Operational
Calculus and
Related Topics
H.-J. Glaeske
Friedrich-Schiller University
Jena, Germany
A.P. Prudnikov
(Deceased)
K.A. Skòrnik
Institute of Mathematics
Polish Academy of Sciences
Katowice, Poland
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Glaeske, Hans-Jürgen.
Operational calculus and related topics / Hans-Jürgen Glaeske, Anatoly P. Prudnikov, Krystyna
A. Skórnik.
p. cm. -- (Analytical methods and special functions ; 10)
ISBN 1-58488-649-8 (alk. paper)
1. Calculus, Operational. 2. Transformations (Mathematics) 3. Theory of distributions (Functional analysis) I. Prudnikov, A. P. (Anatolii Platonovich) II. Skórnik, Krystyna. III. Title. IV. Series.
QA432.G56 2006
515’.72--dc22
2006045622
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In memory of
A. P. Prudnikov
January 14, 1927 — January 10, 1999
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Contents
Preface
xi
List of Symbols
xv
1 Integral Transforms
1
1.1
Introduction to Operational Calculus
1.2
Integral Transforms – Introductory Remarks
1.3
The Fourier Transform
1.4
1.5
1.6
. . . . . . . . . . . . . . . . . . . . .
1
. . . . . . . . . . . . . . . . .
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3.1
Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . .
8
1.3.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3.3
Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3.4
The Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.3.5
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.4.1
Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . .
27
1.4.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
1.4.3
Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
33
1.4.4
The Complex Inversion Formula . . . . . . . . . . . . . . . . . . . .
37
1.4.5
Inversion Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
1.4.6
Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
1.4.7
Remarks on the Bilateral Laplace Transform . . . . . . . . . . . . .
47
1.4.8
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
The Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
1.5.1
Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . .
55
1.5.2
Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
58
1.5.3
The Complex Inversion Formula . . . . . . . . . . . . . . . . . . . .
62
1.5.4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
The Stieltjes Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
1.6.1
Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . .
67
1.6.2
Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
70
1.6.3
Asymptotics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
1.6.4
Inversion and Application . . . . . . . . . . . . . . . . . . . . . . . .
75
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viii
1.7
1.8
1.9
The Hilbert Transform
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
1.7.1
Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . .
78
1.7.2
Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
81
1.7.3
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
Bessel Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
1.8.1
The Hankel Transform . . . . . . . . . . . . . . . . . . . . . . . . . .
85
1.8.2
The Meijer (K-) Transform . . . . . . . . . . . . . . . . . . . . . . .
93
1.8.3
The Kontorovich–Lebedev Transform
. . . . . . . . . . . . . . . . .
100
1.8.4
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
The Mehler–Fock Transform
. . . . . . . . . . . . . . . . . . . . . . . . . .
107
. . . . . . . . . . . . . . . . . . . . . . . . . . .
115
1.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
1.10.2 The Chebyshev Transform . . . . . . . . . . . . . . . . . . . . . . . .
116
1.10.3 The Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . .
122
1.10.4 The Gegenbauer Transform . . . . . . . . . . . . . . . . . . . . . . .
131
1.10.5 The Jacobi Transform . . . . . . . . . . . . . . . . . . . . . . . . . .
137
1.10.6 The Laguerre Transform . . . . . . . . . . . . . . . . . . . . . . . . .
144
1.10.7 The Hermite Transform . . . . . . . . . . . . . . . . . . . . . . . . .
153
1.10 Finite Integral Transforms
2 Operational Calculus
163
2.1
Introduction
2.2
Titchmarsh’s Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
167
2.3
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
180
2.3.1
Ring of Functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
180
2.3.2
The Field of Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
185
2.3.3
Finite Parts of Divergent Integrals . . . . . . . . . . . . . . . . . . .
190
2.3.4
Rational Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
201
2.3.5
Laplace Transformable Operators . . . . . . . . . . . . . . . . . . . .
205
2.3.6
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
213
2.3.7
Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
217
Bases of the Operator Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
219
2.4.1
Sequences and Series of Operators . . . . . . . . . . . . . . . . . . .
219
2.4.2
Operator Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
226
2.4.3
The Derivative of an Operator Function . . . . . . . . . . . . . . . .
229
2.4.4
Properties of the Continuous Derivative of an Operator Function . .
229
2.4.5
The Integral of an Operator Function . . . . . . . . . . . . . . . . .
232
2.4
2.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Operators Reducible to Functions
2.5.1
163
. . . . . . . . . . . . . . . . . . . . . . .
236
Regular Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . .
236
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ix
2.6
2.5.2
The Realization of Some Operators . . . . . . . . . . . . . . . . . . .
239
2.5.3
Efros Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
242
Application of Operational Calculus . . . . . . . . . . . . . . . . . . . . . .
247
2.6.1
Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . .
247
2.6.2
Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . .
258
3 Generalized Functions
271
3.1
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
271
3.2
Generalized Functions — Functional Approach . . . . . . . . . . . . . . . .
