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MATH0STAT.

Edward Bright

Mathematics Deptr


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A COURSEOF

PURE MATHEMATICS


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CAMBRIDGE UNIVERSITY PRESSC. F. CLAY, MANAGER

LONDON : FETTER LANE, E.G. 4

NEW YORK : THE MACMILLAN CO.BOMBAY \CALCUTTA L MACMILLAN AND CO., LTD.MADRAS JTORONTO : THE MACMILLAN CO. OFCANADA, LTD.TOKYO : MARUZEN-KABUSHIKI-KAISHAALL RIGHTS RESERVED


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A COURSEOF

PURE MATHEMATICS)

BYG. H. HARDY, M.A., F.R.S.

FELLOW OF NEW COLLEGESAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY

OF OXFORDLATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE

THIRD EDITION

Cambridgeat the University Press

1921


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VU

First Edition 1908Second Edition 1914

Edition 1921


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PREFACE TO THE THIRD EDITIONNO extensive changes have been made in this edition. The mosfcimportant 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 thegenesis of the circular functions. I have made some attempt tomeet this objection in 158 and Appendix III. Appendix IV is alsoan addition.

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

G. H. H.August 1921

EXTRACT FROM THE PREFACE TOTHE SECOND EDITIONTHE principal changes made in this edition are as follows.I have inserted in Chapter I a sketch of Dedekind s theoryof real numbers, and a proof of Weierstrass s theorem concerningpoints of condensation ; in Chapter IV an account of limits ofindetermination and the general principle of convergence ; inChapter V a proof of the Heine-Borel Theorem , Heine s theoremconcerning uniform continuity, and the fundamental theoremconcerning implicit functions; in Chapter VI some additionalmatter concerning the integration of algebraical functions ; andin Chapter VII a section on differentials. I have also rewrittenin a more general form the sections which deal with the definition of the definite integral. In order to find space for theseinsertions I have deleted a good deal of the analytical geometryand formal trigonometry contained in Chapters II and III ofthe first edition. These changes have naturally involved alarge number of minor alterations.

G. H. H.October 1914


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EXTEACT FEOM THE PEEFACE TO THEFIEST EDITIONbook has been designed primarily for the use of first

year students at the Universities whose abilities reach orapproach something like what is usually described as scholarshipstandard . I hope that it may be useful to other classes ofreaders, but it is this class whose wants I have considered first.It is in any case a book for mathematicians: I have nowheremade any attempt to meet the needs of students of engineeringor indeed any class of students whose interests are not primarilymathematical.

Iregard

the book asbeing really elementary. There areplenty of hard examples (mainly at the ends of the chapters) : to

these I have added, wherever space permitted, an outline of thesolution. But I have done my best to avoid the inclusion ofanything that involves really difficult ideas. For instance, I makeno use of the principle of convergence : uniform convergence,double series, infinite products, are never alluded to : and I proveno general theorems whatever concerning the inversion of limit-

d*f d*foperations I never even define 5-%- and =-4-. In the last twocxdy dydxchapters I have occasion once or twice to integrate a power-series,but I have confined myself to the very simplest cases and givena special discussion in each instance. Anyone who has read thisbook will be in a position to read with profit Dr Bromwich sInfinite Series, where a full and adequate discussion of all thesepoints will be found.

September 1908


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CONTENTSCHAPTER IREAL VARIABLES

SECT. *AGE1-2. Rational numbers . . . . . * -.-. , 13-7. Irrational numbers . . . -; * "~8. Real numbers . 139. Relations of magnitude between real numbers . .1510-11. Algebraical operations with real numbers .... 1712. The number x/2 ..*... . 1913-34. Quadratic surds ... . .. ,, . 1915. The continuum ......... 2316. The continuous real variable ; 2617. Sections of the real numbers. Dedekind s Theorem . . 2718. Points of condensation . . . .19. Weierstrass s Theorem . V ." . . . . 30

Miscellaneous Examples 31Decimals, 1. Gauss s Theorem, 6. Graphical solution of quadratic

equations, 20. Important inequalities, 32. Arithmetical and geometricalmeans, 32. Schwarz s Inequality, 33. Cubic and other surds, 34.Algebraical numbers, 36.

CHAPTER IIFUNCTIONS OF REAL VARIABLES

20. The idea of a function 3821. The graphical representation of functions. Coordinates . 4122. Polar coordinates . ........ djfc-23. Polynomials . . . . ... &.24-25. Rational functions 4

"

26-27. Algebraical functions ....- 4928-29. Transcendental functions ... ...30. Graphical solution of equations 5831. Functions of two variables and their graphical repre

sentation ^


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Vlll CONTENTSSECT. PAG!*32. Curves in a plane . . . . . . . 6033. Loci in space . . . . . , . . 61

Miscellaneous Examples ...... 65Trigonometrical functions, 53. Arithmetical functions, 55. Cylinders,62. Contour maps, 62. Cones, 63. Surfaces of revolution, 63. Ruledsurfaces, 64. Geometrical constructions for irrational numbers, 66.Quadrature of the circle, 68.

