Special functions A graduate text 1st Edition by Richard Beals, Roderick Wong ISBN 052119797X 9780521197977 by Beals R., Wong R. 0511789599 instant download after payment.

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ISBN 10: 052119797X 
ISBN 13: 9780521197977
Author: Richard Beals, Roderick Wong

The subject of special functions is often presented as a collection of disparate results, which are rarely organised in a coherent way. This book answers the need for a different approach to the subject. The authors' main goals are to emphasise general unifying principles coherently and to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more, including chapters on discrete orthogonal polynomials and elliptic functions. The authors show how a very large part of the subject traces back to two equations - the hypergeometric equation and the confluent hypergeometric equation - and describe the various ways in which these equations are canonical and special. Providing ready access to theory and formulas, this book serves as an ideal graduate-level textbook as well as a convenient reference.

Special functions A graduate text 1st Table of contents:

1 Orientation
1.1 Power series solutions
1.2 The gamma and beta functions
1.3 Three questions
1.4 Elliptic functions
1.5 Exercises
1.6 Summary
1.6.1 Power series solutions
1.6.2 The gamma and beta functions
1.6.3 Three questions
1.6.4 Elliptic functions
1.7 Remarks
2 Gamma, beta, zeta
2.1 The gamma and beta functions
2.2 Euler's product and reflection formulas
2.3 Formulas of Legendre and Gauss
2.4 Two characterizations of the gamma function
2.5 Asymptotics of the gamma function
2.6 The psi function and the incomplete gamma function
2.7 The Selberg integral
2.8 The zeta function
2.9 Exercises
2.10 Summary
2.10.1 The gamma function
2.10.2 Euler's product and reflection formulas
2.10.3 Formulas of Legendre and Gauss
2.10.4 Two characterizations of the gamma function
2.10.5 Asymptotics of the gamma function
2.10.6 The psi function and the incomplete gamma function
2.10.7 The Selberg integral
2.10.8 The zeta function
2.11 Remarks
3 Second-order differential equations
3.1 Transformations, symmetry
3.2 Existence and uniqueness
3.3 Wronskians, Green's functions, comparison
3.4 Polynomials as eigenfunctions
3.5 Maxima, minima, estimates
3.6 Some equations of mathematical physics
3.7 Equations and transformations
3.8 Exercises
3.9 Summary
3.9.1 Transformations, symmetry
3.9.2 Existence and uniqueness
3.9.3 Wronskians, Green's functions, comparison
3.9.4 Polynomials as eigenfunctions
3.9.5 Maxima, minima, estimates
3.9.6 Some equations of mathematical physics
3.9.7 Equations and transformations
3.10 Remarks
4 Orthogonal polynomials
4.1 General orthogonal polynomials
4.2 Classical polynomials: general properties, I
4.3 Classical polynomials: general properties, II
4.4 Hermite polynomials
4.5 Laguerre polynomials
4.6 Jacobi polynomials
4.7 Legendre and Chebyshev polynomials
4.8 Expansion theorems
4.9 Functions of second kind
4.10 Exercises
4.11 Summary
4.11.1 General orthogonal polynomials
4.11.2 Classical polynomials: general properties, I
4.11.3 Classical polynomials: general properties, II
4.11.4 Hermite polynomials
4.11.5 Laguerre polynomials
4.11.6 Jacobi polynomials
4.11.7 Legendre and Chebyshev polynomials
4.11.8 Expansion theorems
4.11.9 Functions of second kind
4.12 Remarks
5 Discrete orthogonal polynomials
5.1 Discrete weights and difference operators
5.2 The discrete Rodrigues formula
5.3 Charlier polynomials
