fourier transforms of distributions and their inverses: a collection of tables
TRANSCRIPT
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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
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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
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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.
A C A D E M I C P R E S S , I N C . I l l Fifth Avenue, N e w York, N e w York 10003
United Kingdom Edition published by A C A D E M I C P R E S S , I N C . ( L O N D O N ) L T D . 24/28 Oval Road, London N W l
LIBRARY OF CONGRESS CATALOG CARD NUMBER: 73-182604
AMS (MOS) 1970 Subject Classification: 42A68
PRINTED IN THE UNITED STATES O F AMERICA
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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
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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
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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
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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.
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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.
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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
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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/).
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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
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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
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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
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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,
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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
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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.
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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
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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.
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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)-\ x<b 0, x>b N=hg(l+b/a)
0, x<b {a+x)-«, x>b n = 2 , 3 , 4 , . . . ΛΓ= ( o + 6 ) i - " ( n - l ) - >
0, x<b ix{a+x)lr\ x>b 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)
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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=^a- ' sec t?
a^Ca^+2aV cos(2t>)+o*]-i -§ir<t><Jir
N=ira-i
{a+x)-i Ν=2αή
x-^ia^+x")-*
( o — x < a 0, x>a
{a-x)-\ x<b 0, x>b
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, x<b X-l{x-b)-i, x>b 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)^I-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 ) :
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1. Algebraic Functions 19
2Nf{x) Ngiy)
22
23
24
25
26
27
29
30
31
32
0, x<b ix-b)-iix+h)-\ x>b N= (26)-»«·
0, x<b ix-b)-^ix+a)-\ x>b 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, x<b i(^-3?)-*, b<x<a 0, x>a 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-^ihay+hby)
ir(2o)-*e««'Erfc[(ai/)*]
2~ia-Huuiihay)Kiiiay)
-ihr)¥J-^ihay)Y^ihay)
h-'^-^[Joiay)H.i{ay)+H,(ay)May)2 (2a)-iirErfcC(oi,)»]
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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<b - hiy-'~H expC- tXw/2- by) ]7(>'+1, % ) 0, x>b, !>>-! -expp ( i ' jr /2-6i / ) ]7( i '+ l , - % ) } JV=6'+i(^+l)-i
39 3r-\b-x)''-\ x<b i6 '+^- '5(r ,M)[iFi(v;M+i' ;%) 0, x>b
v>0, μ>0 N=b'•^-'B(p,μ)
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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/)
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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 < i ' < § JV=^a-« ' i r-JR(i ' ) r(^-p)
0, x<2a {x^2ax)-'-^, x>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/)]}
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3. Exponential Functions 23
2Nf{x) Ngiy)
59 0, z<a -W^ihiry)* •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 -^ih+ihvM-^ih-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)
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2 4 Table I: Even Functions
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)»]}
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3. Exponential Functions 25
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-^^expi-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]
x^i 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*
x^iexp(—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)
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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}
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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;), x<b 0, x>b
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
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28 Table I: Even Functions
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), x<b 0, x>b
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 ' ) , x<b 0, x>b
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) ]}
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4. Logarithmic Functions 29
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)|]
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30 Table I: Even Functions
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 , . . .
