A Course of Pure Mathematics

587 Pages·2012·3.15 MB·English

by  Godfrey Harold Hardy

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Title: A Course of Pure Mathematics
Third Edition
Author: G. H. (Godfrey Harold) Hardy
Release Date: February 5, 2012 [EBook #38769]
Most recently updated: August 6, 2021
Language: English
Character set encoding: UTF-8
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A COURSE
OF
PURE MATHEMATICS
CAMBRIDGE UNIVERSITY PRESS
Manager
C. F. CLAY,
LONDON: FETTER LANE, E.C. 4
NEW YORK : THE MACMILLAN CO.
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CALCUTTA MACMILLAN AND CO., Ltd.
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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 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-
terIVanaccountof‘limitsofindetermination’andthe‘generalprincipleof
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 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-
∂2f ∂2f
operations—Ineverevendefine and . InthelasttwochaptersI
∂x∂y ∂y∂x
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 ................................................. 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 ............................... 67
CONTENTS viii
SECT. PAGE
31. Functions of two variables and their graphical representation .. 68
32. Curves in a plane ............................................. 69
33. 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
34–38. Displacements ................................................ 81
39–42. Complex numbers ............................................ 92
43. The quadratic equation with real coefficients .................. 96
44. Argand’s diagram ............................................. 100
45. De Moivre’s Theorem ......................................... 101
46. Rational functions of a complex variable ...................... 104
47–49. Roots of complex numbers .................................... 118
Miscellaneous Examples ...................................... 122
Properties of a triangle, 106, 121. Equations with complex coeffi-
cients, 107. Coaxal circles, 110. Bilinear and other transforma-
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
50. Functions of a positive integral variable ....................... 128
51. Interpolation ................................................. 129
52. Finite and infinite classes ..................................... 130