Book:E.T. Whittaker/A Course of Modern Analysis

Subject Matter

 * Analysis

Contents

 * Preface


 * Part I The Processes of Analysis


 * Chapter I: Complex Numbers
 * 1. Real numbers
 * 2. Complex numbers
 * 3. The modulus of a complex quantity
 * 4. The geometrical interpretation of complex numbers
 * Miscellaneous Examples


 * Chapter II: The Theory of Convergence
 * 5. The limit of a sequence of quantities
 * 6. The necessary and sufficient conditions for the existence of a limit
 * 7. Convergence of an infinite series
 * 8. Absolute convergence and semi-convergence
 * 9. The geometric series, and the series $\sum n^{-s}$
 * 10. The comparison-theorem
 * 11. Discussion of a special series of importance
 * 12. A convergency-test which depends on the ratio of successive terms of a series
 * 13. A general theorem on those series for which $\lim_{n \to \infty} \left({ \frac{ u_{n+1} } {u_n} }\right)$ is 1
 * 14. Convergence of the hypergeometric series
 * 15. Effect on changing the order of terms in a series
 * 16. The fundamental property of absolutely convergent series
 * 17. Riemann's theorem of semi-convergent series
 * 18. Cauchy's theorem on the multiplication of absolutely convergent series
 * 19. Merten's theorem on the multiplication of a semi-convergent series by an absolutely convergent series
 * 20. Abel's result on the multiplication of series
 * 21. Power-series
 * 22. Convergence of series derived from a power-series
 * 23. Infinite products
 * 24. Some examples of infinite products
 * 25. Cauchy's theorem on products which are not absolutely convergent
 * 26. Infinite determinants
 * 27. Convergence of an infinite determinant
 * 28. Persistence of convergence when the elements are changed
 * Miscellaneous Examples


 * Chapter III: The fundamental properties of Analytic Functions; Taylor's, Laurent's, and Liouville's Theorems
 * 29. The dependence of one complex number on another
 * 30. Continuity
 * 31. Definite integrals
 * 32. Limit to the value of a definite integral
 * 33. Property of the elementary functions
 * 34. Occasional failure of the property; singularities
 * 35. The analytic function
 * 36. Cauchy's theorem on the integral of a function round a contour
 * 37. The value of a function at a point, expressed as an integral taken round a contour enclosing the point
 * 38. The higher derivates
 * 39. Taylor's theorem
 * 40. Forms of the remainder in Taylor's series
 * 41. The process of continuation
 * 42. The identity of a function
 * 43. Laurent's theorem
 * 44. The nature of the singularities of a one-valued function
 * 45. The point at infinity
 * 46. Many-valued functions
 * 47. Liouville's theorem
 * 48. Functions with no essential singularities
 * Miscellaneous Examples


 * Chapter IV: The Uniform Convergence of Infinite Series
 * 49. Uniform convergence
 * 50. Connexion of discontinuity with non-uniform convergence
 * 51. Distinction between absolute and uniform convergence
 * 52. Condition for uniform convergence
 * 53. Integration of infinite series
 * 54. Differentiation of infinite series
 * 55. Uniform convergence of power-series
 * Miscellaneous Examples


 * Chapter V: The Theory of Residues; application to the evaluation of Real Definite Integrals
 * 56. Residues
 * 57. Evaluation of real definite integrals
 * 58. Evaluation of the definite integral of a rational function
 * 59. Cauchy's integral
 * 60. The numbers of roots of an equation contained within a contour
 * 61. Connexion between the zeros of a function and the zeros of its derivate
 * Miscellaneous Examples