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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=^airJiiab)
(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+^a+hy) Til+^a-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
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32 Table I: Even Functions
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^)]
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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
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34 Table I: Even Functions
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) :
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5. Trigonometric Functions 35
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
a< i r /2
(cosx)-* cos(2o cosx), x<ir /2 0, x>i 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)]
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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=^air
• Σ ( - 1 ) · ^ » ο ο 8 [ ( » η + | ) ( τ / η ) ] m - l
vy ^e~^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) ^ar^w sech(ia~^7ri/)
178 [sech{ax)y ^or^Ty cschi^ar^wy)
179 \ßech{ax)y N=iar'T
ia-^ir{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
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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
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38 Table I: Even Functions
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)
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7. Hyperbolic Functions 39
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<6< i r
iNr=|a-Vsec(f6)
{l+2cosh[(f7r)*x]}-i
(coshx+cos6)~*, 0 < 6 < π
(coshx+cosh6)~* iV=2*sech(i6)ir(tanh|6)
(coshx— cosö)-*, 0 < 6 < 7 Γ
(cosha— coshx)-*, x< a 0, x>a 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(^i//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-^^iy{cosb) sech(7ry)
2-*7r^_i+iy(cosh6) sech(7rt/)
2r^iQ-i+iy{co^)+Q-^iy{cosb) ]
2-^7r^_4+,j,(cosha)
2-*[0-Hti/(cosha) + 0 _ i _ t y (cosha) ]
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40 Table I: Even Functions
Ng{y)
211
212
213
214
215
216
217
218
219
220
[sech(aa:)]', i '>0
(cschoa;)', 0 < i ' < l
(cosha+cosha;)-', i '>0 N= (sinho)-O,M-i(ctnha)
[cos?>+ coshxjr', v>0, 0<6<i r
JV= (|ΙΓ)»Γ(Ι') (sm6)i-'PL7(co86)
(coshar— cosho)-', x> a 0, x<a
0<v<i N=i{2ir)h-'"{8iaha)i-'T{l-v)
•QL";*(cosha)
(cosha—cosha;)-', x<a 0, x>a
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/*)*], y<b
^TJoiaiy^-m y>b
hTloia{l^-m, y<b
2riirWi{haiy-{y'-m] •Ji{haiy+if-m
bKhr)*J-i{haiy-iy'-m} •J-iihiy+iy'-m
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7. Hyperbolic Functions 41
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<b Ar=|6-%csc(oi.- 'T)
(e^- l ) - i s inh(oa ; ) , o < 6 Λ Τ = | [ δ - ν c tn(aT/6)+a-»]
e-"[smh(6a;)]', > ' > - 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.
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42 Table I: Even Functions
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-':)
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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
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44 Table I: Even Functions
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[Jii^a)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-^ivia) -•^-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)K^i,ia) -Ki+n„ia)Ki-nyia)2
2-wi>)r'\mv+my/a)3\' •^I^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)]
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7. Hyperbolic Functions 45
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(2acos^a;), x<ir 0, a;>ir N=2{anmi-iia)y-iliia)7]
Csech(aa;) ] ' • cosh[6 sech(oa;) ] , v> 0
N=2r^a-^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- ' )
(^a)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'^')
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46 Table I: Even Functions
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
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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)^i(2»a)i)_^i(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=^expiW)Ko{W)
heMW) sech{^iry)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-^)
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48 Table I: Even Functions
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
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8. Gamma Functions and Related Functions 49
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.(«)
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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], x<b
{a^V^)-iK{i{a^-x^)/{a^-m], b<x<a
0, x>a
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>^ir N=2ria-Kp+i)-*
•Kmp+D-^l, ρ=(1+α-«)ί
sech(aa;)K[tanh(aa;) ] N=^a-hr-m^)y
h^iMhay)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
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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 < i 0, 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/)
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52 Table I: Even Functions
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, M<i -i<v<h ΛΓ=2Μ-ΜΓ(λ)
.{Γβα+λ-Μ+,')]
. Γ Β ( 1 + λ - Μ - ν ) ] } - ι
P,C(2a:ya«)-l] , x<a 0, x>a
