Fourier Transforms of Distributions and Their Inverses: A Collection of Tables

Probability and Mathematical Statistics A Series of Monographs and Textbooks

Editors Z. W. Birnbaum Ε. Lukacs University of Washington Bowling Green State University Seattle, Washington Bowling Green, Ohio

1. Thomas Ferguson. Mathematical Statistics: A Decision Theoretic Approach. 1967

2. Howard Tucker. A Graduate Course in Probability. 1967

3. K. R. Parthasarathy. Probability Measures on Metric Spaces. 1967

4. P. Revesz. The Laws of Large Numbers. 1968

5. H. P. McKean, Jr. Stochastic Integrals. 1969

6. B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev. Mathematical Methods of Reliability Theory. 1969

7. Demetrios A. Kappos. Probability Algebras and Stochastic Spaces. 1969

8. Ivan N. Pesin. Classical and Modern Integration Theories. 1970 9. S. Vajda. Probabilistic Programming. 1972

10. Sheldon M. Ross. Introduction to Probability Models. 1972

11. Robert B. Ash. Real Analysis and Probability. 1972

12. V. V. Fedorov. Theory of Optimal Experiments. 1972

13. K. V. Mardia. Statistics of Directional Data. 1972

14. H. Dym and H. P. McKean. Fourier Series and Integrals. 1972

15. Tatsuo Kawata. Fourier Analysis in Probability Theory. 1972 16. Fritz Oberhettinger. Fourier Transforms of Distributions and Their Inverses: A

CoUection of Tables. 1973 17. Paul Erdös and Joel Spencer. Probabilistic Methods in Combinatorics. 1973

18. K. Sarkadi and I. Vincze. Mathematical Methods of Statistical Quality Control. 1973

19. Michael R. Anderberg. Cluster Analysis for Applications. 1973

In Preparation L. E. Maistrov. Probability Theory: A Historical Sketch

W. Hengartner and R. Theodorescu. Concentration Functions

William F. Stout. Almost Sure Convergence

L. H. Koopmans. The Spectral Analysis of Time Series

Fourier Transiorms of Distributions and

Their Inverses

A C O L L E C T I O N O F T A B L E S

"ritz O b e r h e t t i n g e r Department of Mathematics

Oregon State University CorvalliSf Oregon

ACADEMIC PRESS New York and London 1973

A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT © 1973, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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PRINTED IN THE UNITED STATES O F AMERICA

PREFACE

The material in this book originated in a report prepared and submit ted by t h e author to the National Bureau of Standards and sponsored by the OiBce of Naval Research. I t was felt t h a t the information gathered there should be made more widely available. The result is this book, a collection of integrals of the Fourier transform type (including their inverses) involving the class of functions which are nonnegative and integrable over the interval {—^, oo).

Most of the results have been extracted from information already available and scattered through the li terature. An earlier publication by this au thor (^Tabellen zur Fourier Transformation," Springer Verlag, 1957) contained many of the Fourier transforms. I n contrast, in this volume we concentrate on the probabili ty densities. I n addition, a number of new examples have been added.

A sizable amount of effort had to be spent over many years to recognize the functions belonging to the class dealt with in this book. While it is t rue t h a t a particular function may not fulfill t he necessary conditions, it is possible t h a t a suitable combination of a number of them may meet the demanded requirement. I t was in the course of these investigations t h a t a number of hi ther to unknown results, particularly involving higher functions, Avere found.

The Author wishes to express his gra t i tude to the insti tutions mentioned above and especially to Professor Eugene Lukacs for a series of helpful discussions.

ix

INTRODUCTION

Fourier transforms of distribution functions are a n impor tan t tool in t h e theory of probability. I n this connection they are usually called ' 'characteristic functions of probabili ty distr ibutions." They are very useful in t h a t pa r t of probabiUty theory which can be studied independently of the measure-theoretic foundations. T h e present book contains tables of distribution functions and of theh- Fourier transforms. This introduction first Usts those properties of characteristic functions which are import a n t in probabiHty theory; i t then describes the tables and their use.

Character is t ic F u n c t i o n s

A real-valued function F(x) of a real variable χ which is

(i) nondecreasing,

(ii) right-continuous,*

and which satisfies the condition

(iii) lim F(x) = 1, lim F(x) = 0

is called a "dis t r ibut ion function.' '

* Some authors postulate instead of (ii) that the function Ε{χ) be left-continuous.

1

2 Introduction

f{x) dx = l. 0

If F(x) is absolutely continuous, then formulas (1) , (2 ) , and (3) reduce to

g(y) = / e'^-fix) dx, ( la )

a* = Γ x^fix) dx, (2a) •'—00

ßk = r \ x\''f{x)dx, (3a) •'—00

respectively.

* Frequency functions are denoted by the letter /; attached to it is the same subscript (if any) which is carried by the corresponding distribution function.

Let F(x) be a distribution function; i ts Fourier transform

9(υ)^Γβ^^άΡ(χ) (1) — 00

is called the "characteristic function of F(x).^^ This is in general a complex-valued function of the real variable y.

Distr ibution functions are denoted here by the let ter F, with or without subscripts, and characteristic functions by the letters g or G with t he corresponding subscripts or without subscripts.

The following terminology and notat ion are used: The integral

ak = r 7*dF{x) (2) •'—00

is called the "/fcth moment of the distribution F{x)J* Similarly,

ßk = r \x\'dF(x) (3) •'—00

is called the "Mh absolute moment of F{x),'" provided t h a t the integrals (2) and (3) are absolutely convergent. If this is the case, then we say t h a t the moments of order k of F{x) exist. I t is easily seen t h a t the existence of the fcth moment of a uistribution function implies the existence of all moments of order no t exceeding k.

In the main tables, only absolutely continuous distr ibution functions are considered. The frequency function* (probability density) of the distribution function F{x) is denoted b y / ( a : ) = F'{x), A function f{x) is a frequency function if and only if it is nonnegative and if

Characteristic Functions 3

If the function φ{χ) is nonnegative and if

j φ{χ)άχ=Ν<οο^ — 00

then f{x) = {1/Ν)φ{χ) is a frequency function. The corresponding characteristic function is

g{y) = ί β^^'φ{χ) dx. ·'—00

We mention next a few properties of characteristic functions.

T h e o r e m 1 Let F(x) be a distribution function and let g(y) be its characteristic function. Then g{y) is uniformly continuous; moreover \ g(y) \ < g(0) = I and gi-y) = g{y)'

Here g{y) is the complex conjugate of g{y). There is a close connection between characteristic functions and moments , which is described by the following s ta tements.

T h e o r e m 2 If the characteristic function G(y) of a distribution function F(x) has a derivative of order a t = 0, then all the moments of F{x) up to order k exist if k is even bu t only up to order fc — 1 if is odd.

T h e o r e m 3 If the moment a* of order of a distribution exists, t hen the corresponding characteristic fimction g(y) can be differentiated s t imes (s < k) and

- W . = ^(.) ^y) = / ^,iy. dF{x) (s = 1, 2, . . . , fc). «2/ • ' - o o

Moreover a, = i-*g^'\Qi) for α = 1, 2, . . . , fc.

T h e o r e m 4 Let F{x) be a distribution function and assume t h a t its n t h moment exists. The characteristic function g{y) of F{x) can then be expanded in the form

g{y) = l + Z ^ ' ( % ) ^ + 0(r) as 2 / ^ 0 .

The following group of theorems account for the importance of characteristic functions in probability theory.

4 Introduction

T h e o r e m 5 (The Inversion Theorem) Let g{y) be the characteristic function of the distribution function F{x). Then

1 — ß-iyh : e-'O'giy) dy,

provided t ha t F{x) is continuous a t the points a and a + h.

As an immediate consequence of Theorem 5 and of Formula (1) , the following result is obtained.

T h e o r e m 6 (The Uniqueness Theorem) Two distribution functions are identical if and only if their characteristic functions are identical.

A particular case of the inversion formula is of great interest.

T h e o r e m 7 Let g{y) be a characteristic function and suppose t h a t it is absolutely integrable over ( - o o , + o o ) . The corresponding distribution function F{x) is then absolutely continuous and

f(x) = F^(x) = ( 2 7 Γ ) - Ι Γ e-^y^g{y) dy (4)

is the frequency function (probabiHty density) of the distribution F{x).

I t should be noted t ha t the condition of Theorem 7 is only sufBcient bu t not necessary. A distribution may be absolutely continuous and its frequency function may be given by (4) even if the corresponding characteristic function is not absolutely integrable. Natural ly, other conditions which insure the validity of the Fourier inversion formula (4) must then be satisfied.

Let Fi(x) and F2{x) be two distribution functions; it is easily seen t h a t the function

Fix) = Fi(x - z) dF,{z) = Γ F,{x - z) dF,{z) (5) •'—00 •'—00

is also a distribution function. Formula (5) defines an operation, called convolution^ between distribution functions. The resulting distribution function F{x) is called the ' 'convolution of Fi and Fg.'' If, in particular, Fi and F2 are absolutely continuous, then F{x) is also absolutely continuous and its frequency function is given by

f{x) = F'{x) = / " f,{x - z)i,{z) dz = r Mx - z)Mz) dz. (5a) • ' - 0 0 • ' - 0 0

We say then also t ha t f{x) is the convolution of / i and /2.

Description and Use of the Tahles 5

T h e o r e m 8 (Convolution Theorem) A distribution function F{x) [respectively, a frequency func t ion / (a : ) ] is the convolution of two distribution functions Fl and F2 [respectively, two frequency functions / i and /2] if and only if t he corresponding characteristic functions satisfy the relation g{y) = gi{y)g2(y)'

The multipUcation of characteristic functions corresponds therefore uniquely to the convolution operation. In probabili ty theory one studies frequently the distribution of the sum of independent random variables. I t is well-known t h a t the distr ibution function of the sum of two independent random variables is the convolution of the distributions of the summands. The direct determinat ion of the distribution of sums will often lead to very complicated integrations while the characteristic function of this distribution can be found easily. I t is, according to Theorem 8, the product of the characteristic functions of the summands. I n view of the uniqueness theorem it is therefore often advantageous to reformulate a problem concerning distribution functions in terms of characteristic functions.

The next theorem is impor tant in connection with the s tudy of limit distributions. I t indicates t h a t the one-to-one correspondence between distribution functions and characteristic functions is continuous.

T h e o r e m 9 (Continuity Theorem) Le t {Fn{x)} be a sequence of distribution functions and denote the corresponding sequence of characteristic functions by {Oniy)}' The sequence {Fn{x)} converges to a distribution function F{x) in all points a t which F{x) is continuous if and only if t he sequence {gniy)} converges to a function g(y) which is continuous a t 1/ = 0. The limiting function g{y) is then the characteristic function of F{x),

Only some of the important properties of characteristic functions have been listed here. For details, see [ 1 , 2, 3 ] .

Descr ipt ion a n d Use of t h e Tables

The first pa r t of this volume, consisting of Tables I, I I , and I I I , gives Fourier transforms of absolutely continuous distribution functions. The transform pairs are numbered consecutively and are arranged systematically according to the analytical character of the frequency function.

The second par t of the volume, consisting of Tables l A and I I IA , gives the inverse transforms of the functions listed in Tables I and I I I , respectively. T h e entries in the second column of Tables IA and I I I A are characteristic functions (Fourier transforms) of absolutely continuous distribution functions, arranged systematically. The corresponding frequency functions can be found in the thi rd column. The number in the first column coincides with the number given to the same pair in

6 Introduction

Tables I and I I I , respectively. The inverse of the characteristic functions Hsted in Table I I can be found by means of Tables I and lA.

Next we discuss in some detail the individual tables. Table I (pp. 15-73) gives the Fourier transforms (characteristic function) of even

frequency functions ( tha t is, of frequency functions belonging to symmetric distributions) .

Let f{x) be an even frequency function. Then S{ — x) = i{x) = f{\x\) has the Fourier transform

g{y) = Γ /(I X l)e» ^ dx = 2 Γ fix) COS xy dx, (6)

Therefore giy) = gi — y) so t h a t the transform of an even frequency function is always real and even.

I t was found convenient not to list the frequency functions and their transforms directly b u t to tabulate instead a suitable constant multiple. The second column of Table I contains in each box the function 2Nfix) and also the normalizing factor N. The third column yields the function Ngiy). Here/(a:) is an even frequency function while gix) is its transform given by (6) . The first column is used to assign (for reference in the other tables) a number to each pair of transforms. I t should be noted t h a t the arguments of the functions in Table I are always positive. Since fix) as well as giy) are even functions it is not necessary to list t hem for negative values of χ and y.

Examples of frequency functions, often used in statistics and probabili ty theory, which can be found in this table, a re :

Uniform (rectangular) distribution over ( — a, + a ) . No. 1 Cauchy distribution. No. 7 Laplace distribution, No. 60 Normal distribution with mean 0 and s tandard deviation (2a)"^, No. 73

E X A M P L E 1 In i tem 1 of Table I, we find

1 if a; < a, 2Nfix) = '

0 if a: > a,

while Ν = a. Taking into account t ha t / ( a : ) is an even function, we obtain

l / 2 a if \x \ <ay fix) = •

0 if I a: I > o.

The corresponding characteristic function is found from the third column: Since

Ngiy) = agiy) = sin (αϊ/).

Description and Use of the Tables 7

we see t h a t

g{y) = {ay)''^8m{ay).

E X A M P L E 2 I n i tem 7 of Table I,

2Nf{x) = {a" + ^)'^ and Ν = 7r(2a)-^-

hence the frequency function is

Since Ng{y) = {2a)'-^T^e'^y,

we see t h a t g{y) = e-^y for ^ > 0;

hence g{y) = e"'''*'' for - 00 < 2/ < 00.

Table I I (pp. 74r-96) permits the determinat ion of the characteristic fimction of frequency functions f{x) t h a t are zero for negative x. The Fourier transform G{y) of fix) then becomes

Giy) = r^f{x)e^dx = Γ f{x)e<^ dx = g{y)+ ih{y), ( 7 )

Here

g{y)= ί fix) cos xy dx, (8) •Ό

A(2/) = / /(a;) sin xy dx, (9 )

•'o and

^ ( 0 ) = ( 7 ( 0 ) = Γ/(χ) da: = l .

The second column of Table I I contains Ν fix); t he third, Nhiy). The expressions for Ν and giy) can be found in Table I under the number indicated in the first column.

Three examples to illustrate the use of Table I I follow.

E X A M P L E 3 I t em 1 of Table I I yields

1 for a; < α Ν fix) = • and Nh iy) = 2y-' sin^ iya/2).

0 for a: > α

8 Introduction

1/a, 0 < a: < a,

0 , otherwise.

This is the frequency function of a rectangular distribution over ( 0 , a). I t s characteristic function is obtained from the tables as

ay

From this it foUows easily t h a t

G(y) = {e^y- l)/iay.

E X A M P L E 4 I t em 6 0 of Table I I contains the functions

Ν fix) = e-«^ and Nhiy) = y{a^ +

while we see from No. 6 0 in Table I t h a t

Ν = and Ngiy) = aia^ + y^)-\

The frequency function

f(x) = αβ-^^ if a; > 0 ,

0 if a: < 0 ,

is the density of the exponential distribution. The corresponding Fourier transform is obtained by put t ing

Giy) = giy) + My) = (a^ + iay)/ia^ + y') = a/ia - iy).

E X A M P L E 5 I t em 6 5 in Table I I contains the functions

Nfix) = aj -ie-**^ and Nhiy) = ^(v) (a^ + 2 /^)"*" sin[i; a r c t a n ( 2 / / a ) ] .

From 6 5 in Table I,

Ν = a-'Tiv) and Ngiy) = a - ^ r ( y ) [ l + (2/7«')]"*' cos[v a r c t a n ( 2 / / a ) ] .

The frequency function for i tem 6 5 is given by

i [ a V r ( 2 ; ) > ^ - V « ^ a: > 0 , f(x) =

0 , a: < 0 .

This is the frequency function of the gamma distribution. The corresponding Fourier transform is obtained from the tables as

Giy) = giy)+ ihiy) = [1 + (i/Va^)]-*^ exp[ti; a r c t a n ( 2 / / a ) ] .

From item 1 in Table I, iV = a, while Ng{y) = y^^ uniay). We are therefore dealing with the frequency function

Description and Use of the Tables 9

G{y) = / + 00

f{x)e^dx.

The first column gives the serial number of the transform. As an example we mention i tem 1, the uniform distribution over the interval (a, G).

Table l A (p. 1 0 5 - 1 4 4 ) contains the inverse transforms of the frequency functions listed in Table I. I t can be used to determine the frequency functions of even characteristic functions. Column 2 of Table l A contains the fimction Ng{y) sts well as the normalizing factor iV; column 3 gives the function 2Nf{x); while column 1 indicates the serial number which identifies the same pair of functions in Table I .

E X A M P L E 6 The function g(y) = 1/(1 + y^) is an even characteristic function (namely, the characteristic function of the Laplace dis t r ibut ion) . To find its frequency function, use Table lA. I t e m 6 0 of this table contains

Ng{y) = α(α2 + Ν = a''

2Nf{x) = e-^\

Note tha t , for α = 1, g{y) is the desired function. H e n c e / ( x ) = for χ > 0 . Since f{x) is necessarily even, we see t h a t the frequency function corresponding to g{y) i s / ( ; r ) = i e - l x l .

Tables l A and I I can also be used to determine the frequency functions t h a t belong to certain characteristic functions G{y) which are not even. This is possible if the corresponding frequency function vanishes for negative values of the argument . If G{y) is an analytic characteristic function, t hen it is possible to decide whether the corresponding frequency function has this property. In this connection we mention the following result.

T h e o r e m 1 0 Let F{x) be a distribution function wi th analytic characteristic function

r+o G{y)=j e'y-dF{x)

This is not the form in which this characteristic function is famihar to statisticians. However, i t can easily be reduced to the customary form by noting t h a t

a + iy = (α' + y^)^exp[i a r c t a n ( t / / a ) ] .

Using this relation, we obtain

G(y)= α ' (α - iy)-\

Table I I I (p. 9 7 - 1 0 2 ) gives Fourier transforms of frequency functions t h a t do not belong to the classes listed in Tables I and I I . The second column of Table I I I gives the function Nf{x) and the normalizing constant iV, the thi rd column gives the transform NG{y). Here

10 Introduction

and suppose t ha t F{x) is bounded to the left. Then

l e x t [ F ] = - l i m sup t"^ log G(it).

Here lext£F^ denotes the left extremity of the distribution function F{x); for details the reader is referred to [ 3 ] and it is noted here only t h a t Tables l A and I I can be used to determine the frequency funct ion/(x) = F^{x) of G{y) if l e x t [ F ] = 0, t h a t is, if

Um sup ir^ log G{it) = 0. (10) <-».oo

The use of the tables is i l lustrated in the following example.

E X A M P L E 7 Let G(y) = 4y"^ sin^ ^t/e*». I t is easily seen t h a t this function satisfies condition (10) so t h a t the corresponding frequency function vanishes for negative values of its argument . Decompose G{y) into its real and imaginary par t s and obtain

g{y) = Ay^sin^ ^y cos y,

h{y)= 42/-2 sin^ I?/sin y.

Ente r Table IA with giy) and find under i tem 2 t h a t Ν = I while

X

2Nf{x) = ^ 2 - X

0

In Table I I , i tem 2, Nh{y) is listed with

X

Nfix) = ' 2 - X

0

I t can therefore be concluded t ha t

0

fix) = X

2 - X

0

if 0 < X < 1,

if 1 < X < 2,

if X > 2.

if X < 1,

if 1 < X < 2,

if X > 2.

if X < 0,

if 0 < X < 1,

if 1 < X < 2,

if X > 2,

Deecription and Use of the Tables 11

fix) = 0 for X < 0,

(27r)-*x-i e x p ( - l / 2 x ) .

Table I I I A (p. 146-150) contains the inverse transforms of the functions given in Table I I I . The second column contains the function NGiy) and the normalizing constant N; the third column gives Nf(x). Here

R+oo

Giy) = J e'y-f(x) dx.

The notations used in the tables are listed following the Appendix, beginning on p . 162.

is the frequency function corresponding to G{y). This is the frequency function of the tr iangular (Simpson's) distribution.

I t should be noted t h a t a frequency function / ( x ) which vanishes for a; < 0 need not have an analytic characteristic function. If G(y) is not an analytic characteristic function, then no simple criterion is known to determine whether / ( x ) = 0 for χ < 0. However, i t may still be worthwhile to t r y using Tables IA and I I . The next example shows a case where G{y) is not an analytic characteristic function b u t where the method is nevertheless appUcable.

E X A M P L E 8 Let

G(2/) = exp{ - | 2 / | * C l - i ( y / | 2 / | ) ] } ;

this is the characteristic function of a stable distribution with exponent | . Decomposing G(y) into real and imaginary par ts , we obtain for t/ > 0,

g{y) = e x p ( - i / * ) cos h{y) = e x p ( - y * ) sin t/*.

Enter ing Table IA with g(y), find in i tem 84

Ngiy) = (ττ/α)* e x p [ - i2ay)^2 cos[(2ay)*]

and

Ν = ( 7 Γ / α ) * , 2ΛΓ/(χ) = x - * e x p ( - a / x ) .

We look up i tem 84 in Table I I and find

Nfix) = x-t e x p ( - a / x ) , Nhiy) = ( V a ) * e x p [ - (2a2/)*] sin[(2a!/)*].

This agrees with the given functions, and pu t t ing α = i , t he desired frequency function is obtained as

12 Introduction

Tables of t h e Appendix

We saw in Example 5 t h a t the tables sometimes give the characteristic function in a form which is unfamihar to the statistician. I t appears therefore desirable to list separately characteristic fimctions which occur frequently in the statistical l i terature and to write those in their customary form.

Tables of these functions can be found in the Appendix which was compiled by R. G. Laha. The appendix consists of three tables which contain characteristic functions of univariate density functions (Table A ) , of discrete distribution functions (Table B ) , and of mult ivariate distributions (Table C ) . The tables of the appendix do not provide a complete coverage of the statistical l i terature; only the sources listed in the References were consulted. I t is certain t h a t many characteristic functions not listed in the tables of the Appendix occur in the l i terature. Nevertheless, it is hoped t h a t the most impor tant distributions are included.

References

1. H. Cramor (1946). ''Mathematical Methods of Statistics." Princeton Univ. Press, Princeton^ New Jersey.

2. M. Loöve (1955). 'TrobabiUty Theory." Van Nostrand-Reinhold, Princeton, New Jersey. 3. E. Lukacs (I960). "Characteristic Functions." Griffin, London.

TABLE I

EVEN FUNCTIONS

Defini t ions

I n these t a b l e s / ( x ) is a n even function of x, i.e., / ( — x ) = f{x), such t h a t

/ " / (I a; I) (te = 2 Γ f{x) dx = 1.

Consequently, i ts Fourier transform is

g(y) = Γ /(I l) "*' = 2 Γ fix) <^osixy) dx.

Therefore 0^(2/) is an even function of y, i.e., gi—y)= giy)* A real func t ion / (x ) of the real variable χ is called a "frequency function" or

"probabil i ty density'* if i t is defined in the interval — » < χ < 00 and if i t satisfies the properties:

(1) fix) is nonnegative in this interval ;

(2) Γ fix)dx = L •'—00

15

16 Table I: Even Functions

Tables of integrals of the form

G(y) = r f(x)e'^dx •'—00

are Hsted here for the cases:

(a) fix) is an even function of χ (Table I, pp . 17-73) ; (b) fix) vanishes identically for negative χ (Table I I , pp . 75 -96 ) ; (c) fix) does not belong to any of the classes α or 6 (Table I I I , pp . 98-102) .

A list of notations can be found on pp. 162-167.

1. Algebraic Functions 17

1. Algebraic F u n c t i o n s

2ΛΓ/(χ) Ng{y)

10

1, x<a 0, x>a N=a

X, x<l 2-x, l<x<2 0, x>2 N=l

x-», x<l 0, x>l N=2

{a+x)-\ xb N=hg(l+b/a)

0, xb n = 2 , 3 , 4 , . . . ΛΓ= ( o + 6 ) i - " ( n - l ) - >

0, xb N=a-nog{l+a/b)

iV=ir(2o)->

[6^+ ( a - χΥΤ'+ \V+ {α+χΥΤ'

C(a +a:»)(&»+a ):-> ΛΓ=π[2οδ(α+6)]-ι

2ν=2Ηπα-'

2/-* sin(ai/)

4j/~^ cos«/ sitf(52/)

(2Vy)»C(2/)

cos(ay) [Ci(a2/+6j/) — Ci{ay) ] +sin(a2/) [8i(ay+6j/) — siiay) ]

( n - 1 ) !

• sin[|^(n— m) — 6i/] - 1) ! ] -Ϊ8ίη (α2/+ |πη) •Ci(ai/+6?/)—οο8(α?/+§πη) si(ay+by)2

o~^[cos(ai/) Ci (a i /+6 i / ) +s in(a2 / ) s i ( a 2 / + 6 t / ) —Ci(62 / ) ]

7Γ6"^ cos{ay)e'~^

^πα-^ exp (—ay2r^) sin ( i i r+ 2^ay)

18 Table Is Even Functions

2Nfix) Ngiy)

11

12

13

14

15

16

17

18

19

20

21

[ a ^ 2 o V cos (2«y)+o*] - i -§5r<^<^ iV=â- ' sec t?

a^Ca^+2aV cos(2t>)+o*]-i -§ir<t><Jir

N=ira-i

{a+x)-i Ν=2αή

x-îa^+x")-*

( o — x < a 0, x>a

{a-x)-\ xb

b<a i V = - l o g [ l - ( 6 / o ) 3

(α^-ατί)-!, x < 6 0, x>b

b<a

N= (2α)- ' IogC(a+6) (o - f t ) " ' ]

0, x < 6

b>a N= (2o)-i log[(6+a) ( f t - a ) " ' ] 0, xb N=b-h-

x'ia^-x^)-*, x<a 0, x>a

ϊΐτα"* csc(2t>) exp{—ay

ixa-'csc(2t>) exp(—aj/cosi^) 'S\n{u—ay shi&)

ira-i{co8(a2/) [ 1 - C(oi,) - Siay) ] +sin(aj/)[C(aj,)-Ä(a2/)]}

2 a - » - (2ir2/)Mcos(a2/)[L-2S(ay)*] -siniay)il-2C{ayy2}

{hry)Î-iihay)may)

(2ΙΓ)*2/-»[οο8(ο2/) C(oy)+sin(o2/) S(oy) ]

cos(o2/) [Ci(a2/) — Ci(ay— by) ] +8ΐη(αι/) [8ί(θί,) - ai{ay- by) ]

(2o)-M co8(a?/) [Ci(o2/+62/) - Ci (o j / - by) 3 +sin(oj,) la{ay+by) - 8i(oä,-62/) ]}

(2o)-i{ sinCay) [ 8 i ( 6 2 / - a y ) + s i ( 6 y + o y ) 3 - cos(ay) CCi(6i / - oy) - Ci(by+ay) ]}

6 - I T [ L - C ( 6 2 / ) - S ( 6 y ) ]

W i : / o ( a i / ) - ( a 2 / ) - V i ( a y ) :

2Nf{x) Ngiy)

22

23

24

25

26

27

29

30

31

32

0, xb N= (26)-»«·

0, xb N=ia+b)-iT

ia^-3?)-*, x<a 0, x>a N=^

xia^j?)-^, x<a 0, x>a

x-iia'-T?)-^, x<a

0, x>a A r = 2 - * T « | T ( i ) r

0, xa N= arccos(i>/e)

ar i [o+a;+(2ax)»]- ' N=vi2a)-*

:r*ia^+3?)-i[x+ ia^+3?)iT* 0, x<a x-iin^-a^-i, x>a Ν=2Γ*πΚτα)Τ' 0, x<a x-i(x*-o«)-*, x>a N=U~^

(α»+α?)-1[ο+(α»+ί?)»]-1 iV=(2o)-»ir

(26)-iir{cos(6j/) [ 1 - C(26i/) - Si2by) ] +8in(62/)[C(2&i/)-S(262/)]}

(α+6)-*τ{ cos(oi/) [1 -C(oy+6i / ) - Siay+by) ]+sin(oi/) [C(aj/+63/) - Ä ( a y + 6 i , ) ] }

^vJoiay)

a [ l - i i r H i ( a i , ) ]

( k ) ¥ C J ' - i ( § 0 2 / ) T

τΣ^«+ί(|θί ,-^62/) N-O

•J-îhay+hby)

ir(2o)-*e««'Erfc[(ai/)*]

2~ia-Huuiihay)Kiiiay)

-ihr)¥J-îhay)Yîhay)

h-'^-^[Joiay)H.i{ay)+H,(ay)May)2 (2a)-iirErfcC(oi,)»]

2 0 Table I: Even Functions

2Ni{x) Ngiy)

33

34

35

36

n,m=0,l,2, ..., n>m

(2TO)!(2n-2m)! 22"+ίη!»η!(η-ίη)!

χ^{3?«+α"')-^ m , n = l , 2 ,3, . . . , 2m<2n—l N= {ν/2η)ά"^^"-' csc[(V2n)(2m+l)]

x-iia^+x')-i\:x+{a'+^)ilr^ N=2na-h

( r ' -a«) -»[a ;+(x ' -aä ) i ] -" , x>a 0, x<a ΛΓ=(ΐ/η)α-"

§5r[(-l)"'+"/nO(<iy<fe") .[«"'-lexp(—j/a*)]

(ir/2n)a''"*-a''-» Σ {exp(,-ay sin[(2ft-1)

• (ir/2n) ] ) sinC(2A;-1) (2m+1) (ir/2n) +02 /cos(2Ä- l)( ir /2n)]}

2-*α-ΊΓβ-^/ο(|α2/)

—|ö~"fl-[sin(|rMr)/„(a2/) + c o s ( W ) F„(a2/)]

η - α - « Σ Ä;!(n+Ä;~l)![(2Ä;)!(n-Äj)!]-i

2 · Arbitrary Powers

2Nf{x) Ng{y)

37 !(Τ-\ x<l, v>0 h-'M"; «Ή-1; iy)+iFi{p;v+i; - iy) 1 0, x>l

38 {b-x)% x'+1, % ) 0, x>b, !>>-! -expp ( i ' jr /2-6i / ) ]7( i '+ l , - % ) } JV=6'+i(^+l)-i

39 3r-\b-x)''-\ xb

v>0, μ>0 N=b'•^-'B(p,μ)

2. Arbitrary Powers 21

2Nf{x) Ng{y)

40

41

42

43

44

45

46

47

48

49

50

x'{a+x)-\ - 1 < . ' < 0 ΛΓ=—IT csc(i/ir)

i^ia^+3?)-\ - 1 < » ' < 1 Ar=io '-»irsec(inr)

(a^+x")-^, v>0 J V = § a - M C r W / r ( . + | ) ]

{τ'+2αχ)-^, 0<v<^ N=h-^'ir-ir{p)T{h-i')

'ia+{a^+m, >'<h Λ Γ = 2 - » α - » τ * [ Γ ( | - | . ) / Γ ( ! - | ν ) :

x-i{a^+3?)-iZ{a'+3?)^-xJ, v>-h

x-iia^+x^)-iZx+{a'+m, N=a'-iiiTKT{i-¥)/Tii-hv)l

{α'+^)-^1(α'+3ΐ?)*+χΤ', v>0

ΛΓ=αΐ-ν(» '2-1)-»

3f{a^+3i?)-i'-\ -1<ν<2μ+1

(α*—a:^)'"*, x<a 0, x>a, v>-i

( 2 α ) ' ( α τ 2 / ) » { ^ | ^ Α _ ^ . » ( α ί / )

Γ(1+|ίΟ_„ , . -2_, , , . S-^,i{ay)

\ΐ(αΓ^ s e c ( ^ ) cosh(oj/) +»»2 ' -v -T( iv -^) [ r ( i - i . ) r

• i F 2 ( l ; l - f v , f - § v ; i a V )

l{2a)-'^/T{v+h)ly'K,{ay)

2-'-Ό-' ίΓ*Γ(|-»') -y'LMay) Biaiay)-Y,iay) cosCaj/)]

2 - » α - ' Γ ( Κ | . ) • jr*l^^,j(a2/) Λί-ί,.-ι(θί/)

a'ihr) ¥l-i-i,{hy) Ki-i,{hay)

a-'ircsc(w)ÖJ,(tO2/) + υ » ( - % ) - / , ( α 3 / ) c o s ( W ) ]

oT'inr cac{vir)y~^{I,{ay) s i n ( | w ) + m ( i o 2 / ) - J , ( - i o j / ) ] }

+ § i r » ( i t / ) ^ ' + ' r ( i . - M - i ) . [ Γ ( μ - | . + 1 ) ^

. : ^ 2 ( μ + 1 ; μ + 1 - | . ' ; μ + | - ^ . ; 1 α ν )

2-Wr(v+i)r'^(«3/)

22 T a b l e I: Even F u n c t i o n s

2Nf{x) Ngiy)

51

52

53

54

55

56

57

58

xia^-3^)'^, x<a 0, x>a,

v>-h N=a^'+K2v+1)-'

{2ax-i?)'-*, x<a 0, x>a,

J V = o M [ r ( r + ^ ) / r ( v + l ) ]

x'ia^-a?)", x<a 0, x>a,

» ' > - l Ν=ία^^Β{μ+ΐΛ+¥)

. { [α+ [a- {a^-m}, x<a

0, x>a N={2a)-^B{\+\v,-^\v)

+ [ (6+x )» - t (6 -x )» ]^ ' | , 0, α:>6 i^=22'6i-};r ' [ r( |+^)r( |-v)R'

( a ^ - o * ) - - x > a 0, ί ;<α

0 2a

Q<v<\ N=\a-^'ifiT{v)T{\-v)

0, x<a, x-^is^a^)-'-*, x>a

A r = i o - ^ V s e c ( w )

ο«^Η2ρ+1) - ' . [ 1 - ( ΐ α ) - Μ Γ ( . + § ) 2ΓΉ^ι(α2/) ]

(2ο) 'π»Γ(.+^) 'y-'J,{ay) cos{ay)

\α^'Β{μ+\Λ+¥) •M+hv;\,¥+\+^;-W)

( 2 a ) - i f i ( I + | . , I - | . ) •iFi(J-ii';^;-ioy) • i i ' i ( I - ^ i ' ; § ; t a j / )

- K 2 a ) - ' T » r ( I - . )

·2/'7,(α2/)

- I ( 2 a ) - ' i r i r ( I - . ) 'y'Uyiay) s\a{ay)+Y.{ay) cos(aj,)]

'{l-\airylJy{ay)^.-x{ay) -H,(o2/)J,_i(aj/)]}

3. Exponential Functions 23

2Nf{x) Ngiy)

59 0, z<a -Wîhiry)* •Ui'>-i{hay)Y-i>-i{iay)

+ [χ-(χ2-α')*]'}, x>a +J-i^{hay)Yi^{hy)l

i V = ( 2 T ) - » a ' - i r ( i - f . ) r ( i - ^ )

3 . Exponent ia l F u n c t i o n s

2ΛΓ/(χ) Ngiy)

60 aii^+y')-' ΛΓ=α-'

61 x-i{e-^-e-<"), a>b h\ogiia^+y')iV^+y')-^l N= log(a/6)

62 x*<r« Wia'+f)~* cos[f arctan(i//o): ΛΓ=ΑΐΓ»ο-|

63 (^) ί (α»+3 /^) -»[α+(«'+Λ*ϊ ΛΓ=(τ/ο)*

64 0, x < 6 Γ(1+.)(α^+2/«)-ί-ί'β--» (x -6 ) 'e -^ , x>i> •cosCiw/+(i'+l) aretan(y/o)]

» ; > - l i V = o - ' - ' R ( l + i ' ) e - *

65 x-»e-<" a-'T iv) [ 1 + (i/yo^) ] - * ' οοδΗι/ arctan(ii/o) ] ΛΓ=ο-'Γ(ί')

66 N=a-nog2 -îh+ihvM-îh-iiy/a)!

67 x ( e « - l ) - i h-'-U^/a)Xc8chiiry/a)J

68 x -^( l - e -« )* aDog(j/2+4a2)-log(2/2+a^)] iV=2olog2 — 2/ arcctoß(y/ o ) ' + f y / o ]

69 0, x < 6 o-'- ie-<*R(L+i') ( x -6 ) ' e -« , x > 6 . (l+j/ya*)-*'-* C08[%+ (^+1)

v > - l •arctan(y/a)] Ν=α-^'^Τ{1+ν)

2Nf{x) Ngiy)

70

71

72

73

74

75

76

77

78

79

80

81

e-^il-e-*')'-\ v>0 N=b-'B{p,a/b)

x'-'i<r+i)-\ ν>0 i V = ( l - 2 i - ' ) r W f W

j . p - l ( e« r_ l ) - l ^ I '>1

Ν=α-'Τ(ν)ζ{ν)

exp(—ox*)

]\Γ=2-^α-*π[Γ(|)]-ι

χ* exp(—aa;2) A r = 2 - i a - i [ r ( i ) r ^

x*" exp(—aV) ΛΓ=(2α)-2' '-Μ[(2η)!/η!]

x'exp(—ox*), j ' > —1 ΛΓ=ΐα-ί«-^->Γ(Η|.)

