순수 수학 純粹數學 . A Course of Pure Mathematics, by G. H. (Godfrey Harold) Hardy
영국 캠브리지대학교수가 쓴 순수수학책. 1914년경 발행.
The Book of A Course of Pure
Mathematics, by G. H. ( Godfrey Harold) Hardy
Title: A Course of Pure Mathematics
Third Edition
Author: G. H. (Godfrey Harold) Hardy
Language: English
*** START OFTHE BOOKA COURSE OF PURE MATHEMATICS ***
A Course of Pure
MathematicsTranscriber’s Note
Minor typographical corrections and presentational changes have been made without comment. Notational modernizations are listed in the transcriber’s note at the end of the book.A COURSE
OF
PURE MATHEMATICSCAMBRIDGE UNIVERSITY PRESS
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LONDON: FETTER LANE, E.C. 4
NEW YORK : THE MACMILLAN CO.
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MACMILLAN AND CO., Ltd.
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ALL RIGHTS RESERVEDA 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
1921First Edition 1908
Second Edition 1914
Third Edition 1921PREFACE TO THE THIRD EDITION
No extensive changes have been made in this edition. The most impor- tant 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 in- serted in Chapter I a sketch of Dedekind’s theory of real numbers, and a proof of Weierstrass’s theorem concerning points of condensation; in Chap- ter 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 con- cerning implicit functions; in Chapter VI some additional matter concern- ing 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 ge- ometry and formal trigonometry contained in Chapters II and III of thefirst 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.


A Course of Pure Mathematics, by G. H. (Godfrey Harold) Hardy
순수 수학의 과정
순수수학(純粹數學)은 전적으로 이론이나 추상에 대한 수학을 의미하며, 응용수학과 대별되는 말이다. 수학 그 자체의 아름다움을 추구하고, 연구하는 수학자들을 흔히 순수수학자들이라고 부른다. 순수수학자들 중에서는 자신의 연구 결과가 사회에 도움이 되기를 바라는 수학자들도 있지만, 자신의 연구 결과가 사회에 도움이 되지 않기를 바라는 수학자들도 더러 있다. 대표적인 순수수학자로는 영국의 저명한 수학자인 고드프리 해럴드 하디, 데카르트 등이 있다.
순수수학의 범주
? 산술
? 대수학
? 해석학
? 기하학
? 위상수학
?
? 응용수학
? 논리학
? 메타논리학
? 메타수학
수학의 주요 분야
수론
? 대수적 수론
? 해석적 수론
대수학
? 선형대수학
? 추상대수학
? 군론
? 환론
? 가환대수학
? 호몰로지 대수학
해석학
? 미적분학
? 실해석학
? 복소해석학
? 수치해석학
? 측도론
? 함수해석학
? 조화해석학
? 비표준 해석학
기하학
? 대수기하학
? 계산기하학
? 해석기하학
? 미분기하학
? 리만 기하학
위상수학
? 일반위상수학
? 대수적 위상수학
? 미분위상수학
? 매듭 이론
수학기초론
? 수리논리학
? 모형 이론
? 증명 이론
? 계산 가능성 이론
? 집합론
? 범주론
이산수학
? 계산 이론
? 계산 복잡도 이론
? 암호학
? 조합론
? 그래프 이론
확률과 통계 ? 확률론
? 통계학
? 확률미적분학
? 게임 이론
? 결정이론

순수 수학 純粹數學 . A Course of Pure Mathematics, by G. H. (Godfrey Harold) Hardy

CONTENTS
CHAPTER I
REAL VARIABLES
SECT.
PAGE
1?2.
Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
3?7.
Irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
8.
Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
9.
Relations of magnitude between real numbers . . . . . . . . . . . . . . . . .
16
10?11.
Algebraical operations with real numbers . . . . . . . . . . . . . . . . . . . . .
18
12.
The number √2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
13?14.
Quadratic surds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
15.
The continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
16.
The continuous real variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
17.
Sections of the real numbers. Dedekind’s Theorem . . . . . . . . . . . .
30
18.
Points of condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
19.
Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Decimals, 1. Gauss’s Theorem, 6. Graphical solution of quadratic equations, 22. Important inequalities, 35. Arithmetical and geomet- rical means, 35. Schwarz’s Inequality, 36. Cubic and other surds, 38. Algebraical numbers, 41.
CHAPTER II
FUNCTIONS OF REAL VARIABLES
20.
The idea of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
21.
The graphical representation of functions. Coordinates . . . . . . . .
46
22.
Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
23.
Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
24?25.
Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
26?27.
Algebraical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
28?29. Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
30. Graphical solution of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67CONTENTS
viii
SECT.
31.
32.
33.
PAGE
Functions of two variables and their graphical representation . .
68
Curves in a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
Loci in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Trigonometrical functions, 60. Arithmetical functions, 63. Cylinders, 72. Contour maps, 72. Cones, 73. Surfaces of revolution, 73. Ruled sur- faces, 74. Geometrical constructions for irrational numbers, 77. Quadra- ture of the circle, 79.