 * Chapter VI: The expansion of functions in Infinite Series
 * 62. Darboux's Formula
 * 63. The Bernoullian numbers and the Bernoullian polynomials
 * 64. The Maclaurin-Bernoullian expansion
 * 65. Burmann's theorem
 * 66. Teixeira's extended form of Burmann's theorem
 * 67. Evaluation of the coefficients
 * 68. Expansion of a function of a root of an equation in terms of a parameter occurring in the equation
 * 69. Lagrange's theorem
 * 70. Rouché's expansion of Lagrange's theorem
 * 71. Teixeira's generalisation of Lagrange's theorem
 * 72. Laplace's extension of Lagrange's theorem
 * 73. A further generalisation of Taylor's theorem
 * 74. The expansion of a function as a series of rational functions
 * 75. Expansion of a function as an infinite product
 * 76. Expansion of a periodic function as a series of cotangents
 * 77. Expansion in inverse factorials
 * Miscellaneous Examples


 * Chapter VII: Fourier Series
 * 78. Definition of Fourier Series; nature of the region within which a Fourier series converges
 * 79. Values of the coefficients in terms of the sum of a Fourier series, when the series converges at all points in a belt of finite breadth in the $z$-plane
 * 80. Fourier's theorem
 * 81. The representation of a function by Fourier series for ranges other than $0$ to $2\pi$
 * 82. The sine and cosine series
 * 83. Alternative proof of Fourier's theorem
 * 84. Nature of the convergence of a Fourier series
 * 85. Determination of points of discontinuity
 * 86. The uniqueness of the Fourier expansion
 * Miscellaneous Examples


 * Chapter VIII: Asymptotic Expansions
 * 87. Simple example of an asymptotic expansion
 * 88. Definition of an asymptotic expansion
 * 89. Another example of an asymptotic expansion
 * 90. Multiplication of asymptotic expansions
 * 91. Integration of asymptotic expansions
 * 92. Uniqueness of an asymptotic expansion
 * Miscellaneous Examples


 * Part II The Transcendental Functions


 * Chapter IX: The Gamma-Function
 * 93. Definition of the Gamma-function, Euler's form
 * 94. The Weierstrassian form for the Gamma-function
 * 95. The difference-equation satisfied by the Gamma-function
 * 96. Evaluation of a general class of infinite products
 * 97. Connexion between the Gamma-function and the circular functions
 * 98. The multiplication-theorem of Gauss and Legendre
 * 99. Expansions for the logarithmic derivates of the Gamma-function
 * 100. Heine's expression of $\Gamma\left({z}\right)$ as a contour integral
 * 101. Expression of $\Gamma\left({z}\right)$ as a definite integral, whose path of integration is real
 * 102. Extension of the definite-integral expression to the case in which the argument of the Gamma-function is negative
 * 103. Gauss' expression of the logarithmic derivate of the Gamma-function as a definite integral
 * 104. Binet's expression of $\log\Gamma\left({z}\right)$ in terms of a definite integral
 * 105. The Eulerian integral of the first kind
 * 106. Expression of the Eulerian integral of the first kind in terms of Gamma-functions
 * 107. Evaluation of trigonometric integrals in terms of the Gamma-function
 * 108. Dirichlet's multiple integrals
 * 109. The asymptotic expansion of the logarithm of the Gamma-function (Stirling's series)
 * 110. Asymptotic expansion of the Gamma-function
 * Miscellaneous Examples


