View source for Book:Earl David Rainville/Special Functions

{{Book|Special Functions|1960|The Macmillan Company, New York||Earl David Rainville}}

===Contents===
:Chapter 1: INFINITE PRODUCTS
::1. Introduction
::2. Definition of an infinite product
::3. A necessary condition for convergence
::4. The associated series of logarithms
::5. Absolute convergence
::6. Uniform convergence
:Chapter 2: THE GAMMA AND BETA FUNCTIONS
::7. The Euler or Mascheroni constant $\gamma$
::8. The Gamma function
::9. A series for $\dfrac{\Gamma'(z)}{\Gamma(z)}$
::10. Evaluation of $\Gamma(1)$ and $\Gamma'(1)$
::11. The Euler product for $\Gamma(z)$
::12. The difference equation $\Gamma(z+1)=z\Gamma(z)$
::13. The order symbols $o$ and $O$
::14. Evaluation of certain infinite products
::15. Euler's integral for $\Gamma(z)$
::16. The Beta function
::17. The value of $\Gamma(z)\Gamma(1-z)$
::18. The factorial function
::19. Legendre's duplication formula
::20. Gauss' multiplication theorem
::21. A summation formula due to Euler
::22. The behavior of $\log \Gamma(z)$ for large $|z|$
:Chapter 3: ASYMPTOTIC SERIES
::23. Definition of an asymptotic expansion
::24. Asymptotic expansions about infinity
::25. Algebraic properties
::26. Term-by-term integration
::27. Uniqueness
::28. Watson's lemma
:Chapter 4: THE HYPERGEOMETRIC FUNCTION
::29. The function $F(a,b;c;z)$
::30. A simple integral form
::31. $F(a,b;c;1)$ as a function of the parameters
::32. Evaluation of $F(a,b;c;1)$
::33. The contiguous function relations
::34. The hypergeometric differential equation
::35. Logarithmic solutions of the hypergeometric equation
::36. $F(a,b;c;z)$ as a function of its parameters
::37. Elementary series manipulations
::38. Simple transformations
::39. Relation between functions of $z$ and $1-z$
::40. A quadratic transformation
::41. Other quadratic transformations
::42. A theorem due to Kummer
::43. Additional properties
:Chapter 5: GENERALIZED HYPERGEOMETRIC FUNCTIONS
::44. The function ${}_pF_q$
::45. The exponential and binomial functions
::46. A differential equation
::47. Other solutions of the differential equation
::48. The contiguous function relations
::49. A simple integral
::50. The ${}_pF_q$ with unit argument
::51. Saalschütz' theorem
::52. Whipple's theorem
::53. Dixon's theorem
::54. Contour integrals of Barnes' type
::55. The Barnes integrals and the function ${}_pF_q$
::56. A useful integral
:Chapter 6: BESSEL FUNCTIONS
::57. Remarks
::58. Definition of $J_n(z)$
::59. Bessel's differential equation
::60. Differential recurrence relations
::61. A pure recurrence relation
::62. A generating function
::63. Bessel's integral
::64. Index half an odd integer
::65. Modified Bessel functions
::66. Neumann Polynomials
::67. Neumann series
:Chapter 7: THE CONFLUENT HYPERGEOMETRIC FUNCTION
::68. Basic properties of ${}_1F_1$
::69. Kummer's first formula
::70. Kummer's second formula
:Chapter 8: GENERATING FUNCTIONS
::71. The generating function concept
::72. Generating functions of the form $G(2xt-t^2)$
::73. Sets generated by $e^t \psi(xt)$
::74. The generating functions $A(t) \exp [-xt/(1-t)]$
::75. Another class of generating functions
::76. Boas and Buck generating functions
::77. An extension
:Chapter 9: ORTHOGONAL POLYNOMIALS
::78. Simple sets of polynomials
::79. Orthogonality
::80. An equivalent condition for orthogonality
::81. Zeros of orthogonal polynomials
::82. Expansion of polynomials
::83. The three-term recurrence relation
::84. The Christoffel-Darboux formula
::85. Normalization; Bessel's inequality
:Chapter 10: LEGENDRE POLYNOMIALS
::86. A generating function
