Rea> Rep> 0 ( ) Joo j3-1 -x £ ( ) d 4 0 x e a, ... , ap: p,. •.• ,p9:xz x = rr csc({3rr) [E(a1 , ••• ,ap: 1-{3, p1, ••• ,p 9: e±i7T z) -z -j3 E (a 1 + {3, ••• , a P + {3: l + {3, p 1 + {3, ••• , p 9 + (3: e ± i7T z)] p?_q+l, He(ar+{3) >0, r=l, ••• ,p, largzl<rr For p::; q the result holds if the integral is convergent. (5) Joo j3-1 -x £ ( -., ) d x e ~ a1, ••• , a : p,. ••• , p : x z x 0 p q (2 )X -Xm !3-X E ( -.. ) = 1T m a1, ••• ,ap+m:p,. .•• ,p 9:m z He(3>0, m=l,2, •.. , ap+k=(f3+k-l)/m, k=l, •.• ,m See MacRobert, T.M., 1953: Proc. Clas{JJW Math. Assoc. 1, 111-114. (7) Joo xj3-l J (x) E (a1 , ••• , a : p1 , ••• ,p : x-2" z) dx 0 v p q = (2rr)-" (2m )f3-l I exp[~ rr (,8-v-l)i] E [ . (2 )-2., -lfi7T i] x a1, ••• ,ap+Zm:p1, ••• ,p 9: m ze + exp [-~ rr({3-J-·-l)i] E[a, •.• ,ap+Zm: p,. ... ,p 9: (2m)-2" ze" 7Ti]l He({3 + v) > 0, Re(2arm- ,8) > -3/2, r = l, .•. , p {3 + v + 2k-2 {3-v+2k-2 a -p+m+k-2m ap+k = 2m m = l, 2, ... , k = l, ... , m 416 INTEGRAL TRANSFOR l\'JS 20.4 MacRobert's £-function (cont'd) = (2rr) 1-m 2{3-z {3-t t.' [ (2 )-2m J m c a, ••• ,ap+Zm:p1, ••• ,pq: m z f3+v+2k-2 (3-v + 2k-2 '2m , ap+m+k = -------'2m Bef3>1Hevl, m=l,2, •.• , k=l, ••• ,m (9) J""xf3-t e"'K (x)H(a1, ••• ,a :p1, ••• ,p :z/x)dx 0 ll p q See Ragab, F .M., 1953: Proc. Glasgow Math. Assoc. 1, 192-195. (10) J""xf3-te-y,"'WK (x)E(a,. ••• ,a :p,. ... ,p :x-mz)dx 0 ,J.L p q = (2rr) ~ -~m f3+K-~ v ( -m ) m r. a 1, ••• , a P + Zm : p 1 , ••• , p q + m : m z He {3 > IRe 111·- ~2, rn = 1, 2, ... ap+k = ({3 + k + 11- ~1:)/m, ap+m +k = ((3-f1 + k-~)/m Pq+k=(fJ-K+k)/m, k=l, ••• ,m (ll) f000 x,\_·-t E (a1, ••• , ap: p 1 , ••• ,p q: xy) E ({31, ••• , /3r: a1, ••• , a5: xz) dx J~ x,\_-t F.; (a 1 , ••• , a P : p 1 , ••• , p q: xy) E ({3 1 , ••• , {3 r= a1 , ••• , a 5: z/x )dx See Hagab, F'.f\~., 1953: /'roc. r;[asgow Math. Assoc. 1, 192-195. 20.5 ( l) (2) HYPERGEOMETRIC FUNCTIONS 20.5. Meijer's G-function Jo 1 x p-1 ( l -x )a--1 G ;; ( x I a 1' ••• ' a P ) dx \ b 1' ••• 'b q =f'(a)Gm,n+1 (all-p,a1, ••• ,ap) p+1,q+1 b b l p•••• q' -p-a First set of conditions of validity: p + q < 2 (m + n ), larg al < (m + n -~ p -~ q) 77 Re(p+b .)>O, j=l, ... ,m, Rea>O } Second set of conditions of validity: p + q:::; 2(m + n), largal:::;(m + n-~p-~q)rr Re (p + b.) > 0, j = l, ... , m, Re a > 0 p } q Re [ :£ a . -:£ b. + (p -q )(p -Yz)] > -~;f j= 1 } j= 1 } Third set of conditions of validity: p<q (orp::;q and lal<l) Re (p + b.)> 0, j = l, ... , m, Re a> 0 } First set of co.nditions of validity: p + q < 2 (m + n ), I arg a I < (m + n -Yz p -~ q) 11 Re (p-a-a ) > -1, j = l, ... , n, Re a> 0 } Second set of conditions of validity: p+q.S2(m+n), largal:::;(m+n-~p-~g)rr Re (p -a -a . ) > -l, j = l, ... , n, Re a > 0 ] p q Re[ :£a.-:£ b +(q-p)(p-a+~)] >-Yz j=1' j=1' 417 418 (2) (3) (4) INTEGRAL TRANSFORMS Meijer's G-function (cont'd) J,oo x-p (x _ 1)o--l c;; (xj a,····· ap) dx \ b, ' ••• 'b q =r(a)G'"+l,n (al a,, ... ,aP,p) p+l,q+l b b p-a, 1' ••• ' q Third set of conditions of validity: q < p (or q ~ p and I a I > 1) Re(p-a-a .)>-1, j=1, ... ,n, Rea>O J • n II r (b + p) II r (l -a -p) i= 1 J j= 1 J =-------------------------- q p II r(l-b .-p) II r<a +p) j=a+1 J j=n+l J 20.5 p + q < 2 (m + n ), largal < (m + n-~p- ~q)rr -min Reb. < Re p < 1-max Re a . l~j~m J l:SJ~n J ~oo xP-1(x+fJ)-a-c;:: r:xla,, ···' aP)dx \ b, ... ,bq fJ p-a- + 1 + 1 ~ 11 -p, a 1 ' ••. ' a ) =-- G"' ·" a~ P r( ) p+l,q+l t-' b b a a-p, 1, ••• , q First set of conditions of validity: p + q < 2(m + n), largal < (m + n-~p- ~q)rr, largfJI < rr Re(p +b.)> 0, j = 1, .