# Book:Richard Beals/Special functions, a graduate text

## Richard Beals and Roderick Wong: Special functions, a graduate text

Published $2010$, Cambridge University Press.

### 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
1.7 Remarks
2 Gamma, beta, zeta
2.1 The gamma and beta functions
$(2.2.1)$
$(2.2.2)$
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
2.11 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
3.10 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
4.12 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
5.9 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
6.10 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
7.7 Fourier transform and Hankel transform
7.8 Integrals of Bessel functions
7.9 Airy functions
7.10 Exercises
7.11 Summary
7.12 Remarks
8 Hypergeometric functions
8.1 Hypergeometric series
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
8.7 Transformations and special values
8.8 Exercises
8.9 Summary
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
9.9 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
10.8 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
11.9 Remarks
Appendix A: Complex analysis
Appendix B: Fourier analysis
Notation
References
Author index
Index