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INTEGRALS
AND
SERIES
VOLUME 4: DIRECT LAPLACE TRANSFORMS
A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev
Gordon and Breach Science Publishers
Digitized by the Internet Archive
in 2022 with funding from
Kahle/Austin Foundation
https://archive.org/details/integralsseriesO004appr
INTEGRALS AND SERIES
INTEGRALS AND SERIES
Volume 4
Direct Laplace Transforms
A.P. Prudnikov,
Yu. A. Brychkov
Computing Center of the USSR Academy of Sciences,
Moscow
O.I. Marichev
Byelorussian State University, Minsk, USSR
and Wolfram Research Inc., Champaign, Illinois, USA
property of
n isher College
Rochester, N.Y.
GORDON AND BREACH SCIENCE PUBLISHERS
Australia « Canada ¢ China « France ¢ Germany « India « Japan e Luxembourg
Malaysia « The Netherlands « Russia « Singapore ¢ Switzerland « Thailand
Copyright © 1992 OPA (Overseas Publishers Association) Amsterdam B.V.
Published under license under the Gordon and Breach Science Publishers imprint.
All rights reserved.
First published 1992
Second printing 1998
No part of this book may be reproduced or utilized in any form or by any
means, electronic or mechanical, including photocopying and recording, or by
any information storage or retrieval system, without permission in writing from
the publisher. Printed in India.
Amsteldijk 166
1st Floor
1079 LH Amsterdam
The Netherlands
Library of Congress Cataloging-in-Publication Data
A Catalogue record for this book
is available from the Library of Congress
CONTENTS
PREFACE
Chapter 1. FORMULAS OF GENERAL FORM
1.1.
1.1.1.
1.1.2.
1.1.3.
1.1.4.
1.1.5.
TRANSFORMS CONTAINING ARBITRARY FUNCTIONS
Basic formulas
/f(A(x)) and algebraic functions
J (~p(x)) and non-algebraic functions
Derivatives of f(x)
Integrals containing f(x)
Chapter 2, ELEMENTARY FUNCTIONS
2.1.
Peel let IE
2.1.2.
ald:
2.1.4.
2.1.5.
2.1.6.
INT
2.2.
2.2.
2.2.2.
22.3.
2.2.4.
2.2.5.
2.2.6.
24 P25I fe
2.3.
rdo1s
2.3.2.
THE POWER AND ALGEBRAIC FUNCTIONS
Functions of the form x’, @(+x = a)x’, [x]”
Functions of the form (x +z)", (a— x)!
Functions of the form x"(x +z)", x#(a— x)!
Functions of the form x#(x!/* +z)’, x#(a—x"/*)" for 1 Ak
Functions containing /x + z
Functions containing V x!/K + 2//k for lAk
Functions of [x]
THE EXPONENTIAL FUNCTION
exp (—ax!/") and the power function
exp (—ax~"/*) and the power function
exp (—ax*"/*) and algebraic functions
Functions of the form f(x, e~%,e7"*,e~, ...)
Functions containing exp (av? a5 P)
Functions containing exp (f(x))
Functions of [x]
HYPERBOLIC FUNCTIONS
Hyperbolic functions of ax
Hyperbolic functions of ax and the power function
xix
20
vi CONTENTS
Hyperbolic functions of
algebraic functions
Hyperbolic functions of
functions
Hyperbolic functions of
functions
Hyperbolic functions of
exponential functions
Hyperbolic functions of
and algebraic functions
Hyperbolic functions of
Hyperbolic functions of
function
Bets for [4k and
2 f
x +xz and algebraic
74 2
aitb”~+x~ and algebraic
ax, the power and
als for /#k, the power
[x]_
f(e ‘) and: the exponential
Functions containing the exponential function of
hyperbolic functions
TRIGONOMETRIC FUNCTIONS
Trigonometric functions
Trigonometric functions
function
Trigonometric functions
algebraic functions
Trigonometric functions
function
Trigonometric functions
functions
Trigonometric functions
functions
Trigonometric functions
exponential function
Trigonometric functions
of ax
of ax and the power
of bee for [#k and
“/k and the power
of 1 x?+xz and algebraic
2: FETE?
of attb Fx
of ax
and algebraic
of ax, the power and
of ig te for. 1k, the
power and exponential functions
Trigonometric functions
Trigonometric functions
exponential function
of [x]_.
of f(e) and the
Trigonometric and hyperbolic functions
THE LOGARITHMIC FUNCTION
In"(ax) and algebraic functions
in" (axt/*
algebraic functions
+b) and algebraic functions
Functions of the form ind x*! ma +4x7! m and
In”x, the power and exponential functions
The logarithmic function of f(e) and the
exponential function
The logarithmic and hyperbolic functions
The logarithmic and trigonometric functions
100
102
105
107
107
112
113
2.6.
2.6.1.
ZO:
2.6.3.
2.6.4.
2.6.5.
2.6.6.
2.6.7.
2a.
CONTENTS
INVERSE TRIGONOMETRIC FUNCTIONS
Inverse trigonometric functions of algebraic
functions
Inverse trigonometric functions of the
exponential function $I/k
Trigonometric functions of arccos(ax_’)
Trigonometric functions of arccos f(e*) and the
exponential function
arctan (ax*/*), arccot(ax*!/*) and the power
function _ =
arctan f(e “i arccot f(e*) and the exponential
function +I/k
Trigonometric functions of arctan(a« ” )
INVERSE HYPERBOLIC FUNCTIONS
Chapter 3. SPECIAL FUNCTIONS
3.1.
WW WWW WwW & nn RRDRD DD
Anh wown —
—
THE GAMMA FUNCTION [(z)
r"(x+a) and the power and exponential functions
The gamma function of [x]
THE RIEMANN ZETA FUNCTION (€(z) AND
THE FUNCTION 6(z,v)
€(n[x]+p) and various functions
€([x]+p,v) and various functions
THE POLYLOGARITHM Li, (2)
Li, (-ax’) and the power function
Li, f(e *)) and the exponential function
THE EXPONENTIAL INTEGRAL Ei(z)
Eitax’S and the power function
Filan) the power and exponential functions
Ei(f(e *) and the exponential functions
Ei(+ax) and trigonometric functions
Ei(tax) and the logarithmic function
Products of Ei(tax) and the power function
THE SINE si(z),Si(z) AND COSINE ci(z) INTEGRALS
sian ae Silat Ni cea) and the power
function
si(f(e*)), Sif(e*)), ci(f(e”)) and the
exponential function
sitaxt”); ciax*!*) and hyperbolic functions
+1/k
sitax””, ci(ax) and trigonometric
functions
Vii
114
114
116
117
119
120
122
124
126
127
127
127
128
130
130
131
131
13]
132
134
134
137
138
142
142
143
144
144
147
149
153
CONTENTS
si(ax™!), Si(ax*'), the exponential and
trigonometric functions
ci(ax) and the logarithmic function
Products of iar) and Biehl
THE HYPERBOLIC SINE shi(z) AND COSINE chi(z)
INTEGRALS
shi(ax!”*, chi(ax!/*) and the power function
shi(f(e*)), chi(f(e*)) and the exponential
function
; 1/k ; 1/k , :
shi(ax ’"), chi(ax ") and hyperbolic functions
shitann chien) and trigonometric
functions
chi(ax) and the logarithmic function
THE ERROR FUNCTIONS erf(z), erfc(z), AND erfi(z)
The error functions of ax+5 and the power
function
Wk +b.
he or of avx + b/Vx
r +1/k
The error “functions of ax and the
exponential function ae
The error functions of e~ and the exponential
function
The error functions and hyperbolic functions
The error functions and trigonometric functions
The error functions and the logarithmic function
The error functions of ax
The error functions of ax
erf (ae sys the exponential function and inverse
trigonometric functions 1/k
Products of the error functions of ax _
Products of the error functions of f(e *)
}The error functions and the exponential
integral
THE FRESNEL INTEGRALS S(z) AND C(z)
stax, ciax 4) and the power function
S(f(e~)), C(f(e~)) and the exponential
function
Stax yy Cast!’ and hyperbolic functions
+1/k +1/k : F ,
S(ax ), Clax ) and trigonometric functions
THE GENERALIZED FRESNEL INTEGRALS S(z,v)
AND C(z,v)
+1/k
S(ax* Vv), Caxt* yy and the power function
S(f(e*),v), C(f(e*),v) and the exponential
function
159
160
160
160
160
163
165
167
168
169
169
171
174
177
180
186
193
200
201
202
204
205
205
205
207
209
213
217
2d
219
3953!
3.9.4.
3.10.
3.10.1.
3.10.2.
SeOrS:
3.10.4.
Sa0:5;
3.10.6.
SHIOSTe
3.11.
Stoll toile
ab Ee
Sel bias:
3.11.4.
Sululeos
Salil:6-
3.11.7.
3.12.
ayaa
SUZ.
el i ato &
3.12.4.
Sm2.5.
JaZ.6;
Bu 2:7:
3.12.8.
3.12.9.
3.12.10.
CONTENTS
Stax yy,Crax!/*y) and hyperbolic functions
stax! * yy, Craxi!*y) and trigonometric
functions
THE INCOMPLETE GAMMA FUNCTIONS [I\v,z)
AND y(v,z)
T,ax), (v,axt 5 and the power function
I'([x]+v,a), y([x]+v,a) and various functions
Tw,axt", (v,axt and the exponential
function
Tv, f(e*)), yw, f(e*)) and the exponential
function
Baas 7). y(v,axt*y and hyperbolic
functions
Tier”. (vax) and trigonometric
functions
Products of Tv,ax and
y(v axel!)
THE PARABOLIC-CYLINDER FUNCTION D(z)
D (aVx) and the power function
D tx) (a) and various functions
D tax!) and the exponential function
Dif (e ‘)) and the exponential function
D (a¥x) and hyperbolic functions
D (aVx) and trigonometric functions
Products of D (ax)
THE BESSEL FUNCTION J (z)
J (ax) and the power function
Jax’) and the power function
J (ax) and the power function
Jax) and the power function
J, (af x? 4x2 } and algebraic functions
J, (al +b 45x 2) and algebraic functions
ey (ax"') and the power function
ne) and the exponential function
Jax”) and hyperbolic functions
J (a(sinh x») and hyperbolic functions
J (ax
4
J (ae
Vv
of
+1/k
+lx/k
+lx/k
e
CONTENTS
) and trigonometric functions
) and trigonometric functions
Jo (x)) and the logarithmic function
J (ae *) and inverse trigonometric functions
J (ax)J (bx) and the power function
J (axe! 7 (ox in) and the power function
+lx/k +lx/k
dj u (ae
function
)J (ae ) and the exponential
THE NEUMANN FUNCTION Y_ (z)
Y (ax) and the power function
Yx(ax
v
) and the power function
-l/k
) and the power function
Y (f(x)) and algebraic functions
-b/x
e€ Y (ax™') and the power function
Y¢f (e “)) and the exponential function
Y (ax) and hyperbolic functions
Y (ax") and trigonometric functions
Ya(ax
Vv
THE HANKEL FUNCTIONS H‘ (2), H (2)
i)
v
H
+
(ax
t1/k +
) J tax*!*) and various functions
Bhs and the power function
HO? @ xan z) and algebraic functions
THE MODIFIED BESSEL FUNCTION 1, @
I (ax) and the power function
I/k
I (ax ) and the power function
1s (af x? 8.5 ") and algebraic functions
2
Is (a Se Sty 2) and algebraic functions
+1/k
exp(-bx™ )I. (ax ) and the power function
Ife") and the exponential function
Te (ax
Vv
m/2
) and hyperbolic functions
I (ae) and hyperbolic functions
I (ax
Vv
m/2
) and trigonometric functions
277
280
281
282
283
293
296
298
298
300
301
302
304
304
305
307
309
311
311
312
313
3
317
321
322
324
326
329
331
331
CONTENTS
4 ety
+1x/k
ee
I,(f(~)) and the logarithmic function
) and trigonometric functions
i; (ae *) anes gRNETSS trigonometric functions
m (ax ES (bx!! ‘) and the power function
je pie yr, (ae) and the exponential function
tat! 1 fax ) and the power function
I (ax)I (bx) and the power function
I (ax!1 (ox! and the power function
1 Ge) Cae") and the exponential function
THE MacDONALD FUNCTION K (2)
K ae? end the power function
Kas! » and the power function
K (ax! as and the power function
Be @ x 4x2 ) and algebraic functions
K,, (al x 125 7 and algebraic functions
exp(tbx"’)K (ax) and the power function
K (f(e*)) and the exponential function
K (ax) and hyperbolic functions
K yf) and hyperbolic functions
K. yar) and trigonometric functions
i stax! OK, (bx :
y (ax OK, (ox a and the power function
. and the power function
1 tax! yk (ox! fy and the power function
I/k I/k
K lax )K. (bx) and the power function
THE STRUVE FUNCTIONS H, (z) AND L, (z)
H (ax!) Ye L ax” alls and the power PariGtion
H(f(e*)), L (f(e*)) ‘and the exponential
function
1/k 1/k ‘ F
H (ax), L (ax) and hyperbolic functions
/kisve
H (ax
Grdctions
H, (ax) and the Bessel function J, (ax)
Ye ‘ax!’ i — H (ax and the power function
»S L (ax!) and trigonometric
xi
334
335
336
337
339
340
341
346
349
349
349
352
353
355
356
357
359
363
364
365
366
368
368
370
370
370
375
377
381
383
384
xii
Soldat:
OnE:
SoL7.9:
3.18.
3.18.1.
Salse2:
3.18.3.
3.19.
Spe he
3.19.2.
5:19.3.
3.19.4.
3319.5.
3.19.6.
S97:
3.19.8.
3.19.9.
3.19.10.
3.20.
320.1,
3.20.2.
3.20.3.
CONTENTS
Yo (e*)) - H(f (e *)) and the exponential
function
Ba > and the modified Bessel function
Tax)
Ue L(f(e *)) and the exponential
function
THE ANGER FUNCTION JZ) AND THE WEBER
FUNCTION E (2)
Jax", B ax") and the power function
J (ax), E, (ax) and hyperbolic functions
J (ax), E(ax) and trigonometric functions
THE KELVIN FUNCTIONS ber (2), bei, (z), ker, (z),
kei_(z)
Uk oo UR :
ber (ax Ne bei, (ax ) and the power function
ber (ae), bei, (ae ™*) and the exponential
function
ber (ax), bei, (ax!/*) and hyperbolic
functions
ber (ax!/*), bei, (ax!) and trigonometric
functions
Products of the functions ber (ax'’),
bei, (ax), ber’ (ax'/, bei’ (ax! /*)
ker (ax), kei (ax!/*) and the power function
ker (ae~”"), kei (ae~””) and the exponential
function
1/k
ker, (ax Ne kei (ax!/*) and hyperbolic
functions
1 : ‘ 2
a kei (ax!) and trigonometric ker, (ax
functions
The Kelvin functions and the logarithmic
function
THE AIRY FUNCTIONS Ai(z) AND Bi(z)
: I/k 2
Aitax” Ns Bi(ax!”* and the power function
; I/k 3
Ai(ax!! ), Bi(ax!*) and the power function
Ai(f(e *)), Bi(f(e*)) and the exponential
function
384
386
387
390
390
391
392
392
392
394
395
397
398
401
402
404
404
405
405
406
407
408
3.20.4.
3.20.5.
3.21.
3.20.1:
521.2.
3221).3.
3.22.
J 22.0.
3.22.2.
3.22.3.
3.22.4.
S220:
3.22.6.
3.23.
