View source for Book:Richard Beals/Special functions, a graduate text

{{Book|Special functions, a graduate text|2010|Cambridge University Press||Richard Beals|author2=Roderick Wong}}

===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
:<i>Appendix A: Complex analysis</i>
:<i>Appendix B: Fourier analysis</i>
:<i>Notation</i>
:<i>References</i>
:<i>Author index</i>
:<i>Index</i>

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