INTEGRALS OF
BESSEL FUNCTIONS
YUDELL L LUKE
Midwest Research Institute
Kansas City, Missouri
McGRAW-HILL BOOK COMPANY, INC.
New York Toronto London

INTEGRALS OF BESSEL FUNCTIONS
Copyright © 1962 by the McGraw-Hill Book Company, Inc. Printed in
the United States of America. All rights reserved. This book, or
parts thereof, may not be reproduced in any form without permission
of the publishers. Library of Congress Catalog Card Number: 62-19765
Dedicated to the Memory of
My Father
39075

PREFACE
INTEGRALS OF BESSEL FUNCTIONS deals with definite and indefinite
integrals involving Bessel functions. In numerous applied problems,
evaluation of such integrals is necessary, and this compendium is de-
designed to provide the research worker with the basic information in this
field.
There exists a considerable body of information on the subject of
Bessel functions. G.N. Watson's book A TREATISE ON THE THEORY OF BESSEL
FUNCTIONS (Cambridge University Press, 1923) is classic, and a survey of
material since introduced may be found in the books HIGHER TRANSCENDENTAL
FUNCTIONS, Vol. 2, Chapter VII, 1953, and TABLES OF INTEGRAL TRANSFORMS,
Vols. 1 and 2, 1954, by A. Erdelyi, W. Magnus, F. Oberhettinger and
F G. Tricomi, all published by the McGrav-Hill Book Company, Inc. These
and other vorks contain much material on integrals which involve Bessel
functions.
However, for the most part, the integrals in these volumes are
definite and, although definite integrals are also treated in this took,
I have placed special emphasis on indefinite_integrals. Except for
original sources there is little in the literature on indefinite integrals.
The aim of the present book is to fill this gap.
This volume actually covers a much wider territory than its title
suggests. Bessel functions are a special case of transcendents known, as
generalized hypergeometric series or generalized hypergeometric functions.
Further, many integrals involving Bessel functions are also functions of
hypergeometric type, or may be expressed in terms of such functions. A
valuable feature of the book is the treatment given to generalized hyper-
hypergeometric functions in Chapter I. With these results at hand, many useful
representations of Bessel functions and their integrals follow at once by
specialization of parameters. Further, these results are useful to delin-
delineate properties of other hypergeometric functions and their integrals.
Some short tables of Bessel functions of fractional and integral
order and some tables of integrals of Bessel functions are also provided.
These tables enhance the usefulness of the compendium since numerous math-
mathematical functions can be expressed in series of Bessel functions.
vii

viii
PREFACE
CONTENTS
In a vork of this kind, special precautions have to Ъе taken to insure
accuracy of the formulas. It is a pleasure to acknowledge vith thanks the
valuable assistance rendered by Mrs. Wanda Chinnery, Mrs. Geraldine Coombs,
Mrs. Betty Kaon, Mrs. Marilyn Kemp, Mrs. Betty Ruhlman, Mrs. Anna Lee
Samuels, and Mrs. Carolann Winslow. I am particularly grateful to
Mrs. Chinnery and Mrs. Kahn for their help in proofreading and in preparing
the bibliography and indices. In spite of all checks imposed to insure
accuracy, it is probably unreasonable to believe that the text is error-free.
I vould appreciate receiving any criticisms of the material and the identifi-
identification of any errors that may exist.
To acknowledge all sources to which some debt is due is virtually im-
impossible. The bibliography is extensive. I appreciate the useful advice
and valuable suggestions given by Professor Arthur Erdelyi. For a critical
reading of the entire manuscript and numerous suggestions leading to improve-
improvement of the text, I am indebted to my colleagues, Mr. Jerry Fields and Mr. Jet
Wimp.
Many results given in this book are based on my government-supported
research vork. In particular, I am indebted to the Aeronautical Research
Laboratory, Aeronautical Systems Division, Air Force Office of Scientific
Research, David Taylor Model Basin and the National Bureau of Standards.
I should also like to thank the administration of the Midwest Research
Institute, especially Dr.-Charles N. Kimball, the President, and Dr. Sheldon
L. Levy, Director of the Mathematics and Physics Division, for their encour-
encouragement and generous support of my vork.
Finally, it is a pleasure to thank the typist Mrs. Louise Weston for
her expert preparation of the manuscript for photo-offset publication.
Yudell L. Luke
Kansas City, Missouri
April, 1962
Preface
Page No.
vii
1.1.
1.2.
1.3.
1.3.1.
1.3.2.
1.3.3.
1.3.4.
1.3.5.
1.3.6.
1.4.
1.4.1.
1.4.2.
1.4.3.
1.4.4.
1.4.5.
1.4.6.
1.4.7.
1.4.8.
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
Chapter I
BASIC FORMULAS
Introduction
The Gamma Function and Related Functions
Generalized Hypergeometric Series
Definition and Basic Properties
Integral Representations
Asymptotic Expansions
The Form of Lp q(z) for Special Values
of the Parameters
Special Values of Hypergeometric Functions
Expansion of Hypergeometric Functions in Series
of Hypergeometric Functions
3essel Functions
Power Series Expansions and Connecting Formulae . . .
Expansions in Series of Bessel Functions
Difference-Differential Properties
Wronskians
Integral Representations
Asymptotic Expansions for Large z
Polynomial Approximations
Description of Mathematical Tables
Chapter II
INTEGRALS OF THE TYPE Г tM?u(t)dt
Definitions and Connecting Formulae
Differential-Difference Properties
Power Series Expansions
Expansions in Series of Bessel Functions
Asymptotic Expansions for Large z
Infinite Integrals
Circular Representations of Jn(z) and I Jn(t)dt
Jo
ix
1
2
4
4
5
7
14
18
19
22
22
25
27
29
30
31
33
40
42
44
44
51
53
56
57

INTEGRALS OF BESSEL FUNCTIONS
2.8.
2.9.
Polynomial Approximations
Description of Mathematical Tables
60
69
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
3.10.
3.11.
3.12.
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
4.9.
4.10.
4.11.
4.12.
Chapter III
pZ
REPRESENTATIONS OF tH^(t)dt IN TERMS OF LOMMEL FUNCTIONS
A Theorem on Indefinite Integrals Involving a
Bessel Function
Lommel Functions . . :
Recurrence Relations
Formulae for S p(z) When s v(z) Is Not Defined . . .
Integral Representations
Expansions in Series of Bessel Functions
Lommel Functions and Struve Functions
Anger-Weber Functions
nZ
J t|JVu(t)dt and Formulae for Tabulated Functions . . . .
Fourier-Bessel Coefficients
Polynomial Approximations
Description of Mathematical Tables
73
74
75
76
77
79
80
83
85
89
92
94
Chapter IV
I e'ttMTCp(t)dt AND AN ASSOCIATED BESSEL FUNCTION
Introduction 95
Power Series Expansions and Connecting Formulae 95
Expansions in Series of Bessel Functions 100
Asymptotic Expansions for Large z 101
Infinite Integrals 106
An Associated Bessel Function 107
Recurrence Relations 110
Integral Representations 110
Formulae for H^u(z) When h v(z) Is Not Defined 102
Expansions of h v(z) and H w(z) in Series of
Bessel Functions ' 115
nZ
Associated Bessel Function Representations for J e" t|1Ku(t)dt
and4-Kelated Integrals 117
Description of Mathematical Tables 119
CONTENTS
XI
5.1.
5.2.
6.1.
6.2.
6.2.1.
6.2.2.
6.2.3.
6.2.4.
6.2.5.
6.2.6.
6.2.7.
6.3.
6.3.1.
6.3.2.
6.3.3.
6.3.4.
6.3.5.
6.4.
6.5.
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
Chapter V
REDUCTION FORMULAS FOR
nz
J e-ptt%(n)dt
n Z
Evaluation of J е"рЧ^(и )dt for Special Values
General Development
r>z
of the Parameters
Chapter VI
AIRY FUNCTIONS
Introduction
Airy Integrals
Definitions
Derivatives
Interrelations
Differential Equation and Wronskian
Power Series
Asymptotic Expansions
Integral Representations
Integrals of Airy Integrals
Relations to Other Functions and Interrelations . .
Power Series Expansions
Convergent Expansions in Terms of Lommel Functions
Expansions in Series of Bessel Functions
Asymptotic Expansions
The Integrals of Gi(z) and Hi(-z)
Description of Mathematical Tables
Chapter VII
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
Introduction
Elementary Properties
Integral Representations
Asymptotic Expansions for Large z
Infinite Integrals .
Expansions in Series of Bessel Functions . . . . ,
120
121
127
127
127
127
128
128
129
129
131
132
132
133
133
135
136
140
141
144
145
146
146
147
148

xii
INTEGRALS OF BESSEL FUNCTIONS
CONTENTS
xiii
7.7.
7.9.
7.1С.
7.11.
7.12.
7.13.
8.1.
8.2.
8.3.
8.4.
8.5.
8.6.
8.7.
9.1.
9.2.
9.3.
9.4.
9.5.
9.6.
9.7:
Rational Approximations, Continued
Fractions, Inequalities
The Exponential Integral
Sine and Cosine Integrals
Error Functions
Fresnel Integrals
Indefinite and Definite Integrals Associated vith the
Incomplete Gamma Function and Related Functions . .
Description of Mathematical Tables
Chapter VIII
REHEATED INTEGRALS OF BESSEL FUNCTIONS
Introduction
Power Series Expansions and Differential Equations .
Recurrence Equations
Asymptotic Expansions for Large z
Infinite Integrals
Further Representations
Asymptotic Expansions for Large Parameters
Exponential Series Representations for Ka v(z) . . .
Chapter DC
INTEGRALS INVOLVING STRUVE FUNCTIONS
Introduction
Power Series Expansions
Asymptotic Expansions for Large z
Infinite Integrals
Reduction Formulas
The Complete Cicala Function
Description of Mathematical Tables
Chapter X
SCHWARZ FUNCTIONS AND GENERALIZATIONS
152
163
168
172
179
182
187
195
199
211
212
216
217
219
221
223
223
224
226
227
231
232
10.1.
10.2.
Introduction
Power Series Expansions
234
234
10.3. Expansions in Series of Bessel Functions
10.4. Representation in Series of Circular Functions
10.5. Asymptotic Expansions for Large z
10.6. Infinite Integrals
10.7. Description of Mathematical Tables
Chapter XI
INTEGRALS INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AND STRUVE FUNCTIONS
11.1. A General Theorem for the Evaluation of
Indefinite Integrals
11.2. Integrals Involving the Product of Two
Bessel Functions
11.3. Integrals Involving the Product of a Bessel
Function and a Struve Function
11.4. Integrals Involving the Product of Two Struve Functions
11.5. Integrals Deduced from Wronskians
11.6. An Integral Involving the Product of Three
Bessel Functions
Chapter XII
MISCELLANEOUS INDEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS
12.1. The Integral J(x,y)
12.1.1. Introduction
12.1.2. Partial Differential Equations
12.1.3. Power Series Expansions and Expansions in
Series of Bessel Functions
12.1.4. Laplace Transform and Integral Representations
12.1.5. Asymptotic Expansions
12.1.6. Integrals Related to j(x,y)
12.1.7. Description of Mathematical Tables and
Approximations
12.2. A General Theorem for Representing an Indefinite
Integral Involving Bessel Functions in Series
of Bessel Functions
12.3. Other Indefinite Integrals
238
241
243
246
251
254
264
266
268
269
271
271
272
275
276
278
280
282
283
289

xiv INTEGRALS OF BESSEL FUNCTIONS
Chapter XIII
DEFINITE INTEGRALS
13.1. Introduction
13.2. Orthogonality Properties of Bessel Functions
13.3. Finite Integrals
13.3.1. Convolution Integrals
13.3.2. Integrals Involving Bessel Functions vith
Trigonometric Argument
13.3.3. Lommel's Functions of Two Variables
13.4. Infinite Integrals
13.4.1. Integrals vith Exponential Functions
13.4.2. Weber-Schafheitlin Type Integrals
13.4.3. Sonine-Gegeribauer Type Integrals
13.4.4. Hankel-Nicholson Type Integrals
13.4.5. Integrals Involving the Products of Three or
More Bessel Functions
13.4.6. Miscellaneous Integrals
13.4.7. Integrals vith Respect to the Order
13.4.8. Dual and Triple Integral Equations
Chapter XIV
TABLES OF BESSEL FUNCTIONS AND INTEGRALS OF BESSEL FUNCTIONS
Introduction
Table I. Jn(x) , Yn(x) , n = 0,1
Table II. e"xIn(x) , eXjCjx) , n = 0,1 , ex
Table III. Jn(x) , n = 2(lN
Table IV. e"xIn(x) , n = 2(lN
Table V. (TT/2xJJn_i(x) , n = 0(lL
Table VI. Jy(x) , v = ±1/4 , +3/4
Table VII. Jv(x) , v = +l/3 , +2/3
Table VIII. x"^n+?^In+i(x) , e"XIn+i(x) , n = 0(lL
Table IX. Iu(x) , v = +1/4 , +3/4
Table X. Iu(x) , v = +l/3 , ±2/3
290
290
292
292
293
308
312
312
324
327
330
331
335
340
342
349
350
353
356
359
362
365
368
371
374
377
CONTENTS
xv
Table XI. Integrals of JQ(x) and Yo(x) 380
Table XII. Integrals of IQ(x) and KQ(x) 383
Bibliography 336
Index of Notations 404
Author Index 4^0
Subject Index 4^4

CHAPTER I
BASIC FORMULAS
1.1. Introduction
The purpose of this chapter is to collect numerous formulas needed to
establish results of subsequent chapters, and to aid in the evaluation of
these and other expressions. Some general sources for this material are
Erde'lyi et al. A953, 1954), Kratzer and Franz (i960), Rainville (i960),
Watson A945) and Whittaker and Watson A927). Also, for tables of inte-
integrals and related data, see Grobner and Hofreiter A949, 1950), Magnus and
Oberhettinger A948, 1954), Meyer zur Capellen A950) and Ryshik and
Gradstein A957). For further material on Bessel functions and integrals
of Bessel functions, see Bowman A958), Gray, Mathevs and MacRobert A952),
McLachlan A955), Petiau A955), Relton A946), Rey Pastor and Castro
Brzezicki A958), Weyrich A937), Wheelon and Robacker A954).
Description of numerical mathematical tables of integrals involving
Bessel functions is usually given at the end of the chapter where the per-
pertinent material is discussed. In some instances, especially in Chapter
XIII, this information is integrated with a discussion of the tabulated
functions. We have attempted to give thorough coverage of tables since
1945 approximately. Some, but not all prior data are also referenced.
For the sake of completeness, some references for tables of Bessel func-
functions are presented in 1.4.8. As most of these items are standard and
well known, usually only an abbreviated description of the contents of
each reference is provided. For description of numerical mathematical
tables, not only of integrals involving Bessel functions, but also of the
entire spectrum of transcendental functions, see Bateman and Archibald
A944), Fletcher, Miller and Rosenhead A946), Lebedev and Feodorova
A956), Burunova A959), the journal Mathematical Tables and Other Aids
to Computation (now Mathematics of Computation) A943-present), National
Bureau of Standards A962) and Schiitte A955).
Collections of analytical descriptions and numerical tables of trans-
transcendental functions are provided in Jahnke and Emde A945), Jahnke, Emde
and Losch (i960) and National Bureau of Standards A962).

INTEGRALS OF BESSEL FUNCTIONS
1.2
1.2
BASIC FORMULAS
m-1
1.2. The Gamma Function and Related Functions
T(z+1) = zr(z)
Г(п+1) = n.'
(a)k = Г(а+к)/Г(а) .
(*) = *•' = (.)k ("П)к
V k.'(n-k): ^ ;
k.'
T(z)r(l-z) = тт esc ttz
rD+z)r(|-z) = тт sec ttz
i, .. , m-1
r(mz)-= Bп)*{1-т)тШ-* fT r(z+r/m) .
r=O
T(z) = Btt)* [exp |-z+(z4)ln z+ ^jj [1+0G 3)J > lar« z'< п
^-т^^-тШ ¦
ф(г+1) = + (z) + l/z
ф(п+1) = X к + t|r(l)
к=1
¦ (z) - К12) = -тт с0* TTZ
A)
B)
C)
D)
E)
F)
G)
(8)
О)
(Ю)
A1)
C2)
y(mz) = m 2! ¦(z+5s/ni) + In m
k=0
[\a-\x.tf-ht - 2 f/2 (sm e^-'Ccos eJ^1,
i/Q ^0
A3)
= IS^J R(a)>0'R(p)>0 ¦
A4)
е^1иттГ
О 2-Г Bl? +1) Г (?Ё2 +1)
f Vve(ein efd6 = n^rfan)
Jo 2°^ rs^i +1") Г CSES2.
, R(a)>-1 . A5)
tt/2
cos u9(cos e)ade = i в"*1"" / e1Ue(sin 9)ade , R(a)>-1 . A6)
ivQ, .a
cos Q) cL9 =
^i(u-a>rT
R(a)>-1
For F. , see 1.3.
Some useful constants are as follows.
тт = 3.14159 26536 ; T(l/2) = тт^ = 1.77245
38509
Bтт)^ = 2.50662 82746 ; Bтт) = 0.39894 22804
(тт/а)* = 1.25331 41373 ; (а/тт)* = 0.79788 45608
A7)
Y = -t(l) = 0.57721 56649
A8)
A9)
B0)
B1)

INTEGRALS OF BESSEL FUNCTIONS
1.3.1
фA/2) = -Y-21n2 = - 1.96351 00260
УA/3) + уB/3) = - 2Y - 3 In 3 = - 4.45026 81958
f'(l) = тт2/6 = 1-64493 40668
B2)
B3)
B4)
1.3. Generalized Hypergecmetric Series
1.3.1. Definition and Basic Properties
We consider the series
pF*C*^+b2Z,i+4lz) =A(V1+V2)
ЙЬ-A+Ъ1)кA+ъ2)к- • • d+\)kk'
A)
(a)Q = 1 , (a^ = a(a+l)...(a+k-l) = r(a+k)/r(a) .
B)
Where no confusion will result, ve notate the above series as F-
is often convenient to employ a contracted notation and write
pFq(ap;l+bq;z)=Zo(i+-Vkk;
It
C)
Thus Г(ар+к) is interpreted as |~[ Г(аг+к) ; and (ap)k , as
r-1
P
ГТ (ar)k > etc- ^ emP'fcy term is interpreted as unity. For example,
r=l
if p = 2 , (ар)к = 1 for p>2 .
1.3.2
BASIC FORMULAS
pFq is not defined if any Ъ- is a negative integer. It terminates
if any ap is a negative integer or zero. If pFq does not terminate, it
converges for all finite z if p s q , converges for |z| < 1 if p = q+1 ,
1 q+1
converges for z = 1 if p = q+1 and Re ^ У (l+b>) - У av |> > 0 , and
С q q+j- ]
= q+1 and ReJy_ (l+bk) - У_ ak I > 0
lk=l k=l J
diverges for all z ^ 0 if p > q+1 .
If 6 is the operator zD , D = d/dz , then u = pFq is a solution
of the differential equation
[6F+b1)F+b2)...F+bq) - zF+a1)F+a2)...F+ap)] u = 0 . D)
If p s q+1 , then
ui(z) = z" Уа(аР"Ъ1Д+?Ъ1;^
i = 0,1, ...,q, Ъо = 0 , i ^ q in 1+1^-1^ ,
E)
are the (q+l) linearly independent solutions of D) provided that no two of
the bi's differ Ъу a negative integer or zero. In the singular cases just
mentioned, solutions can Ъе constructed as in the case of Gauss' series,
Bessel functions, etc. We omit further details, but see Erdelyi et al.
A953, Vol. I, Chs. 2,6 and 7) and the references quoted there. See also
Chapter VIII where a study is made of a particular 3^3 - If p й q+1 ,
t.here are p formal solutions proportional to Ьр„ , t = l,2,...,p, see
1.3.3(8).
.3.2. Integral Representations
.^Vj.CP'V»1».^) " Xw"//'1'""'''''''"'"* ' A)

INTEGRALS OF HESSEL FUNCTIONS
1.3.2
p+lFq+l(p^ap^P+<T^qAz)
_ 2Г(р-Кг)
,"/2
Ц^-J (em efP-Vs ej^-^apip^jxz sin2e)de , (з)
,V2
= |g^yJ (cos efP'^sin в^-^Ор^Э^Хг cos2e)d8 , D)
p s q+1 , R(p)>0 , R(a)>0 ; |Xz|< 1 if p = q+1
p+lF4(ff,apiPq;X/z) = * / e-^t0 F (a^fi„;Xt)dt ,
1 ^CTJ J Q
p s q , R(ct)>0 ; R(z)>0 if p < q. ; R(z )> R(\) if p = q.E)
2ст
p+fiFa(a,a+V2,apipqj4X2/z2) = ^y J e-^t^-^F^Cap^jX2^)^
p<a , R(ff)> 0 j R(z)>0 if p<q-l ; R(z)> |R(X)| if p = q-l . F)
n*j
x 1-Г(.*)Г(а+г)(-г)*
гТэ^Т pF*(VfV2) = 2^г/_1оо г(фъ)
G)
dt
where |arg(-z)|< n and the poles of T(-t) lie to the right of the contour
while those of r(op+t) lie to the left of the contour.
1.3.3
BASIC FORMULAS
1.3.3. Asymptotic Expansions
Consider the divergent series
k=0
-k
A)
and denote the sum of the first (n+l) terms Ъу Sn(z) . With certain re-
restrictions on arg z j (l) is an asymptotic expansion of f(z) if for n
fixed,
111 2%(г) = lim zn (f(z) - Sn(z)} =
B)
z —* <=
That is, we can make lznRn(z)|<e with e arbitrarily small and |z|
sufficiently large. The notation
f(z)~ Z akz
k=0
-k
C)
means that (l) is the asymptotic expansion of f(z)
notation is
An alternative
f(z) = Sn(z) + OOz-11!)
D)
We now present asymptotic expansions of pFq for Izl—> => and
0 s p s q . We follow the work of Meijer A946) which should Ъе consulted
for further details. Assume that no ap is a negative integer or zero.
Let
К^Ы - [exp {u"V}] zvBTT)*^-^u* ? Nkz^ , NQ = 1 ,
k=0
E)

INTEGRALS OF HESSEL FUNCTIONS
1.3.3
)" > v = u <
P 4
u = (q-p+l)"X , v = u \ (q-p)/a + Z ak - Z A+Ък) f1 > F)
t k=l k=l
and for the determination of the N^'s , see the discussion surrounding
A9)-C8) ahead.
Define
t=l
G)
"at
Oz>=
* T(at) TT r(aj-at)
,1=1,,1/t
TTrd+bj-a,.)
ч+1гр
.VlK>4-V1+at-V^ z~ )> (8)
.^^-P,-1^
where aj-at is not an integer or zero for all j ? t . This restriction
may Ъе removed, but further discussion is deferred to 1.3.4. Let
P , 4
MP,q= TT r(aj)/TT rd+bj!
J=l ' j=l
(9)
The asymptotic representations are divided into two cases.
First Case. 0 ? p <: q-1 .
(Mp,q) pVVl+Vz^K^z) ,
Izl—»°» , larg z|< tt
A0)
1.3.3
BASIC FORMULAS
If 0 s. p < q-1 , then
(Mp,a) pF^fapjl+^j-z)- Kp>q(ze1TT) + Kp^(ze"iTT) ,
0< Z^a>
(ID
In the latter two equations, only the dominant expressions are recorded.
There are (q. - p+1) terms of exponential type, i.e., of the form 1С ,. ,
and the omitted ones are of lower order than those given above. To com-
complete this case, we need the result where p = q-1 . In this instance, we
give a single expression which includes both the dominant and subdominant
parts. Thus
^P+iVWv^Vi^-^^P+i^117)+ ^p+i^17) + Lp,P+i(z)^
|z|—><» , larg z|< атт
A2)
For numerous applications, it is convenient to replace z Ъу z2/4 and
write
^p+iVp+i^p^+vi'-^/^^^p+iCf*2^1"]2)
+ wiG*26"*117]2)+ ^p+i(z2/4) -
|z| —»<= , larg z|< tt
A3)

10
INTEGRAIS OF BESSEL FUNCTIONS
1.3.3
1.3.3
BASIC FORMULAS
11
Also
О
|Z|-
, -(a+e)TT/2<arg z<B-s)tt/2 , e = *1
A4)
The apparent discrepancy in A4) when z has a value such that
larg zl< tt/2 is a case of Stoke's phenomenon. See Watson A945, p.201).
Second Case, p = qal .
Again we give the complete representation.
(MP,p)pFP(ap;1+bP;-z)~Icp,p(ze"iTT) + Wz)
|2|_^.o> , - тт/2 < arg z< Зтт/2
(MpjP)pFp(ap;l+bp;-z)~Kp^p(zeiTT) + LpjP(z)
I z| —> => , arg z = - tt/2
A5)
A6)
Also
ien-ч
(Mp,p)pFp(apJl+bp;z)~Kp)P(z) + I^p(ze )
lz|-
.a, , -B+e)TT/2<arg z<B-e)n/2 , e = ±1
A7)
If larg z|< tt/2 , the apparent discrepancy in A7) is again a case of
Stoke's phenomenon.
The following is an important corollary of the above expansions. If
p = qai or if q-1 = p^O , then
lim
ГЫ) ГТ r(aj-am)
xampFq(apJl+bq;-x) = ?Ы&
(Mp^TTrU+VaJ
j=l
R(am+v)<0 ; R(am-at) <; 0 for all t = 1,2,... ,pj
(aj-am) is not an integer or zero for all j ^ m .
A8)
Evaluation of the coefficients Nk is next of interest. In this
connection note that in the Kpq(z) term associated with the asymptotic
representation of Uj_(z) , see 1.3.1E), the coefficients Nk are inde-
independent of i . The Njj's are conveniently found by constructing a formal
series solution of 1.3.1D) using E) and A0). For later use, we record
in complete detail the expansions 1С, (z) for the cases q = p+1 where
p = 0,1 and 2 ; and p = q. , where p = 1 and 2 . These are as follows.
Case I. q. = p+1 .
From E) and (б), with z replaced Ъу z /4 , we have
w(M2) ¦ Ms I
' co
с„ = 1
v = -^i+ ? ak- Z (^k)j =h
k=l k=l J
A9)
B0)

12
INTEGRALS OP BESSEL FUNCTIONS
1.3.3
Case I.I, p = 0 , q = 1
If Ъд_ = v ,
ck
(^v)k(i-v)k
2kk.'
B1)
and this leads to the asymptotic expansions of Bessel functions. See
1.4.6A-6).
Case 1.2. p=l,5=2 ¦
2(k+l)ck+1 = fkck + gkCj,.! , d = 0 , co = 1 ,
B2)
fk - 3k2 + k(9-6p-4gl) + 3p2 - 9p + 3 + 2Э!Bр-1) + 4(Э2 , B3)
Sk = :
-(к-р-1)(к-р-1-2Ъ1)(к-р-1-2Ъ2) ,
B4)
'Pi = Ъ1 + Ъ2 + 3 > Э2 = A+Ъх)A+Ъ2) , p = a-L - &-,_ + 3/2 . B5)
If a-j_ = 1 and Ъ2 - Ъд_ = v , then' ck is given by Bl).
Case 1.5. p = 2 , q = 3 .
2(k+l)ck+1 = fkck + gkck_i + hkck_2 , C-2 = c_i = 0 , co = 1 , B6)
fk = 5k2 + 2k(l2-5p-3e1+a1+a2) + 5p2 - 22p + 15 + 6Вд_(р-1)
+ 4g2 - 2p(l+a1+a2) - 4аха2 ,
B7)
1.3.3
BASIC FORMULAS
13
- gk = 4k3 + 6k2C-2p-g1) + 2к(бр2-18р+6рЭ1-ЗЭ1+4Э2+4)
- 4p3 + 6p2C-31) - 2рD-ЗЭ1+4Э2) + 6p! - 4g2 - 8З3 - 15 , B8)
hk = (к-р-2)(к-р-2-2Ъ1)(к-р-2-2Ъ2)(к-р-2-2Ъ3) ,
B9)
Эх = 6 + Ъх + Ъ2 + Ъ3 , Э2 = ЗЭд_ - 11 + ЪдЪз + ЪХЪ3 + Ъ2Ъ3 , C0)
Э3 = A+Ъ1)A+Ъ2)A+Ъ3) , р = а1 + аз - Эх + 7/2
C1)
Case II. р = q .
Replace 2 Ъу 2z . Then E) and F) yield
_2z,n_,v
-k
Kp;pBz) = e^Bz)v Z ck^"K , co = 1 ,
k=0
C2)
v = Z (ак-1-Ък) •
k=l
C3)
Case II. 1. p = q. = 1 .
A+Ъ1-а1)кA-а1)к
с = }
2Kk.'
C4)
nnii this also leads to asymptotic expansions for Bessel functions. See the
• uii'luding remarks of 1.4.6.

14
INTEGRALS OF HESSEL FUNCTIONS
1.3.4
Case II.2. p = q. = 2
4(k+l)Ck+1 = fkck + gtfb.! > c-l = 0 , co - 1 >
C5)
fk = 2 [2k2 - kBv+a1+a2-l) + v(a1+a2-l) - aft + A+Ъ1)A+Ъ2)] ,C6)
Sk
= - (к-1-т)(к-1-т-Ъ1)(к-1-т-Ъ2)
C7)
If
= 1 , 1 + Ъ1 = ax + v + 1/2 , 1 + Ъ2 = ax - v + l/2 , C8)
then ск is given Ъу B1).
1.3.4. The Form of Lp ^(.z) for Special Values of the Parameters
The definition of L_ „(z) Ъу 1.3.3G-8) conveniently supposes that
a--a.j. is not an integer or zero for all j f t . If this is not so,
Ll „(z) still has meaning and its representation must Ъе found Ъу a limit-
limiting process. We now ottain the representation in a special case. Assume
first that no pair of a. values differ Ъу an integer or zero except a^
and &2 ¦ Let ag = lim (a^+m+e) where m is a positive integer or zero.
е-н»О
Before stating and proving the main result, it is helpful to introduce fur-
further notation. Let
ХР,Ч
н1 (- )k(m-l-k).'r(a1+k)r(ap-ai-k)
k=0 к.'ГA+ЪA-а1-к)гк
(а1'и)(о - z-ai ~tч-/ ч --1—p
(i)
1.3.4
BASIC FORMULAS
к
15
rFs (*p iiYr ii г\ - т ^w L
,t(Yr+k)-^(Yr)-1((l+3a+k)+t(l+Pa)> , B)
where it is understood that the notation is compact as in 1.3.1C), and
r
further that ф(уг) is short for ^1 t(Yj) > etc- Then
j=l
т f , 4" t* f ^ (al'm)/ n (-)m+1^"a2r(a )r(a -a )
v Ч+lpP f a2'a2 II a2^a2-\
¦X q+i p-i V^m+1, l+a2-a II I,m+l,l+a2-a
)Q.-P
z
)
(-Л'^ГСа )Г(а -a ) f
+ _ ? |щ 2-t(a2)+t(ap-a2)+t(l)-t(l+ba-a2)+t(m+l)J
т.ТA+Ъ -a2)
Г a2^a2-\ K-^-P-
XF ( ^ I ' ' l
q+1 p-lVm+l,i+ap-a_l z У
'2"ap'
C)
where m = a2-a^ is a positive integer or zero. Here it is understood that
p fi 1 , p ^ 2 in the compact notation T(ap-a2 ) , etc.
In the applications, it happens often that ag = Ъ„ for some particular
value of q. . In particular, suppose 4 = 1. Then

16
INTEC1AIS OF HESSEL FUNCTIONS
1.3.4
1.3.4
BASIC FCEMULAS
17
P
z
t=3
(a^m)
bp,a(O- Z 4,a(') + Tp*
(O
)
(. fz'I'a2r(a2+l)r(ap-a2-l) , 1,1,1+82, l+«2-\ (-)^p
+ (m+D-TCV^; a+2FP^2,iB+2,2+a2-ap
(-Г2-а2Г(а2)Г(ар-а2) , ^^^ ИЦ+Ъ.-а^(m+l)} - D)
m.Tfl+K-ao) I
Again the
q. = 2,3,...q.
i.T(l+ba-a2)
notation is compact and in ^l+b.-ag) , фA+Ъ -a2) , etc..
To prove C), we write
[г"а1Г(а»с)Г(а1)Г(ар-а1)
/ аРа1Л (-)Ч"^Л
X q+1 P-14l-m-e,l+a1-a z ^
-ai-ia-sr(_m_e)r(a^+m+e)r(ap.al-m-e)
ГA+Ъ,1-а1-п1-е)
Caj_+ni+*,aj_+nH-e-bq^
1+m+e, И-а^т+в-а^
-Д-Р
)
E)
Now
m-1 fa \ zk
+ -bkz*
l,a_+m
Y X'V
(l+ЪAm.' P+l 4+lVm+i,t +m
•)
G)
Apply this to the first q+iFp_i in F)- Then with the aid of the reflec-
reflection formula for gamma functions, we find
(a^m)
1 О (8,T .111 ]
h,^z) + 4^z) = Tv,l (z) + llm A(e) '
(8)
e—>0
where
<->-^«=^te^^p
Ч.1-е,пм-1,
1+а2-ар
Ч-Р
^
z'er(l-e)r(a2+e)r(a -a2-e) , ag+s^g+s-b^
Г(ш+1+е )ГA+Ъ„-а2-е) 4+1 P Vm+i+e,i+a2+e-a
N4-P
0)
Use L'Hospital's theorem to evaluate A@) . Then this result together
with A), B) and E)-F) readily leads to the assertion C).
F)
Put (а2-Ъ1) = 6 in the third expression of C). Note that F)k = 1
! if к = 0 and lim F )k = 0 if k>0 . Also aF) = F )k [tF+k)-i|tF)] = 0
if к = 0 and Un а(б) = Г(к) , k>0 . Thus the a+1^_x readily sim-
6—>0
plifies and D) follows. Alternatively, we can put (а-^+т-Ъ^) = (ад-Ъ^) = О
in (9). Then the n+2Fp is unity and application of L'Hospital's theorem,
etc., gives D).

18
INTEB1A1S OP HESSEL FUNCTIONS
1.3.5
1.3.6
BASIC FORMULAS
19
If another pair of a, 's differ by a positive integer or zero, say
a4-a3 = m1 , but (a4-a1) is not an integer or zero, then the above limiting
analysis can Ъе applied to evaluate I^q(z) + Lp,q.(z) > and further like
extensions for other pairs of aj's are apparent.
If a2-a-j_ and a3-a2 are positive integers or zero, but no ap-Sj >
j = 1,2,3, p j* j is a positive integer or zero, then as before we can de-
3
termine ^1 *v q(z) bye. limiting process, and so obtain a representation
t=l
for Lp q(z) . The approach for more general situations where ap-aj is an
integer or zero is obvious. Results of this type hold no immediate interest
and no further representations have been obtained.
1.3.5. Special Values of Hypergeometric Functions
(-^(b-c+l^
2F1(-n,n+b;c;l) j-y-
/ -n,a,b | \ =
3 2kc,a+b+l-c-nl J (c)n(c-a-b)n
(=-а)п(=-ъ)п
rn,n+a,b . (c-b)n(a-c+l)n
3*2(чс,а+Ъ+1-с1 J (c)n(a+b+l-c)n
a.UU1)-1 lf П = 0 '
A)
C) f
D)
а(а-Ъ)пп-'
(a-b)(a)n(b+l)n
, n> 0
E)
In the above, none of the o^i or 3^2 denominator parameters are
negative integers or zero. Also n is a positive integer or zero. For Я
other special values, see Erdelyi et al.(l953, v. 1; Ch. 2, pp. 104-105;
Ch. 4, pp. 188-191).
1.3.6. Expansion of Hypergeometric Functions in Series of
Hypergeometric Functions
. a_ ,c
p+r
v.(b:;/k)=s
' s
(ap)k(ot)k(-Z)k
?b "(ъа)к(Ри)к^+к)кк.'
\/ F ( k+ap^k+at ,Л
-k,k+y,cr,pu|
X r+u+2Fs+tQ dg,^
-)
• а„,с„
м+БЧл' ^= k?01№"
k+ap,k+at|
A p+tJ!q+ulvk+bA,k+guIV
X
r+u+1 s+t
(
-k,cr,0u
ds'°1:
')
A)
B)
(l) is a generalization of many known expansion formulae and has been
proved by Fields and Wimp A961). B) is a confluent form of (l) and
follows from the latter upon replacing z by yz у w Ъу w/y and letting
Y-*co . Meijer A952, 1953, p. 355, A13)) has studied expansion theorems
for the G-function and B) is a special case of his results. For some
special cases of B) as well as other expansion formulas for hypergeometric
functions, see also Meijer A954, 1955, 1956) and the references given
there.

20
INTEGRALS OF BESSEL FUNCTIONS
1.3.6
1.3.6
BASIC FORMULAS
21
Some special cases of (l) and B) follow.
-api An
» (c).Bk+c)
p<;;i-?)-*>^>\?0-^-j-
(*)
с;г» •
рЧъа
2яЛ = ?
Р+2Ч^ Ъ
,2 с
(V2)CB^)C
2^(l)c(l/2)cIc(z)
+ 2 Z
к=1
(-f(k+c)(k) -к,к+2с,а
к ? W*WFq+1(l/2+c^ P|w
-к. а
C)
D)
а<?-1- ?)¦r(c)B/zrl к?0 ^ wi(-w,C ЧР1v>E)
Неге J\j(z) and Iv(z) are Bessel functions, see 1.4. For special values
of the parameters and variable v , the hypergeometric polynomials in
C)-E) can be expressed as a product of gamma functions, see 1.3.5. In
this manner we get
2F2(a^|2z) = е^|Г!ЬсA)сA/2)с1сB)
\ъ1,ъ21 у (V2)c L
k+cv
" (-)K(k+c)(kJe(bra1)kBa2-b1)k 1
a2 - l/2 , aj_ + 2a2 = bj_ + b2 ,
(8)
2^2 ( ^ I 2^
2 2Vl+a,2bl J
= eZB/z)Cr(c+l)|lc(z)f
The expansion
(-)>c)Bc-al
Bc-a) ^
с = Ъ - 1/2
A+ak
^fcC2)
0)
(-)kBa-D,
A(^h2)= (^f^rWric) Z ? ^Jk+b-l(z)Jk+c-l(O
k=0
с = 2a+l-b
A0)
(d)kBk+d)(b-a)k(c-a)k
ACM"'8'4) " F(d)B/Z)d ? k:(b)k(Ok J2-(z)
d = b + c-a-l ,
(Ь-а)„
F)
1Р2(ъ!с|-22/4) = Г(с)B/2Г1 ? kT(blJ(z/2)kj^c-i(z) ' G)
follows from A) and 1.4.1A8). For a discussion of D) when p = q. = 1
and z is replaced by z/2 , see 7.2C) and 7.6B). Some other expansions
in series of Bessel functions are given in 1.4.2. See 7.6A,3) for two
further special cases of (l).
Elsewhere in this volume we give representations of transcendental
functions under discussion in series of Bessel functions. These expansions
are advantageous since excellent tables of Bessel functions are available
for desk computation, while for automatic computation they are easy to gen-
generate. In this connection, see Stegun and Abramowitz A957), Goldstein and
Thaler A959), Corbato' and Uretsky A959) and Gautschi A961 a,b).

22
INTEGRAIS OF BESSEL FUNCTIONS
1.4
1.4.1
BASIC FORMULAS
23
1.4. Bessel Functions
1.4.1. Power Series Expansions and Connecting Formulae
where A and В are constants, independent of z though they may depend
on i» , is called a cylinder function. It can represent any of the Bessel
functions of the first, second or third kinds.
'¦<»> ¦ (z/2)\i Ш^ - Ш? o*i(—2л)
(i)
k=0
n-1
TTYn(z) - 2{у«л z/2} Jn(z) - Z (n-k-l)J^/2)
ч-п+2к
k=0
is the Bessel function of the first kind; z is called the variable, v the
order.
J"(z) = e"r(^)V A^^+i^iz) .
B)
?Г0 k.'(n+k): V. 2 к 2 n+ky1
Iu(z) = e'2lUTTJ1,(ze^1TT) , -TT<arg z ? тт/2
О)
^(гештт) = eiml'TTJl;(z) , m an integer . C) *
Yu(z) = (esc im)[(cos im) Ju(z) - J_u(z)] D)
= e^lmJv(ze'i5±TJ) , тт/2 < arg z ? тт .
^„(z) = (n/2)(csc utt) [l_u(z) - Iu(z)] .
A0)
(H)
is the Bessel function of ^he second kind.
raj (z) 9j (z)i \ыЫ)]
:n(z) . [_|__ . (.)n _^_^ , (TT/2)Yo(z) . ^!__ju=o . E)
ttX
Yu(zeinirT) = e"lml'TTY1,(z)+2i(sin m\m)(cot utt)Ju(z), m an integer . F)
hJ^Cz) = Ju(z) fiYv(z) ; 42)(z) = Ju(z) - iYu(z) . G)
Ну (z) and Ну (z) are known as Bessel functions of the third kind.
They are also called the first and second Hankel functions, respectively.
Iy(z) and Ku(z). are called the modified Bessel functions of the first
and second kind, respectively.
"dl (z) Э1 (z)
2I^(z) = (-)П ' ~V
Эи Эи
1 lblv^}
^(z) = ^ie^^^dz) .
A3)
/ imrr4 -imWTT / \ irr(sin iqutt ) T / \ . , /,, \
Ku(ze Ш|) = e Ku(z) Ц —<¦ Iu(z) , ш an integer . A4)
sin utt
Cv(z) = AJu(z) + BYv(z) ,
(8)

24
IHTEGRAIS OF HESSEL FUNCTIONS
1.4.1
2K^(z) =2(-)n+1{Y+l^ 2/2}ln(z)+
n-1
z
k=O
(-)Ук-1).'(г/2)-П+2к
k.'
A5)
±z, o л-п 2n-l
Kn(z) - 8 fciV Z Bn-l-k).T(i-n+k)(±2z)li
1 ^; k=O
+ (-)П+1Aл 2z)In(z)
^'Уц^ Z r(^ffz)k{t(^i)-t(n+^H(an+k+i)} . A6)
The last follows from (a), -A0), and A2). In all the above formulae
-TT < arg z s тт .
^+u+l ц+ц+2
/./P^+U Z' 2 ' 2
1Ъ/а| < 1
A7)
A8)
,2 v
j2(z) = (z/2) Р2(^«+1,2и+1;-22) .
[r(i>+i)J2
A9)
1.4.2
BASIC FORMULAS
v2u
J.u(Oiv(Z) = ^sllt)TT) оР3(л+^,1-*»,*;-г4/б4)
25
B0)
2 (sin im)(z/2)':
., ЛЗ+и 3-и 3 . 4/ A
.(i-,2) °4--'-'2-'-z/64;
B1)
1.4.2. Expansions In Series of Bessel Functions
z sin 9
CO 00
= Iq^)*2 21 (-)\^)^°B 2k9f2 Z (-)\k+i(z)sinBk+l)e .
A)
k=l
k=0
z cos 9
I0(z) + 2 Z Ik(z) cos k9
k=l
B)
(z/2)^+1 = X {^^У^1) Ju+2k+1(z), „ not a negative integer . C)
k=0
"Jj d ц+2к+1
1 = Jo(z) + 2 Z J2k(z) = ^) + 2 Z Jfc(z) •
k=l k=l
D)
00 / \k
-2, /ox|i+l Г(ц+1) ^- (-)д-(и+к+1)гBц+к+2)
(z/2) = ш^1 z k; ^Iu+k+i(z) >
k=0
ji not a negative integer
E)

26 IMEGBAIS OF BESSEL FUNCTIONS 1.4.2 ^3 BASIC FCKMUMS 27
e"z
= Io(*) + 2 Z (-^@ ¦ F) a2lw(z)
k=l
a^
v=0
= -2(Y+ln z/2)KQ(z> - [тт2/б+(у+Ш z/2J]lQ(z)
l
k k For tables of A1) to 5d and A2) to 4d for z = 1AI9 ,
, , , , , r ,i , , ' " (z/2) Jk(z) Jl (-) J2k^z) u = (z-l)(O.l)(z+l), see Airey A935). Lee and Radosevich (i960) have
(rr/2)To(z)- (Y+ln z/2] Jo(z) = X ?~— = -2 Z f— ¦ (8) tabulated A1) to 4d for z = 1@.5I5; v = 1/4AI7/4 , 1/3AI3/3 ,
k l k -1 2/3AI4/3 , 3/4AI9/4 . If v = ±l/2 , aJu(z)/9u can be expressed in
terms of sine and cosine integrals, see 7.9A8,19).
K0(z)+ [Yrta z/2} io(z) - - ? <-*¦***<¦>. 2 ? ЦП . (9) ,
^ ^ С^) = ^?й-!Й^Си+кB),|1-Х2|<1 . A4)
k=0
^(z) = -(-f {-t(n+l)+ln z/2} In(zHn.4i.r 2? ^(n^^ If Cu(z) = Jp(z) , the restriction on X can be dropped.
+ (.f ? (°^?w(') . A0) B sin 3) (z cos 6) _ ? (t, B^r2aIT w(O ,
k=1 k(n+k) , k=0 k.T(u+k+l)
1.4.3. Difference-Differential Properties
" ^.Л2/ j- 2 2i Throughout this volume, we let Ww(z) represent any of the Bessel
= n(Y+ln z/2)Yo(z) - [n /6 + (yfln z/2) JjQ(z) functions of the first three kinds or the modified Bessel functions. Ir
auc
v=0
functions of the first three kinds or the modified Bessel functions. In
formulae involving Wy(z) there appear two parameters a and b which
, . .ъ. > are linked with the particular type of Bessel function as delineated in
_ 2 — { ' ' ^ ' /12^ the table below.
k=l k.'k2

28
INTECSAIS OF BESSEL FUNCTIONS
1.4.3
1.4.4
BASIC FORMULAS
29
VZ>
Ju(z), Yv(z), H^z), 42)(z) 1 1
Iv(z) -1 1
Mz) ! -1
We have the following formulae.
aWu+i(z) + Wj.-^z) = Bo/z)W1,(z) .
-aWu+1(z) + bWp.1(z) = 2WJ(z)
zVj(z) + uWv(z) = bzWu.1(z) .
zVJ(z) - Wv(z) = - azWu+1(z) .
(z'Vazf (z\(z)} = bV-'Vjz) .
(z-Vdzf {z-%(z)} = (-afz-v-mW1,+m(Z)
(m). . -m -— . .k/mx к m-k . .
w; '(z) =2 I(-)(k)»b W,,.m+2k(z)
k=0
In F)-(8), m is a positive integer or zero.
[z2!J + zD + (аЪг2-и2)] Wu(z) = 0
Clearly Cv(z) can replace Wu(z) in B)-(9) if a = Ъ = 1
A)
B)
C)
D)
E)
F)
G)
(8)
О)
A general solution of the differential equation
[z2D2+z {(l-2a)+2e6ze} ^2Y2z2^e262z2e+e6(e-2a)ze+a2-P2V2] у = 0 A0)
is
y=exp {-6ze} zaCu(YzP)
(H)
where a, p, Y> 5 and e are constants, Py / 0 . If p ¦ 0 ,
and if у = 0 ,
у = exp |-6zel za(c^+C2 In z) ;
у = expl-ezejza-^tc^z2^)
A2)
A3)
unless pu = 0 . In A2)-A3), C]_ and c2 are constants of integration.
For other results- of this type, see Watson A945, pp. 95-99), and Rey Pastor
and Castro Brzezicki A958, pp. 205-222). See also the compendiums of
solutions of ordinary differential equations by Kamke A948) and Murphy
A960).
1.4.4. Wronskians
Define
W (u(z),v(z)} = u(z)v'(z)-u'(Ov(z)
We then have the following relations.
W {ju(z),J.u(z)} = " C2/™) sin u
A)
B)

30
INTEGRALS OF BESSEL FUNCTIONS
W {ju(z)Jyv(z)j = 2/ttz .
W{4l)(z),42)(z)}= - 41/ttz .
w{lu(z),I_v(z)j = - B/ttz) sin m
w{lu(z),K,,(z)} = - 1/z .
1.4.5. Integral Representations
-l f2n
Jn(z) = Bтг) / cos(n9-z sin 6)d6
Jo
tt/2
J2n(z) = (-)пB/тг) / cos 2n9 cos(z cos
Jo
, tt/2
J2n+1(z) = (-)ПB/тг) / cosBn+l)e sin(z cos 8 )d8
J 0
tt/2 .
Г ' ix sec 9
Jo(x) + ilo(x) = -Bi/rr) / eXX Sec sec 9de , x>0
с
e-z cosh tcQsh vt dt ^ R(z)>0 .
1.4.5
C)
D)
E)
F)
I
A)
B)
C)
J/z) = r^ivV^ f cos^z cos 9)sin2V9cLe , R(v)>-i . D)
E)
F)
1.4.6
BASIC FORMULAS
/^ TT со
cos(z sin e-ve)ae-(sin vtt) f e"(z slnh t+vt)
о Jo
dt ,
R(z)>0
ttIv(z) = / sin(z sin e-ve)u9- / (eVt+e'Vtcos VTr)e"z slnh *й ,
v/ 0 Jo
R(z)>0
1.4.6. Asymptotic Expansions for Large z
Ну y(z)— B/тгг)^е ^ ^ 4 ^Fq (j5+v,t-v; — J ,
Izl —>co , -rr<arg z < 2тг
42)(z) -(a/™)».^-*^^ (^^w>-ife) '
Ju(z)~ B/ttz)»
|z|—><» ,-2тг < arg z < it
cos(z-^-^) Z 2k ov 2k
k=0 Bk).'BzJk
+ sin(z-^UTr-^rr) 2!
(-)k(^Jk+1(i-uJk+1
2k+l
k=0 Bk+l).'Bz)'
|z|—»co } |arg z| < it
31
G)
(S)
A)
B)
C)

32
INTEGRAIS OF BESSEL FUNCTIONS
1.4.6
(-)k(^Jk(*-Jk
*иЫ~ B/-f sin(Z-i--^) Zq Bk),Bzfk
^/i
- cos(z-iuiT-5TT)
кГ0 Bk+l).'BzJk+1
|zl—>=> , larg z| < тт
-z-e(u+i)iri
D)
Bttz):
Bttz)*
|Z|_»<» } _B+e)Tr/2<arg z<B-s)tt/2 , e •= ±1
Ku(z)-(rr/2z)*e-Z2FoD+u'i-^- Д) '
|z|—»•=> , larg z|< Зтт/2
E)
F)
If и is an odd multiple of J, each series in (l)-F) terminates. In
this event, we can replace asymptotic equality Ъу equality and omit the
restrictions on arg z . In particular, we have
J i(z) = B/ttz)*cos z j J|(z) = B/TTz)*sin z
I_i(z) = B/TTz)*cosh z ; I|(z) = B/ttz )*sinh z
K|(z) = (tt/2z)VZ
G)
(8)
0)
1.4.7
BASIC FORMULAS
33
Equations (l)-F) readily follow from 1.4.1A), the connecting formulae
given in 1.4.1 and the appropriate asymptotic expansions in 1.3.3. Alter-
Alternatively, we can start with 1.4.1B), use the connecting formulae of 1.4.1
and the pertinent asymptotic formulas of 1.3.3. For a discussion of the
remainders in the asymptotic expansions, see Watson A945, Ch. VII).
1.4.7. Polynomial Approximations
*'4(x) = Z (-^(х/ЗJ1^ + sn(x) , n = 0,1,
k=0
|«o(x)| ? 1-10 , |Sl(x)| ? 5-10"9 , -3 <. x <. 3
A)
ako
kl
0
1
2
3
4
5
6
1.00000
2.24999
1.26562
0.31638
0. 04444
0. 00394
0.00021
00
97
08
66
79
44
00
0.50000
0.56249
0.21093
0.03954
0.00443
0.00031
0.00001
000
985
573
289
319
761
109
кп (уп(х)-B/тт)Aл x/2)Jn(x)} = Z Ъкп(х/3Jк+71п(х) , n = 0,1,
k=0
•4-8
-8
|71o(x)| <. 2-10"a , \\(x)\ ? 5.10"a , 0 ? x ? 3 .
B)

34
INTEGRALS OF BESSEL FUNCTIONS
1.4.7
1.4.7
BASIC FORMULAS
35
Jko
Jkl
uko
ukl
0
1
2
3
4
5
6
0.
0.
-0.
0.
-0.
0.
-0.
36746
60559
74350
25300
04261
00427
00024
691
366
384
117
214
916
846
-0.
0.
2.
-1.
0.
-0.
0.
63661
22120
16827
31648
31239
04=009
00278
98
91
09
27
51
76
73
ч2к
¦nJnW = Z (-)Чп<*Л) + «nM ' n = O'1'
k=0
0
1
2
3
4
5
6
7
8
0.36746
1.07661
-2.34985
1.42164
-0.42617
0.07729
-0.00948
0.00085
-0.00005
69052
15157
19931
21221
37419
75809
55882
99770
67433
-0.63661
0.39325
6.85292
-7.39802
3.12613
-0.72689
0.10765
-0.01081
0.00065
97726
62018
36342
41381
99273
45577
76060
75626
35773
^(xW^xW-^Vx)/^^
-10
|eo(x)| s 10-100 , |в1(х)| ? 3-10-iu , -4 s x s 4
C)
Vх) = Z =kn(x/3)
k=0
-k
en(x)
ako
akl
0
1
2
3
4
5
6
7
1- 00000
3.99999
3.99999
1.77775
0.44435
0.07092
0.00767
0.00050
00000
98721
73021
60599
84263
53492
71855
14415
1.99999
3.99999
2.66666
0.88888
0.17775
0.02366
0.00220
0.00012
99998
99710
60544
39649
82922
16773
69155
89769
Xn {хп(х)-B/П)(Лл x/2)Jn(x)} = Z 4n(x/4Jk + ^nW > n = 0'1'
k=0
-10
|T)o(x)|s4-100 , ITIiOOI^-IO" ,0 s x s 4 .
6
Фп(х) = Z <Wx/3)-k + Tln(x) ,
k=O
И = 0,1, 3 S ]? S в ,
|«o(x)|sl.l0"8, |Tlo(x)| SS-IO"8; |Sl(x)| S3.10'8, |TI(x)| S8-10"
E)
"ко
\o
(*)
0
1
2
3
4
5
6
0.79788
-0.00000
-0.00552
-0.00009
0.00137
-0.00072
0. 00014
456
077
74 0
512
237
805
476
0.78539
0.04166
0.00003
-0.00262
0. 00054
0.00029
-0.00013
816
397
954
573
125
333
558

36
INTEGRALS OF BESSEL FUNCTIONS
1.4.7
4a
\l
0
1
2
3
4
5
6
0.
0.
0.
0.
-0.
0.
-0.
79788
00000
01659
00017
00249
00113
00020
456
156
667
105
511
653
033
0.78539
-0.12499
-0.00005
0.00637
-0. 00074
-0.00079
0.00029
816
612
650
879
348
824
166
Jn(x) + iYn(x) = B/mc)*eie [pn(x) + iQjx)]
8 = x - птт/2 - тт/4
k=0
-2k
en(x)
k=0
n = 0,1, 4 ? x ?
~-9
|«0(x)| .3.10-9, |.!(x)| ^6.10"9,|71o(x)! .4.10-9, l^x)! se-lCf . (б)
cko
^ko
ckl
ukl
о
1
2
3
4
5
0.99999
-0.00439
0. 00043
-0.00012
0.00004
-0.00000
9997
4275
4725
2226
3506
9285
-0.03124
0.00114
-0.00021
0. 00008
-0.00003
0.00000
9995
4106
8024
5844
5614
8099
1.
0.
-0.
0.
-0.
0.
00000
00732
00055
00014
00005
00001
0004
3931
9487
5575
0363
0632
0.09374
-0.00160
0.00026
-0.00009
0.00004
-0.00000
9994
1836
6891
9941
0658
9173
1.4.7
BASIC FORMULAS
37
If all four of the above quantities are needed, the evaluation of
either P0(x) or P]_(x) may be accomplished using the fact that for all
Р0(х)Рх(х) + ^(xjQ^x) = 1 ,
thus saving the evaluation of one series.
x"nln(x) = X. ekn(x/aJk + en(x) > n = °>1> Osxsa
k=0
G)
a = 2 , m = 6 , |eo(x)| s 1-Kf9, |ei(x)| s l-lO"9
"ко
"kl
0
1
2
3
4
5
6
1.00000
1.00000
0.25000
0.02777
0.00173
0.00006
0. 00000
0000
0000
0008
7730
6247
9246
2072
0.50000
0.25000
0. 04166
0.00347
0.00017
0. 00000
0. 00000
0000
0013
6563
2576
3024
6251
0001
a = 5 , m = 8 , |eQ(x)| s Lio"a , |e1(x)| s 2-10
v-9
(8)
"ко
ckl
0
1
2
3
4
5
6
7
8
1.00000
6.24999
9.76562
6.78163
2.64934
0.66155
0.11618
0. 01350
0.00202
0000
9820
9849
4474
8275
0732
5994
1925
0755
0.50000
1.56249
1.62760
0.84770
0.26492
0. 05514
0. 00828
0.00085
0.00011
0000
9971
4720
6080
8565
1522
5410
1785
0375

38
INTEGRALS OF BESSEL FUNCTIONS
1.4.7
х'П1п(х) = Z ekn(x/3-75Jk + en(x) , n = 0,1,
k=0
-7
|eo(x)| ^ 1.Ю"' , |в1(х)| ^ l-10-a , -3.75
s i s 3.75
O)
=ko
ckl
0
1
2
3
4
5
6
1.00000
3.51562
3.08994
1.20674
0.26597
0.03607
0. 00458
00
29
24
92
32
68
13 -
0.50000
0.87890
0.51498
0.15084
0.02658
0.00301
0. 00032
000
594
869
934
733
532
411
xn {^(x) -к (-)пAл x/2)ln(x)} = ? {1т(х/г)а + 71n(x) , n = 0,1,
k=0
-9
ho(x)| ^ 1-107 > hi(x)l ? 4>1° , о s x s 2 .
A0)
Lko
Lkl
0
1
2
3
4
5
6
-0.57721
0.42278
0.23069
0. 03488
0. 00262
0. 00010
0.00000
566
420
756
590
698
750
740
1.00000
0.15443
-0.67278
-0.18156
-0.01919
-0.00110
-0.00004
000
144
579
897
402
404
686
1.4.7
BASIC FORMULAS
39
x*e-xIn(x) = ? glm(x/3.75)-k + en(x) , n = 0,1,
k=0
.-9
-9
eo(x)| ? ll-10'a , |e1(x)| ? 11-10 , 3.75 й x s <=
A1)
Ski
0
1
2
3
4
5
6
7
8
0.39894
0.01328
0.00225
-0.00157
0.00916
-0.02057
0. 02635
-0.01647
0.00392
2280
5917
3187
5649
2808
7063
5372
6329
3767
0.39894
-0.03988
-0.00362
0.00163
-0.01031
0. 02282
-0.02895
0. 01787
-0.00420
2280
0242
0183
8014
5550
9673
3121
6535
0587
x*exKn(x) = X bkn(x/2)-k + Tln(x) , n = 0,1,
k=0
ho(x)| ? 7-10-8 , |Th(x)| <; 10-10"
2 < X ? со
A2)
\c
\l
0
1
2
3
4
5
6
1.25331
-0.07832
0. 02189
-0.01062
0. 00587
-0.00251
0. 00053
414
358
568
446
872
540
208
1.25331
0.23498
-0.03655
0. 01504
-0.00780
0.00325
-0.00068
414
619
620
268
353
614
245
Equations A), B), E) and (9-12) are due to Allen A954) and Allen
A956 ), respectively. In (9-12) we give maximum absolute errors, whereas
the original source quotes relative errors. Equations C), D) and F)

40
INTEGRALS OF BESSEL FUNCTIONS
1.4.8
come from Hitchcock A957) who also gives approximations for K^x) and
Ki(x) valid for 1 <. x <. => . Formulas G) and (8) follow by rearranging
the truncated expansion of Io(x) in series of Chebyshev polynomials of
the first kind. In this connection, see Wimp (i960, 1961). Further
approximations for the functions of this section can be obtained from the
work of Luke A955, 1960, 1961a).
1.4.8. Description of Mathematical Tables
In addition to the following, see the discussion in 1.1. For those
tables below which are well known, only an abbreviated description is pro-
provided.
Bessel Functions of Integral Order
British Association for the Advancement of Science A950): Functions of
order zero and unity.
British Association for the Advancement of Science A952): Functions of
positive integer order.
Cambi A948): Eleven- and fifteen-place tables of Jn(x) to all signifi-
significant orders.
Faddeeva and Gavurin A950): Jn(x) , n = 0AI20.
Fox A954): Bessel functions of integer order and large arguments.
Harvard Computation Laboratory A947-1951): Jn(x) , n = 0AI35.
National Bureau of Standards A947a): Jo(z) , J^z) for complex z .
National Bureau of Standards A950): Yo(z) , Yj_(z) for complex z .
Olver (Editor)(l960): Zeros and associated values of Bessel functions.
Vinogradov and Cetaev A950): In(x) , B/тг)Кп(х) , n = 0,1 .
Watson A945): See 2.9.
1.4.f
BASIC FORMULAS
41
Bessel Functions of Fractional Order
Cerillo and Kautz A951): f = З^/За^З/^) f | f | } arg f (in radians),
\ 3/3[Ai(-3";L/3B) ± Bi(-3/3B)], В = iBle1^ , |B| = 0@.2L ,
0 = 0G.5°I8O° , 7d. See Ch. VI.
Fox A960): R(z) = 2A^e§Ai(x) , z = Г1 = 3/2 x/2 = 0 @.001H.050 ,
lOd. Similar listings for functions related to Ai(-x) , Ai'(±x) ,
Bi(±x) , Bi'(+x) .
Harvard Computation Laboratory A945): These are essentially tables of
J+./_(z) for complex z . See Ch. VI.
-1/3
C.W. Jones A956): Essentially IQ+i(x) , Kn+i(x) > n = 0(l)l0,x=0@.l)l0.
J.C.P. Miller A946): Ai(±x) , Bi(±x) . See Ch. VI.
National Bureau of Standards (I947t>): (rr/2xJJu(x) , ±2u = 0(lL3 .
National Bureau of Standards A948, 1949): Jv(x) , Iv(x) , ±v = 1/4,
1/3, 2/3, 3/4 .
Olver (Editor)(i960): Zeros and associated values of Bessel functions.
Smirnov A955): U^s^) = Bи)"И (^g) s*J.iv(vsu) , U2(s,a) =
BuJvr(^|)sEJiv(vsu) } u = i(a+2) , uv = 1 , and their first
derivatives, a = 1 , s = - 6@.01I0 , 4d,5d . a = ±l/4 , ±l/3 ,
±1/2 , ±2/3 , ±3/4 , 5/4, 4/3, 3/2,^5/3, 7/4. 2, s = 0@.0lN, 4d .
vl(s>Po) = s*r(p)Jp.1Bsl) , V2 = s*rB-p)J1_pBs4) and first
derivatives, p = 0.1@.1I , s = 0@.01I0, 4d .
Watson A945): See 2.9.
Woodward and Woodward (with the assistance of Hensman, Davies and Gamble)
A946): Ai(z) , Ai'(z) , Bi(z) , Bi'(z) , z = x + iy ,
x = -2.4@.2J.4 , -у в 0@.2J.4, 4d . See Ch. VI.

