## CONTENTS

**CHAPTER I**

**REAL VARIABLES**

9. Relations of magnitude between real numbers

10-11. Algebraical operations with real numbers

16. The continuous real variable

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

**CHAPTER II**

**FUNCTIONS OF REAL VARIABLES**

21. The graphical representation of functions. Coordinates

28-29. Transcendental functions

30. Graphical solution of equations

31. Functions of two variables and their graphical representation

**Chapter III**

**FUNCTIONS OF REAL VARIABLES**

43. The quadratic equation with real coefficients

46. Rational functions of a complex variable

47-49. Roots of complex numbers

**Chapter IV**

**LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE**

50. Functions of a positive integral variable

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

63-68. General theorems concerning limits

69-70. Steadily increasing or decreasing functions

71. Alternative proof of Weierstrass’s Theorem

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

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

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

**Chapter V**

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

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

98-99. Continuous functions of a real variable

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

107. Continuous functions of several variables

108-109. Implicit and inverse functions

**Chapter VI**

**DERIVATIVES AND INTEGRALS**

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

121. General theorems concerning derivatives. Rolle’s Theorem

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

**Chapter VII**

**ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL ****CALCULUS**

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

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

**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

170. Multiplication of series of positive terms

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

176. Cauchy’s condensation test

183. Series of positive and negative terms

184–185. Absolutely convergent series

186–187. Conditionally convergent series

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

195. Multiplication of series in general

**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

204–206. The exponential function

211. Logarithmic tests of convergence

214. The series for $\arctan x$

216. Alternative development of the theory

**Chapter X**

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

217–218. Functions of a complex variable

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

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

234–235. The logarithmic series

**Appendix I.** The proof that every equation has a root

**Appendix II.** A note on double limit problems