-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 < i ' < 0 i V = - j 7 r - « s m ( w )
• [ Τ ( - ^ . ) Γ ( Η | . ) Ϊ
'!β,(ο coshx), α > 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 : }
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9. Elliptic Integrals and Legendre Functions 53
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-^i{r)2, r = ( l - a « ) i
%{1+2α^amk^x), α>1, - 1 < » ' < 0 iV=|Cocos (w ) r*
•m,{ap)J-lO-^i{ap)J], -l<v<0
'!ß,(2a«cosh«a;-l), α > 1 , - l < i ' < 0 , iV=ia-MPir*(«)<3-V(«)
Ο, sinhx< 1 / α 'i5,(2a«sinh«a;-l), smhx>l /a
- 1 < ί ' < 0 N=-^wcsc(vir)l%(ap)J
0 , ( α coshx), α > 1 , ί ' > - 1
N=2r-ia-ii^iT{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(^ir2/) •MPv'Hr)iQiHr)+Q^I!-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+%)]
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54 Table I: Even Functions
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
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9. Elliptic Integrals and Legendre Functions 55
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 < i 0 < a < l
Ν={^π)Κα'-ί)-*''αΤ^Τ{^-μ)
sechiirx) I Τ{μ+ΰ) p •PL-}q...,(a), - l < a < l , μ > 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 < i 0 < α < 1
Λ Γ = ( 1 - α « ) - * 0 - ^ [ ( 1 + α « ) / ( 1 - « * ) ]
leech{nx)JP-i+Ua)P-^U-a), α<1 Ν=τ-^Κ{α)
sech(Ta;)[<ß_j+«(a)J, ο > 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«(^) ]
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56 Table I: Even Functions
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
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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=i^a)'iwa/b)iJ^ihab)
-7ο [6 (αχ -χ« )*1 χ < ο Ο, χ>α
αδ<2ρο,, ΛΓ=-4τ- '6- ΐ [8Ϊη( |οδ) Οί(^α5)
-cos ( ia6) Si(^a6)]
- ( 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
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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) ]
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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
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6 0 Table I: Even Functions
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 ]
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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-^I,iax)K,iax), 0<ν<^ ΛΓ= W^^iu)Ti^-^) [Τ(Η2« ' ) Τ'
x-i'I,,ibx)K,iax), v>-l a>b N= ^^Ψα—hiTil+v)
.[r(i+M)r'2Fi(H^i;M+i; 6«/ο«)
iUax)J N=hi^a-'
Koiax) Koibx) N=-^ia+b)-^Klia-b)Ka+b)-^']
K,iax)K,ibx), -K»'<i iV=Ma6)-*secM
'^^Zia/2b)+ib/2a)2
xiiK,iax)J, -!<.<! ΛΓ=α-ί(27Γ)-»Γ«(|)
· Γ ( | + . ) Γ ( | - ν )
x-iiKXax)J, -\<y<i N=2-Kan)-in\)TiHv)Ti\-u)
xi-^K,ix)K,ix), f>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
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62 Table I: Even Functions
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 ) :
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11. Modified Beseel Functions 63
2Nf{x) Ng{y)
433
434
435
436
437
438
439
440
441
{a-x)^I,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-^amh{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={^ira/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
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64 Table I: Even Functions
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/]
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11. Modified Beseel Functions 65
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-^i 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>{^ab)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,{^ab)y}
I,Za{<^+^)*lK,Lb{<?+^)il a<b N=hrd'-'Iy{ac)K^ibc)
| 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
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66 Table I: Even Functions
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)
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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) ]
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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 cosh^o;)
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(^i/) 7ίν-ρ(α) - / ,v- , (a)F,v+.(ö)]}
Ky+iy{a)K^y(a)
^av-^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)
2r^a-^irW-^,^iy{2a')
Kiy{(wi'-)Kiy(os-^'^)
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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,^iy{a)Ki^iy{b) + Ki^iy{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
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70 Table I: Even Functions
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 ) * ]
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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-]]
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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«^)
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13. Parabolic Cylindrical Functions 73
2Nf{x) Ngiy)
525
526
527
528
529
(cosx)"*^ exp(—a" secx) •D,[2a(l+seca;)*l z < i i r
0, x>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/)»]
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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
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1. Algebraic Functions 75
1. Algebraic F u n c t i o n s
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, x<b ia+x)-^, x>b n=2, 3 , 4 , . . .
