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

7.6 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