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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ yC-NRLF $B 100 5a? \ / Digitized by Google Digitized by Google INTRODUCTION TO THE THEORY OF. FOURIER'S SERIES AND INTEGRALS BY H. S. CARSLAW ScD. (Cambridge), D.Sc. (Glasgow), F.R.S.E. PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF SYDNEY FORMERLY FELLOW OF EMMANUEL COLLEGE, CAMBRIDGE, AND LECTURER IN MATHEMATICS IN THE UNIVERSITY OF GLASGOW SECOND EDITION, COMPLETELY REVISED MACMILLAN AND CO., LIMITED 3 ST. MARTIN'S STREET, LONDON ^ 1921 i Digitized by VjOOQ IC QA + 04 COPYRIGHT. First Published, 1906. OLASOOW: PRINTED AT THE DMIVKRBITV FRB88 PY ROSKRT MACLKHOSK AND CO, I.TP. Digitized by Google PREFACE This book forms the first volume of the new edition of my book on Fourier's Series and Integrals and the Mathematical Theory of the Conduction of Heat, published in 1906; and now for some time out of print. Since 1906 so much advance has been made in the Theory of Fourier's Series and Integrals, as well as in the mathematical discussion of Heat Conduction, that it has seemed advisable to write a completely new work, and to issue the same in two volumes. The first volume, which now appears, is concerned with the Theory of Infinite Series and Integrals, with special reference to Fourier's Series and Integrals. The second volume will be devoted to the Mathematical Theory of the Conduction of Heat. No one can properly understand Fourier's Series and Integrals without a knowledge of what is involved in the convergence of infinite series and integrals. With these questions is bound up the development of the idea of a limit and a function, and both are founded upon the modern theory of real numbers. The first three chapters deal with these matters. In Chapter IV. the Definite Integral is treated from Riemann's point of view, and special attention is given to the question of the convergence of infinite integrals. The theory of series whose terms are functions of a single variable, and the theory of integrals which contain an arbi- trary parameter are discussed in Chapters V. and VI. It will be seen that the two theories are closely related, and can be developed on similar lines. The treatment of Fourier's Series in Chapter VII. depends on Dirichlet's Integrals. There, and elsewhere throughout the book, the Second Theorem of Mean Value will be found an essential part of the argument. In the same chapter the work of Poisson is adapted to modern standards, and a prominent place is given to Fej6r's work, both in the proof of the fundamental theorem Digitized by VjOOQ IC 470606 ■ vi PREFACE and in the discussion of the nature of the convergence of Fourier's Series. Chapter IX. is devoted to Gibbs's Phenomenon, and the last chapter to Fourier's Integrals. In this chapter the work of Pringsheim, who has greatly extended the class of functions to which Fourier's Integral Theorem applies, has been used. Two appendices are added. The first deals with Practical Har- monic Analysis and Periodogram Analysis, In the second a biblio- graphy of the subject is given. The functions treated in this book are "ordinary" functions. An interval (a, 6) for which f{x) is defined can be broken up into a finite number of open partial intervals, in each of which the function is monotonic. If infinities occur in the range, they are isolated and finite in number. Such functions will satisfy most of the demands of the Applied Mathematician. The modern theory of integration, associated chiefly with the name of Lebesgue, has introduced into the Theory of Fourier's Series and Integrals functions of a far more complicated nature. Various writers, notably W. H. Young, are engaged in building up a theory of these and allied series much more advanced than any- thing treated in this book. These developments are in the meantime chiefly interesting to the Pure Mathematician specialising in the Theory of Functions of a Eeal Variable. My purpose has been to remove some of the difficulties of the Applied Mathematician. The preparation of this book has occupied some time, and much of it has been given as a final course in the Infinitesimal Calculus to my students. To them it owes much. For assistance in the revision of the proofs and for many valuable suggestions, I am much indebted to Mr. E. M. Wellish, Mr. E. J. Lyons and Mr. H. H. Thome of the Department of Mathematics in the University of Sydney. H. S. CARSLAW. Emmanuel College, Cambridge, Jan. 1921. Digitized by Google CONTENTS THE THEORY OF FOURIER'S SERIES AND INTEGRALS PAGE Historical Introduction 1 CHAPTER I RATIONAL AND IRRATIONAL NUMBERS SECTION 1-2. Rational Numbers 16 3-5. Irrational Numbers - - - - 17 6-7. Relations of Magnitude for Heal Numbers - - . - 21 8. Dedekind's Theorem - 23 9. The Linear Continuum. Dedekind's Axiom .... 24 10. The Development of the System of Real Numbers - - 25 CHAPTER II INFINITE SEQUENCES AND SERIES 11. Infinite Aggregates - . . . 29 12. The Upper and Lower Bounds of an Aggregate - . . 30 13. Limiting Points of an Aggregate 31 14. Weierstrass's Theorem 32 15. Convergent Sequences ........ 33 16. Divergent and Oscillatory Sequences 37 17. Monotonic Sequences ..--.... 39 18. A Theorem on an Infinite Set of Intervals ... 40 19-23. Infinite Series - 41 vii Digitized by ^ /Google Vlll CONTENTS CHAPTER III FUNCTIONS OF A SINGLE VARIABLE. LIMITS AND CONTINUITY SECTION PAGE 24. The Idea of a Function - - 49 25. Lt f(.v) 50 at— ♦■a 26. Some Theorems on Limits 52 27. Lt /(^) '- 55 28-29. Necessary and Sufficient Conditions for the Existence of a Limit ... 50 30-32. Continuous Functions - - 59 33. Discontinuous Functions - - -. 64 34. Monotonic Functions ------- - 66 35. Inverse Functions - 68 36. The Possibility of Expressing a Function as the Difl'erence of two Positive and Monotonic Increasing Functions - - 69 37. Functions of Several Variables ^ - - - - - - 71 CHAPTER IV THE DEFINITE INTEGRAL 38. Introductory 7(5 39. The Sums /S' and « - - - - - - - - 77 40. Darboux's Theorem 79 41. The Definite Integral of a Bounded Function - - - - 81 42. Necessary and Sufficient Conditions for Integrability - 82 43. Integrable Functions 84 44. A Function integrable in (a, b) has an Infinite Number of Points of Continuity in any Partial Interval of (a, 6) - - - ' 86 45-47. Some Properties of the Definite Integral - - . . 87 48. The First Theorem of Mean Value --.-.. 92 49. The Definite Integral considered as a Function of its Upper Limit 93 50. The Second Theorem of Mean Value 94 51-56. Infinite Integrals. Bounded lotegrand. Infinite Interval 98 57-58. The Mean Value Theorems for Infinite Integrals - - 109 59-61. Infinite Integrals. Integrand Infinite Ill Examples on Chapter IV 120 Digitized by VjOOQIC CONTENTS ix CHAPTER V THE THEORY OF INFINITE SERIES WHOSE TERMS ARE FUNCTIONS OF A SINGLE VARIABLE SECTION PAGB 62. Introductory 122 63. The Sum of a Series of Continuous Functions may be Dis- continuous 124 64. Repeated Limits 127 65-69. Uniform Convergence - - 129 70. Term by Term Integration 140 71. Term by Term Differentiation 143 72. The Power Series 144 73. Extensions of Abel's Theorem on the Power Series - - 149 74-76. Integration of Series. Infinite Integrals - - - - 154 Examples on Chapter V ....... i64 CHAPTER. VI DEFINITE INTEGRALS- CONTAINING AN ARBITRARY PARAMETER 77. Continuity of the Integral / F{x^y)dx .... 1^9 raf 78. Differentiation of the Integral / F{x^y)dx - - - - 170 raf 79. Integration of the Integral / F{x^y)dx 172 -so 80-83. Infinite Integrals i F{x^y)dx. Uniform Convergence - 173 84. Continuity of the Integral / F{x^y)dx 179 r* 85. Integration of the Integral i F{x, y)dx 180 86. Differentiation of the Integral/ F(x,y)dx •. - - 182 ra' 87. Properties of the Infinite Integral / F(x,y)dx - - - 183 88. Applications of the Preceding Theorems 184 89. The Repeated Integral f dx f f(x,y)dy - - - - 190 Examples on Chapter VI 193 Digitized by VjOOQIC X CONTENTS CHAPTER VII FOURIER^S SERIES SBCTION PAGE 90. Introductory 196 91-92. Dirichlet's Integrals. (First Form) 200 93. Dirichlet's Conditions - - 206 94. Dirichlet's Integrals. (Second Form) 207 96. Proof of the Convergence of Fourier's Series - - - - - 210 96. The Cosine Series - - - - • 215 97. The Sine Series 220 98. Other Forms of Fourier's Series 228 99-100. Poisson's Treatment of Fourier's Series . . - . 230 101. Fej6r's Theorem - - . . . . . . . 234 102. Two Theorems on the Arithmetic Means . . _ . 238 103. Fej^r's Theorem and Fourier's Series ^ - - . - 240 Examples on Cliapter VII 243 CHAPTER VIII THE NATURE OF THE CONVERGENCE OF FOURIER'S SERIES 104-106. The Order of the Terms 248 107-108. The Uniform Convergence of Fourier's Series - - 263 109. Differentiation and Integration of Fourier's Series - - 261 110. A more General Theory of Fourier's Series - - - - 262 CHAPTER IX THE APPROXIMATION CURVES AND GIBBS'S PHENOMENON IN FOURIER'S SERIES 111-112. The Approximation Curves 264 113-114. Gibbs's Phenomenon 268 115. The Trigonometrical Sum gf^E^Ll-lliH - - - - 271 r=i 2r - 1 116. Gibbs's Phenomenon for the Series 22 ^^"(^^•~^)'^^ - - 277 * r=i 2r-l 117. Gibbs's Phenomenon for Fourier's Series in general - * 280 Digitized by VjOOQIC CONTENTS xi CHAPTER X FOURIER'S INTEGRALS SSOnON PAGE 118. Introductory 283 119. Fourier's Integral Theorem for the Arbitrary Function f{x) in its Simplest Form 284 120-121. More General Conditions for f{x) ----- 287 122. Fourier's Cosine Integral and Fourier's Sine Integral - - 292 123. Sommerf eld's Discussion of Fourier's Integrals - - - 293 Examples on Chapter X - 294 Appendix I. Practical Harmonic Analysis and Periodogram Analysis 295 Appendix II. Bibliography 302 Digitized by Google Digitized by Google HISTORICAL INTRODUCTION In the middle of the eighteenth century there was a prolonged controversy as to the possibility of the expansion of an arbitrary function of a real variable in a series of sines and cosines of multiples of the variable. The question arose in connection with the problem of the Vibrations of Strings. The theory of these vibrations reduces to the solution of the Differential Equation and the earliest attempts at its solution were made by D'Alem- bert,* Euler;t and Bernoulli. J Both D'Alembert and Euler obtained the solution in the functional form y = (f}(x+at)+\[r(x—at). The principal difference between them lay in the fact that D'Alembert supposed the initial form of the string to be given by a single analytical expression, while Euler regarded it as lying along any arbitrary continuous curve, different parts of which might be given by different analytical expressions. Bernoulli, on the other hand, gave the solution, when the string starts from rest, in the form of a trigonometrical series y = A^sin x cos at+A^ sin 2x cos 2at+ . . . , and he asserted that this solution, being perfectly general, must contain those given by Euler and D'Alembert. The importance of his discovery was immediately recognised, and Euler pointed out that if this statement of the solution were correct, an arbitrary function of a single variable must be developable in an infinite series of sines of multiples of the variable. This he * M4m. de VAcad&mie dt Berlin^ 3, p. 214, 1747. f/oc, cit., 4, p. 69, 1748. Xloc. dt., 9, p. 173, 1753. CI A • . digitized by GoOglC 2 .•- : .•": y: •HiSTo»rcAL* introduction held to be obviously impossible, since a series of sines is both periodic and odd, and he argued that if the arbitrary function had not both of these properties it could not be expanded in such a series. While the debate was at this stage a memoir appeared in 1759* by Lagrange, then a young and unknown mathematician, in which the problem was examined from a totally different point of view. While he accepted £uler's solution as the most general, he objected to the mode of demonstration, and he proposed to obtain a satisfactory solution by first considering the case of a finite number of particles stretched on a weightless string. From the solution of this problem he deduced that of a continuous string by making the number of particles infinite.! In this way he showed that when the initial displacement of the string of unit length is given hy f(x), and the initial velocity by F(x)y the displacement at time t is given by ri » 2/ = 2 I ^ (sin mrx' sin nirx cos mrat)f{x')dx Jo 1 H I 7j - (sin mrx' sin mrx sin mrat)F{x')dx . This result, and the discussion of the problem which Lagrange gave in this and other "memoirs, have prompted some mathe- maticians to deny the importance of Fourier's discoveries, and to attribute to Lagrange the priority in the proof of the development of an arbitrary function in trigonometrical series. It is true that in the formula quoted above it is only necessary to change the order of summation and integration, and to put ^ = 0, in order that we may obtain the development of the function /(i^} in a series of sines, and that the coefficients shall take the definite integral forms with which we are now familiar. Still Lagrange did not take this step, and, as Burkhardt remarks, J the fact that he did not do so is a very instructive example of the ease with which an author omits to draw an almost obvious conclusion from his results, when his investigation has been undertaken with another end in view. Lagrange's purpose was to demon- *Cf. Lagrange, (Euvrea, T. I., p. .37. \loc. cit,, §37. J Burkhardt, " Entwicklungen nach oscillirenden FuDctionen," Jahresber, J), Math. Ver., Leipzig, 10, Hft. II., p. 32, 1901. Digitized by VjOOQ IC HISTORICAL INTKODUCTION 3 strate the truth of Euler's solution, and to defend- iis general conclusions against D'Alembert's attacks. When he had obtained his solution he therefore proceeded to transform it into the func- tional form given by Euler. Having succeeded in this, he held his demonstration to be complete. The further development of the theory of these series was due to the astronomical problem of the expansion of the reciprocal of the distance between two planets in a series of cosines of multiples of the angle between the radii. As early as 1749 and 1754 D'Alembert and Euler had published discussions of this question in which the idea of the definite integral expressions for the coeflScients in Fourier's Series may be traced, and Clairaut, in 1757,* gave his results in a form which practically contained these coefficients. Again, Euler,t in a paper written in 1777 and published in 1793, actually employed the method of multiplying both sides of the equation f{x) = a^ + 2aiCos x + 2a2COs 2a; + . . . 