Operational Calculus: A Theory of Hyperfunctions

181 Pages·1984·10.657 MB·English

by  K. Yosida (auth.)

I
Applied Mathematical Sciences Volume 55
Applied Mathematical Sciences
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(continuedoninsidebackcover)
K. Yosida
Operational Calculus
A Theory of Hyperfunctions
Springer Science+Business Media, LLC
K. Yosida
3-24-4 Kajiwara
Kamakura 247
Japan
AMS Classification: 33A99, 44A40, 44A45
Library of Congress Cataloging in Publication Data
Yoshida Kosaku
Operational calculus.
(Applied mathematical sciences ; v. 55)
Bibliography: p.
Includes index.
1. Calculus, Operational. 2. Hyperfunctions. 1. Tîtle.
n. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 55.
QA432.Y6713 1984 515.7'2 84-10614
Tîtle ofthe original Japanese edition: Enzanshi-oo, Vniversity of
Tokyo Press, Tokyo, © 1982 by Kosaku Yosida.
© 1984 by Springer Science+Business Media New York
Originally published by Springer-Verlag New York Inc. in 1984
AlI rights reserved. No part of this book may be translated or
reproduced in any form without written permission from Springer-Verlag,
175 Fifth Avenue, New York, New York, 10010, V.S.A.
9 8 7 6 543 2
ISBN 978-0-387-96047-0 ISBN 978-1-4612-1118-1 (eBook)
DOI 10.l007/978-1-4612-1118-1
Preface
In the end of the last century, Oliver Heaviside inaugurated an
operational calculus in connection with his researches in electromagnetic
theory. In his operational calculus, the operator of differentiation was
denoted by the symbol "p". The explanation of this operator p as given
by him was difficult to understand and to use, and the range of the valid
ity of his calculus remains unclear still now, although it was widely
noticed that his calculus gives correct results in general.
In the 1930s, Gustav Doetsch and many other mathematicians began to
strive for the mathematical foundation of Heaviside's operational calculus
by virtue of the Laplace transform
-pt
e f(t)dt.
(
However, the use of such integrals naturally confronts restrictions con
cerning the growth behavior of the numerical function f(t) as t ~ ~.
At about the midcentury, Jan Mikusinski invented the theory of con
volution quotients, based upon the Titchmarsh convolution theorem:
If f(t) and get) are continuous functions defined on [O,~)
such that the convolution f~ f(t-u)g(u)du =0, then either
f(t) =0 or get) =0 must hold.
The convolution quotients include the operator of differentiation "s" and
related operators. Mikusinski's operational calculus gives a satisfactory
basis of Heaviside's operational calculus; it can be applied successfully
to linear ordinary differential equations with constant coefficients as
well as to the telegraph equation which includes both the wave and heat equa
tions with constant coefficients.
v
vi Preface
The aim of the present book is to give a simplified exposition as well
as an extension of Mikusinski's operational calculus.
As for the simplification, I should like to mention two points 10
and 20 below.
10• We give a plain proof of the Titchmarsh convolution theorem by
making use of the well-known Liouville Theorem in analytic function theory.
20• For solving linear ordinary differential equations with constant
coefficients, we need not rely upon the Titchmarsh convolution theorem.
We need only a rather trivial theorem:
Let f(t) be continuous for o< t < and ft f(u)du - O.
GO O
Then f(t):: O.
As for the extension, I should like to mention the following point 30•
30• We define the general power (s-a)Y of the operator (s-a) (a
and yare complex numbers), by making use of the general binomial theorem
in analytic function theory:
GO
(l-z)Y L (~) (_z)k (convergent for Izi < 1).
k=O
Then, by virtue of the general power (s-a)Y, we can solve algebraically
the so-called Laplace's differential equation:
where both the a's and b's are complex numbers and a # O. This equa
2
tion includes Bessel's, Laguerre's and confluent hypergeometric differen
tial equations and the like. It is to be noted here that Henri Poincare
and Emile Picard inaugurated the treatment of such differential equations
by the Laplace transform combined with subtle contour integrations in the
complex plane.
The present book is a revised and enlarged (by 10 and 30) English
Edition of the author's book Operational Calculus, written in Japanese and
published by the University of Tokyo Press (1982). The English transla
tion was done by the author.