CHAPTER II
1ИТЕ(ЖА1? OF THE TYPE
J tMWB(t)d.t
2.1. Definitions and Cormecting Formulae
Let
Wi (z) = / t%(t)dt
A)
where Ww(t) represents any of the Bessel functions of the first three
kinds or the modified Bessel functions, see 1.4.3A). If Ww(z) = J,,(z) ,
we write Ji w(z) in place of Wi^ u(z) . Similarly, if Ww(z)= Kw(z) ,
we write Ki w(z) in place of Wiu „(z) , etc. To ensure the existence of
A), R(p,+u)> -1 for Bessel functions of the first kind while R(p,±u)> -1
for Bessel functions of the second and third kinds.
Recurrence formulae for (l) are given in Chapter V. If p, = 1 ± » ,
(l) can be simply evaluated, and this too is considered in the latter
chapter.
The following connecting formulae are simple consequences of the anal-
analogous formulae for Bessel functions. See 1.4.1.
, „ -4±(р,+»+1)тг , iirr. ,
Ii^w(z) = e * ^ ; Л^„Bее ) , -TT<arg z ? тг/2 ,
= е|1(^1+иK^. (ze-43i") } n/2<arg Z?n . B)
n z
Jo
R(|i±u)> -1
42
C)
2.1
IHTEGRAI? OF THE TXFE
Ki,
\J t%(t)dt
Hi^(z) = f t^)(t)dt = Ji^w(z) + iYi^w(z) ,
J 0
R(p,t»)>-1 .
Hi^(-±n)=ei^-")V^(z) .
/i z
t%(t)dt = n/2(csc utt) [li^.B(z)-Ii (z)]
0
R(p.-»)>-l
-¦j^iTT
^(•¦^„(l)/
In all the above relations, -rr<arg z ? тг .
By contour integration,
n°° p<*> exp(i6)
/ t%(t)dt = / t%(t)d.t ,
J z J z
6 real , I6Ktt/2 ; 161 = тг/2 , R(p,)<i .
43
E)
F)
G)
(8)
0)
Here the path of integration lies in the branch of the cut plane determined
by I arg z| < тг , and is the ray p exp(i6) , p—><*> except for an initial
finite path. The origin is usually excluded from the path of integration,
hut may be included if the further condition R(p,±u)>-1 is imposed. It
follows that

44
IHTEGRAIS OF BESSEL FUNCTIONS
2.2
/.
z ехр(--|1тт)
t%(t)dt = in**1^"^ /"Vl^Wt , R(,)<i . A0)
2.2. Differential-Difference Properties
We have the differential equation
[ z2D3 + (l-2p,)zl? + (|i2-u2+abz2)D] Wi „(z) = 0 . (l)
The special recurrence relation
nZ „Z
/ Ju+I(t)dt = / Ju.iCtJdt - 2JB(z) , R(u)>0 ,
JO J 0
B)
follows from 1.4.3C). For general recurrence formulae pertinent to the
integrals of this chapter, "see Chapter V.
2.3. Power Series Expansions
Series representations for 2.1A) and related integrals are given
below. These follow directly from power series expansions of Wy(z) . See
1.4.1.
I
0tJ"(tLt ^к!(^№1)Г(№1)
zM-1
,+u+l
1*2
C1
i+u+1 ц+и+3
2и(р,+и+1)Г(и+1)
R(p.+ »)>-l
,u+l
;-2/4) ,
A)
2.3
IMTECKALS OF THE TYPE/ t^W,,(t)dt
45
/V» {".(.)}
,3-L
dt =
(-)k(z/2Jk
8 k=o Гк+ bi) {(k+1)-'} 2
3"" 2^3 (i, ^ ;2,2, ^ ;-«2/*) ,
z
Цз^
R(n)<3
B)
,.z
- / t^Yn
2Jn П
t- - ^ F (n-k-l).-(z/2)-^2k
2 ^0 k.'(-n+p,+2k+l)
(t)dt= - ^— x ^;1;;4Z^\^ ' -^+1 i; (-)k(z/2)a+2k
k^O k.' (n+k).' (п+р,+2к+1 )c
,^1 ^ (-)k(z/2r2k
+ ^? ГТ
^ k.'(n+k);(n+p,+2k+i)
(in z/2-^(k+l)-^(n+k+l)} C)
u+1 n~1
и - V-1. (ak-(-n:^D +<^ 8/2^n(B,-/ *-1П^п(*,«
7.H+1
sk/ /„чп+2к
2 k=0 k.'(n+k)!(n+p,+2k+l) V к n+k У
n ^ 0 , R(p,)> n-1
D)

46
INTEGRALS OF BESSEL FUNCTIONS
H /ZYo(L)dL J± f Jy(t)dt]
z ? {-)%/^( z/2. _1N
kt0 (k:JBk+l)V 2k+l^
k=l (k.1 JBk+l) V
k/
where
= z(Y+ln z/2)fQ(z) - zf1(z)
nz
zfo(z) = / J0(t)dt ,
f (z) = l- ?_ + _2_-
O4 ' то адп
z2 . z4 - z6 + z«
,10
12 320 16128 13 27104 1622 01600
Д2
2 76037 63200
f,(z) = 1-
79z
156 7 z
10
z2 + 17z4 _ 83z6
9 3200 " 6 77376 ' 477 75744 " 10 70530 56000
73z
12
79 74420 48000
2.3
E)
F)
G)
(8)
2.3
INTEGRALS OF THE TYPE Г t^JWu(t)dt
47
[\%1(Ь)ЛЬ = 2^ "у (-)k(n-k-l).4z/2)
Jq 2 k=Q k.'(-n+p,+2k+l)
-n+2k
1
+ (_)nzM.+l ^ (z/2)
n+2k
k=0 к.' (n+k)! (n+p,+2k+l J
. (-)V+1 f (z/2)n+2k
2 k=Q k.'(n+k).'(n+p,+2k+l)
X {2 In г/2-ф(к+1)-ф(п+к+1) j
0)
u,+l ^1 , Nk
T (-) (n-k-l).'(z/2)
2 k=Q к.'(-п+м,+2к+1)
-n+2k
- (-)n(Y+ln z/2)li^n(z)+(-)n ^t^Ii (t)dt
J0
. (-)V+1 ? (z/2)n+2k
2 k=Q k.'(n+k).'(n+p,+2k+l)
XCi^-.i^...^) ,
n й 0 , R(p,)>n-1
A0)

48
INTEGRALS OF BESSEL FUNCTIONS
2.3
do (.""«'О J u=O
r,2r2k+iU2k+1 J
k=0 (k.'JBk+l)
+ z ? (^Jk л^
k=i (k.'JBk+l)^
= -z(y+1h z/2)fo(iz) + zf1(iz) ,
k^
A1)
where fQ(z) and f]_(z) are given by G) and (8), respectively.
Equations (l), C) and (9) are obviously related in view of 2.1G).
Indeed the first two formulas with v = n follow from the third. An appeal
to the continuity argument produces (l) for general v .
We next establish the following formulae.
/z *Ч(^ = kffi^i) {-^ z/2 + ^(m+1) + ^(u+ffl+1)}
- 2
_,-, °° / чк, /оч2к-2т
ц+1 у (-) (z/2)
?Г0 Bк-2т)к.'Г(и+к+1) '
R((i)<-^ ; (i+u = -2т-1 , т a positive integer or zero . A2)
Ут00 Г z
tJo(t)dt = / t-1 (l-Jo(t)} dt - (у+1п z/2) . A3)
z ^ 0
2.3
nZ
IUTEGRALS OF THE TYPE / t%(t)dt
49
I /^%W* ¦ гЩг [т "* Z -=-2-i Z° ^-2
k=l k=l
- |ln z/2--|i|r(m+l)--ii|r(n+m+l)j
+ z^+1 ^ (n-k-l).'(z/2)
-n+2k
2 ^¦0k.'(-n+2k+M,+l)
,H+1
z^
(-)k(z/2)n+2k
?rQ k.'(n+k).IBk-2m)
jln z/2-^(k+l)-^(n+k+l) - ^Л
|iS-l i p,+n я -2m-l , m a positive integer or zero . A4)
t-^tjdt = g + (y+1h z/2) [-i(Y4-ln z/2)+go(z)J -gl(z) , A5)
7.
go(-)=(/zfi{i. jo(t)}
dt
«o(Z) = #-4+ z'
,8
,10
8 256 13824 11 79648 1474 56000
gl(z) = 5|i.-L+ *6
53 z
8
143 z
10
16 1024 6912 283 11552 88473 60000
A6)
A7)
A8)

50
IHTEGRAIS OF BESSEL FUNCTIONS
2.3
/>-«?•*?
m m+n
— ГТ* ГТ TT ^X / & M /
-2
k=l
{in г/2--^(т+1)-^(п+шН-1)|
21 z^l n- (-)k(n-k-l).'(z/2)-n*2k
2 "t: к.'(-п+ц+2к+1)
iC—U
M.+1 T (z/g)'
(-fz^+1 Z
,n+2k
k70 к.'(п+к).'Bк-2т)
k^m
{1л z/2-^(k+l)-M(n+k+l) - ^ J
^,+n = -2m-l , m a positive integer or zero
A9)
00 О
F t-^CtJdt = g + (v+ln z/2) [|(Y+ln z/2)-go(iz)] +gl(iz) , B0)
where go(z) and gx(z) are given Ъу A7) and A8), respectively.
It is sufficient to prove A9), for with v = n , A2) and A4) follow
with the aid of 2.1A0). Application of the continuity principle gives
A2) for general v . From (9) and 2.6F),
n-1 . .k.
Г ^ ,+ w и ^ i ^+1 V (-)(n-k-i).'(z/2)
/ t4(*)dt - A(e) - ^ Z 'k^.^k+l)
^  1С— U
,-n+2k
(-)nz'+1 ,^„ k.4^L2.) (^ »/*-it(*">-4t(n*n> - i^ } B1)
k=0
k^m
2.4
INTEGRAI5 OF THE TYPE
z
Г ttJWu(t)dt
51
where
A(e) = Azl^ !L_
m.' (n+m).'
_ (z/2)ef sin2 en
ш-'М.-в^т Iflm+l+Ortn+m+T^T ^^
(z/2)esin2eTT
n2e2
f = 2 1л z/2 - ф(т+1) - ф(п+т+1) , е = -?(ц+п+2т+1)
B2)
To compute A@) = A , use L'Hospital's theorem. Then
A= m:(n+m).' 1 ^~ -Ф'(^1)-Ф'(п^1+1)+4 |щ z/2-^(m+l)-^(n+m+l)j J , B3)
and A9) readily follows. Alternative proofs of A2) and A4) can Ъе
obtained by a similar procedure. See Lowan, Blanch and Abramowitz A943)
for an alternative proof of A2). Equations A4) and A9) are generaliza-
generalizations of results given by Oberhettinger A957a).
Л. Expansions in Series of Bessel Functions
2# ^ (U+2k+l) (r^h
/ t^Ju(t)dt = -^—r Z ^
Jo v ^+u+1) kfo С^а^.)
V 2 /
J u+2k+l
(z) ,
R(n+u)> -1
A)

52
INTEGRALS OF HESSEL FUNCTIONS
2.4
/c
' tHj (t)dt - ^+1 у (^/2)к j (z)
t Jw(t)dt (^+u+l) ^ Ju+k(Z)
V 2 У 5- 1г Д1/ J т , ,
R(n+")>-1
B)
See also 5.2B4-27).
/
Ju(t)dt = 2 X Ju+2k+i(z) ^ R(v)>-1
0 k=0
C)
y,Z „Z П-1
J2n(t)dt = / Jo(t)dt - 2 ~Z J2k+1(
0 J0 k=0
z) . D)
/
J2n+l(t)dt = 1 " Jo(z) " 2 Z J2k(z) •
0 k=l
E)
/.
t'1 (l-Jo(t)} dt = 2Z X Bk+3) [ф(к+2)-фA)] J2k+3(z)
0 k=0
l-2z" J1(z)+2z-1 Z Bk+5)[t(k+3)-t(l)-l]j2k+5(z) .
k=0
F)
2.5
IMEEGRAIS OF TBE TTCPE
J t%(t)dt
53
; f t'1 ho(t)-l\ to =2z-X Z (-)kBk+3)[t(k+2H(l)]e-ZI2k+3(z)
JO k=0
e"z [az'^fz).!] -2Z X (-)kBk+5)[t(k+3H(l)-l]e-ZI2k+5(z) . G)
k=O
/"t-1 {i-Jo(t)} dt = * i (^f^)-t(i?] Jk(z) .
^0 k=l
(8)
/Zt-l(lo(t)-l) dt = -i ^ (-)к(г/2)кЬ(к+1)-*AK e-Zl (z) . (g)
JQ L ^ к=1 к.'
Equations (l) and B) arise from 2.3(l), and 1.3.6F-7), respectively.
Equations D) and E) derive from 2.2B). Equation C) is a special case
of (l) and may also Ъе proved from 2.2B) since for z fixed,
lim Ju(z) = 0 . The first of (б) is a consequence of (l) and 1.4.2C)
V —> oo
with p, = 0; (8), of B) and 1.4.2A4) with Сц(г) = J^(z) and
v = X = 0 .
2.5. Asymptotic Expansions for Large z
Consider the right-hand side of 2.1(9). Replace Ky(t) Ъу its
asymptotic expansion and integrate Ъу parts. Then
г,™ exp(i6)
Г XP1 t%(t)dt~e-WJL)* ? (-)
J z Kdzy k=0
к -к
Izl—>oo , larg z| s tt/2 , |arg 61 < тт/2 ;
|arg 61 = tt/2 , R(n)<5 ,
A)

54
where
INTEGRALS OF BESSEL FUNCTIONS
2.5
ii-^U^)
2kk.'
i кък , ък = sF2(-k,i,n-Jct?ji+i;-k,?-i;-kj2) , B)
C)
ъ = 1 + 2(к+1Нц-к4) ъ
k+1 (v-k4)(W+|) k
The aks can also Ъе found Ъу substituting the series expansion on the
right of (l) into 2.2A) and equating like powers of z . In this fashion,
we get the recurrence formula
2(k+l)ak+1 = [3(k4)(k+5/6)-2M,(k+l)-u2] ah.-(k+|+i>)C5:4-i;)Cs-i-M,)a]5._1 ,
ao = 1 , ax = 5/8 - м, - v*/2
D)
If \i, ~ u+1 , the asymptotic representation of 5.2C1) shows that
ak - <±W"*± . E)
2kk.'
Asymptotic expansions for other integrals of the type / t'JWu(t)dt
J z
follow from (l) and the connecting formulas in 2.1. For an alternative
approach, we start with 2.3A), and use the connecting formulas in 2.1, and
the asymptotic expansions in 1.3.3. We then find the following representa-
representations.
z 2M.r fv+y.+l\
/ t^J,,(t)dt = JL_2 C- - B/-rrz)^z^(f cos 8+g sin 8 ) ,
2 У
R(n+U)>-1 > Izl—»oo , |arg z|<tt
F)
2.5
z
IHTEGRAIS OF THE ТХРЕ Г t^W (t)dt
55
/Q4(t)dt = t. Г(^) r(^)sin(,.u)TT/2
- B/TTz)*z^(f sin 9-g cos e) ,
Н(ц±1>)>-1, |z|—»oo ; |arg z| <тт
G)
Here
9 = z - im/2 + tt/4 ,
(8)
*~Z (-)ka2kz-2k,g,-Z (ОЧк.!^-1
k=0
k=0
(9)
'uid the a^s are as in (l).
f t%(t)dt
ezz^ ^- _ ,.k e-zzM.e-1(u+i)eTT » k _k
2_ 1-; akz
l_ akz--
Bnz)*k=0 BrlzJ
4^)
k=0
R(m,+u)>-1, |z|-»», -B + e)n/2<arg z<B-e)n/2, e = +1
A0)

56
IMTEGRAIS OF BESSEL FUNCTIONS
2.6
JO 4 2 У Ч 2 У k=Q
R(n±u)>-1 , |z|-»« , |arg z|<3tt/2 . (ll)
In (Ю) and (ll), the a?s are again as in (l).
Values of the coefficients a^ for p, = v = 0 and p, » -1 , v = 0 ,
each for к = 0A)8 are listed in the table below.
a for p,=u*O
1 5/2^
2 129/27
3 2655/210
4 3 01035/2 ^
5 108 96795/218
6 9613 19205/222
7 5 00465 71575/225
8 2403 53982 61875/231
a, for ц«=-1,и«0
1
13/23
529/27
14887/210
21 47403/215
945 45267/218
98351 09013/222
59 01645 13695/225
32106 25393 55955/231
2.6. Infinite Integrals
/
t%(t)dt = v ; л , r(^+v)> -i, н(ц)<*
о r(^—)
V 2 ^
r00
/ Jy(t)dt = 1 , R(v)>-1
Jo
C2)
A)
B)
2.7
IMTEGRAIS OF THE TYPE / t^,,(t)cLt
J
ri-jo't)J,-r(?)r(?)
« 0
0 t^
\zJ ^ 2 J , 1<R(^)<3
2,{r(^l)}2
57
C)
/ +LJ,V f+ \a+ 211 т-Л^+и+А tVI-I-U+A . TT / v
R(m,±v)>-1 , R(n)<i •
D)
/;
Yu(t)dt = - tan — , |R(u)Kl
E)
J t^%(t)dt B gH-lr^-t-v-H^r^-w-lj , н(ц±1»)>-1 .
F)
/„
Ky(t)dt = I sec H , IH(«)|<1 .
G)
K.iuations (l), D) and F) come from 2.5F, 7 and ll), respectively.
ibf combination 2.3B) and 1.3.3A8) yields C).
• <¦ ¦ Circular
Representations of Jn(z) and / Jn(t )dt
^0
m-1
J2n(z) = - Z c°s(z c°s ^k)cos 2n^k + rm,2n(z)
k=0
A)

58
Here
INTEGRALS OF BESSEL FUWCTIORS
Ш
J2n(z) = - X ekcos(z cos iuk)cos 2пшк + Um,2n(z)
2.7
B)
k=0
m-1
J2n+1(z) = i Y sin(z cos Xk)sinBn+l)Xk + гт^2п+1(?) . C)
k=0
Ш
J2 +1(z) = i Z eksin(z sin «)k)sinBn+l)iuk + u^2n+l(z) • (*)
k=0
\k = Bк+1)тт/4т , oojj. = ктт/2т
E)
,n(z) = -2 ? (-)Гвг,п(*> > %» =  Zer,n(O > F)
r=l r
r,n(z) = * [(-)nj4mr-n(z) + J4mr+n(z)] >
^k=*
if к = 0,m; ek = 1 if к = 1,2,...,m-1 .
G)
(8)
nz m-±
/ J2n(*)d* =\ ^
Jq k=0
m-1 sin(z cos X )cos 2n^k pz
cos Xk
/ rm,2n(t)dt • О)
r,wl№.i|b*^^.^). • <«,
Two other formulas readily follow Ъу integration of B) and D).
2.7
INTEGRALS OF THE ТГРЕ / tMwv(t)dt
59
In the above rm n(z) and i^ n(z) are remainder terms. Integrals
of the latter can also be expressed as a series of Bessel functions in
view of 2.4A,2). To appraise the error, one can use available tables.
However, this is convenient only when z is pure real or pure imaginary.
The ensuing discussion gives an approximate but simple and very effective
means to estimate the remainder without use of tables of Bessel functions.
Fix z and n . Choose m> z and m> n . Since Jq(z) decreases
rapidly as q increases, to approximate the series defining the error,
it is usually sufficient to consider only the first term of the series.
Thus
VnW^-V'2' '
A1)
and unless n = 0 ,
gl,n(z)~-^-J4m-n(z)
{12)
11 nee
|Ja(z)| * Kz/g) 1еУ , z =x + iy , yS 0 ,
A3)
A4)
A5)
! m.i Lar analysis yields
/ r (t)dt
u n
,4[B/2Lт+1|еУ
Dm+l).'
A6)

60
INTEGRALS OF EESSEL FUNCTIONS
<J 0
g|(z/gLm-n+l|ey ^ n>Q
Dm-n+l).'
2.8
A7)
The results of this section are based on trapezoidal integration rules.
For further discussion, see Fettis A955) and Luke A956). Exponential
series (trapezoidal rule) representations of Ку(г) and its repeated inte-
integrals are given in 8.8.
2.8. Polynomial Approximations
/,
Jo(t)dt = Z (-L(xAJk+1 + «to ,
0 k=0
|e(x)| <; 6 • НГ9 , 0 s x s 4
A)
f Yo(t)dt = B/тг)(иГх/2) Г Jo^)^ " Z (-)kbk(x/*Jk+1+Tl(x) >
J 0 Jo k=0
|Tl(x)| <1 • 10"9 , 0 <: x s 4
B)
0
1
2
3
4
5
6
7
я
4.00000
5.33333
3.19999
1.01586
0.19749
0.02579
0.00236
0.00013
_
0000
3161
7842
0606
2634
1036
2211
3718
1.07661
2.56725
2.28731
0.90475
0.20338
0.02960
0.00303
0.00023
0. 00001
1469
0468
7974
5062
0298
0855
4322
5002
3351
2.8
INTEGRALS OF THE TYPE
Pz
J t%(t)dt
r[jo(t)+iYo(t)]dt =x-*ei^-TT/4)[x (ОЧ^АГ2*-1
Jx L k=0
+ i z (-)Ч(*Л)-2к1+ «oo ,
k=0 J
|e(x)| <: 2 • 10"8 , 4 ? x <;
61
C)
0
1
2
3
4
5
6
0.12461
0.03128
0.02364
0.02200
0.01623
0.00739
0.00149
1058
0848
4978
7499
6617
0830
6119
0.79784
0. 04963
0.02366
0.01825
0.01242
0.00543
0. 00107
8790
5633
4841
5209
2640
4851
6103
f [jo(t)+iYo(t)]dt =x-*ei(x-"A)[ ? (_)kak(x/8)-2k-l
« x L k=0
7 "I
+ i 2 (-L(x/8)k + e(x) ,
k=0 J
|e(x)| s 6 • 100 , 8 ?x
S oo
D)

62
INTEGRAIS OF BESSEL FUNCTIONS
2.8
2.8
IMTEGRAIS OF THE TYPE / t^Ct )dt
63
0
1
2
3
4
5
6
7
0.06233
0.00404
0.00100
0. 00053
0.00039
0.00027
0.00012
0. 00002
47304
03539
89872
66169
92825
55037
70039
68482
0.79788
0.01256
0.00178
0.00067
0.00041
0. 00025
0.00011
0.00002
45600
42405
70944
40148
00676
43955
07299
26238
00
See 3.11A) for another approximation to / [<Jo(t )+iYo(t )]dt
J x
fV^l-Jott^at = Z (-)Ч(*ЛJк+2 + e(x) ,
w 0 k*0
.-9
|e(x)| ? 3 • 10"э , 0 ^ x s 4
E)
f f^ltjtt = n/6-(Y+ln х/2J/п+B/п)(у+Ш x/2) J f1 (l-Jo(t)} dt
Jx 0
- Z (-L(xAJk+2 + n(*) ,
k=0
|T](x)| ? 6 • 10"9 , 0 ? X ? 4
F)
0
1
2
3
4
5
6
7
1.99999
0.99999
0.29629
0. 05554
0.00709
0.00063
0. 00003
_
9936
9326
2677
4803
2535
9765
5817
1.90985
1.11408
0.37725
0.07810
0.01078
0. 00105
0. 00007
0.00000
9297
4491
5736
2710
7555
9499
6217
3546
/;
^[jJtKTjtjjdt =x/2ei(W4)[|- (_)*ак(х/4Г2к
Lk=o
- i X (-)\(x/4)-2k-ll + e(x) ,
b=0 J
|e(x)| s 1.4 • 10 , 4 ^ x
0
1
2
3
4
5
6
0.79775
0.20215
0.16088
0.16477
0.13415
0.06662
0.01453
06
47
74
97
51
97
69
0.32358
0.17027
0.17974
0.19601
0.15761
0. 07593
0.01606
19
78
57
54
16
39
72
CO
J t[jo(t)+iYo(t)]dt
x-wmoFJ: fk(x/8)-k-i|: gk(x/8)-k] +
L k=0 k=l J
|e(x)| s 1.5 • 10 , 8 s x s cc
e(x) ,
G)
(8)

64
INTEGRAIS OF BESSEL FUNCTIONS
«k
0
1
2
3
4
5
6
0.79788
-0.00000
-0.05144
-0.00093
0.01703
-0.00919
0.00181
46
11
50
94
30
09
18
0.
0.
-0.
0.
0.
-0.
-
16206
00005
02331
00244
00598
00237
95
95
78
37
42
31
n °°
See 3.11B) for another approximation to / t" [Jo(t )+iYo(t )Jdt
/
%2k+l
Io(t)dt = X ck(x/2fK^+ e(x) ,
0 k=0
-9
|e(x)| ? 1 • 10"э , 0 <; x ? 2
X X b
Г Ko(t)dt = -Aл x/2) I Io(t)dt+ X dk(x/2Jk+1+n(x)
Jo * 0 k=0
|Т](х)| s 2 • Ю'И , 0 ? x s 2
ck
0
1
2
3
4
5
6
2.00000
0.66666.
0.10000
0.00793
0.00038
0. 00001
0. 00000
0000
6667
0003
6494
5833
2590
0319
0.84556
0.50407
0.11227
0.01110
0.00062
0.00002
0.00000
868
836
902
118
664
069
116
2.8
Si
2.8
(9)
A0)
INTEGRALS OF THE TYPE
J t%(t)dt
ГХ 8
/ Io(t)dt = X ck(x/5Jk+1 + e(x) ,
|e(x)| s 5 • 10"9 , 0 s x s 5
ck
0
1
2
3
4
5
6
7
8
5.00000
10.41666
9.76562
4.84402
1.47186
0.30070
0.04468
0.00450
0.00059
0000
6367
9849
4624
0153
4878
6921
0642
4340
nX 4
:*e"x / Io(t)dt = X ak(x/5)"k+ e(x) ,
J0 k=0
|e(x)| <. 2.4 • 10 , 5 s x
rx 6
x*e"x / Io(t)dt = X Ък(х/8)"к + П(х) ,
«0 k=0
|T)(x)| s 2 • 10 , 8 s x <;
65
A1)
A2)
A3)

66
INTEGRALS OF BESSEL FUNCTIONS
О
1
2
3
4
5
6
0.41612 24
-0.03029 12
0.12941 22
-0.02022 92
-0.01516 60
0.39894 23
0.03117 34
0.00591 91
0.00559 56
-0.01148 58
0.01774 40
-0.00739 95
/,00 4
' K0(t)dt = Z (-)kak(V2)-k+ c(x) ,
x k=0
.-6
xze
|e(x)| <; 1.2 • 10'b , 2 <; x <; 4
f\{t)at = X (-)Ч(х/*Гк + iW >
Jx k=0
|T)(x)| <; 1.3 • lO"b , 4 <; x <; »
Jx k=0
-7
|6(x)| <; 2 • io"' , 7 s x <; »
0
1
2
3
4
5
6
1.24949
0.35846
0.18598
0.07817
0.01603
-
34
41
40
15
95
1.25331
0.19582
0. 07872
0.04814
0.03205
0.01584
0.00371
41
73
84
55
04
49
28
1.25331
0.11190
0. 02576
0.00933
0.00417
0.00163
0.00033
414
289
646
994
454
271
934
2.8
A4)
A5)
A6)
2.8
INTEGRALS OF THE TYPE / tMwu(t)dt
^f 1[l0(t)-l]dt = Z =k(*/2Jk+2 + e(x) ,
k=0
|e(x)| <; 1 • 10"a , 0 <; x <; 2
/ t-1Ko(t)dt = n2/24+i(Y+ln x/2J +(Y+ln x/2) / t-1 {lQ(t )-l} dt
^x J0
Z dk(x/2Jk+2 + Tl(x) , |T)(x)| <; 8 • 10"8 , 0 s; x s 2 .
k=0
/
0
1
2
3
4
5
iX
.-lr.
0.50000
0.06250
0.00462
0.00021
0. 00000
0. 00000
000
000
962
703
692
017
7
0.74999
0.10937
0. 00925
0. 00048
0.00001
0.00000
, /_ ч2к+2
993
537
821
077
544
077
-8
|e(x)| s 1 • 10"a , 0 <; x <; 5
67
A7)
A8)
A9)

68
INTEGRALS OF BESSEL FUNCTIONS
2.8
0
1
2
3
4
5
6
7
3.12499
2.44140
1.13027
0.33116
0. 06615
0.00968
0. 00096
0. 00012
991
746
241
853
507
217
442
S30
10
x3/2e-x / t-l[Io(t)_i]dt = ? ek(x/5)'k + e(x)
 k=0
|e(x)| <; 1.1 • 10 , 5 s x <; »
B0)
2.9
IHTEQIALS OF THE TYPE
J t%(t)dt
X3/2eX f t-\(b)ib = X (-)Ч(х/4)"к + П(х) ,
J x k=0
.-6
|T)(x)| <; s • lo" , 4 <; x <; »
0
1
2
3
4
5
6
1.234684
0.850013
0.590944
0.280367
0.060840
-
1.25331
0.50913
0.32191
0. 26214
0.20601
0.11103
0.02724
41
39
84
46
26
96
00
69
B2)
0
1.
2
3
4
5
6
7
8
9
10
0.39893 14
0.13320 55
-0.04938 43
1.47800 44
-8.65560 13
28.12214 78
-48.05241 15
40.39473 40
-11.90943 95
-3.51Э50 09
2.19454 64
3/2 x
x ' e
Г -l 4
/ t ^(iOat = 2
Jx k=0
(-)kak(x/2)-k+ e(x) ,
Except as noted 'below, the equations of this section follow by inte-
integration of appropriate formulas in 1.4.7. Equations C), G), (l2)j A4)
and B1) are curve fits of known tabular data. Equation D) is due to
rx
Hitchcock A957) who also gives an approximation for / KQ(t )dt valid
J0
for 1 s x <. o> . I am indebted to the late Dr. Milton Abramowitz for
B0). Equations (8), A3), A5), A6) and B2) were derived using methods
discussed Ъу Luke A961a).
2.9. Description of Mathematical Tables
In the following, we list for the most part tables relating to the
integrals of this chapter. See also 3.12. For tables of Bessel functions
of integral and fractional order, see National Bureau of Standards A962,
Chs. 9 and 10) and the references given there.
|e(x)| <; 9 • 10 , 2 <; x <; 4
B1)

70
INTEGRALS OF BESSEL FUNCTIONS
2.9
Cistova
pZ „Z .
Integrals of the Form I t^J^t^t , I t^CtJdt
A958): / t'^^dt , / t^Y^t^t , n = 0,1, x = 0@.001I5
«x Jx
@.01I00, 7d. Also tabulated are auxiliary expressions to facilitate
interpolation near the origin. This volume also gives tables of Jn(x)
and Yn(x) for same range as above.
„со „со „со „X
Note:/ tJ1(t)dt = / JQ(t )dt + Jx(x) J t"xyl(t)dt=Y1(x)- / YQ(t)dt.
Jx Jx x ^x Jo
Ferentz and Harrison A957): x I J0(t)dt , x = 0@.01K1, 4d.
J0
nx nx
Horton A950Ъ): / tnJn(t)dt , / tnHn(t)dt , n = l(lL, x = 0@.1I0 , 4d.
«Jo "o
Here Нц(х) is Struve's function, see 3.7-
nx
Knudsen A953): / Jn(t)dt , n = 0(l)8, x = o@.01I0, 5d.
J0
CO
Longman A959): i t"mJn(t)dt , n = 0,1, m= 0AJ2, 6d.
Jx
ЛХ лХ
Lowan and Abramowitz A943): I JQ(t)dt , / YQ(t)dt , x = 0@.01I0, lOd.
J0 J0
'0 °0
See also National Bureau of Standards A954Ъ).
Lcwan, Blanch and Abramowitz A943): Г tJQ(t )dt , x = 0@.1I0AJ2, lOd;
Г™ С 1
F(x) = / t" J0(t)dt + ln(x/2) , x = 0@.1K, lOd; Fkn;(x)/n: ,
V X
n = 0AI3, x = 10AJ2, 12d. See also National Bureau of Standards
A954Ъ).
nX nX
National Bureau of Standards A962):/ JQ(t)dt , / YQ(t)dt , x = 0@.1I0,
Jo Jo
nx nx
lOd. / t (l-J0(t)j dt , / t" YQ(t)dt , x = 0@.1M, 8d.
Jo Jq
INTEGRALS OF THE TYPE f t%u(t)dt
У^х nx
J0(t)dt , / Y0(t)dt ,
0 
0@.5M0, lOd.
ЛХ nX
Osterberg and Walker A955): / ^^(tjdt = / J0(t )dt - Jx(x) ,
Jo Jo
71
National
x = 0@.5M0, lOd.
f 0 JO
x = 0.01@.01K.85, 4AJ5, 5s or 6s accuracy.
Schmidt A955): / JQ(t)dt , x = 10@.2L0, 6d.
'0
«x -x
A945): i / J0(t)dt , i / Y0(t)dt , x = 0.02@.02 )l , 7d.
Jo u О
a px
Watson
'0 ^0
The first 16 maxima and minima of these integrals are tabulated
to 7d. The following are also tabulated.
Jn(x) , Yn(x) , e"xIn(x) , e\(x) , ex , Hn(x) , n = 0,1 ,
x = 0@.02I6 , 7d; ^/3(х) , Y-^x) , ехКф(х) , x = 0@.02I6, 7d;
Jn(x) > Yn(x) > e"Xln(x) > Kn(x) > J+(n+^)(x) for various x and
integer n
Integrals of the Form / tM'lu(t)dt , / t^Ky(t)dt
hursian and Fock A931): / KQ(t )dt , x = 0@.1I2 , 7d; ex / K^,(t)dt ,
Jx ^x
Px x Px
x = 0@.1I6, 7d; / I0(t)dt , x = 0@.1N , 7d; e / I0(t)dt ,
Jn Jo
x = 0@.1I6 , 7d.

72
INTEGRALS OF BESSEL FUNCTIONS
2.9
Harvard Computation Laboratory (l952t>): S(x) = | J I0(t)dt - Ii(x)j ,
nX
F (x) = . Я - x + Kx(x) I K0(t)dt , x = 0@.01I0, 8d. There are
2 Jo
Ko(t)dt ,
also other tables. See 9.6, 9.7.
Karmazina and Cistova A958): e"x I IQ(t)dt , ex /
 "x
x = 0@.00lM@.005)l5@.0l)l00 , 74. This volume also gives tables
of e"xIn(x) , ехКц(х) and ex , n = 0.1 for same range as above,
except that ex is given to 7s.
ЛХ
Luke and Ufford A951a): I KQ(t)dt = - [у+1п(х/2)] A-^x) + AgCx) ,
Jo
x = 0@.01H.5@.05I , 8d.
Luke and Ufford A953): See 9.6, 9.7.
PX nco
Mack and Castle A953): I I0(t)dt , / ^(tjdt , x = 0@.02 J@.lL , 94.
^0 Jx
ГХ
Muller A939): / KQ(t)dt,-x = 0.02, 0.04, 0.l@.lJ.6@.2)l3.S@.4)l6, 6-9s.
0
рх
Muller A954): x"n I tnKQ(t)dt , n = 0(lK0 , x = 0@.01J@.02M , 8s.
Jo
OX pco
National Bureau of Standards A962): e'X I IQ(t)dt , ex / KQ(t)dt ,
J0 Jx
x = 0@.1I0 , 74. e'X Г t {lQ(t)-l] dt, 8d; xeX / tKQ(t)dt ,
J0 Jx
Sdj x = 0@.1M .
Pearson, Stouffer and David A932): Tables computed Ъу F.N. David and
Г 1-1 ГХ
E.C. Fieller. Sm(x) = тт2т(Ш t%(t )dt , m= 0@.5I1.5 ,
J0
x = 0@.1L@.5I8 , 64.
px
Rothman A949): IQ(t)dt , x = 0@.1J0AJ5 , 8-9s.
CHAPTER III
REPRESENTATIOMS OF
Лч
(tLt IN TERMS OF LOMMEL FUNCTIONS
3.1. A Theorem on In4efinite Integrals Involving a Bessel Function
Let v(z) be a particular solution of
[z2D2+zD+(z2-u2)Jv(z) = f(z) .
A)
Then
/
t-l.
t"±f(t)Cu(tLt = zCu(z)v-(z)-zv(z)C^(z) ,
B)
where Cv(z) is a cylinder function, see 1.4.1(8). v(z) шау Ъе called an
associated Bessel function. An equivalent statement is
J tu+1{E"(t)+t-1Bu+l)E'(t)+E(t)} Ctf(t)dt
= zu+1{e'(z)Cu(z) + E(z)Cu+1(z)}
C)
It is helpful to give a brief survey of the material in the balance
"Г this chapter. Sections 3.2-3.6 delineate properties of Lommel functions.
For particular values of the parameters, Lcmmel functions are essentially
fil.ruve functions. Properties of the latter are consi4ere4 in 3.7. Anger-
Weber functions are also particular Lommel functions, an4 are treate4 in
'.M. Section 3.9 is 4evoted to representations of / t^Wv(t)dt with
• ¦xplicit formulae for tabulated functions. In 3.10 previous results are
•i|.j>lie4 to evaluate a certain class of Fourier-Bessel coefficients. Same
I" .lyriamial approximations are given in 3.11 ,an4 3.12 4escribes some
i*wtinent mathematical tables.
73

74
3.2. Lommel Functions
INTEGRALS OF BESSEL FUNCTIONS
3.2
I 3.3 REPRESENTATIONS OF / t^v(t)dt IN TERMS OF LOMMEL FUNCTIONS 75
**,*& - ^Ч1" {(^-lJ-2} *-2+ {(."IJ-2} {(.-3J-»2} z'*- ...]
If f(z) = zM>+1 , a solution of 3.l(l) is
("A1
к u,+l+2k
(z) = у l-J z"
k=O [(ц+lJ-»2] [(ц+3J-»2].. . [(^+2к+1J-»2]
,+1 . (.)k(z/2J-rC^)rC^)
Й, г(й=§?*)г(ь
^+k)r(^+k)
.M.+1
(M,-u+l)(M,+u+l)
1F2
(^
J.-U+3 Ll+U+3 .
;-z2/4) , A)
provided neither of the nunibers utu is an odd negative integer. Another
particular integral of 3.l(l) with f(z) = z^+1 is
^u(O " ^uB) + [2^r(^|ti) Г(И1|11) esc m]
X [cos[(M,-u)TT/2]j.u(z)-cos[(M,+1;)n/2]jl;(z)J
X|sin[(n-u)n/2]ju(z)-cos[(M,-u)n/2]Yl;(z)j . B)
If either of the numbers ц,±и is an odd positive integer, S v(z) has
the following finite series representation.
Mo»4
= «^V
o^ -f~ > -f— ;-*/г ; •
C)
If either of the numbers ц±1> is an odd negative integer, s (z)
is not defined, but S^ v(z) approaches a limit. See 3.4. '
If C) does not terminate, it is an asymptotic representation of
S^ u(z) valid for lz|—>« and |arg z|< тт . This readily follows from
(l), B) and the pertinent asymptotic expansions given in 1.3.3.
3.3. Recurrence Relations
Both s^}V(z) and SpijU(z) satisfy the following recurrence formulae.
s;,u(z) + (VZ)S^,U(Z) = (^-DVl,»-l(z) •
%,»(*) - (VOS^v(z) = (^-DS^u+1(z) .
Bu/z)S^v(z) = (^u-DSll_1>l;.1(z)-(li.u.l)Sll_1^+1(z)
S^,,;(z) = Wz) •
A)
B)
C)
D)
E)
F)

76
IMECSALS OF BESSEL FUNCTIONS
3.4
3.4. Formulae for S^ v(z) When s „(z) Is Not Defined
If either of the numbers ц,1и is an odd negative integer, s v(z) is
not defined. The same is true for S v(z) as given Ъу 3.2B). However,
S y(z) has a limit when s „(z) is not defined. Now S^ v(z) is an
even function in и , and it is sufficient to consider Su_2p-l, u(z) where
p, = v-2p-l , p a positive integer or zero. From 3.3(l),
22Pp.T(p+l-u)
u-2p P
Z (iO (р-к-1)!Г'(р-к-и) , A)
4p.T(p-i>+l) j^r0
and so knowledge of Sv_2p-i(z) depends on that of Sv_1}V(z) . Using
3.3A) and 3.2B),
u—*-u-i L
(m,-u+1)(m,+u+1) J
и-2
= 2U-±r(u)(lni2)Ju(z)-2i;-%r(U)Yu(z)
- / ^О,ч2Ы.»
-2U-2F(u) Z ('jr SL) {¦(»»*»1H(*1)} .
k=0
и is not a negative integer or zero
B)
If v is a negative integer, use
S-n-l,-n(z) = S-n-l,n(z)
C)
3.5 REPRESENTATIONS OF / tHwv(t)dt IN TERMS OF LCMMEL FUNCniONS
and B). If и = 0 ,
S.ljO(z) = lim Z^1"V,O(Z)
m.-»-i (m,+iJ
3.5. Integral Representations
вц,1;(*0 = ?tt(csc m)L(z) f t*J_v(t)a.t-J_v{z) f t%(t)dt ,
L Jo J о J
и not an integer, R(p,±u)> -1
s^(z) = П ["Yu(z) P t^J^tJdt-J^z) P t%(t)at] ,
2 L i/q " 0 J
K(h±i>)> -1
For results concerning the above integrals, see Chapter II.
77
- If J0(zL Z IllifeL-/^.^!,]2^.^,] . D)
k=0 k.1 L J
A)
B)
I I
„u(Z)=2M^)(Z/2)*(^+1)/2(cos e)^+u(sin e)*<^+1^_u+1)(z sin e)de,
R(m.+v)>-1
C)

78
IUTEGRAIS OF BESSEL FUNCTIONS
,tt/2
^1 "/ z-
cos2ue sin(z sin e)a.e , k(i>)>-4 •
о
S^(z) = Ц,1иB) J VYu(t)dt-Yu(z) ? t4(t)dtj ,
ВЫ<-? •
8,,u(«)-^/V»VlC^ '?ИЙ '
R(z)>0
uS
Sv „(z) = zv fe-z Sinh ^osh^tdt , R(z)>0 .
So „(z) = / e"z slnh ^osh utdt , R(z)>0 .
5 u(z) = z / e'Z Slnh ^inh ut cosh tdt , R(z)>0
-co
Sx „(z) = z / e'Z Sinh ЬсовЪ. vt cosh tdt , R(z)>0
3.5
D)
E)
F)
G)
(8)
(9)
A0)
3.6 REPRESENTATIONS OF
J tWw(t)dt
IN TERMS OF LOMMEL FUNCTIONS 79
3.6. Expemsions in Series of Bessel Functions
- z^hu^)
sn r(z) в 2 Цц*2) ,т f.to^ly Bк+и4-1)Г(ц^к+1) т ,.
^ (n+u+D^-u+1) ^ ; ktl k.'Bk+ju+l)Bk^-u+i) J^^l(z) >
|j,±u is not a negative odd integer .
A)
(z) = 2*(^-3) i(^-wl) Л+u+lN " (z/2)k
^U ^2 JL(^+k) ^"+1)(Z) '
|j,+u is not a negative odd integer .
B)
Since s u(z) is an even function in v , v can Ъе replaced Ъу -и in
the right-hand side of B).
- ^Yu(z) + iJv(z) {t(»+l) - Kl)}] ,
и is not a negative integer or zero .
3-i,o('z^ - 2 2. 5- + г
1 T 1_^-
d»c
Л2
u=0
+ TJo(Z)
C)
D)
Kquations (l) and B) derive from 3.2A) and 1.3.6F-7), respectively. Use
(l), 3.2B) and limit processes to get C) and D) . For expansion of the
[jartial derivatives in C) and D) in series of Bessel functions, see
1 .4.2A1-12).

80
INTEGRALS OF BESSEL FUNCTIONS
3.7
3.7. Lommel Functions and Struve Functions
When |j, = v , Lommel functions and Struve functions are identical ex-
except for a multiplicative constant. We present, below, a partial list of
results which follow Ъу specializing previous equations of this chapter.
H»(Z) " a»-^rL) = г(з/Щ/1) г^^/^^М • CD
Lu(z) -_ e-^+14(ze^) .
^'¦'¦'¦'¦'%-w.)
>v-l
B)
C)
(*)
E)
Hu(z) - Y^z)--M?-T3F0(l,ii-u;-4/z2) , izl-*- , larg z|<n .F)
f(Sr(^u)
If m terms of the hypergeometric series F) are used to approximate
Hv(z) - Yy(z) , then the remainder is of the same sign and numerically less
than the first term neglected provided v is real, z > 0 and m-t-^-v a 0 .
I.u(z)-Ll;(z)^ r[|fr^) 3*О(^-»;Ф2) > 14-*-, largz|<n/2 . G)
3.7 EETRESEMTATIOKS OF / tMvv(t)dt IN TERMS OF LOMMEL FUNCTIONS 81
Hn+*(z) = \ф) + Щ^1 3f0(-^i;-4/z2) ,
n2n:
, 3*0^
n a positive integer or zero . (8)
H.n-i(z) = (-)njn+i(z) ; L_n.i(z) = In+iC") , n an integer . (9)
H_i(z) = B/TTzJsin z ; ^.(z) = B/ttz J(i_COs z)
л4„
L_i(z) = B/nzJsinh z ; L^(z) = B/nzJ(cosh z-1)
W'W'l-fwJfe;
Hy_!(z) - Hu+1(z) = 2Hj(z) - -Tj-
ГD)Г(и+3/2)
zH^(z) + ^(z) = zH^z) .
zH-(z) - uHu(Z) - г(^^5/2) - ^+1B)
Ь {¦¦"*•'¦'} "??й^;---^"
(ю)
(ii)
A2)
A3)
A4)
A5)
A6)
A7)
d-{z4(z)} = zV1(Z) •
A8)

82
d_
dz
IHTEGRAIS OF BBSSEL FUNCTIONS
1 J 2ur(i)r(v+3/2)
H ( 1 = 4 5" Bk+v-KL)r(v+k+l) , .
v r(i)r(u+f) ^To k.'Bk+2v+l)Bk+l) dv+2k+l^ ' •
°()="^ 2k+l
Hl(z),2-2Jo(z) + iZ ^
11 " " k=l 4k2-!
к—и к—и
чк 2
H0(z) =2 Z (") Jk+i(z/2) •
k=O
H (z) = B/тт)-B/ттг)Б1п z+2Z (-) Jk+i(z/2)Jk+3/2(z/2 )
k=O
3.7
A9)
B0)
B1)
B2)
B3)
B4)
B5)
Equation B4) follows from A) and 1.3.6A0). Differentiate B4) to get
B5).
3.8 REPRESENTATIONS OF
J t'Vl;(t)dt
IN TERMS OF LCMMEL FUNCTIONS 83
3.8. Anger-Weber Functions
Anger's function Jv(z) and Weber's function Bv(z) are defined Ъу
the equation
Jo
ei(i>e-z sin e)dQ
(i)
and in view of 1.4.5G-8),
n^(z) - Jv(z)] = (sin vtt) / e
Jo
-z sinh t-vt
dt , R(z)>0 , B)
п[ЕиB) + Yv(z)] = - f e"z slnh *(еУЧе-^соЕ vn)dt , R(z)>0 . C)
From B) we have
Jn(z) = Jn(z) , n = 0,±l,±2,.
(*)
If we expand the integrand.of (l) in powers of z and integrate term
by term, then with the aid of 1.2A5), we can derive power series expansions
for the Anger-Weter functions, and indeed show that they are essentially
Lommel functions. We have
ttJv(z) = (sin vtt)[so^(z) - VB_lfV(z)] ,
E)
ttEv(z) = -(l+cos vtt)s0 v(z) - v(l-cos vtt)s_-l v(z)
F)

84
INTEGRALS OF HESSEL FUNCTIONS
3.8
Since s „(z) is an even function in v , the Anger-Weber functions
are connected by the relations
(sin imty^z) = (cos итт)Еи(г) - E_v(z)
(sin utt)Ev(z) = J_v(z) - (cos utt)Jv(z)
G)
(8)
Difference-differential properties follow from those of Lcramel func-
functions. We omit details, but see Watson A945, p. 311).
The combination of E-6) with 3.2B) yields
"|») " Ju(z)] = (sin ™)[So,i;(z) - uS-l,u(z)] > <9>
tt[bv(z) + Yv(z)] = -A+cos utt)S0^(z) - u(l-cos utt)S.1j1;(z) , A0)
and asymptotic expansions for the Anger-Weber functions follow from the
concluding remarks of 3.2. When n is a positive integer or zero, we de-
deduce from A0) and 3.7E,6)
<J»
»h,w * t«] ¦ i Tj^m
Nn-2k-l
k=0
Г(п+#-к)
.^^¦¦„W]M-,-|^^j—
A1)
A2)
I
3.9 REPRESENTATIONS OF
J t*Vt)at
IN TERMS OF LOMMEL FUNCTIONS 85
3.9. / t|JWl;(t)dt and Formulae for Tabulated Functions
J t%(t)ut = (h+»-1)zCu(z)b11.1j1;_1(z)-zCu.1(z)b11jI;(z) . A)
e^^J t^(t)dt = (^-I)z3l;(z)s^1;l;.1(ze^)
+ ize,,.1(z)B^l;(ze*lTT) ,
iUTT
3v(z) = А1и(г) + Be Ky(z) , A and В are constants
In A) and B), s „(z) can Ъе replaced Ъу S v(z) .
J tUCv(t)dt = 2V-1r^)T(^v)z [cu(z)Hu.1(z)-Cl;_1(z)Hl;(z)} ,
R(u)>-i .
B)
C)
f CQ(t)dt = zC0(z)+(ttz/2) (c1(z)Ho(z)-Co(z)H1(z)} . D)
Jo
J t\(t)dt = 2V-XT(hn?v)z {3u(z)Li;.1(z)-3i;_1(z)L0(z)} ,
R(u)>-| . E)

86
INTEGRALS OF BESSEL FUNCTIOHS
3.9
/ Bo(t)dt = zBo(z)+(ttz/2)/L1(z)Bo(z)-Lo(z)B1(z)} . (б)
Jo L
Let S u(z) denote the sum of the first m terms of the series
3.2C). Then repeated integration Ъу parts yields
/ t\(t)dt = zCu.1(z)S^u(z)-z(^-l)Cu(z)S™.ljU.1(z)+Rm , G)
d z
u z
larg z|<tt , R(n)<-5
(8)
If p, , и and z are real and M = max 10-^A;) | , then
t>z
/"-1-H+vN A-H-v^ ] Mfm X+\i.-v\
ii,., ^2m-l V 2 /Л 2 Л[ V 2 У
К|< i^B/z)'
V 2 J
('
2 у
О)
and except possibly for sign the part in curly brackets is the last term
of the series which appears as the coefficient of С,,(г) in G). The con-
conditions on \i , и and z can be relaxed but we emit details. If ц, = -1 ,
и = 0 , the above result was derived by Smith A943).
3.9 REEKESIMTATIONS OF
/ t^(t)dt
IN TERMS OF LCMMEL FUNCTIONS 87
/Vh^W = ^е^^)"г(^=|^)г(^)
+ (n+v-l)z4l)(z)8ll.1<v.1(z)-zHjy(z)Sli^(z) , R(n±u)>-1 .
A0)
f H0X)(t)dt = l+z41)(z)8 (z)-zhJ1)(z)8_1 ,(z) , A1)
Jo
во,оЫ = / S.1A(t)dt
J z
A2)
z2S.iA(z)-3F0(l,i3/2;-4/z2) j zS0^0(z) ^ 3F0(l,i,i;-4/z2) ,
|Z|—>m , |arg Z|<TT
A3)
/ f^^OOdt =2zS.2A(z)Hol)(z)-zS.lj0(z)Hil)(z) , A4)
S-l,o(z) = 2J S-2,l00ctt ,
A5)
z°S.2A(z)-3F0(l,l,2;-4/zd) j z^S_^0(z) - 3F0(l,l,l;-4/z2) ,
|z|—>co ) |arg z| <тт
A6)

88
IHTEGRAIS OF BESSEL FUHCTIOUS
3.9
/Zt%(t)dt = 2^-ir(Hz|li)r(H^)
- (,+.-l)ze-*1^-2\.1^.1(ze*ln)Ku(z)
- ze-4i^-1\,u(zei1")Kw.1(z) .
A7)
/ Jn(t)dt = (n-lJzs.-L _1(z)Jn(z)-zs0 пЫЛп.-^г) , A8)
s^n(z) = _S_ ^Q.; 3|S , ?|5 ;-z2/4=) , n even or zero , A9)
J-l,n-l<z) = Т^ l^C1' ? ' ^ ;"z2/4) ' П 6Ven OT Zer° • B°)
f Jn(t)dt = l+^-lJzS.! ^(zJjJzJ-zS0,n(z)Jn_1(z) , B1)
zSo,n(z) = 3*0 (^ ^f1 * ^f1 J-4/z2) , n odd , B2)
2S-l,n-l(*) = 3^0 (l, ^ ^ i-4/^2) , n odd .
B3)
If n is odd, B2-23) are exact. If n is even or zero, these equations
axe an asymptotic representation valid for |z|>n , |z|—» oo and
|arg z|<tt .
3.10 RETEEBSENTATIONS OF / t^v(t)dt IN TERMS OF LOMMEL FUNCTIONS 89
m+2
(m+l):
K^z^F^l; 5|3, 5|3 ;z2/4)
B4)
J\\(t)at = 2m-1[rB|i)}2 -(m-Dz-^Jz^Fo^l, ^f , ^1 ^4/z2)
- znK1(zKF0(l, ^S, —^^A2) , m odd . B5)
If m is even, B5) is an asymptotic expansion valid for |z|>m ,
| z | —> m and | arg z | < тт .
3.10. Fourier-Bessel Coefficients
Let Yr ^e ^^e r-th. positive zero of J0(z) , i.e., J0(yr) = 0 .
Suppose we consider the Fourier-Bessel expansion
f(x) = Z Vo<?)
k=l
A)
In view of 13.2E),
/ xJ0(Ymx)J0(Ynx)dx = 0 if m ^ n ,
= iJx(Yn) , m = n ,
B)

90
and so
IHTEGRAIS OF BESSEL FUNCTIONS
3.10
- 2 f1
Jl(Yk)^O
t f(t)J0(Ykt)dt
C)
The latter may Ъе called a Fourier-Bessel coefficient. For a discussion of
Fourier-Bessel series, see Watson A945, Ch. XVIII).
In the following,formulas are presented to facilitate the evaluation
of a^. when f(t) has a power series representation. Some related items
are also considered. Most of the expressions are in a report Ъу Butler and
Pohlhausen A954) where tables are also available. All formulas are immedi-
immediate consequences of results in this and previous chapters.
/ t%(Yrt)dt = Yr^[(^+n-l)Jn(Yr)s x _1(Yr)-Jn.1(Yr)s (Yr)] ,
^0
R(li)>-n-l .
D)
JPn(Y.) =
-2,
2nVIr
4F1(l-n,l-n,n+l,n+l;3/2;-Y;c:) . E)
J2n+l(Yr) = (-)nJi(Yr) 4F1(-n,-n,n+l,n+l;|;-Yr ) . F)
fV^Y^dt = ^> lF2(l, ^ , S» ;.Щ . G)
J n (m+l Y v d d '
f
tX(Yrt)dt - Ъ^1 ^Q.. Tf , Ш. ..Щ , m>0 ,
= Yr > m = °
(8)
z
3.10 REPRESENTATIONS OF j t^W,,(t)dt IN TERMS OF LOMMEL FUHCTIONS 91
The latter two formulae are convenient if m»Yr
Г
Jo
Ло(у^)« =
Jl^r) Bm_iJ P1 Pm.2
Yr
Y?
if Г
% Jo
t"m-"jo(Yrt)dt
Jl(Yr) L ^ (-)kBm-lJBm-3J...Bm-2k,lJ
1+ 2
Yr k»i
Yf
42,^ ^N2 „2 л1
+ (-)mBm-lJBm-3J...32 Г ^^ )ft # m>0
Yr ^0
O)
Jn(Yvt)dt = ^- Z J2k+l(Yr) •
rr k=0
A0)
/
Lt2m+1J0(Yrt)dt = J-±?- 3F0(-m,-m,l;-4/Y2)
A1)
z1
^^(Yrt^t = ^- 3F0(l-m,l-m,l;-4/Y2) , m>0 . A2)
vf
Г\**\{УТ* Lt = ^^ f VmJo(Yrt )dt .
Jq Yr J о
A3)
Kor integrals of the a"bove type involving the product of two Bessel
functions, see 11.2C0-44).

92
INTEGRALS OF EESSEL FUNCTIONS
3.11
3.11 REPRESENTATIONS OF
J t%(t)dt
IN TERMS OF LOMMEL FUNCTIONS 93
3.11. Polynomial Approximations
/ [jo(t)+iYo(t)]dt = xS.1A(x)[jo(x)+iYo(x)]-xSo^o(x)[j1(x)+iY1(x)] ,
J x
xS0>0(x) = (ttx/2)[H0(x)-Y0(x)] , х23.1Л(х) = (пх2/2) [Hl(x)-Y1(x)-2/n] ,
So,o(x) = / e_i l(t)dt >
J x
9 9
xSO,o(x)= X (") ak(x/5)-2k+e0(x), x2S_1A(x)= ? (-)kbk(x/5)-2k+e1(x)
k=O k=O
|eo(x)|<l • 1O, |ei(x)|<0.5 • 1O, 5 <; x <; » . (l)
See 3.9A2,13).
ak
0
1
2
3
4
5
6
7
8
9
1. OOOOO
0.03999
0. 01434
0. 01329
0.01768
0.02206
0.02063
0.01279
0. 00462
0.00073
0000
9434
4735
5944
5262
0880
3981
8010
7212
4559
1.00000
0.11999
0. 07172
0.09307
0.15916
0.24266
0.26824
0.19197
0. 07866
0.01395
0000
8301
3673
1607
7356
9680
1758
0155
2606
6617
See
2.8C,4) for seme other approximations to / [jQ(t)+iY0(t)] dt
J x
00 ,
J t[jo(t)+iYo(t)]dt=2xS.2jl(x)[jo(x)+iYo(x)]-xS.1^o(x)[j1(x)+iY1(x)] ,
J x
S-l,o(x) = 2 / S-2,l№ >
« x
9 9
\k_ /__/,_^2k, _ /._^ __3_ /„\_ ^- / \k.
,-2k.
«8.1,o= Z (") ck(x/5) +^0(x), x°S.2>1(x)= X (-L(x/5) ^i(x) ,
k=0 k=0
n-7
-7
|eo(x)| <2 • 10"', |ei(x)|<2 • 10"', 5 <; x s » .
B)
See 3.9A4,15).
ck
dk
0
1
2
3
4
5
6
7
8
9
1.00000
0.15999
0.10161
0.13081
0.20740
0.28330
0.27902
0.17891
0.06622
0. 01070
0000
2815
9385
1585
4022
0508
9488
5710
8328
2234
1.00000
0.31998
0.30485
0.52324
1.03702
1.69980
1.95320
1.43132
0.59605
0.10702
0000
5629
8155
6341
0112
3050
6413
5684
4956
2336
л ^
See 2.8G,8) for some other approximations to / t" [jQ(t )+iYQ(t)] dt .
"x
The procedure for deriving the approximations to x S_-j_ ]_(x) and
x3S_2 ]_(x) are given in Luke A955). See also Luke (I960).' The approxi-
approximations for xS0 0(x) and x S_]_ 0(x) follow Ъу integration. Using the
approximation to xSQ 0(x) , we can readily derive an approximation for the
Integral of H0(x) . 'see 9.3C). We have

94
INTEGRALS OF BESSEL FUNCTIONS
3.12
«i 8
(и/2)/ [^(t^Yjt^dt = (Y+ln2x)+(x/5)-2 Z (-)Ч(х/5)'2к+е(х) >
Jo k=O
fk = |ak+1/(k+l), |e(x)|<0.5 • 10"', 5 <. x <; a.
C)
0
1
2
3
4
5
6
7
8
0.01999
0.00358
0. 00221
0. 00221
0. 00220
0. 00171
0. 00091
0. 00028
0. 00004
9717
6184
5991
0658
6088
9498
4144
9201
0809
3.12. Description of Mathematical Tables
For tables relating to / tl-V^t )dt , see 2.9. For tables of Struve
functions and their integrals, see national Bureau of Standards A962).
See also 9.7. The following are pertinent to the material of this chapter.
Butler and Polhausen A954)
^0
(Vb)cLt , n = 0,1, r = 1AI0 ,
m = 0AI0 , 5d where yr is the r-th zero of J0(x) , i.e.,
J0(Yr) = 0 . Also given to 5d are the following: yr, J]_(yr) >
r = 1AI0 ; Jn(Yr) , r = 1AI0, all n such that Jn(Yr) ^ 0.5
10
,-5
National Bureau of Standards A946): Hn(x) , L^x) , n = 0,-1,-2,
x = 0@.1I0 , 7 or 8d.
Watson A945): Hn(x) , n = 0,1, x = 0@.02I6 , 7d.
CHAPTER IV
/ e"ttMTCu(t)dt AND AN ASSOCIATED BESSEL FUNCTION
4.1. Introducti on
/
In the first part of the chapter, we use 1.4. lB) as a starting point
to develop convergent and asymptotic expansions for the evaluation of
z
e t^Kp(t)dt and related integrals. Here the material is analogous to
that of Chapter II. The latter part of the chapter is devoted to repre-
representations of the above integral in terms of an associated Bessel function,
see Luke A952), and so is analogous to the contents of Chapter III.
Reduction formulae for a more general form of the above integral are
presented in Chapter V. If p, = +i> , the above integral can be simply
evaluated, and this too is considered in Chapter V.
4.2. Power Series Expansions and Connecting Formulae
f eiVju(t)dt = **)&? Z Г&2?}
Jo " T(i) kt ГBи+1+к)к!
k)Bizf
(ц+и+1+к) '
^+1(z/2)"
(l^l)r(Ll) 2*2(^^2-^-2;2iz)
R(|i+u)>-l
A)
95

96
INTEGRALS OF BESSEL FUNCTIONS
4.2
tt Г it+HY (f)df _ (-f+1Bz)-nz^+1 2|I1 r^-n+k)Bn-l-k).'(-2iz)k
2j0 * V ] ' T(i) J-Q (ц+1+к-п)
z
+ (In 2z) / e^t^J (t)dt
Bz)V+1 ? r(n-Hj+k)Biz)k
~ТШ ^ Bn+k).1k.1(ti+l+n+k)
X |t(k+l)+tBn+k+l)-t(n+k+i)+ ^+1^k I ,
E(ti±n)>-1 . B)
ze?iiTT
^0 ' О
R(n+u)>-l • C)
J e4^u(tLt = (^fcffii) 2F2(^^+l;2.+l^+.+2;±2z) ,
В(ц+»)>-1 . D)
ze-^n . z
/ B-\\(t)it = Ane^(l;^)n Г e^t^^t)^ , B(n±u)>-1 . E)
Jo Jo
f|Z
4.2 / е-Ч^Ку^)^
AND AN ASSOCIATED BESSEL FUNCTION
97
in
-1(ц+1)п Р e-\^(t)dt = e-ivn /« e\^(t)dt.in[Vt\(t)dt ,
Е(ц+1>)>-1 .
F)
_5in
/ e\%(t)dt -ine^1^1')" f e"t^2)(t)clt , В (цЪ)>-1 . G)
X
>t4(tLt = (-)n^j"V+1 2i"X r(b+k)Bn-l-k).4;2z)k
^ T(i) kt (ц+1+к-п)
J0
ЛП^о^П^+1 -
F(i) kf0 Bn+k).'k.4^1+k+n) |_tCk+l)+tC2n+k+l)
-*(n+k+i)+^lfe^} ^R^)>-1 •
(S)
Additional connecting formulae for the above integrals may be written after
the manner of those in 2.1.
We next establish the following relations.

98
/.
INTEGRALS OF BESSEL FUNCTIONS
4.2
e tJv^'aX r(i)m'rBu+l+m)
Jt— \)
R(n)<-i > n+w+l = -m , m a positive integer or zero . (9)
- |ф(т+1)+фBп+т+1)-ф(]§+п+т)-1п 2z|
+ (-)nz-2n-m ^T1 r(jr-n+k)Bn-l-k).'(-2iz)k
2nr(i) И) (k-2n-m)
+ -L. - in 2zl ,
k-m J
R(^)<-i > ц+n+l = -m , m a positive integer or zero . A0)
4.2 Г е~Н^(Ь)д± AND
AN ASSOCIATED BESSEL FUNCTION
99
/.
,16
— . / ,im(l-e)+n n+m-1 ., .
eet ну ,. ча+ _ (-J 2 Г(^п+т)
e t^(t)dt - ГAК.Bп+т).'
X [(e+l/3Зт^/г-ф ' (т+1)-ф ' Bп+т+1)+ф ' (n+m+^)
+ fф(т+1)+фBп+т+1)-ф(п+т+^)-1п 2z] 1
. (-)nz'2n'm 2-|"X rfWkKgn-l-kllf-aez)''
2nr(i) k=0 (k-2n-m)
(-)n2nz-m у r(n+j+k)Bez)*
ТЩ j^0 Bn+k).'k.4k-m)
k^i
X Гф(к+1)+фBп+к+1)-ф(п+к+^) + — - In 2zl ,
L k-m J
ц+n+l = -m , m a positive integer or zero
A1)
Here e = il and the path of integration is as in 2.l(9). For conditions
of validity, К(ц)<-? if e = 1 . If € = -1 , |б| <п/2 and |6| ^ n/2
provided R(n)<--2 • The proof is similar to 2.3B1-23), and it is
sufficient to prove A1) for e ~ 1 and 6 = 0. Using (8) and 4.5E),
we have
f "eVXOOdt " BOO- ^V2n"m T r^)^-l-k)-4-2Z)k
К ^ 2ПГ(*) Sb (k=2n^
" ^F |0 g^SS^j^iW^^D-t^^?. ^ -^ 2я],(л2,
k=0
kAi

100
INTEBtALS OF BESSEL FUNCTIONS
4.3
where R(n)<-i and
B(a) = (-^n+m'a Г п2а2Г(п+т+|-а)
ГD)а2 L (sin2na )Г(т+1-а )ГBп+т+1-а)
+ lM=ti2^?(eln2z.eg-l)l ,
m.'Bn+m).' J
Use L'Hospital's theorem to evaluate B@) and A1) readily follows for
e = 1 and 6 = 0.
4.3. Expansions in Series of Bessel Functions
\»+\ (z)
(и+к+1)(и-ц)к
/i z e z1^ 1 iz) ,. - \u-r&.Tj.)\v-y>,b
I,,+v+1(z) '
L
e"%(t)dt =
R(n+u)>-l •
szl(z) 2e"zzl,_(z)
A)
I,,xvxo@
.-t-г /j.^4. - _H + lli +2ue zz 2^
u+1
u+2
R(u)>-1 •
k-Q (u+k+l)(i;+k+3) '
B)
Equation (l) comes from 4.2D) and 1.3.6(9). In connection vith (l) and
B), see also 5.2G-9), 10.3 and 12.2.
4.4 J e%(t)dt
AND AN ASSOCIATED BESSEL FUNCTION
Let
u+1 " (") (v+2k+l)(v-^)
a " 2Z" ^ —T^Tl Jv+2k+l(^) ,
k=0
(^"+lJk+2
z^+1
H+u+1
J^z^z^1 Z r—tt — W(^)
I=1 (^+u+lJk+l
g = ф(т+1-а) + фBп+т+1-а) - ф(п+т+^а) , ^+п+1+т = а . A3) I ^^ (see ^^ 5_2(м_21))
nz
/ (cos t)t^Jy(t)dt = a sin z+p cos z ,
/ (sin t)t^Jv(t)dt = - a cos z+Э sin z ,
^0
nZ n Z
/ cos(z-t)ttiju(t)dt = p , / sin (z-t)t^Jy(t)dt = а
Jo J о
101
C)
D)
E)
F)
G)
In C-7), R(n+v)>-l , If |i= -3/2 ,
а = (/-^-^(z) , p = (v-i)-1z-%u(z)-(U24)z*Ji;+1(z) . (8)
4.4. Asymptotic Expansions for Large z
The results of this section follow from 4.2, the connecting formulae
for Bessel functions and the pertinent asymptotic expansions in 1.3.3 and
1 ..'». 4.

