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The Hypergeometric Approach to Integral Transforms and Convolutions : MATHEMATICS AND ITS APPLICATIONS (KLUWER ) - Semen B. IAkubovich

The Hypergeometric Approach to Integral Transforms and Convolutions



Published: 31st May 1994
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The aim of this book is to develop a new approach which we called the hyper­ geometric one to the theory of various integral transforms, convolutions, and their applications to solutions of integro-differential equations, operational calculus, and evaluation of integrals. We hope that this simple approach, which will be explained below, allows students, post graduates in mathematics, physicists and technicians, and serious mathematicians and researchers to find in this book new interesting results in the theory of integral transforms, special functions, and convolutions. The idea of this approach can be found in various papers of many authors, but systematic discussion and development is realized in this book for the first time. Let us explain briefly the basic points of this approach. As it is known, in the theory of special functions and its applications, the hypergeometric functions play the main role. Besides known elementary functions, this class includes the Gauss's, Bessel's, Kummer's, functions et c. In general case, the hypergeometric functions are defined as a linear combinations of the Mellin-Barnes integrals. These ques­ tions are extensively discussed in Chapter 1. Moreover, the Mellin-Barnes type integrals can be understood as an inversion Mellin transform from the quotient of products of Euler's gamma-functions. Thus we are led to the general construc­ tions like the Meijer's G-function and the Fox's H-function.

Preliminariesp. 1
Mellin Convolution Type Transforms With Arbitrary Kernelsp. 15
H- and G-transformsp. 41
The Generalized H- and G-transformsp. 59
The Generating Operators of Generalized H-transformsp. 69
The Kontorovich-Lebedev Transformp. 79
General W-transform and its Particular Casesp. 109
Composition Theorems of Plancherel Type for Index Transformsp. 139
Some Examples of Index Transforms and Their New Propertiesp. 149
Applications to Evaluation of Index Integralsp. 167
Convolutions of Generalized H-transformsp. 173
Generalization of the Notion of Convolutionp. 183
Leibniz Rules and Their Integral Analoguesp. 189
Convolutions of Generating Operatorsp. 205
Convolution of the Kontorovich-Lebedev Transformp. 213
Convolutions of the General Index Transformsp. 229
Applications of the Kontorovich-Lebedev type Convolutions to Integral Equationsp. 241
Convolutional Ring [actual symbol not reproducible]p. 253
The Fields of the Convolution Quotientsp. 265
The Cauchy Problem for Erdelyi-Kober Operatorsp. 277
Operational Method of Solution of some Convolution Equationsp. 287
Referencesp. 295
Author Indexp. 311
Subject Indexp. 317
Notationsp. 321
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792328568
ISBN-10: 0792328566
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 324
Published: 31st May 1994
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 25.4 x 17.15  x 2.54
Weight (kg): 0.75