0, x<b [ x ( a + x ) ] - i , x>b
ia'+x^)-'
x-^ia+x)''
ia+x)-i
x-^ia'+:x^)-^
ia-x)-i, x<a 0, x>a
ia-x)-\ x<b 0, x>b, b<a
2y-^ m?iya/2)
4i/-2 uny ur^i^y)
i2T/y)^Siy)
smiay) [C\iay+by)'-C\iay) ] +cos(a2/) {ß\iay+by) — uiay) ]
Σ 7 7 7 7 ia+b)-^i-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' ^iiay)-^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
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76 Table II: Functions Vanishing Identically for Negative Values
Nfix) Nhiy)
18 (aä-r*)-! , x<b 0, x>b, b<a
^a-if eoaiay) [aiiay+by) - 2 si(ay) +si(a2/—6^)] -8Ϊη(αι/) [Ci(a2/+62/) - 2 C\iay) +Giiay-by)-\\
19 0, x<b ix'-a^)-\ x>b, b>a
eoaiay) [siiby—ay)—aiiby+ay) ] - s in (ay) [Ci(62/-at/)+Ci(62/-a2/) ]
20 0, x<b X-l(x-b)-i, X>b
6-iTCC(&i/)-S(6j/)]
22 0, x<b ix-b)-iix+b)-\ x>b
(26)-»T{ [Ci2by) - Si2by) ] cos(62/) -Cl-C(262/)-Ä(262/)]sm(62/)}
23 0, x<b ix-b)-^ix+a)-\ x>b
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
^oirJi(ay)
26 x-^icf-x")-^, x<a 0, x>a
( i ^ ) ¥ [ / j ( i a 2 / ) ?
29 x-^io?+:^)-^[_x^- (o2+r')*]-i 2r^a-^Te-^''Iii^ay)
30 0, x<a x-^i3?-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-^ia^-\-x')-*lx+ (a«+a^)i]-i 2*α-'sinh (lay) iCodaz/)
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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)', x<b Ο, x>b, ν>-1
3r-^ib-x)i^\ x<b Ο, x>b, 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
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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
x^ia^+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-^ia^-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'}, x<b
Ο, x>b
a'ihi-)¥h-i,iiay) KM.ihay)
ατ'π C8civjr){^I,iay) sin(|w) +hijyiiay)-hiJi-iay) 1
y~h~'[ Ι+ΐΊΓ csciw) ZIyiay) cosC xc) -^Uiay)-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)
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3. Exponential Functions 79
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-^ix'-a?)-^, x>a
~h<''<h
0, x<a x-^ix^-a^-^lix+in^-a?)*!'
+ Zx-i:^-a^iJ], x>a
2-'->α-ν*Γ (§-».) y'
•May)
Or-^a-'niTi^-v) •[J.iay) coaay—Y,iay) sm(oj/)]
yaecivir)a-'"y • [Η,(α2/) Yy-iiay) - y,(oj/)H^i(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 iv^a^+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) ]
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80 Table II: Functione Vanishing Identically for Negative Values
Nfix) Nhiy)
70
71
72
73
74
75
77
79
e-^il-e-^)--\ v>0
r-\^+i)-\ v>0
expi—ax')
x~* expi—aa^)
x^expi—ou^)
x" expi — ajc'), v> — l
expi-ax-l^x^)
80
81
x^^exp{—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) ErfC^iy/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
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3. Exponential Functions 81
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-'-^expi-\ayx), v>0
x-i exp[— ax— ib/x)^
a;~* exp[— ax— (6/ x) ]
x'-^expZ-ax- (6^«) ]
x-^expi-a^x-^)
exp(—ox*)
x~^expi—ax^)
x~^expi—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-^iia^+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/)]}
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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/)]
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4. Logarithmic Functions 83
Nfix) Nhiy)
107
111
\ogia-\-x), x<b 0, x>b
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<b 0, x > 6
6 < a < l
—log(o'—x^), x<a 0, x>a
a<l
- l o g ( a ' ' - r ' ) , x<b 0, x>b
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
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84 Table II: Functions Vanishing Identically for Negative Values
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
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5. Trigonometric Functions 85
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-^iha)I-Myiha) -h-^i\a)h^iha)-\
(§a)M sin(ix2/) U^iha)U^iha) +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 ) ]
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86 Table I I : Functions Vanishing Identically for Negative Values
6. Inverse Tr igonometr i c F u n c t i o n s
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;) -^a-M7rtanh(ia-i7r2/)
+imi+h-\) -φ^-Ια-'ν) ]}
178 [sech(aa:)]2 h'W(iia-'y)+^i-lia-'y) -φih+l^a-'y)-φii-ϊ^a-'y):\
182 χ csch (αχ) ^iα-W(H^iα-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)
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7. Hyperbolic Functions 87
