# Book:Ian N. Sneddon/Special Functions of Mathematical Physics and Chemistry

## Ian N. Sneddon: Special Functions of Mathematical Physics and Chemistry

Published $1956$, Oliver and Boyd.

### Contents

Preface
CHAPTER I INTRODUCTION
1. The origin of special functions
2. Ordinary points of a linear differential equation
3. Regular singular points
4. The point at infinity
5. The gamma function and related functions
$(5.1)$
$(5.2)$
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$(5.10)$ (and $(5.10)$)
$(5.11)$
CHAPTER II HYPERGEOMETRIC FUNCTIONS
6. The hypergeometric series
7. The integral formula for the hypergeometric series
8. The hypergeometric equation
9. Linear relations between the solutions of the hypergeometric equation
10. Relations of contiguity
11. The confluent hypergeometric function
12. Generalised hypergeometric series
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$(12.4)$
CHAPTER III LEGENDRE FUNCTIONS
13. Legendre polynomials
14. Recurrence relations for the Legendre polynomials
15. The formulae of Murphy and Rodrigues
16. Series of Legendre polynomials
17. Legendre's differential equation
18. Neumann's formula for the Legendre functions
19. Recurrence relations for the function $Q_n(\mu)$
20. The use of Legendre functions in potential theory
21. Legendre's associated functions
22. Integral expressions for the associated Legendre function
23. Surface spherical harmonics
24. Use of associated Legendre functions in wave mechanics
CHAPTER IV BESSEL FUNCTIONS
25. The origin of Bessel functions
26. Recurrence relations for the Bessel coefficients
27. Series expansion for the Bessel coefficients
28. Integral expressions for the Bessel coefficients
29. The addition formula for the Bessel coefficients
30. Bessel's differential equation
31. Spherical Bessel Functions
32. Integrals involving Bessel functions
33. The modified Bessel functions
34. The Ber and Bei functions
35. Expansions in series of Bessel functions
36. The use of Bessel functions in potential theory
37. Asymptotic expansions of Bessel functions
CHAPTER V THE FUNCTIONS OF HERMITE AND LAGUERRE
38. The Hermite polynomials
39. Hermite's differential equation
40. Hermite functions
41. The occurence of Hermite functions in wave mechanics
42. The Laguerre polynomials
43. Laguerre's differential equation
44. The associated Laguerre polynomials and functions
45. The wave functions for the hydrogen atom
Appendix: The Dirac Delta Function
46. The Dirac delta function
INDEX