272
3.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
272
3.2.2
Distributions of One Variable . . . . . . . . . . . . . . . . . . . . . .
274
3.2.3
Distributional Convergence . . . . . . . . . . . . . . . . . . . . . . .
279
3.2.4
Algebraic Operations on Distributions . . . . . . . . . . . . . . . . .
280
Generalized Functions — Sequential Approach . . . . . . . . . . . . . . . .
287
3.3.1
The Identification Principle . . . . . . . . . . . . . . . . . . . . . . .
287
3.3.2
Fundamental Sequences . . . . . . . . . . . . . . . . . . . . . . . . .
289
3.3.3
Definition of Distributions . . . . . . . . . . . . . . . . . . . . . . . .
297
3.3.4
Operations with Distributions . . . . . . . . . . . . . . . . . . . . . .
300
3.3.5
Regular Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
303
3.3
3.4
3.5
Delta Sequences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
311
3.4.1
Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . .
311
3.4.2
Distributions as a Generalization of Continuous Functions . . . . . .
317
3.4.3
Distributions as a Generalization of Locally Integrable Functions . .
320
3.4.4
Remarks about Distributional Derivatives . . . . . . . . . . . . . . .
322
3.4.5
Functions with Poles . . . . . . . . . . . . . . . . . . . . . . . . . . .
325
3.4.6
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
326
Convergent Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
332
3.5.1
Sequences of Distributions . . . . . . . . . . . . . . . . . . . . . . . .
332
3.5.2
Convergence and Regular Operations . . . . . . . . . . . . . . . . . .
339
3.5.3
Distributionally Convergent Sequences of Smooth Functions . . . . .
341
3.5.4
344
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
346
Local Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
347
3.6.1
Inner Product of Two Functions . . . . . . . . . . . . . . . . . . . .
347
3.6.2
Distributions of Finite Order . . . . . . . . . . . . . . . . . . . . . .
350
3.6.3
The Value of a Distribution at a Point . . . . . . . . . . . . . . . . .
352
3.6.4
The Value of a Distribution at Infinity . . . . . . . . . . . . . . . . .
355
3.5.5
3.6
Convolution of Distribution with a Smooth Function of Bounded Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.6.5
3.7
3.8
Support of a Distribution . . . . . . . . . . . . . . . . . . . . . . . .
355
Irregular Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
355
3.7.1
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
355
3.7.2
The Integral of Distributions . . . . . . . . . . . . . . . . . . . . . .
357
3.7.3
Convolution of Distributions . . . . . . . . . . . . . . . . . . . . . .
369
3.7.4
Multiplication of Distributions . . . . . . . . . . . . . . . . . . . . .
375
3.7.5
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
379
Hilbert Transform and Multiplication Forms
. . . . . . . . . . . . . . . . .
381
3.8.1
Definition of the Hilbert Transform . . . . . . . . . . . . . . . . . . .
381
3.8.2
Applications and Examples . . . . . . . . . . . . . . . . . . . . . . .
383
References
389
Index
401
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Preface
The aim of this book is to provide an introduction to operational calculus and related
topics: integral transforms of functions and generalized functions. This book is a cross between a textbook for students of mathematics, physics and engineering and a monograph on
this subject. It is well known that integral transforms, operational calculus and generalized
functions are the backbone of many branches of pure and applied mathematics. Although
centuries old, these subjects are still under intensive development because they are useful in
various problems of mathematics and other disciplines. This stimulates continuous interest
in research in this field.
Chapter 1 deals with integral transforms (of functions), historically the first method to
justify Oliver Heaviside’s (algebraic) operational calculus in the first quarter of the twentieth
century. Methods connected with the use of integral transforms have gained wide acceptance
in mathematical analysis. They have been sucessfully applied to the solution of differential
and integral equations, the study of special functions, the evaluation of integrals and the
summation of series.
The sections deal with conditions for the existence of the integral transforms in consideration, inversion formulas, operational rules, as for example, differentiation rule, integration
rules and especially the definition of a convolution f ∗ g of two functions f and g, such that
for the transform T it holds that
T[f ∗ g] = T[f ] · T[g].
Sometimes applications are given. Because of the special nature of this book some extensive
proofs are only sketched. The reader interested in more detail is referred for example to the
textbooks of R.V. Churchill [CH.2], I.W. Sneddon [Sn.2], and A.H. Zemanian [Ze.1]. Short
versions of many integral transforms can be found in A.I. Zayed’s handbook Function and
Generalized Function Transformations [Za]. For tables of integral transforms we refer to
[EMOT], [O.1]-[O.3], [OB], [OH], and [PBM], vol. IV, V.
In this book we deal only with integral transforms for R1 - functions. The reader interested
in the multidimensional case is referred to [BGPV].
