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The Interrelations of the Fundamental Solutions of the Hypergeometric Equation
The Confluent Hypergeometric Function
Author: Herbert Buchholz
Publisher: Springer Science & Business Media
Published: 2013-11-22
Total Pages: 255
ISBN-13: 3642883966
DOWNLOAD EBOOK →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.
Advanced Methods for the Solution of Differential Equations
Author: Marvin E. Goldstein
Publisher:
Published: 1973
Total Pages: 364
ISBN-13:
DOWNLOAD EBOOK →This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs.
Gröbner Deformations of Hypergeometric Differential Equations
Author: Mutsumi Saito
Publisher: Springer Science & Business Media
Published: 2013-03-09
Total Pages: 261
ISBN-13: 366204112X
DOWNLOAD EBOOK →The theory of Gröbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gröbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric PDE's introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.
The Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory
An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions
Author: Xiao-Jun Yang
Publisher: Academic Press
Published: 2021-01-23
Total Pages: 504
ISBN-13: 0323852823
DOWNLOAD EBOOK →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
Confluent Hypergeometric Functions
Formulas and Theorems for the Special Functions of Mathematical Physics
Author: Wilhelm Magnus
Publisher: Springer Science & Business Media
Published: 2013-11-11
Total Pages: 516
ISBN-13: 3662117614
DOWNLOAD EBOOK →This is a new and enlarged English edition of the book which, under the title "Formeln und Satze fur die Speziellen Funktionen der mathe matischen Physik" appeared in German in 1946. Much of the material (part of it unpublished) did not appear in the earlier editions. We hope that these additions will be useful and yet not too numerous for the purpose of locating .with ease any particular result. Compared to the first two (German) editions a change has taken place as far as the list of references is concerned. They are generally restricted to books and monographs and accomodated at the end of each individual chapter. Occasional references to papers follow those results to which they apply. The authors felt a certain justification for this change. At the time of the appearance of the previous edition nearly twenty years ago much of the material was scattered over a number of single contributions. Since then most of it has been included in books and monographs with quite exhaustive bibliographies. For information about numerical tables the reader is referred to "Mathematics of Computation", a periodical publis hed by the American Mathematical Society; "Handbook of Mathe matical Functions" with formulas, graphs and mathematical tables National Bureau of Standards Applied Mathematics Series, 55, 1964, 1046 pp., Government Printing Office, Washington, D.C., and FLETCHER, MILLER, ROSENHEAD, Index of Mathematical Tables, Addison-Wesley, Reading, Mass.) .. There is a list of symbols and abbreviations at the end of the book.
Special Functions and Orthogonal Polynomials
Author: Richard Beals
Publisher: Cambridge University Press
Published: 2016-05-17
Total Pages: 489
ISBN-13: 1316578054
DOWNLOAD EBOOK →The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painlevé transcendents, which have been termed the 'special functions of the twenty-first century'.
