Cambridge at the University Press

1921

### Read the preface

**Preface to the third edition**

No extensive changes have been made in this edition. The most important are in §§ 80–82, which I have rewritten in accordance with suggestions made by Mr S. Pollard.

The earlier editions contained no satisfactory account of the genesis of the circular functions. I have made some attempt to meet this objection in § 158 and Appendix III. Appendix IV is also an addition.

It is curious to note how the character of the criticisms I have had to meet has changed. I was too meticulous and pedantic for my pupils of fifteen years ago: I am altogether too popular for the Trinity scholar of to-day. I need hardly say that I find such criticisms very gratifying, as the best evidence that the book has to some extent fulfilled the purpose with which it was written.

G. H. H.

*August* 1921

**Extract from the preface to the second edition**

The principal changes made in this edition are as follows. I have inserted in Chapter I a sketch of Dedekind’s theory of real numbers, and a proof of Weierstrass’s theorem concerning points of condensation; in Chapter IV an account of ‘limits of indetermination’ and the ‘general principle of convergence’; in Chapter V a proof of the ‘Heine-Borel Theorem’, Heine’s theorem concerning uniform continuity, and the fundamental theorem concerning implicit functions; in Chapter VI some additional matter concerning the integration of algebraical functions; and in Chapter VII a section on differentials. I have also rewritten in a more general form the sections which deal with the definition of the definite integral. In order to find space for these insertions I have deleted a good deal of the analytical geometry and formal trigonometry contained in Chapters II and III of the first edition. These changes have naturally involved a large number of minor alterations.

G. H. H.

*October* 1914

**Extract from the preface to the first edition**

This book has been designed primarily for the use of first year students at the Universities whose abilities reach or approach something like what is usually described as ‘scholarship standard’. I hope that it may be useful to other classes of readers, but it is this class whose wants I have considered first. It is in any case a book for mathematicians: I have nowhere made any attempt to meet the needs of students of engineering or indeed any class of students whose interests are not primarily mathematical.

I regard the book as being really elementary. There are plenty of hard examples (mainly at the ends of the chapters): to these I have added, wherever space permitted, an outline of the solution. But I have done my best to avoid the inclusion of anything that involves really difficult ideas. For instance, I make no use of the `principle of convergence’: uniform convergence, double series, infinite products, are never alluded to: and I prove no general theorems whatever concerning the inversion of limit-operations—I never even define $\dfrac{\partial^{2} f}{\partial x\, \partial y}$ and $\dfrac{\partial^{2} f}{\partial y\, \partial x}$. In the last two chapters I have occasion once or twice to integrate a power-series, but I have confined myself to the very simplest cases and given a special discussion in each instance. Anyone who has read this book will be in a position to read with profit Dr Bromwich’s *Infinite Series*, where a full and adequate discussion of all these

points will be found.

*September *1908

## 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