G. H. (Godfrey Harold) Hardy.

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A COURSE

OF

PURE MATHEMATICS



CAMBRIDGE UNIVERSITY PRESS

C. F. CLAY, Manager

LONDON : FETTER LANE, E.C. 4




NEW YORK : THE MACMILLAN CO.

BOMBAY \

CALCUTTA I MACMILLAN AND CO., Ltd.

MADRAS )

TORONTO : THE MACMILLAN CO. OF

CANADA, Ltd.
TOKYO : MARUZEN-KABUSHIKI-KAISHA



ALL RIGHTS RESERVED



A COURSE



OF



PURE MATHEMATICS



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



First Edition 1908
Second Edition 1914
Third Edition 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 accord-
ance 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 defini-
tion 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



EXTEACT 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 ~L- and tt^-. In the last two
r oxoy oyox

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

SECT. PAGE

1-2. Rational numbers . ••..... 1

3-7. Irrational numbers ■ 3

8. Real numbers 13

9. Relations of magnitude between real numbers . .15
10-11. Algebraical operations with real numbers .... 17

12. The number ,/2 19

13-14. Quadratic surds . . 19

15. The continuum 23

16. The continuous real variable 26

17. Sections of the real numbers. Dedekind's Theorem . . 27

18. Points of condensation . 29

19. Weierstrass's Theorem . . . . . . . . 30

Miscellaneous Examples 31

Decimals, 1. Gauss's Theorem, 6. Graphical solution of quadratic
equations, 20. Important inequalities, 32. Arithmetical and geometrical
meaus, 32. Schwarz's Inequality, 33. Cubic and other surds, 34.
Algebraical numbers, 36.



CHAPTER II



FUNCTIONS OF REAL VARIABLES



20.

21.

22.

23.

24-25.

26-27.

2S-29.

30.

31.



The idea of a function

The graphical representation of functions. Coordinates
Polar coordinates .
Polynomials .



Rational functions
Algebraical functions .
Transcendental functions
Graphical solution of equations
Functions of two variables and their graphical repre
sentation



38
41
43
44
47
49
52
58

59



Vlll



CONTENTS



SECT.

32.
33.



Curves in a plane
Loci in space

Miscellaneous Examples



PAGE

60
61



65



Trigonometrical functions, 53. Arithmetical functions, 55. Cylinders,
62. Contour maps, 62. Cones, 63. Surfaces of revolution, 63. Ruled
surfaces, 64. Geometrical constructions for irrational numbers, 66.
Quadrature of the circle, 68.



CHAPTER III



COMPLEX NUMBERS



34-38.

39-42.

43.

44.

45.

46.

47-49.



Displacements

Complex numbers

The quadratic equation with real coefficients

Argand's diagram

de Moivre's Theorem

Rational functions of a complex variable .
Roots of complex numbers ....

Miscellaneous Examples



69

78
81
84
86



101



Properties of a triangle, 90, 101. Equations with complex coefficients,
91. Coaxal circles, 93. Bilinear and other transformations, 94, 97, 104.
Cross ratios, 96. Condition that four points should be concyclic, 97.
Complex functions of a real variable, 97. Construction of regular polygons
by Euclidean methods, 100. Imaginary points and lines, 103.



50.

51.

52.

53-57.

58-61.

62.

63-68.

69-70.

71.

72

73.

74.
75.

76-77.
78.



allies of n



CHAPTER IV

LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE

Functions of a positive integral variable .

Interpolation

Finite and infinite classes ....
Properties possessed by a function of n for large
Definition of a limit and other definitions .

Oscillating functions

General theorems concerning limits
Steadily increasing or decreasing functions
Alternative proof of Weierstrass's Theorem
The limit of .v n

1



The limit of ( 1 +



Some algebraical lemmas
The limit of n(#a?-l).
Infinite series
The infinite sreometrical series



106
107
108
109
116
121
125
131
134
134

137

138
139
140
143



CONTENTS



IX



SECT

79.



PAGE



The representation of functions of a continuous real variable

by means of limits 147

80. The bounds of a bounded aggregate 149

81. The bounds of a bounded function 149

82. The limits of indetermination of a bounded function . . 150

83-84. The general principle of convergence 151

85-86. Limits of complex functions and series of complex terms . 153

87-88. Applications to z n and the geometrical series . . . 156,

Miscellaneous Examples 157

Oscillation of sin n6ir, 121, 123, 151. Limits of n k x n , Z]x, ^n, #(n!),

X n / 771 \

— , I ) x n , 136, 139. Decimals, 143. Arithmetical series, 146. Harmonical
n\ \n J

series, 147. Equation x n+1 =f(x n ), 158. Expansions of rational functions,
159. Limit of a mean value, 160.