3.23015
D.23.2.
323.3.
3.23.4.
3.23.5.
3.23.0.
3.24.
3.24.1.
3.24.2.
3.24.3.
CONTENTS
Products of the Airy functions and the power
function
Products of the Airy functions and the
exponential function
THE INTEGRAL BESSEL FUNCTIONS Ji,(2), Yi,(),
Ki (z)
Ji (ax tl/k
v
+1/k
M Yi (ax ), Ki fax", and the
power function
Jitax™), Yi(ax”), Ki,(ax””) and
hyperbolic functions
Ji (ax'"'), Yi (ax), Ki (ax) and trigonometric
functions
THE LEGENDRE POLYNOMIALS P, (z)
P (axe) and the power function
P(f(x)) and algebraic functions
BM (e *)) and the exponential function
Pig (y) and various functions
P (cosh ax) and P (cos ax)
Products of Pig (x)) and the power function
THE CHEBYSHEV POLYNOMIALS 7 (z) AND U_ (2)
Tee) and algebraic functions
T ,F@)) and algebraic functions
pm (e ~)) and the exponential function
U eax) and algebraic functions
U_(f(x)) and algebraic functions
U fe *)) and the exponential function
THE LAGUERRE POLYNOMIALS Li@)
Ly (ax’) and the power function
L*(ax.’) and the power function
+1/k
L\ (ax ) and the exponential function
xiii
411
412
413
412
416
416
419
419
420
423
424
425
425
425
425
427
428
429
430
431
431
431
435
437
Xiv
3.24.4.
3.24.5.
3.24.6.
3.24.7.
3.24.8.
3.25.
Si apie
Sion
e293:
3.25.4.
S250).
325.0.
SPITE
3.25.8.
3.25.9:
3.26.
320.1.
32622:
3:20.35;
3.26.4.
SEZOr0:
3.27.
S271.
ent. 2.
3.28.
3.28.1.
CONTENTS
Lae) and hyperbolic functions
v +m . . :
L (ax ) and trigonometric functions
L} (ax) and Bessel functions
+m/2
Products of L\ (ax ) and the power function
ie (y) and various functions
THE HERMITE POLYNOMIALS H @)
H (ax”’) and the power function
+1/k : ‘
H (ax) and the exponential function
H, (aes and hyperbolic functions
H ax”), the exponential and hyperbolic
>
functions
+m/2 : ‘ ;
H (ax ) and trigonometric functions
+m/2 : . ,
H (ax ), the exponential and trigonometric
functions
Products of H (avx) and the power function
A ngen™ and various functions
Products of Aig (y) and various functions
THE GEGENBAUER POLYNOMIALS C’ (2)
Z
Can) and the power function
CVF) and algebraic functions.
CUE) and the exponentialfunction
Vv
[x]
Products of Ci)
C..(y) and various functions
THE JACOBI POLYNOMIALS Be (a)
pit)
n
(pu, i 5
Pra (y) and various functions
(f(x)) and algebraic functions
THE BERNOULLI B (2), EULER E,@ AND NEUMANN
O,,(2@) POLYNOMIALS
+
B tax’), Big (y) and various functions
438
44]
443
444
446
448
448
450
453
455
455
458
460
461
461
461
462
465
467
468
468
468
474
476
476
3.28.2.
3.28.3.
3.29.
3.29.1.
O.L9.2.
3129.3.
3.29.4.
o:29..
3.29.6.
3.29.7.
3.30.
3.30.1.
390.2.
3.30;3;
3.30.4.
3.31.
33h.1.
3201.2.
$.31.3.
3.31.4.
Sah Os
3.32.
oz.)
5 Poy ie
3732.35
3.32.4.
3.32.5.
CONTENTS
Ez. (ax* oy. Ex 1 and various functions
O. (ax* ") and the power function
THE BATEMAN FUNCTION _&_ (z)
k, oe a the power function
hy (ax! ‘ and the exponential function x
ra +1x/k
ae
k (ax) and hyperbolic functions
) and the exponential function
k (ax) and trigonometric functions
Products of k (ax) and the power function
Products of k (ae)
THE LAGUERRE FUNCTION L (2)
L, (ax) and the power function
L tax*! 5 and the exponential function
L (ax) and hyperbolic functions
L (ax) and trigonometric functions
COMPLETE ELLIPTIC INTEGRALS D(z), E(z)
AND K(z)
Diax*), Bax
function
D(f(x)), E(f(x)), K((x)) and algebraic
functions
Dif(e*)), E¢(e)y, Kf(e*)) and the
exponential function
D(f(x)), E(f(x)), K(f(x)) and hyperbolic
functions
D(x), Ef), K(f(~)) and trigonometric
functions
THE LEGENDRE FUNCTIONS OF THE FIRST
KIND P'(z)
Ba, (x)) and algebraic functions
ene K (ax*”* and the power
Bog (e *)) and the exponential function
Py (e~*) and various functions
P* (cosh x), the exponential and hyperbolic
functions
Pe re and various. functions
XV
477
477
478
478
479
480
481
481
482
483
483
483
484
484
485
485
485
487
489
489
491
492
493
497
499
501
502
Xvi
3.32.6.
3.33.
Sy avoulle
3r3322.
BRA Bs
3.34.
aay baile
3.34.2.
3.34.3.
3.35.
Shayla
Ska) eae
Shobak
3.35.4.
Saher
3.35.6.
Dont
3.36.
3.36.1.
a 00r2:
3.36.3.
3.36.4.
3.56:9.
3.36.6.
CONTENTS
Products of Pi(f(x))
THE LEGENDRE FUNCTIONS OF THE SECOND
KIND Q'(z)
O: (f(x)) and algebraic functions
Q" (f(e*)) and the exponential function
Q" (f(x)) and various functions
THE LOMMEL FUNCTIONS sy) AND s. y?)
taxt!4. s axtll#
B,v bv
function
S lax), S (ax!/*) and hyperbolic
pv Hv
functions
Ss lax), S fax! and trigonometric
pv Hv
) and the power
functions
THE KUMMER CONFLUENT HYPERGEOMETRIC
FUNCTION iF, (asbs2)
iF (a;b;0x/) and the power function
FP (Gosh), the power and exponential
functions
|, (atm[x] ;b+m[x];o) and various functions
F (ajbj0xe?) and hyperbolic functions
40 (a;b;0x"”"”?) and trigonometric functions
jf (G;0x) and various functions
Products of pF, (45b30x)
THE TRICOMI CONFLUENT HYPERGEOMETRIC
FUNCTION YW (a,;z)
V(absoxt! and the power function
YW (a,b,f(x)) and the exponential function
¥(a,b;f(e*)) and the exponential function
W(a,b,0x~”) and hyperbolic functions
W(a,box"”), the exponential and hyperbolic
functions
+
W(a,b;ox”) and trigonometric functions
502
502
503
503
504
504
504
506
507
508
508
512
513
514
S15
516
syle)
517
S17
520
522
527
528
529
CONTENTS
+
W(a,b;wx"") the exponential and trigonometric
functions
Products iF 1 (asbsax” \¥ (a,b;-0x!" and the
power function
Products iba (a;b;-we*) ¥ (a,b;we*")
Products of ¥(a,b;0x!", the power and
exponential functions
Products of W(a,b;we~*) and the exponential
function
THE GAUSS HYPERGEOMETRIC FUNCTION
F (a,b3c;z)
z +1/k
oF (a,b3¢;-ox ) and the power function
pt oss (a,b;c;f(x)) and algebraic. functions
oF (a,bicif(e *)) and the exponential function
THE GENERALIZED HYPERGEOMETRIC FUNCTION
zit and the power function F ((a_)3;(b_);wx
mn m n Lg
mk n(4,,)3,)sf(€ “)) and the exponential
function
mf 164, + [4]; ,)+ [4] 50) and various functions
THE MacROBERT E-FUNCTION E(u;a 036 :2)
(a)
(b,)
G-function and the power function
G-function and the exponential function
G-function with [x] in parameters
Products of G-functions
THE MEIJER G-FUNCTION Ge f
THETA-FUNCTIONS 6((2,9), 8 (2,9)
8(aVx,9), 8(v,e *)
8,(v,ax)
THE FUNCTIONS v(z), v(z,Q), m(z,A), “(z,A,Q),
v(ax”"“),v(e ), the power and exponential
functions
vax? 9), v(e ”,9) and the power function
XVii
530
530
531
532
532
533
533
535
ays)
546
546.
554
557
558
559
559
560
562
563
564
564
565
566
566
566
XViii
3.42.3.
3.42.4.
3.42.5.
3.43.
3.43.1.
APPENDIX. ELEMENTS OF THE THEORY OF THE LAPLACE
CONTENTS
a a) and the power function
Gar 20S and the power function
A(ax',o) and the power function
THE CONFLUENT HYPERGEOMETRIC FUNCTIONS
OF TWO VARIABLES
Confluent hypergeometric functions and the power
function
TRANSFORMATION
The Laplace transform and its basic properties
The application of the Laplace transformation to the
solution of differential and integral equations
Some comments and references
BIBLIOGRAPHY
LIST OF NOTATIONS OF FUNCTIONS AND CONSTANTS
LIST OF MATHEMATICAL SYMBOLS
567
568
568
568
568
571
571
584
599
601
607
619
PREFACE
This is Volume 4 of the series Integrals and Series. It contains tables of
direct Laplace transforms
co
F(p) = [r (x)e *dx
0
and includes results. previously published in similar books and journals.
However, many of the results were obtained by the authors and are being
published for the first time.
Volume 5 of this series contains tables of inverse Laplace transforms
and tables of factorization of various integral transforms.
The Laplace transformation is used extensively in various problems of
pure and applied mathematics. Particularly widespread and effective is its
application to problems arising in the theory of operational calculus and in
its applications, embracing the most diverse branches of science and
technology. An important advantage of methods using the Laplace
transformation lies in the possibility of compiling tables of direct and
inverse Laplace transforms of various elementary and special functions
¢ommonly encountered in applications.
In this volume the tables are arranged in two columns. The left-hand
column of each page shows original f(x) and the right-hand column gives the
corresponding image F(p). The main text is introduced by a fairly detailed
list of contents, from which the required formulas can be easily found.
A number of Larlace transforms are expressed in terms of the Meijer
G-function. When combined with the table of special cases of the G-function
[82], these formulas make it possible to obtain Laplace transforms of
various elementary and special functions of mathematical physics. Some other
PREFACE XX
formulas, in particular Laplace transforms of general form and those of
piecewise-continuous functions, can be found in [80-82].
For the sake of compactness, abbreviated notation is used. For example,
the formula
erf (ax) 2 [Oh =ao(elew(ta] erfc (ax) P 4a
Re p>0; |arga|<n/4
jarg a|<n/4
is a contraction of the two formulas
2
erf (ax) A exp ae
4a
Pp 2 2a
erfc [2]
[Re p>0; |arg a|<n/4]
(in which only the upper sign and the upper expression in the curly brackets
are taken) and
D
re | Bs ps, erfc (ax) D teo(2| erte( 55]
[|arg a|<n/4)
(in which only the lower sign and the lower expression in the curly brackets
are taken).
References to formulas written in the form 3.7.1.1. denote Formula 1 of
Subsection 3.7.1.; unless other conditions are indicated, k,/,m,n=0,1,2....
The Appendix contains a short survey of the theory of the Laplace
transformation, and examples of its applications in problems of differential
and integral equations.
The bibliographicsources, notations of functions, constants, and
mathematical symbols are listed at the end. of the book.
We would be extremely grateful to any readers who draw our attention to
oversights, which are inevitable in a work of this size.
Chapter 1. FORMULAS OF GENERAL FORM
1.1. TRANSFORMS CONTAINING ARBITRARY FUNCTIONS
1.1.1. Basic formulas
1. F(p) = feMrooae
0
ytioco
1 px
Ih ora J e F(p)dp F(p)
Ye
1.1.2. f(A(x)) and algebraic functions
1 pfe 1. f (ax) a F(2}
[a>0]
a
2. f(xta) e”? [F (p) vale, “F nds]
0
{a>0]
b
3. f(ax+b) cee fa [F [2] - foxes cds]
0
[a,b>0]
4. @(x-a)f (x-a) e PF (p)
[a>0]
. 0, x<b/a ate (2)
f(ax+b), x>b/a a
[a,b>0}
10.
11.
12.
13.
14.
15.
16.
. f (x+a)=f (x)
- 7 \x+a)=-f (x)
- F([x])
ee
x "f (x)
8(a—x)x’ f (x-a)
f (x?)
xf (x*)
at (x?)
xi (x”)
fix!)
FORMULAS OF GENERAL FORM
a
(eee a fePreax
n-1
[-[Fo@an" - {4-2} Fydu
Dp ?p
v+1 0 (v+l) ,,(-ap)/ath,
Cd et he (v+2) , Bu
wal
co
[re v>-l; f-S hx |x\|<r, r>00|
k=0
4u
a 2
f= [u??exp (? a F(u)du
4vn 4 4u
l/2 (one op a ea pu l
BG fu e H,( 9) F i}
0
co
po ae (2Vpu)F(u)du
0
TRANSFORMS CONTAINING ARBITRARY FUNCTIONS 3
ie tae) [4,2vPwFwdu
0
18. x°f(x"') pee i uy Ovpu)Fwdu
a 0
[Re v>-2]
19. vals [Pp reoax
n=1 . 0
yric
20. f, (f(x) — [ F,@F-2dz
2ni pe
aie f(p(x)) and non-algebraic functions
1. e F(x) F (pta)
2. e *F (bx) $F [254]
[b>0]
(0) 2 Key) “a (=v) OEE)
3. O(x-a) X ere) es ah
P hig (Pt1) i yy j! k
Se (ler) FO)
co
E v>-1; Re p>0; fi) he’ Je ‘<r, roe)
k=0
co
5s 1 p —un{u
4. f(ae -a) aripety |e (4) du
0
[a>0]
5. f(a sinh x) [4,(qurF w du
0
[a>0]
4 FORMULAS OF GENERAL FORM
6. sinh ax f(x) 4 LF (p-a) ~ Fip+a)]
7. cosh ax f(x) 4 [F (p-a) + F(p+a)]
8. sin ax f(x) 7lF (-ia) - F(p+ia)}|
9. cos ax f(x) iF (p-ia) + F (p+ia)]
quite MbI2)"
(y+vt+1) 0 (v4+1)
oo CRY Te) oe eet
10. 8(a—x)x" x x
(=ap)’
jecing PTVF2ZY oy
k Dare 1
a (-a’b°/4)
= iis wae
co
[Retwrn>-t fo) ne |x| <r, r>00|
k=0
(Gids Vee
11. 8(x-a) x ~(p+vy Preval) *
- oo (pt+v) CS) =
x d-e*)" x x Bt aN f__ hn (p+v+1) Vanenou (v+1) 1
x J (be F(x) (lata hed PE ipneme “7 4), ee k
[Retr fn-V Hoe le ‘I<r, r>e°; oo]
k=0
x j
12. (1-e™*)" x cl pa pee es
prutl|,e-o (PUTT) 5,4) j7!
X oF (esbe Ff) [re u>-1; Re p>0; fio) he eee “ler, r>e’: »o|
k= 0
TRANSFORMS CONTAINING ARBITRARY FUNCTIONS .