2a„co8 nx+,,, by cos nx and integrating the series term by term between the limits and x. In this way he found that cos nx dx. It is curious that tllOSS pSpfeft '"Seetn to have had no effect upon the discussion of the problem of the Vibrations of Strings in which, as we have seen, D'Alembert, Euler, Bernoulli, and Lagrange were about the same time engaged. The explanation is probably to be found in the fact that these results were not accepted with confidence, and that they were only used in deter- mining the coefficients of expansions whose existence could be demonstrated by other means. It was left to Fourier to place our knowledge of the theory of trigonometrical series on a firmer foundation, and, owing to the material advance made by him in this subject the most important of these expansions are now generally associated with his name ancj called Fourier's Series. His treatment was suggested by the problems he met in the Mathematical Theory of the Conduction of Heat. It is to be * Paris, Hist. Acad, sci., 1754 [59], Art. iv. (July 1757). iPetrop. iV. Acta., 11, 1793 [98], p. 94 (May 1777). digitized by VjOOQIC 4 HISTORICAL INTRODUCTION found in various memoirs, the most important having been presented to the Paris Academy in 1811, although it was not printed till 1824-6. These memoirs are practically contained in his book, Theorie Tnathematique de la chaleur (1822). In this treatise several discussions of the problem of the expansion of a function in trigonometrical series are to be found. Some of them fail in rigour. One is the same as that given by Euler. However, it is a mistake to suppose that Fourier did not estab- lish in a rigorous and conclusive manner that a quite arbitrary function (meaning by this any function capable of being re- presented by an arc of a continuous curve or by successive portions of different continuous curves), could be represented by the series we now associate with his name, and it is equally wrong to attribute the first rigorous demonstration of this theorem to Dirichlet, whose earliest memoir was published in 1829.* A closer examination of Fourier's work will show that the importance of his investigations merits the fullest recogni- tion, and Darboux, in the latest complete edition of Fourier s mathematical works,! points out that the method he employed in the final discussion of the general case is perfectly sound and practically identical with that used later by Dirichlet in his classical memoir. In this discussion Fourier followed the line of argument which is now customary in dealing with infinite series. He proved that wlJen the values «o = 2:^J fix')dx\ If" 1 a„ = — I f(x') cos nx'dx\ TJ'J -IT 1 fir bn=-\ f(oc') sin nx' dx\ 7rJ_^ ) are inserted in the terms of the series %+{^i cosx+b^ sin x) + {a^ cos 2flj-f 63 sin 2cc) -h . . . , the sum of the terms up to cos nx and sin nx is If" sinK2^+l)(a^;~a^)^^. He then discussed the limiting value of this sum as n becomes * Dirichlet, J. Math., BerHn, 4, p. 157, 1829. t (Envres de Fourier, T. I., p. 512, 1888. Digitized by VjOOQ IC n^\, HISTORICAL INTRODUCTION infinite, and thus obtained the sum of the series now called Fourier's Series. Fourier made no claim to the discovery of the values of the coefficients 1 C^ • a,, = - 1 f{x') cos nfhxdxy Ifir ' ft„ = — I /(aj') sin nx dx\ 71^1. We have already seen that they were employed both by Clairaut and Euler before this time. Still there is an important differ- ence between Fourier's interpretation of these integrals and that which was current among the mathematicians of the eighteenth century. The earlier writers by whom they were employed (with the possible exception of Clairaut) applied them to the determination of the coefficients of series whose existence had been demonstrated by other means. Fourier was the first to apply them to the representation of an entirely arbitrary function, in the sense in which we have explained this term. In this he made a distinct advance upon his predecessors. Indeed Riemann* asserts that when Fourier, in his first paper to the Paris Academy in 1807, stated that a completely arbitrary function could be expressed in such a series, his statement so surprised Lagrange that he denied the possibility in the most definite terms. It should also be noted that he was the first to allow that the arbitrary function might be given by different analytical expressions in different parts of the interval ; also that he asserted that the sine series could be used for other functions than odd ones, and the cosine series for other functions than even. ones. Further, he was the first to see that when a function is defined for a given range of the variable, its value outside that range' is in no way determined, and it follows that no one before him can have properly understood the representation of an arbitrary function by a trigonometrical series. The treatment which his work received from the Paris Academy is evidence of the doubt with which his contemporaries viewed * Cf . Riemann, *'Uber die Darstellbarkeit einer Function durch eine trigono- metrische Reihe," Gdtlingen, Ahlu Gea. Wif^s., 13, §2, 1867. Digitized by VjOOQ IC 6 HISTORICAL INTRODUCTION his arguments and results. His first paper upon the Theory of Heat was presented in 1807. The Academy, wishing to en- courage the author to extend and improve his theory, made the question of the propagation of heat the subject of the grand prix de Tnath^matiques for 1812. Fourier submitted his M^TYioire sur la propagation de la Chaleur at the end of 1811 as a candidate for the prize. The memoir was referred to Laplace, Lagrange, Legendre, and the other adjudicators ; but, while awarding him the prize, they qualified their praise with criticisms of the rigour of his analysis and methods,* and the paper was not published at the time in the M^moires de VAcademie des Sciences, Fourier always resented the treatment he had received. When publishing his treatise in 1822, he incorporated in it, practically without change, the first part of this memoir ; and two years later, having become Secretary of the Academy on the death of Delambre, he caused his original paper, in the form in which it had been communicated in 1811, to be published in these M^inoiresA Probably this step was taken to secure to himself the priority in his important discoveries, in consequence of the attention the subject was receiving at the hands of other mathematicians. It is also possible that he wished to show the injustice of the criticisms which had been passed upon his work. After the publication of his treatise, when the results of his different memoirs had become known, it was recognised that real advance had been made by him in the treatment of the subject and the substantial accuracy of his reasoning was admitted. J * Their report is quoted by Darboux in his Introduction (p. vii) to (Euvrat de Fourier, T. I. : — ** Cette pi6ce renferme les v6ri tables Equations difiP^rentielles de la transmission de la chaleur, soit k Tint^rieur des corps, soit k leur surface ; et la nouveaut^ du sujet, jointe k son importance, a determine la Classe a couronner cet Ouvrage, en observant cependant que la maniere dont I'Auteur parvient a ses Equations n'est pas exempte de difiicultes, et que son analyse, pour les int^grer, laisse encore quelque chose k desirer, soit relativement k la gendralite, soit m^me du cote de la rigueur. " t Mimoires de VAcad. des Sc, 4, p. 185, and 5, p. 153. J It is interesting to note the following references to his work in the writings of modern mathematicians : Kelvin, Coll. Works, Vol. III., p. 192 (Article on '* Heat," Enc, Brit., 1878). ** Returning to the question of the Conduction of Heat, we have first of all to say that the th^ry of it was discovered by Fourier, and given to the world through the French Academy in his TMorie analytique de la Chaleur, with Oigitized by VjOOQ IC HISTORICAL INTRODUCTION 7 The next writer upon the Theory of Heat was Poisson. He employed an altogether different method in his discussion of the question of the representation of an arbitrary function by a trigonometrical series in his papers from 1820 onwards, which are practically contained in his books, Traits de Mecanique (T. I. (2*^ ed.) 1833),and Th^orie '^natUmatique de la Gfudeur (1S'S5). He began with the equation 1 - /i^ * 1 — oi 7-^ STTTs = 1 + 2 >)^"cos n{x'—x), l — 2hcos{x—x)+h^ ^ \ /' solutions of problems naturally arising from it, of which it is difficult to say whether their uniquely original quality, or their transcendently intense mathe- matical interest, or their perennially important instructiveness for physical science, is most to be praised." Darboux, Introduction, (Euvrea de Fourier, T. I., p. v, 1888. ** Par Timportance de ses d^couvertes, par I'influence decisive qu'il a exercee sur le d6veloppement de la Physique mathematique, Fourier meritait I'hommage qui est rendu aujourd'hui a ses travaux et ii sa m^moire. Son nom figurera digne- ment k c6t4 des noms, illustres entre tous, dont la liste, destin^e a s'accroltre avec les anntes, constitue d6s a present un veritable titre d'honneur pour notre pays. La Th^rie analytique de la CJialeur . . . , que Ton pent placer sans injustice a cdt^ des ecrits scientifiques les plus parfaits de tous les temps, se recommande par une exposition int^ressante et originale des principes fonda- mentaux ; il ^claire de la lumiere la plus vive et la plus penetrante toutes les idees essentielles que nous devons k Fourier et sur lesquelles doit reposer d^sormais la Philosophic naturelle ; mais il contient, nous devons le reconnaitre, beaucoup de negligences, des erreurs de calcul et de detail que Fourier a su ^viter dans d'autres ecrits." Poincar(5, Thiorie ancUytique de la propagation de la Clialeur, p. 1, § 1, 1891. * ** La theorie de la chaleur de Fourier est un des premiers exemples de I'appli- cation de Tanalyse a la physique ; en partant d'hypoth^ses simples qui ne sont autre chose que des faits exp^rimentaux generalises, Fourier en a deduit une serie de consequences dont I'ensemble constitue une theorie complete et coherente. Les resultats qu'il a obtenus sont certes interessants par eux-mSmes, mais ce qui Test plus encore est la methode qu'il a employee pour y parvenir et qui servira toujours de modeie h tous ceux qui voudront cultiver une branche quelconque de la physique mathematique. J'ajouterai que le livre de Fourier a une importance capitale dans I'histoire des mathematiques et que I'analyse pure lui doit peut-^tre plus encore que I'analyse appliquee." Boussinesq, Thdorie analytique de la Chaleur, T. I., p. 4, 1901. ** Les admirables applications qu'il fit de cette methode {i.e. his method of inte- grating the equations of Conduction of Heat) sont, k la fois, assez simples et assez generales, pour avoir servi de module aux geom^tres de la premiere moitie de ce ^iede ; et elles leur ont ete d'autant plus utiles, qu'elles ont pu, avec de leg^res modifications tout au plus, 6tre transportees dans d'autres branches de la Physique mathematique, notamment dans I'Hydrodynamique et dans la Theorie del'eiasticite." Digitized by VjOOQ IC 8 HISTORICAL INTRODUCTION h being numerically less than unity, and he obtained, by integration, J _/(^')i:r27tcos(aj'-ic)+A2'^^' = r f{x')dx +^^h''[ f{x)co8n{x'-x)dx\ J -n I J\- IT While it is true that by proceeding to the limit we may deduce that m or hU(<^-0)+f(x + 0)] is equal to LtL-f f(x')dx'+-f]h4' nx)cosn(x-x)dx'\ we are not entitled to assert that this holds for ilte value h=l, unless we have already proved that the series converges for this value. This is the real diflSculty of Fourier's Series, and this limitation on Poisson's discussion has been lost sight of in some presentations of Fourier's Series. There are, however, other directions in which Poisson's method has led to most notable results. The importance of his work cannot be exaggerated.* After Poisson, Cauchy attacked the subject in different memoirs published from 1826 onwards,! using his method of residues, but his treatment did not attract so much attention as that given about the same time by Dirichlet, to which we now turn. Dirichlet's investigation is contained in two memoirs which appeared in 1829 J and 1837. § The method which he employed we have already referred to in speaking of Fourier's work. He based his proof upon a careful discussion of the limiting values of the integrals "' .. V sin )ux , ^ ^ fix) . ^ cto...a>0. i ♦ For a fuU treatment of Poisson's method, reference may be made to Bocher's paper, ** Introduction to the Theory of Fourier's Series," Ann. Math., Princeton, N. J. (Ser. 2), 7, 1906. tSee Bibliography, p. 303. t«^. Math., Berlin, 4, 1829. % Dove's Bepertorium der Physik, Bd. I., p. 152, 1837. Digitized by Google HISTORICAL INTRODUCTION 9 as fii increases indefinitely. By this means he showed that the sum of the series ^0 + (<^iCos x+h^ sin x) + (agcos 2a; + 62 sin 2x) -t- . . . , where the coefllcients a^, etc., are those given by Fourier, is K/(^-0)+/(a: + 0)]...-7r ^V ^2» • • • *^n - 1 > ^71 ~ ^* The sum 8={x^-X^)f{x^) + {x^-X^)f{x^) + ..^+{Xn-X^.