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.

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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