Special FunctionsCambridge University Press, 1999 - 664 Seiten 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. |
Inhalt
The Hypergeometric Functions | 61 |
Hypergeometric Transformations and Identities | 124 |
Bessel Functions and Confluent Hypergeometric Functions | 187 |
Orthogonal Polynomials | 240 |
Special Orthogonal Polynomials | 277 |
Topics in Orthogonal Polynomials | 355 |
The Selberg Integral and Its Applications | 401 |
Spherical Harmonics | 445 |
Bailey Chains | 577 |
A Infinite Products | 595 |
The Cesaro Means C α | 602 |
Asymptotic Expansions | 611 |
EulerMaclaurin Summation Formula | 617 |
Lagrange Inversion Formula | 629 |
F Series Solutions of Differential Equations | 637 |
655 | |
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Häufige Begriffe und Wortgruppen
analytic apply asymptotic expansion b₁ Bessel functions beta integral Chapter Chebyshev polynomials coefficients contiguous relations continuous function converges Corollary Deduce defined denote derive differential equation equal Euler's evaluation Exercise F₁ finite formula Fourier functional equation gamma function Gauss given gives harmonic polynomials Hermite polynomials hypergeometric functions hypergeometric series identity implies inequality integer integrand Ja(x Jacobi polynomials Laguerre polynomials Legendre Lemma linear Mellin transform method Note number of partitions Observe obtain orthogonal polynomials orthogonal with respect parameters Pn(x polynomial of degree positive integer proof proves the theorem q-analog q-binomial theorem quadratic transformation Ramanujan recurrence relation result right side satisfies Selberg's sequence Show sin² solution spherical harmonics summation Suppose term ultraspherical polynomials zeros
Verweise auf dieses Buch
Symmetric Functions and Combinatorial Operators on Polynomials, Ausgabe 99 Alain Lascoux Eingeschränkte Leseprobe |
Asymptotic Combinatorics with Application to Mathematical Physics V.A. Malyshev,A.M. Vershik Eingeschränkte Leseprobe - 2002 |