^)f{xJ is formed. And the integral I f{x) dx is defined as the limit of J a *./. Math., BerHn, 17, 1887. Digitized by VjOOQ IC 10 HISTORICAL INTRODUCTION this sum when the number of parts is increased indefinitely and their length diminished indefinitely. On this understanding every continuous function has an integral. For discontinuous functions, he proceeded as follows : If a function f{x) is continuous in an interval (a, h) except at the point c, in the neighbourhood of which f{x) may be bounded or not, the integral of /(a?) in (a, h) is defined as the sum of the two limits Cc-h cb Lt I f{x)dXy Lt I f{x)dx, when these limits exist. Riemann dismisses altogether the requirement of continuity, and in forming the sum S multiplies each interval (Xr — x^^i) by the value of f{x), not necessarily at the beginning (or end) of the interval, but at a point ^r arbitrarily chosen between these, or by a number intermediate between the lower and upper bounds of f(x) in (Xr-u Xr). The integral I f(x)dx is defined as the limit of this sum, if such exists, when the number of the partial intervals is increased indefinitely and their length tends to zero. Riemann's treatment, given in the text in a slightly modified form, is now generally adopted in a scientific treatment of the Calctilus. It is true that a more general theory of integration has been developed in recent years, chiefly due to the writings of Lebesgue,* de la Vall(^e Poussin and Young; that theory is mainly for the specialist in certain branches of Pure Mathe- matics. But no mathematician can neglect the concept of the definite integral which Riemann introduced. One of the immediate advances it brought was to bring within the integrable functions a class of discontinuous functions whose discontinuities were infinitely numerous in any finite interval. An example, now classical, given by Riemann, was the function defined by the convergent series : M [2^ [nx] "^12"^ 22 "^••' 7l2 where [nx] denotes the positive or negative difference between nx and the nearest integer, unless nx falls midway between two consecutive integers, when the value of [nx] is to be taken a<^ * See footnote, p. 77. Digitized by VjOOQ IC HISTORICAL INTRODUCTION 11 zero. The sum of this series is discontinuous for every rational value of X of the form jpj^n, where p is an odd integer prime to n. With Riemann's definition the restrictions which Dirichlet had placed upon the function f{x) were considerably relaxed. To this Riemann contributed much, and the numerous writers who have carried out similar investigations since his time have, still further widened the bounds, while the original idea that every continuous function admitted of such an expansion has been shown to be false. Still it remains true that for all practical purposes, and for all ordinary functions, Dirichlet s investigation established the convergence of the expansions. Simplifications have been introduced in his proof by the introduction of the Second Theorem of Mean Value, and the use of a modified form of Dirichlet's Integral, but the method which he employed is still the basis of most rigorous discussions of Fourier s Series. The nature of the convergence of the series began to be ex- amined after the discovery by Stokes (1847) and Seidel (1848) of the property of Uniform Convergence. It had been known since Dirichlet's time that the series were, in general, only conditionally convergent, their convergence depending upon the presence of both positive and negative terms. It was not till 1870: that Heine showed that, if the function is finite and satisfies Dirichlet's Conditions in the interval ( — x, x), the Fourier's Series converges uniformly in any interval lying within an interval in which f{x) is continuous. This condition has, like the other conditions of that time, since been somewhat modified. In the last thirty or forty years quite a large literature has arisen dealing with Fourier's Series. The object of many of the investigations has been to determine sufficient conditions to be satisfied by the function f{x\ in order that its Fourier's Series may converge, either throughout the interval ( — tt, x), or at particular points of the interval. It appears that the convergence or non-convergence of the series for a particular value of x really depends only upon the nature of the function in an arbitrarily small neighbourhood of that point, and is independent of the general character of the function throughout the interval, this general character being limited only by the necessity for the . existence of the coefficients of the series. These memoirs — Digitized by VjOOQ IC 12 HISTORICAL INTRODUCTION associated chiefly with the names of Du Bois-Reymond, Lipschitz, Dini, Heine, Cantor, Jordan, Lebesgue, de la Vallee Poussin, Hobson and Young — have resulted in the discovery of sufficient conditions of wide scope, which suffice for the convergence of the series, either at particular points, or, generally, throughout the interval. The necessary and sufficient conditions for the con- vergence of the series at a point of the interval, or throughout any portion of it, have not been obtained. In view of the general character of the problem, this is not surprising. Indeed it is not improbable that no such necessary and sufficient conditions may be obtainable. In many of the works referred to above, written after the discovery by Lebesgue (1902) of his general theory of integra- tion, series whose terms did not exist under the old definition of the integral are included in the discussion. The fact that divergent series may be utilised in various ways in analysis has also widened the field of investigation, and indeed one of the most fruitful advances recently made arises from the discussion of Fourier's Series which diverge. The word " sum," when applied to a divergent series, has, of course, to be defined afresh ; but all methods of treatment agree in this, that when the same process is applied to a convergent series the "sum," according to the new definition, is to agree with the " sum " obtained in the ordinary way. One of the most important methods of " summation " is due to Cesaro, and in its simplest form is as follows : Denote by Sn the sum of the first n terms of the series u^ + U2 + n^+.... Let ^ 8, + 8,-f...+8 ,^ " n When Lt Sn = S, we say that the series is "summable," and n->cc that its " sum " is S. It is not difllcult to show that if tlie series is convergent, then Lt Sn= Lt s^, SO that the " condition of consistency " is satisfied. [Cf. § 102.] Digitized by VjOOQIC HISTORICAL INTRODUCTION 13 Fejer was the first to consider this sequence of Arithmetic Means, ?ljj~^ ^1^ ^2 "'" *^3 Si, 2 ' " 3 * ••• for the Fourier's Series. He established the remarkable theorem* that the sequence is convergent, and its limit is i(/(a!+0)+/(a;-0)) at every point in ( — x, tt) where f{x+0) and f(x — 0) exist, the only conditions attached to f(x) being that, if bounded, it shall be integrable in ( — tt, x), and that, if it is unbounded, I f(x) dx shall be absolutely convergent. '"" Later, Hardy showed that if a series ^1 + ^2 + ^3+ ••• is summable by this method, and the general term Un is of the order I/ti, the series is convergent in the ordinary sense, and thus the sum jS ( = Lt 8^) and the sum s ( = Lt 8^) will be the same. [Cf. § 102.] .Hardy's theorem, combined with Fej^r's, leads at once to a new proof of the convergence of Fourier's Series, and it can also be applied to the question of its uniform convergence. Many of the results obtained by earlier investigators follow directly from the application to Fourier's Series of the general theory of summable series.! Recent investigations show that the coefficients in Fourier's Series, now frequently called Fourier's Constants, have im- portant properties, independent of whether the series converges or not. For example, it is now known that if f{x) and (x) are two functions, bounded and integrable in ( — x, x), and a^, a„, b^ are Fourier's Constants for f(x), and a^, a/, 6/ those for (x\ the series 00 1 1 f '^ converges, and its sum is -I f{x){x)dx. To this theorem, *Mnth. Ann., Leipzig, 58, 1904. t Chapman, Q, J. Math., Londm, 43, 1912 ; Hardy, London, Proc. Math. Soc. (Ser. 2), 12, 1913. Digitized by VjOOQIC 14 HISTORICAL INTRODUCTION and to the results which follow from it, much attention has recently been given, and it must be regarded as one of the most important in the whole of the theory of Fourier's Series.* The question of the approximation to an arbitrary function by a finite trigonometrical series was examined by Weierstrass in I885.t He proved that if f(x) is a continuous and periodic function, given the arbitrary small positive quantity e, a finite Fourier's Series can be found in a variety of ways, such that the absolute value of the difference of its sum and f(x) will be less than e for any value of x in the interval. This theorem was also discussed by Picard, and it has been generalised in recent memoirs by Stekloff and Kneser. In the same connection, it should be noted that the application of the method of least squares to the determination of the coefficients of a finite trigonometrical series leads to the Fourier I coefficients. / This result was given by Topler in 1876.}: | As many applications of Fourier's Series really only deal with a finite number of terms, these results are of special interest. From the discussion of the Fourier's Series it was a natural step to turn to the theory of the Trigonometrical Series % + (ttiCos x + b^ sin x) + PgCos 2a? + bgSin 2x) + . . . , where the coefficients are no longer the Fourier coefficients. The most important question to be answered was whether such .an expansion was unique ; in other words, whether a function could be represented by more than one such trigonometrical series. This reduces to the question of whether zero can be represented by a trigonometrical series in which the coefficients do not all vanish. The discussion of this and similar problems was carried on chiefly by Heine and Cantor ,§ from 1870 onwards, in a series of papers which gave rise to the modern Theory of Sets of Point^ another instance of the remarkable influence Fourier's Series hav had upon the development of mathematics. |! In this place it wili ♦Of. Young, London, Proc. /?. Soc. (A), 85, 1911. fj. Math., Berlin, 71, 1870. t Topler, Wien, Anz, AL Wias., 13, 1876. § Bibliography, p. 305. II Van Vleok, "The Influence of Fourier's Series upon the Development of Mathe- matics," American Association for the Advancement of Science (Atlanta), 1913. See also a paper with a similar title by Jourdain, Scient-iu, Bologna (Ser. 2), 22, 1917. Digitized by VjOOQ IC HISTORICAL INTRODUCTION 15 be suflBcient to state that Cantor showed that all the coefficients of the trigonometrical series must vanish, if it is to be zero for all values of x in the interval ( — tt, tt), with the exception of those which correspond to a set of points constituting, in the language of the Theory of Sets of Points, a set of the n^ order, for which points we know nothing about the value of the series. REFERENCES. BuRKHARDT, " Entwicklungen imch oscillirenden Functionen," Jahresber. D. Math. Ver., Leipzig, 10, 1901. Gibson, " On the History of the Fourier Series," Edinburgh, Proc. Math, Soc, 11, 1893. KiEMANN, " Uber die Darstellbarkeit einer Function. durch eiue trigono- metrische Reihe," Gottingen, Ahh. Ges, Wiss., 13, 1867 ; also Werke, pp. 213-250. Sachse, Versuch einer Gesehichte der Daratellung willkiirlwheo' Functionen durch trigonometrische Reihen, Diss. Gottingen, 1879 ; also Zs. Math., Leipzig, 25, 1880, and, in French, BuL sci, math., Paris (S6r. 3), 4, 1880. Digitized by Google CHAPTER I RATIONAL AND IRRATIONAL NUMBERS THE SYSTEM OF REAL NUMBERS 1. Rational Numbers. The question of the convergence of Infinite Series is only capable of satisfactory treatment, when the difficulties underlying the conception of irrational number have been overcome. For this reason we shall first of all give a short discussion of that subject. The idea of number is formed by a series of generalisa- tions. We begin with the positive integers. The operations of addition and multiplication upon these numbers are always possible; but if a and b are two positive integers, we cannot determine positive integers x and y, so that the equations a = b+x and a = by are satisfied, unless, in the first case, a is greater than 6, and, in the second case, a is a multiple of 6. To overcome this difficulty fractional and negative numbers are introduced, and the system of rational numbers placed at our disposal.* The system of rational numbers is ordered^ i.e, if we have two different numbers a and b of this system, one of them is greater than the other. Also, ii a^b and fc^c, then a^c, when a, b and c are numbers of the system. Further, if two different rational numbers a and 6 are given, we can always find another rational number greater than the one and less than the other. It follows from this that between ♦The reader who wishes an extended treatment of the system of rational numbers is referred to Stolz und Gmeinier, Theoreiiache Ariihmetik^ Leipzig, 1900-1902, and Pringaheim, Vorlesungen iiher Zahlen- und Ftmktionenlehre, Leipzig, 1916. 16 Digitized by Google RATIONAL AND IRRATIONAL NUMBERS 17 any two different rational numbers there are an infinite number of rational numbers.* 2. The introduction of fractional and negative rational num- bers may be justified from two points of view. The fractional numbers are necessary for the representation of the subdivision of a unit magnitude into several equal parts, and the negative numbers form a valuable instrument for the measurement of magnitudes which may be counted in opposite directions. This may be taken as the argument of the applied mathematician. On the other hand there is the argument of the pure mathe- matician, with whom the notion of number, positive and negative, integral and fractional, rests upon a foundation independent of measurable magnitude, and in whose eyes analysis is a scheme which deals with numbers only, and has no concern per 86 with measurable quantity. It is possible to found mathe- matical analysis upon the notion of positive integral number. Thereafter the successive definitions of the different kinds of num- ber, of equality and inequality among these numbers, and of the four fundamental operations, may be presented abstractly.! 