Special Functions

by George E. Andrews, Richard Askey, Ranjan Roy

Overview

Special functions, which include the trigonometric functions, have been used for centuries. Their role in the solution of differential equations was exploited by Newton and Leibniz, and the subject of special functions has been in continuous development ever since. In just the past thirty years several new special functions and applications have been discovered. This treatise presents an overview of the area of special functions, focusing primarily on the hypergeometric functions and the associated hypergeometric series. It includes both important historical results and recent developments and shows how these arise from several areas of mathematics and mathematical physics. Particular emphasis is placed on formulas that can be used in computation. The book begins with a thorough treatment of the gamma and beta functions that are essential to understanding hypergeometric functions. Later chapters discuss Bessel functions, orthogonal polynomials and transformations, the Selberg integral and its applications, spherical harmonics, q-series, partitions, and Bailey chains. This clear, authoritative work will be a lasting reference for students and researchers in number theory, algebra, combinatorics, differential equations, applied mathematics, mathematical computing, and mathematical physics.

Product Details

ISBN-13:	9780521623216
Publisher:	Cambridge University Press
Publication date:	01/13/1999
Series:	Encyclopedia of Mathematics and its Applications , #71
Pages:	684
Product dimensions:	6.46(w) x 9.61(h) x 1.77(d)

Table of Contents

1. The Gamma and Beta functions; 2. The hypergeometric functions; 3. Hypergeometric transformations and identities; 4. Bessel functions and confluent hypergeometric functions; 5. Orthogonal polynomials; 6. Special orthogonal transformations; 7. Topics in orthogonal polynomials; 8. The Selberg integral and its applications; 9. Spherical harmonics; 10. Introduction to q-series; 11. Partitions; 12. Bailey chains; Appendix 1. Infinite products; Appendix 2. Summability and fractional integration; Appendix 3. Asymptotic expansions; Appendix 4. Euler-Maclaurin summation formula; Appendix 5. Lagrange inversion formula; Appendix 6. Series solutions of differential equations.

Editorial Reviews

‘Occasionally there is published a mathematics book that one is compelled to describe as, well, let us say, special. Special Functions is certainly one of those rare books. … this treatise … should become a classic. Every student, user, and researcher in analysis will want to have it close at hand as she/he works.’ The Mathematical Intelligencer

‘… the material is written in an excellent manner … I recommend this book warmly as a rich source of information to everybody who is interested in ‘Special Functions’.’ Zentralblatt MATH

‘ … this book contains a wealth of fascinating material which is presented in a user-friendly way. If you want to extend your knowledge of special functions, this is a good place to start. Even if your interests are in number theory or combinatorics, there is something for you too … the book can be warmly recommended and should be in all good libraries.’ Adam McBride, The Mathematical Gazette

‘… it comes into the range of affordable books that you want to (and probably should have on your desk’. Jean Mawhin, Bulletin of the Belgian Mathematical Society

'The book is full of beautiful and interesting formulae, as was always the case with mathematics centred around special functions. It is written in the spirit of the old masters, with mathemtics developed in terms of formulas. There are many historical comments in the book. It can be recommended as a very useful reference.' European Mathematical Society

'… full of beautiful and interesting formulae … It can be recommended as a very useful reference.' EMS Newsletter

‘a very erudite text and reference in special functions' Allen Stenger, MAA Reviews

From the Publisher