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In this paper, the author presents a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, the author gets identities for the hypergeometric series and shows that values of at some points can be expressed in terms of gamma functions, together with certain elementary functions. The author tabulates the values of that can be obtained with this method and finds that this set includes almost all previously known values and many previously unknown values.
Book Synopsis Special Values of the Hypergeometric Series by : Akihito Ebisu
Download or read book Special Values of the Hypergeometric Series written by Akihito Ebisu and published by American Mathematical Soc.. This book was released on 2017-07-13 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, the author presents a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, the author gets identities for the hypergeometric series and shows that values of at some points can be expressed in terms of gamma functions, together with certain elementary functions. The author tabulates the values of that can be obtained with this method and finds that this set includes almost all previously known values and many previously unknown values.
The subject of this book is the higher transcendental function known as the confluent hypergeometric function. In the last two decades this function has taken on an ever increasing significance because of its use in the application of mathematics to physical and technical problems. There is no doubt that this trend will continue until the general theory of confluent hypergeometric functions becomes familiar to the majority of physicists in much the same way as the cylinder functions, which were previously less well known, are now used in many engineering and physical problems. This book is intended to further this development. The important practical significance of the functions which are treated hardly demands an involved discussion since they include, as special cases, a number of simpler special functions which have long been the everyday tool of the physicist. It is sufficient to mention that these include, among others, the logarithmic integral, the integral sine and cosine, the error integral, the Fresnel integral, the cylinder functions and the cylinder function in parabolic cylindrical coordinates. For anyone who puts forth the effort to study the confluent hypergeometric function in more detail there is the inestimable advantage of being able to understand the properties of other functions derivable from it. This gen eral point of view is particularly useful in connection with series ex pansions valid for values of the argument near zero or infinity and in connection with the various integral representations.
Book Synopsis The Confluent Hypergeometric Function by : Herbert Buchholz
Download or read book The Confluent Hypergeometric Function written by Herbert Buchholz and published by Springer Science & Business Media. This book was released on 2013-11-22 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of this book is the higher transcendental function known as the confluent hypergeometric function. In the last two decades this function has taken on an ever increasing significance because of its use in the application of mathematics to physical and technical problems. There is no doubt that this trend will continue until the general theory of confluent hypergeometric functions becomes familiar to the majority of physicists in much the same way as the cylinder functions, which were previously less well known, are now used in many engineering and physical problems. This book is intended to further this development. The important practical significance of the functions which are treated hardly demands an involved discussion since they include, as special cases, a number of simpler special functions which have long been the everyday tool of the physicist. It is sufficient to mention that these include, among others, the logarithmic integral, the integral sine and cosine, the error integral, the Fresnel integral, the cylinder functions and the cylinder function in parabolic cylindrical coordinates. For anyone who puts forth the effort to study the confluent hypergeometric function in more detail there is the inestimable advantage of being able to understand the properties of other functions derivable from it. This gen eral point of view is particularly useful in connection with series ex pansions valid for values of the argument near zero or infinity and in connection with the various integral representations.
Modern algorithmic techniques for summation, most of which were introduced in the 1990s, are developed here and carefully implemented in the computer algebra system MapleTM. The algorithms of Fasenmyer, Gosper, Zeilberger, Petkovšek and van Hoeij for hypergeometric summation and recurrence equations, efficient multivariate summation as well as q-analogues of the above algorithms are covered. Similar algorithms concerning differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the book. The combination of these results gives orthogonal polynomials and (hypergeometric and q-hypergeometric) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given. The materials covered are suitable for an introductory course on algorithmic summation and will appeal to students and researchers alike.
Book Synopsis Hypergeometric Summation by : Wolfram Koepf
Download or read book Hypergeometric Summation written by Wolfram Koepf and published by Springer. This book was released on 2014-06-10 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern algorithmic techniques for summation, most of which were introduced in the 1990s, are developed here and carefully implemented in the computer algebra system MapleTM. The algorithms of Fasenmyer, Gosper, Zeilberger, Petkovšek and van Hoeij for hypergeometric summation and recurrence equations, efficient multivariate summation as well as q-analogues of the above algorithms are covered. Similar algorithms concerning differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the book. The combination of these results gives orthogonal polynomials and (hypergeometric and q-hypergeometric) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given. The materials covered are suitable for an introductory course on algorithmic summation and will appeal to students and researchers alike.
