UID:
edocfu_9959235956502883
Format:
1 online resource (xvi, 664 pages) :
,
digital, PDF file(s).
ISBN:
1-316-08754-9
,
1-107-26686-6
,
1-107-26330-1
,
1-107-26994-6
,
1-107-32593-5
Series Statement:
Encyclopedia of mathematics and its applications ;
Content:
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, including Bessel functions and classical orthogonal polynomials, using the basic building block of the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. Other topics include q-extensions of beta integrals and of hypergeometric series, Bailey chains, spherical harmonics, and applications to combinatorial problems. 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.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Preface; 1 The Gamma and Beta Functions; 1.1 The Gamma and Beta Integrals and Functions; 1.2 The Euler Reflection Formula; 1.3 The Hurwitz and Riemann Zeta Functions; 1.4 Stirling's Asymptotic Formula; 1.5 Gauss's Multiplication Formula for Г(mx); 1.6 Integral Representations for Log Г(x) and Ψ(x); 1.7 Kummer's Fourier Expansion of Log Г(x); 1.8 Integrals of Dirichlet and Volumes of Ellipsoids; 1.9 The Bohr-Mollerup Theorem; 1.10 Gauss and Jacobi Sums; 1.11 A Probabilistic Evaluation of the Beta Function
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1.12 The p-adic Gamma FunctionExercises; 2 The Hypergeometric Functions; 2.1 The Hypergeometric Series; 2.2 Euler's Integral Representation; 2.3 The Hypergeometric Equation; 2.4 The Barnes Integral for the Hypergeometric Function; 2.5 Contiguous Relations; 2.6 Dilogarithms; 2.7 Binomial Sums; 2.8 Dougall's Bilateral Sum; 2.9 Fractional Integration by Parts and Hypergeometric Integrals; Exercises; 3 Hypergeometric Transformations and Identities; 3.1 Quadratic Transformations; 3.2 The Arithmetic-Geometric Mean and Elliptic Integrals; 3.3 Transformations of Balanced Series
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3.4 Whipple's Transformation3.5 Dougall's Formula and Hypergeometric Identities; 3.6 Integral Analogs of Hypergeometric Sums; 3.7 Contiguous Relations; 3.8 The Wilson Polynomials; 3.9 Quadratic Transformations - Riemann's View; 3.10 Indefinite Hypergeometric Summation; 3.11 The W-Z Method; 3.12 Contiguous Relations and Summation Methods; Exercises; 4 Bessel Functions and Confluent Hypergeometric Functions; 4.1 The Confluent Hypergeometric Equation; 4.2 Barnes's Integral for 1F1; 4.3 Whittaker Functions; 4.4 Examples of 1F1 and Whittaker Functions; 4.5 Bessel's Equation and Bessel Functions
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4.6 Recurrence Relations4.7 Integral Representations of Bessel Functions; 4.8 Asymptotic Expansions; 4.9 Fourier Transforms and Bessel Functions; 4.10 Addition Theorems; 4.11 Integrals of Bessel Functions; 4.12 The Modified Bessel Functions; 4.13 Nicholson's Integral; 4.14 Zeros of Bessel Functions; 4.15 Monotonicity Properties of Bessel Functions; 4.16 Zero-Free Regions for 1F1 Functions; Exercises; 5 Orthogonal Polynomials; 5.1 Chebyshev Polynomials; 5.2 Recurrence; 5.3 Gauss Quadrature; 5.4 Zeros of Orthogonal Polynomials; 5.5 Continued Fractions; 5.6 Kernel Polynomials
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5.7 Parseval's Formula5.8 The Moment-Generating Function; Exercises; 6 Special Orthogonal Polynomials; 6.1 Hermite Polynomials; 6.2 Laguerre Polynomials; 6.3 Jacobi Polynomials and Gram Determinants; 6.4 Generating Functions for Jacobi Polynomials; 6.5 Completeness of Orthogonal Polynomials; 6.6 Asymptotic Behavior of Pn(α,β)(x) for Large n; 6.7 Integral Representations of Jacobi Polynomials; 6.8 Linearization of Products of Orthogonal Polynomials; 6.9 Matching Polynomials; 6.10 The Hypergeometric Orthogonal Polynomials; 6.11 An Extension of the Ultraspherical Polynomials; Exercises
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7 Topics in Orthogonal Polynomials
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English
Additional Edition:
ISBN 0-521-78988-5
Additional Edition:
ISBN 0-521-62321-9
Language:
English
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