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TRANSFORM ANALYSIS OF GENERALIZED FUNCTIONS
NORTH-HOLLAND MATHEMATICS STUDIES 119 Notas de Matematica (106)
Editor: Leopoldo Nachbin
Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD
TRANSFORM ANALYSIS OF GENERALIZED FUNCTIONS
0. P. MISRA Indian Institute of Technology New Delhi India
and
J. L. LAVOINE Maitre de Recherche au C. N. R.S. de France
1986
NORTH-HOLLAND -AMSTERDAM NEW YORK 0 OXFORD
@ Elsevier Science Publishers B.V., 1986
All rights reserved. No part of this publication may be reproduced, storedin a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.
ISBN: 0 444 87885 8
Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands
Sole distributors for the U.S.A. and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VanderbiltAvenue New York, N.Y. 10017 U.S.A.
Library of Congrerrs Cetdo&g-inPublicatiin Data Misra, 0. P.
Transform analysis of generalized functions.
(North-Holland mathematics studies ; v. 119) Bibliography: p. Includes index. 1. Distributions, Theory of (Functional analysis)
2. Transformetions (Mathematics) I. Iavoine, J. L. (Jean I,.) 11. Title. 111. Series. Q&324,M57 1986 515.7'82 05-27389 ISBN 0-444-87885-0 (U.S. )
PRINTED IN THE NETHERLANDS
PREFACE
It is a well known fact that the creation of the theory of
distributions by the French mathematician Laurent Schwartz (see
Schwartz L11) is an event of great significance in the history of Modern mathematics. (The numbers in square brackets indicate the
reference of works given by author mentioned in the bibliography at
the end of the book.) In particular, this theory provides a rigorous
justification for a number of manipulations that have become quite
common in technical literature and also it has opened a new era of
mathematical research which, in turn, provides an impetus to the
development of mathematical disciplines such as ordinary and partial
differential equations, operational calculus, transformation theory,
functional analysis, locally compact lie groups, probability and
statistics etc. However, in recent years the mathematization of all sciences and impact of computer technology have created the need to
the further developments of distribution theory in applied analysis.
In order to shed light on this work we confine ourselves to the study
of generalized functions and distributions in transform analysis
which constitutes the bulk of the present book. It conveniently
brings together information scattered in the literature, for examples
distributional solutions of differential, partial differential and
integral equations.
The book is intended to serve as introductory and reference
material suitable for the user of mathematics, the mathematicians
interested in applications, and the students of physics and
engineering. In an effort to make the book more useful as a text
book for students of applied mathematics each chapter of transform
analysis contains an account of applications of the theory of
integral transforms in a distributional setting to the solution of
problems arising in mathematical physics.
V
vi Preface
We wish to thank Gujar Ma1 Modi Science Foundation, University
Grants Commission, New Delhi and C.N.R.S. De France for providing
the financial assistance during the preparation of the book.
We express our gratitude to Professor Laurent Schwartz whose
valuable advice and encouragement to do the collaboration work which
has resulted finally in the form of present book. The constructive
criticism and suggestions of Dr. John S.Lew and Dr. Richard Carmichael on which this book is based were of great value and are gratefully
acknowledged. In addition, we are grateful to Professor H.G.Garnir
and Late Professor B.R.Seth who assisted us in preparing this book.
Our thanks are due also to Miss Rama Misra for her assistance in the
preparation of the symbols and author indices. We are also indebted
to Professor L.Nachbin for his interest in this book and finally its
inclusion in the series. We wish to thank Chaudhary Mehar La1 who
typed the manuscript with great competence and care.
0. P .Mi sra
Jean Lavoine
TABLE OF CONTENTS
CHAPTER 0 PRELIMINARIES 1
0.1. Notations and Terminology
0.2. Vector Spaces
0.3. Sequences
0.4. Some Results of Integration
0.4.1. Set of measure zero on the line IR
0.4.2. The saw-tooth function
CHAPTER 1 FINITE PARTS OF INTEGRALS
1.1. Definition
1.2. Extensions of the Definition
1.3. Integration by Parts
1.4. Analytic Continuation
1.5. Representations of Finite Parts on the Real
1.6. Change of Variable
Axis by Analytic Functions in the Complex Plane
CHAPTER 2 BASE SPACES
2.1. Base Spaces
2.2. The Space ID
2.3. The Space IDk (k 0)
2.4. The Space $ (Functions of Rapid Descent)
2.5. The Space 8 2.6. The Space ZZ (of Entire Functions)
2.7. Inclusions
2.8. The Space 8
2.9. The Space 8 (JRn)
CHAPTER 3 DEFINITION OF DISTRIBUTIONS
3.1. Generalized Functions
3.1.1. Inclusion of @ ' 3.2.. Distributions
7
9
10
12
15
17
19
19
19
2 0
20
21
21
21
22
23
25
25
26
27
vi i
viii Table of Contents
3.2.1. Inclusions
3.3. Examples of Distributions 3.3.1. Regular distributions
3.3.2. Irregular distributions
3.3.3. Pseudo functions
3.3.4. Regular tempered distributions
3.3.5. Tempered pseudo functions
3.3.6. Analytic functionals (ultradistributions)
CHAPTER 4 PROPERTIES OF GENERALIZED FUNCTIONS AND DISTRIBUTIONS
4.1. Support
4.1.1. Point support
4.1.2. Distributions with lower bounded support
4.1.3. Distributions with bounded support
4.2.1. Boundedness
4.2. Properties
4.3. Convergence
4.3.1. Completeness and limit
4.3.2. Particular cases of convergence in D'
4.3.3. Convergence in $I 4.3.4. Convergence to 6 (x)
4.4. Approximation of Distributions by Regular
4.5. Distributions in Several Variables
Functions
CHAPTER 5 OPERATIONS ON GENERALIZED FUNCTIONS AND DISTRIBUTIONS
5.1. Transpose of an Operation
5.2. Translation
5.3. Product by a Function
5.3.1. The space M(@) and the general
5.3.2. Distributions belonging to ID'
5.3.3. Tempered distributions
5.3.4. Ultradistribution
definition of product
or 6' 5.3.2.1. Distributions of finite order
5.4. Differentiation
5.4.1. General outline
5.4.2. Remark
5.4.3. Distributions of finite order having bounded support
27
27
27
28
29
30
30
31
35
35
36
36
37
3 1
37
38
39
39
40
40
41
42
47
47
48
49
50
50
51
51
51
52
52
52
53
Table of Contents ix
5.4.4. Derivatives of the Dirac distribution
5.4.5. Derivatives of a regular distribution
5.4.6. Derivatives of pseudo functions
5.4.7. Derivatives of ultradistributions
5.5. Differentiation of Product
5.6. Differentiation of Limit and Series 5.7. Derivatives in the Case of Several Variables
5.7.1. Generalization of 6' (x)
5.7.2. The Laplacian
5.8.1. General definition
5.8.2. Convolution in ID'
5.8.3. Examples
5.8.4. Convolution in ID;
5.8.5. Convolution in $
5.8.6. Convolution equations
5.8.7. Fundamental solution
5.9. Transformation of the Variable
5.8. Convolution
5.8.5.1. Convolution in $:
5.9.1. Definition of Tu(x)
5.9.2. Examples
5.9.3. Bibliography
CHAPTER 6 OTHER OPERATIONS ON DISTRIBUTIONS
6.1. Division n 6.1.1. Division by x (n>O, an integer)
6.1.2. Division by a function
6.1.3. multiplier for o
6.2.1. Antiderivative in ID:
ZT 6.2. Antidifferentiation
6.3. Value and Limit at a Point of a Distribution
6.3.1. Value at a point
6.3.2. Right and left hand limits at a point
6.3.3. Limit at infinity
6.4. Equivalence
6.4.1, Equivalence at the origin
6.4.2. Equivalence at infinity
CHAPTER 7 THE FOURIER TRANSFORMATION
7.1. Fourier Transformation on 22
7.2. Fourier Transformation on ID
53
54
57
59
59
61
62
63
64
65
65
65
66
68
70
71
71
71
72
72
74
75
77
77
77
78
79
80
81
82
82
83
84
85
a5
88
91
91
93
X Table of Contents
7.3. Fourier Transformation on ID' and Z'
7.4. Inversion and Convergence
7.4.1. Inversion of Fourier transformation
7.4.2. Convergence 7.5. Rules
7.6. Fourier Transformation on E' 7.7. Examples
7.8. Fourier Transformation on j! and $ ' 7.9. Particular Cases
7.10.Examples
7.ll.The Spaces Cf$? and M($) of Fourier
7.12. The Fourier Transformation of Convolution
7.13. Applications
7.14. Bibliography
on ID' and Z'
Transformation
and Multiplication
CHAPTER 8 THE LAPLACE TRANSFORMATION
8.1. Laplace Transformability
8.2. Laplace Transform
8.2.1. Case for functions
8-30 Characterization of Laplace Transform 8.4. Relation with the Fourier Transformation
8.5. Principal Rules
8.5.1. Case for functions
8.6. Convergence and Series
8.7. Inversion of the Laplace Transformation
8.8. Reciprocity of the Convergence
8.6.1. Examples
8.7.1. Example
8.8.1. Corollary in series
8.8.2. Examples
8.9. Differentiation with Respect to a Parameter
8.10.Laplace Transformation of Pseudo Functions
8.10.1. Derivative and primitive
8.10.2. Use of analytic continuation 8.10.3. Change of x to ax, a being complex
8.10.4. Change of x to ix
8.10.5. Convergence
8.11. Abelian Theorems
8.11.1.Behaviour of the transform at infinity
94
94
94
95 96
96
97
99
100
101
101
102
103
10 5
107
10 8
109
110
110
113
113
115
116
117
118
120
120
121
121
122
124
124
125
127
129
131
132
132
Table of Contents xi
8.11.2. Behaviour of the transform near a singular point 134
8.12. Tauberian Theorems 136
of the support 136
8.13. The n-Dimensional Laplace Transformation 138
8.13.1. The Laplace transformation in n variables 139
8.13.2. Convolution 140
8.14. Bibliography 143
CHAPTER 9 APPLICATIONS OF THE LAPLACE TRANSFORMATION 145
8.12.1. Behaviour near the lower bound
9.1. Convolution Equations
9.1.1. Examples
145
146
9.2. Differential Equations with Constant Coefficients 148
9.2.1. Solving distribution-derivative equations 148
9.2.2. Solving traditional differential equations 151
9.2.3. Single differential equations (Cauchy problems) 152
9.2.4. Systems of differential equations 154
9.3. Differential Equations with Polynomial Coefficients
9.3.1. Reduction of order
9.4. Integral Equations
9.4.1. Special Volterra equations
9.4.2. Resolvent series
9.4.3. Remark on uniqueness
9.4.4. Integral equations with polynomial coefficients
9.5. Integro-Differential Equations
9.6. General Concept of Green's Functions
9.6.1. Statement
9.6.2. Green's kernel
9.6.3. Examples
9.6.4. Integral equations
155
156
160
161
162
164
164
166
168
168
169
173
17 6
9.7. Partial Differential Equations 177
9.7.1. Diffusion of heat flow in rods 177
9.7.1.1. Infinite conductor without
9.7.1.2. The cooling of a rod of finite
radiation 17 7
length 17 9
xii Table of Contents
9.7.1.3. Rod heated at an extremity 180
9.7.2. Vibrating strings 182
9.7.3. The telegraph equation 187
9.7.3.1. The lines without leakage which are closed by a resistance 187
9.7.3.2. The infinite line which is perfectly isolated 189
9.8. Convolution Formulae 19 0
9.9. Expansion in Series 193
9,g.l. Function B ( v , z ) 193
9 9.2 Function $ ( 2 ) 194
9 9.3. Fourier series 194
9.9.4. Asymptotic expansions 196
9.10. Derivatives and Anti-Derivatives of Complex Order 198
transformation 198 9.10.1. Definition by the Laplace
9.10.2. Examples 200
9.10.3. Extension of the definition 203
CHAPTER 10 THE STIELTJES TRANSFORMATION 207
10.1.
10.2.
10.3.
10.4.
10.5.
10.6,
10.7.
10.8.
The Spaces E (r) and JI' (r)
10.1.1. The space E (r)
10.1.2. The space JI' (r)
The Stieltjes Transformation
Iteration of the Laplace Transformation
Characterization of Stieltjes Transforms
Examples of Stieltjes Transforms
10.5.1. Examples when Tt E JI' (r)
10.5.2. Examples when Tt E JI'
Inversion
Abelian Theorems
10.7.1. Behaviour of the transform near
10.7.2. Behaviour of the transform at
the origin
infinity
The n-Dimensional Stieltjes Transformation
10.8.1. The space J,l(r)
10.8.2. The Stieltjes transformation in n
10.8.3. The iteration of the Laplace
10.8.4. Inversion
variables
transformation
201
207
209
209
210
211
213
213
215
2 16
219
219
220
221
221
222
222
224
Table of Contents xiii
10.9. Applications 10.10. Bibliography
CHAPTER 11 THE MELLIN TRANSFORMATION
11.1. Mellin Transformation of Functions
11.2. The Spaces E a # w
11.3. The Spaces EA l a
11.3.1. The multiplication in E'
11.3.2. The differentiation in E'
11.3.3. Comparison with Zemanian spaces
a t o
a1w
11.4. The Mellin Transformation
11.5. Examples of Mellin Transforms
11.6. Characterization of Mellin Transformation
11.7. Rules of Calculus
11.8. Mellin and Laplace Transformations
11.9. Mellin and Fourier Transformations
11.10. Inversion of the Mellin Transformation
11.11. The Mellin Convolution
11.11.1. Examples and particular cases
11.11.2. Relation with the Mellin transformation
11.11.3. Relation with the ordinary convolution
11.11.4. The operator (tD)'
11.12. Abelian Theorems
11.13. Solution of Some Integral Equations
11.14. Euler-Cauchy Differential Equations
11.15. Potential Problems in Wedge Shaped Regions
11.16. Bibliography
CHAPTER 12 HANKEL TRANSFORMATION AND BESSEL SERIES
12.1. Hankel Transformation of Functions
12.2. The Spaces Hv and H$
12.3. Operations on Hv and H$
12.4. Hankel Transformation of Distributions
12.4.1. The Hankel transformation on E' (I)
12.5. Some Rules
12.5.1. Transform formulae for Hv
12.5.2. Transform formulae for H:
12.6.1. Remarks
12.6. Inversion
12.7. The n-Dimensional Hankel Transformation
224
224
227
228
230
232
234
234
235
236
237
238
241
242
244
245
249
250
251
251
252
253
258
261
265
268
269
269
272
274
276
280
282
282
283
284
286
287
xiv Table of Contents
12.7.1. The spaces of h and h' u lJ 12.7.2. Operations on h and h'
lJ ?J
12.7.3. The Hankel transformation in n- variables
12.8. Variable Flow of Heat in Circular Cylinder
12.9. Bessel Series for Generalized Functions
12.9.1. Statement
12.10. The Space B
12.11. Representation of a Distribution by its
12.12. Other Properties of the Fourier-Bessel
12.13. The Subspace Bm of Bm 12.14. Sessel-Dini Series
12.14.1. Statement
m I v
Fourier Bessel Series
Series
I V
E k I m I v 12.15. The Space
12.16. Representation of a Distribution by its Bessel-Dini Series
12.16.1. The subspace Bm of B
12.16.2. Another subspace of B H l m l v
H l m , V 12.17. An Application ot the BesselWDini Series
12.18. Bibliography
288
290
291
295
297
297
298
300
302
304
307
307
309
310
311
311
311
314
BIBLIOGRAPHY 315
INDEX OF SYMBOLS 329
AUTHOR INDEX 331
CHAPTER 0
PRELIMINARIES
summary
In our presentation of generalized functions and distributions
and its setting with transform analysis in this book it will be
presumed that same basic knowledge of real and complex analysis and
a first course in advanced calculus are known to the reader, Some
rudimentary knowledge of functional analysis is also assumed. We
also freely use the classical transform analysis and its various
properties which appear in standard references cited in the biblio-
graphy. The purpose of this chapter is to explain certain notations
and terminology used throughout the book. These are related to set
theory, linear spaces, sequences and some results on integration.The
body of the text begins with Chapter 1.
0.1,Notations and Terminology
In this section we state terminology and notations which will be
used throughout this book.
the real and complex n dimensional euclidean spaces. Any number X in
Iff will be denoted by (xl,x2, ..., x ) or occasionally by X. n r will be used to signify the distance f l = A, + x2 + . . . + x Often f(X) will denote f(x1,x2,...,xn) and I
We let Rn and Cn denote, respectively,
The letter 2 2 2
n* f(X)dX will mean
Bn
j//...j f(x1,x2,...,x n )dxl dx2, .... dx,. IRn
In this notation it is sometimes convenient to write r =
partial derivative
1x1. The
will be abridged on occasions. P1 +P2+. .+P, a axl p1 ax2 p2 ... axn pn
1 We recall that IR (IR = IR ) is the line of real numbers and
C(C = C ) the plane of complex numbers. By the symbols IN and INn
we denote the set of non-negative integers in one variable and n
1
1
2 Chapter 0
variables, respectively.
The set theory notations used are as follows:
A C B or B 3 A - the set A is included in the set B; i.e. x E A
then x E B.
A U B - the union of the sets A and B; i.e. the set of elements
belonging to A or €3.
A n B - the intersection of the sets A and B; i.e. the set of
elements belonging both to A and B.
]a,b[ - the open interval from a to b; i.e. the set of points x
such that a < x b.
Ca,bl - the close interval from a to b; i.e. the set of points x such that a 5 x 5 b.
A x B - the direct product of sets A and B; i.e. the set of
pairs (x,y) where x E A and y E B.
lRx\(a 5 x 5 b) - the axis IRx without the interval [a,b] V - denotes for every.
0.2.Vector Spaces
Recall that C denotes the set of complex numbers.
A set E is said to be a vector space (or linear space)provided
E E for
that any finite linear i.e. provided that if cl, c2, ...., cn E C and fl, f2,...,f
any finite n then
combination of elements of E is an element of E,
n
n Clfl + c2f2 +...+ c i = 1 Cifi
n i=l
is an element of E.
The properties of a vector space can be verified by the linear
combination of complex numbers. We outline these properties as
follows :
1. We have
Preliminaries 3
(i) If = f, Y f E E,
(ii) of = og, Y f,g E E.
2 . One can exchange and group arbitrarily the terms of a linear
combinat'ion; if (v1,v2,...,vn) denotes an arbitrary permutation of
(1,. . . ,n) and if 1 < nl < . . . < nk < n, we have
n n
i=l i-1 i=n k +1 "i 1
3 . Any linear combination of linear combinations of elements of
E is again a linear combination; that is,
This unique formula yields the following elementary formulas;
(i) c(clf) = (cc')f,
n n (ii) 1 c.f = ( 1 ci)f,
i=l 1 i=l n n
(iii) c( 1 c!f.) = 1 c cjfi. i=l i=l 1 1
We note that cc'f is the known value of (cc')f and c(c'f) which
enables us equally to write-cf instead of+(-cf) in the linear
combination.
From the preceding axiomsfone can deduce easily all the usual
properties of linear combinations.
We say that E possesses a neutral element 0 such that
(i) f + 0 = f,
(ii) c0 = 0, 'd c E C ,
(iii) of = 0.
Furthermore, each element f E E has a opposite element, denoted
by -f such that
f + (-f) = 0.
0.3. Sequences
Let N be the set of natural integers and let P be a subset of N.
4 Chapter 0
A family (Unln of elements of a set E! indexed by n is said to be a sequence of elements of E. If P is finite (or infinite) then
we say that the sequence is finite (or infinite).
If P = N, then we say simply sequence and denote it by (Un), or often, U1,U2,...,un.
Convergence and uniform convergence. Let E be a vector space over the
complex numbers C. A mapping x + 1 1x1 I of E into IR is said to be
norm if it possesses the following properties:
A sequence {$,(x) 1 is said to convergence to + (x) in E if I ldn(X)-d(x) I I * 0 as n * m if for each TI > 0 there corresponds an integer p such that I l$n(x)-$(x) 1 I < q if n 3 p.
A sequence of functions ($,(x)I tends towards the function W(X)
uniformly on a domain D if to each number 0 > 0 there corresponds an
integer p such that I 10 (x) -w(x) I I < TI for n > p and all x of D. n
n A sequence { $ (x)) is said to be a Cauchy sequence if
1 l$m(x)-$n(x) I I * 0 as m,n + m, i.e. for any E > 0, there exists an integer n such that, for any pair of integer m,n both greater than
0
"0'
I l+m(x)-+n(x) 1 I 5 E .
If every Cauchy sequence converges to a point in E, it is then
said E is complete for the topology defined by this norm. In a
complete normed space a Cauchy sequence 14 (x) I has a limit w(x) in E. n
0.4.Some Results of Integration
The Lebesgue integration is a generalization of the Riemann
integral. It is a functional on a certain class of real or complex functions of the variable x, called the class of summable functions,
and assigns to each summable function f(x) a real or complex number called its integral and denoted by
Preliminaries 5
OD
f(x)dx or 1 f(x)dx or If -m IR
Locally summable function. A function is said to be locally
summable if it is summable over any bounded set of IR.
0.4.1.Set of measure zero on the line IR
For a set E S R, the function $E, which is equal to 1 at each
point x E E and to 0 at each point x p E, is known as the character- istic function of E.
Definition 0.4.1. The measure of an open set is defined as the
least upper bound of the integrals of the continuous functions 2 0, which are zero outside a finite interval and bounded above by the
characteristic function.
For example, if E is the interval ]arb[, then its measure is
(b-a) . Definition 0.4.2. A set E on the line is said to be measure zero
if for any E > 0, there exists an open set of measure 5 E which
contains the set E.
Example. A point is of measure zero.
0.4.2,The saw-tooth function
Let us consider the period function of period 1 that varies linearly from 0 to a inside any interval n < x < n+l (n being any
integer). (When plotted in a diagram, the graph shows the character of the teeth of a saw. Therefore, a convenient notation for this saw
-tooth function is S(x).) In the operational treatment we shall be
concerned mainly with
drawn in Figure 0.4.1
( 0 . 4 .I) S(x) =
We have
if j =
if j =
if j =
the corresponding one-sided function, which is
and is represented by the following formula
x < o
a(x-j+l) , j-1 5 x 5 j, j=l,2,3,... .
1, S(x) = ax, for 0 2 x 5 1; 2 , S(X) = a(x-1) , for 1 5 x 5 2; 3, s ( x ) = a(x-2) , for 2 5 x 5 3 ;
......................................
.......................................
6
etc.
Chapter 0
S W
a
Figure 0.4.1
Thus, we see that the saw-tooth function has an infinite number of
jumps. From (0.4.1), we have S ' ( x ) = 0, x < 0 and S'(x) = a, x > 0
and x # j . Also, we have from (0.4.1),
(0.4.2) S ( X ) = a(x-j), if j 5 x < j+ l .
1 I f O < E < - then we have from ( 0 . 4 . 2 ) ,
S ( j + E ) = a(j+c-j) = aE and as E + 0, S(j+) = 0.
Moreover, from (0.4.1) we have
S ( j - c ) = a(j-E-j+l) and as E + 0, S ( j - ) = a.
CHAPTER 1
FINITE PARTS OF INTEGRALS
Summary
The finite part of a divergent integral, a notion introduced by
Hadamard Cll , is a generalization of the definite or indefinite integral which has wide application to partial differential equations.
Also, Schwartz [l] has shown that finite parts have great interest
in the formulation of distribution theory. Hence, before we discuss
the concept of a distribution, we treat briefly here the finite parts
of divergent integrals. The full scope of the finite part notion
will be an essential tool in the later chapters.
For readers unfamiliar with this topic, we begin with a basic
definition.
1.1.Definition
The meromorphic function Cz/z for Re z > 0 has the integral C
representation I xZ-'dx. C z / z for Re z < 0 has some generalized integral representation.
following remarks answer this question.
Thus one might ask whether the function 0
The
For simplicity, let x be a real variable and let y(x) be the
function
where c is real, Re v > 1, v # 1, and s(x) is integrable on Cc,C]. Choosing any r~ such that c < c + n < C, set J ( n ) = I y(x)dx; then
C
c+n term by term integration yields the result,
s(x)dx. C+ n
7
8 Chapter 1
The function J ( n ) , as r( + 0, approaches no finite limit because -v+l
of the terms .* - b log r( , but the remaining terms on the right
side of (1.l.l)possess a limit which is called the finite part of the C integral I y(x)dx as n + 0.
c c+n C ~p ] y(x)dx to represent this finite part. elation (1.1.1) shows that FPJ y(x)dx
takes the f m , C (1.1.2)
For brevity, we use the notation
C C C - a
Fp I y(x)dx = - ( C - c ) ‘+’+b log(C-c) + I s(x)dx C C
c; = lim{ ] C a(x-c)-‘+ b(x-c)-l+s(x) 1 dx
n+O c+n -v+l
- %’+ b log 111.
If the integrand is (x-c)-’logj(x-c) (’) where j E IN, Re v 2 1, and v # 1, then we obtain after integration by parts j times
j-1 1 j! log (c-c) C -v+l
(1.1.3) Fp ] (x-c)-”logj(x-c)dx = - (‘-‘) v-1 1 C i=O (j-i):(v-l)
where the sum is zero if j=O. Alternatively, taking v = 1 we have
(1.1.4)
Hence, formulae (1.1.2,3,4) permit us easily to define Fp jg(x)dx
when g(x) is a linear combination of the functions y(x)
and (x-c)-vlogl (x-c) . The previous examples motivate the more general, and quite
frequent, case where the integrand g(x) is the derivative of a
function gl(x) admitting, for c < x < C
(1.1.5) gl(x) = K ? 1 . L C ak+ajklogj(x-c) 1 (x-c)-’k
k=l j=l
Here all Re A k > 0 but the Ak are not integers; also some of the
numbers ak, a,k, bk, p j k may be zero, and h(x) is a continuous and
bounded function on Cc,Cl. Then, if we put
(1.1.6) FP gl(X) = h(c+), x = c
we have the easy formula
F i n i t e P a r t s 9
It i s ev ident t h a t i f g ( x ) = s ( x ) i s an i n t e g r a b l e func t ion on Cc , C 1, then
C C (1.1.8) Fp I s ( x ) d x = s ( x ) d x .
This las t r e s u l t shows t h a t t h e ope ra t ion Fpl is a proper gene ra l i za - t i o n of t h e i n t e g r a l .
C C
P u t t i n g s ( x ) = 0 , a = 1, b = c = 0 , and -u = 2-1, R e z < 0 , i n ( 1 . 1 . 2 ) , w e ob ta in
2-1 C Fp 1 xZ-ldx = C / z ;
C
and consequently, t h i s solves t h e o r i g i n a l l y posed problem.
The remainder of t h i s s e c t i o n broadens t h e d e f i n i t i o n of a f i n i t e p a r t to inc lude i n t e g r a l s , wi th many s i n g u l a r p o i n t s and o b t a i n s a number of r e s u l t s t h a t w i l l be needed la te r .
1 . 2 , Extensions of t h e Def in i t i on
Extension 1. The i n t e r v a l s f o r t h e previous f i n i t e p a r t s have a s i n g u l a r i t y a t t h e left end p o i n t c, b u t i f g ( x ) i s r egu la r i n
c C ' , c [, then
C 2c-C' ( 1 . 2 . 1 ) Fp I g ( x ) d x = Fp 1 g(2c-x)dx
C ' C
where w e assume t h e ex i s t ence of t h e r i g h t hand side.
Ex tens im 2, Furthermore, i f g(x) i s r e g u l a r i n C C ' , C l b u t has a s i n g u l a r i t y only a t x = c, C ' < c < C , then
-.If g ( x ) i s of t h e type
(1.2.3) g (x) = a (x-c) -2k+1+s (x )
where k E IN and s ( x ) i s an i n t e g r a b l e func t ion on [ C ' ,C 1 , t hen t h e d e f i n i t i o n (1 .2 .2) is equ iva len t t o t h e d e f i n i t i o n of a Cauchy p r i n c i p a l value. Consequently, w e o b t a i n t h e r e s u l t s ,
10 Chapter 1
(1.2.4) C C c-n C
where pv denotes the Cauchy principal value.
Extention 3 .
c1 < c2 <. . . < CN,
(1.2.5) FP
If g(x) has several singularities on CC',Cl, say
the definition (1.2.2) has the generalization
'n+l
'n
C I g(x)dx = 1 Fp g(x)dx, C' n= 1
c1 < c- < c2 < c <...a2 N <c N <c N+lrC' 1 2 where the Cn are such that C' ==
Extension 4 . The definition (1.2.5) does not apply when g(x) has
infinitely many singularities,but then a finite part can still be
defined when g(x) obeys the following conditions:
1. g(x) is a continuous on [ C' , m [ except on a countable set
{c1,c2, ... 1 where C' < c < C2". . 2.
n = 1,2,.... such that each In contains c
each
The domain [C',mC includes disjoint intervals In= {crn,Bn},
as an interior point, and n
Bn fn = Fp g(x)dx
anm is well defined. A l s o , 1 f is a convergent series. n n=l
3 .
B > 1.
If no I contains x, and x 2 some xo, then Ig(x) I < where n
Obviously, these conditions on g(x) permit the following
extension of (1.2.5) :
(1.2.6) Fp I g(x)dx = lim Fp g(x)dx 5
C' 6 - t - C'
m
where no In contains 5 .
1.3 Integration by Parts
This section generalizes the technique of integration by parts
to include formulae for finite parts of integrals. Later we shall
use this technique to obtain the derivatives of pseudo functions
(see Section 5.4.6 of Chapter 5).
If f(x) has a derivative f'(x) which is integrable on [c-q, C],
and if g(x) has primitives g ( - ' ) (x) which is integrable on [c+q,C]
Finite Parts 11
such that g(-') (x) is of the type (1.1.5) , then we have by the definition (1.1.6) ,
C (1.3 -1) Fpjg (x) f (x) dx = g ( - ' ) (C) f (C) -Fpg(-') (x) f (X)
x=c C C
- Fpj g('') (x) f (x) dx. C
This formula can also be obtained by taking g
(1.1.7).
= gf+g(-')f' in 1
Formula (1.3.1) yields results of great interest when f(x) has
sufficiently many derivatives at x = c, since the Taylor-Lagrange
theorem then gives explicit expressions for the finite parts. The
following examples illustrate the procedure.
Examples. If f(x) has n-1 derivatives at x = c, where n 2,
-n+l then the Taylor-Lagrange expression yields Fp [(x-c) f(x)l = x=c
(n-l) (c) , so that integration by parts gives (n-1) !
If, alternatively, X = n-a, 0 < Re a 1, then we obtain C -X+1
(1.3.3) Fp/(x-c)-Af(x)dx = - f ( C ) C
Even if f(x) has only one derivative integrable over [c,Cl, then
still one obtains C C
(1.3.4) FpJ (x-c)-'f (x)dx = f (C) log(C-c) - lOg(C-C) f'(x)dx. C C
These examples will be used often in the subsequent work. A l s o
the reader should note the considerable difference between (1.3.2)
and (1.3.3) which will have several consequences.
Remark. If f(x) has many derivatives, then repeated integration C
by parts in (1.3.2) expresses Fpl g(x) f (x)dx in terms of an ordinary
integral. This property has an important role; because the theory
of distributions, takes f(x) to be an infinitely differentiable
function.
c
12 Chapter 1
1.4,Analytic Continuation
In this section we work out the finite parts of integrals by
means of analytic continutation.
We let j be a non-negative integer, z be a complex variable,
By D1 we and v be a complex parameter such that Re v 2 1, v # 1. mean the domain of the complex half-plane, Re z > Re v-1; while D2
denotes the domain of the complex z-plane, excluding the points
z = v-1, "the origin" z = 0 belongs to D2. For z in D1, we get
C (1.4.1) MV(z) = [(x-c) Z-vlogj(x-c)dx
C
1-11 j (c-c) 2-v+l j c = (C-Cl i=o (j-i) I ( z - v + l ) l z-v+l
This function M (z) is obviously analytic in D2i it appears as the
analytic continuation in D2 of the integral (1.4.1). Also, we can
set
(1.4.2)
where Ac signifies << analytic continuation of >>.
C Mv(z) = Ac I (x-~)~-~logJ(x-c)dx, z E D2
C
Since z = 0 is a regular point for MV(z), we have
then, according to (1.1.3), we finally obtain
C (1.4.3) Fp [ (x-C) -'log' (z-c) dx = Mv (0) .
C
But the formula (1.4.3) fails if v = 1, because z = 0 is a pole for
i. j-i (C-c) z-i-l j
M1(z) = (C-c)' 1 (-I) 'ijlog -1) ! i=O
In this situation, we first work out the explicit expansion for M1(z) inorder to formulate another definition. Indee'd, the identity
M1(z) yields the form
j -i (C-c) z-i-l 1 (-1) j ! log
(j-i) : j
M1(z) = [1+ z k l [ 1 k= 1 i-0
Finite Parts 13
2 +... 1
I-&- - j log j-1 (c-c) + j (j-1) logJ-2(c-c) +. . .+ (-;iij: = [ 1+ 10$C-c) z + log2 (C-c) z
2:
Z 2 Z Z Z
+ ..... if we rearrange the terms and use the identity Co+C2+C4+ ... = C +C + 1 3 C5+ ... . From this expansion of M1(z), we may infer that M1(z) has a pole of order (j+l).
Laurent expansion in the neighbourhood of origin z = 0 of the form:
Consequently, we conclude that M1(z) has a
where B1,B Z,..., denote coefficients.
We now obtain by definition (1.1.6)
and according to (1.1.4)
Finally, for v = I, we obtain a adequate definition,
= Fp [ A c ~ z=o c
Generalization of (1.4.1) and (1.4.4)
Let g(x) be singular at x=c and for z in
/ g(x) (x-c)’dx exists and equals a function
meromorphic function (denoted also by M(z))
A and the origin. Then
C
C
1
C (1.4.4) Fp/(x-c)-’logJ(x-c)dx = Fp M1(z)
C z=o 2-1 (x-c) log’ (x-c) dxl .
a danain Al, assume that
M(z) continuable to a
in a domain containing
(1.4.5)
where either
(1.4.6) Fp M(z) = M ( 0 )
when M(z) is regular at the origin
Fpl g(x)dx = Fp M(z) C z=O
z=O
14 Chapter 1
or
(1.4.7) Fp M ( z ) = Fp M(x)
when M(z) has a representation of the kind (1.1.5) and (1.1.6).
z=o x=o
Finally, we may infer from these results that the analytic
continuation method is a very fruitful method for calculating
finite parts. We now discuss a simple class of examples which
illustrates more clearly the concepts of this method, concepts
treated above in rather vague and general terms.
Examples. Let Re v > 0 but v # 1,2,. . . If Re(z-v) > 0 and
r(.) denotes the gamma function, then
m
I e-x xz-w-ldx = r (z-v) . 0
However r(z-v) is meromorphic in the entire z-plane, and it has no pole at the origin. Hence, (1.4.5) gives
Thus using the substitution a = -v, we get m -Xxa-ldx
(1.4.8) r ( a ) = Fple
for every a # 0, -1, -2,..., (Fp is not needed here if Re a > 0).
0
If a = -n = 0, -1, -2, -,.... the case is radically different. Then, by formulae (1.4.5) and (1.4.7), we obtain
m n (1.4.9) Fpl e-xx-n-ldx = Fp r(z-n) = $(n+l) n! 0 z=o
where $ ( z ) = (d/dz) logT(z) , and log C is the Euler (Mascheroni) constant which is approximately 0.577... . Moreover, the sum is
zero when n = 0.
Problem 1.4.1
The Bessel function Jv has the property (see Jahnke,Emde and
Losch C11 p . 134)
Prove that this equality also holds true in the sense of Fp if
Finite Parts 15
Re v > 0, i.e.
(-1) "4-" v+2n-1 I v # -1, - 2 , . . .
m
Fp $ Jv(x)+ = v2-' 1 n=O n! r (v+l+n) Fp x+
1.5.Representatiom of Finite Parts on the Real Axis by Analytic Functions in the Complex Plane
The theory of finite parts and the theory of analytic functions
have several common areas of interest. The following work further
develops these areas.
Let z be a complex variable such that z = x+iy and let C'<c<C
be three real numbers. The adjoining diagram (Figure 1.6.1) shows
two paths L, and L- from C' to C.
F I G U R E 1.6.1.
We choose arg (z-c) such that this argument varies from 'TI to 0
along the path L, and from -n to 0 along the path L-.
One could evaluate the analytic function by using this represen-
tation in the complex plane; it is simpler however, to illustrate
this process by means of an example.
dz. The theory of complex For any v # 1, let us find [ (Z-C)-' L+
variables letsus evaluate this quaEtity as a line integral.
cally, integration yields
Specifi-
-v+l + eTivn -v+l C(C-C) IC'-cl 1. -1 (1.5.1) (z-c)-Vdz = - v-1
L+ -
L+ -
Similarly ,
(1.5.2) [ (z-c)-'dz = log(C-c)-loglC'-cl 7 in
We now use finite parts of integrals to give another representa-
tion of these integrals. For this purpose we first note, by the
definitions (1.1.2) and (1.2.2),
16 Chapter 1
L
and
Hence, we may conclude from the formulae (1.5.2) and (1.5.1)
C C (1.5.6) eFivnFpl)x-c/-'dx+Fpf (x-c)-"dx = I (z-c)-"dz, v # 1.
C C L+ - Putting v = n, integer n 2 2 , in (1.5.6), we further obtain
- But if x < c then (-l)"lx-cl-" = (X-C)-" and (1.5.7) takes the form
- (This formula is also valuable if n is zero or a negative integer,
but then Fp is not needed.)
Finally, we may infer from these formulae that if g(x) fulfills
certain conditions, then
C Fp/ g(x)dx = g(z)dz + constant depending on g, C' L
where L is a suitable path joining C' to C in the complex z-plane,
and containing no singular points of g(z).
This representation lets us use the finite parts of integrals
in the theory of analytic functions.
good account of this work.
Lavoine C 5 3 and C 6 1 gives a
Often, the integrals in the right side of formulae (1.5.5,6,8)
are represented by
(1.5.9) (x-c+io)-'dx, A = 1, n or u.
This notation is useful sometimes. (See Gelfand and Shilov [I], Vol.1.)
C
C'
Finite Parts 17
1.6.Change of Variable
For any v # 1, we have
The translation x = 5 + B , for real B , changes the preceding formula
into
The same translation yields,
C- B
C C- B dx = Fp I (c+B-c)-ldg = log(C-c) . -1
FpI (x-c)
Clearly, the finite parts of all other integrals would give analogous
results on translation. Hence, we may conclude that the finite part
of an integral is invariant under translation. But the finite part
of an integral is not invariant under more general changes of variable. For example, if x = ax', then we get
0
because the left hand side is equal to log 2 but the right hand side
equals log 2/a. Section 5.9 of Chapter 5 contains some more peculiar
cases. (See also Lavoine [6] and Di Pasquantonio and Lavoine [l] .)
Foot note
(1) logj (x-c) = (log (x-c) ) j ,
This Page Intentionally Left Blank
CHAPTER 2
BASE SPACES
summary
Before formulating the concept of distributions we need spaces
on which the distributions (or generalized functions) will act. This
chapter deals with precise definitions of these spaces and we shall
call them the base spaces.
Let us now formulate the exact definitions.
2.1.Base Spaces
The base space will be a vector space of functions on which is
defined an appropriate type of convergence(’) for sequences.
functions of this space are said to be base functions (or test functions).
The
Before describing the different kinds of base spaces which will
play fundamental roles in the subsequent work, it is appropriate to
begin with a definition of what we mean by support of a function.
The support of a function of a real variable x (or complex variable z) is the closure in IR (or C) of the set of points x (or z) where
the function is different from zero.
The support of a function can be unbounded or the whole of the
line IR (or the whole of the plane C) . Bounded support (or Compact support). If the support is cont-
ained in a bounded interval of the real line IR (or in a bounded
square of the complex plane C), then we say, that it is a bounded
support and therefore a compact support.
2.2,The Space m
By we mean the space of functions $(x) (real or complex
19
20 Chapter 2
valued) of the real variable x which are infinitely differentiable
(that is, differentiable to every order) and which have bounded
support (that is, there exists a bounded interval outside of which
$(XI = 0 ) .
Concept of convergence. We say that an infinite sequence
{$n(x)}, n E IN, converges to 0 in the sense of ID as n + -, if
(i) each $,(XI E ID;
(ii) a l l the supports of $,(x) are contained in the same
bounded interval;
(iii) $,(x) -+ 0 uniformly, and all the derivatives 0;) (x) -f 0
uniformly, k = 1,2,3,....
Example. The function E(x) defined by:
1x1 2 1 1
belongs to ID. Its support, which is the interval 1x1 5 1 is obviously bounded.
but the sequence {$(x/n)
because of infinite growth of the supports.
1 The sequence (5 E(x)l + 0 in the sense of ID, 1 doe5 not converge in the sense of ID
k 2.3.The Space ID (k L 0)
k The symbol ID denotes the space complex valued) with bounded support
tives of order at least equal to k. k in the sense of ID is analogous to that of ID.
of functions $(XI (real or and having continuous deriva-
The convergence of the sequences
2 .4 . The Space $ (Functions of Rapid Descent)
By $ we denote the space of functions $(x ) which are infinitely continuously differentiable and which decrease in modulus together
with a l l their derivatives more rapidly than any positive power of 1x1 , as 1x1 -f -. (That is, for every set of non-negative integers j, k, I x ’ $ ( ~ ) (x) I -+ 0, as 1x1 + 4
Concept of convergence. We say that an infinite sequence
i$n(x) 1 , n E , converges to 0 in the sense of $, as n -+ -, if
spaces 2 1
(i) each Cpn(x) belongs t o $;
(ii) f o r every set of non-negative in t ege r s j , k , 1x1 4;) (x) I + 0 uniformly on IR.
2.5. The Space E
The symbol E denotes t h e space of func t ions $ ( x ) which are i n f i n i t e l y cont inuously d i f f e r e n t i a b l e and which have a r b i t r a r y support .
Concept of convergence. W e say t h a t an i n f i n i t e sequence { $,(x) 1 , n E IN ,converges t o 0 i n t h e sense of 6 , a s n t m , i f
(i) each $,(XI E 6
(ii) gn(x) + 0 and 41;) (x) -+ 0 uniformly on every bounded i n t e r v a l of IR f o r k = 1,2,3,...
2.6, The Space Z (of En t i r e Funct ions)
By 22 w e denote the space of e n t i r e a n a l y t i c func t ions of t h e complex v a r i a b l e z = x+iy such t h a t f o r every i n t e g e r j > 0 t h e r e e x i s t numbers a and C ( C j > 0) for which l z j $ ( z ) I < C ealyl f o r every z.
j j
Concept of convergence. W e say t h a t t h e i n f i n i t e sequence {$,(z) 1 , n E IN, converges t o zero i n t h e sense of P; , as n + m , i f
(i) each $,(z) E Z;
(ii) t h e r e e x i s t real cons t an t s a and C (j = 0 , 1 , 2 , . . .) , which Is
are independent of n, such t h a t
(iii) $,(z) + 0 uniformly on every bounded domain of t h e complex z-plane.
1z $,(z) 1 < C eaIY1; j
- (1) o ) z j Remarks. I f $(z) E 22 , then i t s Taylor series + $(z)
i n t h e sense of ZZ . t h a t even such simple e n t i r e func t ions l i k e ez and e-z2 do no t belong t o P,. Some o t h e r types of base spaces are d iscussed i n Gelfand and Shi lov c11 Vols. 1 and 2 and Gutt inger Ell.
I f $(z) E Z , then $ ( X I E $. J=l itshouldbenoted
2.7. I nc lus ions
The r e s u l t s of t h e preceding s e c t i o n s enable us t o make t h e
22 Chapter 2
following inclusions.
k Theorem 2.7.1. D is dense in ID . Proof. The proof can be formulated as indicated in Schwartz c11, -
k Chapter 1, Theorem 1, by taking n = 1 and replacing C by ID.
Theorem 2.7.2. ID is dense in $.
Proof. See Schwartz c11, Chapter VII.3, Theorem 111, where - the several variables case has been discussed.
Theorem 2.7.3. ID is dense in e . Proof. The proof can be formulated as indicated in Schwartz 111,
Chapter 111.7, where the case of several variables has been discussed.
Theorem 2.7.4. Z is dense in E . Proof. See the proof of Theorem 7 . 6 . 4 of Zemanian 111 p. 197. -
2.8.The Space 4
k In the subsequent work, 4 denotes any one of the spaces ID , ID, $, E and 4 = ZZ in C, provided that all these spaces have their
own characteristic convergence.
Consequently, from the
vector space. Moreover, if
tend towards 0 in the sense
numbers, then the sequence
sense of 8 . Further, we ca
even if the constants a and are called multipliers (2) .
structure of 8 we may infer that 8 is a
the infinite sequences {$,I and {I) 1 of 4 and if a and b are real or complex
aen + bI) 1 also converges to zero in the
b are replaced by certain functions which
n
n say that this property is satisfied
We say that the sequence {(Bn> in 8 converges to in 4 for the
characteristic convergence defined in b, if ((Bn-+) -+ 0 as n + m; and
this is written $ -+ (B in the sense of 8 . Also , 8 is complete
because every Cauchy sequence has its limit in 8 . This means that
0 has the following property : if ($n-$nl) + 0 as n, nl-+ sense of the characteristic convergence in 8 , then there exists an
element (B E 4 such that 4n -f 6 as n -+ m in the sense of 8 .
n
in the
Spaces 23
2.9. The Space @ (IRn)
The foregoing concept for base spaces with functions in a single
variable have straightforward extension to base spaces which have
functions in n independent variables. Briefly, we outline this
extension in the present section in the following manner.
If we take x = (x1,x2,. . . ,x ) and y = (yl, yz,.. . ,yn) belong to n IRnand z = ( z 1, z 2 , . . . , zn) E Cn instead of x,y eIR and z E C in the
1.
structure of the base spaces ID, I D K , $, E and Z', then we denote these spaces by ID(Wn), IDk (IR"), $(IR"),
ZZ ( Cn) , having functions in n variables. The concept of convergence
and other properties of these spaces will be analogous to those of
the base spaces defined in the preceding sections by replacing IRand
C to IRn and Cn, respectively. Moreover, we illustrate this remarks
by taking the case of ID (IR" ) .
E (IR"), and in Cn,
The space ID(IRn ) denotes a vector subspace of the vector space
of infinitely differentiable and complex valued functions defined on n
IRn . If x = (x1,x2,. . . ,xn) E IR , then we define the space ID( IR") as follows:
A function $(x) on IRn belongs to ID (IR") if and only if it is infinitely differentiable and there exists a bounded set K of IRn
outside of which it is identically zero. For each function +(x), if
K is the smallest closed set outside of which $(x) is zero, then K
is called the supporting set or support of $. This structure of
ID (IR") permits us to make the following definition:
By ID (IR") we mean the space of complex valued functions on IRn which are infinitely differentiable and have bounded support.
Concept of convergence in ID( IR") . We say that an infinite n sequence t+n (x) 3 , n E
in the sense of ID( IR") as n -+ - if of functions in ID( IR" converges to zero
(i) all the supports of $n are contained in the same bounded set,
independently of $n,
(ii) $,(x) -+ 0 uniformly, and all the derivatives of 4:) (x) -+ 0
uniformly for k = 1,2,....
Example. The function
2 4 Chapter 2
r = 1x1 I i=l g(x) = exp ( - -+ if r < l r = Jn g(x) = 0 if r 2 1
1-r
S(x) =
belongs to ID (nn) which is an analogous example to that of ID given in Section 2 . 2 .
As 0, the space 0 (IRn ) denotes any one of the spaces ID(Rn), IDk (IRn), $(IRn), 6 (IR”) , and 4 (C”) = Z (C”) in C” provided that all
these spaces have their own characteristic convergence.
Problem 2.9.1
(1) Define the concept of convergence in the following spaces:
ID^ (mn), $ ( I R n ) , 6 (mn) and P, (‘2”).
Footnotes
(1) the concept of convergence of base space is called the
characteristic type of onvergence (or characteristic convergence) .
(2) see Section 5.3 of Chapter 5.
CHAPTER 3
DEFINITION OF DISTRIBUTIONS
The theory of distributions extends the concept of a numerical
function. To do this conveniently requires an indirect and artificial
definition. Specifically, a numerical function associates a number
with each admissible point of, say, the real line, whereas distribu-
tion, or generalized function, associates a number with each function
in a base space.
shall see that every numerical function can be considered as a
generalized function, and that the usual algebraic operations on
functions have immediate analogous for generalized functions.
These may seem quite different approaches, but we
(Mikusinski 111 and Silva C11 give other, more natural definitions of a generalized function.)
This chapter presents a brief introduction to the theory of
generalized functions and distributions. Other books give a more
thorough discussions; see for example, Garnir, Wilde and Schmets Ell,
Vol. 111, Friedman, A [ll, Schwartz Cll, and Zemanian C1l and C31.
Sections 3.1 and 3.2 of this chapter introduce the concept of
generalized function and distribution, while the rest of the chapter
contains a study of different spaces of distributions and generalized
functions; which we need in Chapters 7 and 8, where we generalize the
Fourier and Laplace transformations.
3.1,Generalized Functions
The base spaces constructed in the preceding chapter enable us in
this section to formulate the structure of generalized functions in
the following manner.
Let Q be a base space as in Section 2.8 of Chapter 2. A funct-
ional F on 0 is an operator which assigns a real or complex number to
25
26 Chapter 3
each function 41 E #. This number will be denoted by <F,$> (or
<FX, 4 (x) >, <Ft, $ (t) > if one must make the variable precise) :
The dual 0' of # is the space of functional F on B which are
linear and sequentially continuous. Recall that F is linear if
F E Q' if and only if
and F is sequentially continuous if
<F,$ > -t 0 as n -t m n
for each infinite sequence t$nl which converges to 0 as n + - in the sense of B .
A l s o , we make # I into a vector-space by defining vector addition
and scalar-multiplication: if F,G E @ I then aF + bG is the functional such that, V $ E 0, <aF + bG, $ > = a <F,$> + b < G I $ > .
The null element of this space is the functional such that, v $ E 8 I < o r + > = 0.
The equality in 8 ' can be defined immediately : if F-G=O then
F = G:
Here, the elements of the space 9' are called generalized functions.
3.1.1. Inclusion of 0'
Let Q1 and Q2 be two base spaces. If B 1 c Q2 algebraically and
topologically (the convergence of sequences is weaker in Q than in
a1), then we have 2
'#; c ". Indeed, for each F E #$, if F is sequentially continuous in the sense
of a$ , then it is likewise in the sense of Qi. Therefore, F E B i .
n Remarks. If we take x = (x1,x2,...,x 1 e IR , a,b E Cn and n Q(lRn) (see Section 2 .9 of Chapter 2 ) in the structure of F on B ,
Definition 27
then Fx will have n-independent variables x and FX E CP' (Eln), the
dual of @(IRn).
generalized functions in n independent variables. One can easily
Here, the elements of @' (IRn) are said to be
obtain the results of this section for the
on 0 ( IRn ) . Moreover, for the benefit of a
remark by taking the case of Fx on ID (IRn )
Section 4.5 of Chapter 4 .
3.2. Distributions
functional F~ E C P ~
reader we illustrate this
in the forthcoming
We term in this section the particular name of generalized
functions by the structure of generalized functions in @'(or @'(I€?))
formulated in the preceding section. This may be stated as follows.
The elements of ID' (or ID' (IR")) are said to be distributions
(or in ID' k(lRn)), are the
of Schwaxtz in IR (many other authors call them simply
generalized functions) ; those of ID'
distributions of order k in IR (or in IR") : those of $' (or $ I (IRn)) , are the tempered distributions or distributions of slow growth in IR
(or in IR") ; those of 6 ' (or 8' (IR")) are the distributions with
bounded support in IR (or in IR"); those of ZZ' (ox ZZ' ((2")) , are the ultradistributions or analytic functionals in C (or in Cn) .
(or in IR") )
3.2.1. Inclusions
Section 3.2 and Section 2.7 of Chapter 2 yield the following
results, we shall need these latex:
3.3, Examples of Distributions
This section provides some useful concept about the properties
of'distributions and, in particular, states that all distributions
are of a particular type.
3.3.1,Regular distributions
Let f(x) be a locally summable function in IR. Then the mapping
ID + C defined by
28 Chapter 3
(3.3.1) 9 .+ {f(x) $(x)dx.
(the integral is taken on the intersection of the supports'') of f
and $ ) is a distribution and is denoted by,
(3.3.2) <f(x), @(x)> = jf(x) $(x) dx, V + € I D .
Every distribution definable through (3.3.2) is called a regular
distribution. A distribution T is called regular if some locally
summable function f (x) satisfies
<TI$> = f(x) Q(x)dx, V Q E ID. IR
Then T is the regular distribution corresponding to f(x), and we can
identify the two, writing T = f(x). This identification is an
essential point in the theory of generalized functions.
The elements of ID and $ are all locally summable functions,so
that each, as above, defines a regular distribution. But we have
identified these functions with the corresponding distributions, so
that we may consider Dand $ linear subspaces of Dl . In symbols
ID c ID' and S C ID'.
Examples. The following functions representregular distributions, -x x2 xk (integer k z 0), Ix("(Re v > -1) , cos x, e , e ,
3.3.2,Irregular distributions
There are other kinds of functionals. For examples, the functional f (x) which associates with every $ (x) its value at x = 0
is obviously linear and continuous.
functional can not have the form of (3.3.2) and locally summable
function f (x) . It can be easily shown that this
Indeed, if some locally summable function f(x) satisfies
(3.3.3) I f(x) $(XI dx = $ ( O )
for every $(x) in 9) , then it satisfies IR
f(x) e(x) dx = 0 IR
for every e(x) E IR such that e ( 0 ) = 0 and e(x) .f(x) 2 0. Hence the
theory of Lebesgue integration implies that f(x) = 0 almost everywhere and any f (x) with this last property satisfies
Definition 29
f(x) @(x)dx = 0 IR
for all Q(x) in ID, even when Q ( 0 ) # 0. This is a contradiction.
All distributions that are not regular are called irregular
(or singular) distributions. An example of an irregular distribution
is the Dirac distribution, defined as follows:
Here we use the customary notation, which might erroneous suggests
that 6(x) was a function.
(3.3.4) has no meaning other than that given by the right side. (See
also Misra [ a ] . ) The following additional cases will.illustrate this
notion.
Hence we emphasize that the left side of
If c is a real constant then 6(x-c) is the functional which
assigns to a function Q its value Q(c). This is a distribution of
order zero:
(3.3.5)
(IDo denotes the space of continuous functions with bounded supporC2)).
6(x-c) also belogs to I D g k , ID' and $ I .
If k is a positive integer, then 6 ( k ) (x-c) is the functional This is a
<S(x-c),$(x)> = $(C), Y $ E IDo,
which assigns to a function Q the number ( - l ) k Q ( k ) (c) . distribution of order k because
(3.3.6)
A l s o , 6 ( k ) (x-c) belongs to Id, j > k, ID' , $ I but not to ld if j < k .
3.3.3.Pseudo functions
If the function g(x) is not locally summable, then it may happen
that the divergent integral Ig(x) $(x)dx has a finite part as defined
in Chapter 1, designated by Fpfg (x) Q (x) dx. We can consider this an
integral on a finite interval because Q(x) is zero out side a baunded
set. Accordingly, we have m
<FP g(x),$(x)> = FP fg(x) $(x)dx, V cp e ID
where the finite part of this integral is a di~tribution(~) and is
denoted by Fp g(x) .
-m
30 Chapter 3
This kind of distribution is called a pseudo function by
L.Schwartz C 11 . Examples. We give below a few examples of pseudo functions:
3.3.4.Regular tempered distributions
We say that f(x) is a tempered function, or a function of slow
growth, if xnf ( x ) , for some positive integer n, is bounded as I X I + w .
If f(x) is a locally suable tempered function, then as in Section 3.3.1, it defines a corresponding distribution in $ I , which we also
write f(x), and which satisfies W
df(X),$(X)> = J f(X)$(X)dX, -+ 0 E OI. -m
The examples in Section 3.3.1 are a l l tempered distributions except
e 2
-X and ex ; however U(X)~-~ E $ I .
3.3.5.Tempered pseudo functions
If g(x) is a function such that x-"f(x), for some positive
integer n, is bounded as 1x1 + m, then W
<Fp g(x),b> = FP /g(x)$(x)dx, Y cp E S -m
defines a distribution Fp g(x) in S t whenever the right side is a well-defined finite part for all 0. Any such Fpg(x) is called a
tempered pseudo function.
The examples of Section 3.3.3 are
except Fp I x 1 -ve-X .But Fp U (x) x-Ve-X is
If g(x) has an infinite number of
definition of Fp g(x) involves further
we assume that g(x) satisfies also the
also tempered pseudo functiow
a tempered pseudo function.
singularities, then the
complications. In this case
following two conditions:
1. g(x)is continuous on I - m , m [ except at a countably infinite
number of points cn, where -m < n In a neighbour-
hood of each cn, g(x) = 1 ank(x-cn)-k+a no loglx-c n I + h n (x) where the
function hn(x) I s continuous in this neighbourhood, K is a fixed
m and c ~ + ~ > cn. K
k=l
Definition 31
positive integer and lankl < Mlxl' for In1
integer and 13 a positive number. no, no a fixed positive
2 . If In is the interval such that (x-cnl < a, then there exists
a positive number a which yields non intersecting intervals Inout-
side of which
and in side of which
If g(x) fulfills these two conditions, then the procedure of Section
1.2, Extension 4 of Chapter 1, may be applied to Fp<g(x),4(x)>.
Hence, by formula (1.2.6) of Chapter 1, for all $ in $, we obtain
5 m
<Fp s(x),$> = FP I g(x)$(x)dx = lim Fp I
where the limits 5 and -5' of the integral take values in the
intervals [ c ~ , c ~ + ~ ] but outside the intervals In.
g(x)$Ix)dx -m 5 -Frn - 5 '
[ ' + m
An example of this case is Fp . sin x
3 . 3 . 6 . Analytic functionals (ultradistributions)
Let z be a complex number such that z = x+iy and if $(z) E 23,
then evidently,
( 3 . 3 . 7 ) <6(z-a), +(z)> = +(a); a E c
( 3 . 3 . 8 ) <6(k)(z-a),$(z)> = (-l)k$(k)(a).
where k is a positive integer.
If f(z) is a analytic function and L a path in the complex
plane, then the mapping:
4 E Z+ 1 f(z)+(z)dz L
belongs to 23' and is said to be a regular analytic functional (or
regular ultradistribution).
Let L = ra be a closed path going around the point a and drawn in the positive direction. Then by ( 3 . 3 . 7 ) and the Cauchy integral
formula, we have
32 Chapter 3
Hence, we obtain the identity
1 6(z-a) = Zni(2-a), .
'a Using these last results, we can rewrite Fp(x-c)-l (see Section
3 . 3 . 3 ) in the following form.
Let r+ be equivalent to the paths, -m to C', L+and C to -, where L+is the s&ne path as described in Section 1.5 of Ciiapter 1.
$ ( z ) E 2 2 , hence, by Taylor's expansion, $ ( z ) = @(c)+(z-c)Y(z) in
some neighbourhood of c where Y(z) is an entire function. Also, note
thatl- is analytic along the paths -- to C' and C to -.
If -
x-c
By making use of the above hypothesis, we have
C' C ldx
C' F~ Jm ? A x = I w x + I m m x + Jy(x)dx + $ (c ) Fp , f l x-c
x-c x-c x-c -m -m
The function @ ( x ) is holomorphic in the neighbourhood of c 1 and C. Thus, by Cauchy's theorem, we have
- Hence, we get
C '
Finally, we obtain
= J q 2 z c
r+ - where r+ = x & i E , E > 0 is
find -
+ 90,x + QIO,~ inlp(c) L+x-c x-c
drawn towards x > 0. Consequently, we
( 3 . 3 . 9 )
Also, according to ( 3 . 3 . 9 ) we have
~p(x-c) -' = (z-c) -' I r+ 2 ins (x-c) . -
Definition 33
Adding above two formulae, we get
The formula (3.3.9) can also be written further as
( 3.3.10) ~p(x-c)-’ = (x-c t i c ) -’ 5 ins (x-c)
of symbolically (since E is arbitrary)
~p(x-c)-’ = (x-c 2 io)-’ f ire (x-c) . This is a formula of Lippmann-Schwinger (or of Sokhotsky).
