TRANSFORM ANALYSIS OF GENERALIZED FUNCTIONS
NORTHHOLLAND MATHEMATICS STUDIES Notas de Matematica (106)
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TRANSFORM ANALYSIS OF GENERALIZED FUNCTIONS
NORTHHOLLAND MATHEMATICS STUDIES Notas de Matematica (106)
Editor: Leopoldo Nachbin CentroBrasileiro de Pesquisas Fisicas, Rio de Janeiro and Universityof Rochester
NORTHHOLLAND AMSTERDAM
NEW YORK
OXFORD
119
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
NORTHHOLLAND 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 thepriorpermission 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&ginPublicatiin Data
Misra, 0. P. Transform analysis of generalized functions. (NorthHolland 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 0527389 ISBN 0444878850 (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 .Misra
Jean Lavoine
TABLE OF CONTENTS
CHAPTER 0
0.1. 0.2. 0.3. 0.4.
CHAPTER 1
1
PRELIMINARIES Notations and Terminology Vector Spaces Sequences Some Results of Integration 0.4.1. Set of measure zero on the line IR 0.4.2. The sawtooth function
FINITE PARTS OF INTEGRALS Definition Extensions of the Definition Integration by Parts Analytic Continuation Representations of Finite Parts on the Real Axis by Analytic Functions in the Complex Plane 1.6. Change of Variable
7 9 10 12
BASE SPACES
19
2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.
19 19
1.1. 1.2. 1.3. 1.4. 1.5.
CHAPTER 2
Base Spaces The Space ID The Space IDk (k 0) The Space $ (Functions of Rapid Descent) The Space 8 The Space ZZ (of Entire Functions) Inclusions The Space 8 The Space 8 (JRn)
vii
17
20
20 21 21 21 22 23 25
CHAPTER 3 DEFINITION OF DISTRIBUTIONS 3.1. Generalized Functions 3.1.1. Inclusion of 3.2.. Distributions
15
@'
25 26 27
Table of Contents
viii
Inclusions 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) 3.2.1.
3.3.
CHAPTER 4
PROPERTIES OF GENERALIZED FUNCTIONS AND DISTRIBUTIONS 4.1.
Support
Point support Distributions with lower bounded support 4.1.3. Distributions with bounded support 4.2. Properties 4.2.1. Boundedness 4.3. Convergence 4.1.1. 4.1.2.
Completeness and limit Particular cases of convergence in D' Convergence in $I Convergence to 6 (x) 4.4. Approximation of Distributions by Regular Functions 4.5. Distributions in Several Variables 4.3.1. 4.3.2. 4.3.3. 4.3.4.
CHAPTER 5
OPERATIONS ON GENERALIZED FUNCTIONS AND DISTRIBUTIONS 5.1. 5.2. 5.3.
5.4.
Transpose of an Operation Translation Product by a Function 5.3.1. The space M(@) and the general definition of product 5.3.2. Distributions belonging to ID' or 6' 5.3.2.1. Distributions of finite order 5.3.3. Tempered distributions 5.3.4. Ultradistribution 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 31 37 38 39 39 40 40 41 42 47 47 48 49 50 50 51 51 51 52 52 52 53
Table of Contents
5.5. 5.6. 5.7.
5.8.
5.9.
CHAPTER 6
CHAPTER 7
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 Differentiation of Product Differentiation of Limit and Series Derivatives in the Case of Several Variables 5.7.1. Generalization of 6' (x) 5.7.2. The Laplacian Convolution 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 $ 5.8.5.1. Convolution in $: 5.8.6. Convolution equations 5.8.7. Fundamental solution Transformation of the Variable 5.9.1. Definition of Tu(x) 5.9.2. Examples 5.9.3. Bibliography
ix 53 54 57 59 59 61 62 63 64 65 65 65 66 68 70 71 71 71 72 72 74 75
OTHER OPERATIONS ON DISTRIBUTIONS
77
6.1. Division 6.1.1. Division by xn (n>O, an integer) 6.1.2. Division by a function 6.1.3. Z T multiplier for o 6.2. Antidifferentiation 6.2.1. Antiderivative in I D :
77 77 78 79 80 81
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
82 82 83 84 85 a5 88
THE FOURIER TRANSFORMATION
91
7.1. Fourier Transformation on 22 7.2. Fourier Transformation on ID
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 on ID' and Z' 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 Transformation 7.12. The Fourier Transformation of Convolution and Multiplication 7.13. Applications 7.14. Bibliography CHAPTER 8 THE LAPLACE TRANSFORMATION
94 94 94 95 96 96 97 99 100 101 101
102 103 105 107
8.1. Laplace Transformability 8.2. Laplace Transform 8.2.1. Case for functions 830 Characterization of Laplace Transform 8.4. Relation with the Fourier Transformation 8.5. Principal Rules
108 109 110 110 113 113
8.5.1. Case for functions 8.6. Convergence and Series 8.6.1. Examples 8.7. Inversion of the Laplace Transformation 8.7.1. Example 8.8. Reciprocity of the Convergence 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
115 116 117 118 120 120 121 121 122 124 124 125 127 129 131
8.11. Abelian Theorems 8.11.1.Behaviour of the transform at infinity
132 132
Table of Contents 8.11.2. Behaviour of the transform near a singular point 8.12. Tauberian Theorems 8.12.1. Behaviour near the lower bound of the support 8.13. The nDimensional Laplace Transformation 8.13.1. The Laplace transformation in n variables 8.13.2. Convolution 8.14. Bibliography CHAPTER 9 APPLICATIONS OF THE LAPLACE TRANSFORMATION 9.1. Convolution Equations 9.1.1. Examples 9.2. Differential Equations with Constant Coefficients 9.2.1. Solving distributionderivative equations 9.2.2. Solving traditional differential equations 9.2.3. Single differential equations (Cauchy problems) 9.2.4. Systems of differential equations 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. IntegroDifferential Equations
xi
134 136 136 138 139 140 143 145 145 146 148 148 151 152 154 155 156 160 161 162 164 164 166
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
168 168 169 173 176
9.7. Partial Differential Equations 9.7.1. Diffusion of heat flow in rods
177 177
9.7.1.1.
Infinite conductor without radiation 9.7.1.2. The cooling of a rod of finite length
177 179
Table of Contents
xii
9.7.1.3. Rod heated at an extremity 9.7.2. Vibrating strings 9.7.3. The telegraph equation 9.7.3.1. The lines without leakage which 9.7.3.2.
are closed by a resistance The infinite line which is perfectly isolated
9.8. Convolution Formulae 9.9. Expansion in Series 9,g.l. Function B ( v , z ) 9 9.2 Function $ ( 2 ) 9 9.3. Fourier series 9.9.4. Asymptotic expansions 9.10.
Derivatives and AntiDerivatives of Complex Order Definition by the Laplace 9.10.1. transformation 9.10.2. Examples 9.10.3. Extension of the definition
CHAPTER 10 THE STIELTJES TRANSFORMATION 10.1.
The Spaces E (r) and JI' (r) 10.1.1. 10.1.2.
The space E (r) The space JI' (r)
10.2. The Stieltjes Transformation 10.3. Iteration of the Laplace Transformation 10.4. Characterization of Stieltjes Transforms 10.5. Examples of Stieltjes Transforms 10.5.1. Examples when Tt E JI' (r) 10.5.2. Examples when Tt E JI' 10.6, Inversion 10.7. Abelian Theorems 10.7.1. Behaviour of the transform near
the origin Behaviour of the transform at infinity The nDimensional Stieltjes Transformation 10.8.1. The space J,l(r) 10.8.2. The Stieltjes transformation in n variables 10.8.3. The iteration of the Laplace transformation 10.8.4. Inversion 10.7.2.
10.8.
180 182 187 187 189 19 0 193 193 194 194 196 198 198 200 203 207 201
207 209 209 210 211 213 213 215 216 219 219 220 221 221 222 222 224
Table of Contents 10.9. Applications 10.10. Bibliography
CHAPTER 11 THE MELLIN TRANSFORMATION
xiii 224 224 227
11.1. Mellin Transformation of Functions 11.2. The Spaces E a#w 11.3. The Spaces EA la 11.3.1. The multiplication in E' ato 11.3.2. The differentiation in E' a1w 11.3.3. Comparison with Zemanian spaces
228 230 232 234 234 235
The Mellin Transformation Examples of Mellin Transforms Characterization of Mellin Transformation Rules of Calculus Mellin and Laplace Transformations Mellin and Fourier Transformations 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. EulerCauchy Differential Equations 11.15. Potential Problems in Wedge Shaped Regions 11.16. Bibliography
236 237 238 241 242 244 245 249 250
11.4. 11.5. 11.6. 11.7. 11.8. 11.9. 11.10.
CHAPTER 12 HANKEL TRANSFORMATION AND BESSEL SERIES Hankel Transformation of Functions The Spaces H v and H$ Operations on Hv and H$ 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. Inversion 12.6.1. Remarks 12.7. The nDimensional Hankel Transformation 12.1. 12.2. 12.3. 12.4.
251 251 252 253 258 261 265 268 269 269 272 274 276 280 282 282 283 284 286 287
Table of Contents
xiv
12.8. 12.9. 12.10. 12.11. 12.12. 12.13. 12.14. 12.15. 12.16.
12.17. 12.18.
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 nvariables Variable Flow of Heat in Circular Cylinder Bessel Series for Generalized Functions 12.9.1. Statement The Space B mIv Representation of a Distribution by its Fourier Bessel Series Other Properties of the FourierBessel Series The Subspace Bm of Bm I V SesselDini Series 12.14.1. Statement The Space EkImIv Representation of a Distribution by its BesselDini Series 12.16.1. The subspace Bm of B Hlmlv 12.16.2. Another subspace of B Hlm,V An Application ot the BesselWDini Series 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 bibliography. 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. We let Rn and Cn denote, respectively, the real and complex n dimensional euclidean spaces. Any number X in Iff will be denoted by (xl,x2, xn) or occasionally by X. The letter 2 2 r will be used to signify the distance = + x22 + + xn* Often f(X) will denote f(x1,x2,...,xn) and I f(X)dX will mean Bn
...,
j//...j
IRn
f l A,
.. .
.... dx,.
f(x1,x2,...,x n)dxl dx2,
In this notation it is sometimes convenient to write r = 1x1. The P1 +P2+. .+P, partial derivative a will be abridged on occasions. axl p1 ax2 p2 axn pn
...
We recall that IR (IR = IR 1) is the line of real numbers and C(C = C1) the plane of complex numbers. By the symbols IN and INn we denote the set of nonnegative integers in one variable and n 1
Chapter 0
2
variables, respectively. The set theory notations used are as follows: A C B or B 3 A
then x
E

the set A is included in the set B; i.e. x
E
A
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 b. such that a < x Ca,bl  the close interval from a to b; i.e. the set of points x such that a 5 x 5 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. A x B
lRx\(a V
5 x 5 b)

the axis IRx without the interval [a,b]
 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 that any finite linear combination of elements of E is an element of E, i.e. provided that if cl, c2, cn E C and fl, f2,...,f n E E for any finite n then 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 (i) If (ii) of
=
f, Y f
= og, Y
E
3
E,
f,g
E.
E
2 . One can exchange and group arbitrarily the terms of a linear
combinat'ion; if (v1,v2,...,vn) (1,. ,n) and if 1 < nl
1, v # 1, and s(x) is integrable on Cc,C]. C Choosing any r~ such that c < c + n < C, set J ( n ) = I y(x)dx; then c+n term by term integration yields the result,
C+n
s(x)dx.
7
Chapter 1
8
The function J ( n ) , as r( v+l  b log of the terms *.
+
0, approaches no finite limit because
r(,
but the remaining terms on the right
side of (1.l.l)possess a limit which is called the finite part of the integral ICy(x)dx as n + 0. For brevity, we use the notation c c+n C ~p ] y(x)dx to represent this f i n i t e part. elation (1.1.1) shows that FPJ y(x)dx C
takes the f m , (1.1.2)
C Fp y(x)dx =
I
C

a
( C  c ) ‘+’+b
log(Cc) +
I
C
C
s(x)dx
C
c;
lim{ ] C a(xc)‘+ b(xc)l+s(x) 1 dx n+O c+n v+l  %’+ b log 111.
=
If the integrand is (xc)’logj(xc) (’) where j E IN, Re v 2 1, and v # 1, then we obtain after integration by parts j times C (1.1.3)
Fp ] (xc)”logj(xc)dx = C
where the sum is zero if j=O.

(‘v1 ‘)
1 j!
logj1 (cc) i=O (ji):(vl) 1
v+l
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 (xc)vlogl (xc)
.
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) =
L
K1 .? C ak+ajklogj(xc) 1 (xc)’k
k=lj=l
Here all Re A k > 0 but the Ak are not integers; also some of the numbers ak, a,k, bk, pjk 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
9
Finite Parts
I t i s e v i d e n t t h a t i f g ( x ) = s ( x ) i s an i n t e g r a b l e f u n c t i o n on Cc ,C 1, t h e n C
(1.1.8)
Fp
C
I s(x)dx =
s(x)dx. C
C
T h i s l a s t r e s u l t shows t h a t t h e o p e r a t i o n F p l i s a p r o p e r g e n e r a l i z a t i o n of t h e i n t e g r a l . P u t t i n g s ( x ) = 0 , a = 1, b = c = 0 , and u = 21, R e z
0, i n
1), cos x, ex , ex2 , 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. It can be easily shown that this functional can not have the form of (3.3.2) and locally summable function f (x)
.
Indeed, if some locally summable function f(x) satisfies (3.3.3)
I
f(x) $(XI
IR for every $(x) in IR
dx = $ ( O )
, then it
9)
satisfies
f(x) e(x) dx = 0
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. Hence we emphasize that the left side of (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. If c is a real constant then 6(xc) is the functional which assigns to a function Q its value Q(c). This is a distribution of order zero: (3.3.5)
<S(xc),$(x)> =
$(C),
Y
$
E
IDo,
(IDo denotes the space of continuous functions with bounded supporC2)). 6(xc) also belogs to I D g k , ID' and $ I .
If k is a positive integer, then 6 ( k ) (xc) is the functional which assigns to a function Q the number (  l ) k Q ( k ) (c) This is a distribution of order k because
.
(3.3.6) Also,
6(k)
(xc) 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 fg(x) $(x)dx,
V cp e ID
m
where the finite part of this integral is a di~tribution(~)and is denoted by Fp g(x)
.
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 +w. If f(x) is a locally s u a b l e 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
IX
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
2
eX and ex ; however U ( X ) ~ E~ $ I . 3.3.5.Tempered pseudo functions If g(x) is a function such that x"f(x), integer n, is bounded as 1x1 + m , then
for some positive
W
= 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 welldefined finite part for all 0. Any such Fpg(x) is called a tempered pseudo function. The examples of Section 3.3.3 are also tempered pseudo functiow except Fp I x 1 veX.But Fp U ( x ) xVeX is a tempered pseudo function. If g(x) has an infinite number of singularities, then the definition of Fp g(x) involves further complications. In this case we assume that g(x) satisfies also the following two conditions: 1. g(x)is continuous on I  m , m [ except at a countably infinite number of points cn, where m < n m and c ~ + cn. ~ > In a neighbourK hood of each cn, g(x) = 1 ank(xcn)k+anologlxcn I + hn (x) where the k=l function hn(x) I s continuous in this neighbourhood, K is a fixed
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 (xcnl < a , then there exists a positive number a which yields non intersecting intervals Inoutside 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. Hence, by formula (1.2.6) of Chapter 1, for all $ in $, we obtain 5
m
=
FP
I
g(x)$(x)dx
lim Fp I g(x)$Ix)dx 5 Frn 5'
=
m
['+m
where the limits 5 and 5' of the integral take values in the intervals [ c ~ , c ~ +but ~ ]outside the intervals In. An
example of this case is Fp sin x
3 . 3 . 6 . Analytic functionals
.
(ultradistributions)
Let z be a complex number such that z then evidently, +(a); a
(3.3.8)
= (l)k$(k)(a).
E
x+iy and if $(z)
E
23,
c
(3.3.7)
=
=
where k is a positive integer. If f(z) is a analytic function and L a path in the complex plane, then the mapping: Z + 1 f(z)+(z)dz L belongs to 23' and is said to be a regular analytic functional (or regular ultradistribution). 4
E
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
Chapter 3
32
Hence, we obtain the identity 6(za)
=
.
1 Zni(2a),
' a Using these last results, we can rewrite Fp(xc)l (see Section 3 . 3 . 3 ) in the following form. Let r+ be equivalent to the paths,  m to C', L+and C to L+is the s&ne path as described in Section 1.5 of Ciiapter 1.

, where If
hence, by Taylor's expansion, $ ( z ) = @(c)+(zc)Y(z) in some neighbourhood of c where Y(z) is an entire function. Also, note thatl xc is analytic along the paths  to C' and C to $(z)
E
22,
.
By making use of the above hypothesis, we have
xc
C I m m x + Jy(x)dx + $ ( c ) Fp , f l ldx xc C' The function @ ( x ) is holomorphic in the neighbourhood of c 1 and C. Thus, by Cauchy's theorem, we have F~
Jm
m
? A x = C' I wxc x xc m
+

Hence, we get C'
Finally, we obtain
+ =
J qz c
9 0,x xc
+
QIO,~ inlp(c)
L+xc
2
r+where r+ = x & find

(3.3.9)
i E ,
E
> 0
is drawn towards x > 0.
'(zc)' I r+ 2 ins (xc).
~p(xc)
=
Also, according to ( 3 . 3 . 9 ) we have
Consequently, we
Definition
33
Adding above two formulae, we get
The formula (3.3.9) can also be written further as ~p(xc)’ = (xc t i c )
( 3.3.10)
of symbolically (since
E
~p(xc)’
’5
ins (xc)
is arbitrary) =
(xc 2 io)’
f
.
ire (xc)
This is a formula of LippmannSchwinger (or of Sokhotsky). Problem 3.3.1 (1) where ra is the path 2ni k! (2a) 1 r a described in section 3.3.6 and k is a positive integer.
(i) Prove that 6(k) (2a) =
(ii) Find the expansion of F p ( ~  c ) where ~ n is a positive integer. Footnotes
(1) see Section 4.1 of Chapter 4 . (2) abusively, some take 6(xc) as a true function and give m ( 3 . 3 . 5 ) under the integral form I G(xc)@(x)dx =@(c), 6(xc) is a measure of mass 1 at; = c and 0 elsewhere. ( 3 ) of an order which depends upon the singularities of g(x)
.
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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 = 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 = 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

Chapter 4
36
Examples. We give below a few examples of the support of a distribution. The support of the distribution f(x) defined by
i
llxj, 1 < x < 1
f(x) =
0
I
1x1 5 1:
is obviously C1,11. 6(xc) has its support at the point c. support.
Fp x" has
SR
as its
We shall say that two distributions F and G E ID' (or $ I ) are equal on a closed interval[c,c'] of IR if = 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 common then = 0.
$
have no points in
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(xc) This type of support is called a and 6 (k)(xc)I k = 1,2,3,. point support.
.. .
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
x+,
where
n x U(x1) I Fp x" U(x+l)
Properties
37
10 , elsewhere,
U(X+l) =
(1, {
1 0,
x > 1
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 I D : , These also belong to 8' This property is important due to the simple structure of the base space 6 : reciprocally, each element of 6' is a distribution with bounded support. $' or 27,'
.
.
We list below some important properties: 1. The value of 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, on an open bounded set U,
to a distribution having its support in a bounded neighbourhood of the set U. 4.2.Properties 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:
Chapter 4
38
1. Any distribution F in ID'and any finite closed interval I determine a nonnegative integer k and a positive constant M with that I < F , $ > II M
where
$
sup^+(^) (XII X
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/I M SUP/(l+X2 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 then F V converges(2), as v + vo, if the numerical function < F v , $ > converges when $ is any element of 0.
Fv in Q',
We deduce from this definition the following facts. 1. Let F1,F 2 , . . . I in Q', be an infinite sequence of generalized functions. Then this sequence is convergent as n + m if the numerical sequence is convergent for every $ in Q. In particular, the sequence IS (xnl)1 is convergent as n + a, m
2 . The series
F. is convergent (on j=j 7
0),
if for every
$I
E
0
the numerical series
1 is convergent. j=jo m
Examples. The series
1 6 (xj) is
j=O
E < 6(xj), $ > = E +(j) converges.
convergent on ID , because
This sum has only a finite
0
Properties
39
number of nonzero terms because 4 has bounded support. m
.
ch'6 (xj) is convergent on $ (and on ID) , j=1 because C < h16 (xj),$ > = Chj$ (j) converges, 4 being bounded. (This series of distributions does not converge on & .) Moreover, If h > 1, then
m
1 6") (za)/j! is convergent on P, j=O 4.3.1.Completeness and limit
the series
.
With the proceding notion of convergence, the spaces ID' , $',and ZZ' are complete : if a family Fv, respectively belonging to ID' , $ ' or P,' , is convergent as v t v 0' then this family has a limit F belonging to 3D' , $ ' or 22' , respectively, and satisfying lim =
.
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 that Fn=n(Gx+l/nGx) 1 Then Fn E IDIkand lim Fn = lim 1 and take h = n h (Gx+hGx)= nh+O Thus DG does not always belong to m l k , in particular, DG E ID I k + l G = 6 ( k ) (x) The space C' becomes complete when we slightly modify the above definition of convergence as follows. An infinite family Fv of distributions having bounded support is convergent and has its support in e' if the numerical family is convergent for every $ E 6 and if all the F v have their supports in the same bounded interval.
.
.
.
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, finite or infinite, and let each value v determine a function fv(x). Then the regular distribution fv(x) , as v + v has the limit 6 (x) 0' if: C
(1)
J fv(x)dX
+
1 for every c > 0;
C
fv(x) + o uniformly on every set, O < E 5 1x1 5 1 b (3) I (fv(x)Idx, for some positive number b, has a bounded b independent of v .
c;
(2)
Example. We construct an example of sequences which converge to If fv(x) = v 1 exp(rx2/v 2) , then 0 2 f (x) < vl, which b m proves (21, and Ifv(x) Idx 5 fv(x)dx = 1, which proves ( 3 ) . Also b m m C m c m 1 2 I fv(x)dx =/'fv(x)dx [f + If,(x)dx = 1(2/v)Jexp(rx 2/ U 2)dx
&(XI.
I
C

m m
2
c
2
> 1(2/v) lexp(r(c +2ct)/v )dt 0
C
=
2 2 l(l/nc)exp(rc /v )
and the last expression has limit 1, which proves (1). Problem 4.3.1 Let
Properties
where
1 w = 21
exp
+
41
dx
1x 1 and show that = a + b X V 9,s EID (IRn), a,b E Cn, and TX is sequentially continuous if
+
0 as n +

