Operational calculus and related topics

408 Pages·2006·2.204 MB·English

by  A. P. Prudnikov, K.A. Skórnik

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 (Func-
tional 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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© 2006 by Taylor & Francis Group, LLC
C6498_Discl.indd 1 5/5/06 7:08:15 PM
In memory of
A. P. Prudnikov
January 14, 1927 — January 10, 1999
© 2006 by Taylor & Francis Group, LLC
Contents
Preface xi
List of Symbols xv
1 Integral Transforms 1
1.1 Introduction to Operational Calculus . . . . . . . . . . . . . . . . . . . . . 1
1.2 Integral Transforms – Introductory Remarks . . . . . . . . . . . . . . . . . 5
1.3 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
1.4 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
1.5 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
1.6 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
© 2006 by Taylor & Francis Group, LLC
viii
1.7 The Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.7.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . 78
1.7.2 Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 81
1.7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
1.8 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
1.9 The Mehler–Fock Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 107
1.10 Finite Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
2 Operational Calculus 163
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
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
2.4 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
2.5 Operators Reducible to Functions . . . . . . . . . . . . . . . . . . . . . . . 236
2.5.1 Regular Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
© 2006 by Taylor & Francis Group, LLC
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2.5.2 The Realization of Some Operators . . . . . . . . . . . . . . . . . . . 239
2.5.3 Efros Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
2.6 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
3.3 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
3.4 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
3.5 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 ConvolutionofDistributionwithaSmoothFunctionofBoundedSup-
port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
3.5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
3.6 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
© 2006 by Taylor & Francis Group, LLC
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3.6.5 Support of a Distribution . . . . . . . . . . . . . . . . . . . . . . . . 355
3.7 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
3.8 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
© 2006 by Taylor & Francis Group, LLC
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 be-
tweenatextbookforstudentsofmathematics,physicsandengineeringandamonographon
thissubject. Itiswellknownthatintegraltransforms, operationalcalculusandgeneralized
functions are the backbone of many branches of pure and applied mathematics. Although
centuriesold,thesesubjectsarestillunderintensivedevelopmentbecausetheyareusefulin
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
justifyOliverHeaviside’s(algebraic)operationalcalculusinthefirstquarterofthetwentieth
century. Methodsconnectedwiththeuseofintegraltransformshavegainedwideacceptance
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.
Thesectionsdealwithconditionsfortheexistenceoftheintegraltransformsinconsider-
ation, 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].
Sometimesapplicationsaregiven. Becauseofthespecialnatureofthisbooksomeextensive
proofsareonlysketched. Thereaderinterestedinmoredetailisreferredforexampletothe
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.
InthisbookwedealonlywithintegraltransformsforR1-functions. Thereaderinterested
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
Mikusin´ski; 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
© 2006 by Taylor & Francis Group, LLC
xii
of functions f and g, continuous on [0,∞) given by J. Mikusin´ski,
Z 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 mul-
tiplication is defined by means of
d Z t
(2) (f ∗g)(t)= f(x)g(t−x)dx.
dt
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, g(x)=b, x∈[0,∞)
equals a function h with h(x) = ab, 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 Mikusin´ski 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.
Chapter3consistsofthetheoryofgeneralizedfunctions. Variousinvestigationshavebeen
put forward in the middle of the last century. The mathematical problems encountered are
twofold: first,tofindananalyticalinterpretationfortheoperationsperformedandtojustify
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
Z +∞
δ(x)=0, x6=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. Mikusin´ski 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.
© 2006 by Taylor & Francis Group, LLC
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
becauseintegralsareunderstoodasLebesgueintegrals. Finally,thereadershouldbefamil-
iar 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 opera-
tional 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 application-
relevant 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 di-
verse 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
wasenvisionedastheculminationofalengthycollaboration. 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
© 2006 by Taylor & Francis Group, LLC