CHAPTER III COMPLEX NUMBERS
Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34?38.
Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39?42.
The quadratic equation with real coefficients . . . . . . . . . . . . . . . . . .
43.
81
92
96
44. Argand’s diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
45.
46.
De Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
Rational functions of a complex variable . . . . . . . . . . . . . . . . . . . . . .
104
Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47?49.
118
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Properties of a triangle, 106, 121. Equations with complex coeffi-
cients, 107.
Coaxal circles, 110.
Bilinear and other transforma-
50.
51.
52.
tions, 111, 116, 125. Cross ratios, 115. Condition that four points should be concyclic, 116. Complex functions of a real variable, 116. Construction of regular polygons by Euclidean methods, 120. Imaginary points and lines, 124.
CHAPTER IV
LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE
Functions of a positive integral variable . . . . . . . . . . . . . . . . . . . . . . .
128
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
Finite and infinite classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130CONTENTS
ix
SECT.
53?57. 58?61.
PAGE
Properties possessed by a function of n for large values of n . . .
131
Definition of a limit and other definitions . . . . . . . . . . . . . . . . . . . . .
138
62.
Oscillating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145
General theorems concerning limits . . . . . . . . . . . . . . . . . . . . . . . . . . .
63?68.
149
69?70. Steadily increasing or decreasing functions . . . . . . . . . . . . . . . . . . . . 157
71.
Alternative proof of Weierstrass’s Theorem . . . . . . . . . . . . . . . . . . .
159
72. The limit of xn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
73. The limit of 1 +
1 n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
n
75. The limit of n( √n x ? 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
76?77. Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
74.
Some algebraical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
165
78.
79.
The infinite geometrical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
172
The representation of functions of a continuous real variable by
means of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
80. The bounds of a bounded aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
81.
The bounds of a bounded function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
180
82. The limits of indetermination of a bounded function . . . . . . . . . . 180
83?84.
The general principle of convergence . . . . . . . . . . . . . . . . . . . . . . . . . .
183
85?86. Limits of complex functions and series of complex terms . . . . . . 185
87?88. Applications to zn and the geometrical series . . . . . . . . . . . . . . . . . 188
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
k
Oscillation of sin nθπ, 144, 146, 181. Limits of n xn ,
√nx,
√nn,
n

m
n
n
√n!,x
, n!
x n, 162, 166. Decimals, 171. Arithmetical series, 175. Harmonical
series, 176. Equation xn+1 = f (x ), 190. Expansions of rational func-
n
tions, 191. Limit of a mean value, 193.
CHAPTER V
LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS
89?92.
Limits as x → ∞ or x → ?∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197CONTENTS
x
SECT.
PAGE
93?97. Limits as x → a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
98?99. Continuous functions of a real variable . . . . . . . . . . . . . . . . . . . . . . . . 210
100?104. Properties of continuous functions. Bounded functions. The
oscillation of a function in an interval . . . . . . . . . . . . . . . . . . . . 216
105?106. Sets of intervals on a line. The Heine-Borel Theorem . . . . . . . . . . 223
107. Continuous functions of several variables . . . . . . . . . . . . . . . . . . . . . . 228
108?109. Implicit and inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Limits and continuity of polynomials and rational functions, 204, 212.
Limit of
x m ? am
, 206. Orders of smallness and greatness, 207. Limit of
sin x
x
Classification of discontinuities, 214.
CHAPTER VI
DERIVATIVES AND INTEGRALS
110?112. Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
x?a
, 209. Infinity of a function, 213. Continuity of cos x and sin x, 213.
113.
General rules for differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
244
114. Derivatives of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
115.
The notation of the differential calculus . . . . . . . . . . . . . . . . . . . . . . .
246
116. Differentiation of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
117.
118.
Differentiation of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
251
Differentiation of algebraical functions . . . . . . . . . . . . . . . . . . . . . . . .
253
119. Differentiation of transcendental functions . . . . . . . . . . . . . . . . . . . . 255
120.
121.
Repeated differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258
General theorems concerning derivatives. Rolle’s Theorem . . . .
262
122?124. Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
125?126. The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
127?128. Integration. The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . 277
129.
Integration of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
281
130?131. Integration of rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281CONTENTS
xi
SECT.
132?139. Integration of algebraical functions. Integration by rationalisa-
PAGE
tion. Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
140?144.
Integration of transcendental functions . . . . . . . . . . . . . . . . . . . . . . . .
298
145.
Areas of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
302
146.
Lengths of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
304
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
308
147.
148.
Derivative of xm, 241. Derivatives of cos x and sin x, 241. Tangent and normal to a curve, 241, 257. Multiple roots of equations, 249, 309. Rolle’s Theorem for polynomials, 251. Leibniz’ Theorem, 259. Maxima and min- ima of the quotient of two quadratics, 269, 310. Axes of a conic, 273. Lengths and areas in polar coordinates, 307. Differentiation of a deter- minant, 308. Extensions of the Mean Value Theorem, 313. Formulae of reduction, 314.