In Chapter 2 (algebraic) operational calculus is considered. This complete return to
the original operator point of view of Heaviside’s operational calculus was done by Jan
Mikusi´
nski; see [Mi.7]. He provided a strict operator basis without any references to the
theory of the Laplace transform. His theory of convolution quotients provides a clear and
simple basis for an operational calculus. In contrast to the definition of the multiplication
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xii
of functions f and g, continuous on [0, ∞) given by J. Mikusi´
nski,
t
(1)
(f ∗ g)(t) =
f (x) g(t − x) dx,
0
in Chapter 2 functions with a continuous derivative on [0, ∞) are considered and the multiplication is defined by means of
(2)
(f ∗ g)(t) =
d
dt
t
f (x) g(t − x) dx.
0
Both definitions have advantages and disadvantages. Some formulas are simpler in the one
case, otherwise in the case of definition (2). In the case of definition (2) the ∗-product of
two functions constant on [0, ∞),
f (x) = a,
equals a function h with h(x) = ab,
g(x) = b,
x ∈ [0, ∞)
x ∈ [0, ∞), such that the ∗-product of two numbers
equals their usual product. In the case of definition (1) this product equals abt. In both
cases the field of operators generated by the original space of functions is the same; the
field of Mikusi´
nski operators. For our version of the starting point we refer to L. Berg,
[Be.1] and [DP]. After an introduction a proof of Titchmarsh’s theorem is given. Then the
operator calculus is derived and the basis of the analysis of operators is developed. Finally,
applications to the solution of ordinary and partial differential equations are given.
Chapter 3 consists of the theory of generalized functions. Various investigations have been
put forward in the middle of the last century. The mathematical problems encountered are
twofold: first, to find an analytical interpretation for the operations performed and to justify
these operations in terms of the interpretation and, second, to provide an adequate theory
of Dirac’s, δ-function, which is frequently used in physics. This “function” is often defined
by means of
+∞
δ(x) = 0,
x = 0,
δ(x) ϕ(x) dx = ϕ(0),
−∞
for an arbitrary continuous function ϕ. It was introduced by the English physicist Paul
Dirac in his quantum mechanics in 1927; see [Dir]. It was soon pointed out that from the
purely mathematical point of view this definition is meaningless. It was of course clear to
Dirac himself that the δ-function is not a function in the classical meaning and, what is
important, that it operates as an operator (more precisely as a functional), that related to
each continuous function ϕ its value at the point zero, ϕ(0); see Laurent Schwartz [S.2].
Similar to the case of operational calculus of Chapter 2, J. Mikusi´
nski together with R.
Sikorski developed an elementary approach to generalized functions, a so-called sequential
approach; see [MiS.1] and [AMS]. They did not use results of functional analysis, but only
basic results of algebra and analysis. In Chapter 3 we follow this same line.
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Preface
xiii
Because this book is not a monograph, the reference list at the end of the book is not
complete.
We assume that the reader is familiar with the elements of the theory of algebra and
analysis. We also assume a knowledge of the standard theorems on the interchange of limit
processes. Some knowledge of Lebesgue integration, such Fubini’s theorem, is necessary
because integrals are understood as Lebesgue integrals. Finally, the reader should be familiar with the basic subject matter of a one-semester course in the theory of functions of a
complex variable, including the theory of residues. Formulas for special functions are taken
from textbooks on special functions, such as [E.1], [PBM] vols. I-III, [NU], and [Le].
The advantage of this book is that both the analytical and algebraic aspects of operational calculus are considered equally valuable. We hope that the most important topics
of this book may be of interest to mathematicians and physicists interested in applicationrelevant questions; scientists and engineers working outside of the field of mathematics
who apply mathematical methods in other disciplines, such as electrical engineering; and
undergraduate- and graduate-level students researching a wide range of applications in diverse areas of science and technology.
The idea for this book began in December 1994 during A.P. Prudnikov’s visit to the
Mathematical Institute of the Friedrich Schiller University in Jena, Germany. The work
was envisioned as the culmination of a lengthy collaboration. Unfortunately, Dr. Prudnikov
passed away on January 10, 1999. After some consideration, we decided to finish our joint
work in his memory. This was somewhat difficult because Dr. Prudnikov’s work is very
extensive and is only available in Russian. We were forced to be selective. We hope that
our efforts accurately reflect and respect the memory of our colleague.