3. Irrational Numbers. The extension of the idea of number from the rational to the irrational is as natural, if not as easy, as is that from the positive integers to the fractional and negative rational numbers. Let a and b be any two positive integers. The equation x^ = a cannot be solved in terms of positive integers unless a is a perfect 6*** power. To make the solution possible in general the irrational numbers are introduced. But it will be seen below that the system of irrational numbers is not confined to numbers which arise as the roots of algebraical equations whose coefiicients are integers. So much for the desirability of the extension from the abstract side. From the concrete the need for the extension is also evident. We have only to consider the measurement of any quantity to ♦ When we say that a set of things has a finite number of members, we mean that there is a positive integer w, such that the total number of members of the set is less than n. When we say that it has an infinite number of members, we mean that it has not a finite number. In other words, however large n may be, there are more members of the set than n, tCf. Hobson, Lfrndoiif Proc. Math. Soc. (Ser. 1), 35, p. 126, 1913 ; also the same author's Theory of Functions of a Real Variable^ p. 13. CI B Digitized by Google 18 RATIONAL AND IRRATIONAL NUMBERS which the property of unlimited divisibility is assigned, e.g. a straight line L produced indefinitely. Take any segment of this line as unit of length, a definite point of the line as origin or zero point, and the directions of right and left for the positive and negative senses. To every rational number corresponds a Fig. 1. definite point on the line. If the number is an integer, the point is obtained by taking the required number of unit segments one after the other in the proper direction. If it is a fraction dzp/q, it is obtained by dividing the unit of length into q equal parts and taking p of these to the right or left according as the sign is positive or negative. These numbers are called the Tneasures of the corresponding segments, and the segments are said to be corriTnensurable with the unit of length. The points correspond- ing to rational numbers may be called rational points. There are, however, an infinite number of points on the line L which are not rational points. Although we may approach them as nearly as we please by choosing more and more rational points on the line, we can never quite reach them in this way. The simplest example is the case of the points coin- ciding with one end of the diagonal of a square, the sides of which are the unit of length, when the diagonal lies along the line L and its other end coincides with any rational point. Thus, without considering any other case of incommensur- ability, we see that the line L is infinitely richer in points than the system of rational numbers in numbers. Hence it \^ clear that if we desire to follow arithmetically all the properties of the straight line, the rational numbers are insufficient, and it will be necessary to extend this system by the creation of other numbers. 4. Returning to the point of view of the pure mathematician, we shall now describe Dedekind's method of introducing the irrational number, in its most general form, into analysis.* ♦Dedekind (1831-1916) published his theory in Sfetigkeit mid irt^atiovnle Zahlen, Braunschweig, 1872 (Englisli translation in Dedekind's Ensai/s on NumheVy Chicago, 1901). Digitized by VjOOQ IC THE SYSTEM OF REAL NUMBERS 19 Let us suppose that by some method or other we have divided all the rational numbers into two classes, a lower class A and an upper class B, such that every number a of the lower class is less than every number ^ of the upper class. When this division has been made, if a number a belongs to the class A, every number less than a does so also; and if a number fi belongs to the class B, every number greater than j8 does so also. Three different cases can arise : (I) The lower class can have a greatest number and the upper cUlss no 87)iallest number. This would occur if, for example, we put the number 5 and every number less than 5 in the lower class, and if we put in the upper class all the numbers greater than -6. (II) The upper class can have a sm/xllest number and the lower class no greatest number. This would occur if, for example, we put the number 5 and all the numbers greater than 5 in the upper class, while in the lower class we put all the numbers less than 5. It is impossible that the lower class can have a greatest number m, and the upper class a smallest number ti, in the same division of the rational numbers ; for between the rational numbers m and n there are rational numbers, so that our hypo- thesis that the two classes contain all the rational numbers is contradicted. But a third case can arise : (III) The lower class can have no greatest number and the upper class no smallest number. For example, let us arrange the positive integers and their s(iuares in two rows, so that the squares are underneath the numbers to which they correspond. Since the square of a frac- tion in its lowest terms is a fraction whose numerator and deno- minator are perfect squares,* we see that there are not rational numbers whose squares are 2, 3, 5, 6, 7, 8, 10, 11, ... , 1 r 3 4... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .... *If a/formal proof of this statement is needed, see Dedekind, loc. cil., English translation, p. 14, or Hardy, Course of Pure Mathematics (2nd Ed. ), p. 6. Digitized by VjOOQIC 20 RATIONAL AND IRRATIONAL NUMBERS However, there are rational numbers whose squares are as near these numbers as we please. For instance, the numbers 2, 1-5, 1-42, 1-415, 1-4143, ..., 1, 1-4, 1-41, 1-414, 1-4142, ..., form an upper and a lower set in which the squares of the terms in the lower are less than 2, and the squares of the terms in the upper are greater than 2. We can find a number in the upper set and a number in the lower set such that their squares differ from 2 by as little as we please.* Now form a lower class, as described above, containing all negative rational numbers, zero and all the positive rational numbers whose squares are less than 2 ; and an upper class containing all the positive rational numbers whose squares are greater than 2. Then every rational number belongs to one class or the other. Also every number in the lower class is less than every number in the upper. The lower class has no greatest number and the upper class has no smallest number. 5. When by any means we have obtained a division of all the rational numbers into two classes of this kind, the lower class having no greatest number and the upper class no smallest number, we create a new number defined by this division. We dall it an irrational number, and we say that it is greater than all the rational numbers of its lower class, and less than all the rational numbers of its upper class. Such divisions are usually called sections A The irrational number ^2 is defined by the section of the rational numbers described above. Similar sections would define the irrational numbers ^S, t^o, etc. The system of irrational numbers is given by all the possible divisions of the rational numbers into a lower class A and an upper class B, such that every rational number is in one class or the other, the numbers of the lower class being less than the numbers of the upper class, while the lower class has no greatest number, and the upper class no smallest number. Jn other words, every irrational number is defined by its section (A, B). It may be said to " cor^respond " to this section. ♦Cf. Hardy, foe. cit., p. 8. t French, couptire ; German, Schnitt. Digitized by Google THE SYSTEM OF REAL NUMBERS 21 The system of rational numlers and ii^ational numbers together make up the system of real numbers. The rational numbers themselves "correspond" to divisions of rational numbers. For instance, take the rational number m. In the lower class A put the rational numbers less than m, and m itself. In the upper class B put all the rational numbers greater than m. Then m corresponds to this division of the rational numbers. Extending the meaning of the term section, as used above in the definition of the irrational number, to divisions in which the lower and upper classes have greatest or smallest numbers, we may say that the rational number m corresponds to a rationed section (A» B),* and that the irrational numbers correspond to irrational sections. When the rational and irrational numbers are defined in this way, and together form the system of real numbers, the real number which corresponds to the rational number m (to save confusion it is sometimes called the rational-real number) is conceptually distinct from m. However, the relations of magnitude, and the fundamental operations for the real numbers, are defined in such a way that this rational-real number has no properties distinct from those of wi, and it is usually denoted by the same symbol. 6. Belations of Magnitude for Beal Numbers. We have extended our conception of number. We must now arrange the system of real numbers in order ; i.e. we must say when two numbers are equal or unequal to, greater or less than, each other. In this place we need only deal with cases where at least one of the numbers is irrational. An irrational number is never equal to a rational number. They are always different or unequal. Next, in § 5, we have seen that the irrational number given by the section (A, B) is said to be greater than the rational number m, when m is a member of the lower class A, and that the rational number m is said to be greater than the irrational number given by the section (A, B), when m is a member of the upper class B. Two irrational numbers are equal, when they are both given by the same section. They are different or unequal, when they are given by different sections. The irrational number a given by the section (A, B) is greater than the irrational number a' given by the section (A', B'), when the class A contains numbers of the class B'. Now the class A has no greatest number. But if a certain number of the class A belongs to the class B', all the numbers of A * The rational number m could correspond to two sections : the one named in the text, and that in which the lower class A contains all the rational numbers less than m, and the upper class B, m and all the rational numbers greater than wi. To save ambiguity, one of these sections only must be chosen. Digitized by VjOOQ IC 22 RATIONAL AND IRRATIONAL NUMBERS greater than this number also belong to B'. The class A thus contains an infinite number of members of the class B', when a > a. If a real number a is greater than another real number a', then a is less than a. It will be observed that the notation >, =, < is used in dealing with real numbers as in dealing with rational numbers. The real number /3 is said to lie between the real numbers a and y, when one of them is greater than /? and the other less. With these definitions the system of real numbers is ordered. If we have two different real numbers, one of them is greater than the other ; and if we have three real numbers such that a> fS and f^>y, then a>'y. These definitions can be simplified when the rational numbers themselves are given by sections, as explained at the end of § 5. 7. Between any two different rational numbers there is an infinite number of rational numbers. A similar property holds for the system of real numbers, as will now be shown : (I) Betioeen any two different real nurtibers a, a' there are an infinite number of rational numbers. If a and a' are rational, the property is known. If a is rational and a irrational, let us assume a>a'. Let a' be given by the section (A', B'). Then the rational number a is a member of the upper class B', and B' has no least number. Therefore an infinite number of members of the class B' are less than a. It follows from the definitions of i^ 5 that there are an infinite number of rational numbers greater than a and less than a. A similar proof applies to the case when the irrational number a is greater than the rational number a. There remains the case when a and a are both irrational. Let a be given by the section (A, B) and a' by the section (A', B'). Also let a>a'. Then the class A of a contains an infinite number of members of the class B' of tt' ; and these numbers are less than a and greater than a'. A similar proof applies to the case when a < a. The result which has just been proved can be made more general : — (II) Betweeii any ttvo different real numbers there are an infinite number of irrational numbers. Let a, a' be the two given numbers, and suppose a0). Then the number M+^h, which is also greater than Jf, would belong to the class A, and M would not separate the two classes A and B. In the second place, whatever the positive number e may be, the number M—e belongs to the class A. It follows from the w^ay in which the class A is defined that there is at least one number of (E) greater than if — e. This number M is called the upper bound of the aggregate (E). ' It may belong to the aggregate. This occurs when the aggregate contains a finite number of terms. But when the aggregate contains an infinite number of terms, the upper bound need not belong to it. For example, consider the rational numbers whose * This notation is convenient, the letter £J being the first letter of the French term ennemhle. Digitized by VjOOQIC INFINITE SEQUENCES AND SERIES 31 squares are not greater than 2. This aggregate is bounded on the right, its upper bound being the irrational number ^% which does not belong to the aggregate. On the other hand, the aggre- gate of real numbers whose squares are not greater than 2 is also bounded on the right, and has the same upper bound. But ;^2 belongs to this aggregate. If the upper bound M of the aggregate {E) does not belong to it, there must be an infinite number of terms of the aggregate between M and J/— e, however small the positive number e may be. If there were only a finite number of such terms, there w^ould be no term oi (E) between the greatest of them and M, which is contrary to our hypothesis. It can be shown in the same way that when an aggregate (E) is hounded on the lefty there is a number m possessivg the follow- ing properties : no number of {E) is smaller than m ; however small the positive number e may be, there is a number of (E) less than m + e. The number 7>i defined in this way is called the lower bound of the aggregate (E). As above, it may, or may not, belong to the aggregate when it has an infinite number of terms. But when the aggregate has only a finite number of terms it must belong to it. 13. Limiting Points of an Aggregate. Consider the aggregate 111 There are an infinite number of points of this aggregate in any interval, however small, extending from the origin to the right. Such a point round which an infinite number of points of an aggregate cluster, is called a limiting point * of the aggregate. More definitely, a will be a Ivmiting point of the aggregate (E) if, however small the positive number e may be, there is in (E) a point other than a whose distance from a is less than e. If there be one such point within the interval (a — e, a + e), there will be an infinite number, since, if there were only n of them, * Sometimes the term point of condensation is used ; French, point-limite, point d'accnmiUation ; German, Havfungspnnht, VerdicIUungspnnkf. Digitized by VjOOQ IC 32 INFINITE SEQUENCES AND SERIES and a,j were the nearest to a, there would not be in (E) a point other than a whose distance from a was less than |a — a^|.