Aaa es) (u+1)
uel, j=o HTT)
CB): j+k+
CoE
13. 6(a—-x)x" x
j+ke+l
x PF, (5¢3 wx) f(x)
De! Cody
Gaaipy mad
x jit! Ay
co
[re p>-l; fixe) hx |x| <r, ron]
k=0
a Powell oo Cae Pech Ww aoe
14 41 -e 7)" | | ‘gdh j k
i,
ce
p+ptl wo (PTLD, geo ae!
x | F)(bsc;we “fF (x)
foe}
[re p>-1; Re p>0; f-y ne*|
k=0
eae fe CBee rs hy ST TESORO SP SRE TF PP
Dare ena yen pee 15. 6(x-a) (1-e *)*«
ay as (b;c;we >) f (x) x (-e (we fot hy
x
co
[re p>0; fl-V tes vmleml<rjnse ns a)
k=0
P p, eth] eo? Cp), , Cad 2b oth
16. (l-e *)#x | | poe j j k
i,
mY
pt+ptl py Cpr prt). ey .7:
PS;
x oF | (asb;c; we yx ae
[re u>-1; Re p>0; foo=V he |
k= 0
aac)
l-e ? kp
17. f([x]) f([x]=k)e
: P ah
1.1.4. Derivatives of f(x)
ete) pF (p) -—f (0)
6 FORMULAS OF GENERAL FORM
(n-1) 2. fa p"F(p) ~ p" 'f 0) - p" f'O) -...- f"" O)
a)" ‘a n
3. [* $5] f (x) | F(p)
dee\e : 4 (a5 x| f (x) [-» $5] F(p)
eral eens fo J-eform an" a ae oe x), Pp |..-P| PF (p) (dp
Pp Dp Dp
Todsale
if Le f(x) hes
k=0,1,2,...,2-1
6. xf" (x) (-5)"t "F(p)] : dp pip
for mn,
da ee er 5 ie
A (PF ()] + (Di sax
CH= 1 ) ! n-m-1 (n-2)!
‘ Ges yr? 10) + Ti=m=2) 1 *
x De O) aa Hf eae ©]
for m<n
d” m m_n_(m)
Te sae f(x)] (-1) pF (p)
x
for m>n,
_ yy npn) n-m-1
(-1l) pF (p) =mlp f() -
ij =? > F. on ! Ay m 26 i ee
me (ral)! (n—m-1)
(n-m-1)! f 0)
for m<n
FLA
8. Ges f(x), (p-n) F(p-n)
if f” ©) -0,
k=0,1,2,...,2-1
TRANSFORMS CONTAINING ARBITRARY FUNCTIONS
9, am f (x,a)
1.1.5. Integrals containing f(x)
1. [rexurdu
a
iY)
2. [rood
4. jae ot No
5. fu raodu
é | uf (udu
1. [F,wor,-wau
i
8. a |r, (u)f,(x-u)du
0
Ja F(p,a)
a
[Fowdu
a
10)
0
= ah a pire
es F(p)
[Re p>0]
p “F(p)
[Re a,Re p>0]
1
ae p)dp
p
Dp
l
— |F(p)d, i [Fierdp
0
F (p)F,(p)
PF (p)F,(P)
9.
10
12.
13.
14
15.
16.
17.
FORMULAS OF GENERAL FORM
fe
iat le (u) du
0
x
; e™ [ox-we™/wrdu
0
x
see fo-w 2s (u)du
0
2
x 3/2 fue™ I du
0
_ 2
: [exw exp - u rendu
0
x-Uu
35
fuce-w >? o,
0
2
x exp| ~ as! f(u)du
x
[sinh” ew (u)du
0
co
[u?sinn Vxu f(u)du
0
JE ravp
[Re p>0]
vn F(2Vp)
[Re p>0]
J Fp+2¥p)
2vn F(p+vp)
RQ LC
a
[-1<Re v<Re p]
[Re p>0}
=) 028)
P(1+(pt+v)/2)
F(p)
TRANSFORMS CONTAINING ARBITRARY FUNCTIONS 9
co
-1/2 I 1
18. x [cosh Vxu f(u)du EA ered
0
[Re p>0]
19. for ?sin Vxu f(u)du 5 = Fla
Pp
0 p
[Re p>0]
-1/2 It 1
20. x [cos vxu f(udu [2 Fi
0
{Re p>0]
21 See aNd seria P)
i IEE) p
0
[Re p>0]
FS & au-1
22; eae OE F(a In p)
0
[a,Re p>0]
co i .
xf Cw) AS 23. WO. du + 0) 5 In p Fln p)
0
[Re p>0]
i 2 24, x r)/2 fexe[- 4] g 2” Vap 2p VyTp)
0
[Re p>0]
x D [4] rena
YAYX.
x
25. Gap x Sx cpea)” - -FiptV 2 p+ da)
0
i 2 (Re v<1/2; Re(p+a)>0]
x exp [-2-0- ga] x
x D, | u ) nau
x-Uu
10
26
27.
28
29.
30.
31.
32. forwy av x—u)f(u)du
FORMULAS OF GENERAL FORM
co
[rows wau
0
aL Ca a xu)f(u)du
0
x
; [2 ec-wofwdu
0
x
fonwy (acx-w yf udu
0
x
[o-wy a (x-u))f (udu
0
XJ (av (x-u) (x-ut+b))x
x f(u)du
x
0
1 1 i onl fees
Pp te
[Re p>0]
—VvE-v— ll 1
2 Dp F(in]
[Re v>-1; Re p>0]
F(p) cer ene
Pelee poe
[Re v>-1; Re p>|Im a|]
Vv
+ Eas F(p)
pre par
[Re v>0; Re p>|Im a|]
(2a) PRiv+1/29
F(p)
Vip Pacey eer
[Re v>-1/2; Re p>|Im a|]
Vv
pel
£xp[b(p-4p* +a’
| 2 D2
Petea
[Re v>-1; Re p>|Ima|; |arg b|<n]J
4 \ oat a’
(5) p exp(- 45] F(p)
[Re v>-1; Re p>0]
)/2) Fp)
33.
34.
35:
37.
38.
39.
TRANSFORMS CONTAINING ARBITRARY FUNCTIONS
x
[raced x? —u *)F(u)du
0
x+uU
xX.
v/2
eS) 1 (al CM OY
0
ul (pines —u a
ail ee
0 em
- , (4 a )
f(x)-ax{
7 2
0 Be i)
xJ (avu (x-u))f(u)du
co
i | (x+u) we
0
x J (av¥x (x+u))f(u)du
(u)du
f(u)du
(p- ayn /2
F¢ pa”)
[Re p>|Im a}]
11
ae p?+a?)
Gieap* +a 2
[Re v>-1; Re p>|Im a|]
F( pear)
[Re p>|Im a|]
——P__- FA p*+a*)
2 2
Peaa
(Re p>|Im a|]
1
[Re p>0]
[Re v>-1;
COLA
Tap? +a
1
alan
Re p>0)
bday? +a + 2B). x
2
2 2
4p +a -p
[Re v>-1; Re p>|Im a|/2)
12
40.
41
42
43
44.
FORMULAS OF GENERAL FORM
co
-bu
ee pee x
0 xX+uU
XJ (av x (x+u))f(udu
~
a [1,@e-w)r du
0
x
; fo-w" x
0
x I (avx —u)f(u)du
x:
: [o-w"x
0
x I (a(x—u))f (u)du
XI (a¥ (x-u) (x-ut+b))x
x f (udu
x
45. fo-w"? x
0
x1 (aVx-u)f(u)du
x
46. Jagat x? =u?) x
0
x f(u)du
2 2
lew -#(? 4p 4742-2]
(Re p>|Im a|/2]
Free greece
[Re v>-1; Re p>|Re a|]
leg F(p)
gy pea
[Re v>0; Re p>|Re a|]
(2a)” T(vt+1/2)
Sa a)
Vu woe
[Re v>-1/2; Re p>|Re a|]
Vv
Ded pa"
2 2 ny ae x £xPlb (p-4 pa )/2) rw)
2 Z
Daa
[Re v>-1; Re p>|Re a|;- |arg 6|<nx]
Vv 9 ooo te
{Re v>-1; Re p>0]
praa)Y pdgeg 2)
[Re p>|Re a|]
TRANSFORMS CONTAINING ARBITRARY FUNCTIONS
* Pa Fy v/2
47. jie x
0
x L,(ad x? uf wdu
48. f(x) +
ered Cade on
+af
0 oa
49. f(x) +
ek (a a) vox | tebe
0 ee
50. [iave (xu) )f(wdu
0
x
Tl os
0
x I (aVu (x-u))f(u)du
a
52. | gla
x1 (av x (x+u))f(u)du
-1/2
2p a8) ed Ya?)
(pray: -a ay
[Re v>-1; Re p>|Re a}]
F( p? Sar}
[Re p>|Re a|]
[Re p>|Re a|]
1
[Re p>0]
[Re v>-1;
<a
[Re v>-1;
ves
Re p>0]
oe
Re p>|Re a|/2]
13
14
53. x
54. x
SD.
56.
37.
58.
FORMULAS OF GENERAL FORM
x I (avx (x+u))f(u)du
- 1)/2 (n+ y
«fs “I Xp) ae A) ide
x
[o-w" x
0
x ae (a;c;A(x—-u)) f (u)du
IL 2
oKn Foss
om 8
ah iesu) faodua
2 2 4 | eee as ] -ro|
[Re p>|Re a|/2]
Sep FQVp)
[Re p>0]
T(c)p* EGS
Cp= ye
[Re c>0; Re p>max(0,Re A)]
(Re p>0]
ale Gg" n+l] u
p Gra :9| Pp
0
[r<q; Im+2n>r+q; Re b 2-1,
(a
seas
(6)
k=1,2,...,m; Re p>0]
k (-1)*
(-1) —[coth (lvp)] F(p) =
p
[k=0,1; Re p>0]
Chapter 2. ELEMENTARY FUNCTIONS
2.1. THE POWER AND ALGEBRAIC FUNCTIONS
2.1.1. Functions of the form x, 0(¢x¥a)x", [x]
4. @(a-x)x”
5. 8(a-x)x”
6. Oca=xin 2
n
T(v+l1)
v+1
[Re v>-1; Re p>0]
n!
n+1
[Re p>0]
(2n-1)!!Vn
non lee:
2 P
[Re p>0]
1
a heap)
[Re v>-1; a@>0]
n+1 n+1
p
n k
n! n! —-ap (ap)
peri S te » k!
k=0
[a>0]
lie
cp hap lay SAR On= oo
Frais i pe) oes Cd a
[a>0]}
Vap)
16 ELEMENTARY FUNCTIONS
1
v+1
7. 0(x-a)x” T(v+1,ap)
{a,Re p>0]
A nt a n Ca so
8. 6(x-a)x © ——e oar
k=0 ‘
nal
{a,Re p>0)
-n-1/2 Bas ~ k 1) k-n-1/2_k
9. O(x-a)x — hn LG ae r{n-k+3) a p+
(tye aN YR
+ Tit+l/2) erfc(v ap)
[a,Re p>0]
1 Bae
n
== e k k-n-1_k
10. 6(x-a)x nip DED (n-k)!a~” pe
n
- 2) Ei(—ap)
[a,Re p>0]
11. [x] A.
pte —1)
{Re p>0]
p
12. S[x17 1
p(e?-1)
[Re p>0]
n Kuk
itd" eb | co" (nt
P ad le" d =
[Re p>0]
2.1.2. Functions of the form (x+z)”, (a-x).
pz
1. (x+z)” P(v+l, pz) v+l
[Re p>0; larg z|<n]
THE POWER AND ALGEBRAIC FUNCTIONS 174
! n k
ie y (pz)
n+1 k!
p =
[Re p>0]
e pees n-1 1 k és
Si kesh 4) (ee Sie gil bt a e”’Ei(—pz)
[Re p>0; |arg z|<n]
Jz eerfc(Vp z)
[Re p>0; |arg z|<x]
= e”Ei(—-pz)
[Re p>0; |arg z|<x]
ee aia Vn pe erfc(Vpz)
Vz
[Re p>0;_ |arg z|<zxJ
+ ae pe’ Ei(-pz)
{Re p>0;, |arg z|<zx]
-ap
y(v+1,—ap)
v+1
[Re v>-l; a>0]
e”* [Ei(—ap—pz) — Ei(-pz)]
[Jarg z|<n or z>a, a>0]
T(v+1) eo?
v+1
[a,Re p>0)
ser? Ei(bp—ap)
{a>b>0]
18 ELEMENTARY FUNCTIONS
be Jy PEED (—apy* - &In(ap)
k=0 :
[a,Re p>0}
2.1.3. Functions of the form x" (x+z)”, xP (a>x),
ptv+l
1. x? (x+z)” T(pt+l)z Wut, wt+v+2; pz)
[Re p>-1; Re p>0; |arg z|<nx)
+1/2
PBeesleai a TGs (2) P22 B
ae p v+1/2{ 2
[Re v>-1; Re p>0; |arg z|<n]
: +1/2 Si ae ~ivn Zz) siap , ((141)/2)
SEX CEL) Fa T(w+1) a e Ase (pz)
[Re v>-1; Re p>0; -(2nt+n)/2<arg z<(2n¥n)/2]
ca 2” +1/2 pz/2
4. Pareienla - T(v+l)e D_,,_,6V2P2)
[Re v>-1; Re p>0; |arg z|<x]
v v +1
5 92 pz/2
2 ee Te Be Dew Doe cre
[Re v>-1; Re p>0; |arg zj<x]
papi r ree ee (v+1)z e T(-v,pz)
[Re v>-1; Re p>0; |arg z|<zx]
5 Be 1 n+l on pz. " (250 " es Gls zee EiGpa+ ) Ely
k=] p
[Re p>0; |arg z|<x]
8 ge n__n-1/2 pz
ean (-1) nz e erfc(v¥pz) +
Yn 1/2-n<
Renter AD, (2n-2k-3)!!(—2pz)"
. k=]
[Re p>0; |arg z|<z]
THE POWER AND ALGEBRAIC FUNCTIONS
9. ee se) -1/2
p v
10. x (a-x),
v v
Il. x (a-x),
12. x2 (q-x) 1”?
p v
Kaa x)
(ee ae
(x+z)°
6 (a-x)
Vx (x+a)
14.
pb v
15.. x (x-a),
v Vv
16. x (x-a)
-v-1/2 : v 47. x! (x-a),,
-v-3/2
18. i(e= a).
sf) (4
[Re p>0; |arg z|<n]
pt+v+l B(u+l,v+l)a fF, (ut, wt+v+2; —ap)
[Re p,Re v>-1; a>0]
v+1/2
VnT(v+1) (<}
(Re v>-1; a>0]
[a>0]
+v+1
# Zz B(ptl,v+l)a °O, (2,0; p+v+2;— 2 ap)
[Re p,Re v>-1; |arg(1+a/z)|<n; a>0]
aay etl = erf’(Yap)]
2Va
{a>0]
+v+1
. e€ Tv+la ~@ PW y+], utv+2; ap)
[Re v>-1; a,Re p>0]
r(v+l1) (2) mM? aol
vx VP
[Re v>-1; a@,Re p>0}
ap
v+1/2 2
EO gt p iy Tap)
2° ¥ pi
[Re v<1/2; a,Re p>0]
T(-1/2-v) -ap/2
avril? is2© D
[Re v<-1/2; a,Re p>0]
ave § 2ap)
19
20
19. x! (x-a),
1/2
ELEMENTARY FUNCTIONS
na erte(Vap)
(a,Re p>0]
PQ) 2 Got cap) nave "cot vn
p
[Re v>-1, v0; a,Re p>0]
l/k.v
2.1.4. Functions of the form sliney Medes x" (a-x es for 1Ak
Werte
Zs xP (xt 2?y”
eae : ee
[Re p,Re z>0] .
