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The Hypergeometric Approach to Integral Transforms and Convolutions

  • Book
  • © 1994

Overview

Authors:
  1. Semen B. Yakubovich

    1. Department of Mathematics and Mechanics, Beylorussian State University, Minsk, Byelorussia, Russia

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  2. Yurii F. Luchko

    1. Department of Mathematics and Mechanics, Beylorussian State University, Minsk, Byelorussia, Russia

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Part of the book series: Mathematics and Its Applications (MAIA, volume 287)

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Preliminaries

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 1-14

  3. Mellin Convolution Type Transforms With Arbitrary Kernels

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 15-40

  4. H- and G-transforms

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 41-58

  5. The Generalized H- and G-transforms

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 59-68

  6. The Generating Operators of Generalized H-transforms

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 69-78

  7. The Kontorovich-Lebedev Transform

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 79-108

  8. General W-transform and its Particular Cases

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 109-138

  9. Composition Theorems of Plancherel Type for Index Transforms

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 139-148

  10. Some Examples of Index Transforms and Their New Properties

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 149-166

  11. Applications to Evaluation of Index Integrals

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 167-172

  12. Convolutions of Generalized H-transforms

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 173-182

  13. Generalization of the Notion of Convolution

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 183-188

  14. Leibniz Rules and Their Integral Analogues

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 189-204

  15. Convolutions of Generating Operators

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 205-212

  16. Convolution of the Kontorovich-Lebedev Transform

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 213-228

  17. Convolutions of the General Index Transforms

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 229-240

  18. Applications of the Kontorovich-Lebedev type Convolutions to Integral Equations

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 241-252

  19. Convolutional Ring Cα

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 253-264

  20. The Fields of the Convolution Quotients

    • Semen B. Yakubovich, Yurii F. Luchko
    Pages 265-276
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics and Mechanics, Beylorussian State University, Minsk, Byelorussia, Russia

    Semen B. Yakubovich, Yurii F. Luchko

Bibliographic Information

  • Book Title: The Hypergeometric Approach to Integral Transforms and Convolutions

  • Authors: Semen B. Yakubovich, Yurii F. Luchko

  • Series Title: Mathematics and Its Applications

  • DOI: https://doi.org/10.1007/978-94-011-1196-6

  • Publisher: Springer Dordrecht

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media Dordrecht 1994

  • Hardcover ISBN: 978-0-7923-2856-8Published: 31 May 1994

  • Softcover ISBN: 978-94-010-4523-0Published: 08 October 2012

  • eBook ISBN: 978-94-011-1196-6Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: XI, 324

  • Topics: Integral Transforms, Operational Calculus, Special Functions, Integral Equations

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