Synopses & Reviews
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.
Review
"...the authors demonstrate a superb familiarity with the historical roots of their subject...All of the chapters are beautifully written...wonderful historical insights...Special Functions will certainly emerge as the chief textbook and reference on special functions for the next several years...This book joins F. W. J. Olver's Asymptotics and Special Functions, first published in 1974 [Academic Press, New York; MR 55 #8655], as the only general books on special functions during the past three decades that belong 'in the Hobbs class,' to quote G. H. Hardy." Mathematical Reviews"The book is packed with brief, challenging superveniences that make it a browser's delight. One of the delightful features of this book is how the sense of history, of mathematics being created and savored, informs the text. This is a splendid work, and I predict that it will be a bestseller as well." Bulletin of the American Mathematical Society
Synopsis
An overview of special functions, focusing on the hypergeometric functions and the associated hypergeometric series.
Synopsis
Presents an overview of the area of special functions, focusing on the hypergeometric functions and the associated hypergeometric series, such as the gamma and beta functions, Bessel functions, orthogonal polynomials, the Selberg integral and its applications, spherical harmonics, q-series, partitions, and Bailey chains. Particular emphasis is placed on formulas that can be used in computation.
Synopsis
Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series. In addition to relatively new work, such as Selbergâs multidimensional integrals, many important but relatively unknown nineteenth century results are included. The authors provide organizing ideas, motivation, and historical background for the study and application of some important special functions. This clearly expressed and readable work can serve as a learning tool and lasting reference for students and researchers in special functions, mathematical physics, differential equations, mathematical computing, number theory, and combinatorics.
Description
Includes bibliographical references (p. 641-653) and indexes.
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.