1, p+l pt del Ae oad eae 3 wtl, 1
2 - 2 2
Pay 84
p+3 Be “fale ~y-Ha 1} 20H?
yO er |
epee prim T(n+2v+1)
sae bast 225 EVES 4 -) + oe x
[Re p>-1; Re p,Re z>0]
T(v) ( d lee Gaal
(-2)""!athedz
EU pore ef Srey |
[Re v>-1; Re p,Re z>0]
Sy AN tee .
mre Pars es ee
— COS pz si(p2))}
(Re p,Re z>0]
m+n (S10 ind sad)" (3/2 1
nt2" iT dp m “ {: [eospe( 3 - Siva] =
“an(t-cvo
[Re p,Re z>0]}
THE POWER AND ALGEBRAIC FUNCTIONS 21
m-1/2 m+n_3/2 3m n
x G2 NN) I d d 2) | pe.
6. Ceoeec ye 4(2n-1)!! eas {ve|77,.[ 4 t
2 (pz
tial 5\]}
{Re p,Re z>0]
mt
ale vd Pa [sin [o:-"5] ci(pz) — cos [p25] si(pa)| t
[m/ 2]
+t on 1-p72)*"
Pp k=1
[Re p,Re z>0]
Reith ae
(any ew
! Fagard Ray te
aml Te
p Zz
8. lh Nes rare:
lr (=v) p"
A(l,-p), iin)
A(k,0)
[Re p>-1; Re p>0; |arg z|<z])
v+1/2
9. (a?-x?)* orev) (28) 7260) ~ Ly 1/260)
[Re v>-1; a>0]
p+2v+]
10. x!(a’-x")” a —B| sa
(Re p,Re v>-1; a>0]
-1/2 (-1) "vn d" ap ap
a aia pT. YP Ki p4(—3) [A1val- 2} *
4 2=1/2 a
xX (a -x'), +1444(2)]}
[a>0]
22
12.
13.
14.
15.
x" (a-x
Why
abe
ELEMENTARY FUNCTIONS
(vel uae Lt?
; x
On ee ish conan
k+l,k
fag 1! A(l,-p), A(k,v+1)
xKGe Sarraary
a p |Atk,0)
[Re p,Re v>-1;_a>0]
(ap)
/2
T(v+l1) (2a vi
Ee K isa Vn p
[Re v>-1; a,Re p>0]
Fle” Ei(-ap-bp) —e “PEi(ap—bp)]
[O0<a<b; Re p>0]
a ai” KS 2 » in
ae ] 9) B[v+1-v- -1} tao
Sot CD PGut2v 1) bw
pps ss preyed Clee ae”
(Re v>-1; a,Re p>0]
v+3/2 n
ah ee ae:
2Vx dp"
{Re v>-1; a,Re p>0]
Bbw’ BY 2 2ealea 2 f(a
“4? i [#5] “Ki74(“5| |} ¥V2n dp
[a,Re p>0]
Pp ap a
a| In Ky B\K/4(25|
[a,Re p>0]
-v-1/2
K .3/2(4P)]
THE POWER AND ALGEBRAIC FUNCTIONS 23
HM, 2) PEND [z= 2 fap
19. x (ea ey) sd 5)
[a,Re p>0]
v,p+l/2
20, alta) Sea) al ee
(20) pa
ee 1! ACU =p), Atkyv+1)
PS Ew aE
Nar NCE)
[Re v>-1; a,Re p>0}
v = 2am,
x veils) SSV We Ge jal
ae 272-32 eat Lae pay 4 )+
ae 2 Pe -a
oes (ce? — @ “cos vn)
[Re v>-1; a,Re p>0]
on alts 1 eae Pop
ee [ip Cee ap) se Eicap)| ate
7 2
x -a
{m/ 2]
l 2k-2
jose car y (m-2k)! (ap)
D k=]
[a,Re p>0]
m aak 3 ay ap
23. ———. Se [e PRi(ap) — (-1)""e“PEi(-ap) +
x -a
+ 2 sin [av-"5) ci(ap) — 2 cos [av-"5} sicap)] 35
[mn / 4] 1 4k-4 aca), (m-4k)!(ap)
p k=]
[a,Re p>0]
ey iy ee -v-1
24. TTk > CL any eS
l-ax (207)
ght! aki! A(1,-v) ,A(k,0) , A(k, 1/2)
x 2k+1,2k [!
p A(k,0) ,A(k,1/2)
[Re v>-1; .a,Re p>Q]
24 ELEMENTARY FUNCTIONS
ph Rrperil2 -p-l vn ki kel k
IA, = eh ee sec G,’ ax
v - k+1,2k eax (ny U2 ey 2 ~ 2k+
am x 7 :
p |Atk,0) ,ACk,(1-v)/2)
[Re u>-1; Re v<1l; a@,Re p>0]
! Osh, set
2.1.5. Functions containing ¥x+z
1. <a a erfc*(¥pz)
(x+2z)Vx+z 2vz
[Re p>0; |arg z|<n]
2 2
(V¥xt+z-vz) nf{d [pz
Ze ray ie 5 dp erfc(V pz)
{Re p>0; |arg z|<x]
s 1 “1/2, -1/2 Re ee
= meer (x+w) + ——..—— erfic(pw) erfc(pz)
WZ
gs Beye [Re p>0; |arg w],|arg z|<n]
eu 2pz 2 {a [2 ro+ve PED (/2pz)
(x+2z) ¥x+z . eve
[Re v>-1; Re p>0; |arg z|<nx]
v/2 /
5. ((Vx+z+Vx)*- RATELY
p v/2| 2
= (Vx+z-Vx)"] {Re p>0; |arg z|<x]
6. yal ( AY . T(v+l1) pr/2p Vips
Vere Vee ee 2 DTR Cyst T2© ~y-(%2 Pz)
[Re v>-1; Re p>0; |arg z|<x]
Vere+V¥z]" 1/4 pz/2
i: ge Vaz Ki(25)
[Re p>0; |arg z|<n]
THE POWER AND ALGEBRAIC FUNCTIONS 25
2.1.6. Functions containing a Meer te for 1+k
1. (xtdx 2422 i
v
x" eee Le
(ieee ee)
4.
{r-0 or 1/2]
x"
5. —————;
Gis i
x(fi+zx! x ree ‘
x
[r-0 or 1/2]
De Vv
za SUV Ze
p psinvn
[Re p,Re z>0]
[J_ (pz) — J_,(pz)]
Vv
Iz
sinvn [_,(p2) - J_(p2)]
[Re p,Re z>0)
(5} bere iz -2v)/4 (25) a
AY (ia wld A ~ Jew /4 (*3] .
XY ua /4 Gl
[Re p,Re z>0]
-2r,2r-1/2 p+i/2 (CRW amen
Van t2ny** Fe T2 per.
ye Gk Bat! : : l :
2k+1, 2k l
p
x
A(l,-p) ’
Atk, (v¥v)/2),
Atk, (v+1)/2), ae oe
Atk, (v+v)/2)
[Re p>-1-(1¥1)//(2k); Re p>0; |arg z|<n]
-2r,2r-1/2ppri/2 (ENP Doe ok
= x
A(l,-p),
XG
2k+1, 2k 1
p
A(k,l-rtv/2), Ack, al
ak kel [2 "0!
A(k,0),
A(k,1/2)
[Re p>-1; Re p>0; |arg z|<x]
ELEMENTARY FUNCTIONS
6. 6(x-a) [ (x+4 ends lee
x [ (+4 leax" ‘ Dee
a(t hia) |
{r=0 or 1/2]
10. ae ibe x
< | (eee le ples
Ze? ( apaies
ees
[r-0 or 1/2]
2va”~
K (ap)
[a,Re p>0]
2a K . (ap)
{a,Re p>0]
§ [22 aP | K. ap
g eat Aa | 5)
[a,Re p>0)
yin 2r cag 2rd epntil2Qayis! Pate
2k, 1 !
a
a (/5 =O)
X Gore] 1
p A(k,0), Atk,v)
A(k, (v+1)/2), anid
[Re(lv+kp)>-k; Re w>-1; a>0]
nega 2 tee
~ x
(Qn i? 1 pee
Delano ia sgh
XG wm) OT p |Ak,0), Atk,1/2)
A(k,1-r-v/2), a
j2.Re p>0)
THE POWER AND ALGEBRAIC FUNCTIONS 27
2.1.7. Functions of [x]
1 —— a
({x]+a) *
aa te
({x]+1)”
cael ER
[x]+l
1
2[(x]+1
4[x]+3
leale!
ie
(2[x])!
(+1) 1
MP ixlti)!
l-e "4 A-P
D (e “, S, a)
[Re p>0]
p
e- = (2, -p
“pe Li (e )
[Re p>0]
_— p -_
Ppa P)
[Re p>0]
-p/2
sinh £ fete — ee
p 2 pees?
[Re p>0]
y PE yD -p/2
? sinh ) arctan e
[Re p>0]
eo) -p/l4 =
gpl4 1 ens 7" (in Une FATT Ape Gion € sia
P lea?! 4
(Re p>0]
IP -p/l4 oS e3P/4 1 e ia Ife => actinic p/4
NG lee
[Re p>0]
_ mH J
ai exp(e i)
weit ol 4 xt l-e , cosh, eh,
p cos
2... p fsinh,-p/2
2 sinn {$30 (e r}
28 ELEMENTARY FUNCTIONS
1
ll. aTx])!
[x] (le
eres Pe
1
13. TaTxj+D!
eae beet.
14. TaTxI+1)! &
Olin)
15. ails
{x] 1
16. > eR
Oy k+l
k
2.2. THE EXPONENTIAL
[=eme —p/4 Zp )] feosh (er + cos(e
v2
—p/4 p/4 Le ) + sin(e
p
[sinh(e
=e = =
ip oe e) T(n,e P)
= * Inte”)
[Re p>0}
FUNCTION
Dp 43 be exp(-ax"/%) and the power function
3. 6(b-x)xe
1
pta
(Re(p+a)>0]
T(v+1)
(pta ) v+l
[Re v>-1; Re(pt+a)>0]
Re Er y y(v+1, ab+bp)
[Re v>-1; 5>0)
—p/4
ela anes
cosh (# Jos
v2
)]
10.
11.
12.
13.
. O(x-b)x"e
5 exp(-ax’)
b xexp(—ax’)
i x"exp(—ax’)
5 2s exp(—ax’)
6(x-b)exp(-ax’)
(x-b) Yexp( -ax’)
exp(-ax”)
THE EXPONENTIAL FUNCTION
1
ae ee T'(v+1, ab+bp)
[b,Re(p+a)>0)
2
ads ea) ene] 7 |— exp| 4s] erfc
[Re a>0]
2
Cet s 73 Fa) 2 [ Ae oe a exp. JON as (Qa wd 2 8a seat My Ie
[Re v>-1; Re a>0]
2 PE fol) st exp erfc
[Re a>0]
mea exp( Fa 3) ete(—2- }]
aa [ ple Va.
[Re a>0]
1{p tee De pa
Hb exp [$5] [«sra( Ba] -Kia(Ea}]
[Re a>0]
1 ey Pele Ae.
Wa Pl ga}*i/4| 8a
[Re a>0]
ab ale exo(4 Bete a}
[b,Re a>0]
2
T(v+1) Diao)
Paani tee exp [Fa prabry)|D_,_,[
[Re v>-1;
3
2
3 Tha sual? 27a
[Re a>0]
b,Re a>0)
2ab+p
¥2a
29
}
30
14. xexp(—ax>)
15.
16.
UTX
18.
19.
20.
21.
exp(-avVx)
x’exp(—aVx)
(n-1)/2
exp(-avx)
x eexp(-a¥x)
ji exp(—avVx)
ELEMENTARY FUNCTIONS
ii. (V1) yeWen/3> ee (veka T 29 eit pi are ee nah a
era) yee oc) Art ae Hest ie Wile oe 4
ni
oe Lae ig e ape ee oa 6 (32) Jae Bele? (Se caesar
(Re v>-1; Re a>0]
le ™ exo( $5) ere ] p 2) 3 de Vp
(Re p>0]
2
20 (v+2) E a )
SE Sm AT O40 Do. ||
(tay ae Pap
[Re v>-1; Re p>0]
2
(-1) |e a = [exe($5 5) ert
Pda” 2
(Re p>0]
mae +4— “22 Fe e(t5 sy ete(—4
2p? 2Vp
{Re p>0}
4\p) “*P\8p) |*3/4| 8p 1/4(8p
[Re p>0]
Felten
[Re p>0]
2 2 ES Se OR is a”
(spy ot esl a) Kval 80)
[Re p>0]
Al
THE EXPONENTIAL FUNCTION 31
1/2,v+1/2_ -v-1 k,1)AC,-v)
22. x*exp(-ax") oom (k+l) 72-1 Gi, oe I
(20) kept ACK-0)
[Re v>-1; Re p>O for ‘kk, Re a>O for l>k,
Re(pt+a)>0 for /=k]
pg Ns exp(-ax and the power function
nei a (v+1)/2
l. xe 2($] K.,,(2¥ap)
[Re a,Re p>0)
n
2. xt l/2-alx ees d ape *F,)
dp
[Re a,Re p>0]
-n-1/2 -a/x n[{nd" -2vap
ge X e (-1) et
{Re a,Re p>0)
4. xl/2e-alx apex A rapye2¥ 2?
Dp
(Re a,Re p>0]
-3/2_-a/x 0 -2v ap
S35 x e JEe
[Re a,Re p>0]
2
Ay -| l-
6. x"exp(-a/x’) pl Po+DgF,[- oo pa | id
(v+1)/2 2
a v+1 JU wees) 2 vay)
= r{- FAG: a |
[Re a,Re p>0)
32 ELEMENTARY FUNCTIONS
2
7. x exp(—a/V¥x) p 'Tw+lF, (3. aye 24) *
2
-v-1/2 1 Op OE
— ap r[v+3] oF vA apr | }+
+2a*r(-2-2) Ff, [S4y, 24y;- 22 a AAA) BIG ae |
[Re a,Re p>0]
NLR pores) Sam ~ -I/k k l Bh Seoqees *) (ke Dye - 10
(2x)
Spee
k_1
ap:
[Re a,Re p>0]
XGiito
A(l,-v), 4)
IIPABY. exp(-ax*/*y and algebraic functions
I OE 20 (v+1) Ce Ly / E:
. Ve CEI i ew a x (2p)
xD. (Bee
Vv2p
[Re v>-1; 5,Re p>d!
6 (b-x) -a/x lf{a | 2Vap a 2: = e 22 [e ert ¥Bp+]$ |-
-e? “Pert ¥bp-[¢ 27°? +e 4
[6,Re a>0]
6(x-b) -a/x lf [ 2vap a 2 eb Af [oF Ffotr-
or “Pert ([$-¥bp) -e” Oe ere ma
[b,Re p>0] .
etx * /
4, ——____ adh QPF PRS ie Givpz
Vx (x+z) , vz ll? je
[Re a,Re p>0; |arg z|<n]
THE EXPONENTIAL FUNCTION
2.2.4. Functions of the form f(x, e”, eet early
ey” 1 pie lis (LSS 4) + B(2, v1]
[Re v>-1; Re a,Re p>0}
v -ax,m v(m (Sa) E 2. x"(1l-e ) OEM Nd toon mane rare
k=0 (pt+ak)
[Re v>--1; Re a,Re p>0)
noon
Si xtc)" ted OL ey" a v+1]
a aaa
dp
[Re v>-n-1; Re a,Re p>0]
~aXxyv 1 pfe P esl 4.-x(-e ©.) 185041] [v[B+vet] »(2}]
[Re v>-2; Re a,Re p>0]
Vv
x (Veta) Pp
5. Ter 2* See ¢(v+1, 2}
[Re v,Re a,Re p>0]
n n+l
x 2 1h I (n) { p
ears, keane
(Re a,Re p>0; n=1,2,...]
i Civ+l al oa p+a Ge Ie] v4, — | v+l,
‘Poesy epeeek 2a 2a
[Re v>-1; Re a,Re p>0]
yee ted 3” Pp
8. Teenee wea a
[Re a,Re p>0)
1 Ab P 2 9 BS - 1B
(is a x 2 a a a
(Re a,Re p>0]
33
34
10.