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
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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<b
e"^[sinh(6x)]", v>-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 /
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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)*]
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90 Table II: Functions Vanishing Identically for Negative Values
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 ( < u 2 )
S(ax->)
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 i^iTii)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) B^iO-^iiyiP)l,
ρ=α+α-ψ
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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<i '<0
$,(l+2a''sinh2x), e > l ,
-1<«'<0
0, s inhx< l / a
^(2a*sinh2x-l), smlia;< 1/o, - l<i '<0
2^-ν*(1-μ->') (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, - i<i ;<0
Jo[6(ox-x^)»], x < o 0, x>a
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/) ] ,
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92 Table II: Functions Vanishing Identically for Negative Values
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(^ay)
'Q^+f)'^-^J.^W^+m
-2τΓ^ύη{\αν)ζ-^ ^[un{\az)[Ci{\oz+\ay) +Ci(ia2— ^ay) ]— 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{^az)[Ci{az)-^ Ci{haz+hay) - \ Qi{\az-\ay)+\og(\ah)-\ogz'\ -cos{\az) [Si(a2) - I ^\{\az^\ay) -\^i{\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«)
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11. Modified Beseel Functions 93
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<i
xt-'Kyix)K,ix), ί>\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
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94 Table II: Functions Vanishing Identically for Negative Values
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'-iJy^iW-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=^a(y*-62)4
y > 6 y < 6
—iir^ gec(iflT,) sin(^ay) G/jv(i2i) I'-jXiau) + Γ},(§2ι)/-,,(*22)1 ί />6
zx=ha[3,±iy'-m
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11. Modified Beseel Functions 95
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)~^Im{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)]} -^aa-^ sin{ay) {sina[Ci(2i)+Ci(22)] —cosa[si(2i)+si(22) ]}
a=a{fji'+f)i, zi=a±ay
K62+2/2)-*{cosaCCi(^i)-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{^iry) . [ /_^(a )J_ ,+ i j , ( a ) - / ^ ( α ) / . 4 4 ^ ( α ) ]
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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_^ii,(a) —/_,^iy(a) Yv-iy{a) —Jy+iy{a) Y-.^iy{a) +J-^iy{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')
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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
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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)
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Functione Not Belonging to Either of These Classes 99
Nfix) NGiy)
10
11
12
13
3r-^ii+x')-\ o<x<oo 0, — <»<x<0 0 < i ' < 2 N=^Tcacihirv)
ix-b)'-^ix+b)-'-*, b<x 0, otherwise
Λ Γ = ( 2 δ Α ) - ί Γ Μ | Τ ( Η . ) ] - ι
14
15
16
xr-^ia+x)~^, 0<x<«> 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)^^ix-b)^\ b<x<oa Ο, otherwise 0<ν<^-μ N= ia+b)^i^^i2»)
'ηί-2μ-2ρ)ίηΐ-2μ)Τ' ix-a)^-\b-x)^-\ a<x<b 0, otherwise M,>'>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=iTc^a^-^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 / | )
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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<b 0, otherwise N=c-^ie-^'-e-^')
0 < λ < 1 Ν=τα^-^ csc(irX)
xia+e-')'^e-^ 0 < λ < 1 iV=7r'a^"^ csc(irX) (loga—7Γ ctnTrX)
ia+e-^)-'ib+e-^)-h->^^ 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/)]
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Functione Not Belonging to Eitlier of These Classes 101
Nfix) NGiy)
27
29
30
31
32
33
34
35
36
e-^ia+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=i^a-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-^e-^l' sßchi^y/a)
2^'a-^iTiv) 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)
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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*)
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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