CHAPTER VII
ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS
Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
319
Taylor’s Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
149. Applications of Taylor’s Theorem to maxima and minima . . . . . 326
150.
151.
Applications of Taylor’s Theorem to the calculation of limits . .
327
The contact of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
330
152?154. Differentiation of functions of several variables . . . . . . . . . . . . . . . . 335
155.
Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
342
156?161. Definite Integrals. Areas of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
162.
Alternative proof of Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . .
367
163. Application to the binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
164. Integrals of complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
Newton’s method of approximation to the roots of equations, 322. Se- ries for cos x and sin x, 325. Binomial series, 325. Tangent to a curve, 331, 346, 374. Points of inflexion, 331. Curvature, 333, 372. OsculatingCONTENTS
conics, 334, 372. Differentiation of implicit functions, 346. Fourier’s inte- grals, 355, 360. The second mean value theorem, 364. Homogeneous func- tions, 372. Euler’s Theorem, 372. Jacobians, 374. Schwarz’s inequality for integrals, 378. Approximate values of definite integrals, 380. Simpson’s Rule, 380.
CHAPTER VIII
THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS
xii
SECT.
165?168. Series of positive terms. Cauchy’s and d’Alembert’s tests of con-
PAGE
vergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
169.
Dirichlet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
388
170. Multiplication of series of positive terms . . . . . . . . . . . . . . . . . . . . . . 388
171?174. Further tests of convergence. Abel’s Theorem. Maclaurin’s inte-
gral test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
390
175.
The series P n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
?s
395
176.
Cauchy’s condensation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
397
177?182.
Infinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
398
183.
Series of positive and negative terms . . . . . . . . . . . . . . . . . . . . . . . . . .
416
184?185.
Absolutely convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
418
186?187.
Conditionally convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
420
188.
Alternating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
422
189.
Abel’s and Dirichlet’s tests of convergence . . . . . . . . . . . . . . . . . . . .
425
190.
Series of complex terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
427
191?194.
Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
428
195.
Multiplication of series in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
433
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
435
The series P
nr
k n and allied series, 385. Transformation of infinite inte- grals by substitution and integration by parts, 404, 406, 413. The series P an cos nθ, P an sin nθ, 419, 425, 427. Alteration of the sum of a series by rearrangement, 423. Logarithmic series, 431. Binomial series, 431, 433. Multiplication of conditionally convergent series, 434, 439. Recurring se- ries, 437. Difference equations, 438. Definite integrals, 441. Schwarz’s inequality for infinite integrals, 442.CONTENTS
CHAPTER IX
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL
VARIABLE
xiii
SECT.
PAGE
196?197. The logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
198.
The functional equation satisfied by log x
. . . . . . . . . . . . . . . . . . . . . 447
199?201. The behaviour of log x as x tends to infinity or to zero . . . . . . . . 448
202. The logarithmic scale of infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
203.
The number e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
452
204?206.
The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
453
207.
The general power a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
456
208.
The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
457
209.
The logarithmic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
459
210.
Common logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
460
211.
Logarithmic tests of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
466
212.
The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
471
213.
The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
475
214.
The series for arc tan x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
476
215.
The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
480
216.
Alternative development of the theory . . . . . . . . . . . . . . . . . . . . . . . .
482
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
484
Integrals containing the exponential function, 460. The hyperbolic func-
tions, 463. Integrals of certain algebraical functions, 464. Euler’s con-
stant, 469, 486. Irrationality of e, 473. Approximation to surds by the bi-
nomial theorem, 480. Irrationality of log10 n, 483. Definite integrals, 491.
CHAPTER X
THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND
CIRCULAR FUNCTIONS
217?218. Functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
219.
220.
221.
Curvilinear integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
496
Definition of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . . .
497
The values of the logarithmic function . . . . . . . . . . . . . . . . . . . . . . . .
499CONTENTS
xiv
SECT.
PAGE
222?224. The exponential function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
225?226. The general power az . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
227?230. The trigonometrical and hyperbolic functions . . . . . . . . . . . . . . . . . 512
231.
232.
233.
The connection between the logarithmic and inverse trigonomet-
rical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
The exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
520
The series for cos z and sin z
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
234?235. The logarithmic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
236.
237.
The exponential limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
529
The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
531
Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
The functional equation satisfied by Log z, 503. The function e , 509.
z
Logarithms to any base, 510. The inverse cosine, sine, and tangent of
a complex number, 516. Trigonometrical series, 523, 527, 540. Roots of transcendental equations, 534. Transformations, 535, 538. Stereographic projection, 537. Mercator’s projection, 538. Level curves, 539. Definite integrals, 543.
Appendix I. The proof that every equation has a root . . . . . . . . . . . . . . . 545
Appendix II. A note on double limit problems . . . . . . . . . . . . . . . . . . . . . . . . 553
Appendix III. The circular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
Appendix IV. The infinite in analysis and geometry . . . . . . . . . . . . . . . . . . . 560