* In that case a would not be a limiting point, contrary to our hypothesis. An aggregate may have more than one limiting point. The rational numbers between zero and unity form an aggregate with an infinite number of limiting points, since every point of the segment (0, 1) is a limiting point. It will be noticed that some of the limiting points of this aggregate belong to it, and some, namely the irrational points of the segment and its end-points, do not. In the example at the beginning of this section, 1 1 1 1 the lower bound, zero, is a limiting point, and does not belong to the aggregate. The upper bound, unity, belongs to the aggregate, and is not a limiting point. The set of real numbers from to 1, inclusive, is an ag^egate which is identical with its limiting points. 14. Weierstrass's Theorem, An infinite aggregate, bounded above and below, has at least one limiting point. Let the infinite aggregate (E) be bounded, and have M and m for its upper and lower bounds. We can arrange all the real numbers in two classes relative to the aggregate (E), A number x will be said to belong to the class A when an infinite number of terms of (E) are greater than X. It will be said to belong to the class B in the contrary case. Since m belongs to the class A and M to the class B, there are members of both classes. Also any number in the class A is less than any number in the class B. By Dedekind's Theorem, there is a number jul separating the two classes. However small the positive number e may he, jm — e belongs to the class A, and /jl + c to the class B. Thus the interval (m — e, M + e) contains an infinite number of terms of the aggregate. * It isjusual to denote the difference between two real numbers a and /), taken positive, by |a-6|, and to call it the absohUe ixilne or modxdus of (a-?>). With this notation |ir + y|^|aT| + |y|, and |ify| = |i»||y|- Digitized by Google INFINITE SEQUENCES AND SERIES 33 Hence /x is a limiting point. As will be seen from the example of § 13, this point may coincide with M or m. An infinite aggregate, when unbounded, need not have a limit- ing point ; e.g. the set of integers, positive or negative. But if the aggregate has an infinite number of points in an interval of finite length, then it must have at least one limiting point. 15. Convergent Sequences. We speak of an infinite sequence of numbers u u^ u^ u when some law is given according to which the general term u„ may be written down. The sequence u^ , Ug , u^y ... is said to he convergent and to have the limit A, when, by indefinitely increasing n, the difference between A and Un becomes, and thereafter reinains, as small as we please. This property is so fundamental that it is well to put it more precisely, as follows : The sequence is said to be convergent and to have the limit A, when, any positive number e having been chosen, as small as we please, there is a positive integer v such that \K-u^\<» and we say that as n tends to infinity, u^ has the limit A* The letter e is usually employed to denote an arbitrarily small positive number, as in the above definition of convergence to a limit as n tends to infinity. Strictly speaking, the words as small as we please are unnecessary in the definition, but they are inserted as making clearer the property that is being defined. We shall Very frequently have to employ the form of words which occurs in this definition, or words analogous to them, and * The phrase ** ?t„ tends to the limit A as n tends to infinity " is also used. Digitized by VjOOQIC 34 INFINITE SEQUENCES AND SERIES the beginner is advised to make himself familiar with them by formally testing whether the following sequences are convergent or not : (^0 1> 2' 2"^' ^^^ ^' ^'^r ^ + 2"'"22'";' (b) 1, -^, ^, .... (d) 1, -1, 1, -1, A sequence cannot converge to two distinct limits A and B. If this were possible, let e<' — -, — '. Then there are only a finite number of terms of the sequence outside the interval (A — €, A+€)y since the sequence converges to the value A. This contradicts the statement that the sequence has also the limit B, for we would only have a finite number of terms in the interval of the same length with B as centre. The application of the test of convergency contained in the definition involves the knowledge of the limit A. Thus it will frequently be impossible to use it. The required criterion for the convergence of a sequence, when we are not simply asked to test whether a given number is or is not the limit, is contained in the fundamental general principle of convergence : — * A necessary and sufficient condition for the existence of a limit to the sequence ^ y^ u ,.. is that a positive integer v exists such that |iXn+p — '?^w| becomes as small as we please when n ^ v, for every positive integer p. More exactly : A necessary and sufficient condition for the existence of a liinit to the sequence ^ ^ u ... is thaty if any positive number e ha^ been chosen, as small as we please, there shall be a positive integer v such tliat I 'W'n+p — 'W'n I when n^v. But (Un+p - Un) = (Un+p -A) + (A- Un). Therefore \tVn+p -iin\ = {"^n+p -A\ + \A-Un\ < ie + y, it n^p, for every positive integer p, <^ (ii) The condition is siijfficient. We must examine two cases ; first, when the sequence contains ' an infinite number of terms equal to one another ; second, when it does not. (a) Let there be an infinite number of terms equal to A. Then, if \\unMp — Un\ < e, when n^p, and p is any positive integer, we may take Un^p = A for some value oi p, and we have \A-'Un\ '^iV+SJ ••• lie within the interval whose end-points are Uy— Je and t%-f ^.e. Digitized by VjOOQ IC 36 INFINITE SEQUENCES AND SERIES There must be an infinite number of distinct terms in this sequence. Otherwise we would have an infinite number of terras equal to one another. Consider the infinite aggregate (E) formed by the distinct terms in ,,^^ ^^^ ^^^.... This aggregate is bounded and must have at least one limiting point A within, or at an end of, the above interval. (Cf.§14.) There cannot be another limiting point A\ for if there were, we could choose e equal to J |J.— ^'( say, and the formula \un+p—Un\ <€, when n^v, for every positive integer p, shows that all the terms of the sequence 'M'j, U2i U^f ,,. , except a finite number, would lie within an interval of length J 1^ —A'\. This is impossible if Ay A' are limiting points of the aggregate. Thus the aggregate (E) has one and only one limiting point A. We shall now show that the sequence u^, u^i u^f ... converges to ^ as ti tends to oo . We have Un— A = {Un— u^) 4- (-?% —A). Therefore | u„— A\^\ Un— ^2% | + 1 n^ —A \ < ie . + |e, when n^N, < e, when'M — iV^. Thus the sequence converges, and has A for its limit. We have therefore proved this theorem : A necessary and sufficient condition for the convergence of the sequence ^^^ ^^^ ^^ ..^ is that, to the arbitrary positive number e, there shall correspond a positive integer v such that \Un+p — Un\ -4, when 7^ = k In this case we say that the sequence is divergent, and that it diverges to + oo , and we write this Lt Un = + 00 . (ii) In the second place, the terms may have the property that if any negative number — ^ is chosen, 'however large A may be, there is a positive integer i> such that Unco has an infinite limit, i.e. Lt Ua = + oc or Lt tt^ = — oo . But it must be remembered that the symbol oo , and the terms infinite, infinity and tend to infinity, have purely conventional meanings. There is no number infinity. Ehrases in which the term is used have only a meaning for us when we have previously, by definition, attached a meaning to them. When we say that n tends to infinity, we are using a short and convenient phrase to express the fact that n assumes an endless series of values which eventually become and remain greater than any arbitrary (large) positive number. So far we have supposed n, in this connection, to advance* through integral values only. This restriction will be removed later. A similar remark applies to the phrases divergence to +00 or to — 00 , and oscillating infinitely, as well as to our earlier use of the terms an infinite number, infinite sequence and infinite aggregates. In each case a definite meaning has been attached to the term, and it is employed only with that meaning. It is true that much of our work might be simplified by the introduction of new numbers + x , — x , and by assuming the existence of corresponding points upon the line which we have used as the domain of the numbers. But the creation of these numbers, and the introduction of these points, would be a matter for separate definition. Digitized by VjOOQ IC INFINITE SEQUENCES AND SERIES 39 17. Monotonic Sequences. // the tenns of the sequence M'l, ^2> '^^3» ••• satisfy either of the following relations u^^u^^u^ ... ^u„, ... or u^^u^^u^ ... ^u„, ..., the sequence is said to be monotonic. In the first case, the terms never decrease, and the se(|uence may be called "inonotonic increasing] in the second case, the terms never increase, and the sequence may be called monotonic decreasing* Obviously, when we are concerned with the convergence or divergence of a sequence, the monotonic property, if such exist, need not enter till after a certain stage. The tests for convergence or divergence are extremely simple in the case of monotonic sequences. // the sequence ^^ ^ ^^^^ i^.^ ^ . . . is monotonic increasing, and its terins are all less titan some fixed number B, the sequence is convergent and has for its limit a number ^ such tJiat Un^^ = B for every positive integer n. Consider the aggregate formed by distinct terms of the sequence. It is bounded by u^ on the left and by B on the right. Thus it must have an upper bound ft (cf. § 12) equal to or less than B, and, however small the positive number € may be there will be a term of the sequence greater than j8 — e. Let this term be u„. Then all the terms after u„_i are to the right of j8— € and not to the right of ft. If any of them coincide with j8, from that stage on the terms must be equal. Thus we have shown that |j8 — u„|«i- Thus the sequence of end-points %, (^2* <^s ; (1) has a limit, say a, and ct„ = a for every positive integer n (§ 17). Fig. 3. Also the sequence of end-points &i, K &3-- (2) has a limit, say /S, and bn = ^ for every positive integer n (§ 17). Now it is clear that, under the given conditions, /8 cannot be less than a. Therefore, for every value of n, But Lt (6,-a„) = 0. It follows that a = ^.* '"^^ * This result also follows at once from the fact that, if Lt o„ = a and Lt 6n = /3, then Lt (a„ - 6„) = a - /3. (Cf . § 26, Theorem I. ) Digitized by Google INFINITE SEQUENCES AND SERIES 41 Therefore this common limit of the sequences (1) and (2) satisfies the inequalities (in = a = bn for every positive integer 7i, and thus belongs to all the intervals. Further, no other point (e.g, y) can satisfy an = y = 6„ for all values of n. Since we would have at the same time Lt a„ = y and Lt 6» = y, w->Qo n-><» which is impossible unless y = a. 19. The Sam of an Infinite Series. Let Ui, ^2, Ug, ... be an infinite sequence, and let the successive sums Si = u^y S^^u^ + u.^, Sn = U^ + U2 + U^+..^ + ltn be formed. If the sequence S^, S^, S^, ... is convergent and has the limit S, then S is called the sum of the infinite series Ui + u^ + u^+.., and t\is series is said to be convergent. It must be carefully noted that what we call the sum of the infinite series is a limits the limit of the sum of n terms of u^ + u^ + u^+.^.y as n tends to infinity. Thus we have no right to assume without proof that familiar properties of finite sums are necessarily true for sums such as S. When Lt Sn=+ or Lt S«=— oo, we shall say that the infinite series is divergent, or diverges to +oo or — x , as the case may be. If Sn does not tend to a limit, or to H- x or to — x , then it oscillates finitely or infinitely according to the definitions of these terms in § 16. In this case we shall say that the series oscillates finitely or infinitely.* ♦ Cf. footnote, p. 37. Digitized by VjOOQ IC 42 INFINITE SEQUENCES AND SERIES The conditions obtained in §15 for the convergence of a sequence allow us to state the criteria for the convergence of the series in either of the following ways : ^nA (i) The aeriik converges and has S for its sum, if, any ^^^i^ositive number e having been chosen, as small as we please, there is a positive integer v such that \S—Sn\<^€, when n^v. (ii) A necessary and sufficient condition for the convergence of the series is that, if any positive number e has been chosen, small as we please, there shall be a positive integer v such that l^n+p — ^n| CO tained in the second criterion. It is a necessary condition for convergence, but it is not a sufficient condition ; e.g. the series is divergent, though Lt Un = 0. n->oo If we denote ^n+i + 'W'n+2+--- + ^n+p, or S^,.