102
IHTEGRAIS OF EESSEL FUNCTIONS
f VVl^tJdf- Г(ц+р+1)Г(ц-1»1)вШтт(ц.-10
<J 0 2^C/2) TT COS UTT
4.4
#+1
Bnz)*(n+i)
3F1(^+u,i-u,-i-ti>2-M.; ^)
,e-2ze-ic(^)n - k
z^e e
2Bttz)? k=0
Z <-)'
-ka >
R(n+U)>-1 > H+i? is not a positive integer or zero ,
|z|—»» , -B+€)TT/2<arg z<B-c)n/2 , e = ±1 ,
A)
where the coefficients notated ck are used throughout this section and are
given by the following relations.
4(k+l)ck+1 = fkck + gfcCt.-L , c.-l = 0 , cQ = 1 ,
fk = 2[2k2 - к(ц-5/2) - u2 - p, + I] ,
gk = -(k+|+i;)(k4-i;)(k-n-|)
B)
C)
D)
(i+O.Clr-u),
k^2 "^k
2kk.'
dk , dk = 3F2(-k,l,n-k+|;2+u-k,i-u-k;l) , E)
dk+1 = 1 + (k+lK^-k-i) d
F)
4.4 f e^t^^Jdt
AND AM ASSOCIATED BESSEL FUNCTION
103
Equations B-4) come from 4.2D) and 1.3.3C5-37). Equation E) is
most easily found Ъу inserting the asymptotic expansion of Iy(z) in
n m
/ e"tt^u(t )dt and integrating Ъу parts.
J z
/
If (ц+i) is a positive integer, say m , then
e"ttm"%(t)dt'-'-i ?i S Гщ 2z+t(BH)-t(m4n))
BттJ2тт.'
- ф(т+-|--и) - тт tan iml
m-l /1
z- i ^ (^)k(i-")k (J-^)C/2^)mC/2-,)m
BttJ k=0 k.'(m-k)Bz)k Bz )Bтг)г2т(т+1).'
v ^1,1^3/2+и-На,3/2-и+т J^N
A42V 2,m+2 2z)~
m-l -2zo-ie(u+i)n »
zee
к -к
2Btt)'
Z (-rv
k=0
R(u+m+|-)>0 ^ m a positive integer or zero ,
|г|->ш } -B+e)TT/2<arg z<B-c)tt/2 , e = ±1
G)
rZe-VK,(t)dt^ Г(И+и+1)Г(,-и+1) .i(n/2z)»z^e-2z J- (.}k -к ^
J0 2^C/2) k=0
M1
R(p,±i;)>-1, |z|—»«. , |arg z|< Зтт/2 .
(S)

(эт)
(II)
901
'@l) ptre (z)S"f ihojj 'Лх^1ТШ]<
ug>z Э^в>д- ' co<—|Z| < i-<(n+ii)a
0=4 g(zug)
(s/S^S
^/^3 u n Or
ээлхЗ (8) рте (s)g-f jo uo-fq-BUTqinao эти;
g/u? > |z Sjb| ' ш<—|z| ' 0<(«+f+ra)a
< С— -\ г+Ж'г ^)^af
V I 1та+л-г/?'та+п+г/?'х'х7 A ш, _
4(zg)(^-ra);5i 0=4
,-(l+ra)TIIg(zg)
s/sn'+s/sjtjo^s/u),!-)
4/ ?чЧ/ ^,s -^ ш2г(гЛ) + Г(л-|+ш)ф-(Л+|+ш)Ф-
# л ^ ^ \# i f^ л / . л ~т L
'(«-f)^(«+!),(-) т-ш
(l+m)/|i+zg щ] —
•I (a-.
.'V
¦(«-f) (л^Ыг/")™(-)
-w(^%.rav j
шномая lassiHH aawioossv kv ohv
*7 V
@1)
g/n? >|z SjbI ' ш<—|z|
' - J° sidT^xtim ¦в osib sf n sssitm -| jo ат<3:тч.1тш в q.ou sf т1 'о <A+л+т|)Я
г+Ti
(T -fTi-ffTi-f-^-f 'л+|) V ^(g/zu) +
utI soo ^C/2)^3
un soo(x+«-Ti)J(i+n+Ti)j
•WD-
Of
Li4s J
UT+SZ
F)
jCq pso^idsj z qq-ТЛ (a) ss^ 'ojsz jo Ja3sq.uf sAxq.fsod: в sx (f+fi) JI
1+ = э < g/uC-g)>z SJB>g/uC+s)- ' ш<—|z|
oJaz jo л:аЭач.ит SAxq.fsod: ¦s q.ou sf S+tI « Т-<(л+т1)д
(L. -<ч\-г<т\-г-<л-г<а^у? X
(S+Ti)s(zug) uTi soo ^C/2)^3
0=4 3(znsK
4- ^ ^zzsa
On
• ч-р(ч-)п1н^а J
ft
snoiiomm lassaH до етудаядм!
тот

106
INTEGRALS OF BESSEL FUNCTIONS
1 Г2
•5 I 4.6 / e-4l%,(t)e.t
cos vn
J0 П 2^C/2 )^cos ид
R(m±u)>-1 , p, is not a multiple of \ unless v is also a multiple of \ ,
|z|—><x> , -2n<arg z<n
A3)
The sum and difference of A2) and A3) gives asymptotic representations
eittM'Jy(t)dt and / elttMYu(t)dt , respectively.
0 ^0
4.5. Infinite Integrals
/;
2^1r(|)r(v-^)
R(ti.+w)>-l ^ R(n)<-i •
A)
е^(^1>ТГ(^^1)Г(ц-^1)
/* e"t\(t)dt =
" 0 тт • 2^C/2 )^cos цд
^ < »2 cos p,n cos vn + i sin(p,+u)nj ,
R(M±V)>-1 , -KE(n)<4
B)
AND AN ASS0CIA3ED BESSEL FUNCTION
107
f eMlu(t)dt = r(tvHH-l?r(-ti-»? ,
J 0 2^+lr(i)r(v-M,)
R(ti.+w)>-l , R(m.)<4 • C)
D)
Г e-4%(t)dt = Г(и*р»1?Г(^-^1) ^ r(^±u)>.1 .
Jq 2^C/2)^
/ ettM'Ku(t)dt = - Г(ц+У»1)Г(ц-«+1)еов vn
^0 2^C/2 )^cos ^n
R(M±»)>-1 , -KR(m,)<4 • E)
4.6. An Associated Bessel Function
With appropriate change of notation, a representation for the evalua-
nz
tion of / e^Ky(t)dt and related integrals follows from 3.1B"). Thus,
for example,
nZ
J e"tt%(t)dt = z[(M,-U)(M,+U+l)-1Ku(z)h^u+1(z)+Ku+1(z)h^u(z)]
+ (^+l)-1e-zz^1Ku(z) . A)
Further formulae of this type are given in 4.11. For the present, ve
delineate properties of h y(z) and a related function notated H „(z)
Here

108
\,v^
м»
INTEGRALS OF EESSEL FUNCTIONS
e'Z у 2^кН-1B^1)Bцн-5). . . Bn+gfcfl)
<2^) k=0 [(^lJ-»2] [(,+2J-U2]...[(,+k+lJ.,2]
ег^1Г(и.-''+1)Г(м.+1'+1) J" Bz)kr(tx>k-f3/2)
Г(ц+3/2 ) k=0 Г(р,-У+к+2)Г(р,+1Я-к+2)
2F2 A, p,+3/2;p,-v+2, p,+v+2 ;2z )
4.6
(m,-u+i)(m,+"+i)
B)
satisfies the differential equation
[z2l? + zD - (z2+v2)]w(z) = e"zz^+1 .
C)
The series B) is finite if p, is a positive odd multiple of -\ (except
-\) provided that both the numbers p+v are not negative integers. If p,
is not a positive odd multiple of -\ (except -\), the expansion is not
defined if either of the" numbers p+u is a negative integer.
In (l), h y(z) maybe replaced by another particular integral of C)
defined as*
И tz) - h (zi Г(*)Г(ц-1>+1)Г(ц+»+1) L / y,
^()'V()" 2^Г(.+3/2) Г( 0
Ku(z)sin(i;-M,)n
TT COS (j,TT
D)
If either of the numbers p+u is a positive integer or zero, H „(z) may
be represented by the terminating series
* Formulae in Iuke A952) corresponding to D) above and B), (8) and A0)
of 4.9 contain typographical errors.
4.6 / e~tt^Ku(t)dt AND AN ASSOCIATED BESSEL
FUNCTION
109
Г \l^\ (,2-/j {(,-D2-v2}
1 + ¦? -Г- + А iJe +
[ C2^1)* B^-l)B^3)z2
~^r 3Fi(^-M.+^-M.-«;i-M.;-i/2z) .
(,2р,+1;
E)
The latter is not defined if p, = -\ . The same is true if p, is an odd
multiple of 5 , unless both the numbers p+v are positive integers or
zero.
If E) is defined but does not terminate, it is an asymptotic repre-
representation of Нц „(z) valid for |z|—»¦» and |arg z|<3tt/2 . To prove
this use B), D; and the pertinent asymptotic expansions in 1.3.3.
If p, = m--g where m is a positive integer or zero, we construct the
asymptotic expansion of hm_i y(z) using results in 1.3.3 and 1.3.4. It
is convenient to change the argument and write
„m+1 ./ \ -iiim 1. , .
2 m (cos ш)е 2 . 4in> j_ H(l)f 4
mm-i(z-mTT/2+un/2+TT/4) m-1 , ..>.,_. ,-k
k=0 r(m+4+u-k )r(m+i- и -к )
Bz) e ч v x x y 5- (m-l-k).'Biz)
П+i- U -
BnzJ
. . -i(z+un/2+Tr/4) » (-)kE+u) (i-u)
+ (cos ц")е у v к42 у:
nBnz)"
z
k=0 к.'B1г)К
X f-ln 2z+in/2+t(|+i;+k)+i)r(|-i;+k)-t(k+l)"[ ,
|z|—>« , I arg z| < n
F)

110
INTEGBAU3 OF EESSEL FUNCTIONS
4.7
If m = 0 , the left-hand side of F) is a representation for Whittaker's
integral and is notated Ъу Watson A945, p. 339) as Wv(z) . In Watson's
asymptotic expansion, there is a typographical error since the term
following -In 2z should read +in/2 .
In those situations vhere h „(z) is not defined, ve can show that
IL „(z) approaches a limit. See 4.9.
4.7. Recurrence Relations
Both h y(z) and Н„ „(z) satisfy the following recurrence formulae.
V^ + iV^-^^VW'**
2Ц.+1
2ц+1
-z u
e zp
B)
*^z> - \ \,^) ¦ W^ Vi,~i<">+ fcg1 • C)
^f^ н^/z) = (^^(^-DH^^^CzbC-^C-^DH^^Cz) . D)
2BM,+l)H^u(z)=(M,+i;)(M,+1;-l)HM|.1^_1(z)+(i;-M,)A;-M,+l)HM|.1^+1(z)+2e-zz^ . E)
\,^z)=^,-^z)
F)
4.8. Integral Representations
Pz r>z
h „(a) = Iw(a) / e-tt%(t)dt-Ku(z) / e-4%(t)dt ,
H+u is not a negative integer
A)
2
4.8 / e"ttM'Ku(t)dt AND AN ASSOCIATED EESSEL FUNCTION
111
e"zz^+1
M-j" M>-u+l
/ (l-t)^" F1(M,+3/2jM,-u+2;2zt)dt
J0
B)
R(n+u+l)> 0 , p,-u is not a negative integer
"*•<•>" ^S^nU)!^1-^'^^-^^ ¦
R(p,)>-3/2 , R(u)>-i , p+u is not a negative integer . C)
Л nl
= П. BnJe2 W];(z) , ±u is not an odd multiple of \ , D)
where Р„(г) is Legendre's function. Also Wy(z) is Whittaker's integral.
See the remarks following 4.6F).
"¦\k,v
e-4*lv(t)aA-lv(z) I e-4%(t)dt , R(n)<4 • E)
Ha,v(z) = - % !1 / e-Zt2Fl(-^-^;i-^-t/2)dt ,
p, ^ n-J , n a positive integer or zero, R(z)>0 . F)

112
INTEGRALS OF BESSEL FUNCTIONS
4.9
4.9 / e^tM-KyCtJdt AND AN ASSOCIATED BESSEL ПЖСЯПШ
113
H^z> =' ТЩШЬТ) //zV^-Vi(x,-,-;i-^-t/2)dt ,
Ц ^ п~Ь > n a positive integer or zero, R(v-p,)> 0 , R(z)>0 . (?)
sufficient to consider Ни__ „(z) vhere p = u-p, is a positive integer.
Employing 4.7(l),
H (z) - gPr(p+7-v)rBv-p) д (¦„•>
n °°
HQ u(z) = -e'Z-u / e"Z COSh tsinh utdt , R(z)>0
' ^0
(8)
- e z
-z u-1 rCp+j-iQrCgv-p)^-1 k.'Bz)
\-k
^ Г(к+з/2-и)ГBи-к) '
A)
,2,.., -z2/2 z2/2 -1 P™ -t2,. -z2/2 z2/2 -1 _ . ,n»
fiQ l(z /2) = -e ' -e ' z / e dt = -e ' -e ' z Erf с z . (9)
Jz
a.^y(z) = и Г e"z COSh гв1пЬ utdt , R(z)>0 .
te"z coshtdt , R(z)>0 .
A0)
A1)
H_3/2^(z) = e"Zz2 /' e"ztln(l+t/2)dt = -eZz'2Ei(-2z) , R(z)>0 . A2)
See also 4.9E-10).
4.9. Formulae for Hn y(z) When п„ y(z) Is Not Defined
п„ „(z) is not defined if ц is not an odd multiple of -¦§¦ (except
-¦§¦) and either of the numbers ц+и is a negative integer; likewise for
H ^(z) as given Ъу 4.6D). However, H y(z) has a limit when h (z)
%u
¦\i,,^
is not defined. Since HL u(z) ^s an even function in в , it is
M...
and so the left-hand side can Ъе found once Hu_1 y(z) is known. Using
4.7(l) and 4.6D),
B^+l)H rz)+e-Zz^
= 2U'1r(i;)lu(z) 1л 2z +
cos un
+ 2"-1r(
'(")e"Z T (gz)"^^^) /t(UH.k4)-t(k4.i)-.BUH.kH.i) "I ,
T) tTrs k.TBi;+k+l) I J
ТЩ кГ0 k.TBU+k+l)
v is not a negative integer or zero, tv is not an odd multiple of -g . B)
H-n-l,-n(z) = H_n.i,n(z)
C)

114
INTEGRALS OF EESSEL FUNCTIONS
4.9
Bц+1)Н 0(z)+e-zzH
H.^z^lim tt|2
p,->0 H
= 4lo(z)(ln 2zJ - К0(г)Aл 2z)
2T(i)
j Bг) T(kH-g) L.(k+i).2^(k+l)+rKk+i)-2Kk+l)l [
:=O (k!J I L J
D)
For an integral representation of the latter, see 4.8A1).
To evaluate H v(z) for ±v an odd multiple of \, ve consider
ЯГ1_Х n+i(z) or H_n_3/2 n-i^(z) where n is a positive integer or zero.
The former cannot Ъе defined Ъу 4.6D) or B). The latter cannot Ъе de-
defined Ъу B) and 4.7F), but 4.6E) yields the asymptotic expansion
-z -n-3/2 ^ , N
H ,/o nii(z)~LJ F.( l,l,2n+2;n+2;- A) ,
-n-3/2,n+2V ' 2(n+l) 3 1Ч- 2z^
|z|—>=> , |arg z| < Зтг/2
E)
From 4.7A) and 4.7D),
^4,n+3/2(z) = - ^~ \-hn^W^K-3/2,n-i^
-Z П-i -Z П+5
e z г е z
2n 2(n+l)
F)
and so Ну_д_ y(z) for ±u an odd multiple of ¦§¦ can Ъе found Ъу recursion
vith the aid of the following formulae.
H X l(z) = ^_ Г-e-zin z+eZEi(-2z)l
-2^2 1 L J
2z2
G)
115
4.10 / e"tt^Ku(t)dt AND AN ASSOCIATED BESSEL FUNCTION
1 [e'ZBz2+3Z-l)-2e"Z(z+l)ln z-2eZ(z-l)Ei(-2z)] . (8)
HI,3/2(Z)
4z
3/2
H-3/2,i(z) = - \Ei(-2z)
2
H-5/2,3/2(z) = -372 [e-ZH-2ez(z-l)Ei(-2z)]
6z '
(9)
A0)
Here Ei(-z) is the exponential integral, see 7.8. To prove these ex-
expressions use the Bessel function representations for v half an odd
integer (see 1.4.6(8-9)),and 4.8E) in the form
H^(z) = lim L(z) Г e-tt^(t)dt-Iu(z) Г e-4%(t)dtl . (ll)
d—>mL J z v z -I
4.10. Expansions of ^„(z) and Н^„(г) in Series of Bessel Functions
h (Z) - gM'H'1r(^l) 4г (-)к(ц+к+1)ГBм,+к+2)
V"vK ' ГBр,+2) kt-Q к.'(м.+"+к+1)(ц-и+к+1) XM.+k+l1'z> '
litu is not a negative integer, ц ^ -1,-2,-3,...
A)
i_L,u(z) = -U~2IO(Z)+2 21 -^— xk(z) > ±v is not ^ integer or zero. B)
k=l k2-u2

116
IHTEGRAIS OF BESSEL FUNCTIONS
2m-1
4.10
b^,,<«W-sJ-<an>W'f b:BMy.{J+;l2]^-i-2^) Vk-^)
2®-+1B/Ti)?(m-l): V (-)kBm-l-2k) K_ k i(a)
W ^ ^ Ak.1Bm-l-k)!Bm-l-2k+2U)Bm-l-2k-2U) ^-k-2'
m>0 , ±2u ^ 1,3,. ..,2m-l
C)
hiv(O
4z
± sinh Z+2Bn)* Z , HV+l) (z)
Й. Bk+l+2u)Bk+l-2u) K^
4u2-l
*2v is not an odd integer .
|^ 9u cos vtt
+ iijz) [¦(»+iM(wi)-t(i)-t(i)] > ^
и is not a negative integer or zero, tu is not an odd multiple of s •
D)
H-l,o(*) =2 Z
- (-)Ч(О , АД2)
k=l
u=0 6
+ =-IoU) •
E)
F)
4.11
JZe%(
t )dt AND AN ASSOCIATED HBSSEL ПШСП0К
117
To prove (l> and B) use 4.6B) and 1.3.6(8). An alternative proof
follows from 4.6C), 1.4.2E,6) and recursive properties of Iv(z) . Use
(l) and B), 4.7A), and limit processes to get E) and (б), respectively.
Finally, (з) and D) follow directly from (l). For expansions in series
of Bessel functions of the partial derivatives in E) and (б), see
1.4.2A1-13).
4.11. Associated Bessel Function Representations for / e~"tt^Kv(t)dt
and Related Integrals
y>Vitt%(t)dt = Ze±^n[(^-U)(^+l)-1Cu(Z)h^u+1(Ze;^ln)
± icWOV^e**1")] +Avfu+l)-1etlazti+1Cu(8) ,
R(n,±u)>-1
A)
e ^ / e
'o
J e^2e-1)tthRu(t)dt = z [(^-,)(^U+l)-\(z)h^u+1(Zeien)
з1ем,тт Г oBe-l)t
6Bc-l)Ru+1(Z)h^u(zeie")] +(li+u+l)-1e1«M"eB«-1)a^+\(z) ,
R(m,±u)>-1 , e = 0,1,
Rv(z) = 1„(г) if 6 = 1 , Ru(z) = К„(г) if 6 = -1 .
B)
If R(p,±u)> -1 , then

118
INTEGRALS OF BESSEL FUNCTIONS
4.11
/"X^W - fx(z) * 2~ ^(^-")ПГ(^^1)Г(,-^1) f E)
Jo П 2^C/2 V
Jo " 2' " 2^C/2) cos атг
COS VTT
, D)
where
fm(z) = ze*i^[(^U)(^+l)-1H(m)(Z)H^u+1(ze-^)
+ ±4^l(z)^>v(ze-^)] * (^Hl)-1eiZZ"+1HW(J) , m = 1,2 . E)
zeis")
а1«мд Г eB«-i)tt^(t)dt = zr(ti.u)(^+1)-1Iu(z)H^u+1(
Jo . L
+ Bс-1Iи+1B)Н^иBе1ет)] ¦(^v+D-V^e^-^S^^z)
+ e-ic(u+l)TT r(p.-|-"-|-l)r(M.-»-|-l)sin(p.-»)TT
2^C/2) n cos р,п
€ = 0,1, R(p,+u)> -1
F)
4-12 «i e"ttMKu(t)dt AMD AN ASSOCIATED BESSEL FUUCTION
119
eie^ Г eBc-l)tt4Ct)dt w гГи,)(^1Г1^(г) ^(ze1-)
J0 L ^^
- Bc-l)Ku+1(z)H^u(zeiOT)] +(^U+l)-1eie^eBe-1)Zz^1Ku(z)
+ е1€м,тт Г_е_ е cos vtt r(p,+w+l)r(p,-w+l)
L =os Ц.ТТ J 2^C/2)^
6 = 0,1, R(p,+u)> -1
G)
In C)-G), p, is not an odd multiple of \ unless v also is an odd
multiple of \ .
4.12. Description of Mathematical Tables
Howarth A950): Column 2 of Table I, p. 137, tabulates tt~5+Btt)//2A(R) ,
A(R) = / t~ e"VKi(v)dt , v = t2/8 , for R = 0.01, 0.05, 0.1, 0.2
Jr
@.2J.0@.5L.0, 00, 3d. For the same range, the third column gives
Btt)/2 ( A(t)dt .
Jo
oee also 10.7.

CHAPTER V
REDUCTION
5.1. General Development
Let
FORMULAS FOR J e'^t^CXt^t
{»Mz)=J e'ptt4(^>tt
vhere Wy(z) is defined by 1.4.3A). Then
pf^(z) = -е"Р\\(\2)+(ц+и)^.1?иB)-аХ^и+1B)
pf^»+l(z) = -e"P8zlVu+1(Xa)+(^«-l)f(i.1)U+1(a>lAf(iI,(O
LMo"
A)
B)
C)
a(u-M,)\f^u+1(z) =--2Ue-pZzX(^z)-2ypfM,,u(z)+'b(M.+")^M,jl,.i(z) , D=)
(p2+X2ab)f „(a) = a\e-pzzX+l(^)+^-«-l)e"PZz^4(^)
- pe"PaaMWi;(Xa)+pB|t.i)f(i_1^(z>[i;a.(li-lJ]f^2,u(z) • E)
To prove B) integrate (l) by parts and use 1.4.3E). Integrate (l) by
parts vith v replaced by u+1 , use 1.4.3D) and obtain C). If in C)
и is replaced Ъу v-1 and this is combined with B), then D) results.
Finally, use B) as is and also vith p, replaced Ъу ц-1 , and combine
with C) to get E). The above is based on a paper by Luke A950a).
120
5.2 REDUCTION FORMULAS FOR Г e"Ptt^y(Xt)dt
5.2. Evaluation of / e"pttMWu(Xt)dt for Special Values
121
of the Parameters
Case I. Let p2+X2ab = 0 and и = ±(ц-1) . Then 5.1E) yields
-pz u+1 r -1
f^z> = ?I^T— |>^) - f wu+1(xz)J ,
f-^u(z) = ¦ elZiv+1[Vv{xz) + ~ w^-i^z)]
We now take X = 1 .
X
u ' 2u+l
eltt"l;Ju(t)dt =
eizz"u+1
ifT- ^(z)+iJ^(z)] + ^I7Z
2и-хBи-1)Г(и)
f eltt-4ji.(t)dt = B/пL Г
e^sin t
dt
= Bn)'2 fsiBz)+i {y+ln 2z-CiBz)Jl
A)
B)
[^(а)-и„+1(г)] , R(u)>-i . C)
D)
E)
J>*X№-4^й- [mzmwz)]-^ЛГ^-R(u)>4• F)

122
IHTEGRAIS OF BESSEL FUNCTIONS
5.2
[VHvtjdt =
'\±t+»T ,+ W = egg^ [lu(z)*Iu+1(O] , B(u)>4
/|Ze-tIn(t)dt = ze-^I^zJ+I^z)] +n[e-zIo(z)-l]
JO
n-1
+ 2e"z 2 (n-k)lk(z)
k=l
G)
= ze-z[ln(z)+In+1(z)]+n[e-zIo(z)-l]+2ne-z J Ik(z) . (8)
k=l
The latter follows from G), 5.1D) and 1.4.3B). For an expansion of
„z
integrals like / e-t cos 6Iu(t)dt in series of Bessel functions, see
10.3.
I
>t"%(t)dt = - ^^[l^I^z)]
, *n
+ 2и-ХB1;-1)Г(и)
(9)
fZettt-ili(t)dt = B/tt)* fZ e±tsinh г dt
J0 2 J0 *
= Btt) [±Ei(±2z);(Y+ln 2z)] . A0)
5.2
REDUCTION FORMULAS FOB Г e"P4^u(Xt )dt
123
±z u+1
JoeiVKu(t)dt=^|r-[Ku(z)±Ku+1(z)],^±l ,R(,)>-i . A1)
King's A914) integral is
f etKQ(t)dt = zez [Ko(z) + Kx(z)] -1 .
«0
A2)
f e*f \(t)dt = ^ [^^(z)] , B(u»i . A3)
о z
do
f
t" cos(z-t)jy(t)dt = -
z-"+1J>)
sin z
2u-l 2"-1Bи-1)Г(и)
t"usin(z-t)Ju(t)dt =
_ z"u+4.i(z)
COS Z
2u-l 2и-1Bи-1)Г(и)
, « ?4 • A4)
, » ^ i • (is)
/o
t"cos(z-t)Ju(t)dt = z^ , R(u)> 4 .
t"sin(z-t)Ju(t)dt = Z Zvll*l{Z) , B(u)>-i •
A6)
A7)
filiations A4)-A7)аге convolution integrals and follow from C)-D). For
Pz
.•valuation of / t^e1^ )ju(t )dt , see 4.3G). The convolution integrals
^O
I..'low can be deduced from (8). For other convolution integrals, see 13.3.1.

124
INTEGRALS OF EESSEL FUNCTIONS
5.2
5.2
,z г n-1 -i
/ cos(Z-t)J2n(t)dt = zJ2n(z)-2n(-)n sin г-21 (-)J2k+l(z) ¦ A8)
Jo L k=o J
z Г n 1
f sin(z-t)J2n(t)dt = zJ2n+1(Z)+2n(-)n cos z-Jo(z)-2 ]T (-)kJ2k(z 1A9)
Jo L b=l J
Г cos(z-t)J2n+1(t)dt = zJ2n+1(z)-Bn+l)(-)n Jo(z)-cos z
+ 2 Z (-)kJ2k(z)l ¦
k=l -1
Г sin(z-t)J2n+1(t)dt = zJ2n+2(z)+Bn+l)(-)n sin z
- 2 Z (-)kJ2k+l(z)l
k=0 J
Case II. Let p = O, p, = tv and X = 1 . Then from 5.lD) ,
B0)
B1)
bf^u.1(z) = zX(z) ,
af-v,v+l(z) = ~z~ Wu(z)
nz
/ t%.1(t)dt = z"jy(z) , R(u)>0
J0
B2)
B3)
B4)
2
REDUCTION FORMULAS FOR Г e"vtt^yv(\t )dt
z
/ f uJu+1(t)dt = —-i z-%(z) .
Jo 2ur(v+l)
z n ¦»
2n / t'1J2n(t)dt=l-2z'1 Y. Bk-l)J2k_1(z)=2z Z Bk-l)J2k.1(z)
0 0 k=l k=n+l
l+Jo(z)+J2n(z)-2 ? J2k(z) = J2n(z)+2 ? J2k(z) , n>0 .
k=0 k=n+l
rz rz n
Bn+l) / t-XJ2n+1(t)dt = / Jo(t)dt-J1(z)-4z'1 X kJ2k(z)
U0 J0 k=l
125
B5)
B6)
¦I,
z n
Jo(t)dt+J1(z)-2J2n+1(z)+4 X J2k+l^z) ' B7)
0 k=0
Equations B6) and B7) follow from 5.1C-4), 1.4.2C-4) and 1.4.3B). An
alternative proof for the first equation of B6) has Ъееп given Ъу Abtiott
A949). He also gives another expansion for the left-hand side of B6) in
series of Bessel functions. In connection vith B6)-B7),see 2.4(l).
Г t"Yu.1(t)dt = *"*„(*) + S!Ki) , к(и)>0 .
 "
J t%.1(t)dt = zulv(z) , R(u)>0 .
B8)
B9)

126
INTEGRALS OF BESSEL FUNCTIONS
5.2
CHAPTER VI
[ f»Iwl(t)dt = z-%(z) - i
Jo 2vr(u+l)
nco
/ f vKu+1(t)dt = z"%(z) .
C0)
f t\ x(t)dt = a^rOO-zVz) , r(v)>o . Ci)
C2)
AIRY FUNCTIONS
6.1. Introduction
In this chapter we consider Airy integrals, their integrals and
associated Airy integrals. Virtually all the material can be readily
deduced from the results given in Chapters I-III.
6.2. Airy Integrals
6.2.1. Definitions
Let
§ = B/3)z3/2 }
Ai(z) = (z*/3)(l_l/3(S)-Il/3(§)} =n(z/3)%l/3(§) ,
Bi(z) = U/3)*{i_l/3(S)+il/3(S)} ,
Ai(-z) = (z*/3)|j.l/3(§)+Jl/3(§)|
Bi(-z) = (z/3)*[j_l/3(§)-Jl/3(§)} .
)',.;>.2. Derivatives
Ai'(z)=-(z/3){l_2/3(§)-I2/3(§)} =|- l^/3(S)
A)
B)
C)
D)
E)
A)
127

128
INTEGRALS OF BESSEL FUNCTIONS
Bi'(z) = 3-*z {1.2/з(§)+12/з(§)}
Ai'(-z) = -(a/5){j.2/3E)-J2/3E>}
Bi'(-z) = 3'*z {j.2/3(§)+J2/3E)}
6.2.3. Interrelations
Ai(z)+e2iTr/3 Ai(ze2lTT/3) + e-21"/3 A^ze1"/3) = 0
Bi(Z)+e2iTr/3Bi(ze21"/3) + e-2i"/3Bi(ze-2i"/3) = 0
Bi(z) = eiTr/6Ai(ze2i"/3) + e-^Ai^e1^3) .
Bi(z) = 2etl^6A±(zet2i^5) - е*1"/2^) .
6.2.4. Differential Equation and Wronsklan
Ai(±z) and Bi(tz) are independent solutions of
ddy = +,
zy
dz^
and
Ai(z)Bi'(z) - Ai'(z)Bi(z) = l/тт
6.2.3
B)
C)
D)
A)
B)
C)
D)
A)
B)
6.2.5
6.2.5. Power Series
AIRY FUNCTIONS
Ai(±z) = а Л(^2/3;±§2/4)+Ъ2 0?1(;4/3;±?2/4 ) .
Bi(tz) = 3*a Л(;2/з;±?2/4)±3*Ь2 ^(-^/Sit^/i)
Ai'(±z) = -Ъ 0F1(;l/3J±?2/4Laz20F1(;5/3j±?2/4) .
Bir(±z) = 3*b ^(A/Sjt^AHs^^CjS/SjtS2^)
3-2/3 -1/3
a = ¦— = 0.35502 80539 , Ъ = °. = 0.25881 94038
ГB/3)
6.2.6. Asymptotic Expansions
ГA/3)
нЫ~Ъ«*г*0(±,1;-±) ,
Izl —> со } | arg zl < тт
«•с,--*,М-«Л(.1, i ,.i) ,
Izl—»» , I arg z| < я
Bi(z)^K-*z-Ve§2F0(l , 5 . 1_\ .ja.en-h-h%Y0(l , 5 ;. ±\ ,
\6 6 2fy Vr e or У
-6 6 2§
6 6 2§/
029
A)
B)
C)
D)
E)
A)
B)
I z|—5»oo > -B+e)n/3<arg z<B-e)n/3 , e = +1
C)

130
INTEGRALS OF BESSEL FUNCTIONS
Bi(
|z|-»o> , -B+e)rr/3<arg z<B-s)tt/3 , e = ±1
6.2.6
L'(z)- тЛЦ*0 (- i , 1 ; ^^cTT-^e-VoO \>l >' 2?) >
D)
[CO
cos(§-?n) Z
k=0
Ц. (-)k(l/6JkE/6Jk
Bk).'B§):
,2k
+ sin(§-irr) X
(-)k(l/6Jk+1E/6Jk+1
k=0 Bk+l).'B§)
|z|—»« , larg z|< 2tt/3
2k+l
Г
Ai'(-z)^TTz^ sin(§-Jn) Z
L . k=0
- (.)k(-l/6M.G/6).
'2кч ' У2к
Bk).'B§Jk
. , . " (-)k(-V6Jk+1G/6Jk+1
- cos(§-4tt) 21
,2k+l
k=0 Bk+l).'B§)
|Z|__»a. , larg zl< 2tt/3
E)
F)
6.2.7
AIRY FUNCTIONS
131
Bi(-z)~ тт2"^ -81п(§-?тг) Y.
L k=f
- С-П1/бJкE/бJк
k=0 Bk).1 B§>
2k
k=0 Bk+l)!B§rk+1
|z|—»•« , |arg z|< 2tt/3
G)
L k=0 Bk).'B§Jk
+ sin(§-irr) 2;
(-)K(-l/6)p^G/6)g
Тгк+Г
;2к+1н/°;2к+1
k=0 Bk+l):B§f
|zl—»oo , |arg z| < 2tt/3
(8)
0.2.7. Integral Representations
TiAi(x) = / cos(t3/3+xt)dt
A)
3i(x) = Г в"*'
-t3/3+xtdt +
TiBiCx) = / e " /""¦"¦"dt + / sin(t /3
'O 
i+xt )dt
B)

132
IMEGRAIS OF BESSEL FUNCTIONS
6.3. Integrals of Airy Integrals
6.3.1. Relations to Other Functions and Interrelations
3'i
f Ai(t)dt = i/3 [Ио^х/з^ИЧ^Я =irV/3(§)
f Ai(-t)dt = 1/3 [J1O1/3(S)*J41/3^5] •

[ Bi(t)dt = 3"i [lio,-i/3^)+I1o,l/3(§)] •
fZBi(-t)dt = 3-* [jiQj _l/3(§ )-Jio,l/3(S)] •
etiTT/3 ,. etin/3 Я*/* -|
Г Bi(t)dt = (±i) / Ai(t)dt+2j Ai(t)dt|
±iTi/3 r zetiTT/3 u
„ ze ' oU Г n ze ли
f / Bi(t)dtdu = (±1) / / Ai(t)dtdu
J0 J0 LJ0 J0
+iTi/3
n zeT ' „u
+ 2e±2in/3 / / Ai(t)dtdi
J0 J0
6.3
6.3.2
A)
B)
C)
D)
E)
F)
AIRY FUHCTIONS
133
6.3.2. Power Series Expansions
/ЛоиМ(«)№.^Л(^|,|я|!)
t/jBi(«)«.t3J.Iir2(I;|,i;l|!)
f [ Bi(+t)dtdu = !3*az2 F2(J ; t ,| ;+ Si)
+ X 'i--3
1Л^Л^|Л|,|,;Е)
ilrre a and Ъ are given Ъу 6.2.5E).
¦ ' • 3. Convergent Expansions in Terms of Lommel Functions
/ Ai(t)dt = тт [Ai(z)Ti'(z)-Ai'(z)ali(z)]
J0
= 1/3 + тт [Al'(z)Gi(z)-Al(z)Gi(z)]
C)
D)
A)

134
Here
Also
INTEGRAIfi OF BESSEL FUNCTIONS
nZ
/ Ai(-t)dt = тт [Ai'(-z)Ti(-z)-Ai(-z)Ti'(-z)]
^0
= 2/3 + тт [Ai'(-z)Hi(-z)-Ai(-z)Hi'(-z)]
nZ
/ Bi(t)dt = тт [Bi(z)Ti'(z)-Bl'(z)Ti(z)]
J0
= тт [Bi'(z)Gi(z)-Bi(z)Gi'(z)]
/ Bi(-t)dt = 17 [Bi'(-z)Ti(-z)-Bi(-z)Ti'(-z)]
J0
[Bi'(-z)Hi(-z)-Bi(-z)Hi'(-z)]
= TT
nTi(-z) - h\?2 (l; I , § >- t) - I Ao,l/3(§)
Hi(-z) = 2 Bi(-z)+Ti(-z) = (§/ttz)S^1^3(§) ,
Gi(z) = 1/3 Bi(z)-Ti(z) = Bi(z)-Hi(z)
6.3.3
B)
C)
(O
E)
F)
G)
Gi(ze±iTT/3>et2iTT/3Hi(-z) = ie±iTT/6 [m(-z )tlBi(-z)J
EzJ „(m). . , -.
where m is 1 or 2 according as the sign is - or + , respectively.
6.3.4
AIRY FUNCTIONS
Gi(z) is a particular integral of
#3..
dz2
zy = -тт
-1
and we have the integral representations
TTHi(z) = fVt3/^tdt ;
n со
TTGi(x) = / sin(t3/3+xt)dt
6.3.4. Expansions in Series of Bessel Functions
/Z г со со -1
Ai(-t)dt = B/3) Z J2k+2/3(§)+ Z J2k+4/3(§)
0 Lk=o k=o J
[ Bi(t)dt =3*B/3)[z (-)W3(§> T (O^kWsC?)!
Jo Lk=o ' k=o -I
Г Bi(-t)dt = 3^B/3) Г Z J2k+2/3(§)- 21 J2k+4/3(§)l
Jo Lk=o k=o J
.i-t.hcr expansions for (l)-D) follow from 6.3.1A-4) and 2.4B).
135
0)
A0)
A1)
/>ZAi(t)dt = B/3) [i (-)\к+2/5(г)- ? (-)kI2k+4/3(§)l • A)
Jo Lk=o k=o ' -I
B)
• C)
D)

136
nTi(z
INTEGRALS OF BESSEL FUNCTIONS
n Jy (-)kBk+l) T ,*)
kt Ck+l)Ck+2)
Tffii'(z) = zi°(g)+2z|oCJ)()L)i2^(§) •
^(-Z) = ^zoEJ^+2)J2k+1(g) .
rrTi'(-z) = -zJQ(§)+2z Z
k=0 Ck+2)Ck+4)
J2k+2(?)
6.3.5
E)
F)
G)
(8)
Further expansions for E)-(8) follow from 6.3.3E) and 3.6B).
6.3.5. Asymptotic Ехрапэ-ions
X
7 -F °°
JAi(t)dt-l/3 - -Z—r Z ("L?"k
Ftt§J k=0
|z| —>t° , I arg z| < тт ,
A)
where the a^'s follow from 2.5B-4) with p, = 0 and и = 1/3 and are
given below for к = 0A)8 .
6.3.5
AIRY FUNCTIONS
137
0 1
1 41/23 • 32
2 9241/27 • 34
3 50 75225/210 • 37
4 51530 08945/2^ • 39
5 167 49663 09205/218 • 311
6 3 98556 96316 33205/222 • 314
7 1038 30537 66658 Э8275/224 • 316
8 25 69132 94549 69752 11365/229 • 318
B)
/;
2e^
-s
Bi(t)dt-^_^ X акГк + i^x Z (-L§"k-iS ,
Ftt§J k=0
.-k , iee
Ftt§ J k=0
Izl^oo , -B+e)Tr/3<arg z<B-e)n/3 , e = ±1
C)
J Ai(-t)dt-| - Ctt§)"^ [(ui+uajcos S-^-UgJeln g]
(O
/ Bi(-t)dt ~ CTT§)[(u1+u2)sin g+(u1-u2)cos g 1 ,
E)
whure |z|—>oo } | arg z|< 2тт/3 . Also
ui = Z (-Г*2Ь5 , «2 = Z (")ka2k+l§
k=0 k=0
-2k-l
F)
i\nd the eu's are as in B).

138
IHTEGRALS OF BESSEL FUNCTIONS
6.3.5
Another set of asymptotic expansions follows from the representations
in terms of Lommel functions, see 6.3.3. It is sufficient to quote the
following expressions.
Hi(-z)^(nz)3F0(^l, з ; з ;- ~)
[z|—><= , | arg zl< 2tt/3
Hi'(-zwV23F0(i, §, f ;-У ,
|z|—>« , |arg z| <2tt/3
,z|-»« , -B+e)n/3<arg z<B-e)n/3 , e = +1 •
л ? /^ 2 4 4 \ _,_ iez^e"? /¦ 1 7 1\
Qi-(z)--и-^-VoC1' I' з; ^-) ^f~2 °v~ e' 6 ^ 2§; '
z|—»•« , -B+e>T/3<arg z<B-e)n/3 , e = +1
For the repeated integrals of Ai(±t) and Bi(tt) , we have
G)
(8)
(9)
A0)
nz />u 3. 1 e'gz/4 ^- ( vk -k
/ Ai(t)dtdu-i z. __ + r- 2. (-) =k§
'0 u0
k=0
|z|—>« , larg z|<n ,
A1)
6.3.5
AIRY FUNCTIONS
139
where
2(k+l)ck+1 = fkck + gkck_-L , c.-l = 0 , co = 1
fk = 3k2 + 6k + 101/36 , gk = -(k+7/6)(k+l/2)(k-l/6)
A2)
The coefficients ck are tabulated below for к = 0(l)8
0
1
2
3
4
5
6
7
8
101/23 • 32
35905/27 • 34
270 49085/210 • 37
3 50151 98785/215 • 39
1386 02554 86385/218 • 311
.22 . ^14
38 91341 34573 23405/222 • 3
21011 91371 80388 18575/225 • 316
1028 71658 49974 61731 12075/231 • 318
A3)
rz Ги 2 1
/ / Ai(-t)dtdu~| z-
Jo Jo
,-5A
3 31/3ГA/з) Bтт)?
[(vi-v2)cos g + Cv-L+VgJsin §] ,
|z|—>« , |arg z|< 2tt/3 ,
A4)
Z, -.k _-2k -sr- / \k _-2k-l
(-) =2k? > V2 = A (-) =2k+l5
A5)
k=0
k=0
urnl the ck's are defined by A2).

140
INTECSAIS OF EESSEL FUNCTIONS
6.4
-k l*e±_ ? ^^-k
J> P e* sr- г-к lee
Jo Jo z5/ tt2 k=0 2z ' n2 k=0
3V6
- iez + ——r—r >
ГA/3)
|z|-
, -B+e)n/3<arg z <B-е)тт/3 > e ¦= ±1 •
A6)
,z ftu
f " [V^dtdu- ^ * ^[(v^sin |-(v1+v2)=os S]
'0 
r(i/3) BTT)f
|Z|_>a> , |arg z|<2n/3
A7)
In A6)-A7), the ck's are as in A2), and v± and v2 are defined by
A5).
6.4. The Integrals of Gi(z) and Hi(-z)
/W^t = 1/3 /W)dt - ЩМ 8^Q;y^) • (i)
/ Vt)at - 2/з /V(-t)dt + ^/±1 2^С2%5}5/Ъ\Щ • B)
Г Hi(t)dt = f Bi(t)dt - Г Gi(t)dt .
Jo Jo Jo
C)
tin/3
Г fz rz rz
Gi(t)dt = Hi(-t)dt - i / Bi(-t)dt + |i / Ai(-
J0 J0 Jq J0
t)dt . D)
6.5
AIRY FUNCTIONS
141
tt f Gi(t)dt-ln В+A/3)Aл 3+2Y) -—i-^/1'1'4/3'5/3^/?2')
Jo 27§2 42 ' ^
iTTee
-§
Ftt§ J k=0
Z/ \k _-k
(-) ak? *
|z|^= , -B+e)TT/3<arg z<B-e)rr/3 , e - ±1 ,
E)
where ak is defined by 6.3.5B ).
tt [ZHi(-t)dt ~m z+(i/3)(m 3+2Y) + -§- ^г^>Ф>Ф _4/?2N ^
JO 27§2 V2 У
|z|—»oo , |arg z|<2tt/3
F)
i". .5. Description of Mathematical Tables
For tables of Airy integrals and related functions, see 1.4.7 and
National Bureau of Standards A962, (Six. 10). See also the following.
Hay A948)
f 2
: Let F(z) = / e-ztAl(t-o1)dt , where i± = -2.3381 is the
first zero of Ai(x). F(z),F'(z),z= x+iy , x = 0( 0.2L ,
у = 0@.2K.2 , 5d.
w
I.In A945): let 3(z) = [l-F(z)] = w [ai'(-w)]'1[l/3+ j Ai(-t)dt] ,
w = zelTT/6 . 3'(z) = z3(z) + weiTT/6[Ai'(-w)]Ai(-w)[l-3(z)] .
Here 3(z) and F(z) are tabulated to 5d for z = 1.0@.2L.8.
In Ldn A955, p. 42) the 3(z) table is repeated. In addition,
3'(z) is given, mostly to 4s, for z = 1.2@.1L.4. The real
part of 3'(z) is also given to 4d for z = 4.5@.1L.8 .

142
INTEGRALS OF BESSEL FUNCTIONS
6.5
Miles (i960): Tables computed Ъу D. Giedt. For notation, see the Lin item
above. 3(z) , 3'(z) > F(z) , z times the imaginary part of F(z) ,
z = -6@.1I0 , 4s. Aside from a few differences of one or two units
in the last figure, the tables of Lin and Miles agree, except that
entries for z =4.6 and 4.8 are in serious disagreement. The need
for 3(z) arises in problems of hydrodynamic stability. See Lin and
Miles and the references quoted there.
National Bureau of Standards A958b): All tables are to 8d.
f(x) = / Ai(-t)dt , F(x) = / f(t)dt , x = -2@.01M .
J0 d0
A0(x) = nHi(-x) , -A^(x) , x = 0@.01I0@.05I1. x = 0.01@.01).1.
«x
G(x) = / Ao(t)dt , x = 0@.5I1 .
^0
Riley and Billings A959): Let A = Ai , В = Bi , F2 = A^B2 ,
CD
(F1J = (A'J+(B'J . / xV2 (a,A2,AB or b) dx ,
^0
pen
/ x^(F') (а'ДА'^А'В' or B1} dx , X = 0(lJ0 , 5s.
J0 L
Rothman A954a): All tables are to 7d. / Ai(t)dt , x = 0@.1O.5 .
Jo
/ Bi(t)dt , x = 0@.1J . / Ai(-t)dt , / Bi(-t)dt ,
Jo ^0 ^0
x = 0@.1I0 . The values for the last two items should read with
opposite sign.
Пх ЛХ
Rothman A954b): / Gi(t)dt , Gi'(x) , / Hi(-t)dt , dHi(-x)/dx = -Hi'(-x) >
«0 ^0
x = 0@.1I0 , 7d.
6.5
AIRY FUNCTIONS
143
u n
nx
Rothman A955a): / Ai(t)dt , / Ai(-t)dt , x = 0@.01J@.1I0 , 8d.
'0 ^0
The values for the latter integral should read with opposite sign.
nX nX
Rothman A955b): / Bi(t)dt , x = 0@.01J , 8d. / Bi(-t)dt ,
J0 ^0
x = 0@.01J@.1I0 , 8d. The values for the latter integral should
read with opposite sign.
Rothman A955c): Gi(x) , Gi'(x) , x = 0@.1J5AO5 , 8d.
Rothman (I955d): Hl(-x) , Hi'(-x) = -dHi(-x)/dx , x = 0@.1J5AO5 , 8d.
The values for the latter function should read with opposite sign.
Rothman
(I955e): / Gi(t)dt , / Hi(-t)dt , x = 0@.1J0 , 8d. The
d0 Jo
values given for the former integral for x = 10.1@.1J0 are wrong.
The source of the mistake is not completely known. However, the value
„10.1
of / Gi(t)dt as deduced from the table is about ten times too
«10
large.
.Scorer A950): Gi(x) , Hi(-x) , x = 0@.1I0 , 7d.
.¦inlrnov A955): V11(s) = j uf(s,l) - Г |2/5i ds ,
Jo L 2ns2 J
Vl2(e) = / ^(8,1)^(8,1) i—r ds ,
J0 L 2CsJJ
J0 L 2 • S-^tts2
ds . For the definitions of
Uj 2(s,l) , see the Smirnov item in 1.4.8. s = 0@.01I0 , 5d.
. ••!¦ und Christiano A952): Ai(x) , Ai'(x) , Bi(x) , Bi'(x) , nGi(x) ,
nGl'(x) , x = -10@.1M , 5d or 5s.

CHAPTER VII
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
7.1. Introduction
The incomplete gamma function may Ъе defined Ъу
Y(a,z) = B/ttJ f ta-2Ki.(t)dt = f e'V^dt , R(a)>0 . (l)
JO 2 JO
Г(а,г) = / e-4a-1dt = Г(а) - Y(a,z) ,
J z
I6K tt/2 , R(a)>0 ;
|6I = tt/2 , 0<R(a)<l . B)
For the path of integration, see 2.1(9). If z^O, 161 < tt/2 , the inte-
integral in B) exists without the restriction on a . If a—>0 , we are led
to the exponential integral, see 7.8(l). In this event exclude the origin
in the path of integration.
In view of (l), many properties of the incomplete gamma function
follow from the material developed in the first four chapters. In addition
to these, we give other important results. Unless stated otherwise, the
reader may assume that the representations in this chapter are essentially
direct consequences of Chapters I-IV, or are given in Erdelyi et al.
A953, Vol. 2, Ch. IX). The incomplete gamma function is a special case
of the confluent hypergeometric function. For the latter, see Erdelyi
et al. A953, Vol. 1, Ch. VI), Buchholz A953), Kratzer and Franz (i960,
Ch. VI), Tricomi A954) and Slater (i960). For references to mathematical
tables of the important functions considered here, see 7.13.
144
7.2
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
145
7.2. Elementary Properties
Partial integration of 7.1A,2) produces the recurrence equations
y(a+l,z) = aY(a,z) - zae"z ,
Г(а+1, z) = аГ(а, z) + zae"z
The connection to the confluent hypergeometric function is given by
y(a,z) = azae"z$(l,l+ajz) = aza$(a, l+a;-z) ,
$(a,c,z) = -^(ujCjZ) ,
Г(а, z) = zae"zf(l,l+a;z) = e"zf (l-a,l-ajz )
The power series representation
/ \ -z a. ^~
Y(a, z) = e z 2.
*k _ za ' (-)kzk
kt0 (a)k+l >=nk;(a+k)
k=0
is the statement C). The functions
Ci(z,a)+iSi(z,a)
^0
-«.«** = »iA-a)i\(l-a,ze-iin)
bdt = e2
A)
B)
C)
(O
E)
F)
nnve been studied by Kreyszig A951, 1953) whom the reader should consult
I'nr tables and an extensive bibliography.

146
INTEGRALS OF BESSEL FUNCTIONS
7.3
7.3. Integral Representations
¦y(a,z) = zacsc тга / е
r
gZ cos eoos(a9+z sln e)de
a not an integer
Y(a,z) = za/2 [VS^jjW]" , R(a)>0 .
-za п"^ (a)<1
о z+t
-*->-f^?
["e-4-^Ka[2(zt)*]dt , в(а)<1 •
^0
A)
B)
C)
D)
7.4. Asymptotic Expansions for Large z
r(a,z)-za-1e-z2F.
¦od^a;-.) = Ba"V» Z (-)*(!-)*
^, „v.z"k
k=0
|z|-
, larg z| <3n/2
A)
m-1
r(a,z) = z^ie Z~ (-)k(l-a)kz-k + \KO
k=0
^(a.z) = (-f(l-a)m rt^V^t , larg z| < 3n/2 .
B)
7.4
INCOMPLETE GAMMA. FUNCTION AND RELATED FUNCTIONS
147
If a and z are real, R^a, z) is less in absolute value than the
first neglected term and of the same sign.
Г.
it, a-1,, a iz
e t dt "-v z e
со I Vе(л я \ со
-z (")( Wl^z
(-)k(l-a),
'2k
k=l
,2k
k=0
^2k+l
|z|—>co , |arg z| <n
C)
If a and z are real and a finite aumber of terms is used in each series,
then error for each series is less in magnitude than the first neglected
term and is of the same sign. For an alternative discussion of the error,
see 3.9G-9) with p, = a--| and v = t\ .
7.5. Infinite Integrals
,-tj.a-l.
e-^'-Sit = Г(а) , R(a)>0
/;
-t
dt = ^ .
/ e^t^dt = е21ттаГ(а) , R(a)>0
J n
/;
e^t'^cLt = (l+i)(n/2)'
A)
B)
C)
(O

148
INTEGRALS OF BESSEL FUNCTIONS
7.6
7.6.
It is convenient to follow Luke A959) and Luke and Coleman A961). We
first record a multiplication formula for the Gaussian hypergeometric func-
function. By confluence, we obtain an expansion for the confluent hypergeometric
function in series of functions of the same kind. Specialization of a param-
parameter leads to an expansion for the confluent function in series of Bessel
functions. An expansion for y(a,z) then follows from 7.2C). We have
k=0
A)
Here it is assumed that the parameters and variables are restricted so that
the resulting expressions are meaningful. In particular, the parameters a ,
fj and у are free save that у must not be a negative odd integer. Let
b = p , replace z by z/b and let b—¦>¦» . Then
$(a,c;Xz) = Zq fc,(^ зГгСУЮ ^а+к>^1+2к>2> ' B>
Equation (l) generalizes a statement of Chaundy (Erdelyi et al., 1953,
Vol. I, p. 187) and if a = a , B) is a multiplication theorem due to
Erdelyi (Erdelyl et al., 1953, Vol. I, p. 285). If a = с , B) gives an
expansion for e^z . A more general expression for the exponential can be
derived from (l). Put Ъ = с , replace X by \/a and let a—*•<» .
Then
Xz _ Z- (-)k(a)k(g)kzk oFq (-Ъ,*У\Л Л(а+к,е+к;у+1+2к;2) .
1 "Jo k!(^+k)k ^ ^ ' J
C)
7.6
INCCMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
Set Y+1 = 2a ln B) ^^ use 1-4.1B). Then
,(a,c;X.)-?fe^ I6(z/2)
149
2e'
&/ k=l
П №№) ^л,0,6.АК?бЫг) , D)
where 6 = a-J is arbitrary except that it must not be a negative odd
multiple of ¦§", and
Rk(a,c,a;X) = 3F2(-|;k^'a|x)
E)
We take X = 1 and write R^(a,c, 6;l) = Rk(a,c,6) . Some properties of
t.he latter are as follows.
Rk(a>°>6) = (-)\(c-e.,c,6); Rk(a,a,6) = (-)k .
-1a(k+6)Rk(a+l, g+1,6) = -c[kRk.1(a, c,6)-2(k+6)Rk(a,c,6)
F)
tk + c:)(k+26)Rk+1(a,c,6) = 2(c-2a)(k+6)Rk(a,c,6)+k(k+26-c)Rk_1(a,c,6) . G)
+ (k+26)Rk+1(a,c,6)]
, D)kF-a+i)k
^^(a^a^) = 0 ; 1^,2^6) = (^&) ^
(8)
(9)

150 IHTEGRALS OF BESSEL FUNCTIONS
(°-2a)
(-)kBa-c)
H k(a,c,a-i) = _ ?
In D), put 6 = o-a-i and use A0). Now let с = a+1 and use
7.1C). Then
7.6
A0)
A1)
7.6
Y(a,z)=a-122az*e-*Zr(a-|)Z(-):
(k+a-|)Ba-l) (a-l).
V ^'k
k=0
k.'(a+l)k
Iv+a-iC2/2) >
J-k+a-2
R(a)>0
A2)
Similarly^ 6=0 gives
¦*z jl+2 X Rk(a)lk(z/2)f '
L k=l J
Y(a,z) = a-Ve-az]l+2 X Rk(a)lk(z/2) [ » R(a)>0 , A3)
¦where
Rk(a)= Rk(a,a+l,0)
/ \K , o и _, ak(cos ап)ГBа)Г(к-а)
"("(, С pF (l-2a,2jk-a+2;i) + ^ /s
Гк-aVk-a+l) 2 1 ога±гъ+а.+1 }
• A4)
2(k-a)(k-a+l)
2'::а"хГ(к+а+1)
The expansions A5) and A9) below are due to Tricomi A954,P. 37-42).
*(a,o,z) = T(c)ehZ ^ Ak(a,c,h)zkE0+k_1(-az) , h a 0 , A5)
k=0
EU(O = s-*4Bz*) , Eu(-z) = z-*UIuBzi) , A6)
INCCMPIETE GAMMA FUNCH?ION AMD RELATED FUNCTIONS
151
where the coefficients Aj^a,c,h) are defined Ъу the generating function
X Ak(a,c,h)zk = e-az[l+(h-l)z]-a(l+hz)a-° , |z|< 1 ,
k=0
and satisfy the recurrence system
(k+l)Ak+1 = [A-2Ь)к-Ьс]Ак+ [a(l-2h)-h(h-l)(c+k-l)]Ak_1-ah(h-l)Ali._2 ,
Ao = 1 , Ax = -he , Аз = |h2c(c+l)+a(|-h) .
A7)
A8)
§(a,o,z) = Г(с)е^ X Bk(K^c/2)(z/2)kE0+li._1(Kz) , К = c/2-a , A9)
k=0
Z ^(^0/2}zk = e2Kz(l-z)-a(l+z)a-° , |z|<l ,
k=0
B0)
(k+1)^! = (k+c-l)^.! - 2^.2 , Bo = 1, BL = 0 , Bg = g/2 . B1)
In A5), put a = 1 , replace с Ъу a+1 and set h = 0 . Then
rn>m 7.2C) ,
Y(a,z) = r(a)e-Zz^ ? ^(-Dz^VaC2^) , ek(-l) = ? LL ,
k=0
m=0
R(a)>0
B2)

152
INTEGRALS OF BESSEL FUNCTIONS
The combination 7.l(l), 3.9B) and 3.6A) gives
,(.„) - м*^ ? '-&?$??#' W"
7.7
k=0
¦ ^w**- ? '-'irtfK^w*' ¦ *"»° ¦B5>
k=0
k.'Bk+a+l)Bk+a)
Alternative expansions
follow in a similar manner from 3.6B). The formulas
sin tut = ¦—г— 2- ,ъ±а n 2k+5/d
Jo
R(a)>-1 ,
B4)
I
Vcos tdt . <тт/а8)Уг(а/2
„ Bk+i)r^+k)
rCf) kT0 r(^+k)
2k+*4
R(a)>0
B5)
come from 2.4A) with ц = a--J and v = tj- , respectively. Further ex-
expansions follow from 2.4B).
7.7.
il Approximati
sns, r.nntinued Fractions. Inequalities
Most of the material in this section is based on a paper by Luke
A958). See also the references quoted there.
7.7
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
153
' У V-Vit - g^ + Rn(z) , Rn(z) =
Vz)
R(a)>0 ,
A)
where
n (-n) (n+a+1)
An(z) = T ,k . . . . k z\T-
kto (a+1)k^-'
3^1
/-n+k,n+a+l+k,l|.l/z-\ B)
V 1+k I ' J ' K '
and B^z) is the к = 0 term in B) whence the 3Fi_ becomes a 2FQ .
Also
/ <,U -Z pZ
_ / ¦, (-) e / / ..n, n+a t,.
za(a+l)
/
J0
mid
/ \n , -z 2n+l
= " ( I'A Z 1F1(n+a+l;2n+a+2;z) ,
(.a+1>2n+l
^j . . (-)We-z(z/2)^ [l+0(l/n)]
2a(a+lJn L J
Timi:, for z fixed, lim Rn(z) = 0 .
П—»<»
Both Ад(г) and Bn(z) satisfy the same recurrence relation.
C)
D)

154
INTEGRALS OF BESSEL FUNCTIONS
7.7
(n+a+l) (z)=ll+
a+l)Bn+a+2) VlU; |_X
az
Bn+a)Bn+a+2)
VO
nzc
Bn+a)Bn+a+l)
ViW
E)
We also have
Bn+a)B^(z) - nBjz) + nzB^z) = 0
F)
Bn+a)A1!1(z)+(n+a)An(z)+nzAI1_1(z)+Z-1aBn+a) |An(z)-Bn(z)j = 0 , G)
[zD2 - (z+2n+a)D + n] Bn(z) =
(8)
The polynomials Bn(z) are known as Bessel polynomials. We have the
following orthogonality relations. If С is the path of a circle in the
complex plane with center at the origin, then
x г ezBn(-2)dz
/ \n t
(-) m-
2i7i J „n+m+a+2 (m-n).'r(m+n+a+2) '
0
(9)
1
2ni
e4("z)Bm(-Z)dz
„n+m+a+2
= 0 if m fi n ,
= (-)V
Bn+a+l)r(n+a+l)
, if m = n , R(a)>-2 . A0)
7.7
INCCMPIETE GAMMA FUNCTION AND RELATED FUNCTIONS
155
Another representation for the left-hand side of (l) is
az"VZ
/•-^¦^¦M.».*.,.^ '
R(a)> 0
A1)
where
n-1 (-n) ,(n+a) ^
-n+l+r,n+a+l+r,l
r+2
-1/^ • A2)
and Dn(z) is zn times the 3F]_ in A2) with r+1 set to 0 . Thus,
Dn(z) is Bn(z) if in the latter we replace a by a-1 . Also
q^z) = (-)naz-ae-z rZ(z.t)ntn+a-letdt
(a)n Jo
/ л , -z 2n
_ (-) n.'e z
(a+
~-\, ^(n+a^n+a+ljz)
¦+J-'l9n
J2n
A3)
and
(z) = (-) (тт/пJе (z/2) Г1+0A/п)-1
2&(a+1Jn-l
A4]
whence for z fixed, lim Sn(z) = 0 .
П—><x,
Both Cn(z) and Dn(z) satisfy the recurrence relation E) if a is
replaced by a-1 . Also, Dn(z) satisfies F) and (8) with a replaced by
a-1 . The relation analogous to G) reads

156
INTEGRALS OF BESSEL FUNCTIONS
If a->0 , the left-hand side of (l) becomes e"z and
.-.3&!.v> -
where
Gn(z)
Gn(z) = zn2F0(-n,n+l;-l/z) = (z/nFzne\+i(z/2) ,
Rn(z) = "
е^ф/2)
Some useful properties of G^z) follow.
Gn+l(z) = 2Bn+l)Gn(z)+z2Gn.1(z) ,
2Gn(z) = Gn(z) - zG^Cz) ,
[zD2 - (z+2n)D + n] Gn(z) = 0
7.7
Bn+a-l)C^(z)+(n+a-l)Cn(z)+nzCn_1(z)+z-1aBn+a-l) {cn(z)-Dn(z)} = 0 . A5)
A6)
A7)
A8)
i
Rn(z) . . j-g^{n?a e-Z(z/2Jn+1 [l+0(l/n)] . A9)
B0)
B1)
B2)
7.7
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
We now consider rational approximations for Г(а,z) .
157
16
Z1-aezr(a,z) = z1-V/' t'-V*» - ^fi. + Tn(z) ,
Jz t^z)
where the path of integration is described in 7.1B). Here
B3)
B4)
and Fn(z) is the 2F2 ln ^24:) with к = 0 whence the 2F2 becomes a
1F1 , a generalized Laguerre polynomial. Also
16
Un(z) = -(l-aJz^V / (t-z)^
</ z
= -(l-a)n.'t(n+l,a;z)
,a-n-2e-tdt
,.„-п-1
= -A-а)п.'г-п-х2Р0(п+1,п+2-а;-1/2) .
B5)
Define К = n+l-a/2 . If a is bounded, |z| = |KIP , 0 s p<l/3 , and
¦l is bounded away from zero, then
Tn(z).
1-a z-4(Kz)'
2ttz- e
ГA-а)
, |K|-»od , |arg(Kz)|<n ,
lim Tn(z) = 0 .
П-»а>
B6)

158 INTEGRALS OF BESSEL FUNCTIONS
Both En(z) and Fn(z) obey the same recurrence formula.
(n+2-a)En+1(z) = (z+2n+2-a)En(z) - пЕд.^г)
We also have
zFn(z) = nFn(z) - nFn_i(z) ,
zE^(z) = (z+n+l-a^z) - nEn_1(z) - zFn(z) ,
[zD2 + (z+2-z)D - n] Fn(z) =
A second rational approximation for Г(а,z) follows from
^7f15^vt-=|R+Vn(zbVn(z)=l
wn(z)
(O '
where
Mn(z) = -z X
n-1 (-n)
k+1
k-Q (k+l).'(l-a+k)
7.7
B7)
B8)
B9)
C0)
C1)
^ ^ V2+k,2+k-a| J s '
and Nn(z) is the 2F2 in C2) with k+1 = 0 whence the 2F2 reduces
to a -j^F-l . Nn(z) is Fn(z) of B3) with a replaced Ъу a+1 , and so
Nn(z) also satisfies B7), B8) and C0) with a replaced Ъу a+1 . Again
M^z) satisfies B7) with a replaced Ъу a+1 , while Мц(г) and Nn(z)
can replace Еп(г) and Fn(z) , respectively, in B9) as it stands. Now
7.7
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
159
i6
,. / v 1-a z / .
Wn(z) = z e / (t-
Jz
%n.a-n-l -t,,
z) t e dt
= n.'zi|((n+l;a+l;z)
= n.'z" 2F0(n+l,n+l-a;-l/z)
C3)
Define Kx = n+(l-a)/2 . Again if a is Ъounded, |z| = |KIP ,
0 s. p < l/3 and z is Ъошк)^ away from the origin, then
1-a z-4(K2_zJ
Vn(z)~ nZ r^_&) , IK^I-x», |arg(K1z)l<TT
lim Vn(z) = 0 .
C4)
Continued fraction developments are as follows.
whore
-a -z
az e
[V-Vdt = i
^0 1 + u-j_z
1 + V-j^Z
1 + UgZ
1 + v2z
1 + .
. , R(a)>0 , C5)
(k+a-l)
> vv =
-k
k Bk+a-2)Bk+a-l) ' к Bk+a-l)Bk+a)
C6)

160
INTEGRALS OF BESSEL FUNCTIONS
7.7
If a = 0 , C5) is a continued fraction development for e"z . The odd
part of C5) is
1 - u, z
1+{\1j+Vj)z - VjUgZ
l+(ug+v2)z - v2u3z'1
l+(u3+v3)z -
C7)
and the n-th convergent of this expansion is An(z)/Btl(z) , see (l). The
even part of C5) is
1+U-j^Z - U-jV-j^
1+(V1+U2)Z " UgVVjZ
l+(v2+u3)z -
and its n-th convergent is Cn(z)/Dn(z) , see (ll).
16
zl-aez Г ta-le-tdt = ^_
Jz z +
A-a)
1 + 1
z + B-a)
1+2
z + E-a)
1 + 3
z + .
C8)
. . C9)
7.7 INCOMPLETE GAMMA. FUNCTION AND RELATED FUNCTIONS
161
See B3) for conditions of validity. The odd part of C9) is
1- (±±1
z+B-a) - B-a)
z+D-a) - 2E-a)
z+F-a) - 5D-a)
z+(8-a) - .
D0)
and the n-th convergent of D0) is En(z)/Fn(z) , see B3). The even part
of C9) is
z+(l-a) - A-a)
z+C-a) - 2B-a)
z+E-a) - 5E-a)
z+G-a) - .
D1)
and its n-th convergent is Mn(z)/Nn(z) , see C1).
Combination of the representations (l) and (ll) leads to some useful
inequalities; likewise, for the union of B3) and C1).
DnW «/o
eudt \ -ii—- , x>0 , a>0 ,
D2)
where > or < sign is chosen according as n is odd or even, respectively.
Thus, for n = 1 ,
(a+l)(a+2)-x -& -:
(a+l)(a+2+x) <ax e
nx
¦x /> ta-let
Jn
a+1
dt<i7Tz: > x>° > a>° > («)
a+l+x

162
INTEGRALS OF BESSEL FUNCTIONS
— <e'X <-i- , x>0 ,
2+x x+1
i^?<x-1e"x2 /1Vdt<
1 5+Kx2 J П
2х2+Ъ
, x>0
In D3)-D5), we have equality if x =
= 0 .
\(x)
sl-aex /> ta-l
Ux
E (x)
"V^t < -Vt * 0<х<ш , a<l
FnW
7.7
D4)
D5)
D6)
7.8
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
x2
f e"t2dt :?^тт[(х2+4/тт)*-х] ,
<Jx
x s 0
i [(xP^/P.x] < exp(xP) rexp(-tp)dt S ^[(x^c;1I7^] ,
"X
= [r(l+p
_1n1p/(p-D
, p>l , 0 <. x <
Using 7.1B) with 6 = 0, E2) can Ъе transformed into
163
E1)
E2)
Equality obtains if x-*«>. If n = l,
i[(X+2)VP.xl/P] s exr(p-l^x) fi pOp[(x+c-l)l/P.xl/P] ,
x < x1"aex F t^^^dt < ^t^ , 0<x<co , a<l ,
x+l-a Jx x+2-a
x x2
x <ехГ
2x2+l
^x
2 v2+1
dt < -i-Ii , 0<xO ,
xBx2+3)
D7)
D8)
p> 0 , 0 <. x < od ,
and if p—*•¦» ,
i ln(l+2/x) <. eX
/ t e
Jx
E3)
dt & ЗлA+1/х) , 0<х<ш . E4)
CO
-1- < eX / t^e^dt < ,X^. , 0<x<oo
x+1 Jx x(x+2)
D9)
Equation E0) is due to Kcmatu A955), E1) to Pollak A956), and E2)-
E4) to Gautschi A959a).
We also have the following inequalities.
Q A CO Q
|[(x2+2)i-x] < ex / e"* dt < (x^lp-x , 0 s x <
E0)
7.8. The Exponential Integral
i6
Ex(z) = -Ei(-z) = / е"Ч" ^-dt = T@,z) = e"zy(l,l;z) ,
larg z|<tt
A)

164
INTEGRALS OP BESSEL FUNCTIONS
7.8
For the path of integration, see the discussion surrounding 2.1(9) and
7.1B).
Ei(x) = - (Г et = (Г e*f 4t , x>0
J-x "-<»
B)
li(x) = Ei(ln x) = ф (In t^dt , x>l
^0
C)
In B) and C) the integral is a Cauchy principle value.
f (l-e^t^dt = lim {a-1za-Y(a,z)}
^0 a—»0
D)
-Ei(-z) + (Y+ln z) = / (l-e^"^
Уо
чк к
- Z ^r- = e"z Z (i+l+...+i/k)zk/k: . E)
k=l K-K k=l
Ei(z) - (Y+ln z) = / (e1-!^!
J0
¦dt
= Z ^— = -ez Z (-)k(l+?+---+l/k)zk/k.1
k=l kik k=1
Ei(z) = Ei(-ze±lTT) J in .
F)
G)
7.8
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
Ei(z) = \ [Bi(-zelTT) + Ei(-ze-lTT)]
E1(z)^z-Ie-Z 21 i'^! , |z|^od , |arg z|< 3n/2
k=0 z
165
(8)
(9)
See 7.4B) with a = 0 .
The combination of 7.6A2) and D) produces
J^l-e-Ъt-1dtЧпz)*e-^(z/2)+(пz)*i^ki2l_ e-*zIk+3/2(z/2) , (l0)
and also
/o'(et.l)t-4t.(n.)^(^)-(m,4| {rfelfj e-Hk+3/2(z/2 ).(!!)
Similarly, 7.6A3) and D) yield
„z ш
/ (l-e^^^dt =2 Y. fkezIk(z/2)
WO k=l
= 2[l-e-*ZI0(z/2)]+4 X gk+2e"*2Ik+2(z/2) , A2)
k=0
where

166 INTEGRALS OF BESSEL FUNCTIONS
= ? fk - 1 , f0 = 0 , f-L = 2
(k+l)fk+1 = 2fk + (k-l)^.! + 4 ,
f2k = 4A+1/3+. . .+l/2k-l) ,
7.8
f2k+l - f2k + gf+T ' f2k+2 f2k+l + 2^+1
Another expression for the left-hand side of A2) can Ъе deduced from
7.6B4,25). See 7.9(lO,ll).
We now derive another expansion for the exponential integral as a
series of Bessel functions. First, we show that
A3)
aru(O
3Ky(z)
i_ = Bttz) [eZEi(-2z) ; e"ZEiBz)]
Эи
„*t?
= + Bz/-n)ezEi(-2z)
A4)
A5)
To prove A4), multiply both sides of 5.2(9) Ъу 2i>-l , differentiate
partially with respect to и , set и = \ and use 5.2A0). Equation A5)
then readily follows from 1.4.l(ll). For an alternative proof of A4) and
A5), see Oberhettinger A958). Similarly, from 9.5A5,16),
(nz/2)"
,^M')
аи
= -T](Y+ln z/2) + ? e~z (EiBz)-2Ei(z)|
- bZ {Ei(-2z)-2Ei(-z)}
7] = 1 if v = ^ ; 7] = 0 if и = -
A6)
7.8
INCOMPLETE GAMMA FUNCTION AMD RELATED FUNCTIONS
167
Now use 1.4.2G) to evaluate the left-hand side of A4). Then
PZ i
/ t-1(l-e"t)dt = 2e-2ZC0Sh z/2 + zh-^z) - 2hg(z) ,
Jo
A7)
z
Г t(et-l)dt = -2e-^zeosh z/2 + zh^z) + 2h2(z) , A8)
Jo
where
k=0 к.'Bк+3-2т)г
, m = 1,2
A9)
Of all the expansions in series of Bessel functions for the exponential
integral given, the latter converges the best.
/.Z
If 7.6B3) is used to form / ta(l-e"t )dt , and the limit of this
^0
as a->0 is evaluated using L'Hospital's theorem, then with the aid of
A4), we get the duplication formula
n2z _z
/ f1(l-e"t)dt - / t'1(l-e"t)dt = e"zsinh
"V2 ) /4 (k+l).'(k+l)Bk+l) Х2к+3/2^;
k=0
i(m^ f (-)k(^5)C/2)k o.Zt
^^"^ kt0 (k+l).'(k+l)Bk+3) ^+5/2lZ; •
B0)
Rational approximations, continued fractions and inequalities follow
from the material in 7.7.