 * Chapter X: Legendre Functions
 * 111. Definition of Legendre polynomials
 * 112. Schläfli's integral for $P_n\left({z}\right)$
 * 113. Rodrigues' formula for the Legendre polynomials
 * 114. Legendre's differential equation
 * 115. The integral-properties of the Legendre polynomials
 * 116. Legendre functions
 * 117. The recurrence-formulae
 * 118. Evaluation of the integral-expression for $P_n\left({z}\right)$, as a power-series
 * 119. Laplace's integral-expression for $P_n\left({z}\right)$
 * 120. The Mehler-Dirichlet definite integral for $P_n\left({z}\right)$
 * 121. Expansion of $P_n\left({z}\right)$ as a series of powers of $1/z$
 * 122. The Legendre functions of the second kind
 * 123. Expansion of $Q_n\left({z}\right)$ as a power-series
 * 124. The recurrence-formulae for the Legendre function of the second kind
 * 125. Laplace's integral for the Legendre function of the second kind
 * 126. Relation between $P_n\left({z}\right)$ and $Q_n\left({z}\right)$, when $n$ is an integer
 * 127. Expansion of $\left({t - x}\right)^{-1}$ as a series of Legendre polynomials
 * 128. Neumann's expansion of an arbitrary function as a series of Legendre polynomials
 * 129. The associated functions $P_n^m\left({z}\right)$ and $Q_n^m\left({z}\right)$
 * 130. The definite integrals of the associated Legendre functions
 * 131. Expansion of $P_n^m\left({z}\right)$ as a definite integral of Laplace's type
 * 132. Alternative expansion of $P_n^m\left({z}\right)$ as a definite integral of Laplace's type
 * 133. The function $C_n^v\left({z}\right)$
 * Miscellaneous Examples


 * Chapter XI: Hypergeometric Functions
 * 134. The hypergeometric series
 * 135. Value of the series $F\left({a,b,c,1}\right)$
 * 136. The differential equation satisfied by the hypergeometric series
 * 137. The differential equation of the general hypergeometric function
 * 138. The Legendre functions as a particular case of the hypergeometric function
 * 139. Transformations of the general hypergeometric function
 * 140. The twenty-four particular solution of the hypergeometric differential equation
 * 141. Relations between the particular solutions of the hypergeometric differential equation
 * 142. Solution of the general hypergeometric differential equation by a definite integral
 * 143. Determination of the integral which represents $P^{\left({a}\right)}$
 * 144. Evaluation of a double-contour integrla
 * 145. Relations between contiguous hypergeometric functions.
 * Miscellaneous Examples


 * Chapter XII: Bessel Functions
 * 146. The Bessel coefficients
 * 147. Bessel's differential equations
 * 148. Bessel's equation as a case of the hypergeometric equation
 * 149. The general solution of Bessel's equation by Bessel functions whose order is not necessarily an integer
 * 150. The recurrence-formulae for the Bessel functions
 * 151. Relation between two Bessel functions whose orders differ by an integer
 * 152. The roots of Bessel functions
 * 153. Expression of the Bessel coefficients as trigonometric integrals
 * 154. Expression of the integral-formula to the case in which $n$ is not an integer
 * 155. A second expression of $J_n\left({z}\right)$ as a definite integral whose path of integration is real.
 * 156. Hankel's definite-integral solution of Bessel's differential equation
 * 157. Expression of $J_n\left({z}\right)$, for all values of $n$ and $z$, by an integral of Hankel's type
 * 158. Bessel functions as a limiting case of Legendre functions
 * 159. Bessel functions whose order is half an odd integer
 * 160. Expression of $J_n\left({z}\right)$ in a form which furnishes an approximate value to $J_n\left({z}\right)$ for large real positive values of $z$
 * 161. The asymptotic expansion of the Bessel functions.
 * 162. The second solution of Bessel's equation when the order is an integer
 * 163. Neumann's expansion; determination of the coefficients
 * 164. Proof of Neumann's expansion
 * 165. Schlömilch's expansion of an arbitrary function in terms of Bessel functions of order zero
 * 166. Tabulation of the Bessel functions
 * Miscellaneous Examples


 * Chapter XIII: Applications to the Equations of Mathematical Physics
 * 167. Introduction: illustration of the general method
 * 168. Laplace's equation; the general solution; certain particular solutions
 * 169. The series-solution of Laplace's equation
 * 170. Determination of a solution of Laplace's equation which satisfies given boundary-conditions
 * 171. Particular solutions of Laplace's equation which depend on Bessel functions
 * 172. Solution of the equation $\frac {\partial^2 V} {\partial^2 x^2} + \frac {\partial^2 V} {\partial^2 y^2} + V = 0$
 * 173. Solution of the equation $\frac {\partial^2 V} {\partial^2 x^2} + \frac {\partial^2 V} {\partial^2 y^2} + \frac {\partial^2 V} {\partial^2 z^2} + V = 0$
 * Miscellaneous Examples