^p - S^^, by pR,, the above necessary apd suflScient condition for convergence of the 'Series may be written \pRn\ < €, when n ^ j/, for every positive integer p. Again, if the series u^ + U2+tL^+ ... converges and has S for its sum, the series converges and has S—S^for its sum. For we have SnA.p = Sn+pR,i. Also keeping n lixed, it is clear that Lt Sn^p = S. Therefore Lt (j,R„)=:S-S„. ♦As remarked in §15, this condition can be replaced by: 7b the arbitrary positive number e there must correspond a positive integer n such that I S„+p - iS„ I < e for every positive integer p. Digitized by Google INFINITE SEQUENCES AND SERIES 43 Thus if we write Rn for the sum of the series we have 8=Sn+ Rn . ^ The first criterion for convergence can now be put in the form I jBn| < €, when n^i/, Rn is usually called the reinainder of the series after n terms, and pR^ a partial remainder, 20. Series whose' Terms are all Positive. Let U1+U2+U3+... he a sei'ies whose terms are all positive. The sum of n terms of this series either tends to a limity or it diverges to +qc . Since the terms are all positive, the successive sums ^1 = ^1, form a monotonic increasing sequence, and the theorem stated above follows from § 17. When a series whose tervis are all positive is convergent^ the series we obtain when we take the tervis in any order we please 18 also convergent and has the same swm. This change of the order of the terms is to be such that there will be a one-one correspondence between the terms of the old series and the new. The term in any assigned place in the one series is to have a definite place in the other. Let • Si = u^, 82 = u^+u^, i Then the aggregate ( U), which corresponds to the sequence '^l* ^^2' ^S* • • • > is bounded and its upper bound S is the sum of the series. Let ({/') be the correisponding aggregate for the series ob- tained by taking the terms in any order we please, on the understanding we have explained above. Every number in Digitized by VjOOQ IC 44 INFINITE SEQUENCES AND SERIES {U') is less^than 8. In addition, if A is any number less than S, there must be a number of ( U) greater than A, and a fortiori a number of (J^') greater than A. The aggregate (U') is thus bounded on the ri^ht, and its upper bound is 8. The sum of the new series is therefore the same as the sum of the old. It follows that if the series whose terms are all positive, diverges^ the series we obtain by changing the order of the terms must also diverge. The following theorems may be proved at once by the use of the second condition for convergence (§19): If the series ^^ + ^^ + ^^+... is convergent and all its terms are positive, the series we obtain from this, either 0) by keeping only a part of its terms, or (2) by replacing certain of its terms by others, either i positive or zero, which are respectively equal or in ferior to them, or (3) by civanging the signs of some of its terins, are also convergent 21. Absolute and Conditional Convergence. The trigono- metrical series, whose properties we shall investigate later, belong to the class of series whose convergence is due to the presence of both positive and negative terms, in the sense that the series would diverge if all the terms were taken with the same sign. A series with positive and negative terms is said to be y. absolutely convergent, when the series in which all the terras are taken with the same sign converges. In other words, the series u^ + u.^ + u^+... is absolutely convergent when the series of absolute values is convergent. It is obvious that an absolutely convergent series is also con- vergent in the ordinary sense, since the absolute values of the partial remainders of the original series cannot be greater than Digitized by VjOOQ IC INFINITE SEQUENCES AND SERIES 45 those of the second series. There are, however, (vonvergent series which are not absolutely convergent : e,g, 1 — -J- + 4 ... is convergent. 1 + 1 + J ... is divergent. Series in which the convergence depends upon the presence ^-^ of hath positive and negative temns are said to he conditionally convergent. ^ The reason for this name is that, as we shall now prove, an ' absolutely convergent series remains convergent, and has the same sum, even although we alter the order in which its terms are taken; while a conditionally convergent series may con- verge for one arrangement of the terms and diverge for another. Indeed we shall see that we can make a conditionally convergent series have any sum we please, or be greater than any number we care to name-, by changing the order of its terms. There is nothing very extraordinary in this statement. The rearrange- ment of the terms introduces a new function of n, say S'„, instead of the old function S^, as the sum of the first oi terms. There is no a priori reason why this function S'n should have a limit as n tends to infinity, or, if it has a limit, that this should be the same as the limit of /Sf^.* ♦ 22. Absolutely Convergent Series. The sum of an absolutely convergent series remains the same when the order of the^ '^" tet'Tns is changed. Let (S) be the given absolutely convergent series; (8') the series formed with the positive terms of (S) in the order in which they appear; (>S") the series formed with the absolute values of the negative terms of (S), also in the order in which they appear. Jf the number of terms either in (8') or (8'') is limited, the theorem requires no proof, since we can change the order of the terms in the finite sum, which includes the terms of (8) up to the last of the class which is limited in number, without altering its sum, and we have just seen that when the terms are of the same sign, as in those which follow, the alteration in the order in the convergent series does not aflfect its sum. *Cf. Osgood, Introdiiction to Infinite Series, p. 44, 1897. Digitized by VjOOQ IC 46 INFINITE SEQUENCES AND SERIES Let 2 Be the sum of the infinite series formed by the absolute values of the terms of (S). Let Sn be the sum of the first n terms of (S). In this sum let n' terms be positive and n'' negative. Let Sn' be the sum- of these n' terms. Let Sn" be the sum of the absolute values of these n'' terms, taking in each case these terms in the order in which they appear in (S). Then >S; = S„.-;Sf„., Sn"<^. Now, as n increases S^', S^'^ never diminish. Thus, as n increases without limit, the successive values of S^', Sn" form two infinite monotonic sequences such as we have examined in § 17, whose terms do not exceed the fixed number 2. These sequences, therefore, tend to fixed limits, say, 8' and S'\ Th^^« Lt {8n) = S'^S'\ Hence the sum of the absolutely convergent series (S) is equal to the difference between the sums of the two infinite series formed one with the positive tei^ms in the order in which they appear^ and the other with the absolute values of the negative terms, also in the order in which they appear in {S), Now any alteration in the order of the terms of {S) does not change the values of S' and S" ; since we have seen that in the case of a convergent series whose terms are all positive we do not alter the sum by rearranging the terms. It follows that {S) re^nains convergent and has the same sum when the order of its terms is changed in any way we please, provided that a one- one correspondence exists between the terms of the old series and the new. We add some other results with regard to absolutely con- vergent series which admit of simple demonstration : Any series whose terms are either equal or inferior hi absolute value to the corresponding terms of an absolutely convergent series is also absolutely convergent An absolutely convergent series rem/iins absolutely convergent when we suppress a certain number of its tei^is. Digitized by Google INFINITE SEQUENCES AND SERIES 47 // 1^1 + ^2+..., are two absolutely convergent series whose surtis are U and F, theseHes („^+„^)+(^^+^^)+... and (Uj — v{)+ (^2 — ^2) + . . . are also absolutely convergent and their sumsi are equal to U± V respectively. 23. Conditionally Convergent Series. The sum of a condi- tionally convergent series depends essentially on the order of its terms. Let (8) he such a series. The positive and negative terms must both be infinite in number, since otherwise the series would converge absolutely. Further, the series formed by the positive terms in the order in which they occur in (S\ and the series formed in the same way by the negative terms, must both be divergent Both could not converge, since in that case our series would be equal to the difference of two absolutely convergent series, some of whose terms might be zero, and therefore would be absolutely convergent (§ 22). Also (8) could not converge, if one of these series converged and the other diverged. We can therefore take sufficient terms from the positive terms to make their sum exceed any positive number we care to name. In the same way we can take sufficient terms from the negative terms to make the sum of their absolute values exceed any number we care to name.' Let a be any positive number. First take positive numbers from (8) in the order in which they appear, stopping when»ve*i the sum is greater than a. Then take negative terms from (S\ in the order in which they appear, stopping whenever the combined sum is less than a. Then add on as many from the remaining positive terms as will make the sum exceed a, stopping when the sum first exceeds a ; and then proceed to the negative terms ; and so on. In this way we form a new series (8') composed of the same terms as (fi»), in which the sum of n terms is sometimes greater than a. and sometimes less than a. Digitized by Google 48 INFINITE SEQUENCES AND SERIES Now the series (S) converges. Let its terms be u^, u^, u.^, .... Then, with the usual notation, Kn| = a, A similar argument holds for the case of a negative number, the only difference being that now we begin with the negative terms of the series. We have thus established the following theorem : // a conditionally convergent series is given, we can so arrange the oi^der of the terms as to nmke the sum of the new series converge to any vahve we care to name, REFERENCES. Bromwicti, loc. cit., Ch. I. -IV. De la Valli^e Poussin, loc. cit., T. I. (3* ed.), Introduction, § 2, Ch. XL, §§ h 2. GoURSAT, loc. ciL, T. I. (3« ed.), Ch. I. and VIII. Hardy, Course of Pure Matheinatics (2nd. Ed.), Ch. IV. and VIII., Cam- bridge, 1914. Pringsheim, loc. ci't.y Bd. I., Absch. I., Kap. III., V. Absch. II., Kap. I.-III. Stolz u. Gmeiner, loc. cit., Abth. II., Absch. IX. And Pringsheim, " Irrationalzalilen u. Konvergenz uneudlicher Prozesse," Enc. d. math. Wiss., Bd. 1., Tl. I., Leipzig, 1898. Digitized by Google CHAPTER III FUNCTIONS OF A SINGLE VAKIABLE LIMITS AND CONTINUITY 24. The Idea of a Function. In Elementary Mathematics, when we speak of a function of x, we usually me«hi a real expression obtained by certain operations, e.g, x^, y/x, logo?, sin '^05. In some cases, from the nature of the operations, the range of the variable x is indicated. In the first of the atove examples, the range is unlimited; in the second, 05 = 0; in the third aj > ; and in the last 0^a? = 1. In Higher Mathematics the term " function of x " has a much more general meaning. Let a and b be any two real nu^nbers, where by^a. If to every value of x in the interval a = x^b there corresponds a {real) number y, then we say that y is a function ofx in the interval (a, 6), a:nd we wHte y=f(x). Sometimes the end-points of the interval are excluded from the domain of ic, which is then given by a -< a? -< 6. In this case the interval is said to be open at both ends; when both ends are included (i,e. a = a? = 6) it is said to be closed. An interval may be open at one end and closed at the other (e.g, a < cc = 6). Unless otherwise stated, when we speak of an interval in the rest of this work, we shall refer to an interval closed at both ends. And when we say that x lies in the interval {a,b\ we mean that a^x^b, but when x is to lie between a and &, and not to coincide with either, we shall say that x lies in the open interval (a, b)* *ln Ch. II., when a point x lies between a and h, and does not coincide with either, we have referred to it as within the interval (a, h). This form of words is convenient, and not likely to give rise to confusion. c. I 49 D Digitized by Google -^ 50 FUNCTIONS OF A SINGLE VARIABLE Consider the aggregate formed by the values of a function f{x)y given in an interval ((t, h). If this aggregate is bounded (cf. § 11), we say that the function f{x) is hounded in the interval. The numbers M and 77i, the upper and lower hounds of the aggregate (cf. § 12), are called the upper and lower hounds of the function in the interval. And a function can have an upper bound and no lower bound, and vice versa. The difference (M—m) is called the oscillation of the function in the interval. It should be noticed that a function may be determinate in an interval, and yet not bounded in the interval. E,g, let /(0) = 0, and f{x)=- when a;>0. Then f{x) has a definite value for every x in the interval O^ajga, where a is any given positive number. But f{x) is not bounded in this interval, for we can make f{x) exceed any number we care to name, by letting x approach sufficiently near to zero. Further, a bounded function need not attain its upper and lower bounds ; in other words, M and m need not be members of the aggregate formed by the values of f{x) in the inte^rval. ^.gr. let /(0) = 0, and /(aj) = l-flj when a limit when ti-^oo of a sequence u^, u^, u^, In other words, we have been dealing with a function {n\ where ti is a positive integer, and we have considered the limit of this function as We pass now to the function of the real variable x and the limit of f{x) when x-^a. The idea is familiar enough. The Differential Calculus rests upon it. But for our purpose we must put the matter on a precise arithmetical footing, and a definition of what exactly is meant by the limit of a function of x^ as x tends to a definite value, must be given. f{x) is said to have the limit h as x tends to a, when, any positive nwinher e having heen chosen, as small as we please, Digitized by VjOOQIC LIMITS AND