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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 ' ' ) ,
1. Algebraic F u n c t i o n s
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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-^
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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=^a-'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
^iry-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-^ix"^—cschx)
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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-'^amiay) N=a
1, x<a 0, x>a
172 iry~^e~^ 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 ^a~^ 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
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110 Table ΙΑ: Even Functions
Ngiy) 2Nfix)
380
87
2 coai^ay)z~'^ smi^az) N=%-^ami^ab), 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— ]
6. Inverse Tr igonometr i c F u n c t i o n s
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, x<b ix-b)'e-'"^, x>b
e-"'(logx)='
exp(—6x) sinh(aa;*)
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7. Hyperbolic Functions 111
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-^Eii-bx)
523 -ihr)*'r'iy+ia+m • cos{2i' arctan[j/»/(a+2/*) ]}
N=-i^)h-'a\ v<0
X ' ^ expi^ah) 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(^a-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-^ia'+f) sech(irj//2a) [sech (ax)] '
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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^^ cschi^iry) smh^i^y arcsina) iV'= (arcsina) 2, a < l
^iry"^ 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)
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7. Hyperbolic Functione 113
Ngiy) 2Niix)
183
225
357
182
67
191
192
271
362
435
442
516
«•6-1 co8(^a/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?)-^aulW-m, y>b
Ar=26-ismh(^a6)
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)
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114 Table ΙΑ: Even Functione
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) Icoai^a/c)]
c>a>b
1 , .
i V = - l o g
'cosh(^iry/&) +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?-\)-^oι^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)
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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 , . . .
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116 Table ΙΑ: Even Functione
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+^a)T\ a>-l
2-«jrcos(i2/)r(l+a) •iTil+^a+h)Til+ha-h)lr\
a>-l iV= 2 - v r ( l + a ) [TCl+^a)]-"
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=^ian)~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
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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 < i . ' ± 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
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118 Table ΙΑ: Eren Functions
Ngiy) 2Nfix)
237
153
152
184
187
294
281
2'-^iaTip)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'[^o(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'-'smi^irs)}
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;)
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8. Gamma Functions and Related Functions 119
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)
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120 Table ΙΑ: Even Functions
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
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8. Gamma Functions and Related Functions 121
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), x<b 0, x>b
log(a+x) , x<b 0, x>b
- log(o*-x*) , x<b 0, x>b
Ö-C(ax*)]cos(ox*) + β - 5 ( ο χ * ) ] 8 ί η ( α χ * )
Koias^)
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122 Table ΙΑ: Even Functions
Ngiy) 2Nfix)
385
379
386
383
451
456
-42 -Msm ( ia2)CCi(a2)- | Cii^az+^ay) Ci(§o«-Jay)+log(|f l*)-log2]
-cosi^az) [Si(o«) Sii^az+^ay) —I Siiiaz—^ay)} 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—^oy) ]—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
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8. Gamma Functions and Related Functions 123
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, x<b x-^ix-b)-*, x>b
(α—x)-*, x<a 0, x>a
x-iia+x)-'
0, x<b ix-b)-iix+b)-\ X>b
0, x<b ix-b)-*ia+x)-\ x>b
—sin(ax*) si(ax*) —cos(oai) Ciiaa?)