The author gratefully acknowledges fine help from many friends;
Hiroshi Fujita, Heinz Gotze, Shuichi Okamoto, Shigetake Matsuura, KyUya
Masuda and Jan Mikusinski. Fujita kindly invited the author to write the
Japanese Edition for the U.T.P. Gotze of the Springer-Verlag kindly sug
gested the English edition. The above mentioned 10 and 20 were obtained
Preface vii
by the author jointly with Matsuura and Okamoto, respectively. Masuda
0
was a fine critic on 3 during its maturity. Mikusinski's fine work
aroused the author's interest towards the operational calculus. To them
all, including the U.T.P. and the Springer-Verlag, Inc., I express my
sincere thanks.
Kosaku Yosida
Kamakura
July 1983
Contents
Page
Preface v
Part I. INTEGRATION OPERATOR h AND DIFFERENTIATION OPERATOR s
(CLASSES OF HYPERFUNCTIONS: CAND C)
H
Chapter I. INTRODUCTION OF THE OPERATOR h THROUGH THE
CONVOLUTION RING C
1. Convolution Ring 1
2. Operator of Integration h 3
Chapter II. INTRODUCTION OF THE OPERATOR s THROUGH THE RING C 5
H
3. The Ring C and the Identity Operator I =h/h 5
H
4. C as a Class of Generalized Functions of
H
~yperfunctions 8
s. Operator of Differentiation s and Operator
of Scalar Multiplication [a) 9
6. The Theorem :-h,- = eat 12
S-taJ
Chapter III. LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH
CONSTANT COEFFICIENTS 14
7. The Conversion of the Initial Value Problem
for the Differential Equation into a
Hyperfunction Equation 14
8. The Polynomial Ring of Polynomials in s
has no Zero Factors 15
9. The Partial Fraction Decomposition of a
Rational Function of s 18
10. Hyperfunction Solution of the Ordinary
Differential Equation (The Operational
Calculus) 22
11. Boundary Value Problems for Ordinary
Differential Equations 29
Chapter IV. FRACTIONAL POWERS OF HYPERFUNCTIONS h, s AND __1__ 32
s-a
12. Euler's Integrals - The Gamma Function and
Beta Function 32
-1
13. Fractional Powers of h, of (s-a) ,and
of (s-a) 34
Chapter V. HYPERFUNCTIONS REPRESENTED BY INFINITE POWER
SERIES IN h 39
14. The Binomial Theorem 39
15. Bessel's Function In(t) 40
viii
Table o{ Contents ix
Page
CHAPTER V. (cont.)
16. Hyperfunctions Represented by Power Series
in h 42
PART II. LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH LINEAR
COEFFICIENTS (THE CLASS C/C OF HYPERFUNCTIONS) 47
CHAPTER VI. THE TITCHMARSH CONVOLUTION THEOREM AND THE
CLASS C/C 47
17. Proof of the Titchmarsh Convolution Theorem 47
18. The Class C/C of Hyperfunctions 50
CHAPTER VII. THE ALGEBRAIC DERIVATIVE APPLIED TO LAPLACE'S
DIFFERENTIAL EQUATION 53
19. The Algebraic Derivative 53
20. Laplace's Differential Equation 60
21. Supplements. I: Weierstrass' Polynomial
Approximation Theorem. II: Mikusinski's
Theorem of Moments 70
l 2
PART III. SHIFT OPERATOR exp(-As) AND DIFFUSION OPERATOR exp(_As / ) 74
l 2
CHAPTER VIII. EXPONENTIAL HYPERFUNCTIONS exp(-As) AND exp(_As / ) 74
22. Shift Operator exp(-As) =e-AS.
Function Space K=K[O,oo) 74
23. Hyperfunction-Valued Function f(A) and
Generalized Derivative d~ f(A) = f'(A) 79
AS
24. Exponential Hyperfunction exp(As) =e 83
25. Examples of Generalized Limit. Power Series
in eAS 86
r
26. o e-AsfCA)dA = {f(t)} For {f(t)} E C l 2 92
1/2 _As /
27. Exponential Hyperfunction exp(-As ) e 94
28. Logarithmic Hyperfunction wand Exponential
Hyperfunction exp(Aw) 99
PART IV. APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS 106
CHAPTER IX. ONE DIMENSIONAL WAVE EQUATION 108
29. Hyperfunction Equation of the Form
2
fllCA) - w f(A) = gCA), wE C/C 108
30. The Vibration of a String 113
31. D'Alembert's Method 118
32. The Vibration of an Infinitely Long String 122
CHAPTER X. TELEGRAPH EQUATION 124
33. The Hyperfunction Equation of the Telegraph
Equation 124
34. ACable With Infinitely Small Loss 125
x Table of Contents
Page
CHAPTER X. (cont.)
35. Conductance Without Deformation 126
36. The Thomson Cable 128
37. Concrete Representations of exp(-A~) 132
38. ACable Without Self-Induction 138
39. ACable Without Leak-Conductance 140
40. The Case Where All the Four Parameters Are
Positive 143
CHAPTER XI. HEAT EQUATION 145
41. The Temperature of a Heat-Conducting Bar 145
42. An Infinitely Long Bar 147
43. ABar Without an Outgoing Flow of Heat 149
44. The Temperature in a Bar with a Given Initial
Temperature 150
45. AHeat-Conducting Ring 153
46. Non-Insulated Heat Conduction 156
ANSWERS TO EXERCISES 157
FORMULAS AND TABLES 160
REFERENCES 166
PROPOSITIONS AND THEOREMS IN SECTIONS 167
INDEX 169