Book:Earl David Rainville/Special Functions

Earl David Rainville: Special Functions

Published $1960$, The Macmillan Company, New York.

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 $\gamma$

$7.(1)$

$7.(2)$

7.(3)

8. The Gamma function

$8.(1)$

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 $\Gamma(z)$

$15.(1)$

16. The Beta function

17. The value of $\Gamma(z)\Gamma(1-z)$

18. The factorial function

$(1)$

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 $F_n(z)$

$(1)$

$(2)$

$(3)$

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

$(1)$

$(2)$

$(3)$

$(4)$

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