Tables of Fourier Transforms and Fourier Transforms of Distributions

260 Pages·1990·8.86 MB·English

by  Fritz Oberhettinger (auth.)

Fritz Oberhettinger
Tables of
Fourier Transforms
and
Fourier Transforms
of Distributions
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo Hong Kong
Fritz Oberhettinger
Professor emeritus. Oregon State University
P. O. Box 84. Seal Rock. OR 97376/USA
Mathematics Subject Classification (1980): 42A38. 44A1 0. 44A1 5
ISBN-13: 978-3-540-50630-0 e-ISBN-13: 978-3-642-74349-8
001: 10.1007/978-3-642-74349-8
Library of Congress Cataloging-in-Publication Data
Oberhettinger, Fritz. (Tabellen zur Fourier Transformation. English) Tables of Fourier
transforms and Fourier transforms of distribution/Fritz Oberhettinger. p. cm.
Rev. and enl. translation of: Tabellen zur Fourier Transformation. 1957.
1. Fourier transformations. 2. Mathematics-Tables. I. Title. QA404.0213 1990
515'.723-dc20 90-9507
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of
illustrations, recitation, broadcasting, reproduction on microfilms or in other ways,
and storage in data banks. Duplication of this publication or parts thereof is only
permitted under the provisions of the German Copyright Law of September 9, 1965,
in its current version, and a copyright fee must always be paid. Violations fall under
the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1990
2141/3140-543210- Printed on acid-free paper
Preface
These tables represent a new, revised and enlarged version of the
previously published book by this author, entitled "Tabellen zur Fourier
Transformation" (Springer Verlag 1957). Known errors have been correc-
ted, apart from the addition of a considerable number of new results,
which involve almost exclusively higher functions. Again, the follow-
ing tables contain a collection of integrals of the form
(Al J f(x)cos(xy)dx Fourier Cosine Transform
o
(B) J f(x)sin(xy)dx Fourier Sine Transform
o
(C) ge(y) = J f(x)eixYdx Exponential Fourier Transform
-00
Clearly, (A) and (B) are special cases of (C) if f(x) is respec-
tively an even or an odd function. The transform parameter y in (A)
and (B) is assumed to be positive, while in (C) negative values are
also included. A possible analytic continuation to complex parameters
y* should present no difficulties. In some cases the result function
g(y) is given over a partial range of y only. This means that g(y)
for the remaining part of y cannot be given in a reasonably simple
form. Under certain conditions the following inversion formulas for
(A), (B), (C) hold:
(A' ) f(x) = 2 J g (y)cos(xy)dy
11 0 c
(B') f (x) 2 J gs(y)sin(xy)dy
11 0
-1 00 -ix
(C' ) f(x) = (211) J ge(y)e Ydy
In the following parts I, II, III tables for the transforms (A),
(B) and (C) are given. The parts I and II are subdivided into 23 sec-
tions each involving the same class of functions. The first and the
second column (in parenthesis) refers to the location of the corre-
spondent page number for the cosine- and sine transform respectively.
*The domain of analyticity is the strip in the direction of the real
axis of the complex y plane.
VI Preface
Compared with the before-mentioned previous edition, a new part IV
titled "Fourier Transforms of Distributions" has been added. In this,
those functions f(x) occuring in the parts I-III have been singled
out which represent so-called probability density (or frequency dis-
tribution) functions. The corresponding normalization factors are
likewise listed.
The author wishes to express his thanks for the expertise of
Mrs. Jolan Eross in the completion of this book.
Seal Rock. January 1990 Fritz Oberhettinger
Contents
Part I. Fourier Cosine Transforms (Tables I) 1
1. 1 Algebraic Functions .......................................... 3
1.2 Arbitrary Powers ............................................. 8
1.3 Exponential Functions 11
1.4 Logarithmic Functions 16
1.5 Trigonometric Functions ...................................... 20
1.6 Inverse Trigonometric Functions .............................. 32
1.7 Hyperbolic Functions ......................................... 33
1. 8 Orthogonal Polynomials ....................................... 44
1.9 Gamma- and Related Functions ................................. 47
1.10 The Error- and the Fresnel Integrals ......................... 48
1.11 The Exponential- and Related Integrals ....................... 53
1.12 Legendre Functions ........................................... 56
1.13 Bessel Functions of Arguments x, x' and l/x ................... 66
l:;
1.14 Bessel Functions of Argument (ax' + bx + c) ................... 77
