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Cornell University Library QA 303.H26C8 1921 A course of pure mathematics. 3 1924 001 573 488 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001573488 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 450A3 First Edition 1908 Second Edition 1934 Third Edition 1921 rWNTIin IN OKSAT ■BBCTAIN 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 dififerentials. 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 ;exteaot 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 ;r^- and 7r4^. In the last two ■^ cxoy oyax 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 SECT. 1-2. 3-7. 8. 9. 10-11. 12. 13-14. 15. 16. 17. 18. 19. CONTENTS CHAPTER I EEAL VARIABLES Rational numbers . .... Irrational numbers ... . . Real numbers Relations of magnitude between real numbers . Algebraical operations with real numbers . The number v/2 Quadratic surds The continuum The continuous real variable .... Sections of the real numbers. Dedekind's Theorem Points of condensation Weierstrass's Theorem Miscellaneous Examples PAGE 1 3 13 15 17 19 19 23 26 27 29 30 31 Decimals, 1. Gauss's Theorem, 6. Graphical solution of quadratic equations, 20. Important inequalities, 32. Arithmetical and geometrical means, 32. Schwarz's Inequality, 33. Cubic and other surds, 34. Algebraical numbers, 36. 20. 21. 22. 23. 24-25. 26-27. 28-29. 30. 31. CHAPTER II FUNCTIONS OF REAL VARIABLES The idea of a function The graphical representation of functions. Coordinates Polar coordinates Polynomials Rational functions Algebraical functions . Transcendental functions Graphical solution of equations rvmctions of two variables and their graphical repre- sentation 38 41 43 44 47 49 52 58 59 Vlll CONTENTS SECT, 32. 33. PAGE Curves in a plane 60 Loci in space 61 Miscellaneous Examples 65 Trigonometrical functions, 5.S. Arithmetical functions, 55. Cylinders, 62. Contour maps, 62. Cones, 63. Surfaces of revolution, 63. Euled surfaces, 64. Geometrical constructions for ■ irrational numbers, 66. Quadrature of the circle, 68. CHAPTER III COMPLEX NUMBERS 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 69 78 81 84 86 101 Properties of a triangle, 90, 101. Equations with complex coefficients, 91. Coaxal circles, ^3. Bilinear and other transformations, 94, 97, 104. Cross ratios, 96. Condition that fonr 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. CHAPTER IV 50. 51. 52. 53-57. 58-61. 62. 63-68. 69-70. 71. 72 73. 74. 75. 76-77. 78. LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE Functiops of a positive integral variable . Interpolation Finite and infinite classes Properties possessed by a function of n for large values of 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 x'^ . The limit of (-3" Some algebraical lemmas The limit of n{ijx-\). Infinite series The infinite geometrical series 106 107 108 109 ^116 121 125 131 134 134 137 138 139 140 143 CONTENTS IX SECT. PAGE 79. 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 2" and the geometrical series . . . 156 Miscellaneous Examples 157 Oscillation of sinnflx, 121, 123, 151. Limits of n'l", ;yx, ;^n, i/{n\), -7,1 la;", 136, 139. Decimals, 143. Arithmetical series, 146. Harmonical series, 147. Equation x,^^=f{x„), 158. Expansions of rational functions, 159. Limit of a mean value, 160. CHAPTER V LIMITS OP FUNCTIONS OP A CONTINUOUS VARIABLE. AND DISCONTINUOUS FUNCTIONS Continuous 89-92. Limits as a;-»-QO or x-^—<x> 162 93-97. Limits as x-i^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, 172t Limit of ——,173. Infinity of a function, 177. Continuity of cos? 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 diflPerential calculus 116. Differentiation of polynomials 117. Differentiation of rational functions . 118. Differentiation of algebraical functions 197 203 205 205 207 209 210 X 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 Derivative of a"", 201. Derivatives of cos a; and sinx, 201. Tangent and normal to a curve, 201, 214. Multiple roots of equations, 208, 255. Eolle'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, 803. Points of inflexion, 272. Curvature, 278, 802. 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. Jaoobiaus, 303. Schwarz's in- equality for integrals, 306. Approximate values of definite integrals 307 Simpson's Eule, 307. ' CONTENTS XI CHAPTER VIII THE CONVERGENCE OF INFINITE SEEIES AND INFINITE INTEGRALS SECT. PAGE 165-168. Series of positive terms. Cauohy's and d'Alembert's tests of convergence . 308 169. Diriohlet's Theorem . . • 313 170. Multiplication of series of positive terms .... 313 171-174. Further tests of convergence. Abel's Theorem. Maclaurin's integral test 3x5 175. The series Sn.-' 3x9 176. Cauohy's condensation test 320 177-182. Infinite integrals 321 183. Series of positive and negative terms 335 184-185. Absolutely convergent series 336 186-187. Conditionally convergent series 338 188. Alternating series 34O 189. Abel's and Diriohlet's tests of convergence . . . 342 190. Series of complex terms 344 191-194. Power series 345 195. Multiplication of series in general 349 Miscellaneous Examples 350 The series Sn*?" and allied series, 311. Transformation of infinite integrals by substitution and integration by parts, 327, 328, 383. The series Sa„oosnS, Sa„sin«9, 338, 343, 344. Alteration of the sum of a series by rearrangement, 841. Logarithmic series, 348. Binomial series, 348, 349. Multiplication of conditionally convergent series, 350, 834. Beourring series, 852. Difference equations, 353, Definite integrals, 355. Schwarz's inequality for infinite integrals, 856. CHAPTER IX THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL VARIABLE 196-197. The logarithmic function 357 198. The functional equation satisfied by log^ .... 360 199-201. The behaviour of log« as x tends to infinity or to zero . 360 202. The logarithmic scale of infinity 362 203. The number e 363 204-206. The exponential function 364 207. The general power a" 366 208. The exponential limit 368 209. The logarithmic limit 369 210. Common logarithms 369 211. Logarithmic tests of convergence 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 PJGE 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 logio n, 387. Definite integrals, 393. CHAPTER X THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIHCULAH FUNCTIONS 217-218. Functions of a complex variable 395 219. Curvilinear integrals 396 220. Definition of the logarithmic function .... 397 221. The values of the logarithmic function .... 399 222-224. The exponential function 403 225-226. The general power a' 404 227-230. The trigonometrical and hyperbolic functions . . . 409 231. The connection between the logarithmic and inverse trigonometrical functions .... . 413 232. The exponential series 414 233. The series for cose and sinz 416 234-235. The logarithmic series 417 236. The exponential limit 421 237. The binomial series 422 Miscellaneous Examples 425 The functional equation satisfied by Log z, 402. The function e", 407. Logarithms to any base, 408. The inverse cosine, sine, and tangent of a complex number, 412. Trigonometrical series, 417, 420, 431. Eoots 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(-q) = (-p)/q, (-p)K-q)=plq- To the rational numbers thus defined we may add the ' rational number ' obtained by taking ^ = 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. I. If r and s are rational numbers, then r + s, r~s, rs, and rjs are rational numbers, unless in the last case s = (when r/s is of course meaningless). 2. If X, m, and n are positive rational numbers, and m > n, then X(m^-n^), 2\m.n, and \{m^ + n'') 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 terminateddecimal represents a rational number whose denomi- nator contains niTfactors 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 series as follows : 1' 1121 1) 21 3', 1' 21 3> 41 •■•• Show that p/q is the [^(p + q-l){p + q-2) + q]th term of the series. [In this series every rational number is repeated indefinitely. Thus 1 occiirs as J, |, f , .... We can of course avoid this by omitting every number H. 1 -2 REAL VARIABLES [l which has already occurred in a simpler form, but then the problem of deter- mining the precise position of piq 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 maybe 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^Ai of any length. We call ^o the m-igin, or the point 0, and A^ the paint 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 Ar such that AfiAr/Ai)Ai = r, A„Ar being a stretch of the line extending in the same direction along the line as A^A^, 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 1 1 ! A -3 A_i Ao A], 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 A^A^), and negative if measured in the other, so that AB = -BA ; and to take as the point representing r the point A_g such that A„A_s = - A_gA = - A^A,. l-'S] REAL VARIABLES 3 We thus obtain a point ^^ on the line corresponding to every rational value of r, positive or negative, and such that and if, as is natural, we take ^o'^i as our unit of length, and write A„Ai= 1, then we have AoAr = r. We shall call the points A^ 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-^^A^. It is evident that if we choose a positive integer k so that k.BG>l (1),* and divide A1A2 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 A1A2 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 'are infinity of, in such sentences as ' BC includes infinitely many rational points ' or ' there are an infinity of rational points on BC 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 15 known as the Axiom of Archimedes. 1—2 4 BEAL VARIABLES [l whatever n may be, e.ff. for ?i= 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 rhj 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 r 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 Ijn, where n is any positive integer assigned beforehand. From these considerations the reader might be tempted to infer that an adequate vievv^ 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 ia 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 lengtli, 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 + 6. 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 h, 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-\- 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 composed of a series of points, each corresponding to a rational number, cannot possibly satisfy all these requirements. There are various elementary geometrical constructions, for example, which purport to construct a length x such that x^ = 2. For instance, we Fig. 2. may construct an isosceles right-angled triangle ABC such that AB = AG=\. Then if BG=x, as^ = 2. Or we may determine the length x by means of Euclid's construction (Euc. vi. 13) for a mean proportional to 1 and 2, as indicated in the figure. Our requirements therefore involve the existence of a length measured by a number x, and a point P on A such that Ai,P = x, x'=2. 6 REAL VARIABLES L^ But it is easy to see that there is no rational number such that its square is 2. In fact we may go further and say that there is no rational number whose square is m/w, where mjn is any positive fraction in its lowest terms, unless m and n axe both perfect squares. . For suppose, if possible, that p^jc^ = mjn. p having no factor in common with q, and m no factor m common with n. Then np'' = mq^ Every factor of q^ must divide wp^ and as p and q have no common factor, every factor of q^ must divide n. Hence n = \q^, where X is an integer. But this involves m = \p^ : and as m and n have no common factor, \ must be unity. Thus m =p^, n = q^,as was to be proved. In particular it follows, by taking n = 1, that an integer cannot be the square of a rational number, unless that rational number is itself integral. It appears then that our requirements involve the existence of a number x and a point P, not one of the rational points already constructed, such that A^P = oi;, x'' = 2; and (as the reader will remember from elementary algebra) we write x = ^2. The following alternative proof that no rational number can have its square equal to 2 is interesting. Suppose, if possible, that p/q is a positive fraction, in its lowest terms, such that {pjqf='2, or p^ = 9.q^. It is easy to see that this involves {iq—py=%{p-qY; and so (2q-p)/(p!-q) is another fraction having the same property. But clearly q<p<2q, and so p — q<q-\ Hence there is another fraction equal to pjq and having a smaller denominator, which contradicts the assumption that pjq is in its lowest ^tgfms. Examples II, 1. Show that no rational number can have its cube equal to 2. 2. Prove generally that a rational fraction pjq in its lowest terms cannot be the cube of a rational number unless p and q are both perfect cubes. 3. A more general proposition, which is due to Gauss and includes those which precede as particular cases, is the following : an algebraical eqvxUion ^''+jDi.r''-i+P2a;»-2+...+p„=0, with integral coefficients, cannot have a rational but non-integral root. [For suppose that the equation has a root ajb, where a and b are integers 3, 4] REAL VAEIABLES 7 ■without a common factor, and 6 is positive. Writing a/b for x, and multiply- ing by 6""!, we obtain a fraction in its lowest terms equal to an integer, which is absurd. Thus 6 = 1, and the root is a. It is evident that a must be a divisor of jo„.] 4. Show that if />„= 1 and neither of is zero, then the equation cannot have a rational root. 5. Find the rational roots (if any) of t /' ^-4^-8^2 + 13^'+ 10 = 0. [The roots can only be integral, and so +1, +2, +5, +10 are the only possibilities : whether these are roots can be determined by trial. It is clear that we can in this way determine the rational roots of any such equation.] 4. Irrational numbers {continued). The result of our geometrical representation of the rational numbers is therefore to suggest the desirability of enlarging our conception of ' number ' by the introduction of further numbers of a new kind. The same conclusion might have been reached without the use of geometrical language. One of the central problems of algebra is that of the solution of equations, such as The first equation has the two rational roots 1 and — 1. But, if our conception of number is to be limited to the rational numbers, we can only say that the second equation has no roots ; and the same is the case with such equations as a? =2, of = 7. These facts are plainly sufficient to make some generalisation of our idea of number desirable, if it should prove to be possible. Let us consider more closely the equation a:^ = 2. We have already seen that there is no rational number x which satisfies this equation. The square of any rational number is either less than or greater than 2. We can therefore divide the rational numbers into two classes, one containing the numbers whose squares are less than 2, and the other those whose squares are greater than 2. We shall confine our attention to the positive rational numbers, and we shall call these two classes the class L, or the lower class, or the left-hand class, and the class R, or the upper 8 KEAL VARIABLES [l class, or the right-hand class. It is obvious that every member of R is greater than all the members of L. Moreover it is easy to convince ourselves that we can find a member of the class L whose square, though less than 2, differs from 2 by as little as we please, and a member of R whose square, though greater than 2, also differs from 2 by as little as we please. In fact, if we carry out the ordinary arithmetical process for the extraction of the square root of 2, we obtain a series of rational numbers, viz. 1, 1-4, 1-41. 1-414, 1-4142,... whose squares 1, 1-96, 1-9881, 1-999396, 1-99996164,... are all less than 2, but approach nearer and nearer to it ; and by taking a sufficient number of the figures given by the process we can obtain as close an approximation as we want. And if we increase the last figure, in each of the approximations given above, by unity, we obtain a series of rational numbers 2, 1-5, 1-42, 1-415, 1-4143,... whose squares 4, 2-25, 2-0164, 2-002225, 2-00024449,... are all greater than 2 but approximate to 2 as closely as we please. The reasoning which precedes, although it will probably convince the reader, is hardly of the precise character required by modem mathematics. We can supply a formal proof as follows. In the first place, we can find a member of L and a member of R, difiering by as little as we please. For we saw in § 3 that, given any two rational numbers a and b, we can construct a chain of rational numbers, of which a and h ai-e the first and last, and in which any two consecutive numbers differ by as little as we please. Let us then take a member x oi L and a member y of R, and interpolate between them a chain of rational numbers of which x is the first and y the last, and in which any two consecutive numbers dififer by less than 8, fi being any positive rational number as small as we please, such as -01 or -0001 or -000001. In this chain there must be a last which belongs to L and a first which belongs to R, and these two numbers difibr by less than 8. We can now prove that an x can he found in'L and a y in R such that 2 — a'^ and y^-'i are as small as we please, say less than 8. Substituting J 8 for 8 in the argument which precedes, we see that we can choose x and y so that y — x<^h; and we may plainly suppose that both x and y are less than 2. Thus y + x<i, y'^-x^={y-x){y+x)<4:{y-x)<8; 4, 5] EEAL VABIABLES 9 and since a;2<2 and 2/^>2 it follows a fortiori that 2-x^ and 2/^-2 are each less than S. It follows also that there can be no largest member of L or smallest member of R. For if x is any member of L, then x' < 2. Suppose that a^=2 — 8. Then we can iind a member Xi of L such that Xi^ differs from 2 by less than 8, and so x^'' > x^ or x^ > x. Thus there are larger members of L than x; and as x is any member of i, it follows that no member of L can be larger than all the rest. Hence L has no largest member, and similarly R has no smallest. 5. Irrational numbers (continued). We have thus divided the positive rational numbers into two classes, L and R, such that (i) every member of R is greater than every member of L, (ii) we can find a member of L and a member of R whose difference is as small as we please, (iii) L has no greatest and R no least member. Our common-sense notion of the attributes of a straight line, the requirements of our elementary geometry and our elementary algebra, alike demand the existence of a niumber x greater than all the members of L and less than all the members of R, and of a corresponding point P on A svch that P divides the points which correspond to members of L from those which correspond to members ofR. L LL R R Fig. 3. Let us suppose for a moment that there is such a number x, and that it may be operated upon in accordance with the laws of algebra, so that, for example, x'^ has a definite meaning. Then x^ cannot be either less than or greater than 2. For suppose, for example, that x^ is less than 2. Then it follows from what pre- cedes that we can find a positive rational number ^ such that 1^ lies 10 REAL VAEIABLES [^ between a^ and 2. That is to say, we can iind a member of L greater than x ; and this contradicts the supposition that x divides the members of L from those of B,. Thus a? cannot be less than 2, and similarly it cannot be greater than 2. We are therefore driven to the conclusion that a? = % and that x is the number which in algebra we denote by \/2. And of course this number v'2 is not rational, for no rational number has its square equal to 2. It is the simplest example of what is called an irrational number. But the precediug argument may be applied to equations other than a? = 2, almost word for word ; for example to x- = iV, where N is any integer which is not a perfect square, or to af = 3, «==7, a;* = 23, or, as we shall see later on, to a? = 3x + 8. We are thus led to believe in the existence of irrational numbers x and points P on A such that x satisfies equations such as these, even when these lengths cannot (as \/2 can) be constructed by means of elementary geometrical methods. TBe reader will no doubt remember that in treatises on elementary algebra the root of such an equation as afi=n is denoted by ^n or n^li, and that a meaning is attached to such symbols as by means of the equations And he will remember how, in virtue of these definitions, the ' laws of indices' such as are extended so as to cover the case in which r and s are any rational numbers whatever. The reader may now follow one or other of two alternative courses. He may, if he pleases, be content to assume that 'irrational numbers' such as V2, -^3, ... exist and are amenable to the algebraical laws with which he is familiar*. If he does this he will be able to avoid the more abstract discussions of the next few sections, and may pass on at once to §§ 13 e< seq. If, on the other hand, he is not disposed to adopt so naive an * Thia is the point of view which was adopted in the first edition of this book. 5, 6] REAL VARIABLES 11 attitude, he will be well advised to pay careful attention to the sections which follow, in which these questions receive fuller consideration *. Examples III. 1. Find the difference between 2 and the squares of the decimals given in § 4 as approximations to ^2, 2. Find the differences between 2 and the squares of h h h a, th n- 3. Show that if m[n is a good approximation to J2, then (m + 2n)l{m + n) is a better one, and that the errors in the two caseS are in opposite directions. Apply this result to continue the series of approximations in the last example. 4. If .r and y are approximations to J2, by defect and by excess respec- tively, and 2-^2<-g^ y^-2<8, then y-.»<S. 5. The equation a;2=4 is satisfied by x=2. Examine how far the argu- ment of the preceding sections applies to this equation (writing 4 for 2 throughout). [If we define the classes X, R as before, they do not include all rational numbers. The rational number 2 is an exception, since 2^ is neither less than or greater than 4.] 6. Irrational numbers (continued). In § 4 we discussed a special mode of division of the positive rational numbers x into two classes, such that a}^<2 for the members of one class and a^ > 2 for those of the others. Such a mode of division is called a section of the numbers in question. It is plain that we could equally well construct a section in which the numbers of the two classes were characterised by the inequalities oc^ <2 and a;' > 2, or 01* < 7 and af^ >7. Let us now attempt to state the principles of the construction of such 'sections' of the positive rational numbers in quite general terms. Suppose that P and Q stand for two properties which are mutually exclusive and one of which must be possessed by every positive rational number. Further, suppose that every such number which possesses P is less than any such number which possesses Q. Thus P might be the property ' a;^ < 2 ' and Q the property ' as" > 2.' Then we call the numbers which possess P th6 lower or left-hand class L and those which possess Q the upper or * In these sections I have borrowed freely from Appendix I of Bromwich's Infinite Series. 12 EEAL VARIABLES L^ right-hand class R. In general both classes will exist ; but it may happen m special cases that one is non-existent and that every number belongs to the other. This would obviously happen, for example, if P (or Q) were the property of being rational, or of being positive. For the present, however, we shall confine ourselves to cases in which both classes do exist; and then it follows, as in § 4, that we can find a member of L and a member of R whose difference is as small as we please. In the particular case which we considered in § 4, Z had no greatest member and R no least. This question of the existence of greatest or least members of the classes is of the utmost im- portance. We observe first that it is impossible in any case that L should have a greatest member and R a least. For if I were the greatest member of L, and r the least of R, so that I <r, then \{l + r) would be a positive rational number lying between I and r, and so could belong neither to L nor to R ; and this contradicts our assumption that every such number belongs to one class or to the other. This being so, there are but three possibilities, which are mutually exclusive. Either (i) L has a greatest member I, or (ii) R has a least member r, or (iii) L has no greatest member and R no least. The section of § 4 gives an example of the last possibility. An example of the. first is obtained by taking P to be 'a.'2<l' and Q to be ^x^>l'; here l = \. If P is 'a-^ < 1 ' and § is '^^ > 1,' we have an example of the second possibility, with r=l. It should be observed that we do not obtain a section at all by taking P to be ' ^^ < 1 ' and § to be ' a;^> 1 ' ; for the special number 1 escapes classification (cf. Ex. in. 5). 7. Irrational numbers {continued). In the first two cases we say that the section corresponds to a positive rational number a, which is I in the one case and r in the other. Conversely, it is clear that to any such number a corresponds a section which we shall denote by a*. For we might take P and Q to be the properties expressed by X S^a, x> a respectively, or by a; < a and x^a. In the first case a would be the greatest member of L, and in the second case the least member • It will be convenient to denote a section, corresponding to a rational number denoted by an English letter, by the corresponding Greek letter. 6-8] REAL VAEIABLES 13 of R. There are in fact just two sections corresponding to any positive rational number. In order to avoid ambiguity we select one of them ; let us select that in which the number itself belongs to the upper class. In other words, let us agree that we will consider only sections in which the lower class L has no greatest number. There being this correspondence between the positive rational numbers and the sections defined by means of them, it would be perfectly legitimate, for mathematical purposes, to replace the numbers by the sections, and to regard the symbols which occur in our formulae as standing for the sections instead of for the numbers. Thus, for example, a > a' would mean the same as a > a, if a and a' are the sections which correspond to a and a'. But when we have in this way substituted sections of rational numbers for the rational numbers themselves, we are almost forced to a generalisation of our number system. For there are sections (such as that of § 4) which do not correspond to any rational number. The aggregate of sections is a larger aggregate than that of the positive rational numbers; it includes sections corresponding to all these numbers, and more besides. It is this fact which we make the basis of our generalisation of the idea of number. We accordingly frame the following definitions, which will however be modified in the next section, and must therefore be regarded as temporary and provisional. A section of the positive rational numbers, in which both classes exist and the lower class has no greatest member, is called a positive real number. A positive real number which does not correspond to a positive rational number is called a positive irrational number, 8. Real numbers. We have confined ourselves so far to certain sections of the positive rational numbers, which we have agreed provisionally to call 'positive real numbers.' Before we frame our final definitions, we must alter our point of view a little. We shall consider sections, or divisions into two classes, not merely of the positive rational numbers, but of all rational numbers, including zero. \^e may then repeat all that we have said about sections of the positive rational numbers in §§ 6, 7, merely omitting the word positive occasionally. 14 REAL VAKTABLES [l Definitions. A section of the rational numbers, in which both classes exist and the lower class has no greatest member, is called a real number, or simply a number. A real number which does not correspond to a rational number is called an irrational number. If the real number does correspond to a rational number, we shall use the term ' rational ' as applying to the real number also. The term 'rational number' will, as a result of our definitions, be ambiguous; it may mean the rational number of § 1, or the corresponding real number. If we say that J > ^, we may be asserting either of two different propositions, one a proposition of elementary arithmetic, the other a proposition concerning sections of the rational numbers. Ambiguities of this kind are common in mathematics, and are perfectly harmless, since the relations between different propositions are exactly the same whichever interpretation is attached to the propositions themselves. From ^>i and J>J we can infer i>i ; the inference is in no way affected by any doubt as to whether J, ^, and J are arithmetical fractions or real numbers. Sometimes, of course, the context in wKch (e.g.) '^' occurs is sufficient to fix its interpretation. When we say (see § 9) that ■^< V(i)i "^^ ™''*' mean by '|' the real number §. The reader should observe, moreover, that no particular logical importance is to be attached to the precise form of definition of a ' real number ' that we have adopted. We defined a ' real number ' as being a section, i.e. a pair of classes. We might equally well have defined it as being the lower, or the upper, class ; indeed it would be easy to define an infinity of classes of entities each of which would possess the properties of the class of real numbers. What is essential in mathematics is that its symbols should be capable of soine interpretation; generally they are capable of many, and then, so far as mathematics is concerned, it does not matter which we adopt. Mr Bertrand Russell has said that 'mathematics is the science in which we do not know what we are talking about, and do not care whether what we say about it is true', a remark which is expressed in the form of a paradox but which in reality embodies a number of important truths. It would take too long to analyse the meaning of Mr Russell's epigram in detail, but one at any rate of its implications is this, that the symbols of mathe- matics are capable of varying interpretations, and that we are in general at liberty to adopt whichever we prefer. There are now three cases to distinguish. It may happen that all negative rational numbers belong to the lower class and zero and all positive rational numbers to the upper. We describe this section as the real number zero. Or again it may happen that the lower class includes some positive numbers. Such a section 8, 9] REAL VAEIABLES 15 we describe as a positive real number. Finally it may happen that some negative numbers belong to the upper class. Such a section we describe as a negative real number* The difference between our present definition of a positive real number a and that of § 7 amounts to the addition to the lower class of zero and all the negative rational numbers. An example of a negative real number is given by taking the property P of § 6 to be ^k+KO and Q to be x+l-^O. This section plainly corresponds to the negative rational number - 1. If we took P to be x^<-2 and Q to be afl> - 2, we should obtain a negative real number which is not rational. 9. Relations of magnitude between real numbers. It is plain that, now that we have extended our conception of number, we are bound to make corresponding extensions of our conceptions of equality, inequality, addition, multiplication, and so on. We have to show that these ideas can be applied to the new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of § 1.. To do all this systematically would occupy a considerable space, and we shall be' content to indicate summarily how a more systematic discussion would proceed. We denote a real number by a Greek letter such as a, j3,y, ...; the rational numbers of its lower and upper classes by the corre- sponding English letters a, A; b, B; c, C; .... The classes them- selves we denote by (a), (A), ...., If a and ^ are two real numbers, there are three possibilities : (i) every a is a 6 and every AaB; in this case (a) is identical with (6) and (A) with (5); * There are also Bections in which every number belongs to the lower or to the upper class. The reader may be tempted to ask why we do not regard these sections also as defining numbers, which we might call the real numbers positive and negative infinity. There is no logical objection to such a procedure, but it proves to be incon- venient in practice. The most natural definitions of addition and multiplication do not work in a satisfactory way. Moreover, for a beginner, the chief difficulty in the elements of analysis is that of learning to attach precise senses to phrases containing the word ' infinity'; and experience seems to show that he is likely to be confused by any addition to their number. 16 REAL VARIABLES [l (ii) every aisab, but not all A's are B's ; in this case (a) is a proper part of (&)*, and (B) a proper part of (^) ; (iii) every A is Sk B, but not all a's are b's. These three cases may be indicated graphically as in Fig. 4. In case (i) we write a = /3, in case (ii) a < /S, and in case (iii) a > /3. It is clear that, when a and /8 are both rational, these ? (i) definitions agTce with the ideas of equality and inequality between — + ' (") rational numbers which we began by taking for granted; and that ■ 1 ■ '■■•' a any positive number is greater Fig. 4. than any negative number. It will be convenient to define at this stage the negative — a ot a positive number a. If (a), (A) are the classes which consti- tute a, we can define another section of the rational numbers by putting all numbers —Am the lower class and all numbers — a in the upper. The real number thus defined, which is clearly negative, we denote by — a. Similarly we can define — a when a is negative or zero ; if a is negative, — a is positive. It is plain also that —(— a) = a. Of the two numbers a and — « one is always positive (unless a = 0). The one which is positive we denote by I a I and call the modulus of a. Examples IV. 1. Prove that 0=-0. 2. Prove that = a, |3<a, or /3>a according as a=/3, a>/3, or a</3. 3. If Q=/3 and /3 = y, then a=y. 4. If a < j3, P<y, or a</3, /3 g y, then a<y. 5. Prove that -/3= -a -0< -a, or -j3> -a, according as n=i3, a</3, or a>p. 6. Prove that a>0 if a is positive, and a<0 if a is negative. 7. Prove that ag I a |. 8. Prove that l<V2<x/3<2. 9. Prove that, if a and /3 are two different real numbers, we can always find an infinity of rational numbers lying between a and ^. [All these results are immediate consequences of oiu: definitions.] * I.e. is included in but not identical with (6). 9, 10] EEAL VARIABLES 17 10. Algebraical operations with real numbers. We now proceed to define the meaning of the elementary algebraical opera- tions such as addition, as applied to real numbers in general. (i) Addition. In order to define the sum of two numbers a and ^, we consider the following two classes : (i) the class (c) formed by all sums c = a + b, (ii) the class (C) formed by all sums G = A+B. Plainly c < in all cases. Again, there cannot be more than one rational number which does not belong either to (c) or to (C). For suppose there were two, say r and s, arid let s be the greater. Then both r and s must be greater than every c and less than every G; and so G — c cannot be less than s — r. But G-c = {A-a) + (B-b); and we can choose a, h, A, B so that both A— a and B—h are as small as we like; and this plainly contradicts our hypothesis. __- If every rational number belongs to (c) or to ((7), the classes (c), (0) form a section of the rational numbers, that is to say, a number 7. If there is one which does not, we add it to (0). "We have now a section or real number 7, which must clearly be rational, since it corresponds to the least member of {G). In any case we call 7 the sum of a and /3, and write 7 = a + ;8. If both a and ^ are rational, they are the least members of the upper classes {A) and {B). In this case it is clear that a+^ is the least member of (6% so that our definition agrees with our previous ideas of addition. (ii) Subtraction. We define a - ,8 by the equation a_^ = a + (-^). The idea of subtraction accordingly presents no fresh difficulties. Examples V. 1 . Prove that a + ( - a) = 0. 2. Prove that a-\-0=0-\-a=a. 3. Prove that a+^=/3+a. [This follows at once from the fact that the classes (a+6) and (6+a), or {A+E) and {B-\-A), are the same, since, e.g., a-^h=h-\-a when a and h are rational.] 4. Prove that a + 03+ 7)= (a + /J) +y. 2 18 EEAL VARIABLES [l 5. Prove that a - a = 0. 6. Prove that a - /3 = - ((3 - a). 7. From the definition of subtraction, and Exs. 4, 1, and 2 above, it follows that (a-/3) + /3 = {a + (-0)} + (3 = a + {(-^) + /3} = a + O = a. We might therefore define the difference a-j3 = 7 by the equation y+/3 = a. 8. Prove that a- (/3-y)=a-0 + y. 9. Give a definition of subtraction which does not depend upon a previous definition of addition. [To define y=a-/3, form the classes (c), (C) for which c=a-B, C=A-b. It is easy to show that this definition is equivalent to that which we adopted in the text.] 10. Prove that ||a|-|/3||<|a±0|S|aH-|/3l. 11. Algebraical operations with real numbers (con- tinued), (in) Multiplication. When we come to multiplication, it is most convenient to confine ourselves to positive numbers (among which we may include 0) in the first instance, and to go back for a moment to the sections of positive rational numbers only which we considered in §§ 4 — 7. We may then follow practi- cally the same road as in the case of addition, taking (c) to be (ab) and (G) to be (AB). The argument is the same, except when we are proving that all rational numbers with at most one exception must belong to (c) or (G). This depends, as in the case of addi- tion, on showing that we can choose a, A,b, and B so that C—c is as small as we please. Here we use the identity G-c = AB-ah = (A-a)B + a(B-b). Finally we include negative numbers within the scope of our definition by agreeing that, if a and /3 are positive, then (-a)y3 = -«A «(-/3) = -«/3, (-a)(-/3) = a;8. (iv) Division. In order to define division, we begin by de- fining the reciprocal 1/a of a number a (other than zero). Con- fining ourselves in the first instance to positive numbers and sections of positive rational numbers, we define the reciprocal of a positive number a by means of the lower class (1/-4) and the upper class (1/a). We then define the reciprocal of a negative number -a by the equation l/(- a) = -(l/a). Finally we define a/^ by the equation a//3=ax(l//3). lt>-13] REAL VARIABLES ■'•'•■'^mV 19 We are then in a position to apply to all real numbers, rational or irrational, the whole of the ideas and methods of elementary algebra. Naturally we do not propose to carry out this task in detail. It will be more profitable and more interesting to turn our attention to some special, but particularly important, classes of irrational numbers. Examples VI. Prove the theorems expressed by the following formulae : 1. axO=Oxa=0. 2. aXl = lXa=a. 3. ax(l/a) = l. 4. ai3=^a. o. a (/3y) = (ajS) y. 6. a(/3 + y)=a0 + ay. 7. (a + /3)y=ay + /3y. 8. | a3 | = |a| |^|. 12. The number ^Q. Let us now return for a moment to the particular irrational number which we discussed in §§ 4 — 5. We there constructed a section by means of the inequalities a? < 2, a;^ > 2. This was a section of the positive rational numbers only ; but we replace it (as was explained in § 8) by a section of all the rational numbers. We denote the section or number thus defined by the symbol \/2, The classes by means of which the product of \/2 by itself is defined are (i) (aa'), where a and a' are positive rational numbers whose squares are less than 2, (ii) (AA'), where A and A' are positive rational numbers whose squares are greater than 2. These classes exhaust all positive rational numbers save one, which can only be 2 itself. Thus (V2)==V2V2 = 2. -Again (- V2)^ = (- V2) (- V2) = V2 V2 = W2y = 2. Thus the equation x^=2 has the two roots ^/2 and - V2. Similarly we could discuss the equations x'' = 3, ai^=*I, ... and the corre- sponding irrational numbers V3, — V^, \^7, — 13. Quadratic surds. A number of the form + >Ja, where a is a positive rational number which is not the square of another rational number, is called a pure quadratic surd. A number of the form a ± \/h, where a is rational, and V^ is a pure quadratic surd, is sometimes called a mixed quadratic surd. 2—2 20 REAL VARIABLES [l The two numbers a + ^h are the roots of the quadratic equation" Conversely, the equation x^+ipx+q-^O, where p and q are rational, and p^ — 2>0, has as its roots the two quadratic surds —p±\f(p^ — q)- The only kind of irrational numbers whose existence was suggested by the geometrical considerations of § 3 are these quadratic surds, pure and mixed, and the more complicated irrationals which may be expressed in a form involving the repeated extraction of square roots, such as V2 + V(2 + V2) + V{2 + V(2 + V2)}. It is easy to construct geometrically a line whose length is equal to any number of this form, as the reader will easily see for himself. That irrational numbers of these kinds onli/ can be con- structed by Euclidean methods (i.e. by geometrical constructions with ruler and compasses) is a point the proof of which must be deferred for the present*. This property of quadratic surds makes them especially interesting. Examples VU. l. Give geometrical constructions for J2 V(2+V2), V{2+V(2 + V2)}. 2. The quadratic equation axl+2b3i;+c—0 has two real roots t if 62_ac>0. Suppose a, b c rational. Nothing is lost by taking all three to be integers, for we can multiply the equation by the least common multiple of their denominators. The reader will remember that the roots are {-b±xf(,b^-ac)}Ja. It is easy to construct these lengths geometrically, first constructing ^filfi-ac). A much more elegant, though less straightforward, construction is the following. * See Ch. II, Misc. Exs. 22. + I.e. there are two values of x for which aj?+2hx + c = 0. If h^-ac^O there are no such values of x. The reader wiU remember that in books on elementary algebra the equation is said to have two 'complex' roots. The meaning to be attached to this statement will be explained in Ch. III. When V'=ac the equation has only one root. For the sake of uniformity it is generally said in this case to have ' two equal ' roots, but this is a mere convention. 18, 14] REAL VAEIABLES 21 Draw a circle of unit radius, a diameter PQ, and the tangents at the ends of the diameters. Fig. 5. Take PP'= - 2a/b and QQ'= -cj2b, having regard to sign*. Join P'Q', cutting the circle in M and N. Draw PM and PN, cutting QQ' in X and T. Then QX and QT are the roots of the eqiiation with their proper signsf. The proof is simple and we leave it as an exercise to the reader. Another, perhaps even simpler, construction is the following. Take a line AB of unit length. Draw BG= -ibja perpendicular to AB, and CD=c/a perpendicular to BG and in the same direction as BA. On AD as diameter describe a circle cutting BG in X and T. Then BX and BY are the roots. 3. If ac is positive PP' and QQ' will be drawn in the same direction. Verify that P'Q' will not meet the circle ii b''<ac, while if b'^=aa it will be a tangent. Verify also that if b^=ac the circle in the second construction will touch BG. 4. Prove that ^{pq) = Vp X ^q, ^f(p-'q) =pjq. 14. Some theorems concerning quadratic surds. Two pure quadratic surds are said to be similar if they can be ex- pressed as rational multiples of the same surd, and otherwise to be dissimilar. Thus V8 = 2 ^J2, V-¥-=|V2, and so \/8, V-^ are similar surds. On the other hand, if M and N ar e integers w hich have no common factor, and neither of which is a perfect square, ijM and i^N are^issimilar surds. For suppose, if possible, q \/ u s V M where all the letters denote integers. * The figure is drawn to suit the case in which h and e have the same and a the opposite sign. The reader should draw figures for other cases. t I have taken this construction from Klein's Legons sur certaines questions dc giomUrie SUmentaire (French translation by J. Griess, Paris, 1896). 22 REAL VARIABLES [l Then ^JMN is evidently rational, and therefore (Ex. ii. 3) integral. Thus MN = P^ where P is an integer. Let a,h,c,... be the prime factors of P, so that where a, /3, 7, . . . are positive integers. Then MN is divisible by a^, and therefore either (1) M is divisible by a^, or (2) N is divisible by a?°-, or (3) M and N are both divisible by a. The last case may be ruled out, since M and N have no common factor. This argument may be applied to each of the factors a^", 6^^, c-"*, ..., so that M must be divisible by some of these factors and N by the remainder. Thus l/ = Pl^ N=Pi, where P^ denotes the product of some of the factors a^% b^, d''', . . . and P^ the product of the rest. Hence M and N are both perfect squares, which is contrary to our hypothesis. Theorem. If A, B, C, D are rational and A + >/B = C + ^D, then either (i) A = G, B = D or (ii) B and B are both squares of rational numbers. For B — I) is rational, and so is ^/B->JD=G-A. If B is not equal to D (in which case it is obvious that A is also equal to G), it follows that ^B + ^D=^{B- I))I(^B - VD) is also rational. Hence V-B and ^/B are rational. Corollary. If A + >^B = G + ^/B, then A-^B = G-^/B (unless V-B and >JB are both rational). Examples Vin. l. Prove ah initio that ^2 and V3 are not similar surds. 2. Prove that ^a and V(l/o), where a is rational, are similar surds (unless both are rational). 3. If a and h are rational, then s/a+s/b cannot be rational unless ^a and ^b are rational. The same is true of ^a — Jb, unless a = 6. 14, 15] EEAL VARIABLES 23 then either (a) A = G a,nd B=D, or (b) A=D and £=0,or (c) V^, V-B, ^/C, V-D are all rational or all similar surds. [Square the given equation and apply the theorem above.] 5. Neither (a + ^^6)3 nor (a - Jb)^ can be rational unless ^6 is rational. 6. Prove that if a;=p+^q, where p and g are rational, then x^, where m is any integer, can be expressed in the form P+ QJq, where P and Q are rational. For example, {p + 'Jq?==p^+q+%p^q, (^p + ^qf=p^ + Zpq+{Zp2j^q) ^q. Deduce that any polynomial in ss with rational coefficients {i.e. any expression of the form ao«"+ai «"-'+. ..+«„, where ag, ... a„ are rational numbers) can be expressed in the form P+Q^q. 7. If a + ^b, where b is not a perfect square, is the root of an algebraical equation with rational coefficients, then a~yjb is another root of the same equation. 8. Express lj(p+^fq) in the form prescribed in Ex. 6. [Multiply numerator and denominator by jo - «/?■] 9. Deduce from Exs. 6 and 8 that any expression of the form O {x)IH (x), where 0{x) and H{io) are polynomials in x with rational coefficients, can be expressed in the form P+Q^Jq, where P and Q are rational. 10. If p, q, and p'^ — q are positive, we can express -Jip+s/q) in the form -Jx-\-^y, where *-=Hp+V(p'-?)}. y=i{p~'Jip''-<i)}- 11. Determine the conditions that it may be possible to express slip +'Ji), where ^ and q are rational, in the form sJx-^-s/y, where x and y are rational. 12. If a^- 6 is positive, the necessary and sufficient conditions that V(«+V6)+v/(«-v/6) should be rational are that a?-b and ^{a+V(a^-&)} should both be squares of rational numbers. 15. The continuum. The aggregate of all real numbers, rational and irrational, is called the arithmetical continuum. It is convenient to suppose that the straight line A of § 2 is composed of points corresponding to all the numbers of the arithmetical continuum, and of no others*. The points of the * This supposition is merely a hypothesis adopted (i) because it suffices for the purposes of our geometry and (ii) because it provides us with convenient geometrical illustrations of analytical processes. As we use geometrical language only for purposes of illustration, it is not part of our business to study the foundations of geometry. 24 REAL VARIABLES [l line, the aggregate of which may be said to constitute the linear continuum, then supply us with a convenient image of the arithmetical continuum. We have considered in some detail the chief properties of a few classes of real numbers, such, for example, as rational numbers or quadratic surds. We add a few further examples to show how very special these particular classes of numbers are, and how, to put it roughly, they comprise only a minute fraction of the infinite variety of numbers which constitute the continuum, (i) Let us consider a more complicated surd expression such as 2=4'(4+v/15) + 4'(4-v/15). Our argument for supposing that the expression for z has a meaning might be as follows. We first show, as in §12, that there is a number _y = V15 such that y2=15, and we can then, as in § 10, define the numbers 4+V15, 4-^15. Now consider the equation in Zj, Zi3=4 + Vl5. The right-hand side- of this equation is not rational : but exactly the same reasoning which leads us to suppose that there is a real number x such that 3? ='2, (or any other rational number) also leads us to the conclusion that there is a number z^ such that 0i'=4+;^15. We thus define 01 = 4/(4+^15), and similarly we can define 02=\/(4—;^15); and then, as in §10, we define 0=01+02. Now it is easy to verify that 03=30 + 8. And we might have given a direct proof of the existence of a unique number such that 2^=30+8. It is easy to see that there cannot be two such numbers. For if 0/ = 30i + 8 and 03' =3% + 8, we find on subtracting and dividing by 01-02 that 01^+0102+03^=3. But if 0i and 02 are positive z^>^, 02' > 8 and therefore 0i>2, 02 > 2, zi^+ZiZ2 + zi^>12, and so the equation just found is impossible. And it is easy to see that neither 01 nor z^ can be negative. For if 01 is negative and equal to — f, f is positive and f3-3f+8 = 0, or 3-f2=8/f. Hence 3-f2>0, and so f<2. But then 8/f >4, and so 8/f cannot be equal to 3- f^, which is less than 3. Hence there is at most one such that 05=30 + 8. And it cannot be rational. For any rational root of this equation must be integral and a factor of 8 (Ex. n. 3), and it is easy to verify that no one of 1, 2, 4, 8 is a root. Thus 03=30+8 has at most one root and that root, if it exists, is positive and not rational. We can now divide the positive rational numbers x into two classes L, R according as .^ < 3:;; + 8 or x^ > 3.r + 8. It is easy to see that if ^ > 3.K + 8 and ^ is any number greater than x, then also y^>Zy + 8. For suppose if possible y^^Zy+%. Then since afl>Zx-\-fi we obtain on sub- tracting y^ - x^ <Z {y - x), or y'' + xy + x'^ <Z, which is impossible; for y is 15] REAL VARIABLES 25 positive and a; > 2 (since sfi>8). Similarly we can show that it i«><2x + 8 and ^<a; then also j/^ < 3y + 8. Finally, it is evident that the classes L and R both exist ; and they form a section of the positive rational numbers or positive real number z which satisfied the equation z^=Sz + 8. The reader who knows how to solve cubic equations by Cardan's method will be able to obtain the explicit expression of e directly from the equation. (ii) The direct argument applied above to the equation a^ = 3a! + 8 could be applied (though the application vfould be a little more difficult) to the equation a;* = a? + 16. and would lead us to the conclusion that a unique positive real number exists which satisfies this equation. In this case, how- ever, it is not possible to obtain a simple explicit expression for X composed of any combination of surds. It can in fact be proved (though the proof is difficult) that it is generally impossible to find such an expression for the root of an equation of higher degree than 4. Thus, besides irrational numbers which can be expressed as pure or mixed quadratic or other surds, or combinations of such surds, there are others which are roots of algebraical equations but cannot be so expressed. It is only in very special cases that such expressions can be found. (iii) But even when we have added to our list of irrational numbers roots of equations (such as a^ = x+ 16) which cannot be explicitly expressed as surds, we have not exhausted the different kinds of irrational numbers contained in the continuum. Let us draw a circle whose diameter is equal to A^A^, i.e. to unity. It is natural to suppose * that the circumference of such a circle has a length capable of numerical measurement. This length is usually denoted by tt. And it has been shownf (though the proof is un- fortunately long and difficult) that this number tt is not the root of any algebraical equation with integral coefficients, such, for example, as 7r^ = n, tt' = m, tt" = tt -I- n, * A proof will be found in Ch. VII. t See Hobson'a Trigonometry (3rd edition), pp. 305 et seq., or the same writer's Squaring the Circle (Cambridge, 1913). 26 REAL VARIABLES [l ■where n is an integer. In this way it is possible to define a number which is not rational nor yet belongs to any of the classes of irrational numbers which we have so far considered. And this number tt is no isolated or exceptional case. Any number of other examples can be constructed. In fact it is only special classes of irrational numbers which are roots of equations of this kind, just as it is only a still smaller class which can be expressed by means of surds. 16. The continuous real variable. The 'real numbers' may be regarded from two points of view. We may think of them as an aggregate, the 'arithmetical continuum' defined in the preceding section, or individually. And when we think of them individually, we may think either of a particular specified number (such as 1, -^, \/2, or tt) or we may think of any number, an unspecified number, the number x. This last is our point of view when we make such assertions as 'x is a number', 'x is the measure of a length', 'x may be rational or irrational', The x which occurs in propositions such as these is called the continuous real variable : and the individual numbers are called the values of the variable. A 'variable', however, need not necessarily be continuous. Instead of considering the aggregate of all real numbers, we might consider some partial aggregate contained in the former aggregate, such as the aggregate of rational numbers, or the aggregate of positive integers. Let us take the last case. Then in statements about any positive integer, or an unspecified positive integer, such as.'n is either odd or even', n is called the variable, a positive integral variable, and the individual positive integers are its values. Naturally ' x' and 'n' are only examples of variables, the variable whose 'field of variation' is formed by all the real numbers, and that whose field is formed by the positive integers. These are the most important examples, but we have often to consider other cases. In the theory of decimals, for instance, we may denote by x any figure in the expression of any number as a decimal. Then « is a variable, but a variable which has only ten different values, viz. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The reader should ■'^5-17] REAL VARIABLES 27 think of Other examples of variables with different fields of varia- tion. He will find interesting examples in ordinary life: policeman X, the driver of cab x, the year x, the icth day of the week. The values of these variables are naturally not numbers. 17. Sections of the real numbers. In §§ 4—7 we con- sidered 'sections' of the rational numbers, i.e. modes of division of the rational numbers (or of the positive rational numbers only) into two classes L and B possessing the following characteristic properties: (i) that every number of the type considered belongs to one and only one of the two classes ; (ii) that both classes exist ; (iii) that any member of L is less than any member of R. It is plainly possible to apply the same idea to the aggregate of all real numbers, and the process is, as the reader will find in later chapters, of very great importance. Let us then suppose * that P and Q are two properties which are mutually exclusive, and one of which is possessed by every real number. Further let us suppose that any number which possesses P is less than any which possesses Q. We call the^ numbers which possess P the loiuer or left-hand class L, and those which possess Q the upper or right-hand class R, Thus P might be ^ g ^/2 and Qhe x> J% It is important to observe that a pair of properties which suffice to define a section of the rational numbers may not suffice to define one of the real numbers. This is so, for example, with the pair 'x < JV and ^x > J2' or (if we confine ourselves . to positive numbers) with ' it;^ < 2 ' and ' .r^ > 2 '. Every rational number possesses one or other of the properties, but not every real number, since in either case ^2 escapes classification. There are now two possibilities f. Either L has a greatest member I, or R has a least member r. Roth of these events * The aiscussion which follows is in many ways similar to that of § 6. We have not attempted to avoid a certain amount of repetition. The idea of a ' section,' first brought into prominence in Dedekind's famous pamphlet Stetigkeit und irrationale Zahlen, is one which can, and indeed must, be grasped by every reader of this book, even if he be one of those who prefer to omit the discussion of the notion of an irrational number contained in §§ 6 — 12. + There were three in § 6. 28 REAL VARIABLES [l cannot occur. For if L had a greatest member I, and R a least member r, the number \{l+r) would be greater than all members of L and less than all members of R, and so could not belong to either class. On the other hand one event must occur*. For let ii and R^ denote the classes formed from L and R by- taking only the rational members of L and R. Then the classes Zi and jRi form a section of the rational numbers. There are now two cases to distinguish. It may happen that Li has a greatest member a. In this case a must be also the greatest member of L. For if not, we could find a greater, say yS. There are rational numbers lying between a and /3, and these, being less than P, belong to L, and therefore to L^; and this is plainly a contradiction. Hence a is the greatest member of L. On the other hand it may happen that L^ has no greatest member. In this case the section of the rational numbers formed by Xi and R^ is a real number a. This number a must belong I to i or to R. If it belongs to L we can shew, precisely, as before, \ that it is the greatest member of L, and similarly, if it belongs to R, it is the least member of R. Thus in any case either L has a greatest member ot R a, least. Any section of the real numbers therefore 'corresponds' to a real number in the sense in which a section of the rational numbers sometimes, but not always, corresponds to a rational number. This conclusion is of very great importance; for it shows that the consideration of sections of all the real numbers does not lead to any further generalisation of our idea of number. Starting from the rational numbers, we found that the idea of a section of the rational numbers led us to a new conception of a number, that of a real number, more general than that of a rational number; and it might have been expected that the idea of a section of the real numbers would have led us to a conception more general still. The discussion which precedes shows that this is not the case, and that the aggregate of real numbers, or the continuum, has a kind of completeness which the aggregate of the rational numbers lacked, a completeness which is expressed in technical language by saying that the continuum is closed. * This wa3 not the case in § 6. 17, 18] REAL VARIABLES 29 The result which we have just proved may be stated as follows: Dedekind's Theorem. If the Hal numbers are divided into two classes L and R in such a way that (i) every number belongs to one or other of the two classes, (ii) each class contains at least one number, (iii) any member of L is less than any member of R, then there is a number a, which has the property that all the numbers less than it belong to L and all the numbers greater than it to R. The number a itself may belong to either class. In applications we have often to consider sections not of all numbers but of all those contained in an interval {8, y), that is to say of all numbers x such that ^ < .r g y. A ' section ' of such numbers is of course a division of them into two classes possessing the properties (i), (ii), and (iii). Such a section may be converted into a section of all numbers by adding to Z all numbers less than ^ and to R all numbers greater than y. It is clear that the conclusion stated in Dedekind's Theorem still holds if we substitute ' the real numbers of the interval (ft y) ' for ' the real numbers ', and that the number a in this case satisfies the inequalities /3 gagy. 18. Points of accumulation. A system of real numbers, or of the points on a straight line corresponding to them, defined in any way whatever, is called an aggregate or set of numbers or points. The set might consist, for example, of all the positive integers, or of all the rational points. It is most convenient here to use the language of geometry*. Suppose then that we are given a set of points, which we will denote by S. Take any poipt ^, which may or may not belong to S. Then there are two possibilities. Either (i) it ij possible to choose a positive number S so that the interval (f — S, f + S) does not con- tain any point of S, other than | itself f,'0r (ii) this is not possible. Suppose, for example, that S consists of the points corresponding to all the positive integers. If $ is itself a positive integer, we can take S to be any number less than 1, and (i) will be true; or, if | is halfway between two positive integers, we can take S to be any number less than |. On the other hand, if S consists of all the rational points, then, whatever the value of ^, (ii) is true ; for any interval whatever contains an infinity of rational points. * The reader will hardly require to be reminded that this course is adopted solely for reasons of linguistic convenience. + This clause is of course unnecessary if f does not itself belong to S, 30 EEAL VAKIABLES [l Let US suppose that (ii) is true. Then any interval (^ - S, f + S), however small its length, contains at least one point ^i which belongs to S and does not coincide with f ; and this whether ^ itself be a member of 8 or not. In this case we shall say that | is a point of accumulation of 8. It is easy to see that the interval i^-S, ^+S) must contain, not merely one, but infinitely many points of 8. For, when we have determined ^i, we can take an interval (^ — S^, ^+ Sj) surrounding f but not reaching as far as ^i. But this interval also must contain a point, say ^j, which is a member of 8 and does not coincide with ^. Obviously we may repeat this argument, with fa in the place of fi; and so on indefinitely. In this way we can determine as many points ?1> fe2j 63) ••■ as we please, all belonging to 8, and all lying inside the interval A point of accumulation of 8 may or may not be itself a point of 8. The examples which follow illustrate the various possibilities. Examples IX. 1. If S consists of the points corresponding to the positive integers, or all the integers, there are no points of accumulation. 2. If S consists of all the rational points, every point of the line is a point of accumulation. 3. If S consists of the points 1, J, ^, ..., there is one point of accumula- tion, viz. the origin. 4. If S consists of all the positive rational points, the points of accumula- tion are the origin and all positive points of the line. 19. Weierstrass's Theorem. The general theory of sets of points is of the utmost interest and importance in the higher branches of analysis ; but it is for the most part too difficult to be included in a book such as this. There is however one funda- mental theorem which is easily deduced from Dedekind's Theorem and which we shall require later. Theorem. If a set 8 contains infinitely many points, and is entirely situated in an interval {a, /S), th&n at least one point of the interval is a point of accumulation of 8. We divide the points of the line A into two classes in the following manner. The point P belongs to L if there are an 18, 19] REAL VARIABLES 31 infinity of points of 8 to the right of P, and to i? in the contrary case. Then it is evident that conditions (i) and (iii) of Dedekind's Theorem are satisfied; and since a belongs to L and ^ to R, condition (ii) is satisfied also. Hence there is a point f such that, however small be S, f — S belongs to Z and f+S to R, so that the interval (f-S, f +S) contains^an infinity_of points of' /S. Hence ^ is a point of accumu- lation of *S. This point may of course coincide with a or /3, as for instance when a=0, /3 = 1, and S consists of the points 1, ^, J, .... In this case is the sole point of accumulation. MISCELLANEOUS EXAMPLES ON CHAPTER L 1. What are the conditions that ax+bi/ + cz=0, (1) for all values of X, y, z; (2) for all values of x, y, z subject to aa;+/3^ + y0=O; (3) for all values of .37, y, z subject to both ax-^^y-iryz — Q and Ax-irBy-^Cz—QI 2. Any positive rational number can be expressed in one and only one way in the form ■ ■^1.2.3...,!;' where Ui, a^, ...,ai are integers, and 0<ai, 0<a2<2, < 03O, ... 0<a;t</5r. 3. Any positive rational number can be expressed in one and one way only as a simple , continued fraction ^ 1 1 1 where cti, 02, ... are positive integers, of which the first only may be zero. [Accounts of the theory of such continued fractions will be found in text- books of algebra. For further information as to modes of representation of rational and irrational numbers, see Hobson, Theory of Functions of a Real Variable, pp. 45 — 49.] 4. Find the rational roots (if any) of Qx^ - C^'"* + 15;i; - 10 =0. 5. A line AB is divided at C in aurea sectione (Euc. 11. 11) — i.e. so that .42? . AC=BC^. Show that the ratio A GjAB is irrational. [A direct geometrical proof will be found in Bromwich's Infinite Series, § 143, p. 363.] 6. A is irrational. In what circumstances can , , , , where a, b, c, d cA + ol are rational, be rational? 32 REAL VARIABLES [l 7. Some elementary inequalities. In what follows aj, aj, ... de- note positive numbers (including zero) and p, q, ... positive integers. Since ajP — a^ and ai« - 02' have the same sign, we have {af - a^'') (ai'' - 02') =0, or ajP+a + ajP + 'gaiPaa'+^i^" (^)' an inequality which may also be written in the form a^P+i+c^P + i ^/ g^P + a^p \ / a^i + a^i \ By repeated application of this formula we obtain 2 '=\—r-)\r^-)\r^~) ^^^' and in particular ^f^^(^y (4). When p = q = \ in (1), or ^ = 2 in (4), the inequalities are merely different forms of the inequality a-^-'ra^^^'iaia^, which expresses the fact that the arithmetic mean of two positive numbers is not less than their geometric mean. 8. Generalisations for n numbers. If we write down the \n{n-V) inequalities of the type (1) which can be fonned with n numbers a^, a^,..., a„, and add the results, we obtain the inequality raSaP + '^SaPSa' (5), or (2aP + «)/»> {(2aP)/m} {(2a«)/re} (6). Hence we can deduce an obvious extension of (3) which the reader may formulate for himself, and in particular the inequality (Sa^VreSpaVw}" (7). 9. The general form of the theorem concerning the arithmetic and geometric means. An inequality of a slightly different character is that which asserts that the arithmetic mean of a^, aj, ..., a„ is not less than their geometric mean. Suppose that a,, and Ug are the greatest and least of the a's (if there are several greatest or least a's we may choose any of them indifferently), and let O be their geometric mean. We may suppose (? > 0, as the truth of the proposition is obvious when 6=0. Ifnowwe replace a, and a, by we do not alter the value of the geometric mean ; and, since ar' + a,'-a,.-ai={ar-6)(a,-6)IG<0, we certainly do not increase the arithmetic mean. It is clear that we may repeat this argument until we have replaced each of a,, a^, ..., a„hj G; at most n repetitions will be necessary. As the final value of the arithmetic mean is G, the initial value cannot have been less. REAL VARIABLES 33 10. Scliwarz's inequality. Suppose that a^, a^, ..., a„ and b^, 62, ..., K are any two sets of numbers positive or negative. It is easy to verify the identity {Sa^br)^ = 201,2 2a,2 - 2 {a,.b, - afi-rf, where r and s assume the values 1, 2, ..., «. It follows that {2arbrfS:'S.ar^-S,br\ an inequality usually known as Schwarz's (though due originally to Cauchy). 11. If %, aa, ..., «„ are all positive, and Sn=^i + <^i + --- + <X'nt then (H-ai)(l + a2)...(l+a„)Sl + «„+|^ + ...+^2!. {Math. Trip. 1909.) 12. If csi, 02, ..., o!„ and 6j, 621 •••! ^n s-re two sets of positive numbers, arranged in descending order of magnitude, then (ai+a2+ ■■. + «») (61+ &2+-"+&n)S»(ai 61+3262+ •••+a„6»). 13. If a, b, 0, ... h and A, B, C, ... K are two sets of numbers, and all of the first set are positive, then aA + bB + ...+kK a + b+...+k lies between the algebraically least and greatest of A, B, ..., K. 14. If Jp, sjq are dissimilar surds, and a + b ^p + c ^q+d .J(pq)=Q, where a, b, e, d are rational, then a=0, 6 = 0, c=0, d=Q. [Express Jp in the form M+ N s/q, where Ma,nd N are rational, and apply the theorem of § 14.] 1 5. Show that if a ;v^2 + 6 ^3 + c i;/5 = 0, where a, 6, c are rational numbers, then a=0, 6=0, c=0. 16. Any polynomial in s/p and Jq, with rational coefficients (i.e. any sum of a finite number of terms of the form A ijjp)^ (v/?)", where m and a are integers, and A rational), can be expressed in the form a+b ,Jp-^CsJq + ds!pq, where a, 6, c, d are rational. 17 Express , "^ . , where a, 6, etc. are rational, in the form ^ d+Bs/p+f^q' A+BJp + O^q + D^pq, where A, B, G, D are rational. [Evidently a + bjp + c^q (a + bs/p + ey/q)(d+ejp-fs/q) ^ a+^ -Jp + y Jq + B Jpq d+e^p+f^q {d+e^pf-Pq f + Up where a, ^, etc. are rational numbers which can easily be found. The required 3 34 EEAL VARIABLES [l reduction may now be easily completed by multiplication of numerator and denominator by e—^/Jp. For example, prove that 1+^2 + ^3 2 ' 4^ 4 18. If a, b, X, y are rational numbers such that (oy - 6.2;)2 + 4 (a - .r) (6 - y ) = 0, then either (i) x=a, y=h or (ii) \ — db and \ — scy are squares of rational numbers. {Math. Trip. 1903.) 19. If all the values of x and y given by a.r^ + 2/wy + 6y^ = 1, a V + 2h'xy + h'y'^ = 1 (where a, h, b, a', k', b' are rational) are rational, then (Ji-h:f-{a-a'){b-b'), {ab' - a'bf + 4 {ah' - a'h) (bh' - b'h) are both squares of rational numbers. {Math. Trip. 1899.) 20. Show that v/2 and ^^3 are cubic functions of iJ'i. + iJS, with rational coefficients, and that ^^2 — \'6 + 3 is the ratio of two linear functions of V2 + V3. {Math. Trip. 1905.) 21. The expression v/{a + 2m v/(a - TO^)} + %/{« - 2 w V(a - m^)} is equal to 2m if 2m2 > a > in', and to 2 ^(a - m^) if a > '2,m\ 22. Show that any polynomial in 4/2, with rational coefficients, can be expressed in the form a + 64/2+c4'4, where a, b, c are rational. More generally, if ^ is any rational number, any polynomial in "^p with rational coefficients can be expressed in the form ao + aia+aja^ ... +a!,„_ia'»-i, where a„, a^, ... are rational and a=^p. For any such polynomial is of the form 60+610 + 620^+ ••.+6ta*, where the 6's are rational. lfk&m-\, this is already of the form required. If k>m - 1, let a'' be any power of a higher than the (m - l)th. Then r=\m + «, where X is an integer and Og,s&m-\ ; and a''=a*^'"+*=p''a''. Hence we can get rid of aU powers of a higher than the (m- l)th. 23. Express (4/2-1)5 and (^2-l)/(4/2 + l) in the form a+64/2 + c4/4, where a, b, e are rational. [Multiply numerator and denominator of the second exi^rassion by 4/4-4/2 + 1-] 24. If a + 64/2+c4/4=0, wlicre a, b, c are rational, then a = 0, 6 = 0, c=0. REAL VARIABLES 35 [Let y=il% Then 2/3=2 and Hence 2cy2+26y+a^'=0 or ay2 + 2cy+26=0. Multiplying these two quadratic equations by a and c and subtracting, we obtain (a6-2c2)y+a^-26c=0, or 2/= -(a^-26c)/(a6-2o2), a rational number, which is impossible. The only alternative is that a6 — 2c^=0, a2-26c = 0. Hence a& = 2c^, a*=45V. If neither a nor 6 is zero, we can divide the second equation by the first, which gives a? = W: and this is impossible, since 4^2 cannot be equal to the rational number a/6. Hence ab=0, c=0, and it follows from the original equation that a, b, and c are all zero. As a corollary, if a+ 6 4^2+0 4/4= c?+ 6^/2 +/4'4, then a=d, b = e, c=f. It may be proved, more generally, that if . i/w ■ , (m-i)/™ „ p not being a perfect mth power, then ao=fl!i=...=am-i=0; but the proof is less simple.] 25. If 4 + 4/5= C+ 4/ A then either A = C,B=D,ov B axA B are both cubes of rational numbers. 26. If 4/4 4- 4/5 + 4/0=0, then either one (AA,B,C\s zero, and the other two equal and opposite, or ^ A, ^B, ^/O are rational multiples of the same surd 4/X 27. Find rational numbers a, ^ such that 4/(7 + 5v/2) = a+i3V2. 28. If (a- 6') 6 >0, then is rational. [Each of the numbers under a cube root is of the foi-m where a and ^ are rational.] 29. If a=^p, any polynomial in a is the root of an equation of degree n, with rational coeflScients. [We can express the polynomial {x say) in the form a?=?i+mia+...+?-ia'""^ where li, mj, ... are rational, as in Ex. 22. 3—2 36 BEAL VARIABLES [l Similarly a;^ = Zj + ma a + • ■ • + ''"2 a '"""'' » Hence Zi^+Z2«2 + ... + Z„a^=A, where A is the determinant .^1 »ll . •'•i ^2 '"h ■ ■ r2 ?n m„. ■''•„ and Zi, Li, ... the minors of li, l^, ....] 30. Apply this process to x:=p+^q, and deduce the theorem of § 14. 31. Show that y = a + bp''^ + cp''' satisfies the equation y3 _ 3a f+ 31/ (a2 -bcp)-a?-b^p- ^p^ + Zabcp = 0. 32. Algebraical numbers. We have seen that some irrational numbers (such as V2) are roots of equations of the type where a^, csj, ..., o„ are integers. Such irrational numbers are called alge- braical numbers : aU other irrational numbers, such as t (§ 15), are called transcendental numbers. Show that if a; is an algebraical number, then so are kx, where k is any rational number, and ^'"'", where m, and n are any integers. 33. \i X and y are algebraical numbers, then so are x-\-y,x—y,xy and xjy. [We have equations aoA-"" + ai^" ~ ' + ... + a„, = 0, where the a's and 6's are integers. Write x+y=z,y=z-x in the second, and eliminate x. We thus get an equation of similar form Co0P + Ci2P-l + ...+Cj, = O, satisfied by z. Similarly for the other cases.] 34 If ao^+ai*'»-i + ...+a„=0, where o^, aj, ..., a„ are any algebraical numbers, then a; is an algebraical number. [We have n+1 equations of the type «0,rar'^ + ai, rar™'-''+. ..+«».,., r = {r = Q, 1, ..., n), in which the coefficients ao, r> «i, r> ••• are integers Eliminate qj. a^, ..., a^ between these and the original equation for x.] 35. Apply this process to the equation x'^-'2,XsJ%+^3=0. [The result is afi-\Qxfi + b?,x^-4&x'^ + Q=0.'] KEAL VARIABLES 37 36. Find equations, with rational coefficients, satisfied by l+x/2+v/3, ^1^, Vy3+V2} + V{V3-V2}, 4/2 + ^3. 37. If x^=3; + l, then ^'''=a„a;+6„+c„/^, where 38. If xfi+afi-2x^ — xfi+.v^ + l=0 and y = .r''-;e2^,;_l^ then y satisfies a quadratic equation with rational coefficients. ' {Math, Trip. 1903.) [It will be found that f+i/ + l=0.] y^fx- ■■) -^ [y - ^^ (-!-> Vr'/ ^- CHAPTER II FUNCTIONS OF EBAL VARIABLES 20. The idea of a function. Suppose that x and y are two continuous real variables, which we may suppose to be repre- sented geometrically by distances AoP = x, B^Q = y measured from fixed points A^, B, along two straight lines A, M. And let us suppose that the positions of the points P and Q are not independent, but connected by a relation which we can imagine to be expressed as a relation between x and y: so that, when P and X axe known, Q and y are also known. We might, for example, suppose that y = x, or y= 2x, or ^x, ov x^+1. In all of these cases the value of x determines that of y. Or again, we might suppose that the relation between x and y is ■given, not by means of an explicit formula for y in terms of x, but by means of a geometrical construction which enables us to determine Q when P is known. In these circumstances y is said to be a function of x. This notion of functional dependence of one variable upon another is perhaps the most important in the whole range of higher mathe- matics. In order to enable the reader to be certain that he understands it clearly, we shall, in this chapter, illustrate it by means of a large number of examples. But before we proceed to do this, we must point out that the simple examples of functions mentioned above possess three characteristics which are by no means involved in the general idea of a function, viz.: (1) y is determined /or every value ofx; (2) to each value of x for which y is given corresponds one and only one value of y; (3) the relation between x and y is expressed by means of an analytical formula, from which the value of y corresponding to a given value of x can be calculated by direct substitution of the latter. 20] FUNCTIONS OF REAL VARIABLES 39 It is indeed the case that these parbicular characteristics are possessed by many of the most important functions. But the con- sideration of the following examples will make it clear that they are by no means essential to a function. All that is essential is that there should be some relation between a; and y such that to some values of x at any rate correspond values of y. Examples X, 1. Let y =x or 2a; or J^ or x'^ + l Nothing further need be said at present about cases such as these. 2. Let^=0 whatever be the value of x. Then ?/ is a function of x, for we can give x any value, and the corresponding value of y (viz. 0) is Itnown. In this case the functional relation makes the same value of i/ correspond to all values of x. The same would be true were y equal to 1 or - 1 or ^2 instead of 0. Such a function of x is called a constant. 3. Let if = x. Then if x is positive this equation defines two values of ^/ corresponding to each value of x, viz. ±yjx. If x=0, y=0. Hence to the particular value of ^ corresponds one and only one value of y. But if x is negative there is no value of y which satisfies the equation. That is to say, the function y is not defined for negative values of x. This function therefore possesses the characteristic (3), but neither (1) nor (2). 4. Consider a volume of gas maintained at a constant temperature and contained in a cylinder closed by a sliding piston*. Let A be the area of the cross section of the piston and W its weight. The gas, held in a state of compression by the piston, exerts a certain pressure Pa per unit of area on the piston, which balances the weight W, so that Let Va be the volume of the gas when the system is thus in equilibrium. If additional weight is placed upon the piston the latter is forced downwards. The volume («) of the gas diminishes ; the pressure {p) which it exerts upon unit area of the piston increases. Boyle's experimental law asserts that the product of p and v is very nearly constant, a correspondence which, if exact, would be represented by an equation of the type Tpv=a (i), where a is a number which can be determined approximately by experiment. Boyle's law, however, only gives a reasonable approximation to the facts provided the gas is not compressed too much. When v is decreased and p increased beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation (i). It is known that a * I borrow this instructive example from Prof. H. S. Carslaw's Introduction to the Calculus. 40 FUNCTIONS OF REAL VARIABLES [n much better approximation to the true relation can then be found by means of what is known as ' van der Waals' law', expressed by the equation (?'+p)(^-^)=v (")' where u, /3, y are numbers which can also be determined approximately by experiment. Of coui'se the two equations, even taken together, do not give anything like a complete account of the relation between p and v. This relation is no doubt in reality much more complicated, and its form changes, as v varies, from a form nearly equivalent to (i) to a form nearly equivalent to (ii). But, from a, mathematical point of view, there is nothing to prevent us from con- templating an ideal state of things in which, for all values of v not less than a certain value V, (i) would be exactly true, and (ii) exactly true for all values of v less than V. And then we might regard the two equations as together defining p as a fimction of v. It is an example of a function which for some values of v is defined by one formula and for other values of « is defined by another. This function possesses the characteristic (2) . to any value of v only one value of f corresponds : but it does not possess (1). "For p is not defined as a function of v for negative values of v; a 'negative volume' means nothing, and so negative values of v do not present themselves for considera- tion at all. 5. Suppose that a perfectly elastic ball is dropped (without rotation) from a height Jgr^ on to a fixed horizontal plane, and rebounds continually. The ordinary foi-mulae of elementary dynamics, with which the reader is probably familiar, show that A= Jg-i^ if Q)^t&T, h=\g ^r-tf if t<<<3t, and generally h=\g{'Lnr-tf if (2?i-l)Tg<g(2?» + l)r, Abeing the depth of the ball, at time t, below its original position. Obviously A is a function of t which is only defined for positive values of t. 6. Suppose that y is defined as being the largest prime factor of x. This is an instance of a definition which only applies to a particular class of values of se, viz. integral values. ' The largest prime factor of i^ or of ^/2 or of n- ' means nothing, and so our defining relation fails to define for such values of x as these. Thus this function does not possess the characteristic (1). It does possess (2), but not (3), as there is no simple formula which expresses y in terms of x. 7. Let y be defined as the denominator of x when x is expressed in its lowest terms. This is an example of a function which is defined if and only if X is rational. Thus.y=7 if x= - 11/7 : but y is not defined for x=^2, 'the denominator of y/2 ' being a meaningless form of words. 20, 21] FUNCTIONS OF BEAL VARIABLES 41 8. Let y be defined as the height in inches of policeman Cx, in the Metropolitan Police, at 5.30 ^.m. on 8 Aug. 1907. Then y is defined for a certain number of integral values of x, viz. 1, 2, ...,N, where N is' the total number of policemen in division C at that particular moment of time. 21. The graphical representation of functions. Sup- pose that the variable j/ is a function of the variable x. It will generally be open to us also to regard a; as a function of y, in virtue of the functional relation between x and y. But for the present we shall look at this relation from the first point of view. We shall then call x the independent variahle and y the dependent variable; and, when the particular form of the functional relation is not specified, we shall express it by writing y =/(«) (or F{x), <f)(x), '^{x), ..., as the case may be). The nature of particular functions may, in very many cases, be illustrated and made easily intelligible as follows. Draw two lines OX, OF at right angles to one another and produced indefinitely in both direc- tions. We can represent values of x and y by distances measured from along the lines OX, OY respectively, regard being paid, of course, to sign, and the positive directions of measure- ment being those indicated by arrows in Fig. 6. Let a be any value of x for which y is defined and has (let us suppose) the single value h. Take OA = a, OB = b, and complete the rectangle OAPB. Imagine- the point P marked on the diagram. This marking of the point P may be regarded as showing that the value of 2/ for a; = a is h. If to the value a oi x correspond several values of y (say h, h', h"), we have, instead of the single point P, a number of points P, P', P". We shall call P the point (a, h) ; a and h the coordinates of P referred to the axes OX, OT; a the abscissa, b the ordinate of P; OX and OY the axis of x and the axis of y, or together the Fig. 6. 42 FUNCTIONS OF REAL VARIABLES [ll axes of coordinates, and the origin of coordinates, or simply the origin. Let us now suppose that for all values a oi x for which y is defined, the value h (or values h, V, h", ...) of y, and the corre- sponding point P (or points P, P', P", ...), have been determined. We call the aggregate of all these points the graph of the function y. To take a very simple example, suppose that y is defined as a function of x by the equation Ax + By + G=Q (1), where A, B,C are any fixed numbers*. Then y is a function of x which possesses all the characteristics (1), (2), (3) of § 20. It is easy to show that the graph of y is a straight line. The reader is in all probability familiar with one or other of the various proofs of this proposition which are given in text-books of Analytical Geometry. We shall sometimes use another mode of expression. We shall say that when x and y vary in such a way that equation (1) is always true, the locus of the point (x, y) is a straight line, and we shall call (1) the equation of the Zooms, and say that the equation represents the locus. This use of the terms 'locus', 'equation of the locus' is quite general, and may be applied whenever the relation between x and y is capable of being represented by an analytical formula. The equation Ax + By+G = Q is the general equation of the first degree, for Ax + By +G is the most general polynomial in x and y which does not involve any terms of degree higher than the first ■ in X and y. Hence the general equation of the first degree repre- sents a straight line. It is equally easy to prove the converse proposition that the equation of any straight line is of the first degree. We may mention a few further examples of interesting geo- metrical loci defined by equations. An equation of the form {x-ay + {y-^y = p^, * If B = 0, y does not oeour in the equation. We must then regard y && a. fnnoliou of x defined for one value only of x, viz. x= - CjA, and then having all values. 23, 22] FUNCTIONS OF EEAL VARIABLES 43 or a:'' + y^+20x+2Fy+G = 0, where G^ + F''-G>0, represents a circle. The equation Ax^ + 2Hxy + By' + 2Gx + 2Fy + = (the general equation of the second degree) represents, assuming that the coefficients satisfy certain inequalities, a conic section, i.e. an ellipse, parabola, or hyperbola. For further discussion of these loci we must refer to books on Analytical Geometry. 22. Polar coordinates. In what precedes we have determined the position of P by the lengths of its coordinates OM=x, MP = y. If OP = r and MOP = 6, 6 being an angle between and 27r (measured in the positive direction), it is evident that a; = r cos ^, y — r sin 0, r = 'ij{x^ + y^), cosd:sind:l::x:y:r, and that the position of P is equally well determined by a knowledge of r and d. 6 ^ We call r and the polar coordinates ^ig- 7. of P. The former, it should be observed, is essentially positive*. If P moves on a locus there will be some relation between r and 0, say r =f(0) or = F (r). This we call the polar equation of the locus. The polar equation may be deduced from the (x, y) equation (or vice versa) by means of the formulae above. Thus the polar equation of a straight line is of the form rcos{6 — a)=p, where p and « are constants. The equation r=2a cos represents a circle passing through the origin ; and the general equation of a circle is of the form ^2 + c2 - 2rc cos {0-a) = A% where A, c, and a are constants. * Polar coordinates are sometimes defined so that r may be positive or negative. In this case two pairs of coordinates — e.g. (1, 0) and ( - 1, ir) — correspond to the same point. The distinction between the two systems may be illustrated by means of the equation ljr=l-ecoa0, where Z>0, e>l. According to our definitions r must be positive and therefore cos 9<l/e: the equation represents one branch only of a hyperbola, the other having the ' equation -ljr=l-e cos 9. With the system of coordinates which admits negative values of r, the equation represents the whole hyperbola. 44 FUNCTIONS OF REAL VARIABLES [II 23. Further examples of fiinctions and their graphical representation. The examples which follow will give the reader a better notion of the infinite variety of possible types of functions. A. Polynomials. A polynomial in a; is a function of the form a„a;'" + ai a?™-' +...+«», = X2 where a„(h., ..., a„ are constants. The simplest polynomials are the simple powers y = sc, x\ a^, ...,«'",... . The graph of the function x^ is of two distinct types, according as vi is even or odd. First let m = 2. Then three points on the graph are (0, 0), (1, 1), (- 1, 1). Any number of additional points on the graph may be found by assigning other special values to x: thus the values x = \, 2, 3,-i -2, 3 give 2/ = i 4, 9, J, 4, 9. If the reader will plot off a fair number of points on the graph, he will be led to conjecture that the form of the graph is something like that shown in Fig. 8. If he draws a curve through the special points which he has proved to lie on the graph and then tests its accuracy by giving x new values, and calculating the cor- responding values of y, he will find that they lie as near to the curve as it is reasonable to expect, when the inevitable inaccuracies of drawing are considered. The curve is of course a parabola. There is, however, one fundamental question which we cannot answer adequately at present. The reader has no doubt some notion as to what is meant by a continuous curve, a curve without breaks or jumps ; such a curve, in fact, as is roughly represented in Fig. 8. The question is whether the graph of the function 2/ = a;^ is in fact such a curve. This cannot be proved by merely (0,0) Fig. 8. 23] FUNCTIONS OF EEAL VAEIABLES 45 constructing any number of isolated points on the curve, although the more such points we construct the more probable it will appear. This question cannot be discussed properly until Ch. V. In that chapter we shall consider in detail what our common sense idea of continuity really means, and how we can prove that such graphs as the one now considered, and others which we shall consider later on in this chapter, are really continuous curves. For the present the reader may be content to draw his curves as common sense dictates. It is easy to see that the curve y = ^2 jg everywhere convex to the axis of x. Let Po, Pi (Fig- 8) be the points (^j, x^), {x^, x^). Then the coordinates of a point on the chord Pa Pi are x — Xx^+jiXi, y = \xi^^ + fiXi\ where X and fi are positive numbers whose sum is 1. And so that the chord lies entirely above the curve. The curve y^af'is similar to y = a;^ in general appearance, but flatter near 0, and steeper beyond the points A, A' (Fig. 9), and y = a;"*, where m is even and greater than 4, is still more so. As m gets larger and larger the flatness and steepness grow more and more pronounced, until the curve is practically indis- tinguishable from the thick line in the figure. y^x Fig. 9. I'ig- 10. The reader should next consider the curves given by y=x^, when TO is odd. The fundamental difference between the two cases is that whereas when m is even (- a;)™ = a)"*, so that the curve is symmetrical about OY, when m is odd (- a;)"* = - a;'", so 46 FUNCTIONS OF REAL VARIABLES [ll that y is negative when x is negative. Fig. 10 shows the curves y = x, y = a?, and the form to which y — x^ approximates for larger odd values of m It is now easy to see how (theoretically at any rate) the graph of any polynomial may be constructed. In the first place, from the graph of 1/ = «"* we can at once derive that of Cx™, where G is a constant, by multiplying the ordinate of every point of the curve by G. And if we know the graphs of f(x) and F(x), we can find that o{f(x) + F(x) by taking the ordinate of every point to be the sum of the ordinates of the corresponding points on the two original curves. The drawing of graphs of polynomials is however so much facilitated by the use of more advanced methods, which will be explained later on, that we shall not pursue the subject further here. Examples XI. 1. Trace the curves y = 7xfi, y — Zx^, y = x">. [The reader should draw the curves carefully, and all three should be drawn in one figure*. He will then realise how rapidly the higher powers of X increase, as x gets larger and larger, and wiU see that, in such a polynomial as (or even x^^-^ZO^ + IOOxi^), it is the first term which is of reaUy preponderant importance when x is fairly large. Thus even when x=i, a;i''> 1,000,000, while 30^ < 35,000 and 700:2;^ < 180,000; while if x=\0 the preponderance of the first term is still more marked.] 2. Compare the relative magnitudes of x^^, 1,000,000.^8, l,000,000,000,000.f when x=\, 10, 100, etc. [The reader should make up a number of examples of this type for himself. This idea of the relative rate of growth of difierent functions of x is one with which we shall often be concerned in the following chapters.] 3. Draw the graph oi ax^+'ibx + c [Here y-{{ac- b^)la} =a{x+ (bla)Y. If we take new axes parallel to the old and passing through the foint x= -bja, y=(ae-l^)/a, the new equation is y' = ax'^. The curve is a parabola.] 4. Trace the curvesy=^-3^ + l, y=x^(x-l), y=x{x-\f. * It will be found convenient to take the scale of meaeurement along the axis of y a good deal smaller than that along the axis of x, in order to prevent the figure becoming of an awkward size. 23, 24] FUNCTIONS OF REAL VARIABLES 47 24. B. Rational Functions. The class of functions which ranks next to that of polynomials in simplicity and importance is that of rational functions. A rational function is the quotient of one polynomial by another : thus if P (x), Q (x) are polynomials, we may denote the general rational function by In the particular case when Q (x) reduces to unity or any other constant (i.e. does not involve cc), R (x) reduces to a polynomial : thus the class of rational functions includes that of polynomials as a sub-class. The following points concerning the definition should be noticed. (1) We usually suppose that P{x) and Q {x) ht/ve no common faotor« + a or A'P + «.»*' " 1 + 6a''' ~ ^ + . . . + ^, all such factors being removed by division. (2) It should however be observed that this removal of common factors does as a rule change the function. Consider for example the function ^/^, which is a rational function. On removing the common factor x we obtain 1/1 = 1. But the original function is not always equal to 1: it is equal to 1 only so long as « 4=0. If ;r = it takes the form 0/0, which is meaningless. Thus the function xjx is equal to 1 if a; 4=0 and is imdefined when a; = 0. It therefore diflfers from the function 1, which is always equal to 1. (3) Such a function as \x-^\'^ x-\)l \a;^ x-y may be reduced, by the ordinary rules of algebra, to the form x'^{x-1) {x-\f{x+\y which is a rational function of the standard form. But here again it must be noticed that the reduction is not always legitimate. In order to calculate the value of a function for a given value of oi we must substitute the value for x in the function m the form in which it is given. In the case of this function the values ;;;= -1, 1, 0, 2 all lead to a meaningless expression, and so the function is not defined for these values. The same is true of the reduced form, so far as the values - 1 and 1 are concerned. But .i;=0 and x = 2 give the value 0. Thus once more the two functions are not the same. (4) But, as appears from the particular example considered under (3), there will generally be a certain number of values of x for which the function is not defined even when it has been reduced to a rational function of the standard form. These are the values of x (if any) for which the de- nominator vanishes. Thus {x^-7y{x^-Zx + 2) is not defined when x=l or 2. 4.8 FUNCTIONS OF KEAL VARIABLES [ll (5) Generally we agree, in dealing with expressions such as those con- sidered in (2) and (3), to disregard the exceptional values of x for which such processes of simplification as were used there are illegitimate, and to reduce our function to the standard form of rational function. The reader will easily verify that (on this understanding) the sum, product, or quotient of two rational functions may themselves be reduced to rational functions of the standard type. And generally a rational function of a rational function, is itself a rational function: i.e. if in z = P{y)IQi]/), where P and Q are polynomials, we substitute y=^Pi{x)IQ^{ai), we obtain on simplification an equation of the form z=P^{x)/Q2{a;). (6) It is in no way presupposed in the definition of a rational function that the constants which occur as coef&cients should be rational numbers. The word rational has reference solely to the way in which the variable a; appears in the function. Thus a^ + x+^/S :r4/2-7r is a rational function The use of the word rational ari.ses as follows. The rational function P {x)IQ {x) may be generated from a; by a finite number of operations upon X, including only multiplication of x by itself or a constant, addition of terms thus obtained, and division of one function, obtained by such multiplications and additions, by another. In so far as the variable x is concerned, this pro- cedure is very much like that by which all rational numbers can be obtained from unity, a procedure exempUfied in the equation 5^1+1+1+1+1 3 1+1+1 • Again, any function which can be deduced from x by the elementary operations mentioned above, using at each stage of the process functions which have already been obtained from x in the same way, can be reduced to the standard type of rational function. The most general kind of function which can be obtained in this way is sufficiently illustrated by the example ^■Ci + JjT^^sTfj /(^^+ J ' 9.2;+] which can obviously be redticed to the standard type of rational function. 25. The drawing of graphs of rational functions, even more than that of polynomials, is immensely facilitated by the use of methods depending upon the differential calculus. We shall therefore content ourselves at present with a very few examples. Examples XII. 1. Draw the graphs oiy=l/x, y=\jx\ y=\/x^, .... [The figures show the graphs of the first two curves. It should be observed that, since 1/0, 1/0^, ... are meaningless expressions, these functions are not defined for ^'=0.] 24-26] FUNCTIONS OF REAL VARIABLES 49 2. Trace y=x+{\lic), x-(lj.v), ^'2+(l/^2), x^-^l/x^) and ax+(b/x} taking various values, positive and negative, for a and b. 3. Trace ^^1 /.y+iy 1 x^ + l " x-V \x-\)' {x-Vf' ^2-1- 4. Trace y = \\{x-a){x-h), l/{x-a) (x-b) (x-c), where a<b<c. 5. Sketcli the general form assumed by the curves ^=l/«»> as m becomes larger and larger, considering separately the cases in which m is odd or even. (-l.-l) y = llx y = 1jx> Pig. 11. Fig. 12. 26. 0. Explicit Algebraical Functions. The next im- portant class of functions is that of explicit algebraical functions. These are functions which can be generated from a; by a finite number of operations such as those used in generating rational functions, together with a finite number of operations of root extraction. Thus \ x^2-ir ) are explicit algebraical functions, and so is a;™'" (i.e. J^x™), where m and n are any integers. It should be noticed that there is an ambiguity of notation involved in such an equation as y = slx. We have, up to the present, regarded {e.g.) -v/2 as denoting the positive square root of 2, and it would be natural to denote by ^/x, where x is any H. 4 50 FUNCTIONS OF EEAL VAEIABLES [ll positive number, the positive square root of x, in which case y = t^/x would be a one-valued function of x. It is however often more convenient to regard \/x as standing for the two-valued function whose two values are the positive and negative square roots of x. The reader will observe that, when this course is adopted, the function aJx differs fundamentally from rational functions in two respects. In the first place a rational function is always defined for all values of x with a certain number of isolated exceptions. But ijx is undefined for a whole range of values of x {i.e. all negative values). Secondly the function, when x has a value for which it is defined, has generally two values of opposite signs. The function ^/x, on the other hand, is one-valued and defined for all values of x. Examples XIII. 1. i.J{{x-a){h-x)}, where a<h, is defined only for a&x &b. If a<x<h it has two values : if x=a or h only one, viz. 0. 2. Consider similarly ^J{{x -a){x- h) {x - c)} (a < 6 < c), .Jixix-'-a')), y[{x-af{b-x)} (a<b), V( l +.,.)- V(l -^-) 3. Trace the oiu'ves y^=x, y^=x, y^=x?. 4. Draw the graphs of the functions y=v'(a^-a;2), y='b ^t{].-{x^la^)}. 27. D. Implicit Algebraical Functions. It is easy to verify that if •^ V(l + «) + 'v/(l-«)' then fl+^y=(l+^. \l -yl (1 - xy ' or if y = \/x + a/(« -I- 'Jx), then y*-(4.f + 4,y + l)x = 0. Each of these equations may be expressed in the form y'^+R,y^^ + ...+B^==0 (1), where It^, B^, ..., R^ are rational functions of x: and the reader will easily verify that, if y is any one of the functions considered m the last set of examples, y satisfies an equation of this form. 26, 27] FUNCTIONS OP REAL VARIABLES 51 It is naturally suggested that the same is true of any explicit algebraic function. And this is in fact true, and indeed not diflScult to prove, though we shall not delay to write out a formal proof here. An example should make clear to the reader the lines on which such a proof would proceed. Let _ a)+ 'Jx + V(a; + Va;} + ^/(l + x) ^ ~ a; - V« + Vi« + V"'} - -V^Cl + *■) ' Then we have the equations x+ u+v+w V= . X — U +V — W u^ = x, v^ = x+u, w' = 1 + a?, and we have only to eliminate u, v, w between these equations in order to obtain an equation of the form desired. We are therefore led to give the following definition : a function y=f{x) will he said to he an algehraical function of x if it is the root of an equation such as (1), i.e. the root of an equation of the wi** degree in y, luhose coefficients are rational functions of x. There is plainly no loss of generality in supposing the first coefficient to be unity. This class of functions includes all the explicit algebraical functions considered in § 26. But it also includes other functions which cannot be expressed as explicit algebraical functions. For it is known that in general such an equation as (1) cannot be solved explicitly for y in terms of x, when m is greater than 4, though such a solution is always possible if m = 1, 2, 3, or 4 and in special cases for higher values of m. The definition of an algebraical function should be compared with that of an algebraical number given in the last chapter (Misc. Exs. 32). Examples XIV. 1. If »i=l,y is a rational function. 2. If m=2, the equation is y^+Riy+Ri=0, so that 2/=H-^i±VW-4i?2)}- This function is defined for all values of a; for which li^^^ill^. It has two values if Iii'^>4R2 and one if Ri^=4R2. If OT = 3 or 4, we can use the methods explained in treatises on Algebra for the sohition of cubic and biquadratic equations. But as a rule the process is complicated and the results inconvenient in form, and we can generally study the properties of the function better by means of the original equation. 4—2 52 FUNCTIONS OF KEAL VARIABLES [ll 3. Consider the functions defined by the equations in each case obtaining 2^ as an explicit function of x, and stating for what values of x it is defined. 4. Find algebraical equations, with coefficients rational in x, satisfied by each of the functions 'Jx+sJ{llx), ijx+il{\lx), VCx' + Va'), ^l{x+'J{x+^Jx)}. 5. Consider the equation y*=xK [Herej/2=+x If ^ is positive, ^=«y a;: if negative, y=V(- .j;). Thus the function has two values for all values of x save ir=0.] 6. An algebraical function of an algebraical function of x is itself an algebraical function of x. [For we have 3^'» + /Ji(2)y'»-i + ...+i?„(2) = 0, where 2"+*?! {x) 2"-^ + ... + (S„ (a;) =0. Ehininating z we find an equation of the form yJ' + Ti{x)f--^ + ... + T^{x)=0. Here all the capital letters denote rational functions.] 7. An example should perhaps be given of an algebraical function which cannot be expressed in an explicit algebraical form. Such an examj)le is the function y defined by the equation y^ — y — x=0- But the proof that we cannot find an explicit algebraical expression for y in terms of a; is difficult, and cannot be attempted here. 28. Transcendental functions. All functions of x which <^' are not rational or even algebraical are called transcendental functions. This class of functions, being defined in so purely negative a manner, naturally includes an infinite variety of whole kinds of functions of varying degrees of simplicity and importance. Among these we can at present distinguish two kinds which are particularly interesting. E. The direct and inverse trigonometrical or circular functions. These are the sine and cosine functions of elementary trigonometry, and their inverses, and the functions derived from them. We may assume provisionally that the reader is familiar with their most important properties *. * The definitions of the circular functions given in elementary trigonometry pre- suppose that any sector of a circle has associated with it a definite number called its area. How this assumption is justified will appear in Ch. VII. 27, 28] FUNCTIONS OF REAL VARIABLES 53 Examples XV. 1. Draw the graphs of cos x, sin x, and a cos a; + 6 sin x. [Since acosa; + 6sina?