The Hypergeometric Approach to Integral Transforms and Convolutions / Edition 1

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This volume deals with the theory and applications of integral transforms and convolutions of certain classes of integral, integrodifferential equations, and operational calculus. An extensive discussion is presented, based on the universal hypergeometric approach, i.e. many constructions of convolution and integral transforms are obtained using the theory of Mellin—Barnes integrals and the Mellin transforms of hypergeometric type functions. This approach is spread on so-called index transforms, in which the Kontorovich—Lebedev and the Mehler—Fock transforms play a very important part. The general constructions of index transforms are given and application to the evaluation of improper integral with respect to a parameter of special function (index) is considered. The operational calculus for general integrodifferential operators is constructed for both new types of convolutions. The book is self-contained, and includes a list of symbols with definitions, author and subject indices, and an up-to-date bibliography.
This work will be of interest to researchers and graduate students in the mathematical and physical sciences whose work involves integral transforms and convolutions.

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Editorial Reviews

The authors, both at the Research Scientific Laboratory of Applied Methods of Mathematical Analysis, Byelorussian State University, Minsk, develop a new approach--the hypergeometric approach--to the theory of various integral transforms, convolutions, and their applications to solutions of integro-differential equations, operational calculus, and evaluation of integrals. This approach allows students, postgraduates in mathematics, physicists and technicians, and serious mathematicians and researchers to find new interesting results in the theory of integral transforms, special functions, and convolutions. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780792328568
  • Publisher: Springer Netherlands
  • Publication date: 5/31/1994
  • Series: Mathematics and Its Applications (closed) Series, #287
  • Edition description: 1994
  • Edition number: 1
  • Pages: 340
  • Product dimensions: 0.88 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Preface. 1. Preliminaries. 2. Mellin Convolution Type Transforms with Arbitrary Kernels. 3. H- and G-Transforms. 4. The Generalized H- and G-Transforms. 5. The Generating Operators of Generalized H-Transforms. 6. The Kontorovich—Lebedev Transform. 7. General W-Transform and its Particular Cases. 8. Composition Theorems of Plancherel Type for Index Transforms. 9. Some Examples of Index Transforms and their New Properties. 10. Applications to Evaluation of Index Integrals. 11. Convolutions of Generalized H-Transforms. 12. Generalization of the Notion of Convolution. 13. Leibniz Rules and their Integral Analogues. 14. Convolutions of Generating Operators. 15. Convolution of the Kontorovich—Lebedev Transform. 16. Convolutions of the General Index Transforms. 17. Applications of the Kontorovich—Lebedev Type Convolutions to Integral Equations. 18. Convolutional Ring Calpha. 19. The Fields of the Convolution Quotients. 20. The Cauchy Problem for Erdelyi—Kober Operators. 21. Operational Method of Solution of Some Convolution Equations. References. Author Index. Subject Index. Notations.

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