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

Subject Matter

 * Special Functions

Contents



 * $\text {I}$: 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
 * Examples $\text {I}$


 * $\text {II}$: HYPERGEOMETRIC FUNCTIONS
 * 6. The hypergeometric series
 * 7. An 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
 * Examples $\text {II}$


 * $\text {III}$: LEGENDRE FUNCTIONS
 * 13. Legendre polynomials
 * 14. Recurrence relations for the Legendre polynomial
 * 15. The formulae of Murphy and Rodrigues
 * 16. Series of Legendre polynomials
 * 17. Legendre's difference equation
 * 18. Neumann's formula for the Legendre functions
 * 19. Recurrence relations for the function $\map {Q_n} \mu$
 * 20. The use of Legendre functions in potential theory
 * 21. Legendre's associated functions
 * 22. Integral expression for the associated Legendre function
 * 23. Surface spherical harmonics
 * 24. Use of associated Legendre functions in wave mechanics
 * Examples $\text {III}$


 * $\text {IV}$: 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
 * Examples $\text {IV}$


 * $\text {V}$: THE FUNCTIONS OF HERMITE AND LAGUERRE
 * 38. The Hermite polynomials
 * 39. Hermite's differential equation
 * 40. Hermite functions
 * 41. The occurrence 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
 * Examples $\text {V}$


 * : THE DIRAC DELTA FUNCTION
 * 46. The Dirac delta function


 * INDEX



Source work progress
* : Chapter $\text I$: Introduction: $\S 1$. The origin of special functions: $(1.5)$