Hans-Juergen Glaeske and Krystyna A. Sk´ornik
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List of Symbols
F
8
˜ν
H
90
L1 (R) = L1
8
L1,ρν (R+ )
91
C(R) = C
8
Lr,ρν (R+ )
92
C0 (R) = C0
10
KT
93
Fc
11
Kν
93
Fs
11
Kν
93
96
13
Kν−1
−1
18
Kν
97
Fc−1
Fs−1
19
KL
100
19
L−1,1 (R+ ) = L−1,1 100
L
27, 205
KL−1
102
Ea
27
MF
107
Lloc
1 (R)
27
Pν
107
28
Pν−n
C
k
F
¯a
H
108
−1
Ha
28
MF
σac
29
T
116
σc
110
30
Tn
116
−1
37
116
(II)
L
¯0
H
47
116
44
L01 (−1, 1)
L0p
−1
T
118
H0
45
P
122
Eab
¯ ab
H
Hab
48
Pn
122
48
L1 (−1; 1) = L1
123
48
Lp
123
MT
55
C[−1; 1]
123
L
P
−1
124
55
P
λ
131
M
62
S
67
M
Pab
−1
55
H
78, 381
Pnλ
Cnλ
Lλ1 (−1; 1)
Hν
85
C[−1; 1]
132
λ −1
131
131
=
Lλ1
132
85
(P )
133
Hν−1
86
133
Jν
89
Lλp (−1; 1)
(α,β)
Jν
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P
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137
xvi
Rn
(α,β)
137
Lloc
1 (R+ ) = L
180
(α,β)
Pn
(α,β)
Lp
137
Ln
184
138
M
185
139
M(M )
185
N0
190
S
205
L
205
S∗
205
δ
271, 278
(P
(α,β) −1
)
Laα
144
Lα
n
144
Lp,wα
144
(Laα )−1
145
He
153
Hn
153
Hn
153
C∞
272
L1,exp (R)
154
C0∞
273
(He)
155
D
275
M
180
D
276
−1
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Chapter 1
Integral Transforms
1.1
Introduction to Operational Calculus
In the nineteenth century mathematicians developed a “symbolic calculus,” a system of
rules for calculation with the operator of differentiation D :=
d
dt .
The papers of Oliver Heaviside (1850–1925) were instrumental in promoting operational
calculus methods. Heaviside applied his calculus in the solution of differential equations,
especially in the theory of electricity. He had a brilliant feel for operator calculus, but
because he did not consider the conditions for the validity of his calculations his results
were sometimes wrong. Heaviside published his results in some papers about operators
in mathematical physics in 1892–1894 and also in his books Electrical Papers (1892) and
Electromagnetic Theory (1899); see [H.1]–[H.3].
Heaviside used the operator D and calculated with D in an algebraic manner, defining
D0 := I,
Dk :=
dk
,
dtk
k ∈ N,
where I is the identity. This method seems to be clear because of the following rules of
calculus
D(cf )(t) = cDf (t)
(1.1.1)
D(f + g)(t) = Df (t) + Dg(t),
k
l
D (D f )(t) = D
k+l
f (t),
k, l ∈ No .
(1.1.2)
(1.1.3)
If one replaces derivatives in differential equations by means of the operator D, then there
certain functions of D appear. Because of (1.1.1) through (1.1.3) it is easy to understand
the meaning of Pn (D), where Pn is a polynomial of degree n ∈ No . Because of
∞
f (t + h) =
∞
(hD)n
hn (n)
f (t) =
f (t) =: ehD f (t)
n!
n!
n=0
n=0
one can interpret the operator ehD as the translation operator:
ehD f (t) = f (t + h).
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2
Integral Transforms
According to the rules of algebra one has to define D−1 :=
1
D
as
t
D−1 f (t) =
f (τ )dτ.
0
Example 1.1.1 Look for a solution of
y (t) + y(t) = t2 .
(1.1.4)
The solution of this first-order linear differential equation is well known as
y(t) = c e−t + (t2 − 2t + 2),
(1.1.5)
with some arbitrary constant c. By means of the operator D, equation (1.1.4) can be
rewritten as
(1 + D)y = t2
and therefore,
y(t) =
There are various interpretations of
1
1+D .
1
t2 .
1+D
(1.1.6)
Quite formally one has, for example,
1
= 1 − D + D2 ∓ · · · ,
1+D
and applying the right-hand side to (1.1.6) one has
y(t) = t2 − 2t + 2;
this is the solution (1.1.5) of (1.1.4) with y(0) = 2. On the other hand one can write
1
1
=
1+D
D 1+
1
D
=
1
1
−
± ··· .
D D2
Interpreting
1
1
1
f=
f ,
Dk
D Dk−1
k∈N
as k-time integration of f from 0 to t we obtain from (1.1.6)
t3
t4
t5
−
+
∓ ···
3
3·4 3·4·5
t0
t1
t2
t3
= −2
− + − ± · · · + t2 − 2t + 2
0! 1! 2! 3!
= −2e−t + t2 − 2t + 2,
y(t) =
and this is the solution (1.1.5) of (1.1.4) with y(0) = 0. So depending on the interpretation
of the expression
1
1+D ,
one obtains different solutions of (1.1.4). In this manner one can
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Introduction to Operational Calculus
3
develop an elementary legitimate calculus for the solution of linear ordinary differential
equations with constant coefficients. Let us look for a solution of
Ln [y](t) := y (n) (t) + a1 y (n−1) (t) + · · · + an y(t) = h(t)
ak ∈ R,
k = 0, 1, · · · , n,
(1.1.7)
with the initial value conditions
y(0) = y (0) = · · · = y (n−1) (0) = 0.