11.
12.
13.
14, 1
x
iss SS
16.
17.
18.
1 -ax
lice )x
xa Sees)
~ax.m
(ie yy se
—bx.n
—e x (1 )
ELEMENTARY FUNCTIONS
-bp -ab
e o(-£ hi
az z a
[b.Re a,Re p>0; |arg z|<n)
Fuel) ofl, v+1,2|
vtl
a Z
[Re v>-1; Re a,Re p>0; z¢[0,1]}]
- 1 ml
+ In P(p) + =) Inptpt qin(2n)
[Re p>0]
p(p+b) — In(p+a)
[Re(p+a) ,Re(p+b)>0)
pt+b
In pta
[Re p>-Re a,-Re }]
2Vn(Vp+b-vVp+a)
{Re p>-Re a,-Re }]
(p+2a) In (p+2a) + (p+26) In (p+2b) —
— 2(p+a+b)In(p+at+b)
[Re p>-2Re a,-2Re b}
in (29) (pt 5)
p(pt+at+b)
[Re p>0,-Re a,-Re b,-Re(a+d)}
Cy ) co'(a) S co’(") 52
k=0 k j=0 ]
x In [p+ (m—-j)a+(n-k) 6]
[Re p>0,-mRe a,-nRe b,-Re(matnb)}
19. 1
20.
a. 4
22.
23.
x
24.
toa -~e™) m x
x
~bx.n
x(d-e -)
m
x
nl
x (1 ~exp(—ax)]
THE EXPONENTIAL FUNCTION 35
p \In p-(p+a)|n(p+a) —(p+6)In(p+b)+
+(p+a+b)|In(p+a+b)
[Re p>0,-Re a,-Re b,-Re(a+b)]
n mt a
» co*(7) 2 c1y(") [p+ (m-j)a+(n—-k)b]x
k=0 j=0 )
xX In[p+ (m-j)at+(n-k) 6]
[Re p>0,-Re a,-—nRe b,-Re(ma+nb)]
tf el A
nN
(NOE odin AWS 7g hen (CAEL! dp" Sa no 1) [pra,+a, +.
n-1
+a, | In [aja tata |
Lin Peds i,
[o<m<n Re p>0,Re p>-Re [- eo Tyee, :
j
iy Fe k
the notation means that the kth member of the
sum contains i terms which differ by the subsets
of indices is i, Seen i, from the set j=1,2,...,7
Iho (222) — » (ete
Cc c ¢c
[Re c>0; Re p>-Re a,-Re 6]
(pta)/(2c), (pt+b+c)/ (2c)
InT
(p+b)/(2c), (p+at+c)/ (2c)
[Re c>0; Re p>-Re a,-Re 6]
1/,,(pt@ p+b)_,(p)_ [erated “eleaalt Sacelatle| <8
[Re c>0; Re p>0,-Re a,-Re 6,-Re(a+b)]}
plc, (ptatb)/c
InT
(pta)/c, (ptb)/c
[Re c>0; Re p>0,-Re a,-Re b,-Re(a+b)]
36
26.
27.
28.
29.
30.
31.
32.
(i-e ~)
x
[z+(e*-1)
p
l /k j
ELEMENTARY FUNCTIONS
n
+ _ yep har(2 na ka}
k=0
(Re c>0; Re p>0,-nRe a]
i Ipi2 1- Hl. ne PRET ba [2.1—v) Fy .-am ws p-vtls wo)
[Re v<l; Re a>0; |arg(l—u)|,|arg(1—-v) |<m]
B(utl, p/a) | ee 2 Ml
v oF a’?™? qthtl; zy
az -
[Re p>-1; Rea>O; |arg(l+z )|<m]
Vv po
(k/z) é rex
v
A(l,1-p), i,
x Gk ke! rigs
A(k,0) , A(l,-—p-v)
[Re p>-1; Re p>0; |arg z|<m]
klay Fr f?] x
v k-1 v
(20)
kee |
iol:
A(l,-p) , Ak, ual
A(k,0) ,A(1,-p-v)
[Re »>-1; Re p>0; |arg z|<m]
pet eee
k+1-2 x
(20) z*T(v) P(pt+ptl)
xG
k+l,k+l} —k
k+1,k+l
A(,1-p), al
A(l,u+1), A(k,0)
(Re p>-i; Re p>0; |arg z|<m]
A(l,1-p), P (pt) wk ok, k+l -k
peas 2k+ 1,2k+1
A(k,0) ,
A(k,0), A(k,1/2) |
A(k,1/2), A(l,-p-p)
[Re p>-1; a,Re p>0]
THE EXPONENTIAL FUNCTION 37
A(l,-p), Lae Ke =
33, lac fh evs ee roe lat
aie A(k,0),
2k+1,2k+l
A(k,0), ACk,1/2)
A(k,1/2), A(l,-p-p)
[Re p>-1; a@,Re p>0]
34, joel Oot 2 ae UF 6 9 eee roe a
a=€er A) © / (20) )~ ar(ptpt1) 2k+1l,2K+1
A(i,u+1), A(k,0), A(k,1/2)
A(l,1-p), A(k,0), ee
[Re p>-1; a,Re p>0)
Beats ,
5 ae — thle) p[wt] x
ly Le” cos (vn/2)
A(1,1—-p), ky k+l -k
x Gore 1,2k+l f
A(k,0) ,
Atk, lv), Atk, (1-v)/2)
A(k,(1-v)/2), ACL,-p-p))
[Re p>-1; Re v<l; a,Re p>0]
as he n(k/a)~ r[e] x
LP-cos(vn/2)
A(d,-p),
2k+ 1 ,2k+l A(k,0),
x Gi‘ k+l [."
A(k,l-v), Atk, (l1-v)/2)
A(k,(1-v)/2),A(l,-p-p)
[Re p>-1; Re v<l; a@,Re p>v)
38 ELEMENTARY FUNCTIONS
qa Roa? secnh2)
esi x
2(2n) Pv) P(pt+ptt)
INU M=o)- Gt"! k+l -k
X Go kel, 2kel A(,p+)),
A(k,l-v), Ak, (A-v)/2)
A(k,0), A(k, (1-v)/2)
[Re p>-1; Re v<l; a@,Re p>0}
Vv
38. Gime" ax a)
SOUUGR al y
k kane
xX (a-e ) xG ;
-lx/k.v kyl 1
+ k+l,k+1
A(,1-p), Hie
a |A(k,0) , A(1,-p-p)
[Re p>0; Re v>-1 for O<a<i, Re w>-1 for
a>l, Re(u+v)>-1 for a=1)
v
39. (l-e *)"« a Eee b ata
a= =e) xXxG rn
-x.l/k,v kyl 1
+ k+1,k+l
A(dl,-p), A(k,v+l1) |
a |A(k,0), A(l,—p-p)
[Re p>-1; Re p>O for a>l, Rev>-1 for
O<a<1, Re(p+v)>0 for a=1]
2y v,nt+D
40. (l-e Dex Gaal r| v+l1 |x
(ame ee utpt+l
_ Xtal ANE fy
SER sh. ee
Ad,l-p), A(k,v+1)
A(i,u+1), A(k,0)
[Re v>-1; a@,Re p>0]
THE EXPONENTIAL FUNCTION
41. (l-e *)*x (2) P (n+l) (v4+1) %
hae
-lx/k v 0, k+l ie Ad, l=p), A(k,v+l)
ee eee XG). k+l
A(k,0) , A(Z,-p-p)
[Re p,Re v>-1; O0<a<1]
42. (1-e *)*x (¢) POISE ps
1?
Sle 5 acy 0, k+l ie NCU AU) ame GK paves)
CMe sal) OE ver A(k,0), ACL, -p-p)
[Re v>-1; 9<a<1l; Re p>0]
v,pt+p
- 1 v+1 43. (l-e *)"x a ae | ]x (Qn) ! Pee: p+pt+l
et oe ret (1 A(,1-p), A(k,v+1)
(e-bay, XGuy a
A(,u+1), A(k,0)
(Re p,Re v>-1; a>0]}
: 72 Cen / 2 PEP Gel) 44. (1-e *)*x Seer a WF Lane a we Cay 1/2 ,u+t t/2 PAT
Vv
yidieze 17 +1] ye Gk tke! | ok A(l,1-p),
Clive oe r 2k+1, 2k+/ Atk, (v¥v)/2),
([r-0 or 1/2] A(k, (1+v)/2), Atk, 1-2r+v/2)
Atk, (vtv)/2), A(l,-p-p)
[Re p>-1; 2kRe p>-(1¥1)/; |arg z|<x]
40 ELEMENTARY FUNCTIONS
45. (en x
Vv
Udez ine *) 1" +}
rel keer
[l+z(l-e ~) ]
(r=0 or 1/2]
x u
Fen ca all
[l+z(l-e ~) ]
Nc Ae eee punts
seGize ye:
w/e) [r=0 or
a at i
x [[rdi-e-*)"-
aerial te heer) |
[r-0 or 1/2)
VD (ELD) ep) e
k= 2 2
(22) i
Aoi) ’ k,2k+1 k
XG ake! :
A(k, (v¥v)/2),
Atk, (1+v)/2), ins
Atk, (vtv)/2), A(l,-p-p)
[2kRe p>-(1¥1)/-2k, Re p>0; |arg z|<x]
V2 Gey) 2) PGs)
(any Ro F/B ypwe ly l 2-23
AN Wi sil=fD). Pasi Nf
XG) kel c
A(k,0) ,
Atk,l-rtv/2), A(k,1-r¥v/2)
A(k,1/2), Ad,-p-p)
[Re p>-1; Re p>0; |arg z|<z]
[IED eel)
Mien Sie et
kizkel | ok
X Gop) 24! c
NCE |),
A(k,0),
A(k,1-r+v/2), pera
A(k,1/2), Ad,-p-p)
[Re p>-1; Re p>0; |arg z|<z]
nviDp 22h, Eten
2p+v+1-2r
[Re p,Re(p+v)>0]
THE EXPONENTIAL FUNCTION 4]
49. dae alii l+e *+
itce Ve
es id eae eae
[r=-0 or 1/2]
{r=0 or 1/2]
-lx/k.-r
+ 52, Gee )' a=e'). x
Sl (Faxda mTR) is
OM Valse)
[r-0 or 1/2),
v+2p+2r-2,, a Be
p+v/2+1-2r
[Re p,Re(p+v)>0]
QP rep (221) 2PHB-D/4 7
1/2-v
x ea aS
[Re p>0]
OE atin Stage
DA) me | x
v
1=p, sits
Gas Gq
{ OWsaP
[Re p, Re(pt+v)>0; O<a<1 or a>l, Re p>-1]J
2. v/2-r
2 ¥nk v/2-r
Cae
v 1
x Gk! we A(l,1—p),
2k+1,2k+1 A(k,0) ,
A(k, (1+v) /2), A(k,1-2r+v/2)
Atk,V) » ACL, =p=py
[Re p,Re(kptlv)>0;, 0<a<1 or a=l,
Re p>2r-3/2 or a>l, Re p>-1]
42
53.(1-e *)"x
x Pe ees ox
< varlaecincmt yee
eeiyel ee
eeat leer ee
[r=-0 or 1/2]
ELEMENTARY FUNCTIONS
2k, -k
x ae
A(k,0),
Atk, (1+v) /2), A(k,1-2r+v/2)
INCE). NY)
(Re p>-1; Re(kptiv)>-k,; O<a<1 or
a=1, Re p>2r-1/2 or a>l, Re p>0]
2.2.5. Functions containing exp (—cl x +5 a
1/2
Notation: ued? (5a4 2) ,
: =aelpe prea
+
- exp(—b eet)
rer
1/2
3. exp(-b1 x40") x
2
x Hay
x
1
(ale cane Hh ff 2
ae ue
P as i Sie
[ex,,| 2 |x, 2 ]
(Re a,Re b,Re(p+b)>0}
yuT2 1 exp(-ad b?-p”) erfew_)-
fee
= i. exp (af b*-p 2) erfotu,)}
+
(Re a,Re b,Re(p+b)>0)
— eerfc(u erfc(u_)
Vv2a
[Re a,Re b,Re(p+b)>0]
THE EXPONENTIAL FUNCTION 43
PSeciisa CA ae 1 ie ee ee [2 r{>~»] Dj y(V2u)D,_, (V7 2u)
ty Sac
x exp( <b x a 2) [Re v<1/2; Re.a,Re b,Re(p+b)>0]
Sy (eeay 1? x b zal exp(-al p? 252) erfcw_)—
b z
; 2 v Ss
x exp(- x? 4") 1 exp (ol p?=0Jenew,)
+
[a,Re b,Re(p+b)>0]
2 2
2 2-3/4 (2 ae u
oa a) ex Be | 2 \x,,| 2 |
x exp (-s x ges a z) {a,Re b,Re(p+b)>0]
-1/2
(x-a) ,
It ap
LD sear remem 75 —— e erfc(v, erfc(v_)
a
exp |= Peal [a,Re b,Re(p+b)>0)
zh 2v+3
8. (xtay"(x’-a?) * x 2s D. 070, 47,020)
ae he ara ee GP v-1/2 ives ff) =
scetp|_-t4 me) \ [Re v<1/2; Re a,Re b,Re(p+b)>0]
v/2
= p 24 2
9. Gag, x 2a | 4 K (al p -b }
p-b
x [eke [a>0; Re p>0|Re b|)
Mert t=? |*
i aa.
exp <b x72) |
2.2.6. Functions containing exp(f(x))
1. exp(—ae”) a’T'(-p, a)
[Re a>0]
ELEMENTARY FUNCTIONS
2. exp(-ae *)
. (l-e *)exp(—ae”)
. (l-e *)’exp(-ae *)
Sed ~e *)"exp(-ae*")
. d-e *)’x
X exp( <gers
. (l-e*)*x
—x, l/k
<exp[—a(l—e")) J
(lene)
x exp[-a(l-e *) Rule
a ’y(p, a)
[Re p>0]
Pewee? P/,42y
ay
x Pare: (p+1)/2,-p/2
[Re v>-1; Re p>0]
(a)
B(p,v+1) F, (p;p+v+1; —a)
[Re v>-1; Re p>0]
Bip,vt lz "®, [Pwr ia ia)
[Re v>-1; Re p>0, Jarg(l+z")|<m]
Costly ke ote
(ln) tere
x
bal
k
a
xG
Ad,1-p), ie
A(T) —p =v)
[Re v>-1; Re a>0]
a x
(22) Ck =) 9/82
kyl yf A(1,1-p)
“Sino PTE
‘ k A(k,0) ’ A(l,—p-v)
{Re v>-1; Re p>0]
1/2,-p
(p)k l x
(2n) CK aD
k | A(l,-v)
XG cee ona
: kK} Ak,0), Ad,—p—v)
[Re v>-1; Re p>0]
r( Wee Lae
(Quy eee
k|AW@-v), AD _ 70, k+l) k ’ ? ’
a |
A(l,—p-v)
[Re a,Re p>0]
THE EXPONENTIAL FUNCTION 45
» 1/2,ptv
10. (1-e "x Gee ~
(20) I (p+v+l )
k1AC,1-p), A(k,1)
x exp[-a(e*—1) OE A
“ta |A@,v+1)
[Re v>-1; Re a>0]
1/2,pt+v
(21) I (pt+v+l1)
x exp[—a(e*-1) oe RG £-
. k- | Ad,v+1), A(k,0)
[Re a,Re p>0]
12. (1-e*)"exp| - Pipya” ew (a) : EXE eee P -p-v/2,(1+v)/2
[Re v>-1; Re p>0]
2.2.7. Functions of [x]
-_ ToL - -
1. a re (t-ae?)"
