P. 206, equation (14), righthand side: Read E (. .. u : x ) instead of E (... 0 : x). P. 209, equation (lo), righthand side: Read 22p'2q x 2 instead of 22p'2* x. .... measure by the considerable reduction in size of the book, and by the ...... R'e can describe in the zplane a circle of radius r as large as we please .... (10) B(x, y) = 2.
H I G H E R T R A N S C E N D E N T AL F U N C T 1'0 N S Volume 1 Rased, in part, on notes left by
Harry Bateman Late Professor of illalhemalics, Theorelical Physics, and Aeronautics rtl the Califor n.ia I nst it uf e of Technology and compiled by the
Staff of the Bateman Manuscript Project Prepared at the California Institute of l'eclinology uiider Coiif.ract#
No. N6onr244 Task Ordcr XIV with the Offict! of Naval Research Project Desigiiation Number: NR 043045
Printed and Published by
ROBERT E. KRIEGER PIJBLISHIWQ;C O V P A N Y , INC. KR I F . r r F R DRIVE M,41,ABAR, FLORIDA 3 2 W J Copyright @ 1953 by McGKAPrHICI, BOOK COWPAMY, INC. Reprinted by Arrangement
Rlf rights reserved. No part of this book may be reproduced in any form or b y any electronic or mechanical means inrlridina information storage and retrieud systems without permission in writing.frorn the publisher.
Printed in the United States of America
I
Library of Congress Cataloging in Publication Data‘
Bateman Manuscript Project, California Institute of Technology. Higher transcendental functions. “Prepared at the California Institute of Technology under contract no. N6onr211 task order XIV with the Office of Naval Research, project designation number: NR 043.015.’’ Reprint of the edition published by McGrawHill, New York. Includes bibliographies. I. Bateman, Harry, 1. Fiinctions, Transcendental. 18821946. 11. Erddlyi, Arthur. 111. United States. Office of Naval Research. IV. Title. 515’.5 7926544 [QA353.T7B37 19811 ISBN 0898742064 (V .I) 10 9 8 7 6 5 4 3
This work is dedicated to the memory of
HARRYBATEMAN
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 Erd6lyi
Res earch Associates

W ilheh Magnus (1948 50) Fritz Oberhettinger (1948 51) Francesco G. Tricomi (194851) Research Assistants

David Bertin (1951 52)
W. B*Fu&s (194950) A. R Harvey (194849) D. L. Thomsea, Jr. (195051) Maria A. Weber (194451) E. L. Whitney (194849) Yarityp ist
Rosemarie Stampfel
,
PREFACE The late Professor Harry Batenran of the California Institute of Technology w a s one of those rare scientists who, responding t o the interplay between mathematical analysis and physical understandin&, made outstanding contributions to American applied mathematics. His contributions to aero and fluid mechanics, to electromagnetic theory, to thermodynamics, t o geophysics, and to a host of other fields in which h i s adroit mathematical skills were applied, resulted in significant advances in these field$. During his last years he had embarked upon a project whose successful completion, h e believed, would prove of great value t o scientists in all fields. He planned an extensive compilation of ‘‘special functions” solutions of a wide c l a s s of mathematically and physically relevant functional equations. He intended to investigate and to tabulate properties of such functions, interrelations between such functions, their representations in various forms, their macro and microscopic behavior, and to construct tables of important definite integrals involving such functions. It is true that much of this material was already in existence. However, anyone who h a s been faced with the task of handling and discussing and understanding in detail the solution to an applied problem which is described by a differential equation is painfully familiar with the disproportionately large amount of scattered research on special functions one must wade through in the hope of extracting the desired information. Professor Bateman was eminently qualified to embark on such a compilation, for h e was unusually familiar and systematically so with existing mathematical literature on the subject; he was exceptionally adept in mathematical analysis; and h e was ever conscious of the needs of the scientist who m u s t so often u s e these functions. When his death cut short h i s work, the California Institute of Technology, in recognition of one of i t s great scientists, and the Office of Naval Research, in recognition of the extremely useful service such a compilation could render to both basic and applied science, pooled their efforts to continue the task initiated by Professor Cat.eman.


