Chapter 14 Legendre and Related Functions

T. M. Dunster Department of Mathematics and Statistics, San Diego State University, San Diego, California.

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Acknowledgements:

This chapter is based in part on Abramowitz and Stegun (1964, Chapter 8) by Irene A. Stegun. The author is indebted to Richard Paris for correcting a long-standing error in Eq. (14.18.3) in previous literature.

Notes:

The main reference used in writing this chapter is Olver (1997b). For additional bibliographic reading see Erdélyi et al. (1953a, Chapter III), Hobson (1931), Jeffreys and Jeffreys (1956), MacRobert (1967), Magnus et al. (1966), Robin (1957, 1958, 1959), Snow (1952), Szegő (1967), Temme (1996b), and Wong (1989).

Referenced by:

§18.3, Profile T. Mark Dunster, § ‣ Chapter Authors, § ‣ Associate Editors, Erratum (V1.0.28) for Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function, Erratum (V1.1.6) for Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions, § ‣ Software Cross Index

Permalink:

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• Notation
  1. 14.1 Special Notation
• Real Arguments
  1. 14.2 Differential Equations
  2. 14.3 Definitions and Hypergeometric Representations
  3. 14.4 Graphics
  4. 14.5 Special Values
  5. 14.6 Integer Order
  6. 14.7 Integer Degree and Order
  7. 14.8 Behavior at Singularities
  8. 14.9 Connection Formulas
  9. 14.10 Recurrence Relations and Derivatives
  10. 14.11 Derivatives with Respect to Degree or Order
  11. 14.12 Integral Representations
  12. 14.13 Trigonometric Expansions
  13. 14.14 Continued Fractions
  14. 14.15 Uniform Asymptotic Approximations
  15. 14.16 Zeros
  16. 14.17 Integrals
  17. 14.18 Sums
  18. 14.19 Toroidal (or Ring) Functions
  19. 14.20 Conical (or Mehler) Functions

• Complex Arguments
  1. 14.21 Definitions and Basic Properties
  2. 14.22 Graphics
  3. 14.23 Values on the Cut
  4. 14.24 Analytic Continuation
  5. 14.25 Integral Representations
  6. 14.26 Uniform Asymptotic Expansions
  7. 14.27 Zeros
  8. 14.28 Sums
  9. 14.29 Generalizations
• Applications
  1. 14.30 Spherical and Spheroidal Harmonics
  2. 14.31 Other Applications
• Computation
  1. 14.32 Methods of Computation
  2. 14.33 Tables
  3. 14.34 Software

13.32 Software14.1 Special Notation

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