[Re p>In|a|]
Se LY 8
2. a“ en—x) ize l-ae—
B l-ae
se ny
3 [xja"*! L ; ae ’(1-ae ”) <
[Re p>In|a|]}
[x] Spl i
A ae l-€ o(ae P 's,b)
({x]+)°
[Re p>in|a|; |arg 5|<z)
be l-e ? -p h-ay (na-P
5, —4£—_—. §(n-x) ————-[M(ae “,s,b)-a e “(ae “,s,n+b)]
([x]+5)° P
ex} Lda =
6. —_4+—__ £1 Lita P)
([x]+1)
[Re p>In|a|]
46
[x]+m
az lk)
9, ———___
2(xi+1
i | :
10. AC sl) AE qe!
2 (x) +1
Ae]
ie 2
4[x]+1
4[x]
Za
4[x]+3
tx]
a
13. ean:
fea]
14, Txpee
Lexa)
a
arg CEA OT
[x]
(ET) 2[x] 16. (ZIxp)t 2
xa
id (= = 0 eye)
a C2ir eink
ELEMENTARY FUNCTIONS
ieeu
Indl -ae ”)
[Re p>inja|]
nt
“2 ie eS m=1 k mys
en? & inc -ae P) - Lae e€ 4
k
[Re p>In|a|]
-p/2
a) sinh 8 in +42 —__
ap 2) lop eee 2
[Re p>2In|a|]
/2
2. sinh 5 arctan(ae °’“)
ap
{Re p>2In|a|]
pisdieee:. if rae Gy
4ap -p/4
+2 arctan?)
[Re p>4In|a]]
‘ 3 = -p/4 : SYN ten In itae —___4 arctan(ae "|
{Re p>4In|a|]
m= /P. =
oes nae ey
[|arg a|<n]
eee = 7
Tra ty Tp exPlae Pn, ae?)
Dp
Cia = Pe
ap [exp(ae “) - 1]
[larg a|<n]
=e" {cosh -p/2
D bee (ae )
[larg a|<x]}
2, dp (sink, =pi2
ap ae 2 1a =< }
[|arg a|<n]
18.
19.
20.
21.
22.
23.
24.
25.
26.
THE EXPONENTIAL FUNCTION
mesa
(4[x])!
(x ]
(St) 4[x]
(4 fee”
ae!
(4[x]+1)!
[x]
(-1) 4[x]
(Atxieiyi”
qisn)
(4[x]+2)!
tex)
(1b) 4[x]
CATH eR
4[x “ene
(4[x]+3)!
[x]
(-1) 4[x]
© WESE hee
(ety le
(erttteteey#
B _ -
l [cosh(ae play + cos(ae pA,
[arg a|<x}
Pe sah
Ine cosh 7 eth) cos [+ oy]
P Ua Sp id
[larg a|<z]
Pak |- A =p © —p
tae Sa ) + sin(ae “)]
[|arg a|<zx)
-p
eo 1-2 I sin Be | cos = | +
¥2ap v2 v2
+ cosh (-£ | sin (2 "||
v2 v2
{larg a|<n]
sinh P fcosh(ae ”’*) — cos ae ?'4y)
1
2
a
[larg a|<zx]
2
2
sinh 7 § sinh {2 e a sin(—2 an]
a’ p v2 v0
[|arg a|<x]
Bes te z,
Pls ee sinh ie 4
Zigh sp
[|arg a|<x]
pipe sin(ae?’*)]
47
—se 4 -
ert eo [sin ie e? ) cosh [4 f ‘" =
Y2a° p v2 v2
— cos & e | sinh (Ze oy]
{larg a|<x]
-p .
e. ine _ y(v, ae Py
ap
[|arg a|<x]
48
27.
28.
29.
30.
31.
32.
33:
34.
Shy
ELEMENTARY FUNCTIONS
(1p §21 Qt)
Qes Ee Dy aliens)
qgi*!
[x] ! ([x]+n)
[exo qzi*)
ROAEIES®
(£10 ee
GEES yee)
1) 9213
(2x42)! Cl x41)
(+1) 1%)
(ixleiyt *
qzi*)
XTHx1ED
ee
COTEDP RECT EEI OD
Cae 4[x]
PAEVOMT LUGS
ei (ex)
QixsteptT™
qi kel
EBSD
lees ee ee
+ ap [C + Ina — p~ Ei(tae “)]
{larg a|<zx]
Kou.
x(t - exp(ae ”) D Spee et)
(larg a|<x]
icy
au sinh erfi (a6
ap 2 \ erf
A We -p/2 erfi(, ~p/2, -
| mae orf (2e 5
{larg a|<z]
a“p
F exp(ta’e ?) + 1
{larg a|<zx]
Da .
+ 2£ —* 5 : [{ CF ae
a” p
[|arg a|<z]
{arg a|<n]
Vn MH Pl4iy
4ap ad > [erf(ae ?
{larg a|<z]
n p/4l-e- ~p/2,
[Be i C(a’e
{larg a|<n]
% 3p/4l—-e ° ee
Re” a Se of,
a p
(larg a|<x]
EK + erfi(ae °/4 )]
36.
Sil
38.
39.
40.
41.
42.
43.
THE EXPONENTIAL FUNCTION 49
(41) '*) any
[x]! ([x]+n) %
(+1) (*) “Als
WON Ss aoa
(Ca ks a ae |
“Coane ode
ree
(2ix])! [x] 1q71
(ry)
Qi a1p)s!
[x] 1q2l41
(2[x])! oid)
Claisive
(2[x]+1)! [1
2
Cl xc ale)
(2[x ytql*)
LonuGialaips
Qixtria
Cis) (isla 1)
2 A & "(2ae? ues
ail ce
{larg a|<n]
eet? J, (dae?!) +1, (2ae?/*)
2P \2 ber (dae?! 4)
[|arg a|<x]
\ i (Qae~?!) J (ae?!)
sas sinh 7 ora
a’ p 2 bei (2ae )
[Jarg a|<x]
area + |B ae Pleexp [= ’) x
(alse)
{larg a|<x]
5
2 a sinh exp [egre#] x
{ls}
[|arg a|<nx]
_ ply - -
Hees ae he
(Re p>In(4]a|)]
fe -3/2 (1-4ae ”)
[Re p>In(4]a|)]
p a
hae ii—(i-4ae")
[Re p>In(4]a])]
-p
e ‘-1 -p,1/2 Sis eid F) ( ae“)
[Re p>In(4|a])]
50
45.
46.
47.
48.
49.
50.
Se
$2.
53.
ELEMENTARY FUNCTIONS
(+1) ! EG eek.
Clecla!s) PPIX a1)
Gen) xepet)
See !
x 2
Del»
eae 2 ots)
Chilechoiie
{x ]
-1)
isnt
Chatty: 2[x]
TONES EN BUS:
-phly
2
[all 2[x]
icine 2
21x) I
Glxtiye
yx A Qalecile)
CUPS) ~
x a2)
(eTEONE ey ts
Chali s
UESLEBLS Y
Che rts=
qzi*)
Su ES ET
lee arcsin -p/2
ap inh Abie io »}
(Re p>2In(2|a|)]
5 e?-1 farcsin (poe \
ss oe arcsinh | 2
[Re p>2In(2/|a|)]
= -p/2
l-e ? _ nae g a_-p/2
p [ 4 H (Je
[larg a|<n]
ean)
[larg a|<x]
2n p -p/2 ap sinh 5H es
[larg a|<zx]
lee ?
p
{larg a|<n]
exp(2ae ")I,(2ae ”)
=Hy
2 Fe Qae?! :
Dp
{larg a|<n}
2 Ne -p/2
x ? K(4a )
[Re p>2In(4|a|)]}
i ea 1 -p/2
K D E(4 )
[Re p>2In(4}a|)]
HYPERBOLIC FUNCTIONS
-p
4. n [x] l-e —p.n
5 (eg x D (1+ae *)
x 6(n+1-x)
n.)?-1 el? -pn, ler+a 55. a x (ae ?)"py | £2
[x] p n
e'-a
x 6(n+1-x)
2.3. HYPERBOLIC FUNCTIONS
2.3.1. Hyperbolic functions of ax
sinh ax 1 a
cosh ax Fiche Dp
[Re p>|Re a|]
—
mila [n/ 2]
oT] te? 2k+ 09707)
Pp k=0
E boat Re p>n|Re a]
. n
2. sinh ax
[(n-1)/2] # 1
3. cosh"ax ee [i]
2 k=0 —(n-2k)
1+(-1)" n
ets ee Re)
[Re p>n|Re a|]
sinh ax 2 1 2a”
4.
cosh ax p(p?-4a’) p’-2a*
(Re p>2|Re a|]
5 L i ,(ere
“ coshax a 2a
(Re p>-|Re a|]
l joe 1 7" 2 (84) -2
cosh7ax 2¢ .
[Re p>-2|Re a|]
52 ELEMENTARY FUNCTIONS
sinh ax 1 a ete sinh ab
7. 8(b-x) Memon pews) a
cosh ax Da— Ga) Da a cosh ab
cosh ab
+a
sinh ab
(6>0]
sinh ax sinh ab
ss 2 2 ae
+
cosh ax pa=a cosh ab
are cosh ab
- sinhac
a (a -e Pp 4
0 [0<x<b.or x>c] sinh ab
cosh ac
cosh ac
+a
sinh ac
9. sinh ax sinh bx oy: ea pees 2 Fon
[p?-(a+b) 7) [p?-(a-b) 7]
[Re p>|Re a|+|Re 5|]
2 2 2
10. sini ax cosh bx tb a th)
[Data Dy) lip tab) s)
[Re p>|Re a|+|Re b})
11. cosh ax cosh bx aL Peete 8) eee
[p°-(a+b)“] [p°-(a-b)*}
[Re p>|Re a|+|Re 6|]
Sinhax 1 .({ptat+b p-a+b
12. Sinhbx 75|¥| 2b el 2b =I
[Re p>|Re a|-|Re b|)
13. tanh 85 ( 2 el nh ax aba 5
[Re p>0]
14. (cosh ax-1)” 2 3(2=4 2ve1] a ’
[Re a>0; Re v>-1/2; Re p>Re(va)]
15. (cosh x—cosh by,
HYPERBOLIC FUNCTIONS 53
P'(v+1)P(p-v)sinh”b P’? (coth 5)
[-1<Re v<Re p; 6>0]
2.3.2. Hyperbolic functions of ax and the power function
sinh ax
| ee, 3
cosh ax
<3 sinh ax
2s \X
cosh ax
sinh ax
Shy bs
cosh ax
2 sinh ax
4. x
cosh ax
n-1/2{ Sinh ax
53 x.
cosh ax
_1/2{Simh ax
x
cosh ax
1k 1 sinh ax
a
Piya ) lp=a z Gay]
[Re v>-(3+1)/2; Re p>|Re al]
n+l i eae
2 2
Daa
{ (n+1-6)/2j {n+] a 2k+5
x F) Ds 2k+6] \?
[re p>|Re al; {5}
1 2ap
(peat)? ee a
[Re p>|Re a|]
2 a(3p? +a’)
Cp a4) : p(p?+3a’)
[Re p>|Re a|]
ane re =] =|"
2 n 2 2
dp p —a@
[Re p>|Re a|]
[Re p>|Re a|]
1), peta 2 In o=a
[Re p>|Re a|]
54
=f y/7)
my BS sinh ax
ns sinh ax)n
9, x
cosh ax
me. n
x sinh ax
ret
= sinh ax x S
ELEMENTARY FUNCTIONS
Tepe pe -a2y!/?
[RE p>|Re al]
Wer ob. reyes!
a cap ist (Z| x
2 k=0
x [(p-na+2ka)” 'F (ptna—2ka)"7} +
1+(-1)" n
a Na tht rob | inf]
[Re v>-1-(1+1)n/2; Re p>n|Re al)
m! n {n oe
? m+
DS PSO (p-nat+2ka)
[Re p>n|Re a]
n=1
-2n k-1{2n Foe)
ah k
x In [p?-4(n-k) a7] = Fl aleaae
2
[Re p>2n|Re a|]
n
-2n-] k(2n+1)
2 (-1) ( x
me k J
Dar (a= 94 ete) )) Gi
so a aay HS Dw
[Re p>(2n+1)|Re a|]
2
l 4a -4in(i 423)
p
[Re p>2|Re a|]
2
2
@arccoth = ie
p 4 p
(Re p>2|Re a|]
a) arccoth 4 + 1 arccoth 3a
p 4 p
[Re p>3|Re a}]}
16.
as
18.
19.
20.
21.
22.
23.
24.
HYPERBOLIC FUNCTIONS 55
1 sinh?ax
v
x
sinhax
v
x
coshax
Vv
x tanh ax
x coth ax
a sinh ax
6(b-x) x
cosh ax
sinh ax
Ne sinh ax
cosh ax
6(b-x)
x
aye arccoth 3a + 3p arccoth 4 -
4 4 p
2
p -a
[Re p>3|Re a]
GSB SE vets
v wea
[Re v>0; Re p>-|Re a|]
[(v+1) ar@t\) +3a FOL ffoontis) -e(oone24]
[Re v>-1; Re p>-|Re a|]
-v-]1
CS ga ee pta Ssearrorh|[s[v+1 4a] eve, za\|
[Re v>-2; Re p>0]
eee in D2 | 2 ine ee
4a 4a 4a
[Re p>0)
-v-1 a 1
2” ¢(v+1. 85] at
[Re v,Re p>0]
Tv+l) E
Fltp-ay yw, bp-ab) ¥
F (pta) ” 'y(v+l, bp+ab)]
[Re v>-(1+1)/2; 6>0]
i 5) Ei(—bp-ab) yin BES + 5 Bi(-bp+ab) -
[b>0}
Flip-ay el, bp-ab) ¥
¥ (pta)” 'T(v+l, bp+ab)
[Re p>|Re a|; 5>0)
56
25.
26.
27
28.
29.
30.
31.
32.
33:
ELEMENTARY FUNCTIONS
Mesa sinh ax
= cosh ax
McComas
x
l-coshax
2
a5
ax-sinhax
2
x
ax-sinhax
2
2
x
coshax-coshbx
x
coshax-coshdbx
2
x
sinhax-axcoshax
x
Sinhax-axcoshax
2
x
fel
2
(Re p>|Re al; 5>0)
1 a’
x in{1-25}
2 ne
[Re p>|Re a|]
Ei(-bp+ab) + 4 Ei(-bp-ab)
ea = a, pra
pin p 7 IN =a
(Re p>|Re a|]}
a oe Py, Pa ine
2)
[Re p>|Re a|]
| ae eK a
[er in[t- $5} «Pe in B*2 — ap]
[Re p>|Re a|]
ib 2»?