Problem 3.3.1
where ra is the path (i) Prove that 6(k) (2-a) = (-1)
2ni k! (2-a) 1 r a
described in section 3.3.6 and k is a positive integer.
(ii) Find the expansion of Fp(~-c)-~ where n is a positive integer.
Foot notes
(1) see Section 4.1 of Chapter 4 .
(2) abusively, some take 6(x-c) as a true function and give
( 3 . 3 . 5 ) under the integral form I G(x-c)@(x)dx =@(c),
6(x-c) is a measure of mass 1 at-; = c and 0 elsewhere.
m
( 3 ) of an order which depends upon the singularities of g(x) .
This Page Intentionally Left Blank
CHAPTER 4
PROPERTIES OF GENERALIZED FUNCTIONS AND DISTRIBUTIONS
summary
Our subsequent work does not require a complete theory of
generalized functions and distributions. Hence, in this chapter,
without proof, merely quotes some deeper results which govern limits,
convergence and completeness, or concern approximation by regular
functions. The end of this chapter briefly considers distributions
in several variables.
4.1,Support
Let us first note the definition of equality for generalized
functions.
Definition 4.1. Two generalized functions F,G E 4 ' are said to
be equal on an open set in IR (or in C if 0' = 2Z')if <F,I+> = <G,I+>
for every function $ E @ and which has its support in this open set.
Since integrable functions which are equal almost everywhere
give rise to the same generalized functions, this definition implies
that such functions are to be regarded as the same. Hence values of
a generalized function can be specified only in a interval and not at
a point.
If F E a ' , then our last remark requires us to define the statement that F is zero on an open subset of IR.
By Definition 4.1, a generalized function F E 0' is said to be
zero on an open set if <F,$> = 0 for every function I+ E 4 and which
has its support in this open set.
The support of a generalized function (or distribution) is the
complement of the biggest open set on which it is zero. (This can
be a point or all of IR.) - 35
36 Chapter 4
Examples. We give below a few examples of the support of a
distribution.
The support of the distribution f(x) defined by
l-lxj, -1 < x < 1 i 0 I 1x1 5 1: f(x) =
is obviously C-1,11.
6(x-c) has its support at the point c. Fp x-" has SR as its
support.
We shall say that two distributions F and G E ID' (or $ I ) are
equal on a closed interval[c,c'] of IR if <F,$> = <G,$> for every
+ E ID (or $) with support in a (open) neighbourhood of Cc,c'l.
If S1 and S2 are the supports of distributions F1 and F2, then
the support of a distribution aF1 + bF2 is contained in the union S I U S2 but may not be equal to it.
N o t e . If the support of F and the support of $ have no points in common then <F,$> = 0.
4.1.1. Point support
If a distribution F has a single point, say c, as its support,
then this F is a finite linear combination of distributions 6(x-c)
and 6 (k) (x-c) I k = 1,2,3,. . . . point support.
This type of support is called a
In the following sections we now classify the distributions by
means of a support of a distribution.
4.1.2. Distributions with lower bounded support
By ID: we mean the space of distributions having lower bounded support('' . (For each element F E ID; I there exists a finite number c
such that the support of F is contained in the half line x c.)
Example. The following distributions belong to ID;:
n n x+, x U(x-1) I Fp x-" U(x+l)
where
Properties 37
10 , elsewhere,
(1, x > -1
U(X+l) = { 1 0 , elsewhere;
and n is a positive integer.
ID; will have an important role when, subsequently, we study
convolution and Laplace transformation (see Chapters 5 and 8).
4.1.3. Distributions with bounded support
The distributions with bounded support evidently belong to ID:,
$' or 27,' . to the simple structure of the base space 6 : reciprocally, each element of 6' is a distribution with bounded support.
These also belong to 8' . This property is important due
We list below some important properties:
1. The value of <F,Q> depends only on the values of +(x)Q'(x)...
Q (k) (x) taken over the support S when k is the order of F.
2. Section 5.4 of Chapter 5 introduces the distributional
derivatives. Here, anticipating this definition, we note that any
distribution with bounded support has infinitely many representations
as a finite sum of distributional derivatives of continuous €unctions,
the support of each function being an arbitrary open neighbourhood of
the support S.
3 . Each distribution in ID' is equal,
to a distribution having its support in a
the set U.
4.2.Properties
on an open bounded set U,
bounded neighbourhood of
This section contains further useful information about the
properties of generalized functions and distributions.
4.2.1.Boundednes.s
We state some very important properties of distributions:
38 Chapter 4
1. Any distribution F in ID'and any finite closed interval I determine a non-negative integer k and a positive constant M with
that
I<F,$>I I M sup^+(^) (XI I X
where $ is any function in ID.
2 . The property 1 is no longer valid for all F E ID'if we allow
I to be an infinite interval. However, all tempered distributions
do possess a property 1 that holds over an infinite interval as
stated in the following.
Let F E $' be a tempered distribution. There exist an integer
k 2 0, real p, and constant M > 0 such that for every (B E $,
I<F,$>/ I M SUP/(l+X 2 I p / 2 p 1 X
where k, M and p depend only on F.
Zemanian (Cll, Sections 3 . 3 and 4 . 4 ) gives proofs of 1 and 2 .
4.3.Convergence
This section provides an account of the convergence in the
different spaces of generalized functions and distributions.
If each value v of a parameter determines a generalized function
Fv in Q', then F converges(2) , as v -+ vo, if the numerical function
< F v , $ > converges when $ is any element of 0. V
We deduce from this definition the following facts.
1. Let F1,F 2 , . . . I in Q', be an infinite sequence of generalized
In particular,
functions. Then this sequence is convergent as n + m if the numeri-
cal sequence <Fnr$> is convergent for every $ in Q.
the sequence IS (x-n-l) 1 is convergent as n + a,
m
2 . The series F. is convergent (on 0), if for every $I E 0 j=j 7
0-
the numerical series 1 <Fj,(B> is convergent. j=jo
m
Examples. The series 1 6 (x-j) is convergent on ID , because j=O
E < 6(x-j), $ > = E +(j) converges. This sum has only a finite
Properties 39
number of nonzero terms because 4 has bounded support. m .
ch-'6 (x-j) is convergent on $ (and on ID) , If h > 1, then j=1
because C < h-16 (x-j) , $ > = Ch-j$ (j) converges, 4 being bounded. (This series of distributions does not converge on & .) Moreover,
the series
4.3.1.Completeness and limit
m
1 6") (z-a)/j! is convergent on P, . j=O
With the proceding notion of convergence, the spaces ID' , $',and are complete : if a family Fv, respectively belonging to ID' , $' ZZ'
or P,' , is convergent as v -t v then this family has a limit F
belonging to 3D' , $' or 22' , respectively, and satisfying 0'
lim <F $ > = <F,$>. v - t v
0
The proof of this result is given in Zemanian [I] . However, ID' is not complete. Indeed, Sections 5.2 and 5 . 4 of
Chapter 5 define translation and differentiation. Thus, given
n = 1,2,... and any distribution G in ID' such 1 n and take h = -.
DG E ID I k + l . Then Fn E IDIkand lim Fn = lim
n- h+O Thus DG does not always belong to
G = 6 ( k ) (x) . The space C' becomes complete when the above definition of convergence as follows.
that Fn=n(Gx+l/n-Gx)
h (Gx+h-Gx)=
m l k , in particular, we slightly modify
An infinite family
-1
Fv of distributions having bounded support is convergent and has its
support in e' if the numerical family <F $ E 6 and if all the F v have their supports in the same bounded
interval.
$ > is convergent for every V ,
4.3.2,Particular cases of convergence in ID'
Let an infinite sequence {fn(x)) of locally summable functions
converge to the function f(x) almost everywhere. If all Ifn(x) I are bounded by the same positive locally summable function, then f(x) is
also locally summable,andregular distributions fn(x) E ID1 tend to
the regular distribution f(x). This is a consequence of Lebesgue's
theorem on integral convergence. Thus we see that the convergence
of distributions and of generalized functions generalizes that of
functions.
Definition. We say that the sequence of functions{fn(x)l
40 Chapter 4
converges in the sense of distributions (or in lD')if the sequence
of distributions {fn(x) 1 (which are identified with these functions) converges.
If the functions f(v;x), which are locally summable with respect
to x, converge uniformly on every bounded interval of IRto the limit function f(x) as v + vo, then the functions f(v;x) also converge to
f(x) in the sense of distributions as v + v0.
4.3.3.Convergence in $'
The convergence in $' is analogous to that in ID' , if one replaces ID' by $' in Section 4.3.2 and replaces 'locally summable
functions' by 'functions with slow growth'.
4.3.4.Convergence to 6(x)
Let the variable v , integer or non integer, have the limit uo,
has the limit 6 (x)
finite or infinite, and let each value v determine a function fv(x).
Then the regular distribution fv(x) , as v + v
if: 0'
C
(1)
(2)
(3)
J fv(x)dX + 1 for every c > 0;
fv(x) -+ o uniformly on every set, O < E 5 1x1 5 c; I (fv(x) Idx, for some positive number b, has a bounded independent of v .
-C 1
b
-b
Example. We construct an example of sequences which converge to -1 2 2 &(XI. If fv(x) = v
proves (21, and
exp(-rx /v ) , then 0 2 f (x) - < v-l, which b m
Ifv(x) Idx 5 I fv(x)dx = 1, which proves ( 3 ) . Also -b -m
m 2 2
C m c m 1 2 I fv(x)dx =/'fv(x)dx -[f + If,(x)dx = 1-(2/v)Jexp(-rx /U )dx
-- c C -m -C
2 2 2 2 m
- > 1-(2/v) lexp(-r(c +2ct)/v )dt = l-(l/nc)exp(-rc /v ) 0
and the last expression has limit 1, which proves (1).
Problem 4.3.1
Let
Properties 41
where 1
0 1-x w = 2 1 exp + dx
1 and show that <,(XI -+ 6(x), as v + 0.
n E IN, we further obtain clIn(x) -+ 6 (x) as n + -. Then, by replacing v by ;i,
We shall see later that the Laplace transformation (see Chapter
8) gives other expressions having limit 6(x).
4.4.Approximation of Distributions by Regular Functions
The distribution 6(x), by the preceding section 4.3.1, is a limit
of very regular (indeed, infinitely differentiable) functions. This
result enables us in this section to show that all other distribu-
tions can be approximated by regular functions in the
manner.
Let F E ID' and let rn(x), n f IN, be an infinite
functions whose supports tend uniformly to the origin
integrals satisfy m
f o 1 lowing
sequence of
and whose
rn(x)dx = 1. -m
If we put
then we have the following results from Schwartz [I1 , Chapter VI.
1. The @ (x) have infinitely many continuous derivatives.
2 . The $,(x), as n + m, converge to F in the sense of
n
distributions.
These @,(x) are called regularizations of the distribution F.
Both the r (x) and clIn(x), by Section 4.3.4, are regularizations n of 6(x). With respect to 2 . , we note
gn(x) = rn(x) x F + 6(x) x F = F
(see Section 5.8 of Chapter 5 ) .
42 Chapter 4
Indeed, 3 distribution F satisfies the relation rn(x)*F=$n(x). If we can divide this formally by rn(x), then we can express F
formally as a convolution quotient. This remark confirms Mikusinski's
construction C13 . 4.5.Distributions in Several Variables
The foregoing concept for distributions in a single variable
have a straightforward extension to distributions in n independent
variables. Briefly, we outline in this section the basic theory of
this extension.
As remarked in Section 3.1 of Chapter 3 , we first €ormulate the
structure of functionalson ID(*). coordinates (x1,x2,. . . ,x ) of a point in lRn and dx = dx
We recall that x is the set of dx2. . . . .dxn. n
on ID (lan) A functional T is an operator which assigns a real
This number will be X
or complex number to each function $ E ID(#).
denoted by <TX, $ (X) > I
TX : ID(IRn) -+ Cn
ID
TX
ID (lRn) + <TX,$(X)>.
The dual ID' (lRn) of ID( En) is the space of functionals TXon
3Rn) which are linear and sequentially continuous. Recall that
is linear and if TX E ID'(d ) if and only if
cTX,a$(X)+bY(X)> = a <T ,$(XI> + b <TX,Y(X)>
V 9,s EID (IRn), a,b E Cn,
X
and TX is sequentially continuous if
<TX,$n> + 0 as n -+ - for each infinite sequence i$nl converges
sense of ID( IR") . elements of ID' (IRn) are called the distributionsin n independent
variables.
to zero as n + 0) in the
As termed in Section 3.2 of Chapter 3 , the
Support of a distribution in ID' (IRn)
A distribution TX in ID' D") is said to be zero in an open set R of IRn if <TX,$(X) > = 0 for any function $(X) of ID (IRn) which has
its Support in fl. The support of T is the complement of the biggest X
Properties 43
open set il on which T is zero. X
Now we state below a few examples of distributions.
Example 1. Let f(X) be a locally summable function which is
identified with a distribution TX. Then we define
Here TX is the regular distribution corresponding to f(X) as in the
case of one variable (see Section 3.3.1 of Chapter 3). The integral
indeed exists, for the domain of integration which is not JRnbut the
bounded support of Q on this support f is sununable and Q continuous
so that fQ is summable. Also, the value of the integral is obviously
a linear functional of 4 .
Example 2. Let Qn(c) be the domain whose each point
X = (X~,X~,...,X ) is such that x.> c, i 5 i 5 n. In particular,
(i) G3(c) is the half-axis x 2 c; and (iv) Qn(O,m) is the first orthant of IRn (see Section 12.7 of Chapter 12).
1 1- 1 " 3 of IR ; (ii) Q2(c) is 7 of the plane; (iii) Q1(c) is
By ID; (IRn) we mean the space of distributions in IRn having
lower bounded support.
there exists a real number c such that the support of TX is contained
in Qn(c).
(4.5.1) <TX,O(X)> = f f(X)O(X)dX
This means that TX belongs to ID: (IR") if
If TX = f(X), a summable function in Qn(c), then we have
Qn(c) m m
= 1 . . . j f (XI $(XI ax1,. . . ,axn. C C
2 2 4 For instance, if n=2, we can take f(X) = 1 in the disk (x1+x2) 5 a, and f (X)= 0 elsewhere; in this case, the support of TX = f (X) is
contained in every Q,(c) such that c -a.
Moreover
<Tx,Q(X)> = J f(x)$(x)dx Kn
where Kn denotes the support of f(X)
n = 3, a hyper-volume if n > 3). Obviously, the domain of this
integration is the intersection of K with the support of Q.
(a surface if n = 2, a volume if
n
If c = 0 in above space, then we denote the space IDb+(lRn )
44 Chapter 4
n instead of ID; (IR") . takes the form
(4.5.1') <TXl$(X)' = ...f TX $(X)dX, V .$ E ID(IRn).
The space IDA+ (IRn) will play an important role, when subsequently in Chapter 8 we study the Laplace transformation of distributions in
In this case if TX E IDA+ (IR ) then (4.5.1)
a (I)
0 0
IRn . Example 3. The Dirac distribution 6 ( X ) is defined by
<6(X)I$(X)> = O ( 0 ) I v 9 E ID(JRn).
The point distribution 6 at the point c of IRn is defined by C
< 6 (x1-c1'x2-c2'.. . IXn-cn) I 0 (x11x2'. . . I X n ' = 0 ( c 1 I c 2 I . . I en) 4 E ID (IRn), c = (C1'".'Cn).
Example 4. The n dimensional spaces admit surface distributions
whose definition can be formulated in the following form.
Let S be a regular surface and let k(X) be a piecewise
continuous function on S. Then we denote a surface distribution by
k6 and define it as
(4.5.2)
Occasionally, we call ksS a distribution of single layor of density
S
<k 6s,$(X)> = f k(X)+(X)dS. S
k(x)
The surface distribution 6 carried on the sphere Sn(a,R) Sn(alR)
having equation (x -a ) 2 +(x2-a2)2+...+ (xn-an)2 = R 2 is defined as 1 1
(4.5.3)
11
whe're An(R) is the area of the sphere and 9 E ID (Illn ) . In order that the sphere Sn(a,R) is contained in Qn(0), it is
necessary that R < sup 1 3 In
6Sn(alR) f%+(lRn ) . In other wordsl we have
(aj). If this condition is satisfied, then
Propert ies 4 5
Footnotes
(1) c e r t a i n other authors denote ID:, t he space of d i s t r i b u t i o r s having support i n t h e half l i n e x 2 0.
( 2 ) weakly convergent, i n t he rigorous sense of topology.
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CHAPTER 5
OPERATIONS OF GENERALIZED FUNCTIONS AND DISTRIBUTIONS
This chapter considers some standard operations on functions,
and extends them to distributions and generalized functions. To
define these operations on the larger domain, we use the method of
transposition, which plays an important role in the theory.
Briefly, this chapter has the following structure. Sections
5.1 through 5.8 use transposes of standard operations in the base
space to define analogous operations on generalized functions:
translation, product by a function, derivative and partial derivative,
convolution. Finally, the end of this chapter discusses transforma-
tions of the independent variables.
Let us first introduce the notion of transpose.
5.1.Transpose of an Operation
Let B be a mapping of the base space 8 into itself which is
continuous for sequences in Q: that is, if {$,(x)} + 0, as n + m
n E IN, in the sense of 8 , then the sequence {Q $,(x)} also converges
to zero in the sense of 0. To B, there corresponds an operation
defined on the dual 8' of Q. This operation is called the transpose
of Q , and will always be denoted by Q' in this book, Thus, B' is
defined by the formula,
If arb E C, we have according to (5.1.1),
47
48 Chapter 5
which yields ,
n 1 (aFl+bF2) = an1Fl+bn'F2.
Consequently, R 1 is linear. Also, n' is continuous for convergence in 8 ' ; that is, if the sequence {FnI converges in Q' then the
sequence t n ' Fn) also converges in 0'.
In addition, if 0 is an isomorphism on +, which implies that its
inverse is continuous for sequences, then 0' is an isomorphism
on C p l and (n ' ) - ' is also the transpose of n-l(see Treves c11 ) . The preceding are rather general remarks, but we now treat some
particular operations which more clearly illustrate the concept of
transpose. Usually, in the following sections, we shall be able to
consider 0 a subspace of Q 1 and consider Q' an extension of n, so
that the transpose, in such cases, will have the same name as the
defining operation n.
5.2.Translation
Let rC be an operator of translation where c is real defined by
(5.2.1) Tc$ (x) = $(X+C) , Y $ E ID.
It is evident if the sequence of functions {$nl converges to zero
in ID, then the sequence of translations t r 4 1 also converges to zero in ID.
we denote by T~ is also called translation. According to (5.1.1), we
obtain
c n Take T - ~ for il (see example 2); its transpose Q ' which
<T F,$> = <FIT +> V 4 E ID. C -C
(5.2.2)
Usually, we write F
Examples. We list below a few examples of translation.
instead of T ~ F . x+c
1. According to (5.2.2), V + F ID
Hence, we obtain
rc6 (x) = 6 (x+c).
Operations 49
2 . If F = f(x) is a locally summable function, we have according
to (5.2.2) for every Q, E ID
<TCf (X) ,'$ (X) > = <f (X) I (X) >
= If (x) Q, (x-C) dx = I f(xtc) '$ (x)dx lR m
= <f (xtc) , Q, (XI >
which yields the identity
Tcf(x) = Fx+c*
The cited examples show that the translation T in ID' properly
We list below some particular C
generalizes the concept of T~ in ID.
types of distributions which will be utilized later in our study.
If for every positive or negative integer n
<Fx,Q,(x-np)> = <Fx:Q,(x)>, VQ, E ID,
which yields the relation,
= Fx+np
then the distribution
The reflection of
by the relation
FX'
F is a periodic distribution with period p.
Fx is a distribution F-x (or Fx) and is defined
X
L
<F-X~ 4 (XI > = <Fx, Q, (-XI >.
The distribution Fx is said to be symmetric (or even) if F-, = Fx.
X' Also, F, is anti-symmetric (or odd) if Fex = -F
As for functions, we can say that Fx is symmetric (resp., anti- symmetric) with respect to c, if F-(x-c) - - FX+C; therefore F,x=Fx+2c
(resp. F-(x+c)= -FX+C and hence F-x = - Fx+2c) * Applying these definitions to periodic distributions, we obtain
periodic symmetric and anti-symmetric distributions.
5.3. Product bya Function
In general the product of two locally summable functions is not
locally summable, hence, it is not possible to give a meaning to the
product FT of two arbitrary distributions F and T. However, if f(x)
50 Chapter 5
is a locally summable function and a(x) is a continuous function,
then the product a(x)f(x) is locally summable. We see next that the
transpose has an important role in generalizing such restricted
products.
5.3.1. The space M ( 8 ) and the general definition of product
Let ci be a function such that a+ E 0 whenever + E # and a $n -+ 0
in the sense of 0 whenever 4n -+ 0 in the sense of Q. called a multiplier for 9.
Then a is
Clearly, such multipliers form a vector space M(#). The product
of a generalized function F e 4 ' by a function CL EM(#) is the element
of # I , denoted by aF, which is defined according to (5.1.1) by
5.3.2. Distributions belonging to ID' or E'
The definition of M ( @ ) immediately implies that M ( I D ) = , the space of infinitely differentiable functions having arbitrary support.
Therefore, if F E ID' (resp. E ' ) and if a E e I then aF belongs to
ID' (resp. 6 ' ) where we have by (5.3.1)
Examples. The following examples will illustrate this definition.
1. The product of a regular distribution f(x) by a function
a E 8 is the regular distribution a(x) f ( x ) E D' defined by
(5.3.3) <a(x)f(x),+(x)> = a(x)f(x)+(x)dx, v o E ID.
Hence the product defined by (5.3.2) properly generalizes the product
of functions.
lR
2. As a(x)Fp 'g(x) = Fp a(x)g(x) we conclude that some products
can drop the symbol Fp.
if j is an integer s 0, then xjFpx-' = Fpxj-' (here Fp is not needed
if Re(j-v) > -1).
For instance, xFpx-' = 1; on the other hand;
3. According to (5.3.2) v I$ E m ,
and hence we obtain the identity
Operations 51
(5.3.4) x6(x) = 0.
Similarly, if a(x) c 8 , then
which yields the result
(5,3.5) a(x)6(x-a) = a(a)6(x-a).
(See also below Section 5.3.2.1.)
5.3.2.1,Distributions of finite order
k By M ( I D ) we mean the space of k-times continuously differentia- k ble functions.
IDgk, and in accordance with (5.3.5). If a(x) c M ( I D ) and ~(x-c)EID'~, then a(x)d(x-c) E
a(x)d(x-c) = a(c)6(x-c).
This also holds even if a(x) continuous only.
5.3.3. Tempered distributions
By M($) we denote the space of infinitely differentiable functions h with slow growth such that [ a(k) (x) I < Alxl , as 1x1 + m when A and
h > 0 may depend on k but are finite.
If F E $ I and if a c M ( $ ) , then the product aF also belongs to $'
and according to (5.3.1)
(5.3.6) <aF,@> = <F,a $>, V @ c $.
To make precise the difference between the formulae (5.3.2) and
(5.3.6) , we remark that ex belongs to (i but not to M ( $ ) . F E 9' (for example, if F = f(x) a locally summable function with
slow growth) , then eXF belongs to D' but not to $ I .
Thus, if
5.3.4. Ultradistribution
By M ( Z ) we denote the space of entire analytic functions of the complex variable z , z = x+iy, whose modulus is bounded by A I z I hea l y I as I z [ + m , where A and a are real constants, and h is an integer.
If F E Z 1 and a E M ( Z ) , then the product aF also belongs to Z'.
52 Chapter 5
We have according to (5.3.1) ,
5.4.Differentiation
In this section the notion of transpose enables us to define what we mean by the derivative of a distribution so far as this is possi-
ble. Then, for later use, we obtain a number of results concerning derivatives of distributions.
5.4.1. General outline
Let d/d denote the usual differential operation on functions (1) , and let h be an integer 21. Let 9 denote any one of the base spaces, ID , 8 , $, or Zz . Then the mapping
of 0 into itself is continuous for sequences. Its transpose,
operating on the dual a ' , is the mapping denoted by D (D, if h-1) called differentiation (more precisely, distributional differentia- tion) of order h which is defined by
(5.4.1)
h
h <D F,$> = (-11~ <F,+(~)> , Y + E o and F E 0 1 .
DF is the derivative of the generalized function (or distributiod h F, and D F is its derivative of order h.
By (5.4.1) we deduce that Dh is linear. Also,
(5.4.2) h h+lF D D F = D ,
h and if F b ID' then D F E ID1k+h*
Section 5.8.3.4 will give another aspect of differentiation.
5.4.2. Remark
Prior to showing that D is indeed a generalization of different-
iation, we first need to show that the name differentiation given to
D is consistent with translation.
The notation and terminology of Section 5.2 express the derivative
of a function:
Operations 53
TcO (X) -0 (X) = lim
C c + o
Similarly, we obtain for the derivative of a distribution
DF = l h 'cFWF c + o c
because V 0 E ID, we have by (5.2.2) and (5.4.1)
= l h <F, - 0 ' (x) > + lh C <F, O1(x) > c + o c + o
= <F, -c$'(X)> = <DF, $>.
2 Taylor's formula illustrates to write +(x-c)-$(x) = - c$'(x)+c $l(x),
o1 E ID.
In particular, we observe that 6'(x) = D6(x) (see Section 5.4.4)
represents a unit dipole since
D6(x) = lirn 6(x+c)-6(x) c + o C
5.4.3.Distributions of finite order having bounded support
If F is a distribution of order 5 k and has bounded support, then the following exist:
1. a continuous function h(x) and an integer p, p 5 k+2, such that F = DPh(x) ;
2. a regular distribution f(x) and an integer qr g 5 k + l , such
. that F = Dqf (x) ;
r 3. a measure m and an integer r, r 5 k, such that F = D m.
The proof can be found in Schwartz [11 , Chapter 111.7.
5.4.4. Derivatives of the Dirac distribution
h Using (5.4.1) and (3.3.6) of Chapter3, we have V 0 E ID (whence
54 Chapter 5
which yields the result
h D 6 (x-c) = 6(h) (x-c) . Usually 6") (x-c) is denoted by 6' (x-c) . 5.4.5,Derivatives of a regular distribution
Let f(x) be a function such that : f has a locally summable d derivative (in the ordinary sense) =f(x) except at the isolated
points, c , where f has a left hand limit f(c.-) and a right hand limit f (c .+) . Then we have a derivative of the regular distribution f(x)
(5.4.3)
d where -f(x) is considered as a regular distribution. dx
j 7
3
d Df(x) = Ef(X) + 1 [f (C.+) - f (C.-) 1 6(X-Cj) 3 3 j
proof. For simplicity, assume that f (x) has a discontinuity at
X=C. Using (5.4.1) and integration by parts together with the fact
that (c+) = Q (c-) = Q (c) (since + is continuous), we have Ti Q c ID,
<Df(x),Q(x)> = - / f(x) & +(x)dx IR
md + / (f(x))Q(x)dx
= <Cf(c+)-f(C-)lS(x-c), +(XI>
+ I a;; (f(x))Q(x)dx
C
- d -m
which yields the result
Df (x) = [: f (c+)-f (c-)I 6 (x-c) + & f (x) . Replacing c by c in the above formula we.get (5.4.3). The proof is
j
Operations 55
thus completed.
Note that Df (x) given by (5.4.3) or (5.4 .l) is the distributioral
derivative of the function.
If f(x) is differentiable (everywhere) then there is no dis-
continuity and (5.4.3) leads to
d Df (x) = a;; f(x)
which asserts that the distributional derivative D is a proper
generalization of the ordinary derivative. Moreover Df(x) almost
everywhere, by (5.4.3) , is the usual derivative f (x) for the
previously specified functions.
d
dx
If f'(x) = f(x) also satisfies condition mentioned above and d has discontinuities at ci, then applying (5.4.3) to
obtain
(5.4.3')
f (x) , we
D G f d (X) = T f d2 (X) + 1 [f' (Ci+)-f ' (Ci-)l 6 (X-Cj) . dx i
Also, applying D to (5.4.3), we find
( 5.4 .3" )
By putting %f(x) from (5.4.3') in (5.4.3"), we finally obtain
D2f(x) = &(x) + ~Cf(cj+)-f(C.-)l 6'(x-c.). 3 3 I
n
Here cf include the previous c
points.
but may also include additional 1'
Repetition of the arguments for (5.4.4) yields a general result
for higher derivatives of f(x).
If p(x) is a polynomial of degree < h, then
h (5.4.5) D p(x) = 0.
The following examples will illustrate the above results.
Examples. In classical analysis, ordinary differentiation gives
1. DeaX = ae ax
56 Chapter 5
On t h e o the r hand, we have according t o (5.4.3), ( 2 ) ax Dey = ae+ + 6(x)
2 ax e+ = a e+ ax + a s ( x ) + 6 ' ( x ) .
2. By applying (5.4.3) t o U(x) ( t h e Heaviside func t ion) w e ob ta in
DU(x) = 6 (x) . Also, DU(x-c) ( 3 ) = 6(x-c). w e g e t
Fur ther , by making use of Sec t ion 5.4.4,
DhU(x-c) = 6 (h-l) (x-c) , h 2 1.
3 . The saw-tooth func t ion S (x ) (4) can also be w r i t t e n a s
(5.4.6) S ( X ) = a U ( X ) 1 x (x;j-l,j)(x-j+l) j=1
where j-1 c x < j
elsewhere. x (x; 1-1, j) =
By applying (5.4.3) t o (5.4.6) w e ob ta in
m
D S ( X ) = a U ( X ) + a 1 6(x-j) j=1
m and 2 D S ( X ) = a s ( x ) + a 1 & ' ( x - j ) .
j=1
Also, according t o Sec t ion 5.4.3, t h e saw-tooth func t ion S (x ) is the 2
m
d e r i v a t i v e of t h e continuous func t ion a 1 [ (x - j+ l ) + j - l l x x ( X i l - 1 r J ~ 'j=l
Problem 5.4.1
Prove t h a t Y $ E ID
(i) D [ (x-c)U(x-c)l = U(x-c)
and hence deduce,
(ii) 6(x-c) = D 2 [ (x-c)U(x-c) l (5) .
Operations
For $ E $, prove that
57
(iii) DU(a;x) = 6(x)-a(x-a)
where
O i x i a
elsewhere.
5.4.6,Derivatives of pseudo functions
In this section we compute the derivatives of some functions
which are not differentiable (globally) in the ordinary sense but
can be differentiated in the distributional sense. To do so, we
first find the derivative of (log x)+.
Accordiqg to (5.4.1), we have Tf @ E ID,
(5.4.7) <D(lOg X)+,@(X)> = - <(log X)+,$'(X)> m
= - I log x +'(x)dx. 0
By virtue of (1.3.4) of chapter 1 together with b ( m ) = 0, (5.4.7)
takes the form,
(5.4.7') m
<D(lOg x)+,@(x)> = - I log x $'(x)dx 0
= Fp [-log X $ (x)] + Fp b(x)dx 0
-1 = <FP x+ I @(XI>
which yields the result,
(5.4.8) -1 D(log XI+ = Fp x+ . . Next, we describe the derivative of DFpx;" where n is a positive
number.
According to (5.4.1) together with the technique of (1.3.2) of
chapter 1, we have Tf 4 E ID,
-n -n <DFpX+ , $ (x) > = -Fp<x+ , $ ' (x) > =
- - - n! I @(")(O) - nFp
58 Chapter 5
which y i e l d s t h e r e s u l t
(5.4.9) DFpx;" = n! 6 ( n ) ( x ) - nFpx+ . -n-1
Furthermore, i f R e v > 0 then w e have i n t h e ord inary sense
v - 1 (5.4.10) Dxi = vx+ . I f v # 0, -1, -2 , ---- , t hen Fpxi and Fpxi-', wi th r e spec t t o v , a r e continuous d i s t r i b u t i o n s , so t h a t a n a l y t i c con t inua t ion of (5.4.10) y i e l d s
v - 1 (5.4.11) DF~X: = V F ~ X+ . Here t h e Fp a r e no t needed i f R e v > -1 and R e v > 0 on t h e l e f t and r i g h t s ides , r e spec t ive ly .
Las t ly , l e t g ( x ) be a func t ion which i s zero f o r x < c, continuous wi th a d e r i v a t i v e g ' ( x ) f o r x > c, and also admits a r ep resen ta t ion of t h e type (1.1.5) of chapter 1. Then t h e d e r i v a t i v e of g ( x ) can be obtained by applying (5.4.11) o r (5.4.9) t o each term of g ( x ) and us ing t h e f a c t Dh(x) = h' (x)+h(c+) x 6 (x-c) . Accordingly, w e ob ta in
K k DFpg(x) = Fpg' (x )+h(c+) 6 (x-c) + 1 (-l) 6 ( k ) (x-c). k l bk (5.4.12)
Problem 5.4.2 k = l
Prove t h a t t h e fol lowing formulae are v a l i d on t h e space ID:
-1 (i) D(1oglxl) = Fpx ;
-n-l - L , ( n ) (x) ; n! (ii) DFpx-" = -nFpx
where n i s a p o s i t i v e in t ege r :
(iii) D(log(x-c)x>c) = F ~ ( x - c ) ~ > ~ -1 I . - A - 1 ( i v ) ~ ~ p ( ~ ( x - c ) ( x - c ~ - ~ ) = - A F ~ u(x-c) (x-c)
(v) DFp ( U (x-c) (x-c) -k) = -kFpU (x-c) (x-c)
,x z 01112r3,. . .; k! I
-k-l+ (-1) k & ( k ) (x-c)
k = 0,1,2,3, . . . .
Operations 59
5.4.7.Derivatives of ultradistributions
If f ( z ) is an analytic function and ab is a path from a to b which avoids the singularities of f(z), then according to (5.4.1) we have Y $ E ZZ
which yields the result
(5.4.13)
where the path ab must be on a single sheet of the Riemann surface.
Df ( z ) ab = f (z)+f (a) 6 (2-a) -f (b) 6 (z-b)
If f(z) is a meranarphic function and r is a closed path avoiding poles 5, of f ( z ) , we obtain
Df(z)r = f'(~)~.
(Recall that in this case, f(zIr is equal to a linear combination of
6 ( k ) (2-5,) for the poles 5, which are inside of r . )
If ab or r goes through singular points, finite parts of integrals or pseudo functions also occur. (See Lavoine C5l and C6l.)
Problem 5.4.3. (Taylor's series)
If F E 23 , show that m n
F = 1 2 D"F. a n=O
5.5,Differentiation of Product
The study made in Section 5.3 for the existence of the product
of distributions by a function enables us in this section to show
that the derivative of a product also holds:
According to (5.4.1), we have Y + E $,
60 Chapter 5
which yields the result
(5.5.1) D(aF) = aDF + a'F.
Differentiating again, we obtain
(5.5.2) D (aF) = aD F + 2a'DF + a"F. 2 2
More generally, successive differentiation according to the
Leibnitz rule yields
(5.5.3) hl a(h-j) j D F. h D (aF) = 1 -
J ! ('h-j)! j=O
Examples. 1. According to (5.5.1) , we have V 0 E
= -a (0) 0 I (0) -a' (0) Q (0)
= < a ( 0 ) 6 ' - a ' ( O ) s , ' $ >
which enables us to conclude
(5.5.4) a6' = a(0) 6'-a' (0) 6.
In particular, one finds
L X 6 ' = -6, X 6' = O....
If j is a positive integer, we have according to (5.5.1) for
every Q E 0,
and hence we get
(5.5.5) D (x 6(x)) = 0. k j
Problem 5.5.1
Operations 61
where k is any positive integer.
5.6.Differentiation of Limit and Series
Differentiation of series of generalized functions can be
defined in a similar way to that of ordinary functions but has some
different properties due to the differentiation of the limit of
generalized functions defined in the following manner.
If Fv -+ F in $ I as v -+ v then according to (5.4.1), we have 0'
for every $ E a ,
<D h FV,Q> = <Fv,(-l)h$h> -+ <F,(-l) h h $ > = <D h F,$>
which yields the result
D ~ F ~ -+ D h F . (5.6.1)
Example. Consider
O , X ( O 1 nx, 0 < x 2 - n Un(x)=
1 1, x 2 ii. 1; Also, as n -+ m , Un(x) -+ U(x)
(5.6.1) together by the example 2, of Section 5 . 4 , we obtain
(Heaviside function). According to
DUn(x) -+ DU(x) = 6(x) in ID' ,
We remark here that the derivative of Un(x) does not exist in the
ordinary sense.
Keeping in mind the differentiation of the limit of generalized
functions, we now define the differentiation of series.
A convergent series in 8' can be differentiated term by term an
infinite number of timesI and the series thus obtained is convergent
m h in a ' ; that is, Dh Fn = 1 D Fn, Fn E 8 ' . n=O n= 0
62 Chapter 5
m cos nx Example The series 1 ,7
sense and hence in the distributional n=l n
m m sin nx
n entiated series 1 - and butional sense.
cos nx are convergent in the distri- n= 1 n= 1
is convergent in the ordinary
sense. Therefore, the differ-
More generally, we have the following:
Criterion of convergence for trignometric series
If the numbers an are such that, for [nl sufficiently large m
lanl < Alnl', A and X being fixed, then the series 1 sin
an n=-m
nwx are convergent in the sense of distributions. m cos
inwx = h
inwx is a convergent series in the ordinary where S = 1 w . e
sense.
Indeed, for h being a positive integer 2 A+2, 1 an e D S I m n=-m
-hn-h n=-m
5.7.Derivatives in the Case of Several Variables
The aim of this section is to define the derivatives for
distributions in several real variables with the proceeding notion
of derivatives for distributions in one variable.
Let DiT denote the partial derivative of a distribution
TX E ID' (IR") , V Q E ID (IRn ) , we have from the general rule,
(5.7.1)
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . . <Di TI$> = 'TI j = 1,2,...,n
j where & is the ordinary partial derivative of the function @(XI with respect to xi.
The interest of this case arises when TX is identifiable with a
locally summable function f(X) but is discontinuous on a surface S .
In this case we have
Operations 63
Now, proceeding with similar calculations as made for (5.4.3), we
have m
-b (XI 6 (xldx +s (Xp) 6 (Xp) dxn ,,axl x1
(5.7.3) <Dlf , @> =in-l dx2.. . a where - f(x) is the ordinary partial derivative of the function
f(X) and sx (Xp) is the jump of f(X) when S is crossed to the point
of XDI evaluated at the point of intersection of S with a line
axl
1
c
parallel to the x1 axis with coordinates x2,x3,...,xn passing
through X
at the same point. Hence
the function Q in the product s (Xp)$(X ) is evaluated P; x1 P
As the function s (X ) is null outside of S, the surface integral
is equivalent to x1 p
el being the angle made by the x1 axis with the normal to S and taken in the sense in which the surface is crossed (i.e. increasing
xl). Hence, according to (4.5.2) of Chapter 4 we obtain,
(5.7.5) a D f = - f + s cos 1 axl x1
where one can evidently replace the index 1 by the index i=1,2,3,
..., n. Note that (5.7.5) is a generalization of the formula (5.4.3).
Differentiating again we get
2 a2 D.f = -f + D.(S cos ei6s) + sL1)cos 8.6 1 s
i 1 x. (5.7.6)
where
surface of discontinuities S' (which may inciude S).
ax2i 1 af (x) (X) denotes the jump of the function axi on crossing its
i X
5.7.l.Generalization of 6 (x)
Let S be a regular, and let ? denote a normal to S to the point X. Also, let k(X)be a continuous function on S. Then the distribu-
tion K ( k 6 S ) operates according to the following formula: a
64 Chapter 5
a a (5.7.7) < x ( k s S ) ,$> - / k ( x ) s $(X)ds, if $ E Cmn)
(See Vladimirov [1 I3
This represents a <double layer> a spatial density of charges which
is formed by dipoles directed along the normal
density of moment k(X) .
S
with a surface
5.7.2.The Laplacian
The distributional Laplacian is the operator of the form
" 2 Ad =i&l Di. One can calculate Adf (X) by means of (5.7.6) i but for
any distribution TX we have
(5.7.8) <AdT,$> = <TIP$>.
It is often interesting if we utilize the Green's formula. For
example, if f(X) has a support V(vo1ume) which is limited by the
regular surface S , and if f ( X ) is differentiable on S and also two
times differentiable inside V, then the formula of Green gives
-
a f /(fA@-@Af)dX + I (f# - @ -) avi dS = 0 V S i
+ where vi denotes the inward normal to S . Hence
af -@dS - 1 f%S. IfA@dX = I(Af)$dX + V V s avi s avi
Consequently, by (5.7.8), ( 4 . 5 . 2 ) of Chapter 4, and (5.7.7) we have
a f a (5.7.9) Adf = Af + 6s + -(f6s). a vi
Thus, the distributional Laplacian of the function f(X) is equal to
its ordinary Laplacian augmented of a single layer and of a double
layer on S .
In addition the formula of Green leads to the following results 2 2 by taking r = /(x, + ... + xn):
(5.7 .lo)
The descriptions of these results can be found in Schwartz C11,
Chapter 11.3, and [21. p. 98, and Vladimirov 111, pp. 40-41 .
Operations 65
The formula (5.7.10) gives, after being multiplied by some
factor, the fundamental solution of the equation AdTX = BX i.e.the
distribution E X
5 .8 , Convolution
is such that AdEX = 6(X) (see Section 5.8.7).
In this section we shall be concerned with the convolution of
distributions which depends on the notion of transpose discussed in
the previous sections. To present the results in a neat form we
shall start with a general definition of convolution. The remaining
sections will deal with definitions and properties of the convolution
and will present some examples of its application.
5.8.1,General definition
2 In classical analysis the convolution of two functions f and f
1 is defined by
(5.8 .l) fl*f2 = fl (Y) f 2 (x-y)dy IR
if this integral exists.
In particular, if fl = f, f being a locally summable function,
and f2 = $ E ID we obtain
It is clear that f*$ is a function o f x, we generalize this by
defining the convolution of $ by F E ID'to be the function of x in
the following definition:
F * I$ = <F ,+(X-y)>. Y
(5.8.2)
5 . 8 . 2 . Convolution in ID'
Let k' denote the reflection of the distribution F: that is
L <k',+. = <F,+> = <Ft,+(-t)>, V @ E ID.
Let G E E' (distribution with bounded support). Then k*$, (<GY,$(x+y)>) is a function of x which belongs to ID because it is infinitely differentiable with bounded support. Consequently, the
operation
$ -t 6 * $
66 Chapter 5
is a mapping of ID into itself. We can easily verify that this
mapping is continuous for sequence. Hence, by Section, 5.1, this
operation has a transpose in ID1 which we term as convolution by G.
Explicitly, the convolution of F E ID'by G is an element of ID' , denoted by G * F, and is defined as follows:
(5 .8 -3) <G * F,+> = <F,k * +>
= <Fx,FGy,$(x+y)>]>, V 9 E ID.
In a similar manner, we can show that !$ * + belongs to E . G E 8' we can define
(5.8.3')
Since
<F * GI+> = <G,k * + >
<Gx [ <F y' 9 (X+Y) >I >
= <Gy, C <Fx, 9 (X+y) >I >.
Now, according to the generalization of Fubini's theorem (see
Schwartz C11 Chapter IV.3 and Treves [11 pp. 292, 416(6)), (5.8.3')
takes the form,
( 5 . 8 . 4 ) <Gy I C <Fx, 0 (X+Y 1 > I > = <FX , C <Gy, 0 (x+Y 1 > I >
= <Fx @ Gy, +(x+y)> .
Therefore, G * F = F * G, and the convolution of two distributions is commutative. Also,
(5.8.3")
Certain authors use this relation to define convolution of distribu-
tions straightforwardly without using transposition.
<G * F, + > = <F * GI 9' = <Fx @ Gy, +(x+y)>.
Associativity. The convolution of three or more distributions is
associative when the supports of all these distributions, except for
at most one of them, are bounded.
Continuity.If the family Gvtends towards G ins' as v + vo, then
we have
(5.8.5) G v * F + G * F i n I D ' .
5.8.3. Examples
We give the following examples to illustrate the above results.
Operations 67
1. If F = f(x), f being a locally summable function, and G = h(x),
h being an integrable function with bounded Support, then (5.8.3)
or (5.8.3') yields the result
Hence, we may infer that
h(x) * f(x) = h(y)f(x-y)dy IR
which brings us back to (5.8.1). It follows that the convolution
defined by (5.8.3) is a proper generalization of the convolution of
functions which is defined by (5.8.1).
2. If F E ID' , we have according to (5.8.3)
< 6 ( X ) * F,$> = <F, k(X) * $ > = <Fr$>
which yields the result
(5.8.6) 6(x) * F = F.
This formula and (5.8.5) serve the purpose of regularization of the
preceding Section 4 . 4 of Chapter 4 .
3. If F E ID' , we have according to (5.8.4)
(5.8.7) 6(x-c) * F = T-~F.
which expresses translation.
4 . If F E ID' , we have
<6(h)(x) * F, $ > = <6(h)(y) @ Fx,$(x+y)>
= <Fxr(-l)h$(h) (x)> = <D h F,$>
which yields the result
(5.8.8) h 6(h) (x) * F = D F.
68 Chapter 5
This expression is very useful for the derivative of order h of a
distribution.
5. Derivative of a convolution. If F E ID' and G E E' , then we have h h (5.8.9) D~(G * F) = (D GI * F = G * D F.
Indeed, according to (5.8.8) I
Hence (5.8.9) with the associativity being allowed here.
6. Example of non-associativity. If 1 and xn are distributions
having non bounded supports. Then
tl * 6'(x)] * xn is not associative, since on the one hand [l * &'(X)] * xn = 0 xn = 0,
and on the other hand
which is a divergent integral.
5.8.4. Convolution in ID
Let the strip S be the set of pairs (xIy) such that C( 5 x+y - < fj
as shwon in the Figure 5.8.1. If +(x) E ID and if its support
contains in a certain interval Ca,B 1, then the support of function $(x+y) having two variables is contained in the strip S (see
Figure 5.8.1) . 1 2 Let Fx and Fx E ID, and let there exist a half axis x 2 c which
1 2 is in the quarter of the plane denoted by Q which is the Y
contains the supports of these distributions. Therefore, the support
of Fx @ F
set of pairs (x,y) such that x >c,y'c. (See, Example 2 of the
Section 4.5 of Chapter 4.)
bounded but the Figure 5.8.1 shows that the intersection of S and Q
is bounded. Therefore, we may infer that the intersection of the
supports of F: @ F2 I and $(x+y) is bounded; that assures the
can be performed as given in (5.8.3"). Accordingly, we have
The strip S and the quarter Q are not
existence of <Fx 1 U3 y 2 FYI + (x+y) > and hence the convolution of F1 and F 2 X Y
Operations 69
(5.8.10)
The symbol Q denotes the 2, of the plane limited by dotted lines,
The coordinates of the apex (A) of Q are x = y = c.
FIGURE 5.8.1
1 Hence, we may infer that FX * F: E ID'+.
(5.8.6) we have:
Therefore, by virtue of
ID; is an algebra on the field of complex numbers with the
multiplicative law being the convolution and 6(x) being the unit
element.
This property also holds in the following spaces:
70 Chapter 5
ID'_ (the space of distributions with upper bounded support)
and E ' .
These three algebras are without a divisor of zero. Thus
F E ID:
and 1 => G = 0, or G does not belong to ID:
G * F = O
(see Section 5.8.3,6).
Examples. I f f (x) and f2(x) are summable functions for x > 0, 1 then the distributions fl(x)+ and f2 (x)+ belong to ID; , and
(5.8.11) fl(x)+ * f2(X)+ = f3(X)+
with
Indeed, remembering that f 2 ( x ) + = f2(x) U(x), we obtain
fl (XI + * f2 (XI + = 1 fl (y) f2 (X-Y) U (x-y) dy. m
0
5.8.5. Convolution in $'
The convolution of tempered distributions can be achieved as in
the preceding Section 5.8.1. We describe another particular rule
by means of a new space.
The space C($') of distributions with slow growth
By C($') we mean the vector space of distributions in $' which
K k are of the form 1 continuous functions such that the products (1+x2) n'2f,(x) are
bounded on IR for every n 2 0.
D fk(x), where K is finite and f (x) are k k= 0
If F E $ I and G E C($'), then the convolution G * F exists in $'
and is defined by
(5.8.12) <G * F, Cp> = <F * G,$> = <Fx @ Gy,Cp(x+Y)>, V Cp E $.
Its calculation can be performed as in the formula ( 5 . 8 . 4 ) .
The convolution of several distributions belonging to $' all of
Operations 71
which except at most one belong to C($') is associative and
cummutative (see Schwartz c11 Chapter VI1,S).
5.8.5.1, Convolution in $1
Let $: be the space of tempered distributions with lower bounded
support.
also belong to $:.
If the distributions belong to SJ., then their convolution
$; is an algebra on the field of complex numbers with the
multiplicative law being convolution and 6(x) being the unit element.
In this algebra there is no divisor of zero.
5.8.6.Convolution equations
An equation of the form
(5.8.13) A * X = B
is known as a convolution equation. In this equation A and B are
given distributions and X is an unknown distribution for which the
equation is to be solved. (See also Section 9.1 of Chapter 9.)
Example. The difference equation
can be written as
c I aj 6(x+cj) I * x = B
(see Section 7.13 of Chapter 7).
The integral equation m
f(x) + IK(x-t)f(t)dt = g(x), -zr,
where K ( x ) and g(x) are given functions, can be written as
(see also Section 6.2.1 of Chapter 6).
5.8.7. Fundamental solution
We call a solution E of the equation (5.8.13)a fundamental
72 Chapter 5
solution if E is a distribution which is a solution of the equation
(5.8.14) A * E = 6(x).
This solution does not always exist. If there exists one then
there exist infinitely many solutions. The difference between any
two of these solutions is a solution of the equation A*X = 0.
If the convolution A * E * B is associative, then X = E * B is a solution of the equation (5.8.13), because
The fundamental solution is concerned with the study of Green's
functions (see Section 9.6 of Chapter 9).
For a detailed study of the fundamental solution, (see Schwartz
111 and Garnir 111 1 .
5.9.Transformation of the Variable
In classical analysis one often replaces the variable x by a
function u(x) in an ordinary function f(x) and obtain a new function
f(u(x)). There may exist an analogous operation for generalized
functions. For example, let u(x) be a monotonic function on the
whole of IR and let v(x) denote its inverse such that v(x) = 0 for x
outside of u(lR). For any $ belonging to a convenient base space 0,
this leads to
or
Generalizing this notion, we can associate a generalized function
T t to the generalized function Tu(x) E 0' defined by X
In the following sections, we extend this definition to the case
when u(x) is not monotonic on IR.
u (XI 5.9.1. Definition of T
Let T be a generalized function belonging to a space 0' and let X
Operations 73
S be the bounded or unbounded support of Tx.
single-valued real function defined on IR or a part of IR where it
is continuously differentiable a sufficient number of times and
possesses the following properties:
Also, let u(x) be a
(1) There exists N(NF1) closed intervals Xn which further
satisfy the two conditions:
C1. u(Xn) contains S
c 2 . u'(x) does not vanish on Xn (therefore u(x) is strictly
monotonic on Xn).
( 2 ) There is also a one to one correspondence between Xn and
u(xn). Hence to each x E S there corresponds only one X'E X which
is denoted by v (x ) such that u(x') = u(vn(x)) = x when vn(x)=un (x),
u-l(x) being the inverse of u for the monotonic branch described by
u(x) when x belongs to Xn.
function belonging to 0.
-1 n
n Outside of u(Xn), vn(x) is extended by a
Let a S ( x ) E 0 be equal to 1 on S and zero outside of an interval
containing S. We put
(5.9.1) N
From these preliminary conditions imposed on u(x), we may infer that
the operation
is a sequentially continuous mapping of 0 into itself.
By transposition we then obtain the generalized function Q'Tx~Q'
and defined by u(x) which is denoted by T
( 5.9.. 2 1
The support of Tu(x) is
(5 .9.3)
Note that ul vn
countable monotonic branch, as for example u(x) = sin x ) , then
(5 .9 .2 ) is still valid if the convergence of the series is assured
su = u vn(s) = uix E XnlU(X) E SI.
may be substituted for v;(x) in (5.9.2). 1
If there is a countably infinite number of Xn (if u(x) has a
74 Chapter 5
for each $ B Q.
If we require (5.9.2) to be valid for all distributions having
arbitrary support then u(x) must be continuously infinitely differe-
ntiable on IR and its derivative u' (x) should not vanish on IR . u (XI Let Dx(TU(x)) denote the derivative of the distribution T
and let (DT)u(x) denote the distribution by changing of x to u(x) in
the derivative of DTx. Then, we have according to (5.4.1) V $ E 8 ,
A l s o ,
(5.9.4')
We explain the definition (5.9.2) with the aid of some examples.
1. If TX = f (x) , locally summable function and if u(x) has a nonzero derivative on IR , then (5.9.2) together with v(x) = u-l(x)
yields ff + E ID
<T ,$(XI > = 1 f (x) Iv' (XI 16 (v(x) )ax JR
IR
u (XI
= 1 f(u(x))$(x)dx = <f(u(x)),$(x)>.
Here we obtain Tu(x) = f(u(x)) as expected.
2. Let Tx = f(x), a summable function with support S = [c,c'], 2 and let u(x) = x +b where b < c. Hence the conditions C1 and C2 are
satisfied by two Xn:X1 = C-H,-n] and X
such that 0 < n < y = Jc-b, Hz y ' = &;. Then we have v1(x)=-6,
v,(x) = F b , and (5.9.2) yields
- [ n,H] where n and H are
Operations 75
n
= < f l (x'+b) , $ (x) >
where
f (x2+b) i f G b 5 1x1 5 &b
otherwise.
2 f l ( x +b)=
Hence w e conclude
which can be obtained immediately. On t h e other-hand, i f b > c
then Tu(x) n s a t i s f y i n g t h e condi t ions C 1 and C2.
does n o t e x i s t because there does n o t e x i s t any X
It i s t o be remarked h e r e t h a t t h e examples 1 and 2 demonstrate t h a t t h e formula (5.9.2) gene ra l i zes t h e change of v a r i a b l e i n t h e theory of func t ions .
5.9.3. Bibliography
A more comprehensive d i scuss ion on change of v a r i a b l e f o r d i s t r i b u t i o n s can be found i n Albertoni and Cugiani Ell, F i she r c11, Gfitt inger C11, Jones Ell, Schwartz Ell, Chapter I X .
Footnotes
(1) d/d = dldx o r d/dz, according t o t h e v a r i a b l e .
1 , x > c ( 3 ) U(x-C) = 'c o , x < c
(4) see Sec t ion 0 . 4 . 2 of Chapter 0. 1 (5) more gene ra l ly , 6(x-c) = D [(x-c)U(x-c) + ax+b1.
This Page Intentionally Left Blank
CHAPTER 6
OTHER OPERATIONS ON DISTRIBUTIONS
Summary
The motivation of this chapter is to define the division and
antidifferentiation of generalized functions.
method of transposition as discussed in the preceding Chapter 5 but
as inverses of multiplication and differentiation. Further, we
shall discuss the limit and value at a point of a distribution as
well as the notion of equivalence. The results presented herein will
suffice for many applications, but those readers who are interested
in a more complete treatment are referred to sources on the topic.
We do this not by the
6.1. Division
If S is a given distribution, there evidently exists in every
open set where H E E does not vanish, a distribution T which satis- fies HT = S. This is because - is infinitely differentiable, and we obtain T = S by multiplying on both sides by B. longer the case when H has zeros. From now on we shall be concerned
with the case when H has zeros. The problem of division has wide
importance in the theory of integral equations and the theory of
partial differential equations. Also, the division is often used in
quantum mechanics.
6.1.1,Division by xn (n>O, an integer)
1 H 1 1 Such is no
For a given generalized function G E Q', let us first consider
the problem of division by xn.
such that G =F/Xn. In general F can not be considered as being
multiplied by l/xn in the sense of Section 5.3 of Chapter 5, because
To do so, we seek to determine G E 0 '
l/xn may not be a
as the inverse of
(6.1.1) G =
multiplier for Q. Therefore, we define division
multiplication by
1 n - F , i f x G = F X n
77
Chapter 6
that is, if
(6.1.2) <G,xn$> = <F,$>.
In addition to this there can exist a divisor of zero in 0'. For
instance in ID' (see (ii) of Problem 5.5.1 of Chapter 5),
(6.1.3) (x) = 0, k = 0,1,2,.. ., (n-1). xn6 (k)
Consequently, there is a certain arbitrariness in the determination
of G . If we take G in the space ID' of distributions, we have the
following theorem.
Theorem 6.1.1. Let F E ID' . Then the equation xnG = F has
infinitely many solutions G belonging to ID' ; the difference between
any two of them is of the form
constants.
n-1
k= 0 1 a b(k) (x) where ak are arbitrary k
- Proof. The proof is given in Schwartz [11 , Chapter V,4, Giitt inger [l] and ChoquetBruhat c11 , pp. 124-129.
Examples. If x3 E ID' , we have 1 (1) 7 x3 = x + ao6(x) + a161 (x) .
Here xn = x2, F = x , and hence according to Theorem 6.1.1 we have
x G = x (x+ao6(x)+a16'(x)) = x3 = F which shows the existence of (1)
X 3
2 2
by making use of (6.1.3).
(2) If F = f (x) is locally summable on lR , then -F = Fp- + a 6 (x) . (Here the Fp is not needed if fo is integrable in the neighbourhood of the origin.)
1 f (x) X X 0
X
The verification of this example can be made easily by taking a
similar existence technique of the preceding example.
6.1.'2. Division by a function
Let F E 8' and a(x)E M(8) (see Section 5.3.1 of Chapter 51, then
I Q = 7 F, if a(x)Q = F. x)
Also, there is a certain arbitrariness in this situation. If we take
Q in the space ID' of distributions, we mention the following theorem.
Other Operations 79
Theorem 6.1.2. If F E ID' , and if a(x) E E has roots x of P
order n on IR then the equation a(x)Q = F has infinitely many
solutions Q belonging to ID' ; the difference between any two of them
is of the form G c a 6(k) (x-xp) , where a are arbitrary
constants.
P
np- 1
p k-0 PIk Plk
It is obvious that Q is unique if a(x) does not vanish on IR.
The following examples will illustrate this theorem.
Examples.
associated) to
have in ID' .
where a and a'
1 - 2 * x2,c2 lx -
Let lX be the distribution which is identical (or
the function equal to 1, and let c E IR. Then, we
~p + + a 6 (x-c) +a1 6 (x+c)
are arbitrary constants; and
1 -1'
-C
x +c
But in the space ZZ' of ultradistributions (see Section 3.3.6 of
Chapter 3) , we have 1 2 + a G(z-ic) + a'd(z+ic) 3. -1 = 1
+c l r = 2 + c
where the path II does not pass through the points ic.
Proofs. Here a(x) = x2-c2 has two simple roots c and -c.
-1
Let
Then according to Theorem 6.1.2, Q== x'
Q = Q + a6(x-c) + a'd(x+c) 1 2 2
where Q1 is a distribution such that (x -c )Q, = lx; hence we can
take Q1 = Fp r2. 1 Consequently, l.is established. x -c
2 . Here a(x) = x2+c2 which does not vanish on the real axis. 1 1 Let Q = --2.1x; hence Q is unique and is equal to -2 since
x +c x +c 2 2 1 (x +c ) 'n = lx'
x +c
3. This is left as an exercise for the reader.
1 6.1.3. multiplier €or 0
In this case Q(x)/a(x) E Q , V + E Q. Then for F in Q',
Q=A F is unique and defined by, a (XI
80 Chapter 6
By IDo we mean the space of continuous functions $(XI having
bounded support; while IDo(b) denotes the space of continuous functions $(x) whose supports are bounded below by b 0. Consequen-
tly, JDA , IDA (b) , and IDA (0) are duals of the spaces Do , mD0 (b) and Do (0) , respectively.
Example. For v > 1, we have V Q, E Do (b) , 1 (6.1.4) <';;[lX+6 (x-2b)l , Q, (x) > = <x-v+~-v6 (x-2b) , + (x) >
-V X = <x Q, (X) > + <X-"6 (x-2b) f $ (x) >.
The second term on the right hand side of (6.1.4) can be written as,
Consequently, (6.1.4) takes the form,
<-[Cx+6(x-2b)] 1 ,$(X)> = <x-vf$(x)> + (2b)-' x X V
< 6 (x-2b) , Q, (x) which yields the result in lDb(b)
1 - [ lX+6 (~-2b)I = x-' + (2b)-' 6 (~-2b) . (6.1.5) XV
However, this division is impossible in JD' 0
and D L ( 0 ) .