•• , m, Re (p-a+ a.)< 1, j = 1, ... , n J J 20.5 (4) (5) HYPERGEOMETRIC FUNCTIONS 419 Meijer's G-function (cont'd) J:oo Xp-1(x + {3)-a c;qn (.X ,a,''"' ap) dx \ b,, ... ,bq (3p-a +1 +1 ( ~1-p, a,, ... ' a) =--G"' ,n a(3 P r( ) p+1,q+1 b b a a-p, 1, ... , 9 Second set of conditions of validity: p ~ q, p + q ~ 2(m + n), \arga\.S (m+n-~p-~q)77, \arg{3\ < 77 Re(p +b.)> 0, j = 1, ... , m, Re(p-a+ a.)< 1, j = 1, ... , n 1 1 p q Re[ ~a .-~ b.-(q-p)(p-a-~)] >1 j=1 1 j=1 1 Third set of conditions of validity: p ~ q, p + q .S 2(m + n), \arga\.S (m+n- ~p-~q)77, \arg{3\ < 77 Re (p + b ) > 0, j = 1, ... , m, Re (p-a+ a .) < 1, j = 1, ... , n 1 1 p q Re[ ~a .-~ b.+(p-q)(p-~)] >1 j=11 j=11 Jooo x-pe-f3x c;; (xla,, ... 'ap) dx \ b,, ... ,bq = (3p-1 G"'•n+1 (.!:_IP' a,, ... ' ap) p+l,q (3 b b 1' ••• ' q p + q < 2(m + n), \arga\ < (m + n- ~~p-~q)77 , \arg{3\ < ~277 He(b.-p) > -1, j = 1, ... , m 1 (6) J 000 e-f3x c;:; (ax2la,' ... 'aP)dx b, ' ... ' b q = 77_~ _1 G"•n+ 2(4a ,0, Jt2, a1 , ... , ap) (3 p + 2, q (l 2 b b ,.... 1 ' ••• ' q p + q < 2(m + n), \arga\ < (m + n-~p-~q)77 , \arg/3\ < ~277 Heb1>-~, j=1, ... ,m 420 INTEGRAL TRANSFORMS 20.5 Meijer's G-functions (cont'd) (7) ~oo sin(cx) G~~ (ax21 a,,"'' ap) dx \' b,' ... 'b q = 7Ty, c_, cm,n+1 (4a 10, a,, ... ' ap, )12) p +2,q 2 b b c 1' ••• ' q p + q < 2(m + n), [arga[ < (m + n-!lzp-?zq)rr, c > 0 Hebj>-1, j=l, ... ,m, Beaj<~, j=l, ... ,n (8) f000 cos(cx) c;; (x21 a,' ... ' aP) dx \ b,, ... ,bq = 7T y, c -I G m, n + 1 ( 1 a I ~ 2, a 1 ' • • • ' a P ' 0) p + 2, q 2 b b c 1' ... ' q p+q<2(m+n), [arga[ <(m+n- Y:;p-~1:/q)rr, c>O l1ebj >-~·~, j=l, ... ,m, Heaj <~~. j=l, ... ,n (9) Joo X-p J ( 2 X y,) C m n ~X I a 1 ' .. • ' a P) dx 0 tl pq b b 1' ••• ' q + 1 (a I p -~~ v, a 1 , ... , a P, p + ~12 v) = C"• n p +2, q b b 1' ... ' q p + q < 2 (m + n ), [arg a[ < (m + n -~ p -~ q )rr -~·4 + rnax He a . < He p < l + ~2 He v + rnm He b. 1_$j'S':n 1 1$j$m 1 Joo x-p Y (:2xy,) G"" (ax Ia,' ··· 'ap) dx 0 v ~ ~ b b ( I I 1 :! "• ' q ~ ~· ) = C"• n+2 a p-;: v, p + 2 v, a 1' ... 'a P' p + : + ~ v p + 3, q + 1 b b IJo I ' 1'"'' 9,p+.2+/2V ( 10) I' + q < 2 (m + n ), [ arg a [ < (m + n -!~ p -!~ q )rr -31, + max He a.< lte p <:: r.-:1n He b.+ ~2[11ev[ + l 1_$j,$n 1 1_$j_$m 1 20.5 (12) (13) HYPERGEOMETRIC FUNCTIONS 421 Meijer's G-function (cont'd) I m n +2 01 p -~2 v, p + ~ v, a 1 ' .. • ' a ) =~G · a P p+2, q b b 1' ••• ' q p + q < 2(m + n), iargal < (m + n-1!.lp-lfq)rr Hep<1-/21Hevl+ mm Reb. 1:Si:Sm 1 J""x-pH)2xX)c~; (axla1, ... ,aP)dx 0 b1, ... ,bq (a lp -~ -~ v, a,. ... , a P, p + ~·~ v, p -~ v) I' ~ b b p-/2- ."v, 1, ••• , q p + q < 2 (m + n ), iarg al < (m + n -~ p -~~ q )rr ( J v-1) 1 3 max --:.Re-- + max Rea.<Rep< min Reb.+-Rev+- 4 2 1:SiS.n 1 1:Sj:Sm 1 2 2 Joo 1 0 la1, ••• ,a ) x-P(x-l)a- F (K+a-p A+a-p· a·1-x) cmn ax p dx 1 2 1 ' ' ' pq b b 1''' ., q +2 ( Ia 1, ••• , a , K +A+ a-p, p) =l(a)Gm ,n a P p+2, q + 2 ~ b b K, '" 1' ••• ' q First set of conditions of validity: p + q < 2(m + n), iargal < (m + n-lzp-/2q)rr He a> 0, He K > lle A> Re a -1, ;· = 1, ... , n - } Second set of conditions of validity: p + q :S 2(m + n,) !argal :S (m + n- ~1zp-l2qm Hea>O, HeK>ReA>Ilea. -1, j=1, .•. ,n p q - } He[ La- L b.