168
INTEGRALS OF BESSEL FUNCTIONS
7.9
The repeated integral
<x>

B1)
occurs in a wide variety of applied ргоЪ1ешй. Blanch (see National Bureau
of Standards A954Ъ), р. 61) developed four terms of an asymptotic ex-
expansion for lar.ge n which is uniform in x ? 0 . She also gives the re-
remainder term. Her result has Ъееп generalized Ъу Gautschi A959Ъ) who
proves that
^w^IAt^s^H '
(х+п)[_к=0 (x+nJk
fk+1(x) = (n-2kx)fk(x)+x(n+x)fk(x) , f0 = 1 ,
°* - a^) * 4i+ ~i)
B2)
B3)
B4)
where the c^'s and Pj^'s are lower and upper Ъоип<3.8, respectively, of
xFf (x)/(x+nJm in the interval x ? 0 . The first eight polynomials
fk(x) and the corresponding values of ak and gk are listed in the
reference cited. There hk(u) = п"т'к(х) , u = x/n .
7.9. Sine and Cosine Integrals
si(z) = Tt^sintdt = Bi)-1[K1(Ze^)-E1(ze-i1")]
= Bi)-1[Ei(-ze-4iTT)-Ei(-ZeiliT)] . (D
7.9 INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
169
/ 11
t^cos tut = -i[E1(ze2iTT)+E1(zelTT)]
= i[Ei(-zeiliT)+Ei(-Ze-5liT)]
Si(z) = / t" sin tut = tt/2 + si(z) .
B)
C)
Ci(z)-(Y+ln z) = - / t'1(l-cos t)dt
D)
S1(Z) = JQ Bk+l).'Bk+l) '
E)
F)
Ci(z)+i si(z) = f eXtt-X<& ~-z'h1* |Z
Jc= [k=0
(-)KBk^l).'
„2k+l
+ 1 Y (-) (^); \. } |Z|_»oo , |arg z| <тт
k=0 z^k
G)

170
INTEGRALS OF BESSEL FUNCTIONS
Jo
"^dt = Y+ln z-d(z)+iSi(z)
= (ttz
¦where fv is given Ъу 7.8A3).
/
Jn V2zV k=0 C/2^k
5 i °° (i)kDk+1)
t'V-eoe t)dt = i(|rJ 2! ?
k=l
X {t(w)-tdW(^)-t(i)} J2k+i(z) "
- (z/4)kJk(z/2)
^(а) = ?) k.'Bk+3-2mJ
, m = 1,2
7.9
)^[^^Wl/8))'(e)
Jo k=l
A0)
(П)
fZt-l(l-e-")dt - 2e-^=os z/2 + e"^^ {iZgl(z)-2g2(z)} , C2)
A3)
Of all the e^sions given for the sine and cosine integrals in series of
Bessel functions, A2) converges the best.
7.9
where
Dk+3)(i)
A(Z) = JQ (k+l)!(k+l)Bk+l) J2k+3/2(Z)
« Dк+5)C/2)ъ
B(Z) " Jn (k+l)!(k+l)Bk+S) J2k+5/2
(Z) •
k=0
Results analogous to 7.8A4-16) are as follows.
ajw(z)|
ajw(z)
Эи
^ = (ttz/2)'2 [sin z CiBz)-cos z SiBz)]
1 = (ttz/2) [cos z CiBz)+sin z SiBz)]
8Yv(z)
SJy(z)
v=5- Эи
i - ttJi.(z)
V=-2 2
Эи
dJu(z)
u=_i Эи
U=5
- ttJ_\{z)
ATZ/2)i^(o
Эи
1 = (y+ln z/2)+cos z[ciBz)-2Ci(z)]
+ sin z [SiBz) - 2Si(z)]
171
INCOMPLETE GAMMA FUMCTION AND RELATED FUNCTIONS
i
SiBz) - Si(z) = i sin 2z + i^— >2 |sin zA(z)+cos zB(z)^ , A4)
CiBz)-Ci(z) = In 2-sin2z+irH5^2 ^cos zA(z)-sin zB(z)) , A5)
A6)
A7)
A8)
A9)
B0)
B1)
B2)

172
i ЭН (z)
INTEStALS OF HESSEL FUNCTIONS
v=_i = - sin г [ciBz)-2Ci(z)]
+ cos z [siBz)-2Si(z)]
Rational approximations and continued fractions can Ъе deduced from 7.7.
Some iterated sine and cosine integrals have Ъееп studied and tabu-
tabulated Ъу Halle'n A947, 1955) and Bouwkamp A948),see 7.13.
7.10. Error Functions
7.10
B3)
Erf(z) = ( e"* dt = Ыт>>^) = z$(| , | J-z2)
= ze-z2§(l,3/2;z2) . (l)
Erfc(z) = f e"* dt = in2 - Erf(z) = 4r(i,z2)
J7.
= Kz2*(i,i^2)
The notation erf(z) = B/тт2)ЕгГ (z ) is often used.
Erfi(z) = -i Erf(iz) = / e° dt
iz) = f
B)
C)
Erfc(z) = Btt)'2 / Ki.(t)dt , |arg z| ? тт/4 .
и z
D)
7.10
INCOMPLETE GAMMA FUNCTION AMD RELATED FUNCTIONS
173
i 2 2 n°° -at11
Erfc(az) = nae"a z / e dt , larg a|<n/2 , |arg z| <. n/4
J 0 t^+a^
E)
00 , -.к 2k+l 2 °° 2k+l
kto k-'^2k+1)
2k+l
t, C/2),
k=0 '¦-'^•'k
2 °° / чк 2к+1
J«i(O-Z ^^T-e"" Z^^
Гп k!Bk+l)
k=0
k=0
W^O
(e)
G)
, л , Z2 - (-)k(i)k
Erfc(z)~^e Z X
k=0 z'
2k+l
|z|—»-oo , jarg z|<3tt/4
(8)
Erfi(z)~"=b
z2 f (^ i
kto z^+l " 2
|z|—>oo ^ -5-n/4<arg z<tt/4
(9)
For the error term, see the discussion surrounding 7.4B).
Erf(z) = -z X ' ' k
k=0
(-)k€ke-^2Ik(z2/2)
4k2-l
e = 1 ; efc = 2 , k>0
A0)
&" Erfi(z) = -z
00 „ P-iz x / 2 у
к4
-Z2_ , ^ вке-** I^/j,}
k=0
4k2-!
A1)

174
INTE(BALS OF BESSEL FUNCTIONS
7.10
Erf (z sin 6 ) = iz "Z.
к
у , ГвШBк*1)8 . Binigk-ll9L-b2T r,2/P^ . A2)
— k I 2k+l 2k-1 [ K
:=0 L J
(_чк_ _-z% ,„2
Erf(z) = -z X о
k=0 16k2-!
\к,„,_., \_-z^_ ^ 2\
*iz 21 —(«икай)
l-fv~ ht^ ^ i (-Л^>" W' » . (B)
k=0
In A1)-A3), ek is given Ъу A0).
Erf(z) = i(nz)*e-z2 Z ek(-l)Ak+iBz) .
k=0
l l 2 °°
Erf(z) = пге-52 X Ik+i(z2/2)
Using
k=0
i °° Гк/г! 2
Erf(z) = (tt/2J X ("Г Ik.i(^ )
k=l
l °° fk/2l о
Erfi(z)= (tt/2)' Z ("Г' JIk+i(Z )
k=0
D) and 8.8A) with a = 1 and v = b
A4)
A5)
A6)
A7)
7.10 INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
175
Brfc(z) =
h у e-z2cOshB^1)h/2coshBk+lW4 , fl,^z2>
^~jl k=0 coshBk+l)h/2
BttJ
h у е
Btt)* k=0
-z cosh kh
cosh kh/2 . he'Z + gl,^z ^
?Btt)* Btt)
cosh kh
|2 ('о„\2
larg z| й tt/4 ,
A8)
where
f,^2)
^k /2,
-2 X (-fgk(z > ' 8X iLz ) =  Z gk(z )
k=l ' k=l
gk(z2) = * Kiqi(Z2>K1;lqe(z2)] ,
q1 = p+i/2 , q^ = p-i/2 , p = 2iTk/h ,
A9)
¦w*-/
00 -z2cosh t
cos qt
cosh t
dt
,-i/2a2
sin e+ I- cos e+ -z-
4q
32^^
B4-z4)sine|j [l+0(l/q3)] ,
9 = q In S3 - тт/4 - 13/l2q
ez
B0)

176
INTEGRALS OF EESSEL FUNCTIONS
We also have
where
-a2z2 - -[Bk+l)hz/2]2
Erfc(az) = ahe± 2 2 + к
tt2 k=o [Bк+1)ь/2] +a^
ahe^ * e<^f b^ff_ + .
т— 2. -— 1— + s >
2 2
tt2 k=O (kh) +a 2атт
larg z| <; тт/4 , larg a|< тг/2 ,
E = -2 X (")\ > s =  Z &k ,
k=l k=l
-a2,2
Gk =
a z nu
~^F~ Jo
z2 2
e"z cos yt , _ 2тгк
? 2 dt ' У " ~ '
t +a
7.10
B1)
B2)
B3)
7.10
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
177
For further results on the computation of error functions and related
functions, see Eosser A948) and Salzer A951). For rational approxima-
approximations, continued fractions and inequalities, see 7.7.
Salzer A956) tabulated complex zeros of the error function. Luke
A961Ъ) used 7.7A,11) with a = -| to generate polynomials rrn(y) such
that if yaa is the smallest zero of rrn(y) , then lim yon = p , the
n—>°o
ftZ
radius of univalence of the function exp(z2) / exp(-t2)dt . The repre-
J0
sentations in 7.7 are also useful to compute zeros of the incomplete gamma
function.
For repeated integrals of the error function, we follow Hartree
A936) and write
i°erfc(z) = 2TT~2Erfc z = 2n"i / e"* dt ,
CO
inerfc(z) = i inerfc(t)dt
B6)
Gk = i [eayErfc(az+y/2z)+e'ayErfc(az-y/2z)J , B4)
A partial list of results follows.
and with a and z fixed,
V
)|A-ay. -A-c^W
2.24
у -4а z
B5)
Equations B1)-B5) follow from E) and the work of Fettis A955) and Luke
A956). In the latter reference, the formula for Gjj. corresponding to
B5) contains a typographical error.
°° / чП
inerfc(z) = 2тг~2
J, n:
-Z} e"* dt
inerfc@) = i
2пГ(п/2+1)
(l?+2zD-2n) [inerfc(z)j = 0
B7)
B8)
B9)

178
INTEGRALS OF HESSEL FUNCTIONS
i-(inerfc(z)) = -i^WcCz) , n>0
dz I J
7.10
C0)
inerfc(z) = - | i^^rfcCz) + ^ in-2erfc(z) . C1)
An effective method for the computation of inerfe(z) is to use C1) in
the backward direction. See Gautschi A961Ъ).
,k к
inerfc(z) = X ^^
k=0 2п"кк.'гГ1+ SdE")
2
e"z /• n+1 1 2\
Vr(»+i) Ч-т-'^О
Ч2 '
<§ «.! -2)
Д-irrSti^
-^(
2 ^
-7.2
--W-Jgsi^oC^.I—"О
|z|-»<» , larg z| <3tt/4
C2)
inerfc(z) = Bn-±TT)-2HanB2z) ,
where Hhj^z) is given in British Association for the Advancement of
Science A951).
C3)
C4)
i_2
-ipd -zBn)'
inerfc(z) =e e
2nr(|+l)
[l+0(n)l , z bounded, n—>&> ¦ C5)
7.11 INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
179
7.11. Fresnel Integrals
C(z) = i / J_i(t)dt = Bn)'? / t'2cos tdt; C(^ttz ) = / cos ^rrt2dt . (l)
Jo 2 «o "o
nz l rz l rz
S(z) = | / Jr(t)dt = Bn)"^ / f^sin tdt; S(|nz2) = / sin ^rrt2dt . B)
Jo 2 Jo ^o
C(z)+iS(z) = BTT)'^nY(i, ze"*117)
2Bn)"*e^iTTErf(z2e-^iTT) .
C(z) = BnL[e-4f(z^)>Erf(Zi^n)]
S(z) = iBn)-*[e-^iTrErf(z*e^iiT)-e^1TErf(z*e-bin)j
C(z) = Bz/n)^ Z T^fe
k=0
Bk).'Dk+l)
S(z) = Bz/ttJ Y.
,4f Ы
к 2к+1
^¦0 Bk+l)!Dk-K5)
C)
D)
E)
F)
G)
l . г " / чк 2к „. °° f -,k 2k+l "I
C(z)+iS(z) = Bz/n)Vz X Щ- - f Y. ^l" ¦ (8)
Lk=O C/2Jk 3 k=0 E/2Jk J

180
INTEGRALS OF BESSEL FUNCTIONS
7.11
C(z)~i-BTfz)"^ ? cos z
II. ^ (")k(V2Jk
- sin z 2.
- ("f(V2).
'2k
k=0 z'
,2k+l
k=0 z1
,2k
|z|—>» , |arg z|<tt
(9)
S(z)~i-BTfz)~^ U sin z X — + cos z
(-)k(l/2Jk-
k=0
2k+l
k=0 z'
,2k
|z|—»<» , |arg z| <тг
C(z)+iS(z) = -Bz/rrJe
^z ^ «A<a/2>
fc=0 4k2-!
A0)
A1)
C(z>iS(z) = -Bz/rrJeiZ[2: ^|1
Lk=o i6k^-i
.JoJO - (Sk+DJpb.^2)
41 Z
2k+lv
кГ0 Dk+l)Dk+3)
]'
so= 1 , ek =2 , k>0 .
C(z) = Z J2k+i(z)
k=0
S(z) - Z Jp>+3/p(z)
k=0
2к+3/2^
A2)
A3)
A4)
Further representations follow from C) and the material in 7.10. For
rational approximations and continued fractions, see 7.7.
7.11 INCOMPLETE GAMMA FUNCTION AND BELATED FUNCTIONS
181
The so-called "rocket functions", see Rosser, Newton and Gross A947),
are related to Fresnel integrals since
л°° l
rc(z) = rr(z) + i ri(z) = ieiz / t'^e'^dt
J z
(l+i)(TT/2Lelz - BTr)iieiz[c(z)-iS(z)] . A5)
=(z) = Btt)*RcQ2z/tt} 2) ; Rc(z) = ^ f
1 - (I,it2)e-^z2t2
dt ,
1+z4
Rc(z) = Rr(z) + iRi(z) ; R(z2)>0
(IB)
Another function is
ic(z) = ir(z) + i ii(z) = / trc(t)dt
= / t-x
Jo
A7)
ic(z) = Tr(i+i)[c(z)+is(z)]-2iz г^Сг^/г!12)
A8)
An expansion in series of Bessel functions for ic(z) follows, for
example, from (ll). For ic(z) , we have
ic(z) = Tr(l+i)[c(z)+iS(z)]+4Z akJ2k+2(z)i "^ t)kJ2k+l(z) >
k=0 k=0
2k+l 2k
= ? (гт+i)-1 , ък = X (-)VD'1
m=0
A9)
m=0

182
INTEGRALS OF BESSEL FUNCTIONS
7.12
For additional references to rocket functions and related items, see
Eosser A948), Eankin A949), Rosser and Walker A953) and Roberts and
Calctwell A959). Each source contains tables, see 7.13.
7.12. Indefinite and Definite Integrals Associated with the Incomplete
Gamma Function and Related Function^
(^¦^J'^<.?V) ¦
R(a)>0 , Е(Ъ) ? 0 , R(ua+M,+l)>0
Jo
A)
R(a)>0 , Е(Ъ) й 0 , R(ua+u)>0
B)
I
Z И.+1
t^i(-t>tU)dt " Z
^»(-M>)-^^<^'^ '
Е(Ъ) ? 0 , E(p.)>-1 > E(»)>0
nZ
/ [Ci(t)+i si(t)]dt = z[Ci(z)+i si(z)J-i(l-<
Jo
eiz) •
I
Erf(t)dt = z Erf(z) - i(l-e'Z )
C)
D)
E)
7.12 INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
183
/ [c(t)+iS(t)]dt = (z-i/2)[c(z)+iS(z)]+i(z/2rrJeiz
Jo
F)
re-(at2+2tt+c)dt = а4е(.Ъ2-*с)/*^Х+в.-Щ>)-Ж(э.-Ъ)] , G)
J 0
Г e-(at2^t+c)dt m а-^^-ас)/^^^-^) }
R(a)>0 ; R(a) = 0 , Е(Ъ)> О
f e'at cos 2bt dt = |(тг/аLе"Ъ /а , R(a)>0 .
Jo
/ e'at sin 2bt dt = &'^e~b /^fi(аъ) , R(a)> 0
J0
Г e-a2t2-b2/t2dt = Ba)-ire2a1:)Erf(az+Vz)+e-2a1:)Erf(az-Vz)
i i 2аЪл1_^ -2аЪ 1 .,.2, .
- fie -Kfn^e I , R(b ) ? 0
/;
ra2t2-b2/t2dt = ^Bа)-1е-2аЪ ^ E(a2)>Q ^ е(ъ2)>0
(8)
О)
A0)
A1)
A2)
Jo
2^2
>,e-a^dt _ Ti0, f E(a2)> Q } H(D)> ^ _
A3)

184
INTEGRALS OF EESSEL FUNCTIONS
7.12
I
a-pz
е"В*у(а,глLЬ = - ^-— Y(a,bz) + if—") v(a, [р+Ъ] z)
0 P рЧр+Ъ/ ч
К(р)>0 , Е(а)>0
Г e-ptEi(-btU)dt = i [yA-v)-1b р7ъ1+ - Ei(-pz)
Jo PL J P
L^Ei(-bZu) + ^ (\-\
p pjo
z v
-1 -pt, -bt ., \,,
e r (e -l)dt ,
A4)
E(p)>0 , Е(Ъ)>0 , E(u)>0 .
00 / vk.
Ге-^Е1(-Ъ^ = i ГуA-р)-Лд P>1+ 1 Z ^^TT
Jo pL J Pk=ik.'(p7b
*1
'A)k
E(p)>0 , Е(Ъ)>0 , 0<Е(и)<1
/;
/;
e"ptEi(bt)dt = - - 1п(р/Ъ-1) , Е(р)>Ъ>0
_ 00
/ e"I)tCi(t)dt = -(гр)^^) , E(p)>0
^0
A5)
A6)
e-PtEi(-bt)dt = - i 1пA+р/ъ) > R(p)> ° j larg ЪКп . A7)
P
A8)
A9)
7.12 INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
185
/ e"ptsi(t)dt = -(arc tan p)/p , E(p)> 0
^0
B0)
X
>2
e"PtErf(t)dt = ?!L_ Erfc(p/2) , E(p)> 0 .
0 P
B1)
/ e-PtErfcOOdt = Ei A-еЬ ) + e^_ Ег^р/г) , E(p) a 0 .
Jo 2p p
B2)
f
I'D
-t sinh.8
[c(t)+iS(t)]dt = (csch 2e)(e2e+ie-2e) , E(sinh e)> 0 . B3)
f e-PVf^t^dt = - 2-H Erf(bz)* + i(^^JErf fZ (Ъ+Р) V . B4)
/.
i Л , -ц 4i
e-ptKrf(btJdt = 5-(rJ-) , E(p)>0 , Е(Ъ) й 0 .
q dp \ D+p У
B5)
/;
e-ptKrfc(b/tJdt = g е'2(ЪрJ , E(p)>0 , Е(Ъ) ^ 0 . B6)
X
.-pt
dt
0 (t+z J
= 2pepzErfc(pzJ , E(p)>0 , z ? 0 ,
I arg z |< ТГ
B7)
I
e-pt i. i_
-I dt = 2(n/zJep Erfc(pzJ , E(p)>0 , z ^ 0 , larg z|< тт . B8)
0 t?(t+z)

186
INTEGRALS OF EESSEL FUNCTIONS
7.12
7.13
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
187
f iSL^l dt = ttB/zJ [{|-C(pz)] cos pz + (i-S(pz)} sin pzl
Jo t2+z2 Ц. J I J
R(p)>0 , z f 0 , |arg z|<n/2
Г f2e"pt dt = (tt/z)B/z)* r{i-S(pz)} cos pz- (i-C(pz)} sin pzl
Jo t2+z2 L J
B9)
R(p)>0 , z f 0 , larg z|<tt/2
C0)
- fctt (- [pze-*111 }2)j , R(P2)> 0 , z { 0 . C1)
The latter has been studied and tabulated by Goodwin and Staton
A948). See also Ritchie A950) and Erde'lyi A950). In Ritchie, the term
involving Ei(x2) in f(x) should be negative.
Г sinxt dt = Ba)-lr eaxEi(-aJc)+e-aXEi(ax)]
Jo a2+t2
x>0 , a ? 0 , larg al< n/2
C2)
Г * cos rt dt = 4 ГеахЕ1(-ах)+е-ахЕ1(ахI
Jo a2+t2
x>0 , a ? 0 , larg a| <n/2
C3)
For a list of indefinite integrals which are expressible in terms of
the exponential integral, see Corrington A961).
i
7.13. Description of Mathematical Tables
In addition to the following, see Jahnke and Emde A945), Jahnke,
Emde and Losch (i960) and National Bureau of Standards A961, Chs. 5,7).
See also the remarks in 1.1.
Exponential Integrals for Real Argument
Akademiia. Nauk, SSSR A954a): Ei(x) , Ex(x) , x = 0@.0001I.3@.001)
3@.0005I0AI5 , 7d.
British Association for the Advancement of Science A951): Ei(x) - In x ,
-[Ei(x)+ln x] , x = 0@.1M , lid. Ei(x) , x = 5@.1I5 , 10-lls.
Ex(x) , x = 5@.1I5 , 13-14d.
Harris A957): E1(x) , exE]_(x) , Ei(x) , e'xEi(x) , x = l(lL@.4)
8AM0 , 18s or 19d.
Karpov and Razumovskii A956): li(x) , x = 0@.0001 J.5@. 001 J0@. 01)
200@.1M00AI0000A0J5000 , 7s. li(x) - In |l-x| , x = 0.95
@.0001I.05 , 6d.
National Bureau of Standards A940): Vol. I, Ei(x) , Ex(x) , x = 0@.0001)
2@.1I0 , 9d. Vol. II, Ei(x) , Ex(x) , x t= 0@.001I0@.1I5 ,
7-lls.
Exponential Integrals for Complex Argument
(see also sine and cosine integrals)
Hershey A959): Ei(z) , z = x+iy , x = -20AJ0 , у = 0AJ0 , 13s.
Also approximations to 13s of the coefficients of two polynomials of
degree fourteen whose quotient gives ze"zEi(z) to within a maximum
relative error in the absolute value of 2.2 x 10 . Approximation
is valid over the left half plane outside the unit circle.
Mashiko
CO
A953): Let l(z) = Г t^e^dt = С (g)-iS (g), where z = §e
z
for 0 <. I <. 5 and z = TT1ela for T) ? 0.2 . Put l(TT1eia) =
z-VzAa(Tl) exp (i§a(Tl)} . Ca(|) + In g , Sa(|) , | = 0@.05M ,
a = 0°BoN0°(lo)90° , 6d. Aa(T]) to 6d and §a(T]) to 5d for
T] = 0@.01H.2 and a as above.
ia

188
INTEGRALS OF EESSEL FUNCTIONS
7.13
National Bureau of Standards A958a): All tables to 6d. z = x+iy .
E-l(z) + Ш z ; x,y = 0@.02I ; -x,y = 0@.l)l . Ех(г) ; x = 0@.02L,
у = 0@.02K@.05I0 ; x,y = 0(lJ0 ; -x,y = 0@.1K.1 ;
-x = 0@.5L.5 , у = 0@.1L@.5I0 ; -x = 4.5@.5I0 , у = 0@.5I0 ;
-x,y = 0AJ0 . exE!(z) ; x = 4@.l)l0 , у = 0@.5I0.
Sine and Cosine Integrals and Iterated Sine and Cosine Integrals
(see also exponential integrals for complex argument)
Akademiia Nauk, SSSR(l954b): Si(x) , Ci(x) , x «= 0@.0001J@.001I0
@.005I00 , 7d. Ci(x) - Ln x , x = 0@.0001H.0099 , 7d.
Bouwkamp A948): E(x) =f^f1(l-e-±t) , Ex(x) =/Q tE(t)dt ,
x = 0@.2J0 , 6d.
British Association for the Advancement of Science A951): Si(x) ,
Ci(x) - 1л x , x = 0@.1M , lid.
Gerbes, Reynolds, Hoes and Drane A958): S(x) = 2 [si(x)-x(l-cos x)] and
its first eleven derivatives, x - 0°(l°I8OOO° , 6d.
nx Px
Hallen A947): Let L(x) *= / t-1(l-e-it)dt . L1;L(x) - / t-1L(t)dt ,
J0 0
Px
x = 0@.2J8 , 5d. Lg-^x) = / telt[LBt)-L(t)Jdt , x = 0@.2I4 ,
J0
5d.
Hallen A955): Let L(x) = / t-1(l-e-lt)dt . L01(x) = / t-1L(t)dt ,
J0 J0
x = 0@.01H.2 , 6d. Li;L(x) = f f 1eltL(t)dt , L001(x) =
fXt-1LO1(t)dt , L011(x) = /lXt-1L11(t)dt , L101(x) =
/lXt-1eitL01(t)dt , Lm(x) = />Xt-1e-ltL11(t)dt ,
Jo Jo
x = 0@.01H.2@.2I4 , 6s or 6d.
7.13
INCCMPLEJTE GAMMA FUNCTION AND RELATED FUNCTIONS
189
National Bureau of Standards A940): Vol. I, Si(x) , Ci(x) ,
x = 0@.0001J@.1I0 , 9d. Vol. II, Si(x) , Ci(x),
x = 0@.001I0@.1L0 , lOd.
National Bureau of Standards A954a): Si(x) , Ci(x) , x = 10@.01I00 ,
lOd.
Tai A951): Y + Ш x - Ci(x) , x = 0@.001I0@.01M0 , 6d.
Generalized Sine, Cosine and Exponential Integrals
Bleick A953): Let Sia(x,y) = / (t2+y2)-Lt sin tdt , Cia(x,y) =
Jo
fix л x+iy
/ (t2+y2) cos tdt , Si (x+iy) = / fl-sin tdt , Ci(x+iy) =
"со +iy
лх+iy
/ t -'-cos tdt . For x, у = 0@.1K.1 , the origin excluded,
Joo+iy
tables of the above integrals are given to 12d though only 10d are
guaranteed.
Harvard University Computation laboratory A949a): Let u = (t2+a2 )~Z .
nx „x
S(a,x) = / u'-'-sin udt , C(a,x) = / u(l-cos u)dt , Ss(a,x) =
J0 ^0
/x nx
u'-'-sin u sin tdt , Sc(a,x) = / u'-'-sin u cos tdt , Cs(a,x) =
О J0
nx nx
/ u'-'-cos u sin tdt , Cc(a,x) = / u^cos u(l-cos t)dt , are tabu-
J0 J0
lated to 6d. Tabulation extends over 0 s a s 25 , о s x s 25 .

190
INTEGRALS OF EESSEL FUNCTIONS
7.13
7.13
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
191
Harvard University Computation Laboratory A949b): Let u = (t +a J .
nX ЛХ
E(a,x) = / u(l-e-u)dt , Es(a,x) = / e-^-^in udt , Ec(a,x) =
0 u0
u(l-e-ucos u)dt , a,x = 0(hL9h , h = 0.001, 0.002, 0.005, 0.01,
0.02, 0.05, 0.1, 0.2, 6d.
Error Function for Real Argument, Its Repeated Integrals, Error
Function for Pure Imaginary Argument, and Related Functions
British Association for the Advancement of Science A951): Hhn(x) ,
n = -7(l)-4 , (n+13) decimals; n = -3(lM , x = -7@.l) to point
where functions vanish identically to lOd.
2
Carslaw and Jaeger A959, p. 485): ex erf с x , erf x , erf с х ,
x = 0@.05I.0@.1K.0 , the first to 4d, the latter two to 6d.
Bn)inerfc x , x = 0@.05I.0@.1) to point where 4d entries vanish.
Gawlik A955): e"^ / e^ dt , x = 0@.01I0 , lOd.
0 -
Hartree A936): 2 ierfc x , 4i2erfc x , x = 0@.01I@.02J , 4d. A few
other entries are also given.
Harvard University Computation laboratory A952a):
Ф(}(х) - B.Г* Ге-^dt , cp(n>(x) - B.)"* *^±!) ,
J0 dx11
n = -1AI0 , x = 0@.004) to point where values to 6d do not change;
n = 11AJ0 , x = 0@.002) to point where 6d values vanish identically.
Kave A955): inerfc(x) , x = 0@.01) to point where functions vanish
identically. The number of decimals corresponding to n are as
follows, n = 0,1,2 , 6d; n ¦= 3,4 , 7d; n = 5,6,8 , 8d; n = 7,9 , 9d;
n = 10,11 , lOd.
I
National Bureau of Standards A953): g(x) = Bn)
g(t)dt ,
x = 0@.0001I@.001O.8 , 15d. g(x) , 2 / g(t)dt , x = 6@.01I0 ,
7s.
National Bureau of Standards A954c): f(x) = 2rf2e~X , / f(t)dt ,
¦Г
x = 0@.0001I@.001M.6 , 15d. Also x = 4@.01I0 , 8s.
к X Гх i.
1 Radio Corporation of America A946): Bтт)"а / tcosh tdt ,
Btt)~2 / tsinh tdt , 2tt'2 /
i "° i
,-tc
i ЛА t2
 / ег г
t~2sinh tdt , 2n'2 / e"° dt , 2n / e° dt ,
'0 ^0 "О
nX
x = 0@.001H.02@.01J , 8d. n'2 / f2A(x-t)dt , where A(x) is
^0
cosh x or sinh x , x = 0@.02I@.1M , 8d.
p nx о
Rosser A948): e^ / еЪ dt , x = -0.2@.05L@.1N.5@.5I2.5 , lOd.
J0
J2 fx t2 Гх .2 V-
e / e" dt , x = -0.2@.05K.8@.1N.3 , lOd. I e"' dy /
J0 ^0 ^0
/ e^dy / e~b dt , x = -0.2@.05K.5@.1N , lOd.
'У t2
e dt ,

192 INTEGRALS OF EESSEL FUNCTIONS
Error Function for Complex Argument
(see also Fresnel integrals and error function for pure
imaginary argument above)
7.13
Carslaw and Jaeger A959, p. 486): e"z < l+2iTT'2 / e° dt } , z = x+iy ,
x = 0@.1I.0@.2J@.5KAM , у = 0@.1J@.2M , 4d.
V>} '
Clemmov and Munf ord A952): (тт/2Jе51пр / e"1^ dt , |p| = 0@.01H.8 ,
v/p
arg p = 0°(l°L5° , 4d.
Faddeeva and Terent'ev A954): e"z < l+2iTT" / e* dt > , z = x+iy ,
x,y = 0@.02K ; x = 3@.1M , у = 0@.1K ; x = 0@.1M ,
у = 3@.1M ; 6d.
2 Г1сР +2
Fried and Conte (i960): Z(cp) = 2ie'cp / e dt , cp = x+iy , x = 0@.l)l0,
у = -10@.1I0 , 6s mostly.
2 Г°° 2
Hensman and Jenkins A955): B/n)ez / e-t dt , z = x+iy , x = 0@.02J ,
/
у = 0@.02L , and x,y = 0@.l)l0 , 6d.
/ e* dt ,
J0
Karpov A954): e"z" / e^dt , z = pe19 , p = 0@.0002M ,
9 = 0@.625° L5° , 5d; 9 = 90° , p as above, 5s.
Karpov
Pz 2
A958): / et dt , z = pe19 , 0 ? p ? p0 , тт/4 s 9 ? n/2 and
6=0, 5d or 5s. p0 depends on 9 and decreases from p0 = 5 for
6 = tt/4 to p0 = 3 for 9 = n/2 except for 9 = 0 and in this case
p0 = 10 .
7.13
INCOMPLETE GAMMA FUNCTION AND RELATED FUNCTIONS
193
Fresnel Integrals
(see also error function for complex argument
and rocket functions)
Akademiia. Nauk, SSSR A953): С(-§ттх2) , S(^ttx2) , x = 0@.001J5 , 7d.
С^ттх2) , x = 0@.001H.101 , 7s. S(^rrx2) , x = 0@.001H.58 , 7s.
Pearcey A956): C(x) , S(x) , x = 0@.01M0 , 6-7d.
Radio Corporation of America A946): C(x) , S(x) , x = 0@.001H.02@.01 J,
8d. n / tA(x-t)dt where A(x) is cos x or sin x ,
J0
x = 0@.02I@.1M , 8d.
van Wijngaarden and Scheen(l949): Ci^nx2) , S^rrx2) , x = 0@.01J0 , 5d.
Rocket Functions
(see also Fresnel integrals)
Rankin A949): A(x)+iB(x) = Rc(x) , see 7.11A6); Z(x) = n / A(t)dt ;
^0
z(x) - In x ; x = 0@.01I@.05I.5@.1I5 , 4d. A*(x) =
/ t" A(t)dt , B*(x) = / t"T(t)dt , A*(x) + \ In x , B*(x) + 5 In x,
^x ^x
x = 0@.1I , 4d. A*(x) , B*(x) , x = 1.1@.1K , 4d.
Roberts and Caldwell A959): rc(x) , Rc(x) , see 7.11A6), x = 0@.01J0 ,
6d; rr(x) , x = 0@.001H.15 , 6d; ri(x) , x = 0@.001H.380 , 6d.
2 2 /)X
Rosser A948): Rr(x) , Ri(x) , Rr (x) + Ri (x) , / Rr(t)dt ,
J0
x = -0.06@.02K.5 , x = 3@.05M , 12d. The first three are also
given to 12d for x = 5.05 , 5.10 and 5.15.

194
INTEGRALS OF BESSEL FUNCTIONS
7.13
Г 1 —
TOijis A947): rc(x) , ir(x) , ra(x) =[n-2ii(x)j2 =
l}? , rt(x) = arc tan ri(x)/rr(x), x <= 0@.0001H.003
Rosser, Newton and Grc
(rr2(x)+ri2(x)]? , rt(x) = arc tan ri(x)/rr(x), x - 0@.0001H.0010
@.0002H.0030@.0005H.010@. 001H.020@.002H. 050@. 005H.10@. 01)
0.20@.02H.60@.05I.50@.1K.0@.2)8.0@.5M1 , 6d
Rosser and Walker A953): / Rr(t)cos a(t-x)dt , / Ri(t)sin a(t-x)dt
Jx ^x
(see 7.11A6)), x = 0@.1M , a/n = 0.1@.1J , 5d. For a lesser
range of x , values of the integrals are given for a/n = 2@.1)8 .
Miscellaneous Incomplete Gamma Functions
Abramowitz A951): / e-t dt , x = 0@.01J.5 , 8d.
J0
Kreyszig A951): Si(x,a) , a = 0.25@.25I.75 , x = 0@.2L@.5J0 , 3d.
Si(z,a) , z = x+iy , x = 0AJ0 , у = 0AM , a = 0.25@.25I.75 ,
2d or 3s.
Kreyszig A953): Ci(x,a) , a = 0.25,0.5,0.75 , x = 0@.2L@.5J0 , 3d.
Ci(z,a) , z = x+iy , x = 0(lJ0 , у = 0AM , a = 0.25,0.5,0.75 ,
2d or 3s.
nm
G.F. Miller A960): (x+n)ex / e"xtt"ndt , n = l(l)8 , x = 0@.0l)l ;
Jl
n = 1AJ4 , x = 0@.1J0 ; n = 1AJ4 , x'1 = 0@.001H.05 ; 8d.
nco
National Bureau of Standards (I954"b): E^x) = / t~ne~xtdt , n = 0AJ0 ,
^1
x = 0@.01J , 7d; x = 2@.l)l0 , 7-10d. Eg(x) - x 1л х ,
x = 0@.01H.5 , 7d. %(x) + i х21л x , x = 0@.01H.1 , 7d.
CO ,
Pagurova A959): Е„(х) = Г t"Ue dt . ТаЪ1е I is a reproduction of
к
National Bureau of Standards (I954"b) above. exEn(x) , n = 2(l)lO , 7d
and exEu(x) , v = 0@.l)l all for x = 0.01@. 01O@.05I2@.1J0 .
Added in Proof
Fox (I960): F(±z) = e+XEi(±x) , z = x = 0@.001H.1 , lOd. Similar list-
listings for functions related to Si(x) , Ci(x) and /V^ dt .
CHAPTER VIII
REPEATED INTEGRALS OF BESSEL FUNCTIONS
8.1. Introduction
Suppose f(z) is analytic and g(z) is integrable in the domain
|z| s с . Let
f_r(z) = fihl, fo(z) = f(z) t
dzr
A)
and
go(z) = g(z) , gx(z) = / go(t)dt,..., gr(z) = / gr_1(t)dt . B)
That is, gr(z) is the r-th repeated integral of g(z) . Then the repre-
representation
/ f(t)g(t)dt = X (-) f.k(z)gk+1(z>(-)m / f.m(t)gm(t)dt C)
J0 k=0 Jo
affords a method for the evaluation of the integral on the left. Thus, to
evaluate integrals involving Bessel functions, an analysis of repeated
integrals of Bessel functions is useful.
It is convenient to generalize and treat the fractional integral
-1 fiz
¦afV{z) ={r(o)}' J (z-t)a"\(t)dt , R(a)>
0 ,
D)
where Wv(t) is defined Ъу 1.4.3A), and R(u)>-l( IR(u )l< 1) for Bessel
nmctions of the first (second) kind. The path of integration is the
19b

196
INTEGRALS OF BESSEL FUNCTIONS
8.1
segment t = zt , T>0 or any equivalent path. If Wy(z) » Jy(z) , we
write Ja v(z) in place of wa y(z) , etc. If v = 0 , write wa 0(z)
wa(z) . If a = 0 , we define
Repeated integration Ъу parts of D) yields
E)
vajV(z) •{ГИ1))/^-*^»1*14' ' R(a)>r-
„z nz
1 , F)
G)
so that wr y(z) is the r-th repeated integral of W^(z)
of D) yields the functional, representation
l = wa-l,v(z) •
Differentiation
(8)
dz
If r is a positive integer, we write
»-r,»<2
dzr
(9)
see 1.4.3(8). For a general discussion of fractional integrals and deriva-
derivatives, and a glossary of tables, see Erdelyi et al.A954, Vol. 2, Ch. XIII)
and the references given there.
8.1
REPEATED INTEGRALS OF BESSEL FUNCTIONS
197
We also consider
Ko,u(z) ={r(a)}"V (t-O^VtJdt ,
R(z)>0 , R(a)>0 ; z = 0 , R(a) > | R(w) | ,
where the path of integration is one of the rays t = zt , т>1 or
t = z+t , т > 0 . We can write
A0)
16
KajU(z) ={r(o)}" f (t-z)a"\(t)dt ,
6 real, 16 К it/2 , R(a)> 0 ; 161 = тт/2 , 0<R(a)<3/2 . (ll)
Here the path of integration lies in the branch of the cut plane deter-
determined Ъу -тт< arg z <. tt and is the ray pe , p—»» , except for an
initial finite path. Thus,
*aM*'*") - bi^iV-a){r(*)}~1 />m(t-xr141)(t)dt ,
"x
x>0 , 0<R(a)<3/2 . A2)
The formulas analogous to F)-G) are
Ka,v(z) ={r(a-r+l)}'X f (t-z)a"rKr.ljl>(t)ut , R(a)>r-1
J 7.
A3)

198
INTEGRALS OF BESSEL FUNCTIONS
r16
О г
16
Xt,u(z) = / ^-1,1,(*)й* > |arg 6I<TT/2 >
u z
where the path of Integration is as in A1). Also
^-Ka,u(z) = "Ka-l,v(z) >
CLZ
^ ,yr d%(z)
The equations 1.4.5F) and A4) and the continuity principle give
*-»=/0*
,-z cosh t(sech t)acosh vt dt ^
8.1
A4)
A5)
A6)
A7)
R(z)>0 ; R(z) = 0 , R(a)> |R(i>)|
The formulae telovf follow from the relations connecting Bessel
functions.
7a>v{z) = (esc utt)[(cos »TT)jajU(z)-Jaj_u(z)] , |H(w)|<l • A8)
8.2
REPEATED INTEGRALS OF BESSEL FUNCTIONS
199
la,v(z) = e"*L(U+a>TJf,..,,(zeiLTT) > -"<аг8 z W2 ,
"a,uv
= e^Ki^aVj^^^-iSiTTj t n/2<arg
z s n .
ka^(z) = |n(csc m)[iaj.B(z)-lM(8)] , |H(OI<1 •
L du J u=o
B0)
B1)
B2)
Ч^е41П) = ^l("+1-a)n ba,V(z)+i^,v^] ' |H(U)|<1 . B3)
In A8)-B3), we take -n<arg z s n .
8.2. Power Series Expansions and Differential Equations
The combination 8.1D), 1.4.1A), and 1.2A4) produces
± (z) - 2a у (z/2)^2k+ar(,+2k+l)
a, u
kt k•' Г[u+k+l)r(u+2k+a+l)
2 (z/2) ^u+1 u+2 v+a+1 u+a+2 z
T(u+a+l) 2 3 (^— ' — 'u+1' ^— ' —Г— ¦¦ —
4 r7-) - za F /'I . a+1 a+2 . z2>.
'f) ' ^
B)
and from the former, if n is a positive integer or zero,
Ч-пB) =ta,n(z) + ia,n(z) '
C)

200
where t Jz) <= 0 , and
Cl, U
integrals of bessel functions
8.2
n-r
L 2 J
t
. .k. / .a-n+2k. , ..
(v\ = 2a V (-) (z/2) (n-k-1).'
a,n^ /f- k.1(n-2k-l).T(a-n+2k+l)
(O
It follows that i y(z) and i _v(z) are independent unless v = 0 , or
unless и is a positive integer and a is a negative integer or zero. If
we use the series expansion for ia^(z) to define kajU(z) by 8.lBl),
the resulting series representation is valid for all v provided v is not
an integer. Similarly, a series representation for ya v(z) follows from
A) and 8.1A8). The formula
ka(z) = - (у+1л z/2)ia(z)+2a ? (Z/2Jk+aBk)i /t(?k+a+lHBk+1)}
+ 2a ? (z/2Jk^Bk).' |1+4+...+l/k}
k=l-(к.1 JГBк+а+1)
E)
follows from (l) and 8.1B2). Also
2ПУ
' (z)=(Y«* z/2)JfT(z)-2a2: (-) (z/g) aBk)' (фBк+а+1)-фB^1)}
a a k=0 (к.1 JГBк+а+1) L
2a ^ (-)k(z/2Jk+aBk).' Г1ф -+1/к| .
k=l (к.1 JГBк+а+1) L
Using 1.3.1D-5) and previous results of this chapter; we find
F)
8.2
REPEATED INTECEAIS OF BESSEL FUNCTIONS
201
[z2D4-Ba-5)zD3+ ((a-2J-u2+abz2) D2-2ab(a-2 )zD
+ ab(a-l)(a-2)]wajU(z) = 0 .
Ka^y(z) is also a solution of this equation if ab <= -1 . If v = 0 ,
A , A]_ , and В are arbitrary constants, then
G)
[zD3-(o-2)r?+zD-(o-l)] {AJa(z)+Bya(z)j -
2Bz'
a-2
(8)
[zD3-(a-2)D2-zDf(a-D] {Aijz )+AlKa(z)+Bka(z)} = - ?gL . (9)
If ab = -1 , and u = 0 , then ia y(z) and ia (z) coalesce and
a second solution is k^z) • From 1.3.1E), two other solutions of G)
are conveniently taken in the form
fa4v(z)=fa2F3(^f , ^ ; Ц*
2+u-a 1 . z?
' 2 ' 4
)
fa-l4v(z) = fa-lz 2F3(^ , 3lu - 3^-a
3+v-q 3 . z2\
2 2 4^
A0)
A1)
where
fa = 2a'X*f)r(^)/r(a)
A2)
In series form, we have
raaa,»(z)=
2атт sin an
r(l-a+2k)(z/2)
,2k
sin(a+u) E sin(a-u) | k=0 г(^ +к) rQ^ +k) Bk).'
, A3)

202
INTEGRALS OF BESSEL FUNCTIONS
8.2
2an sin(g-l)n
rB-a-b2k)(z/2)'
,2k+l
**-**>»М~ sin(a-I+u)|sin(a-l-u)| k=0 гE=р *) г(^ +k) Bk+l).'
A4)
Note that
|D»w}-ww
A5)
Here we assume the parameters are such that the series A3)-A4) have
meaning. Singular cases and situations where i „(z) , V-u^z^ ' aa v^z)
and Ъ „(z) are not linearly independent are later studied in connection
with series representations of Ka u(z) .
It is convenient to introduce the formal series
с (z
a, i>
, _, /3-a 2-a 2+u 2-u . 3 . 4J\
) = A^T" ' ~2~ ' 2 ' 2 ' 2 ' Z2j
, ч /2-a 1-a l+i> l^u . 1 . 4Л
V»(z) = л^т"j т' т' 2 ' 2' Z2;
A6)
A7)
Then, if r is an even positive integer,
r-2
frar,u(z) = J^)T (™/2)(csc UTT/2)cr^(Z) ,
v is not an even integer ,
A8)
„r-1
fr-lbr,u(Z) = |ТГГ)Т C"/2)(eeo UTT/2)dr^(z)
v is not an odd integer
A9)
8.2
REPEATED IHTECKALS OF BESSEL FUNCTIONS
203
If r is an odd positive integer, then
_r-l
frar,u(z) = J^JT (n/2)(sec UTr/2)dr^(z) ,
v is not an odd integer ,
fr-lbr,»(z) = fJ)T ("tt/2)(csc vn/2)crjU(z) ,
и is not an even integer
B0)
B1)
We now suppose that (L5), A4) and ia ±v(z) are defined and linearly-
independent . Let
Ka,»(z) = aia^(z)+bia,-v(z)+cfaaa,u(z)+dfa-lba,^z) ' B2)
where а , Ъ , с and d are constants of integration. Take
Re(a-l)>|Re(i>)| . Then 8.1A5) and B2) with z = 0 and 8.5A) shew-
that с = 1 and d = -1 . To find a and Ъ , we construct the asymptotic
expansion of Ka u(z) using B2) and the appropriate results of 1.3.з\
Equate this to the asymptotic expansion of Ka v(z) which follows from
8.1A3), see 8.4C). Then '
2a = -TT csc(o,+i>)tt , 2Ъ = -тт csc(a-u)TT ,
B3)
no that
>WZ) =2-[slg^- h^} + V«,^' - f-lbe,u<«>
• B4)

204
INTEGRALS OF BESSEL FUNCTIONS
8.2
In
particular, if a is a positive integer, say a ¦» r , then
r r-2
. (-f^ (тг/2)(вес ^/2)^^@ ,
v is not ал integer
B5)
Equation B4) is valid for all values of the parameters for which the
functions on the right have meaning. If a is not an integer and (a+v) is
a positive even integer, then faaa u(z) and тт/2 ia v(z )/sin(a+u)rr are
not defined though their difference can be defined using L'Hospital's
theorem. Thus, in this and other similar singular cases, we can derive
series expansions for К (z) by limit processes. These singular cases
are delineated below, and from the analyses it is apparent how one con-
constructs independent solutions of 8.2G) for all possible combinations of
the parameters a and v .
We first suppose that" a is a negative integer or zero. Let
a = -r . Then
K-r,u(z) = ant-fcsc vn [i_r^_u(z) - l_r>u(z)] ,
v is not an integer . B6)
Let v = n+e , n a positive integer or zero. Then 8.2A) with appropriate
use of 1.2E) gives
8.2
REPEATED INTEGRALS OF BESSEL FUNCTIONS
205
*-r,n+e(z)
f—1
-r-1 Д J(-)k(z/2Jk-n-r-er(n+e-k)r(n+e-2k+r)
k=0
k.T(n+e-2k)
Гп+r-ll
+ (-)П2-Г-181п en |- (-)k(z/2Jk-n-r'er(n-i-e-k)r(n4-e-2k-fr)rBk-fl-n-e)
L 2 J
+ (-)Г2-Г-1
T1 (-)k(s/2Jk-n-r-e
к_ГпТг+1-| к!ГBк+1-п-г-е)
L 2 J
Г(п-»е-к)ГBк+1-п-е)
^(-ГУ1-1! Z (z/2Jk^-r-ert2k^!-e)
sin етт lv=o Г(к+1-е)(п+к)'ГBк+п-г+1-е)
- i (z/2Jk+n-^erBk+n.l.eI
k=0 к.'Г(п+к+1+е)ГBк+п-г+1+е) J
B7)
Thus

206
INTEGRALS OF BESSEL FUNCTIONS
8.2
№
-r-l ? Jf-^Z/2Jk-n-r(n-k-l^.[(n+r-2k-l).'
K-r,n<z):=2 ?n k.'(n-2k-l):
* (-('г"' 1;1 (-)t('/gi^'°'rU-^i)'(a^°).'
-m
k.1 Bk-n-r)!
+ (-Г ^
П+Г -Г-1
f fz/2r-rB№).'
^ k:(n+k).'Bk+n-r).1
X {-2 1пB/2)-2фBк+п+1)+2фBк+п+1-г)+ф(к+1И(п+к+1)}
B8)
provided r<n-l . If r s n-1 , the second series in B8) disappears. If
r S n+1 , the last series in B8) must be modified in view of the singu-
singularities arising from Bkin-r)! and \|i Bk+n+l-r) for certain values of к,
Let r = n+l+m , m s 0 . Then with the aid of 1.2E) and 1.2A2), the last
series of B8) remains as it is, save that the summation begins with
к = -— instead of к = 0; and to this we must add the finite series
[f]
2"r Z
k=0
( z/2 )п+'г*-т Bk+n): (m-2k) ¦'
k.'(n+k)!
B9)
Note that К r n(z) is essentially the r-th derivative of K^z) , see
8.1A6).
We next consider the situation where a and (a-v) are not integers,
but (a+u) is a positive even integer or zero. It is convenient to define
the auxiliary functions
8.2
REPEATED INTEGRALS OF BESSEL FUNCTIONS
207
,(z)=f a „(z) -ZJhl^L
эа,ич ' а а,и4 ' 2 sin(a+u)TT
i (z)
pa,v(z) = -fa-lba,v(z) " I sin(I-v)TT '
C0)
C1)
so that
\,v^ =Sa,v(z) + Pa,^z) •
C2)
Under the above hypotheses, the series for p „(z) is defined, but a
series representation for ga ^(z) must be derived by a limit process.
Let a+v = 2m+2e , m a positive integer or zero. We can write
S2m.-v+2e,v^z'
_ ^^2m-2-v+2e
sin(e-u )tt
sinBe-v)n ^- (-) (z/2) Г(т+е-к)ГBк+1+У-2т-2е)
k=0
Bk).T(k+l+u-m-e)
. TT22m-2-V+2e[sinBe-v)TT у (z/2Jk+2mrBk+l+v-2e)
sin етт |sin(e-v)TT k=0 Г(к+1-е)Г(к+1+и-е)Bк+2т).'
00 , , ,2k+2m+2e , , ч
- sec етт У (z/2) Tg rBk+l+v)
к=0 к!Г(к+1+и )rBk+2m+l+2e)
C3)
whence

208
§2ш-Ь"B)
INTEGRALS OF BESSEL FUNCTIONS
, ^2m-2-» ^ (-^(z/2Jk(m-l-k).TBk+l+»-2m)
{->^ 2- Bk).T(k+l+v-m)
+ 22m-2-, у (z/2Jm+2krBk+l+l>) f CQt m . 2 ^(z/2)
8.2
k=0
- 2фBk+l+v)+ф(k+1)++(k+l+u)+2фBk+l+2m)}
C4)
Let us retain the hypothesis that a+v =2m , but now assume a-v Is
an odd integer. Thus a and v are same (but, of course, not necessarily
the same) odd multiples of -g. As Ka ^(z) is an even function in v , it
is sufficient to put и = n+^- , n a positive integer or zero. In this
event g2m_y „(z) is defined, but a series representation for pa v must
be developed by the usual limit process. The analysis runs along the same
lines as in the derivation of series representations for K_r n(z) and
ggjjj.y y(z) . We therefore omit details and state only the final results.
P2m-n-!,n+i(z)
m+n 2m-n-5/2 ^ f- )k(z/2Jk+1rBk+n-2m+5/2)(m-n-2-k).-
= (-) 2 2. Г(к-т+3/2)Bк+1):
k=0
+ 22m-n-5/2 " (z/2Jk+?nl-2n-1rBk-n^)
*-_ к'.Г(к-п+?)Bк+2т-2п-1):
k=0
X {2 lu(z/2)+2\|iBk+|-n)-i|r(k+l)-i|r(k-n-t4)-2i|rBk+2m-2n)|
m г n+1
C5)
8.2
REPEATED INTEGRALS OF BESSEL FUNCTIONS
209
P2m-n-in+i(z)
= 22m-n-3/2 ^ (г/2Jк+2ш-2п-1ГBк-п^)Bп-2т-2к->:
k=0
k.T(k-n+^)
2k+l
+ 22m-n-5/2 у (z/2 )'::J''r"LrBk+n-2m+5/2 )
. _n Г(к-т+з/2)(k+n+l-m).' Bk+l).'
a.— U
X{2 m(z/2)+2^Bk+n-2m+5/2)-iKk-m+3/2H(k+n+2-m)-2tBk+2)j >
m <. n+1
C6)
Now suppose that a+v = 2m , m a positive integer or zero and that
v is a positive integer or zero. Then neither C0) nor C1) is defined.
Here let и = n+e , and express K2m_n_e n+e(z) in series form using C2)
along with the combination (l), A4) and C4). Then a series representa-
representation for )<2m_n n(z) follows by applying the familiar limit process.
Again we spare details and state the final results.
K2m-n,n(z>
= P,
2m-n, n
, . 2m-n-2 m'^-'1 (z/2Jk(m-l-k).'(m-l-n-k).'
KZ) kt0 Bk).'Bm-l-n-2k).'
m
- - - , vk. , ,2k+2m-2n, .
+ 22m-n-2 у (-) (z/2) (n-l-k).'
j^Tq k.' Bk+2m-2n).' (n-l-2k).'
X {-2 lti(z/2)-2i|r(n-2k)+i|((k+l)+i|r(n-k)+2i|rBk+l+2m-2n)} ,
m s n
C7)

210
K2m-n,n(z>
IHTEGRAIS OP BESSEL FUNCTIONS
8.2
, , Pm-n-1 ^^ fOk(Z/2Jk+2m-2n(n-k-lVBn-2m-2k-l).'
= P2m-n,n(z>+2 4- k.'(n-2k-l).'
л.— U
rgm-n-li
, ,n+in 2m-n-2L J- J Mk(Z/2Jk(m-k-l).'
+ (-> 2 ?0 Bk).'(k+n-m)!Bm-n-2k-l).'
X {- 2 1иB/2)-2фBт-п-2к)+ф(т-к)+ф(п+к+1-т)+2фBк+1)] ,
where
„2m-n-l
?2m-n,n(z) - (-JV^ Z
m ? n ,
n-1 , л. , <2k+2m-2n
(- ^(z/a^^'^tafn)! (n-k-1).'
k.'Bk+2m-2n):
L 2 J
. .m 2m-n-2 ^ 2,- J (-)k(Z/2Jk+1r(m-n-k-j)
^"; ^ п ^- Bk.+l).T(k-m+3/2)Bm-n-2k-2)!
+ 22ш-п-1 f (Z/2Jm+2k(n+2k):
A, k!(n+k).'Bk.+2m):
X (-2 1пB/2)-2фBк.+п+1)+г|((к+1)+ф(п+к+1)+2И2к+1+2т)}
C8)
C9)
8>г
. О
REPEATED IMCEffiALS OF HBSSEL FUNCTIONS
211
There remain other singular cases, but as these can be readily related
to results already obtained, we sketch the manner of derivation and omit
details. Suppose, for example, that a and a-v are not integers, but
a+v is a positive odd integer. Put a+u = 2m-l , m a positive integer.
Then a series representation for ^2m.i.v viz) follows from that of
K2m_v y(z) by differentiation in view of 8.1A5). Again, suppose
a+v = 2m-l and v = n+g- where n is a positive integer or zero. Then
differentiation of the series expansion for K^m_n_i n+i(z) leads to our
desired result. Series representations for K^ , n n(z) follow in a
similar manner from C7)-C9). '
8.3. Recurrence Equations
Using 8.1(8,15), the differential equations 8.2G-9) are easily con-
converted to the following difference equations.
аЪа(а-1>а+1^(г) = 2ab(a-l)zw^v(z)- [(a-lJ-u2+abz2} wa_^u(z )
+ Ba-3)zwa_2^(Z)-z2wa_3^(Z) .
a(a-l)Ka+l,v(z) = -2(a-l)zKa^(z)+ {(a-lJ-u2-z2} K^^^z )
A)
+ Ba-3)ZKa_2^(Z)+Z2Ka_3^(Z) .
aJa+l(z) = zJa(z) " (a-l)ya-l(z) + zJa-2(z) •
aVi(z) = *ya(z) - (a-Dv^z) + zya.2(z) - i^±
aKa+l(z) = -zKa(z) + (a-l)Ka-i(z) + zKa-2(z> •
B)
C)
D)
E)

212
INTEGRALS OF BESSEL FUNCTIONS
8.4
Note that each of the four components of Ka v{z) as given by 8.2B4) also
satisfy B).
We also have the contiguous relations
F)
G)
8.4. Asymptotic Expansions for Large z
We start with 8.1A1). Replace Kv(t) by its asymptotic expansion
1.4.6F), and noting that(Erde^yi, et al., 1954, Vol. 2, p. 202)
_1
Г(
-Г (t-zf-W^dt = е-К"Ъ-Ч)ЪB) ,
A)
where W& b(z) is Whittaker's function and
Wa,b(z) ~ e~^za2Fo(i-a+b,!-a-b;-l/Z) ,
4a = l-2a-2k , 4b = l-2a+2k , |z|—>« , I arg z|<3tt/2 . B)
We find
XaU(Z)~(TT/2zFe-z Z (-L^"k >
' k=0
|z|—>oo , larg z|<3tt/2 ,
C)
8.4
REPEATED INTEGRALS OF HESSEL FUNCTIONS
213
and (see also (б))
ck
{hv)^-v)* 3F2(-k,i-k,a;i-^,i-*-,;2)
2kk.f
D)
In the following, E)-(8) come from 8.2(l), 8.2A0-11) and the results
of 1.3.3. The connecting formulae 8.1A8-23) produce (9)-(l2).
i (z) —
a, uv '
-z
kz " + ~^—I e
BttzJ k=0 BnzJ
Z_ ckz-k+-^e-ie^+a+i)"i: (-)\z"k
k=0
* Г(а-1) Ca,^(z) *
=--5ге(и+1)тт a-1
da.,(z) ,
a, u
|z|—»oo , -B+e)TT/2<arg z<B-e)n/2 , e = tl ,
E)
where c^u(z) and ^„(z) are given by 8.2A6-17) and
2(k+l)ck+1 = fkck+gkc]5..1+h]5.c]5._2 , c_2 = c.x = 0 , cQ = 1 ,
fk = 5к2+Dа-1)к-ф-а-и2 ,
gk = "(k-i) [4к2+(ба-7)к+7/2+2а2-5а-2и2] ,
hk = (к-3/2)(к-^)(к+а+и-3/2)(к+а-и-3/2) .
F)

214
aa,v(z
INTEGRALS OF BESSEL FUNCTIONS
8.4
}..rC-F-;rC 2 ;VL z CkZ-*-iee-* z (-)Ч
,-k
k=0
1 „(о_1)Г?2±р) rB=p) (z/2)a-Ce
kpO
,a-2jh.e(a-2)n
tt(csc im/2)(sec an/2)
ca,v(z)
rC^X^)^/2)'
a-1 iie(a-l)n
rr(sec im/2)(csc arr/2)
\v^ '
|Z|-»oo , -B+е)тт/2<аг8 z<B-e)n/2 , e = ±1
z Z ckz-k+iee-Z Z (-L^"k
=0 k=0
ur / 3+v^N r (i-v-a\ (z/2 )a-2eiie(a-3 )тт
V 2 > V 2 У c (z)
tt(csc vtt/2)(csc атт/2) a>v
TT(l-a)(sec итт/2)(зес агт/2)
da,v(z) '
|z|-
, -B+e)n/2<arg г<B-е)тт/2 , e = ±1
G)
(8)
8.4
REPEATED INTEGRALS OF BESSEL FUNCTIONS
215
Wz>~ (TT/2z)WieOT X ("Lz-k
k^O
" Ы^^/2)^ c^{z)+ jnCsec Wa^a-l
UI-J-oo , -B+e)TT/2<arg г<B-е)тт/2 , e = ±1
L K u k=0 J
da,v(z) *
O)
??l)Ca^(iZ)+?Sda'u(lZ) '
|z|—»oo , |arg z|< тт ^
A0)
>*r
L k-0 k=0 J
. у (cot итт/2Jа-2 (tan №/2)za-1 „ , .
Г(а-1) ca,vt") - r(g; ^ 4a,v(lz) *
|z|—>oo , |arg z|<tt ,
A1)
where
9 = z - Bи+2а+1)тт/4
A2)

216
8.5. Infinite Integrals
INTEGRALS OF HESSKL FUNCTIONS
8.5
K (о) =^— rfsiMrfe") , h(o)>|h(u
*aM0) Г(о) А2/ ^2 J
KX „@) =(тт/2)зес vtt/2 , |H(v)|<l •
Kg „@) =(utt/2)csc utt/2 , |R(v)|<2 •
)l
K (o)=H^4^sec^fT Г1"-^! ' |R(U)I<
*2г+1У0> 2 г7г(?) 2 ^=1 L Bк-1J J
A)
B)
C)
2Г+1 . D)
Kg;
f0^ - vtt ir-ll|r(ii csc vtt TT ["i-JzLl, |H(»)I
¦'vl0) "Г 2Г(г^) 2 ^ L 4k2 J
<2r . E)
Formula A) comes from 8.1A0) and 2.6F), while B)-E) readily
follow using 1.2A,5,6).
From 8.4A0-11), we get
(t)dt = 1 , R(u)>-1 ,
J0
f"yv(t)dt - - tan|H , |H(v)|<l ,
Jo
n<*> n t
/ / Yo(u)du dt = -2/tt
Jo ^o
F)
G)
(8)
8.6
REPEATED INTEGRAI5 OF BESSEL FUNCTIONS
217
8.6. Further Representations
Circular representations for jn r(z) readily follow from 2.7.
Exponential series representations for Ka v{z) are given in 8.8.
The formulas
Jr(z) = ^(zJjJzJ+Brfz)^B)+СгB) f Jo(t)dt ,
A)
yr(z) = Ar(z)Yo(z)+Br(z)Y1(z)+Cr(z) / Yo(t)dt - 2/тт ^(z) , B)
Jo
ft °°
Kr(z) = irAr(iz)Ko(z)+ir-1Br(iz)K1(z)+ir-1Cr(iz) I Ko(t)dt , C)
J z
where (see 8.2A6-17)),
r-1
(„\ _ z д (л„\ _ z' /2-r 1-r 1. 4\
(O
M*) - (Й C^o(lz)" (йтт лС?' ? ^ Iг - ^) ' E)
und A?.(z) and Br(z) are also polynomials in z , are easily derived by-
repeated applications of the recurrence equations 8.3C and 5), respec-
respectively. A?.(z) and Br(z) are obviously solutions of 8.3C). The above
polynomials are tabulated below for r = 1AN .

218
INTEGRALS OF EESSEL FUNCTIONS
8.6
Vz)
Br(z)
cM)
Dr(z)
1
2
0
0
24 8
?_ _ 15 z*
120 120
z
2
z^ + 2z_
6 3
^ + z?
24 24
. llz3
120 120
8z
15
z
2~
3
z_
6
z
24
zf__
120
1
2
z_
2
? + =
51 + 3z
12 8
z
24
z
6
4 „2
2
3
2z
3
15
F)
Repeated integration of 1.4.5E) yields
(-i)r+1 [^e±X Se° ecosr-1ede = тг/2 (jr(x)-Cr(x)j +i rr/2 {yr(x)+2/rr Dr(x)}*
•/о
x>0
G)
In particular,
P2n(x) = (-)П Г sin(x sec e)cos2ned9 = -тт/2 |у2п+1(х)+2/тг D2n+1(x)|
Jo
x>0
(8)
8.7
REPEATED IHTEGRALS OF BESSEL FUNCTIONS
219
,^
P2n+l(x) = (")П / cos(x sec 9)cos2n+1ed9 = тг/2 {у^+г^У^/ъ D2n+2(x)} >
(9)
x>0 ,
which have Ъееп discussed Ъу Havelock A923, 1925).
Repeated integration of 2.4(l) with ц = 0 and appeal to the con-
continuity principle yields the expansion
лв,„(.) = ^ Хо^Ч+а+2кЫ ' ^>-^
A0)
A similar type formula for repeated differentiation of any Bessel function
is given Ъу 1.4.3(8).
8.7. Asymptotic Expansions for Large Parameters
If z is fixed, |u|—>m and/or |a| —><*> , then the various ascending
series in z afford convenient expressions for computational purposes.
The combination 8.2(l) and 1.2(8) gives
i^vl2) = : " 7T jexp |(и+аIл ez
[2rr(i;+a)P
L-rW Lo[-i
2(u+a) 12(u+a)JJ [ \(u+aK
\y v /'w+l u+2 .1,+t u+a+1 u+a+2 . z2 N
|u+a|—><» , |arg(u+a)l<n
A)
Note that if z is fixed, the 2^3 in (l) is essentially a descending
series in v . With v = q exp(^in) , a and a fixed, we have

220
INTECSALS OF BESSEL FUNCTIONS
8.7
^¦^¦"•[^^{^ьег)]
|u|->oo , |arg(a+iq.)l<TT ,
where
= 2TT a(a+l)Bq+l)
a p о '
2 12cf
Using 8.2B4) and arguments similar to those above, then
B)
C)
D)
>Wz>
. Unfafo/if^ Г .n i_ CQS J_ (8Bа+1).Л sin J
| sinh qn L а 4Ч а 32q2 I- J J
L 2 {q2+(a-2Jj J L б{<12+(а-3J} JJ L V-
Iql^oo ? |arg(a+iq.)l<TT
Here
9a = ea- от
-a-1
a Ш
exp
/-q.TT/2 + a(a-3-)(a-g) 1
I 6q.2 J
E)
F)
G)
8.8
REPEATED IMTEGRALS OF EESSEL FUNCTIONS
221
If a •= 0 , Kq i (z) = K. (z) and E) essentially agrees with a known
result, see Erdelyi et al. A953, Vol. II, p.88).
8.8. Exponential Series Representations for Ka „(z)
Employ 8.1A7) and the theory of Fettis A955) and Luke A956). Then
» -z coshBk+l)h/2 . . ,
^„(O-hZ2 , cosh "Bk+1)h/2 * %...(O , (i)
k=0 coshaBk+l)h/2
a, uv
00 -z cosh kh
/i , v e cosh ukh i , -z / \
Ka)U(Z)=hZ 4be +g (z) ,
k=0 coshTdi
B)
where
fa,u(z) =  2 (-) gt(O , gajU(O = -2 Z gt(O
k=l k=l
et(z)=i[KO)lqi(z)+KOjlqs(z)] ,
q.i = p+i" ^ ч2 = p-i» ^ p = —
h
C)
D)
Note that (l) and B) are trapezoidal rule representations of 8.1A7) with
remainder terms fa ^(z) and ga y{z) , respectively. With h suffici-
sufficiently small, p and so also I Q-i g | are large. Thus, an easy appraisal
of the error terms follows frcm 8.7E).