CONTINUITY 51 there is a positive mvmber tj such that \f{x) — h\<^€y for all valttes of X for which < |a3 — a | = >/. In other words \f(x)'-b\ must be less than c for all points in the interval {a — vj, a + vj) except the point a. When this condition is satisfied, we employ the notation Lt f(x) = 6, for the phrase the limit of f(x\ as x tends to a, is h, and we say that f(x) converges to 6 as a: tends to a. One advantage of this notation, as opposed to Lt/(aj) = 6, is that it brings out the fact that we say nothing about what happens when x is equal to a. In the definition it will be observed that a statement is made about the behaviour of f(x) for all values of x such that < |aj — a| ^i;. The first of these inequalities is inserted expressly to exclude x = a. Sometimes x tends to a from the right hand only (i.e. x > a), or from the left hand only {i,e, x < a). In these cases, instead of < | a? — /, we have -<(a; — a) = ly (light hand) and <(«— a?) = J7 (left hand), in the definition. The notation adopted for these right-hand and left-hand limits ^^ Lt fix) and Lt f{x). The assertion that Lt f{x) = h thus includes Lt f{x)= Lt f(x)=h. It is convenient to use /(a-|-0) for Lt f{x) when this limit exists, and similarly /(a — 0) for Lt f{x) when this limit exists. x->a - When f{x) has not a limit as x->a, it may happen that it diverges to -|- oo , or to — oo , in the sense in which these terms were used in § 16. Or, more precisely^ it may happen that if any positive number A, however large, is chosen, there corre- sponds to it a positive number rj such that I f(x)^A, when 0<|a:— a|^i;. In this case we say that Lt f{x)= + oo . x-^a Again, it may happen that if any negative number ^A is chosen, however large A may be, there corresponds to it a positive 'immber rj such that f(x) < — -4, when < |«; — a| ^ jy. Digitized by Google 52 FUNCTIONS OF A SINGLE VARIABLE In this case we say that Lt f(x) = — qo . x—>-a The modifications when f(a zt 0) = ± oo are obvious. When Lt f{x) does not exist, and when f{x) does not diverge to +ac , or to — X , as x-^a, it is said to oscillate as x->a. It oscillates finitely if f{x) is boujided in some neighbourhood of that point.* It oscillates infinitely if there is no neighbourhood of a in which f{x) is bounded. (Cf. § 16.) The modifications to be made in these definitions when aj-x' only from the right, or only from the left, are obvious. 26. Some General Theorems on Limits. I. The Limit of a Stun. // Lt f(x) = a and Lt g(x) = ^, then Lt [f(x)+g(x)] = a + ^.'\ x-^a 3c-^a x-^a Let the positive number e be chosen, as small as we please. Then to e/2 there correspond the positive numbers rj^ , j/g such that \f{x)-a |<|, when < |aj-a | < jy^, lfl^(^)-/5|<|. whenO<|aj-a|^j72- Thus, if ri is not greater than tj^ or ti^, < I + |, when 0<|aj-a| = »/, << e, when 0<;|aj — a|=»/- Therefore Lt {f{x)+g{x)]=-a + ^. x->a This result can be extended to the sum of any number of functions. The Limit of a Sum is equal to the Sum of th^ Limits. *f{x) is said to satisfy a certain condition in the neighbourhood of X'^ when there is a positive number h such that the condition is satisfied when 0<\x-a\^h. Sometimes the neighbourhood is meant to include the point x—a itself. I" this case it is defined by \x-a\^h, t The corresponding theorem for functions of the positive integer n, as ?i->*' is proved in the same way, and is useful in the argument of certain sections "t the previous chapter. Digitized by Google LIMITS AND CONTINUITY 53 11. The Limit of a Product. // Lt f(x) = a and Lt g(x) = fi, x->rt X— ►a then Lt [fix)g(x)'] = a^. x->a Xet f(x) = a'+(x) \lr(x). From Theorem I. our result follows if Lt [(x)\lr(x)] = 0. Since (p(x) tends to zero as iC-Ht and ^(a:;) tends to zero as iC->a, a proof of this might appear unnecessary. But if a formal proof is required, it could run as follows : Given the arbitrary position number e, we have, as in I., \(^)\<\/€y when 0<|a;-a|^j;i, IV^(^)I<^^» when 0< |x--a|=j;2. Thus, if fj is not greater than pj^ or j/g, \{x)+a and examining the expression 1 1 a 0(u?)+>a (ii) If Lt /(aj) = a, and Lt g{x) = ^^0, then Lt Pv^^^V?.. ar->a x-^a jc->a \-g{X)J p This follows from II. and III. (i). This result can obviously be generalised as above. IV. The Limit of a Function of a Function. Lt f[<^(x)]. x—^a Let Lt (.r) = 6 and IX f{u)=-f{b), X— ►a . u—*b Then . Lt /[(^)l=/[ Lt (/»(.r)]. X— ►o X— ►a We are given that Lt f{u)=f(b). Digitized by Google 54 FUNCTIONS OF A SINGLE VARIABLE Therefore to the arbitrary positive number c there corresponds a positive number r/i such that l/[<^W]-/(6)l<«, when \{x)-b\^vi 0) Also we are given that Lt ^{x)=h, Therefoi-e to this positive number »/i there corresponds a positive number 7; such that Ic^W-^Kt;!, when 0<|^-a|^7; (2) Combining (1) and (2), to the arbitrary positive number c there corre- sponds a positive number 77 such that |/[<^(^)]-/(fc)|<€, when 0<|^-a|^r;. Thus Lt m{x)-\=f{h)=f[ Lt c^W]. EXAMPLES. 1. If 7i is a positive integer, Lt x^' = 0. 2. If n is a negative integer, Lt .r"=+Go; and Lt ^"=-x or +oc according as n is odd or even. *"*'*'^ '~^~° [If w=0, then c^=\ and Lt ^=1.] X— M) 3. Lt (a,yp"+aia;"-^ + ...+a„_i.r+a„) = a„. x-*0 4. Lt(^q±^^:;±i4^;;i=i^) = ^», unless 6„=0. 5. Lt 07**= a", if n is any positive or negative integer. • X — ►a 6. If P{x)^a^''+a^ar-^-\- ,.. + arn-iX-\-am, then Lt P(.r) = P(a). X— >a 7. Let P(a') = a^rf* + a^x^~^ + . . . + am-i^ + «m, and §(a') = V"+^'^*"^ + --+^»-i-^+^.- itm-wy " «<''>=^«- 8. If Lt /(.v) exists, it is the same as Lt f{x+a), x—*-a x—*0 9. If f{^)a X— X* n. If Lt/(^) = ^50, then Lt |/W| = K|. X— ►a ^ X— X* The converse does not hold. Digitized by Google LIMITS AND CONTINUITY 55 12. Let f(,v) be defined as follows : /*(.r)=a7 8inl/:r, when ji'^0\ /(o)= r Then Lt f(x)=f{a) for all values of a, 27. Lt f(x). A precise definition of the meaning of the term " the limit of f{x) when x tends to + oo (or to — x ) " is also needed. f{x) is said to have the limit b a^ x tends to +oo , if, any positive number e having been chosen, as small as we please, there is a positive nuTtiber X such that |/(a?)--6| <;e, when x^X. When this condition is satisfied, we write Lt f{x) = b. A similar notation, Lt f{x) = 6, A'-> -00 is used when f{x) has the limit 6 as a; tends to — oo , and the precise definition of the term can be obtained by substituting " a negative number — X " and '' x = ^X'* in the corresponding places in the above. When it is clear that only positive values of x are in question the notation Lt f{x) is used instead of Lt f{x). From the definition of the limit of f{x) as x tends to ± oo , it follows that Lt f{x)=h carries with it Lt fO-) = h. And, conversely, if Lt f{x) = 6, then Lt /(-) = 6. Similarly we have Lt f{x)= Lt /(-.)• X— >-x) x-^-0 V-*/ The modifications in the above definitions when (i) Lt f(x) = + 00 or - X , ac— *•+<» and (ii) Lt /(a;)=-f-x or -oo, X — ►— 00 will be obvious, on referring to § 25. And oscillation, finite or infinite, as x tends to -f-x or to — oo, is treatej as before. Digitized by Google 56 FUNCTIONS OF A SINGLE VARIABLE 28. A necessary and sufficient condition for the existence of a limit to f (x) as x tends to a. The general principle of convergence.'*' A necessary and sufficient condition for the existence of a limit to f(x) as x tends to a, is that, when any positive number € has been chosen, as small as we please, there shall be a positive number j; sv^ch that \f(x')—f(x)\/. (i) The condition is necessary. r~ Let Lt/(a;) = fe. Let c be a positive number, as small as we please, Then to e/2 there corresponds a positive numbei* >/ such that |/(a;)-6|<|, when 0Qo Let J7j, 1/2, J73, ... be corresponding positive numbers such that ■|/(0-/(^')| we can obviously assume that »;„ = »7„+i. Now take e^ and the corresponding ri^. In the inequalities (1) put x' — a+ti^ and x"=x. Then we have 0<|/(«)-/(a+';i)| ^3f ••■ ' each lying entirely within the preceding, or lying within it and having with it a common end-point; and, since the length of '4„~2e„, we have Lt -4,j = 0, for we are given that Lt en = 0. If we denote the end-points of these intervals by a^, a2, ag, ... a»d j8i, jSg, /Sg, ... , where ^n'> cm, then we know from 18 that Lt an= Lt I3n' n— >oo n— >-oo Denote this common limit by a. We shall now show that a is the limit of f{x) as x-^i. Digitized by Google 58 FUNCTIONS OF A SESTGLE VARIABLE We can choose €n in the sequence e^, eg? ^3> ••• s^ *'h*^ 2€n<;e, where e is any given positive number. Then we have, as above in (2) and (3), an a As a matter of fact, we have not obtained |/(^)-a| < €, when < |ci,'-a|"S?;„ in the above, but when < |A-a| < >;„. However, we need only take rj smaller than this tj^, and we obtain the inequalities used in our definition of a limit. 29. In the previous section we have supposed that x tends to a from both sides. The slight modification in the condition for convergence when it tends to a from one side only can easily be made. Similarly, a necessary and sujfficient condition for the existence of a limit to f{x) as x tends to +oo , is that, if the positive number e has been chosen, as small as ive please, there shall be a positive number X such that \f{x")-f{x')\x'^X. In the case of Lt f{x), we have; in the same way, the condition \f{x")-f(x')\ < e, when «" '^^^'^ l« - ^ol = »/• When f{x) is defined in. an interval (a, &), we shall say that it is continuous in the interval (a, 6), if it is continuous for every value of x between a and b (a 0. 31. Properties of Continuous Functions.* We shall now prove several important theorems on continuous functions, to ♦ This section follows closely the treatment given by Goursat {loc, ciL), T. I. (3-^d.),§8. Digitized by VjOOQ IC 60 FUNCTIONS OF A SINGLE VARIABLE which reference will frequently be made later. It will be seen, that in these proofs we rely only on the definition of continuity and the results obtained in the previous pages. Theorem I. Let f(x) be continuous in the interval (a, 6)*, and let the positive number e be chosen, as smaU as we please. Then the interval (a, b) can always be broken up into a finite number of partial intervals, such that \f(x')—f(x'')\► 0. We shall show that for at least one value of X between a and 6, f{x) = 0. From the continuity of f(x), we see that it is negative in the neighbourhood of a and positive in the neighbourhood of b. Consider the set of values of x between a and b which make f(x) positive. Let X be the lower bound of this aggregate. Then a < X < 6. From the definition of the lower bound f(x) is negative or zero in a = x when lo^-Xl^i;, since f(x) is continuous when x = X. The function f(x) would then be negative for the values of x in (a, b) between X and X + jy Digitized by VjOOQ IC LIMITS AND CONTINUITY 63 and X would not be the lower lx)und of the above aggi'egate. We must therefore have /(X) = 0. Now let N be any number between f(a) and /(6), which may be of the same or different signs. The continuous function (P(x) =f{x) — N has opposite signs when x = a and x = b. By the case we have just discussed, (x) vanishes for at least one value of X between a and 6, i.e. in the open interval (a, b). Thus our theorem is established. Again, \if{x) is continuous -in (a, fc), we know from Corollary I. above that it is bounded in that interval. In the next theorem we show that it attains these bounds. Theorem III. If f{x) is continuous in tlie interval (a, 6), aiid M, m are its upper and lower bounds^ then f(x) takes the value M and the value m at least once in the interval. We shall show first that f{x) = M at least once in the interval. Let c = (a+6)/2 ; the upper bound of f{x) is equal to ilf, for at least one of the intervals (a, c), (c, b). Replacing (a, b) by. this interval, we bisect it, and proceed as before. In this way, as in Theorem I., we obtain an infinite set of intervals (a, 6), {a^, b^), ((Xg, 62), ... tending to zero in the limit, each lying entirely within the preceding, or lying within it and having with it a common end-point, the upper bound oif{x) in each being M. Let X be the common limit of the sequences a, a^, ag, ... and i, 61 , 62, . . . . We shall show that /(X) = M. For suppose f(X) = M—h, where h'^0. Since f{x) is continuous at cc = X, there is a positive number t^ such that l/(^)-/WI<| when \x-\\^r,. Thus /(aj)< Jtf — ^, when {x — Xl^rj. Now take n so large that (bn — an) will be less than rj. The interval (a„, bn) will be contained wholly within (X — >;, X + >;). The upper bound of f(x) in the interval (an, bn) would then be different from Af, contrary to our hypothesis. Combining this theorem with the preceding we obtain the following additional result : Theorem IV. If f(x) is continuous in the interval (a, 6), and My m are its upper and lower bounds, then it takes at least Digitized by Google 64 FUNCTIONS OF A SINGLE VARIABLE once in this interval the rahws M, iiiy itnd every value between M and vi. Also, since the oscillation of a function in an interval \s^as defined as the difference between its upper and lower bounds {cf. § 24), and since the function attains its boiinds at least once in the interval, we can state Theorem I. afresh as follows : If f{x) is continuous in the interval (a, 6), then w^ can divide ((X, 6) into a finite number of partial intervals (a, Xi), (a^i, a?2), ... {x,,.^,b\ in each of which the oscillation of f(x) is less than any given positive number. And a similar change can be made in the statement of the property known as uniform continuity. 