(1.1.8)
Setting Ln (D) = Dn + a1 Dn−1 + · · · + an one has
Ln (D)y(t) = h(t)
or
y(t) =
For the interpretation of
1
Ln (D)
1
h(t).
Ln (D)
we start with L1 (D) = D. Then we have the equation
Dy = h
and
t
1
y(t) = h(t) =
D
h(τ )dτ.
(1.1.9)
0
In the case of L1 (D) = D − λ we have (D − λ)y = h or
eλt D e−λt y = (D − λ)y = h.
Therefore,
D e−λt y(t) = e−λt h(t),
and according to (1.1.9),
t
e−λτ h(τ )dτ =:
λt
y(t) = e
1
h(t)
D−λ
(1.1.10)
0
Similarly one can perform the case of a polynomial Ln (λ) with a degree n > 1 and n zeros
λj ,
j = 1, 2, · · · , n, where λi = λj if i = j. Then
Ln (λ) = (λ − λ1 )(λ − λ2 ) · · · (λ − λn ),
and
1
A1
A2
An
=
+
+ ··· +
Ln (λ)
λ − λ1
λ − λ2
λ − λn
with
Ak =
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1
,
Ln (λk )
k = 1, 2, · · · , n.
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4
Integral Transforms
From (1.1.7) we obtain, applying (1.1.10)
n
y(t) =
k=1
Ak
h(t) =
D − λk
t
n
e−λk τ h(τ )dτ.
λk t
Ak e
k=1
(1.1.11)
0
One can easily verify that (1.1.11) is the solution of (1.1.7) with vanishing initial values
(1.1.8).
This method can also be extended to polynomials Ln with multiple zeros.
Problems arose applying this method to partial differential equations. Then one has to
“translate” for example functions of the type Dn−1/2 ,
√
n ∈ N or e−x
D
. Heaviside gave
a translation rule for such functions in the so-called “Expansion Theorem.” The solutions
often took the form of asymptotic series, often better suited for applications than convergent series. Sometimes incorrect results appeared because conditions for the validity were
missing. In Heaviside’s opinion:
“It is better to learn the nature of and the application of the expansion theorem by actual
experience and practice.”
There were various attempts to justify Heaviside’s quite formal operational methods. At
the beginning of the twentieth century mathematicians such as Wagner (1916), Bromwich
(1916), Carson (1922), and Doetsch used a combination of algebraic and analytic methods.
They used two different spaces: A space of originals f and a space of images F , connected
with the so-called Laplace transform (see section 1.4)
∞
e−pt f (t)dt,
F (p) = L[f ](p) =
p ∈ C,
(1.1.12)
0
provided that the integral exists. Integrating by part one has
L[f ](p) = pF (p) − f (0).
(1.1.13)
This formula can be extended to higher derivatives. So Heaviside’s “mystique” multiplication with the operator D is replaced by the multiplication of the image F with the complex
variable p. From (1.1.13) we see that nonvannishing initial values also can be taken into
consideration. In the space of images the methods of the theory of functions of a complex
variable can be used. Of course one needs a formula for the transform of the images into
the space of originals. This is explained in section 1.4. The disadvantage of this method
is that it is a mixture of analysis and algebra. Because of the convergence of the integral
2
(1.1.12) quite unnatural restrictions appear. So, for example, L[et ] does not exist, and
Dirac’s δ also cannot be included in this theory. Nevertheless, the Laplace transform was
used and is still used today in many applications in electrotechnics, physics, and engineering. In the following, similar to the Laplace transform, many other integral transforms are
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Integral Transforms – Introductory Remarks
5
investigated and constructed for the solution of linear differential equations with respect to
special differential operators of first or second order.
A radical return to the algebraic methods was given by J. Mikusi´
nski. His theory is free
of the convergence–restrictions of integral transforms, and Dirac’s δ appears as a special
operator of the field of Mikusi´
nski operators. This is explained in Chapter 2.
Chapter 3 introduces spaces of generalized functions. Their elements have derivatives of
arbitrary order and infinite series can be differentiated term-wise. Moreover, they include
subspaces of “ordinary” functions. They are linear spaces in which a multiplication of its
elements, called convolution “∗” is defined, such that, for example,
(Dn δ) ∗ f = Dn f
is valid. So one again has an operational calculus for the solution of linear differential
equations with constant coefficients.
1.2
Integral Transforms – Introductory Remarks
In Chapter 1 we deal with (one-dimensional) linear integral transforms. These are mappings of the form
b
F (x) = T[f ](x) =
f (t)K(x, t)dt.