222
IHTEffiALS OF BESSEL FUNCTIONS
8.9
8.9. Description of Mathematical Tables
n00 n00
Bickley and Naylor A935): Ki-^x) = / KQ(t)dt , Kin(x) = / Kij^CtOdt =
"x ^x
Kn Q(x) , see 8.1A4), n = 1AI6 , x = 0@.05H.2@.1J,3, 9d.
Havelock A925): Pn(x) , see 8.6(8-9). n = 3,4,5, x = 0@.4L.8,5AI0 ,
4d.
Jaeger A948): 2~njn(x) , see 8.1D-7). n = 1AO , x = 0AJ4 , 8d.
CHAPTER IX
INTEGRALS INVOLVING STRUVE FUNCTIONS
9.1. Introduction
The results of this chapter axe much akin to those of Chapters I and
V, and we omit all proofs. Definitions and connecting formula are as
follows.
A)
LV»(Z> = /lZ*4(t)dt = -e^^+V\ (Ze^) , Н(ц+и)>-а . B)
U a
In (l)-B), -n<arg z s: n .
We have the differential equation
,M.+"+l
2г05+A-2цJ0г+(Ди2+22H] Hi „(z) = -^
*' 2"-1Г(*)Г(^)
C)
9.2. Power Series Expansions
pz oo k, . .u+2k+l
/ t^E ftUt = z^1 T (-) (z/2)
Jo ^ ^ ; " kt0 (^+2k+2 )Г(к+3/2 )r(u+k+3/2)
a+v+2
2и+±(ц+и+2 )ГC/2 )Г(и+3/2 )
2F3\ 3
1,1+ i±iz
2
1 ,u+ 1 Q+ttll
2 ' 2 2
z
4~
R(n+")>-2
A)
523

224
INTEffiALS OF BESSEL FUNCTIONS
9.3
[ t№.(t)dt = (iff* . [in z/2-4t(m+i)-4t(v*B4)]
-2^Z
(-)k(z/2Jk+2-2m
(к+1-т)Г(к+3/2)r(v+k+3/2)
k=O
k/m-1
Е(ц)<? > M*+u = m > m a positive integer
B)
I
4k/ /„\2k
t-2^,(t)dt = B/n)B-Y-ln 2z)-(l/n) Z ^"? (z/2),. • C)
{C/2^}'
k=l k'
9.3. Asymptotic Expansions for large z
f t^tby^t^dt-/-
^е^)ге=р>о.™
tt sin(n+u)n-/2
7H+u
a^^n+vjr^jrlv+j)
ЛA, I, I -«,- f ,* Ъ-% ¦
|z|—->-oo , |arg z|<n , R(n±v)>-1 ,
H+u is not a positive even integer or zero
A)
9.3
IHTESIAIS INVOLVING KERUVE FUNCTIONS
225
«z .m+1 u+l , т ^ i
/ t>1[H|;(t)-Yv(t)]dt~.^ g r(m^)r(m-^-p)cos vn
Jo tt2
kySm
|z|—»oo , |arg z|< п , Е(ц-и)>-1 , ц+и = 2m ,
m a positive even integer or zero .
B)
nZ oo (
/ [Ho(t)-Yo(t)]dt~ B/tt)(Y+1h 2z)+(l/n) Z"
J0 k=l
_ _J U_ B/zJk ^
|z|—>oo , |arg z| <n
See 3.11C) for a polynomial approximation to C)
/ t^[i u(t)^(t)]dt~ - ^ ; r x,2 J
Jn L ~v v J n tan(u,+uW2
C)
z^
,+u
2v-L(n+v)r(i)r(v+J)
—aOM-"^'1-*?'?)
2ie cos
noo
^-^ / t^CtJdt , [zl-^-oo , -B+e)rT/2<arg z<B-e)n/2 ,
U z
e = ±1 , R(n-u)> -1 , R(li+w)> -2 ,
H+u is not a positive even integer or zero .
D)

226
INTEGRALS OF EESSEL FUNCTIONS
9.4
L
V[i (t)-Vt)]at~ 2"+1r(^)r(^-v)cos vn
z2mcos vp. ^ Г(к^)Г(к4-У) ,„, ,2k
2V k=0
X {m z/2-i* &*?)%&**¦»)} - g ^ X (k.m) B/z)
2ie cos un
Г00
/ t\(t)dt , |z|—»co , -B+е)тт/2<аге z<B-e)n/2 ,
«7.
e = ±1 , R((j,-u)>-l , (j,+v = 2m , m a positive integer or zero . E)
9.4. Infinite Integrals
f tH'H (t)dt = ^1
,+v >t/2
-2<R(|j,+u)<0 , R(n)<i •
ft-V-\(t)at = ff , R(,)>-3/2 .
Jo 2u+1r(v+l)
A)
B)
I t*4 [Hu(t)-Yu(t)] dt
Jo
тт sin(|j,+i; )tt/2
R(p,+u)< 0 , R(n±v)>-1
C)
9.5
INTEGRALS INVOLVING STRUVE FUNCTIONS
227
/ t^i.^tM^Ct^dt = -
u П
2^rC 2 ;rCV-)cos m
тт tan((j,+u)n/2
-2<Н(й+и)<0 , R(M,-v)>-l
D)
/со 17+1
^ e-VLu(t)dt « - 2(g1>gy? , -KH(u)<-i • E)
/ e"tt-%(t)dt = Ov-l^rC^Mi)]'1 , R(v)>* . F)
9.5. Reduction Formulas
Let
z
g^(z) =J e'ptt4(t)dt ,
A)
where Vy(z) represents the Struve function or the modified Struve
function. The parameter Ъ introduced in the reduction formulae is
associated with the particular Struve function as noted in the following
table.
VyU) Ъ
VO 1
Lu(z) -1 . B)
I*H,v(s) = -e-pZZ4(z) + (^)Vl)V(Z) - tg^u+1(z)
[г^гСу+з/г)^!^)]'1/1 e-^t^dt
C)

228
INTECKALS OF HESSEL FUNCTIONS
9.5
9.5
(P2^)g^(z) = te-pZZ*Vu+1(Z)H-(Ml-v-l)e-pzz^1Vu(Z)-pe-I)ZZ*Vu(Z)
+ рBц-1^и(*)+ [»2-U-iJ]v2,»(z)+ [^«^MVa)]
K"-H0g^u+l(z) = -2"e'pZZtV1,(z)-2uP8^v(z)+(^)gML,v.i(z)
+ (u-n)[2Vr(u+3/2)r(V2)]f e"*V+udt .
Z) • D)
Evaluation of g^ u(z) for Special Values of the Parameters
Case I
let р2+Ъ = 0 and v = ±(ц-1) in E). Then
-pz v+1 _ , -.
- [BU+lJur(v+3/2)r(l/2)]'1J e-ptt2u+1dt
. [P2BU-lJU-1r(U4)r(i)]e"PZ •
E)
F)
G)
(8)
INTEGRALS INVOLVING STRUVE FUNCTIONS
/Q e^ttjdt - sgg^O-lW.)]
- [Bи+1JиГ(и+3/2)ГA/2)]" /
eitt2u+ldt ^ j,^^,! } v f _±
If v = 4 ,
/ e^tH i(t)dt = Г eltt-2ji(t)dt ,
J0 ^0 2
and the latter is given Ъу 5.2E).
f- A, iZ -u+1
Jo -"fXttjat = - 2_^__ [Hu(z)+4.l(z)]
+ [Bu-lJ1'-1r(v4)r(i)](elz.l) , u ^ * .
[ e1V*5i(t)dt = B/n)* /VV^l-cos t)dt
^0 JO
= BTT)'^[{2Ci(Z)-CiBZ)-Y-ln Z/2J +i[2Si(z)-SiBz)}]
/
e-4\(t)dt = ^з- [4(z) * 4+1(z)]
229
O)
A0)
(ID
A2)
- [Bи+1JиГ(^3/2)ГA/2)]'1 f е±ЬЬ2]}+1аь , R(u)>-1 , u ^ -i . A3)

230
IHTEGRALS OF EESSEL FUNCTIONS
If v - -\ ,
[>b\ l(t)dt = f e^t-
Jo  Jo
-4Ii(t)dt ,
and the latter is given Ъу 5.2A0).
I
s , ±z -u+1
e-V\(t)dt = - e~^— [Lu(z) , Vl(.}]
/ etttL1(t)dt = B/ttJ / e'V^cosh t-l)dt
Jo 2 ^o
= BттГ? [Ei(±2z)-2Ei(tz)+Y+ln z/2]
Case II
let p = 0 and м, = tv in F). Then
5-v,v+l(z) = "z'4(z) + ^
2иГ(и+3/2)ГA/2)
/* t\.1(t)dt = z\(z) , R(v)>4
Jo
9.5
A4)
+ [Bu-lJU-1r(v4)rD)](l-e±Z) , u ^ 4 • A5)
A6)
A7)
A8)
A9)
9.6
INTEGRALS INVOLVING STRUVE FUNCTIONS
f t-"H (t)dt = 5
J0 2иГ(и+3/2)ГA/2)
- -(z)
^ t\_x(t)dt = z\(z) , R(u)>-| .
/" t"%+1(t)dt = z-\(z) -
2иГ(У+3/2)ГA/2)
9.6. The Complete Cicala Function
Let
F(z) = f e"zt [l+t-i-d+t)*] dt , R(z)> 0 .
J0
Then, see Lake and Ufford A951b),
F(z) = i-Y+m 2z+z-1+n/2[Y1(z)-H1(z)] +п/г f [ Ho(t)-YQ(t)]dt
Also, let
F(ix) = FR(x)+iFl(x) = J e-^fl+fi-d+t-S^dt , x>0 ,
G(x) = FR(x) - A-Y-ln 2x) = S(x) - R(x) .
231
B0)
B1)
B2)
A)
B)
C)

232
Then
INTEGRALS OF BESSEL FUNCTIONS
9.7
R(x) - tt/2 [/Xbo(t)dt-LL(x)] , S(x) = tt/2 j^lJOdt-I^x)] , D)
(x) = .„/2 - x + Kx(x) + / Ko(t)dt
Jo
E)
We also have the representation
F(ix) = ix-l[T(x)-l] , T(x) = i™ f t-1[HL(it)-Y1(it)-2/TT]
2 Jx
dt . F)
9.7. Description of Mathematical TatxLes
For integrals of Struve functions, see National Bureau of Standards
A962, Ch. 12). See also the following.
«x „x
Abramowitz A950): / H^t^t , / Ln(t)dt , n = 0,1, x = 0@.1I0 ,
«0 
6d or 6s at least.
Harvard University Computation laboratory A952Ъ): FR(x) , Fj(x) , R(x) ,
S(x) (see 9.6C-5)),x = 0@.01I0 , 8d.
Horton A950b): / t"H (t)dt , n = l(lL , x = 0@.1I0 , 4d.
J0
W. P. Jones A952): F(z) (see 9.6A-2)). z = 0@.02H.04@.04H.2@.2)
0.4@.1H.6@.2I.2,1.5,2AI5 , 4d.
Kussner A940): T(x) , see 9.6F). x = 0@.02H.6@.1I@.2NAI0B)
14,20 , 5d. The same is also tabulated in Dingle and Kussner A947).
In t3oth of these sources, the notation used is 8(x) .
9.7
INTEGRAIS INVOLVING STRUVE FUNCTIONS
233
Luke and Ufford A953): G(x) , FR(x) , Fj(x) (see 9.6C-5)),
x = 0@.0lH.l@.lL@.2N@.5)l0 , 6d. Comparison with the Harvard
tables above shows that there are a few errors of at most two units
in the sixth decimal place.
nx
Struve A882): J - (l/n) / tHxBt )dt =i+ (i/2ttx)H1Bx)
Jo
n2x
- (l/тт) / tHo(t)dt . x = 0@.1L@.2O@.4I5 , 4d. Godfrey
JO
A948) tabulates (l/
'") f f\(
Jo
2t)dt . This is essentially a copy
of 8truve's tatsle with some additional entries apparently olatained Ъу
linear interpolation.

CHAFFER X
SCHWARZ FUNCTIONS AND GENERALIZATIONS
10.1. Introduction
The material of this chapter is related to that of Chapter IV. We
define
Jp(\,z) = Jc(\,z)+iJs(\
nZ
,*) = / el4
Jo
(H )dt ,
Jo
it.
Y (\,z) = Y(X,z)+iY4(\,z) = / e"YQ(Xt)dt ,
A)
B)
which are knovm as Schwarz A944) functions. We also consider the more
general function, see 13.3.2C3),
nl
T6,^,u^'(u) = / e1(*rt(l-tNt^Ju(et)dt , H(,i+u)>-l , RF)>-1 . C)
10.2. Povrer Series Expansions
T6,n,ufe'u>) " r(u+i) Jn г^ц+б+Пгкн-г") ak,u(e^)
i(g/2)V у rFH-l)r(^H-2kH-2) ( }
r("+i) j^o r(n+e+wefc«) k,^p' ;
A)
234
10.2
where
SCHWARZ FUNCTIONS AND GENERALIZATIONS
235
к 2к
vk+1, /„ч2к+1
\»(э'в) s Ч»" еA'Же? Л^-й;^/^)
(з)
For the development of (l)-C) and other related results, see Luke, Constant
and Ruhlman A956) and Luke and Fettis A958). Jordan A955) gives the
equivalent of (l)-C) except for some typographical errors.
We have the following contiguous relations
ЭЪ,
a, = 'k'" • Ъ = bB"^1>v
(*)
Bk+l)Bk+2)Bk+2u+l)Bk+2u+2)ak+1 v = - | cu2Dk+2u+l)Dk+2u+3)
(р2-Ш2JDк-*-2ин-5)
Dk+2u-l) ^-1*" '
Bк+2)Bк+3)Bк+21Л-2)Bк+21Л-3)Ък+1^ = - L2Dk+2u+3 )Dk+2u+5 )
+ (P2-,.,2) Г BkH-l)Bkf2^l)Dk^2^5) x Bk+2 )Bk+2u+2 I1 Ък „
'I Dk+2u+l) JJ ^'и
E)
(p2-UJJDk-t-2i;-f5)
Dk+2u+l) k-l,u
F)

236
IHTESIALS OF HESSEL FUNCTIONS
10.2
(k+u+l)Bk+2u+l)(e2/^)ak x = (u+l)[Dk+4u+l)(e2/^)-2U]akjl;
2/ 2,
+ 2u(u+l)(l-eV*'i)ak^.1 .
(к+У+1)Bк+2и+3)(р2/ш2)ЪкI,+1 = (и+1)[Dк+4У+3)(е2/ш2)-2и]ък>и
(V)
,2 /...2 -
+ 2и(и+1)A-ес1/ш?;)Ък^_1
(8)
If и = p = О , then
'....ОС0'") = %SS^ Л(^6+2;1Щ)
16,ц,<^
and so
T. ...„№,«> -a*»Z
Г(ц+6+2)
= Г(ц+1)ГF+1) е1ш F F+иц+6+2;-1«>) ,
Г(м,+ 6+2) -1- -1-
> ^ (-)k(p/2)"+2krFtl)r(f^2k+l) (ш)
(9)
Here
l6,n,»"""" " * ^ к.'Г(и+к+1)Г(г+2к) k'r^
Cj,. r(u))S =kjr = 1F1F+ljr+2k;-ii»)) , r = ,j,+ 6-HM-2 ,
A0)
A1)
and
ilu(r+2k-6-l)ckjr+1 = -(r+2k)(r+2k-l-iiu)ck^r+(r+2k)(r+2k-l)ckjI,_1 ,
ck,r+2 = ck+l,r > 5 fixed • A2)
10.2
SCHWARZ FOKCTIONS AND GENERALIZATIONS
237
In a number of applications 6
currence relations
= 0 . In this event, we have the re-
jw°k,r+l = Bk+r)(l-cj^r) >
*2ck+l,r = Bk+r)Bk+r+l)(l-ck^r)-iiuBk+r+l)
re"t%(xt)dt = z^leiz X (-)k(XZ/2)V+2k
Jo k=0 к.'(и+У+2к+1)Г(и+к+1)
X 1F1(l;M,+u+2k+2;-iz)
- (-)kz2k
Jc(X,z) = z Z )-> z. gF^-k,-^;!^2)
k=Q Bk+l).
2 ? (-)Vk
jB(x,z) = z^ 21 )f^fTr2Fi(-k,-Mf;i;^)
k=o ^K+?i>-
iz.
Je(O,z) = i(l-e") .
Je(l,z) = zeiz[jo(z)-ij1(z)]
A3)
A4)
A5)
Je(x,z). z -^F feltt2kdt
k=0 k! Jo
- ieiZ Z (?) U'iZ- T ^~ U/2Jk • A6)
k=0 L r=0 r- J
A7)
A8)
A9)
B0)

238
IlWEffiAIS OF EESSEL FUNCTIONS
10.3
(n/2)Ye(\,z) = ln(Xz/2)Je(X,z)+ielz[^(Xz/2)Jo(Xz)-(n/2)Yo(Xz)]
+ ielz ? (X/2JkBkk) [tBkfl)-*(Wl)] U
k=0 L
2k-l
-iz_ V (-lz)r
r=0
r! j
+ie'
iz
k=o Kky L r=o • -i
(n/2)Ye(l,z) = (nz/2)e1Z[Yo(z)-iY1(z)]-i . B2)
lim [(n/2)Ye(X,z)-lji(Xz)Je(X,z)] = i [(у-Зл 2 )(l-elz)
X->0
+ Ci(z) - (y+ln z) + iSi(z)] . B3)
10.3. Expansions In Series of Bessel Functions
/oV\Ut)at - zeiz Z[^;f2l:(fk[wk(z)-lWk41(z)]
,(i-x2r^[i^MiH|, Ii-x21<i ,
- ±al
A)
where Wk(z) now stands for any of the Bessel functions of the first three
kinds only and cr depends on the particular Bessel function under consid-
consideration as tatmlated Ъе1о>г.
10.3
SCHWARZ FUNCTIONS AND GENERAUZATIONS
239
Wk(z)
Jk(z)
Yk(z)
42)(Z)
0
2/tt
21/tt
-21/tt
B)
If Wk(z) = Jk(z) , the restriction on X can Ъе removed. The аЪоуе
follows from 1.4.2A4), 5.2C,6) and 1.4.1G).
/„
e"« C0S eJu(t)dt = 2e-iZ cos 9 Z l4<°« в^1+и(«) ,
0 k=0
where
R(u)>-1
Uk(cos 9) = sin(kH)e
k sin 9
is the СЬеЪузЬеу polynomial of the second kind, see Erdelyi et al.
A953, Vol. 2, Ch. 10).
C)
D)
I
¦Jv(t)dt = 2e'lz X i (k+l)Jk+1+u(z) , R(u)>-1 .
0 k=0
E)
f e-"ju(t)dt = e-±z\zJv(Z)+±ZJv+1(z)-2v ? ikJk+u+1(z)l . F
J0 L k=0 J

240
INTECSALS OF EESSEL FUNCTIONS
10.3
10.4
SCHWARZ FUNCTIONS AND GEUERAUZATIONS
241
e-t cos eIv(t)dt = 2e
-z cos 9 ^_ Uk(cos 9)lk+1+u(z) , R(u)>-1 . G)
k=0
r\-t cos e^j^ + (Y+ln z/2) Te-t cos 9Iq
On "n
(t)dt
I
z _
^„(tfet = 2e*z Z (t)VDlk+i+u(O •
0 k=0
(8)
= 2e
-z cos 8
" / /оч2т+к+1
Z ^(=0. e) Z^tol?, [K^+2)-t(i)]
[Ze?tlu(t)dt = e*zfzlu(z) t zlu+1(z)-2v ? (^Ч^^2)! • (9) i
J0 L k=0 -I
nZ Г2
n/2 / e"" oos 9Y (t)dt-(y+ln z/2) / e"" oos 9JQ(t)dt
Jo Jo
,m, /„ ч2т+к+1
k=0
m=0
_2e-iz cos 9 j- ^(м e) [ф(к+1).фA)] Jk+1(z)
k=l
.2e-izcose Ji4(-e)(Z/2)]
,k+l oo / „ /rs \m
(z/2)mJm(z)
k=0
k' m=0 m.'(m+k+lJ
A0)
2e-z cos 9 ^ Uk(cQS e) ^(k+l).t(l)J Ik+i(z)
k=l
+ 2e
-z cos 9 ? VCOS 9)(z/2)k+1 Л (-fC^^Jz)
m
k=0
k.'
m=0 m.1 (m+k+1J
A1)
Equations C)-(9) follow from the material in 12.2. To prove A0), use
C), 1.4.1A,5) and 1.4.2G,11). The equations (ll) arise in a similar
fashion.
10.4. Representation in Series of Circular Functions
/
'- .+ m+1 I r r ,
e^t'VU^dt = ^— Y (=°s 2n ^k) l{Aak(z)+1*mb(z)\
k=0
{Cmk(z)+1IW(z)}] + Rq,2n(z) '
A)
/ ^Vj^^XtJdt =^T (einBn+l)Xk)[.{Ank(z)+iE^(z)}
Jo dq- k=o L L
+ {cnk(z)+inmk(z)}]+ R^2n+1(z) , B)

242
INTEGRALS OF BESSEL FUNCTIONS
10.4
10.5
SCHWARZ FUNCTIONS AND GENERALIZATIONS
243
where
Amk(z)+i^Dk(z) = z
-m-1 I Z.laktim.
/iz
tludt
Cnk(z)+inmk(z) = z-m-1 Г
e^Vdt ,
C)
D)
E)
ak = l+iDjj , Ък = l-afe , (% = X oosBk+l)Tr/4q. ,
T!n|(Xz/2L<1-nzm+1|eXy
|R (Z)|S_?L _ , Xz = Xx+iXy , Xy :> 0 ,
i q.,n4 " Dq.-n).'Dq.-n+m+l)
110 = 2 , 11n = 1 for n a 1 • F)
The following relations are useful for the computation of Aj^z) ,
R (z) , etc. It is sufficient to omit the к subscript and write
Then
V») - ^(z) = z-1 (V^t-at
Ao(z) = sin e/e , bo(z) = (i-cos e)/e , e = az ,
G)
z) + mBm_1(z) = sin 9 ,
z) - mV^z) = - cos 9 , m>
0
(8)
0)
A0)
The above statements generalize those of Lake and Fettis A956) who
note that the results are useful to solve an integro-dlfferential equation
arising in panel flutter. The latter source contains some typographical
errors. There, in Kq.. E), for zm~^ read zm+;l- . In each line of F),
for z1" read z11 . Also, in Eqs. (8)-(lO), к should Ъе replaced
Ъу m .
10.5. Asymptotic Expansions for Large z
16
/ ePtt^(Xt)dt~ ("Mg)a z^e-(X-P^z X a.
Jz CX-p) k=0
-k
|z|—>oo , 6 real , 161 <; тт/2 , X>0 , X>R(p) ,
where the path of integration is defined as in 2.1(9) and
ak =
к.' BХ )k
Sk(S) ,
SjjCe) = 3F2(-k,M,-k+i,i;u-k+i,-u-k+i;e) , e = Щ- ,
X-p
о fni - l t 9(кн-1)(и,-к-^) , ,
/;
16
e4%(t)dt~ - i^Zfi! zVi^»,^», 4-^*-^-i/2O ,
|z|—>« , 6 real , 161 <; тт/2 , R(jj,) <-¦§¦
A)
B)
C)
D)
E)

244
INTEGRALS OF BESSEL FUNCTIONS
Jx ^ Wx^ (X+p)
Xj 2_ l-J a2]5X +i 2, С") a2k+lx f
I k=O k=O J
0<x-»=° , X>0 , p real , X+p ^ О , R(n)<i ,
where ak is given Ъу B)-D) with 9 = SL. .
¦ к -2k . ^- , .к -2k-ll
Xi I (") ^k* -1 Z (") a2k+lx j '
k=0
k=0
0<x-».co , \>o , p real , \ ? p , R(n)<i >
where ak is given Ъу B).
/VVhJ1^ )dt ~ {2™)\»е±[2*^-^
Lk=O
k=0
10.5
F)
G)
X(Z (-)ka2kx'2k+1 i- (-)ка2к+1х'2к"Н^ °<x-^-, R(^)<4 , (8)
where ak is given Ъу B)-D) with X = 6 = 1
10.5
SCHWARZ FUNCTIONS AND GEHKRAUZATIONS
245
X 3Fi(i-u^^-i-M.;i-|j.;-i/2ix) , o<x-».co , r(m,)<4 . (9)
«/x ° U+^ -^П^ [ kt0 [4i(l+X)x]kr=0^rA8X ; J
0<x->co , X>o
A0)
/'
"x
e«B<a)(XtLt
VTixi Lk=O [4i(l-X)x]k r=0 8X J
0<X->co } \>0
A1)
/;
>t%(xt)dt^^-(p:x?' j(-Lz-k
BttXzJ(p-\) k=0
|z|—>-co , R(p-X)> 0 , |arg z|<tt/2 ,
A2)
where a^ is given Ъу B).
/;
e-4%(t)dt.
,H+1
BnXzJ(n+i)
3F1(i-i»,i+i»,-i-M.ji-M,;l/2z) ,
|z| —*.» , R(M,)<-i .
A3)

246
INTEGRALS OF BESSEL FUNCTIONS
10.6
10.6
SCHWARZ FUNCTIONS AND GENERALIZATIONS
247
10.6. Infinite Integrals
See also 2.6 and 4.5.
fe^t^T (Xt)dt = (V^V^+D 2FYHiZli , ^1 ;v+l;-X2/p2)
Jq р^+1Г(и+1) V 2 2 У
R(p,+u)>-l , R(p)>|l(X)| , |X/p|<l
A)
2 2 -
R(n+v)>-l , R(p)> |I(X)| , |X/r|<l , r = (p+X f ¦ B)
rC°e-ptt%(Xt)dt - Г(^+1)г-»-\У(Ф) > H(m.+i;)>-1 , H(p)>|l(X)| , C)
where
A*
^(Z) = F^I)(S) 2^-^^-^-^ ^
is the Legendre function of the first kind, see Erdelyi et al. A953,
Vol. I, Ch. 3). Here, z lies in the complex plane cut along the real axis
from -1 to 1. If z = x , -1<х<1 , replace E+l\zVl Ъу /l+x^11 .
Vz-1/ \l-xS
rVptti;jl,(\t)dt = (ax)"^)^-2^1 , h(i>)> 4 , r(p)> |i(\)| •
Jo
E)
f e-pttU+1Jv(Xt)dt = pBX)UC/2)l,r-2l;-3 , R(u)>-1 , R(p)>|l(X)| . (б)
^0
/ e-p\(Xt)dt = r-VCp+r)-" , R(u)>-1 , R(p)>|l(X)| . (?)
J0
ЛСО
/ e-ptt-1Ju(Xt)dt = iTV(pfr)7 , R(u)>0 , R(p)> |I(X)| .
(8)
/>t4(t),t = е^^Г^ВгС-Ь) , ^„j^, f ЕЫ<4 . (9)
J0 r(iJ^+1r(u-ti)
fe-4%(t)dt = Г(^1)ГD-р) f !,(,„)>.! , R(li)<.l
0 r(iJ^r(u-^)
A0)
ПС»
/ e-ptY0(Xt)dt = - i- arc sinh p/X
Jn nr
= ^m(^?), R(p)>|i(x)|
(ii)
J e-PV^Xt^t = r-1-2i(nr)-1arc sinh(p/X) , H(p)> |l(X)| . A2)
In C)-(8) and A1)-@2), r is defined as in B).

248
INTEGRALS OF EESSEL FUNCTIONS
10.6
f e-Ffct\(Xt)dt
Jo
r(u,-m-l)sin рл ov
s^sin^+iOn
^(р/в)
r(m,+u)>-1 , R(p+X)>0 , s = (p2-X2J ,
A3)
where
«!<o
-fe? [*•>-№;•<¦>]
A4)
is the Legendre function of the second kind, see Erdelyi et al. A953,
Vol. I, Ch. 3), and z is in the complex plane cut along the real axis from
-1 to 1. See the remark after D).
Ге-РЧ*-Ч+±(>Л)dt = (n/2X)* Г(^)Г(^+1) р-,(р/х} ?
Jo sM/
R(n+v)>-l , R(n-v)>0 , R(p+X)>0 .
A5)
e±i;t%(t)dt , see 4.5D-5).
-t cosh 9,,
**№ - (sin^ 9 - l*(»)l<l,R(eosh9)>-l • (IS)
/* e"PtKo(Xt)dt = s
Jo
= s arc sinh s/X
Г11л(^),В(^)>0 .
A7)
10.6
SCHWARZ FUHCTIONS AND GENERALIZATIONS
249
/ (cos pt)K0(t)dt = ^n(l+p^)-2 , |l(p)|<l •
J n
A8)
/;
(sin pt)K^(t)dt = (l+p2)arc sinh p , |l(p)|<l . A9)
Some special cases of the a"bove results are found in sections 2.6 and
4.5. Further special cases are given Ъе1сяг.
Throughout the remainder of this section, s = (p -X2)^ , u = (X2-p2)?
Л со
/ t" Jy(Xt)sin pt dt = v~ sinfv arc sin(p/X)l , 0 s p s X ,
= vU(p+s)' sin un/2 , p 2: X>0 , R(u)>-1 . B0)
/CO
t Jv(Xt)cos pt dt = v~ cos ly arc sin(p/x)] , 0 s p s X ,
= vXU(p+s)' cos un/2 , p ;> X>0 , R(u)>0 . B1)
/ Ju(Xt)sin pt dt = u" sinfv arc sin(p/XI , 0 ? p<X ,
^0
= s-1Xu(p+s)-u cos un/2 , p>X>0 , R(v)>-2 . B2)
„CO
/ Jv(\t)cos pt dt = u cos[u arc sin(p/X)] , 0 <. p <. X ,
= -sXU(p+s)' sin un/2 , p>X> 0 , R(v)> -1 . B3)

250
IHTEGRALS OF BESSEL FUNCTIONS
10.6
/;
eiptJo(Xt)dt = u , 0 ? p<X ,
= is , 0<\<p
B4)
/ eiptYo(Xt)dt = 2i^- arc sin p/X , 0 s p<X ,
^0 П
= .s-l + 2i?ilnCPliV 0<X<p
n V X J
f
Jo
2u
-1
eiPtHA)(Xt)dt = ^— arc cos p/X , 0 ? p<X ,
-1
f
J0
= -2B_iaCEz?V 0<X<p
n V X У
e^V^ptJdt = B/ттр) , p>0 .
/;
Equations B0)-B8) are special cases of the discontinuous №еЪег-
Schafheitlin integral, see 13.4.2.
B5)
B6)
B7)
iptH02)(Xt)<lt = 2U'1 - ^— arc cos p/X , 0 s p<X ,
= 2is + 2?± Дл(Е=Г\ , 0<X<p .
n V 1 /
B8)
10.7
SCHWARZ FUNCTIONS AND GENERAUZATIONS
251
10.7. Description of Mathematical Ta"bles
Crowley A954): / eitJQ(Xt)dt , X = 0@.1I0 , x = 0@.02I0 , Xx S 15 ,
«'O
5d.
Garrick and RuMnow A946): w~l I eltJ0(Mt )dt , M'1 = 0.2@.1H.9 ,
</o
5 ranges from 0.02 to 20 , 5d.
oL
Hoyt A947): Q*(R) = l-Q(R) , Q(R) = (l-Ъ2) / е'Х10(ЪХ)й\, (l-b2)L = R2 ,
J0
Ъ = 0,0.3@.1H.9,0.95,1; R = 0.2@.2H.8,1.6,2.0 . ТаЪ1еэ are
accurate to аЪо^ 3d. See the author's remarks and compare with
Rice's ta"bles described below for ("b,R) = @.6,0.8), @.8,0.6).
л1
Huckel A956): / tXe'lS5tJ0(M'1ujt )dt , («(M2-!) = 2км2 , X = 0(l)ll ,
^0
M = 1.2@.1I.6@.2J.0@.5L.0,5.0 , к = 0@.005 H.15@.01H.2
@.025H.35@.05I@.01J , 6d.
Luke A950b): / extJ0(Xt)dt , X = 0@.l)l , x = 0@.02J@.1I0 ,
'0
5d at least.
/>x
Mathematics Center, Amsterdam, Computation Department A951): / eictJQ(t)dt,
^0
/ elctY (t)dt , с = 0.3@.1H.8 , x = 0@.1N.1 , 8d.
u n

252
INTEGRALS OF HESSEL FUNCTIONS
10.7
nx
Rice A948): / e^Ost )dt , к = 0@.2I , x = 0@.2 M@.4 )9(l)l5,со , 4d.
^0
Also к = 0.9 , x = 10AI5,со , 4d; к = 0.86,0.9,0.96,1.0 , x = 15A)
20,co , 4d.
„x
Schvarz A944): / el1;J0(H )dt , X = 0@.l)l , x = o(O.O2J , 6d;
^0
x
x = 0@.1M , 8d. / eitY0(Xt)dt , X = 0@. l)l , x = 0@.02J@.1M ,
Jo
6d. Auxiliary ta"bles axe also provided to facilitate interpolation
of the latter integral near the origin. For errata, see Math. Ta"bles
Aids Сотр. 4, p. 100.
Zartarian and Voss A953): Let M = iu/p and v = 0 in 10.2A-3).
a^M) = (-)n«4,0 > W*> = (-f+V2n\o , n = 0AN ,
M = 5/4, 10/7, 3/2, 5/3, 2, 5/2, 9d.
See also 2.9 and 3.12.
CHAPTER XI
INTEGRALS INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AND STRUVE FUNCTIONS
11.1. A General Theorem for the Evaluation of Indefinite Integrals
Let
Л, (z )
*%(z)
X a^z) ±— = Fx(z) , Y. 4(z) ПГ~ = ^(Z) ' Ъ2B) = *гМ > C1)
k=0 dz^ k=0 dz^
W(z) = f^zjf^z) - f{(z)f2(z)
B)
Then
a2(z)W'(z)+a1(z)w(z) = fx(z )F2(z )-f2(z )F1(z )+ [aQ(z )-Ъ0(г )} ^(г)^(г)
+ {а1B)-Ъ1B)} fi(z)f^(z) ,
and
.* ff (t)F (t)-fp(t)F (t)]
W(z)v(z)=J X 2 aJA 1 [v(t?dt
a2(t)
rz fa (t)-b (t)l
h/(-^Fr]^(t)f2(t)v(t)dt
rzfa (t)-b (t)l z a (t)
h/ tJ-^fe-]fi(t)^(t)v(t)dt' ^Нг) -J 4o
dt
C)
D)
?53

254
INTEGRALS OF EESSEL FUNCTIONS
11.2
This is a generalization of 3.1B-3), 4.6(l), and results of Chapter V, and
includes statements previously given Ъу many authors. See Coulmy A954),
Horton A950a), Luke (l950a,1953), MacLachlan A955), Maclachlan and Meyers
A936), Maximon A955), Maximon and Morgan A955), Petiau A955), Picht
A949), Schubert A953), Straubel A941,1942), and Watson A945).
Except where noted, proof of virtually all the expressions in sections
11.2-11.4 follows from D) and the difference-differential properties of
Bessel functions and Struve functions. For the most part, the integrals of
this chapter are indefinite, though some definite integrals are given in
11.2C0-44). For definite integrals involving the product of two or more
Bessel functions, see Chapter XIII.
11.2. Integrals Involving the Product of Two Bessel Functions
In the following equations Cv(z) and Du(z) are cylinder functions,
see 1.4.1(8).
J {(k2-X2 )t- (f-v2 )t"X} C^kt )DuUt )dt
f tCu(kt)Du(Xt)dt = —^— fkCu+1(kz)D1)Uz)-XC1,(kz)D1,+1(Xz))
k2-*2
|cu(kz) SL DB(Az)-DB(Az) ^L Cu(k)) . B)
pz
J tCu(kt)Du(kt)dt
= ^ z2 BCu(kz)Du(kz)-C1,_1(kz)D1,+1(kz)-C1,+1(kz)D1,.1(kz)} . C)
11.2 INTEGRAIS INVOLVING PRODUCTS OF EESSEL FUNCTIONS
AND STRUVE FUNCTIONS
J tC^(kt)dt = i z2 |cy(kz)-Cu_1(kz)Cu+1(kz)|
= *z2[(^-^)^(kzK2(kz)| .
z
J tCMi(kt)Du(kt)dt = - -^_ |cMi+1(kz)Du(kz)-CMi(kz)Du+1(kz)]
\ir-v
С (kz)D (kz)
+ j±—_j;
^i+U
J tCu(kt)Du(kt)dt = g |cu+1(kz) ^L - Cu(kz) aDp^(kZ) j
Cu(kz)Du(kz)
2y
(p+H+y)/ tPCMi(t)Du(t)dt+(p-|i-u-2)J tp-1C11+1(t)Du+1(t)dt
= zp{c^z)Du(z)+C^+1(z)Du+1(z)} .
J Jy(t)Jy+1(t)dt = ^ Jy+k+l(z) •
k=0
255
D)
E)
F)
G)
/*"ii"';wt)D«+i(t)dt= g(^I)b(z)Bv(z)v(z)vi(z)} • (e)
/ t^+\(t)Du(t)dt = ^^ {c^(z)Du(z)+C^+1(z)Du+1(z)} . (9)
A0)

256
INTEGRALS OF EESSEL FUNCTIONS
11.2
f jn(t)jn+1(t)dt = i fi-jf(z)} -i4)= z 4(z) > n>° • (id
JO l -J k=l k=n+l
nz z
C2)
n-1
C1AB)DI,(z)+C(A+n(z)Du+n(z) + 2 21 Vk(z)Du+k(z) .
k=l
A3)
nz °°
2uJ f^CtJdt = jf,(z) + 2 Z Ju+k(z) • A4)
k=l
n-1
k=l
/Z П oo
f^OOdt = 1+J^(z)+J^(z) - 2 Z 4(z) = Jn(z) + 2 Z 4(z) *
0 k=0 k=n+l
n>0
A6)
(^2)J t^+2(^(t)dt = („+1) {y2-J(^lJ]jVc^(t)dt
+ | z^+1^{zCi;(zL(li+l)Cu(z)|2+ {z2-y24(^lJ}c^(z)l . A7)
-2nJ t-^CtJlJnCtJdt = C0(z)D0(z)+Cn(z)Dn(z) + 2 Z Ck(z)Dk(z) , n>0.(l5) |
11.2 INTEGRALS INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AND STRUVE FUNCTIONS
257
2By+l)J1Vy+1C^(t)dt = z2y+2[c^B)+C^+1(z)] ,,^ . Aв)
2A-2, )|Zt-2u+1C2(t)dt = zU+2 [^(zJ+C^.^z)] , i; ^ | . A9)
2z(v2-i)J t(^(t)dt = F{zCIJ(z)+icl7(z)}2+ {z2-y2+i}(^(z)l ,
v f±\ .
B0)
/'
t'2Jn(t)Jm(t)dt =
-1
z'xJ (z)j (z) J . (z)J (z)+J (z)j Лг)
nv ' mv ' n-14 ' mv ' nv ' m-lv '
(m+n+l)
(m+nJ-l
2zJn-l(z)Jm-l(Z) A zJn-2(z)Jm(z)
[(m+nJ-l] [(m-nJ-l] (m+n+l)[m2-(n-lJJ
zJn(z)Jm-2(z)
(m+n+l) [n2-(m-lJ]
, |n-m| f 1
B1)
J t- Jn-i(t)Jn(t)dt = -i z Jn-i(z)Jn(z)
Г n~2 1
¦^4) + г1 4B)-ИLB)-И^г) -"МД. B2)

258
INTEGRALS OF BESSEL FUNCTIONS
J t-aJn(t)Jm(t)dt
,-1
±_ ZJnB )Jm(Z > тг-т[Jn_i(z )Jn(z )+Jn(Z)Jm.x(Z)]
(m+n+2)
(m+n)
+ L-^~ [jn-2(z)Jm(O+2Jn.l(z)Jm-l(z)+Jn(z)Jm-2(z)]
(m+n-ir-1
, z | Jn-5(ZV2) ,
Snr:4)_,^jn-2(z)jm-i(z)
(т+п)[ m2.(n.2J (n-m)[m2-(n-2J]
(n-a44) , u , x , Jn(Z>WZ)l
+ <«UoLfi J-(z)Jm-2(z) + -??zr\
n^m, |n±ml^2
Jzt-34(t)dt =. _^_TL-24(Z)+z-iJn.i(z)jn(z)
1 Г 2 П'2 2 2
+ i z CJn-3(Z^n(z)-Jn-2(z)Jn-l(z))j I ' n ^ °'1 •
11.2
B3)
B4)
fZt-34(t)dt = - -?l_(i^4(z)+ 21 k24(z)l , n ^ 0,1 . B5)
J n(nd-l) L ^ k=l J
11.2 INTEGRALS INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AND STRUVE FUNCTIONS
259
/ t-3Jn.2(t)jn(t)dt= - ?!jn_2(Z)jn(z)- ?i Jn.i(z){jn.2(z)-Jn(z)}
- 8n(n-l)(n-2) |Jo(z) + 2 Z 4(z) - (ПГ) Jn-2(z)
- (n-2L_1(zL(n-l)(n-2)j^(z)l , n^ 0,1,2 . B6)
We have noted typographical errors in the forms for B4)-B6) given Ъу
Petiau A953); likewise, for the form of B4) given Ъу Picht A949). For
further reduction formulae and results of the type B1)-B6) relating to
Я7,
J Л (t)Ju(t)dt , see Coulmy A954), Picht A949), Schubert A953) and
Straubel A941, 1942).
Equations B7)-B9) come from 1.4.1A7,18,20).
X
0 J^at)Jv(^)dt - г(ц+1)Г(и+1)
- l-ffaz) F(-k' -^k; и+1;Ъ /а }
X Z „,, ч/ ,,ч / R(n+v+p)>-l
k=0
k!(M,+y+p+2k+l)(M,+l)k
B7)
nz /l vM,+U p+1
(|i+u+p+l)r(|i+l)r(u+l)
X 3^4 ^
,+U+l U.+U+2 U,+U+D+l U+U+D+3
2\
R(|l+U+p)> -1
B8)

260
INTEGRALS PF BESSEL FUNCTIONS
11.2
pz fl ,2u p+1
/ tVtOi^tOdt = tsLLJ
J0 Bu+p+l) |r(u+l))'
w t, /21M-P+1 . v+1 u+2 2y+p+5 z4\ „, , w /OQ^
Let yr Ъе the r-th positive zero of J0(x) , i.e., J0(vr) = 0 . The
equations C0)-D4) below axe essentially due to Butler and Pohlhausen A954).
See also 3.10.
I
JO(Yrt)Jo(Yst)dt = 2Jl(Ys) Z
0 k=0
- (-)kj2fcfl^8)PkB^-D
2k+l
ш = Yr/Ys s x >
where
C0)
PkBou2-1) = (-)\Fx(-b,b+lil;m2)
C1)
is the Legendre polynomial.
Г1 2
J J0(Yrt)dt = 2J1(Y^)
k=0
(-Lk+l(Vr)
2k+l
C2)
11.2 IMTEffiALS INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AND STRUVE FUNCTIONS
261
/;
j1(Yrt)j0(Yst)dt
-1_ . , f (")k+1J2k(Ys)
= ® ji(ys) 2. ¦
k=l
2k
|ркBш2-1)+Рк.1BиJ-1)|
= <«Jl(Ys) Z J2k+2(Ys) 2Fl(-k^k+2^2^ ) •
k=0
C3)
Here u) = Yr/Ys fi 1 • If Yr/Ys - ^ > use tlie relation
Ys / J1(Yst)J0(Yrt)dt + Yr / Ji(Yrt)Jo(Yst)dt = x •
J 0 J0
C4)
In F1) of the cited reference, which corresponds to our C3), the sign
of &2п_1_ should Ъе positive.
\(yTt)Jo(yTt),t = Jx(yT) f;.^ WYr>_ l
k=0
k+1
2Yr
C5)
J j^tjj^tjdt = лх(У8)? "JB?1)У5){Рк+1Bш2'1)+2РкBш2)
Johi(yJ
Рк.хBцJ-1)}= uuJx(Ys) Z 2k^1S 5Fg(-k,kH-l,5/2;l/2,2;ou2) . C6)
J k=0 ^k -1

262
INTEGRALS OF BESSEL FUNCTIONS
11.2
(Vl(Yrt)dt. Jl(Yr) z (")kyk;r(Vr) - -?b*> /^(Yrt^ ,
= 1 ; ek = 2 , k> О
C7)
f V^(Yrt)dt = i- fjf (Yr)- ^=^ f tm-2^(Yrt)dt} , m ? 0 . C8)
f tmj2(Yrt)dt = 1 fjf(Yr)-(m+l) / tmJ2(Yrt)dtj ,
<J0 №-!-) I J0 J
pi
/ t jf(Yrt)dt = \ jf(Yr) .
Г tmJ0(Yrt)J0(Yst)dt =-iH^L[2YrYg [^(Y^Ye)
- (m-3)y tm-2J1(Yrt)J1(Yst)dt]
m / 1 . C9)
D0)
- (m-l)(y^y2e) J tm-2Jo(Yrt)Jo(Yst)dtl .
D1)
11.2 INTEGRALS INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AND STRUVE FUNCTIONS
263
и n
tmJ1(Yrt)J1(Yst)dt = -(l^_i^^2
(Yr+Ys) [ji(Yr)Jl(Ye)
- (m-3) f tm-2J1(Yrt)J1(Yst)dt]
- 2(m-l)YrYs J tm-2Jo(Yrt)Jo(Yst)dt I
/ Л0(Уг*^1(УВ*L* = "рЦ Vr rjl(Yr)Jl(Ys)
J0 Y?-Y| L L
Г1
- (m-2) / ^^(^^(YgtjdtI
Jo J
/Vvo(
- mYs / t- "J0(Yrt)J0(Yst)dt
0
Yr / tmJ1(Yrt)J0(Yst)dt+Ys Г tmJ0(Yrt)J1(Yst)dt
 J 0
Л1
= m / tm-1Jo(Yrt)Jo(Yst)dt .
J0
D2)
D3)
D4)
The following are tabulated by Butler and Pohlhausen A954). All
entries are to 5D.

264
л ~
INTEGRALS OF EESSEL FUNCTIONS
11.3
Ли(у^^„(уг*L* J ц - 0,1 ; в - 0,1 j г - 1AI0 ; m = O(l)lO
Г1
/ ^(Ypt^^Yst)^ i|,= 0,lit=O,li r,s = 1AM ; m = 0AM
Jo
11.3. Integrals Involving the Product of a Bessel Function and a
Struve Function
tJu(kt)Hy(jet)dt = z |JeJu(kz)Hl;.1Uz)-kJi,.1(kz)Hl;(Xz)|
^u+l pkz
-ЩпЩI tJ^(t)dt •
(^2-v2) J t^CtJJ^tJut = z (ju(z)H^.1(z)-Ju_1(z)H^(z)J
z
i Г t%(t)dt-(n-u)H^(z)Ju(z) .
-1r(i)r(u-ti)J
2^-1Г(|)Г(^4)
A)
B)
(PV")/ tP-^^t^^tJdt + (P-^-y-2)J tp"XH^+1(t)Ju+1(t)dt
= zP
{H,(z)Ju(z)+H^+1(z)Ju+1(z)) - HI7iX^M/\^Ju(t)dt . C)
11.3 INTEGRALS INVOLVING PRODUCTS OF EESSEL FUNCTIONS
AND STRUVE FUNCTIONS
265
„z
2(H+I>+1)J t"|1"I;-:lB11+1(t)J1,+1(t)dt =
2-M,
ГA/2)Г(м,+3/2)
J t-uju(t)dt
^^K(z)jv(z)+vi(z)j^iB)} •
2b+v+l)J ^+^\(t)Jv(t)it = -
2-1*
Г (l/2X|i+3/2)
./>+y+2Ju(t)dt
+ 2^и+2{^B)^B)+Н11+1(О^+1(^} •
D)
E)
у Г t'\(t)Ju(t)dt - (v+l)J tHu+1(t)Ju+1(t)dt
¦ (i) {^(^„(^w^io} - 2^tl/l,T(v.s/JZ*v*№ ¦ ^
z n-1 /\kJ (t)dt
J v=n 2krfl/2 ^rfk+3/:
k=0 2кГA/2)г(к+3/2)
г 1 п"х
4H0(z)J0(z)H-Hn(z)Jn(z)[ - 2 Z Hk(z)Jk(z) , n>0 . G)
L J k=l
z z
J tH1(t)J1(t)dt = (!/tt)J Jo(t)dt-(i)|Ho(z)Jo(z)+H1(z)J1(z)J . (8)

266
INTEGRALS OF BESSEL FUNCTIONS
11.4
11.4. Integrals Involving the Product of Two Struve Functions
(k2-/)J tHv(kt)HvUt)dt = z [xHu(kz)Hv.1(*z)-kH1,_1(kz)H1,Uz)j
r(t)r(v+t)
[(к/2/Г1 J^tX(t)dt-(V2k)U+1/ \%(t)dt} . A)
z »
(H2-)J tHMi(t)H1,(t)dt = z JH^iOH^zHl^ztev.!^)!
1-1» nz
л1"^ PZ
rD)r(»+i)
J t\(t)dt - r(i2)r(^i)J ^(*)«-(^-»)^(а)н»(г) • B)
In Horton A950a), the last term of B) is missing.
(р+ц+и)/ tpH^(t)Hv(t)dt + (p-n-u-2)Jtp~1Hti+1(t)Hv+1(t)dt
= zp (H^^tz)^^)^^)} - r(l/22)r(,+5/2)/ t^+PHu(t)dt
r(l/2)r(v+3/2)
•JZtu+\(t)dt . (з)
11.4 INTEB1AI? INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AND STRUVE FUNCTIONS
267
2(^fl)J t^-U-1^+1(t)Hu+1(t)dt
D-M.
r(V2)r(ii+3/2).
JbJT)! *"Ч(*L*
,-u
ГB/2)ГA»+3/2)
/Zt~4(
t)dt
- Z"^-
" {V2»»(z4+i(zWo} •
D)
= ^+U+2 {^(z^Cz^^z^^z)}
«j1 t"^(t)dt - (v+l)J t"^+1(t)dt
= i{H^(z)+H^+1(z)}-
ГA/2)Г(и+3/2)
¦I t Hy(t)dt
F)
2nJ t"^(t)dt = - [^(z)f^(z) J
n-1
-- - о П f tkH. & )dt
- 2 Z ^@ + 2 X ^T ~ ' n>0
k=l k=0 2кГA/2)Г(к+3/2)
G)

268
INTEGRALS OF BESSEL FUNCTIONS
11.5
11.5. Integrals Deduced from Wronskians
Let gx(z) and g2(z) Ъе independent solutions of the differential
equation
a2(z)E^+ai(z)^+ao(z)g(z) = 0 ,
and put
Then
dz'
W(z) = gl(z)g2(z) - gi(z)g2(z) = ^y ,
A , a known constant, In
{V(z)} =|
,z ax(t)
a2(t)
dt
'/
dt
_g2(z)
V(t)g^(t)
gx(O
A)
B)
C)
'/"
dt
gx(O
V(t)gl(t) S2(z)
'/"
V(t)gl(t)g2(t) ш[б1(г)
= In
(O
E)
The above also follow from 11.1C).
tt J-v(z)
j2(t) " ' 2 sinuiT Jv(z)
PZ dt _
J tJ^(t)
F)
11. 6 IMTEORALS INVOLVING PRODUCTS OF BESSEL FUNCTIONS
AMD STRUVE FUNCTIONS
269
/;
dt tt v
W
'I
dt V
«,,(O
j?(t) 2 vz)'J ti2u(t)": iv(o
G)
/
dt
- .. ]-M
W,,(t)J.i,(t) 2 sin wr Jv(z)
(8)
/'
dt
-^^i.r
= In
I (z)
tJv(t)Yu(t) 2 Jy(z)JJ «„(t^Ct) K,(z)
(9)
Г dt jt JjM]_ . Г
•/ t^(t) " 2 Yu(z) ' J
J,,(O rz dt Iv(z)
tK2(t) Ku(z)
A0)
dt
• HB)(z)
in v v '
J t[H«(t)]^4 ^«(z)
'/¦
dt
iTT^1^)
t[HB)(tf 4 HB)(z)
A1)
/:
dt
= — In
tn^(t)ni2ht) 4 ^^(z)
A2)
11.6. An Integral Involving the Product of Three Bessel Functions
Fettis A957a) studied the integral
I(o,P,y,x) = / tJo(at)Jo(pt)Jo(Yt)dt ,
Jo
A)

270
INTEGRALS OF BESSEL FUNCTIONS
11.6
and gave a short table of 1(уг,ув,у^,1) for all combinations of r,s,t s 3
where the Yr's for г = I*2;3 B^e the first three positive zeros of
Jo« •
CHAPTER XII
MISCELLANEOUS INDEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS
12.1. The Integral j(x,y)
12.1.1. Introduction
We consider
J(x,y) = 1 - e"y I е'\ J2(yt)H dt .
A)
A related function is
l 2 P 3 2
ttVf^p) = 2e"p / te-t IoBpt)dt = 1 - j(x,y) , x = 02 , у = p2 . B)
« n
J(x^y) appears in a wide variety of applied problems. For the subsequent
material 12.1.1 through 12.1.5, we closely follow the work of Goldstein
A953) who should Ъе consulted for further details and an extensive bibliog-
bibliography. See also the work of Goldstein and Murray A959). For additional
references and description of tabular material, see 12.1.7.
We introduce the notation
2(xyJ = 5 , (y/x)* = П , (Ax=J = z
C)
Some elementary properties of (l) follow.
J@,y) = 1 ; J(x,0) = e"
J(x,y) + J(y,x) = 1 + e-(x+y^IoE)
D)
E)
271

272
INTEGRALS OF BESSEL FUNCTIONS
J(x,x) = i [l+e-2XIoBx)]
lim J(x,y) = 1 5 lim J(x^) = °
12.1.2. Partial Differential Equations
aj(x,y) = .e-(x+y)io(O
Эх
Ъу
fjibll + Ml&ll + ti2 ^Ibil = о .
ЭХ2 ЭХ ЪУ
b2ju,y); (y-i+i) m^ii+г2 щь*1=°
Эу2
Эу
S2j(xjy) + -1 9J(x,y) _ ^-2 fjix.yl =
ЭУ2
Эу
Эх2
fjix^ + bJj&Yl + Э?(х^ = о .
ЭхЭу Эх Эу
Consider
92K(r,t) + 1 bK&tl - j: 5K(r;t) в Q ,
2 г Эг s 9t
К(г,О) = 1 for
г<а , К(г,О) = 0 for r>a
12.1.2
F)
G)
A)
B)
C)
D)
E)
F)
G)
12.1.2
MISCELLANEOUS IMDEFIKITE INTEGRALS
INVOLVING BESSEL FUNCTIONS
273
where a and s are constants, and K(r,t) is finite for all values of
r and t . Then
*<*'*> = 1-Кй'й)
(8)
The solution of the telegraphy equation, see Goldstein A953, p. 169),
LC
g^L^ + d^cR) av(x,t) + RSV(x^t) = s2v(x,t) >
dt Эх2
(9)
dtc
where L, C, R ' and S are the self-inductance, capacitance, resistance
and leakage resistance of the line per unit length, respectively, шау Ъе
expressed in terms of J(x,y) . At t = 0 , suppose the end x = 0 is
raised to unit potential (V=l) and is thereafter maintained at that poten-
potential. If V = 0 for x>vt , then for x<vt ,
V(x,t) - e"P* [е^Л^у^^ (l-Jty^)}] , A0)
where
-1
p = |(R/L+S/G) , v2 = (LC)
a = [(R/LJ-(S/CL] , Э = [(R/LJ+(S/CJ] ,
x-l = ia(t+x/v) , yx = &(t-x/v) ,
xg = ig(t+x/v) , y2 = ia(t-x/v)
A1)

274
INTEGRALS OF BESSEL FUNCTIONS
Again, consider
.2_ .2„ч Ъ2В
*< %х2 Ь^} ^ 9z2
where к]_ and Ъз are constants, subject to the conditions:
At t = 0 : s = 0 vhen z = 0 and x +y s a
2 2 2
s = oo vhen z = 0 and x +y < a
s = 0 vhen z ^ 0 ;
со oo оэ
for t г О
^ -co « -oo u -
s dxdydz = Q , a constant
Then
where F(P,p) is given by 12 .1.1B). See Wilson A951).
If
X =
J(rs,t) >
j(rs,t)+e(r-1)(t-s){l-J(s,rt)}
12.1.2
A2)
(И)
f}=|a(k,t)-*, P = i(ML(^V)= , A4)
A5)
12.1.3
MISCELLAMEOUS INDEFINITE INTEGRAIS
INVOLVING EESSEL FUNCTIONS
1-J(t,rs)
j(rs,t)+e(r-1^t-S){l-J(s,rt)}
then
See Opler and Hiester A954).
12.1.3. Power Series Expansions and Expansions in Seri(
of__Bessel Functions
J(x,y) = 1-х X •Цр-2Р1(-к,.к;2Л)-2) , 7)>1
k=0
sk к
J(x,y) = 1-х X ijjL^. Л(-к,-1-к;1;7J) , 7)<1
k=0
J(x,y) = е-(х+У) 21 7)^E), T)<1
k=0
J(x,y) = 1-е-(х+У) X Ti-\(S), 7)>1
k=l
A generalization of D) is given Ъу 12.2B0).
275
A6)
-(tl)t - (fOs - X(l-«) - r»(l-O . A7)
A)
B)
C)
D)

276
INTEGRALS OP BESSEL FUNCTIONS
12.1.4
! xe
J(x,y) ¦= 2 -
-(x+y)
(yx)
ioE) "
-(x+y),
2(y-x) 1^;
а к. ,2k-1 -(х+у) л
у И (у-х) е jlk(x+y)+Ik.i(x+y)f- E)
?b k.I2kBk-l)(x+y)k L
« e?-1I Bx)
j(x+e,x) - i {l+e-2XIoBx)} -e-2x Z ^— ' ^
>- J k=l
vhere Д is the forward difference operator, and
AIn(z) = In+1(z)-In(Z) , Ak+1In(z) = u*Tn^(z)-b\{z)
Г Pv 1 ?v Д екДк11Bх)
J(x,x+e) = i |l+e-2xIoBx)J +ex Z jr •
12.1.4. Laplace Transform and Integral Representations
f e-Pyj(x,y)dy = p-^xp (- ?Ь.Л , R(p)>0 .
Г е-РУ0Г(у,х)ау = p^Jl-Cp+lJ-^xp (- ЖЛ\ , R(p)>-1
n c+i<»
J(x,y) = gij- / P'^xp (py - E_) dp , c> 0
G)
(8)
A)
B)
C)
12.1.4
MISCELLAKEOUS INDEFINITE INTEGRALS
INVOLVING BESSEL FUNCTIONS
Let
Then
we ъ) = e^ Г4 e-^t(l-n^TTt)dt
" J0 t2D-tJ[(l-nJ+nt]
°^; an Jo tiD.t)i[A.,J^t]
277
J(x,y) = е-(х+УI(§,Т|) for T|<1 ,
= 1 + e-(x+y)l(§,T|) for T)>X . D)
« 0 1-2T) cos e+T]2
dTT J о 1-2Т) cos e+T]^
F)
J(x,y) = 1-х2 I e-2J0 |2(yt)H ^^(xtjijdt . G)

278
INTEGRALS OF BESSEL FUNCTIONS
12.1.5
12.1.5. Asymptotic Expansions
j(x,y) ~KZ {e-5io(.)+ (g*) (ans)"* Zo *k4B§rk}
e"Z^) ? akckB5)"k , 71 «1 , S-»« >
1"T1 k=0
where
ak =
k.'
> 4 = 2Fi (-k,i;*-k;- -^i)
=k = i( i-Tl Hr( i+il L » akck = ^4A+11 ^V^)
J(x,y) -1Ф'2 [eIoE)- (^) Bn5L ^Z акЪкB5Гк]
-"Wf*
* ^ (*Wz)
eIoE)+(z/n)* Z -
k=0 k.'
B5+z)b
71<1 , z/g small, §->» ,
A)
B)
C)
^x.e ^s; ^ алB5) , ti »i, s-*« * D)
where ak , Ък and ck are defined by B) and C).
E)
12.1.5
where
MISCELLANEOUS INDEFINITE INTEGRALS
INVOLVING BESSEL FUNCTIONS
Eo(z) = 2zez(^ - Erf z*) ,
E^z) = zm2F0(-m,h-z~L) = (|)m ^(-md-miz)
= (i)m^zA(i^-m;-z) .
J(x,y)~l + KZ(e-5loE)-(z/n)i ? ?ЬЩ } ,
- L k=0 k.'B5+z)k J
T)>1 , z/g small, 5—*oo ,
where Em(z) is defined Ъу (б) and G).
j(x,y)~KZ e-§IoE)+-^Z
(i)kFk(z)
Bтт§)гк=0 k.'Bg)k
T1<1 , S-
j(x,y)~i + Kz eioE) - iacsL 2:
,XJ ^ (i)A(z)
Bn5 J k=0 k.' B? )k
T)>i , 5->co ,
279
F)
G)
(8)
(9)
A0)

280
where
INTEGRALS OF EESSEL FUNCTIONS
Fo(z) = Eo(z)
is given Ъу (б), and
Fk+1(z) + zFk(z) = (i)k ,
k-1
Fk(Z)-z (-)maifm)-m+(-)kAo(Z) ,
m=0
r(i)
Fk(z) = (i)keZzk^r(i-k,z) .
12.1.6. Integrals Related to J(x,y)
Let
f(X
,X,T) = I
Te-^lJCv2-»2)*}
dw ,
;X,T)=^VWIO{(-2^L}^ >
g(X
h(\,X,T) = / we'Xwl
o{(v2-^)*}
dw
Then
12.1.6
A1)
A2)
A3)
A4)
A)
B)
C)
(l-\2)h(\,X,T) = XTe'XTIo(Z)+e-XTZI1(Z)-\Xe-XX-Xg(X,X,T)-XX2f(\,X,T) , D)
12.1.6
MISCELLANEOUS INDEFINITE INTEGRALS
INVOLVING BESSEL FUNCTIONS
281
where
Z2 = T2 - X2 .
f(X,X,T) — + x |Io(Z)+e [1J\2 ' 2ZX
)]
-^HM)]) '
where
Zx = {x+(X2-lJ j (T.X) . Z2 = ^-(x2-!)^ | (T.x) .
g(X,X,T) = (X2-l)-V^<!e*(Zl+Zk;/Zl)^^Zl Z2
i(Z2+Z2/Z2)
H^t)]}-
ГA,х,т)=.^^1о(г)+|AЧ?,^)}
g(l,X,T) = (T-X)e-TIo(z)+e-TZI1(Z)-2x|l-j(^ , ^)j
h(l,X,T) = l/3(T-X)(T+X-X2)e'TIo(Z)+l/3(T+l-X2)Ze'TI1(Z)
E)
F)
G)
(8)
(9)
A0)
2^(h(?;?))
A1)

282
INTEGRALS OF BESSEL FUNCTIONS
X2g(l,X,T)+3h(l,X,T) = Z2e-TIo(Z)+(T+l)Ze-iI1(Z)
Х^A,Х,Т)+8A,Х,Т) = -Xe-X-HTe-TIo(Z)+e-TZI1(Z) .
12.1.7
A2)
A3)
A7)
See Ford A958) for (l)-(l3). Binnie and Miller A955) have shown that
со к 2k -T ,
g(i,x,T) = -x - Z {~Г e vn RdO + ^-iW > <14>
k=0 к.|2кBк-1)'Г;'1 L J
nT
Q(X,T) = 2 I R(X,w)dw = 2BT+l)g(l,X,T) - 4h(l,X,T) , A5)
Jx
where Н(Х,т) = e^Iw(Z) + 2g(l,X,T) . A6)
Blanusa A948, 1950) gives reduction formulas for
[W-xV/2in[(v2-x2)*}dw ¦
Jx L
12.1.7. Description of Mathematical Tables and Approximations
Admiralty Research laboratory A953): 104F(p,p) , see 12.1.1B).
Э = 0@.05J.50 , p = 0@.05J.50 , 4d.
Binnie and J.C.P. Miller A955): Tables computed Ъу G.F. Miller. R(X,T) ,
Q(X,T) , see 12.1.6A5-16). X = 0@.2M.0 , T = X(o.2M(lJO , 4 or 5s.
P l / 2 2 \
Bose A947): P(L) = I t^W2^ +X ^Iq(H)dt , q. = fe-1 , where
Jo
2L2 = nprf[ , D2 = if - 2/n , 2\2 = прД2 , and Д2 and D2 are the
population and estimated squared distances of two p-variate samples
whose harmonic mean size is n . Values of L are tabulated for
P = 0.99, 0.95, 0.05 and 0.01 for p = l(l)l0 , X = 0@.5KAN,
8,12FJ4,36,54,72,108,216,432 , 2d.
12.2
MISpELLANEOUS INDEFINITE INTEGRALS
INVOLVING BESSEL FUNCTIONS
283
т>г P + i
Brinkley and Brinkley A947): e"K / e"tIQBRt2)dt , r = 0@.1M ,
Jo
R = 0@.1M , 5s.
/ix l
e'tIoB[yt]2)dt , x = у =
0
0@.1M@.2I0@.5J0AM0BI00EJ00A0M00 , 6d.
Hastings and Wong A953): See this source for numerous analytical approxi-
approximations to / е~г(Р +х )plo(px)dp .
Jr
Opler and Hiester A954): See 12.1.2^15-17). This report tabulates X
and (i) to 4d as follows, r = 0.2@.2I,2, s = 1,2B)8 , t/s =
0.2,0.5,1,2,5 . r = 0.2@.1I,1.2@.1I.5,2AM , s = 10EI00A0)
1000 , t/s = 0.1@.1H.4@.2IAL,6 . r = 0.2@.1H.9 , s = 10E)
100A0M00 , t/s is selected so that X = 0.1 and 0.9 .
П °° i/t2+x2 \
Rand Corporation A951): I te 2V 'lo(xt)dt , R = 0.1@.1J0 ,
Jr
x = 0@.05I , 6d.
Wilson A951): 104F(P,p) , see 12.1.1B) , @ = 0@.25L , p = 0@.25M ,
4d.
12.2. A General Theorem for Representing an Indefinite Integral Involving
Bessel Functions in Series of Bessel Functions
The following results are due to Maximon A956). Let
su(f,g) = Z fk+uWg) >
A)
k=0

284
INTEGRALS OF BESSEL FUNCTIONS
12.2
where f = f(z) and g = g(z) . Then differentiation of (l) and use of
the difference-differential properties of <J\,(z) leads to the formula
ijzexp [-i(fg-gA)] [(fg)'fwju-1(g>+(e/f)'*Ч(в)]dz
= exp [-Htg-g/t)] S,,(f,g) •
B)
Now put
6(о^+Э) = yf > 2a(af2+P) = yit2-!) ,
C)
vhere a , Э and у are independent of z and и . Then with
,z
we have
Fv = Y re"a(of2+3)"afi;+1Ju(g)af ,
gF^ - oF,, = e~*Bv(t,g)
and solving this as a difference equation, we find
jVvAerVj^^df = g^ |^ (f/P)k(ek+1-ak+1)jk+u(g) ,
(*)
E)
k=0
Y ^ 0 , a f Э ^
F)
\U+n,
L'u+n
provided that lim (a/p)""rilF))+rl = 0 . We retain this assumption throughout.
n—»<»
If a = Э ,
12.2
MISCELLANEOUS INDEFINITE INTEB?ALS
INVOLVING HESSEL FUNCTIONS
f"e^d+^rVj^itgJaf = (a/Y)e"V Z (к+1)Лк+иF)
J k=0
= (a/YJe'V-1 [(l-i;+Y/aa)f Z Л^(в)
L k=0
+ ig{fju.1(g) - Jv(g)}l , y ^ о .
If y= O, since g = 0 gives trivial results, put
f2 = - p/o , f - f = 2h
Then from B),
A'^K-lCsW»]^ = 2e-^ Z fk+1Jk+u(s) >
k=0
whence
fV^j,,.^^ = fj^ ? (f/p)k(Pk+1-ak+1)Jk+u(g) , a / Э ,
285
G)
(8)
(9)
A0)
k=0
provided that lim f"nGu+ = 0 where G^ is the left-hand side of
n—><=
A0).