6.2,Antidifferentiation (2)
Let JI be a base space containing derivatives of base functions
An antideri~ative'~) of F E B ' is a generalized function of 0.
P E 6' such that its derivative is F; that is
In general, there is a certain arbitrariness in the deteminatior
of P. In this situation, we have the following theorem.
Theorem 6.2.1. If F E ID' (Or I D n k , k 2 01, then the equation
Other Operations 81
DP = F has infinitely many solutions P belonging to ID' (or ID 1k-1)(4).
the difference between any two of them is of the form a1 a being
an arbitrary constant.
I
X I
- Proof. The proof is available in Schwartz [11 I Chapters I 1 , I V
and V where the several variables case has been studied.
Besides this proof, we can obtain a very simple proof of this
theorem by employing the Fourier transformation of Chapter 7. For
this purpose, let $ and P denote the Fourier transformation of
P and F respectively. By employing the Fourier transformation
on (6.2.1), we have
5 5
5 5'
2nis 6 = i 5 5'
(6.2.3)
According to Theorem 6.1.1 (5 playing the role of x), the equation (6.2.3) has infinitely many solutions P and the difference between
any two of them is equal to (a 6(x)), a being an arbitrary constant.
By taking inverse Fourier transformation of (6.2.3), we conclude
that the equation DP = F has infinitely many solutions P, and the
difference between any two of them is equal to a (or also denoted
by alx).
5
This completes the proof of theorem.
Examples.1. The antiderivatives of F = f(x), a locally summable
function on IR, are the distributions identical (or associated) to
primitives of the function f(x) defined by
X C I f(x)dx + a, - I f(x)dx + a, C X
where c is real and a is complex. However both a and c are arbitrary.
2.
6.2.1 that the primitive of 6(k) (x) is 6(k-1) (x) + alx or 6 (k-l) (x) +a. Since D 6 (k-l) (x) = 6(k) (x) (kl) I we deduce by applying Theorem
Problem 6.2.1
Prove that the antiderivatives of
1 (i) FP
(ii) 6(x) is U(x) + alx.
6.2.1. Antiderivative in ID'+
is log (XI + alx;
Let F be a distribution having the support bounded below by c.
8 2 Chapter 6
Hence it has a unique antiderivative in ID: dudh that P = U(x) * F with the support of P also being bounded below by c.
Indeed, according to the formula (5.8.8) of Chapter 5, (6.2.1)
can be written as
(6.2.4) 6'(~) * P = F.
As these distributions belong to D:, we can convolute both,members
of (6.2.4) by U(x) , and because of
(6.2.4) takes the final form
P = U(x) * F.
(6.2.4) has no other solution in ID' i since 23' is an algebra of
convolution without divisors of zero.
In Chapter 9 (see Section 9.10) we shall present a more detailed
discussion of derivatives and antiderivatives of any order.
6.3,Value and Limit at a Point of a Distribution
According to Section 4.1 of Chapter 4 a distribution is an
operator which acts globally and not point by point on each function
belonging to ID. Therefore, it would be improper to talk about the
value at a point or limit at a point of a distribution. These
abuses in languages are however justified by a similarity with
functions.
6.3.1. Value at a point
If h(x) is a function continuous in a neighbourhood of a point c, the limit of h(c+Ax) as X + 0 is h(c), the value of the function h
at'the point c. Similarly, we have the following definition accord-
ing to p1o j asiewicz C 11.
Let F be a distribution belong to ID' . If the limit lim Fc+Xx a-+o
exists in 23' and if this limit is equal to the distribution (5) alx
where a is a real or complex number, we say that a is the value at c
of F and we write F = a. The justification of this definition is
given below. (C)
Other Operations 83
According to this definition, if a = F then F(c)lx= limFc+xx. (C)
Hence, according to the rule of translation (see Section A-co
5.2 of Chapter 5) we have V 4 E ID,
ConseWentlY, taking for 4 a $ such that 1 $(x)dx = 1, (6.3.1) takes
the form IR
1 x-c (6.3.2) F = lim <Fx, $ ( + ' I Tf JI E ID.
(c) A + 0
Theorem 6.3.1!6) A necessary and sufficient condition for a to
be the value of a distribution F at c is that there exists a non- negative integer k and a function h(x) continuous in a neighbourhood of c such that F = D k h(x) in this neighbourhood and that 7 k!h(x) + a
(in the ordinary sense) as x -f c. (x-c)
Corollary. If F = G in a neighbourhood of c, then these two
distributions have the same value at c.
Examples. Let F = h(x) in a neighbourhood of c and let h(x) be continuous at the point c. Then we have F = h(c). For instance
in the neighbourhood of x = c # 0, Sfx) is equal to zero; hence by the Theorem 6.3.1, the value of 6(x) at c is zero. On the other
hand, if 6(x) = D x+, x+ is continuous and x+/x2 has no limit as x + 0. Consequently, according to Theorem 6.3.1, the value of 6(x)
at the origin does not exist.
(C)
2
A function f ( x ) can have a value at c in the sense of distribu- For instance,by tions without f(c)xexisting in the ordinary sense.
putting h(x) = 2 / t sin sin -, we have cos ;; = D h(x). As 0
x + o , - h(x) + 0; then according to Theorem 6.3.1, the value of the 1 distribution cos - is zero at x = 0.
6.3.2.Right and left hand limits at a point
1 2 1 1 dt-x X
X
X
Right hand limit
Let h(x) be a continuous function for x > c; its limit from the
right at c is h(c+) = l i m h(y) if this limit exists. Similarly,
we have according to Eojasiewicz. y + c+
84 Chapter 6
If a distribution F has the value F = a(y) for y i c and if
a(y) has the limit a+ as y 3 c+, then we say that the right hand
limit at c of the distribution F is a+.
This can be justified in the following manner:
Y
We denote a+ by lirn F. c+
According to this definition, if lim Fy = a+, then lim F = a+. C+ y + c + Y
Hence, by Section 5 .2 of Chapter 5, we have Tf 9 E ID,
(6.3.3) lim F i +(x)dx = lim lim <Fy+Ax , + (XI > 6 + c + y I R y + c + x + o +
= lim lim <FX, 1 +(,)>. x-y
y - + c + x + o +
Left hand limit
Eefinition for the left hand limit lim F will be analogous to C- that of the right hand limit.
Examples. We state below the following examples of limits:
l i m eyx = 1, lim eyX = 0. But Fp x-l has no right or left hand limit
at the origin. O+ 0-
Theorem 6.3.2. A necessary and sufficient condition for a+ to be
the right hand limit of a distribution F at c is that there exist a
number 5 > 0, an integer k F 0, and a continuous function h(x) on
(in the ordinary sense)as x + c+.
k ]c,r] such that F = D h(x) on ]c,r[ and that k : h(x)/(x-c)k + a +
Proof. The proof can be carried out in a similar manner as in the - proofs of Lojasiewicz C11 and Silva [4].
Note: The theorem for the left hand limit will be analogous to - the above theorem.
6.3.3.Limit at infinity
eojasiewicz does not define the limit at infinity of a distribu-
tion. We shall follow the definition of Silva [4] . (See also
Lavoine and Misra Cl] for this limit.)
A distribution F has the number a for the limit at infinity
(i.e. we write limF = a) if there exist a number 5 >O, a non-negative
integer k, and a continuous function h(x), such that F=D h(x) on
[ r , - [ and that k: h(x) /xk + a (in the ordinary sense) . k
m
Other Operations 85
Note: The definition will be analogous for limit at -- i.e. lim F. -a
Examples. We list below some examples for limit at a.
If F = f(x) is a function such that (in the ordinary sense)
lim f(x) = a, we have lim F = a. m m
cos x X
lim sin x = 0, because sin x = -D cos x and -+ 0 as x + a. m
Thus lim eiwx = 0. If F has a bounded support, then lim F=limF = 0. m m -a
The notions of value at a point, limit at a point, and limit at
infinity of a distribution are discussed in the integral calculus
of distributions (see Antosik, Mikusinski and Sikorski C11, Campos
Ferreira J. [1J and C2l , Silva [ 4 1 , Mikusinski and Sikorski [l],
Zemanian c11 , p. 71,Lavoine and Misra ill , Misra [6])and in
Abelian theorems (see Section 8.11 of Chapter 8 ) .
6.4. Equivalence
This section provides a brief account of the equivalence of
distributions which will play a central role in establishing the
behaviour of Laplace and Stieltjes transformations in Chapters 8
and 10 respectively.
6.4.1, Equivalence at the origin ( 7 )
Let ID;o be the space of distributions in D' having support in
CO,-C. For each F E ID' , we have the decomposition of F such that L
F = G+G where G E Dloand k1 is the reflection of a G1 also belonging to
differentiable functions which are equal to functions of on [O,-[.
Further, we put
1 By E-U $+ we mean the space of infinitely
xvlogJx, for x > o
0 , for x < 0
x-n-llogjx, for x > o
0 , for x < 0
I i
(xVlogJx)+ =
and
(x -n-1 log'x)+= '
where j,n E IN. When v < -1 and j,n E IN , the distributions
86 Chapter 6
-n-1 j generated by (xvlogjx)+ and (x
the sense of finite parts of Hadamard (see Chapter l), and we denote
these distributions by Fp(xvlogJx)
ively. Let us now study the behaviour at the origin for distribut-
ions belonging to D:o.
log x)+ are understood to be in
and Fp(x-"-'logjx) + respect-
In the style of tojasiewic~(~) we define the following.
1. G E Diois equivalent at the origin to aFp(x'logjx)+(and we
denote G .. aFp(x'log'x)+ as x + O + ) , v # -1,-2,..., j = O I l 1 2 , . . . ,
if and only if there exist a number 5 > 8 and a distribution R E 6' with support in [0,53, such that
G = aFp(xvlogjx)+ + R on[ 0,c J
and that
(6.4.1)
as h + O+.
2.
write G .. aFp(x-"-'logJx)+ as x + 0+) for a set of non negative
integers jrn, if and only if there exist a number 5 > 0 and a distribution Q E E' with support in c0 ,51 , such that
G E m i o is equivalent at the origin to aFp(x-"-'logjx)+ (and we
-n-1 G = aFp(x logjx)+ + G on C 0 , ~ l and that
.n
as X -+ O+.
We remark that if v = j = 0, (6.4.1) takes the form
1 7 <Rxr@(x/A)> + 0 as A + O + ;
consequently , according to (6.3.2) , we have
These results enable us to state the following theorems.
Theorem 6.4.1. A necessary and sufficient condition for G E ID;,
to be'equivalent at the origin' to aFp(xVlog3x)+, where v is a non negative integer and j is a positive integer, is that there exist a
Other Operations 87
number 5 > 0, a non negative integer k for which Re(v+k)>O, and a function h(x) which is continuous in a neighbourhood of [0,73 such
that
k G = D h(x) on C0,sl
and
(V+lIk. v+ h'x) + a (in the ordinary sense) as x + O+, x klogjx
( ~ + l ) ~ = ( v + l ) ( v + 2 ) ...( v+k) I k 2 1, fv+l)o = 1.
This theorem would serve as a basis for the equivalence definition
in the style of Silva C41 . Theorem 6.4.2. A necessary and sufficient condition for G E D;
be equivalent at the origin to (aFp(x-n-llogjx)+) (for a set of non
negative integers j,n) is that there exist a number 5 7 0, a non
negative integer k, and a function h(x) which is continuous in a
neighbourhood of [O,S] such that
G = D h(x) on C0,sl n+ k+ 2
and
+ a (-l)"n! ( k + l ) ! (j+l) k+ h (x)
x llogJ+lx
(in the ordinary sense) as x -+ O+.
The following theorem is a consequence of l., defined above.
Theorem 6.4.3. If f(x) is a locally summable function and v is
such that Re v > -1 and if we have f(x) I a xvlog7x as x -+ O+ in the
ordinary sense, then we have f (x)+ sense of distributions.
a(xvloglx)+ as x + O+ in the
The following result is less apparent.
Theorem 6.4.4. Let a be such that Re a L 1. Suppose that for
x c C O , 5 1 we have
with the conditions that certain constants a must be zero, Pq
0 2 Re 6 < Re a-1, and P -
88 Chapter 6
h (XI where - is bounded on [O ,g ] for a certain w > -1. Then we have
j Fp g(x)+ .. a Fp(x-'log XI+, as x -+ O+, in the sense of I, or 2 ,
defined above.
Examples. We give the following examples.
1. sense) where a >-1, then we have G
in the ordinary sense, f(x) is equivalent to (ax') as x -f O+ if
If G = f (x)+ is equivalent to the function (ax:) (in the ordinary
ax:, as x + O+. We recall that,
f (x) = xv(a+r(x)) , r (x) tending to zero as x + O+.
2 .
to (ax-Blogjx) for B 2 1 as x -f O+, then we have
G ~ a Fp(x-BlogJx)+, as x -t o+.
3 .
have according to (6.4.1)
If G = Fp g(x)+, and if g(x) is equivalent in the
In 6(x) + Fpx x:, S ( x ) plays the role of R (see.1
ordinary sense
) . Then we
-+ <~(x),$(x/~)> = y+ Oash -f o if v < -1.
V Hence, S(x)+Fpx+ ~ Fpx: as x + 0 + , if v c -1.
6(x)+xi is not equivalent to x: as x -+ O + ; because then I p ( O ) / X does not tend to 0 as X + 0.
But if v > -1, v+l
6.4.2.Equivalence at infinity
1 Let m(x) be a function such that is continuous on ]c.,- [ for 5 > 0. Then we say that a distribution F is o(n(x)) at infinity
if there exist a non negative integer k and a continuous function
h(x) such that
k F = D h(x) on I<,-[
and that h(x) xkmo+ O
in the ordinary sense x + a.
Note. This is like the criterion for F = o(xa) , a > -1 given in - Silva C41 .
Other Operations 89
We say that a distribution F is equivalent to the function m(x)
at infinity (i.e. we write F m(x) as x +- -) if there exist a
number 5 > 0 and a distribution Q such that F = m(x)+Q on l r r - C and such that Q is o(m(x)) at infinity.
Example. If F = f(x) on Is , - [ and if the function f(x) is
equivalent to xvlogjx at infinity in the ordinary sense, then
F ,. xvlogjx, as x +- m.
Theorem 6.4.5. A necessary and sufficient condition for F,axVas
x -+ -, where v is not a negative integer, is that there exist a non
negative integer k, a number 5 > O r and a continuous function h(x)
such that F = D h(x) on 1 5 , - C and such that ( v + l ) kh(x)/xv+k -+ a (in
the ordinary sense) as x + m+.
k
Note. The definition and results for -- will be analogous to that of +-.
Footnotes
see, Schwartz c 11 , Chapter, V.4. primitivation ( 3 ) or primitive.
by ID'lwe mean the space of integrable functions Cp which have
bounded support. A sequence in ID converges to zero if
$n the duals (ID') -'c we do not say <<equal to the distributional constants>> just to
avoid confusion between this distribution and the number a.
see Lojasiewicz c'll , p.7 and Constantinesco C 11 , pp. 7-12. one can reduce to this case by means of translation.
Fp is not needed here if Re v > -1.
we complete here the definition proposed in Lavoine and Misra
-1
+- 0 almost everywhere. Since ID C ID C ID'l , we have for ( I D ' I k C ID'.
c11.
This Page Intentionally Left Blank
CHAPTER 7
THE FOURIER TRANSFORMATION
Summary
In Chapter 5 we have seen the importance of the notion of
transposition which extends some basic operations of analysis from
functions to distributions. A similar procedure is followed in this
chapter in defining the Fourier transformation.
For convenience, we divide this chapter into two parts. The
first part, which consists of Sections7.1 to 7.7, presents the
Fourier transformation as an isomorphism from the space ID (or Z)
onto Z (or ID) ; and consequently, its transpose is then taken as an
isomorphism from the topological space ID' (or Z') onto Z' (or ID')
as is done in the theory of Gelfand and Shilov 111, Vol.1.
A similar technique to that of the first part is followed in the
second part, which extends from Sections 7.8 to 7.13 and deals with
the Fourier transformation as an automorphism on $I . The second part
is consistent with the theory of L.Schwartz L11 . Notations. We will use the following notation and terminology.
We write
(7.0 .l)
where IF denotes the Fourier transformation.
Fourier transform of f. We also define
(7.0.2)
where the notation Pi1 denotes the inverse Fourier transformation.
F;' (g(x)) is called the inverse Fourier transform of g(x) .
co
g(x) = IFx f (t) = ff,(t)e-2nixtdt -m
IFx (f(t)) is called the
m
P;l(g(x)) = I g(x) e 2nixtdt -m
7.1.Fourier Transformation on Z
This Section provides the structure of Fourier transformation on
91
9% Chapter 7
8 in the following manner.
Definition 7.1.1. Let J l ( z ) E Z. Then we define the Fourier
transformation by the relation
(7.1.11
Here the integral is taken along a path of the complex plane going
from -m to +-, particularly along the real axis.
Theorem 7.1.1. If $(z) E 8. Then the Fourier transform of a
function of complex z is the function of real x belonging to IDwhich we denote by l F x J l ( z ) = $ ( x ) , $ ( x ) E ID.
Proof. To show $(x) belongs to ID we recall that for every z -
If yJR) denotes the semi-circle I z 1 = R, n 2 0 then we have making use of (7.1.2) for x a
03
I $ ( X ) 1 = ’ I 1 $ ( Z ) e-2nixz dz I -m m
-2rixzdzI 5 lim J OR -2 5 J I$(x) I le -03 ~ + m YAW
Rd 0
R-l l’e-(X-a)R sin BdO =
ea I n I .-xRsin e
< lim - It).- 0
We can also obtain the same result for x 5 -a by the semi-circle 1.1 = R, n 0 . From these results we may infer that $(x) has its
support in [-a,al; therefore, it has bounded support. Further,
according to (7.1.1) and the properties of + ( z ) , it is evident that $(XI is infinitely differentiable. Thus, we conclude that $(XI &longs to ID. This proves the theorem.
Now we quote connections which will be useful for our subsequent discussiont
(7.1.3) dk k I F . 2 $ ( z ) = (2nix) I F ~ $ ( Z ) ,
(7.1.4)
Also, $(z) is said to be the anti-transform (or inverse transform
Fourier Transform 93
of $(x) = IFx $(z) , and it is denoted by $(z) = I F - l $ ( x ) .
(7.0.2) we may infer that
(7.1.5)
7.2,Fourier Transformation on ID
From 2
m
Ez -1 $(XI = J $ W e 2nizxdx. -m
A similar technique to that of Section 7.1 is followed in this
section to define the Fourier transformation on ID.
Theorem 7.2.1. Let O(x) E ID. Then the Fourier transformation
of a function of real x is the function of complex z belonging to
ZZ defined by the relation
(7.2.1) $ ( z ) = IFz$(x) = J $(XI e dx . m -2nizx
--m
- Proof. Since +(x)EID, it has bounded support. Let the support
of $(x) be equal to [-b,b] . Then we have according to (7.2.1)
-2nizx b
-b $ ( z ) = I $(XI e ax.
It is evident that +(z) is infinitely differentiable and therefore
analytic. A l s o ,
b (7.2.2) (2ri)JzJ +(z) = I +(J)(x) e-’*iZX ax.
-b
Further, by making use of (7.2.2), we have for n = Imz
b b
-b -b IzJ$(z) ]<(2s)-j J
If we put a = 2nb and
(x) (e2‘nxdx <(2n)-Je2nb1n1/l$(J) (x) (dx.
b = (2n)-j I I$(’) (x) Idx c
-b j
then
IZJ$(Z) 1 < cj e w
Consequently, +(z )be longs to 2 2 . A l s o , we may infer that the SF
maps ID to !iZ and is a continuous operation for sequences. The
formulae for IFz are analogous to those of (7.1.3,4,5), and we
remark here that the interested readers can obtain these results
easily.
2
Conclusion. The spaces of functions ID and 22 can be mapped into
94 Chapter 7
one another by the Fourier transformation.
Remark. By comparing (7.1.5) and (7.2.1) , we obtain
(7.2.3) IF;l $(XI = X z @ ( X ) = lFz +(-x) ,
Consequently, we may infer from the above results that the Fourier
transformation is an isomorphism from 22 (or ID ) onto ID (or Z) . 7.3.Fourier Transformation on ID' and 22'
Throughout this section the results of preceding sections enable
us to take the Fourier transformation as a transpose on ID' and 22' . Theorem 7.3.1. Let Tx E ID'.
distribution Tx is an ultradistribution in El.
IFzTx and define it by the relation
Then the Fourier transform of a
We denote it by
(7.3.1) <IFZTxj $ ( Z ) > = <Tx,IFxJI(Z)> I Y + E 22
Reciprocally, we have
Theorem 7.3.2. Let UZ€ 22'. Then the Fourier transform of is I F x U z E ID' , and we define it by the relation uZ E Z?,I
(7.3.2) <lFXUZ,$(X)> = < U Z , I F Z $ ( X ) > , v 0 E ID.
From these results, we may conclude that the spaces of generali-
zed functions ID1 and 22' can be transformed from one into the other
by the Fourier transformation.
7.4.Inversion and Convergence
In this section we describe inversion and convergence of the
Fourier transformation on ID' and . 7.4.1. Inversion of Fourier transformation on lD' and Z'
Theorem 7.4.1. Let Uz E zt' . Then it has an anti-transform L
IFi1UZ on ID' which is equal to ( I F x U z ) . - Proof.
e ( z ) = IFz $(XI. IF,x$(z) = (IFx$) and IF-z$(x) = (IFz$) .
By virtue of (7.2.3), I F z ~ ( - x ) = lFz1$qx) = $(z), where Also, note that according to relations (7.2.3) ,
L L
Fourier Transform 95
Moreover, according to Theorem 7.3.2, EXUz belongsto ID' , hence its reflection (IFxUzk also belongs to ID' . By making use of the above relations, we have for every J, E ZZ
which yields the relation
L IFz (IFx UZ) = uz.
This relation can be written further as,
IFz (IFXUZk = ";I UZ'
Since IFi1PZ gives identity, we finally obtain
-1 (IFXUZ3y = IFx UZ
Theorem 7.4.2. Let Tx E D1. Then it has an anti-transform L FilTx on Z' which is equal to (IFz Tx) .
Proof. The proof is very similar to that of the proof of - Theorem 7.4.1.
7.4.2. Convergence
1. It can be easily seen that
u, = 0 e==> u = 0 x z
2. By virtue of Theorems7.3.1 and 7.4.1, if T is an infinite xtv
family in ID' depending on a parameter v then
TxIv -c Tx in D' IFZTxIu + IFZTx in Z1 ] ee==> [ 0'
as v + v a s v + v 0
We have a similar result for U -+ Uz in Z' by virtue of ZI V
Theorem 7.4.2.
96 Chapter 7
From these results we may infer that the Fourier transformation
is an isomorphism between the topological spaces ID' and Z' . 7.5.Rules
We state below a few rules of calculus for the Fourier
transformation which will. be utilized later in our study.
1. According to formula (7.3.1) we have for T, E 3D' ,k=0,1,2,...,
Consequently, we obtain the result
2. By virtue of the formula (7.3.1) and making use of (7.1.31, we have for T, E ID'
< I F z X k Txr$(Z)> = <X k TxrlFx$(Z)> = <Tx,X k D?,$(Z)>
-k = (2ni) <Tx,Ex $(k) (z) >
= (2ni)-k<~Z Tx (z) >
= (2ri) -k (-I)~<D~ IF^%,+(^)>
which yields the result
(7.5.2)
There may also exist analogous formulae €or IFx operating in ZZ' and we remark here that the interested readers can find out these formulae
easily.
7.6. Fourier Transformation on E'
k i k k lFz X Tx = -7- D IF T z X. (Zr)
The results of Section 7.3 enable us in this section to formulate the distributional setting of Fourier transforms on E'.
If Tx = f ( x ) is a summable function, we obtain by applying Fubinis theorem to (7.3.1) that
(7.6.1)
Fourier Transform 97
This result, as well as results of Section 7.5, show that we have a proper generalization of the ordinary transformation.
If T is a distribution with bounded support, then we.have X
according to (7.1.1) and (7.3.1) together with Fubini's theorem
(7.6.2) <Tx @ $ ( z ) , e -2nizx> = -2nizx,, <Tx < $ (z 1 , e
= <TX,IFX$JZ)> = <IFZTx,J,(Z)>.
Also,
-2nizx> -2nizx (7.6.3) <Tx @ J,(z),e < C<Tx,e >I , J , ( z ) > . By comparing (7.6.2) and (7.6.31, we obtain
(7.6.4) e- 2 n izx > IFz Tx = <TX,
In this case, an important generalization of the Paley-Wiener
theorem (see Paley and Wiener Cll, pp. 12-13) holds:
I I
I
I z x Tx E 6' I I IF T is an entire mytic fun^
I ' I=ZTxI I Z I
I 1 r l = I r n z .
I tion such that there exists a with support contained I o==> in the interval [-c,c] 1
I I
nonnegative integer m so that -m e-2ncl~l
I I I is bounded as (zI + m, where I
I I
The proof of this result is available in Treves C13, Chapter 29, Theorem 29.2, and another related statement is given in Schwartz [l]
Chapter VII,I, Theorem X V I .
7.7. Examples
We give below some examples in order to illustrate the results
of the preceding sections.
1. Let c be a real number and k = 0,1,2,3,... . Then, according to (7.6.4) we have
-2aizx, k -2nizx> IF^ 6 (k) (x-c) = < 6 ('1 (x-c) ,e < 6 (x-c) , (-D) e
which yields the result
(7.7.1) IF^ 6 (k) (x-c) = (Zriz) ' e-2sicz.
98 Chapter 7
2. By v i r t u e of (7.7.1) , w e have IFz 6 (x) = 1. F u r t h e r , making
use of t h e Theorem 7.4.2, w e o b t a i n IF;' 6 ( x ) = (IFz 6 ( x ) ) = k = 1. Consequently, 1 = IF;' 6 (x) and f i n a l l y IFx 1 = 6 (x) . F u r t h e r , by means of t h e r u l e (7.5.2), w e g e t
(7.7.2) IFxz -3 ik 6 ( k ) ( ~ ) , k = 0,1,2, . . . . 3. According t o (7.3.2) w e have
which y i e l d s t h e r e s u l t
(7.7.3)
i n ID'.
2ncx IFx 6 (2-ic) = e
More g e n e r a l l y , i f 5 i s complex, t h e n w e have
- 2 n i ~ x IFx6(z-&) = e
. F u r t h e r , making u s e of Theorem 7.4.1, 3-l G(z+iC) = (IFx G(z+ic) ) = (e -2ncx)
= e2ncx, which y i e l d s e 2ncx = IF;16 ( z + i c ) . IF eZncx = 6 ( z + i c ) . t h e n w e g e t
(7.7.4)
If w e t a k e c = 0, t h e n (7.7.4) y i e l d s t h e r e s u l t IFz lX = 6 ( 2 ) . Furthermore, by means of t h e analogous r u l e (7.5.2) , w e f i n a l l y o b t a i n
-2ncx 4. According t o (7.7,.3) , we have IFx 6 ( z + i c ) = e
Consequent ly , w e o b t a i n I f we r e p l a c e c by c / 2 n i n t h e las t r e s u l t ,
2
E~ ecx =6 ( z + i c / 2 n ) i n 8' .
which correspcnds t o t h e formula (7.7.2) . 5. Le t 5 b e a p o i n t i n t h e complex p l a n e and l e t y be a closed
5 p a t h going around 5 i n t h e p o s i t i v e d i r e c t i o n . Now, a c c o r d i n g t o (7.3.2) t o g e t h e r w i t h t h e r e s i d u e theorem as g iven i n t h e t h e o r y of f u n c t i o n s of a complex variable, w e o b t a i n ,
Fourier Transform 99
m
cp (x) dx - - J e-2ni5x
-m
which yields the result
(7.7.6) 1 -2niSx
in ID'. This formula is consistent with the Section 3.3.6 of
Chapter 3.
7.8.Fourier Transformation on $ and $'
We have seen in Section 7.3 that the Fourier transformation
defined by (7.3.1) transforms the space ID' into the space Z' . We shall present in this section the Fourier transformation as an
automorphism on $I .
For real x, the Fourier transformation of a function cp(x)e$ is
defined by,
(7.8.1) = 1 e-2nix5
and its conjugate by,
m
cp(S)dC, real 5 -0
(7.8.2)
The following results can be easily obtained:
1.
2.
I F x $ and rxcp are functions of x which belong to $.
If a sequence {$,I + 0 in the sense of $, then IFxcpn + 0 and
Pxcpn -+ 0 in the sense of $.
Fx+ isthe anti-transform of 9; that is, if wxcp = $(x), we have
IF $ = $ o r I F IFx cp = 4 (x) . Therefore,
From these results we may conclude that IF and Fare reciprocal
- 3.
IF;l =Tx. - X
topological automorphisms on $.
By transposition (see Section 5.1 of Chapter 5 ) , the Fourier
transform of a distribution Txc$' is the element of $' denoted by
IFxTx (Or simply P x T or IFT) and defined by
(7.8.3)
Similarly, we define its conjugate F T by
<IF T,+(x)> <Tx,lFx cp>, V cp e $. X
100 Chapter 7
- (7.8.4) < T X T , $ ( X ) > = <Tx, IFx$> , Y 4 E $.
I n (7.8.31, t h e s u b s t i t u t i o n of F T for Tx l e a d s t o
<Ex ( T x T ) , @ ( x ) > = <FxT,IFX@> 5 <Tx,Fx IF$>=<Txr$(X)>.
- Hence IFx F X T X = Tx and
(7.8.5) IF-1 E
(7.8.6) F -1s
Also,
I t f o l l o w s t h a t
T = 0 +=>IF T = 0 ,
- T = 0 -=>'IF T = 0.
It is easy t o show t h a t i f T t h e n w e have t h e fo l lowing:
i s an i n f i n i t e f a m i l y be longing t o $' X I V
Tx,v * Tx I n t h e sense of $'
a s v * v 0
IFxTx,v + DxTx
i n t h e s e n s e of $'
0' a s v + v
Hence, w e conclude t h a t 'IF and automorphisms on $ I .
= IF'l are r e c i p r o c a l topological
7.9. P a r t i c u l a r Cases
The resu l t s of S e c t i o n 7.6 and t h e preceding section e n a b l e u s i n t h i s s e c t i o n t o make t h e f o l l o w i n g p a r t i c u l a r cases.
As w e have seen i n S e c t i o n 7.6, i f T sE',
5' -i2nxf
>; (7 .9 .1) E x T = <T
and if T = f ( x ) is a summable f u n c t i o n , t h e n w e f i n d a g a i n t h e ordinary F o u r i e r t ransform
( 7 . 9 . 2 )
Fourier Transform 101
We denote this function by g(x). Since 1;(x) 1 is bounded, if we set Ex$ = g(x) then (7.8.3) takes the form
m m . . f(x)g(x)dx = f(x)i(x)dx.
-m -m
Thus (7.8.3) constitutes a generalization of Parseval's formula.
7.10. Examples
We list below a few examples in order to illustrate the results
of the preceding sections.
Let c be real and k = 0,1,2,.... Then we obtain according to
(7.9.11,
-2nix2, IF 6 ( k ) (x-c) = < 6 ( k ) (x-c) , e
AlsoI from (7.10.1) we deduce that
Further, after applying IF to both sides and making use of (7.8.6),
we find
7.11.The Spaces C($')and M($) of Fourier Transformation
Let C ( $ ' ) (the space of rapidly decreasing distributions) denote -_ the space of generalized
finite and the fk(x) are
lim [XIn
1x1 + -
Also, C(3') is contained
functions of the form K k 1 D fx(x) I where K is k= 0
continuous functions on IR such that
f (x) = 0, for every n 2 0.
in $', (see Section 5.8.5 of Chapter 5).
k
For example, distributions associated (identical) with continuous
functions which are rapidly decreasing at infinity and distributions
with bounded support ( E 5 ' ) belong to C($').
We have introduced this space because of the following results
which we state without proofs. However, the proofs may be established
102 Chapter 7
by the same kind of arguments used in the proofs of theorems given
by (Schwartz 111 Chapter VII and Treves C11 Chapter 30).
Theorem 7.11.1. Let T E $' and U E C($'), then the convolution
T * U E $ !
Theorem 7.11.2. If U E C($'), then I F U and F U belong to M($),
(see Section 5.3.3 of Chapter 5).
Conversely, we have the following theorem . Theorem 7.11.3. If a function a(x) belongs to M($), then IF(@) ."
and r ( a ) belong to C($').
Hence, we conclude that C($') and M($) are isomorphic by IF.
7.12,The Fourier Transformation of Convolution and Multiplication
In this section we establish some important relations between
Fourier transformation and convolution as well as multiplication of
Chapter 5.
The principal property of the Fourier transformation is given
in the next theorem.
Theorem 7.12.1 [Exchange theorem).If T E $ I , U E C($') and
a E M ( $ ) , then we have
(7.12 .l)
(7.12.2) IFx (a(x)T) = (IFxa(x) * IFxT),
and similar formulae can be obtained for F= IF-',
E X ( U * T) = (IFxU) EXT,
Proof. For each 4 E $, we have
<IFx (U * T), $(x) > = <U * T, IF $ > = <TX,u(x)> X
(7.12.3)
with
u(x) = <u y' Ex+y $ > *
Since U E C($'), we can write
Fourier Transform 103
One can interchange the order of integrations by virtue of properties
of fk(x) and thus we obtain
m
O(5) IFcU dE k -i2nx~ a3 K
$ ( E ) 1 r e D fk(x)dS = / e = e-i2+xc -m k-0 -m
which is justified by the Theorem 7.11.2. Consequently,
and we finally get
<TX,U(X)> = <IFxT,$(X) F X U > = <(IFxU)IFxT,$(X)>.
Comparing this with (7.12.3) we obtain (7.12.1).
We also have IFx (U*T) = (ZxU)FxT. Applying IF to both sides
we get
IF [(SU) F T l = U * T; 1 and taking F T = T E $' we obtain
(7.12.4) lF[FU T 1 = U * IFT . If F u = a(x) E M($), U = IFa(x) E C($'), then (7.12.4) yields
1 1
which is (7.12.2).
Hence, we may infer that the Fourier transform changes convolu-
tion into multiplication and multiplication into convolution, in
accordance with equations (7.12.1) and (7.12.2) (provided that the
requisite conditions are satisfied).
7.13, Applications
We mention below some applications of the preceding results.
1. By making use of D T = 6 (k) (x) * T (see equation (5.8.8)of k
Chapter 5) together with (7.12.1), fo r k = 0,1,2,...., we obtain
104 Chapter 7
k k (7.13.1) E D T = (2six) I F T .
Furthermore, according to (7.12.2), we have
SF (xkTx) = (F xk) x IF T 5 F 5 X*
Next, replacing IF xk by the given value of (7.10.2) , we finally obtain
2. Let ‘ I ~ be the translation of amplitude c, rcT=6(x-c)*T. Then
3. Differential equation
Let P(x) be a polynomial of degree d and let A be a distribution
determined by the equation
(7.13.4) P(D)X = A
where D still denotes the operatioh of distributional derivative.
Set X = IF X and A = IF5Ax. Employing the Fourier transformation
on (7.13.41, we obtain the algebraic equation
(7.13.5) ~ ( 2 n i ~ ) ~ ~ = ii
If B is a particular solution of (7.13.5) (i.e. P(2niS)B
then the general solution of (7.13.4) can be given as
L. A
5 E X 5
n
5 ’
5 5
where the ak are arbitrary numbers.
L.
Let Xx be the anti-transform of X and F x B E denotes the anti- 5
d-1
k= 0 transform of B Also, let anti-transform of 1 ak6(k) ( 5 ) be the
5’ .~
d-1. Thenin tennS of these relations, the polynomial P1(x) of degree
inversion of (7.13.6) is
- X x = I F B +
x 5
where the polynomail P1 (x)
p1 (XI
is arbitrary but of a degree 5 d-1.
Fourier Transform 105
4. Convolution equation
Let
(7.13.7) Q *'X = A
where A and Q are known distributions such that A E $', Q E C($')
and X is to be determined.
By employing the Fourier transformation on (7.13.7), we have
IF (Q * X) = F A = A .
Next, according to (7.12.11, we obtain
q(x)X = A 0 . 6
where q ( x ) = IFQ and X = IFX.
If Y is such that
.. q(X)Y = A ,
A
then we get X = Y which yields the result
x = S Y .
5. Primitivation (see Section 6.2 of Chapter 6).
Note. The foregoing study made on the Fourier transformation of
ID' ( IRn) ,Z (Cn) , a single variable can be extended to the case of several variables
by replacing the spaces ID , ID' , Z I E' to ID (IRn 1 6'(IRn) (see Section 2.9 of Chapter 2 and Sections 3.1 and 3.2 of
Chapter 3 ) and letting x = (x1,x2 , .. . ,xn) , z = (zl , z2 , .. . ,zn) be denoted by x and z respectively.
7.14.Bibliography
A comprehensive account of work on the Fourier transformation of distributions can be obtained in the following references: Arsac [l],
Bremermann C11 , Bremermann and Durand [11 , Choquet Bruhat [l] , Chapter IV, Gristescu and Marinescu C13 , de Jager C11 Chapter 1.6, Ehrenpreis C11 , Gelfand and Shilov [l] , Vol.2, Chapters 3 and 4, Lavoine c6l , Milton c 2 1 , Rudin c11 , Schwartz C21 , Silva [41 , Vo-Khac-Khoan [11 , Zemanian 111 ,Chapter 7. Footnote
(1) because p(21riE)s(~)(E) = 0 for k < d (degree ofplynanial p(2aiS)),
This Page Intentionally Left Blank
CHAPTER 8
THE LAPLACE TRANSFORMATION
Summary
As remarked in Chapter 4 , this chapter presents the setting of
generalized functions especially distributions having lower bounded
support with Laplace transformation and we shall see later that this
setting has an important role to enlarge and simplify the theory of
Laplace transformation.
Briefly, the organization of this chapter is as follows. In
Sections 8.1 to 8.9 we carry the development of the Laplace transfor-
mation in a distributional setting as mentioned above; while the
next section covers the work of Laplace transformation of pseudo
functions. Moreover, Sections 8.11 and 8.12 provide a condensed
exposition of the asymptotic behaviour of the Laplace transformation
by working with equivalence of distributions as discussed in Chapter
6 . Finally, Section 8.13 bears a brief exposition of an outline of
the basic theory in a distributional setting of Laplace transforma-
tion in n variables.
In above setting, the Laplace transformation permits a great
flexibility in applications. In the next two chapters, the distri- butional Laplace transformation will be used as an essential tool.
Notations. Recall that ID; denotes the space of distributions
(If Tx E ID;, then there exists a real with lower bounded support.
number c such that the support of Tx is contained in Cc, mC.)
By $'nIDi we denote the space of tempered distributions with lower bounded support and by the symbol $: the space of functions
belonging to $ with lower bounded support.
Let x, 5 and ri denote real numbers belonging to IR and set
z = c+in.
107
108 Chapter 8
8.1.Laplace Transformability
A definition of the Laplace transforms based upon the Fourier
transformation of distributions can be found in Schwartz [lJ . An alternative definition based upon a subspace of "tempered distribu- tions" is also introduced by Schwartz c21 . The most of the basic
idea in this section is due to Schwartz C21, although the present
version contains'many differences.
T o prepare for this section we first need the following result.
Lemma 8.1.1. Let Tx E ID:. If there exists 5' E IR such that
.-5'x Tx E $In ID; ,then for every z satisfying Re z > c ' , we have TX E $'.
Proof.
e- zx
For every + c $, <e-'IXTx,+ (x) > exists and is equal to
+ on the support of T. TxrOl(x) > with $1 E $+ and equal to
As e - (z-s')x +,(x) belongs to $+, we have
+,(XI > = <e-ZX T,, $,(XI> <e-S'X - ( z - c ' ) x
Txfe - ZX =<e Tx, $(x)>.
Definition 8.1.1. A distribution T E ID; is called Laplace
Then by virtue of transformable for Re z > 5 ' if e's'x Tx E $:
Lemma 8.1.1, e-zx Tx c $' for every z such that Re z > 5'.
The number S(T) denotes the lower bound of these 5' and is
called the abscissa of convergence. Thus for every Re z > 5 (T) , - zx e Tx c $'.
As T c ID: is Laplace transformable for every z if e-'lXTx c $If
V 5' E IR , then C(T) becomes --. Examples. We mention below a few examples in order to illustrate
the above results.
2 1. T = e: is not Laplace transformable for any 5' E IRand e-'IXT
does not belong to $'.
2. 6(x+c) is Laplace transformable for every z .
Problem 8.1.1
If T = U(x+c)xv eax is Laplace transformable for Re v > -1 where
Laplace Transform 109
1, x > 'C,
0, x < 'C,
U(x+c) =
find its abscissa of convergence.
or not?
If 5 < Re a, then e-5x T E $I
We have the following theorem.
Theorem 8.1.1. Every distribution with bounded support is
Laplace transformable for every z because it is a tempered distri-
bution (see Section 4.1.3 of Chapter 4 ) .
8.2.Laplace Transform
The results of the preceding section enable us in th is section to
construct the distributional setting of Laplace transform in the
following manner.
Definition 8.2.1. Let T c ID; be Laplace transformable for
Re z > C(T). Then its Laplace transform is the function of the
complex variable z defined in the half-plane Re z > E(T) and
denoted by
(8.2.1)
where IL denotes the Laplace transformation. A l s o , (8.2.1) is
equal to
(8.2.2) <e
ILT = G ( z ) = <T e-ZX> X'
-'Ix e-(z-5')x> , Re z > 5 ' 7 c(T). TX'
The existence of (8.2.2) can be justified because e-S'XTxE $'
and because of the fact that we can also find functions n ID; belonging to $ which equal to e-(z-s')x on the support of T.
Morsover (8.2.2) does not depend upon the choice of 5' in the
interval 1 5 (T) , Re z C; indeed, if 6 ' < 5" < Re z , we have
-5"x
Since e
- ( Z - - S " ) X > = <e- ( 5 " - 5 ' )x .-5'XTxIe (S"-S')x e-(z-5' )X> <e Txte
is a multiplier in $+ we finally obtain - ( 5 11- 5 I ) X
-C"X - ( z - s " ) x > = <e-s'x - ( 2 - s '1 X > * <e TXi e TXi e
This independence on 5 ' justifies the notation (8.2.1). Moreover,
mostly in practice, <Tx, e zx> is well defined immediately without
having to use the decomposition (8.2.2).
-
110 Chapter 8
Consequently, if T has a bounded support, then we have by
Theorem 8.1.1,
(8.2.3) ,. , V z e C . -ZX> ILT = T(z) = <Tx, e
Examples. We list below some standard formulae concerning the
Laplace transform:
(8.2.4) lLIL(x) = I, I L d k ) ( X ) = z k ,
k e - ~ ~ IL dk) (x-c) = 2 *
where c is a real constant belonging to IR and k is a non-negative
integer.
8.2.1. Case for functions
If T = f(x) is a locally summable function having c as lower
bound for its support and such that, for certain ~ ' E I R ,e-"f (x) -+ 0
as x + m , then (8.2.1) yields
(8.2.5)
where S(f) = lower bound of 6'. We thus obtain the definition of
Laplace transformation and its abscissa of convergence(2) in the
ordinary sense. The more general cases of pseudo functions will be
examined in the subsequent Section 8.10.
m
ILf(x) = z ( z ) = I f(x) e-zxdx, Re z S(f) C
8.3. Characterization of Laplace Transform
In this section we shall show how Laplace transforms can be
characterized. For this purpose, we first have
Theorem 8.3.1. Let T E ID: with c being a lower bound for the
support of T and with T being Laplace transformable for Re z > 5 (T) , (respectively for every z E C). Then
* 1. the function T ( z ) is analytic in the half plane Re z > 5 (T) ,
(respectively in the whole plane(3) ) ;
2 . there exists a non-negative integer k such that l $ ( z ) 1 . I z I-k is bounded as z + - in the half plane Re z > C(T) ec Re z
(respectively in the whole plane).
- Proof. 1. e-zx is an analytic function with respect to z whose
Laplace Transform 111
-2x de r iva t ive i s -xe , it follows t h a t
e x i s t s f o r every z where T i s Laplace transformable, t h i s exis tence can be j u s t i f i e d i n a similar manner t o t h a t of (8.2.1).
More generally,
2. L e t 5 ' be such t h a t R e z > 5 ' > S ( T ) and e-'IxT E $'n m;. W e then observe t h a t e-(z-5')x coincides on [ c,*[ with a function belonging t o $.
4, t he re ex i s t two non-negative integers j , k and a number M such t h a t According t o property 2 of Section 4 . 2 . 1 of Chapter
> I - S I X - ( z - 5 ' ) x (8.3.2) l G ( z ) 1 =l<e Txt e
- < M sup(l+x')j/21 (z-5') ' e - ( z - " ) X I XLC
When x > c and R e z i s s u f f i c i e n t l y l a rge (i.e. R e z > c ) , w e have
Further, according t o a property of the exponential funct ion, t he re e x i s t s P 0 such t h a t
( I + X ~ ) ~ / ~ e-(z-5')(x-c)< p when x 2 A.
Now, by multiplying e '-") on both s i d e s , w e have
( l+x 2 1 1/2e-(z-s1)x < -c R e z
by taking P ec5' = M1.
Next, w e choose A > I c l ; then f o r c 2 x 2 A, w e have
1 1 2 Chapter 8
sup ( 1+x2) j /2 I e- ('-' 'I I < Mle-c Re '+M2 (1+A 2 ) J/ae-c Rez. X'C
Consequently, w e may i n f e r t h a t (8.3.2) can be dominated for s u f f i c i e n t l y l a r g e R e z > 5 such t h a t
Moreover, t h e r e ex is t s a number M3 such t h a t 1; ( z ) 1 < M3 I z I t h e re fo re , l & ( z ) I J z I - ~ ec Re i s proved.
e-c Re
Hence t h e 2 of Theorem 8.3.1 < M3.
t A
Theorem 8.3.2. If TX E E , then T ( z ) i s an e n t i r e a n a l y t i c func t ion , and if T has i t s support contained i n t h e bounded i n t e r v a l C-b,bl, then t h e r e ex is t s a non-negative i n t e g e r k such t h a t I;(Z) I I Z I - ~ e - b l R e '[is bounded a s IzI .+ a.
X
Z - Proof. Since Tx has a bounded suppor t G ( z ) = <TX, e zx> = e(-) 2 r 1
The above theorem -
where 0 ( z ) = IF T , t h e Four ie r t ransform of Tx. i s a consequence of t h e Sec t ion 7 .6 of Chapter 7 .
z x
Remark. Often 2 of t h e Theorem 8.3.1 g i v e s a more p r e c i s e - es t imat ion f o r I T ( z ) ( t han t h e Theorem 8.3.2.
k Theorem 8 . 3 . 3 . L e t T = D s ( x ) , where s ( x ) i s a l o c a l l y swnmable func t ion such t h a t c is t h e lower bound of t h e suppor t of s ( x ) and f o r which e-Sxs(x) i s bounded a s x -+ m. Then
A
1. T ( z ) i s a n a l y t i c i n t h e ha l f -p lane R e z > c , 2. i ( z ) z-kecz+ 0 a s R e z .+ -. Proof. 1. The proof of 1 fol lows by i t e r a t i o n of 1 i n Theorem
2. Since
- 8.3.1.
m
G ( z ) = ( - z J k I s ( x ) e-ZXdx, R e z > 5 , C
2 by p u t t i n g R e z = 215) + w , w > 0,
C
Laplace Transform 113
The las t two terms tend t o zero as R e z -+ m, because a s w -+ - then l/w + 0.
This theorem w i l l be u s e f u l i n Sec t ion 8.11. I n add i t ion , i f k = 0 , one can ob ta in here a w e l l known r e s u l t of t h e ord inary Laplace t ransformation.
8 .4 .Relat ion wi th t h e Four i e r Transformation
I n t h i s s e c t i o n an important r e s u l t concerning t h e r e l a t i o n between Four ie r t ransforms and Laplace t ransforms w i l l be e s t a b l i s h e d . To do so, w e f i r s t have.
Theorem 8.4.1. L e t T E Dl be Laplace t ransformable f o r R e z > S ( T ) and l e t F ( E ; x ) be t h e Four i e r t ransform ( i n t h e sense of s ec t ion 7.8 of Chapter 7 of e-" Tx f o r 5 > €, (T) . (8.4.1)
Then w e have
G ( z ) = F ( C ; Q , i f z = E + i q . 2 n
Proof. According t o Theorem 8.3.1, f o r a f i x e d 5 > C ( T ) , - T ( < + i 2 + n ) is a continuous func t ion of t h e r e a l v a r i a b l e n and i t s modulus is dominated by a power of I q I , as 1 nl +. -. Hence, w e conclude 1 T(S+i l rq)$(n)dr i ex is t s f o r every + E $. L e t
(8.4.2) H ( $ ; S ) = T(S+iZ+n)$(n)dn = < T ( € , + i Z r n ) ,+(n)>.
Then ( a s
t enso r p r o d u c t ( 4 ) , (8.4.2) t akes t h e form
m -
-m - * A
-m
T, E $ I ) , by making use of t h e commutative p rope r ty of
>I > 42s qx
-i2rnx,
= <e-SXTx, [<$(n) , e -i2nxn>] > = <e-CXTx, IFx $ >.
H ( + ; s ) = < $ ( a ) , [<e-Sx TX,e
= < b ( q ) e-SX T x , e
Now, by v i r t u e of (7.8.3) of Chapter 7, w e f i n a l l y o b t a i n
H ( $ ; S ) = < F ( S ; n ) t $(n)>* Consequently, w e g e t by means of (8.4.2)
and hence, (8.4.1) fol lows.
8.5. P r i n c i p a l Rules
Each of t h e fol lowing r u l e s is v a l i d f o r every z belonging t o every ha l f -p lane R e z > €, where t h e r ight-hand s i d e is an a n a l y t i c
114 Chapter 8
function.
1. Addition:
(8.5.1) IL (T+aU) = T(z) + aU(z), a E C.
2. Translation:
- cz (8.5.2) I L T ~ T ~ - ILTx+= = e T(z), C E IR.
3. Change, of scale (or homothesis):
l A z ILTbx = i; TO;) I b > 0.
k
(8.5.3)
4 . Multiplication by x I k E IN:
dk G ( z ) , in the sense of ordinary differentiation.
k
2 (8.5.4) ILx Tx = (-1)
5. Multiplication by eaXI a E C:
(8.5.5) lLeaXTx = G(2-a).
6. Differentiation:
(8.5.6) ILDk Tx = zk G ( z ) .
7. Antidifferentiation:
I L D - ~ T ~ = z -k ~ ( 2 ) . A (8.5.7)
Here D-kT is the distribution in DL such that Dk(D-kT) = T.
T = f(x) is a locally summable function having c as a lower bound
for its support and if we set F(x) = IXf (t)dt, then we have
If
ILF(x) = z-lILf(x). C
8. Convolution:
A
(8.5.8) IL(Tx * Ux) = T(z) U ( z ) .
9. Multiplication by a function a ( x ) E M($):
5+in Z (8.5.9) IL [~(XIT~] = v ( ~ ) = v(%)
1
where V(c+in) = T(2r(5+in)) * A of a ( x ) in the sense of Section 7.8 of Chapter 7.
with Ax being the Fourier transforn P n
10. Division by xk:
Laplace Transform 115
For Tx having bounded support i n [b,-[ , b > 0 , then
(8.5.10) IL> Tx = Wk(z)
1
X where
satisfies t h e e q u a t i o n v k ( z ) = T ( z ) wi th t h e condi t ion Wk(z) -+ 0 a s R e z + -.
T i s t h e quo t i en t having support i n [b , - [ , and Wk(z) dk
dz
- Proofs . l . , 3 . , and 5., fol low from t h e d e f i n i t i o n (8.2.1) of T ( z ) ,
2., and 6 . , fol low by v i r t u e of (8 .2 .4) , (5 .8 .7) and (5.8.8) of Chapter 5.
4 . , (8.5.4) fol lows from ( 8 . 3 . 1 ) .
7., l e t X ( z ) = ILD-kTx. = T ( z ) . Consequently, w e f i n a l l y zk X ( z ) = ILD D T = ILTx
ob ta in X ( z ) = z - ~ P ( z ) which proves (8 .5 .7 ) .
Then, according t o r u l e (8.5.6) w e have . t . k -k
8 . , By making use of (5.8.4) of Chapter 5, w e have
IL (T*u) = < T ~ e uy, e - z ( x + Y ) > = ( ILT)(ILU).
9., According to Theorem 8.4.1, i f V(z) s a t i s f i e s (8.5.8) and remembering t h a t f o r s u i t a b l e S , e- 2ncxTx E $' t oge the r wi th t h e use of (7.12.2) of Chapter 7 , w e have
a(x)Tx] = IF [ a ( x ) .e-2ncxT 3 - 2 n c x V ( S + i q ) = IF [ e n n X
rl - - [r .-ZnEx T~ I * C IF^ a ( x ) 1.
1 0 . If t h e quo t i en t T/xk i s forced t o have i t s suppor t i n [ b,-[ ,
then it i s unique. According t o (8.5.4) and Theorem 8.3.1, dk A - W (z) = T ( z ) ; Wk(z) is a n a l y t i c i n t h e ha l f -p lane R e z> E(T)
dz k k __ and Wk(z) + 0 as R e z + -. 1 k T(z) i n a unique way.
Thus one can determine W (z) f r o m
8.5.1. Case f o r func t ions
I f T and U are i d e n t i c a l ( a s soc ia t ed ) with t h e Laplace t r a n s f o r - mable func t ions , then w e f i n d again t h e above r u l e s of ord inary Laplace t ransformation.
I n p a r t i c u l a r , i f T = f ( x ) is a l o c a l l y summable func t ion which is continuous f o r x > c , where c i s a lower bound f o r i t s suppor t , and having d e r i v a t i v e f (x) ( i n t h e ord inary sense) , then (5.4.3) of
116 Chapter 8
Chapter 5 and (8.5.6) yield the result
Hence, we get the well known rule
If T = f(x) is a locally summable function having c as lower
bound for its support, and if we get r
I f(t)dt, x > c F(x) = i" c
then (8.5.7) yields
8.6.Convergence and Series
In this section we compute the Laplace transforms by means of
the convergence of sequences and series of distributions belonging
to ID:. To do so, we first need the following result.
Theorem 8.6.1. Let T be an infinite family of distributions
belonging to IDJ depending upon the parameter v and having their
supports in the same half-line [c,m[ . If there exists a number 5 '
such that €or every 5 > 5 ' , e-5x TXtV -t e-sxTx,u, I in the sense $ I
as v + v I , then
XI v
A
TV(z) + Tvi (2 )
for Re z > 6 ' with v' possibly being w .
Proof. Indeed, IF e-SXT + IF e-sxTx, I and by making use of X I V -
Theorem 8.4.1, the proof follows.
Corollary 8.6.1. Let T , n = 0,1,2 , .. . , be a sequence of distributions belonging to IDJand having their supports in the same
half-line [c,-[ . If there exists a number 5 ' such that for every
6 > 5' the series
xln
m
e-sx T converges in $', then xIn n= 0
=CTx,n - - E T n ( Z )
for Re z > 5 ' .
The following result is more useful.
Laplace Transform 117
Corol la ry 8.6.2. L e t TXln , n=0,1,2, . . . , be a sequence of d i s t r i - bu t ions such t h a t Tx cons t an t s , K is f i n i t e , and t h e f n ( x ) are l o c a l l y summable func t ions having support i n t h e h a l f - l i n e C c , m C . If t h e series l I f n ( x ) I converges uniformly on every f i n i t e i n t e r v a l and admits a majora t ion of t h e form xm e S t x (m being a non negat ive i n t e g e r ) , a s x -t m , t hen w e have
k = (ao+alD+ ...+ akD ) f (x ) where t h e ak a r e ,n n
IL ITx,., = I T n ( z ) .
f o r R e z > 5'.
W e expla in
8.6.1.Examples
these r e s u l t s by means of a f e w examples.
According t o Corol la ry 8.6.1, w e have m m
1 1. IL 1 6(x-n) = 1 e-nz = -. n= 0 n= 0 l-e-'
More gene ra l ly , by making use of Sec t ion 5.6 of Chapter 5, w e have
m k IL 1 6 ( k ) (x-n) = -.
n=O l-e-' Z
2. The Corol la ry 8.6.2 a l s o c o n s t i t u t e s a method f o r ob ta in ing We i l l u s t r a t e t h i s w i th t h e h e l p of t h e express ion i n t h e series.
an example.
I f v is r e a l and no t equal t o 0 , -1, -2, . . . , t hen w e have (see Erde ly i (Ed.) [ n ] , vol.1, p.182 ( 5 ) 1
ILFp - J y ( x ) + = ( z + (8.6.1)
Now w e t r y t o f i n d a series equal t o t h e r i g h t hand s i d e . For t h i s purpose, w e f i r s t have (see Problem 1 . 4 . 1 of Chapter 1).
V 17- 'V ( z +1)) , R e z > 0.
X
n=O H e r e t h e 5' of Corol la ry 8.6.2 is equal t o 1(5) . By t ak ing Laplace t ransform of each term of t h i s series and comparing (8.6.11, w e ob ta in
t h e with
which is extendable i n t h e domain IzI > 1. Next, by changing z i n t o 1 ;, w e have
118 Chapter 8
This is a hypergeometric series with a multiplier u2-'.
8.7.Inversion of the Laplace Transformation
In the preceding sections we have derived the results of the
Laplace transformation T(z) when the distribution Tx is prescribed.
In this section, these results are considered in the inverse orienta-
tion; that is, we begin with some specific knowledge of T(z) and seek
information about the distribution Tx.
A
Definition 8.7.1. Let v(z) be a function of the complex variable
Then, we call the distribution Tx the Laplace anti-transform (or Z.
Laplace inverse transform) of v(z) if ILzTx = v(z) and denote it by qlv(z1 - .
Further, by virtue of the relation (8.4.21, we have
(8.7.1) ~;lv(z) = e'x~v(c+2rix).
We remark here that the interest of this formula is more
theoretical than practical. Now we state the main result of this
section.
Theorem 8,7.1(Existence theorem), If v(z) is analytical in a
is bounded as
half-plane Re z > 5' > 0 and if there exist a nonnegative integer k
and a real number c' such that Iv(z)l. 1.1 -k ecIRe
z * in [c',-[ .
, then ILilv(z) exists in ID; and has its support contained (7)
Further ILilv(z) is unique and satisfies,
which is the distributional derivative of the function
(8.7.3)
where the integral is taken in the complex plane along the line
parallel to the imaginary axis passing through 5, or along any
equivalent path.
Proof. The hypothesis on v(z) assures the existence of the integral and, by means of Cauchy's theorem, its independence can be
justified with respect to 5. To do so, we remark first that w(x) is
Laplace Transform 119
e v i d e n t l y d e f i n e d by z i n t h e o r d i n a r y (8.7.3). Furthermore
-k-2 . 8.7.3) i s t h e Laplace a n t i - t r a n s f o r m of v ( z ) sense. Note t h a t (8.7.2) is a consequence of
one can show e a s i l y t h a t w(x) is cont inuous i .e. w(x+q)-w(x) -f 0 as q -+ 0. (Also , t h e c o n t i n u i t y of w(x) can be j u s t i f i e d by means of a g e n e r a l theorem on i n t e g r a t i o n . )
L e t x ' > 0 and choose i n t h e complex-2-plane a c i rcumference c c e n t e r e d a t t h e o r i g i n and of r a d i u s R > 6. F u r t h e r assume c p a s s e s through t h e p o i n t s c - i Y , c + i Y of t h e s t r a i g h t l i n e ( 5 - i - , c + i m ) . L e t A be a n arc of C on t h e r i g h t hand s i d e of ( 6 - i m , c+im). Cauchy's theorem, w e have
Then by
(8.7.4)
A s R + m, t h e l e f t s i d e of (8 .7 .4) + w(c ' -x ' ) , and t h e r i g h t s i d e + 0 ,
more r a p i d l y t h a n 1/R. Therefore , w e have w(c ' -x ' ) = 0 which y i e l d s w ( x ) = 0 i f x 5 c' . It follows t h a t t h e s u p p o r t of w ( x ) i s c o n t a i n e d i n Cc' , -[ . Hence, w e conclude w(x) E ID: .