::87. Differential recurrence relation
::88. The pure recurrence relation
::89. Legendre's differential equation
::90. The Rodrigues formula
::91. Bateman's generating function
::92. Additional generating functions
::93. Hypergeometric forms of $P_n(x)$
::94. Brafman's generating functions
::95. Special properties of $P_n(x)$
::96. More generating functions
::97. Laplace's first integral form
::98. Some bounds on $P_n(x)$
::99. Orthogonality
::100. An expansion theorem
::101. Expansion of $x^n$
::102. Expansion of analytic functions
:Chapter 11: HERMITE POLYNOMIALS
::103. Definition of $H_n(x)$
::104. Recurrence relations
::105. The Rodrigues formula
::106. Other generating functions
::107. Integrals
::108. The Hermite polynomial as a ${}_2F_0$
::109. Orthogonality
::110. Expansion of polynomials
::111. More generating functions
:Chapter 12: LAGUERRE POLYNOMIALS
::112. The polynomial $L_n^{(\alpha)}(x)$
::113. Generating functions
::114. Recurrence relations
::115. The Rodrigues formula
::116. The differential equation
::117. Orthogonality
::118. Expansion of polynomials
::119. Special properties
::120. Other generating functions
::121. The simple Laguerre polynomials
:Chapter 13: THE SHEFFER CLASSIFICATION AND RELATED TOPICS
::122. Differential operators and polynomial sets
::123. Sheffer's $A$-type classification
::124. Polynomials of Sheffer $A$-type zero
::125. An extension of Sheffer's classification
::126. Polynomials of $\sigma$-type zero
:Chapter 14: PURE RECURRENCE RELATIONS
::127. Sister Celine's technique
::128. A mild extension
:Chapter 15: SYMBOLIC RELATIONS
::129. Notation
::130. Symbolic relations among classical polynomials
::131. Polynomials of symbolic form $L_n(y(x))$
:Chapter 16: JACOBI POLYNOMIALS
::132. The Jacobi polynomials
::133. Bateman's generating function
::134. The Rodrigues formula
::135. Orthogonality
::136. Differential recurrence relations
::137. The pure recurrence relation
::138. Mixed relations
::139. Appell's functions of two variables
::140. An elementary generating function
::141. Brafman's generating functions
::142. Expansion in series of polynomials
:Chapter 17: ULTRASPHERICAL AND GEGENBAUER POLYNOMIALS
::143. Definitions
::144. The Gegenbauer polynomials
::145. The ultraspherical polynomials
:Chapter 18: OTHER POLYNOMIAL SETS
::146. Bateman's $Z_n(x)$
::147. Rice's $H_n(\zeta,p,v)$
::148. Bateman's $F_n(z)$
::149. Sister Celine's polynomials
::150. Bessel polynomials
::151. Bedient's polynomials
::152. Shively's pseudo-Laguerre and other polynomials
::153. Bernoulli polynomials
::154. Euler polynomials
::155. Tchebicheff polynomials
:Chapter 19: ELLIPTIC FUNCTIONS
::156. Doubly periodic functions
::157. Elliptic functions
::158. Elementary properties
::159. Order of an elliptic function
::160. The Weierstrass function $P(z)$
::161. Other elliptic functions
::162. A differential equation for $P(z)$
::163. Connections with elliptic integrals
:Chapter 20: THETA FUNCTIONS
::164. Definitions
::165. Elementary properties
::166. The basic property table
::167. Location of zeros
::168. Relations among squares of theta functions
::169. Pseudo addition theorems
::170. Relation to the heat equation
::171. The relation $\theta_1'=\theta_2 \theta_3 \theta_4$
::172. Infinite products
::173. The value of $G$
:Chapter 21: JACOBIAN ELLIPTIC FUNCTIONS
::174. A differential equation involving theta functions
::175. The function $\mathrm{sn}(u)$
::176. The functions $\mathrm{cn}(u)$ and $\mathrm{dn}(u)$
::177. Relations involving squares
::178. Relations involving derivatives
::179. Addition theorems
:Bibliography
:Index

[[Category:Book]]