ix

X
SPECIAL FUNCTIONS
In 1948 arrangements were completed between the California Institute of Technology and the Office of Naval Research to employ at the California Institute of Technology four mathematical analysts of international reputation to complete Professor Bateman’s work: Professors Arthur Erdelyi of the University of Edinburgh; Wilhelm Magnus of the University of Gsttingen; Fritz Oberhettinger of the University of Mainz; and Francesco Tricomi of the University of Torino. It was not long after this team began work that it became apparent that not only would Professor Bateman’s original project find its completion in their unusually competent hands, but that the activities of such a group would lead to significant mathematical investigations and advances in the general field of mathematical analysis, as well a s in the more particular field of special functions. The present compendia bear undeniable witness t o the s u c c e s s of the undertaking. The Office of Naval Research is proud of i t s collaboration with the California Institute of Technology, not only for erecting this lasting memorial to Professor Bateman, but also for producing what i t considers a significant contribution to general science. These compendia, which have taken their roots in Professor Bateman’s “shoe boxes” (his repository for card fiIes) have been nurtured into mathematical maturity under the deft minds and penetrating work of the members of the international team of Erdklyi, Magnus, Oberhettinger, and Tricomi. In addition, we are pleased to have been able to render support to several young American mathematicians who have not only contributed to these compendia but were able to avail themselves of the opportunity to work and study under the direction of distinguished scientists in a field that is sorely in need of young recruits. We feel that special thanks should be extended to both Dean E. C. Watson of the CaIifornia Institute of Technology and to Professor Erddyi; to the former, for his extremely helpful and untiring interest in seeing to the establishment and completion of this task; to the latter, for assuming, in addition to scientific participation, both the scientific administrative duties of the project and the general editorial responsibilites for the publication of these compendia. MINA REES, Director Mathematical Sciences Division Office of Naval Research
FOREWORD The late Professor Harry Bateman w a s one of the greatest authorities in that part of mathematics now usually described as classical analysis. H i s knowledge of the literature w a s encyclopedic and probably unsurpassed and his ability to utilize this knowledge for specific problems w a s extraordinary. Research workers in difficulties would often write to him and receive, by return mail, detailed answers to their questions together with a list of references which in many c a s e s amounted to a complete bibliography. It w a s natural for Bateman to want to make accessible in a systematic form the tremendous amount of material which he had collected in the course of the years. His book on Partial Differential Equations (1932) w a s an attempt to carry out this task in a restricted field. Although the book was received with enthusiasm, and, after twenty years, is still one of the most important books on i t s subject, Bateman w a s not satisfied with this method of providing information. For a number of years he made plan after plan to organize and prepare for publication his material, a task made extremely difficult by the very breadth of the field which he intended to cover. At the time of Bateman’s death (1946) his notes amounted to a veritable mountain of paper. His cardcatalogue alone filled several dozen cardboard boxes (the famous ‘‘shoeboxe”’). His family, his friends, and h i s colleagues a t the California Institute of Technology very naturally wished to have some of this material prepared for posthumous publication, thereby erecting a monument to one of the most distinguished and most versatile members of the faculty of the Institute. Professor A. D. Michal, for many years a friend and colleague of the deceased, undertook the sifting of Bateman’s notes. He spent several months in this herculean task, sorted out those notes which might be considered for publication and made recommendations for proceeding further with the matter. Dr.A.Erde’lyi, then of the University of Edinburgh i n Scotland, was invited to prepare a detailed report and proposals, and spent the academic year 194748 in Pasadena for this purpose. xi
xii
SPECIAL FUNCTIONS
It turned out that Bateman’s notes ranged over a wider field than even h i s friends had suspected and also that no single section of this wide field w a s in a state sufficiently advanced for immediate publication. Indeed the field was s o wide that it appeared imperative to narrow it down if anything useful was to be accomplished. Notes for books on functional equations, integrals in potential theory, binomial coefficients and factorials, and many other matters had to be laid aside entireIy. Of the remaining material the most important part w a s a projected trilogy on the higher transcendental functions, on definite integrals (especiaIIy those containing higher functions), and on numerical tables of functions occurring in applied mathematics. Since the appearance of the Index of iGiathematicaZ T a b l e s by Fletcher, hliller, and R osenhead, adequate information has been available on numerical tables, and s o it was decided to concentrate on the first two parts, and these came to be called the handbook and the i n t e p a l tables. The Office of Naval Research recognized the great importance of such a work by giving generous financial support to it. T h u s originated what at the California Institute came to Le called the Bateman Manuscript Project. The Institute was fortunate indeed, not only in being able to persuade Professor E d e l y i to remain as i t s Director and as Editor of the forthcoming publications, but a l s o in securing the services of Professor Rilhelm Magnus of the University of G6ttingen (now of New York University), of Dr. Fritz Oberhettinger of the University of h3ainz (now Professor at the American University, Washington, D.C.) and of Professor Francesco G. Tricomi of the University of Turin. These distinguished and internationally known scholars were assisted by a staff of younger mathematicians. The technical preparation of the vari typescript suitable for reproduction by a photooffset process was in the capable hands of Miss Rosemarie Stanipfel. The present volume is the first of three projected volumes on the higher transcendental functions. These three volumes will be followed by two volumes of integral tables. The California Institute of Technology wishes to express i t s thanks both to the family of the late Professor Bateman for the gift of his notes and of h i s library, and to the Office of Naval Research and especially to Dr. Mina Rees, the Director of i t s Mathematical Sciences Division, for the generous support they have given to this work and for the understanding they have constantly shown for the difficulties encountered. The Institute a l s o wishes to record i t s appreciation and thanks to the following persons and organizations: to Professor Rlichal for his preliminary survey of Eateman’s notes; to the University of Edinburgh for