I - + O o n A because accord ing t o t h e h y p o t h e s i s I v ( z ) z -k-2 e ( ~ ' - ~ ' ) ~
Now, set z = 5 + 2 n i q . For f ixed 5 > c ' , w e have m
v ( c + 2 n i a ) 2nixn (8.7.5) e-cXw(x) = J k+2 dq
-m ( ~ + 2 n i q ) By making u s e of (7.8.2) of Chapter 7 , (8 .7 .5) t a k e s t h e form
Consequently, by v i r t u e of (7.8.4) of Chapter 7 w e deduce t h a t w(x) is unique and
v ( <+2niq) IFe-cXw(x) = +2 ( c + 2 r i n )
(8.7.7)
A l s o , by (8.4.2) w e have
(8.7.8) ! i ( < + 2 i n q ) = IF e-cXTx.
L e t I L w ( x ) = w ( z ) . Then, by t a k i n g Tx = w(x) , (8.7.8) g i v e s
Ze-cXw(x) = & ( c + 2 i n q ) .
F u r t h e r ,
A
Hence, w
by making u s e of (8.7.7), w e have
120
i.e.
Chapter 8
which proves the theorem.
8.7.1. Example
Take v(z) = zk eCZ, k E IN, c E IR. Then according to (8.7.3),
we have
0 x(-c -k-2 - sL-lz-2e~~ -
X - c x+c, x -c. - w(x) = IL;lv(z)z
That is,
w(x) = U(X+C) (x+c) . Consequently, by means of (i) and (ii) of Problem 5.4.1 of
Chapter 5,
Dw(x) = U(x+C) , DLw(X) = b(X+C)
and finally, according to (8.7.2) we have
IL-' zk ecz - - &(k) (x+c). X
8.8. Reciprocity of the Convergence
This section provides an account about the convergence and series
of Laplace inverse transformation. To do so, we first need:
Theorem 8.8.1. Let V(V;Z) be an infinite family of functions of
the variable z depending on a parameter v and satisfying conditions
of the Existence Theorem 8.7.1 in a half-plane Re z > 5 ' independent
of V. on every compact subset of the half-plane Re z > c ' , then
If v(v;z) - v(vo; z) -+ 0 uniformly as v + v 0
IF^ -1 v(v;z) -+ ~~;'v(v ; z ) in ID'. 0
(vo may be infinity).
Proof. Let 5 > 5 ' . By considering v(v;E+2*ix) as a distribution - in x, we have
Laplace Transform 121
Hence, according to the sense of Section 7.8 of Chapter 7.
1~v(v;~+2nix) -f ~ v ( v *S+2mix) in 8'; 0'
and therefore in ID1 . Thus, by (8.7.1) we have e-'xLi'v(v;z) -+ .-Ex IL~V(V~;Z) -' in D' .
This proves the theorem.
8.8.1. Corollary in series
This section contains the following result.
corollary 8.8.1. Let v,(z), n = 0,1,2,..., be a sequence of
functions satisfying the condition of the Theorem z.7.1 in a half-
plane Re z > 5 ' independent of n. If the series 1 vn(z) converges n= 0
uniformly on each compact subset of the half-plane Re z > < I , then
we have
This corollary generalizes the Heaviside method in which v ( 2 ) is of
the form an z . n -n
Remark. In Theorem 8.8.1 and its Corollary 8.8.1, the uniform
convergence is a sufficient condition but not necessary (see example
9.1. ,2,3).
8.8.2. Examples -
The following examples will illustrate the above results.
1. Representation of the Dirac functional and its derivatives. Let
k be a non-negative integer.
compact subset of the half-plane Re z > 0. Then, Theorem 8.8.1 yields
As v +O, zk+"- zk + 0 uniformly on each
n"
(z+n) 2 . As the integer n -f -, 2 -1 -+ 0 uniformly on every compact
subset of the plane. Then, Theorem 8.8.1 yields (n and take the
role v and vo, v(v;z) = nn ez/(z-n)", (vo;z) = 1)
n
122 Chapter 8
3 . The function "exponential integral" admits the representation by
Section 9.4.4 of Chapter 9,
( - 2 ) -n Ei(-l/z) = log c/z 1 n.n! . m
n=l
This series converges uniformly on every compact subset of the half-
plane Re z > 1. By (8.10.2') we have
-1 -1 lLx log C/Z = -Fp X+
and on the other hand (see Erdelyi (Ed.) C2l Vol.1, p. 182(5))
then, by Corollary 8.8.1 we have
x: . Consequently, we obtain
E;'Ei(-l/z) = (210g C)6(x) + Fp;; 1 J0(2&)+.
8.9.Differentiation with Respect to a Parameter
From now on we shall derive the Laplace transformation by means
of a variable parameter. For this purposer we first state.
Definition 8.9.1. Let Tv be a distribution depending upon a ;X
real or complex parameter v which varies continuously in domain E.
Then Tu
<T v,x
sense).
is differentiable with respect to v if for any $ E ID, ;x
,$(x) > is a function of v differentiable in E (in the ordinary
a Furthermore, the derivative Tv;X is the distribution defined
by
< 3v;x,$(x)> = a < T v ; ~ h ) > l I TT 6 E ID.
If TVFx = Dkf (v;x) , with f (v;x) being a locally summable function of x and having for almost all x a partial derivative 8; a f(v;x) which is
continuous with respect to v and satisfying for every v E E,
function, then we have
a f(v;x) I < g(x), where g(x) is a positive locally summable
= D~ & f(v;x) in E. a av _1
Laplace Transform 123
Example. If v varies in the closed domain of the complex plane
not including the integers 2 1, i.e. (the points v = 1,2,..., do not
belong to this domain), then we have
a -V (8.9.1) FpX, = -Fp(x-' log XI+.
a a k x+ = Dk a x+ k- v k-v
a v mk, where k is an Indeed Fpx;' = -h av FWk .. ~~
integer, such that k- Re v > -1 and (-v+lJk = (-v+l) (-v+2) ,...,(- v+k).
(If Re v ~ 1 , the Fp is not needed here.)
are distributions which a ix av Tv;x Theorem 8.9.1. 1. If Tv and -
are Laplace transformable in the same half-plane Re z > 5' independ-
ently of v in E, then we have
(8.9.2) JL- a av Tv;x av - - a &TViX
a 2. If the functions of z , v(v;z) and v(v;z) satisfy the
conditions of the Theorem 8.7.1 in the same half-plane of Re z > 5'
independently of v in E, then we have
(8.9.3) I?& v(v;z) = & E,-' v(v;z) in ID' , - Proof. (8.9.2) and (8.9.3) are, respectively, the consequences
of the formulae (8.4.1) and (8.7.1) which proves the theorem.
Example. If v varies in the complex plane not including the
integers 21, then the relation
v-1 ILFpxIV = T(-v+l)z , Re z > 0,
holds and this result can be established by the method of Section
8.10.1. Further, by utilizing the Theorem 8.9.1 and the formula
(8.9.11, we get
(8.9.4) ILFp(x-'log x)+= -I'(-v+a)z'-l [log z-@(-v+l)],Re z > 0.
One can also obtain this relation by the analytic continuation method
of Section 8.10.2. More generally, by differentiating (1-1) times in
(8.9.4) with respect to v, we obtain
(8.9.5)
Re z > 0,
where (;) = &(Fp is not needed if Re v < 1).
124 Chapter 8
8.10.Laplace Transformation of Pseudo Functions
The results of the preceding sections applied to pseudo functiols
(see chapters 1,3,5) enable us to obtain the analysis of this
section. To do so, we first formulate the explicit definition m
ILFpf (x) = CFpf (x) , = Fpl e-zxf (x)dx C
where
c = lower bound of the support.
Moreover, we have the following particular rules.
8.10.1. Derivative and primitive
Let g(x) be a function such that
if x < 0, g(x) = 0
if 0 < x < E , g(x) admits a representation of the type (1.1.5)
if x > E , g(x) e-ZX is continuous and integrable for Re z > E l ,
if x > 0, g(x) has an ordinary derivative g' (x).
(8)
of Chapter 1,
We set m
G(z) = ILFpg(x) = Fp 1 e-ZX g(x)dx, Re z > 6 ' 0
and X
Fp 1 g(t)dt, x > 0
, x < o .
g(-l) (x) =
Next, by making use (5.4.12) of Chapter 5, we have for Re z > 5'
If g(x) has the expansion of the form (1.1.5) of Chapter 1 with
c = 0, then we have
K' J' - xi K' J' g-1 (x) = [ 1 ~ai+a;~logjxl x + 1 1 B;k xl-klogjx +
k=l j=1 k=l j=1
where hl(0) = 0 and h i are not integers: g(") (x) has therefore an expansion of the form (1.1.5). Let
Laplace Transform 125
ILFp g(”) (x) = G 1 ( z ) .
NOW, taking into account the conditions imposed on g(x), we then
apply formula (8.10.1) to g(-l) (x) . Accordingly, we have ILFp g”’) ’ (x) = ILFp g(x) = G ( z ) ,
or
where
Consequently,
(8.10.2)
Example.
we obtain
From IL (log x)+ = - (see Erdelyi (Ed.) c 2 1, Vol.1, p. 218(1)) We deduce by virtue of (8.10.1),
(8.10.2’)
and differentiating again we obtain
ILFp xT1 = - log Cz, Re z > 0 1
-ILFp xi2 s - z log Cz + z .
More generally if n E IN, we finally obtain
n n n -n-l - - (log Cz - 1 I/]) , Re z > 0. j =1 IL FPX+ n! (8.10.2”)
8.10.2. Use of analytic continuation
The notion of analytic continuation (see Section 1.4 of Chapter
1) enables us now to obtain the Laplace transform of pseudo functions in the following manner.
Theorem 8.10.1. Let g(a;x) be a function of the real variable x
and the complex variable a which varies in a domain E C C. We
supposeg(a;x) and E are such that if x < O , g(a;x) = 0; if x 20,
where x ( O t l ) = 1, if x E
h(a;x), ak(a) and vk(a) are holom~rphic(~) in E, vk(a) # -1,-2,-3,..., for any a in E.
0,l , ak(a) and v k ( a ) are bounded:
Moreover, Re vk(a) > -1 in a part El of E. If
126 Chapter 8
a Re z > L(g) , Ih(a;x)e'ZXI and IK-h(a;x)e-ZXI are majored when a is
in E, by an integrable function y(x) 2 0, then
1. if a E El, the function g(a;x) has a Laplace transformation
in the ordinary sense given by
- zx ~ ( a ; z ) = 1 g(a;x)e dx, Re z > E ( g ) ;
0 2. as a function of a ,G(a ;z ) has an analytic continuation with
respect to a in E and also is holomorphic in E;
3 . for every a E E , we have
ILFp g(a;x) = G ( a ; z ) , Re z > [(g).
(here Fp is not needed if a E El).
Proof. Set - vk ( a )
Nk(a;z) = ILFp x X(OI1)
Let us assume that there exists \ E lN such that, for every a E E , Re v,(a) + Mk> -1, then we have
lv (a)+m m m lvk(a)+m =%c' (-z)mFp xk dx+l(-Z) f x dx
= c [vk(a)+m+l] m! m' 0 Mk m= 0 a' 0
( -zlm m
m= 0
Therefore, as a function of a , Nk(a;z) is holomorphic in E. Set
-2X H ( U ; Z ) =rr,h(a;X) = j h(a;x) e dx, Re z > s(g)
0
where H(a;z) is holomorphic in E. Consequently, we obtain by (i)
K ILFp g(a;x) = H(a;z) + 1 ak(a) Nk(a;z), Re z > c ( g )
m k= 0 - (which is equal to g(a;x)e ZXdx, if a E El) is holomorphic in E.
0
This theorem has very useful applications and we illustrate this
remark with the help of a few examples.
Examples. For a > -1 and b>O, the ordinary Laplace transformation
yields (see Erdelyi (Ed.) C7.l , Vol.1, p. 133, 4.2(3) and p. 182(1))
Laplace Transform 127
and ba 1T-T -a ILJa(bx)+ = -[ z + z +b 1 , R e z > 0.
r r - After changing a t o -a, w e deduce from Theorem 8.10.2, f o r a-1 g! IN,
(8.10.3) ILFpx;" = r(-a+l)za-' , R e z > 0 ,
and
(8.10.4) ILFpJ_,(bx)+ = - b-' [z+ JZ2+bz]&, R e z > 0 . m 8.10.3. Change of x t o ax, a being complex
The change of x t o ax permits us t o deduce lLFp g ( a x ) from ILFpg(x) by means of Sec t ions 1 . 6 and 5.9 of Chapters 1 and 5 , r e spec t ive ly . For t h i s purpose, w e f i r s t have
Theorem 8.10.2. L e t
0 , x < o
x-"h(x) I X > 0 g ( x ) =
where v is not an i n t e g e r 1. 1 and h ( x ) is ana ly t i c . S e t G ( z ) =
ILFpg(x) and Ga(z) = ILFpg(ax) where a E C is f ixed .
W e suppose t h a t t h e r e e x i s t s an angle A i n t h e complex w plane having i t s v e r t e x a t t h e o r i g i n and conta in ing ha l f l i n e s w 1. 0 and w = ax (x varying from 0 t o m) and i n t h e z-plane a domain B, such t h a t h ( w ) is holomorphic i n A and such t h a t I w a g ( w ) e w -+ i n A, z remaining i n B and Re(a-v) 1. -1.
I as - z w/a
Then w e have
1 G a ( z ) = a G ( z / a ) , z E B.
From t h i s r e s u l t obtained i n B , w e deduce, by a n a l y t i c cont inua t ion , Ga(z) i n every half-plane where Fp g (ax ) is Laplace-transformable.
- Proof. L e t a be a complex v a r i a b l e , and set g ( a ;x) = xag(x ) Qlence
g(x ) = g ( 0 ; x ) ) . Then
G(a;z) = ILFp g ( a ; x )
128
and
Chapter 8
Ga(a;z) = IL Fp g ( a ; a x ) .
(The Fp are n o t needed i f R e ( a - v ) > -1.)
I f L denotes t h e h a l f l i n e w = ax, provided t h a t Re(a-v) > -1 and z E B , w e have
m
Ga(a;z) = g ( a ; a x ) e-zxdx 0
1 a = - G(a;z/a)
where t h e t h i r d e q u a l i t y follows by v i r t u e of Cauchy's theorem. Hence, by Theorem 8.10.1, w e have
Ga(O,z) = g I G(O,z/a)
which y i e l d s t h e r e s u l t ,
Example. Take g ( x ) = J - v ( x ) + , v being a non-integer 1, and a Then, by (8.10.1) , being a complex number such t h a t 0 5 a r g a 5 3.
w e have
(8.10.5) G( z) = IL FpJ-" ( x ) + = [z+ (z2+1) ( ~ ~ + l ) - " ~ .
Furthermore, t a k i n g t h e a n g l e A and domain B d e f i n e d by
--E < a r g w < ' + E ( O < E < $ ) and I z I > 21al w i t h
- i + E < a r g z - a r g a
T 3 a 71-E,
r e s p e c t i v e l y . Now by Theorem 8.10.2, w e o b t a i n t h a t t h e (10.8.5) t a k e s t h e form
(8.10.6) I L F P J - , ( ~ ~ ) + = a-"[z+(z 2 +a 2 1 / 2 1 ~ ( z ~ + a 2 ) - ~ / ~ , R e z > I m a.
Remark. The Theorem 8.10.2 is n o t a p p l i c a b l e i f v is a p o s i t i v e - i n t e g e r and hence f o r v = n = 1,2,3. . . , t h e formula (8.10.6) does n o t hold t r u e . S i n c e J-,(ax) = ( - l lnJn(ax) , w e have,
ILJ-n(ax) = (-l)nan~z+(a2+z2)1/2]-n(a2+z2)-1~2, R e z > I m a.
Laplace Transform 129
Thus, w e see t h a t t h i s r e s u l t is d i f f e r e n t t o t h a t of (8.10.6). Moreover, i f w e want t o keep t h e same terms a s i n t h e r i g h t s i d e of (8.10.6) , then w e need t o add c n-l (&) ($)n-2J6(n-1-2j)(x) i n
05 - 2 t h e l e f t s i d e of (8.10.6) as fol lows:
= a -n ~ z + ( z ~ + a ~ ) ~ / ~ 1 ~ ( z ~ + a ~ ) - ~ / 2 1 R e z > I m a.
8.10.4. Change of x t o i x
Taking a = i b , b > 0 , i n t h e formula (8.10.6) and bear ing i n mind t h a t i - ' JV(ibx) = I v ( b x ) , which i s t h e modified Bessel func t ion , w e o b t a i n
f o r every non-integer v 2 1.
Another Example. From
1 2 2 1/2 ILFpl/x J o ( b x ) + = - log T [ Z + ( Z +b ) 3 - l og C,
w e deduce
Jo (bx) 1 ILC- X - l /x]+ = - log ~ [ z + ( z ~ + b ~ ) ~ / ~ l + l og E.
The l e f t s i d e is a r e g u l a r f u n c t i o n of x and has t h e ord inary Laplace t ransformat ion . t h i s change y i e l d s t h e r e s u l t
The change of x t o i x can be performed e a s i l y , and
Consequently, w e f i n a l l y ob ta in
- l o g c I R e z b. (8.10.8) ILFFp-I 1 (bx)+ = - l og ~ [ Z + ( Z 1 2 -b 2 ) 1 / 2 ] x o
These two methods are n o t app l i cab le when t h e change x t o i x g ives r i se t o an i n f i n i t e number of po le s a s one can f i n d i n t h e fol lowing problem.
Problem 8.10.1
Deduce ILFp 1 from 3 L F p r 1 . + x+ s i n x
130 Chapter 8
Theorem 8.10.3. Let g(Z) be a function of Z = x+iy(x,y E IR)
satisfying the three conditions:
1. g(Z) is holomorphic in the neighbourhood of the origin or has the representation
K 1 (akZ-"k + bkZ-k)
k= 1 g(Z) = h(Z) + f? log Z +
where h(Z) is holomorphic, vk is not an integer
coefficients B,ak,bk may be zero:
1 and some of the
2. g(Z) is holomorphic in a quarter plane (Q): x > -a, y > - a ,
a > 0, except at the points (making a countable set) 2 = iy,,
n = 1,2, ...,Y,+~ > yn > 0, in the neighbourhood of which it can be
written as
J g(Z) = hn(Z) + BnlOg(Z-iyn) + 1 C .(Z-iy n 1-1.
j=1 nJ
In this neighbourhood, hn(Z) is holomorphic and satisfies Ihn(Z) I <
A12 I m I J is finite, I BnJ and IC . 1 are bounded by Ay:, and the
constants A and m are the same for fixed n 2 N. n3
3 . There exist non intersecting circles (Cn):IZ-iynl<an< c t I such
that Ig(Z) I < AIZjm when Z is in ( a ) but lies outside of (Cn).
With these conditions, we have the following properties:
a.
mable for Re z > 0, and these transforms will be denoted by G ( z ) and
G (z) respectively.
b. The function G ( z ) has an analytic continuation in the half-plane
Im z > 0.
The pseudo functions Fp g(x)+ and Fp g(ix)+ are Laplace transfor-
1
c. If z is in the quarter plane Re z > 0, Im
where rn being the residue of g(Z)e-izZ at z =
at Z = 0.
z > 0 , then we have
iyn and ro its residue
- Proof. The proof of this theorem is quite complicated and can be
found in Lavoine c11 , p . 991, and [ 2 1.
Remarks: According to (1.2.6) of Chapter 1, we have
Laplace Transform 131
5 G ( 2 ) = lim Fp I g(iy)e-zYdy
5 -+,= 0 1
where 5 > 0 is such that the point is is exterior to the circles (Cn). m
The conditions 2 and 3 assure the convergence of the series 1 rn. n=l
If g(Z) eizz has poles in the quarter plane [ Re z > 0 , Im z > 0 1 then the theorem remains valid provided we add to the right hand side
of (8.10.8) the product of 2s and the sum of residues at these poles.
The combination of the Theorem 8.10.3 and the rule (8.5.5) permits
us to consider the case where g(Z) is singular elsewhere excluding the
origin, because (8.5.5) has singularity at the origin.
Example: We start from (see Lavoine C2l , p. 76)
ILFp[b/sh bx1+ = -+(z/2b++)-log 2cb, Re z > -b.
with b>O,b/sh bZ satisfying the conditions of the previous theorem with poles at the origin and the points z = inn/b,n = 1,2,3,...,
Here, the rn series converges for Re z > 0 and (8.10.8) leads to
-nz/b n 1 . where residues of beizZ/sh bZ are respectively ro=l, and r n =(-e
n z th r
1 ILFp[b/sin bxl+ = -$(z/2bi+q) - log2Cb - i* 2
which can also be rewritten in the following form without imaginary i;
8.10.5.Convergence
If Tx is the limit of the sequence of distributior&(Fpg,(x)) as
n
preceding Theorem 8.6.1 are fulfilled. But it must be note3 that, if
these conditions are not fulfilled, then gn(x) + g(x) does not lead
always to ILFp gn(x) -+ ILFp g(x) . r ( l/n) z -'In does not converge to lLg$ = -log Cz: while x Indeed, here the condition of Theorem 8.6.1 are not fulfilled,
-'+'In = n(DemSX + x:/") does not converge because e-sx x+
in $', as n -+ -.
-+ m , then ILTx = lim ILFpgn(x) provided that conditions of the n + m
-l+l/n - - For instance, E x + -l+l/n -+ x-l
x+
From now on we shallbe concerned with the Abelian and Tauberian
theorems for the Laplace transformation. The procedure in obtaining
132 Chapter 8
these theorems is similar as indicated in Lavoine C91.
8.11. Abelian Theorems
In this section we shall present the theorens which deal with the
behaviour of the Laplace transformation of a distribution from the
behaviour of the distribution as discussed in the preceding section
8.3 (see also Milton [l] ) .
8.11.1.1&haviaur of the transform at infinity
Here we establish the behaviour of the Laplace transformation by
working with the equivalence of the distribution at the origin
(Section 6.4.1 of Chapter 6).
(10) Theorem 8.11.1. Let Tx be a distribution which has the origin
for lower bound of its support and which is Laplace transformable for
R e z > c > O . 0
If according to the sense of 1. of Section 6.4.1 of Chapter 6,
T~ - mp(xviogjx)+, as x + o+,
with j = 0,1,2,..., and v # -1, - 2 , . . . , then we have
(8.11.1) I L T ~ ~ (-i)j~r ( v + i ) z-v-liogJz
(in the ordinary sense) as z + - in the half-plane Re z > 5,.
- Proof. By virtue of Section 6.4.1 of Chapter 6, we can set
Tx = AF'p(x"log1 x)+ + Rx + Sx
where R is a distribution having support contained in C 0x1 such
that for every + E
X
E- U $+,
(8.11.2) C<RXI+(x/X)>I + 0, if + O + , XVfllogJ A
and Sx is a distribution having support in C s , m l and Laplace trans-
formable for Re z > 5,. We further set
A
T ( z ) = ILTx, P ( Z ) = lLFp(~"log'~)+,
A L
R(Z) = LRX, S ( Z ) = asx.
Consequently, we can write
Laplace Transform 1 3 3
Fur the r , by making use of ( 8 . 9 . 3 ) , w e have
1 Denoting a r g z by 8 and IzI by 5; ( i n such a way t h a t A -+ O+ as z -+ m) , w e have by v i r t u e of ( 8 . 1 1 . 2 ) ,
( 8 . 1 1 . 5 ) Iz"+llog-Jz &) I 2 A-v-llog-Jh l & Z ) I i ex -e
= I A-"-'log-j "RX, exp A >I I + 0
a s z -+ m. F i n a l l y , according t o t h e Theorem 8 . 3 . 1 , s i n c e 5 > 0 , w e have
as z -+ m i n t h e ha l f plane R e z > 5,. ( 8 . 1 1 . 4 1 , (8.11.5) and ( 8 . 1 1 . 6 1 , w e deduce ( 8 . 1 1 . 1 ) .
Consequently, from ( 8 . 1 1 . 3 ) ,
Theorem 8 . 1 1 . 2 . L e t Tx be a d i s t r i b u t i o n which has t h e o r i g i n a s lower bound f o r i t s suppor t and which is Laplace t ransformable f o r R e z > Lo > 0.
I f i n t h e sense of 2 . of Sec t ion 6 . 4 . 1 of Chapter 6 ,
-n-1 * Tx - AFPCX log'xl+ , as x + o+
f o r j,n = O , l , 2 , . . . . , then w e have
a s z -+ - i n t h e ha l f plane R e z > 5,.
Proof. W e proceed here i n t h e same manner as i n t h e proof of t h e previous theorem and make use of t h e approximation. Accordingly w e have
-
n+j+l ILFp[x-n-llogj x]+ ,. (-1) n; ( j + l j zn1ogj+'z, as z 3 m.
Example. L e t No(x) be t h e Bessel func t ion of second kind and b > 0. Then w e have (see Lavoine [ Z ] , p. 92),
134 Chapter 8
Consequently, making use of the Theorem 8.11.2, we have
2 - ILFp Cx-'N0(bx) 1, -IT '2 log z
as z + w with Re z > 5 . 8.11.2. Behaviour of the transform near a singular point
0
When a distribution Tx is Laplace transformable with abscissa
of convergence E(T), then its Laplace transform is holomorphic in
the half-plane Re z > C(T) and possesses one or several singular
Points a such that Re a = C(T). This notion permits us to obtain
the following result.
Theorem 8.11.3. Let Tx E ID; be Laplace transformable with E ( T )
as abscissa of convergence and be equal to ewx"logjxCA+f2 (x) +w (xw
on the interval x > X > 1, with the conditions that a,A,v are numbers
such that Re a = C(T), Re v > -1, and j is a non negative integer.
Also ~ ( x ) is a function tending to zero as x + -, and Q(x) is a continuous function such that
X'
X I j n(x) e-lXIm(z-a)dxl < M
where M > 0 is independent of X' and z for every X' > X when z
belongs to a certain neighbourhood (V) of a. Then
as z +. a in the intersection of (V) with the half plane Re z > C ( T ) .
Proof. For simplicity, we first deal the case'when j = O . We can - write T, as
T =Ae ax X+ v + Bx + x v a x e Q(x)x(X,-)+X V e ax ~(X)X(XI~)I X
where Bx is a distribution whose support is bounded in l-m,X] and
x(X,-) is the characteristic function of the interval [XI=[. Further, we set
E (2) =(Z-a) '+lIL Tx-Ar ("+I) = (z-a) "+I [IL Bx+IL xveaxf2 (x) x (X, m )
v ax + ILX e w ( x ) X ( ~ , m ) l . Now, according to Abel's theorem, we have
Laplace Transform 135
m
dx I -ix Im(z-a) 52 (x) e
v ax v -x Re(z-a) IILX e n(x)x(x,-) I = I J x e
X -X Re(z-a) < xv. < M X ' ~
Let (W) denote the intersection of (V) with the half-plane Re z>c(T).
Then, for given arbitrary E > 0, we can choose X such that
- < e l < n/2.
Consequently, if larg(z-a) I < e l , we have
v a x V m
l ~ x e u(x)x(x,-) I < SUP Iw(x) I 1 x x,x X
It follows that if z E (W), then we have
I E ( z ) I < Iz-a
If there exists a number r
MX' I z-a I "+' 12-1 IL BX
v + l
- v - 1 z-a1 .
B~+MX' I z-a I 5 . > 0 such that for Iz-al < r, then we have
E
5 '
< E 7
because ILBx is an entire function (see Theorem 8.3.2). Consequently,
if z E (W) and if I (z-a) I < r, then I E ( z ) I < E; hence we conclude
(8.11.8) with j=O.
When j is an integer 21, the proof can be developed in a similar
manner. By making use of the formula (8.9.5), we obtain
(z-alv+l v ax j
log3 (z-a) IL (x e log x)+ - (-1)' r(v+l)
k=l
as z -+ a.
Example. Let b > 0. Then we have for the largest value of x (see Jahnke, Emde and Losch [I] , pp. 134 and 147)
(8.11.9)
From (6,11.9), we may infer that FpJa(bx)+ is Laplace transformable
with abscissa of convergence equal to 0. Also,
n - 4 Ja(bx) = (2/abx)' [cos(bx - a 2 - a)].
Ja(bx) = e ibxx-f ri-a-1/2 (2ab)-*+ n(x) + w(x)]
136
where
Chapter 8
w(x)+ 0 as x + m
n(x) = i a+1/2 (2+b) -1/2 .-i2bx
The conditions of the theorem are satisfied by taking (V) in a half-
plane, Im z b', Ib'I < b. Hence, according to (8.11.8), we have
-1/2 IL Fp Jcr (bx) + ., i-' (2ib) -'I2 (z-ib)
as z + ib in the half-plane Re z > 0. This result is consistent
with the formula (8.10.4).
Remark. In the Theorem 8.11.3 we cannot drop the condition - Re v > -1. Furthermore, taking Tx = 6(x) + FPX;~'~, we have
ILTx = l-2dsz ., 1, as Z + 0,
though Tx = x -3/2 on 1 1 , m ~ . 8.12.Tauberian Theorems
Tauberian theorems appear as the converse of the Abelian theorems.
These yield the behaviour of a distribution whose behaviour of the
Laplace transform is known.
8.12.1.Behaviour near the lower bound of the support
Theorem 8.12.1. If there exists 6, > 0 such that in the half- plane Re z > 5 , the function v ( z ) is holomorphic and
0
v(z) ., AeCZ z-' logjz, as I z I +. 03
with a non-negative integer j, real c, complex A and v ( v # 0,1,2,...,) and if Tx = IL-'v(z) , then we have in the sense of the Sections 5.2 and 6.4.1 of Chapters 5 and 6,
A (8.12 .l)
(Fp is not needed if Re v > 0).
'I -c T x = TX-C ., (-1)J rFp(xv-llogj r V ) x)+ as x -+ O+,
Proof. We put - w(z) = e-"v
1 w(z) = [A+r
Laplace Transform 137
where rl(z) -+ 0 as I z ( -+ -. Let k be a non negative integer such that Re (v+k) > 1. Also, we set
~ ( z ) = CA+rl(z)l z-v-kiogjz
and h(x) = IL-lH(z), which is continuous.
(8.12.2) IL-'w(z) = DXh(x).
Consequently, we obtain
Now making use of the Theorem 8.11.1, we have
A z-v-klogjz = r(v+k) (-1)' A IL(x v+k-llogjx) ++ r2 ( 2 ) z-v-klogj z
where r2(z) -P
where r(z) a0
our setting,
(8.12.3)
Putting z = m, we obtain X
m
-v-k lo ('+in) - 1)1 eisdn ( 1:g x ~ ( x ) = e' J r ( a ) ('+in)
-m x
where TI is a real variable. Therefore,by virtue of the properties of
r(z), p(x) + 0 as x -t O+. As x + O+, (8.12.3) takes the form
Consquently, by (8.12.2) together with Theorem 6.4.1 of Chapter 6, we
may infer
-1 (-1) j v-1 j IL w(z) ., r(v)- DP(X log XI+,
hence, the result (8.12.1) is obtained.
Theorem 8.12.2. If there exists 5 , > 0 such that in the half-plane
Re z > 5, the function v(z) is holornorphic and satisfies
v(z) - Aecz znlogj+lz, as 121 + m
where j and n are non-negative integers, c is real, and A is complex,
138 Chapter 8
and if Tx = IL-'v(z), then we have in the sense of Sections 5.2 and
6.4.1 of Chapters 5 and 6.
(8.12.4) T'-cTx = Tx-,= - (-1) j+ntln! (j+l)AFp(x-"-llogJx) +
as x -+ O+.
Proof. The proof is similar to the proof of Theorem 8.12.1 if
we change k into n+2 and use Theorem 6.4.2 in the place of Theorem
6.4.1 of Chapter 6.
-
Remark. The Theorem 8.12.1 is not applicable to U ( z ) = Az3/*+
B e - b T b >O, because we do not have U ( z ) I Az~'~, as Iz I + -, Re z > 0, since lu(i0) I is of the order of B I n I as 1111 +. m. However,
according to (8.10.3) , IL-l 2O-l = - Fpx;", which yields
IL'l z3j2 = 3 Fpxi5j2 by putting a = 5/2; while according to (8.2.4),
2
r (1-a)
4 6 IL-le-bz 2
- 3A Fp~;~/~+B6"(x-b) , which is equivalent at the origin to ~ F p x , 3A
z = 6"(x-b). Hence, we obtain from these results IL-lU(z) = -5/2 .
4 6
We can deal with this kind of questions in the following way:
Let v(z) be an inverse Laplace transform that can be written in
the form
-bz v(z) = vl(z) + e v2(z)
with the inequality c1 < c2.
denote the lower bounds of the supports of IL-'vl(z) and IL
Then, the behaviour of the distribution J.L-'v(z) near c1 would be
the same as that of sL-lvl(z).
8.12.2 is applicable to v (z) then the behaviour of IL-lvl(z) can be
easily determined.
AlSO, by Theorem 8.7.1, if c and c2 I- 1
v2(z).
If any one of the Theorems 8.12.1 and
1
8.13.The n-Dimensional Laplace Transformation
In the preceding sections we have studied the Laplace transforms
in a distributional setting of one variable. The present section
developes the n-variables case corresponding to the preceding work
Here we use the following notations and terminology (see also
Section 4.5 of Chapter 4).
Here, we shall restrict x = (xl,x 21....I~n) to the domain Qn(O
of mn which is defined by Qn(0) = fx E IRn , 0 5 xv <-,v=1,2,...,n
Laplace Transform 139
and also we denote 5+i2rq = (51+i2~~l,...,5 +i27rq ) where for every
j=1,2,. . . ,n, 5 . and n space C" is given by as z = (z1,z2,. . . ,z ) and xz = x z +x z +.. .+ x z By Dx we mean the differentiation with respect to x in the
distributional sense.
distributions of lRn whose support contains in Qn(0).
n n E lRn . The point of an n-dimensional complex
3 j
n 1 1 2 2
j j n n'
Recall that by lDb+(IRn) we mean the space of
If f is a locally summable function f(x) such that f(x) = 0 for
Then, according to Example 1 of Section 4 . 5 of Chapter all xy < 0.
4, f (x) generates a regular distribution in IDA+ (mn) deflned by m m
(see also equation (4.5.11) of Chapter 4 ) .
To each locally summable complex function f(x) of the real
variable x which is zero for all xy < 0 and in addition obeys
certain appropriate restrictive conditions, we assigns a correspond-
ing holomorphic function T(z) of the complex variable z , defined by
the integral
h
- m m
(8.13.1) G ( z ) = I, ...., e Zxf(x)dxl dx2 ..... dxn. 0 0
If z = S+in, then the modulus of f (XI e-zx is (f (x) Thus, the abscissa of absolute convergence of the integral (8.13.1) depends
only on the real part of 5 of z.
8.13.1.The Laplace transformation in n variables
n Let Tx E lDb+(IRn) and let there exist 5, E IR (~o=(SO,l,...r~ ))
O,n such that for 6 > 5, (i.e. si >
e-SXTx t $' (E? ) . complex
for all j) such that
i=1,2,...,n) we have
Then its Laplace transform is the function of the
variable z defined in the domain Re z > c0 (i.e. Re z > 5 0,j
(8.13.2) - m m
ILTx = G ( z ) = <Tx,e-zx> = I , . . . . , I e ZXTx dxl , .. . ,dxn 0 0
where IL denotes the Laplace transform in n variables.
Let a(x) (i.e. a(x) = a(xlIx2, ..., x ) ) be an infinitely differen- n tiable function having support in a neighbourhood of Qn(0) and equal
to 1 on a neighbourhood of the support of Tx. For Re z > C0, there
140 Chapter 8
for each j . Then, Re
exists 5 ' = ( t i , ...., 5;) such that C0 ,j < 5; <
Tx E $'(IRn) and u(x) e-(z-s')x E. $'(IR ) it follows that -5'X since e
<e-s'x e-(z-5')x> TX'
has a meaning. This expression is independent of 5' and gives by
definition what we denote by <TX, e-zx>.
terminology we shall say that every distribution belonging to
IDA+ (IR" ) is Laplace transformable in n variables.
To conform with established
Theorem 8.13.1. If the function T ( z ) is holomorphic in the domain
Re z > C0 then there exist positive &S ki, 1 5 j 5 n, such that n -k ( i ( z ) I .TI l z j l j is bounded as Iz.1 + =.
3 J =1
The proof can be carried out by the iteration of the very well
known result of Laplace transformation in one variable given in
Section 8.3.
Examples. If c = (c1,c2, ..., c ) is any number of IRn such that n % >O, i = 1,2,. . . ,n and k is any non-negative integers of IRn . Then we have
(8.13.3) I L k a k 6(x-c) = z k e - ~ ~ j
I
axi J
and
ILd(x) = 1.
These results follow strictly by the definition (8.13.2).
8.13.2.Convolution
n Let T and S be two distributions in lDb+(IR ) and possessing
Laplace transforms for 5 > bl and 5 > b2, respectively where bl and
and b2 = (82,1,B2,2,...,B2,n) for each j = 1,2,...,n.
every 5 > b = max (bl,b2) (i.e.
(8.13.4) i ( z ) = < T ~ , e-'Y>
b2 denote the points of lRn such that bl = B1 , j-( - B1, 1, . . , Bl,n) Then, for
> max (Bl,jf@2,j) '1
A - ZX S ( z ) = <Sx, e >,
Hence
Laplace Transform 1 4 1
Theorem 8.13.2. If S and T have a Laplace t ransform f o r 5 > b =
max(b1,b2) then t h e Laplace t ransform of S*T is e q u a l t o t h e product of t h e Laplace t ransforms of S and T.
Remark. I f w e r e p l a c e t h e t u b e s r + i IRn of Schwartz 11, Chapter vIII ,Section 3, by t h e domain R e z > a 1 5 j 5 n. Then one can t a k e
j j ' t h e Or thant Qn (a) , a = (al ,a2 '.. . 1 f j 5 n. I n t h i s manner w e can re la te t h e p r e s e n t s t r u c t u r e of Laplace t ransforms w i t h t h e t h e o r y of Laplace t ransforms given i n Schwartz C11. I n t h i s work Schwartz does n o t d i s c u s s t h e i n v e r s i o n of Laplace t ransform b u t w e cover i t s i n v e r s i o n i n t h e fo l lowing manner.
) fo r t h e domain r where 5 > a n j 1'
L e t V ( z ) be a f u n c t i o n of t h e complex v a r i a b l e z . Then w e c a l l t h e d i s t r i b u t i o n Tx t h e Laplace i n v e r s e (or a n t i - ) t r a n s f o r m of V ( z ) and w e denote it by I L i l V ( z ) if ILzTx = V ( z ) . The main r e s u l t of this s e c t i o n is t h e fo l lowing i n v e r s i o n theorem.
Theorem 8.13.3. I f t h e f u n c t i o n V(z) be holomorphic i n t h e domain il where R e z > 0 (b. = (bl ,b2, ... 'b ) f o r j = l ,21 . . . ,n ) and i f j j 3 n t h e r e ex is t non-negative i n t e g e r s m.and a p o s i t i v e number B such t h a t
(8.13.7)
> b
7 -m n 3
2 -ml -m Izl z 2 , . . . ' z % ( z ) I < B for a l l 1z.I + m w i t h
then K:V(z) e x i s t s i n IDA+ (IR") . s a t i s f i e s
F u r t h e r I L I I V ( z ) i s unique and X
m +2 m2+2 mn+2
1 2 n Dx . . . Dx w(x) (8.13.8) I L i l V ( z ) = Dx 1
w i t h - - - Cl+i- c2+i- cn+im
(8.13.9) w(x) = ( 2 ~ i ) - ~ dzl dz Zr...l I U ( z ) e X Z dzn - - - ~ ~ - 1 - C 2 - i - 5 -i- n
142 Chapter 8
n -m -2 where U ( z ) = V ( z ) II z 1 . Here Ilf;'denotes the inverse Fourier
transform of the variables 17
j=1 j
j' and the numbers 5 are greater than b
j j
Proof. The relation (8.13.9) can be rewritten as -
with
- +i- x z u(z) e n n
(8.13.12) Un(zl,z 2,...,zn) = I dznt 5, > bn- - ~
[,-im
Let and 5; be two numbers such that 5; > FA > bn and let
Un(p; z1,z2, ..., z
of integral (8.13.2) at the points and t i , respectively. Also, - ih, Q ' = 5' + ih, PI' = c: - ih, let the four points PI = 5;
Q" = 6" + ih be in the complex zn plane and denote
- ) and Un(tTr; zl, z2, ..., znml) denote the value
n-1
n
n
I1 = I G(z)dzn, I2 = I G(z)dzn, I3 = I G(z)dzn, I4 = I G(z)dzn Q'Q" P"Q" P"P'
x z P ' Q ' where G(z) = U ( z ) e n.
Now, by applying Cauchy's theorem to G(z) along the contour
P 1Q1Q'8P l lP we obtain
(8.13.13)
Since I G ( z ) I < B e
I4 + 0 with 5; > 5: by letting h -+ -. ( 8.13.13) that
I1 = I3 - I2 - 14. xnc; - 2
z on Q'Q" and P"P', one can get I2 + 0 and n Hence, we may infer from
lim I1 = lim I3
Therefore, we may conclude that Un(zl,z2,...,z ) does not depend
upon the 5, and consequently by (8.13.11) w(x) does not depend sn. In a similar manner we can show that w(x) does not depend upon the
choice of ? which justifies its notation.
n-1
Thus, we may infer that w(x) is a function of x only j'
NOW, we have to show that Un(z1,z2,...,zn-l) = 0 when xn < 0;
this means that xn = -(xnl. For this purpose, let y denote the arc
Laplace Transform 143
of circle whose center is the origin and extremities are the points
P = 5 -ih and Q = En+ih, with in > sup(b,O).
Cauchy's theorem to G ( z ) along the path PQyP, we obtain
Then, by applying n
x z -Ixnlzn (8.13.14) / U(z) e "dz, = - 1 U ( z ) e dz,.
PQ QY P By the hypothesis made on V(z) there exists B' such that lu(z) 1 <
B' Iznl-' if Re zn 2 5,. right side of (8.13.14) tends to zero as h -+ -. Also , the left side
of (8.13.14) tends to Un(z1,z21...,zn-l) by means of (8.13.12). This
enables us to conclude that Un(z1,z2,...,zn-l)= 0 if x
consequently w(x) = 0 for xn< 0.
w(x) = 0 for all x < 0. From this property of w(x) we may infer
that its support is contained in Qn(0) where x
Finally, by (8.13.8) we may conclude that the support of IL-lV(z) is
contained in Qn(0) for all x and hence ILx V(z) E IDA+ (IR ) .
- If zn is on QyP then one can show that the
< 0 and n Similarly, one can show that
> 0 for all j . j
j -
-1 nx j
If we put z = 5 . + 2nin in (8.13.9) then we obtain (8.13.10).
Now, we have to show that ILzILx V(z) = V ( z ) is true if IL;'V(x) is
given by (8.13.8). This leads us to verify that
1 1 j -1
m +2 m2+2 mn+2
x1 x2 n IL D D ..... Dx w(x) = V ( z )
or n m.+2 11 z J ILzw(x) = V(Z)
j=1 j n m .+2 n (~.+i2*a.)~ I L ( ~ + ~ ~ ~ ~ ) w(x) = V(E+i2nq).
7 (8.13.15)
Since the support of w(x) lies in Qn(0), we have
j=1 J
m m m -xS-i 2 nxn w(x) = / dxl 1 dx2.. . . . / dxn e w (XI
0 0 0 IL (~+i2nq)
= IF e-X'w(x) = I F I F - ~ U ( E + ~ ~ ~ ~ ) = ~(5+i2aq) n n x by (8.13.10). Hence we obtain (8.13.15) in view of the definition
of U(z) . This proves our theorem.
When n = 1, the present theorem reduces to Theorem 8.7.1 with
c' = 0.
8.14. Bibliography
Complete and partial work on the Laplace transformation of
144 Chapter 8
distributions and pseudo functions can also be found in the following
references.
Benedetto [11 , Churchill [I], Colombo and Lavoine [l], Ditkin and Prudnikov [11, Doetsch c21, Erdelyi 111, Erdelyi (Ed.) [2l, Vol.
1, Garnir and Munster [l], Ghosh El1 , Jones [I], Korevaar C11, Krabbe c11, Livermann c11, Mikusinski [ 2 3 , Silva [ 3 J , Vander P o l and Bremmer c11, Zemanian Clland C31.
Footnotes
A
other notation : T] T ( z ) . Often one employs p in place of z .
(8.2.5) shows that the assumption T belong to ID; is not
necessary in order that its Laplace transform exists, for example
ILe-X = T-% e22/4 but by restricting the Laplace transform to
3p l , the theory is more coherent. T ( z ) is then an entire function.
or an extension of Fubini's theorem (see Section 5.8.2 of
Chapter 5) .
the series of moduli is equal to vx-'i-'Jv (ix) = vx 'Iv (x) and
other notation : v ( z ) [T . in order for c' to become the lower bound for the support, we
ought to complicate uselessly our assumptions, but the calcula-
tion described by (8.7.1) determined exactly the support of anti-
transform of v ( z ) . when necessary one can apply the rule (8.5.2).
it can happen that h(a;x) is not continuous with respect to x.
2
- Iv(x) is equivalent to (2nx) -'I2 ex as x + a.
(10) we can reduce to this case by translation; See rule (8.5.2).
CHAPTER 9
APPLICATIONS OF THE LAF'LACE TRANSFORMATION
Summary
As pointed out in the previous chapter, we show how the distri-
butional Laplace transformation permits a great flexibility in
numerous applications. For this purpose we give some of these
applications, and in particular we apply the Laplace transformation
to convolution equations, difference equations, differential and
integral equations. Moreover, this chapter applies the Laplace
transformation to Green's functions and partial differential equat-
ions, including the heat equation, the wave equation, and the
telegraph equation. Further sections construct series and asymptotic
expansions. Finally, this chapter uses the Laplace transformation
to consider derivatives and anti-derivatives of complex order.
We treat each application briefly, then give some examples which
should illustrate sufficiently how to tackle other problems in the
same category.
9.1. Convolution Equations
We have seen the importance of convolution equations and their
fundamental solutions in Sections 5.8.6 and 5.8.7 of Chapter 5. Using
formula (5.8.5) of Chapter 5, this section shows that the Laplace
transformation offers an effective way to solve these equations. We
shall see that such equations, in this general context, include many
problems of more familiar types.
Let U and V be two given distributions belonging to ID:. We X X
seek a distribution Xx E ID: such that
(9.1.1) u * xx = vx. X
Furthermore, assuming that these three distributions are Laplace
145
146 Chapter 9
transformable, we put
A
U ( z ) = l L U , V(Z) = ILV, X ( Z ) = I L X .
Now we obtain according to (8.5.8) of Chapter 8,
A A .L
(9.1.2) U ( Z ) X ( Z ) = V(z)
and
= .G(z)/G(z)
in a certain half plane Re z > 5 . Hence, by inversion
(9.1.3) xx = IL-lG(z)/i(z)
A A
provided that V(z)/U(z) has an inverse transform. Consequently, by
virtue of Section 5.8.4 of Chapter 5, the convolution in ID: does
not admit any divisor of zero.
Fundamental solution
This implies that Xx is unique in ID;.
If
(9.1.4) Ex = c1 l/j(z) , the equality (9.1.3) can be written
an expression which no longer requires that Vx is Laplace transform-
able or lie in ID:.
of Chapter 5.) Accordingly, E is the fundamental solution of the
equation U * X = V and we conclude that the Laplace transformation
is an effective tool to find it. (see also the method for the
resolvent series in Section 9.4.24
(Evidently, Ex * Vx must exist; see Section 5.8.;
9.1.1. Examples
The convolution equation
-1 2 x+ (9.1.5) Fp x+ * X = log
is mapped by the Laplace transformation into
A (log CZ) 2+7r2/6 Z
-(log CZ) X ( Z ) =
Applications 147
where C = 0,577.... is Euler’s constant. From this we deduce
Hence, by inversion
2 x = log x+ - + V(X/C)+
where
where I’ denoting the gamma function. The fundamental solution of
(9.1.5), according to (9.1.4) , is given by
where m t+a
dt. X v ( x ; a ) = J r(t+a+l)
2 . The difference equation
can be rewritten as the convolution equation
- [ G(x-a)-k G(x-b) 3 * Xx - Vx.
Its solution according to (9.1.3’) is X
tal solution E is given by
= E * Vx, and the fundamen- X
E = L-l[ e-az - ke-b7
according to (9.1.4). But we have
This series converges uniformly in the half-plane Re z > . NOW, we conclude by inversion that the series
m .
E = 1 k’G(x-[jb-(j+l)a]) j = O
converges in ID;.
a solution if V belongs to DD;.
Consequently, we may infer that (9.1.6) will have
Accordingly, the solution is
m . - k’vx+ (j+l) a-jb. xx = E * vx -
j =O
148 Chapter 9
3 . The differential - difference equation - (9.1.7) Xx-a - k DXx-b - Vxi b > a,
is also the convolution equation
[6(x-a)-kat(x-b)] * Xx = Vx.
Its fundamental solution according to (9.1.4) is given by
E = n-l[e-az - k~e-~']-' . But, we put
provided that
This problem does not satisfy the convergence criteria of
Corollary 8.8.1 of Chapter 8, but these are not necessary conditions,
hence we invert term by term and we obtain m
E = 1 kJ&(')(x-jb+ja+a). j=O
This formula is exact, because we verify that this series is
convergent in ID; and that its convolution with 6 (x-a) -k6 (x-b) is
given by 6 (x) . Therefore, Xx = E * Vx is the solution of (9.1.7).
9.2,Differential Equations with Constant Coefficients
In the space of distributions, the derivative must be replaced
by the distributional derivative and, consequently, differential
equations by "distribution-derivative equations". First we consider
these broader problems: then we treat more familiar differential
equations.
9.2.1. Solving distribution - derivative equations
Let Vx be a given distribution belonging to ID;, and let coIcl,
... ,C derivativ? equation with constant coefficients has the form
be given constants with cn # 0, n 2 1. A distribution- n
Applications 149
It has a unique solution in ID: given by
(9.2.2) XA = E(x)
where
* vx
n
j=O p(z) = 1 CjZj.
Also, the equation (9.2.1) has many solutions in ID’ and these
solutions are of the form
(9.2.3) Xx = Xi + h(x)
where the function h(x) is any solution of the equation
(9.2.4)
To show (9.2.3) , we note that according to ( 5 . 8 . 8 ) of Chapter 5, (9.2.1) can be written in the convolution form
cnh(”)(x) + ... + c 1 h’(x) + coh(x) = 0.
It is evident that (9.2.1) has many solutions which have the same
support as that of V.
Therefore X‘ c ID; and satisfies
n (9.2.1’) [ cj S(j)(x)l * X’ = V.
The convolution algebra constructed over ID: has no divisor. Hence
we conclude X’ is unique.
Let its one.solution be denoted by X!
J=o
In addition X-XI is the solution of
C f c.Dj I(X-X’) = 0. j=o J
Hence (Schwartz Ell , Chapter V, 6, Theorem IX) X-X’ is equal to h(x) , which is an infinitely differentiable function with an unbounded
support and satisfies (9.2.4). By employing Laplace transformation
on (9.2.1’) and continuing the same processes to that of Section
9.1, we obtain
X1 = E(x) * Vx. X
Hence the results (9.2.2) and (9.2.3) are established.
150 Chapter 9
As usual, let V ( z ) denote the Laplace transform of V. Then
according to (9.1.3) XI can be written in the form
(9.2.5) X' = C1v(z)/p(z).
Because p(x) is a polynomial, the fundamental solution is a
function, and this justifies the notation E(x). Further, E(x) has
continuous derivatives upto order (n-2) and a derivative of order
(n-1) which is discontinuous at x=O. Indeed, according to Theorem
8.7.1 of Chapter 8 if 0 (q 211-1, we have
(see also Vo-fiac-Khoan [ l l , p. 109.) Moreover, E(') (0) = 0, 0 5 q 5 n-2.
Calculation of E(x)
If the polynomial p(z) has n distinct roots (real or complex)
rk, k = 1,2,...,n, and if ak = cel II (r -r )-', we have n jfk k j
and
Consequently, according to (9.2.2)
(9.2.6) n
X4 = 1 k=l
r x akVx * e+k .
If some locally summable function v(x) is a representative of V(x) and some real number a is the lower bound of its support, then we find,
by (5.8.1) of Chapter 5, a well known result
n r x --I: t X' = 1 ak e v ( t ) e dt, for x > a
k=l a
X' = 0, for x < a .
If V = 6(k) (x) , q 5 n-1, then XI is represented by a function. if v = 6 ( q ) (x ) with q 2 n, Xi is no longer represented by a function since E (n-l) (x) is not continuous.
But
When p(z) has one or several multiple roots the operations are
more complicated. For instance, take the case where
Applications 151
Hence, putting b = (r-s)-', we find
E(x) = b [xerx - b(erx - eSX,3 + -
It is evident that E(0) = 0 and
E1(x) = b [r+erx + erx - b(rerx - seSX)]+,
E'(O+) = b [l-b(r-s)] = 0;
hence E1(x) is also a continuous function. If a function v(x)
represents V, as above, then, combining (5.8.1) of Chapter 5 with
(9.2.2), we find a well known result X
XI = b(x-b)e rx /v(t)e-rtdt + b2esx v(t)e-stdt - a a
X berX j tv(t) e-lrtdt.
a
9.2.2.Solving traditional differential equations
1. We seek a function f(x) determined by the equation
n cnD f(x)+ ...+ clDf(x)+c f ( x ) = g(x) , n 2 1, 0
where D denotes, as always, the distributional derivative and g(x)
is a given function.
2. Moreover, we impose a set (C) of conditions. According to
(9.2.3)
f (x) = E(x) * g(x) + h(x)
where the function h(x) is still a solution of the equation (9.2.4),
but it is no longer indeterminate that it is determined by the
conditions (C). Also, suppose that g(x) admits the convolution with
E(x)
More frequently, the following cases occur:
152 Chapter 9
9.2.3.Single differential equations (Cauchy problems)
Let f(1) (x) denote the ordinary derivative of order j. We seek a function f(x) satisfying the equation
(9.2.7)
and the conditions
(9.2.8) f(x) = 0, x < a
cnf(%x) + ... + clf'(x) + cof(x) = g(x), x > a,
f(a+) = woI
f(j)(a+) = wjI j = 1,2,...,n-l, (9.2.9)
where c
contained in [a,-[ . Then this problem has a a,wj are given numbers and the function g(x) has support
j'
by the formula
X n- 1
a q=o (9.2.10) f (x) = I E(x-t) g(t)dt + 1 Q
where E(x) is a function of Section 9.2.1 and
unique solution given
~(9) (x-a)
n Q = c cjwj-q-l' q j=q+l
We now verify our contention that (9.2.10) is the solution of
(9.2.7).
- Proof. According to ( 5 . 4 . 3 ) of Chapter 5, we have
f'(x) = Df (x) - wo6 (x-a) ,
NOW, the equation (9.2.10) takes the form
n n-1 1 c.DJf = g(x) + 1 nq 6(q) (x-a) = Vx (say) , on lR. (9.2.10 ' )
j = o J q=o
A l s o , according to Section 9.2.1, its solution is
f(x) = E(x) * Vx
because f(x) E ID' , and consequently f(x) takes the form n-1
f(X) = E(x) * g(x) + 1 f2 6'q)(x-a) * E(x). q=o q
Hence,we obtain the formula (9.2.10) ; because E(x) is a function
whose first in-2) derivatives are continuous.
Applications 153
* If the Laplace transform g ( z ) of g(x) is known, then E(x)*g(x)
is sometimes more easily obtained by using the formula
(9.2.11) E(x)*g(x) = I?i(z)/p(z).
Remark. If one of the w is not given and if on the other hand
we impose on f(x) (or one of its derivatives) a condition at
z = z > a, then we replace w by a parameter which will be finally
determined by (9.2.10) and this condition at zo.
j'
0 9
with the conditions
1' f(x) = 0 for x < 0, f(O+) = wo, fl(O+) = w
2 -1 Solution. Here p ( z ) = z + A 2 . Hence E(x) = A sin Ax+. As
ILeiVX = (z+v)-' and
(9.2.11) gives
x - e VtE(x-t)dt = 2-2 1 [ e;"'-(D-V) A-1 Sin Ax,]. 0 v + A
- On the other hand, no - wl, Ql = w * hence according to (9.2.10)
we have 0'
+ ;i- sin Ax, for x > 0.
2. Solve
with the conditions
f(x)=O for x < 0, f(O+) = w0, f'(n/A) = w i .
1 as Solution. Returning to the preceding example, consider w
a parameter in (9.2.12) which is determined by using the condition
fl(n/A) = w i in the form
154 Chapter 9
- (e-vn/A++l) - w1 = w i , .2-_>
-1 -v"/h+l) Then it suffices to replace w1 in (9.2.12) by -wi-v(v2-X2) (e
3. Solve
f"(x)-3f1(x) + 2f(x) = x for x > 0
with the conditions
f(x)=O, for x < 0, f(O+) = w0, f'(o+) = wl.
Solution. Here p(z) = Z2-32+2 = (2-2) (2-1) and
1 - 1 1 m-2-2-xi
~ ( x ) = e y - e: . hence
- AS fto = -3w +w 1, ftl - too, (9.2.11) gives
1 X 3 f (x) = ( wl-wo+a) e y - ( wl-2w0+1) e+ + (x/2+7) u (x) . 9.2.4. Systems of differential equations
In the preceding sections, we have seen that the Laplace
transformations changes a differential equation with constant
coefficients into an algebraic equation. It transforms similarly a
system of differential equations into a system of algebraic equations
and often offers an advantageous way of solution. The following
example illustrates the method.
Let f(x) and g(x) be two functions satisfying the system
f"(x) - g(x) = ax+b
for .x > 0 (9.2.13)
f'(x) - g'(x)+f(x)-g(x) = 2b,
f(x) = g(x) = 0 for x < 0
f(O) = 0, f'(O+) = 1, g(o+) = 1.
According to (5.4.3) of Chapter 5, we have
n
f'(x) = Df, f"(x)=D'f-S(x) and g' (x) = Dg-6 (x) ,
Applications 1 5 5
the system (9.2.13) can be written in terms of distributions as
D'f-g = (ax+b)+ + 6(x), for x E IR.
D (f -9) +f -g = 2bU (x) -6 (x)
The new system is changed by the Laplace transformation into the
algebraic system
2 2 Z F(z)-G(z) = z-2(a+bz+z )
(z+l) F ( z ) - (z+l) G ( z ) = ~ ~ ~ ( 2 b - z )
where F(z) = ILf, G ( z ) = ILg, Re z > 1. We define
bz+a+zL + 2b-z F(z) = - z (z2-1) ( z + l ) z2 ( z2-1)
Hence, inverting the last equation and recalling that 6'(Z2-1)-' = cash X+ and IL -1 z(z2-1)'l = sinh x+, we obtain
1 2 f = ~(2b-D) ( 2-e-x-xe-x-cosh x) +- (a+bD+D ) (x-sinh x) +
which gives
f(x) = b-a~-[(b+~)x+b] 1 e-x(a+T)sinh 3 x, x > 0.
By the first of the equations (9.2.13) we further have
, x > 0. g(x) = -ax-b- [ (b+-)x-b-l] e-x+(a+3)sinh x
To appreciate the value of the Laplace transformation we observe
that the solution of the system (9.2.13) by the direct method leads
to the third order differential equation
1 2 T
fBB'(x) + fBB(x) - fB(x) - f(x) = ax + a-b.
9 . 3 . Differential Equations with Polynomial Coefficients
The Laplace transform of a "distribution-derivative" equation
with polynomial coefficients is an algebraic equation which is again
a distribution derivative equation. But the new equation may have
a simpler form, for example, it may have lower order. Then the
transformation yields advantage. We see this point in the following.
156 Chapter 9
9.3.1. Reduction of order,
Consider the equation
19.3.1)
where Xx is a unknown distribution, Vx is a given Laplace transform-
able distribution, and the p. (x) are polynomials where pn(x) is not
identically zero. 3
By employing the Laplace transformation on(9.3.1) and putting A * X ( z ) = I L X and V(z) = lLVx we obtain
(9.3.2)
and the equation is valid in a certain half plane Re z > 5 .