(1.2.1)
a
Here K is some given kernel, f : R → C is the original function and F is the image of f
under the transform T. Sometimes x belongs to an interval on the real line, sometimes it
belongs to a domain in the complex plane C. In these cases the transform T is called a
continuous transform; see sections 1.3 through 1.9. If the domain of definition of the images
F is a subset of the set of integers Z the transform T is sometimes called discrete, sometimes
finite; see section 1.10. We prefer the latter. Sometimes the variable of the images appears
in the kernel as an index of a special function. Yakubovich [Ya] called these transforms
index transforms. Index transforms can be continuous transforms (see sections 1.8.3 and
1.9) or finite transforms (see section 1.10).
In the following chapters we deal with transforms of interest for applications in mathematical physics, engineering, and mathematics. The kernels K(x, t) “fall down from heaven,”
since otherwise the sections would become too voluminous. The kernels can be determined
by means of the differential operators in which one is interested. For example, to find a
kernel for the operator D with Df = f on R+ one has
∞
F[Df ](x) =
∞
f (t)K(x, t)dt =
∞
[f (t)K(x, t)]0
0
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−
f (t)
0
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∂
K(x, t)dt.
∂t
6
Integral Transforms
To obtain a kernel K such that the operation of differentiation is transformed into multiplication with the variables of images one can choose
∂
K(x, t) = −xK(x, t)
∂t
and
lim K(x, t) = 1,
lim K(x, t) = 0.
t→+∞
t→0+
A special solution is
K(x, t) = e−xt ,
x, t ∈ R+ ,
and so we derived the kernel of the Laplace transform (1.1.12), with the differentiation rule
(1.1.13). This transform is considered in detail in section 1.4.
Another problem is as follows: Let u(x, y) be a solution of the Laplace equation on the
upper half plane
2 u(x, y)
= uxx (x, y) + uyy (x, y) = 0,
x ∈ R, y ∈ R+
(1.2.2)
with the boundary conditions
u(x, 0) = eiξx ,
lim
|x|,y→+∞
ξ, x ∈ R,
u(x, y) = 0.
(1.2.3)
(1.2.4)
One can easily verify that
u(x, y) = eiξx−|ξ|y
is a solution of the problem. To solve the problem under a more general condition than
(1.2.3)
u(x, 0) = f (x)
one can choose the superposition principle since
2
(1.2.5)
is a linear differential operator. This
leads to the attempt to set
∞
F (ξ)eiξx−|ξ|y dξ
u(x, y) =
(1.2.6)
−∞
for some function F . Condition (1.2.5) yields
∞
F (ξ)eiξx dξ,
f (x) =
x ∈ R.
(1.2.7)
−∞
This is an integral equation for the function F and the solution leads to an integral transform
F = T[f ].
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(1.2.8)
Integral Transforms – Introductory Remarks
7
The formulas (1.2.8) and (1.2.7) are a pair consisting of an integral transform and its
inversion. In this special example we have the Fourier transform, investigated in section
1.3.
Readers interested in the derivation of the kernel of an integral transform are referred to
Sneddon [Sn.2], Churchill [Ch.2], and especially to [AKV].
The sections that follow start with the definition of a transform, conditions of the existence, inversion formulas and operational rules for the application of the transforms, such
as differentiation rules considered in the examples above. A convolution theorem plays an
important part. Here a relation f, g → f ∗ g has to be defined such that
T[f ∗ g] = T[f ] · T[g].
All these operational rules are derived under relatively simple conditions, since in applications one has to use the rules in the sense of Heaviside; see section 1.1. One applies the
rule, not taking note of the conditions of their validity (pure formally), and afterward one
has to verify the result and state the conditions under which the formally derived solution
solves this problem. Here often the conditions are much less restrictive than the set of
conditions for the validity of the operational rules that have been used for the calculation
of the solution.
Remark 1.2.1 For every transformation there is a special definition of the convolution.
Because there are few unique signs “∗” sometimes the same sign is used for different transforms and therefore for different convolutions. In this case this sign is valid for the transform
discussed in the section under consideration. If in such a section the convolution of another
transform is used, then we will make additional remarks.
Notations.
In the following, N is the set of natural numbers N = {1, 2, 3, · · · },
No = N ∪ {0}, Z is the set of integers, Q the field of rational numbers, R the field of
¯ + = R+ ∪ {0}, and C is the set of
real numbers, R+ the set of positive real numbers, R
complex numbers. All other notations are defined, when they first appear; see also the
“List of Symbols” at the beginning of this volume.
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8
Integral Transforms
1.3
1.3.1
The Fourier Transform
Definition and Basic Properties
Definition 1.3.1 The Fourier transform (F T ) of a function f : R → C is the function f ∧
defined by
∞
∧
f (t)e−iτ t dt,
f (τ ) = F[f ](τ ) =
τ ∈ R,
(1.3.1)
−∞
provided that the integral exists.
Remark 1.3.2 Instead of the kernel e−iτ t sometimes eiτ t , e−2πiτ t , (2π)−1/2 e±iτ t are chosen and in certain instances these kernels are more convenient.
Remark 1.3.3 The convergence of the integral (1.3.1) can be considered in a different
manner: As pointwise convergence, as uniformely convergence, in the sense of the principal
value of Cauchy, in the sense of Lp –spaces or others.
