CONTENTS

CHAPTER I

REAL VARIABLES

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 $\sqrt{2}$

13-14. Quadratic surds

15. The continuum

16. The continuous real variable

17. Sections of the real numbers. Dedekind’s Theorem

18. Points of condensation

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. Polynomials

24-25. Rational functions

26-27. Algebraical functions

28-29. Transcendental functions

30. Graphical solution of equations

31. Functions of two variables and their graphical representation

32. Curves in a plane

33. Loci in space

Miscellaneous Examples

Chapter III

FUNCTIONS OF REAL VARIABLES

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 $x^n$

73. The limit of $(1 + \frac{1}{n})^n$

74. Some algebraical lemmas

75. The limit of $(n(\sqrt[n]{x}-1)$

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 $z^n$ and the geometrical series

Miscellaneous Examples

Chapter V

LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS

89-92. Limits as $x \to \infty$ or $x \to −\infty$

93-97. Limits as $x \to a$

98-99. Continuous functions of a real variable

100-104. Properties of continuous functions. Bounded functions. The oscillation of a function in an interval

105-106. Sets of intervals on a line. The Heine-Borel Theorem

107. Continuous functions of several variables

108-109. Implicit and inverse functions

Miscellaneous Examples

Chapter VI

DERIVATIVES AND INTEGRALS

110–112. Derivatives

113. General rules for differentiation

114. Derivatives of complex functions

115. The notation of the differential calculus

116. Differentiation of polynomials

117. Differentiation of rational functions

118. Differentiation of algebraical functions

119. Differentiation of transcendental functions

120. Repeated differentiation

121. General theorems concerning derivatives. Rolle’s Theorem

122–124. Maxima and minima

125–126. The Mean Value Theorem

127–128. Integration. The logarithmic function

129. Integration of polynomials

130–131. Integration of rational functions

132–139. Integration of algebraical functions. Integration by rationalisation. Integration by parts

140–144. Integration of transcendental functions

145. Areas of plane curves

146. Lengths of plane curves

Miscellaneous Examples

Chapter VII

ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS

147. Taylor’s Theorem

148. Taylor’s Series

149. Applications of Taylor’s Theorem to maxima and minima

150. Applications of Taylor’s Theorem to the calculation of limits

151. The contact of plane curves

152–154. Differentiation of functions of several variables

155. Differentials

156–161. Definite Integrals. Areas of curves

162. Alternative proof of Taylor’s Theorem

163. Application to the binomial series

164. Integrals of complex functions

Miscellaneous Examples

Chapter VIII

THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS

165–168. Series of positive terms. Cauchy’s and d’Alembert’s tests of convergence

169. Dirichlet’s Theorem

170. Multiplication of series of positive terms

171–174. Further tests of convergence. Abel’s Theorem. Maclaurin’s integral test

175. The series $\sum n^{-s}$

176. Cauchy’s condensation test

177–182. Infinite integrals

183. Series of positive and negative terms

184–185. Absolutely convergent series

186–187. Conditionally convergent series

188. Alternating series

189. Abel’s and Dirichlet’s tests of convergence

190. Series of complex terms

191–194. Power series

195. Multiplication of series in general

Miscellaneous Examples

Chapter IX

THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL VARIABLE

196–197. The logarithmic function

198. The functional equation satisfied by $\log x$

199–201. The behaviour of $\log x$ as $x$ tends to infinity or to zero

202. The logarithmic scale of infinity

203. The number $e$

204–206. The exponential function

207. The general power $a^x$

208. The exponential limit

209. The logarithmic limit

210. Common logarithms

211. Logarithmic tests of convergence

212. The exponential series

213. The logarithmic series

214. The series for $\arctan x$

215. The binomial series

216. Alternative development of the theory

Miscellaneous Examples

Chapter X

THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS

217–218. Functions of a complex variable

219. Curvilinear integrals

220. Definition of the logarithmic function

221. The values of the logarithmic function

222–224. The exponential function

225–226. The general power $a^z$

227–230. The trigonometrical and hyperbolic functions

231. The connection between the logarithmic and inverse trigonometrical functions

232. The exponential series

233. The series for $\cos z$ and $\sin z$

234–235. The logarithmic series

236. The exponential limit

237. The binomial series

Miscellaneous Examples

 

Appendix I. The proof that every equation has a root

Appendix II. A note on double limit problems

Appendix III. The circular functions

Appendix IV. The infinite in analysis and geometry

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