Book:Richard Beals/Special functions, a graduate text
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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 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
- 7.6 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
- 7.12 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
- 8.6 Quadratic transformations
- 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