FOREWORD
xi ii
granting leave of absence t o Dr. Erdklyi; to the Rockefeller Foundation for defraying rravelling expenses for Dr. and Mrs. Erde?yi on their visit in 194748; to the University of Turin for granting leave of absence to Professor Tricomi; to Professors T. M. Apostol of the California Institute, R. C. Archibald of Brown University, E. D. Rainville of the University of Michigan, Mr. S. 0. Rice of the Bell Telephone Laboratories, and Professor C. A. Truesdell of Indiana University for information or consultations in connection with the work; and to the McGrawHill Company for technical advice and publication.Last but not least, acknowledgments should be expressed to Dr. Erddlyi and the staff of the Bateman Manuscript Project for the faithful and highly competent performance of a difficult task. E . C . WATSON Dean of bhe Faculty California Institute of Technology
ERRATA
HIGHER TRANSCENDENTAL FUNCTIONS. VOL. I. P. 5, line 10: Read [ 1  2/(2n
+

1)Il i n s t e a d o f [ 1 2/(2n
+
1)'].
P. 12, equation (32): Read tan' ( q / p ) instead of tanm1( p / q ) .
P. 13, equation (40): Add
a
> 0, p > 1.
P . 24, line 6 up: Read R e s > 1 instead of Re s > 0. P. 34, line 6 up: Formula should read
i n s t e a d of e 2 ( x  X )
P. 41, l i n e 5: Read
P. 52, equation (4): Read b" i n s t e a d of b a  ' . P. 57, s e c o n d equation (4): Read r ! i n s t e a d of m !
.
P. 61, l i n e 6: Read from instead of form. P. 63, line 2 up: Insert
I
after Iarg(z).
P. 64, equation (25): Read 1+ a  b i n s t e a d of 1  a + b. P. 65, l i n e 2: Insert
I
before t h e l a s t comma.
P. 70, equation (6): Read
r
+ 1  rn
i n s t e a d of rn
P. 76, equation (11): On the righthand s i d e read r ( a + n).
D. 86, l i n e s 7 up and 9 up:
+

