product iamge

Operational Calculus and Related Topics

This document was uploaded by one of our users. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.

Download
Search on Amazon

Simply click on the Download Book button.

Yes, Book downloads on Ebookily are 100% Free.

Sometimes the book is free on Amazon As well, so go ahead and hit "Search on Amazon"

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 Mikusiński’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 theorems

Because 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.

Author(s): A. P. Prudnikov, K.A. Skórnik
Series: Analytical Methods and Special Functions
Edition: 1
Publisher: Chapman and Hall/CRC
Year: 2006

Language: English
Pages: 424

H.-J. Glaeske, A. P. Prudnikov, K.A. Skórnik - Operational Calculus and Related Topics (Analytical Methods and Special Functions) (Chapman ,2006).jpg......Page 1
Operational Calculus and Related Topics......Page 2
Dedication......Page 4
Contents......Page 5
Preface......Page 9
List of Symbols......Page 12
Table of Contents......Page 0
1.1 Introduction to Operational Calculus......Page 14
1.2 Integral Transforms – Introductory Remarks......Page 18
1.3.1 Definition and Basic Properties......Page 21
1.3.2 Examples......Page 24
1.3.3 Operational Properties......Page 26
1.3.4 The Inversion Formula......Page 30
1.3.5 Applications......Page 34
1.4.1 Definition and Basic Properties......Page 40
1.4.2 Examples......Page 44
1.4.3 Operational Properties......Page 46
1.4.4 The Complex Inversion Formula......Page 50
1.4.5 Inversion Methods......Page 53
1.4.6 Asymptotic Behavior......Page 57
1.4.7 Remarks on the Bilateral Laplace Transform......Page 60
1.4.8 Applications......Page 62
1.5.1 Definition and Basic Properties......Page 68
1.5.2 Operational Properties......Page 71
1.5.3 The Complex Inversion Formula......Page 75
1.5.4 Applications......Page 76
1.6.1 Definition and Basic Properties......Page 80
1.6.2 Operational Properties......Page 83
1.6.3 Asymptotics......Page 86
1.6.4 Inversion and Application......Page 88
1.7.1 Definition and Basic Properties......Page 91
1.7.2 Operational Properties......Page 94
1.7.3 Applications......Page 96
1.8 Bessel Transforms......Page 97
1.8.1 The Hankel Transform......Page 98
1.8.2 The Meijer (K-) Transform......Page 106
1.8.3 The Kontorovich–Lebedev Transform......Page 113
1.8.4 Application......Page 119
1.9 The Mehler–Fock Transform......Page 120
1.10.1 Introduction......Page 128
1.10.2 The Chebyshev Transform......Page 129
1.10.3 The Legendre Transform......Page 135
1.10.4 The Gegenbauer Transform......Page 144
1.10.5 The Jacobi Transform......Page 150
1.10.6 The Laguerre Transform......Page 157
1.10.7 The Hermite Transform......Page 166
2.1 Introduction......Page 175
2.2 Titchmarsh’s Theorem......Page 179
2.3.1 Ring of Functions......Page 192
2.3.2 The Field of Operators......Page 197
2.3.3 Finite Parts of Divergent Integrals......Page 202
2.3.4 Rational Operators......Page 213
2.3.5 Laplace Transformable Operators......Page 217
2.3.6 Examples......Page 225
2.3.7 Periodic Functions......Page 229
2.4.1 Sequences and Series of Operators......Page 231
2.4.2 Operator Functions......Page 238
2.4.4 Properties of the Continuous Derivative of an Operator Function......Page 241
2.4.5 The Integral of an Operator Function......Page 244
2.5.1 Regular Operators......Page 248
2.5.2 The Realization of Some Operators......Page 251
2.5.3 Efros Transforms......Page 254
2.6.1 Ordinary Differential Equations......Page 259
2.6.2 Partial Differential Equations......Page 270
3.1 Introduction......Page 282
3.2.1 Introduction......Page 283
3.2.2 Distributions of One Variable......Page 285
The Space D(R)......Page 286
Definition and Examples of Distributions......Page 287
3.2.3 Distributional Convergence......Page 290
3.2.4 Algebraic Operations on Distributions......Page 291
Differentiation of Distributions......Page 292
Linear Transformations of an Independent Variable......Page 296
The Antiderivative of a Distribution......Page 297
3.3.1 The Identification Principle......Page 298
3.3.2 Fundamental Sequences......Page 300
3.3.3 Definition of Distributions......Page 308
3.3.4 Operations with Distributions......Page 311
Properties of Algebraic Operations on Distributions......Page 313
3.3.5 Regular Operations......Page 314
The Following Operations Are Regular:......Page 315
3.4.1 Definition and Properties......Page 322
3.4.2 Distributions as a Generalization of Continuous Functions......Page 328
3.4.3 Distributions as a Generalization of Locally Integrable Functions......Page 331
3.4.4 Remarks about Distributional Derivatives......Page 333
3.4.5 Functions with Poles......Page 336
3.4.6 Applications......Page 337
3.5.1 Sequences of Distributions......Page 343
3.5.2 Convergence and Regular Operations......Page 350
3.5.3 Distributionally Convergent Sequences of Smooth Functions......Page 352
3.5.4 Convolution of Distribution with a Smooth Function of Bounded Support......Page 355
3.5.5 Applications......Page 357
3.6.1 Inner Product of Two Functions......Page 358
3.6.2 Distributions of Finite Order......Page 361
3.6.3 The Value of a Distribution at a Point......Page 363
3.7.1 Definition......Page 366
Historical Remarks......Page 367
3.7.2 The Integral of Distributions......Page 368
Differentiation of a Piecewise Continuous Function......Page 375
Distributions with a One-Point Support......Page 378
Applications......Page 379
1. Convolution of Two Smooth Functions......Page 380
2. Convolution of Two Distributions......Page 381
3. Properties of Convolution of Distributions......Page 383
3.7.4 Multiplication of Distributions......Page 386
Properties of the Product......Page 387
On the Associativity of the Product......Page 389
Nonexistence of δ2......Page 390
The Product x · 1x......Page 391
3.8.1 Definition of the Hilbert Transform......Page 392
3.8.2 Applications and Examples......Page 394
References......Page 399