(6*+x*) - iexp( -oV) iV=i6-'irexp(o«6«) Erfc(o6)

exp(—ox—6V) ΛΓ=^δ-Μβχρ(ον46*)

•Erfc(o/26)

x^'exp(—ox—6x*), i '>0 iV= (2δ)-'/«βχρ(-ο*/86)

.r( .)D_,[o(2&)-i]

e x p [ - (ox)»] iV=r ( i ) ( 3o ) - i

hb-HBi,', ia-iy)/b2+BZp, (o+ij/)/&]}

Γ W ( y - cos(iw)+§(2o)-'{|-[v, i + ( V 2 o ) ] + f C", §-ϊ·(2//2α) : - f C ^ iiy/2a) ] - f C ' ' , - i ( y / 2 a ) ] } )

i o - T W { f C . , l + ( V « ) ] +?[»',!-(%/«):}

ΚτΑ )*βχρ(-2/* /4ο)

2-ίο-ίπ2/ί exp(-2/y8o)/_i( t /y8o)

2-»ο-*5Γ2/ίβχρ(-2/*/8α) .C/_, (yy8o)-7i( i /y8o)]

(-1)"2-»-ΐα-«"-Μ .exp(-yy4o«)ffe2n(2-»y/o)

^ο-ία-Η·)Γ(Η^.) · ι ί Ί [ Η ^ . ; § ; - ( 2 / Υ 4 ο ) ]

i&-»ir exp(o*6*){exp(-6y) Erfc[o6- (y/2a)] +exp{by) ErfcCoH-(2//2o)]}

K2&)-''* exp[(o*-2/*)/86]r(v) . {exp(-toi,/46)i>-,C(o-t2/) (26)"»: +exp(my/46)Z)_,C(o+i2/) (26)"»]}

(3o)-i2/Mexp(iiir)Ao.,C(2//3o)» •2 exp(ijir)]+exp(-ii9r) .Äo , j [2exp( - I IT) ( I , /3o)»]}

2ΛΓ/(χ) Ngiy)

82

83

84

85

86

87

88

89

90

91

92

93

xfexpi-caf), μ>-1, 0 < c < l

N= ο-ΐα-<^»'«Γ[(μ+1) c-i]

afexp(-ax«), m > - 1 ,

N= c->a-t^«/«rC(M+1) c-»]

x-*e-"' ΛΓ=(«/α)»

x-^êxpi-aVix), v>0 ΛΓ=2*Ό-'Γ(ν)

x~* exp[— ox— ib/x) ] ΛΓ=ο-*ΐΓ»6χρ[-2(αδ)*]

_ x~* exp[— o x - (6/a;) ] 'ΛΓ=(τ/6)»βχρ[-2(ο6)ί]

xî e x p [ - o x - (ί>*/χ) ] iV=26O-'"'Z,(2a»i>)

x-*exp(-o*x-«) iV=§o-M

exp(—ox*) JV=2o-«

x~*exp(—ox*) ΛΓ=2α-'

x~*exp(—ox*) ΛΓ=2ο-*ΐΓ*

xîexp(—ox*), i '>0 N=2a-^Ti2v)

- Σ { ( - a )" (« ! ) - ' r (M+H-nc)

·8ίη[ίτ(μ+ηο)]2Γ*-ι-«}

e-ΐΣ { ( - l ) ' O - » ' ^ « / « C ( 2 n ) ! r ' N-O

•Γ[(2η+1+μ)ο-'>*»}

(π/ο)* e x p [ - (2oy)*] C08C(2oy)»]

2O-'y*'{ expP(Mr/4) 2K,iae^yi) +expC-i-(v7r/4) ]i:,(ae-<*V)}

ir*(o«+i/«)-*exp(-2«6*) • [« cos(26*») - V sm(26*») ] Μ=2-*[(ο*+Λ*+ο]» »=2-*[(ο*+ί/*)»-ο]*

(IR/i>)*exp(-26*M) co8(2i>*») w=2-*C(o*+i/*)*+o]* «;=2-*[(o*+i/*)*-o]*

6*{(ο+ί2/)-'/«ί:,[2&(ο+ίί,)*] + (ο-ίί /)- ' /*ί: ,[26(ο-%)*]}

N-O n ! r ( H i n )

2-»oir*y^ . { c o 8 ( o y 4 y ) Ö - C ( a y 4 j / ) ] + s i n ( o y 4 y ) Ö - Ä ( o y 4 y ) ] }

(2Vy)MCO8( V / 3 / ) Ö - 5 ( i a y y ) ] - s i n ( i o y y ) C | - C ( i o y y ) ] }

io*iry-*{/j(ay8y) s in(oy8y+iir) - y i ( o y 8 y ) c o 3 ( a y 8 y + i T ) }

Γ(2,') ( 2 y ) - ' ( e x p { - i Ö w + ( o y 8 y ) ] } .D_2 ,Co(- i /2y)*]

+exp{iÖ«r+(oy8y)]|i)-i,Ca(v'%)*J)

26 Table I : Even Functions

2Nfix) Ngiy)

94

95

96

97

98

99

100

101

102

103

exp[-6(a*+x*)i: N=aKiiab)

(a«+x*)-l expC-6(a*+a?)»] N=Koiab)

x-i{l^+^)-i exp[-o(62+x*)i] N= 2ia^b-iir*Kiiab) [ Γ ( | ) Jr'

ΛΓ=6ί(2ΐΓ)-»[ίΓι(^α6)Τ

•exp[-a(6*+x*)i] iV=(Vo-»)e-'*

+ [(a*+x*)»- x]'} e x p [ - 6(a*+x*) *] Ar=2a'Ä:,(o6)

x'-i(o2+x*)-i

•exp[-6(o*+x*)i] , v > - J ΛΓ= 2(''«-ia-»r ( H i " ) D_^[2(o6) *]

(o*+x*)-iCx+(a*+x*)iJ •exp[-6(a*+x*)l]

ΛΓ= a' csc(Mr) {ir7_,(o6)

— / exp(oi>cosi)cos(i'i) cK}

(a*+x*)-i[(ffl*+x*)*+a]-i •exp[-b(a*+x*)*]

ΛΓ= (2α)-*πβ<*ΕΓίο[(2αδ)»]

abi¥+f)-iK,[,aib'+m

i:oCa(ft*+2/*)*]

(W)*/-l{i&C(a^+2/^)»-a]} •i!^i{i6[(«'+^*+a]}

δΗ2τ ) -*ί: ί{Κ (6^+2/*)*-2/]} 'maZiV^+y^)i+y2}

2»6»T-J{Z5Bo((6*+i/*)J-i /)] •Ä,Öo((i'^+i /^)»+2/)]

.ÄiÖa((6*+3/*)i+y)]}

(^π)*(α«+2/*)^ . [ α + (α*+/) ί ]» exp [ -6 (a*+2/*)»]

2α' coslv arctan(i//6)] . ί: ,[α(6*+2/*)»]

2('/»-Ja-lr(i+i^)2/-i .[(6*+y2)l+6]lZ)_^ • iC2a((6*+^i+6)]»} •Mw2),_i{Ca(6*+j/*)*-6]}

α" csc(w) {π coa[y arctan(y/6) ] -I-laiV^+m

— j exp{ab cost) cosh (ay smt)

-cosivt) dt}

{2α)-^7Γ^ •Erfc{a*[(62+2/2)j+5-]i}

4. Logarithmic Functions 27

2Nf(x) ΛΓ!/(2/)

104 ( a * - r ' ) - l e x p [ - 6 ( o * - x * ) i ] , x<a 0, x>a . [ /_i(«i)Fi(2^)+F_l(0i) /}( i2) :

2

105 x-^(i){a^-3?)-i{ [a- (a«-x*)»p a- 'r(H ' ' )r( i-v)(2Tj/)-» . exp[6(a*-x*) i ]+[a+ (a^-a?)*? ' ·βχρ[-6(α*-χ2)»:) , idz i '>0 .M_,._i{ap.-(6*-i/*)»]}, b>y

N=iia)W

•r(i+.)r(i-.)M,._i(2ai.)

4. L o g a i i t h m i c F u n c t i o n s

2ΛΓ/(χ) Ngiy)

106

107

108

109

110

—hgx, x<l 0, x>l N=\

log(a+a;), xb

a>l N= ia+b) log(o+6) — a logo— 6

-il-x^logx, x<l 0, x>l J\r=|Tlog2

(a*+a^)-ilog(a*+a?), a > l i\r=o-Vlog(2o)

(a*+a^)-»-*log(a*+x*), a > l , n = l , 2 , 3 ,

Ar=o-*"B(i n) 2»-l

. D o g ( a / 2 ) - Σ m - 1

2/-i{sin(6j/) log(a+6) — cosiay) \ßiiay+by)—aiiay) ] +sin(ay) [Ci(ay+62/) - Ciiay) ]}

iπlog2Jo(^/)+i^ΓΣ«-V2„(y) n - 1

- K ' T { e - « « ' [ 7 - l o g ( 2 a / y ) ] -e« ' 'Ei( -2a2/)}

-η![(2η)!]-Η22//α) ' · Η

• { Χ „ ( α ί / ) [ γ - 2 Σ ( 2 m - l ) - I + l o g ( 2 y / a ) A

2Nf{x) Ngiy)

111

112

113

114

115

116

117

118

119

120

N=ia-b)ir

(o*+x*)-i log{x-'Za+ (α*+χ*)»]} JVr=iir*

N=a

ix'-(^)-nogix/a)

log(l+a*a;-*) N=ira

—\ogia—x), x<a 0, x>a

a<l N=ail-loga)

— logia—x), xb

b<a<l N=—aloga+6+ (a— b) log(a— b)

- log(o*-a^) , x<a 0, x>a

0<a<l Λ Γ = 2 ο ( 1 - logo)

-(o ' -x*)-»log(o*-a;*) , x<a 0, x>a

0 < o < l JV=irlog(2/o)

- log(o*-ar ' ) , xb

b<a<l N=-ia+b)logia+b)

+ ia-b) l og (o -6 )+26

iry-\e-^-e-^) ± as 0 ^ 6

Jr*[7o(oj / ) -LoM]

i » 2 / - T H - L o ( o i / ) - 7 o M ]

|o~V[sm(oi/) Ci(a2/) — cos(o!/) s i (o3 / ) ]

^ - i ( l _ e - » )

- y-i{ sm(oj/) [Ci(o2/) - γ - l o g y ] — cos(o3/) Si(oy)}

—y-'{sm(6j/) log(o—6) +sm(oy) [Ci(oy) - Ciiay-by) ] - cos(oj/) CSi(oy) - S i (oy- by) ]}

y-^eoaiay) Si(2oy) +sin(oy) [7+log(y/2o) -C i (2ay) ]}

^T{C7+log(2y/o)l /o(ay) -hrYoiay)]

- r M s i n ( 6 y ) log(o*-6*) - cos(oj/) [si(oy+6j/) - 8 i (oy- 63/) ] +sin(oy) [Ci(oy+6y) - Ci(oy-6y) ]}

2NS{x) Ngiy)

121

122

123

124

125

126

-Ca ! ( l -a ; ) ] - J l og [a ; ( l -x ) l a;<l 0, x > l iV=4irlog2

-(o«-x*)»-*log(o*-x*), x<a 0, x>a a<l, n = l , 2 , 3 , . J V = - a * » f i ( i H n )

. D o g ( a / 2 ) - £ ( - 1 ) " ^ - » ]

- (2aa ; -x ' )» - i log(2ax-x*) , x < 2 a 0, x > 2 a o < l , n = 0 , 1 , 2 , . . . Ν 2 a * ' ' 5 ( i H n )

. D o g ( 2 a ) + E (-1)"·»»- ' :

ra-l

n-1 F o r n = 0 , Σ ( ) = 0

0, x<a - l og{§x-»C(x+a )»+(x -o ) i : } , x > a

e-~(loga;)* N=a-^\:W+iy+loga)^l

( l + r ' ) - M o g [ x + ( l + x ä ) i ] •{[x+(l+x*)»]» -Cx+(H-x*)»r") .expC-6(l+x*)»]

n-1

'Kn,ib)Zm\in-m)T'}, n = l , 2 ,3 ,

-(7+log4y)/o(i l ,)]

-M2n)\in\)-K2y/a)--

• {In! Σ (§α2/)'»-»Λ.(α2/) Cm!(n-m) r

Η

+/„(oy)C2 Σ ( 2 m - l ) - i - 7 - l o g ( 2 i , / o ) ]

+| irr„(ai / ) )

-ir(2n)!(w!)-' cos(o2/) (22//o)-» n-1

• { § η ! Σ (ia2/)»-»[:»»!(n-m)r'/«(ai/) m-O

n-1 +Jniay)i2Z ( 2 m + l ) - » - 7

m-O

- log(22/ /a ) ]+ |TF„(ay)} n-1

F o r n = 0 , Σ ( ) = 0

hr'ihrMay)+niay)l

(o«+j/*)-i{ioir^+23/ arctan(3//a) + o [ 7 + | l o g ( a * + y * ) T —o[arctan(y/o) J }

n! cos[n arctan(2//6) ]

m-O

• C m ! ( n — 2 arctan(y/6) •sinCn arctan(t//6)]2i:nC(62+t/2)|]

2ΛΓ/(χ) Ng{y)

127 l og ( l+e -« ) |[ay-*—in/-' csch(o->iry) ] ΛΓ=(12α) -ν

128 - l o g ( l - e - « ) - JCo2/-*-irj/-i ctnh(o-iir2/) ] Λ Γ = ( 6 ο ) - ν

5. T r i g o n o m e t r i c F u n c t i o n s

2Nf{x) Ngiy)

129 χ-*[8Ϊη(αχ) J Ma-h), y<2a 0, y>2a

130

131

(smaa/a;)2»«, m= 1,2,3, , . . iV=(-l)"»2i-2*«m7r

• £ ( - 1 ) · , ί ^ Γ ' V ^ 1 ( m + n ) ! ( w - n ) !

iV=iX-l)'^2-2'»-2(2^+l)-i - 1 - 1

. \ 2 n + l y.

132

133

134

lo—cosx)-*, x<ir 0, x>ir iV=xcscho

(α*+χ')->(1-2δοο8χ+6*)-ι, 6 < 1 N= o-V(e"+6)/2(e»- 6 ) ( 1 - 6*)

(a*+r')-»(l-26cosa;+6*)-», 6 < 1 ΛΓ= «•(β·+6)/2ο(ί?·-6) (1-6*)

(-1)"·2-*·>ΜΓ (m!)-V"-i

+ Σ n-l

y<2am 0, y>2am

(-1)"[(2οη+2/)*·»-ι + ( |2αη-ί / | )* '»- ΐ]

(m+n)! (m—η)!

ί ( -1)»2-* ' -*(2η+1)- ι

V 2 η + 1 ) (n+h-ihaW-' \ 2η+1 I

sinha ^

| a -»7r ( l -6*) - ' ( e" -6) - i

§ 0 - ^ ( 1 - 6 * ) - » • [e-^-\- {be-^- δ"+ 'β-*·) / (β-- 6) + (6e-«''-°»+6»+»e°«)/(e"-6)], y=n+5

0<δ<1, n = l , 2 , 3 , . . .

5. Trigonometric Functions 3 1

2Nf{x) Ngiy)

135

136

137

138

139

140

141

142

(cosa:—cosS)-*, χ<δ 0, χ>δ

δ < π N=^Kismi8)

(cosx—cosS)*"*, x<S 0, χ>δ 0<δ<ΐΓ, v > - i

N= ihr)* sin'8r(i'+i)P=i(cos5)

(sinx)", x<ir 0, x>ir

a>-l N= 2 -«7ΓΓ(1+α) [ Γ ( 1 + | α )

(coar)", χ < 5 7 Γ

Ο, x>^ α > - 1

N= 2-^Vr(l+a) [ Γ ( 1 + | α ) J" '

sm[6(o*-r ' )»] , x<a 0, x>a

ab<K N=âirJiiab)

(o2_jj!)-lsin[6(o*-ar')*], χ < α 0, x>a

ab<w N=ihr)ibmhab)J

(o*- a^)-i cos[6(a*- ar») *] , a;< a 0, x > o

ai.<iir

(o*-x*)-icos[6(o*-x*)i], x < a 0, x > a

N=bKhr)*iJ-iihab)J

2Γ-hP-i+yicoaδ)

(iir)isin'5r(v+J)Pr2j(co8«)

2-«jrcos(j2/)r(l+a) . \Til+â+hy) Til+â-hy)

2-«-ΐΐΓΓ(1+α) . [ T ( l + i a + i y ) r ( l + i a - | 2 / ) r '

δΚ§'Γ)ί/ι{Κ(6^+ί/^)*-?/]1

2Ni{x) Ngiy)

143

144

145

146

147

148

149

150

151

152

x~*ib—x)~^ ooa[ax*ib—x)*2> 0, a;>6

ab<ir N=TJoi^ab)

a;-*(o*-a^)-*cos[6(o*-x*)»l x<a 0, x>a

Λ Γ = 2 - * ( 6 / α ) Μ 7 _ ι ( α δ ) + [ Γ ( | ) ] - ι

exp(o cosx), x<ir 0, χ>π N=Tloia)

(cosa;)-*exp(—acosa;), x<^ 0, x>hr N=aKbr)HiI-iih)J-ima)J}

(cosx)"* exp(a cosa;), x<^ 0, χ>^π N=aKiirmi-i{h)J+ih(ha)7}

(siiia:)~*exp(—asiiLc), χ<π 0, x>w i V = ( ^ a ) M { [ / _ j ( | a ) r i ^ - [ 7 i ( | a ) T }

(ainx) -* exp (α sinx), χ<·π 0, x>ir i V = ( | a ) M { [ 7 _ i ( i a ) T + C / i ( i a ) T !

log[sec(|ira;) ] , x< 1 0, x > l ΛΓ= log2

logCcsc(iri;)], x<l 0, a ;>l N= log2

[cos(ira;/2) ] · - ' log[sec(ira;/2) ] , x< 1 0, .τ>1

i ' > 0

ΛΓ=2ΐ-'Γ(>') [ 1 ( 1 + 5 " ) 1"' . C l o g 2 - ^ ( v ) + ^ ( H i ^ ) ]

1Γ cos(|6j/)

( i i r ) ¥ . J _ i { K ( 6 * + i / * ) i - 6 ] }

»r/y(α)+sin(ir2/) exp(— α coshi— i/i) dt

aKhr)* •iI-i-iyiha)I-HUha) -/ΐΗΗ,,(|α)7Μ»(*α)]

a H k ) » 'U^iia)I-Myiia) +7ι_ΐν( |α)7}+, . ( |α) ]

( l a ) M c o s ( i ^ y ) .[7_i-}„(|a)7_jHH,(|a) - 7 j _ i . ( | a ) 7 j + 4 . ( i a ) ]

( J a ) M cos(ix2/) .C7_j_j.(ia)7_i+j,(ia) +7 i_} . ( i a )7 j+ j , ( i a ) ]

2 / - ' s i n y • { 7 + l o g 2 + i f C l + ( 2 / A ) : + | ψ [ 1 - ( 2 / Α ) ] }

//-' siny • { 7 + l o g 2 + | f [ l + ( 2 ^ ) - ' 2 / ] +i lACl - (2 / / 2x ) ]}

2 '-Τ(ν)ίΓβ+§.+ ( 2 / Α ) ] } - ' • { Γ Ο + έ . - ( 2 / Α ) : } - ' • { l o g 2 + | ^ [ H i " + ( 2 / A ) ] +m+¥-iyMl-H^)]

5. Trigonometric Functions 33

2Nf{x)

153 (sinira;) log[csc(πa;) ] , x< 1 0, x>l ' {rih+iv+ ( ί / /2χ ) ]Γ[Η*' ' - (y/2r)2

• { l o g 2 + i ^ [ H | . + (2//2T)] Λ Γ = 2 ' - ' Γ ( . ) [ Γ ( Η | ρ ) ] - ' ' + i i A [ H I " - ( y / 2 T ) ] - ^ W )

• D o g 2 - ^ ( v ) + ^ ( H i . ) ]

154 -(o*+x*)-ilog(C*sm*6x), C<1, -a-iir{cosh(o2/) log( l -e -*^) b>0 m

Λ Γ = - α - ' π + Σ s inh[a (y-26n) ] • log{§C[l -exp(-2aö) :} n -1

+ log(iC)e-«'), ' » < ( 2 / / 2 6 ) < » i + l , m = l , 2 , 3 , . . .

m

F o r m = 0 , Σ ( ) = 0 n -1

155 -(o*+a^)-ilog((7*sm*&t), C < 1 , -a-V{cosh(a2/) log(l+e-2«6) &>0 + l o g ( i C ) e - ^

Λ Γ : a-h- m

• logÖC(l+e-*^)] + Σ ( - l )"n - ismhCa (3/ -26n)]} , n - 1

» i < ( 2 / / 2 6 ) < m + l , m = 0 , 1 , 2, 3, . . .

m F o r m = 0 , Σ ( ) = 0

n -1

156 - (α*+χ*) - ι - α V{cosh(ai/) l o g ( l z t e - ^ ) • logßc* cosh8±|c* cos(6a:) ] , + (iö+logic)e-««'

c cosh (5 /2 )< l TO + Σ (=Fl)"w->e-^sinh[a(y-6n)] | ,

+ l o g Ö c ( l ± e - ^ ) : } n -1

wi<(2 /A )<OT+l , TO=0,1,2,3, . . .

F o r m = 0 , Σ ( ) = 0 n -1

157 (α*+χ*)-ι a-V{cosh(a2/) l o g ( l d b e - ^ ) •logßc* coshSzbfc* co8(6a;) ] ,

c 8inh( |5) > 1 m + Σ (=F)'»n-ie--«sinh[a(2/-6n)]},

+ l o g Ö c ( l ± e - ^ ) ] n -1

m<{y/b)<m+l, m = 0 , 1 , 2, 3, . . .

F o r m = 0 , Σ ( ) = 0 n -1

2Nf{x) Ng{y)

158

159

160

161

162

163

—a;-*log[cos*(ax)], o > 0 Ν=ατ

JV=Ta-Mog[cosh (αδ) ]

(cosx—cos5)-* • log(cosx— cos5), x< δ

0, χ>δ 0<δ<2ν

Λ^=7Γ2-*[27Γ-»ί:(8ίηδ/2) log(sini) -ii(cos|a)]

sin(asinx), χ<π Ο, χ>τ

α<ΐΓ iV=irHo(a)

sin(ocosa;), χ<π/2 Ο, χ>τ/2

0<7Γ ΛΓ=|πΗο(α)

cos(acosa;), χ<ν/2 Ο, χ>ιγ/2

α<|ΐΓ

ox{ l - (o - i log2)3/ τη

- α - » Σ (-1)"η-Η2 / -2αη)1

m<y/2a<m+l, ?η=0,1 , 2, 3, . . . η»

F o r m = 0 , Σ ( ) = 0 η-1

ira-»{ai>-log2(ai/+e-"') +cosh(oy) log(l+e-*»»)

τη + Σ {-l)'*n-i-Csinh(ai/-2o6n)

n -1

+sinh(2a6n) - ay cosh(2aön) ]} m<y/2h<m+l, m = 0 , 1 , 2, . . .

F o r m = 0 , Σ ( ) = 0 n - 1

«-2-i{P_}+,(cos5) • Dog(sm5)- 7 - l o g 4 ^ ^ ( H 2 / ) ] -Q-i^icoaδ)}

| π ctn(|7ry) [J ,(a) - J-n,(a) ]

iir cosec(|ir2/) CJv(a) - J_„(a) ]

| i r sec( iT2 / ) [J«(a )+J - , (a) :

2Nf{x)

164

165

166

167

168

169

170

171

cos(asinx), χ<τ 0, x>r

Λ Γ = τ Λ ( α )

(sinx)-* sm(2o sinx), x<ir 0, x>ir

a < | i r N=aWiia)J

(sinx)"* co8(2o sinx), x<ir 0, x>v

a<7r /4 ΛΓ=αΜ0λ_ι(ο)?

(cosx)~*sin(2acosx), χ < π / 2 0, x>ir /2

ai r /2

α<τ/4

(sinx)-* sm(2o sinx), x < x 0, x>ir

a<v/2 ΛΓ=2(ατ)*{[/_ι(α)] '+[7ί(α)Τ)

(cosjx)"^ sin(2a cos^x), χ<π

0, Χ > 7 Γ

α<π/2 N=2{aT)mj^{a)J+[Ji(a)J]

(co&r)"* exp(—^α^ secx), χ < ^ τ γ

0, χ > ^ π

Ng{y)

kCJi / (a)+J- . (a) ]

|αΜ/_^44^(α)/ΗΜν(α)

2(α7Γ)*οο8(|π2/) • [./-i-4i/(«).^-i-Hv(«)+.^!-ii/(a)^fHv(ö) ]

2(απ)»[./_ι_,(α)/_4+ν(α) +/}+^(α)./}^ι;(α)]

36 Table Is Even Functions

2Nf(x) Ngiy)

172

173

174

175

176

arctan(2aya:*) N=air

arcsina;, x< 1 0, x>l i V = J i r - l

arccosa;, x< 1 0, x>l N=l

x~^ arcsina;, x< 1 0, a;>l iV=|Tlog2

arctan(a"a;-"), n = 2, 4, 6, . . . N=âir

• Σ ( - 1 ) · ^ » ο ο 8 [ ( » η + | ) ( τ / η ) ] m - l

vy ê~^sm(ay)

|π2/->[8Ϊην-Ηο(ί/)]

|ΐΓΐ/-Ήο(2/)

KCi(2/)-Jio(2/)

-hry-' η

• Σ ( - l ) ' » e x p { - a 2 / 8 i n [ ( m - | ) ( V n ) ] }

•sinfoi/ c o s [ ( w - i ) (π/η)]}

7· Hyperbol ic F u n c t i o n s

2Nf{x) Ngiy)

177 sech(ax) âr^w sech(ia~^7ri/)

178 [sech{ax)y ôr^Ty cschiâr^wy)

179 \ßech{ax)y N=iar'T

ia-îr{a^+y^) sech(^t//2a)

180 [sech(aa:)]2^ n = 2 , 3 , 4 , . . . Ν=2^^-χα(2η-1)\Τ'

•C(n -1 ) !T

22'»-ia-V[(2n-1) csch(7ry/2a)

m - l

6. Inverse Tr igonometr i c F u n c t i o n s

7. Hyperbolic Functions 37

2Nf{x) Ngiy)

181

182

183

184

185

[sech(ax) ]*"+', » = 1 , 2 , 3 , ΛΓ=2-«"-'α-ΊΓ(2η)!(η!)-*

X csch(aa;) N=iWa)'

co8h(aa;) sech(6a;), a<b N= (ir/26) 8ec(air/2&)

siiih(oä:) sech(6x), a<b

N= -6-» 2

IT tan (1)

sinhiax) csch(6a;), a<b iV=i6-'irtan(air/26)

186 l - tanh(aa;) iV=a-ilog2

187

188

189

a^Ka;"'-cscha:), - l < s < 1 ΛΓ=2Γ(8) ( 2 - - 1 )

a;-*sinh(aa;) sech(6a:), O<6

•l+sm(OIR/26)

^ 2^*'^Ll-sm(OIR/2&)J

a;-'[sinh(aa5)Jcsch(6x), a < | 6 iV=-§log[cos(OIR/6)]

2 « " - ' a - V [ ( 2 n ) s e c h ( 5 R 2 / / 2 a )

• ri iirn-h)'+iy'/^')l m - l

(ir/&) cos(air/26) cosh(iiV26) • [cos(air/6)+cosh(iry/6) ]- i

4

+27rsin

|6-%-sin(oir/6) + [cos(air/&)+cosh(iry/i>)]-'

i a - ' [ ^ ( V 4 a ) + ^ ( - V 4 a ) -^ih+na-^)-^ih-ila-'y)l

- ( s - l ) - I R « • {2 - (s-1) [f(8, Hik)+r(s, | - % ) ] -j/'-'sin(|ir8)}

•co8h(ir2//2&)+sin(oir/2ft) 2 Lcosh(ir!//2&) - sin(oir/25) J

1 Γ l+C08h(iry/6)

4 °^Lco8h(iii//6)+co8(2airA)J

2NJ{x) Ng(y)

190

191

192

193

194

195

196

197

198

iV=log2

(α*+χ2) sech(7rV2a)

x{a^+3i?) csai(Tx/a)

(l+x2)-^sech(7rx) Λ Γ = 2 - | π

{l+x'y'seMhrx) iV=2-^[π-21og(2*+l) ]

( l+r^)- isech( | i rx) Ar=log2

xiM)-' cscHirx) iV=i (21og2- l )

(ar^+l)~^sinh(ax) csch(7rx), α < π N= sina log (2 cos Jo)

—|a cosa

(χ2+1)-ι cosh(ax) sech(jTa:), α<^τΓ

N= a sina +cosa log(2 cosa)

199 (l+x^)~^ sinh(aa:) csch(iπx), α < ^ 7 Γ

Λ^=^π sina — cosa log[ctn( Jtt— a) ]

log(H-e-'«')

2a3[sech(ay)]3

fa^[sech(|ai/)]4

2 cosh(Jt/)-ei' arctan(e-*») —e"^arctan(e*i')

2~*{πβ"^+2 sinhi/ arctan(2~* cschi/) — (coshy) log[(cosh2/+2-*) . ( c o s h 2 / - 2 - i ) - i ] }

hye-y-h+GOshy log(l+e-^)

he~y{y siny—a cosa) + i sina cosht/ log(l+2e-^ cosa+e'^v) —cosa sinhy arctan[sina(e*'+cosa)"'^]

ye~y cosa+ae~y sina

+s ina sinhy arctan e-^y sin(2a)

Ll+e-2ycos(2a)J

+ J cosa cosh?/ • log[ l+2 cos(2a)e-2i'+e-4«']

Jtt sina6~*'

— - cosa cosh?/ log 2

coshi/+sina Lcoshi/—sina.

+s ina sinh?/ arctan (cosa cschy)

2Nf{x) Ng{y)

200

201

202

203

204

205

206

207

208

209

210

x"^ csch(cx) [cosh(aa;) — cosh(6a;)], c>a>b

N=\og cos(|7r6/c)

Lcos(iira/c)J

[cosh(aa:)+cos6]-S 0<6<i r N= ar^h csc6

[cosh (oa;)+cosh6]"^ iV^=a~^6csch6

cosh ( l ax ) [cosh (αχ)+cosh6]"^ i \ r= |a -% sech(i6)

cosh(|ax) • [cosh(ax)+cos&]-\ 0<6a iV=2V sech(Ja)Z[ tanh( ia ) ]

0, x<a (coshx— cosha) - i , x> a i \r=2*sech(ia)/i:[sech(ia)]

cosiirb/c)+cosh (wy/c)

2 Lcos(Ta/c)+cosh(7r2//c) J

a csch smh(by/a) csch(7ri//a)

a~V cscho sin(6i//a) csch(î//a)

|a"Vsech( |6) cosh{by/a) •sech(n7//a)

Ja~Vsec( |6) cosh(a""%) •sech(a~Vy)

( |7r)Ml+2cosh[(f^)V]}-i

2-^TP-^îy{cosb) sech(7ry)

2-*7r^_i+iy(cosh6) sech(7rt/)

2rîQ-i+iy{co^)+Q-îy{cosb) ]

2-^7r^_4+,j,(cosha)

2-*[0-Hti/(cosha) + 0 _ i _ t y (cosha) ]

Ng{y)

211

212

213

214

215

216

217

218

219

220

[sech(aa:)]', i '>0

(cschoa;)', 0 0 N= (sinho)-O,M-i(ctnha)

[cos?>+ coshxjr', v>0, 0<6 a 0, x<a

0<va

v<l N= (sinha)-Q_,(ctnha)

8ϊώ[6(α*-χ*)*], x<a 0, x>a Ν=^ατΙι{<ώ)

(a*-ar')-icosh[6(a*-x*)»], x<a 0, x>a

(a^-3?)-ismh[b(a^-m x<a 0, x>a N=hKh^Kh{hob)J

(a*-r')-lcoshC6(a*-x*)*], x<a 0, x>a iV=6i (k) i [7_i( iat )T

i a - M C r ( i . ) r ( H J ' ' ) r •n\v+iha-'v)Vl\v-i{y/2a):\

2 'a-%sin(§w)r( l - ; ' ) . [ r ( i - i . + i i 3 / / « ) r ( i - * ' ' - I V « ) ] - ' • cosh {Ty/ 2a) [cosh(ir2//a)—cos( w )

(ir /2)»[r(i')rHsmho)i-' • r(^+ti/) r(^-%)^Llii.i„(cosha)

(k )» [ r (v ) rHs in6 ) i - ' 'T{v+iy)nv-iy) ·Ρίΐίί..„(οο8δ)

i ( 2 i r ) - i e - ' T ( l - i ' ) (sinha)*-• [OL-jl,„(cosha)+0'Ll*+<„(cosha) ]

(|7Γ)*Γ(1-Ι') (sinha)*-' .^Lli..„(cosha)

\aiyK{f--H')-^JW-m, y>b ifl-a6(6*-j/*)-*/i[a(6*-2/*)*], yb

hTloia{l^-m, y<b

2riirWi{haiy-{y'-m] •Ji{haiy+if-m

bKhr)*J-i{haiy-iy'-m} •J-iihiy+iy'-m

2NS{x) Ng{y) 221

222

223

224

225

226

227

228

229

230

231

232

x-«(o«-x«)-*cosh[6(a«-x2)*], x<a 0, x>a ΛΓ= 2 -»(6/o)M|T(f) T'I-i(ab) exp(— a cosha;) cosh(6 sinha;), o > b

x-h-^ 8mb{bx) N=hlogi{a+b)/ia-b)2

(e^+l)-»sinh(oa;), a ' > - l , bv<a Λ Γ = 2 - ^ ' 6 - Τ ( ΐ ' + 1 )

•ΓίΙίΛ-ΐ-Ιΐ') •crd+i . '+ iafe -or'

exp(—fea; ) cosh(oa;) Λ Τ = | ( τ / 6 ) * β χ ρ ( - ν / 6 )

a;-*exp(—a;^) smh(a;*)

x - " * e x p ( - o V ) smh(oV) Ar=2*OT-i

exp(—6a;) smh(aa;*)

ctnh(6a;*) tanh(6a;»)

χ-* sech(ax*)

ibr)¥l-uhaib-{l^-m} -I-il^aib+{l^-m]

Kiyi{a^-m cosCi/ i&nh-Kb/a) ]

1 [y'+ia+by 4 °^Lj /*+(a-6)»J

6-*T sm(air*ir) cosh(6-V2/) • [cos(2&-Vj,) -cos(2o6->ir) ]"> -haia'+y")-'

§6-V sm(2ao->ir) [cosh (26-^2/) - cos(2ai»-V) T'+Ma^+y")-'

2-^''b-'T{v+l)

mb-\a+h+

Γ [ | 6 - ' ( α -Ha-bp+iy)3 ] a+bp+iy)+l^ mb-Ko

|(«/ft)*cos(|a2//6) exp[Ka«-2/*)/6]

(iir) Μ e x p ( - - 2-*π% Erfc(2-»j,)}

24a«- i{exp(- ia -V) -2-*o-'yexp(-2/yi6a=') ΕΓίο(2-»ο-Ί/))

W(ö^+2 /^)i βχρ{α^6[4(δ^+ί/^) Τ'} •cos[f arctan(6->3/)-ia^(62+2/2)-i]

See Morden, L. J. (1920). Messenger Math. 49, 65-72.

Äee Morden, L. J. (1939). Acta Math. 61, 323-360.