=0co8(a;-a), where /3=V(a2 + 6^), and a is an angle whose cosine and sine are ajJifl^-^W) and bj^{a^+h'^), the graphs of these three functions are similar in character.] 2. Draw the graphs of cos^ x, sin^x, acos^x+h sin^ x. 3. Suppose the graphs of f{x) and F{x) drawn. Then the graph of f{x)co&'^x+F{x)sw?'x is a wavy curve which oscillates between the curves y=f{x), y=F{x). Draw the graph when /(;«;) = a-, F{x)=x\ 4. Show that the graph of Qospx+cosqx lies between those of '2iCoa\{p-q)x and -2cos^(^ + g') A', touching each in turn. Sketch the graph when (p-q)l{p + q) is small. (Math. Trip. 1908.) 5. Draw the graphs of .»+ sin a;, (l/.s;)+sin.j;, «sin.», {aiD.x)lx. 6. Draw the graph of sin (l/x). [If y =sin (!/«)> then y = when x—ljmir, where m is any integer. Similarly i/ = l when «=l/(2m+^)7r and t/=—l when x=ll(2m-^) tt. The curve is entirely comprised between the lines y= —1 and y=l (Fig. 13). It oscillates up and down, the rapidity of the oscillations becoming greater and greater as X approaches 0. For x=0 the function is undefined. When x is large y is small*. The negative half of the curve is similar in character to the positive half.] 7. Draw the graph of x sin (l/«). [This curve is comprised between the lines y= - x and y=x just as the last curve is comprised between the lines y= — 1 and y=l (Fig. 14).] Fig. 13. Fig- 14. * See Chs. IV and V for explanations as to the precise meaning of this phrase. 54 FUNCTIONS OF REAL VARIABILES [ll 8. Draw the graphs of x^ sin (1/^), (1/^) sin (l/o;), sin^ (1/^), {x sin (l/^)}^, acos2(l/a;) + 6sin2(l/x'), sin^ + sin(l/^), sin ^ sin (1/^). 9. Draw the graphs of cos^', sinai^, acosx^-VhAnx': 10. Draw the graphs of arc cos x and arc sin x. [If y = arccos«, ^=cosy. This enables us to draw the graph of x, con- sidered as a function of y, and the same curve shows y as a function of x. It is clear that y is only defined for — lgA'<l, and is infinitely many- valued for these values of x. As the reader no doubt remembers, there is, when - 1<^<1, a value of y between and tt, say a, and the other values of y are given by the formula 2mjr + a, where n is any integer, positive or negative.] 11. Draw the graphs of tana;, cot a;, seoar, coseca-, \,&v?x, cot^a;, sec^a?, cosec^ir. 12. Draw the graphs of arc tan «, arc cot a;, arc sec .r, arocoseca;. Give formulae (as in Ex. 10) expressing all the values of each of these functions in terms of any particular value. 13. Draw the graphs of tan (1 /a;), cot(l/a;), seo(l/«), cosec(l/a;) 14. Show that cos x and sin x are not rational functions of x. [A function is said to be periodic, with period a,ii f{x)=f (x+ a) tor sM values of x for which f{x) is defined. Thus cos x and sin x have the period 2jr. It is easy to see that no periodic function can be a rational function, unless it is a constant. For suppose that f{x)=P{x)lQix), where F and Q are polynomials, and that/(a;) =f(x+a), each of these equations^ holding for all values of x. Let /(O) = k. Then the equation P {x) - IQ (x) = is satisfied by an infinite number of values of x, viz. x=0, a, 2a, etc., and therefore for all values of x. Thus f{x) = k for all values of x, i.e. f{x) is a constant.] 15. Show, more generally, that no function with a period can be an algebraical function of x. [Let the equation which defines the algebraical function be g^ + R^r,^-^ + ...+R^=Q (1) where R^, ... are rational functions of x. This may be put in the form -Poy'"+Ay"-i+...+p„=o, where P,, Pi, — are polynomials in x. Arguing as above, we see that Po^-f-Pi^™-! -t- ... -)-P„=0 28, 29] FUNCTIONS. OF KEAL VAEIABLES 55 for all values of x. Hence y = k satisfies the equation (1) for all values of x, and one set of values of our algebraical function reduces to a constant. Now divide (1) hj y — k and repeat the argument. Our final conclusion is that our algebraical function has, for any value of x, the same set of values h, K, ... ; I.e. it is composed of a certain number of constants.] .16. The inverse sine and inverse cosine are not rational or algebraical functions. [This follows from the fact that, for any value of x between - 1 and + 1, arc sin x and arc cos x have infinitely many values.] 29. F. Other classes of transcendental functions. Next in importance to the trigonometrical functions come the expo- nential and logarithmic functions, which will be discussed in Chs. IX and X, But these functions are beyond our range at present. And most of the other classes of transcendental func- tions whose properties have been studied, such as the elliptic functions, Bessel's and Legendre's functions. Gamma-functions, and so forth, lie altogether beyond the scope of this book. There are however some elementary types of functions which, though of much less importance theoretically than the rational, algebraical, or trigonometrical functions, are particularly instruc- tive as illustrations of the possible varieties of the functional relation. Examples XVI. 1. Let y=[x], where \x'\ denotes the greatest integer not greater than x. The graph is shown in Fig. 15 a. The left-hand end points of the thick lines, but not the right-hand ones, belong to the graph. 2. y=x-[x'\. (Fig. 15 6.) 1 2 12 Fig. 15(1. Fig. 156. 56 FUNCTIONS OF REAL VARIABLES [ll 3. y=V{^-M}. (7ig-15c.) 4. y=M + V{^-M}- (Fig. 15 c?.) 5. y={x-[x-\f, M + (.r-M)2. 6. y=y=oi M, v^-[V4 ^'-[n [1-n Fig. 15 c. Fig. 15 d. 7. Let y be defined as the largest prime factor of x (cf. Exs. x. 6). Then y is defined only for integral values of x. If ^=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ... , then y=l, 2, 3, 2, 5, 3, 7, 2, 3, 5,11, 3,13,.... The graph consists of a number of isolated points. 8. Let y be the denominator of x (Exs. x. 7). In this case y is defined only for rational values of x. We can mark ofi' as many points on the graph as we please, but the result is not in any ordinary sense of the word a curve, and there are no points corresponding to any irrational values of x. Draw the straight lino joining the points {N- 1, N), {N, N), where iV is a positive integer. Show that the number of points of the locus which lie on this line is equal to the number of positive integers less than and prime to N. 9. Let !/ = when x is an integer, y=x when x is not an integer. The graph is derived from the straight line j(=a; by taking out the points ...(-1, -1), (0,0), (1,1), (2,2),... and adding the points (-1, 0), (0, 0), (1, 0), ... or^the axis otx. The reader may possibly regard this as an unreasonable function. Why, he may ask, if y is equal to x for all values oi x save integral values, should it not be equal to x for integral values too 1 The answer is simply, why should U1 The function y does in point of fact answer to the definition of a function : there is a relation between x and y such that when x Ls known y is known^. We are perfectly at liberty to take this relation to be what we please, however arbitrary and apparently futile. This function y is, of course, a quite different function from that one which is always equal to x, whatever value, integral or otherwise, x may have. 29] FUNCTIONS OF REAL VARIABLES 57 10. Let y = lwhen.» is rational, but y=0 when ^ is irrational. Thegraph consists of two series of points arranged upon the lines y=l and y=0. To the eye it is not distinguishable from two continuous straight lines, but in reality an infinite number of points are missing from each line. 11. Let y=x when x is irrational and ?/= V{(1 +P^)/(1 +2^)} when a; is a rational fraction 'plq^. Fig. 16. The irrational values of x contribute to the graph a curve in reality dis- continuous, but apparently nottobedisti nguished from the straight line y = x. Now consider the rational values of x. First let x be positive. Then V{(l+i'^)/(^ + ?^)} cannot be equal to piq unless p = q, i.e. x=l. Thus all the points which correspond to rational values of x lie off the line, except the one point (1, 1). Again, if p<q, \/{(l+p^)/(l+2^)} >i'/2' 5 if i? > ?> »J{(l+p^)/(J+q^)] <plq. Thus the points lie above the liney=a; if 0<a;<l, below if a; > 1. lip and q are large, ^{{1 +p^)l{l + q^)} is nearly equal to pIq. Near any value of x we can find any number of rational fractions with large numerators and denominators. Hence the graph contains a large number of points which crowd round the Hne ■i/=x. Its general appearance (for positive values of x) is that of a line surrounded by a swarm of isolated points which gets denser and denser as the points approach the line. The part of the graph which corresponds to negative values of x consists of the rest of the discontinuous line together with the reflections of all these isolated points in the axis of y. Thus to the left of the axis of y the swarm of points is not round y=^ but round y=—x, which is not itself part of the graph. See Fig. 16. 58 FUNCTIONS OF REAL VARIABLES . [ll 30. Graphical solution of equations containing^ a single unknown number. Many equations can be expressed in the form /(^) = </'W (1), where /(a;) and (f) (x) are functions whose graphs are easy to draw. And if the curves intersect in a point P whose abscissa is |, then ^ is a root of the equation (1). Examples XVII. 1. The c[uadratic equation ax'^+2bx+c=0. This may be solved graphically in a variety of ways. For instance we may draw the graphs of i/=a.v+2b, y=-c/^, whose intersections, if any, give the roots. Or we may take y=.v^, y = — (25a; + c)ja. But the most elementary method is probably to draw the circle a(x''+f)+2bx + c=0, whose centre is { — h/a, 0) and radius {sj{h'^ — ac)}la. The abscissae of its intersections with the axis of x are the roots of the equation, 2. Solve by any of these methods 3. The ectuation x!^+ax+h=0. This may be solved by constructing the curves y — x'^, y——ax—h. Verify the following table for the number of roots of a;™ + ax + 6=0: f 5 positive, two or noiie, (a) m even < , ,. \_o negative, two • ,,. , y t a positive, one, (6) m. odd \ ^ . • ' > la negative, three or one. Construct numerical examples to illustrate all possible cases. 4. Show that the equation tana;=a;!; + 6 has always an infinite number of roots. 5. Determine the number of roots of sin^=;r, sina;=J^, sina; = |:!;, smx=^^^x. 6. Show that if a is small and positive (e.g. a = -01), the equation x — a=^Tram^x has three roots. Consider also the case in which a is small and negative. Explain how the number of roots varies as a varies. 30, 31] FUNCTIONS OF BEAL VARIABLES 69 31. Functions of two variables and their graphical representation. In § 20 we considered two variables connected by a relation. We may similarly consider three variables (x, y, and z) connected by a relation such that when the values of x and y are both given, the value or values of z are known. In this case we call z a function of the two variables x and y ; x and y the independent variables, z the dependent variable; and we express this dependence of z upon x and y by writing The remarks of § 20 may all be applied, mutatis mutandis, to this more complicated case. The method of representing such functions of two variables graphically is exactly the same in principle as in the case of functions of a single variable. We must take three axes, OX, OY, OZ in space of three dimensions, each axis being perpendicular to the other two. The point (a, h, c) is the point whose distances from the planes YOZ, ZOX, XOY, measured parallel to OX, OY, OZ, are a, b, and c. Regard must of course be paid to sign, lengths measured in the directions OX, OY, OZ being regarded as positive. The definitions of coordinates, axes, origin are the same as before. Now let z =f(x, y). As X and y vary, the point {x, y, z) will move in space. The aggregate of all the positions it assumes is called the locus of the point {x, y, z) or the graph of the function z =f(x, y). When the relation between x, y, and z which defines z can be expressed in an analytical formula, this formula is called the equation of the locus. It is easy to show, for example, that the equation Ax + By+Gz + I) = (the general equation of the first degree) represents a plane, and that the equation of any plane is of this form. The equation (X- ay + (y - ^y+ (z - yy = p'', or x'' + y'' + z^ + 2Fx+2Gy + 2Hz + G=0, where F" + G^ + H^ — G > 0, represents a sphere ; and so on. For proofs of these propositions we must again refer to text-books of Analytical Geometry. 60 FUNCTIONS OF EEAL VARIABLES [ll 32. Curves in a plane. We have hitherto used the notation y=f{.o=) (1) to express functional dependence of y upon x. It is evident that this notation is most appropriate in the case in which y is ex- pressed explicitly in terms of x by means of a formula, as when for example y = os^, sin x, a cos'' x + h sitf x. We have however very often to deal with functional relations which it is impossible or inconvenient to express ia this form. If, for example, y^-y-x = or c(? + y^-ay = 0, it is known to be impossible to express y explicitly as an algebraical function oi X. If a,2 + 2/2 + 2Gx + 2Fy + C = 0, y can indeed be so expressed, viz. by the formula y = -F+>^{F^ - «= - 2Gx - G)\ but the functional dependence of y upon x is better and more simply expressed by the original equation. It will be observed that in these two cases the functional relation is fully expressed hy equating a function of the two variables x and y to zero, i.e. by means of an equation ./(^,2/) = (2). We shall adopt this equation as the standard method of expressing the functional relation. It includes the equation (1) as a special case, since y —f{x) is a special form of a function of x and y. We can then speak of the locus of the point (x, y) subject tof(x, y) = 0, the graph of the function y defined by f{x, y) = 0, the curve or locus f{x, y) = 0, and the equation of this curve or locus. There is another method of representing curves which is often useful. Suppose that x and y are both functions of a third variable t, which is to be regarded as essentially auxiliary and devoid of any particular geometrical significance. We may write x=fit), y = F{t) (3). If a particular value is assigned to t, the corresponding values of X and of y are known. Each pair of such values defines a point 32, 33] FUNCTIONS OF REAL VARIABLES 61 {x, y). If we construct all the points which correspond in this way to different values of t, we obtain the graph of the locus defined hy the equations (3). Suppose for example « = a cos t, y = a sin t. Let t vary from to 277. Then it is easy to see that the point (x, y) describes the circle whose centre is the origin and whose radius is a. If t varies beyond these limits, (x, y) describes the circle over and over again. We can in this case at once obtain a direct relation between x and y by squaring and adding: we find that x^ + y^ = a^, t being now eliminated. Examples XVIII. 1. The points of intersection of the two curves whose equations are f{x, y)=0, (f> (x, y)=0, where / and (^ are polynomials, can be determined if these equations can be solved as a pair of simultaneous equations in X and y. The solution generally consists of a finite number of pairs of values of so and y. The two equations therefore generally represent a finite number of isolated points. 2. Trace the ciu'ves {x +y)2 =^\, xy=\, x'^—y^=\. 3. The curve fix, y)-\-'K<j>{x, y)=0 represents a curve passing through the points of intersection of /=0 and ^=0, 4. What loci are represented by (a) x=at+b, y=ot+d, (/3) x/a=2tl{l + t^), yla={l-t^)l{l + fi), when t varies through all real values ? 33. Loci in space. In space of three dimensions there are two fundamentally different kinds of loci, of which the simplest examples are the plane and the straight line. A particle which moves along a straight line has only one degree of freedom. Its direction of motion is fixed; its position can be completely fixed by one measurement of position, e.g. by its distance from a fixed point on the line. If we take the line as our fundamental line A of Chap. I, the position of any of its points is deteripined by a single coordinate x. A particle which moves in a plane, on the other hand, has two degrees of freedom ; its position can only be fixed by the determination of two coordinates,. A locus represented by a single equation z =fip, y) plainly belongs to the second of these two classes of loci, and is called a surface. It may or may not (in the obvious simple cases 62 FUNCTIONS OF REAL VARIABLES [ll it will) satisfy our common-sense notion of what a surface should be. The considerations of § 31 may evidently be generalised so as to give definitions of a function / {x, y, z) of three variables (or of functions of any number of variables). And as in § 32 we agreed to adopt /(«, y) = as the standard form of the equation of a plane curve, so now we shall agree to adopt f{x, y,z) = as the standard form of equation of a surface. The locus represented by two equations of the form z =/(«, y) or fix, y, z) = belongs to the first class of loci, and is called a curve. Thus a straight line may be represented by two equations of the type Ax + By+Gz + D=Q. A circle in space may be regarded as the intersection of a sphere and a plane; it may therefore be represented by two equations of the forms {x- of + {y - ^y + {z- r^y = p\ Ax + By+Cz + D=0. Examples XIX. 1. What is represented by three equations of the type /(^,y,2)=0? 2. Three linear equations in general represent a single point. What are the exceptional cases ? 3. What are the equations of a plane curve /(^, y) =0 in the plane XOY, when regarded as a curve in space? [/(», 2/)=0i z=0.] 4. Cylinders. What is the meaning of a single equation /(^, y)=0, considered as a locus in space of three dimensions ? [ AU points on the surface satisfy / (x, y) = 0, whatever be the value of z. The curve f{x, y)=0, 0=0 is the curve in which the locus cuts the plane XOY. The locus is the surface formed by drawing lines parallel to OZ through all points of this curve. Such a surface is called a cylinder.^ 5 Graphical representation of a surface on a plane. Contolir Maps. It might seem to be impossible to represent a surface adequately by a drawing on a plane ; and so indeed it is : but a very fair notion of the nature of the surface may often be obtained as follows. Let the equation of the surface be z=f{x, y). If we give z a particular value a, we have an equation f{x, y) = a, which we may regard as determining a plane curve on the paper. We trace this curve and mark it (a). Actually the curve (a) is the projection on the plane 33] FUNCTIONS OF REAL VARIABLES 63 XOY of the section of the surface by the plane 2 = a. We do this for all values of a (practically, of course, for a selection of values of a). We obtain some such figure as is shown in Fig. 17. It will at once suggest a contoured Ordnance Survey map ; and in fact this is the principle on which such maps are constructed. The contour line 1000 is the projection, on the plane of the sea level, of the section of the surface of the land by the plane parallel to the plane of the sea level and 1000 ft. above it*. 300O 6. Draw a series of contour lines to illustrate the form, of the surface 7. Bight circular cones. Take the origin of coordinates at the vertex of the cone and the axis of z along the axis of the cone ; and let a be the semi-vertical angle of the cone. The equation of the cone (which must be regarded as extending both ways from its vertex) is x^+y^ — z'^ta.-a.^ a=0. 8. Surfaces of revolution in general. The cone of Ex. 7 cuts ZOX in two lines whose equations may be combined in the equation x^=z^tstn^a. That is to say, the equation of the surface generated by the revolution of the curve y=0, x^=z^ta.n^ a roxmd the axis of z is derived from the second of these equations by changing x^ into x^+t/K Show generally that the equation of the surface generated by the revolution of the curve y=0, x=f{z), round thq axis of z, is V(^-H2/=)=/(^). 9. Cones in general. A surface formed by straight lines passing through a fixed point is called a cone: the point is called the vertex. A particular case is given by the right circular cone of Ex. 7. Show that the equation of a cone whose vertex is is of the form fizjx, zjy) =0, and that any equation of this form' represents a cone. [If {x, y, z) lies on the cone, so must (X^, Xy, \z), for any value of X.] * We assume that the effects of the earth's curvature may be neglected. 64 FUNCTIONS OF BEAL VARIABLES [II 10. Buled surfaces. composed of straight lines. The two equations Cylinders and cones are special cases of surfaces Such surfaces are called ruled surfaces. a; = az+h, y = cz + d.. ■(1) represent the intersection of two planes, i.e. a straight line. Now suppose that a, b, c, d instead of being fixed are functions of an auxiliary variable t. For any particular value of t the equations (1) give a line. As t varies, this line moves and generates a surface, whose equation may be found by eliminating t between the two equations (1). For instance, in Ex. 7 the equations of the line which generates the cone are x=zta,nacost, y=«tanasin<, where t is the angle between the plane XOZ and a plane through the line and the axis of s. Another simple example of a ruled surface may be constructed as follows. Take two sections of a right circular cylinder perpendicular to the axis and at a distance I apart (Fig. 18 a). We can imagine the surface of the cylinder to be made up of a number of thin parallel rigid rods of length I, such as PQ, the ends of the rods being fastened to two circular rods of radius a. Now let us take a third circular I'od of the same radius and place it round the surface of the cylinder at a distance k from one of the first two rods (see Fig. 18 as, where Pq=h). Unfasten the end Q of the rod PQ and turn PQ about P until Q can be fastened to the third circular rod in the position §'. The angle qOQ' — a in the figure is evidently given by Z2-A2=jg2=(2asin^a)2. Let all the other rods of which the cylinder was composed be treated in the same way. We obtain a ruled surface whose form is indicated in Fig. 18 6. It is entirely built up of straight lines ; but the siirface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings (Fig. 18 c). f^~7^ p \ o \--- 3 '-S:^' Idl" ~~^ Q Fig. 18 a. Fig. 186. Fig. 18 c. FUNCTIONS OF REAL VARIABLES 65 MISCELLANEOUS EXAMPLES ON CHAPTER II 1. Show that a y=f {x) = {ax + h)j{cx - a) then x=f{^). 2. li f(x)=f{-x) for all values of x,f{x) is called an even function. lif{x)= —f{ — x\ it is called an odd function. Show that any function of x, defined for all values of x, is the sum of an even and an odd function of x. [Use the identity/(^) = i {/(^)+/( -^)}+H/(*-) -/(-^)}-] 3. Draw the graphs of the functions 3sin« + 4cos.a;, sin ( -^ sin a; J . (i/a^A. TVjp. 1896.) 4. Draw the graphs of the functions SI w 2t' / sm ^\ smx{aoos^x-'rhsiv?x), (cscos^^r + ftsin^^), ( ). X \ X J 5. Draw the graphs of the functions x [1/*'], [xllx. 6. Draw the graphs of the functions (i) arc cos (2^2— 1) — 2 arc cos a;, (ii) arc tan ;; ^^ — arc tan a — arc tan x, where the symbols arc cos a, arc tan a denote, for any value of u, the least positive (or zero) angle, whose cosine or tangent is a. 7. Verify the following method of constructing the graph of /{<^ (x)} by means of the line if = x and the graphs of f{x) and (j> (x) ; take OA = x along OX, draw AB parallel to OF to meet y = <f){x) in B, BO parallel to OX to meet ^=^ in C, CD parallel to OF to meet y=f{x) in i), and BP parallel to OX to meet AB in P; then P is a point on the graph required. 8. Show that the roots of x^-\-px-^q = Q are the abscissae uf the points of intersection (other than the origin) of the parabola y=«2 and the circle ir2+/+(^-l)y+2'.»=0. 9. The roots of a;* + no? ■\-px'^ +qx+r=0 si,re the abscissae of the points of intersection of the parabola x''=y — \nx and the circle ^^+3/^+(^w2--^^7i+^+2)^+Cp-l-im^)y+»-=0. 10. Discuss the graphical solution of the equation by means of the curves y=x'^, y= -ax^-bx-o. Draw up a table of the various possible numbers of roots. 11. Solve the equation sec 5 + cosec 19 = 2^2; and show that the equation sec ^ + cosec d = c ha^ two roots between and 27r if c^<8 and four if c^>8. 5 6Q FUNCTIONS OF KEAL VARIABLES [ll 12. Show that the equation 2;l,' = (2« + 1 ) 7r (1 - cos ^), where ?i is a positive integer, has 2m + 3 roots and no more, indicating their localities roughly. (Math. Trip. 1896.) 13. Show that the equation §^sin^ = l has four roots between -tt and TT. 14. Discuss the number and values of the roots of the equations (1) cota;+.r-|7r = 0, (2) x^-Vwi^ x=\, (3) tan^ = 2x/(l+a;2), (4) sin«-^+Ja;'=0, (5) (1-cos.r) tana-a;+sina;=0. 15. The polynomial of the second degree which assumes, when x=a, b, c the values a, ft y is (x-b){x-c ) (x-c){x-a) (x-a)ix-b ) "(a-6)(a-c)'^''^ (6-c}(6-a) ^^ (c-a){c-b)- Give a similar formula for the polynomial of the (»-l)th degree which assumes, when ^ = ai, 02, ... ««, the values ai, aj, ... u„. 16. Find a polynomial in x of the second degree which for the values 0, 1, 2 ot X takes the values 1/c, l/(c + l), l/(c + 2); and show that when x=c+2 its value is l/(e + l). {Math. Trip. 1911.) 17. Show that if a; is a rational function of y, and y is a rational function of a;, then Axy + Bx + Ci/+I)=0. 18. If y is an algebraical function of x, then x is an algebraical function of y. 19. Verify that the equation COS^TT^-l //o_„^ is approximately true for all values of x between and 1. [Take x=0, J, J, J-, §, f , 1, and use tables. For which of these values is the formula exact?] 20. What is the form of the graph of the functions 2=W + [>]> s = ^ + »/-M-[>]? 21. What is the form of the graph of the functions 2=sina- + siny, 2 = sin.j;siny, z=smxif, z=Bm{x^+y')1 22. Geometrical constructions for irrational numbers. In Chapter I we indicated one or two simple geometrical constructions for a length equal to aJ2, starting from a given unit length. We also showed how to construct the roots of any quadratic equation ax^ + 2bx+c = 0, it being supposed that we can construct lines whose lengths are equal to any of the ratios of the coefficients a, b, c, as is certainly the case if a, b, c are rational. AH these con- structions were what may be called Euclidean constructions ; they depended on the ruler and compasses only. FUNCTIONS OF REAL VARIABLES 67 It is fairly obvious that we can construct by these methods the length measured by any irrational number which is defined by any combination of square roots, however compHcated. Thus 7/ // 17 + 3x/ll N /m-3VllM IS a case in point. This expression contains a fourth root, but this is of course the square root of a square root. We should begin by constructing Vll, e.cf. as the mean between 1 and 11 : then 17 + 3^11 and 17-3^11, and so on. Or these two mixed surds might be constructed directly as the roots of ^2_34;i;4.190 = 0. Conversely, onli/ irrationals of this kind can be constructed by Euclidean methods. Starting from a unit length we can construct any rational length. And hence we can construct the line Aa; + Bi/ + C==0, provided that the ratios oi A, B, C are rational, and the circle (ov x^+f + 2gx + 2jT/+c = 0), provided that a, 13, p are rational, a condition which implies that g, /, c are rational. Now in any Euclidean construction each new point introduced into the figure is determined as the intersection of two hnos or circles, or a line and a circle. But if the coefficients are rational, such a pair of equations as Ax + By-\-C=0, x^+y^+2gix+2fy+c = Q give, on solution, values of x and y of the form m-i-n,Jp, where m, n, p are rational : for if we substitute for x in terms of y in the second equation we obtain a quadratic in y with rational coefficients. Hence the coordinates of all points obtained by means of lines and circles with rational coefficients are expressible by rational numbers and quadratic surds. And so the same is true of the distance \/{(^i-'*'2)^+(yi-y2)^} between any two points so obtained. With the irrational distances thus constructed we may proceed to construct a number of lines and circles whose coefficients may now themselves involve quadratic surds. It is evident, however, that all the lengths which we can construct by the use of such lines and circles are still expressible by square roots only, though our surd expressions may now be of a more complicated form. And this remains true however often our constructions are repeated. Hence Euclidean methods will construct any surd expression involving square roots only, and no others. One of the famous problems of antiquity was that of the duplication of the cube, that is to say of the construction by Euclidean methods of a length measured by 4^2. It can be shown that ^2 cannot be expressed by means of any finite combination of rational numbers and square roots, and so that the problem is an impossible one. See Hobson, Squaring the Circle, pp. 47 et seq. ; the first stage of the proof, viz. the proof that ^2 cannot be a root of a quadratic equation ax'^ + 2hx-\-c=0 with rational coefficients, was given in Ch. I (Misc. Exs. 24). 5—2 68 FUiiCTIONS OF REAL VARIABLES [ll 23. Approximate quadrature of the circle. Let be the centre of a circle of radius R. On the tangent at A take AP='^^R and .4 §= J-/ /J, in the same direction. On AO take AN=OP and draw NM parallel to OQ and cutting AP in M. Show that AMlR=^^gJ\4e, and that to take AM as being equal to the circumference of the circle would lead to a value of jr correct to five places of decimals. If R is the earth's radius, the error in supposing .4 J/ to be its circumference is less than 11 yards. 24. Show that the only lengths which can be constructed with the ruler only, starting from a given unit length, are rational lengths. 25. Constructions for 4^2. is the vertex and S the focus of the parabola y^ = 'ix, and P is one of its points of intersection with the parabola «2=2y. Show that OP meets the latus rectum of the first parabola in a point $ such that ^§=4/2. 26. Take a circle of unit diameter, a diameter OA and the tangent at A. Draw a chord OBC cutting the circle at B and the tangent at G. On this line take OM=BG. Taking as origin and OA as axis of x, show that the locus of if is the curve (the Cissoid of Diodes). Sketch the curve. Take along the axis oiy a, length 0D = 2. Let AD cut the curve in P and OP out the tangent to the circle at A in Q. Show that AQ = ^2. CHAPTER III COMPLEX NUMBERS 34. Displacements along a line and in a plane. The ' real number ' x, with which we have been concerned in the two preceding chapters, may be regarded from many different points of view. It may be regarded as a pure number, destitute of geometrical significance, or a geometrical significance may be attached to it in at least three different ways. It may be re- garded as the measure of a length, viz. the length A^P along the line A of Chap. I. It may be regarded as the mark of a point, viz. the point P whose distance from Ao is x. Or it may be regarded as the measure of a displacement or change of position on the line A. It is on this last point of view that we shall now concentrate our attention. Imagine a small particle placed at P on the line A and then displaced to Q. We shall call the displacement or change of position which is needed to transfer the particle from P to Q the displacement PQ. To specify a displacement completely three things are needed, its magnitude, its sense forwards or backwards along the line, and what may be called its point of application, i.e. the original position P of the particle. But, when we are thinking merely of the change of position produced by the dis- placement, it is natural to disregard the point of application and to consider all displacements as equivalent whose lengths and senses are the same. Then the displacement is completely speci- fied by the length PQ = x, the sense of the displacement being fixed by the sign of x. We may therefore, without ambiguity, speak of the displacement [x]*, and we may write PQ = [«]. * It is hardly necessary to caution the reader against confusing this use of the symbol [re] and that of Chap. II (Bxs. xvi. and Misc. Exs.). 70 COMPLEX NUMBERS [III We use the square bracket to distinguish the displacement [x] from the length or number x*. If the coordinate of P is a, that of Q will be a + x; the displacement [x] therefore transfers a particle from the point a to the point a + x. We come now to consider displacements in a plane. We may define the displacement PQ as before. But now more data are' required in order to specify it completely. We require to know : (i) the magnitude of the displacement, i.e. the length of the straight line PQ ; (ii) the direction of the displacement, which is determined by the angle which PQ makes with some fixed line in the plane-; (iii) the sense of the displacement; and (iv) its point of application. Of these requirements we may disregard the fourth, if we consider two displacements as equivalent if they are the same in magnitude, direction, and sense. In other words, if PQ and RS are equal and parallel, and the sense of motion from P to Q is the same as that of motion from R to S, we regard the displacements PQ and RS as equivalent, and write PQ = RS. Now let us take any pair of coordinate axes in the plane (such as OX, OF in Fig. 19). Draw a line OA equal and parallel to PQ, the sense of motion from to A being the same as that from P to Q. Then PQ and OA are equivalent displacements. Let x and y be the coordinates of A. Then it is evident that OA is completely specified if X and y are given. We call OA the displacement [x, y] and Fig. 19. write OA = PQ = RS=[x,y]. * Strictly speaking we ought, by some similar difference of notation, to dis- tinguish the actual length x from the number x which measures it. The reader will perhaps be inclined to consider such distinctions futile and pedantic. But increasing experience of mathematics will reveal to him the great importance of distinguishing clearly between things which, however intimately connected, are not the same. If cricket were a mathematical science, it would be very important to distinguish between the motion of the batsman between the wickets, the run which he scores, and the mark which is put down in the score-book. S't-Se] COMPLEX NUMBERS 71 35. Equivalence of displacements. Multiplication of displacements by numbers. If ^ and t) are the coordinates of P, and f and t}' those of Q, it is evident that «=^'-l^> y = v'-v- The displacement from (^, r)) to (f , ^') is therefore It is clear that two displacements [on, y], [x', y'] are equivalent if, and only if, x = x',y = y'. Thus [x, y] = [x, y'] if and only if x = x', y = y' (1). The reverse displacement QP would be [^ - f '> t) — r)'], and it is natural to agree that [^-f. v-v'] = -[^'-^, v'-vl QP=-PQ, these equations being really definitions of the meaning of the symbols - [^' - ^, rj' - r}], - PQ. Having thus agreed that - [^, y] = [-«:,- yl it is natural to agree further that a[x, y] = [ax, ay] (2), where a is any real number, positive or negative. Thus (Fig. 19) iiOB = -^OA then OB = -^OA = -^[x,y]=[-^x,-^y]. The equations (1) and (2) define the first two important ideas connected with displacements, viz. equivalence of displacements, and multiplication of displacements hy numbers. 36. Addition of displacements. We have not yet given any definition which enables us to attach any meaning to the expressions PQ + PV, [oo,y] + \x',y'l Common sense at once suggests that we should define the sum of two displacements as the displacement which is the result of the successive application of the two given displacements. In 72 COMPLEX NUMBERS [III other words, it suggests that if QQi be drawn equal and parallel to P'Q', so that the result of successive displacements PQ, PQ' on a particle at P is to transfer it first to Q and then to Qi, then we should define the sum of PQ and WQ' as being PQ^. If then we draw OA equal and parallel to PQ, and OB equal and parallel to P'Q', and complete the parallelogram OACB, we have PQ + PQ=PQ,= OA + OB=OC. AV .-■■% ..■'c Fig. 20. Let us consider the consequences of adopting this definition. If the coordinates of B are x, y', then those of the middle point of AB are \{x + x), \ (y+y), and those of G are x+x', y+y. Hence [x,y]+[x',y']^[x + x',y + y'] (3), which may be regarded as the symbolic definition of addition of displacements. We observe that [x, y'] + [x, y] = [x' + x,y' + y] = [x+x',y + y'] = [x, y] + [x, y'] In other words, addition of displacements obeys the commutative laiu expressed in ordinary algebra by the equation a + 6 = 6 + a. This law expresses the obvious geometrical fact that if we move from P first through a distance PQ2 equal and parallel to PQ', and then through a distance equal and parallel to PQ, we shall arrive at the same point Qi as before. 36] COMPLEX NUMBERS 73 In particular [x,y]=[x,0] + [0,y] (4). Here [oc, 0] denotes a displacement through a distance x in a direction parallel to OX. It is in fact what we previously denoted by [x], when we were considering only displacements along a line. ^ We call [x, 0] and [0, y] the components of [x, y], and [x, 2/] their resultant. When we have once defined addition of two displacements, there is no further difficulty in the way of defining addition of any number. Thus, by definition, \x, y] + [x', y'] + [x", y"] = {[x, y] + [x', y']) + [x'\ y"] = [x + x',y+ y'] + [x", y"] = [x + x' + x", y + y' + y"]. We define subtraction of displacements by the equation [x, y] - [x', y'] = [x, y] + (- [x', y']) (5), which is the same thing as [x, y] + [- x', - y'] or as [x -x',y- y"\. In particular \.x,y-\-[_x.y-\ = [(),Q\ The displacement [0, 0] leaves the particle where it was ; it is the zero displacement, and we agree to write [0, 0] = 0. Examples XX. 1. Prove that (i) a\px,0y] = ^ [ax, ay] = [a^x, afiy], (ii) ([x y-]+[a/, y])+K, y"]=[^, y]+{[A ^J+K /]). (iii) [x, y] + [x', y ] = [x', y ] + [x, y], (iv) ia + l3)lx,y} = a[x,y]+^lx,yl (v) a {[x, y] + [x', y ]} = a[x,y-] + a [x', y ]. [We have already proved (iii). The remaining equations follow with equal ease from the definitions.. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).] 2. If Jfis the middle point of f§, then 0!S'=^(OP+0§). More generally, if M divides PQ in the ratio ;x : A, then 3. If G is the centre of mass of equal particles at Pi, P2, ..., /"„, then 0G^{0Pi + 0P2 + ...+0Pn)/n. 74 COMPLEX NUMBERS [III 4. If P, Q, R are coUinear points in the plane, then it is possible to find real numbers a, /3, y, not all zero, and such that a.OP+^.OQ + y.OR^O; and conversely. [This is really only another way of stating Ex. 2.] 5. If AB and A C are two displacements not in the same straight line, and a . AB + ^. 'AC=y . AB + 8. AC, then a = y and |3=S. [Take ABi=a . AB, AGi=^ . AC. Complete the parallelogram ABj^PiCi. Then AP^ = a. AB + ^ . AC. It is evident that .4 P^ can only be expressed in this form in one way, whence the theorem follows.] 6. ABCD is a parallelogram. Through Q, a point inside the paral- lelogram, RQS and TQU are drawn parallel to the sides. Show that RU, TS intersect on AC. [Let the ratios ^7: AB, AR : AD be denoted by a, 0. Then AT=a.AB, 'AR=^.AD, AU=a.AB + AD, AS=AB+P.A3. Let RU meet AC in P. Then, since R, U, P are collinear. Fig. 21. AP = ^AR+^AU, where /i/X is the ratio in which P divides R U. That is to say AP=^AB+^AD. But since P lies on AC, AP is a numerical multiple of AC; say AP=k . AC=k . iB+k . AD. Hence (Ex. 5) a^=^X+/Lt = (X + ;i) k, from which we deduce The symmetry of this result shows that a similar argument would also give rlC, AP = - "^ a+jS-l if P' is the point where TS meets AC. Hence P and P' are the same point.] 7. ABCD is a parallelogram, and M the middle point of AB. Show that i) J/ trisects and is trisected by AC*. * The two preceding examples are taken from WiUard Gibbs' Vector Analysis. 36, 37] COMPLEX NUMBERS 75 37. Multiplication of displacements. So far we have made no attempt to attach any meaning whatever to the notion of the product of two displacements. The only kind of multipli- cation which we have considered is that in which a displacement is multiplied by a number. The expression \x, y] X [x', y'] so far means nothing, and we are at liberty to define it to mean anything we like. It is, however, fairly clear that if any definition of such a product is to be of any use, the product of two displace- ments must itself be a displacement. We might, for example, define it as being equal to [x + x, y + y'] ; in other words, we might agree that the product of two displace- ments was to be always equal to their sum. But there would be two serious objections to such a definition. In the first place our definition would be futile. We should only be introducing a new method of expressing something which we can perfectly well express without it. In the second place our definition would be inconvenient and misleading for the following reasons. If a is a real number, we have already defined a [oc, y] as [ax, ay]. Now, as we saw in § 34, the real number a may itself from one point of view be regarded as a displacement, viz. the displacement [a] along the axis OX, or, in our later notation, the displacement [a, 0]. It is therefore, if not absolutely necessary, at any rate most desirable, that our definition should be such that [a,0][x,y] = [ax,ayl and the suggested definition does not give this result. A more reasonable definition might appear to be [x, y] [x, y'] = [xx', yy']. But this would give [a,0][x,y]=[ax,0]; and so this definition also would be open to the second objection. In fact, it is by no means obvious what is the best meaning to attach to the product [x, y] [x, y']. All that is clear is (1) that, if our definition is to be of any use, this product must itself be 76 COMPLEX NUMBERS [III a displacement whose coordinates depend on x and y, or in other words that we must have \x,y-\\p',y'-\ = \_X,Y\ where X and F are functions of x, y, x, and y' ; (2) that the definition must be such as to agree with the equation [x, 0] [x', y'] = [xx', xy] ; and (3) that the definition must obey the ordinary commutative, distributive, and associative laws of multiplication, so that [«> 2/] K y'] = W. y'] b; yl {[x, y] + [x', y']) [x", y"] = [x, y] [x", y"] + [x', y'] [x", y"\ \_x, y-\ {\x', y'-\ + \_x', y"-\) = {x, y] [^', y'^ + \x, y] \x" , y"\ and \_x, y-] {[x , y'] \x", fj) = {{x, y] [x', 2/]) [x", y"\ 38. The right definition to take is suggested as follows. We know that, if OAB, OGD are two similar triangles, the angles corresponding in the order in which they are written, then OBjOA = OD/OG, or OB .OG=OA . OD. This suggests that we should try to define multiplication and division of displacements in such a way that OB I OA = 01)100, OB.OG=OA.OD. Now let OB = ix,yl OG=[x',y'], OD = [X,Yl o 37, 38] COMPLEX NUMBERS 77 and suppose that A is the point (1, 0), so that OA = [1, 0]. Then OA .m = [1, 0][X, Y] = [X, Y], and so {x,y-}[x',y'] = [X,Y]. The product OB . OG is therefore to be defined as OB, D being obtained by constructing on OG a triangle similar to OAJB. In order to free this definition from ambiguity, it should be observed that on OG we can describe two such triangles, OGI) and OGD'. We choose that for which the angle GOD is equal to A OB in sign as well as in magnitude. We say that the two triangles are then similar in the same sense. If the polar coordinates of B and G are {p, 6) and (jr, <f), so that x = p cos 6, y = p sin 6, x' = <t cos 0, y' = (t sin j>, then the polar coordinates of B are evidently pa and d+<\). Hence X = per cos {9 + (j)) = xx' — yy', Y= pa- sin {d + ^) = xy' + yx'. The required definition is therefore \p, y\ W, y'] = [««' - yy', «^y' + vA (6). We observe (1) that if y = 0, then X = xx, Y=xy', as we desired ; (2) that the right-hand side is not altered if we inter- change X and x', and y and y', so that \x,y'][x',y''] = [x',y"]lx,y']; and (3) that {[x, y-\ + \x', 2/']} K, 2/"] ^\x^x',y^ y] {x", y"-\ = \ix + a;') a;" - (y + y') y" , {x + a;') y" + (2/ + 2/') a;"] = [^a,'" - 2/2/", «^" + K] + [*■'*" - 2/y'. o^y + y'^'"] = [^,2/][«'",2/"] + [^'>2/']K.2/"]- Similarly we can verify that all the equations at the end of § 37 are satisfied. Thus the definition (6) fulfils all the requirements which we made of it in § 37. Example. Show directly from the geometrical definition given above that multiplication of displacements obeys the commutative and distributive laws. [Take the commutative law for example. The product OB . ~0C is OP (Fig. 22), COD being similar to AOB. To construct the product OC . OB we 78 COMPLEX NUMBERS [ill should have to construct on OB a triangle BODi similar to ^OC; and so what we want to prove is that D and 2>i coincide, or that BOD is similar to AOC. This is an easy piece of elementary geometry.] 39. Complex numbers. Just as to a displacement [po] along OX correspond a point («) and a real number x, so to a displace- ment [x, y] in the plane correspond a point {x, y) and a pair of real numbers x, y. We shall find it convenient to denote this pair of real numbers X, y by the symbol X + yi. The reason for the choice of this notation will appear later. For the present the reader must regard x + yi as simply another '-* way of writing [x, y}. The expression x + yi is called a complex number. We proceed next to define equivalence, addition, and m,ultiplica- tion of complex numbers. To every complex number corresponds a displacement. Two complex numbers are equivalent if the corresponding displacements are equivalent. The sum or product of two complex numbers is the complex number which corresponds to the sum or product of the two corresponding displacements. Thus x + yi = x' + y'i (1), if and only if x = x', y = y' ; {!c+yi) + ix+y'{) = (x + x') + (y + 7/)i (2); (a; + yi) (x + yi) = xx' - yy' + {xy' + yx) i (3). In particular we have, as special cases of (2) and (3), a; + 2/i = (a; + Oz) + (0 + yi), {x + Oi) {x' + ifi) = xx' + xy'i; and these equations suggest that there will be no danger of confusion if, when dealing with complex numbers, we write x for X + Oi and yi for + yi, as we shall henceforth. Positive integral powers and polynomials of complex numbers are then defined as in ordinary algebra. Thus, by putting x = x, y = y' in (3), we obtain {x + yif = {x + yi) {x + yi) = x'-y^+ 2xyi. 38, 39] COMPLEX NUMBERS 79 The reader will easily verify for himself that addition and multiplication of complex numbers obey the laws of algebra expressed by the equations x + yi + {x' + y'i) = («' + y'i) + (aj + yi), {(.00 + yi) + {x' + y'i)] + {x" + y"i) = {x + yi) + [{x' + y'i) + {x" + y"i)], (x + yi) (x' + y'i) = (x' + y'i) {x + yi), (x+yi) {(x' +y'i)+{oo" + y"i)] ={x + yi) {x' + y'i) + {x + yi) {x" + y"i), {{x+yi)+{oo'+y'i)] (x"+y"i)=(x + yi) (x" + y"i) + (»■' + y'i) (x" + y"i), (x + yi) [(x' + y'i) (x" + y"i)} = {(x + yi) (x' + y'i)} {x" + y"i), the proofs of these equations being practically the same as those" of the corresponding equations for the corresponding displace- ments. Subtraction and division of complex numbers are defined as in ordinary algebra. Thus we may define {x + yi) — {x' + y'i) as {x + yi) + {- {x + y'i)} = x + yi + (-x'- y'i) = (x- x') + {y- y') i ; or again, as the number ^ + rji such that {x' + y'i) + (^ + rii) = x + yi, which leads to the same result. And {x + yi)/{x' + y'i) is defined as being the complex number f + rji such that {x' + y'i)(^ + vi) = x+yi, or x'^ - y't} + (.x't) + y'^) i = x + yi, or x'^-y'7] = x, x'7] + y'^=y (4). Solving these equations for ^ and tj, we obtain xx' + yy' _ yx' — xy x'^ + y'^ ' '^~ x'^ + y'^ This solution fails if x' and y' are both zero, i.e. if x' + y'i = 0. Thus subtraction is always possible; division is always possible unless the divisor is zero. 80 COMPLEX NUMBERS [III Fig. 23. Examples. (1) From a geometrical point of view, the problem of the division of the displacement ftB by OC is that of finding D so that the triangles COB, AOD are similar, and this is evidently possible (and the solution unique) imless C coincides with 0, or 0C=0. (2) The numbers x+yi, x-yi are said to be conjugate. Verify that {x+yi){oo-yi)=x^-\-y\ so that the product of two conjugate numbers is real, and that x+yi _ {x+yi) (^'-/»') ai+y'i \iif+y'i){^-y''i) _ xaf+yy'->r{^ y-xjf)i if^+y''^ 40. One most important property of real numbers is that known as the factor theorem, which asserts that the product of two numbers cannot he zero unless one of the two is itself zero. To prove that this is also true of complex numbers we put x = 0, 2/ = in the equations (4) of the preceding section. Then These equations give f = 0, t; = 0, i.e. ^ + vi = 0, unless «' = and 2/ = 0, or x' + y'i = 0. Thus x + yi cannot vanish unless either x' + y'i or ^ + r)i vanishes. 41. The equation i^ = — 1. We agreed to simplify our notation by writing x instead of a; + Oi and yi instead of + yi. The particular complex number \i we shall denote simply by i. It is the number which corresponds to a unit displacement along OY. Also {2 = M = (0 + li) (0 + li) = (0 . - 1 . 1) + (0 . 1 + 1 . 0) i = - 1. Similarly {—iy = — l. Thus the complex numbers i and —i satisfy the equation a;^ = — 1. The reader will now easily satisfy himself that the upshot of the rules for addition and multiplication of complex numbers is this, that we operate with comjdex numbers in exactly the same way as with real numbers, treating the symbol i as itself a number, 39-43] COMPLEX NUMBERS 81 but replacing the product ii = i^ by — i whenever it occurs. Thus, for example, (x + yi) (»' + y'i) = xx' + xy'i + yx'i + yy'i'^ = (««' - yy) + {xy' + ?/*■') i. 42. The geometrical interpretation of multiplication by i. Since {x + yi) i= — y + xi, it follows that if a; + yi corresponds to OF, and OQ is drawn equal to OP and so that POQ is a positive right angle, then {x + yi) i corresponds to OQ. In other words, multiplication of a complex number by i turns the corresponding displacement through a right angle. We might have developed the whole theory of complex numbers from this point of view. Starting with the ideas of X as representing a displacement along OX, and of i as a symbol of operation equivalent to turning x through a right angle, we should have been led to regard yi as a displacement of magnitude y along OY. It would then have been natural to define x + yi as m §§ 37 and 40, and (x + yi) i would have represented the dis- placement obtained by turning x + yi through a right angle, ie. -y + xi. Finally, we should naturally have defined (x + yi) x' as xx' + yx'i, (x + yi) y'i as — yy' + xy'i, and (x + yi) (x' + y'i) as the sum of these displacements, i.e. as xx' — yy' + (xy' + yx) i. 43. The equations ^'' + 1 = 0, az' + 2bz + c = 0. There is no real number z such that ^^^ + 1 = ; this is expressed by saying that the equation has no real roots. But, as we have just seen, the two complex numbers i and -i satisfy this equation. We express this by saying that the equation has the two complex roots i and — i. Since i satisfies z^ = — l, it is sometimes written in the form VC— !)• Complex numbers are sometimes called imaginary*. The expression is by no means a happily chosen one, but it is firmly * The phrase ' real number ' was introduced as au antithesis to ' imaginary number '. 82 COMPLEX NUMBERS [ill established and has to be accepted. It cannot, however, be too strongly impressed upon the reader that an 'imaginary number' is no more ' imaginary ', in any ordinary sense of the word, than a ' real ' number ; and that it is not a number at all, in the sense in which the 'real' numbers are numbers, but, as should be clear from the preceding discussion, a jjatV of numbers (x, y), united symbolically, for purposes of technical convenience, in the form x + yi. Such a pair of numbers is no less 'real' than any ordinary number such as ^, or than the paper on which this is printed, or than the Solar System. Thus i = + li stands for the pair of numbers (0, 1), and may be represented geometrically by a point or by the displacement [0, 1]. And when we say that i is a root of the equation a^ + 1 = 0, what we mean is simply that we have defined a method of combining such pairs of numbers (or displacements) which we call 'multiplica- tion ', and which, when we so combine (0, 1) with itself, gives the result (- 1, 0). Now let us consider the more general equation 02^ + 26^ + c = 0, where a, h, c are real numbers. If b^ > ac, the ordinary method of solution gives two real roots {-6 + V(6'-ac)}/a. If ¥ < ac, the equation has no real roots. It may be written ia the form {z + (bla)Y=^-(ac-¥)la^ an equation which is evidently satisfied if we substitute for z + {b/a) either of the complex numbers ±i'J{ac — h')/a*. We express this by saying that the equation has the two complex roots {-b±i^/(ac-¥)}/a. If we agree as a matter of convention to say that when b'' = ac (in which case the equation is satisfied by one value of x only, viz. —bja), the equation has two equal roots, we can say that a quadratic equation with real coefficients has two roots in all cases, either two distinct real roots, or two equal real roots, or two distinct complex roots. * We shall sometimes write x + iy instead oi x + yi for convenience in printing. 43] COMPLEX NUMBERS 83 The question is naturally suggested whether a quadratic equation may not, when complex roots are once admitted, have more than two roots. It is easy to see that this is not possible. Its impossibility may in fact be proved by precisely the same chain of reasoning as is used in elementary algebra to prove that an equation of the nth degree cannot have more than n real roots. Let us denote the complex number a; + yi by the single letter z, a convention which we may express by writing z=x + yi. Let / {z) denote any polynomial in z, with real or complex coefficients. Then we prove in succession : (1) that the remainder, when /(«) is divided hy z — a, a being any real or complex number, is /(a) ; (2) that if a is a root of the equation f{z) = 0, then f{z) is divisible hy z — a; (3) that if f{z) is of the ?ith degree, and f{z) = has the n roots Oi, a^, ..., a„, then f{z) = A{z-a,) (z-a^)... (z-a„), where .4 is a constant, real or complex, in fact the coefficient of z'^ in f(z). From the last result, and the theorem of § 40, it follows that f(z) cannot have more than n roots. We conclude that a quadratic equation with real coefficients has exactly two roots. We shall see later on that a similar theorem is true for an equation of any degree and with either real or complex coefficients : an equation of the nth degree has exactly n roots. The only point in the proof which presents any difficulty is the first, viz. the proof that any equation must have" at least one root. This we must postpone for the present*. We may, however, at once call attention to one very interesting result of this theorem. In the theory of number we start from the positive integers and from the ideas of addition and multiplication and the converse operations of subtraction and division. We find that these operations are not always posiible unless we admit new kinds of numbers. We can only attach a meaning to 3 — 7 if we admit negative numbers, or to f if we admit rational fractions. When we extend our list of arithmetical operations so as to include root extraction and the solution of equations, we find that some of * See Appendix I. 84 COMPLEX NUMBERS [ill them, such as that of the extraction of the square root of a number which (like 2) is not a perfect square, are not possible unless we widen our conception of a number, and admit the irrational numbers of Chap. L Others, such as the extraction of the square root of — 1, are not possible unless we go still further, and admit the complex numbers of this chapter. And it would not be unnatural to suppose that, when we come to consider equations of higher degree, some might prove to be insoluble even by the aid of complex numbers, and that thus we might be led to the con- siderations of higher and higher types of, so to say, hyper-complex numbers. The fact that the roots of any algebraical equation whatever are ordinary complex numbers shows that this is not the case. The application of any of the ordinary algebraical operations to complex numbers will yield only complex numbers. In technical language ' the field of the complex numbers is closed for algebraical operations '. Before we pass on to other matters, let us add that all theorems of elementary algebra which are proved merely by the application of the rules of addition and multiplication are true whether the numbers which occur in them are real or com- plex, since the rules referred to apply to complex as well as real numbers. For example, we know that, if a and /3 are the roots of az^ + 2bz + c = 0, then a + /3 = -(2b/a), a^ = (c/a). Similarly, if a, /3, 7 are the roots of az^ + 3bz^ + 3cz + d = 0, then a + ;e+7=-(36/a), ^87 + 7a + a/3 = (3c/a), ci0y = -{d/a). All such theorems as these are true whether a,b, ... a, /3, ... are real or complex. 44. Argand's diagram. Let P (Fig. 24) be the point (x, y), r the length OP, and 6 the angle XOP, so that x = rcQse, y = rsme, r=^{af + y% cosd -.smd -.l-.-.x-.y ir. 43, 44] COMPLEX NUMBERS 85 We denote the complex number x + yi by z, as in § 43, and we call z the complex variable. We call P the point z, or the point corresponding to z; z the argument of P, x the real part, y the imaginary part, r the modulus, and 6 the amplitude of z ; and we write « = R {z), y=I (z), r=\z\, 6 = a,mz. Fig. 24. When 2/ = we say that z is real, when « = that ^ is purely imaginary. Two numbers a; + yi, x — yi which differ only in the signs of their imaginary parts, we call conjugate. It will be observed that the sum 2a; of two conjugate numbers and their product x' + y'^ are both real, that they have the same modulus \/{x^ + y'^) and that their product is equal to the square of the modulus of either. The roots of a quadratic with real coeflScients, for example, are conjugate, when not real. It must be observed that 9 or am z \s a, many- valued function of X and y, having an infinity of values, which are angles differing by multiples of 27r*. A line originally lying along OX will, if turned through any of these angles, come to lie along OP. We shall describe that one of these angles which lies between — tt and TT as the principal value of the amplitude of z. This de- finition is unambiguous except when one of the values is vr, in which case — tt is also a value. In this case we must make some special provision as to which value is to be regarded as the principal value. In general, when we speak of the amplitude of z we shall, unless the contrary is stated, mean the principal value of the amplitude. Fig. 24 is usually known as Argand's diagram. * It is evident that | z | Is identical with the polar coordinate r of P, and that the other polar coordinate d is one value of am«. This value is not necessarily the principal value, as defined below, for the polar coordinate of § 22 lies between and 27r, and the principal value between - tt and tt. 86 COMPLEX ■ NUMBERS [ill 45. De Moivre's Theorem. The following statements follow immediately from the definitions of addition and multi- plication. (1) The real (or imaginary) part of the sum of two complex numbers is equal to the sum of their real (or imaginary) parts. (2) The modulus of the product of two complex numbers is equal to the product of their moduli. (3) The amplitude of the product of two complex numbers is either equal to the sum of their amplitudes, or differs from it by 27r. It should be observed that it is not always true that the principal value of am (««') is the sum of the principal values of am z and am ^. For example, if z=^= — \+i, then the principal values of the amplitudes of z and / are each JTT. But z^= - 2i, and the principal value of am (0/) is - ^tt and not f n-. The two last theorems may be expressed in the equation r (cos + i sin 6) x p (cos ^ + i sin <f>) = rp {cos {6 + (f) + i sin {6 + ^)}, which may be proved at once by multiplying out and using the ordinary trigonometrical formulae for cos {6 + ^) and sin {d + ^). More generally ri (cos ^1 + i sin ^i) x r^ (cos ^2 + i sin ^2) x ... x r„ (cos 6^ + i sin 6^ = nr^ ... r„ {cos (01 + 612+ ... + 6'„) +tsin (6'i + 6I2+ ... + 0„)}. A particularly interesting case is that in which }-i = rs = ... = r„ = 1, 01 = 02 = ... = 0„ = e. We then obtain the equation (cos 6 +ism dy = cos nd + i sin nO, where n is any positive integer: a result known as Be Moivre's Theorem*. Again, if z = r (cos 0+i sin 0) then 1/2 = (cos - i sin 0)/r. Thus the modulus of the reciprocal of z is the reciprocal of the modulus of z, and the amplitude of the reciprocal is the negative of the amplitude of z. We can now state the theorems for quotients which correspond to (2) and (3). * It will sometimes be convenient, for the sake of brevity, to denote cos 9 + i sin 9 by CisS: in this notation, suggested by Profs. Harkness and Moriey, De Moivre's theorem is expressed by the equation (Cis9)» = Cis7!9. 45] COMPLEX NUMBERS 87 (4) The modulus of the quotient of two complex numbers is equal to the quotient of their moduli. (5) The amplitude of the quotient of two complex numbers either is equal to the difference of their amplitudes, or differs from it by 27r. Again (cos + i sin 6)-^ = (cos d — i sin 6Y = {cos (- e) + i sin (- 6)]^ = cos (— nd) + i sin (— n6). Hence De Moivre's Theorem holds for all integral values of n, positive or negative. To the theorems (1) — (5) we may add the following theorem, which is also of very great importance. (6) The modulus of the sum of any number of complex numbers is not greater than the sum of their moduli. Fig. 25. Let 7)P, OP', ... be the displacements corresponding to the various complex numbers. Draw PQ equal and parallel to OP', QR equal and parallel to OP", and so on. Finally we reach a point U, such that oF=ap + op' + aP + .... The length OU is the modulus of the sum of the complex numbers, whereas the sum of their moduli is the total length of the broken line OPQB...U, which is not less than OU. A purely arithmetical proof of this theorem is outlined in Exs. XXI. 1. 88 COMPLEX NUMBERS [ill 46. We add some theorems concerning rational functions of complex numbers. A rational function of the complex variable z is defined exactly as is a rational function of a real variable a, viz. as the quotient of two polynomials in z. Theorem 1. Any rational function R {z) can he reduced to the form X+ Yi, where X and Y are rational functions of x and y with real coefficients. In the first place it is evident that any polynomial P (a; + yi) can be reduced, in virtue of the definitions of addition and multi- plication, to the form A + Bi, where A and B are polynomials in X and y with real coefficients. Similarly Q{x + yi) can be reduced to the form G + Di. Hence B(x+ yi) = P(x + yi)/Q (x + yi) can be expressed in the form (A + Bi)/{G + Di) = (A + Bi) {G - Di)l{G + Bi) {G - Bi) AG+BB BG-AB . - C' + D^ + G^ + B^ *' which proves the theorem. Theorem 2. If R{x + yi) = X+ Yi, E denoting a rational function as before, hut with real coefficients, then R{x—yi)=X—Yi. In the first place this is easily verified for a power {x + yiy by actual expansion. It follows by addition that the theorem is true for any polynomial with real coefficients. Hence, in the notation used above, P . Aj-Bi_AG + BB B G-AB . ±C{x y^)-'Q_j)^ CH-i)='~ G' + B' '' the reduction being the same as before except that the sign of i is changed throughout. It is evident that results similar to those of Theorems 1 and 2 hold for functions of any number of complex variables. Theorem 3. The roots of an equation a^'^+a^z"--^+ ...+an=0, whose coefficients are real, may, in so far as they are not themselves real, he arranged in conjugate pairs. 46] COMPLEX NUMBERS 89 For it follows 'from Theorem 2 that if ^ + yi is a root then so is X — yi. A particular case of this theorem is the result (§ 43) that the roots of a quadratic equation with real coefficients are either real or conjugate. This theorem is sometimes stated as follows : in an equation with real coefficients complex roots occur in conjugate pairs. It should be compared with the result of Exs. viii. 7, which may be stated as follows : in an equation with rational coefficients irrational roots occur in conjugate pairs*. Examples XXI. 1. Prove theorem (6) of § 45 directly from the definitions and without the aid of geometrical considerations. [First, to prove that | 2+2' | g | 1 + | 2* | is to prove that The theorem is then easily extended to the general case.] 2. The one and only case in which is that in which the numbers z, zf, ... have all the same amplitude. Prove this both geometrically and analytically. 3. The modulus of the sum of any number of complex numbers is not less than the sum of their real (or imaginary) parts. 4. If the sum and product of two complex numbers are both real, then the two numbers must either be real or conjugate. 5. If a + b.j2 + {o+d^2)i=A + BJZ + {C+D^2){, ■where a, h, c, d, A, B, 0, D are real rational numbers, then a=A, b=B, c=C, d=D. 6. Express the following numbers in the form A + Bi, where A and B are real numbers : (!+»)'. [iZi)' ViTI-j' \^i' \\^i) Vm-"^7' where X and /n are real numbers. 7. Express the following functions of z=x + i/i in the form X+ Yi, where Xand Fare real functions of ^ and ?/: z^ z\ 0", Ijz, 0+(l/0), (a + ^0)/(y+80), ■where a, ft y, 8 are real numbers. 8. Find the moduli of the numbers and functions in the two preceding examples. * The numbers a + Jb, a-Jb, where a, b are rational, are sometimes said to be • conjugate '. 90 COMPLEX NUMBERS [HI 9. The two lines joining the points z = a, z — b and z = o, z=d will be perpendicular if i.e. if (a — b)/(c-d) is purely imaginary. What is the condition that the lines should be parallel 1 10. The three angular points of a triangle are given by 0=0, 2=0, 2= y, where a, ft y axe complex numbers. Establish the following propositions : (i) the centre of gravity is given 6y z= J (a+/3+y) ; (ii) the circum-cenire is given by \z—a\ = \z-^ I = I ■^^ V 1 > (iii) the three perpendiculars from the angular points on the opposite sides meet in a point given by (iv) there is a point P inside the triangle such that CBP=ACP=BAP=a, and cot 6) = cot jd+ cot 5+ cot (7. [To prove (iii) we observe that ii A, J3, G are the vertices, and P any point z, then the condition that AP should be perpendicular to BC is (Ex. 9) that {z — a)/(fi—y) should be purely imaginary, or that R(2-a)RO-y) + I(2-a)IO-y) = 0. This equation, and the two similar equations obtained by permuting a, /3, y cyclically, are satisfied by the same value of z, as appears from the fact that the sum of the three left-hand sides is zero. To prove (iv), take BC parallel to the positive direction of the axis of x. Then* y-/3 = a, a-y=-6Cis(-(7), ^-a=-cCisB. We have to determine z and a> from the equations ( 2-a)(/3o-ao) ^ (z-fi)(yo-^o) ^ (2-y)(ao-yo) ^ p;_ 2,. (2o-ao)(,3-a) (2o-;3o)(y-/3) {zo-yo){a-y) where z„, oq, /Sj, yo denote the conjugates of z, a, ft y. Adding the numerators and denominators of the three equal Iractiona, and using the equation i cot <B = (1 + Cis 2(b)/(1 - Ois 2(o), we find that ,-<,ot„= (/3-y) Oo-yo) + (y-°) (yo-ao)-Ka-ff) (ao-i3o) ^ 0yo - 0oy + y "0 - 7o° + "^0 - "o^ From this it is easily deduced that the value of cot <» is' (a^+b^ + cr^)li A, where A is the area of the triangle ; and this is equivalent to the result given. * We suppose that as we go round the triangle in the direction ABC we leave it on our left. 46] COMPLEX NUMBERS 91 To determine 2, we multiply the numerators and denominators of the equal fractions by (yo-^o)/(/3-a), (ao-yo)/(y-/3), iPa-<'a)l{a-y), and add to form a new fraction. It wiU be found that _ aaOis^+fc^O is B + oy Cis , ^ a Cis ^ + 6 Cis B-\-oOisC~' "-' 11. The two triangles whose vertices are the points a, h, and x, y, z respectively will be similar if = 1 1 1 a h c X y z [The condition required is that AB\AG=XYIXZ (large letters denoting the points whose arguments . are the corresponding small letters), or (b-a)l{a-a) = (i/-a:)/{z — a:), which is the same as the condition given.] 12. Deduce from the last example that if the points x, y, z are coUinear then we can find reaZ numbers a, /3, 7 such that a + /3 + 'y=0 and aii;+^2'+7^=0, and conversely (cf. Exs. xx. 4). [Use the fact that in this case the triangle formed by x, y, z is similar to a certain line-triangle on the axis OX, and apply the result of the last example.] 13. The general linear equation with complex coefficients. The equation a0+/3 = O has the one solution z= - (fi/a), unless a = 0. If we put a—a+Ai, ^ — h-\-Bi, z=x+yi, and equate real and imaginary parts, we obtain two equations to determine the two real numbers a; and y. The equation will have a real root if y=0, which gives ax+b=0, Ax+B=0, and the condition that these equations should be consistent is aB — bA=0. 14. The general quadratic equation with complex coefficients. This equation is {a+Ai) «2+2 (b + Bi) z+(c+C{)=0. Unless a and A are both zero we can divide through by a+iA. Hence we may consider z^+2(b+Bi)z-^(c+Ci)=0 ....'. (1) as the standard form of our equation. Putting z=x+yi and equating real and imaginary parts, we obtain a pair of simultaneous equations for x and y, viz. x^-y^+2(bis-By) + c=0 2xy+2 (by+Ba;)+ C=0. If we put x+b=^, y+B=r,, b^-B^-c=h, ibB-C=Jc, these equations become ^-n^=K 2^7=^. 92 COMPLEX NUMBERS [HI Sqviaring and adding we obtain We must choose the signs so that ^ has the sign of Je : i.e. if k is positive we must take lilce signs, if h is negative unlilie signs. Conditions for equal roots. The two roots can only be equal if both the square roots above vanish, i.e. if h=0, h=0, or if c = lfi-B\ C=^hB. These conditions are equivalent to the single condition c + Ci = (6+5j)2, which obviously expresses the fact that the left-hand side of (1) is a perfect square. Condition for a real root. If x'^ + '2,{h + Bi) x+{o-\-Ci)^Q, where x is real, then x^ + 'ihx-^c = Q, 2Bx+C=0. Eliminating x we find that the required condition is C'^-ibBC+icB^^O. Condition for a purely imaginary root. This is easily found to be 02 -4650- 46V =0. Conditions for a pair of conjugate complex roots. Since the sum and the product of two conjugate complex numbers are both real, h + Bi and c + Ci must both be real, i.e. B=0, 0=0. Thus the equation (1) can have a pair of conjugate complex roots only if its coefficients are real. The reader should verify this conclusion by means of the explicit expressions of the roots. Moreover, if V^Z.c, the roots will be real even in this case. Hence for a pair of conjugate roots we must have B=0, 0=0, b^<c. 15. The Cubic equation. Consider the cubic equation ^ + 3IIz+G=0, where G and H are complex numbers, it being given that the equation has (a) a real root, (6) a purely imaginary root, (c) a pair of conjugate roots If ir=\+fii, = p + a-i, we arrive at the following conclusions. (a) Conditions for a real root. If /i is not zero, then the real root is - (r/3fi, and (T^ + 21\fi'(T — 27ii^p = 0. On the other hand, if fi = then we paust also have o- = 0, so that the coefficients of the equation are real. In this case there may be three real roots. (6) Conditions for a purely imaginary 'root. If /i is not zero then the purely imaginary root is {p/5fi)i^ and p^-%TKiJp-2,7iJ.^a- = 0. If /x = then 'also p=0, and the root is yi, where y is given by the equation y^ — 3Xy — o- = 0, which has real coefficients. In this case there may be three purely imaginary roots. (c) Conditions for a pair of conjugate complex roots. Let these be x+yi and x—yi. Then since the sum of the three roots is zero the third root must be — 'i^x. From the relations between the coefficients and the roots of an equation we deduce y'^-Zx'i = ZH, 2x{,x'^+y^) = a. Hence O and H must both be real. In each case we can either find a root (in which case the equation can be reduced to a quadratic by dividing by a known factor) or we can reduce the solution of the equation to the solution of a cubic equation with real coefficients. 46] COMPLEX NUMBERS 93 16. The cubic equation x^ + oj.r^ + a^x + aj = 0, where a{=Ai+Aii, . . . , has a pair of conjugate complex roots. Prove that the remaining root is -Ai'oslAs, unless A3'=0. Examine the case in which A^'^O. 17. Prove that if 2^ + 3-^2 + (?=0 has two complex roots then the equation 8a^ + 6aH-0=Q has one real root which is the real part a of the complex roots of the original equation ; and show that a has the same sign as O. 18. An equation of any order with complex coefficients will in general have no real roots nor pairs of conjugate complex roots. How many con- ditions must be satisfied by the coefficients in order that the equation should have (a) a real root, (6) a pair of conjugate roots ? , 19. Coaxal circles. In Fig. 26, let a, b, z be the arguments oi A, B, P z-h Then ■ = APB, if the principal value of the amplitude is chosen. If the two circles shown in the figure are equal, and z', zi, z{ are the arguments of P', Pi, P{, and APB = 5, it is easy to see that 2'-6 = ir — B am ■ -_b_ -a -e. and am —, — — -ir + 6. The locus defined by the equation am = 8, z-a where 6 is constant, is the arc APB. By writing v — d, -6, — tt + 6 toi 8, we obtain the other three arcs shown. The system of equations obtained by supposing that 5 is a parameter, varying from -jr to +n-, represents the system of circles which can he drawn through the points A, B. It should however be ob- served that each circle has to be divided into two parts to which correspond difierent values of 6. Fig. 26. =X ■(1), 20. Now let us consider the equation \ z-b \z — a\ where X is a constant. Let K be the point in which the tangent to the circle ABP at P meets AB. Then the triangles KPA, EBP are similar, and so APIPB=PKjBK=KAIKF=-\. 94 COMPLEX NUMBERS [III Hence KAIKB^XS and therefore iT is a fixed point for aU positions of P which satisfy the equation (1). Also KP'=KA.KB, and so is constant. Hence the locus of P is a circle whose centre is E. The system of equations obtained by varying X represents a system of circles, and every circle of this system cuts at right angles every circle of the system of Ex. 19. The system of Ex. 19 is called a system of coaxal circles of the common point kind. The system of Ex. 20 is called a system of coaxal circles of the limiting point land, A and B being the limiting points of the system. If X is very large or very small then the circle is a very small circle containing A or B in its interior. 21. Bilinear Transformations. Consider the equation z=Z+a (1), where z=x^yi and Z=X-\-Yi are two complex variables which we may suppose to be represented in two planes xoy, XOY. To every value of z corresponds one of Z, and conversely. If a=a+3i then x=X^a, y=Y+p, and to the point {x, y) corresponds the point (X, Y). If {x, y) describes a curve of any kind in its plane, {X, Y) describes a curve in its plane. Thus to any figure in one plane corresponds a figure in the other. A passage of this kind from a figure in the plane xoy to a figure in the plane XOY by means of a relation such as (1) between z and Z is called a transformation. In this particular case the relation between corresponding figures is very easily defined. The {X, Y) figure is the same in size, shape, and orientation as the {x, y) figure, but is shifted a distance a to the left, and a distance /3 downwards. Such a transformation is called a translation. Now consider the equation ^^pZ (2). where p is real. This gives x=pX,y=pY. The two figures are similar and similarly situated about their respective origins, but the scale of the {x, y) figure is p times that of the {X, Y) fig-ure. Such a transformation is called a magnification. Finally consider the equation «=(cos (^+isin <^) Z (3). It is clear that 1 2 1 = | ^ | and that one value of am s is am Z+ (j), and that the two figures differ only in that the (x, y) figure is the {X, Y) figure turned about the origin through an angle (^ in the positive direction. Such a trans- formation is called a rotation. The general linear transformation z = aZ+h (4) 46] COMPLEX NUMBERS 95 is a combination of the three transformations (1), (2), (3). For, if | a l=p and am a = <p, we can replace (4) by the three equations 2=2' + 6, z'=pZ', Z' = (ooa<j>+isin(j)) Z. Thus the general linear transformation is equivalent to the combination of a translation, a magnification, and a rotation. Next let us consider the transformation z=\IZ (5). If \Z\ = R and am^=e, then \z\ = \jR and am 2= -9, and to pass from the {x, y) figure to the {X, Y) figure we invert the former with respect to o, with unit radius of inversion, and then construct the image of the new figure in the axis ox {i.e. the symmetrical figure on the other side of ox). Finally consider the transformation '=^Z+d (^)- This is equivalent to the combination of the transformations z = {alc) + {bo-ad){s'/c), z' = l/Z', Z' = cZ+d, i.e. to a certain combination of transformations of the types already con- sidered. The transformation (6) is called the general bilinear transformation. Solving for Z we obtain Z=^^. ez — a The general bilinear transformation is the most general type of trans- formation for which one and only one value of z corresponds to each value of Z, and conversely. 22. The general bilinear transformation transforms circles into circles. This may be proved in a variety of ways. We may assume the well-known theorem in pure geometry, that inversion transforms circles into circles (which may of course in particular cases be straight lines). Or we may use the results of Exs. 19 and 20. If, e.g., the (x, y) circle is \{z-^)l{z-p)\=X, and we substitute for z in terms of Z, we obtain |(^-cr')/(^-p')l=X', , b-a-d , b-pd , la — pel where "■ = , p = 1 « = a. a-ac' '^ a — pc \a-a-o\ 23. Consider the transformations z^lfZ, z = {l+Z)/(l~ Z), and draw the (X, 7) curves which correspond to (1) circles whose centre is the origin, (2) straight lines through the origin. 96 COMPLEX NUMBERS [UI 24. The condition that the transformation z = {aZ+b)l{cZ+d) should make the circle x''+y^ = l correspond to a straight line in the (X, Y) plane is |a| = |c|. 25. Cross ratios. The cross ratio (0122, 23^4) is defined to be (z,- 23) (22-24) (%-%)(22-%)' If the four points z^, z^, z^, 24 are on the same line, this definition agrees with that adopted in elementary geometry. There are 24 cross ratios which can be formed from z^, 02, %, 24 by permuting the suffixes. These consist of six groups of four equal cross ratios. If one ratio is X, then the six distinct cross ratios are X, 1 -X, 1/X, 1/(1 —X), (X- 1)/X, X/(X- 1). The four points are said to be harmonic or harmonically related if any one of these is equal to — 1. In this case the six ratios are —1, 2, -1, ^, 2, J. If any cross ratio is real then all are real and the four points lie on a circle. For in this case j^,^ (^l-Z3)(Z2-g4) (^1- 24) (22-^3) must have one of the three values -n-, 0, ir, so that 3.m{{z^-z^l{z^-z^} and am {(Z2 - Z3)/(z2 - 24)} must either be equal or diflfer by n- (cf. Ex. 19). If (2i02, 2324)= - 1, we have the two equations Zi-Z3 ,!i^^=±^ + am^^^ l?1^3 2i - ^i ^2 - ^4 I Zi - z^ The four points ^,, Jj, -^s. ^i He on a circle, A^ and A2 being separated by ^3 and ^4. A\&o AiA^jAiAi=AiA3lAiA^. Let be the middle point of ^^3^4. The equation (Zl-%)(Z2-Z4 )_ ^ {Zi-Zi){z^-Z3) may be put in the form [Z^+Zi) (23+24) = 2(2i 22 + 2324), or, what is the same thing, fe-i (%+24)} {^3-i(23 + 24)} = {H%-«4)P. But this is equivalent to OAi . 'OT^^^OA^^'OAi. Hence OA^ and OAi make equal angles with A^A^, and 0^1.0^2=0^3^=0.442. jt ^ju ^g qJ^ served that the relation between the pairs ^1, A^ and A^, A^ is symmetrical. Hence, if 0' is the middle point of A^A^, O'As and O'A^ are equally inclined to .41^2, and O'Jj. 0'^4=0'^ 12=0^22. 26. If the points Ai, A^ are given by az^ + 2bz+c=0, and the points ^3, Ai by a'z'^ + 2b'z+c'=0, and is the middle point of ^3^4, and ac' + a'c- 266' = 0, then OAi, OA^ are equally inclined to Aul and OA^ . OA, = OAi^ OA,K ^j^^^^^ y,,J,_ \^^^^ 46] COMPLEX NUMBERS 97 27. AB, CD are two intersecting lines in Argaud's diagram, and P, Q their middle points. Prove that, ii AB bisects the angle CPD and PA^^PB'=FG.PD, then CD bisects the angle^§5 and QC^= QD'^QA . QB. {Math. Trip. 1909.) 28. Tlie condition that four points should lie on a circle. A sufficient condition is that one (and therefore all) of the cross ratios should be real (Ex. 25) ; this condition is also necessary. Another form of the condition is that it should be possible to choose real numbers a, /3, y such that 1 1 1 =0. a y [To prove this we observe that the transformation Z= \j{z — z^j is equivalent to an inversion with respect to the point zi, coupled with a certain reflexion (Ex. 21). If «i, 22, Z3 lie on a circle through z^, the corre.sponding points Z■^ = \|{^l- Zij, Z2=ll{z2 — Zi), Zi=\j{zz — z^ lie on a straight line. Hence (Ex. 12) we can find real numbers a', /S', y' such that a'4-/3' + y = and a'l{zi — Zi)+^'/{z2 — z^)+y'/{z3-Zi) = 0, and it is easy to prove that this is equivalent to the given condition.] 29. Prove the following analogue of De Moivre's Theorem for real numbers : if ^1, <l>2, ^3, ... is a series of positive acute angles such that tan (^„+i=tan (j)^ sec <^i +sec 0,„ tan cj)i, then ta,n<pm + n= tan ^^sec(^„ + sec(/)„tan(/)„, sec ^,„ + „=sec c^m sec ^„+tan </>„, tan <^„, and tan </>,„ + sec (^„ = (tan 0i + sec ^i)"*. [Use the method of mathematical induction.] 30. The transformation z=Z". In this case r^R'", and d and me differ by a multiple of 2ir. If Z describes a circle round the origin then z describes a circle round the origin m times. The whole {x, y) plane corresponds to any one of m sectors in the {X, T) plane, each of angle 2jr/m. To each point in the {x, y) plane correspond m points in the {X, Y) plane. 31. Complex functions of a real variable. If f{t), 4> {() are two real functions of a real variable t defined for a certain range of values of t, we call z=f(t) + i<f>{t) (1) a complex function of *. We can represent it graphically by drawing the curve 7 98 COMPLEX NUMBERS [ll[ the equation of the curve may be obtained by eliminating t between these equations. If is a polynomial in t, or rational function of t, with complex coefficients, we can express it in the form (1) and so determine the curve represented by the function. (1) Let z=a + {b-a)t, where a and 6 are complex numbers. If a=a+a'i, h=^+ffi, then .r=a + (/3-a)<, Jf=a' + (^'-a')<. The curve is the straight line joining the points z=a and 0=6. The seg- ment between the points corresponds to the range of values of t from to 1. Find the values of t which correspond to the two produced segments of the line. where p is positive, then the curve is the circle of centre a and radius p. As t varies through all real values 2 describes the circle once. (iii) In general the equation z = {a-\-ht)j{c-\-dt) represents a circle. This can be proved by calculating x and y and eliminating : but this process is rather cumbrous. A simpler method is obtained by using the result of Ex. 22. Let z={a+bZ)/(c+dZ), Z=t. As t varies Z describes a straight line, viz. the axis of X. Hence z describes a circle. (iv) The equation «=a + 2fe + c«2 represents a parabola generally, a straight line if 6/c is real. (v) The equation = (a+2fo + c«2)/(a + 2i3i; + yi!2)^ wliere a, ft 7 are real, represents a conic section. [Ehminate t from a;=(4 + 25<-|-Ci;2)/(a + 2^i;-f-y«2), y=(4' + 2fi'i!+C"0/(a + 23<+yK2), where 4 + 4'i=a, B + Bi=b, C+C'i=c.] 47. Roots of complex numbers. We have not, up to the present, attributed any meaning to symbols such as Va, «'""'. when a is a complex number, and m and n integers. It is, however, natural to adopt the definitions which are given in elementary algebra for real values of a. Thus we define ^a or a''", where n is a positive integer, as a number z which satisfies the equation «" = a ; and a""", where m is an integer, as (a"")'". These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation. 48. Solution of the equation 2» = a. Let a = p (cos (j>+i sin (ft), where p is positive and 4> is an angle such that - tt < S tt. If 46-48] COMPLEX NUMBERS 99 we put z = r (cos 6 +i sin 6), the equation takes the form r-" (cos nd + i sin nO) = p (cos (\) + i sin 0) ; so that '"' = /', cos 71^ = cos ^, sinr!.0 = sin^ (1). The only possible value of r is \/p, the ordinary arithmetical jith root of p ; and in order that the last two equations should be satisfied it is necessary and sufficient that 7i0 = ^ + 2i7r, where k is an integer, or 6 = {(^ + 2hir)jn. If k=pn + q, where p and q are integei's, and 0&q<n, the value of 6 is 2p7r + (</> + 2qir)/n, and in this the value of p is a matter of indifference. Hence the equation z^ = a=p (cos ^ + i sin <})) has n roots and n only, given hy z = r (cos 6 + i sin 6), where r==^p, d = {<l> + 2qir)ln, (<?= 0, 1, 2, ... n-1). That these n roots are in reality all distinct is easily seen by plotting them on Argand's diagram. The particular root ^p {cos (</)/n) + i sin ((^/ji)} is called the pnncipal value of a/ a. The case in which a = l, p = l, (j> = is of particular interest. The 71 roots of the equation a?" = 1 are cos (2qTr/n) + i sin {2qTr/n), {q=0, 1, ... n — 1). These numbers are called the »ith roots of unity; the principal value is unity itself If we write Wn for cos {2Tr/n) + i sin (27r/w), we see that the nth. roots of unity are 1. <"«, «"«,•■• <"'• Examples XXII. 1. The two square roots of 1 are 1, - 1 ; the three cube roots are 1, ^{-l + i^3), J(-l-iV3); the four fourth roots are 1, I, —1, -i; and the five fifth roots are 1, il v/fl-l+»V{10+2V5}], i[-V5-l+iV{10-2^/5}], J[-v/5-l-*x/{10-V5}l i[ V5-l-»V{10 + 2v/5}]. 2. Prove that l + (a„+<B° + ... + (a^"'=0. 3. Prove that (x+ya^ + zcol){x+2/al+za>s)=x^+y^+z^-i/z-zx-xy. 4. The Jith roots of a are the products of the nth. roots of unity by the principal value of !i/a. 7—2 100 COMPLEX NUMBERS [ill 5. It follows from Exs. xxi. 14 that the roots of are ± v'B (VC"^ +/3^) +<>}] ± i M M"' +^') " °}1 like or unlike signs being chosen according as /3 is positive or negative. Show that this result agrees with the result of § 48. 6. Show that (^^m _ a'^)l{x''' - a^ is equal to (x"'- %ax cos - + a^ (x^ - 2ax cos — + aA...(x^- lax cos '^^aiZ^JL + a^ . [The factors of ^2n>_ a2m are {x-a), {x-am^J, {x-awlJ, ... (^-aco^""'). The factor x - am"„ is x + a. The factors {x - aa>lJ, (x - a<o^"„") taken together give a factor x^ — 2axcoa{s7rlm) + a\] 7. Kesolve a;2"»+i-a2-» + ij x^"' + a^'^, and a;2"' + i + a2m + i into factors in a similar way. 8. Show that x^" - 2»"a!'' cos 6 + a^" is equal to lx''-2xaoos — \-aA (x^-2xacoa l-a^ ) ... ...(x^'-2x, 2,acos'±^''-^^ + a^ [Use the formula ^2n _ 2^n„n COS 6 + a^" = {a;" - a" (cos 6 + 1 sin fl)} {a,'" - a" (cos fl - 1 Sin 5)}, and split up each of the last two expressions into n factors.] 9. Find all the roots of the equation .r» -- 2x^ + 2 = 0. {Math. Trip. 1910.) 10. The problem of finding the accurate value of <»„ in a numerical form involving square roots only, as in the formula 0)3 = ^ ( — l + iV3), is the algebraical equivalent of the geometrical problem of inscribing a regular polygon of n sides in a circle of unit radius by Euclidean methods, i.e. by ruler and compasses. For this construction will be possible if and only if we can construct lengths measured by cos (2n-/M) and sin (2n/n) ; and this is possible (Ch. II, Misc. Exs. 22) if and only if these numbers are expressible in a form involving square roots only. Euclid gives constructions for « = 3, 4, 5, 6, 8, 10, 12, and 15. It is evident that the construction is possible for any value of n which can be found from these by multiplication by any power of 2. There are other special values of n for which such constructions are possible, the most inter- esting being n = 17. 48, 49] COMPLEX NUMBERS 101 49. The general form of De Moivre's Theorem. It follows from the results of the last section that if g is a positive integer then one of the values of (cos + i sin dy^ is cos (0/q) + i sin (6/q). Eaising each of these expressions to the power p (where p is any integer positive or negative), we obtain the theorem that one of the values of (cos d + isin dyii is cos {pOlq) + i sin {pOjq), or that if a is any rational number then one of the values of (cos 9-\-i sin df- is cos ad + i sin a^. This is a generalised form of De Moivre's Theorem (§ 45). MISCELLANEOUS EXAMPLES ON CHAPTER III. 1. The condition that a triangle (ays) should be equilateral is that [Let XFZ be the triangle. The displacement ZX'is YZ turned through an angle fn- in the positive or negative direction. Since Q\s,\Tr = a^^ Cis(-|n-) = l/(Bg={i>j, we have x — z={z-y)<ii^ or x-z = {z—y) J:^ Hence ^+!/a)5+z(»3=0 or ^+y<B3+ 20)3=0. The result follows from Exs. xxii. 3.] 2. If Z YZ, X' Y'Z are two triangles, and YZ. Y'Z'=ZX.Z'X' = XY.X'r, then both triangles are equilateral. [From the equations say, we deduce 2 l/(y' - z') = 0, or Sjc'^ - Sj/z' = 0. Now apply the result of the last example.] 3. Similar triangles BCX, CAY, ABZ are described on the sides of a triangle ABC. Show that the centres of gravity oi ABC, XYZ are coincident. [We have {x-c)/{b — c) = (t/-a)/{o-a) = {z-b)l{a-b) = \, say. Express J (x + T/+z) in terms of a, b, c] 4. If X, Y, Z are points on the sides of the triangle ABC, such that BXIXC=CYIYA = AZIZB=r, and if ABC, XYZ are similar, then either »•=! or both triangles are equilateral. 5. li A, B, C, D are foiu; points in a plane, then AD.BCS.BD.CA + CD.AB. 102 COMPLEX NUMBERS [ill [Let 2i, 02, «3, ■^4 be the complex numbers corresponding to A, B, C, D. Then we have identically Hence I (-1^1 - ^i) («2 - ^3) ! = I (^2 - ^4) (.^a - ^i) + (^3 - ■^4) (•»! - ^'2) I < I {X^-Xi) (.*-3-^l) l + l (^3-a'4) (•^1-^2) 10 6. Deduce Ptolemy's Theorem concerning cyclic quadrilaterals from the fact that the cross ratios of four concyclic points are real. [Use the same identity as in the last example.] 7. If 22 + 0'2= 1, then the points z, 2' are ends of conjugate diameters of an ellipse whose foci are the points 1, - 1. [If CP, CD are conjugate semi- diameters of an ellipse and S, H its foci, then GD is parallel to the external bisector of the angle SPH, and SP . HP= CDKI 8. Prove that |a-)-6|2+|a-J|2=2{|a|2 + |5|2}. [This is the analytical equivalent of the geometrical theorem that, if M is the middle point of PQ, then 0P^+0Q^=20M'^+2MPK] 9. Deduce from Ex. 8 that |a + V(a'-&2)| + |a-V(a2-62)| = |a + 6| + |a-6|. \li a-\- sj{a^ -h^) = Z\, a-iJ{a?'-V) = z2, we have Nl|' + l^2|2 = i|Zl + Z2|H4|01-22|'=2|a|2 + 2|a2_62|^ and so (|2i| + |z2|)'=2{|aP+|a2-62| + |6|2} = |a + 6|2 + |a-6|2 + 2|a2-52|. Another way of stating the result is : if z^ and z^ are the roots of 022+2/3?+ y=0, then |2,| + |02| = (l/|a|){(|-^ + Vay|) + (|-^-Va7|)}.] 10. Show that the necessary and sufficient conditions that both the roots of the equation z^ + az + b = should be of unit modulus are |a|g2, 151 = 1, am6 = 2ama. [The amplitudes have not necessarily their principal values.] 11. If x^+iaiX^ + Qa2x'^ + ia3X-\-ai=0 is an equation with real coefficients and has two real and two complex roots, concyclic in the Argand diagram, then a^ + a^ai + ai - 020(4 - 2ai02a3 = 0. 12. The four roots of a^"^ + Aa^x^ + Qa^x"^ + ^a^x + aj = will be harmonic- ally related if ttffl^ + a^Ui + a^ - Uffl^ai — ^aia^a^ = 0. [Express ^23,14^31,24^12,34, where ^23,i4=(si-22) (23-a)-I-(2i-23) (jj-Zi) and 2i, 22, %, 24 are the roots of the equation, in terms of the coefficients.] COMPLEX NUMBERS 103 13. Imaginary points and straight lines. Let cu;+hi/ + c = be an equation with complex coeflBcients (which of course may be real in special cases). If we give a; any particular real or complex value, we can find the corre- sponding value of 2/. The aggregate of pairs of real or complex values of x and y which satisfy the equation is called an imaginary straight line ; the pairs of values are caUed imaginary points, and are said to lie on the line. The values of x and y are called the coordinates of the point (.r, y). When X and y are real, the point is called a real point : when a, b, c are all real (or can be made all real by division by a common factor), the line is called a real line. The points x=a+^i, y = y + di and x=a — pi, 3/ = y - Si are said to be conjugate ; and so are the lines {A + A'i)x+{B+Ei)y-vC+C'i=0, {A-A'i)x+{B-B'i)y+C-C'i=0. Verify the following assertions : — every real line contains infinitely many pairs of conjugate imaginary points ; an imaginary line in general contains one and only one real point ; an imaginary line cannot contain a pair of conjugate imaginary points : — and find the conditions (a) that the line joining two given imaginary points shovild be real, and (6) that the point of intersection of two imaginary lines should be real. 14. Prove the identities {x+y + z){x+yco3+zal)(x+yal+sa>^)=x^+y^ + ^-Sxi/z, (x+y+s) {x+ya>^+za^ {x+yiol+za>l) {x+y<4+z<4) i^+V<4+^''5) =x^+y^ + 2^— bx^yz + 5xyh^. 15. Solve the equations ^3 _ 3a.c+ (a3 + 1) = 0, x^- 5aJ^ + 5a\v + (a^ + 1) = 0. 16. If/(a;) = ao + ai^+". + afAthen {f{x)+f{wx) + ...+/ ((»" - ^x)}/n = ao + a„.r'' + a2u«^'' + • • • + O'Kn^''"; 0) being any root of ,r" = l (except x=l), and Xre the greatest multiple of n contained in t Find a similar formula for a^ + a^+„^'' + %+2»'=»^'' + •••• 17. If (l+.''^''=Po+PiX+P^^+--, n being a positive integer, then 18. Sum the series x x^ ^ x"'^ 2!w-2!"^5!n-5! 8!)i-8! «-l!' n being a miUtiple of 3. UVat^- Trip. 1899.) 19. It < is a complex number such that |i!| = l, then the point x={at-\-h)l{t-c) describes a circle as t varies, unless |c| = l, when it describes a straight line. 104 COMPLEX NUMBERS [ill 20. If t varies as in the last example then the point x=\{at-^-(blt)) in general describes an ellipse whose foci are given \yj x'^=ah, and whose axes are I a I + 1 6 1 and | a | - 1 6 |. But if | a | = | 6 1 then ic describes the finite straight line joining the points - s/{ab), J{ab). 21. Prove that if t is real and z=fi-l+^(t^-fi), then, when t^<l, zis represented by a point which lies on the circle x^+^^+a;=0. Assuming that, when t'>l, J(t^-t^) denotes the positive square root of i*-fi, discuss the motion of the point which represents z, as t diminishes from a large positive value to a large negative value. {Math. Trip. 1912.) 22. The coefficients of the transformation z= (aZ+ h)j{eZ+d) are subject to the condition ad—hc=\. Show that, if c4=0, there are t^o fixed points a, /3, i.e. points unaltered by the transformation, except when (a + c^)^=4, when there is only one fixed point a ; and that in these two cases the transforma- tion may be expressed in the forms Z — a_„Z—a 1 _ ^ , r' Show further that, if c=0, there will be one fixed point a unless a=d, and that in these two oases the transformation may be expressed in the forms z-a=K{Z-a), z=Z-^K. Finally, if a, b, e, d are further restricted to positive integral values (in- cluding zero), show that the only transformations with less than two fixed points are of the forms {llz) = (llZ)+K, z = Z+K. {Math. Trip. 1911.) 2.3. Prove that the relation z={l + Zi)j(Z+i) transforms the part of the axis of X between the points z=l and z= — l into a semicircle passing through the points Z= 1 and Z= - 1. Find all the figures that can be obtained from the originally selected part of the axis of x by successive applications of the transformation. (Math. Trip. 1912.) 24. If z=2Z+Z' then the circle \Z\ = 1 corresponds to a cardioid in the plane of z. 25. Discuss the transformation z=^{Z+(ljZ)}, showing in particular that to the circles X^+ Y^=a'' correspond the confooal ellipses iK-i)}' {K-;)f =1. 26. If (z4- 1)2 = 4/.^ then the unit circle in the z-plane corresponds to the parabola Jicoa^^& = l in the .2-plane, and the inside of the circle to the outside of the parabola. 27. Show that, by means of the transformation z = {{Z—ei)l{Z+ci)Y, the upper half of the 0-plane may be made to correspond to the interior of a certain semicircle in the .2^-plane. COMPLEX NUMBERS 105 28. If z=Z^-\, then as 2 describes the circle |2| = k, the two corre- sponding positions of Z each describe the Cassinian oval pip2=K, where pi, p2 are the distances of Z from the points -1, 1. Trace the ovals for different values of k. ' 29. Consider the relation az^ + %hzZ+hZ^-'r'igz + '2,fZ-\-c=0. Show that there are two values of Z for which the corresponding values of z are equal, and vice versa. We call these the branch points in the Z and z-planes re- spectively. Show that, if z describes an ellipse whose foci are the branch points, then so does Z. [We can, without loss of generality, take the given relation in the form z^ + 2zZ cos <o+Z^=l: the reader should satisfy himself that this is the case. The branch points in either plane are coseo <b and — cosec m. An ellipse of the form specified is given by |2-|-oosecei)|-l-|2-coseoci) \ = C, where C is a constant. This is equivalent (Ex. 9) to I z + ^(32 _ cosec^ (o) 1 4- 1 z - V(2^ - cosec2 <a) | = C. Express this in terms of Z.] 30 If z=aZ"'-i-bZ", where m, n are positive integers and a, h real, then as .^ describes the unit circle, z describes a hypo- or epi-cycloid. 31. Show that the transformation { a-\-di)Zo + h " cZu -{a- di) ' where a, b, c, d are real and a^^d^^-^bo 0, and Z^ denotes the conjugate of Z, is equivalent to an inversion with respect to the circle c (a-2 -t-y2) - 2a^ - 2rfy - 6 = 0. What is the geometrical interpretation of the transformation when a2-i-rfH6«<0? 32. The transformation \-z ^ /l-.Z'Y YVz^V'rZ) ' where c is rational and < c < 1 , transforms the circle \z\ = \ into the boundary of a circular lune of angle tr\c. CHAPTER IV LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE 50. Functions of a positive integral variable. In Chapter II we discussed the notion of a function of a real variable x, and illustrated the discussion by a large number of examples of such functions. And the reader will remember that there was one important particular with regard to which the functions which we took as illustrations differed very widely. Some were defined for all values' of cc, some for rational values only, some for integral values only, and so on. Consider, for example, the following functions : (i) x, (ii) ijx, (iii) the denominator of x, (iv) the square root of the product of the numerator and the denominator of x, (v) the largest prime factor of x, (vi) the product of ,Jx and the largest prime factor of x, (vii) the irth prime number, (viii) the height measured in inches of convict x in Dartmoor prison. Then the aggregates of values of x for which these functions are defined or, as we may say, the fields of defimxtion of the functions, consist of (i) all values of x, (ii) all •positive values of x, (iii) all rational values of x, (iv) all positive r'ational values of x, (v) aU integral values of x, (vi), (vii) all positive integral values of x, (viii) a certain number of positive integral values of x, viz., 1, 2, ..., N, where N is the total number of convicts at Dartmoor at a given moment of time*. Now let us consider a function, such as (vii) above, which is defined for all positive integral values of x and no others. This * In the last case N depends on the time, and convict x, where x has a definite value, is a different individual at different moments of time. Thus if we take different moments of time into consideration we have a simple example of a function y=F (x, t) of two variables, defined for a certain range of values of t, viz. from the time of the establishment of Dartmoor prison to the time of its abandon- ment, and for a certain number of positive integral values of x, this number varying with t. 50, 51] FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE 107 function may be regarded from two slightly different points of view. We may consider it, as has so far been our custom, as a function of the real variable x defined for some only of the values of X, viz. positive integral values, and say that for all other values of X the definition fails. Or we may leave values of x other than positive integral values entirely out of account, and regard our function as a function of the positive integral variable n, whose values are the positive integers 1,2,3,4,.... In this case we may write y = <ti (n) and regard y now as a function of n defined for all values of n. It is obvious that any function of x defined for all values of x gives rise to a function of n defined for all values of n. Thus from the function y = x" we deduce the function ?/ = n^ by merely omitting from consideration all values of x other than positive integers, and the corresponding values of y. On the other hand from any function of n we can deduce any number of functions of X by merely assigning values to y, corresponding to values of x other than positive integral values, in any way we please. 51. Interpolation. The problem of determining a function of x which shall assume, for all positive integral values of x, values agreeing with those of a given function of n, is of extreme importance in higher mathematics. It is called the problem of functional interpolation. Were the problem however merely that of finding smne function of x to fulfil the condition stated, it would of course present no difficulty whatever. We could, as explained above, simply fill in the missing values as we pleased : we might indeed simply regard the given values of the function of n as all the values of the function of x and say that the definition of the latter function failed for all other values of x. But such purely theoretical solutions are obviously not what is usually wanted. What is usually wanted is some formula involving x (of as simple a kind as possible) which assumes the given values for ^=1, 2, .... In some cases, especially when the function of n is itself defined by a formula, there is an obvious solution. If for example y = 4> ('Oi whei-e (f) (n) is a function of «, such as n^ or cos nn, which would have a meaning even were n not a positive integer, we naturally take o\n' function of x to be j/ = ^ (x). But even in this very simple case it is easy to write down other almost equally obvious solutions of the problem. For example 2/ = 4> (.J;) + sina-7r assumes the value (p (/i) for x = n, since sin ra7r = 0. 108 LIMITS OF FUNCTIONS OF A [iV In other oases (/) (n) may be defined by a formula, such as (-rl)", which ceases to define for some values of a? (as here in the case of fractional values of a; with even denominators, or irrational values). But it may be possible to transform the formula in such a way that it does define for all values of x. In this case, for example, ( — l)" = C0S9Hr, if n is an integer, and the problem of interpolation is solved by the function In other cases <j}(x) may be defined for some values of .a;. other than positive integers, but not for all. Thus from y=ra" we are led to y=^. This expression has a meaning for some only of the remaining values of x. If for simplicity we confine ourselves to positive values of a;, then /x^ has a meaning for all rational values of .^■, in virtue of the definitions of fractional powers adopted in elementary algebra. But when x is irrational a?' has (so far as we are in a position to say at the present moment) no meaning at all. Thus in this case the problem of interpolation at once leads us to consider the question of extending our definitions in such a way that os^ shall have a meaning even when x is irrational. We shall see later on how the desired extension may be efiected. Again, consider the case in which y=\ . 2 ... n'=n ! . In this case there is no obvious formula in x which reduces to n\ for a;=m, as X ! means nothing for values of x other than the positive integers. This is a case in which attempts to solve the problem of interpolation have led to important advances in mathematics. For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides. 52. Finite and infinite classes. Before we proceed further it is necessary to make a few remarks about certain ideas of an abstract and logical nature which are of constant occurrence in Pure Mathematics. In the first place, the reader is probably familiar with the notion of a class. It is unnecessary to discuss here any logical difficulties which may be involved in the notion of a 'class': roughly speaking we may say that a class is the aggregate or collection of all the entities or objects which possess a certain property, simple or complex. Thus we have the class of British subjects, or members of Parliament, or positive integers, or real numbers. 