CHAPTER V

LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE.
AND DISCONTINUOUS FUNCTIONS



Continuous



89-92. Limits as x-*-cc or #-*- — oo 162

93-97. Limits as x-*-a 165-

98-99. Continuous functions of a real variable .... 174

100-104. Properties of continuous functions. Bounded functions.

The oscillation of a function in an interval . . 179

105-106. Sets of intervals on a line. The Heine-Borel Theorem . 185

107. Continuous functions of several variables . . . .190

108-109. Implicit and inverse functions 191

Miscellaneous Examples 194

Limits and continuity of polynomials and rational functions, 169, 176.



Limit of



, 171. Orders of smallness and greatness, 172. Limit of



■ , 173. Infinity of a function, 177. Continuity of cos x and sin a:, 177.

x

Classification of discontinuities, 178.



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



197
203

205
205
207
209
210



CONTENTS



SECT.

119. Differentiation of transcendental functions

120. Eepeated differentiation

121. General theorems concerning derivatives.
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.

rationalisation. Integration by parts

140-144. Integration of transcendental functions

145. Areas of plane curves ....

146. Lengths of plane curves

Miscellaneous Examples



Rolle's Theorem



Integration by



PAGE

212
214
217
219
226
228
232
233

236
245
249
251

253



Derivative of x m , 201. Derivatives of cos.r and sin.r, 201. Tangent
and normal to a curve, 201, 214. Multiple roots of equations, 208, 255.
Rolle's Theorem for polynomials, 209. Leibniz' Theorem, 215. Maxima
and minima of the quotient of two quadratics, 223, 256. Axes of a conic,
226. Lengths and areas in polar coordinates, 253. Differentiation of a
determinant, 254. Extensions of the Mean Value Theorem, 258. Formulae
of reduction, 259.



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



262
266

268
268
270
274
280
283
298
299
299
300



Newton's method of approximation to the roots of equations, 265.
Series for cos x and sin x, 267. Binomial series, 267. Tangent to a curve,
272, 283, 303. Points of inflexion, 272. Curvature, 273, 302. Osculating
conies, 274, 302. Differentiation of implicit functions, 283. Fourier's
integrals, 290, 294. The second mean value theorem, 296. Homogeneous
functions, 302. Euler's Theorem, 302. Jacobiaus, 303. Schwarz's in-
equality for integrals, 306. Approximate values of definite integrals, 307.
Simpson's Rule, 307.



CONTENTS



XI



CHAPTER VIII

THE CONVERGENCE OP INFINITE SERIES AND INFINITE INTEGRALS



SECT.

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 2n~ 8

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



PAGE



sos-
sis

313



315

319

320

321

335

336

338

340

342

344.

345.

349

350



The series 2n l r" and allied series, 311. Transformation of infinite
integrals by substitution and integration by parts, 327, 328, 333. The
series 2a n cos?i0, 2a„sin?i0, 338, 343, 344. Alteration of the sum of a
series by rearrangement, 341. Logarithmic series, 348. Binomial series,
348, 349. Multiplication of conditionally convergent series, 350, 354.
Recurring series, 352. Difference equations, 353. Definite integrals, 355.
Schwarz's inequality for infinite integrals, 356.



CHAPTER IX

THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL VARIABLE



196-197. The logarithmic function ....

198. The functional equation satisfied by log.r .

199-201. The behaviour of log x as x tends to infinity or

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. Losrarithmic tests of convergence .



to zero



357
360
300
362
363
364
366
368
369
369
374



Xll



CONTENTS



SECT.

212.
213.

214.
215.
216.



The exponential series . . . .
The logarithmic series . . . .
The series for arc tan. v
The binomial series .
Alternative development of the theory

Miscellaneous Examples



IMGE

378
381
382
384
386

387



Integrals containing the exponential function, 370. The hyperbolic
functions, 372. Integrals of certain algebraical functions, 373. Euler's
constant, 377, 389. Irrationality of e, 380. Approximation to surds by the
binomial theorem, 385. Irrationality of logjon, 387. Definite integrals, 393.



CHAPTER X

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



217-218.
219.

220.

221.

222-224.

225-226.

227-230.

231.

232.

233.

234-235.

236.

237.



Functions of a complex variable .

Curvilinear integrals ....