2Nfix) Ngiy)

233

234

235

236

237

238

239

240

241

(a^+x^)-^ smh[6(o«+r ' ) i : •8ech[c(a2+a:2)i], b<c

Ν=ΐΓ0-'Σ { ( - l ) " s i n [ ( n + | ) 6 T / c ] n-O

'i{n+h)'W)+a'T^]

(o«+a2)-icos[6(o''+x2)»] csch[c(a' '+r ') i] , b<c

i V = | w - i £ { ( - l )»e„ n-O

• ia^+ (nV/<?) yi co8[nir{b/c) ]}

coab.[b{a^+x')i2 •sech[c(a2+x!')i], b<c

iV=c-VE{(-l)"(n+|)

.οο8[(6Α)(η+|)π]}

—log(l—e-'^) coshx ΛΓ=1

[cosh(aa;) Jr' log[cosh(oa;) ] , v> 0

log( l + c o 8 a sechx), o<ir

coshx log (2 coshx)—X sinhx N=hr

log[ctnh(aa;) ]

log(l+a-*sech*x)

N={logi{l+a-^+a-'T

{(-1)"8ίη[(«+|)(6Α)τ] n-O

•exp(-2/[(n+|)»(»yc»)+a»]i)}

^ n-O

• C08C(i.A) wt] exp( - yla^+c-Wy) j

c-VE((-l)"(n+|)[a^+(n+i)» n-O

.(^y(^)]-*co8[(n+§)T(6A): •exp(-2/[a^+(n+i)»(,ry(i)J)}

(1-^(1+2/^)-* +W(l+2 /»)-itanh(|7r2/)

2 ' - T a r w r M m.+iK2/A)] I' . ( i A W - i o g 2 - R e { ^ ß . + i K y A ) ] } )

IT«/-' csch(ir2 /)[cosh(Jiri/)—cosh(a2/)]

i7r(l+2/2)-igech(|T2/)

2irj/-' C8ch(|in/) •8m«{i2/logC(l+a-»)»+o-':)

7 . Hyperbolic Functions 43

2Nf{x) Ng{y)

242

243

244

245

246

247

248

249

—log( 1— sech^x), a< 1 iV= (arcsina) 2

log[_{h+a sechx) (6—α sechx)"^] iV=irarcsin(a/6)

(coshx+cos5)~* • log(coshx+cos5), 0<δ<τΓ

iV=2-^7r[log(sin5) •27r-iü:(sin|5)+^(cos|6)]

(coshx+cosha) "^log(coshx+ cosha) N= 2-^7r[sech(ia)ü:(sechia)

+log(sinha) •(2/7Γ) sech(Ja)ü:( tanh|a)]

— (cosha—coshx)"* • log( cosha— coshx), x < α

0, χ > α cosha< 2

iSr= 2-*7Γ{ 27r-Mog(sinha) •sech( |a)i i :[ tanh(aj)] -sech(Ja)ü:[sech(Ja)]}

log cosh(ax)+sin6

Lcosh(ax)—sin6j

N=Tar'h

cosh(ax)+coso log

L cosh (αχ)+cose J

Ar=i(c2-62)/a, c>6, c,6<7r

(coshx—a) ~*

, Γ (coshx+1) *+ (coshx— a) * * los ~—

L (coshx+1) *— (coshx— a) * J

27Γί/-ι csch^TTi/)

•sinh^di/arcsina)

ΤΓΙ/"^ sech.(^Ty) sinh[y arcsin(a/6) ]

2-*7Γ sech(7ri/) {P-4+.v(cos6) • C - 7 + l o g ( s i n 5 ) - l o g 4 - J ^ ( H t 2 / ) - J ^ ( J - t 2 / ) ] +iQ-i+.y(cosö)+iQ_i_v(cosö)}

2r^T sechiiry) {^_j+t, , ( cosha) • C"~ y~ log4+log sinha -m+iy)-mh-iy):\ +iQ_i+.y (cosha)+ iO_i_tv (cosha)}

—2"*7Γ sech(π2/) { $ _ j + i y ( cosha) • [log(secha)—7— log4

—|Q_j^-ty(cosha) —|Q_j_ty(cosha)}

wy-^ smh(hy/a) sech{^y/a)

vy ^csch(π^//α) • [cosh(ci / / a)—cosh(62// a) ]

2"i7r2[sech(7r2/)]2p_j+,,(a)

ΛΓ=2*πϋ:[( | - |α)*] a < l

2Nf{x) Ngiy)

250

251

252

253

254

255

256

257

258

exp(—osinhx) Λ Γ = | π [ Η ο ( α ) - 1 Ό ( α ) ]

exp(—ocosha;) N=Koia)

(sinha;)-* exp(—2a sinha;) Λ Γ = | τ ( α π ) *

.C / i (a ) r_ i (a ) - / _ j (a )F j (a ) ]

(coshx)-* exp(—2o coshx) N=ia/T)iZKiia)J

(coshx)-* exp(— 2o coshx) N^2ir-iaimia)J-iKiia)J]

[sech(ax) ] ' exp {- 6[sech (ox) J } N=2'-''a-'Bi^v,^v)

· ι ί Ί ( Ι " ; Η | ν ; - 6 )

(sinhx)-* exp (—a cschx) N=aihr)H[Jiiâ)J+iYiiha)7}

sinh (o cosx), x < Jir 0, x> i i r iV=|^Lo(a)

sinh(osinx), x<v 0, x>ir Ar=7rLo(a)

2591 cosh(ο sinx), χ<π 0, x>ir N=irIoia)

260 cosh(ocosx), x < | i r 0, x> i i r N=Whia)

5ο,<»(α) = — csch(ir2/) • [Λν(α) - / -<ν(α) - J<„(a)+J_,„(o) ]

Kiyia)

Mair)KJi-iiyi(^)Y-îvia) -•^-i+<ii/(a) I'i+h/Ca) +/i+<}v(a) Y-i+iM-J-i-iiiiia) 'Yi-iivia)l

ia/ir)iK^MKi-ihi<^)

2ir-iailKi+ii,ia)Kî,ia) -Ki+n„ia)Ki-nyia)2

2-wi>)r'\mv+my/a)3\' •Î^p+ihiv/a), l ^ - i M i z / a ) ; h", H I " ; - 6 3

2*Re{r(Hii/)ö-i-.»C(2m)*: .D_i_„[(-2ia)*]}

-itir csc(|ir2/)CJ,(Mi)-J_»(ia)]

- | i i r ctn(|ir?/) υ,(ίο) - J_,(io) ]

|^CJ»(ta)+J-„(io)]

Jirsech(|ir3/)CJv(«»)+J-v(ia)]

2Nf{x) Ng{y)

261

262

263

264

265

266

267

268

269

270

(sina;)-* sinh (o sina;), a;<ir 0, x>T

(sinx)"* cosh (a sinx), χ<τγ 0, χ>τΓ

i V = ( | a ) M [ / _ i ( i a ) T

(secx)* sinh (a cosx), χ < | π 0, x> | i r N=aKhrKh{h)J

(seca;) * cosh (o cosa;), x< |ir 0, x>hi-

(csca;)*sinh(2asinx), χ<τ 0, x>x JV=2(air)»{ [/-!(«) T - [ I j ( o ) T }

(seca;)*sinh(2acosâ;), x<ir 0, a;>ir N=2{anmi-iia)y-iliia)7]

Csech(aa;) ] ' • cosh[6 sech(oa;) ] , v> 0

N=2râ-^B{h,h)

[8ech(ox) ] ' • sinh[6 sech (ax) ], v> 0

iV=2'- 'a->6B(i+iv , i+iv)

arctan[sinho sech(6x) ] N=hra/b

(l+cäsinh«a;)-i' • coshQj» arctan(c sinhx) ] , 0<i'< 1

ΛΓ=(Ν«)»Γ(ν ) • ( i -c*)J- i 'PL7(c- ' )

(â)M c o s ( W )

Ί-i-iyiha)

2(air)*cos(§irj/) •C^-i-i»(o)/-i+|»(o) -7|+j,(a)7l-ii/(a)]

2(αΐΓ)«{7_ι-,(α)7_Ηι,(α) -7i_,(o)7}+,(o)}

2 ' - ΐδ[αΓ(1+ι/)]-ι I r ( i + § v + % / a ) |*

sm{ay/b) sech(|irj//6)

(Wc)»[r(.')]->cosh(W) • | r ("+ iy ) |Mi-c^)*-»'P*-lW'^')

2Nf{x) iVi/(2/)

271 |Γ(α+Μ;)Ρ, α>0 2ν=2-2·πΓ(2α)

2-^r(2a)[sech(ii/)J»

272 \Τ{α+αχ)τα-α+ιΙχ)\·^, 0 < ο < ^ ΛΓ=[6ΐΓ8ίη(2αΐΓ)]-ι

[6ir 8ίη(2αΛ·) ]-i^2<_iCc08h(ijr/6) ]

273 I T{\+ibx) \* Ν= ( 6 ι γ ) - '

26-%-« aeoh{ly/b)Klt&MWb) 1

274 I T{b+iax)Tic+iax) Ρ Λ Γ = 2 ΐ - ^ Μ ο - Τ ( 2 6 )

.r(2c)r(&+c)[r(H6+c)r

2»-*-«ο-Ίγ*Γ(26)Γ(2ο) •r(6+c)[sinh(W«):*~^ .^tc_iCcosb(Va)]

275 I Tia+icx) ΡI r(6+ica:) Μ, 6 -ο>^ ΛΓ=2»-2^ν-Τ(2ο)

· 5 { 6 - α - ϋ ) | Τ ( 2 6 - 1 ) ] - '

i(2ir)i2^<ri[r(6-a)r' .Csinh( 2//c)T-<^ •e-"("-W)iD:;Ji|Cco8h( V c ) ]

276 CI Γ(Ηίχ) i c o s h M r JV=2»T-ilog(l+2»)

2»ΐΓ-ί(οο8%)-» .logC(l+co8hy)»+(co8hi,)*]

277 a; Erfc(oa;) ih-'+r') eM-hJ'/a^-y-'

278 Erf c (αχ) ΛΓ=ο-ΐπ-*

a->x-*iFi( l;f;- ia-V) or -i2/-"exp(-2/y4a«) Erf(|ia-'2/)

279 x->CErfc(ox)-Erfc(6x)l o < 6 ΛΓ= log(V«)

iCEi(-2/y4a«)-Ei(-2/y46*)]

280 x^iErfc(ax), i '>0

ΛΓ=ο-ν-ν^Γ(Η^'')

a-V-V-»r(Hiv)

281 —ia;-iexp(—α^χ^) Erf(iax) iirErfc(ia-V)

282 Erfc[(ax)»] ΛΓ=§α-ι

(ia)»C(«*+l/*)*+a]-i(a«+i/«)-»

8. G a m m a F u n c t i o n s ( Inc lud ing I n c o m p l e t e G a m m a F u n c t i o n s ) a n d R e l a t e d F u n c t i o n s

8. Gamma Functions and Related Functions 47

2Nf{x) Ngiy)

283 Erfcfa[6+(B«+x2)»J} i\r=|a-«exp(-2o»6)

2-*oexp(-a«6)(o<+2r')-* . [a«+ (0*+^)*]-» e x p C - K o H - ^ * ]

284 Erfc / l + c o s x V '

. \ cosx / . , x < t / 2 exp(-a«)Z)î(2»a)i)_î(2»a)

0, i\r=i7rCErfc(a)]

χ > π / 2

285 Erfc(o coshx) ΛΓ=ΑΕί(-α*)

h-'eM-W)W-i.iiy{a^

286 exp[(a coshx)«] Erfc(o coshx) N=êxpiW)Ko{W)

heMW) sech{îry)Kiiy{W)

287 —i exp(—0« cosh«x) Erf(Mi coshx) N=hireM-W)Io{W)

2 - V e x p ( - | a « ) s e c h ( ^ ) • [ / i . (W )+ / - .v (^«^ ) ]

288 Erf[6sech(ax)] iV=eri6ir»2F2(iJ;!

a->6irisechßir(2//a)] • sF iÖ+ÜCs/a) , i - ^ i ( i / / a ) ; 1 , 1 ; - δ ' ]

289 (sechx)* exp(o* sechx) •Erfc[a(H-sechx)i]

N={2ir)-iaiKiiW)7

sech(ir2/)D_H.<v(2*a)i)-4-.v(2*a)

290 exp(aV) Erfc(ax+6) Ν ^ -»o-»Ei ( -62)

-ix-»a-iexp(i2/ya«) Ei[-(6«+ij /ya«) ]

291 - E i ( - a x ) y~^ arctan(2//a)

292 - E i ( - 6 x ) , x < o 0, x>a ΛΓ=6- ΐ -οΕ ϊ ( -αό )

-2/-i[sin(ay) E i ( - a 6 ) —arctan(2//6)—Ei{—ab—iay) +iiEi{-ab+iay)2

293 - β - « Έ ϊ ( - 6 χ ) i \r=a-»log(l+o /6) , a>-b

{a^+y')-Hy arctan[y(a+i.)-i] + ia logC(l+a /6 )«+2/y6«]}

294 -Έϊ{-ω?) N=iw/a)i

vy-' Erf (12/0-^)

2Nf{x) Ng{y)

295

296

297

298

299

300

301

302

303

304

-exp(aa^) ΕϊΖ-{ω?+1)·} Ν=^αή Ετίφ)

ΕΪ(αχ) - e " Ei{-ax) ]

exp{—aa?) Ei(ox«) -exp(ox«) Ei(—ca?)

-€ i "E i{ -a [ (6«+r ' ) i+a ; ]} - e - « E i { - a [ ( 6 « + a ? ) i - x ] }

N=2a-^Koiab)

— exp(o cosha;) Ei(—acosha;) Ar=Jir«[7„(a)-Lo(a)]

+i [S_ i .o ( i a )+S_ i ,o ( - i a ) ]

exp(—acosha;) Ei(acoshx) JV=|ir«[7„(a)-Lo(a)]

- i [ S - i . o ( i a ) + S - i , o ( - t a ) ]

exp(—acoshx) Ei(a coshx) — exp(acoshx) Ei(—acoshx)

Si(6x), x<a 0, x > a iV=oSi(a6)-26- 'sin«(ia6)

e-^SKax) N=b~^ arctan(a/6)

[s i (ox)J

^Λ-*βχρ(ν/α) Erfc(ii/a-l+6i)

•arctan(o/j/)

§A-l[exp(i2 /ya) Encihya-i) +texp ( - i2 /ya )Er f ( i i2 , a - i ) ]

2α(α«+2/«)->ΧοΡ>(α«+ι/«)»]

-iT«[csch(Ti/) Τ[7.ν(α)+7_ν(α) -exp(iir2/)J,„(ia) -exp(-i ir2 / )J_<,(m)]

^7r«Ccsch(xj/) T{cosh(^i/) [7.ν(α)+7_„(α) ] -exp(iiry)J . - , (-«i) -exp(-|iri/)J--<v(-Mi)}

ir«[csch(7r2/) ]« sinhiJirj/) .{8inh(W) [/.·»(«)-/-.·.(«)] +J .»(ia)-J- .v(ia)}

§ν->{2 8ίη(02/) Si(a6) +Ciiay+ab)-Ci{\ ay-ab |) +log( | (2/-6)/(2/+&) I)}, y^^b

i6->[2sin(at) Si(oi.)+Ci(2a6) - T - l o g ( 2 a 6 ) l y=b

i(62+i/«)-i{6 a rc tan[ (a+y) /6 ] — 6 arctan[(i/—a)/&]—52/ •log[ft«+(2/+a)«] +i2 / log[6*+(«-2/) ' ]}

^2rMog(l+2/)a-', 2/<2a ^TT^ log[(2/+a) (y-α)"»] , 2/>2o

2Nf{x) Ng(y)

305

306

307

308

309

310

311

312

313

314

315

[Ci(ax)T

— [sin(oa;) ai{ax)—coaiax) Ci(ax)]

ainiaa?) Ci(aa^)-cos(ox*) si(ar«) N={hir)*a-i

— [sin(aa:*) u{(u?) +cos(or ' ) Ci{a3?)2

N=a-i{hry

sm(o coshx) Ci(a coshx) —cos(o coshx) si(o coshx)

ΛΓ=ΐΐΓ«[Ηο(α)-7ο(α)]

— cos(o coshx) Ci(a coshx) — sin(a coshx) si(a coshx)

ΛΓ=5-ι,ο(β)

Ih-C {0x^)2 coaita?) + β-5(αχ«)]8Ϊη(οχ«)

ΛΓ=Κπ/2α)*

S(ax-i) N=a

x-Mcos(aa^) [0(αχ«) - S(ax«) ] +sin(ox') ZC{ax')+S{ax?) -1]|

N=iir

cos(a coshx) ß — S{a coshx) ] — sm(o coshx) [§— c(o coshx) ]

i V = W P ( i ) Ä j . o ( a )

cos(a coshx) ß — C(o coshx) ] +s in (o coshx) [|— S (o coshx) ]

iV= |T -»P( i )Ä_ i .o ( a )

§irir ' log(H-iia->), y<2o hnr'logiy'a-^-l), y>2a

+ c o s ( i j / y a ) Ö - C ( i i / y a ) ] }

(2α)-»π*{ cos(2/y4o) ß - Siy'/ia) ] - s i n ( 2 / y 4 a ) ß - C ( j / y 4 a ) ] }

^sech(^ir2/)So,.»(a)

iiri/csch(§5ry)iS_i,.v(a)

i(2aT)-»Csm(yy4a) Οΐ(ί/«/4α) -cos(2/y4a) si(2/y4a)]

ii/-i{sm[2(aj/)»]-cos[2(a2/)*] +expC-2(ay)»]}

k { H C « ( | i / y a ) + S « ( i j / y a ) - W / a ) - Ä ( | 2 ^ / a ) }

^r-T(i-i^j/)r(Hi^y)5j.,-,(a)

k - r ( i - ihy)ni+ih/) S-i.i.(«)

50 Table I: Even Functione

2Nfix) Ngiy)

316 x-^y{i>, 03?) wm-*m-pAf/a)

317

318 ex^{ax')T{v,ca?), -l<v<Q Ar=iir(ir /o)»lT(l-i ' ) c o s M ] - '

i ( V a ) - » r ( H . ) | T ( l - . ) r •exp{Wa)Tih-P,Wa)

9. EUiptic Integra ls a n d Legendre F u n c t i o n s

Mfix) Ng{y)

319

320

321

322

323

324

ü : [ ( l - a - V ) i l x<a 0, x>a

0, x>l Ν=^ίτα)ΐΓ^

ia^-^)-iK{aa^V^)/{a^x^)y], xa

K[eo8{^x)2, x<ir 0, x>v Λ Γ = Κ Γ α ) ϊ [ Γ ( ΐ ) : - «

(l+o«cos«x)-* • Κ[μ cosx ( 1 + a* cos'x) x < §ir

0, x>îr N=2ria-Kp+i)-*

•Kmp+D-^l, ρ=(1+α-«)ί

sech(aa;)K[tanh(aa;) ] N=â-hr-m^)y

hîMhay)J

Απ'[Γ(ί)]-=ν-»Μ<.ο(»)

h^Joih{a+b)Voih{a-b)2

lircosiiry)Ti^+h)Ta-h) •Cr(f+|2/)r(f-i2/)r

• 0 - M « ( p )

Α(απ)-Μ Γ(Ηϋ2//α) Ρ

9. Elliptic Integrals and Legendre Functions 51

2Nfix) Ngiy)

325

326

327

328

329

330

331

332

333

secha; Kisechx) N=hrimjmi)T'

sechx K{a sechx), a< 1

sechx ü:[( l-o2sech2a:)*] , α < 1

Ο, smhx<a"^

cschx l i : [ ( l -a-2csch2x)»] , sinhx> α" i \ r = ( i + « ) - > { i : [ ( H i « ) - » ] P

( 1 + 0 cosha;)-*JiC{2lCl+a coshx]"*} ΛΓ=Κ2πα)-ί[Γ(1)?

(l+a^sinh^x)-* .Ä:[(l+o''smh«x)-i], a>\

i v = a - > z { B - i ( i - 0 » ] M • ir{CHi( i -a-«)*]M

(1+α« cosh«x)-* •Ä:[a coshx(l+o« cosh«x)-l]

ΛΓ= 2α->(ρ+ 1)->ί:[(ρ- 1)»(P+1)-*] • i :C2i (p+l ) - i ] , p = ( l + a - « ) i

/ , . Ν 1 , Γ/β coshx—iV (acoshx+l)-*«: ( )

L \ a c o s h x + l / . a>\

N=2n{m)-^{T{\)J . { X C 2 ^ ( l - s ) ] + X [ 2 ^ ( l + s ) ] } ,

8=(l-a-«)»

(i-x«)-J' 'Pi;(^), x l

-h<v<h M < i ΛΓ=2^Μ

· ί Γ [ Κ 3 - μ + . ) ] Γ Ο ( 2 - μ - . ) ] } - '

^v{\+\iy)n\-m m\+\iy)m-\iy)

^mAi{hni){P-Ukivir)Y, r = ( l - a « ) i

iπ«(sechiπy)«P_4+,.„(r) .P_H.i . . (r) , r = ( l - a « ) i

+ [ 0 - 4 - ί , , ( 2 ) ϊ } , «= ( l+a -« )*

^(^/a)iT{\+\iy)V{\-\iy) .PH+.v[ ( l -a-«) i ]

^ o - i s e c h ( ^ )

sech(§ir2/) •15-W.v(p)[Ci-Wiv(p) + 0 - j - j < v ( p ) ]

2-v\T/a)*mAv{'Ky)n\+\iy) -m-hw)

.[P-i+.v(8)+P-+t-.v(-s)]

2 ^ - ' « - Μ Γ β ( 3 - Μ - . ) ] Γ β ( 2 - Μ - . ) ] } - » •(M+v)(M-v-1)j/^-» •«-M-i.M (2/)

2Nf(x) Ng{y)

334

335

336

337

338

339

340

341

342

χ^-1(1-χ^)-^Ρ^(χ), x<l 0, z>l

λ>ο, Ma

-h<v<h Ar=o (2i '+l ) -»cosM

%lia^+lr'-3?)/2ab2, x<a-b 0, x>a—b

a>b

• [ ( α / 6 ) ' ^ - ( 6 / ο ) · ^ :

Λ;-ιφ,(2χ-«-1), a;<l 0, a;>l

- l < v < 0 ΛΓ=—|ircsc(wr)

ΛΓ=2-*ιγ(2ρ+1)->

0 , [ (χ«+ο«+6«)/2α61 P > - k iV=a-6'+V(2>'+l)->, o > 6

^,(l+2a-2cos«a;), χ < | ι γ

0, x>^ir

^»(cosha;), - l 1 , - 1 < > ' < 0

iV=-2'-*(air)-»cot(>'ir/2) •Cr(Hi . )T{P i r*[ ( i -0* ] +ΡΐΓ*[ - (1 -α -« )» ]}

2^'ν»Γ(λ) • {Γ[1+Κλ-μ+.')]Γ[Κΐ+λ-μ-.)])-ι ·ΛΒλ ,Κι+λ ) ; ϋ ( ι+λ -μ- . ) , Ι+Κλ-μ+ί-);-2/^/4]

i a i r /^ (a2/ /2) / -M (« i / /2 )

Μαδ)*τ[/^(6ί/)7,+ί(θ2/)

-/^(α2/)7Η4(&2/)]

-§TCSc(w)iPi(l+i'; l;iy) 'iFi{l+p; 1; - % )

2-iTl,+i{2riy)K^{riy)

{ah)^irI,^{hy)K^{ay)

-ix-«smMr(-|;;+|ii/) •V{-\v-\iy)m-^hv+\vy) • r ( R i P - i t V )

-2'-*(oir)-*sm(w) . Ccosh(xi/) -cos(Mr) T'nh+b+hw) •r(Hi.-iiy) {PI'j+U(l-a-«)i] + p i | ; ; U - ( i - a - « ) i : }

2Nfix) my)

343

344

345

346

347

348

349

350

(sinha;)''^/(cosha;), ν+μ<0, μ-ν<1

ΛΓ=2-^-ν-ί{ΓΟ(1+.-μ)]Ρ •[Τ{-ν-μ)Τ{1+ν-μ)Τ' 'ίΤα-μ)Τ'{ΤίΗ-^-μ)Τ

$,(l+2a«sinh«x), 0 < ο < 1 , -ί<ρ<0

N=^P,{r)ZQ,{r) +Q-î{r)2, r = ( l - a « ) i

%{1+2αâmk^x), α>1, - 1 < » ' < 0 iV=|Cocos (w ) r*

•m,{ap)J-lO-î{ap)J], -l<v<0

'!ß,(2a«cosh«a;-l), α > 1 , - l l /a

- 1 < ί ' < 0 N=-^wcsc(vir)l%(ap)J

0 , ( α coshx), α > 1 , ί ' > - 1

N=2r-ia-iiîT{Hh)J . P i r * C ( l - a - ^ ) » ]

e-*'(sinha;)-* •OiJCcoshx), μ + μ + 1 > 0 , μ<^

JV=2^-V*r(HM) •{Γ0(1+ν+μ)]}1

.{Γ[1+Κ.-μ)])-^

2^-ν-*Γβ(1+>'-μ+ί2/)] τliil+v-μ~^y)2m(-f-^χ+iy)l

•ΓΟ(-' '-μ-%)] • [ Γ { - . - μ ) Γ ( ΐ + . - μ ) Γ ( | - μ ) ] - '

I sech(îr2/) •MPv'Hr)iQiHr)+QÎ!-i{r)2},

i [ o c o s ( w ) ] ~ ' .{piU.i«WRei:Q=4i!i.«]

8 = ( i - O '

.{[|Γ(ΐ+.+§ί3/)|φι'ίΐΙ.„(ρ)ί -CI r ( - . + | i 2 / ) i r - 1 W p ) i } ,

p=( l+a -« )»

iCacosh(Jiri/)]-i

. { P I A W R e C Q l U i i ^ W : +P!l+lV,}„(8) Re[QI|;!},(s):},

8 = ( l - o - « ) »

(απ)-ι Ite[Ol^;Ji,(p)OLY+ij,(p) ] , p= ( l+a-«) i

2'-i(Va)ir(Hi.+ii2/)r(Hi.-|iy) •p iSUd-«- ' )* ]

2^-ΜΓ(ί-μ)ΓΟ(1+;/+μ+ίι/)] •ΓΟ (1+.+μ - ί2 / ) ] •{Γ[1+Κ^-μ+%)]

2Ni{x) my)

351

352

353

354

355

356

357

358

e-'^-Csinha;)-"^! •0!;(ctnha;), (>'±m)>-1

ΛΓ=2'-2[Γ(1+ΐ')Γ(1+ί.-μ) ]-i · Ρ [ Κ ΐ + ' ' - μ ) Π Γ Ώ ( 1 + ν + Α . ) ]

{a^coah^x-l)ii'e^ •Or^iacoshx), μ - ΐ ' - 1 < 0 , a>l

-mh+h-h)jim+^)T'

Q,(l+2o«sinh«x), 0<a<l , i '>-l N={^/8)\:P,{r)J+m{r)7

O,(l+2o«sinh«x), a>l, v>-l Ν i i r C a s m M : - ' P i r * ( s )

.e+*W, s=(l-a-«)J

Q,(l+2o«cosh«x), )'>-l ΛΓ=ΚΟ,(αρ)Τ

O,(2a«cosh2x-1), o> l , i/>-l iV=ia-V[r(l+r) ? [ P i r * ( s ) Τ

sech(xx)P_i+<x(a), - l < a < l iV=2-l(l+a)-l

. ; 'P-i+iM, - l < a < l , sinh (αχ)

^ ' α>π

359

ΛΓ=«->Σ (-l)"i„Q-H(„x/.)(a) n-O

(sechirx)2P_j+«(o), - l < o < l i V = 2 - V - H l - o ) - i

• l o g { [ 2 » + ( l - a ) i ] y ( l + a ) }

2 ' -« [Γ (1+^)Γ (1+^-μ)] - ' • m { l + v - M + i y ) T O ( l + . - M - % ) ] •m(l+v+M+il/)TO(l+.+M-t2/)]

2 - ' - V — Μ [ Γ ( § + ν ) ] - ' • | r ( H l v - § M + i i y ) P

hrcBchihry)iPi*''{T)QrHr) -Pr"'(r)Qi"'{r)l r=(l-o«)i

-KasinWrPIji i i . is) .Re[Q!lY+<l«(«)]

^a-4r |r(l+.+%) P [ ^ I ? ; U p ) T . p=(l+rt-«)*

ia-V|r(l+.+i§2/) p • [ P l i ; y s ) T , s=(l-a-^)i

2-*(a+cosh2/)-*

α-' Σ (-1)"«« cos[(n7r/a)2/:<2(„,/«)-}(o),

2-*ir-Hcosh2/-a)-*

i[(cosh2/+1)»+ (coshy- a)ij •log

1+a

2Nf{x) Ng{y)

360

361

362

363

364

365

366

367

368

369

370

\Ti\+qx)\'P-i+Ua), 0<a<l

I Τ(μ+ίχ) p Ρ ΐ γ + i M ) , -l<a< 1, M > 0

Λ Γ = ( ί ΐ Γ ) ί Γ ( μ ) (1 -α )*^(1+α)- ίΜ -1

\m-y+iix) Ρ -Pt^Ua), 0 < α < 1 , M<§

cosh(W) I Τϋ-μ+ΐχ) p 'PU+M, M 0

N=2-^Kl+a)^-*il-a)-iT{2μ) . β χ ρ [ - ί τ ( § - Μ ) ] 0 * = ί [ 2 » ( 1 - α ) - » :

sech(ira;)[P_j+,-;„(a)J, 0 < α < 1

\nh-^+ix)\'iPti+My, M 1

|ra-M+a:)ppßW«)J, α > 1 ,

ΛΓ= ( α « - 1 ) - 1 0 _ ^ [ 1 + 2 ( α « - 1 ) - ' ]

2*ir»(r+cosh2/)-* . Χ[2*( l + r - ' coshj/)-*], r = ( 1 - α«)»

2 - ' ' - » ( α + l)»''(o+coshy)-H^ . ' i}i[(l+coshj,)Ka+cosh2 /)-i]

( ^ * Γ ( μ ) ( 1 - α « ) ί Μ - 1 • (a+coshy)^

ΐΓ»2^ν-*0-μ-ι(ι^ι coshy), r= ( l -o«)*

( ^ χ ) Κ ο « - 1 ) - » ^ Γ ( | - μ )

• (a '+s inhV)*^ •cos[(5—Α») arctan(a~^ sinhy)]

2 -Μ+ι ( ΐ -α« )*τ4Γ (2μ)

.OjlfC2i cosh(i3/) (cosh3/-a)-»]

sech(|y) •i(:[(l-a«)»sech(|j/)]

( l - a « ) - i O _ ^ [ 2 ( l - a « ) - > cosh«(i2/) - 1 ]

' sech( iy)J i : { [ l - ( l -a«) sech«(i2,) J |

• ä:{ a ( l - o-«) »Ca«+sinh«(^)

(a«- l ) -JCU_i[ l+2(a«-1) -> co8h«(^) ]

10. Besse l F u n c t i o n s

2Nf{x) Ngiy)

371

372

373

374

375

376

x-iiJo{ax)J i V = ( 2 a A ) - i r ( J ) [ r ( | ) r '

x-iiMax)J, ,>-l N= (2αΑ)-»Γ(1+;')

•[r«(f)r(f+.)r

[x'J,{ax)J, -\<v<0 iV=|C7rr (J - . ) ] -

· Γ ( - ^ ) Γ ( Η 2 . ' )

377

378

. [ Γ ( Η . ) Γ ( Η 2 ν ) ] - '

xi[/_i(ax«)T ΛΓ=2-5π-»Γ( | ) /Γ( | )

ΛΕδία^-χ^)*], x<a 0, x>a

ab<To.i

N=b-^ siniab)

(o2 - r ' ) - i ^ [6 (a« -x« ) i ] , x<a 0, x > a

. > - l , ai<r,,i

N^h^iMiab)!

(α«-χ«)*'Λ[6(α2-χ«)»1 χ < α Ο, x>a

ah<T,,i, v> - 1 ,

N= {hTa/b)^J^{ab)a'

(W2/)Mi'-}[(l-4aV-^)*]}', 2/>2a

( | τΑ )*Γ (Η . ) [Γ ( | - . ) ] - ι •{Pl![[(l-4a«r^)i]}«, 2/>2a

2 '-*7Γ-ία--*2/-'-*[Γ(Α-' ')]-' • (4α«-Λ- '{ΐΓφί_}[(αΑ)+ (2/A«)] -2e-'"Oi-i[(aA)+(2/A«)]}, 2/<2a

- 2"+*7Γ-ί sin(w) a'-iCr (H»-) •i/~' (2/«—40«)"' exp(—tw) ·ΟΪ-}[(αΑ)+(2/Α«)], 2/>2α

27Γ-*οΐ-2ΐΓ(2,;)]-ΐΓ(Η2^)

/ST

COS«'« (4α« sin«<- y«) - * di, ' ar<»in(t//26)

0, y<2a y>2a

- Ja-H|ir2/)iJ_i(2/yi6a) Fjiyyiea)

«-'sin(a2), z=il^+i^)i

Wi-'iiai^+y) Vi^ihiz- y) 1 z={W+y^)i

10. Beseel Functions 57

2Nfix) Ngiy)

379

380

381

382

383

384

385

- I o g ( a « - x 2 ) / o [ 6 ( a 2 - x « ) i ] , x<a 0, x>a

aö<To,i, a<l Ar=-26-i{sm{o5)[Ci(2oft)

- C i ( a b ) + l o g a ] -cos(ai))[Si(2a6)-Si(ab)]}

JoCftCex-x")*], x<a 0, x>a

a6<2ro,i iV=26-isin(io6)

0, x>a «*<2r,.i, " > - !