+(q-p)(K+/2)]>-12 j=1 } j=1 } p q He [ ": a . -L b . + (q -p) (A + ~ )] > -~ j= 1 } j= 1 } 422 (14) i 00 0 G mn pq INTEGRAL TRANSFORMS Meijer's G-fwtction (cont'd) ~ I a 1, ... , a ) kl G I c 1' '1X P G {3x b 1 ' .. • ' b q rs d 1 ' ' c,)dx 'd s = a-1 Gk+n, l+m~{3~-b,. ... ,-bm, c,. ... , c,, -bm+1''"'-bq) q+r, p+s d d a -a1, ••• ,--an, 1, ••• , s'-an+1, ••• ,-ap 20.5 For (five sets of) conditions of validity see Meijer, C.S., 1941: Nederl. Akad. Wetensch ., Proc. 44, 82-92. APPENDIX NOT A TIONS AND DEFINITIONS OF HIGHER TRANSCENDENTAL FUNCTIONS H.T.F. I refers to volume I, and H.T.F. II to volume II, of Higher transcendental functions by the same authors as the present work. Miscellaneous notations Ad hoc notations are explained where they occur. Notations occurring several times on a page are explained at the bottont of the page. In general, real variables and parameters are denoted by Latin letters, and complex variables and parameters by Greek letters, Exceptions are made to preserve traditional notations (such as y in chat~ter XIV)o The letters m, n denote integers mostly. Re z, Im z, Real and imaginary parts of a complex quantity z, izl, arg z, Modulus and argument (phase) of a complex quantity. Cauchy Principal Value, If the integrand has a singularity at c, a< c < b, the Cauchy Principal Value of t f(x) dx a IS b Jc -E J b ) ] :f f(x)dx= lim [ f(x)dx + {(x dx a c +€ a ( > o, ( .... o. Empty sums are to be interpreted as zero, and empty products as unity. b b 2. , I l are empty if b < a, n= a n= a [x] largest integer:;; x, (a).,= r(a + v)/r (a) (a)0 = 1 (a) = a(a + 1) ••• (a+ n-1) n n = 1, 2, .•. 423 • • 424 INTEGRAL TRANSFORMS (a) =(-1)"(1-a-n) n n (a) = (-1)"/(1-a) -n n· Binomial coefficient (a) r (a+ 1) (3 = r ((3 + l) r<a-,8 + 1) • { -1 sgn x = ~ x<O x=O x>O Euler-Mas~heroni constant. C = lim ~ f 1/n -log m) = 0.5772156649 ... m-. oo n= 1 y = ec • n integer n integer Note that in H.T.F. and many other hooks Cis denoted by y. Orthogonal polynomials See also If .T .F. II Chapter X and PP• 265-269 of the present volume. Legendre polynomial 1 d" P (x) =----(x2-1)" n 2 n n! dx" • Gegenbauer polynomial ( 2)" (v) d" Cv(x) - n (1-xz) ~-v __ (1-xz)n+v- ~. n n ! (n + 2 v) dx n n Tchebichef polynomials T <x)= cos(n cos-1 x) n sin [(n + 1) cos -1 x] Un(x)= _1 sin (cos x) Jacobi polynomial ( 1)" d" p<a.,f3)(x )= ----(1-x )-a (1 + x)-!3 __ l(1 -x )n+a (1 + x )" +f3]. n 2 n n! dx n NOTATIONS Laguerre polynomial e z z -a d" La(z)=-----(e-z zri+a) n n! dz n Hermite polynomials y, 2 d" y, 2 He (x) = (-1)" e •x --(e-•x ) n dx" · 2 d" H (x) = (-1)" ex - n dx" Charlier polynomial 2 -x e p n (x; a)= n! a-n L ~-n (a). The gamma function and related functions See also H.T.F. I Chapter I. Gamma function r() Joo -t z-ld z = e t t 0 Logarithmic derivative of the gamma function r '(z) !j;(z)=--' r(z) Beta function d!j; t/l'(z)=--, dz r(x) r(y) B(x, y) = • r(x + y) Euler's dilogarithm etc. oo n 1 z L 2 (z) = L : 2 = -logO-z) dz. n=l 0 z 425 fie z > 0. Incomplete gamma functions. See under Confluent hypergeometric functions. Incomplete beta function. See under Hypergeornetric functions. 426 INTEGRAL THANSFORMS Riemann's zeta function and related functions ((z, a)= }; (n +a)-', n= 0 00 <I> (z, s, v) = l zn n= 0 (v + n) s Legendre functions See also I1.T .F. I Chapter III. For expressions of products of Legendre functions as hypergeometric series see Meijer, C.S., 1936: Math. Ann. 112, 469-489 and Proc. Nederl. Akad. Wetensch. 39, 394-403 and 519-527; 1938: Nieuw Arch. Wiskunde (2) 19, 207-234. 