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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 < i ' < 0
x»(2a)'cr(-i.)rr'-'
Λ Γ = τ ί 2 ί ' - ^ Γ ( - § - | ν ) ο - > .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)
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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 )
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126 Table ΙΑ: Even Functions
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 < « ' < i
N=U(^)-iOy-iihMWa), a>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<i
hr8ecQlriμ-y)2Til+μ+v) .(a*+j/*)-*''-i{Pr[i/(oH-2/*)-»: +ΡΜ"Τ^ί/(«Η-ί^)-*]}
μ ± ΐ ' > - 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)
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9. Elliptic Integrale and Legendre Functione 127
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(^i,/6) ] JV=[6irsm(2air)]-S 0 < o < i
2*-«-<'α-ΜΓ (26) Γ (2c) Γ (H-c) . Csmh(ij//a) ]»-^iJ!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/)-*-*' ' •^iC(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
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128 Table ΙΑ: Even Functions
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
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9. Elliptic Integrals and Legendre Functions 129
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
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130 Table ΙΑ: Even Functione
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)
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9. Elliptic Integrale and Legendre Functione 131
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 )
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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-^i(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 ^ir'iWoiay)+8iiay)2 N=^a
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
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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-^i^Tiv)Ti\-v), v<\
[ 7 + log(22//a) y^iay)-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 ) -» , - | < ' ' < i
i7?-a')-'-i, x>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-^iV'-7?)-*[lih+xy+iib-xyj' -\-lib^x)*-iih-xyj'], x<b
Ο, x>b
P,i2i^/a'-\), x<a 0, x>a
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134 Table ΙΑ: Even Functions
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) -J^iay)Y^i{by)2
N= ( α δ ) Η 2 ί ' + 1 ) - ΐ ( α / 6 ) ' ^ -a>b
- iaV(§ir i / )§ •iJiy-i{hciy)Y-i.^i{hay) +J^^{lay)Yi^ihay)2
N={2r)-ia-iril-iv)T{\+hv),
-K«'<i hra-'y{ Uiiif/a) J+ iY^iW/a) 7] Λ Γ = Κ 2 α ) - ί Γ ( | ) [ Γ ( | ) ] - '
hra-'y{ i J i i m J- [F_i(ij/»/o) y\ iV=K2a)-*r(i)Cr(|)r
-h-KhryW-My'/a) Yi/,{^f/a) iV=2-«/V-»r(i)[r(i)]->
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-^ix^-a^-i
'{[x+ix'-a^ij + \:x-ix'-a^ij},
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
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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
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136 Table ΙΑ: Even Functions
Ngiy) 2Nfix)
392
503
502
391
478
396
(ia)/i(^n.j,) i^a) +JHy+i-v) (i«)«^i(H-i+i/) (i^) ]
N= hrav-'{ iJ^.-^iia) 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~^ Qiosi^iry)
(iö)«A(H-i+i/) ] N=\^av-'\lJ^^iha)J
+ [^iv+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 ^ < i
*π[/,,(^ΐ) Yiyiz2) -Jiyiz2) YiyiZl) ] Yoiz2)-Joiz2) Fo(2l)],
α<6<το,ι 2 i = i [ ( H - a ) * ± ( 6 - a ) i ]
sec(ix) •.7„(a cos|x), χ<7Γ
0, Χ>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
11. Modified Besse l F u n c t i o n s
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
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11. Modified Beseel Functions 137
Ngiy) 2Nfix)
29
339
46
426
74
422
497
75
431
438
443
2-*o-* smh(ioj/) Kii^ay) 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) N^ihr)*
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)-^iKiiW/a)J Ν=^π-Κ2α)-*[Τα)7
l{2a/y)-i expi-^a) U-iiWa) -hiW/a)!