to zero as n + for each infinite sequence i$nl converges in the sense of ID( IR") As termed in Section 3.2 of Chapter 3 , the elements of ID' (IRn) are called the distributionsin n independent variables.
.
0)
Support of a distribution in ID' (IRn) A distribution TX in ID' D")is said to be zero in an open set of IRn if = 0 for any function $(X) of ID (IRn) which has its Support in fl. The support of T is the complement of the biggest X
R
Properties
43
open set il on which T is zero. X Now we state below a few examples of distributions. Example 1. Let f(X) be a locally summable function which is identified with a distribution TX. Then we define
Here TX is the regular distribution corresponding to f(X) as in the case of one variable (see Section 3.3.1 of Chapter 3). The integral indeed exists, for the domain of integration which is not JRnbut the bounded support of Q on this support f is sununable and Q continuous so that fQ is summable. A l s o , the value of the integral is obviously a linear functional of 4 . Example 2. Let Qn(c) be the domain whose each point X = (X~,X~,...,X ) is such that x.> c, i 5 i 5 n. In particular, 11 " 3 1 (i) G3(c) is of IR ; (ii) Q2(c) is 7 of the plane; (iii) Q1(c) is the halfaxis x 2 c; and (iv) Qn(O,m) is the first orthant of IRn (see Section 12.7 of Chapter 12). By ID; (IRn) we mean the space of distributions in IRn having lower bounded support. This means that TX belongs to ID: (IR") if there exists a real number c such that the support of TX is contained in Qn(c). If TX = f(X), a summable function in Qn(c), then we have (4.5.1)
=
f
Qn(c) m
=
f(X)O(X)dX
... j
1 C
m
f (XI$(XI ax1,.
C
..,axn.
2 2 45 a, For instance, if n=2, we can take f(X) = 1 in the disk (x1+x2) 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 =
J
f(x)$(x)dx
Kn
where Kn denotes the support of f(X) (a surface if n = 2, a volume if n = 3, a hypervolume if n > 3). Obviously, the domain of this integration is the intersection of Kn with the support of Q. If c = 0 in above space, then we denote the space IDb+(lRn )
44
Chapter 4
instead of ID; (IR") takes the form
.
In this case if TX a
(4.5.1')
1 and R e v > 0 on t h e l e f t and r i g h t sides, respectively. L a s t l y , l e t g ( x ) be a f u n c t i o n which i s zero f o r x < c, c o n t i n u o u s w i t h a d e r i v a t i v e g ' ( x ) f o r x > c, and also a d m i t s a r e p r e s e n t a t i o n of t h e t y p e (1.1.5) o f c h a p t e r 1. Then t h e d e r i v a t i v e of g ( x ) c a n be o b t a i n e d by a p p l y i n g (5.4.11)
o r (5.4.9)
of g ( x ) and u s i n g t h e f a c t Dh(x) = h ' ( x ) + h ( c + ) x 6 (xc)
.
t o each term Accordingly,
w e obtain
K
(5.4.12)
DFpg(x) = Fpg' ( x ) + h ( c + )6 (xc)
k
+ 1
(l) 6 ( k ) (xc). k = l k l bk
Problem 5.4.2 Prove t h a t t h e f o l l o w i n g f o r m u l a e are v a l i d on t h e s p a c e ID: ( i ) D ( 1 o g l x l ) = Fpx (ii)DFpx"
1
= nFpx nl
;
 L ,(n) (x) ; n!
where n i s a p o s i t i v e i n t e g e r :
.
(iii)D ( l o g ( x  c ) x > c ) = F ~ ( x 1 c ) ~I > ~
( i v ) ~ ~ p ( ~ ( x( x  c c) ~  ~=)  A F u(xc) ~ (xc) ( v ) DFp ( U (xc) (xc) k) = kFpU ( x  c ) (xc)
A1
,x z
kl+
01112r3,.
..;
(1)k & ( k )(xc) k!
k = 0,1,2,3,....
I
Operations 5.4.7.Derivatives
59
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 Df ( z ) ab = f (z)+f (a)6 (2a)f (b)6 (zb)
(5.4.13)
where the path ab must be on a single sheet of the Riemann surface. 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 ) (25,) for the poles 5, which are inside of r . ) If ab or r goes through singular points, finite parts of integrals or pseudo functions a l s o occur. (See Lavoine C5l and C6l.)
Problem 5.4.3. (Taylor's series) If F
E
23
,
show that m n F = 1 D"F. a n=O
2
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
$,
Chapter 5
60
which yields the result (5.5.1)
D(aF) = aDF
+
a'F.
Differentiating again, we obtain (5.5.2)
D2 (aF) = aD2 F
+
2a'DF + a"F.
More generally, successive differentiation according to the Leibnitz rule yields Dh (aF) =

a(hj) D jF. hl 1 j=O J !('hj)! Examples. 1. According to (5.5.1) , we have V 0
(5.5.3)
E
= a (0)0 I (0)a' (0)Q (0) = 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)
=
,C I>
, 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. If the distributions belong to SJ., then their convolution .: also belong to $ $; 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
cI
aj 6(x+cj)
I* x
= B
(see Section 7.13 of Chapter 7). The integral equation m
f(x)
+
IK(xt)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
Chapter 5
72
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 solution of the equation (5.8.13), because
=
E
*
B is a
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 TX t to the generalized function Tu(x) E 0' defined by
In the following sections, we extend this definition to the case when u(x) is not monotonic on IR. 5.9.1. Definition of T u (XI Let TX be a generalized function belonging to a space
0'
and let
73
Operations
S be the bounded or unbounded support of Tx. A l s o , let u(x) be a singlevalued 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:
(1) There exists N(NF1) closed intervals Xn which further satisfy the two conditions: C1. u(Xn) contains S c2.
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 vn ( x ) such that u(x') = u(vn(x)) = x when vn(x)=un1 (x), ul(x) being the inverse of u for the monotonic branch described by n u(x) when x belongs to Xn. Outside of u(Xn), vn(x) is extended by a function belonging to 0. Let a S ( x ) E 0 be equal to 1 on S and zero outside of an interval containing S. We put N
(5.9.1)
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 which is denoted by T u(x) ( 5.9..
21
The support of Tu(x) is (5.9.3)
su = u
1 Note that u l vn
vn(s) = uix
E
XnlU(X)
E
SI.
may be substituted for v;(x)
in (5.9.2).
If there is a countably infinite number of Xn (if u(x) has a 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
Chapter 5
74 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 differentiable on IR and its derivative u' (x) should not vanish on IR
.
Let Dx(TU(x)) denote the derivative of the distribution Tu (XI 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 ,
Also,
(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) = ul(x) yields ff + E ID =
1 f ( x ) Iv' (XI
JR
16 (v(x) )ax
1 f(u(x))$(x)dx = . IR Here we obtain Tu(x) = f(u(x)) as expected. =
2. Let Tx = f(x), a summable function with support S = [c,c'], and let u(x) = x2+b where b < c. Hence the conditions C1 and C2 are satisfied by two Xn:X1 = CH,n] and X  [ n,H] where n and H are such that 0 < n < y = Jcb, Hz y ' = Then we have v 1 ( x ) =  6 , v,(x) = F b , and (5.9.2) yields
&.;
Operations
n
=
< f l (x'+b)
75
,$ ( x ) >
where f(x2+b)
i f G b 5 1x1 5 & b
2
f l ( x +b)= otherwise. Hence w e conclude
which can be o b t a i n e d immediately. On t h e otherhand, i f b > c does n o t e x i s t because t h e r e does n o t e x i s t any X then T u ( x ) n s a t i s f y i n g t h e c o n d i t i o n s C 1 and C 2 . I t i s t o be remarked h e r e t h a t t h e examples 1 and 2 d e m o n s t r a t e t h a t t h e formula (5.9.2) g e n e r a l i z e s t h e change of v a r i a b l e i n t h e t h e o r y of f u n c t i o n s .
5.9.3. Bibliography
A more comprehensive d i s c u s s i o n 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 A l b e r t o n i and Cugiani Ell, F i s h e r c11, Gfittinger C11, J o n e s Ell, Schwartz Ell, Chapter I X . Footnotes (1) d/d = d l d x o r d/dz, a c c o r d i n g t o t h e v a r i a b l e .
'c
( 3 ) U(xC) =
1 , x > c o , x < c
(4) see S e c t i o n 0 . 4 . 2 of Chapter 0. ( 5 ) more g e n e r a l l y , 6(xc) = D
1
[(xc)U(xc)
+
ax+b1.
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CHAPTER 6
OTHER OPERATIONS ON DISTRIBUTIONS
Summary The motivation of this chapter is to define the division and antidifferentiation of generalized functions. We do this not by the 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. 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 satis1 is infinitely differentiable, and fies HT = S. This is because H 1 1 Such is no S by multiplying on both sides by B. we obtain T = 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) For a given generalized function G E Q', let us first consider the problem of division by xn. To do so, we seek to determine G E 0 ' 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 l/xn may not be a multiplier for Q. Therefore, we define division as the inverse of multiplication by (6.1.1)
G =
1F , i f x n G=F n X
77
Chapter 6 that is, if (6.1.2)
= .
In addition to this there can exist a divisor of zero in 0'. instance in ID' (see (ii) of Problem 5.5.1 of Chapter 5), xn6 (k)(x) = 0, k = 0,1,2,..
(6.1.3)
For
.,(n1).
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
n1
any two of them is of the form 1 a b(k) (x) where ak are arbitrary k=0 k constants.
Proof. Giittinger
The proof is given in Schwartz [11 , Chapter V,4, [l] and ChoquetBruhat c11 , pp. 124129.
Examples. If x3
1 x3 (1) 7 X
=
E
ID'
, we
have
x + ao6(x) + a161 (x)
.
Here xn = x2, F = x3 , and hence according to Theorem 6.1.1 we have x2G = x2 (x+ao6(x)+a16'(x)) = x3 = F which shows the existence of (1) by making use of (6.1.3).
.
1F = Fp f (x) + a 6 (x) (2) If F = f (x) is locally summable on lR , then X X 0 (Here the Fp is not needed if fo is integrable in the neighbourhood X of the origin.)
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 Q =
I 7 F, x)
if a(x)Q = F.
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 P solutions Q belonging to ID' ; the difference between any two of them np1 is of the form G c a 6(k) (xxp), where a are arbitrary p k0 PIk Plk constants. It is obvious that Q is unique if a(x) does not vanish on IR. The following examples will illustrate this theorem. Examples. Let lX be the distribution which is identical (or associated) to the function equal to 1, and let c E IR. Then, we have in ID'
.
~p
++
a
6
(xc)+a1 6 (x+c)
C
where a and a' are arbitrary constants; and
1
2*
1 1 ' x2,c2 lx  x +c
But in the space ZZ' of ultradistributions (see Section 3.3.6 Chapter 3) , we have 3.
11 +c
lr
=
=
of
1 + a G(zic) + a'd(z+ic) 2 2+c
where the path II does not pass through the points
ic.
Proofs. Here a(x) = x2c2 has two simple roots c and c. 1 Then according to Theorem 6.1.2, Q = x' = Q = Q
1
Let
+ a6(xc) + a'd(x+c) 2
2
where Q1 is a distribution such that (x c )Q, = lx; hence we can 1 Consequently, l.is established. take Q1 = Fp r2. 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 ) = lx' x +c 3. This is left as an exercise for the reader.
'n
6.1.3.
1
multiplier €or 0
In this case Q(x)/a(x)
E
Q
,V +
Qa = FA is unique and defined by, (XI
E
Q.
Then for F in Q',
Chapter 6
80
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. Consequently, 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, 1
(6.1.4)
c; its limit from the right at c is h(c+) = l i m h(y) if this limit exists. Similarly, y + c+ we have according to Eojasiewicz.
Chapter 6
84
If a distribution F has the value F = a(y) for y i c and if Y 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+. We denote a+ by lirn F. c+ This can be justified in the following manner: According to this definition, if lim F y C+ Hence, by Section 5 . 2 of Chapter 5, we have
= Tf
a+, then lim F y+c+Y 9 E ID,
=
a+.
+
lim F i +(x)dx = lim lim 6+c+yIR y+c+ x + o + = lim lim ),(+ xy y+c+x+o+ Left hand limit
(6.3.3)
Eefinition for the left hand limit lim F will be analogous to Cthat of the right hand limit. Examples.
We state below the following examples of limits:
l i m eyx = 1, lim eyX = 0. But Fp xl has no right or left hand limit O+
0
at the origin.
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 k ]c,r] such that F = D h(x) on ]c,r[ and that k : h(x)/(xc)k + a+ (in the ordinary sense)as x + c+.

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 distribution. We shall follow the definition of Silva [4] (See a l s o 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 nonnegative m
integer k, and a continuous function h(x), such that F=Dkh(x) on [ r , [ and that k: h(x) /xk + a (in the ordinary sense)

.
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 x lim sin x = 0, because sin x = D cos x and cos + 0 as x +
a.
X
m
Thus lim eiwx = 0.
If F has a bounded support, then lim F=limF
m
m
= 0.
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 For each F E ID' , we have the decomposition of F such that L F = G+G where G E Dloand is the reflection of a G1 also 1 belonging to By EU $+ we mean the space of infinitely differentiable functions which are equal to functions of on [O,[. Further, we put CO,C.
k1
(xVlogJx)+ = and ' (xn1log'x)+=
where j,n
E
IN.
I
xvlogJx, for x
>
o
,
0 and a distribution Q E E' with support in c 0 , 5 1 , such that
2. G write
E
.
G = aFp(x n1logjx)+
+
G
on C 0 , ~ l
and that .n as X + O+.
We remark that if v = j 1 7
= 0, +
(6.4.1) takes the form
0 as A
consequently, according to (6.3.2)
, we
+
O+;
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+ 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.
(V+lIk.
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(xnllogjx)+) (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
n+k+2
h(x) on C0,sl
and (l)"n! ( k + l ) ! (j+l) k+h (x) + a x llogJ+lx (in the ordinary sense) as x + O+. The following theorem is a consequence of l.,
defined above.
If f(x) is a locally summable function and v is Theorem 6.4.3. such that Re v > 1 and if we have f(x) a xvlog7x as x + O+ in the ordinary sense, then we have f (x)+ a(xvloglx)+ as x + O+ in the sense of distributions. I
The following result is less apparent.
x
c
Theorem 6.4.4. Let a be such that Re a L 1. Suppose that for C O , 5 1we have
with the conditions that certain constants a must be zero, Pq 0 2 Re 6
1, as x + O + ; because then I p ( O ) / X v+l
0+,
6.4.2.Equivalence at infinity Let m(x) be a function such that 1 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
F = Dkh(x) on I < ,  [ and that
h(x)
xkmo+ O
in the ordinary sense x
+ a.
Note. This is like the criterion for F Silva . C41
=
o(xa) ,
a >
1 given in
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 = Dkh(x) on 1 5 ,  C and such that ( v + l ) kh(x)/xv+k + a (in the ordinary sense) as x + m+.
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 ID1 converges to zero if C I D ' l , we have for $n + 0 almost everywhere. Since ID C ID the duals (ID') 'c ( I D ' I k C ID'.
we do not say > just to avoid confusion between this distribution and the number a. see Lojasiewicz c'll , p.7 and Constantinesco C 1 1
, pp.
712.
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 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 co
(7.0.l)
g(x) = IFx f (t) = ff,(t)e2nixtdt m
where IF denotes the Fourier transformation. IFx (f(t)) is called the Fourier transform of f. We also define m
(7.0.2)
P;l(g(x))
=
I g(x) e2nixtdt
m
where the notation Pi1denotes the inverse Fourier transformation. F ; ' (g(x)) is called the inverse Fourier transform of g(x)
.
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 ) transformation by the relation
E
Z. Then we define the Fourier
(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 $(x) belongs to ID we recall that for every 
z
If yJR) denotes the semicircle I z 1 = R, n 2 0 then we have making use of (7.1.2) for x a 03
I$(X)
1
=
I
’
1
m
$(Z)
e2nixz dz I
m
5
OR2 I$(x) I le2rixzdzI 5 lim J ~ + mYAW ea I n I .xRsin e Rd 0 J
03
< lim It).
Rll’e(Xa)R
sin BdO
=
0
We can also obtain the same result for x 5 a by the semicircle 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
IF.2
k
$ ( z ) = (2nix) I F ~ $ ( Z )
,
(7.1.4) Also,
$(z) is said to be the antitransform (or inverse transform
Fourier Transform of $(x) = IFx $(z) , and it is denoted by $(z) (7.0.2) we may infer that (7.1.5)
93
= IFl$(x). 2
From
m
Ez 1 $(XI
= J $ W e2nizxdx. m
7.2,Fourier Transformation on ID similar technique to that of Section 7.1 is followed in this section to define the Fourier transformation on ID. A
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 m
(7.2.1)
$ ( z ) = IFz$(x) = J
$(XI
e2nizx dx
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)
.
b
2nizx e ax. b It is evident that +(z) is infinitely differentiable and therefore analytic. A l s o , b (2ri)JzJ +(z) = I +(J)(x) e’*iZX ax. (7.2.2) $(z) =
I $(XI
b
Further, by making use of (7.2.2), we have for n = Imz b b (x)(e2‘nxdx
I
I
1 I
I
'I
I I
I
1
I
I
I
IF T is an entire m y t i c fun^ z x tion such that there exists a nonnegative integer m so that m e2ncl~l I=ZTxI IZI is bounded as (zI + m , where rl=Irnz.
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,... to (7.6.4) we have
IF^ 6 (k)(xc) =
< 6 (xc) (D) e
98
Chapter 7
, we
2. By v i r t u e of (7.7.1)
have IFz 6 (x) = 1.
F u r t h e r , making
k
w e o b t a i n IF;' 6 ( x ) = (IFz 6 ( x ) ) = = 1. F u r t h e r , by Consequently, 1 = I F ; ' 6 (x) and f i n a l l y IFx 1 = 6 ( x ) means of t h e r u l e (7.5.2), w e g e t
u s e of t h e Theorem 7.4.2,
(7.7.2)
.
3
IFxz
6 ( k ) ( ~ ) , k = 0,1,2,...
ik
.
w e have
3. According t o (7.3.2)
which y i e l d s t h e r e s u l t (7.7.3)
IFx 6 (2ic) = e 2ncx
i n ID'. More g e n e r a l l y , i f 5 i s complex, t h e n w e have IFx6(z&)
=
e
4. According t o (7.7,.3) making u s e of Theorem 7.4.1,
2ni~x
, we
= e2ncx, which y i e l d s e 2ncx =
IF eZncx= 6 ( z + i c ) 2 then we get (7.7.4)
.
.
h a v e IFx 6 ( z + i c ) = e2ncx Further, = (IFx G ( z + i c ) ) = ( e2ncx)
3lG(z+iC)
IF;16 ( z + i c ) .
Consequently, w e o b t a i n
I f we r e p l a c e c b y c / 2 n i n t h e l a s t r e s u l t ,
E~ ecx =6 ( z + i c / 2 n )
i n 8'
. .
y i e l d s t h e r e s u l t IFz lX= 6 ( 2 ) F u r t h e r m o r e , by means of t h e a n a l o g o u s r u l e ( 7 . 5 . 2 ) , w e f i n a l l y obtain If w e t a k e c = 0 , t h e n (7.7.4)
which c o r r e s p c n d s t o t h e f o r m u l a (7.7.2)
.
5. L e 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 g o i n g 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 i v e n 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

m
J
99
e2ni5x
cp (x)dx
m
which yields the result 2niSx
1
(7.7.6)
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)
m
1 e2nix5cp(S)dC,
=
real 5
0
and its conjugate by, (7.8.2) The following results can be easily obtained: 1. 2.
IFx$
and
functions of x which belong to $.
If a sequence {$,I
Pxcpn
+
3.
r x c pare
0 in the sense of $, then IFxcpn
+
0 and
0 in the sense of $.
Fx+ isthe
antitransform of 9; that is, if
IF $ = $ o r I F X
+

.
IFx cp = 4 (x)
Therefore, IF;l
wxcp =
$(x), we have
=Tx.
From these results we may conclude that IF and F a r e reciprocal 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)
,
V cp e $.
Similarly, we define its conjugate F T by
100
Chapter 7

(7.8.4)
< T X T , $ ( X ) > =
4
E
$.
t h e s u b s t i t u t i o n of F T f o r Tx l e a d s t o
I n (7.8.31,

,Y
<Ex( T x T ) , @ ( x ) > = 5 =.
Hence IFx F X T X = Tx and (7.8.5)
IF1
E
Also, (7.8.6)
F 1s
It follows t h a t
T = 0 +=>IF
T = 0,
T = 0 =>'IF
T = 0.

I t is e a s y t o show t h a t i f T
t h e n w e have t h e f o l l o w i n g :
XIV
Tx,v * Tx I n t h e sense of $ ' a s v * v0
i s an i n f i n i t e f a m i l y b e l o n g i n g t o $ '
IFxTx,v
+
DxTx
i n t h e s e n s e of $'
a s v + v
Hence, w e c o n c l u d e t h a t 'IF and automorphisms on $ I .
0 '
= 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 r e s u l t s of S e c t i o n 7.6 and t h e p r e c e d i n g 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', (7.9.1)
E x T =
= c ' , we have e zx TX E $'.
+
Proof. For every c $, <e'IXTx,+ (x)> exists and is equal to TxrOl(x)> with $1 E $+ and equal to + on the support of T.
+,(x)
As e(zs ' )x
belongs to $+, we have
<eS'X
(zc')x
Txfe
+,(XI > = <eZX T ,,
= < e ZX
$,(XI>
Tx, $(x)>.
Definition 8.1.1. A distribution T E ID; is called Laplace transformable for Re z > 5 ' if e's'x Tx E $: Then by virtue of Lemma 8.1.1, ezx 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), e zx Tx c $'.

As T V 5'
E
IR
c
is Laplace transformable for every z if e'lXTx C(T) becomes
ID:
, then
.
c
$If
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' does not belong to $'.
E
IRand e'IXT
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 1, x
109
> 'C,
U(x+c) = 0, x
where IL denotes the Laplace transformation. A l s o , (8.2.1) is equal to (8.2.2)
<exI'
TX'
e(z5')x>
, Re
z > 5'
7
c(T).
The existence of (8.2.2) can be justified because eS'XTxE $' n ID; and because of the fact that we can also find functions belonging to $ which equal to e(zs')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  ( Z   S " ) X > = <e ( 5 "  5 ' )x .5'XTxIe ( S "  S ' ) x e(z5' )X> <e Txte
Since e ( 5 11 5 I ) X is a multiplier
in $+ we finally obtain
<e C " X TXi e ( z  s " ) x > = <es'x TXi e ( 2  s '1 X > * Moreover, This independence on 5 ' justifies the notation (8.2.1).  zx> is well defined immediately without mostly in practice, 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) ) ;
. I Ik
2 . there exists a nonnegative integer k such that l $ ( z ) 1 z ec Re z is bounded as z + in the half plane Re z > C(T)

(respectively in the whole plane).
Proof. 1. ezx is an analytic function with respect to
z whose
Laplace Transform d e r i v a t i v e i s xe 2x
, it
111
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 e x i s t e n c e 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 g e n e r a l l y ,
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 c o i n c i d e s on [ c,*[ with a f u n c t i o n
belonging t o $. According t o property 2 of Section 4 . 2 . 1 of Chapter 4 , t h e r e e x i s t two nonnegative i n t e g e r s j , k and a number M such t h a t (8.3.2)
lG(z)
SIX
1 =l<e
< M
 (z5')x >I
Txt e
sup(l+x')j/21 (z5')'
XLC
e(z")X
When x > c and R e z i s s u f f i c i e n t l y l a r g e (i.e. R e z >
I c),
w e have
Further, according t o a property of t h e exponential f u n c t i o n , t h e r e 0 such t h a t exists P
( 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
c R e z
( l + x2 1 1 / 2 e  ( z  s 1 ) x < by t a k i n g P ec5' = M1.
Next, w e choose
A >
I c l ; then f o r c 2 x 2
A , w e have
Chapter 8
112
s u p ( 1+x2)j / 2 I e (''
I
'I
< Mlec
Re '+M2
(1+A 2 ) J/aec Re z .
X C'
Consequently, w e may i n f e r t h a t (8.3.2)
c a n b e dominated f o r
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
I
ec Re
Moreover, t h e r e e x i s t s a number M3 s u c h t h a t ;1 ( z ) 1 < M3 z I t h e r e f o r e , l & ( z ) I J z I  ~ec Re Hence t h e 2 of Theorem 8.3.1 < M3. i s proved. t
A
Theorem 8.3.2. If TX E E , t h e n T ( z ) i s an e n t i r e a n a l y t i c f u n c t i o n , and if T h a s i t s s u p p o r t c o n t a i n e d i n t h e bounded i n t e r v a l X
t h e n t h e r e e x i s t s a nonnegative i n t e g e r k such t h a t
Cb,bl,
I
I;(Z)
I Z I  ~
eblRe
'[is bounded a s IzI
.+ a.