If the highest degree of all the Polynanials p . ( . ) is m < n, the 3
equation (9.3.2) is of order 5 m, which is lower than that of equation (9.3.1). Therefore, there is a reduction of order.
Still, this reduction, except in special cases, may insufficient-
ly offset the increased complexity of the transformed equation.
However, as we see in example 3 given below, the Laplace
transformation is a new efficient way to discover distributions
having point supports which are solutions of (9.3.1) and which are
obtained with difficulty by the direct method.
Example. 1. Consider the second order equation
2 2 -x 2 (9.3.3) xf" (x1-f' (x)-af(x) = a xe-X = a xe -ax , for x > 0,
with the conditions
f(x) = 0, for x < 0,
f(O+) = 1, f'(O+) = -1.
Solution. Here,
f'(x) = Df-G(x)
and f"(x) = D 2 2 f-6 (X)+6(X).
Making use of x 6 ' = -6, x6 = 0, (9.3.3) takes the form
Applications 157
2 2 (9.3.4)
which is an equation in the sense of distributions in ID;.
xD f - Df - af = [a xe-= - ax2 ]++ 26(x)
By employing the Laplace transformation on (9.3.4) and putting
F ( z ) = Z f , we have
z ~ ' ( 2 ) + (32+a)F(Z) = 2a - - + 2 2 2 a
7 (z+a)2
which is a first order equation whose solution is
2 A + a k 7 eaz z+a F ( z ) = 3 +
Z Z
where k is temporarily arbitrary but can be found by the condition
f'(O+) = -1. Hence, inverting by a formula of Erdelyi (Ed.) [21 , Vol.1, 245 (35), (where 12(.) being the modified Bessel function
of order 2) we have
2 f (x) = x + kx 12(2E) , for x > 0.
And finally the condition f'(O+) = -1 implies k = 3/2.
2. We obtain the distribution X E ID; satisfying the second
order equation
2 2 (9.3.5) xD X - DX - ax X = b6'(x).
By employing the Laplace transformation on (9.3.5) and putting L.
X ( z ) = ILX, (9.3.5) is transformed into the first order equation
whose solution is
-b 2 2 -3/2 3 $ ( z ) = - + kl(z -a )
where kl is arbitrary.
Hence inverting by a formula of (Erdelyi (Ed.) C21, Vol.1, p.239
(19) , or Colombo and Lavoine C11 , p. 98) , we have
X = -b 6(x) + kx Il(ax)+ 3 (9.3.61
where k is arbitrary.
Next, we seek the solution of (9.3.5) in ID'. We put X = xY,
158 Chapter 9
which enables us to write (9.3.5) as
2 2 1 x [D Y + - DY - (a2+$)Yl = b6'(x) X
X
in ID'. Equating the square bracket to zero, we obtain the
modified Bessel equation of order 1, whose general solution is
Y = k I (ax) + k K (ax), x # 0 (see Watson Cll), we thus have
X = - b-S(x) + kxIl(ax)+ + klxIl(ax) + k2xKl(ax)
1 1 2 1
3
in ID'.
3. Consider the second order equation
2 (9.3.7) x D X + (x+ 3) D X + X = 0
n
First we find its solution in a);. Putting X(z) = I L X , we are led to the first order equation
2 - 6
(z +z) X'(Z) - ZX(2) = 0
whose solution is
* X ( z ) = k(z+l).
Hence with k arbitrary
(9.3.8) X = k6 (x) + k6 (x)
in ID;. Now we obtain the solution of (9.3.7) in $' by making use
of the Fourier transformation. If W denotes the Fourier transforma-
tion of X, then Section 7.13 of Chapter 7 transforms (9.3.7) into
C(2inF + 1) DW - 2incW = 0
in $'.
Section 6.1 of Chapter 6 that Dividing by C , we have (kl being arbitrary) by making use of
(2in5+1) DW + 2irW = 2nik16(5)
whose solution is
W = klinc(E) (2irE+l) + k(2ir5+1)
where
Applications 159
Performing the inverse Fourier transformation on W (by Formulae of
Lavoine C61, p. 85) we have
(9.3.9) -
X = kl (x 2-x-1) + k[ 6' ( x ) + 6 (x) 1
We observe that (9.3.9) contains the solution (9.3.8). But
through the general theory of differential equations of second order
we know that for x # 0, the solution of (9.3.8) depends on two independent functions. As only one function appears in (9.3.9), we
conclude that a non tempered distribution is associated with the
other function. If X denotes this non tempered distribution, then it satisfies
1
xD2X1 + (x+3) DX1 + X1 = 0.
We attempt to take X1 = e-bxY, where we must determine the nonzero
number b and the tempered distribution Y. Here also, Y is associated
with the equation
xD2Y - (2b-1)xDY + 3DY + b(b-1)xY - (3b-l)Y = 0.
If 2 denotes the Fourier transform of Y, then this equation becomes,
according to Section 7.13 of Chapter 7,
(2in5-b) C 2i~c-b+l] DZ -2in(2inE-b)Z = 0 5 5
which gives after division by (2in5-b) (this factor does not vanish
for any real 5 )
(ZinC-b+l)DZ - 2irZ = 0. 5 5 (9.3 .lo)
if b # 1, we get
Z = k(2inE-b+l)
where k is arbitrary. Hence by means of the inverse Fourier
transformation
Theref ore
(9.3.11)
160
I f b = 1
Chapter 9
Z = Pinkc + k2ins(c)(2inc)
is the solution of (9.3.10). Hence by inversion
-2 Y = k&' (x) + k2Fpx
and consequently,
(9.3.12)
We observe that (9.3.11) is not different from (9.3.8). But (9.3.12)
is not contained in (9.3.9). Adjoining these two formulae, we have
the general solution of (9.3.7) in ID' which is
X1 = e-XY = k2Fpxm2 e-x + k[&'(x)+S(x)l,
(9.3.13) X = k[&'(x)+&(x)] t klFp(x-1-x-2)+k2Fpx -2 e -x . The last result has been derived by Bredimas [ a ] , pp. 339-340,
in a quite different and original way.
We remark that the usual theory of differential equations gives
a general solution for equation (9.3.7) involving linear combinations
of two independent functions. But we find that the theory of
distributions yields a general solution for the same equation
requiring combinations of three independent distributions.
9.4. Integral Equations
We distinguish two well known types of linear integral equations
by describing their upper and lower limits of integrations. A
Fredholm equation has the form
b f(x) = F(x) + X I K(x,y) f(y)dy
where F and K are given functions, X a,b are finite constants, and f(x) is the unknown function. If the upper limit is the variable x
rather than a constant, then the equation takes the form
a
X f(x) = F(x) + X /K(x,y)f(y)dy
a
and this is called a Volterra equation of the second kind. using
the Laplace transformation, this section will solve some volterra
equations.
Applications 161
9.4.1. Special Volterra equations (1)
Consider the "problem" to determine the function f such that
(9.4.1) k(x-t)f(t)dt + Af(x) = h(x), a > 0 a
(9.4.2) f(x) = 0, x < a
where the numbers a and A are given, the function k(x) is null if
x 0 and Laplace transformable, and the function (or distribution)
h has support in C a,=[. Note that (9.4.1) can be written in the
f orm
Ck(x) + AS(x)l * f(x) = h(x)
in ID;.
According to Section 9.1 (see remark 9.4.3), f is unique and
given by
(9.4.3) f(x) = E * h(x)
where
-1 1 E = Z - k(z)+A
with
k(z) = ILk(x).
If h(x) is Laplace transformable and if H(z) = ILh(x), (9.4.3)
can be presented eventually in the simple form
(9.4.4)
Note that the fundamental solution(*) is not necessarily
representable by a function.
Examples. 1. Solve
f(t)dt + Xf(x) = h(x), x > a a
f(x) = 0, x < a.
1 Solution. Here k(x) = U(x) (Heaviside-function), K ( z ) = z, and
1 6 2
Thus
Chapter 9
and further, if h(x) is a locally summable function whose support
is bounded below by a, we have
( 9 . 4 . 5 ) f (x) = X-’h(x) - A - 2 I h(t) et/’dt. x
a
2. Solve
-2 -vx
f ( x ) = 0, x a.
X (9.4.6) 1 (x-t)e-vtf(t)dt + w e f(x) = h(x), x > a ,
a
Solution. After multiplication by evx, (9.4.6)takes the form
[xe: + w - ~ ~ ( x ) ] * f(x) = evxh(x).
vx Here k ( x ) = xe+ , K ( z ) = ( z - v ) ’ ~ and
4 1 -1 2 - E = =-I w ( 2 - v ) = JL L w
2 2
( 2 - v ) 2+w2 ( 2 - v ) 2+w2
3 vx = w2.s(x) - w e f1 -2+ z + w
evx sin wx+; 2 = w &(X) - w
hence
f(x) = w evxh(x) - w 3 [ evx sin ax]+ * [ xvxh(x)]
evx Ch(x) - w sin wx * h(x)l + = w
by ( 5 . 8 . 1 0 ) of Chapter 5 .
If h(x) is a locally summable function whose support is bounded
below by a , we have
X sin ax+ * h(x) = 1 h(t) sin w(x-t)dt.
a -2 -2 If we replace w by - w in ( 9 . 4 . 6 ) , we get
f (x) = - w 2 evx C h(x)- w sinhwx, * h(x)l . 9 . 4 . 2 . Resolvent series
If k(x), in the preceding section, is not only Laplace transfor-
Applications 163
mable but also a locally summable function (and still has support
in CO,-[) then Theorem 8.3.3 of Chapter 8 tell us that IK(z)/hl 1
when Re z is large enough say when Re z>E1. It follows that
1’ The convergence of this series is uniform in the half-plane Re
We can invert term by term and obtain according to Corollary 8.8.1
of Chapter 8 that
(9.4.7)
where
-1 j k.(x) = IL K ( 2 ) ; 7
that is
kl(X) = k(X) r
k2(X) = k(x)*k(x) = 1 k(x-t)k(t)dt,
kj(x) = kj-l(~)*k(x) = k(x-t)kj-l(t)dt.
X
O x
0
The series (9.4.7) is called the resolvent series and it converges
in ID’. Now (9.4.3) takes the form
(9.4.8)
(See also Lew [11 .I
m
f (x) = X-’h(x) + 1-l 1 ( - A ) - ] kj(x) * h(x), j =1
If h(x) is a locally summable function whose support is bounded
below by a , we get m . x
f(x) = A-‘h(x) + X-’ 1 ( - A ) - ’ I kj(x-t)h(t)dt j =1 a
which is the usual solution of the Volterra type equation (9.4.1) in
the sense of functions.
This method of resolving series can be utilized to solve a
convolution equation of the type
u * x + AX = [U+A6(x)] * x = v
in ID; if there exists a half plane Re z > 5 in which 1 U ( Z ) I < (See also Yosida [ 21, Chapter VIII.)
1 .
164 Chapter 9
9.4.3.Remark on uniqueness
In the absence of the condition (9.4.2), if the support of f is
not bounded below and if (9.4.1) is replaced by
X I k(x-t)f(t)dt + Xf(x) = h(x), -a
then the solution is not unique. Accordingly, we have
(9.4.9) f(x) = E * h(x) + P(k:x)
where P(h:x) is an arbitrary linear combination of the solutions of
the equation
X I k(x-t)p(t)dt + Xp(x) = 0.
-a
These p(x) are eigen functions of the convolution operator k(x)+.
Of course, if there is only one solution p(x) , then P(Aix) = c p(x) , where c is an arbitrary number.
Examples. Consider the equation
X i f(t)dt - w-lf(x) = h(x) -m
where w > 0 and h(x) is a locally summable
bounded below by a. According to (9.4.9)
is
function whose support is
and (9.4.5) the solution
X f (x) = -wh(x) - w2ewx I h(t) e-wtdt + ceWX
U
with c arbitrary: indeed, ewx is the solution of
- X p(t)dt - w 'p(x) = 0.
-m
9.4.4. Integral equations with polynomial coefficients
Consider the equation
X n (9.4.10)
with f(x) = 0 if x c a and cn # 0.
k(x-t)f(t)dt + 1 c.xjf(x) = h(x) U j=o J
If F ( z ) = ILf(x),K(z) = ILk(x), and H ( z ) = ILh(x), (9.4.10) is
changed by the Laplace transformation to
n 1 (-l)JcjF(J\z) + K(z) F ( z ) = H ( z ) .
j = U
Applications 165
This is an ordinary differential equation of order n whose domain
is certain half-plane Re z > 5 and whose solution is easier than the
solution of (9.4.10) . We remark here that the transformation is disadvantageous when
k(x) is a polynomial of degree m n-1; because one can obtain a
differential equation of order (m+l) < n by differentiating (9.4.10),
(m+l) times.
Example.1. Find the solution of
(9.4.11)
and (Erdelyi (Ed.) C 2 I, Z Solution. We have IL cosh x+ = l5 Vol. I, p. 239 (13))
(9.4.12) lLXVIY (x) = 2 v (v++ 7T-l/* (z2-1) -v-1'2.
Using these results, we can rewrite (9.4.11) in the form:
(2v+1) z v 1 -1/2 2 -v-1/2 F'(z) + '7 F ( z ) = a 2 (v+T)~T ( z -1) I
z -1 and whose solution is given by
with c an arbitrary number. Hence, using (9.4.12) to perform the
inversion, we obtain
(9.4.13)
Consequently, (9.4.11) has the solution
f(x) = [as'(x) + CS(X)I * X'I~(X)+.
f (XI = xVCaIv-l (XI + CI,, (XI I,.
f = aI1 (x)+ + a6 (x) + cIo (XI+ . If we had put v = 0, this last formula would yield
This is no longer a function but it still satisfies the convolution
equation
cosh x+ * f - xf = a1 (x)+ 0
which can also be obtained by taking v = 0 in (9.4.11). Indeed, the
initial form of (9.4.11) no longer holds in the sense of functions
since its first member vanishes at x = 0; whereas Io(0) # 0. other words, the equation
In
X a V (9.4.14) coshx+ * - 2v+T = 2v+l IV(X)+
166 Chapter 9
can be written in the form (9.4.11) only when v S O . If v < -1 and
v is not an integer, then we replace X'I~(X)+ by FP X'I~(X)+. Hcnce,we
conclude for every v # 0, -1, -2,...., that (9.4.13) is the solution
of (9.4.14).
9.5, In- - Differential Equations The aim of this section is to solve integro-differential
equations by means of the fundamental solution and Laplace
transformation.
Let us determine the function f such that
X n k(x-t)f(t)dt +
a j=O (9.5.1) cjf(j) (x) = g(x) for x > a
(9.5.2) f ( x ) = 0, for x < a
0 f (a+) = w
(9.5.3)
given the numbers a , w . , c
able function k(x)+, and the locally W l e function g(x) wia support in [a,$ .
f(j) (a+) = wjl 1 5 j 5 n-1
(cn # 0, n 1. l), the Laplace transform- 1 1
This problem has a unique solution given by the formula
X n- 1
a q=o (9.5.4) f(x) = E(x-t)g(t)dt + c Qq E(q) (x -a ) , x > a
where
and n
We now verify our contention that (9.5.4) is the solution of
(9.5.1).
Proof. By the conditions imposed on f, the equation (9.5.1) can - be written
n n- 1 X
I k(x)+ + 1 c.6") (x)l * f(x) = g(x) + 1 ng6(q)(x-a) = V j=o J q= 0
Applications 167
in ID:.
equation is
According to Section 9.1, the unique solution of this
(9.5.6) f(X) = E(x) * Vx B
E(x) is given by (9.5.5). But k(x)+ is locally sumable, so that
k(z) + 0 as Re z -+ a; and
One can show as in Section 9.2.1 that E(x) is a continuous function
having (n-1) derivatives in the sense of functions, of which the
first n-2 are continuous. It follows that (9.5.6) takes the form
is of the same order as z'~. 1 K(z)+p(z)
(9.5.4).
Example. Consider the equation
X (9.5.7) a J f(t)dt + cf(x) + f'(x) = g(x)
where g(x) is a locally summable function which is null for x < 0
but it satisfies the conditions:
0
f(x) = 0, for x < 0
f(O+) = W.
a Here n = 1, no = W , k(x) = aU(x), K ( z ) = z , and
Z k E(x) = If1[: + c + z 1-l = z1 - 1 ' &'(x) * f l ( 1 - 1
2 - A FF-1 - x - A
2 where A and A' are the two roots of the polynomial z +cz+a. We have
1 X'X E ( X ) = &'(XI * ( e y - e+ I
- A Ax A ' A'x - = e + - m e +
and (9.5.4) gives
X A f(x) = eAx C w + e-Atg(t)dtl - 0
A't X A eAIx C W + e y(t)dtl x > 0.
0 A-h'
Particular case
1. If c = 0 and a = - v 2 , then A = v , A' = - v , and
X f(x) = g(t)cosh v x dt + w cosh V X , x > 0.
0
168 Chapter 9
2. xf a = h 2 and c = -21, then h is the double root of the 2 polynomial z2-2hz+h2 = ( z - h ) . We have
AX = (Ax+l) e+
and (9.5.4) gives X
f(x) = elx [ W+ (hx-At+l) .-It g(t)dtl , x z 0. 0
Remark. The method is still valid if the initial conditions
(9.5.3) are replaced with other suitable conditions. The following
example permits us to understand easily how it can be adapted.
Consider the system
X (9.5.8)
(9.5.9) f(0) = 0
a2 I f(t)dt - f'(x) = -h(x) a
where h(x) is a locally summable function having support
a 5 6 < 0 B ' - < - and a is bounded.
[ B I B ' 1 ,
Denoting by w the value (temporarily unknown) of f ( a + ) , (9.5.5)
gives
and (9.5.4) leads to
X f (x) = u(x-B) J h(t) cosh a(x-t)dt+w U(x-a)cosh a(x-a).
B
If we put
0 A = J h(t) cosh at dt
B
then (9.5.9) requires
9.6. General Concept of Green's Functions
9.6.1. Statement
Let Lx be a differential operator subject to boundary conditions
Applications 169
and L f(x) = h(x) be the differential equationwhere h(x) is a given
function.
the formula for the solution of the'differential equation in the
form of an integral
X We recall that the method of Green's function(3) provides
where g(x,t) is a Green's function of the operator L . X -
If L denotes the adjoint operator of Lx, g(x,t) satisfies the X equation 1 4 '
Ltg(x,t) = 6(t-x).
Indeed, we have
We present here a method which leads to an analogous expression
for f(x) and does not involve the adjoint operator.
9.6.2. Green's kernel
We denote the interval Q < x < $ by Ix and the closed interval - 01 (x 5 8 by Ix. the operator L(x,d/dx) of order N 2 1 is defined by
The symbol Jx denotes a neighbourhood of Ix and
(9.6.1) N
L(x,d/dx)f(x) = k(x) * f(x) + 1 n=l
an(x)f(n) (x)
where k(x) and the a (x) are given functions and the an(x) are n
times continuously differentiable on Jx but, may vanish on Ix. n
The problem which we want to solve is the following:
Find the function f(x) which is N-1 times continuously differen-
tiable on a neighbourhood of [ a , $ ] and which satisfies the equation
and the conditions
(9.6.3)
170 Chapter 9
where the xn’s belong to [cr,BI and may or may not be distinct.
Since the derivatives are continuous up to order N-1, (9.6.2) is
equivalent to
Xf (9.6.4) L(xlDx)f (x) = h(x) , x E I
in the sense of distributions.
Put B
H = h(t)dt = <x(it),h(t)>
where x (It) denotes the characteristics function of It. CL
Now let G for the two variables x and t, be a distribution in Xft -
ID; x ID‘
equation
which is zero on IRt - It and which satisfies the t
and the following two conditions.
n n t 1. on Jxf D: Gxft = yt(x) where y (x) is continuous at x for
almost all t and
(9.6.6) w n -
y n (x ) = X(It)f 0 I n I N-1 t n
2. on J we have G = D:r(x,t) , where (a/ax)”r(x,t) , 0 5 n I N-1, X x,t
are continuous for almost all t and some positive summable function
p (t) majorizes all their moduli on It. n
Then
f(x) = <G ,h(t)> X I t
(9.6.7)
is the solution of the system (9.6.2) - (9.6.3).
If G is representable by a summable function of t on It, then Xlt
f(x) takes the integral form
(9.6.8) f(x) = G’(x,t) h(t)dt.
Then we say that G
%
a
is Green’s(5) kernel of the problem. x,t
We now verify our contention that the function (9.6.7) solves
the system (9.6.2) - (9.6.3).
Applications 171
Proof. The general solution of (9.6.5) depends on at. least N
arbitrary functions tnat will be determined with the help of condit-
ions 1 and 2.
-
According to (9.6.7) and (9.6.5), we have
which implies (9.6.4) and hence (9.6.2).
The derivatives f (n) (x) , n < N-1, are continuous because
= (-l)q I -r(x,t)h(')(t)dt an a axn
which is a continuous function at x.
n
B
r(x,t) = Dx I'(x,t) because of continuity and In addition, - an a xn
f In) (x) = (-l)q I$I'(x,t)h(q) (t)dt a
= <D:D:r(x,t),h(t)> = <D: DZ r(x,t),h(t)>
= 'DE Gxlt,h(t) > = <yn(x) t ,h(t) >.
Therefore, we have by (9.6.6) that
f(n)(~n) = <yi(xn),h(t)> = w n
and the equations (9.6.3) are verified.
Remark I. Actually (see examples) the calculation of H is not
needed.
Remark 11. If (9.6.5) can be written as the convolution equation
( 9.6.9) Ux * Gx,t = 6(x-t)
and if U ( z ) is the Laplace transform of Ux then by putting
G(z,t) = I L G we have x,t
U(z) G(z,t) = e'tz
and hence by inversion
17 2 Chapter 9
Gx,t = t t IL''l/u(Z).
Thus, if y(x) = <'l/U(z) (or, more generally, if Ux * y ( x ) =6(x-t))
then we have
G = y(x-t) + g(x,t) x,t
(9.6.10)
where g(x,t) is the solution of
ux * g(x,t) = 0
such that (9.6.10) satisfies the conditions 1 and 2.
Remark 111. Instead of conditions (9.6.3) we can take
f (X ) = an, 0 5 n 5 N-1, distinct xn n
or
f(n)(xo) = on.
Then (9.6.6) is replaced by
or w n n
Yt(X0) = ff- (It).
More generally, if (9.6.3) is replaced by the conditions
then (9.6.6) is replaced by
&k (9.6.11) nk(Gx,t) = J, (It)
We suppose that the number of operatorsilk and their properties are
such that the system (9.6.11) is compatible and we are able to
determine some or all of the constants which occur in the general solution of the equation (9.6.5).
Applications 173
9.6.3. Examples
We begin with an example which is directly solvable and whose
simplicity permits us to understand the method of Section 9.6.2.
1. Find the solution of the following system:
xf"(x) = h(x) on C-2,31
f(0) = 0, f'(1) = w (9.6.12)
where h(x) has a derivative h'(x) which is summable on C-2,31 . Solution. Put
3 H = I h(t)dt.
-2
The Green's kernel is defined by
= 6 (x-t) , 2 (9.6.13) DXGxIt
(9.6.14) Y , ( W = 0, 0
(9.6.14') y;(l) = ; x(-2 5 t 5 3 ) .
1 1 t We deduce from (9.6.13) and the equality ;;6(x-t) = --6(x-t) that
Next
1 G = Fp T(X-t)+ + CX+ + CIX + C2r x,t
(9.6.15)
where C, C1, and C2 are independent of x.
But C = 0 because G must be a continuous function of x for x,t 1 almost all t.
by (9.6.14) and (9.6.14'1, since
Also C2 = U(-t) and C1 = -Fp $I(l-t)+ x(-2 < t < 3)
G = 0 - Fpl! t U(1-t) + Fp(&)U(x-t)+U(-t). x,t H
x,t One can write G in the following form:
174 Chapter 9
We therefore have the solution of the system (9.6.12):
3 f(x) = < x ( - 2 5 t 5 3 ) GxIt,h(t)> = Fp GxIt h(t)dt
- 2
or explicitly,
0 WX-x ~p I t lh (t)dt + J h(t)dt for x 5 -2
-2 -2
X wx-x I' dt - I h(t)dt for 0 5 X 5 1
X 0 X
WX+X w t - h(t)dt for 1 5 x 5 3 1 0
3 wx+x I 3 w t - I h(t)dt for x 2 3 .
1 0
Remark. Suppose that the problem is replaced by - 1 3 Xf"(x) = h(x) I X E [ -13 I,
f(+) = 0, f'(1) = w .
Then, according to (9.6.15)
(C,= C+C1). The functions of t, C2, and C3 are determined by
G(+,t) = 0
a 3 -1 ax
2 . consider the problem
- G(1,t) = W [ I h(t)dtl ,
1/3
X a2 1 f(u)du-f'(x) = h(x), x E [ a BI I c1 < 0 < B ,
- 0 )
(9.6.16)
f(0) = 0
where h(x) is a summable function on c a , B ] . The Green's kernel G
is null on lRt\ (a
(9.6.17) a U(X) * G(x,t) - Dxg(x,t) = 6(x-t)
(9.6.18) G(0,t) = 0.
x,t t < 6 ) and defined by the system
2
Applications 175
If Gl(xIt) is a particular solution of (9.6.17) having a Laplace
transformation G1 (z ,t) , (9.6.17) is transformed to
which gives
G1(xIt) = -U(x-t)cosh a(x-t).
On the other hand
a2 g(u,t)du - g;(x,t) = 0 -m
has the solution
where C(t) is an arbitrary function of t only. Consequently,we have
G(x,t)=
and finally
Gx , t=
-U(x-t)cosh a(x-t)+e ax C(t), a < t < B
elsewhere; !.. using condition (9.6.18) we have
a(x-t)+eaxU(-t)cosh at, a < t < B
elsewhere.
Thus the solution of the system (9.6.16) is
or, by putting 0
A = h(t) cosh at dt, a
we obtain
x c a X h(t)cosh a(x-t)dt a 5 x 5 8
Aeax - 1 h(t)cosh a(x-t)dt x 1. B.
f(x) =
We ask the reader to compare this method with that employed in
176 chapter 9
solving the equation (9.5.8).
9 .6 .4 . Integral equations
We consider the case where the operator L(x,d/dx) defined in
Section 9.6.2 is of order 0. In this case we have the integral
equation
(9.6.19)
where h(x) is a continuous function on this interval.
X k(x-u)f(u)du f ao(x)f(x) = h(x), x E [aIBl
-m
As in Section 9.6.2, Green's kernel G is a distribution (not XIt
necessarily unique) which is null on IRt\(a 5 t 5 B ) and which satisfies the equation
k(x) * GxIt + ao(x)Gx = 6(x-t). I t
(9i6.20)
Then we have
f(x) = <G ,h(t)>. x,t
(9.6.21)
Example. Consider the integral
X (9.6.22) a I f(u)du + (x-b)f(x) = h(x), x E [a,@] ,
-m
where this interval does not contain b.
The Green kernel is given by
aU(x) * GxIt + (x-b) Gx,t = 6(x-t).
Considering the equation deduced by differentiating
we obtain
elsewhereI Gx,t = t o t
where K1 is arbitrary if a > 0 and K = 0 if a 2 0. Consequently, 1
Applications 177
a+lI x < I k (x-b)
(9.6.23) =
where K is arbitrary if a > 0 and null if a 5
We observe that if b E [a,B] the integral
longer defined for x > b. But we verify that
satisfies the equation
X
0.
in (.9.6.22) is no
for a > 0, (9.6.23)
(9.6.24) a Fp I f(u)du + (x-b)f(x) = h(a), x E Ca,f31.
If a < 0, (9.6.24) is satisfied by
-OD
9.7.Partial Differential Equations
The Laplace transformation permits us to simplify partial
differential equations by reducing the number of variables with
respect to which derivatives are taken. We give some
or more well known examples of the use of the Laplace
in solving partial differential equations in order to
mechanics of this method to the reader.
Recall that the Laplace transform of the summable
is the function of the complex variable p defined by 00
lLpf(t) = f(t) e-dt.
9.7.1. Diffusion of heat flow in rods
0
of the simpler
transformation
show the
function f (t)
The rods considered here are homogeneous and of constant
section(6). There is no radiation. We take
thermal conductivity heat capacity by unit of length' k =
9.7.1.1. Infinite conductor without radiation
Consider the conductor to be an axis having a variable point
denoted by x. The temperature at x and at the instant t of the
17 8 Chapter 9
conductor is denoted by the function u(x,t). The temperature at the
initial instant is a continuous bounded function a(x) which is
given. The temperature function u(x,t) satisfies
U(X,t) = S(x,t), t > 0, a a2 ax2
- (9.7.1) at u(x,t) - k
where S(x,t) is a function (and eventually a distribution) which
characterizes the source of heat and is subjected to further
conditions.
Let ;(x,t) and S(x,t) be equal to u(x,t) and S(x,t) for t 0
and null for t < 0. Let us denote u(x,t) o IL ;(x,t). Applying
the Laplace transform to (9.7.1) we have P
(9.7.2)
a Suppose that u(x,t) and - u(x,t) are continuous with respect A a ax
to x; hence u(x,p) and ax u(x,p) are also continuous. physically evident that u(x,t) is bounded as 1x1 + -, it follows that u(x,p) is Fourier transformable in the sense of distributions; and from (9.7.2) we deduce
Since it is
A 2 2 -1 IF u(x,p) = (4r ky +p) IF C a(x) + IL z(x,t)l. Y Y P
Hence by means of the inversion of the Fourier transformation
Also, by the inversion of the Laplace transfornation,
(71 (9.7.3) ii(x,t) = - 1 u(t) t-112 e-x2/4kt ; ; [a(x) A(t)+~(x,t)] provided that S(x,t) permits the convolution to hold with respect to
x. We thus conclude that if a(x) has support in (a,@) and if S(x,t)
is a suitable function, then
2 m
1 fBds .-(X-C) 2 /4kta (9.7.4) u(x,t) = -
2 K t a
W
-1/2 e-52/4kw - S(S-x,w-t) l t + - f dw f d5u 2 G O --
for t > 0.
Applications 1 7 9
The formula ( 9 . 7 . 3 ) illustrates to deal with the theoretical case
where the initial temperature is null and the source is at a point.
In this case S(x,t) is of the form S(t)G(x-X) and
( 9 . 7 . 5 ) u (x
where
S(t
9 . 7 . 1 . 2 . The cooling of a rod of finite length
The extremities of the rod of length L are maintained at 0' and
there is no radiation. The temperature u(x,t) satisfies the system
of equations
( 9 . 7 . 7 ) u(0,t) = u(L,t) = 0
( 9 . 7 . 8 ) u(x,O) = a(x). 0 < x < L,
where the intial temperature a(x) is a continuous function with
bounded variation in the interval ( O I L ) .
Let a(x) be any bounded function which extends a(x) to IR. Then,by
( 9 . 7 . 4 ) we have
( 9 . 7 . 9 ) m
u(x,t) = - 1 f e -E2/4kt a(x-c)dc, t > 0, 2 m -m
which satisfies ( 9 . 7 . 6 ) and ( 9 . 7 . 8 ) . In order that u(x,t) satisfy
( 9 . 7 . 7 ) it is sufficient that
Hence g(x) must be anti-symmetrical of period 2L and equal to a(x)
on the half period ( O I L ) . Such a function a(x) has the Fourier
series representation m
( 9 . 7 .lo) X a(x) = 1 a sin nn - n L n=l
with
2 L X a = J a(x) sin nn - d ~ . L 0 n
rao Chapter 9
BY Putting (9.7.10) in (9.7.9) we obtain
E 0) - 2 X - 1 ancos nn - e-' /4ktsin nn dg:).
n= 1 L -a
The last integral is null by the anti-symmetry of the integrand.
On the other part we have
2 2-2 - (X-1'2~OS = 2 (nkt) 1/2e-n II L kt, - IL1/4kt
and (9.7.11) yields
X m 2 2-2 -n n L ktsin n* (9.7.12) u(x,t) = 1 an e 1;' t > 0,
n= 1
which is the famous solution of Joseph Fourier,
9.7.1.3. Rod heated at an extremity
The rod of length L is initially at the temperature Oo. The
extremity choosen as origin is maintained at 0'; the other extremity
has the temperature f(t). The temperature u(x,t) of the rod
satisfies the system of equations
u(x,O) = 0,
U(O,t) = 0, U(L,t) = f(t), f(t) = 0 if t < 0.
I
If we put u(x,p) = ILu(x,t) and f(p) = 1Lf(t), the system (9.7.13)
transforms to
whose solution is
A *I s h x m U(X,P) = f(P)
sh L m
Hence by the inversion of the Laplace transformation, we have
Applications 181
(9.7.14) u(x,t) = f(t) * E(x,t)
where
-1 sh X-
sh L m E(x,t) = lLt
Since
s h x Jp/k=a c h x Jp/k sh L Jp/k ax sh L m
we have
a (9.7.15) E(x,t) = El(xIt)
where
-1 ch x Jp/k El(x,t) = IL sh L m
which can be obtained more easily than E(x,t). A l S O , we have
-xL/4ktB LX -L' 2(- I 7 - 1 2nikt ikt = U(t)e
where 8 (. I .)is a Jacobi elliptical function (Erdelyi (Ed.) [l] , V01.2, p. 355). With the aid of transformation formulae associated
with Jacobi's theta-functions (Erdelyi (Ed.) [l], Vo1.2, p. 370,
formula 8 ) , we deduce
2
Consequently, making use of (9.7.151,
X m 2 2 2
E(x,t) = U(t) 1 (-l)"+'ne-" kt/L sin nr - . L L n=l
(Erdelyi (Ed.) c21 , Vol.1, p. 258, formula 31) , and finally by(9.7.14)
182 Chapter 9
we have
X 2 2-2k(t-w)
dw sin nn- L’ -n n L m t 2nk
L n-1 0 (9.7.16) u(x,t) = 1 (-1in+ln I f(w)e
Let f(t) = u
2 n -
t)T where T is a constant. Recall that
NOW, from (9.7.161, we deduce
X m 2 2 2 x 2T (-1ln -n n kt/L sin nn- L (9.7.17) u(x,t) = T~ + - 1 e
n=l
for t > 0, 0 2 x L.
(See Carslaw and Jaeger L21 , p. 185 and Colombo c21.1
9.7.2. Vibrating strings
The elastic string of length L is stretched by its extremities.
At rest it is laid on an axis whose variable points we denote by x.
At the initial instant we displace the string from equilibrium. The
displacement from equilibrium at the point x along the string and at
time t is denoted by the function u(x,t) which satisfies the system
of equations:
U(X,t) = 0, t < 0.
Here
BY a(x) we mean a continuous function defined on (0,L) with
a(0) = a(L) = 0; while b(x) is defined on [OIL] and is a function.
(Latter b(x) will be a distribution of order zero in which case we
have b(x) = DB(x) where B(x) is a bounded function.)
We shall denote the extensions of a(x) and b(x) to IR by a ( x ) and
Applications 183
g(x) I respectively. will be made below.
The precise way in which we form these extensicns
Consider u(x,t) as a distribution in x and a function of t. Let n
u(x,p) = IL u(x,t), Rep > 6 > 0. Then the system (9.7.18) transforms
to
(9.7.19)
P
2 2 - c DX U(X,p)-p2 i(x,p) = -p,a(x) - E ( x ) , x E IR,
The equation
1 has a fundamental solution of the form(-) [ U(-x) eXPiC+U (x) e-xp/c 1. Hence (9.7.19) has the solution
(9.7.21) -
:(x,p) = kc u(-x)exP’c+u(x)e-xP/C 1 * a(x)
+ c,(p) explc + c2(p) e-xP/c . The functions C (p) and C (p) must satisfy (9.7.20). We find that
these are null if 1 2
E(-x) = -E(x) , E(L-x) = -i;(L+x);
that is a(x) and E(x) are to be defined as:
a(x) = A(x), and anti-symmetrical periodic function of period
(9.7.22) 2~ which extends a(x) ,
E(x) = B(x), an anti-symmetrical periodic function or distribution of period 2L which extends b(x).
By the inversion of the Laplace transforms in (9.7.21) we obtain
1 2c + -[U(x+ct) + U(-x+ct)l * B(x) for t > 0
184 Chapter 9
or
u(x,t) = z [ 1 A(x+ct)+A(x-ct) ] + 1 U(X+Ct)+U(-X+Ct)l* (9.7.231
When b(x) is an integrable function, we get d'Alembert's
solution:
(9.7.241 u(x,t) = $ [A(x+ct) + A(x-ct) 1 + L XjctB(S) dc. 2c x-ct
When b(x) = DB(X) , we have B(x) = 6 ' (XI * B1(X) I where B1(X) is
is a symmetrical function of period 2L which extends B(x); and
(9.7.23) gives
(9.7.25) u(x,t) = 7': 1 A(x+ct)+A(x-ct)l +- 1 CB1(X+Ct)-B1(X-Ct)].
2c
On account of the process employed to determine C,(p) and C,(p)
in (9.7.21), we must verify that (9.7.23) and (9.7.25) give the
unique solution of the system (9.7.18). For this purpose, suppose
that there exists another solution u1 (x,t) . Then v(x,t) = u,(x,t)
-u (x , t) satisfies the system L - a L 9 v(x,t)-c2 a_ v(x,t) = 0,
at" axL a
v(x,O) = 0, ;i"s v(x,O) = 0,
v(0,t) = v(L,t) = 0, t ' 0,
This system is transformed by the Laplace transformation into n
P 2* V(X,P) - c -$ j(x,p) = 0,
A A
v(O,P) = V(LIP) = 0 1
* for which v(x,p) = 0 is the unique solution. Hence, we conclude
v(x,t) = 0.
It is easy to see from (9.7.25
respect to t of period
C * T = - 2L
Indeed, if n is a positive integer
that u(x,t) is periodic with
we have
Applications 185
u(x,t+mT) = 1 [ A(x+ct+2mL)+A(x-ct-2mL)lt~ 1 [ B1 (x+ct+2mL)]
- B1 (x-Ct-2mL) = u(x,t) , because A(x) and B (x) have period 2L. 1
In order for the function u(x,t) given by (9.7.25) to be the
solution of the system (9.7.18) in the sense of functions, it is
necessary that a(x) be twice continuously differentiable and b(x) be
a continuously differentiable function.
in the sense of distributions, we can impose weaker conditions on
a(x) and b(x) . For example we can take b(x) = 0 and
If we consider the problem
h F , O(X<X,
a (x) = i" where the derivative is not continuous in a neighbourhood of x = x, 0 < X < L. Note that (9.7.25) is also valid when b(x) is a point
distribution as in the case of the struck string, which we shall
consider below.
Let A(x) and B (x) be periodic anti-symmetrical distributions
which are represented by the Fourier series m
A(X) = 1 an sin nn 5 L n=l (9.7.26) - ..,
B(x) = 1 b sin n ?I - X L n=l n L
in the sense of distributional convergence, where
2 L an = i; a(x) sin nn f dx,
0
2 L X (9.7.27) bn = b(x) sin nn i; dx.
0
By putting (9.7.26) in (9.7.24) we have OD
ct 1 L n n L L u(x,t) = 1 c a cos nn- + -b sin n n G I sin nn 5 (9.7.28)
n=l
which is a well known result in harmonic analysis. The fundamental
frequency is given by & = $ where c is the propagation speed of the waves along the string.
According to Abel's rule the series in (9.7.28) converges to a
functionif a(x) has a bounded derivative and if b(x) is the sum of a
186 Chapter 9
bounded function and of a finite number of point distributions of
zero order. We note that a(x) will almost always have a bounded
derivative because of the physical origin of the problem.
Example. Struck string
Assume a string is at rest and then it is struck at a point X,
0 < X < L, at the initial instant. Then a(x) = 0 and b(x)=IG(x-X) where I measures the intensity of the impact. We have
W
B (XI = I 1 c 6 (x-x+z~L) -6 (x+x+~~L) 1 n=-w
which is an anti-symmetrical distribution that is the derivative of
f., (2n-1)L < x < 2nL-x
B1(x)= 0 , 2nL-X < x < 2nL+X 1. 2nL+X < x < (2n+l)L
Also, the equation (9.7.25) gives
’ I U(xrt) = 2~ [B1(X+Ct)-B1(X-Ct)l .
It is now easy to obtain the vibration of the middle of the string. 2L X , $ = C. Then L If < X < LI T = -
1 4 i 0, (m+$T+$ < t < (m+T)T-@,
0, 0 < t < - T-$, I, (m++)T-$ < t < (m+T)T+$, 1
-I, (rn++~-@ < t < (m+T)~++r 5
1 3 0, (m+$T+$ < t < (m+;r)T-$,
3 5
u(L/2,t) =
where m is a positive integer.
Moreover, (9.7.27) gives
21 X b = - sin nn - n nc L
and (9.7.28) gives
21 1 X X ct u(x,t) = - 1 - sin nn?; sin n y sin nr- ““n=l” L
with the series being convergent according to Abel’s rule.
Applications 187
9.7.3. The telegraph equation
The telegraph equation is encountered in the theory of electric
transmission lines and in other branches of science where media
capable of oscillation are investigated. The present section deals
with such equations in electric transmission.
In the homogeneous transmission line which has R resistance per
unit length, inductance L, capacitance C, and leakage G, the tension
E(x,t) at the point x and at the time t satisfies the equation
a 2 a 2 a ax at
(9.7.29) TE(x,t) = LC 7 E(x,t)+(RC+LG) at E(x,t) + RGE(x,t)
with the line being taken along the x axis.
If L = 0, we get the heat diffusion equation with radiation (or
without radiation if G = 0).
If R = G = 0, we get the equation of the vibrating string.
In Sections 9.7.3.1 and 9.7.3.2 we put a = &.
9.7.3.1. The lines without leakage which are closed by a resistance
In a line without leakage we have R = G = 0. We switch on an
electromotive force Eo(t), which is null for t < 0 and at the origin
x = 0. The extremity x = 1 is joined to earth (of potential zero)
by the resistance r ohms. Also, we suppose that the line is neutra-
lly maintained up to the instant t = 0. Then, we have the system
of equations
2 - a 2 E(x,t) = LC TE(X,t) a 2 = a2 %E(x,t), O < X < l , at at 2 (9.7.30)
ax
E(O,t) = Eo(t) I
E(x,t) = - E(x,t) = 0 for t 5 0. a at
A 6.
If we put E(x,p) = IL E(x,t) and Eo(p) = IL E (t), then the P P O
system gives
188 Chapter 9
Now, the equation (9.7.31) has the solution
i(x,p) = A(p)eapX + B(p)e - apx . To determine A(p) and B(p) we need a condition at x = 1; and the
laws of electricity will be supplied. Let I(x,t) denote the
intensity on the line. We have
consequently, we obtain
f(x,p) = -- I d A - E(x,p) = - :[ A(p)eaPX-B(p)e-apX1. LP dx
(9.7.33)
Here Ohm's law applied to the closing resistance gives
E(1,t) = r I(1,t) or
A * E(l,p) = r I(1,p).
Hence, according to (9.7.33) , we get
= - - ra [ A(p) - B (p) e-apll L A(p)eapl + B(p)e
which along with (9.7.32) permits us to determine Afp) and B(p) by
putting
L-r a B = - L+ra
Finally, we obtain
when Re p i s large enough, and one can expand the denominator into a
series. ' ly, we have
Hence by inverting the Laplace transformation we get m
E(x,t) = Eo(t) * 1 0"~~(t-a[2nl+xI)-BG(t-a[2(n+l)l-x]) 1 , n= 0
and finally we have for t > O f
N
n= 0 (9.7.34)
where N denotes the largest integer such that N < t+ax - 1. E(x,t) = 1 Bn{Eo(t-a[2nl+X1) - BEo(t-aC2(n+l)l-xl) 1
Applications 189
Further (9.7.34) shows that E(x,t) is a superposition of waves which are direct or reflected and which are propagating with the
speed - = 1/m. 1
If r = m, then B = 0; and (9.7.34) gives simply
0 if t < aa
Eo(t-ax) if t > ax.
1
JaC
(9.7.35) E(x,t) =
The crest of the wave is propagated at the speed a = - . 9.7.3.2. The infinite line which is perfectly isolated
In this case, G = 0. The line is assumed to be neutral up to the
instant t = 0 at which time the electromotive force Eo(t) is turned
on at the origin. We put
The system of equations which govern this problem is as follows:
32 a 2 a ax - E(X,t) = LC ,E(X,t) + RCE E(x,t), X > 0,
at 2 (9.7.36)
E(x,t) +. 0 as x +. -. The later condition is due to the presence of the resistance R.
A
By putting E(x,p) = IL E(x,t) we transform the system (9.7.36) to P
d2 2 A 2 2 6
--p(x,P) = C(Lp +Rp)E(x,p) = a (P +2~p)E(x,p)~ dx A n
E(O,p) = Eo(P) = ILp Eo(t)r
A
E(x,p) -f 0 as x -+ -,
whose solution is
But we would have (see Erdelyi (Ed.) [21, Vol.1, p.249 ( 3 5 ) )
190 Chapter 9
2- 2 1/2 Il(a(t b 1 1
t ’ e x p - b m = 6 (t-b) +abU (t-b) (t2-b2) 1/2
where I1(.) is a modified Bessel function of the first kind and a
and b are positive. We deduce by (8.5.5) of Chapter 8
Clexp-b- = 6(t-b)e-by+abU(t-b)e-YtIl(a(t2-b2)3 (t 2- b 2 ) -%
With the help of this result, we use the inverse Laplace transform
in (9.7.37) and obtain
t e ayxEo(t-ax)+ayx J e-YW~l(y(w2-a2x2)‘) i - ax
E(x,t) =
L (w2-a2x2)-’E(t-w)dw if t > ax.
The crest of the wave is still propagated at the speed ; 1 1 = - . 6
Other applications of the Laplace transformation to the telegraph
equation are given in Maclachlan [ 11 and Doetsch C 3 1 .
9.8. Convolution Formulae
The formula (8 .5 .8 ) of Chapter 8 permits us to find the convolu-
tion product of distributions for which the direct calculation is
difficult, as is the situation in the case o f pseudo functions. In
this section we utilize the formulae of derivatives of pseudo
functions without any special mention about them (see Section 5.4.6
of Chapter 5).
Let us proceed to explain these convolution formulae with the
aid of a few examples.
Examples. 1. Since LFpx;’ = - log Cz, then we have IL[FPX;~I*~ = log 2 Cz.
(9.8.1)
But according to Lavoine C21, p. 69
YLCFp %I+ = $(log2 CZ + n2/6)
Hence, by means of inversion
c110g2Cz = 2Fp [ e l + - (7 r2 /6 ) 6(x)
and (9.8.1) gives
Appl ica t ions 1 9 1
2 (9.8.2) [ FPx;1]*2 = 2[Fp x- l log XI+ - $6(x).
-2 2. Since 1LFpx+ = z log Cz-z, then w e have
IL [Fpxi2 x Fpx;' ] = z log Cz-z log 2 Cz. (9.8.3)
But, by means of i nve r s ion
I L ' ~ log c z = - D F ~ X ; ~ = ~px;' + 6 1 ( x )
2 and IL -1 z log 2 Cz = - 2 F p [ ~ - ~ l o g X I + + 2Fpxi2- +6' (x ) ;
hence according t o (9.8.3) , w e ob ta in
3. W e have
l o c z ILFpCx;' * U(x)] = - = IL[log x ] +
which y i e l d s t h e r e s u l t
(9.8.5) Fpx;' * U(x) = log x+.
W e remark he re t h a t t h i s r e s u l t i s e a s i l y obta ined d i r e c t l y a s fol lows :
Fpx;' = D log x+ = log x+* 6 ( x )
and consequently
-1 Fpx+ * U ( X ) = log x * ~ ' ( x ) *U(x) = lOgx+*6(x) = logx+.
4. Since (see Lavoine [ 2 1, p. 69)
ILF px;"-l - n+l zn (9.8.6) - (-1) Clog z - J l (n+ l ) l
where n is a nonnegative i n t e g e r , then w e have f o r an i n t e g e r k 2 1, k- 1
IL[U(x) * Fp~;~- ' l = (-1) k+l 5 [ logz - $ (k+l ) 1 k k-1 k k-1
(k-1) . , [ logz -$ (k ) ]+ (-l) k.k!
Hence wi th t h e h e l p of (9.8.6) w e g e t
192 Chapter 9
k (XI. 1 -k + (-1) &(k-l)
k.k! I - 1; Fpx+ -k-1 (9.8.7) U (X) *Fpx+
Remark. Multiplying by -k and differentiating both members of - (9.8.7), we obtain
k- 1 k DFpxik - 6 (k) (x) = -k& ’ (x) *U (x) *Fpx; k!
= -k6(x)*Fpxik-’= -k Fpx+ -k-1
which is in accordance with the formula (5.4.10) of Chapter 5 . The formula (8.5.8) of Chapter 8 also permits us to express
certain functions or distributions as convolution products.
following examples illustrate to see this,
The
Examples. From
ILx;~’~ cos h 6 = (n/z) 1/2 ea/4z
we deduce
x+’l2cosh - fl(Id2, = x + I
= f’(~/z)”’~* f1 ea/4zl
-1/2 -1/2
I E ~ ~ ~ / ~ ~ = 5 all2 x-’i2 I l ( ~ ) + b (x) . + L
Hence, we conclude
(9.8.8)
where I1 denotes the modified
I1 ( K u ) du + - 1 Ju(x-u) J;;
Bessel function of order 1.
i n By putting a = -1 = e we obtain
where J1 denotes the Bessel function of order 1.
Another example. We put
r = (z2+1) 1/21 R = z+r.
If v # -1,-2,..., then we have (see Lavoine C21 , p.84) (9.8.10) Fp Jv (x) + = r-lRmV I R-’r-’R-(’-’)
Applications 193
where Fp is not needed if Re v > 1.
But
If v # 0,-1,-2,..., then we deduce from (9.8.10)
where Fp is not needed if Re v > 0.
Moreover, from
.-lR-2 -1 -1 -1 ILJ2(x)+ = = R r R
we deduce
(9.8.12) -1 x J1 (u) J1 (x-u) lu, x > 0. x-u J2 (x) =x J1 (x) +*J1 (XI + = I
0
9.9. Expansion in Series
The Corollaries 8.6.2 and 8.8.1 of Chapter 8 are very interesting
from the point of view of applications. The following examples may
be added to the examples of Section 8.6.1 of Chapter 8.
9.9.1.Function B(v,z)
If Re v > 0 and Re z > 0 , then we have (see Lavoine [ 21 I p.70
or Erdelyi (Ed.) c21 , Vol. I, p. 144(8))
Hence
or
(9.9.11
This series is convergent in the considered domain i.e. Re v > 0,
Re z >O. If v is a positive integer, then the series reduces to a
194 Chapter 9
finite number of terms.
In particular, if v = + , (9.9.1) gives
Remark. If Re v 5 0, the series in (9.9.1) is not convergent. - 9.9.2. Function $ ( z )
If Re z > 0, then we have (see Lavoine C2l , p. 76)
According to the Euler formula m
coth x = 1 + 2 1 . e g ; j=1 x +J n
X
and by Maclachlan and Humbert [11 , p. 14,
Consequently, we deduce
(9.9.2)
Recall that si(z) and ci(z) are the complex extension of;
00
1 $ ( z ) = log z - 22'2 1 si(2jnz)sin(2jnz)+ci(2jnz)cos(2jnz).
j=1
00 m sin u - du, ci(x) = - f cos Ud u, x > 0.
U si(x) = - f
X U
X
9.9.3. Fourier series
The Laplace transformation permits us to obtain the Fourier
expansion of certain distributions. We have (see Lavoine C21, p.81)
z1oglsin XI+ = - + lo 2 -,i 7 1 ,+2 , j=1 z +4j
and
Z 5' 7 2 = cos 2jx+, z +4J
which yields the result,
logisin X I + = - (log~)~(x) - T 1 cos 2jx+ . ( 9 . 9 . 3 ) j=1 J
Also
Applications 1 9 5
(9.9.4) m
cos 2jx log(sin x ( = - log 2 - 1 j=l j *
The Abel rule assures the convergences of (9.9.3) in the sense of
functions and hence (9.9.3) is consistent in the sense of distribu-
tions. It follows that by differentiating (9.9.4) term by term, we
obtain
(9.9.5) FP cot gx = 2 1 sin 2jx
in the sense of distributions. Multiplying this equality by sin x
we obtain an equality such that both members are equal to cos x ,
which constitutes a partial verification.
m
j=1
Changing x to x + n/2 in (9.9.5), we obtain
(9.9.6)
Also, differentiating (9.9.5) and (9.9.61, we obtain
(9.9.7) m
2 FP l/sin x = -4 1 j cos 2jx j=1
~p l/cos x = -4 1 (-1) J j cos 2jx. j=1
co 2
(9.9.8)
Multiplying (9.9.7) by sin x and (9.9.8) by cos x , we obtain
m
(9.9.9) FP l/sin x = 2 1 sin (2j+1)x
(9.9.10)
j =O m
~p l/cos x = 2 1 ( -1) jcos ( 2 j + l ) x . j=O
Remark. From (9.9.5) and (9.9.9) one cannot deduce that - Fp cotg x+ and Fp l/sin x+ are represented by
00 m
2 1 sin 2jx+ and 2 1 sin (2j+l )x+ j=1 j = O
because these series are not convergent in the sense of distributiors.
But differentiating (9.9.3), we obtain
(9.9.11) FP cot gx+ = -(log 2 ) 6 (x )+2 1 [ cos 2 jx+ - Moreover, we have (see Lavoine 121, p. 7 9 )
m
6(x)1. ?3 j=1
2 m 1 2 .
J=o lLFp l/sin x+ = log 2 - Hence, we conclude by means of inversion
196 Chapter 9
or explicitly
(9.9.12) 1 m
FP l/sin x+ = (log Z)S(X)+Z 1 [sin(zj+l)x+ - m6(x)]. j=O
9.9.4. Asymptotic expansions
The Laplace transformation also permits us to find asymptotic
expansions. In this section we shall present these results.
Let Re z > 0. Ei(-z) denotes the complex extension of the
function
m --u Ei(-%) = - I +u, x > 0,
X
which is called the "Exponential integral". We have (see Lavoine
c21 , P. 66) 1 eZEi(-z) - log cz = ILFP c m l + ,
or
z (9.9.13) e Ei(-z) = ILFp(4) X+ +' Let
j J
j=1
Z i ( z ) = e Ei(-z) - 1 (-1) (j-l)!z-j.
If we put
we have from (9.9.13)
A
A(z) = ILa(x)+.
But, when 1x1 < 1, we have
therefore,
Hence, as x + O+
Applicat ions 197
J+1 J a ( x ) + .. (-1) x+
which y i e l d s , according t o Sec t ion 8.11.1 of Chapter 8 t h e r e s u l t
Consequently, w e deduce t h e asymptot ic expansion
31 Z
(9.9.14)
This formula is s t i l l t r u e i f z i s pure imaginary. For example l e t t i n g z = x / i and remembering Ei ( -x / i ) = c i ( x ) + i s i ( x ) , w e o b t a i n (see Magnus, Oberhet t inger and Soni C11 , p. 349)
(9.9.15) ix 1 l! 3 ! c i ( x ) + isi(x) .. e (z + 2 + --J + . . . I
f o r l a r g e va lues of x.
Another example. I f v # 0 , -1,-2,..., and if DeV denotes t h e pa rabo l i c c y l i n d r i c a l func t ion (or Weber f u n c t i o n ) , w e have f o r R e z > 0 (see Erde ly i (Ed.) C2l , Vol.1, p. 289(1) )
Put
where
As x + O+, w e have
Hence, according t o Sec t ion 8.11.1 of Chapter 8, J+l ('1 2 j + 2 -v- 5-2
Z , R e z -+ m . &z) " (-1)
2' j ! Consequently, w e deduce t h e asymptot ic expansion
f o r l a r g e va lues of R e z .
198 Chapter 9
9.1O.Derivatives and Anti-Derivatives of Complex Order
For two centuries non-integer derivatives have been of interest
to numerous mathematicians. Oldham and Spanier C1l , and R0as.B C11 have given a complete discussion of this topic. We give an
approach here to complex differentiation on the Laplace transforma-
tion of pseudo functions.
9.10.1. Definition by the Laplace transformation
Let Tx E ID; be Laplace transformable. Then according to A
Section 8.5.6 of Chapter 8 and 7, if n 8 IN and T ( z ) = ILTx, we have
-1 n A IL z
z-lZ-n .. of order n of Tx.
X ' T(z) is the derivative DnTx of order n of T
T ( z ) is the anti-derivative (in ID;) D-"Tx
We shall generalize these results by defining differentiation
of complex order v and define it as
-1 v A (9.10.1) D'T~ = IL 2 T ( z ) , Y v E c.
When v = - v ' where Re v ' > 0, we must say that D-"'Tx is the
anti-derivative (the primitive) in ID: of order v l .
We deduce from (9.10.1) that
DOT = T i D-'D'T = T
DADVT = D'+VT (9.10.2)
and
(9 .lo. 3 ) D'(T*s) = (D'T) * s = T * (D's)
which are the basic relations of differentiation. To show these relations it is sufficient to remark that
x -1 A v A -1 x+v* D D'T = IL z z T ( Z ) = IL z T(Z)
D'(T*s) = ~ ~ z ~ i ( z ) i ( z ) = ~ n ~ ' - z ~ i ( z ) l * s. and
In order to make DvTx explicit, put
Applicat ions 1 9 9
W e have
( X I i f v = n s I N
(x) = Fp x iV- ' /F ( -v ) i f v > 0 , v k
f6 (n )
-v-1 x+ / r t - v ) R e v < O , [ (9.10.4)
and (9.10.1) can be obtained by a convolut ion:
(9.10.5) D ' T ~ = 6 i V ) ( x ) * T ~ , Y v E c .
I f v j! IN and R e v > 0 , one can w r i t e v = n+a, where n E IN and a-1 n + l 0 2 R e a < 1, and w e have according t o (9.10.2) , DVT = D D
(Here D"'l p lays t h e r o l e of DV and Dn+lT p lays t h e r o l e of T.) Consequently, by (9.10.5)
T.
DvT = Da-'Dn+lT = 6 (a-1) * Dn+lT + which y i e l d s t h e r e s u l t according t o ( 9 . 1 0 . 4 )
( 9 .lo. 6 ) 1 -a* Dn+lT DVTx = x+ X.
I f T is represented by a s u f f i c i e n t r egu la r func t ion f (x) having suppor t i n [a , - [ , w e have t h e p r i m i t i v e of o rde r v ' :
(9.10.7) l X V l - f ( - ' I ) (x) = I (x-u) ' f (u)du , R e v ' > 0 . r v i a
On t h e o t h e r hand, i f f ( x ) i s continuous and i s such t h a t i t s f i r s t n d e r i v a t i v e s are cont inuous, w e have according t o ( 9 . 1 0 . 6 ) , t h e d e r i v a t i v e s of order (n+a):
(9.10.8)
W e now f i n d t h e p r i m i t i v e s and d e r i v a t i v e s of complex order s tud ied by L i o u v i l l e C11 and [Zl , Riemann, Riesz and o t h e r s .
Another express ion f o r S:"' (x ) can be obta ined as fol lows. Since f o r R e z > 0 and h > 0 ,
m z V = l i r n h -v( l -e -hz)v = l i m h-' 1 (-1)'(i) e-phz
h+O h+O p=O where
(9.10.9) m
("(x) = l i m h-' 1 (-1) P V G(x-ph). h-4 p=O &+
200 chapter 9
Also, by (9.10.5) m
(9 10.10) DvTx = l im h-' 1 (-l)P(i)Tx-phI T E ID;, h+O p=O
which is a result corresponding to those of Grunwald (1867) cited by Lavoie, Osler and Tremblay Ell . The formula (9.10.10) shows
that the operator Dv is analogous to the operator It:') of Bredimas
Ell (see also Section 9.10.3). If v = n is a positive integer, then
the sum in (9.10.10) contains only (n+ l ) terms.
Moreover, for Re z > 0, we have
z v = lim h-"(ehz-l) = lim h - V e vhz (l-emhz)"
h+O h -+O W
= lim h-' 1 (-l)P(i)e(v-P)hz. h+O p= 0
Hence, we conclude 03
6:") (x) = lim h-" 1 (-1)'(i) 6 (x+(v-p) h) , h-+O p=O
and according to (9.10.5)
(9.10.11) D"T~ = lim h+O
for Tx E ID:.