51-53] POSITIVE INTEGRAL VAEIABLE 109 Moreover, the reader has probably an idea of what is meant by a finite or infinite class. Thus the class of British subjects is a finite class: the aggregate of all British subjects, past, present, and future, has a finite number n, though of course we cannot tell at present the actual value of n. The class of present British subjects, on the other hand, has a number n which could be ascertained by counting, were the methods of the census effective enough. On the other hand the class of positive integers is not finite but infinite. This may be expressed more precisely as follows. If n is any positive integer, such as 1000, 1,000,000 or any number we like to think of, then there are more than n positive integers. Thus, if the number we think of is 1,000,000, there are obviously at least 1,000,001 positive integers. Similarly the class of rational numbers, or of real numbers, is infinite. It is convenient to express this by saying that there are an infinite number of positive integers, or rational numbers, or real numbers. But the reader must be careful always to remember that by saying this we mean simply that the class in question has not a finite number of members such as 1000 or 1,000,000. 53. Properties possessed by a function of n for large values of n. We may now return to the ' functions of n ' which we were discussing in §§ 50 — 51. They have many points of difference from the functions of x which we discussed in Chap. II. But there is one fundamental characteristic which the two classes of func- tions have in common : the values of the variable for which they are defined form an infinite class. It is this fact which forms the basis of all the considerations which follow and which, as we shall see in the next chapter, apply, mutatis mutandis, to functions of x as well. Suppose that ^ (n) is any function of n, and that P is any property which <^ (n) may or may not have, such as that of being a positive integer or of being greater than 1. Consider, for each of the values n=\, 2, 3, ..., whether ^(?i) has the property P or not. Then there are three possibilities: — (a) (f> (n) may have the property P for all values of n, or for all values of n except a finite number N of such values : 110 LIMITS OF FUNCTIONS OF A [iV (6) (j) (n) may have the property for no values of n, or only for a finite number N of such values : (c) neither (a) nor (b) may be true. If (b) is true, the values of n for which (/> (n) has the property form a finite class. If (a) is true, the values of n for which (f> (n) has not the property form a finite class. In the third case neither class is finite. Let us consider some particular cases. (1) Let (j) {n)=n, and let P be the property of being a positive integer. Then <^ (w) has the property P for all values of n. If on the other hand P denotes the property of being a positive integer greater than or equal to 1000, then (f) (n) has the property for all values of n except a finite number of values of ?i, viz. 1, 2, 3, ..., 999. In either of these cases (a) is true. (2) If (^ {n) = n, and P is the property of being less than 1000, then (b) is true. (3) If (j) (n) = n, and P is the property of being odd, then (c) is true. For <j) {n) is odd if n is odd and even if »!. is even, and both the odd and the even values of n form an infinite class. Example. Consider, in each of the following cases, whether (a), (6), or (c) is true : (i) (f) {n) = n, P being the property of being a perfect square, (ii) <j){n)=pa, where pn denotes the ?ith prime number, P being the property of being odd, (iii) (j) W=f m P being the property of being even, (iv) <f> {n)=pa, Pbeing the property i^ (n)>7i, (v) <j){n) = l-{-l)^{ljn), P being the property ^(n)<l, (vi) <^(?i) = 1-(-1)»(1/b), P being the property cj)(n)<2, (vii) ^(M) = 1000{l + (-l)''}/n, P being the property 0(»)<1, (viii) (j) (n) = 1/m, P being the property <^ (n) < -001, (ix) (f,(n) = (- 1)"/to, P being the property | <^ (m) | < -001, (x) ^(m) = 10000/«, or (-l)''10000/», P being either of the properties 4>{n)<-001 or |0(«)|<-OO1, (xi) <ji{n)=(n-l)l(n+\), P being the property 1-(^(to)<-0001. 54. Let us now suppose that ^ (?i) and P are such that the assertion (a) is true, i.e. that (/> (w) has the property P, if not for all values of n, at any rate for all values of n except a finite number N of such values. "We may denote these exceptional values by n„ 112, ...,Vif. 53, 54] POSITIVE INTEGRAL VARIABLE 111 There is of course no reason why these N values should be the first N values 1, 2, ..., If, though, as the preceding examples show, this is frequently the case in practice. But whether this is so or not we know that ^ (n) has the property P if ji > nj^. Thus the nth prime is odd it n>2,n = 2 being the only exception to the statement; and 1/n < -001 if n > 1000, the first 1000 values of n being the exceptions ; and 1000{l + (-l)"}/n<l if Ji> 2000, the exceptional values being 2, 4, 6, .... 2000. That is to say, in each of these cases the property is possessed for all values of n from a definite value onwards, "We shall frequently express this by saying that (n) has the property for large, or very large, or all sufficiently large values oLn. Thus when we say that ^ (n) has the property P (which will as a rule be a property expressed by some relation of inequality) for large values of n, what we mean is that we can determine some definite number, n^ say, such that (^ (n) has the property for all values of n greater than or equal to n^. This number n,,, in the examples considered above, may be taken to be any number greater than n^, the greatest of the exceptional numbers : it is most natural to take it to be n^^ + l. Thus we may say that ' all large primes are odd', or that ' 1/n is less than "001 for large values of n '. And the reader must make himself familiar with the use of the word large in statements of this kind. Large is in fact a word which, standing by itself, has no more absolute meaning in mathematics than in the language of common life. It is a truism that in common life a number which iS' large in one connection is small in another ; 6 goals is a large score in a football match, but 6 runs is not a large score in a cricket match; and 400 runs is a large score, but £400 is not a large income : and so of course in mathematics large generally means large enough, and what is large enough for one purpose may not be large enough for another. We know now what is meant by the assertion ' <p (n) has the property P for large values of n '. It is with assertions of this kind that we shall be concerned throughout this chapter. 112 LIMITS OF FUNCTIONS OF A [iV 55. The phrase 're tends to infinity'. There is a some- what different way of looking at the matter which it is natural to adopt. Suppose that n assumes successively the values 1, 2, 3, .... The word ' successively ' naturally suggests succession in time, and we may suppose n, if we like, to assume these values at successive moments of time (e.g. at the beginnings of successive seconds). Then as the seconds pass n gets larger and larger and there is no limit to the extent of its increase. However large a number we may think of (e.g. 2147483647), a time will come when n has become larger than this number. It is convenient to have a short phrase to express this unending growth of n, and we shall say that n tends to infinity, or w -* oo , this last S3n2ibol being usually employed as an abbreviation for 'infinity'. The phrase 'tends to' like the word 'successively' naturally suggests the idea of change in time, and it is convenient to think of the variation of Ji as accomplished in time in the manner described above. This however is a mere matter of con- venience. The variable n is a purely logical entity which has in itself nothing to do with time. The reader cannot too strongly impress upon himself that when we say that n ' tends to oo ' we mean simply that n is supposed to assume a series of values which increase continually and without limit, There is no number 'infinity': such an equation as Jl = 00 is as it stands absolutely meaningless : n cannot be equal to oo , because ' equal to oo ' means nothing. So far in fact the symbol 00 means nothing at all except in the one phrase 'tends to oo', the meaning of which we have explained above. Later on we shall learn how to attach a meaning to other phrases involving the symbol co , but the reader will always have to bear in mind (1) that 00 by itself means nothing, although phrases con- taining it sometimes mean something, (2) that in every case in which a phrase containing the symbol oo means something it will do so simply because we have previously attached a meaning to this particular phrase by means of a special definition. 55, 56] POSITIVE INTEGRAL VARIABLE 113 Now it is clear that if (f> (n) has the property P for large values of n, and if n 'tends to oo ', in the sense which we have just explained, then n will ultimately assume values large enough to ensure that <p(n) has the property P. And so another way of putting the question ' what properties has <f) (n) for sufficiently large values of w ? ' is ' how does (j) (n) behave as n tends to oo ? ' 56. The behaviour of a function of n as n tends to infinity. We shall now proceed, in the light of the remarks made in the preceding sections, to consider the meaning of some kinds of statements which are perpetually occurring in higher mathematics. Let us consider, for example, the two following statements : (a) 1/n is small for large valves of n, (b) 1 — (l/n) is nearly equal to 1 for large values of n. Obvious as they may seem, there is a good deal ia them which will repay the reader's attention. Let us take (a) first, as being slightly the simpler. We have already considered the statement ' l/n is less than '01 for large values of n'. This, we saw, means that the inequality l/w<'01 is true for all values of n greater than some definite value, in fact greater than 100. Similarly it is true that ' \jn is less than '0001 for large values of n' : in fact 1/w. < '0001 if n > 10000. And instead of -01 or -0001 we might take -000001 or ■00000001, or indeed any positive number we like. It is obviously convenient to have some way of expressing the fact that any such statement as 'Ijn is less than '01 for large values of n' is true, when we substitute for "01 any smaller number, such as -0001 or -000001 or any other number we care to choose. And clearly we can do this by saying that ' however small h may he (provided of course it is positive), then Ijn < B for sufficiently large values of n '. That this is true is obvious. For ^ l/n<B if n> 1/S, so that our ' sufficiently large' values of n need only all be greater than 1/S. The assertion is however a complex one, in that it really stands for the whole class of assertions which vre obtain by giving to S special values such as -01. And of course the smaller 8 is, and the larger 1/8, the larger must be the least of the ' sufficiently large ' values of n : values which are sufficiently large when 8 has one value are inadequate when it has a smaller. The last statement italicised is what is really meant by the statement (a), that l/n is small when n is large. Similarly 8 114 LIMITS OF FUNCTIONS OF A [iV (b) really means "if ^(»i)=l — (1/n), then the statement 'l — <f>{n)<S for sufficiently large values of n' is true whatever positive value (such as 01 or '0001) we attribute to 8 ". That the statement (b) is true is obvious from the fact that 1 — ^ (n) = Ijn. There is another way in which it is common to state the facts expressed by the assertions (a) and (b). This is suggested at once by § 65. Instead of saying ' 1/n is small for large values of n ' we say ' Ijn tends to as n tends to oo ', Similarly we say that ' 1 — (1/n) tends to 1 as m tends to oo ' : and these statements are to be regarded as precisely equivalent to (a) and (b). Thus the statements ' 1/ji is small when n is large ', ' 1/n tends to as w tends to oo ', are equivalent to one another and to the more formal statement 'if 8 is any positive number, however small, then 1/n < S for sufficiently large values of n ', or to the still more formal statement ' if S IS any positive number, however small, then we can find a number «o such that 1/n < S for all values of n greater than or equal to n^'. The number no which occurs in the last statement is of course a function of S. We shall sometimes emphasize this fact by writing no in the form no (S). The reader should imagine himself confronted by an opponent who questions the truth of the statement. He would name a series of numbers growing smaller and smaller. He might begin with '001. The reader would reply that l/?i<'001 as soon as »i>1000. The opponent would be bound to admit this, but would try again with some smaller number, such as -0000001. The reader would reply that 1/to< 0000001 as soon as m> 10000000: and so on. In this simple case it is evident that the reader would always have the better of the argument. We shall now introduce, yet another way of expressing this property of the function 1/n. We shall say that ' the limit of 1/n as n tends to oo is ', a statement which we may express symboli- cally in the form lim - = 0, 56, 57] POSITIVE INTEGRAL VARIABLE 115 or simply lim (1/n) = 0. We shall also sometimes write as M -* 00 ', which may be read ' 1/n tends to as w tends to oo ' ; or simply 'Ijn^O '. In the same way we shall write lim fl--) = l, lirafl-i) = l, orl-(l/n)-*l. 57. Now let us consider a different example : let <f) (n) = n^. Then ' n^ is large when n is large ', This statement is equivalent to the more formal statements ' if A is any positive number, however large, then n'> ^ for sufficiently large values of n ', ' we can find a number r?o (^) such that n^ > A for all values of n greater than or equal to Wq (A) '. And it is natural in this case to say that 'n" tends to oo as w tends to oo ', or ' n^ tends to oo with n ', and to write Finally consider the function (f>(n) = — m?. In this case <^ (n) is large, but negative, when n is large, and we naturally say that ' — n^ tends to — x as n. tends to oo ' and write — n^ -»- — 00 . And the use of the sjrmbol — oo in this sense suggests that it will sometimes be convenient to write n^ ^ + oo for n^^co and generally to use + oo instead of oo , in order to secure greater uniformity of notation. But we must once more repeat that in all these statements the symbols oo , + oo , — oo mean nothing whatever by themselves, and only acquire a meaning when they occur in certain special connections in virtue of the explanations which we have just given. ^ 8—2 116 LIMITS OF FUNCTIONS OF A [iV 58. Definition of a limit. After the discussion which precedes the reader should be in a position to appreciate the general notion of a limit. Roughly we may say that ^ (w) tends to a limit I as n tends to co if cp (n) is nearly equal to I when n is large. But although the meaning of this statement should be clear enough after the preceding explanations, it is not, as it stands, precise enough to serve as a strict mathematical definition. It is, in fact, equivalent to a whole class of statements of the type 'for sufficiently large values of n, <f>{n) differs from I by less than S '. This statement has to be true for 8 = "01 or '0001 or any positive number ; and for any such value of B it has to be true for any value of n after a certain definite value Mq (S), though the smaller 8 is, the larger, as a rule, will be this value n^ (8). We accordingly frame the following formal definition : Definition I. The function ^ (n) is said to tend to the Ihnit I as n tends toco,if, however small he the positive number S, cf) (n) differs from I by less than S for sufficiently large values of n; that is to say if, however small he the positive number S, we can determine a number n„ (8) corresponding to S, such that </> (n) differs from I by less than 8 for all values of n greater than or equal to n,, (S). It is usual to denote the difference between <p (n) and I, taken positively, by | ^ (n) — Z |. It is equal to <f) (n) — I or to I — (j) (n), whichever is positive, and agrees with the definition of the modulus of (f) (n) — I, as given in Chap. Ill, though at present we are only considering real values, positive or negative. With this notation the definition may be stated more shortly as follows : ' if, given any positive number, S, however small, we can find «, (8) so that | <j} (n) — I \ < S when n ^n^ (8), then we say that ^ (n) tends to the limit I as n tends to oo , and write lim <j){n) = l '. Sometimes we may omit the 'w-»-oo ' ; and sometimes it is convenient, for brevity, to write (%)-»■?. The reader will find it instructive to work out, in a few simple cases, the explicit expression of % as a function oS^ Thus if ^ (x) = \jn then l = Q, and - the condition redvices to \jn<b for mgra,,, which is satisfied if »io=l + [l/S]f. ^ There is one and only one case in which the same Wq will do for all values of 8:- ^ * Here and henceforward we shall use {x\ in the sense of Chap. II, i.e. as the greatest integer not greater than x. 68-60] POSITIVE INTEGEAL VARIABLE 117 If, from a certaii\ value N of m onwards, <f) (n) is constant, say equal to C, then it is evident that (») — C= for n^ff, so that the inequality \(ji{n) — C\ < 8 is satisfied for n>N' and all positive values of S. And if \(f>(n) — l\ <8 for n>N and all positive values of 8, then it is evident that (f> {n)=l when m2iV, so that {n) is constant for all such values of n. 59. The definition of a limit may be illustrated geometrically as follows. The graph of jj (n) consists of a number of points corresponding to the values n = l, 2, 3, .... Draw the line y = l, and the parallel lines y = 1 — B, y = l + B at distance B from it. Then lim <f) (n) = I, Fig. 27. if, when once these lines have been drawn, no matter how close they may be together, we can always draw a line x = n^, as in the figure, in such a way that the point of the graph on this line, and all points to the right of it, lie between them. We shall find this geometrical way of looking at our definition particularly useful when we come to deal with functions defined for all values of a real variable and not merely for positive integral values. 60. So much for functions of n which tend to a limit as n tends to X . We must now frame corresponding definitions for functions which, like the functions n^ or -n\ tend to positive or negative infinity. The reader should by now find no difficulty in appreciating the point of Definition II. The function (j>(n) is said to tend to +x {positive infinity) with n, if, when any number A, however large, is assigned, we can determine n„ (A) so that <j>{n)>A when n S n, (A); 118 LIMITS OF FUNCTIONS OF A [iV that is to say if, however large A may be, </> (n) > A for sufficiently large values ofn. Another, less precise, form of statement is 'if we can make <f) (n) as large as we please by sufficiently increasing n '. This is open to the objection that it obscures a fundamental point, viz. that (^ («) must be greater than A for all values of n such that w S «o (A), and not merely for some such values. But there is no harm in using this form of expression if we are clear what it means. When <^ (n) tends to + oo we write <^ (m) -* + 00 . We may leave it to the reader to frame the corresponding definition for functions which tend to negative infinity. 61. Some points concerning the definitions. The reader should be careful to observe the following points. (1) We may obviously alter the values of <^(n) for any finite number of values of n, in any way we please, without in the least affecting the behaviour of ^ (n) as n tends to oo . For example Ijn tends to as ?i tends to co . We may deduce any number of new functions from Ijn by altering a finite number of its values. For instance we may consider the function ^ (n) which is equal to 3 for n=l, 2, 7, 11, 101, 107, 109, 237 and equal to 1/n for all other values of n. For this function, just as for the original function 1/n, lim <f> (w) = 0. Similarly, for the function (f, {n) which is equal to 3 if w = 1, 2, 7, 11, 101, 107, 109, 237, and to n^ otherwise, it is true that (/> (n) -* + oo . (2) On the other hand we cannot as a rule alter an infinite number of the values of <^ (n) without affecting fundamentally its behaviour as n tends to oo . If for example we altered the function Ijn by changing its value to 1 whenever n is a multiple of 100, it would no longer be true that lim <f) (n) = 0. So long as a finite number of values only were affected we could always choose the number n^ of the definition so as to be greater than the greatest of the values of n for which <^ (n) was altered. In the examples above, for instance, we could always take n^ > 237, and indeed we should be compelled to do so as soon as oiir imaginary opponent 60, 61] POSITIVE INTKGKAL VARIABLE 119 of § 56 had assigned a value of 8 as small as 3 (in the first example) or a value of A as great as 3 (in the second). But now however large n^ may he there will be greater values of n for which (n) has been altered. (3) In applying the test of Definition I it is of course absolutely essential that we should have \^{n)—l\<8 not merely when n = Wo but when n S «„, i.e. for n^ and for all larger values ofn. It is obvious, for example, that, if ^ (n) is the function last considered, then given S we can choose n^ so that | </> (n) | < S when w = Ho : we have only to choose a sufficiently large value of n which is not a multiple of 100. But, when n^ is thus chosen, it is not true that | ^ (n) | < S when ?i g n„ : all the multiples of 100 which are greater than ??o are exceptions to this statement. (4) If <^(n) is always greater than I, we can replace \^{n) — l\ by ^(n) — I. Thus the test whether 1/n tends to the limit as n tends to oo is simply whether l/n<8 when n = no. If however </> (n) = (— l)"/n, then I is again 0, but (?i) — lis some- times positive and sometimes negative. In such a case we must state the condition in the form \(}}{n) — l\<S, for example, in this particular case, in the form | ^ (n) | < S. (5) The limit I may itself be one of the actual values of (f) (n). Thus if <j>{n) = for all values of n, it is obvious that lim (j) (n) = 0. Again, if we had, in (2) and (3) above, altered the value of the function, when n is a multiple of 100, to instead of to 1, we should have obtained a function (f> (n) which is equal to when n is a multiple of 100 and to 1/n otherwise. The limit of this function as n tends to x is still obviously zero. This limit is itself the value of the function for an infinite number of values of n, viz. all multiples of 100. On the other hand the limit itself need not (and in general will not) be the value of the function for any value of n. This is sufficiently obvious in the case of <}> (n) = 1/n. The limit is zero ; but the function is never equal to zero for any value of n. The reader cannot impress these facts too strongly on his mind. A limit is not a value of the function : it is something quite distinct from these values, though it is defined by its relations 120 LIMITS OF FUNCTIONS OF A [iV to them and may possibly be equal to some of them. For the functions <p{n) = 0, I, the limit is equal to all the values of (n) : for <j>{n) = l/n, (-irin, l + (l/n), 1 + {(- 1)"/^ it is not equal to awy value of <j> (n) : for <j} (n) = (sin \mr)ln, 1 + {(sin \nv)jn} (whose limits as n tends to oo are easily seen to be and 1, since sin \nnT is never numerically greater than 1) the limit is equal to the value which ^ (n) assumes for all even values of n, but the values assumed for odd values of n are all different from the limit and from one another. (6) A function may be always numerically very large when n is very large without tending either to + oo or to — oo . A sufficient illustration of this is given by (^ (w) =(—!)''«, A function can only tend to + oo or to — x if, after a certain value of n, it maintains a constant sign. Examples XXIII. Consider the behaviour of the followiug functions of .r as m tends to qo : 1. (^ (m)=?i*, where ^ is a positive or negative integer or rational fraction. If k is positive, then # tends to + x with n. If h is negative, then lim#=0. \ik=0, then #=1 for all values of ?i. Hence limH*=l. The reader will find it instructive, even in so simple a case as this, to write down a formal proof that the conditions of our definitions are satisfied. Take for instance the case of h>Q. Let A be any assigned number, however large. We wish to choose Mq so that #>A when JigjiQ. We have in fact only to take for Kq any number greater than 4/A. If e.g. ^ = 4, then m*> 10000 when m>ll, 5i*>100000000 when mglOl, and so on. 2. (t>{n)=p„, where pn is the rath prime number. If there were only a finite number of primes then <f) (n) would be defined only for a finite number of values of n. There are however, as was first shown by Euclid, infinitely many primes. Euclid's proof is as follows. If there are only a finite number of primes, let them be 1, 2, 3, 5, 7, 11, ... JV. Consider the number ! + (! . 2 . 3 . 5. 7 . 11 ... JV). This number is evidently not divisible by any of 2, 3, 5, ... iV, since the remainder when it is divided by any of these numbers is 1. It is therefore not divisible by any prime save 1, and is therefore itself prime, which is contrary to our hypothesis. It is moreover obvious that {n)>n for all values of n (save-« = l, 2, 3). Hence 0(ra)^-+QO. 61, 62] POSITIVE INTEGRAL VARIABLE 121 3. Let (p (n) be the number of primes less than n. Here again (n) -* + a> . 4. (n) = [an'\, where a is any positive number. Here 0(w)=O (Og»<l/a), 4,(11) = ] {lla^n<2la), and so on; and 0(«)-»-+qo. 5. If (n) = 1000000/m, then lim (n) = : and if ^ (n) ■-= ?i/1000000, then !//■ («)-*■+ 00 . These conclusions are in no way affected by the fact that at first <f> (n) is much larger than \jf (m), being in fact larger until m = 1000000. 6. 0(») = l/{m-(-l)»}, »-(-l)» «{!-(- 1)"}. The first function tends to 0, the second to + oo , the third does not tend either to a limit or to + oo . 7. (m) = (sin 91^77 )/m, where fl is any real number. Here | (»} |<l/ra, since |sinm57r|gl, and lim(j){n)=0. 8. (f) (n) = (sin n6ir)jjn, (a oos^ n6 + b sin^ nd)ln, where a and 6 are any real numbers. 9. 0(?i) = sin7ifl7r. If 6 is integral then cj){n) = for all values of n, and therefore lim (ra) =0. Next let 6 be rational, e.g. 6=plq, where p and q are positive integers. Let n = aq + b where a is the quotient and b the remainder when n is divided by q. Then sin {np-n-jq) = ( - 1)"'' sin {bpnjq). Suppose, for example, p even ; then, as n increases from Otoq-l,cj) {n) takes the values 0, sin {pnlq), sin {^pnlq), ... sin {{q - 1) pwjq). When n increases from q to 2j - 1 these values are repeated ; and so also as n goes from 2q to Zq — 1, Zq to 42" — 1, and so on. Thus the values of (?i) form a perpetual cyclic repetition of a finite series of different values. It is e^-ident that when this is the case (?i) cannot tend to a limit, nor to + oo , nor to - 00 , as » tends to infinity. The case in which 6 is irrational is a little more difiicult. It is discussed in the next set of examples. 62. Oscillating Functions. Definition. When ^ {n) does not tend to a limit, nor to + cc , nor to -co , as n tends to <x> ,we say that (f) (n) oscillates as n tends to oo . A function <j>{n) certainly oscillates if its values form, as in the case considered in the last example above, a continual repetition of a cycle of values. But of course it may oscillate without possessing this peculiarity. Oscillation is defined in a purely negative manner : a function oscillates when it does not do certain other things. 122 LIMITS OF FUNCTIONS OF A [iV The simplest example of an oscillatory function is given by ,/,(«) = (-!)», which is equal to + 1 when n is even and to — 1 when n is odd. In this case the values recur cyclically. But consider <^(n)=(-l)»+(lM the values of which are -1 + 1, l + (l/2), -l+(l/3), l + (l/4), -l + (l/5), .... When n is large every value is nearly equal to +1 or — 1, and obviously j> (n) does not tend to a limit or to + oo or to — oo , and therefore it oscillates: but the values do not recur. It is to be observed that in this case every value of </> (n) is numerically less than or equal to 3/2. Similarly ^(TO) = (-l)»100 + (1000/ft) oscillates. When n is large, every value is nearly equal to 100 or to —100. The numerically greatest value is 900 (for n = \). But now consider ^ (w) = (— 1)" n, the values of which are — 1, 2, — 3, 4, — 5, .... This function oscillates, for it does not tend to a limit, nor to + oo , nor to — oo . And in this case we cannot assign any limit beyond which the numerical value of the terms does not rise. The distinction between these two examples suggests a further definition. Definition. If (f> (n) oscillates as n tends to oo , then <^ (n) will be said to oscillate finitely or infinitely according as it is or is not possible to assign a number K such that all the values of (p (n) are numerically less than K, i.e. | 0(«) | < K for all values of n. These definitions, as well as those of §§ 58 and 60, are further illustrated in the following examples. Examples XXIV. Consider the behaviour as n tends to oo of the following functions: 1. (-1)"-, 5 + 3(-l)'', (1000000/?i) + (-l)", 1000000 (-l)'' + (l/»). 2. {-lyn, 1000000 + (-l)"«. 3. 1000000 -TO, ( -1)" (1000000 -m). 4. m {1 + ( - 1)"}. In this case the values of (^ {n) are 0, 4, 0, 8, 0, 12, 0, 16, .... The odd terms are all zero and the even terms tend to + oo : <^ (») oscillates infinitely. 62] POSITIVE INTEGRAL VARIABLE 123 5. ji2 + ( — 1)"2k. The second term oscillates infinitely, but the first is very much larger than the second when n is large. In fact (jj (n) gm^ _ 3,1 and n^-2n = {n-iy-l is greater than any assigned value A if n>l + J(A+l). Thus (»)-»-+ CO. It should be observed that in this case <^(2^+l) is always less than <j> {2k), so that the function progresses to infinity by a con- tinual series of steps forwards and backwards. It does not however 'oscillate' according to our definition of the term. 6. n^{l + {-l)^}, i-l)''nHn, n^ + {-l)''n^ 7. sinm^jT. We have already seen (Exs. xxiii. 9) that 0(?i) oscillates finitely when 6 is rational, unless 6 is an integer, when <f> {n) = 0, (j) (»)-•■ 0. The case in which is irrational is a little more difficult. But it is not difficult to see that <^ (n) still oscillates finitely. We can without loss of generality suppose 0<5<1. In the first place |<^(to)|<1. Hence <f>(n) must oscillate finitely or tend to a limit. We shall consider whether the second alternative is really possible. Let us suppose that lim smndn = l. Then, however small 8 may be, we can choose n^ so that sin ndn- lies between 1-8 and l+S for all values of n greater than or equal to ?Iq. Hence sin (ji+l) ^TT-sinm^TT is numerically less than 28 for all such values of n, and so | sin^^Tr cos (w-|-J)0;r |<S. Hence cos {n + ^) dir =coa ndjr cos^Att — sinm^Tr sin^^^ir must be numerically less than 8/ 1 sin ^ 6n \. Similarly cos (n — i)6T!- = cos ndn cos ^dn + sin ndrr sin ^ Stt must be numerically less than 8/ 1 sin ^dir \; and so each of cos nSir cos ^ 6n, sin nBn sin \ 6it must be numerically less than 8/ 1 sin \ 6n \. That is to say, cos n6iT cos \ Btt is very small if n is large, and this can only be the case if cos jiBtt is very small. Similarly sin ndn must be very small; so that I must be zero. But it is impossible that cos iiStt and sin ndir can both be very small, as the sum of their squares. is unity. Thus the hypothesis that sin n6n tends to a limit I is impossible, and therefore sin ndn oscillates as n tends to 00 . The reader should consider with particular care the argument ' cos nOir cos I dir is very small, and this can only be the case if cos n6ir is very small'. Why, he may ask, should it not be the other factor cos^^w which is 'very small'? The answer is to be found, of course, in the meaning of the phrase ' very small ' as used in this connection. When we say ' (j) (?i) is very small' for large values of n, we mean that wo can choose «o so that (^ (j?,) is numerically smaller than a»y assigned number, if n is sufficiently large. Such an assertion is palpably absurd when made of a Jixed number such as cosi^TT, which is not zero. Prove similarly that cos Jidv oscillates finitely, unless 6 is an even integer. 8. siandTT+illn), sinm^Tr + l, sin»^7r+M, ( - 1)" sin ?i,<9n-. 9. a cos n^JT + J sin m^TT, sin^w^ir, a cos^ m^n- + 6 sin^ n^Tr. 124- LIMITS OF FUNCTIONS OF A [iV 10. a + &re + ( - 1)" (c + dn)+e cos n6ir +/.sin ndir. 11. ?i sin ndir. If n is integral, then </> («)=0, (ra)-»-0. If 5 is rational but not integral, or irrational, then (^ {n) oscillates infinitely. 12. n{acos^n6n + hs\rfin6n). In this case 0(w) tends to +qo if a and h are both positive, but to — oo if both are negative. Consider the special cases in vrhich a = 0, 6>0, or a>0, 6=0, or a=0, 6 = 0. If u and 6 have opposite signs <j> («) generally oscillates infinitely. Consider any excep- tional cases. • 13. sm{n^6'K). If ^ is integral, then (^(m)-»-0. Otherwise ^ (tj) oscillates finitely, as may be shown by arguments similar to though more complex than those used in Exs. xxill. 9 and xxiv. 7*. 14. sin {n ! Btt). If 6 has a rational value pjq, then n\ 6 is, certainly integral for all values of n greater than or equal to q. Hence (n) -»-0. The case iu which 6 is irrational cannot be dealt with without the aid of considera- tions of a much more difficult character. 15. cos {n ! dv), a cos^ {n ! 5rr) -1-6 sin^ (ra ! dn), where 6 is rational. 16. an-\hnl {-\Y{an-[hn-\). IV. [V»], (-!)"[>/»], V»J-[V»]. 18. The smallest p-ime factor of n. When re is a prime, 0(m)=ji. - When n is even, [n) = 2. Thus <^ (w) oscillates infinitely. 19. The largest prime factor of n. 20. The number of days in the year n a.b. Examples XXV. 1. If <^(»)-*-hoo and ^{n)>4>{n) for all values of n, then t|/- (ra) -»- 4- oo . 2. If <^ (m)-*0, and I a/, (m) I g I («) | for all values of n, then ^r {n)^0. 3. If lim I (^ (to) I = 0, then lim ^ (to) = 0. 4. If (to) tends to a limit or oscillates finitely, and | V' (w) | s| (to) | when to Stoo, then \/f (to) tends to a limit or oscillates finitely. 5. If (to) tends to -f co , or to - oo , or oscillates infinitely, and l^(»)IS|0(™)| when TO>no, then ^{n) tends to -f oo or to - oo or oscillates infinitely. 6. ' If (to) oscillates and, however great be to,,, we can find values of n greater than TOj for which i^ (to) > (?i), and values of to greater than «„ for which i|^ (to) < (»), then -^ (to) oscillates '. Is this true ? If not give an example to the contrary. 7. If {n)^l as TO^oo , then also {n+p)^l, p being any fixed integer. [This follows at once from' the definition. Similarly we see that if (to) tends to -J- CO or - 00 or oscillates so also does {n-\-p).] 8. The same conclusions hold (except in the case of oscillation) iip varies with TO but is always numerically less than a fixed positive integer N ; or if p varies with to in any way, so long as it is always positive. * See Bromwieh's Infinite Series, p. 485. 62, 63] POSITIVE INTEGRAL VARIABLE 125 9. Determine the least value of »„ for which it is true that (a) m2+2?i>999999 (»i>«o)> (&) JiH2?i> 1000000 («>?io). 10. Determine the least value of n^ for which it is true that (a) « + (-l)">1000 {n>no), (b) ?i + (-l)''> 1000000 (»i>mo). 11. Determine the least value of Mq for which it is true that (a) n^+2n>A {n>n^\ (6) m + (-l)''>A (/i>n„), A being any positive number. [(«) «o=[v/(A + l)]: {b) •n(,= l + [A] or 2+[a], according as [A] is odd or even, i.e. «„=l + [A] + ni + (-l/'^'}-] .12. Determine the least value of n^ such that (a) ?i/(?i2+l)<-0001, (6) (l/ra) + {(-l)»/»2}< -00001, wheuM^jEj. [Let us take the latter case. In the first place (l/m) + {(-l)»/m2}<(„ + l)/M2, and it is easy to see that the least value of n^, such that (m+ l)/m2< -000001 when m^Ho, is 1000002. But the inequality given is satisfied by n= 1000001, and this is the value of m,) required.] 63. Some general theorems with regard to limits. A. The beha-viour of the sum of two functions whose behaviour is known. Theorem 1. If ^ (n) and yjr (n) tend to limits a, b, then (n) + -\|r (?(,) tends to the limit a + b. This is almost obvious*. The argument which the reader will * There is a certain ambiguity in this phrase which the reader will do well to notice. When one says ' such and such a theorem is almost obvious ' one may mean one or other of two things. One may mean ' it is diCBcult to doubt the truth of the theorem', ' the theorem is such as common-sense instinctively accepts', as it accepts, for example, the truth of the propositions ' 2 + 2 = 4 ' or ' the base-angles of an isosceles triangle are equal'. That a theorem is 'obvious ' in this sense does not prove that it is true, since the most confident of the intuitive judgments of common sense are often found to be mistaken; and even if the theorem is true, the fact that it is also ' obvious ' is no reason for not proving it, if a proof can be found. The object of mathematics is to prove that certain premises imply certain conclusions; and the fact that the conclusions may be as 'obvious' as the premises never detracts from the necessity, and often not even from the interest of the proof. But sometimes (as for example here) we mean by ' this is almost obvious ' something quite different from this. We mean ' a moment's reflection should not only convince the reader of the truth of what is stated, but should also suggest to him the general lines of a rigorous proof. And often, when a statement is ' obvious ' in this sense, one may well omit the proof, not because the proof is in any sense unnecessary, but because it is a waste of time and space to state in detail what the reader can easily supply for himself. 126 LIMITS OF FUNCTIONS OF A [iV at once form in his mind is roughly this : ' when n is large, <f) (n) is nearly equal to a and yjr (n) to b, and therefore their sum is nearly equal to a + 6 '. It is well to state the argument quite formally, however. Let S be any assigned positive number (e.g. -001, '0000001, ...). We require to show that a number Kq can be found such that |^(«) + -«|r(w)-a-fe|<S (1), when n S rio. Now by a proposition proved in Chap. Ill (more / generally indeed than we need here) the modulus of the sum oi^/J two numbers is less than or equal to the sum of their moduli. Thus I <}) (n) + yjr (n) -a-&ls|<^(n)-a| + |i//- (n) -b\. It follows that the desired condition will certainly be satisfied if Wj can be so chosen that \<f>in)-a\ + \f(n)-b\<S (2), when w S «o- But this is certainly the case. For since lim <j}{n) = a we can, by the definition of a limit, find rii so that | ^ (n) — a | < S' when n ~ n^, and this however small 8' may be. Nothing prevents our taking 8' = ^S, so that \(\>{n) — a\<\h when n S n^. Similarly we can find n^ so that | i|r (n) —b\<\h when n^n^. Now take n^ to be the greater of the two numbers «i, n^. Then | ^ (n) — a | < ^8 and I ilr (w) — 6 1 < -18 when n S ??o. and therefore (2) is satisfied and the theorem is proved. The argument may be concisely stated thus: since lim0(n)=as and \\m.-<\r {n)=b, we can choose %, ^2 so that |0(»i)-a|<i8 (n.>Mi), \^{n)-h\<^b {n^n^); and then, if n is not less than either n^ or n^, |^(m)+x/.(m)-a-6i<|0(m)-a| + |0(n)-6|<S; and therefore hm {^ (to) + t/^ (k)} = a + 6. 64. Results subsidiary to Theorem I. The reader should - have no difficulty in verifying the following subsidiary results. 1. If <j){n) tends to a limit, but ■yfr (n) tends to + oo or to — oo or oscillates finitely or infinitely, then (f) (n) + ^)r (n) behaves like f (n)- 2. If <j) (n) ^ + CO , and -i|r (n) -* + oo or oscillates finitely, then <ji (w) + i/r (?i) -3. + 00 . 63, 64] POSITIVE INTEGEAL VARIABLE 127 In this statement we may obviously change + oo into — oo throughout. 3. If <j){n)-*<X! and -v/r (n) -* — oo , then ^(n,) + y{r (n) may tend either to a limit or to + oo or to — so or may oscillate either finitely or infinitely. These five possibilities are illustrated in order by (i) (n)=n, ■i}f(n)= —n, (ii) ^{n) = v?, ■^{n)=-n, (iii) <\> {n) = n, ■^{n)=-n\ (iv) ^(m)=TO + (-l)», ^{n)=- n, {v) ^{n)=rfi+{-\Yn, ^{n)= - n\ The reader should construct additional examples of each case. 4. If ^ (n) -^ + cc and yjr (n) oscillates infinitely, then <l){n) + ylr (n) may tend to + oo or oscillate infinitely, but cannot tend to a limit, or to — oo , or oscillate finitely. For ^{n)= {(^ (n) + ^ {n)) - (») ; and, if ^{n) + ^ {%) behaved in any of the three last ways, it would follow, from the previous results, that -v/f (?i) -*- — oo , which is not the ease. As examples of the two cases which are possible, consider (i) 4>{n) = n\ i/r (»i) = (-l)"m, (ii) (f>(n)=n, i/f (»i) = (-l)"?i2. Here again the signs of +oo and -oo may be permuted throughout. 5. If <f> (n) and i|r (n) both oscillate finitely, then ^ (n) + i/r (n) must tend to a limit or oscillate finitely. As examples take (i) <^W = (-1)", ^\r{n) = {-lY*\ (ii) <i,{n)=^{n) = {-\Y. 6. // (w) oscillates finitely, and y}r (n) infinitely, then <f) (n) + yjr (n) oscillates infinitely. For (j) (n) is in absolute value always less than a certain constant, say K. On the other hand ■</'(ji), since it oscillates. infinitely, must assume values numerically greater than any assignable number (e.g. lOK, lOO.fi', ...). Hence <ji{n)+yjr(n) must assume values numerically greater than any assignable number (e.g. 9K, 99K, ...). Hence <l)(n)+-\jf(n) must either tend to +oo or - 00 or oscillate infinitely. But if it tended to + oo then would also tend to + oo , in virtue of the preceding results. Thus (^ (m)+\/^ (n) cannot tend to + oo , nor, for similar reasons, to — oo : hence it oscillates infinitely. 7. If both ^ (n) and a/^ (w) oscillate infinitely, then ^(n) + y}r (n) may tend to a limit, or to -{■ oo , or to — oo , or oscillate either finitely or infinitely. Suppose, for instance, that (») = (-!)"», while \jf{n) is in turn each of the functions (-l)''+»m, {l + (-l)"+i}?i, - {l + (- 1)"} w, (-l)''^^(n+l), ( — l)"re. We thus obtain examples of all five possibilities. 128 LIMITS OF FUNCTIONS OF A [iV The results 1 — 7 cover all the cases which are really distinct. Before passing on to consider the product of two functions, we may point out that the result of Theorem I may be immediately extended to the sum of three or more functions which tend to limits as w -* 00 . 65. B. The behaviour of the product of two functions whose behaviour is known. We can now prove a similar set of theorems concerning the product of two functions. The principal result is the following. Theorem II. If lim ^(n) = a and lim yjr (n) = b, then lim (j} (n) \/r (n) = ah. Let ^ (w) = « + (/>! {n), ->|r (m) = 6 + i/ti {n), so that lim ^i {n) = and lim i|ri (n) = 0. Then (/) (m) ■y^ (m) -ah + a^fri (n) + btf)^ (n) + ^i (n) i/ri (n). Hence the numerical value of the difference <f) (n) yjr (n) — ab is certainly not greater than the sum of the numerical values of ai|ri (n), b(f>i («), ^1 (m) i|fi (m). From this it follows that lim {(j){n)\lr (n) — ab] = 0, which proves the theorem. The following is a strictly formal proof. We have I {n)^{n) -ab\^\a^j.^(n)\ + \ 6<^, {n)\ + \ <t>i(n)\\f^(n) \. Assuming that neither a nor b is zero, we may choose n^ so that |<#.i(«)|<J8/|6|, rV'jWI<iS/|a|, when 7ig»(|. Then \ct,{n)^{n)-ab[<l8 + id + {i8V{\a\\b\)}, which is certainly less than 8 if 8<J|a||6|. That is to say we can choose Ho so that I (j) (?i) i/<- (ji) — a6 1 < 8 when n^n^, and so the theorem follows. The reader should supply a proof for the case in which at least one of a and b is zero. We need hardly point out that this theorem, like Theorem I, may be immediately extended to the product of any number of functions of n. There is also a series of subsidiary theorems concerning products analogous to those stated in § 64 for sums. We must distinguish now sioo different ways in which ^ (n) may behave as n tends to co . It may (1) tend to a limit other than 64-66] POSITIVE INTEGRAL VARIABLE 129 zero, (2) tend to zero, (3a) tend to + oo , (36) tend to — oc , (4) oscillate finitely, (5) oscillate infinitely. It is not necessary, as a rule, to take account separately of (3a) and (36), as the results for one case may be deduced from those for the other by a change of sign. To state these subsidiary theorems at length would occupy more space than we can afford. We select the two which follow as examples, leaving the -verification of them to the reader. He will find it an instructive exercise to formulate some of the remaining theorems himself. (i) If (f> (n) -*- + CO and yjf (n) osoillates finitely, then <p (n) r//' (ra) mitst tend io + 00 or to — 00 or oscillate infinitely. Examples of these three possibilities may be obtained by taking <^ in) to be n and ^ (n) to be one of the three functions 2 + ( - 1)", - 2 - ( - 1)", ( - 1)*. (ii) If (f> (n) and i//- (m) oscillate finitely, then cj) (n) \jr (n) must tend to a limit (which may be zero) or oscillate finitely. For examples, take (a) ^(m) = ^/^(^l) = (-l)^ (6) <^ (m) = l + (-l)*, •\|/- (?i) = 1 - ( - 1)", and (c) (^(?i) = cos Jmn-, T^(m) = sin Jmtt. A particular case of Theorem II which is important is that in which yfr (n) is constant. The theorem then asserts simply that lim k4> (n) = ka if lim ^ (n) = a. To this we may join the subsidiary theorem that if <^(m)^+oo then A;(/>(n)-^+oo or k <j) (n) -^ — oo , according as k is positive or negative, unless k = 0, when of course kcj) (n) = for all values of n and lim k<p (n) = 0. And if (f) (n) oscillates finitely or infinitely, then so does kcj) (n), unless k—0. 66. C. The behaviour of the difference or quotient of two functions \rhose behaviour is known. There is, of course, a similar set of theorems for the difference of two given functions, which are obvious corollaries from what precedes. In order to deal with the quotient fin)' we begin with the following theorem. Theorem III. If lim <f) (n) = a, and a is not zero, then hm j-r-z = - . ^(n) a Let ^ (w) = a + ^1 (w), 9 130 LIMITS OF FUNCTIONS OF A [iV SO that lim 0i (n) = 0. Then 1 1 4>{n) a \<l>An)\ \a\\a + (j)i{n)\' and it is plain, since lim ^i (n) = 0, that we can choose n^ so that this is smaller than any assigned number 8 when w S «„ . From Theorems II and III we can at once deduce the principal theorem for quotients, viz. Theorem IV. If lim <^{n) = a and lim ■^ (n) = b, and b is not zero, then lim 7-V^ = r. Y (n) The reader will again find it instructive to formulate, prove, and illustrate by examples some of the 'subsidiary theorems' corresponding to Theorems III and IV. 67. Theorem V. IfR {</) (n), yjr (n), x («-), • • •} m o,ny rational function of <j){n), yfr (n), %(«), ... , i.e. any function of the form P{cf>(n),fin),x(n), ...}/Q {</.(«), yjrin), x(n), ...], where P and Q denote polynomials in (ft (n), ■yfr (n), % (n), ...: and if lim <f) (n) = a, lim yjr (w) = b, lim % (m) = c, . . . , and Q(a,b,c,...)=^0; then lim R {<f) (n), yfr (n), x («•)> ■•.}=R(a,b, c, . . .). For P is a sum of a finite number of terms of the type A{^in)}v{f(n)}i..., where .4 is a constant and p, q, ... positive integers. This term, by Theorem II (or rather by its obvious extension to the product of any number of functions) tends to the limit Aa^b^ ..., and so P tends to the limit P (a, &. c, ...), by the similar extension of Theorem I. Similarly Q tends to Q {a,b, c, ...); and the result then follows from Theorem IV. 68. The preceding general theorem may be applied to the following very important particular problem : what is the behaviour of the most general rational function ofn, viz. o/ - _ aonP + a^jiP-'^ + . . . + g^ as n tends to oo* 1 * We naturally suppose that neither ffj nor &„ is zero. 66-69] POSITIVE INTEGRAL VARIABLE 131 In order to apply the theorem we transform S (n) by writing it in the form The function in curly brackets is of the form i?{<^(n)), where ^ (w) = 1/n, and therefore tends, as n tends to oo , to the limit B, (0) = a„/6o. Now nv-^-* if ^ < 5 ; n^-i = 1 and n^-*-* 1 if V = q\ wAn^-i-^ +00 \iip>q. Hence, by Theorem II, lim a {n) = (p < 5), lim >S(n) = ao/6„ (^ = 3), S (?i) ^ + 00 (p > g, tto/^o positive), S{n)-*— CO (p>q, ao/ba negative). Examples XXVI. 1. What is the behaviour of the functions \n+l) ' ^ '■> \n+lj ' n ' ^^^ ^T" as n-*-a>'i 2. Which (if any) of the functions l/(cos2^m7r+?j sitf^wn-), !/{« (cos^ Jjur+nsin^^nTr)}, (w oos^ ^wjT +sin2 ^?j7r)/{« (cos^ ^nn + n sin^ ^»n-)} tend to a limit as n -*- 00 ? 3. Denoting by (^(ra) the general rational function of n considered above, show that in all cases 69. Functions of n which increase steadily with n. A special but particularly important class of functions of n is formed by those whose variation as n tends to 00 is always in the same direction, that is to say those which always increase (or always decrease) as n increases. Since — </> (n) always increases if <^ (n) always decreases, it is not necessary to consider the two kinds of functions separately; for theorems proved for one kind can at once be extended to the other. Definition. The function {n) will he said to increase steadily with nif <^{n+l)^<f> {n)for all values of n. 9—2 132 LIMITS OF FUNCTIONS OF A [iV It is to be observed that we do not exclude the case in which <p (n) has the same value for several values of n ; all we exclude is possible decrease. Thus the function ,j)(n) = 2n+(-l)^, whose values for n = 0, 1, 2, 3, 4, . . . are 1, 1, 5, 5, y, y, ... is said to increase steadily with n. Our definition would indeed include even functions which remain constant from some value of n onwards; thus <j)(ri)=l steadily increases according to our definition. However, as these functions are extremely special ones, and as there can be no doubt as to their behaviour as n tends to oo , this apparent incongruity in the definition is not a serious defect. There is one exceedingly important theorem concerning functions of this class. Theorem. If ^ (n) steadily increases with n, then either (i) (^ (n) tends to a limit as n tends to oo , or (ii) (^ (w) -* + oo . That is to say, while there are in general five alternatives as to the behaviour of a function, there are two only for this special kind of function. This theorem is a simple corollary of Dedekind's Theorem (§ 17). We divide the real numbers ^ into two classes L and R, putting ^ in i or i? according as (n) S | for some value of n (and so of course for all greater values), or (/> (m) < f for all values of n. The class L certainly exists; the class R may or may not. If it does not, then, given any number A, however large, ^ («) > A for all sufficiently large values of n, and so <ji (n) -* + CO . If on the other hand R exists, the classes L and R form a section of the real numbers in the sense of § 17. Let a be the number corresponding to the section, and let 8 be any positive number. Then ^ (w) < a + g for all values of n, and so, since S is arbitrary, ^ (n) S a. On the other hand ^(n)>a-B for some value of n, and so for all sufficiently large values. Thus a-B<(f>{n)^a 69, 70] POSITIVE INTEGRAL VARIAELE 133 for all sufficiently large values of n ; i.e. <p (n) -* a. It should be observed that in general <^ {n) < a for all values of n ; for if <j) (n) is equal to a for any value of n it must be equal to a for all greater values of n. Thus <f) (») can never be equal to a except in the case in which the values of (n) are ultimately all the same. If this is so, a is the largest member of L ; otherwise L has no largest member. Cor. 1. If <}) {n) increases steadily with n, then it will tend to a limit or to + <x> according as it is or is not possible to find a number K such that ^ (n) < K for all values of n. We shall find this corollary exceedingly useful later on. Cor. 2. If ^ (n) increases steadily with n, and <j)(n) < K for all values of n, then ^ (n) tends to a limit and this limit is less than or equal to K. It should be noticed that the limit may be equal to ^ : if e.g. ^ (n) = 3 — (1/w), then every value of ^ (to) is less than 3, but the limit is equal to 3. Cor. 3. If ^ (n) increases steadily tvith n, and tends to a limit, then <f> (n) & lim (^ (to) for all values of n. The reader should wi'ite out for himself the corresponding theorems and corollaries for the case in vrhich </> (w) decreases as n increases. 70. The great importance of these theorems lies in the fact that they give us (what we have so far been without) a means of deciding, in a great many cases, whether a given function of n does or does not tend to a limit as n->-cc , without requiring us to be able to guess or otherwise infer beforehand what the limit is. If we know what the limit, if there is one, must be, we can use the test \<\>{n)-l\<h (n^Uo): as for example in the case of <^ (n) = 1/n, where it is obvious that the limit can only be zero. But suppose we have to determine whether <^(n) = ( 1+- 134 LIMITS OF FUNCTIONS OF A [iV tends to a limit. In this case it is not obvious what the limit, if there is one, will be : and it is evident that the test above, which involves I, cannot be used, at any rate directly, to decide whether I exists or not. Of course the test can sometimes be used indirectly, to prove by means of a reductio ad ahsurdum that I cannot exist. If e.g. (j){n) = (- 1)", it is clear that I would have to be equal to 1 and also equal to — 1, which is obviously impossible. 71. Alternative proof of Weierstrass's Theorem of § 19. The results of § 69 enable us to give an alternative proof of the important theorem proved in § 19. If we divide PQ into two equal parts, one at least of them must contain infinitely many points of S. We select the one which does, or, if both do, we select the left-hand half; and we denote the selected half by PiQi (Fig. 28). If Pj §1 is the left-hand half. Pi is the same point as P. Pj I T Qi Q2 Q Fig. 28. Similarly, if we divide Pj §1 into two halves, one at least of them must contain infinitely many points of S. We select the half Pg §2 which does so, or, if both do so, we select the left-hand half. Proceeding in this way we can define a sequence of intervals PQ, PiQu P2Q2, P3Q3, ..., each of which is a half of its predecessor, and each of which contains infinitely many points of S. The points P, Pj, Pj, ... progress steadily from left to right, and so P„ tends to a limiting position T. Similarly §„ tends to a limiting position 7". But TT' is plainly less than PnQn, whatever the value of w; and P„§„, being equal to P§/2» tends to zero. Hence T' coincides with T, and P„ and §„ both tend to T. Then y is a point of accumulation of S. For suppose that | is its coordinate, and consider any interval of the type (|-8, $+S). If n is sufficiently large, P„§„ will lie entii'ely inside this interval*. Hence (^-S, ^ + d) contains infinitely many points of S. 72. The limit of x'^ as n tends to 00 . Let us apply the results of § 69 to the particularly important case in which </) (n) = a;». Iix = l then <f> (n) = 1, lim (f> (n) = 1, and if a; = then ^(?i) = 0,lim^('rt)=0, so that these special cases need not detain us. * This will certainly be the case as soon as PQ/2'» < 5. 70-72] POSITIVE INTEGRAL VARIABLE 135 First, suppose x positive. Then, since ^ (ji + 1) = «</> (n), <^ (n) increases with m if a; > 1, decreases as n increases if a; < 1. If a; > 1, then «» must tend either to a limit (which must obviously be greater than 1) or to + oo . Suppose it tends to a limit I. Then lim ^ (?i + 1) = lim ^ (n) = I, by Exs. xxv. 7 ; but lim ^ (n + 1) = lim X(^ (n) = x lim (f) (n) = xl, and therefore l=xl: and as x and i are both greater than 1, this is impossible. Hence a;" -* + 00 (a; > 1 ). Example. The reader may give an alternative proof, showing by the binomial theorem that ^'''>l + ?i8 if 8 is positive and x=\ + h, and so that a;" -»- + 00 . On the other hand a;" is a decreasing function if x<\, and must therefore tend to a limit or to — x . Since a;" is positive the second alternative may be ignored. Thus lim a;" = I, say, and as above I = xl, so that I must be zero. Hence lima;« = (0<a;<l). Example. Prove as in the preceding example that (1/^')" tends to + oo if Q<x<\, and deduce that x'^ tends to 0. We have finally to consider the case in which x is negative. If — 1 < a; < and a; = — y, so that < y < 1, then it follows from what precedes that lim y" = and therefore lim a;" = 0. If a; = — 1 it is obvious that a;" oscillates, taking the values — 1, I alterna- tively. Finally if a; < — 1, and x = —y,so that y > 1, then y" tends to + 00 , and therefore a;" takes values, both positive and negative, numerically greater than any assigned number. Hence a;" oscillates infinitely. To sum up : </) (?i) = a;" ^ + 00 (« > 1), lim^(w) = l (a!=l), lim (^ (re) = (- 1 < a; < 1), </) (n) oscillates finitely . (a; = — 1), {n) oscillates infinitely (a; < — 1). Examples XXVII*. 1. If {n) is positive and {nJr\)>K<l> (ri), where /f >1, for all values of n, then ^ (w) -9-4- oo . * These examples are particularly important and several of them will be made use of later in the text. They should therefore be studied very carefully. 136 LIMITS OF FUNCTIONS OF A. [iV .[For (ji(n)>K(t>{n-l)>K^(j)in-2) ... >/f"-i(^(l), from which the couolusion follows at once, as jff'''-»-a) .] 2. The same result is true if the conditions above stated are satisfied only when jigwo- 3. If (t>{n) is positive and (j>{h + l)<K(f){n), where 0<K<1, then lim (j) (n) = 0. This result also is true if the conditions are satisfied only when 4. If |^(» + l)|<ir|<^(»)| when m>TOo, and 0</i<l, thenlim(^(»)=0. 5. If <^ (n) is positive and lim {0 (w + 1 )}/{<^ (n)} = 1>1, then <^ (m) -»- + oo . [For we can determine jio.so that {<f> {n + l)}l{(t) {n)}>K>l when wgWj: we may, ^.g., take K half-way between 1 and I. Now apply Ex. 1.] t 6. If lim {(^(w + !)}/{</> (»)} = ?, where I is numerically less than unity, then lim<l>(n) = 0. [This follows from Ex. 4 as Ex. 5 follows from Ex. 1.] 7. Determine the behaviour, as «-*-x, of (j}(n)=n''x^, where r is any positive integer. [If /i;=0 then (^ (m) = for all values of n, and ^ (?i)-*-0. In all other casesj <t>(n+l) _/n + l\r^,^^ <P{n) \ n ) First suppose x positive. Then <^(?i)-*- + oo if x>l (Ex. 5) and <^(re)-»-0 if x<\ (Ex. 6). If x=\, then <^ (%) = «.''-*--(- (». Next suppose ^- negative. Then | <^(m) |=?i'"| « !"■ tends to +oo if |a;|>l and to if \x\<\. Hence (m) oscillates infinitely if x&- 1 and (^ (to)-*0 if - 1<;!;<0.-] 8. Discuss W^al^ in the same way. [The results are the same, except that 0(m)-»-O when x = \ or —1.] 9. Draw up a table to show howlwVjbehaves as m-*-QO, for all real values of x, and all positive and negative integral values of k. [The reader will observe that the value of k is immaterial except in the j special cases when ^=1 or —1. Since lim {()i + l)/»}*=l, whether k bel positive or negative, the limit of the ratio (f){n + \)j(\>{n) depends only on , X, and the behaviour of (/> (m) is in general dominated by the factor x\ The ' factor # only asserts' itself when x is numerically eqvial to 1.] 10. Provethat if .dispositive then^*'-»-l as7i-s-oo. [Suppose, e.5'.,5;>l. Then x, ^Jx, ^x, ... is a decreasing sequence, and ^x>l for all values of n. Thus ^x-^l, where ?S1. But if 1>1 we can find values of m, as large as we please, for which Sjx>l or «>?" ; and, since ?''-*--l-oo as «-»-oo, this is impossible.] 11. Xln-*\. [For '' + V(n + l)<;/m if (« + l)''<m''+i or {1 + (1/m)}"<?i, which is certainly satisfied if mgS (see § 73 for a proof). Thus ^n decreases as n increases from 3 onwards, and, as it is always greater than unity, it tends to a limit which is greater than or equal to unity. But if ^n-»-l, where l>l, then ?!■>?", which is certainly untrue for sufficiently large values of n, since I'^jn^ + tn with n (Exs. 7, 8).] 72, 73] POSITIVE INTEGRAL VARIABLE 137 12. XI {^ !)"*■ + '»• [However large A may be, m ! >A" if « is large enough. For if Un=A''/nl then m„ + i/m„=A/(to + 1), which tends to zero as n^oo, so thatM„ does the same (Ex. 6).] 13. Show that if - 1< .» < 1 then ^^^m(m-l)...(m-^+l) ^^^ /mN ^.„ n\ \nj tends to zero as «-*•<» . ' [If »» is a positive integer, m^=0 for n>m. Otherwise tt„ ?i + 1 ' unless x = 0.] 73. The limit of (l + - j . A more difficult problem which can be solved by the help_ of § 69 arises when (}>(n) = {1 + 1/n)". It follows from the binomial theorem * that + ,-^— (l-i)(l-?)...(l-!^'). 1.2...?i\ n/ \ nj \ n J The (p + l)th term in this expression, viz. x:^,(-3('-.i)-(-^-i^). is positive and an increasing function of n, and the number / 1\" of terms also increases, with n. Hence f 1 +-J increases with n, and so tends to a limit or to + oo , as w ^- oo . But (^-iJ , , 11 1 1.2 ' 1.2.3 1.2.3. <l + l + 2 + 2i + ... + 2S3i<3- (1\" 1 + - j cannot tend to + oo , and so lim f 1 + -V = e, where e is a number such that 2 < e S 3. * The binomial theorem for a positive integral exponent, which is what is used here, is a theorem of elementary algebra. The other cases of the theorem belong to the theory of infinite series, and will be considered later. 138 LIMITS OF FUNCTIONS OF A [iV 74. Some algebraical lemmas. It will be convenient to prove at this stage a number of elementary inequalities which will be useful to us later on. (i) It is evident that if a>l and ?• is a positive integer then ?-a''>a'-l + a'-2+... + l. Multiplying both sides of this ineqiiality by a- 1, we obtain and adding r{a^-l) to each side, and dividing by r (r+l), we obtain ^— -^>^^— i (a>]) (1). r + 1 r ' Similarly we can prove that l_«r+l \-Rr h^<hr- (0<'3<1) (2)- It follows that if r and s are positive integers, and r>s, then a^^a^-1^ lZ^<k:^l (3). r s r s Here 0</3<l<a." In particular, when s = l, we have a'--l>r{a-l), l-^<r{l-fi) (4). (ii) The inequalities (3) and (4) have been proved on the supposition that r and s are positive integers. But it is easy to see that they hold under the more general hypothesis that r and s are any positive rational numbers. Let us consider, for example, the first of the inequalities (3). Let r=ajh, s = o/d, where a, b, c, d are positive integers; so that ad>bc. If we put 0=7** the inequality takes the form (v'"«-l)/ac?>(/"-l)/6c; and this we have proved already. The same argument applies to the re- maining inequalities ; and it can evidently be proved in a similar manner that a'-l<s(a-l), l-^«>s(l-/3) (5), if s is a positive rational number less than 1. (iii) In what follows it is to be understood that all the letters denote positive nu7nbers, that r and s are rational, and that a and r are greater than 1, ^ and s less than 1. Writing 1/0 for a, and 1/a for ft in (4), we obtain a'--l<ra'-i(a-l), \-^>r^'-^^{\-^) (6). Similarly, from (5), we deduce a»-l>sa«-i(a-l), l-/3»<si3»-i (1 -|3) (7). Combining (4) and (6), we see that «-a'-'(a-l)>a'--l>r(a-l) (8). 74, 75] POSITIVE INTEGRAL VARIAlJLE 139 Writing x/y for a, we obtain rx^-'^ {x-y)>x''-yf'>ry''-'^{x-y) (9) if x>y>0. And the same argument, applied to (5) and (7), leads to s«'-i {x-y)<x'~y'<sy'-^ (x-y) (10). Examples XXVIII. 1. Verify (9) for r=2, 3, and (10) for s=i, ^. 2. Show that (9) and (10) are also true if y>x>0. 3. Show that (9) also holds for r<0. [See Chrystal's Algebra, vol. ii, pp. 43—45.] 4. If <i){n)-»-l, where l>0, as ji-»-oo, then 0*-».i* k being any rational number. [We may suppose that k>0, in virtue of Theorem III of § 66; and that ^l<^<'i,l, as is certainly the case from a certain value of n onwards. If k>\, /5;0*-l ((^ - I) ></)«= -Z* >yi;(!*-l (^ - 1) or /i;Z*-i {l-(l))>l''-(t>''>k(j,''-i {l-<p), according as <}>>l or (p<l. It follows that the ratio of |(^*— Z*| and \4> — ^ lies between k {^lf~^ and k (2?)*-i. The proof is similar when <^ <1. The result is still true when 1=0, if X; > 0.] 5. Extend the results of Exs. xxvii. 7, 8, 9 to the case in which r ox k are any rational numbers. 75. The limit of n (i/x - 1). If in the first inequality (3) of § 74 we put r=l/(?i— 1), s='l/n, we see that («-l)("-i/a-l)>«(;/a-l) when a>l. Thus if {ii)=n {^a-1) then (f> (n) decreases steadily as n in- creases. Also (j>(n} is always positive. Hence <^(;i) tends to a limit I as «-»-oo , and l^Q. Again if, in the first inequality (7) of § 74, we put s = 1jn, we obtain .(;/a-l)>;/a(l-l)>l-l. Thus Z>1 -(l/a)>0. Hence, if a>l, we have lim W(^a-l)=/(a), where f(a)>0. Next suppose /3<1, andlet;8=l/a; thcn«(^^-l)= —n(,!^a-l)/^a. Now w(C/a-l)-a-/(a), and (Exs. XXVII. 10) Hence, if |3=l/a<l, we have Finally, if x=l, then n (;!^x-l)=0 for all values of n. 140 LIMITS OF FUNCTIONS OF A [iV Thus we arrive at the result : the limit lna.n{!!/x-l) defines a function of x for all positive values of x. This function f {x) possesses the properties /(!/*)= -/(^), /(1)=0, and is positive or negative according as a;>l or a; < 1. Later on we shall be able to identify this function with the Napierian logarithm of .^' Example. Prove that/(a;y)=/(«)+/(y). [Use the equations f{m/)=\im n {Xjxy - l)=lim {n {^x-\)yy+n{;:iy- 1)}.] 76. Infinite Series. Suppose that u {n) is any function of n defined for all values of n. If we add up the values of u {v) for V = 1, 2, ... n, we obtain another function of w, viz. s(n)=M(l)+M (2) + ... + «(«), also defined for all values of n. It is generally most convenient to alter our notation slightly and write this equation in the form «» = Ml + M2+ •••+«„, n or, more shortly, Sn = S m„. If now we suppose that s„ tends to a limit s when n tends to 00 , we have n lim 2 w„ = s. n-*- oo 1/=! This equation is usually written in one of the forms 00 2 M„ = S, Ml + M2 + Ma + . . . = S, v=\ the dots denoting the indefinite continuance of the series of m's. The meaning of the above equations, expressed roughly, is that by adding more and more of the m's together we get nearer and nearer to the limit s. More precisely, if any small positive number S is chosen, we can choose n^ (S) so that the sum of the first «o (8) terms, or any of greater number of terms, lies between s—h and s + 8 ; or in symbols s - S < s„ < 5 + S, if n S Wo (8). In these circumstances we shall call the series iti + Mu + . . . a convergent infinite series, and we shall call s the sum of the series, or the sum of all the terns of the series. 75-77] POSITIVE INTEGRAL VARIABLE 141 Thus to say that the series Mj + Mj + . . . converges and has the sum s, or converges to the sum s or simply converges to s, is merely another way of stating that the sum s„ = Uj + Mg + ... + m„ of the first n terms tends to the limit s as w -»- oo , and the consideration of such infinite series introduces no new ideas beyond those with which the early part of this chapter should already have made the reader familiar. In fact the sum s„ is merely a function (f) (n), such as we have been considering, expressed in a particular form. Any function <f> (n) may be expressed in this form, by writing <t>{n) = 4> (1) + (</. (2) - <}> (1)} + ... + {<l>(n)-<p{n-l)}; and it is sometimes convenient to say that </> (n) converges (instead of ' tends ') to the limit I, say, as n ^- oo . If s„ -* + 00 or s„ -»- — 00 , we shall say that the series Mj + Mj + . . . is divergent or diverges to + oo , or — oo , as the case may be. These phrases too may be applied to any function <p (n) : thus if (^ (n) -* + 00 we may say that ^ (n) diverges to + oo . If «» does not tend to a limit or to + oo or to — oo , then it oscillates finitely or infinitely : in this case we say that the series Mi + Wg + . . . oscillates finitely or infinitely*. 77. General theorems concerning infinite series. When we are dealing with infinite series we shall constantly have occasion to use the following general theorems. (1) If i<i + M2 + ... is convergent, and has the sum s, then a + u^+ u^ + ... is convergent and has the sum a + s. Similarly a + h + c + ...+k + ih + Ui + ■•• is convergent and has the sum a+6 + c4-... + ^ + s. (2) If Mi + i(2+... is convergent and has the sum s, then u m+i + t(m+2 + • • • is convergent and has the sum S — V,^ — U2 — ...—Um- (3) If any series considered in (1) or (2) diverges or oscillates, then so do the others. (4) If ■M1+M2+... is convergent and has the sum s, then Icu^ + hu^ + ... is convergent and has the sum ks. * The reader should be warned that the words ' divergent ' and ' oscillatory ' are used differently by different writers. The use of the words here agrees with that of Bromwich's Infinite Series. In Hobson's Theory of Functions of a Real Variable » series is said to oscillate only if it oscillates finitely, series which oscillate infinitely being classed as ' divergent'. Many foreign writers use 'divergent' as meaning merely ' not convergent '. 142 LIMITS OF FUNCTIONS OF A [iV (5) If the first series considered in (4) diverges or oscillates, then so does the second, unless h = 0, (6) If Ui + U2 + ... and Vi + Vs+ ... are both convergent, then the series (mi + v^) + (mj + 1^2) + • • • is convergent and its sum is the sum of the first two series. All these theorems are almost obvious and may be proved at once from the definitions or by applying the results of §§ 63 — 66 to the sum «„ = Mj + zta + ... + m„. Those which follow are of a some- what different character. (7) //" Ml + Ma + . . . is convergent, then lim m„ = 0. For «„ = s» — Sn-i, and s„ and s„_i have the same limit s. Hence lim m„ = s — s = 0. The reader may be tempted to think that the converse of the theorem is true and that if hm u„=0 then the series 2m„ must be convergent. That this is not the case is easily seen from an example. Let the series be so that Un=lln. The sum of the first four terms is The sum of the next four terms is i+J^ + ^ + i>|=i ; the sum of |he next eight terms is greater than ■^=i, and so on. The sum of the first 4+4 + 8 + 16+.. .+2" = 2''+i terms is greater than 2+Hi+i+-+i=i(»+3), and this increases beyond all limit with n : hence the series diverges to + 00 . (8) //■ Ml + Ma + M3 + . . . is convergent, then so is any series formed by grouping the terms in brackets in any way to form new single terms, and the sums of the two series are the same. The reader will be able to supply the proof of this theorem. Here again the converse is not true. Thus 1-1 + 1-1 + .. . oscillates, while (1-1)+(1-1)+... or + + 0+ ... converges to 0. (9) If every term m„- is positive {or zero), then the series 2i{» must either converge or diverge to + <xi . If it converges, its sum must be positive (unless all the terms are zero, when of course its sum is zero). For s„ is an increasing function of n, according to the definition of § 69, and we can apply the results of that section to s„. 77, 78] POSITIVE INTEGRAL VARIABLE 143 (10) If every term. w„ is positive (or zero), then the necessary and sufficient condition that the series 2m„ shoidd be convergent is that it should be possible to find a number K such that the sum of any number of terms is less than K; and, if K can be so found, then the sum of the series is not greater than K. This also follows at once from § 69. It is perhaps hardly necessary to point out that the theorem is not true if the condition that every Un is positive is not fulfilled. For example 1-1 + 1-1 + ... obviously oscillates, s„ being alternately equal to 1 and to 0. (11) If u^-\-Ui+ ..., Vi + 'V2 + ... are two series of positive (or zero) terms, and the second series is convergent, and if Un S Kv^, where K is a constant, for all values of n, then the first series is also convergent, and its sum is less than or equal to that of the second. For \i V1+V2+ ...=t then Vi+Vi + ...+Vn = tioT all values of n, and so jfj + M2 + . . . + m,» S Kt ; which proves the theorem. Conversely, if Sttn is divergent, and Vn = Kun, then S^n is divergent. 78. The infinite geometrical series. We shall now con- sider the ' geometrical ' series, whose general term is m„ = r"~'. In this case s„ = 1 + r + r^ + . . . + 7'"-i = (1 - r")/(l - r), except in the special case in which r = 1, when s„= 1 + 1 + ... + 1=«. In the last case s„ ^ + 00 . In the general case s„ will tend to a limit if and only if r" does so. Referring to the results of § 72 we see that the series 1 + r + r''' + . . . is convergent and has the sum 1/(1 — r) if and only if—\<r< 1. If r- S 1, then s„ S n, and so s„ -^ + 00 ; i.e. the series diverges to + 00. If r = — 1, then s„ = l or s„ = according as n is odd or even : i.e. Sn oscillates finitely. If >• < — l,,then s„ oscillates infinitely. Thus, to sum up, the series l + r + r'' + ... diverges to+<x> ifr^l, converges to 1/(1 -r) if -l<r<l, oscillates finitely if r = -l, and oscillates infinitely ifr< — \. Examples XXIX. l. Recurring decimals. The commonest example of an mfiiiite geometric series is given by an ordinary recurring decimal. V lO^y 12375 • 144 LIMITS OF FUNCTIONS OF A [jV Consider, for example, the decimal -21713 This stands, according to the ordinary rules of arithmetic, for A J_ ^ J_ 3_ J_ 3_ ^217 13 10 "*" 102 ■*" 103 + 10* 106 + 108 ■•■ lO'' 1000 ■*" 106^ The reader should consider where and how any of the general theorems of § 77 have been used in this reduction. 2. Show that in general ■aia,...a„a,a2...a„= 99...yu0...0 ' the denominator containing n 9's and m O's. 3. Show that a pure recurring decimal is always equal to a proper fraction whose denominator does not contain 2 or 5 as a factor. 4. A decimal with m non-recurring and n recurring decimal figures is equal to a proper fraction whose denominator is divisible by 2™ or 5" but by no higher power of either. 5. The converses of Exs. 3, 4 are also true. Let ?'=p/5', and suppose first that q is prime to 10. If, we divide all powers of 10 by g' we can obtain at most q different remainders. It is therefore possible to find two numbers »i and 112, where M2>™i) such that 10"' and 10"^ give the same remainder. Hence 10"' - 10"^= 10"= (10"'-"=- 1) is divisible by q, and so 10»- 1, where »=mj-«2, is divisible by q. Hence r may be expressed in the form /"/(lO" — 1), or in the form ion +102.1 + ■••> i.e. as a pure recurring decimal with n figures. If on the other hand q=2''5^'Q, where Q is prime to 10, and m is the greater of a and fi, then 10™ »• has a de- nominator prime to 10, and is therefore expressible as the sum of an integer and a pure recurring decimal. But this is not true of lO**?-, for any value of /x less than m ; hence the decimal for r has exactly m non-recurring figures. 6. To the results of Exs. 2 — 5 we must add that of Ex. i. 3. Finally, if we observe that 9 9 9 '^"T0"'"l02+103'*"""'"-^' we see that every terminating decimal can also be expressed as a mixed recurring decimal whose recurring part is composed entirely of 9's. For example, "217 = '2169. Thus every proper fraction can be expressed as a recurring decimal, and conversely. 7. Decimals in general. The expression of irrational numbers as non-recurring decimals. Any decimal, whether recurring or not, corresponds to a definite number between and 1. For the decimal -aiaiasai... stands for the series 10"^102"^103+""" 78] POSITIVE INTEGRAL VARIABLE 145 Since all the digits a, are positive, the sum «„ of the first n terms of this series increases with n, and it is certainly not greater than •£) or 1. Hence s„ tends to a limit between and 1. Moreover no two decimals can correspond to the same number (except in the special case noticed in Ex. 6). For suppose that -aia^as ..., ■h^h.^h^... are two decimals which agree as far as the figm'cs «,_!, 6r-ii while Or>^r- Then ar^bj.-\-\>hr ■ 6r + i6r + 2-" (unless 6^+1) 6r + 2i ••• are all 9's), and so •aia2...ar«r + l ... > ■6162- •• ^r^r 4-1 ••• • It follows that the expression of a rational fraction as a recurring decimal (Exs. 2^6) is unique. It also follows that every decimal which does not recur represents some irrational number between and 1. Conversely, any such number can be expressed as such a decimal. For it must lie in one of the intervals 0, 1/10; 1/10, 2/10; ... ; 9/10, 1. If it lies between r/10 and (?'+l)/10, then the first figure is r. By subdividing this interval into 10 parts we can determine the second figure; and so on. But (Exs. 3, 4) the decimal cannot recur. Thus, for example, the decimal 1-414..., obtained by the ordinary process for the extraction of V2, cannot recur. 8. The decimals -1010010001000010... and -2020020002000020..., in which the number of zeros between two I's or 2's increases by one at each stage, represent irrational numbers. 9. The decimal -IIIOIOIOOOIOIO..., in which the nth figure is 1 if w is prime, and zero otherwise, represents an irrational number. [Since the number of primes is infinite the decimal does not terminate. Nor can it recur: for if it did we could determine m and p so that m, m+p, m + 2p, m+3p, ... are all prime nuinbers ; and this is absurd, since the series includes m + mp."]* Examples XXX. 1. Theseriesj-^+j-^+1 + ...isconvergentif -1<)-<1, audits sum is l/(l-r)-l -r- ... -r'"-! (§ 77,(2)). 2. The series j-m+r^ + iH- ... is convergent if - 1< ?• < 1, and its sum is »'"V(1 - »') (§ 77, (4)). Verify that the results of Exs. 1 and 2 are in agreement. 3. Prove that the series l + 2r+2r^+ ... is convergent, and that its sum is (l+>-)/(l-»-). (a) by writing it in the form -l + 2(l+?-+r'+ ...), (fi) by writing it in the form l + 2{r + r^+ ...), (y) by adding the two series l^r + r^+ ..., r+r'+.... In each case mention which of the theorems of § 77 are used in your proof. * All the results of Exs. xxix may be extended, with suitable modifications, to decimals in any scale of notation. For a fuller discussion see Bromwioh, Infinite Series, Appendix I. H. 10 146 LIMITS OF FUNCTIONS OF A [iV 4. Prove that the ' arithmetic ' series a + {a+b) + {a + 2b)+ ... is always divergent, unless both a and b are zero. Show that, if 5 is not zero, the series diverges to + co or to — co according to the sign of b, while if J = it diverges to +oo or - oo according to the sign of a. 5. What is the sum of the series {1 -r) + {r -r^) + (r^-r^)+ ... when the series is convergent ? [The series converges only if - Krgl. Its sum is 1, except when r=l, when its sum is 0.] J.2 J.2 6. Sum the series yHrx^+ ,. ^v^ + ... . [The series is always con- vergent. Its sum is l+r\ except when '/• = 0, when its sum is 0.] 7. If we assume that 1 + ?• -f r^ + . . . is convergent then we can prove that its sum is 1/(1 -r) by means of § 77, (1) and (4). For if l + !-+?-2+ ... =s then s = l-l-)-(l + r2+ ..•) = !+'■«• . r r 8. Sum the series »"+ r": — ^ 7r~, — r9 + "' l+r {l+rf when it is convergent. [The series is convergent if -l<l/(l + j')<l, i.e. if r< - 2 or if j'>0, and its sum is 1 +r. It is also convergent when i'=0, when its sum is 0.] 9. Answer the same question for the series l+»- ' (l+r)2 ••■' ' 1-r ' (l-?-)2^"'' l+r \l+»7 l-r \l — rj 10. Consider the convergence of the series (l+?-) + (?-2 + j'3) + ..., {i + r + r') + (r^+r^-i-r^) + ..., l-2r+r2 + !-S-2)-*+?^ + ..., (l-2r-t-?-2)-t-(»-3-2r«-t-!-5) + ..., and find their sums when they are convergent. 11. If Oga„<l then the series ao + air+a2r^+ ... is convergent for 0£»'<1, and its sum is not greater than 1/(1-?-). 12. If in addition the series ao + ai + a2+... is convergent, then the series ao + »i'' + «2'"^ + -- is convergent for Ogrsl, and its sum is not greater than the lesser of ao+»i + a2 + ... and 1/(1— ?•). 13. The series l+i+t^ + r 1 '1.2 1.2.3 is convergent. [For 1/(1 . 2 ... m) g 1/2" - 1.] 78, 79] POSITIVE INTEGRAL VARIABLE 14.7 14. The series ^■'"r72'''i.2.3.4"'"-' I + OTs + iTaTsTT.s"''- are convergent. 15. The general harmonic series where a and b are positive, diverges to + oo . [For M„=l/(a+w5)>l/{n(a+6)}. Now compare with 1+1 + J+... .] 16. Show that the series («o-«:) + («i-«2)+(m2-«3)+ .-. is convergent if and only if m„ tends to a limit as n-»-Qo . 17. If U1 + U2 + U3 + ... is divergent then so is any series formed by grouping the terms in brackets in any way to form new single terms. 18. Any series, formed by taking a selection of the terms of a convergent series of positive terms, is itself convergent. 79. The representation of functions of a continuous real variable by means of limits. In the preceding sections we have frequently been concerned with limits such as lim 4)n (x), and series such as Wj («) + W2 {x) + ... = lim {mi {x) + M2 (a;) + ... + w„ (00)}, in which the function of n whose limit we are seeking involves, besides n, another variable x. In such cases the limit is of course a function of x. Thus in § 75 we encountered the function f(x) = lim n {HJx — 1): and the sum of the geometrical series l+a; + a;^ + ...isa function of X, viz. the function which is equal to 1/(1 -a!)if-l<a;<l and is undefined for all other values of x. Many of the apparently ' arbitrary ' or ' unnatural ' functions considered in Ch. II are capable of a simple representation of this kind, as will appear from the following examples. 10—2 148 LIMITS OF FUNCTIONS OF A [iV Examples XXXI. 1. (t>n{^)=^- Here ?i does not appear at all in the expression of ^„ (x), and (j) (a;)=lim 0^ {x) = x: for all values of x 2. 0„ {x)=xln. Here (jb {x) =lim ^„ (a?) =0 for all values of x. 3. ^,,{x) = nss. If ^>0, <^„(«)-s- + oo ; if .j;<0, ^„(A-)-a--co : only when ;E=0 has 0„(a;) a limit (viz. 0) as n^^. Thus ^(a;) = when a;=0 and is not defined for any other value of x. 4. (f)n{x) = llnx, nx/{nx+l)i 5. (p„{x)=x''. B.ere(l>{x)=0,{-l<x<l); (t>{x) = l,{x=l); and (t){x) is not defined for any other value of x. 6. <f>,i(x) = x'^(l-x). Here (l>{x) differs from the (pix) of Ex. 5 in that it has the value when x=l. 7. 0a i^) = x'^jn. Here {x) differs from the {x) of Ex. 6 in that it has the value when x= — l as well as when x = l. 8. 0„ (a;) =*•"/(*•»+ 1). [0(^) = O,(-l<a-<l);0(:c) = ^,(^=l);0(ii;) = l, (^'<-l or x>\); and <p{x) is not defined when x= —1.] 9. 0„(^)=:i-»/(^-l), l/(^-"+l), l/(«^"-l), l/(^''+^-»), l/(«''-a;-"). 10. 0„(^) = (;j;»-l)/(^"+l), (tu.'" -!)/(««" + 1), («»-ii)/(^+m). [In the first case (a;) = l when |a;|>l, (;r)= -1 when \x\ <1, (p{x)=() when ^=1 and (.r) is not defined when x= — 1. The second and third functions differ from the first in that they are defined both when x=\ and whenj;= —1 : the second has the value 1 and the third the value - 1 for both these values of x.l 11. Construct an example in which 0(«) = 1, (|i'|>l); 0(,3;)=-l, (|-:!;|<1); and 0(a;) = O, (j;=l and .^-= — 1). 12. 0„(^) = ^{(^2n_i)y(^.27.+ l)}2^ njix'^+x-'^+n). 13. (j>n{=o) = {x^f{x)+g{x)}l{x"+\). [Here <\> {x)=f {x),{\x\>\); 4i{x)=g{x), (|:j;|<l); <i>{x) = \{f{x)-\-g{x)}, ix=\) ; and (^) is undefined when x= — 1.] 14. 0„(x') = (2/ff) arc tan («,»). [0(«) = 1, {x>0); 0(a;) = O, (^=0); 0(.j;)=-l, (*'<0). This function is important in the Theory of Numbers, and is usually denoted by sgn x.'\ 15. 0„(*) = sin»i.i;7r. [0(.j;) = O when x is an integer; and 0(^) is otherwise undefined (Ex. xxiv. 7).] 16. If 4,n{x) = ain{n\xTr) then 0(.i;) = O for all rational values of x (Ex. XXIV. 14). [The consideration of irrational values presents greater difiiculties.] 17. 0,i(x) = (cos2a'jr)". [0(a:)=O except when x is integral, when 0(a,') = l.] 18. If N^ 1752 then the number of days in the year N a.d. is lim {365 + (cos2 JjVn-)" - (cos^ j^NnY + (cos^ ji^.Vn-)"}.- 79-81] POSITIVE INTEGRAL VARIABLE 149 80. The bounds of a bounded aggregate. Let S be any system or aggregate of real numbers s. If there is a number K such that s^K for every s of S, we say that S is hounded above. If there is a number k such that s'^h for every s, we say that S is bounded below. If S is botli bounded above and bounded below, we say simply that S is bomided. Suppose first that S is bounded above (but not necessarily below). There will be an infinity of numbers which possess the property possessed by K ; any number greater than K, for example, possesses it. We shall prove that among these numbers there is a least*, which we shall call M. This number if is not exceeded by any member of S, but every number less than Mis exceeded by at least one member of /S. We divide the real numbers | into two classes Z and R, putting | into L or R according as it is or is not exceeded by members of S. Then every | belongs to one and one only of the classes L and R. Each class exists ; for any number less than any member of S belongs to L, while K belongs to R. Finally, any member of L is less than some member of S, and therefore less than any member of R. Thus the three conditions of Dedekind's Theorem (§ 17) are satisfied, and there is a number J/' dividing the classes. The number M is the number whose existence we had to prove. In the first place, J/" cannot be exceeded by any member of S. For if there were such a member s of S, we could write s= if+rj, where rj is positive. The number M+^Tj would then belong to L, because it is less than s, and to R, because it is greater than M ; and this is impossible. On the other hand, any number less than M belongs to Z, and is therefore exceeded by at least one member of S. Thus M has all the properties required. This number M we call the upper bound of S, and we may enunciate the following theorem. Ani/ aggregate S tohich is bounded above has an upper bound M. No membei' of S exceeds M; but any number less than M is exceeded by at least one member of S. In exactly the same way we can prove the corresponding theorem for an aggregate bounded below (but not necessarily above). Any aggregate S which is bounded below has a lower bound m. No member of S is less than m; but there is at least one member of S which is less than any number greater than m. It will be observed that, when S is bounded above, Jf < K, and when S is bounded below, m^L When S is bounded, ^ g m < 1/ < K. 81. The bounds of a bounded function. Suppose that (f> (n) is a func- tion of the positive integral variable n. The aggregate of all the values <^ (n) defines a set S, to which we may apply all the arguments of § 80. If..S is bounded above, or bounded below, or bounded, we say that (j> (n) is bounded * An infinite aggregate of numbers does not necessarily possess a least member. The set consisting of the numbers ill 1 • 2' 3 n' ■■■' for example, has no least member. 150 LIMITS OF FUNCTIONS OF A [iV above, or bounded below, or bounded. If 4> (n) is bounded above, that is to say if there is a number K such that <p{n)^K for all values of n, then there is a number M such that (i) <^ («•) S M for all values of n ; (ii) if S is any positive number then <^{n)> M-& for at least one value of n. This number M we call the upper bound of cj) {n). Similarly, if <}> (n) is bounded below, that is to say if there is a number k such that (^ («) g A for all values of n, then there is a number m such that (i) (ji{n)^m for all values of n ; (ii) if 8 is any positive number then ^ {n)<m + S for atleast one value of n. This number m we call the lower bound of ^ (»). If K exists, M ^ K ; \i h exists, m.'Z.h; and if both h and K exist then k<m^Ms,K. 82. The limits of indetermination of a bounded function. Suppose that ^ (ra) is a bounded function, and M and in its upper and lower bounds. Let us take any real number |, and consider now the relations of inequality which may hold between | and the values assumed by ^ (m) for large values of n. There are three mutually exclusive possibilities ; (1) I S ^ (m) for all sufficiently large values of n ; (2) I S <^ in) for all sufficiently large values of n \ (3) J < 1^ (k) for an infinity of values of n, and also ^ > <^ (») for an infinity of values of n. In case (1) we shall say that ^ is a superior number, in case (2) that it is an inferior number, and in case (3) that it is an intermediate number. It is plain that no superior number can be less than m, and no inferior number greater than M. Let us consider the aggregate of all superior numbers. It is bounded below, since none of its members are less than to, and has therefore a lower bound, which we shall denote by A. Similarly the aggregate of inferior numbers has an upper bound, which we denote by X. We call A and \ respectively the upper and lower limits of indetermination of (f) (b) as n tends to infinity ; and write A = lim (^ (n), X = lim {n). These numbers have the following properties : (1) m^\^k.^M; (2) A and X are the upper and lower bounds of the aggregate of intermediate numbers, if any such exist ; (3) if S is any positive number, then (^ (m) < A + 8 for all sufficiently large values of n, and {n) > A — 8 for an infinity of values of n ; (4) similarly i^ (m) > X - 8 for all sufficiently large values of n, and (^(ft)<X + 8 for an infinity of values of n; 81, 82] POSITIVE INTEGRAL VARIABLE 151 (5) the necessary and sufficient condition that (m) should tend to a limit is that A = X, and in this case the limit is I, the common value of \ and A. Of these properties, (1) is an immediate consequence of the definitions ; and we can prove (2) as follows. If A=X = ?, there can be at most one inter- mediate number, viz. I, and there is nothing to prove. Suppose then that A >X. Any intermediate number | is less than any superior and greater than any inferior number, so that X g | g A. But if X < ^ < A then ^ must be intermediate, since it is plainly neither superior nor inferior. Hence there are intermediate numbers as near as we please to either X or .\. To prove (3) we observe that A + 8 is superior and A - 8 intermediate or inferior. The result is then an immediate consequence of the definitions ; and the proof of (4) is substantially the same. Finally (5) may be proved as follows. If A = X = Z, then l-h<(i){n)<l-\-h for every positive value of S and all sufficiently large values of n, so that (^ (?i) -9- 1. Conversely, if ^ (n) -»- 1, then the inequalities above written hold for all sufficiently large values of n. Hence l-his, inferior and ^ + 8 superior, so that X>Z-S, A<Z + 8, and therefore A - X g 28. As A - X g 0, this can only be true if A = X. Examples XXXII. 1. Neither a nor X is affected by any alteration in any finite number of values of <^ (»). 2. If (^ (m) = a f or all values of », then m='\ = A = M=a. 3. If (^(a) = l/w, then to = X = A = and M=l. 4. If 0(re) = (-l)", then to = X=-1 and A = i/"=1. 6. If ^()i) = (- !)"/«, then to=-1, X = A = 0, M=\. 6. If ^(w) = ( -!)"{! -I-(1/m)}, then »i=-2, X= -1, A = l, J/=f. 7. Let (^(?i) = sin M^TT, where 5>0. If ^isanintegerthenTO = X = A = i/=0. If 6 is rational but not integral a variety of cases arise. Suppose, e.g., that 6=plq, p and q being positive, odd, and prime to one another, and 2'>1. Then <i> {ti) assumes the cyclical sequence of values sm{pnjq), sin (2j07r/j), , sin{(22'- l)^7r/9}, sm{2qpnlq), It is easily verified that the numerically greatest and least values of (fj (n) are cos {7r/22') and - cos (nj-Iq), so that m = \=- cos (n/2q), A = M= cos (5r/2j). The reader may discuss similarly the cases which arise when p and q are not both odd. The case in which 6 is irrational is more difficult : it may be shown that in this case mi=X= - 1 and A=i/"= 1. It may also be shown that the values of </) («) are scattered all over the interval ( - 1, 1) in such a way that, if | is 152 LIMITS OF FUNCTIONS OF A [iV any number of the interval, then there is a sequence nj, ?i2, such that <^(%)-*-| as ^-»-oo.* The results are very similar when ^ (m) is the fractional part of n6. 83. The general principle of convergence for a bounded function. The results of the preceding sections enable us to formulate a very important necessary and sufficient condition that a bounded function (m) should tend to a limit, a condition usually referred to as the general pri-nciple of convergence to a limit. Theorem 1. The necessary and sufficient condition that a hounded function (f> (n) should tend to a limit is that, when any positive number S is given, it should be possible to find a number % (S) «mcA that I'^(m2)-<^(»i)l<8 for all values of «.[ and n^ such that ra2 > ™i S «o (S)- In the first place, the condition is necessary. For if <j){n)-a~l then we can find n^ so that l-iS«j>{n)<l + i8 when 71 > Wo 1 ^^^ so |«^(«2)-<>K)I<8 (1) when Ml S lio and Kj 2 »?o- In the second place, the condition is sufficient. In order to prove this we have only to show that it involves X = A. But if X < A then there are, however small 8 may be, infinitely many values of n such that 0(«)<X + 8 and infinitely many such that (j>(n)>A-B; and therefore we can find values of Ui and M2) each greater than any assigned number n^, and such that <^ («2) - ^ («i) > A - X - 2S, which is greater than -^ (A - X) if 8 is small enough. This plainly contradicts the inequality (1). Hence X = A, and so (f> [n) tends to a limit. 84. Unbounded functions. So far we have restricted ourselves to bounded functions; but the 'general principle of convergence' is the same for unbounded as for bounded functions, and the words 'a bounded function' may be omitted from the enunciation of Theorem 1. In the first place, if (f> (n) tends to a limit I then it is certainly bounded ; for all but a finite number of its values are less than 1 + 8 and greater than 1-8. In the second place, if the condition of Theorem 1 is satisfied, we have whenever iii^n^ and ji2S»io- ^^^ ^s choose some particular value n^ greater than Mo- Then (>(%)- 8 < (kj) < (»i) + S when »2 S»o- Hence (f> (n) is bounded ; and so the second part of the proof of the last section applies also. * A number of simple proofs of this result are given by Hardy and Littlewood, " Some Problems of Diophantine Approximation", Acta Mathematica, vol. xxxvii. 83-85] POSITIVE INTEGRAL VARIABLE 153 The theoretical importance of the ' general principle of convergence ' can hardly be overestimated. Like the theorems of § 69, it gi%-es us a means o{ deciding whether a function (j> (n) tends to a limit or not, without requiring us to be able to tell beforehand what the limit, if it exists, must be ; and it has not the limitations inevitable in theorems of such a special character as those of § 69. But in elementary work it is generally possible to dispense ■with it, and to obtain all we want from these special theorems. And it will be foimd that, in spite of the importance of the principle, practically no applications are made of it in the chapters which follow.* We will only remark that, if we suppose that ^(jl) = S„ = Mi + M2+---+«ll. we obtain at once a necessary and sufficient condition for the convergence of an infinite series, viz : Theorem 2. The necessary and sufficient condition for the convergence of the series U1 + U2+... is thai, given any positive number S, it should he possible to find Uq so that I «n,+l + M„,+2 + ...+M„J< S for all values o/kj and n^ such that n2>MiS»!o- 85. Iiimits of complex functions and series of complex terms. In this chapter we have, up to the present, concerned ourselves only with real functions of n and series all of whose terms are real. There is however no difficulty in extending our ideas and definitions to the case in which the functions or the terms of the series are complex. Suppose that ^ (n) is complex and equal to p (n) + ia- (w), where p (n), a- (n) are real functions of n. Then if p (n) and a (n) converge respectively to limits r and s as n^>-co ,we shall say that d> (n) converges to the limit l = r + is, and write lim <^ (m) = I. Similarly, when Un is complex and equal to Vn + iWn, we shall say that the series «i + Wa + «3 + ■ • • is convergent and has the sum l = r + is, if the series V1 + V.2 + V3+ ..., W1+W2 + W3+... are convergent and have the sums r, s respectively. * A few proofs given in Ch. VIII can be simplified by the use of the principle. 154 LIMITS OF FUNCTIONS OF A [iV To say that Ui + Ui + Us+ ... is convergent and has the sum I is of course the same as to say that the sum Sn = Ui + U3+ ... +Un = (Vi+V2 + ...+V„) + i(Wi + W2+ ... + Wn) converges to the limit I as n^-oo . In the case of real functions and series we also gave definitions of divergence and oscillation, finite or infinite. But in the case of complex functions and series, where we have to consider the behaviour both of p (n) and of a (n), there are so many possibilities that this is hardly worth while. When it is necessary to make further distinctions of this kind, we shall make them by stating the way in which the real or imaginary parts behave when taken separately. 86. The reader will find no difficulty in proving such theorems as the following, which are obvious exbensions of theorems already proved for real functions and series. (1) If lim (fi{n)=l then lim <j){n + p)=l for any fixed value of p. (2) If U1 + U2 + ... is convergent and has the sum I, then a + b + c + ... + k + Ui + U2 + ... is convergent and has the sum a + b + c+ ... +k + l, and «j,+i + Up^^ + ... is convergent and has the sum l — Ui — u^—... —Up. (3) If lim (j)(n) — l and lim yjr (n) = m, then lim {(j) (n) + i|r (n)} =l + m. (4) If lim <j) (n) = I, then lim k(j) (n) = M. (5) If lim ^ (n) = I and lim \{r (n) = m, then lim ^ (n) ■^ (n) = Im. (6) If III + U2+ ... converges to the sum I, and V1 + V2+ ... to the sum m, then (itj + Vi) + (W2+ ^2) + ■•■ converges to the sum l+m. (7) If Ml + ^2 + . . . converges to the sum I then hui + ku2 + ■■■ converges to the sum kl. (8) It U1 + U2+U3+ ... is convergent then lim i<„ = 0. (9) If ifi + Mo + M3 + . . . is convergent, then so is any series formed by grouping the terms in brackets, and the sums of the two series are the same. 85, 86] POSITIVE INTEGRAL VARIABLE 155 As an example, let us prove theorem (5). Let <j>{n) = p(n) + icr(n), ■•^{n) = p' {n) + i(T' {n), l = r + is, m = r' + is'. Then p{n)-^r, cr{n)^-s, p'{n)^r', o-' (»)-*«'. But (j) (n) yjf (ra) = pp - (Tcr' + i (pa-' + pa), and pp' -<T<r'-i- }'!•' — ss', pa-' + p'o'-^rs' + r's ; so that (^ (w) >//■ (m) -»- ) v' - ss' + i{rs' + r's), i.e. <j) (n) yJA {n)-»-(r+is) (r' + is') = lni. The following theorems are of a somewhat different character. (10) In order that ^(n) = p (n) + ia- (n) should converge to zero as n-* QC , it is necessary and sufficient that I</>W1=n/[{pW+{'^W] should converge to zero. If p{n) and o-(m) both converge to zero then it is plain that Jip^-^-a-'^) does so. The converse follows from the fact that the numerical value of p or o- cannot be greater than ^/(p^ + c^). (11) More generally, in order that <f)(n) should converge to a limit I, it is necessary and sufficient that \^(n)-l\ should converge to zero. For (j){n) — l converges to zero, and we can apply (10). (12) Theorems 1 and 2 of §§ 83 — 84 are still true when <f) (n) and Un are complex. We have to show that the necessary and sufficient condition that ^{n) should tend to I is that |0(«2)-*(«i)l<8 (1) when «2>»i 2 Wo- lf <i> {n)-*l then p in)-*-r and o- {n)->-s, and so we can find numbers Mq' and jjq" depending on 8 and such that |p(»i2)-p(mi)|<P. k(«2)-<r(«i)l<JS, the first inequality holding when «2 > ki g «o'i and the second when % > «i S %"• Hence I ^ {n^)-<t>{n,) I g I p {ni)-p{ny) \ + \a- (nj) - (r (»i) |<8 when ?i2>?ii S«oi where Kq is *lie greater of %' and mo". Thus the condition (1) is necessary. To prove that it is sufficient we have only to observe that I P («2) - p (»i) I ^ I <^ ("2) - * ('h) I < S when n2>ni>no. Thus p (n) tends to a limit r, and in the same way it may be shown that o- (n) tends to a limit s. 156 LIMITS OF FUNCTIONS OF A [iV 87. The limit of ^" as n, ^ oo , « being any complex number. Let us consider the important case in which ^ (m) = z\ This problem has already been discussed for real values of z in §72. If z'" -* I then a"+i ^ I, by (1) of § 86. But, by (4) of § 86, 0»+l = zz^ -* zl, and therefore I = zl, which is only possible if {a) 1 = or (6) z=l. If = 1 then lira ^" = 1. Apart from this special case the limit, if it exists, can only be zero. Now \i z = r (cos 6 + i sin 6), where r is positive, then so that I «" I = r'". Thus | «"■ | tends to zero if and only if ?• < 1 ; and it follows from (10) of § 86 that lim «" = if and only if r < 1. In no other case does z'"' converge to a limit, except when z=\ and ^" -»- 1. 88. The geometric series I + z + z'^ + ... when z is complex. Since S„= l+« + ^2+ ... + 2"-l= (1 -2")/(l -«), unless z=\, when the value of s„ is n, it follows that the series 1 + z + z''+ ... is convergent if and only if r = \z\< 1. And its sum when convergent is 1/(1 — z). Thus ii z = r (cos +i sin 6) = r Cis 9, and j* < 1, we have 1 + z + z^+... = 1/(1 -rCise), or 1 + r- Cis 6) + ?-^ Cis 2(9 + ... = 1/(1 - r Cis ^) = (1 — r cos 6 + ir sin ^)/(l — 2r cos ^ + r^). Separating the real and imaginary parts, we obtain' l+rcos^ + r''cos20+ ... = (1 - j- cos ^)/(l - 2r- cos + r^, r sin ^ + r^ sm2d + ...=r sin 6/(1 - 2r cos 6 + r% provided r < 1. If we change into ^ + tt, we see that these results hold also for negative values of r numerically less than 1. Thus they hold when — 1 < r < 1. 87, 88] POSITIVE INTEGEAL VARIABLE 157 Examples XXXIII. 1. Prove directly that (t>{n) = ]-'^coand converges to when r< 1 and to 1 when »• = 1 and tf is a multiple of Stt. Prove further that if r = l and is not a multiple of 2n-, then <^(«) oscillates finitely; if r>l and 5 is a multiple of Ztt, then (p (?i)-*- + oo ; and if r>l and 5 is not a multiple of 2ir, then <ji (n) oscillates infinitely. 2. Establish a similar series of results for 4>{n)= »•" sin nd. 3. Prove that 2'"+2'" + i+ ... =a»'/(l-z), a'» + 22'» + i + 20"' + 2+ ... =z»»(l+2)/(l-0), if and only if 1 2 1 < 1. Which of the theorems of § 86 do you use ? 4. Prove that if — 1<»'<1 then l + 2,rcosd + 2r^coa2d+...=(l-r^)j{l-2rcose + r^). 5. The series '^ + T^- + ( -rr-) +••• 1+z \l+zj converges to the sum 1 / ( 1 - yj^ j = 1 + 2 if 1 2/(1 + 2) | < 1. Show that this condition is equivalent to the condition that 2 has a real part greater than —-J. MISCELLANEOUS EXAItlPLES ON CHAPTER IV. 1. The function 4){n) takes the values 1, 0, 0, 0, 1, 0, 0, 0, 1, ... when w=0, 1, 2, .... Express 0(m) in terms of m by a formula which does not involve trigonometrical functions. [4>{n) = i{l + { — l)" + i'^+{-i)"}.'] 2. If (j) (n) steadily increases, and \jr (n) steadily decreases, as n tends to 00 , and if ^(ri)><f> {n) for all values of n, then both <p (n) and iJa (n) tend to limits, and lim(^(?i)glimi/f (ra). [This is an intermediate corollary from §69.] 3. Prove that, if -^« = C4)"' ^W = (l-9'"' then (^ (» + 1) ><^ (re) and 1/^ (?i + 1) < 1/' (»)• [The first result has already been proved in § 73.] 4. Prove also that i/^ (m) >(^ (n) for all values of n : and deduce (by means of the preceding examples) that both cf) (n) and -^ (ra) tend to limits as n tends to 00 . * 5. The arithmetic mean of the products of all distinct pairs of positive integers whose sum is n is denoted by S,^.. Show that lim (SJn^) = ljQ. {Math. Trip. 1903.) * A proof that lim {\p (n) - (ji)} = 0, and that therefore each function tends to the limit e, will be found in Chrystal's Algehra, vol. ii, p. 78. We shall however prove this in Ch. IX by a different method. 158 LIMITS OF FUNCTIONS OF A [iV 6. Prove that ii Xi=i{ji;+{Alx)}, X2=i{xi+{Alxi)], and so on, .v and A being positive, then \imx„=s/A. [Prove first that ^^^^ = f^v^ Y" •] 7. If (j) (n) is a positive integer for all values of n, and tends to oo with n, then x^^"^ tends to if 0<a;<l and to +oo if .r>l. Discuss the behaviour pf ^*(«)^ as m-*-co , for other values of x 8 * If a„ increases or decreases steadily as n increases, then the same is true of (ai + 02 + . . . + Hn)!^- 9. If x;n + i = iJ{k+3:„), and k and Xi are positive, then the sequence xi, x^, Xi, ... is an increasing or decreasing sequence according as Xi is less than or greater than a, the positive root of the equation x''=x + k; and in either case .Tn-s-o as m-*-oo . 10. If Xn + i = hl(l+.Vn), and k and x^ are positive, then the sequences Xi,X3,x^, ... and X2,x^,x^, ... are one an increasing and the other a decreasing sequence, and each sequence tends to the limit a, the positive root of the equation x''+x=k. 11. The function f(x) is increasing and continuous (see Oh. V) for all values of x, and a sequence Xi, x<i, X3, ... is defined by the equation Xn + i=f{x„). Discuss on general graphical grounds the question as to whether a?„ tends to a root of the equation x=f{x). Consider in particular the case in which this equation has only one root, distinguishing the cases in which the curve y=f{x) crosses the line y=x from above to below and from below to above. 12. If .«;i , a;2 are positive and x„+i=\ (x„ + a;„ _ 1), then the sequences XijX^, xs, ... and X2, Xi, x^, ... are one a decreasing and the other an increasing sequence, and they have the common limit ^{xi + ^x^). 13. Draw a graph of the function y defined by the equation ,. x^ainiirx+x^ ,,y,T m- -.n/M \ y= lim gi— . (Math. Tnp. 1901.) 14. The function y= lim r-= — «-*.co l+nam^TTX is equal to except when *■ is an integer, and then equal to 1. The function lim ^(x) + n(t>{x)sm^nx n-a-x l+Jisin^n-^ is equal to (^ {x) vmless x is an integer, and then equal to >\r {x). 15. Show that the graph of the function y:» lim ^'''^W+^""'/^W * Exs. 8—12 are taken from Bromwich's Infinite Series* POSITIVE INTEGRAL VARIABLE 159 is composed of parts of tlie graphs of <f){x) and ■\/'(a.'), together with (as a rule) two isolated points. Is y defined when (a) a;=l, (b) x= -1, (c) « = 0? 16. Prove that the function y which is equal to when n; is rational, and to 1 when x is irrational, may be represented in the form i/= lim s^?i{sin^(m! 7r.r)}, where sgnx— lim (S/tt) arc tan ijix), as in Ex. xxxi. 14. [If x is rational then sin^ (m ! tt*'), and therefore sgn {sin^ (m ! nx)}, is equal to zero from a certain value of m onwards : if X is irrational then sin^ {m ! nx) is always positive, and so sgn {sin^ (m ! nx)} is always equal to 1.] Prove that y may also be represented in the form 1 — lim [lim {cos (m ! nx)}^"'}. 