P r o o f . S i n c e Tx h a s a bounded s u p p o r t G ( z ) = = e()
where 0 ( z ) = IFz Tx , t h e F o u r i e r t r a n s f o r m of Tx. i s a consequence of t h e S e c t i o n 7 . 6 of C h a p t e r 7 .
Z
2r1 The above theorem

Remark. O f t e n 2 of t h e Theorem 8.3.1 g i v e s a more p r e c i s e
e s t i m a t i o n f o r I T ( z ) ( t h a n 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 f u n c t i o n such t h a t c i s t h e lower bound of t h e s u p p o r t of s ( x ) and f o r which eSxs(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 h a l f  p l a n e R e z > 2. i ( z ) zkecz+ 0 a s R e z .+
.
c,

Proof. 1. The proof of 1 f o l l o w s by i t e r a t i o n o f 1 i n Theorem
8.3.1.
2.
Since G(z) = (zJk
I
m
s ( x ) eZXdx, R e z > 5 ,
C
by p u t t i n g R e z = 2 1 5 )
C
+
w
2
,
w > 0,
L a p l a c e Transform The l a s t two terms t e n d t o z e r o as R e z
113
b e c a u s e a s w +
+ m,
l/w + 0. T h i s theorem w i l l be u s e f u l i n S e c t i o n 8.11.

then
I n addition, i f
k = 0 , one can o b t a i n h e r e a w e l l known r e s u l t o f t h e o r d i n a r y Laplace transformation. 8.4.Relation with t h e F o u r i e r Transformation I n t h i s s e c t i o n an i m p o r t a n t r e s u l t c o n c e r n i n g t h e r e l a t i o n between F o u r i e r t r a n s f o r m s and L a p l a c e t r a n s f o r m s 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. L e t T E Dl b e L a p l a c e t r a n s f o r m a b l e f o r Theorem 8.4.1. R e z > S ( T ) and l e t F ( E ; x ) be t h e F o u r i e r t r a n s f o r m ( i n t h e s e n s e o f Then w e have s e c t i o n 7.8 of C h a p t e r 7 of e" Tx f o r 5 > ,€ ( T )
.
G ( z ) = F(C;Q, 2n
(8.4.1)
if z = E
P r o o f . According t o Theorem 
+
iq.
f o r a fixed 5 > C(T),
8.3.1,
T ( < + i 2 + n ) is a c o n t i n u o u s f u n c t i o n of t h e r e a l v a r i a b l e n and i t s
modulus i s dominated by a power of m
conclude
1

I q I,
as
1 nl
+ .
Hence, w e
+
.
E
$.
Let
T(S+ilrq)$(n)dri exists f o r every
m
*
(8.4.2)
H($;S)
A
T ( S + i Z + n ) $ ( n ) d n= < T ( € , + i Z r n ),+(n)>.
= m
Then ( a s
T,
E
by making u s e of t h e commutative p r o p e r t y of
$ I ) ,
t e n s o r p r o d u c t ( 4 ) , (8.4.2) H(+;s)
t a k e s t h e form 4 2 s qx
= < $ ( a ) , [<eSx TX,e =
i2rnx,
, ei2nxn>]
> = <eCXTx, IFx $ >.
of C h a p t e r 7, w e f i n a l l y o b t a i n
H ( $ ; S ) = < F ( S ; n ) t $(n)>* Consequently, w e g e t by means o f ( 8 . 4 . 2 )
and hence,
(8.4.1)
follows.
8.5. P r i n c i p a l Rules Each o f t h e f o l l o w i n g r u l e s is v a l i d f o r e v e r y z b e l o n g i n g t o e v e r y h a l f  p l a n e R e z > €, where t h e r i g h t  h a n d s i d e i s an a n a l y t i c
Chapter 8
114
function. 1. Addition: (8.5.1)
+
IL (T+aU) = T(z)
aU(z),
a
C.
E
2. Translation:
(8.5.2)
T ~ILTx+= = ecz T(z), I L T ~
C
E
IR.
3 . Change,of scale (or homothesis):
(8.5.3)
l A z
ILTbx = i; TO;) I b > 0.
4 . Multiplication by x k I k
(8.5.4)
kTx = (1)
ILx
dk G ( z ) , in the sense of ordinary 2 differentiation.
5. Multiplication by eaXI a (8.5.5)
IN:
E
E
C:
lLeaXTx = G(2a).
6. Differentiation: (8.5.6)
ILDk Tx
=
zk G ( z ) .
7. Antidifferentiation: (8.5.7)
I L D  ~ T= ~z k
~ A ( 2 ) .
Here DkT is the distribution in DL such that Dk(DkT) = T. If T = f(x) is a locally summable function having c as a lower bound for its support and if we set F(x) = IXf (t)dt, then we have I L F ( x ) = zlILf(x). C 8. Convolution:
(8.5.8)
IL(Tx
*
A
Ux) = T(z) U ( z ) .
9. Multiplication by a function a ( x ) (8.5.9)
IL [~(XIT~]= v 1
5+in (
E
M($):
Z
~ =) v(%)
where V(c+in) = T(2r(5+in)) * A with Ax being the Fourier transforn P n of a ( x ) in the sense of Section 7.8 of Chapter 7. 10. Division by xk:
Laplace Transform
For Tx having bounded s u p p o r t i n [b,[
IL>
(8.5.10)
115
,b
> 0 , then
Tx = Wk(z)
where 1
T i s t h e q u o t i e n t having s u p p o r t i n [ b ,  [ , and Wk(z) satisfies t h e e q u a t i o ndk v k ( z ) = T ( z ) w i t h t h e c o n d i t i o n Wk(z) + 0 dz as Re z + X
.

Proofs.l., 2.,
3.,
and 6 . ,
and 5.,
of T ( z ) ,
f o l l o w from t h e d e f i n i t i o n (8.2.1)
f o l l o w 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)
f o l l o w s from ( 8 . 3 . 1 ) .
l e t X ( z ) = ILDkTx. Then, according t o r u l e (8.5.6) w e have k k zk X ( z ) = ILD D T = ILTx = T ( z ) . Consequently, w e f i n a l l y o b t a i n X ( z ) = z  ~P ( z ) which proves ( 8 . 5 . 7 ) . 7.,
.
8.,
.
of Chapter 5, w e have
By making u s e of (5.8.4)
IL (T*u) = < T e~ 9.,
t
uy,
ez(x+Y)
According to Theorem 8.4.1,
> = (ILT)(ILU).
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 t h e u s e of (7. 12. 2) of Chapter 7 , w e have V(S+iq)
= IF [ e
n
E
$' t o g e t h e r w i t h
 2 n c x a ( x ) T x ] = IF [ a ( x ) . e  2 n c x T
 [rrl .ZnEx
n
T~
I *C
X
3
IF^ a ( x ) 1.
1 0 . If t h e q u o t i e n t T/xk i s f o r c e d t o have i t s s u p p o r t i n [ b ,  [ , t h e n it i s unique. According t o (8.5.4) and Theorem 8.3.1,
dk W (z) __ k k
A
= T(z);
dz
and Wk(z)
+
Wk(z) is a n a l y t i c i n t h e h a l f  p l a n e R e z > E ( T )
0 as R e z
1
T ( z ) i n a unique way. 8.5.1.
+
.
Thus one can determine W (z) f r o m k
Case f o r f u n c t i o n s
I f T and U are i d e n t i c a l ( a s s o c i a t e d ) w i t h t h e Laplace t r a n s f o r mable f u n c t i o n s , t h e n w e f i n d a g a i n t h e above r u l e s of o r d i n a r y Laplace t r a n s f o r m a t i o n . I n p a r t i c u l a r , i f T = f ( x ) i s a l o c a l l y summable f u n c t i o n which i s continuous f o r x > c , where c i s a lower bound f o r i t s s u p p o r t , and having d e r i v a t i v e f (x) ( i n t h e o r d i n a r y s e n s e ) , t h e n (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
i" r
I
F(x)
c
=
f(t)dt, x > 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 do so, we first need the following result. to .:DI Theorem 8.6.1. Let T be an infinite family of distributions XI v belonging to IDJ depending upon the parameter v and having their supports in the same halfline [c,m[ If there exists a number 5 ' such that €or every 5 > 5' , e5x TXtV t esxTx,u, I in the sense $ I as v + v I , then
.
A
TV(z)
+
Tvi ( 2 )
for Re z > 6 ' with v' possibly being Proof. Indeed, IF eSXT 
X IV
+
w.
IF esxTx,
I
and by making use of
Theorem 8.4.1, the proof follows.
...
Corollary 8.6.1. Let T , n = 0,1,2 , , be a sequence of xln distributions belonging to IDJand having their supports in the same halfline [c,[ If there exists a number 5 ' such that for every
.
m
6 > 5' the series
n=0
converges in $', then esx T xIn
=CTx,n 
E
Tn(Z)
for Re z > 5 ' . The following result is more useful.
L a p l a c e Transform
117
b e a sequence of d i s t r i C o r o l l a r y 8.6.2. L e t T X l n , n = 0 , 1 , 2 , . . . , b u t i o n s such t h a t Tx = (ao+alD+ akDk ) f ( x ) where t h e ak a r e ,n n c o n s t a n t s , K i s f i n i t e , and t h e f n ( x ) are l o c a l l y summable f u n c t i o n s
...+
.
If t h e series l I f n ( x ) I having support i n t h e h a l f  l i n e C c , m C converges u n i f o r m l y on e v e r y f i n i t e i n t e r v a l and a d m i t s a m a j o r a t i o n of t h e form xm e S t x (m b e i n g a non n e g a t i v e i n t e g e r ) , a s x t m , t h e n
w e have
IL ITx,.,
= ITn(z).
f o r R e z > 5'. W e e x p l a i n t h e s e r e s u l t s by means of a f e w examples.
8.6.1.Examples
w e have
According t o C o r o l l a r y 8.6.1,
1. IL
. 1
m
m
1 6(xn) n= 0
= 1 enz = n= 0 le'
More g e n e r a l l y , by making u s e of S e c t i o n 5.6 of C h a p t e r 5, w e have m
IL
1 6(k)
(xn) =
n=O 2. The C o r o l l a r y 8.6.2
.le'k Z
a l s o c o n s t i t u t e s a method f o r o b t a i n i n g
t h e e x p r e s s i o n i n t h e series.
We i l l u s t r a t e t h i s w i t h t h e h e l p of
an example. I f v i s r e a l and n o t e q u a l t o 0 , 1, 2,..., E r d e l y i (Ed.)
[n],
vol.1, p.182 ( 5 ) 1
(8.6.1)
ILFp
 Jy(x)+= V
X
(z
+
17+1)) (z
'V
t h e n w e have (see
, Re
z > 0.
Now w e t r y t o f i n d a series e q u a l t o t h e r i g h t hand s i d e . purpose, w e f i r s t have (see Problem 1 . 4 . 1 o f C h a p t e r 1).
H e r e t h e 5' o f C o r o l l a r y 8.6.2
n=O i s e q u a l t o 1(5)
.
For t h i s
By t a k i n g t h e
L a p l a c e t r a n s f o r m of e a c h term of t h i s series and comparing w i t h (8.6.11,
we obtain
which is e x t e n d a b l e i n t h e domain IzI > 1.
,;1
w e have
Next, by changing z i n t o
Chapter 8
118
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 orientation; 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 antitransform (or Laplace inverse transform) of v(z) if ILzTx = v(z) and denote it by qlv(z1 Z.
.
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 halfplane Re z > 5' > 0 and if there exist a nonnegative integer k is bounded as and a real number c' such that Iv(z)l. 1.1 k ecIRe z * , then ILilv(z) exists in ID; and has its support contained (7) Further ILilv(z) is unique and satisfies, in [c',[
.
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
L a p l a c e Transform
119
e v i d e n t l y d e f i n e d by 8.7.3) i s t h e L a p l a c e a n t i  t r a n s f o r m of v ( z ) k2 i.n t h e o r d i n a r y z sense. Note t h a t (8.7.2) i s a consequence of (8.7.3). F u r t h e r m o r e one c a n show e a s i l y t h a t w ( x ) i s c o n t i n u o u s
i . e . w(x+q)w(x)
f
0 as q
+
0.
( A l s o , t h e c o n t i n u i t y of w(x) c a n 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 complex2plane a c i r c u m f e r e n c e 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 of t h e s t r a i g h t l i n e ( 5  i  ,
through t h e p o i n t s c  i Y , c + i Y
A b e a n arc of C on t h e r i g h t hand s i d e of
(6im,
c+im).
c+im).
Let
Then by
Cauchy's theorem, w e have (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 , b e c a u s e a c c o r d i n g t o t h e h y p o t h e s i s I v ( z ) zk2 e ( ~ '  ~ ' ) ~ I +OonA more r a p i d l y t h a n 1/R. T h e r e f o r e , w e have w ( c '  x ' ) = 0 which y i e l d s w ( x ) = 0 i f x 5 c ' . I t 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,
Now, s e t z = 5
w e conclude w(x)
+
I D :
E
ecXw(x) =
J m
By making u s e of (7.8.2)
c ' , we
For fixed 5 >
2niq. m
(8.7.5)
.
v(c+2nia)
k+2
2nixn
(~+2niq)
of C h a p t e r 7 ,
Consequently, by v i r t u e of (7.8.4) is u n i q u e and
(8.7.5)
dq
t a k e s t h e form
of C h a p t e r 7 w e d e d u c e
t h a t w(x)
v ( c ' , then
~~;'v(v
; z ) in ID'. 0
(vo may be infinity).

Proof. Let 5 in x, we have
>
5'.
By considering v(v;E+2*ix) as a distribution
Laplace Transform
121
Hence, according to the sense of Section 7.8 of Chapter 7. 1~v(v;~+2nix) ~ v ( v*S+2mix) in 8'; f
and therefore in ID1
.
0'
Thus, by (8.7.1) we have
e'xLi'v(v;z)
+
.Ex
'
I L ~ V ( V ~ ; Zin ) 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 halfplane Re z > 5 ' independent of n. If the series vn(z) converges n=0 uniformly on each compact subset of the halfplane Re z > < I , then we have
1
This corollary generalizes the Heaviside method in which vn ( 2 ) is of the form an z 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 As v +O, zk+" zk + 0 uniformly on each compact subset of the halfplane Re z > 0. Then, Theorem 8.8.1 yields
k be a nonnegative integer.
As the integer n
n" , 2 n 1
+ 0 uniformly on every compact (z+n) subset of the plane. Then, Theorem 8.8.1 yields (n and take the role v and vo, v(v;z) = nn ez/(zn)", (vo;z) = 1)
2.
f
Chapter 8
122
The function "exponential integral" admits the representation by Section 9.4.4 of Chapter 9, 3.
m
Ei(l/z)
=
log c/z
(  2 ) n 1 n .n! n=l
.
This series converges uniformly on every compact subset of the halfplane Re z > 1. By (8.10.2') we have 1 log C/Z = Fp
lLx
1
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. is differentiable with respect to v if for any $ E ID, Then Tu ;x ,$(x)> is a function of v differentiable in E (in the ordinary Il TT 6
E
ID.
If T V F x= 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, a f(v;x) I < g(x), where g(x) is a positive locally summable function, then we have a
_ 1
av
= D~
& f(v;x) in E.
Laplace Transform
123
Example. If v varies in the closed domain of the complex plane do not not including the integers 2 1, i.e. (the points v = 1,2,..., belong to this domain), then we have
a
(8.9.1)
Indeed
a
Fpx;'
V
FpX, =
= Fp(x' log XI+.
kv
a k x+ h
= Dk
FWk
kv a x+ a v mk, where k is an
av .. 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.) ~~
Theorem 8.9.1. 1. If Tvix and aa v Tv;x are distributions which are Laplace transformable in the same halfplane Re z > 5' independently of v in E, then we have

JLa
(8.9.2)
a v Tv;x

a &TViX av 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 halfplane of Re z > 5' independently of v in E, then we have
I?& v(v;z) = & E,' v(v;z)
(8.9.3)
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 ILFpxIV = T(v+l)z
v1 , 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 ILFp(x'log
(8.9.4)
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 (11) times in (8.9.4) with respect to v, we obtain (8.9.5)
Re z where );(
= & (Fp
is not needed if Re v < 1).
> 0,
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 ezxf (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(8) if 0 < x < E , g(x) admits a representation of the type (1.1.5) of Chapter 1, if x >
E,
g(x) eZX is continuous and integrable for Re z >
E l ,
if x > 0, g(x) has an ordinary derivative g' (x). We set m
G(z)
=
ILFpg(x) = Fp
1 eZX
g(x)dx, Re z > 6 '
0
and
X
Fp
1 g(t)dt,
x > 0
g(l) (x) =
, Next, by making use (5.4.12)
c
x
5'
If g(x) has the expansion of the form (1.1.5) of Chapter 1 with then we have K' J' K' J' g1 (x) = [ 1 ~ai+a;~logjxlx + 1 1 B;k xlklogjx + k=l j=1 k=l j=1
= 0,
 xi
where hl(0) = 0 and h i are not integers: g(") expansion of the form (1.1.5). Let
(x) has therefore an
Laplace Transform ILFp g(”)
125
(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
.
’ (x) =
ILFp g”’)
ILFp g(x) = G ( z ) ,
or
where
Consequently, we obtain (8.10.2)
Example.
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’)
ILFp xT1 =

log Cz, Re z > 0 1
and differentiating again we obtain ILFp xi2 More generally if n (8.10.2”) 8.10.2.
s
 z log Cz
+
z.
IN, we finally obtain n n n nl (log Cz 1 I/]) IL FPX+ n! j=1 E

, Re
z > 0.
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 0,l , ak(a) and v k ( a ) are bounded: h(a;x), ak(a) and vk(a) are holom~rphic(~)in E, vk(a) # 1,2,3,..., for any a in E. 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 IKh(a;x)eZXI are majored when in E, by an integrable function y(x) 2 0, then
,
1. if
a
is
El, the function g(a;x) has a Laplace transformation in the ordinary sense given by a E
1 g(a;x)e zxdx,
~ ( a ; z )=
0
Re z > E ( g ) ;
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) = (here Fp is not needed if
Re z > [(g).
G(a;z),
El).
a E
Proof. Set Nk(a;z)
= ILFp
x
vk
Let us assume that there exists Re v,(a) + Mk> 1, then we have
(a)
X(OI1)
\
E
lN such that, for every
a
E
E,
=%c'
lv (a)+m m m lvk(a)+m (z)mFp xk dx+l(Z) f x dx m= 0 a' 0 m' 0 Mk m ( zlm m= 0 [vk(a)+m+l] m!
= c
Therefore, as a function of H(U;Z)
a,
Nk(a;z) is holomorphic in E.
Set
=rr,h(a;X) = j h(a;x) e2Xdx, Re z > s(g) 0
where H(a;z) is holomorphic in E. ILFp g(a;x) = H(a;z) m
(which is equal to
Consequently, we obtain by (i)
+
K
1
k=0 g(a;x)e ZXdx, if a E
ak(a) Nk(a;z), Re z > c ( g ) 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
and I L J a ( b x ) + = [ ba rr

z
+ 1TT z +b 1a ,
127
R e z > 0.
A f t e r changing a t o a, w e deduce from Theorem 8.10.2,
(8.10.3)
Re z > 0,
= r(a+l)za',
ILFpx;"
f o r a1 g! IN,
and
(8.10.4) 8.10.3.
ILFpJ_,(bx)+ =
[z+ JZ2+bz]&, b'
m
z > 0.
Re
Change o f x t o ax, a being complex
The change of x t o ax p e r m i t s u s t o deduce lLFp g ( a x ) from ILFpg(x) by means of S e c t i o n s 1 . 6 and 5.9 of Chapters 1 and 5 , For t h i s purpose, w e f i r s t have
respectively.
Theorem 8.10.2.
Let 0,
x < o
g(x) = x"h(x)
I
X
> 0
where v i s n o t a n i n t e g e r 1. 1 and h ( x ) i s a n a l y t i c . ILFpg(x) and Ga(z) = ILFpg(ax) where a
E
Set G(z) =
C is f i x e d .
W e suppose t h a t t h e r e e x i s t s an a n g l e A i n t h e complex w p l a n e having i t s v e r t e x a t t h e o r i g i n and c o n t a i n i n g h a l f l i n e s w 1. 0 and w = ax ( x v a r y i n g from 0 t o m ) and i n t h e zplane a domain B , such t h a t h ( w ) i s holomorphic i n A and such t h a t I w a g ( w ) e zw/a

w +
I
as
i n A, z remaining i n B and Re(av) 1. 1. Then w e have Ga(z) =
a1 G ( z / a ) , z
E
B.
From t h i s r e s u l t o b t a i n e d i n B , w e deduce, by a n a l y t i c c o n t i n u a t i o n , Ga(z) i n every h a l f  p l a n e where Fp g ( a x ) is Laplacetransformable.
Proof. L e t
a be a complex v a r i a b l e , and s e t g ( a ; x ) = x a g ( x ) Qlence
g(x) = g(0;x)).
Then
G ( a ; z ) = ILFp g ( a ; x )
128
Chapter 8
and 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 d e n o t e s t h e h a l f l i n e w = ax, p r o v i d e d t h a t Re(av) z E B , w e have
> 1 and
m
Ga(a;z) =
g ( a ; a x ) ezxdx 0
=
a1 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
I G(O,z/a) Ga(O,z) = g
which y i e l d s t h e r e s u l t ,
Take g ( x ) = J  v ( x ) + , v b e i n g a n o n  i n t e g e r
Example.
b e i n g a complex number such t h a t 0 5 a r g a 5 w e have
G( z) = IL FpJ" ( x )
(8.10.5)
+
= [z+ (z2+1)
3.
1, and a
Then, by (8.10.1)
,
(~~+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
< arg w
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' I j n(x) elXIm(za)dxl < M X 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

Proof. For simplicity, we first deal the case'when j = O . write T, as
TX=AeaxX+v + Bx
+
> C(T).
We can
xv ea x Q(x)x(X,)+XV eax~ ( X ) X ( X I ~ ) I
where Bx is a distribution whose support is bounded in lm,X] and x(X,) is the characteristic function of the interval [XI=[. Further, we set
E (2)=(Za) '+lIL TxAr ("+I)= ( z a ) "+I [IL Bx+IL xveaxf2(x)x (X,m )
+ ILXv eaxw ( x ) X ( ~ , m ) l Now, according to Abel's theorem, we have
.
Laplace Transform
v ax IILX e n(x)x(x,)
I
135
m
ix Im(za) v x Re(za) 52 (x)e x e dx X xv. < M X ' X ~ Re(za)
c(T). Then, for given arbitrary E > 0, we can choose X such that
< el
n/2.
1. Also, we set +
~(z) = CA+rl(z)l zvkiogjz and h(x) = ILlH(z), which is continuous.
IL'w(z)
(8.12.2)
=
Consequently, we obtain
DXh(x).
Now making use of the Theorem 8.11.1, we have A zvklogjz = r (1)' (v+kA)IL(xv+kllogjx)++ r2 ( 2 ) zvklogjz
where r2(z)
P
where r(z) a0 our setting, (8.12.3)
Putting z =
m, we obtain X m
('+in) ~ ( x )= e' J r ( a ) ('+in) vk ( lo g:1 x m x

1)1 eisdn
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 IL w(z)
j v1 ., r(1) (v) D P ( X
j log XI+,
hence, the result (8.12.1) is obtained. Theorem 8.12.2. If there exists 5 , > 0 such that in the halfplane Re z > 5, the function v(z) is holornorphic and satisfies v(z)
 Aecz znlogj+lz, as
121
+
m
where j and n are nonnegative 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) as x
+
T'cTx = Tx,=

(1)j+ntln! (j+l)AFp(x"llogJx)
+
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 ) Az~'~, as Iz I + , Re z > 0, since lu(i0) I is of the order of B I n I 2 as 1111 +. m. However, according to (8.10.3) , ILl 2Ol = r (1a) Fpx;", which yields I

z3j2 = 3 Fpxi5j2 by putting a = 5/2; while according to (8.2.4), 4 6 ILlebzz2= 6"(xb). Hence, we obtain from these results I L  l U ( z ) = 3A Fp~;~/~+B6"(xb) , which is equivalent at the origin to ~3AF p x5/2 , 4 6
IL'l
.

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 v(z) = vl(z) + ebz v2(z) with the inequality c1 < c2. AlSO, by Theorem 8.7.1, if c and c2 denote the lower bounds of the supports of IL'vl(z) and ILI1 v2(z). Then, the behaviour of the distribution J.L'v(z) near c1 would be the same as that of sLlvl(z). If any one of the Theorems 8.12.1 and 8.12.2 is applicable to v (z) then the behaviour of ILlvl(z) can be 1 easily determined. 8.13.The nDimensional Laplace Transformation In the preceding sections we have studied the Laplace transforms in a distributional setting of one variable. The present section developes the nvariables 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,x21....I~n) to the domain Qn(O of mn which is defined by Qn(0) = fx E IRn , 0 5 xv C0 then there exist positive &S + =. ( i ( z )I .TIn l z j l k j is bounded as Iz.1 3
ki, 1 5 j 5 n, such that
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, cn) is any number of IRn such that = 1,2,. ,n and k is any nonnegative integers of IRn Then we have
..
% >O, i
(8.13.3)
I L akk 6(xc) = zk e
j
axi
.

