Operational Calculus and Related Topics (Analytical Methods and Special Functions)

420 Pages·2006·2.25 MB·English

by  H.-J. Glaeske

Operational
Calculus and
Related Topics
ANALYTICAL METHODS AND SPECIAL FUNCTIONS
An International Series of Monographs in Mathematics
FOUNDING EDITOR: A.P. Prudnikov (Russia)
SERIES EDITORS: C.F. Dunkl (USA), H.-J. Glaeske (Germany) and M. Saigo (Japan)
(cid:3)
∫∫
Volume 1
Series of Faber Polynomials, P.K. Suetin
Volume 2
Inverse Spectral Problems for Linear Differential Operators
and Their Applications, V.A. Yurko
Volume 3
Orthogonal Polynomials in Two Variables, P.K. Suetin
Volume 4
Fourier Transforms and Approximations, A.M. Sedletskii
Volume 5
Hypersingular Integrals and Their Applications, S. Samko
Volume 6
Methods of the Theory of Generalized Functions, V.S. Vladimirov
Volume 7
Distributions, Integral Transforms and Applications, W. Kierat
and U. Sztaba
Volume 8
Bessel Functions and Their Applications, B. Korenev
Volume 9
H-Transforms: Theory and Applications, A.A. Kilbas and M. Saigo
Volume 10
Operational Calculus and Related Topics, H.-J. Glaeske,
A.P. Prudnikov, and 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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In memory of
A. P. Prudnikov
January 14, 1927 — January 10, 1999

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
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
ix
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