ΛΓ=2-ία-1[Γ( |)]- '
-Ta-'yYili2ay)i2Kiii2ay)i2
ihra)Kob)'ilr'-v')-i^IMt^-m y<b
ikira)Kab)'iy'-l^)-i'-iJMy'-m y>b
N= i^na/b) io'7,+i(o6), v>-l
iira)i{hab)'cosi^ay) 'ii^-y')-^iHiihif^-m y<b .iy'-i^)-*'-^J.+iiW-m, y>b
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
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138 Table ΙΑ: Even Functione
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.^iab)
2a' coaiv arctan(j//6) ] KK [ O ( 6 H - 2 / ' ) * ] N=2a'Kyiab)
^Kyizi)Kyiz2) N=^iK,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'+x^i-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*)*]}
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11. Modified Beseel Functions 139
Ngiy) 2Nfix)
m
m 468
467
472
471
475
474
287
251
222
^Κο[α (2 coshj/)*] N=-h^Koi^a)
firK2,(2o cosh|j/) N=^irK2,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>-m^ia)J+U,^ia)y}, u>0
hr" csc(2in') [Ι_,^(α)7-Η^(α) -I^ia)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)
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140 Table lA: Even Functione
Ngiy) 2Nfix)
286 ^expiW)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 K^iyia)K,-iyia) N=iK,ia)J
Ki,i2a coshfx)
480 ^av-^lKi^y+iyia) K^p-iyia) —Ki-,+iyia) Ki-,-^yia) ]
N=>-M [Ä4+.(«) Ί - LKi-M J)
sech(5x) •X2»(2acosh5x)
488 Ki^iyia)Ki^yib) + Ki^iyib)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-'T^oiy) 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 ^o-^-V ββοίπν) {l-^airy[Jyiay)'B,^iiay) -H,(a?/)/,^i(oj/)])
0, x<a x-Hx^-a*) -*, x > o
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12· Functione Related to Beseel Functions 141
Ngiy) 2Nfix)
51 .Η^ι(οί,)3
xia'-χ')·^, x<a 0, x>a
112 h^Uoiay)-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{K^ia^)J
423 hra-Khry)iU-iiy'/a)-L^im2 JV=r l (o /T ) -» |T ( f ) r
x^Ktiax")
424 iT'iMKl-iilv'M-L-iiWa)! JV=2-*/Vo-»/tr(i)]-*
^K^ia^)
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,
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142 Table ΙΑ: Even Functions
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^,,^iy)
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!^^a)
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,^ia3?)
Klaiix)i2K,\:ai-ix)*l
/ ,{a[(6*+x*)i-x]} .ί:,{α[(ο*+χ*)1+χ]}
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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 » ' < i
K26)-»'exp[i(a»-t/)/6]r(.) • {exp(-itayA)ö-C(2&)-*(a-t2/) ] +expi\iay/b)D.,Zi2b)-iia+iy) 2}
N= (26)-»' exp(|ay6)r(»')Z)_,Ca(26)il i '>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)
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144 Table ΙΑ: Even Functione
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)
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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
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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
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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)]'
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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 < i ' < l
πδ-" csc(7n/) [Γ{ν) y'Tiv, iby) N=Trb-^''csciTv), 0 < v < l
{iy)^'Vil-v,iby) Ν=(ν-1)-ψ-% v > 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-^^
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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)'-iK^ia \ 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-fe^e-*)
expia^eH-B^e-*)!« •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
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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
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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, a<b
4 (2a)-i sin(a<)/o< (4) — a<x<a, a>0
5 l-\x\ (2/<«)( l -cosi ) -l<x<+l
153
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154 Appendix
Probability density function No. i{x)
Characteristic function
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 .0, 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)
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Table A: Univariate Density Functione 155
No. Probability density function
fix)
Characteristic function
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)
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156 Appendix
Probability density function No. fix)
Characteristic function
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)
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Table A: Univariate Density Functione 157
Probability density function No. fix)
Characteristic function
<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.
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TABLE Β
UNIVARIATE DISCRETE DISTRIBUTIONS
Probability (discrete) No. Vx
Characteristic function
^(0=Σβ*'^Ρχ Notes
p y
x=Q,l, ...,n; 0 < P < 1 , n > 0 (integer)
a :=0,1, oo; m>0
x = 0 , 1 , oo; 0 < P < 1 , n > 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
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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)""^
Characteristic function
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).
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TABLE C
MULTIVARIATE DENSITY FUNCTIONS
Characteristic function
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
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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
Characteristic function
= ί expCi(iiXi, •'—00
Notes
(3)
Notes: (1) Multivariate normal distribution. (3) Multinomial distribution.
(2) Wishart distribution.
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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
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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 ) ]
6. Modified Besse l F u n c t i o n s
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)
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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)
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Λ (αϊ, 02, . . . , «u; ίΊ. ί > 2 . . . . , f»n; Ζ )
List of Abbreviations, Symbols, and Notations 165
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)
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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
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List of Abbreviations, Symbols, and Notations 167
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