I
~
~
J
and
ILd(x) = 1. These results follow strictly by the definition (8.13.2). 8.13.2.Convolution Let T and S be two distributions in lDb+(IRn ) and possessing Laplace transforms for 5 > bl and 5 > b2, respectively where bl and b2 denote the points of lRn such that bl = B1 ,j(  B1, 1, ,Bl,n) Then, for and b2 = (82,1,B2,2,...,B2,n) for each j = 1,2,...,n. every 5 > b = max (bl,b2) (i.e. > max (Bl,jf@2,j)
..
'1
(8.13.4)
i(z) = a n j 1' 1 f j 5 n. I n t h i s manner w e can r e l a t e t h e p r e s e n t s t r u c t u r e of Laplace t r a n s f o r m s w i t h t h e t h e o r y of Laplace t r a n s f o r m s g i v e n i n Schwartz C11. I n t h i s work Schwartz d o e s n o t d i s c u s s t h e i n v e r s i o n
'...
of L a p l a c e t r a n s f o r m b u t w e c o v e r i t s i n v e r s i o n i n t h e f o l l o w i n g manner. Then w e c a l l
L e t V ( z ) b e a f u n c t i o n of t h e complex v a r i a b l e z .
t h e d i s t r i b u t i o n Tx t h e L a p l a c e i n v e r s e ( o r a n t i  ) t r a n s f o r m of V ( z ) and w e d e n o t e it by I L i l V ( z )
if ILzTx = V ( z ) .
The main r e s u l t o f
this s e c t i o n i s t h e f o l l o w i n g i n v e r s i o n theorem. Theorem 8.13.3. il where R e z
> b
I f t h e f u n c t i o n V ( z ) b e holomorphic i n t h e domain > 0 ( b . = (bl,b2,
...
'b ) f o r j = l , 2 1 . . . , n ) and i f j j 3 n t h e r e e x i s t nonnegative i n t e g e r s m.and a p o s i t i v e number B s u c h t h a t 7 ml m m 2 Izl z2 ,...'z % ( z ) I < B for a l l 1z.I + m wi t h (8.13.7) n 3
t h e n K:V(z)
e x i s t s i n IDA+ (IR")
.
Further ILIIV(z) X
i s u n i q u e and
satisfies (8.13.8)
ILilV(z)
m +2
m2+2 Dx 2
= Dx 1
1
with (8.13.9)
Cl+iw(x) = ( 2 ~ i )  ~
...
mn+2 Dx w(x) n
c 2 + i 
~ ~ dzl1 C 2  i 
d z Zr...l
I5
cn+im
n
U ( z ) e X Z dzn i
142
Chapter 8
n m 2 where U ( z ) = V ( z ) II z 1 j=1 j transform of the variables 17
.
Proof. 
Here Ilf;'denotes and the numbers
j
the inverse Fourier
5j
are greater than b
j'
The relation (8.13.9) can be rewritten as
with (8.13.12) Un(zl,z 2,...,zn)

+ix z u(z) e n n dznt 5, > bn[,im
I
=
~
Let and 5; be two numbers such that 5; > FA > bn and let znml) denote the value z n1) and Un(tTr;zl, z2, Un(p; z1,z2, 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; n Q" = 6" + ih be in the complex zn plane and denote n

...,
...,

G(z)dzn, I2 = I G(z)dzn, I3 = I G(z)dzn, I4 Q'Q" P"Q" x z where G(z) = U ( z ) e n. I1 =
I
P'Q'
=
I
G(z)dzn
P"P'
Now, by applying Cauchy's theorem to G(z) along the contour P 1 Q 1 Q ' 8 P l l P we obtain
(8.13.13)
I1 = I3
xnc;

I2
 14.
2
Since I G ( z ) I < B e zn on Q ' Q " and P"P', one can get I2 + 0 and I4 + 0 with 5; > :5 by letting h + Hence, we may infer from ( 8.13.13) that
.
lim I1 = lim I3
Therefore, we may conclude that Un(zl,z2,...,z n1) does not depend upon the 5, and consequently by (8.13.11) w(x) does not depend In a similar manner we can show that w(x) does not depend upon the choice of ? Thus, we may infer that w(x) is a function of x only j' which justifies its notation.
sn.
NOW, we have to show that Un(z1,z2,...,znl) = 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 = 5nih and Q = En+ih, with in > sup(b,O). Then, by applying Cauchy's theorem to G ( z ) along the path PQyP, we obtain (8.13.14)
x z
/
U(z) e
"dz, =
 1 U(z)
Ixnlzn e dz,.
PQ QY P By the hypothesis made on V(z) there exists B' such that lu(z) 1 < ,5 If zn is on QyP then one can show that the B' Iznl' if Re zn 2 . right side of (8.13.14) tends to zero as h + A l s o , the left side of (8.13.14) tends to Un(z1,z21...,znl) by means of (8.13.12). This enables us to conclude that Un(z1,z2,...,znl)= 0 if x < 0 and n consequently w(x) = 0 for xn< 0. Similarly, one can show that w(x) = 0 for all x < 0. From this property of w(x) we may infer j that its support is contained in Qn(0) where x > 0 for all j . j Finally, by (8.13.8) we may conclude that the support of ILlV(z) is 1V(z) E IDA+ (IRnx) contained in Qn(0) for all x and hence ILx
.
.
j
If we put z = 5 . + 2nin in (8.13.9) then we obtain (8.13.10). 1 1 j 1 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 m +2 m2+2 mn+2 IL D D Dx w(x) = V ( z ) x1 x2 n
.....
or
n 11
(8.13.15)
j=1 n
m.+2 z J ILzw(x) = V(Z) j
m .+2
n (~.+i2*a.)~ I L ( ~ + w(x) ~ ~ = ~ V(E+i2nq). ~ ) 7
j=1
J
Since the support of w(x) lies in Qn(0), we have m
m
=
IF eX'w(x)
n
... / dxn exSi nxnw m
w(x) = / dxl 1 dx2.. IL (~+i2nq) 0 0 =
2
0
(XI
I F I F  ~ U ( E + =~ ~(5+i2aq) ~~~)
n x
by (8.13.10). Hence we obtain (8.13.15) in view of the definition This proves our theorem. of U(z)
.
When n = 1, the present theorem reduces to Theorem 8.7.1 c' = 0.
with
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 . shows that the assumption T belong to ID; is not (8.2.5) necessary in order that its Laplace transform exists, for example 2 ILeX = T% e22/4 but by restricting the Laplace transform to 3pl , 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 Iv(x) is equivalent to (2nx)'I2 ex as x + a. 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 calculation described by (8.7.1) determined exactly the support of antitransform 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. (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 distributional 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 equations, 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 antiderivatives 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 X and V X be two given distributions belonging to I D : . seek a distribution Xx E ID: such that (9.1.1)
uX * xx
=
We
vx.
Furthermore, assuming that these three distributions are Laplace 145
Chapter 9
146
transformable, we put A
U ( z ) = l L U , V(Z) = ILV,
X ( Z )
= ILX.
Now we obtain according to (8.5.8) of Chapter 8, A
A
(9.1.2)
U(Z)
.L
X ( Z ) =
V(z)
and = .G(z)/G(z)
in a certain half plane Re z > 5 . (9.1.3)
xx
=
A
Hence, by inversion
ILlG(z)/i(z) 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. This implies that Xx is unique in ID;. Fundamental solution If (9.1.4)
Ex =
c1l/j(z) ,
the equality (9.1.3)
can be written
an expression which no longer requires that Vx is Laplace transformable or lie in ID:. (Evidently, Ex * Vx must exist; see Section 5.8.; 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 9.1.1.
Examples
The convolution equation (9.1.5)
Fp x+1
*
X = log2 x+
is mapped by the Laplace transformation into (log CZ)
A
X(Z) =
(log CZ)2+7r2/6 Z
Applications where C
is Euler’s constant.
= 0,577....
147
From this we deduce
Hence, by inversion
x
=
log x+
 +2 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 v(x;a)
J
=
m
t+a r(t+a+l) dt. X
2 . The difference equation
can be rewritten as the convolution equation [
G(xa)k G(xb) 3
Xx  Vx.
*
Its solution according to (9.1.3’) is XX tal solution E is given by E
=
Ll[ eaz
according to (9.1.4).

=
E
*
Vx, and the fundamen
keb7
But we have
This series converges uniformly in the halfplane 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;. Consequently, we may infer that (9.1.6) will have a solution if V belongs to DD;. Accordingly, the solution is
xx
= E
* vx 
m
.
j=O k’vx+ (j+l)ajb.
Chapter 9
148
3 . The differential
(9.1.7)
Xxa

 difference equation
k DXxb

Vxi b > a,
is also the convolution equation [6(xa)kat(xb)]
* Xx
= Vx.
Its fundamental solution according to (9.1.4) is given by E = nl[eaz

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
1 kJ&(')(xjb+ja+a). j=O This formula is exact, because we verify that this series is convergent in ID; and that its convolution with 6 (xa)k6 (xb) is given by 6 (x) Therefore, Xx = E * Vx is the solution of (9.1.7). E
=
.
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 "distributionderivative 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 be given constants with cn # 0, n 2 1. A distributionn derivativ? equation with constant coefficients has the form
...
Applications It has a unique solution in ID: XA = E(x)
(9.2.2)
*
149
given by
vx
where
n
1 CjZj. j=O Also, the equation (9.2.1) has many solutions in ID’ and these solutions are of the form (9.2.3)
p(z)
=
Xx
Xi
=
+ h(x)
where the function h(x) is any solution of the equation cnh(”)(x)
(9.2.4)
+
... + c1h’(x) + coh(x)
= 0.
To show (9.2.3) , we note that according to ( 5 . 8 . 8 ) (9.2.1) can be written in the convolution form
of Chapter 5,
It is evident that (9.2.1) has many solutions which have the same support as that of V. Let its one.solutionbe denoted by X! Therefore X‘ c ID; and satisfies n (9.2.1’) [ cj S(j)(x)l * X’ = V.
J=o
The convolution algebra constructed over ID: we conclude X’ is unique.
has no divisor. Hence
In addition XXI is the solution of
f
c.Dj I(XX’) = 0. j=o J Hence (Schwartz Ell , Chapter V, 6, Theorem IX) XX’ 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 C
X1 = E(x) X Hence the results (9.2.2)
*
Vx. and (9.2.3)
are established.
Chapter 9
150
As usual, let V ( z ) denote the Laplace transform of V. according to (9.1.3) XI can be written in the form (9.2.5)
Then
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 (n2) and a derivative of order (n1) which is discontinuous at x=O. Indeed, according to Theorem 8.7.1 of Chapter 8 if 0 (q 2111, we have
Moreover, ) ' ( E
[ l l , p. 109.)
(see also VofiacKhoan
(0) = 0, 0
5 q 5 n2.
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) n r x X4 = 1 akVx * e+k (9.2.6) k=l 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
X' = 0, for x
a, then we replace w 0 9 by a parameter which will be finally determined by (9.2.10) and this condition at zo.
with the conditions f(x)
= 0
for x
Solution. Here p ( z ) ILeiVX = (z+v)' and
=
< 0,
f(O+)
= wo,
fl(O+) =
w
1' 1 Hence E(x) = A sin Ax+.
z2+ A 2 .
As
(9.2.11) gives xe VtE(xt)dt
0
=
212 [ e;"'(DV) v +A
On the other hand, no  wl, Ql = we have
w
*
0'
A1
Sin Ax,].
hence according to (9.2.10)
+ ;i sin Ax, for x
> 0.
2. Solve
with the conditions f(x)=O for x < 0, f(O+)
= w0,
f'(n/A)
= wi
.
Solution. Returning to the preceding example, consider w 1 as a parameter in (9.2.12) which is determined by using the condition fl(n/A) = w i in the form
Chapter 9
154

(evn/A++l)
.2_>
Then it suffices to replace
w1
w1
= wi,
in (9.2.12) by wiv(v2X2) 1(ev"/h+l)
3. Solve f"(x)3f1(x)
+
2f(x)
=
x for x > 0
with the conditions f(x)=O, for x Solution. Here p(z) 1
=
1

+
bz+a+zL z2 ( z21)
Hence, inverting the last equation and recalling that 6'(Z21)' = cash X+ and IL1 z(z21)'l = sinh x+, we obtain 1
f = ~(2bD) ( 2exxexcosh x)
+
(a+bD+D2) (xsinh x)
+
which gives f(x) = ba~[(b+~)x+b] 1 ex(a+T)sinh 3 x, x > 0. By the first of the equations (9.2.13) we further have g(x) = axb
[
1 (b+)xbl] ex+(a+3)sinh x , x > 0. 2 T
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 fBB'(x)+ fBB(x) fB(x)  f(x) = ax + ab. 9 . 3 . Differential Equations with Polynomial Coefficients
The Laplace transform of a "distributionderivative" 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.
Chapter 9
156
9.3.1.
Reduction of order,
Consider the equation 19.3.1) where Xx is a unknown distribution, Vx is a given Laplace transformable distribution, and the p. (x) are polynomials where pn(x) is not 3 identically zero. A
X(z)
By employing the Laplace transformation on(9.3.1) * = I L X and V(z) = lLVx we obtain
and putting
(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 insufficiently 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. (9.3.3)
Consider the second order equation
xf" (x1f'(x)af(x) = a2xeX
=
a2xexax2 , for x > 0,
with the conditions f(x)
=
f(O+)
0, for x < 0,
=
1, f'(O+) = 1.
f'(x)
=
DfG(x)
f"(x)
=
D2f62 (X)+6(X).
Solution. Here,
and
Making use of
x 6 ' = 6,
x6
= 0,
(9.3.3) takes the form
Applications
(9.3.4)
xD2f  Df

af
=
[a2xe=
157

ax2
]++ 26(x)
which is an equation in the sense of distributions in ID;. By employing the Laplace transformation on (9.3.4)
and putting
F ( z ) = Z f , we have z 2~ ' ( 2 )+ (32+a)F(Z) = 2a
7

a2 + 2 (z+a)2
which is a first order equation whose solution is F(z) =
2 + 3 Z
A + a keaz 7 z+a 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 f (x) = x
2
+
kx 1 2 ( 2 E ) , for x > 0.
And finally the condition f'(O+) = 1 implies k = 3/2. 2. We obtain the distribution X order equation (9.3.5)
xD2X

DX

E
ID; satisfying the second
ax2X = b6'(x).
By employing the Laplace transformation on (9.3.5) and putting X ( z ) = ILX, (9.3.5) is transformed into the first order equation L.
whose solution is $(z) =
b 2 2 3/2 3 + kl(z a )
where kl is arbitrary. Hence inverting by a formula of (Erdelyi (Ed.) C21, Vol.1, (19), or Colombo and Lavoine C11 , p. 98) , we have (9.3.61
X =
b 3
6(x)
+
p.239
kx Il(ax)+
where k is arbitrary. Next, we seek the solution of (9.3.5)
in ID'. We put X = xY,
Chapter 9
158
which enables us to write (9.3.5) as x2 [D2Y
+
X1 DY 
(a2+$)Yl
=
b6'(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 1 1 2 1 X
=

+ kxIl(ax)+ + klxIl(ax) + k2xKl(ax)
bS(x)
3
in ID'. 3 . Consider the second order equation
(9.3.7)
x D2 X + ( x + 3) D X + X = 0
First we find its solution in led to the first order equation (z2+z) X'(Z)

n
Putting X(z)
a);.
= ILX,
we are
6
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 transformation of X, then Section 7.13 of Chapter 7 transforms (9.3.7) into C(2inF + 1) DW

2incW
=
0
in $'. Dividing by C , we have (kl being arbitrary) by making use of Section 6.1 of Chapter 6 that (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. 8 5 ) we have (9.3.9)

X = kl (x 2x1)
+
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 X1 denotes this non tempered distribution, then it satisfies xD2X1
+
(x+3)DX1
+ X1
= 0.
We attempt to take X1 = ebxY, where we must determine the nonzero number b and the tempered distribution Y. Here also, Y is associated with the equation xD2Y

(2b1)xDY + 3DY + b(b1)xY

(3bl)Y = 0.
If 2 denotes the Fourier transform of Y, then this equation becomes, according to Section 7.13 of Chapter 7, (2in5b)C 2i~cb+l]DZ 2in(2inEb)Z 5 5
= 0
which gives after division by (2in5b) (this factor does not vanish for any real 5 ) (9.3 .lo)
(ZinCb+l)DZ
5

2irZ = 0. 5
if b # 1, we get Z = k(2inEb+l)
where k is arbitrary. Hence by means of the inverse Fourier transformation
Therefore (9.3.11)
160
Chapter 9
Ifb=1 Z = Pinkc
+ k2ins(c)(2inc)
is the solution of (9.3.10). Y
=
k&' (x)
Hence by inversion
+
k2Fpx2
and consequently, (9.3.12)
X1 = eXY = k2Fpxm2 ex
+
k[&'(x)+S(x)l,
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 (9.3.13)
X
=
k[&'(x)+&(x)]
t
klFp(x1x2)+k2Fpx2ex
.
The last result has been derived by Bredimas [ a ] , pp. 339340, 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 a 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 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 9.4.1.
161
Special Volterra equations (1)
Consider the "problem" to determine the function f such that (9.4.1) (9.4.2)
a
+ Af(x)
k(xt)f(t)dt
= h(x), a > 0
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 form Ck(x) + AS(x)l
*
f(x) = h(x)
in ID;. According to Section 9.1 (see remark 9.4.3), given by (9.4.3)
f(x)
=
E
f is unique and
* h(x)
where E = Z1  1 k(z)+A with k(z) = ILk(x). If h(x) is Laplace transformable and if H(z) can be presented eventually in the simple form
=
ILh(x), (9.4.3)
(9.4.4)
Note that the fundamental solution(*) is not necessarily representable by a function. Examples.1. Solve f(t)dt a
+
Xf(x) = h(x), x > a
f(x) = 0, x < a.
1 Solution. Here k(x) = U(x) (Heavisidefunction), K ( z ) = z, and
Chapter 9
162
Thus
and further, if h(x) is a locally summable function whose support is bounded below by a, we have x (9.4.5) f (x) = X’h(x) A2 I h(t) et/’dt.

a
2 . Solve X
1
(9.4.6)
f(x) Solution.
+
(xt)evtf(t)dt
a
w
= 0,
2 vx
e
x
f(x) = h(x), x > a ,
a.
After multiplication by evx, (9.4.6)takes the form
+
[ x e :
w  ~ ~ ( x ) *] f(x) = evxh(x).
vx Here k ( x ) = xe+ , K ( z ) E
=
2
=I
and
= (zv)’~
2
w (2v)
1 L w2 = JL

( 2  v ) 2+w2
= w2.s(x)


2
= w &(X)
w w
3
4
1
( 2  v ) 2+w2
vx e
f12+
z +w evx sin wx+;
hence f(x)

= w
evxh(x)
= w
evx Ch(x)
w3 [
evx sin ax]+

sin wx+
w
*
*
[
xvxh(x)]
h(x)l
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 sin ax+
*
h(x) =
X
1 h(t)
sin w(xt)dt.
a
If we replace w 2 by
w
2
f (x) =  w 2 9.4.2.
in ( 9 . 4 . 6 ) evx C h(x)
, we w
get
sinhwx,
*
h(x)l
.
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 1 in CO,[) then Theorem 8.3.3 of Chapter 8 tell us that IK(z)/hl It follows that when Re z is large enough say when Re z>E1.
The convergence of this series is uniform in the halfplane Re 1’ We can invert term by term and obtain according to Corollary 8.8.1 of Chapter 8 that (9.4.7)
where k.(x) = IL1K j( 2 ) ; 7
that is kl(X) = k(X) r k2(X) = k(x)*k(x) =
X
1 k(xt)k(t)dt, O
kj(x) = kjl(~)*k(x) =
x 0
k(xt)kjl(t)dt.
The series (9.4.7) is called the resolvent series and it converges in ID’. Now (9.4.3) takes the form m
(9.4.8)
f (x) = X’h(x)
+
1l 1 (  A )  ] j=1
kj(x)
*
h(x),
(See also Lew [11 .I
If h(x) is a locally summable function whose support is bounded below by a , we get m . x I kj(xt)h(t)dt f(x) = A‘h(x) + X’ 1 (  A )  ’ 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 (See also Yosida
[ 21,
Chapter VIII.)
1U(Z)
I < 1.
Chapter 9
164
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(xt)f(t)dt + Xf(x) = h(x),
a
then the solution is not unique. (9.4.9)
f(x)
= E
*
Accordingly, we have
h(x) + P(k:x)
where P(h:x) is an arbitrary linear combination of the solutions of the equation X
I
k(xt)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
m

wlf(x)
=
h(x)
where w > 0 and h(x) is a locally summable function whose support is bounded below by a. According to (9.4.9) and (9.4.5) the solution is f (x) = wh(x)

w2ewx
I
X
h(t) ewtdt
+
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
k(xt)f(t)dt
(9.4.10) U
with f(x)
=
0 if x
c
a
n
+ 1 c.xjf(x) j=o J
= h(x)
and cn # 0.
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 halfplane 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 n1; 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) Solution. We have IL cosh x+ = Vol. I, p. 239 (13)) (9.4.12)
Z
(Erdelyi (Ed.) C 2 I,
l 5 and
lLXVIY(x) = 2 v (v++ 7Tl/* (z21) v1'2.
Using these results, we can rewrite (9.4.11) F'(z) +
in the form:
v 1 1/2 2 v1/2 (2v+1)z ' 7 F ( z ) = a 2 (v+T)~T ( z 1)
z 1
I
and whose solution is given by
with c an arbitrary number. inversion, we obtain
Hence, using (9.4.12) to perform the
(9.4.13) f(x) = [as'(x) + CS(X)I * X'I~(X)+. Consequently, (9.4.11) has the solution f (XI = xVCaIvl(XI + CI,,(XII,. If we had put v = 0, this last formula would yield f = aI1 (x)+ + a6 (x) + cIo (XI+ This is no longer a function but it still satisfies the convolution equation
.
cosh x+
*
f

xf = a10 (x)+
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. In other words, the equation X a V (9.4.14) coshx+ * 2v+T = 2v+l IV(X)+