1 r (four times).
r ( a + n + 1) i n s t e a d of
Omit ( I)".
P. 88, line 8 up: Read 2.1(24) i n s t e a d of 1.5(24).
Lrrata. Higher T r a n s c r n d e n t a l Functions, vol. I
2
P. 107, equation (36): Read r(6)instead
of [  ( c ) .
'.
P. 108, line 1.5 up: Head D f D ' instead o f D = fl
I
P. 108, e q u a t i o n s (1) and (2): Insert P. 110, l a s t line: Read (1  z
before arg ( 1
)  ~ instead of (1
P. 112, equation (17): Head z ' I 2 instead of z
P. 112, equation (29): Read z 2 ( 2 
 z)'.
.
112
t2/(2 z)~.
z )  ~ instead of
P. 113, equation (34): On the righthand s i d e rend a
P. 116, line
 z )1 .
 b + 1 instead of a  b  1.
4 up: Delete 2.1 (15).
P. 126: Insert
a horizontal rule midway between (20) and (21),
P. 145, equation
(23): Read
r(!/2+ ! $ v  % p ) instead of r(l+ 5;v % p ) .

P. 154, line 10: Read Re p < 1 instead of He v > 1. P. 166, line 9: Read (sin u)lL instead P. 168, l i n e 8 up: Read
of (sin u ) " .
r (v + m + 1) instead of r (v + m).
P. 169, line 5: Read 0 5 8' 0,
where
(4)
y = lim ( m+m
5
l/n
 log r n ) = OS772156649
$
0
n=l
denotes Euler's or hlascheroni's constant. The definition (1)w a s used by Euler, (2)(in a slightly different notation)by Gauss, and(3)by Weierstrass. Replacing t by st in (1) (s real and positive) we get He
2
> 0.
It can be s h o w n [cf. 1.5(34)] that this formula holds for complex values of s and for a path of integration along the straight line from the origin to m e i s . Thus we have . C
 (:in+ 8) < arg s T h i s equation holds for arg s
+8= k 1
n provided 0
< x n
 8,
< He z < 1.
Ile z
> 0.
SPECIAL FUNCTIONS
2
L1
From (2) and (3) i t is seen that the gamma function is an analytic function of whose only finite singularities are z = 0, 1, 2, , From (1) i t follows that
.. .
(7)
=
4'
ett2I
dt+
t
e't''dt=P(z)+Q(z),
Q ( z ) being an integral function. Expanding e  t in a power s e r i e s and inte

grating term by term: m
(8) P
(2)
= n= 0
(1)" [n ! (z + n)]".
Ilence i t follows that (l)"/n! i s the residue of r ( z ) at the simple pole n, (n = 0, 1, 2, ) [cf. 1.17(11)1. It will be shown 'that the expressions (11, (2), (3) represent the same function. For a positive integer n and Re z > 0 repeated integration by parts yields
z=
...