1 (z+1)~J..L P~(z)= --2FT(-v, v+1; 1-{.l;~-~z) ['(1-{.l) z-1 ef.L7Ti 7T~ r(u+ v+ 1) QJ-L() r z-J..L-v-T(z2_1)~J..L V z = 2v+T r(v+ 3/2) 0f.l+V+1 f.l+ll+2 3 1) X 2FT ' ; II + -; -2-2 2 2 z z in the complex plane cut along the real axis from -1 to l. 1 p~(x)= 1 ( l+x)~J..L 2FT (-v, v+ 1; l-{.l; ~-YzX) r(1-f.i) l-x -1<x<1 Qf.L(x )= ~ e-iJ-L7T[e -~J.L7Ti Qf.L(x + i 0 )+ e '/,J-L7T i QJ-L(x -i 0 )] v v v -1<x<1 P )z) = P ~ (z ), Q )z) = Q ~ (z ). Bessel functions and related functions See also H .T .F. II Chapter VII, and PP• 331-332 of the present volume. Dessel functions 00 (-1 )m (~ Z )v+2m m!r(v+m+1) Y )z) = cosec V7T [J )z) cos V7T-J _)z )] NOTATIONS H(ll(z)= J (z)+ iY (z) v v v H(21(z)=J (z)-iY (z) v v v % dt J i (x) = J J (t) -• v 00 v t Modified Bessel functions 00 I)z) = L m = 0 m! [' (v + m + l) rr I_)z) -~ (z) K (z) =- • v 2 sin vrr Kelvin's and related functions her)z) + i hei)z) = Jv(ze l-:7Ti) her (z)-i hei (z)=J (ze-l-:7Ti) v v v ker (z) + ikei (z) = K (ze !47Ti) v v v kerv(z.)-i kei)z)= Kv(ze-!47Ti) her(z) = her 0 (z ), hei (z) = hei 0 (z ), ker(z) = ker0 (z), kei (z) = kei0 (z). 427 Note that the definition of kerv(z) and keiv(z) differs from that given in H.T.F. II sec. 7.2.3. x~b)(z)= her~(z)+ hei~(z) V ~b)(z) = [he< (z W + [bei~ (z )] 2 w~b)(z)= berv(z) bei~(z)- beiv(z) ber~(z) }'2Z~b)(z) = herv (z) hei;(z) + bei)z) ber;(z). Neumann polynomials <Xn l l " n (n -m -l) ! 00(x)=-; On(x)=-4 f.. 1 (}'2 )n-2m+1 X m=Om. 2X n = l, 2, ••• n=l,2, ••• 428 INTEGRAL TRANSFORMS Anger-Weber functions 7T J)z·)= 11-1 ~ cos(vtl- z sine) de Struve's functions 00 H)z)= L f'(m + 3/2) f'(v+ m + 3/2) m=O (z/2)v+l ['(3/2) f'(v + 3/2) 1F2 (l; 3/2, v + 3/2;-z2/4) Lommel 's functions S (z) = s (z) + 2J.L-1 [' J..L, 111 J..L' ll (JL- 2v+l) r(Jl+V 2+l) Lommel's functions of two variables ( )v+z.. U)w,z)=.~0(-l)"' : Jv+2 .. (z) (w z 2 V1T) V )w, z)= cos -+ --+- + U 2 (w, z). 2 2w 2 -v NOTATIONS Hypergeometric functions See also H. T .F. I Chapters II, IV. Generalized hypergeometric series 429 2F1 (a, b; c; z) is Gauss' hypergeometric series and is often (for instance in H. T .F. I Chapter II) denoted by F (a, b; c; z ). 1F1(a; c; z) is Kummer's confluent hypergeometric series and is some­ times (for instance in H .T .F. I Chapter VI) denoted by <IJ (a; c; z ) • • Fn(a1, ••• , a.; y1, ••• , yn; z) is sometimes written as .,Fn [a,. ... , am; z J Y1•···•Yn Incomplete beta function 13.,(p; q) = ]0" tp-1 (1-t)q-1 dt = p-1 xP 2F1 (p, 1-q; p + 1; x). I ( ) = B .,(p, q) " p, q B ( ) 1 p, q n r<bj-bh) n II' 2: j= 1 l + 2bh S n (b,. b2, b3, b 4 ; Z) = z 4 h = 1 II r(l+bh-bj) j= n+1 (1 + bh-61' ... * •.. , 1+bh-b4;(-1)"z2) The prime m II' and the asterisk in 0~ mean tit at the term contammg b h -b h is to be omitted. For n = 1 the product II in the nun1erator, for n = 4 that in the denominator is to be replaced by unity. 430 INTEGRAL TRANSFORMS Confluent hypergeometric functions See also li.T.F. I Chapter VI and H.T.F. II Chapters VIII and l.X. See also under f!ypergeometric functions, Orthogonal polynomials, E -function, C-function. Whittaker's functions •1 ( ) -y, + J.L -y,' F ( 1 ' 2 ) t•. K, J.1. Z -Z e 1 1 i'2 + f1-K; f1 + l; Z Parabolic cylinder functions D ( ) 2y, v+\> -Y, W (ll 2) Z = < Z 1/ +'/ I/ /2 Z 11 IJ v 74, /4 Bateman's function The exponential integral and related functions dt -Fi(-x) = E 1 (x) = J"" e -1-= I'(O, x) -TT < arg X < TT X f Ei+(x) = Ei(x + iO), Ei-(x) = Ei(x-iO) The last function is denoted by E*(x) in 1-I.T.F. II sec. 9. 7. li(z) =1' .