Chapter 9
166
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 f o r 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 integrodifferential equations by means of the fundamental solution and Laplace transformation. Let us determine the function f such that X n cjf(j) (x) = g(x) for x > a k(xt)f(t)dt + (9.5.1) a j=O (9.5.2)
f ( x ) = 0, for x < a
f (a+) = (9.5.3)
w
0
f(j) (a+) = wjl 1 5 j 5 n1
given the numbers a , w . , c (cn # 0, n 1. l), the Laplace transform1 1 able function k(x)+, and the locally W l e function g(x) w a i support in [a,$
.
This problem has a unique solution given by the formula X n1 E(xt)g(t)dt + Qq E(q) ( x  a ) , x > a (9.5.4) f(x) = a q=o where
c
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 n1 Ik(x)+ + 1 c.6") (x)l * f(x) = g(x) + 1 ng6(q)(xa) = V X j=o J q=0
Applications
167
According to Section 9.1, the unique solution of this in ID:. equation is (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 1 k(z) + 0 as Re z + a; and is of the same order as z'~. K(z)+p(z) One can show as in Section 9.2.1 that E(x) is a continuous function having (n1) derivatives in the sense of functions, of which the first n2 are continuous. It follows that (9.5.6) takes the form (9.5.4). Example. Consider the equation (9.5.7)
X
a J f(t)dt
+
+
cf(x)
f'(x)
g(x)
=
0
where g(x) is a locally summable function which is null for x but it satisfies the conditions: f(x) = 0, for x f(O+) = Here n = 1, no
=
W.
z
E(x) =
0
0
0.
168
Chapter 9
2. xf a = h 2 and c = 21, then h is the double root of the polynomial z22hz+h2 = ( z  h ) 2 We have
.
AX
= (Ax+l) e+
and (9.5.4)
gives f(x) = elx
X [
W+
(hxAt+l) .It
g(t)dtl
,x
z 0.
0
Remark. The method is still valid if the initial conditions are replaced with other suitable conditions. The following example permits us to understand easily how it can be adapted. (9.5.3)
Consider the system a2
(9.5.8)
I
X
f(t)dt
a

f'(x) = h(x)
f(0) = 0
(9.5.9)
where h(x) is a locally summable function having support [ B I B ' a 5 6 < 0 B' < and a is bounded. Denoting by gives
and (9.5.4)
1,


w
the value (temporarily unknown) of f ( a + ) , (9.5.5)
leads to f (x) = u(xB)
X
J h(t) cosh a(xt)dt+w U(xa)cosh a(xa). B
If we put 0
A =
then (9.5.9)
J h(t) cosh at dt B
requires
General Concept of Green's Functions
9.6.
9.6.1.
Statement
Let Lx be a differential operator subject to boundary conditions
Applications
169
and LX f(x) = h(x) be the differential equationwhere h(x) is a given function. We recall that the method of Green's function(3) provides the formula for the solution of the'differential equation in the form of an integral
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 14 ' Ltg(x,t) = 6(tx). 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 (x 5 8 by Ix. The symbol Jx denotes a neighbourhood of Ix and the operator L(x,d/dx) of order N 2 1 is defined by 01
L(x,d/dx)f(x) = k(x)
*
f(x)
N
+ 1
an(x)f(n) (x) n=l where k(x) and the an (x) are given functions and the an(x) are n times continuously differentiable on Jx but, may vanish on Ix. (9.6.1)
The problem which we want to solve is the following: Find the function f(x) which is N1 times continuously differentiable 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 N1, (9.6.2) is equivalent to L(xlDx)f(x) = h(x) , x
(9.6.4)
E
IX f
in the sense of distributions. Put
H
B =
h(t)dt = <x(it),h(t)>
CL
where
x (It)denotes
the characteristics function of It.
Now let G for the two variables x and t, be a distribution in Xft which is zero on IRt  It and which satisfies the ID; x ID‘ t equation
and the following two conditions. n n 1. on Jxf :D Gxft = yt(x) where y t (x) is continuous at x for almost all t and wn (9.6.6) X(It)f 0 I n I N1 y n (x ) = t n 2. on J we have G = D:r(x,t) , where (a/ax)”r(x,t) , 0 5 n I N1, X x,t are continuous for almost all t and some positive summable function p (t) majorizes all their moduli on It. n
Then (9.6.7)
f(x) =
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.
a
Then we say that G 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 Proof. The general solution of 
171
depends on at. least N arbitrary functions tnat will be determined with the help of conditions 1 and 2. According to (9.6.7)
(9.6.5)
and (9.6.5), we have
which implies (9.6.4) and hence (9.6.2). The derivatives f (n)(x), n < N1, are continuous because
I
an r(x,t)h(')(t)dt a axn which is a continuous function at x. = (l)q
In addition,
an r(x,t) a xn
n
= Dx I'(x,t) because of continuity and B
f In) (x) = (  l ) q I$I'(x,t)h(q) (t)dt a
= = 'DE
that
f(n)(~n) = = (9.6.3)
DZ r(x,t),h(t)>
Gxlt,h(t)> = .
Therefore, we have by (9.6.6)
and the equations
= 0 and K1
= 0
if a 2 0.
Consequently,
Applications
(9.6.23) =
Ik
a+lI
x
177
0 and null if a 5 0. We observe that if b longer defined for x > b. satisfies the equation (9.6.24)
a Fp
I
X
E
[a,B] the integral in (.9.6.22) is no But we verify that for a > 0, (9.6.23)
f(u)du + (xb)f(x) = h(a), x
E
Ca,f31.
OD
If a
< 0,
(9.6.24) is satisfied by
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 of the simpler or more well known examples of the use of the Laplace transformation in solving partial differential equations in order to show the mechanics of this method to the reader. Recall that the Laplace transform of the summable function f (t) is the function of the complex variable p defined by 00
lLpf(t) 9.7.1.
=
0
f(t) edt.
Diffusion of heat flow in rods
The rods considered here are homogeneous and of constant section(6). There is no radiation. We take thermal conductivity
k = heat capacity by unit of length'
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
178
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
a at u(x,t)
(9.7.1)

k
a2 U(X,t)
ax2
= S(x,t), t > 0,
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 P the Laplace transform to (9.7.1) we have (9.7.2)

Suppose that u(x,t) and a u(x,t) are continuous with respect a ax u(x,p) are also continuous. Since it is to x; hence u(x,p) and 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
ax
A
2 2 1 IF u(x,p) = (4r ky +p) IF C a(x) + IL z(x,t)l. Y Y P A
Hence by means of the inversion of the Fourier transformation
Also, by the inversion of the Laplace transfornation,

(71 u(t) t112 ex2/4kt [a(x) A(t)+~(x,t)] 1 2 m 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 1 fBds .(XC) 2 /4kta (9.7.4) u(x,t) = 2 K t a (9.7.3)
ii(x,t) =
;;

+for t > 0.
l f t dw 2 G O
W
f
1/2 e52/4kw
 d5u

S(Sx,wt)
Applications
179
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(xX) 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)
(9.7.8)
u(x,O)
=
= 0
a(x). 0 < x
0, m
which satisfies ( 9 . 7 . 6 ) and ( 9 . 7 . 8 ) . ( 9 . 7 . 7 ) it is sufficient that
In order that u(x,t) satisfy
Hence g(x) must be antisymmetrical 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)
a(x)
=
an
2

an sin nn X L n=l
1
with =
X J La(x) sin nn L 0
d ~ .
rao
Chapter 9
BY Putting (9.7.10)
in (9.7.9)
we obtain
n=11ancos nn L

X
0)
e'
2
/4ktsin nn E dg:).
a
The last integral is null by the antisymmetry of the integrand. On the other part we have


IL1/4kt
(X1'2~OS
= 2 (nkt)1/2en
2 22 II
L kt,
and (9.7.11) yields m L ktsin n* X n 2 n 22 1;' t > 0, u(x,t) = 1 an e n= 1 which is the famous solution of Joseph Fourier,
(9.7.12)
9.7.1.3.
Rod heated at an extremity
The The rod of length L is initially at the temperature Oo. 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
U(X,P) = f(P)
s h x m sh L
m
Hence by the inversion of the Laplace transformation, we have
Applications
(9.7.14)
u(x,t) = f(t)
*
181
E(x,t)
where
1 sh XE(x,t) = lLt sh L
m
Since s h x Jp/k=ac h x Jp/k sh L Jp/k ax sh L m we have (9.7.15)
E(x,t) =
a El(xIt)
where El(x,t) = IL1 ch x Jp/k sh L
m
which can be obtained more easily than E(x,t).
=
LX U(t)e xL/4ktB 2( 2nikt
I
AlSO,
we have
L' 71 ikt
where 8 2 (. I .)is a Jacobi elliptical function (Erdelyi (Ed.) [l] , V01.2, p. 355). With the aid of transformation formulae associated with Jacobi's thetafunctions (Erdelyi (Ed.) [l], Vo1.2, p. 370, formula 8 ) , we deduce
Consequently, making use of (9.7.151, m
E(x,t) =
L
(Erdelyi (Ed.) c21 , Vol.1,
U(t)
1 (l)"+'ne"
2 2 2 X kt/L sin nr L
.
n=l p. 258, formula 31) , and finally by(9.7.14)
Chapter 9
182
we have (9.7.16)
u(x,t)
Let f(t) =
u
=
2nk
m
1 (1in+ln I
n1
L
t X n2n 22k(tw) L dw sin nnf(w)e L’
0
t)T where T is a constant.
Recall that
n2 NOW, from (9.7.161,
we deduce
m 2 2 2 X u(x,t) = T~x + 2T 1 (1ln en n kt/L sin nnL n=l for t > 0, 0 2 x L.

(9.7.17)
(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
g(x) I respectively. will be made below.
183
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 P to (9.7.19)
2 2 c DX U(X,p)p2 i(x,p) = p,a(x)

E(x), x
E
IR,
The equation
has a fundamental solution of the form() Hence (9.7.19) has the solution (9.7.21)
:(x,p)
=
1
[
U(x) eXPiC+U (x)exp/c 1.

kc u(x)exP’c+u(x)exP/C
+ c,(p)
explc
+
1 * a(x)
c2(p) exP/c
.
The functions C1 (p) and C2 (p) must satisfy (9.7.20). these are null if
We find that
E(x) = E(x) , E(Lx) = i;(L+x); that is a(x) and E(x) are to be defined as:
a(x) = A(x), and antisymmetrical periodic function of period
(9.7.22)
2~ which extends a(x) ,
E(x)
=
B(x), an antisymmetrical 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 + [U(x+ct) 2c
+
U(x+ct)l
*
B(x) for t > 0
184
Chapter 9
or u(x,t) = z 1 [ A(x+ct)+A(xct)
(9.7.231
+ 1
]
U(X+Ct)+U(X+Ct)l*
When b(x) is an integrable function, we get d'Alembert's solution:
$ [A(x+ct)
+ A(xct) 1 + L XjctB(S) dc.
(9.7.241
u(x,t) =
(9.7.25)
u(x,t) = 7': 1 A(x+ct)+A(xct)l
2c xct 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
+1 CB1(X+Ct)B1(XCt)]. 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
.
a L 9 v(x,t)c2 a_ v(x,t) L
at"
= 0,
axL
a v(x,O) = 0, ;i"s v(x,O)
=
0,
' 0,
v(0,t) = v(L,t) = 0, t
This system is transformed by the Laplace transformation into
$ n
P2* V(X,P) A

c
j(x,p)
=
0,
A
v(O,P)
=
V(LIP) = 0 1
*
for which v(x,p) = 0 is the unique solution. v(x,t) = 0. It is easy to see from (9.7.25 respect to t of period
that u(x,t) is periodic with
T = 2L C
Hence, we conclude
*
Indeed, if n is a positive integer we have
Applications
185
u(x,t+mT) = 1 [ A(x+ct+2mL)+A(xct2mL)lt~ 1 [ B1 (x+ct+2mL)]
 B1 (xCt2mL)
= u(x,t)
,
because A(x) and B1 (x) have period 2L. 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. If we consider the problem 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
.
i"
hF,
a (x)=
O(X<X,
where the derivative is not continuous in a neighbourhood of 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.
x,
Let A(x) and B (x) be periodic antisymmetrical distributions which are represented by the Fourier series m
A(X) =
1
n=l
(9.7.26)
B(x) =
an sin nn 5 L
,.
1 b sin n L n=l n
?I
X L
in the sense of distributional convergence, where
(9.7.27)
2 L an = i; 0 a(x) sin nn f dx, 2 L X b(x) sin nn i; dx. bn = 0
By putting (9.7.26) in (9.7.24) we have OD
ct 1 u(x,t) = 1 c a cos nnL + b n nsin n nLG I sin nn 5 L 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. (9.7.28)
& $
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
Chapter 9
186
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, Then a(x) = 0 and b(x)=IG(xX) where I measures the intensity of the impact. We have 0 < X < L, at the initial instant. W
6 ( x + x + ~ ~1L ) 1 c 6 (xx+z~L) n=w which is an antisymmetrical distribution that is the derivative of
B (XI = I
f.,
B1(x)=
(2n1)L < x < 2nLx
1. 0,
2nLX < x
0. U
X
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
j=1
71 2+,
,
z +4j
and
5' 7
Z
2 = cos 2jx+, z +4J
which yields the result, (9.9.3)
Also
logisin
XI+
=

 j=1 1 cos 2jx+ .
(log~)~(x)
T
J
Applications
log(sin x (
(9.9.4)
m

=
1 95
log 2
cos 2 j x
 j=l1
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 distributions. It follows that by differentiating (9.9.4) term by term, we obtain m
FP cot gx
(9.9.5)
1
= 2
sin 2jx
j=1
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. Changing x to x + n / 2 in (9.9.5),
we obtain
(9.9.6)
Also, differentiating (9.9.5)
and (9.9.61,
(9.9.7)
FP l/sin 2 x = 4
(9.9.8)
~p l/cos 2x = 4
we obtain
m
1 j cos 2jx j=1 co
1
(1)J j cos 2jx.
j=1
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
(9.9.10)
~p l/cos x = 2
1
j =O
sin ( 2 j + 1 ) x
m
1
(1)jcos
(2j+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 m
00
2
1
sin 2jx+ and 2
1 sin
(2j+l)x+
j=O
j=1
because these series are not convergent in the sense of distributiors. But differentiating (9.9.3), we obtain m
(9.9.11)
FP cot gx+
=
(log 2 ) 6 ( x ) + 2
1
[
cos 2 jx+
j=1
Moreover, we have (see Lavoine lLFp l/sin x+
= log 2

121, p. 7 9 ) m
1
J=o
Hence, we conclude by means of inversion
2
2
.
 ?3 6(x)1.
196
Chapter 9
or explicitly m
FP l/sin x+ = (log Z)S(X)+Z 1 [sin(zj+l)x+ j=O Asymptotic expansions
(9.9.12) 9.9.4.
1  m6(x)].
The Laplace transformation also permits us to find asymptotic expansions. In this section we shall present these results. Let Re z > 0. function
Ei(z) denotes the complex extension of the
Ei(%) =
I
m
u
+u,
x > 0,
X
which is called the "Exponential integral". We have (see Lavoine c 2 1 , P. 66)
eZEi(z)

log
cz
= ILFP
c m1
l
+
or (9.9.13)
z
e Ei(z) = ILFp(4) X+
+'
Let i ( z ) =
eZ Ei(z)
If we put
we have from (9.9.13) A
A(z) But, when 1x1
=
0
(see E r d e l y i (Ed.) C2l
, Vol.1,
p. 2 8 9 ( 1 ) )
Put
where
As
x
+
O+, w e have
Hence, a c c o r d i n g t o S e c t i o n 8.11.1 of C h a p t e r 8 ,
&z)
"
(1)J+l ('1 2 j + 2 2'
Z
v 52
, Re
j!
Consequently, w e deduce t h e a s y m p t o t i c e x p a n s i o n
f o r l a r g e v a l u e s of R e z .
z
+ m .
Chapter 9
198
9.1O.Derivatives and AntiDerivatives of Complex Order For two centuries noninteger 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 transformation of pseudo functions. 9.10.1.
Definition by the Laplace transformation
Let Tx Section 8.5.6
E
ID; be Laplace transformable. Then according to of Chapter 8 and 7, if n 8 IN and T ( z ) = ILTx, we have A
1 n
IL z
zlZn
T(z) is the derivative DnTx of order n of TX ' A
.
T ( z ) is the antiderivative (in ID;)
D"Tx
of order n of Tx. We shall generalize these results by defining differentiation of complex order v and define it as (9.10.1)
D'T~
1 v A = IL 2 T ( z ) ,
Y v
E
c.
When v =  v ' where Re v ' > 0, we must say that D"'Tx is the antiderivative (the primitive) in I D : of order v l . We deduce from (9.10.1) that
i
DOT
(9.10.2)
= T DADVT = D'+VT
D'D'T
and (9.lo. 3 )
= T
D'(T*s)
= (D'T)
*
s = T * (D's)
which are the basic relations of differentiation. T o show these relations it is sufficient to remark that D xT ' D
=
IL1 z A z v TA( Z ) = IL1z x+v*T(Z)
and D'(T*s)
= ~ ~ z ~ i ( z ) i =( ~z n) ~ '  z ~ i ( z*) s. l
In order to make DvTx explicit, put
199
Applications
W e have f6 ( n )
(x)
(9.10.4)
(XI
i f v = n s I N
Fp x i V  ' / F (  v ) v1
i f v > 0, v k
,[ =
/rtv)
x+
and ( 9 . 1 0 . 1 )
can b e o b t a i n e d by a c o n v o l u t i o n : D ' T ~ = 6 i V ) ( x )* T ~ , Y v
(9.10.5)
E
c.
IN and R e v > 0 , one can w r i t e v = n+a, where n E IN and a1 n+l R e a < 1, and w e have a c c o r d i n g t o ( 9 . 1 0 . 2 ) , DVT = D D T.
If v
0 2
R e v < O
j!
(Here D"'l p l a y s t h e r o l e of D V and D n + l T p l a y s t h e r o l e of T.) Consequently, by (9.10.5)
*
= 6 (a1)
DvT = Da'Dn+lT
+
Dn+lT
which y i e l d s t h e r e s u l t a c c o r d i n g t o ( 9 . 1 0 . 4 ) ( 9 .lo. 6 )
a* Dn+lT X.
1 DVTx =
x+
I f T i s r e p r e s e n t e d by a s u f f i c i e n t r e g u l a r f u n c t i o n f (x) having
s u p p o r t i n [ a ,  [ , w e have t h e p r i m i t i v e of o r d e r v ' : (9.10.7)
(x) =
f('I)
l
I X(xu)
Vl
r v i a
'f(u)du,
Re v'
> 0.
On t h e o t h e r hand, i f f ( x ) i s c o n t i n u o u s 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 c o n t i n u o u s , w e have a c c o r d i n g 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 o r d e r
s t u d i e d by L i o u v i l l e C11 and
, Riemann,
[Zl
Another e x p r e s s i o n f o r S:"'
R i e s z and o t h e r s .
( x ) can b e o b t a i n e d as f o l l o w s .
S i n c e f o r R e z > 0 and h > 0 , zV = l i r n h  v ( l  e  h z ) v = l i m h' h+O
h+O
1
m
(1)'(i)ephz
p=O
where
m
(9.10.9)
("(x) &+
= l i m h'
h4
1
p=O
(1)P V
G(xph).
chapter
200
9
Also, by (9.10.5) m
DvTx = l i m h' 1 (l)P(i)TxphI T h+O p=O
(9 10.10)
E
ID;,
which is a result corresponding to those of Grunwald (1867) cited The formula (9.10.10) shows by Lavoie, Osler and Tremblay Ell that the operator Dv is analogous to the operator )':tI of Bredimas If v = n is a positive integer, then Ell (see also Section 9.10.3). the sum in (9.10.10) contains only ( n + l ) terms.
.
Moreover, for Re z > 0, we have z v = lim h"(ehzl) = lim h V evhz (lemhz)"
h+O
h+O W
1
= lim h'
h+O
(l)P(i)e(vP)hz.
p= 0
Hence, we conclude 03
(x) = lim h"
)":6
h+O
1 (1)'(i)
p=O
6 (x+(vp) h) ,
and according to (9.10.5)
D " T ~= lim h+O
(9.10.11) for Tx
E
I D : .
If TX is represented by a function, we get a formula of Liouville 111
.
9.10.2. Examples
1. If Re
A >
"x+A
(9.10.12)
If vA1
1, then according to (9.10.1)
= n, n
E
=
, we
have
~'~(A+~)z~+'~
IN, we deduce
:X'D
=
r (!,+I) 6 'n) (x);
and if v31 # n, (9.10.12) can be written as 1
r(x+l)
D x+ = r ( A  v + l )

z
&(A"+l)
Hence, we conclude according to (8.10.3) v A
D x+ =
r(A+l)
AV
FP x+

)i+v1
of Chapter 8 ,
Applications
(Fp is not needed if Re(Av) Dnx:
>
1).
In particular
,...,(Antl) FpX+A w .
= (A1)
When Re X 5 1, A+l # n, wA1
g!
IN, (9.10.12) leads to
r ( ~ + i ) ~ p xA  +W
D'FPX:
201
= r(Av+l)
.
A
The process also gives DWFpx+ in other cases. 2. If w # 1,2,3,...,
we have
I L F ~ X  ~ / ~ J  ~ (=~ J ~;) z vlel/z = n1/2zv1/2J7;/z el/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  ~ / ~ c2 of i s+ = ~
/
Hence, we have according to (9.10.1) (9.10.13) Putting
,
,
v/2 w =
z
Jw/2 1 n + 7, n
(2J;;) = n1/2Dv1/2 cos 2 G , for 0. Jj;: E lN, we obtain the well known formula
2
cos 2& , x > o . (2&) = n 1/2xn/2+1/4 Jn1/2 dxn & We ask the reader to compare these formulae to those given in Section 9.10.3. (9.10.14)
3. Abel'S integral equations. These are the equations of the type X
I (xu)'lf(u)du ma
= h(x), x > a, Re w
where h(x) is a given function having support in Ca,m[. equation is equivalent to D'f(x)
=
h(x);
hence, according to (9.10.5)
we have
f(x) = DV h(x) = )":6
(x) * h(x).
If w = n is a positive integer, we obtain f (x) = h(n) ( x )
.
t
0,
This
202
Chapter 9
Here we suppose that h(x) is a (n1) times continuously differentiable function for x > a. If v = n+a, 0 5 Re (9.10.8)
f(x)
a
2 1, n
1
E
v1
IN, we obtain according to
*
=
r(j)Fpx+
=
‘r(l.7x(xu) a
h(x) h(n+l) (u)du
provided that h(x) is ntimes 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) = ehtw(x,t), the system (9.10.15) becomes
(9.10.16)
w(x,O) = 0, w(0,t) = Teht, t > 0.
Using the 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 axa w(x,t) + D;”w(x,t)
w(x,O)
= 0,
=
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

T w(0,z) = 2h whose solution is e X G w(x,z) = T 2h A
A
Hence, by means of inversion u(x,t) = Teht f’(2h) lexJk/Z ;
, VOl.1,
and by Erdelyi (Ed.) [2]
p. 246(10),
r
u(x,t) = ” {eeX h/k erfc c
(9.10.17)
z
X GI+ eX & 2
G
61 1 2G t which is the result given in Carslaw and Jaeger C2l I where  eu2du erfc x = 2 s 1/2 erfc
[
2+
X
5. We find an application of the derivatives of complex order in Lavoine C31, pp. 439441 and 648650.
9.10.3.
Extension of the definition let us consider the distribution 61v)(x)
As f o r (9.10.4),
defined by
I:
6 (n)(x)
(9.10.18)
6:”)
(XI=
ifv=nslN
ivsFpxIV’/r(v)
if v # n and Re v > 0
ivs xZv’/r(v)
if Re v
4
0.
where we use the notation
1x1’
if x
c
o
if x > 0. The formula (9.10.5) induction: (9.10.19)
D_VT
then suggests the following definition by
= )’:6
(x) * T.
DI is exactly the operator
IIv
of Bredimas C 11.
Chapter 9
204
Of course in (9.10.19) we suppose that the distribution T Possesses some properties which permit convolution. If T is represented bq' a locally summable function f(x) such that xOf(x) + 0 for some real number n as x + a, then we have for Re v > and v ,d IN the equality Dl)f(x) =
(9.10.20)
ivn
m
r o X(ux)'lf Fp
(u)du.
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 m ivn v1 eaudu = Fp (Ux)
,h
D:
=
X
eivn ax v1 F j JL,FpX+
eivn a v e  ~ ,
Hence, we conclude
v ax = eivnav De
(9.10.21)
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 Hence, f o r
a >
eiux = eivn/2 wveiwx
0,
D_*sin wx = m a sin(wx
+
a+)
(9.10.22) cos wx = w'cos
: D
(x +
a$).
These results are consistent with those of Zygmund C11 3 . The operator : D
.
a l s o permits us to write formulae concerning
special functions. We shall give a few examples. Bessel function: (9.10.23)
( 2 ~ ) ieivnv1/2xv/2 Dv1/2 sin 2&, V

J;;
+ ,
E C
Applications
205
1 we get the well known formula and by putting v= n+, 2
We ask the reader to compare (9.10.23) to (9.10.13). Cylinder parabolic function (or Weberfunction) : ivn x2/4 ex2/2 (9.10.24) Dv(x) = e e
DI
Kummer function: (9.10.25)
F(a,c;.l/x) = ei(ca)n r(c) XaDac
X
c ellx,Re a > 0.
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 Stakgold C11 (4) Schwartz C11 , Chapter V.6,FriedmannI B.Cl1, Ohapter 3 and VoKhacKhoan [: 11 (5) note again that the majority of authors denote by this term a kernel having different properties. (6) see also 4. of Section 9.10.2. (7) here as usual U(t) is the Heaviside function.
.
.
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 transformation 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 'r 1+a for which I jt't f(t)dt( is bounded by a number which is indepent' dent of t' and t" with t" > t' 2 A.

