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

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## E.T. Whittaker and G.N. Watson:

## E.T. Whittaker and G.N. Watson: *A Course of Modern Analysis*

Published $\text {1902}$, **Cambridge University Press**

### Subject Matter

### 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 I: Complex Numbers

- 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 II: The Theory of Convergence

- 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 III: The fundamental properties of Analytic Functions; Taylor's, Laurent's, and Liouville's Theorems

- 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 IV: The Uniform Convergence of Infinite Series

- 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 V: The Theory of Residues; application to the evaluation of Real Definite Integrals

- 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 VI: The expansion of functions in Infinite Series

- 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 VII: Fourier Series

- 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

- Chapter VIII: Asymptotic Expansions

**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 IX: The Gamma-Function

- 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 X: Legendre Functions

- 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 XI: Hypergeometric Functions

- 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 XII: Bessel Functions

- 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 XIII: Applications to the Equations of Mathematical Physics

- 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 XIV: The Elliptic Function $\wp\left({z}\right)$

- 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 XV: The Elliptic Functions $\operatorname{sn} z$, $\operatorname{cn} z$, $\operatorname{dn} z$

- 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

- Chapter XVI: Elliptic Functions; General Theorems

- Index