5.4 Krawtchouk polynomials
5.5 Meixner polynomials
5.6 Chebyshev–Hahn polynomials
5.7 Exercises
5.8 Summary
5.8.1 Discrete weights and difference operators
5.8.2 The discrete Rodrigues formula
5.8.3 Charlier polynomials
5.8.4 Krawtchouk polynomials
5.8.5 Meixner polynomials
5.8.6 Chebyshev–Hahn polynomials
5.9 Remarks
6 Confluent hypergeometric functions
6.1 Kummer functions
6.2 Kummer functions of the second kind
6.3 Solutions when c is an integer
6.4 Special cases
6.5 Contiguous functions
6.6 Parabolic cylinder functions
6.7 Whittaker functions
6.8 Exercises
6.9 Summary
6.9.1 Kummer functions
6.9.2 Kummer functions of the second kind
6.9.3 Solutions when c is an integer
6.9.4 Special cases
6.9.5 Contiguous functions
6.9.6 Parabolic cylinder functions
6.9.7 Whittaker functions
6.10 Remarks
7 Cylinder functions
7.1 Bessel functions
7.2 Zeros of real cylinder functions
7.3 Integral representations
7.4 Hankel functions
7.5 Modified Bessel functions
7.6 Addition theorems
7.7 Fourier transform and Hankel transform
7.8 Integrals of Bessel functions
7.9 Airy functions
7.10 Exercises
7.11 Summary
7.11.1 Bessel functions
7.11.2 Zeros of real cylinder functions
7.11.3 Integral representations
7.11.4 Hankel functions
7.11.5 Modified Bessel functions
7.11.6 Addition theorems
7.11.7 Fourier transform and Hankel transform
7.11.8 Integrals of Bessel functions
7.11.9 Airy functions
7.12 Remarks
8 Hypergeometric functions
8.1 Hypergeometric series
8.1.1 Examples of generalized hypergeometric functions
8.2 Solutions of the hypergeometric equation
8.3 Linear relations of solutions
8.4 Solutions when c is an integer
8.5 Contiguous functions
8.6 Quadratic transformations
8.7 Transformations and special values
8.8 Exercises
8.9 Summary
8.9.1 Hypergeometric series
8.9.2 Solutions of the hypergeometric equation
8.9.3 Linear relations of solutions
8.9.4 Solutions when c is an integer
8.9.5 Contiguous functions
8.9.6 Quadratic transformations
8.9.7 Transformations and special values
8.10 Remarks
9 Spherical functions
9.1 Harmonic polynomials; surface harmonics
9.2 Legendre functions
9.3 Relations among the Legendre functions
9.4 Series expansions and asymptotics
9.5 Associated Legendre functions
9.6 Relations among associated functions
9.7 Exercises
9.8 Summary
9.8.1 Harmonic polynomials; surface harmonics
9.8.2 Legendre functions
9.8.3 Relations among Legendre functions
9.8.4 Series expansions and asymptotics
9.8.5 Associated Legendre functions
9.8.6 Relations among associated functions
9.9 Remarks
10 Asymptotics
10.1 Hermite and parabolic cylinder functions
10.2 Confluent hypergeometric functions
10.3 Hypergeometric functions, Jacobi polynomials
10.4 Legendre functions
10.5 Steepest descents and stationary phase
10.6 Exercises
10.7 Summary
10.7.1 Hermite and parabolic cylinder functions
10.7.2 Confluent hypergeometric functions
10.7.3 Hypergeometric functions, Jacobi polynomials
10.7.4 Legendre functions
10.7.5 Steepest descents and stationary phase
10.8 Remarks
11 Elliptic functions
11.1 Integration
11.2 Elliptic integrals
11.3 Jacobi elliptic functions
11.4 Theta functions
11.5 Jacobi theta functions and integration
11.6 Weierstrass elliptic functions
11.7 Exercises
11.8 Summary
11.8.1 Integration
11.8.2 Elliptic integrals
11.8.3 Jacobi elliptic functions
11.8.4 Theta functions
11.8.5 Jacobi theta functions and integration
11.8.6 Weierstrass elliptic functions
11.9 Remarks

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Tags: Richard Beals, Roderick Wong, Special, functions