(οχ-χ2)* 'Λ[6(οχ-χ«)»: , χ < α Ο, χ > α

c*<2r,,i, ν> — ί N=iâ)'iwa/b)iJîhab)

-7ο [6 (αχ -χ« )*1 χ < ο Ο, χ>α

αδ<2ρο,, ΛΓ=-4τ- '6- ΐ [8Ϊη( |οδ) Οί(^α5)

-cos ( ia6) Si(â6)]

- ( a x - x « ) - i F , [ 6 ( a x - r ' ) » ] , χ < α Ο, χ > ο

β*<2ρ,.ι, - 1 < ί ' < 1 iV=7r{2cos ( f7 rv ) /} , ( | aö )

• F j . a a o ) - s i i i ( i w ) • [ / | , ( i a 6 ) + F | , ( i a 6 ) ] }

- log(ax-x2)JoC&(ax-x ' )* l χ < α Ο, x > o

α6<2το,ι, α < 2 iV=46-i{sin(ia6)

• [Ci (a6) -Ci ( ia i>)+log( ia ) : -cos(|afc)[Si(aft)-Si(|(rf>):}

- 2 z - ' { s i n ( a « ) [ C i ( 2 a i ) - | Ciias+ay) - | Ci (ai ; -ai / )+log (a5)- log2] -cos (a2 ) [Si (2a2 ) - f Si(az+a2/) -hSiiaz-ay)3}, «=(δ«+2/«)*

2 cos(|o2/) a-i sin(|o«), 3 = (6«+2/') *

7Γ cos(ioy)/},[ia(2+2/) ] / } , [ ϊα (0 - ?/) ] ,

Μ Kioi))'008(^02/)

-27r~ico8(^02/)2-i .{sin(iaz)[Ci(|(K+io2/) +Cidog— lay) ] — co8(ia«) . [ S i ( | a 2 + i a y ) + S i ( | a 2 - i a y ) ] } ,

-IT cosCiay) {cosdirv) •iM\a2+\ay)YU\az-lay) + Yi,iW+hy)M\(^-hy) ] - s in( |w) Jo2+|oy) J} , ( io0 - iay) + y},( ia2+iay) F j , 0 - J a y ) ] } ,

2=(62+y«)l

-42-Msin(|a2) [ C i ( a 2 ) - | Ci( ia2+iay) - I Ci(§a2-iay)+log(|ai>) - l oga] -cos( laz) CSi(a2) S i ( io2+|ay) - | S i ( i o a - | o y ) ] } cos(|ay),

2=(6«+y«)i

58 Table Ii Even Functions

2Nfix) Ngiy)

386

387

388

389

390

391

392

0, x>a αδ<ρο,ι

i\r=-2jr-i6-i[sm(a&) Οί(αδ) -cos(aö) Si(<rf>)]

-ia^-x')-iY,[bia^-3^)*l, x<a 0, x>a

ab<p,,u -\<v<\ N- caaiW)Mhob) Y^,ihob)

ΛΓ= (2δ«)'-4ΓΜ[Γ(Η2»')]-' ^ιF2iv·Λ^-v,2v+\•,-aΨ)

J,iacosx), χ<ΐΓ Ο, x>w

α<·Τρ,ι, ν> — ί N^TlMm

J,ia amx), χ<τ Ο, χ>τ

0'<τ,,\, ν> — 1 N=TlJi,iha)J

CSCX J,ia sinx), x<ir Ο, χ > χ

α<'Ί·,ι, ' ' > 0

•{[^}(-i)(§«)J+[/i(M-i)Q«)T}

sec(ix)/ , [e cosCix) ]], x < x Ο, χ>τ

α<'',.ι, ' ' > 0 ΛΓ=|ΐΓαΐ'-ι

•{[7j(-i)(§a)J+[/«H-i)(i«)?}

— (jrz)-i{sin((w) [Ci(<K!+a2/) +Ci(o2— ay) ]— cos(az) • [Si(a0+ay)+Si(aa-ay)]},

-^{cOsCiflT) [/},(2i) F},(Z2) + F},(2i)/i,(22)]-8in(iirv) •[/j,(2l)/iv(22) + Fiv(2l)y},(Z2)]),

21=^(6^+2/^)»^^:

(2α)»-'[Γ(Η'')Τ-'

• / cos'''<(46«sin2<-y')i'-} •'anieinCWa)

• / , ^ [α (46 ' sin«<- y«) dt, y< 26 0, y>26

ir/j^(|a)/jH^(ia)

7Γ cos(§7ry) /},^( |a) /j^(ia)

iirac"' cos(§iry) • Uh(.v-i-v) (§«)' i('-i+K) + < ^ } ( · 4 - ι - ϊ ) (2α)·^}(·4-ι+») (z«) ]

^ταχ-'Ε/κ,^ι-ν) (§a)/j(,^i+») ( |α) }(ι4-ι-ϊ) (i«)«^i(>+i+v) (i 'i) ]

11. Modified Beseel Functions 59

2Nfix) Ngiy)

393 icosx)~™Jmia cosx), χ<π/2 m

i7r(ia)-m! Σ € n [ ( m + n ) l{m-n) \T' 0, x>ir/2 n - O

a<rm,i, m=0,1,2, . . . •«^n- iy ( fa ) /nH- i l / (2a)

ΛΓ=Κ^«)»»ί-·[Η»Μ(α)+Η„Μ4(α) ]

394 (sinx)-™/»,(α sinx), x<ir 0, Χ>7Γ m

a<T„,i, m = 0 , 1 , 2 , . . . • Σ i n [ ( m + n ) ! ( m - n ) !]-'/„-},(§«) N= (firo)»OT-> n - O

•CH„_,(a)+H,H4(«)] •<^n+}i,(|a)

395 Λ(2ο οο8χ)/μ(2α cosx), x < x / 2 iτiha)'^''Tiv+μ+l) 0, Χ>7Γ/2 .{Γ(1+.)Γ(1+μ)Γ[1+Κ^+μ+2/)]

ο<Μίη(τ,,ι;τμ,ι), ν+μ> — ί •Γ[1+§(''+μ-2/)]}- ' Ν=Μ^)·^ην+μ+1) ·4 ί ·6β(1+»'+μ) ,Κΐ+>'+μ) ,

• {Γ(1+ν)Γ(1+μ)|:Γ(1+|.+|μ)ί}-ι l+iC^+M), 1 + | ( « ' + μ ) ; Γ + 1 , μ + 1 , ·2Ρ30(>'+μ+1),Κ»'+μ+ΐ); ν+μ+1,1+α,+μ+ν), ^ + 1 , μ + 1 , ί . + μ + 1 ; - 4 α η ^+hiι'+μ-y)•,-m

396 JoZib— α coshx)*], coshx<ö/a 0, coshx>6/a ·2ι=Κ(6+α)»±(6-α)»]

a<b<To,i 2

N= ix[Jo(2i) Yoizi) -Joiz2) Yoizi) ]

397 (coshx)-^J',(o sechx), ί'+μ> 0, 2Μ-ν[Γ(ρ+1)Γ(μ+ί ' )]-ι a<Ty,i • I ΓΜμ+»'+%) Ρ

iV=2^VCr(^+l)]-' •2Fz[_Uß+y+iy, hiμ+y-iy); •Bih+^.y+h) Κ μ + ν ) , § ( 1 + μ + . ) , 1 + ν ; - ν ) : ι ^lmi^^+'')•Λil+μ+»),

398 aechiJ^){[JUa)J+iYUa)J] Ηο(2α cosh^y) — Fo(2o cosh^y) ΛΓ=Ηο(2α)-Γο(2α)

11. Modified Besse l F u n c t i o n s

2Ay(x) Ngiy)

399 e-^/o(ax), b>a 2-i[(6«-a«-2 /«)«+46V]-* iV=(ft«-a«)-i • {[( i ' ' -a«-y«)«+4&y]i+6«-a«-2 /«}i

2Nf{x) Ngiy)

400

401

402

403

404

405

406

407

408

409

410

Koiax) N=hra-'

x^'Koiax), n = 0 , 1 , 2 , . . . iV=ir(2o)-2"-i[(2n)l/n!T

χ2»+>Κο(αχ), n = 0 , l , 2 , . . ΛΓ=α-«»-22«»(η!)«

xiRoiax) N=i2a)-ia-mm

x-^Koiax) N=2ria-iim)J

K,iax), - 1 < » ' < 1 JV=§ira-isec(§7n')

x'Kyiax), v>-h ΛΓ=2' - ΐΛ- ' -Τ (Η' ' )

xi'Kyiax), μ+ν>-1 N=a-''-'2f'-'

'Ti^+h+h'')Tih+h-h^)

x-'ilfi+x')-'K,iax), y<§ .V=lir«6-'~»sec(w)

. [Η,(α6)-η(α6)]

loibx) Koiax), a>b N= (ο+6)-ιχ[2(ο6)Κα+6)-']

Iyibx)K,iax), a>b, v>-^ -ν=Καδ)-*0^(«/2δ+6/2α)

kria'+y^-i

(-1)·' |χ(2η)!(α«+Λ-»-* •P2n[2/(a«+2/«)-i]

(-1) ' ·+Κ2η+1)! •ia^+y')-^-'QMa^+f)-^l

i(27r)M{2i?[(f-i2/2)i] +2iC(Hiy2)i:-X[(i-iyz)i:

2'-VV(a«+y«)-^r(H^)

ixsecÖ7r(M-;')]r(l+M+v)

* · {PZ\via'+f)-^W7l-yio^+fTn

i x i 6 - H 2 a ) - T ( | - v )

j'e-^Ka'+e)'-^ dt

+6-*» p e-*\a^+^)'^ dt

.Α{2(αδ)*[(α+6)«+2/«]-1}

Κα6)-*θΜ[(2αδ)-Ηα '+6 '+Λ ]

11. Modified Beesel Functione 61

2Nf{x) Ngiy)

411

412

413

414

415

416

417

418

419

420

x-iIyiax)Kyiax), v>-\ N=^^Ti\)Tii+y)

•cr( f )r( f+ . )r

x-Î,iax)K,iax), 0<ν<^ ΛΓ= W^îu)Ti^-^) [Τ(Η2« ' ) Τ'

x-i'I,,ibx)K,iax), v>-l a>b N= ^^Ψα—hiTil+v)

.[r(i+M)r'2Fi(Hî;M+i; 6«/ο«)

iUax)J N=hiâ-'

Koiax) Koibx) N=-îa+b)-^Klia-b)Ka+b)-^']

K,iax)K,ibx), -K»'UI+IH iV=2f-'[r(f)rm(f+M+.)l

•rcKf+M-»')]rß(f-M+^)] •rcMr-M-")]

r''exp(-a^)/,(ar'), - i < « ' < 0 iV=^- i2 - ' - i r ( -v ) r (§+2v)

r(H^)Cr(H»')rKk/2/)* •e<"Ol!iC(l+4oV-*)»]^Z5[(l+4aV^)»]

2-"-ΜΓ(Η'')(2//α)' . (y«+4a«)'-*Pr-'}C(2/'-4a«)u/«+4a«)-':

Γ(4α2+2/«)-1ί:[2/(4α«+2/«)-*]

• i i i [2/«+(α-6)«Κί/«+(o+6)«]-»)

ix^secMCoi)-* .φ [(2α6)-Η2/ +« +& )]

r(f+«')Cr(-i-»')rexp(i2w) .(ix3/)H4a«+2/«)-i •Ol!;[(l+4aV-')l]Ol?C(l+4aV-')i]

rCH") [Vi\-v) r βχρ(ί27Π') (|ir/2/)i .{Oll[C(l+4a«2/-«)i]}«

2f-'[r(f) :-Tß(f+M+»') ] Γ [ | ( Γ - μ + ; ' ) ]

• 4F30(r+M+»'), Kf+M-"), hii-l^+v),

• ( Γ - μ - ί Ί ^ , If, i(iTl-f);-12/^1

^-12— ir(-^)r(i+2,') exp(-i2/«)

+ 2 « - * r ( . ) l T ( | - . ) r

2Nf{x) Ng{y)

421 2n'y'-' exp(-2/yi6)TF_,M.(i2/»)

422 iV=2-»o-»P(J)

hra-'yK^{\0a) •C/l( i2/ya)+/- l ( l i /ya)]

423 x^Kiiax") ΛΓ=2-1(αΑ)-»|Τ( |)]- '

hra-^^y)* .C7_i(i2/ya)-L_i(j2/yo)]

424 xiKiiaa?) Ν=2-^*α-*ίΤ{\)Τ'

\a-^(.hn/)^ •[7_i(i2/yo)-L_i(i2/ya)]

425 α?'Κ,{αα?), v>-\ N=2'-^a •^r(H'')U(f)r

2·~%α-'-»{Γ(Η,' ) [Γ( | ) ] - ' 'M-^v;hhitf/a') -a2 /ya )r ( f+ . ) [ r ( f ) ] - i • i F 2 ( f + . ; f , f ; i f e / y « ' ) }

426 exp{-=^)Ko{a?) N=ihr)i

(|x)iexp(-2/yi6)7o(2/yi6)

427 3?'eM-ax')K,ia3?), v>-\ ΛΓ=Κ2α)--*πΓ(Η2^) U d + v ) T'

|π(2α) ' - * r ( H 2 . ) [ T ( l + . ) r • iFi ( | - . ; l+ . ' ; i2 /ya)

428 x-^'exp{x^)K,ix'), 0<v<l N=2'-i cosMrWr(f-2v)

x 2 - * T ( i - 2 . ) [ r (H . ) r .2/->exp(2/yi6)Tr,,.-iX|j/«)

429 χ>'βχρ(-χ')Κ,{2?), - 1 - 2 μ < μ < - 1 + 2 » '

N=2-ii-h*T{^+y-v) · Γ ( Η § μ + ν ) | Τ ( 1 + | μ ) ] - ·

2-ί''-*«·ίΓ(Η|μ-;') Γ ( Η Ι μ + ' ' ) • [ Γ ( 1 + | μ ) : - ι · 2 ί ' 2 ( Η | μ + ^ , Η | μ - ' ' ;

430 xiKi{ax?)J iV=2-*A-»

2-Ma-'[7o( i i /ya)-Lo(it /ya):

431 χ-^Κο(αχ-') N=a-^

-πο->2/Κι[(2α2/)*]ί:ι[(2αί/)»:

432 Ko{ax^) N=2a-^

Ir'CCidayy) sin(iay2/) - s i ( i a y y ) co sdayy ) :

2Nf{x) Ng{y)

433

434

435

436

437

438

439

440

441

{a-x)Î,g>{a-x)i2, x<a 0, x>a

v>-l ΛΓ=26-Μ'+*/Η.ι(6α4)

/Ο[6(Α2-Χ«)»1 x<a 0, x>a N=b-âmh{ab)

(o«-a?)-*72,[6(a«-ai)»l x<a 0, x>a

v>-l

[Χ(ο -Α; ) ] -* /2 , {6 [Α; (ο -Χ)J} , x<a 0, X > A

v>-i

(α«-Χ«)»'/,[6(α«-Χ2)1], x<a 0, x>a

v>-l

N={îra/b)ia''I^{ab)

Koibia^+7?)i2

Χ2„[δ(α«+Χ«)»], n = l , 2 , 3 , . . . η

N=vib-hiΣ (n+k-l)\ik\(n-k)!]"' t-o

cosCv ΜΘΐΕΗ(Χ/6)]Α,[ο(62+Χ*)»1 ΛΓ=|ΐΓΑ-»Β-* - 1 < ί ' < 1

—IIR ' sec (FLT)^ •iamihru-hr-WMMW/y) +coa{hn>-i,r-My) YriW/y) ]

{y'-l?)-iaiaZa{y'-m, y>b

*ΙΓΛ(2Ι)Λ(22) zi^haivMy'-m 2

JT οο8(|α2/)Λ(«ι)Λ(ίί!)

./H^[O(2/«-6«)J], y>b

hril^+fJ-iexpl-aiV'+M

( Ι χ ο ) ί η Σ (n+Ä-l)ICÄ!(re-ft)!]-i t-o

• (Wa)->'iV'+y')i>^K^la{V>+y')i2

2NJ{x) Ngiy)

442

443

444

445

446

447

448

449

450

7„[6(ox-x«)*], x<a 0, x>a JV=26-isinh(ia6)

iax~x')i'Iyibiax-3?)i2, x<a 0, x>a

v>-l N=iira/b)Kh)'Iy+i{hob)

Koib{ax-x')*2, x<a 0, x>a

.CS i ( - i§ao ) -S i ( t | a6 ) ] -smh(|a5)[Ci(i |a i>) + C i ( - i § a 6 ) ] l

icix-x')-iK,ib{ax-3?)*2, x<a 0, x>a

-l<v<l

ίχ{1+χ)2-*Κφ[.χ{1+χ)^}, -h<''<h

JV=57r«sec(iiT)

ia'+a?)-iKilb{a'+3?)i2 ΛΓ=§5Γ(αδ)-»β-^

ia^+7?)-iK2lbia'+x')i2

N=U2m/b)ia-'K,^iab)

(α«+χ«)-ΐί:ο[δ(α«+χ«)»] Λ Γ = - | π α - Έ ί ( - α δ )

2 cos(|aj/) i(&«-2/«)-isiiihÖa(6«-2/«)*], 2/<6

(ira)»(|fl*)'cos(|ay)

-cos(ia2/)(2/«-6«)-i • { s m a [ C i ( 0 i ) + C i ( 2 2 ) ] - c o s a [ S i ( 2 i ) - S i ( « 2 ) ] }

2

a=W-b^ Jo, y>b

arga=

—3ir« secCsTiT) cosday)

|ir«sec(irj'){cos(|2/) • [ Λ ( 2 ι ) Λ ( « 2 ) + F , ( 2 i ) F , ( 2 2 ) ] + s m ( k ) • [ ^ ( 2 2 ) F , ( 2 i ) - ^ ( 2 l ) F , ( 2 2 ) ] l

2

JJ i : , (2 i )J i [ , (22)

2

- Jira-Te-"" E i ( - 2 2 ) + e " ' ' E i ( - 2 i ) ]

2i=a[(ö«+2/«)*±2/]

2Nfix) Ngiy)

451

452

453

454

455

456

457

458

KJibi2ax+a?)i2 N=b-'Zam(ab) Ci{ab)

— cos(o6) si(ai>)]

\ogia^+T')Koib{a^+3?)i2 N=irb-^le-'* loga-e»» Ei(-2ai>) ]

/ίο[δ(α«-3^)*:, x<a 0, x>a N=ib-î co8Hab)[Si(-iab)-Si(iab)2

- smh(ao) lCiiiab)+Ci{-iab) ]}

{a'-xt)-iK,[bia'-^)*l, a;<o 0, x>a

-l<v<l N=hr sec{^)Ki,{hob)

'UUiob)+I-i,acib)2

N=\h>-mKM>{âb)J -iKi^,{hab)J]

0, x<a Kolbix'-a^J, x>a ΛΓ=α->[8ίη(ο6) Ci(o6)

—cos(ai>) si(o6)]

0, x<a (χ«-α«)-1Χ,[δ(ζ«-α«)*1 x>a

-l<v<l N=Waec{^)

•{iMhab)J+iYi,{âb)y}

I,Za{<^+^)*lK,Lb{<?+^)il a<b N=hrd'-'Iy{ac)Kîbc)

| αα - ι eoaiay) {smoi[Ci(2i)+Ci(2:2) ] — cosa[si(2i)+si(2:2) ] }+§αα- ' aiaiay) • {cosa[Ci(2i)—Ci(z2)]+sma

.Csi(2i)-Si(%)]}

2

7r(6«+i/«)-*{exp[-a(i)2+2/«)*]Dog(a&) - § log(ö2+2/«):-exp[a(i>^+2/«)i] •Ei[-2a(6«+2/2)i]}

-i(^-t^)-i{smaiCi(zi)+Ci{z2)l -cosa[Si(2i)-81(22)])

a=a(f—b^)^, Zi=aydza 2

— iir^secdav) • [/},(2i) r-,,(22)+ Yi,izi)J-i,{z2) ]

2i=§a[y±(y«-6ä) l ] , 2/>6

ibu-WM^hazi)KM.ih<^) -K^y{hazi)Ki^.{ho^)l

2ι=(6«+Λ»±2/ 2

K&'+^-»{sma[Ci(2i)+Ci(22) ] —cosa[s i (2 i )+s i (22) ] }

a=o(6='+y2)i, 2 ι=α±α2/

Sir^secClnT) • [/ίκ(2ι)/},(22)+ Fi,(2i) r,,(22) ]

2i=ia[(6«+2/«)i±2/]

Γ V^-^'J,{at)J,{U)

.(i '+c«)-iexp[-y(c«+<«)ij<ß

2Nfix) Ngiy)

459

460

461

462

463

464

465

466

467

468

N= hra-'Uo{2ab) - Lo(2oo) ]

(a«+x«)l'X,[6(a«+a«)»] N= {hra/b)ia-Ky+^{ab)

· ί :„ία[χ-(χ«-6«)»]1 N= |7Γ'α-ΐ[Ηο(2αδ) -Yo(2ab)2

•Ky{aai^+^)i+x:\} N=a-'liKsec{m>)h,{2ab)

+|iso,2,(i2fl*)3

K,\alx+i:,?-m

ΛΓ=ΐ7Γθ-ΐΑο,2κ(2ο6)

• i :4a [ (ö«+x«) i+x] | ΛΓ= 1x0-1^2,(206)

ί : ο [ α ( ύ ) ί ] ϋ : ο [ ο ( - ώ ) ί : ΛΓ=Ατο-«

Χ,[α (ώ)» ]Κ, [α ( -Μ; )*1 - 1 < ^ < 1 iV=i7ro-2sec(|w)

72,(2osinx), Χ<7Γ Ο, χ>π

N=tUMJ

72,(2ο COS5X), χ < τ Ο, χ > ΐ Γ

iV=7r[7 , (a)J

ix(4o«+2/«)-* . {7o[6(4o«+2/«)»]-L,[6(4o«+j/«)*]}

( W ) i ( a 6 ) ' . ( ί^+2^) -}^ ί :^ [ο (6«+2/« )* ]

ix«(4o«+3/«)-i . {Ho[6(4o«+i /«)i]- yo[6(4a«+y«)*]}

(4o«+2/«)-l{|ir s^{Ti>)h,U)i4a^+f)i2

7Γ(4α«+2/«)-ί5ο,2κ[6(4ο«+2/^)ί:

x(4o«+2/«)-»Ä2,C6(4a«+2/«)i]

lirV-iCHodoV-') - FodoVO ]

Vr'sec(|in')So,,(|o«2/-0

TT cos{^Ty)Iy-iy{a)Iy+iy{a)

vly-y{a)l,+yia)

11. Modified Beesel Functions 67

2Ni{x) Naiy)

469

470

471

472

473

474

475

(sinx)~^/,n(2a sinx), Χ < 7 Γ

0, χ>τΓ m = 0 , l , 2 , . . .

m

Ν=παΤτη\Σ ( - 1 ) " ί „ [ ( » ί + η ) ! n - O

'{τη-η)\Ττΐη{α)7

{eos^x)~"Imi2a cos^x), x<v 0, x>ir m = 0 , l , 2 , . . .

m

N=ira^m\ Σ ( - l)»i«C(m+n)! n - O

cscx h v (2a sinx), x < π 0, χ > π

ϊ/>0 iV= Απα,-Μ C/.-i(a) Τ+C7.+i(a) ϊ }

secf(|x)/2v(2a cos^x), x<7r 0, χ > π

ί/>0 iV=iira.-MC/.-i(a)T-C7.+i(a)T}

(sechx)'*/p(a sechx), μ + ϊ ' > 0 N= 2 Μ - ν [ Γ ( 1 + ϊ . ) ^ 5 0 ( μ + ν ) ,

Μ μ + ν ) ] ι ^ 2 0 ( Μ + ϊ ' ) ;

i ( i+M+^) ,^+i ;V)] Ä'2v(2a sinx), χ < χ Ο, Χ > 7 Γ

-\<ν<\ Ν= W csc(27r.) { [ /_ . (α ) [ / . (α) J}

ϋΓ2κ(2α cos|x), χ < 7 Γ Ο, χ > π

iV= CSC(2TV) { [ /_ . (α ) Τ - [ / , (α)

Tfl'^m! cosC^TTi/) Σ (—!)**€»» η - Ο

. [ ( m + n ) ! (m-n) ly^n-iyia) Ιη^{α)

πα^τηΐΣ ( - 1 ) % η - Ο

. [ ( m + n ) ! (m-n) !]-^/η-,(α)7η+,(α)

|παν-^ cos(i7r2/) •C^M-h/(«)^M+ii/(«)

— 7,-4-ι/(α)/μ4 .^^(α)]

2 Μ - ν [ Γ ( 1 + ^ ) Γ ( μ + ί / ) ] - ι

i ( M + ^ ) , i ( l + M + v ) , . + l;ia2]

^ΙΓ2 csc(27n') cos(^iri/) •[7_^j,(a) /_H4t,(a) - - / , - 4 i / ( a ) 7 H - i y ( a ) ]

*7Γ2 csc(27n,)C/_^(a)7_.+,(a) -7 ,_ ί , (α )7 ,+^(α) ]

68 Table I: Even Functione

2Nf{x) Ngiy)

476

477

478

479

480

481

482

483

484

Koiia'+V'-2ab coax)i2, x<ir 0, x>r N=Tlo{b)Koia), a>b

(secxms^x)* •exp{—a^ secx)Ki£a^{l+aecx) ],

x<iir 0, x>hr N= 2ήπα-* expi-o") [2)_i(2a) J

Ki,{2a8iahix), -^KvKi iV=4ir«sec(7rv)

• { [Λ(α)Τ+[η(α )?}

K2v(2a cosh^x)

N=iKMJ

sechdx) 7},(2a coshô;)

m = 0 , 1 , 2 , . . . m

Ν={-1)"'τη\α'^Σ (-1)"«» n-O

'am+n)Km-n)ir'iKnia)7

(coshx)* exp[(a sinhx)«] •iijCCa coshx)«]

N= 2-^l*a-ilT(l) ?TFi.o(2a«)

(coshx)* exp[— (a sinhx)«] •iilC(a coshx)«]

N=2ria-iTW-i,o{2a^)

iLoCa(2 coshx)»] iV=Xo(aei")ifo(ae-^")

2/sin(x2/) Σ ( - l )%(y«-n«)- i7„(6) i :„ (a ) n=0

(2α)-^7Γ exp(-a2)D_j+,(2a)Z)-i-,(2a)

+tan(î/) 7ίν-ρ(α) - / ,v- , (a)F,v+.(ö)]}

Ky+iy{a)K^y(a)

âv-^lKi+y+iy{a) Ki+,^y{a) — ( a ) K^y-iy{a) ]

. [ ( m + n ) ! ( m - n ) ! ] " i K n + . , ( a ) i ^ n - i i / ( a )

2-ö%-*|Γ(H§^2/) ΡΤΓ^.·.(2α2)

2râ-îrW-^,îy{2a')

Kiy{(wi'-)Kiy(os-^'^)

12 . Functione Related to Beseel Functions 69

2Nfix) Ng{y)

486

487

488

489

490

491

-Kom-iti'-m]

Koi(a^+ly'+2abcoahx)i:\ N=Ko{a)Ko{b)

{[ (a+6e-) (6+ae-) - 'J ' + [(6+αβ-)(α+6β-)-':*Ί 'K,l(a'+b'+2abcoshx)*2

N=2KUa)Ki,{b)

Ν=^τΚο{2α)

Ν=^Κ,{2*α)

N=hrK2,{2a)

•Kiiym-ib'-aηi2}

miymci'+^)*+bi}

Kiyia)K,y{b)

Ki,îy{a)Kiîy{b) + Kiîy{b)Ki^yia)

ΙπΚο(2α coshiy)

Jirü:o[a(2cosh3/)»]

^πΚ2,{2α coshiy)

12. F u n c t i o n s Re la ted t o Besse l F u n c t i o n s

2Nf(x) Ngiy)

492 χ-Ήο(αχ) arccos(y/a), y<a Ν=^π 0, y>a

493 χ — Ή , ( χ ) , p>h (2χ)Ηΐ-2/«)*'+»ΡΓ-Τ*(2/)- 2 /<l JV=2—ΐπ[Γ(1+^)] • 0, y>i

494 x'[H,{ax)-Yy{ax)2, - | < ' ' < 0 i coa{wv)a''T{i+2v)y iV= -2 'π» 050{πν)α-'-^ΖΤ(^-ν) \{a'-^)-^^ZlZ {a/y), y<a

(y/a), y>a

2Nf(x) Ngiy)

495

496

497

499

500

501

502

503

504

505

506

x—>[7_,(ax)-L,(ax)], v < 0 i V = 2 - ^ V cosMrC-i»)

Η„(αχ«) Λ Τ = Κ 2 α ) - * Γ α ) ( Τ ( | ) ] - ι

Ioiaa?)-Uiax') ΛΓ=Ατ->(2α)-1|Τ(1)Τ

N=iir\:Mhab)J

ΛΓ=·Η ·2-^ΐ6- ' [Τ(ΐ '+1)]- ι

(α«+χ«)-*{7ο[6(α«+χ«)»] -LoC6(a«+x«)i:}

2\Γ=7ο(|α6)-ί:ο(|αδ)

Ηο(2α coshja;) ΛΓ=ΐΓ[/ο(α)ϊ

Ηο(2α cosh|a;) — Fo(2a cosh§a;) N=hr\iMa)7+iYoia)J}

7o(2o coshjx)—Lo(2a coshja;) ΛΓ=27ο(ο)Χο(α)

Ηο(αχ«)-Γο(ατ') i V = 2 - i a - i r ( i ) [ r ( f ) r

cscMCJ,(aa:«) ΛΓ=2-ία- ίΓ(Η§μ)

-π-ΐ2- ' ->α- ' ' 0θ8(ιπ')»'-'Γ(|-»')

2n{2ira)-m(y'/8a)J

0, y>b

0, y>b

^-rKoihaiy+iv^-m) •maZy-iy'-m]

maliV'+y')i-y2}Ko{hiiW)*+yl]

A t sech(T2/) {Q7i,(a) 7+iJ-,y{a) J}

hr sechM { i M a ) J+LYiyia) J}

aech(iry)Kiy{a) [7<„(ο)+7_<»(ο) ]

hra-'yi Zmy'/a) J+iYiUl^/a) J]

{2ra)-iT{^+v) • D _ ^ [ 2 / ( 2 a i ) » P - , ^ [ 2 / ( - 2 a i ) * ]

12. Functions Related to Bessel Functions 71

2Nf{x) Ngiy)

507

508

509

510

511

512

513

514

N=Tia-i{T{l-iv) • [ r ( i - i . ) ] - ' + c o s M •r(Hi.)cr(f+§.)r}

caciiTp) • [Ji-Ce coshx) —Λ(ο coshx) ]

N=-mo)

— [F,(a coshx) +J5?,(a coshx)]

N=-W'{0)

χ-ΐ'-^ΒμΑαχ), μ±ΐ'<0 N=hra'^T{-y-^v)T{-h+h>')

• [ Γ ( | - | μ + | . ) Γ ( | - | μ - | ν ) ] - '

χ'5μ.,(αχ), v>-h M - l " K l , -2<μ+»'<0

N= 0 8 θ β ΐ Γ ( ν + μ ) ] · Γ ( Η . ) [ Γ ( | - | ν - * μ ) •m+h-y)T'

Ä^.,(coshx), μ < | ί Ί + 1 ΛΓ=§(2α)'^ΜΓΟ(1-μ-»')]

· Γ Η ( 1 - μ + ^ ) ] ! -

• Γ «-"(α^+ί^)-^/!:},^*)^ dt

(coshx) *Αμ,}(α coshx), μ < | ΛΓ= (2ο)-*2-^-ΐΓ(|-μ)^[Γ(1-|μ)]-'

• θμ+ι,ο(α)

(α«+χ«)-*{|χ sec(|xi;)/,C6(a«+x«)i] +tso.,p6(«'+a^)»])

N=hrlUhob)Ki,{^ab)

(2xa)-»{r(i-.)2)^[y(2a0-*] • D ^ [ 2 / ( - 2ai)-»]+cos(x>') rd+J ' ) . D_^[2/(2oO-*]i)-M[2/(-2at)-*]}

—jix csch{iry)f(y) fiv) =Juy+iy) i'iC.-V) ( 2 0 )

- J ix csch(xy)/(i/) fiy) =J-iHiivii(i) r_^(n4,v)(|o)

(I«) Y-ii^-iy) ( ia)+cos(xi')

•l^K-+ii/>(5ffl)]

1 ( | χ / α ) * 2 - * ' Γ ( - | . - | μ ) Γ ( | . - | μ ) i ( 2 / « - a « ) i ' ^ i i i ( y / a ) , 2/>a

• | (a2_j^)Mpi+M ( j , /«) , 2/<a

χΗ2ΐ'+1)-'2"+'Ό-'-ι Γ ( 1 + | . + | μ ) [ Γ ( § - § . - | μ ) ^

•sFid+v, 1 + | ν + | μ ; 1+^; l-y'/a^)

2''α''+> I ΓΟ(1-μ-»' ) ] Γ ^ ( 1 - μ + . ) ]!->

• Γ t-'^{a'+i')-^Kiy+ii,mKi,-iyiU) dt

(2«)-ί2-^->[Γ(|-μ)]-ί •T{.\-y-¥y)n\-y+¥y)

WiMam+v')^-yl} •K.Aha{.(V^-{-f)*+y-]]

72 Tablet: Even Functions

2Nf{x) Ng{y)

515 (o«+x«)-5 iKiAhiv+iv'-mi

516 So.ixia) | irexp(—asinhy)

517 aechi^x) So.izia) N= sina Ci(a)—cosa si(a)

sin(acoshi/) Ci(acosh2/) — cos(a coshy) si(o coshy)

518 X csch(7ri;) S-i,ix(a) N= — [ cosa Ci(a)+sina si(a) ]

— [cos(a coshy) Ci(acosh2/) +s in (a coshy) si(a coshy) ]

519 χ8μ,^{ω^), - 2 < μ < 0 N= —|ΐΓ*α~' cos(§7rμ)

. Γ ( - 1 - μ ) Γ ( 2 + μ ) [ Γ ( | - μ ) ] - '

ϊΐΓ*α-ίΓ(2+μ) •inh-ß)T'yS-,-i.iiWa)

13. Parabol ic Cyl indrical F u n c t i o n s

2Nf(x) Ngiy)

520

521

522

523

524

exp(iaV)D-2(ox) ΛΤ=ο( |π)*

exp(iaV)D,(aa:), » ' < - l iV=i r*2 i ' -* r ( - | - | ) ; ) a - i

exp(- iaV)D,(aa ; ) ΛΓ=2*'-*7Γ*α-ΐΓ(1- |> ' ) : - ' , v<l

expi\ah)Diy-iiax^), v<0 N=-iiT)h-W

x''eM-Wx')D,iax), μ > - 1 , »'<1 ΛΓ=2*('-^->ΜΓ(1+μ)

• [ Γ ( 1 + | μ - | . ) ] - ΐ ο - - - »

ihr)hexpilf/a^D-,iy/a)

^i2a)'lTi-hv)T'y-'-' •expiiv'/a^m+^u.hy'/a^

2ί·~*ΐΓ»[Γ(1-|ΐ ')3-»α-ι uF,il;l-h;-W/<i')

- ( V 2 ) V - ' [ 2 / + ( a + 2 / i ) « i • cos{ 21/ arctanCyy (a+y*) ]}

2«-Μ-ΐ )^»Γ(1+μ) [ Γ ( 1 + | μ - | ν )

-i2/y«^)

13. Parabolic Cylindrical Functions 73

2Nf{x) Ngiy)

525

526

527

528

529

(cosx)"*^ exp(—a" secx) •D,[2a(l+seca;)*l z hr p<l

N ^ T ^ i ' e x p ( - a«) [D}M(2O) J

D,\:axii)miaxi-i)il, u<-i N=-hrh-^iiv+h)Ti-p)T'

exp[^(a sinha;) «]I),(2a coshx), i'< 0 ΛΓ=2*'^ο-ν*Γ(- | ΐ ' )

•Cr(|-|v):-W},^,o(2o«)

expf—(asinhx)«]D»(2acoshx), v < l ΛΓ=2»·-ίπ»α-'ΤΤ},,ο(2α«)

sech (irx) D-i+ixia) D ^ , ( a ) ΛΓ=§ΐΓ*βχρ(|α«) Erfc(o)

ir»2i' e x p ( - a « ) A ^ ( 2 a ) 2 ) , ^ ( 2 a )

M a r ( - i ' ) ] - ' s e c ( w ) . [ J _ ^ ( | 3 / y a « ) - J _ ^ ( i j / y a « ) ]

ϊ - ί - ί α - ΐ Γ ί - . ) r T ( - i . ' + t § i / ) . r ( -§ , ' -%)TFj^ , j .v (2a«)

2i'~Ma-W},,}.v(2a«)

5ir*(sechj/)* exp(|o« sechy) •Erfc[2-io(l+sech2/)»]

TABLE I I

FUNCTIONS VANISHING IDENTICALLY FOR NEGATIVE VALUES OF THE ARGUMENT

Defini t ions

Here/(a : ) is assumed to vanish identically for negative x. The Fourier transform G{y) o f / ( x ) then becomes

G{y) = f f{x)e^ydx = ί f{x)e^^ydx = g{y) + ih{y), • ' - 0 0 •'O

g{y) = ί fix) cos{xy) dx, h{y) = ί f{x) sm{xy) dx,

9(0)= Γ fix) dx = 1. •Ό

Table I I , which follows, gives Nh{y), while the properties Ν and g{y) are the same as those listed in Table I under the same number.

74

Nfix) Nhiy)

7

13

14

15

16

1, x<a 0, x>a

Xy X<1 2-X, \<x<2 0, x>2

x-^y x<\ 0, x>\

ia+x)-\ x<h 0, x>h

0, xb n=2, 3 , 4 , . . .

0, xb

ia'+x^)-'

x-îa+x)''

ia+x)-i

x-îa'+:x^)-^

ia-x)-i, x<a 0, x>a

ia-x)-\ xb, b<a

2y-^ m?iya/2)

4i/-2 uny urî^y)

i2T/y)^Siy)

smiay) [C\iay+by)'-C\iay) ] +cos(a2/) {ß\iay+by) — uiay) ]

Σ 7 7 7 7 ia+b)-î-y)^-^-'^

• c o s ß 7 r ( n — m ) —62/] — ( - 2 / ) ^ - ΐ ( η - 1 ) ! ] - ^ c o s ( a 2 / + | 7 m ) •Ci (a2/+6t / )+sin (ai /+ |7rn) si(a2/+%)]

a-i[cos(a?/) si(a2/+62/) —si(62/) — sin(a2/) C i ( a 2 / + % ) ]

(2α)-ΐ[β-^ί' îiay)-^y E i ( - a i / ) ]

7ra-*{ cos(a2/) [Ciay) — Siay) ] - s in (a2 / ) [ l -C (a2 / ) -Ä (a2 / ) ] }

i27ry)nsmiay)il-2Siay)^:\ +cos (a2 / ) [ l -2C(a2 / ) i ]}

( i^ )¥ / i ( i a2 / ) i ^ i ( i a i / )

(27r)*i/-*[sin(ai/)C(a2/) — cos(ai/)AS(a2/)]

siniay) [Ciiay) —Ciiay—by) ] — cos(a^) [siiay) — siiay— by) ]

17

76 Table II: Functions Vanishing Identically for Negative Values

Nfix) Nhiy)

18 (aä-r*)-! , xb, b<a

â-if eoaiay) [aiiay+by) - 2 si(ay) +si(a2/—6^)] -8Ϊη(αι/) [Ci(a2/+62/) - 2 C\iay) +Giiay-by)-\\

19 0, xb, b>a

eoaiay) [siiby—ay)—aiiby+ay) ] - s in (ay) [Ci(62/-at/)+Ci(62/-a2/) ]

20 0, xb

6-iTCC(&i/)-S(6j/)]

22 0, xb

(26)-»T{ [Ci2by) - Si2by) ] cos(62/) -Cl-C(262/)-Ä(262/)]sm(62/)}

23 0, xb

ir(a+6)-»{cos(ai,)[C(ai/+62/)-S(ai/+6i/)] -sin(ai /) [ 1 - Ciay+by) - Ä(aj/+6t/) ]}

24 ia'-3?)-i, x<a 0, x>a

lirHoCay)

25 xid^-^)-^, x<a 0, x>a

ôirJi(ay)

26 x-îcf-x")-^, x<a 0, x>a

( i ^ ) ¥ [ / j ( i a 2 / ) ?