286
INTEGRALS OF HESSEL FUNCTIONS
12.2
If a = P ,
fe-*\_x(t)lt =2e"iz Z (k+a)Ak+u(z)
J k=O
= e-lz[2(l-y) Z ikWz)+z {ju.1(O+iJu(O}l
¦ A1)
See also 10.3C-6,10).
If, in the above equations, f and g are replaced Ъу -if and ig ,
respectively, we get the corresponding results involving Iu(z) . It is
also convenient to replace a and у Ъу -a and -y , respectively. Let
Tu(f,g) = Z fk+uWs)
k=0
A2)
where f and g are defined as in (l). Then
i|Zexp[-|(fg+g/f)] [(fg)'fU'1Iu.1(g)-(g/fIfUIu(g)]d2
= exp[-|(fg+g/f)] Tu(f,g) •
A3)
Let
gCoAp) = yt , 2с(а^+р) = 7(^+1) ,
A4)
12.2
MISCELIAHEOUS IHDEFINITE IMTEORALS
INVOLVING BESSEL FUNCTIONS
287
where а , P and у do not depend on z or и . Then
/V^f^rVv^df - -^ % (?/?? {ek+1-(-a)k+1} Wg) ,
Y^0,a + P^0 .
A5)
/
e-c(l+f2)-2f%.1(g)df = -(a/Y)e-V Z (к+1)Лк+и(е)
k=0
= -(a/Y)e-V-1|(l-U+Y/2a)f Z Лк+и(й)
k=0
i g [fiy-i(g)+iu(g)}
Y Ф 0 .
If у = 0 , we put
f2 = -p/a , f + f = 2p
A6)
A7)
Then
[V^^teJde = ^i i (f/g)kf gk+1-(-a)k+1)lk+u(g) , a+g^ 0 , A8)
J a S k=0 L J

288
INTEGRALS OF BESSEL FUNCTIONS
12.2
/
e;tlu.!(t)dt = 2e+Z ? (±)k(k+l)lk+u(z)
k=O
xt i_2
f "е* x-U+1Iu(Xx) f e^ /+1I>y)dya.
Jo «o
= exp
[i(^?)W ? %(^)u+kiu+k(^%)
L ^ i+t2 ^J k=o vi+t2y
s0 = t<7(t^+l), ek = 1, k>0, R(u)>-1
B1)
= e;z fad-v) ? (±L+y(z)+z K-i(z) J I^Z)}1 • ^19)
L k=0 L J -1 i
See also 10.3G-9,11).
Far an application, put f = z/X and g = \z in A3). Then
f Ze"*t2tU+1Iu(H)dt = xV*z2 ? (zA)k+"+1Ik+»+l^z) > R(u)>° ' B0)
Jo k=0
and-with, appropriate change of notation, this is the statement 12.1.3D) if
v = 0 . Finally, in the cited reference, Maximon shows that
12.3
MLSCELIAKEOUS INDEFINITE INTEGRALS
INVOLVING BESSEL FUNCTIONS
289
12.3. Other Indefinite Integrals
fe-\(t)Tn(l- |i)dt = §g [ln(z)+In+1(Z)] . A)
M>) = f
F(a,-b) = (td+bd)-2J1(t)dt
B)
Tabulated. See Froberg and Wilhelmsson A957). a = 0.1@.1J@.2 )lO ,
Ъ = 0@.1J@.2I0 , 6d. To facilitate interpolation for Мах(а,ъ) s 1 ,
an auxiliary function f(a,u) defined by F(a,b) = Jr[(a2+b2 )г-ъ] -
[f(a,b)]3 is also tabulated for a = 0.1@.1I , Ъ = 0@.1I , 6d.
<ln(R/r)
л1п(й/г)
/ (cosh ut)(z-w)'pJp(z-w)dt , -w2 = R2+r2-2Rr cosh t .
C)
See Buchholz A949).
I
X Jn(t)
dt
D)
4d.
Tabulated. See Kinizer and Wilson A947). n = 1AK , x = 0@.1)9.9,

CHAPTER XIII
DEFINITE INTEGRALS
13.1. Introduction
In this chapter ve list integrals over finite and infinite intervals.
Far the most part, these are not covered in previous chapters. The list is
by no means complete, but is representative of the type of results known in
closed form. For more extensive tables, consult Erdelyi et al. A953,
1954). See also Campbell and Foster A948), Oberhettinger A957b) and the
references given in 1.1. We give a rather thorough coverage of numerical
and analytical material which has appeared since about 1945 and 1950,
respectively.
13.2. Orthogonality Properties of Bessel Functions
Let Cy(z) be a cylinder function of order v . That is,
Cv(z) = AJy(z) + BJy(z)
where A and В are constants independent of v . Then
fb
/ tCu(\mt)C,,(\nt)dt = 0 if m^ n
Ja
= f*t2{A--^)^nt)+Ci2(^)]|
L L *-nt J J
(i)
if m = n , 0<a<b , B)
provided the following two conditions are satisfied.
1. \n is a real zero of
h1XCu+1(Xb) - hgCyUb) = 0
290
C)
13.2
DEFINITE INTEGRALS
291
2. There exist numbers кд_ and kg (both not zero) so that for all
WW^na) - k2CuUna) = 0 . D)
If a - 0 , the above relations axe valid if В = 0 . This case is covered
Ъу the following result.
/ tJ^t)^
J0
(ont)dt = 0 if m ? n , v>-l ,
= i [^(ап)] if m = n , b = 0 , u>-l ,
1 Га2, 2 21,2, ,
2an Lb<
if m = n,b^0,va-l ,
E)
where аро^, • • • are the positive zeros of aJv(x)-t-bxJ,J(x) = 0 , and a
and b are real constants. The expressions (l)-E) follow from 11.2B,4).
If b = 0 , some results in connection with E) are given in 3.10. From
13.4.2E),
Г°° -1
= DП+217+2) if m = n , H(u )+n+m> -1 , (б)
J t'1Jn+|(t)Jm+|(t)dt = 0 if m/ n
** —00
>-l
= 2Bn+l)~ if m = n
(V)

292
INTEGRALS CF BESSEL FUNCTIONS
13.3. Finite Integrals
13.3.1. Convolution Integrals
f 3Jt)Jv(z-t)it = 2 ? (-)kVu+2k+l(z) > R(^)>-! > *(»)>-! •
J 0 k=0
/ Jy(t)J.u(z-t)dt = sin z , |R(u)|<l
Jo
/
L
13.3
A)
Л z
/ J,,(t)!!.,,(z-t)dt = J0(z) - cos z , -l<R(u)<2 . B)
C)
Г t-^OOJ^z-t^t = ^'\+иB) , R(m.)>0 , R(u)>-1 . D)
^0
Z J^(t)J,(z-t) ^ = (^z)-l(^+u)j (z) , R(^)>0 , R(u)>0 . E)
t(z-t)
V+u'
Z(z-t)'%(t)dt = tt(z/2)^4^(8)J^,-J(») , R(»)>-1 • F)
The above expressions are easily proved Ъу applying the convolution
theorem of the Laplace transform calculus. If
Fi(p) = I e-5zfi(z)dz = L[fi(z)] ,
Jo
G)
13.3.2
then
DEFINITE INTEGRALS
293
5(P) = Fi(p)F2(p) = L[f3(z)] = L| I fX(u)fa(z-u)dul (8)
provided F^(p) and Fg(p) are absolutely convergent. Convolution
integrals can Ъе converted to finite definite integrals with trigonometric
argument. Thus
iTt/2
P ' 2 2
f3(z) = 2z / f-[_(z sin u)f2(z cos u) sin u cos u du
U П
O)
See 13.3.2. For example, E) above is a special case of 13.3.2C6). For
other convolution integrals, see 1.3.2A), 4.3G), 5.2A4-21), 8.1D), and
13.3.2A5-18, 36-46). See also Bailey A930a, 1931) and Rutgers A931).
13.3.2. Integrals Involving Bessel Functions with Trigonometric Argument
Most of the integrals in this section may Ъе proved as follows. The
Bessel function(s) in the integrand are expanded using 1.4.1A) or
1.4.1A7-19) as appropriate, and termwise integration is performed with
the aid of 1.2A4-16) as appropriate. The derived expansions can Ъе ex-
expressed as either a hypergeometric function or as a series of such func-
functions. Further results are obtained Ъу specializing parameters, variaЪles
or Ъо№, and using the formulas in 1.3.5, 1.3.6 and 1.4.1. Many, ЪиЬ not
all, special cases which lead to rather simplified expressions are de-
delineated. For integrals not covered Ъу these remarks, we usually give
proof or references, or the reader should refer to 13.1.

294
IHTEGRAIS OF BESSEL FUNCTIONS
13.3.2
/
тг/2
J2uBz cos t)cos2at cos 2gt dt
/iff
- к1пр f.•
Jo
J~Bz sin t)sin2ate2letdt =
TTBu+lJaz2u
2u
22а+2и+1Г(а+и+е+1)Г(а+и-е+1)
X oF,
Г J2uBz sin t)e21gtdt = neineju.g(z)Ju+p(z) , R(u)>4 . B)
Jo
ntr/2
/ J.+UBz cos t)cos(n-i;>t dt = ^(z )Ju(z) , R(ti+u)>-l . C)
Jo
,rr/2
Ju(z sin t)sinU+1t cos 2nt dt
n
(-)%/2zfe/z)nf (-)kO(z/2)k , г (z) h^w.,
D)
( Jy(z sint)sinU+1+2nt dt
^0
1 n (-)k(^) (z/2)k
In(n/2zJB/z)n Z ^ J»+i+n+k(z) . B(O>-n-l • E)
= (»+DD
13.3.
DEFINITE INTEGFtAIS
X
tt/2
Ju(z sin t)sinut sinBn+l)t dt
The combination B) and 1.4.1E) gives
iir/2
/ Y0Bz sin t)cos 2nt dt = ^nJn(z)Yn(z)
Jn
Similarly B) with p = i and 7.9A8) yields
f"/2
/ YQ(z sin t)sin t dt = (nz)~ [sin z Ci(z)-cos z Si(z)l
Jo
/ [J2mBz cos ^ ) " J2mBz)]cos 2(m"nH csc "t d^
« n
= 4п(-)П+1 Z (-)kJk+n(z)Jk+n.2m(z) , m = 0
or m = n
k=l
where n is a positive integer or zero. See Fettis A957b).
295
(-)n(n/2z)^2/z)n f (-)k(D (z/2)\ , , ww , ..
" (i+*)-n Ib "^^ J^n+k(O,H(u)>^ . F)
G)
(8)
O)

296
IHTEORAIS OF BESSEL FUNCTIONS
13.3.2
/,
тг/2
J (z sin t)sin2a cos2^1 t dt
lf-/n\V
*(z/grr(B)r(o+v/2) TfcL+v/2 \-z2/i) , R(v+2a)>0 , R(g)>O.(lO)
.тг/2
ГП/\(В -in t)Binv+1+fint со-8?-1 t*
n (-)k (n) (z/2)k
= ir(p)(v+l)nB/z)P+n Z 7^) Jg+vWz) '
k=0
R(u)>-n-l , R(P)>O
A1)
P uf *- „ т ОЛ-Л
j Jn(z sin tjcsc11'1 t cos P 4dt
Jo
r n-1 . .k, , >p-n+2k I
= i(-)nr(e)B/Z)pLn(z)- z ^jrl;^) J ^r^)>
0 . C2)
.тт/2
rn^Ju+1(z sin t)sinu+1 t cos2u t dt = |tt(|)vB/z)uJu(z/2)Ju+1(z/2) ,
Jn
R(v)>4 •
A3)
13.3.2
DEFINITE 1НТЕ(ЖА1Б
297
.тг/2
Г JV(Z sin t)sin^ t cos2^ t dt = iir(i)tlB/z)^(z/2)Ju(z/2) ,
«0
li = ±V,K(li)>-5;K(li+v)>-l . A4)
X
"/2
1,с^,)„М,^,„^№№1;;Ч
a+v q+v+1
v „ , 2 ' 2
2 3i ,n a+B+v a+P+v+1
^+1' 2 ' 2
.тг/2
-2/4
, R(u+a)> 0 , R(g)>0 . A5)
Jv(z sin^ t)sin t dt = -^ Jiu+i.(z)J|v4(O ,
JO 2 '
Jo
R(v)>-1 .
П/2е12 Sin2t(s.n t cQs tf^t=^^lz-%^Ja{z/2) f R(a)>.i
'a
¦m/2
A6)
A7)
Г Jy(z sin2t)sin2v+3t cos t dt = Bг)'^и+1(г) , R(u)>-1 . A8)
The integrals (l5)-(l8) are of the convolution type. See the concluding
remarks of 13.3.1.

298
IHTEGRALS OF HESSEL FUNCTIONS
13.3.2
I
tt/2
ЗЛт. sin t)Jv(z sin t)slj?a~\ cos2^ t dt
(z/2f+VT(t)rQf- V)
J.+V+1 Ц.+Ц+2 u+V. + „
rVl)r(v+l)r(^ +a+p) 3 *\n+i;+l,n+l,u+l, ^ +a+g
Е(ц,+и+2а)>0 , R(p)>0
A9)
If [i = m+| , v = n+i , 2a = -(m+n) and p = 3/2 , Levine and Schwinger
A948) denote the latter integral Ъу 1^ and show that
(z/2I
,m+n-l
^n 2 (m+n)r(m4)r(n+i)
i л "/2
- тт / cos(m-n.)t(z cos t y1 %+n_^Bz cos t )dt , (m+n)>0 . B0)

In particular,
Xl,l = (z/2Tr)-DTTz)+(8z2)'1HoBz)+Dz)H;LBz)
n2z
- A6z3)-1Dz2+l) / H0(t)dt
B1)
and similar type expressions are given for 1д_ 2(z) an^- ^2 2(z) *
Equation A9) has also Ъееп studied Ъу de Hoop A955) for the cases
а = -i , g = 3/2 and а = 0 = i • If p = v = 1 , а = i , g = 3/a ,
A9) has Ъееп taЪulated Ъу the National Physical Laboratory A953Ъ) to 4d
for z = 0@.1J0 .
13.3.2
DEFINITE INTEffiALS
299
irr/2
I J,,_i(z sin t)J,,(z sin t)tan2u t dt = i(i)_uzu'1Jl;Bz) ,
^0
0<R(u)<| . B2)
,tt/2
J (z sin t)Ju(z sin t)sin t tan ^ t dt = 2~M'(|)_M:Bz)"'
0
/i2z
ttlJl;(t)dt , p, = ±u , R(ti)>-i , RC^+i;)>-2
n
,тг/2
I j|(z sin t)csc t dt = Bz)'1[z-J1Bz)]
Jo
I
тг/2
Jp (z sin t)jp (w cos t)sin2at cos2^ t dt
J2^
i(z/2J^(v/2J"r(a+u)r(B^) f (-)k(z/2Jk(a+^k
ГB^+1)ГB1;+1)Г(а+Э+н.-н>) ^TQ k.1 B^+l)k(a+e+ti+u)k
• -k,-2p,-k,p+u
X 3F2^2v+l,l-a-p,-k
»tt/2
OTT/^
/ J (z sin t )J,,(w cos t)sinM'+;L t cosu+;L t dt
B3)
B4)
-w2/z2^ , R(a+u)>0 , R(P+m,)>0 . B5)
= у'\*/у)»Ь/у)\+]1+х(у) > У2 = z ^ , Н(ц)>-1 , R(v)>-1 . B6)

300
INTEGRAIS OF BESSEL FUNCTIONS
«тг/2
If v = 0 , the integral becomes
Г(ц-*2)
13.3.2
I 3v{z sin t)Jv(z cos t) sin20 t cos2^ t dt
Jo
= A -ST r(v/2+q+k)r(i;/2+g+k) , , *v+2k , .
2 ,±-n k.TA;+k+l)r(v+a+p+2k) K ' ' Ji;+2k^ '
R(v+2a)>0 , R(u+2g)> 0 ,
follows from 1.4.2A5) and 1.2A4). See also Bailey A938).
I J2m,(z sin t)I2i;^z cos 't)sin ^ ^ cos t dt
Jo
= ?(z/2) ^ Г(В+у) F /u+l-e I 2/4Л
rBv+l)r(g+2M,+v+l) X 2Ч2и+1,Э+2р,+1;+1Г У '
R(n)>4 , R(u+3)>0 .
лТт/2
/ J (z sin t)lu(z cos t)sinM:+1 t cos1'1' t dt
B7)
B8)
B9)
13.3.2
DEFINITE INTEGRAIS
301
in/2
/ J0(z sin t)Ja(z cos t)sin t cosa t dt = |B/z )a (i)ai2a+1(z) ,
Jo
R(a)>-i • C0)
itt/2
/ J2 (z sin t)!2v(z cos t)tan2M:+1 t dt
^(z/2J^r(v-u)
Г(и,+и+1)
J2u(zj , R(n)>-| , R("-n)>0 .
ПТт/2 +i
/ J2u,-|(z sltl t)Iu(z cos ¦t)sinM' 2 t tan^ t dt
Jn
(n/2zJ(z/2)^
iir/2
Г J (z sin2 t)Ju(w sin2 t)sin2a'1 t cos23 t dt
^0 ^
, fz/2ftw/2)VB)r(^%) f (-} B/2) (^+"+aJk
2 Г(и.+1)Г(и+1)Г(р,+и+а+В) j^-q k.'(n+l)k(ti+i;+a+3Jk
2/ 2
X 2F1(-k,-M,-k;i;+ljwVz ) , R(p,+i;+a)>0 , R(g)>0
If \ь = ±b > *>he latter is easily transformed into 10.1C).
C1)
/„ Jli(z/2)J^z/2^ > * = tv > ЕЫ>-? > R(w-n)>-l • C2)
C3)

302
IHTEGRAIS OF EESSEL FUNCTIONS
13.3.2
L
т/2
J (z sin2 tU (z sin2 t)sin2-1 t cos2* t dt= ЦФГут^Ъ)
X „F,
U.+u+a n+u+g+1 u+1 v+2
/2 2 2 2
**5\ц+1,и+1,ц+и+1, И±Е±2*1 , й+^а+Р+1
-Z2/4
C4)
nrr/2
J0
. 2
,2 +Л=,п2а-1 + _я2Э-1
J (z sin t)Ju(w cos^ t)sin'da"-1- t cos^P"-1- t dt
1 (z/2)^w/2)';r(a+a)r(B+n ^ ^^2 ^k
2 Г(м.+1)Г(и+1)Г(а+е+ц,+и) ?\ k'fu+l1) /^a+B+U+Ц N ^a+B+u+u+lN
k-0 к.(.ц,+х;к^—с-^_^к^__с_к jk
-к -п-к Ё1Н. S+^+l
X ,F;
2 2
v 2 2.
.2 /_2
w /z ) , R(m,+cx)>0 , R(u+p)>0 . C5)
птт/2
/ J (z sin2 t)jy(z cos2 t) sin2" t cos2^1 t dt
^0
_ x r(a+u)r(B+,)(Z/2)-a-@ f (") (РУе+^к
2 Г(ц+1)Г(и+1) ^
k=0
k.-("+l)kk ^+^^
-k,-u-k, a^a+^L
^lA-P-y-k.l-p-kl1) Vv+a+e+2k(z) > R(^+a)>0 , R(U+p)>0 . C6)
13.3.2
DEFINITE INTECBAIB
303
JW\b sin2 t)Jv(z cos2 t)sin2^+1 t cos2^1 t dt = ^^g^fr^
^- (-)kr(u,+v+k-4)r(u+k+i)r(u-P+2k+l)(u+u-^k+i)
4" k.T(u+k+l)rBp,+u+g+2k+l) Jp,+u+2k+i(>Z''
V V 2 У
(z/2)i(e-^) ^o k.T(v+k+i)r(^i +ц+к) ^+^+к
(О ,
вЫ>-4 > k(v+p)>o
C7)
пт/2
рп/г
I J (z sin2 t)Jv(z cos2 t)sin2^+1 t cos t dt
Jo
ГBр,+1) ^- (-)k(u+i>+2k+l)r(u,+^+k+l)r(u,+k+l)r(u-^k+l) ,,
?- i.ir/,.ii.j.i ^r^/,.-ю..J-ol,JO^ tJ(j,+u+2k+l^Zl'' >
z[r(tx+l)]2 k=0
к.'Г(и+к+1)Г(и-^ц,+2к+2)
R(m,)>-| , R(u)>-1 .
C8)
,tt/2
/ J (z sin t)Jv(z cos t) sin t cos t dt
J0
, , sa " (-)к(а)кГ(а+ц+к)
k=0
R(^,+a)> 0 , R(u)>-1
C9)

304
INTEGRATE OF BESSEL FUNCTIONS
13.3.2
13.3.2
DEFINITE IHTEGRAIS
305
г.тт/2
nil/ С
/ J^(z sin2 t)ju(z cos2 t)cot2u+3 t dt
Jo
22UC/2)ur(^-v-l)(z/2)U
.tt/2
/ J (z sin t)Ju(z cos t)sin ^ t cos t dt
Jo
Г(м.+и+1)
[v^-^vH -
2 C/2W3/2U2/W
,3/2
TT *• ' 'U*- I JV
16 Г(ц+и+2)
)-&21
V»«/2(Z)-^V^/2(B)] '
R(m.-")>1 , R(i>)> -1
D0)
R(n)>-1 , R(u)>-1 •
(«)
i tt/2
I J (z sin2 t)jy(z cos2 t)cot2l;+;L t dt =
Jn
2 (i)ur(^)(z/2)"
Г(м,+и+1)
Vz)'
,tt/2
r(h.-v)>o , r(u)>-| .
D1)
I J (z sin2 t)Ju(z cos2 t)sin2^+1 t cos2u+1 t dt
Jo
n (i) (i)uB/Ttz)*
4 r(ti+u+l) V+^+t
J.,+u+i(z) , R(ti)>-i , R(w)>-5 •
D6)
,n/2
Г ' 2 2 1
/ J0(z sin t)j,,(z cos t)tan t dt = Bu) Ju(z) , R(u)>0 . D2)
J0
The integrals C6)-D6) are of the convolution kind. See the concluding
remarks of 13.3.1.
J JQ(z sin2 t)Ju(z cos2 t)tan3 t dt = [2v(i>-l)J Jju(z)- -^ Jv+1(z)J , Г j^B sln 2t)Ju(z sin 2t)sin2a t cos2^ t dt
^ Jo
R(u)>l
D3)
-i
\ 2 /42 /
Г J (z sin2 t)Ju(z cos2 t)sin2^+1 t
Jo
2.^^+1. cos2U+3t dt
A/2) C/2)uB/ttz)=
i—iWij V^Z) ' RU)>4, R(O>-i
D4) ;
X 4F5
r(^+l)r(v+l)r(n,+v+a+p)
Jiii
j,+i>+2
2
R(^+a)>0,R(^+P)>0 .
D7)

306
IHTEGRAIS OF BESSEL FUNCTIONS
13.3.2
,tt/2
^J (z sin 2t)Jv(z sin 2t)tanM'' t dt
0 ^
= BzrV ( V^J +u(t)dt , R(n)>-1 , R(u)>0 . D8)
mt/2
/ К +l)Bz cos t)cos(|j,-u)t dt
^0
= (|пJовс(ц+«)тт[1 BI.„Ы-1^I„(О] , |R(n+«)|<l • D9)
itt/2
f KyBz cos t)sin2" t cos" t dt = iTTz'U(|)uIv(z)Ku(z) , |R(v)|<i . E0)
itt/2
/ Jy(z sin t)sin(pz .sin t )dt
= * X (") J2k+1(MJiu+k+i(V2)J^-k-i(z/2) , R(«)>-2 • E1)
k=0
,tt/2
/ Jy(z sin t)cos(pz sin t)dt = |n Jo(pz)Jiy(z/2 )
у n
+ tt X (-W^W^^-k^/2) ' R(«)>-1
k=l
E2)
13.3.2
DEFINITE IHTEGRAIS
307
rr/2
/ Jy(z sin t)cos (Pz cos t)dt = ^rrJil)(izeu)jil)(|ze"u) ,
Jo 22
I
= sinh u ^ R(u)>-1
, V Ш 2u-VT(v)(wz)"
Tl2 = z2+w2-2zw cos t , R(v)>-| ,
E3)
E4)
where (^(z) is the Gegeribauer polynomial, see Watson A945, p. 367).
,n/2
[ (\z)'|I-\+u(\z)eit(|i-U)(Cos t)|I+l'dt=nBaz)^Bbz)-V(az)Ju(bz) ,
J-rr/2
\ = [2 cos t(a2elt+b2e"it)]2 , R(^+u) >-1 .
For proof of E5), see Erde'lyi et al A953, v. 2, p. 48).
nn/2
/ [?-±(х csc "t)-Li(x csc t)J(sin t cos t) dt
Jo
Tabulated. See Pollard and Present A947). x = 0.5,1,2,4,00.
For evaluation of
E5)
E6)

308
INTEGRAIS OF BESSEL FUNCTIONS
13.3.3
лтг/2
I f(a,t)g(b,t)dt ,
f(a,t) = (esc t)fcos(a cos t)-cos a] or sec t sin t sin (a cos t) ,
g(b,t) = J0(b sin t) or cos(b cos t) , E7)
see Khudsen A952).
13.3.3. Lommel's Functions of Two Variables
These functions arise in a wide variety of applied problems, and should
assume added importance in view of seme tables by Dekanasidze (i960). See
also Boersma A962). The tables are described at the end of the section.
Except for F)-(8), the following formulae are in Watson A945, pp. 537-550)
where other expressions are also recorded.
Uu(w,z)+iUu+1(w,z) = z(w/z)U / Ju.1(zt)exp(iiw(l-t2)j tUdt

= z(w/z)U Г J^.xCz cos e)exp j|iw cos2e| sin" 9 cos 9 de , R(v)>0 . (l)
See 13.4.6A8) for evaluation of the first integral with upper limit equal
to infinity.
13.3.3
DEFINITE INTEGRAIS
309
/CO
J1.u(zt)exp |-^iw(l-t2)} t2"Udt
in/2
= -wtz/w)" / Ji_u(z csc 9)exp |-g-iw cot2e|(csc 9 K"Ucot 9 d9 ,
w >0 , z > 0 , R(u)>-|
/1
Jx_u (zt)exp ||iw(l-t2 )| t"dt
= -U2.u(z2/w,z)+iU1.u(z2/wjZ)+e^1W[u2.l;(z2/w30)-iU-L_l;(z2/w,0)]
B)
Uu(v,z) = Z (-)k(v/z)^kJu+2k(z)
k=0
• C)
D)
vn(w>z) = (') 51 (") (z/w) Jn+2k^z) ' n ^ integer or zero. E)
k=0
Uv(wjZ)+iUu+1(w,z) = z(w/z)Ue*1WBrr/wJ Z (-i )kBk+l)AkJk+i(w/4)
k=0
лк. W.w.^ Z (-k)»W2) J—M
m=0
m;r(u+m-k)
, R(u)>0 . F)
If v is a positive integer n and k>n , then

310
INTEGRAIS OF BESSEL FUNCTIONS
Ak 2 H2/ZJ ( ) J^ m.T(m+k+2-n)
13.3.3
G)
In particular,
U1(w,z)+iU2(w,z) = (w/z)Bn/wJe1P-w ? ikBk+l)Jk+i(w/4)J2k+1(z) . (8)
k=0
To prove F) start with (l). Use 1.3.6D) with p = q. = 0 , с = •§ ,
w = 1-t2 and z = iw/4 to represent the exponential function in the inte-
integrand. Use 1.4.l(l) to expand the Bessel function and evaluate the integral
and series with the aid of 1.2A4) and 1.3.5A). Then Ak is recognized
as a jj?2 > an(^ the expansion in series of Bessel functions follows from
1.3.6G). Equation (б) generalizes a statement of Zernike and Hijboer
A949).
For unrestricted v , w and z , Vv(v,z) can Ъе defined by
2 / , i,,
Vv(w,z) = cos(^w+^ z /w+|w) + U2.u(wjZ) .
Uu(w,z) + Uu+2(w,z) = (w/z)UJu(z) .
Vv(v,z) + Vu+2(w,z) = (w/z)"%(z) .
(9)
A0)
(H)
^ Uu(v,z) = -(z/v)Uu+1(w,z) , ^ Vu(v,z) = -(z/wty^Kz) . A2)
Both Uv(wjz) and V2_l,(w,z) are particular integrals of
3z2 z az w2
A3)
13.3.3
DEFINITE INTEC3RAIB
311
Uu(Vj0) - i cffr/s?;4*'
k=0 Г(и+2к+1)
A4)
k=o k-
A5)
vy(v,o-? (-)k(:!/2v)kvu-k(v,0)
k=0
A6)
Wv,0)M-)n[coB^- Z^-^^]
A7)
U^KOM-flsin^Z^-)^)^1
k=0
Bk+l)!
A8)
U-n(w.0) = cosDw+AriTT) j Vo(w,0) = 1 ; Vn(w,0) = 0,n>0
A9)
n , ,k, / s2k+m
V-an-m^O) = (-)П Z ''HI , m = 0 or 1
k=0
(гк+ш):
B0)
Un(w,z) = (-)\(z2/w,z) ; Vn(w,z) = (- )nUn(Z2/w, z)
B1)
In A7)-B1) , n is a positive integer or zero.
Dekanasidze (I960) tabulates Un(w,z) and Vn(w, z) for n = 1,2 to
6d. The w and z range is 0.5<:w<6.25,wSzs 4w and
6.25 sv slO , y s г slO . The spacing in z is always 0.01 while that
of w varies. For 0.5 <. w <. 1.2 , 1.2 <ws;4,4swslO, the spacing

312
INTEGRALS OP EESSEL FUNCTIONS
13.4
in w is 0.02, 0.05 and 0.1, respectively. Equation D) on page 2 of this
reference contains several typographical errors. Boersma A962) tabulates
B/w)Un(w,z) for n = 1,2 to 6d where w = тг(ттI0гт , z = 0,1,2 .
13.4. Infinite Integrals
13.4.1. Integrals with Exponential Functions
The integrals of this section and their special cases may Ъе viewed as
Laplace transforms, Fourier transforms, etc. For references to more ex-
extensive tables, see the remarks in 13.1.
/ e";PttH1Wl;(\t)dt . See 2.6,4.5 and 10.6 .
Ге-Р2^-^)*; = W2p)"r(^) 1Р1(^;и+1;-а2/4р2
J0 2pMT(u+l)
. e-a2/^(a/2P)"r(^) lBiDv^+li1,+ljeW) ,
2pMT(u+l)
Е(ц,+и)>0 , R(p2)>0 .
A)
B)
If v = n = -2(^J and jj, = m-n+1 , m = -?DI , Heatley A943) gives
tables of B) for various values of r = ia/p , i = л/^Т . For other tables
of the confluent hypergeometric function -^(a, c;x) see the general re-
remarks in 1.1 and Slater (i960). See also Chapter VII.
re-ttia-lJa[2(zt)i]dt
Jo
= z"a/2Y(a,z) , R(a)>0 , R(z)>0
C)
13.4.1
DEFINITE INTEGRALS
313
(V^V^at^t - a»Bp2r-Va2/4*2 , R(U)>-1 , R(p2)>o .
J0
0°° 2.2 i =2/Rt,2 p о
/ e P Ъ Ju(at)dt = T,42v)e /8P 11и(а2/8р ) , B(»)>-1 , R(p2)> 0 .
J 0
Г e"P2t2Yu(at)dt
(O
E)
2/„ 2,
= _ g e-a /sP^l4u(a2/8p2)tan Urr/2+rr-Ix^(a2/8p2)sec т/г] ,
|B(i»)|<l , R(p2)>0 .
J">e-^t2Ku(at)dt = g ea2/8^(a2/8p2)sec Urr/2 ,
F)
|B(i»)|<l , R(p2)>0 .
G)
Г tXe"^2J (at)ju(bt)dt
J0 ^
l(a/2p)^(b/2p)"r^+^+1>) oo (-)k(a/2pJkCl±±2±MiA
рХ+1Г(ц+1)Г(и+1)
Z
k=0
к-'(ц+1)к
,2,2
X 2F1(-k,-M,-k;u+l;b /вГ) , R(|il+u+\)>-l , R(p^)>0
(S)

314
/;
INTEGRALS OF EESSEL FUHCTIOHS
„2.1.2 ч/.„2
13.4.1
te-^2Ju(at)Ju(bt)dt - -^ e-(^V^Iu(^)
R(y)> -1 , R(p2)> О
О)
/ tVP t J (at)Ju(at)dt = —— ^—- -
Jo ^ PX+1r(^+i)r(u+i)
i+V+1 LL+V+2 A+LL+U+1
X F
3 SV^+i^u+i^+u+i
2/2
-a/jT) , R(\+n+y+l)>0 , R(pci)>0 . A0)
Let
pco
l(n^;A) = / e'pttXJ (at)Ju(bt)dt , a>0 , Ъ>0 ;
^0
R(p) > 0 , R(n,+u+X)> -1 ;
if p = 0 , R(|i,+u+l)> -R(X) > -1 for a f Ъ ;
if p = 0 , R(n,+u+l)> -R(\)> 0 for a = Ъ .
A1)
If p=O, see 13.4.2. If A = -g , see Cl). For a discussion of tables
of A1), see the remarks after A8) and B5).
13.4.1
DEFINITE INTEGRAIS
315
2И-%Ц-»+*+1Г(ц-„+1) Vo
хл(^,^^ -f)™\ ,
r2 = а2+Ъ2-2аЪ cos 9 , R(p,-v+l)>-l , v is a positive integer or zero. A2)
l(|i.^;A) = B/n) / tXK (at)lu(bt)cos (pt^p.-w+X )n) dt , а>Ъ . A3)
For discussion of (ll)-(l3) including power series representations, re-
recurrence formulae and special cases, see Eason, Noble and Sneddon A955).*
We now give some special cases of (ll). The following notation is
used. Let
k2 =
4ab
p2+(a+bJ
, к = sin a , sin {3 = p
[(a-bJ+P2]
_i_
A4)
K(k) and E(k) are the complete elliptic integrals of the first and
second kinds, respectively. Л0(а,Р) is essentially the complete
elliptic integral of the third kind introduced and tabulated by Heuman
A941).
* In Eq.s. B.3), B.4) and B.6) of this reference, only the real parts
of the integrals should be taken.

316
IMTEGRAIS OF BESSEL FUNCTIONS
l(n,n;O) =
к -
,42 J(
tt/2
cos 2n8 de
тт(аЪJ Jo A-k2 sin2 bf
13.4.1
A5)
For conditions of validity for A5) and the following, see the remarks
surrounding (ll)-(l3).
l@,0;0) = ^Щ ,
тт(аЪ)г
l(l,l;O) = 2—т ГA4к2)К(к)-Е(кI ,
ттк(аЪJ L
I(n,n;l) = ^-r- Gn(k2) , Gn(k2) = (-)П f
т/2
4тт(аЪ)'
cos 2n6 d9
(i-k2 sin2 eM/2
A6)
A7)
A8)
Riegels A950) has shown *that Gn(k ) can be expressed in the form
(l-k2)'1in[fnK(k2)-gnK(k2)] where fn and gn are polynomials of
degree n in (l-k2) . The exact coefficients in these polynomials are
tabulated for n = 0AO . He derives an expansion for Gn(k ) in powers
of 1st valid for кг<<1 and also an expansion valid near к = 1 .
(l-k2)Gn(k2) is tabulated to 4d for n = 0(lK , k2 = 0@.0lH.9@.001)
0.999. There are seme errors (see Math. Rev., v. 11, 1950, p. 617).
l@,0;l) =
pk5E(k)
4тт(аЪK/2A-к2)
A9)
i(i,i;i) =
pk
2n(abK/2(l-k2)
[(l-|k2)E(k)-(l-k2)K(k)J . B0)
13.4.1
DEFINITE INTEGRAI?
317
.3,2,2 2
J(l n-n - kJ(ag-^V)E(k) kK(k)
8rr(abK'/2a(l-k2) 2тт(аЪJа
B1)
тnлл^ Pk к(к) Ло(а^} , 1 я^.
I(l,0j0) = - *¦ *-{ + - , a>b ,
2тт(аЪ)^а 2а а
2тта
2а
, а = Ъ ,
ркК(к) , Ло(а^}
2тг(аЪ)^а 2а
, а<Ъ ,
B2)
An equivalent statement by Sura-Bura A950) is
1A,0;0) = a [l-AoCkbBi)} ,
2 X2(X!-2P2) г л\
ki= ~—~tv: 'sln 3i= LV^i^^J '
Xl(x2+2pd)
xx = f-a2+b2+p2, Xg = f+a2-b2-p2, f2 = (а2+Ъ2+р2 J-4а2Ъ2
B3)
EquationB2) has also been studied by Fettis (i960) who notes that the Byrd
and Friedman A954, p. 251) statement for l(l,0j0) does not give correct
results. In this connection, Fettis' result for B2) and the expression
given for B3) by Eason, Noble and Sneddon, Sura-Bura and Byrd and Friedman
contain typographical errors.

318
IMTEGRAIS OF BESSEL FUNCTIONS
13.4.1
l(lj0;.1? _- *М?Ш + I***** *00 + ?b?!l . 2 , a>b ,
2a
2E(k) p_
nk 2a
Pfab^E(k) , (a2-b2)kK(k) *Ло(а'Р)
пак
2n(abJa
2a
, a = b ,
, а<Ъ
B4)
2 V2.
l(l,l;-l) =
к) ,k(a2+b2V/2)K(k) , (а-Ъ >Ло^> , Ъ
Л " _ , . ,3/2 4аЪ 2а '
п(аъJк 2п(аЪ)'
,3/2
а >Ъ
2^.2,
pS(k) . ?kBaf+?/2}_KaO + 1
пак
гтта0
ГИ ^Га2+Ъ2+^/2) К(к) (а2-Ь2)Л0(а,Р) , a
п(аъ)гк
2п(аЪKА
2^2.
4аъ
2Ъ
, а = Ъ ,
, а< Ъ .
B5)
Eason, Hoble and Sneddon A955) have composed tables of l(^,v;\) for
(l*,"iX) = @,0;0) , (l,l;O) , A,1;±1), (l,Oj±l) , (l,0;0) , (O,l;±l) ,
@,l;0) , (O,Ojl) ; Ъ/а,р/а = 0@.2J,3; 4d. To facilitate interpolation
in the vicinity of singular points, auxiliary functions are also tabulated.
Nomura A940a,Ъ, 1941) also has tables of l(^,u;\) for X = -1 ,
a = b = 1 and all pairs of |i,u such that ц = n , v = 0(l)n , n = 0AL ,
and (ц,«) = E,0),E,l),E,2),E,3),F,0),F,l) . Also ^ and v are
halves of integers varying from 1 to 8 . p = 0.4@.2I.2,1.5@.5K,5 and
10,20 in some cases. 5d, 7d.
If p = 1 , Weeg A959) has tabulated l(O,ljO) and l@,l;-l) to 5d
for (a,b) = (l,l),C,l),F,l),(l/3,l/3),(l,l/3),(l/6,l/6) and A,1/6).
13.4.1
V
DEFINITE IHTEGRAI?
319
For other integrals involving Bessel functions which can be expressed
in terms of complete elliptic integrals, see Byrd and Friedman A954,
pp. 249-251) and 13.4.2A1-14).
Some other representations of (ll) follow.
l(n,v;X)
^^i/г./o^^;y
. (а/2Г(ъ/2Гг(ц+1;+\+1) j-
r(^+l)r(v+l)^+V+X+1 k=0
(_ )к(я /pJk/ X+M.+U+1 N ^Х+ц,+и+2'\
4 2 AV 2 Jb
k.'(n+ll
X 2Fx(-k,-^-k;u+l;b^/a^) , R(|i+u+X+l) > 0 , R(p+ia + ib)>0 . B6)
l(^,u;X)
(a/2f (b/2) Г(м.+и+Х+1)
z
(-^f^rYV^^) Q^f^-)
a2+p2/ V 2
r(,+l)r(u+l)(a2+p2)^W1>k=0
k.1 (»+!)„
x v (^izk -k, ^2^1 +k^+l;
2'IV 2
a2+p2
)
2, 2i
R(b)>0, |R(ib)|<p, |b| < (a +pJ-p , R(n+v+X+l)>0 . B7)
1(ц^Д)|а=ъ
Г(ц+1)г(и+1)р^+и+Х+1
4F3'
i+u+\+l u,+u+X+2 li+u+I u,+u+2
^+l,U+l,|i+U+l
4a?
P2.
2/ 2i
| a /p |<1, R(p+2ia)>0, R(^+y+X+l) > 0; if R(p+2ia) = 0,
also require R(x)<0 . B8)

320
INTEGRALS OF BESSEL FUNCTIONS
13.4.1
fVl)tt*J_i(at)Jt(at)dt = [np(p2+4a2)/2] , R(p)>2l(a) . B9)
I
н(р2+4а2)а
'e'&tk x(at)J ... (at)dt = vH^^ Г i } R(p)> 2I(a) . C0)
0 -* /4 a^pW)]*
f e"pttl (atjl^btjdt = C2r(n+u4)P'^(cosh a)P'|\(cosh p) ,
Jn ^ 2
sinh a = ас , sinh p = t>c , cosh о cosh p = pc ,
R(p±a+b)>0 , |1(а)|<тт/2 , |l(P)|<n/2 , R(n+«)>-i • C1)
For similar type integrals involving other kinds of Bessel functions, see
Cooke A956a). For integrals of the type
/
<Ve-p2tkKi(ata)Ku(btf3)at ,
C2)
special cases and generalizations, see Ragab A952-1953, 1954, 1954-1956),
MacRobert A953, 1954-1956, 1957) and Rathie A954, 1954-1956) and the
references given there. See also 13.4.5A4).
rVptt-3/2Kl/,Bt-4)dt =2Texp(-3p1/3) , R(p)>0 . C3)
Jn ' 32
13.4.1
DEFINITE INTEGRALS
321
n со
/ e-pttK2/3Bt-2)dt =HlV-1/5exV(-3V1/5) , R(p)>0 . C4)
For C3^-C4), see Klamkin A957). See also 13.4.1C2).
( ">e-iu)tJn(t)dt =
J -co
2(-i)nTn^) 2
A-uT)
1— , uj <1 ,
d\2
= 0
, or > 1 ,
C5)
where Tn(uj) is the Chebyshev polynomial of the first kind.
?
t-VluJtjn(t)dt = f- (-1)пA-ш )kn_xM , ш2<1 ,
, щ2>1 , C6)
= 0
where Un(w) is the Chebyshev polynomial of the second kind. For a
generalization of C5) and C6) to the case where the integrand also con-
contains as a factor a periodic continuous function of period 2n , see
Pinney A958).
Г t-2e-iwtJn+i(t)dt = (-ОПBтт)*Рп(Ш) , ш2<1 ,
J -co
= 0
, w2 >1 ,
C7)
where Pn(ou) is the Legendre polynomial.

322
INTEGRALS OF BESSEL FUNCTIONS
A(r) = г Г tet"t2/4TJo(rt)dt .
Jo
13.4.1
C8)
Tabulated
. See Bullaxd and Cooper A948). т = 1 . Various r , 4d. Also
4d values of / A(t)dt .
0
i:
f(x>y) = I e-t (cosh yt Jo(xt)-l] csch t
J 0
dt
C9)
Tabulated. See (Admiralty Computing Service, 1945). x = 0@.1M ,
у = 0@.1I . f(x,y) is the solution of
tl + I M + S?f = 0
D0)
for
ax2 x Sx ay2
and satisfies the conditions — =0 on у = 0 for x / 0 , —- = (x +l) '
dy 5у
on у = 1 . and (—) +( — "} ~» 0|(x +y ) near the origin and also
large values of x .
[V^tV^E2)]"^ , Ге^Е^+^Е2)]'^ ,
Jo Jo
D = K0(t)-Kx(t) , E = Io(t)+I-L(t) . D1)
Tabulated. See Sears A940). x = 0@.25)l@.5JB IOEKO,4O . 3d.
v = 1
2 Г e'^2 [Jo(t )Yn(Xt )-Yp(t )Jo(\t )]
"Jo t[jo(t)+Yo(t)]
dt
D2)
13.4.1
DEFINITE INTEGRAIS
323
Tabulated. See Jaeger A956).
X=i@.lJ, 71=0.001@.001H.01@.01H.1@.1IAI0A0I00A00I000 . 3d.
\ = 2AI0 , 1] = 0.1@.1IAI0A0I00A00I000 . 3d.
\ = 10A0I00 , 71 = 10A0I00A00I000 . 3d.
Kp
,<i;x) = /
-l=-xtc
t хе
0 [pt Jx(t )+qJQ(t )] 2+ [ptY1(t )+qY0(t)] 2
, R(x)>0 . D3)
For expansions of l(p,q.;x) and a table of l@,l;x) , see Jaeger A942)
and Jaeger and Clarke A942). x = 0@.0l)l(l)l0(l0I00(l00)l000 , 3d.
I(a,x) = B/rr)
/;
t'2(l-e-xt2){j1(t)Yo(xt)-Y1(t)J0(xt)}
J2(t)+T^(t)
dt ,
R(x)>0 .
D4)
See Zonneveld and Berghuis A955)
W(x) =
h 4(t)+n2i2(t)
dt , R(x) > 0
D5)
For rational approximations to W(x) valid for 0 s x < <» , see Hastings
A955,, pp. 193-194).
Equations D2)-D5) are typical of a large number of integrals which
occur in applied problems. The integrals usually arise by solving a partial

324
INTEGRAIS OF BESSEL FUNCTIONS
13.4.2
differential equation using the Laplace transform, and inverting this trans-
transform by contour integration. In this connection, see Carslaw and Jaeger
A959); Miles A959), Ward A955) and the references given there.
13.4.2. Weber-Schafheitlin Type Integrals
X>
t"xJ (at)Ju(bt)dt
(b/a)»(a/2)^r(^f±l)
2r(v+l)r^^
2F1V 2
,+u-X+l и-ц-X+l .1}+1.
2
;v+l;b2/a2j ,
2 J
R(n+v-A+l)>0 , R(x)>-1 , 0<b<a . (l)
t"XJ (at)Ju(bt)dt
i&nb/rt-h Q*f^)
>-\i+\+± \
,+v-X+l ц-tf-X+l
2rlV 2
JH+l;a2/b2J ,
2Г(,+1)Г(^ 2 J
R(m,+"-X+1)>0 , R(X)>-1 , 0<а<Ъ . B)
R(^+u+l)>R(X)> 0 , a>0
C)
Note that the expressions on the right of (l) and B) are not continuous at
a = Ъ . For some special cases of (l)-C), see 10.6B0-28).
13.4.2
DEFINITE IHTEGRAIS
325
/ tU+1"^J (at)Ju(bt)dt = 0 if 0<а<Ъ ,
_ bV(s.2-b2r-V+1
i-v-1.
2^"l;--La^r(n-u)
Rdi-u )> 0 , R(u)> -1 .
if 0< Ъ< а ,
JoV\(at)Ju(at)dt = \ Sl^-y , H(^)>0 , a>0 .
/ tJo(at) [l-Jo(bt)j dt = 0 if Ъ s a
= 1л(ъ/а) if Ъ ? a .
Jo
t-XK^(at)l);(bt)dt
Д-1. . v
(a/2) (ъ/а) г^У^гЛ-М^
V 2 / V 2 /
D)
E)
F)
4Г(и+1) v
R(w±|j,-X+1)> 0 , а>Ъ
2Fl(^-^ * 2 i^ibV^J ,
G)
t'XK (at )Ky (bt )dt
(а/2)Х-1(ъ/а)^г(ИН1^) г^11^1) Г (^f^) г(^у.)
8ГA-Х)
X gFl Л+v+n-X ^ itH^i л-хЛ-Ъ2/а2) , R(-A±^±u+l)>0, R(a+b)>0. (8)

326
INTEGRAIS OF BESSEL FUNCTIONS
13.4.2
13.4.3
DEFINITE INTEGRAIS
327
Г t-XYu(at)ju(tt)dt = ? sin(u-n-\W2 / t'\(at)lu(ht)dt ,
Jo n JO
R(u±|i-\+l)>0 , а>Ъ . (9)
fV'+\(at)Jv(H)dt = (^/Wr(^+1)
Jo " (а2+Ъ2)и^+1
R(u+l)>|R(p,)| , R(a)>|l(b)| .
Г J0(at)J0(ht)dt = B/тта) К(Ъ/а) , |Ъ/а| < 1 .
Jo
/ t'1J0(at)J1^t)dt = B/п)Е(а/ъ) , |а/Ъ|<1
Jo
/ K0(at)l0(bt)dt = A/a) K(b/a) , |ъ/а|<1 .
J0
(Ю)
A1)
A2)
A3)
f t2Ko(at)lo^t)dt = l& Е(ъ/а) - \-— К(Ъ/а) , |ъ/а| < 1 . A4)
Jn (а2-Ъ2J а(а2-Ъ2)
More generally, if in G) the numbers v , -|-(и-Х+ц) and ^-(u-X-ц) are
positive integers or zero, and |ъ/а|<1 , then G) may Ъе expressed as a
linear combination of complete elliptic integrals of the first and second
kind. See, Muller A955). A similar statement applies to (8) if X is a
negative integer or zero, -g-(v±|j,-?i) are positive integers and R(a+h)>0 .
13.4.3. Sonine-Gegenbauer Type Integrals
I
P (t^+Z2JU
а<Ъ ,
[ z ] J^-l(^(a -Ъ J] , а>Ъ ,
R(u)>R(|j,)> -1 if а^Ъ , E(u)>E(p,+l)>0 if a = Ъ
Г" / J (a(t2+z2)*}t^ 2^(^+1)^ (az)
u n
dt =
(t2+z2)*u T^+V
Ъ > a , r(u+1)>r(m,)> -1
For a generalization of B), see 13.4.5A3).
s:
ytt)
{a(t2+z2)*}t^+1
(t2+z2)h
dt
- Г/я2+>,2ч|1«^-1
= ^ (a +^ )г
av I z
> Ки.^.1{2(а2+Ъ2)г| , R(n)>-1 , R(z)
/;
-a(t2-y2Ft е-1у(а2+Ъ2)*
J (Tat) г
dt =
,2 2
(а2+Ъ2L
A)
B)
>0 . C)
— , arg(t -y ) = 4tt if t<y . D)

328
INTEGRAIS OF BESSEL FUNCTIONS
13.4.3
r *A«**f}№ dt = rJ>t)(t2.z2ft-dt
J0 (t2+z2)^ к
au,+iz"-n
X
a*0, r(^)>R<h)>-1 •
KJa(t2+z2)^t2^ a№aj_y
Q (t2+z2)^ a,+V-,-l »"^
l> 0 , R(p,)> -1 •
Jo (t2+z2)"
2U-2
dt
r4(t)(t2-z2f5/2
dt
= _K^ii_Hl,BZ) , R(u)>*
i0 (t2+b2)^/2+"/2
2r(i)z"+1 "
dtx
= ln^Lli J LLil^ (y/x)kj (by)Ju.k.i^x) ,
ъ^+и k=o
R(p,+u)>0 ,Ъ>0,х>0,у>0
E)
F)
G)
(8)
13.4.3
DEFINITE INTEGRAIS
329
/Q\ {(t2+z2)i} jv {{t^)i}t2«-\t2^f-ht
= l t2p-1(t2-z2)a-1J^(t)Ju(t)dt
(z/2)^+1z2a+2^3
Г(а)Г^+^)з1п^+^)п
Г(й+1)г(и+1)Г (a+P+ *??) sin (a+p+ J^) „
J.+V+1 M.+U+2 Ll+V
"^J 2 'Э+ 2
X F I
3 4\ц+1^+1,М,+и+1,а+3+ ^~
(tt/2 J2a+2p-2rC-2a-2P )cso (W ^) тт
гB-а-р- ИСЛ) гB-о-р- И**) Г B-О-Э+ ^) гB-о-Э+ ^)
^3/2-a-g,2-a-g,l-a
X »F,
3Г4
ч2.о.Э. МС? ,2.a.g. l|v ^_а„р+ ^ ^.а.р+ ^
R(a)>0 , R(a+g)<3/2
О)
The latter is a generalization of results given Ъу de Hoop A955). It may
be proved Ъу contour integration using 1.4.1A8), 1.3.2G) and 1.2A4).
See Pritchard A951) for (8).

330
IMTEGRAIS OF HESSEL FUNCTIONS
13.4.4
13.4.5
DEFINITE INTEGRAIS
331
13.4.4. Hankel-Nicholson Type Integrals
u+1
» t" X(at) a^z^
dt =
0 (t2+z2)^+1 2^Г(ц+1)
-KR(u)<2R(p,)+3/2
K._.,(az) , a>0 , R(z)>0 ,
¦4>-ц
CL"C ~~
0 t2+z2
2z
v+1
[^(az^L^az)] ,
a>0 , R(z)> 0 , R(v)> -5/2
I
'V,,(t)-
dt =¦
0 t2+z2 z4Binm
[jv(z)-Jv(z)] * R(z)>0 , R(
I
Yjat)
0 t2+z2
dt
= -z'^az) , a>0 , R(z)>0 .
I
ctx ~
p p
0 1T+zd
^[H.u(az)-Y.u(aZ)]
4 cos mjr
i'
R(a)>0 , R(z)>0 , R(u)>-i
I
A)
B)
u)>-l • C)
(O
E)
° J^dt = I1 P)^ (^),a>0)R(Z)>0)R(»)>-l • F)
0 (t2+z2)i ^А2У ^V2>
Рш t"VJ (at) B2/r»
a>0 , R(z)> 0 , R(v)> -\
P» tU+1J (at) (82/rv
a> 0 , |arg z|< tt/4 , R(i>)> -\
I
00 tu+5Ju(at)
a >0 , |arg z| < tt/4 , R(u) > l/б .
13.4.5. Integrals Involving the Products of Three or
More Bessel Functions
I
>%(at)j>t)Ju(ct)dt = (ЪС)Й11П"А Pti(cos A) ,
Bтг)га^ 2
R(p.)>-i , R(v)>-i ,
G)
(8)
(9)
A)
if a , Ъ , с are the sides of a triangle of area Д, and A is 0 ,
2 2 2
arc cos c ~a or ТГ according as a^ is less than, hetween, or greater
2Ъс
than the two numbers (Ъ-сJ, (Ъ+сJ . P^(z) is defined Ъу 10.6D). Note
t.hat Д = |Ъс sin A . Thus

332
INTEGRALS OF EESSEL FUNCTIONS
13.4.5
13.4.5
DEFINITE INTEGRALS
333
f>4(at)Ju(bt)Ju(ct)at - -^^ , R(v)>4 • B) /\(at)Ju(bt)Kx(ct)at
If а , b , с are not the sides of a triangle, the latter integral is zero.
Jx(at)J0(bt)J0(ct)dt = ^ . C)
Г t1"^,! (at^bt^ct^t
J0
. (bc)^-1ie1^sin(v-,)nsinh^i ЦЦ
A^
Q2.b2_02 = 2Ъс cosh x, a >b+c , b>0 , o>0 , R(p,)> -\ , R(")>
D)
2bo cosh у = a2+b2+o2 , R(a±ib±ic)>0 , Н(й+«)>-1 , R(«)>-1 • E)
Q^(z) is defined by 10.6A4). See Watson A94^, p. 411 ff) for the above
and other results pertinent to this section. See also Erde'lyi et al A954,
v. 2, Ch. 19). For the definition of OjJ(z) , we follow the latter refer-
reference and not that of Watson.
Let a , b and с be positive. Then
Г(ц+1)Г(и+1)с^+и+1
FMWFb,W '
(e)
provided R(|j,+u±x )> -1 . Here, with a = a2 _, p = b2 , у = с2 ,
= v-q+B- f(у-^^+^Л;g
i-g- 1(у+а-РJ+43у}г
2Y
= Y+a-g- Kv+q-g) +43y.
2Y
For extension of this to complex с and
related matters, see Henrici A957). See also Erdelyi et al A954, v. 2,
pp. 52-53).
I tK0(at )J0(bt)J0(ct)dt = [a4+b4+c4-2b2c2+2a2c2+2a2b2] ,
Jo
R(a)>|l(b)| + |l(c)| .
f^(at)dt ¦ 2Г(./2+1/6) a>Q f R(W)>.V3
JO 3ar(u/2+5/6Wrf2/3U2
ЗаГ(и/2+5/б) (гB/3)}
G)
(8)
I
'J^(at)J.u(at)dt = ^(v/2+l/6)sin n(u+l/5) >Q> R{v)>.x . (9)
33/2аГ(и/2+5/б) (ГB/3)}

334
IHTEGRA.IS OF BESSEL FUNCTIONS
13.4.5
j>^[JvMJv(bt)]\t - SiXlth) ^Л^Ъ*2/ъ*) ,
R(u)> 0 , 0<а?Ъ
I
j^zt^txi^dOdt
A0)
(n)
Tabulated. See Bennett A948). n = 6(l)lO , z = l(l)lO ; n = 100 ,
z = 4CI0EK0 ; (n,z) = F,5.8), G,6.6), (8,7.2), (8,7.5) . 4d, 6cL.
f [jo(^)]nJl(
Jo
rt )dt
C2)
Tabulated. See
AI7 , 4d.
Greenwood and Durand A955). n = 6AJ4= , r =/6.5@.5I2
Г [w^
-_ 1 dt
-\-±=l-
(t2^2)^
2^14 u,+l) TT fT (я _,\1
^+1znu 1=1
b> Z ai > R(Win+|)>R(iJ,)>0
i=l
Г
J0
rpttk-XK,](axte)Ku(xte)f(t)dt
A3)
A4)
13.4.6
DEFINITE INTEGRA.IS
335
For A4) and generalizations of same, see Ragab A955a, 1955Ъ, 1956)
and the references given there. In A4), the parameters and associated
f(t) are as follows. e = ±1 , p = 1 , a = ±1 , f (t) = 1 ; e = -1 ,
p = 0 , a = ±1 , f(t) = Kf|Bt) or Jf]Bt) ;e = l,p = 0,a = ±l,
f(t) = Kf|(t) ; e = ±1 , p = 1 , a = ±1 , f(t) = I^t) . See also
13.4.1C2).
/ tnI1(at)l1(bt)K0(ct)dt . See Bowkamp A947) . A5)
CO
csc |tt(h+u-\) Г t"^ (t)jy(t)Jx(pt)dt
A6)
Tabulated. See Kobayashi A939). p = 2(o.2K@.5L,5,7,10 ; \ = o(lM ;
ц, = 0.5AJ.5 ; v = 0.5AM.5 ,\i^v,\j, + v<6; \ and p,+u are not
at the same time even or odd. Otherwise all combinations of X , ц , v . 7d.
1&Ю
r^(t) dt г
L ^t)*' Jo
tK
K0(xt)
dt
A7)
Tabulated. See Ollendorff A926, 1929). x = 0.001,0. 01,0.05,0.1@.1H.5 .
3d. For the given range on x , the first integral is given with the limits
0.1 to 3.
13 .4 .6. Miscellaneous Integrals
r
Jo(at)sin Ы
z2+t2
dt = z^sinh ЪгК0(аг) , R(z)>0 , a>0 , 0<Ъ<а . (l)

336
INTEGRALS OP BESSEL FUNCTIONS
13.4.6
13.4.6
DEFINITE INTEGRAIS
337
П- Jo(at)cos Ы ^ Е Ь-1е-Ъ»1о(м) , R(z)>0 , a>0 , а,Ъ«
Jo z2+t2
f
Jo
- «„(at^Ort) ^ _ Iu(,z)Ku(az) f R(z)>0 , а*ъ>0 , r(v)>-1
z2+t2
Jo
•1(t+X)jo(tLtB ^1)^,^0
B)
C) J
D)
x+t
F(z,n) = 2z
f "^B2 eluhtjeoeh 2rt dt = ^ [4(O+^(O] > R(z)> ° • ^5) \
["^B2 sinh t)e'2ntdt . See Olver (l95l) , (б)
Jo ° /^
where % and dz/dn = F(z,n) are tatulated to lOd for n = 12DI5 .
Г"к +uB? oosht)oosh(p,-u)t dt = ^(z^^z) , R(z)>0 . G)
J о ^ ч\
f"... f\ —V rf tf/p-4(ts)-s = v^1/pb
Jo Jo 4*2'--Vl в=1
z>
0 , R(v)>i - 2/p , p = 2,3,
(8)
/„"••• />(tA±vl)n »ГЛ-Ч(*,)«а ¦ »-Vp^) •
R(z)>0 , p = 2,5,...
For proof of (8)-(9), see Ragab A952-1953, 115-117).
J -J^~ « = 2 [Ho(az)_Yo(az)] , a>0 , |arg z|<n .
0)
f
J n
"^— dt = \ [H0(ax)+Y0(ax)] , a>0 , x>0 .
A0)
A1)
For proof of (lo) and A1), and short tatles of same, see Lamb A917).
J^(at)
tt/2
I —7— dt = - - Jn(az)Yn(az)+(-) H0Baz cos t )oos 2nt dt ,
Jo Z^ 2 Jo
j2(at)
a > 0 , | arg z |< тт .
n/2
(Л2)
<f — dt = - Jn(ax)Yn(ax)+(-)n / H0Bax cos t )oos 2nt dt ,
Jo x-t 2 Jo
a > 0 , x > 0 .
l''ur proof of A2) and A3), see Dorr A953).
A3)

338
IHTEGRA.IS OF BESSEL FUNCTIONS
13.4.6
[".^(x cosh t)(sinh tJX(oosh t^dt = 2 XVM JV-XW '
x>0 , R(X)>0 , R(u-2\)>-|
For a generalization of A4), .see MacRobert A956).
A4 )
1+LL-2U
Г -A- x [2(at)*V r2(ab+at)ildt = гB,-,-1)ъ^-
Jq (b+t)u ^L J^"L J 2a" V2
a / О ., Ъ / 0 , |arg a|< тт , I arg ЪКтт , R(n)> -1 , RBu-|j,-l)>0
\
\
See Rathie A954a) far a generalization of A5).
A5)
a-tiTT
^i o^e **
tIl;(t2)K2UBzt)dt = ^(z2) , z / 0 , larg z|<tt/4 . A6)
ooe*iTr
5(,+l)in/2 Jte4it2Ku(zt)H(D(zt)dt = ^^2) , z{o , larg zl< n/4 . A7)
Here L is the path «e5111 to бе^117 , 6>0 , a quarter circle of radius
6 with center at the origin to 6 and then to =° . For other formulas
of the type (l6)-(l7), see Meijer A951).
J"Ju.1(-t)exp [±4iV(l-t2)} t»« = z-^z/w)^ {± f t Ц2- T ^] ,
0<R(u)<3/2 , w>0 , z>0 . A8)
13.4.6
DEFINITE INTEGRAIS
00
/ Ju.1(zt)expf+|iwt2} tudt = z(z/w)uexp /± i^? T ±M 1 .
Jo L J (_ 2w 2 j
For a discussion of the integral A8) with lower limit equal to unity,
see 13.3.3.
>=» (t2-62JJi(at)dt
339
A9)
'o Bt2-lJ-4t2[(t2-l)(t2-62)]1
B0)
See Reissner A952).
*n(z) = / Jo f2(ztJ} Jn(t)dt .
Jo ^ J
B1)
Tabulated. See Jaeger A948). §n(z) , $^(z) , n = 0(lO , z = 0AJ4= , 5d.
tJx(xt)
/
t + sinh t
dt
B2)
Tabulated, x = 0AM , 6d. For this and discussion relating to the evalu-
evaluation of
F(
Nelson A961).
(sinh Jt+Jt cosh ^^(^^(yt )
y(sinh t+t)
dt
B3)

340
INTEGRAIS OF BESSEL FUHCTIONS
13.4.7
13.4.7. Integrals vlth Respect to the Order
/.:
°° 2i9t Ju+t(x)lVt(y) dt J 2 еов_9_
X J,,
^-le^ie
*U+U) i(-,)e
v+J{2(x2e-19+y2e19)cos 8 } "] , R(^)>1 ,
provided 9 is real and |ЭК п . If 181 > тт , the integral,-"is zero.
A)
I
2,-
Jt(zt)j.t(zt)oos TTt dt = i(l-z )'г , |z|<l
t [jt(zt)J_t(zt)cos TTt-l)]dt = -|tt'
Jo
B)
C)
I Jlt(x)seoh |nt cos yt dt = 2 sin(x cosh y) , x>0 , у * о . D)
|J -CO
Л CO
(t Jlt(x)osoh |nt cos yt dt = -21 oos(x cosh y) , x>0 , у г 0 . E)
J-OD
/:
ei9tJ,(X)dt ^ elxSln 9 , -n<e<n ¦,
\
= тт ;
, 191 > тт , 9 real
F)
13.4.7
HEFIWITE 1Е1ТЕСЖА1?
fC°it(z)dt = У" - f{^^)-\-Z °°Sh t dt , R(z)>0
For F) and G), see Cooke A954).
f Kj_t(x)eoB yt dt = |тте-х oosh У , |l(y)| ^ |тт , x> 0
i(n+")
X V. H (ea+e-a)(x2eVe-a)} 2] , x / 0 , у / 0 .
For (9), see Conolly A956).
341
G)
(8)
(9)
I Kit+iM,(x)Kit+ii;(y)dt = "Ki^-iuCx+y) , larg x| + largyKn . A0)
^|CD TTt
e Kit+i^^^t+iuCy)" = ^""Xn-iu^-y) > x>y>0 . A1)
-CO
/" ela\+lt(x)Ku.lt(y)tt = „(Si^y^ ^^ oosh a)i] ,
d-cx, \xela+y/ L J
larg x| + larg у I + |l(a)|< тт . A2)
" Ui'c Oberhettinger and Higgins A961) for an extensive table of the
J/'bedev transform
p со
r(t)KIt
(x)dt . See also the references in 13.1.

