by

## G. H. Hardy, M.A., F.R.S.

FELLOW OF NEW COLLEGE
SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD
LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE

Third Edition

Cambridge at the University Press

1921

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

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 $\sqrt{2}$

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