29 x-îo?+:^)-^[_x^- (o2+r')*]-i 2râ-^Te-^''Iiiây)

30 0, x<a x-î3?-a^)-^, x>a

-ih^)¥Jiihay)Yiiiay)

31 0, x<a x-Hri-a^)-*, x>a

-Wy[Hoiay)Y,iay) + 7ο(α2/)Η_ι(αι/)]

32 - w(2o)-* Erfp(ay)i] Erfc[(a2/)»]

35 x-îa^-\-x')-*lx+ (a«+a^)i]-i 2*α-'sinh (lay) iCodaz/)

2. Arbitrary Powers 77

Nfix) ΛΓΑ(2/)

36 0, x<a iira-»[cos(i>Mr)/„(o2/) — s i n ( ^ i r ) y „ ( a 2 / ) ]

n = l , 2 , 3 , . . . η -ύΤ^Σ k\{n+k-1) ![(2Α;) !(η-Α;)

2. Arbitrary Powers

ΛΓ/(χ) ΛΓΑ(2/)

37

38

39

40

41

42

43

44

45

3Τ-\ χ<1 ο, χ>1, ν>0

ib-x)', xb, ν>-1

3r-^ib-x)i^\ xb, ν,μ>0

χ ' ( α + χ ) - ' , - 1 < ι ; < 0

»'(o^+x?)-!, - 1 < ΐ ' < 1

ia'+x')^, ν>0

(χ2+2αχ)- '-», 0 < ΐ ' < |

x - -*(a2+x»)-»[(o2+a?) i+a] ' , ,-<§

ari(a«+x«)-»C(a«+a?)i-x]'.

-§«'-ΐιίΊ(ΐ ' ;»'+1;ί2/) - i F i ( v ; i / + l ; - t 2 / ) ]

-52r '"MexpC - t ' ( iw -62 / ) :7( i '+ l , % ) -explii^w-by)2yip+l, -iby)}

• iiFiiv; ρ+μ;-iby)-iFiiv; ν+μ; iby) ]

2 _ . . , /S-.H.i(«y)

Γ(ι+*.) Γ ( * - | . )

2 ' - M r ( i . ) i T ( f - i . ) r y - ' • i i ' 2 ( i ; f - i « ' , i - 5 ' ' ; i a V ) —Jfl-o'"' csc(iirj') siiih(oy)

2 — · π ί Γ ( | - ν ) (2//α)'[7,(αί/) -L_,(a2/) ]

T-'-^a-'ir^Ti^-v)y' •iJ,iay) coaiay)+Yyiay) sin(ay)]

2»o-T (I-1^) r*F},,}(02/) M-},.j iay)

a'ihir)¥lM'ihay)Ki-i,iM

7 8 Table II: Functions Vanishing Identically for Negative Values

Nfix) Nhiy)

46

47

48

49

50

51

52

53

54

55

χ-»(α«+χ«)-ί[χ+ ia^+x')ij, v<^

'\:ia'+:^)i+xT', »'>0

ix+ia'+3?)iT', i'>l

xîa^+x')-'-^\ v>-l, ν-2μ<1

(α*-χ2)-5 , χ<:α Ο, χ>α

xia^—a?)"^, χ<α Ο, χ>α

i2ax-x')'^, χ<2α Ο, χ>2α

x'ia^-:!?)", χ<α Ο, χ> α, ν,μ> — 1

x-îa^-x^)-* •{Za+ia'-A^)iJ+ia- ia'-I?)*!],

χ<α Ο, χ>α

-\<ν<\

X-iiV-3?)-i{ [ ( 6 + x ) i + t ( 6 - x ) i ] ^ ' + lih+x)^-iih-x)^J'}, xb

a'ihi-)¥h-i,iiay) KM.ihay)

ατ'π C8civjr){Î,iay) sin(|w) +hijyiiay)-hiJi-iay) 1

y~h~'[ Ι+ΐΊΓ csciw) ZIyiay) cosC xc) -Ûiay)-hU-iay)l]

ίar^Bil+hv,μ~ip) •yiF2ii+ip;i+h»-n,hiay) +2'-^-^lΓiTi^P-μ) | Τ ( μ - | ν + § ) • y^·^hF2iμ+1; μ - i P + l μ-h+1; HV)

2'-Vr(Hi')7rV-'H,(a2/)

2-νΓ(Ην)π*2/-'/κ+ι(α2/)

(2a)'xir(H»')2/~''sin(oy) 'May)

ha·^^'Biμ+l, l+iv)y •iF,il+hu;l 2 + J v + M ; - | a V )

i2a)iBii+h'',^-¥)y •iFiil-hvA;-iay) •iFiii-hv;hiay)

•Jv+iihby)J-y+iihby)

Nfix) Nhiy)

56

57

58

59

ix'-a^)-'-*, x>a 0, x<a

0<p<h

0, x<a ix'-2ax)-'-*, x>a

0, x<a x-îx'-a?)-^, x>a

~h<''<h

0, x<a x-îx^-a^-^lix+in^-a?)*!'

+ Zx-i:^-aîJ], x>a

2-'->α-ν*Γ (§-».) y'

•May)

Or-â-'niTi^-v) •[J.iay) coaay—Y,iay) sm(oj/)]

yaecivir)a-'"y • [Η,(α2/) Yy-iiay) - y,(oj/)Hî(aj/) ]

'UM'i\'^y)Y'>r^'i\ay) ^J^,i\ay)Y^^,i\ay)^

3 . Exponent ia l F u n c t i o n s

Nfix) Nhiy)

60 yia'+y')-'

61 j ; - i ( e -*^ -e -^ ) , o>6 arctanC(a— 6) y/ if+ab) ]

62 ivâ^+y^)-^ sm[f arctan(2//o) ]

63

65 Tiv) (a2+i/')-»/2 ginj-p arctan(y/a) ]

66 ( e - + l ) - ' 52/"*- |TO-'csch(in//a)

68 x-2(l-e-~)2 2a arctanCoi//(2/'+2o'') ]

logiy'iy'+Aa^/ia^+m

69 0, x < 6 ix-h)'e-^, x>b, p>-l

e-<*r(y+l)(o«+j/2)-i('+» • sin[62/+ (1/+1) arctan(y/a) ]

80 Table II: Functione Vanishing Identically for Negative Values

Nfix) Nhiy)

70

71

72

73

74

75

77

79

e-îl-e-^)--\ v>0

r-\^+i)-\ v>0

expi—ax')

x~* expi—aa^)

xêxpi—ou^)

x" expi — ajc'), v> — l

expi-ax-l^x^)

80

81

x^êxp{—ax—bx^), c > 0

exp[- (ar)3]

-§i6- i{5[v, ia-iy)/b2-B[y, ( a + % ) / 6 ] }

rW{r'sm(Aw) +ii(2a)-'[f(^H|%/a) - f (> ' , 5 -5%A)- f ( ' ' , 5%/«)

|ta-TW{rCv, 1+(ΐ2 / /α) ] - f ( ^ 1 -%/a )}

- |Mi-»ir»exp(-i /y4a) ErfCîy/o*)

2-ia-i7r2/i e x p ( - | a - y ) 7 i ( | a - V )

2 - " * a - i 7 r y ' e x p ( - | a - V ) . [ 7 _ j ( | a - y ) - 7 , ( i a - y ) i l

- -iw^b-' 4

i;f;

\fa-iy\n exp LUJJ

— exp [m"]-c-?)i

exp I

^ • ( 2 6 ) - » ' e x p ( ^ ) r W

( t ^ ) i ) _ . [ ( a + i 2 / ) ( 2 6 ) - i ]

- e x p ( - i ^ ) D-^lia-iy) (2&)-»]

ί(3α)-ί2/ί{β'*'Αο,,[2(|2//α)¥·'] -e-ä'-'Äo.,r2(V«)'e-**']l

Nfix) Nhiy)

82

83

84

85

86

87

89

90

91

92

xi'expi-ax"), μ>~1, 0 < c < l

xi'expi—ax"), μ > — 1 , c > l

x~^ expi—a/x)

x-'-êxpi-\ayx), v>0

x-i exp[— ax— ib/x)^

a;~* exp[— ax— (6/ x) ]

x'-êxpZ-ax- (6^«) ]

x-êxpi-a^x-^)

exp(—ox*)

x~êxpi—ax^)

x~êxpi—ax^)

Σ ( - a ) » r ( M + l + w c ) ( w ! ) - ' n-O

c-'Σi- 1 ) ' ' α - ' ^ » > ' ΐ ( 2 η + 1 ) ΙΤ' n-O

•Γ[(μ+2η+2)/ο]2/«»+ι

(π/α)» expC- (2a2/)i3sin[(2aj/)»]

i2'a-'2/»'{ei'"u:,[a(i2/)»] -e-<i"ü: ,Ca(-t i /) i]}

χ1(ο2+^ΐ)-ίβχρ(-26*Μ) • [u sin(26»»)+t) cos(26*w) ] ,

«=2-i[(a»+2/«)i+a]», v=2-îia^+f)i-ay

6-»ir»exp(-2fe*w) sin(26»i;), «=2 - i [ ( a '+ l / ^ ) i+a ] i t ; = 2 - i [ ( a ' + ^ i _ „ - ) i

i6'{ (α+%)-ί'ίί:ν[26(α+ί2/)*] - (a- i i / ) - i ' iC,[26(a- i2/)»]}

Σ C ^ ( 2 m + 2 ) + # ( m + l )

- l o g ( a y ) ]

r '+a(iT)*2/-'{sin(o/42/) . [ i - C ( a / 4 y ) ] - cos (ay42 / )B -S (ay42 / ) ]}

(2T)»2/-l{cos(ay42/)Ö-C(ay42/)] + s m ( a y 4 y ) ß - S ( a y 4 y ) : }

-|aJiry-l{/}(ay82/) ο ο 8 [ | τ + ( a y S y ) ] + rj(ay82 /)sm[iir+(ay8i/)]}

8 2 Table I I : Functions Vanishing Identically for Negative Values

Nfix) Nhiy)

93

95

96

101

102

103

105

x"'^ exp (—ax^), Ϊ^> 0

•exp [ -6(a2+i2 ) l ]

^-}(62+^) - i

•C(a2+a^)»+a]- βχρ[-6(α2+χ2)»3,

•exp[-6(o2+r')i]

(aä+a:^)-J[(a2+a^)i+a]-l ·βχρ[-6(α2+χ2)1]

• { [ α - (a2-r')J]2, exp[6(a2-r')}] + C o + (α>'-χ2)ί]2' βχρ[-6(α2-χ2) ί]} ,

x<a Ο, x>a

iT{2p) (22/)-'{exp[-t-(ivir+ (aySy) ) ] •D_2,[a(-t/22/)i] - e x p P ( i ^ + (aV&y) m^^MV^y)*!}

arctan(2//6)Zo[a(ft2+2/2)i]

- ( l A ) /" exp(o6cosi)< •Ό

•sinh(a2/sin<) dt

2-»TV/i{§6[(a^+2/^)*-a]! 'Ki{hKia'+y')i+a2]

2W 'a-lr(|+iv)j/-l

.D_^{(2a)*[(&2+2/^)i+6i} •Mw{a[(6^+2/^)»-6]!

o' csc(w) {τΓ sinCi/ arctaii(2//6) ]

— / Θχρ(αδ cosi) smh(o2/ sini)

•Ό •ain{vt) dt\

-t(2a)-lire"» Erfc{ai[(6!'4.j^)j-j_ft j } •Erf{m»[(62+2/2)i-6]i}

2»a-%-*r(f+j') •m-p)y-i.

•M_„i{aC6-(6»-2/2)i]}, 6>j/

4 . L o g a i i t h m j ic F u n c t i o n s

Nfix)

106 — logo;, x<l 0, a;>l

-Ci(t/)]

Nfix) Nhiy)

107

111

\ogia-\-x), xb

o > l

±logC(a2+a:^)/(6»+i^)] ± according as α 6

113 log{Ca;+(a2+a?)»]/2x}

115

116

117

118

120

121

123

log(l+a2a!-«)

—log(a—x), x<a 0, x>a

a<l

—log(o—x), x 6

6 < a < l

—log(o'—x^), x<a 0, x>a

a<l

- l o g ( a ' ' - r ' ) , xb

b<a<l

- [ x ( l - x ) ] - i l o g [ x ( l - x ) : , x < l 0, x > l

- ( 2 a x - x ' ) ' ' - i l o g ( 2 a x - x * ) , x<2a 0, x > 2 a

a < l n = 0 , l , 2 , . . .

2/-i{logo—cos{%) log(o+i>) +sin(oi/) lsiiay+by)—aiiay) ] +eo8iay) ZCiiay+by)-Ciiay) ]}

±r*C2 log(a/6)+e»» E i ( - % ) - e ^ E i ( -oy)+e"*» Ei(6y) - e - ' E i C a y ) :

rTi :o (a2 / )+7+log( | a i / ) ]

y - i [ 2 7 + 2 log(oy)-e«'' E i ( -a j / ) - « - " E i C a y ) ]

—i/~'{logo—sin(a2/) Si(a2/) - cosiay) [Ci(ay) - γ - logy]}

— JTM logo— cos(6y) log(o— &) +cos(oy) [Ci(oy— by)—Ci(a2/) ] +sin(ay) CSi(oy-6y) -SiCay) ]}

-y-^coaiay) [Ci (2ay)+7+log(y /2a) - 2 Ci(ay) ]+sin(ay) CSi(2oy) -2S i (oy ) ]+21oga}

-y-M21oga-cos(6y) log(o ' ' -y) —cos(ay) [2 Ci(ay)—Ci(ay— by) —Ci (ay+ by) ] — sin(oy) • [2 si(ay)—si(oy—öy) —si(ay+i>y) ])

-rs in( iy)Ö7rFo( |y) - ( 7 + l o g 4 y ) / o ( § y ) ]

- τ ( 2 η ) !(η!)-' sin(ay) (2y /o)-{iirF„(ay)

+hn\ Σ ( |ay)""' [m!(n-m ) rV„(oy) „-i

+ / „ ( a y ) C 2 i : ( 2 m + l ) - > - 7 m-O

- log (2y / a ) ] } n - 1

Forn=0, Σ ( ) = 0 m-O

Nfix) Nhiy)

124

125

127

128

0, x<a - log{C(x+a) i+(a ; -a ) i ] /2x i} , x>a

e-~(logx)='

log(l+e-"')

- l o g ( l - e - « ' )

h-'iirYoiay)+]o^-Ciiay) ]

ia'+f)-H ^-yiaTctaniy/a) J +iy+hiogia'+f)liyy+h ' logia'+y')-2a arctan(2//a) ])

r M l o g 2 - i ^ C l + ( i 2 / / 2 a ) ] -m-iiy/2a) (ί2//2α) ] +Uih-iiy/2a)2]

y-'{y+hl'ii+ iiy/a) ( V « ) : i

5. Tr igonometr i c F u n c t i o n s

Nfix) Nhiy)

129

130

131

132

x'^iainax)^

x-^'"iamax)^'", m= 1,2,3,

«-«(βΐηχ)"», n= 0,1,2,

(cosha—coac)"', .ι;<π 0, x>T

Uiy+2a)logiy+2a)+iy-2a) •logiy—2a)—2ylogy}

(mOVlog -

+ Σ ( - 1 ) " n -1

+ Σ ( - 1 ) " n -1

(2/-2an)2"-MogC2n-(i//a)] (m—n)!(m+n)!

(y+2any^-^ logC2n+ (y/a)] (m—n) ! (m+n)!

- ( - y ) -

2 n + l

+ Λ 2 n + l / .

2/[cos(ir3/)/sinha][ir2/ ' csc(irj/)]

+ Σ ( - Ο Μ η ' - Λ - ' β - » » n-O

Nfix) Nhiy)

137

144

148

149

151

153

161

164

165

166

169

(sinz)", x<ir 0, χ>π

a>-l

x-i{ai-xi)-icos[bia^-3?)*l x<a 0, x>&

(siiu;)~*exp(—asiiu;), x<v 0, x>T

(sinx)"* exp(o sina;), x<ir 0, χ>π

log[csc(irx)], x < l 0, x>l

(sinirx) log(cscirx), x< 1 0, x > l

i '>0

sin(osinx), x<ir, a<v 0, χ>τ

cos(esinx), x<5ir, a<J i r 0, x>hir

(sinx)~* sin(2o sinx), x<ir 0, x>ir

(sinx)~* cos(2a sinx), x<ir 0, x>ir

a < | i r

(sinx)"* sin(2a sinx), x<ir 0, Χ>ΊΓ

α < | π

2-«irsin(§2/)r(H-a)

r(H-ia+i2/)r(l+ia-i2/)

•/ι{Κ(&^+Λ*-6]1

(§o)M Mhry) U-îha)I-Myiha) -h-î\a)hîha)-\

(§a)M sin(ix2/) Uîha)Uîha) +7l_j,(|a)7}+}v(i«):

22Γ^ sin^(§2/) { 7 + l o g 2 + | ^ [ l + iy/2x) ] +Mi-KyA)]}

2>-T(v) s i n ( | 2 / ) { r ß + i p + ( 3 / / 2 » ) ]

•r[Hi''-(y/27r)]}-' •{log2+i^CHi^+(y/2x): +ΐΨ[Η§''-(ν/2π)]-^(^)1

k C J „ ( a ) - J _ , ( a ) ]

\π tan(|iry) • [J , ( a )+J_ , (o ) ]

ir(a7r)*sin(|ir2/) •/j_},(a)/j+}v(o)

7Γ(απ)*8ίη(§ΐΓ2/)

••^-i-i»(«)'^-l+l»(e)

2(o5r)»sin(|ir2/) •[•^-}+ίϊ(«)·^-+-Ι»(α) +Jn4 , ( a ) / }_ i , ( a ) ]

86 Table I I : Functions Vanishing Identically for Negative Values


Nfix) Nhiy)

173 arcsina;, x< 1 ' ΐ Λ ( ^ ) - 0 Ο 8 2 / ] 0, x>l

174 arccosa;, x< 1 - ΐ ΐ - Λ ( 2 / ) ] 0, a;>l

175 x"^ arcsinx, x< 1 1 - π ο

8ί(2/)+Γ<-Ήο(<)ώ 0, x>l

Δ

7. Hyperbol ic F u n c t i o n s

Nfix) Nhiy)

177 sech(aa;) -â-M7rtanh(ia-i7r2/)

+imi+h-\) -φ^-Ια-'ν) ]}

178 [sech(aa:)]2 h'W(iia-'y)+î-lia-'y) -φih+lâ-'y)-φii-ϊâ-'y):\

182 χ csch (αχ) îα-W(Hîα-V) -φ\^-ι^-'ν) ]

183 cosh(aa;) sech(6a;), a<h

27rsinh(6-Vi/) cosh(6~^7r2/)+cos(a6~V).

184 sinh(oa;) sech(6a;), a<h 7Γ6"^ sin(ia6~^7r) smhi^b~^wy) . [cosh(6-i7ri/)+cos(a6-V)

Nfix) Nhiy)

185

186

187

188

sinh (ox) csch(6x), a<h

1—tanh(ax)

x*-^ (χ-^— cschx), —1< «< 1

x"^ sinh(ax) sech(6x), a<b

190

201

202

χ"^^""^"" cschx)

[cosh (αχ)+cos5]~^, — π < 6< π

[cosh (αχ)+cosh6]"^

- 6~V jsinh ( ^ J cosh ( — ) 2 I \ h j l \bj

+ C 0 S (^-jj

2 / - ^ - | a -V csch(|a-V?/)

t 2 - r ( s ) [f (s, i - z i y ) (s, H % ) ] -cos(is7r)r(s-l)i/-*^i

arctan[tan(iao~V) tanh(i6-V2/)]

' 3 6 ~ α + ^ \

-logr

logr

/ 3 6 - ο - ΐ Α "

46 / . _ , ^ ^ ^ ^ 3 6 + α + ^ Λ

-2/Dog(i2 /)-l]

00

(csc6)22/Σ (-l)"+ie„(2r'+fflV)-isin(6n)

08Λ6{2 /Σ ( - l )"€„(2 / ' +nV ) - i e -»*

— a~% csch(o~'iri/) cos(a~*6y) j

8 8 Table I I : Functions Vanishing Identically for Negative Values

Nfix) Nhiy)

210

212

215

223

225

226

0, x<a i coshx— cosha) x> a

(cschax)", 0<v<l

0, x<a (coshx—cosha)"", x > a

0<v<l

x-^e-^ sinhibx)

( e^ - l ) -^ s inh(ax) , a-l, bv<a

231

240

ictnh(6x*) ^ [tanh(6xi)

log[ctnh(ax)]

2~*π tanh iiry) P-^+iy (cosha)

cos(| ϊ /π)Γ(l-^/)

-('-i'-ä)r-(2) • cosh ( — )—cos(w)

\ 2 a /

2^5r - i r ( l - i ' ) ( c scho ) - i • siiih(jrj/) V{v-\-iy)V {v— iy) .pLr+,„(cosha)

|arctan[262/(2/2+o2_52)-i-)

. [cosh(26-V2/) -cos(2ai>-%) ]-> +ΐ |6-"{ψ[1+(«+ί2/) /&] - ψ [ 1 + ( α - ί 2 / ) / 6 ] }

- i 2 - ' -26 -T(y+ l )

r ß 6 - » ( a + i » ' - i y ) + i :

ΓΟ&-Κα-δ»'+φ)]

rß&-i(a+i»'+t2/)+ll See Mordell, L. J. (1920). Mess. Math.

49, 65-72.

7+21og2+

+

2^\2 4 a /

8. Gamma and Related Functions 89

Nfix) Nhiy)

250 exp(—asinhx) yS-uy{a) = ^T cschivy) Uivia) +J-iyia)-Ma)-J-iMl

251 exp (—acoshx) csch(7r2/) j exp(o cosi) cosh(«/<) dt • a

-^TUiy{a)+I-iy{a):\

252 (sinhx)"* exp (—2α sinhx)

W-i+%(a) i^ i+%(ö) ]

256 (sinhx) ~* exp (—α cschx) - 2 * I m { r ( H % ) .D_J_·,[(2α^)*]i)-i-^·.C(-2α^)*]l

261 (cscx)* sinh(a sinx), χ<τ 0, Χ > 7 Γ

2~*oM sini^wy) 'Ii^»ih)lMika)

262 (cscx)* cosh(a sinx), χ<τ 0, x>T

(ia)hismihry) 'I-i-iyih)I-Mih)

265 (cscx) * sinh (2a sinx), x < ττ 0, x>T

2(απ)ί sin[(7r/2)2/][7_i_j,(o)7_H4,(a) -7i_jv(o)7i+j„(a)]

8. G a m m a a n d R e l a t e d F u n c t i o n s

Nf(x) Nhiy)

277 X Erfc (αχ) ία-'τ-^νΜ-Λ-,-^/α^)

278 Erfc(ax) , V - T l - e x p ( - i a - V ) ]

280 x»^! Erfc (αχ) §a ' - ' Γ ( 1 + ^ ν ) Γ ( Η ^ ν ) π - 1 [ Γ ( § + Α . ) 3 -ν

282 Erfc[(ax)*]

Nfix) Nhiy)

291

292

293

302

303

311

312

-Eii-ax)

-Eii-bx), x<a 0, x>a

-e-"Eii-bx), a>-b

Biibx), x<a 0, x>a

e-^Siiax)

ß - C ( « c 2 ) ] c o s ( )

iy-''iogil+fa-η

-hy-HEi\:-aib+iy)2+EiZ-aib-iy)2 -logil+y'b-^-2 cosiay) Ε ϊ ( - α δ ) }

- (ο2+2/)- '{α a r c t anC2 / (a+6 ) - i ] -hϊog[b-y+il+ab-ψ2}

§2/-i[Si(ai+ o y ) + S i ( a 6 - ay) -2 cosiay) Si(a6)]

+y arctan

V'+ia+y)' V'+ia-y)^

( ^ ) - y arctan ( ? ^ )

+

\ 4 a / \ 4 a / J \ 4 o /

C{ — ]—S[ — I cosl — U \ 4 a / \ 4 a / J \ 4 a / J

i2 / -M2-exp[-2(ai /)*] - cos[(2a?/) i ] - sin[(2a2/) *]}

9. El l ipt ic Integrals a n d Legendre F u n c t i o n s

Nfix) Nhiy)

320 Kiih-kx)^l, x<l iîTii)r'y-^si.oiy) 0, x>l

328 0, sinhx< α~' -§irtanh(§iri/) cscha;Ä:C( 1 - α-2 csch^x) i ] , sinhx> a"' .φ_5+%(Ρ) BîO-îiyiP)l,

ρ=α+α-ψ

10. Beesel Functions 91

Nfix) Nhiy)

333

344

345

348

il-x')-i^Piix), x < l 0, x > l

μ- v<3, μ+ν<2,

<ß,(H-2a''sinh2a;), 0<o<l ,

- l l ,

-1<«'<0

0, s inhx< l / a

^(2a*sinh2x-l), smlia;< 1/o, - l') (2+ί—μ) · ί Γ Ο ( 3 - Μ - . ) ] Γ Β ( 4 - μ + . ) ] ! -'y'^-*Si-μ,i+,iy)

~h sech(iiry) Im{Pr'*''(r) • CQi*''(r)+Q*:*2i-i(r)]), r= (l-o^)*

[2asm(w)]-itanh(iiri/) •{PIi;Us)RereL+|V,j,(s)] +PlY+,i„(s)Rerei][;!}„(s)]},

s = (1-0-2)»

( ο τ ) - ι Ι ι η { Ο Ι Ϊ ΐ ^ ( 1 + ο - 2 ) * ] •QLV+,,,C(l+o-2)i]l

10. Besse l f u n c t i o n s

W ^ ) Nhiy)

371

372

373

380

381

x-i[Joiax)J

x-iiMax)J, v>-\

lx'Max)J, - ia

αό<2το,ι

(ox-x=')- iJ , [6(ax-r ' )»] , χ < ο Ο, χ>α

ab<2r,,i, ρ>-1

(V2/)Mi'-iC(i-4oV-^)*]P, y>2o

(W2/)»r(H.)[r(f-.)r' .{P=5;C(l-4aV-^)i]p, ί/>2α

22-π-Τ (Η^) [1(1-2.;) ]-Η2ο)· •y- '^(4o ' ' -2 / ' ) - 'e '"Oli_} •[(2/2+4aV4o2/], y<2o

22'ir-i 008(2^) ?(!+") (20)· •2 / - ' -K2/ ' -4o«)- 'e-"Oi_i •[(y2+4a2)/4a2/], y>2o

2sin(ioy)2-'sin(^a?), 2=(62+2/')i

7Γ sm(io2/) J} , [ io (0+y) J /4v[ io(2-2/) ] ,

Nfix) Nh(y)

382

383

384

385

390

391

394

{ax-x')i''Jy[b{ox-x')^2. x<a 0, x>a

α6<2τΜ, P>-1

-Yo[b{ax-3?)^2y x<a 0, x>a

a6<2fo.i

0, x>a a^<2fM, - - 1 < ϊ ' < 1

— \og{ax—x^)Jo[b{ax—Qi?)^2) 0, x>a

aö<2ro,i, α < 2

Λ (α sinx), χ < 7 Γ Ο, Χ > 7 Γ

ϊ/> — 1 , α<τ„.ι

cscx Λ(ο sinx), χ < 7Γ Ο, Χ > 7 Γ ϊ/>0, α<Τν , ι

(sinx)~^Jm(a sinx), Χ < 7 Γ

Ο, Χ > 7 Γ m = 0 , 1 , 2 , . . . , α<Τη»,ι

(xa)*(ia6)''sin(ây)

'Q^+f)'^-^J.^W^+m

-2τΓ^ύη{\αν)ζ-^ ^[un{\az)[Ci{\oz+\ay) +Ci(ia2— ây) ]— cos(f o«) . [Si( |a^+|a2 / )+Si(ia0- ia2/)]) ,

^=(6^+2/^)*

—7Γ sin(iay) {cos(|7n/) •[/ iv(ia2+ia2 /)yi . ( ia0-ia2/) + Yky{\az+\ay)JU\az-\ay) ] - s i n ( ^ ) [/i.(ia2+ia2/)./j.(ia2-ia2/) + Y^.{\oz+\ay) YUl^^z-lay) ]},

^=(6^+2/2)*

-^z-^[un{âz)[Ci{az)-^ Ci{haz+hay) - \ Qi{\az-\ay)+\og(\ah)-\ogz'\ -cos{\az) [Si(a2) - I ^\{\az^\ay) -\î{\az-\ay)~\] un{\ay),

7Γ sindTTi/) J j ^ i , ( i a ) JiK+iyda)

TrCia)'"! !! sin(^7ri/)

• Σ €nC(m+n) ! (m-n ) ! ] - i /n- i , ( |a)

·Λ4-*ι/(2«)

11. Modified Besse l F u n c t i o n e

Nfix) Nhiy)

399 ίΓ*=Ίο(αχ), 6 < o

Ktiax)

s^Koiax), n = 0 , 1 , 2 , . . .

x"^^Koiax), n = 0 , l , 2 , . . .

x^Koiax)

x-iKoiax)

K,iax), -l<v<l

xfKyiax), μ±ρ>-1

x-iI,iax)K,iax), ν>-\

x-iiKyiax)7, -\<v\μ\+\>'\

(a2+2/»)-J log{a-»[y+ ia'+M

( - l ) » ( 2 n ) \ i a ' + y ' ) ^ Q M a ' + y ' ) - * l

(-l)»i^(2n+l)!(a2+j/ä)-"-i

i ( 2 T ) ¥ { 2 5 C ( i - | 2 / 2 ) i ] -2m^+hß)y+Kiih+hyz)*l

ihrz)HKiih+hz)*l -mi-hz)*l}. z=ia'+y>)-*

hr C8c(^w) (a»+2/*)-l{a-'Ci/+ ia'+m -a'iy+ia^+y')^T'\

hr cscίiτiμ->')lTil+μ+v) ia'+y')-^ • lP7iyi^'+i/^-^l-P7i-yi<^'+f)-^l}

m+p)im-y)rKWy)* •e"-QlJ[(H-4oV')»]^I|[[(l+4a*2/-2)l:

r ( f + v ) c r ( | - p ) r

2f-21T(l+r)]-> r [ | ( f + j u + v + l ) ] · Γ [ Κ Γ + μ - ' ' + 1 ) ] Γ β ( ί - μ + . - + ΐ ) ]

• m ( f - M - i ' - i ) ] y 4 i ' 8 B ( r + M + ' ' + i ) , K f + M - « ' + l ) , K f - / . + ' ' + l ) , Kr - M - v + l ) ; § , i ( f + l ) , l +r / 2 ;

400

401

402

403

404

405

407

411

418

419

Nfix) Nhiy)

420

423

429

431

432

433

437

442

443

444

445

3?'expi-x')Iyi3?), -\<P<0

x^Kiiaa?)

xi'expi-3?)K,i3?), - 1 - 2 ν < μ < - 1 + 2 ί .

X-'Koiax-^)

Koiaxi)

x-*K2yiaa*), -h<v<h

ixia-x)lril^lxia-x)y], x<a 0, x>a

7ο[6(αχ-χ^)*], x<a 0, x>a

iax-:^)i'I,g)iax-3?)i-], x<a 0, x>a

v>-l

KoU)icuc-3?)i^, x<a 0, x>a

ica-x')-iK,[biax-x^)i2, x<a 0, x>a

- ! < ! / < 1

^m+p)iTii-v)TKh)-''

•r'' e x p ( - y y 8 ) i F i ( J - 2 ^ ; 1- . - ; W)

hra-Khry)mwa)-mm ]

2r^hriTii+h-p)Tii+h+p) ·CΓ(f+iμ)r'2/

•2F,il+y+p, 1+y-p; | + * μ ι | ; - iy*)

Ta -V£(2ay ) i ]X i [ (2ay ) i ]

-hr'iCiiW/y) oosiWv)

+siiW/v) s m ( i a y y ) ] —Jir' seeiin>)y^

• [cos( W - i ^ - W/y)MW/y) -sinihrp-hr-W/y)YyiW/y)l

i r sm ( iay ) / , ( s i ) / , ( z2) , zi=la[,y±iy'-m 2

2 sin(§oy) i (62_2^)- is inhßa(6»-y2) i l y < 6

' iy^-V^)-isinlW-m, y>b

(7ra)»(§o6)'sin(ioy) i (62-y2)- i - l /H4Öa(&2-y2)»l 2/<6 \if-l^)-i'-iJyîW-m y>h

- s i n ( l a y ) (2/2-6*)-» •{sina[Ci(2i)+Ci(22)] -cosaCSi(2i)-81(22)]}

2 i=ia[y±( i /2-62)»l a=â(y*-62)4

y > 6 y < 6

—iir^ gec(iflT,) sin(ây) G/jv(i2i) I'-jXiau) + Γ},(§2ι)/-,,(*22)1 ί />6

zx=ha[3,±iy'-m

Nfix) Nhiy)

446

451

456

457

465

466

467

m

471

474

Zxii+x)yiK^[xii+x)y\,

Xo[6(a^-e*)*l x>a 0, x<a

ia^-a^-iKlHa^-a^-], x>a 0, x<a

-l<p<l

KJiaiix)i2Koiai-ix)i:i

K,laiix)i2KM-ix)*l - ! < « ' < 1

72,(2a siiu), x<v 0, x>ir

v>-h

{8mx)~Îm{2a sinx), x<ir 0, χ>π m = 0 , l , 2 , . . .

cscx l2p{2a sinx), Χ < 7 Γ

0, x>T

p>0

"2,» (2a sinx), χ < π 0, x > i r

iir2sec(7n/){cos(^l/) . [Λ(ε2) Y.izi) -Mzi) ] -s in(i2/) •[Λ(^ΐ)Λ(^2)+Γ.(.ΐ)Γ.(02)]!,

Ζι=ϊί(ν^+Ί/')^±ν1 2

|αα"^ cos(ay) {cosa[Ci(2;i)—Ci(«2) ] +sina[si(0i) —si(22)]} -âa-^ sin{ay) {sina[Ci(2i)+Ci(22)] —cosa[si(2i)+si(22) ]}

a=a{fji'+f)i, zi=a±ay

K62+2/2)-*{cosaCCi(î)-Ci(22)] +sinQ:[si(2?i) —si(02)]}

a==a{y^+f)^, zi=a±ay 2

|7r2sec(j7n/) 'iM^2)YU^l)-Ji.MYU^2)l

^ ι = Κ ( 6 ^ + Λ * ± 2 / ] 2

^2/-^θ_ι.ο(^αν^)

i7n .csc( |7 r i ; )2 / - ' 'S - i , . ( iaV')

7Γ sin(|7n/)/„-4i,(a)7v+iy(a)

n-O

. [ ( m + n ) ! ( m - n ) ! ] - i / n - i y ( a ) / n + i « ( a )

jTraj/-^ sin( |πί/) 'ily-i-^yla)l^+iy{a) — / v + M i / ( « ) ^H-i-Hi/(c^) ]

^^^080(271^) sin{îry) . [ /_^(a )J_ ,+ i j , ( a ) - / ^ ( α ) / . 4 4 ^ ( α ) ]

96 Table I I : Functions Vanishing Identically for Negative Values

Nfix) Nhiy)

476 Koi{a^+h^'-2ab cosx)^2> x<'^ y Σ enif-n^-'il- ( - 1 ) - cos(π2/) ] 0, χ > π n-O

. / n ( 6 ) i i n ( a ) , 6 < a

478 K2v{2a sinh^x) itV^ csc(27n/) Q/^j,(ö) F_îi,(a) —/_,îy(a) Yv-iy{a) —Jy+iy{a) Y-.îy{a) +J-îy{a)Y,+iy{a)2

487 üCo[(a2+62+2a6 coshx)*] Σ {-iren{n'+y')-Un{h)Kn{a) n-O

—§7Γ csch(7r?/)iiLij/(a)

. [ 7 , , ( 6 ) + / _ , , ( 6 ) l a>b

12. Parabol ic Cyl indrical F u n c t i o n s

Nfix) Nhiy)

520 expi\aV)D-2iax) -h-'v^MWM) Eii-hv'/a')

521 exp(|aV)D,(oa;), . ' < - l τ ΐ Γ ( § - § . ) ] - Η 2 α ) '

524 x''expi—Wx')D,iax), v>-l 2}(-Μ-«^ΐΓ(μ+2) [ Γ ( ^ μ - ^ ^ + § )

-Wa')

T A B L E I I I

FUNCTIONS NOT BELONGING TO EITHER OF THESE CLASSES

Defini t ion

Here

Gm = Γ fix) dx = 1. •'—00

The following tables give NG(y).

97

98 Table III: Functione Not Belonging to Either of These Classes

Nfix) NG{y)

1, 0<x<h 0, otherwise N=b-a

χ'», 0<x<h 0, otherwise n = 0 , l , 2 , . . .

