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

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

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

### 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