CHAPTER IIICOMPLEX NUMBERS

34-38. Displacements * 6939-42. Complex numbers -. .... . 7843. The quadratic equation with real coefficients . . . 8144. Argand s diagram . . . . . .. . . 8445. de Moivre s Theorem . . . .? ., . ... . 8646. Rational functions of a complex variable . .... . 8847-49. Roots of complex numbers . . . . . * 98

Miscellaneous Examples . . . * . .... 101Properties 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 polygonsby Euclidean methods, 100. Imaginary points and lines, 103.

CHAPTER IVLIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE

50. Functions of a positive integral variable . . . .10651. Interpolation . 10752. Finite and infinite classes 10853-57. Properties possessed by a function of n for large values of n 10958-61. Definition of a limit and other definitions . . . .11662. Oscillating functions . 12163-68. General theorems concerning limits 12569-70. Steadily increasing or decreasing functions . . . l.il71. Alternative proof of Weierstrass s Theorem . . . 1, 34l ........................ (1),*and divide A^A^ into k equal parts, then at least one of the points jof division (say P) must fall inside BC, without coinciding witheither B or C. For if this were not so, BC would be entirelyincluded in one of the k parts into which A 1A 2 has been divided,which contradicts the supposition (1). But P obviously corresponds to a rational number whose denominator is k. Thus atleast one rational point P lies between B and C. But then wecan find another such point Q between B and P, another betweenB and Q, and so on indefinitely ; i.e., as we asserted above, we canfind as many as we please. We may express this by saying thatBC includes infinitely many rational points.

The meaning of such phrases as * infinitely many or an infinity of\ insuch sentences as BC includes infinitely many rational points or there arean infinity of rational points on C } or there are an infinity of positiveintegers , will be considered more closely in Ch. IV. The assertion there arean infinity of positive integers means given any positive integer n, howeverlarge, we can find more than n positive integers . This is plainly true

* The assumption that this is possible is equivalent to the assumption of whatis known as the Axiom of Archimedes. 12


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4 REAL VARIABLES [lwhatever n may be, e.g. for n= 100,000 or 100,000,000. The assertion meansexactly the same as we can find as many positive integers as we please .

The reader will easily convince himself of the truth of the follo\vingassertion, which is substantially equivalent to what was proved in the secondparagraph of this section : given any rational number r, and any positiveinteger ?i, we can find another rational number lying on either side of r anddiffering from r by less than l/n. It is merely to express this differently tosay that we can find a rational number lying on either side of r and differingfrom r by as little as we please. Again, given any two rational numbers-r and s, we can interpolate between them a chain of rational numbers inwhich any two consecutive terms differ by as little as we please, that is tosay by less than l/?i, where n is any positive integer assigned beforehand.

From these considerations the reader might be tempted toinfer that an adequate view of the nature of the line could beobtained by imagining it to be formed simply by the rationalpoints which lie on it. And it is certainly the case that if weimagine 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 likea line.

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

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

The straight line must be composed of points, and any segmentof it by all the points which lie between its end points. Withany such segment must be associated a certain entity called itslength, which must be a quantity capable of numerical measurement in terms of any standard or unit length, and these lengthsmust be capable of combination with one another, according to-the ordinary rules of algebra, by means of addition or multiplication. Again, it must be possible to construct a line whoselength is the sum or product of any two given lengths. If thelength PQ t along a given line, is a, and the length QR, alongthe same straight line, is b, the length PR must be a + 6.


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3] REAL VARIABLESMoreover, if the lengths OP, OQ, along one straight line, are1 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, thislength must be ab, the algebraical fourth proportional to 1, a, b.And it is hardly necessary to remark that the sums and productsthus 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 obviouslaws concerning inequalities as well as equalities : thus ifA, B, C are three points lying along A from left to right, we musthave AB< AC, and so on. Moreover it must be possible, on ourfundamental line A, to find a point P such that A QP is equal toany segment whatever taken along A or along any other straightline. All these properties of a line, and more, are involved in thepresuppositions of our elementary geometry.

Now it is very easy to see that the idea of a straight line ascomposed of a series of points, each corresponding to a rationalnumber, cannot possibly satisfy all these requirements. There arevarious elementary geometrical constructions, for example, whichpurport to construct a length x such that x* = 2. For instance, we

MFig. 2.

may construct an isosceles right-angled triangle ABC such thatAB = AC=1. Then if BC=oc, #2 = 2. Or we may determinethe length x by means of Euclid s construction (Euc. vi. 13) fora mean proportional to 1 and 2, as indicated in the figure. Ourrequirements therefore involve the existence of a length measuredby a number x, and a point P on A such that


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6 EEAL VARIABLES [lBut it is easy to see that there is no rational number such thatits square is 2. In fact we may go further and say that thereis no rational number whose square is m/n, where m/n is anypositive fraction in its lowest terms, unless m and n are bothperfect squares.

For suppose, if possible, that

p having no factor in common with q, and m no factor in commonwith n. Then np2 = mqz. Every factor of

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