Jia sp
pea
(Re p>|Re a|,|Re 5|)
2 2
H On poy ni is paby Nant pa
pie 2 2 eb” 2 bee
{Re p>|Re a|,|Re 5]
A pi de td Jee
2 D=a 2?
Pa
[Re p>|Re a|]
a =f jyypete
2° p-a
[Re p>|Re a}]
35.
36.
37.
38.
39.
40.
41.
42.
43.
HYPERBOLIC FUNCTIONS
sSinhax-axcoshax
3
x
sinhax-2axcoshax
2
x
ey list «
x xcoshax
a 1
x sinhax
L —acoth ax
2
2 ee: 6 a
2 F 2
x sinh ax
sinhaxsinhdbx
x
sinhaxsinhbx
2
x
sinhaxcoshbx
x
3 2
sinh ax
xcosh(2ax)
il 7 Zak PHS
aK @ yin =a 2ap|
[Re p>|Re a]
2 2
B jn B44
p
[Re p>|Re a|}
2inr(2434) -2inr(2*2) - nF
4 4a
{a,Re p>0]
Pea Aa pes
| a Ino
(a,Re p>0)
[a,Re p>0]
Igo fa-)”
tos i) 2
Dima (Cat0))
(Re p>|Re a|+|Re 5|]
1
4
57
ee Dig, 2
4 jy (PO) ~a- , Big (pray db ,
Cp-h ya (p-a) *-5"
2 2
+f in? —(at+b)
poe Ca by -
(Re p>|Re a|+|Re 5|]
1, (pt+a)~-b
4h MIPS)
(p-a) -6b
{a,Re p>0]
58 ELEMENTARY FUNCTIONS
2.3.3. Hyperbolic functions of ax s for Zk and algebraic functions
1. sinh aVvx
2. cosh aVvx
» sinhavx
SK 3%
cosh avx
4. x"sinh aV¥x
7. x sinh aVx
8. x cosh aVvx
2
a\nx a
a oe (45)
[Re p>0]
an + ae exp fa ne }]
P 24p 4p Wn
[Re p>0]
2
me2 =
2} exp ($5 | Pear . B
(2p) 2Vvp
=D CS
-2v-2 2Vp
[Re v>-(5+1)/4; Re p>0]
(C1) var en ae ia
pe Le Saas |e
[Re p>0]
PES Ip
(Re p>0}
2 [Sem
25[5+$5+ "ae (3+ 3a pce )x
4p - ee)
2 eee) 2¥p.
{Re p>0]
Yna (6p+a*sexp (47)
Pre 4p
[Re p>0]
1 [i+ + gf ex0( 22) et a )]
2 4 yrs
[Re p>0]
HYPERBOLIC FUNCTIONS 59
9 1/2¥ a 22 “ie Tt at a
. x “sinh av¥x 5 gees 5 exp} Z| ert
2p Pp p 2vp.
[Re p>0]
2 2
10. x! cosh aVx 2p ia Foal
» 4p
[Re p>0]
1/4 sinhavx Kika 3/2 a2 ac
11. Se se Gs.
‘ fatiin® lb exo $5] Bawa
2
eeiell
[Re p>0}
12 bal seinhiav x FF exo (45) ex( a
Pp 4p
[Re p>0}
2
-1/2 EL ame
13. x ‘“coshavx JF exe($5}
[Re p>0]
aA sinh avx am ( ) (35)2 we eral nf 5 ex? 8p) +1/4|8p
[Re p>0]
15 1 sinh ax n erfi[
x 2vp
[Re p>0]
sinh a -2/3 3 ni/2 3ni/2, _
16. x ses oak aale S.1/3 42 )F 551/30
-1/2 {u-2p *(a/3)>”: ; Re p>0]
60 ELEMENTARY FUNCTIONS
sinh yey vti/2_-v-t
‘ 2k
17. x" (ax) Bet Clot): a2 ie Ez) as
cosh i(2n)
] 1)}AC,-v)
x (5) P) | A(k,5/2),A(k, (1-8) /2)
[pe Re v>- 1-58, Re p>0; {i \]
18. —- sinh avx aE AG inh ay amean ert pues +
xX+zZ 2 2Vp.
” orf Ee }]
2Vp.
[Re p,Re z>0]
yr i/2 avi
19. == cosh avx x 2 [2 cosh ave" “ent [¥pz+44 i|=
2vz 2vp
- €™ ert{ voz a }]
2Vp
[Re p,Re z>0]
20. se coshavx vn AP sinh’B
cosh bx vx ne ba cosh B
2
a
[+-- =a 4B-in 2* 5 -2Ab; Re pine bi|
a1, Sinh[(2n+1) avx] 74 Free ea
¥xsinhavx Fl ext 0:
{Re p>0]
22. cosh{[(2n+1) avx]
(<1) at 22
¥xcoshavx 34 PX 1) ‘expika"/p)}
{Re p>0]}
HYPERBOLIC FUNCTIONS
2.3.4. Hyperbolic functions of dee? Hee and algebraic functions
Notation: z,= 2 zped p 7)
an
- — cosh(
i (x+z)! f x
F sinh (as x 2 4+xz)
| ¥74x2z)
x
2
sinh(a4 x ~+xz)
x
cosntae x ae z)
-1/2
. see hal x7+x2z)
ted yeas
v
x
en Le
sinh(a ~ 74x27)
x 2
cosh(a4 x +xz)
yee sinh(a@ x7+xz)
i cosh( x? +xz)
[Re p>|Re a]; |arg z|<n}
HZ,
IO
Cp —a~)z
[Re p>|Re al; |arg z|<nx]
72
IE asn/2(PttP -@)
ae
ee
p“-a
[Re p>|Rea|; |arg z|<n]
2x, (§ pone
[Re p>|Re al; |arg z|<nx)
G2 18) / 2
2
zva
<M (z_)W
v+3/4,+1/4 —v-3/4,1/4
¥#1/2
U[vesge| ox
(z,)
61
exp(z_)
[Re v>-(5+1)/4; Re p>|Rea|; |arg z|<nx]
erf(¥z_)
2 ete¥=
1 Zz
[Re p>|Re a]; |arg z|<x)
62 ELEMENTARY FUNCTIONS
2... =3/4 ee 2/2 Fie Faas
TOG RRAN 52s “re heb ne
sinh(a x2+xz)
< 7} [Re p>|Re a|; |arg z|<nx]
cosh(@aax +xz)
1 v 2)" ays v/2 ype —v/2 pel
. ae [(+3 é tx2] ¥ 3] ie Sleed) | 2
XX: Z
# (x45 1x? 4x2] | x xK 4 [#e?- p a’
sinh(aa x Le 2
x (Re p>|Re a|;_ |arg z|<x]
cosh(ad x2 +Z .)
-1/2 GES 7) 9) Af AO)
oe Seal ue ly ise 4
sinh(a x 7427)
2 2 94
\cosh(atx +z")
[Re p>|Re a|; Re z>0; ved Dp 2 —a ? +ia)]
2.3.5. Hyperbolic functions of oleh eax? and algebraic functions
Notation: u,= aod p?+a 2+),
= biptd ) 2 —a a
2, - ri 1. (bx-x yy me gals pr+az
x cosh(at bx -x 7) [b>0}
HYPERBOLIC FUNCTIONS 63
=)
x costed bx-x aN
. O(x-b) x’ sinh(a pee ea
» by? x
X cosh(a rh}
; 6(x-b) x
Vx+b
ee x 2 5 |
x
cosh(ad x 4 -—b a)
(x-b) |
(eeb) Sue
sinh(a 6p 2 al OF
x ae
cosh(atx~-5b”)
2
Gaby ats
x x+b
sinh(a -h 5
x
coshta x ny zy
3
_ANN p
EU gt ;4%) ~
Sfp)
[5>0]
head “be bd 2a
(prema?) (oot) 72 wont PeReS?
[o=0 or 1; 6>0; Re p>|Re al]
2 2
p -a
2 2)1/2
[2 eee seh ored)
[b>0; Re p>|Re a}}
qeitly/2, 21/2 F1/2
It soe
2 x
pasa
X exp(-Hi p ‘ =a 25
[b>0; Re p>|Re a}]
4 ) M,,3/4,41/40 X
AW 3/4ipae
[b>0; Re v>-1; Re p>|Re a|]
erfc(Vvv_)
x ePeric(V ,) *
¥26 1
[b>0; Re p>|Re a|]
64 ELEMENTARY FUNCTIONS
o
= bp) 2g?
8. x0 (x?-B?) x > 3 s72 Ko! D— a.)
(p= —a™)
xX cosh (a eb) [o-0 or 1; 6>0; Re p>jRe al]
eye : ”), o
9. ve ae Ort 1/4| 2 1/4|2
xX cosh( eSB) [b>0; Re p>|Re al]
v v
20 2 =3 14 a _- Mae
he ae hil )Kunalz
7) 7) [b>0; Re p>|Re a|)
cosh(aix -b”)
es at
x
2.3.6. Hyperbolic functions of ax, the power and exponential functions
_by. y{ Simh ax 1 ye a
1. (l-e) F518 [v+t, 254] -B(v+1, 2¢4)]
(cosh ax 26 b b
[Re v>-(3+1)/2; Re p>|Re a|,|Re a|-Re(bv)]
nebx ie. ee pra) >. | psa 2. (l-e’”) sinh ax 5 ¥( ; vf b }]
[Re p>|Re a|,|Re a|+Re 5)
85 -cx 2 2
oh a cosh ax doi pte) as aa
(p+b)~-a
[Re p>|Re a|-Re b,|Re a|-Re c]
= 2 a4
sinh ax Sy eee By aed In erate + : (ped) eee 2 p-atb
a bse in Ptare
p-atc
(Re p>|Re a|-Re 6,|Re a|-Re c]
HYPERBOLIC FUNCTIONS 65
5. ehh Se oo shax 1, (pte)? =a”
. (p+b)?
[Re(p+b)>0; Re(ptc)>|Re al]
6. x (ae *sinh cx - Ly oe, + fi ptd) ; ptatd _
(oes) 2 22 p-a+d
Ere ainian) _a(prtb) ) ptbte
2) p+b-c
[Re(p+b)>|Re c|; Re(p+d)>|Re a|]
2 sinh ax
7. exp(—bx’) LF exw 252) erfe{2=2] +
cosh ax 2WVd
2 .
Pexb ane ae ae)
2vd.
[Re 4>0]
sinh ax 2 2
(v+l1) +a a 8. x*exp(—bx”) ee EXD (2552-] [exe(- $3) x
eashiae 2(2b) 6% t!)/2 85 46
x. (55) #0 [75]>_.. (Z| —~v-1 YIb 46 ~-l YI5.
[Re 6>0; Re v>-(3+1)/2]
sinh ax 2
9. x exp(-bx’) ee (p-a)exp Cer ertcl ao =
8} ,.3 4b
cosh ax b 2Vvb
+ (pt+a)exp (‘eze2-) erfc bal + ie
2vb
[Re 5>0)
sinh ax J 10. x vb/x pre ipa) eT Rs (bp-ab)
cosh ax
= (tay eK (2¥b p+ab))
[Re b>0; Re p>|Re a|]
66 ELEMENTARY FUNCTIONS
sinh ax a : =
He 7/2678! ee = [o-o 3/4,-20bp-ab ~
cosh ax 26 :
¥ (pta) (o-3)/4 -2¥ bpt+ab |
{[o-0 or 1; Re d>0; Re p>|Re al]
l+ax-e"* p+a)_ p_)_..(pta
12. xsinhax 2inr [Fe 2int 2a a! 2a
[a,Re p>0}
2.3.7. Hyperbolic functions of Bee for I#k, the power and algebraic
functions
a sinh avx ape 2 e
lie oe 3 [85 [@-wer (5S Jerte( 4 2 +
cosh avx Pp P 2Vvp.
2 =
+ tarexp( PFO] erte( +4} ve eas
[Re p>0]
2, Pert * grat POMD op (5) [exe(- $2] x
cosh avx (Qn; | SP #
b-a) — ab b+a Dav a(2=2) #ex0(24) 0, (24) 2v-2 von 4p} -2v-2 YIp
[Re v>-(5+1)/4; Re p>0]
inh
Gel mG 2] [exo( 2 erte(®=2) 7
vx cosh avx 24 p 4p Vp.
2 of ot)
[Re p>0]
HYPERBOLIC FUNCTIONS 67
_b/x | Sinh(a/x)
4. xen seme aA (2Vbp-ap) ¥
cosh(a/x) Becks My
¥ (b+a) CK avbprap)|
[Re b>|Re a|; Re p>0]
5, x 0/22 b/ xy : —- = [o-a (o-3)/4,-2V°b p-ap ~
2p o- r
sinh(a/x)
x begs (o-3)/4 -2¥ bpt+ap |
cosh(a/x)
[o=1 or 3; 6>0; Re b>|Rea|; Re p>0}
% : sinhd
6. x ee DLR YO exp(—u,)
coshd
sinh(ax+c/x) , fei
x [a- mr in2 tt wars ((b+c) (pta) / (b-c) (p-a)) “4+
cosh(ax+c/x) F a Fe
+rs((b-c) (pa) / (b+c) (p+a))
re(prea®y'4; sa(b-c?)"”4;
Re b>|Rec|; Re p>|Re ai]
2.3.8. Hyperbolic functions of [x]
aD =p.
‘is 6"! sinh ax] l-e Be sinha —
P 1-2be "coshat+b“e “?
[Re p>In|b|+|Re a|)
-_ ae = me 2. bl cosh a[x] l-e l-be ‘“cosha
Dy toane Ceosharh e 7"
[Re p>In|b|+|Re a|]
He ne
e t-2.0.
3. sayin a(x] 474 — arctanh —¢ S
t aha
x] 1-be ?cosha
(Re p>In|b|+|Re a|]
68 ELEMENTARY FUNCTIONS
[x] ca ee = x
4. b cosh a[x] “ if u In(1-2be Pcosh atb’e 2» )
[Re p>In|b|+|Re aj]
—£ exp (be cosh a) X
Mee a
y,
5.