342
IMTEGRAIS OF BESSEL FUNCTIONS
Jo
^С^^-х-У
tKit(x)Klt(y)tanh TTt ut = -5тт(х+у)~ (*yJe~ ~y ,
larg х|<тт,х^0,у>0
ftK^WK^sinh nt at = -^f- e-(-2/^)
Jq 2 ' У '
I
|arg х|<ч/4,х/0,у>0
ntKlt(x)K2it(y)sinrtat=^7Fe-(^/8x)
О / у '
2"/ x
|arg х|<п/2,х/0, у>0
13.4.8. Dual ana Triple Integral Equations
The pair of equations
f(t)g(t)Ju(xt)at = A(x) , 0<x<l
I
f(t)ju(xt)at = в(х) , x>i ,
13.4.8
A3)
A4) j
A5)
A)
where g(x) , A(x) ana B(x) are known, ana f(x) is to be aeterminea,
are oallea aual integral equations. They are important in a number of ap-
applied fieias ana arise in the solution of Ъоипаагу value problems where the
oonaitions on one Ъоипаагу are of "mixed." type. Suppose
13.4.8
where
DEFINITE INTEGRA.IS
f(x) = f (x) + fp(x)
I f2(t)Ju(xt) = 0 if 0<x<l ,
«0
= B(x) if x>1 .
Then by the Hankel inversion formula
f2(x) = x I tB(t)ju(xt)at ,
Ji
ana the pair (l) reauces to
/ f1(t)g(t)ju(xt)at =a(x) - Г f2(t)g(t)Ju(xt)at , o<x<i , ¦
Jn «n
^0
(xt )at = о , x > i
343
B)
C)
D)
E)
Thus it is sufficient to oonsiaer (l) with B(x) = 0 . To avoia confusion,
we write
/ f(t)h(t)Ju(xt)at = a(x) , 0<x<l
J0
Г f(t)jy(xt)cLt = 0 , x>l .
Jo
F)

344
INTEGRAIS OF BESSEL FUNCTIONS
13.4.8
If h(t) = t2a , then
f(x)=
Г(
u+a4
ds Jo
ra*l-a f V^J (xs) *- f'"a(t)tl;+1(s2-t2)adtds , -К а< О ,
a+1) Jo
ьо+Э+1,
Г(x) = ". Г а.„ч / ^^W*** ,
Jo
2агЛ+а+1±]Л
V 2 /
|R(a)|<l , RBa+P)>-3/2 , R(a+w)>-l , n(v)>-1
and with g = и ,
f(x) =
Г(и+1)
Bx)ar(i;+a+l)
Same special oases of (8) are given in the table below.
Hi)
f(x)=J^t! fV^^V^xs) fSa(t)tU+1(s2-t2r1dtds , 0<a<l .-
Г\а) Jo Jo
Further^ if a(x) = xP , then 1.2A4=) and 2.3(l) give
гЛ+&±?>х"Bа+Э+1)
V 2 /
(8)
JB+a+1(x),|R(a)|<l)E(a+»)>-lJBW>-l • (9)
13.4.8
-i
-i
1_
1_
1
I
0
1
0
0
0
V
0
1
1
1
0
EEFINITE IMEGRAIS
f(x)
B/n)sin x ,
D/nx)(sin x-x cos x) ,
1 - COS X
Bx2)-1B-2 cos x-x sin x) ,
B/nx2)(sin x-x cos x) .
345
A0)
For further discussion and applications of (б)-(ю), see Chong A953),
Erdelyi et al A953, v. 2, pp. 1&-П), Gordon A954=), Noble A956, 1958),
Tranter A956), Sneddon (i960) and the references given in these sources.
In the special case given by Erdelyi et al,which corresponds to the last
equation of (Ю) above, for v = 1 read v = 0 .
To solve (б) in the general case. Tranter A956) assumes
f(x) = x1'6 X Vu+eW3^
k=0
(n)
In this event, the second equation of (б) is automatically satisfied in
view of 13.4.2B), and the ak's can be found by solving an infinite set
of linear equations. If a(x) = xu , Tranter develops an iterative scheme
to compute the a-^'s and illustrates with some examples. Cooke A956b)
assumes
p p+1 pl
Г(Х) = ГA^Г)] U^(u)Ji;..g(xu)du > °<R(P)<1 , A2)

346
INTEGRALS OF BESSEL FUNCTIONS
13.4.8
13.4.8
DEFINITE INTEGRALS
347
where b(u) is to Ъе determined. Again the second equation of (б) is
automatically satisfied and with suitable restrictions, b(u) satisfies an
integral equation of Fredholm type.
Anders A954) discusses the pair
I
tf(t)Jo(xt)dt = 1 , 0<x<l ,
t6(t)f(t)j (xt)dt = 0 , x >1 ,
A3)
where 6(t) = (l-t2J , t ? 1 ; 6(t) = i(t2-!J , t >1
\
tf(t)j (xt)dt = A(x) , 0<x<l ,
A4)
I t(t2-a2)kf(t)JQ(xt)dt = 0 , x>l ,
where a ^ 0 and 0<l^< 1 has Ъееп studied Ъу Achiezer A954). He also
considers the system
f(t)J (xt)dt = A(x) , 0<x<l ,
r03
f(t)J_ (xt)dt = 0 , x >1 ,
A5)
for a >0 and -1<и<1 .
Lebedev and Ufliand A958) study the pair
pea
I f(t)J (xt)dt = A(x) , x<a
J0
f"tf(t) ^2ht+slnh2tb0(xt)dt = 0, x>a .
 sinh2 t
Study of a flow pattern at high subsonic speeds leads Helliwell
A961) to consider
A6)
Л CO
cosh 2y-t-cosh 2ut
sinh 2vt
-)j.x/3(xt)dt = 2x'1/3k(u)
2K(x1,x2) J [xf/5J2/5 (xxt )-x2/5 Jg/5 (xgt)] If2 l^l J.l/3(xt)dt
Isinh 2vt
Xg < X< X-^
A7)
I f(t)J_w3(xt)dt = 0 , 0<x<x2 , x >x-, .
The cases Xg = Xj_ and x2 = 0 are treated in detail.
Peters A961) applies Sonine integrals (see 13.3.2B6) and 13.4.3A))
to solve
n со
I tef(t)J
(xt)dt = A(x) , 0<x<l
p со
(t2-k2)af(t)Ju(xt)dt = B(x) , x>l , k>0
J0
A8)

348
INTEGRAIS OF HESSEL FUNCTIONS
13.4.8
This generalizes (б) (if h(t) = t2a ) and A3)-A7). He also notes that
13.4.3A) may Ъе used to solve
f tSf(t)J (x(t2+l32J}dt =A(x) , 0<x<l ,
f f(t)J (x(t2+b2)*]dt = B(x) , x>l .
Tranter (i960) treats the triple integral equation system
I f(t)Jy(xt)dt = A(x) , 0<x<a ,
Jo
I t2af(t)Jy(xt)dt = B(x) , а<х<Ъ ,
Jo
Г f(t)J (xt)dt = 0 , х>Ъ ,
.Jo
where A(x) and B(x) are known and a = *i • He assumes
A9)
B0)
CHAPTER XIV
TABLES OF BESSEL FUNCTIONS AMD IHTEGRAIS OF BESSEL FUNCTIONS
Introduction
Here we give tables of Bessel functions of integral and fractional
order, and some integrals of Bessel functions. The tables are designed
to enable the user to check expansions and make trial computations as
usually required in exploratory studies. The tables are not necessarily
intended for further table making and so are not too extensive. In many
applications linear interpolation should give sufficient accuracy. For
the most part, sufficient tabular data are provided so that interpolation
to virtually the full accuracy of the tables is readily accomplished using
Taylor's theorem. Other than this, we provide no interpolatory analyses.
Most of the tables are available to greater accuracy in the cited refer-
references, and these sources should be consulted where necessary.
The entries in each table are accurate to within 0.52 units of the
last decimal. An (n-l) decimal value obtained by rounding off an n
decimal table which is accurate to within 0.52 units of the n-th place may
contain errors up to 0.55 units of the (n-l)-th place. This could happen
if the n-th figure is a five. In all such instances, we recomputed to
insure the accuracy stated for our tables.
f(t) = t"a X Ck^+a+lW13*)
k=0
B1)
so that the third equation of B0) is automatically satisfied. With the aid
of 13.4.2A), the ck's are determined by a dual series. Closed form solu-
solutions are known only for the cases и = ±i , see Tranter A959). In partic-
particular, if A(x) = 0 , B(x) = 1 , a = -i and v = \ , then
ck = &(-) Pk(cos e)/K(co= 6/2) , a = b cos e/2 ,
where Pjj is the Legendre polynomial and К is the complete elliptic
integral of the first kind.
B2)
349

350
INTEGRALS OF BESSEL FUNCTIONS
TABLE I. Jn(x) , Yn(x) , n = 0,1
Jo«
Y0(x)
Jt(x)
Yl(x)
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
1.0000000
0.9975016
.9900250
.9776262
.9603982
0.9384698
.912 0049
.8812 009
.8462874
.8075238
0.7651977
.7196220
.6711327
.6200860
.5668551
0.5118277
.4554022
.3979849
.3399864
.2818186
0.2238908
.1666070
.1103623
. 0555398
.0025077
-0.0483838
- .0968050
- .1424494
- .1850360
- .2243115
V
-CO
-1.5342387
-1.0811053
-0.8072736
- .6060246
-0.4445187
- .3085099
- .1906649
- .0868023
+ .0056283
0.0882570
.1621632
.2280835
.2865354
.3378951
0.3824489
.4204269
• .4520270
.4774317
.49682 00
0.5103757
.5182937
.5207843
.5180754
.5104147
0.4980704
.4813306
.4605035
.4359160
.4079118
0. 0000000
.0499375
.0995008
.1483188
.1960266
0.2422685
.2867010
.3289957
.368842 0
.4059495
0.4400506
.4709024
.4982891
.5220232
.5419477
0.5579365
.5698959
.5777652
.5815170
.5811571
0.5767248
.5682921
.5559630
.5398725
.52 01853
0.497 0941
.4708183
.4416014
.4097092
.3754275
-6
-3
-2
-1
-1
-1
-1
-0
-
-0
-
-
-
-
-0
-
-
-
-
-0
-
+
0
-co
.4589511
.3238250
.2931051
.7808720
.4714724
.2603913
.1032499
.9781442
.8731266
.7812128
.6981196
.6211364
.5485197
.4791470
.4123086
.3475780
.2847262
.2236649
.1644058
.1070324
. 0516786
. 0014878
.0522773
.1004889
.1459181
.1883635
.2276324
.2635454
.2959401
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
* Extracted from British Association for the Advancement of Science
MATHEMATICAL TABLES - Volume VI, BESSEL FUNCTIONS PART I - FUNCTIONS
OF ORDER ZERO AND UHITY, Cambridge University Press, London and New
York, 1950, with permission of the Royal Society and the publisher.
' 1 i,
1
TABLES OF BESSEL FUNCTIONS
TABLE I. Jn(x) , Yn(x) , n = 0,1
351
Jo«
*oW
Jx(x)
*l(x)
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
b.5
b.6
b. 7
b.8
b.9
G.O
C.I
G.2
C.3
0.4
-0.2600520
- .2920643
- .32 01882
- .3442963
- .3642956
-0.3801277
- .3917690
- .3992302
- .4025564
- .4018260
-0.3971498
- .3886697
- .3765571
- .3610111
- .3422568
-0.32 05425
- .2961378
- .2693308
- .2404253
- .2097383
-0.1775968
- .1443347
- .1102904
- .0758031
- .0412101
-0.0068439
+ .0269709
.05992 00
.0917 026
.1220334
0.1506453
.1772914
.2017472
.223812 0
.2433106
0.37685 00
.3431029
.307 0533
.269092 0
.2296153
0.1890219
.1477100
.1060743
.0645032
.0233759
-0.0169407
- .0560946
- .0937512
- .1295959
- .1633365
-0.1947050
- .2234600
- .2493876
- .2723038
- .2920546
-0.3085176
- .3216024
- .3312509
- .3374373
- .3401679
-0.3394806
- .3354442
- .3281571
- .3177464
- .3043659
-0.2881947
- .2694349
- .2483100
- .2250617
- .1999486
0.3390590
.3009211
.2613432
.2206635
.1792259
0.1373775
. 0954655
.0538340
. 0128210
- .0272440
-0.0660433
- .1032733
- .1386469
- .1718966
- .2027755
-0.2310604
- .2565528
- .2790807
- .2984999
- .3146947
-0.3275791
- .3370972
- .3432230
- .3459608
- .3453448
-0.3414382
- .3343328
- .3241477
- .3110277
- .2951424
-0.2766839
- .2558648
- .2329166
- .2080869
- .1816375
0.3246744
.3496295
.3707113
.3878529
.4 010153
0.4101884
.4153918
.4166744
.4141147
.40782 00
0.3979257
.3845940
.3680128
.3483938
.3259707
0.3009973
.2737452
.2445013
.2135652
.1812467
0.1478631
.1137364
.0791903
.0445476
.0101273
-0.0237582
- .0568056
- .0887233
- .1192341
- .1480772
-0.1750103
- .1998122
- .2222836
- .2422495
- .2595599
3.0
3.1
3.2
3.3
3.4
3.5
3. 6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4

352
INTEGRALS OF BESSEL FUNCTIONS
TABLES OF BESSEL FUNCTIONS
353
TABLE I. Jn(x) , Yn(x) , n = 0,1
TABLE II. e-xIn(x) , e^x) , n = 0,1 , ex *
J0(x)
Y0(x)
Ji(x)
Yl(x)
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
0.2600946
.2740434
.2850647
.2930956
.2981020
0.3000793
.2990514
.2950707
.2882169
.2785962
0.2663397
.2516018
.2345591
.2154078
.1943618
0.1716508
.1475175
.1222153
. 0960061
.0691573
0.0419393
.0146230
- .0125227
- .0392338
- .0652532
-0.0903336
- .1142392
- .1367484
- .1576552
- .1767716
-0.1939287
- .2089787
- .2217955
- .2322760
- .2403411
-0.245 9358
-0.1732424
- .1452262
- .1161911
- .0864339
- .0562537
-0.0259497
+ .0041818
.0338504
.0627739
.0906809
0.1173133
.1424285
.1658016
.1872272
.2065209
0.2235215
.2380913
.25 01180
.2595150
.2662219
0.2702 051
.2714577
.2699992
.2658749
.2591558
0.2499367
.2383360
.2244937
.2085701
.1907439
0.1712106
.1501801
.1278748
.1045271
.0803773
0.0556712
-0.1538413
- .1249802
- .0953421
- .0652187
- .0349021
-0.0046828
+ .0251533
.0543274
.0825704
.1096251
0.1352484
.1592138
.1813127
.2013569
.2191794
0.2346363
.2476078
.2579986
.2657393
.2707863
0.2731220
.2727548
.2697190
.2640737
.2559024
0.2453118
.2324307
.2174087
.2004139
.1816322
0.1612644
.1395248
.1166386
.0928401
.0683698
0.0434727
-0.2740913
- .2857473
- .2944593
- .3001869
- .3029176
-0.3026672
- .2994789
- .2934226
- .2845944
- .2731150
-0.2591285
- .2428010
- .2243185
- .2038851
- .1817211
-0.1580605
- .1331488
- .1072407
- .0805975
- .0534845
-0.0261687
+ .0010840
. 0280110
.0543556
.0798694
0.1043146
.1274659
.1491128
.1690613
.1871357
0.2031799
.2170590
.2286600
.2378932
.2446924
0.2490154
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10. С
e"XI0(x)
%»
%(x)
ХКд.(х)
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
1.0000000
0.9071009
.8269386
.7575806
.6974022
0.6450353
.5993272
.5593055
.5241489
.4931630
0.4657596
.4414404
.4197821
.4004249
.3830625
0.3674336
.353315 0
.3405157
.3288719
.3182432
0.3085083
.2995631
.2913173
.2836930
.2766223
0.2700464
.2639140
.2581801
.2528055
.2477557
0.0000000
.0452984
.0822831
.1123776
.1367632
0.1564208
.1721644
.1846700
.1944987
.2021165
0.2079104
.2122016
.2152569
.2172976
.2185076
0.2190394
.2190195
.2185528
.2177263
.2166119
0.2152693
.2137477
.2120877
.2103230
.2084811
0.2065846
.2046523
.2026991
.2007374
.1987773
CO
2.6823261
2.1407573
1.8526273
1.6626821
1.5241094
1.4167376
1.3301237
1.2582031
1.1971634
1.1444631
1.0983303
1. 0574845
1.02 09732
0.9880700
0.9582101
. 9309460
.9059181
.8828335
.8614506
0.8415682
.8230172
.8056540
.7893561
.7740181
0.7595487
.7458682
.7329072
.7206041
.7089050
CO
10.8901827
5.8333860
4.1251578
3.2586739
2.7310097
2.37392 00
2.1150113
1.9179303
1.7623882
1.6361535
1.5314038
1.4428976
1.3669873
1.3010537
1.2431659
1.1918676
1.1460392
1.1048054
1.0674709
1.0334768
1.0023681
0.9737702
.9473722
.9229137
0.9001744
.8789673
.8591319
.8405301
.823042 0
1.0000000
1.1051709
1.2214028
1.3498588
1.4918247
1.6487213
1.8221188
2.0137527
2.2255409
2.4596031
2.7182818
3.0041660
3.3201169
3.6692967
4.0552000
4.4816891
4.9530324
5.4739474
6.0496475
6.6858944
7.3890561
8.1661699
9.0250135
9.9741825
1.1023176A)
1.2182494A)
1.3463738A)
1.4879732A)
1.6444647A)
1.8174145A)
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Extracted from Watson, G.N., A TREATISE ON THE THECBY OF BESSEL FUNCTIONS,
Cambridge University Press, London and New York, 1945, with permission
of the publisher.

354
INTEGRALS OF BESSEL FUNCTIONS
TABLES OF BESSEL FUNCTIONS
355
X
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
TABLE
e"XI0(x)
0.2430004
.2385126
.2342688
.2302480
.2264314
0.2228024
.2193462
.2160494
.2129001
.2098875
0.2070019
.2042345
.2015774
.1990233
.1965656
0.1941983
.1919159
.1897134
.1875862
.1855300
0.1835408
.1816151
.1797495
.177 9409
.1761863
0.1744833
.1728291
.1712215
.1696584
.1681377
0.1666574
.1652159
.1638114
.1624424
.1611073
II. e "Inlx,
e'XIX(x)
0.1968267
.1948921
.1929786
.1910902
.1892299 '
0.1874000
.1856022
.1838379
.1821076
.1804119
0.1787508
.1771245
.1755325
.1739746
.1724502
0.1709588
.1694997
.1680723
.1666757
.1653093
0.1639723
.1626639
.1613833
.1601298
.1589026
0.1577010
.1565242
.1553716
.1542424
.1531359
0.1520515
.1509885
.1499463
. 1489243
.1479220
) , e"Kn^x; ,
eXK0(x)
0.6977616
.6871311
.6769751
.6672592
.6579523
0.6490263
.6404560
.6322181
.6242916
.6166573
0.6092977
.6021965
.5953390
.5887114
.5823013
0.5760968
.5700872
. 5642625
.5586133
.5531310
0.5478076
.54263^4
.53760G4
.53271^0
.52795ф
0.5233247Ч
.5188116 \
.5144136
.5101258
.5059438
0.5018631
.4978799
.4939902
.4901905
.4864773
n = u,x ,
e\(x)
0.8065635
.7910030
.7762803
.7623243
.7490721
0.7364675
.7244607
.713 0065
.7020647
.6915988
0.6815759
.6719662
.6627424
.6538798
.6453559
0.6371498
.6292426
.6216169
.6142566
.6071468
0.6002739
.5936250
.5871886
.5809536
.5749099
0.5690480
.5633590
4 .5578348
45524676
.5472503
0.5421759
.5372382
.5324313
.5277494
.5231874
2.0085537A) 3.0
2.2197951A) 3.1
2.4532530A) 3.2
2.7112639A) 3.3
2.9964100A) 3.4
3.3115452A) 3.5
3.6598234A) 3.6
4.0447304A) 3.7
4.4701184A) 3.8
4.9402449A) 3.9
5.4598150A) 4.0
6.0340288A) 4.1
6.6686331A) 4.2
7.3699794A) 4.3
8.1450869A) 4.4
9.0017131A) 4.5
9.9484316A) 4.6
1.0994717B) 4.7
1.2151042B) 4.8
1.3428978B) 4.9
1.4841316B) 5.0
1.6402191B) 5.1
1.8127224B) 5.2
2.0033681B) 5.3
2.2140642B) 5.4
2.4469193B) 5.5
2.7042641B) 5.6
2.9886740B) 5.7
3.3029956B) 5.8
3.6503747B) 5.9
4.0342879B) 'б.'о
4.4585777B) W.I
4.9274904B) 6.2
5.4457191B) 6.3
6.0184504B) 6.4
X
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
j 7.5
7.6
7. 7
7.8
7. 9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
D.G
U. 7
'.1.8
'. I. 'J
i о. о
TABLE
e"XI0(x)
0.1598048
.1585337
.1572925
.1560802
.1548956
0.1537377
.1526056
.1514982
.1504147
.1493541
0.1483158
.1472990
.1463028
.1453267
.1443699
0.1434318
.1425118
.1416094
.1407239
.1398549
0.1390018
.1381642
.1373417
.1365336
.1357397
0.1349595
.1341927
.1334388
.1326975
.1319684
0.1312513
.1305457
.1298514
.1291681
.1284955
0.1278333
II. e"XIn(x
е-Х1д.(х)
0.1469386
.1459738
.1450270
.1440976
.1431852
0.1422892
. 1414093
.1405450
.1396958
.1388613
0.1380412
.1372350
.1364424
.1356630
.1348965
0.1341425
.1334007
.1326708
.1319524
.1312454
0.1305494
.1298641
.1291892
.1285246
.1278699
0.1272250
.1265895
.1259634
.1253462
.1247379
0.1241382
.1235470
.1229640
.1223891
.1218220
0.1212627
) , eXW ,
eXK0(x)
0.4828474
.47 92978
.4758254
.4724276
.4691016
0.4658451
.4626556
.4595308
.4564686
.4534669
0.4505237
.4476372
.4448056
.4420271
.4393001
0.4366230
.4339944
.4314127
.4288766
.4263848
0.4239360
.4215289
.4191625
.4168355
.4145468
0.4122955
.4100806
.4079010
.4057558
.4036441
0.4015651
.3995180
.3975018
.3955159
.3935596
0.3916319
n
0
0
0
0
0
•
0.
•
0.
•
0.
= 0,1 , eX
еХКд.(х)
.5187402
.5144032
.5101719
.5060421
.5020099
.4980716
.4942235
.4904623
.4867848
.4831880
.4796689
.4762249
.4728534
.4695518
.4663178
4631491
4600436
4569992
454 0139
451085 9
4482134
4453946
4426280
4399119
4372448
4346252
4320519
42 95234
4270385
4245 960
4221945
4198332
4175107
4152261
4129784
4107666
6
7
8
8
9
1
1
1
1
1
1
1
2
2
2
2
3
3
4
4
4
5
6.
6.
7.
8.
8.
9.
1.
1.
1.
1.
1.
1.
1.
2.
ex
.6514163B)
.3509519B)
.1240583B)
.9784729B)
.9227472B)
.0966332C)
.2119671C)
.3394308C)
.4802999C)
.6359844C)
.8080424C)
.9981959C)
.2083480C)
4406020C)
6972823C)
9809580C)
2944681C)
6409503C)
0238724C)
4470667C)
9147688C)
4316596C)
0029122C)
6342440C)
3319735C)
1030839C)
9552927C)
8971291C)
0938019D)
2088381D)
3359727D)
4764782D)
6317607D)
8033745D)
9930370D)
2026466 D-} -
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8.
8.
8.
9.
9.
9.
9.
9.
9.
9.
9.
9.
9.
П.
X
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
n

356
INTEGRALS OF BESSEL FUNCTIONS
X
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.
0.
0.
0,
0
0
J2(x)
0000000
0012490
0049834
0111659
0197347
0306040
0436651
,0587869
,0758178
,0945863
,1149035
.1365642
.1593490
.1830267
.2073559
.2320877
.2569678
.2817389
.3061435
.3299257
.3528340
.3746236
.3950587
.4139146
.4309800
.4460591
.4589729
.4695615
.4776855
.4832271
0.
•
0.
0.
0,
0
0
TABLE
J3(x)
0000000
0000208
0001663
0005593 •
0013201
0025637
0043997
,0069297
, 0102468
, 0144340
. 0195634
.0256945
.0328743
. 0411358
.0504977
.0609640
.0725234
.0851499
.098802 0
.1134234
.1289432
.1452767
.1623255
.1799789
.1981148
.2166004
.2352938
.2540453
.2726986
.2910926
III. Jn(x)
J4(x)
0.0000000
.0000003
.0000042
. 0000210
. 0000661
0.0001607
.0003315
.0006101
.0010330
. 0016406
0.0024766
.0035878
.0050227
.0068310
.0090629
0.0117681
. 0149952
.0187902
. 0231965
.0282535
0.0339957
.0404526
.0476471
.0565957
.0643070
0.0737819
.,0840129
.0949836
.1066687
.1190335
>
0.
¦
0.
0,
0
0
0
n = 2AN
J5(x)
0000000
0000000
0000001
0000006
0000026
0000081
0000199
,0000429
, 0000831
. 0001487
.0002498
.0003987
. 0006101
.0009008
. 0012901
. 0017994
. 0024524
.0032746
.0042936
. 0055385
.0070396
.0088284
. 0109369
.0133973
. 0162417
.0195016
.0232073
.02 73876
.0320690
.0372756
*
J6W
0. 0000000
.0000000
.0000000
.,0000000
.0000001
0. 0000003
.0000010
.0000025
.0000056
.0000112
0.00002 09
.0000368
.0000615
.0000986
.0001523
0.0002280
.0003321
.0004721
.0006569
.0008965
0.0012024
.0015875
.0020660
.0026534
.0033669
0.0042246
. 0052461
. 0064518
.0078634
.0095032
X
0.
•
•
•
•
0.
•
1.
1.
1.
1.
1.
1.
1,
1,
1
1,
2
2
2
2
2
2
2
2
2
2
0
1
2
3
4
5
6
7
8
9
0
1
2
,3
,4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
AWfclU OCU. И'иШ L/CUI1UX j ЛИ . у 1-) ¦ I'-' V.UJ.4 -TV1.W-/ J- J-J. ¦¦¦ i jii.li j. -i_m.VJ_. ^-.^ ^ „^
FUNCTIONS OF THE FIRST KIND, TO ALL SIGNIFICANT ORDERS, Dover Publi-
Publications, New York, 1948, with permission of the publisher.
TABLES OF BESSEL FUNCTIONS
357
TABLE III. Jn(x) , n = 2AN
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
C.I
0.2
C.3
6.4
Jg(x)
0.4860913
.4862070
.4835277
.4780317
.4697226
0.4586292
.4448054
.4283297
.4093043
.3878547
0.3641281
.3382925
.3105347
.2810592
.2500861
0.2178490
.1845931
.1505730
.1160504
' .0812915
0.0465651
.0121398
- .0217184
- .0547481
- .0866954
-0.1173155
- .1463755
- .1736560
- .1989535
- .2220816
-0.2428732
- .2611815
- .2768816
- .2898714
- .3000723
J3(x)
0.3090627
.3264428
.3430664
.3587689
.3733889
0.3867701
.3987627
.4092251
.4180256
.4250437
0.4301715
.4333147
.4343943
.4333470
.4301265
0.4247040
.4170686
.4072280
.3952085
.3810551
0.3648312
.3466186
.3265165
.3046415
.2811260
0.2561179
.2297789
.2022838
.1738184
.1445786
0.1147684
.0845982
. 0542833
. 0240416
¦ .0059077
J4(x)
0.1320342
.1456177
.1597218
:1742754
.1891991
0.2044053
.2197991
.2352786
.2507362
.2660587
0.2811291
.2958266
.3100286
.3236110
.3364501
0.3484230
.3594094
.3692925
.3779603
.3853066
0.3912324
.3956468
.3984683
.3996253
.3990576
0.3967168
.3925672
.3865863
.3787657
.3691107
0.3576416
.3443929
.3294138
.3127681
.2945339
J5U)
0.0430284
.0493448
.0562380
.0637169
.0717854
0.080442 0
.0896797
. 0994854
.10984 00
.1207178
0.132086 7
.1439079
.1561363
.1687200
.1816009
0.1947147
.2079912
.2213550
.2347252
.2480168
0.2611405
.2740039
.2865116
.2985665
.31007 04
0.3209247
.3310313
.3402935
.3486170
.3559105
0.3620871
.3670646
.3707668
.3731243
.3740750
J6(x)
0.0113939
.0135591
.0160220
. 0188061
.0219344
0.0254290
.0293112
.0336009
.0383164
.0434740
0.0490876
. 0551683
.0617245
.0687611
.0762792
0.0842763
.0927455
.1016755
.1110507
.1208502
0.1310487
.1416157
.1525155
.1637077
.1751469
0.1867827
.1985602
.2104199
.2222981
.2341274
0.2458369
.2573523
.2685972
.2794926
.2899584
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4

358
INTEGRALS OP BESSEL FUHCTIOKS
TABLE III. Jn(x) , n = 2AN
!
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
J2U)
-0.3074304
- .3119161
- .3135251
- .3122776
- .3082186
-0.3014172
- .2919660
- .2799797
- .2655949
- .2489678
-0.2302734
- .2097035
- .1874649
- .1637778
- .1388734
-0.1129917
- .0863797
- .0592888
- .0319725
- .0046843
+0.0223247
.0488084
.0745271
.0992506
.1227594
0.1448473
.1653229
.1840111
.2007550
.2154167
0.2278792
.2380464
.2458447
.2512230
.2541532
0.2546303
-0.0353466
- .0640599
- .0918370
- .1184740
- .1437753'
-0.1675556
- .1896411
- .2098717
- .2281019
- .2442023
-0.2580609
- .2695840
- .2786971
- .2853455
- .2894950
-0.2911322
- .2902644
- .2869200
- .2*811478
- .2730169
-0.2626162
- .2500533
- .2354537
- .2189598
- .2007296
-0.1809352
- .1597613
- .1374038
- .1140677
- .0899655
-0.0653153
- .0403388
- .0152594
+ .0096999
.0343183
J4(x)
0.2748027
.2536798
.2312830
.2077417
. 1831965
0.1577981
.1317058
.1050866
.0781139
.0509660
0. 0238247
- .0031260
- .0297016
- .0557187
- .0809963
-0.1053574
- .1286310
- .1506526
- .1712668
- .1903277
-0.2077009
- .2232641
- .2369090
- .2485413
- .2580827
-0.2654708
- .2706601
- .2736223
\- .2743470
- .2728415
-0J691309
- .Й632581
- .2552835
- .2452843
- .2333542
J5W
0.3735654
.3715506
.3679958
.3628760
.3561771
0.3478963
.3380421
.3266347
.3137062
.2993006
0.2834739
.2662935
.2478382
.2281981
.2074735
0.1857748
. 1632215
.1399418
.1160713
.0917524
0.0671330
.0423657
. 0176064
- .0069869
- .0312549
-0.0550389
- .0781816
- .1005286
- .1219297
- .1422400
-0.1613213
- .1790430
- .1952837
- .2099320
- .2228874
J6(*)
0.2999132
.3092757
.3179645
.3258995
.3330022
0.3391966
.3444098
.3485726
.3516206
.3534944
0.3541405
.3535122
.3515695
.3482804
.3436210
0.3375759
.3301390
.3213134
.3111118
.2995568
0.2866809
.2725266
.2571462
.2406017
.2229649
0.2043165
.1847462
.1643521
.1432398
.1215223
0.0993191
.0767551
.0539601
.0310680
.0082154
0.0583794 -0.2196027 xO. 2340615 -0.0144588
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
TABLES OF BESSEL FUNCTIONS
TABLE IV. e" In(x) , n = 2(lN *
359
0.0
.1
.2
.3
.4
0.5
.6
.7
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
¦г.2
:.'.3
;>.4
~%M e-XI3(x)
0. 0000000
.0011320
.0041073
.0083969
.0135860
0.0193521
.0254458
.0316770
.0379022
. 0440151
0.0499388
. 0556193
.0610206
.0661209
.0709088
0.0753811
.0795406
.0833947
.0869539
.0902306
0.0932390
.0959939
. 0985103
.1008034
.1028881
0.1047787
.1064892
.1080327
.1094217
.1106680
0.0000000
. 0000189
.0001368
. 0004191
.0009027
0.0016043
.0025257
. 0036585
.0049877
.0064938
0.0081553
. 0099497
.0118547
. 0138486
.0159110
0.0180231
. 0201679
. 0223299
. 0244955
.0266527
0.0287912
.0309022
.0329781
.0350127
.0370010
0.0389387
.0408227
.0426507
. 0444207
. 0461318
e"XI4(x)
0.0000000
. 0000002
. 0000034
.0000157
.0000450
0.0001000
.0001886
.0003182
.0004948
. 0007233
0.0010069
.0013479
. 0017471
.0022045
.0027189
0.0032885
.0039110
.0045834
.0053023
.006 0642
0.0068654
.0077019
. 0085701
.0094659
.0103857
0.0113259
.0122829
. 0132534
.0142344
.0152228
e'^sCx)
0.0000000
.0000000
.0000001
.0000005
.0000018
0.0000050
.0000113
.0000222
.0000394
.0000647
0.0000999
.0001468
.0002072
.0002 826
.0003746
0.0004843
.0006129
.0007611
.0009298
.0011192
0.0013298
.0015615
.0018142
.0020879
.0023819
0.0026960
.00302 93
.0033813
.0037511
.0041380
e"XI6(x)
0. 0000000
.0000000
.0000000
.0000000
.0000001
0.0000002
.0000006
.0000013
.0000026
.0000048
0.0000083
.0000134
.0000205
.0000303
. 0000432
0.0000597
.0000805
.0001060
.0001369
.0001735
0.0002166
. 0002664
. 0003235
. 0003882
.0004610
0. 0005420
.0006317
.0007301
.0008375
.0009539
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Kxtracted from Watson, G.N., A TREATISE ON THE THEORY OF BESSEL
FUNCTIONS," Cambridge University Press, London and New York, 1945, with
permission of the publisher, and British Association for the Advance-
Advancement of Science, MATHEMATICAL TABLES - VOLUME X, BESSEL FUNCTIONS
PART II - FUNCTIONS OF POSITIVE INTEGER ORDER, Cambridge University
Ргчч;и, New York and London, 1952, with permission of the Royal Society
•mil the publ.i.:,her. The valuer
I.MbLo:; of .L(;(x) K'ven in the
of e"xIg(x) were deduced from the
.Latter reference.

360
INTEGRALS OF BESSEL FUNCTIONS
!
TABLES OF BESSEL FUNCTIONS
361
X
3.
3.
3.
3.
3.
3.
3.
3.
3.
3.
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
e"xI2(x)
0.1117825
.1127758
.1136572
. 1144358
.1151197
0.1157167
.1162339
.1166776
.117 0540
.1173686
0.1176265
.1178323
.1179905
.1181048
.1181791
0.1182166
.1182204
.1181933
.1181380
.1180568
0.1179519
.1178254
.1176790
.1175145
.1173335
0.1171374
.1169276
.1167 052
.1164714
.1162272
0.1159736
.1157115
.1154416
.1151648
,- .1148817
/
TABLE IV.
— V / \
e xI3(x)
0.0477833
.0493750
.0509071
. 0523802 •
.0537949
0. 0551523
.0564535
.0576999
.0588928
.0600338
0. 0611243
. 0621661
.0631607
. 0641096
. 0650147
0.0658774
.0666994
.0674822
.0682274
. 0689364
0.0696107
.0702518
.0708610
.0714396
.0719889
0.0725101
.0730045
. 0734.732
.0759173
.0743378
0.0747357
. 0751121
.0754678
. 0758038
. 0761209
e"XIn(x) , n
— X-r / \
e xl4(x)
0.0162159
.0172112
.0182063
.0191910
.0201876
0.0211700
. 0221447
.0231102
.0240654
. 0250090
0.0259400
.0268576
.0277610
. 0286495
. 0295227
0.0303800
.0312212
.0320458
. 0328538
. 0336449
0.0344190
. 0351762
. 0359163
. 0366395
.0373459
0.0380355
.0387084
.0393650
.0400052
. 0406295
0.0412379
. 0418307
. 0424083
.0429707
. 0435184
= 2AN
e"XI5(x)
0.0045409
.0049590
. 0053913
.0058369
.0062947
0.0067638
.0072431
.0077318
.0082288
.0087333
0.0092443
.0097611
. 0102826
.0108082
.0113371
0.0118685
.0124017
.012 9361
.0134711
. 0140060
0.0145403
.0150735
.0156051
. 0161346
.0166617
0.0171858
.0177068
. 0182241
.0187376
.0192470
0.0197519
.02 02521
.0207475
.0212378
. 0217229
e"xI6(x)
0.0010796
.0012144
.0013584
. 0015116
.0016738
0.0018449
.002 0249
.0022135
.0024106
.0026159
0.0028291
.0030501
.0032785
. 0035141
.0037566
0.0040056
.0042609
. 0045221
.0047890
.0050612
0. 0053384
.0056203
.0059065
.0061968
.0064909
0.0067885
.0070892
.0073928
.0076990
.0080075
0.0083181
.00863 06
. 0089445
.0092599
.0095763
X
3.
3.
3.
3.
3.
3.
3.
3.
3.
3.
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
I
0
1
2
•7 i
3 1
4
5
6 !
8 ']''
9 kg
fj
0 J9
1 Я
2 11
3 II
4 ^Ш
И
5 ш
6 31
7 U
. 8 ,^ffi
• 9 Ц
H
•° flj
.1 mm
•2 Я
•3 Hj
•7 |HJ
¦8 ,^и
.9 Ш1
.0 BJ
.1 HJ
X
6
6
6
6
6
7.
7.
7.
7.
7.
7.
7.
7.
7.
7.
8.
8.
1 8.
8.
8.
8.
8.
8.
8.
8.
9.
9.
9.
9.
9.
9.
9.
9.
9.
9.
1.0.
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
Ь
6
7
8
У
0
е
0
0
0
0
0
0
0
0
-Х12(х)
.1145930
.1142992
.1140008
.1136985
.1133927
.113 0837
.1127720
.1124579
.1121418
.1118240
.1115048
.1111845
.1108632
.1105413
.1102189
.1098962
.1095734
.1092506
.1089281
.1086060
1082843
1079633
107643 0
1073235
1070049
1066873
1063708
1060554
1057413
1054284
105116 9
1048067
1044980
1041908
103885 0
1035808
TABLE IV.
е"Х13(х)
0
0.
0.
0.
,
t
ш
4
0.
0.
4
0.
0.
0764199
0767016
0769668
0772161
0774503
0776700
0778758
0780684
0782482
0784159
0785720
0787169
0788511
0789752
0790895
0791944
0792904
0793778
0794569
0795282
0795920
0796486
0796982
0797412
0797778
0798084
0798331
0798523
0798661
0798748
0798785
0798775
0798721
0798623
0798483
0798304
е"х1п(х) , п
е"Х14(х)
0.0440515
.0445704
.0450754
. 0455667
.0460446
0.0465094
. 0469614
.0474009
.0478282
. 0482436
0.0486473
.0490396
.0494208
.0497911
.0501509
0.0505003
.0508398
. 0511693
.0514894
. 0518001
0.0521017
.0523945
.0526787
.0529545
. 0532221
0.0534817
.0537336
.0539779
. 0542148
.0544445
0.0546673
.0548833
. 0550926
.0552955
.0554921
0.0556826
= 2AN
е"х15(х)
0.0222 027
.0226769
.0231454
.0236083
.0240653
0.0245164
.0249615
.0254006
.0258337
.02626 07
0.0266815
.0270963
.0275049
.027 9074
.0283038
0.0286 941
.0290783
. 0294565
.0298286
.0301948
0.0305551
.0309094
.0312580
.0316 007
.0319378
0.0322691
.0325948
.032 9150
.0332297
.0335390
0.0338429
.0341415
.0344349
.0347231
.035 0062
0.0352843
е"Х16(х)
0.0098936
.0102115
.0105299
.0108486
.0111673
0.0114860
.0118043
. 0121223
.01243 96
. 0127562
0.0130719
.0133866
.0137 001
.0140124
. 0143233
0.0146328
.0149406
. 0152468
.0155513
.0158539
0.0161546
.0164533
.0167500
.0170446
.017337 0
0.0176271
.0179151
.0182007
.018483 9
.0187648
0.0190432
. 0193192
.0195928
.0198638
.0201323
0.0203983
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0

362
INTEGRALS OF BESSEL FOTCTIOHS
TABLE V. (n/2xJJn_i(x) , n = 0(lL *
an(x) = (n/2x)\ i(x)
-1
aQ(x)=x'icos x a}_(x)=x sin x a2(x)
*ъЫ
a4(x)
0.
0.
1.
1.
1.
1.
1.
1.
1.
1.
1,
1,
2,
2
2
2
2
2
2
2
2
2
0
1
2
3
4
5
6
7
8
9
0
1
,2
,3
,4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
CO
9.9500417
4.9003329
3.1844550
2.3026525
1.7551651
1.3755594
1.0926317
0.8708834
.6906777
0.5403023
.4123601
.3019648
.2057683
.1214051
0.0471581
- .0182497
- .0757909
- .1262234
- .1701524
-0.2080734
- .2404029
- .2675 005
- .2896852
- .3072474
-0.3204574
- .3295726
-^ .3348415
- .3365080
- .3348132
1.0000000
.9983342
.9933467 •
.9850674
.9735459
0. 9588511
.9410708
.9203110
.8966951
.8703632
0.8414710
.8101885
.7766992
. 7411986
.7038927
0.6649967
.6247335
.5833322
.5410265
.4980527
0.4546487
.4110521
.3674984
.3242197
.2814430
0.2393889
.1982698
.1582888
.1196386
. 0824 9S8
0.
•
0.
0.
0.
0,
0
0000000
0333000
0664004
0991029
1312122
1625370
1928920
2220983
2499855
2763925
3011687
3241749
3452846
,3643844
,3813754
,3961730
,4087081
,4189275
.4267936
,4322854
.4353978
.4361420
.4345452
.4306503
.4245153
.4162130
.4058302
.3934670
.3792361
.3632614
0.
0.
-
0.
•
0.
•
0.
0.
0000000
0006662
0026591
0059615
0105453
0163711
0233890
0315388
0407505
0509452
0620351
0739248
0865122
0996886
1133403
1273493
1415943
1559516
17 02963
1845032
1984479
2120079
2250633
2374981
2492 Oil
2600667
2699959
2788967
2866857
2 932878
0.
•
0.
0.
0.
0.
0.
0000000
0000095
0000760
0002559
0006041
0011740
002 0163
0031787
0047053
0066361
0090066
0118471
0151829
0190331
0234113
0283246
0337739
0397536
0462516
0532494
0607221
,0686387
,0769623
08565 00
,0946537
.1039205
.1133926
.1230084
.1327027
.1424073
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Extracted from National Bureay of Standards, TABLES OF SPHERICAL BESSEL
FUNCTIONS, v. 1, Columbia University Press, 1947, with permission of
the National Bureau of Standards and the publisher.
TABLES OF BESSEL FUNCTIONS
TABLE V. (tt/2x)^J x(x) , n = 0(lL
n-p
363
an(x) = (n/2xJJn,i(x)
x aQ(x)=x~ cosx a2_(x)=x sinx a2(x)
,3(x)
3.0 -0.3299975
3.1 - .3223017
3.2 - .3119671
3.3 - .2992363
3.4 - .2843524
3.5
3.6
3.7
3.8
3.9
4.5
4.6
4.7
4.8
4.9
-0.2675591
- .2490996
- .2292162
- .2081494
- .1861365
4.0 -0.1634109
4.1 - .1402010
4.2 - .1167288
4.3 - .0932091
4.4 - .0698484
-0.0468435
- .0243810
- .0026359
+ .0182290
.0380637
5.0
5.1
b.2
S.3
j.4
c). 5
,.&
,.1
,.8
..9
0.0567324
. 0741133
.0900994
.1045989
.1175357
0.1288490
.1384939
.1464408
.1526758
.1571997
I-.. 1.
',.¦1
0.1600284
.1611915
. 1607326
.1587077
.1551851
0.0470400
. 0134131
- .0182419
- .0478017
- .0751591
-0.1002238
- .1229223
- .1431990
- .1610152
- .1763503
-0.1892006
- .1995798
- .2075180
- .2130618
- .2162732
-0.2172289
- .2160198
- .2127496
- .2075343
- .2005005
-0.1917849
- .1815323
- .1698951
- .1570316
- .1431045
-0.1282801
- .1127262
- .0966115
- .0801038
- .0633689
-0.0465692
- .0298627
- .0134015
+ .0026689
.0182108
0.3456775
.3266285
.3062665
.2847509
.2622468
0.2389237
.2149545
.1905138
.1657770
.1409185
0.1161107
. 0915230
.0673197
.0436598
.0206954
-0.0014296
- .0225798
- .0426300
- . 0614653
- .0789822
-0.0950894
- .1097078
- .1227715
- .1342275
- .1440366
-0.1521727
- .1586236
- .1633902
- .1664868
- .1679402
-0.1677899
- .1660871
- .1628941
- .1582841
- .1523397
0.2986375
.3026790
.3053668
.3066662
.3065534
0.3050155
.302 0511
.2976696
.2918918
.2847491
0.2762837
.2665478
.2556035
.2435222
.2303837
0.2162759
.2 012938
.18553 90
.1691185
.1521441
0.1347312
.1169983
.0990654
.0810537
.0630842
0.0452768
.0277493
.0106166
- .0060100
- .0220244
-0.0373257
- .0518195
- .0654182
- .0780422
- .0896200
0.1520517
.1615634
.1708691
.1798948
.1885670
0.1968128
.2045609
.2117424
.2182912
.2241445
0.2292439
.2335353
.2369702
.2395055
.2411043
0.2417361
.2413774
.2400119
.2376304
.2342313
0.2298206
.2244120
.2180267
.2106933
.2024479
0.1933334
.1833997
.1727031
.1613057
.1492755
0.1366852
.1236121
.1101375
.0963458
.0823240
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4

364
INTEGRALS OF BESSEL FUNCTIONS
TABLE V. (tt/2xJJ _i(x) , n = 0AL
an(x) = (TT/2xJJn_^(x)
x aQ(x)=x~ cos x а-^(х)=х sinx ag(x) аз(х)
a4(x)
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
.0.0
0.1502443
.1439746
.1364751
.1278526
.1182210
0.1077003
.0964150
.0844932
.072 0654
. 0592632
0.0462180
.0330605
.0199187
.0069174
- .0058231
-0.0181875
- .0300672
- .0413603
- .0519731
- .0618201
-0.0708249
- .0789209
- .0860513
- .0921697
- .0972399
-0.1012367
- .1041452
- .1059613
- .1066909
- .1063503
-0.1049655
- .1025717
- .0992129
- .0949415
- .0898173
-0.0839072
0.0330954
.0472032-
.0604254
.0726637
. 0838318
0.0938552
.1026717
.1102316
.1164982
.1214470
0.1250667
.1273579
.1283335
.1280184
.1264483
0.1236698
.1197395
.1147232
.1086954
.1017380
0.0939397
.0853950
. 0762034
,0664679
. 0^62945
0.045X909
.0350658
.0242272
.01338Ё2
.00263^7
-0.0079106
- .0181590
- .0280166
- .0373958
- .0462157
-0.0544021
-0.1451527
- .1368226
- .1274564
- .1171667
- .1060715
-0.0942924
- .0819542
- .0691833
- .0561068
- .0428514
-0.0295425
- .0163029
- .0032520
+ .0094953
. 0218292
0.0336462
. 0448498
. 0553510
.0650689
. 0739317
0.0818767
. 0888506
.0948103
.0997228
.1035651
0.1063246
.1079986
.1085947
.1081298
.1066307
0.1041328
.1006801
.0963246
.0911256
. 0851490
0.0784669
-0.1000889
- .1093953
- .1174954
- .1243549
- .1299499
-0.1342663
- .1373002
- .1390580
- .1395557
- .1388192
-0.1368837
- .1337932
- .1296005
- .1243664
- .1181587
-0.1110524
- .1031284
- .0944729
- .0851765
- .0753338
-0.0650420
- .0544006
- . 0435101
- .0324714
- .0213849
-0.0103494
+ .0005382
. 0111841
.0214984
. 0313954
0.0407947
.0496216
.0578077
.0652914
. 0720185,
0.0779422
0.
•
-0.
- .
- .
- .
- .
-0.
- .
- .
- .
- .
-0.
- .
- .
- .
- .
-0.
- .
- .
- .
- .
-0.
- .
- .
- .
- .
-0.
- .
- .
- .
-0.
0681612
0539474
0397733
0257293
0119049
0016120
0147361
0273848
0394794
0509454
0617133
0717190
0809042
0892173
0966132
1030540
1085094
1129564
1163801
1187733
1201367
12 04789
1198162
1181725
1155791
1120743
1077 02 9
1025163
0965716
0899310
0826619
0748355
0665268
0578136
0487761
0394958
6.
6.
6.
6.
6.
7.
T.
7.
7.
7.
7.
7.
7.
7.
7.
8.
8.
8.
8.
8.
8.
8.
8.
8.
8.
9.
9.
9.
9.
9.
9.
9.
9.
9.
9.
10.
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
.0
1
.2
.3
Л
.5
,6
,7
.8
.9
,0
TABLES OF BESSEL FUNCTIONS
365
TABLE VI. Jv{x) , v = ±1/4 , ±3/4 *
0.0
.1
.2
.3
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2-6
2.7
P.8
P.9
J-l/4(x)
1.7199851
1.4318694
1.2721878
1.1559365
1.0595996
0.9736936
.8936461
.817 0352
.7425250
0.6693848
.5972478
.5259769
.4555864
.3861925
0.317 9807
.2511829
. 1860614
.1228962
. 0619752
0.0035869
- .0519863
- .1044733
- .1536191
- .1991881
-0.2409679
- .2787715
- .3124404
- .3418454
- -3668893
0.0000000
.520657 9
.6154579
.6742996
.7143957
0.7416566
.7588823
.7676603
. 7690144
.7636791
0.7522313
. 7351612
.7129110
.6858985
.6545312
0.619213 9
.5803540
.5383637
.4936604
.4466673
0.3978111
.3475212
.2962272
.2443567
.1923325
0.1405701
.0894748
.0394390
- .0091605
- .0559651
J-5/4(x)
2.5824445
1.4892322
1.0422622
0.7770330
0.5899242
.4441712
.3234194
.2193486
.1273451
0. 0447011
- .0302308
- .0984742
- .1606700
- .2172184
-0.2683659
- .3142624
- .3549975
- .3906266
- .4211891
-0.4467207
- .4672628
- .4828696
- .4936124
- .4995833
-0.5008973
- .4976941
- .4901382
- .4784194
- .4627520
0.0000000
.1148846
. 1923848
.2588967
.3180234
0.3711055
.4187434
.4612204
.4986670
.5311378
0.5586525
.5812195
.5988503
.6115694
. 6194199
0.6224676
.6208042
.6145479
.6038450
.5888693
0.5698218
.5469301
.5204462
.4906460
.4578261
0.4223024
.3844077
.3444884
.3029022
.2600155
0.0
.1
.2
.3
.4
0.5
.6
.7
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Extracted from National Bureau of Standards, TABLES OF BESSEL FUNCTIONS
OF FRACTIONAL ORDER, Vol. 1, Columbia University Press, New York,
1948, with permission of the National Bureau of Standards and the
publisher.

366
INTEGRALS OF HESSEL FUNCTIONS
X
3.
3.
3.
3.
3.
3.
3.
3.
3.
3.
4.
4.
4.
4.
4.
4.
4.
4.
4.
4.
5
\5
\
5-
5
5
5
5
5
5
5
6
6
6
6
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
,4
,5
.6
.7
.8
.9
.0
.1
^
.3\
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
J_!/4(x)
-0.3875067
- .4036657
- .4153678
- .4226480
- .4255746
-0.4242480
- .4188006
- .4093949
- .3962223
- .3795010
-0.3594744
- .3364087
- .3105902
- .2823231
- .2519268
-0.2197326
- .1860813
- .1513199
- .1157985
- .0798675
-0.0438745
- .0081614
+ .0269388
. 0611044
\ .0940279
^ 1254183
> 1550037
.1825335
.2077799
.23%402
0.2506379
.2679240
.2822782
.2936A)93
.3018Й59
TABLE VI. Jylx)
Jl/4(x)
-0.1006371
- .1428618
- .1823501
- .2188400
- .2520987
-0.2819241
- .3081460
- .3306271
- .3492638
- .3639867
-0.3747606
- .3815845
- .3844909
- .3835457
- .3788467
-0.3705224
- .3587308
- .3436574
- .3255132
"- .3045328
-0.2809721
- .2551051
- .2272223
- .1976271
- .1666332
-0.1345618
- .1017387
- .0684910
- .0351444
- .0020204
+0.0305669
. 0623131
.0929266
.1221307
.1496660
, v = il/4 , i
J_3/4(x)
-0.4433741
- .4205463
- .3945498
- .3656847
- .3342679
-0.3006304
- .2651149
- .2280732
- .1898633
- .1508465
-0.1113845
- .0718365
- .0325562
+ .0061109
.0438305
0.0802817
.1151604
.1481808
.1790781
.2076106
0.2335612
.2567392
.2769815
.2941539
.3081519
0.3189011
.3263574
.3305077
.3313687
.3289871
0.3234387
.3148269
.3032821
.2889600
.2720394
3/4
J5/4(x)
0.2161998
.1718289
.1272764
.0829116
.0390976
-0.0038123
- .0454774
- .0855718
- .1237878
- .1598376
-0.1934559
- .2244019
- .2524608
- .2774459
- .2991995
-0.3175939
- .3325325
- .3439502
- .3518135
- .3561204
-0.3569003
- .3542133
- .3481494
- .3388270
- .3263921
-0.3110163
- .2928951
- .2722460
- .2493062
- .2243300
-0.1975868
- .1693584
- .1399359
- .1096174
- .0787052
X
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
TABLES OF BESSEL FUNCTIONS
367
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
'J.5
9.6
9.7
9.8
9.9
1). 0
TABLE VI. Ju(x)
J-1/4W
0.3069865
.3089994
.3079221
.3038110
.2967506
0.2868519
.2742514
.2591094
.2416082
.2219503
0.2003559
.1770611
.1523152
.1263780
.0995179
0.0720085
.0441266
. 0161489
- .0116498
- .0390000
-0.0656395
- .0913159
- .1157892
- .1388333
- .1602389
-0.1798143
- .1973881
- .2128099
- .2259516
- .2367088
-0.2450012
- .2507733
- .2539946
- .2546597
- .2527880
-0.2484237
0.1752928
.1987928
.2199708
.2386565
.2547054
0.2680000
.2784506
.2859956
.2906021
.2922654
0.2910090
.2868840
.2799683
.2703654
.2582033
0.2436331
.2268271
.2079769
.1872916
.1649956
0.1413261
.1165308
.0908655
.0645915
.0379733
0. 0112754
- .0152394
- .0413131
- .0666947
- .0911424
-0.1144262
- .1363297
- .1566520
- .1752096
- .1918382
-0.2063938
, v = +1/4 , +3/4
J-3/4W
0.2527211
.2312252
.2077892
.1826654
.1561186
0.1284234
.0998616
.0707194
.0412852
.0118460
-0.0173143
- .0459182
- .0736969
- .1003924
- .1257603
-0.1495717
- .1716151
- .1916984
- .2096503
- .2253220
-0.2385880
- .2493473
- .2575241
- .2630679
- .2659544
-0.2661849
- .2637862
- .2588103
- .2513332
- .2414545
-0.2292959
- .2149996
- .1987275
- .1806586
- .1609877
-0.1399232
J5/4(*)
-0.0475027
- .0163118
+ .0145696
. 0448509
. 0742505
0.1024993
.1293421
.1545405
.1778744
.1991442
0.2181723
.2348045
.2489112
.2603883
.2691578
0.2751687
.2783965
.2788438
.2765395
.2715387
0.2639214
.2537920
.2412777
.2265272
.2097091
0.1910101
.1706328
.1487942
.1257228
.1016568
0.0768416
.0515272
.0259661
.0004106
- .0248895
-0.0496893
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0

368
IHTEGBAIS OF HESSEL FUNCTIONS
X
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
J-i/зМ
CO
1.997 0537
1.5672305
1.3432 949
1.1879338
1.06442 05
.9582049
. 8623192
.7730526
.6883111
0.6068875
.5280981
.4515854
.3772032
.3049452
0.2348995
.1672171
. 102 0895
.0397330
- .0196239
-0.0757500
- .1284188
- .1774149
- .2225387
- .2636108
-0.3004752
- .3330020
- .3610897
- .3846672
- .4036942
TABLE VII. Ju(x)
Jl/3(*)
0.0000000
.4117819
.5158967
.5850148
.6354112
0.6728308
.7000271
.7185627
.7294377
.7333598
0.7308764
.7224452
.7084752
.6893506
.6654453
0.6371326
.6047900
.568802 0
.5295619
.4874713
0.4429398
.3963830
.3482210
.2988759
.2487696
0.1983209
.1479429
.0980398
.0490046
.0012161
, v = ±1/3 , :
J-2/3(x)
CO
2.7297582
1.6808383
1.2337379
0.9624591
0.7683442
.6155527
.4878613
.3769031
.2779879
0.1883403
.1062622
.0306972
- .0390135
- .1032744
-0.1623263
- .2163034
- .2652732
- .3092625
- .3482771
-0.3823156
- .4113791
- .4354791
- .4546428
- .4689175
-0.4783731
- .4831043
- .4832317
- .4789024
- .4702899
±2/3 *
J2/3(x)
0.0000000
.1501170
.2372244
.3085208
.3698149
0.4233108
.4700616
.5106313
.5453450
.5744049
0.5979500
.6160899
.6289258
.6365631
.6391200
0.6367323
.6295577
.6177769
.6015950
.5812414
0.5569697
.5290559
.4977977
.4635121
.4265338
0.3872124
.3459100
.3029983
.2588562
.2138663
X
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Extracted froirf National Bureau of Standards, TABLES OF BESSEL FUNCTIONS
OF FRACTIONAL ORDER, Vol. 1, Columbia University Press, New York,
1948, with permission of the National Bureau of Standards and the
publisher.
M
TABLES OF BESSEL FUNCTIONS
TABLE VII. Ju(x) , v = ±1/3 , +2/3
369
X
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
r>.3
j.4
>.5
..6
>.7
..8
!>.9
r,.O
I-..1
Г..2
Г..З
¦..4
J
-0
-
-
-
-
-0
-
-
-
-
-0
-
-
-
-
-0
-
-
-
-
+0
0
0
-1/3 M
.4181629
.4280978
.4335560
.4346268
.4314312
.4241205
.4128753
.3979042
.3794411
.3577443
.3330932
.3057864
.2761389
.2444796
.2111478
.1764910
.1408614
.1046131
.0680988
.0316673
.0043399
.0395913
.0737681
.1065667
.1377016
.1669071
.1939397
.2185799
.2406339
.2599349
2763443
.2897525
30007 97
3072758
3113210
-0.0449638
- .0891928
- .1311505
- .1705405
- .2070929
-0.2405659
- .2707474
- .2974564
- .3205442
- .3398952
-0.3554274
- .3670927
- .3748770
- .3787997
- .3789131
-0.3753019
- .3680815
- .3573972
- .3434221
- .3263554
-0.3064205
- .2838623
- .2589454
- .2319509
- .2031741
-0.1729216
- .1415082
- .1092543
- .0764830
- .0435167
-0.0106747
+ .0217298
.0533927
.0840213
.1133370
J-2/3W
-0.4575938
- .4410386
- .4208724
- .3973657
- .3708092
-0.3415113
- .3097965
- .2760023
- .2404767
- .2035752
-0.1656584
- .1270886
- .0882268
- .0494304
- .0110495
+0.0265755
. 0631165
.0982608
.1317132
.1631983
0.1924629
.2192778
.2434392
.2647706
.2831239
0.2983801
.3104502
.3192756
.3248283
.3271107
0.3261551
.3220231
.3148050
.3046181
.2916057
J2/3(x)
0.1684122
.1228753
.0776320
.0330505
- .0105118
-0.0527116
- .0932219
- .1317348
- .1679638
- .2016460
-0.2325441
- .2604479
- .2851760
- .3065771
- .3245307
-0.3389481
- .3497725
- .3569792
- .3605758
- .3606012
-0.3571253
- .3502480
- .3400977
- .3268302
- .3106267
-0.2916921
- .2702528
- .2465545
- .2208595
- .1934448
-0.1645987
- .1346185
- .1038076
- .0724727
- .0409208
X
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4

370
INTEGRALS OF HESSEL FUNCTIONS
X
6.
6.
6.
6.
6.
7.
7.
7.
7.
7.
7.
7.
7.
7.
7.
8.
8.
8.
8.
8.
8.
8,
8
8,
8,
9
9
9
9
9
9
9
9
9
9
10
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9 V
0
l
2
,3
,4
.5
.6
.7
-8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
J_l/3(x)
0.3122253
.3100280
.3047971
.2966283
.2856436
0.2719900
.2558377
.2373786
.2168236
.1944010
0.1703540
.1449380
. 1184182
. 0910673
- .0631623
OL 0349823
/. 0068056
- .0210928
- .0484439
( - . 0749871
-0.1004725
- .1246633
- .1473379
- .1682918
- .1873392
-0.2043150
- .2190759
- .2315013
- .2414948
- .2489845
-0.2539233
- .2562897
- .2560872
- .2533441
- .2481132
-0.2404711
TABLE
J
0.
#
•
0.
¦
0.
¦
0.
0.
0
_
_
-
-0
-
-0
VII. Jv{x)
1/3(*)
1410775
1669992
1908793
2125171
2317362
2483853
2623395
2735015
2818013
2871974
2896766
2892534
2859699
,2798952
.2711241
,2597762
,2459942
.2299425
.2118057
.1917857
.1701008
.1469824
.1226734
.0974252
.0714958
. 0451467
.0186408
.0077604
.0337991
.0592239
.0837918
.1072706
.1294411
.1500990
.1690567
.1861452
J.2/3(x)
0.2759353
.2577969
.2374008
.2149757
.1907661
0.1650301
.1380369
.1100636
.0813934
.0523123
0.0231065
- .0059399
- .0345481
- .0624467
- .0893744
-0.1150824
- .1393361
- .1619180
0 .1826287
- .2012892
-0.2177421
- .2318529
- .2435112
- .2526312
- .2591526
-0.26304 07
- .2642863
- .2629062
- .2589421
- .25246 01
-0.2435501
- .2323246
- .2189172
- .2034815
- .1861890
-0.1672277
12/3
-0.0094569
+ .0216194
.0520162
. 0814518
.1096576
0.1363798
.1613822
.1844477
.2053806
.2240074
0.2401790
.2537713
.2646862
.2728523
.2782252
0.2807877
.2805496
.2775473
.2718431
.2635248
0.2527039
.2395147
.2241130
.2066739
.1873903
0.1664707
.1441371
.1206230
.0961707
.0710291
0.0454511
.0196914
- .0059961
- .0313608
- .0561577
-0.0801496
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
TABLE
TABLES OF BESSEL FUNCTIONS
VIII. x"(n+*}In+i(x) , e"\+i(x) , n = 0AL *
Ъ» = x-^I^x)
371
X
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
¦?. 2
¦?.ъ
? .4
;\5
,¦.6
,\7
,'.0
,'.9
' . 0
'..1
'.4
0
0
0,
1,
1,
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
2.
2.
2.
2.
2.
2.
3.
3.
3.
.7978846
.7992150
.8032144
.8099068
.8193323
.8315478
.8466268
.86466 03
.8857576
.9100465
.9376749
. 9688114
.0036466
.0423947
.0852948
.1326128
.1846431
.2417111
,3041752
,3724299
4469080
5280845
6164797
7126633
8172583
930946
054471
188646
334359
492581
664369
850880
053376
273231
511951
0
0,
0.
•
0.
.
0.
.
•
0.
0.
•
-пч — •
4(x)
.2659615
.2662276
.2670269
.2683629
.2702413
.2726702
.2756601
.2792238
.2833768
.2881370
.2935253
.2995652
.3062830
,3137085
,3218744
3308171
3405766
3511966
3627252
3752148
3887225
4033104
4190462
4360033
4542611
4739061
4950318
5177393
5421385
5683479
5964959
6267214
6591748
6940184
7314282
-П+Г"'
10- Ъ2(х) Ю2-Ъ3(х)
0
0
0
0.
¦
0.
•
0.
•
0.
1.
1.
1.
1.
.5319230
.5323031
.5334445
.5353511
.5380292
.5414879
.5457386
.5507956
.5566760
.5633996
.57 09891
.5794703
.5888720
.5992265
.6105694
.6229398
.6363807
,6509389
,6666656
.6836164
7 018514
7214358
7424401
76494 05
7890190
8147641
8422712
8716428
9029893
9364294
9720907
0101104
0506359
0938256
1398496
0.7598901
.7603123
.7615802
.7636973
.7666692
0.7705042
.7752128
.7808078
.7873046
.7947212
0.8030780
.8123983
.8227080
.8340358
.8464136
0.8598763
.8744621
.8902123
.9071722
.9253905
0.9449199
.9658175
.9881445
1.0119666
1.0373548
1.0643850
1.0931386
1.1237029
1.1561714
1.1906440
1.2272279
1.2660376
1.3071954
1.3508322
1.3970879
103-Ъ4(х)
0
0
0
0.
•
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
.8443223
.8447061
.8458586
.8477823
.8504817
.8539631
. 8582345
.8633058
.8691889
.8758973
.8834469
.8918552
.9011422
.9113296
. 9224416
.9345047
.9475477
.9616020
.9767013
,9928824
0101847
0286507
0483260
0692594
0915033
1151136
1401501
1666767
1947616
2244774
2559016
2891168
3242109
3612775
4004163
0
0
1
1
1
1
1
1
1
1,
1,
1,
2.
2.
2.
2.
2.
2.
2.
2.
2.
2.
3.
3.
3.
3.
3.
X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
,3
,4
.5
6
7
8
9
0
1
2
3
4
Extracted from Jones, C.W., A SH03T TABLE FOR THE BESSEL FUNCTIONS
In+i(x) > 2/ti К +i(x) , Camtaidge University Press, London and New York,
1956, with permission оГ the Royal Society and the publisher.

