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Special Functions

Book Description

The subject of special functions is often presented as a collection of disparate results, which are rarely organised in a coherent way. This book answers the need for a different approach to the subject. The authors' main goals are to emphasise general unifying principles coherently and to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more, including chapters on discrete orthogonal polynomials and elliptic functions. The authors show how a very large part of the subject traces back to two equations - the hypergeometric equation and the confluent hypergeometric equation - and describe the various ways in which these equations are canonical and special. Providing ready access to theory and formulas, this book serves as an ideal graduate-level textbook as well as a convenient reference.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Orientation
    1. 1.1 Power series solutions
    2. 1.2 The gamma and beta functions
    3. 1.3 Three questions
    4. 1.4 Elliptic functions
    5. 1.5 Exercises
    6. 1.6 Summary
    7. 1.7 Remarks
  8. 2. Gamma, beta, zeta
    1. 2.1 The gamma and beta functions
    2. 2.2 Euler’s product and reflection formulas
    3. 2.3 Formulas of Legendre and Gauss
    4. 2.4 Two characterizations of the gamma function
    5. 2.5 Asymptotics of the gamma function
    6. 2.6 The psi function and the incomplete gamma function
    7. 2.7 The Selberg integral
    8. 2.8 The zeta function
    9. 2.9 Exercises
    10. 2.10 Summary
    11. 2.11 Remarks
  9. 3. Second-order differential equations
    1. 3.1 Transformations, symmetry
    2. 3.2 Existence and uniqueness
    3. 3.3 Wronskians, Green’s functions, comparison
    4. 3.4 Polynomials as eigenfunctions
    5. 3.5 Maxima, minima, estimates
    6. 3.6 Some equations of mathematical physics
    7. 3.7 Equations and transformations
    8. 3.8 Exercises
    9. 3.9 Summary
    10. 3.10 Remarks
  10. 4. Orthogonal polynomials
    1. 4.1 General orthogonal polynomials
    2. 4.2 Classical polynomials: general properties, I
    3. 4.3 Classical polynomials: general properties, II
    4. 4.4 Hermite polynomials
    5. 4.5 Laguerre polynomials
    6. 4.6 Jacobi polynomials
    7. 4.7 Legendre and Chebyshev polynomials
    8. 4.8 Expansion theorems
    9. 4.9 Functions of second kind
    10. 4.10 Exercises
    11. 4.11 Summary
    12. 4.12 Remarks
  11. 5. Discrete orthogonal polynomials
    1. 5.1 Discrete weights and difference operators
    2. 5.2 The discrete Rodrigues formula
    3. 5.3 Charlier polynomials
    4. 5.4 Krawtchouk polynomials
    5. 5.5 Meixner polynomials
    6. 5.6 Chebyshev–Hahn polynomials
    7. 5.7 Exercises
    8. 5.8 Summary
    9. 5.9 Remarks
  12. 6. Confluent hypergeometric functions
    1. 6.1 Kummer functions
    2. 6.2 Kummer functions of the second kind
    3. 6.3 Solutions when c is an integer
    4. 6.4 Special cases
    5. 6.5 Contiguous functions
    6. 6.6 Parabolic cylinder functions
    7. 6.7 Whittaker functions
    8. 6.8 Exercises
    9. 6.9 Summary
    10. 6.10 Remarks
  13. 7. Cylinder functions
    1. 7.1 Bessel functions
    2. 7.2 Zeros of real cylinder functions
    3. 7.3 Integral representations
    4. 7.4 Hankel functions
    5. 7.5 Modified Bessel functions
    6. 7.6 Addition theorems
    7. 7.7 Fourier transform and Hankel transform
    8. 7.8 Integrals of Bessel functions
    9. 7.9 Airy functions
    10. 7.10 Exercises
    11. 7.11 Summary
    12. 7.12 Remarks
  14. 8. Hypergeometric functions
    1. 8.1 Hypergeometric series
    2. 8.2 Solutions of the hypergeometric equation
    3. 8.3 Linear relations of solutions
    4. 8.4 Solutions when c is an integer
    5. 8.5 Contiguous functions
    6. 8.6 Quadratic transformations
    7. 8.7 Transformations and special values
    8. 8.8 Exercises
    9. 8.9 Summary
    10. 8.10 Remarks
  15. 9. Spherical functions
    1. 9.1 Harmonic polynomials; surface harmonics
    2. 9.2 Legendre functions
    3. 9.3 Relations among the Legendre functions
    4. 9.4 Series expansions and asymptotics
    5. 9.5 Associated Legendre functions
    6. 9.6 Relations among associated functions
    7. 9.7 Exercises
    8. 9.8 Summary
    9. 9.9 Remarks
  16. 10. Asymptotics
    1. 10.1 Hermite and parabolic cylinder functions
    2. 10.2 Confluent hypergeometric functions
    3. 10.3 Hypergeometric functions, Jacobi polynomials
    4. 10.4 Legendre functions
    5. 10.5 Steepest descents and stationary phase
    6. 10.6 Exercises
    7. 10.7 Summary
    8. 10.8 Remarks
  17. 11. Elliptic functions
    1. 11.1 Integration
    2. 11.2 Elliptic integrals
    3. 11.3 Jacobi elliptic functions
    4. 11.4 Theta functions
    5. 11.5 Jacobi theta functions and integration
    6. 11.6 Weierstrass elliptic functions
    7. 11.7 Exercises
    8. 11.8 Summary
    9. 11.9 Remarks
  18. Appendix A: Complex analysis
  19. Appendix B: Fourier analysis
  20. Notation
  21. References
  22. Author index
  23. Index