1.15 Bessel Functions of Trigonometric and Hyperbolic
Arguments .................................................... 83
1.16 Bessel Functions of Variable Order ........................... 86
1.17 Modified Bessel Functions of Arguments x, x' and l/x .......... 89
l:;
1.18 Modified Bessel Functions of Argument (ax' + bx + c) .......... 93
1.19 Modified Bessel Functions of Trigonometric and Hyperbolic
Arguments .................................................... 97
1.20 Modified Bessel Functions of Variable Order .................. 100
1.21 Functions Related to Bessel Functions 102
1.22 Parabolic Cylinder- and Whittaker Functions .................. 106
1.23 Elliptic Integrals ........................................... 109
VIII Contents
Part II. Fourier Sine Transforms (Tables II) ....................... 113
2.1 Algebraic Functions ........................................... 115
2.2 Arbitrary Powers .............................................. 120
2.3 Exponential Functions 124
2.4 Logarithmic Functions 129
2.5 Trigonometric Functions ....................................... 133
2.6 Inverse Trigonometric Functions ............................... 144
2.7 Hyperbolic Functions .......................................... 145
2.8 Orthogonal Polynomials ........................................ 152
2.9 Gamma- and Related Functions .................................. 155
2.10 The Error- and the Fresnel Integrals .......................... 155
2.11 The Exponential- and Related Integrals ........................ 158
2.12 Legendre Functions .....................................•...... 160
2.13 Bessel Functions of Arguments x, x 2 and l/x .................... 164
!:i
2.14 Bessel Functions of Argument (ax 2 + bx + c) .................... 173
2.15 Bessel Functions of Trigonometric and Hyperbolic
Arguments ..................................................... 178
2.16 Bessel Functions of Variable Order ....................................................... 179
2.17 Modified Bessel Functions of Arguments x, x 2 and l/x ...................... 180
!:i
2.18 Modified Bessel Functions of Argument (ax 2 + bx + c) .... , ...... 183
2.19 Modified Bessel Functions of Trigonometric and Hyperbolic
Arguments ..................................................... 187
2.20 Modified Bessel Functions of Variable Order ................... 188
2.21 Functions Related to Bessel Functions 1.89
2.22 Parabolic Cylinder- and Whittaker Functions ................... 192
2.23 Elliptic Int~grals ............................................ 194
Part III. Exponential Fourier Transforms (Tables III) .............. 197
Part IV. Fourier Transforms of Distributions (Tables IV and V) ...... 209
Appendix ........................................................... 249
List of Functions ..............................................•... 258
Part I
Fourier Cosine Transforms
(Tables I)
1.1 Algebraic Functions 3
1.1 Algebraic Functions
f (x) gc(y)
-1
1.1 1 x<a y sin(ay)
0 x>a
1.2 x a<x<b y-1(b sin(by)-a Sin(ay )]
0 x>b
+ y-2(COS(bY)-COS(ay )]
1.3 x x<a 4y-2cos(ay)sin2(~ay)
2a-x a<x<2a
0 x>2a
1.4 0 x<a -Ci(ay)
-1
x x>a
1.5 x-~ (~1T/Y) ~
1.6 x-~ x<a (21T/Y) ~C (ay)
0 x>a
1.7 0 x<a (21T/Y) ~{~-C (ay) }
x-~ x>a
-1
1.8 (a+x) x<b cos (ay) (Ci (ay+by)-Ci (ay) ]
0 x>b + sin(ay) (Si (ay+by)-si (ay) ]
1.9 0 x<b -sin(ay)si(ay+by)+cos(ay)Ci(ay+by)
-1
(a+x) x>b
-1
1.10 (a-x) x<b cos (ay) (Ci (ay)-Ci (ay-by) ]
0 x>b + sin (ay) (si(aY)-Si(aY-bY) J
a>b
loll 0 x<b a -1 ( cos(ay)Ci(ay+by)
-1 -1 + sin(ay)si (ay+by)-Ci (by) 1
x (a+x) x>b
J
4 I. Fourier Cosine Transforms
f (x) gc(y)
-n ( -y )h-l
1.12 (a+x) (cos(aY+~TIn)si(ay)
(n-l) !
n=2,3, •••
- sin(aY+~TIn)Ci(ay»)
n-l
( m-l)') -m n-m-l
+ I (n-l) i a sin (~TIn-~TIm) (-y)
m=l
(_y)n-l
1.13 0 x<b (cos(aY+~TIn)si(ay+by)
(n-l) !
-n -sin(aY+~TIn)Ci(ay+by»)
(a+x) x>b
n-l
n=2,3, ••. (m-l) ! -m n-m-l
+ I
(n-l) ! (a+b) sin (~TIn-~TIm-by) (-y)
m=l
1.14 x~(a+x)-l (2Y/TI)-~-TIa~cos(aY){1-C(ay)-s(ay)}
- TIa~sin(ay){c(ay)-s(ay) }
-~ -1
1.15 x (a+x) TIa-~(cos(ay) {l-C(ay)-S(ay)}
+ sin (ay) {C (ay) -S (ay) })
1.16 (a+x) -~ x<b (2TI/y)~(cos(aY){C(aY+bY)-C(ay)}
0 x>b + sin(ay) {S(ay+hy)-S(ay)})
1.17 0 x<b (2TI/y)~(sin(ay){~-S(ay+by)}
-~ + cos(ay){~-C)ay+by)})
(a+x) x>b
1.18 (a+x)-3/2 2a-~-(2TIy)~(cos(aY){1-2S(ay) }
- sin z{1-2C (ay) })
-~
1.19 0 x<a (2Y/TI) {cos (ay) -sin (ay) }
-~
(x-a) x>a
-1 ~
1.20 x (x-a) x>a ~ ~
(~TI/y) {cos(ay)-sin(ay) }-TIa {l-C(ay)-S(ay)}
0 x<a
-1
1.21 (a-x) cos(ay)Ci(ay)+sin(aY){~TI+Si(ay)}
(Cauchy principal value)
-1<
1.22 0 x<a TIa 2{l-C (ay) -S (ay) }
-1 -~
x (x-a) x>a