N=b^+^{n+l)-'

x\ 0 < x < 6 0, otherwise v>-l ΛΓ=δ-+·ι(^+ΐ)-ι

χ-" , 6 < x < o o 0, - ο ο < α : < 6

iV= ( ί , - ΐ ) - ΐ δ ΐ -

( α + χ ) - " , 0 < χ < ο ο Ο, - ο ο < χ < 0 ν>\

( 6 - x ) ^ 0<x<h ο, otherwise Ϊ ; > - 1

χ'''(α+χ)-\ 0 < χ < ο ο Ο, - ο ο < χ < 0 0<ν<1 Ν^ΊΓΰΤ" csciTir)

χ - ^ χ - δ ) " , 6 < χ < ο ο Ο, otherwise 0 < Ϊ ; < 1 N=Th-'' csc(7n/)

iy-^{e-^-e-^y)

m—ο

( i 2 / ) - ' ^ i 7 ( ^ + l , % )

{iy)^'T{l-p,iby)

{iy)^'e^yT(l-v,iay)

{iyy'-'e-'^y{p+l,-iby)

war' cscMlTMT^ 'e^yV(v,iay)

irb-' C8c(tv) [Γ(μ) T^T(v, iby)

Functione Not Belonging to Either of These Classes 99

Nfix) NGiy)

10

11

12

13

3r-îi+x')-\ o<x<oo 0, — <»<x<0 0 0, - ο ο < χ < 0

iV=(Va)»rW[T(H,/)r

x'ia+x)!", 0<x<<» 0, - o o < j ; < 0 -1<ν<-μ-1 ΛΓ== ο · ^ Τ ( 1 + ν ) Γ ( - Ι - ν - μ )

. [ T ( - M ) r

ix+a)^îx-b)^\ b<x<oa Ο, otherwise 0<ν<^-μ N= ia+b)î^î2»)

'ηί-2μ-2ρ)ίηΐ-2μ)Τ' ix-a)^-\b-x)^-\ a<x'>0 ΛΓ= (&-α)2 ' '+«'-Τ(2μ)

·Γ (2 , / ) [Γ (2μ+2ν) ] -

il-x)-Kl+x)'^\ - l < a : < l 0, otherwise »',μ>0 N=2·^-^Biμ,v)

ia^+ixdzb)^!--

N=iTcâ^-^Tiv-^)[Tiv)T'

ircsc(«T)7,(2i2/, 0)

2 - i H r ( v ) D _ 2 i 2 ( % ) i ]

2Ό-*Γ iv) e<i<*D_2v[(2Mi2/)*]

α«'+^>Γ(1+0(ί2/)-*('+^>

( o + 6 ) ' ^ - T ( 2 v ) (%)-"- ' • expp |2 / (a - b) ]Tfμ_,,μ+,^[Ϊ2/(α+6) ]

( 6 - α ) · ^ - Η ί ί / ) - ' - ' β χ ρ [ - ϋ ι / ( α + 6 ) ]

· ι ί Ί ( μ ; ί ' + μ ; - 2 % )

π»2^««'[Γ(.')]-' . ( § | 3 , | / α ) ' - ί ί : ^ ( α | 2 / | )

100 Table III: Functions Not Belonging to Either of These Classes

Nfix) NGiy)

17

18

19

20

21

22

23

24

25

26

Zia'+x^+xlr^ (}<x<OO 0, - o o < x < 0 v>l ΛΓ=α'+ν(ϊ/2_ι)-ι

(a2+x2)-i[a:+ (α2+χ2)ί]-.^ 0 < χ < oo 0, - o o < a ; < 0 v>0 N=v-^ar'

e-«^, a<x^^ 0<\<2

N=irib-a)-' CSCCTTX) (a^-i-6^-i)

0<α<ϊ/ /ο

iV=c exp[6(a—i'/c)]ß(ac, v—ac) e-^^log(l+e-^) - 1 < λ < 0

( l+e- ) -Uog( l+e- ) N=t'/6

(l+e-)-Mog(H-e-)

. [ Ι . ( % ) - Λ ( % ) ] - 1 }

πα "CSCCTTV) • [ Ι . ( % ) - Λ ( ι α ? / ) ]

- e x p [ - 6 ( c - i V ) ] }

^^λ-1+»ν CSc(7rX+l7r?/)

• [loga—7Γ ctn(^X+*7r^) ]

7Γ(Ο—c)~^ csc(7rX+i7r?/) .(^λ-ι+.ν-5λ-ι+ίν)

c exp[6(a—i//c)] •exp (%)J5 [c (a+i? / ) , j / - c ( a + % ) ]

7Γ(λ+%) ^ CSCITX+iwy)

ITT csch(7r2/) [ 7 + ^ ( 1 — ]

— ίτΓ csch(7r2/)[^(i/)—ψ(ϊ/—1*2/)]

Functione Not Belonging to Eitlier of These Classes 101

Nfix) NGiy)

27

29

30

31

32

33

34

35

36

e-îa+e-')-' log(o+e-*) o > l , v>\>0 N=a^-'Bi\v-\)

• C l ^ W - ^ ( ^ - X ) + l o g a ]

e"exp(—oe*) N=cr'Tiv)

ve"exp(—oe*) N=a-'Tiv)ZΨiv)-loga2

Cexp(e-»)-ir*e-^ λ > 1 ΛΓ=Γ(λ)Γ(λ)

[exp(e-«) + l ] - ' e - ^ λ > 0 iV=(l -2 i -^)r (X)f(X)

N=2Koi2ab)

sech (oa;+6) Ν=τα-^

[8echiax+b)2' v>0

N=2-'a-mhp)JiTiv)T'

exp[—i> tanh (ox) ]Csech(aa;) 0<a;<<»

0, - « > < a ; < 0 i '>0 N=iâ-Khb)^'Tihv)Ii,-iib)

exp(oV+62e-^) EMaei'+be-^) N=2Koi2ab)

ο^-·+*«'5(λ+%, v-\-iy) ' ifiv) —fiv—\— iy)+logai

'"Tiv+iy)

a-'-*>>Tiv+iy)[.4'it>+iy)-logai

Ti\+iy)n\+iy)

(1_2ΐ-λ-<»)Γ(λ+ίί/)τ(λ+ί2/)

2(6/ο)*Κ<,(2α6)

m-ê-^l' sßchi^y/a)

2^'a-îTiv) exp (%/a ) 'm>'-iiy/2a)2mp+iiy/2a)2

a-i^'-ib-^'lTiv)T' • T[iv+iiy/2a) mp-iiy/2a) ] •M,»/2o,m(2&)

2(6/0)·«'sech (ιγ2/)α:<,(2ο6)

102 Table III: Functions Not Belonging to Either of These Classes

Nfix) NGiy)

37

38

39

40

41

]:ia+x)ia-x)-'y

0, | a ; | > a Ar=46-'smh2(Aa6)

exp(-ae*)7,(6e'), a>b, v>0 N=v-ψla+ia'-V')iT'

x^Kiiaxi) Ν=2π3ήα-'

[ (a+6e - ) (ae -+6 ) - i ] i -•K,iia'+l^+2abcoshx)i2

N=2Ki,ia)KUb)

expC— (o sinha;) 2]D,(2o sinhx), 0 > . ' > - l

iV=2i'(2jra*)-iexp(a2) .cos(i7n')[rKl+v)TTF_i,,o(2a2)

26-' {cos[a(2/*- 6*)»]- cos(o^) +i2/(2/*-6*)-*sin[a(2/ä-62)i] — isin(o2/)}

Γ(μ+ί2/)(α*-6*)-*·''' . ^ l U a ( a * - ö * ) - i ]

2 7 Γ 3 - ί α - » β χ ρ ( - ί ι ^ / α * )

2ii:j„_<,(o)iCjH-.,,(6)

2i'(2ira*)-* •exp(a*)r[Kl+i '+%):rC5( l+»'-%): • cos[iir(i/- %) ]ΐΓ_},,,·}„(2α*)

T A B L E ΙΑ

EVEN FUNCTIONS

Defini t ion

These tables contain the inverse transforms of the tables from pp. 15-73. The numbers a t the beginning of each formula coincide with those of the corresponding formula pair of P a r t I.

105

106 Table Ι Α : Even Functions

Ngiy) 2Nfix)

60

129

306

400

282

63

399

131

k ( o - ^ y ) , y<2a 0, v>2a Ν=\πα

hria+y)-'

W+f)-^ N=\wa-'

iWid^+mi€?w)*+ay N=iT/a)i

• { [ ( 6 * - a * - 2 / » ) 2 + 4 W

N= il^-a^-i

i ( _ l ) » 2 - 2 n - 2 ( 2 „ + l ) - I 'My+iiaV]-^ L\ 2n+l )

_ '(n+h-iha\Y \ 2 n + l ) \ J - l ) n 2 - 2 n - 2 ( 2 n + l | -1

- 1

L\2n+1 / . \ 2 n + l / .

—sin(aa;) si (ox) —cos (ox) Ci(aa;)

Erfc[(ax)*]

η = 0 , 1 , 2 , . . .

2· Arbitrary Powers

Ngiy) 2ΛΓ/(χ)

406 2 - W ( a * + 2 / * ) - ' - i r ( H ' ' ) ,

3. Exponential Functione 107

Ngiy) 2Nfix)

405 ^8eoihn>)ia^+y^)-* •{a-'[y+ia^+m +a'iy+ia'+m']

iV=^o-»sec(iirv), - l < y < l

K,iax)

3 . Exponent ia l F u n c t i o n e

Ngiy) 2N}ix)

7 ir(2a)-ie-»«' ΛΓ=τ(2α)-»

115 ^-i(l_e-^») N=7ra

log(l+o%-«)

111 T y - \ e ^ - e ^ )

N=ia-b)ir, a>b

9 ^ia'-P)-'lb-'e-*»-cr'e-^2 ΛΓ=ΐΓ[2α6(ο+6)]->

73 KVa)*exp(-i/ /4a) ΛΓ=ΚΐΓ/α)»

exp(—oa:*)

277 ih-^+y-') expi-W)-y-' a; Erfc((ia;)

33 l ^ ( _ l ) m + « ( „ t ) - l

• (d»/<fe") [ 2 " - » exp(-2/«i): N= (-1)'"+»π(2»η) ! (2n-2m)!

•2-2"->Cn!m!(w-TO)!]-*

n, m = 0 , 1 , 2, . . . , w > m

447 |π(οδ)-ΐβχρ[-α(62+2/2)»] ΛΓ=ΐ7Γ(αί>)-ΐβ-»»

439 iT(6»+i/^)-lexpC-a(6»+j/2)»] N=hrb-'e-^

108 Table ΙΑ: Even Functions

Ngiy) 2Nfix)

99 ihT)Ka'+f)-*ia+ia'+m (62+x=')-i[(6^+a^)»+&]i

•expi-bia'+m .expC-a(6='+a;2)i-]

N=iv/a)ir^

283 2riaexpi-a^b) ia*+y')-i EMaib+i^+x^)*y} • W+ (a^+2/^)»]-i expi-bia*+f)i2

N=ia-^ expi-2a^b)

441 hra-'ia^+f)-i e x p [ - 6 ( a « + ^ * ] cos[y arctan (a;/6) ] •{iia'+y')i+yj+iia^+y')i-yj}

N=â-'e-^, - ! < ! ' < 1

4. L o g a r i t h m i c F u n c t i o n s

Ngiy) 2Nfix)

304

30S

61

223

190

îry-Hogil+y/a), y<2a ^ j r M o g [ ( i / + a ) ( y - a ) - i ] , y-^2a N=hra-'

hry-'logil+y/a), y<2a \iry-nogif/a^-\), y>2a

k\ogiia'+f)iV^+f)-'l N= logia/b), a>b

\log\ V+ia+b)'

ly'+ia-b)^]

log(l+e-'>') N= log2

a>h

[si(ax)T

[Ci(ax)T

x-h-^ 8mh{hx)

x-îx"^—cschx)

5. Trigonometric Functione 109

5. T r i g o n o m e t r i c F u n c t i o n s

Ngiy) 2Nfix)

1 y-'âmiay) N=a

1, x<a 0, x>a

172 iry~ê~^ sm(o2/) ΛΓ=α5Γ

arctan(2aVa;2)

2 42/-^ cosy sm*(|y) N=l

X, x<l 2-x, l<x<2

0, x>2

10 |πα-'βχρ(—02/2"*) sin(iπ+2-*αy) (α«+χ*)-'

8 Ίώ~^ <iOsiay)e^ N=irb-^

[ft»+(a-x)*]-i +U^+ia+x)'r'

12 §iro-' CSC(2Y>) expi—ay cos ) ·siniφ—ay sin )

N= ira-^ secY>, —^ΐΓ<φ<^π

x2[x*+2oV cos(2v)+a«r

11 â~^ cos(2i>) expi—ay cos ) 'smiφ+ayam.φ)

Ar=ixa-'sec<p, - | π < < ρ < § 5 Γ

[a:<+2oVcos(2v>)+a^r*

227 i(V6)*cos(W/&) exp[Ka*-i )/6] Ar=§(V6)»exp(ioV6)

exp(—6s*) cosh(oa:)

312 irM8inC2(oi/)»]-cos[2(oi/)i] +exp[-2(ai,)»]l

N=a

Siax-')

84 (τ/α)» exp[- (2a2/)*: cosC(2ai/)i] N= (ir/a)*

376 2-* sin(az) «=(62+j/2)l, N=b-^smiab)

ab<To.i

JoW-x')*!, x<a 0, x>a

Ngiy) 2Nfix)

380

87

2 coaiây)z~'^ smiâz) N=%-âmiâb), a6<2ro.i

iT/b)i expi-2bhi) co8(26»«;) N= (7Γ/δ)»βχρ:-2(α6)»] u V

= 2-*C(ai+2/«)»±a]»

Jo[biax-x')^2, x<a 0, x>a

χ"* exp[—αχ— ]


Ngiy) 2Nfix)

492

291

2 %

65

64

125

230

arccos(i//a), i /<o 0, y>a N=h^

y~^ arctan(2//a)

τ arctan (a/y)

α-'Γ(ν) (l+j/^/a*)-*' cos(i' arctany/a) ΛΓ=α-Τ(ΐ'), i '>0

TH+v)ia'+y')-i-i'e-^ •cos[6äi+(i'+l) arctan(2//a)]

Λ Γ = α — ^ ( l + O e - ^ , i ' > - l

ia^+f)-H kaTr'+2y arctan(y/o) + a [ 7 + ^ l o g ( a ' + / ) T —a[arctan(j//o) f]

N=a-'ih'^+iy+loga)^2

§απ*(6»+2/*)ί exp{a*6[4(6^+2/2) •cosCf arctan(j / /6)-ia^(6*+2/*)-']

Ar=iai>Mexp(-iaV6)

χ-Ήο(οχ)

- Ε ϊ ( - α χ )

x- i [e-« Eiiax) _ e « E i ( - a x ) ]

0, xb

e-"'(logx)='

exp(—6x) sinh(aa;*)

Ngiy) 2ΛΓ/(χ)

303 1 (62+y2)-i{6 arctan[(a+2/)/6] —b arctan[(j/—o)/6] -hlogU^+ia+y)'! +hi\ogU^+ia-y)'l}

Ar=6-iarctaii(o/6)

e-*» Siiax)

293 (α*+ί/»)-Μ2/ arctan[y(a+6)-'] +ialogC(l+a/6)«+2/«/6»]}

N=a-Hogil+a/b), a>-b

-e-Êii-bx)

523 -ihr)*'r'iy+ia+m • cos{2i' arctan[j/»/(a+2/*) ]}

N=-i^)h-'a\ v<0

X ' ^ expiâh) D2y-iiax^)

7. Hyperbol ic F u n c t i o n e

Ngiy) 2Nfix)

240 ^-^t&nhila-hry) log[ctnh(aa;)]

177 ia'hr sech(â-iirj/) iV=§a->ir

sech (αχ)

178 ha-^vy cschdo-Vy) [sech(aa;) J

127 | [ Ο 2 Γ ' ' - ' Π Γ ' csch(o-iiry)] ΛΓ=(12ο)-ν

l o g ( l + e - ' )

128 - | [ 0 2 / - * - j T j r * ctnh(o-%2/) ] N= ( 6 o ) - V

- l o g ( l - e - ^ )

239 iir(H-y2)-> sech(W) coshx log (2 coshx) —X sinhx

179 la-îa'+f) sech(irj//2a) [sech (ax)] '

112 Table ΙΑ: Even Functione

Ngiv) 2Nfix)

236

205

243

247

242

269

202

201

204

203

238

248

185

(l-j/*)(l+t/»)-*

N=l

(i7r)Ml+2cosh[(2ir/3)%]}-'

iry"^ sech(fπί/) siiih[^ arcsin(a/6) ] N=7Γ arcsin(a/6), a< b

7ry~^8mh{by/a) sech(j7rt//a) N=wa-% b<^T

2π^^ cschiîry) smhî^y arcsina) iV'= (arcsina) 2, a < l

îry"^ sm{ay/b) sech(j7ry/6) N=iira/b

a~V csch6 un{by/a) csch(iri//a) N= ar^b cscho

a~% csc6 sinh(62//a) csch(ir2//a) ΛΓ= α-^δ csc6, 6 < π

|a~%sec( |6) cosh(a~%) sech(a~%i/) i \ r=ia-Vsec( j6) , 6 < π

ia""% sech(j6) cosh(%/a) sech(7r2//a) iV= ia -V sech(i6)

Tty-^ csch (πι/) [cosh(^2/) — cosh(ai/) ] ΛΓ=|7Γ2-|α2, α<7Γ

πί/-ι csch(7ri//a) [cosh(q//a) — cosh(62//a) ] ^=Κ<^-δ')Α, c>6, c,6<7r

i6""Vsin(a7r/6) • [cos(α7Γ/6)+cosh(πί//6)

i \ r=i6-Vtan(faT/6) , a < 6

- l o g ( l - e - 2 x ) coshx

( l+2cosh[(27r/3)*a ; ]}-i

iog[{b+a sechrc) (6—α secha;)"^]

log cosh(aa:)+sin&

Lcosh(aa:)—sin6j

—log(l—asecho;)

arctan[sinha sech(6x) ]

[cosh (aa;)+cosh6]-i

[cosh(aa;)+cos5]~

cosh (|αα;) + [cosh (αα;)+cos6]~^

cosh(iax) • [cosh (αχ)+cosh6]~^

log (1+cosa sechx)

log cosh(ax)+cos6 Lcosh(ax)+cosc J

sinh (αχ) csch(6x)

7. Hyperbolic Functione 113

Ngiy) 2Niix)

183

225

357

182

67

191

192

271

362

435

442

516

«•6-1 co8(â/6) c o 8 h ( ^ / 6 ) • [cos(air/6)+cosh(iry/6) ]"»

A r = ( ^ / 6 ) s e c ( W ö ) , a<h

i6-V8m(2oir/6) . [cosh(2iry/6) -οο8 (2ατ /6) ]">

N= ^ [6-ΊΓ c tn (a i r /6 )+O-»] , O < 6

2-»(a+co8hj/)-* iV=2-i(14-a)-*, - l < a < l

K ' r / a ) T s e c h ( W a ) ?

i ir*-KVa)Tcsch(»j / /a)T

2a'[sech(oy)]» ΛΓ=2ο»

fCa8ech(ioj/)]*

2-^τΓ(2α)[8βΛ(ί^)Τ· ΛΓ=2-*νΓ(2α)

(^)*Γ(μ) ( l - a 2 ) * ^ ( a + c 0 8 h y ) ^ ΛΓ= (|π)»Γ(μ) ( l - a ) b - i ( l + a ) - » ' - l

- 1 < 0 < 1 , μ > 0

if-l^)-iamlaiv'-m, y>h (6S_j^)- lginh[a(6ä-j /2) i l 2/<6 iV=6-'smh(a6)

2 co8(foy) '(6»-2/«)-isinhßa(6»-j/*)»], 2/<6 [iy'-l?)-âulW-m, y>b

Ar=26-ismh(â6)

irexpC—osinhy)

cosh(aa;) sech(6x)

(e»'- l)- isinh(aa;)

sechiirx) P-ii-Ua)

X caehiax)

xi^-1)-^

(a^+a?) aech{^x/a)

x{a^+:x?) cach{^x/a)

I T(a+ix) p

\Τ(μ+ιχ) I PLlWa)

0, x>a

Ο, χ > ο

Äo,.x(a)

Ngiy) 2Nfix)

189

200

188

276

359

364

194

j l og 4

l+ooahiiry/b)

Lcosh(in//6)+cos(2air/ö)J

Ν ilogCco8(air/&)l a<^b

^log

N= log

cos iwb/c)+cosh jvy/c) Lcos(ira/c)+cosh(iri//c) J

coai^b/c) Icoaiâ/c)]

c>a>b

1 , .

i V = - l o g

'cosh(îry/&) +sin(|oir/6)

L c o s h ( W 6 ) - s m ( W 6 ) J

l+sinCV/i»)

L l - s m ( W 6 ) J ' a<b

ir-Hf»" coshy)-» •log[(l+cosh2/)»+ (coshj/)*]

JV=fl- i (k)-*log(l+2i)

T-i(2cosh2/-2a)-»

C(l+coshy)*+ (coshy-o)*J •log

ΛΓ= (2-2o)-*«-»log

1+a

[2»+(1 -α)»?· 1+α

- 1 < α < 1

(^)»(α»-1 ) -*' 'Γ( | -μ) • (o'+smh«y)i^ cos{ i\-μ) •arctan[(l/a) sinhy]}

ΛΓ= ikκ)Kc?-\)-ôι^τi\-μ) μ<\, 0 < α < 1

2-§{a-ir«'+2 sinhy arctan(2-* cschy) —coshy Iog[(coshy+2-*) .(cosh2/-2-»)-»]}

iV=2-iCir-21og(2*+l)]

a;-i[smh(aa;) csch(6x)

χ-ι csch (ex) • [cosh(ax)—cosh(6x) ]

χ -1 sinh(ax) sech(6x)

Π Γ { | + ΐ χ ) I c o s h M r *

[sech(Tx)JP_}+.x(a)

cosh(lirx) I Γ ( ^ - μ + ί χ ) I* •PiL}+ix(«)

( l+x«)- 'sech(iTx)

7a. Orthogonal Polynomials 115

Ngiy) 2Nfix)

199

196

195

224

241

193

1 . 1 - τ sina e" cosa cosht/ 2 2

•log coshiz+sina

Lcosh /—smaj

+sina smhy arctan (cosa cschy) iV=^sina

—cosa log[ctn(j7r— a) ], a< ^

iye"*'—i+coshy l o g ( l + ^ ) iSr=§(21og2-l)

ye'~^+coshy log(l+e"" *') iV=log2

b'hr smiaw/h) cosh(^2//6) • Ccos(27r2//6) - cos(2ax/o) ^ -§α(α2+2/2)-ι

iV=i6-Vcsc(aV6)-^a-S a<b

2Try'^ QSQhi^y) •sm2{ii/log[(l+a-2)*+a-i]}

iV={log[(l+a-2)i+a-i]}2

2 cosh(^y) -e*' arctan(e-^) —arctan(e*^)

iV=2 - i 7r

( l+x^)-^ sinh(air) csch(§^x)

a:(l+r^)-icsch(irx)

(l+ar^)-isech(j7rx)

( e ^ + l ) - i sinh(ax)

log(l+a-2sech2a;)

(l+x*)-isech(7rx)

7a. Orthogona l P o l y n o m i a l s

ΛΓί;(ι/) 2iV/(x)

76 (_ΐ )η2 -η -1^-2»-1^} x 2 ' ' e x p ( - o V ) •exp(-i2/2/a2)ffe2„(2-iy/a)

ΛΓ= (2α)-2' '-Μ(2η) !/n! n = 0 , l , 2 , . . .

401 (- l)-M2n)!(a»+2 /*)-»-i

iV=7r(2a)-2"->[(2n)!/n!J n = 0 , l , 2 , . . .

Ngiy) 2Nfix)

138

137

211

322

324

70

341

325

343

2 — W ( l + a ) [ r ( l + i a + i i / ) •r(l+i«-ii/)r

N=2-^hrTil+a)[Til+â)T\ a>-l

2-«jrcos(i2/)r(l+a) •iTil+â+h)Til+ha-h)lr\

a>-l iV= 2 - v r ( l + a ) [TCl+â)]-"

h-'^LTih)m+b)T' 'mv+ihy/a)Tihu-ihy/a)

iV=ia-Mrai;)[r(§+|;')rS ' ' > 0

hro08iTy)ril+hy)Vi\-h) 'im+h)ni-hi)r'

N=i>riTi\)JimT'

Mar)-'\n\+ib/a)Y N=îan)~mm

hb-nBiu, ia-iy)/b2+Bip. ia+iy)/b2] N=b-^Biv,a/b), v>0

-^-HmiPT)Ti-h+¥y) •Ti-hv-hiy)Tihp+i+¥y) 'TiM-¥y)

N= -hr-' smipr)iTi-h)Tih+h)7, -l<v<0

hrm+\iy)ril-iiy) •iTii+\iy)Til-hv)T'

N=hriTi\)7ini)T' 2->^hriTii+h'-h+¥y)

-Ti^+h-h-hmi-h-hß+hy) 'Ti-h-y-¥y) . [ Γ ( - ν - Μ ) Γ ( 1 + ν - μ ) Γ ( | - μ ) ] - '

N=2-i-hc-i[Tii+h-hß)n-¥-hß)J

'ίη-ν-μ)ηΐ+Ρ-μ)η^-μ)Τ\ Η - μ < 0 , μ - » ' < 1

(cosx)", x<¥r Ο, x>¥r

(sinx)", x < x 0, x>ir

[sech(aa;) ] '

Ä:CCOS(|X)], x < v

0, x>ir

sech(ox) i i [ tanh(ox)3

e«(l_e-»«)v-i

<ß,(coshx)

sechx X(sechx)

(sinhx)'"!ß5;(cosha;)

8. G a m m a F u n c t i o n s ( I n c l u d i n g I n c o m p l e t e G a m m a F u n c t i o n s ) a n d R e l a t e d F u n c t i o n s

8. Gamma Functione and Related Functions 117

Ngiy) 2Nfix)

350 2r^Tii-ß)mil+p+ß+iy)-] (sinhx)-"

-mil+p+ß-iy)! •e-**»Oi!(coshx)

•{m+uu-ß+iy):\ -Tii+hip-ß-iy)!]-' ΛΓ=2^-ΜΓ(^-μ)

• {ΓΟ (1+Η-Μ)]}ΜΓ [1+έ(>'-Μ)]}Λ μ + Η - 1 > 0 , M < |

351 2^2[Τ (1+»')Γ (1+ΐ' -μ)]- ' (s inhx)-^!

'mii+p-ß+iy)mii+p-ß-iy)l •e-*"Oi(ctnhx)

•mii+p+ß+iy)mii+y+ß-iy)l i\r= 2 ' - ' [Τ(1+ί ' ) r i l + v - μ ) ] - i . { Γ Ο ( ΐ + ν - μ ) ] Γ Ο ( 1 + Η - μ ) ] Ρ ,

μ - 1

226 2 - - » 6 - Τ ( 1 + . ) mh-Ka-J.-iy)l ^ ' mb-Ka+bu-iy)+ll

e-<«[sinh(6x)]'

mb-'ia-bu+iy)2 mb-Ka+bH-iy)+l2' mb-'ia-bu+iy)2

mb-Ka+bH-iy)+l2' i '> — 1 , 6i'<o

N= 2-'->6-'Γ(1+»') r d o i - i - ^ v ) '[Tihab-'+hp+Dy

212 2O-Vsm( iw ) r ( l - i ' ) . iTil-^p+Wa)Til-h-hiy/a) • c o s h ( ^ / o ) [cosh(iry/o) — cos(w)

0<i'<l N=h-'Tih)Tih-h)

(cschax)'

66 \cr'ίφiih//a)+φi-^h/a) ~Ψih+iWa)-ψih-ih/a)l

J\r=o-Mog2

(e"+l)->

151 2Γ' s in2/ [7+log2+i^( l+iyA) log[csc(irx)], a;<l +*Ψ(1-Ιί/Α): 0, x>l

N= log2

ISO log[sec(i5ra:)], x< l •CT+log2+i^(l+i , /x)+^(l-y/T)] 0, x>l

iV=log2

118 Table ΙΑ: Eren Functions

Ngiy) 2Nfix)

237

153

152

184

187

294

281

2'-îaTip)T'\TihH-¥y/a)\' ' {ψip)-hg2-•Remhp+¥y/a)2}

N=2-^''a-^Bih,h) ^ίΨi^+h)-Φih>')l, p>o

2^-'rip) cosih)

• im+^v+hMrih+h-hM r +m+h-h/T)-fiv)l

N=2'-'Tiv)[Tih+b)lr^ •Dog2-4'iv)+4'iH¥)l, p>o

2^-'Tiv)im+h+yMT' ^ίm+h-yMr'[ô(ß-Φip) +m+h+yM+m+h-yM 1

Ar=2 ' -TW [ r ( |+ |^ ) r ' 'Dog2-φip)+Ψihl·m P>0

- K ^ ' ) - K ^ ) Kf)K?)+"*(?)-+2ir sinl

i V = § 6 - ' [ x t a n ( W ö ) +φil-Wb)-4'iί+ϊa/b)2, a<b

-Tis-1) Γ is) {2-(β- mis, HiiV) +iis,h-¥y)l-y'-'smiîrs)}

J V = 2 r ( s ) f ( s ) ( 2 - - l ) , - l < s < l

Ty-'Eni^ya-i) N= (x/a)»

hr Enciy/a) Λ Γ = | χ

[sech(ax)] ' •log[cosh(cKi;)]

[sinixx)]"-! • log[csc (xx) ] , z < 1

0, χ > 1

•log[sec(^x)3, x<l 0, x>l

smh(aa:) sech(6a;)

a;»-i(a;-i—cschx)

-Eii-ax")

- ύ - ' e x p ( - a V ) Erf (MM;)

Ng{y) 2Nfix)

295

229

297

78

79

32

228

28

103

529

k(T /a)»exp(i2/ä/a) Erfc(|i/a-i+i>*) i\r=^(ir/a)iErfc(6*)

- 2 - * a - » y e x p ( - J ^ / a * ) Erfc(2-»a->i/) ] N=ihT)-*a

hrir/aKexpih/^/a) Erfc(iya-i) +iexTpi-W/a) Erf(t§|/a-i)]

N=hriir/a)i

ib-^TexpiaΨ)lexpi-by) Erfc(oft-Jy/a) +exp(6i/) E r f c ( a 6 + ^ / a ) ]

N=^b-hr 6χρ{αΨ) Erfc(a6)

4 exp [ \ 26 jj

i V = j 6 - V e x p ( i a V 6 2 ) Erfc(ia/6)

(2a)-»7rErfc[(a2/)»] ΛΓ= (2α)-ν

( ix) i{exp ( - i2 /*) - i ( ix)» i / Erfc(2-»2/)}

(2a)-^7re"''Erfc[(a2/)i] iV=ir(2a)-i

(2a)-lxe°»Erfc{a»C(62+i/2)j+5-|jj

ΛΓ= (2a)-iire<* ErfcC(2a6)i]

| ( ί Γ sechy)* βχρ(^α* sechj/) • E r f c [ o ( H 5 s e c h j / ) » ]

JV=|7r»Erfc(o)

-expCaar») E i [ - ( α τ ^ + δ ) ]

x - » ' « e x p ( - a V ) s i n h ( a V )

exp(—αχ*) Εί(αχ*) -exp(oirä) Ε ΐ ί -οχ*)

(6*+x*) - ' exp ( - aV)

exp{—αχ—6*x*)

(α*+χ*)-ί •[a+(a*+x*)»r*

x-*exp(—x*) sinhx*

χ-*[α+χ+(2αχ)»]- ι

(α*+χ*)-*[α+(α*+χ*)»]-* •exp[-6(a*+x*)»]

sech(irx) •D_j+.v.(a)Z)_i-te(a)

Ngiy) 2Nfix)

109

292

279

290

450

452

106

175

114

116

118

-h-^{g-">Zy-logi2a/y) ] -β««Έΐ(-2α2/)1

i\r=o-Vlog(2o), o > l

y-^[smiay) E i ( - a 6 ) —arctan(i//6) —|t Ei(—o6—fay) + i i E i ( - ( r f ) + i o y ) ]

Λ Γ = δ - ΐ - α Ε ί ( - α δ )

mi-\v'/a')-Eii-W)l N=logib/a), a<b

-hr-ia-^ expiW) E i [ - H^+W/a") ] i V = - i ^ - i a - ' E i ( - 6 * )

- Ιτα- ΐβ-»* Eii-z,)+<^ E i ( - 2 i ) ] Λ Γ = - ^ α - ΐ Ε ϊ ( - α ό )

xih'+y')-i{expi-aiV^+m .Dog(a6)- | log(6*+2/*)] -expCa(&*+3/*)i] Ei[-2a(6*+2/*)i]}

ΛΓ=χ6-ΐ[β-* l o g a - e ^ Ε ΐ ( -2αδ) ]

y-'Siiy) N=l

k[Ci(2 / ) - J to(3 / ) ] iV=ijrlog2

ia-%[sin(a2/) Ci(ay) —cos(ay) si(ay)]

-jri{sin(a2/)CCi(o2/)-7-log2/] —cos(a2/) Si(ay))

JV=o( l - logo) , o < l

y-^{cosiay) Si(2o2/)+sin(oy) • [ 7 + l o g ( k / « ) - C i ( 2 a y ) ] }

Ar=2a( l - logo) , o < l

(a*+x*)-Mog(o*+x*)

E i ( - 6 x ) , a;<a 0, x>a

x-XEnoiax)-ETfcibx)2

exp(oV) Erfc((M;+6)

(a*+x*)-i

log(a*+x*)i:o[6(a*+x«)»]

—logx, a;<l 0, x>l

x~^ arcsinx, x< 1 0, x > l

( r ' - a*) - i log(x /a )

—log(o—x), x < o 0, x>a

—log(o*—x*), x < o 0, x > o

Ngiy) 2Nfix)

19

17

18

117

107

120

311

432

cos(ay) [Ci(oy+%) — Ci(oy) ] +sm(o2/) Isiiay+by)—siiay) ]

Ar=log(l+6/a)

a'^lcoaiay) Ciiay+by) +8m(aj/) 8iiay+by)—Ciiby)^

JV=o-Mog(l+o/6)

(2a)-i{ sin(ai/) Csi(6y— ay) +aiiby+ay) ] — cos(ay) ICiiby- ay) - Ciihy+ay) ]}

ΛΓ= (2α)-ι logC(H-a) 6>α

cos(ai/) CCi(a3/)—Ci(a2/— hy) ] +sin(a2/) [si(oy) —si(oj/—6j/) ]

JV=-logCl-(Va)] , «'<α

(2a)-Mcos(a2/) [Ci(ai/+6y) -Ci(o3/-6j ,) ] +sin(aj/) [si(ai/+%)—si(a2/—fty) ]j

ΛΓ= (2o)-i logC(a+fc) ( α - 6 < ο

— s i n ( ö y ) log(a— 6) +sm(oi/) CCi(oy) — Ci(ay— by) ]— cos(oy) [Si(ay) -S i (ay -6y ) : }

ΛΓ= 6 - ο loga+ ( α - 6) log(o- 6), 6< o< 1

y-*{sin(6y) log(o+6)—cos(oy) [si(ay+6y) — si(oy) ]+8Ϊη(οι/) [Ci(oy+by) -Ci(ay)]}

ΛΓ= (α+δ) log(o+&)-α loga-6, o > 1

—y-i{sin(6y) log (o*—6 ) —cos(ay) • [si(oy+6y)—si(oy— 6y) ]+sm(oy) . [Ci(ay+6y ) -Ci(ay -6y) :}

iV=26+(a -6 ) log(a-&)-(a+6) •log(o+6), 6 < o < l

§(2ax)-»[sin(iy*/a) Ci(iy*/a) -cos(iy*/a) 8i(i2/*/a)]

JV=i(Wa)»

iirTCi(iaVy) sin(V/y) -8i(iaVy) C08(iaVy):

ΛΓ=2ο-*

(α+χ)->, a;<6 0, x > 6

0, x<h lxia+x)r\ x>h

0, x<h (χ*-α*)-ι, x > 6

( a - x ) - i , a;<6 0, x>b

ia'-x')-\ x<h 0, x>b

— log(o—x), xb

log(a+x) , xb

- log(o*-x*) , xb

Ö-C(ax*)]cos(ox*) + β - 5 ( ο χ * ) ] 8 ί η ( α χ * )

Koias^)

Ngiy) 2Nfix)

385

379

386

383

451

456

-42 -Msm ( ia2)CCi(a2)- | Ciiâz+ây) Ci(§o«-Jay)+log(|f l*)-log2]

-cosiâz) [Si(o«) Siiâz+ây) —I Siiiaz—ây)} cosdoy)

N= - 46-i{ sinCioft) [Οί(αό) - Ci(|o6) +log(ia)]-cos( iai»)[Si(at) - S i ( | a 6 ) ] } , a < 2 , o6<2ro.i

-22-»{sm(a«)[Ci(2as)-§ Ci(<K+ay) —J Ci(a2—aj/)+log(o6)—logz] -cos(ae)[Si(2cK;) - J Si(a2+ai/) - i S i ( a 2 - o i / ) ] )

iV= -26-i{sm(a6) CCi(2ai) - Ci(a6) +loga]-cos(oö)[Si(2ot) - S i ( a 6 ) ] } , a < l , αό<το,ι,

2 = ( δ * + Λ ί

- M-ifsinCo«) [Ci(a2+oi /)+Ci(aä!-ay)] -cos(as) [Si(og+ay)+Si(oz-03/) ]}

ΛΓ=-2(6ΐΓ)-»[8Ϊιι(αδ) Ci(a6) -cos (a6) Si(a6)3, οδ<Γο,ι,

- 2 ( ΐ Γ 2 ) - ι cosdoy) {sin(§a«)CCi(§02+io2/) +Ci(ia2—ôy) ]—cos(ia2) .CSi(|a2+ia2/)+Si(^<M:-|aj/)]}

iV=_4(6^)- i [s in( |o6) Ci( ia6)-cos( io6) •8iihab)2, a6<2ro,i, 2=(6'+i/*)»

| o a - i coaiay) {smaCCi(2i)+Ci(22) ] -cosa[si(2i)+si(22) ]}+ |αα-» sm(ay) • {cosa[Ci(2i)—Ci(22) ]+s ina [s i (2 i ) -Si(22)]}

«=o(6*+2/*)», 2i=aC(6*+y*)»±y]

ΛΓ=&-ΐ[8ίη(ο6) Ci(ai))-cos(ai>) 8i(ai>)]

|aa- i{sma[Ci(2i)+Ci(22)] —cosa[8i(2i) +81(22) ]}

ΛΓ=δ-»[8ίη(ο6) Ci(o6)-co8(a6) 81(06)] a=a(6*+2/ä)J, 2ι=α[(6*+2/«)*±?/]

—log(aa;—r')/oC6(<M;—X*)*], x < a 0, x>a

- log(a*-x*) •Λ[6(α ' -χ*)»1 χ < α

Ο, x>a

-Yoibia'-^)*!, x<a 0, x>a

-FoC6(ox-x«)*] , x<a 0, x>a

ί:ο[6(2αχ+χ«)»]

0, x < o ίΓο[6(χ'-α')*], χ > α

Ngiy) 2Nfix)

453

517

518

20

16

13

22

23

-§aa-Msma[Ci(2i)+Ci(z2) ] -co8aCSi(zi)-Si(22)]}

N= hb-'ii coaHab) [ 8 ί ( - Μ ΐ δ ) -Siiiab) ] -8mh(e6) [Ciiiab) +Cii-iab)2\

a=aif-V')i, 2i=a(j/dz(j/*-6*)i]

üTgif-V") = 0, y>b y<b

308

sin(ocoshi/) Ci(ocoshy) —cos(o coshy) si(a coshj/)

N= sin« Ci(a)—coso si(o)

—cos(ocoshy) Ci(acoshj/) —sin(a coshj/) si(o coshj/)

N=—cosa Ci(a)—sino si(o)

i2x/y)iCiy) N=2

b-iirii-Ciby)-Siby):\ iV=Hi r

(2ir/i/)*[cos(oi/) Ciay)+smiay)Siay) ] ΛΓ=2ο*

πο-*{cos(o2/) il-Ciay) - Siay) ] +sin(a i / ) [C(ay)-S(a i / ) ]}

(2ί>)-*«·{ cos(6y) [ 1 - C(262/) - 5(2%) ] +8in(6y)CC(2i>y)-S(2bi,)]}

ΛΓ= (26)^ir

(o+6)-»ir{cos(o2/)Cl-C(ai/+6j/) - Siay+by) ]+sin(oy) ICiay+by) -Siay+by)2}

N=ia+b)-h

i2a/r)-i{cosiW/a) ß - Sily'/a) ] -Bmi\y^/a)Zh-Cilf/a)2]

i\r=i(2«/T)-*

Ο, x>a

sech(^a;) 5ο,<χ(α)

a;csch(ira;)/S_i,,x(o)

a;-*, a;<l 0, x > l

0, xb

(α—x)-*, x<a 0, x>a

x-iia+x)-'

0, xb

0, xb

—sin(ax*) si(ax*) —cos(oai) Ciiaa?)