J " ( 1  t/n)"
dt =
tZ'
0
so that
n!n' z(z
+
l)(z
+ 2)
* * * (z
+n)
'
by Tannery's theorem
T h u s (I) is equivalent to (2). Equation (3) can be deduced from (2) as follows. By (2) w e have ~ / I  ( Z ) = Iim z ( l + z ) ( t ~ ! ~ z )* e * (1 + z/n) e' l w n nm
mz) = zeYL ii
n= 1
,
[ ( 1+ z / n )
.
of z i s negative, and n + 1> R e d z ) > n, (n = 0, 1, 2 , . .), r ( z ) .can be represented by a n integral due to Cauchy and SaalschGtz (RhittakerR'atson, 1927, p. 243);
If the real part
(9) I'(z) =
Tow [ e  '  ?= o A
(t)'/rn
!I
(t'' d t
(n+ l)%
R e y > 0.
Substituting t = v / ( l+v), the relation
(2) B ( x , y)
=
p" v '  ' 0
( 1 + v)'Y du
Rey>o
is obtained, and from this
(3) B ( x , y) =
S,' ( v '  '
+ v Y  ' ) ( 1+ v)'Y
Re x
dv
> 0,
Re y
> 0.
can be deduced. It follows that
(4) B ( x , y) If
=
B (y, XI.
[cf. 1.1(4)] i s multiplied by v '  l , integrated with respect to v between 00, and if the order of integration is inverted, we have
0 and
dt
txtY"
v X  l dv = T ' ( x + y)
v'I
(1 + v)"ydv
or
(9)
1
E(n,
m)
= m
(
n+m

') ( =
n
n t m 

')
n , m, positive integers.
1.5.1. Definite integrals expressible i n terms of the beta function
Ry means of suitable substitutions, a number of definite integrals, such as the following, are reducible to the beta function :
SPECIAL FUNCTIONS
10
(10) B(x, y ) = 2
Jol
[(l+t)"' (1t)Y''
1.5.1
+(l+~)Y~(lt)''] Re x
> 0,
dt
Re y > 0,
J b
Rex>O,
Rey>O,
cO,
b 0,
b > 1,
GAMMA FUNCT!ON
1.5.1
11
R e x > 0,
hrn
(3) (sinh t)" (cosh t)P dt = % B ( % a + %,
Re y > 0,
X p  %a)
Rea>l,
(24)
I," e  X ' ( l  e")Y"
(28)
COS
(at)sech (nt)d t = !; sech z
s" cosh ( 2 z t ) sech ( n t ) dt 0
0,
21
< K n,
Il{ez(
< Yi TI.
11.
Forniuln (27)i s known as Rarnanujan's formula. Formulas (E), (13), (17), (19) originate from (1); (11) from (2); (10) and (26) from (3); (14), (15), (20),and (21) from (11); (16) and (22) from (12); (18)froni (16);(24) froni (17);(23) from (22); (25) from (24); (27)and (28) from (26); all are obtained by easily recognizable substitutions or specializations of parameters. Evidently the range of validity of the fornlulas ( l l ) , (20), and (121, (16), (2X), (22) with respect to b can be extended to a n y values of 6 in the complex 6plane supposed cut along the real axis from  1 to  00 and from 0 to  00 respectively. I!y complex intepation it i s possible to express some further trigononketric integrals in tern;s of the gamma function. Consider (z'

z ) z p~  ' dz
where C i s a contour consisting of t h e upper seniicircle (zI = 1 and its diameter. l'he contour i s indented at z = 0, k 1, a n d the radius of each
SPEQAL FUNCTIONS
I2
LS. 1
indentation i s 6. On letting 6 approach zero, one obtains ( c f . Nielsen, 1906, p. 158) the following result:
Rea>l.
If C is
a contour consisting of the semicircle IzI = 1 in the righthalf plane and the straight line joining the points z = rt i, with indentations at z 0, f i, and if the radii of indentation are made to approach 0, the
evaluation of
J^,(z'+ z ) a z J  ' d z gives
Rea>l.
For other similar integrals s e e 2.4(6) to 2.4(10). Next consider
I,
2"'
e''
dz
c>
0,
where the contour C consists of the real axis from + E to + R , the arc of the circle z = R e i d f r o m 6 0 to $=/3(j/2n1 p 5 Xn), the straight line from z = R e @ to c e @, and the arc of the circle z = r e i & from 4 = p to 4 = 0. Since the value of the contour integral i s zero, o n making 6 + 0 and R + 00 it follows that
(31)
t a  ' ,et
cospictsin
a dt: = r(a)c 
a e  iaD
0
%~