Even though the theories of operational calculus and integral transforms are centuries old, these topics are constantly developing, due to their use in the fields of mathematics, physics, and electrical and radio engineering. Operational Calculus and Related Topics highlights the classical methods and applications as well as the recent advances in the field.Combining the best features of a textbook and a monograph, this volume presents an introduction to operational calculus, integral transforms, and generalized functions, the backbones of pure and applied mathematics. The text examines both the analytical and algebraic aspects of operational calculus and includes a comprehensive survey of classical results while stressing new developments in the field. Among the historical methods considered are Oliver Heaviside's algebraic operational calculus and Paul Dirac's delta function. Other discussions deal with the conditions for the existence of integral transforms, Jan Mikusinski's theory of convolution quotients, operator functions, and the sequential approach to the theory of generalized functions.Benefits…· Discusses theory and applications of integral transforms· Gives inversion, complex-inversion, and Dirac's delta distribution formulas, among others· Offers a short survey of actual results of finite integral transforms, in particular convolution theoremsBecause Operational Calculus and Related Topics provides examples and illustrates the applications to various disciplines, it is an ideal reference for mathematicians, physicists, scientists, engineers, and students.

H.-J. Glaeske is a professor of mathematics at Friedrich-Schiller University, Jena, Germany K.A. Skornik, doctor of mathematics, is a scientist working at the Institute of Mathematics, Polish Academy of Sciences, Katowice, Poland

Preface

xi

List of Symbols

xv

1 Integral Transforms

1

(162)

1.1 Introduction to Operational Calculus

1

(4)

1.2 Integral Transforms — Introductory Remarks

5

(3)

1.3 The Fourier Transform

8

(19)

1.3.1 Definition and Basic Properties

8

(3)

1.3.2 Examples

11

(2)

1.3.3 Operational Properties

13

(4)

1.3.4 The Inversion Formula

17

(4)

1.3.5 Applications

21

(6)

1.4 The Laplace Transform

27

(28)

1.4.1 Definition and Basic Properties

27

(4)

1.4.2 Examples

31

(2)

1.4.3 Operational Properties

33

(4)

1.4.4 The Complex Inversion Formula

37

(3)

1.4.5 Inversion Methods

40

(4)

1.4.6 Asymptotic Behavior

44

(3)

1.4.7 Remarks on the Bilateral Laplace Transform

47

(2)

1.4.8 Applications

49

(6)

1.5 The Mellin Transform

55

(12)

1.5.1 Definition and Basic Properties

55

(3)

1.5.2 Operational Properties

58

(4)

1.5.3 The Complex Inversion Formula

62

(1)

1.5.4 Applications

63

(4)

1.6 The Stieltjes Transform

67

(11)

1.6.1 Definition and Basic Properties

67

(3)

1.6.2 Operational Properties

70

(3)

1.6.3 Asymptotics

73

(2)

1.6.4 Inversion and Application

75

(3)

1.7 The Hilbert Transform

78

(6)

1.7.1 Definition and Basic Properties

78

(3)

1.7.2 Operational Properties

81

(2)

1.7.3 Applications

83

(1)

1.8 Bessel Transforms

84

(23)

1.8.1 The Hankel Transform

85

(8)

1.8.2 The Meijer (K-) Transform

93

(7)

1.8.3 The Kontorovich—Lebedev Transform

100

(6)

1.8.4 Application

106

(1)

1.9 The Mehler—Fock Transform

107

(8)

1.10 Finite Integral Transforms

115

(48)

1.10.1 Introduction

115

(1)

1.10.2 The Chebyshev Transform

116

(6)

1.10.3 The Legendre Transform

122

(9)

1.10.4 The Gegenbauer Transform

131

(6)

1.10.5 The Jacobi Transform

137

(7)

1.10.6 The Laguerre Transform

144

(9)

1.10.7 The Hermite Transform

153

(10)

2 Operational Calculus

163

(108)

2.1 Introduction

163

(4)

2.2 Titchmarsh's Theorem

167

(13)

2.3 Operators

180

(39)

2.3.1 Ring of Functions

180

(5)

2.3.2 The Field of Operators

185

(5)

2.3.3 Finite Parts of Divergent Integrals

190

(11)

2.3.4 Rational Operators

201

(4)

2.3.5 Laplace Transformable Operators

205

(8)

2.3.6 Examples

213

(4)

2.3.7 Periodic Functions

217

(2)

2.4 Bases of the Operator Analysis

219

(17)

2.4.1 Sequences and Series of Operators

219

(7)

2.4.2 Operator Functions

226

(3)

2.4.3 The Derivative of an Operator Function

229

(1)

2.4.4 Properties of the Continuous Derivative of an Operator Function

229

(3)

2.4.5 The Integral of an Operator Function

232

(4)

2.5 Operators Reducible to Functions

236

(11)

2.5.1 Regular Operators

236

(3)

2.5.2 The Realization of Some Operators

239

(3)

2.5.3 Efros Transforms

242

(5)

2.6 Application of Operational Calculus

247

(24)

2.6.1 Ordinary Differential Equations

247

(11)

2.6.2 Partial Differential Equations

258

(13)

3 Generalized Functions

271

(118)

3.1 Introduction

271

(1)

3.2 Generalized Functions — Functional Approach

272

(15)

3.2.1 Introduction

272

(2)

3.2.2 Distributions of One Variable

274

(5)

3.2.3 Distributional Convergence

279

(1)

3.2.4 Algebraic Operations on Distributions

280

(7)

3.3 Generalized Functions — Sequential Approach

287

(24)

3.3.1 The Identification Principle

287

(2)

3.3.2 Fundamental Sequences

289

(8)

3.3.3 Definition of Distributions

297

(3)

3.3.4 Operations with Distributions

300

(3)

3.3.5 Regular Operations

303

(8)

3.4 Delta Sequences

311

(21)

3.4.1 Definition and Properties

311

(6)

3.4.2 Distributions as a Generalization of Continuous Functions

317

(3)

3.4.3 Distributions as a Generalization of Locally Integrable Functions

320

(2)

3.4.4 Remarks about Distributional Derivatives

322

(3)

3.4.5 Functions with Poles

325

(1)

3.4.6 Applications

326

(6)

3.5 Convergent Sequences

332

(15)

3.5.1 Sequences of Distributions

332

(7)

3.5.2 Convergence and Regular Operations

339

(2)

3.5.3 Distributionally Convergent Sequences of Smooth Functions

341

(3)

3.5.4 Convolution of Distribution with a Smooth Function of Bounded Support

344

(2)

3.5.5 Applications

346

(1)

3.6 Local Properties

347

(8)

3.6.1 Inner Product of Two Functions

347

(3)

3.6.2 Distributions of Finite Order

350

(2)

3.6.3 The Value of a Distribution at a Point

352

(3)

3.6.4 The Value of a Distribution at Infinity

355

(1)

3.6.5 Support of a Distribution

355

(1)

3.7 Irregular Operations

355

(26)

3.7.1 Definition

355

(2)

3.7.2 The Integral of Distributions

357

(12)

3.7.3 Convolution of Distributions

369

(6)

3.7.4 Multiplication of Distributions

375

(4)

3.7.5 Applications

379

(2)

3.8 Hilbert Transform and Multiplication Forms

381

(8)

3.8.1 Definition of the Hilbert Transform

381

(2)

3.8.2 Applications and Examples

383

(6)

References

389

(12)

Index

401

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