124 Table lAs Even Functione

Ngiy) 2Nfix)

307

313

90

91

14

38

317

316

318

521

xi2aM-i{smi\y'/a)ih-SiW/a)l + c o s ( i i / * / a ) Ö - C ( V / a ) ] }

ΛΓ= ihr)h-i

^Mi+iCiy/a)J+ZSi\y^/a)J -W/a)-Silf/a)}

N=hr

arH2yA)-*{ cos(iaVy) β - CiW/v) 1 +smiW/y)ih-Si\am

N=2a-'

i2w/y)HcosiW/y)ih-siW/y)l -MW/y)ih-siia^/y)l}

ΛΓ=2α-»

2a-*- i27ry)H cos(at/) [ 1 - S(ay)»] -sm(aj/)Cl-2C(a2/)J]}

N=2a-i

-hr-^Hexpi-iihn>-by)2yiv+l, iby) -exp[iiiin>-by)2yiv+l, -iby)}

N=b'^'iv+l)-\ v>-l

h^i\y')'-*yih-'',h'/a) ΛΓ=χ»ο--*(1-2ι /)-ι , v < §

hKii\f)'-*Tih-p,\m N=iria'-ii2v-l)-\

i(Va)-»r(H .)[r(l-.)r

ΛΓ=^π(7Γ/ο)ί(Τ(1-«') 0 θ 8 ( π ν ) ] - ' , - l .Cr(-i.)r(J-§.)r\ "<- !

— cos(ax*) 8i(aa:*)

x-M οο8(αίΓ*) [C(ax*) - S(ax*) ] +8in(ax*) [C(ax*)+S(ox*) - 1 ] }

exp(—ox*)

x-*exp(—ox*)

(a+x)-»

( δ - χ ) ' , x < 6 0, x>b

x-'^'Tiv, ax")

χ-^'yiv, aa?)

exp(ox*)r(v, ox*)

exp(ioV) i) ,(ox)

9. Elliptic Integrale and Legendre Functione 125

9. El l ipt ic In tegra l s a n d Legendre F u n c t i o n e

Ngiy) 2Nfix)

414

415

409

403

404

273

366

368

369

360

493

τ(4αΗ-Λ-*Χ[ί /(4αΗ-Λ-*] iV=iir*a->

xiia+b)^-y'lri •K{\j/'+ia-b)'Jm-ia+b)n

N=xia-\-b)-^Kiia-b)Ka+b)-i2

[(a+&)*+i/*]-i 'K{2iab)iZia+b)'+fT*]

ΛΓ= (α+6)-ιΚ[2(ο6)»(ο+&)-ι1 a>b

hi2xz)H2Eiih-hlz)H-2Eiih+hz)*l -Kiih-iyz)*l-Kiih+Wl}

N= (2a)-ia-i[T(f) J, z= ia^f)-*

ibrz)Wit+W¥rmh-hz)n

N=\i2a)-*[Tim, 2=(a*+2/)-»

26-ΊΓ-* sech(ij//6) J!:Ctanh(Jj//6)] ΛΓ=(6«-)-ι

π-1 sech(iy)2i:[(l-a*)» sßMh) ] ΛΓ=Χ-ΙΑ:[ (1-Ο*)»] , α < 1

π-» sech(§i/)ir{Cl- (l-a*)sech(§2/)*J} ΛΓ=ΐΓ-'Χ(<ι), α < 1

«-'CaH-sinhHi2/)J^ •Ä:{a(l-o-*)»CaH-sinh*(^) ]"»}

ΛΓ= (oir)-»X[(l-o-*)-i] , α> 1

2(2ir)i[(l-a*)H-coshj/]-i •-K{[Hi(l-a')-*coshy]-i}

ΛΓ=2(2ΐΓ)»[1+(1-α*)ί]-1 • Ä { c m ( i - « ^ ) - » r » ) , α < 1

(2π ) ί (1 -2 / * ) ίΗίρ - -» (2 , ) , 2 ,<ι 0. y > l ΛΓ=2—ν [Γ (1+ρ) ] - ' , ν > |

[ϋΓο(αχ)ϊ

/oCM^oCoa;)

x^Koiax)

x-^Kaiax)

I r(i+i&^) 1

sech(irx)[P_j+,-,(o)]«

[sech(Ta;)J . Ρ _ } + ώ ( β ) Ρ - ί + ί χ ( - α )

sech(7rx) .[φ_Η,·χ(α)ϊ

I n\^¥x) \'P-^isia)

X — Ή , ( χ )

Ngiy) 2Nfix)

510

416

410

494

373

412

407

i ( W a ) » 2 - ' ' r ( - i . - i ^ ) r ( i . - j M )

| ( a 2 - j ^ ) J M 4 i p j ; + iy/a), y>a iy/o), y<a

N=hra>'Ti-hß-b)Ti-y+¥) •iTih~y+h)Til-h-h)T\

μ±ν<0

i i r * s e c M ( a 6 ) - *

N= h'^iab)-^ 8 β ο ( π » ' ) φ _ ΐ ( ^ Η - W « ) , - i < « ' b,

(JIT)-* C08iin>)a'Til+2v)y-^ •ia"-f)-i^ZlZ\ia/y), y<a iy'-a^-i-^PZl^Zlia/y), y>a

N = - 2'ii* CSCCHT) O - - ' | T ( i - V ) ] " ' ,

-K ' '<o

-ixfUia/y+h/a) -2e-<"OUia/y+h/a)l y<2a -^•^-*smirv)ar-iy-'-iiTih-v)T' . ( ί /*-4α*)- 'β-·"θ:^(α/2Η-ΐ2//α),

2/>2α

-K ' '<o

2r'^hriTii-v)iy/a)' . (3/*+4α*)-ίΡΓΑ[(ί^-4α*) {2Η-4α*)->]

ΛΓ= ν - Τ ( ν ) Γ ( § - ν ) !Τ(Η2>') Τ\ 0<v - 1

K,iax)K,ibx)

I,ibx)K,iax)

3f[H,iax)-Y,iax)2

ix'J,iax)7

x-H,iax)K,ittx)

x'Kyiax)

Ngiy) 2Nfix)

402

418

411

417

272

274

361

370

367

(-l)»(2n+l)!(a*+j/*)-"-'Q2„Ci/(a*+i/*)-*] N= o-*"-^2*''(n!)S n= 0 ,1 ,2 , . . .

.{Ol5C(l+4o*i/-*)»]}* ΛΓ=Κ2απ ) - ίΓ*α )Γα+.)Γα- . ) ,

-Ην<ϊ

r(H»')|T(i-v)]-KWy)»e*" .OIliC(l+4aVi/*)i]?l5[(l+4oVj/*)»]

Ar=M§ )*r(i)r{i+.) ud )Γ(|+.) v>-i

Tii+v) m-l-") rV^"(4a*+2/*)-i .(W)»OLi[(l+4aVi/*)*] .OIi;C(l+4aVj/*)»]

N= α - ΐ ( 2 τ α ) - » Ρ ( | ) Γ ( | + . ) Γ ( | - ν ) , -i<v<i

[br 8Ϊη(2οτ) ^'P2a_i[cosh(î,/6) ] JV=[6irsm(2air)]-S 0 < o < i

2*-«-<'α-ΜΓ (26) Γ (2c) Γ (H-c) . Csmh(ij//a) ]»-îJ!l.'lf Ccosh(iy/a) ]

N= 2i-»-Mo-ir(26) r(2c) r(6+c)

• [T ( i+6+c ) r '

2^(o+l)i''(o+cosh2/)-*-*' ' •îC(l+cosh2/)i(a+ coshi/)-*]

ΛΓ=2-"-*(1+α)-Κβ |::(Η |α)-*], - 1 < α < 1 , μ < 1

(α*- 1 ) -»0_,_ι [1+2(ο*-1)- · cosh*(|y) ] Ν= (α«- 1)-*0_,_4[1+2(ο»- 1)- ΐ ] ,

α > 1 , μ < |

(l-a*)-iO^_jC2(l-a«)-i cosh*(iy)-l] iV= ( 1 - α * ) - » 0 _ ^ [ ( 1 + α * ) ( Ι - α * ) - ! ] ,

α < 1 , μ < §

x*»+>ii:o(ac)

x-i[i:,(ax)J

x-iI,iax)K,iax)

^lK.iax)J

τia+ibx)Ti^-a+^L·) p

I Tib+iax)Tio+iax) |*

sechx Plj+,vc(o)

I η^-μ+ίχ) ρ

| Γ ( ^ - μ + ί τ ) Ρ •CPiLi+«(o)T

2Nfix)

363

275

365

135

136

160

323

340

206

249

ΛΓ=ΐΓ*2"+Κΐ-α*)-10_Μ_}[(1-α*)-»], α < 1 , μ < §

|(27r)»2*-«c-»[r(6-o) sinh(|j,/c) •e-<'(«-*^'02;^|Ccosh(Wc) ]

N= 2**-^*c-T(2a) [Γ (26 -1 ) · 5 ( & - α - ϋ ) , 6 - a > i

2-(rt-i(i-o2)i»^r(2M) (coshy-a)-*""* .^.>(i-rtOjl |[cosh(| j /) coshj/-^a)-i]

ΛΓ=2ΐ- ' '(1+ο)*'^(1-Λ)-»Γ(2μ)

- l < a < l , M>0

2-»xP_i+„(cosS) ΛΓ=2ίΧ{8Ϊη^δ), δ<7Γ

N= (k)i(sin5)'r(H-i)PIi[(cosa), δ<π , » - > - §

ir2- i{P_H.(cos5)[-7- log4 -^ (H2/ )+ log(sm5)] -Q_j - , (cos5)}

Ar=ir2^C27r-"X(sini5) log(sm5) -XCcos^S)], δ<7Γ

|a-» cos(iin/)0-4+}j(p)0-+4v(p) JV=2-*o- i ( rM)-* i^ [ ( ip+ i ) -^ ] p = ( l + a - * ) »

2-*a-P_n.<„(cosi>) sech(iry) iV=2*i:(smi&), 6<ir

2-*T*P_H<,(a)[sech(7ri/)]* JV=2»irJ!r[(i-io)»], o < l

r( i - iM+|u) p •p:ii+.-.(o)

r(a+tcz)r(6+ica;) |-*

s e c h M I Γ{μ+ίχ) |*

(cosx—cos5)"*, χ < δ 0, χ > δ

(cosx—cosS)*^, Χ < δ 0, χ > δ

(cosx— •log(cosx—

0, , χ < δ

χ > δ

(l+a^cos^x)-* • K[a cosx (1+ cos^x)"^], x < §ir

0, x>hr

^,( l+2a-2cos2x) , x<hr 0, χ > | π

(coshx+cos5)~*

(coshx—a)"*

•(cosha>4-l)*+(coshx-a)i' •log

L(coshx+l)*- (coshx-a)*J

Ngiy) 2Nfix)

329

214

349

332

270

342

209

207

216

i(Wa)*r(Hiij/)ra-iiy) .P^+.-,C(l-a-^)»]

JV=K2ira)-iCra)TX{Ö-Kl-Oii}

ihr)*iTip) T-'isinb)i-'Tip+iy)Tiv-iy) -Ρίγ+φοώ)

N= (ΐτ)*Γ(ν) (8Ϊη6)ί-Ρ»_1'(οο8δ), 6<ΐΓ, i '>0

2'-iiir/a)iTih+hy+hiy)Tih+h-¥y) •Pli+Ud-a-»)*]

ΛΓ=2-ί( ΐΓ/α)ί[Τ(Η^.)Τ •p=r*c(i-«"'):, «>i. »'>-i

^ " * ( τ / α ) 1 sechiiry)TiH¥y)Til-¥y) •[p_}+iv(8)+p-»+i»(-8):

+ ί : [ 2 - ΐ ( ι + 8 ) ] } 8 = (1 -0 -* )» , 0 > 1

(Wc)*[T(«;)rcosh(W) . I Tiv+iy) Kl-t?)i-»'P».-A,„(c-i)

ΛΓ= (^c)»r(«') (l-c*)»-»'Pl-j '(c-0, 1

-2 '-i(2ojr)-» sm(in') Ccosh(irj/)-cos(in')]-i

•TihhhH-hiy)nh+b-¥y) '{PZ\+liil-a-')il +PZ\+ii-il-a-^2]

ΛΓ=-2'- ΐ (2ατ)-» cot ( i i rv) |T(H^« ' )T .{Pir*[(l-a-*)»] +pir*[-(i-«-^)»]),

o>l, - l < i ' < 0 2-»x'!ß_j+,„(cosho) ΛΓ= 2» sech(io) KCtanhdo) ]

2-»ir'!ß_4+,v(cosh6) sech(ir3/) N= 2» 8ech(§6) Kltanhi^b) ]

(^)»r(l-.)(smha)»-15'_-j*+,,(cosho) ΛΓ= (smho)-0- , (c t i iho) , v< 1

( l + o coshx)-»

(cos6+coshx)"

£X,ia coshx)

( 1 + 0 coshx)-»

(ocoshx—1\» o c o s h x + 1 / .

(l+c*smh*x)-»' • coshfi/ arctan (c sinhx) ]

^ , ( 0 coshx)

(cosha—coshx)-», χ < α Ο, x>a

(coshx+cosh6)-»

(cosho—coshx)-', x < o 0, x > o

Ngiy) 2Nfix)

213

208

210

215

348

244

245

246

326

•r(v+iy)r(^-t2/)^L-j;..,(cosh«) ΛΓ= (smha)-O ,^ i(ctnha) , v> 0

2-»K_j+<v(cos6)+Q_i_-,(cos&) ] iV=2iÄ(cosi&), b<T

2-*[0}+.v(cosho) +O}_„(cosha) ] Ar=2»8ech(ia)Ä:[sech(ia)]

i2n)-iTil-v) (sinha)»-e-<'('-*> . [Qll».,.(cosha)+OLV+.-.(cosha) ]

N= (|ir)-»r(l->') (sinha)»-' .e-.>(M)Ql-»(cosha), p< 1

(απ)-> R^[0ii; | ,„(p)0'4Vi,„(p): N=-hTCSciin>)[3.iap)J, -l<v<Q V= ( l+a-*)»

2-»ir sßcHxy) {P_}+.v(cos5) [ - 7 - I o g 4

+iog(s inS) -Uih+iy)-mh-iy) ] +iO-H-iv(cos6)+^Ö_^,(cos5)}

iV=2-»irDog(sin5)2ir->X(sin^6)+ii(cosi3):, δ<1Γ

2-»ir sech(iry) {^_i+i„( cosha) [log(8inho) - 7 - l o g 4 - i ^ ( H i 2 / ) - # ( i - i i / ) ] +iO-}+i»(cosha)+|0_}-<i(cosha))

N= 2^x|sech(io)ii;:[sech(|a) ] + (2/ir) log(sinha) sech(^a) •Z[tanh(ia)]}

— 2-»7Γ sech(7rj/) {^_j+i^(cosha) Pog(sinha) - 7 - i o g 4 - | i A ( i + % ) - h 4 ' i h - i y ) ] —|0_j+,„(cosha) —iCl-j_i„(cosha)}

iV=2-»5r{2x-ilog(sinha) sech(ia) •K[tanh( |o)]-sech(io)K[sech(ia)]},

cosho<2

ΐπ2 8βοΗ(ίπ2/){Ρ_Η-^·,[(1-α2)*]Ρ Ν={Κ[_2ή{1-{1-α^)ψ-]]\ α < 1

(cosha+coshx)""

(coshx—cos6)"*

0, x < a (coshx— cosha) x > α

(coshx— c o s h a ) x > α 0, χ < α

Ο, sinhx<a"^ ^ , (2a2sinh2x- l ) , sinhx>a-i

(coshx+coso)"* •log(coshx+cos^)

(coshx+cosha)"* • log (coshx+ cosha)

— (cosha—coshx)"* •log (cosha—coshx), x < a

0, x>a

sechx jK;(a sechx)

Ngiy) 2Nfix)

356

330

327

355

346

331

354

345

Ja-^r I r(I+H-ity) Ρ {PIi;t , , [ ( l -a-*)i]}* N=\a-hr[Til+v)J

•{ΡΐΓ*[(1-α-»)»]}«, α>1, v>-l

•p^i,Zii-<r^)iy>^yi- (1-α-*)»: ΛΓ=α-ΐίΓ{0-Μΐ-α-')»]*}

• ·Κ{ [ΗΙ (1 -α -^ )* ϊ ) , α>1

iT*(sechiTi/)»P_M..[(l-a*)»] .P_ l^ . ,C- ( l -a* )» ]

i V = X { 2 - i [ l - ( l - a * ) » J ) 'K{2rXl+ii-an, « < 1

Λ Γ = i { i O , C α ( l + α - ä ) * ] ) ^ . ' > - l

K p * - l ) - * t a n M

'{i\Tii+v+¥y)m\+l,ip)7 -i\Ti-H-¥y)\W^Aiiy(p)J]

• { [ Q v ( p ( p ^ - i ) ^ ) J -CO- .^ ICPCP ' - I ) -* )?} , - I < . < O

P > 1

ia-%sech(iirj/)$_}+i.v(p) •[Q_Hi.»(p)+CL+4,„(p)]

JV= 2<rKin-i)-^Kiip- ι)Κίή-ι)-*1 .Z[2»(H-1)-*1 P = ( l + a - ' ) »

- i x [ a s m M r P = i ; t . v ( s ) • ReKL+iVj, ,(8):

N= - i x [ a sin(xv) T'P-fHs) Q'^Hs) s = ( l - o - 2 ) i , a > l , v > - l

|Cacos(xv):-MP'-V+j.-„(8) M Q I i+|t»(s)]

-pij4,,(e) R«KLU».V(«):!

-pir*woiY«} « = ( 1 - 0 - " ) » , a>l, -l<v<0

O,(2o''cosh'a;-1)

( l+o 's inh^)-* 'KZil+aHmhh)-*2

aechxKlil-a'aechh)*']

£l,il+2aHosh?x)

^ , { l + [ 2 / ( p 2 - l ) ] c o s h ^ }

(l+a^cosh'^)-* 'KZa cosha;(l+aä cosh««)"»]

Q,(l+2o2sinh«a;)

^ , ( l + 2 o ' ' s m h 2 x )

132 Table lAs Even Functione

Ngiy) 2Nfix)

328

347

344

353

N=ii+z)-HKah+h)-*l} z= ( 1 + 0 - * ) *

A [ a c o s h ( W ) r ' . {PIj ; | ,»Rere '4Vj ,„(s ) ] +PL+,Vi.v(«) ΜΟ=ίΐ|.ν(β)]}

ΛΓ=ΚΐΡ=Γ*(«)ο-Ϋ(«) +P'_+j»(s)Qir*(s)l

s=(l-o-'')*, o>l , - 1 < » ' < 0

\ sech(^y) • ReiPr*''"« COi'nO+Ql'i-xCr) ]1

r = ( l - a 2 ) * , o<l , - 1 < ) ' < 0 ^r=iP.(r)[QXr)+Q-î(r)]

TT C8ch(^xj/) • CPi''''(r)Qr*'''('-) -Pr*'''(r)Qi*''(r) ]

iV=Mi'.( '-)T+KQ.WT r=(l-o«)*, a<l , i ' > - l

0, siiilu;<o-' cschx K\_i\-a-' csch^x)*], sinhx> a"'

^,(l+2o''sinh2x)

0,( l+2a2sinh«x)

10. Bessel F u n c t i o n s

Ngiy) 2ΛΓ/(χ)

24 hrJoiay) (α2-χ2)-ί , χ<α 0, x>a

124 îr'iWoiay)+8iiay)2 N=â

0,

—log "(r+a)*+(x-a)*' L 2x* J

x<a

, x>a

50 2 ' -wr(H-^)r' '^(ay) iV=Vv*r(H^)|T(l+,.)]-i. . . > - i

(α''-r')'-*, χ<α 0, x>a

52 (2α)'χ*Γ(Η^)2/-"'Λ(θ2/) cos(ai/) ΛΓ=α^'π*Γ(Η»'):υ(1+ΐ')]-»,

(2οχ-0,

X * ) - * , x<a

x>a

10. Beseel Functions 133

Ngiy) 2Nfix)

56

43

57

119

121

319

26

321

-hi2a)-'inTih-v)y'Y,iay) iV=ia -^^rWr ( i -v) , 0 < i ' < ^

§(2α ) -ΊΓίΓ(^- . ) •fiJyiay) s\niay) — Y,iay) cos(ay)]

N=h-''ir-iTiv)Ti^-v), P < k

- § ( 2 α ) - ' τ * Γ ( | - ν ) 'y'U.iay) sva.iay)+Y,iay) cos(oy)]

N=ha-î^Tiv)Ti\-v), v<\

[ 7 + log(22//a) yîay)-hnYoiay)} iV=irlog(2/o), o < l

—K emi\y)[^7cYf>i\y) - ( 7 + l o g 4 2 / ) / , ( ^ ) ]

iV=4irlog2

\a-^[Joihay)J

^(W)»C^-l(W)T ]ν=τ( | τ )* [Γ( | ) ] -^

^Λβί / (α+6)Μ2/ (α -6 ) :

55

335

ΛΓ= 2«'ir(x/6)i[r(i+.) r ( i - ν )

iV=acosM(2H - l ) -» , - | < ' ' a 0, a:<o

(a:2+2aa;)-—i

0, x<2a (r ' -2aa!)- ' -», χ>2α

- ( o ' ' - r ' ) - i l o g ( o ' ' - x ' ) , x<a 0, x > o

- [ x ( l - x ) ] - » l o g [ x ( l - a ; ) l x < l 0, a;>l

ί [ [ (1-α-^χ2)* : , χ < α Ο, x>a

χ-*(α2-χ«)-», χ < ο Ο, a;>a

ο, χ>α

x-îV'-7?)-*[lih+xy+iib-xyj' -\-lib^x)*-iih-xyj'], xb

P,i2i^/a'-\), x<a 0, x>a

Ng{y) 2Nf(x)

30

336

59

505

496

375

92

433

141

143

139

-T{hryW-iihy)Y-iihy) iV=7r(ix)»Cr(f)r

hr{abKJ^{byy¥,+i{ay) -Jîay)Yî{by)2

N= ( α δ ) Η 2 ί ' + 1 ) - ΐ ( α / 6 ) ' ^ -a>b

- iaV(§ir i / )§ •iJiy-i{hciy)Y-i.î{hay) +J^^{lay)Yiîhay)2

N={2r)-ia-iril-iv)T{\+hv),

-K«'

iir(a/y)HJi{W/y) siniW/y+h) -YiiW/y) coaiW/in-W)}

iV=2 (x /a) i

—IxseciflT) (π/ί/)*

+ c o s ( § « ' - i x - i a V t / ) r , ( i a V y ) ]

Wo[a(62+t/ ')*] ΛΓ=|π/ο(οί>), αδ<|ΐΓ

xcos(i62/)/oß&(aH-2/^)»] JV=x/o(ia6), α6<π

ia6ir(6H-i/=)-*/i[a(6H-^»] ΛΓ=^αΐΓ/ι(α6), a6<ir

Ο, x<a x-iix'-a')-^, x>a

φ,[(α='+ο2-χ2)/2οδ], x<a-b 0, x>a-b

0, x<a x-îx^-a^-i

'{[x+ix'-aîj + \:x-ix'-aîj},

Ho(ar')-Fo(aar')

χ- ' exp (—ox*)

X-1Ä:2,(OX*)

(θ2-χ2)-»0Ο8[&(θ«-χί)»], X < 0 0, x > o

x-*(6-x)- i •cos[ox*(6—x)*], x < 6

0, x > 6

amlbia^-x'y], a;<o 0, x > o

10. Beseel Functione 135

Ng{y) 2Nfix)

378 I {hra)K(>by{l^+f)-*'-*J,+iia{l^+m ab<T,,i

382 I (Ta)»( |a5) 'cos(W) • ( δ » + Λ - » ' - * / ^ β α ( 6 ^ ί / ' ) » ]

ΛΓ= (^a ) ' (Wi>)*^K+}( ia6) , - 1 , α6<2τ,,ι

377 \hrJiM)Jiv{zi)

N=hriMhab)J, i'>-l, ah<r,,i 381 |ircos(|oy)/j,( |2i)/j ,(i22)

N=T\:Ji,ilab)J, p>-l, ο6<2τΜ

457 liir*sec(|w)[J,,(2i)J},(22)

436 Wy{zi)Mzi) N=hTiU^ab)y, p>-h

2

437

389

390

τ οο8(|α2/)Λ(2ι/2)Λ(22/2) N=TU,i\ab)J,

2

τ / } . ^ ( | β ) / } Η - Ι / ( | θ )

IT cos(iir2/) J } , ^ ( i a ) J},^^^(|a)

•Jyib{a'-c^yi, x<a 0, i > o

'J,[b{ax-3?y2, x<a 0, x>a

(α2-χ»)-»Λ[6(α«-χ«)*:, x<a 0, x>o

(ox-x*)-i •ΛΕδίοχ-χ")*], x<a

0, a;>o

0, x<o {x^-a^-i

{a'-x^-i -Ι,Μα'-χψ^, x<a

0, x>a

{ax-x")-* 'hAiHax-mi, x>a

0, x<a

Jy{a cos^x), x<ir 0, χ>τ

Jp{a sinx), x<ir 0, x>v

Ngiy) 2Nfix)

392

503

502

391

478

396

(ia)/i(^n.j,) iâ) +JHy+i-v) (i«)«î(H-i+i/) (i^) ]

N= hrav-'{ iJ^.-îia) J+ [/^^(ία) J}, v>0, a<r,,i

\ir sech(7r2/) {Uiyia) J+lYiyia) J] N=\k{[JMJ+IYMJ]

hir sech(7r2/) {Uiyia) ] ^ [J_-,(a) J] iV=π[Jo(α)T

\irav~^ Qiosiîry)

(iö)«A(H-i+i/) ] N=\âv-'\lJ^îha)J

+ [îv+i(|a)]2}, υ>0, α < Τ Μ

|7Γ2{Λ·^_,(α)Λ·ϊ,+ν(α) + Yiy-M y»v^,(a)+tan(7n/) • iy+via) Yiy-vio) -J iy-via) Yiy^vio) ]}

iV=ix2sec(7iT)

•{[Λ(α)]Μ-[η(α)]}, - K ^ 7Γ

Ηο(2α cosh^x) - 7 ο ( 2 α coshlx)

Ho (2α cosh^x)

cscx •Jviasinx), χ < τ γ

0, Χ>7Γ

K2vi2a sinhjx)

J o [ ( α coshx) coshx< 6/a 0, coshx>6/a


Ng{y) 2iV/(x)

35 2-1α-'τβ-^''/ο(^α2/)

42 xi(2a)- ' [ r(H-i)]-ViC,(a2/) (a'+rt^) - *

iV=ia -Mr (v ) |T (H' ' )rS ' ' >0

Ngiy) 2Nfix)

29

339

46

426

74

422

497

75

431

438

443

2-*o-* smh(ioj/) Kiiây) Ν=ΐ-*α-^

(οί>)ίπ7^(ΜΖ^(α3/) Ν=α-Ί/ΐΓ(2ν+1)-\ v>-h a>b

N= α'(2α/π)-^τα-^)ίΤ{1-^)ΐΓ\ "<*

(^x)»exp(-i/»/16)7o(i/^/16) Nîhr)*

hri2a/y)-i exp(-Wa)I-i{Wa) ΛΓ=2-*ο-νΐΤ(|)Τ-^

hra-'ymy'/a) UiiWa)+I-i(W/a) ] ΛΓ=Κ2α)-*Ρ(1)

2-i{2ira)-îKiiW/a)J Ν=^π-Κ2α)-*[Τα)7

l{2a/y)-i expi-â) U-iiWa) -hiW/a)!

ΛΓ=2-ία-1[Γ( |)]- '

-Ta-'yYili2ay)i2Kiii2ay)i2

ihra)Kob)'ilr'-v')-iÎMt^-m yb

N= i^na/b) io'7,+i(o6), v>-l

iira)i{hab)'cosiây) 'ii^-y')-îHiihif^-m yb

N=iira/b)Kh)'I,+iihab), v>-l

x-iia^-x")-* .[a^f-(a^x^)ir*

0, [ ( : ι«+αΗ-6 ' ) /2οδ:

'ix+{a^c^)ij

expi-x')W)

x~* expi—ax")

Koiax")

Ioiaa?)-Uiax?)

x* exp (—ox*)

x-'Koiax-η

•7 , [6 (a ' - r ' )» ] , x<a 0, x>a

iax-x")^' •I,\T}iax-x")i2, x<a

0, x>a


Ngiy) 2Nfix)

88

298

464

449

100

448

501

514

455

500

b'{ia+iy)-i'Kl2bia+iy)i2 + ia-iy)-i'Kl2bia-iy)i2}

N=2b'a-i'K,i2aib)

2aia^-y")-'K£bia^-m N'=2a-^Koiab)

Ti4a'+y')-iK„ibi4a^-y")i2 ΛΓ=§χα-'Χ2,(2αί>)

N=ii2Ta/b)icr'K.îab)

2a' coaiv arctan(j//6) ] KK [ O ( 6 H - 2 / ' ) * ] N=2a'Kyiab)

^Kyizi)Kyiz2) N=îK,iiab)J 2ι=Κ(δΗ-2/^)»±2/:

2

hiz2)Kcizi) N=hihab)-Koihab)

iwUzi)Kyizi) N=hrLihob)K,iiab)

2

Ibp-^ZK^MKMyizt) -Ki-i,iz,)Ki^riz,):\

iV=j6.-> {[XHiXioft) J - iKi-i,ihab) J}

2

A^=^r{[/o(l«6)T+Cro(iao)T}

x""' exp i—ax—¥/x)

-e—Ei{-aC(i^fx»)*-a;]}

+ iia'+xî-xj} exp[-6(a^x*)J:

·Χ2,[6(α^χ^)*]

(αΗ-χ*)-*

-Lo[6(aH-x')»:}

(aM-x')-*{ksecM

+iso.2,p6(aH-x*)*]}

(αΗ-χ*)-^ .ί:,[6(αΗ-χ^)*]

(α^χ*)-» .{Ηο[6(α«+χ2)*] -7ο[δ(α'+χ*)*]}

Ngiy) 2Nfix)

m

m 468

467

472

471

475

474

287

251

222

^Κο[α (2 coshj/)*] N=-h^Koiâ)

firK2,(2o cosh|j/) N=îrK2,i2a)

irl,-yia)ly+yia) iV=ir[/v(a)T, «'>-*

1Γ cos(|irj/) 1,-iyia) /H4»(«) ΛΓ=π[Λ(α)?, ^ > - |

4 i r a , . - ' [ 7 ^ ( a ) 7 ^ ( o ) -JHi-»(o)J^H+f»(e)]

ΛΓ=^αρ-ΐ{[ /Μ(«)Τ-[/^(«)] '} .