17. Sum the series 00 1 00 1 [Since 1 =.1 / 1 1 I n 1 _ 1 f 1 1 1 we have 2^(^^i)...(^+^) -^|x . 2.../(; in + l){n + 2)...(n + k)j •^"•^"^ !.(. + l)...(.+>t-) = ^l!)-^ 18. If |0|<|ai,then — =--('! + - + -„ + ...) ; andif |0|>|a|,then ^= ^ (l + f +|'+ •••) • 19. Expansion of {Az+B)/{az^+2hz+e) in powers of 2. Let a, /3 be. the roots of az^ + 2,bz + c=0, so that az^ + 2bz+c=a{z- a){z-fi). We shall suppose that A, B,a,b, c are all real, and a and /3 unequal. It is then easy to verify that Az+B ^ 1 / Aa+B _ A^ + B\ az^ + 2bz + c a{a-^) \ z-a 2-/3 / There are two cases, according as b^>ac or 6^ < ac. (1) li b^>ac then the roots a, ^ are real and distinct. If |2| is less than either \a\ or |(3| we can expand 1/(2 -a) and 1/(2-0) in ascending powers of 2 (Ex. 18). If 1 2 1 is greater than either | a | or | /3 1 we must expand in descending powers of z; while if |2| lies between \a\ and |/3| one fraction must be ex- panded in ascending and one in descending powers of z. The reader should write down the actual results. If |2| is equal to \a\ or |0| then no such expansion is possible. 160 LIMITS OF FUNCTIONS OF A [iV (2) If V^Kac then the roots are conjugate complex numbers (Oh. Ill § 43), and we can write a = pCis<^, /3=pCis(-0), where p^ = al3 = c/a, p cos = ;^ (a + 0) = - b/a, so that cos ^ = - ^{pjac\ sin<^=V{l-(iV««)}. If |2|<p then each fraction may be expanded in ascending powers oiz. The coefficient of 2" will be found to be ^psinm<^ + 5sin |(m+l)0} ap^^^ sin (^ If |2|>p we obtain a similar expansion in descending powers, while if |z|=p no such expansion is possible. 20. Show that if | z | < 1 then l + 22 + 3z'''+... + (?H- 1)2"+.. . = 1/(1 -2)2. [The sum to n terms is .-; r„ - = .1 ■■ (1 — 2)'' 1 — 2 -■ 21. Expand Lj{z-af in powers of 2, ascending or descending according as 1 2 1 < I a I or | 2 1 > | a |. 22. Show that if 6^= osc and | as | < | 6 1 then Az + B _" <^2 + 262 + C~o^''^" vi\iQXG pn={{-aYlh^*^] {{n + \)aB-nhA}; and find the corresponding ex- pansion, in descending powers of 2, which holds when |a2|>|6|. 23. Verify the result of Ex. 19 in the case of the fraction 1/(1 +z^). [We have l/(l+2'') = S2»sin{i(re + l)7r} = l-22+s*-...-j 24. Prove that if i2|<l then 25. Expand (1 +2)/(l +22), (1 +z^)j{\ +z^) and (1 +2-|-22)/(l +2*) in ascend- ing powers of 2. For what values of 2 do your results hold ? 26. If a/(a -f 62 -1- cz^) = 1 +piz \'p^z^ -)-... then a-fC2 a^ W'p^z-\-'p.^z^-^...-- a-cz a'- (62 _ 2ac) z -t- d^z^ ' {Math. Trip. 1900.) 27. If lims„ = 2 then )l-*-CO lim ^'+^^+-+^"=;. [Let Sn = l + t„. Then we have to prove that {ti + t2 + ... + Qln tends to zero if t,, does so. POSITIVE INTEGRAL VARIABLE ]61 We divide the numbei'S ti, t^, ... *„ into two sets ti, h, ..., tp and tp + i, 'p + 2> ■••) tn- Here we suppose that /> is a function of n which tends to oo as m-»co, but more slowly than n, so that p-*-oo andjo/w-a-O : e.g. we might suppose p to be the integral part of yjn. Let S be any positive number. However small 8 may be, we can choose «o so that fp + i, <j,+2, ..., «,i are all numerically less than JS when w^Wq, and so But, if A is the greatest of the moduli of all the numbers t-^, t^, ..., we have \(ti + t2 + ... + tp)/n\<pA/n, and this also will be less than JS when to^Mqi if n^ is large enough, since p/n -*- as ?i-»- 00 . Thus \{ti + t2 + ... + t„)jn\^\{ti + t2 + ... + tp)jn\ + \{tp^^+... + t„)ln\<8 when ?i>?io ; which proves the theorem. The reader, if he desires to become expert in dealing with questions about limits, should study the argument above with great care. It is very often necessary, in proving the limit of some given expression to be zero, to split it into two parts which have to be proved to have the limit zero in slightly different ways. When this is the case the proof is never very easy. The point of the proof is this : we have to prove that {ii + i2+... + t„)ln is small when n is large, the fa being small when their suffixes are large. We split up the terms in the bracket into two groups. The terms in the first group are not all small, but their number is small compared with n. The number in the second group is not small compared with n, but the terms are all small, and their number at any rate less than », so that their sum is small compared with n. Hence each of the parts into which {t^+t2+... + t„)JH has been divided is small when n is large.] 28. If (j){n)-(j>{n-l)-*'l as ?i-«-oo, then (ft (n)i7i-»-l. [If cj) (n)=Sj^+S2 + ...+Sn then (w)-0 (ra- l)=s,„ and the theorem re- duces to that proved in the last example.] 29. If «n= i {1 - ( - 1)"}) so that Sn is equal to 1 or according as n is odd or even, then (Sj-l-«2 + -"+*iO/™-*"2 ^ ?i-i-oo. [This example proves that the converse of Ex. 27 is not true : for s„ oscillates as n-^x .] 30. If c„, «„ denote the sums of the first n terms of the series 1 + cos 5+ cos 25+..., sin5+sin25+..., then lim(ci + C2+-+«n)/»=0> lim(si + S2 + ... + 5„)/M = icotii9. 11 CHAPTER V LIMITS OF rUNCTIONS OF A CONTINUOUS \^AIIIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS 89. Limits as a; tends to oo . We shall now return to functions of a continuous real variable. We shall confine our- selves entirely to one-valued functions*, and we shall denote such a function by (p (x). We suppose x to assume successively all values corresponding to points on our fundamental straight line A, starting from some definite point on the line and progressing always to the right. In these circumstances we say that x tends to infinity, oi to oo , and write x^-oo . The only difference between the ' tending of m to oo ' discussed in the last chapter, and this ' tending of a; to oo ', is that x assumes all values as it tends to 00 , i.e. that the point P which corresponds to x coincides in turn with every point of A to the right of its initial position, whereas n tended to oo by a series of jumps. We can express this distinction by saying that x tends continuously to oo . As we explained at the beginning of the last chapter, there is a very close correspondence between functions of x and functions of n. Every function of n may be regarded as a selection from the values of a function of x. In the last chapter we discussed the peculiarities which may characterise the behaviour of a function (^ (n) as n tends to oo . Now we are concerned with the same problem for a function ^ («) ; and the definitions and theorems to which we are led are practically repetitions of those of the last chapter. Thus corresponding to Def 1 of § 58 we have : * Thus Jx stands in this chapter for the one-valued function + ^x and not (as in § 26) for the two-valued function whose values are + Jx and - ^Ix. 89] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 163 Definition 1. The function («) is said to tend to the limit I as X tends to co if, when any positive number B, however small, is assigned, a number «„ (S) can be chosen such that, for all values of X equal to or greater than x^ (8), ^ {x) differs from I by less than S, i.e. if \^{x)-l\<S when x = Xq (S). When this is the case we may write lim (j) (x) = I, or, when there is no risk of ambiguity, simply lim <p {x) = I, or 4> {x) -*■ I. Similarly we have : Definition 2. The function 4> («) is said to tend to oo with X if, when any number A, hoiuever large, is assigned, we can choose a number a;o(A) such that ^ («) > A whe'ii X = Xo (A). We then write <j> (x) -^ 00 . Similarly we define ^(x)^- — <x>*. Finally we have : Definition 3. If the conditions of neither of the two preceding definitions are satisfied, then <f> (x) is said to oscillate as x tends to CO . If \^ (x) I is less than some constant K when x = x^\, then 4>{p) is said to oscillate finitely, and otherwise infinitely. The reader will remember that in the last chapter we con- sidered very carefully various less formal ways of expressing the facts represented by the formulae <^ (n) -* Z, <f)(n)-^<x> . Similar modes of expression may of course be used in the present case. Thus we may say that <j) (x) is small or nearly equal to I or large when X is large, using the words ' small ', ' nearly ', ' large ' in a sense similar to that in which they were used in Ch. 17. * We shall sometimes find it convenient to write +00, a;^»- + oo,0(a;)^»- + a) instead of 00 , x-^-x , (a;) -»■ co . t In the corresponding definition of § 62, we postulated that | (j> (n) | <.K for all values of n, and not merely when n > Wj . But then the two hypotheses would have been equivalent; for if \<t>(n)\<K when ng«o> *lisn \<)i(n)\<K' for all values of n, where K' is the greatest of 0(1), 0(2), ... , 0(?io-l) and K. Here the matter is not quite so simple, as there are infinitely many values of x less than x^. 11—2 164 LIMITS OF FUNCTIONS [v Examples XXXIV. 1. Consider the behaviour of the following functions as A-^x : Ijx, H-(l/«), x\ ^, [4 a;-[xl M + Vk" Ml- The first four functions correspond exactly to functions of n fully dis- cussed in Ch. IV. The graphs of the last three were constructed in Ch. II (Exs. XVI. 1, 2, 4), and the reader will see at once that [«]-»- oo , a; - [x] oscillates finitely, and [x] + sj{x - [a;]} -*■ oo . One simple remark may be inserted here. The function <l>{x)=x-{x'\ oscillates between and 1, as is obvious from the form of its graph. It is equal to zero whenever x is an integer, so that the function <^(») derived from it is always zero and so tends to the limit zero. The same is true if <^ {x) = sin XTT, ^ (n) = sin Kff = 0. It is evident that 4>{x)^^l or <^(.r)-»-Qo or <^{x)^--'x> involves the corre- sponding property for («), but that the converse is by no means always true. 2. Consider in the same way the functions : {sva.XTt)jx, ^sin^TT, (^sin^Tr)^, tan^ir, acos'^xir-^-haiv^XTr, illustrating your remarks by means of the graphs of the functions. 3. Give a geometrical explanation of Def. 1, analogous to the geometrical explanation of Ch. IV, § 59. 4. If (^ {x) -*■ I, and I is not zero, then (.r) cos x-n- and <j) (x) sin xn oscillate finitely. If (f>{x)-»-co or ^ («)-»-— oo, then they oscillate infinitely. The graph of either function is a wavy curve oscillating between the curves y=(^ (^) and y= — (^). 5. Discuss the behaviour, as x-^m , of the function y=f{x)C0Si^XTr+F{x)sVD^XTr, where f{x) and F{x) are some pair of simple functions {e.g. x and x^). [The graph of y is a curve oscillating between the curves y=f{p:), y=F{x).'] 90. Limits as x tends to — oo . The reader vfill have no difficulty in framing for himself definitions of the meaning of the assertions ' x tends to — oo ', or ' a; ^- — co ' and lim </)(«) = ?, <^ (a;) -* 00 , ^ («)-»■ — oo . a:-*- — oo In fact, if x = -y and cj) (x) = <j) (- y) = yjr {y), then y tends to oo as « tends to — co , and the question of the behaviour of j) {x) as X tends to — oo is the same as that of the behaviour of x/r (y) as y tends to oo . 89-93] OF A CONTINUOUS VARIABLE 165 91. Theorems corresponding to those of Ch. IV, §§ 63—67. The theorems concerning the sums, products, and quotients of functions proved in Ch. IV are all true (with obvious verbal alterations which the reader will have no difficulty in supplying) for functions of the continuous variable x. Not only the enunciations but the proofs remain substantially the same. 92. Steadily increasing or decreasing functions. The definition ■which corresponds to that of § 69 is as follows : the function cf) {x) will he said to increase steadily with x if 4> (^i) =4' i^i) whenever x.i>Xi. In many cases, of course, this condition is only satisfied from a definite value of X onwards, i.e. when x.i>Xy =x^. The theorem which follows in that section requires no alteration but that of n into x : and the proof is the same, except for obvious verbal changes. If (ji{x^>(j){xi), the possibility of equality being excluded, whenever X2>Xi, then ^(a;) will be said to be steadily increasing in the stricter sense. We shall find that the distinction is often important (cf. §§ 108 — 109). The reader should consider whether or no the following functions increase steadily with x (or at any rate increase steadily from a certain value of X onwards); x^-x, .r+sin^*, ^ + 2sini!;, x^ + 2smx, [x], [a;]+sin,t', M 4. ^y^ - [x]} . All these functions tend to qo as a; ^- 00 . 93. Limits as x tends to 0. Let (cc) be such a function of a) that lim ^ {x) = I, and let y = l/co. Then say. As x tends to cc , y tends to the limit 0, and yjr (y) tends to the limit I. Let us now dismiss x and consider 1^ (y) simply as a function of y. We are for the moment concerned only with those values of y which correspond to large positive values of x, that is to say with small positive values of y. And ■^ (y) has the property that by making y sufficiently small we can make -^{y) differ by as little as we please from h To put the matter more precisely, the statement expressed by lim <f,(x) = l means that, when any positive number B, however small, is assigned, we can choose Xo so that \<f)(x) — l\<S for all values of x greater than or equal to Xo. But this is the same thing as saying that we can choose y^ — ija;^ so that I -i/r (2/) - Z I < S for all positive values of y less than or equal to yo. We are thus led to the following definitions : 166 LIMITS OF FUNCTIONS [v A. //, when any positive number S, however small, is assigned, we can choose y„ (8) so that \<}>(y)-l\<B when < y S j/o {^), if^i^ ^^ 5*2/ 1^^^ 4> iv) t^^^^ to ^'^6 limit I as y tends to by positive values, and we write Km 4> (y) = I. B. If, when any number A, however large, is assigned, we can choose 2/o (A) so that </)(y)>A when 0<y ^yo (A), then we say that <f> (y) tends to co as y tends to by positive values, and we write (^ (2/) -* CO . We define in a similar way the meaning of ' (j) (y) tends to the limit I as y tends to by negative values ', or ' lim ^(y) = l when y~^—0'. We have in fact only to alter < y g yo (S) to — y„{B) ^y<0 in definition A. There is of course a corresponding analogue of definition B, and similar definitions in which ^(2/)-*- 00 as 2/ -^ + or y -»- — 0. If lim ^(y) = l and lim <f){y) = l, we ^vrite simply lim <f> (y) = I. This case is so important that it is worth while to give a formal definition. If, when any positive number S, however small, is assigned, we can choose y^ (8) so that, for all values of y different from zero but numerically less than or equal to y^ (S), ^ (y) differs from I by less than h, then we say that ^ {y) tends to the limit I as y tends to 0, and write lim (/> iy) = I. So also, if ^ (t/) -* 00 as y -* + and also as y -* — 0, we say that (^ {y) ^-00 as y -»- 0. We define in a similar manner the statement that <f>(y)-^— <x> as y -* 0. 93, 94] OF A CONTINUOUS VARIABLE 167 Finally, if <},(y) does not tend to a limit, or to oo , or to - 00 , as y •-*• + 0, we say that (f> (y) oscillates as y-^+0, finitely or infinitely as the "case may be; and we define oscillation as y -* — in a similar manner. The preceding definitions have been stated in terms of a variable denoted by y : what letter is used is of course immaterial, and we may suppose x written instead of y throughout them. 94 Limits as x tends to a. Suppose that if>{y)^l as y -^0, and write y = x-a, <f)(y) = ^(x-a)=y{r (x). If 3/ -* then x-^a and i/r (x) -* I, and we are naturally led to write lim sfr (x) = I, or simply lim yp- (x) = I or -^ (x) ^ I, and to say that i^ (x) tends to the limit I as x tends to a. The meaning of this equation may be formally and directly defined as follows: if, given S, we can always determine e(S) so that \<li(x)-l\<S when 0<\x — a\^ e (S), then lim (f> (x) = I. By restricting ourselves to values of x greater than a, i.e. by replacing 0<\x - a\s.e{h) hy a<xS.a + e (S), we define ' ^ (x) tends to I when x approaches a from the right', which we may write as lim ^ (x) = I. In the same way we can define the meaning of lim ^ (x) = I, -ffl-O Thus lim (j)(x) = l is equivalent to the two assertions x-*'a lim (f> {x) = I, lim <^ (x) = I, -a+O We can give similar definitions referring to the cases in which 4>(x)-^<xi or (j)(x)-^—(x> as x-^a through values greater or less than a; but it is probably unnecessary to dwell further on these definitions, since they are exactly similar to those stated above in 168 LIMITS OF FUNCTIONS [v the special case when a=0, and we can always discuss the behaviour of <}> (x) as x-*a by putting x — a = y and supposing that 2/-*0. 95. Steadily increasing or decreasing functions. If there is a number E such that (j) {x') ^(j) {x") whenever a-e<ic'<x"<a + f, then (ji{x) will be said to increase steadily in the neighbourhood of x=a. Suppose first that x<a, and put y=l/(a-^). Then y-»-» as x-»-a—<), and (l>ix) = yjf(2/) is a steadily increasing function of y, never greater than {/>(«). It fohows from § 92 that (x) tends to a Umit not greater than ^ (a). We shall write lim (j>{x) = <l){a+0). We can define <j)(a-0) in a similar manner ; and it is clear that (^(a-O)S0(a)<0(a + O). It is obvious that similar considerations may be applied to decreasing functions. If if>{!x^)<(\>{x"), the possibility of equality being excluded, whenever a — e<a,''<a;''<a + e, then ^{x) will be said to be steadily increasing in the stricter sense. 96. Limits of indetermination and the principle of convergence. All of the argument of §§ 80 — 84 may be applied to functions of a con- tinuous variable x which tends to a limit a. In particular, if (\){x) is hounded in an interval including a {i.e. if we can find c-, H, and K so that . H<4i (a;)<^ when a — e<a;ga+ e)*, then we can define X and A, the lower and upper limits of indetermination of <^ (x) a,s x-»-a, and prove that the necessary and sufficient condition that {x)-»-l as x-»-a is that \ = A = l. We can also establish the analogue of the principle of convergence, i.e. prove that the necessary and suficient condition that {x) should tend to a limit as x-*-a is that, when S is given, we can choose e (8) so that \ (.^2) - (■^'1) I < ^ when 0<\x2-a\<\xi-a\S:f{8). Examples XXXV. 1. If 4>{x)-^l, f{x)-»-l', a,s x-i-a, then (l)(x) + yjf(x)-^l + l', <l>{x)y^{x)^-ll', and (\i{x)l-<^{x)-^ljl\ unless in the last case l' = 0. [We saw in § 91 that the theorems of Oh. IV, §§ 63 et seq. hold also for functions of X' when x^^ 00 or«-»- — 00. By putting .*'=l/y we may extend them to functions of y, when y-s-O, and by putting ?/ =2 — a to functions of z, when z-»-a. * For some further discussion of the notion of a function bounded in an interval see § 102. 94-97] OF A CONTINUOUS VARIABLE 169 The reader should however try to prove them directly from the formal definition given above. Thus, in order to obtain a strict direct proof of the first result he need only take the proof of Theorem I of § 63 and write throughout x for n, a for qo and 0<\x - a\<e for n^Uf,.'] 2. If OT is a positive integer then ^'"^s-O as a;-9-0. 3. If TO is a negative integer then .^;"-»-+co as a--»-+0, while a,-™-»--oo or ^™^»- + co asa;-»--0, according as m is odd or even. If ■»i=0 then a;™=l and a^-fl. 4. Mm {a + hx + cx'^+ ... +kx'^) = a. !e-»-0 5. lim {(a + 6^ +...+ ^^/(a + 0a; +...+K^)| = a/a, unless (1 = 0. Ifa = and a 4=0, |3=|=0, then the function tends to +qo or — oo, as i!;-»- + 0, according as a and have like or unlike signs; the case is reversed if .^•-»--0. The case in which both a and a vanish is considered in Ex. xxxvi. 5. Discuss the cases which arise when a =1=0 and more than one of the first coefiicients in the denominator vanish. 6. lim ^'" = a"', if to is any positive or negative integer, except when a = x~*-a And m is negative. [If m>0, put x=y + a and apply Ex. 4. When m<0, the result follows from Ex. 1 above. It follows at once that lim P {x) = P (a), if P (x) is any polynomial.] 7. lim R (x) = R (o), if R denotes any rational function and a is not one x-*-a of the roots of its denominator. 8. Show that lim x"^=a"' for all rational values of m, except when a=0 x->-a and m is negative. [This follows at once, when a is positive, from the in- equaUties (9) or (10) of § 74. For \x!^-a'^\<H\x-a\, where E is the greater of the absolute values of mx/"^~'^ and ma"*~' (cf. Ex. xxviii. 4). If a is negative we write x= -y and a= -b. Then lim :i;»»=lim ( - l)™y'"= ( - l)'"5'» = a'».] 97. The reader will probably fail to see at first that any proof of such results as those of Exs. 4, 5, 6, 7, 8 above is necessary. He may ask ' why not simply put a; = 0, or a; = a ? Of course we then get a, a/a, a™, P (a), R (a) ' It is very important that he should see exactly where he is wrong. We shall therefore consider this point carefully before passing on to any further examples. The statement lim </> (x) = I a;-»-0 is a statement about the values of ej){x) when x has any value 170 LIMITS OF FUNCTIONS [v distinct from but differing by little from zero *. It is not a statement about the value of ^ (x) when x = 0. When we make the state- ment we assert that, when x is nearly equal to zero, ^ (x) is nearly- equal to I. We assert nothing whatever about what happens when X is actually equal to 0. So far as we know, (j>{x) may not be defined at all for x = 0; or it may have some value other than I. For example, consider the function defined for all values of x by the equation (J3 (x) = 0. It is obvious that lim^(a;) = (1). Now consider the function i|r (x) which differs from ^ (x) only in that 1^ («) = 1 when x = 0. Then limi|r(a;) = (2), for, when x is nearly equal to zero, yjr (x) is not only nearly but exactly equal to zero. But yfr{0)=l. The gra.ph of this function consists of the axis of x, with the point x = left out, and one isolated point, viz. the point (0, 1). The equation (2) expresses the fact that if we move along the graph towards the axis of y, from either side, then the ordinate of the curve, being always equal to zero, tends to the limit zero. This fact is in no way affected by the position of the isolated point (0, 1). The reader may object to this example on the score of artificiality : but it is easy to write down simple formulae repre- senting functions which behave precisely like this near x = 0. One is ^}r(x) = [l-x■^l where [1 — a;^] denotes as usual the greatest integer not greater than 1-x^. For if « = then i/r (a;) = [1] = 1 ; while if 0<a;<l, or - 1 < a; < 0, then < 1 — i?;^ < 1 and so yjr (x) = [1 - x^] = 0. Or again, let us consider the function y = xjx already discussed in Ch. II, § 24, (2). This function is equal to 1 for all values of x save x = 0. It is not equal to 1 when x = 0: it is in fact not defined at all for x = 0. For when we say * Thus in Def. A of § 93 we make a statement about values of y such that 0<2/S2/o) tlie first of these inequalities being inserted expressly in order to exclude the value j/ = 0. 97] OF A CONTINUOUS VARIABLE 171 that <fi (x) is defined for a; = we mean (as we explained in Ch. II, I.e.) that we can calculate its value for x = by putting a; = in the actual expression of ^ («). In this case we cannot. When we put x = in <j)(x) we obtain 0/0, which is a meaningless expression. The reader may object 'divide numerator and de- nominator by X '. But he must admit that when x = this is impossible. Thus y = x/x is a function which differs from y=l solely in that it is not defined for x=0. None the less lim (x/x) = 1, for x/x is equal to 1 so long as x differs from zero, however small the difference may be. Similarly (f> (x) = {{x -(- 1)^ — l}/a; = a; + 2 so long as x is not equal to zero, but is undefined when a; = 0. None the less lim (x) = 2. On the other hand there is of course nothing to prevent the limit of <f) (x) as x tends to zero from being equal to <f) (0), the value of ^ (x) for a; = 0. Thus if <p{x)=x then (0) = and lim ^ {x) = 0. This is in fact, from a practical point of view, i.e. from the point of view of what most frequently occurs in applications, the ordinary case. Examples XXXVI. 1. lim {x' - a?)l{x -a) = 2a. .2. lim {af^ — a'^)l{x — a) = ma,™'~\ if to is any integer (zero included). x-*-a 3. Show that the result of Ex. 2 remains true for all rational values of m, provided a is positive. [This follows at once from the inequalities (9) and (10) of § 74.] 4. lim (.r^-2^+l)/(«3-3^2+2) = l. [Observe that x-l is a factor of both numerator and denominator.] 5. Discuss the behaviour of (^(^) = (a„«'''+ai^+i+ ... +aiA-"'+*)/(6o^" + &i *•""''+■■. +&!^'""0 as X tends to by positive or negative values. \li m>n,\\m^{x)'=0. If m = n, lim (j}{x)='aol bg. If ot<?i and n-m is even, (x) -► + oo or <^ (^) -s- - oo according as a„/6o > or a-o/bo < 0. lim<n and ?i-TOisodd, 0(.v)^ + oo as^^ + Oand<^(.r)^ — oo as.j;^-0, or ^ (.■!;) ^^ — co a,s x^ + a,nd (i)(x)^+'X> as x-^ — 0, according as ao/&o>0 or ao/*o<0-] 172 LIMITS OF FUNCTIONS [V 6. Orders of smallness. When x is small x^ is very much smaller, x^ much smaller still, and so on : in other words lim(.r2te) = 0, lim (.r8/^2) = o, .... a;-*-0 a;-»-0 Another way of stating the matter is to say that, when x tends to 0, x\ x^, ... all also tend to 0, but x^ tends to more rapidly than x, ifi than x"^, and so on. It is convenient to have some scale by which to measure the rapidity with which a function, whose limit, as x tends to 0, is 0, diminishes with x, and it is natural to take the simple functions x, ic\ x\ ... as the measures of our scale. We say, therefore, that <^ {x) is of the first order of smallness if <^ {x)jx tends to a limit other than as x tends to 0. Thus 2x+3x^+x'' is of the first order of smallness, since lim {2x + 3x^ + x'')lx = % Similarly we define the second, third, fourth, ... orders of smallness. It must not be imagined that this scale of orders of smallness is in any way complete. If it were complete, then every function <^ {x) which tends to zero with x would be of either the first or second or some higher order of smallness. This is obviously not the case. For example (f>(x) = x'"^ tends to zero more rapidly than x and less rapidly than x^. The reader may not unnaturally think that our scale might be made complete by including in it fractional orders of smallness. Thus we might say that x''!^ was of the |^th order of smallness. We shall however see later on that such a scale of orders would still be altogether incomplete. And as a matter of fact the integral orders of smallness defined above are so much more important in applications than any others that it is hardly necessary to attempt to make our definitions more precise. Orders of greatness. Similar definitions are at once suggested to meet the case in which (j) (x) is large (positively or negatively) when x is small. We shall say that (x) is of the ^'th order of greatness when x is small if (j) (^)/a;~*=«* (x) tends to a limit different from as ^ tends to 0. These definitions have reference to the case in which x-^0. There are of course corresponding definitions relating to the cases in which a;-*- oo or ^-»-a. Thus if a^<\){x) tends to a limit other than zero, as a^-s-oo , then we say that <f>{x) is of the ^th order of smallness when x is large: while if (x - a)'' (ji (x) tends to a limit other than zero, as x-^a, then we say that {x} is of the kth order of greatness when x is nearly equal to a. *7. limV(l+A')=l™>/(l--'''") = l- [Put l+x=7/ or l-x=i/, and use Ex. XXXV. 8.] 8. lim{^(l+«)-^(l-^)}/^=l. [Multiply numerator and denominator hy^{l+.v) + J{l-x).] * In the examples which follow it is to be assumed that limits as .t -* are required, unless (as iu Exs. 19, 22) the contrary is explicitly stated. 97] OF A CONTINUOUS VARIABLE 173 9. Consider the behaviour of {^/(l +x'^) - ^{1 - x'^)]lx^ as x-*-0, m and n being positive integers. 10. \imy{l+x+x'^)-l}lx=^. J{l+x)-J{l+x^^) ^^- ^''^^{i-x^)-J{l-x) ^• 12. Draw a graph of the function Has it a hmit as x-*-0'i [Here «/ = l except for x=l, ^, ^, J, when y is not defined, and y-^l as .«-*-0.] .„ ,. sin^i; 13. lim = 1. X [It may be deduced from the definitions of the trigonometrical ratios* that if X is positive and less than ^tt then sin:K<:c<tan^ sin a; , or cos.r< <1 X OT 0<1 <l-cos:i;=2sin2Ja'. X (sin,»\ „ , ,. sin^ 1--— =0,and hm -—=1- X J a;^+o ^ T^g smjK j^ ^^ ^^^^ function, the result follows.] X 14 limi— ^®-=i. 15. lim?^J^=a. Is this true if a = 0? x^ '■ x arc sin .r ■, tt. x • n 16. lim = 1. [Put^ = smy.] , . tan ax , . arc tan cum 17. hm = a, hm =a. X X ,„ ,. COSeCK-COt^U , ,„ i;„ l+C0 3"'^_]L 18. hm = i. 19. ]i^^^^,— _i. * The proofs of the inequalities which are used here depend on certain pro- perties of the area of a sector of a circle which are usually taken as geometrically intuitive ; for example, that the area of the sector is greater than that of the triangle inscribed in the sector. The justification of these assumptions must be postponed to Ch. VII. 174 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [V 20. How do the functions sm(l/a:), (l/x)sin(l/^), a;ain{llx) behave as x^-0 ? [The first oscillates finitely, the second infinitely, the third tends to the limit 0. None is defined when ^=0. See Exs. xv. 6, 7, 8.] 21. Does the function ^=('"'3/(''°9 tend to a limit as x tends to ? [No. The function is equal to 1 except when sin(l/.j;) = 0; i.e. wheua;=l/n-, 1 /2n-, ...,- I/tt, - I/Stt For these values the formula for y assumes the meaningless form 0/0, and y is therefore not defined for an infinity of values of x near x = 0.'] 22. Prove that if m is any integer then [x]-*-m and .^'-[*]-i>-0 as .j:-*-m+0, ai;d [j;]^*-m- 1, x — \x\-»-\ as a--*-m-0. 98. Continuous functions of a real variable. The reader has no doubt some idea as to what is meant by a continuous curve. Thus he would call the curve G in Fig. 29 continuous, the curve C generally continuous but discontinuous for x = ^' and X = f. Either of these curves may be regarded as the graph of a function ^ («). It is natural to call a function continuous if its graph is a continuous curve, and otherwise discontinuous. Let us take this as a provisional definition and try to distinguish more precisely some of the properties which are involved in it. In the first place it is evident that the property of the function y = <j}(x) of which C is the graph may be analysed into some property possessed by the curve at each of its points. To be able to define continuity for all values of x we must first define continuity for any particular value of x. Let us there- fore fix on some particular value of x, say the value x = ^ 97, 98] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 175 corresponding to the point P of the graph. What are the characteristic properties of <j) (sc) associated with this value of x ? In the first place <^ (x) is defined for x = ^. This is obviously essential. If ^ (^) were not defined there would be a point missing from the curve. Secondly ^ («) is defined for all values of x near x = ^\ i.e. we can find an interval, including a; = ^ in its interior, for all points of which ^ {x) is defined. Thirdly if x approaches the value ^ from either side then ^ (x) approaches the limit (f> (^). The properties thus defined are far from exhausting those which are possessed by the curve as pictured by the eye of common sense. This picture of a curve is a generalisation from particular curves such as straight lines and circles. But they are the simplest and most fundamental properties : and the graph of any function which has these properties would, so far as drawing it is practically possible, satisfy our geometrical feeling of what a continuous curve should be. We therefore select these properties as embodying the mathematical notion of continuity. We are thus led to the following Definition. The function ^ (x) is said to be continuous for x = ^ if it tends to a limit as x tends to f fivin either side, and each of these limits is equal to <f> (^). We can now define continuity throughout an interval. The function ^(x) is said to be continuous throughout a certain interval of values of x if it is continuous for all values of x in that interval. It is said to be continuous everywhere if it is continuous for every value of x. Thus \oc\ is continuous in the interval (e, 1 — e), where e is any positive number less than \ ; and 1 and x are continuous everywhere. If we recur to the definitions of a limit we see that our definition is equivalent to ' ^ {x) is continuous for x=^ if, given S, we can choose e (8) so that | <^ («) — ^(f) I < S t/O S | a; — ^ | fi^ (S)'. We have often to consider functions defined only in an interval (a, b). In this case it is convenient to make a slight and obvious 176 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [V change in our definition of continuity in so far as it concerns the particular points a and b. We shall then say that <f> (x) is con- tinuous for a; = a if ^ (a + 0) exists and is equal to (f) (a), and for a; = 6 if (/) (6 — 0) exists and is equal to <f) (b). 99. The definition of continuity given in the last section may be illustrated geometrically as follows. Draw the two horizontal lines y = <f) {^) - S and y = cf> (^) + S. Then | (^ («)-</> (|) | < S ex- presses the fact that the point on the curve corresponding to a; lies -y=<p{i)-s f + e Fig. 30. between these two lines. Similarly \x— ^\ ^e expresses the fact that X lies in the interval (^— e, ^-l-e). Thus our definition asserts that if we draw two such horizontal lines, no matter how close together, we can always cut off a vertical strip of the plane by two vertical lines in such a way that all that part of the curve which is contained in the strip lies between the two horizontal lines. This is evidently true of the curve G (Fig. 29), whatever value ^ may have. We shall now discuss the continuity of some special types of functions. Some of the results which follow were (as we pointed out at the time) tacitly assumed in Ch. II. Examples XXXVII. 1. The sum or product of two functions continuous at a point is continuous at that point. The quotient is also continuous unless the denominator vanishes at the point. [This follows at once from Ex. XXXV. 1.] 2. Any polynomial is continuous for all values of x. Any rational fraction is continuous except for values of x for which the denominator vanishes. [This follows from Exs. xsxv. 6, 7.] 99] CONriNUOlIS AND DISCONTINUOUS FUNCTIONS 177 3. ^x is continuous for all positive values of x (Ex. xxxv. 8). It is not deiined when x < 0, bnt is continuous for .j; = in virtue of the remark made at the end of g 98. The same is true of a;"*'", where m and n are any positive integers of which n is even. 4. The function «■"/", where n is odd, is continuous for all values of x. 5. l/,r is not continuous for x=0. It has no vakie for .i' = 0, nor does it tend to a limit as .r-»-0. In fact 1/a;-*- +aoor 1/^-*- -oo according as a,'-9-0 by positive or negative values. 6. Discuss the continuity of x~™l^, where m, and n are positive integers, iox x=(i. 7. The standard rational function Rix) = P {x)jQ{x) is discontinuous for x=a, where a is any root of § (x) = 0. Thus {x'^-\-\)j{x'^ — Zx + 'i) is discon- tinuous for x = \. It will be noticed that in the case of rational functions a discontinuity is always associated with (a) a failure of the definition for a particular value of x and (6) a tending of the function to + qo or — oo as .*- approaches this value from either side. Such a particular kind of point of discontinuity is usually described as an infinity of the function. An 'infinity' is the kind of discontinuity of most common occurrence in ordinary work. 8. Discuss the continuity of s.l[{,x-a){b-x)), m{x-a){h-x)}, J{{x-a)l{h-x)}, y{[x-a)l{h-x)} 9. sin X and cos x are continuous for all values of x. [We have sin {x + h) — sin ^= 2 sin ^h cos (x + \h). which is numerically less than the numerical value of A.] 10. For what values of x are tan^, cot x, sec x, and coscc .» continuous or discontinuous ? 11. If /(y) is continuous iov y = r], and ^ (x) is a continuous function of X which is equal to rj when .r = |, theuf{<f> {x)} is continuous for x=^. 12. If (x) is continuous for any particular value of x, then any poly- nomial in (ji (x), such as a {(f> (x)}"'+..., is so too. 13. Disciiss the continuity of IKaco&'x + bam^x), V(2 + cosa-), ^'(l-»-sin.r), llj(l+amx). 14. sin(l/a-), ^sin(l/.r), and .r^sin (1/.k) are continuous except for x = 0. 15. The function which is equal to a; sin (l/x) oxcei^t when .^■ = 0, and to zero when x=0, is continuous for all values of x. 16. [r] and x-[x] are discontinuous for all integral values of x. 17. For what (if any) values of x are the following functions discon- tinuous : [x^l Wxl J(,x-[x-]), [x] + ^(x-[x]), [2x1 M-l-[-a-]? II. 12 ]78 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [v 18. Classification of discontinuities. Some of the preceding examples suggest a classifioatiou of difi'ereut types of discontinuity. (1) Suppose that <^ {x) tends to a limit as x^~a either by values less than or by values greater than a. Denote these limits, as iu § 95, by i^ (a - 0) and (^ (a + 0) respectively. Then, for continuity, it is necessary and sufficient that <i> {x) should be defined for x = a, and that qb (a - 0) = (^ (a) = (^ (a + 0). Dis- continuity may arise in a variety of ways. (a) (a — 0) may be equal to (\> (as+O), but <j) {a) may not be defined, or may differ from (^ (a - 0) and <f>{a+Q). Thus if (^ (.j;) = .3;sin (1/j;) and a=0, (^ (0 - 0) = ^ (0 + 0) =0, but 4> {x) is not defined for x = 0. Or if {x) = [1 - x^ anda = 0, (0-0) = <^ (0 + 0) = 0, but</)(0) = l. O) (d-O) and (^(a + 0) may be unequal. In this case 0(a) may be equal to one or to neither, or be undefined; The first case is illustrated by {x) = [x], for which (0 - 0) = - 1, (0 + 0) = (0) = ; the second by {x) = \x\ -[-xl, for which (0 - 0) = - 1, (0 + 0) = 1, (0) = ; and the third by {x) = \.v'\+xsin (l/.r), for which (0-0)= - 1, (0 + 0) = 0, and (0) is undefined. Iu any of these cases we say that {x) has a simple discontinuity at x = a. And to these oases we may add those in which {x) is defined only on one side of x=a, and <p{a — 0) or (a + O), as the case may be, exists, but {x) is either not defined when x=a or has when x=a a value diS'erent from 0(a-O) or 0(a + O). It is plain from § 95 that a function which increases or dea-eases steadily in the neighbourhood of x = a can have at most a simple discontinuity for x=a. (2) It may be the case that only one (or neither) of (as - 0) and (a + 0) exists, but that, supposing for example (a + O) not to exist, cf>(x)-^ + (xi or {x)-i>~- 00 as a;-*-a + 0, so that {x) tends to a limit or to +oo or to - co as X approaches a from either side. Such is the case, for instance, if (x) = llx or {x) = \jx^, anda = 0. In such cases we say (cf. Ex. 7) that x=a\s,B.Ti infinity of ix). And again we may add to these cases those in which (.r)-»- +oo or {x)~»- — a:)a,iix-»-a from one side, but {x) is not defined at all on the other side oi x=a. (3) Any point of discontinuity which is not a point of simple discoji- tinuity nor an infinity is called a point of oscillatory discontinuity. Such is the point x=0 for the functions sin (l/;t.■)^ (1/*) sin (1/^'). 19. What is the nature of the discontinuities at a; = of the functions {sinx)jx,[x'] + \_- x\ coseo X, •Jiljx), il{\jx), cosec (l/jc), sin (l/.r)/sin (l/.r) ? ' 20. The function which is equal to 1 when x is rational and to when X is irrational (Ch. II, Ex. xvi. 10) is discontinuous for all values of x. So too is any function which is defined only for rational or for irrational values of x: 99, 100] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 179 21. The funotiou which is equal to x when x is irrational and to iJ{0-+2J-)IO--\-q^)} when a; is a rational fraction pjq (Oh. II, Ex. xvi. 11) is discontinuous for all negative and for positive rational values of x, but continuous for positive irrational values. 22. For what points are the functions considered in Ch. IV, Exs. xxxi discontinuous, and what is the nature of their discontinuities ? [Consider, e.g., the function y = lim a;" (Ex. 5). Here y is only defined when — 1 <a'£l : it is equal to when - \<x<\ and to 1 when x=\. The points x = l and x= — 1 are points of simple disoontinuitj'.] 100. The fundamental property of a continuous function. It may perhaps be thought that the analysis of the idea of a con- tinuous curve given in § 98 is not the simplest or most natural possible. Another method of analysing our idea of continuity is the following. Let A and B be two points on the graph of j> {x) whose coordinates are a'o, ^{x^) and x^, (p{Xi) respectively. Draw any straight line X which passes between A and B. Then common sense certainly declares that if the graph of (j) («) is continuous it must cut \. If we consider this property as an intrinsic geometrical property of continuous curves it is clear that there is no real loss of generality in supposing A, to be parallel to the axis of x. In this case the ordinates of A and B cannot be equal : let us suppose, for definiteness, that ^ {x^) > (j) (xo). And let X be the line y = •»;, where (f> (xo) < t) <cj) (xi). Then to say that the graph of d> (x) must cut \ is the same thing as to say that there is a value of X between x^ and x^ for which <f> (x) = r). We conclude then that a continuous function ^ (x) must possess the following property : if and ya<V<yi> *^*^" there is a value of x between Xo and x.^for which A (x) = 1]. In other words as x varies from x^ to Xy, y must assume at least once every value between t/o and y^. We shall now prove that if 4> («) is a continuous function of x in the sense defined in § 98 then it does in fact possess this property. There is a certain range of values of x, to the right of «„, for which <f) h) < 71. For <i> (a-'o) < 17, and so (/> {x) is certainly less than 7? if 12—2 180 CONTIGUOUS AND DISCONTINUOUS FUNCTIONS [v <p (x) — <fi (a;,,) is numerically less than i] — (j) («„). But since (f> (x) is continuous for x = x^, this condition is certainly satisfied if x is near enough to «,,. Similarly there is a certain range of values, to the left of i^i, for which <j} (x) > tj. Let us divide the values of x between x^ and Xi into two classes L, JR, as follows : (1) in the class L we put all values ^ of a; such that <f>{x)<ri when « = ^ and for all values of x between «„ and ^ ; (2) in the class R we put all the other values of x, i.e. all numbers f such that either ^ (f ) S ?; or there is a value of x between Xo and f for which ^ (x) S rj. Then it is evident that these two classes satisfy all the conditions imposed upon the classes Z, i? of § 17, and so constitute a section of the real numbers. Let ^o be the number corresponding to the section. First suppose ^ (fo) > V> so that ^j belongs to the upper class : and let <f> (|o) = i? + /c, say. Then 4> i^')<V ^nd so for all values of ^' less than fo; which contradicts the condition of continuity for x = ^o- Next suppose ^ (^o) = v ^ k<V- Then, if ^' is any number greater than |^„ , either </> (^') S ?; or we can find a number f " between ^^ and ^' such that (f> (|") S ij. In either case we can find a number as near to ^o as we please and such that the corre- sponding values of <p (x) dififer by more than k. And this again contradicts the hypothesis that <j) (x) is continuous for x = Sa- lience 0(|^o) = '7> and the theorem is established. It should be observed that we have proved more than is asserted explicitly in the theorem ; we have proved in fact that ^o is the least value of X for which ^ (x) = rj. It is not obvious, or indeed generally true, that there is a least among the values of x for which a function assumes a given value, though this is true for continuous functions. It is easy to see that the converse of the theorem just proved is not true. Thus such a function as the function <j) {x) whose graph is represented 100-102] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 181 by Fig. 31 obviously assumes at least onoe every value between (f> (X(,) and 4> (^i) '■ y®* "^ (''') i^ discontinuous. Indeed it is 'not even true that (j) (so) must be continuous when it assumes each value once and once only. Thus let (^ {x) be defined as follows from x—0 to x= 1. If x!=0 let <f>{x) = 0; if < a- < 1 let ^{x) = \-x; and ]i x=\ let ^{x) = \. The graph of the function is shown in Fig. 32 ; it includes the points 0, C but not the points A, B. It is clear that, as x varies from to 1, <i>{x) assumes once and once only every value between <^ (0)=0 and <^ (1) = 1 ; but ^ (x) is discontinuous for ^ = and x=l. Fig. 31. Fig. 32. As a matter of fact, however, the curves which usually occur in elementary mathematics are composed of a finite number of pieces along which y always varies in the same direction. It is easy to show that if y = (fi(x) always varies in the same direction, i.e. steadily increases or decreases, as x varies from ^0 to Xi, then the two notions of continuity are really equivalent, i.e. that if <j> {x) takes every value between (j) (xo) and cf> (xi) then it must be a continuous function in the sense of § 98 For let $ be any value of x between Xo and Xi. As x-*-§ through values less than ^, <f>(x) tends to the limit <^(| — 0) (§ 95). Similarly as x-a-^ through values greater than $, (x) tends to the limit (j) (1+0). The function will be continuous for x = ^ii and only if <^(l-o)=0 (!)=<> (1+0) But if either of these equations is untrue, say the first, then it is evident that (^(^) never assumes any value which lies between (^ (^ — 0) and (f> (^), which is contrary to our assumption. Thus <p (x) must be continuous. The net result of this and the last section is consequently to show that our common- sense notion of what we moan by continuity is substantially accurate, and capable of precise statement in mathematical terms. 101. In this and the following paragraphs we shall state and prove some general theorems concerning continuous functions. 182 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [v Theorem 1 . Suppose that </> (x) is continuous for x = ^, and that 4>{^) is positive. Then we can determine a positive number e such that (j) (^) is positive throughout the interval (^ - e, f + e). For, taking B = ^^ (^) in the fundamental inequality of p. 175, we can choose e so that |</'(«')-^(f)i<#(?) throughout (f — e, ^+e), and then so that (j) (x) is positive. There is plainly a corresponding theorem referring to negative values of </> («). Theoreji 2. If (j) (x) is continuous for x = ^, and (f) (x) vanishes for values of x as near to ^ as we please, or assumes, for values of X as near to ^ as ive please, both positive and negative values, then </>(|) = 0. This is an obvious corollary of Theorem 1. If (^ (^) is not zero, it must be positive or negative ; and if it were, for example, positive, it would be positive for all values of x sufficiently near to ^, which contradicts the hypotheses of the theorem. 102. The range of values of a continuous function. Let us consider a function </> {x) about which we shall only assume at present that it is defined for every value of x in an interval (a, b). The values assumed by <^ (x) for values of x in (a, b) form an aggregate S to which we can apply the arguments of § 80, as we applied them in § 81 to the aggregate of values of a function of n. If there is a number K such that (f) (x) ^ K, for all values of x in question, we say that (f> (x) is bounded above. In this case ^ («) possesses an upper bound M : no value of <f) (x) exceeds M, but any number less than M is exceeded by at least one value of ^{x). Similarly we define 'bounded below', 'lower bound', 'bounded', as applied to functions of a continuous variable x. Theorem 1. //" («) is continuous throughout (a, b), then it is hounded in (a, b). 102] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 183 We can certainly determine an interval (a, f), extending to the right from a, in which ^ {x) is bounded. For since ^ {x) is continuous for a; = a, we can, given auy positive number 8 however small, determine an interval (a, ^) throughout which </> {x) lies between (f>(a)— B and <f)(a) + B; and obviously («) is bounded in ihis interval. Now divide the points f of the interval (a, b) into two classes L, R, putting f in Z if ^ (^) is bounded in (a, ^), and in R if this is not the case. It follows from what precedes that L certainly exists : what we propose to prove is that R does not. Suppose that R does exist, and let yS be the number corresponding to the section whose lower and upper classes are L and R. Since <j) (x) is continuous for a; = /3, we can, however small S may be, determine an interval {^ — rj, ^ + ri)* throughout which Tims (j) (x) is bounded in (yS — ■»;, /3 + t?). Now y3 — t? belongs to L. Therefore (/>(«) is bounded in (a, ^ — rj): and therefore it is bounded in the whole interval (a, /3 + rj). But ,8 + t; belongs to R and so ^ (x) is not bounded in {a, ^ + rj). This contradiction shows that R does not exist. And so (f) (x) is bounded in the whole interval (a, 6). Theobem 2. If (f> (x) is continuous throughout (a, b), and M and m are its upper and lower bounds, then <j> (x) assumes the values M and m at least once each in the interval. For, given any positive number B, we can find a value of x for which M-(^{x)<B ox l/{il/ - 4> {x)] > l/B. Hence lj{M-^ (x)} is not bounded, and therefore, by Theorem 1, is not continuous. But M—^(x) is a continuous function, and so l/{M'—<j)(x)} is continuous at any point at which its denominator does not vanish (Ex.- XXXVII. 1). There must therefore be one point at which the denominator vanishes : at this point (f) (x) = M. Similarly it may be shown that there is a point at which <j> (x) = m. The proof just given is somewhat subtle and indirect, and it may be well, in view of the great importance of the theorem, to indicate alternative lines of proof. It will however be con- venient to postpone these for a momentf. * If ;8 = 6 we must replace this interval by (S-ij, /3), and /3 + jj by |3, throughout the arRument which follows. + See § 101. 184 CUKTINUOUS AND DISCONTINUOUS FUNCTIONS [v Examples XXXVIII. 1. If 4>(x)=llx except when x=0, and <f)(x)=0 when x=0, then (^(x) has neither an upper nor a lower bound in any interval which includes x=0 in its interior, as e.g. the interval ( — 1, +1). 2. If (ji{x) = l/x^ except when j; = 0, and (f>{x)=0 when ^ = 0, then (ji(x) has the lower bound 0, but no upjjer bound, in the interval (— 1, + 1). 3. Lot ^ (a;) = sin (1/a?) except when a; = 0, and (^ (^) = when .■!;=0. Then <^ {x) is discontinuous for a; = 0. In any interval ( - S, +6) the lower bound is — 1 and the upper bound +1, and each of these values is assumed by 0(j;) an infinity of times. 4. Let (t>(x)=x!-[.v]. This function is discontinuous for all integral values of x. In the interval (0, 1) its lower bound is and its upper bound 1. It is equal to when ^-=0 or x=l, but it is never equal to I. Thus (j>{x) never assumes a value equal to its upper bound. 5. Let ^ix) = when x is irrational, and ^ {x) = q when x is a rational fraction pjq. Then <p (x) has the lower bound 0, but no upper bound, in any interval {a,h). But if (^{x) = {-^\y q when x =pjq, then <^ {x) has neither an upper nor a lower bound in any interval. 103. The oscillation of a function in an interval. Let ^ {x) be any function bounded throughout (a, h), and M and m its upper and lower bounds. We shall now use the notation M (a, b), m (a, h) for M, m, in order to exhibit explicitly the de- pendence of Al and m on a and b, and we shall write (a, /)) = 31 (a, b) - m {a, b). This number (a, b), the difference between the upper and lower bounds of <p {x) in (a, b), we shall call the oscillation of (p (x) in (a, b). The simplest of the properties of the functions M{a, b), VI (a, b), (a, b) are as follows. (1) If aS: c S.b then M (a, b) is equal to the greater of M (a, c) and M{c, b), and m (a, b) to the lesser of m (a, c) and m (c, b). (2) M (a, b) is an increasing, m (a, b) a decreasing, and (a, b) an increasing function of b. (3) 0(a,b)SO(a,c)+0(c,b). The first two theorems are almost immediate consequences of our definitions. Let /x be the greater of M (a, c) and M (c, 6), and let 8 be any positive number. Then cj) (x) S fi throughout {a, c) and (c, b), and therefore throughout (a, b) ; and <f)(x)> fi-B somewhere in (a, c) or in (c, b), and therefore somewhere in (a, b). 102-105] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 185 Hence M {a, h) = /i. The proposition concerning m may be proved similarly. Thus (1) is proved, and (2) is an obvious corollary. Suppose now that /l/, is the greater and M^ the less oi M{a, c) and M (c, b), and that m^ is the less and m^ the greater of m (a, c) and m (c, h). Then, since c belongs to both intervals, ^ (c) is not greater than il/g nor less than m^. Hence M^ = m^, whether these numbers correspond to the same one of the intervals (a, c) and (c, h) or not, and (a, b} = Mi-mi^ Mi+ M.i-mi- m^. But (a, c) + (c, 6) = ilfi + i¥2 - 7^1 - m^ ; and (3)- follows. 104. Alternative proofs of Theorem 2 of § 102. The most straight- forward proof of Theorem 2 of § 102 is as follows. Let | be any number of the interval (a, 6). The function M (a, f ) increases steadily with | and never exceeds M. We can therefore construct a section of the numbers | by putting I in Z or in A according as M{a, |) <M or M{a, ^) = M. Lot fi be the number corresponding to the section. If a<^<b, we havo for all positive values of 17, and so by (1) of § 103. Hence ^ {x) assumes, for values of x as near as wo please to j3, values as near as we please to M, and so, since {x) is continuous, (/3) must be equal to M. If ^=a then M{a, a + r{) = M. And if |3 = 6'then M{a, b-ri) <M, and so M{b-rj, h)=M. In either case the argument may be completed as before. The theorem may also be proved by the method of repeated bisection used in § 71. If M is the upper bound of (/> {x) in an interval PQ, and PQ is divided into two equal parts, then it is possible to find a half P] 9i i"^ which the upper bound of (f> (x) is also M. Proceeding as in § 71, we construct a sequence of intervals PQ, P^Qi, P2Q2, ■■• in each of which the upper bound of (x) is M. These intervals, as in § 71, converge to a point T, and it is easily proved that the value of {x) at this point isjJ/. 105. Sets of intervals on a line. The Heine-Borel Theorem. We shall now proceed to prove some theorems con- cerning the oscillation of a function which are of a somewhat abstract character but of very great importance, particularly, as we shall see later, in the theory of integration. These theorems depend upon a general theorem concerning intervals on a line. 186 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [v Suppose that we are given a set of intervals in a straight line, that is to say an aggregate each of whose members is an interval (a, /3). We make no restriction as to the nature of these intervals; they may be finite or infinite in number; they may or may not overlap*; and any number of them may bo included in others. It is worth while in passing to give a few examples of sets of intervals to which we shall have occasion to return later. (i) If the interval (0, 1) is divided into n equal parts then the n intervals thus formed define a finite set of non-overlapping intervals which just cover up the line. (ii) We take every point ^ of the interval (0, 1), and associate with | the interval (| — e, l+c), where f is a positive number less than 1, except that with we associate (0, e) and with 1 we associate (1 — f, I), and iu genei'al we reject any part of any interval which projects outside the interval (0, 1). We thus define an infinite set of intervals, and it is obvious that many of thoni overlap with one another. (iii) We take the rational points pjq of the interval (0, 1), and associate with piq the interval (p_l ■P,L\ \q 2^' q^f)' where c is positive and less than 1. We regard as 0/1 and 1 as 1/1 ; in these two cases we reject the part of the interval which lies outside (0, 1). We obtain thus an infinite set of intervals, which plainly overlap, with one another, since there are an infinity of rational points, other than pIq, in the interval associated with pjq. The Heine-Borel Theorem. Suppose that we are given an interval (a, h), and a set of intervals I each of whose members is included in {a, b). Suppose further that I possesses the following properties : (i) every point of (a, b), other than a and b, lies insidej at least one interval of I ; (ii) a is the left-hand end point, and b the right-hand end point, of at least one interval of I. Then it is possible to choose a finite number of intervals from the set I which form a set of intervals possessing the properties (i) and (ii). * The word overlap is used in its obvious sense : two intervals overlap if they have points in common which are not end points of either. Thus (0, ^) and (|, 1) overlap. A pair of intervals such as (0, 4) and (J, 1) may be said to ahut. t That is to say ' in and not at an end of. 105] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 187 We know that a is the left-hand end point of at least one interval of /, say (a, a^. We know also that a-^ lies inside at least one interval of /, say (a/, a,). Similarly ag lies inside an interval {a^, as) of /. It is plain that this argument may be repeated in- definitely, unless after a finite number of steps a„ coincides with b. If an does coincide with b after a finite number of steps then there is nothing further to prove, for wo have obtained a finite set of intervals, selected from the intervals of /, and possessing the properties required. If «„ never coincides with b, then the points tt], tta, tts, ... must (since each lies to the right of its predecessor) tend to a limiting position, but this limiting position may, so far as we can tell, lie anywhere in (a, b). Let lis suppose now that the process just indicated, starting from a, is performed in all possible ways, so that we obtain all possible sequences of the type a^, a^, a^, .... Then we can prove that there must be at least one such sequence which arrives at b after a finite number of steps. a a\ fli a'2 Qj I 03 k' i^ I" ^^ b Fig. 33. There are two possibilities with regard to any point ^ between a and b. Either (i) ^ lies to the left of some point a„ of some sequence or (ii) it does not. We divide the points | into two classes L and R according as to whether (i) or (ii) is true. The class L certainly exists, since all points of the interval (a, aO belong to L. We shall now prove that R does not exist, so that every point ^ belongs to L. If R exists then L lies entirely to the left of R, and the classes L, R form a section of the real numbers between a and b, to which corresponds a number ^o- The point ^o lies inside an interval of /, say (^', ^"), and |' belongs to L, and so lies to the left of some term a„ of some sequence. But then we can take (^', ^") as the interval («„', an+i) associated with a„ in our construction of the sequence a„ a^, a^, ...; and all points to the left of ^" lie to the left of a„+i. There are therefore points of L to the right of ^0, and this contradicts the definition of R. It is therefore impossible that R should exist. 188 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [y Thus every point f belongs to L. Now h is the right-hand end point of an interval of /, say (61, h), and &i belongs to L. Hence there is a member a„ of a sequence Oj, a^, a^, ... such that a„ > 61. But then we may take the interval (a„', ctn+i) corre- sponding to a„ to be (61, 6), and so we obtain a sequence in which the term after the nth coincides with h, and therefore a finite set of intervals having the properties required. Thus the theorem is proved. It is instructive to consider the examples of p. 186 in the light of this theorem. (i) Here the conditions of the theorem are not satisfied the points 1/m, 2/n, Zjn, ... do not He inside any interval of / (ii) Here the conditions of the theorem are satisfied. The set of intervals (0, 24 (f, 3f), (2e, 4f), ..., (1-2., 1), associated with the points e, 2f, 3c, ..., 1-c, po.ssesses the properties re- quired. (iii) In this case we can prove, by using the theorem, that there are, if c is small enough, points of (0, 1) which do not lie in any interval of /. If every point of (0, 1) lay inside an interval of / (with the obvious reservation as to the end points), then we could find a finite number of intervals of / possessing the same property and having therefore a total length greater than 1. Now there are two intervals,of total length 2f, for which g' = l, and q-\ intervals, of total length 2e{q—l)/q^, associated with any other value of q. The sum of any finite number of intervals of / can therefore not be greater than 2. times that of the scries 12 3 1 + 23 + 35 + 4=+-' which will be shown to be convergent in Ch. VIII. Hence it follows that, if e is small enough, the supposition that every point of (0, 1) lies inside an interval of / leads to a contradiction. The reader may be tempted to think that this proof is needlessly elaborate, and that the existence of points of the interval, not in any interval of /, follows at once from the fact that the sum of all these intervals is less than 1. But the theorem to which he would be appealing is (when the set of intervals is infinite) far from obvious, and can only be proved rigorously by some such use of the Heine-Borel Theorem as is made in the text. 108. We shall now apply the Heine-Borel Theorem to the proof of two important theorems concerning the oscillation of a continuous function. 105, 106] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 189 Theorem 1. If <^ (x) is continuous throughout the interval (a, b), then we can divide (a, b) into a finite number of sub -intervals (a, Xi), (xi, X2), ... (xn, b), in each of which the oscillation of cj) {x) is less than an assigtied positive number S, Let ^ bo any number between a and b. Since (f> {x) is con- tinuous for 3; = ^, we can determine an interval (^ — e, f + e) such that the oscillation of <^ {x) in this interval is less than 8. It is indeed obvious that there are an infinity of such intervals corre- sponding to every ^ and every S, for if the condition is satisfied for any particular value of e, then it is satisfied a fortiori for any smaller value. What values of e are admissible will naturally depend upon ^; we have at present no reason for supposing that a value of e admissible for one value of f will be admissible for another. Wo shall call the intervals thus associated with ^ the ^-intervals of ^. If ^ = a then we can determine an interval {a, a + e), and so an infinity of such intervals, having the same property. These we call the S-intervals of a, and we can define in a similar manner the S-intervals of b. Consider now the set / of intervals formed by taking all the S-intervals of all points of (a, b). It is plain that this set satisfies the conditions of the Heine-Borel Theorem ; every point interior to the interval is interior to at least one interval of /, and a and b are end points of at least one such interval. We can therefore determine a set /' which is formed by a finite number of intervals of /, and which possesses the same property as I itself. The intervals which compose the set I' will in general overlap as in Fig. 34. But their end points obviously divide up Zli:^::^ ^^_^_____ (a, b) into a finite set of in- " >' tervals I" each of which is '^' included in an interval of I', and in each of which the oscillation of ^ (x) is less than 8. Thus Theorem I is proved. Theorem II. Given any positive number S, we can find a number 7} such that, if the interval (a, b) is divided in any manner into sab-intervals of length less than tj, then the oscillation of <f> (x) in each of them will be less than S. 190 CONTINUOUS AND DISCONTINUOUS BUNOTIONS [v Take Si < ^8, and construct, as in Theorem I, a finite set of sub- intervals j in each of which the oscillation of ^ («) is less than Sj. Let ri be the length of the least of these sub-intervals j. If now we divide {a, b) into parts each of length less than ij, then any such part must lie entirely within at most two successive sub- intervals j. Hence, in virtue of (3) of § 103, the oscillation of </> (a), in one of the parts of length less than ?;, cannot exceed twice the greatest oscillation of (p («) in a sub-interval j, and is therefore less than 2Si, and therefore than S. This theorem is of fundamental importance in the theory of definite integrals (Ch. VII). It is impossible, without the use of this or some similar theorem, to prove that a function continuous throughout an interval necessarily possesses an integral over that interval. 107. Continuous functions of several variables. The notions of continuity and discontinuity may be extended to functions of several independent variables (Ch. II, §§ 31 et seg.). Their Application to such functions, however, raises questions much more complicated and difficult than those which we have considered in this chapter. It would be impossible for us to discuss these questions in any detail here ; but we shall, in the sequel, require to know what is meant by a continuous function of two variables, and we accordingly give the following definition. It is a straightforward generalisation of the last form of the de- finition of § 98. The function (j) (x, y) of the two variables x and y is said to be continuous for x= ^, y = 7] if, given any positive number 8, how- ever small, we can choose e (S) so that i<^(*, 2/)- </>(?, '?)!<S when S I a; - ^ I ^ 6 (S) and ^ | y - j; | ^ e (8); that is to say if we can draw a square, whose sides are parallel to the axes of coordinates and of length 2e (S), whose centre is the point (^, rj), and luhichis such that the value of (f> {x, y) at any point inside it or on its boundary differs from (p (^, tj) by less than B* This definition of course presupposes that </>(», y) is defined at all points of the square in question, and in particular at the point * The reader should draw a figure to illustrate the defiaition. 106-lOS] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 191 (^, Tj). Another method of stating the definition is this : </> (x, y) is continuous for x=^, y=7] if ^{t«,y)^<j)(^,rj) when x^^, y~^r\ in any manner. This statement is apparently simpler; but it contains phrases the precise meaning of which has not yet been explained and can only be explained by the help of inequalities like those which occur in our original statement. It is easy to prove that the sums, the products, and in general the quotients of continuous functions of two variabljjs are them- selves continuous. A polynomial in two variables is continuous for all values of the variables ; and the ordinary functions of x and y which occur in every-day analysis are generally continuous, i.e. are continuous except for pairs of values of x and y connected by special relations. The reader should observe carefully that to assert the continuity of <^ {x\ y) with respect to the two variables x and y is to assert much more than its continuity witli respect to each variable considered separately. It is plain that if (^ {x, y) is continuous with respect to x and y then it is certainly continuous with respect to x (or y) when any fixed value is assigned to y (or .v). But the converse is by no means true. Suppose, for example, that (b (x, y) = -., ,, when neither x nor y is zero, and {x, y)=0 when either .r ory is zero. Then if y has any fixed value, zero or not, ^ {x, y) is a continuous function of x, and in particular continuous for x = 0; for its value when ^ = is zero, and it tends to the limit zero as x^O. In the same way it may be shown that <h {.r, y) is a continuous function of y. But <j> {x, y) is not a. continuous function of X andy for A-=0, ?/ = 0. Its value when ^' = 0, y = is zero ; bat if x and y tend to zero along the straight line y = or, then </> (■^•. y) = J-+-. . ii«i "^ G^. y) = Y^i. . which may have any value between — 1 and 1. 108. Implicit functions. We have already, in Oh. II, met with the idea of an implicit fimction. Thus, if x and y are connected by the relation yf.^yy-y-X = Q (1), then y is an 'implicit function' of x. But it is far from obvious that such an equation as this does really define a function y of x, or several such functions. In Ch. II we were content to take this for gi-anted. We are now in a position to consider whether the assumption we made then was justified. 192 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [V We shall find the following terminology useful. Suppose that it is possible to surround a point (a, b), as in § 107, with a square throughout which a certain condition is satisfied. We shall call such a square a neighbourhood of {a, b), and say that the condition in question is satisfied in the neighbour- hood of (a, b), or near (a, 6), meaning by this simply that it is possible to find some square throughout which the condition is satisfied. It is obvious that similar language may be used when we are dealing with a single variable, the square being replaced by an interval on a line. Theorem. If (i) f{x, y) is a continuous fuTiction of x and y in the neighbourhood of {a, b), (ii) f{a,b) = 0, (iii) / (,'(?, y) is, for all values of x in the neighbourhood of a, a steadily increasing function of y, in the stricter sense of % 95, then (1) there is a unique function y=<j> {x) which, when suhstituted in the ecp.iation f{x, y)=0, satisfies it identically for all values of x in the neighbour- hood of a, (2) <^ {x) is continuous for all values of x in the neighbourhood of a. In the figure the square represents a ' neighbourhood ' of (a, 6) tlirough- out which the conditions (i) and (iii) are satisfied, and P the point {a, b). If we take Q and R as in the figure, it follows from (iii) that/(^, y) is positive at Q and negative at R. This being so, and f{x, y) being con- tinuous at Q and at R, we can draw lines QQ' and RIt' parallel to OX, so that SQ' is parallel to OF and f(x, y) is positive at all points of QQ and negative at all points of RR'. In par- ticular f{x, y) is positive at Q^ and negative at R, and therefore, in virtue of (iii) and § 100, vanishes once and only once at a point P' on A'y. The same construction gives us a unique point at which f{x,y) = on each ordinate between RQ and R'Q'. It is obvious, moreover, that the same construction can be carried out to the left of RQ. The aggregate of points such as P' gives us the graph of the required function y=<^ (x). It remains to prove that (j> (x) is continuous. This is most simply effected by using the idea of the 'limits of indetermination ' of ^{x) as xs-a (§ 96). Suppose that x-»-a, and let X and A be the limits of indetermination of ^ (x) as x-^a. It is evident that the points (a, X) and (a. A) lie on QR. Moreover, we can find a sequence of values of x such that (x)-i~\ when x-^a through the values of the sequence; and aince f {x,<f>{x)]=0, and f(x,y) is a con- tinuous function of x and y, we have /(a,X) = 0. Hence \ = b; and similarly A=6. Thus <^ (x) tends to the limit h as x-^a, and so ^(.i;) is continuous for x=a. It is evident that we can show in Q Q' P' (a, 6) 'P R' R Fig. 35. 108-109] CONTINUOUS AND DISCONTINUOUS FUNCTIONS 193 exactly the same way that (f> (lo) is continuous for any value of x in the neighbourhood of a. It is clear that the truth of the theorem would not be aflfected if we were to change 'increasing' to 'decreasing' in condition (iii). As an example, let us consider the equation (1), taking a=0, 6=0. It ia evident that the conditions (i) and (ii) are satisfied. Moreover /(^, y) -/(«. y) = (y -y) (/ +yy +yyHyyH.?/'^ - ^ - 1) has, when x, y, and 3/' are sufficiently small, the sign opposite to that of y-y. Hence condition (iii) (with 'decreasing' for 'increasing') is satisfied. It follows that there is one and only one continuous function y which satisfies the equation (1) identically and vanishes with x. The same conclusion would follow if the equation were y'^ — xy — y - x=<). The function in question is in this case 2/ = i{H-^-v/(l + 6:i; + «2)}, where the square root is positive. The second root, in which the sign of the square root is changed, does not satisfy the condition of vanishing with x. There is one point in the proof which the reader should be careful to ob- serve. We supposed that the hypotheses of the theorem were satisfied 'in the neighbourhood of (a, 6)', that is to say throughout a certain square | — f <a;£^ + £, rj — e ^y £,r] + f. The conclusion holds 'in the neighbourhood oi x — a', that is to say throughout a certain interval | — €iga;^|-l-fi. Thei'c is nothing to show that the ei of the conclusion is the e of the hypotheses, and indeed this is generally untrue. 109. Inverse Functions. Suppose in particular thai f{x,y) is of the form Fijf) — x. We then obtain the following theorem. If F{y) is a function of y, continuous and steadily increasing {or decreasing), in the stricter sense of § 95, in the neighbourhood of y = h, and F(b) = a, then there is a unique continuous function y = (f>(x) which is equal to b when x = a and satisfies the equation F{y) = x identically in the neighbourhood of x=a. The function thus defined is called the inverse function of F{y). Suppose for example that y^ = x, a = 0, 6 = 0. Then all the conditions of the theorem are satisfied. The inverse function is x=^y. If we had supposed that y'''=x then the conditions of the theorem would not have been satisfied, for y"^ is not a steadily increasing function of y in any interval which includes 3/ = : it decreases when y is negative and increases when y is positive. And in this case the conclusion of the theorem does not hold, for y'^=x defines two functions of x, viz. y = -Jx and y=-Jx, both of which vanish when x = 0, and each of which is defined only for positive values of *' so that the equation has sometimes two solutions and sometimes none. Tlie reader should consider the more general equations y2» = ^, ^2/. + ! = ^ 1.3 194 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [v in the same way. Another interesting example is given by the equation ah-eady considered in Ex. xiv. 7. Similarly the equation sniy = se has just one solution which vanishes with a;, viz. the value of arc sin x which vanishes with x. There are of course an infinity of solutions, given by the other values of arc sin x (c£ Ex. xv. 10), which do not satisfy this condition. So far we have considered only what happens in the neighbourhood of a particular value of x. Let us suppose now that F(jj) is positive and steadily increasing (or decreasing) throughout an interval (a, b). Given any point ^ of (a, b), we can determine an interval i including |, and a unique and con- tinuous inverse function (^n (x) defined throughout i. From the set I of intervals i we can, in virtue of the Heine-Borel Theorem, pick out a finite sub-set covering up the whole interval (a, b) ; and it is plain that the finite set of functions (^^ {x), corresponding to the sub-set of intervals i thus selected, define together a unique inverse function <j){x) continuous throughout (a, 6). We thus obtain the theorem : if x=F{y), where F(j/) is continuous and increases steadily and strictly from A to B as x increases from a to b, then there is a unique inverse function y = (j)(x) which is continuous and increases steadily and strictly from a tob as x increases from A to B. It is worth while to show how this theorem can be obtained directly with- out the help of the more difficult theorem of § 108. Suppose that A <^<B, and consider the class of values of y such that (i) a<y<b and (ii) F(i/) g |. This class has an upper bound rj, and plainly F{j))^^. If F{r)) were less than ^, we could find a value of y such that y>-q and F{y) <§, and t; would not be the upper bound of the class considered. Hence F{t)) = ^. The equation F(j/) = ^ has therefore a unique solution y = r) = cf>{^), say; and plainly i; increases steadily and continuously with |, which proves the theorem. MISCELLANEOUS EXAMPLES ON CHAPTER V. 1. Show that, if neither a nor b is zero, then a^" -h 6:r" - 1 -t- . . . -H ^ = a^" (1-)- f J,), where e^ is of the first order of smallness when x is large. 2. If P{x) = ax^ + bx"~'^ + ...+]c, and a is not zero, then as x increases P(x) has ultimately the sign of a ; and so has P{x-\-\)- P{x), where X is any constant. 3. Show that in general (a»» -(- 6.i;» - 1 -I- . . . -f i)/( J .r" -f 5a;" - 1 4- . . . -1- .fl') = a -I- 0/a;) (1 + f ^), where a = ajA, IB = {bA -aB)/A^, and e^ is of the first order of smallness when c is large. Indicate any exceptional cases. CONTINUOUS AND DISCONTINUOUS FUNCTIONS 195 4. Express {ax^+bx+c)/{Ax^ + Bx-\-C) in the form a + ((3/:e) + {y/x^) (1 + e^), where e^ is of the first order of smallness when x is large. 5." Show that liui y/x{^(x+a) — Jx} = ^a. [Use the formula ^{x + a) — ^'x = al{yf(x + a) + ^x}.'] 6. Show that y/(x+a) = ^/x + l(a/,Jx) (1 +ea;), where e^ is of the first order of smallness. when x is large. 7. Find values of a and p such that ^/(ax^ + 2bx +a) — ax — (3 has the limit zero as X-*- 00 ; and prove that lim .» {J{ax^ + 2bx + c) — ax — fi} = {aa- b^)l2a. 8. Evaluate lim xy[x'^ + ^f{x'>' + l)]-x^/2}. 9. Prove that (sec *■ — tan *')-»- as x-r-^n. 10. Prove that (j){x) = l- cos (1 - cos x) is of the fourth order of smallness when X is small ; and find the limit of (j) {x)lx^ as x~*-0. 11. Prove that (j) (x) = x sin (sin x) - sin^ x is of the sixth order of smallness when X is small ; and find the limit of ip {x)lafi as x-»-Q. 12. From a point P on a radius OA of a circle, produced beyond the circle, a tangent PT is drawn to the circle, touching it in T, and TN is drawn per- pendicular to OA. Show that NAjAP-»-\ as P moves up to .4. 13. Tangents are drawn to a circular arc at its middle point and its extremities ; A is. the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and A' the area of that formed by the three tangents. Show that A/A'-»-4 as the length of the arc tends to zero. 14. For what values of a does {a + sva.{llx)}jx tend to (1) oo , (2) -oo, as x^-O'i [To 00 if a>l, to — oo if a<-l: the function oscillates if -l^a<l.] 15. If (p(x) = l/q when x=plq, and <f>(x) = when x is irrational, then (f) {x) is continuous for all irrational and discontinuous for all rational values of X. 16. Show that the function whose graph is drawn in Fig. 32 may be repre- sented by either of the formulae \-x + [x'\-\\-x\ \-x- lim (cos2" + i7r.»). 17. Show that the function 4)(x) which is equal to when x=0, to \ — x when 0<x<^, to \ when x = \, to f-a- when ^<x<\, and to 1 when .r = l, assumes every value between and 1 once and once only as x increases from to 1, but is discontinuous for x = Q, x=h, and x=\. Show also that the function may be represented by the formula l-x + ^[2x]-i[\-2x\ 13—2 196 CONTINUOUS AND DISCONTINUOUS FUNCTIONS [v 18. Let <j){x) = a: when x is rational and ^{x) = \-x when x is irrational. Show that ^ {x) assumes every value between and 1 once and once only as x increases from to 1, but is discontinuous for every value oi x except x=\. 19. As X increases from --^n- to ^ir, y=sva.x is continuous and steadily increases, in the stricter sense, from - 1 to 1. Deduce the existence of a function .j;=arc siny which is a continuous and steadily increasing function of y from y= —1 to y — \. 20. Show that the numerically least value of arc tan y is continuous for all values of y and increases steadily from -^tt to Jn- as ?/ varies through all real values. 21. Discuss, on the lines of §§ 108 — 109, the solution of the equations y^-y-x=0, y^-y'^-x'^ = 0, y'^-y^ + .'>f-=0 in the neighbourhood oi x=0,y=0. 22. If ax^ + '2,hxy + cy^ + 'it.dx + '2.ey = Q and A = 'ihde — ae'^ — cd\ then one value of 2/ is given by y = ax + Px^ + {y-\-e^)s(fi, where a=-d/e, (3 = A/2e3, y = {cd- be) A/2e«, and «j. is of the first order of smallness when x is small. [U y-ax=Ti then -2eri = ax^ + 2bx{ri + M:) + c{ri + axf = Ax^+2Bxri-\-Cr]% say. It is evident that r; is of the second order of smallness, xtj of the third, and 7/2 of the fourth ; and - 2e?y =Ax^ — (ABje) a?, the error being of the fourth order.] 23. lix = ay + hy'^ + cy^ then one value oi y is given by y = ax + /3i'2 + {y + c^) x?, where a — \la, /3= —hja^, y = {2b^-ac)la\ and e^ is of the first order of small- ness when X is small. 24. If x = ay + by'\ where n is an integer greater than unity, then one value of y is given by y = ca' + l3x'* + (y + ej:) x'^^''-, where a=l/a, ^ — 6/a''"^^, y = rifely a2'' + i, and t^; is of the {n- l)th order of smallness when x is small. 25. Show that the least positive root of the equation xy=sm a; is a con- tinuous function of y throughout the interval (0, 1), and decreases steadily from jr to as i/ increases from to 1. [The function is the inverse of (sin x)jx : apply § 109.] 26. The least positive root of xy=ta.nx is a continuous function of y throughout the interval (1, oo ), and increases steadily from to Jtt as y increases from 1 towards oo . CHAPTEE VI DERIVATIVES AND INTEGRALS 110. Derivatives or Difierential Coefficients. Let us return to the consideration of the properties which we naturally associate with the notion of a curve. The first and most obvious property is, as we saw in the last chapter, that which gives a curve its appearance of connectedness, and which we embodied in our defini- tion of a continuous function. The ordinary curves which occur in elementary geometry, such as straight lines, circles and conic sections, have of course many other properties of a general character. The simplest and most noteworthy of these is perhaps that they have a definite direction at every point, or what is the same thing, that at every point of the curve we can draw a tangent to it. The reader will probably remember that in elementary geometry the tangent to a curve at P is defined to be ' the limiting position of the chord PQ, when Q moves up towards coincidence with P ' Let us consider what is implied in the assumption of the existence of such a limiting position. In the figure (Fig. 36) P is a fixed point on the curve, and Q a variable point ; PM, QN are parallel to F and PR to OX. We denote the coordinates of P hy^ x, y and those of Q by a + h, y + k: h will be positive or negative according as N lies to the right or left of M. We have assumed that there is a tangent to the curve at P, or that there is a definite ' limiting position ' of the chord PQ. Suppose that PT, the tangent at P, makes an angle yjr with OX. Then to say that PT is the limiting position of PQ is equivalent to saying that the limit of the angle QPR is •\/r, when Q approaches 198 DERIVATIVES AND INTEGRALS [VI P along the curve from either side. We have now to distinguish two cases, a general case and an exceptional one. The general case is that in which yjr is not equal to Jtt, so that PT is not parallel to Y. In this case BPQ tends to the hmit yjr, and RQ/PR= tan RPQ tends to the limit tan i|r. Now RQjPR = (NQ - MP)jMN = {<f> {x + h) - <f> (x)]/h ; y d)(x + h)-6{x) and so hm ^-^ ( — ^— ^^ = tan T/r (1). The reader should be careful to note that in all these equa- tions all lengths are regarded as affected with the proper sign, so that (e.g.) RQ is negative in the figure when Q lies to the left of P ; and that the convergence to the limit is unaffected by the sign of h. Thus the assumption that the curve which is the graph of <f> (x) has a tangent at P, which is not perpendicular to the axis of X, implies that <^ («) has, for the particular value of a; corresponding to P, the property that {0 (x+h) — (j>{x)]/h tends to a limit when h tends to zero. This of course implies that both of {^ {X + h)-4, {x)]lh, {^{x-h)-<i, (^)}/( - h) tend to limits when A-»-0 by positive values only, and that the two limits are equal. If these limits exist but are not equal, then the curve y = <i>{x) has an angle at the particular point considered, as in Pig. 37. Now let us suppose that the curve has (like the circle or ellipse) a tangent at every point of its length, or at any rate every 110, 111] DERIVATIVES AND INTEGRALS 199 portion of its length which corresponds to a certain range of variation of x. Further let us suppose this tangent never per- pendicular to the axis oi x: in the case of a circle this would of course restrict us to considering an arc less than a semicircle. Then an equation such as (1) holds for all values of x which fall inside this range. To each such value of x corresponds a value of tan yfr : tan •</r is a function of x, which is defined for all values of X in the range of values under consideration, and which may be calculated or derived fi-om the original function </>(«). We shall call this function the derivative or derived function of (ft (x), and we shall denote it by 4>'(x). Another name for the derived function of ^ (x) is the difiFer- ential coefficient of ^ (x) ; and the operation of calculating <f>' (x) from (f)(x) is generally known as differentiation. This terminology is firmly established for historical reasons : see § 116. Before we proceed to consider the special case mentioned above, in which •\|r = ^tt, we shall illustrate our definition by some general remarks and particular illustrations. 111. Some general remarks. (1) The existence of a derived function ^' (x) for all values of x in the interval a^x&h implies that <f> (x) is continuous at every point of this interval. For it is evident that {<f> (x + h) — (ji (x)]/h cannot tend to a limit unless lim (f>(x + h) = (f) (x), and it is this which is the property denoted by continuity. (2) It is natural to ask whether the converse is true, i.e. whether every continuous curve has a definite tangent at every point, and every function a differential coefficient for "every value of x for which it is continuous.* The answer is obviously No: it is sufficient to consider the curve formed by two straight lines pig 37^ meeting to form an angle (Fig. 37). * .We leave ont of account the exceptional case (which we have still to examine) in which the curve is supposed to have a tangent perpendicular to OX : apart from this possibility the two forms of the question stated above are equivalent. 200 DERIVATIVES AND INTEGKALS [VI The reader will see at once that in this case {(j) (x + h) — <f> {x)]jh has the limit tanyS when A-*0 by positive values and the limit tan a when K -»-0 by negative values. This is of course a case in which a curve might reasonably be said to have two directions at a point. But the following example, although a little more difficult, shows conclusively that there are cases in which a continuous curve cannot be said to have either one direction or several directions at one of its points. Draw the graph (Fig. 14, p. 53) of the function x sin (!/:»). The function is not defined for x=0, and so is discontinuous for lo—O. On the other hand the function defined by the equations ^{x) = a;am{ljx) (.i;=t=0), ^(a') = («=0) is continuous for x=0 (Exs. xxxvil. 14, 15), and the graph of this function is a continuous curve. But {x) has no derivative for x = 0. For ^' (0) would be, by definition, lim {(^ {h) - </) (0)}/A or lim sin (1/A) ; and no such limit exists. It has even been shown that a function of x may be continuous and yet have no derivative for any value of x, but the proof of this is much more difficult. The reader who is interested in the question may be referred to Bromwich's Infinite Series, pp. 490-1, or Hobson's Theory of Functions of a Real Variable, pp. 620-5. (3) The notion of a derivative or differential coefficient was suggested to us by geometrical considerations. But there is nothing geometrical in the notion itself. The derivative ^' (x) of a function <^ («) may be defined, without any reference to any kind of geometrical representation of 4> («), by the equation and ^ {x) has or has not a derivative, for any particular value of x, according as this limit does or does not exist. The geometry of curves is merely one of many departments of mathematics in which . the idea of a derivative finds an application. Another important application is in dynamics. Suppose that a particle is moving in a straight line in such away that at time t its distance from a fixed point on the line is s = (i). Then the 'velocity of the particle at time t' is by definition the limit of 0(< + A)-<j)(O h as A-*-0. The notion of ' velocity ' is in fact merely a special case of that of the derivative of a function. Ill, 112] DERIVATIVES AND INTEGRALS 201 Examples XXXIX. 1. If (^ (;;;) is a constant then (^' {x)=0. Interpret this result geometrically. 2. If <^ (^) = aw; + 6 then <f)' {x) = a. Prove this (i) from the formal de- finition and (ii) by geometrical considerations. 3. If (j} {a:) = 3;™, where m is a positive integer, then 0' (a;) = »i,»'"~i. [For (^' (x) = lim (^+^)J-^ =lim |OT.t""-' + '"'-"^^ a;"-Vt + ... + /t'»-A The reader should observe that this method cannot be applied to ai"!'', where pjq is a rational fraction, as we have no means of expressing {x + A)"/'' as a finite series of powers of L We shall show later on (§ 118) that the result of this example holds for all rational values, of m. Meanwhile the reader will find it instructive to determine <^' (x) when »i has some special fractional value {e.g. ^), by means of some special device.] 4. If (x) = sin X, then <j)' {x) = cos x ; and if {x) = cos x, then <fi (x) = - sin X. [For example, if <^ (;;;) = sin ^, we have {0 {x+h) - (j) {x)}jh={2 sin JA cos {x + ^h)}lh, the limit of which, when A-»-0, is cos x, since lim cos (x + ^h) = cos x (the cosine being a continuous function) and lim {(sin ^A)/^A} = 1 (Ex. xxxvi. 13).] 5. Equations of the tangent and normal to a curve y=^{x). The tangent to the curve at the point (^qi ^o) ^^ the line through {x^, y,,) which makes with OX an angle ^//■, where ta,n\j/ = <p' (xd). Its equation is therefore .y-yo=(^-^o)0'(.»o) ; and the equation of the normal (the perpendicular to the tangent at the point of contact) is (S/-yo)(l>'{'Vo)+x-Xa=0. We have assumed that the tangent is not parallel to the axis of y. In this special case it is obvious that the tangent and normal are x=xo and y=yo respectively. 6. Write down the equations of the tangent and normal at any point of the parabola x^=iay. Show that if Xo = 2ajm, y^ = alnfi, then the tangent at (^0, J/o) is x=mi/ + (a/m). 112. We have seen that if (j) (x) is not continuous for a value of X then it cannot possibly have a derivative for that value of cc. Thus such functions as 1/x or sin (1/x), which are not defined for x=0, and so necessarily discontinuous for x = 0, cannot have derivatives for x = 0. Or again the function [x], which is discon- tinuous for every integral value of x, has no derivative for any such value of x. 202 DERIVATIVES AND INTEGRALS [VI Example. Since [x] is constant between every two integral values of x, its derivative, whenever it exists, has the value zero. Thus the deriva- tive of fa;], which we may represent by [;»]', is a function equal to zero for all values of x save integral values and undefined for integral values. It sin ttx is interesting to note that the function 1 - Sin TTUi/ properties. has exactly the same We saw also in Ex. xxxvil. 7 that the types of discontinuity which occur most commonly, when we are dealing with the very simplest and most obvious kinds of functions, such as polynomials or rational or trigonometrical functions, are associated with a relation of the type ■ ^ («)-*+ 00 or rf) (a;) ^- — 00 . In all these cases, as in such cases as those con- sidered above, there is no derivative for certain special values of x. / Q Q ^ /\ El Q K R 1 — R R P 1 R p R \ -,R Rr R P C I a \ 'a Q / W (6) (c) (d) Fig. 38. In fact, as was pointed out in § 111, (1), all discontinuities of^(x) are also discontinuities of <^' (x). But the converse is not true, as we may easily see if we return to the geometrical point of view of § 110 and consider the special case, hitherto left aside, in which the graph of ^ {x) has a tangent parallel to OY. This case may be subdivided into a number of cases, of which the most typical are shown in Fig. 38. In cases (c) and {d) the function is two valued on one side of P and not defined on the other. In such cases we may consider the two sets of values of ^ («), which occur on one side of P or the other, as defining distinct functions ^i(«) and ^^{x), the -upper part of the curve corresponding to ^i («). 112, 113] DERIVATIVES AND INTEGRALS 203 The reader will easily convince himself that in (a) [(j>(x + h)-(j> {x)]lh-*+ 00 , ash-^0, and in (b) {^(x + h) — (j){x)}jh-^—oo ; while in (c) {^1 (x + h)-^i (x)]/h^+ 00 , {(^2 (x+h)- ^2 {x)]/h-> — oo , and in (d) {4>i (x + h)- ^1 (x)]/h-- — CO , {^2 (x + h)- <^2 (x)}lh-*+ oo , though of course in (c) only positive and in (d) only negative values of h can be considered, a fact which by itself would preclude the existence of a derivative. We can obtain examples of these four cases by considering the functions defined by the equations (a) y^ = x, (b) y^=-x, (c) if = x, (d) y^ = -x, the special value of x under consideration being a; = 0. 113. Some general rules for differentiation. Through- out the theorems which follow we assume that the functions f(x) and F(x) have derivatives /'(a;) and F'(x) for the values of X considered. (1) // ^ (x) =f(x) + F (x), then <f> (x) has a derivative i>'(x)^f'(x) + F'(x). (2) If <f) (x) = kf (x), where h is a constant, then <^ (x) has a derivative <l>'(x) = kf(x). We leave it as an exercise to the reader to deduce these results from the general theorems stated in Ex. xxxv. 1. (3) If <j) (x) =f(x)F(x), then ^ (x) has a derivative <\>'(x)=f(x)F'(x)+f(x)F(x). For f(.) = lim/(-^^)^<-Y^--^^-)-^^-) .Hm [f(x + h) ^^^'-^l-^^"'\ F(x)f^^^\-f^^^] =f(x)F'(x) + F(x)f(x). 204 DERIVATIVES AND INTEGRALS [VI (4) If<ji{x) = j--^ , then (jt («) has a derivative ^^'^^- {/(*)r In this theorem we of course suppose that f{x) is not equal to zero for the particular value of x under cotisideration. Then ,,, , ,. l{f{x)-f{x+h)\ f'{x) fix) (5) If <f>(x)= p , then ^ (x) has a derivative f(x)F(x)-f(x)F'(x) •^ ^''^ WW ■ This follows at once from (3) and (4). (6) If ^(x) = F {/(«)}, then ^ («) has a derivative cf>'(x)=F'{f(x)}f'(x). For let /(«) = 2/. f{'>^ + h) = y + k. Then A;-*0 as ^ — 0, and k/h^f{x). And = F'(y)f'(x). This theorem includes (2) and (4) as special cases, as we see on taking F(a!) = kx or F(x) = 1/x. Another interesting special case is that in which f(x) = ax+b: the theorem then shows that the derivative of F (ax + b) is aF' (ax + b). Our last theorem requires a few words of preliminary explana- tion. Suppose that x = y^ (y), where i|r (y) is continuous and steadily increasing (or decreasing), in the stricter sense of § 95, in a certain interval of values of y. Then we may write y = <^ («), where </> is the 'inverse' function (§ 109) of i^. (7) If y = (f} (x), where j> is the inverse function ofyfr, so that x = yjf (y), and yjr (y) has a derivative y}r' (y) which is not equal to zero, then ^ (x) has a derivative 1 <^'(x) ■■ >'(2/)' 113-115] DERIVATIVES AND INTEGRALS 205 For if <f>{x + h) = y + k, then A -» as h^O, and ft^o {a; + h)-a; t-*o ir iy + k) - ^ (y) ^'(y)' The last function may now be expressed in terms of os by means of the relation y = (f> (x), so that 4>'{x) is the reciprocal of ■^'{4> («)}• This theorem enables us to differentiate any function if we know the derivative of the inverse function. 114. Derivatives of complex fHinctions. So far we have supposed that y = <p(x) is a purely real function of x. If 2/ is a complex function (jb (x) + iyjr (x), then we define the derivative of y as being <f)'{x) + iylr'{x). The reader will have no difficulty in seeing that Theorems (1) — (5) above retain their validity when <f) (x) is complex. Theorems (6) and (7) have also analogues for complex functions, but these depend upon the general notion of a ' function of a complex variable ', a notion which we have en- countered at present only in a few particular cases. 115. The notation of the'difiFerential calculus. We have already explained that what we call a derivative is often called a differential coefficient. Not only a different name but a different notation is often used ; the derivative of the function y = j>{x) is often denoted by one or other of the expressions ^.^. I- Of these the last is the most usual and convenient : the reader must however be careful to remember that dy/dx does not mean ' a certain number dy divided by another number dx' : it means ' the result of a certain operation B^ or djdx applied to y = (fy (x) ', the operation being that of forming the quotient {(j) {x + h) — <j} (x)]/h and making h~^0. Of course a notation at first sight so peculiar would not have been adopted without some reason, and the reason was as follows. The denomi- nator h of the fraction {0 {x + h)-(f> {x)}jh is the diflference of the values x + h, X of the independent variable x ; similarly the numerator is the difference of the corresponding values (f){x + h), (f) {x) of the dependent variable y. These differences may be called the increments of x and y respectively, and denoted by bx and by. Then the fraction is byjbx, and it is for many purposes convenient to denote the limit of the fraction, which is the same thing as 206 DERIVATIVES AND INTEGRALS [VI 0'(^), by dyldx. But this notation must for the present be regarded as purely symbolical. The dy and dx which occur in it cannot be separated and standing by themselves they would mean nothing : in particular dy and dx do not mean lim by and lim bx, these limits being simply equal to zero. The reader will have to become familiar with this notation, but so long as it puzzles him he will be wise to avoid it by writing the differential coefficientin the form D^^y, or using the notation <i>{x), <i>'{x), as we have done in the preceding sections of this chapter. In Ch. VII, however, we shall show how it is possible to define the symbols dx and dy in such a way that they have an independent meaning and that the derivative dyjdx is actually their quotient. The theorems of § 113 may of course at once be translated into this notation. They may be stated as follows : (1) ify^y^^y., tken 1 = 1^ + ^^ (2) ify^hy^, then dx dx ' (3) i/'2/ = 2/i2/3. <^«e« dx y^dx^y^dx' (4) ify=-, then dy _ 1 dy^_ dx y^ dx ' (5) 2/2 dx [y^dx y^dxily^' (6) ify is a function of x, mid z a function of y, then dz dz dy dx dy dx ' 0) dy^^'/dx\ dx 1 \dy) ' Bzan iplesXL. 1. Ify= =y\yiyz then t=y^^^t^y^y'fx-y^y'fx' if.y= =,?'iy2...3'™then dx r=\ • Viyi-yr-xyr^x-yn-^. In particular, if y=z\ then dyldx=nz''-'^{dzldx) ; and if y=x^, then dyjdx=nce^-\ as was proved otherwise in Ex. xxxix. 3. 115, 116] DERIVATIVES AND INTEGRALS 207 y dx y^dx y^~dx^"' yn dx ' In particular, if « = 2'' then I'^^Vl't. y dx z dx 116. Standard forms. We shall now investigate more systematically the forms of the derivatives of a few of the simplest types of functions. A. Polynomials. If </> (x) = «„«" + a^x^'^ + . . . + a„, then ^' {x) = wrao*""' + (n - 1) aia;»-2 + . . . + a„_i. It is sometimes more convenient to use for the standard form of a polynomial of degree n in x what is known as the binomial form, viz. /ti\ fii\ ffoa;" + L ) a^x''-^+ Lj a2«"-'+ ... + a„. In this case The binomial form of ^ (a;) is often written symbolically as (a„, tti, ...,a,i][a;, l)"; and then <^' {x) = n{ao, a^, ..., aji-{^x, 1)"~^ We shall see later that ^ (a;) can always be expressed as the product of n factors in the form ^ («) = a„ (« - tti) (a; - «2) ...(a^-o-n). where the a's are real or complex numbers. Then ^' {x) = tto S (a; - aa) {x — a^... (x - «„), the notation implying that we form all possible products of n — 1 factors, and add them all together. This form of the result holds even if several of the numbers a are equal ; but of course then some of the terms on the right-hand side are repeated. The reader will easily verify that if ^ {x) = a,{x- a,)"'- {x - a,y^ ...(x- a,)'"', then <f>'(x)=a,l,nh(x-a,)'^-'{x-a^)"'' . . . (a; - a,)™". 208 DERIVATIVES AND INTEGRALS [Vi Examples XLI. 1. Show that if {x) is a polynomial then <ji' (x) is the coefi&cient of h in the expansion of (p (x + h) in powers of h. 2. If (j) (x) is divisible by (x — a)% then <f)' (x) is divisible by :!,• — a : and generally, if cp {x) is divisible by (x — a)'^, then ^'(.r) is divisible by (a.' -a)""'. 3. Conversely, if <^ (.r) and (j>' {x) are both divisible by x-a, then {x) is divisible by {x-af ; and if (^ (a;) is divisible by ^-a and 0' (a;) by (:);-a)"'-i, then (^ (^) is divisible by {x - a)™. 4. Show how to determine as completely as possible the multiple roots of P {x) = 0, where P{x) is a polynomial, with their degrees of multiplicity, by mer.ns of the elementary algebraical operations. [If H^ is the highest common factor of P and i", H^ the highest common factor of Hi and P", H^ that of H^ and P", and so on, then the roots of HiHilHi=Q are the double roots of P=0, the roots of Il2ffi/Hg' = the treble roots, and so on. But it may not be possible to complete the solution of HiH3lHi=Q, H^H^lHi=0, .... Thus if P{x)={x-lf{x^-x-7y then HiH3lH^=x^-x-'J and JI„ffi/ff3^=x-l ; and we cannot solve the first equation.] 5. Find all the roots, with their degrees of multiplicity, of x* + 33,^-3x^^-Ux-6 = 0, a;8 + 2^-5 _ 8.^4 _ 14^ + lXa,.2 + 28.^ + 12=0. 6. If ax^ + 2bx+c has a double root, i.e. is of the form a{x-a.y, then 2 (ax + b) must be divisible by a; - a, so that a= — bja. This value of x must satisfy ax'^ + %bx+c=^0. Verify that the condition thus arrived at is ac-62 = o. 7. The equation \l{x-a) + \l{x-b) + \l{x — c) = can have a pair of equal roots only if a=b = o. {Math. Trip. 1905.) 8. Show that ax^ + Zb.v^ + Zcx-\-d=0 has a double root if (?2 + 453 = 0, where H=ac-b\ 0=aH-Zabo + ibK [Put ax+b=y, when the equation reduces to y3 + 35y + (?=0. This must have a root in common with y^ + 3=0.1 9. The reader may verify that if a, 0, y, S are the roots of ax*+ibx^ + 6o.v^ + 4dx+e= 0, then the equation whose roots are A«{(a-i3)(y-8)-(y-a)(/3-8)}, and two similar expressions formed by permuting a, ft y cyclically, is where 5'2 = ««- ^Sii+Sc^, gs = ace + 2bad-ad'^-eb^-<^. It is clear that if two of a, j3, y, 8 are equal then two of the roots of this cubic will be equal. Using the result of Ex. 8 we deduce that g^-iTg^=0. 116, 117] DERIVATIVES AND INTEGRALS 209 10. Bolle's Theorem for polynomials. If ^{x) is any polynomial, then between any pair of roots of (^{x) = lies a root of (j>' {x) = 0. A general proof of this theorem, applying not only to polynomials but to other classes of functions, will be given later. The following is an algebraical proof valid for polynomials only. We suppose that a, /3 are two aviccessive roots, repeated respectively m and n times, so that <i> {x) = (:i; - a)™ {x - /3)» 6 {x), where 6 {x) is a polynomial which has the same sign, say the positive sign, for aS^S/3- Then <^' (;;;) = (^ - a)"* (ic - j3)" (5' (.r) + {to (^ - a)™ - 1 (a- - /3)" + « (.e - a)™ (« - /3)" - 1} fi (a-) = {x-aY-'^{x-^Y-'^[{x-a){x-^)6'{x) + {m{x-^)-^n{x-a)}6{x)'] = {x-a)^-'^{x-^Y-'>-F{x), say. Now F {a)=m{a- P) 6 (a) and F{l3) = n(^-a)8(fi), which have opposite signs. Hence F{x), and so (j) (x), vanishes for some value of x between u and /3 117. B. Rational Functions. If where P and Q are polynomials, it follov^s at once from § 113, (5) that _ P'(x)Q(x)-P{a:)q(x) and this formula enables us to write down the derivative of any rational function. The form in which we obtain it, however, may or may not be the simplest possible. It will be the simplest possible if Q {x) and Q' (x) have no common factor, i.e. if Q («) has no repeated factor. But if Q (x) has a repeated factor then the expression which we obtain for R' (x) will be capable of further reduction. It is very often convenient, in differentiating a rational function, to employ the method of partial fractions. We shall suppose that Q (x), as in § 116, is expressed in the form ao{x- Hi)"'' {^^ - »2T°- ■■■(«:- a.T". Then it is proved in treatises on Algebra* that R{x) can be expressed in the form (x) H — + -, t;; + ... + -, ^— T-r '^x-a,{x-a,y'^ -'^ (x-a,)'"^^ ' * See, e.g., Chrystal's Algebra, vol. i, pp. 151 et seq. ■u 210 DERIVATIVES AND INTEGRALS [VI where 11 («) is a polynomial; i.e. as the sum of a polynomial and the sum of a number of terms of the type A (x-ay where a is a root of Q {x) = 0. We know already how to find the derivative of the polynomial: and it follows at once from Theorem (4) of § 113, or, if a is complex, from its extension indicated in § 114, that the derivative of the rational function last written is pA {x - ay-'' pA {x - oifp (a;-a)P+i' We are now able to write down the derivative of the general rational function R {x), in the form ^*^ («-«,)' (a;-«i)' ■■■ {x-Oi^y {x-a^y •■■■ Incidentally we have proved that the derivative of x™ is »na;'"~^, for all integral values of m positive or negative. The method explained in this section is particularly useful when we have to differentiate a rational function several times (see Exs. XLv). Examples XLII. 1. Prove that 4x dx \l +x^) ~ (1 +a;Y ' dx [l +x') ~ ' dxXl+x^j {l+x'^'f dx\\+x'/ (l+a;2)2" 2. Prove that d^ ( ax^+'ibx+c \ _(ax-\-h){Bx-\-(J)-{hxJrc){Ax-^E) dx \Ax^'+2Bx + c) " '' (ZJH^S^TTT ■ 3. If Q has a factor {x — a)™, then the denommator of R' (when R' is reduced to its lowest terms) is divisible by (57 — 0)™+' but by no higher power of ^ — a. 4. In no case can the denominator of R' have a simple factor x—a. Hence no rational function (such as 1/^) whose denominator contains any simple factor can be the derivative of another rational function. 