Special Functions and Orthogonal Polynomials 2nd Edition by Richard Beals, Roderick Wong ISBN 978-1107106987 1107106982
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ISBN 10: 1107106982
ISBN 13: 978-1107106987
Author: Richard Beals, Roderick Wong
The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors’ main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations – the hypergeometric equation and confluent hypergeometric equation – and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painlevé transcendents, which have been termed the ‘special functions of the twenty-first century’.
Table of contents:
1 Orientation
1.1 Power series solutions
1.2 The gamma and beta functions
1.3 Three questions
1.4 Other special functions
1.5 Exercises
1.6 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 Remarks
3 Second-order differential equations
3.1 Transformations and symmetry
3.2 Existence and uniqueness
3.3 Wronskians, Green’s functions, and comparison
3.4 Polynomials as Eigenfunctions
3.5 Maxima, minima, and estimates
3.6 Some equations of mathematical physics
3.7 Equations and transformations
3.8 Exercises
3.9 Remarks
4 Orthogonal polynomials on an interval
4.1 Weight functions and orthogonality
4.2 Stieltjes transform and Padé approximants
4.3 Padé approximants and continued fractions
4.4 Generalization: measures
4.5 Favard’s theorem and the moment problem
4.6 Asymptotic distribution of zeros
4.7 Exercises
4.8 Remarks
5 The classical orthogonal polynomials
5.1 Classical polynomials: general properties, I
5.2 Classical polynomials: general properties, II
5.3 Hermite Polynomials
5.4 Laguerre Polynomials
5.5 Jacobi polynomials
5.6 Legendre and Chebyshev polynomials
5.7 Distribution of zeros and electrostatics
5.8 Expansion theorems
5.9 Functions of the second kind
5.10 Exercises
5.11 Remarks
6 Semi-classical orthogonal polynomials
6.1 Discrete weights and difference operators
6.2 The discrete Rodrigues formula
6.3 Charlier polynomials
6.4 Krawtchouk polynomials
6.5 Meixner polynomials
6.6 Chebyshev-Hahn polynomials
6.7 Neo-classical polynomials
6.8 Exercises
6.9 Remarks
7 Asymptotics of orthogonal polynomials: two methods
7.1 Approximation away from the real line
7.2 Asymptotics by matching
7.3 The Riemann-Hilbert formulation
7.4 The Riemann-Hilbert problem in the Hermite case, I
7.5 The Riemann-Hilbert problem in the Hermite case, II
7.6 Hermite asymptotics
7.7 Exercises
7.8 Remarks
8 Confluent hypergeometric functions
8.1 Kummer functions
8.2 Kummer functions of the second kind
8.3 Solutions when c is an integer
8.4 Special cases
8.5 Contiguous functions
8.6 Parabolic cylinder functions
8.7 Whittaker functions
8.8 Exercises
8.9 Remarks
9 Cylinder functions
9.1 Bessel functions
9.2 Zeros of real cylinder functions
9.3 Integral representations
9.4 Hankel functions
9.5 Modified Bessel functions
9.6 Addition theorems
9.7 Fourier transform and Hankel transform
9.8 Integrals of Bessel functions
9.9 Airy functions
9.10 Exercises
9.11 Remarks
10 Hypergeometric functions
10.1 Solutions of the hypergeometric equation
10.2 Linear relations of solutions
10.3 Solutions when e is an integer
10.4 Contiguous functions
10.5 Quadratic transformations
10.6 Integral transformations and special values
10.7 Exercises
10.8 Remarks
11 Spherical functions
11.1 Harmonic polynomials and surface harmonics
11.2 Legendre functions
11.3 Relations among the Legendre functions
11.4 Series expansions and asymptotics
11.5 Associated Legendre functions
11.6 Relations among associated functions
11.7 Exercises
11.8 Remarks
12 Generalized hypergeometric functions; G-functions
12.1 Generalized hypergeometric series
12.2 The generalized hypergeometric equation
12.3 Meijer G-functions
12.4 Choices of contour of integration
12.5 Expansions and asymptotics
12.6 The Mellin transform and G-functions
12.7 Exercises
12.8 Remarks
13 Asymptotics
13.1 Hermite and parabolic cylinder functions
13.2 Confluent hypergeometric functions
13.3 Hypergeometric functions and Jacobi polynomials
13.4 Legendre functions
13.5 Steepest descents and stationary phase
13.6 Exercises
13.7 Remarks
14 Elliptic functions
14.1 Integration
14.2 Elliptic integrals
14.3 Jacobi elliptic functions
14.4 Theta functions
14.5 Jacobi theta functions and integration
14.6 Weierstrass elliptic functions
14.7 Exercises
14.8 Remarks
15 Transcendental Painlevé
15.1 The Painlevé method
15.2 Derivation of PII
15.3 Solutions of PII
15.4 Compatibility conditions and Bäcklund transformations
15.5 Construction of Ψ
15.6 Monodromy and isomonodromy
15.7 The inverse problem and the Painlevé property
15.8 Asymptotics of PII(0)
15.9 Exercises
15.10 Remarks
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Tags: Richard Beals, Roderick Wong, Special Functions, Orthogonal Polynomials