372
INTEGRALS 0У BESSEL FUNCTIONS
TABLE VIII. x-(n+i)ln+A(x) , e"xIn+|(x) , n = 0AL
Ъп(х) = x-(n+4)ln+A(x)
Ъх(х) 10-Ъ2(х) 10^-Ъ3(х) 103-Ъ4(х)
Ь0(х)
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
X
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
3.
4.
4.
4.
5.
5.
5.
6.
6.
7.
7.
8.
9.
10.
10.
11.
0.
0.
0.
771173
052690
358453
690597
051449
443551
869678
332861
836407
383931
979380
627063
331691
098408
932837
841119
1784043
1766480
1749424
1732851
1716739
1701067
1685816
1670967
.1656502
. 1642407
, 1628665
. 1615262
.1602184
,1589419
.1576953
0.7715941
.8147218
.8610336
.9107701
.9641915
1.0215792
1.0832377
1.1494965
1.2207123
1.2972708
1.3795897
1.4681213
1.5633551
1.6658211
1.7760935
1.8947941
cx(x)
0.1427396
.1420243
.1413103
.1405985
.1398895
0.1391839
.1384823
.1377852
.1370929
.1364058
0.1357241
.1350481
.1343781
.1337141
.1330563
1.1888907
1.2411453
1.2968242
1.3561541
1.4193783
1.4867584
1.5585753
1.6351310
1.7167498
1.8037804
1.8965977
1.9956045
2.1012339
2.2139518
2.3342592
2.4626947
(x) = e"xIn+i
10 • c2(x)
0.9276052
.9310434
.9341726
.9370110
.9395756
0.9418822
.9439461
. 9457814
.9474013
.9488184
0.9500446
.9510909
.9519678
.9526850
.9532520
1.4461120
1.4980642
1.5531154
1.6114479
1.6732565
1.7387497
1.8081500
1.8816952
1.9596396
2.0422549
2.1298315
2.2226799
2.3211322
2.4255433
2.5362929
2.6537870
(x)
10 • c3(x)
0.4997913
.5074555
.5148602
.5220120
.5289172
0.5355825
.5420143
.5482192
.5542037
.5599742
0.5655370
.5708985
.5760646
.5810415
.5858351
1.4417335
1.4853419
1.5313618
1.5799209
1.6311554
1.6852100
1.7422387
1.8024054
1.8658845
1. 9328616-
2.0035340
2.0781120
2.1568192
2.2398938
2.3275892
2.4201754
10 • C4(x)
0.2278974
.2345359
.2410915
.2475612
.2539421
0.2602318
.2664282
.2725297
.2785347
.2844422
0.2902514
.2959615
.3015722
.3070833
.3124948
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
X
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.X
6.2
6.3
6.4
TABLES OF BESSEL FUNCTIONS
TABLE VIII. x-(n+*)ln+4(x) , e'X^Cx) , n
cn(x) = e-XIn+|(x)
373
= 0AL
=oW
=l(x)
Ю- c2(x) 10-c3(x) 10-c4(x)
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.0
9.6
9.7
9.8
9.9
0.0
0.1564777
.1552878
.1541246
.1529872
.1518746
0.1507859
.1497203
.1486769
.1476551
.1466540
0.1456731
.1447115
.1437688
.1428442
.1419373
0.1410474
.1401740
.1393167
.1384749
.1376482
0.1368361
.1360382
.1352541
.1344834
.1337258
0.1329808
.1322481
.1315274
.1308183
.13012 06
0.1294340
.1287581
.1280927
.1274374
.1267922
0.1261566
0.1324049
.1317599
.1311214
.1304895
.1298641
0.1292453
. 1286331
.1280275
.1274285
.1268360
0.1262501
.1256706
.1250976
.1245309
.1239706
0.1234165
. 1228686
.1223269
.1217912
.1212615
0.1207377
.1202198
.1197077
.1192012
.1187004
0.1182051
.1177153
.1172309
.1167518
.1162780
0.1158093
.1153458
. 1148872
.1144336
.1139849
0.1135410
0.9536773
.9539692
. 9541354
. 9541831
.9541193
0.9539504
.9536825
.9533213
.9528722
. 9523403
0.9517304
.9510472
.9502947
.9494772
. 9485984
0.9476620
.9466712
.9456295
.9445397
.9434048
0.9422276
.9410104
.9397559
.9384664
.9371439
0.9357905
.9344083
.9329990
.9315644
.9301062
0.9286260
.9271251
.9256052
. Э240674
. 9225130
0.9209434
0.5904511
.5948951
.5991728
.6 032893
.6072499
0.6110598
.6147237
.6182465
.6216329
.6248872
0.6280140
.6310173
.6339013
.6366699
.6393269
0.6418762
.6443211
.6466653
.6489119
.6510644
0.6531257
.6550989
.6569869
.6587926
.6605186
0.6621676
.6637421
.6652446
.6666773
.6680427
0.6693429
.6705800
.6717561
.6728732
.6739332
0.6749380
0.3178069
.3230198
.3281340
.3331501
.3380687
0.3428907
.3476169
.3522483
.3567858
.3612307
0.3655841
.369847 0
.3740209
.3781068
.3821062
0.3860203
.3898505
.3935982
.3972646
.4008512
0.4043593
.4077904
.4111458
.4144268
.4176348
0.42 07713
.4238374
.4268347
.4297643
.4326276
0.4354259
.4381605
.4408327
.4434436
.4459946
0.4484868
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0

374
I-l/4(x)
INTEGRALS OF BESSEL FUNCTIONS
TABLE IX. 1„(х) , v = ±1/4 , ±3/4 *
[-3/4(x)
0.0
.1
.2
.3
.4
0.5
.6
.7
.8/
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
X?
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
CO
1.7314900
1.4705672
1.3508656
1.2861067
1.250.9702
Д;2'384027
--" 1.24 04066
1.2552377
1.2813195
1.3177529
1.3640690
1.42 00957
1.4858805
1.5616453
1.6477593
1.7447224
1.8531569
1.9738038
,' J.1075223
2.2552929
2.4182222
2.5975496
2.7946563
3.0110754
3.2485040
3.5088168
3.7940817
4.1065770
4.4488103
0.0000000
.5227447
.6253845
.6990174
.7616164
0.8196760
.8764808
.934 0401
.9937580
1.0567231
1.1238519
1.1959691
1.2738563
1.3582835
1.4500325
1.5499137
1.6587808
1.7775426
1.9071749
" 2.0487314
2.2033545
2.3722866
2.5568820
2.7586193
2.9791147
3.22 01363
3.48362 00
3.7716865
4.0866593
4.4310859
CO
2.6346146
1.6133150
1.2482474
1.0721795
0.9800768
.9343098
.9179580
.9223276
.9425725
0.9758675
1.0205489
1.0756719
1.1407679
1.2157032
1.3 005955
1.3957616
1.5016863
1.6190031
1.7484842
1.8910354
2.0476965
2.2196442
2.4081979
2.6148279
2.8411660
3.0890172
3.3603746
3.6574351
3.9826179
0.0000000
.1152133
.1945961
.2656404
.3328996
0.3985851
.4641153
.5305533
.5987802
.6695802
0.7436871
.8218142
.9046745
.9929953
1. 0875313
1.1890745
1.2984643
1.4165970
1.5444345
1.6830142
1.8334588
1.9969864
2.1749219
2.3687083
2.5799203
2.8102772
3.0616593
3.3361232
3.6359204
3.9635170
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Extracted from National Bureau of Standards, TABLES OF BESSEL FUNCTIONS
OF FRACTIONAL ORDER, Vol. 2, Columbia University Press, New York,
1949, with permission of the National Bureau of Standards and the
publisher.
TABLES OF BESSEL FUNCTIONS
375
TABLE IX. Iv(x) , v = ±1/4 , ±3/4
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
.3
.4
.5
.6
.7
.8
:,.O
с .0
(i.4
^1/4 (*)
4.8235404
5.2338004
5.6829239
6.1745739
6.7127745
7.3019457
7.9469418
8.6530940
9.4262569
10.2728597
11.1999627
12.2153193
13.3274442
14.5456886
15.8803227
17.3426264
18.9449893
20.70102 07
22.6256699
24.7353597
27.0481323
29.5838100
32.3641724
35.4131504
38.7570408
42.4247418
46.4480114
50.8617531
55.7043291
61.0179054
66.8488315
73.2480581
80.2715966
87.9810251
96.4440448
4.8077592
5.2197421
5.6703946
6.1634023
6.7028092
7.2930528
7.9390030
8.6460044
9.4199236
10.2672001
11.1949036
12.2107957
13.3233983
14.5420690
15.877 0836
17.3397272
18.9423937
20.6986964
22.6235881
24.7334947
27.0464612
29.5823124
32.3628299
35.4119468
38.7559617
42.4237740
46.4471433
50.8609743
55.7036304
61.0172784
66.8482688
73.2475530
80.2711432
87.9806180
96.4436792
I-3/4(x)
4.3385847
4.7282628
5.1548699
5.6219419
6.1333640
6.6934043
7.3067513
7.9785549
8.7144722
9.5207167
10.4041137
11.3721607
12.433 0937
13.5959603
14.8707003
16.2682343
17.8005608
19.4808635
21.3236290
23.3447767
25.5618012
27.9939293
30.6622926
33.5901176
36.8029346
40.3288077
44.1985879
48.4461916
53.1089070
58.2277313
63.8477420
7 0.0185052
76.7945251
84.2357390
92.4080617
4.3216154
4.7131775
5.1414515
5.6099996
6.1227299
6.6839305
7.2983 072
7.9710254
8.7077554
9.5147226
10.3987626
11.3673820
12.4288247
13.5921454
14.8672903
16.2651852
17.7978338
19.4784238
21.3214459
23.3428227
25.5600518
27.9923628
30.6608895
33.5888607
36.8018084
40.3277984
44.1976833
48.4453806
53.1081798
58.2270792
63.8471571
7 0.0179805
76.7940544
84.2353166
92.4076826
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4

376
INTEGRALS OF BESSEL FUNCTIOHS
TABLE IX. I (x) , v = ±1/4 , +3/4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7^3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
^1/4 (*)
105.7350927
115.9360159
127.1368131
139.4364514
152.9437650
167.7784444
184.0721253
201.9695870
""21-.-63QQ714
243.228735?
266.9582510
293.0305620
321.6788244
353.1595385
387.7548970
425.7753709
467.5625559
513.4923073
563.9781906
619.4752819
680.4843525
747.5564761
821.2981032
902.3766486
991.5266444
1089.5565164
1197.3560468
1315.9045931
1446.2801398
1589.6692680
105.7347645
115.9357211
127.1365483
139.4362135
152.9435512
167.7782524
184.0719528
201.9694319
221.6299320
ч 243.2286105
266.9581384
293.0304607
321.6787334
353.1594566
387.7548234
425.7753047
467.5624964
513.4922538
563.9781424
619.4752386
680.4843135
747.5564410
821.2980717
902.3766202
991.5266189
1089.5564934
1197.3560261
1315.9045745
1446.2801231
1589.6692529
I-5/4(x)
101.3839848
111.2432367
122.0735086
133.9712541
147.0425692
161.4041609
177.1844139
194.5245646
213.5799939
234.5216500
257.5376156
282.8348332
310.6410041
341.2066804
374.8075681
411.7470628
452.3590422
497.0109418
546.1071406
600.0926912
659.4574260
724.7404811
796.5352771
875.4950057
962.3386721
1057.8577504
1162.9235137
1278.4951081
1405.6284460
1545.4860021
101.3836446
111.2429313
122.0732344
133.9710079
147.0423481
161.4039623
177.1842355
194.5244044
213.5798499
234.5215207
257.5374994
282.8347288
310.6409102
341.2065961
374.8074923
411.7469946
452.3589809
497.0108867
546.1070911
600.0926466
659.4573860
724.7404451
796.5352447
875.4949765
962.3386459
1057.8577268
1162.9234925
1278.4950890
1405.6284288
1545.4859867
1747.
1920.
2111.
2321.
2552.
3781350
8445688
6513884
5410757
4319375
1747.
1920.
2111.
2321,
2552.
3781214
8445566
6513774
5410658
4319286
1699,
1868.
2054,
2259,
2485.
3476039
6223176
8615420
7734319
2387875
1699.
1868.
2054.
2259.
2485.
3475900
6223051
8615307
7734218
2387784
10.0 2806.4359071 2806.4358991 2733.3285584 2733.3285502 10.0
TABLES OF BESSEL FUNCTIONS
377
TABLE X. 1и(х) , v = ±1/3 , ±2/3
[-1/з(*)
-2/3 (x)
0.
0
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
2 m
2.
2.
2-
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2
1
1
1
1
1
1
1
1
1.
1.
1.
1.
1.
1.
1.
1.
1.
2.
2.
2.
2.
2.
2.
3.
3.
3.
4.
4.
CO
0120879
6149615
4371141
3394785
2842546
2561471
2476400
2546101
2746558
3063509
3488729
4018033
4650122
5385906
6228083
7180892
8249957
9442209
0765853
2230372
3846561
5626584
7584051
9734111
2093571
4681025
7517 007
0624157
4027413
0. 0000000
.4133289
.5236935
.6050965
.6747060
0.7389732
.8012491
. 8636228
.9275727
. 9942516
1.0646314
1.1395838
1.2199303
1.3064753
1.4000298
1.5014290
1.6115477
1.7313128
1.8617151
2.0038208
2.1587826
2.3278511
2.5123870
2.7138733
2.9339291
3.1743242
3.4369943
3.7240585
4.0378376
4.3808746
CO
2.7710137
1.7847962
1.4122449
1.2244377
1.1211475
1.0655452
1.0407379
1.0378964
1.0520188
1.0801397
1.1204768
1.1719880
1.2341247
1.3066883
1.3897433
1.4835637
1.5885996
1.7054569
1.8348860
1.9777766
2.1351566
2.3081949
2.4982070
2.7066626
2.9351960
3.1856181
3.4599304
3.7603414
4.0892847
0.0000000
. 1505680
.2400883
.3169644
.3879996
0.4562832
.5236824
.5915239
.6608584
. 7325854
0.8075213
. 8864402
.9701013
1.0592689
1.1547268
1.2572918
1.3678244
1.4872393
1.6165154
1.7567061
1.9089493
2.0744783
2.2546333
2.4508730
2.6647883
2.8981162
3.1527548
3.4307805
3.7344659
4.0662997
0.
0.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
2.
2.
2.
2.
2.
2.
2.
2.
2.
2.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
Extracted from National Bureau of Standards, TABLES OF BESSEL
FUNCTIONS OF FRACTIONAL ORDER, Vol. 2, Columbia University Press,
New York, 1949, with permission of the National Bureau of Standards
and the publisher.

378
E-l/5
С*)
INTEGRALS OF BESSEL FUNCTIONS
TABLE X. Iy(x) , v = ±1/3 , ±2/3
l-2/3^
3.
3.
3.
3.
3.
3.
3.
3.
3.
3.
4.
4.
4.
4.
4.
4.
4.
4,
4.
4,
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
4.
5.
5.
6.
6.
7.
7.
8.
9.
10.
11.
12.
13.
14.
15.
17.
18.
20.
22.
24,
26,
29
32
35
38
42
46
50
55
60
66
72
79
87
96
7754221
1834770
6302249
1193134
6547504
2409387
8827145
5853895
3547974
1973450
1200684
1306947
2377104
4504360
.77 91086
,2349722
.8303773
.5788908
.4954161
.5963262
.8996089
.4250277
.1942980
.2312814
.5621989
.2158664
.2239525
.6212634
.4460555
.7403797
.5504602
.9271114
. 9261963
.6091316
.0434425
4.
5.
5.
6.
6.
7.
7.
8.
9.
10.
11.
12.
13.
14.
15.
17.
18,
20,
-22,
24
26
29
32
35
38
42
46
50
55
60
66
72
79
87
96
7559569
1661406
6147769
1055418
642-4679
2299799
8729329
5766556
3469962
1903747
1138384
1251248
2327292
,4459802
.7751217
.2314040
.8271830
.5760306
.4928546
.5940316
.8975531
.4231854
.1926468
.2298011
.5608717
.2146762
.2228850
.6203058
.4451964
.7396089
.5497685
.9264905
.9256390
.6086312
.0429933
4.4494394
4.8437528
5.2754659
5.7481411
6.2656931
6.8324230
7.4530560
8.1327826
8.8773044
9.6928839
10.5864003
11.5654097
12.6382122
13.8139258
15.1025666
16.5151386
18.0637311
19.7616267
21.6234198
23.6651468
25.9044301
28.3606352
31.0550449
34.0110500
37.2543598
40.8132328
44.7187320
49.0050042
53.7095885
58.8737551
64.5428786
70.7668487
77.6005218
85.1042186
93.3442715
4.
4.
5.
5.
6.
6.
7.
8.
8.
9.
10.
11.
12.
13.
15.
16,
18,
19
21
23
25
28
31
34
37
40
44
49
53
58
64
70
77
85
93
4290087
8255817
2592953
7337431
2528669
8209918
4428636
1236909
8691913
6856415
5799328
5596322
6330496
.8093111
. 0984405
.5114484
.0604298
.7586726
.6207758
.6627798
.9023106
.3587369
.0533444
.0095264
.2529943
.8120089
.7176348
.0040204
.7087 062
.8729638
.5421687
.7662118
.5999503
.1037058
.3438112
3.
3.
3.
3.
3.
3.
3.
3.
3.
3.
4.
4.
4.
4.
4.
4.
4,
4,
4,
4
5
5
5
c;
5
5
5
5
5
5
6
6
6
6
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
У.З
J.4
Kb
1.6
i.7
).8
). У
TABLES OF BESSEL FUNCTIONS
TABLE X. Iy(x) , v = +1/3 , ±2/3
379
105.3033750
115.47 05673
126.6347914
138.8947672
152.3590596
167.147 0658
183.3 901017
201.2325984
220.8334184
242.3673051
266.0264787
292.0223923
320.5876658
351.9782150
386.4755953
424.3895827
466.0610150
511.8649202
562.2139619
617.5622319
678.4094272
745.3054496
818.85547 04
899.7255080
988.6485705
1086.4314197
1193.962 0213
1312.2177493
1442.2744223
1585.3162557
1742.6468236
1915.7011320
2106.0589174
2315.4592944
2545.8168907
105.3029715
115.4702050
126.6344660
138.8944749
152.3587970
167.1468298
183.3898897
201.2324079
220.8332471
242.3671512
266.0263404
292.0222679
320.5875540
351.9781144
386.4755049
424.3895014
466.0609419
511.8648545
562.2139028
617.5621787
678.4093793
745.3054065
818.8554316
899.7254731
988.6485391
1086.4313915
1193.9619959
1312.2177265
1442.2744017
1585.3162372
I-2/5(x)
102.3936269
112.3325085
123.2491469
135.2405835
148.4135545
162.8854648
178.7854594
196.2556034
215.4521809
236.5471247
259.7295911
285.2076928
313.2104077
343.9896798
377.8227330
415.0146190
455.9010221
500.8513487
550.2721283
604.6107601
664.3596377
730.0606924
802.3103958
881.7652704
969.1479576
1065.2539018
1170.9587113
1287.2262664
1415.1176499
1555.8009838
102.3932138
112.3321375
123.2488138
135.2402844
148.4132859
162.8852235
178.7852427
196.2554087
215.4520059
236.5469675
259.7294498
285.2075659
313.2102936
343.9895772
377.8226408
415.0145361
455.9009476
500.8512816
550.2720680
604.6107059
664.3595890
730.0606485
802.3103564
881.7652349
969.1479257
1065.2538731
117 0.9586855
1287.2262431
1415.1176290
1555.8009650
1742
1915
2106
2315
2545
.6468069
.7011170
.0589039
.4592822
.8168797
1710.
1880,
2068.
2274.
2500.
5622641
8172956
12483 90
2010928
9356459
1710
1880
2068
2274
2500
5622471
8172803
1248253
2010804
9356348
x
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
lu.O
2799.2396196 2799.2396097 2750.4090524 2750.4090423 10.0

380
INTEGRALS OF BESSEL FUNCTIONS
TABLE XI. INTEGRALS OF Jo(x) AMD YQ(x) *
TABLES OF BESSEL FUNCTIONS
381
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Jo
Jo
(t)dt
0.0000000
.0999167
.1993343
.2977576
.3946986
0.4896805
.5822413
.6719368
.7583444
.8410659
0.9197304
.9939971
1.0635577
1.1281384
1.1875020
1.2414495
1.2898210
1.3324969
1.3693986
1.4004885
1.4257703
1.4452882
1.4591263
1.4674080
1.4702 940
1.4679809
1.4606 996
1.4487125
1.4323117
1.4118157
/ Y (t)dt
Jo
0. 0000000
- .2174306
- .3457088
- .4392832
- .5095248
-0.5617955
- .5992716
- .6240996
- .6378689
- .6418402
-0.6370694
- .6244792
- .6049027
- .5791113
- .5478319
-0.5117590
- .4715613
- .4278862
- .3813624
- .3326004
-0.2821929
- .2307132
- .1787150
- .1267297
- .0752650
-0.0248029
+ .0242025
.0713269
.1161778
.1583962
Г t^fl-Jjt^dt Г ЛоA)й1
JO Jx
0.0000000
.0012496
. 0049938
.0112184
. 0199003
0.0310070
. 0444971
.0603206
.0784188
.0987252
0.1211652
.1456572
.1721124
.2004357
.2305261
0.2622772
.2955780
.3303129
.3663631
.4036067
0.4419194
.4811754
.5212478
.5620091
.6033325
0.6450916
.6871619
.7294208
.7717484
.8140279
— CO
-1.3413838
-0.4342307
- .0510783
+ .1523804
0.2696885
.3383921
.3768981
.3954387
.400223 0
0.3952729
.3833291
.3663369
.3457240
.3225670
0.2976970
.2717671
.2452990
.2187136
.1923541
0.1665014
.1413859
.1171968
. 0940880
.0721837
0.0515823
.0323599
.0145725
- .0017414
- .0165593
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
\.i
1/8
li.9
2l0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
The first two columns are extracted from National Bureau of Standards,
TABLES OF FUNCTIONS AND ZEROS OF FUNCTIONS, AMS 37, 1954, pp. 21-31.
The third column is easily derived from data given in the latter refer-
reference, pp. 33-39. The entire ta"ble is availatole in National Bureau of
Standards, HANDBOOK OF MATHEMATICAL TABLES, 1962. All the material is
reprinted here with permission of the National Bureau of Standards.
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
:,.O
:,.l
:..2
:,.3
..4
.Г)
.'.»
. 0
. I.
.1
TABLE XI. INTEGRALS OF JQ(x) AND YQ(x)
f Jo(t)dt f Yo(t)dt f fX{\-Jo(t)}dt f"tYo(t)dt
Jo Jo Jo Jx
1.3875673
1.3599297
1.3292840
1.2960259
1.2605618
1.2233057
1.1846760
1.1450913
1.1049678
1.0647153
1.0247342
0.9854122
.9471213
. 9102152
.8750261
0.8418625
. 8110073
.7827151
.7572111
.7346894
0.7153119
.6992074
.6864710
.6771641
.6713139
0.6689145
.6699268
.6742798
.6818719
.6925719
0.7062212
.7226354
.7416065
.7629051
.7862833
0.1976583
.2336767
.2662021
.2950236
.3199700
0.3409095
.3577504
.3704407
.3789674
.3833561
0.3836696
.3800068
.3725007
.3613169
.3466516
0.3287288
.3077978
.2841310
.2580207
.2297758
0.1997194
.1681849
.1355135
.1020502
. 0681412
0.0341306
.0003568
- .0328499
- .0651705
- . 0963001
-0.1259506
- .1538528
- .1797587
- .2034440
- .2247089
0.8561467
.8979960
.9394719
.9804757
1.0209143
1.0607003
1.0997528
1.1379971
1.1753654
1.2117967
1.2472371
1.2816397
1.3149650
1.3471804
1.3782606
1.4081872
1.4369487
1.4645405
1.4909645
1.5162286
1.5403472
1.5633401
1.5852324
1.6060544
1.6258411
1.6446314
1.6624685
1.6793984
1.6954706
1.7107367
1.7252504
1.7390671
1.7522434
1.7648364
1.7769034
-0.0298727
- .0416861
- .0520155
- .0608874
- .0683376
-0.0744103
- .0791572
- .0826368
- .0849132
- .0860555
-0.0861371
- .0852346
- .0834276
- .0807977
- .0774277
-0.0734012
- .0688020
- .0637132
- .0582169
- .0523937
-0.0463221
- .0400778
- .0337340
- .0273600
- .0210218
-0.0147811
- .0086955
- .0028178
+ .0028036
.0081253
0.0131091
. 0177215
. 0219345
. 0257247
.0290741
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4.
4.
4.
4.
4.
5.
5.
5.
5.
5.
5.
5.
5.
5.
5.
6.
6.
6.
6.
6.
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
,9
,0
.1
2
3
4
5
6
7
8
9
0
1
2
3
4

382
INTEGRALS OF BESSEL FUNCTIONS
TABLE XI. INTEGRALS OF Jo(x) AND Yo(x)
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
i
0.
0.
#
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1,
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
X
Jo(t )dt
8114767
8382077
8661878
8951209
9247061
9546403
9846217
0143521
0435401
0719033
0991714
1250885
1494149
1719300
1924333
,2107468
,2267161
.2402114
.2511289
.2593913
.2649480
.2677758
.2678783
.2652858
.2600546
.2522664
.2420271
.2294652
.2147308
.1979938
.1794418
.1592783
.1377206
.1149972
.0913459
.0670113
nX
Yo(t)dt
Jo
-0.2433806
- .2593137
- .2723918
- .2825279
- .2896645
-0.2937745
- .2948602
- .2929536
- .2881150
- .2804327
-0.2700213
- .2570206
- .2415937
- .2239252
- .2042194
-0.1826975
-. .1595961
-* .1351640
- .1096602
- .0833508
-0.0565066
- .0294008
- .0023055
+ .0245102
.0507830
0.0762580
.1006909
.1238504
.14552 02
.1655010
0.1836121
.1996932
.2136056
.2252334
.2344843
0.2412903
f t^fL-JoCt)}
Jo L
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1,
1,
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
7885019
7996885
8105190
8210478
8313276
8414092
8513411
8611691
8709363
8806828
8904454
9002577
9101495
9201475
9302743
9405491
9509876
, 9616015
, 9723992
, 9833855
.9945620
.0059270
.0174757
.0292004
.0410908
.0531340
. 0653150
.0776168
.0900204
.1025056
.1150507
.1276333
.1402299
.1528170
.16537 04
.1778664
dt f t'^CtJdt
Jx
0.0319695
.0344026
.0363698
.0378721
. 0389145
0.0395063
.03966 09
.0393947
.0387277
.0376829
0.0362858
. 0345642
\ .032547 9
\ .0302684
\ .0277583
) 0.0250514
/ .0221818
- .0191841
.0160928
.012942 0
0.0097652
.0065949
.0034624
.0003977
- .0025711
-0.0054176
- .0081175
- .0106488
- .0129916
- .0151288
-0.0170455
- .0187296
- .0201715
- .0213643
- .0223037
-0.0229880
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
• 7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
TABLES OF BESSEL FUNCTIONS
TABLE XII. INTEGRALS OF Io(x) AND
383
Ko(x) *
fo(x) = e"x f I0(t)dt, gQ(x) = ex f K0(t)dt,
 Jx
fi(x) = 10-e"X f t^h^t)-!} dt, Sl(x) = xex f V^Jt)dt
JO J Jx
foW
SoW
fl(x)
8l(x)
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
L.5
1.6
L.7
1.8
L.9
~>. 0
\l
'.2
?.3
>.4
• О
'.6
'. V
'. !!
'. 9
0.0000000
.0905592
.1642928
.2239179
.2717246
0.3096429
.3392999
.3620671
.3791005
.3913742
0.3997088
.4047952
.4072152
.4074578
.4059339
0.4029885
.3989109
.3939429
.3882868
.3821111
0.3755557
.3687367
.3617498
.3546738
.3475729
0.3404993
.3334948
.3265930
.3198199
.3131959
1.5707963
1.3578482
1.2503254
1.1728009
1.1117128
1.0612717
1.0183648
0.9810970
.9482180
.9188556
0.8923752
.8682997
.8462610
.8259689
.8071904
0.7897357
.7734480
.7581962
.7438697
.7303744
0.7176295
.7055650
.6941202
.6832416
.6728826
0.6630015
.6535616
.6445298
.6358768
.6275760
0
0
0.
0.
¦
0.
•
0.
0000000
0113140
0409877
0835768
1347363
1910285
2497622
3088584
3667383
4222295
4744889
5229376
5672080
6070995
6425420
6735663
7002797
7228458
7414688
7563806
7678298
7760744
7813746
7839884
7841674
7821544
7781809
7724664
7652168
7566245
0
0.
0.
•
0.
0.
t
•
0.
*
000000
368126
460111
506394
532910
548819
558366
563828
566545
567355
566811
565291
563058
5603 02
557163
553745
550126
546364
542506
538587
534635
530670
526711
522768
518854
514976
511139
507350
503610
499924
0.0
.1
.2
.3
.4
0.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Columns 1 and 2 and that part of Columns 3 and 4 for x = 0@.lM are
taken from National Bureau of Standards, HANDBOOK OF MATHEMATICAL
FUNCTIONS, 1962, with permission of the National Bureau of Standards.

384
INTEGRALS OF BESSEL FUNCTIONS
TABLE XII. INTEGRALS OF Io(x) AND Ko(x)
fo(x) = e"x / Io(t)dt, go(x) = ex / Ko(t)dt,
Jo Jx
fx(x) = 10-e'x f t^lljt)-!} dt, gl(x) = xex Г t-1Ko(t)dt
Jo L Jx
TABLES OF BESSEL FUNCTIONS
385
fo(*)
BoOO
fl(x)
BlOO
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5/
.6
.7
.8
.9
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
.0
.1
.2
.3
.4
0.3067362
.3004518
.2943504
.2884367
.2827131
0.2771802
.2718370
.2666811
.2617094
.2569178
0.2523018
.2478561
.2435756
.2394546
/"-2554874
\
0.2316683
.2279915
.2244513
.2210421
.2177583
0.2145946
.2115458
.2086068
.2 057728
.203 0389
0.2004008
.1978540
.1953944
.1930181
.1907213
0.1885002
. 1863516
.1842720
.1822584
.1803078
0.
0.
0.
0,
0,
0
0
6196034
6119374
6045584
5974484
5905911
5839714
5775757
5713913
5654066
,5596109
,5539942
,5485472
.5432615
,5381291
.5331427
.5282952
.5235803
.5189919
.5145243
.5101724
.5059310
.5017955
.4977616
.4938250
.4899819
.4862286
.4825616
.4789775
.4754734
.4720460
.468692 9
.4654111
.4621983
.4590520
.4559699
0.
0.
0.
0,
0
0
0
7468681
7361124
7245090
7121963
6993006
6859360
6722060
6582 033
,6440109
,6297029
,6153450
.6009952
.5867042
.5725166
.5584708
,5446000
.5309325
.5174921
.5042 989
.4913691
.4787161
.4663501
. 4542 7 9 0"
.4425085
.4310421
.4198818
.4090280
.3984797
.3882349
.3782905
.3686426
.3592865
.3502172
.3414289
.332 9154
0.496292
.492717
.489198
.485736
.482332
0.478984
.475694
.472459
.469280
.466155
0.463085
.460067
.457100
.454185
.451320
0.448503
.445734
.443012
.440335
.437703
0.435115
.432569
.430065
.427601
.425177
0.422792
.420445
.418134
.415860
.413621
0.411416
.409245
.407107
.405001
.402926
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
TABLE XII. INTEGRALS OF IQ(x) AND KQ(x)
XX со
Io(t)dt, go(x) = ex I Ko(t)dt,
0 Jx
fx(x) = 10.e"X Г^1 [lQ(t)-l\ dt, gl(x) = xeX [ "Ло(
JO J Jx
t)dt
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
'J.I
'J.2
!).3
'3.4
;).5
¦J.6
'.1.7
:t.8
'.|.Э
1 ii. 0
fo(x)
0.1784174
.1765844
.1748064
.1730809
.1714055
0.1697782
.1681968
.1666593
.1651639
.1637089
0.1622 924
.1609130
.1595691
.1582593
.1569821
0.1557364
.1545208
.1533342
.1521755
.1510436
0.1499374
.1488561
.1477988
.1467644
.1457523
0.1447616
.1437916
.1428416
.1419108
.1409987
0.1401046
.1392278
.1383679
.1375243
.1366965
0.1358840
goW
0.4529498
.4499897
.4470876
.4442415
.4414497
0.4387105
.4360222
.4333834
.4307923
.4282476
0.4257481
.423292 0
.4208786
.4185063
.4161740
0.4138807
.4116252
.4094065
.4072237
.4050756
0.4029615
.4008804
.3988315
.3968140
.3948269
0.3928697
.3909415
.3890417
.3871695
. 3853241
0.3835053
.3817120
.3799439
.3782003
.3764806
0.3747843
0.3246704
.3166872
.3089589
.3014786
.2942392
0.2872336
.2804546
.2738954
.2675487
.2614077
0.2554657
.2497158
.2441515
.2387665
.2335544
0.2285091
.2236247
.2188954
.2143156
.2098800
0.2055832
.2014201
.1973859
.1934759
.1896854
0.1860100
.1824456
.1789879
.1756332
.1723775
0.1692173
.1661490
.1631693
.1602749
.1574628
0.1547299
Sl(x)
0.400881
.398867
.396882
.394925
.392997
0.391095
.389221
.387372
.385549
.383751
0.381977
.380228
.378501
.376798
.375117
0.373458
.371820
.3702 03
.368607
.367032
0.365475
.363939
.362421
.360922
.359441
0.357978
.356532
.355104
.353692
.352297
0.350919
.349556
.348209
.346877
.345560
0.344258
X
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0

/ BIBLIOGRAPHY
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CO
Tabulation of the function f(x,y) = / e-t {cosh yt J0(xt )-l|csch t dt .
Admiralty Research Laboratory, 1953: ТаЪ1е of
p P"P2 i> Э T|2
F = -7= z-— j I BpT))e ц T\ dT) , Teddington, Middlesex.
NT" g2 JQ
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Bennett, W.R., 1948: Quart. Appl. Math. 5, 385-393.
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Bose, P.K., 1947: Sankhya 8, 235-248
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Brinkley, S.R., H.E. Edwards and R.W. Smith, 1952: Table of the temperature
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British Association for the Advancement of Science, 1950: Bessel functions,
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Ik-Ltish Association for the Advancement of Science, 1951: Mathematical
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Hritish Association for the Advancement of Science, 1952: Bessel functions,
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Vol. X, Cambridge.
386

388
INTEGRALS OF BESSEL FUNCTIONS
BIBLIOGRAPHY
389
Buchholz, H., 1949: J. Angew. Math. Mech. 29, 356-367.
Buchholz, H., 1953: Die konfluente hypergecmetrische Funktion, Springer.
Bullard, E.G. and R.I.B. Cooper, 1948: Proc. Roy. Soc. London, 194A, 332-347.j
Bursian, V.R. and V. Fock, 1931: Akad. Nauk. Leningrad, Fiziko-Mat.
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Burunova, N.M., 1959: Handbook on mathematical tables, supplement no. 1
(in Russian), Moscow. Also in English, Pergamon Press, 1960.
Butler, T. and K. Pohlhausen, 1954: Tables of definite integrals involving
Bessel functions of the first kind, Wright Air Development Center,
Wright-Patterson Air Force Base, Ohio, Technical Report 54-420.
Byrd, P.F. and M.D. Friedman, 1954: Handbook of elliptic integrals for
engineers and physicists, Springer-Verlag.
Cambi, Е.,Л548: Eleven- and fifteen-place tables of Bessel functions of
the first kind to all significant orders, Dover.
Campbell, G. and R. Foster, 1948: Fourier integrals for practical applica-
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Carslaw, H.S. and J.C. Jaeger, 1959: Conduction of heat in solids, 2nd
edition, Oxford. •
Cerrillo, M.V. and W.H. Kautz, 1951: Properties and tables of the extended
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Chong, F., 1953: Iowa State Coll. Sci. 27, 321-334.
Cistova, E.A., 1958: Tables of Bessel functions with real arguments and
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Clemmow, P.С and CM. Munford, 1952: Philos. Trans. Roy. Soc. London A,
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Coulmy, G., 1954: Ann. Telecommun. 9, 305-312.
Crowley, Т.Н., 1954: Tables of integrals of certain Bessel functions,
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Erde'lyi, A., 1950: Math. Tables Aids Comput. 4, 179.
Erde'lyi, A. See Erde'ly1 et al.
Erdelyi et al. A953), that is,
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Erde'lyi, A., W. Magnus, F. Oberhettinger and F.G. Tricomi, 1954: Tables
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Faddeeva, V.N. and M.K. Gavurin, 1950: Tables of the Bessel function Jn(x)
for orders 0 to 120 (in Russian), Moscow.
Faddeeva, V.N. and N.M. Terent'ev, 1954: Tables of values of the function
w(z) = e"z (l+2i
l fz 2
tt-2 / ex dx)
J0
for complex argument (in Russian),
Moscow. Also available in English, Pergamon Press, 1961.
rtz, M. and С Harrison, 1957: A tabulation of the function
-1 fX
x I J0(y)dy , Argonne National Laboratory, Lemonte, Illinois.
J0

390
INTEGRALS OF HESSEL FUNCTIONS
Fettis, H.E., 1955: Math. Tables Aids Comput. 9, 85-92.
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function and its first twenty derivatives, Cambridge, Mass.

392
IHTECKALS OF BESSEL FUNCTIONS
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394
INTEGRALS OF HESSEL FUNCTIONS
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395
KLamkin, M.S., 1957: Amer. Math. Monthly 64, 661-663.
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396
INTEGRALS OF BESSELNfUNCTIONS
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Mack, С and M. Castle, 1953: Tables of I I0(x)dx and / Ko(x)dx ,
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398
INTEGRALS OF HESSEL FUNCTIONS
BIBLIOGRAPHY
399
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400
INTEGRALS OF BESSEL FUNCTIONS
Pritchard, R.L., 1951: J. Acoust. Soc. Amer. 23, 591.
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Schmidt, P.W., 1955: J. Math. Phys. 34, 169-172.

402
INTEGRALS OF BESSEL FUNCTIONS
Schubert, A., 1953: Wiss. Z. Tech. Hochsch. Dresden, 2, 437-440.
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Vinogradov, I.M. and N.G. Cetaev, 1950: Tables of Bessel functions of
imaginary argument (in Russian), Moscow.
Ward, G.N., 1955: Linearized theory of steady high-speed flow, Cambridge.
Watson, G.N., 1945: A treatise on the theory of Bessel functions,
Cambridge.
Weeg, G.P., 1959: Math. Tables Aids Comput. 13, 312-313.
Weyrich, R., 1937: Zylinderfunktionen und ihre Anwendungen, Teubner.
Wheelon, A.D. and J.T. Robacker, 1954: A table of integrals involving
Bessel functions, Ramo-Wooldridge Corp., Los Angeles, Calif, and Rand
Corp., Santa Monica, Calif.
Whittaker, E.T. and G.N. Watson, 1927: A course of modern analysis,
Cambridge.
van Wijngaarden, A. and W.L. Scheen, 1949: Verb. Nederl. Akad. Wetensch.
Afd. Natuurk. (l), 19, No. 4.
Wilson, E.M., 1951: Solutions of the equation (y")^ = yy' and two other
equations, Admiralty Research Laboratory, Teddington, Middlesex.
Wimp, J., 1960: Polynomial expansions of Bessel functions and some
associated Bessel functions, Midwest Research Institute, Kansas City,
Mo.
Wimp, J., 1961: Math, of Comput. 15, 174-178.
Woodward, P.M. and A.M.Woodward (with the assistance of R. Hensman,
H.H. Davies and N.Gamble), 1946: Phil. Mag. G) 37, 236-261.
Zartarian, G. and H.M. Voss, 1953: J. Aero. Sci. 20, 781-782.
Zernike, F. and B.R.A. Nijboer, 1949: La theorie des images optiques.
Centre National de la Recherche Scientifique, Paris, 227-235.
Zonneveld, J.A. and J. Berghuis, 1955: The asymptotic expansion of a
special function and some relations with Bessel functions, Mathe-
matisch Centrum, Amsterdam.

INDEX OF NOTATIONS
The system of notation used to locate specific material lists, in the
following order, chapter, section and subsection (if any).
Ex. 1.4.3 means Chapter I, fourth section, third subsection.
A number in parentheses refers to an equation.
Ex. 1.4.3D) refers to the fourth equation in 1.4.3.
An equation number in parentheses, standing Ъу itself, refers to the
equation of the particular section or subsection in vhich the reference
occurs.
Ex. A reference to D) in 1.4.3 means the fourth equation of 1.4.3.
There is much ad hoc notation vhich is explained in the text near where
it occurs. These data are excluded in this index.
In the listings below, the numbers refer to pages on which the func-
functions are defined.
Ai(z), Airy integral, 127
An(z), polynomial used in rational
approximation to у(а>2).> 153
aa i>(z)-> а particular 2F3> 201
B
Bi(z), Airy integral, 127
Bn(z), polynomial used in rational
approximation to у(а,х), 153
Ъд „(z), a particular 2F3, 201
С
C(z), Fresnel integral, 179
Ci(z), cosine integral, 169
Ci(z,a), a generalized cosine inte-
integral, 145
(^(z), polynomial used in rational
approximation to у{е.^), 155
Су(г), cylinder function, 22
ca u(z), a particular 4^, 202
D
Dn(z), polynomial used in rational
approximation to y(a>z)> 155
Dy(z), cylinder function, 254
da „(z), a particular 4FX, 202
E
E^(z), exponential integral, 163
En(x), repeated integral of Ei(x),
168
En(z), polynomial used in rational
approximation to r(a,z), 157
404
IITOEX OF NOTATIOHS
405
Ei(x), exponential integral, 164
Ei(-z), exponential integral, 163
Erf(z), error function, 172
Erfc(z), complementary error func-
function, 172
Erfi(z), modified error function,
172
E(k), complete elliptic integral of
the second kind, 315
Ey(z), 150
Ey(z), Weber's function, 83
erf z, error function, 172
F
Fn(z), polynomial used in rational
approximation to T(a,z), 157
pFq; pFq(V1+1Vz) ,
/ al>a2>---.>ap I N
eralized hypergeometric function, 4
Xv II \
0
15
p^Opjiiv*
3u(z), 85
f(\,X,T), 280
^u(z), 120
Gi(z), associated Airy integral, 134
Gn(z), polynomial used in rational
approximation to e"z, 156
g(X,X,T), 280
ga,u(z)> function related to repeated
integral of Bessel function, 207
g^^z), 227
Hlln(z); repeated integral of error
function, 178
Hi(z), associated Airy integral, 134
Hi^(z),Hi^(z), indefinite integral
of z^hJ^z) and z^S2)(z),
respectively, 43
H^^z), an associated Bessel func-
function, 108
H^ '(z),E^2'(z), Bessel functions of
the third kind or Hankel functions
of the first and second kind, re-
respectively, 22
Hu(z), Struve function, 80
Hi|_^i;(z).> indefinite integral of
z^(z), 223
h(X,X,T), 280
n(j,,i>(z)j an associated Bessel func-
function, 108
Ii(Ji^l,(z), indefinite integral of
z%(z), 42
Iu(z), modified Bessel function of
the first kind, 23
IW;X), 314

406
INTEGRATE OF BESSEL FUNCTIOHS
INDEX OF NOTATIONS
407
inerfc(z), repeated integral of
error function, 177
'ic(z), ii(z), ir(z), rocket func-
functions, 181
i-a „(z), fractional or repeated inte-
integral of Iy(z), 199
J
J(x,y), 271
Jc(\,z), Je(\,z), Js(\,z), Schwarz
functions, 234
JL y(z), indefinite integral of
zWy(z), 42
Jy(z), Bessel function of the first
kind, 22
Ju(z), Anger's function, 83
jr(z), r-th repeated integral of
J0(z), 196
j „(z)j fractional or repeated inte-
integral of Jy(z), 196
К
Ki „(z), indefinite integral of
zM-Ky(z), 42
Kp n(z); a function relate'd to asymp-
asymptotic expansion of pFq for large
z, 7
Kr(z), r-th repeated integral of
K0(z), 217
Ku(z), modified Bessel function of
the second kind, 23
Ka u(z), fractional or repeated in-
integral of Ky(z), 197
K(k), complete elliptic integral of
the first kind, 315
кц y(z), fractional or repeated inte-
integral of Ky(z), 199
Lp,q(z)^ ^,q(z)^ functions related
to asymptotic expansion of -pFg
for large z, 8
Ly(z), modified Struve function, 80
Li ^(z), indefinite integral of
zM-L^z), 223
li(x), logarithmic integral, 164
M
Mn(z )> polynomial used in rational
approximation to r(a,z), 158
"p,*' 8
Nn(z)> polynomial used in rational
approximation to r(a,z), 158
Рп(ш), Legendre polynomial, 260
Pn(z), function used in error repre-
representation for a rational approxima-
approximation to y{e.,z), 153
P^z), Legendre function, 111, 246
Py(z), associated Legendre function,
246
p_ „(z), function related to repeated
integral of Bessel function, 207
Q Tn(z), error for rational approxima-
approximation to r(a,z), 157
0ц(г), function used in error repre-
representation for a rational approxima- Тп(ш), Chebyshev polynomial of the
tion to y(a^z), 155
Qy(z), associated Legendre function ±
--=¦'- ,..._._ p,q
* of the second kind, 248
R
Rc(z), Ri(z), Rr(z), rocket func-
functions, 181
Rk(a,c,6;\), 149
Rn(z), error for a rational approxi-
approximation to у(а,г), 153
rc(z), ri(z), rr(z), rocket func-
functions, 181
S(z), Fresnel integral, 179
Si(z), sine integral, 169
first kind, 321
(a,m).
(z), 14
function, 234
'a , ,,(Р^ш), generalized Schwarz
и
Un(z), function used in error repre-
representation for a rational approxima-
approximation to r(a,z), 157
ип(ш), Chebyshev polynomial of the
second kind, 321
Uu(w,z), Lommel function of two
variables, 3 08
V
Vn(z), error for a rational approxi-
approximation to r(a,z), 158
Si(z,a)> a generalized sine integral, Vv (z), 227
145
Vu(w, z), Lommel function of two
Sk(e), 243 variables, 3 09
Sn(z), error for a rational approxi- W
mation to y(a^z), 155
S y(z)> Lommel function, 74
si(z), sine integral, 168
s v(z)i Lommel function, 74
W <u(z), v(z)| , Wronskian of u(z)
and v(z), 29
Wi y(z), indefinite integral of
zW,(z), 42
W,
i,b^z)^ Whittaker's function, 212
m. /\ .j_jn- • л. т it л Wy,(z), function used in error repre-
Ti(z), associated Airy integral, 134 n f л. • „ х- -, ¦
4 /; sentation for a rational approxima-
approximation to T(a,z), 158

408
IHTEGRAIS OF BESSEL FUNCTIONS
INDEX OF NOTATIONS
409
?v{z), function used to represent any
of the Bessel functions of the first
Three kinds or the modified Bessel
functions of the first and second
kind, 28
Wy(z), Whittaker's integral, 110
wa i>(z)> factional or repeated in-
integral of Wy(z), 195
Yc(A,z), Ye(\,z), Ys(A,z), Schwarz
functions, 234
Yi y(z), indefinite integral of
%v{z), 42
Yy(z), Bessel function of the second
kind, 22
yr(z), r-th repeated integral of
Y0(z), 217
ya p(z), fractional or repeated in-
integral of Yy(z), 198
GREEK LETTERS
F(a,z), complementary incomplete
gamma function, 144
F(z), gamma function, 2
Y, Euler's constant, 3
y(8->z)i incomplete gamma function,
144
\ji(z), logarithmic derivative of the
gamma function, 2
ф (a, c,z), confluent hypergeometric
function, 145
MISCELLANEOUS NOTATIONS
z = x+iy, i = (-1J , is a complex
number.
R(z) = x = real part of z.
I(z) = У = imaginary part of z.
|z| = absolute value of z = (x +y )^.
arg z = argument of z, tan(arg z) =
(y/x).
In z = principal value of the natural
logarithm of z.
In z = In |z| + i arg z, -n<arg z Stt.
with In z defined as
za = ea In z
above.
(a)k = Г(а+к)/г(а).
D = d/dx.
m;
(m) = binomial coefficient = —n
~j means approximate or asymptotic
equality.
§ means Cauchy principal value of an
integral.
Yr, r-th positive zero of J0(x), 89 (_jn = (_±fi^ д ^ lnteger or zero_
Л0(а,Р), Heuman's complete elliptic
integral of the third kind, 315
§(a,c,z), confluent hypergeometric
function, 145
[x] = largest integer contained in x, j
x >0.
d as in 6d means 6 decimals•
s as in 5s means ,5 significant
figures.
The notation x:0@. 02)l, for example,
means that entry values of function^
tabulated range from x = 0 to ^
x = 1 in steps of 0.02.
A number in parentheses following a
numerical number indicates the
power of ten by which the numeri-
numerical number must be multiplied.
Thus, for example, 3.6503742B)
means 3.6503742 • 102. See pages
353-355.

N&UTHOR 1ЖРЕХ
(Numbers Refer to Pages)
AUTHOR INDEX
411
Abbott, W.R. 125
Abramowitz, M. 21,51,69,70,194,232
Achiezer, N.I. 346
Admiralty Computing Service 322
Admiralty Research Laboratory _ 282
Airey, J.R. 27
Akademiia. Hauk 187,188,193
Allen, E.E. ,39
Anders, T. 346
Archibald, R.C. 1
В
Bailey, W.N. 293,300
Bateman, H. 1
Bennett, W.R. 334
Berghius, J. 323
Bickley, W.G. 222
Billings, C. 142
Binnie, A.M. 282
Blanch, G. 51,70,168
Blanuea, D. 282
Bleick, W.E. 189
Boersma, J. 308,312
Bose, P.K. 282
Bouwkamp, C.J. 172,188,335
Bowman, F. 1
Brinkley, R.F. 283
Brinkley, S.R. 283
British Association for the Advance-
Advancement of Science 40,178,187,188,
190,350,359
Buchholz, H. 144,289
Bullard, E.C. 322
Bursian, V.R. 71
Burunova, N.M. 1
Butler, T. 90,94,260,263
Byrd, P.F. 317,319
Caldwell, G.C. 182,193
Cambi, E. 40,356
Campbell, G. 290
Carslaw, H.S. 190,192,324
Castle, M. 72
de Castro Brzezicki, A. 1,29
Cerrillo, M.V. 41
Cetaev, N.G. 40
Chaundy, T.W. 148
Chong, F. 345
Christiano, J.G. 143
Cistova, E.A. 70,72
Clarke, M. 323
Clemmow, P. С 192
Coleman, R.L. 148
Conolly, B.W. 341
Cooke, J.C. 320,341,345
Cooper, R.I.B. 322
Constant, P.C. 235
Conte, S.D. 192
Corbato', F.J. 21
Corrington, M.S. 186
Coulmy, G. 254,259
Crowley, Т.Н. 251
David, F.H. 72
Davies, H.H. 41
Dekanasidze, E.N. 308,311
Dingle, M. 232
Dorr, J. 337
Drane, C.J., Jr. 188
Durand, D. 334
Eason, G. 315,317,318
Edwards, H.E. 283
Emde, F. 1,187
Erde'lyi, A. 186(see also Erdelyi et
Erde'lyi et al., i.e., Erde'lyi, A., W.
Magnus, F. Oberhettinger, and F.G.
Tricomi 1,5,19,144,148,196,212,221, (
239,246,248,290,307,332,333,345
Faddeeva, V.N. 40,192
Feodorova, R.M. 1
Ferentz, M. 70
Fettis, H.E. 60,176,221,235,243,
269,295,317
Fields, J.L. 19
Fieller, E.C. 72
Fletcher, A. 1
Fock, V. 71
Ford, F.A.J. 282
Foster, R. 290
Fox, L. 40,41,194
Franz, W. 1,144
Fried, B.D. 192
Friedman, M.D. 317,319
Froberg, C.E. 289
G
Gamble, N. 41
Garrick, I.E. 251
Gautschi, W. 21,163,168,178
Gavurin, M.K. 40
Gawlik, H.J. 190
Gerbes, W.W. 188
Giedt, D. 142
Godfrey, G.H. 233
Goldstein, M. 21
Goldstein, S. 271,273
Goodwin, E.T. 186
Gordon, A.N. 345
Gradstein, I.S. 1
Gray, A. 1
Greenwood, J.A. 334
Grobner, W. 1
Gross, G.L. 181,194
H
Hallen, E. 172,188
Harris, F.E. 187
Harrison, C. 70
Hartree, D.R. 177,190
Harvard University Computation
laboratory 40,41,72,189,190,232
Hastings, C, Jr. 283,323
Havelock, Т.Н. 219,222
Hay, H.G. 141
Heatley, A.H. 312
Helliwell, J.B. 347
Henrici, P. 333
Hensman, R. 41,192
Hershey, A.V. 187
Heuman, C. 315
Hiester, H.K. 275,283
Higgins, T.P. 341
Hitchcock, A.J.M. 40,69
Hoes, M.R. 188
Hofreiter, N. 1
de Hoop, A.T. 298,329
Horton, C.W. 70,232,254,266
Howarth, L. 119
Hoyt, R.S. 251
Huckel, V. 251
Jaeger, J.C. 190,192,222,323,324,339
Jahnke, E. 1,187
Jenkins, D.P. 192
Jones, C.W. 41,371
Jones, W.P. 232
Jordan, P.F. 235
К
Кашке, Е. 29
Karmazina, L.H. 72
Karpov, K.A. 187,192
Kautz, W.H. 41
Kaye, J. 190
King, L.V. Л23
Kinizer, J.P. 289
Klamkin, M.S. 321
Knudsen, H.L. 70,308
Kobayashi,
Komatu, Y.
I. 335
163
Kratzer, A. 1,144
Kreyszig, E. 145,194
Kussner, H.G. 232
410

412
IHTEGRALS OF BESSEL FUNCTIOHS
AUTHOR INDEX'
413
Lamb, H. 337
Lebedev, A.V, 1
Lebedev, H.H: 347
Lee, K. 27 \
Levine, H. 29§
Lin, С. С. 141Д42
Longman, I.M. fo
Losch, F. 1,187 \
Lowan, A.N. 51,7O4
Luke, Y.L. 40,60,6^,72,93, 95,108,
120,148,152,176,177,221,231,233,
235,243,251,254
M
Mack, C. 72
MacRobert, T.M. 1,320,338
Magnus, W. 1 (see also Erdelyi
et al. )
Mashiko, M. 187
Mathematical Review 316
Mathematical Tables and Other Aids
to Computation (now Mathematics
of Computation) 1
Mathematics Center, Amsterdam, Com-
Computation Department 251
Mathews, G.B. 1
Maximon, L. С 270,283,288
McLachlan, N.W. 1,254
Meijer, C.S. 7,19,338
Meyers, A.L. 254
Meyer zur Cape lien, W. 1
Miles, J.W. 142,324
Miller, G.F. 194,282
Miller, J.C.P. 1,41,282
Morgan, G.W. 254
Muller, G.M. 72,326
Muller, R. 72
Munford, CM. 192
Murphy, G.D. 29
Murray, J.D. 271
N
National Bureau of Standards 1,40,41
69,70,72,94,141,142,168,187,188,189,
191,194,232,362,365,368,374,377,380,
383
National Physical Laboratory 71,298
Haylor, J. 222
Helson, C.W. 339
Hewton, R.R. 181,194
Hijboer, B.R.A. 310
Noble, B. 315,317,318,345
Nomura, Y. 318
0
Oberhettinger, F. 1,51,166,290,341
(see also Erdelyi et al. )
Ollendorff, F. 335
Olver, F.W., Jr. 40,41,336
Opler, A. 275,283
Osterberg, H. 71
Pagurova, V.I. 194
Pearcey, T. 193
Pearson, K. 72
Peters, A.S. 347
Petiau, G. 1,254,259
Picht, J. 254,259
Pinney, E. 321
Pohlhausen, K. 90,94,260,263
Pollak, H.O. 163
Pollard, W.G. 307
Present, R.D. 307
Pritchard, R.L. 328
R
Radio Corporation of America 191,193
Radosevich, L.G. 27
F.M. 320,335,337
Rainville, E.D. 1
Rand Corporation 283
Rankin, R.A. 182,193
Rathie, C.B. 320,338
Razumovskii,S.H. 187
Reissner, E. 339
Relton, F.E. 1
Reynolds, G.E. 188
Rey Pastor, J. 1,29
Rice, S.O. 252
Riegels, F. 316
Riley, J.A. 142
Ritchie, R.H. 186
Robacker, J.T. 1
Roberts, J.A. 182,193
Rosenhead, L. 1
Rosser, J.B. 177,181,182,191,193,
194
Rothman, M. 72,142,143
Rubinow, S.I. 251
Ruhlman, B. 235
Rutgers, J.G. 293
Ryshik, I.M. 1
Salzer, H.E. 177
Scheen, W.L. 193
Schmidt, P.W. 71
Schubert, A. 254,259
Schutte, K. 1
Schwarz, L. 234,256
Schwinger, J. 298
Scorer, R.S. 143
Sears, W.R. 322
Slater, L.J. 144,312
Smirnov, A.D. 41,143
Smith, R.W. 283
Smith, V.G. 86
Sneddon, I.N. 315,317,318,345
Staton, J. 186
Stegun, I.A. 21
Stouffer, S.A. 72
Straubel, R. 254,259
Struve, H. 233
Sura-Bura, M.R. 317
Tai, C.T. 189
Terent'ev, H.M. 192
Thaler, R.M. 21
Tranter, C.J. 345,348
Tricomi, F.G. 144,150 (see also
Erdelyi et al.)
Tyler, СМ., Jr. 143
U
Ufford, D. 72,231,233
Ufliand, Ia.S. 347
Uretsky, J.L. 21
Vinogradov, I.M.
Voss, H.M. 252
40
W
Walker, G.L. 71,194
Walker, R.J. 182
Ward, G.H. 324
Watson, G.N. 1,10,29,33,40,41,71,
84,90,94,110,254,307,308,332,353,
359
Weeg, G.P. 318
Weyrich, R. 1
Wheelon, A.D. 1
Whittaker, E.T." 1
тал Wijngaarden, A. 193
Wilhelmsson, H. 289
Wilson, E.M. 274,283
Wilson, I.G. 289
Wimp, J.J. 19,40
Wong, J.R., Jr. 283
Woodward, A.M. 41
Woodward, P.M. 41
Zartarian, G. 252
Zernike, F. 310
Zonneveld, J.A. 323

SUBJECT INDEX
(Numbers "pefer to Pages)
Airy functions, 127 ff." .See also Airy'integrals and Integrals of Airy
integrals. — ~
Airy integrals, 127 ff.; asymptotic expansions, 129; definitions, 127; de-
derivatives, 128; integral representations, 131; interrelations, 128;
power series, 129. See also Bessel functions.
Anger function, 73,83.
Approximations, polynomial, see under modifiers such as Bessel functions,
etc.; rational, for Incomplete gamma function and related functions,
152 ff.
Associated Bessel functions, 73,107. See also Associated Bessel functions
\,№ > Hm,,^z) > and Lammel functions s^U) , S^y(z) .
Associated Bessel functions h u(z) ; Яц v(z) > Ю7 ff.; asymptotic expan-
expansion for large z, 109; basic properties, 108; connection with derivatives
of Bessel functions with respect to the order, 116; difference-differen-
difference-differential properties, 108,110; expansionsin series of Bessel functions, 115;
formulae for H^y(z) when h^v(z) is not defined, 112; integral
representations, 110; relation to Whittaker's integral, 110; used to
/z
e""ttM'Ku(t)dt and related integrals, 117.
Asymptotic expansions (general), 7. See also under modifiers such as
Bessel functions, Hypergeometriс functions (generalized), etc.
Bessel functions, 22 ff.; asymptotic expansions for large z , 31; basic
properties, 22; circular representations of Jn(z ) , 57; connection with
Anger-Weber functions, 84; cylinder functions, 22; derivatives with
respect to the order, 26,116,166,171; difference-differential proper-
properties, 27; expansions in series of Bessel functions, 25; exponential
series representations for Ky(z) , 221; generation of, by automatic
computers, references to, 21; integral representations, 30; order, half
an odd integer, 32; orthogonality properties, 290; polynomial approxi-
approximations, 33; products, 24,27; tables of Jn(z) , Yn(z) , n = 0,1 , 350-
352; tables of e"xIn(x) , exKn(x) , eX , 353-355; tables of Jn(x) ,
n = 2AN, 356-358; tables of e"xIn(x) , 359-361; tables of x cos x ,
x" sin x , (rr/2x)%n_i(x) , n = 2,3,4, 362-364; tables of Jy(x) ,
v = ±1/4 , ±3/4, 365-367; tables of Ju(x) , v = ±l/3 , ±2/3 , 368-370;
tables of x"(n+?)ln+A(x) , e"xIn+i(x) , n = 0AL= , 371-373; tables of
1„(х) , и = ±1/4 , +3^4 , 374-376;2tables of 1„(х) , v = ±l/3 , +2/3 ,
377-379; tables of, references to, 27,40; Wronskians, 29. See also
Airy integrals.
Bessel polynomials, 154.
Cicala function, 231.
Confluent hypergeometric function, 144; expansions in series of Bessel
functions, 148; integral representation, 312; relation to incomplete
gamma function, 145. See also Hypergeometric functions (generalized).
414
SUBJECT INDEX
415
Constants, 3,129. v
Continued fractions, for the Incomplete gamma function arid related functions,
161.
Convolution integrals, 123,191,195,292,297,30b.
Cylinder functions, 23,73,254,290.
Elliptic integrals, complete, used to exprcuu integrals of Bessel functions,
315 ff., 326.
Error functions, 172 ff.; complex zeros, 177; continued fractions, 159; 'ex-
'expansions in series of Bessel functions, 173; inequalities, 161; inte-
integrals involving, 182 ff.; integral representations, 173,183 ff.; radius
of univalence, 177; rational approximations, 152; repeated integrals,
177,190,191; representations in series of exponential functions, 175;
tables of, references to, 190 ff. See also Incomplete gamma function.
Expansions in series of Bessel functions, a general theorem for represent-
representing an indefinite integral involving Bessel functions, 283; ez sln 9,
{z/2)^+1 , e"z(z/2)^+1 , 25; e"z , 26. See also under modifiers such
as Bessel functions, expansions in series of Bessel functions, etc.
See also Fourier-Bessel coefficients.
Exponential function, expansion in series of Bessel functions, 25; expan-
expansion in series of hypergeometric functions, 148; rational approximation,
156.
Exponential integral, 163 ff.; continued fractions, 159; expansions in
series of Bessel functions, 165; generalized, 189; inequalities, 161;
integral representations, 112,186; integrals involving, 182 ff.;
rational approximations, 152; relation to derivatives of Bessel func-
functions and Struve functions with respect to the order (half an odd
integer), 166; repeated integral, 168; tables of, references to, 187,
194. See also Incomplete gamma function.
Fourier-Bessel coefficients, 73,89,94,260,291.
Fractional integrals, see Repeated integrals and Convolution integrals.
Fresnel integrals, 179 ff.; continued fractions, 159; expansions in series
of Bessel functions, 180; inequalities, 161; integrals involving, 183 ff. ;
rational approximations, 152; relation to rocket functions, 181; tables
of, references to, 192,193. See also Incomplete gamma function.
G-function, of Meijer, 19.
Gamma function and related functions, 2; mathematical constants, related
to, 3.
Gaussian hypergeometric function BF1), 18,148. See also Hypergeometric
functions (generalized).
G-egenbauer integrals, 327.
Generalized hypergeometric series, see Hypergeometric functions
(generalized).

416
INTEGRALS OF BESSEL FUNCTIONS
SUBJECT IHDEX
417
Hankel functions1) 22.
Hankel-Nichols on Чуре integrals, 330 ff>
Hypergeometric functions (generalized), 4 ff. ; asymptotic expansion
(general) for large z , 7; asymptotic expansion of pFp+i for large ¦'
z , 9; asymptotic expansion «f pFp for large z , 10,13; differential
equation satisfied Ъу, 5; expansions in series of Bessel functions, 20;.
expansions in series of other hyperge(metric functions, 19; integral
representations, 5; special values, 11,18. See also Confluent hypergeo-
hypergeometric function and Gaussian hypergecmetriс function.
Incomplete gamma function, 144 ff.; asymptotic expansions for large z ,
146; basic properties, 144; connection with confluent hypergeometric
function, 145; continued fractions, 159; expansions in series of Bessel
functions, 148; inequalities, 161; infinite integrals, 147; integral
representations, 146, 182 ff.,312; integrals involving, 182 ff. ; rational
approximations, 152 ff.; tables of, references to, 187 ff., 194. See
also Error functions, Exponential integral, Fresnel integrals and Sine
and cosine integrals.
Inequalities, for the Incomplete gamma function and related functions, 161.
Integral equations, dual and triple, 342 ff.
Integrals of Airy integrals, 132 ff.; asymptotic expansions, 136; expansions
in series of Bessel functions, 135; expansions in terms of Lommel func-
functions, 133,138; interrelations, 132; power series, 133; repeated inte-
integrals, 138; tables of, references to, 141. See also Integrals of Bessel
/z
tMVy(t)dt .
Integrals of Bessel functions, definite, with trigonometric argument, for
/tt/2 • 2a.-1
J2,|i(z sin t)J2u(w cos t)sin +
See also Convolution integrals, Fourier-Bessel coefficients and Lommel's
functions of two variables.
Integrals of Bessel functions (indefinite); general theorems, 73,195,253,2831
involving the product of two Bessel functions, 254 ff.,268; involving thel
product of a Bessel function and a Struve function, 264; involving• the
product of three Bessel functions, 269; miscellaneous, 289.
Integrals of Bessel functions, J tM?l,(t)dt , 42 ff., 73 ff.; asymptotic
expansions for large z , 53,243; basic properties, 42; circular repre-
Jn(t)dt , 57; difference-differential properties, 44,
120; evaluation of, for special values of the parameters, 124; expansic
in series of Bessel functions, 51,125; infinite integrals, 56, 246 ff.;
polynomial approximations, 60,92; power series, 44; representations of,
in terms of Lommel functions and Struve functions, 73, 85; tables of
/0XJo(t)dt , /QXYo(t)dt , J^f1 {l-Jo(t)} dt , /xVVt)dt , 380-
382; tables of e'x f*IQ(t)dt , ex ?°KQ{t )dt ,e"x /JV1 {lQ(t)-l} dt
See
t cos
2P-lt
dt , 293-308.
xex / f1Ko(t)dt, , 383-385; tables чof, references to, 69,94,232.
also Incomplete gamma function, Integrals of Airy integrals, Repeated
integrals of Bessel functions, Schwarz functions and generalizations .
/z .
e~ t|1Kl;(t)dt and related integrals,
95 ff.; asymptotic expansions for large z , 101,243; difference-differ-
difference-differential properties, 120; evaluation of, for special values of the param-
parameters, 121; expansions in series of Bessel functions, 100,122,124,238,
285; infinite integrals, 106,246 ff., 312 ff., 321; power series, 95;
representations of, in terms of an associated Bessel function, 107;
tables of, references to, 119,251. See also Schwarz functions and gen-
generalizations.
Integrals of Bessel functions, I e"p"ttl1Kl,(t )dt , reduction formulas,
120 ff. J nX ,
Integrals of Bessel functions, JQ e~ IQ |2(ytJ| dt , 271 ff.; asymptotic
expansions, 278; elementary properties, 271; expansions in series of
Bessel functions, 275; generalization, 288; integral representations,
276; partial differential equations satisfied by, 272; power series,
275; related integrals, 280; tables of, references to, 282.
Integrals of Bessel functions, infinite, J e t^Ky(\t)dt and related
integrals, 56,106,246 ff.,312; J
grals, 212; Jo
316 and 318 (tables of, references to), 321,324 ff. ; J t^e"P t К (ata)
X Kytbt6)dt and related integrals, 320,321,335,338; of the type
2 2^
e-P t tt-t-^-j^at )dt and related inte-
t\g-pc:t j (a-tjj^-b-t),^ and related integrals, 314-320,
So vbt)
J, {a(tg+zg)i}t^
the type
J0
ta
(t^z^
'+4(at)
dt and related integrals, 327 ff.; of
dt , 330 ff.; involving the products of three or
(t2+z2)n+l
more Bessel functions, 331 ff.; with respect to the order, 340 ff. ;
miscellaneous, 322-324,331,335 ff. See also under the names of various
integrals, e.g., Hankel-Nicholson type integrals.
Integrals of incomplete gamma functions and related functions, 182 ff.
Integrals of Lommel functions, see Lommel functions.
Integrals of Struve functions, /¦ t|J'Hl,(t)dt , 223 ff.; asymptotic expansions
for large z , 224; basic properties, 223; difference-differential prop-
properties, 223,227; evaluation of, for special values of the parameters,
228; infinite integrals, 226; polynomial approximation for
J L^o^^^o^M -dt > 34' relation to the complete Cicala function,
231; tables of, references to, 70,232.

418
MTEGRAIS OF BESSEL FUNCTIOHS
SUBJECT INDEX
I
419
Integrals of StruVe functions, J e"p t^Hy(t)dt ; evaluation of, for special
values of the parameters and variable, 228 ff.; reduction formulas, 227.
Integrals of Struve functions (indefinite), involving the product of a Struve
function and a Bessel.. function, 264; involving the product of two Struve
functions, 266.
Legendre's function, 111,246,248.
Legendre's polynomial, 260,321.
Lcmmel functions, s^ y(z) and s^,v(z) > 73 ff • ; asymptotic expansion for
large z , 74; basic properties, 74; connection with Anger-Weber functions,
83; connection with Fourier-Bessel coefficients, 90; connection with
Struve functions, 80; differential-difference properties, 75; expansions
in series of Bessel functions, 79; formulae for S t(z) when s „(z) '
is not defined, 76; integral representations, 77; relation to Ti(z) , }
Gi(z) and Hi(z) , 134; used to represent / tMWl,(t)dt , 85; used to ,
represent integrals of Airy integrals, 133,138.
Lcmmel functions, Ti(z) , Gi(z) and Hi(z) , 133 ff.; asymptotic expansions,
138; differential equation, 135; expansions in series of Bessel functions,
136; integrals of, 140; integral representation, 135; tables of, refer-
references to, 143; used to represent Integrals of Airy integrals, 133. See
also Lommel functions, в„ y(z) and S^ u(z) .
Lcmmel's functions of two variables, 308 ff.; basic properties, 308; ex-
expansion in series of Bessel functions, 309,310; partial differential
equation satisfied by, 310; tables of, references to, 311.
Nicholson integrals, ЗЗО ff.
Repeated or iterated integrals of, Airy integrals, 138; Bessel functions,
195 ff.; error function, 177,190,191; exponential integral, 168; sine and
cosine integrals, 172,188.
Repeated integrals of Bessel functions, 195 ff.; application to evaluation
of integrals involving Bessel functions, 195; asymptotic expansions for
large parameter, 219; asymptotic expansions for large z , 203,212; basic
properties, 195; circular representations for On,r(z) > 217; difference-
differential properties, 201,211; expansion of ja y(z) in series of
Bessel functions, 219; exponential series representations for KQ „(z) ,
221;further representations, 217; generalization to fractional inte-
integrals, 195; infinite integrals, 216; power series, 199; relation to
integrals of Havelock, 219; tables of, references to, 222. See also
Integrals of Bessel functions.
Eocket functions, 181.
Schwarz functions and generalizations, 234 ff.; asymptotic expansions for
large z , 243; definitions, 234; expansions in series of Bessel func-
functions, 238,285; expansions in series of confluent hypergeometric func-
functions, 236; infinite integrals, 246 ff.; power series, 234; representa-
representations in series of circular functions, 241; tables of, references to,
25|. Sec ul.:;o Integrals of Bessel functions, Г e"Ptt^Ky(t)dt ,
J e'4%(t)dt , J tl%(t)dt . J
Sine and cosine integrals, 168 f f.; continued fractions, 159; expansions in
series of Bessel functions, 170; generalized, 145,152,189; inequalities,
161; integrals involving, 182 ff.; iterated integrals, 172,188; rational
approximations, 152; relation to derivatives of Bessel functions and
Struve functions with respect to the order (half an odd integer), 171;
tables of, references to, 188,194. See also Incomplete gamma function.
Sonine-Gegenbauer type integrals, 327 ff.
Stoke's phenomenon, 10.
Struve functions, 73,80; basic properties, 80; derivatives with respect to
the order (half an odd integer), 166,171; expansions in series of Bessel
functions, 82; polynomial approximations, 92,94; tables of, references
to, 94; used to represent Г tuWy(t)dt , 85.
Tables of mathematical functions and integrals, references to, 1.
under modifiers such as Bessel functions, tables of, etc.
Weber function, 73,83.
Weber-Schafheitlin integrals, 250,324 ff.
Whittaker's function, 212.
Whittaker's integral Wy(z) , 110,111.
See also