118. C. Algebraical Functions. The results of the pre- ceding sections, together with Theorem (6) of § 113, enable us to obtain the derivative of any explicit algebraical function whatsoever. The most important such function is a;™, where m is a rational number. We have seen already (§ 117) that the derivative of this 117, 118] DEEIVATIVKS AND INTEGRALS 211 function is mx"^-^ when in is an integer positive or negative ; and we shall now prove that this result is true for all rational values of m. Suppose that y = x^ = ocP'i, where p and q are integers and q positive ; and let z = x^'i, so that x = zi and y = zP. Then dx \dzjl \dzj q This result may also be deduced as a corollary from Ex. XXXA'I. 3. For, if (j) (x) = a;™, we have f(a;) = lim^'' + ^y~'^'"' = lim ^ = mx^^'K It is clear that the more general formula T- (ax + b)"" = ma {ax + by~^ holds also for all rational values of in. The differentiation of implicit algebraical functions involves certain theoretical difficulties to which we shall return in Ch. VII. But there is no practical difficulty in the actual calculation of the derivative of such a function : the method to be adopted will be illustrated sufficiently by an example. Suppose that y is given by the equation !ffij^ys_ Qaxy = 0. Differentiating with respect to x we find dy of — ay and so t^ = - -„ . ax y^ — ax Examples XLIII. 1. Find the derivatives of 2. Prove that 3. Find the differential coefficient of y when (i) ax"' + 2/uy+bf+2ffx + 2f^ + c = 0, (ii) :v^+f-5axy=0. 14—2 212 DERIVATIVES AND INTEGRALS [VI 119. D. Transcendental Functions. We have already proved (Ex. xxxix. 4) that Dx sin X = cos x, D^ cos x — — sin x. By means of Theorems (4) and (5) of § 113, the reader will easily verify that Dx tan X = sec'^ x, D^ cot x = — cosec^ x, Dx sec X = tan x sec x, Dx cosec x = — cot x cosec x. And by means of Theorem (7) we can determine the derivatives of the ordinary inverse trigonometrical functions. The reader should verify the following formulae: Dx arc sin a; = + l/VCl — «''^), J^x arc cos x= + 1/V(1 — «°), Dx arc tan« = 1/\1 + x% Dx arc cot x= — 1/(1 + x-), Dx arc sec x= ± l/{x^/{x'' — 1)}, Dx arc cosec x= + l/{x>^(x'' — 1)}. In the case of the inverse sine and cosecant the ambiguous sign is the same as that of cos (arc sin «), iii the case of the inverse cosine and secant the same as that of sin (arc cos x). The more general formulae Dx arc sin (x/a) = + l/>J(a^ - x% D^ arc tan (x/a) = a/(x'^ + a?), which are also easily derived from Theorem (7) of § 113, are also of considerable importance. In the first of them the ambiguous sign is the same as that of a cos {arc sin (x/a)}, since a V{1 - («V«')} = ± \/(a' - «') according as a is positive or negative. Finally, by means of TJieorem (6) of § 113, we are enabled to differentiate composite functions involving symbols both of alge- braical and trigonometrical functionality, and so to write down the derivative of any such function as occurs in the following examples. Examples XLIV.* 1. Find the derivatives of cos"*;!;, sin"" a;, cos.*™, sin*"*, cos (sin a;), sin (cos .t), I/O a , 7o • , , COS a; sin ^ si {a' cos^ x + lflsn\^ x). a; arc 8\nx+J{l-x^), (1 +a.') arc tan ijx-.jx. * In these examples m is a rational number and a, b, ... , a, j3, ... hn,ve such values that the functions which involve them are real. 119] DERIVATIVES AND INTEGRALS 213 2. Verify by differentiation that arc sin x 4- arc cos ^ is constant for all values of x between and 1, and arc tan^r+arocota; for all positive values of X. 3. Find the derivatives of ) ■ How do you explain the simplicity of the results ? 4. Differentiate 1 , ax+h 1 . flfjr+i : arc tan —r, roi , — r, i arc sm - 5. Show that each of the functions fx-fi\ „ , /f^-&\ ,„2J{(a-^)(^-g)} .....ye-^. -"-^(-f: lias the derivative 6 Prove that d J{{a-x){x-^)y {arccosy(^)}=yi de\ V Vcos^^/J \ \oosecoaZ6j {Math. Trip. 1904.) 7 Show that 1 dV /.f g(tM,-2+fl) ) • c-aG)}dx\_ ^''° °°^ V \c {Ax'- + (7)1 . J{G(Ac-aC)}dx 8. Each of the functions 1 (Ax^+0)^{ax^-+c) , arc cos /acosA'+6\ 2 , f / fa-h\, , 1 — TT . ,, o T,s arc tan \ ^ / ( -— -r tan \x\ ^/(a2-62) has the derivative l/(a+6cos^). 9. If Z'=a + 6cosi!;+csin:!;, and 1 aX-a^ + Jf' + c^ y~ ^{a^-h^-c'f '"''"' X^{b^ + c^) ' then dyldx=\IX. 10. Prove that the derivative of F\_f {(f) (x)}] is F'[f {^ (,r)}] / {(j) (x)} 4>' (x), and extend the result to still more complicated cases. 11. If « and i; are functions of x, then Dx arc tan (u/v) = {vDxU — uDxV)/{u^ -^-v^). 12. The derivative of y = (tan x + sec x)^ is wiy sec x. 13. The derivative of y= cos X-+1 sin iris ly. 14. Differentiate x cos x, {smx)l.v. Show that the values of x for which the tangents to the curvesy=,5; cos x, y=(sin x)/x are parallel to the axis of x are roots of cot.j;=a', tana:=.j; respectively. 214 DERIVATIVES AND INTEGRALS [VI 15. It is easy to see (cf. Ex. xvii. 5) that the equation sin.r=a.r, where a is positive, has no real roots except x=Oii as>l, and if a< 1 a finite numher of roots which increases as a diminishes. Prove that the values of a for which the number of roots changes are the values of cos |, where ^ is a positive root of the equation tan |=|. [The values required are the values of a fur which y — ax touches y=sin x.] 16. If 0(a;)=a;2sin(l/«) when^' + O, and (^(0) = 0, then <}>' (x) = 2x sin (1/x) - cos (l/x) when x=^0, and <j>' (0)=0. And ^' (.r) is discontinuous for .•!;=0 (of. § 111, (2)). 17. Find the equations of the tangent and normal at the point {Xf^, y^) of the circle x^+y'^=a^. [Here y = ij(a''-x^), dy/dx= —x/^^a'-x^), and the tangent is y - yo = (^ - ^o) { - ^ol-Jia^ - V)}> which maybe reduced to the form xxo+yyo = a^- The normal is xyQ-yx^=Q, which of course passes through the origin.] 18. Find the equations of the tangent and normal at any point of the ellipse {xlaf + iylby=l and the hyperbola {xjaf-iylbf^l. 19. The equations of the tangent and normal to the curve x = (j>{t), y = -\[r {t), at the point whose parameter is t, are ^'tr = '^^^^' {^-^«}<^'W+{y-V'«}V''(0=o. 120. Repeated differentiation. We may form a new function </)"(«) from <f>' (x) just as we formed (f)' (x) from (f) (x). This function is called the second derivative or second differential coefficient of (f) (x). The second derivative oi y= ^ {x) may also be written in fl,ny of the forms ^"^' W/^' dx^- In exactly the same way we may define the nth derivative or nth differential coefficient oi y = <j> (x), which may be written in any of the forms ^-(.), B.^y, {^Jy, g. But it is only in a few cases that it is easy to write down a general formula for the nth. differential coefficient of a given function. Some of these cases will be found in the examples which follow. 119, 120] DERIVATIVES AND INTEGRALS 215 Examples XLV. 1. If (a;) = a;™ then (|)(") {a;) = m{m-l)...{m-n + l):c"'-». This result enables us to write down the rath derivative of any polynomial. 2. If (j) (x) = (ax + 6)'" then (f)i'')(a;)=m(m-l)... {m- 7i + l)a?'(ax + b)"'-". In these two examples m may have any rational value. If m is a positive integer, and n>m, then <^W (.r) = 0. 3. The formula (AX ^ ( iw P(y + l)-(?'+rt-l)^ \dxj {x-af ^ ' (a^-a)P + » enables us to write down the «th derivative of any rational function expressed in the standard form as a sum of partial fractions. 4. Prove that the ?ith derivative of 1/(1 — x^) is |(m!){(l-^)-"-i + (-l)''(l+^) — -'}. 5. Leibniz' Theorem. If y is a product uv, and we can form the first n derivatiA'es of u and v, then we can form the ?ith derivative of y by means of Leibni£ Theorem, which gives the rule 'n\ V V, /n\ V M. where suf&xes indicate differentiations, so that ii„, for example, denotes the nth derivative of u. To prove the theorem we observe that (UV)2 = U^V + 2 Ml «! + M2'2 1 and so on. It is obvious that by repeating this process we arrive at a formula of the type (M«)„ = 1t„V+a„,iM„_iVi+a,^2tt„_2^'2 + •••+«», r«n-rt'r+---+M"n- Let us assume that a„, r= (' j for ''= !> % ••• n-\, and show that if this is so then a,i + i, ,. = ("■ j for r = \, 2, ... n. It will then follow by the principle of mathematical induction that a,,, ,. = ( ] for all values of n and r in question. When we form (mw)„ + i by differentiating (««)„ it is clear that the coefficient of M„+i_rVris This establishes the theorem. 216 DERIVATIVES AND INTEGRALS [VI e. The nth derivative of x^f{x) is (m-n)\ •'^•'^ (m-n + 1)! + nL2^(m-« + 2)!'' •^ W + -. tlie series being continued for n + 1 terms or until it terminates. 7. Prove that Z)j;''cosi- = cos (^■4-|7i7r), Z)j;"sin^=sin(.s;+iji7r) 8. If y=4cosma; + 5sinTO^ then 7)/y+m^y=0. Andif 2/ = 4 cos ma- + B sin mx + P„ {x\ where Z',i(a;) is a polynomial of degree n, then i>j;" + 3y + m2i)a." + iy = 0. 9. If ;!;22)^2y + ^Z)^y+y = then [Diflferentiate m times by Leibnitz' Theorem.] 10. If 6',j denotes the wth derivative of {Lx+3I)/{x^—2Bx + C), then [First obtain the equation when n = ; then differentiate n times by Leibnitz' Theorem.] 11. Thenth deiivaXives ot al{a^+x^) and xl{a^+x^). Since we have a _ 1 / 1 ^\ a: - V -^ 1 N a^ + .r^ 2i\a; — at x + aij' a^ + x^ '■2\x — ai x + aij ' / g \_ (-irn\ ( 1 1 I and a similar formula for Z)j.''{.5;/(a2 + a;i!)j_ jf p = ^(a;2 + a2^j and 6 is the numerically smallest angle whobe cosine and .sine are x/p and a/p, then x + ai=p Cis 6 and x-ai=p Cis (- fl), and so i)^"{a/(a2 + «2)} = {(-l)''»!/2t}p-''-i[Cis{(»+l)6}-Cis{-(n + l)i9}] = (-l)"»!(a;2 + a2)-('' + i)'2sin{(B + l)arotan(a/a;)}. Similarly i;^» {a;/(a2 + ^2)} = (- 1)"M I (•^2 + a2)-(»+i)/2 COS {(ji+ 1) arc tan {ajx)). 12. Prove that i)/{(oos^)/x} = {P„cos(a; + |mff) + §„sin(a; + ^mn-)}/a»+i, 2)j," {(sin 5;)/^} = {P„sin (x + ^nn) - §„ cos (a;+ J?i7r)}/a;'» + 1, where P„ and §„ are polynomials in x of degree m and n-1 respectively. 13. Establish the formulae ^:^\/f'^\ ^^_rf^//c^Y d?x^_id^dy__ (dYWIfdyV' dy I \dx) ' df dxy \dx) ' df \dx^ dx '^ \dxy\/ \di) ■ 120, 121] DERIVATIVES AND INTEGRALS 217 14. If y2 = 1 and yr={\lr !) Djy, z, = {\js !) D^'z, then 1 Z Zl 22 -ll^a ys S3 h «2 ^3 2/^ 'ys ^4 «2 ^3 24 (i/aitA. THp. 1905. 15. If IT {y, z, u)=- y z ii , dashes denoting diflferentiations with y' 2' * y'' 2" t respect to a;, then W{y, z, u)=y^W (l, -, -). 16. If ax^ + 2hxy + by'' + 2gx + 2fy + c=Q, then c?y/rfa? = - (ax + hy +g!)/{hx + by+f) and dh/ldx'' = {ahc + 2fgh - af - hg^ - ch^)j{hx + hy+ff. 121. Some general theorems concerning derived func- tions. In all that follows we suppose that j> (x) is a function of x which has a derivative <^' (x) for all values of x in question. This assumption of course involves the continuity of (j& {x). The meaning of the sign of ^'(a;). Theorem A. If <fi' (xq) > then <j) (x) < (j> (xq) for all values of x less than x^ hut sufficiently near to x^, and (j) (x) > (^ (xq) for all values of x greater than Xo but sufficiently near to x^. For {(j) {xa + h)— (f) (Xf,)}jh converges to a positive limit ^' {x^) as h^-O. This can only be the case if <f) {x„ + h) — <j> {x^ and h have the same sign for sufficiently small values of h, and this is precisely what the theorem states. Of course from a geometrical point of view the result is intuitive, the inequality ^' («) > expressing the fact that the tangent to the curve y = <^{x) makes a positive acute angle with the axis of x. The reader should formulate for himself the corresponding theorem for the case in which </>' (x) < 0. An immediate deduction from Theorem A is the following important theorem, generally known as Rolle's Theorem. In view of the great importance of this theorem it may be well to repeat that its truth depends on the assumption of the existence of the derivative ^' {x) for all values of x in question. Theorem B. If cf){a) = and <j) (b) = 0, then there must be at least one value of x which lies bettveen a and b and for which 4>' (x) = 0. There are two possibilities : the first is that ^ (x) is equal to 218 DERIVATIVES AND INTEGRALS [VI zero throughout the whole interval (a, b). In this case <^' (x) is also equal to zero throughout the interval. If on the other hand 4> (x) is not always equal to zero, then there must be values of X for which <^ (x) is positive or negative. Let us suppose, for example, that <f) (x) is sometimes positive. Then, by Theorem 2 of § 102, there is a value ^ of x, not equal to a or b, and such that (j) (A is at least as great as the value of (f> (x) at any other point in the interval. And <j>' (f) must be equal to zero. For if it were positive then (/> (x) would, by Theorem A, be greater than ^ (^) for values of x greater than f but sufificiently near to ^, so that there would certainly be values of ^ (x) greater than ^ (^). Similarly we can show that <^' (^) cannot be negative. Cor. 1. If ^ ip,) = <f>{b) = k, then there must be a value of x between a and b such that (f) (x) = 0. We have only to put <j) (x) — k = -yjr {x) and apply Theorem B to -v|r (x). CoE. 2. If (f>' (x) > for all values of x in a certain interval, then <f) (x) is an increasing function of x, in the stricter sense of § 95, throughout that interval. Let «i and x^ be two values of x in the interval in question, and «! < x^. We have to show that (/> {x^ < ^ {x^). In the first place (aji) cannot be equal to </> (x^ ; for, if this were so, there would, by Theorem B, be a value of x between x-^ and x,^ for Avhich <^' (a;) = 0. Nor can ^ {x^ be greater than ^ (x^. For, since <;£)' {x^ is positive, {x) is, by Theorem A, greater than {x^ when x is greater than x^ and sufficiently near to x^. It follows that there is a 'value x^ of x between x-^ and X2 such that <^ {x^ = (j) (x^) ; and so, by Theorem B, that there is a value of x between Xi and x^ for which (f>' (x) — 0. Cor. 3. The conclusion of Cor. 2 still holds if the interval (a, b) considered includes a finite number of exceptional values of x for which <{>' (x) does not exist, or is not positive, provided j> (x) is continuous even for these exceptional values of x. It is plainly sufficient to consider the case in which there is one exceptional value of x only, and that corresponding to an end of the interval, say to a. If a < «i < aJa < b, we can choose a+e so that a + € <Xi, and (f) (x) > throughout (a + e, b), so that (j) (i»i) < <ji (xs), by Cor. 2. All that remains is to prove that 121, 122] DERIVATIVES AND INTEGRALS 219 <j)(a)< <l> (Xi). Now </) (xi) decreases steadily, and in the stricter sense, as x^ decreases towards a, and so <f) (a) = (f> (a + 0) = lim <j} (ajj) < (j) (x^). a;i-*-a+0 Cor. 4. //"(/>' (x) > throughout the interval (a, h), and (f> (a) S 0, then <f) (x) is positive throughout the interval (a, b). The reader should compare the secoud of these corollaries very carefully with Theorem A. If, as in Theorem A, we assume only that <f)'{a;) is positive at a single point x=x^, then we can prove that (j) {xi) < (j) (^-2) when Xi and x'2 are sufficiently near to ss^ and Xi<x:„< x^. For <f>{x{)<<p {xg) and (^ (x^) > (f> (xq), by Theorem A. But this does not prove that there is any interval including Xg throughout which <^ (x) is a steadily increasing function, for the assumption that Xi and X2 lie on opposite sides of Xg is essential to our conclusion. We shall return to this point, and illustrate it by an actual example, in a moment (§ 124). 122. Maxima and Minima. We shall say that the value <j)(^) assumed by <^ (x) when a; = f is a maximum if <f) (f ) is greater than any other value assumed by <j) (x) in the immediate neighbourhood of x=^, i.e. if we can find an interval (^ — e, f + e) of values of X such that (j>{^)><f) (x) when ^— e<x<^ and when ^<x< f 4- e; and we define a minimum in a similar manner. Thus in the figure the points A correspond to maxima, the points B to minima of Fig. 39. the function whose graph is there shown. It is to be observed that the fact that ^s corresponds to a maximum and Bi to a minimum is in no way inconsistent with the fact that the value of the function is greater at £1 than at -4 3. Theorem 0. A necessary condition for a maxivium or minimum value of <f>(x) at x=^ is that ^' (^) = 0.* * A function which is continuous but has no derivative may have maxima and minima. We are of course assuming the existence of the derivative. 220 DERIVATIVES AND INTEGRALS [VI This follows at once from Theorem A. That the condition is not sufficient is evident from a glance at the point C in the figure. Thus ii y = ai' then </>'(«) = Sfl;^ which vanishes when x = 0. But ^ = does not give either a maximum or a minimum of a^, as is obvious from the form of the graph of of (Fig. 10, p. 45). But there will certainly be a maximum at x = ^ if ^' (^) = 0, <^' {x) > for all values of x less than but near to ^, and (j)' (x) < for all valves of x greater than but near to f : and if the signs of these two inequalities are reversed there will certainly be a minimum. For then we can (by Cor. 3 of § 121) determine an interval (f — e, ^) throughout which ^ {x) increases with x, and an interval (f , ^ + e) throughout which it decreases as x increases : and obviously this ensures that (/> (f) shall be a maximum. This result may also be stated thus. If the sign of ^ {x) changes at x = ^ from positive to negative, then x = ^ gives a maximum of ^ {x) : and if the sign of <j}' (x) changes in the opposite sense, then x= ^ gives a minimum. 123. There is another way of stating the conditions for a maximum or minimum which is often useful. Let us assume that (j) (x) has a second derivative ^" (x) : this of course does not follow from the existence of ^' (x), any more than the existence of <j)' (x) follows from that of (j) (x). But in such cases as we are likely to meet with at present the condition is generally satisfied. Theorem D. If <^'(?)=0 and <^"(^)4=0, then 4)(x) has a maximum or minimum at a;=^, a maximum if (j>"{^)<0, a ininimum if <f>" (^) > 0. Suppose, e.g., that 0"(^)<O. Then, by Theorem A, <f>'{x) is negative when x is less than ^ but sufficiently near to f, and positive when x is greater than J but sufficiently near to ^. Thus x = ^ gives a maximum. 124. In "what has preceded (apart from the last paragraph) we have assumed simply that (x) has a derivative for all values of x in the interval imder consideration. If this condition is not fulfilled the theorems cease to be true. Thus Theorem B fails in the case of the function .y=l-V(.V''), 122-124] DERIVATIVES AND INTEGRALS 221 where the square root is to he taken positive. The graph of this function is shown in Fig. 40. Here <^(- 1) = ^(1) = 0: but <l)'{x), as is evident from the figure, is equal to 1 if ^ is negative and to — 1 if ^ is positive, and never vanishes. There is no derivative for ^ = 0, and no tangent to the graph at P. And in this case x=0 obviously gives a maximum of <|>(^), but <f>' (0), as it does not exist, cannot be equal to zero, so that the test for a maximum fails. The bare existence of the derivative (p' {x), however, is all that we have as- sumed. And there is one assumption iu particular that we have not made, and that is that <^' {x) itself is a con- tinuous function. This raises a rather subtle but still a very interesting point. Can a, function (f> (x) have a derivative for all values of x which is not itself continuous ? In , other words can a curve have a tangent at every point, and yet the direction of the tangent not vary contiimously ? The reader, if he considers what the question means and tries to answer it in the light of common sense, will probably incline to the answer No. It is, however, not difficult to see that this answer is wrong. Consider the function (l>{x) defined, when x^O, by the equation <f>{x)=x^aiii{llx) ; and suppose that ^(0) = 0. Then (j){x) is continuous for all values of x. If x=¥0 then (p' {x) = 2xsm(}lx) — cos(l/.r) ; while d)'(0) = lim h^Bin(l//i) = 0. Thus <^'(.v) exists for all values of x. But cf)' (x) is discontinuo>is for x=0; for ixam(llx) tends to as xs-O, and cos(l/.r) oscillates between the limits of indetermination —1 and 1, so that <f>'(x) oscillates between the same limits. What is practically the same example enables us also to illustrate the point referred to at the end of § 121. Let <j) [x) =.r2sin {l/x) + a.v, where 0<a<l, when a;=t=0, and 0(O) = O. Then 0'(O) = a>O. Thus the conditions of Theorem A of § 121 are satisfied. But if x4=0 then (f> (x) = 2x sin (1/x) — cos (Ijx) + a, which oscillates between the limits of indetermination a-1 and a + 1 as.r-^0. As a - 1< 0, we can find values of x, as near to as we like, for which (l)'{x)<0; and it is therefore impossible to find any interval, including x=0, throughout which (f> (x) is a steadily increasing function of x. 222 DERIVATIVES AND INTEGRALS [VI It is, however, impossible that (j>'(x) should have what was called in Ch. V (Ex. XXXVII. 18) a 'simple' discontinuity; e.ff. that <^'{x)-»-a when a;^s- + 0, <j)'{x)-^b when x^' — O, and <^'(0) = c, unless a=b = c, in which case <i)'{x) is continuous for x=0. For a proof see § 125, Ex. xlvii. 3. Examples XL VI. 1. Verify Theorem B when <}> (x) = (a; - a)™ (x- - 6)» or (j) (x) = {x- a)™ {x - 6)» (x - c)P where m, n, p are positive integers and a<b<c. [The first function vanishes for x=a and x=b. And (j)'{x) = {x-a)'^-^{x~b)''-'^{{m+n)x-mb-na} vanishes for x={mb+na)l{m+n), which lies between a and b. In the second case we have to verify that the quadratic equation (m+n+p)x^-{m,{b + c)+n(c + a)+p{a+b)}x + mbc + nca+pab=0 has roots between a and 6 and between 6 and c] 2. Show that the polynomials 2x^ + dx^-12x + 7, 3x* + 8x^-6x^-2Ax+19 are positive when x>l. 3. Show that x — sin x is an increasing function throughout any interval of values of x, and that tan j;-a- increases as x increases from -^n- to in-. For what values of a is ax — sin x a steadily increasing or decreasing function of a,-? 4. Show that tana; — .3; also increases from x = ^'n- to x=^ir, from x=?^Tr to a; = f 7!-, and so on, and deduce that there is one and only one root of the equation tana;=a; in each of these intervals (cf. Ex. xvii. 4). 5. Deduce from Ex. 3 that avcix-x<0 if x>0, from this that cosx-\ + \x'^>0, and from, this that simx-x+\sfi>0. And, generally, prove that if C2,„ = cosa;-l+-2i-...-(-l)'»2^, ^ ^.2m + 1 S„„,,^Anx-x+^^-...-{-ir^^^^^y, and .r>0, then C2,„ and (S2m+i S'l'^ positive or negative according as m is odd or even. 6. If f{x) and /" {x) are continuous and have the same sign at every point of an interval (a, 6), then this interval can include at most one root of either of the equations / {x) = 0, /' {x) = 0. 7. The functions u, v and their derivatives u', v' are continuous throughout a certain interval of values of x, and uv' — u'v never vanishes at any point of the interval. Show that between any two roots of m=0 lies one of ■y = 0, and conversely. Verify the theorem when u = cosx, v=saix. [If V does not vanish between two roots of m = 0, say a and ft then the function ujv is continuous throughout the interval (a, ^) and vanishes at its extremities. Hence {ulv)'={u'v — wv')jv^ must vanish between a and j3, which contradicts our hypothesis.] 124] DERIVATIVES AND INTEGRALS 223 8. Determine the maxima and minima (if any) of (x- 1)^ (^.• + 2), x^-Zx, case sketch the form of the graph of the function. [Consider the last function, for example. Here (j>'{x) = 5x^{x''-Q), which vanishes for x=—3, x=0, and ^=3. It is easy to see that a;= -3 gives a maximum and a;=3 a minimum, while a;=0 gives neither, as (ji'ix) is negative on both sides of a;=0.] 9. Discuss the maxima and minima of the function {x - a)" {x - 6)", where m and n are any positive integers, considering the different cases which occur according as m and n are odd or even. Sketch the graph of the function. 10. Discuss similarly the function (x - a) (x - b)^ (x - c)', distinguishing the different forms of the graph which correspond to different hypotheses as to the relative magnitudes of a, h, c. 11. Show that {ax + h)l{cx-\r<^ has no maxima or minima, whatever values a, b, c, d may have. Draw a graph of the function. 12. Discuss the maxima and minima of the function y = {ax^ +^bx+ c)j{Ax^ + Wx + c), when the denominator has complex roots. [We may suppose a and A positive. The derivative vanishes if {ax + b) {Bx+ O) - {Ax+B) {bx+c) = (i (1). This equation must have real roots. For if not the derivative would always have the same sign, and this is impossible, since y is continuous for all values of X, and y^—ajA as a;-»- + oo or x-^— oo. It is easy to verify that the curve cuts the line y = ajA in one and only one point, and that it lies above this line for large positive values of x, and below it for large negative values, or rice versa, according as bja>BjA or bja<BIA. Thus the algebraically greater root of (1) gives a maximum if bla>BjA, a minimum in the contrary case.] 13. The maximum and minimum values themselves are the values of X ' for which ax^ + 'ibx + c — \{Ax'^ + 2Bx+C) is a perfect square. [This is the condition that y=\ should touch the curve.] 14. In general the maxima and minima of R{x) = P{x)jQ {x) are among the values of X obtained by expressing the condition thaX P{x) — \Q{x) = should have a pair of equal roots. 15. If Ax^-\-^Bx + C=0 has real roots then it is convenient to proceed as follows. We have .^ y-(a/.4)-(X^+fL)/{4(4^2+2Z?.^;+0)}, where \ = bA-aB, ii = cA-aC. Writing further ^ for \x + ii and r) for (A/X^) {Ay - a), we obtain an equation of the form 224 DERIVATIVES AND INTEGRALS [VI This transformation from {x, y) to (|, r)) amounts only to a shifting of the origin, keeping the axes parallel to themselves, a change of scale along each axis, and (if X < 0) a reversal in direction of the axis of abscissae ; and so a minimum of y, considered as a function of x, corresponds to a minimum of r\ considered as a function of |, and vice versa, and similarly for a maximum. The derivative of r) with respect to | vanishes if (|-i')(|-2)-|(|-p)-|(|-?) = 0, or if ^=pq. Thus there are two roots of the derivative if p and q have the same sign, none if they have opposite signs. In the latter case the form of the gi-aph of t] is as shown in Fig. 41 a. 7 J p ,V 4-- y o f] « Fig. 41a. Fig. 416. Fig. 41c. Y J v_ O A «2 X Fig. 42. When p and q are positive the general form of the graph is as shown in Fig. 416, and it is easy to see that ^=^{pq) gives a maximum and f = ~^l{pq) a minimum.* In the particular case in which p = q the function is and its graph is of the form shown in J^ig. 41c. The preceding discussion fails if X = 0, i.e. if a/A = b/B. But in this case we have y - (a/A) =tij{A {Ax^+^Dx+ C)} ^t^j{A^ {x - ^i) {x - a^z)}, - say, and dyjdx=0 gives the single value .^■ = ^ {xi + x^). On drawing a graph it becomes clear that this value gives a maximum or minimum according as fi is positive or negative. The graph shown in Fig. 42 corresponds to the former case. [A full discussion of the general function y = {ax"^ + ibx-^- c)j{Ax'^ + 2Bx-\rC), by purely algebraical methods, will be found in CLrystal's Algebra, vol i, pp. 464-7.] 16. Show that {x - a) (x — P)/(x - y) assumes all real values as x varies, if y lies between a and /3, and otherwise assumes all values except those included in an interval of length 4^{|a-y| |j3 — yl). * The maximum is -Vi-Jp-sJi)', the minimum -l/(vV + v/?)^ °f which the latter is the greater. -(; 124] DERIVATIVES AND INTEGRALS 225 17. Show that _ x^ + 2x + c can assume any real value if <c < 1, and draw a graph of the function in this case. {Math. Trip. 1910.) 18. Determine the function of the form {aa;' + 2bx+o)j{Ax'^+2Bx + C) which has turning values (i.e. maxima or minima) 2 and 3 when x = l and a;= -1 respectively, and has the value 2-5 when x = 0. (Math. Trip. 1908.) 19. The maximum and minimum of (x + a) {x + b)l{.v-a) {x-b), where a and b are positive, are f^ a+^b y _ f^/a-y/by Wa-JbJ' Wa + s/bJ 20. The maximum value of (x - lfl{x + 1)3 is ^. 21. Discuss the maxima and minima of x{x-\)l{x'i + Zx + 2.), ^/(■■s-l) (.r-3)5, {x-iy{Zx'^-2x-Z-)j{x + bf{Zi:'^-lix-\). {Math. Trip. 1898.) [If the last function be denoted by P{x)jQ{x), it will be found that P'Q-Pq'^12{x-l){x-Z){x-\){x + \){x + 2){x + b).-\ 22. Find the maxima and minima of acos.» + 6sin^. Verify the result by expressing the function in the form A cos {x — a). 23. Find the maxima and minima of a^ cos^ x + b"^ sin^ x, A cos^ x + 2H cos x sin ^ + 2? sin^ x. 24. Show that sin (.rH-a)/sin (a; + 6) has no maxima or minima. Draw a gi'aph of the function. 25. Show that the function sin^.a; .,,,.,, ,, (0<fl!<6<ir) sin{x + a)sm{x + b) ' has an infinity of minima equal to and of maxima equal to - 4 sin a sin b/sm'^{a - b). {Math. Trip. 1909.) 26. The least value of a,^ sec^ x + b"^ coseo^ x is {a + 6)^. 27. Show that tan Zx cot Zx cannot lie between \ and f . 28. Show that, if the sum of the lengths of the hypothenuse and another side of a right-angled triangle is given, then the area of the triangle is a maximum when the angle between those sides is 60°. {Math. Trip. 1909.) 29. A line is drawn through a fixed point (a, b) to meet the axes OX, OT in P and Q. Show that the minimxim values of PQ, OP+0(i, and OP. OQ are respectively {a^l^ + b^'^fl\ {^a+^b)\ and 4a6. H. 13 226 DERIVATIVES AND INTEGRALS [VI 30. A tangent to an ellipse meets the axes in P and §. Show that the least value of P§ is equal to the sum of the semiaxes of the ellipse. 31. Find the lengths and directions of the axes of the conic a.»2 + 2A^y + 6y2 = 1 . [The length r of the semidiameter which makes an angle 6 with the axis of X is given by l/?-2 = a cos^e+SA cos 6 sin 5 + 6 sin^ 6. The condition for a maximum or minimum value of r is tan2fl = 2A/(a-&). Eliminating 6 between these two equations we find {a-(l/0}{6-(l/0} = /A] 32. The greatest value of ^"'y", where x and v are positive and x+y = lc, is 33. The greatest value of ax + hy, where x and y are positive and x'^ + xy-k-y''^ = ZK^, is 2KV(a^-a6 + 62). [If ax-\-by is a maximum then a + b{dy/dx)=0. The relation between ;;; and y gives {2x+y) + (x+2y) (dyldx) = 0. Equate the two values of dyldx.] 34. If 5 and (/> are acute angles connected by the relation a sec 5 + 6 sec = c, where a, b, c are positive, then a cos 5 + 6 cos ^ is a minimum when 6 = (f>. 125. The Mean Value Theorem. We can proceed now to the proof of another general theorem of extreme importance, a theorem commonly known as ' The Mean Value Theorem ' or ' The Theorem of the Mean'. Theorem. If ^ (x) has a derivative for all values of x in the interval (a, b), then there is a value ^ of X between a and h, such that <l>(b)-cf>(a) = (b-a)ct>'(^). Before we give a strict proof of this theorem, which is perhaps the most important theorem in the Differential Calculus, it will be well to point out its obvious geometrical meaning. This is simply (see Fig. 43) that if the <p{a) m Fig. 43. curve APB has a tangent at all points of its length then there 124, 125] DERIVATIVES AND INTEGRALS 227 must be a point, such as P, where the tangent is parallel to AB. For ^'(f) is the tangent of the angle which the tangent at P makes with OX, and {^ {h) - (\> (a)}/{b - a) the tangent of the angle which AB makes with OX. It is easy to give a strict analytical proof. Consider the function ^(b)-<f>(x)-l^^^{cl>{b)-cf>(a)}, which vanishes when x = a and x = b. It follows from Theorem B of § 121 that there is. a value f for which its derivative vanishes. But this derivative is i>{b)-<l^{a) b- a -<f>'(x); which proves the theorem. It should be observed that it has not been assumed in this proof that (f>' (x) is continuous. It is often convenient to express the Mean Value Theorem in the form 4>(b) = ^ (a) + (b-a)<^'{a + e{b- a)], where 6 is a number lying between and 1. Of course a-\-6{b—a) is merely another way of writing 'some number ^ between a and b'. If we put 6 = a + A we obtain <^ (a + A) = (^ (a) + h^' (a + dh), which is the form in which the theorem is most often quoted. Examples XL VII. 1. Show that <^(6)-.^(^)-|^'{<^(6)-<^(a)} is the difference between the ordinates of a point on the curve and the corresponding point on the chord. 2. Verify the theorem when (p (x) =x^ and when <^ (x) = x^. [In the latter case we have to prove that (b^ — a^)j(b — a) = 3^^, where a<^<b ; i.e. that if J (b'' + ab + a^)=^^ then | lies between a and 6.] ,3. Establish the theorem stated at the end of § 124 by means of the Mean Value Theorem. [Since <^'(0) = c, we can find a small positive value of .v such that {(j){x)-(li{0)}/x is nearly equal to c; and therefore, by the theorem, a small positive value of ^ such that 0' (|) is nearly equal to u, which is inconsistent with lim 0'(a;) = a, unless a=c. Similarly ft = c.] a-»-+0 15—2 228 DERIVATIVES AND INTEGRALS [VI 4. Use the Mean Value Theorem to prove Theorem (6) of § 113, assuming that the derivatives which occur are continuous. [The derivative oiF{f{x)} is by definition j.^ F{f{x+h)}-F{f{x)} h But, by the Mean Value Theorem, f{x + h) =/ {x) + hf (|), where J is a number lying between x and x-\-h. And F{f{x)+hf{^)}^F{f{x)}+hf {^)F' (10, where |i is a number lying between/(^) and/(«) + A/' (^). Hence the deriva- tive of i^ {/(a;)} ia lim/'(|)i?"(|:)=/'(^)i^'{/W}, since ^^-x and ^\-^f{x) as A-»-0.] 126. The Mean Value Theorem furnishes us with a proof of a result which is of great importance in what follows : if <^' {x) = 0, throughout a certain interval of values of x, then ^ (x) is constant throughout that interval. For, if a and h are any two values of x in the interval, then 4>{h)-4> (a) = {h-a)4>'{a + e{h- a)\ = 0. An immediate corollary is that if ^' {x) = i/r' (x), throughout a certain interval, then the functions <^ {x) and -v/r {x) differ through- out that interval by a constant. 127. Integration. We have in this chapter seen how we can find the derivative of a given function (^ («) in a variety of cases, including all those of the commonest occurrence. It is natural to consider the converse question, that of determining a function whose derivative is a given function. Suppose that i/r (*•) is the given function. Then we wish to determine a function such that <^' («) = ■'^ (x). A little reflection shows us that this question may really be analysed into three parts. (1) In the first place we want to know whether such a function as (x) actually exists. This question must be carefully distinguished from the question as to whether (supposing that there is such a function) we can find any simple formula to express it. (2) We want to know whether it is possible that more than one such function should exist, i.e. we want to know whether our 125-127] DERIVATIVES AND INTEGRALS 229 problem is one which admits of a unique solution or not; and if not, we want to know whether there is any simple relation between the different solutions which will enable us to express all of them in terms of any particular one. (3) If there is a solution, we want to know how to find an actual expression for it. It will throw light on the nature of these three distinct ques- tions if we compare them with the three corresponding questions which arise with regard to the differentiation of functions. (1) A function <f) (x) may have a derivative for all values of x, like X™, where m is a positive integer, or sin x. It may generally, but not always have one, like ^x or tana; or sec a;. Or again it may never have one : for example, the function considered in Ex. xxxvii. 20, which is nowhere continuous, has obviously no derivative for any value of x. Of course during this chapter we have confined ourselves to functions which are continuous except for some special values of x. The example of the function ^/x, how- ever, shows that a continuous function may not have a derivative for some special value of x, in this case a; = 0. Whether there are continuous functions which never have derivatives, or con- tinuous curves which never have tangents, is a further question which is at present beyond us. Common-sense says No : but, as we have already stated in § 111, this is one of the cases in which higher mathematics has proved common-sense to be mistaken. But at any rate it is clear enough that the question ' has (x) a derivative (/>' (x) ? ' is one which has to be answered differently in different circumstances. And we may expect that the converse question ' is there a function (f) (x) of which yjr (x) is the deriva- tive ? ' will have different answers too. We have already seen that there are cases in which the answer is No : thus if i/r (x) is the function which is equal to a, b, or c according as as is less than, equal to, or greater than 0, then the answer is No (Ex. XLVii. 3), unless a = h =c. This is a case in which the given function is discontinuous. In what follows, however, we shall always suppose ■^{x) continuous. And then the answer is Yes : if ■>^{x) is continuous then there is always a function (j) (x) such that ^' (x) = i/r (x). The proof of this will be given in Ch. VII. 230 DERIVATIVES AND INTEGRALS [VI (2) The second question presents no difficulties. In the case of differentiation we have a direct definition of the derivative which makes it clear from the beginning that there cannot possibly be more than one. In the case of the converse problem the answer is almost equally simple. It is that if (j) (x) is one solution of the problem then ^{x) + G is another, for any value of the constant 0, and that all possible solutions are comprised in the form <f) {x) + G. This follows at once from § 126. (3) The practical problem of actually finding ^' {x) is a fairly simple one in the case of any function defined by some finite com- bination of the ordinary functional symbols. The converse problem is much more difficult. The nature of the difficulties will appear more clearly later on. Definitions. If ^ {x) is the derivative of 4> (*). *^*«w we call <f) (x) an integral or integral function of yjr (x). The operation of forming ijr {x)from <f> (x) we call integration. We shall use the notation ^ (x) = I yjr (x) dx. It is hardly necessary to point out that \...dx like djdx must, at present at any rate, be regarded purely as a symbol of operation : the I and the dx no more mean anything when taken by them- selves than do the d and dx of the other operative symbol djdx. 128. The practical problem of integration. The results of the earlier part of this chapter enable us to write down at once the integrals of some of the commonest functions. Thus I x™dx = =- , I cos xdx = sin x, I sin xdx = — cos x. . .(1). J m+ 1 J J These formulae must be understood as meaning that the function on the right-hand side is one integral of that under the sign of integration. The Tnost general integral is of course obtained by adding to the former a constant C, known as the arbitrary constant of integration. 127, 128] DERIVATIVES AND INTEGRALS 231 There is however one case of exception to the first formula, that in which m= — 1. In this case the formula becomes meaningless, as is only to be expected, since we have seen already (Ex. XLil. 4) that l/x cannot be the derivative of any polynomial or rational fraction. That there really is a function F{a;) such that BxF{x) = Ijx will be proved in the next chapter. For the present we shall be content to assume its existence. This function F(x) is certainly not a polynomial or rational function ; and it can be proved that it is not an algebraical function. It can indeed be proved that F{x) is an essentially new function, independent of any of the classes of functions which we have considered yet, that is to say incapable of expression by means of any finite combination of the functional symbols corresponding to them. The proof of this is unfortunately too detailed and tedious to be inserted in this book; but some further discussion of the subject will be found in Ch. IX, where the properties of F{x) are investigated systematically. Suppose first that x is positive. Then we shall write ^^ = log^ (2), and we shall call the function on the right-hand side of this equation the logarithmic function : it is defined so far only for positive values of x. Next suppose x negative. Then -xis positive, and so log {-x) is defined by what precedes. Also so that, when x is negative, ''dx !'- I' ^=^ogi-x) (3). The formulae (2) and (3) may be united in the formulae / — = log(+a;) = log \x\ (4), where the ambiguous sign is to be chosen so that + a; is positive : these formulae hold for all real values of x other than x = 0. 232 DEKIVATIVES AND INTEGRALS [VI The most fundamental of the properties of log x which will be proved in Ch. IX are expressed by the equations log 1 = 0, log(l/a;)=-log«, log a;y = log ^ + logy, of which the second is an obvious deduction from the first and third. It is not really necessary, for the purposes of this chapter, to assume the truth of any of these formulae ; but they sometimes enable us to write our formulae in a more compact form than would otherwise be possible. It follows from the last of the formulae that log.t:^ is equal to 21og^ if x>0 and to 2 log ( - a;) if a; < 0, and in either case to 2 log | a; | . Either of the formulae (4) is therefore equivalent to the formula /' -^ = \\ogx^ (5). The five formulae (1) — (3) are the five most fundamental standard forms of the Integral Calculus. To them should be added two more, viz. I = arc tan x, \ -jrz ;r = + arc sin x* (6). 129. Polynomials. All the general theorems of § 113 may of course also be stated as theorems in integration. Thus we have, to begin with, the formulae j{f(x) + F{x)]dx= ff(a;)dx + JF{x)dx (1), [kf(x)dx = Jcjf(x)dx (2). Here it is assumed, of course, that the arbitrary constants are adjusted properly. Thus the formula (1) asserts that the sum of any integral of f(x) and a?iy integral of F (x) is an integral of f{x) + F{x). These theorems enable us to write down at once the integral of any function of the form 2 A^f^ (x), the sum of a finite number of constant multiples of functions whose integrals are known. In particular we can write down the integral of any polynomial: thus / ^■11+1 a,x" (aoa;" + ai«"-i + ... + a„) dx= ' +-^—-+ ... + a^x. * See § 119 for the rule for determining the ambiguous sign. 128-130] DERIVATIVES AND INTEGRALS 233 130. Rational Functions. After integrating polynomials it is natural to turn our attention next to rational functions. Let us suppose R (x) to be any rational function expressed in the standard form of § 117, viz. as the sum of a polynomial 11 (x) and a number of terms of the form A/(x — a)P. We can at once write down the integrals of the polynomial and of all the other terms except those for which p = l, since /( A J ^ 1 ax = — {x-a)P p-1 {.X - a)P-' ' whether a be real or complex (§ 117). The terms for which p=l present rather more difficulty. It follows immediately from Theorem (6) of § 113 that JF'{f(x)}f'(x)dx = F{fix)} (3). In particular, if we take f{x) = ax + b, where a and b are real, and write cf) (x) for F (x) and yjr {x) for F' (x), so that tp (x) is an integral of yjr (x), we obtain / ■\Jr{ax + b) dx = -(j){ax + 6) (4). Thus, for example, <^^ 111 , 1. 1 5- = - losr \ax-\- b\ ax + o a ° ' ' I: and in particular, if a is real, dx h = log I a? — a I X — a We can therefore write down the integrals of all the terms in i?. (x) for which p= 1 and a is real. There remain the terms for which ^ = 1 and a is complex. In order to deal with these we shall introduce a restrictive hypothesis, viz. that all the coefficients in R (x) are real. Then if a = <y + Bi is a root of Q (x) = 0, of multiplicity m, so is its con- jugate a = 'y — Si; and if a partial fraction Apj{x—a.y occurs in the expression of R (x), so does Ap/{x — tt)P, where Ap is conjugate to Ap. This follows from the nature of the algebraical processes by means of which the partial fractions can be found, and which are explained at length in treatises on Algebra*. * See, for example, Chrystal's Algebra, vol. i, pp. 151-9. 234 DERIVATIVES AND INTEGRALS [VI Thus, if a term (X, + fii)l{oii — y — Bi) occurs in the expression of R{ic) in partial fractions, so will a term {\ — /j,i)/{x — y + Si); and the sum of these two terms is 2{\(x-y)-fiS} {x-yy+B'^ • This fraction is in reality the most general fraction of the form Ax + B ax^ + 2bx + c' where ¥ < ac. The reader will easily verify the equivalence of the two forms, the formulae which express X, /m, y, B in terms of A, B, a,h,c being A, = J/2a, /i = - D/(2aVA), y^-hja, B = >J^/a, where A = ac — b'', and D—aB—hA. If in (3) we suppose F {f{x)] to be log \f{x) |, we obtain \' -dx = \og\f{x)\ (5); and if we further suppose that f{x) = {x — xy + fi^, we obtain 2{x-k) /< .dx = logl(x-\y + /x^]. (x - xy + fi? And, in virtue of the equations (6) of § 128 and (4) above, we have /< dx=-—2B arc tan ' {x—\f+ij? \ fl These two formulae enable us to integrate the sum of the two terms which we have been considering in the expression of R («); and we are thus enabled to write down the integral of any real rational function, if all the factors of its denominator can be deter- mined. The integral of any such function is composed of the sum of a polynomial, a number of rational functions of the type A 1 ^ - 1 {x- a)P-^ ' a viimber of logarithmic functions, and a number of inverse tangents. It only remains to add that if a. is complex then the rational function just written always occurs in conjunction with another in which A and a are replaced by the complex numbers conjugate to them, and that the sum of the two functions is a real rational function. 130] DERIVATIVES AND INTEGRALS 235 Examples XL VIII. 1. Prove that h ■dx=-~ log A 4- „ r, r l02 ax + 6+V(-A) ^aa^ + ibx+c 2a ^ ' ' 2aV(-A) (where X=ax'^+^hx + c) if A<0, and C Ax + B A , . „. D , /ax + b\ if A > 0, A and D having the same meanings as on p. 234. 2. In the particular case in which ac = b'^ the integral is D A, , ., ; , , H — log \ax + b\. 3. Show that if the roots of Q(x)=0 are all real and distinct, and P{x) is of lower degree than Q {x), then \R{x)dx='2.-J^\oz\x-a\, the summation applying to all the roots a of § {x) = 0. [The form of the fraction corresponding to u may be deduced from the facts that as a;-*- a.] 4. If all the roots of Q (x) are real and a is a double root, the other roots being simple roots, and P{x) is of lower degree than Q(x), then the integral is 4/(^-a) + ^'log|^--a| + 2i?log \x-^\, where 2PW „ 2{3P-(a)e"(a)-P(«)r(a)} P(/3) and the summation applies to all roots /3 of § {x) = other than a. Calculate / -r. TTTTTTTfa • [The expression in partial fractions is 11? 2-1 i 2+1 4 (a;- 1)2 ~ 2(.i;-l) ~ 8(;»-i7"^87S^ "*" 8"(^ + t)^ "*" Slx+i)' and the integral is 1 t). '4 (*•-!) 4(A-2+l) -Jlog|:j;- 1 1 +^log(a;2+i) + ^arctan.2;.] 6. Integrate X {x-a){x-b){x-cy {x-afix-by {x-af{x-bf' {x-af' (a:2 + a2)(^2 + 62)' {x' + a?){x'^+bf' x'^{3? + a?)' x{x'- + a''f 236 DEEIVATIVES AND INTEGRALS [Vl 7. Prove the formulae : 131. Note on the practical integration of rational functions. The analysis of § 130 gives us a general method by which we can find the integral of any real rational function R (x), provided we can solve the equation Q {x) = 0. In simple oases (as in Ex. 5 above) the application of the method is fairly simple. In more complicated cases the labour involved is some- times prohibitive, and other devices have to be used. It is not part of the purpose of this book to go into practical problems of integration in detail. The reader who desires fuller information may be referred to Goursat's Cows (^Analyse, second ed., vol. i, pp. 246 et seq., Bertrand's Calcul Integral, and Dr Bromwich's tract Elementary Integrals (Bowes and Bowes, 1911). If the equation Q(x)=0 cannot be solved algebraically, then the method of partial fractions naturally fails and recourse must be had to other methods*. 132. Algebraical Functions. We naturally pass on next to the question of the integration of algebraical functions. We have to consider the problem of integrating y, where y is an algebraical function of x. It is however convenient to consider an apparently more general integral, viz. j R (x, y) dx, where R (x, y) is any rational function of x and y. The greater generality of this form is only apparent, since (Ex. xiv. 6) the function R {x, y) is itself an algebraical function of x. The choice of this form is in fact dictated simply by motives of convenience : such a function as px + q + \/(ax^ + 2bx + c) px + q — \/(ax^ -h 26a; + c) is far more conveniently regarded as a rational function of x and the simple algebraical function i\/{ax^ + 2hx + c), than directly as itself an algebraical function of x. * See the author's tract "The integration of functions of a single variable" (Cambridge Tracts in Mathematics, No. 2, second edition, 1915). This does not often happen in practice. 130-134] DERIVATIVES AND INTEGRALS 237 133. Integration by substitution and rationalisation. It follows from equation (3) of § 130 that if 1 -f (x) dx=(j)(x) then 'f{f(t)}f'{t)dt = cl>{f(t)] (1). /• This equation supplies us with a method for determining the integral of yjr (x) in a large number of cases in which the form of the integral is not directly obvious. It may be stated as a rule as follows: put x=f{t), where f(t) is any function of a new variable t which it may he convenient to choose ; multiply by f {t), and determine {if possible) the integral of yjr {f(t)}f' (t); express the result in terms of x. It will often be found that the function of t to which we are led by the application of this rule is one whose integral can easily be calculated. This is always so, for example, if it is a rational function, and it is very often possible to choose the relation between x and t so that this shall be the case. Thus the integral of R (^/x), where R denotes a rational function, is reduced by the substitution x=t'' to the integral of 2tR{P), i.e. to the integral of a rational function of t. This method of integration is called integration by rationalisation, and is of extremely wide application. Its application to the problem immediately under consideration is obvious. If we can find a variable t such that x and y are both rational functions oft, say x = Ri(t), y = R^ii), then Jr (x, y) dx = JR {R,(t), R^)} R/(t)dt, and the latter integral, being that of a rational function of t, can be calculated by the methods of^ 130. It would carry us beyond our present range to enter upon any general discussion as to when it is and when it is not possible to find an auxiliary variable t connected with x and y in the manner indicated above. We shall consider only a few simple and inter- esting special cases. 134. Integrals connected with conies. Let us suppose that x and y are connected by an equation of the form ax" + 2hxy + by'' + 2gx + %fy + c = Q; in other words that the graph of y, considered as a function of x 238 DERIVATIVES AND INTEGRALS [VI is a conic. Suppose that (^, ri) is any point on the conic, and lei X— ^= X, y — r]=Y. If the relation between x and y is expressed in terms of X and F, it assumes the form aX' + 2hXY + 6P + 2GX + 2FY = 0, where F=h^ + br]+f, Q = a^ + hr) + g. In this equation put Y=tX. It will then be found that X and Y can both be expressed as rational functions of t, and therefore x and y can be so expressed, the actual formulae being 2(G + Ft) _ 2t(G + F t) ^ ^~ a+2ht + bt'' y '' a + 2ht + bt'' Hence the process of rationalisation described in the last section can be carried out. The reader should verify that hx + by+f=-^ia + 2ht + bf) ^ , ^°*^^* ]hx + by+f^~^la+2ht + bf-- When fi} > ab it is in some ways advantageous to proceed as follows The conic is a hyperbola whose asymptotes are parallel to the lines ax"" + 2hxy + by^ = 0, or b{y- ixx) (y - im'x) = 0, say If we put y — fix = t, we obtain . , 2ot + 2fy + c y-fix = t, y-fj,x = --- j-^ , and it is clear that x and y can be calculated from these equations as rational functions of t. We shall illustrate this process by an application to an important special case. 135. The integral \-r. — ^ , . . Suppose in particular that y2 = ax2 + 26i' + c, where a>0. It will be found that, if we put y-\-XiJa=^t, we obtain dt {t^aJrbf ' ^^ t^a + b ' ' and so tdx ( dt 1,1, 6 1 134-136] DERIVATIVES AND INTEGRALS 289 If in particular a=i, 6=0, c=a.^, or a=\, 6 = 0, c= -a% we obtain /;/(^)=iog^+V(-^+''% j;j(^)='°s\^+^(^'-a^)\ ...(2), equations whose truth may be verified immediately by differentiation. With these formulae should be associated the third formula I: = arc sin (a?/a) (3), /. which corresponds to a case of the general integral of this section in which a < 0. In (3) it is supposed that a > ; if a < then the integral is arc sin (a;/ 1 a () (of § 119). In practice we should evaluate the general integral by reducing it (as in the next section) to one or other of these standard forms. The formula (3) appears very different from the formulae (2) : the reader will hardly be in a position to appreciate the connection between them until he has read Ch. X. 136. The integral I -j-, — ; — ^rj- r dx. This integral can be integrated in all cases by means of the results of the preceding sections. It is most convenient to proceed as follows. Since \x + [1 = Q^/a) (ax -i-b) + f/,- (Xb/a), -7r-^~- oi. . \ '^^ = '■^("''^ + 2&* + c), V(a* +26a;+c) ^ we have f (\x + u.)dx X ,, „ , o, , s , f Xb\ [ dx -^ ^ -s = - v(o^* + 2oa; + c) + { u, -r, — -— ^t r . In the last integral a may be positive or negative. If a is positive we put x \fa + (b/n/a) = t, when we obtain ^Jaj ^/(t^+K)' where K = {ac — P)la. If a is negative we write A for — a and put x^/A -(b/'^A) = t, when we obtain 1 r dt V(- a) iv(- «-«')■ It thus appears that in any case the calculation of the integral may be made to depend on that of the integral considered in § 135, and that this integral may be reduced to one or other of the three forms r dt f dt f dt jV¥+^y iV(«'-n')' i\/(a^-«^)' 240 DERIVATIVES AND INTEGRALS [VI 137. The integral j{\x+tJ.)^{ax^+2bx+c)dx. In exactly the same way we find and the last integral may be reduced to one or other of the three forms (sj(fi + a2) dt, h{fi- o») dt, [V(a2 - fl) dt. In order to obtain these integrals it is convenient to introduce at this point another general theorem in integration. 138. Integration by parts. The theorem of integration by parts is merely another way of stating the rule for the differentia- tion of a product proved in § 113. It follows at once from Theorem (3) of § 113 that jf (00) Fix) dx =f(x) F (x) - jf{x) F' {x) dm. It may happen that the function which we wish to integrate is expressible in the form f {x)F{x), and that f (x) F' (x) can be integrated. Suppose, for example, that <j) (x) — x-^fr (x), where i/r (x) is the second derivative of a known function p^ (x). Then J4>(x)dx=j xx" (x) dx = XX («) - j x {x) dx = xx {x) - x («)• We can illustrate the working of this method of integration by applying it to the integrals of the last section. Taking f{x) = ax\h, F{x)=^{aa^ + ibx+o)=y, we obtain a jydx=(a.v+b)i/- p-^^^y^dx=(ax + b)i/-aLdx + {cw-h')f—, so that LdxJ^^^+^ + '^^rEf^i and we have seen already (§ 135) how to determine the last integral Examples XLIX. 1 Prove that if a>0 then j J{xi + a'^)dx = ix ^f(x^ + a^)+i a2 log {x + ^(x^ + a^)} , jj(x^-a^)dx = ix^f{x^-a'')-ianog\x + ^{x^-a^)\, Ll{a^ - a;2) dx = ix V(a2 - x^) + i a2 arc sin (ji'/a). 137, 138] DERIVATIVES AND INTEGRALS 241 2. Calculate the integrals I -^^ — j., \.J{a^-x'^)dx by means of the substitution x=asm.6, and verify that the results agree with those obtained in § 135 and Ex. 1. 3. Calculate \x{x-\-d)™dx, where m is any rational number, in three ways, viz. (i) by integration by parts, (ii) by the substitution ix+a)'"^=t, and (iii) by writing {x+a)-a for x ; and verify that the results agree. 4. Prove, by means of the substitutions ax+b = lji and x=l/u, that (in the notation of §§ 130 and 138) b.v+0 fdx _ax + h fxdx _ Ay t dx 5. Calculate \ -rn ttt ^, where 6>a, in three ways, viz. (i) by J sj{{x — a){b — X)) the methods of the preceding sections, (ii) by the substitution (b — x)j(x — a) = fi, and (iii) by the substitution x=acos''d + bsin^d; and verify that the results agree. 6. Integrate J{(x — a){b- x)} and y/{(b — x)l(x-d)}. 7. Show, by means of the substitution 2x + a + b = ^{a — b) {fi + {l/t)^}, or by multiplying numerator and denominator by x/(A- + a)-;^(i' + 6), that if a>b then 8. Find a substitution which will reduce I , ..,., ' , r^^s to the ] {x-\- afi^ + {x — afi^ integral of a rational function. {Math. Trip. 1899.) 9. Show that \ R{x, Xj{ax + b)]dx is reduced, by the substitution a.c + b=y", to the integral of a rational function. 10. Prove that jf" (i)F{x)dx=f (.r) F{x) -fix) F' (x) + jfix) F"{x)d.v and generally ffW(x)F(x)dx=fi''-n{x)F{x)-fi"-^)(x)F'{x) + ... + (-l)"lf{x)FM(x)dx. 11. The integral Ul+xyx^dx, where jd and q are rational, can be found in three cases, viz. (i) if p is an integer, (ii) if g is an integer, and (iii) if p + qia an integer. [In case (i) put x=u^, where s is the denominator of g ; in case (ii) put 1 +x=t', where s is the denominator of p ; and in case (iii) put 1 +x=xt', where s is the denominator of p.] a. 16 242 DERIVATIVES AND INTEGRALS [VI 12. The iutegvsd f ^^(ax^ + bydx can ba reduced to the precedmg integral by the substitution ax''=bt. [In practice it is often most oon- venieut to calculate a particular integral of this kind by a 'formula of reduction' (of. Misc. Ex. 39).] 13. The integral iR{x,y/(ax + b), J{cx+d)} dss ca,nhsied\xcc6itotha,to{ a rational function by the substitution 4*= - (6/a) {t+{llt)}^ - {die) {t- {llt)}\ 14. Reduce Ii2(;!!;,y)(^a;, where 2/2(^_y)=^2j to the integral of a rational function. [Putting j/= to we obtain x=l/{fi(l—t)}, i/=l/{t(\.-t)}.] 15. Eeduce the integral in the same way when (a) y{a;-y)'^=x, (6) {x^-'ry^Y—a^{x^—y^). [In case (a) put x — y=t: in case (6) put a;2 +,{/2 = « {x -y), when we obtain x = aH {fi + a?)l{t^ + a% y = aH (fi - a^)l{t* + a*).] 16. Ify(^-y)2=^then J^ =i\og{{x-yf-l}. 17. If (x^+y^y=2c^(afl-y^) then f , , '^% — =-, = - 4log (~^) . 139. The general integral I R {x, y) dx, where y^'=ax'^+'ihx-\-c. The most general integral, of the type considered in § 134, and associated with the special conic y''=ax'^ + '2,bx+c, is {R{x,^X)dx (1), where X=y^=ax''->r2bx+c. We suppose that ^ is a read function. The subject of integration is of the form PjQ, where P and Q are poly- nomials in X and ^X. It may therefore be reduced to the form A+B^X _ {A + Bsjr){C-DJX) c+Djx JF^nwx ^+^V-I, where A, B, ... are rational functions of x. The only new problem which arises is that of the integration of a function of the form F s/X, or, what is the same thing, Gj^X, where (? is a rational function of x. And the integral G I. VA-* « can always be evaluated by splitting up G into partial fractions. When we do this, integrals of three different types may arise. (i) In the first place there may be integrals of the type ■ X" / -^'^^ •.••■•(3). 138, 139] DERIVATIVES AND INTEGRALS 243 where m ia a, positive integer. The cases in which m = or »i = l have been disposed of in § 136. In order to calculate the integrals corresponding to larger values of m we observe that where a, (3, y are constants whose values may be easily calculated. It is clear that, when we integrate this equation, we obtain a relation between three successive integrals of the type (3). As we know the values of the integral form=0 andm = l, we can calculate in turn its values for all other values of m. (ii) In the second place there may be integrals of the type I. •(4), where p is real. If we make the substitution x — p=\jt then this integral is reduced to an integral in t of the type (3). (iii) Finally, there may be integrals corresponding to complex roots of the denominator of G. We shall confine ourselves to the simplest case, that in which all such roots are simple roots. In this case (cf. § 130) a pair of con- jugate complex roots of G gives rise to an integral of the type LxJrM k dx (5). ^{Ax'^ + 'iBx+C)^ (cu;2 + 26a- + c) In order to evaluate this integral we put where ^ and v are so chosen that anv-\-h{jj. + v) + c=Q,. AiJ.v+B(jj.+v) + C=0; so that /i and v are the roots of the equation {aB-hA)^^-{cA-aO)^ + {hC-cB) = 0. This equation has certainly real roots, for it is the same equation , as equation (1) of Ex. xlvi. 12 ; and it is therefore certainly possible to find real values of it and v fulfilling our requirements. It will be found, on carrying out the substitution, that the integral (5) assumes the form { tdt „ r dt .(6). The second of these integrals is rationalised by the substitution t _ •which gives /■ iM ^ r du J (afi + fi) s'{y^ + 8) ~ j ,3 + (aS - l3y) tt2 ■ 16- 244 DERIVATIVES AND INTEGRALS [VI Finally, if we put < = 1/m in the first of the integrals (6), it is transformed into an integral of the se6ond type, and may therefore be calculated in the manner just explained, viz. by, putting ■ul^{y+bu^) = u, i.e. l/^{yfi + 8)=v.* Examples L. 1. Evaluate f dx ( dx f dx Jx^{x^ + 2x + 3y ]{x-l)^{x^ + l)' ]{x+l)^{l+2x-x^)- 2. Prove that r dx ^ 2 //^-g\ J{x-p)J{{x -p) (x-q)}~ q-p \ \x-pj' 3. If ag'^ + ch^=-v<0 then I ,-; ^-, ^ ^= - -r arc tan ^^-^ . J{/ix+g)J{ax^ + c) V" \_ ch-agx J [ dx 4. Show that \-, —, where «2=cu.''H2te + c, may be expressed in one J{x-Xo)2/ or other of the forms _ 2^ I axxo + b(x+Xf)) + a+i/i/o 1 , (axxti + b(x + xo) + o] - arc tan \ ^ ~ — V 2o I y^o J according as axo' + 2hX(, + c is positive and equal to yo' or negative and equal to —00^. 5. Show by means of the substitution y=sJ{pix^-\-ibx + c)l{x-p) that /■ dx^ /" dy j {x-p)^{ax^ + '2.bx+e) " J ^{'ky^-i>) ' where X = ap^ + ibp + c, fi = ao-b\ [This method of reduction is elegant but less straightforward than that explained in § 139.] 6. Show that the integral dx h lx^(3x'^ + 2x+l) is rationalised by the substitution x={l+y^)i{3-y^). (Math. Trip. 1911.) 7. Calculate /: (x + l)dx (x2+4K/p + 9)" * The method of integiation explained here fails if a/4 = 6/B; but then the integral may be reduced by the substitution ax + b = t. For further information concerning the integration of algebraical functions see Stolz, Grundziige der Differential-und-integralrechnung, vol. i, pp. 331 etseq.; Bromwich, Elementary Integrals (Bowes and Bowes, 1911). An alternative method of reduction has been given by Sir G. Greenhill: see his A Chapter in the Integral Calculus, pp. 12 et seq., and the author's tract quoted on p. 236. 139-141] DERIVATIVES AND INTEGRALS 245 8. Calculate /: dx (5^2 4- 12^ 4- 8) ^{bx^ + 2.17 - 7) ' [Apply the method of § 139. The equation satisfied by ^ and v is |2+3| + 2=0, so that ^=—2, j/= — 1, and the appropriate substitution is X— -(2<4- !)/(<+ 1). This reduces the integral to j(4<2 + l)V(9«2-4) j(4i!2 + i tdt The first of these integrals may be rationalised by putting tjj(9fi — 4:) = u and the second by putting l/^{9fi-4:) = v.] 9. Calculate [ {x + l)d.v r (x-l)dx {Math. Trip. 1911.) 10. Show that the integral I /i (.j;, y) (f.j7, where y2=a.j;2 + 25^ 4. (.^ ig ration- alised by the substitution t = (x — p)l(j/+q), where {p, q) is any point on the conic y'^ = <xx'^ + '2hx + c. [The integral is of .course also rationalised by the substitution t = {x—p)j(j) — q) : cf. § 134.] 140. Transcendental Functions. Owing to the immense variety of the different classes of transcendental functions, the theory of their integration is a good deal less systematic than that of the integration of rational or algebraical functions. We shall consider in order a few classes of transcendental functions whose integrals can always be found. 141. Polynomials in cosines and sines of multiples of x. We can always integrate any function which is the sum of a finite number of terms such as A cos™ ax sin™'aa; cos"6« sin"' 6a;..., where m, m, n, n, ... are positive integers and a, b, ... any real numbers whatever. For such a term can be expressed as -the sum of a finite number of terms of the types acos {{pa + qb + ...)«}, ^ sin {{pa + qb+ ...)x} and the integrals of these terms can be written down at once. 246 DERIVATIVES AND INTEGRALS [VI Examples LI. 1. Integrate sin^ ^ oos^ S.-!?. In this case we use the formulae sin^ a; = J (3 sin x — sin Zx), cos^ 2a;= -^ (1 + cos 4^). Multiplying these two expressions and replacing sin s cos 4^, for example, by ^ (sin 5.r; — sin3.r), we obtain iVJC'sin^- - 5 sin 3^ + 3 sin hx — sin 7.^) dx = - j^ cos X + /g- cos Zx - ^ cos 5.r + y} j cos 7«. The integral may of course be obtained in different forms by different methods. For example I sin' a; cos^ 2a; c?.j; = I (4 cos* .» — 4 cos^ .1- + 1 ) ( 1 — cos'^ ,1;) sin a; dx, which reduces, on making the substitution cos.r=i, to /(4i;6-8(:* + 5«2-l)rfi;=f cos^^-f cos^^ + ^cos^a-cos^a;. It may be verified that this expression and that obtained above differ only by a constant. 2. Integrate by any method cos ax cos hx, sin ax sin hx, cos ax sin hx, cos^.r, sin^.^, cos*«, cos x cos %x cos 3x, cos' 2.t; sin^ S.r, cos^ x waf x. [In cases of this kind it is sometimes convenient to use a formula of reduction (Misc. Ex. 39).] 142. The integrals I *■" cos x dx, I x"- sin x dx and associated integrals. The method of integration by parts enables us to generalise the preceding results. For I «" cos xdx= a;" sin a; — n I »""' sin x dx, I x" sin xdx = — «" cos x + n I «"""' cos x dx, and clearly the integrals can be calculated completely by a repetition of this process whenever n is a positive integer. It follows that we can always calculate I a;" cos ax dx and I a;" sin axdx if w is a positive integer; and so, by a process similar to that of the preceding paragraph, we can calculate /^ P (x, cos ax, sin ax, cos bx, sin bx, ...) dx, where P is any polynomial. 141-143] DEHIVATIVES AND INTEGRALS 247 Examples LII. 1. Integrate ^sin^, .i^cosa-, x^oos'^h;, ^2sin2^gin2 2^^ X sin'^ X cos'' X, x^ sin^ ^x. 2. Find polynomials P and Q suoli that j{(3x-l) cosa.'+(l -2^) sin x} dx = Pcosx+Q sin x. 3. Prove that I ^' cos ;rc?^=P,t cos a;+§„ sin ^, where 143. Rational Functions of cos a; and sin «. The integral of any rational function of cos x and sin x may be calculated by the substitution tan ^x = t. For 1-t^ . 2t dx 2 cos X — , sin«= , -;- = , \+f' \+f' dt l+t^' so that the substitution reduces the integral to that of a rational function of t. Examples LIU. 1. Prove that lseca;<^.j;=log | sec .r+ tana; |, /cosec.rc?.j;=log|tan|^|. [Another form of the first integral is log | tan (J7r + ^.r ) | ; a third form is ^log|(l+sin.v)/(l-sina')|.] 2. /tan.rrf^= -log|cos.j;|, /cot A'c;?,j; = log|sin^|, \&q<? xdx=\:!kn x, I coseo^ xdx= — cot x, j tan x sec ^c&j;=sec x, I cot x ooseo xdx= — cosec x. [These integrals are included in the general form, but there is no need to use a substitution, as the results follow at once from § 119 and equation (5) of § 130.] 3. Show that the integral of l/(a + bcoax), where a + 6 is positive, may be expressed in one or other of the forms ^— arctanf* /(^W -^ , ^{b + a) + tj(,b-a) where <=tan J.r, according as a^>b^ or a^<b^. If a^=b'^ then the integral reduces to a constant multiple of that of sec^ ix or coseo^ J.r, and its value may at once be written down. Deduce the forms of the integral when a + 6 is negative. 4. Show that if y is defined in terms of x by means of the equation {a + b cos x)(a-b cos i/) = a' — b^, where a is positive and a->b\ then as x varies from to tt one value of y also varies from to n. Show also that _ J (a^-b^) sin 1/ sin a; dx _ siny " a-bcoay ' a + bcosx dy a—bcosy' 248 DERIVATIVES AND INTEGRALS [VI and deduce that if < ^ < tt then [ dx 1 faooax+lA ; = -n—r, — rw, arc cos — — j , ja + bcosx! ij{a' — h^) \a + booaxJ Show that this result agrees with that of Ex. 3. 5. Show how to integrate l/(a + 6 cos « + c sin A'). [Express 6 cos ,r + c sin a; in the form ^{b'^ + c^) cos (x-a).] 6. Integrate (a + 6 cos ^ + c sin x)l{a + /3 cos « + y sin x) [Deteimine X, fj., v so that o! + 6cos*'+csin^=X+/i(a + /3cos:!; + 'y sin^) + j/ (-^sina;+y oosa;). Then the integral is U.X + V log|a+/3cos:!;+ysina;l+xl— -^ ; -. — .] 7. Integrate l/(a 003^^ + 26 cos ^ sin ^ + csin2;!;). [The subject of inte- gration may be expressed in . the form 1/(4 -1-5 cos 2^4- Csin2.r), where A=\{a + c), B=\{a — c), C=b : but the integral may be calculated more simply by putting tan a;=t, when we obtain f sec^ xdx _ r "^^ 1 Jo-l-26 tan^-l-ctan^a- ~ J a + 2bt+cfi "-' 144. Integrals involvingf arc sin x, arc tan on, and log x. The integrals of the inverse sine and tangent and of the logarithm caa easily be calculated by integration by parts. Thus r r ccdoc I arc sin xdx — x arc sin x — I —rr^ rr = x arc sin x + iJ{\ — «-), j JV(i-«') I / T Cm T I arc tan xdx = x arc tan x — I ^ = x arc tan x — \ log (1 + x^), I log xdx = X log X— \ dx = X (log x — 1). It is easy to see that if we can find the integral of y =/(«) then we can always find that of « = ^ {y), where is the function inverse tof. For on making the substitution y =f{x) we obtain J4'(y)<^y=j ¥' i'") ^^^ = '«/{«') - j /(*) <^*- The reader should evaluate the integrals of arc sin y and arc tan y in this way. Integrals of the form I P {x, arc sin x) dx, I P {x, log x) dx, 143-145] DERIVATIVES AND INTEGRALS 249 where P is a polynomial, can always be calculated. Take the first form, for example. We have to calculate a number of integrals of the type I «'" (arc sin a;)" dx. Making the substitution x = sin y, we obtain 1 3/" sin*" y cos ydy, which can be found by the method of § 142. In the case of the second form we have to calculate a number of integrals of the type I a;"* (log xY dx. Integrating by parts we obtain [ a;™ (log xY dx = ^'""^' (log '^)" n f ^^ ^^^^ ,^^„_j ^^^ and it is evident that by repeating this process often enough we shall always arrive finally at the complete value of the integral. 145. Areas of plane curves. One of the most important applications of the processes of integration which have been explained in the preceding sections is to the calculation of areas of plane curves. Suppose that P^PP' (Fig. 44) is the graph of a continuous curve y = <f) (x) which lies wholly above the axis of x, P being the point {x, y) and P' the point {x + h, y + k), and h being either positive or negative (positive in the figure). P' R Pi P R Fig. a a. N. N N' Fig. 44. The reader is of course familiar with the idea of an ' area ', and in particular with that of an area such as ONPP^. This idea we shall at present take for granted. It is indeed one which needs and has received the most careful mathematical analysis : later on we shall return to it and explain precisely what is meant by 250 DERIVATIVES AND INTEGRALS [VI ascribing an ' area ' to such a region of space as ONFP„. For the present we shall simply assume that any such region has associated with it a definite positive number {ONPPa) which we call its area, and that these areas possess the obvious properties indicated by common sense, e.g. that {PRP') + (NN'RP) = {I^N'P'P), (N.NPP,) < (ONPP,), and so on. Taking all this for granted it is obvious that the area ONPP^ is a function of a; ; we denote it by * {x). Also <1> {x) is a continuous function. For $ (a; + A) - * {x) = (NN'P'P) = (NN'RP) + (PRP') = h<p (x) + (PRP'). As the figure is drawn, the area PRP' is less than hk. This is not however necessarily true in general, because it is not neces- sarily the case (see for example Fig. 44 a) that the arc PF should rise or fall steadily from P to P'. But the area PRP' is always less than \h\\ (h), where X, (h) is the greatest distance of any point of the arc PP' from PR. Moreover, since ^ (x) is a continuous function, X (h)-^Q as h-^0. Thus we have ^(x+h)-^(x)^h {<p(x) + /x(/i.)}, where \fj. (h)\<X (h) and X. (h) -^0 as h ^0. From this it follows at once that <1> (x) is continuous. Moreover ^(x + h)-(i> (x) *' (x) = lim ^^^ ^ ' = lim {(}> (x) + fi (h)] = <i> (x). Thus the ordinate of the curve is the derivative of the area, and the area is the integral of the ordinate. We are thus able to formulate a rule for determining the Area, ONPP„. Calculate ^(x), the integral of (j)(x). This involves an arbitrary constant, which we suppose so chosen that $ (0) = 0. Then the area required is <I> (x). If it were the area N^NPPx which was wanted, we should of course deter- mine the constant so that * {x-^ = 0, where x-y is the abscissa of Pi. If the curve lay below the axis of x, * {x) would be negative, and the area would be the absolute value of * {x). 145, 146] DERIVATIVES AND INTEGRALS 251 146. Lengths of plane curves. The notion of the length of a curve, other than a straight line, is in reality a more difficult one even than that of ,an area. In fact the assumption that PqP (Fig. 44) has a definite length, which we may denote by S(x), does not suffice for our purposes, as did the corresponding as- sumption about areas. We cannot even prove that S (x) is con- tinuous, i.e. that lim {S (P') — S (P)] = 0. This looks obvious enough in the larger figure, but less so in such a case as is shown in the smaller figure. Indeed it is not possible to proceed further, with any degree of rigour, without a careful analysis of precisely what is meant by the length of a curve. It is however easy to see what the formula must be. Let us suppose that the curve has a tangent whose direction varies continuously, so that </>' (x) is continuous. Then the assumption that the curve has a length leads to the equation {S(x+}i) -S(x)}/h = {PP'}lh = {PP'/h) X {{PP'}jPP'), where {PP'} is the arc whose chord ii3 PP'. Now pp' = ^{PB' .+ RF'^) = h y (i + 1:) , and k = (j}(x + h)-<j) (x) = hij)' (|), where f lies between x and x + h. Hence lim {PP'ih) = lim V{1 + [4>'{m = V{1 + [</>' («')?}• If also we assume that lim [PP'yPP' = 1, we obtain the result 8' (x) = lim {S(x + h)-S {x)}/h = V{1 + [<!>' (^)]1 and so 8 (x) = J V{1 + [<^' («')]'} dx. Examples LIV. 1. Calculate the area of the segment cut off from the parabola y=x-jia by the ordinate x=^, and the length of the arc which bounds it. 2. Answer the same questions for the curve ay'^=x^, showing that the length of the arc is 3. Calculate the areas and lengths of the circles x'^+y^ = a\ x^+y- = iax by means of the formulae of §§ 145—146. 252 DERIVATIVES AND INTEGRALS [VI 4. Show that the area of the eUipse {x^jaF) + (3/^/6^) = 1 is jra6. 5. Knd the area bounded by the curve y = sina; and the segment of the axis of X from x=0 to x='iir. [Here *(^)— — cosj;, and the difference between the values of -cos a: for ^=0 and x=in is zero. The explanation of this is of course that between x=n and x=in the curve lies below the axis of X, and so the corresponding part of the area is counted negative in applying -the method. The area from ^=0 to x=n is -cos7r + cos0=2; and the whole area required, when every part is counted positive, is twice this, i.e. is 4] 6. Suppose that the coordinates of any point on a curve are expressed as functions of a parameter t by equations of the type x=(j){t), y=^{t), <f) and yJA being functions of t with' continuous derivatives. Prove that if X steadily increases as t varies from t^ to ti, then the area of the region bounded by the corresponding portion of the curve, the axis of x, and the two ordinates corresponding to to and <j, is, apart from sign, A {ti)—A (Iq), where A{t) = j^{t)4>'{t)dt^jy^^dt. 7. Suppose that C is a closed curve formed of a single loop and not met by any parallel to either axis in more than two points. And suppose that the coordinates of any point P on the curve can be expressed as in Ex. 6 in terms of t, and that, as t varies from to to «i, P moves in the same direction round the curve and returns after a single circuit to its original position. Show that the area of the loop is equal to the difference of the initial and final values of any one of the integrals this difference being of course taken positively. 8. Apply the result of Ex. ■ 7 to determine the areas of the curves given by ... X \-t^ y It ,.., , , . „ W a = lT?' a rT7^' ("^ ^=«cos3«, y=6sm3*. 9. Find the area of the loop of the curve 3?+i^=Zaxy. [Putting y=tx we obtain x=Zatl(l + t'^), y = 3at^l(l + t^). As t varies from towards 00 the loop is described once. Also #S-4f)-=-4H(i)*-4/<IS-.-=.w,. which tends to as t-^ x . Thus the area of the loop is fa^.] 10. Find the area of the loop of the curve x^+y'^ = bax'^y'^. 11. Prove that the area of a loop of the curve a; = asin2<, y = asiniis ^«^- {Math. Trip. 1908.) 146] DERIVATIVES AND INTEGRALS 253 12. The arc of the ellipse given by x=acoat, i/=bamt, between the points t=ti and t=t2, is F(t^—F{ti), where Fit) = aU(l-e^ sin2 1) dt. e being the eccentricity. [This integral cannot however be evaluated in terms of such functions as are at present at our disposal.] 13. Polar coordinates. Show that the area bounded by the curve r=f{6),vihere f{d) is a one- valued function of ^, and the radii B = 6i, 6=62, is F{6^-Fifii), where F{e) = ^\r'de. And the length of the corresponding arc of the curve is * (^2) - * (^i), where ^vHs/{"<WV'- Hence determine (i) the area and perimeter of the circle r='iawa.6; (ii) the area between the parabola r=|^Zseo2^0 and its latus rectum, and the length of the corresponding arc of the parabola ; (iii) the area of the limagon r = a + hGoad, distinguishing the cases in which a>h, a=b, and a<b; and (iv) the areas of the ellipses l/r^=acos^d + 2kcos6sind + bsiB.^6 and f dQ llr^Y + ecoaO. fin the last case we are led to the integral (tt r-, , "■ ] (\ + e aos 6Y which may be calculated (cf. Ex. Lili. 4) by the help of the substitution (l + ecos^)(l-ecos^) = l-«2.] 14. Trace the curve 'i6 = {ajr) + {rja), and show that the area bounded by the radius vector 5 = /3, and the two branches which touch at the point r=a, 6 = 1, is |a2(/32- 1)3/2. [Math. Trip. 1900.) 15. A curve is given by an equation p = f{r), r being the radius vector and p the perpendicular from the origin on to the tangent. Show that the calculation of the area of the region bounded by an arc of the curve and two radii vectores depends upon thai of the integral ^ j -~ j MISCELLANEOUS EXAMPLES ON CHAPTER VL 1. A function/(^) is defined as being equal to 1 +x; when x^Q, to x when 0<a;<l, to 2-.r when l£a;g2, and to 3x: — x^ when ^>2. Discuss the continuity oi f{x) and the existence and continuity of /'(.») for x=Q, x='i, and .«=2. {Math. Trip. 1908.) 2. Denoting a, ax+h, ax^ + 2bx + e, ... by v^, %, Mjj ••■, show that ■Mo^!(3 — 3moMi!<2 + 2mi^ and UqVi — AuiUs + 'Sv^^ are independent of x. 254 DERIVATIVES AND INTEGRALS [vi 3. Uoq, «!, ..., 02,1 are constants and U,.= (ao, csj, ..., a,^x, ly, then 1.2 is independent of x. {Math. Trip. 1896.) [Difierentiate and use the relation Ur' = rUr-i.] 4. The first three derivatives of the function arc sin (^ sin .r) - x, where fi>l, are positive when QS-x^^n. 5. The constituents of a determinant are functions of x. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered. 6. If /i, /a, /s, fi are polynomials of degree not greater than 4, then /i /2 /s fi K fi fi fi Jl /2 73 Ji j: III ^ III j: III ^ /// /l /2 /3 74 is also a polynomial of degree not greater than 4. [Differentiate five times, using the result of Ex. 5, and rejecting vanishing determinants.] 7. liy^ + Zyx+iJi? = Qt\iQnx'^{\+afi)y"-^X!/+y=0. {Math. Trip. nOi.) 8. Verify that the differential equation J' = ^{i/'(y])} + <^{«-i|f (i/j)}, where y^ is the derivative of y, and -^ is the function inverse to 0', is satisfied by y — ^{c) + <p{x-c) or by y = '2,<f){^x). 9. Verify that the differential equation y= {a;/\^ (yj)} ^ {\^ (yj}, where the notation is the same as that of Ex. 8, is satisfied by y=c(f>{xlo) or hy y=^x, where p=(j>{a)!a and a is any root of the equation 0(a) - a(f>'{a)=0. 10. If ax + by + c=0 then ,^2=0 (suffixes denoting differentiations with respect to x). We may express this by saying that the greneml differential equation of all straight lines is y2=0. Find the general differential equations of (i) all circles with their centres on the axis of x, (ii) all parabolas with their axes along the axis of x, (iii) all parabolas with their axes parallel to the axis of y, (iv) all circles, (v) all parabolas, (vi) all conies. [The equations are (i) l+yi^+yy2=0, (ii) yi^+yyi^^O, (iii) ^3=0, (iv) (l+^/l2)y3=3yl2/2^ (v) ^yz^^Zy^yi, (vi) Oya^yj- 45^2^3?/^ +40^38=0. In each case we have only to write down the general equation of the cxirves in question, and differentiate until we have enough equations to eliminate all the arbitrary constants.] 11. Show that the general differential equations of all parabolas and of all conies are respectively DERIVATIVES AND INTEGBALS 255 [The equation of a conic may be put in the form From this we deduce If the conic is a parabola then jo=0.] 12. Denoting ^. - ^, _ ^^, _ ^, ... by t, a, h, c, ... and dx 1 d^x 1 d^x 1 d^x , „ , j^, , ^' 2! ^' 3! ^3- T! rf/' - ^y '■■ °' ^' y' ■••' '^'°^ *^^^* 4ac - 562 = (4„y _ 5/33)/t8, 6< - a^ = _ (^^ _ „2)y^a. Establish similar formulae for the functions a-d-3abc — 2b^, {l + fi)l>-2a-t, 2ct — bah. 13. Prove that, if y^ is the kth derivative of y=sin (marc sin «), then {\-x^)y^^^-{2k + \)xy^^-^-V{n^-B)y„=0. [Prove first when ^=0, and differentiate k times by Leibniz' Theorem.] 14. Prove the formula where n is any positive integer. [Use the method of induction.] 15. A curve is given by x=a{2 cos i + oos 2t), y=a{2 sin t - sin 2t). Prove (i) that the equations of .the tangent and normal, at the point P whose parameter is *, are a;sin^i+yoos^< = fl!sinf <, a;cosJ< — y sin4<=3aoos§i ; (ii) that the tangent at P meets the curve in the points §, R whose para- meters are —\t and ■ir — \t\ (iii) that §/J=4a; (iv) that the tangents at $ and R are at right angles and intersect on the circle ^2+y2=a2. (y) that the normals at P, §, and R are concurrent and intersect on the circle x^->ry'''=^a^ ; (vi) that the equation of the curve is (a;2 +/ + 12«A--1- 9a2)2= 4a (2.v + Sa)'. Sketch the form of the curve. 16. Show that the equations which define the curve of Ex. 15 may be replaced by ^/a = 2u + {llu^), r]/a={2ju) + u^, where ^=x+yi, r]=x-yi, M=Cis t. Show that the tangent and normal, at the point defined by u, are v,^-uri = aiii?-l), u^+uri = 3a{u^ + l), and deduce the properties (ii) — (v) of Ex. 15. 17. Show that the condition that x*+4tpx^-iqx-l = should have equal roots may be expressed in the form {p+qy/^ — {p — qYi^=l. {Math. Trip. 1898.) 256 DERIVATIVES AND INTEGRALS [Vi 18. The roots of a cubic f{x) = are a, ft y in ascending order of magni- tude. Show that if (a, fi) and (/3, y) are each divided into six equal sub-intervals, then a root of /' (x) = will fall in the fourth interval from ^ on each side. What will be the nature of the cubic in the two cases when a root of/' (a')=0 falls at a point of division? {Math. Trip. 1907.) 19. Investigate the maxima and minima oi f{x), and the real roots of f{x) = 0, f{x) being either of the functions X — sin X — tan a (1 — cos x\ x — sin x — {a — sin a) — tan ^a (cos a - cos x), and o an angle between and jr. Show that in the first case the condition for a double root is that tan a — a should be a multiple of w. 20. Show that hj choice of the ratio X : /i we can make the roots of \(ax^ + bx+o)+ii{a'x^ + b'x + c')=0 real and having a diiferenoe of any mag- nitude, unless the roots of the two quadratics are all real and interlace; and that in the excepted case the roots are always real, but there is a lower limit for the magnitude of their difference. {Math. Trip. 1895.) [Consider the form of the graph of the function {ax^ + hx -f c)j{a'x'^ + h'x -)- c'): cf. Exs. XLVI. 12 et seq.'\ T, .-, , sin jra- 21. Prove that n- <-7.; — , g 4 x{l — x) when < .i; < 1, and draw the graph of the function. 22. Draw the graph of the function , 1 1 TT cot irX r . X X- 1 23. Sketch the general form of the graph of y, given that |^(6.^+.-l)(.-l)V.+l)3^ (ifa.;i. rn>. 1908.) 24. A sheet of paper is folded over so that one corner just reaches the opposite side. Show how the paper must be folded to make the length of the crease a maximum. 25. The greatest acute angle at which the ellipse {x^ja'^)-'r{y'^IW) = \ can be cut by a concentric circle is arc tan {{a^ — b'^)jiah}. {Math. Trip. 1900.) 26. In a triangle the area A and the semi-perimeter s are fixed. Show that any maximum or minimum of one of the sides is a root of the equation s{x — s)x^ + 4.A'^ = 0. Discuss the reality of the roots of this equation, and whether they correspond to maxima or minima. [The equations a4-6-|-c=2s, s{s- a){s — b){s — c) = i^^ determine a and 6 as functions of c. Differentiate with respect to c, and suppose that dajdc=Q. It will be found that 6 = c, s~b = s — c=^a, from which we deduce that s{a-s)a^+iA^ = (i. DERIVATIVES AND INTEGRALS 257 This equation has three real roots if s*>27a^, and one in the contrary case. In an equilateral triangle (the triangle of minimum perimeter for a given area) «* = 27A^; thus it is impossible that «*<27a^. Hence the equation in a has three real roots, and, since their sum is positive and their product negative, two roots are positive and the third negative. Of the two positive roots one corresponds to a maximum and one to a minimum.] 27. The area of the greatest equilateral triangle which can bo drawn with its sides passing through three given points A, B, C is a^ + lfi + c^ a, b, c being the sides and A the area of A DC. {Math. Trip. 1809.) 28. If A, A' are the areas of the two maximum isosceles triangles which can be described with their vertices at the origin and their base angles on the cardioid r=a (1+cos 6), then 256AA' = 25a*^5. {Mat/i. Trip. 1907.) 29. Find the limiting values which (x'2-4y + 8)/(y2- 6.2; + 3) approaches as the point [x, y) on the curve a;^y — 4.r2-4r^+y- + 16a; — 2j/-7 = ap- proaches the position (2, 3). (Math. Trip. 1903.) [If we take (2, 3) as a new origin, the equation of the curve becomes ^2,j_^2^.^2=o, and the function given becomes (|H4|-4i7)/(ijH 67-61). If we put T) = t^, we obtain ^ = (1 - t^)lt, r] = l — t^. The curve has a loop branching at the origin, which corresponds to the two values t= — 1' and i = 1 . Expressing the given function in terms of t, and making t tend to 1 or 1, we obtain the limiting values - f , — §■] 30. If fi^) = -- —■ 7 \ > sin x — sm a {x-a) cos a then J- { 1™ / (■^')} - lim /' {x) = j sec^ a - f ^ sec a. (Math. Trip. 1896.) 31. Show that if <j)(x) = l/(l+x^) then (jy^") (x) = Q„(a;)l(l+xY'^\ where Q,i(x) is a polynomial of degree n. Show also that (i) §„ + i = (l+^)««'-2(/H-l)^§„, (ii) §n + 2 + 2(ra + 2)a-& + i + (?i + 2)(»+l)(l + ^2jft,=0, (iii) (l+x-^) Q^'-inxQ,;+n(n + l) §„=0, (iv) §„=(-l)»«!{(» + l).'.-^^^±2M!^ ."- + ...}, (v) all the roots of §„=0 are real and separated by those of §„_i = 0. 32. If f(x), cj) (x), ^|r (x) have derivatives when a ^ x ^b, then there is a value of ^ lying between a and 6 and such that f(a) 4,(a) ^(a) =0. f(b) i>(b) ^(b) f'i$) <^'«) f'(© H. 17 258 DERIVATIVES AND INTEGRALS [VI [Consider the function formed by replacing the constituents of the third row hjf(x), (^ (x), i//- (x). This theorem reduces to the Mean Value Theorem (§ 125) when <^{x) = x and i\r{x) = \.'\ 33. Deduce from Ex. 32 the formula f{h)-f{a) _ /'(!) <l>{b)-<t>(a) (^'(f) 34. If (j)'(x)^~a as x^^ao, then <\){x)jx^a. If <^'(:r)-^Qo tlion (^ (a;) -»- 00 . [Use the formula 4>'{^) - <j> (x^) = {x — x^ <j}' (|), where a^o < | < x.] 35. If (j){x)-»-a aia x-»-<x>, then <j)' (x) cannot tend to any limit other than zero. 36. If (^ {x) + <f>' {x)-»-a as x-»-co, then (f)(x)-»-a and <f)' {x)-»-0. [Let <j){x) = a + ^(x), so that •\//- (^) + !//■' (a;) -»-0. If i/r' (a;) is of constant sign, say positive, for all sufficiently large values of x, then i//- (a;) steadily increases and must tend to a limit ? or to oo . It \jf (x)-^ ao then \//-' (a;) -»- - oo , which contradicts our hypothesis. If ■\lr{x)-^l then yjf' (x)-^—l, and this is impossible (Ex. 35) unless l==0. Similarly we may dispose of the case in which i//-' {x) is ultimately negative. If i/r (x) changes sign for values of a; which surpass all limit, then these are the maxima and minima of yjr (x). If x has a large value corresponding to a maximum or minimum of yjr{x), then \lf{x) + -\jr'(x)ia small and i/^' (a;) = 0, so that ^{x) is small. A fortiori are the other values of \|f ix) small when x is large. For generalisations of this theorem, and alternative lines of proof, see a paper by the author entitled " Generalisations of a limit theorem of Mr Mercer," in volume 43 of the (Quarterly Journal of Mathematics. The simple proof sketched above was suggested by Prof. E. W. Hobson.] 37. Show how to reduce \r \x, ^ /f-^^') , , /f-^^l^^l dx to / I V ymx + nj V \mx+nj) the integral of a rational function. [Put mx+n= \jt and use Ex. XLix. 13.] 38. Calculate the integrals : r dx f /fx-r\dx r xdx j{l+x^r j\/\x+l)x' J^(l+x)-^{l+x)' UU'+./(b' + -)\dx, (coseo'^xdx, [„ 5cosa;+6 JV { \/ \ xj) J ' J2cosa; + sina;+3 ' /dx ' /"cos a? sin a; c?a; f „ v , (2-siu^^)(2 + sina;-sin2a,-)' j cos* ^ + sin^ *■ ' j ''osec a; V(seo 2a;) .^Ja;, f dx fx+ainx , [ , f )j{{l+sinx){2+smx)y ju^^^x'^''^ j arc sec a; rfa;, j (arc sm a;)2<^a;, [ . J jx arc sin x , /"arc sin x , /"arc sin x , ja.arcsma;<^a;, j^jj^~^ dx, j—^~dx, j^^^^dx, /• arc tan a; [ arc tan a; /• log(aHi3V ) ,, flogia+fix) , }—^<^^' JOT^^-'^-^' j x^ ''^' J ^a + bx)^ ^^- DERIVATIVES AND INTEGRALS 259 39. Formulae of reduction, (i) Show that [Put .jr+-|p = «, q-lp' = \: then we obtain _i [__dt__ 1 r t^ f 1 ] ~ X j (<2+X)"-i '^2X(n-l)j'dt \(fi + \)'^-'j ' and the result follows on integrating by parts. A formula such as this is called a formula of reduction. It is most useful /dx . .^ — jr~Vi ^'^ terms r clx °^ I r"2 . _i , and so evaluate the integral for every value of n in turn.] (ii) Show that if /p, ,= I^vp (1 +a:)''dx then and obtain a similar formula connecting /p, , with /p_i, , + 1. Show also, by means of the substitution x= — y/(l +y), that (iii) Show that if X=a + hx then LA'-'/3rfar= -3 (3a-26a-)-f2«/io62, L^X- 1/3 dx = 3 (9a2 - Bahx ^W^x"^) X ^I^j4:0b\ jxX - '/4 dx^ - 4 (4a - 36.») X^i*j2W, (x'^X-^l^dx = 4 (32a2 - 2'iabx + 216%2) A'3/''/23] fi^. (iv) If /„,„= j^^^j;^ then 2(«-l)4,,„=-^-Hl+-^')-'''-'' + (™-l)^.-2,»-i. ( v) If /„ = / «" cos /3.r c?.r and J„ — I x'^ sin ^x dx then /34=.j;"sin/3a;-?i/„_i, ^/„= -^cos/3a-+?i/„_i. 17—2 260 DERIVATIVES AND INTEGRALS [Vl (vi) If /n= Icos'^xdx and J„= Isin^xdx then »4=sin:!;oos"-ia;+(«-l)/„_2, nJ„= -cos«sm»-i^+(?i-l)/„_2. ( vii) If /„ = / tan" a; c^jc then (m - 1 ) ( /,i + /„ _ 2) = ^^an" " ' x. ( viii) If /m, „ = / cos"* j; sin" x dx then (?ji+?i)/to,„= -cos"' + ia;sin''-i^ + (?i-l)7^„_2 = cos"'-ia;sm''+'^-+(m-l)iOT-2,n- [We have (m + l)/„,,„= - / sin"- 1 « -^ (008"*+^. -!;)££» = -cos'"+'^sin"-ia;+(7i — 1) \sos^'^^xsa\^-^xdx = - 008™+ 1 X Sin"-1 ^+ (» - 1) (im,»_2 - I,n,n), which leads to the first reduction formula.] (ix) Connect /,„,„= I sin™ a; sin ?m;o?^ with /,„_ 2,, I- {Math. Trip. \%^1.) (x) If Im,n= \^ cosec" X dx then (»-l)(?i-2)/„^„=(ra-2)2/„,„_2 + OT(m-l)7^_2,„_2 -^""'cosec''~ia;{msin;!; + (?i — 2)^cosa;}. {Math. Trip.\&%.) (si) If /„= \{a + h cos x)~'^dx then (»i-l)(a2-62)7^=_6sin^(a + 6cos^)-(''->) + (2re-3)a7„_i-(ra-2)/„_2. (xii) If /„= j(acos^:«; + 2Acosa;sin^+6sin^«)-''c?.5; then (72/ (J/a«A. Trip. 1898.) (xiii) If 7^„= Ix'^QogxYdx then (m + l)7^„=ii;'» + i(log^)»-mi„^„_i. 40. If w is a positive integer then the value oi \xS^ (log xY dx is \m + \ {m + \f "^ (ni + 1)^ •■•■^(ni + l)» + "J ' 41. Show that the most general function {x), such that 0" + a^0 = O for all values of x, may be expressed in either of the forms .4cosafj;+5sina,r, p cos(a.^- + 6), where A, B, p, e are constants. [Multiplying by 20' and DERIVATIVES AND INTEGRALS 261 integrating we obtain <f)"^+a^(l)^ = a^b^, where 6 is a constant, from which we deduce that ax= i ,,,,. ,„^ .] 42. Determine the most general functions y and z such that y + 0)2 = 0, and z" — coy = 0, where w is a constant and dashes denote differentiation with respect to a>. 43. The area of the curve given by , sinasind) . ^ sinacosrf) ^ l-cos2asm2(^' -^ ^ 1 - cos^a sin^^ ' where a is a positive acute angle, is Jtt (1 +sin aYjain a. (Math. Trip. 1904.) 44. The projection of a chord of a circle of radius a on a diameter is of constant length 2a cos fi ; show that the locus of the middle point of the chord consists of two loops, and that the area of either is a^ O — cos sin j3). (Math. Trip. 1903.) 45. Show that the length of a quadrant of the curve {xja)i +{yjVf^ =\ is (a2 + a6 + 62)/(ffi + 6). (Math. Trip. 1911.) 46. A point A is inside a circle of radius a, at a distance 6 from the centre. Show that the locus of the foot of the perpendicular drawn from .4 to a tangent to the circle encloses an area n (a'^+^b^). (Math. Trip. 1909.) 47. Prove that if (a, b, c, f, g, h\x, y, 1)^=0 is the equation of a conic, then h dx ■, PT , ^ ' (Ix + my + n) (hx + by +/) & py where PT, PT' are the perpendiculars from a point P of the conic on the tangents at the ends of the chord lx + my + n=0, and u, j3 are constants. (Math. Trip. 1902.) .n oi, 4.1, 4. ax' + 2bx+o 48. Show that n 2T-0D— ; /vw "^ will be a rational function of x if and only if one or other of AC -IP and aC+cA - 2bB is zero.* 49. Show that the necessary and sufficient condition that I {F(x}}^'^'^' where / and F are polynomials of which the latter has no repeated factor, should be a rational function of x, is that f'F' —fF" should be divisible by F. (Math. Trip. 1910.) .„ mi j.i_ J. /■acos.r+/3sinr + y , 60. Show that / — r^ — - dx J (1 — e cos .'c)' is a rational function of cos^ and sin a; if andonly if ae + 'y=0; and determine the integral when this condition is satisfied. (Math. Trip. 1910.) * See the author's tract quoted on p. 236. CHAPTEE VII ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS 147. Higher Mean Value Theorems. In the preceding chapter (§ 125) we proved that if f{x) has a derivative /'(a) throughout the interval {a, h) then f{h)-f{a)^{b-a)f'{^), where a<^<h; or that, if /(«) has a derivative throughout (a, a + h), then f(a + h)-fia) = hf(a + e,h) (1), where 0<6i<l. This we proved by considering the function /(6)-/(*)-^ {/(&)-/(«)} which vanishes when x= a and when x = h. Let us now suppose that / («) has also a second derivative /" {x) throughout (a, 6), an assumption which of course involves the continuity of the first derivative f'(x), and consider the function m - f{«=) -(b- «=)/' {x) - (I^J {/(b) -f{a) -{b- a)f (a)]. This function also vanishes when x = a and when x = b; and its derivative is ^^^} {/(b) -/(«) -{b- a)/' (a) - i (6 - aYf" (x)}, and this must vanish (§ 121) for some value of x between a and b (exclusive of a and b). Hence there is a value ^ of x, between 147] ADDITIONAL THEOKEMS IN THE CALCUIiUS 263 a and h, and therefore capable of representation in the form a + 6i{h — a), where < ^^ < 1, for which f{h) ^f{a) + {b- a)/' (a) +^(b- ajf (?). If we put 6 = a + A. we obtain the equation f{a + h)=f{a) + hf (a) + py" {a + e,h) (2), which is the standard form of what may be called the Mean Value Theorem of the second order. The analogy suggested by (1) and (2) at once leads us to formulate the following theorem : Taylor's or the General Mean Value Theorem. If f(w) is a function of x which has derivatives of the first n orders throughout the interval (a, b), then /(6)=/(o) + (6-o)/'(«)+fc^/"(o) + ... where a<^<b; and ifb = a+h then f(a + h) =/(a) + hf (a) + \hY' {a)+ ... where < ^„ < 1. The proof proceeds on precisely the same liaes as were adopted before in the special cases in which n=l and w = 2. We consider the function Fn{x)-(^^^^F^{a), where F^, {x) =f{b) -f(x) -(b- x)f (x) - ^^^ f" {x)- ... This function vanishes for « = a and a; = 6 ; its derivative is and there must be some value of x between a and b for which the derivative vanishes. This leads at once to the desired result. 264 ADDITIONAL THEOREMS IN THE CALCULUS [vn In view of the great importance of this theorem we shall give at the end of this chapter another proof, not essentially distinct from that given above, but different in form and depending on the method of integration by parts. Examples LV. 1. Suppose that /(;;;) is a polynomial of degree r. Then /'"'(a:) is identically zero when n>r, and the theorem leads to the algebraical identity /(a + A)=/(a.) + ¥'(«) + J/"(«) + - + J|/"->W- 2. By applying the theorem to f{x) = \lx, and supposing x and x+k positive, obtain the result 1 1 A A3 (-l^-iA"-! (-1)"A» x + h X x''- a? '" 3i^ {x-\-t 1 1 h /i2 (-l)"-iA"-i (-l^A" [Since — -r = 5 + ^.-...+^^ '-— +^ '— iM + l" x->rh X x"- x^ " 3^ x^{x-\-hy we can verify the result by showing that x'^{x-\-K) can be put in the form (.r + fl„A)''+S or that .r^ + i <a''" (.r+A) <(a; + A)''+i, as is evidently the case.] 3. Obtain the formula . , ,, . , A2 . A3 sni (^ + A) = sui.j; + A cos a-- ~ sm.?; — — oosii;-!-... /j2m-l /,2i. the corresponding formula for oos(a;+A), and similar formulae involving powers of A extending up to A^^ + i. 4. Show that if m is a positive integer, and n a positive integer not greater than m, then Show also that, if the interval (x, x + h) does not include x=0, the formula holds for all real values of m and all positive integral values of n ; and that, even if x<0<x + h or x+h<0<x, the formula still holds if m-n is positive. 5. The ioTmu\a,f(x + h)=f{x)+hf'{x + eih) is not true if /(.■!;) = l/a; and x<0<x + h. [Forf{x + h)-f{x)>Oa.nd hf {x-^'6-Ji)= -hl(x + 6i/iy<0 At is evident that the conditions for the truth of the Mean Value Theorem aro not satisfied.] 6. If x=-a, h^'ia, f{x) = x^i^, then the equation f{x + h)=f{x) + hf{x + e^h) is satisfied by 6i = \ + ^^iZ. [This example shows that the result of the theorem may hold even if the conditions under which it was proved are not satisfied.] 147] ADDITIONAL THEOEEMS IN THE CALCULUS 265 7. Newton's method of approximation to the roots of equations. Let ^ be an approximation to a root of an algebraical equation /(:r) = 0, the actual root being I + A. Then so that h- ^(^) u'^rM±^2h) so that ''--/^-*'' -Tl^V It follows that in general a better approximation than a;=f is If the root is a simple root, so that /'(^+ A) =1=0, we can, when h is small enough, find a positive constant K such that |/' (.«) | > Z' for all the values of X which we are considering, and then, if h is regarded as of the first order of smallness, /(|) is of the first order of smallness, and the error in taking ^ — {/(|)//'(|)} as the root is of the second order. 8. Apply this process to the equation «2=2, taking |=3/2 as the first approximation. [We find h= - 1/12, |+A= 17/12= 1-417..., which is quite a good approximation, in spite of the roughness of the first. If now we repeat the process, taking |=17/12, we obtain ^ + A=577/408 = 1-414215..., which is correct to 5 places of decimals. 9. By considering in this way the equation x^ — \—y=0, where y is small, show that ^(1 +y) = 1 + iy - {i^^/(2 +y)} approximately, the error being of the fourth order. 10. Show that the error in taking the root to be k-{flf')-\{ff''IP\ where | is the argument of every function, is in general of the third order. 11. The equation aiiix=ax, where a is small, has a root nearly equal to n-. Show that (1 — a) tt is a better approximation, and (1 -a + a^) ir a better still. [The method of Exs. 7 — 10 does not depend on f{x;) = being an algebraical equation, so long as/' and/" are continuous.] 12. Show that the limit when h-*-0 of the number 6^ which occurs in the general Mean Value Theorem is l/(m + l), provided that /(" + ') (a;) is continuous. [For f{x + h) is equal to each of A" „.w . ^ ,v ^, ^ . .A" „„> , , , A" + ' where 6„ + i as well as 3n lies between and 1. Hence /w(.+5./0=/<"'(-)+^^^^^^^^^^ But if we apply the original Mean Value Theorem to the function /(") {x), taking 6„A in place of h, we find /(") (x + e„h)=fW (a;) + 6„hf^'' + ') {x + Senh), f(x) + ... + ^/(") (x + dnh), f{x) + ...+ ^/W (x) + £^^/(»^i)(a,- + fl„^i/0. 266 ADDITIONAL THEOREMS IN THE CALCULUS [vil where 6 also lies between and 1. Hence from which the result follows, since/(» + i>(^ + 6^„A) and/(" + i)(a; + 5„+iA) tend to the same limit /(''+^)(:!;) as A->-0.] 13. Prove that {/(^ + 2A)-2/(a;+A)+/(a;)}/A2-*-/"(x) as ^.-e-0, provided that/" {x) is continuous. [Use equation (2) of § 147.] 14. Show that, if the/(") {x) is continuous for x = 0, then /(^) = ao + aia; + a2^H... + (a„ + fx)^"> where ar=/('')(0)/r! and fj.^-0 as x^^Q* 15. Show that if af^ + aiX+aiX^ + ... + {a^+e^)x^=bfi + b^x + h2X^+... + {h^+T,^)x^, where e^ and i/j, ten d to zero as .r -»- 0, then «„ = 6,,, aj = 6i , . . . , a„ = 6„ . [Making x-^0 we see that a^=h^. Now divide by x and afterwards make x-^-Q. We thus obtain a^ = bi; and this process may be repeated as often as is necessary. It follows that if /(^)=a!o + «i^+«2'«H... + (o„ + ej;)a^, and the first n derivatives oif{x) are continuous, then a^=f^''){0)lr !.] 148. Taylor's Series. Suppose that /(«) is a function all of whose differential coefficients are continuous in an interval (a — Tj,a + r)) surrounding the point x = a. Then, if h is numeri- cally less than rj, we have f(a + h) =f(a) + hf («) + ... + (-^^^/<"-" (a) + ^^-j/<«) (« +0„/.), where < 0„< 1, for all values of n. Or, if we have f(a + A) - /S„ = i?„. Now let us suppose, in addition, that we can prove that -K„^0 as n.^-00 . Then f(a + h) = Urn Sr, =f{a) + hf (a) + ~.f" («) + .... This expansion of/(a+A) is known as Taylor's Series. When a = the formula reduces to /W=/(o)+A/(o)+|:/"(0) + ..., * It is in fact sufficient to suppose that/W (0) exists. See E. H. Fowler, "The elementary differential geometry of plane curves" (Cambridge Tracts in Mathe- matics, No. 20, p. 104). 147, 148] ADDITIONAL THEOREMS IN THE CALCULUS 267 which is known as Maclaurin's Series. The function ii„ is known as Lagrange's form of the remainder. The reader should be careful to guard himself against supposing that the continuity of all the derivatives oif{x) is a sufficient condition for the validity of Tayloi-'s series. A direct discussion of the behaviour of Rn is always essential. Examples LVI. 1. Let/(^)=sin :r. Then all the derivatives of /(.r) are continuous for all values of x. Also I/" (^) | S 1 for all values of x and n. Hence in this case | iJ» | < A"/m ! , which tends to zero as m^- x (Ex. xxvii. 12) whatever value h may have. It follows that sm {x-\-h,) = sax x+h cos ^ - ^ sm ^ - --j cos .K+--i sm.2;+ ..., for all values of x and A. In particular smA = A-3^ + --..., for all values.of h. Similarly we can prove that cos {x + A) = coax — /is\nx — — cos.»+—sm .»; + ..., cosA=l — — + — —.... ^ 1 ' o ! ^ ! 4 1 2. The Binomial Series. Let /(^) = (1+^)™, where m is any rational number, positive or negative. Then/(")(^)=m(TO- l)...(m — m + l)(l+a;)™~'' and Maclaurin's Series takes the form (i+^r=i+Q^+Q^H.... When m is a positive integer the series terminates, and we obtain the ordinary formula for the Binomial Theorem with a positive integral exponent. In the general case /f» = JJ/'''' (5„.r) = (™)^'"(l + ^»^)"-", and in order to show that Maclaurin's Series really reijresents (1 +x)™ for any range of values of x when m is not a positive integer, we must show that R„-a~0 for every value of ;;; in that range. This is so in fact if - 1 <.j; < 1, and may be proved, when < :» < 1, by means of the expression given above for Rn, since (1 + ^k^)"*-" < 1 if » > ot, and (jX^-^Osisn-^oa (Ex. x.wii. 13). But a difficulty arises if -1<^<0, since 1 + 5„.2;<1 and (l + 5„*')"'-"> 1 if n>m; knowing only that < 5n < 1, we cannot be assured that 1 + 0„.c is not quite small and (l + 5„a;)'"-'' quite large. In fact, in order to prove the Binomial Theorem by means of Taylor's Theorem, we need some different form for Rn, such as will be given later (§ 162). </-(?+ A)- <^ (a = fl -^ <"'(?+ ^«'^)- 268 ADDITIONAL THEOREMS IN THE CALCULUS [vil 149. Applications of Taylor's Theorem. A. Maxima and minima. Taylor's Theorem may be applied to give greater theoretical completeness to the tests of Ch. VI, || 122 — 123, though the results are not of much practical importance. It will be remembered that, assuming that <f> («) has derivatives of the first two orders, we stated the following as being sufficient conditions for a maximum or minimum of <f> (cc) at a; = f : for a maximum, <^'(^)=0, 0"(f)<O; for a minimum, 0'(|^)=O, ^"(f)>0. It is evident that these tests fail if <j>" (f) as well as <j>'{^) is zero. Let us suppose that the first n derivatives <P'{x), <}>"(x), ..., </,'") (^) are continuous, and that all save the last vanish when x—^. Then, for sufficiently small values of h, '"n\ In order that there should be a maximum or a miflimum this expression must be of constant sign for all sufficiently small values of h, positive or negative. This evidently requires that w should be even. And if n is even there will be a maximum or a minimum according as <^'"' (f) is negative or positive. Thus we obtain the test : if there is to he a maodmum or minimum the first derivative which does not vanish must be an even derivative, and there will be a maximum if it is negative, a minimum if it is positive. Examples L VII. 1. A''erify the result when (\>{3c) = ia:-a)'^, mhwaga, positive integer, and ^=a. 2. Test the function {x - a)'" (x - 6)", where m and n are positive integers, for maxima and minima at the points x—a, x=b. Draw graphs .of the diflerent possible forms of the curve y = {x — a)'^(x-b)'^. 3. Teat the functions sina; — i-, sin^-a; + — , am. x — x+-x — j^t ■■■■, cos^— 1, COST— 1+ — , cos^-lH-- — — , ... for maxima or minima at a;=0. 