If T is represented X
Liouville 111 . by a function, we get a formula of
9.10.2. Examples
1. If Re A > -1, then according to (9.10.1) , we have " A = ~'~(A+~)z-~+'-~
(9.10.12) x+
If v-A-1 = n, n E IN, we deduce
D'X: = r (!,+I) 6 'n) (x) ;
and if v-3-1 # n, (9.10.12) can be written as
1 r ( x + l ) & ( A - " + l ) z - )i+v-1 r ( A - v + l ) D x+ =
Hence, we conclude according to (8.10.3) of Chapter 8 ,
v A r ( A + l ) A-V D x+ = FP x+ -
Applications 201
(Fp is not needed if Re(A-v) > -1). In particular
A- w Dnx: = ( A - 1 ) ,..., (A-ntl) FpX+ .
When Re X 5 -1, A+l # -n, w-A-1 g! IN, (9.10.12) leads to
r ( ~ + i ) A - W D'FPX: = r(A-v+l) ~px+ . A The process also gives DWFpx+ in other cases.
2. If w # 1,2,3,..., we have
ILF~X-~/~J-~(~J~;) = z v-le-l/z = n-1/2zv-1/2J7;/z e-l/z
by Lavoine [2], p.90 (see also Erdelyi (Ed.) [21, Vol.1, p.185(30)).
But (see Lavoine [2], p.82; Erdelyi (Ed.) [21, Vol.l,p.158(63))
~ x - ~ / ~ c o s 2 f i + = ~ / z
Hence, we have according to (9.10.1) ,
(9.10.13) -v/2 (2J;;) = n-1/2Dv-1/2 cos 2 G , for , 0. J-w/2 Jj;: 1 Putting w = n + 7, n E lN, we obtain the well known formula
, x > o . (2&) = n -1/2xn/2+1/4 2 cos 2& J-n-1/2 dxn -&
(9.10.14)
We ask the reader to compare these formulae to those given in
Section 9.10.3.
3. Abel'S integral equations. These are the equations of the
type X I (x-u)'-lf(u)du = h(x), x > a, Re w t 0, ma
where h(x) is a given function having support in Ca,m[. This
equation is equivalent to
D-'f(x) = h(x);
hence, according to (9.10.5) we have
V f(x) = D h(x) = 6:") (x) * h(x).
If w = n is a positive integer, we obtain
f (x) = h(n) (x ) .
202 Chapter 9
Here we suppose that h(x) is a (n-1) times continuously different-
iable function for x > a.
If v = n+a, 0 5 Re a 2 1, n E IN, we obtain according to
(9.10.8)
1 - v - 1 f(x) = r(-j) Fpx+ * h(x)
x(x-u) a h(n+l) (u)du = ‘r(l-.7
provided that h(x) is n-times continuously differentiable for x > a.
4 . Heat flow problem. We consider a homogeneous rod whose
initial temperature is null and one of whose extremities (origin
of x ) is maintained at the temperature T. The temperature u(x,t)
at x and at time t > 0 satisfies the system
where k is a constant and h is a radiant characteristic coefficient.
Putting
u(x,t) = e-htw(x,t),
the system (9.10.15) becomes
(9.10.16)
Using the
w(x,O) = 0, w(0,t) = Teht, t > 0.
idea of Doetsch [111 p. 1 2 , the differential equation
(9.10.16) takes the form
Also, we obtain a first order system in x given by
k a ax w(x,t) + D;”w(x,t) = 0
w(x,O) = 0, w(0,t) = Teht, t > 0.
Further, taking the Laplace transformation with respect to t and
Applications 203
CI
putting w(x,z) = ILw(x,t), we get
d A w(x,z) + 6 &X,Z) = 0
A T w(0,z) = - 2-h
whose solution is
A e - X G w(x,z) = T 2-h
Hence, by means of inversion
u(x,t) = Te-ht f’(2-h) -le-xJk/Z ;
and by Erdelyi (Ed.) [2] , VOl.1, p. 246(10),
(9.10.17) u(x,t) = ” {eeX r- h/k erfc c X -GI+ e X&
2 G z
erfc [ 2 + 61 1 2 G t
which is the result given in Carslaw and Jaeger C2l I where
- 2 -1/2 e-u du erfc x = 2 s X
5. We find an application of the derivatives of complex order
in Lavoine C31, pp. 439-441 and 648-650.
9.10.3. Extension of the definition
As for (9.10.4), let us consider the distribution 61v)(x)
defined by
6 (n) (x) i f v = n s l N
ivs FpxIV-’/r(-v) if v # n and Re v > 0 I: ivs xZv-’/r(-v) if Re v 4 0.
(9.10.18) 6:”) (XI=
where we use the notation
1x1’ if x c o
if x > 0.
The formula (9.10.5) then suggests the following definition by
induction:
(9.10.19)
DI is exactly the operator IIv
D_VT = 6:’) (x) * T. of Bredimas C 11.
204 Chapter 9
Of course in (9.10.19) we suppose that the distribution T
Possesses some properties which permit convolution. If T is repre-
sented bq' a locally summable function f(x) such that x-Of(x) + 0 for
some real number n as x + a, then we have for Re v > and v ,d IN
the equality ivn m
Dl)f(x) = Fp (u-x)-'-lf (u)du. (9.10.20) r o X Eventually for negative values of v, Fp is not needed and the
definition is analogous to that of Weyl.
Examples.1. If Re a > 0, (9.10.20) gives ivn m
-v-1 e-audu D: = ,h Fp (U-x) X
eivn -ax -v-1 eivn a v e - ~ , = F j JL,FpX+
Hence, we conclude
v -ax ivn v (9.10.21) D-e = e a for every complex v.
The successful proposed result is given by Liouville Ill, p.3 .
2. If w > 0 and Re v > 0, the preceding process gives
D: eiux = e ivn/2 wveiwx
Hence, for a > 0,
D_* sin wx = ma sin(wx + a+)
D: cos wx = w'cos (x + a $ ) .
These results are consistent with those of Zygmund C11 .
(9.10.22)
3 . The operator D: also permits us to write formulae concerning
special functions. We shall give a few examples.
Bessel function:
( 2 ~ ) ie-ivnv-1/2xv/2 Dv-1/2 sin 2&, ,+ E C V J;;
(9.10.23) -
Applications 205
1 2
and by putting v= n+--, we get the well known formula
We ask the reader to compare (9.10.23) to (9.10.13).
Cylinder parabolic function (or Weber-function) :
(9.10.24) Dv(x) = e e
Kummer function:
(9.10.25) F(a,c;.l/x) = e X D- X ellx-, Re a > 0.
-ivn x 2 /4 DI e-x2/2
i(c-a)n r(c) a a-c -c
Footnotes
(1) simplified Volterra type of second kind. (See Yosida [l].)
(2) we call here resolvent kernel of (9.4.1).
(3) see Roach [11 , Courant and Hilbert c11, p.352, Yosida [l] and
(4) Schwartz C11 , Chapter V.6,FriedmannI B.Cl1, Ohapter 3 and
(5) note again that the majority of authors denote by this term
(6) see also 4. of Section 9.10.2. (7) here as usual U(t) is the Heaviside function.
Stakgold C11 . Vo-Khac-Khoan [: 11 . a kernel having different properties.
This Page Intentionally Left Blank
CHAPTER 10
THE STIELTJES TRANSFORMATION
Summary
The reader can study this chapter directly after Chapter 8.
It is well known in classical transform theory that the
Stieltjes transform exists naturally as an iteration of the Laplace
transform (see Widder C11 ) . By making use of this notion we present
in this chapter the theory of Stieltjes transformation by working
with distributions by means of an iteration of the Laplace transfor-
mation of Chapter 8. Prior to formulate the distributional setting
of Stieltjes transformation we need the structure of a distribution within the periphery of present work which we describe below as
follows.
10.1. The Spaces E(r) and Jrr'tr)
Our study made on the spaces of base functions and distributions
(see Chapters2 to 5)enbles us to construct some spaces of functions
and distributions in the following manner.
10.1.1. The space E(r)
Let E(r) (r being any real number greater than -1) denote
the space of functions f(t) which are null for t < 0, summable on
[O,A] ( A 1) and such that there exists a positive number a < r+l
for which I j t dent of t' and t" with t" > t' 2 A.
f(t)dt( is bounded by a number which is indepen- t'' -r - 1 +a
t'
Examples. If g(t) is null for t < 0, locally summable, and if
there exists a number a ' > 0 such that t-r+a' g(t) is bounded as t -+ m,
then g(t) E E(r).
If g(t) is such that is bounded independently of t'
2 0 8 Chapter 1 0
m
and t", then g ( t ) belongs t o E(r). Since I s in t l d t and 0 2
0
t l s i n t l d t a r e d ivergent i n t e g r a l s , consequently, s i n t+ and
But, t 0
(t s i n t ) + a r e not summable.
and I I t s i n t d t I = zlcos tnI2-cos t 'fizl which enable us t o conclude
t h a t s in t , and (t s i n t ) + belong t o E ( r ) . number v # 0 , (tW-'sin tV)+ and ( t v - l c o s tu)+ belong t o
2 s i n t d t l = [ c o s t '-cos t " 1 5 2 2 1 t
t ' 2 A l s o , f o r any real E ( r ) .
I f f ( t ) E E ( r ) , then f ( t ) F E ( r ' ) , r ' > t. Thus E ( r ) c E ( r ' ) i f r < r ' . I f f (t) c E ( r ) , then i ts p r imi t ives belong t o E ( r + l ) .
L e t f ( t ) E E ( r ) . Then i t s St ie l t jes t ransformation of index r , which w e denote by $if(t) ,is def ined as
(10.1.1)
where s i s a complex parameter. The ex i s t ence of (10.1.1) can be assured by t h e Abel's r u l e .
m
gff (t) = I f (t) (t+s)-r- ldt , ) a r g s ) < 0, 0
Now w e s t a t e t h e following r e s u l t which w i l l be used i n t h e subsequent work.
Theorem 10.1.1. L e t f ( t ) E E ( r ) and l e t f o ( t ) = 0 f o r t < 0 wi th f o ( t ) = j 'f(x)dx f o r t > 0 , then f o ( t ) is a continuous func t ion
and such t h a t t-r-l+a f o ( t ) i s bounded a s t -+ - f o r any a < r+l. 0
Proof. The con t inu i ty of f ( t) fol lows from t h e con t inu i ty of 0
an i n t e g r a l wi th r e spec t t o i t s supremum l i m i t . Hence, by t h e d e f i n i t i o n of f o ( t ) , w e have
-
A
Also, if t > A w e have
NOW, by t h e Bonnet's formula f o r t h e mean, t h e r e may e x i s t t" > t such t h a t
j t:-r-l+a f (x) dx. A-r-l+a I f (x)dx = t
A A
Stieltjes Transform 209
Finally, we obtain
f(x)dxl t"-r-l+a
fo(A)+I/ x t-r-l+a t-r-l+a
fo(t) t
which is bounded as t -+ m. Hence the proof follows.
We, therefore, have from the above result that
(10.1.2)
and according to (10.1.1) ,
(10.1.2 )
10.1.2. The space Jr'(r)
f(t) = & fo(t) = Dfo(t) ,
m
j%if(t) = (r+l) f fo(t) (t+s)-r-2dt. 0
Let S'(r) denote the space of distributions Tt of the form
k' (10.1.3) Tt = D f,(t)
where Dk denotes the differential operator of order k' E lN, fl(t) is
a locally summable function, null for t < 0 and such that
t-r-k'+af (t) is bounded as t + - for a certain a 0. 1
Note that, according to (10.1.2) E(r) C 9 ' ( r). The space
~ ' ( r ) also contains all the distributions Bt having bounded support in the halfdaxis r o t - [ , because according to Section 4.1.3 of
Chapter 4, Bt can be written into the form (10.1.3).
If r = 0 we write 9' instead of 9' (0). Further, we remark
here that every distribution belonging to S'(r) (or JI I; is tempered
by virtue of (10.1.3).
The related spaces of this kind can also be found in Lavoine
and Misra [ 11 , C 21 and C 31 . (See also Benedetto C 21 .)
10.2. The Stieltjes Transformation
The structure of a distribution in JI' (r) given in the preceding
section enables us in this section to formulate the setting of
Stieltjes transformation with distributions which we describe as
follows.
Let Tt E 9 ' (r). Then its Stieltjes transformation of index r
is defined according to (10.1.3) as
210 Chapter 10
(10.2.1) m
-r-1 > = -r-k'-ldt r (r+ $iTt = <Tt, (t+s)
where s is a complex variable such that larg s I < TI.
When Tt = f(t) E E(r), we obtain the ordinary Stieltjes
transformation and by virtue of (10.1.2') with fl(t) = fo(t) and
k' = 1.
When Tt = Bt, we have
r -r-1 (10.2.1 1 ) $, Bt = at, (t+s) >.
One can easily verify that (10.2.1') exists. To do so, we see that
coincides on the the right side of (10.2.1') exists since (t+s)
support of B with some functions of ID. We remark here that in all
the cases, if $: Tt exists, then $3, exists for r'
-r-1
t ' r.
For brevity, we shall write $5 (or $ )-transfornation instead S
of Stieltjes transformation of index r (orindex 0). To conform with
established terminology, we shall say that every distribution
belonging to JI' ( r) (or 55') is Stieltjes transformable with index
r (or index 0).
10.3. Iteration of the Laplace Transformation
The structure of a distribution in JI' (r) (see Section 10.1.2)
states that any element Ttin JI' (r) is the kth distributional deri-
vative of a tempered locally summable function. Hence, by Theorem
8.1.1 of Chapter 8, we conclude that every element Tt E JI'(r)
the Laplace transformable. We use this notion in the present
section and show that the Stieltjes transformation of a distribution
in LU' (r) can be obtained by means of an iteration of the distribu-
tional setting of Laplace transformation in 55' (r).
is
This section contains the following result.
Theorem 10.3.1. If Re s > 0 and if we put F,(s) = $5 Tt with Tt E J I 1 (r) , then we have (10.3.1) Fr(s) = $iTt =<Ttt(t+S)-r-l> = 1 lLsU(x)xrlLxTt
where ~ ( x ) is the Heavisi.de function and II, denotes the Laplace
transformation.
Stieltjes Transform 211
k' - Proof. Since Tt = D fl(t), we have ILxTt = xk'G(x) with L
F(x)= lLxfl(t). Also, we have
(10.3.2) m
-sx r+kl- ILsU(x)Xr lLxTt = I e x F(x)dX 0
m m
= I I e -(tcs)x xr+k'fl(t)dt dx. 0 0
To show the existence of (10.3.2) and the possibility of inverting
the integrations, we need to show that X~+~';(X) I s summable in the
neighbourhood of the origin. For this purpose, by the condition
is bounded as t + which enables us to find out two numbers N and
M such that
-t-k ' +a imposed on f 1 (t) in Theorem 10.1.1 we may infer that t fl(t)
If,(t) I < Mtr+k'-a if t N.
Hence m m
dt IG(x) I = e-xtfl(t)dtl < I e -xt Ifl(t) Idt + I Me-xttr+k'-a 0 0 0 N -r-k'-l+a I Ifl(t) Idt + Mr(r+k'+l-a)x 0
Therefore, we may conclude that x~+~';(x) is summable in theneighhour-
hood of the origin. A l s o , by using Theorem 8.3.1 with c = 0 of
Chapter 8 we assure the convergence of f e -sxxr+k'; (x) dx. Finally, by Fubini's theorem we can rewrite (10.3.2) in the form,in accordance
m
0
with (10.2.1) I m m
ILsU(x)xrlLxTt = I fl(t) [ j e-(t+s)x ~ ~ + ~ ' d x ] dt 0 0
m
dt -r-k'-1 = r(r+k'+l) f fl(t)(t+s)
= r(r+l)gz T ~ .
0
Note. There may exist a similar theorem for $ T when Tt c 9'. s t - 10.4. Characterization of Stieltjes Transforms
We shall show in this section that the Stieltjes transform of
distributions satisfies several desirable properties.
Theorem 10.4.1!2) If Tt c J I ' (r) and Fr(s) = $3,, we have the
following properties
212 Chapter 10
(ii) Fr(s) is a holomorphic function of s in the region larg s( < n
8 # 0,
(iii) there exists a number 0 >
1s1@IFr(s) I is bounded as
Proof. We have - dm (.- 1 - F,(s) = - asm
0 ( 0 may be dependent on r) such that
181 + m in the region larg sI < n.
dt dk' -r -m- 1 m
1 fp) T( t + S ) m+k' r (r+l+m) 0 dt r (r+l)
m r r+l+m) -r-m-1, (-1) -i$yq~- <Ttf(t+s)
and hence we conclude
which proves (i) ,
2. (ii) is a consequence of (i) . 3. Put s = wei? 191 < TI (5 = 1.1 and u = tlw. By the conditions
imposed on fl(t) in Theorem 10.1.1, we can find out three
positive numbers, M, N and a ' , a ' < r+k'+l such that
If,(t) I < Mtr+k"a' if t 1 . N .
If 0 < t < N < W , then It+wei4f = ut$ + ei91 < 2 W.
-r-kv-l < 2-r-k' -lW-r-kl-l -r-k'-l Because r+k'+l > O f we have I t+eib I <o
Since w > N, we have by (10.2.1)
The last two terms are bounded by taking 0 such that 0 < 0 < a ' .
This completes the demonstration of (iii). Moreover, to illustrate
Stieltjes Transform 213
(iii) we ask the reader to note the following examples.
(a) If Bt is a distribution having bounded support in [O,m[
we have
r+l r (10.4.1) s SdsBt is bounded as I s [ + -
(b) If in the decomposition (10.1.3), If(t) I < Mt' as t + - and if A 5 k'-lt we have
(10.4.2) s $, Tt is bounded as Is1 + -. 10.5. Examples of Stieltjes Transforms
r+l r
We list below some standard formulae for the Stieltjes transfor-
mation when Tt E JI'( r) and Tt E JI' . 10.5.1. Examples when Tt"St' ( r)
For Tt E JI' ( r) , we have
(10.5.1)
(10.5.2)
(10.5.2')
(10.5.3)
(10.5.4)
(10.5.5) $:(FptI) = r-:
SS r Tt-c - - $s+ct r c > 0, 13)
r (r+l+m) r (r+l) $E 6 (m) (t-c) =
$: Tct = c r $:, Ttt c > 0 ; (3)
$: Tt = .m-/ 1
gSD r m Tt = 'r(r+l) r (r+l+m) $i r+mT tr m > O r 131
-r-m-1 c > 0, m=Otlt2,...
m
xr(ILxTt)e-SXdxt Re s > 0;
Re v r, v # -n; n=1,2,3t... B(r-v v+l)
S
(Fp is not needed here if Re v > -1);
(10.5.6)
r > -n;
where B(,) denotes Euler's function and r and ij denote the gamma
function and logarithmic derivative of the gamma function
respectively.
Proofs. 1. The proofs of (10.5.1) (10.5.2) and (10.5.2') - follow directly from the definition (10.2.1).
214 Chapter 10
2. The proof::of (10.5.3) can be obtained by using Theorem 10.3.1
and the proof (10.5.4) follows from (i) of Theorem 10.4.1.
3. First note that for Re u r we have
which can be analytically continued to complex v #'-n. obtain (10.5.5) . Thus we
By differentiation in (10.5.7) with respect to u and putting
v = 0, we have
-r -r-1 s gf;(log t+) = <log t+,(t+s) > = --=[log s-q(r)+q(ln = z(r,s).
Let D be a distributional derivative and Tt c JI' (r). We have
(10.5.8) gs(DTt) = <DTt, (t+s) > = (r+l) <Tt, (t+s)'r'2> r -r-1
Thus
r gS(D log t+) = (r+l) Z(r+l,s);
That is
Further calculations in (10.5.8) yield
$(D-"-~FP t;') = r (r+n) ept+ -1 , (t+s) -r-n> r r+
r (r+
(10.5.10)
= 7 r (r+n) Y(r+n-l,s)
in accordance with (10.5.9) after replacing r by r+n-1.
Moreover, we have
1 1 where Sn = 1 + +...+ - and (10.5.11)
l+n'
l'(r+n) -r-n
Therefore, according to (10.5.10) and (10.5.11) we have
gr CFptin1 = n-lr r+n) C Y(r+n-l,s)+Sn s-r-nl S
where $(1)+Sn=$(n+2). Hence (10.5.6) is also obtained.
Stieltjes Transform 215
10.5.2. Examples when Tt E JI'
We have for a > 0
&Is [u(a;t)] = log s s+a ;
1 $, [ 6 ] = - , if a 2 0;
's [&a
$, (Tat) = gaS(Tt) :
gs [ t-1'2 e-at~ = n s -'I2 eas erfc(af sf)
(10.5.12)
a a+s
(m)l = m!(a+s)-m-l, if a 1. 0, m = 0,1,2 ,...: (10.5.13)
(10.5.14)
(10.5.15)
(10.5.16)
where Erfc is the complementary error function.
Proofs. 1. (10.5.12), (10.5.13) and (10.5.14) follow directly - from the definition (10.2.1).
2.
3 . than every power of t as t + a. Hence it belongs to JI' as well as
to JI' (r) even if r < 0. (10.5.16) is obtained by calculating the
Stieltjes transform as a repeated Laplace transform. For instance
(10.5.15) follows from the definition <Tat, (t+s)-'>=<Tt, (t+as)-'>.
For (10.5.16) we first note that t-' decreases more rapidly
nx[t-' e-atl = r 4 (x+a)+ and
zs c (x+a)-' I = -' eas Erfc (af s') s-'
so that we immediately have the formula
gs [ t-f e-atl = n s -4 eas Erfc (a' s ' ) .
Extensive tables of the Stieltjes transform are given in Erdelyi
(Ed.) C21 , Vol. 2, pp. 216-232.
Note. We mention below an intereshg relation concerning the derivative of the Stieltjes transformation when Tt behaves as a
function which decreases at infinity more rapidly than l/tn. We have
Then
216 Chapter 10
10.6. Inversion
In the preceding sections we have derived certain results
concerning, the Stieltjes transform $f of a distribution T all of these results give information about the transform gS when the distribution Tt is prescribed. In this section we shall consider the
converse problem, that of deriving information about the distribution
Tt when we have some knowledge of its Stieltjes transform.
is prescribed any formula enabling us to derive the form of the
distribution Tt is called an inversion formula for the Stieltjes
transform.
and it is defined by $E(Sr);l F(s) = F(s).
Almost t'r
When $:
We denote the inverse Stieltjes transformation by ($r) ; I I
Now we state the main results of this section and our discussion
of these results are entirely equivalent as indicated in Lavoine
and Misra C31.
Theorem 10.6.1 (Inversion theorem). If the function F(s), in the
domain of the complex plane larg sl < r , s # 0, satisfies the properties ( 4 )
(i) F ( s ) is holomorphic
and
(ii) there exists a number f3 > 0 such that Is181F(s) I is bounded as
191 + - 1
then the inverse Stieltjes transform of F(s) exists and is unique.
Set f (x) = xer lLil F(s) , where the function f (x) can be continued analytically in the half z-plane Re z > 0 and let it be
denoted by f(z).
function or a distribution having support in C 0 , m C , then we have ILL1 f ( 2 ) , T being a Further, if we put Tt =
t
(10.6.1) ($r);l~(s) = r(r+i)Tt.
- Proof. We proceed here according to the properties of (i) and
(ii) of Theorem 10.4.1 in order for F(s) to be a Stieltjes transform
in the sense of distributions in S'(r) . By virtue of (i) and (ii) and from Theorem 8.7.1 of Chapter 8,
S I F ( s ) exists and is a unique function h(x) which is null for x < 0.
Ftrther, I$ F(s) is locally sununable because this function is a
Stieltjes Transform 217
derivative of a continuous function. Hence, we have
(10.6.2) F(s) = I h(x) e-sXdx, Re s > 0.
Set
0
0
g(x) = 6; s-'F(s) ,
and
f (XI = x-rg' (x).
Then
(10.6.2 I ) h(x) = xr f(x)
and
Let 0' = n-q? e l 1 = - e l , o < 17 < */loo. Since F(s) is holomorphic
for larg S I < IT, Cauchy's theorem gives
where w is a real variable. If x > O t one can differentiate with
respect to x in the above integral and multiplying by x-I
ie' c'x = -in w f(x) = e e jF(c'+eie'w) e-(xe dw
2ni xr 0
By substituting z in place
ation we obtain
(10.6.3) e f(z) = -c'z
According to property (ii)
of x and employing the Laplace
we obtain
tr an sf o m -
and from Theorem 8.3.1 of Chapter 8, the
function e-clzf ( z ) is holomorphic in the angle (arg z I < 5 - n and there exists an integer k L 0 such that I z I - ~ le-C'Zf(z) I is bounded as 1.1 + m. (More precisely, Theorem 8.3.1 of Chapter 8, employed
218 Chapter 10
here with c = 0 and c(T) = 0; and F(c'+efiE'x) plays the role of Tx
in this theoren,.) Since 11 can be arbitrarily small, one can replace
the angle by the half plane Re z > 0 from which we deduce that f(z)
is bounded as I z I -+ -. Consequently, we may infer that f (z) satisf- ies the conditions of Theorem 8.7.1 of Chapter 8 from which we obtain
a unique distribution T having support in [-c',m[.such that
is holomorphic in the half-plane Re z > 0 wher I z I - k e'c'Re I f ( z ) I
t
(10.6.4)
Since c' is arbitrary but positive, Tt has support in [O,-[ . according to (10.6.2) , (10.6.3) and '110.6.4) , we have
f(z) = ILZTt, Re z > 0.
Now,
F ( s ) = I xr(lLxTt) e-sxdx. 0
Further, by making use of (10.5.3) we finally get
Consequently,
and hence (10.6.1) is obtained.
The following examples will illustrate to understand this
theorem.
Examples. Let F ( s ) = s-', Re s > 0. Then we have
v - 1 -1 -v x = T , Rev > 0. -1 r (v ILxF(s) = ILxs
Also , according to (10.6.1) we obtain
In other words
- tt-" if Re v < 1 1
B(v,r-v+l)
(r+l) (r+2). . . (r+n) 1 r-v
6 (n) (t) if v=r+l+n, n EN
in all other cases.
1
v,r-v+l) Fp t+
($) ;w=
Remark. F ( 8 ) must satisfy the condition (i)(5) in Theorem 10.6.1.
I n d e e d l ) - l , which is not holomorphic in the domain larg .s1 < r,
S t i e l t j e s Transform 219
is not S t i e l t j e s inversible because IL -1 (s 2 +1) = s i n t+ and z-rsin z+ t
i s not Laplace inve r s ib l e s ince nei ther c ' nor k e x i s t f o r which
Re z > 6 .
ls in zl i s bounded(6) as z -+ m i n any half-plane I I -ke-c ' R e 2
Therefore, Theorem 8.7.1 of Chapter 8 is applicable to z-r s i n z .
10.7. Abelian Theorems
Recall t h a t every d i s t r i b u t i o n Tt belonging t o S ' ( r ) i s S t i e l t j e s transformable of index r. W e def ine
(10.7.1)
(see Section 1 0 . 2 ) .
Throughout t h i s sect ion w e r e s t r i c t ourselves t o r e a l and pos i t i ve s i n the de f in i t i on of t he S t i e l t j e s transformation of index r.
The theorems with which w e s h a l l be concerned i n t h i s sect ion r e l a t e the behaviour of a S t i e l t j e s transformation a s s approaches zero o r i n f i n i t y t o the behaviour of Tt a s t approaches zero o r i n f i n i t y . Theorems of t h i s nature a r e ca l l ed Abelian theorems (see Misra E l 3 ) . W e study two types of Abelian theorems. The f i r s t theorem r e l a t e s the asymptotic behaviour of Tt as t -f O+ t o t h e behaviour of $: a s s -+ O+. of the transform near t he o r i g i n ' ( i n i t i a l value theorem) s ince it i s t h e i n i t i a l behaviour of T t h a t is considered. The second theorem discussed i s cal led 'behaviour of the transform a t i n f i n i t y ' ( f i n a l value theorem)since it relates the behaviour of Tt as t -+ - t o the behaviour of $f as s -+ m. Other proofs of these theorems can a l s o be found i n Lavoine and Misra C21 .
S
This r e s u l t i s r e fe r r ed t o as 'behaviour
t
10 .7 .1 . Behaviour of the transform near the o r ig in
W e now s t a t e our main theorems concerning the behaviour of the d i s t r i b u t i o n a l S t i e l t j e s transformation' near the or igin.
Theorem 10.7.1. Tt E JI' ( r ) and i f i n the sense of 1, Section 6.4.1 of Chapter 6,
(10 .7 .2 ) T~ m p ( t v l o g j t ) + a s t + o+
with R e v < r , v # -n-1, n E JN , then w e have r v - r (10.7.3) B, T~ - m(v+I,v-r) s log's a s s -+ O+.
220 Chapter 10
- Proof. According to Theorem 8.11.1 of Chapter 8, we have
xr I L ~ T ~ - A(-U j r(v+i)x r-v-1 logJx as x + m .
Hence (11.7.3) is obtained by formula (10.5.3) with the use Of
Theorem 8.11.1 of Chapter 8.
Corollary 10.7.1. If Vt = DmT .;and if Tt has the property t (10.7.21, then
Ar (v+l) . r (r+m-v) sv-r-m j log s as s -+ O+. (10.7.4) r (r+l) This is a consequence of formulas (10.5.4) and (10.7.3).
Theorem 10.7.2. If T~ E J I ' ( r ) and if in the sense of 2 . ,
Section 6.4.1 of chapter 6,
(10.7.5) Tt I MP(t -n-l logjt)+ as t + o+, n E IN.
then
(10.7.6) r ( - i l n r (r+l+n) s-n-r-l log j s as s -+ O + .
gs Tt - A nl (j+iTr (r+lr Proof. The proof of this theorem is analogous to that of the
preceding theorem.
8.11.1 of Chapter 8.
Here we use Theorem 8.11.2 h a of Theorem
Corollary 10.7.2. If DmTt = VtI m E lN and if Tt has the
property (10.7.5) , then
(10.7 7) r Vt - A
This is a consequence of
-m-n-r-llogj+ls as ~ o+.
formulas (10.5.4) and (10.7.6).
10.7.2. Behaviour of the transform at infinity
In this section we establish the behaviour of the distributional
Stieltjes transformation at infinity.
Theorem 10.7.3. If Tt E JI' (r) and if
(10.7.8) Tt = At"1og't
for t > tl > 1 and -1 < Re v rr then
(10.7.9) ii$~~ ~~(v+l~r-v)s~-~log j s as s + m I larg s l < n/2.
Stieltjes Transform 221
Proof. According to Theorem 8.11.3 of Chapter 8, we have - r lLxTt - A(-l)jr(v+l)x r-v-llogjx as x -+ o+.
Hence (10.7.9) is obtained by virtue of the formula (10.5.3) and
using Theorem 8.11.1 of Chapter 8.
Corollary 10.7.3. If DmTt = VtI m E IN I aqd if Tt has the
property (10.7.9) I then we have
(10.7.9) log's as s -+ m I larg sl+ . $iVt ., Ar (;;ii;;r+m-w) sv-r-m
This is a consequence of formulas (10.5.4) and (10.7.9).
10.8. The n-Dimensional Stieltjes Transformation
The results obtained in the preceding sections enable us in this
Here, we use the following notations and terminology (see also
section to give the structure of n-dimensional Stieltjes transforma-
tion.
Section 4.5 of Chapter 4).
As customary we denote the point of an n-dimensional real space
lRn by X = (X~~X~~...,X~) and by Qn(0) we mean the set of point X
such that all the x > 0 for j = 1121...,n. The point of an n-
dimensional complex space Cn is given as s = ( S ~ , S ~ ~ . . . ~ S ~ ) and
k = (klIk2,...rkn)I k = kl.k 2.....k n where kn is a non-negative
j -
-
integer. The symbol Dk stands for D' = Dkl :2....D> in the x1 x2 n
distributional sense.
r = r1 r2....r
Throughout this work the notation A denotes the set of 8 E Cn such
that s $ I - m , O [ for all j, i 5 j - < n.
10.8.1. The space J;(r)
By r we mean r =(rlIr 2r...Irn) and - where the complex numbers r are such thar Re r > -1.
n j j
j
By J;(r) we denote the space of distributions in n variables Tx
which can be expressed in the form T = D f(X) where f(X) i s a locally X summable function in IRn and zero outside of Qn(0) such that for-any
1 conventional positive number a = (alIa 2 1 . . . , a 1 I f(X) =
as 1x1 -+ m for each k E INn.
i;
r+k-a O( 1x1 n
If we put r = 0 in J;(r) then we write JA instead of JA(0).
222 Chapter 10
10.8.2. The Stieltjes transformation in n variables
Let TX E JA(r). Then its Stieltjes transformation of index (or
multi-index) r for s E A which we denote by $ETx is defined as
(10.8.1) m m
BETx = A(r,k) I.. .If (X) (xl+sl)-rl-kl-l.. . (x +sn) -r n -kn-l n
&n
0 0 axl ax 2....
where
r (rl+kl+l). . . r (rn+kn+l) r (rl+l) . . . . r (rn+l) A(r,k) =
If k=(O,. . . ,0) and TX = f (x) , then we obtain ordinary Stieltjes transformation in n variables:
(10.8.2) $,f (X) = I.. . . . If (XI (xl+sl) ... (xn+sn) n m m r -rl-l -r -1
0 0 dxl dx 2... axn.
If TX = BX (1.e. any distribution having bounded support in
Qn(0) ) then we get
(10.8.3) $=B = <B ( s x x' x1 1 +s ) -rl-l ...(x,+s,~-~n-~> s E A .
If TX E JA, then we obtain Stieltjes transformation of index
zero such that m m
-k2-1 (10.8.4) $STX = A(0,k) I.....'If(X) (xl+s1)-kl-1(x2+s2) .... 0 0
(xn+sn) -kn-l dxl dx2.. . dxn where A(0,k) = r(kl+l) r(k2+l) ,.... r(kn+l). 10.8.3, The iteration of the Laplace transformation
The results obtained in Section 8.13 of Chapter 8 permit us in
this section to obtain Stieltjes transformation in n variables by
means of an iteration of Laplace transformation in JA(r).
Theorem 10.8.1. If s E A and if TX E JA(r), then we have
where t = (tl,t2, ..., tn) (all real t.) and w(r;t) is the function 1
such that rn tl',ti ,..., t if all the t. are positive n 3
w(r;t) = (' elsewhere,
Stieltjes Transform 223
and IL as usual denotes the Laplace transformation in n variables.
r -(s.+x.)t Proof. Since (s.+x.) -r j -1 = ~+ <u(tj)tjj8 e - 3 3 r r + )
3 J j,
J
with u(t.) is the characteristic function of the half axis t > 0,
we have
(10.8.6) n (s.+x.) 3 = <Y(r) w(r;t), e >
where
7 j
-- n -1: .-1 -xt-st
j=1 3 3
j;F = Xltl + x t +. . . .+x t 2 2 n n - st = s t + s t +".+ sntn.
1 1 2 2
Now, by making use of (10.8.6), the second term of (10.8.5) takes the
form by tensor product of Schwartz (see Section 5.8 of Chapter 5): -- -xt-sr
S ~ T s x = eX,C<Y(r)w(r;t) > I > -- -xt-st
= <TX @ Y(r)w(r;t)8 e
= Y(r)<w(r;t),(<TX8e >) e >
> - - -xt -st
- -st
= Y(r) w(r;t),ILtTt e
= Y ( r ) <w(r;t).ILtTX, e-st> = Y(r)IL w(r;t)ILtTX. -
S
Hence, we obtain our desired result (10.8.5).
w(r;t)3LtTX is the Laplace transformable in tl t2,....t n can be shown
easily as in the case of one variable (see Section 10.3).
Here, we remark that
Theorem 10.8.2. If Tt E J;(r), then we have the following:
where r(r,m) = (rl,r2 ,..., r.+m, rj+18 ...., rn);
3
(ii) $: is a holomorphic function of s in the region s E A ;
(iii)
lsll
there exist n positive numbers ( B1, B2,. . .p 8,) such that
61 B2 Is2( ....? /snlBn $: Tt is bounded as Is1 -f - in the region SEA.
Chapter 10 224
- Proof. The proof can be formulated easily by the iteration of
the case of one variable given in Section 10.4.
10.8.4. Inversion
In this section we prove the following inversion problem.
Let F(s1,s2,...,sn) i.e. F(s) be a given function. NOW, we have
to find out a distribution Rx such that $: RX = F(s1,s2 , .. . ,s 1 . this case we call the distribution Rx as Stieltjes inverse of F ( s )
and denote it by % = ($ )x
In n
r -1 F ( s ) .
Theorem 10.8.3. Let F ( s ) be a function of n complex variables
satisfying the properties (ii) and (iii) of Theorem 10.8.2. If there
exists a function O(t) = $(t1,t2,..,tn) of n real variables t
that IL$(t) = F(s) and if there exists a distribution RX having
support in Q (0) belonging to JA(r) such that
such j
n r (rl+l) . . . . . r (rn+l) 0 (t)
I L R = (where w(r;t) is given in
Theorem 10.8.1) then we have (&fr)il F(s) = RX.
t X w(r;t)
- Proof. By making use of the given hypothesis on #(t) and by means of (10.8.5) we get
ILsw(r;t) IL R S: % = r(rl+l~.ser(rn~l~ = I L ~ o ( ~ ) = F(s).
Consequently, we obtain
(Sf');' F ( s ) = % which is our desired result.
10.9. Applications
The applications of the Stieltjes transformation in a distribu- tional setting have been studied by Tuan and Elliott [11 and McClure
and Wong [11 . of Stieltjes transformation may play an important role for such
applications and we leave the use of this setting to interested
readers .
We remark here that the present distributional setting
10.10. Bibliography
In addition to the works cited in the text w e should like to
mention the following references which also contain the works of
Stieltjes Transform 225
StieLtjes transformation in a distributional setting:
Camichael [11 , Carmichael and Hayashi [11 , Carmichael and Milton C11, Erdelyi C21, Misra C 61, Pandey Ell, Silva [ 51 and
Zemanian C31.
Footnotes
similar results can be obtained if r is any complex number such that Re r > -1.
there may exist a similar theorem for SsTt when Tt E JI' . rules of calculus.
see Theorem 10.4.1.
the condition (ii) is also necessary. For instance take F(s)=l
which satisfies (i) but not (ii). From this, we may infer that
F ( s ) = 1 is not Stieltjes inversible. To see this note that
If: 1 = 6(x) and X-~&(X) is not a defined function.
analytically continued to the half plane Re z > 5 > 0. This
yields f (2 ) = 0 and hence ILt f ( z ) = 0. yields (#);'(l) = 0 which contradicts the fact $ z ( O ) = 0.
put z = cl+iy, 5 ' > 5 and let y tend towards a.
But it is
-1 Now the present theorem
This Page Intentionally Left Blank
CHAPTER 11
THE MELLIN TRANSFORMATION
Summary
It is well known fact that the Mellin transforms occur in many
branches of applied mathematics and engineering. They play an
important role in electrical engineering and the theory of integral
equations. (See for examples, Gerardi [l], Fox c11, Handelsman and Lew C11 and C21.)
The theory of Mellin transforms has previously been studied in
a distributional setting by Zemanian in, €or instant, Chapter 4 of
C31 . The object of the present chapter is to show how the theory
can be extended further by working with distributions in a form
suitable for those whose interest lies in applications.
The organization of this chapter is as follows. Section 11.1
presents some classical results of the Mellin transformation and we
summarize the requisite construction and properties of the spaces
E and El in Sections 11.2 and 11.3, respectively. Further, by
using these results we formulate the distributional setting of Mellin a,w C L I W
transformation in Section 11.4 and also with respect to this setting
we obtain its examples, characterizations, rules of calculus, and its
relations to the Laplace and Fourier transformations, inversion and
convolution in Sections 11.5 to 11.11, respectively. Moreover,
Section 11.12 provides an account of the asymptotic behaviour in terms
of Abelian theorems for the Mellin and inverse Mellin transformations
in a distributional setting.
Finally, we present in Section 11.3 and 11.4 the use of Mellin
transformation €or obtaining the solutions of some integral and
Euler-Cauchy differential equations in a distributional setting and
the chapter ends by describing the use of Mellin transformation to
solve some problems in potential theory having generalized functions
boundary condition.
2 2 7
228 Chapter 11
11.1.Mellin Transformation of Functions
In this section we give some classical results of Mellin
transforms which we need subsequently in the distributional setting
of Mellin transformation.
Let us consider the function f(t) which is absolutely integrable over 0 < t m and which satisfies the following conditions:
(a) f(t) is defined for t > 0;
(b) there exists a strip al < Re s < a2 in the complex s-plane such that ts-’f (t) is absolutely integrable with respect to t over 10,-[.
The Mellin transformation is an operation I that assigns a S
function F(s) of the complex variable s = a +iw to each locally
summable function f(t) that satisfies conditions (a) and (b). The
operation lMs is defined by
(11.1.1) F(s) = Isf(t) = f(t) ts-’dt.
By the condition (b), (11.1.1) converges absolutely for all s in the
strip al <Res< a2, which is called the strip of definition for Isf.
W
0
It is useful to remark that if we put t = e-u in (11.1.1) , we obtain
ca
F ( s ) = Is f (e-U) = f (e-U) e-sudu -W
which is the bilateral Laplace transfornation.
2’ If we put t = e-2nu and s = a+iy for real y, al < a < a
we have
(11.1.2) m
du -2nu -2nu1 .-2nau e-i2nyu F ( S ) = mSf(e = 2n j f(e -m
-2nu -2rau = 2n IF f(e ) e
where IFdenotes the Fourier transformation as indicated in Chapter 7.
Now we state the inversion theorem for the Mellin transformation.
Theorem 11.1.1. Assume that over the strip a < Res< a2, F(s) 1 is analytic and satisfies the inequality.
(11.1.3) I F ( s ) ( 5 Klsl
where K is a constant.
-2
Then IMi1F(s) (inverse Mellin transform) is
Mellin Transform
given by
229
and IM;'F(s) converges to a continuous function f (t) for t > 0 whose
Mellin transform is F ( s ) . Proof. Note that f(t) does not depend on 2, a fact which
follows directly from Cauchy's theorem and the bound on F(s). -
Since F(s) is analytic in s = a+iw and IF(s) I 5 K ~ s I - ~ , we have m m
(11.1 i 5)
Also, we may put
fIF(a+iw)t-iW)dw 5 I IF(a+iw) Idw < -. -m -m
1 - =-oo
(11.1.6) taf (t) = 2 1 F(a+iw)t-@dw
Thus the integral in (11.1.6) converges uniformly for all t > 0,
which implies the continuity of f(t) (see Apostol ClJ, p. 438 and p. 441).
Finally (11.1.5) and (11.1.6) demonstrate that for each choice
- a in the interval al < a < a2 , taf (t) is bounded on 0 < t < m. Hence,
we conclude that (11.1.4) converges to a function f (t).
Now we show that (11.1.4) gives the inverse of F(s), i.e.
f (t) = IM['F(s). For this purpose we need to show that
Nsf(t) = F(s)
if
. We have -2nu Put t = e
If we put s = a+iy in (11.1.8) , we get
(11.1.9)
where denotes the inverse Fourier transformation as indicated in
Chapter 7. Furthermore, by making use of (11.1.2) we finally obtain
(11.1.10) IMf(t) = 2slE'f(e
m
2sf (e-2ru) e-2nau = JF(a+iy)ei2nUYdy = H F(a+iy). -m
IF F F (a+iy) =F (a+iy) =F ( s ) . -2nu) e-2nau =
230 Chapter 11
It is important to remark that the inverse Mellin transform
depends on the analytic strip where F ( s ) is considered.
- Remark. In order for IM;lF(s) to be represented by a function
it is not necessary I F ( s ) 1 < Kls12should be satisfied. Thus
(discontinuous)
(discontinuous) -ls-l - Re s < 0, IMt - -U(t-l)
where
(11.1.11)
Let us now
1, O Z t L l
0 , elsewhere. i U(1;t) =
ist a few relations of this Me in transformation.
Examples. If F ( s ) = lMsf(t), for a1 c Re 8 a 2 # then
(11.1.12) lMs[tvf(t)l = F(s+v), al - Rev<Re s < a2 - Re v ,
and
(11.1.12') ms[emkt tVl = r0, ks+w s + - v , -v-~, -v -2 ,.... Proofs. (11.1.12) follows directly from the definition (11.1.11).
We now derive the formula (11.1.12'). Taking k = 1, we have m
ws[e-t] = f e-t ts-ldt = r ( s ) , Re s > 0. 0
Similarly, operating with tw e-kt, we obtain m
, Re s >-Rev. .-kt ts+~-ldt - l ' (s+v) ms = J ks+v 0
Now by analytic continuation,we have m
, s # -v, -u-ls -v-2. I 0
tv .-kt ts-ldt - r(s+v) ks+v
In the next sections, we shall generalize this transformation
to functionals (generalized functions) belonging to E' which are
spaces that contain, among many others, the distributions of bounded
support in 1 0 , m C .
a,u
11.2. The Spaces E a l u
Our study made on the spaces of base functions (see Chapter 2)
Mellin Transform 231
enables us to construct the spaces Ea in the following manner. I W
(p,q are finite real numbers with p q) we denote the
linear space of infinitely differentiable functions Q(t) defined on
1 0 , d and such that there exist two strictly positive numbers 5 and
5 ' for which
By EP,q
(Q(t) will be null for t < 0.) We set
, t'O
tS-l, + [; t < 0,
s-1 so that t+ belongs to E if p < Re s < q. Put Ptq
t-P,
t-9, t 2 1,
0 < t 5 1, k (t) = PI9
and
(11.2.1)
is a norm. k P I q
The ykIPIq($) are all bounded and are semi-norms;
We now provide the following topolocJy in E PI9.
A sequence I $ , 1 + 0 in E if and only if the yk (Q j) + 0 1 Prq IPIq for each k c IN. Thus, E is provided with a structure of a
countably multinormed space (see Zemanian [ 3 ] ) . A l s o , we have
algebraically and topologically
P,q
(11.2.2) E c EpIIqlI if P' 5 P < q 2 q'* PIq
Note that, E-m is the inductive limit of E as p + -m (see Plq Iq
Garnir, Wilde and Schmets, [I] , Vol.1, p. 121). This means that
a sequence {$ j l -+ 0 in E--
such that
if and only if there exists a p < q I 9
and {Q.) + 0 in E (i.e. Y ~ , ~ , ~ ( Q ~ ) + 0 as j + -). 'j ' Ep,q 3 PIq
In a similar manner, we can show that E is the inductive limit of P I r n
232 Chapter 11
E as q + 0 , and E-- is the inductive limit of E as p + -- and q -+ -. Plq 1 - PI4
From these results, we have the following inclusions:
E C P?q
E C E C P?q PI"
We denote all these spaces by E where a can be finite or -- and a,u
w can be finite or - . Moreover if a' 5 a < w 2 u'. (11.2.3) Ea,u= Ea'"''
Also, we have ID (I) C E where ID(1) is the space of infinitely
differentiable functions q(t) having compact support in I where I
denotes the open interval 10,- C : a sequence {%'.I + 0 in I D ( 1 ) if and
only if the { Y i k ) l + 0 uniformly on every compact subset of I.
a,u
7
Let {aj(t) I e ID (I) , j E IN , be a sequence of functions such that a.(t)-1 -+ 0 uniformly as j + - on I with the same being true for each of their derivatives.
3 ID (I) and converges to $(t) in the sense of Ealu. conclude that ID is dense in E
11.3. The Spaces El
7 If + E E a F W , then a. (t) @ ft) belongs to
Consequently, we
a,u'
a I u
By using the properties of generalized functions and distribu-
tions (see Chapters 3 to 5), this section provides the structure of
distributions in El as follows. a , u
By EArw we denote the linear space of continuous linear
functionals on E which vanish on I--,O [ . Consequently I E;,u is a.w the dual of Earu.
More precisely, V E E' if and only if the following four a l u
conditions are satisfied:
1.
2.
3.
4.
<Vt,4 (t) > is a complex number defined for each 4 E E
<vt,$(t) > = 0 if $(t) = 0 for t > 0;
<v ,c 0 + c $ > <=>
: a ? u
c I t 1 [<V ,Q >] + C2[<Vtr02>] , c1,c2 E c. t 1 1 2 2
<Vtt$j> + 0 if the sequence b$.} -+ 0 in Ea as j + - . 3 I W
Mellin Transform 233
Inclusions. Froin (11.2.3) we conclude
where A and n are numbers.
Boundedness property. Let E' denote the dual of E If P19 p,q'
V E E' (p,q being finite real numbers such that p < q), then we
have PI9
(11.3.2)
where M > 0 and the integer K depends upon V but not on 4 . (See
Zemanian C31, pp. 18-19, and Garnir, Wilde and Schmets Ell, Vol.1, p. 159.)
Examples. Every distribution B with support contained in La,b],
0 < a < b < - belongs to all E' and therefore it belongs to El, . PI9
When W,$> is given by an integral, that is m
<V,$> = 1 f(t)@(t)dt, 0
then we can identify (as in the distributions of ID') the function
f(t) with the functional with which it is associated and hence we
denote Vt by f (t) . If P(m,a;t) is a function of t with support [O,al such that
t-mP(m,a;t) is summable and bounded as t -f 0+, then P(m,a;t) belongs
to Elmlm and therefore it belongs to all ELm with q > -m.
If Q(-n,b;t) is a function of t which is null for t b with
support in Cb,mC such that t"Q(-n;b;t) is locally summable and
bounded as t + m , then Q(-n,b;t) belongs to E-m - and therefore it belongs to all El with p < n.
en'
P rn
If Vt = Bt + P(m,a;t) + Q(-n,b;t) with n > -m, then Vt E
Note that eYt E E; *further all the summable functions with a ,a'
bounded support in [O,-[ belong to E; . I -
The distributions 6 (t) and 6 ( k ) (t) do not belong to all the because the functions of E are not all continuous at the
0 a l w origin.
234 Chapter 11
The pseudo functions Fp t-'eTt, r 2 1, do not belong to E ' if PI"
p < r because in general the derivatives of order > r-1 of $ E E
do not exist at the origin. But Fp = t e+ is in E' PI" if -r -t
PI"
P L =. For real or complex z , tf, which belongs to $' does not belong
to any E:,w.
z > q which violates the condition p < 9.)
(If it did then tf would belong to E' with p > Re PI9
For each V E E' and for each JI E I D ( 1 ) , <vt,JI(t)> exists and defines a linear functional which is continuous for the topology of
m(1) ; therefore V belongs to I D ' ( 1 ) , the space of distributions on the open half axis C o r m C. Hence we conclude that E ' are spaces of
distributions. As we shall see later, the spaces E ' are the spaces
of Mellin transformable distributions.
11.3.1. The multiplication in E '
a l w
t
a,w
a ? W
a,w
If z is a real or a complex number, one can see easily that if
then tz$(t) E Eafu. a- 5 , w-5 C = R e z a n d $ E E
Let V E E: Then its product by tZ is the element of E:-Erw-E I
denoted by tZVt and defined by
(11.3.3) <t Vt,$(t)> = <Vt,tZ $(t)> Y 0
For example, tSeit E Eic , m , since eYt E E;, function such that the mapping
z
. Moreover if m (t) is a ? m
with a+A<w+v, that is, if a On Ea+A ,w+v is an isomorphism of Ea+A,w+v
sequence($.(t)) + 0 in Ea then this implies that the sequence 3 ? w
(t) = C $ (t) /m (t) 3 + 0 in E and reciprocally, that the product a,w
and is defined by m(t) Vt E:+A,w+v
(11.3.4) <m(t) Vtt $(t)> = <Vt, m(t) $(t)> , Y $ E E ~ + ~ , ~ + , , .
11.3.2,The differentiation in E' a l w
' E IN, then $ ( j ) (t) E a+ j ,w+ j ' 3 We can show easily that if $ E E
This leads to the following definition: %a,w'
If Vt E ELlw, then its derivative of order j is element of
Mellin Transform 235
denoted by D3Vt and defined by 'A+ j , w+ j
1 ' @ EEa+j,u+j. (11.3.5) <D(J)Vt,@(t) > = (-,)I <Vt,@ (j) (t) >
Examples. Let
forO(t(a
elsewhere.
1 -
U(a;t) =
Then, we have for every + E El that
Hence,
(11.3.6) DU(a;t) = -6(t-a) in Ei
Problem 11.3.1
I - *
Prove that if c 2 0 and real A I then
-ct D eTct = -c e+ in E; i
D [ u (a;t) e-"'j =-cU(a;t)e -Ct-e-cag(t-a) in E;,-~ and also
on $ n E~ * , m t
I - (1)
(ii)
(iii) D Cu(a;t) e -ct t A 1 = U(a;t)(A-ct)t '-1 e -ct-a'e-cag (t-a) in
Ei-A,-*
11.3.3. Comparison with Zemanian spaces (see Zemanian C3l )
Let MaPb denote the space of all smooth complex valued functions
e ( t ) on lo , - [ such that for each non-negative integer k
(11.3.7)
where
and consequently M'
p. 103.)
denotes the dual of Malb. (See Zemanian C31, arb
Note that the norms y k defined in Section 11.2 corresponds rP19
236 Chapter 11
of (11.3.7). L e t ,+,(t) E E Then t-P+k+l'c,+,(k) (t) 3 0 to 'p,q,k Pl9'
a s t + O+ f o r c e r t a i n c > 0 r equ i r e s t -p+k+l,+, (k) (t) + 0 as t -+ O + ;
+ 0 i s bounded f o r 0 < t < 1. Thus, E i s a smaller space than M and hence we conclude t h a t El i s l a r g e r than MI Also
while f o r every e E M it is s u f f i c i e n t t o remark t -P+k+le (k) ( t ) P?9
P?q Prq' P?9 P?q '
O , < t < l [: elsewhere g ( t ) =
does not belong t o M' w e have
* while g ( t ) E EA
1
, because i f ,+, E Eo O , - . ' I - I - ,
< g ( t ) , , + , ( t ) > = O(t)dt . 0
11.4. The Mellin Transformation
The r e s u l t s obtained i n t h e preceding s e c t i o n s enable u s i n t h i s s ec t ion t o formulate t h e d i s t r i b u t i o n a l s e t t i n g of Mellin transforma- t i o n i n t h e following manner.
L e t Vt E Ei
complex variable s denoted by lMsV and def ined i n t h e s t r i p S = { s , a < R e s<w} by
(11.4.1)
Then i t s Mellin t ransform is t h e func t ion of a I W '
a l u
It%! v = <v ts-1 S t ' + '-
For b rev i ty , w e use t h e no ta t ion v ( s ) i n s t ead of IMsV. T o conform with e s t ab l i shed terminology, we s h a l l say t h a t every d i s t r i b u t i o n belonging t o El i s Mell in t ransformable.
a,w
L e t Sv denote t h e widest of t h e s t r i p s S corresponding t o V ; a,w
then w e cal l Sv t h e (maximal) s t r i p of existence of IMsV.
absc issae which de f ine t h i s s t r i p w i l l be c a l l e d t h e absc i s sae of ex is tence ( i n f e r i o r o r supe r io r ) . W e s h a l l see i n Theorem 1 1 . 6 . 1 t h a t Sv i s the widest s t r i p p a r a l l e l t o t h e imaginary a x i s on which v ( s ) i s holomorphic.
The t w o
A s i n t h e examples of Sec t ion 11.3; w e have
i f V = B , Sv is t h e whole plane;
i f Vt = Bt + P ( m , a ; t ) , Sv i s t h e half-plane R e s > -m;
i f Vt = Bt + Q(-n ,b ; t ) , Sv is t h e half-plane R e s < n;
Mellin Transform 237
if Vt = Bt + P(m,a;t) + Q(-n,b;t), Sv is the strip -m<Re s < n.
When Vt = f (t), a function of the form P(m,a;t) + Q(-n,b;t), then the definition (11.4.1) gives
m
(11.4.2) Wsf = 1 f(t) ts-ldt, -m < Re s < n 0
and we get (11.1.1). Hence P(m,a;t) is null for t > a and for t < 0
and Q(-n,b;t) is null for t < b. (See also Lavoine and Misra [ 4 1 . )
Remark. tf does not belong to any E' and therefore is not a,w
Mellin transformable. This is evident from the divergent integral
in (11.4.2).
11.5. Examples of Mellin Transforms
In this section we state a number of standard formulae for the
distributional Mellin transformation which will be utilized in the
subsequent study.
If a > O,?. # O,-l,-2, ..., v # -1,-2,..., we have
msU(t-a)tz = -a s+z (s+z)-l, Re s < - Re z ;
msU(a;t)tZ = a s+z (s+z)-', Re s > - Re z; (11.5.1)
(11.5.2)
(11.5.3) IMs 6 (t-a) = as-' in the whole plane;
(11.5.4) k s-k-1 in ms &(k) (t-a)= (-1) (s-1) (s-2) ,... , (s-k) a
the whole plane;
lMs Fp [: U(t-a) (t-a) 1 = aS+'B(v+l, -s-v) ,Re s+Re v (2). (11.5.5) ,
Proofs. 1. Formulae (11.5.1) to (11.5.4) follow directly from
the definition (11.4.1) . 2. Derivation of formula (11.5.5):
Consider first, Re v > -1, then Fp becomes classical. Thus m
Ms c U (t-a) (t-a) 1 = ( t-a)v ts-'dt a
dY s+v v -s-v-1
= a j (1-y) Y 0
s+v = a B(-s-v, u+l) , Re s > - Re v
t which follows from the change of variable, y = fi and the use of the
238 Chapter 11
definition of the Euler’s function (see Erdelyi (Ed.)[ll , Vol.1, p.9, Equation (5)). Now by analytic continuation (see Section 1.4 Of
Chapter 1) we obtain (11.5.5).
Other formulae can be found in Colombo and Lavoine c11, Laughlin
[ 11 and Sneddon C 21.
Problem 11.5.1
Prove that
IMsFpCU(a;t) (a-t)” 1 = a S+V (i) B(v+l,s), Re s > 0;
‘(ii) IMsFp CU(a;t)(a-t)-ll = a s-1 [Jl(s)+log C/a3; Re s > 0 (3) ;
(H) IMs [U[a;t)log(a-t)l = -as ~-~[J,(s+l)+logC/al ; Re s > 0(3):
(iv) MsFp [U(l;t)t-zllogtlA-ll = r(X) ( s - z ) - ~ ; Re s
(v)
Re z ;
IMsFp CU(t-l)t-z(log t)”lI = r(A)(z-~)-~; Re s > Re z.
In (iv) and (v), Fp is connected with t = 1 and not needed if Re A > 0.
11.6. Characterization of Mellin Transformation
In this section we shall obtain the characterization of the
Mellin transformation. For this purpose we first have the following.
Theorem 11.6.1.(Analyticty theorem). If V E Ellw, then v(s)=IMsV
is holomorphic in the strip S and therefore a t w
d p-1 (11.6.1) v(s) = <Vt, + log t>
sa,u’
is valid in S and hence v ( s ) is holomorphic Pt9
Proof. Because - d t ~ - l - - t ~ - l log t E E ds a,w -
right side of (11.6.1) exists.
in Sv. Also
in the substrip S of P19
if a Re s c w , the
Now we show that (11.6.1) is valid in each substrip S of S
(The equilities are P!4 a,w
where p and q are finite with a 5 p to be considered if a and w are finite.)
q 5 W .
Let As be a complex increment whose modulus 6 + 0. We put
Mellin Transform 239
V(s+As) - V(S) - <Vt' p-1 + log t>l As H = lim [ 6+ 0
= lim at, ts-'(tAs-l - AS log t)> 6+0
= lim C <vtth(~s,t) >I 6 +O
where
We write
Hence
2 6"Ilog t p
n! n= 0
PI9' It follows that, as 6 + 0, sup k (t) Ih(As,t) I + 0 for scS
t,O prq Therefore yo(h) -+ 0.
obtain yk(h) + 0.
S
By analogous process continuing k times, we
Hence we conclude H = 0 and (11.6.1) is valid in
Pl9'
On the other hand, it is evident that if s goes round a closed
then v(s) preserves the same determination.
From the above results we may infer that v(s) is holomorphic
Pl9' path in S
in the substrip S of S O I W . Pf9
Corollary 11.6.1. If B is a distribution having support in
[arb] , 0 < a < b < m , then lMsB is an entire-analytic function.
Proof. Indeed, B belongs to Elm and the preceding theorem is I m -
valid in SB which is the whole plane.