 * Chapter XIV: The Elliptic Function $\wp\left({z}\right)$
 * 174. Introduction
 * 175. Definition of $\wp\left({z}\right)$
 * 176. Periodicity, and other properties, of $\wp\left({z}\right)$
 * 177. The period-parallelograms
 * 178. Expression of the functions $\wp\left({z}\right)$ by means of an integral
 * 179. The homogeneity of the function $\wp\left({z}\right)$
 * 180. The addition theorem for the function $\wp\left({z}\right)$
 * 181. Another form of the addition theorem
 * 182. The roots $e_1, \, e_2, \, e_3$
 * 183. Addition of a half-period to the argument of $\wp\left({z}\right)$
 * 184. Integration of $\left({ax^4 + 4bx^3 + 6cx^2 + 4dx + e}\right)^{-\frac 1 2}$
 * 185. Another solution of the integration-problem
 * 186. Uniformisation of curves of genus unity
 * Miscellaneous Examples


 * Chapter XV: The Elliptic Functions $\operatorname{sn} z$, $\operatorname{cn} z$, $\operatorname{dn} z$
 * 187. Construction of a doubly-periodic function with two simple poles in each period-parallelogram
 * 188. Expression of the function $f\left({z}\right)$ by means of an integral
 * 189. The function $\operatorname{sn} z$
 * 190. The functions $\operatorname{cn} z$ and $\operatorname{dn} z$
 * 191. Expressions of $\operatorname{cn} z$ and $\operatorname{dn} z$ by means of an integral
 * 192. The addition-theorem for the function $\operatorname{dn} z$
 * 193. The addition-theorems for the functions $\operatorname{sn} z$ and $\operatorname{cn} z$
 * 194. The constant $K$
 * 195. The periodicity of the elliptic functions with respect to $K$
 * 196. The constant $K'$
 * 197. The periodicity of the elliptic functions with respect to $K + iK'$
 * 198. The periodicity of the elliptic functions with respect to $iK'$
 * 199. The behaviour of the functions $\operatorname{sn} z$, $\operatorname{cn} z$, $\operatorname{dn} z$ at the point $z = iK'$
 * 200. General description of the functions $\operatorname{sn} z$, $\operatorname{cn} z$, $\operatorname{dn} z$
 * 201. A geometrical illustration of the functions $\operatorname{sn} z$, $\operatorname{cn} z$, $\operatorname{dn} z$
 * 202. Connexion of the function $\operatorname{sn} z$ with the function $\wp\left({z}\right)$
 * 203. Expansion of $\operatorname{sn} z$ as a trigonometric series
 * Miscellaneous Examples


 * Chapter XVI: Elliptic Functions; General Theorems
 * 204. Relation between the residues of an elliptic function
 * 205. The order of an elliptic function
 * 206. Expression of any elliptic function in terms of $\wp\left({z}\right)$ and $\wp'\left({z}\right)$
 * 207. Relation between any two elliptic functions which admit the same periods
 * 208. Relation between the zeros and poles of an elliptic function
 * 209. The function $\zeta\left({z}\right)$
 * 210. The quasi-periodicity of the function $\zeta\left({z}\right)$
 * 211. Expression of an elliptic function, when the principal part of its expansion at each of its singularities is given
 * 212. The function $\sigma\left({z}\right)$
 * 213. The quasi-periodicity of the function $\sigma\left({z}\right)$
 * 214. The integration of an elliptic function
 * 215. Expression of an elliptic function whose zeros and poles are known.
 * Miscellaneous Examples


 * Index