We consider the Fourier transform in the space
∞
L1 (R) = L1 = {f : f measurable on
R,
f
1
|f (t)|dt < ∞}.
=
−∞
The space L1 is obviously suited as the space of originals for the Fourier transform. The
Fourier transforms of L1 –functions are proved to belong to the space
C(R) = C = {f : f continuous on R,
f = sup |f (t)| < ∞}.
t∈R
Theorem 1.3.1 Let f ∈ L1 , then f ∧ = F[f ] ∈ C. The F T is a continuous linear transformation, i.e.,
F[αf + βg] = αf ∧ + βg ∧ ,
α, β ∈ C,
f, g ∈ L1
(1.3.2)
and if a sequence (fn )n∈N is convergent with the limit f in L1 then the sequence (fn∧ )n∈N
of their Fourier transforms is convergent with the limit f ∧ in C.
Proof. We have
∞
∧
|f (τ )| ≤
|f (t)|dt = f
−∞
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1
The Fourier Transform
9
and, therefore, there exists f ∧ = supτ ∈R |f ∧ (τ )|. If h ∈ R and T > 0 then
∞
∧
∧
|e−iht − 1| |f (t)|dt
|f (τ + h) − f (τ )| ≤
−∞
T
−iht
≤
|e
−T
− 1| |f (t)|dt + 2
∞
|f (t)|dt +
−∞
−T
|f (t)|dt < ε,
T
since |e−iht − 1| becomes arbitrarily small if |h| is sufficient small and the last two integrals
become arbitrary small if T is sufficiently large. As such we have f ∧ ∈ C. The F T is
obviously linear. From
f ∧ − fn∧ = sup |(f − fn )∧ (τ )| ≤ f − fn
1
τ ∈R
we obtain the continuity of the F T .
Example 1.3.2 If f ∈ L1 then f ∧ ∈ C, but the image f ∧ must not belong to L1 . Let
f (t) =
|t| ≤ 1
|t| > 1.
1,
0,
Then f ∈ L1 , but
1
∧
e−iτ t dt = 2τ −1 sin τ
f (τ ) =
−1
does not belong to L1 . But there holds
Theorem 1.3.2 Let f ∈ L1 . Then f ∧ (τ ) = F[f ](τ ) tends to zero as τ tends to ±∞.
Proof.
Step 1. Let f be the characteristic function of an interval [a, b], −∞ < a < b < ∞, i.e.,
f (t) = χ[a,b] (t) =
1,
0,
t ∈ [a, b]
t ∈ R \ [a, b].
Then we have
b
∧
e−iτ t dt = i
f (τ ) =
e−ibτ − e−iaτ
τ
a
∧
and f (τ ) tends to zero as τ → ±∞.
Step 2. Let f be a “simple function,” i.e.,
n
f (t) =
αj χ[aj ,bj ] ,
αj ∈ C,
j=1
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j = 1, 2, . . . , n,
10
Integral Transforms
where the intervals [aj , bj ] are disjointed. Then
n
f ∧ (τ ) =
j=1
i
αj (e−ibj τ − e−iaj τ )
τ
and f ∧ (τ ) tends also to zero as τ → ±∞.
Step 3. The set of simple functions are dense in L1 . Therefore, for every ε > 0 there
exists a simple function fo such that
f − f0
1
< ε/2,
and there exists a number T > 0 such that
|fo∧ (τ )| < ε/2,
|τ | > T
according to step 2.
Therefore,
|f ∧ (τ )| = |(f − fo )∧ (τ ) + fo∧ (τ )| ≤ |(f − fo )∧ (τ )| + |fo∧ (τ )|
ε ε
≤ f − fo 1 + |fo∧ (τ )| < + = ε
2 2
if |τ | > T .
Remark 1.3.4 Not every function g, continuous on R, uniformly bounded with g(τ ) → 0
as τ → ±∞ is an image of an L1 –function f under the F T . One can prove that the function
g which is defined by means of
g(τ ) =
1/ log τ,
τ /e,
τ >e
0≤τ ≤e
and g(−τ ) = −g(τ ), is not a F T of a function f ∈ L1 (see [Ob.], pp. 22–24).
Remark 1.3.5 Let Co be the Banach space
Co (R) = Co = {f : f ∈ C : lim f (τ ) = 0}.
τ →±∞
Then because of Remark 1.3.4 we have:
Theorem 1.3.3 The F T is a continuous linear mapping of L1 into Co .
Finally we obtain by straightforward calculation:
Proposition 1.3.1 If f is even (respectively odd), then f ∧ is even (respectively odd) and
we have
∞
∧
∧
f (−τ ) = f (τ ) = 2
f (t) cos τ tdt
0
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(1.3.3)
The Fourier Transform
and
11
∞
∧
∧
f (−τ ) = −f (τ ) = −2i
f (t) sin τ tdt.