Examples. If g(t) is null for t < 0, locally summable, and if there exists a number a ' > 0 such that tr+a'g(t) is bounded as t + then g(t) E E(r). If g(t) is such that
is bounded independently of t'
m,
Chapter 1 0
208
m
E(r).
and t", t h e n g ( t ) belongs t o 0
Since
2
0
I s i n t l d t and
t l s i n t l d t a r e d i v e r g e n t i n t e g r a l s , consequently, s i n t+ and 0 t ( t s i n t 2 ) + a r e n o t summable. But, s i n t d t l = [ c o s t'cos t " 1 5 2 t 2 1 and I t s i n t d t I = zlcos tnI2cos which e n a b l e us t o conclude
I
t'fizl
t' 2 t h a t s i n t , and ( t s i n t ) + belong t o E ( r ) . A l s o , f o r any r e a l number v # 0 , (tW'sin t V ) + and ( t v  l c o s t u ) + belong t o E(r).
I f f ( t ) E E ( r ) , t h e n f ( t ) F E ( r ' ) , r ' > t. Thus E ( r ) c E ( r ' ) i f r < r ' . I f f ( t ) c E ( r ) , t h e n i t s p r i m i t i v e s belong t o E ( r + l ) . Let f ( t ) E E(r). r , which w e d e n o t e by
Then i t s S t i e l t j e s t r a n s f o r m a t i o n of index
$if(t) , i s d e f i n e d as m
(10.1.1)
g f f ( t ) = I f ( t )( t + s )  r  l d t , 0
)arg s )
< 0,
The e x i s t e n c e of (10.1.1) can be
where s i s a complex parameter. a s s u r e d by t h e Abel's r u l e .
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 w i t h f o ( t ) = j'f(x)dx f o r t > 0 , t h e n f o ( t ) is a continuous f u n c t i o n 0
and such t h a t trl+af o ( t ) i s bounded a s t
+

f o r any
a
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 Arl+a
t I f (x)dx =
A
A
j t:rl+a
f
(x)dx.
Stieltjes Transform
209
Finally, we obtain trl+a fo(t) which is bounded as t
+
m.
t"rl+a trl+a fo(A)+I/ x f(x)dxl t Hence the proof follows.
We, therefore, have from the above result that
(10.1.2)
f(t)
=
& fo(t) = Dfo(t) ,
and according to (10.1.1)
, m
(10.1.2
)
10.1.2.
The space Jr'(r)
Let S'(r) (10.1.3)
j%if(t)
=
(r+l) f fo(t) (t+s)r2dt. 0
denote the space of distributions Tt of the form Tt = Dk' 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 trk'+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)
$iTt = =
m
rk'ldt
r (r+
where s is a complex variable such that larg s I
.
One can easily verify that (10.2.1') exists. To do so, we see that the right side of (10.2.1') exists since (t+s)r1 coincides on the support of B with some functions of ID. We remark here that in all t ' the cases, if $: Tt exists, then exists for r' r.
$3, 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 derivative of a tempered locally summable function. Hence, by Theorem 8.1.1 of Chapter 8, we conclude that every element Tt E JI'(r) is 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 distributional setting of Laplace transformation in 55' (r). This section contains the following result.
Tt
E
Theorem 10.3.1. If Re (r), then we have
s > 0
and if we put F,(s)
=
$5 Tt with
JI1
(10.3.1)
Fr(s) = $iTt ==
1
lLsU(x)xrlLxTt
where ~ ( x ) is the Heavisi.de function and II, denotes the Laplace transformation.
211
Stieltjes Transform
L
k' Proof. Since Tt = D fl(t), we have ILxTt = xk'G(x)
F(x)=
with
Also, we have
lLxfl(t).
I
ILsU(x)Xr lLxTt =
(10.3.2)
m
sx r+kle x F(x)dX
0
=I
m
0
m
I e(tcs)x xr+k'fl(t)dt dx.
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 imposed on f1 (t) in Theorem 10.1.1 we may infer that ttk ' +a fl(t) is bounded as t + which enables us to find out two numbers N and M such that If,(t)
I
0 ( 0 may be dependent on r) such that 1s1@IFr(s) I is bounded as 181 + m in the region larg sI < n. Proof. We have dm F,(s) asm
=

m
fp)
dk' rm 1dt 1 m+k' r (r+l+m) 1 T(t+S) r (r+l) 0 dt m r r+l+m) rm1, (1) i$yq~ O f we have I t+eibI rkvl< 2rk' lWrkll
r r+ 7 r (r+n) Y(r+nl,s) r (r+
after replacing r by r+n1.
Moreover, we have
where Sn = 1
+
1
+...+l+n' 1
and l'(r+n)
(10.5.11)
Therefore, according to (10.5.10)
grS
CFptin1 =
where $(1)+Sn=$(n+2).
rn
and (10.5.11) we have
nlr
Hence (10.5.6)
r+n) C Y(r+nl,s)+Sn srnl is also obtained.
215
Stieltjes Transform
10.5.2.
Examples when Tt
E
JI'
We have for a > 0 (10.5.12)
&Is[u(a;t)]
s
= log s+a ;
a
1 a+s , if a 2 0;
(m)l
= m!(a+s)ml,
(10.5.13)
$,
[ 6 ]=
(10.5.14)
's
[&a
(10.5.15)
$,
(Tat) = gaS(Tt) :
(10.5.16)
gs
[
t1'2
eat~= n s 'I2
if a 1. 0, m = 0,1,2
,...:
eas erfc(af sf)
where Erfc is the complementary error function. Proofs. 1. (10.5.12), (10.5.13) from the definition (10.2.1). 2.
(10.5.15)
and (10.5.14) follow directly
follows from the definition .
decreases more rapidly For (10.5.16) we first note that t' 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 3.
nx[t'
eatl
= r
and
zsc (x+a)' I =
4 (x+a)+
'
eas Erfc (af s') s'
so that we immediately have the formula
gs
[ tf
eatl = n s4 eas Erfc (a'
s').
Extensive tables of the Stieltjes transform are given in Erdelyi (Ed.) C21 , Vol. 2, pp. 216232.
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
Chapter 10
216
10.6. Inversion
In the preceding sections we have derived certain results concerning,the Stieltjes transform $f of a distribution T Almost t ' r 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. When $: is prescribed any formula enabling us to derive the form of the distribution Tt is called an inversion formula for the Stieltjes transform. We denote the inverse Stieltjes transformation by ($r); I I and it is defined by $E(Sr);l F(s) = F(s). Now we state the main results of this section and our discussion of these results are entirely equivalent as indicated in Lavoine and Misra C31. Theorem 10.6.1 (Inversion theorem). If the function F(s), in the domain of the complex plane larg sl < r , s # 0, satisfies the properties ( 4 ) (i) F ( s ) is holomorphic and (ii) there exists a number f3 > 0 such that Is181F(s)I is bounded as 191
+
1
then the inverse Stieltjes transform of F(s) exists and is unique. Set f ( x ) = xer lLil F(s) , where the function f (x) can be continued analytically in the half zplane Re z > 0 and let it be denoted by f(z). Further, if we put Tt = ILL1 f ( 2 ) , T being a t function or a distribution having support in C 0 , m C , then we have ($r);l~(s) = r(r+i)Tt.
(10.6.1)
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
0
0
h(x) esXdx, Re s > 0.
Set g(x) =
6; s'F(s)
f (XI
xrg' (x).
,
and =
Then (10.6.2
I
h(x) = xr f(x)
)
and
Let 0' = for larg
nq?
SI
el1 =  e l ,
17 < */loo. Since F(s) is holomorphic theorem gives
o
0.
(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(s) =
I
0
.
Now,
xr(lLxTt) esxdx.
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.
1 1 v = T x v, 1 ILxF(s) = ILxs
r (v
Then we have R e v > 0.
Also , according to (10.6.1) we obtain
In other words
($)
1 B(v,rv+l) 1 (r+l)(r+2).
;w=

tt"
if Re v < 1
..(r+n) 6 (n)(t) if v=r+l+n, n
EN
1 rv in all other cases. v,rv+l) Fp t+ 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
z+
i s not S t i e l t j e s i n v e r s i b l e because IL1 (s2 +1) = s i n t+ and zrsin t i s not Laplace i n v e r s i b l e s i n c e n e i t h e r c ' nor k e x i s t f o r which I kec ' R e 2 l s i n zl i s bounded(6) a s z + m i n any halfplane
I
Re z > 6 .
Therefore, Theorem 8.7.1
of Chapter 8 i s applicable to zr 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 d e f i n e (10.7.1)
(see Section 10.2). Throughout t h i s s e c t i o n w e r e s t r i c t ourselves t o r e a l and p o s i t i v e s i n t h e d e f i n i t i o n of t h e 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 s e c t i o n r e l a t e t h e behaviour of a S t i e l t j e s transformation a s s approaches S zero o r i n f i n i t y t o t h e behaviour of Tt a s t approaches zero o r infinity.
Theorems of t h i s n a t u r e a r e c a l l e d Abelian theorems
( s e e Misra E l 3 ) . W e study two t y p e s of Abelian theorems. The f i r s t theorem r e l a t e s t h e asymptotic behaviour of Tt as t f O+ t o t h e This r e s u l t i s r e f e r r e d t o as 'behaviour behaviour of $: a s s + O+. of t h e transform near t h e o r i g i n ' ( i n i t i a l value theorem) s i n c e it
i s t h e i n i t i a l behaviour of T t h a t i s considered. The second t theorem discussed i s c a l l e d 'behaviour of t h e transform a t i n f i n i t y ' ( f i n a l v a l u e theorem)since it relates t h e behaviour of Tt as t + Other proofs of t h e s e theorems can t o t h e behaviour of $f as s + m. a l s o be found i n Lavoine and Misra C21

.
10.7.1.
Behaviour of t h e transform near t h e o r i g i n
W e now s t a t e our main theorems concerning t h e behaviour of t h e
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 t h e o r i g i n .
6.4.1
Theorem 10.7.1. of Chapter 6,
(10.7.2)
with R e v (10.7.3)
r , v # n1,
B,r
E
JI' ( r ) and i f i n t h e s e n s e of 1, S e c t i o n
mp(tvlogjt)+as t
T~
(10.7.9)
If Tt
E
JI' (r) and if
Tt = At"1og't tl > 1 and 1 < Re v ii$~~
r r then
~ ~ ( v + l ~ r  v ) s ~ js~ as l o gs
+ m
I
larg s l
< n/2.
Stieltjes Transform Proof. According to Theorem 8.11.3 r lLxTt

221
of Chapter 8, we have
A(l)jr(v+l)x rvllogjx 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. If DmTt = VtI m Corollary 10.7.3. property (10.7.9) I then we have $iVt
(10.7.9)
., Ar (;;ii;;r+mw)
E
IN I aqd if Tt has the
svrmlog's
as s
+
mI
larg s l + .
This is a consequence of formulas (10.5.4) and (10.7.9). 10.8. The nDimensional Stieltjes Transformation
The results obtained in the preceding sections enable us in this section to give the structure of ndimensional Stieltjes transformation. Here, we use the following notations and terminology (see also Section 4.5 of Chapter 4). As customary we denote the point of an ndimensional real space lRn by X = ( X ~ ~ X ~ ~ . . . ,and X ~ )by Qn(0) we mean the set of point X such that all the x > 0 for j = 1121...,n. The point of an nj dimensional complex space Cn is given as s = ( S ~ , S ~ ~ . . . ~and S~) k = (klIk2,...rkn)I k = kl.k 2.....k n where kn is a nonnegative
integer.
The symbol Dk stands for 'D
=
Dkl :2....D> x1 x2
n
in the
distributional sense. By r we mean r =(rlIr2r...Irn) and where the complex numbers r are such thar Re r > 1. r = r1 r2....r n j j 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.

j
10.8.1.
The space J;(r)
By J;(r) we denote the space of distributions in n variables Tx which can be expressed in the form T = D i;f(X) where f(X) i s a locally X summable function in IRn and zero outside of Qn(0) such that forany conventional positive number a = ( a l I a2 1 . . . , a n1 I f(X) = O( 1x1 r+ka1 as 1x1 m for each k E INn. +
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 multiindex) r for s E A which we denote by $ETx is defined as BETx
(10.8.1)
=
where A(r,k)
.
A(r,k) I.. .If (X)(xl+sl)rlkll.. (x +sn)rnknl n 0 0 axl ax2.... &n m
=
..
m
..
r (rl+kl+l). r (rn+kn+l) r (rl+l) r (rn+l)
....
If k=(O,. ,0) and TX = f (x), then we obtain ordinary Stieltjes transformation in n variables: m
r
(10.8.2)
$,f (X) =
I..
0
...0If (XI (xl+sl) m
rll
... (xn+sn)rn1 dxl dx2...
axn.
If TX = BX (1.e. any distribution having bounded support in Qn(0) ) then we get $=B = s x x' x1 1
(10.8.3)
s
E
A.
If TX E JA, then we obtain Stieltjes transformation of index zero such that m
(10.8.4)
m
$STX = A(0,k) I.....'If(X) 0
0
....
k21 (xl+s1)kl1(x2+s2)
(xn+sn)knl dxl dx2..
where A(0,k) 10.8.3,
=
r(kl+l) r(k2+l)
,....r(kn+l).
. dxn
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
...,tn)
where t = (tl,t2, such that
('
tl',ti
w(r;t)
=
E
A
and if TX
E
JA(r), then we have
(all real t.) and w(r;t) is the function 1
,...,trnn if all the t. are positive 3
elsewhere,
Stieltjes Transform
223
and IL as usual denotes the Laplace transformation in n variables. r (s.+x.)t 3 J j, Proof. Since (s.+x.) rj1 = r r + ) 0, 7 j we have n 1: .1 xtst (10.8.6) n (s.+x.) 3 = j=1 3 3

where
..
+ x2t2+. .+xntn s t + s t +".+ sntn. 11 2 2
j;F = Xltl
st =
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):

Ss~xT= eX,CI>
xtst
=

st e >
st w(r;t),ILtTt e 
<w(r;t).ILtTX, est> = Y(r)IL S w(r;t)ILtTX.
Here, we remark that 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). Theorem 10.8.2.
If Tt
where r(r,m) (ii) $:
=
E
J;(r),
then we have the following:
,...,r.+m, rj+18....,rn); 3
(rl,r2
is a holomorphic function of s in the region s
.
E
A;
(iii) there exist n positive numbers ( B1, B2,. .p 8,) such that 61 B2 in the region SEA. lsll Is2( ....?/snlBn$: Tt is bounded as Is1 f

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 , ,sn1 In this case we call the distribution Rx as Stieltjes inverse of F ( s ) 1 and denote it by % = ( $r)x 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) = $(t1,t2,..,tn) of n real variables t such j that IL$(t) = F(s) and if there exists a distribution RX having support in Qn (0) belonging to JA(r) such that r (rl+l) r (rn+l)0 (t) I L R = (where w(r;t) is given in t X w(r;t) F(s) = RX. Theorem 10.8.1) then we have
.....
(&fr)il

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~ = IL~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 distributional setting have been studied by Tuan and Elliott [11 and McClure and Wong [11 We remark here that the present distributional setting of Stieltjes transformation may play an important role for such applications and we leave the use of this setting to interested readers
.
.
10.10. Bibliography
In addition to the works cited in the text w e should like to mention the following references which a l s o 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. But it is analytically continued to the half plane Re z > 5 > 0. This 1 yields f ( 2 ) = 0 and hence ILt f ( z ) = 0. Now the present theorem yields (#);'(l) = 0 which contradicts the fact $ z ( O ) = 0. put z = cl+iy, 5 ' > 5 and let y tend towards
a.
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 a,w CLIW using these results we formulate the distributional setting of Mellin 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 EulerCauchy 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. 227
Chapter 11
228
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 splane such that ts’f (t) is absolutely integrable with respect to t over 10,[. The Mellin transformation is an operation IS that assigns a 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 W
(11.1.1)
F(s)
=
Isf(t) =
0
f(t) ts’dt.
By the condition (b), (11.1.1) converges absolutely for all s in the strip al 0.
we obtain
.kt
0
ts+~ldt l'(s+v) ks+v
, Re
s
>Rev.
Now by analytic continuation,we have m
I 0
tv .kt
tsldt  r(s+v) , s # v, uls v2. ks+v
In the next sections, we shall generalize this transformation to functionals (generalized functions) belonging to E' which are a,u spaces that contain, among many others, the distributions of bounded support in 1 0 , m C . 11.2. The Spaces E
alu
Our study made on the spaces of base functions (see Chapter 2)
Mellin Transform enables us to construct the spaces Ea
231
in the following manner.
I W
(p,q are finite real numbers with p q) we denote the By EP,q 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
(Q(t) will be null for t < 0.)
+ tSl, so
[;
,
We set t'O t < 0,
s1 belongs to E that t+ if p Ptq
tP, 0 k
PI9
(t) =
0 and the integer K depends upon V but not on 4 . (See Zemanian C31, pp. 1819, 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
=
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 0+, then P(m,a;t) belongs tmP(m,a;t) is summable and bounded as t with q > m. to Elmlm and therefore it belongs to all ELm f
If support bounded belongs
Q(n,b;t) is a function of t which is null for t b with in Cb,mC such that t"Q(n;b;t) is locally summable and as t + m , then Q(n,b;t) belongs to Em and therefore it en' to all El with p < n. P rn
If Vt = Bt

+ P(m,a;t) + Q(n,b;t) with n
Note that eYt E E; bounded support in [O,[
> m, then Vt
E
*further all the summable functions with a belong to E;
,a'
I
.
The distributions 6 (t) and 6 ( k ) (t) do not belong to all the are not all continuous at the because the functions of E 0 alw origin.
Chapter 11
234
r 2 1, do not belong to E ' if PI" p < r because in general the derivatives of order > r1 of $ E E PI" r t if = t e+ is in E' do not exist at the origin. But Fp The pseudo functions Fp t'eTt,
P
L
=.
PI"
For real or complex z , tf, which belongs to $' does not belong (If it did then tf would belong to E' with p > Re to any E:,w. PI9 z > q which violates the condition p < 9.) For each V E E' and for each JI E I D ( 1 ) , exists and alw defines a linear functional which is continuous for the topology of m ( 1 ) ; therefore Vt belongs to I D ' ( 1 ) , the space of distributions on are spaces of the open half axis C o r m C. Hence we conclude that E ' a,w distributions. As we shall see later, the spaces E ' are the spaces a ? W of Mellin transformable distributions. 11.3.1. The multiplication in E '
a,w
If z is a real or a complex number, one can see easily that if C = R e z a n d $ E Ea 5 ,w5 then tz$(t) E Eafu.
Then its product by tZ is the element of E:ErwE Let V E E: I denoted by tZVt and defined by z
(11.3.3)
=
Y 0
For example, tSeit E Eic, m , since eYt function such that the mapping
E
E;, ?m
.
Moreover if m (t) is a
is an isomorphism of Ea+A,w+vOn Ea+A ,w+v with a+A<w+v, that is, if a then this implies that the sequence sequence($.(t)) + 0 in Ea 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)> =
,Y
$
E
E~+~,~+,,.
11.3.2,The differentiation in E'
alw
%a,w'
We can show easily that if $ E E a + j , w + j ' 3' This leads to the following definition: If Vt
E
E
IN, then $ ( j ) (t)
ELlw, then its derivative of order j is element of
E
235
Mellin Transform 'A+ j,w+ j denoted by D3Vt and defined by = (,)I
(11.3.5)
1
'
@
EEa+j,u+j.
Examples. Let forO(t(a U(a;t) = elsewhere. Then, we have for every
+
E
El
that 1
Hence, DU(a;t) = 6(ta) in Ei
(11.3.6)
I
*
Problem 11.3.1 Prove that if c 2 0 and real A
I
then
(1)
ct in E; D eTct = c e+
(ii)
D [ u (a;t)e"'j =cU(a;t)eCtecag(ta) in E;,~ and also
i I 
on $ (iii)
n
E~
*
,mt
D Cu(a;t) ecttA 1 = U(a;t)(Act)t '1 ecta'ecag
(ta) in
EiA,* 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 nonnegative integer k (11.3.7) where
denotes the dual of Malb. and consequently M' arb p. 103.) Note that the norms y k
rP19
(See Zemanian C31,
defined in Section 11.2 corresponds
236
Chapter 11
to 'p,q,k
Then tP+k+l'c,+,(k) (t)3 0 Pl9' > 0 r e q u i r e s t p+k+l,+, ( k ) ( t ) + 0 as t + O+;
L e t , + , (Et )E
of (11.3.7).
a s t + O+ f o r c e r t a i n c while f o r every e E M it i s s u f f i c i e n t t o remark t P+k+le ( k ) ( t ) P?9 + 0 i s bounded f o r 0 < t < 1. Thus, E i s a smaller space t h a n P?q M and hence we conclude t h a t El i s l a r g e r t h a n MI Also Prq' P?9 P?q'
[:
O , < t < l
g(t)=
does n o t belong t o M'
O,.'
w e have
*
elsewhere
while g ( t )
E
, because
EA
i f ,+,
E
Eo
I
I  ,
1
< g ( t ) , , + , ( t ) >=
O(t)dt. 0
11.4.
The Mellin Transformation
The r e s u l t s o b t a i n e d i n t h e preceding s e c t i o n s e n a b l e u s i n t h i s s e c t i o n 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 transformat i o n i n t h e following manner. L e t Vt
E
Ei
Then i t s Mellin t r a n s f o r m i s t h e f u n c t i o n of a denoted by lMsV and d e f i n e d i n t h e s t r i p
I W'
complex variable s S
alu
= { s , a < R e s<w} by
(11.4.1)
It%!
S
v
=
m;
, Sv
i s t h e h a l f  p l a n e R e s < n;
Mellin Transform if Vt = Bt
237
+ P(m,a;t) + Q(n,b;t), Sv is the strip m
(11.5.3)
IMs 6 (ta)
(11.5.4)
ms &(k) (ta)= (1)k (s1)(s2)
= as'

 Re v
which follows from the change of variable, y = fi t 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 S+V B(v+l,s), Re s > 0; (i) IMsFpCU(a;t) (at)” 1 = a
‘(ii) IMsFp CU(a;t)(at)ll = as1[Jl(s)+log C/a3; Re s > 0(3); (H) IMs
[U[a;t)log(at)l
=
as ~~[J,(s+l)+logC/al ; Re s > 0(3):
(iv) M s F p [U(l;t)tzllogtlAll = r(X) ( s  z )  ~
;
Re s
Re z ;
= r(A)(z~)~; Re s > Re z.
(v) IMsFp CU(tl)tz(log t)”lI
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 is holomorphic in the strip S
atw
d
(11.6.1) is valid in
theorem). If V E E l l w , then v(s)=IMsV and therefore in Sv. Also
v(s) = +
S
Pt9
and hence v ( s ) is holomorphic in the substrip
S
of P19
sa,u’
Proof. Because d t ~  l t ~  log l t ds right side of (11.6.1) exists.
E
E
a,w
if
Re
a
s c w,
the
of S Now we show that (11.6.1) is valid in each substrip S P!4 a,w (The equilities are where p and q are finite with a 5 p q 5 W. to be considered if a and w are finite.) Let As be a complex increment whose modulus 6
+
0.
We put
Mellin Transform
H = lim [ V(s+As) As 6+ 0 = lim a t ,
 V(S) 
ts'(tAsl
239
J.
Proof. Let Y(t) E ID having support I = [an, b+nl and be equal n Sine B E ID' J I there exists a number M > 0 such to t s K [a,b] that (see Bremermann [l], Section 4.4, Lemma 1, p. 30)
.
241
Mellin Transform
(11.6.6) is arbitrarily small. where of Y(t) Yy(k)
Put Bk
=
sup tcI
(t) = (s1)(s2).
I Y ( k ) (t)I n
Now, we have by the proposed structure
..( s  k ) l'kst
if t
c
In.
and if Re s > k+l, then we have
k sup Its'k'll tEI
< Is1
Bk = , e = . e is often easy to utilize the following definitions:
8 ( l o g t)/t
butions (11.8.2) (11.8.3)
Now, it
E
E,,,u.
$(eX) eX> = , Y
= , Y E E  ~ , ~ e where P x and Q have their supports bounded below by log b and e e log b respectively. G:Q
If we put $(x) = eSX with Re then (11.8.2) and (11.8.3) yield
s > a
and e ( x ) = esx with Re sew,

mspt  ILSP,,XI Re s > a, e (11.8.5) IMs Qt  ILs Q Re s < W. e Where IL denotes the Laplace transformation. virtue of (11.8.1) that (11.8.4)

