内容简介

自从1908年出版以来，这本书已经成为一部经典之著。一代又一代崭露头角的数学家正是通过这本书的指引，步入了数学的殿堂。

在本书中，作者怀着对教育工作的无限热忱，以一种严格的纯粹学者的态度，揭示了微积分的基本思想、无穷级数的性质以及包括极限概念在内的其他题材。

作品目录

CHAPTER I

REAL VARIABLES

SECT.

1-2. Rational numbers

3-7. Irrational numbers

8. Real numbers

9. Relations of magnitude between real numbers

10-11. Algebraical operations with real numbers

12. The number 2

13-14. Quadratic surds

15. The continum

16. The continuous real variable

17. Sections of the real numbers. Dedekind's theorem

18. Points of accumulation

19. Weierstrass's theorem .

Miscellaneous examples

CHAPTER II

FUNCTIONS OF REAL VARIABLES

20. The idea of a function

21. The graphical representation of functions. Coordinates

22. Polar coordinates

23. Polynomias

24-25. Rational functions

26-27. Aigebraical functious

28-29. Transcendental functions

30. Graphical solution of equations

31. Functions of two variables and their graphical repre-

sentation

32. Curves in a plane

33. Loci in space

Miscellaneous examples

CHAPTER III

COMPLEX NUMBERS

SECT.

34-38. Displacements

39-42. Complex numbers

43. The quadratic equation with real coefficients

44. Argand's diagram

45. De Moivre's theorem

46. Rational functions of a complex variable

47-49. Roots of complex numbers

Miscellaneous examples

CHAPTER IV

LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE

50. Functions of a positive integral variable

51. Interpolation

52. Finite and infinite classes

53-57. Properties possessed by a function of n for large values

of n

58-61. Definition of a limit and other definitions

62. Oscillating functions

63-68. General theorems concerning limits

69-70. Steadily increasing or decreasing functions

71. Alternative proof of Weierstrass's theorem

72. The limit of xn

73. The limit of(1+

74. Some algebraical lemmas

75. The limit of n(nX-1)

76-77. Infinite series

78. The infinite geometrical series

79. The representation of functions of a continuous real

variable by means of limits

80. The bounds of a bounded aggregate

81. The bounds of a bounded function

82. The limits of indetermination of a bounded function

83-84. The general principle of convergence

85-86. Limits of complex functions and series of complex terms

87-88. Applications to zn and the geometrical series

89. The symbols O, o,

Miscellaneous examples

CHAPTER V

LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS

AND DISCONTINUOUS FUNCTIONS

90-92. Limits as x-- or x---

93-97. Limits as z-, a

98. The symbols O, o,～: orders of smallness and greatness

99-100. Continuous functions of a real variable

101-105. Properties of continuous functions. Bounded functions.

The oscillation of a function in an interval

106-107. Sets of intervals on a line. The Heine-Borel theorem

108. Continuous functions of several variables

109-110. Implicit and inverse functions

Miscellaneous examples

CHAPTER VI

DERIVATIVES AND INTEGRALS

111-113. Derivatives

114. General rules for differentiation

115. Derivatives of complex functions

116. The notation of the differential calculus

117. Differentiation of polynomials

118. Differentiation of rational functions

119. Differentiation of algebraical functions

120. Differentiation of transcendental functions

121. Repeated differentiation

122. General theorems concerning derivatives, Rolle's

theorem

123-125. Maxima and minima

126-127. The mean value theorem

128. Cauchy's mean value theorem

SECT.

129. A theorem of Darboux

130-131. Integration. The logarithmic function

132. Integration of polynomials

133-134. Integration of rational functions

135-142. Integration of algebraical functions. Integration by

rationalisation. Integration by parts

143-147. Integration of transcendental functions

148. Areas of plane curves

149. Lengths of plane curves

Miscellaneous examples

CHAPTER VII

ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS

150-151. Taylor's theorem

152. Taylor's series

153. Applications of Taylor's theorem to maxima and

minima

154. The calculation of certain limits

155. The contact of plane curves

156-158. Differentiation of functions of several variables

159. The mean value theorem for functions of two variables

160. Differentials

161-162. Definite integrals

163. The circular functions

164. Calculation of the definite integral as the limit of a sum

165. General properties of the definite integral

166. Integration by parts and by substitution

167. Alternative proof of Taylor's theorem

168. Application to the binomial series

169. Approximate formulae for definite integrals. Simpson's

rule

170. Integrals of complex functions

Miscellaneous examples

CHAPTER VIII

THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS

SECT. PAGE

171-174. Series of positive terms. Cauchy's and d'Alembert's

tests of convergence

175. Ratio tests

176. Dirichlet's theorem

177. Multiplication of series of positive terms

178-180. Further tests for convergence. Abel's theorem. Mac-

laurin's integral test

181. The series n-s

182. Cauchy's condensation test

183. Further ratio tests

184-189. Infinite integrals

190. Series of positive and negative terms

191-192. Absolutely convergent series

193-194. Conditionally convergent series

195. Alternating series

196. Abel's and Dirichlet's tests of convergence

197. Series of complex terms

198-201. Power series

202. Multiplication of series

203. Absolutely and conditionally convergent infinite

integrals

Miscellaneous examples

CHAPTER IX

THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS

OF A REAL VARIABLE

204-205. The logarithmic function

206. The functional equation satisfied by log x

207-209. The behaviour of log x as x tends to infinity or to zero

210. The logarithmic scale of infinity

211. The number e

212-213. The exponential function

214. The general power ax

215. The exponential limit

216. The logarithmic limit

SECT.

217. Common logarithms

218. Logarithmic tests of convergence

219. The exponential series

220. The logarithmic series

221. The series for arc tan x

222. The binomial series

223. Alternative development of the theory

224-226. The analytical theory of the circular functions

Miscellaneous examples

CHAPTER X

THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL，

AND CIRCULAR FUNCTIONS

227-228. Functions of a complex variable

229. Curvilinear integrals

230. Definition of the logarithmic function

231. The values of the logarithmic function

232-234. The exponential function

235-236. The general power a

237-240. The trigonometrical and hyperbolic functions

241. The connection between the logarithmic and inverse

trigonometrical functions

242. The exponential series

243. The series for cos z and sin z

244-245. The logarithmic series

246. The exponential limit

247. The binomial series

Miscellaneous examples

The functional equation satisfied by Log z, 454. The function e, 460.

Logarithms to any base, 461. The inverse cosine, sine, and tangent of a

complex number, 464. Trigonometrical series, 470, 472-474, 484, 485.

Roots of transcendental equations, 479, 480. Transformations, 480-483.

Stereographic projection, 482. Mercator's projection, 482. Level curves,

484-485. Definite integrals, 486.

APPENDIX I. The proof that every equation has a root

APPENDIX II. A note on double limit problems

APPENDIX III. The infinite in analysis and geometry

APPENDIX IV. The infinite in analysis and geometry

INDEX