Theory of Hypergeometric Functions
Theory of Hypergeometric Functions

Author: Kazuhiko Aomoto

Publisher: Springer Science & Business Media

Published: 2011-05-21

Total Pages: 327

ISBN-13: 4431539387

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This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne’s rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff’s classical theory on analytic difference equations on the other.

Basic Hypergeometric Series and Applications
Basic Hypergeometric Series and Applications

Author: Nathan Jacob Fine

Publisher: American Mathematical Soc.

Published: 1988

Total Pages: 142

ISBN-13: 0821815245

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The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. This book provides a simple approach to basic hypergeometric series.

An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions
An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions

Author: Xiao-Jun Yang

Publisher: Academic Press

Published: 2021-02-11

Total Pages: 502

ISBN-13: 0128241543

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An Introduction to Hypergeometric, Supertigonometric, and Superhyperbolic Functions gives a basic introduction to the newly established hypergeometric, supertrigonometric, and superhyperbolic functions from the special functions viewpoint. The special functions, such as the Euler Gamma function, the Euler Beta function, the Clausen hypergeometric series, and the Gauss hypergeometric have been successfully applied to describe the real-world phenomena that involve complex behaviors arising in mathematics, physics, chemistry, and engineering. Provides a historical overview for a family of the special polynomials Presents a logical investigation of a family of the hypergeometric series Proposes a new family of the hypergeometric supertrigonometric functions Presents a new family of the hypergeometric superhyperbolic functions

Hypergeometric Functions and Their Applications
Hypergeometric Functions and Their Applications

Author: James B. Seaborn

Publisher: Springer Science & Business Media

Published: 2013-04-09

Total Pages: 261

ISBN-13: 1475754434

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Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface A wide range of problems exists in classical and quantum physics, engi neering, and applied mathematics in which special functions arise. The procedure followed in most texts on these topics (e. g. , quantum mechanics, electrodynamics, modern physics, classical mechanics, etc. ) is to formu late the problem as a differential equation that is related to one of several special differential equations (Hermite's, Bessel's, Laguerre's, Legendre's, etc. ).

A = B
A = B

Author: Marko Petkovsek

Publisher: CRC Press

Published: 1996-01-01

Total Pages: 231

ISBN-13: 1439864500

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This book is of interest to mathematicians and computer scientists working in finite mathematics and combinatorics. It presents a breakthrough method for analyzing complex summations. Beautifully written, the book contains practical applications as well as conceptual developments that will have applications in other areas of mathematics.From the ta

An Introduction to Special Functions
An Introduction to Special Functions

Author: Carlo Viola

Publisher: Springer

Published: 2016-10-31

Total Pages: 172

ISBN-13: 3319413457

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The subjects treated in this book have been especially chosen to represent a bridge connecting the content of a first course on the elementary theory of analytic functions with a rigorous treatment of some of the most important special functions: the Euler gamma function, the Gauss hypergeometric function, and the Kummer confluent hypergeometric function. Such special functions are indispensable tools in "higher calculus" and are frequently encountered in almost all branches of pure and applied mathematics. The only knowledge assumed on the part of the reader is an understanding of basic concepts to the level of an elementary course covering the residue theorem, Cauchy's integral formula, the Taylor and Laurent series expansions, poles and essential singularities, branch points, etc. The book addresses the needs of advanced undergraduate and graduate students in mathematics or physics.

Equations and Analytical Tools in Mathematical Physics
Equations and Analytical Tools in Mathematical Physics

Author: Yichao Zhu

Publisher: Springer Nature

Published: 2021-10-04

Total Pages: 255

ISBN-13: 9811654417

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​This book highlights a concise and readable introduction to typical treatments of partial differential equations in mathematical physics. Mathematical physics is regarded by many as a profound discipline. In conventional textbooks of mathematical physics, the known and the new pieces of knowledge often intertwine with each other. The book aims to ease readers' struggle by facilitating a smooth transition to new knowledge. To achieve so, the author designs knowledge maps before each chapter and provides comparative summaries in each chapter whenever appropriate. Through these unique ways, readers can clarify the underlying structures among different equations and extend one's vision to the big picture. The book also emphasizes applications of the knowledge by providing practical examples. The book is intended for all those interested in mathematical physics, enabling them to develop a solid command in using partial differential equations to solve physics and engineering problems in a not-so-painful learning experience.

Special Functions & Their Applications
Special Functions & Their Applications

Author: N. N. Lebedev

Publisher: Courier Corporation

Published: 2012-04-30

Total Pages: 340

ISBN-13: 0486139891

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Famous Russian work discusses the application of cylinder functions and spherical harmonics; gamma function; probability integral and related functions; Airy functions; hyper-geometric functions; more. Translated by Richard Silverman.