5 fx) ieee Taeae
[x]! cosh(a[x] +c)
cosh(be "sinh a+c)
2.3.9. Hyperbolic functions of f (e*) and the exponential function
1
Notation: 6=
0
sinh 6 2 : -x a (DEP), NN i DEO a ag
f oc } p+6 ey pay Saban y ae 27
[Re p>-(1+1)/2}
Sinn
I (lee (ae *) B ¢ VED) | FO; pt+v+l1; a) ¥
cosh
+ \F (p; ptv+1;-a)]
(Re v>-1; Re p>-(1+1)/2]
% sinh _ 5
3) eam (ae) 5-8 (255,v+1] x
cosh
2
p+6,1 45 p+6 ae x, 9] 37 +6, 9) tv+1; 7
[Re v>-1; Re p>-(1+1)/2]
4. (en) ¥nkT(v+1) ct! ta 2k
i 6 1 v+l 1,2k+l k
sinh _ A(i,1-p) , fos mal P
cosh A(k, 5/2) ,A(k, 1-8) /2) ,A(l,-p—v)
(Re v>-1; Re p>-/8/(2k)]
wr
eet fe) a (oe) me a ‘SJ = a
=
& ~ —
| bss)
1 *
~
ye
10. tanh 2
)
HYPERBOLIC FUNCTIONS
-x, 1/ (2k) |
—e J
Pp hy Seas
[Re p>0]
v¥n(2 as ) PVT had el
a i 2) P
+1/2
[Re p>0]
a°B [F+v+1.7] x
5 8 lind x ,F, (5 tv+]; 7 +v+p+1,6+5; +}
(Re v>-(5+1)/4; Re p>0]
f 2k
Y¥UKT(p) Gk! (52)
re pe 1,2k+1] | 2k
A(,-v)
A(k, 5/2), A(k, (1-6) /2) ,A(,-p-v)
[Re v>-1-/5/(2k); Re p>0]
(1+1)/2 2
hal F (151+ Reed 2)
p+ pe: 4
70 ELEMENTARY FUNCTIONS
2.3.10. Functions containing the exponential function of hyperbolic
functions
1. exp(-a sinh x)
2. exp(-— a cosh x)
ee exp(—a cosh Vx)
vx
ie eS exp(—a sinh x)
¥vVsinhx
ee exp|-
¥Vsinhx sinhx
6. sinh“bx exp(-a coth bx)
qt CSC prt [J (a) - J (a)
(Re a>0]
Tt
CSC pr f cos (px)exp(a cos x)dx —
0
- nt (a)
(Re a>0]
120 aa R
e'sec FKaia(3|
[Re a>0}
| 3
ua a a
8 ly (1-2p)/4 [5] J s2p)/4 [5] ss
a a
* OEE Y “(sop /4 (5)]
(Re a>0]
v2T [> 5 Den ay 2a) xX
-ni/4
XD_) 1/2 ¥2a)
{Re a>0; Re p>-1/2]
neh
46
-—vb
x Roos oe ae
x W
—(b+p) / (2b) ,v/ 120)|
(v-2)/4
lear): TeJES)
[Re a>0; Re p>Re(vb)]
TRIGONOMETRIC FUNCTIONS
2.4. TRIGONOMETRIC FUNCTIONS
2.4.1. Trigonometric functions of ax
a
Notation: a
jee
' (|
1‘
are
cos ax
ny ap
[Re p>|Im al]
2. |sin ax|
se = coth 32
Deira
[Re p>|Im a|]
3. | cos ax| 7} 5 [0+ acsch 32
p +a
[Re p>|Im a}]
Prantl [n_/ 2]
-
4. sin"ax
sie: I] [p24
(ka) 202] 1
24-1
Pp k=0
[Re p>n|Im a}]
[(n-1)/2]
5. cos"ax
a (;)
bd
2 rhe po+(n-2k)
“a
Eo | n
9 htehy [n/2]
[Re p>n|Im a|}
2
sin ax 4
; oe
6.
por es ae
COS ax
p(p?+4a’) pie2e
[Re p>2|Im a|}
x inh 22 7. |sin ax|” 2
——[sint .
| | 7 Seta
Va
xB( 12 gee yaa ia)”
a
[Re p>Re v Im a]
71
10. 6 [3--] cos x
sinx
11. 6(n-x)
cos x
. sin ax
2:
COS ax,
[b<x<c]
0 [0<x<b or x>c]
13. 6 [5-5] sin” x
ELEMENTARY FUNCTIONS
1 a gaoP sinab
ren - f
pape p p ag cos ab
cos ab
a @
sin ab
(=) / 2 eee
D ——____
p?+l
sinab
cos ab
cos ab sinac
+a
sinab cos ac
cos ac
ta
sin ac
[n/2]
n! 2 P<
Pa [p +4(k+))"] {1- pz 1 abl
n-1 [n/2)
ine, *exp| (- 1) os] [i y (p* ANA)
X (p or fers Ns )...(p 2401)" ix
x [(21+2A)!] “yi
TRIGONOMETRIC FUNCTIONS 73
14. 8 (3-+) cos”"x
15.
16.
17:
18.
19.
20.
6(n-x)sin’x
eEEew,
6(m-x)sin xcos x
-_ tt
6(mn—x) sin ‘ax
sin ax sin bx
sin ax cos bx
cos ax cos bx
[n/2]
! =
a I] [p? + 4(k+a)7] 7! x
k'=0
2-1
Pp
{terns pi ag
[n/ 2]
+ Y (p2+4n) (p?+4(041)%)...x
1
X wn (D-+4(A41-1)7) [(21+24) 1] “y}
z -pxn/2F¢ a ae -1
ue ——_[p(2452 + 1, 2-2, 1)
2” (v+1)
[Re v>-1]
heey -pn/2 -= =
D : mpe : [a(t 4 =|,
Pp +(V+1)
[Re v>-1]
n! 2m _-mpx
ear) e )x
p
[n/ 2] i
x |] 4ceyy 7
k=0
2abp
[p= (AHRe ) [p74 (apes
(Re p>|Im a|+|Im 5}]}
api +a"—b7)
[p?+(a+b) 7] [p*+(a-b) 7]
[Re p>|Im a|+|Im 5}]
p(p’+a*+b")
(paar) 1 tp +(a-6) °}
[Re p>|Im a|+|Im 5}]
74
21.
22.
23.
24.
25.
26.
27.
28.
ELEMENTARY FUNCTIONS
sin bx -n-2 ae
Lon 2 1¥#142n_.
sin of ‘| PES IP exp aes ni} x
b+ipt+na ,|ociptna -
x 2a + (-1) 2a
n+l n+1
[a,6,Re p>0]
sinax Le les O_o ((W=O=a ie
sinbx 25|¥| 25 vl 36
[b0; a,Re p>0]
. [n/ 2] ny 24-1
sinnx 2 _ Ise Geil}
sinx 2p py [p+ (2k 2 \] m
esl (a Ts
Fea Te
[Re p>0]
2n+1 foe ae eel cos [(2n+1)x]tan a Se Se ae y C2 eet)
p +(2n+1) k=0 p Det Dy
[Re p>0]
co = k -kb
1 i (Fl) e
a Te csch o[ 4+ 2p = coshbtsinax P ay one
[a,Re p>0}
3 OO: yn k kb sinax a (F1) “ke
Goshhisa nae + 2a — coshdb+sinax ye eee
[a,Re p>0]
oe eee 1 [2s y (-b)*
1+2bcosaxt+b’ Bi ba k=0 p +kig-
(15|<1; @,Re p>0]
oo k-1 sinax = error.) a 1+2bcosax+b k=1 p'+ka
L15|<1; @,Re p>0]
TRIGONOMETRIC FUNCTIONS
2.4.2. Trigonometric functions of ax and the power function
sin ax
I. 5 Fixet ) fe ten tia) a
COS ax
¥ (p-ia) |) =
: Piv+l) sinu
On ag 5 aah Re ah eae
[u=(v+l)arctan(a/p); Re v>-(3+1)/2; Re p>|Im a|]
sin ax n+]
Dax nt rear x
COS ax pas
[ (n+1-5)/2] k n+] e 2k+5
x y (-1) [<]
k=0 2k+6] \?
[5=(1+1)/2; Re p>|Im a]
sin ax 2ap
3% _Lin bi infos:
7 2 DRAB) 2 2
COS ax (peta =) Daa
{Re p>|Im a|]
2 Sin ax 2 a(3p?-a’)
4. x a Te
COS ax (p +a’) 4 p(p?-3a’)
{Re p>|Im a|]
ifs sin ax Cyn | 2 at 1/2
SX Me 2 2
cOS ax "ey eae!
[Re p>|Im a}]
i = 1/2
cOS ax p +a
(Re p>|Im a}]
7 Af in a: arctan 4 - Sina: Pp
[Re p>|Im a|]}
76
8.
2:
=3/ 2m
x sin ax
simax)n
Vv
x
cOS ax
ELEMENTARY FUNCTIONS
Via peepee
(Re p>|Im a|]
((n-1)/2]
T(v+l1) —,. (n/2]-k
ale Ae (+1) x
[p24 (n-2k)2y OP? 5
sinu n
n Nata Ga) n
(ie. } 2 epee oD (72)
(u=(v+1)arctan((n-2k)a/p); Re v>-1-(1+1)n/2;
Re p>n|Im a}]
n+1n-1
oo Esler
2 k=0
2n
x In preain-ntt | | 7) In p
n
ipy| 2
[Re p>2n|Im a]
(-1)" ee
(-1) x a2 Dy. k
x arctan 22=2k+1) @
p
[Re p>(2n+1) |{m a]
2
qia(1 +42)
p
[Re p>2\Im a}]}
2a_p 4a’ @ arctan — -* In} 1 + pein(t +425)
[Re p>2|Im a}]
farctan 2 - } arctan 3a
a)
[Re p>3|Im a|]
15.
16.
17.
18.
19.
20.
21.
TRIGONOMETRIC FUNCTIONS
Liaw.
—> sin ax
x? (26-x)1/? x
sin(ax—ab)
x
cos (ax—ab)
Oe: sin ax
COS ax
a[x- Ab sin x
PB 3a_3p a 4 arctan P 4 arctan D +
Zz 2
3a, p_+3a + In
8 DED
p ta
[Re p>3|Im a|]
jAtelyi2 +
3 | pia)” yp(v+1, bptiab) F
¥ (p-ia) gl wrt, bp-iab)}
[Re v>-(3+1)/2; 6>0]
j (ll /2,2 Fe ‘
— aoe [1 (p—iab) rs I, (optiab)]
[b>0]
Ses ewe: a
Sg Kae I'(v+1, bp+iab) +
¥ (p-ia) “'T(v+l, bp-iad)]
[Re p>|Im a|; 5>0}
-pxn/2
£230" p+9"-1]
(p +1)
(Re p>0]
-pxn/2
£ 7) 5[Fu*+D+29]
(p +1)
[Re p>0]
2
in{1 |
p
[Re p>|Im a|]
brRo|—
77
78
22.
23.
24.
25.
26.
27.
28.
29.
30.
ELEMENTARY FUNCTIONS
l=cosax
2
x
39S INGE
2
x
ax-sinax
3
x
cosax-cosbx
x
cosax-cosbx
2
x
SUNGX=@xC O'S'G%
x
Sinax-axcosax
24
x
Sinax-axcosax
sinax
X (2ax cos ax-sin ax)
2
[@ias Pi a_ a arccot pao in{1 + “|
{Re p>|Im a|]
2
a as Le $ in( +5] + p arccot © a
{Re p>|Im a}]
1 ap Inj 1+ a + (p*+a’)arccot oe ap
Bee Ae a
(Re p>|Im a|]
is pith?
pees 2
pero,
(Re p>|im a|,|Im 5]
Dee?
|, yy es arctan “ + b arctan 4
2 ee aed P
{Re p>|Im a|,|Im 5]
arccot 1g a
Dea
[Re p>|Im a|]
= 12 a — parccot -
[Re p>|Im a]]
1 DED
AKG +a) arccot = = ap|
[Re p>|Im a|]
2
Pia| tees
p
[Re p>|Im a|]
TRIGONOMETRIC FUNCTIONS 79
. E 2 2
31. Sinaxsinbx qinb thar b)
Dt GaeD))
[Re p>|Im a|+|Im |]
32. Soe 4 arctan —2bp__ + arctan “22+ 2 Dy 2 2
pta —b Vat Si
p +(atb)
[Re p>|Im a|+|Im }|}
33. singxcoshx F arctan $22 eels
p-a a is
[Re p>|Im a|+|Im 5]; +p +b'¥a>0]
: ; : 1/k : ,
2.4.3. Trigonometric functions of ax ~ for /4#k and algebraic functions
F 22
sin ax sinu cos u
1. 2 [z. [7 -sw| + [2
COS ax COS uu sinu
x [5 = zu
(u=p’/ (4a); a,Re p>0]
7 Z
sin ax a itale) ey) 2 2
v i iv+l) Victor De
2. x 2 aay 2 {exp -i{ 4 n=) | x
cos ax 2(2a)
xD | p ew") s
Pe) a
(Re v>-2¥1; a,Re p>0]
ELEMENTARY FUNCTIONS
2 5
sin ax 1 sinu
3.x 2 : ,—Vap
[2 -cw] =
a 3/2 2
cos ax 0 (2a) cos u
cos u 1
F{ E - sw
Sinu
[u=p’/ (4a); a.Re p>0]
Sie anyl De
a n| 1 p? 2
yx SS pera Eolas eal
[a,Re p>0)
2
. sinavx 3[23 e(- nl
Pp
{Re p>0]
2
- cos avx ari ks [Fs ( a ] p erfi
Pape We 2vp
[Re p>0]
4 2,2
- lanavx 2/5 yy Dk exp(-4 K )
p” k=) P
[Re p>0]
10.
ex
13.
14.
16.
Ww.
x sinavx
n-1/2
cos aVx
é x?! "sin avx
x sin avx
x cos avx
Wiran
» x Sin av x
x! 2695 avx
TRIGONOMETRIC FUNCTIONS 81
a sec vit vi,@
9 Rel es, 9% CAP 8p .
CSC VIt
a
x Ean ee Rese (- =i)
[Re v>-(5+1)/4; Re p>0]
(-1) "va ( -] ( a
A a ONY Roe pel ==
7 ae 4p) 2n+1 2p.
[Re p>0]
2
[Re p>0]
(-1) "Va a’ a
2n AytH 2? |” ap H,,
Pp 2vp
2
=F Seegyt Tela 3a" AR
4p° P P bag?
2
xX exp - | erfi Gal
4p 2Vp
[Re p>0]
e 2
ts (6p =a “exp - ss]
p
[Re p>0]
[Re p>0]
2 2
4, + Pee a [Fs exp(- $5} erfi[ a }]
4 p> 4p 2VD.
2p?
[Re p>0]
[Re p>0]
82
sinavx
ieee
cos aVx
18. pa avx
19. x W205 avx
sinavx
20. 3/4
bese avx
Di ei avx
x
sin ax!
ip, gel wt
cos ax '~)
23.
sin
Fu es C
cos
ELEMENTARY FUNCTIONS
[Re p>0]
It a 2 a a — 4 _| erfi [5 exe ie ee
[Re p>0]
2 Freo(- 43 Pp 4p
[Re p>0]
2 2
a a a
n] $5 exp(- D ler
[Re p>0]
x erf ES
2Vp
[Re p>0]
+ 3, (itl /2
ni/4 3ni/4. _
2a [e So1jg4e) F
{u=2(a/3)"p ”*; Re p>0]
G 4 ate
(2n) 2k EI) r 2 1,2k
. (4) L AC, v)
\P) | A(k,6/2),A(k, 1-8) /2)
[S5=(1+1)/2; Re v>-1-/5/(2k); a,Re p>0]
vaE eos 2 Stays (5 yx
TRIGONOMETRIC FUNCTIONS
24. a sin aVx
Pa
25. ao = COSY
sin bx VE
26. cosavx
cos bx Vx
D7 oS,
x Sin[(2n+1) avx]
sinaVvx
oR)
y £08 [(2n+1) avx]
cosavx
ie el 2
29.
1-2bcosavx+b"
ia i g |i:
2Vp.
~ 6 ert[¥pz- Z } - 2sinn av7]
Vp 2Vvp
[Re p,Re z>0]
= fle ert[¥pe+ 3 )+
2Vz 2Vp
+ oe" erte| ¥pz- Z }]
2vp
[Re p,Re z>0]
Vie oA sinB
CN eal SE cos B
[4(p +6) A=a’, 2B=arctan(b/p)-Ab;
a>0; Re p>|Im d{]
felt +2 Y exe(- ms ]]
{Re p>0]
n
(-1)" alt 2) -rytexp(-452]]
P ei P
{Re p>0]
2 iss 2
2 | 1 k ka
saatale oe exp(- Zp
(|d|<1; a,Re p>0)
83
84
2.4.4. Trigonometric functions of ax
sin(a/x)
cos
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