Definition of the logarithmic function

The values of the logarithmic function

The exponential function

The general power a 2 ....

The trigonometrical and hyperbolic functions

The connection between the logarithmic and

trigonometrical functions
The exponential series
The series for cos 2 and sin z
The logarithmic series .
The exponential limit .
The binomial series



Miscellaneous Examples



395
396
397
399
403
404
409

413
414
416
417
421
422

425



The functional equation satisfied by Log z, 402. The function e 2 , 407.
Logarithms to any base, 408. The inverse cosine, sine, and tangent of a
complex number, 412. Trigonometrical series, 417, 420, 431. Boots of
transcendental equations, 425. Transformations, 426, 428. Stereographic
projection, 427. Mercator's projection, 428. Level curves, 429. Definite
integrals, 432.



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



433
439

443
445






CHAPTER I

REAL VARIABLES

1. Rational numbers. A fraction r =p/q, where p and q
are positive or negative integers, is called a rational number. We
can suppose (i) that p and q have no common factor, as if they
have a common factor we can divide each of them by it, and
(ii) that q is positive, since

pl(r ?) = (-p)/q> (-p)K- q)=p/q-

To the rational numbers thus denned we may add the ' rational
number ' obtained by taking p = 0.

We assume that the reader is familiar with the ordinary
arithmetical rules for the manipulation of rational numbers. The
examples which follow demand no knowledge beyond this.

Examples I. 1. If r and s are rational numbers, then r + s, r - s, rs, and
rjs are rational numbers, unless in the last case s=0 (when rjs is of course
meaningless).

2. If X, m, and n are positive rational numbers, and m > n, then
X(m 2 — ?i 2 ), 2X«m, and X(m 2 + ;i 2 ) are positive rational numbers. Hence show
how to determine any number of right-angled triangles the lengths of all of
whose sides are rational.

3. Any terminated decimal represents a rational number whose denomi-
nator contains no factors other than 2 or 5. Conversely, any such rational
number can be expressed, and in one way only, as a terminated decimal.

[The general theory of decimals will be considered in Ch. IV.]

4. The positive rational numbers may be arranged in the form of a simple
scries as follows :

l i l J 2 14 3 2 1

1) 1> 2 5 D 2» 3' 1) 2> 3 5 it""

Show that p/q is the [^ (p + q - 1) (p + q - 2) + q]th term of the series.

[In this series every rational number is repeated indefinitely. Thus 1
occurs as {, f , % , .... We can of course avoid this by omitting every number

H. 1



2 REAL VARIABLES [i

which has already occurred in a simpler form, but then the problem of deter-
mining the precise position of pjq becomes more complicated.]

2. The representation of rational numbers by points
on a line. It is convenient, in many branches of mathematical
analysis, to make a good deal of use of geometrical illustrations.

The use of geometrical illustrations in this way does not, of
course, imply that analysis has any sort of dependence upon
geometry : they are illustrations and nothing more, and are em-
ployed merely for the sake of clearness of exposition. This being
so, it is not necessary that we should attempt any logical analysis
of the ordinary notions of elementary geometry; we may be content
to suppose, however far it may be from the truth, that we know
what they mean.

Assuming, then, that we know what is meant by a straight
line, a segment of a line, and the length of a segment, let us take
a straight line A, produced indefinitely in both directions, and a
segment A (> A 1 of any length. We call A the origin, or the point
0, and J.! the point 1, and we regard these points as representing
the numbers and 1.

In order to obtain a point which shall represent a positive
rational number r=p/q, we choose the point A r such that

A A r jA A 1 = r,

A A r being a stretch of the line extending in the same direction
along the line as A A U a direction which we shall suppose to be
from left to right when, as in Fig. 1, the line is drawn horizontally
across the paper. In order to obtain a point to represent a

1 j 1 1 !

A- s A_i A A x A,

Fig. 1.

negative rational number r = — s, it is natural to regard length as
a magnitude capable of sign, positive if the length is measured in
one direction (that of ^o^), and negative if measured in the
other, so that AB = — BA ; and to take as the point representing
r the point A_ s such that

A A_ S = — A_ S A = — A A 8 ,



1-3] REAL VARIABLES 3

We thus obtain a point A r on the line corresponding to every
rational value of r, positive or negative, and such that

A A r = r . AqAx)

and if, as is natural, we take A A t as our unit of length, and write
A A 1 = 1, then we have

A A r = r.

We shall call the points A r the rational points of the line.