-..!!.!._ = Fi (log z) log t 100 sin l l si(x) =---dt =-[Ei(ix)- Ei(-ix)] t 2i X Jx sin t Si (x) = - 1-dt = Yz TT + si (x) 0 !"" cost Ci(x)=- x - 1-dt=-ci(x)=~2[Ei(ix)+Ei(-ix)] x>O X> 0. NOTATIONS 431 Error functions and related functions ~;:x 2 2x Erf (x) = 2 rr- e -t dt = --t Ft o yrr (~ ~. -xz) 2' 2' These functions differ by the factor 2 rr -X from the functions introduced in I!.T.F. II sec, 9.9. S( ) -X "rx -~ X = 2 1 Tr-n Jo t 2 sin t dt , Incomplete gamma functions ( ) rx -t a-t d -t a F. ( l ) y a, x = Jo e t t = a x t t a; a + ; -x r (a, x) = .foo e -t ta-t dt = I~(a)-y(a, x) % ~(a-t) -~xm () =X e "Y,(a-t),~ax. Particular cases of \\'hittaker's functions 'I ( )-ti. -~x!4 e~xErf(x~) lY _ ~ , ~ X -, :l TT (-1)".2-n- l\ Mn+~,!4(x)= (3/2) n 432 INTEGRAL TRANSFORMS W (x) = (_:_ \~ K (_:__) o, J.l. 7T ) J.l. 2 (rrx)~ 2 (x) W0 (ix) = --exp [-(~ v + ~) rri] H < 1 - ,J.l. 2 J.l. 2 • ( 7T X)~ . 1 (X ) W0 (-Lx)=-- exp[(~v+~)rr£]1-! <l- ,J.l. 2 J.l. 2 W + +'/ ± (x)=(-l)"n!xJ.J.+ X e-~xL2 J.l.(x) J.L n /2, J.L n NOTATIONS 433 MacRobert's E-function See also H .T .F. I Chapter V. If p :2: q + 1, p p II' r (a -a ) 2: s= 1 s r . a E (p; ar: q; p s: x) = r(ar)x r q r= 1 II r (p t-ar) t= 1 where lx I < 1 when p = q + l. If p ~ q + 1, E (p ; a r: q ; p s : x) = ----- pFq (a 1 , ••• , a P ; p , ••• , p q;-1/ x) q II r<p) s = 1 where x f. 0 and lxl > 1 if p = q + 1. If p > q + 1, the last relation gives the asymptotic expansion of the £-function for large x. E(Y:; + v, Y:;-v:·:zx) = sec(vrr) (2rrx) y, e"' Kv(x) E(a, {3: :x)= r(a)r({J)x-k ey,"' Wk (x) • m k = Y:; (1-a-{3), m = Y:; (a-{3) 134 INTEGHAL THANSFOHI\1S ( a+(3 a+{J+l x2 ) E a, (3, --, : a + (3 :-2 2 1\. =77y,I'(a)l'((3) _:_ If/ (ix)lfl (-ix) 0 )-2k 2 k, m k, m Meijer's G-function See also li.T .F. I Chapter Y. G"·n (xI a, ..... ap) p,q b b 1' ••• ' q fi [' (b -s) ft 1~0-a .+s) 1 11 =--j=1 1 j=1 ---------------------------------- xsds ') . ~ 77 t L q II j= m +1 1(1-b.+ s) 1 p 11 ['(a -s) j = n+1 1 where L is a path separating the poles of 1 (b 1 -s) ·•• l' (b m -s) from those of 1 ( 1 -a 1 + s) ... 1 (1 -an + s ). For a more detailed definition see H.T.F. J sec. 5.3. Formulas involving the G-function may be used as key formulas from which many integrals with Bessel functions, Legendre functions, and other higher transcendental functions follow by specializing parameters. The following two lists give expressions of certain special G-functions in terms of well-known higher transcendental functions, and, conversely, expressions for higher transcendental functions in tern,s of G-funcLions. The list is not complete. See also ll.T.F. I sec. 5,6. Particular cases of the C-function G20(xJa b)= 2xY,(a+b)K (2xy,) 02 ' a-b NOTATIONS 435 1(1-a+b) b x FU-a+b·l+b-c·-x) f'(l+b-c) 1 1 ' ' G20 (xI ~ )= 77-Y, e-Y,x K (~x' 12 b, -b b J = XY,(b+c-1) e -Y.x W (x) k, m k = ~ ( l + b + c) -a, m = ~ b -~ c G~~ (x ib,~b) co::77 eY.x Kb(~x) G~! (x\b,ac) =f'(b-a+l)f'(c-a+l)x Y,(b+c-l)eY,xwk,m(x) k =a-% (b + c + l), m = ~ b-~ c G10(x\a b 2b-a b+~)=77-Y.xbi (2312x114)J (2312x114) 04 ' ' ' 2 2 (a-b) 2(a-b) G: (x\ a, a+ ~. b, 2a-b)=~ 77-y, sec (b-a) 77 X X a [J (23/2 X 1/4) I (2 3/2 X 1/4) 2(a-b) 2(b-a) +I (2 3/2 X 1/4) J (2 3/2 X 1/4)] 2(a-b) 2(b-a) c:(x\a+ ~.a, b, 2a-b)= ~77-v.Lsin(a-b)77r1 X X a [J (2 3/2 X 1/4) I (2 3/2 X 1/4) 2(a-b) 2(b-a) -I (2 3/2 X 1/4) J (2 3/2 X 1/4)) 2(a-b) 2(b-a) G20(x\a a+~ b b+'<)=xY,(a+b)J (4x114) 04 ' 2• ' 12 2(a-b) 436 INTEGRAL TRANSFORMS G~(xiO, Yz, a, -a)= 77X i-1 (sin 2a77)-1 x [e 2a1Ti J (ze -7Ti/4) J (ze 7Ti/4) _ e -2a1Ti J (ze 7Ti/4) 2a -za 2a z = 23/2 X 1/4 G~ (xl3a- Yz, a, -a-Yz, a-Yz) = 277X (cos 2a 77)-1 xxa-1/2K (23/2x114)[J (23/2x1/4)+J (23/2x1/4)] 4a 4a -4a G 30 ( I 0 1/ 1/ I/) X -X 04x ,a-n,-a- /2,-/2 =477 x G 30 ( I 1/ 1/ 1/ ) X -Y, x -/2 a-/2 -a-/22 0 =-477 x ' 04 ' .... , ' G~(xl a, a+ Yz, b, 2a-b)= 23 77X xa x K (23/2 x 1-/4 e 7Ti/4) K (23/2 x 1/4 e -7Ti/4) 2(b-a) 2(b-a) rtl ( I ) -x ( x G04 