150. B. The calculation of certain limits. Suppose that_/(a;) and cp (x) are two functions of x whose derivatives /' (x) and (j)' (x) are continuous for x=^- and that /(^) and ^(|) are both equal to zero. Then the function ir{x)=f{x)l(l>{x) 149, 150] ADDITIONAL THEOREMS IN THE CALCULUS 269 is not defined when a; = f. But of course it may well tend to a limit as a;-*-^. Now f{x) =fix) -/(f) = {x- f)/ {X,), where ^i lies between ^ and x ; and similarly ^ (x) = (x — ^) <f>' {x^, where x^ also lies between f and x. Thus We must now distinguish four cases. (1) If neither/' (f) nor <^' (f) is zero, then f{x)l<l>{x)-^f'{^)l4>'{^). (2) If/(f) = O,f(f)+0,then f{x)l4>{x)^0. (3) If /' (f) =t= 0, 0' (?) =0, then f{x)/(j) (x) becomes numerically very large as «-*?: but whether /(«)/^ («) tends to oo or — oo , or is sometimes large and positive and sometimes large and negative, we cannot say, without further information as to the way in which (j)' (x)^~0 as x-^^. (4) lf/'(f) = 0, (j)'{^)=0, then we can as yet saynothing about the behaviour of /(a;)/^ (x) as x-^0. But in either of the last two cases it may happen that f(x} and <!> (x) have continuous second derivatives. And then fix) = fix) -/(?) -ix- 1)/ (0 = H^ - ^Tf" ('^O. <f>ix) = <f> ix) -^i^)-ix- ?) </.' (I) = i (a. - a^ ,/," ix,), where again Xi and x^ lie between ? and x ; so that t(*)=/"(*O/0"(«^2). We can now distinguish a variety of cases similar to those considered above. In particular, if neither second derivative vanishes for x=^, we have fix)/cf>ix)^f"i^)/<f>"i^). It is obvious that this argument can be repeated indefinitely, and we obtain the following theorem: stippose that fix) and ^ix) and their derivatives, so far as may he wanted, are continuous for x = ^. Suppose further that /'^' («) and <f>^i^ ix) are the first derivatives of fix) and ^ ix) which do not vanish when x = ^. Then (1) ifp = q, f{x)l4>ix)-*f^^im'''{^)\ (2) ifp>q, fix)/<l>ix)^0; 270 ADDITIONAL THEOREMS IN THE CALCULUS [VII (3) if p<q, and q—p is even, either /(«)/</>(«)-* +00 or f(x)l4> (x)^- CO , the sign being the same as that ofpv) (f )/^(9i (|); (4) if p<q and q—p is odd, either f(x)l<f) (x) -^ + 00 or f(x)l^(x) -^ — <X) as x^~^ + 0, the sign being the same as that of /'^' (f)/<^'9' (f), while if x->-^ — the sign must be reversed. This theorem is in fact an immediate corollary from the equations Examples LVIII. 1. Find the limit of as .r-^1. [Here the functions and their first derivatives vanish for x=l, and/"(l) = ?i(«+l), <^"(1) = 2.] 2. Find the limits as x-»-0 of (tau X — x)l(x — sin x), (tan nx-n tan .v)i{n sm x— sin nx). 3. Find the limit of A-{x/(*2 + a^)-a'} as a;-»-oo. [Put^=l/y.] 4. Prove that r / s (-1)" r 1 f (-1)"! (-1)"^ hmtj?— ?i)cosec^7r = =^ —, lim JcosecaiTr- 7^ r— (■ = --,, — , x^n f ar^n^-n \ {x-n)nj 6 n being any integer ; and evaluate the corresiJonding limits involving cot xv. 5. Find the limits as x-^0 of 1 / 1 ^\ 1 / , 1 x\ —„ I oosec X 5, -^cot« ^5- x^\ X 6J ' x^ \ X dj 6. (sin^arcsin^ — a;2)/;2;^^9-Jg^, (tanj^arctana'-^p^)/*'^-*-!, as x-*-0. 151. C. The contact of plane curves. Two curves are said to intersect (or cut) at a point if the point lies on each of them. They are said to touch at the point if they have the same tangent at the point. Let us suppose now that f(x), (f> (x) are two functions which possess derivatives of all orders continuous for x = ^, and let us consider the curves y =f{x), y = <i> (x). In general /(^) and <^ (f ) will not be equal. In this case the abscissa x = ^ does not corre- spond to a point of intersection of the curves. If however 150, 151] ADDITIONAL THEOREMS IN THE CALCULUS 271 fi^) = ^ (?). the curves intersect in the point x=^, 2/=/(f)=<^(f). Let us suppose this to be the case. Then in order that the curves should not only cut but touch at this point it is obviously necessary and sufficient that the first derivatives /'(a?), ^'(a;) should also have the same value when x = ^. The contact of the curves in this case may be regarded from a different point of view. In the figure the two curves are drawn touching at P, and QR '^' is equal to <^ (f + h) -f(^ + h), or, since </.(^)=/(f), <^'(^)=/(a to ih^<j>"(^+eh)-f"(^+eh)}, where d lies between and 1. Hence lim^ = i{f' (r)-/"(?)}, when h^~0. In other words, when the curves touch at the point whose abscissa is ^, the difference of their ordinates at the point whose abscissa is ^ + h is at least of the second order of smallness when h is small. The reader will easily verify that Km {QRjh) = (^' (|) -/' (|) when the curves cut and do not touch, so that QR is then of the first order of smallness only. It is evident that the degree of smallness of QR may be taken as a kind of measure of the closeness of the contact of the curves. It is at once suggested that if the first ?i — 1 derivatives of / and ^ have equal values when x = ^, then QR will be of the ?ith order of smallness; and the reader will have no difficulty in proving that this is so and that lim^ = ^{r'(f)-/<"'(^)}. We are therefore led to frame the following definition : Contact of the mth order. // f{^) = (/> (f ), /' (^) = 0' (|), ..., y(»>(^) = ^{»)(^), hut /'"+"(l') + ^'"+"(l), then the curves y =/(«), y = <f)(x) will be said to have contact of the nth order at the point whose abscissa is ^. The preceding discussion makes the notion of contact of the nth order dependent on the choice of axes, and fails entirely 272 ADDITIONAL THEOREMS IN THE CALCULUS [vH when the tangent to the curves is parallel to the axis of y. We can deal with this case by taking y as the independent and' x as the dependent variable. It is better, however, to consider x and y as functions of a parameter t. An excellent account of the theory will be found in Mr Fowler's tract referred to on p. 266, or in de la Vallee Poussin's Cours d' Analyse, vol. ii, pp. 396 et seq. Examples LIX. 1 . Let <j> (x) =ax+b, so that 2/=(j){x) is a, straight line. The conditions for contact at the point for which x=^ are f(^)=:a^+b f {^) = c- If ■^fi determine a and b so as to satisfy these equations we find «=/' (S), b=f{& - kf (?), and the equation of the tangent to y=f{x) at the point ^•=^ is y=^/'(l)+ {/«)-!/' (I)}, or y-/(?) = (^-?)/' «)• Cf. Ex. XXXIX. 5. 2. The fact that the hne is to have simple contact with the curve completely determines the line. In order that the tangent should have contact of the second order with the curve we must have /" (^) = 0" (|), i.e. y"(|) = 0. A point at which the tangent to a curve has contact of the second order is called a point of inflexion. 3. Find the points of inflexion on the graphs of the functions 3a"* - G.r^ + 1, 2x'/(l+^^), %mx, acos?x-\-bs\n^x, taxix, arctanx 4. Show that the conic ax^ + 'ihxt/-\-by'^-\-'igx+^fy-{-c=0 cannot have a point of inflexion. [Here ax + hy +g + {hx + by +f)yi = and a + 2Ayi + 6y ,2 + {hx + hy+f)y^ = 0, suffixes denoting difierentiations. Thus at a point of inflexion a + 'ihyi + byi^ = 0, or a{hx+by+ff-2h{ax+hy+g){hx+by+f) + b{ax + hy+gf=Q, or {ab-h?){ax^+ihxy + by'^+igx+'2.fy}-irap-'2.fgh + bg'^=0. But this is inconsistent with the equation of the conic unless af'-2fgh+bg^'=c(ab- h?) or abc + 2fgk-af^-bg^-ch?=0; and this is the condition that the conic should degenerate into two straight lines.] 5. The curve y={a3:^+2hx+c)l{ax^+2^x+y) has one or three points of inflexion according as the roots of ax^ + 'i^x + y=0 are real or complex. [The equation of the curve can, by a change of origin (cf. Ex. xlvi. 15), be reduced to the form 1 = ^KM' + 2^1 + G) = ^j{A (I -p){^- q)}, where p, q are real or conjugate. The condition for a point of inflexion will be found to be ^'-3j09l+?'g'(p+?) = 0, which has one or three real roots according as {pq{p-q)\ is positive or negative, i.e. according as p and q are real or conjugate.] 151] ADDITIONAL THEOREMS IN THE CALCULUS 273 6. Discuss in particular the cvavea y = {l-x)/{l+x^), y = {l-x^l{l+x^), 2/=(l+^)/(l-^'2). 7. Show that when the curve of Ex. 5 has three points of inflexion, they lie on a straight line. [The equation $^ — Spq^+pq (p + g) = can be put in the form (| -p) (^ - j) (^ +jo + ?) + (^ - 2)^ I = 0, so that the points of inflexion lie on the line i + A {p-q)''ri+p + q = or A^-4:{AC-B^)ri^2B.] 8. Show that the curves y=a'sin.r, y = {amx)lx have each infinitely many points of inflexion. 9. Contact of a circle with a curve. Curvature*. The general equation of a circle, viz. {x-ay + (j/-by=r^ (1), contains three arbitrary constants. Let us attempt to determine them so that the circle has contact of as high an order as possible with the curve y=f{p) at the pomt (|, i;), where n=f{$)- We write i;,, 1,2 for /' (^),/" (|). Differentiating the equation of the circle twice we obtain (^-a) + (y- 6)^^1 = (2), H-yiH(y-6)y2=o (3). If the circle touches the curve then the equations (1) and (2) are satisfied when 0;=^, y = i;, yi = 7;i. This gives {i-a)lr,i= -{r,-b) = rl^/{l+rii^). If the contact is of the second order then the equation (3) must also be satisfied wheny2 = 7;2- Thus 6=?) + {(l+7;i^)/))2} ; and hence we find The circle which has contact of the second order with the curve at the point (I, i;) is called the circle of curvature, and its radius the radius of curvature. The measure of curvature (or simply the curvature) is the reciprocal of the radius : thus the measure of curvature is /" (^)/ {1 + [/' (|)]^} ^% or m del I \dt 10. Verify that the curvature of a circle is constant and equal to the reciprocal of the radius ; and show that the circle is the only curve whose curvature is constant. 11. Find the centre and radius of curvature at any point of the conies y'^=4:ax,{xlaf + {ylbf=\. 12. In an ellipse the radius of curvature at P is CD^jab, where CD is the semi-diameter conjugate to CP. * A much fuller disonssion of the theory of curvature will be found in Mr Fowler's tract referred to on p. 272. H. 18 274 ADDITIONAL THEOREMS IN THE CALCULUS [vn 13. Show that in general a conic can be drawn to have contact of the fourth order with the curve y=f{x)) at a given point P. [Take the general equation of a conic, viz. ax^ + '2,hxy+hy'^ + 'ilgx + '2.fy-\-o=0, and differentiate four times with respect to x. Using suffixes to denote differentiation we obtain ax + hy+g-^{hx + hy+f)yi = 0, a + 2hyi + byi'+{hx + by+f)y2=0, 3 (A + byi) 2/2 + {ftx + by +/) y^ = 0, 4:{h + by,)ys + 3byi' + {hx+by+f)yi=0. It the conic has contact of the fourth order, then these five equations must be satisfied by writing |, rj, rjumtnii Vi foi' ^> ^i yii 3/2, ysi yi- We have thus just enough equations to determine the ratios a : b : c :f : g : h.] 14. An infinity of conies can be drawn having contact of the third order with the curve at F. Show that their centres all lie on a straight line. [Take the tangent and normal as axes. Then the equation of the conic is of the form 2y=ax^ + 2hxy+by'', and when x is small one value of y may be expressed (Ch. V, Misc. Ex. 22) in the form y=^ax^ + {^ah + e^)x\ where e^-^Q with x. But this expression must be the same as y=if"(.o)x^+{if"iO)+e,'}x^, where fa;'-*-0 with x, and so a=f" (0), h=f"' {Q)IZf" (0), in virtue of the result of Ex. LV. 15. But the centre lies on the line ax + hy = Q.'] 15. Determine a parabola which has contact of the third order with the ellipse («/a)2+(2//6)'=l at the extremity of the major axis. 16.' The locus of the centres of conies which have contact of the third order with the ellipse (x/ay-\-(j/lby = l at the point (a cos a, 6 sin a) is the diameter .r/(a cos a) =yl{b sin a). [For the ellipse itself is one such conic] 152. DifTerentiation of functions of several variables. So far we have been concerned exclusively with functions of a single variable x, but there is nothing to prevent us applying the notion of differentiation to functions of several variables x,y, .... Suppose then that/(a;, y) is a function of two* real variables X and y, and that the limits Ijjjj fi«'+h,y)-f(oo,y) ^ J. j^ f(x,y + Ic)-f{x,y) * The new points which arise when we consider functions of several variables are illustrated sufficiently when there are two variables only. The generalisations of our theorems for three or more variables are in general of an obvious character. 151, 152] ADDITIONAL THEOREMS IN THE CALCULUS 275 exist for all values of x and y in question, that is to say that f{x,y) possesses a derivative dfjdx or Dxf{x,y) with respect to x and a derivative dfjdy or Dyf{x, y) with respect to y. It is usual to call these derivatives the partial differential coefficients off, and to denote them by df df dx'' dy or fx{p,y), fy'{x,y) or simply /a,', /y' or fx,fy. The reader must not suppose, however, that these new notations imply any essential novelty of idea: ' partial differentiation ' with respect to x is exactly the same process as ordinary differentiation, the only novelty lying in the presence in / of a second variable y independent of x. In what precedes we have supposed x and y to be two real variables entirely independent of one another. If x a,nd y were connected by a relation the state of affairs would be very different. In this case our definition oi fx would fail entirely, as we could not change x into x + h without at the same time changing y. But then f{x,y) would not really be a function of two variables at all. A function of two variables, as we defined it in Ch. II, is essentially a function of two independent variables. If y depends on a), 2/ is a function of x, say y=<f> (^) > ^'iid then /(«.2/)=/{«>0(«)} is really a function of the single variable x. Of course we may also represent it as a function of the single variable y. Or, as is often most convenient, we may regard x and y as functions of a third variable t, and then f{x, y), which is of the form /{cj} (t), i/^(0}. is a function of the single variable t. EzamplesLX. 1. FiOYeth&tii x=rcoa6,i/ = rsmd,soth.a.tr=^(x^+y^), fl=arc tan (2/ /a;), then dr _ X dr _ y 95 _ y dd _ x dx . 9v . „ 'dx ■ „ dy =- = cosfl, ^ = s\ad, 5t:=— rsmfl, ^=rcos5. dr or dS ' ad 2. Account for the fact that 5~=t=l/(5~) ^^^ 2"^^ Km)' [When we were considering a function y of one variable x it followed from the definitions that dyjdx and dxjdy were reciprocals. This is no longer the 18—2 276 ADDITIONAL THEOREMS IN THE CALCULUS [VII ease when we are dealing with functions of two variables. Let P (Fig. 46) \>e the point {x, y) or (r, 6). To find 'dr\'bx we must increase x, say by an increment MMi = 8x, while keepmg y constant. This brings P to Pj. If along OPi we take OP' = OP, the increment of r is P'Pi = hr, say; and 3r/3.r=lim(6Y/8i!;). If on the other hand we want to calculate 3j;/3r, x and y being now regarded as functions of r and 6, we must increase r by A>-, say, keeping 5 constant. This brings P to Pj, where PP2=Ar: the corresponding increment of :j; is MMi = Ax, say; and 3^/3r=lim (AxJAr). Now Ax=8it;* : but A?-4=8r. Indeed it is easy to see from the figure that lim {8r 1 8x) =lim {P' Pi I PPi) = cos 0, but lim (A»-/ A^) = lim (PPj / P^i) = sec 0, so that lim (dr/Ar) = cos^ 6. The fact ia of course that 3^/3?' and =' ^°- 3r/3.j; are not formed upon the same hypothesis as to the variation of P.] 3. Prove that if z =f {ax + by) then 6 {dzjdx) = a (dzfdy). 4. Find dXIdx, dXjdy, ... when X+Y=x, Y=xy. Express x, y as functions of Z, Y and find 3.j;/3X, 3.;!;/3F, .... 5. Find dX/dx, ... when X+Y+Z=x, Y+Z=xy, Z=xyz; express iO, y, z in terms of X, Y, Z and find dxjdX, .... [There is of course no difficulty in extending the ideas of the last section to functions of any number of variables. But the reader mxist be careful to impress on his mind that the notion of the partial derivative of a function of several variables is only determinate when all the independent variables are specified. Thus ii u=x+y + z, x, y, and z being the independent variables, then 3m/3^ = 1. But if we regard m as a function of the variables x, x+y=ij, and x+y + z = ^.,s.o that M = f, then dujdx=0'.'\ 153. DifTerentiation of a flinction of two functions. There is a theorem concerning the differentiation of a function of one variable, known generally as the Theorem of the Total Differential Coefficient, which is of very great importance and depends on the notions explained in the preceding section re- garding functions of two variables. This theorem gives us a rule for differentiating with respect to t. * Of course the fact that Aa; = Sa; is due merely'to the particular value of Ar that we have chosen (viz. PP2)- -A-ny other choice would give us values of Ax, Ar proportional to those used'here. 152, 153] ADDITIONAL THEOREMS IN THE CALCULUS 277 Let us suppose, in the first instance, that /(«, y) is a function of the two variables x and y, and that fx, fy are continuous functions of both variables (§ 107) for all of their values which come in question. And now let us suppose that the variation of X and y is restricted in that {x, y) lies on a curve x = (}>{t), y = ^{t), where ^ and ■^ are functions of t with continuous differential coefficients <^' {t), yjr' (t). Then f(x, y) reduces to a function of the single variable t, say F(t). The problem is to determine F'{t}. Suppose that, when t changes to t + r, x and y change to x+ ^ and y + 'r]- Then by definition ^ = lim I [/ {<^ (t + t), ^ (i + t)} -/ {<!> (t), yjr (t)]] at T^o T = lim -{/(« + f, 2/ + 7?) -f{x, y)} T ^l ? T 9; T But, by the Mean Value Theorem, {/('« +ly + v) -fi^B, y + r)M =U («= + ^^ 2/ + v). {/(^, y+v) -/(^, y)}/'? =/; (^. y + ^'n), where 6 and 6' each lie between and 1. As t^O, ^^0 and i?*0, and ^/r-*<j>'(t), rj/r-^yjr'it): also fx (^ + d^,y + v)^fx (^, y), fy (^. y + o'v)^fy' {x, y). Hence jf" (t) = A/{<^ (0. t (<)} =/»' («'. 2/) </>' (0 +// (^' y) ^' (*)• where we are to put x = (f){t), y = 'f{t) after carrying out the differentiations with respect to x and y. This result may also be expressed in the form df^dldx^dfd^ dt dx dt dy dt Examples LXI. 1. Suppose <^ (t) = (1 - f^W + 0, ^ (0 = 2«/(l + 1'), so that the locus of (x, y) is the circle x^+y^=\. Then ^' (0 = - 4</(l + <2)2, V' (<) = 2 (1 - «')/(! + <')^ i^-" (<)={- 4</(i + <2)2}/; + {2 (1 - <2)/(i + <2)2}/;, where .r and y are to be put equal to (1-0/(1 + ^^) and 2«/(l + after carrying out the differentiations. 278 ADDITIONAL THEOREMS IN THE CALCULUS [vil We can easily verify this formula in particular cases. Suppose, e.g., that f{x, y) = x'^ + y^. Then fj = '2.x, fy = 2y, and it is easily verified that F' {t) = %x<^' (t) + '2,y\lr' {t)='0, which is obviously correct, since F{t)=\. 2. Verify the theorem in the same way when (a) x=t^, y—X — c^^ f{x,y) = x+y ; (b) x = acost, y = asm.t, f{x,y) = x'^+y^. 3. One of the most important cases is that in which t is x itself. We then obtain B^fix, ^ {x)} = DJ {x, y) + Dyfix, y) ^' {x). where y is to be replaced by y\r (x) after differentiation. It was this case which led to the introduction of the notation dfjdx, dfjdy. For it would seem natural to use the notation dfldx for either of the fmictions I>xf{^t V' (^)} 8.nd Dxfix, y), in one of which y is put equal to -^{x) before and in the other after differentiation. Suppose for example that y=\-x siaA f{x,y) = x+y. Then B^f{x, \-x) = Dx^ = ^, but D^f{x,y) = l. The distinction between the two functions is adequately shown by denoting the first by dfjdx and the second by 'dfjdx, in which case the theorem takes the form ^Jl + ^Idy. dx dx dy dx ' though this notation is also open to objection, in that it is a little misleading to denote the functions f{x, \Ja (x)} and f(x, y), whose forms as functions of x are quite different from one another, by the same letter /in dfjdx and dfjdx. 4. If the result of eliminating t between .»=^ {t), y=-^{,t) is f(x, y) = 0, then dx dt dy dt 5. If X and y are functions of t, and r and 6 are the polar coordinates of {x,y),t}ieiir'={xx'+yy')jr, d'={xy'-y.v')/r'^,da,shea denoting differentiations with respect to t. 154. The Mean Value Theorem for functions of two variables. Many of the results of the last chapter depended upon the Mean Value Theorem, expressed by the equation 4)(x + h)-<j} (x) = hf {x + eh), or as it may be written, it y = (f> (x), Sy=f'(x + eSx)Sx. Now suppose that z =f(x, y) is a function of the two inde- pendent variables x and y, and that x and y receive increments h, k or Sx, By respectively : and let us attempt to express the corresponding increment of 2, viz. 8z =f{x + h,y + h)-f{x, y), in terms of h, k and the derivatives of z with respect to x and y. 153, 154] ADDITIONAL THEOREMS IN THE CALCULUS 279 Let f{x + ht,y + Id) = F{t). Then f{x + h,y + k) -f{x, y) = F{1) -F{0) = F' (d), where < ^ < 1. But, by § 153, F'(t) = Dtf(x + ht,y + kt) = hf-,' {x + ht,y + U) + kfy {x + ht,y + kt). Hence finally g^=/(«+A, y+k) -f(x, y)=hf^'(x + eh, y+ 0k) + kf,J {x + 0h, y+dk), which is the formula desired. Since /„', // are supposed to be continuous functions of x and y, we have /,' {x + eh,y + 6k) =// (x, y) + €n,k, fy (x +0h,y + 0k) =fy {x, y) + 17^,4, where e^,^ and i;^^^ tend to zero as h and k tend to zero. Hence the theorem may be written in the form s^ = (/,' + 6) s^ +(/; + ,?) §2/ (1), where e and rj are small when hx and hy are small. The result embodied in (1) may be expressed by saying that the equation hz^f^hx+fy'hy is approximately true ; i.e. that the difference between the two sides of the equation is small in comparison with the larger of hx and 8y*- We must say 'the larger of Bx and By' because one of them might be small in comparison with the other; we might indeed have Sx=0 or Sy= 0. It should be observed that if any equation of the form 8z=XSx+fi8i/ is 'approximately true' in this sense, we must have X =/,.', 11= fy. For we have dz -f^' S.V -fy' 8i/=fdx+r,8i/, Bz-\8a;- fiby = e' bx + rj' by where «, i;, e, t] all tend to zero as 8^ and by tend to zero ; and so (X-/;) bx + {i,-f^) by^p bx+p'by where p and p' tend to zero. Hence, if f is any assigned positive number, we can choose a- so that I (X -/,') bx+(,.-f,') 8y I < f (I 8^ l + l 8j/ 1) for all values of bx and by numerically less than a-. Taking 8y=0 we obtain \(>--fx')S^\<i\^^\i 01 \\ -fx I < f , and, as f may be as small as we please, this can only be the case if X=/a.'. Siinilarly p.=fy'. * Or with \5x\ + \Sy\ or J(5x''' + 5y% 280 ADDITIONAL THEOREMS IN THE CALCULUS [VII 155. DifTerentials. In the applications of the Calculus, especially in geometry, it is usually most convenient to work with equations expressed not, like equation (1) of § 154, in terms of the increments Sx, Sy, Sz of the functions x, y, z, but in terms of what are called their differentials dos, dy, dz. Let us return for a moment to a function y=f(x) of a single variable x. If/' {x) is continuous then hy={f'{x)+e]hx (1), where e-^O as Zx-^0: in other words the equation hy^f'{x)hx (2) is ' approximately ' true. We have up to the present attributed no meaning of any kind to the symbol dy standing by itself We now agree to define dy by the equation dy=fix)Sx (3). If we choose for y the particular function x, we obtain dx= Sx (4), so that dy=f'(x)dx (5). If we divide both aides of (5) by dx we obtain l=/'(^) "• («)' where dyjdx denotes not, as heretofore, the differential coefficient of y, but the quotient of the differentials dy, dx. The symbol dy/dx thus acquires a double meaning; but there is no incon- venience in this, since (6) is true whichever meaning we choose. The equation (5) has two apparent advantages over (2). It is exact and not merely approximate, and its truth does not depend on any assumption aa to the continuity of/' (x). On the other hand it is precisely the fact that we can, under certain conditions, pass from the exact equation (5) to the approxi- mate equation (2), which gives the former its importance. The advantages of the ' differential ' notation are in reality of a purely technical character. These technical advantages are however so great, especially when we come to deal with functions of several variables, that the use of the notation is almost inevitable. When /' (x) is continuous, we have lim^ = l when 8.r-*-0. This is sometimes expressed by saying that dy is the prinidpal part of 8y when 8x is small, just aa we might say that ax; is the ' principal part ' of cuc + bx^ when x is small. 155] ADDITIONAL THEOREMS IN THE CALCULUS 281 We pass now to the corresponding definitions connected with a function z of two independent variables x and y. We define the differential dz by the equation dz=fjhx+fy^y (7). Putting z = x and z — yva. turn, we obtain dx=hx, dy = 8y (8), so that dz=f^'dx+f,;dy (9), which is the exact equation corresponding to the approximate equation (1) of § 154. Here again it is to be observed that the former is of importance only for reasons of practical convenience in working and because the latter can in certain circumstances be deduced from it. One property of the equation (9) deserves special remark. We saw in § ] 53 that if z=f{a;, y), x and y being not independent but functions of a single variable t, so that z is also a function of t alone, then dz _ df dx df dy dt dx dt dy dt ' Multiplying this equation by dt and observing that dx='^^-dt, dyJj-^dt, dzJ^^dt, we obtain dz=fJdx-\-fy'dy, which is the same in form as (9). Thus the formula which expresses dz in terms of dx and dy is the same whether the variables x and y are independent or not. This remark is of great importance in applications. It should also be observed that if z is a function of the two independent variables x and y, and dz=\dx+iidy, then X=/j,', t'^=fy'- This follows at once from the last paragraph of § 154. It is obvious that the theorems and definitions of the last three sections are capable of immediate extension to functions of any number of variables. Examples LXII. 1. The area of an ellipse is given by A — nab, where a, b are the semiaxes. Prove that dA _ da db 'A^l^'^'b' and state the corresponding approximate equation connecting the increments of the axes and the area. 282 ADDITIONAL THEOREMS IN THE CALCULUS [vil 2. Express A, the area of a triangle ABC, as a function of (i) a, B, C, (ii) A, b, c, and (iii) a, b, c, and establish the formulae dA „ da cdB bdO dA . . , . db da A a asoiB asm 6 A _ be dA=R (cos ^ c?a + cos Bdb + coa Cdc), where R is the radius of the circumcircle. 3. The sides of a triangle vary in such a way that the area remains constant, so that a may be regarded as a function of 6 and c. Prove that 3a cos B da _ cos C db ~ cos A ' da cos A ' [This follows from the equations da=^rrdb ^ TT-dc, cos Ada + ooa Bdb + cob Gda=0. db do 4. If a, b, a vary so that R remains constant, then da db da _ and so cos A cos B cos C da _ cos A da _ cos A db cos B ' da cos C ' [Use the formulae a = 2R sin A, ..., and the facts that R and ^ + 5 + Care constant.] 5. If z is a function of u and v, which are functions of x and y, then dz dz du dz dv dz _ dz du dz dv dx du dx &« 3* ' dy du dy dv dy ' [We have . dz , dz , , du , dii J , 3i; , Sw , az=/^dii + ^dv, du==^dx+^^dy, dv=7^dx + ;r- dy. du dv dx dy " dx 3y Substitute for du and dv in the first equation and compare the result with the equation '^^=3^^^+|'^^-] 6. Let be a function of x and y, and let X, Y, Z be defined by the equations x=axX-\-bj:'-^a-^Z, y^aiX+biV+c^Z, z=a3X+biY+aiZ. Then Z may be expressed as a function of X and Y. Express dZjdX, dZjdY in terms of dzfdx, dzjdy. [Let these differential coefficients be denoted by F, Q and p, q. Then dz - pdx — qdy = 0, or {CiP + C2q-Cz)dZ-\-{a^pJraiq-a3)dX+{b^p + h2q-bi)dY=0. 155, 156] ADDITIONAL THEOREMS IN THE CALCULUS 283 •] Comparing this equation with dZ—PdX~ §(iF=0 we see that p_ <»ip + a2g-a3 ci 6i£+62£-63 ■C3 7. If then CijB + Cjg'-Ca CijB + C22'-C3 (ai:); + 6iy + Ci3)jo + (a2^ + 62,y + C23)g' = a3^+63y + «32i (J/aiA. Tnp. 1899.) 8. Diflferentiation of implicit functions. Suppose that/(.j;, y) and its derivative fy {x, y) are continuous in the neighbourhood of the point (a, 6), and that f(a,b)=0, f,'{a,b)^0. Then we can find a neighbourhood of (a, 6) throughout which /„' {x, y) has always the same sign. Let \is suppose, for example, that /„' {x, y) is positive near {a, h). Then f{ic, y) is, for any value of x sufficiently near to a, and for values of y sufficiently near to 6, an increasing function of y in the stricter sense of § 95. It follows, by the theorem of § 108, that there is a unique continuous function y which is equal to 6 when x = a and which satisfies the equation /(«,_y) = for all values of ^ sufficiently near to a. Let us now suppose that f{x,y) possesses a derivative f^ (x,y) which is also continuous near (a, 6). If f{x, y)=0, x=a + h, y=b+k, we have where and •) tend to zero with h and k. Thus A fl' + r}'^ /i" ^ = -fl. dx /(,' 9. The equation of the tangent to the curve fix, y) = 0, at tho point .(.•(,, yQ, is Definite Integrals and Areas. It will be remembered 156. that, in Ch. VI, § 145, we assumed that, if f(x) is a continuous function of x, and PQ is the graph of y =/(«), then the region PpqQ shown in Fig. 47 has associated with it a definite number which we call its area. It is clear that, if we denote Op and Oq by a and so, and allow X to vary, this area is a function of x, which we denote by^(^')- Fig. 47. 284 ADDITIONAL THEOREMS IN THE CALCULUS [vil Making this assumption, we proved in § 145 that F' ((c)=f(x) and we showed how this result might be used in the calculation of the areas of particular curves. But we have still to justify the fundamental assumption that, there is such a number as the area F{x). We know indeed what is meant by the area of a rectangle, and that it is measured by the product of its sides. Also the properties of triangles, parallelograms, and polygons proved by Euclid enable us to attach a definite meaning to the areas of such figures. But nothing which we know so far provides us with a direct definition of the area of a figure bounded by curved lines. We shall now show how to give a definition of F{iiii) which will enable us to prove its existence.* Let us suppose /(«) continuous throughout the interval {a, h), and let us divide up the interval into a number of sub-intervals by means of the points of division x^, !i\, x^, ..., a?„, where a = Xg<x^< ... < «„_! <Xn=b. Further, let us denote by S„ the interval (x^, x^+j), and by m„ the lower bound (§ 102) o{f(x) in S„, and let us write s = m„S|, + miSi-|- ... +TO„8„ = Sm„8„, say. It is evident that, if M is the upper bound of /(«) in (a, b), then s^M (b — a). The aggregate of values of s is therefore, in the language of § 80, bounded above, and possesses an upper bound which we will denote by j. No value of s exceeds j, but there are values of s which exceed any number less than j. In the same way, if Jf„ is the upper bound of /(a;) in 8„, we can define the sum It is evident that, if m is the lower bound o(f(x) in (a, b), then S ~m{b — a). The aggregate of values of S is therefore bounded below, and possesses a lower bound which we will denote by J. No value of S is less than /, but there are values of S less than any number greater than J. * The argument which follows is modelled on that given in Goursat's Gours d' Analyse (second edition), vol. i, pp. 171 et seq. ; but Goursat's treatment is much more general. 156] ADDITIONAL THEOREMS IN THE CALCULUS 285 It -will help to make clear the significance of the sums s and S if we observe that, in the simple case in which f(x) increases steadily from x=a to x=b, niy is f{Xy) and My is f{Xy + {j. lu this case s is the total area of the rectangles shaded in Fig. 48, and S is the area bounded by a thick line. In general s and S will still be areas, composed of rectangles, respectively included in and including the curvi- linear region whose area we are trying to define. We shall now show that no sum such as s can exceed any Fig. 48. su7n such as S. Let s, S be the sums corresponding to one mode of subdivision, and s', S' those corresponding to another. We have to .show that s & S' and s' S S. We can form a third mode of subdivision by taking as dividing points all points which are such for either s, S or s, S'. Let s, S be the sums corresponding to this third mode of subdivision. Then it is easy to see that sSs, sSs', Ss,Sf, SS/S' (1). For example, s differs from s in that at least one interval S„ which occurs in s is divided into a number of smaller intervals so that a term m^S„ of s is replaced in s by a sum in^,! K,i + '>np,i Sv,2 + . . . + m^^p S^p, where m^,i, m„,2, ... are the lower bounds of /(«) in S„^i, B^^^, •••• But evidently m„,i = m„, m„,2 = "to, . . ., so that the sum just written is not less than m„8„. Hence s Ss; and the other inequalities (1) can be established in the same way. But, since s S S, it follows that s&sSS&S', which is what we wanted to prove. It also follows that j S J. For we can find an s as near to j as we please and an S as near to J as we please*, and so j >J would involve the existence of an s and an S for which s >S. * The s and the S do not iu general correspond to the same mode of subdivision. 286 ADDITIONAL THEOREMS IN THE CALCULUS [vil So far we have made no use of the fact that/(a;) is continuous. We shall now show that j = J, and that the sums s, S tend to the limit J when the points of division «„ are multiplied indefinitely in such a way that all the intervals S, tend to zero. More pre- cisely, we shall show thsit, given any positive number e, it is possible to find S so that O^J-s<e, 0^S-J<6 whenever 8„ < S for all values of v. There is, by Theorem II of § 106, a number S such that lf„ - in„ < ejQ) — a), whenever every S„ is less than 8. Hence 8-s = t{M^-m,)h,<e. But S-s^{S-J) + {J-j) + {j-s); and all the three terms on the right-hand side are positive, and therefore all less than e. As J — j is a constant, it must be zero. Hence j = J and 0^j — s<e, O^S — J<€, as was to be proved. We define the area of PpqQ as being the common limit of s and 8, that is to say J. It is easy to give a more general form to this definition. Consider the sum where /^ denotes the value of f(w) at any point in S„. Then o- plainly lies between s and S, and so tends to the limit / when the intervals B^ tend to zero. We may therefore define the area as the limit of cr. 157. The definite integral. Let us now suppose that/(iB) is a continuous function, so that the region bounded by the curve y =/(«), the ordinates x = a and x = b, and the axis of x, has a definite area. We proved in Ch. VI, § 145, that if F{x) is an ' integral function ' oif(x), i.e. if F'{x)=f{x), F(x) = jf{x)dx, then the area in question is F{b) — F(a). As it is not always practicable actually to determine the form of F{x), it is convenient to have a formula which represents the area PpqQ and contains no explicit reference to F (x). We shall write „ r* iPpqQ)= f(a^)dx. J a 156, 157] ADDITIONAL THEOREMS IN THE CALCULUS 287 The expression on the right-hand side of this equation may then be regarded as being defined in either of two ways. We may regard it as simply an abbreviation for F{b) — F(a), where F (x) is some integral function of /(«), whether an actual formula expressing it is known or not ; or we may regard it as the value of the area PpqQ, as directly defined in § 156. rb The number f(x) dx J a is called a definite integral; a and h are called its lower and upper limits; f{x) is called the subject of integration or integrand; and the interval (a, 6) the range of integration. The definite integral depends on a and h and the form of the function /(«) only, and is not a function of x. On the other hand the integral function F{x)={f{x)dx is sometimes called the indefinite integral of/ (a;). The distinction between the definite and the indefinite integral is merely rh one of point of view. The definite integral / f{x)dx=F{b)-F{a) is a J a, function of 6, and may be regarded as a particular integral function of /(6). On the other hand the indefinite integral F{x) can always be expressed by means of a definite integral, since F(x)=F(a)+rf{t)dt. J a But when we are considering ' indefinite integrals ' or ' integral functions ' we are usually thinking of a relation between two functions, in virtue of which one is the derivative of the other. And when we are considering a ' definite integral ' we are not as a rule concerned with any possible variation of the limits. Usually the limits are constants such as and 1 ; and ^^f{x)dx = F{\)-F(fi) /; ' is not a function at all, but a mere number. It should be observed that the integral / f{t)dt, having a differential J a coefficient /(a:), is a fortiori a continuous function of .».• Since Ifx is continuous for all positive values of x, the investigations of the preceding paragraphs supply us with a proof of the actual existence of the function log.a;, which we agreed to assume provisionally in § 128. 288 ADDITIONAL THEOREMS IN THE CALCULUS [VII Fig. 49, 158. Area of a sector of a circle. The circular functions. The theory of the trigonometrical functions cos x, sin x, etc., as usually presented in text-books of elementary trigonometry, rests on an unproved assumption. An angle is the configuration formed by two straight lines OA, OP; there is no particular difficulty in translating this ' geometrical ' definition into purely analytical terms. The assumption comes at the next stage, when it is assumed that angles are capable of numerical measurement, that is to say that there is a real number x associated with the configuration, just as there is a real number associated with the region PpqQ of Fig. 47. This point once ad- mitted, cos a; and sin a; may be defined in the ordinary way, and there is no further difficulty of principle in the elaboration of the theory. The whole difficulty lies in the question, what is the X which occurs in cos x and sin x1 To answer this question, we must define the measure of an angle, and we are now in a position to do so. The most natural definition would be this: suppose that AP is an arc of a circle whose centre is and whose radius is unity, so that OA = OP = 1. Then x, the measure of the angle, is the length of the arc AP. This is, in substance, the definition adopted in the text-books, in the accounts which they give of the theory of ' circular measure '. It has however, for our present pur- pose, a fatal defect; for we have not proved that the arc of a curve, even of a circle, possesses a length. The notion of the length of a curve is capable of precise mathematical analysis just as much as that of an area; but the analysis, although of the same general character as that of the preceding sections, is decidedly more difficult, and it is impossible that we should give any general treatment of the subject here. We must therefore found our definition on the notion not of length but of area. We define the measure of the angle AOP as twice the area of the sector A OP of the unit circle. Suppose, in particular, that OA is y = and that OP is y = mx, where m > 0. The area is a function of m, which we may denote by ^ {m). If we write fi, for (1 -I- m^) ~ ^ P is the point {im, mjj), and 158] AND INTEGRAL CALCULUS 289 we have (m) = |m/i= + [ VCl - «') dx. J ft Differentiating with respect to m, we find 1 f"^ dt Thus the analytical equivalent of our definition would be to define arc tan m by the equation f" dt arc tan m = : Jo !+<=' and the whole theory of the circular functions could be worked out from this starting point, just as the theory of the logarithm is worked out from a similar definition in Ch. IX. See Appendix III. Examples LXIII. Calculation of the definite from the indefinite integral. 1. Show that /, ai^dx= ; , /•I 1 and in particular that / x"dx = ;■ ^ Jo n+l „ f' , sin mb - sin ma f ' 2. I cosf)ixax= . / J a "1 J a f' dx , , i /"' <^^ 1 3. / :; ;= arc tan 6 — arc tan a, / =— — r,=ii _ I '' J sinm6-sin»ia p. , oosma-coamb 2. I cos'lHxdx= . / siD-mxdx= — , /: [There is an apparent difficulty here owing to the fact that arc tan ^ is a many valued functiou. The difficulty may be avoided by observing that, in the equation I --"., = arc tana;, fol+i!' arc tan x must denote an angle lying between - ^n- and ^tt. For the integi-al vanishes when x=Q and increases steadily and continuously as x increases. Thus the same is true of arc tan ^, which therefore tends to \n as a;-*-x. In the same way we can show that arctano;-*-— Jtt as a-*- — cc. Similarly, in the equation dt — arc sin x, i: 'on/(1-<^) where — 1<;^<1, arc sin a? denotes an angle lying between — ^jt and Jn- Thus, if a and b are both numerically less than unity, we have dx /: x/(l-^) dx = arc sin 6 — arc sin a.] P dx ^ tiT /"»_ H, 19 290 ADDITIONAL THEOREMS IN THE DIFFERENTIAL [vil dx =■ /;r ; — a — r, ■ if -7r<a<7r, except when a=0, when the L+2^cosa+a;2 2 sin a Value of the integral is \, which is the limit of ^a ooseca as a-»-0. 6. [ J{l-x'^)dx=ln, ry/{a^-x')dx=iird^ (a>0). jo 7 7. I " y— — = ,, r i.9> , if a > 1 6 1. [For the form of the indefinite j a+b cos X ij(cfi-b^)^ integral see Exs. Liil. 3, 4. If | a |<| 6 1 then the subject of integration has an infinity between and tt. What is the value of the integral when a is negative and - ct>| 6 1 ?] --: s ,„ . o = ^^ , if a and 6 are positive. "What is the a^cos^iK + o^sm^a; 2ao value of the integral when a and 6 have opposite signs, or when both are negative ? 9. Fourier's integrals. Prove that if in and n are positive integers then (■2n- /, /, cos mx sin nx dx is always equal to zero, and /cos mx cos nx dx, I sin mx sin nx dx Jo are equal to zero unless m=n, when each is eqvial to tt. 10. Prove that I co%m.x.cosnxdxa,ndi I sin «i^ sin raa; <& are each equal Jo Jo to zero except when 'm=n, when each is equal to ^n ; and that I cos mx sin nx dx = -= :, , | cos mx sin nx dx = 0, Jo n^-m^' Jo according as »i — m is odd or even. 159. Calculation of the definite integral from its defini- tion as the limit of a sum. In a few cases we ca,ii evaluate a, definite integral by direct calculation, starting from the definitions of §§ 156 and 157. As a rule it is much simpler to use the indefinite integral, but the reader will find it instructive to work through a few examples. fh Examples LXIV, 1. Evaluate / xdx by dividing {a, V) into n equal J a parts by the points of division a=x^, x^, x^, ..., Xn—h, and calculating the limit as TO-*- 00 of ' («i - ^o) / K) + (*^2 - ^i) f{x^^...-V K - %- 1)/ K - 1)- 158-160] AND INTEGRAL CALCULUS 291 [This sum is -'-?[-+'-?<'+^+-+("->»]-('— )f+<»-)^~>}. which tends to the limit | (b^-a^) asn-^as. "Verify the result by graphical reasoning.] fb 2. Calculate / x^dx in the same way. J a ■ \ 3. Calculate I xdx, where 0<a<6, by dividing (a, b) into n parts by the points of division a, ar, af', ... ar^~\ ar", where r^=b/a. Apply the same method to the more general integral / x'" dx. J a fb • fb 4. Calculate / cos mx dx and | sin mx dx by the method of Ex. 1. J a J a -1 1 ^ -5— [This follows from the fact that 5. Prove that n 2 —r, i-*-i7r as ?i-»-oo . ?i2^m2+1^ »H(?l-l)2 r=0 1-1-(W»)^' i -„ as K -*- 00 , in virtue of the direct definition of the integral.] 6. Prove that -„ "s ' V(«^ - 1^) ^ i^- [The limit is / V(l - «^) o?'»-] 'I r=0 J 160. Creneral properties of the definite integral. The definite integral possesses the important properties expressed by the following equations.*. (1) [" f{x)dx = -\''f{x)dx. J a J b This follows at once from the definition of the integral by means of the integral function F{x), since F{b)-F{a)= -{F(a)-F{b)}. It should be observed that in the direct definition it was presupposed that the upper limit is greater than the lower ; thus this method of definition does not apply to the integral I f(x) dx when a<b. If we adopt this definition as fundamental we must extend it to such cases by regarding the equation (1) as a definition of 'its right-hand side. * All functions mentioned in these equations are of course continuous, as the definite integral has been defined for continuous functions only. 19—2 292 ADDITIONAL THEOKEMS IN THE DIFFERENTIAL [VH (2) rf(a;)dx=0. J a (3) ['/(a,) dx + {"fix) dx = {'fix) dx. J a J b J a (4) f kf{x)dx = k\ f{x)dx. (5) f {/(«) + <l> (<«)] dx=f f(x) dx+[ (f) (x) dx. J a J a •'a The reader will find it an instructive exercise to write out formal proofs of these properties, in each case giving a proof starting from (a) the definition by means of the integral function and (/3) the direct definition. The follow^ing theorems are also important, (6) Iff{x) S when a&x&h, then ( f(x) dx > 0. J a We have only to observe that the sum s of § 156 cannot be negative. It will be shown later (Misc. Ex. 41) that the value of the integral cannot be zero unless /{x) is always equal to zero ; this may also be deduced from the second corollary of § 121. (7) If H S:f{x)^K when a&x&h, then H(b-a)^[ f{x)dx&K{h-a). J a This follows at once if we apply (6) to/(a;) -.ff and K-f{a;). (8) \''f{'^)dx = {h-a)fi^), J a where ^ lies between a and h. This follows from (7). For we can take H to be the least and K the greatest value of f{x) in (o, 6). Then the integral is equal to j; (6 - a), where Tj lies between H and K. But, since f{x) is continuous, there must be a value of I for which /(^) = i; (§ 100). liF{x) is the integral function, we can write the resiUt of (8) in the form F{h)-F{a) = {h-a)F'{^), so that (8) appears now to be only another way of stating the Mean Value Theorem of § 125. We may caU (8) the First Mean Value Theorem for Integrals. 160] AND INTEGRAL CALCULUS 293 (9) The Generalised Mean Value Theorem for inte- grals. If <j) (x) is positive, and H and K are defined as in (7), then h{ <l>{x)dx& I f{x)(j){x)dx^Kl (j>(x)dx; J a J a J a and I f(x)(j){x)dx=f{^) I (j>(x)dx, J a J a where ^ is defined as in (8). This follows at once by applying Theorem (6) to the integrals f \f{x) -H}^ (*•) do!, f \K-f{x)} <i> {X) dx. J a J a The reader should formulate for himself the corresponding result which holds when <^ {x) is always negative. (10) The Fundamental Theorem of the Integral Cal- culus. The function Fix)=rfit)dt J a has a derivative equal to f{x). This has been proved already in § 145, but it is convenient to restate the result here as a formal theorem. It follows as a corollary, as was pointed out in § 157, that F {x) is a continuous function of x. Examples LXV. 1. Show, by means of the direct definition of the definite integral, and equations (1) — (5) above, that (i) i" ^{x^)dx^^l"'^{x^)dx, l" x^{x'^)dx=0; J -a Jo J -a /Jit r Jtt f IT cj) (cos x) dx— (fi (sin x)dx=^ \ (j) (sin .3;) dx ; Jo Jo (iii) ( (j) (cost x)dx=m I (j) (coa^ x) dx, Jo Jo m being an integer. [The truth of these equations will appear geometrically intuitive, if the graphs of the functions under the sign of integration are sketched.] 2. Prove that / ^ dx is equal to tt or to according as n is odd or J sin X an. [Use the formula (sinraii;)/(sin.j;) = 2 cos{(n — 1)^} + 2 cos{(?i-3)^} + ..., 3 last term being 1 or 2 cos x.'\ 3. Prove that I sin ?m; cot ^ote is equal to or to tt according as n is odd Jo or even. 294 ADDITIONAL THEOREMS IN THE DIFFERENTIAL [vil 4. If (j) {x) = cto + «! COS X + fti'sin » + aj cos 2^ + . . . + a„ cos nx + 5„ sin n«, ■and ^ is a positive integer not greater than n, then /■ 2)7 /■ 2"- /■ 2ir I 0(:!;)rf^=27rao, I .cos]cx<l>{x)d«—Traji,, I sinte(^ (^)(fa'=n-fej. If /i;>»i then the value of each of the last two integrals is zero. [Use Ex. LXiii. 9.] 5. If <^{x)=ao + ai00sx + a20oa'ix+...-\-an00snx, and h is a positive integer not greater than m, then I (j}{x)clx=nao, I coskxct)(x)cb;=^Tra,:. Jo Jo lik>n then the value of the last integral is zero. [Use Ex..lxiii. lO.j /* 2"" cdx 2ir 6. Prove that if a and h are positive then I --s — -= — , ,„ ■ „ =-y- [Use Ex. LXIII. 8 and Ex. 1 above.] . . 7. lif{x)^^{x) when a <^g6, then / fdx £, I :(j>dx. . .,. , ,■.,.„ -.1 J a J a 8. Prove that ; ; •; </■''' sin» + i.j;dr < / ''sin".rrf^, 0< I tanVi^c?^<, I tm''xdx.. jo y j'o ■ ' '1 '. JO :'. . • -— jSS < '^^- [The first iueftu j,lity -fpUows from the fact that V(l - «^'')< 1, the second from the fact that ' ' /I (^^ ' " ' . 11. Prove that {3x+8)ll6<lls/(4:-Sx + x^)<ll^{i-2x) if 0<:»<1, and hence that ^| < / _ -^^^-^-^-^ < 12. Prove that -573 < ,,, T^ , ■^ <-595. [Put :s=H-m: then re- place 2 + 3m2+m3 by 2 + 4^2 and by 2 + 3^2.] 13. If a and are positive acute angles then U dx ^ (t> Jo n/(1 -sin2asin2.j;) ^(1 — sin^gsin^^)* If a=0 = ^7r, then the integral lies between '523 and •541. 14. Prove that f{x)dx\sl \f{x)\dx. ] J a I J a [If n- is the sum considered at the end of § 156, and o-' the corresponding sum formed from the function \f{x) \, then | o- 1 < o-'.] 15. li\f{x)\^M',then I f{x)cf,{x)dx\sif \<j}{x)\dx. ' J a ■ ' J a * Exs. 9—13 are taken from Prof. Gibson's Elementary Treatise on the Calculus. 160, 161] AND INTEGRAL CALCULUa 295 161. Integration by parts and by substitution. It follows from § 138 that f /(*) f (*) dx =f{h) 4> (b) -f(a) 4>{a)-( V' {<^) <i> {.«=) dx. ■I a J a This formula is known as the formula for integration of a definite integral by parts. Again, we know (§ 133) that if F(t) is the integral function of f(t), then \f{<f>(x)}<l,'(x)dx = F{4>(x)}. h Hence, if ij) (a) = c, (j) (b) = d, we have f V(0 dt=F(d)-F(c) = F{cf> (6)} -F{<j, (a)}= f /{<^ (*)} f («>) dx; J c J a which is the formula for the transformation of a definite integral by substitution. The formulae for integration by parts and for transformation often enable us to evaluate a definite integral without the labour of actually finding the integral function of the subject of integra- tion, and sometimes even when the integral function cannot be found. Some instances of this will be found in the following examples. That the value of a definite integral may sometimes be found without a knowledge of the integral function is only to be expected, for the fact that we cannot determine the general form of a function F(x) in no way precludes the possibility that we may be able to determine the difference F(b) — F(a) between two of its particular values. But as a rule this can only be effected by the use of more advanced methods than are at present at our disposal. Examples LXVI. 1. Prove that j\f" {x) dx={bf (6) -/(6)} - {af {a) -f{a)}. 2. More generally, / .'K"/(™ + ')(a;)(£^=i'''(6)-i^(a), where i?'(^-) = ^"'/<"''(^) -»»■*''"" V'""''^.+ »i(™-l)*'"'~V'"'~^' ■»-•■• + (-1)'" »»'/(.■»)■ 3. Prove that /arcsina!i3?.j;=^7r — 1, I xa.vais.nxdx;=\ir — \. Jo 296 ADDITIONAL THEOREMS IN THE DIFFERENTIAL [vil 4. Prove that if a and h are positive then 'i" xcce,xs).vi.xdx n /.' fo {cficos^x + b'^sin^x)^ 4ab^{a + b)' [Integrate by parts and use Ex. lxiii. 8.] 5. If Mx)=jy{t)dt, f2{^)=jyi{t)dt,...,Mx)=jy,.,{t)dt, then f,(x)= ^^^y^ J 7(0 (x-tf-^dt. [Integrate repeatedly by parts.] 6. Prove by integration by parts that if u„^n= j ^- (1 - xfdx, where m and n are positive integers, then (m+w + 1) «OT,ii = 't^'in,7i-i) and deduce that (m + ra + l)!' 7. Prove that if t«„= I tan».^•&■ then «„ + m„_2 = !/(»- !)• Hence 1 evaluate the integral for all positive integral values of n. [Put tan" x=\,s,-ti!^~'''x{s,%(?x-\) and integrate by parts.] 8. Deduce from the last example that m„ lies between l/{2 (» — !)} and l/{2(» + l)}. 9. Prove that if M„= I sin"^cfe then «„= {(ra -!)/«} m„_2. [Write } sin""' « sin .j: for sin"^ and integrate by parts.] 10. Deduce that «„ is equal to 2.4.6..(?i-l) , 1. 3.5..(?t-l) 3.5. 7. .n ' ^'^ 2.4.6..W ' according as n is odd or even. 11. The Second Mean Value Theorem. If /(^) is a function of x which has a differential coeflScient of constant sign for all Vcalues of x from «=a to ^=6, then there is a number ^ between a and h such that rf{x)(l)(x)dx=f{a)l^(t>{x)dx+f(b) l^ ^{x)dx. [Let r(j>{t)dt = ^(x). Then [ f{x) (*) rf.r= \ f{x) *' (^) rf^=/(6) ^{b)- r f (x) * (^) c?4' J a J a J a =/(6)*(6)-*(^) f'/'W^*-, by the generalised Mean Value Theorem of § 160 : i.e. ^f{x)4> (x) dx=f{b) * (6) + {/(a) -f{b)] * (I), /: wjiich is equivalent to the result given.] 161] AND INTEGRAL CALCULUS 297 12. Bonnet's form of the Second Mean Value Theorem. If/' {x) is of constant sign, and/ (6) and /(a) -/(6) have the same sign, then I /(^) W dx=f{a) / " <^ {x)dx, J a J a where X hes between a and ^. [For f{b)^(b) + {f(a)-f{b)}^{i) = nf{a), where fi lies between * (^) and * (6), and so is the value of * (x) for a value of ^ such as X. The important case is that in which 0^f(b)^{x)^f{a).] Prove similarly that if /(a) and/ (6) -/(a) have the same sign, then jj{x) 4, {X) dx=f{b) r,#, (x) dx, where X lies between a and b. [Use the function * (|) = / (^ (^) dx. It will be found that the integral can be expressed in the form /(«)^(«) + {/W-/(«)}*(l)- The important case is that in which Og/((x)g/(ii;) </(6).] 13. Prove that! f^?^cf^ <^ if X'>X>0. [Apply the first formula of Ex. 12, and note that the integral of sin x over any interval what- ever is numerically less than 2.] 14. Establish the results of Ex. lxv. 1 by means of the rule for sub- stitution. [In (i) divide the range of integration into the two parts {—a, 0), (0, a), and put x= —y in the first. In (ii) use the substitution x=\ir — y'M obtain the first equation : to obtain the second divide the range (0, tt) into two equal parts and use the substitution x=\Tv'ry. In (iii) divide the range into m equal parts and use the substitutions .j;=7r-f y, x='in-^y, ....] /■ft (h 15. Prove that I P{x)dx=\ F{a + b-x)dx. J a J a fiir /■iir 1 6. Prove that / cos™ xsm'^xdx=2-'^ I coB'^xdx. Jo Jo 17. Prove that / x<l}(smx)dx=iir I <p{ainx)dx. [Put^=7r— y.] Jo Jo 18. Prove that f" -^-^?-f- dx=in\ Jo l-|-COS''ar 19. Show by means of the transformation x^acos^ 6 + bam^ 6 that "^{{x -a){b-x)}dx=i7r(b- af. L 20. Show by means of the substitution {a + bcosx) {a -b con y) = a' — b^ that /"''(a+*cos«)-"«fo;=(a2_62)-(n-i) f ' (a -b cos y)''-^dy, when 71 is a positive integer and a>\b\, and evaluate the integral when n=l, 2, 3. 298 ADDITIONAL THEOREMS IN THE DIFFERENTIAL [vu 21. 11' TO and M are positive integers then f''(^-a)'»(6-;B)"c?^=(6-a)"' + "^i '"''^L. . J a. (TO+m+l)!' [Put 30=a + {b — a)y, and use Ex. 6.] 162. Proof of Taylor's Theorem by Integration by Farts. We shall now give the alternative form of the proof of Taylor's Theorem to which we alluded in § 147. Let/(a;) be a function whose first n derivatives are continuous, and let Fn {x) =f{b) -fix) - (6 - x)f' (x)-...- ^(;"J"i)7 /'"'" H- Then F,: (x) = - ^^~_''f)7 /'"' (*>' and so Fr,(a) = Fr,(b)-rFn'(x)dx=^-l—r{b-xr-^fir')(x)dx. If now we write a + h for b, and transform the integral by putting x = a + th, we obtain f{a + h) =f{a)+hf'(a) + ... + (^^j/'"-" (a) + -R« • • -(1), where E„ = , ^" , C (1 - <)"-i/(«> (a + th)dt (2). (n — 1)1.1 Now, if p is any positive integer not greater than n, we have, by Theorem (9) of §160, [\l - «)"-'/'"' (a + iA.) dt= [ (1 - 0"-^(l - 0^~'/'"' (a + th) dt Jo -0 . = (1 - 6l)"-^/(") (a + 61^) f \l - ty-'dt, Jo where < < 1. Hence (i-g)"-i'/(»)(c^ + gA);^" -""- p(n-l)! *^^^- If we take p = 7i we obtain Lagrange's form of i?,, (§ 148). If on the other hand we take j3 = 1 we obtain Cauchy's form, viz. (l-^)"-^/'")(a+gA)fe" ^" Oi^^l)! -^ ^ " * The method used in § 147 can also be modified so as to obtain these alternative forms of the remainder. 16J.-164] ..... AND INTEGRAL CALCULUS 299 163. Application of Cauchy's form to the Binomial Series. If f{x) = {\ + xy^, where m is not a positive integer, then Cauchy's form of the remainder is „ _ m(m-l)...(m-w + l) (l-5)"-iir" " 1.2...(m-1) (l + to)"-"' Now {\ — d)!{\ + 6ic) is less than unity, so long as -l<,:r<l, whether X is positive or negative; and (l + te)™-! is less than a constant K for all values of n, being in fact less than (l + l^i)'""* if m>l and than (l-l^rD^-Mf m<l Hence \Rn\<K\m\ (::;) say But p^-^O as n-»-oo , by Ex. xxvii. 13, and so iJ„-*-0. The truth of the Binomial Theorem is thus established for all rational. values of m and all values of x between — 1 and 1. It will be remembered that the difficulty in using Lagrange's form, in Ex. lvi. 2, arose in connection with negative values oi X. 164. Integrals of complex functions of a real variable. So far we have always supposed that the subject of integration in a definite integi-alis real. We" define the integral of a complex function f {x) = 9{x) + iyjr (so) of the real variable a;,- between the limits a and b, by the equations rb rb rb _ rb f{x)dx=\ [j> {x) + i^ {x)] dx = \ <^{x)dx + i\ ■<^{x)dx; J a J a J a J a and it is evident that the properties of such integrals may be deduced from those of the real integrals already considered. There is one of these properties that we shall make use of later on. It is expressed by the inequality (''f(x)dx ^f\f{x)\dx (1)^ This inequality may be deduced without difficulty from the definitions of §| 156 and 157. If 8„ has the same meaning as in I 156, (f>y and i^, are the values of <f) and ilr at a point of 8„, and f„ = 4>u + i'^>', then we have I fdx=\ (jidx + il ilrcia;— limS<^^S„ + ilim 2-v/f„S^ J a J a •' 1 = lim 2 (</)„ + i'^v) S„ = lim 2/^fi^, rb and so I fdx - \ lim 2/„S„ | = lim | 2/^S„ | ; J a * The corresponding inequality for a real integral was proved in Ex. lxv. 14. 300 ADDITIONAL THEOREMS IN THE DIFFERENTIAL [vil rb while I |/|c?ir = lim S |/„ I S„. J a The result now follows at once from the inequality |2/,S.|g2|/„|8.. It is evident that the formulae (1) and (2) of § 162 remain true when /is a complex function <]} + i-t{r. MISCELLANEOUS EXAMPLES ON" CHAPTER VII. 1. Verify the terms given of the following Taylor's Series : (1) t3.nx = x + ^x^+^afi + ..., (2) sec^=H-i^H-5^^+..., (3) xcosecx=l-[-ix^ + g^x'^+..., (4) xcotx=l — ^x^ — -^x^—.... 2. Show that if f(x) and its first k + 2 derivatives are continuous, and y()i + i) (0)=(=0, and 6„ is the value of d which occurs in Lagrange's form of the remainder after n terms of Taylor's Series, then _J_ n f /(" + q(0) I "~m+l'^2(» + l)^(« + 2) l/("-"')(0)'*'^"=J ^' where Ca, -»- as x-^0. [Follow the method- of Ex. lv. 12.] 3. Verify the last result when / (a;) = 1 /( 1 + x). [Here (1 + fl^:!;)" + 1 = 1 +.r.] 4. Show that iif{x) has derivatives of the first three orders then f{b) =/(a) +i (6 - a) {/' (a) +/' (6)} -^{b- aff" (a), where a<a<h. [Apply to the function / W -/(a) - i (^ - «) {/' (») +/' (^)} - (f£-y [/(*) -/(«) - K& - «) {/' («)+/' (6)}] arguments similar to those of § 147.] 5. Show that under the same conditions f{b)=fia) + {b-a)f' a [a + b)} + ^{h-aff"{a). 6. Show that i(f{x) has derivatives of the first five orders then /(6)=/(a)+^(6-a)[/' («)+/' (6) + 4/'{i(a + 6)}]-5g»5i5(6-a)»/("(a). 7. Show that under the same conditions f(b) =f{a) + i(b-a){f'ia)+f{b)}-^,(b-ay {/" (6) -/" (a)} + ^i^{b - afp)(a). AND INTEGRAL CALCULUS 8. Establish the formulae 9 {a) 9Q>) where /3 lies between a and b, and 301 (i) =(6 -a.) (ii) /(«) /(6) /(c) 9 {a) g{h) g{c) h{a) h{b) hie) -\{b-c){c-a){a-h) /(«) Z'O) /"(v) g{a) g'{l3) g" {y) h{a) A'(/3) A"(v) where /3 and y lie between the least and greatest of a, 6, c. [To prove (ii) consider the function 0(^) = /(«) /(6) /(^) 9{o) g{b) g{x) h(a) h{h) h{x) (x — a) (x — b) (c-a) (c—b) /(«) fib) f(c) 9 {a) gib) 9i") h{a) hih) h{c) which vanishes when x=a,x=b, and x=c. Its first derivative, by Theorem B of § 121, must vanish for two distinct values of x lying between the least and greatest of a,b, c; and its second derivative must therefore vanish for a value 7 of iS satisfying the same condition. We thus obtain the formula /(a) fiV) fie) 9ia) 9ib) gie) h{a) h{b) h{c) = ^{o-a){c-b) /(«) fib) f'iy) 9i") 9ib) 9"iy) h{a) h{b) h"{y) The reader will now complete the proof without difficulty.] 9. If F(x) is a function which has continuous derivatives of the first n orders, of which the first n—1 vanish when x = 0, and A ^ /"(") {x) ^ B when S,x^h, then A {x'^jn \)^F{x)^B {x'^jn \) when g .r < h. Apply this result to /(^)-/(0)-^/'(0)-" and deduce Taylor's Theorem. 10. If A^<^(^) = </)(^)-<#)(.v + ^), Aft2^(a;) = A;.{A^(^(.^;)}, and so on, and (^ {x) has derivatives of the first n orders, then {n-\)\ /(»-!) (0), A*"<#>(^)= S(-l) (^(^+?-A) = (-A)"<^(")(^), where ^ lies between x and x+nh. Deduce that if ^W {x) is continuous then {Ai"rf)(ir)}/A"-*-( — 1)"0("'(*) as A-»-0. [This result has been stated already when w=2, in Ex. lv. 13.] 11. Deduce from Ex. 10 that x''-'^Ah''af^^-m{m-\) ...{m-n + \)h^ as j; ^3- 00 , m being any rational number and n any positive integer. In Ijarticular prove that x^xyx-'2.sJix + \)+^ix + %)}^-^. 3fl2 ADDITIONAI, THEOREMS IN THE DIFFERENTIAL [vil 12. Suppose that y = (j){x) is a function of a with continuous derivatives of at least the first four orders, and that (0) =0, <^' (0) = 1, so that where Ej^ -a- as a;-»-0. Establish the formula .r = f (y) =2/ - a^y^ + {%a^ - Oj) y^ - (ba^ - Saj oss + a4 + f ,,) j^, where Cj, -»- as y-*-0, for that value of x which vanishes with y; and prove that — as-:r-^0. ■ ■ 13. The coordinates (§, rj) of the centre of curvature of the ciure io=f{t), y = F{t), at the point {x, y), are given by - (I - ^)/y = (-? -y)/^' = i.^" +y'^Wy" - ^Y) ; and the radius of curvature of the curve is dashes denoting differentiations with respect to f. 14. The coordinates (^, if) of the centre of curvature of the curve 27a2/2=4a^, at the point (a;, y), are given by 3a(|+a;) + 2^2^0, r,=iy + (Siay)jx. {Math. Trip. 1899.) 15. Prove that the circle of curvature at a point {x, y) will have contact of the third order with the curve if {l+y-^)y3 = ^yiy^ at that point. Prove also that the circle is the only curve which possesses this property at every point ; and that the only points on a conic which possess the property are the extremities of the axes. [Cf. Ch. VI, Misc. Ex. 10 (iv).] ' 16. The conic of closest contact with the curve y = ax'^ + bjfi + Cis* + . . . + &", at the origin, is a?y = a'^x'^-'ra%xy-\-{ac-h'^)y'^. Deduce that the conic of closest contact at the point (|, rj) of the curve y=f(jx) is \Sr,iT^Q,,t^{x-k? + Qr,ir,ilx-^) ^+(3,274-^) T\ where T={y-r,)—ni{x-k). {Math. Trip. 1907.) 17. Homogeneous functions.* If u=x'^f{ylx, zjx, ...) then u is un- altered, save for a factor \", when x,y,z, ... are all increased^in the ratio A ; 1. In these circumstance? u is called a homogeneous function of degree n in the variables x, y, z, .... Prove that if u is homogeneous and of degree n then 'du :du du This result is known as Euler's Theorem on homogeneous functions! 18. If w is horhogeheous and of degree n then dujdx, dufdy, ... are homogeneous and of degree n—1. * In this and the following examples the reader is to assume the continuity of all the derivatives whioh-ocour. AND INTEGRAL CALCULUS 3oa 19. Let /(a;, y)=0 be an equation in x and y (e.g. ai^ + i/^-x=0), and let F{x, y, 2)=0 be the form it assumes when made homogeneous by the intro- duction of a third variable z in place of unity (e.g. ic"-!-^"— ^2''~i = 0). Show that the equation of the tangent at the point (^, ?;) of the curve/(^, y) = is xFi+yF,,+zFi=0, where Fi, Fr,, F( denote the values oi F^, Fy, F^-whea x^^, y=rf, z=i^ X. 20. Dependent and independent functions. Jacobians or functional determinants. Suppose that u and v are functions of x and y connected by: an identical relation _ , <i>('u,v)=Q (1). Difierentiating (1) with respect to a;, and y, we obtain 3<^ OM 9c^ dv du dx dv dx ' and, eliminating the derivatives of <j), 30 du 30 3*' _ f^ du dy dv dy .(?)o J= •(3), where u^, Uy, v^, Vy are the derivatives of u and v with respect toia; andy. This condition is therefore necessary for the existence of a. relation such as (1). It can be proved that the condition is also sufficient ; for this we mvist , refer to Goursat's Cours d! Analyse, vol. i, pp. 125 et seq. Two functions u and v are said to be dependent or independent according as they are or are not connected by such a relation as (1). It is usual to call J the Jacobian or functional determinant of u and v with respect to x and y, and to write jjd (u, v) d(x,y)' Similar results hold for functions of any number of variables. Thus three functions u, v, w of three variables x, y, z are or are not connected by a relation 0(m, V, w)=0 according as , u^ d (u, V, w) does or does not vanish for all values of x, y, z. 21. Show that ax^ + 'i.hxy-irby^ and Ax^'-^'HIxy^rBy''' are independent mi\essalA=/ilS=b/B. 22. Show that ax'^ + by'' + cz^ + 2fyz +2gzx +2hxy oa.n be expressed as a product of two linear functions of x, y, and z if and only if ahc+^fgh- af - 65-2 - ch? = 0. [Write down the condition that px+qy-\-rz and p'x+^y + r'z should be connected with the given function by a functional relation.] 304 ADDITIONAL THEOREMS IN THE DIFFERENTIAL [vn 23. If u and v are functions of ^ and ;;, which are themselves functions of X and y, then 8 («, '«) ^ 3 (m. «) 8 (I, v) 8 (^, y) 8 (I, ij) 3 (^, y) ' Extend the result to any number of variables. 24. Let/(^) be a function of x whose derivative is \lx and which vanishes when x=l. Show that if u=f{ic)+f{y),v=xy,ih.&a «iS;„-m„i)j,=0, and hence that v, and v are connected by a functional relation. By putting y=l, show that this relation must 'bef{x)+f(y)=f{xy). Prove in a similar manner that if the derivative oi f{x) is lj{\+x^), and /(0)=0, then f{x) must satisfy the equation /(-)+/(y)=/(Sj). 25. Prove that if / (^) = f " „ "' ^,, then 26. Show that if a functional relation exists between '*=/W+/(y)+/W, v=f(y)f{z)+f{z)f{x)+f{x)f(j,), w=f{x)f(j!)f{z), then / must be a constant. [The condition for a functional relation will be found to be /'W/'(y)/'W{/(y)-/(^)l{/(^)-/(^)}{/W-/W}=o.] 27. If fii/, z), f{z, x), and f{x, y) are connected by a functional relation then fix, x) is independent of x. [Math. Trip. 1909.) 28. If M=0, ■j) = 0, «;=0 are the equations of three circles, rendered homogeneous as in Ex. 19, then the equation 8 (^, y, z) represents the circle which cuts them all orthogonally. {Math. Trip. 1900.) 29. If A, B, C are three functions of x such that A A' A" B B B" C C G" vanishes identically, then wecan find constants X, fi, v such that \A +fiB + vC vanishes identically ; and conversely. [The converse is almost obvious. To prove the direct theorem let a = BC'-B'C, .... Then a' = BC"-B"0, ..., and it follows from the vanishing of the determinant that ^y' — ffy=Q, ... ; and so that the ratios a : |3 : y are constant. But aA +;35 + -yC=0.] 30. Suppose that three variables x, y, z are connected by a relation in virtue of which (i) 2 is a function of x and y, with derivatives Zx %> ^^^ (") ^ is a function of ^ and z, with derivatives Xy, x^. Prove that Xy= ~ ZyjZgy ^*j = I /^a, . AND INTEGRAL CALCULUS 305 [We have dz =Zxdx-\- Zy dy, dx =Xydy-{- x, dz The result of substituting for dx in the first equation ia dz=(ZxXy-^Zy) dy + ZxX^dz, which can be true only \iz^Xy-\-Zy = 0, 0^.^^=!.] 31. Four variables x, y, z, u are connected by two relations in virtue of which any two can be expressed as functions of the others. Show that y^z^Xy^= -2//«/V=l. ««"2/+2/b"V=1, where y^ denotes the derivative of y, when expressed as a function of z and jt, with respect to z. {Math. Trip. 1897.) 32. Find A, B, (7, X so that the first four derivatives of fit) dt-x [Af{a) + Bf{a + ~Kx) + Cf{a^ x)] J a vanish when x=0 ; and A, B, C, D, X, /x so that the first six derivatives of ra+x I f{t) dt-xiAf{a)+Bf{a+'Kx) + Cf{a+,,j;)+Df{a+x)'\ J a vanish when x = 0. 33. If a > 0, aa — lfl> G, and Xi> Xo, then /, ^•? = 1 arc tan [ (^-i-^o)x/(«c-5') } -^^ ax^+2bx + G iJ{ac — V') \aa;i^o + & (■^i+^o) + <^J ' the inverse tangent lying between and tt.* 34. Evaluate the integral | = — = '- 7. For what values of o is ° y _i l-2^C0Sa + a"' the integral a discontinuous function of a ? {Math. Trip. 1904.) [The value of the integral is Jn- if 2«;r <a<(2«. + l) tt, and --^n- if (2)1 - 1) TT < a < 2n7r, n being any integer ; and if a is a multiple of n-.] 35. li ax'^+ibx + oO when Xo^x&Xi,f{x) = ^{ax'^ + %x+c), and y=f{x), 2/o=/W. yi=/(^i), -^'=(^i-^o)/(yi+yo). according as a is positive or negative. In the latter case the inverse tangent lies between and ^tt. [It will be found that the substitution X ^ sc I dt t= -" reduces the integral to the form 2 ; -..l y+yo Jo !-««' /a d^ ^ ., a i^ = i■^■ i^ath. Trip. 1913.) x+ ^i(a —X) 37. Ifa>lthen T '^^^£|5c?^=^{a_^(a2-l)}. * In connection with Exs. 33—35, 38, and 40 see a paper by Dr Bromwioh in vol. XXXV of the Messenger of Mathematics. H. 20 306 ADDITIONAL THEOREMS IN THE DIFFERENTIAL [vil 38. If^>l, 0<2'<1, then fi dx 1m i: 'ov/[{l+(p''-l)4"{l-a-?')4] {p + q)smco' where <» is the positive acute angle whose cosine is (1 +pq)Kp + q). 39. If a > 6 > 0, then ^ b = TTt {» - V(«^ - b^)}- {Math. Trip. 1904.) 40. Prove that ita>J{b^+o^) then [" de _ 2 fv/(a2-63_c2)] j„ a + 5cos5 + <;sin5"V(«'-63-c2)^'"'=^''''\ c J' the inverse tangent lying between and n. 41. lif{x) is continuous and never negative, and I f{x)dx=0, then 7 a f(x) = for all values of .» between a and h. [If/ (a;) were equal to a positive number /i; when x==^, say, then we could, in virtue of the continuity oi f{x), find an interval (^-S, 1 + 8) throughout which /(a-) >p ; and then the value of the integral would be greater than 8^.] 42. Schwarz's inequality for integrals. Prove that ([ ^y\rdx\ < I 4>'^dx i ^dx. [Use the definitions of §§ 156 and 157, and the inequahty (2<^,^/',8,)2g2<^,^8,2f,^8„ (Ch. I, Misc. Ex. 10).] 43. If Pn (x) - (g _ ^)n „ 1 {ff {(^ - °) (^ - ^)} "' then P^{x) m a poly- nomial of degree n, which possesses the property that /: Pn{x)e(x)d^fi=0 if 6{x) is any polynomial of degree less than n. [Integrate by parts m + 1 times, where m is the degree of 6 {x), and observe that 6^™*'^) (^) = 0.] 44. Prove that I P,„ («) P„ (x) dx = it m=^ n, but that if m = n then the value of the integral is (/3 - a)/(2n + 1). 45. If Qn {x) is a polynomial of degree m, which possesses the property that I §^ (ir) 6 {x) dx=0\i 6 {x) is any polynomial of degree less than n, then Qn {^) is a constant multiple of P„ (a;). [We can choose k so that §„— kP„ is of degree n-l: then f Qn{Qn-<Pn)dx = 0, j ^P^{Q„- kP^) dx = 0, AND INTEGRAL CALCULUS 307 and so I {Qn-KP„)^d3; = 0. Now apply Ex. 41. J 46. Approximate Values of definite integrals. Show that the error fb in taking ^{b — a){4)(a) + (j){b)} as the value of the integral / <p (x) dx is less J a than ^M(f> — aY, where Mm the maximum of | <\)"{x) \ in the interval (a, h) ; and that the error in taking (b — a)(j) {^ (a + b)} is less than ^M (6 - a)'. [Write /' {x) = {x) in Exs. 4 and 5.] Show that the error in taking H6 - ») [<^ («) + <#. (6) + 40 {H« + 6)}] as the value is less than -^jsM {b - aY, where M is the maximum of c^W (.r). [Use Ex. 6. This rule, which gives a very good approximation, is known as Simpson's Bule. It amounts to taking one-third of the first approximation given above and two-thirds of the second.] Show that the approximation assigned by Simpson's Rule is the area bounded by the lines x=a, x=b, y=0, and a parabola with its axis parallel io OY and passing through the three points on the curve y=<^{x) whose abscissae are a, \{a-\r b), b. It should be observed that if (/> {x) is any cubic polynomial then (^(*) {x) — 0, and Simpson's Rule is exact. That is to say, given three points whose abscissae are a, ^(a + b), b, we can draw through them an infinity of curves of the type y^a + fix + yx^ + dx^ ; and all such curves give the same area. For one curve S = 0, and this curve is a parabola. 47. If (j) (x) is a polynomial of the fifth degree, then r ct,{x)dx = ^{5cl,{a) + 8<j>{i) + 5<t, (/3)}, J a and /3 being the roots of the equation x^-x + ^=0. {Math. Trip. 1909.) 48. Apply Simpson's Rule to the calculation of ir from the formula /•I (£^ ^TT = I " ., . [The result is '7833.... If we divide the integral into two, from to ^ and J to 1, and apply Simpson's Rule to the two integrals separately, we obtain -7853916.... The correct value is •7853981....] 49. Show that 8-9 < j" y/{4: + x^)dx<9. (Math. Trip. 190^ 50. Calculate the integrals to two places of decimals. [In the last integral the subject of integration is not defined when ^-=0 : but if we assign to it, when x=0, the value 1, it becomes continuous throughout the range of integration.] 20—2 CHAPTER VIII THE CONVERGENCE OF INFINITE SERIES AND INFINITE INTEGRALS 165. In Ch. IV we explained what was meant by saying that an infinite series is convergent, divergent, or oscillatory, and illustrated our definitions by a few simple examples, mainly derived from the geometrical series \+x + x^-\- ... and other series closely connected with it. In this chapter we shall pursue the subject in a more systematic manner, and prove a number of theorems which enable us to determine when the simplest series which commonly occur in analysis are convergent. We shall often use the notation n m 00 and write 2m,i , or simply Sif^ , for the infinite series «„ + Mj + Mg + . . . .* 166. Series of Positive Terms. The theory of the con- vergence of series is comparatively simple when all the terms of the series considered are positivef. We shall consider such series * It is of course a matter of indiffereuoe whether we denote our series by «i + M2+-.- (as in Ch. IV) or by Mq + % + ... (as here). Later in this chapter we shall be concerned with series of the type a(i + a^x + a2x'+...: for these the latter notation is clearly more convenient. We shall therefore adopt this as our standard notation. But we shall not adhere to it systematically, and we shall suppose that tij is the first term whenever this course is more convenient. It is more convenient, for example, when dealing with the series 1 + 4 + J + ... , to suppose that ti„=l/)t and that the series begins with n-^ , than to suppose that m„=1/{?i + 1) and that tha series begins with «(,. This remark applies, e.g., to Ex. lxvii. 4. t Here and in what follows ' positive ' is to be regarded as including zero. 165-168] THE CONVERGENCE OF INFINITE SERIES, ETC. 309 first, not only because they are the easiest to deal with, but also because the discussion of the convergence of a series containing negative or complex terms can often be made to depend upon a similar discussion of a series of positive terras only. When we are discussing the convergence or divergence of a series we may disregard any finite number of terms. Thus, when a series contains a finite number only of negative or complex terms, we may omit them and apply the theorems which follow to the remainder. 167 It will be well to recall the following fundamental theorems established in § 77. A. A series of positive terms must he convergent or diverge to 00 , amd cannot oscillate. B. The necessary and sufficient condition that 2m» should he convergent is that there shoidd he a number K such that Mo +■"! + ••■+ ^m < -^ for all values of n, C. The comparison theorem. If 2m„ is convergent, and Vn^ Unfor all values of n, then %Vn is convergent, and 2?;„g Sm„. More generally, if Vn^Kun, where K is a constant, then 2% is convergent and 1v„^KXun. And if 2m„ is divergent, and Vn = Kun, then ^v„ is divergent* Moreover, in inferring the convergence or divergence of %Vn by means of one of these tests, it is sufficient to know that the test is satisfied for sufficiently large values of n, i.e. for all values of n greater than a definite value n„. But of course the con- clusion that 'ivn S Ktun does not necessarily hold in this case. A particularly useful case of this theorem is D. If Xun is convergent (divergent) and Unjvn tends to a limit other than zero asn^ao, then Xvn is convergent (divergent). 168. First applications of these tests. The one important fact which we know at present, as regards the convergence of any * The last part of this theorem was not actually stated in § 77, but the reader will have no difficulty in supplying the proof. 310 THE CONVERGENCE OF INFINITE SERIES [vill special class of series, is that Sr" is convergent if r < 1 and divergent if r = l.* It is therefore natural to try to apply Theorem C, taking M„ = r". We at once find 1. The series Xvn is convergent if Vn & Kr^, where r < 1, for all sufficiently large values of n. When K = \, this condition may be written in the form v„^/" g r. Hence we obtain what is known as Cauchy's test for the con- vergence of a series of positive terms ; viz. 2. The series "Zvn is convergent if Vn"^ g r, where r <\, for all sufficiently large values of n. There is a corresponding test for divergence, viz. 2a. The series l,Vn is divergent if 'y„^'" = 1 for an infinity of values ofn. This hardly requires proof, for v„^'" S 1 involves Vn = 1, The two theorems 2 and 2a are of very wide application, but for some purposes it is more convenient to use a different test of convergence, viz. 3. The series Xvn is convergent if Vn+ijVn^r, where r<\,for all suffciently large values of n. To prove this we observe that if Vn+i/vn = r when « = «„ then and the result follows by comparison with the convergent series 2r". This test is known as d'Alembert's test. We shall see later that it is less general, theoretically, than Cauchy's, in that Cauchy's test can be applied v/henever d'Alembert's can, and sometimes when the latter cannot. Moreover the test for divergence which corresponds to d'Alembert's test for convergence is much less general than the test given by Theorem 2a. It is true, as the reader will easily prove for himself, that if Vn+ijvn = r = 1 for all values of n, or all sufficiently large values, then Xv^ is divergent. But it is not true (see Ex. Lxvii. 9) that this is so if only Vn+il^n = ^ = 1 for aw infinity o/ values of n, whereas in Theorem 2a * We shall use r in this chapter to denote a number which is always positive or zero. 168] AND INFINITE INTEGRALS 311 our test had only to be satisfied for such an infinity of values. None the less d'Alembert's test is very useful in practice, because when Vn is a complicated function Vn+ijvn is often much less complicated and so easier to work with. In the simplest cases which occur in analysis it often happens that Vn+ijvn or t;„"" tends to a limit as n ^- oo .* When this limit is less than 1, it is evident that the conditions of Theorems 2 or 3 above are satisfied. Thus 4. If V" or Vn+ilvn tends to a limit less than unity as n-^x , then the series S-y.^ is convergent. It is almost obvious that if either function tend to a limit greater than unity, then liVn is divergent. We leave the formal proof of this as an exercise to the reader. But when v„^"' or 'Vn+i/vn tends to 1 these tests generally fail completely, and they fail also when w„"" or Vn+i/vn oscillates in such a way that, while always less than 1, it assumes for an infinity of values of n values approaching indefinitely near to 1. And the tests which involve Vn+i/Vn fail even when that ratio oscillates so as to be sometimes less than and sometimes greater than 1. When Vn'"' behaves in this way Theorem 2a is sufficient to prove the divergence of the series. But it is clear that there is a wide margin of cases in wliich some more subtle tests will be needed. Examples LXVII. 1. Apply Cauchy's and d'Alembert's tests (as specialised in 4 above) to the series SjiV", where A is a positive rational number. [Here ■!'„^.i/y„={(M + l)/?i}'';'-^r, so that d'Alembert's test shows at ouce that the series is convergent if »■ < 1 and divergent if •/■ > 1. The test fails if r=l : but the series is then obviously divergent. Since lim «!'"■ = I (Ex. xxvii. 11), Cauchy's test leads at once to the same conclusions.] 2. Consider the series S(^l# + 5«*^ "' + ...+ if) r". [We may suppose 4 positive. If the coefficient of r" is denoted by P{n), then P{n)/n''-3~A and, by D of § 167, the series behaves like 2)iV".] 3. Consider ^ '^''''iV!fi~-il"'t^^ '''^ {A>0,a>0). an' + fin' '^+ ...+ K [The series behaves like 2w'-'r". The case in which >•=!, k<l requires further consideration.] * It will be proved in Ch. IX (Ex. lxxxvii. 36) that if v^+Jv^-^l then v^^l" -* I. That the converse is not true may be seen by supposing that v^=l when n is odd and f,, = 2 when n is even. 812 THE CONVERGENCE OF INFINITE SERIES [vill 4. We have seen (Ch. IV, Misc. Ex. 17) that the series 2 V n{n + l)' n(n + l)...{n+p) are convergent. Show that Cauohy's and d'Alembert's tests both fail when applied to them. [For lim «„!'" = lim (m„ + j/m J = 1.] 5. Show that the series 2m "J", where p is an integer not less than 2, is convergent. [Since Mm. {n{n+l)...{n+p-l)}jn''=l, this follows from the convergence of the series considered in Ex. 4. It has already been shown in § 77, (7) that the series is divergent if ^= 1, and it is obviously divergent if PSO.] 6. Show that the series An'' + Bn''~'^ + ... + E is convergent iil>k + l and divergent ifls,k + l. 7. If m„ is a positive integer, and m„ + i>OT„, then the series 2r™.i is con- vergent if j'<l and divergent if r 21. For example the series l-t-?'+r*-|-»''-l-... is convergent if ?'<1 and divergent if J-Sl. 8. Sum the series l+2r-|-2r*+... to 24 places of decimals when r='l and to 2 places when r=-9. [If »•=•!, then the first 5 terms give the sum 1-2002000020000002, and the error is 2^26 + 2,.38+ . . . <2r25 + 2r38 + 2r« + . . . = 2r26/(l - r") O/IO^'. If r=-9, then the first 8 terms give the sum 5'458..., and the error is less than 2?-M/(l -'•")< -003.] 9 If 0<a<5<l, then the series a + b + a^ + b^-\-a^ + ... is convergent. Show that Cauchy's test may be applied to this series, but that d'Alembert's test fails. [For 10. The series i.-hr+^ + ^i-... a.nd l+r-|--2 + -j-|-... are convergent for all positive values of ?•. 11. If 2m„ is convergent then so are 2uJ and 2m„/(1 + m„). 12. If 2Un^ is convergent then so is 2tt„/ra. [For 2UnJn<Un^ + {ljn^) and 2 (1/w^) is convergent.] 13. Show that 1 + i +^+... = 7 ('l + ^2 + p + ...) and 32-^52-^— 4\^^22 I + 22+32 + 52 + 62 + 72 + 92 + —-16 y-'^2^'^si + —)' 168-170] AND INFINITE INTEGRALS 313 [To prove the first result we note that i4+^,+...=(i4)+(^,+j)+... = l+p + 52 + — + p(^l + 22 + 32 + ...j, by Theorems (8) and (6) of § 77.] 14. Prove by a reductio ad ahsurdum that 2 (l/«) is divergent. [If the series were convergent we should have, by the argument used in Ex. 13, i+HH-=(i+H*+-)+Hi+i+J+-), which is obviously absm-d, since every term of the first series is less than the corresponding term of the second.] 169. Before proceeding further in the investigation of tests of convergence and divergence, we shall prove an important general theorem concerning series of positive terms. Dirichlet's Theorem.* The sum of a series of positive terms is the same in whatever order the terms are taken. This theorem asserts that if we have a convergent series of positive terms, Ui, + Ui + u.2+ ... say, and form any other series Va + Vi + V2+ ... out of the same terms, by taking them in any new order, then the second series is convergent and has the same sum as the first. Of course no terms must be omitted : every u must come some- where among the v's, and vice versa. The proof is extremely simple. Let s be the sum of the series of us. Then the sum of any number of terms, selected from the m's, is not greater than s. But every v is a u, and therefore the sum of any number of terms selected from the v's is not greater than s. Hence Svm is convergent, and its sum t is not greater than s. But we can show in exactly the same way that s ^t. Thus s = t. 170. Multiplication of Series of Positive Terms. An immediate corollary from Dirichlet's Theorem is the following theorem : t/" t<o + «! + «2 + • • • <^'*^ «» + ^i + "2 + • • ■ os^^ ttvo convergent * This theorem seems to have first been stated explicitly by Dirielilet in 1837. It was no doubt known to earlier writers, and in particular to Cauchy, 814 THE CONVERGENCE OF INFINITE SERIES [VIII series of positive terms, and s and t are their respective sums, then the series Mo Vo + (Wi^o + UoVi) + (U^V^ + iti«i + UoVi) + ... is convergent and has the sum st. Arrange all the possible producbs of pairs UmVn in the form of a doubly infinite array UiVo U,Vi U^Va U^V-, U2V2 U3V1, UsV, UsV^ U3V3 ■Mo ■"2 UlV^ Mo^s lh% U2V3 We can rearrange these terms in the form of a simply infinite series in a variety of ways. Among these are the following. (1) We begin with the single term u^v^ for which m+n = 0; then we take the two terms u^v^, Uf,Vi for which m + n=l; then the three terms Ur^Vo, u^v^, Wo^a for which m + ?i = 2 ; and so on. We thus obtain the series U^Vo + (Mif + Mo^i) + (UiVo + UiVi + UoV-i) + ... of the theorem. (2) We begin with the single term 1*0^0 for which both suffixes are zero; then we take the terms u^Vo, u^v^, UdV^ which involve a suffix 1 but no higher suffix; then the terms Ma^oi ^^i, u-iV-i, u{V2, UqV2 which involve a suffix 2 but no higher suffix ; and so on. The sums of these groups of terms are respectively equal to UoV„, {u„+Ui){Vo+Vi)-UoV„, (Uo + Ml + U.2) {Vo +Vi + V2) - (Uo + Wi) (Vo +V,), ... and the sum of the first n. + 1 groups is (Mo+ Ml + ... +Un) (Vo + Vi + ... +V„), and tends to s* as ?i ^ 00 . When the sum of the series is formed in this manner the sum of the first one, two, three, ... groups comprises all the terms in the first, second, third, ... rectangles indicated in the diagram above. The sum of the series formed in the second manner is st. But the first series is (when the brackets are removed) a rearrange- ment of the second ; and therefore, by Dirichlet's Theorem, it con- verges to the sum st. Thus the theorem is proved. 170, 171] AND INFINITE INTEGRALS 315 Examples LXVIII. 1 Verify that if r < 1 then l+r^+r + ri + r» + r^ + ... = l+r + rs+r^ + r^+r'' + ... = ll{l-r). 2* If either of the series Mo+Mi + ..., Va + Vi + ... is divergent, then so is the series WoV^ + {tiii'^ + it^vi) + {u2Vf, + UiVi+Uf)V2) + ..., except in the trivial (jase in which every term of one series is zero. 3. If the series Mo + m, + ..., v^+Vi + ..., ^Of) + Wi + ... converge to sums r, s, t, then the series 2\, where Xj. = 2M^(;„Wp, the summation being extended to all sets of values of m, n, p such that m + 7i+p = k, converges to the sum rst. 4. If 2w„ and 2vn converge to sums s and t, then the series 2w„, where io„=2uiVm, the summation extending to all pairs I, m for which lm = n, converges to the sum st. 171. Further tests for convergence and divergence. The examples on pp. 311 — 313 suffice to show that there are simple and interesting types of series of positive terms which cannot be dealt with by the general tests of § 168. In fact, if we consider the simplest type of series, in which m„+i/m„ tends to a limit as n-^cc , the tests o/ § 168 generally fail when this limit is 1. Thus in Ex. Lxvii. 5 these tests failed, and we had to fall back upon a special device, which was in essence that of using the series of Ex. lxvii. 4 as our comparison series, instead of ■the geometric series. The fact is that the geometric series, by comparison with which the tests of § 168 were obtained, is not only convergent but veiy rapidly convergent, far more rapidly than is necessary in order to ensure convergence. The tests derived from comparison with it are therefore naturally very crude, and much more delicate tests are often wanted. We proved in Ex. xxvil. 7 that n^r^-s-O as n-»-oo, provided r<l, what- ever value k may have; and in Ex. lxvii. 1 we proved more than this, viz. that the series S^?'" is convergent. It follows that the sequence r, r', r^, ..., r"; ..., where ?•<!, diminishes more rapidly than the sequence 1 ~ *, 2 " *, 3 ~ *, . . . , n ~ *•', This seems at first paradoxical if r is not much less than unity, and k is large. Thus of the tw6 sequences 3J y ' ^Tj '■• J ^) 409TiJ T3T44T) "■ whose general terms are (§)" and m~'^, the second seems at first sight to decrease far more rapidly. But this is far from being the case ; if only we go far enough into the sequences we shall find the terms of the first sequence very much the smaller. For example, (2/3)*=16/81<l/5, (2/3)i2<(l/5)3<(l/10)2, (2/3)io'>''<(l/10)ie«, while 1000-12 = 10-30; * lu Exs. 2 — 4 the series considered are of course series of positive terms. 31(5 THE CONVERGENCE OF INFINITE SERIES [viu SO that the 1000th term of the first sequence is less than the lO'Hh part of the corresponding term of the second sequence. Thus the series 2 (2/3)" is far more rapidly convergent than the series Sw^^ and even this series is very much more rapidly convergent than Sn~^.* 172. We shall proceed to establish two further tests for the convergence or divergence of series of positive terms, Maclaurin's (or Cauchy's) Integral Test and Cauchy's Condensation Test, which, though very far from being completely general, are sufficiently general for our needs in this chapter. In applying either of these tests we make a further assumption as to the nature of the function m„, about which we have so far assumed only that it is positive. We assume that m„ decreases steadily with n: i.e. that w,i+i S m„ for all values of w. or at any rate all sufficiently large values. This condition is satisfied in all the most important cases. From one point of view it may be regarded as no restriction at all, so long as we are dealing with series of positive terms : for in virtue of Dirichlet's theorem above we may rearrange the terms without affecting the question of con- vergence or divergence ; and there is nothing to prevent us rearranging the terms in descending order of magnitude, and applying our tests to the series of decreasing terms thus obtained. But before we proceed to the statement of these two tests, we shall state and prove a simple and important theorem, which we shall call Abel's Theoremf. This is a one-sided theorem in that it gives a sufficient test for divergence only and not for convergence, but it is essentially of a more elementary character than the two theorems mentioned above. 173. Abel's (or Pringsheim's) Theorem. If 2Un is a convergent series of positive and decreasing terms, then lim »m„=0. Suppose that nUn does not tend to zero. Then it is possible to find a positive number 8 such that ?iM„gS for an infinity of values of n. Let n^ be the first such value of n ; 712 the next such value of n which is more than * Five terms suffice to give the sum of 2?i-'2 correctly to 7 places of decimals, whereas some 10,000,000 are needed to give an equally good approximation to Sn-^. A large number of numerical results of this character will be found in Appendix III (compiled by Mr J. Jackson) to the author's tract ' Orders of Infinity' (Cambridge. Math. Tracts, "Sio. 12). t This theorem was discovered by Abel but forgotten, and rediscovered by Pringsheim. 171-173] AND INFINITE INTEGRALS 317 twice as large as n, ; 113 the next such vahie of n which is more than twice as large as «2 ; and so on. Then we have a sequence of numbers W], 712, n^, ... such that %>2rei, 513 > 2«2, ... and so W2-«i>|m2, ^s- \>in3, ...; and also «iM„, S8, Kj^nj >8, .... But, since «„ decreases as n increases, we have W„, + ... +M„j_i >(W2-Mi)m«j> i/J2''n,SiS, M„, + ...+W„3_l>(?i3-»2)«n3>i«3Mn3Si8, and so on. Thus we can bracket the terms of the series 2un so as to obtain a new series whose terms are severally greater than those of the divergent series 8+i8+p + ...; and therefore 2Mn is divergent. Examples LXIX. 1. Use Abel's theorem to show that 2 (1/m) and S{l/(a«+6)} are divergent. [Here nun-^l or nu^-^lja.'] % Show that Abel's theorem is not true if we omit the condition that m„ decreases as n increases. [The series l + 2ii + 3^+4+52+62 + 72+ 82 ''■9 + 102 "''■■■' in which M„=l/m or l/ji^, according as w is or is not a perfect square, is convergent, since it may be reai'ranged in the form 11111 /, 11\ + 52+62+75 +8^ + W2+-+(l+4 + 9 + -). Jl 1.1.1.1.1.1 22 "''32 and each of these series is convergent. But, since mM,i = l whenever m is a perfect square, it is clearly not true that jiMn-^0.] 3. The converse of Abel's theorem is not true, i.e. it is not true that, if m„ decreases with n and lim nUn—0, then 2m„ is convergent. [Take the sei'ies 2 (l/w) and multiply the first term by 1, the second by |, the next two by J, the next four by ^, the next eight by J, and so on. On grouping in brackets the terms of the new series thus formed we obtain i+i.HJ(J-+i)+i(*+*+f+^)+-; and this series is divergent, since its terms are greater than those of i+i-Hi-i+i-i+-. which is divergent. But it is easy to see that the terms of the series i+i-i+J-H*-i+i-*+^*+- satisfy the condition that n.M„-*0. In fact nun=llv if 2"~^<)j<2"~\ and v-»-x) as ?i-»-co.] 318 THE CONVERGBNCE OF INFINITE SERIES [vm 174. Maclaurin's (or Cauchy's) Integral Test. * If «„ decreases steadily as n increases, we can write Un = <j} (n) and suppose that ^ (n) is the value assumed, when x = n, by a con- tinuous and steadily decreasing function ^ (x) of the continuous variable x. Then, if v is any positive integer, we have when v—l&x^v. Let v, = <jiiv-l)-r <f>(x)dx=r {<f>(v-l)-4>(x)}dx, J v-l J v-1 so that 0^v,^^{v-l)-^ (v). Then l,v„ is a series of positive terms, and Hence 2v„ is convergent, and so V2 + V3+ ...+VnOr n-l /•« l,6{v)— i d> (x) dx 1 Ji tends to a positive limit as n-*oo . Let us write * (?) = | <A (^) '^^> so that $ (?) is a continuous and steadily increasing function of f. Then «! + J«2 + . . . + lln-z — ^ (n) tends to a positive limit, not greater than ^ (1), as 91 -^ 00 . Hence Sw^ is convergent or divergent according as $ (n) tends to a limit or to infinity as n ^ 00 , and therefore, since $ (n) increases steadily, according as <!>(?) tends to a limit or to infinity as f -^ 00 . Hence if <p (x) is a function of x which is positive and continuous for all values of x greater than unity, and decreases steadily as x increases, then the series ./.(l) + <^(2)+... does or does not converge according as ^(^)=j j>{x)dx does or does not tend to a limit I as ? -* 00 ; and, in the first case, the sum of the series is not greater than <^ (1) + 1. * The test was discovered by Maclaurin and rediscovered by Cauohy, to whom it is usually attributed. 174, 175] AND INFINITE INTEGRALS 319 The sum must in fact be less than (f)(1) + 1. For it follows from (6) of § 160, and Ch. VII, Misc. Ex. 41, that «)„<0(^-l)-0(:/), unless (^(.») = 0W throughout the interval (v -l,v); and this cannot be true for all values of v. Examples LXX. l. Prove that 2. Prove that -Jff<l ^^— 2<|7r. (Math. Trip. 1909.) 3. Prove that if w > then ni2^(m+l)2^(m + 2)2^'"^ m 175. The series ^n~^. By far the most important applica- tion of the Integral Test is to the series 1-s + 2-« + 3-" + . . . + n~' + ..., where s is any rational number. We have seen already (§77 and Exs. Lxvii. 14, LXix. 1) that the series is divergent when s = 1. If s S then it is obvious that the series is divergent. If 5 > then Un decreases as n increases, and we can apply the test. Here ^^^ J lOf 1-s ' unless s = 1. If s > 1 then ^'-*-* as ^-*- oo , and say. And if s<l then ^^~*-»-co as ^-^oo, and so 0(^)-^-oo. Thus the series 2w~* is convergent ifs>l, divergent if s ^ 1, and in the first case its sum is less than s/(s — l). So far as divergence for s<l is concerned, this result might have been derived at ouce from comparison with 2(l/?i), which we already know to be divergent. It is however interesting to see how the Integral Test may be applied to the series 2 (l/»), when the preceding analysis fails. In this case and it is easy to see that * (|) -»- oo as |-9- oo . For if ^>2'' then ,^, C^'dx f^tlv (id.v [^"-'d.v ji, * Jl X J2 X J 2" X 320 THE CONVERGENCE OF INFINITE SERIES [vill But by putting x = 2^u we obtain j ar X ~ J 1 u' and so *(!)>» ( — , which shows that * (|) -»- oo as | -3-00 . J 1 " Examples LXXI. 1. Prove by an argument similar to that used above, /Idx —, where «< 1, tends to infinity with |. 2. The series Sn~\ 2n~^l^, Sra"'^"" are convergent, and their sums are not greater than 2, 3, 11 respectively. The series ^n-^'% 2n-i"/" are divergent. 3. The series ^n'Kn' + a), where a>0, is convergent or divergent aocord- ng as «> 1 +a or < S 1 +s. [Compare with 2 Ji'~'.] 4. Discuss the convergence or divergence of the series 2 (aim^i + 0271*2+ ... +at?i'*)/(6i?i*i + 62W*2 + ... +bin\ where all the letters denote positive numbers and the «'s and fa are rational and arranged in descending order of magnitude. 5. Prove that {Math. Trip. 1911.) 6. If (») -9- ? > 1 then the series 2ra ~ '^ (") is convergent, li (j)(n)-^l<l then it is divergent. 176. Cauchy's Condensation Test. The second of the two tests mentioned in § 172 is as follows : if Un = 4>{n) is a decreasing function of n, then the series 2^(n) I's convergent or divergent according as 22" ^ (2") is convergent or divergent. We can prove this by an argument which we have used already (§ 77) in the special case of the series 2(l/n). In the first place .^(3) + <^(4)S2</>(4), <;!,(5) + <^(6)+...+</>(8)S4</>(8), <|) (2" + 1) + (^ (2" + 2) + . . . + (/> (2"+0 S 2" (2»+0. If 2 2" ^(2") diverges then so do 22"+i^(2"+0 and 2 2"<^(2"+0, and then the inequalities just obtained show that % <j)(n) diverges. I75rl77] AND INFINITE INTEGEALS 821 On the other hand <f>(2) + <}>(B)£2(j>(2), </.(4) + <^(5)+...+<^(7)s4(/,(4), and so on. And from this set of inequalities it follows that 'if S 2" (2") converges then so does 1 4> (n). Thus the theorem is established. For our present purposes the field of application of this -test is practically the same as that of the Integral Test. It enables us to discuss the series S n~^ with equal ease. For 2 n~^ -will converge or diverge according as X2''2"~"* converges or diverges, i.e. ac- cording as s > 1 or s ^ 1. Examples LXXII. l. Show that if a is any positive integer greater than 1 then ^<f>{n) is convergent or divergent according as 2a''(f){a'^) is convergent or divergent. [Use the same arguments as above, taking groups of a, a^, a^, ... terms.] 2. If 22"'0 (2") converges then it is obvious that hm2»0(2'') = O. Hence deduce Abel's Theorem of § 173. 177. Infinite Integrals. The Integral Test of § 174 shows that, if (f) (x) is a positive and decreasing function of x, then the series % </> (n) is convergent or divergent according as the integral function <I> {x) does or does not tend to a limit as a; ^ oo . Lefc us suppose that it does tend to a limit, and that lim \ ^{t)dt = l. a:->.oo J 1 Then we shall say that the integral (j) (t) dt i: is convergfent, and has the value I ; and we shall call the integral an infinite integral. So far we have supposed </> (i) positive and decreasing. But it is natural to extend our definition to other cases. Nor is there any special point in supposing the lower limit to be unity. We are accordingly led to formulate the following definition : If (j) (t) is a function oft continuous when t = a, and lim \ <\){t)dt =1, n. 21 322 THE CONVERGENCE OF INFINITE SERIES [vill then we shall say that the infinite integral j'"<j>{t)dt (1) is convergent and has the value I. The ordinary integral between limits a and A, as defined in Ch. VII, we shall sometimes call in contrast a finite integral. On the other hand, when \<l>{t)dt- -00, we shall say that the integral diverges to oo , and we can give a similar definition of divergence to — oo . Finally, when none of these alternatives occur, we shall say that the integral oscillates, finitely or infinitely, as a; -* oo . These definitions suggest the following remarks. (i) If we write / "^ ^ (<) (^< = * {x), J a then the integral converges, diverges, or oscillates according as * {x) tends to a limit, tends to oo (or to — oo ), or oscillates, as :» -*- oo . If * (.■») tends to a limit, which we may denote by * (oo ), then the value of the integral is $ (oo ). More generally, if * {x) is any integral function of (j> (x), then the value of the integral is * (oo ) — * (a). (ii) In the special case in which <^(«) is always positive it is clear that *(a;) is an increasing function of x. Hence the only alternatives are convergence and divergence to oo . (iii) The integral (1) of course depends on a, but is quite independent of t, and is in no way altered by the substitution of any other letter for t (cf. § 157). (iv) Of course the reader will not be puzzled by the use of the term infinite integral to denote something which has a definite value such as 2 or ^TT. The distinction between an infinite integral and a .finite integral is similar to that between an infinite series and a finite series: no one supposes that an infinite series is necessarily divergent. fx (v) The integral I <^ (t) dt was defined in §§ 156 and 157 as a simple limit, i.e. the limit of a certain finite sum. The infinite integral is therefore the limit of a limit, or what is known as a repeated limit. The notion of the infinite integral is in fact essentially more complex than that of the finite integral, of which it is a development. 177, 178] AND INFINITE INTEGRALS 323 (vi) The Integral Test of § 174 may now be stated in the form : if <f> (x) is positive and steadily decreases as .v increases, then the infinite series 2 {n) and the infinite integral I {x) dx converge or diverge together. (vii) The reader will find no difficulty in formulating and proving theorems for infinite integrals analogous to those stated in (1) — (6) of § 77. Thus the result analogous to (2) is that if j 4> (x) dx is convergent, and b>a, then (j) (x) dx is convergent and /, I (j) {x) dx = j (l)(x)dx+ j <f) {x) dx. J a J a J b 178. The case in which <ji (a) is positive. It is natural to consider what are the general theorems, concerning the con- vergence, or divergence of the infinite integral (1) of § 177, analogous to theorems A — D of § 167. That A is true of integrals as well as of series we have already seen in § 177, (ii). Corre- sponding to B we have the theorem that the necessary and sufficient condition for the convergence of the integral (1) is that it should he possible to find a constant K such that r^{t)dt<K J a for all valves of x greater than a. Similarly, corresponding to C, we have the theorem : if I (fi (so) dx is convergent, and i/r (x) S Z"^ (x) for all values of x J a greater than a, then I ifr (x) dx is convergent and J a Too r 00 I y]f(x)dxS: K I 4' (*) '^*- J a J a We leave it to the reader to formulate the corresponding test for divergence. We may observe that D'Alembert's test (§ 168), depending as it does on the notion of successive terms, has no analogue for integrals ; and that the analogue of Cauchy's test is not of much importance, and in any case could only be formulated when we have investigated in greater detail the theory of the function 21—2 324 THE CONVERGENCE OF INFINITE SERIES [VlII (j) (cc) = r", as we shall do in Ch. IX. The most important special tests are obtained by comparison with the integral '$ («>0). /■ J a whose convergence or divergence we have investigated in § 175, and are as follows : if <f> (x) < Kx~^, where s>\, when x ^ a, then I («) dx is convergent ; and if <f> (x) > Kx^^, where s g 1, when a X = a, then the integral is divergent ; and in particular, if lim afi (f> (x) = I, where I >0, then the integral is convergent or divergent according as s>l or ss 1. There is one fuudamental property of a convergent infinite series in regard to which the analogy between infinite series and infinite integrals breaks down. If 20 (ra) is convergent then 0(7i)-»-O; but it is not always true, even when (p (x) is always positive, that if I (x) dx is convergent J a then <f>(x)~a-0. Consider for example the function (x) whose graph is indicated by the thick line in the figure. Here the height of the peaks corresponding to the points x=\, 2, 3, ... is in each case unity, and the breadth of the peak corre- sponding to x=n is 2/(m + 1)^. The area of the peak is l/(m-f 1)^, and it is- evident that, for any value of ^, />' 'x) dx<2- JO "o(»+l)^' so that I (j) {x) dx is convergent ; but it is not true that d> (a;) — Jo Examples LXXIII. I. The integral /: dx. j„^.^■«+5a;'-l-^-... + i ' where a and A are positive and a is greater than the greatest root of the denominator, is convergent if s>?' + l and otherwise divergent. 178] AND INFINITE INTEGRALS 325 2. Which of the integrals Ja-J^' jaX^I^' JaC^ + X^' jaC' + x'' jaCH^^' j„ a + 20X^ + yx' are convergent? In the first two integrals it is supposed that a>0, and in the last that a is greater than the greatest root (if any) of the de- nominator. 3. The integrals I coaxdx, I sin^c^j;, I coa{ax + ^)dx J a J a J a oscillate finitely as |-s- oo . 4. The integrals I X coaxdx, j x'^ainxdx, I ,i^ cos (ax + P) dx, J a J a J a where n is any positive integer, oscillate infinitely as |-^qo . ra 5. Integrals to - CO . If I <^(.j;)(f^tendstoalimitZasf-»--oo,thenwe ra say that I (x) dx is convergent and equal to I. Such integrals possess / -co properties in every respect analogous to those of the integrals discussed in the preceding sections : the reader will find no difficulty in formulating them. C. Integrals from - to to + oo . If the integrals r a r» I <j) (x) dx, I (j) (x) dx j -m J a are both convergent, and have the values k, I respectively, then we say that f " I <f> {x)dx is convergent and has the value k + l. 7. Prove that [° dx _ f dx _ f dx _ 8. Prove generally that /• 00 roo i 4>(x^)dx=2 j (j}{x^)dx, provided that the integral I (x^) dx is convergent. xcl){x^)dxia convergent then I x(j)(x^)dx=0. ,/ -oo 326 THE CONVERGENCE OF INFINITE SERIES [vill 10. Analogue of Abel's Theorem of § 173. If <p {se) is positive and steadily decreases, atid I (^{x)dx is convergent, then x(p{x:)-»-Q. Prove this J a (a) by means of Abel's Theorem and the Integral Test and (6) directly, by arguments analogous to those of § 173. 11. If a = XQ<xi<a;'^<... and a;„-i>-oo, and m„= | (^) (&•, then the convergence of I (x) dx involves that of 2 m„. If (x) is always positive J a the converse statement is also true. [That the converse is not true in general is shown by the example in which ^{x) = q,o^x, ii;„=m7r.] 179. Application to infinite integrals of the rules for substitution and integration by parts. The rules for the transformation of a definite integral which were discussed in 1 161 may be extended so as to apply to infinite integrals. (1) Transformation by substitution. Suppose that j <J3(x)dx (1) is convergent. Further suppose that, for any value of f greater than a, we have, as in § 161, (^<f>(x)dx= \\{f{t)]f'{t)dt (2), where a=f{b), |^=/(t). Finally suppose that the functional relation x=f{t) is such that x-^co asi-*oo. Then, making t and so f tend to oo in (2), we see that the integral J b , <l>{f(t)}f{t)dt (3) is convergent and equal to the integral (1). On the other hand it may happen that f -*- oo as t -* — to or as T^-c. In the first case we obtain r 4>{x)dx= lim r<i>[f{t)]f'{t)dt = - lim fV {/(«)}/' («)rf<=-f <l>{f(t)}f'(t)dt. T-*- — 00 J T J —CO In the second case we obtain r <t>(x)dx = lim r(j>{f{t)}f'(t) dt (iX J a T-*-cJ b We shall return to this equation in § 181. 