(1.3.4)
0
The integrals in (1.3.3) respectively (1.3.4) are called the Fourier–cosine respectively
Fourier–sine transform:
∞
Fc [f ](τ ) =
f (t) cos τ tdt,
τ >0
(1.3.5)
f (t) sin τ tdt,
τ > 0.
(1.3.6)
0
∞
Fs [f ](τ ) =
0
1.3.2
Examples
Example 1.3.3 Let 1+ be Heaviside’s step function:
1
0
1+ (t) =
if t > 0
if t < 0.
(1.3.7)
Then we obtain
F[1+ (T − |t|)](τ ) = 2
sin T τ
,
τ
T > 0.
(1.3.8)
Example 1.3.4 Let α > 0. Then we have
∞
∞
e−[α|t|+iτ t] dt =
−∞
[e−(α−iτ )t + e−(α+iτ )t ]dt =
1
2α
1
+
= 2
,
α − iτ
α + iτ
τ + α2
0
i.e.,
F[e−α|t| ](τ ) =
2α
,
τ 2 + α2
α > 0.
(1.3.9)
Example 1.3.5 Using the Fresnel integral
∞
2
eix dx =
√
πeπi/4
(1.3.10)
−∞
(see [PBM], vol. I, 2.3.15, 2) we obtain
∞
e
−∞
∞
b
it2 −iτ t
dt = lim
a,b→∞
−a
i(t− τ2 )2 − 4i τ 2
e
2
eix dx =
dt = e
√
i
πe− 4 (τ
2
−π)
,
−∞
i.e.,
2
F[eit ](τ ) =
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− 4i τ 2
√
i
πe− 4 (τ
2
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−π)
.
(1.3.11)
12
Integral Transforms
Remark 1.3.6 The original exp(it2 ) does not belong to L1 . So one should not wonder that
the right-hand side of (1.3.11) tends not to zero as τ tends to ±∞.
Example 1.3.6 Let α > 0. Then we have
∞
−α2 t2
F[e
]τ = α
∞
−(x2 + iτ
α x)
−1
e
dx = α
−1 −τ 2 /4α2
iτ
−∞
2
e−(x+ 2α ) dx
e
−∞
iτ /2α+∞
= α−1 e−τ
2
/4α2
2
e−z dz.
−∞+iτ /2α
By means of the theory of residues one can easily prove that the integral on the right-hand
side is equal to
∞
2
e−x dx =
√
π.
−∞
Therefore we have
√
2 2
F[e−α
t
](τ ) =
π −τ 2 /4α2
e
,
α
α > 0.
(1.3.12)
Example 1.3.7 Now we are going to prove that
F[|t|p−1 ](τ ) = 2 cos πp/2 Γ(p)|τ |−p ,
0 < p < 1.
For the proof of formula (1.3.13) we consider the function
f : z → z p−1 e−αz ,
0 < p < 1, α > 0, 0 ≤ arg(z) ≤ π/2.
Figure 1
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(1.3.13)
The Fourier Transform
13
Figure 1
If L is the closed contour of Figure 1 by means of the theorem of residues we have
R
R
ε
L1
f (z) dz − i
f (x)dx +
f (z)dz +
f (z) z = 0 =
L
f (iy)dy.
ε
L2
The integrals on L1 respectively L2 tend to zero as ε → +0 respectively R → +∞. Therefore,
we have
∞
πip/2
e
∞
y
p−1 −iαy
e
xp−1 e−αx dx = α−p Γ(p)
dy =
0
0
by means of the integral representation of the Gamma function. Furthermore, we obtain
∞
0
−πip/2
e
p−1 −iαy
|y|
e
−πip/2
y p−1 eiαy dy = α−p Γ(p)
dy = e
−∞
0
because this is the conjugate complex value of the upper integral and the result is real-valued.
Adding the last two formulas and substituting y → t, α → τ leads to the result (1.3.13).
Analogously taking the difference of the last two formulas we obtain by means of 1.3.1,
Proposition 1.3.1:
F[|t|p−1 sgn t](τ ) = −2i sin πp/2 Γ(p)|τ |−p sgn τ,
0 < p < 1.
(1.3.14)
For many examples of Fourier transforms we refer to the tables [O.1], [EMOT], vol. I.
1.3.3
Operational Properties
For the application of the F T we need certain operational properties. By straightforward
calculation we obtain:
Proposition 1.3.2 Let f ∈ L1 , a, b ∈ R, b = 0. Then
F[f (t − a)](τ ) = e−iaτ f ∧ (τ )
(1.3.15)
F[eiat f (t)](τ ) = f ∧ (τ − a)
(1.3.16)
F[f (bt)](τ ) = |b|−1 f ∧ (τ /b).
(1.3.17)
For the application on differential equations the F T of derivatives is of interest. Let as
usual
C k = {f : f
k − times continuous differentiable on
Then the following holds:
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R},
k ∈ N.