X a, b, c, d =X sn a, b, c, d; X ) n = l, 2, 3, 4 c21 ( I a+~ ) 13 x a+~. b, a 21 ( I a+~ ) G13 x a, a+~. b 1TX~(a+b) ----- llb_a(2x ~)-La_b(2x ~)J cos (a -b) rr 437 G31 (xl a \=2-2a+2r(l-a-b)r(l-a+b ·)S2_12b(2xy,) 13 a, b, -b) a ' 31 ( I a+~ \ G 13 x b, 2a-b, a) G12(x\-c1,-c2) 22 a-1,-b =rr512 2-1lcos(b-a)rrr1 X xa H~~)a(x ~) Hb(:)a(x ~) r (a + c 1) r (a + c 2) r (a+ b) 438 INTEGRAL TRANSFORMS c~ (xl 0 '~ ) =i2-2rrl{ 24 a, b, -b, -a ( I~+ a,~-a) G31 X 24 0, ~. b, -b rrl{r(~-a+b)x-~ ( l{) ( y,) --------W 2x M 2x' r (l + 2 a) a, b -a, b 40 ( ~~+a, ~-a) G24 X I 0, ~. b, -b ( I a, a+~ ) G~ X b + c, b-c, b + ~ + c, b + ~-c _ Y, 2-k b-\( -xl{ W (2 y,) -7T .x e k,2c x k=~+2b-2c C41 X ' 2 = 7T ( 0 ~ ) -2 -2 5/2 24 I a, b, -b, -a i sinarr sinbrr ( I ~. 0 ) G41 X 24 a, b, -b, -a cos arr cos brr (X I~~ a,~-a) = x-y, rrl{ r(~ + b-a) r(~-b-a) o, ~. b, -b X Wa,b(2ixl{) Wa,b(-2ix~) NOTATIONS 439 G~ (xl a,a+~ 1 ) b + c, b -c, b + ~ + c, b + ~-c = 2k+1 rr312 r(l-2a + 2b + 2c) r(l-2a + 2b-2c) k=2a-2b- ~ 4 G!: (x I a-l, -c 1, -c 2' -c 3 ) -b1, -b2, -b3, -64 n r(a + bh) h= 1 -------x a-1 3 II r (a+ c h) h= 1 x 4F3 (a+ b 1, a+ b 2, a+ b 3, a+ b 4; a+ c 1, a+ c 2, a+ c 3; -x) p 1 ( I a 1, ••• , a ) G p X p pq b 1' ••• , b q ll r(l+b1-a .) j= 1 1 b 1 X q II r<l+b1-b.) j = 2 1 x pFq_1(l+b1-a1, ••• , l+b1-ap; l + b 1-b 2, ••• , l + b 1-b q; -x) 1 ( Ia,. ... , a ) G n X. p pq b1, ••• ,bq n b 1 n r(l+b1-a )x j= 1 1 q p n r (l + b 1 -b . ) n r (a . -b 1) j=2 1 j=n+1 1 x pFq_1 (l + b 1 -a,. .•. , l + b 1 -a P; l + b 1-b 2, ••• , l + b 1 -b q; -x) Gq1 (x\a1, ••• ,ap) pq b1, ••• ,bq a -1 xx 1 E(l-a1+b,. ••• ,l-a1+bq:l-a1+a2, ... ,l-a1+ap:x) 440 INTEGRAL TRANSFORMS Functions expressible in terms of the G-function x J.L J )x) = 2 J.L G ~~ (~ x 2 1 X! v + X! IL• X! 11 -X! v) xJ.L K (x) = 4J.J.-I rr-1 v X G:(4-" x"l ~V+ ~II> ~2 + ~v+ ~II> -~v + ~/L> 72-~v+ ~/L) e -z K)x) = rr~ G 20 (2x I X! ) 12 v, -v ez K (x) = rr-~ cos vrr G21 (2x\ X! ) v 12 v,-v J.L ( ) 2J.I. G II ( u 21 X! + X!v + Xl/1 ) X H X = /4X v 13 X + 72v + X11, X11 -X!v, XliL + X!v -2 · ( 21 Xl+Xv ). H (x)-Y v(x) = rr cosvrr G31 ~x v 13 X + X!v, -X!v, ~2v NOTATIONS 441 xiL [ I_ 1}x)-L,/x)J -1 2JL c21 =TT COSVTT 13 l S (x) = 211--1 ----------,---,--- .,....,------ JL,V r(~-}iiJ.-72v) r(~-~/1 + ~V) C31 X 13 J)x)J_)x)=rr- )4~:~(x2 1 ~ ) 0, v, -v ~ • It 10 X II }I 1 ( 41 ) -TT sm (i'2 vrr) G 04 64 12, 0, 12v, -~v I (x)K (x)= 2-1 TT-)4 C21 (x21 ~ ) v v 13 v,O,-v 442 INTEGRAL TRANSFORMS x G: ( 6~ x 4 \ ~fl. + Xv, ~fL + X, ~p., ~fL -Xv) 22 [ 21 X a, X a + ~~ J X G24 X X (v + fL +a), X (v +a-p.), X (p. +a-v), X (a-v-p.) 31 ( 21 X + Xp. ) X G 13 X Xp. + v, Xp.-v, Xp. [ 21 Xa,Xa+X J xG40 x 24 X(v+p.+a) , X(v+a-p.), X(p.+a-v), X(a-v-p.) 2J.J. K ( 7T i/4) K ( -7Ti/4) = 2 3J.J.-3 -y, x 21.1 xe 2v xe TT (I l-k+l ) X l e -Y,x IT' (x) = G 20 X II k ,m 12 m + l + X, l -m + /1 l Y,x X - G 21 ;~:: 1 C I k+l+l ) X e wk,m( )-I'(X+m-k)l'(X-m-k) 12 l-m+Y.,m+l+X NOTATIONS 443 40 ( -2 21 ~-lf?.k, %-lf?.k ) x G 24 2 x If?. + If?. m, If?.-If?. r.J., If?. m, -If?. m X li 2-(k+ 1) 1T -3/2 e X W k ( 2 X) = =-;::-;------:-:--=-=-:-----:-:- ,11! 1 (X + m -k) [' (If?. -m -k) rr-lir(l+2m) ( I l+k 1-k ) wk (x)M k (x)= G2341 ~x2 I I ' I ,m -,m ['(If?.-k + m) If?., l, X+ m, X-m -li x 1 W ( 2 ix) W (-2 ix) = -=-:-:-;--- x-17 -:-:-=-:,...,.----....,....,... k,m k,m ['(lf?.+m-k)['(lf?.-m-k) ( 21 If?.+ lf?.l + k, If?.+ lf?.l-k ) G41 X X 1 1 I I I 24 lf?.l,lf?.