Theorem 11.6.2. (Boundedness theorem). If V E
exists a polynomial P(ls1) in each substrip having
s c S such that PI9 a t w
240 Chapter 11
(11.6.2) ImsvI < p(lslj.
There also exist a number A > 0 and an integer K 2 0 such that
(11.6.3) ImsvI < A M K
for s E S with Is1 > K. PI9
Proof. According to (11.3.1), V also belongs to E' Now by P19' -
(11.3.2) we have
Since
which is a polynomial in 151. Hence we get (11.6.2).
On the other hand, when K < Is!, we have
(lsl+l)(ls1+2), ..., (Isl+K) < ((sl+KIK < ZKIsIK.
Hence, IIMsVI < M 2K lslx
which show that (11.6.3) is valid with A = 2 M. K
Remark. In the statement of the Theorem 11.6.2 one can not - delete the condition on the finite width of the substrip. For
example, ms6(t-a) = as-' and msU(a,t) = as s-' whose growth is
exponential as Re s -t -. Theorem 11.6.3. (Distributions having bounded support). If B is
a distribution of order J with support contained in [a,b3 with
lzaLb< -, then there exist two positive numbers A and A' such that
(11 -6.4) (msBI < A I S I J bRe if Re s > J+1
(11.6.5) IIMsBI < A ' l s l J aRe if Re s < 1 with Is1 > J.
Proof. Let Y(t) E ID having support I = [a-n, b+nl and be equal n to t s K [a,b] . that (see Bremermann [l], Section 4.4, Lemma 1, p. 30)
Sine B E ID' J I there exists a number M > 0 such
Mellin Transform 241
(11.6.6)
where
of Y(t)
is arbitrarily small. Now, we have by the proposed structure
Y y ( k ) (t) = (s-1) (s-2). . . ( s - k ) ts'k'l if t c In.
Put Bk = sup I Y ( k ) (t) I and if Re s > k+l, then we have tcI
k bRe s-k-1 , since b 1. 1, k n Bk <Is1 sup Its'k'll < Is1
tEI
J bRe s Bk < 1s I bRe and consequently sup Bk s I OLki J
and putting this value in (11.6.6) we can estimate (11.6.4).
On the other hand, if Re s < k + l and if I s 1 c k, then we have
s-k-1 I < Is1 aRe S-k-l,since a 2 1, k Bk < I s I I t
tcIn
Bk < lslk aRe and hence sup < Is1 J aRe s OLkLJ
which gives (11.6.5) by making use of (11.6.6).
11.7. Rules of Calculus
In this section we give a number of operation rules that are
applicable to the members of E' and then show how these operations
can be transformed under the Mellin transform. Throughout this
section we assume that
For simplicity, we set for k E lN
Then for a > 0 and complex z we have
IMs (2 v +z v ) = z v (s)+z2v2(s), s E S" fl s ; 1 v2 1 1 2 2 1 1 (11.7.1)
(11.7.2) MsVat = a v(s) in Sv;
(11.7.3)
(11.7.4)
(11.7.5)
-S
IMsVl,t = v( - s ) , - W < Re s < -a;
mstZV = v(s+z), a - Re z < Re s < w- Re z;
I M ~ (log t) v = v(k) (s) in sV; k
242 Chapter 11
lMsD k V = (-1) k (s-k)kv(s-k) , a+k < Re s < w+k;
IMsD k k t V = (-1) k (s-k)kv(s) in Sv;
m s t k k D v = (-1) k (s)~v(s) in Sv;
Pls (tD)kv = (-1) k k s v(s) in Sv;
(11.7.6)
(11.7.7)
(11.7.8)
(11.7.9)
k k (11.7.10) ms (Dtlkv = (-1) (s-1) v(s) in sV.
, a3 = sup(a ,a ) and 1 2 Proofs. 1. Let V1 E El , V2 E E' al'wl "2fW2
- w3 = inf (u1,w2). According to (11.3.1), V1 and V2 belong to EE,
and also (zlV1+z2V2) belongs to Ei since this space is linear.
Further, by employing the Mellin transformation on (zlVl+z2V2)r we
obtain (11.7.1).
3tw3
3tw3
1 2. (11.7.2) follows from the definition <Vat,$(t) >=-Vt,p(t/a). ,
3. (11.7.3) follows from the definition <Vl,t,+(t)> =
- <Vt, t-2$ (l/t) >.
4. (11.7.4) follows from the definition (11.3.3).
5. (11.7.5) can be obtained from the definition (11.6.1) by
repeating this process k times.
6. (11.7.6) follows
7. (11.7.7) and (11
and (11.7.6).
8. For k=l, (11.7.8
from the definition (11.3.5).
7.8) can be obtained by combining (11.7.4)
2 gives lMstDV = -sv(s). Next IMs (tD) v =
s v ( s ) ? and by continuing this process k-times we obtain (11.7.9). Also, (11.7.10) can be obtained by an analogous
process fron! (11.7.7).
2
11.8. Mellin and Laplace Transformations
The structure of a distribution in El defined in Section 11.3 U I W
enables us in this section to establish the following relations between Mellin and Laplace transformations.
If b is a strictly positive number, then every V E E' admits a r w
the decomposition
Mellin Transform 243
(11.8.1)
where P E E'
its support contained in Cb,-C. That is
Vt = Pt + 0,.
has its support contained in C0,bl and Q E Elm,whas a I -
<PtI+(t)> = 0 for every + null on C0,bI and
<Qt,+(t)> = 0 for every + null on Cb,mC.
If a (a > 0) is the lower bound for the support of V, and if
b = a; then P = 0 and Q = V.
Let $(XI and e ( x ) be functions such that $(-log t)/t E E and a,u
8(log t)/t E E,,,u.
butions P -x and Q e e
By the Section 5.9.1 of Chapter 5, the distri-
are defined by
(11.8.2)
(11.8.3) <Q e ( x ) > = <Qtr e(log t)/t>.
<p -x' $ ( X I > = <PtI $(-log t)/t>, e
e
Now, it is often easy to utilize the following definitions:
<P
G:Q xI+(e 1 e > = <Qt, O(t)> , Y + E E - ~ , ~
$(e-X) e-X> = <Ptr+(t)> , Y + E E ~ , ~ e
x x
e
where P -x and Q
log b respectively.
have their supports bounded below by -log b and e e
If we put $(x) = e-SX with Re s > a and e(x) = esx with Re sew,
then (11.8.2) and (11.8.3) yield
Re s > a,
(11.8.5) IMs Qt - IL-s Q Re s < W.
Where IL denotes the Laplace transformation. Now, we deduce by
virtue of (11.8.1) that
- (11.8.4) mspt - ILSP, ,XI
e -
e
(11.8.6) MsVt - - ILsPe-x + L - s P a < Re s < w. e
With the help of this formula and the theorems of Section 8.3
of Chapter 8, we can get the results of the preceding Section 11.7.
Remark. The use of the bilateral Laplace transformation (which
244 Chapter 11
is not studied in this book) yields, instead of (11.8.61, a simpler
formula in which sL-s does not occur and which does not require a
decomposition into the form (11.8.1). On this topic, see Colombo
C11 and Zemanian C31, Chapter 4.
11.9.Mellin and Fourier Transformations
In this section the structure of a distribution defined in
Section 11.3.and the results of Chapter 7 enable us to establish the following relations between Mellin and Fourier transformations.
A distribution Vt and the interval I a l w C give rise to the
family of distributions denoted(4) by e'rxV -x and defined by e
-rx ,r-l (11.9.1) <e V - x l $ ( x ) > = <VtI $(-log t)>, r E Ia,wC ,
and
(11.9.2) <e-l% ,e(emx)> = at, tr-'e(t),
where each $(XI and each e ( x ) be such that tr-'$(-1og t) and tr-'8(t)
belong to the functions space on which Vt is defined. the main result of this section.
e
e"x
Now, we give
Theorem 11.9.1. distribution in the strip S a I w l it is necessary and sufficient that
r E 1 a , W[ , and e-r% therefore Fourier-transformable).
(11.9.3) v(r+2nic) = rc e"xV --x, 5 E IR.
In order for Vt to be a Mellin transformable
should be a tempered distribution (and e-x
If v(s) = MSVt, then we have
e
Proof. Let p,q be finite such that a 5 p < r < q 2 w (with
equalities being possible if a and w are finite). Let 6 and c 1 be such that 0 < < S-p and O < 5 ' < q - 5. Put
r-1 (3.1.9.4) q t ) = t $(-log t) I t ' 0,
$(XI = e (r-l)x$r(e-x) , x E IR.
If $ ( x ) E $ then we have that for all k E IN
tk+l-P-C
tk+l-q-~'@~) It) -+ 0 as t + m ,
(k)(t) + o as t + o+, @r
Therefore @,(t) belongs to E and hence it belongs to Ea PI4 I
Mellin Transform 245
Conversely, if $,(t) E Ea,u, then $(XI E $. To see this, we consider
their topologies and find that the spaces $ and E are isomorphic
by (11.9.4) and hence the first part of the theorem follows from
(11.9.1).
a,w
According to Theorem 11.6.2, v(r+2siC) can be majored by a
polynomial in 16 I as 151 + m. Hence the integral 1 v(r+2sic)$(E)d5 exists for all JI E $3. NQW, by virtue of the formui; (11.9.1) and
the commutative property of the tensor product, we have
m
tr-l 2nic <v(r+2nic) ,$(S) > = <C<Vt, t > I , $ ( E l >
> I , $ ( E l > -2sixc
= <[<e "XV -x' e e
eCx
e
r $ ( S ) > I > - 2 TI ix& = <e"3 C -
= <e-rXV -x, rX $ ( 5 ) >
= <F e-rXV -x, $(c)> e
which proves (11.9.3) and also confirms the first part of theorem.
We shall see in Section 11.11 that the distributions e-rXV -x
5
e constitute a convolution algebra.
11.10.Inversion of the Mellin Transformation
In the preceding sections we have derived the results of the
Mellin transformation v(s) when the distribution Vt is prescribed.
In this section, these results are considered in the inverse orien-
tation, that is, we begin with some knowledge of v(s) and seek
information about the distribution V t'
Theorem 11.10.1. If the function v(s) is holomorphic in the
strip S P?q
integer K 2 0 as 191 + m , then there exists a unique distribution
in E' called the anti-transform (or inverse transform) of Mellin
of v(s) and denoted by IM;lv(s), such that
(11.10.1)
of finite width where s-~+~v(s) is bounded for a certain
P rq
m milv(s) = v(s), p < Re s < q. S
Moreover,
K tK g(t) (11.10.2)
and
246 Chapter 11
(11.10.3)
where
lM;lv(s) = ( - l ) X ( t D ) K h ( t )
and g ( t ) and h ( t ) are taken equal t o zero f o r t < 0. Before g iv ing the proo€ of t h i s theorem, w e g ive t h e fol lowing needed lemma.
Lemma 11.10.1.If t he func t ion F ( s ) i s holomorphic i n t h e s t r i p - then 2 S (p,q a r e f i n i t e ) and i f s F ( s ) is bounded i n S PIq P?q '
r + i m [& L-im F ( s ) t-'ds, t > 0
(11 .10 .4 ) f ( t ) =
I t 0,
PIq i s t h e unique d i s t r i b u t i o n E E' with p < r q which is t h e a n t i - transform of F ( s ) for S ? t h a t i s
(11.10.5) Ex F ( s ) = f ( t ) .
Prq - Proof. By v i r t u e of t h e condi t ions on F f s ) , f ( t ) e x i s t s every
I f w e put t = eex and s = r+2niC; -
where and i s independent of r . then (11.10.4) g ives
m
d5 2nixs f (e-X) = erx I F(r+2niE) e -m
Hence
- (11.10.5') e rxf (e-X) = p x F ( r + 2 n i C ) .
Also, by t h e r e l a t i o n (11.9.3) w e have
F(r+ZniE) = IF e-rxf (eeX), 5
(11.10.6)
Since F(r+ZniC) belongs t o $ w e deduce from (11.10.5') t h a t e-rxf(e-x) belongs t o $;, and it fol lows by means of (11.10.6) and from t h e Theorem 1 1 . 9 . 1 t h a t f ( t ) belongs t o El and F ( s ) is t h e M e l l i n t ransformation of f (t) .
5 '
PI9
Now w e show t h e uniqueness of f ( t ) . For t h i s purpose, l e t us Put Yt = Wt-f (t) . Then suppose t h a t Wte E'
(11.10.7)
f o r any I M s W = F ( s ) . PIq
lM Y = IM w - lMsf( t ) S s t
Mellin Transform 247
which gives
(11.10.7 ' )
This implies Y = 0 i n E' Now, by Theorem 11.9 .1 , (11.10.7)
implies t h a t 1Fe ' lxY -x = 0.
Yt = 0 i n E ' the Section 11.9.
l M s Y = 0 on S p,q'
P,9* Therefore, e-rXY -x = 0 i n $ I . Hence
by v i r t u e of (11.9.1) and the explanations given i n e e
PI9 Hence w e conclude t h a t l M s W = F ( s ) .
Proof (of t he theorem 11.10.1). L e t r ' be a r e a l number e x t e r i o r -1 t o Cp,q].
t o Lemma 11.10.1 and is equal t o f ( t ) . Now, by the r u l e (11.7.9) w e have
P u t F ( s ) = v ( s ) (~+ r ' ) -~ . A l s o , lMt F ( s ) e x i s t s according
s J F ( s ) = (-1)' lMs ( t D ) j f ( t ) ;
hence by inversion,
IM-lsJF(s) t = ( - 1 ) ' ( t D ) J f ( t ) .
Since F ( s ) = v ( s ) / ( s + r ' I K , w e have K
v ( s ) = ( s + r l ) K F ( s ) = 1 (g ) rvK-1s jF ( s ) ; j=O
hence
K
j-0 = 1 (-1) 1 (j) r S K - j ( t D ) j f ( t ) .
-1 This proves the existence of t he anti-transform IMt v ( s ) , and the f i r s t p a r t of t he theorem follows.
L e t Q be an open i n t e r v a l contained i n ]p,q[ , and does not contain any of t he numbers 1,2,...,K. i n t h e complex plane such t h a t R e s B 0 , and put G ( s ) = ~ ( s ) / ( s - K ) ~ . Now, by lemma 11.10.1, G ( s ) has the anti-transform q l t ) . Consequently, by the r u l e (11.7.7), we have
A l s o , l e t S, denote the s t r i p
K K K (-1) YsD t g ( t ) = (s-K), G ( s ) = v ( s ) on S,,
hence
K K K (-1) D t g ( t ) = lMilv(s)
which proves (11.10.2) .
2 4 8 Chapter 11
Similarly, formula (11.10.3) can be established with the help
of rule (11.7.9). Also, the rule (11.7.6) can lead to a simple
inversion formula.
Remark. In the above theorem, if p has no lower limit, then
lMt v(s) EE-L,~; if q has no superior limit then IMt v(s) E EL,,; if these two conditions are simultaneously satisfied, then lMt v ( s ) E
- -1 -1
-1
E’ . -.v,m
An important remark on uniqueness. The above theorem states
But if the function v(s) is holomorphic in
the uniqueness of lMi1v(s) with respect to a properly determined
strip of holomorphia.
various strips parallel to the imaginary axis, which are separated
by the singular points, and in these strips v(s) is majorized by a
power of lsl,then there exist as many distinct anti-transforms
lMt .v(s) as strips.
function v(s), one should take care to choose the strip in which it
is holomorphic.
-1 A l s o , when we say the anti-transform of a
Problem 11.10.1
(i)
(ii)
(in)
(iv)
(v)
(vi)
(vii) lM~ls-l(l+s)-l = -(t-l)U(t-l) , Re s < -1;
Eli1 (s+z ) - ’ = -U(t-l)tz for the half-plane Re s < -Re 2 ;
lMil (s+z)’’ = U(l;t)tz for the half-plane Re s > -Re z;
lM;’r(A)(s-z)-’ = Fp U(l;t)t-zllog tl’”, Re s > Re z;
IMi’r(A)(s-z)-’ = e-i’’Fp U(t-l)t-z(log t)‘“, Re s < Re z ;
mi1 s-’(l+s)-l = u(1;t) (t-1) , Re s > 0;
IM;’ s-’ (l+s)-’ = U(1;t)t + U(t-11, -1 < Re s < 0;
= U(1;t) sin llog t[ , Re s > 0;
= U(t-1) sin log t, Re s < 0.
Remarks. If Re s > 0, (11.10.4) does not exist for t = 0. For
t < 0, f(t) = 0 due to extension, and we will not obtain a continuous
function at the origin. This is the case of (v) which is discontin- uous at the origin.
(viii) mt -1 (s2-1)-’
(ix) (s2-1)-’
If Re s < 0, the integral in (11.10.4) has a value for t = 0,
and we obtain an everywhere continuous function f(t) which are the
cases of (vi) and (vii).
Mellin Transform 249
11.11. The Mellin Convolution
We have already seen the convolution of Fourier and Laplace
transformations in Chapters 7 and 8. type of convolution (which we shall call Mellin type) that can be
readily analysed by means of the Mellin transformation in the
following manner.
In this section we show another
Definition 11.11.1. Let V E E' ? P = SUP(Pl'P2) pl'ql' E' p2 '92 and q = inf (qlIq2) where p < q.
is the distribution belonging to E' defined by
(11.11.1)
Then the Mellin convolution W \ V
P?q
P'q' <(W\ V),,$(t)> = < WU,"Wt,$(ut)>l>, Y $ e E
To prove the existence of (11.11.1) it is sufficient to show
that W(U) = <VtI .$ (ut) > belongs to E C E when u > 0. P'q p2 4 2
Since $ E E there exist and n u r 0 < 'I < I P'9'
0 < 11' < 7 , and bounded functions bk(t) such that tk+l$(k) (t) = k (t) bk(t). -p- 'I, -q+ 'I'
Put
Similarly, we can show that there exists< > 0 such that
250 Chapter 11
Hence
P19' w (u) EE
As an illustration of these ideas, consider the following.
11.11.1. Examples and particular cases
1. By making use of (11.11.1), we have
which yields the identity
(11.11.2) at\ 6(t-a) ,o(t)> = aU,[<6(t-a),+(tu)>l>
1 = <vu,O(au) > = <~Vu/a,~(u) >.
Consequently, we obtain
(a > 0). vtL 6 (t-a) = ; vtIa, 1 (11.11.3)
2. By (ll.ll.l), we have
(vLf)u = at,rf(U/t)>, 1 Y f E (1)
which yields
VLf E ID' (f)(5).
If g is a locally summable function, then we have
Theorem 11.11.1. The Mellin convolution is associative and
commutative (see Section 11.11.2).
Theorem 11.11.2. The space E' is an algebra whose multiplica- PI9
tion law is the Mellin convolution and whose unit element is 6(t-1).
Proof. This is a consequence of Theorem 11.11.1 and formula
(11.11.2) because from the definition 11.11.1, we have
W\V E EiIq and W E E' P,9'
Mellin Transform 2 51
11.11.2. Relation with the Mellin transformation
With the notations being the same as that of Definition 11.11.1,
let MsV = v(s) and lMsW = w(s).
(11.11.4) mS CWLV) = w(s)v(s) , p < Re s < q
which can be obtained easily by putting +(t) = t,
can be related with the relations which exist between the ordinary
convolution and the transforms of Fourier as well as Laplace.
Then, we have
s-1 . This relation
Problem 11.11.1.
(1) Prove that for k E IN
k k (i)
(ii) G(t-a)\ 6(t-a') = G(t-a.a'), a and a' > 0,
(iii) Vt\ t 6'(t-1) = tDVt.
vLs(k) (t-1) = D t vt
( 2 ) Prove that
11.11.3. Relation with the ordinary convolution
Let V and W E EAIw. Then P = W L V E EA
Let $ ( x ) E $. Then according to the statements given in r-1
.
Section 11.9, we have +,(t) = t
fixed number in the interval l a , w [ . Moreover, by (11.11.1) we have $(-log t) E Ea for r being a
r a - r-1 r-1
at,t +(-log t)> = <w ,P ht , t $(-log t - log u ) > l > U
Now, according to the statements given in Definition 11.11.1, we have
which belongs to Ea , W.
given in the proof of Theorem 11.9.1.
Hence, $,(y) E $, according to the statements
Further, by making use of
252 Chapter 11
Section 5.9 of Chapter 5, (11.11.5) can be rewritten as
(11.11.6) <e'rxp - x , $ ( x ) > = <e'=YW , c <e-rXv -,,$(x+y)>~> e e- e
= <(e-3 _,~*(e-~~v -x ,$(x) > e e
by virtue of (5.8.3) of Chapter 5.
The existence of the first member of (11.11.6) assures us that
the convolution exists and hence we conclude
which illustrates the reason for calling the operation\ defined in
Definition 11.11.1 the Mellin convolution.
11.11.4. The operator (tD) " According to (iii) of Problem 11.11.1, we have
tDV = [: t S' (t-1)1\ V.
By repetition, we further have
(tD)2V = [ t&'(t-1) I\ k&'(t-l)l L V = [t&'(t-l)J12\V=K2\V
L2 where K2 = (tG'(t-1)) . Iterating this (n-1) times yields, we obtain
(11.11.8) (tD)"V = [ t S'(t-1) A n L V = Kn\ V
where
Kn = (t b'(t-1) L n . i.e. K is the n-th power of Mellin COnVOlUtiOn of t 6'(t-1)* Since,
Ws t 6' (t-1) = -s I then (11.11.4) transforms (11.11.8) to n
(11.11.9) ms (tD)"V = (-l)n S~V(S)
in accordance with (11.7.9). Also, we deduce from (11.11.9)
(tD)"V = (-1)"(m;'Sn) \ V. Further, we generalize this process by putting
Mell in Transform 253
(11.11.10)
w i t h
where each v i s a r e a l or complex number. If A # 0,1,2,3, . . . , w e have according t o ( i v ) and (v) of Problem 11.5.1,
-A-1 i n E',,~. KX = -FpU(t-1) 1 ( log t) r ( - A
A It fol lows t h a t when V c Eh
represented by t h e couple K t \ V and K ; \ - v ~ When o a < w, ( t D ) 'v = K l L V .
wi th a < 0 < w, t hen ( t D ) V i s
And when a < w 5 0 , ( t D ) V = IZ1 V. x
I f A i s r e a l , then ( t D ) 'V w i l l be real (and equal t o KX\ V)
only i f V E E' with a 5 0 , which occurs i n p a r t i c u l a r ; when V is represented by a func t ion belonging t o I D ( 1 ) . (For t h e c a l c u l a t i o n , see p a r t i c u l a r cases of Sec t ion 11.11.1.)
a , w
Simi la r ly , one can gene ra l i ze t h e ope ra t ion ( D t ) 'and n o t e aga in t h a t t h e formula (11.7.10) sugges ts t h e d e f i n i t i o n
( D t ) % = eivn [m;'(s-i)" J\ v.
11 .12 . Abelian Theorems
I n s e c t i o n 1 1 . 4 w e have introduced t h e absc i s sae of e x i s t e n c e of v(s)=IMsV, absc i s sae t h a t w e denote by a and u which a r e l i m i t e d by t h e wides t s t r i p S a i n which v ( s ) is holomorphic. Hence, i f an a b s c i s s a of ex i s t ence i s f i n i t e , then t h i s is t h e real p a r t of t h e a f f i x of a s i n g u l a r po in t f o r t h e func t ion v ( s ) . L e t s = A and s * 2 be t h e s i n g u l a r p o i n t s corresponding ( 6 ) t o a and W . W e now show t h e behaviour of v ( s ) i n a neighbourhood of t h e s e p o i n t s and c a l l t h e r e s u l t s of t h i s behaviour a s Abelian theorems f o r t h e Mell in t r ans - formation. The r e s u l t s presented h e r e i n are q u i t e equ iva len t as i nd ica t ed i n Lavoine and Misra [ 4 1
I W
Theorem 1 1 . 1 2 . 1 [ f o r t h e i n f e r i o r absc i s sa ) . I f Vt c E' is a,w
equal t o
t - A \ l o g tIV [: H+h(t)+g(t)]
254 Chapter 11
on 10,TC , T < l/e, where
(if A,H,V are numbers such that Re A = a and Re v > -1,
(ii) h(t) is a function tending to 0 as t + 0+,
(iii) g(t) is continuous function such that
dtl < M -l+i Im(s-A) T
T' I J g(t) e
with M being independent of T' and s when 0 < TI < T and ls-AI < n, then
-v-1 (11.12.1) lMsVt - Hr(v+l) (s-A) as s + A, with - f + E 2 arg(s-~) 2 4j - c , E > 0.
- Proof. Here the distribution P -x defined in Section 11.8 is e equal to
eAx xv [ H+h (e-X)+$e-x)lon]( log t I ,- [. Section 8.11.2 of Chapter 8 is applicable here and the formula
(8.11.8) of Chapter 8 and (11.8.6) give (11.12.1).
Theorem 11.12.2 (for the superior abscissa). If Vt€ El a r u
is equal to
t-'(log t)' CHl+hl(t)l + gl(t)
on ]TII-[ ,T1 > e, where
(i)
(ii) hl(t) is a function tending to 0 as t -+ m,
0 , HII v are numbers such that Re n = w and Re v > -1,
(iii) g (t) is a continuous function such that 1
.L
with M being independent of T' and s when T' > T1 and Is+nl < n, then
-v-1 MsVt - H~ r(v+i) (n-s) (11.12.2)
3* as s -+ n, with f + E 5 arg(s-n) 5 - - 2 € *
Proof. The proof is similar to that of the previous theorem but
instead of P -x we consider the distribution Q 11.8. e e
defined in Section
Mellin Transform 255
In the following two theorems we now show the behaviour of v(s)
at infinity.
Theorem 11.12.3 (for Re s + a ) . Let Pt E E' be such that in a?- k the sense of Section 11.3.2 , Pt = D f(t), with the function f(t)
satisfying the conditions:
(i) f(t) has its support in Cola] and a belongs to this support, (ii) tamkf (t) is summable,
(iii) f(t) H(log a/t)" as t + a-0, where H is a number and
Re v > -1. Then
k (11.12.3)
as s + m in an angle where larg 81 2 5 - E .
mSpt - (-1) H r ( v + l ) as-k sk-'-"
Proof. We set -
with w(t) -f 0 as t + a-0. Hence
where
w(ae-x) + o as x -+ o+.
Now, by virtue of the Theorem 8.11.1 of Chapter 8, we have a m
m f (t) = I f (t) ts-'dt = as! f (ae-x)e'ax dx 0 0 S
S = a ~ ~ ~ f ( a e - ~ ) ~ H r ( u + l ) ass-'-'
as s -+ w in an angle where larg s I 2 5 - E. Finally, we deduce
(11.12.3) by means of (11.7.6).
Theorem 11.12.4 (for Re s + -a). Let Qt E Elmlube such that
Qt = Dk fl(t), with the function fl(t) satisfying the conditions:
(i) fl(t) has its support in [bra[ with b>O belonging to this
support I
(ii) tw-kfl(t) is summable,
(iii) fl(t)- H(log t/b)v as t+b-0, where H is a number and Re v > - l .
256 Chapter 11
Then
(11.12.4)
3n as s + - in an angle where $ t E 5 arg s 2 - E.
It is necessary here that if s is the half-plane Re s<inf(w,O),
then its argument is included between n/2 and 3n/2 in such a way
that emins is real and positive as s runs along the negative half-
axis. (See (v) of Problem 11.5.1.)
The proof of this theorem is very similar to that of the
previous theorem.
Finally, we are concerned in the following theorems with the
asymptotic behaviour of the inverse Mellin transformation at infinity.
Hence, we may refer to these results as Abelian theorems for the inverse Mellin transformation according to the usual definition of
such theorems for the Mellin transformation in a distributional
setting.
Theorem 11.12.5 (for Re s + 0 ) . If the function v ( s ) is holomorphic in the half-plane Re s > a and satisfies
as I s 1 + m , with a > 0, k E lN, Re X 2 2, and number (real or complex) G independent of s , then we have
(11.12.5)
where p(t) is a continuous function for t > 0 which is null for t > a
and such that
-1 k lMt v(s) = D p(t).
(11.12.6)
k Proof. We set w(s) = (-1) v ( s + k ) ( s ) ~ and wl(s) = asw(s) with - ( s ) ~ = s(s+l) ,.... ,(s+k-l), k = 1,2,3, .... holomorphic in the half-plane Re s > sup(0,ci-k) and satisfies
(11.12.7) wl(s) .. (-a) G s as 1.1 -+ - in this half plane.
is a continuous function p,(t) for t > 0.
we have for real r
In this setting wl(s) is
k -1
Now by lemma 11.10.1, we conclude that lM;lwl(s)
Moreover, by (11.12.7),
Mellin Transform 257
This can be rewritten as
(11.12.8)
In (11.12.81, the first integral tends to zero as r + - and when t > 1, tr-1 grows with r. Hence, (11.12.8) requires
1 m
1 P,(t) tr-'dt + 0 1
p,(t) tr-'dt + 0 as 1: + m .
pl(t) = 0 for t > 1.
On the other hand, by (11.8.6) and making use of (11.12.7)
we have
I L ~ P ~ ( ~ - ~ ) = mspl(t) = wl(s) .. (-a) k G s-Aas + m .
Hence, by a Tauberian theorem well known for Laplace transformation
(see Theorem 8.12.1 of Chapter 8) we have
k pl(e-x) ~ ,w xA-l as x + o+.
It follows that
S NOW, we put p(t) = pl(tla). Since w(s) = a wl(s) and k
v(s) = (-1) (s-k)k we have by the rules of Calculus (11.7.2) and
(11.7.6) that It w(s) = p(t), and finally, mt v(s) = D p(t) which
proves the theorem.
-1 -1 k
Theorem 11.12.6 (for Re s + -a). If the function v ( s ) is
holomorphic in the half-plane Re s w and satisfies
v(s) I G bs(e'i"s)k'X
as Is1 + m , with b > 0, k E H I and Re s 2 2, then we have
where q(t) is a continuous function which is null for t < b and
which satisfies
The proof of this theorem is similar to that of the previous
theorem.
258 Chapter 11
11.13. Solution of Some Integral Equations
In this section we shall derive briefly how the preceding
theory of Mellin transforms may be used to determine the solution of certain integral equations.
Consider the integral equation m
(11.13.1) V(x)P(xt)dx = Q(t), (t > 0).
The Mellin transformation of a distribution (Section 11.4) can be
applied to solve such equations. For this purpose by applying the
Mellin transform of a distribution on both sides of (11.13.1), we
get
0
m m m
(11.13.2) I V(x)P(xt)dx I tS-ldt = I ts-’Q(t)dt, 0 0 0
Taking y = xt and x as a variable in the left hand side of (11.13.2)
and changing the order of integration by Fubini‘s theorem, we have
where
IMs[V] = V(S) for V E EAlw,
IMs[P] = p(s) for P L EkIU,
MsCQI = q(S) for Q E EkIU#
and their strip of definitions are represented by Sv, Sp, SQ,
respectively. Here v(s), V ( x ) and Sv are unknown. The equation
forces Sv to contain 1-s as s belonging to a conventional subset of
Sp n SQ. 1-s E Sv, then the equation
In other words, if pt denotes the set of s such that
(11.13.3) q ( s ) = v(l-s)p(s)
holds for s E % n Sp n SQ.
Replacing 8 by (1-s) on both sides of (11.13.3), we have
1 1 Put k ( s ) = -j= and IMs[K(x)] = k(s) = po. Therefore P( -s
Mellin Transform 259
If Bt = IM;' q(1-s) , then we have IMsBt = q(1-s) .
(11.11.4) we have
By making use of
IMs (B\ K) = q(l-s)k(s)
and by (11.13.4)? Ms ( B L K) = v(s) = IM V which gives
(11.13.5) V(X) = ( B I K)x.
S
Since q(s) = NsQ(t), we have Bt = t -1 Q(,) 1; and by (11.11.4) and
(11.13.5) we get
- 1 -2 V(x) = I Q(y)K(;)y dy.
0
Now by putting t = I in above integral, we finally obtain
(11.13.6) V(x) = I Q(t)K(xt)dt provided, of course? the inverse Mellin transform
m Y
0
~ ( x ) = mi1 [pol 1 , s + x) exists;
i.e. (11.13.6) is the solution of integral equation (11.13.1).
In particular, the equation (11.13.1) will have the solution m
V ( x ) = I Q(t)P(xt)dt 0
if
(11.13.7) P(S)P(l-s) = 1;
that is, the equation (11.13.7) is a necessary condition of p to be
a Fourier kernel (see Colombo C11).
Exam le. We mention below an example of such an equation. Take P(x) -+ = x Y,(x) where Yv(x) is the Bessel function of the second kind
of order v , Then (see Sneddon 121, Problem 2.37 (b) ) we have
s-1/2 r(1/4 + s/2 + v/2) cot(3" - p(s) = 2 r (3/4 - s/2 + v/2) 4 2 2) + Vfl
and hence
so that
260 Chapter 11
Now making use of Sneddon C2 1, Problem 2.38 (b) w e see t h a t
where Hv is thes t ruve func t ion .
I n o the r words, w e have shown t h a t t h e i n t e g r a l equat ion m
( x t ) % ( x ) Yv(xt )dx = Q(t) 0
has t h e so lu t ion
V(x) = m 4 (x t ) Q(t) H, , ( tx)d t . 0
Now, w e can de r ive the s o l u t i o n of t h e i n t e g r a l equat ion
i n a s imi l a r manner. If w e consider W,R and G t o be i n E' , then atw
l e m 11.11.1 w e can w r i t e (11.13.8) i n t h e form by (2) of Pro
(11.13.9)
By v i r t u e of
(11.13.10)
where
1 1 . 1 1 . 4 ) w e ob ta in
and t h e i r s t r i p of d e f i n i t i o n s are represented by Sw, SR and SG respec t ive ly . Here R , r (s) and S a r e unknown. Since s E Sw, and SG, w e can say t h a t (11.13.10) holds f o r s E Sw n
sR R SR n SG.
The equat ion (11.13.10) can be w r i t t e n as
Mellin Transform 261
conventional function.
then (11.11.4) yields If Gl(t) = lMt -1 gl(s) and H(t) = lMilh(s),
R(t) =(GIL HIt.
Now, by virtue of ( 2 ) of Problem 11.11.1, the above can be written
under the form of the integral
(11.13.12) R(t) = !G,(y)H(t/y)y-'dt. m
0
The integral equation
dx 1 (11.13.13)
is a more interesting form of the above equation. If we make the
substitution
I Wl(t/x)Rl(x)r = Gl(t), 0 < t < 1, W1,Rland G 1 ~ E h , w , t
R(t) = Rl(t) [U(t)-U(l-t)], G(t) = Gl(t) [U(t)-U(1-t)] and
W(t) = Wl(t) [ U(t)-U(1-t)l where
O < t < l
elsewhere, U(t)-U(l-t) = [,
then we see that (11.13.13) is equivalent to (11.13.8) and hence
to (11.13.10).
11.14. Euler-Cauchy Differential Equations
In this section we also illustrate the use of Mellin transform-
ation in the following differential equations in a distributional
setting.
Let Xt be a distribution having support in ]O,-[ satisfying
N (11.14.1) 1 AntnDnXt = Vt
n= 0
where Vt is a distribution whose Mellin transform is v(s) in the
strip Sv (see Section 11.4) and An # 0. at times Euler-Cauchy differential equations. The Mellin transforma-
tion generates an operational calculus by means of which (11.14.1)
may be solved for the unknown X
able distribution.
Such equations are called
when Vt is a known Mellin transform- t
- Put x(s) = IMs [ Xtl . According to the Section 11.7 (formula
(11.7.8) ,we have
262 Chapter 11
and (11.14.1) transforms to
(11.14.2) P,(S)X(S) = v(s)
where
Hence
Let ak, k = 1,2,3,...,K (N, be the roots (real or complex) of
If the polynomial PN(s) and let mk be their orders of multiplicity.
we suppose that a corresponds to the root which has largest real
part. Then, l/PN(s) can be written in the form:
Also, we have
mi1 (s-ak)-j = IL (j ,a ;t) , Re s > Re a where (11.14.4) k
as can be seen in Section 11.5, formula (iv) of Problem 11.5.1.
(See also Colombo and Lavoine [11 , p. 152.)
The inversion of (11.14.3) by means of Mellin convolution
yields, (see Section 11.11.4)
provided that Sv contains a substrip in which Re s z ak. Consequently
(11.14.5) is a solution of (11.14.1).
such that Re s > ak.
S
Re al < Re a2
(11.14.6)
If Sv does not contain any s
Then the case is more complicated. Suppose be the strip a < Re s < w and let the roots be arranged such that V
Re a3,. . . , Then, we have
IMt (s-a)-j = E(j,a;t), Re s < Re a, -1
Mellin Transform 263
where
- (-1) 1 IL(j,a;t) = ~T U(t-a)t-a log1-lt, j = 1,2,3 ,... (7 - !
then the K'+1' by (v) of Problem 11.5.1.
inversion of (11.14.3) yields
If Re aKl < w < Re a
K' mk K m k (11.14.7) Xt = 1 1 B. V L JL(j,ak;t) + 1
k=l lk k=K'+1 j=1
where the B 's can be obtained by decomposition of l/PN(x) jk
Particular case. Taking N = 1, (11.14.1) reduces to
AO 1
A1 AltDXt + A X = V or tDXt + - Xt = V A1 t' o t t
That is
(11.14.8) tDXt + AXt = Vt
AO 1 A, A, t t' by denoting - by A and -V by V (11.14.3) will now take the I I
form
(11.14.9) v(s) x(s) = - - S-A
If w > Re A, (11.14.5) gives
\ IL l,A;t), t (11.14.10) Xt = -v
and if w < Re A, (11.14.7) yields,
\ (1,A;t). t (11.14.11) Xt = -v
On the other hand if Vt is represented by a function h(t)
having support [ f.! ,y] contained in [ 0,-[ , then (11.14.10) gives
(11.14.12)
If y = +m and if w < ReA, we have by (11.14.11)
(11.14.13)
Y Xt = +-A f h(u)uA-'du, B < t < y .
B
t Xt = t-A f h(u)uA-'du, t > B .
B
It can be easily verified that (11.14.12) and (11.14.13) give
the solutions of (11.14.8). If v(s) is such that
(11.14.14) V(S) = (s-A)g(s)
then (11.14.9) gives
264 Chapter 11
-1 (11.14.15) Xt = mt g ( s ) .
We remark here that (11.14.14) implies that the existence of
X such that t
V = -tDGt - AGtr msG = g ( S ) , t
and the equation (11.14.8) can be written as
tD(Xt+Gt) + A(Xt+Gt) = 0.
Hence the solution Xt = -G
(11.14.15).
is obtained which is identical to t
If (11.14.13) and (11.14.12) do not give computable results,
then one can consider (11.14.9) in the form of a series
Hence by (11.7.9) we have m
- 1 (-l)n A-n-l(tD)%t 't - n=O
under the condition that the series converges.
The solution of an Euler-Cauchy differential equation for
functions can be obtained in a different but very similar way to
that of distributions. For instance, we seek a function h(t) continuous on [ 0 , y l (y is bounded) such that
d (11.14.16) t;ir h(t) + Ah(t) = f(t), t E [0 r y ]
where f(t) is a Mellin transformable function having support in
LO, Yl (7).
Denoting h(y-) by W, and making use of (5.4.3) of Chapter 5,
we have
d h(t) = Dh(t) + W6 (t-y)-h(O+) 6 (t)
and (11.14.16) yields with th(O+)b(t) = 0
Mellin Transform 265
which is similar to equation (11.14.8).
F(s) = Ms f (t) , and applying the Mellin transformation to (11.14.16') we obtain
By putting H ( s ) = IMs h(t),
and its inversion is
where f(t) is similar to that given by (11.14.10) and
0 I t 2 Y
elsewhere. x(O,y;t) = 1
11.15.Potential Problems in Wedge Shaped Regions
In this section we shall describe briefly how Mellin transfor-
mation in a distributional setting may be used to determine the
solution of a physical problem which occurs in mathematical physics.
We deal this work with a simple problem in potential theory.
Consider an infinite two dimensional wedge as indicated in
Figure 11.15.1. We choose a polar coordinate system u(r,e) with the
origin at the apex of the wedge and the side of the wedge along the
radial lines 8 = -a and 8 = a ( 0 < a < 2 r ) . Specially, the problem
we wish to solve is the following: Find a function u(r,e) (which is
a function of r and 8) in the interior of this wedge such that
(i) it satisfies the partial differential equation
where 0 5 r 5 a and -a 5 0 - < a . The equation (11.15.1) is Laplace's
equation in polar coordinates multiplied by r ; 2
(ii) it satisfies the boundary condition
O z r l a
r > a (11.15.2) u(r, +a) =
(iii) u(r,e) is bounded as r is bounded.
266 Chapter 11
Figure 11.15.1
To solve this problem we identify u(r,e) with a distribution
in r.
hence one can take u(r,e) = 0 for r > a, then (see Section 11.3)
u(r,e) E E;
Re s > 0.
AS we see above that u(r,e) is defined only for 0 2 r 5 a and
i.e. the Mellin transformation of u(r18) exists for I -
From the structure of u(r,8) E E; I -
we may conclude that u(r,e)
is bounded as r is bounded. Also, u(r,+a) may be identified as
U(a;t) and hence according to (11.3.6) we have
Consequently, we obtain
Du(r,+a) = - 6(r-a) in Ei ,-• Hence, we may infer that u(r,B) satisfies the conditions (11.15.2)
and (iii) . When applying the Mellin transformation we shall treat r as
the independent variable and 6 as a fixed parameter:
s-1 M u(r,e) = (u(r,e), r > = u(s,O), Re s > 0. S
Now, by the operation transform formula (11.7.8) of Section 11.7,
Mellin Transform 267
ms transforms (11.15.1) to
a 2
a e 2 if we assume that - can be interchanged with Ms. Therefore, we obtain
-is8 + B ( s ) e is 8 (11.15.3) U ( s , e ) = A ( s ) e
where the unknown functions A ( s ) and B ( s ) do not depend upon 8. To
determine A ( s ) and B ( s ) we first operate Mswith (11.15.2) and
accordingly, we get
i.e.
Thus, if M [u(r,ta)] = U(s,ta) for s E Qu = Is: a
then we obtain
< Re s < a 1, u1 u2 S
so that
as 2 s cos s a
A ( s ) = B ( s ) =
Consequently, (11.15.3) takes the form
(11.15.4)
If s = a+iw, we have (since -a < 0 < a)
2 Thus, we may conclude that s u(s,8) is bounded as 1.1 -+ - and valid in the strip 71 < Re s <
holomorphic in this strip. Consequently, U ( s , B ) in the strip 7F - < Re s < satisfies all the needed conditions for the results 2a of Section 11.10. Thus upon invoking Theorem 11.10.1 with K = 0 or
simply Lemma 11.10.1, we obtain our desired solutions:
and hence one can say that U ( s , e ) is 4 a 2a
268 Chapter 11
where h is a number lying between 71 and
according to Lemma 11.10.1, u(r,e) is the unique distribution in
E;l,- for 4a u(r,e) of our proposed problem.
i.e. z . c h c 5. Thus, 4a 2 a 4a 2a
< h < 2a. Consequently, we obtain the unique solution
11.16.Bibliography
In addition to the works cited in the text, we mention the
following references dealing with material of the present chapter.
Fox c21, Fung Kang C11 I Jeanquartier, P [ 11.
Footnotes
other authors call this the abscissae of convergence under the form of integral (11.4.2). One deduces the abscissae of
existance directly form the structure of V, but it is easy to determine the singular points which fix the widest strip in
which v ( s ) is holomorphic.
Fp is not needed here if Re v F -1.
$ ( s ) = and C is Euler's constant.
see Section 5.9 of Chapter 5.
recalling that if Vt = f(t) conventional function, we have
e-% -x = e-rxf (em%).
Zemanian C 3 1 , p. 118.
several singular points may correspond to the same abscissa
of existence as in the case of NsU(l;t) sin !log ti = (S2-1)-'
for Re s > 0. Such a plurality leads to introduce the
functions g(t) and gl(t) in Theorem 11.12.1 and 11.12.2.
r s
We employ this notation by
On the other hand e'rx6 (e-X-1)=6 (x) . e
a function of support [ O , y ] is Mellin transformable if its
behaviour at the origin is known.
CHAPTER 12
HANKEL TRANSFORMATION AND BESSEL SERIES
Summary
In recent years there has been considerable work on the use of
Hankel transformation and Bessel series for functions in the solut-
ion of problems arising in mathematical physics. In this chapter
we further develope this work in the distributional setting suitable
for those whose interest lies in applications.
For convenience we divide the chapter in two parts. The first
part treats the Hankel transformation of functions in base spaces
to distributions as in the definition of Fourier transformation in
Chapter 7 and the formation of this distributional setting and its
extension to several variables contain in Sections 12.2. to 12.1.
Finally, we conclude this part by using this distributional setting
of one variable in solving the heat conduction problem of circular
cylinder.
The second part deals with the work of Bessel series for
generalized functions and the analysis of this topic including its
application contains in Sections 12.9 to 12.17.
12.1. Hankel Transformation of Functions
In this section we present those classical results of the
Hankel transformation by means of Mellin transformation (see Section
11.1 of Chapter 11) which we need subsequently in the distributional
setting of Hankel transformation.
Let us take the function @(t) which satisfies the following
conditions:
(a) $(t) is defined for t > 0;
(b) $(t)EL(O,m) (i.e. space of equivalence classes of functions that are Lebesgue integrable on ( 0 , ~ ) ) .
2 6 9
270 Chapter 12
If $(t) satisfies the above conditions, then we define the
relation
(12.1.1)
where y > 0, v > - 7 is a real number, and Jv denotes the Bessel function of the first kind and of order v . In this relation we
shall say that the function q(y) defined by (12.1.1) is the Hankel
transform Of the function b(t).
mation instead of Hankel transformation of order v.
W
$(Y) = Mf:$(t) = 1 $(t) F t Jv,(yt)dt 0
1
For brevity, we write Mv-transfor-
A standard result concerning (12.1.1) is the following
inversion theorem.
Theorem 12.1.1. If $(t) E L(0,m) , $(t) is of bounded variation in a neighbourhood of the point t = y, and $(y) is defined by
(12.1.1), then
(12 .l. 2) OJ
b(t) = M;'&(Y) = M~$(Y) = 1 m(y)Gt Jv(yt)dy, 1 0
T ' v 2 -
where Mi1 denotes the inverse Hankel transformation.
defined by (12.1.2) is called the inverse Hankel transform of $(y).
Note that, when v 2 - 7, this inverse Hankel transformation M i 1 is
defined precisely the same formula as is the direct Hankel transfor-
mation mV ; in symbols, m,, = m~~ .
Here $(t)
1
Now we establish the following result which will play an
important role to obtain our main results.
then we want to prove y (y) = $ (y) . We now prove our desired result.
that (see Erdelyi (Ed.) [a], V01.2, p. 227(7))
For this purpose, we recall
(12 .l. 4 )
2s-1 v+s 1 v-s 3 I r ( T + $ / r (T + $ , v > -1, s > 0,
where Ws denotes the Mellin transformation as indicated in
Chapter 11.
Hankel Transform 271
Further, we impose the ccaditians (a) and (b) of Section 11.1 of sapter 11 of the Mell in t ransformat ion of func t ions on $ and 4, and set F(s)=IMs $ ( y ) and f ( s ) = lMs $ ( y ) . Now w e have according t o (12.1.1) and by means of Fubin i ’s theorem,
m m
F ( s ) = IMs$ = J yS-’dy $ ( t )q t J v ( y t ) d t 0 0
m m
= t - s $ ( t ) d t 1 u s-1 u4 J v ( u ) d u 0 0
by p u t t i n g y t = u. Then w e have
(12.1.5) F ( s ) = f ( l - s ) B v ( s ) .
Also, l e t g ( s ) = lMsy (y ) . t h e manner as explained above, w e g e t
Then ope ra t ing s i m i l a r l y wi th (12.1.3) i n
Hence, w e g e t according t o (12.1.5)
g ( s ) = f ( s ) B v ( s ) By(l-s).
Because By(s) B V ( l - s ) = 1 (see Colombo C11 ) , w e have
g f s f = f(s).
Now,by t h e Theorem U.1.1 of Chapter 11 of t h e Mel l in t ransformat ion w e g e t y (y) = $ (y ) . Therefore , w e f i n a l l y o b t a i n
(12.1.6) $ ( Y ) = M\ $(t) = $(t)qt J y ( y t ) d t ,
and i t s inve r s ion according t o (12.1.2) i s
m
0
(12.1.7)
I f w e r ep lace t by y and y t o x i n t h e above i n t e g r a l , then w e o b t a i n
(12.1.8)
I n t h e above i n t e g r a l t can a l s o be rep laced by x. by t and y t o x i n t h e i n t e g r a l of (12 .1 .6) , then w e g e t
If w e r e p l a c e y
m
(12.1.9) $(XI = EI;’$(y) = M: $ ( y ) = $(t)&t J v ( x t ) d t . 0
272 Chapter 12
Also, in this integral, t can be replaced by y.
12.2. The Spaces Hv& H;
Our study made on the spaces of base functions, generalized
functions and distributions (see Chapters 2 to 5) enable us in this
section to construct the spaces Hv and H; which provide the structure
of a distribution to formulate the distributional setting of Hankel
transformation in our subsequent work.
Let I denote the open interval 10I-C and v be a real number.
By HV (denotes HV(x) whenever the variable needs to be specified) we
denote the space of infinitely differentiable and complex valued
functions
(12.2.1)
is finite
The
following
$(x) defined on I such that
for all non-negative integers k and m.
space Hv is provided with a topology defined in the
manner :
A sequence { $ . I , j E s N I converges to zero in H as j -+ m l if 7
and only if
- l d k - XEI
as j -+ - for a set of non-negative integers k and m. Also , the
space HV is complete, (see Zemanian C31, Theorem 1.8.3).
Lemma 12.2.1. $(x) is a member of Hv if and only if satisfies
the following conditions:
(i) $(x) is an infinitely differentiable and complex valued
function on 0 < x < m;
(ii) for each non-negative integer k,
(12.2.2) 2 2k Cao+a2x +...+ aZkx + R2k(x)] v+1/2
$(XI = x
where the a's are constants given by
(12.2.3) a2k '7 l im (x D) 1 -1 k x-~-+L$(x) k! 2 x+O+
and the remainder term RZk(x) satisfies
Hankel Transform 273
-1 k (12.2.4) (X D) RZk(X) = O(1J x -c O+;
k (iii) for each non-negative integer k, D Q (x) is of rapid k decrease as x -c - (i.e. D Q (x) tends to zero faster than any power
of l/x as x + m ) .
- Proof. Assume that $(x) E Hv. Condition (i) is satisfied by
definition. The proof of (ii) and (iii) can be carried out as
indicated in Zemanian [ 3 ] , pp. 130-131. The conditions (i) , (ii) , (iii) imply that Q is in Hy ,
Note that, for any fixed y L 1,Ky Jv(xy) as a function of x satisfies conditions (i) and (ii) of above lemma. However, it does
not satisfy condition (iii) since
(See Jahnke, Emde and Losch [ 11 , pp. 134 and 147.) is not a member of Hv.
Hence Jxy Jv (xy)
The space H: (H:(x)) is the dual of Hv, and it is the space of
distrlbutions(continuous linear functionals) on Hw.
V E H;
The value of
on Q E Hy is usually denoted by
By I D ( 1 ) we denote the space of infinitely differentiable and complex valued functions $(x) with bounded support properly contained in I.
The topology of this space is defined in the following manner:
a sequence { $ . I , j E IN tends to zero in 3
and only if
as j -F m, for each k E IN.
Also, D(1)is a subspace of HV and convergence
I D ( 1 ) as j 3 m , if
in D(I) implies
convergence in Hv (see Zemanian 1 3 1 ) .
of any element of H: to ID(1)is an element of ID'(1) , the dual of D(1) and the space of distributions with support in I.
Consequently, the restriction
By E ' ( 1 ) we denote the space of distributions having bounded We remark here that the spaces defined support with respect to I.
274 Chapter 12
herein bear the close resemblance to those of Zemanian C31.
12.3,Operations on H,, and H:,
In this section we establish some important results of
operations on H and H: . Multiplication
h For any real number h and v I the mapping $(x) + x $ ( x ) is an
isomorphism of HV onto H v + h . It follows that V + xhV defined by
A x (12.3.1) <x v, $(XI> = a, x +(XI>
is an isomorphism of H:+hon H:. (For the proof see Zemanian C3] p.135.)
Operators
We shall use the following differentiation operations:
(12.3.2)
(12.3.3)
k -v-1/2 d 2v+l $- x-~-1/2)km
k -1 d k x-~-1/2
sx dx sv = (x I
ad Rv = (x I
where k is a non-negative integer.
The transpose of (12.3.2) is
-v-1/2 Dx2v+l Dx-v-1/2 ) k in H: . (12.3.2 ' ) = (x
and the transpose of Rt is
(12.3.3 ' ) ^k -~-1/2 (Dx-l) k Rv = x
where as usual, D denotes the distributional derivative. Here we
call S v and Rv as transpose differential operators in the sense of
Section 5.4 of Chapter 5.
^k *k
Let Mv I Nv I N i l be the operators on Hv defined by
-~-1/2 d ~+1/2+(~) (12.3.4) Mv$(x) = x E X
(12.3.5) 4 (XI v+1/2 d - ~ - 1 / 2 N,,$(x) = x azx
Hankel Transform 27 5
A
Further M and N are defined by
Thus,
A -v-1/2D xv+1/2 Mv = x
A v+1/2D x-v-1/2
X ( 1 2 . 3 . 7 )
Nv = x X
( 1 2 . 3 . 8 )
We summarize these results by:
N is an isomorphism of HV onto Hv+l whose inverse is N;' ; V
M is a continuous linear mappinq from Hv+l into Hv; V -
N is a continuous linear mappine of H$ into H$+l; V
A
M is an isomorphism from H' into H{ . V v + l
Moreover LI
2 4 v L - 1 M v Nv - Dx - - 4x
A , . - 2
Also, we denote
, . a ': = M~ Mv+l"' 'v+k-l'
Note that we have the following equivalence relations between
these operators:
C I A ik = (Mu N v ) k ;
A k k+v+1/2 - ^k - Pv . Rv
We end this section by giving an important differentiation
formula
( 1 2 . 3 . 9 )
(see Koh [13 ) .
276 Chapter 12
12.4, Hankel Transformation of Distributions
By the results of Section 12.1 and the structure of a
distribution described in the preceeding sections we formulate in
this section the distributional setting of Hankel transformation by
means of Hankel transformation Of functions in Hv to distributions
in H: in the following manner.
If Q E Hv, then it has according to equation (12.1.8) , the Hankel transformation of order v as
(12.4.1) 4 (y) = M: 4 (XI = 4 (t) Jv(yt)dt:
and by (12.1.9) we have
(12.4.2)
m
0
m -1 $(XI = Mv i(y) = lH:m(y) = Q(t) 6 Jv(yt)dt.
0
By (12.4.1) and (12.4.2), we can state the following results:
1. Mu 4 and
2.
4 are functions of y and x which belong to Hv;
if a sequence {Qn} + 0 in the sense of Hv, then MvQn + 0 and
M i 1 $n +. 0 in the sense of H,;
3. in (12.4.1) and (12.4.2), Q(x) and $(y) are called the anti- transforms (or inverse-transforms) of 4 (y) and Q (x)
respectively.
Also, by (12.4.1) and (12.4.2), we may write
Mt M; $ ( X I = $ ( X I ;
M; MZ O ( Y ) = O(Y).
Hence, we may conclude that M: or M: is self reciprocal. Therefore,
Zit and M:
Q E H, , then Mv 4 E H, . are reciprocal automorphism of each other on H,: if
The above result permits us to make the following definition:
The Hankel transformation of order v of a distribution V E H:(x) is
the distribution belonging to H:(y) which is denoted by M,V (M:Vx
whenever we desire to make the variable precise) and is defined by
Hankel Transform 277
This is a definition by transposition (Section 5.1 of Chapter
5). To conform with established terminology, we shall say that
every generalized function (or distribution) belonging to H: is a
My-transformable generalized function (or distribution).
A l s o , according to (12.4.2), (12.4.3) is equivalent to
(12.4.3') <M:v~, M: 4(x)> = (vxI~(x)>, for every E H".
Since MV 4 belongs to Hv, we deduce from (12.4.3') that Mu v belongs to H{ and hence the transformation defined by mv in (12.4.3) is an automorphism on H:.
If the distribution V is associated with a summable function
h(x) on 0 < x < -, then (12.4.3) leads to the equality
(12.4.4) MVh(x) =I h(t) fl Jv(yt)dt = Mzh(x) m
0
in accordance with (12.4.1). Thus, we may infer that the present
setting of Hankel transformation of distributions properly generalizes
the Hankel transformation of functions.
-1 (x+a) . Then,we have by v-1/2 Examples. (i) If h(x) = x
(12.4.4) that m
MVh(x) = I t v-1/2(t+a)-1 c t Jv(yt)dt 0
= 3. avsec(vr)y1/2 [~-~(ay) - ray)^ 1 7, larg a1 < r where - T < Rev< 7, v #
Struve function and the Bessel function of the second kind. (See
Erdelyi (Ed.) [2] : V01.2, p. 22.)
and H-v(ayI and Y-v (ay) are the 1 3
(ii) Let h(x) = x '+'I2 (x2+a2)'I. Then we have
m
Mvh(x) = 1 t v+1/2(t2+a2)-1 F t Jv(yt)dt 0
= Irav-1sec(vr)y1/2CIv(ay) 2 - ~-~(ay) I
1 5 where Re a > 0, - Tv < Re v < z, Iv and L-v are the modified Bessel
function of the first kind and the modified Struve function. (See
Erdelyi (Ed.) [21 , Vol. 2, p. 23.)
Chapter 12
Another particular case
As we see in the proof of Lemma 12.2.1, Jv(xy) as a
function of x does not belongs to Hv, and therefore the statement
(12.4.5)
is not well defined for every V E H:.
restrictions on V, (12.4.5) will possess a meaning and will agree
with the definition (12.4.3’).
contains the distributions of the form:
MvVx = <V Jv(xy)> XI
However, under certain
For this purpose note that H;
k if Vx = D s(x) where s(x) is zero for x < 0, and locally
summable for x > 0 and x a+1/2-k+1s(x) is bounded as x + O+ for a c v . A l s o , if there exists n > 0 such that x-~s(x) is bounded as x -+ -.
Then we have under these conditions:
(12.4.6) ax, $ (x) > = < s (x) , (-1) ’$ ( k ) (x) > = (-1) klms (XI $ (k) (x)dx. 0
we now verify the existence of (12.4.6).
Proof. Indeed, by our proposed structure of Vx, we have - m
(12.4.7) <VxI$(x)> = ( - l ) k j s(x)+(~) (x)dx
(XI dx = ( - 1 ) k j 01 xa-k+3’2s(x) x k-a-3/2$ (k)
+ (-1) k 1 x -n ~(x)x~$(~)(x)dx. 0
1
The second integral in (12.4.7) is bounded according to our
hypothesis and the relation (12.2.3) of Lemma 12.2.1. Also, the
third integral in (12.4.7) is bounded according to our hypothesis as
well as condition (iii) of Lemma 12.2.1 and by relation (12.2.1).
This proves (12.4.6).
If v has a bounded support contained in I, then M v V defined by
M V = <V X’ G J V ( x y ) > , y > 0. V
(12.4.8)
This case will be discussed in detail in the Section 12.4.1.
Remark. The relation (12.4.3’) enables us to assign a Hankel - transformation to distributions equal to certain increasing functions
such as xn, n > 0, because $ (x) decreases more rapidly than every
Hankel Transform 279
1 power of - as x + -. X
Examples. If IL denotes the Laplace transformation, then show
that
(i) xi: e-PxVx = IL Jyx J"(YX)V~
when V is represented by a function $(x) E H ~ . X
Proof. We recall that - Mt Q (y) = ~6 Jv (XY) ,$ (Y) >.