178, 179] AND INFINITE INTEGRALS 327 There are of course corresponding results for the integrals fa roo I <^ (x) dx, I <\) (x) dx, J —CO J —00 which it is not worth while to set out in detail : the reader will be able to formulate them for himself. Examples LXXIV. 1. Show, by means of the substitution x=f', that if s > 1 and a > then r CO /• oo and verify the result by caleulating the value of each integral directly. 2. If / (^ (.i;) dx is convergent then it is equal to one or other of J a f [(.a-f.)lcL a\ (ii{at+fi)dt, -a] 4){at + ^)dt, J (a-/3)/a J -00 according as a is positive or negative. 3. If <!> {x) is a positive and steadily decreasing function of x, and n and P are any positive numbers, then the convergence of the series 2 (p (n) implies and is implied by that of the series 2(/)(q71+/3). [It follows at once, on making the substitution x=at+^, that the integrals roo /•» I (i>{x)dx, <f>{at + ^)dt J a j (a-/3)/a converge or diverge together. Now use the Integral Test.] 4. Show that [" ,, f", , =\ir. [Put^-C^.] r°° jx 5. Show that | 7^— — ^dx = \'!T. Jo (1+^)^ [Put x=fi and integrate by parts.] 6. If {x)^3-h as a'-»-QO , and {x)-^Tc as x-*- — a:i , then [ {(P{x-a)-<i>{x-h)}dx=-{a-'b){h-k). [For/ {4,{x-a)-^{x-h)}dx= \ ^(x-a)dx- f <f)(x-b)dx = f ^"" <l>(t)dt- ( ^'^<i,{t)dt= ('^''\(t)di- P"V(Oc?«- J -i'-a J -^-b J -t-a J S-a 328 THE CONVERGENCE OF INFINITE SERIES [vil] The first of these two integrals may be expressed in the form {a-b)k+ I pdt, \ J -f'-ft where p-».0 as |'-»-oo, and the modulus of the last integral is less than or equal to | a - 6 1 k, where .k is the greatest value of p throughout the interval (-g'-a, -|'-6). Hence r-S-i \ (\){t)dt-^{a-h)k. The second integral may be discussed similarly.] (2) Integration by parts. The formula for integration by parts (§ 161) is \^f{x) </.' (*) dx =/(f) <^ (f) -/(a) <f> (a) - f V' (^J) <i> (oo) dec. J a ' J a Suppose now that ^-^oo . Then if any two of the three terms in the above equation which involve | tend to limits, so does the third, and we obtain the result J a /(cc) cj,' (x) dx = lim /(^) <p (0 -f(a) .^ (a) - / (x) </, (w) dx. There are of course similar results for integrals to — oo , or from — 00 to 00 . 1'° X f" dx Examples LXXV. 1. Show that I j- 7^dx=^ I .-=— — ^=i. jo (1+^)* ^ Jo {i+^y /^ sf^dx (1 -rx) Im,n={^nj{m+n-\)} I^_i^n. Hence prove that i„,„=TO ! (»- 2) !/(m +»-!)!. 4. Show similarly that if /„,,„= I nj^£i\m+n ^^^n ■4.,.i = {W(™ + »-l)}4i-i,u, 2/^,„ = »i!(m-2) !/(m + w-l)!. Verify the result by applying the substitution x = t^ to the result of Ex. 3. 180. Other types of infinite integrals. It was assumed, in the definition of the ordinary or finite integral given in Ch. VII, that (1) the range of integration is finite and (2) the subject of integration is continuous. It is possible, however, to extend the notion of the ' definite integral ' so as to apply to many cases in which these conditions 179, 180] AND INFINITE INTEGRALS 329 are not satisfied. The 'infinite' integrals which we have discussed in the preceding sections, for example, differ from those of Ch. VII in that the range of integration is infinite. We shall now suppose that it is the second of the conditions (1), (2) that is not satisfied. It is natural to try to frame definitions applicable to some such cases at any rate. There is only one such case which we shall consider here. We shall suppose that </> (x) is continuous throughout the range of integration (a, A) except for a finite number of values of X, say a; = fi, ^2, . . ., and that ^ (a;) ^ ao or (x) -^ — oo as a; tends to any of these exceptional values from either side. It is evident that we need only consider the case in which (a, A) contains one such point f. When there is more than one such point we can divide up (a, A) into a finite number of sub-intervals each of which contains only one ; and, if the value of the integral over each of these sub-intervals has been defined, we can then define the integral over the whole interval as being the sum of the integrals over each sub-interval. Further, we can suppose that the one point ^ in (a, A) comes at one or other of the limits a, A. For, if it comes between a and A, we can then define I S (x) dx as \^<l>{x), J a I (f>(x)dx+ j <p (x) dx, J a J I assuming each of these integrals to have been satisfactorily de- fined. We shall suppose, then, that f = a ; it is evident that the definitions to which we are led will apply, with trifling changes, to the case in which ^ = A. Let us then suppose <j) (x) to be continuous throughout (a, A) except for x = a, while ^(a;)-«-oo as x^^a through values greater than a. A typical example of such a function is given by <j) (x) = (x- a)-\ where s > ; or, in particular, if a = 0, by ^ (x) = x~\ Let us therefore consider how we can define f^dx Jo ^ ^'^' when s > 0. 330 THE CONVERGENCE OF INFINITE SERIES [vill The integral I y^-^ dy is convergent if s < 1 (§ 175) and means J\IA lim 1 2/'"" dy. But if we make the substitution y = 1 /«, we ri-^xJyA obtain 'A 1 y^~^ dy = I a;~* da;. i l/A J 1/7, Thus lim / x~^dx, or, what is the same thing, l)-*-ao J l/ri lim I «"' cia;, exists provided that s < 1 ; and it is natural to define the value of the integral (1) as being equal to this limit. Similar considerations lead us to define I (x — a)~' dx by the equation J a rA rA I (x — a)~' dx = lim I (x — a)~^ dx. J a e~-*--i-oJ a+€ We are thus led to the following general definition : if the integral I (f} (x) dx J a+e tends to a limit I as e-^+0, we shall say that the integral M I (f) (x) dx J a is convergent and has the value I. Similarly, when ^{x)-^<x as a; tends to the upper limit A, we define I (f>(x)dx as being J a rA-e lim / (^ {x) dx : e-*-+0-'a and then, as we explained above, we can extend our definitions to cover the case in which the interval (a, A) contains any finite number of infinities of (^ {x). An integral in which the subject of integration tends to oo or to — 00 as a; tends to some value or values included in the range of integration will be called an infinite integral of the second land : the first kind of infinite integrals being the class discussed in §1 177 et seq. Nearly all the remarks (i) — (vii) made at the end of § 177 apply to infinite integrals of the second kind as well as to those of the first. 180, 181] AND INFINITE INTEGRALS 331 181. We may now write the equation (4) of § 179 in the form f <i>{x)dx=\'',i>{f{t)}f'{t)dt (1). J a J b The integral on the right-hand side is defined as the limit, as t-^-c, of the corresponding integral over the range (6, t), i.e. as an infinite integral of the second kind. And when ^{/(O} /' (t) has an infinity at t = e the integral is essentially an infinite integral. Suppose for example, that </)(it;) = (l + «)-"*, where l<m<2, and a=0, and that/(<) = «/(l-«). Then 6 = 0, c=l, and (1) becomes /o"(iSp=/;(^-)"-'^' ■(2); and the integral on the right-hand side is an infinite integral of the second kind. On the other hand it may happen that <^ {/(<)}/' (0 is continuous for t=o. In this case Jo is a finite integral, and lim /"'</, {f(t)}f'(t)dt= f °0 {f{t)}f{t)dt, T-*-c Jo J b in virtue of the corollary to Theorem (10) of § 160. In this case the substitution sc=f{t) transforms an infinite into a finite integral. This case arises if m = 2 in the example considered a moment ago. Examples LXXVI. 1. If ^(x) is continuous except for x=a, while [A (j) {x)^>-cD a.HX-a-a, then the necessary and sufiicient condition that / <j) (x) dx J a should be convergent is that we can find a constant K such that rA <!> (x) dx<K for all values of e, however small (of. § 178). It is clear that we can choose a number A' between a and A, such that (^ {x) is positive throughout (a, A'). If <^ {x) is positive throughout the whole interval (a. A) then we can of course identify A' and A. Now (A fA' [A I <^ {x) dx = I (j){x) dx+ I (f) (x) dx. J a-' J a—e J A' The first integi'al on the right-hand side of the above equation increases as e decreases, and therefore tends to a limit or to oo ; and the truth of the result stated becomes evident. fA If the condition is not satisfied then I (f>(x)d.v^-co . We shall then say J a—e fA that the integral / <p (x)dx diverges to oo. It is clear that, if (f){x)^-oD J a a.s x-»-a+0, then convergence and divergence to oo are the only alternatives for the integral. We may discuss similarly the case in which <j){x)-^-aa. i: 382 THE CONVERGENCE. OF INFINITE SERIES [vill 2. Prove that / {x-a)-^dx = ^- '- — \-s if s < 1, while the integral is divergent if s S 1. 3. If <j() (^) -=>- 00 as x^-a + and (a;) <A' (^ — a)~', where a<\, then I (j)(x)dx is convergent; and if (f>{x)> K{x — a) ', where «21, then the J a integral is divergent. [This is merely a particular case of a general com- parison theorem analogous to that stated in g 178.] 4. Are the integrals M dx M dx M dx }a^l{{^-a){A-x)}- ]a.{A-x)ij'{x-ay j a {A-x)ll{A-xy M dx M dx M dx [^ dx convergent or divergent ? /I (Jj; fa+l J-c sT" 1 I sjr^' — \ ^1"^ convergent, and the value of -1 V^ j a-1 VO*'^'^) each is zero. f TT f2f^ 6. The integral / -7-4 — - is convergent. [The subject of integration J V (sm .37) tends to 00 as ^ tends to either limit.] y-. ^ is convergent if and only if s < 1. (sin xy / C\^ x' V 8. The integral / -j—. r- dx is convergent if «< s + 1. j (sm x) 9. Show that I — —rf.», where A >0, is convergent if p< 2. Show also that, if < p < 2, the integrals /f sin .r , /" 2ir sin « , T S" sin .a; , dx, I dx, I — -—dx,... t> X" ' }„ XP ' J2,r X" alternate in sign and steadily decrease in absolute value. [Transform the integral whose limits are kw and (^•-|-l)7rby the substitution x = /cTr + i/.] /[ Sin i^/ 10. Show that I dx, where 0<jo<2, attains its greatest value when A = 57. " {Math. Trip. 1911.) x/ 11. The integral I (cos x)' (sin x)™dx is convergent if and only if Z> - 1, Jo m > - 1. /■» x^'^dx — j , where s < 1, does not fall directly 1+x under any of our previous definitions. For the range of integration is infinite 181] AND INFINITE INTEGRALS 333 and the subject of integration tends to oo asa.--»- + 0. It is natural to define this integral as being equal to the sum jo 1+x ji l+x ' provided that these two integrals are both convergent. The first integral is a convergent infinite integral of the second kind if 0<s<l. The second is a convergent infinite integral of the first kind if s <l. It should be noted that when s > I the first integral is an ordinary finite integral ; but then the second is divergent. Thus the integral from to on is convergent if and only if < s < 1. 13. Prove that I ^^ -, dx is convergent if and only if < s < <. ; l + « — = dx is convergent if and only if < s < 1, 1 ~x < i < 1. [It should be noticed that the subject of integration is undefined when ^=1; but (a^-i -«'"')/(!- a-) -»-Z — s as ^-»-l from either side; so that the subject of integration becomes a continuous function of x if we assign to it the value t-s when x=l. It often happens that the subject of integration has a discontinuity which is due simply to a failure in its definition at a particular point in the range of integration, and can be removed by attaching a particular value to it at that point. In this case it is usual to suppose the definition of the subject of integration completed in this way. Thus the integrJtls [ i" sin mx , fi" sin mx I CCX^ I ' ", clx jo X 'Jo amx are ordinary finite integrals, if the subjects of integration are regarded as having the value in when x = 0.] 15. Substitution and integration by parts. The formulae for trans- formation by substitution and integration by parts may of course be extended to infinite integrals of the second as well as of the first kind. The reader should formulate the general theorems for himself, on the lines of § 179. 16. Prove by integration by parts that if s > 0, t>l, then [^x'-'^{l-xy-'^dx = ~ rx'{i-xy-^d.v. 17. If«>Othenj^^^=J^-^^. [Put.= lA.] 18. IfO<.<lthenj^-3-^^--c^.=j^-^ = j^^^. 19. Ifct + 6>0then ji, {x-ira)^l{x-b) ^/{a + b) ^ ' 334 THE CONVERGENCE OF INFINITE SERIES [vill 20. Show, by means of the substitution x=t/{l -t), that if I and m are both positive then Jo (1+^y*™ jo 21. Show, by means of the substitution a;=pt/(p + l — t), that if I, m, and p are all positive then „ .-^ (1 -.r- (,-^^),.„ = j^:^^]^ .-• (1 - 0™- dt. 22. Prove that/^ ^^(,.g(6_^)} -- ''"'^ AV{(^W} =*'^ ^"+*^' (i) bymeans of the substitution x=a + {h — a)t'^, (ii) by means of the substitu- tion {b—x)j{x — a) = t, and (iii) by means of the substitution a:= a cos^ < + 6 siu^ t. 23. If s > - 1 then , dx 24. Establish the formulae f^ f(x)dx /■*"■,, . a\ja ['' /|a/(^)} '^^•^^a P^/ftan d) cos 5 sin 5 dd- 25. Prove that [Put .r=sin2 (9 and use Ex. lxiii. 8.] {Math Trip. 1912.) 182. Some care has occasionally to be exercised in applying the rule for transformation by substitution. The following example affords a good illustration of this. Let J= [ ' {x^-6x+ 13) dx. We find by direct integration that J=i8. Now let us apply the substitution y=a;2-6^ + 13, which gives x=3±^(y — 4:). Since y = 8 when x=l and 3/ = 20 when a; = 7, we appear to be led to the result J. [^ dx , , , po ydv The indefinite integral is J(3/_4)3/2+4(y-4)>/2, and so we obtain the value + ^, which is certainly wrong whichever sign we choose. 181-183] AND INFINITE INTEGRALS 335 The explanation is to be found in a closer consideration of the relation between x and y. The function a;'^ - 6:i;+ 13 has a minimum for x=Z, when y=4. As a; increases from 1 to 3, y decreases from 8 to 4, and dxjdy is negative, so that dx _ 1 As X increases from 3 to 7, y increases from 4 to 20, and the other sign must be chosen. Thus a formula which will be found to lead to the correct result. Similarly, if we transform the integral I dx=v by the substitution Jo x=a.TO sin y, we must observe that dxldy = ll^{l -y^) or dxjdy = - 1/V(1 -if) according as g a; < ^tt or ^n <x^n. Example. Verify the results of transforming the integrals /{ix^—x+^)dx, j coa^xdx Jo by the substitutions 4x^ — x+^=y, a;=arc sin 2^ respectively. 183. Series of positive and negative terms. Our defini- tions of the sum of an infinite series, and the value of an infinite integral, whether of the first or the second kind, apply to series of terms or integrals of functions whose values may be either positive or negative. But the special tests for convergence or divergence which we have established in this chapter, and the examples by which we have illustrated them, have had reference almost entirely to the case in which all these values are positive. Of course the case in which they are all negative is not essentially different, as it can be reduced to the former by changing w„ into — M„ or ^ (x) into — ^ (x). In the case of a series it has always been explicitly or tacitly assumed that any conditions imposed upon m„ may be violated for a finite number of terms : all that is necessary is that such a condition (e.ff. that all the terms are positive) should be satisfied from some definite term onwards. Similarly in the case of an infinite integral the conditions have been supposed to be satisfied for all values of x greater than some definite value, or for all values of X within some definite interval (a, a + B) which includes the 336 THE CONVERGENCE OF INFINITE SERIES [vin value a near which the subject of integration tends to infinity. Thus our tests apply to such a series as ^ 71" - 10 ^ n' ' since n^ — 10 > when m S 4, and to such integrals as Sec -7 , [n-2x r 3^-7 [i ii (^qny^^*' Jo dx, {x + Vf '. Jo Va; since 3a; - 7 > when no > ^, and 1 - 2a; > when < a; < ^. But when the changes of sign of m„ persist throughout the series, i.e. when the number of both positive and negative terms is in- finite, as in the series l-i + ^-i+-..; or when ^ (x) continually changes sign as a; ^ oo , as in the integral 1"°° sin X , or as x^^a, where a is a point of discontinuity of </)(«), as in the integral /, ^ . / 1 \ da; sm \x — aj X — a then the problem of discussing convergence or divergence becomes more difficult. For now we have to consider the possibility of oscillation as well as of convergence or divergence. We shall not, in this volume, have to consider the more general problem for integrals. But we shall, in the ensuing chapters, have to consider certain simple examples of series con- taining an infinite number of both positive and negative terms. 184. Absolutely Convergent Series. Let us then consider a series Sm» in which any term may be either positive or negative. Let I W» I — "n ) SO that «„ = Un if Un is positive and ««=-«» if «■» is negative. Further, let «;„ = t/„ or Vn = 0, according as m„ is positive or negative, and Wn = — Un or w„ = 0, according as m„ is negative or positive ; or, what is the same thing, let Vn or w„ be equal to a„ according as Un is positive or negative, the other being in either case equal to zero. Then it is evident that v„ and w„ are always positive, and that Un=Vn-Wn, «„ = !'„ -f- W„. 183, 184] AND INFINITE INTEGRALS 337 If, for example, our series is l-(l/2)2 + (I/3)2- ..,, then «„ = (-l)»-V»' and a„ = l/)i2, while v„ = lln^ or v^=0 according as n is odd or even and «'„ = 1/to2 or w„ = according as n is even or odd. We can now distinguish two cases. A. Suppose that the series Sa,i is convergent. This is the case, for instance, in the example above, where 2a„ is 1+ (1/2)^ +(1/3)=+.... Then both S^n and 2w„ are convergent : for (Ex. xxx. 18) any series selected from the terms of a convergent series of positive terms is convergent. And hence, by theorem (6) of § 77, Sm„ or S (v„ — Wn) is convergent and equal to Sw^ — 2w,i- We are thus led to formulate the following definition. Definition. When 2«,i or 2 | «•„ | is convergent, the series 2u„ is said to be absolutely convergent. And what we have proved above amounts to this : if Sm^ is absolutely convergent then it is convergent ; so are the series formed hy its positive and negative terms taken separately ; and the sunn of the series is equal to the sum of the positive terms plus the sum of the negative terms. The reader should carefully guard himself against supposing that the statement ' an absolutely convergent series is convergent ' is a mere tautology. AVhen we say that 2 tf„ is ' absolutely convergent ' we do not assert directly that 2m„ is convergent: we assert the convergence of another series 2|m„|, and it is by no means evident a priori that this precludes oscillation on the part of 2e(n- Examples LXXVII. 1. Employ the 'general principle of convergence' (§ 84) to prove the theorem that an absolutely convergent series is con- vergent. [Since S | m,, | is convergent, we can, when any positive number S is assigned, choose kq so that l"«i + l|+l«n,+2| + ---+ |«nj<8 when m2 > «i £ ?!o- -^ fortiori Kn,+l+«.H + 2+-"+««J<S, and therefore 2i«„ is convergent.] 2. If 2a„ is a convergent series of positive terms, and 1 6„ | S Ka„, then 26,1 is absolutely convergent. 3. If 2 ctn is a convergent series of positive terms, then the series 2ci„ r" is absolutely convergent when -l^.r<l. H. 22 338 THE CONVERGENCE OF INFINITE SERIES [vin 4. If 2 a„ is a convergent series of positive terms, then the series 2 a,i cosn$, 2 Ok sin m5 are absolutely convergent for all values of 6. [Examples are afforded by the series Sr'^cosnB, Sr'^sinnd of § 88.] 5. Anj' series selected from the terms of an absolutely convergent series is absolutely convergent. [For the series of the moduli of its terms is a selection from the series of the moduli of the terms of the original series,] 6. Prove that if 2 1 «« | is convergent then |2?J„|S2|Mi,|, and that the only case to which the sign of equality can apply is that in which every term has the same sign. 185. Extension of Dirichlet's Theorem to absolutely convergent series. Dirichlet's Tlieorem (§ 169) shows that the terms of a series of positive terms may be rearranged in any way without affecting its sum. It is now easy to see that any abso- lutely convergent series has the same property. For let Sm„ be so rearranged as to become 2m„', and let «,/, Vn, w„' be formed from Un as «„, w,j, w„ were formed from Un. Then Sa^' is con- vergent, as it is a rearrangement of Sff,i, and so are S«n', Sw^', which are rearrangements of 2%, 2w,i. Also, by Dirichlet's Theorem, Xvn = "Zvn and 2w„' = 2wn, and so 2m„' = XVn — 2ty„' = XVn — 2w« = "Zun. 186. Conditionally convergent series. B. We have now to consider the second case indicated above, viz. that in which the series of moduli 2a„ diverges to oo. Definition. If Xiin is convergent, hut 2 | «« | divergent, the original series is said to be conditionally convergent. In the first place we note that, if Xun is conditionally con- vergent, then the series 2i)„, 2w„ of § 184 must both diverge to oo . For they obviously cannot both converge, as this would involve the convergence of 2 (f» + Wn) or 2«„. And if one of them, say 2w„, is convergent, and 2v„ divergent, then N N iV ZUn = XVn -2w„ ■(1), and therefore tends to oo with N, which is contrary to the hypothesis that 2wn is convergent. Hence 1v,i, l,w„ are both divergent. It is clear from equa- tion (1) above that the sum of a conditionally convergent series 184-187] anC initinite integrals 339 is the limit of the difference of two functions each of which tends to 00 with n. It is obvious too that 2u» no longer possesses the property of convergent series of positive terms (Ex. xxx. 18), and all absolutely convergent series (Ex. Lxxvil. 5), that any selection from the terms itself forms a convergent series. And it seems more than likely that the property prescribed by Dirichlet's Theorem will not be possessed by conditionally convergent series ; at any rate the proof of § 185 fails completely, as it depended essentially on the convergence of %Vn and Sw„ separately. We shall see in a moment that this conjecture is well founded, and that the theorem is not true for series such as we are now considering. 187. Tests of convergence for conditionally convergent series. It is not to be expected that we should be able to find tests for conditional convergence as simple and general as those of §§ 167 et seq. It is naturally a much more difficult matter to formulate tests of convergence for series whose convergence, as is shown by equation (1) above, depends essentially on the cancelling of the positive by the negative terms. In the first instance there are no comparison tests for convergence of conditionally convergent series. For suppose we wish to infer the convergence of S% from that of Xun We have to compare V0 + V1+ ... +Vn, Mo + Mi + ••■ +M)i- If every u and every v were positive, and every v less than the corresponding u, we could at once infer that Vq+ Vi+ ... +Vn< Uo + ■■■ + Un, and so that Iv^ is convergent. If the u's only were positive and every v numerically less than the corresponding u, we could infer that \%\ + \Vi\ + ... + \Vn\<Ua+ ...+Un, and so that %Vn is absolutely convergent. But in the general case, when the u's and v's are both unrestricted as to sign, all that we can infer is that \v,\+\Vi\ + ...+\Vn\<\Uo\+-..+\Un\. This would enable us to infer the absolute convergence of Sv„ from the absolute convergence of 'Zun', but if S«„ is only con- ditionally convergent we can draw no inference at all. 22—2 340 THE CONVERGENCE OF INFINITE SERIES [vill Example. We shall see shortly that the series 1 — i+J — i + ... is con- vergent. But the series J-l-J+j + i + ... is divergent, although each of its terms is numerically less than the corresponding term of the former series. It is therefore only natural that such tests as we can obtain should be of a much more special character than those given in the early part of this chapter. 188. Alternating Series. The simplest and most common conditionally convergent series are what is known as alternating series, series whose terms are alternately positive and negative. The convergence of the most important series of this type is established by the following theorem. If <^{n) is a positive function of n which tends steadily to zero as n-^cc , then the series </.(0)-,^(l)+,/,(2)-... is convergent, and its sum lies between <j) (0) and (^ (0) — ^ (1). Let us write ^o; ^i. ••• for ^ (0)> <j> (1)) ••■ ; and let s«=^o-<^i+02---- +(-1)" ^«. Then *2»i+i S^n—i ^ Yin Y2n+i = ^> S^n Sam_2 = — \(p2ii—i — <p2ii) = 0- Hence s^, s^, s^, ..., s^n, ■•• is a decreasing sequence, and therefore tends to a limit or to -co, and Sj, Sg., ^s, ..., San+i, •••is an in- creasing sequence, and therefore tends to a limit or to oo . But lim {s^n+i — Sm) = lim (- l)«'+i ^^^^j = 0, from which it follows that both sequences must tend to limits, and that the two limits must be the same. That is to say, the sequence So, s^, ...,Sn, ... tends to a limit. Since So = ^O) ^i = ^q— 4>i> it is clear that this limit lies between (^o and ^o — <^i- Examples LXXVIII. 1. The series ,_1 1_1 n_J_ , 1 1 2 lull! ; (-1)" ; (-1)" s^(-i)" '{n+a)' ^{n + ay {Jn+^a)' {s/n + ^af where a>0, are conditionally convergent. 2. The series 2 (-1)" (»-|-a)-«, where a>0, is absolutely convergent if « > 1, conditionally convergent if < s ^ 1, and oscillatory if s < 0. 18f , 188] AND INFINITE INTEGRALS 341 3. The sum of the series of § 188 lies between s„ and s„ + i for all values of n ; and the error committed by taking the sum of the first n terms instead of the sum of the whole series is numerically not greater than the modulus of the (»i + l)th term. 4. Consider the series (-1)" ^7«+(-i)"' which we suppose to begin with the term for which m=2, to avoid any difficulty as to the definitions of the first few terms. This series may be written in the form say. The series S \|/„ is convergent ; but 2 Xn is divergent, as all its terms are positive, and lim»X"=l- Hence the original series is divergent, although it is of the form i^2-<^3 + 04- •••j where <^„^0. This example shows that the condition that (j>n should tend steadily to zero is essential to the truth of the theorem. The reader will easily verify that iJ{in+V)-\<sl{^n) + l, so that this condition is not satisfied. 5. If the conditions of § 188 are satisfied except that 0„ fends steadily to a positive limit I, then the series 2 (- 1)"^„ oscillates finitely. 6. Alteration of the sum of a conditionally convergent series by rearrangement of the terms. Let s be the sum of the series 1-4+J-J + ..., and S2„ the sum of its first 2(t terms, so that lims2n=s- Now consider the series i+4-4+Hi-i+ W in which two positive terms are followed by one negative term, and let <3„ denote the sum of the first 3n terms. Then 1 JL 1_1_ _J_ i!3u-l + 3 + — + 4„_i 2 4 ■■■ 2?t 1 1 1 2)1 + 1 2n + 3 ■■■ 4?i-l Now lim \j^^ - 2^^qp2 + 2-^^:^ - - +4^-1 - 4^]=°. since the sum of the terms inside the bracket is clearly less than m/(2n + l)(2«+2); and /I 1 1\ 11- 1 ^ 1 _il'^dx l™U + 2 + 2^T4 + - + W^* ».-!.rTW^-^ji cc ' by SS 156 and 158. Hence lim«3„=s + i j^— , 342 THE CONVERGENCE OF INFINITE SERIES [vill and it follows that the sum of the series (1) is not s, but the right-hand side of the last equation. Later on we shall give the actual values of the sums of the two series : see § 213 and Ch. IX, Misc. Ex. 19. It can indeed be proved that a conditionally convergent series can always be so rearranged as to converge to any sum whatever, or to diverge to co or to - 00 . For a proof we may refer to Bromwicli's Infinite Series, p. 68. 7. The series l + -y-- -^ + -Tr+ -7= --T| + ... diverges to 00. [Here 1 1 , 1 '3n = «2n+ /,o„ , i n + /ra„ , on + — + . //^^ _ 1 \ > «2n + where S2n= 1 — 75 + • •• — To~ > "wliich tends to a limit as n-a~ 00 .] 189. Abel's and DiricUet's Tests of Convergence. A more general test, which includes the test of § 188 as a. particular test case, is the following. Dirichlet's Test. // <^» satisfies the same conditions as in § 188, and 2 a„ is any series xohich converges or oscillates finitely, then the series «0 1^0 + '^i <^i + '''2 02 + . • • is convergent. The reader will easily verify the identity where s„=ao + »i + -" + «ii- Now the series (0o~0i) + (0i~02) + -" i* oo'^ vergent, since the sum to n terms is 0o — 0n and lim^„=0; and all its terms are positive. Also since 2a,„ if not actually convergent, at any rate oscillates finitely, we can determine a constant K so that | s„ | < ^ for aU values of v. Hence the series 2s^(0„ — 0^ + i) is absolutely convergent, and so «o(0O-0l)+Sl(01-02) + "-+Sn-l(0>l-l-0n) tends to a limit as m-a-oo. Also 0„, and therefore s„^„, tends to zero And therefore «O0O + «101+-"+«ii0» tends to a limit, i.e. the series 2a^0„ is convergent. Abel's Test. There is another test, due to Abel, which, though of less frequent application than Dirichlet's, is sometimes useful. Suppose that (^„, as in Dirichlet's Test, is a positive and decreasing function of n, but that its limit as m -3- 00 is not necessarily zero. Thus we postulate less about 0k, but to make up for this we postulate more about 2an, viz. that it is convergent. Then we have the theorem : if ct>n is a positive and decreasing function of n, and 'S.an is convergent, then 'S,an<i>n. is convergent. For (f)n has a limit as m -»- 00 , say I : and lim (0n -0 = 0' Hence, by Dirichlet's Test, 2a„{(j)n-l) is convergent; and as Sa^ is convergent it follows that 2a,j0„ is convergent. 188, 189] AND INFINITE INTEGRALS 343 This theorem may be stated as follows: a convergent series remains con- vergent if we multiply its terms ly any sequence of positive and decreasing factors. Examples LXXIX. 1. Dirichlet's and Abel's Tests may also be established by means of the general principle of convergence (§ 84). Let us suppose, for example, that the conditions of Abel's Test are satisfied. We have identically + "-+«m.n-l(<^n-l-0n)+«m,n'#>n (1), The left-hand side of (1) therefore lies between /i0„ and H^)^, where h and H are the algebraically least and greatest of v,m, «m.«.+i, •••, Vn- But, given any positive number S, we can choose OTq so that | v " I <S ^h^a m i OTq, and so when n>m>mQ. Thus the series 2a„(^„ is convergent. 2. The series 2cosw5 and 2sin?i^ oscillate finitely when 6 is not a multiple of n. For, if we denote the sums of the first n terms of the two series by s„ and i„, and write z==Cis.6, so that \z\ = l and «4=1, we have |«n+!<„| l-z"|_^H-|£''J^_ 2 1_2|=|1_3| =|I-2|' and so | s„ | and | <„ | are also not greater than 2/ 1 1 - « | . That the series are not actually convergent follows from the fact that their wth terms do not tend to zero (Exs. xxiv. 7, 8). The sine series converges to zero if ^ is a multiple of tt. The cosine series oscillates finitely if 6 is an odd multiple of n and diverges if 6 is an even multiple of n. It follows that if (^„ is a positive function of n which tends steadily to aero as n-^aa , t/ien the sm'ies ScpnOoand, Scftn sin nd are convergent, except perhaps the first series when 5 is a multiple of Stt. In this case the first series reduces to S^ni which may or may not be conver- gent: the second series vanishes identically. If 2<^„ is convergent then both series are absolutely convergent (Ex. lxxvii. 4) for all values of d, and the whole interest of the result lies in its application to the case in which 20„ is divergent. And in this case the series above written are con- ditionally and not absolutely convergent, as will be proved in Ex. lxxix. 6. If we put 5 = IT in the cosine series we are led back to the result of § 188, since cos tctt = ( - 1)". 3. The series Sn~'coand, Sn~'smnd are convergent if s>0, unless (in the case of the first series) is a multiple of 27r and 0<sS,l. 34,4 THE CONVERGENCE OF INFINITE SERIES [vill 4. The series of Ex. 3 are in general absolutely convergent if s>l, conditionally convergent if 0<sg 1, and oscillatory if s g (finitely if s=0 and infinitely if s<0). Mention any exceptional cases 5. If 2a,iK.~» is convergent or oscillates finitely, then 2ffl„?i~' is convergent when t>s. 6. If 4>n is a positive function of n which tends steadily to as m ->- co , and S<f>n is divergent, then the series 2(^„cosm5, S^nSinmS are not absolutely convergent, except the sine-series when 5 is a multiple of jr. [For suppose, e.g., that 2 (^„ | cos nd \ is convergent. Since cos^ n8 g | cos nS \ , it follows that 2 0„ 008'"^ n6 or |2<^,i(l + cos2?ifl) is convergent. But this is impossible, since 2<^» is divergent and 20„ cos 2n6, by Dirichlet's Test, convergent, unless 5 is a multiple of n. And in this case it is obvious that 2 0„ | cos nd | is divergent. The reader should write out the corresponding argument for the sine-series, noting where it fails when ^ is a multiple of vr.j 190. Series of complex terms. So far we have confined ourselves to series all of whose terms are real. We shall now consider the series tUn = S (Vn + nUn), where Vn and Wn are real. The consideration of such series does not, of course, introduce anything really novel. The series is convergent if, and only if, the series are separately convergent, There is however one class of such series so important as to require special treatment. Accordingly we give the following definition, which is an obvious extension of that of § 184. Definition. The series 2it„, where Un = Vn + iWn, is said to he absolutely convergent if the series tvn and twn are absolutely convergent. Theorem. The necessary and sufficient condition for the absolute convergence of Xun is the convergence of 2 | w„ | or S //(Vn + Wn)- For if 2it„ is absolutely convergent, then both of the series 2 1 1;„ 1 , 2 1 w„ I are convergent, and so 2 { | ?^„ | + 1 w„ | } is con- vergent : but 189-191] AND INFINITE INTEGRALS 345 and therefore S | m„ | is convergent. On the other hand SO that 2 I ■y» I and 2 | w» | are convergent whenever S | Mn ( is con- vergent. It is obvious that an absolutely convergent series is convergent, since its real and imaginary parts converge separately. And Dirichlet's Theorem (§§ 169, 185) may be extended at once to absolutely convergent complex series by applying it to the separate series 2vn and Sw»„. The convergence of an absolutely convergent series may also be deduced •directly from the general principle of convergence (cf. Ex. lxxvii. 1). We leave this as an exercise to the reader. 191. Power Series. One of the most important parts of the theory of the ordinary functions which occur in elementary analysis (such as the sine and cosine, and the logarithm and exponential, which will be discussed in the next chapter) is that which is concerned with their expansion in series of the form 'Za^x'"'. Such a series is called a power series in x. We have already come across some cases of expansion in series of this kind in connection with Taylor's and Maclaurin's series (§ 148). There, however, we were concerned only with a real variable x. We shall now consider a few general properties of power series in z, where z is & complex variable. A. A power series XanZ'^ may he convergent for all values of z, for a certain region of values, or for no values except z = Q. It is sufficient to give an example of each possibility. 1. The series 2 —7 is convergent for all values of x. For if m,,= — ; then |«»H-i|/|2«J = hl/(''»+l)-*0 as m -»- 00 , whatever value z may have. Hence, by d'Alembert's Test, 2 | «,, | is convergent for all values of z, and the original series is absolutely con- vergent for all values of z. We shall see later on that a power series, when convergent, is generally absolutely convergent. 2. The series 2n!z" is not convergent for any value of z except z=0. For if v,n = n\ z^ then | Wn + 1 I/I «u 1 = (» + 1) I ^ 1 1 which tends to 00 with n, unless ^ = 0. Hence (cf. Exs. xxvii. 1, 2, 5) the modulus of the nVa. term tends to oo with n; and so the series cannot converge, except when 2=0. It is obvious that any power series converges when 2=0. 346 THE CONVEKGENCE OF INFINITE SERIES [VIII 3. The series Sz" is always convergent when |2|<1, and never convergent when \z\>l. This was proved in g 88. Thus we have an actual example of each of the three possibilities. 192. B. If a power series 2 anZ^ is convergent for a par- ticular value of z, say % = r^ (cos d-^ + i sin ^i), then it is absolutely convergent for all values of z such that \z\ <r-y. For lim anZi" = 0, since 2a„«i'' is convergent, and therefore we can certainly find a constant K such that | a,A"' | < Z' for all values of n. But, if | ^ | = r < ri, we have and the result follows at once by comparison with the convergent geometrical series 2 (»'/i'i)"- In other words, if the series converges at P then it converges absolutely- at all points nearer to the origin than P. Example. Show that the result is true even if the series oscillates finitely when s=z^. [If s„=ao + ai2i + ... +««%" then we can find K so that I s„ I < /f for all values of n. But | a^z^" I = | s,. - Sn-i | ^ I s»-i I + 1 Sn I < 2^, and the argument can be completed as before.] 193. The region of convei'gence of a power series. The circle of convergence. Let z = r he any point on the positive real axis. If the power series converges when z = r then it converges absolutely at all points inside the circle | ^ | = r. In particular it converges for all real values of z less than r. Now let us divide the points r of the positive real axis into two classes, the class at Avhich the series converges and the class at which it does not. The first class must contain at least the one point z = (). The second class, on the other hand, need not exist, as the series may converge for all values of z. Suppose however that it does exist, and that the first class of points does include points besides z = Q. Then it is clear that every point of the first class lies to the left of every point of the second class. Hence there is a point, say the point z = R, which divides the two classes, and may itself belong to either one or the other. Then the series is absolutely convergent at all 'points inside the circle \z\ = R. 191-193] AND INFINITE INTEGRALS 347 For let P be any such point. We can draw a circle, whose centre is and whose radius is less than R, so as to include P inside it. Let this circle cut OA in Q. Then the series is con- vergent at Q, and therefore, by Theorem B, absolutely conver- ° gent at P. On the other hand the series cannot converge at any point P' ^^^' ^^' outside the circle. For if it converged at P' it would converge absolutely at all points nearer to than P ; and this is absurd, as it does not converge at any point between A and Q' (Fig. 51). So far we have excepted the cases in which the power series (1) does not converge at any point on the positive real axis except z = or (2) converges at all points on the positive real axis. It is clear that in case (1) the power series converges nowhere except when z = Q, and that in case (2) it is absolutely convergent everywhere. Thus we obtain the following result: a power series either (1) converges for z = and for no other value of z; or (2) converges absolutely for all values of z; or (3) converges absolutely for all values of z within a certain circle of radius R, and does not converge for any value of z outside this circle. In case (3) the circle is called the circle of convergence and its radius the radius of convergence of the power series. It should be observed that this general result gives absolutely no information about the behaviour of the series on the circle of convergence. The examples which follow show that as a matter of fact there are very diverse possibilities as to this. Examples LXXX. 1. The series H-a2+a^a^+... , where a > 0, has a radius of convergence equal to 1/a. It does not converge anywhere on its circle of convergence, diverging when 2=l/a and oscillating finitely at all other points on the circle. 2. The series t2 + H2 + o2 + '" ^^^ ^*^ radius of convergence equal to 1 ; it converges absolutely at all points on its circle of convergence. 348 THE CONVERGENCE OF INFINITE SERIES [vill 3. More generally, if | a„ + 1 1 / ] a„ | ^- X, or | a„ |''" -»- X, as ?i -»- co , then the series ao + «i ^ + fflj^^ + . • • has 1/X as its radius of convergence. In the first case lim|ci„+is''+M/l«n2"l = >^|2|. which is less or greater than unity according as | 2 | is less or greater than 1/X, so that we can use D'Alembert's Test (§ 168, 3). In the second case we can use Cauchy's Test (§ 168, 2) similarly. 4. The logarithmic series. The series is called (for reasons which will appear later) the 'logarithmic' series. It follows from Ex. 3 that its radius of convergence is unity. "When z is on the circle of convergence we may write z = cos5+isinfl, and the series assumes the form cos 5-| cos 25 + J cos 35-. ..+i (sin 5-^ sin 25+ J sin 35 -...). The real and imaginary parts are both convergent, though not absolutely convergent, unless 6 is an odd multiple of w (Exs. lxxix. 3, 4). If 5 is an odd multiple of tr then 2= —1, and the series assumes the form — 1-| — J-..., and so diverges to — qo . Thus the logarithmic series converges at all points of its circle of convergence except the point s= — 1. 5. The hinomial series. Consider the series m(m— 1) „ TO(TO-l)(»n — 2) , l+m2:+— ^2i — -z^-^-^ g^p ^zH... If m IS a positive integer then the series terminates. In general \an\ « + l ' so that the radius of convergence is unity. W6 shall not discuss here the question of its convergence on the circle, which is a little more difficult. * 194. Uniqueness of a power series. If SanZ" is a power series which is convergent for some values of 2 at any rate besides 2 = 0, and /(«) is its sum, then it is easy to see that/(2) can be expressed in the form fl!i)+ai2 + cr232+... + (a„ + c2)2'', where Ej-s-O as 1 2 1-3-0. For if /n is any number less than the radius of con- vergence of the series, and |2|</x, then |am|/i"<A'', where AT is a constant (c£ § 192), and so |/(2)-2a,2''|slo„ + i||2"-^i| + |a„ + 2||2''-''| + ... ^^ Z|2h+i -©"'(-^-^■-)%-? (m-I^I • * Seo Bromwich, Infinite Series, pp. 225 et seq. ; Hobsou, Plane Trigonometry (3rd edition), pp. 268 et seq. 193-195] AND INFINITE INTEGRALS 349 where Z is a number independent of z. It follows from Ex. lv. 15 that if 2a„2" = s6n2" for all values of z whose modulus is less than some number ji, then «» = 6„ for all values of n. This result is capable of considerable generalisations into which we cannot enter now. It shows that the same function f{z) cannot be represented hy two different power series. 195. Multiplication of Series. We saw in § 170 that if 2m,j and Ivn are two convergent series of positive terms, then 1<Un X 2i)„= XiUn> where Wn = U^Vn + UiVn-1 + . . . + Un'l'o- We can now extend this result to all cases in which Srt,j and l,Vn are absolutely convergent ; for our proof was merely a simple application of Dirichlet's Theorem, which we have already ex- tended to all absolutely convergent series. Examples LXXXI. 1. If |0| is less than the radius of convergence of either of the series 2a„c", 26„2", then the product of the two series is Sc^z", where e„=a|,6„ + ai6„_i + ... + a„65. 2. If the radius of convergence of 2a„2" is R, and /(a) is the sum of the series when \z\<E, and \z\ is less than either B or unity, then /(2)/(l-2) = 2s„2", where .s„=ao + ai + . .. + «„. 3. Prove, by squaring the series for l/(l-s),tha,tl/(l—zf = l + -2z+3z^ + ... il\z\<l. 4. Prove similarly that 1/(1 -2)^ = 1+32+62^+..., the general term being ^(•/i + l)(ji + 2)2". 5. The Binomial Theorem for a negative integral exponent. If 1 2 1 < 1, and m is a positive integer, then 1 m(m+l) „ , m(m + l)...(m + n-l) (1-2)"' 1.2 1.2...m [Assume the truth of the theorem for all indices up to m. Then, by Ex. 2, 1/(1 - 2)™ + 1 = 2 s^z", where m{m + l) , , m(m + \)...{m + n-'l) _(m+l)(m + 2)...(m+n) s,= l+m+ ^2 +...+ ^yy--— ^^^-^^^^^^ , as is easily proved by induction.] 6. Prove by miiltiplication of series that if /(«,2) = l + (™)2+(™)2H..., and 1 2 1 < 1, then/('/n, z)f{m', 2) =/(m+m', 2). [This equation forms the basis of Euler's proof of the Binomial Theorem. The coefficient of 2" in the product series is (:')KT) (»-.)+©(»"; +-+1,". The 350 THE CONVERGENCE OF INFINITE SEEIES [vill This is a polynomial in m and m' : but when m and m! are positive integers this polynomial must reduce to ( , ) , in virtue of the Binomial Theorem for a positive integral exponent, and if two such polynomials are equal for all positive integral values of m and m' then they must be equal identically.] 7. If /(0) = l + 2+ |^+... then f{z)f{^)=f{z+z!). [For the series for f{z) is absolutely convergent for all values of z : and it is easy to see that if .f! ., _^" .^„„ „., _(i±^ t««=-, , '»n=—r then w„=^— — - .] 8. If c(.)=l-ij + |j-..., 5 (.)=.-!-, + g^-..., then ClzJrz') = C (z) C (s') - ^S (2) -S (2!), Si2 + z') = S (z) C {z') + C(,z)jS (,/), and {C(2)P + {S(2)}2 = 1. 9. Failure of the Multiplication Theorem. That the theorem is not always true when Sm„ and Sf^ a^^e not absolutely/ convei'gent may be seen by considering the case in which (-1)" Then n 1 But ^{(r + 1) (ra + 1 - )■)} S i (m + 2), and so | w„ | > (2» + 2)/(9i + 2), which tends to 2 ; so that 2w„ is certainly not convergent. MISCELLANEOUS EXAMPLES ON CHAPTER VIII. 1. Discuss the convergence of the series Sn'' {^{n + 1) - 2^n+^(n-l)} , ■where k is real. {Math. Trip. 1890.) 2. Show that 2re''A*(?i>'), where Am„ = m„ - m„ + 1 , a2«„ = A ( A«Oi and so on, is convergent if and only if ^ >?• + « + 1, except when s is a positive integer less than k, when every term of the series is zero. [The result of Ch. VII, Misc. Ex. 11, shows that A*(?i») is in general of order m«~'.] 3. Show that " ?t'+9w + 5 _5_ i(m + l)(2?i + 3)(2?H-5)(TO+4)~36'- {Math. Trip. 1912.) [Resolve the general term into partial fractions.] AND INFINITE INTEGRALS 351 4. Show that, if R {n) is any rational function of n, we can determine a polynomial -P(m) and a constant A such that 'S,{R{n) — P{n)-{Aln)} is convergent. Consider in particular the cases in which R{n) is one of the functions ll(an + b), {a9i^ + 2b>i + c)/(an^ + 2l3n + y). 5. Show that the series l+z'^'2 2 + 2+3 3 + 2'^'" is convergent provided only that z is not a negative integer. 6. Investigate the convergence or divergence of the series .a 1 . a / IN., ■ « /, o,\ , ,^ /, o,\ Ssm-, 2-sm-, 2(-l)"sm-, 2 1-oos- , 2(-l)"?i 1-cos- , n n m n ' \ ?i/ ' \ nj' where a is real. 7. Discuss the convergence of the sei-ies 1 \ 2 3 nJ 11 where B and a are real. {Math. Trip. 1899.) 8. Prove that the series in which successive terms of the same sign form groups of 1, 2, 3, 4, ... terms, is convergent ; but that the corresponding series in which the groups contain 1, 2, 4, 8, ... terms oscillates finitely. {Math. Trip. 1908.) 9. If 1*1, «2i ''S) •■•is ^ decreasing sequence of positive numbers whose limit is zero, then the series Ml--^(Ml + «2) + iK + ''2+«3)-..., Mi-J(Mi+M3)+J(Mi + M3 + M5)--.- are convergent. [For if (tii + !(2 + -.- +'«.i)/« = !'n then Dj, v^, V3, ... is also a decreasing sequence whose limit is zero (Ch. IV, Misc. Exs. 8, 27). This shows that the first series is convergent ; the second we leave to the reader. In particular the series i-Hi+4)+Hi+i+J)--- i-ja+i)+Ki+Hi)-- are convergent.] 10. If iio + Ui + U2+... is a divergent series of positive and decreasing terms, then (Uo+U2 + ...+y^i)l{Ui + Us + ...+'U2n + l)^'i- 11. Provethat if a>0 then lim S {p + n)~^~''=0. 12 Prove that lim a 2 to~^~'"=1. [It follows from § 174 that a^+O ] <l-'-" + 2-'"''+... + (« - 1)"'""- j oT^-'-dx^l, and it is easy to deduce that 2?r'"'" lies between 1/a and (l/a) + l.] 352 THE CONVEEGENCE OF INFINITE SERIES [viU oo 13. Find the sum of the series 2 ?<„, where 1 _ a;"-a;-"-i 1 / 1 _ 1 \ ""~ (*■" + :»-") (xo + l + A--"-!) ~ X^l \.V" + X-'' ~ a;" + l+^-"-lj ' for all real values of x for which the series is convergent. (Math. Trip. 1901.) ' [If I X I is not equal to unity then the series has the sum x/{(x-l) («2+l)}. If a;=l then iCn=0 and the sum is 0. It x=-l then i«„==^(- i)"+i and the series oscillates finitely.] 14. Find the sums of the series z 2z^ ie* z z^ ^ T+i''"l + z'''"'"l+2*' 1 -2^ + 1 -24 + 13^+- (in which all the indices are powers of 2), whenever they are convergent. [The first series converges only if | z | < 1, its sum then being 2/(1 —z) ; the second series converges to z/(l -z) if 1 2 1 < 1 and to 1/(1 — 2) if 1 2 1 > 1.] 15. If I a„ I ^ 1 for all values of n then the equation 0=1+ a-^z + a^^ -V ... cannot have a root whose modulus is less than \, and the only case in which it can have a root whose modulus is equal to \ is that in which «„= - Cis(n5), when z = \ Cis ( - 5) is a root. 16. Recurring Series. A power series 'S.a,^ is said to be a recurring series if its coefficients satisfy a relation of the type an+P\an-\+Pta-n-2+ ■.■+Pkan-h=0 (1), where n'Z.Jc and py, p^, ..., p^ are independent of n. Any recurring series is the expansion of a rational function of 2. To prove this we observe in the first place that the series is certainly convergent for values of z whose modulus is suflaciently small. For let G be the greater of the two numbers 1, \pA + \P2\ + - + \Pic\- Then it follows from the equation (1) that \an\£,Gan, where an is the modulus of the numerically greatest of the preceding coefficients ; and from this that I an I < A'&'", where K is independent of n. Thus the recurring series is certainly convergent for values of 2 whose modulus is less than \jQ. But if we multiply the series f(z) = 1anZ^ by pjZ, ^2^^ ■■•Pt^i a"d add the results, we obtain a new series in which all the coefficients after the Qc— l)th vanish in virtue of the relation (1), so that where P,,, Pj, ..., Pj-i are constants. The polynomial 1 +^12+^2^^ + ••• +^'42* is called the scale of relation of the series. Conversely, it follows from the known results as to the expression of any rational function as the sum of a polynomial and certain partial fractions of the type Al{z — a.y, and from the Binomial Theorem for a negative integral AND INFINITE INTEGRALS 353 exponent, that any rational function whose denominator is not divisible by z can be expanded in a power series convergent for values of 2 whose modulus is sufficiently small, in fact if | « | < p, where p is the least of the moduli of the roots of the denominator (of. Ch. IV, Misc. Exs. 18 et seq.}. And it is easy to see, by reversing the argument above, that the series is a recurring series. Thus the necessary and sufficient condition that a power series should be a recurring series is that it should he the expansion of such a rational function of z. 17. Solution of Difference-Equations. A relation of the type of (1) in Ex. 16 is called a linear difference-equation in a„ ivith constant coefficients. Such equations may be solved by a method which will be sufficiently ex- plained by an example. Suppose that the equation is an-a»-i-8a„_2-H12a„_3=0. Consider the recurring power series 2a„2». We find, as in Ex. 16, that its sum is a(,+{ai-ao)z + {a2-ai-%a^)z^ ^ A^ A^ B 1-2-82^122^ 1-22 "*" (1-22)2 "''1-1-32' where Ai, A2, and B are nurabers easily expressible in terms of Oq, «!, and a^. Expanding each fraction separately we see that the coefScient of 2" is a„=2»{^i + (M-|-l)^2}-t-(-3)»5. The values of A^, A^, B depend upon the first three coefficients oiq, ai,ai, which may of course be chosen arbitrarily. 18. The solution of the difference-equation w„— 2cos5tt„_i-f m„_2 = is Un=A cos nd + B sin nd, where A and B are arbitrary constants. 19. If M„ is a polynomial in n of degree k, then 2m„2" is a recurring series whose scale of relation is (1 - z)''*^. (Math. Trip. 1904.) 20. Expand 9/{(2 — l)(z + 2)^} in ascending powers of 2. {Math. Trip. 1913.) 21. Prove that if /(m) is the coefficient of 2" in the expansion of 2/(1 -f2-|-22) in powers of 2, then (1) f{n)+f{n-l)+fin-2y=0, (2) /(m) = (o.3''-a,3=»)/(a.3-<032), where 0)3 is a complex cube root of unity. Deduce that f(n) is equal to or 1 or —1 according as m is of the form 3k or 3^-1-1 or Si: + 2, and verify this by means of the identity zl(l + z+z^)=z {l-z)l{l-z^). 22. A player tossing a coin is to score one point for every head he turns up and two for every tail, and is to play on until his score reaches or passes a total n. Show that his chance of making exactly the total tojsJ{2-|-(-J)»}. (Math. Trip. 1898.) [Itpn is the probability thcnpn=i (Pn-i+Pn-2)- also Po=1-,Pi = i-] H. 23 , dx 854 THE CONVERGENCE OF INFINITE SERIES [vill 23. Prove that if m is a positive integer and a is not one of the numbers —1, —2, ..., —n. [This follows from splitting up each term on the right-hand side into partial fractions. When a> -\, the result may be deduced very simply from the equation J 1—^ J by expanding (l-^'')/(l-iB) and \ — {l-xY in powers of x and integrating each term separately. The result, being merely an algebraical identity, must be true for all values of a save — 1, —2, ..., — m.] 24. Prove by multiplication of series that 2— jS^ —T- =2 1+5 + + — + - — r o?i!i n.n\ i\ 2 3 nj ni [The coefficient of z" will be found to be Now use Ex. 23, taking a! = 0.] 25. If An^'A and Ba-s-B as n-»-oo , then {AiB^+A2B„_i + ... + A^Bi)ln^AB. [Let A„=A + en. Then the expression given is equal to j^ Bi + B2 + ... + B^ fi.B„+f2^n-i + ... + f«-Bi n n The first term tends to ^B (Ch. IV, Misc. Ex. 27). The modulus of the second is less than /3{|€i|-t-|f2| + .-. + | f»i|}/», where /3 is any number greater than the greatest value of 1 5^ | : and this expression tends to zero.] 26. Prove that if c„=ai6„+a26n-i + -.- + a„6i and An = ai-\-a2+...+an, .B„ = 6i + 63+...-|-6m, C„ = Ci-|-C!2 + ...+C,i, then C„=aiB„-fa2-B„_i+...+a„5i = 6i^,j-l-624„_i + ... + 6n^i and Ci+C^ + ... + C^=AiB^+A^B^_i + ...+A^Bi. Hence prove that if the series 2an, 26^ are convergent and have the sums A, B, so that An^'A, Bn-^B, then {Ci+C2 + ... + Cn)ln^AB. Deduce that if 2Ck is convergent then its sum is AB. This result is known as Abel's Theorem on the multiplication of Series. We have already seen that we can multiply the series 2a„, 'Sbn in this way if both series are absolutely convergent : Abel's Theorem shows that we can do so even if one or both are not absolutely convergent, provided only that the product series is convergent. AND INFINITE INTEGRALS 355 27. Prove that i(i-HJ--)^=i-i(i+i)+i(i+H^)--, i(i-i+i-...)^=i-i(i+J)+Hi+J+i)--- [Use Ex. 9 to establish the convergence of the series.] 28. For what vahies of m and n is the integral / sin™ x {I- cos a-)" dx Jo convergent ? [If m + 1 and m + 2m + 1 are positive.] 29. Prove that if a > 1 then fl dX TV /: -i(a-*)V(l-^) Jid'-iy 30. Establish the formulae j^ Fy{x^' + l)-x}dx=ij'^ (l + l)i^(y)rfy. In particular, prgve that if ?i > 1 then [In this and the succeeding examples it is of course supposed that .the arbitrary functions which occur are such that the integrals considered have a meaning in accordance with the definitions of §§ 177 ei seq.] 31. Show that if ^y = ax—{hjx), where a and 6 are positive, then y in- creases steadily from — oo to co as ;r increases from to qo . Hence show that /o"-^{* ('^^ 4)} <i^-lf_JW^<^^)} {^^rK^au}'^ =lf7w(/+a6)Wy. "■ ] 32. Show that if 27/ = ax + (blx), where a and 6 are positive, then two values of x correspond to any value of y greater than J(ab). Denoting the greater of these by Xi and the less by x^, show that, as y increases from J(ab) towards ao, Xi increases from Jib/a) .towards co, and X2 decreases from ■Jibja) to 0. Hence show that 23—2 and that 356 INFINITE SERIES AND INTEGKALS [vill 33. Prove the formula f (sec ix-\- tan * x) -rr-. r = ; / (cosec .v) -tj-. ; , 34. If a and b are positive, then jo(a''' + a2)(^2 + 62)-2a6(a + 6)' jo {x'^-¥d^){x^-\-b^)~ 'i{a + h)' Deduce that if a, ft and y are positive, and ff^ S ay, then dx _ '^ r" .2?^!^^ _ n f a:r« + 2;8^2 + y ~ 27(2y^) ' j a^+2|3.»2 + y ~ 2 V(2a4) ' where .4=3+V(a')')- Also deduce the last result from Ex. 31, by putting f{y) = ll{c^+y^). The last two results remain true when ^^<ay, but their proof is then not quite so simple. 35. Prove that if 6 is positive then C °° x^dx ^jr_ r°° x*dx jr^ jo (x^-a'f + b''x^ 2b' Jo {{x^-a'y' + bKvy~ ib^' 36. Extend Schwarz's inequality (Ch. VII, Misc. Ex. 42) to infinite integrals of the first and second kinds. /: 37. Prove that if <}>{x) is the function considered at the end of § 178 then jo ^^ ' o(»+l)^ 38. Prove that Establish similar results in which the limits of integration are and 1. IJiath. Trip. 1913.) CHAPTER IX THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS OF A REAL VARIABLE 196. The number of essentially different types of functions with which we have been concerned in the foregoing chapters is not very large. Among those which have occurred the most important for ordinary purposes are polynomials, rational functions, algebraical functions, explicit or implicit, and trigonometrical functions, direct or inverse. We are however far from having exhausted the list of functions which are important in mathematics. The gradual expansion of the range of mathematical knowledge has been accompanied by the introduction into analysis of one new class of function after another. These new functions have generally been introduced because it appeared that some problem which was occupying the attention of mathematicians was incapable of solution by means of the functions already known. The process may fairly be compared with that by which the irrational and complex numbers were iirst introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised. One of the most fruitful sources of new functions has been the problem of mtegration. Attempts have been made to integrate some function f(oo) in terms of functions already known. These attempts have failed; and after a certain number of failures it has begun to appear probable that the problem is insoluble. Sometimes it has been proved that this is so ; but as a rule such a strict proof has not been forthcoming until later on. Generally it has happened that mathematicians have taken the impossibility for granted as soon as they have become reasonably convinced of it, and have introduced a new function F (x) defined by its 358 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix possessing the required property, viz. that F' (x) =f(cc). Starting from this definition, they have investigated the properties of F(a;) ; and it has then appeared that F (x) has properties which no finite combination of the functions previously known could possibly have; and thus the correctness of the assumption that the original problem could not possibly be solved has been established. One such case occurred in the preceding pages, when in Ch. VI we defined the function log a; by means of the equation ''dx 1 f' loga;=j - Let us consider what grounds we have for supposing log.j; to he a really new function. We have seen already (Ex. XLii. 4) that it cannot be a rational function, since the derivative of a rational function is a rational function whose denominator contains only repeated factors. The question whether it can be an algebraical or trigonometrical function is more difficult. But it is very easy to become convinced by a few experiments that differentiation will never get rid of algebraical irrationalities. For example, the result of differentiating ^'(1 +^) any number of times is always the product of ^(1+^) by a rational function, and so generally. The reader should test the correctness of the statement by experimenting with a number of examples. Similarly, if we differentiate a function which involves sin x or cos x, one or other of these functions persists in the result. We have, therefore, not indeed a strict proof that log.» is anew function — that we do not profess to give* — but a reasonable presumption that it is. We shall therefore treat it as such, and we shall find on examination that its properties are quite unlike those of any function which we have as yet encountered. 197. Definition of log x. We define log x, the logarithm of x, by the equation logx=j^ -. We must suppose that x is positive, since (Ex. Lxxvi. 2) the integral has no meaning if the range of integration includes the point a; = 0. We might have chosen a lower limit other than 1 ; but 1 proves to be the most convenient. With this definition log 1=0. We shall now consider how log x behaves as x varies from towards oo . It follows at once from the definition that log a; is a * For such a proof see the author's tract quoted on p. 236. 196, 197] OF A KEAL VARIABLE 359 continuous function of x which increases steadily with x and has a derivative Dx\ogx= l/x; and it follows from § 175 that log x tends to oo as a; ^ oo . If a; is positive but less than 1, then log a; is negative. For log. = j^- = -J^-<0. Moreover, if we make the substitution t = 1/u in the integral, we obtain ]oga,=j^- = -J^ _ = _log(l/a.). Thus log X tends steadily to — oo as x decreases from 1 to 0. The general form of the graph of the logarithmic function is shown in Fig. 52. Since the derivative of log x is 1/a;, the slope of Y Fig. 52. the curve is very gentle when x is very large, and very steep when X is very small. Examples LXXXII. 1. Prove from the definition that if it > then ul{\+u) < log (!+«)< u. [For log (1 + m) = / — — , and the subject of integration lies between 1 and It, w 2. Prove that log(H-M) lies between m--^- and m-^-,-^ when u is /■« tdt -. positive. [Use the' fact that log (!+») = «- /^ j^ -J 3. If < M < 1 then m <-log (!-«)< W(l -'*)• 4. Prove that lim ^-^= lim '-SfJllfLl. [Use Ex. 1.] ^X—\ f-»0 360 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix 198. The functional equation satisfied by l6gx. The function log a; satisfies the functional equation f{ooy)=f(x)+f(y) (1). For, making the substitution t = yu, we see that _ ['"' dt _ f"" du _ f'^du _ f^!«du J] t Jl/y u Ji u Ji u = log a; - log (Ijy) = log « + log y, which proves the theorem. Examples LXXXIII. 1. It can be shown that there is no solution of the equation (1) which possesses a differential coefficient and is fundamentally distinct from log m. For when we differentiate the functional equation, first with respect to x and then with respect to y, we obtain the two equations 3'/'(^^)=/'W, 'of'{xy)=f{y); and so,, eliminating /' {xy), xf'.{x) =yf' (y). But if this is true for every pair of values of x and y, then we must have xf (x) = G, or /' (») = C/.r, whore C is a constant. Hence f{x)=\^dx+C' = C\ogx+C', and it is easy to see that C"=0. Thus there is no solution fundamentally distinct from log.r, except the trivial solution /(^) = 0, obtained by taking 0=0. 2. Show in the same way that there is no solution of the equation /w+/(3/)^v(r4) which possesses a differential coefficient and is fundamentally distinct from arc tan x. 199. The manner in which log x tends to infinity with x. It will be remembered that in Ex. xxxvi. 6 we defined certain different ways in which a function of x may tend to infinity with x, distinguishing between functions which, when x is large, are of the first, second, third, . . . orders of greatness. A function / {or) was said to be of the Ath order of greatness when / {x)lx^ tends to a limit different from zero as x tends to infinity. It is easy to define a whole series of functions which tend to infinity with x, but whose order of greatness is smaller than the fir.st. Thus y/x, ^x, \/x, . . . are such functions. We may say generally that «*, where a is any positive rational number, is of the «th order of greatness when x is large. We may suppose a as small 198-201] OF A EEAL VARIABLE 361 as we please, e.g. less than -0000001. And it might be thought that by giving a all possible values we should exhaust the possible 'orders of infinity' of f (x). At any rate it might be supposed that if / (x) tends to infinity with x, however slowly, we could always find a value of a so small that x°- would tend to infinity more slowly still; and, conversely, that ii f(x) tends to infinity with x, however rapidly, we could always find a value of a so great that a;" would tend to infinity more rapidly still. Perhaps the most interesting feature of the function log x is its behaviour as x tends to infinity. It shows that the presupposition stated above, which seems so natural, is unfounded. The logarithm of X tends to infinity with x, but more slowly than a,ny positive power of X, integral or fractional. In other words loga;-*- oo but of- for all positive values of a. This fact is sometimes expressed loosely by saying that ' the order of infinity of log x is infinitely small '; but the reader will hardly require at this stage to be warned against such modes of expression. 200. Proof that (loga;)/a;''^-0 as aj^oo. Let /3 be any positive number. Then 1/^ < l/i'~^ when t>\, and so pdi f^ dt log^ = J^-<j^^,_-^, or log x<{x^-l)\^< aPj^, when x>\. Now if a is any positive number we can choose a smaller positive value of p. And then < (log x)lx'- Kcc^-^/B (a; > 1 ). But, since a > /3, x^~'^/l3-^0 asxs- cc , and therefore (loga;)/a;°'-^0. 201. The behaviour of log a; as a; -* + 0. Since {log x)lx- = -y'' logy ii x = 1/y, it follows from the theorem proved above that lim 2/"logy = - lim (log a;)/*" = 0. j/^ + O x^ + x Thus log a; tends to - oo and log (1/a;) =-loga; to oo as a; tends to zero by positive values, but log (1/a;) tends to oo more slowly than any positive power of 1/a;, integral or fractional. 362 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix 202. Scales of infinity. The logarithmic scale. Let us consider once more the series of functions which possesses the property that, if f{x) and <^ (*•) are any two of the functions contained in it, then/(^) and <^ {x) both tend to m as x-»-xi, while f{x)j(j) (x) tends to or to oo according as f{x) occurs to the right or the left of <f> {x) in the series. We can now continue this series by the insertion of new terms to the right of all those already written down. We can begin with log X, which tends to infinity more slowly than any of the old terms. Then ^/(log^) tends to oomore slowly than log^, i^Q.ogx) than ./(log «), and so on. Thus we obtain a series X, ^Jx, ^x, ..., Xlx, ... \ogx, J{\ogx), 4/(log«), N/(loga;), ... formed of two simply infinite series arranged one after the other. But this is not all. Consider the function log log a;, the logarithm of \ogx. Since (\.ogx)lx°--^-0, for all positive values of a, it follows on putting x=\ogy that (log log y)/(log y)'^= (log x)lx " ^ 0. Thus log logy tends to oo with y, but more slowly than any power of logy. Hence we may continue our series in the form X, ^x, i/x,... log X, v/(log x), il(\.ogx), ... log log X, V(log log x),... ^(log log x),...; and it will by now be obvioiis that by introducing the functions log log log x, log log log log X, ... we can prolong the series to any extent we like. By putting x=llywe obtain a similar scale of infinity for functions of y which tend to cx) as y tends to by positive values. * Examples LXXXIV. 1. Between any two terms /(a;), I'(x) of the series we can insert a new tei-m (x) such that <t> (x) tends to oo more slowly than f{x) and more rapidly than i^(^). [Thus between >Jx and ^x we could insert _^6/i2. between v'(log.j;) and 4^(log ^) we could insert Q.ogxyi'^K And, generally, (^ {x) = ij{f{x) F{x)} satisfies the conditions stated.] 2. Find a function which tends to oo more slowly than ^Ix, but more rapidly than x"-, where a is any rational number less than 1/2. [.Jx/{logx) is such a function; or ^x/Q.ogx)^, where is any positive rational number.] 3. Find a function which tends to oo more slowly than ^x, but more rapidly than slxj{\og x)"-, where a is any rational number. [The function ■Jxl(].oglogx) is such a function. It will be gathered from these examples that incompleteness is an inherent characteristic of the logarithmic scale of infinity.] 4. How does the function f{x) = {*•« (log*-)"' (log log xf]j{x^ (log xf (log log xf'} behave as x tends to oo ? [If a=l=/3 then the behaviour of f{x) = x''-^ {logxf'-^' (loglog A-)*'""^" * For fuller information as to ' scales of_infinity ' see the author's tract ' Orders of Infinity ', Gamb. Math. Tracts, No. 12. 202, 203] OF A HEAL VARIABLE 363 is dominated by that of o,-"'"^. If a=/3 then the power of x disappears and the behaviour of /(x) is dominated bythat of (log^)"*'"^', unless a' = j3', when it is dominated by that of (log log a,-)"*"-^". Thus/(a-)^oo if a > /3, or a=/3, a'>0', or a=/3, a' = 0', a"> 0", and / (:i;) -* if a < ft or a = ft a' < (3', or a = fta' = ^',a"</3".] 5. Arrange the functions aVs/(log a;), ^ ^(log «)/log log x, x log log .r/^/(log x\ (^logloglog;i;)/V(loglog;t;) according to the rapidity with which they tend to infinity as a;-»-QO. 6. Arrange logloga;/(5;log^), (log;;?)/^, ,rloglog^/V(A'2 + l), {v/(a; + 1 )}/.!; (log a.^^ according to the rapidity with which they tend to zero as a;-9-oo . 7. Arrange a;loglog(l/a;), v/-^'/{log (1/a;)}, v'{^sin^log(l/;i;)}, (1 - cos «) log (l/«) according to the rapidity with which they tend to zero as ^-»- + 0. 8. Show that A log log x=lj{x log x), D^ log log log a= ] l{x log ;;; log log x), and so on. 9. Show that B^ (log 0!)'^=-- al{x (log x)^ - % D^ aog log x)''=al{x log X (log log xf - % and so on. 203. The number e We shall now introduce a number, usually denoted by e, which is of immense importance in higher mathematics. It is, like ir, one of the fundamental constants of analysis. We define e as the number whose logarithm is 1. In other words e is defined by the equation t ' Since log x is an increasing function of x, in the stricter sense of § 95, it can only pass once through the value 1. Hence our definition does in fact define one definite number. Now log XT/ = log X + log y and so log ic' = 2 log a;, log«'= 3 log*, ..., log a;" = n log a;, where n is any positive integer. Hence log e" = n log e = n. -i:- 364 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix Again, if p and q are any positive integers, and e^l^ denotes the positive gth root of e^, we have p = log fiP = log (e3'/«)« = 5 log e'Pli, so that log e^'/a = pjq. Thus, if y has any positive rational value, and e^ denotes the positive yih. power of e, we have \ogey = y (1), and log e"" = — log e^ = -y. Hence the equation (1) is true for all rational values of y, positive or negative. In other words the equations y = loga;, a; =6" (2) are consequences of one another so long as y is rational and ey has its positive value. At present we have not given any definition of a power such as e" in which the index is irrational, and the function e" is defined for rational values of y only. Example. Prove that 2 < e < 3. [In the first place it is evident that /: and so 2 < e. Also [^dt_ f^dt [^ dt^ n du n du ^ r du ji T' ji t "*" J2 t ~Jo 2-u jo 2 + M jo 4-t(2> J so that e < 3.] 204. The exponential function. We now define the ex- ponential function e^ for all real values of y as the inverse of the logarithmic function. In other words we write if 2/ = log X. We saw that, as x varies from towards oo , y increases steadily, in the stricter sense, from '— co towards oo . Thus to one value of x corresponds one value of y, and conversely. Also y is a continuous function of x, and it follows from § 109 that x is likewise a continuous function of y. It is easy to give a direct proof of the continuity of the exponential function. For if x = ey and ^■ + |=e''+'' then /•«+s* dt "=]. T- Thus I t; I is greater than |/(a;+ 1) if | > 0, and than |||/a; if | < ; and if i; is very small ^ must also be very small. 203-205] OF A REAL VARIABLE 365 Tkus e" is a positive and continuous function of y which increases steadily from towards x as y increases from — oo towards oo . Moreover e^ is the positive yth power of the number e, in accordance with the elementary definitions, whenever y is a rational number. In particular e" = 1 when y = 0. The general form of the graph of e'' is as shown in Fig. 53. 205. The principal properties of the exponential function. (1) If x = ey, so that y = \ogx, then dyjdx=\lsc and dx dy Thus the derivative of the exponential function is equal to the function itself. More generally, \i x = e^y then dxjdy = ae<^y. (2) The exponential function satisfies the functional equation f{y+^)-f{y)f(f)- This follows, when 2/ and z are rational, from the ordinary rules of indices. If y or z, or both, are irrational then we can choose two sequences yi,y2, ..., y^, ... and z^, z^, ...,^„, ... of rational numbers such that lim2/„ = 2/, lim.^„=^, Then, since the exponential function is continuous, we have g!/ X e^ = lim ei'n x lim e^» = lim e^^+^n = ey*'. In particular s" x e"^ = e» = 1, or e"^ = 1/e^. We may also deduce the functional equation satisfied by e^ from that satisfied by log x. For if y^ = log x^,yi = log x^, so that x^ = ew. , x^ = ey^ then yi + 2/2 = log «i + log x^ = log x^x^ and 366 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix Examples LXXXV. l. If dxjdy—ax then x=Ke'^y, where A" is a constant. 2. There is no solution of the equation f(j/+z)=f{^)f{z) fundamentally distinct from the exponential function. [We assume that /(,y) has a differential coefficient. Differentiating the equation with respect to y and- z in turn, we obtain f'(j/+^)=f'ii/)f(.^), /'(y+2)=/(2/)/'W and so /'()/)//(?/) =/'(«)//( 2), and therefore each is constant. Thna ii a;=f{y) then dxjdy = ax, where a is a constant, so that x = Ke'"i (Ex. 1).] 3. Prove that («"!' -\)jy-»-a as y -^0. [Applying the Mean Value Theorem, we obtain e"" -1 = ay e«i, where < 1 1; | < 1 y | •] 206. (3) The function e" tends to infinity with y more rapidly than any power of y, or lim 2/"/e» = lim e-yy^ = as y ^<x> , for all values of a however great. We saw that (loga;)/a;^-*0 as a; -^00, for any positive value of /3 however small. Writing « for 1//8, we see that (log «)"/«-»■ for any value of a however large. The result follows on putting X = e^. It is clear also that e^^ tends to 00 if 7 > 0, and to if 7 < 0, and in each case more- rapidly than any power of y. From this result it follows that we can construct a ' scale of infinity ' similar to that constructed in § 202, but extending in the opposite direction ; i.e. a scale of functions which tend to co more and more rapidly as x^-x.* The scale is «. 7.2 .V.3 px fi2x px^ /)X3 ^e* where of course e*^, ..., e'', ... denote eW, ..., eM, .... The reader should try to apply the remarks about the logarithmic scale, made in § 202 and Bxs. lxxxiv, to this 'exponential scale' also. The two scales may of course (if the order of one is reversed) be combined into one scale ...log log ^, ... log .■»!•, ... X, ... e", ... e"', .... 207. The general power a^. The function a* has been defined only for rational values of x, except in the particular case ■1' The exponential function was introduced by inverting the equation j/ = log a; into x = ey ; and we have aooordingly, up to the present, used y as the independent and X as the dependent variable in discussing its properties. We shall now revert to the more natural plan of taking x as the independent variable, except when it is necessary to consider apair of equations of the type y= log a;, x = e" simultaneously, or when there is some other special reason to the contrary. 205-207] OF A REAL VARIABLE 367 when a = e. We shall now consider the case in which a is any positive number. Suppose that a; is a positive rational number p/q. Then the positive value y of the power aP/« is given by yf = aP; from which it follows that qlogy=p log a, log y = (pjq) log a = a; log a, and so y = e^'oeo. We take this as our definition of a^ when x is irrational. Thus 10V2 = eV2i<'8:io It ig to ije observed that a", when x is irrational, is defined only for positive values of a, and is itself essentially positive; and that log a'' = x log a. The most important properties of the function a" are as follows. (1) Whatever value a may have, a'^xa'-' = O^+v and {a'^)y = a'^v. In other words the laws of indices hold for irrational no less than for rational indices. For, in the first place, and in the second ((X*)^ = e^lo^"^ = e^!/loBa = Qxy_ (2) If a > 1 then a* = e'^'^'S" = e""', where a is positive. The graph of a* is in this case similar to that of e% and a^^txi as a; -»- CO , more rapidly than any power of x. If a< 1 then a" = e»^'°^<» = e'^", where /8 is positive. The graph of a" is then similar in shape to that of ^, but reversed as regards right and left, and a^-^-0 as x^cc , more rapidly than any power of Ijx. (3) a* is a continuous function of x, and D^ a" = Da^e^'"^" = e^i°s» log a = a^ log a. (4) a^ is also a continuous function of a, and Da a"' = Da e^'"^" = e^'°sa (a;/^) = a;a*-i. (5) (a"'— !)/;»-* logo, as x^^O. This of course is a mere corollary from the fact that Dxa'' = a''\oga, but the particular form of the result is often useful ; it is of course equivalent to the result (Ex. Lxxxv. 3) that (e" — !)/« ^ a as a; ^ 0. In the course of the preceding chapters a great many results involving the function a' have been stated with the limitation that x is rational. The definition and theorems given in this section enable us to remove this restriction. 368 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS. [ix 208. The representation of e"' as a limit. In Ch. IV, §73, we proved that {l+(l/n)}" tends, as n^ao , to a limit which we denoted provisionally by e. We shall now identify this limit with the number e of the preceding sections. We can however establish a more general result, viz. that expressed by the equations lim ('l + -f = limfl --")"" = e» (1). As the result is of very great importance, we shall indicate alter- native lines of proof. (1) Since |log(l + ^0 = r:^^. it follows that ,. log(l +xK) lim ° , = X. h^o ft. If we put h = 1/^, we see that lim ^ log ( 1 + -g 1 = a; as^-*oo or^-^— 00. Since the exponential function is con- tinuous it follows that as ^-^00 or ^-*— cc: i.e. that \im(l + -y= limfl-l-|?=e^ (2). If we suppose that ^^^oo or^^- — oo through integral values only, we obtain the result expressed by the equations (1). (2) If n is any positive integer, however large, and ;«; > 1, we have f" dt f^dt^ [^ dt J 1 <1 +•(!/») < j 1 T "^ j 1 'fl^iVn) ' or TC(l-a;-i/") <log^<»i(a;V''-l) (3). Writing y for logo;, so that y is positive and x = ei, wo obtain, after some simple transformations, H)'<'<H)" «■ i^-t)' Now let 208-210] OF A REAL VARIABLE 869 Then 0<r]i<ri2, at any rate for sufficiently large values of n; and, by (9) of §74, which evidently tends to as n ^ oo . The result now follows from the inequalities (4). The more general result (2) may be proved in the same way, if we replace 1/w by a continuous variable h. 209. The representation of log^ as a limit. We can also prove (cf. § 75) that lim n (1 -^-'/'>)=limM (a.-V''-l)=loga;. For n {x^'" - 1) -?i (1 - a-- v») = ,i (xV^-i) (1 -^-v), which tends to zero as n-^ao, since «(xV''-l) tends to a limit (§ 75) and a-"» to 1 (Ex. XXVII. 10). The result now follows from the inequalities (3) of §208. Examples LXXXVI. 1. Prove, by taking y = l and m=6 in the in- equalities (4) of § 208, that 2 . 5 < e < 2 . 9. 2. Prove that if <> 1 then ((!'/'■_ i-i/«)/(i-_ j-i) < i/„_ and so that if A- > 1 then p dt [=' dt 1 /'"'/,_1\^_1 / 1 o^ Hence deduce the results of § 209. 3. If $n is a function of n such that m|„ -»- ? as m -»- cjo , then (1 +^„)''-»- e'. [Writing ra log (1+|„) in the form and using Ex. lxxxii. 4, we see that n log (1 + |„) -s- 1.} 4. If raf„^-oo, then (l + |n)"^-°o I and if 1 + |„>0 and «|„ -*--«>, then (l + ^„)»^0. 5. Deduce from (1) of § 208 the theoi'em that e" tends to infinity more rapidly than any power of y. 210. Common logarithms. The reader is probably familiar with the idea of a logarithm and its use in numerical calculation. He will remember that in elementary algebra logaa;, the logarithm of X to the base a, is defined by the equations x = ay, y=\ogaX. This definition is of course applicable only when y is rational, though this point is often passed over in silence. H. 24 370 THE LOGARITHMIC AND EXPOXENTIAL FUNCTIONS [ix Our logarithms are therefore logarithms to the base e. For numerical work logarithms to the base 10 are used. If y = \ogx = log, X, z = logic CO, then x = ey and also «= 10^ = e"^"^^", so that logic 00 = (logsa;)/(log, 10). Thus it is easy to pass from one system to the other when once loge 10 has been calculated. It is no part of our purpose in this book to go into details concerning the practical uses of logarithms. If the reader is not familiar with them he should consult some text-book on Elementary Algebra or Trigonometry.* Examples LXXXVII. 1. Show that Dx e°^ cos hx = re"™ cos (6^ + 5), I)xe'"'sin.hx= reF' sin ( J.r + 6) ■where r = v'(a^ + 6^), cos 6 = alr, sin d=blr. Hence determine the nth deri- vatives of the functions e°^ cos &.r, e'^sinhx, and show in particular that 2. Trace the curve y=e~"* sin 6:r, where a and b are positive. Show that y has an infinity of maxima whose values form a geometrical progression and which lie on the curve 3. Integrals containing the exponential function. Prove that /" , , , aooa6;!;-(-6sin6^ f „^ ■ -l , a sin bx-b cos bx , I e<™ cos bxdx = 5 — r^ e™, / e'" sm bx ax = ^ — rs e ■ J a^+b^ J a^ + b^ [Denoting the two integrals by I, J, and integrating by parts, we obtain al = e^^: cos bx + b J, aJ=e^''ainbx-bI. Solve these equations for /and J.] 4. Prove that the successive areas bounded by the curve of Ex. 2 and the positive half of the axis of x form a geometrical progression, and that their sum is b l+e-""'" a^ + b'^l-g-'"'"'' ■ 5. Prove that if o > then I e-'"^oosbxdx = -T—y.,, I e''^'^ sin bxdx = Jo a^ + b" Jo b a' + b^' * See for example Chrystal's Algebra, vol. i, oh. xxi. The value of logj 10 is 2-302... and that of its reciprocal 'iSi... . 210] OF A EEAL VARIABLE 371 6. lll„=je<":x''dx then al^==e'"'a;'^-7il,,_i. [Integi-ate by parts. It follows that /„ can be calculated for all positive integral values of m.] 7. Prove that, if n is a positive integer, then Jo \ ^ 2\ nlj and / e-^^Vch-^^nl. 8. Show how to find the integral of any rational function of e'^. [Put ^=logM, when e*=w, dx/du = l/u, and the integral is transformed into that of a rational function of v.] 9. Integrate e ,2i (C V + «%-»:) (C V + 6% - a:) ' distinguishing the cases in which a is and is not equal to 6. 10. Prove that we can integrate any function of the form P{x, e", ^^, ...), where P denotes a polynomial. [This follows from the fact that P can be expressed as the sum of a number of terms of the type Ax"'e'"^, where m is a positive integer.] 11. Show how to integrate any function of the form P(^, e% e'"^, ,.., ooslx, cosmx, ..., ainlx, ainmx, ...). 12. Prove that I e->^ R (x) dx, where X > and a is greater than the J a, greatest root of the denominator of R {x), is convergent. [This follows from the fact that e^^ tends to infinity more rapidly than any power of x.] /oo e-^'^^+i^'^dx, where X > 0, is convergent for all values of f" fi, and that the same is true of I e-f^x'+ii-xx^dx, where n is any positive integer. 14. Draw the graphs of e^, e"^, xe^, xe'',xe^, xe''^, and .rlog.r, deter- mining any maxima and minima of the functions and any points of inflexion on their graphs. 15. Show that the equation e<"i^ — bx, where a and b are positive, has two real roots, one, or none, according as 6 > ae, 6 = ae, or 6 < ae. [The tangent to the curve y = ea« at the point (^, e'>i) is y — e'^i= ae"^ (x — |), which passes through the origin if a|=l, so that the line y—aex touches the curve at the point (1/a, e). The result now becomes obvious when we draw the line y=hx. The reader should discuss the cases in which a or 6 or both are negative.] 24—2 372 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [iX 16. Show that the equation e'^ = l+x has no real root except a'=0, and that e''=l+x+^x'' has three real roots. 17. Draw the graphs of the functions /l+x\ log {a; + J(x^ + l)}, log (t^^.) ) €-«'«' cos^ bx, 18. Determine roughly the positions of the real roots of the equations log{x + ^{x^ + l)} = :^^, e==-|±| = j^, e-sin^-=7, e^^ sin ^-=10000. 19. The lijrperbolic functions. The hyperbolic functions cosh x* sinha;, ... are defined by the equations cosh «=-^ (e^+g-^), sinh x=^{e^—e~^), tanh .a; = (sinh ^)/(cosh x), coth x = (cosh .j;)/(sinh x), sech x=l /(cosh x), cosech x = l/(sinh x). Draw the graphs of these functions. 20. Establish the formulae cosh (-.?;) = cosh ^, sinh ( — :!?)=— sinh :!;, tanh ( - ^) = - tanh a:, cosh^^ — sinh2^=l, seoh2^+tanh2.r=l, coth^.^; — cosech2^=l, cosh 2.27 = cosh^ X + sinh^ x, sinh 2.» = 2 sinh x cosh x, cosh {x+y) = cosh x cosh y + sinh x sinh y, sinh {x+y) = sinh x cosh y + cosh x sinh y. 21. Verify tha.t these formulae may be deduced from the corresponding formulae in cos a; and sin .j;, by writing cosh x for cos x and i sinh x for sin x. [It follows that the same is true of all the formulae involving cos nx and sinnx which are deduced from the corresponding elementary properties of cos^' and sin.^-. The reason of this analogy will appear in Oh. X.] 22. Express cosh x and sinh x in terms {a) of cosh 2^ (6) of sinh ix. Discuss any ambiguities of sign that may occur. {Math. Trip. 1908.) 23. Prove that Z^a; cosh x= sinh .», Zf^; sinh .^• = cosh .f, i)a:tanh,r=sech2.r, D^CQi\h.x= -cosech^o;, /(j: sech .r = - sech ^ tanh x, D^ cosech x= — cosech x coth x, Z)j. log cosh X = tanh :;;, D^ log | sinh .» | = coth x, B^ arc tan e^=\ sech x, Z)j; log | tanh \x\ = cosech x. [All these formulae may of course be transformed into formulae in inte- gration.] * ' Hyperbolio cosine ' ■ for an explanation of this phrase see Hobson's Trigo- nometry, ch. XVI. 210] OF A Seal variable 373 24. Prove that cosh x>l and - 1 < tanh .r < 1 . 25. Prove that if y=cosha; then x=\og{2/±^/(y''-l)}, if2/ = sinha' then J!=log[i/ + ^(i/^ + l)},a,nd a i/ = ta.ti\ix then x=i\og {(l + i/)/{l-2/)}. Account for the ambiguity of sign in the first case. 26. We shall denote the functions inverse to cosh x, sinh x, tanh x by arg cosh x, arg sinh x, arg tanh x. Show that arg cosh x is defined only when ^ > 1, and is in general two-valued, while arg sinh x is defined for all real values of ^, and arg tanh ^ when -1<a'<1, and both of the two latter functions are one-valued. Sketch the graphs of the functions. 27. Show that it -^n <x<^Tr and y is positive, and cos x cosh y = 1 , then y = log(sec^-|-tan^), I)^y=aeox, D„x=-tmohy. 28. Prove that if a > then f ., f^ ,. = arg sinh {xja), and f „ f " „ is J \l(X' + a') J J[x' — a^) equal to arg cosh {xja) or to - arg cosh {-xja), according as a;> or ^ <0. 29. Prove that if a > then I ^21:^2 '® equal to - (1/a) arg tanh {xja) or to - (1/a) arg coth {xja), according as | a; | is less than or greater than a. [The results of Exs. 28 and 29 furnish us with an alternative method of writing a good many of the formulae of Ch. VI.] 30. Prove that [ dx \- j{{x-t){b-x)r ^^'''^^'' \/(f^) ^'^ <^<*)- 31. Prove that \x\og{\+^x)dx=l-i\ogi<hi\v'^dx = l. Jo Jo {Math. Trip. 1913.) 32. Solve the equation a cosh a'-f-6 sinh x = c, where c > 0, showing that it has no real roots if b^ + c^ — a^<0, while if b^ + c?-a'^>0 it has two, one, or no real roots according SiS a + b and a — b are both positive, of opposite signs, or both negative. Discuss the case in which b^ + c' -a^ = 0. 33. Solve the simultaneous equations cosh x cosh y = a, sinh x sinh y = 6. 34. .v^P'^l as x^cD. [For «i/j:=e(iogj:)/i and (log ^-Va- ^ 0. Of. Ex. XXVII. 11.] Show also that the function x^/" has a maximum when x = e, and draw the graph of the function for positive values of a?. 35. x^^l as x-* + 0. 374 THE LOGABITHMIC AND EXPONENTIAL FUNCTIONS [ix 36. If {f(n + l)}l{f{n)]^l, where l>0, as m-^co, then X/{f{n))-^l. [For log/(« + l)-log/(>i)^log?, and so (l/w)log/(ra)^logZ (Ch. IV, Misc. Ex. 27).] 37. !i/(n \)ln -3-\je as ?i -a- oo . [If/(w)=»-''re! then {/(to+1)}/{/(ii)} = {l + (l/?i)}-"-*l/e. Now use Ex. 36.] 38. ;y{(2w) !/(«-!)-} ^4 asm ^00. 39. Discuss the approximate solution of the equation g«=a'W"«""i. [It is easy to see by general graphical considerations that the equation has two positive roots, one a little greater than 1 and one very large* arid one negative root a little greater than - 1. To determine roughly the size of the large positive root we may proceed as follows. If e'=x^'">''^ then x=\Q'^\ogx, log«=13-82 + logloga;, logloga; = 2-63 + log ('l + ^-^HfV rougUy, since 13'82 and 2'63 are approximate values of log 10° and log log 10' i-espeotively. It is easy to see from these equations that the ratios log x : 13'82 and log log .r : 2-63 do not differ greatly from unity, and that a;=10<'(13-82+logloga;) = 10°(13-82 + 2-63) = 16450000 gives a tolerable approximation to the root, the error involved being roughly measured by lO" (log log .2; -2-63) or (10''logloga;)/13-82 or (10" x 2-63)/13-82, which is less than 200,000. The approximations are of course very rough, but suffice to give us a good idea of the scale of magnitude of the root,] 40. Discuss similarly the equations e^ = 1000000 ^m"""™, e»^= ^1000000000. 211. Logarithmic tests of convergence for series and integrals. We showed in Ch. VIII (§§ 175 et seq.) that U-13 ^'^'^ are convergent if s > 1 and divergent if s S 1. Thus 2 (1/m) is divergent, but 2 w~'""* is convergent for all positive values of a. "We saw however in § 200 that with the aid of logarithms we can construct functions which -tend to zero, as n -^ 00 , more rapidly than l/«, yet less rapidly than n~^~'^, however small a may be, provided of course that it is positive. For example l/(«logm) is such a function, and the question as to whether the series wlog n * The phrase ' very large ' is of course not used here in the techmcal sense explained in Ch. IV, It means ' a good deal larger than the roots of such equations as usually occur in elementary mathematics '. The phrase ' a little greater than ' must be interpreted similarly. 210, 211] OF A REAL VARIABLE 375 is convergent or divergent cannot be settled by comparison with any series of the type S n~K The same is true of such series as 2 i V log log w ?i(logn)^' «V(logw)* It is a question of some interest to find tests which shall enable us to decide whether series such as these are convergent or divergent; and such tests are easily deduced from the Integral Test of § 174. For since D^ (log xy-^ = -^,^^ , D^ log log « =-r^ , x{\ogxy' " ^ » a, log a;' we have [i dx _ (log ^y-^- (log ays r( dx , , . , , j„^(bi^« T^s 'J ^g^ = l°gl«gf-l°gl°g«' if a > 1. The first integral tends to the limit - (log ay-'/(l - s) as ^ -^ 00 , if s > 1, and to oo if s < 1. The second integral tends to 00 . Hence the series and integral Z,n(logn)'' J a x(logxy' where «o <^nd a are greater than unity, are convergent if s>l, divergent ^f s &!. It follows, of course, that S(^(«) is convergent if 0(n) is positive and less than K/{n(logny], where s > 1, for all values of n greater than some definite value, and divergent if ^ (n) is positive and greater than K/(n log n) for all values of n greater than some definite value. And there is a corresponding theorem for integrals which we may leave to the reader. Examples LXXXVIII. 1. The series are convergent. [The convergence of the first series is a direct consequence of the theorem of the preceding section. That of the second follows from the fact that (log nY"" is less than n^ for sufficiently large values of n, how- ever small /3 may be, provided that it is positive. And so, taking /3 = 1/200, {logny" m-iM/ioo is less than n-'iomoo for sufficiently large values of n. The convergence of the third series follows from the comparison test at the end of the last section.] 376 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix 2. The series n (log nfP ' Mwo/wi(logm)i'»' (nlog»)2 + l are divergent. 3. The series (logw)i' (logw)''(l0gIogTO)g (I0gl0g?l)g »! + » ' «! + ' ' m(Iog«)i + «' where s>0, are convergent for all values oi p and q ; similarly the series ni-»(logM)''' »i-«(logre)»(loglogra)«' w(logTO)i-«(loglog?i)!' are divergent. 4. The question of the convergence or divergence of such series aa ^ 1 ^ log log log n n log n log log n ' n log n ^/(log log n) cannot be settled by the theorem of p. 375, since in each case the function under the sign of summation tends to zero more rapidly than l/(?ilog>i) yet less rapidly than ?i~i (logji)"''"'', where u is any positive number however small. For such series we need a still more delicate test. The reader should be able, starting from the equations 1-s i)^(log4x)i-»= X log X logs ^- • -logi-i X (logi xY ' 1 X log X loga a-. . . logs _ 1 :s logj. X ' where log2 ^=logloga;, log3.j;=logloglog«, ..., to prove the following theorem : the series and integral ~ 1 r dx ^ n log n logo w . . . logt _ 1 « (logj re)» ' j „ ^ log :» loga ^ . . . log^ _ j ^ (log^ x)' are convergent if s>l and divergent if sgl, n^ and a being any numbers sufficiently great to ensure that log^ra and logj:;; are positive when n^n^ or x^a. These values of «o and a increase very rapidly as k increases: thus log^>0 requires «>1, log2a;>0 requires x>e, loglog^>0 requires X > e', and so on ; and it is easy to see that e^> 10, e^° > eW > 20,000, e^ > e^o.ooo > io8ooo_ The reader should observe the extreme rapidity with which the higher exponential functions, such as e«°^ and e^ , increase with x. The same remark of course applies to such functions as a'^^ and Ofl'^ , where a has any value greater than unity It has been computed that 99' has 369,693,100 figures, while lOlOW has of course 10,000,000,000. Conversely, the rate of increase of the higher logarithmic functions is extremely slow. Thus to make log log log log a; > 1 we have to suppose x a number with over 8000 figures.* * See the footnote to p. 362. 211] OF A REAL VARIABLE 377 5. Prove that the integral / - jlog f-\\ dec, where < a < 1, is con- vergent if s < - 1 , divergent if s > - 1 . [Consider the behaviour of as f-^+0. This result also may be refined upon by the introduction of higher logarithmic factors.] 6. Prove that / -jlog(-U cl.v has no meaning for any value of s. [The last example shows that s < — 1 is a necessary condition for convergence at the lower limit : but {log(l/.5;)}» tends to oo like (l-^)», as a^-^-l-O^lf « is negative, and so the integral diverges at the upper limit when «<-!.] 7. The necessary and sufficient conditions for the convergence of I aJ^-Mlogf-U da; are a>0, s>-l. Examples LXXXIX. 1. Eiiler's limit. Show that (t> in)=^l+^ + - + ... + — -log n tends to a limit y as »i-*-oo, and that 0<ygl. [This follows at once from § 174. The value- of y is in fact -577..., and y is usually called Euler's constant.] 2. If C6 and b are positive then 111 I 1, , .^ -H — n:+ —TFT. + •■■-' — r? tti - Tlog[a + nb) a a + b a + 2b a + (n-l)b b tends to a limit as »-»-<x) . 3. IfO<s<lthen <^(n) = l+2-'-|-3-»-l-...-l-(»-l)-»-^ tends to a limit as w^- oo . 4. Show that the series 1+2(1+^)^3(1+1+1)^- is divergent. [Compare the general term of the series with l/(»ilog?i).] Show also that the series derived from 2 «~», in the same way that the above series is derived from 2 (l/«), is convergent if s > 1 and otherwise divergent. 5. Prove generally that if 2m„ is a series of positive terms, and Sn = 1ti + 1l2+---+'^n, then 2 («„/«„ -i) is convergent or divergent according as 2?«„ is convergent or 378 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix divergent. [If 2m„ is convergent then »„_i tends to a positive limit I, and so 2(m„/s„_i) is convergent. If 2!{„ is divergent then ,?„_i^»-oo, and M„/s„_i>log{l + (M„/s„_i)} = log(sJs„_,) (Ex. Lxxxil. 1) ; and it is evident that log (S3/S1) + log {S3/S2) + . . . + log (s Js„_ j) = log (sjsi) tends to 00 as m-s-oo.] 6. Prove that the same result holds for the series 2 (ujs^). [The proof is the same in the case of convergence. If Su^ is divergent, and m„<s„_i from a certain value of n onwards, then s„<2s„_i, and the divergence of 2 {ujs„) follows from that of 2 (ujsn-i). If on the other hand % i«„_i for an infinity of values of n, as might happen with a rapidly divergent series, then M„/s„ g|- for all these values of n.] 7. Sum the series 1 — -^ + ^ — . . .. [We have '^+l+- + ^=^°S(^n+l) + y + e^, 2g+^ + ...+l)=log(»+l) + y+f„', by Ex. 1, y denoting Euler's constant, and fn, en being numbers which tend to zero as n-^co . Subtracting and making k— 00 we see that the sum of the given series is log 2. See also § 213.] 8. Prove that the series 2(-l)»(l+^ + ,... + ^-log.-0) oscillates finitely except when C= y, when it converges. 212. Series connected with the exponential and log- arithmic functions. Expansion of e'" by Taylor's Theorem. Since all the derivatives of the exponential function are equal to the function itself, we have 2! {n- 1)! n\ where 0<^<1. But a!'Yw!^-0 as m-* 00 , whatever be the value of a; (Ex. xxvii. 12); and e'" < e^. Hence, making n tend to 00 , we have e. = l + , + |- + ... + |- + , (1). The series on the right-hand side of this equation is known as the exponential series. In particular we have = 1 + 1+1 + ...+ ^, + (2); and so (1 + 1 + 2^ + ... + -,+ ...) =1+^ + -+...+ - + (3), 211, 212] OF A REAL VARIABLE 379 a result known as the exponential theorem. Also a* = e»^i«B«=i + (a;loga)+(^M^+ (4) for all positive values of a. The reader will observe that the exponential series has the property of reproducing itself when every term is differentiated, and that no other series of powers of x would possess this property : for some further remarks in this connection see Appendix II. The power series for ^ is so important that it is worth while to investigate it by an alternative method which does not depend upon Taylor's Theorem. Let and suppose that x>0. Then (^^^\' i_L (A ^nin-V) /x\^ , n{n-\)...l fxy which is less than En {x). And, provided 'n>x, we have also, by the binomial theorem for a negative integral exponent. Thus A •^A"" T , M , n{n + \) fxy (i+l)"<^„(.)<(i-0-: But (§ 208) the first and last functions tend to the limit e^ as ra-a-oo, and therefore Enix) must do the same. From this the equation (1) follows when X is positive ; its truth when x is negative follows from the fact that the exponential series, as was shown in Ex. lxxxi. 7, satisfies the functional equation /(^)/(^)=/(A'+y), so that/(.i-)/(-x)=/(0) = l. Examples XC. 1. Show that , x"^ .«* . , a? sfi cosh«=H-27 + j-, + ..., 8mh^'=.i;+g-| + r-,H-.... 2. If X is positive then the greatest term in the exponential series is the ([^] + l)-th, unless x is an integer, when the preceding term is equal to it. 3. Show that n \>{n\ef: [For n^\n ! is one term in the series for e".] 4. Prove that e" = (»"/?» !) (2 + ^j + >S'2), where l + v (l + i')(l + 2v) and v = l/n; and deduce that n ! lies between 2 (w/a)" and 2 (m + 1) (n/e)% 5. Employ the exponential series to prove that e"^ tends to infinity more rapidly than any power of x. [Use the inequality e'>x'"jn !.] 380 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix 6. Show that e is not a rational number. [If e=pjq, where p ^nd q are integers, we must have or, multiplying up by g ! , ^'V? 2! - q\) ^ + l + (j + l)(2+2)-^- and this is absurd, since the left-hand side is integral, and the right-band side less than {l/(^-f-l)} + {l/(2'-|-l)}2-|-... = l/?.] 7. Sum the series 2 P^ (n) — , where P^ W is a polynomial of degree r ^ • in n. [We can express P^ W in tli© form .4(i-f id i«-l-yl2'''{™ "!) + ••■ + ^>-™ («-!)• •■('»- '■+1)> and 2 Pr (n) ^ =^0 2 — -f- Ai^ , -<-. + . .. + Ar2 . ^ 8. Show that OO «3 °° 71* i?i! ' 1 ml and that if <S„=13-(-23-i-...+9i3 then i 5„ ^ = i (4r .). 14j;H 8X-3 -l-o:*) e"'. 1 m! In particular the last series is equal to zero when x= —2. {Math. Trip. 1904.) 9. Prove that 2 {njn !) = e, 2 {n^jn !) = 2e, 2 {n^jn !) = 5e, and that 2 {rfju !), where k is any positive integer, is a positive integral multiple of e. 10. Prove that 2 ) ti , = {(■^^^ -Sx + 3)e^ + ha;^ - Z}/x\ [Multiply numerator and denominator by m-l- 1, and proceed as in Ex. 7.] 11. Determine a, b, a so that {{x + a)e'^+(bx + c)}/x^ tends to a limit as x-3-0, evaluate the limit, and draw the graph of the function e^-\ . 12. Draw the graphs of 1 + x, l+x+^a;% l + x + ^x'^ + ^ifi, and compare them with that of c==. 13. Prove that e-''-l+x-'-^ + ...-( — iy>''— is positive or negative according as n is odd or even. Deduce the exponential theorem. 212, 213] OF A REAL VARIABLE 381 14. If then dXyjdx = X„ _ j . Hence prove that if <> then X^{t)= f Xadx<te\ Xi{t)= \ X^dx< [^xe='dx<e' {*xdx=t-,e\ Jo Jo Jo Jo 2!' t" and generally X^ {t) < — e'. Deduce the exponential theorem. 15. Show that the expansion in powers of p of the positive root of a;2+»=a2 begins with the terms a{l-J^^loga + J^21oga(2 + logci)}. {Math. Trip. 1909.) 213. The logarithmic series. Another very important eicpansion in powers of x is that for log (1 +x). Since and 1/(1 + 1) = 1 — i + 1^ — . . . if Hs numerically less than unity, it is natural to expect* that log (1 + x) will be equal, when — 1 < a; < 1, to the series obtained by integrating each term of the series 1 — t + £' — ... from t=0 to t=x,i.e. to the series x — ^x^ + ^a?— And this is in fact the case. For 1/(1 +t) = l-t + t' -... + {- l)^Hrn-l + ^ \ , and so, if a; > — 1, log(l+.)=/^lf-, = --|+- + (-l)-|^ + (-l)'"i?«., We require to show that the limit of iJ^, when m tends to oo , is zero. This is almost obvious when < a; S 1 ; for then i?„ is positive and less than where 72™ = ] ^y^^. /. rc^!5 = — — T, m + l and therefore less than l/(m + 1 ). If on the other hand - 1 < a; < 0, we put t = -u and a; = - f, so that • See Appendix II for some further remarks on this subject. 382 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix which shows that R^ has the sign of (- 1)™. Also, since the greatest value of 1/(1 - u) in the range of integration is 1/(1 - ^\ we have l-^i( 0<\E^\< 5 u^du f fcm 1 Uo (»i + i)(i-f)^("^;rrTKrr|)' and so ii„i ^ 0. Hence log(l +x) = x-^x^ + ^os'— ..., provided that -l<a;Sl. If a; lies outside these limits the series • is not convergent. If a; = 1 we obtain log2 = l-K^-..., a result already proved otherwise (Ex. LXXXix. 7). 214. The series for the inverse tangent. It is easy to prove in a similar manner that r^' dt f arc tan «= ,-r^= (1 - f + f - ...) dt J J- + 5 y = « — Ja;^ + la;^ — . . , , provided that — 1 S « S 1. The only difference is that the proof is a little simpler ; for, since arc tan sc is an odd function of oc, we need only consider positive values of x. And the series is convergent when a; = — 1 as well as when a; = 1. We leave the discussion to the reader. The value of arc tan x which is represented by the series is of course that which lies between — ^ir and ^vr when — 1 ^ a; g 1, and which we saw in Ch. VII (Ex. LXIII. 3) to be the value represented by the integral. If a; = 1, we obtain the formula 477— ± s-r-g— — ExamplesXCI. 1. log(^-—A=x+ix^+}x^ + ...it-l£x;<l. 2. argtanlia;=Jlogf Y^ ] = x: + ^.v^+^x^ + ... if -1<^<1. 3. Prove that if x is positive then (Math. Trip. 1911.) 4. Obtain the series for log(l+A') and arc tan .r by means of Taylor's theorem. [A difficulty presents itself in the discussion of the remainder in the 213, 214] OF A REAL VARIABLE 383 first series when a; is negative, if Lagrange's form /J„ = (-l)''-i^"/{»(l + to)''} is used ; Cauchy's form, viz. if„=(- l)«-i (1 - e)"-ix»/(l +5x)>', should be used (cf. the corresponding discussion for the Binomial Series, Ex. LVI. 2 and § 163). In the case of the second series we have D^ arc tan a- = Z>^" - 1 {1 /( 1 + .r^)} , =(-l)"-i(»-l)!(a;2 + l)-''/2sin{»arotan(l/a;)} (Ex. XLV. 11), and there is no difficulty about the remainder, which is obviously not greater in absolute value than 1/m.*] 5. If y>0 then [Use the identity y=\\-V ^-~^ ) / (^ " ^^ry) ■ This series may be used to calculate log 2, a purpose for which the series 1-|+J-..., owing to the slowness of its convergence, is practically useless. Put y = 2 and find log 2 to 3 places of decimals.] 6. Find log 10 to 3 places of decimals from the formula log 10 = 3 log 2 + log (1+ J). 7. Prove that log r^ = 2 j-1- + - J— + —1— + I if ^ > 0, and that (a;-l)H^ + 2) (2 1/ 2 Y 1/ 2 Y, I if ^>2. Given that log 2 = -6931471... and Iog3 = l-O986123..., sliow, by putting .37 = 10 in the second formula, that logll = 2'397895.... (Math. Trip. 1912.) 8. Show that if log 2, log 5, and log 11 are known, then the formula log 13 = 3 log 11 +log 5-9 log 2 gives log 13 with an error practically equal to '00015. (Math. Trip. 1910.) 9. Show that ^log2 = 7a + 56 + 3c, ^log3 = lla + 864-5c, |log5 = iea + 12&+7c, where a=argtanh (1/31), 6=argtanh (1/49), c = argtanh (1/161). [These formulae enable us to find log 2, log 3, and log 5 rapidly and with any degree of accuracy.] * The formula for D/ arc tan x fails when x=0, as arotan(l/a;) is then undefined. It is easy to see (cf. Ex. xlv. 11) that arc tan (l/ic) must then be interpreted as meaning Jir. 384 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [iX 10. Show that J7r = arctan (l/2) + arctan (l/3) = 4arctan (1/5) -arc tan (1/239), and calculate tt to 6 places of decimals. 11. Show that the expansion of (1 +a;)''''* in powers of a; begins with the terms l + .r + a•2 + ^.