+lf?.l,lf?.l+m, lf?.l-m W k ,m (x) W -k ,m (x) -li 40(1 21 k+l,-k+l ) = 1T G 24 ~ X I I lin If?., l, m +X, -m + ,2 2F1 (a, b; c; -x) i(c) x ['(a) r (b) 12 ( ~-a, -b ) G 22 X -1, -c r (e)[' (f) [' (l) F (a b c d· e r l· -x) = X 4 3 ' ' ' ' ' ' ' ['(a) ['(b) ['(c) ['(d) ( 1-a -b -c -d) G14 ' ' ' X 44 X -l, -e, -f, -l 444 INTEGHAL TflANSFOilMS q n r(b) j=1 PF/a1' .•• , ap; bp ••• , bq; -x)= P J n f'(a .) i= 1 J x xG •P x P 1 ( ~-a 1' ••• , -a ) p, q + 1 -1, -b 1' ••• ' -b q 1 ( 11, f3 1 ' •• ' f3 ) E (p · a : q · f3 : x.) = G p, x q ' r ' s q+ 1, p a, ... ,ap p~q+1 For further special functions expressible in terms of the G-function, in particular for combinations of Legendre functions, and also con,bina­ tions of generalized hypergeometric series, see C.S. ~.1eijer, Nederl. Akad. Wetensch., Proc. 43 (1940), 198-210 and 366-378; 44 (1941), 82-92, 186-194, 298-307, 435-451, 590-605, 1062-1070; 49 (1946), 227- 235, 344-356, 457-469, 632-641, 765-772, 936-943, 1063-1072; 1164-1175; 55 (1952), 369-379, 483-487; 56 (1953), 43-49, 187-193. Hypergeometric series of several variables See also fl.T.F. I Chapter V. Hypergeometric series of two variables. In all double sums m and n run for 0 to oo. ~ (a)m+n ({3)., ({3')n F; (a; {3, {3 '; y; x, y) = f..t (y)m+nm!n! F2(a; {3, {3'; y, y'; x, y)= 1 (a)m+n ({3)., ({3 ')n (y)m (y')nm!n! ( , {3 {3' ) 1 (a)., (a')n ({3),. ({3')n F3 a, a , , ; y ; x, y = ( ) 1 1 y m+n m. n. F( {3· '·x )=2: (a)m+n({3)m+n x'" n 4 a, ' y' y '. ' y ( ) ( ') ' f y Y m Y nm. n. <II( {3 ) 2: (a)m+n({3)., m n 1 a, , y; X, y = ( ) l l X y Y m+n m.n. X m y" NOTATIONS , I ({3) m ({3 ')n IJ.> 2 ({3, {3 ' y; x, Y) = ( ) I I X m Y n y m+nm.n. I (a)., +n ({3)'" IJI1 (a, {3, y, Y,; x, Y) = ,-----=---"------"'---- X m Y n (y),. (y')n m! n! I (a)"' +n '1'2(a, y, y'; x, y) = (y) (y') m!n! " n X m y" '\"' (a) (a') ({3) E 1(a,a~{3,y;x,y)=L_, '" n "x"y". (y)., +n m! n! ..., ( {3 ) 2: (a)" ({3)"' X" Y n :::!.2 a, 'y;x,y = (y),.+nm!n! 445 For other hypergeometric seriesoftwo variables see H.T.F. I sec. 5.7.1. Hypergeometric series of several variables. All summations run from 0 to oo. 2: (a)m + ... +m (,'1 1)m "' ({3 )m 1 n 1 n = --,----:....-----,-::...--..:-~ (y ) ... (y ) m 1! ... m n! 1 m 1 n m n m1 z 1 m ... z n ({3 ) "• ({3 ) rn ... ,z)=I 1m 1 n m ___ _,_ ___ _....,___ z7•· .. z n n (y) m !•oom! m1+ ... +mn 1 n (a) ( ) I · m 1 + ••• + m m1 mn '1'2 a; y1, ••• ,yn; z1, ••• ,zn = z ... z (y ) ... (y ) m ! ••• in ! 1 n 1 m 1 n mn 1 n 446 INTEGRAL TRANSFORMS Elliptic functions and integrals See also H.T.F. II Chapter XIII. Complete elliptic integrals K(k)=J~7T(l-k2sin2"-)-~d"-=);277 F()!f Yz·1·k 2) 0 'f" 'f" 2 1 ' ' ' Theta functions eo(vir)=(-ir)-~ ~ n= -oo el (vir)= (-i r)-~ ~ (-l)" e -i rr(v -Y, +n)2 /r n= -oo n=-oo n=-oo The series given here are connected with the definitions given in 1-i.T.F. II equations 13,19(10) to (13) by means of Jacobi's imaginary trans­ formation, see Il.T.F. II equations 13,22(8). Modified theta functions A oo 2 -oo ( )2 tJO(vir)= (-ir)-~ [ L e-i7T(V +Y,+n) /r-L e-i7Tv+Y,+n fr] n=O n=-1 e2 (vir)= (-ir)-~ L I (-l)" e -i7T(v +n)2 /r--r (-l)" e -in(v+n)2 !r] n=O n=-1 A oo ( )2 -oo 2 e (vir)=(-ir)-~[ .l e-i7Tv+n /r_ .l e-i7T(v+n) lr]. 