Now, we deduce by
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.
(11.8.6)
MsVt
ILsPex
+
LsP
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 sLs 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 ,rl (11.9.1) <e V  x l $ ( x ) > = , r E Ia,wC e and
,
,e(emx)> = at, tr'e(t), e"x where each $ ( X Iand each e ( x ) be such that tr'$(1og t) and tr'8(t) belong to the functions space on which Vt is defined. Now, we give the main result of this section. (11.9.2)
<el%
In order for Vt to be a Mellin transformable Theorem 11.9.1. distribution in the strip S a I w l it is necessary and sufficient that should be a tempered distribution (and r E 1 a , W[ , and er% ex therefore Fouriertransformable). If v(s) = MSVt, then we have (11.9.3)
v(r+2nic)
=
rc e"xV
5
x,
E
IR.
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 < < Sp and O < 5 ' < q 5. Put

r1 qt) = t $(log t) I t
(3.1.9.4)
$(XI If $ ( x )
E
' 0,
= e (rl)x$r(ex) , x
$ then we have that for all k
tk+lPC
(k)(t) @r
+
o
IR.
E
E
as t
IN +
tk+lq~'@~) It) + 0 as t + Therefore @,(t)
o+, m,
and hence it belongs to Ea belongs to E 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 a,w by (11.9.4) and hence the first part of the theorem follows from (11.9.1). According to Theorem 11.6.2, v(r+2siC) can be majored by a m 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 trl 2nic = 2sixc >I, $(El> x' e e 2TI ix& = <e"3 Cr $ ( S ) >I> eCx = <erXV x, rX$ ( 5 ) > e = 5 e which proves (11.9.3) and also confirms the first part of theorem. = l>, Y $ e E
P'q'
To prove the existence of (11.11.1) it is sufficient to show C E when u > 0. that W ( U ) = belongs to E P'q p2 4 2 Since $ E E there exist and n u r 0 < 'I < I P'9' , and bounded functions bk(t) such that 0 < 11'
=
aU,[l> 1
= = .
Consequently, we obtain (11.11.3) 2.
1 vtL6 (ta) = ; vtIa,
By (ll.ll.l),
(a > 0).
we have
1 Y f (vLf)u = at,rf(U/t)>,
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 multiplicaPI9 tion law is the Mellin convolution and whose unit element is 6(t1). 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
11.11.2.
2 51
Relation with the Mellin transformation
With the notations being the same as that of Definition 11.11.1, let M s V = v(s) and lMsW = w(s). Then, we have
mS CWLV) = w(s)v(s) , p
(11.11.4)
= <wU
r1 $(log t ,P ht,t
log u ) > l >
Now, according to the statements given in Definition 11.11.1, we have
Hence, $,(y) which belongs to Ea ,W. given in the proof of Theorem 11.9.1.
E
$, according to the statements
Further, by making use of
Chapter 11
252
Section 5.9 of Chapter 5, (11.11.5) can be rewritten as (11.11.6)
<e'rxp
by virtue of (5.8.3)
 x , $ ( x ) > = <e'=YW , c <erXv ,,$(x+y)>~> e ee = < ( e  3 _,~*(e~~v x,$(x) > e e 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' (t1)1\ V. By repetition, we further have (tD)2V = [ t&'(t1) I\
k&'(tl)l
LV
= [t&'(tl)J12\V=K2\V
where K2 = (tG'(t1)) L2 we obtain (11.11.8)
.
Iterating this (n1) times yields,
(tD)"V = [ t S'(t1) A n L V
=
Kn\ V
where
.
Kn = (t b'(t1) L n
i.e. K is the nth power of Mellin COnVOlUtiOn of t 6'(t1)* n Ws t 6' (t1) =  s I then (11.11.4) transforms (11.11.8) to (11.11.9)
ms (tD)"V
= (l)n S~V(S)
in accordance with (11.7.9). (tD)"V
=
Also,
(1)"(m;'Sn)
we deduce from (11.11.9)
\ V.
Further, we generalize this process by putting
Since,
M e l l i n Transform
253
(11.11.10)
with
where e a c h v i s a r e a l or complex number.
# 0,1,2,3,...,
If A
we
have a c c o r d i n g t o ( i v ) and ( v ) of Problem 11.5.1,
KX = FpU(t1) r ( 1 A
in E ' ,~ .
( l o g t )A1
A
I t f o l l o w s t h a t when V c Eh
w i t h a < 0 < w, t h e n ( t D ) V i s r e p r e s e n t e d by t h e c o u p l e K t \ V and K ; \  v ~ When o a < w, ( t D ) =
KlLV.
x
And when a < w 5 0 , ( t D ) V = IZ1
I f A i s r e a l , t h e n ( t D ) 'V only i f V
E
E'
a,w
w i l l be
real
'v
V.
(and e q u a l t o KX\
V)
w i t h a 5 0 , which o c c u r s i n p a r t i c u l a r ; when V is
r e p r e s e n t e d by a f u n c t i o n b e l o n g i n g 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 S e c t i o n 11.11.1.) S i m i l a r l y , one can g e n e r a l i z e t h e o p e r a t i o n ( D t ) 'and t h a t t h e formula (1 1 . 7 . 1 0 ) s u g g e s t s t h e d e f i n i t i o n = eivn
(Dt)%
11.12.
[m;'(si)"
note again
J\ v.
Abelian Theorems I n s e c t i o n 1 1 . 4 w e have i n t r o d u c e d t h e a b s c i s s a e o f e x i s t e n c e
of v(s)=IMsV, a b s c i s s a e t h a t w e d e n o t e by a and u which a r e l i m i t e d by t h e w i d e s t s t r i p S a
i n which v ( s ) i s holomorphic.
Hence, i f an
I W
a b s c i s s a of e x i s t e n c e i s f i n i t e , t h e n t h i s i s t h e r e a l p a r t of t h e a f f i x of a s i n g u l a r p o i n t f o r t h e f u n c t i o n 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 c o r r e s p o n d i n g ( 6 ) t o a and W . W e now show t h e b e h a v i o u r of v ( s ) i n a neighbourhood o f 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 b e h a v i o u r a s Abelian theorems f o r t h e M e l l i n t r a n s f o r m a t i o n . The r e s u l t s p r e s e n t e d h e r e i n are q u i t e e q u i v a l e n t as i n d i c a t e d i n Lavoine and Misra [ 4 Theorem 1 1 . 1 2 . 1
1
[for t h e i n f e r i o r abscissa).
equal t o t  A \ l o g tIV[: H + h ( t ) + g ( t ) ]
I f Vt
c E'
a,w
is
254
Chapter 11
on 10,TC (if
,T
1,
A = a
(ii) h(t) is a function tending to 0 as t
+
0+,
(iii) g(t) is continuous function such that T
l+i Im(sA)dtl < M g(t) e T'
IJ
with M being independent of T' and s when 0 then (11.12.1)
lMsVt
 Hr(v+l) (sA)v1
as s
f+
E
+
A, with
2 arg(s~)2
4j 
Proof. Here the distribution P x e
c,
0.
defined in Section 11.8 is
equal to
[.
eAx xv [ H+h (eX)+$ex)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 is equal to t'(log
(for the superior abscissa). If Vt€ El
aru
t)'
CHl+hl(t)l + gl(t)
on ]TII[ ,T1 > e, where HII v are numbers such that Re n =
w
and Re v > 1,
(ii) hl(t) is a function tending to 0 as t
+
m,
(i)
0,
(iii) g1 (t) is a continuous function such that .L
with M being independent of T' and s when T' > T1 and Is+nl < n, then (11.12.2)
MsVt
as s
f+
+
n, with
E
 H~ r(v+i) (ns) v1 3* 5 arg(sn) 5 2
€*
Proof. The proof is similar to that of the previous theorem but instead of P x we consider the distribution Q defined in Section 11.8. e e
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?the sense of Section 11.3.2 , Pt = Dk 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, H(log a/t)" as t (iii) f(t) Re v > 1. Then (11.12.3)
as s
+
m
mSpt
(1)k H r ( v + l ) ask sk'"
in an angle where larg
Proof. 
with w(t)

f
a0, where H is a number and
+
81
5
2
E.
We set
0 as t + a0.
Hence
where w(aex)
+
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 (aex)e'ax dx S
0
S
= a
0
~ ~ ~ f ( a e H~ r)( u + l ) ass''
as s + w in an angle where larg s I 2 (11.12.3) by means of (11.7.6).
Qt
=
(i)
~
5
E.
Finally, we deduce
Theorem 11.12.4 (for Re s +  a ) . Let Qt E Elmlube such that Dk fl(t), with the function fl(t) satisfying the conditions: fl(t) has its support in [bra[ with b>O belonging to this supportI
(ii) twkfl(t) is summable,
(iii) fl(t) H(log t/b)v as t+b0, where H is a number and Re v >  l .
Chapter 11
256
Then (11.12.4) as
s +

$t
in an angle where
E
3n
5 arg s 2

E.
It is necessary here that if s is the halfplane Re s 0 ) .
as I s 1 + m , with a > 0, k complex) G independent of (11.12.5)
1 lMt v(s)
If the function v ( s ) is a and satisfies
lN, Re X 2 2, and number (real or
E
s,
k
= D
then we have p(t).
where p(t) is a continuous function for t > 0 which is null for t > a and such that (11.12.6) (1)kv ( s + k ) and wl(s) = asw(s) with Proof. We set w(s) s(s+l) ,....,(s+kl), k 1,2,3, .... In this setting wl(s) is =
( s ) ~=
( s ) ~
=
holomorphic in the halfplane Re s > sup(0,cik) and satisfies (11.12.7)
wl(s)
. (a)kG
s 1 as
1.
+

in this half plane. Now by lemma 11.10.1, we conclude that lM;lwl(s) is a continuous function p,(t) for t > 0. Moreover, by (11.12.7), we have for real r
Mellin Transform
257
This can be rewritten as 1 m (11.12.8) 1 P,(t) tr'dt + p,(t) tr'dt + 0 as 0 1 In (11.12.81, the first integral tends to zero as r + t > 1, tr1 grows with r. Hence, (11.12.8) requires
1:

+
m.
and when
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)kG sAas
+ m.
Hence, by a Tauberian theorem well known for Laplace transformation (see Theorem 8.12.1 of Chapter 8) we have k pl(ex) x A  l as x + o+. ~
,w
It follows that
NOW, we put p(t) = pl(tla). Since w(s) = aSwl(s) and k v(s) = (1) (sk)k we have by the rules of Calculus (11.7.2) and 1 1 k (11.7.6) that Itw(s) = p(t), and finally, mt v(s) = D p(t) which proves the theorem. Theorem 11.12.6 (for Re s +  a ) . holomorphic in the halfplane Re s v(s) as
Is 1
+ m,
I
w
If the function v ( s ) is and satisfies
G bs(e'i"s)k'X
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 which satisfies
0).
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
I
m
V(x)P(xt)dx
I
m
tSldt =
I
m
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 (11.13.2)
where for V
E
EAlw,
IMs[P] = p(s) for P
L
EkIU,
MsCQI = q(S) for Q
E
EkIU#
IMs[V]
= V(S)
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 1s as s belonging to a conventional subset of Sp n SQ. In other words, if pt denotes the set of s such that 1s E Sv, then the equation (11.13.3)
q ( s ) = v(ls)p(s)
%n
Sp n
holds for s
E
Replacing
by (1s) on both sides of (11.13.3), we have
Put k ( s )
8
1 P( s
== j
SQ.
and IMs[K(x)]
=
k(s)
=
1 p o. Therefore
Mellin Transform
259
q(1s) , then we have IMsBt = q(1s) . If Bt = ';MI (11.11.4) we have IMs (B\
By making use of
K) = q(ls)k(s)
and by (11.13.4)? M s ( B L K) = v(s) = IM V which gives S
(11.13.5)
V(X) = ( B I K)x.
Since q(s) = NsQ(t), we have Bt = t1Q(,)1; and by (11.11.4) and (11.13.5) we get V(x) =
IQ(y)1K(;)y
0
2 dy.
Now by putting t = I in above integral, we finally obtain Y (11.13.6)
V(x) =
m
I
0
Q(t)K(xt)dt
provided, of course? the inverse Mellin transform ~(x) = i.e.
(11.13.6)
mi1
[pol 1
,s
+
x) exists;
is the solution of integral equation (11.13.1).
In particular, the equation (11.13.1) will have the solution V(x) =
I
m
0
Q(t)P(xt)dt
if (11.13.7)
P(S)P(ls)
=
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
s1/2 r(1/4 + s/2 + v/2) p(s) = 2 r (3/4  s/2 + v/2) and hence
so
that
cot(3" 4
2
+
Vfl 2)
260
Chapter 11
Now making u s e of Sneddon C2 1, Problem 2.38
( b ) w e see t h a t
where Hv i s t h e s t r u v e f u n c t i o n . I n o t h e r words, w e have shown t h a t t h e i n t e g r a l e q u a t i o n m
( x t ) % ( x ) Yv(xt)dx = Q ( t ) 0
has t h e s o l u t i o n
m( x t ) 4Q ( t H ) ,,(tx)dt.
V(x) = 0
Now, w e can d e r i v e t h e s o l u t i o n of t h e i n t e g r a l e q u a t i o n
i n a s i m i l a r manner. If w e c o n s i d e r W,R and G t o be i n E' , t h e n atw by (2) of Pro l e m 11.11.1 w e can w r i t e (11.13.8) i n t h e form (11.13.9) By v i r t u e of
11.11.4)
we obtain
(11.13.10)
where
and t h e i r s t r i p of d e f i n i t i o n s are r e p r e s e n t e d by Sw, SR and SG Here R , r ( s ) and SR a r e unknown. S i n c e s E Sw, respectively. sR and SG, w e can s a y t h a t (11.13.10) h o l d s f o r s E Sw n SR n SG. The e q u a t i o n (11.13.10)
can be w r i t t e n as
Mellin Transform conventional function. then (11.11.4) yields
261
If Gl(t) = lMt 1gl(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 m
(11.13.12)
R(t) = !G,(y)H(t/y)y'dt. 0
The integral equation 1 (11.13.13) I Wl(t/x)Rl(x)rdx = Gl(t), 0 t
Re a where
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). If Sv does not contain any s such that Re s > ak. Then the case is more complicated. Suppose SV be the strip a < Re s < w and let the roots be arranged such that Re al < Re a2 Re a3,. , Then, we have
..
(11.14.6)
1 IMt (sa)j = E(j,a;t), Re
s
B. (11.14.13) (11.14.12)
Xt = +A
f h(u)uA'du,
B
It can be easily verified that (11.14.12) and (11.14.13) give If v(s) is such that the solutions of (11.14.8). (11.14.14)
V(S) =
then (11.14.9) gives
(sA)g(s)
Chapter 11
264
(11.14.15)
X t
Xt =
1 mt g(s).
We remark here that (11.14.14) implies that the existence of such that V
t = tDGt
and the equation (11.14.8)
 AGtr
msG =
g(S)
,
can be written as
+
tD(Xt+Gt)
A(Xt+Gt) = 0.
Hence the solution Xt = G is obtained which is identical to t (11.14.15). 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
t'


1
n=O
(l)n Anl(tD)%t
under the condition that the series converges. The solution of an EulerCauchy 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
(11.14.16)
d 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 and (11.14.16)
h(t) = Dh(t)
+
W6 (ty)h(O+)6 (t)
yields with th(O+)b(t) = 0
Mellin Transform
265
which is similar to equation (11.14.8). By putting H ( s ) = IMs h(t), F(s) = M s f (t), and applying the Mellin transformation to (11.14.16') we obtain
and its inversion is
where f(t) is similar to that given by (11.14.10) and
1
x(O,y;t) =
0 I t 2 Y
elsewhere.
11.15.Potential Problems in Wedge Shaped Regions In this section we shall describe briefly how Mellin transformation 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) 2 equation in polar coordinates multiplied by r ; (ii) it satisfies the boundary condition O z r l a (11.15.2)
u(r, +a)
=
r > a
(iii) u(r,e) is bounded as r is bounded.
is Laplace's
266
Chapter 11
Figure 11.15.1
To solve this problem we identify u(r,e) with a distribution in r. AS we see above that u(r,e) is defined only for 0 2 r 5 a and hence one can take u(r,e) = 0 for r > a, then (see Section 11.3) i.e. the Mellin transformation of u(r18) exists for u(r,e) E E; IRe s > 0. we may conclude that u(r,e) From the structure of u(r,8) E E; I is bounded as r is bounded. A l s o , 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(ra) 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: s1 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

if we assume that a 2 can be interchanged with Ms. Therefore, we a e2 obtain (11.15.3)
U(s,e)
is8
= A(s) e
+
is8
B(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 MS [u(r,ta)] = U(s,ta) for then we obtain
s
E
Qu = Is: a < Re s < a 1, u1 u2
so that A(s) = B(s) =
2s
as cos s a
Consequently, (11.15.3) takes the form (11.15.4)
If
s =
a+iw, we have (since a
0,

1 Tv
0 and xa+1/2k+1s(x) is bounded as x + O+ for a A l s o , if there exists n > 0 such that x~s(x)is bounded as x +
c
.
v.
Then we have under these conditions:
ax, $ (x)>
(12.4.6)
klms(XI
= < s (x),(1)’$ ( k ) (x)> = (1)
$ (k)
(x)dx.
0
we now verify the existence of (12.4.6). Proof. Indeed, by our proposed structure of Vx, we have m
= (  l ) k j s(x)+(~) (x)dx
(12.4.7)
= (  1 ) k01 j xak+3’2s(x)
0
xka3/2$ (k)(XI dx
+ (1)k 1 xn~(x)x~$(~)(x)dx. 1 The second integral in (12.4.7) is bounded according to our A l s o , the hypothesis and the relation (12.2.3) of Lemma 12.2.1. 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
(12.4.8)
M V V = < VX ’ G J V ( x y ) >
,y
> 0.
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 power of X1 as x

Examples.
+
279
.
If IL denotes the Laplace transformation, then show
that
xi:
(i) when V
X
ePxVx
IL
=
Jyx J"(YX)V~
is represented by a function $(x)
E H ~ .
Proof. We recall that Mt Q (y) =
~6 Jv (XY),$
(Y)>.
Now (12.4.3) gives 4
M :

e 'xVX,$(y)>
px x
= l,+(y)
>
where
u (xa)U (bx) =
c
< V x , G Jv(xy)ePX>
Consequently, we have M :
M yFP
my
= aI
Jxy
2n1. 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 2n1/2 = (v21)~v 29)
M V
...(v2(2n1)2)y2n1/2
I
v 2n1. 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) = M w B x
=
.
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 ) i s a smooth f u n c t i o n on, 0 < y < m. I n d e e d , it can be e x t e n d e d i n t o an a n a l y t i c
f u n c t i o n on t h e complex p l a n e whose o n l y s i n g u l a r i t i e s a r e b r a n c h p o i n t s a t t h e 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 s e t (12.4.10)
b ( z ) = IH; I f Bx
Theorem 1 2 . 4 . 1 .
BX = < B x 1 6 J v ( x z ) > . E
&'(I),
t h e n zv1/2b(z)
i s an e n t i r e
f u n c t i o n 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 zplane)
.

E'(I),
t h e n i t s s u p p o r t is c o n t a i n e d i n t h e k i n t e r i o r of lo,[. A l s o , i f w e s e t B = D s ( x ) where s ( x ) i s a X c o n t i n u o u s f u n c t i o n having s u p p o r t i n [ a r i 3 l , 0 < a < 6 m. Then, by making u s e of t h i s s e t t i n g , w e have P r o o f . If Bx'
(12.4.11)
b ( z ) = (l)k 1 s ( x ) ?;[= dk Jv(zx)ldx.
a dx Making u s e of t h e series expansion of J ( z x ) (see Problem 1 . 4 . 1
of
C h a p t e r 1) w e have
or
where
Since
2'jj!u
j f i n i t e zplane.
. Remark
is bounded a s
j
f
m,
t h e series c o n v e r g e s i n t h e
T h i s p r o v e s o u r theorem. Since
b ( z ) z'li2
i s holomorphic i n t h e h a l f  p l a n e
y = R e z > 0 c a n b e s e e n above and c o n s e q u e n t l y w e may i n f e r t h a t b ( y ) i s a smooth f u n c t i o n on 0 < y < m. Theorem 12. 4. 2. d e f i n e d by ( 1 2 . 4 . 9 ) .
I f v 2 1/2 and i f L e t Bx E e'(1). 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
(12.4.12)
1 < y < 
Chapter 1 2
282
where K and p are sufficiently large real numbers.
C31,
Proof, The proof can be carried out as indicated in Zemanian 
pp. 1 4 6  1 4 7 .
1 2 . 5 . Some Rules
This section provides an account of the operational transform formulae for the spaces Hv and H:
.
12.5.1.
Transform formulae for Hv
If 9
E
Hv, we have
(12.5.1)
My+1(X9) =
(12.5.2)
mv +1(Nv9)
and if 4
E
 NvMv9;
= Y xv9;
Hv+l, then
Proof. The proofs of 
to ( 1 2 . 5 . 4 ) and ( 1 2 . 5 . 6 ) to 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: (12.5.1)
( 1 2 . 5 . 7 ) are given in Zemanian C31, pp. 1 3 9  1 4 0 .
(12.5.10)
Jvl(X)
+ Jv+l(X)
(see Sneddon C21, pp. 5 1 0  5 1 1 ) . Making use of (12.5.10) we have
=
2v x
Jv(X)j
Hankel Transform
283
m
=
Hence (12.5.8) is established. (12.5.10) we have R.H.S.
Of (12.5.9)
=
f
0
JV(xy)4 (XI
dx =
MY$(XI.
Further making use of (12.5.11) and m
1
dx
5 { 2 ~~[Jv,l(~y)Jv+l(Xy) ]X+(X) 0
m
= 1 1 2 v /2.J:(xy)x$
4v
12.5.2.
0
(x)
dx+2vj
1
0 xy
Transform formulae for H :
We now state a number of operationtransform 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) =
(12.5.13)
Mv+l(NVV)
= y
(12.5.14)
2 mV (X V)

=

NvMvV; MvV; 1
MvNvMvV;
284
Chapter 12
2
. . A
(12.5.15)
M v (MvNvV) = y M V V ;
(12.5.16)
Mv
and if V
E
H:+lI
cqvxl
= (1)k y2kIHvvx:
then *
(12.5.17)
MV (xV) = M v M V + l V ~
(12.5.18) (12.5.19)
M v (DxVI = gi(2v1) MV+1V(2~+1)Mv1V3..
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 143144. The formula pp.
(12.5.16) can be obtained by applying (12.5.15) successively k times k and making use of S*k 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). A
n
Problem 12.5.1 For k = OIl121....I show that
^k Vx = (1)k y2kM V V x I Vx
(i)
MvSV
'(ii)
M R
^k
v
v
vX
k
H :
k~1/2 Vxt Vx
= y M ~ + ~ x
*k k (iii) M v P v Vx = y Mv+kV x I Vx where
E
ikI ik and P*k
t
;
E
H : ;
H$+k
are defined in Section 12.3.
12.6. Inversion In Section 12.4 we have described the distributional setting of Hankel and inverse Hankel transformations. The present section further work out the inverse Hankel transformation by working with distributions in H : and we term this relation as an inversion of the Hankel transfornation of distributions. The main result of this section is: Theorem 12.6.1.
Let W Y
E
H$(y).
If Vx
E
H:(x)
and such that
285
Hankel Transform
(12.6.1)
= WY'
M:Vx
fo
then Vx i s c a l l e d t h e i n v e r s e Hankel t r a we obtain
vx
(12.6.2)
= M ' :
f W
Y
and con eque t l Y I
wy.
Proof. According t o (12.6.1)
I H :
m
and (12.4.3')
W , $ ( x ) > = cIH:
Mf: V X I J , ( x ) >
Y
=
<myv
VX'
w e have
I
XI;*(%)>
I
Ti J, E
Hy
= wx'm;M;*(X)s
= < V X I $(XI >
which y i e l d s t h e r e l a t i o n Mt W = Vx. Y
Hence (12.6.2)
i s established.
I n addition X
Y
because
The exchange formulae (12.6.1) and (12.6.2) enable u s t o c a l c u l a t e numerous transforms. We mention below a f e w examples of them. Examples. Show t h a t
(i)
M;
(ii) I H :
6(xa) =
Jax
Jay J v ( a y ) ,
J ~ ( ~ x =)
a > 0, y > 0;
6(ya);