"In 1813, Gauss first outlined his studies of the hypergeometric series which has been of great significance in the mathematical modelling of physical phenomena. This detailed monograph outlines the fundamental relationships between the hypergeometric function and special functions. In nine comprehensive chapters, Dr. Rao and Dr. Lakshminarayanan present a unified approach to the study of special functions of mathematics using Group theory. The book offers fresh insight into various aspects of special functions and their relationship, utilizing transformations and group theory and their applications. It will lay the foundation for deeper understanding by both experienced researchers and novice students." -- Prové de l'editor.
Book Synopsis Generalized Hypergeometric Functions by : K. Srinivasa Rao
Download or read book Generalized Hypergeometric Functions written by K. Srinivasa Rao and published by . This book was released on 2018 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "In 1813, Gauss first outlined his studies of the hypergeometric series which has been of great significance in the mathematical modelling of physical phenomena. This detailed monograph outlines the fundamental relationships between the hypergeometric function and special functions. In nine comprehensive chapters, Dr. Rao and Dr. Lakshminarayanan present a unified approach to the study of special functions of mathematics using Group theory. The book offers fresh insight into various aspects of special functions and their relationship, utilizing transformations and group theory and their applications. It will lay the foundation for deeper understanding by both experienced researchers and novice students." -- Prové de l'editor.
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.
Book Synopsis Basic Hypergeometric Series and Applications by : Nathan Jacob Fine
Download or read book Basic Hypergeometric Series and Applications written by Nathan Jacob Fine and published by American Mathematical Soc.. This book was released on 1988 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 overview of special functions, focusing on the hypergeometric functions and the associated hypergeometric series.
Book Synopsis Special Functions by : George E. Andrews
Download or read book Special Functions written by George E. Andrews and published by Cambridge University Press. This book was released on 1999 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: An overview of special functions, focusing on the hypergeometric functions and the associated hypergeometric series.
Zeta and q-Zeta Functions and Associated Series and Integrals is a thoroughly revised, enlarged and updated version of Series Associated with the Zeta and Related Functions. Many of the chapters and sections of the book have been significantly modified or rewritten, and a new chapter on the theory and applications of the basic (or q-) extensions of various special functions is included. This book will be invaluable because it covers not only detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions, but stimulating historical accounts of a large number of problems and well-classified tables of series and integrals. Detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions
Book Synopsis Zeta and Q-Zeta Functions and Associated Series and Integrals by : H. M. Srivastava
Download or read book Zeta and Q-Zeta Functions and Associated Series and Integrals written by H. M. Srivastava and published by Elsevier. This book was released on 2011-10-25 with total page 675 pages. Available in PDF, EPUB and Kindle. Book excerpt: Zeta and q-Zeta Functions and Associated Series and Integrals is a thoroughly revised, enlarged and updated version of Series Associated with the Zeta and Related Functions. Many of the chapters and sections of the book have been significantly modified or rewritten, and a new chapter on the theory and applications of the basic (or q-) extensions of various special functions is included. This book will be invaluable because it covers not only detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions, but stimulating historical accounts of a large number of problems and well-classified tables of series and integrals. Detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions
An extensive summary of mathematical functions that occur in physical and engineering problems
Book Synopsis Handbook of Mathematical Functions by : Milton Abramowitz
Download or read book Handbook of Mathematical Functions written by Milton Abramowitz and published by Courier Corporation. This book was released on 1965-01-01 with total page 1068 pages. Available in PDF, EPUB and Kindle. Book excerpt: An extensive summary of mathematical functions that occur in physical and engineering problems
Book Synopsis Some Special Values for the BC Type Hypergeometric Function by : R. J. Beerends
Download or read book Some Special Values for the BC Type Hypergeometric Function written by R. J. Beerends and published by . This book was released on 1991 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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.
Book Synopsis An Introduction to Special Functions by : Carlo Viola
Download or read book An Introduction to Special Functions written by Carlo Viola and published by Springer. This book was released on 2016-10-31 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