^•=. (Math. Trip. IQIQ.) 12. Show that .logio« \ogwe-^{x (x+1)} logio y—^J = 24^-2 ' approximately, for large values of x. Apply the formula, when a; =10, to obtain an approximate value of logio «, and estimate the accuracy of the result. (Math. Trip. 1910.) 13. Show that T^\og(:r-^')=x + (l+i)x3 + (l+i+i)xi + ..., if - 1< « < 1. [Use Ex. Lxxxi. 2.] 14. Using the logarithmic series and the facts that logio2-3758 = •3758099... and logio e = '4343..., show that an approximate solution of the equation a;=100 logioo; is 237-58121. (Math. Trip. 1910.) 15. Expand log cos ^ and log (sin x/.x:) in powers of x as far as x^, and verify that, to this order, log sin X = log j; - jig log cos x + f J log cos ^x. (Math. Trip. 1908.) 16. Show that :—--^=x-lafi+l:xP- ...ii -l^x £,1. Deduce that l-i + J---- = {'f + 21og(v/2 + l)}/4V2. (Math. JWp. 1896.) [Proceed as in § 214 and use the result of Ex. xlviii. 7.] 17. Prove similarly that 4-i+A--=J„f:r^=k-2iog(v/2+i)}/4V2. 18. Prove generally that if a and b are positive integers then 1 1 1 _ n t'^-'^dt a a + b + 26 "'~lo l + Z* ' and so that the sum of the series can be found. Calculate in this way the sums of 1-| + ^ — ... and ■^- J + J- .... 215. The Binomial Series. We have already (§ 163) investigated the Binomial Theorem 214, 215] OF A REAL VARIABLE 385 assuming that - 1 < a; < 1 and that m is rational. When m is irrational we have (1 +a;)'" = e'"'°s(i+a:)^ D^ (1 + «)'" = {m/(l + x)] e"'''"? »+*) = m (1 + xY-\ so that the rule for the differentiation of (1 + «)"* remains the same, and the proof of the theorem given in § 163 retains its validity. We shall not discuss the question of the convergence of the series when a; = 1 or a; = — 1.* Examples XCII. 1. Prove that if - 1<a< 1 then V(H-.i;2) 2-''+2.4'' •••' V(l-^') 2^+2.4'* +•■•■ 2. Approximation to quadratic and other surds. Let ^M be a quadratic surd whose numerical value is required. Let iV^ be the square nearest to M ; and let M=N^ + x or M=N''-x, x being positive. Since x cannot be gi-eater than N, xjN^ is comparatively small and the surd ^M=N ^{X + ixIN'')} can be expressed in a series -M'*Ki-.)-i-i(F.y*-}. which is at any rate fairly rapidly convergent, and may be very rapidly so. Thus Ve,=.<«.3,.,(,.'(i-)-lJ(l)%...} Let us consider the eiTor committed in taking 8^ (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than 3764^, which is less than -003. 3 If .r is small compared with N^ then the error being of the order af^jN''. Apply the process to ^937. [Expanding by the binomial theorem, we have X x'^ xfi "2^'~8#3 + T6iV6' the error being less than the numerical value of the next term, viz. 5^/128iV'. Also iVx __^ A ■ _^V^_^ __^ . _^ 1 {^m + x) ~ W V 2A^2J -4_/v 8iV3'*'l6iV5' the error being less than a;*/32iV'. The result follows. The same method may be applied to surds other than quadratic surds, e.g. to ^1031.] * See Bromwioh, Infinite Series, pp. 150 et seq. ; Hobson, Plane Trigonometry (3rd edition), p. 271. H. 25 386 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix 4. If M differs from N^ by less than 1 per cent, of either then ^M difters from f iV+J {MjN^) by less than iV/90000. {Math. Trip. 1882.) 5. If M=N^+x, and a; is small compared with N, then a good approxi- mation for li/M is 51 ,^ b M "ilNx 66 ' 56 iV3 14(7i/+5iV«)' Show that when iV=10, x=\, this approximation is accurate to 16 places of decimals. {Math. Trip. 1886.) 6. Show how to sum the series where Py {n) is a polynomial of degree r in n. [Express Py {n) iu the form A(j + Ain+A,^n (m- 1) + ... as in Ex. xc. 7.] 7. Sum the series Sm ( ) ^, Sn^l ) a;" and prove that 2w3 (™) ai'^={m?a^.\.m{Zm-\)x'^+mx){\ + x)'^-K \"/ 216. An alternative method of development of the theory of the exponential and logarithmic functions. We shall now give an outhne of a method of investigation of the properties of e^ and log x entirely different in logical order from that followed in the preceding pages. This method starts from the exponential series \ + x-^-^+.... "We know that this series is convergent for all values of x, and we may therefore define the function exp X by the equation exp.?; = l+a;+|-, + (1). We then prove, as in Ex. lxxxi. 7, that exp:!;xexpy = exp(A'+2/) (2). , . expA — 1 ^ h h^ Agam -^ = l + _ + _ + ...=l + p(/j)_ where p (A) is numerically less than \m+\m'Mm'+-=\kh\i{\-\\h\), so that. p (A) -^0 as A -9-0. And so exp(.r-l-A)-expi!; /expA-l\ — —h ="^p^ (-V-j ^^^p^ as A -»- 0, or Dx expa;=exp« (3). Incidentally we have proved that exp x is a continuous function. We have now a choice of procedure. Writing y = exp x and observing that exp = 1, we have dy _ _ [vdt 215, 216] OF A REAL VAEIABLE 387 and, if we define the logarithmic function as the function inverse to the exponential function, we are brought back to the point of view adopted earlier in this chapter. But we may proceed differently. From (2) it follows that if m is a positive integer then (exp x)" = exp nn;, (exp 1)" = exp n. If a; is a positive rational fraction »i/?i, then {exp (m/ra)}" = exp m = (exp l)"*, and so exp (m/n) is equal to the positive value of (exp ])"»/». This result may be extended to negative rational values of x by means of the equation exp«exp ( — a;) = l ; and so we have exp X = (exjj 1 )'^ = e=^, say, where e = exp 1 = 1 + 1 + — + — +..., for all rational values of x. Finally we define e", when x is irrational, as being equal to exp^. The logarithm is then defined as the function inverse to exp X or e*. Example. Develop the theory of the binomial series i + (7)^+0-^+...=/K--), where - 1 <a; < 1, in a similar manner, starting from the equation f{m,x)f{m\x)=f(7)i-i-m' x) (Ex. Lxxxi. 6). MISCELLANEOUS EXAMPLES ON CHAPTER IX •■. 1 . Given that logjo e = -4343 and that 2i" and Z:^^ are nearly equal to powers of 10, calculate logio2 and logioS to four places of decimals. {Matk. Trip. 1905.) 2. Determine which of {^e)^^ and (^2)*' is the greater. [Take logarithms and observe that V3/( V3 + ivr) < j ^/3 < -6929 < log 2.] 3 Show that log,ore cannot be a rational number if n is any positive integer not a power of 10. [If n is not divisible by 10, and logio»=p/?, we have 10P = ?i«, which is impossible, since 10* ends with and m» does not. If ?i= lOoiV, where N is not divisible by 10, then logioi\^ and therefore logio»=a+logioA'' cannot be rational.] > A considerable number of these examples are taken from Bromwioh's Infinite Series. 25—2 388 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix 4. For what values of x are the functions log x, log log x, log log log x, ... (a) equal to (6) equal to 1 (c) not defined ? Consider also the same question for the functions Ix, llx, lllx, ..., where i!^=log \x\. 5. Show that \ogx-(^^\oi{x+\) + (^^\og{x + i)-.., + {-\Y\og{x+n) is negative and increases steadily towards as a; increases from towards oo . [The derivative of the function is 0^ ' \rjx + r x{x + \)...{x + n)' as is easily seen by splitting up the right-hand side into partial fractions. This expression is positive, and the function itself tends to zero as x-a-co, since log{x+r)=logx+€^, where ,^^0, and 1-Q + (^fj- ...=0.] 6. Prove that {Math. Trip. 1909.) 7. If :i;> - 1 then x^> (1+x) {log (1 +x)}\ (Math. Trip. 1906.) [Put 1+^=6^, and use the fact that sinh ^> | when ^> 0.] 8. Show that {log {l+x)}lx and xj{{l+x) log (1 +a;)}both decrease steadily as X increases from towards od . 9. Show that, as x increases from — 1 towards oo , the function {l + x)~^'^ assumes once and only once every value between and 1. (Math. Trip. 1910.) 10. Show that -. — r ->- t- as ^ -»- 0. log(l + ^) X 2 11. Show that ; — jz < — decreases steadily from 1 to as x increases log(l-l-x) X •' from —1 towards oo. [The function is undefined when x=0, but if we attribute to it the value ^ when a;=0 it becomes continuous for x=0. Use Ex. 7 to show that the derivative is negative.] 12. Show that the function (log f — log ^)/(^ — »), where ^ is positive, decreases steadily as x increases from to ^, and find its limit as x-*-^. 13. Show that ^ > Mx , where M and N are large positive numbers, f X is greater than the greater of 2 log M and IQN'^. [It is easy to prove that log x<i^x ; and so the inequality given is certainly satisfied if x>\ogM+'iN ^x, and therefore certainly satisfied if |«>logj¥, \x>%N Jx."] OF A REAL VARIABLE 389 14. If /(.r) and <^(.r) tend to infinity as a--»-oo, and f {x)j<p'{x)-=-a^, then/(^)/0(A')^-Qo. [Use the result of Ch. VI, Misc. Ex. 33.] By taking f(x) = x% <t)(x) = logx, prove that (log.r)/a;''-*-0 for all positive values of a. 15. lip and q are positive integers then 1 1 J_ /q\ n-s-<x>. [Cf. Ex. Lxxviii. 6.] 16. Prove that iix is positive then »i log {^(l+a;!'")}^- — ^ log a: as m -*- 00 . [We have 7ilog{J(l+a;V'')} = ?ilog {l-*(l-a;V'>)} = jM(l_a;Vn) '°g(^~'"^ where m=^ (1-^""). Now use § 209 and Ex. lxxxii. 4.] 17. Prove that if a and 6 are positive then {Ha!i'» + 6i/«)}»^^(a6). [Take logarithms and use Ex. 16.] 18. Show that ^+3'''5'*"-"''2^::ri=4^os"+ios2+iy+fn, where y is Euler's constant (Ex. lxxxix. 1) and f„-»-0 as »-»-co. 19. Show that the series being formed from the series 1 - i + J - . . . by taking alternately two positive terms and then one negative. [The sum of the first 3m terms is 1 1/, 1 1\ =ilog2rH-log2 + iy + e„-i(logre+y + e„'), where fn and fn' tend to as m ^ oo . (Cf. Ex. lxxviii. 6).] 20. Show that l--^-i+i-i-i^+i-T'i3-...=ilog2. 21. Prove that "1 S ,„-, 2 — r\= — 3 + 323„ + i -S„-<5„ 2 v(36>»'' — 1) where ,S„=H-^ + ...+^, 2»=l + 3+-+2^^3i • Hence prove that the sum of the series when continvied to infinity is - 3 +1 log 3 + 2 log 2. {Math. Trip. 1905.) 22. Show that 2 , \ , =2log2-l, S .^\ ,, =§(log3-l). ,11 1 ^+3 + 5 + - 390 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [iX 23. Prove that the sums of the four series 2 . o , , S \ / , , 2 ,- T-r, — 7 , 2 H^ t4n2_i' I 4n^-l ' 7(2?i+l)^-l' t(2?H- 1)^-1 are ^, ^jt-^, J, ^log2-|- respectively. 24. Prove that n ! (a/re)" tends to or to oo according as a< e or a>e. [If M„=TO!(a/»i)" then ■»„+i/M„=a{l + (l/w)}-''-»-a/e. It can be shown that the function tends to oo when a=e: for a proof, which is rather beyond the scope of the theorems of this chapter, see Bromwich's Infinite Series, pp. 461 et seq.'] 25. Pind the limit as .j;-»- oo of / ao+aiX+...+arX:'\ '"'+''' " distinguishing the different cases which may arise. {Math. Trip. 1886.) 26. Prove that 2log(l+^) (^•>0) diverges to oo . [Compare with S {xjn).] Deduce that if x is positive then (l+a')(2+a;)... (n+a;)/n\-^cc as M ^- 00 . [The logarithm of the function is 2 log ( 1 + - J.] 27. Prove that if .r > - 1 then {.v+lf (^ + l)(^ + 2)^(:!; + l)(A-+2)(^+3) 2J ■•■ {x+1) (x + 2) (x+3) (« + 4) ■^••" {Math. Trip. 1908.) [The difference between l/(a; + l)2 and the sum of the first n terms of the series is 1 n\ , {x+l)i {a;+2){x+3) ...{x + n + 1) '^ 26. No equation of the type Ae'^'' + B^^+...=-0, where A, B, ... are polynomials and a, /3, ... difi'erent real numbers, can hold for all values of x. [If a is the algebraically greatest of a, (3, . . . , then the term Ae"'" outweighs all the rest as «^-*- oo .] 29. Show that the sequence ai = e, a2 = e , as = e^ , ... tends to infinity more rapidly than any member of the exponential scale. [Let ei{x) = e'', e2(a;) = e'iW, and so on. Then, ii ejc{sc) is any member of the exponential scale, ««> et(tt) when n > k.] OF A REAL VARIABLE 391 SO. Prove that where a is to be put equal to -f (^) and ^ to (^(^) after differentiation. Establish a similar rule for the differentiation of <|)(.a;)[^'i''(*)'*^*^J. 31. Prove that if i):t"e'»' = e-^(^„(;!;) then (i) <j,n(x) is a polynomial of degree n, (ii) <^n + i= - 2.r0„+0„', and (iii) all the roots of (>„=0 are real and distinct, and separated by those of (^„_i = 0. [To prove (iii) assume the truth of the result for n = 1, 2, ... <, and consider the signs of (f,^^^ for the n values of iK for which <^^ = and for large (positive or negative) values of x.] 32. The general solution of f{xy)=f{x)f{y), where / is a differentiable function, is x'^, where a is a constant : and that of f{^+y)+f{^-y)=^f{^)f{y) is cosh ass or cos ax, according as/" (0) is positive or negative. [In proving the second result assume that / has derivatives of the first three orders. Then 2/(^) +y' {/" (^) + ^v) = 2/(^) [/(O) +yf' (0) + J.y2 {/" (0) + .;}] , where ^y and f/ tend to zero with y. It follows that /(0) = 1, /'(0) = 0, f"{x)=f"{0)f{x), so that a = V{/"(0)} or a = v/{-/" (0)}.] 33. How do the functions :r'*" <l/^', x'^" "'^\ ^"'^^'^dW behave as ^-* + 02 34. Tracethe curves y = tan^ «'*"•". y = sin;slogtan^a;. 35. The equation e''=ax + b has one real root if a<0 or a = 0, 6>0. If a > then it has two real roots or none, according as a log a >b — a or a log a<b — a. 36. Show by graphical considerations that the equation e^=ax^ + 2bx + c has one, two, or three real roots if a > 0, none, one, or two if a < ; and show how to distinguish between the different cases. I /gx _ 1\ 37. Trace the curve y = - log I I , showing that the point (0, J) is X \ X / a centre of symmetry, and that as x increases through all real values, 3/ steadily increases from to 1. Deduce that the equation has no real root unless 0<n<l, and then one, whose sign is the same as that of a-i. [In the first place " '-t=M'-C-=i-")---"}4i-(T^') is clearly an odd function of x. Also 392 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [ix The function inside the large bracket tends to zero as a; -»- ; and its derivative is X t \sinh ^xj ) which has the sign of x. Hence d!//dx>0 for all values of x.] 38. Trace the curve y = e ^'^ ^(x^ + 2x), and show that the equation has no real roots ifn is negative, one negative root if 0<a<a= e''^^^(2 +2^/2) and two positive roots and one negative if a > a. 39. Show that the equation fn{x) = l+x + -r-T + ...-i — -=0 has one real ^ i 71'. root if n is odd and none if n is even. [Assume this proved for m=l, 2, ... 2k. Then /2i+i(.*) = has at least one real root, since its degree is odd, and it cannot have more since, if it had, /'jt + 1 (x) or fit, (x) would have to vanish once at least. Hence fiu + j (a;) = has just one root, and so f2!c+i{x) = cannot have more than two. If it has two, say a and j3, then /'an. 2 (.:!;) or/2ii + i(x) must vanish once at least between a and 0, say at y. And yii + 2 /21 + 2 (r ) =f2k + 1 (y) + ,2^.^2)1 ^ °' But /27:+2(.r) is also positive when x is large (positively or negatively), and a glance at a figure will show that these results are contradictory. Heuce f2in-i{^) = ^ has no real roots.] 40. Prove that if a and b are positive and nearly equal then , a 1 , ,, /I 1\ approximately, the eiTor being about J{(a — 6)/ap. [Use the logarithmic series. This formula is interesting historically as having been employed by Napier for the numerical calculation of logarithms.] 41. Prove by multiplication of series that if — 1 < ^ < 1 then |{l0g(l+a;)P = ^^2_|(t+^)^+J(l+^ + ^)^,4_...^ ^(arctana;)2 = |.^•2-i(H-i)a,*+i(l + K■l)^-•■•• 42. Prove that (l+axf'' = e' {l-ia^x + ^ (8+3a) a^x'^ {l + e^)}, where e^. -s- with a;. l + x + -^ + ...-] — j-1 in powers of x are ^.^^ iJ "L. . + -^ +(_l)n _^1 ^l n! [n + l l!(» + 2)^2! (71 + 3) "'^^ ' n] {2n + l)\ ' {Math. Trip. 1899.) OF A REAL VARIABLE 393 44. Show that the expansion of in powei's of x begins with the terms 45. Show that if — 1 <x<\ then 1 +Li2.,.+i443V+._/(-+_31 3 ^3.6 ^3.6.9 ^■•" 9(1 -a;)"3' 3^'"'"376^''' +3.6.9'*'^"^-""~27(l-^)'»/3 ' [Use the method of Ex. xcii. 6. The results are more easily obtained by differentiation ; but the problem of the differentiation of an infinite sex-ies is beyond our range.] 46. Prove that J" ^^-^^"J^^ = -i^ log (|) , /„ (^+aK^ + &^) = (TO^ ^"""-^ ^°^ G)} ' provided that a and h are positive. Deduce, and verify independently, that each of the functions a-\-\o%a, aloga-a+1, ^na-loga, ^ir+aloga is positive for all positive values of a. 47. Prove that if a, A y are all positive, and ^^>ay, then J ax^+2l33;+y ■J(.P^-ay) ^\ V(ay) J' while if a is positive and ay>j32 the value of the integral is ___,rctan|-^|, that value of the inverse tangent being chosen which lies between and tt. Are there any other reaUy different cases in which the integral is convergent^ 48. Prove that if a> - 1 then r <^.^ ^ r _j[*_=2 r ^'^ ■ ji (a;+a)v/(«''-l) io cosh<+a Ji u^ + 2au + l' 394 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [iX is —rjz ^\ aiu uctu ^ / { = I and deduce that the value of the integral is 2 -T, arc tan if — l<a<l, and 1 , v'(a+l) + v/(a-l) 2 , , /fa~\\ if a> 1. Discuss the case in which a = \. /" dx . \ // 2 I IN ' ■"'here a>0, in the ways, showing that its value is 1 , ■ a! + l + s/(a2 + l) 2 ^ ,V(aHl) 50. Prove that | arctan.j;rfa:=Jn- — Jlog2. y 51. If 0<a<l, 0</3<l, then 52. Prove that if a> 6> then d6 ■TT i: I cosh 6 + b sinh d ^{a^ - ¥■) 53. Prove that p]^^dx-- r'^^^dx r'^^^dx-0 and deduce that if a>0 then /'" log^ lOg^ TT , I -s-^ — -, dx= -r- log a. jo cfi + x'' 2a ^ [Use the substitutions x=\jt and a!=aM.] 54. Prove that I log ( 1 + ^ j rf.2; = ira if a > 0. [Integrate by parts.] CHAPTER X THE GENERAL THEORy OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS 217. Functions of a complex variable. In Ch. Ill we defined the complex variable z = x + iy*, and we considered a few simple properties of some classes of expressions involving z, such as the polynomial P{z). It is natural to describe such expressions as functions of z, and in fact we did describe the quotient P (z)/Q (z), where P (z) and Q (z) are polynomials, as a ' rational function '. We have however given no general definition of what is meant by a function of z. It might seem natural to define a function of z in the same way as that in which we defined a function of the real variable x, i.e. to say that .^ is a function of z if any relation subsists between z and Z in virtue of which a value or values of Z corre- sponds to some or all values of z. But it will be found, on closer examination, that this definition is not one from which any profit can be derived. For if z is given, so are x and y, and conversely : to assign a value of z is precisely the same thing as to assign a pair of values of x and y. Thus a ' function of z ', according to the definition suggested, is precisely the same thing as a complex function f (^, y) + «7 («. y)' of the two real variables x and y. For example x-iy, xy, \z\ = V(«^ + y""), am ^ = arc tan {yjx) are ' functions of z'. The definition, although perfectly legitimate, * In this chapter we shall generally find it convenient to write x + iy rather than x + yi. 396 THE GENERAL THEORY OF THE LOGARITHMIC, [x is futile because it does not really define a new idea at all. It is therefore more convenient to use the expression ' function of the complex variable ^ ' in a more restricted sense, or in other words to pick out, from the general class of complex functions of the two real variables x and y, a special class to which the expression shall be restricted. But if we were to attempt to explain how this selection is made, and what are the characteristic properties of the special class of functions selected, we should be led far beyond the limits of this book. We shall therefore not attempt to give any general definitions, but shall confine ourselves entirely to special functions defined directly. 218. We have already defined polynomials in z (§ 39), rational functions of z (§ 46), and roots of z (§ 47). There is no difficulty in extending to the complex variable the definitions of algebraical functions, explicit and implicit, which we gave (§§ 26 — 27) in the case of the real variable x. In all these cases we shall call the complex number z, the argument (§ 44) of the point z, the argument of the function / (z) under consideration. The question which will occupy us in this chapter is that of defining and determining the principal properties of the logarithmic, ex- ponential, and trigonometrical or circular functions of z. These functions are of course so far defined for real values of z only, the logarithm indeed for positive values only. We shall begin with the logarithmic function. It is natural to attempt to define it by means of some extension of the definition 'dt loga;= I f (*>0); and in order to do this we shall find it necessary to consider briefly some extensions of the notion of an integral. 219. Real and complex curvilinear integrals. Let AB be an arc of a curve defined by the equations x = 4,(t), y = f(t), where (j) and a^ are functions of t with continuous difi'erential coefficients ^' and yjr' ; and suppose that, as t varies from t^ to t^, the point (x, y) moves along the curve, in the same direction, from A toB. 217-220] EXPONENTIAL, AND CIRCULAE FUNCTIONS 397 Then we define che curvilinear integral \^{9{«',y)dx+h{x,y)dy] (l), where g and h are continuous functions of x and y, as being equi- valent to the ordinary integral obtained by effecting the formal substitutions x= ^ (t), y = '^ (t), i.e. to /, {g(<f>,ylr)cl,'+h{(f,,y{r)y{r']dt. 'to We call C the path of integration. Let us suppose now that 2 = x + iy = (j)(t) + iyjr(t), so that 2 describes the curve C in Argand's diagram as t varies. Further let us suppose that f(z) = u + iv is a polynomial in ^ or rational function of z. Then we define /, ^f(^)dz (2) as meaning I (u + iv) (dx + idy), which is itself defined as meaning / (udx — vdy) + i \ (vdx + udy). J c J c 220. The definition of Log ^. Now let f =1^ + 17; be any complex number. We define Log ^, the general logarithm of f, by the equation Logr=f ^^ .! C ^ where C is a curve which starts from 1 and ends at f and does not pass through the origin. Thus (Fig. 54) the paths (a), (6), (c) are paths such as are contemplated in the definition. The value of Log 2 is thus defined when the particular path of integration has been chosen. But at present it is not clear how far the value of Log 2 resulting from the definition depends upon what path is chosen. Suppose for example that f is real and positive, say 398 THE GENERAL THEORY OF THE LOGARITHMIC, Lx equal to f. Then one possible path of integration is the straight line from 1 to f, a path which we may suppose to be defined by Y the equations x=t,y = 0. In this case, and with this particular choice of the path of integration, we have so that Log ^ is equal to log f, the logarithm of ^ according to the definition given in the last chapter. Thus one value at any rate of Log ^, when ^ is real and positive, is log f . But in this case, as in the general case, the path of integration can be chosen in an infinite variety of different ways. There is nothing to show that every value of Log ^ is equal to log ^ ; and in point of fact we shall see that this is not the case. This is why we have adopted the notation Log ?■, Log ^ instead of log ^, log ^. Log f is (possibly at any rate) a many valued function, and log f is only one of its values. And in the general case, so far as we can see at present, three alternatives are equally possible, viz. that (1) we may always get the same value of Log f, by whatever path we go from 1 to ^; (2) we may get a different value corresponding to every different path ; (3) we may get a number of different values each of which corresponds to a whole class of paths : and the truth or falsehood of any one of these alternatives is in no way implied by our definition. 220, 221] EXPONENTIAL, AND CIRCULAR FUNCTIONS 399 221. The values of Log f. Let us suppose that the polar coordinates of the point z = ^ Ave p and (/>, so that f = jO (cos + i sin 0). We suppose for the present that - tt < ^ < tt, while p may have any positive value. Thus f may have any value other than zero or a real negative value. The coordinates {x, y) of any point on the path G are functions of t, and so also are its polar coordinates {r, 6). Also Jc z Jc oo + iy Jt„ x + iy \dt at J m virtue of the definitions of § 219. But x = r cos 6, y = r sin 0, and dx . ■ dy ( „dr ■ ad6\ , .( . ^dr ^dO " -^ — ' """ ^ - - r sin 6 -37 j + % ( sm 6 -,- + r cos 6 Tt-^'Tt=r'^dt dt dt / a , ■ ■ a\ fdr . d& so that Log? = J Vq r dt " ' 'Ju dt where [log r] denotes the difference between the values of log r at the points corresponding tot = ti and t = to, and [6] has a similar meaning. It is clear that [logr] = logp-logl=log/3; but the value of [6] requires a little more consideration. Let us suppose first that the path of integration is the straight line from 1 to ^. The initial value of 6 is the amplitude of 1, or rather one of the amplitudes of 1, viz. 2^77, where k is any integer. Let us suppose that initially 6 = 2^7r. It is evident from the figure that 6 increases from 2^7r to 2kTr + </> as t moves along the line. Thus [d]=(2kTr+^)-2k7r=cf>, and, when the path of integration is a straight line. Log ^ = log p + i4>- Fig. 55. 400 THE GENERAL THEORY OF THE LOGARITHMIC, [X We shall call this particular value of Log?" the principal value. When f is real and positive, ^= p and ^ = 0, so that the principal value of Log ^ is the ordinary logarithm log f. Hence it will be convenient in general to denote the principal value of Log ^ by log f. Thus logf = log/) + i<^, and the principal value is characterised by the fact that its imaginary part lies between — tt and -jr. Next let us consider any path (such as those shown in Fig. 56) such that the area or areas included between the path and the straight line from 1 to f does not include the origin. It is easy to see that [6] is still equal to cj). Along the curve shown in the figure by a continuous line, for example, 0, initially equal to 2kTr, first de- creases to the value 2Jc-7r-XOP and then increases again, being equal to 2JcTr at Q, and finally to 2kir + (}). The dotted curve shows a similar but slightly more complicated case in which the straight line and the curve bound two areas, neither of which includes the origin. Thus if the path of integration is such that the closed curve formed by it and the line frovi 1 to ^ does not include the origin, then Log^=logf=log/3 + i</). On the other hand it is easy to construct paths of integration such that \6'\ is not equal to <^. Consider, for example, the curve indicated by a continuous line in Fig. 57. If 6 is initially equal to 2kTr, it will have increased by 27r when we get to P and by 47r when we get to Q\ and its final value will be 2k'n- + 4nr + ^, so that [0] = 4i7r + ^ and Fig. 56. Fig. 57. Log f = log p + i (47r + </)). 221] EXPONENTIAL, AND CIRCULAR FUNCTIONS 401 In this case the path of integration winds twice round the origin in the positive sense. If we had taken a path winding k times round the origin we should have found, in a precisely similar way, that [^] = 2kiT + ^ and Log f = log p + 1 {2kTr + <j)). Here k is positive. By making the path wind round the origin in the opposite direction (as shown in the dotted path in Fig. 57), we obtain a similar series of values in which k is negative. Since | ^\ = p, and the different angles 2kjT + (j) are the different values of am ^, we conclude that every value of log | ? ] + * am f is a value of Log f ; and it is clear from the preceding discussion that every value of Log ^ must be of this form. We may summarise our conclusions as follows : the general value of Log ^ is log I f I + 1 am ? = log p + i (ikir + ^), where k is any positive or negative integer. The value of k is determined by the path of integration chosen. If this path is a straight line then k = and Logf=log^=log|0 + i</). In what precedes we have used ^ to denote the argument of the function Log f, and (^, tj) or {p, (j>) to denote the coordinates of f ; and z, (so, y), (r, 6) to denote an arbitrary point on the path of integration and its coordinates. There is however no reason now why we should not revert to the natural notation in which z is used as the argument of the function Log z, and we shall do this in the following examples. Examples XCIII. 1. We supposed above that -ir<6<iT, and so excluded the case in which z is real and negative. In this case the straight line from 1 to 2 passes through 0, and is therefore not admissible as a path of integration. Both n and - n are values of am z, and 6 is equal to one or other of them: also r=-z. The values of Log 2 are still the values of logUl+iamz, viz. ' ' \og{-z) + {ih+l)ni, where k is an integer. The values log {-z) + ni and log {-z)-iri correspond to paths from 1 to z lying respectively entirely above and entirely below the real axis. Either of them may be taken as the principal value of Loge, as convenience dictates. We shall choose the value log {-z) + «' corresponding to the first path. 26 402 THE GENERAL THEOEY OF THE LOGARITHMIC, [X 2. The real and imaginary parts of any value of Log 2 are both continuous functions of so and y, except for ^'=0,^ = 0. 3. The functional eetuation satisfied by Logz. The function Log 2 satisfies the equation Log 0102 = Log 2i + Log 02 (1), in the sense that every value of either side of this equation is one of the values of the other side. This follows at once by putting 01 = j-i (cos ^1 + 1 sin 6i), 02 = '•2 (cos 62 + i sin ^2), and applying the formula of p. 401. It is however not true that Iog0i02 = log0i + log02 (2) in all circumstances. If, e.g., 2i = 22=i (- 1 +«\/3)=cos §77+1 sin fir, then log 01= log 02 =§571, and log 0i+ log 02=^771, which is one of the values of Log 0102, but not the principal value. In fact log 0103= -f n-i. An equation such as (1), in which every value of either side is a value of the other, we shall call a complete equation, or an equation which is completely true. 4. The equation Log«™=mLog0, where m is an integer, is not completely true : every value of the right-hand side is a value of the left-hand side, but the converse is not true. 5. The equation Log (1/0)= -Log is completely true. It is also true that log (1/0)= — log0, except when z is real and negative. 6. The equation is true if z lies outside the region bounded by the line joining the points z = a, = 6, and lines through these points parallel to OX and extending to infinity in the negative direction. 7. The equation ■•«(r:)='°K'-f)-'°<'-!) is true if lies outside the triangle formed by the three points 0, a, h. 8. Draw the graph of the function I (Log a:) of the real variables. [The grajih consists of the positive halves of the lines y = ikn and the negative halves of the lines y=(2^-M) w.] 9. The function f{x) of the real variable x, defined by ■nf {x) =p7r + {q-p)I (log x), is equal to^ when x is positive and to q when x is negativa 221, 222] EXPONENTIAL, AND CIRCULAE FUNCTIONS 403 10. The function / {x) defined by «■/ W = Pn- + (? -i)) I {log («- 1 )}+()•-?) I (log ^) is equal to p when a;>l, to q when 0<5;<1, and to r when «<0. 11. For what values of z is (i) logz (ii) any value of Logz (a) real or (6) purely imaginary ? 12. liz=x+iy then LogLog0=log5+i(e+2^'7r), where and O is the least positive angle determined by the equations cos e : sin 9 : 1 : : log »• : fl + 2;i;jr : V{(log rf+{6 + 'ikirf). Plot roughly the doubly infinite set of values of Log Log (1 + !;\/3), indicating which of them are values#)f logLog(l + iV3) and which of Loglog(l+iV3)- 222. The exponential function. In Ch. IX we defined a function e" of the real variable y as the inverse of the function y = log X. It is naturally suggested that w^e should define a function of the complex variable z which is the inverse of the function Log«. Definition, If any value of Log z is equal to ^, we call z the exponential of f »wd write z = exp f. Thus £1 = exp 5' if ?'=Log2. It is certain that to any given value of z correspond infinitely many different values of f It would not be unnatural to suppose that, conversely, to any given value of f correspond infinitely many values of z, or in other words that exp ^ is an infinitely many-valued function of f. This is however not the case, as is proved by the following theorem. Theorem. The exponential function exp ^ is a one-valued function of ^. For suppose that 0j = ri (cos 01 +i sin ^i), z^ = r^ (cos 6^ + i sin 6^ are both values of exp ?■. Then ^=l,ogZi = 'LogZi, and so log n + i {6^ + 2rmr) = log n + » (^2 + 2w7r), where m and n are integers. This involves log ri = log r^' Oi + Im-TT =6^+ 2w7r. Thus ri = Ti, and d-^ and 6^ differ by a multiple of l-rr. Hence Zi = ^2. 26—2 404 THE GENERAL THEORY OF THE LOGARITHMIC, [x Corollary. If ^is real then expf=ef, the real exponential function of f defined in Ch. IX. For if z = e^ then log z = ^, i.e. one of the values of Log z is f. Hence z = exp 5". 223. The value of exp f. Let ^=^ + in and z = exp ^=r (cos 6 + i sin 6). Then ? + 1»; = Log ^ = log r + i (^ + 2m7r), where m is an integer. Hence ^ = log r, ri = 6 + 2m7r, or r = e^, 6 = 7] — 2tmr ; and accordingly exp (f + 19?) = e^ (cos 77 + 1 sin ■»;). If 7? = then exp ^ = e^, as we have already inferred in § 222. It is clear that both the real and the imaginary parts of exp (f + iri) are continuous functions of ^ and rj for all values of ^ and 97. 224. The functional equation satisfied by exp f. Let ?i = ?i + iVi , ?2 = I2 + iv^ • Then exp ^1 X exp ^2 = e^i (cos r)i+i sin tji) x e^' (cos 172 + i sin 172) = e^'+f^ {cos (771 + 172) + i sin (771 + 772)} = exp(fi+5'2). The exponential function therefore satisfies the functional relation y(^i + ^2) =/(?!) /(Q) ^'^ equation which we have proved already (§ 205) to be true for real values of fi and fa- 225. The general power a^. It might seem natural, as exp f = e^ when f is real, to adopt the same notation when ? is complex and to drop the notation exp f altogether. We shall not follow this course because we shall have to give a more general definition of the meaning of the symbol e^: we shall find then that e^ represents a function with infinitely many values of which exp f is only one. We have already defined the meaning of the symbol a^ in a- considerable variety of cases. It is defined in elementary Algebra in the case in which a is real and positive and f rational, or a real and negative and f a rational fraction whose denominator is odd. According to the definitions there given a^ has at most two values. 222-225] EXPONENTIAL, AND CIRCULAR FUNCTIONS 405 In Ch. Ill we extended our definitions to cover the case in which a IS any real or complex number and f any rational number pjq ■ and m Ch. IX we gave a new definition, expressed by the equation which applies whenever ^ is real and a real and positive. Thus we have, in one way or another, attached a meaning to such expressions as but we have as yet given no definitions which enable us to attach any meaning to such expressions as We shall now give a general definition of a^ which applies to all values of a and f, real or complex, with the one limitation that a must not be equal to zero. Definition. The function a^ is defined by the equation a^ = exp (f Loga) where Log a is any value of the logarithm of a. We must first satisfy ourselves that this definition is consistent with the previous definitions and includes them all as particular cases. (1) If a is positive and f real, then one value of ^ Log a, viz. flog a, is real: and exp (flog a) = e^^°^'^, which agrees with the definition adopted in Ch. IX. The definition of Ch. IX is, as we saw then, consistent with the definition given in elementary Algebra ; and so our new definition is so too. (2) If a = e^ (cos i/r + i sin i|r), then Log a = T + i (t/t + 2rmr), exp {(pjq) Log a} = eP'li Cis {{pjq) {f + 2m7r)}, where m may have any integral value. It is easy to see that if m assumes all possible integral values then this expression assumes q and only q different values, which are precisely the values of a^li found in § 48. Hence our new definition is also consistent with that of Ch. III. 406 THE GENERAL THEORY OF THE LOGARITHMIC, [X 226. The general value of a^. Let ^=^+ ir), a = a (cos i|r + i sin i/r) where — 7r<<|fS7r, so that, in the notation of § 225, o- = e'' or T = log tr. Then f Log a = (f + it)) {log o- + i (''/^ + 2mir)} = L + iM, where i = f logo-— 97(i|p + 2mw), M =7) logo- + ^(\|r + 2m7r); and a^ = exp (f Log a) = e^ (cos M + i sin i/). Thus the general value of a^ is gf log ,-,(*+jm^) [cog 1^ log o- + ^ (i^ + 2m7r)} + i sin {t] log o- + f (^Ir + 2m7r)}]. In general a^ is an infinitely many-valued function. For has a different value for every value of m, unless ■r]=0. If on the other hand i? = 0, then the moduli of all the different values of a^ are the same. But any two values differ unless their amplitudes are the same or differ by a multiple of 2Tr. This requires that ^{y{r + 2imr) and ^(ylr + 2mr), where m and n are different integers, shall differ, if at all, by a multiple of 27r. But if f (i/r + 2m7r) - H'f + 2n7r) = 2IcTr, then f = k/(m — n) is rational. We conclude that a^ is infinitely many-valued unless fis real and rational. On the other hand we have already seen that, when f is real and rational, a^ has but a. finite number of values. The principal value of a^=exp(f Loga) is obtained by giving Logo its principal value, i.e. by supposing ot = in the general formula. Thus the principal value of a^ is e^io^''-n^f'{cos(r,\og,^+^)+^sin{r,\ogo■ + i^|,)}. Two particular cases are of especial interest. If a is real and positive and C I'eal, then cr=a, ■\jr=0, |=f, t; = 0, and the principal value of a^ is eSiog"^ which is the value defined in the last chapter. If |a| = l and f is real, then o-=l, ^=f, ^ = 0, and the principal value of (cosi/r+tsinifr)^ is cosfi/^ + i'sin f\/^. This is a further generalisation of De Moivre's Theorem (§§ 45, 49). 226] EXPONENTIAL, AND CIRCULAR FUNCTIONJS 407 Examples XCIV. l. Find all the values of i<. [By definition i'=exp (t'Logi). But t=COS^7r+lsini7r, Logi = (2/&+J)n-2, where & is any integer. Hence ti=exp{-(2k+i) «•} =e-(2*+i)''. All the values of i* are therefore real and positive.] 2. Find all the values of (1+i)*, i'+», {l+t)^**. 3. The values of a% when plotted in the Argand diagram, are the vertices of an equiangular polygon inscribed in an equiangular spiral whose angle is independent of a. {Math. Trip. 1899.) [If a^= r (cos 6 + i sin ff) we have ^=,llog<r-,(^+2m.)_ e = ,log<r + ^(Vr + 2™;r); and all the points lie on the spiral »-=(r^^'' +i'')/f g-iW] 4. The function e^. If we write « for a in the general iormula, so that log o-=l, ^/'=0, we obtain e^ = e^- ^""^ {cos (i, + 2»i77|) + 1 sin (, + 29)in-|)}. The principal value of e^is e*(coS7;+isin77), which is equal to exp f (§ 223). In pai'ticular, if f is real, so that 7;=0, we obtain e (cos 2mjrf + 1 sin 2»i7rf ) as the general and e^ as the principal value, e^ denoting here the positive value of the exponential defined in Ch. IX. 5. Show that Log e' = (l + 2m7ri) f +2«7ri, where m and n are 'any integers, and that in general Logos^ has a double infinity of values. 6. The equation l/a^=a~^ is completely true (Ex. xcili. 3): it is also true of the principal values. 7. The equation a^x6^=(as6)^ is completely true but not always true of the principal values. 8. The equation a^x a^' =o^^^' is not completely true, but is true of the principal values. [Every value of the right-hand side is a value of the left- hand side, but the general value of cSxcS' , viz. exp {f (log a -)- 2»i7ri) -1- f ' (log a -(- invi)) , is not as a rule a value of cS^^ unless to=?i.] 9. What are the corresponding results as regards the equations Logaf=fLoga, {a<f={a<^)^=cS^''i 10. For what values of f is (a) any value (6) the principal value of / (i) real (ii) purely imaginary (iii) of unit modulus ? 408 THE GENERAL THEORY OF THE LOGARITHMIC, [x 11. The necessary and sufficient conditions that all the values of af should be real are that 2| and {i; log | a | + 1 am a}/ir, where am a denotes any value of the amplitude, should both be integral. What are the corresponding con- ditions that all the values should be of unit modulus ? 12. The general value oi\x*+x~''\, where z>0, is g -(>» - n) ;r^|-2 i^Qgjj 2 (m + re) n- + cos (2 log x)}]. 13. Explain the fallacy in the following argument : since e^'""'»=:g8«'ri_j where m and n are any integers, therefore, raising each side to the power i we obtain e-2'»''=e-2«''. 14. In what circumstances are any of the values of x^, where a; is real themselves real ? [If x> then a^ = exp (.^• Log ^) = exp (a; log x) Cis 2m.irx, the first factor being real. The principal value, for which m=0, is always real. If :» is a rational fraction pl(2q + l), or is irrational, then there is no other real value. But if x is of the form pl2q, then there is one other real value viz. — exp (x log x), given hj m = q. lix=-$<Othen ^=exp{-^Log(-|)} = exp(-|log|)Ois{-(2TO + l),r|}. The only case in which any value is real is that in which |=^/(2j-|-l), when m—q gives the real value exp(-|log|)Cis(-;o,r) = (-l)Pr^. The cases of reality are illustrated by the examples (4)*=4'i, (#=±n/|-, (-|r* = 4/f, (-4)-*= -4^3.] 15. Logarithms to any base. We may define f =Log„2 in two diflferent ways. We may say (i) that f = Log„ z if the principal value of a^ is equal to z ; or we may say (ii) that f = Log^ s if any value of a^ is equal to z. Thus if 05=6 then f=Log,z, according to the first definition, if the principal value of e^ is equal to z, or if expf=z; and so LogeS is identical with Logz. But, according to the second definition, f=Logj« if /=exp(fLoge)=2, fLoge = Log2, or i-= (Log z)l(Log e), any values of the logarithms being talsen. Thus /■_T,^o■ ,_logM + (am0+2mjr)t i-uog,z i + 2re^i ' so that f is a doubly infinitely many- valued function of z. And generally, according to this definition, Loga3=(Log0)/(Loga). 16. Logel = 2m7nV(l-<-2?i7n'), Loge(-l) = (2m-f-l) ^/(l + 2»n-i), where m, and n are any integers. 226-228] EXPONENTIAL, AND CIRCULAR FUNCTIONS 409 227. The exponential values of the sine and cosine. From the formula exp (^ + it]) = exp ^ (cos r^+i sin ??), we can deduce a number of extremely important subsidiary formulae. Taking | = 0, we obtain exp {irj) = cos t? + i sin rj ; and, changing the sign of rj, exp (- ir}) = cos rj- i sin tj. Hence cos 7; = ^ {exp {irj) + exp ( - iri)], sin 7; = — ^ i {exp {ir)) — exp ( - irj)]. We can of course deduce expressions for any of the trigonometrical ratios of r) in terms of exp {irf). 228. Definition of sins' and COBS' for all values of f. We saw in the last section that, when ^ is real, cosS'= h {exp (10 + exp (-if)} (la), sinS'=-ii{exp(iO-exp(-iS')} (16). The left-hand sides of these equations are defined, by the ordinary geometrical definitions adopted in elementary Trigonometry, only for real values of i^. The right-hand sides have, on the other hand, been defined for all values of S", real or complex. We are therefore naturally led to adopt the forrimlae (1) as the definitions of cos S" and sin S' for all values of f. TheSe definitions agree, in virtue of the results of § 227, with the elementary definitions for real values of i^. Having defined cos ?' and sin 5", we define the other trigono- metrical ratios by the equations ^ sin t ^ o cos ^ (,1 s- 1 /ON *^^^=^' ^«*^=shrr ^^•'^=c-^r cosec?=^-^^...(2). It is evident that cos ^ and sec f are even functions of f, and sin f, tan f, cot ^, and cosec f odd functions. Also, if exp (if) = t, we have cos ?= M* + (1/*)}. sin r = - i ^ {^ - 0-lt% cos^r+sin^r=i[{«+(l/01=-{^-(l/0j=]=l (3). We can moreover express the trigonometrical functions of ^+ f' in terms of those of f and f' by precisely the same formulae 410 THE GENERAL THEORY OF THE LOGARITHMIC. [x as those which hold in elementary trigonometry. For if exp (i^) = t, exp {i^') = t', we have cos «,,).i(«.,,i)=i|(,.>)(.4).(<-l)(.4 = cos ii'cos f — sin f sinf (4); and similarly we can prove that sin(?'+ i:')=sin fcos ?' + cosfsin ^' (5). In particular cos(r+4^) = -sinr, sin(r+i7r)=cos? (6). All the ordinary formulae of elementary Trigonometry are algebraical corollaries of the equations (2) — (6) ; and so all such relations hold also for the generalised trigonometrical functions defined in this section. 229. The generalised hyperbolic functions. In Ex. lxxxvii. 19, we defined cosh f and sinh f, for real values of C, by the equations coshf=f {expf+exp(-f)}, sinh f=Hexp f- exp (-f)} (1). We can now extend this definition to complex values of the variable; i.e. we can agree that the equations (1) are to define cosh f and sinh f for all values of f real or complex. The reader will easily verify the following relations : cosif=coshf, sin if =2 sinh f, coshif=oosf, sinh if =i sin f. We have seen that any elementary trigonometrical formula, such as the formxda cos 2f=cos^ f — sin^ f, remains true when f is allowed to assume complex values. It remains true therefore if we write cos if for cos f, sin if for sin f and cos2if for cos 2f ; or, in other words, if we write cosh f for cos f, i sinh f for sin f, and cosh 2f for cos 2f. Hence cosh 2f = cosh^ f + sinV f. The same process of transformation may be applied to any trigonometrical identity. It is of course this fact which explains the correspondence noted in Ex. LXXXVII. 21 between the formulae for the hyperbolic and those for the ordinary trigonometrical functions. 230. Formulae for cos(|+i?)), sind+i'ij), etc. It follows from the addition formulae that cos (^ + II)) = cos I cos i?; - sin I sin irj = cos | cosh ij-i sin ^ sinh ij, sin (^ + »))) = sin ^ cos ir) + cos | sin irj = sin | cosh ij + i cos | sinh rj. These formulae are true for all values of | and rj. The interesting case is that in which | and r] are real. They then give expressions for the real and imaginary parts of the cosine and sine of a complex number. 228-230] EXPONENTIAL, AND CIRCULAR FUNCTIONS 411 Examples XCV. 1. Determine the values of f for which cos f and sin f are (i) real (ii) purely imaginary. [For example cosf is real when iy=0 or when I is any multiple of tt.] 2. I cos (1 + 1,,) I = V(cos2 ^ + sinha t;) = Vtt (cosh St, + cos 2|)}, I sin (^ + !,,) I = V(sin2 ^ + sinh2 ,,) = ^{i (cosh 2r, - cos 2|)}. [Use (e.cf.) the equation |cos(?+j,)) |=V{cos(^4-J'?)cos (^-i.,)}.] 3. tan(^ + t,) = !H^|^ti^H^^ cot(^+t,) = ^i?lM^i!li^^ ^^ '^ cosh 2,+ cos 2^' ^"ne-T";; cosh 2, - cos 2^ • [For example tan (f + in) = ^'" (^ + ^"'?) c°3 (!-'■"'; ) ^ sin 2| + sin 2z, ; cos (I +1),) cos (I -l';;) COS 2g + COS 2l?, ' which leads at once to the result given.] 4. sec r^ + ^-1 - coslcoshrj + isingsinh,; 4. sec (§ + *,,) ^(eosh2,+cos2|) ' ,» . . _ sin£cosh_5— j.' cos | sinh rj cosec(5+t,;-^ . 5. If I cos (I + 11,) 1 = 1 then sin2|=sinh2?,, and if | sin (^ + i),)| = l then cos2^ = sinh^i7. 6. If I cos (^+1,,) 1 = 1, then sin {am cos (| + ii,)} = + sin^ | = + sinh^ i,. 7. Prove that Log cos (| + 1'?,) = .4 + iB, where ^ = i log {J (cosh 27, + cos 2|)} and 5 is any angle such that cos B sin B 1 cos I cosh I, sin I sinh), ^{^ (cosh 2i, + cos 2|)}" Find a similar formula for Log sin (|+ii,). 8. Solution of the equation cosf=a, where a is real. Putting f=|+i),, and equating real and imaginary parts, we obtain cos I cosh), = a, sin I sinh ), = 0. Hence either ), = or | is a multiple of tt. If (i) ), = then cos | = a, which is impossible unless - Igagl. This hypothesis leads to the solution f = 2Z-7r + arc cos a, where arc cos a lies between and Jti-. If (ii) | = mn then cosh ), = ( - 1 )™ a, so that either a S 1 and m is even, or a < - 1 and m is odd. If a = ± 1 then ), = 0, and we are led back to our first case. If | a | > 1 then cosh >, = | a | , and we are led to the solutions f= 2lk,r±i log { a + V(a2-l)} (a>l), f=(2,{; + l)7r±ilog{-a + x/(a2-l)} ('•<-l)- For example, the general solution of cos f= -^ is (=(2k + l)n + nog^. 412 THE GENERAL THEORY OF THE LOGARITHMIC, [x 9. Solve sin f =a, where a is real. 10. Solution of cos f =0+2/3, where /3=t=0. We may suppose |8>0, since the results when /3 <0 may be deduced by merely changing the sign of i. In this case cos |cosh?) = a, sin^sinh);= — j3 (1), and (a/cosh rjf + 0/sinh i;)^ = I. If we put cosh.'^ri=a; we find that «2-(l + aii+/32)a; + a2=0 or x = {Ai±A2y, where Supposea>0. Then ^j>^2>0 a-nd cosh;; = ^i + 42' -A-lso cos| = a/(coshi;) = jli + 42! and since cosh r] > cos | we must take C0shr;=^l+^2) C0S| = j4j — J.2. The general solutions of those equations are | = 2^7r + arccosil/; ri=±log{L+^/{L'^-l)} (2), where L^Ai+A^, M=Ai- A^, and arccos Jflies between and ^n. The values of j) and | thus found above include, however, the solutions of the equations cos|cosh j; = a, sin|sinh?; = (3), as well as those of the equations (1), since we have only used the second of the latter equations after squaring it. To distinguish the two sets of solutions we observe that the sign of sin | is the same as the ambiguous sign in the iirst of the equations (2), and the sign of sinh i; is the same as the ambiguous sign in the second. Since ^ > 0, these two signs must be different. Hence the general solution required is f = 2/t7r ± [arc cos if - 1 log {X + V(-Z^2 - 1 )}]. 11. Work out the cases in which a <0 and a = in the same way. 12. If 0=0 then Z = || a + 1 | +^ | a- 1 1 and J[f=^ | a + 1 |-^| a-1 |. Verify that the results thus obtained agree with those of Ex. 8. 13. Show that if a and are positive then the general solution of sin f=a + i/3 is ^=kir + {-Vf[&xosinM+i\og{L + sJ{L'^-V))], where arc sin M lies between and ^ir. Obtain the solution in the other possible oases. 14. Solve tan f =a, where a is real. [All the roots are real.] 230, 231] EXPONENTIAL, AND CIRCULAR FUNCTIONS 413 ^ 15. Show that the general solution of tan C=a + 0, where /3=j=0, is f ^V + (1-/3)4' where 6 is the numerically least angle such that cos 5 : sin i9 : 1 :: 1 -n2-,32 : 2a : V{(1 -a2-/32)2+4a2}. 16 If = |exp (i,ri), where ^ is real, and o is also real, then the modulus 01 cos 'inz — cos 2jrc is ^[^{1 +C0S 47rc + cos (27r| V2) + cosh (27r| ^2) - 4 cos 2770 cos (tt^ ^2) cosh (tt^ V2)}]. 17. Prove that | exp exp (^+z,) | = exp (exp | cos ,), R {cos cos (^ + i,)} = cos (cos I cosh >,) cosh (sin | sinh i,), I {sin sin (^ + 1,)} = cos (sin | cosh .7) sinh (cos | sinh >/). 18. Prove that |exp f | tends to co if f moves away towards infinity along any straight line through the origin making an angle less than |,r with OX, and to if f moves away along a similar line making an angle greater than \-!t with OX. 19. Prove that |cosf| and isinf| tend to 00 if f moves away towards infinity along any straight line through the origin other than either half of the real axis. 20. Prove that tanf tends to -i or to i if f moves away to infinity along the straight line of Ex. 19, to - i if the line lies above the real axis and to i if it lies below. 231. The connection between the logarithmic and the inverse trigonometrical functions. We found in Ch. VI that the integral of a rational or algebraical function ^{x, a, ff, ...), where a, /3, ... are constants, often assumes different forms according to the values of a, ft ... ; sometimes it can be expressed by means of logarithms, and sometimes by means of inverse trigonometrical functions. Thus, for example. if a>0, but / dx I X ,,. -,- — = -7- arc tan — - (1) /.. ^^— ^— log!^^4z4| (2) .^•2+a 2V(-a) ®U- + V(-a)l if a < 0. These facts suggest the existence of some functional connection between the logarithmic and the inverse circular functions. That there is such a connection may also be inferred from the facts that we have ex- pressed the circular functions of f in terms of exp if, and that the logarithm is the inverse of the exponential function. Let us consider more particularly the equation jx^^ = 2-a^°«[^' 414 THE GENERAL THEORY OF THE LOGARITHMIC, [x which holds when a is real and {a; — a)l(x+a) is positive. If we could write ia instead of a in this equation, we should be led to the formula =-©4^°^(|^)-^ (3). where C is a constant, and the question is suggested whether, now that we have defined the logarithm of a complex number, this equation will not be found to be actually true. Now (§ 221) Log (,r ± ia) = I log {x^ + a^) ±i{4> + 2^jr), where k is an integer and (ji is the numerically least angle such that cos (f) = x/s/(x^ + a?) and sin = aj^{x'^ + a^). Thus TT- Los I ^) = — d)-?7r, 2z ° \x+laj ^ ' where I is an integer, and this does in fact differ by a constant from any value of arctan(a;/a). The standard formula connecting the logarithmic and inverse circular functions is ;tan^=l.Log(l±^) (4), where .•;; is real. It is most easily verified by putting .!i;=tany, when the right- hand side reduces to 1 T /cosv-t-«siny\ 1 ,. , „. , , 2^-^°S (oo^;-rrsi4J = 2iLog(exp2^y)=y + ^., where i; is any integer, so that the equation (4) is ' completely ' true (Ex. xcili. 3). The reader should also verify the formulae arocosa;= —iliog{x±ii^{l — x^)}, arcsin.:!;= — iLog{ia; + ^/(l -.j^)}...(5), where — 1 £a;g 1 : each of these formulae also is ' completely ' true. Example. Solving the equation cosM=^=|{y + (l/y)}, where y = exp (in), with respect to y, we obtain 2/=x±i^{l -x^). Thus: u=-i Log y=-i Log {« ± tV(l - -i^)}, which is equivalent to the first of the equations (5). Obtain the remaining equations (4) and (5) by similar reasoning. 232. The power series for exp^*. We saw in § 212 that when z is real exp^ = H-^+|,+ (1). Moreover we saw in § 191 that the series on the right-hand side * It will be coavenient now to use z instead of f as the argument of the exponential function. 231, 232] EXPONENTIAL, AND CIRCULAR FUNCTIONS 415 remains convergent (indeed absolutely convergent) when . is com- plex. It is naturally suggested that the equation (1) also remains true, and we shall now prove that this is the case. Let the sum of the series (1) be denoted by F{z). The series being absoluteb^ convergent, it follows by direct multiplication (as m Ex. LXXXL 7) that F (z) satisfies the functional equation F{z)F{h) = F{z + h) Now let z = iy, where y is real, and F(z) = f (y). Then and so Ik+^tLlM = y (y) j/Wnl But ■^-^,^=ih+':^+(i^i, Jc i 2 ! ^ 3 ! "^ ■ and so, if | /<; | < 1, .(2). — % ~+^j + ...)|/.j<(e-2)!/.|. k Hence {f{k) - l}/k-*i as A— 0, and so f^y^=UrafJl±l^)=.if^,^ (3). Now f(y) = F{iy) = 1 + (i2/) + MV . . . = (y) + if (y), where 6 (y) is an even and yjr (y) an odd function of y, and so l/(3/)l = V[{<?.(2/)}^ + {^(2/)n = V[{0 (y) + if (y)} {cf> (y) - if (y)}] ^^{F(iy)F(-iy)} = ^{FiO)} = l; and therefore / (y) = cos F + i sin Y, where F is a function of y such that — tt < F S tt. Since / (y) has a differentia] coefficient, its real and imaginary parts cos Fand sin F have differential coefficients, and are a/brfo'ori continuous functions of y. Hence F is a continuous function of y. Suppose that F changes to Y+ K when y changes to y + k. Then K tends to zero with k, and K ^ f cos(F+^)-cosF ) /(c os(F+/0-cosF j Of the two quotients on the right-hand side the first tends to a 416 THE GENERAL THEORY OF THE LOGARITHMIC, [x. limit when k-*0, since cos Y has a differential coefficient with respect to y, and the second tends to the limit — sin Y. Hence K/k tends to a limit, so that Y has a differential coefficient with respect toy. dY Further f'(y) = (-• sin Y+i cos Y) t- . But we have seen already that f'(y) = ifiy) = - sin F+ i cos F. Hence T "'^' ^ = 1/ + '^' where C is a constant, and / (y) = cos (y+(J) + i sin {y + O). But / (0) = 1 when y=0, so that C is a multiple of 27r, and y (y") = cos y + i sin y. Thus F (iy) = cos y +ism y for all real values of y. And, if x also is real, we have F{x + iy) = F{x)F (iy) = exp a; (cos y + isiny) = exp (x + iy), or exp«=l + £r + 2|+ ■•■, for all values of z. 233. The power series for cos z and sin z. From the result of the last section and the equations (1) of § 228 it follows at once that z^ z^ . z^ z^ cos a = 1 - 21 + ^j - •.., sin^ = ^:-g-, + g-| - ... for all values of z. These results were proved for real values of z in Ex. LVL 1. Examples XCVI. 1. Calculate cos i and sin i to two places of decimals by means of the power series for cos z and sin z. 2. Prove that | cos z \ g cosh | z \ and | sin z \ & sinh \z\. 3. Prove that if | 0| <1 then |cos2|<2 and | sin^] <|- 1 z|. 4. Since sin 2z=2 sin z cos z we have Prove by multiplying the two series on the right-hand side (§ 195) and equating coefficients (§ 194) that Verify the result by means of the binomial theorem. Derive similar identities from the equations cos^ z -f sin^ « = 1, cos 2z = 2 cos^ 2 - 1 = 1 - 2 sin^ 2. 232-234] EXPONENTIAL, AND CIRCULAR FUNCTIONS 417 5. Show that exp {(1 + i) 4 = i 2^" exp (i nni) -, . u ! 6. Expand cos 2 cosh in powers of z. [We have coszcoshz + isin0sinh0=oos{(l-i)2} = ^[exp{(l+i>} + esp{-(l + i)2}] =i 2 2*» {1 + ( - 1)"} exp {inni) |J , and similarly cos z cosh z-i sin 2 sinh 2 = cos (1 + 1) 2 = 4i2*»{l + (-l)»}exp(-^««)^. Hence cos2Cosh2 = *22*"{H-(-l)''}cosim7r =- = 1-^ + =^- 1 ii 4!8! -■ 7. Expand sin z sinh 2, cos 2 sinh 2, and sin 2 cosh 2 in powers of 2. 8. Expand sin^2 and sin32 in powers of 2. [Use the formulae sin22=^(l-cos22), sin32=J (3 8in2-sin 32), .... It is clear that the same method may be used to expand cos"z and sin" 2, where n is any integer.] 9. Sum the series cos 2 cos 22 cos 32 „ sin z sin 22 sin 32 C=l+ YJ-+-^p + -3-i- + ..., *= -lT"'"~2^'^T^'^•••• [Here (7+z5 = H-^) + ^-^^... = exp{exp(»2)} = exp (cos 2) {cos (sin 2) + 1 sin (sin 2)}, and similnrly C-iS= exp {exp ( - iz)} = exp (cos 2) {cos (sin 2) - i sin (sin z)}. Hence C= exp (cos 2) cos (sin 2), >S= exp (cos 2) sin (sin 2).] a cos 2 a^ cos 22 , a sin 2 a^ sin 22 10. Sum l + — j-j— H 2^ — +..., ^7- + — 21 — ■ cos 22 cos4z cos 2 cos 32 11. Sum 1-~2F"'"~4! ""■' T'- SV and the corresponding series involving sines. 12. Show that cos 42 cos 82 ^ , ,^ ,^gg^v j,Qgjj ^gjjj 2) ^cog (gin ^j cosh (cos 2)}. '418! ' ••• -i ' ^ 13. Show that the expansions of cos (x+h) and sin (x + h) in powers of A (Ex. Lvr. 1) are valid for all values of x and A, real or complex. 234. The logarithmic series. We found in § 213 that log(l+^) = ^-4^'+i^- (1) when z is real and numerically less than unity. The series on the riffht hand side is convergent, indeed absolutely convergent, when ^ 27 H. 418 THE GENERAL THEORY OF THE LOGARITHMIC, [x z has any complex value whose modulus is less than unity. It is naturally suggested that the equation (1) remains true for such complex values of z. That this is true may be proved by a inodification of the argument of § 213. We shall in fact prove rather more than this, viz. that (1) is true for all values of z such that 1^1 = 1, with the exception of the value - 1. It will be remembered that log (1 + z) is the principal value of Log (1 + z), and that log(l + .) = J^-. where G is the straight line joining the points 1 and 1 + ^ in the plane of the complex variable u. We may suppose that z is not real, as the formula (1) has been proved already for real values of ^. If we put z = r (cos d + i sin 6) = fr, so that I r I g 1, and w = 1 + ?«, then u will describe G ast increases from to r. And ( dM_ r ^dt Jcu ~j„l+ ^t =J k- ?'t+^t'- ... + (- 1)*"-^ f""**"-^ + ^nh^ — I where i?«.=(-in-+^/^'^ (3). It follows from (1) of § 164 that '^"'i=io|i+r«i ^*^- Now 1 1 + f< I or I M I is never less than ct, the perpendicular from on to the line G* Hence dt m+1 1 '"'Jo (m+1) or (m + l)^" * Since z is not real, G cannot pass through vrhen produced. The reader is recommended to draw a figure to illustrate the argument. 234., 235] EXPONENTIAL, AND CIRCULAR FUNCTIONS 419 and so R^ -*- as m -* oo . It follows from (2) that \og{l+z) = z-^z' + ^z'"- (5). We have of course showa in the course of our proof that the series is convergent : this however has been proved already (Ex. LXXX. 4<). The series is in fact absolutely convergent when z\<l and conditionally convergent when | « | = 1. Changing z into — z we obtain log(r^j=-\og{l-z)^z-\-iz' + ^z=+ (6). 235. Now log (1 + ^;) = log {(1 + r cos 6) + ir sin 6] = \ log (1 + 2r cos e + 7-^) + ^ arc tan \ ^^^^^^q ) • That value of the inverse tangent must be taken which lies between - ^tt and ^tt.' For, since 1 + 2 is the vector represented by the line from - 1 to 2, the principal value of am (1 + z) always lies between these limits when z lies within the circle \z\ = \.'' Since ^™ = r^ (cos mQ^-i sin mQ\ we obtain, on equating the real and imaginary parts in equation (5) of § 234-, \ log (1 + 2r cos ^ + r^ = r cos - ^r" cos W + \r^ cos 3(9 - . . ., arctanf,^^^) = rsin^-4r^sin20 + ir'sin3^-..,. \1 + J- cos c// These equations hold when g r S 1, and for all values of Q, except that when r = 1, ^ must not be equal to an odd multiple of -it. It is easy to see that they also hold when - 1 ^ r- S 0, except that, when r = - 1, ^ must not be equal to an even multiple of ir. A particularly interesting case is that in which r = l. In this case we have ^j^^ ^ , log (1 + ^) = log (1 + Ois Q) =4 log (2 + 2 cos^) + i arc tan [^y^^^q = ^log(4cos»^^) + ii^, if - TT < ^ < TT, and so cos^-^cos2^ + icos35-...=ilog(4cosM^), sin^-isin2^ + isin3^-... = i^- * See the preceding footuote. 27—2 420 THE GENERAL THEORY OF THE LOGARITHMIC, [X The sums of the series, for other values of 6, are easily found from the consideration that they are periodic functions of 6 with the period 2-ir. Thus the sum of the cosine series is ^ log (4 cos'' ^6) for all values of 9 save odd multiples of tt (for which values the series is divergent), while the sum of the sine series is \{0 — 2kir) if {2k —l)Tr< 6 < {2k + 1) tt, and zero if 6 is an odd multiple of tt. The graph of the function represented by the sine series is shown in Fig. 58. The function is discontinuous for 6 = {2k + 1) tt. Fig. 58. If we write is and - iz for z in (5), and subtract, we obtain If 2 is real and numerically less than xinity, we are led, by the results of § 231, to the formula arctan2=0— l^s^+^s^- ..., already proved in a different manner in § 214. Examples XCVII. 1. Prove that, in any triangle in which a> 5, b W logc=loga — cos C-^^cos2C-.... [Use the formula log c=\\a% {cfi-\-}fi — ^ah cos C).] 2. Prove that if -\<r<\ and -\t!<6<\t! then »-sin25- Jr2sin45 + Jr3 sin 6fl- ... = d-arc tan K^i^'') tan si , the inverse tangent lying between -\n and \t!. Determine the sum of the series for all other values of 6. 3. Prove, by considering the expansions of log (1 +12) and log (1- 10) in powers of z, that if — 1<»'<1 then >-sin5 + J?-2cos25-^?-3sin35-ir»cos4(9+... = ^log(l+2?-sin5+r2), rcos(9 + i?-2sin25-^?'3cos3fl-ir*sin4^+...=arctan('-^122!i_\ \1— j-sm dj the inverse tangents lying between -^jr and ^w. 235, 236] EXPONENTIAL, AND CIECULAR FUNCTIONS 421 4. Prove that cos^cos^-icos2^cos25 + Jcos35cos35-...=^log(i + 3cos^^), sm^smd)-^sin25sinM+|si„35sin35-... = arcoot(l + oot5 + cot^5), ™!Jr'T "T^""' ^^"^ ^'*"'^" -*'' ^'^'' *-; -d fi-id similar ex- pressions for the sums of the series cos5sin5-icos2^siu25 + ..., sin 5cos 5- Jsin 25cos25 + .... 236. Some applications ofthe logarithmic series. The exponential limit. Let z be any complex number, and h a real number small enough to ensure that \hz\<l. Then log (1 +hz) = hz-^ Qizy + ^ {hzf -..., and so \og{l+hz) 1 =z + <i>{h, z), where {h, z) = - ^hz^ + lh?z^ - Ih^z' + . . . ^ \cj>(h,z)\<\hz^\(l + \hz\ + \h^z^'\+...)=-lI^'A- 1 —\hz\' so that (ji (h, z)~^0 as h-*0. It follows that ,. log{l+hz) lim ^ V '- = z (I) A-»-0 If in particular we suppose h = 1/n, where n is a positive integer, we obtain 1+!)=^, and so lim ralog( This is a generalisation of the result proved in § 20S for real values of ^. From (1) we can deduce some other results which we shall require in the next section. If t and h are real, and h is sufficiently small, we have log (1 + tz + hz)- log (1+tz) ^ 1 J A hz \ h h ^\ 1+tzJ which tends to the limit z/(l + tz) as h^O. Hence |,{log(l4-^.)} = l^ (3). 422 THE GENERAL THEORY OF THE LOGARITHMIC, [X We shall also require a formula for the differentiation of (1 + tz)™, where m is any number real or complex, with respect to t. We observe first that, if (j){t) = yjr (t) + 1% (t) is a complex function of t, whose real and imaginary parts <ji (t) and ^ (t) possess derivatives, then Jf (exp <f>)^jf {(cos X + i sin x) exp f] = {(cos % + i sin x) '^' + (— sin % + * cos %) %'} exp y^ = i-^' + ix) (cos X + *■ sin x) exp -^ = {'^' + *'%') e^P (^ + *X) = <^-' exp ^, so that the rule for differentiating exp ^ is the same as when ^ is real. This being so we have j^ (1 + tz)^ = ^ exp {m log (1 + tz)] = m2^(l+i^)'"-i (4). Here both (1 + tzy^ and (1 + te)'""^ have their principal values 237. The general form of the Binomial Theorem. We have proved already (§ 215) that the sum of the series _ lin\ ^ + (l) is (1 + zj^ = exp {m log (1 + z')\, for all real values of m and all real values of z between — 1 and 1. If a„ is the coefiScient of s" then w+1 whether m is real or complex. Hence (Ex. Lxxx. 3) the series is always convergent if the modulus of z is less than unity, and we shall now prove that its sum is still exp \m log (1 + 2:)}, i.e. the principal value of (1 + zy^. It follows from § 236 that if t is real then ^ (1 + tzY = m^ (1 + tzY-\ 236, 237] EXPONENTIAL, AND CIRCULAR FUNCTIONS 423 z and m having any real or complex values and each side having Its principal value. Hence, if («) = (1 + tzY, we have ^(n) (0 = m (m - l)...(m - n + 1) ^» (1 + tzf-^. This formula still holds if i = 0, so that n\ \n) Now, in virtue of the remark made at the end of § 164, we have where i?„ = j~:Yy\\ ^^ ~ *^"" '^'"' ^^ ^*- But \i z = r (cos d + i sin 6) then \l+tz\ = ^J{\ + 2tr cos + fr") ^1-tr, and therefore \ mim-l)...(m-n + l)\ r^ (1 - Q" -' |m(m- l)...(m-M + l)| (l-Of-^r"' {n-l}\ (1-6'r)"-™' where < ^ < 1 ; so that (cf § 163) I ii;„ I < il (n-l)\ ~ ^'" say. But pn+i J m-n \^ ^^^ Pn n and so (Ex. xxviL 6) pn-^0, and therefore -B„-*0, as n^co. Hence we arrive at the following theorem. Theorem. The sum of the binomial series (m\ 1% i + (7)-+Q-=+" is exp [m log (1+ z)], where the logarithm has its principal value, for all values of m, real or complex, and all values of z such that z\< 1. A more complete discussion of the binomial series, taking account of the more difficult case in which \z\=l, will be found on pp. 225 et seq. of Bromwich's Infinite Series. 424 THE GENERAL THEORY OF THE LOGARITHMIC, [x Examples XOVIII. 1. Suppose m real. Then since log(l+3)=ilog(l + 2)-cos(9+?-2) + iarctanf-?^-^i5_l^\ \i+>-oos5y we obtain i ^'^ j 2'> = exp {\m log (1 + 2r cos fl+r^)} Ois jm arc tan /" '"^'"^ Al = ( 1 + 2?- cos ^ + r'-f'^ Cis jwi arc tan f , '' '^"^ ^ ^ "^ !■ , I \l+?-cos5/J all the inverse tangents lying between —\tt and I^tt. In particular, if we suppose ^=^jr, z=ir, and equate the real and imaginary parts, we obtain l-i^f) '-^+(4) ?-4-... = (H-?-2)i'"cos(marctan?-), (:)'-© '•^ + ( '") '•^ - . . . = (1 + J-^)^'" sin (»i arc tan r). 2. Verify the formulae of Ex. 1 when ot = 1, 2, 3. [Of course when m is a positive integer the series is finite.] 3. Prove that if £)• < 1 then , 1.3 .1.3.5.7 , 2.4 ^2.4.6.8 1 1.3.5 , 1.3.5.7.9, ■ r - - — — — . r^ + - ^ /r v/(l + r') + l ^ •■VI 2(1+?-^) J' = /f N/(l+'-^ )-M •• VI 2(l+»-2) ]■■ 2 2.4.6 2. 4. 6. 8. 10 [Take ot= — ^ in the last two formulae of Ex. 1.] 4 Pro ve that if — J TT < 5 < i TT then cos Mie= oos'"5 \l - T") tan^a + (^ tan«(9 - . . .1 , sin m6 = cos'" d | ("* ) tan 5 - Z™ j tan' 5 + . . . [ , for all real values of m. [These results follow at once from the equations cos m5 + J sin m6 = (cos 5 + i sin 5)" = cos™ 5(1+1 tan d)™.] 5. We jjroved (Ex, lxxxi. 6), by direct multiplication of series, that / (m, «) = 2 ( j 2", where | s | < 1, satisfies the functional equation /K 2)/(™', z)=f{m + m', z). Deduce, by an argument similar to that of § 216, and without assuming the general result of p. 423, that if m is real and rational then / (m, 2) = exp {m log (1 +2)) . 6. If z and fi are real, and — 1< 2 < 1, then 2 ( '^ j «" = oos{/ilog(l + 2)}+isin{/xlog(l+2)}. EXPONENTIAL, AND CIRCULAE FUNCTIONS 425 X. MISCELLANEOUS EXAMPLES ON CHAPTER 1. Show that the real part of i^°« (i+») jg «'**+^^''''^oos{i(«+l)„log2}, where k is any integer. 2. If a cos d + 6sin 6+c=0, where a, 6, e are real and c2>a2 + 62, then where m is any odd or any even integer, according as c is positive or negative and a IS an angle whose cosine and sine are al^{a^ + b^) and blsj{a? + b'^). ' 3. Prove that if 6 is real and sin 5 sin (j!) = 1 then ^ = (^ + i) jr ± I log cot ^ (^-ff + (9), where k is any even or any odd integer, according as sin 6 is positive or negative. 4. Show that if x is real then ^ exp {{a + ib) x) = (a + j6) exp {(a + 16) ^-j, fexp {{a + ib) 4 dr=?2i(^i^l . J a+tb Deduce the results of Ex. lxxxvii. 3. 5. Show that if a>Q then / exp { - (a + ib) x} dx=~^^, and deduce the M. c -c ^ •''' a + ib' results of Ex. lx.xxvii. 5. 6 . Show that if {xjaf + (j/jbf = 1 is the equation of an ellipse, and / {x, y) denotes the terms of highest degree in the equation of any other algebraic curve, then the sum of the eccentric angles of the points of intersection of the ellipse and the curve differs by a multiple of Sn- from - i {log/ (a, ib) - log/ (a, - ib)}. [The eccentric angles are given by/(cscosa, 6sina) + ... — or by where w=expta ; and 2a is equal to one of the values of — iLogP, where P is the product of the roots of this equation.] 7. Determine the number and approximate positions of the roots of the equation tan z = az, where a is real. [We know already (Ex. xvii. 4) that the equation has infinitely many real roots. Now let z = x+iy, and equate real and imaginary parts. AVe obtain sin 2.i^/(cos 2x + cosh 2y) = ax, sinh 2y/(cos 'ix + osh 2y) = ay, so that, iinless x or y- is zero, we have (sin 2a;)/2,i;=(sinh 2y)/2y. 426 THE GENERAL THEORY OF THE LOGARITHMIC, [x This is impossible, the left-hand side being numerically less, and the right- hand side numerically greater than unity. Thus a;=0 or y=0. If y=0 we come back to the real roots of the equation. If ^=0 then tanhy=ay. It is easy to see that this equation has no real root other than zero if a 5 or aSl, and two such roots if <a < 1. Thus there are two purely imaginary roots if 0<a<l ; otherwise all the roots are real.] 8. The equation ta,nz=az + b, where a and 6 are real and 6 is not equal to zero, has no complex roots if agO. If a>0 then the real parts of all the complex roots are numerically greater than | 6/2a | . 9. The equation tan 3= a/2, where a is real, has no complex roots, but has two purely imaginary roots if a<0. 10. The equation tanz=atanh cz, where a and c are real, has an infinity of real and of purely imaginary roots, but no complex roots. 11. Show that if a; is real then e"" cos *^=2 ^ ia»- (fj a''-262+ Q a"-«6*- ...I , where there are ^(w+l) or ^{n + 2) terms inside the large brackets. Find a similar series for e"^ sin hx. 12. If n(f) {z, n) -3- z as n^-00 , then {1 + <^ (z, »)}" -»- exp z. 13. If <^ (f) is a complex function of the real variable t, then [Use the formulae •^ = f + % log (^ = ^ log (,/.2 + ;^2) + i arc tan {x!^).'] 14. Transformations. In Ch. Ill (Exs. xxi. 21 et seq., and Misc. Exs. 22 et seq.) we considered some simple examples of the geometrical relations between figures in the planes of two variables z, Z connected by a relation z=f{Z). We shall now consider some oases in which the relation involves logarithmic, exponential, or circular functions. Suppose firstly that 2 = exp (irZIa), Z= {ajir) Log z where a is positive. To one value of Z corresponds one of z, but to one of z infinitely many of Z. If x, y, r, 6 are the coordinates of z and X, Y, R, e those of Z, we have the relations X = e"^!"' cos (n- Yja), y = e"^''^ sin {n Via), X= {ajn) log r, r= {aeJTt) + 2ia, where h is any integer. If we suppose that - rr< 5 g tt, and that Log 2 has its principal value logz, then ^=0, and Z is confined to a strip of its plane parallel to the axis OX and extending to a distance a from it on each side, one point EXPONENTIAL, AND CIRCULAK FUNCTIONS 4-27 of this strip corresponding to one of the whole z-plane, and conversely. By taking a value of Loge other than the principal value we obtain a similar relation between the z-plane and another strip of breadth 2a in the ^-plane. To the lines in the .Z'-plane for which X and T are constant correspond the circles and radii vectores in the 2-plane for which r and 6 are constant. To one of the latter hnes corresponds the whole of a parallel to OX, but to a circle for which r is constant corresponds only a part, of length 2o, of a parallel to OY. To make Z describe the whole of the latter line we must make z move continually round and round the circle. 15. Show that to a straight line in the Z-plane corresponds an equi- angular spiral in the 2-plane. 16. Discuss similarly the transformation 0=flcosh (wZ/a), showing in particular that the whole s-plane corresponds to any one of an infinite number of strips in the .2'-plane, each parallel to the axis OX and of breadth 2a, Show also that to the line X=X^ corresponds the ellipse J ^ \\[ y. 1^=1 \c cosh {wXola)) [c sinh {irXo/a)) ' and that for different values of Xq these ellipses form a confocal system ; and that the lines F==Yo correspond to the associated system of confocal hyper- bolas. Trace the variation of z as ^ describes the whole of a line X=A'o or Y= Yq How does Z vary as z describes the degenerate ellipse and hyperbola formed by the segment between the foci of the confocal system and the remaining segments of the axis of ^? 17. Verify that the results of Ex. 16 are in agreement with those of Ex. 14 and those of Ch. Ill, Misc. Ex. 25. [The transformation z=c cosh (irZja) may be regarded as compounded from the transformations z = az^, 2i = i{22 + (l/22)}, 22 = exp(7r.Z/a).] 18. Discuss similarly the transformation 0=ctanh(n-^/a), showing that to the lines X=Xq correspond the coaxal circles {« - c coth {2irXola)Y+f = c^ cosech^ {27rX Ja), and to the lines Y= Ya the orthogonal system of coaxal circles. 19. The StereograpMc and Mercator's Projections. The points of a unit sphere whose centre is the origin are projected from the south pole (whose coordinates are 0, 0, - 1) on to the tangent plane at the north pole. The coordinates of a point on the sphere are f, r,, f, and Cartesian axes OX, OY are taken on the tangent plane, parallel to the axes of | and r,. Show that the coordinates of the projection of the point are a;=2|/(l-ff), y=2,/(l-f-a and that x!+iy=2ta.nidCis4,, where <^ is the longitude (measured from the plane 7j = 0) and 6 the north polar distance of the point on the sphere. 428 THE GENERAL THEORY OF THE LOGARITHMIC, [x This projection gives a map of the sphere on the tangent plane, generally known as the Stereographic Projection. If now we introduce a new complex variable Z=X+iT= -i\og\z= —i\og^{x + iy) so that X=<f>, !r=logcot ^5,' we obtain another map in the plane of Z, usually called Meroator's Projection. In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of X and Y respectively. 20. Discuss the transformation given by the equation showing that the straight lines for which x and y are constant correspond to two orthogonal systems of coaxal circles in the ^-plane. 21. Discuss the transformation U{Z-ay^J{Z-h)\ = Log{^ showing that the straight lines for which x and y are constant correspond to sets of confocal ellipses and hyperbolas whose foci are the points Z=a and Z=h. [We have J{Z-a) + ^{Z-h) = ^{b-a)&x^{ x-\-iy), J{Z- a) - ^{Z- h) = V(6 - a) exp {,-x-iy); and it will be found that \Z-a\\\Z-h\=^\h-a\co^'ix, \Z-a\-\Z-h\ = \h-a\wi,'i,y.\ 22. The transformation 2=^*. If z=Z^, where the imaginary power has its principal value, we have exp (log )' + i5)=2 = exp (i log Z)=exp (i log R - e), so that logj'= -e, fl=log fl + 2i7r, where h is an integer. As all values of Ic give the same point z, we shall suppose that ^=0, so that log)-=-e, 6=\o%R (1). The whole plane of Z is covered when R varies through all positive values and 6 from - tt to tt : then r has the range exp ( - tt) to exp tt and 6 ranges through all real values. Thus the ^-plane corresponds to the ring bounded by the circles r=exp(-7r), ?'=exp7r; but this ring is covered infinitely often. If however 6 is allowed to vary only between - tt and tt, so that the ring is covered only once, then R can vary only from exp ( - ir) to exp TT, so that the variation of Z is restricted to a ring similar in all respects to that within which z varies. Each ring, moreover, must be regarded as having a barrier along the negative real axis which z (or Z) must not cross, as its amplitude must not transgress the limits - w and tt. EXPONENTIAL, AND CIRCULAR FUNCTIONS 429 We thus obtain a correspondence between two rings, given by the pair of equations z=Z\ Z=z-\ where each power has its principal value. To circles whose centre is the origin in one plane correspond straight lines through the origin in the other. 23. Trace the variation of z when Z, starting at the point exp tt, moves round the larger circle in the positive direction to the point - exp tt, along the barrier, round the smaller circle in the negative direction, back along the barrier, and round the remainder of the larger circle to its original position. 24. Suppose each plane to be divided up into an infinite series of rings by circles of radii -(2»+l):r -IT 1, Sir (2li+l) ^ Show how to make any ring in one plane correspond to any ring in the other, by taking suitable values of the powers in the equations z = Z'', Z=z~^. 25. If z=Z^, any value of the power being taken, and .2" moves along an equiangular spiral whose pole is the origin in its plane, then z moves along an equiangular spiral whose pole is the origin in its plane. 26. How does Z-^^, where a is real, behave as z approaches the origin along the real axis. [Z moves round and round a circle whose centre is the origin (the unit circle if i"* has its principal value), and the real and imaginary parts of Z both oscillate finitely.] 27. Discuss the same question for Z=z'^*^'', where a and h are any real numbers. 28. Show that the region of convergence of a series of the type 2 a-n^""', - CO where a is real, is an angle, i.e. a region bounded by inequalities of the type 6t, < am z<6i [The angle may reduce to a hne, or cover the whole plane.] 29. Level Curves. It/(«) is a function of the complex variable z, we call the curves for which \f{z)\ is constant the level curves oi f(z). Sketch the forms of the level curves ot z — a. [concentric circles), {z-a){z-b) {Cartesian ovals), (z-a)/{z-b) (coaxal circles), espa (straight lines). 30. Sketch the forms of the level curves of (z-a) (z-b) (z-c), (l+z^3 +z^)/z. [Some of the level curves of the latter function are drawn in Fig. 59, the curves marked l-vil corresponding to the values •10, 2-x/3 = -27, -40, 1-00, 2'00, 2 + ^/3 = 3■73, 4-53 of \f(z)\. The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function ; but to enter into details would carry us into the general theory of functions of a complex variable.] 430 THK GENERAL THEORY OF THE LOGARITHMIC, [X Fig. 59. Pig. 60. Fig. 61. EXPONENTIAL, AND CIRUULAK FUNCTIONS 431 31. Sketch the forms of the level curves of (i) z exp z, (ii) sin z. [See Fig. 60, which represents the level curves of sin z. The curves marked i-viii correspond to /; = -35, -50, •'Zl, 1-00, 1-41, 2-00, 2-83, 4-00.] 32. Sketch the forms of the level curves of exp z-c, where c is a real constant. [Fig. 61 shows the level curves of |exp2-l|, the curves i-vii corresponding to the values of k given by log ^=-1-00, --20, --05, 000, •05, -20, 1-00.] 33. The level curves of sin z-c, where c is a positive constant, are sketched in Figs. 62, 63. [The nature of the curves differs according as to whether c<l or c>l. In Fig. 62 we have taken c=-5, and the curves I-vill correspond to ^=-29, -37, -50, -87, 1-50, 2-60, 4-50, 7-79. Tn Fig. 63 we have taken c=2, and the curves l-vii correspond to ^='58, 1-00, 1'73, 3-00, 5-20, 900, 15'59. If c = \ then the curves are the same as those of Fig. 60, except that the origin and scale are different.] Fig. 62. Fig. 63. 34. Prove that if 0<(9<n- then cos 5+ J cos 35+^ cos 55 + ...= J log cot2^ 5, sin 5 + J sin 35+^ sin 55 + .. .=i7r, and determine the sums of the series for all other values of B for which they are convergent. [Use the equation 2 + ^23 + 1^5+. ..=^ log (l±f) where 0=cos 5+«sin 5. When d is increased by ^ the sum of each series simply changes its sign. It follows that the first formula holds for all values of 6 save multiples of «■ (for which the series diverges), while the sum of the second series is J. if ^k.<6<l?.k+V)., -J- if (2^+1) :r<5<(2^+2) ., and if 5 is a multiple of tt.] 432 THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS [x 35. Prove that if 0<^<^7r then sin 5 -| sin 35 + ^ sin5d-...=ilog(secd+tan5)2; and determine the sums of the series for all other values of 6 for which they are convergent. 36. Prove that cos 6 cos a + ^cos 25cos2a+|cos3fl cos3a+...= — J log {4 (cos 5 - cos a)^}, unless d-a or 6+a is a multiple of 2n. 37. Prove that if neither a nor b is real then dx _ log( — a)— log( — 6) /: [0 {x — a){x-b) a — b each logarithm having its principal value. Verify the result when a = ci, b— — ci, where c is positive. Discuss also the cases in which a or 6 or hoth are real and negative. 38. Prove that if a and ^ are real, and /3>0, then ' dx _ ni /: What is the value of the integral when /3<0 ? 39. Prove that, if the roots of .4^^ + 2A« + C=0 have their imaginary- parts of opposite signs, then dx ni i: the sign of ,J{B^—AC) being so chosen that the real part of is/{B^ — AC)}IAi is positive. APPENDIX I (To Chapters III, IV, V) The Proof that every Equation has a Root Let Z=P(2)=a„2'' + aia"'-i + ...+a„ be a polynomial in z, with real or complex coefficients. We can represent the values of z and Z by points in two planes, which we may call the «-plane and the ^-plane respectively. It is evident that if 2 describes a closed path 7 in the z-plane, then Z describes a corresponding closed path r in the ^-plane. We shall assume for the present that the path r does not pass through the origin. To any value of Z correspond an infinity of values of am Z, differing by multiples of 2n, and each of these values varies continuously as Z describes r.* We can select a particular value of am Z corresponding to each point Fig. A. Fig. B. of r, by first selecting a particular value corresponding to the initial value of Z, and then following the continuous variation of this value as Z moves along r. We shall, in the argument which follows, use the phrase 'the amphtude of Z' and the formula am Z to denote the particular value of the amplitude of Z thus selected. Thus am^' denotes a one- valued and con- tinuous function of X and Y, the real and imaginary parts of Z. * It is here that we assume that T does not pass through the origin. 28 434 APPENDIX I When Z, after describing r, returns to its original position, its amplitude may be the same as before, as will certainly be the case if r does not enclose the origin, like path (a) in Fig. B, or it may differ from its original value by any multiple of Stt. Thus if its path is like (6) in Fig. B, winding once round the origin in the positive direction, then its amplitude will have increased by 27r. These remarks apply, not merely to r, but to any closed contour in the ^-planei, which does not pass through the origin. Associated with any such contour there is a number which we may call ' the increment of am Z when ^describes the contour', a number independent of the initial choice of a particular, value of the amplitude of Z. We shall now prove that if the amplitude of Z is not the same when Z returns to its original position, then the path of z must contain inside or on it at least one point at which Z=0. We can divide y into a number of smaller contours by drawing parallels to the axes at a distance 81 from one another, as in Fig. C* If there is, on the boundary of any one of these contours, a point at which Z==0, what we wish to prove is already established. We may therefore suppose Fig. C. Fig. D. that this is not the case. Then the increment of saaZ, when z describes y, is equal to the sum of all the increments of am Z obtained by supposing 2 to describe each of these smaller contours separately in the same sense as y. For if z describes each of the smaller contours in turn, in the same sense, it will ultimately (see Fig. D) have described the boundary of y once, and each part of each of the dividing parallels twice and in opposite directions. Thus PQ will have been described twice, once from P to § and once from Q to P. As z moves from P to Q, am Z varies continuously, since Z does not pass through the origin ; and if the increment of am Z is in this case 8, then its increment when z moves from Q to P is -6 ; so that, when we add up the increments of am Z due to the description of the various paiis of the smaller contours, all cancel one another, save the increments due to the description of parts of y itself. * There is no difficulty in giving a definite rule for the construction of these parallels : the most obvious course is to draw all the lines x = kdi, y=zh5i, where k la an integer positive or negative. APPENDIX I 435 Hence, if am Z is changed when z describes y, there mtist be at least one of the smaller contours, say -yi, such that a,mZ is changed when z describes yi. This contour may be a square whose sides are parts of the auxiliary parallels, or may be composed of parts of these parallels and parts of the boundary of y. In any case every point of the contour lies in or on the . boundary of a square A; whose sides are parts of the auxiliary parallels and of length Sj. We can now further subdivide y^ by the help of parallels to the axes at a smaller distance Sj from one another, and we can find a contour ■y2i entirely included in a square A2, of side S2 and itself included in Aj, such that am Z is changed when z describes the contour. Now let us take an infinite sequence of decreasing numbers Sj, 821 •••! 8,„, ... , whose limit is zero.* By repeating the argument used above, we can determine a series of squares Ai, Aj, ..., A,„, ... and a series of contours yi, 72> •■■> 7m> ••■ such that (i) A^ + i lies entirely inside A^, (ii) Tm lis** entirely inside A,„, (iii) axaZ is changed when z describes ym- If {iCm, Vm) and (a^m + ^m) i/m-^^ir^ are the lower left-hand and upper right- hand corners of A^, it is clear that x^, x^, ..., x^, ... m B.n increasing and x^ + bi, ^2-1-82, ..., ^m + S«i> '"a decreasing sequence, and that they have a common limit Xq. Similarly y„j and ym+^m have a common limit yoi and i.Hi 2/0) is the one and only point situated inside every square A^. How- ever small 8 may be, we can draw a square which includes {Xo, ^0)1 and whose sides are parallel to the axes and of length 8, and inside this square a closed contour such that am Z is changed when z describes the contour. It can now be shown that P{xo+iyo) = 0. For suppose that P(a;o-J-tyo) = a, where | a| = p>0. Since P{x + i!/) is a con- tinuous function of a; and y, we can draw a square whose centre is {xo, 2/0) and whose sides are parallel to the axes, and which is such that \F (x+iy)- P {xo+iyo)\<ip at all points x+iy inside the square or on its boundary. At all such points P(x+t>/) = a + (l>, where \<j>\<^p. Now let us take any closed contour lying entirely inside this square. As z describes this contour, Z=a + (ji also describes a closed contour. But the latter contour evidently lies inside the circle whose centre is a and whose radius is ^p, and this circle does not include the origin. Hence the amplitude of Z is unchanged. But this contradicts what was proved above, viz. that inside each square A,„ we can find a closed contour the description of which by z changes am Z Hence P(^o+V/o)=0. * We may, e.g., take S„= dJ2"'-K 28—2 436 APPENDIX I f All that remains is to show that we can always find some contour such that a,voiZ is changed when z describes y. Now We can choose R so that kil where 8 is any positive number, however small ; and then, if y is the circle whose centre is the origin and whose radius is R, we have ^■=aoz"(H-p), where | p | < 8, at all points on y. We can then show, by an argument similar to that used above, that am(l + p) is unchanged as z describes y in the positive sense, while am z" on the other hand is increased by inn. Hence am^ is increased by 2?nr, and the proof that Z=0 has a root is completed. We have assumed throughout the argument that neither r, nor any of the smaller contours into which it is resolved, passes through the origin. This assumption is obviously legitimate, for to suppose the contrary, at any stage of the argument, is to admit the truth of the theorem. We leave it as an exercise to the reader to infer, from the discussion which precedes and that of § 43, that wJten z describes any contour y in the positive sense the increment of am Z is Zkjr, where h is the number of roots of Z=0 inside y, multiple roots being counted multiply. There is another proof, proceeding on different lines, which is often given. It depends, however, on an extension to functions of two or more variables of the results of §§ 102 et seq. We define, precisely on the lines of § 102, the wpper and lower bounds of a function /(^, y), for all pairs of values of x and y corresponding to any point of any region in the plane of {x, y) bounded by a closed curve. And we can prove, much as in § 102, that a continuous function / (x, y) attains its upper and lower bounds in any such region. Now \Z\ = \P{x+iy)\^ is a positive and continuous function of x and y. If m is its lower bound for points on and inside y, then there must be a point Zq for which | Z\ = m, and this must be the least value assuined by \Z\. If ot=0, then P(2o) = 0, and we have proved what we want. We may therefore suppose that m>0. The point zq must lie either inside or on the boundary of y : but if y is a circle whose centre is the origin, and whose radius R is large enough, then the last hypothesis is untenable, since \P {z)\^~co as | z | -*- oo . We may therefore suppose that z^ lies inside y. APPENDIX I 437 If we put z = 2o+ f, and rearrange P (2) accordins to powers of f, we obtain say. Let Aj, be the first of the coefficients which does not vanish, and let lj4i| = /i, |f|=p. We can choose p so small that Then 1^ W-P (2o)-^.f*l<iMP^ and |PW|<|P(:o) + ^tfM + iFP*. Now suppose that z moves round the circle whose centre is zq and radius p. Then F{z,) + Aui'' moves k times round the circle whose centre is P(%) and radius | j4if'| = /^p*^, and passes k times through the point in which this circle is intersected by the line joining P (zg) to the origiu. Hence there are k points on the circle described by 2 at which | P (z,,) + ^ ^ f ' | = | P (2,) | — jip'' and so |P(2)|<|P(2o)|-w«^ + iw' = m-ijL.p*<m; and this contradicts the hypothesis that m is the lower bound of | P(z) \. It follows that m must be zero and that P (2o)=0- EXAMPLES ON APPENDIX I 1. Show that the number of roots of /(2) = which lie within a closed contour which does not pass through any root is equal to the increment of {log/(2)}/2,ri when 2 describes the contour. 2. Show that if B is any number such that I«il,|a2l, .la^Ui then all the roots of 2" + 012" -' + ...+«„ =0 are in absolute value less than R. In particular show that all the roots of 25-132-7 = are in absolute value less than 2^. 3. Determine the numbers of the roots of the equation 2^^' + 02 + 6 = where a and 6 are real and p odd, which have their real parts positive and negative. Show that if a>0, 6>0 then the numbers are ^-1 and p + 1 ; if a<0, 6> they are ;? + 1 and jo - 1 ; and if 6 <0 they are p and p. Discuss the particular cases in which a = or 6 = 0. Verify the results when p = l. [Trace the variation of am (z'^P + as + b) as z describes the contour formed by a large semicircle whose centre is the origin and whose radius is R, and the part of the imaginary axis intercepted by the semicircle.] 4 Consider similarly the equations g^'' + az + b = 0, 24»-» + a2 + J = 0, z^'i*^ + az + b = 0. 438 APPENDIX 1 5. Show that if a and (3 are real then the numbers of the roots of the equation «2'' + aV''~^+^^ = which have their real parts positive and negative are n-l and n+1, or n and n, according as n is odd or even. (Math. Trip. 1891.) 6. Show that when z moves along the straight line joining the points 2=«j, z=Z2, from a point near zi to a point near «2, the increment of is nearly equal to ir. 1 1 + — -Zi z~. 7. A contour enclosing the three points 2 = 2i, z=Z2) 2=23 is defined by parts of the sides of the triangle formed by z-^, z^, z^, and the parts exterior to the triangle of three small circles with their centres at those points. Show that when z describes the contour the increment of /I 1 1 \ am + + \Z-Zi z-z^ i-^zl is equal to - 27r. 8. Prove that a closed oval path which surrounds all the roots of a cubic equation / (2) = also surrounds those of the derived equation f(z) = 0. [Use the equation •Z2 Z where %, z^, % are the roots oi f{z)=Q, and the result of Ex. 7.] 9. Show that the roots of/' (2) = are the foci of the ellipse which touches the sides of the triangle {zi, z^, 23) at their middle points. [For a proof see Ceskro's Elementares Lehrhuch der algebraischen Analyds, p. 352.] 10. Extend the result of Ex. 8 to equations of any degree. 11. If /(z) and (^ (2) are two polynomials in z, and y is a contour which does not pass through any root of /(«), and | (^ (2) |<|/(«) | at all points on 7, then the numbers of the roots of the equations /(2) = 0, /(2) + <^(2)=0 which lie inside y are the same. 12. Show that the equations 6^ = a2, ^=az^, e'=az^, where a>e, have respectively (i) one positive root (ii) one positive and one negative root and (iii) one positive and two complex roots within the circle 1 2 1 = 1. {Math. Trip. I^IQ.) APPENDIX II (To Chaptees IX, X) A Note on Double Limit Problems In the course of Chapters IX and X we came on several occasions into contact with problems of a kind which invariably puzzle beginners and are indeed, when treated in their most general forms, problems of great difficulty and of the utmost interest and importance in higher mathematics. Let us consider some special instances. In § 213 we proved that where — 1<«<1, by integrating the equation \l{\ + t) = \-t+t^-... between the limits and x. What we proved amounted to this, that /I if-r />-/>+/>-•' or in other words that the integral of the sum of the infinite series l-t-\-t'^— ..., taken between the limits and x, is equal to the sum of the integrals of its terms taken between the same limits. Another way of expressing this fact is to say that the operations of summation from to oo , and of integration from to X, are commutative when applied to the function (— l)"^'', i.e. that it does not matter in what order they are performed on the function. Again, in § 216, we proved that the differential coefficient of the ex- ponential function ex-px=l+x+-^^ + ... is itself equal to exp x, or that I)^(l+x+^ + ..}j=l)^l+D^x+I)^^,+...; 440 APPENDIX II that is to say that the differential coefficient of the sum of the series is equal to the sum of the differential coefficients of its terms, or that the operations of summation from to oo and of differentiation with respect to x are commu- tative when applied to a;"/ra!. Finally we proved incidentally in the same section that the function exp ^ is a continuous function of x, or in other words that a:-»-f \ ** * / •" • ar-*-§ a;-*"? x~^5 " • i.e. that the limit of the sum of the series is equal to the sum of the limits of the terms, or that the sum of the series is continuous for a'=|, or that the operations of summation from to oo and of making x tend to ^ are com- mutative when applied to x'^jn !. In each of these cases we gave a special proof of the correctness of the result. We have not proved, and in this volume shall not prove, any general theorem from which the truth of any one of them could be inferred im- mediately. In Ex. XXXVII. 1 we saw that the sum of a finite number of con- tinuous terms is itself continuous, and in § 113 that the differential coefiScient of the sum of a finite number of terms is equal to the sum of their differential coefficients ; and in § 160 we stated the corresponding theorem for integrals. Thus we have proved that in certain circumstances the operations symbohsed by lim..., Dx-'-t i...dx are commutative with respect to the operation of summation of a fliiite number of terms. And it is natural to suppose that, in certain circumstances which it should be possible to define precisely, they should be commutative also with respect to the operation of summation of an infinite number. It is natural to suppose so : but that is all that we have a right to say at present. A few further instances of commutative and non-commutative operations may help to elucidate these points. (1) Multiplication by 2 and multiplication by 3 are always commutative, for 2x3i<x = 3x2xx for all values of x. (2) The operation of taking the real part of z is never commutative with that of multiplication by i, except when 2=0 ; for ix.'R{x+ii/)=ix, 'R{ix{x + it/)} — -y. (3) The operations of proceeding to the limit zero with each of two variables x and ^ may or may not be commutative when applied to a function f(x, y). Thus lim {lim {x+y))=^\vra x=0, lim {lim(A-+y)}=limy=0; APPENDIX II 441 but on the other hand lim ( lim ■^) = lim - = lim 1 = 1, ^'"^l ^™ ^)= lit" — = hm(-l)=-l. (4) The operations 2..,, lim... may or may not be commutative. Thus if x-*-l through values less than 1 then lim|s^-^^a;4=lim log(l+;r) = log2, «-»! I 1 ''^ J !C-»1 2 i lim ^—^ x^'l = 2 *^— ^ =log 2 ; but on the other hand limil («''-a;'' + i)| = lim{(l-a;)H-(^-.»2)4-...}=liml=.l, 2 j lim (a?" -a;" + 1)1= l(l-l) = + + + ... = 0. 1 U^l J 1 The preceding examples suggest that there are three possibilities with respect to the commutation of two given operations, viz. : (1) the operations may always be commutative ; (2) they may never be commutative, except in very special circumstances ; (3) they may be commutative in most of the ordinary cases which occur practically. The really important case (as is suggested by the instances which we gave from Ch. IX) is that in which each operation is one which involves a passage to the limit, such as a differentiation or the summation of an infinite series : such operations are called limit operations. The general question as to the circumstances in which two given limit operations are commutative is one of the most important in all mathematics. But to attempt to deal with questions of this character by means of general theorems would carry us far beyond the scope of this volume. We may however remark that the answer to the general question is on the lines suggested by the examples above. If L and L' are two limit operations then the numbers LL'z and L'Lz are not generally equal, in the strict theoretical sense of the word 'general'. We can always, by the exercise of a little ingenuity, i5nd z so that LL'z and L'Lz shall differ from one another. But they are equal generally, if we use the word in a more practical sense, viz. as meaning 'in a great majority of such cases as are likely to occur naturally ' or in ordinary cases. 442 APPENDIX II Of course, in an exact science like pure mathematics, we cannot be satisfied with an answer of this kind ; and in the higher branches of mathematics the detailed investigation of these questions is an absolute necessity. But for the present the reader may be content if he realises the point of the remarks which we have just made. In practice, a result obtained by assuming that two limit-operations are commutative is probably true : it at any rate affords a valuable suggestion as to the answer to the problem vender consideration. But an answer thus obtained must, in default of a further study of the general question or a special investigation of the particular problem, such as we gave in the instances which occurred in Oh. IX, be regarded as suggested only and not proved. Detailed investigations of a large number of important double limit problems will be found in Bromwich's Infinite Series. APPENDIX III (To § 158 AND Chapter IX) The circular functions The reader will find it an instructive exercise to work out the theory of the circular functions, starting from the definition dt [^ c (1) y=y{x)=a,vat&nx= \ — ./o 1- Df.* The equation (1) defines a unique value of y corresponding to every real value of X. As y is continuous and strictly increasing, there is an inverse function ai=x (jj), also continuous and steadily increasing. We write (2) x=a: (y) = ta,ny. Df. If we define jr by the equation (3) i-=r^.> Df- then this function is defined for — ^7r<y<^n-. We write further (4) cosy=-===, siny=— f=, Df. Vl + a;^ Vl+^ where the square root is positive; and we define cosy and smy, when yia - ^ jt or-^TT, so that the functions shall remain continuous for those values of y. Finally we define cos y and sin y, outside the interval ( - ^tt, ^ tr), by (6) tan(y + 7r) = tany, cos (y + 7r)= -cosy, sin (y + »;■)= —sin y. Df. We have thus defined cosy and sin;/ for all values of y, and tany for all values of y other than odd multiples of ^n. The cosine and sine are continuous for all values of y, the tangent except at the points where its definition fails. The further development of the theory depends merely on the addition formulae. Write 1 -XiX^ and transform the equation (1) by the substitution ^l + M t-x^ 6 — .; , W — - X—Xxti' \+xit' We find ■^1 + ^2 f^' <^" /"*' du , Z"*! du arc tan ;; = / ■^—, — r, = / r"; — 5 + / ^r-. — s \-XiX2 } -00,1+u' Jo 1 + M^ Jo 1+M^ = arc tan Xi + arc tan x^ . * These letters at the end of a line indicate that the formulae which it contains are definitions. 444 _ APPENDIX III Prom this we deduce ,„, , , ^ tany.+tanyo (6) tan(y,+2/,) = ^-^L_|, an equation proved in the first instance only when 3/1, j/ji ^nd 2/1+I/2 lie in ( — Jtt, -^jt), but immediately extensible to all values of yi and yj by means of the equations (5). From (4) and (6) we deduce cos (yi +3/2) = ± (cos yj cos ya - sin yjsin yg). To determine the sign put y2=0. The equation reduces to cosyi= +cosyi, which shows that the positive sign must be chosen for at least one value of ya, viz. y2=0. It follows from considerations of continuity that the positive sign must be chosen in all cases. The corresponding formula for sin(yi+y2) may be deduced in a similar manner. The formulae for diflferentiation of the circular functions may now be de- duced in the ordinary way, and the power series derived from Taylor's Theorem. An alternative theory of the circular functions is based on the theory of infinite series. An account of this theory, in which, for example, cos a' is defined by the equation oosa; = l- -, + -,-... will be found in Whittaker and Watson's Modern Analysis (Appendix A). APPENDIX IV The infinite in analysis and geometry Some, though not all, systems of analytical geometry contain 'infinite' elements, the line at infinity, the circular points at infinity, and so on. The object of this brief note is to point out that these concepts are in no way dependent upon the analytical doctrine of limits. In what may be called ' common Cartesian geometry ', a point is a pair of real numbers {x, y). A line is the class of points which satisf3' a linear relation ax + hy + c— 0, in which a and b are not both zero. There are no infinite elements, and two lines may have no point in common. In a system of real homogeneous geometry a point is a class of triads of real numbers {so, y, 2), not all zero, triads being classed together when their constituents are proportional. A line is a class of points which satisfy a linear relation ax + by + cz = 0, where a, b, c are not all zero. In some systems one point or line is on exactly the same footing as another. In others certain ' special ' points and lines are regarded as peculiarly distinguished, and it is on the relations of other elements to these special elements that emphasis is laid. Thus, in what may be called ' real homogeneous Cartesian geometry ', those points are special for which 2=0, and there is one special line, viz. the line 2=0. This special line is called 'the line at infinity'. This is not a treatise on geometry, and there is no occasion to develop the matter in detail. The point of importance is this. The infinite of analysis is a ' limiting ' and not an ' actual ' infinite. The symbol ' 00 ' has, throughout this book, been regarded as an 'incomplete symbol', a symbol to which no independent meaning has been attached, though one has been attached to certain phrases containing it. But the infinite of geometry is an actual and not a limiting infinite. The 'line at infinity' is a line in precisely the same sense in which other lines are lines. It is possible to set up a correlation between 'homogeneous' and 'common' Cartesian geometry in which all elements of the first system, the special elements excepted, have correlates in the second. The line ax + by-\-cz=Q, for example, corresponds to the line ax + by + c = 0. Every point of the first line has a correlate on the second, except one, viz. the point for which 2 = 0. When {x, y, 2) varies on the first line, in such a manner as to tend in the limit to the special point for which 2 = 0, the corresponding point on the second line varies so that its distance from the origin tends to infinity. This correlation is historically important, for it is from it that the vocabulary of the subject has been derived, and it is often useful for purposes of illustration. It is how- ever no more than an illustration, and no rational account of the geometrical infinite can be based upon it. The confusion about these matters so prevalent among students arises from the fact that, in the commonly used text books of analytical geometry, the illustration is taken for the reality. CAMBRIDGE : PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PRESS