Now (12.4.3) gives
- -px x 4 M: e 'xVX,$(y)> = <vx,e my $(Y)>
= < c < v x , G J"(XY) e'Px>l,+(y) >
for all $ E Hv. Consequently, we have
M: e'p"v X = < V x , G Jv(xy)e-PX>
1 (ii) For v 1. -7 and
U (x-a) -U (b-x) a-x M y FP
0 < a < b < -, show that
= I a x-a dx+L+l x-a a+l Jxy J,(xy)-& Jy(ay) b GJ,(x)
1 i f a < x < b
0 elsewhere. c where
u (x-a) -U (b-x) =
Proof. - U (x-a) -U (b-x) =
Fp x-a Fp fbL x-a Jv.(xyjdx a
l i m [ E+O a+€
b Jv (xy) d x + 6 J v (ay) log E]
a+ 1
lim E+O [ a+€ I A&Jv x-a (xy)dx +&J, (ay) log €1
+ b Jxy Jv(xy) I x-a lx .
a+l
280 Chapter 12
a+l
a+ e By replacing log E by - I x-a dx, we obtain
+ 7 G J v (XY) a+l KYJ~ (XY) -KYJ~ (ay) =I U (x-a) -U (b-x)
x-a dx . a+l x-a a Fp x-a
(iii) For v 2 - z, 1 (a) Mf: x
show that in the sense of equality in H::
2n+1/2 = v(v2-4).. . (v2-4n 2 ) y -2n-3/2, , 2n, 2 -2n-1/2 M: x 2n-1/2= (v2-i) (v 2 -9). . . (v2- (2n-1) )y
(b) I
v > 2n-1.
where n is a positive integer.
Proofs. From Magnus, Oberhettinger and Soni [11, we have -
If we put t=y, a=x and u=-X+l, in this integral and consider this
formula as a Hankel transformation, then we get
Now, apply M:to both sides, we have
If X = 2n+11 we get
If X = 2n, we get
Y 2n-1/2 = (v2-1) ~v 2 -9) ... (v2-(2n-1) 2 )y -2n-1/2 M V I
v 2n-1.
Remark. Since y does not belong to Hv(y) and hence (1) - does not exist in the sense of the ordinary Hankel transformation.
12.4.1. The Hankel transformation on 6’ (I)
As mentioned above, the Hankel transformation of certain (but
not all) members of H: takes the form
(12.4.9) b(y) = MwBx = <BXl 6 Jw(xy)>.
Hankel Transform 281
W e s h a l l e s t a b l i s h t h a t when Bx E E' ( I ) , b ( y ) is a smooth func t ion on, 0 < y < m. Indeed, it can be extended i n t o an a n a l y t i c func t ion on t h e complex p lane whose only s i n g u l a r i t i e s a r e branch po in t s a t t he o r i g i n and a t i n f i n i t y . To do t h i s , l e t z = c+ in be a complex v a r i a b l e , and se t
(12.4.10) b ( z ) = IH; BX = < B x 1 6 J v ( x z ) > .
Theorem 12 .4 .1 . I f Bx E &'(I), then z-v-1/2b(z) is an e n t i r e func t ion of t h e complex v a r i a b l e z ( i .e . it i s holomorphic i n t h e f i n i t e z-plane) . - Proof. If Bx' E'(I), then i t s support is conta ined i n t h e
k i n t e r i o r of lo,-[. A l s o , i f w e set B = D s (x ) where s ( x ) is a continuous func t ion having suppor t i n [ a r i 3 l , 0 < a < 6 m. Then, by making use of t h i s s e t t i n g , w e have
X
(12 .4 .11 ) b ( z ) = (-l)k 1 s ( x ) --?;[= dk J v ( z x ) l d x . a dx
Making use of t h e series expansion of J (zx) (see Problem 1 . 4 . 1 of Chapter 1) w e have
o r
where
Since 2'jj!u is bounded a s j -f m , t h e series converges i n t h e
f i n i t e z-plane. This proves our theorem. j
- Remark . Since b ( z ) z-'-li2 i s holomorphic i n t h e half-plane y = R e z > 0 can be seen above and consequent ly w e may i n f e r t h a t b (y ) is a smooth func t ion on 0 < y < m.
Theorem 12.4.2. L e t Bx E e'(1). I f v 2 -1/2 and i f def ined by (12.4.9) . Then, b ( y ) s a t i s f i e s t h e i n e q u a l i t y
V + V 2 0 < y < 1
1 < y < - ( 1 2 . 4 . 1 2 )
282 Chapter 1 2
where K and p are sufficiently large real numbers.
Proof, The proof can be carried out as indicated in Zemanian
C31, pp. 146-147. -
1 2 . 5 . Some Rules
This section provides an account of the operational transform
formulae for the spaces Hv and H: . 1 2 . 5 . 1 . Transform formulae for Hv
If 9 E Hv, we have
( 1 2 . 5 . 1 ) My+1(X9) = - N v M v 9 ;
( 1 2 . 5 . 2 ) mv +1(Nv9) = -Y xv 9;
and if 4 E Hv+l, then
Proof. The proofs of ( 1 2 . 5 . 1 ) to ( 1 2 . 5 . 4 ) and ( 1 2 . 5 . 6 ) to - ( 1 2 . 5 . 7 ) are given in Zemanian C31, pp. 139-140. The formula
( 1 2 . 5 . 5 ) f o l l o w s directly from ( 1 2 . 3 . 9 ) . To prove ( 1 2 . 5 . 8 ) we use
the following recurrence relations:
2v ( 1 2 . 5 . 1 0 ) J v - l ( X ) + J v + l ( X ) = x J v ( X ) j
(see Sneddon C21, pp. 5 1 0 - 5 1 1 ) .
Making use of (12.5.10) we have
Hankel Transform 283
m
= f JV(xy) 4 (XI dx = MY $(XI. 0
Hence (12.5.8) is established. Further making use of (12.5.11) and
(12.5.10) we have
m 1
R.H.S. Of (12.5.9) = 5 { 2 ~ ~[Jv,l(~y)-Jv+l(Xy) ]X+(X) dx 0
m
= 112v /2.J:(xy)x$ (x) dx+2vj 1 4v 0 0 xy
12.5.2. Transform formulae for H:
We now state a number of operation-transform formulae for the
generalized Hankel transformation. These are exactly similar to the
formulae of the preceding section, but deal with generalized
operations.
If V E Hi, we have
(12.5.12) Mv+l(X.V) = - NvMvV;
(12.5.13) Mv+l(NVV) = -y MvV;
- 1 2 (12.5.14) mV (X V) = - MvNvMvV;
284 Chapter 12
2 . . A
(12.5.15) Mv (MvNvV) = -y MVV;
Mv cqvxl = (-1) k y 2k IHvvx: (12.5.16)
and if V E H:+lI then
(12.5.17) MV (xV) = M v M V + l V ~ *
(12.5.18)
(12.5.19) Mv (DxVI = gi(2v-1) MV+1V-(2~+1) Mv-1V3..
Proof. The proofs of (12.5.12) to (12.5.15) and (12.5.17) to - (12.5.18) can be found in Zemanian [ 3 1 1 pp. 143-144. The formula
(12.5.16) can be obtained by applying (12.5.15) successively k times
and making use of S v = (MvNv) which is the equivalence relation given
in Section 12.3. The formula (12.5.19) can also be easily obtained
by using recurrence relations (12.5.10) and (12.5.11).
*k A n k
Problem 12.5.1
For k = OIl121....I show that
^k k 2k (i) M v S V Vx = (-1) y M V V x I Vx E H: ;
-k-~-1/2 Vxt Vx E H:; ^k k '(ii) M R vX = y M ~ + ~ x v v
*k k (iii) M P Vx = y
where ikI ik and P 12.6. Inversion
Mv+k VxI Vx t H$+k
are defined in Section 12.3.
v v
*k
In Section 12.4 we have described the distributional setting of
Hankel and inverse Hankel transformations. The present section
further work out the inverse Hankel transformation by working with
distributions in H: and we term this relation as an inversion of the
Hankel transfornation of distributions.
The main result of this section is:
Theorem 12.6.1. Let W E H$(y). If Vx E H:(x) and such that Y
Hankel Transform 285
(12.6.1) M:Vx = WY'
(12.6.2) vx = M': wy.
then Vx is ca l l ed the inverse Hankel t r a w e obtain
f o m f W and con Y
eque t l Y I
Proof. According t o (12.6.1) and (12.4.3') I w e have
IH: W , $ ( x ) > = cIH: Mf: V X I J , ( x ) > I Ti J, E Hy Y
= <myv VX' XI;*(%)> = wx'm;M;*(X)s
= < V X I $(XI >
which y i e lds the r e l a t i o n Mt W = Vx. Hence (12.6.2) is established. Y
I n addi t ion
X
Y
because
The exchange formulae (12.6.1) and (12.6.2) enable us t o ca l cu la t e numerous transforms. We mention below a f e w examples of them.
Examples. Show t h a t
M; 6(x-a) = Jay J v ( a y ) , a > 0, y > 0;
IH: Jax J ~ ( ~ x ) = 6(y-a);
(i)
(ii)
Proofs. According t o (12.4.8) w e have - IH; 6 (x-a) = < 6 (x-a) ,G Jv(xy) 7 = & Jv , ( ay ) .
and hence (i) is establ ished, Also, by (12.6.2),
IH; Jay ~ " ( a y i = 6 (x-a) I
286 Chapter 12
and by changing y to x, we have
which proves (ii). Now by ( 1 2 . 4 . 8 1 , we have
= - @ Jy(ay) - & (ay) +$ Jy+,(ay) a u-1
by (12.5.10). This proves (iii),
12.6.1. Remarks
The space H: does not contain the Dirac functional 6(x) nor its
derivatives. If we take a semi-closed interval [O,w[: in place of I
then 6(x) and certain of its derivatives would be in H: . this case each of the elements of this space would not have a unique
transform nor a unique inverse because the transform of 6(x) then is
null and hence Mywould not be an automorphism on H: . Let us
illustrate this remark with the help of the following example.
But in
1. According to (12.4.8) , we have
1 IH;~(X) = <6(x) Jxy, J"(xY)> = 0, v > - 2'
If M: Vx = W then we may write IHz [vx+c6 (x) 1 = W for any c. But
in this case, the inverse of W , according to Theorem 12 .6 .1 , is not
unique and it 'would be Vx+c6 (x) . Y r Y
Y
Moreover, in certain cases we see that the distributions in H;
associated with 6 (k) (x) . solution X = 6(x) in ID'
For instance, the equation XX = 0 has the
and the solution X = F in Hd defined by
< F , ~ I > = lim x '"'1/2#I(x) , Q E H, ; X'O+
because
<xF,Q(x) > = <F,xQ(x) > = 0
since
meniber of H: follows directly from the condition (ii) of Lemma 12.2.1.
sup x 'v-1/2#I (x) is finite. XCI
That this truly defines F as a
Hankel Transform 287
(See Zemanian [a], pp. 151-152.)
I 2.
[ S ] has shown that a Hankel transformation of any order can be
defined. Briefly, we see this as follows.
In the preceding case we have taken v 1. - 2. However Zemanian
Let v be a real number and let k be a non-negative integer 1
such that v+k 1. - z for every $(y) E Hv . We now set
Let Vx E H'(x) and if the Hankel transformation of order v+k
is denoted by M:Vx which is a distribution in Hl . of V
M ' V can be defined as
Then X
v x
<M:Vx,$(X)' = <vx, Mv,k$(X)>
1 If we take v 1 - 7 in M:, then M: for every Q E Hv.
follows that M; Vx coincides with the Hankel transformation of distributions defined by (12.4.3).
= My and it
12.7,The n-Dimensional Hankel Transformation
In the preceding sections we have extended the Hankel transfor-
mation to certain generalized functions of one dimension. In the
present section we develope the n-dimensional case corresponding to
the preceding work. Some of the results presented herein are similar
to
of
We
those of Koh [ 2 ] . Here we use the following notations.
For our purpose we shall restrict x and y to the first orthant IRn which we denote by I. Thus I = Ix E IRn , 0 < xv < m v = l r . . ,nl.
2 % shall use the usual euclidean norm, 1x1 = C xvl . A function on v=l
a subset of IRn shall be denoted by f (x) = f (x1,x2,...,x ) . By Cxl
we mean the product x1 x2.. . xn. Thus,Cx j = x1 x2 ... xn where n " m m m1 m2
m = Cml,m2,...,mn3.
xv 5 y and xv < y, ( v = 1 , 2 ,.. . ,n) . non-negative integers in IR
Letting (k) = kl+k2+ ...+ kn, Dx shall denote
The notations x 2 y and x < y mean respectively
The letters k and m shall denote n U
i.e. kv and mv are non-negative integers. k
(k) (12.7.1)
0
k kl Ic2 n axlax2.. . axn while (x-'Dx) denotes
288 Chapter 12
(12.7 - 2 )
12.7.1. The spaces h and h' P - 1-I
Let 1-1 be a fixed number in ( - m , m ) . By hp we mean the
infinitely differentiable and complex valued functions $ ( x ) which
are defined on I and such that €or each pair of non-negative integers
m and k in IRn
(12.7.3)
Since $ is infinitely differentiable, the order of differentiation
in (x-lDx) is immaterial; thus
-1 a -1 a -1 a -1 a (Xi axi) (Xj ax' j = (x j ~ ) ( ~ i
for all i,j = 1,2,...,n.
The space h is a linear space. Since y P are norms, we have P m,o
a separating collection of semi-norms i.e. a multinorm. An equiva-
lent topology for h may be given by the multinorm {p:l with v
As k and m traverse a countable index set, h is, in fact, a counta-
bly multinormed space.
if $, E h 1-I
n -+ m independently.
P We say that a sequence {$,,I is Cauchy in h
P 1-I for all v and for every m,k, ym,k ($,-On) + 0 as v and
k Lemma 12.7.1. If $ ( x ) E hP, then Dx$(x) is of rapid descent for each k.
Proof. Since - -1 a ) k ix-p-1/2
$(Xl,. . . ,xi I . . . ,Xn) (xi 5 i
= x . -*ix-p-1/2 ki 1 b.x. j a (-)I$, '
j=O i 1 i J 1 ax
we have
(12.7.4) (x-lDX) [ x 1- P-1/2 $ (x)
kl kn . .+In -1-1-1'2 ... 1 b.[xj] x 4 = cx ICxl
j,=o j =O J '1 Jn
-2k
a xn n ax l....
where the b. are appropriate constants. N o w , consider $ E h . By 3 P
Hankel Transform 289
i = l,...,n
Fina l ly , by induct ion on k and using (12 .7 .4) w e have
The space h' is t h e dua l of h and is t h e space of d i s t r i b u - ?J P '
lJ' t i o n s (continuous l i n e a r func t iona l s ) on h
The fol lowing p rope r t i e s are immediate ex tens ions of t h e one dimensional case. Using t h e r e l a t i o n (12 .7 .4) whenever c a l l e d f o r :
1. I D ( 1 ) , t h e space of i n f i n i t e l y d i f f e r e n t i a b l e func t ions wi th compact support on I, i s a subspace of h f o r every choice of u. Thus, t h e r e s t r i c t i o n of any f E h' t o ID( I ) is i n I D ' ( 1 ) . However D(1) is not dense i n h . 2. The complex number t h a t f E h' a s s igns t o $ E h is denoted by < f , $ > . W e a s s ign t o h' t h e following topology:
P
-0
v
u P
u
a sequence { f , ) converges t o f E h' i f < f - f j , @ > -+ 0 a s j -+ m ?J J
?J f o r a l l $ E h . For each f E h'
?J non-negative in t ege r r
I <f,$>
t h e r e exists a p o s i t i v e cons t an t C and a such t h a t
lJ m a x y m I k ( $ 1 . Recall t h a t p: = O I m , kz r
3 . L e t f (x) be a Locally summable func t ion on I such t h a t f (x) is of slow growth a s (x I +
on 0 < x y < 1, v = 1 , 2 , . . . ,n. Then f (x) genera tes a r egu la r genera- l i z e d func t ion f i n h ' def ined by
and Cxl u+1/2f (x) i s abso lu te ly i n t e g r a b l e
?J
m m
< f I + > = ..... ( f (x l Ix2 , ... ,xn) $(x1,x2 ,... ,xn)dxldx2 .... .dx n' 0 0
4 E hV.
This s ta tement fol lows from t h e mean va lue theorem f o r n-dimensional
290 Chapter 1 2
i n t e g r a l s (See Fleming CllIp.155) and t h e fact t h a t 9 is of r ap id descent .
12.7.2. Operations on h and h1 ! J - 1.1
I n t h i s s ec t ion w e perform some opera t ions on h and h' i n t h e lJ !J
following manner.
Lemma 12.7.2. For any p o s i t i v e o r negat ive i n t e g e r n and f o r any u I t he mapping $ ( x ) + Cx ln9(x ) is an isomorphism from h
It follows t h a t f ( x ) + Ix lnf (x) def ined by h;l+n*
onto !J
d. Cxlnf (x) I 9 ( X I > = < f ( X I I [ X I " $ ( X I >
i s an isomorphism from h t on to hl u+n !J*
Proof. If 9 E hlJI then
Cxlncp(x) I lJ+n ( [ X I n $ ) = sup I. cx Im (x-lox) kCx 1 -u-1/2-n 'm,k I
- - 1 1 ymIk(9)
2 W e now de f ine t h e following opera tors on h
= x !J+1/2 a x-lJ-1/2 Nt!J i axi i
N = N1!JN21J.. .. , N n U = [ x ] ~ ' + ~ / ~ a n [ -j - u- 1/2 axl. .... ax* lJ I
-u-1/2 a xu+1/2 M = x i y i axi i
Also, w e de f ine an inve r se opera tor t o N as fol lows: v
9 ( t ? X 2 , . . . , x n ) d t p+1/2 !J-1/2 N;: Q = xi I
m
N - l cp = xl+1/2 , x 2 t - p - 1 / 2 ~ ( x 1 1 t I . . . ~ x n ) d t 2lJ m
and so on.
Ctl-!J-1/2Q(t)dtR,.. . ,d tn .
That N i l i s t r u l y the inve r se t o N fo l lows from t h e f a c t t h a t Q is lJ
Hankel Transform 291
infinitely differentiable and of rapid descent.
Lemma 12.7.3. N 0 is an isomorphism from h onto h?J+l. P ?J
U + 1 Lemma 12.7.4. M + is a continuous linear mapping of h
11
!i* on to h
2 p + l ax1,.,.. Lemma 12.7.5. M N = [ x ] - ~ ' - ~ ' *
U P
n an axl.. . [ 1 - ?J-1/2- - = (2--).
a xn i=l
is a continuous mapping of h into itself.
A A
In the dual spaces, we define N and M as transpose differen- u Fi tial operators by
(12.7.5)
(12.7.6)
<NUf,$> = <f,(-l)"M lJ $>, f Ehk, $ E h?J+l,
<ilJf,+> = < f t ( - l ) % lJ $>; f E hL+l, + E hP.
A l s o , we can define
These definitions are consistent with the usual meaning of transpose
differential operators in the sense of Chapter 5. In views of lemmas
12.7.3, 12.7.4 and 12.7.5, we have
.L
Lemma 12.7.6. (I) The transpose differential operator N defined ?J
by (12.7.5) is a continuous linear mapping of h' into hL+l. ?J
(2) The transpose differential operator Mudefined by (12.7.6)
is an isomorphism from h' onto h'. P+1 v
r h
(3) The transpose differential operator M N given by (12.7.7) l J ? J
is a continuous linear mapping of h' into itself.
12.7.3. TheJiankel transformation in n-variables
lJ
The structure of a distribution formulated in Section 12.7.1
enables us in this section to describe the distributional setting of Hankel transformation in n-variables in the following manner.
292 Chapter 12
We define the n-dimensional classical Pth order Hankel trans-
formation by
m m
= 1.. . . . I$ (xl,. . . ,xn) Z'J,, (2) dxl.. . . .dxn 0 0
where Z = x y +...+ xnyn.
for every 9 F h This is due to the fact that $ is infinitely
differentiable and of rapid descent as 1x1 -+ -; while Z J (Z )=O (Z"')
as Z + O+ and it remains bounded as IZI + m. These properties of
$(xl, ..., x ) also ensure the validity of the classical results
(12.1.8) and (12.1.9) when extend to n-dimensional and these results
are given by
For 1.1 2 - $, the Hankel transform exists 1 1
P. f P
n
(12.7.9) U Y , , . ..rYn) = q 4 4 X l,*..,x n 1
% a0 m n
0 0 i= 1 = 1.. . . . J $ Rl,. .. ,tn) n (tiYi)
and
(12.7 .lo)
where y = (y, ,... ,yn) and x = (x, ,... ,x 1 . n
Note that these formulae are also valid if $ E h By (12.7.9) 11.
and (12.7.10) we can make the following inclusions.
1. M $ and IH'l4 are functions of (yl ,..., yn) and (xl ,..., x ) P P n
P* which belong to h
2. If a sequence {$,I -+ 0 in the sense of h then M u $n + 0 P I
P* and + 0 in the sense of h
3 , In (12.7.10) and (12.7.91, $(xl ,..., x,) and $(yl ,..., y,) are called the anti-transforms (or inverse-transforms) of
4 (y ,Y,) and 4 (X ll... #x n 1
Also, by (12.7.9) and (12.7.10) we have
Hankel Transform 293
Y M; M u @ (XI = 9 (XI , x= (xl,.. . ,x n ) and y= (yl,.. . ,Y n 1
Hence, we may conclude that *lJ or My is self reciprocal. Therefore,
IH; and My are reciprocals automorphism of each other on h lJ
if lJ l J ;
9 E hlJ, then M 6 E H . lJ lJ
The above results permit us to make the following definition:
The Hankel transformation of order l~ of a distribution V E h' (x)
is the distribution in H ' (y) which is denoted by M V (My Vx whenever
we desire to make the variable precise ) and defined by
lJ
lJ lJ lJ
(12.7.11) <M; VX, 9 (Y, . . tYn) > = 'Vxi 9 ( ~ 1 r ryn)> V9 E hlJ - This is a definition by transposition given in Section 5.1 of Chapter
5. To conform with established terminology, we shall say that every
generalized function (or distribution) belonging to h' is a M - transformable generalized function (or distribution) in n-variables.
lJ v
A l s o by (12.7.10) , (12.7.11) is equivalent to
< MYV I M: @(x, ,.. . ,X ) > = <Vx, $(xl,. . . ,xn)bV @ E h . lJx n ?J (12 .7.11')
Since M 9 belongs to h we deduce from (12.7.11') that M I J V belongs
to h' and hence we conclude that transformation defined by M in
(12.7.11) is an automorphism on h' . lJ l J r
lJ lJ
lJ
If the distribution V is associated with a summable function
h (xl,.. . ,xn) on I, then (12.7.11) leads to the equality
4 m CO n (12.7.12) ..... Ih(tl,...,tn) ( n (t.y.) 1 1
0 i=l
in accordance with (12.7.9). Thus, we may infer from this distribu-
tional setting that the Hankel transformation of generalized
functions in n-variables generalizes the Hankel transformation of
functions in n-variables.
Problem 12.7.1
denotes the Surface distribution in the sense of If 'Sn(a,R)
294 Chapter 12
Sec t ion 4.5 of Chapter 4, then show t h a t
We now e s t a b l i s h some t ransformat ion formulae on h and h l . P
1 Lemma 12.7.7. L e t 2 - z' I f 0 E h,,, t hen
(12.7.13) M,,+l ( C - X I 0 ) = N,, H,, 0 (XI
(12.7.14) (N,,0) = C-ylM,, 0
(12.7.15) H$Cxl20 (X)) = M,, N,, M,, 0
(12.7.16)
I f 0 E h,,+l, then
(12.7.17) M,, (Cx l0 ) = M,, M ,,+10
(12.7.18) M,, (M,,*) = CYIM, ,+~$ .
M,, (M,, N,,O) = (-1) "Ly12m,, 0.
Proofs. The proofs of these formulae can be seen i n Koh C n l , pp. 432-433.
The above lemma enables us t o prove t h e fol lowing theorem whose proof follows analogous arguments t o those of Sec t ion 12.5.2 us ing t h e appropr ia te d e f i n i t i o n of t r a m p o s e d i f f e r e n t i a l ope ra to r s (12.7.5) , (12.7.6) and (12.7.7).
Theorem 12.7.1. L e t I.I 2 - i. I f V E h;, then
(12.7.19)
(12.7.20)
(12.7.21)
rnJ (-1) " C X l V ) = i+i,,v
m,, +l(i,,v) = (-1) ncyllH ,,v M$(-l)"[xI 2 V) = M N,, H,,V . L A
!J
(12.7.22)
I f v E h;, t hen A
(12.7.23) m,, ICxlV) = M,,M,,+lV
A
(12.7.24) XI,, (M,,V) = C Y I M , , + ~ V .
Hankel Transform 295
12.8,Variable Flow of Heat in Circular Cylinder
In this section we use the preceding theory of generalized
Hankel transformation in one variable to solve flow of heat in
circular cylinder.
generalized function in h' which is approached by the solution at
some particular boundary. Specially, the problem we wish to solve
is the following :
By a generalized boundary condition we mean a
V '
Find a temperature function V(r,t) on i(r,t) , r > 0,
-m < t < -, 0 < 0 < 2t) 1 that satisfies with k thermal conductivity of the diffusion equation:
(12.8.1)
and the following boundary conditions:
(a) As t + O+, V(r,t) converges in some generalized sense to
the distribution f(r). We consider here that r varies from 0 to R
where R is the radius of cylinder. Also, we consider V(r,t) and
f (r) have bounded support (relative to r) and belonging to E ' ( 1 ) where I = 0 < r < R.
The differential equation (12.8.1) can be converted into a
form that can be analysed by our zero order Hankel transformation
by using the change of variable
Here again u(r,t) and g(r) E e'(1). Accordingly, (12.8.1) becomes
applying Mo to (12.8.2) , formally interchanging Mowith and setting U (p ,t) = Mo Cu (r,t) 1 , we can convert (12.8.2) into
2 dU(p,t) + k P U(p,t) = 0. dt
The boundary condition suggests that A (p ) = Mog (r)
Section 12.4.1 we may write
so that by
296 Chapter 12
(12.8.3)
Furthermore, Theorems 12.4.1 and 12.4.2 state that, for each fixed
t > 0, A ( P ) e- kp2tis a smooth function of p in L ( 0 , m ) . Therefore,
we may apply the conventional inverse Hankel transformation to get
our formal solution:
A ( P ) = < g (XI I & J0(xp) > .
(12.8.4)
That (12.8.4) is true the solution can be shown as follows.
First of all, Jo(rp) and J,(rp) are bounded on 0 < rp < - and 2 2
-kp for T -5. t < -, 0 < p < m . These facts and the Theorem -kp t, - e
12.4.1 allow us to interchange the differentiation in (12.8.2) with
the integration in (12.8.4) since at every step, the resulting
integral converges uniformly on every compact subset of 0 < r < m.
Since emkp
for each fixed p, we can conclude that u(r,t) also satisfies (12.8.2).
Hence V (r, t) satisfies (12.8.1) .
2 G p Jo(rp) satisfies the differential equation (12.8.2)
We now prove our boundary condition. A s a function of p,
(12.8.5)
is smooth, and for each fixed t > 0, it is a member of L ( 0 , m ) by
virtue of Section 12.4.1. Thus, (12.8.5) satisfies the conditions
under which the conventional Hankel transformation is a special case
of our generalized Hankel transformation. According to (12.8.4) , its Hankel transform is u (r,t) , so that, for any 0 E Ho and # = Mo 4 , our definition of generalized Hankel transformation yields (see
equation (12.4.3)),
The integral on the right-hand side converges uniformly on OLt< - because its integrand is bounded by
l<g (XI, KP J0(xp) > @ ( P I I E L ( O , m ) .
Thus, we may interchange the limiting process t + O+ with the
integration to get m
lim <u(r,t),O(r)> = j<g(x), EP J0(xp)> Q(p)dp. t+O+ 0
Bessel Series 297
Again by (12.4.3) I the right hand side is equal to <g (r) ,$ (r) >. Thus,
we have shown that, in the sense of convergence in HA,u (r,t) + g (r) as t + O + . In otherwords, V (r,t) + f (r) in a generalized sense i.e.
in the sens,e of HA.
Problem 12.8.1
Prove that (12.8.4) satisfies the differential equation
(12.8.2) for 0 < r < m and 0 < t c a.
12.9. Bessel Series for Generalized Functions
F r m now we shall be concerned with the second part of this
chapter which shows that the distributions having support on Co,a]
can be developed in a Fourier Bessel and Eessel Dini series which
converge to the distributions on the space of conventional functions
contained in a particular space I D ( 1 ' ) where I' denotes the interval
[O,al. The results presented in this part bear the close resemblance
as indicated in Lavoine [ E l .
12.9.1. Statement
The construction of the results presented in this section
depends upon the properties of the Bessel function given in Watson
c11.
Throughout this section we take v - i. The Bessel function
of order v is defined by (see Problem 1.4.1 of Chapter 1)
and A j = 1,2,3 denote the positive roots of J (ax) = 0 in increa-
sing order. We put j ' V
f
elsewhere.
For each v,J1 (A.x) forms an orthogonal system: v 7
a I x Jv(A.x)Jv(Akx)dx = 0 7 [ -. J;+::ja) I k = j .
(12.9.1)
By T we denote a distribution with support contained in 1'. (This
298 Chapter 12
means that < T I $ > = 0 for each function +(XI whose support is
exterior to I!.) By every distribution with bounded support (see
Section 4.1.3 of Chapter 4), T is of finite order, and this order of
T we denote by m, m = 0,1,2,... . We associate the numbers A . T with T as
J
(12.9.2)
and the sum
(12.9.3) n
Tn = 1 A j (T) J,(X.x) j =1 v 3
which is evidently a distribution with support in 1'.
It is important to remark that if v is neither an integer nor
0, then in order that A . ( T ) exists in general, it is necessary to be
v > m-1. This is because the derivatives of xJv(A x) of order m are not continuous at the origin.
3
j
In (12.9.2), the A . (T) are called the coefficients of the 3
Fourier-Bessel series of T. If T = f(x) a summable function with support contained in I', then (12.9.2) gives
a A . (f) = 2 1 f (x)xJv(X.x)dx, l a' J;+~(A~~) o 3
and we find again the coefficients of the Fourier-Bessel series in
the sense of functions. Also, we say that m
is the Fourier-Bessel series of the distribution T. Now we want to
show that it is convergent and equal to T on a certain space of functions.
12.10. The Space B m, v
We construct in this section the space B in the following m,v
manner on which the distributional setting of Fourier Bessel series
will be formulated in the subsequent section.
Let m be a non-negative integer, and let x denote a real variable.
continuously differentiable on a compact neighbourhood of I' and
By Bm,v we denote the space of functions which are m times
Bessel Series 299
which admit on I' a representation of the type 0
(12.10.1)
with the conditions that either v > m - 1 or v = 0,13,..., and in all
these cases the series 1 la. ($1 I A m is convergent. Here a. (0) are
the numbers which do not depend upon the variable x but depend upon
the choice of the functicm Q (x) in B,,,.
m
j=1 3 3 3
For x = 0 we have x Jv(A.x) = 0. Hence, we have by (12.10.1) 3
(i)
A l s o , by the property of A given in Section 12.9.1, we have
J (A.a) = 0. Therefore, we obtain by (12.10.1),
(ii) Q (a) = 0 where Q E Bm
Q (0) = 0 when Q E Bm for all integers m 2 0. I V
j
v 3
, V *
Hence, we conclude that (i) and (ii) are the necessary conditions
for (0 E B and in particular for Q belongs to Bo,v. Let us now
further state the property of Bm m , v
, V *
Consider a function Q [x) of Bm,v. If there exists an interval
1" containing I' , on which Q (x) is .m times continuously differentia- ble. A l s o , if a(x) be an infinitely differentiable function whose
support is a neighbourhood of I" and such that a(x) = 1 on 1'. Then
a (x) 0 (x) is m times continuously differentiable with bounded support and as a consequence belongs to the space IDm of Section 2.3 of
Chapter 2. Hence <T,a (x) Q (x) > exists, since T is a distribution of
order rn with support contianed in I' 3nd its value does not depend
upon the choice of a(x), and hence we denote this value simply by
<T,O (XI >.
Equality on B m , v
Let S , p E IN be a sequence of distributions of order m with P
support contained in 1'. If there exists a distribution T with
support contained in I' of order less than or equal to m such that as
P - + m
<T - S Q(x) > -t 0 for each Q of B,,,, P' (12.10.2)
then we Will say S + T on B , or that lim S = T on Bm,v. P P+- P m,v
300 Chapter 12
12.11. Representation of a Distribution by its Fourier Bessel Series
The results of Section 12.9 and the construction of the space
B given in the preceding section enable us in this section to
formulate the distributional setting of Fourier Bessel series in
terms of the representation of a distribution on Bm
Bessel series which we outline in the following manner.
m,v
by its Fourier I V
Theorem 12.11.1. Let T be a distribution of order m with support contained in 1'. Then we have
lim Tn = T n-
or , more explicity in the sense of series On Bm,vI
ce
T = 1 A . (T) Ja(A.x) j=1 J V I
Before giving the proof of this theorem we will need the
following three lemmas which enable us to formulate the proof of this theorem.
Lemma 12.11.1. Let 4 belong to B and if we put m,v
m
Then
1. the first m derivatives of I$ (x) exist and are continuous on 1 every neighbourhood of 1';
2. we have on I' that $:h) (x) - 4 (h) (x) = 0 where h = 1,2,... ,m.
Proof. By the known properties of Bessel function (see Watson
[l]) and considering certain conditions on a,($) (see Section 12.10) J m .h
we can show easily the uniform convergence of 1 aj ($)a x Jv (A .x) j=1 dxm I
on every bounded interval of 1'. Hence, l., is established. By the
description of B
consequently 2 . , is also established.
given by (12.10.1) , we have d;, (x) = $ (x) and m,v
Lemma 12.11.2. Let m
d;,(x) = 1 a. ($1 x J,,(X~X) , Y 4 E B ~ , ~ . j=n J
Bessel Series 301
Then 0, (x) + 0 as n -+ - and $dh) (x) + 0 uniformly for h = 1 , 2 , .. . ,m. proof.^ It is easy to show that, for h = 1,2,...,m with K is any
positive constant
which tends to zero by virtue of the properties of BmIV.
Lemma 12.11.3. If T of order m has support contained in I', then
we have
<TI$ (XI > = (XI >, v Q Bm,v*
This is a consequence of Lemma 12.11.1 and of (Schwartz C11,
Chapter 111.7, Theorem XXVIII).
Proof of Theorem 12.11.1. It is sufficient to show that
<T-Tnr$(X)> -+ 0, Cp E B m,v
(12.11.1)
By Lemma 12.11.3, we can write
(12.11.2) <T-Tnr+ (XI > = <TI 4, (X) > - <Tn, 4, (X) >. NOW, by (12.9.3) , (12.9.1) and (12.9.2) , we have
n m a
n
Hence, by (12.11.21,
which tends to 0 as n + m . Let a(x) be an infinitely differentiable
function whose support is a compact neighbourhood of I' such that
a(X) = 1 on 1'. Then <TIJn(x)> = q,a(x)J,(x)> . Now, by the. Lemma 12.11.2, a(x) $,(XI -+ 0 in the space of f l o f Section 2.3 of Chapter
2 and consequently T belongs to the topological dual IDfm of IDrn,
302 Chapter 12
hence C T r a (x) qn (x) > -+ 0 as n -+ -. Examples. 1. If 0 5 c 2 a, the Dirac measure 6(x-c) which is
such that < 6 (x-c) ,$ (x) > = $ (c) is a distribution of order zero having support contained in 1'. the Theorem 12.11.1 yields
the representation
For v 2 -
00 J (cX.) (12.11.3) S(x-c) = % 1 J;(Ijx)
a j=l Jv+l(aAj)
defined on B O I v . Moreover, it is easy to verify this equality by
means of Lemma 12.11.3 and formulae of Section 12.9.
Note that (12.11.3) gives 6 (x) = 0 and 6 (x-a) = 0 which do not
yield a contradiction because if $ belongs to B then we have <6 (x) ,$ (x) > = 0 and <6 (x-a) I $ ( x ) > = 0.
0 , v '
2. Let the function g (x) be defined by
xv(1-x2)-a, 0 < x c 1
I elsewhere, ( 0
g (X) =
where v > - $ and a is not an integer (a > 1).
any Fourier Bessel expansion in the sense of functions. But in the
sense of distributions, Fp g(x) (which is of the order a' = integer
part of a) according to the Theorem 12.11.1, is represented on
Also, g (x) has not
Bi, by the series (2) I V
Note that v must either be a non-negative integer or satisfy
v > a'-1.
12.12. Other Properties of the Fourier-Bessel Series
The results of the preceding sections enable us to formulate a
uniqueness theorem of Fourier Bessel series in a distributional
setting which we term as other properties of the Fourier-Bessel series.
To prepare for this section we first need the following result.
Theorem 12.12.1. The representation of T on B by a Fourier- mIv
Bessel series of order v is unique. In other words, if
Bessel Series 303
on Bm then Aj = Aj (T). I V '
Proof. x Jv (Akx) belongs to B for each k 2 1. Now by m, v
Theorem 12.11.1, we have
Hence, by (12.9.1), it follows that Ak = Ak(T). Therefore, Aj=Aj (T).
Magnitude of the coefficients. We give now the following result
by means of which one can conclude the magnitude of the coefficients.
Theorem 12.12.2. We have
where K is any positive constant.
Proof.According to Section 5.4.3 of Chapter 5, T is equal on an
arbitrary neighbourhood of I' to the (m+l)th distributional derivat-
ive of a measurable function f(x) which is bounded on this neighbour-
hood. It follows that A.(T) can be written in the form 3
Now by differentiation and known properties o€ Bessel functions (see
Watson C11) we can obtain the desired result of this theorem.
Properties of a given series. We now give below the following
result which governs the properties of a given series.
Theorem 12.12.3. Let A j = 1,2,..., be a set of numbers with
H any positive constant such that [A. 1 < HAm+', for a large enough
number j.
sense of (12.10.2) ) .
j f
m 7 j Then the series 1 A JA (A . x ) is convergent on Bm (in the
j=l j v 3 f V
Proof. For each 4 in Bmfv, Lemma 12.11.3 gives
m ca
304 Chapter 12
and by (12.9.1) we can show that the modulus of this given series is
majorized by the convergent series
where K is any conventional number.
12.13. The Subspace Bm of B mlv
As remarked in the Section 12.10, we construct in this section
the subspace Bm of Bm
setting of Fourier Eiessel series on Bm.
and present some result of distributional I V
By % we denote the space of functions +(x) which are m+l times continuously differentiable on an interval containing I' and such
that 9 (x) is on 1' and
The importance of the space Bm can be seen because it contains the
space IDm+2 (I1) of functions which are (m+2) times continuously
differentiable with support contained in I' and hence also contains
the space ID (1') of infinitely differentiable functions having
support in 1'.
Theorem 12.13.1. Bm is a subspace of Bm,".
This result can be obtained from the following lemmas.
Lemma 12.13.1. Let the Fourier-Bessel coefficient of $(x)/x be
a I 9 (XI Jv (Ajx) dx.
a2J:+1 (ahj) 0
2 (12.13.1) cj ($/XI =
If 9 belongs to Bml we have
(12.13.2)
where M is any conventional number.
Lemma 12.13.2. If 4 belongs to Bml then we have on I', m
The proofs of these lemmas need many calculations and we give the
Bessel Series 305
outlines of them. (12.13.1) can be written c. ($/x)=
2a'2 Jzl(aX )A ( v , @ ) where 3
j j a
First we take $ in Bo. Then we have
where
and
By the structure Bol we have for 0 5 x -< a
(i) IO"(x)I < H =>lO'(x)I <Hx and 16(x)I <%x2
where H is a conventional number. Also, when 0 5 x 2 X i 6 with large
enough j I we have IJv (Xjx) I < H
tional number. Therefore, with 6 = zvt;6 , we obtain X \ l xv where HI is another conven-
1 3 2b+5
. -6
Hence, we finally get I where M1 = 2v+6HH1.
A < M X-5/2 j 1 j (ii)
By the expansion of Jv (z) (see Watson [ll) , we have
where 8 = v $ + $ and bv (z) is a bounded function for large enough z > Now, by putting the value of Jv (z) with z = h . x from (iii)
to Jv (X.x) in A', we obtain 0. 3
1 j 2 2 1/2 4v -1
(iv) A; < CI1,I + 8 1121 + II,l]
where
a I1 = J cos ( X x-8) $J IX) x-1'2dx,
J .-6 j
306 Chapter 12
I2 = A; 3'2 7 sin (A.x-B) Q (x) x - ~ / ~ ~ x , . -6 7
a
.-6 bv (hjx) 4 (x) x - ~ / ~ ~ x . -5/2
j I3 = A
*j
Further, if we integrate I1 by parts two times by using (i) and
property $ (a) = 0 together with the fact Xi6 + 0 as j + 0, then we
can conclude that there exists
Also, integrating by parts one
(vi)
J
a certain number G1 such that
for j > lo.
time to 12, we obtain
for j > jo
where G2 is a certain number.
If j > j-, then there exists a U
A T 6 5 x 5 a and by (i) I we have 7 a
number K such that bv ( A .x) < K for 3
J
and consequently there may exist a number G3 such that
(vii)
Let us denote M2 =
l f g i < G ~ hj -5 /2 for j > jo.
2 2 4 4v -1 (3 CG1 + -g-- G2 + G31 I
then by making use of (iv) , (v) , (vi) and (vii) , we obtain
A! M A - ~ / ~ for j > jo. I 2 j
(viii)
Pinally, since A. (v,Q) < A.+A! I we get by combining (viii) and (ii) 3 1 7
-5/2 for j > jo. j
IAj (v,Q) I < (M1+M2) h
Take now Q in Bm. If we integrate A.(v,$)mtbes by parts by 7
using the properties of Bessel function (see Watson [l]), then we
obtain
m A. (v,4) = (-1) x - ~ A. (v+m,f)
Hence by (12.13.41 I IAj ( v I Q ) 1 <(M1+M2) Aj -m-5/2 (which
7 j i
where f E B
is also true for v+m). 0'
Finally, we deduce (12.13.2) because Jv+l (axj)
Bessel S e r i e s 307
is of order ,I-% as j -f -. j
The Lemma 12.13.2 i s a consequence of t h e Lemma 12.13.1 and of t h e fol lowing r e s u l t from t h e theory of Fourier-Eessel series of func t ions (see Watson Cll, Sec t ions 18.24 and 18.26):
uniformly on Ca,al,a > 0 , a s n -+ m.
Remark. I f m = 29-2, s 2 1, then (12.13.2) g ives Ic. ($/x) I <
M , I T Z S + 1 / 2 s i n Theorem 1 of (Tolstov Cll, Sec t ion 8.20) , which imposes more r e s t r i c t i v e condi t ions on $ (x) /x than our condi t ions . (See a l s o Khoti Ell.)
3
1
Corol la ry of t he Theorem 12.13.1. I n t h e Theorem 1 2 . 1 1 . 1 and 12.12.3 w e can r ep lace Bm (i) v = 0 , 1 , 2 , ..., i
and v > m-1.
by Bm wi th t h e condi t ion t h a t e i t h e r (ii) m = 0 and v 2 - L, or (iii) m = 1 , 2 ,... I V
2
12.14. Eessel-Dini S e r i e s
I n t h e above sec t ions , w e have shown t h a t t h e d i s t r i b u t i o n s wikh support i n C0,aI a r e represeptab le by a Fourier-Bessel series on t h e space B where w e impose t h e condi t ion t h a t i t s func t ions vanish a t x = a. This condi t ion i s rep laced by another condi t ion
m , v
f o r c e r t a i n problems when w e s tudy Bessel-Dini series as given below.
This cons t ruc t ion and r e s u l t s of t h e p re sen t s e c t i o n are p a r a l l e l t o t h a t of Sec t ion 12.9.
12 .14 .1 . Statement
W e t ake v 2 - i, H is a real parameter; I ' denotes t h e i n t e r v a l CO,a]; x is a r e a l va r i ab le ; and B;, j = 1 ,2 , . . . , are t h e p o s i t i v e
J
r o o t s of t h e equat ion axJ:(ax)-HJv(ax) = 0. We p u t
i f v > H I
i f v = H
Iv (Box) , i f v <H. 1 Ry Iv w e denote the modified Bessel func t ion , I (x) = i-'JV (ix) ,
V
and $, i s the p o s i t i v e number such t h a t + i B o i s a r o o t of t h e equat ion
308 Chapter 12
z-'CazJ: [az) - HJ\, (az) I = 0,
so that
For each V , G ~ @,XI and the J~ ( 8 .XI system on 1' such that
j = 1,2,. . . , form an orthogonal 3
where
,if v > H,
n = , i f v = H, 0
we put
elsewhere G; @,XI =
and
elsewhere.
(12.14.2) JA (6.X) = v 3
To a distribution T with support contained in I' and of order
m (see Sections 4.1.3 and 5.4.3 of Chapters 4 and 5) we associate
the numbers
$(T) = 1, < T, xGv (H,x) > 00 I
0
B. (T) = a <T, XJv (8 .X) > 3 I
(12.14.3)
and the sum
Bessel Series 309
which evidently is a distribution having support 1'.
It is important to remark that if v is .not an integer nor 0,
then in order that 8.(T) exists in general, it is necessary to be
v > m-1. This is because the mth derivatives of xJv ( 8 .x) are not
continuous at the origin.
7
1
In (12.14.3), the B. (T) are called the Eessel-Dini coefficients 7
of T. If T = f(x), a summable function having support in I', then
we obtain the B. (f) to be the Bessel-Dini Coefficients in the sense
of ordinary functions. We further say that 3
m
is the Bessel-Dini series of the distribution T.
is to show that it converges and is equal to T on
of functions,
NOW our purpose
a certain space
12.15. The Space
In this section we construct the space % on which the rmrv
distributional setting of Fourier-Dini series is to be formulated in
the subsequent section.
Let m be a non-negative integer and x a real variable. By
we denote the space of functions $(x) which are m times %,m,v continuously differentiable on a compact neighbourhood of I' and
which admit on I' a representation of the form m
$ (XI = do ($1 xGV (H,x) + 1 d. (0) xJv (Bjx) j=1 J
with the conditions that either v > m-1 or v = 0,1,2,..., and in all
these cases the series
the numbers which depend upon the choice of $(XI in
m
1 Id. (0) lf3m is convergent. Here d. (0) are j=1 3 3 3
rmrv.
We give below (Theoremsl2.6.5 and 12.6.6) definitions of the
independent of Bessel functions.
The existence of <T,@(x) > , Y 0 E %,m,v can be shown as that
a ,m,v subspaces of
BmIv (see section 12.11).
Equality on EkIm Two distributions T and S having support in I' and of order z m
310 Chapter 1 2
Convergence on a I f
Let S I p E IN, be a sequence of distributions with support P
contained in I' and of order -a.
having support on I' of order -a such that
If there exists a distribution T
(12 .15 .1 )
then we will say S + T on
12.16. Representation of a Distribution by its &SEel-Dlni Series
+spI Q (XI > = 0, Y Q E %,m,v, as p -t 0 0 ,
or that lim S = T on P %tmtv P+" P
In this section we obtain results for the distributional setting
of Bessel-Dini series which would be similar to those of Fourier- Bessel series given in the Sections 12.11 to 1 2 . 1 2 .
Theorem 1 2 . 1 6 . 1 . Let T be a distribution of order m having support in 1'. Then we have
- lim Tn = T on %,m,v i n-
or more explicity in the terms of series m
T = B~(T) G;(H,x) + j=1 1 B. 1 ( T ) J ; ( B . x ) 3
Theorem 12 .16 .2 . The representation of T on by a
Eessel-Dini series of order v and parameter H is unique, i.e. if
on %,m,v*
rmr V
m
T = B~ G;IH,X) + B.J~(B.,X) on % j=1 I v 3 rmrv
then B = Bj (T) , where j 2 0. j
Magnitude of the coefficients. We now give the following result
by which one can conclude the magnitude of the coefficients.
Theorem 1 2 . 1 6 . 3 . We have IB. (T) I < K B';+3'2 where K is any 3
positive constant.
Properties of a given series. The preceding results enable us to make the following result.
Theorem 12.16.4. Let B j = 0,1,2,..., be a set of numbers j r
with M any positive constant such that IB. 1 < MBY+l for j being a 3
Bessel S e r i e s 311
m
l a r g e enough number. Then t h e series 1 B. J;(B.x) i s convergent
On % , m , v j=1 7 J ( i n t h e sense of (12.25.1) ) .
12.16.1. The subspace Bm of %,m,v
B i s t h e space a l ready descr ibed i n Sec t ion 12.13.
Theorem 12.16.5. B i s a subspace of % m
m , m , V .
Corol lary. I n t h e Theorems 12.16 .1 and 12.16.4, one can r ep lace
e i t h e r m = 0 and v
by Bm with t h e condi t ion t h a t e i t h e r v = 0 , 1 , 2 , . . . , 1 - T, o r e i t h e r m = 1 ,2 , . . . . , and v > m-1.
N o t e . The proof of Theorem 12.16.5 is analogous t o t h a t of Theorem 12.13.1 of Sec t ion 12.13 by r ep lac ing t h e r e fe rences (Watson Cll, Sec t ions 18.24 and 18.26 by Sec t ions 18.33 and 18.35).
%m,v 12.16.2,Another subspace of
Theorem 12.16.6. By % w e denote t h e space of func t ion $(x) , O , V
which are t w o t i m e s d i f f e r e n t i a b l e on an i n t e r v a l conta in ing I ' and such t h a t Q" (x) /x is bounded on I ' , $ (0) = $ I (0) = 0 , and 0' (a) - (H+L) Q (a) = 0.
Proof. The proof is analogous t o t h a t of Theorem 12.13.1 i f
w e pu t f (x) = and u t i l i z e t h e r e s u l t s of (Tols tov [l], Sec t ions 8.22 and 8.23).
1 2 . 1 7 . An Appl ica t ion of t h e Bessel-Dini S e r i e s
This s e c t i o n provides an account of t h e use of t h e preceding theory of Bessel Dini series t o f i n d o u t t h e s o l u t i o n of t h e problems of hea t flow i n a c y l i n d e r ' o f i n f i n i t e length.
S p e c i a l l y , t h e problem w e wish t o solve i s t h e fol lowing: Find a temperature f (r,t) (which i s a func t ion of r and t) such t h a t
1. it i s def ined f o r r i n I ' = [O,al;
2. it s a t i s f i e s t h e p a r t i a l d i f f e r e n t i a l equat ion
f o r 0 2 r 2 a , where v 2 0 , wi th g (t) being an i n t e g r a b l e func t ion €or t > 0 and S (r) i s a d i s t r i b u t i o n wi th support i n 1';
312 Chapter 12
3. it satisfies the conditions
(12.17.3) f (r,t) is bounded when r is bounded
- Hf (a,t) = 0, H < u. (12.17.4) a ar a- f (r,t) I
r=a
To solve this problem, we identify f(r,t) with a distribution
in r on l D ( 1 ' ) that eliminates the condition 1 and the restriction
0 < r < a in (12.17.1).
By Theorem 12.16.1 and the corollary of Theorem 12.16.5, the
distribution S (r) can be represented by
(12.17.5)
where J; (6 .r) is given by (12.14.2) and S .= 5 < S (r) , r Ju (Bjr) >.
The structure of S(r) given above enables us to consider the 1 ' j
representation of f(r,t) such that m
with F. (0) = 0. Now, we verify that (12.17.6) satisfies the
conditions (12.17.2) , (12.17.3) and (12.17.4) . 3
Since we take F . (0) = 0 and hence f (r,t) given by (12.17.6) 3
satisfies the condition (12.17.2).
From Watson [l], we know that all the J; (8.r) is bounded for 1
u 2 0 (see also Theorem 12.16.4) and hence fw,t) represented by
(12.17.6) satisfies the condition (12.17.3) . By the property of 8 . given in Section 12.14.1, we have
3
for r = a. Hence, f (r,t) given by (12.17.6) satisfies the condition
(12.17.4).
NOW, by putting the values of f (r,t) and S (r) from (12.17.6)
and (12.17.5) in (12.17.1), w e obtain
Bessel Series 313
But, we have by the recurrence formulae of Bessel function (see
Watson [ 11) that
Thus, we obtain
Consequently, by making use of (12.17.8) , (12.17.7) takes the form
m
From this result we observe that (12.17.1) can be verified in a
distributional setting if we solve the equation
2 B 7 7 . F . (t) + k & F j (t) = s 7 .g (t)
with F. (0) = 0 (which can be solved easily). Hence, we finally
obtain 7
2 2 m t B .ku
J;(B .r) dule-@ k' I
f(r,t) = k 1 S . C l g(u)e j=1 1 o
on (1').
If we put v 2 0 in (12.17.1), then the present case is a
problem of heat conduction in a cylinder of infinite length and of
radius a, cooled over its cylinderical surface and heated by spring
of intensity g (t)S (r) when f (r,t) denotes the temperature at time t
at the point whose distance from the axis is r. The use of Bessel
Dini series is already known in the situation of this case when S(r)
314 Chapter 12
is a function (see Carslaw and Jaeger [ 2 ] ) . ( S e e also Section 12.8.)
A theoretically interesting case arises when S(r)= 6(r-c)/2nc, 2 2 % 0 c < a. In this case we recall that r is equal to (x +y )
and hence we have
6 (r-c) 1 2n I 4 (c cos e , c sin e)ae. <- 2nc , cp (x,y) > = 0
12.18, Bibliography
In addition to the works given in the text we should like to
mention some references, which deal with the material of the present
chapter.
Dubey and Pandey Cll, Fenyo Ell, GUY [ l l r W e e [I], Lions [l], Srivastav [ll, Trione C11 and Zemanian [ 2 1 .
Footnotes
(1) Zemanian [ S ] has taken k to be a positive integer, we take
k 0.
(2) for the calculation of the coefficients (see Erdelyi (Ed.)
C21, v01.2, p. 26).
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This Page Intentionally Left Blank
INDEX OF SYMBOLS
Ac, 12
BH,O,v' 311 309 B
Bm, 304, 311
BIll,V' 298
H,rn,V'
C ( $ ' ) r 70
ID, 19
ID' I 27
c", 1
ID,, 80 ID;, 80
ID:, 36
IDo (01, 80
IDA (01, 80
Do (b) t 80
(b) 80 I D ( 1 ) ,232, 273
I D ' ( 1 ) , 273 ID(IR"), 23
ID'( W"), 27
ID; (W"), 43
IDb+(lR"), 43 I D k , 20
m k ( m n ) , 23 I D V k l 27
ID' k( lR" ) , 27
ID'l , 89 (ID') -I, 89
227, 230
227, 232 Ea,lll'
E:,ul \
E 231
E 233
E(r) , 207 Pr9'
E , 21 E' i 27
f i ( I R n ) , 23
E ' ( I R n ) , 27
FP, 8
Fx, 91
..;I, 91
hv, 288
hl, 288
Mp, 292
My', 292
Hu , 272 H: , 272 M v l 270
Mi', 270
JA , 221 JA(r), 221
JI', 209
JI ' ( r l , 207 L ( O , m ) , 269
IL, 109
329
This Page Intentionally Left Blank
AUTHOR INDEX
Albertoni, S., 75, 315
Antosik, P., 85, 315
Apostol, T.M., 229, 315
Arsac, Jacques, 105, 315
Benedetto, J., 144, 209, 315
Berg, L., 315
Bochner, S., 315
Bredimas, A., 160, 203, 315
Bremermann, H.J., 105, 240, 316, 325
Bremmer, H., 144, 326
Campos Ferreira, J., 85, 316
Carmichael, Richard, D., 225, 316
Carslaw, H.S., 182, 203, 314, 316
Chandrasekharan, K., 315
Choquet, Bruhat, Y., 78, 105, 316
Churchill, R.V., 144, 316
Colombo, S., 144, 157, 182, 238, 244,
Constantinesco, 89, 317
Courant, R., 205, 317
Cristescu, R., 105, 317
Cugiani, M., 75, 315
de Jager, E.M., 105, 317, 325
Di Pasquantonio, F., 17, 317
Ditkin, V.A., 144, 317
Doetsch, G., 144, 190, 317
Dubey, L.S., 314, 318
Durand, L., 105, 316
Ehrenpreis, L., 105, 318
Elliott. D., 224, 326
Emde, F., 14, 135, 273, 320
Erdelyi, A., 144, 225, 318
Erdelyi, A,(Ed.), 117, 122, 125, 126,
259, 262, 271, 317
144, 157, 165, 181, 189, 193, 197, 201, 215, 238, 270, 277, 314, 318
Fenyo, I., 314, 318
Fisher, B., 75, 318 Fleming, W.H., 290 , 318
331
FOX, C., 227, 268, 318
Friedman, A., 25, 318
Friedmann, B, 205, 319
Fung, Kang, 268, 319
Garnir, H.G., 25, 72, 144,
Gelfand, I.M. 16, 21, 91,
G e r a d i , F.R, 227, 319
Ghosh, P.K., 144, 319
Giittinger, W., 21, 75, 78,
Guy, D.L., 314, 319
Hadamard, J., 7, 320
Handelsman,Richard,A.,227,320
Hayashi, Elmer K.,225, 316
Hilbert,D, 205, 317
Humbert,P., 194, 322
Ince,E.L., 320
Jaeger,J.C.,182, 203,314,316
Jahnke,E.,14, 135, 273, 320
Jeanquartier,P.,268, 320
Jones, D.S. 75, 144, 320
Kaufmann,H., 324
Khoti,B.P., 307, 320
Koh, E.L.,275, 287, 294, 320
Korevaar, J., 144, 320
Krabbe, G., 144, 321
Kree, PI 314, 321
Laughlin, T.A., 238, 321
Lavoie, J.L. 200, 321
Lavoine,Jean, 16, 17, 59, 84, 85, 89, 105, 130, 131, 132, 133, 144, 157, 190, 191, 192, 193, 194, 196, 201, 203, 209, 216, 219,
317 , 321
322
231, 233, 319
105, 319
319, 325
237, 238, 253, 262, 297,
Lew, John,S., 163, 227,320,
Loins,J.L. 314, 322
332 Author Index
Liouville, J., 199, 204, 322
Livennan,T.P.G., 144, 322, 325
Lojasiewicz, S., 82, 83, 84, 89, 322
LOSChl F a , 14, 135, 273, 320
Maclachlan,N.W.,190, 194, 322
Magnus, W., 197, 280, 323
Marinescu, G., 105, 317
McClure,J.P., 224, 323
Mikusinski,J,,25, 42, 85, 144,315, 323
Milton, E.O., 105, 132, 225, 316, 323
Misra,O.P., 29, 84, 85, 89, 209, 216 219, 225, 237, 253, 321, 323
Munster, M., 144, 319
Oberhettinger,F., 197, 280, 323
Oldham, K.B.,198, 323
Osler,T.J., 200, 321
Paley,R., 97, 324
Pandey,J.N., 225, 314, 318, 324
Prudinkov,A.P., 144, 317
Roach, G.F., 205, 324
Roas, B., 198, 324
ROOS, B.W., 324
Roberts, G.E. 324
Rota, G.C., 324
Rudin,W. , 105 I 324
Sato, 324
Sauer,R., 3171 324
Schmets,J., 25, 231, 233, 319
Schwartz,L., 7, 22, 25, 6 4 , 66, 71, 72, 75, 78, 81, 89, 91, 97, 102 105, 108, 141, 149, 205, 324
Shilov,G.E.,16, 21, 91, 105, 319
Sikorski,R., 85, 315, 323
Silva e Sebastia, 25, 84, 85, 87,
Smith, M., 325 Sneddon,I.N., 238, 259, 260, 282, 325
Soboleff, S.L., 325
Soni,R.P., 197, 280, 323
Spanier,J., 198, 323
105, 144, 225, 324, 325
Srivastav,R.P. 314, 325
Stakgold,I. ,205, 325
SZabO, I., 317, 324
TOlStOV,G.P. 307,3111 326
Tremblay,R.,200, 321
Treves,F., 48, 66197, 102,326
Trione,S.E., 314, 326
Tuan,P.D. 224, 326
Van Der Pol. 144, 326
Vladirnirov,V.S.,64, 326
Vo-Khac-Khoan , 105.,150,205,326 Watson,G.N.,158, 297, 305,
Wiener,W., 97, 324
Widder,D.V., 207, 326
Wilde, M.De., 25, 231,233,319
Wong,R., 224, 323
Yosida,K.,. 163, 205, 326
Zemanian,A.H., 22, 25,38,85,
282,284,287,314,320,325, 326, 327
306, 307, 311,312,313,326
1 0 5 1 144,225,227,231,233, 235, 244,268,272,273,274,
Zygmund,A., 2041 327.