^β*-! cos(|irj/) [/..4_}„(a)/._i+}v(e) - • i p + H » ( a ) J ^ H + * ( « ) ]

N=hrm>-mîa)J+U,îa)y}, u>0

hr" csc(2in') [Ι_,^(α)7-Η^(α) -Iîa)I,+yia)2

N=W csc(2xv) {[/_,(«) J- ίΐΛα) J], - K ' ' < l

Ιττ CSC(2IIT) cos(iirj/)C/_,^(o)/_H4v(a) -7^(α )7 ,+4 , (α ) ] ,

iV=|ir* csc(2x,') {C7_,(o) J- iUa) J]

hr exp(—|a*) sech(§irj/) •UiyiW)+I-iviW)'2

N=hrexpi-W)IoiW)

Kiyia) N=Koia)

7iivC(a*-6*)*] cosCi/ arctan(6/a) ] JV=ZoC(a*-6')*:, a>b

KUo^"'*) 'KUae-'"*)

K,+ixia) Κ,-ixia)

72,(2o cosfx), x<ir 0, x>ir

72,(2a sinx), χ<π 0, x>ir

secCfx) •72,(2acosfx), x<ir

0, x>ir

cscx •72»(2osinx), x<ir

0, x>ir

7ί2,(2α cos^x), χ<π 0, χ > π

ίΓ2»(2α sinx), x<ir Ο, χ>ΐΓ

— i exp[— (α coshx)*] • Erf (io coshx)

exp (—ocoshx)

exp (—ocoshx) cosh (6 sinhx)

140 Table lA: Even Functione

Ngiy) 2Nfix)

286 êxpiW)secHhry)KiiyiW) N=iexpiW)KoiW)

exp[(a cosha;)*] •Erfc (α cosha;)

504 sechiTy) Kiyia) [7.ν(α)+/_<,(α) ] N=2Ioia)Koia)

/o(2o coshja;) — Lo(2ocoshiz)

484 Kiyiae<'l')Kiyiae-"l*) N=Koiai"l*)Koiae-"l*)

Χο[α(2 cosha;)*]

487 Kiyia)Kiyib) N=Koia)Koib)

Χο[(αΗ-ί>Μ-2ο6 cosha;)»]

479 Kîyia)K,-iyia) N=iK,ia)J

Ki,i2a coshfx)

480 âv-^lKi^y+iyia) K^p-iyia) —Ki-,+iyia) Ki-,-^yia) ]

N=>-M [Ä4+.(«) Ί - LKi-M J)

sech(5x) •X2»(2acosh5x)

488 Kiîyia)Ki^yib) + Kiîyib)Ki^yia)

N=2Ki,ia)Ki,ib)

{[(a+6e-)(H-ae')-']»' +[(Η-αβ')(α+6β-)-ΐ]*'} •K,[(oH-6'+2o6 coshx)*]

12. F u n c t i o n s Re la ted t o Bessel F u n c t i o n s

Ngiy) 2ΛΓ/(χ)

174 hry-'Tôiy) N=l

arccosx, x < 1 0, x > l

173 iir2/-'[sin2/-Ho(y)] N=^-l

arcsinx, x < 1 0, x > l

25 a[l- i irHi(aj/)] N=a

xia"-x^)-*, x<a 0, x>a

58 ô-^-V ββοίπν) {l-âiry[Jyiay)'B,îiay) -H,(a?/)/,î(oj/)])

0, x<a x-Hx^-a*) -*, x > o

12· Functione Related to Beseel Functions 141

Ngiy) 2Nfix)

51 .Η^ι(οί,)3

xia'-χ')·^, x<a 0, x>a

112 hÛoiay)-Loiay)2 (aH-x*)-» .logCaA+(l+ayx*)»]

113 hrr'ii+Uiay)-hiay)2 N=a

430 2-Va- ' [7o(i3 /yo)-L , ( i j /ya)] x{Kîa^)J

423 hra-Khry)iU-iiy'/a)-Lîm2 JV=r l (o /T ) -» |T ( f ) r

x^Ktiax")

424 iT'iMKl-iilv'M-L-iiWa)! JV=2-*/Vo-»/tr(i)]-*

^Kîa^)

465 Wy-'öioiW/y)-YoiW/y)l N^hra-'

Kolaiix)*-]

461

JV= iiräo-»CHo(2afc) - 7ο(2αί>) ]

Ko[alx+i^-m} •Ko[alx-ix"-m\

459 hriia^f)-* • {7o[6(4a*+2/«)i]-Lo[6(4aH-^*D}

N= |ΐΓα-ΐ7ο(2ο6) - Lo(2a&) ]

7„{a[(6H-x^)»-x]} .7ίο{α[(6*+χ*)Η-χ]}

398 Ho(2a coshiy) — ΙΌ(2α cosh|j/) ΛΓ=Ηο(2ο)-7ο(2α)

sech(irx)

164 §xCJv(«)+J-,,(a)] N=xJoia), a<hr

cos(asiiix), Χ < 7 Γ

0, x>ir

163 iirsec(§Tj/)CJ»+J_.(o)] N=hrJoia), a<hr

cos(acosa;), χ < | π 0,

Ngiy) 2Nfix)

161

162

259

260

258

257

526

333

519

466

462

§^ctn(^2/)CJ„(a)-J_.(o)] iV=irHo(a), ο < π

iircsc(§iri,)[J,(a)-J_,(a)] iV=iirHo(a), o<ir

hrayiia)+J-yiia)l N=Tloia)

h- sech(iirj/) Uyiia)+J-yiia) 1 ΛΓ=§τ/ο(α)

- Jtx ctn(iiri/) CJ,(MI) -J-yiia) ] N=irUia)

-\iw csc(iiri/) CJV(MI) - J-i,(ia) ] iV=^Lo(a)

i T i [ a r ( - y ) ] - i s e c M

Λ Γ = - § π » α - ΐ ( Η - * ) Γ ( - ν ) ] - ι , ^ < - *

2 ^ ν ί { Γ Ο ( 3 - Μ + . ) ] Γ Ο ( 2 - Μ - . ' ) ] 1 - ' • (μ+μ) (μ-»- -1 ) y^S-i^,,îy)

i v = 2 " - Μ { Γ β ( 3 - μ + Γ ) : Γ [ Κ 2 - μ - » ' )

^ Γ ί α - ί Γ ( 2 + μ ) [ Γ ( | - μ ) ^ •2/Ä_M-j,j(i!^â)

ΛΓ= - § 1 Γ » 0 - 1 0 0 8 ( ^ μ ) Γ ( - 1 - μ ) . Γ ( 2 + μ ) [ Γ ( * - μ ) ] - ' , - 2 < μ < 0

iV=iira-2sec(i7n'), - 1 < » ' < 1

(4aH-2/*)-»iiir sec(T»')72,C&(4aM-2/*)»] +tSo,2,[i6(4a^2/*)»]}

)V= a- i[ i ir 8βο(ΐΓ»')/2κ(2α6) +^Ϊ8ο ,2/ί2α6):

8ίη(α sinx), x<ir Ο, χ>τ

sin(ocosx), x<hr Ο, x>iir

cosh(osinx), x<ir 0, x>v

cosh(acosx), x<iir 0, χ > ϊ τ

8inh(asina;), x<ir 0, x>ir

sinh (a cosx), x< Jir 0, x > ^

D,Cax(i)i]ACax(-i)»]

( l -x*)- i ' 'P ; (x) , x < l 0, x > l

xSμ,îa3?)

Klaiix)i2K,\:ai-ix)*l

/ ,{a[(6*+x*)i-x]} .ί:,{α[(ο*+χ*)1+χ]}

13. Parabolic Cylindrical Functione and Wbittaker Functions 143

Ngiy) 2Nfix)

463 ir(4o«+j/a)-iSo>[6(4a«+2/»)»] KAai^ix'-n N=hra-^So.i.i2ab) •KAaix-ix"-m}

250 So.iyia) = —t§ir csch(iri/) exp(—asinhx) . Uiyia) -J-iyia) -Jiyia)+J-iyia) ]

i V = i » [ H o ( a ) - F o ( a ) ]

513 i2a)-^mh-ß)T' (coshx)*/S^,j(a coshx) •Ti\-h-hiy)n^-hi^¥y) s,+i,iyia)

JV= ( 2 a ) - * 2 - ^ - i | T ( i - M ) ] - i [ T ( i - | M ) r •S^+i,o(a), M < i

13. Parabol ic Cyl indrical F u n c t i o n s a n d W b i t t a k e r F u n c t i o n s

Ngiy) 2Nfix)

506

507

80

525

256

(2τα) - ίΓ (Η ' ' ) •D.^Zyi2(d)i-]D-M-^y2

ΛΓ=|(2α)-»Γ (Η^.)[Τ( |+ | . ) ] ->, ' ' > - ^

(2Ta)-»{r(i- .) i)^[y(2at)*] . Ι>^[2 / ( -2αΟ*Ή- ο ο β Μ Γ ί Η . - ) .i)_,^C2/(2oi)i]i)_MlJ/(-2aO*3}

JV=i (2a ) - i { r ( i - i v ) [ r ( l - i v ) ] - i + c o s M r ( H § v ) [ r ( H i v ) r ' ,

- K » ' 0

τί2ί' e x p ( - a * ) D j ^ ( 2 a ) D } ^ „ ( 2 a ) iV=ir»2i'exp(-a*)Cr>},^(2o)]*, ; '<1

2iR«{r( |+%)D_,_„C(2«i)i] .2 )_4_<, [ ( -2ώ )1]}

iV=a(k) '{C^l(ia)>fCri(§a)T}

csc(in/) . α ( α χ 2 ) - Λ ( α χ 2 ) ]

x""^ exp (—αχ— 6x^)

(secx)**^ exp(—a^ secx) •D , [2a( l+secx)*], x<iir

0, χ > | π

(cschx)* exp(—α cschx)

Ngiy) 2Nfix)

289

44

428

421

527

528

iri sech(ir3/)D_j+.v(2*a)I>-i_<»(2»o) N=i2T)-iaiKiiW)J

2-»o-T i\-hv) r*Wiy.iiay) M-i,,-iiay) iV=(2aA)-ir(i-J.)|T(f-|.)r,

Ι Γ 2 - * Τ ( | - 2 . ) [ Γ (Η . ) ] - Ι .r-'exp(j/yi6)Tr,,._},(i/y8)

JV=2^cos(irv)r(v)r(i-2i '), 0<ν<\

2-*'j/-i exp ( - i / y i6 ) ir_4„i,(2/y8) iV=2'-irW[r(i+2v)J-\ P>0

2n^a-'[Ti-v) T^Ti-li^hy)

•Ti-h-¥y)Wiy^.Hyi2a") N=2^a-hiTi-hv)

•Cr( i- | . )rWi^.o(2a*), p<0

2i-i7ria-W,,..j,(2a*) iV=2»'-»ir»a-W},.o(2o*), . '<1

(sechs;)» exp(o* sechr) •Erfc[o(H-sechs)»]

χ-'-iia^-x")-* •ia+ia^-x")ij

x-''expix")K,ix")

a;-*'exp(-x*)/,(x*)

exp[(o sinhr)*] •Z>,(2a coshx)

exp[— (o sinhx)*] •D,(2o coshx)

T A B L E I I A

FUNCTIONS VANISHING IDENTICALLY FOR NEGATIVE VALUES OF THE ARGUMENT

Defini t ion

Take giy) from the inverse (Table I) before (pp. 105-144) and obtain hiy) under the same number from Table I I on pp. 74-96.

145

TABLE IIIA

FUNCTIONS NOT BELONGING TO EITHER OF THESE CLASSES

Defini t ion

This table contains the inverse transforms of Table I I I from pp. 97-102. Corresponding pairs of formulas have the same number.

146

Functione Not Belonging to Either of These Oasses 147

NGiy) Nfix)

2 η

n!(ii/)-"-'-e-*«' Σ n!&»(tj,)"^-yw! •n-O

n = 0 , l , 2 , . . . . ΛΓ=ί.»+ΐ(η+1)-ι

χ", 0<x<b 0, otherwise

1 N=b-a

1, 0<a;<6 0, otherwise

19 ic- ij/)-i[e-(«-«')- e-*«^*)] JV=c-i(e-«-e-»')

e"", o<a;<6 0, otherwise

39 3-*o-i2Texp(-tiVi/ya«) JV=2ir3-*a-i

xiKiiax*)

24 ir(X+ij/)-i csc(irX+iirj/) JV=irX-icsc(TX), - 1 < λ < 0

e-^log(H-e-')

22 T ( 6 - C ) - I csc(irX+iir3/) (aX-i-Hv_ftx-i+<») iV=ir(6-a)- i csc(xX)(a'*-i-6^-i),

0 < λ < 2

ia+r')-'iin-^)-h-^

20 ιτα^-ι+'» csc(TX+iirt/) JV=iro''-icsc(irX), 0 < λ < 1

(a+e-*)-»e-^

21 ^g\-i+iv csc(wX+iirj/) • [loga—1Γ ctn(irX+iiri/) ]

iV=ira^-icac(TX)(logo-TctiMrX), 0<λ< 1

a;(o+e-')-ie-^

37 26-1 j cos[o(3/2_ j 2 ) » ] _ cosiay) +iy(y^-l^)-i sinCo(i/2-6«)»]-i sin(a2/)|

JV=46-isinh2(ia6)

[ ( α + χ ) ( α - χ ) - · ] » •Ji[6(o2-x»)»l | x | < o

0, ia;|>o

33 a-o-'e*»'" sech(iT3//o) sech (oaH-6)

28 ΛΓ=ο-'Γ(»'), ; '>0

e' 'expC—ae*)

34 2-ΐα-ΐ[Γ(ν)Τ-'β*»"' •rÖ^-Wa)r(iH-itV/a)

ΛΓ=2·^Ό->[Γ(|ν)ϊ |Τ(»')]- ' , i '<0

[sech(aa>+6)]'

148 Table IIIA: Functione Not Belonging to Either of These Classes

NG(y) Nfix)

23

25

26

29

27

30

c^(or.^lc)e^B[cia+iy), v-cia+iy)2 ΛΓ= c^^^-'l'^Bi(w, V- ac), 0 < a< v/c

VK 080Η(π2/) Cr+- ( 1 - ^v) Λ ΛΓ=π2/6

- m csch(7ri/) [ψ(ϊ') - ψ ί ί ' - iy) ] Ν=ψ\ν), v>0

a-v-ivY(j^iy) {\l^{v+iy)-\oga2 iV=a-T(i /)[^W-loga], Ϊ '>0

aX-H-.V5(x+i2/, v-X-iy) • W - ^ ( ί ' - X- i i / )+loga]

N=a^-'Bi\ v-\)[ypiv) -iiv-\)+\oga2, a > l , ν > λ > 0

N = b - ^ ' i i ^ l ) - \ v > - l

( i y y ' ^ i e - ^ y i ^ l ^ ^ i b y ) N = i v + l ) - ' b ^ \ v > - l

TOT' csc(x^) [Tiv) r'e^^r(i/, Ν=τα~'' csc(7n/), 0 l

iiy)'-h^yVil-v, iay) N=iv-l)-'a'-% P>1

Ti\+iy)n^+iy) ΛΓ=Γ(λ)ί (λ) , λ > 1

(H-(f)-Mog(l+e-)

( l+e-)-Mog(l+e*)

i/e*'*exp(—06*)

β-λ*(α+β-*)-'' •log(a+e-*)

x% 0<x<b 0, otherwise

( 6 - x ) ^ 0<x<b 0, otherwise

^ - " (α+χ) -^ 0<a;<oo 0, - o o < x < 0

x-'ix-b)% b<x<oo 0, otherwise

χ-% b<x<oo 0, - o o < a ; < 6

ia+x)-% 0<x<oo 0, - o o < r c < 0

[exp(e-^)-l]- ie-^^

Functione Not Belonging to Either of These Oasses 149

NGiy) Nfix)

31

38

16

36

40

17

18

10

11

15

14

(1_2ΐ-λ-<»)Γ(λ+ί2/)Γ(λ+ί2/) ΛΓ=(1-2ΐ-λ)Γ(λ){·(λ), λ > 0

r(H-iy)(a*-6»)-»"'

N=v-^Zar\-ia'-l^)ilr', a>b, v>0

^^^lTiu)rKh I y \/a)'-iKîa \ y |) N=h'ia^-^Tiv-^)[Tiv)r\ v>i

2ib/a)*''K{yi2ab) N=2Koi2ab)

2ib/a)*ysediiiry)Kiyi2ab) N=2Koi2ab)

2Ki,-iyia) Kif+iyib) N=2Ki.ia)Ki,ib)

ia-'y-Hirv CSCCHT) -J,iiay) ] - 1 } ΛΓ=α·+ν(ι^-1)-ι, v>l

ταΤ· cscinv) CJ,(taj/) —J,iiay) ] N=v-hr', v>0

2-»6-»r(i')2)_2,C2(%)i] iV=(26A)-»rwcr(H'')rS «'>0

2'α-»Γ(ΐ') e^D-i,Zi2iay) *] ΛΓ=(ΙΓΑ)*Γ«[Τ(Η.)Γ' , ' ' >0

2·^- ΐ5( , ί , v)e*'ΊFliμ·, Η-μ; -2 ί ί , ) N=2'+''-^Biß,v), ν,μ>0

(6_a)>^->(%)-M-'e-l*«'(*+*>

•M^_,,H-p-iP2/(&-ffl)] ΛΓ=(&-α) '^ ' -Τ(2Μ)

· Γ ( 2 ν ) [ Γ ( 2 μ + 2 ν ) ^ , μ . ' ' >0

Cexp(e-')+l]-»e-'*

exp(-oe ')/,(6e»)

[αΗ-(χ±6 )Τ'

expC-oV-feê-*)

expiaêH-Bê-*)!« •Erfc(aei*+öe-*»)

[(α+6β')(αβΗ-ί>)-^ϊ' .1Γ,[(οΗ-6*+2ο6 coshx)»]

C(a'+a?)»+a;]-', 0<x<oo 0, - ο ο < ί ; < 0

(αΗ-χ*)-*[(αΗ-χ*) »+χ]-', 0 < χ< «> Ο, - o o < a ; < 0

( χ - 6 ) - Η Η - 6 ) - ^ , b<x Ο, otherwise

ar-Ho+x)-'-», χ > 0 ο, otherwise

( ι -χ ) ' -»( ΐ+χ) ' ' - ' , - 1 < χ < 1 Ο, otherwise

(x-o)V-i(6-a;)s-i, a<x<b Ο, otherwise

150 Table IIIA: Functione Not Belonging to Either of These Classes

NG(y) Nfix)

13 (ο+6 ) ·^ΐΓ (2ν) (%) " ' ix+a)^-'ix-b)^\ b<x<oo

•expCiJi/(a-6) 2W^-y,^^,^[iyia+b) ] 0, otherwise N= ia+b)'i^^i2v)

^Til-2μ-2y)[Til-2μ)r\ 0<ρ<^-μ

3S ο-'2*'-'6-ί'[Γ(^)]-> exp[—6t«nh(aa:)] 'riiu+¥y/a)rih-¥y/a)Müy/a.i,-ii2b) •(sechoKe)', x>0

ΛΓ=Λ-Η^6)*-»'Γ(^^)7,^(6), ν>0 0, x<0

41 2*'(2τα2)-* expC— (β sinhx) *]D,(2ffl sinhx) .exp(a^) r[Kl+.'+iy) ] Γ [ | ( 1 + ρ - ί 2 / ) ] •coslhriv-iy)lW-i,iiyi2a^)

N=!&'i2Ta")-*expia") c o s ^ w )

•Cr(Hi )?Tr_j,.o(2a«), 0<i'<l

TABLE A

UNIVARIATE DENSITY FUNCTIONS

Probability density function No. fix)

Characteristic function

vit)= Γ e<^ix) dx • ' — 0 0

Notes

1 (2ir)-*exp(- |x*) (1) — ' » < x < O O

2 expiibt-ha^) (2)

— O O < x < O O , o > 0

3 (e«'-e<^)/t-(6-o)< (3) a<x<h, a0

5 l-\x\ (2/<«)( l -cosi ) -l<x<+l

153

154 Appendix

Probability density function No. i{x)


Notes

10

11

12

13

14

15

16

17

18

{l—eoax)/v3?

χ>α, a>0

(TCP)]-'»»^!«-* x>0, p>0

x>0, p>0, o > 0

Cr (p+3 ) /r (p )r (g )>^( l+e-) - o o < a ; < o o , p > 0 , g > 0

[2-»»/Γ(|η)>»"-»β-1' a;>0, n > 0 (integer)

Cx(i+x»)r' — < » < a ; < o o

a[waH-Kix-myj-^ — o o < a ; < o o , a>0

{2/τ){ί+χ")-' — 00 < χ < 00

(2Α)χ«(1+χ^)-ι — 00 < x < 00

— 00 <α;< 00

(2a)-ie--"'l*-^» — o o < x < o o , o > 0

Cr(p)]- 'exp(px-e*) - - ο ο < χ < ο ο , p > 0

i - M I , l < l i

ο/(ο-ώ)

ii-ü)-p

T{p+Ü)T{q-Ü)/np)nq)

exp{imt—a\ 11)

(1+| ί | ) «Γ ΐΊ

( l - | < | ) e - l ' i

(1+i^) - '

Tip+Ü)/T{p)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(11)

(12)

(13)

(11)

Table A: Univariate Density Functione 155

No. Probability density function

fix)


φ{ί)=Γ e»fix)dx Notes

19 ?>^i^/ar(rM)Cl+(x/a)? •exp[ -p (a : / a ) ]

- a < a ; < O O ; p > 0 , a > 0

(14)

20 Kl+|x|)e-'*' — ΟΟ<Χ<ΟΟ

(1+i*)-' (11)

21 exp(—X—e"*) — 00 < Χ < 00

r( l -Ä) (15)

22 (α/ΐΓ») exp[2a(6*)>-»{expC-6x-(ayx)]} x>0 ; a>0, 6 > 0

exp{2o[6»-(6-Ö)*]} (11)

23 J sechdirx) — ΟΟ<Χ<ΟΟ

secht (16)

24 Ksech ( | i rx ) i — 00 < Χ < 00

<csch( (16)

25 ix csch(J^x) — OO<X<OO

sech*< (16)

26 Cr(?^fg)/r(p)r(g)>^Hl-^)^^ 0<x<l; p > 0 , g > 0

(17)

27 — ΟΟ<α;<ΟΟ; m > 0

iih\t\)'^/irT(m-i)2K„^{\t\) (18)

28 — O O < X < O O ; m > 0

( l + i ) - - i (11)

29 x>0; m > 0 , p > 0

C2/r(p):(tm<)»''irp[2(-imi)i] (11)

30 [Γ(^^f(^)/Γ(p)Γ((^)>^Hl+^)-'^« x>0 ; p > 0 , (7>0

iFi(p; - g ; - Λ )

31 [T(?^f(z)/r(p)r(i)]e-^(l-6-)^i x>0, p > 0 , q>0

Γ(Η-9)Γ(ρ-ί<)/Γ(ρ)Γ(ρ+3-Λ)

156 Appendix



v{t) = fe'*'fix)dx Notes

32

33

34

35

36

37

38

39

40

41

— oo<a;<oo, p > 0

[2»r (m+n) ]-^ϊ»->ΤΓ„,η-}(23;) — οο<ί;<<», m>0, n>0

• exp(—αχ) /p_i[ö(x)*] x > 0 , α>0, p > 0

^Mp^χip-ir'I^^[2{L·)i2 x>0, b>0, p > 0

Xi-*«e-»^a*'-*e-*'/jn-i(Xx)» x > 0 , λ > 0 , n > 0 (integer)

C2-i^yr(p) > t a ^ W i ^ , . l ^ ^ ( 2 x ) — <»<χ<οο, p > 0 , g > 0

Ζ^/Τ{ρ)π*:ί^^Κ^{χ) — oo<x<oo; p > 0

[T(rH?)/r(p)r(g)](l+e-)-'-»e'» 0<X<oo; p>0 , g > 0

4-1 exp(-i62)2-iiM»a;b'-rt

— oo<a;<oo; p > 0 , g > 0

exp{-W)ir-i{hx)^

• Ε ί ^ ί ϊ ^ ) ^ - * - * ^ ^ )

— » < X < o o , p > 0

| r (p+it<) p [T(p) ] -*

(1+ίί) ·»-"(1-{()-»-«

^ - » ( i / a ) ! ^ •exp{ä(&/2a)«Cl-tX</a)]-i}

( 1 - ι 6 ) - ί β χ ρ [ ί δ < / ( 1 - Λ ) ]

(1-2Λ)-*» exp[i<X/(l-2i«) ]

(1-Λ)-*(1+ΐ<)-«

Γ (ρ+ί<)Γ (3-ώ ) /Γ (ρ )Γ (5)

( 1 - ί ί ) - ' ( 1 + ώ ) - « .expC - i6»iy ( l+i*)]

( l + i » ) - ' e x p [ - W ( l + < » ) ]

(11)

(11)

(19)

(20)

(21)

(22)

(23)

(22)

(21)

(21)

Table A: Univariate Density Functione 157



<pit) = fe'*'fix)dx Notes

42 (3/τ)ίχ-<»/«)'·ΤΓ|,,(Μχ-») e x p { - | f | i [ l - i \ ^ « / | < | ) ] } a;>0

43 i(3/ir)»x-» W-i,i(-^) e x p { - | < | » [ l + i « / | < | ) ] } x>0

44 ( 3 A ) » | x h e - e ' « " ' V j . i ( ^ - * ) e x p ( - | < | ' ) — «><x<«>

45 T-»3-i(2/3x)ii:|[i3»(2/3a;)i] e x p ( - | i | » { l - ( t / V 3 ) [ < / ( | i | ) ] } ) x > 0

46 (2x)-*e-J-x-i e x p ( - | < | i { [ l + C < / ( | < | ) ] } ) x>0

(24)

(24)

(24)

(24)

(25)

Notes: (I) Standardized normal distribution. (2) Normal distribution. (3) Rectangular or uniform distribution. (4) Symmetric rectangular distribution. (5) Khintchine's convex distribution, H. Cramer, "Mathematical Methods of Statistics,**

p. 94. Princeton Univ. Press, Princeton, New Jersey, 1946. (6) Exponential distribution. (7) Gamma distribution. (8) Chi-square distribution with η degrees of freedom. (9) Standardized Cauchy distribution. (10) Cauchy distribution. (II) S. KuUback, "Theory and Application of Characteristic Functions," Lecture notes. (12) Standardized Laplace distribution. (13) Laplace distribution. (14) Pearsonian type III distribution. (15) R. A. Fisher and L. H. C. Tippet, Proc. Cambridge Phil Soc. 24, 180 (1928). (16) J. Bass and P. Levy, C. R. Acad. Set, iPans) 230, 815 (1950). (17) Beta distribution. (18) Pearsonian type VII distribution. (19) R. G. Laha, BvU. Calcutta Math. Soc. 46, 60 (1954). (20) T. A. McKay, Biometrika 24, 39 (1932). (21) Noncentral chi-square with η degrees of freedom and a noncentral parameter λ. See

reference in note (19). (22) S. KuUback, Ann, Math. Stat. 7, 52 (1936). (23) K. Pearson, S. A. Stouffer, and F. N, David, Biometnka 24 (1932). (24) V. M. Zolotarev, Dokl. Akad. Nauk. USSR 98, 715 (1954). (25) B. V. Gnedenko and A. N. Kolmogorov, "Limit Distribution for Sums of Independent

Random Variables," p. 171. Addison-Wesley, Reading, Massachusetts, 1954.

TABLE Β

UNIVARIATE DISCRETE DISTRIBUTIONS

Probability (discrete) No. Vx


^(0=Σβ*'^Ρχ Notes

p y

x=Q,l, ...,n; 0 0 (integer)

a :=0,1, oo; m>0

x = 0 , 1 , oo; 0 0

( X - j / r r ) x = 0 , 1 , 2 , . . A f , Λ , η positive integers

i l + P { e " - l ) l r (1)

expCm(e<'-l)] (2)

C(l -P) / ( l - fe '0 ]» (3)

jFi ( - M , - n ; - M - N ; 1-«*') (4)

158

Table Β: Univariate Discrete Distributions 159

Probability (discrete) No. px

α, real, +{e)=EaJ^ ζ

θ real, x=0,1, 2, oo

- ^ A l o g ( l - ö ) a:=l,2, oo; 0<Θ<1

(20^+1) l o g C ( l W ( l - ö ) ]

a;=l,2, oo; 0<θ<1

(arcsinö)~S x= 1

1 · 3 · · · ( 2 χ - 1 ) 2-4··-(271) 2H-1

α:=1,2,3, . . . ; 0<Θ<1

(arcsinö)""^


log(l-i?e*0/log(l-ö)

log(l+ge*0-log(l-(?6*0 log(l+ö)-log(l-i?)

aresin (öe*0 arcsin(ö)

Notes

(5)

(5)

(5)

(5)

Notes: (1) Binomial distribution. (2) Poisson distribution. (3) Negative binomial distribution. (4) Hypergeometric distribution. (5) Α. Noack, Ann. Math, Stat. 21, 127 (1950).

TABLE C

MULTIVARIATE DENSITY FUNCTIONS


Probability density function No. / ( X l , ...yXk)

= / expp(iia;i, . . . , to*) ] •'—00

•/(xi, ..,,Xk) dxv'dxk Notes

IA |-i {2ir)-^^exvl-\{X-M)A-\X-M')'} X={xi, . . . , ^ λ ) ; ~«><a:y<oo,

i = l , 2 , . . . ,Α; M= {mi, mi, .,,,τπίο) A={aij);i,j= 1,2, . . . , n

symmetric positive definite

^-4*(*-i) / n r ( l n - i i )

ij Xij= Xji and Z = (xij), i,j=l,2,.,.,k

symmetric positive definite A={ai,j); i , i = l , 2 , •••,Α;

symmetric positive definite

expliTM'-iTAr^ T={ti, t2,.,.,tk)

T={eijij), i,j=l,2,.,.,k

" jo, i F^ i

(1)

(2)

160

Table C: Multivariate Density Functione 161

Probability density function No. f{xu...,Xk)

m

Xj> 0 (integer); j=l,2, ...,k

3

i = l , 2 , ... ,Ä;; E p y = l


= ί expCi(iiXi, •'—00

Notes

(3)

Notes: (1) Multivariate normal distribution. (3) Multinomial distribution.

(2) Wishart distribution.

LIST OF ABBREVIATIONS, SYMBOLS, AND NOTATIONS

€n = Neumann 's number, €o = 1, e« = 2, η = 1, 2, 3, . . .

7 = Euler 's constant, y = 0.57721 · · ·

Q = binomial coefficient, = Γ ( a + 1 ) / Γ ( 6 + 1 ) Γ ( α - 6 + 1 )

Τν,ι = first positive root of Jp{x), fv, i = first positive root of Yy{x)

N o t a t i o n s

1. E l e m e n t a r y F u n c t i o n s

Trigonometric and inverse trigonometric functions: sinx, cosx, t anx = sinx/cosx, c t n x = cosx/sinx, secx = l /cosx, cscx = l / s inx ; arcsinx, arccosx, arctanx, arcctnx.

Hyperbolic functions: sinhx = f(e^—e~^), coshx = i ( e * + e " ' ) , t a n h x = s inhx/ coshx, c t n h x = coshx/sinhx, sechx = 1/coshx, cschx = 1/sinhx.

2. G a m m a F u n c t i o n a n d R e l a t e d F u n c t i o n s

Gamma function: Τ{ζ)=ί e-^H'-^ dt, Re z>0

162

List of Abbreviations, Symbols, and Notations 163

Beta function: B{x, y) =T{x)T{y)/T(x+y)

φ function: ψ (ζ) = d[logT (ζ) ]/dz

3· R i e m a n n a n d H u r w i t z Zeta F u n c t i o n

00

ί ( « ) = Σ R e s> l n - 1

i{Syv)= Σ (n+v)"', R e s > l n - 1

4. Legendre F u n c t i o n s (Definit ion after Hobson)

rΛzXT{ί-μ)lΓ^\:(^+l)/{^-l)J'',F^i-y, . + 1; 1-μ;

ο ; (ζ) = 2 - ' -»[Γ ( ^ + § ) ]-ΐβ"ΜΓ (μ+ν+Ι)

·2Γ' ' - ' - Ι (2*-1)* ' 'Λ[Ι (Μ+' '+1) , | ( μ + » ' + 2 ) , v+i;r-^'], 1<ζ<«> P;(x )=[ r ( l -M ) ] - 'C( l+a ; ) / ( l -a ; )J ' ' ,F i ( - . , .+ l ; l -M; | -^x) , - l < a ; < l (ΤΛχ) =h e x p ( - i V M ) [ e x p ( - t W ) O ! ; ( x + i 0 ) + e x p ( i W ) O ! ; ( 3 ; - i ö ) ] ,

- l < a ; < l

5. Beesel F u n c t i o n s

Λ(ζ)= Σ (-l)»[n!r(^+n+l)]-i(i2)'+'» n -1

n(z) =Csin(w)]-»[J,(z) c o s M - / _ , ( 0 ) ]


I,{z)=expi-ihirv)Mze'')= £ [η!Γ(,;+η+1)]-ΐ(|ζ)'+*» n -1

K,{z) = i 7 r ( s m 7 r ^ ) - ^ / - . ( 0 ) - / . ( 2 ) ]

7. Anger-Weber F u n c t i o n s

Jp(z)=T-^ ί cosizsint—d) dt, (2;) = — ττ" ί 8ΐη{ζ sint—vt)

164 Appendix

8. Struve F u n c t i o n s

n-O

L , (z )= - i exp( - i | f lT )H , ( ze<» ' )

9. L o m m e l F u n c t i o n s

s,Az) = i{ß-p+l)iß+>'+l)l-'iF,il;h{ß-y+S), H M + ^ + S ) ; - ^ Z ' ] , μ±ν9έ-1, -2, - 3 , . . .

S . . . ( z ) = V . ( z ) + 2 ' - » r C H M - « ' + l ) ] r [ i ( M + ' ' + l ) ]

'{άnih1r(μ-y)y,{^)-cos[i1r(μ-y)lYΛ^)]

9a. Special Cases of L o m m e l F u n c t i o n s

s , . , ( 2 ) = , r i 2 ' - i r ( i + . ) H , ( 2 )

S,.,(z) = π * 2 ' - ΐ Γ ( | + . ) [ Η , ( 0 ) - Κ , ( 0 ) ]

SoAz) = i ^ c s c ( « ) C J , ( 2 ) - J . , ( z ) ]

So.,(2) =^ir csc(«)CJ,(z) - J . , ( 2 ) - Λ ω

S _ 1 „ ( Z ) = - § X „ - » CSC(X^)CJ , (2 )+J . , (2 ) ]

si,,(z) = 1 - | T V C S C ( X « ' ) C J , ( 2 ) + J . , ( Z ) ]

S I . , ( 2 ) = 1 + J T . C S C ( X . ) [ J , ( 2 ) + J - , ( Z ) - J , ( Z ) - J - , ( 2 ) ]

Si,i(z) =2-*; θι.»(2) =2»

<S-},±}(0) =2r*[sin2 Ci(z) — cosz s i(z)]

-S-l.iiCz) = -2-*Csiiu; si(z) +cos2 Ci(z) ] , 1 ΐ ι η [ Γ ( . — M ) ] - V I . » ( 2 ) = - 2 ' - » Γ ( , - ) / , ( Ζ )

Lommel functions of two variables:

U,{w,z)=i ( - 1 ) » ( Μ ; / Ζ ) ' + « - Λ + 2 „ ( Ζ ) n-O

V,{w, z) = cos (^U)+JzVtü+ivTr) + C / 2 - (« ' , 2)

Λ (αϊ, 02, . . . , «u; ίΊ. ί > 2 . . . . , f»n; Ζ )


10. Genera l ized Hypergeometr i c Series

Γ ( 6 χ ) . . . Γ ( 6 „ ) Γ ( α ι ) . . . Γ ( α „ )

" r(ai+k)'--T(cu+k) 'h T{h+k)"-T{K+k) k\

11. G a u s s i a n Hypergeometr i c Series

ipr κ ^ r(c) ' T{a+k)Tib+k) z" '^^^'^' ^' ^' ^) = näm^) £ r(c+Ä:) fc!

12. Conf luent Hypergeometr i c F u n c t i o n s

r(c) " T{a+k) z* iFiia; c; z) =

Γ(α) r(c+fc) k\

Whiitaker functions:

MkAz) =z''+k-i^F^i^+μ-k•, 2μ+1; ζ)

Parabolic cylindrical functions:

D,{z) =2»'+iz-iTr,^j.±l(|z^)

i)„(z) = (-l)»exp(iz*)(d"M")Cexp(-*2^)] = expC-iz*]iie„(z), n = 0 , 1, 2, . . .

Z)_i(z) = (IT)* exp(iz«) Erf(z2-»)

Ό.^{ζ) = {\ζ/τΥΚ^{\ζ^)

Incomplete gamma functions:

γ (a, z)=( i—ie-' d< = a-'2iFi(a; a + 1 ; - z )

•Ό

Γ(α, ζ) = j " " f-^e-" dt = T{a)-y{a, z) =ζί«-ίβ-ί'ΤΓ}„_}.ϋ<.(ζ)

166 Appendix

2ΐΓ-»2-1 e x p ( - §2«)

Erfc(2) =2ir-» J\xp(-e) d < = l - E r f ( 2 ) = ( i r « ) - * e x p ( - | z ' ) iF_i.±i(2«)

Erf(a;»e«') =2le*i ' [C(x)- i -S(a; ) ]

Erfc(xVJ') = 1-C(a;) - S ( a ; ) - 8{χ)2

Fresnel integrals:

Cix) = {2ir)-i f tricoBtdt

S(x) = (2ir)-» f irisintdt •Ό

Exponentiai integral:

-Ei(-2)=J*r>e-'d< = r(0,2)=2-*e-»'W^_,.o(2) = - 7 l o g 2 - £ ( n ! n ) - K - 2 ) '

Ei(a;) = lim r'e-'d<+/ r^e-' dt=y+logx+ Σ {ηΙη)-^χ''

E i ( - i a ; ) = C i ( a ; ) - i si(a;)

Εΐ(ώ) = C i ( a ; ) + C i r + s i ( x ) ]

x > 0

Sine integral:

Si{x)= ί ir^sintdt •'o

si(x) = - j <-i sini dt = Si(a;) -

Cosine integral:

Ci{x) = - Pr> cosi d< = 7+loga;+ £ [2n(2n) ! ] -» ( - l )"a?" •Ί n-1

Error functions:

Erf(2)=2ir-* Γβχρ(-Ρ) dt--J Λ


13. El l ipt ic Integra l s

K{k)^\ ( l - f c 2 s i n 2 0 - * * = |7r 2 F i ( ü ; l ; Ä : ^ )

E{k) = t {\^¥unH)Ut = \ir 2 F i ( - i , I; 1; fc^) •Ό

14. Besse l Integra l F u n c t i o n

Jio(a:)= -Tt-Uoit) dt = y+log{hx)+h Σ [n(n!)2]-H-l) '»( ia:)2-·'« n -1

Fourier Transforms of Distributions and Their Inverses: A Collection of Tables

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Transcript of Fourier Transforms of Distributions and Their Inverses: A Collection of Tables