Proofs. According t o (12.4.8)
IH; 6 (xa) = < 6 (xa) and hence (i) i s e s t a b l i s h e d ,
IH;
Jay
~ " ( a y i= 6 (xa)
I
w e have
,G J v ( x y )
7
=
Also, by (12.6.2),
& Jv,(ay).
Chapter 12
286
and by changing y to x, we have
which proves (ii).
Now by ( 1 2 . 4 . 8 1 ,
=
by (12.5.10).
12.6.1.
we have
 @ Jy(ay)  & a u1 (ay) +$
Jy+,(ay)
This proves (iii),
Remarks The space H:
does not contain the Dirac functional 6(x) nor its
derivatives. If we take a semiclosed interval [ O , w [ : in place of I But in 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.
.
.
1.
According to (12.4.8) IH;~(X)
= = = 0 since sup x'v1/2#I XCI
meniber of H:
(x) is finite.
That this truly defines F as a
follows directly from the condition (ii) of Lemma 12.2.1.
Hankel Transform (See Zemanian
[a],
287
pp. 151152.)
I In the preceding case we have taken v 1.  2. However Zemanian [ S ] has shown that a Hankel transformation of any order can be defined. Briefly, we see this as follows.
2.
Let v be a real number and let k be a nonnegative integer 1 such that v+k 1.  z for every $(y) E Hv We now set
.
Let Vx E H'(x) and if the Hankel transformation of order v+k is denoted by M:Vx which is a distribution in Hl Then X M ' V can be defined as v x
.
of V
<M:Vx,$(X)'
=

1 in M:, then M: for every Q E Hv. If we take v 1 7 = M y and it follows that M; Vx coincides with the Hankel transformation of distributions defined by (12.4.3). 12.7,The nDimensional Hankel Transformation In the preceding sections we have extended the Hankel transformation to certain generalized functions of one dimension. In the present section we develope the ndimensional case corresponding to the preceding work. Some of the results presented herein are similar to those of Koh [ 2 ] . Here we use the following notations. For our purpose we shall restrict x and y to the first orthant of IRn which we denote by I. Thus I = Ix E IRn , 0 < xv < m v = l r . ,nl. 2 % We 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 " mn m m1 m2 we mean the product x1 x2.. xn. Thus,Cx j = x1 x2 xn where
.
.
.
The notations x m = Cml,m2,...,mn3. xv 5 y U and xv < y, ( v = 1 ,2 ,n) n nonnegative integers in IR i.e. kv k shall Letting (k) = kl+k2+ kn, Dx
,... .
...+
(12.7.1) while (x'Dx)
0
(k)
kn kl Ic2 axlax2.. axn
.
denotes
...
.
2 y and x < y mean respectively The letters k and m shall denote and mv are nonnegative integers. denote
288
Chapter 12
(12.7  2 ) 12.7.1.
The spaces h
and h'
P
1I
Let 11 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 nonnegative integers m and k in IRn (12.7.3) Since $ is infinitely differentiable, the order of differentiation in (xlDx) is immaterial; thus
a
1
1
a
axi) (Xj ax' j =
(Xi
a
1
1
a
(xj ~ ) ( ~ 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 seminorms i.e. a multinorm. An equivawith lent topology for h may be given by the multinorm {p:l v
k and m traverse a countable index set, h is, in fact, a countaP bly multinormed space. We say that a sequence {$,,I is Cauchy in h P 1I if $, E h for all v and for every m , k , ym,k ($,On) + 0 as v and 1I n + m independently.
As
Lemma 12.7.1. for each k.
If $ ( x )
E
k hP, then Dx$(x) is of rapid descent
Proof. Since 1 a k ixp1/2 . $(Xl,. ..,xi (xi 5 i )
,Xn)
I . .
= x *ixp1/2 . 1
i
ki 1 b.x. j ()I$, a j=O J 1 ax i
'
we have (12.7.4)
(xlDX) [ x 1 P1/2 $ (x) = cx
2k
ICxl2'111
kl
j,=o
...jk1=nO b.[xj] n
where the b . are appropriate constants. 3
x
..+In
4
'1 Jn l.... a xn ax N o w , consider $ E h By J
P
.
Hankel Transform
289
i = l,...,n F i n a l l y , by i n d u c t i o n on k and u s i n g ( 1 2 . 7 . 4 )
w e have
The space h' i s t h e d u a l of h and i s t h e s p a c e of d i s t r i b u ?J P' t i o n s (continuous l i n e a r f u n c t i o n a l s ) on h lJ'
The f o l l o w i n g p r o p e r t i e s are immediate e x t e n s i o n s of t h e one dimensional case.
Using t h e r e l a t i o n ( 1 2 . 7 . 4 )
whenever c a l l e d f o r :
1. I D ( 1 ) , t h e s p a c e 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 f u n c t i o n s w i t h compact s u p p o r t on I , i s a subspace of h f o r every c h o i c e of u. P Thus, t h e r e s t r i c t i o n of any f E h ' t o ID( I ) i s i n I D ' ( 1 ) However 0 D ( 1 ) i s n o t dense i n h
v
2.
.
.
The complex number t h a t f
E
assigns to $
h'
u
by < f , $ > . W e a s s i g n t o h' t h e following topology:
E
h
P
i s denoted
u
a sequence { f , ) converges t o f J
for all $
E
h
.
E
h' i f < f  f j , @ > + 0 a s j + ?J
m
?J
t h e r e exists a p o s i t i v e c o n s t a n t C and a For each f E h ' ?J nonnegative i n t e g e r r such t h a t
I Recall t h a t 3.
lJ
max ymIk ($1. O I m ,kz r be a Locally summable f u n c t i o n on I such t h a t f (x) i s and Cxl u+1/2f ( x ) i s a b s o l u t e l y i n t e g r a b l e a s ( xI + v = 1,2,. ,n. Then f ( x ) g e n e r a t e s a r e g u l a r generaf i n h ' d e f i n e d by =
p:
L e t f (x)
of slow growth on 0 < x y < 1, lized function
..
?J
.....( f ( x l I x 2 , ...,xn) $(x1,x2 ,...,xn)dxldx2 .....dx n' m
m
< fI+> = 0
0
4
T h i s s t a t e m e n t f o l l o w s from
E
hV.
t h e mean v a l u e theorem f o r ndimensional
Chapter 1 2
290
i n t e g r a l s (See Fleming CllIp.155) and t h e f a c t t h a t 9 i s of r a p i d descent. Operations on h
12.7.2.
and h 1 1.1
!J
I n t h i s s e c t i o n w e perform some o p e r a t i o n s on h following manner. Lemma 12.7.2.
lJ
and h' i n t h e !J
For any p o s i t i v e o r n e g a t i v e i n t e g e r n and f o r
any u I t h e mapping $ ( x ) + C x l n 9 ( x ) is an isomorphism from h o n t o !J h;l+n* It follows t h a t f ( x ) + I x l n f ( x ) d e f i n e d by . d
Cxlnf ( x ) I 9 ( X I > = < f
i s an isomorphism from h t
u+n
Proof. If 9
E
(XI
I
[XI"$
(XI >
o n t o hl
!J*
hlJI t h e n
I.
= sup cx Im(xlox) kCx 1u1/2n
lJ+n ( [ X I n $ )
'm,k
Cxlncp(x) I
I
 1y 1m I k ( 9 )
W e now d e f i n e t h e following o p e r a t o r s on h
= x !J+1/2 Nt!J
N
M
lJ
i y
i
a xlJ1/2 axi i
= N1!JN21J..
= xu1/2 i
..
,NnU = [ x ] ~ ' + ~ / ~
p+1/2
Q = xi
N  l cp = 2lJ
xl+1/2
axl.
a xu+1/2 axi i
Also, w e d e f i n e an i n v e r s e o p e r a t o r t o N N:;
2
!J1/2
I
v
an
....ax*
 u 1/2
[ j
as f o l l o w s :
9 (t?X2,.
m
,x2tp1/2~(x11t
.., x n ) d t I...~x
n) d t
m
and so on.
Ctl!J1/2Q(t)dtR,.. That
Nil
i s t r u l y the inverse t o N
lJ
., d t n .
f o l l o w s from t h e f a c t t h a t Q i s
I
Hankel Transform
291
infinitely differentiable and of rapid descent. Lemma 12.7.3.
N 0 is an isomorphism from h
Lemma 12.7.4.
M + is a continuous linear mapping of h U+1 11 on to h
Lemma 12.7.5.
M
P
onto h?J+l.
?J
! i *
U
N
P
2p + l
= [x]~'~'*
ax1,.,..
n an axl.. a xn [ 1 ?J1/2 i=l
=
.
is a continuous mapping of h
(2).
into itself. A
A
In the dual spaces, we define N and M as transpose differenu Fi tial operators by (12.7.5)
=
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 lJ v transformable generalized function (or distribution) in nvariables. Also
,
by (12.7.10)
(12.7.11')
< MYV
lJx
I
M :
(12.7.11) is equivalent to
,...
,X ) > =
@(x,
n
m1. This is because the derivatives of xJv(A x) of order m are j not continuous at the origin. In (12.9.2), the A . (T) are called the coefficients of the 3 FourierBessel 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 FourierBessel series in the sense of functions. Also, we say that m
is the FourierBessel 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 nonnegative integer, and let x denote a real variable. By Bm,v we denote the space of functions which are m times continuously differentiable on a compact neighbourhood of I' and
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
m
1 la. ($1 I A 3m is convergent. Here a. (0) are j=1 3 3 the numbers which do not depend upon the variable x but depend upon the choice of the functicm Q (x) in B,,,. these cases the series
For x = 0 we have x Jv(A.x) = 0. 3
Q (0) = 0 when Q
(i)
Bm
E
A l s o , by the property of A
J (A.a) = 0. v
3
(ii)
IV
Hence, we have by (12.10.1)
for all integers m 2 0.
given in Section 12.9.1, we have Therefore, we obtain by (12.10.1),
Q (a) = 0 where Q
E
j
Bm
,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 m,v further state the property of Bm ,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 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 . 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 (12.10.2)
 SP'
t
0 for each Q
, or m,v
T on B
of B,,,, that lim S = T on Bm,v. P P+
Chapter 12
300
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 m,v formulate the distributional setting of Fourier Bessel series in terms of the representation of a distribution on Bm by its Fourier IV Bessel series which we outline in the following manner. Theorem 12.11.1. Let T be a distribution of order m with support contained in 1'. Then we have lim nOn
Tn = T
Bm,vI o r , more explicity in the sense of series ce
T
=
1 AJ. (T) Ja(A.x) V I
j=1
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 every neighbourhood of 1';
2. we have on I' that $:h)
(x)
I$

1
(x) exist and are continuous on
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 a j x Jv (A .x) j=1 dxm I on every bounded interval of 1'. Hence, l., is established. By the description of B given by (12.10.1) , we have d;, (x) = $ (x) and m,v consequently 2 . , is also established.
($)a
Lemma 12.11.2.
Let m
d;,(x)
=
1 a. ($1 J
j=n
x J,,(X~X) , Y 4
E
B ~ , ~ .
Bessel Series Then
0, (x)
+
0 as n
+

and
$dh)
(x)
301
...,m.
0 uniformly for h = 1,2 ,
+
proof.^ It is easy to show that, for h positive constant
=
1,2,...,m
with K is any
which tends to zero by virtue of the properties of BmIV. If T of order m has support contained in I', then
Lemma 12.11.3. we have
<TI$(XI > =
(XI >,
v
Bm,v*
Q
This is a consequence of Lemma 12.11.1 and of (Schwartz C11, Chapter 111.7, Theorem XXVIII). Proof of Theorem 12.11.1. (12.11.1) By
Lemma 12.11.3,
(12.11.2)
+
It is sufficient to show that 0, Cp
B
m,v
we can write
= <TI
NOW, by (12.9.3)
E
,
(12.9.1)
4, (X)>
and (12.9.2)
n
m

.
have
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) $,(XI 0 in the space of f l o f Section 2.3 of Chapter 2 and consequently T belongs to the topological dual IDfm of IDrn,
.
+
302
Chapter 12
hence
CTr a
(x)qn (x)>
+
0 as n
+
.
Examples. 1. If 0 5 c 2 a, the Dirac measure 6(xc) which is such that < 6 (xc),$ (x)> = $ (c) is a distribution of order zero having support contained in 1'. For v 2 the Theorem 12.11.1 yields the representation J (cX.) (12.11.3) S(xc) = 1 J;(Ijx) a j=l Jv+l(aAj)
%
00
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 (xa) = 0 which do not yield a contradiction because if $ belongs to B0 , v ' then we have = 0 and = 0. 2. Let the function g (x) be defined by xv(1x2)a,
0
and a is not an integer (a > 1). Also, g (x) has not 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 Bi, by the series (2) I V
Note that v must either be a nonnegative integer or satisfy v > a'1.
12.12. Other Properties of the FourierBessel 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 FourierBessel series. To prepare for this section we first need the following result. Theorem 12.12.1. The representation of T on B by a FouriermIv Bessel series of order v is unique. In other words, if
Bessel Series
on Bm
IV'
303
then Aj = Aj (T).
Proof. x J v (Akx) belongs to B for each k 2 1. Now by m, v Theorem 12.11.1, we have
it follows that Ak = Ak(T).
Hence, by (12.9.1),
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 derivative of a measurable function f(x) which is bounded on this neighbourhood. 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 j f 1 < HAm+', for a large enough H any positive constant such that [A. m 7 j (in the number j . Then the series 1 A JA (A . x ) is convergent on Bm j=l j v 3 fV sense of (12.10.2) )
.
Proof. For each 4 in Bmfv,Lemma 12.11.3 m
ca
gives
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 fo
B mlv
As remarked in the Section 12.10, we construct in this section and present some result of distributional the subspace Bm of Bm IV setting of Fourier Eiessel series on Bm. By % we denote the space of functions +(x) which are m+l times continuously differentiable on an interval containing I' and such (x) is on 1' and that 9
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. (12.13.1)
Let the FourierBessel coefficient of $(x)/x be
cj ($/XI =
a
2 a2J:+1
I
(ahj) 0
9 (XI J v (Ajx)dx.
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 outlines of them. (12.13.1) 2 a ' 2 Jzl(aX )A ( v , @ ) where j j
305
can be written c. ($/x)= 3
a
First we take
in Bo.
$
Then we have
where
and
By the structure Bol we have for 0 5 x
IO"(x)I < H =>lO'(x)I
(i)
< a < H x and 16(x)I 0. Now, by putting the value of J v(z) with z = h . x from (iii) 3 to J v (X.x) in A', we obtain 1 j 2 1/2 4v21 (iv) A; < CI1,I + 8 1121 + II,l] where a
J cos ( X x8)
I1 = J
.6
j
$J IX)
x1'2dx,
Chapter 12
306
I2 = A;
7
3'2
.
6
sin (A.xB) 7
Q (x)
x~/~~x,
a I3 = A 5/2 bv (hjx)4 (x) x  ~ / ~ ~ x . j .6 *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 J can conclude that there exists a certain number G1 such that for j > lo. Also, integrating by parts one time to 12, we obtain for j > jo
(vi) where
G2
is a certain number.
If j > j, then there exists a number K such that bv ( A .x) < K for U 3 A T 6 5 x 5 a and by (i)I we have 7 a
J
and consequently there may exist a number G3 such that (vii)
lfgi
Let us denote M2 =
< G ~ hj 5 / 2
for j > jo.
4v2 1 CG1 + g
2 4
(3
G2
+ G31 I
then by making use of (iv), (v), (vi) and (vii) , we obtain (viii)
A!
I
M
2
Pinally, since A. (v,Q) 3
IAj (v,Q)
A  ~ / ~ for j > jo.
j
H I i f v = H
1
Iv(Box),if v 00
I a 3 0
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 7 v > m1. This is because the mth derivatives of xJv ( 8 .x) are not 1 continuous at the origin. In (12.14.3), the B. (T) are called the EesselDini coefficients 7 of T. If T = f(x), a summable function having support in I', then we obtain the B. (f) to be the BesselDini Coefficients in the sense 3 of ordinary functions. We further say that m
is the BesselDini series of the distribution T. NOW our purpose is to show that it converges and is equal to T on a certain space of functions, 12.15. The Space on which the In this section we construct the space % rmrv distributional setting of FourierDini series is to be formulated in the subsequent section. Let m be a nonnegative 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 > m1 or v = 0,1,2,...,
and in all
m
1 Id. (0) lf3m3 is convergent. Here d.3 (0) are j=1 3 the numbers which depend upon the choice of $(XI in rmrv.
these cases the series
We give below (Theoremsl2.6.5 and 12.6.6) definitions of the independent of Bessel functions. subspaces of ,m,v
a
The existence of BmIv (see section 12.11). Equality on
,Y
0
E
%,m,v
can be shown as that
EkIm
Two distributions T and
S
having support in I' and of order z m
Chapter 1 2
310
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. If there exists a distribution T having support on I' of order a such that
+spI
(12.15.1)
then we will say S
Q (XI> = 0, Y Q
E
%,m,v,
as p
t 0 0 ,
or that lim S = T on %tmtv P+" P 12.16. Representation of a Distribution by its &SEelDlni +
P
T on
Series
In this section we obtain results for the distributional setting of BesselDini series which would be similar to those of FourierBessel 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
n
=
T on %,m,v
i
or more explicity in the terms of series m
+j=1 1 B.1 ( T ) J ; ( B 3. x ) on %,m,v*
T = B~(T) G;(H,x)
Theorem 1 2 . 1 6 . 2 . The representation of T on by a rmr V EesselDini series of order v and parameter H is unique, i.e. m
T = B~ G;IH,X) +
then B
j
= Bj
B . J ~ ( B . , X )on j=1 I v 3
if
% rmrv
(T) , where j 2 0.
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 . positive constant.
We have IB. (T) I 3
m1. replace

Note.
The proof of Theorem 12.16.5
i s a n a l o g o u s t o t h a t of
Theorem 12.13.1 o f S e c t i o n 12.13 by r e p l a c i n g t h e r e f e r e n c e s
(Watson
Cll, S e c t i o n s 18.24 and 18.26 by S e c t i o n s 18.33 and 1 8 . 3 5 ) . 12.16.2,Another
s u b s p a c e of
Theorem 12.16.6.
By
%m,v
%,O,V
w e d e n o t e t h e s p a c e of f u n c t i o n $(x)
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 a n i n t e r v a l c o n t a i n i n g I ' and such t h a t Q" (x) /x i s bounded on I '
0' (a)  (H+L) Q (a)
,
$ (0)
=
$ I
(0) = 0 , and
= 0.
P r o o f . The proof i s a n a l o g o u s t o t h a t of Theorem 12.13.1
w e p u t f (x) =
and u t i l i z e t h e r e s u l t s of
( T o l s t o v [l],
if
Sections
8.22 and 8.23). 1 2 . 1 7 . An A p p l i c a t i o n of t h e BesselDini
Series
T h i s s e c t i o n p r o v i d e s an a c c o u n t of t h e u s e of t h e p r e c e d i n g t h e o r y of B e s s e l D i n i 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 h e a t flow i n a c y l i n d e r ' o f i n f i n i t e l e n g t h . S p e c i a l l y , t h e problem w e wish t o solve i s t h e f o l l o w i n g : Find a t e m p e r a t u r e f ( r , t ) (which i s a f u n c t i o n of r and t) such t h a t 1. it i s d e f i n e d 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 equation
f o r 0 2 r 2 a , where v 2 0 , w i t h g (t) b e i n g an i n t e g r a b l e f u n c t i o n €or t > 0 and S (r) i s a d i s t r i b u t i o n w i t h s u p p o r t i n 1';
Chapter 12
312
3. it satisfies the conditions
(12.17.3)
f (r,t) is bounded when r is bounded
I
a Hf (a,t) = 0, H < u. aar f (r,t) 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). (12.17.4)

By Theorem 12.16.1 and the corollary of Theorem 12.16.5, distribution S (r) can be represented by
the
(12.17.5) where J; (6 .r) is given by (12.14.2) and 1
S .=
'
5
<S
(r), r J u (Bjr)>.
j
The structure of S(r) given above enables us to consider the representation of f(r,t) such that m
with F . (0) = 0. Now, we verify that (12.17.6) satisfies the 3 conditions (12.17.2) , (12.17.3) and (12.17.4)
.
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
for r = a. (12.17.4).
3
Hence, f (r,t) given by (12.17.6) satisfies the condition
NOW, by putting the values of f (r,t) and and (12.17.5) in (12.17.1), w e obtain
S
(r) from (12.17.6)
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. F7. (t) +
k&
F j (t) =
s 7.g(t)
with F. (0) = 0 (which can be solved easily). Hence, we finally 7 obtain 2 2 m t B .ku dule@ 'k J;(B .r) f(r,t) = k 1 S . C l g(u)e I 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)
Chapter 12
314
is a function (see Carslaw and Jaeger
[2]).
(See
also Section 12.8.)
A theoretically interesting case arises when S(r)= 6(rc)/2nc, c < a. In this case we recall that r is equal to (x2+y 2) % and hence we have
0
=
I
1 2n 4 (c cos e , c sin 0
e)ae.
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
C21,
0.
(2) for the calculation of the coefficients (see Erdelyi (Ed.) v01.2, p. 26).
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..
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1962.
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Bib 1iography
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327
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INDEX OF SYMBOLS Ac,
12
F x , 91
BH,O,v' 311 B 309 H,rn,V' Bm, 304, 311 BIll,V' 298 C($')r
70
hv, 288 hl, 288
My', 292 Hu , 272 : , 272 H M v l270
ID, 19 ID' I 27
80 80 36
Mi',
80 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 IDo (01,
IDA (01,
IDb+(lR"), IDk,
91
M p , 292
c", 1
ID, ID;, ID:,
,I;..
270 JA , 221 JA(r), 221 JI', 209 JI ' ( r l , 207 L(O,m), 269 IL, 109
43
20
I D V k l 27
m k ( m n ) , 23 ID' k ( l R " ) , 27 ID'l , 89 (ID') I, 89 227, 230 Ea,lll' 227, 232 E:,ul E 231 E 233 Pr9' E(r) , 207 E , 21 E' i 27 f i ( I R n ) , 23 E ' ( I R n ) , 27 FP, 8 \
329
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AUTHOR INDEX Albertoni, S., 75, 315 Antosik, P., 85, 315 Apostol, T.M., 229, 315 Arsac, Jacques, 105, 315
FOX, C., 227, 268, 318 Friedman, A., 25, 318 Friedmann, B, 205, 319 Fung, Kang, 268, 319
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
Garnir, H.G., 25, 72, 144, 231, 233, 319 Gelfand, I.M. 16, 21, 91, 105, 319 Geradi, F.R, 227, 319 Ghosh, P.K., 144, 319 Giittinger, W., 21, 75, 78, 319, 325 Guy, D.L., 314, 319
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, 259, 262, 271, 317 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, 144, 157, 165, 181, 189, 193, 197, 201, 215, 238, 270, 277, 314, 318 Fenyo, I., 314, 318 Fisher, B., 75, 318 290 , 318 Fleming, W.H., 331
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, P I 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, 237, 238, 253, 262, 297, 317 , 321 Lew, John,S., 163, 227,320, 322 Loins,J.L. 314, 322
Author Index
332
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, 105, 144, 225, 324, 325 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
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 VoKhacKhoan ,105.,150,205,326 Watson,G.N.,158, 297, 305, 306, 307, 311,312,313,326 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,
1 0 5 1 144,225,227,231,233,
235, 244,268,272,273,274, 282,284,287,314,320,325, 326, 327 Zygmund,A., 2041 327.