3 n=O n=-1 NOTATIONS 447 Miscellaneous functions See also H.T .F. III Chapter XVIll. loo Xs 8a Jl(x,a) = 0 r(s+1)ds v (x) = .Ia oo -r-(-:-~-1-) ds foo xs+a --!00 x• v(x, a)= ds ds r (s + a+ 1) r (s + 1) 0 a INDEX OF NOTATIONS B bei(x), bei)x), ber(x), ber)x) Kelvin's functions, 427 c Cauchy Principal Value, 423 ci (x), Ci (x) Cosine integrals, 430 C (x) Fresnel integral, 431 C~ (x) Gegenbauer polynomial, 424 D Dn(z), Dv(z) Parabolic cylinder functions, 430 E E (k) Complete elliptic integral, 446 E (p; a,: q; p s: x) MacRobert's E -function, 429 Ei(-x), Ei+(x), Ei-(x), Ei(x) Exponential integrals, 430 Erf (x), Erfc (x) Error functions, 431 Ev(z) Weber's function, 428 F F(a, b; c; z), .. Fn(a1, ••• ,am;y1, ••• ,yn;z) Hypergeometric series, 429 F1( ••• ; x, y), ••• , F4 ( ••• ; x, y) Hyper geo­ metric series of two variables, 444 449 FA( ••. ; zl, ••• ,zn) Lauricella's series, 445 S1: , ~ , ~ Fourier transforms, xi Oc 1.... e s G G mn (x) Meijer's G-function; 434 pq H H.T.F. 423 H n (x ), Hen (x) Hermite polynomials, 425 H 11 )(z), H 12) (z) Bessel functions ~f v v the third kind, 427 Hv(z) Struve's function, 428 S2v Hankel transform, 3 Iv (z) modified Bessel function of the first kind, 427 I" (p, q) Incomplete beta function, 429 J Jv (z) Bessel function of the first kind, 426 Ji (xi Bessel integral function, 427 v Jv(z) Anger's function, 428 450 INTEGRAL TRANSFORMS K k2v(z) Bateman's function, 430 kei(x), keiv(x), ker(x), kerv(x) Modified Kelvin functions, 427 K (k) Complete elliptic integral, 446 Kv(z) modified Bessel function of the third kind, 427 ~v K-transform, 121 L li (z) Logarithmic integral, 430 L 2 (z) Euler's dilogarithm, 425 L n (z ), L ~ (z) Laguerre polynomials, 425 L)z) Modified Struve function, 428 l3 Laplace transform, xi M M K ,J-L (z) Whittaker's confluent hyper­ geometric function, 430 WI Mellin transform, xi 0 0 n (x) Neumann's polynomial, 427 p p n (x; a) Charlier polynomial, 425 P n (x)' Legendre polynomial, 424 P (a, f3) (x) Jacobi polynomials, 424 n P v (z ), P~(z ), P~(x) Legendre functions of the first kind, 426 Q Q v (z ), Q~(z ), Q~(x) Legendre functions of the second kind, 426 R 3t J-L Fractional integral, 181 s si (x), Si (x) Sine integrals, 430 s J-L,)z ), S J-L,V (z) Lommel 's functions, 428 S (x) Fresnel integral, 431 S n (b 1 , ••• , b 4 ; z ), 429 q 6 Stieltjes transforms, 213 p T T n (x) Tchebichef polynomial, 424 u U n (x) Tchebichef polynomial, 424 U v (w, z) Lommel 's function of two variables, 428 v V (b) (z) 427 v ' V v(w, z) Lommel's function of two variables, 428 w W K,J-L(z) Whittaker's confluent hyper­ geometric function, 430 W (b) (z) 427 v ' Sffi Fractional integral, 181 J-L X X (b) (z) 427 v ' y Y v (z) Bessel function of the second kind, 426 ['v Y -transforms, 95 z z (b) 427 v ' GREEK LETTERS B {x, y) Beta function, 425 B x (p, q) Incomplete beta function, 429 [' (z) Gamma function, 425 y(a, x), f'(a, x) Incomplete gamma functions, 431 ( (s ), ( (z, a) Zeta f~nctions, INDEX 426 "' 80(vir>, ... , 84(vlr),80(vlrl, ... ,83(vir> Theta functions, 446 fl (x, a), 447 v(x), v(x, a), 447 <1> (z, s , v ), 4 26 <1> (a; c; z) Confluent hyper geometric series, 429 <P,( .•. ;x,y), ••• , <1>3 ( ••• ; x, y) Confluent hypergeometric series of two variables, 444 ff. <1>2( ••• ; zf' ••• , zn) Confluent hyper­ geometric series of n variables, 445 tf; (z) Logarithmic derivative of the gamma function, 425 1J! 1 ( ••• ; x, y), '¥ 2 ( ••• ; x, y) Confluent hypergeometric series of two variables, 445 1J! 2 ( ••• ; z 1, ••• , z n) Confluent hyper­ geometric series of n variables, 445 451 t"<t ), 426 =1 ( ... ; x, y), =2 ( ... ; x, y) Confluent hypergeometric series of two variables, 445 MISCELLANEOUS NOTATIONS (;) binomial coefficient, 424 (a)v= f'(a + v)/['{a), 423 ff. C, y Euler-Mascheroni constant, 424 sgn x, 424 [x] largest integer::::; x Re z real part of z (complex) Im z imaginary part of z (complex) lzl modulus of z (complex) arg z argument (or phase) of z (complex) } Cauchy Principal Value, 423