3. Irrational numbers. If the reader will mark off on the
line all the points corresponding to the rational numbers whose
denominators are 1, 2, 3, ... in succession, he will readily convince
himself that he can cover the line with rational points as closely
as he likes. We can state this more precisely as follows : if we
take any segment BG on A, we can find as many rational points as
we please on BG.

Suppose, for example, that BG falls within the segment A X A,.
It is evident that if we choose a positive integer k so that

k.BC>l (1),*

and divide A 1 A 2 into k equal parts, then at least one of the points
of division (say P) must fall inside BG, without coinciding with
either B or C. For if this were not so, BG would be entirely
included in one of the k parts into which A X A 2 has been divided,
which contradicts the supposition (1). But P obviously corre-
sponds to a rational number whose denominator is k. Thus at
least one rational point P lies between B and G. But then we
can find another such point Q between B and P, another between
B and Q, and so on indefinitely ; i.e., as we asserted above, we can
find as many as we please. We may express this by saying that
BG includes infinitely many rational points.

The meaning of such phrases as '■infinitely many' or '•an infinity of, in
such sentences as ' BG includes infinitely many rational points ' or ' there are
an infinity of rational points on BG' or 'there are an infinity of positive
integers', will be considered more closely in Ch. IV. The assertion 'there are
an infinity of positive integers ' means ' given any positive integer n, however
large, we can find more than n positive integers'. This is plainly true

* The assumption that this is possible is equivalent to the assumption of what
is known as the Axiom of Archimedes.

1—2



4 KEAL VAKIABLES [i

whatever n may be, e.g. for n = 100,000 or 100,000,000. The assertion means
exactly the same as ' we can find as many positive integers as we please '.

The reader will easily convince himself of the truth of the following
assertion, which is substantially equivalent to what was proved in the second
paragraph of this section : given any rational number r, and any positive
integer n, we can find another rational number lying on either side of r and
differing from r by less than 1/n. It is merely to express this differently to
say that we can find a rational number lying on either side of r and differing
from r by as little as we please. Again, given any two rational numbers
}• and s, we can interpolate between them a chain of rational numbers in
which any two consecutive terms differ by as little as we please, that is to
say by less than 1/n, where n is any positive integer assigned beforehand.

From these considerations the reader might be tempted to
infer that an adequate view of the nature of the line could be
obtained by imagining it to be formed simply by the rational
points which lie on it. And it is certainly the case that if we
imagine the line to be made up solely of the rational points,
and all other points (if there are any such) to be eliminated,
the figure which remained would possess most of the properties
which common sense attributes to the straight line, and would,
to put the matter roughly, look and behave very much like
a line.

A little further consideration, however, shows that this view
would involve us in serious difficulties.

Let us look at the matter for a moment with the eye of
common sense, and consider some of the properties which we may
reasonably expect a straight line to possess if it is to satisfy the
idea which we have formed of it in elementary geometry.

The straight line must be composed of points, and any segment
of it by all the points which lie between its end points. With
any such segment must be associated a certain entity called its
length, which must be a quantity capable of numerical measure-
ment in terms of any standard or unit length, and these lengths
must be capable of combination with one another, according to
the ordinary rules of algebra, by means of addition or multipli-
cation. Again, it must be possible to construct a line whose
length is the sum or product of any two given lengths. If the
length PQ, along a given line, is a, and the length QR, along
the same straight line, is b, the length PR must be a + b.



3]



REAL VARIABLES



Moreover, if the lengths OP, OQ, along one straight line, are
1 and a, and the length OR along another straight line is b,
and if we determine the length OS by Euclid's construction (Euc.
VI. 12) for a fourth proportional to the lines OP, OQ, OR, this
length must be ab, the algebraical fourth proportional to 1, a, b.
And it is hardly necessary to remark that the sums and products
thus defined must obey the ordinary ' laws of algebra ' ; viz.

a + b = b + a, a + (b 4- c)==(a+b) + c,
ab = ba, a (be) = (ab) c, a(b + c) = ab + ac.
The lengths of our lines must also obey a number of obvious
laws concerning inequalities as well as equalities : thus if
A, B, C are three points lying along A from left to right, we must
have AB< AC, and so on. Moreover it must be possible, on our
fundamental line A, to find a point P such that A P is equal to
any segment whatever taken along A or along any other straight
line. All these properties of a line, and more, are involved in the
presuppositions of our elementary geometry.

Now it is very easy to see that the idea of a straight line as



Online LibraryG. H. (Godfrey Harold) HardyA course of pure mathematics → online text (page 1 of 37)