32. Continuity in an Infinite Interval. Some of the results of the last section can be extended to. the case when /(a?) is continuous in ^^a, where a is some definite positive number, and Lt /(^) exists. X— ►» Jjet u = a/x. When ^^a, we have 0<%^1. With the values ot uiu 0»o ^^ either side, and it may oscillate on one side or the other. Take in this section the cases in which there is no oscillation. These may be arranged as follows : (i)/((ro+0)=/K-0)=+(X) (or -cc), Ex. f(''^) = ^l(^-^ofi when x^Xq. (ii) f(xQ+0)= +00 (or — x ) and f(xQ—0)= —x> (or +oo ). Ex. /(•^) = l/(^-'--^o)> when x^x^, ^iii)/(a;o+0)=+oo (or -oo)| or f(xQ—0)=+cc (or — oo )\ /(xq— 0) exists J /(a^o+0) exists J* Ex. f{x) = 1 /(^ - ^'o), when x > Xq\ f(x)=x-XQy when x^x^) In these cases we say that the point Xq is an infinity of /(a?), and the same term is used when Xq is an end-point of the interval in which /(a;) is given, and/(a;o+0), or/(a;o— 0), is +oo or — oo , It is usual to say that f{x) becomes infinite at a point Xq of the kind given in (i), and that f{x^ = +oo (or — x ). But this C. I E Digitized by VjOOQ IC 66 FUNCTIONS OF A SINGLE VARIABLE must be regarded as simply a short way of expressing the fact that /(a?) diverges to +<» (or to — x ) as x-^Xq, It will be noticed that tan x has an infinity at ^tt, but that tan Jtt is not defined. On the other hand, tan(^7r— 0)= +00 and tan(j7r— 0) = — x. IV. When f{x) oscillates at Xq on one side or the other, Xq is said to be a point of oscillatory discontinuity. The oscillation is finite when f{x) is bounded in some neighbourhood of Xq ; it is infinite when there is no neighbourhood of Xq in which f(x) is bounded (cf. § 25). Ex. (i) f{x) = sin 1 l(x - x^), when j? 5 ^o • (ii) f{x) = 1 /(^ - x^) sin l/{x - ^o), when ^ 5 ^o • In both these examples Xq is a point of oscillatory discon- tinuity. The first oscillates finitely at Xq, the second oscillates infinitely. The same remark would apply if the function had been given only for one side of Xq. The infinities defined in III. and the points at which f(x) oscillates infinitely are said to be points of infinite discontinuity. 34. Monotonic Functions. The function f{x), given in the -^nterval (a, 6), is said to be monotonic in that interval if either {\) f{x')^f{pf\ when a^i»'-«> (iii) If f{x) is monotonic increasing in an open interval (a, 6), and f{x) is greater than some fixed number A in that open interval, then f(a+0) exists and is greater than or equal to A. (iv) If f{x) is monotonic increasing in an open interval (a, 6), and f{x) is less than some fixed number A in that open interval^ then fih — Q) exists and is less than or equal to A. These results can be readily adapted to the case of monotonic decreasing functions, and it follows at once from (iii) and (iv) that if f(x) is bounded and Tnonotonic in an open interval, it can only have ordinary discontinuities in that interval, or at its ends. It may be worth observing that if f(x) is monotonic in a closed interval, the same result follows, but that if we are only given that it is monotonic in an open interval, and not told that it is bounded, the function may have an infinity at either end. E.g. /(a;) = l/a; is monotonic in the open interval (0, 1), but not bounded. At first one might be inclined to think that a function which is bounded and monotonic in an interval can have only a finite number of points of discontinuity in that interval. The following example shows that this is not the case : Let/(aj) = l, when - it is clear that | ^ - ^q | < c, when |y - ^q | ^ ?/. Therefore l<^(y)-(yo)l < «» when [y-yol^v- Thus {x) defined as follows : In a^xf{x) + ai + ag. At X2 , <^(a?)=/(^)+a, + a2+a3. And so on, ^ij «4> <*3> ••• being definite numbers depending on /(^liO), /(:fi), etc. We can now apply the theorem proved above to the function ^/> ^^ ^^^^ arbitrary positive number e, there corresponds a positive number rj such that \f{Xy 2/)--/(xo, 2/o) K^ /^^ ^^^ values of (x, y) for which |a5--aj^)| = i; and 1^ — yol = '/- In other words, \f{Xy y)—f{xQy y^ \ must be less than e for all points in the square, centre (x^y j/o), whose sides are parallel to the coordinate axes and of length 2ri* It is convenient to speak of a function as continuous at a point (a?Q, 3/0) instead of when x = Xq and y = yQ. Also when a function of two variables is continuous at (x, y), as defined above, for every point of a domain, we shall say that it is a continuous function of (x, y) in the domain. It is easy to see that we can substitute for the square, with centre at {Xq, y^y referred to above, a circle with the same centre. The definition of a limit would then read as follows : f(Xy y) is said to have the limit I as (Xy y) tend to (Xq, 2/o)» if to the arbitrary positive number e, there corresponds a positive number rj such that \f{x, 3/) — ^| y) -/(i»o» 2/0) l< ^ M ^^^ values of (a;, y) for which Every function, which is continuous at (Xq, 2/0) under this definition, is continuous at {x^y 2/0) under the former definition, and conversely. It is importcant to notice that if a function of x and y is continuous with respect to the two variables, as defined above, it is also continuous when con- sidered as a function of x alone, or of y alone. For example, let /(a?, y) be defined as follows : f /(^, .y)= 2/2 * when at least one of the variables is not zero, 1/(0, 0)=0. Then /(^, y) is a continuous function of a?, for all values of x, when y has any fixed value, zero or not ; and it is a continuous function of y, for all values of y, when x has any fixed value, zero or not. But it is not a continuous function of (^, y) in any domain which includes the origin, since /(^'j y) is not continuous when ^=0 and y=0. For, if we put .r=r cos ^, y =r sin ^, we have/(;r, 3^)=sin 2$, which is inde- pendent of r, and varies from - 1 to + L However, it is a continuous function of (^, y) in any domain which does not include the origin. On the other hand, the function defined by f/C-^j y) — K %\_ 2\ > when at least one of the variables is not zero, 1/(0, 0)=0, is a continuous function of (^, y) in any domain which includes the origin. The theorems as to the continuity of the sum, product and, in certain cases, quotient of two or more continuous functions, given in § 30, can be readily extended to the case of functions of two or more variables. A continuous function of one or more continuous functions is also continuous. In particular we have the theorem : Let w=<^(^, y), v = yl/(xyy) be continuous at (^o> yoX ^^ ^^^ ^o^^i^oi 2/o)* Zet z=f{u, v) be continuous in {u, v) at {uq, Vq). Then 2=/[<^(^, y\ i^{^, y)] is continuous in {x, y) at (^o> yo)- Further, the general theorems on continuous functions, proved in §31, hold, with only verbal changes, for functions of two or more variables. Digitized by Google LIMITS AND CONTINUITY 75 For example : If a fii'tictimi oftioo variables w conttnuoiu at every point of a closed domain^ it is uniformly continuous in the domain. In other words, when the positive number e has been chosen, as small as we please, there is a positive number rj such that \ f{^ , y') - f{x" , y")\ < €, when (x\ /) and (pd\ y") are any two points in the domain for which * REFERENCES. De la Vallj^e Poussix, loc. cit., T. I. (3* 6d.), Introduction, § 3. GouRSAT, loc. ciL, T. I. (3* 6d.), Ch. I. Hardy, loc. cit. (2nd Ed.), Ch. V. Osgood, Lehrbuch der Funktionentheorie, Bd. I., Tl. I.,*Kap. I., Leipzig, 1907. PiERPONT, Theory of Functions of Real Variables, Vol. I., Ch. VI.-VIL, Boston, 1905. And Pringsheim, "Grundlagen der allgenieinen Funktionenlehre," Enc. d. math. Wiss., Bd. II., Tl. I., Leipzig, 1899. Digitized by Google CHAPTER IV THE DEFINITE INTEGRAL 38. In the usual elementary treatment of the Definite Integral, defined as the limit of a sum, it is assumed that the function of x considered may be represented by a curve. The limit is the area between the curve, the axis of x and the two bounding ordinates. For long this demonstration was accepted as sufficient. To-day, however, analysis is founded on a more solid basis. No appeal is made to the intuitions of geometry. Further, even among the continuous functions of analysis, there are many which cannot be represented graphically. E.g, let f{x) = aj sin - , when a? < 0, [ and /(0) = 0. i Then fix) is continuous for every value of x, but it has not a differential coefficient when a; = 0. It is continuous at .^•=0, because l/W-/(o)l=l/Wl^kl; and \f{x) -f(0) \ < €, when < | .r | ^ r?, if 7; < €. Also it is continuous when .r^O, since it is the product of two continuous functions [cf. § 30]. It has not a differential coefficient at .v = 0, because and sin 1/A has not a limit as A->0. It has a differential coefficient at every point where .v^O, and at such points 11 1 / Y.t') = sin cos - . 76 Digitized by VjOOQ IC THE DEFINITE INTEGRAL 77 More curious still, Weierstrass discovered a function, which is continuous for every value of cc, while it has not a differential coefficient anywhere.* This function is defined by the sum of the infinite series « 2 ct'*cos b^wx, a being a positive odd integer and b a positive number less than unity, connected with a by the inequality ab > 1 +f x.t Other examples of such extraordinary functions have been given since Weierstrass's time. And for this reason alone it would have been necessary to substitute an exact arith- metical treatment for the traditional discussion of the Definite Integral. Kiemann J was the first to give such a rigorous arithmetical treatment. The definition adopted in this chapter is due to him. The limitations imposed upon the integrand f(x) will be indicated as we proceed. In recent years a more general definition of the integral has been given by Lebesgue,§ and extended by others, notably de la Vallde Poussin and W. H. Young, the chief object of Lebesgue's work being to remove the limitations on the integrand required in Riemann's treatment. 39. The Sums S and s. || Let f{x) be a bounded function, given in the interval (a, b). * It seems impossible to assign an exact date to this discovery. Weierstrass himself did not at once publish it, but communicated it privately, as was his habit, to his pupils and friends. Du Bois-Reymond quotes it in a paper published in 1874. t Hardy has shown that this relation can be replaced by 01 and ab^l [cf. Trans. Amer. McUh. Soc., 17, 1916]. An interesting discussion of Weierstrass's function is to be found in a paper, ** Infinite Derivates," Q. J, Math,^ London^ 47, p- 127, 1916, by Grace Chisholm Young. Jin his classical paper, *^Uher die Dar8teUharkeit\e%ner Function durch eine triganometrische Rtihe" See above, p. 9. But the earlier work of Cauchy and Dirichlet must not be forgotten. §Cf. Lebesgue, Lemons sur V Integration, Paris, 1904; de la Vallee Poussin, Inl4grales de Lebesgue, Paris, 1916. And papers by Bliss and Hildebrandt in Bull. Amer. Math. Soc, 24, 1917. II The argument which follows is taken, with slight modifications, from Goursat's Cmirs d' Analyse, T. I. (3«^.), pp. 171 et seq. Digitized by Google 78 THE DEFINITE INTEGRAL Suppose this interval broken up into n partial intervals {a,x^\ (iCi,iC2), ... (a?„_i, 6), where a < aj^ < ajg • • • < ^n-i ^ ^• Let M, 7n be the upper and lower bounds of f(x) in the whole interval, and Mr, mr those in the interval (aj^.i, Xr), writing a = XQ and 6 = a?„. Let S=M^(x^'-a)+M^{x^--x^)+.,.+M^(b''X^_^)^ and 8 = mi(aJi — a)+m2(aj2— aJi)+...+m„(6— aj„.i)J To every mode of subdivision of (a, b) into such partial intervals, there corresponds a sum S and a sum 8 such that The sums S have a lower bound, since they are all greater than m(b — a), and the sums s have an upper bound, since they are all less than M(b — a), Let the lower bound of the sums 8 be J, and the upper bound of the sums s be /. We shall now show that I=J. Let a, 05^, x^,,.,x^_^, b be the set of points to which a certain S and s correspond. Suppose some or all of the intervals (a, x^\ (x^, x^), ... (i»n-i» ^) to be divided into smaller intervals, and let be the set of points thus obtained. The second mode of division will be called consecutive to the first, when it is obtained from it in this way. Let 2, (T be the sums for the new division. Compare, for example, the parts of 8 and 2 which come from the interval (a, x^). Let M\, m\ be the upper and lower bounds of f(x) in (a, y^), JIf'g, 'W^'2 in (2/1 > 2/2)' *n^ ^^ ^n- The part of S which comes from (a, x^) is then ^\(yi-o.i)+M\(y2-yi) ... +il/'fc(aJi-J/fc-i). But the numbers M\, M\, ... cannot exceed M^, Thus the part of S which we are considering is at most equal to M-J^x^ — a), Similarly the part of 2 which comes from {x^, x^ is at most equal to M^{x^'-x^\ and so on. Digitized by VjOOQ IC THE DEFINITE INTEGRAL 79 Adding these results we have S = ^. Similarly we obtain (r = «. Consider now any two modes of division of (a, 6). Denote them by a, Xi, X2, ... ic^-i, 6, with sums S and «, (1) and a, y^y y^, ... 2/«-i> ^, with sums S' and s' (2) On superposing these two, we obtain a third mode of division (3), consecutive to both (1) and (2). Let the sums for (3) be S and a. Then, since (3) is consecutive to (1), fif^2 and (r^8. Also, since (3) is consecutive to (2), /SrgS and er ^s'. But 2S(r. Therefore fifes' and S'^8. Thus the sum 8 arising from any mode of division of (a, li) is not less than the sum 8 arising from the same, or any other, mode of division. It follows at once that I=J, For we can find a sum « as near / as we please, and a sum 8 (not necessarily from the same mode of division) as near J as we please. If I^J, this would involve the existence of an 8 and an S for which 8^8, The argument of this section will offer less difficulty, if the reader follow it for an ordinary function represented by a curve, when the sums S and « will refer to certain rectangles associated with the curve. 40. Darboux's Theorem. The sunns 8 and s tend respectively to J and I, when the points of division are multiplied inde- finitely, in such a way that all the partial intervals tend to zero. Stated more precisely, the theorem reads as follows : If the positive nuraher e is chosen, as small as we please, there is a positive number rj such that, for all modes of division in which all the partial intervals are less than or equal to tj, the ^.uTTi 8 is greater than J by less than e, and the sum s is smaller -^ f^an I by less than e. Let € be any positive number as small as we please. Digitized by Google 80 THE DEFINITE INTEGRAL Since the sums S and 8 have J and / for lower and upper bounds respectively, there is a mode of division such that the sum S for it exceeds J by less than Jc. Let this mode of division be a, a{, ttg, ... ctp.if h, with sums 8^ and s^ (1) Then S^Si '" in^ aiiy values of x in the partial intervals (a, x^l (a?!, x^\ ... {Xr-u Xrl ... (x^-iy b) respectively. The sum /(fl)(^l-a)+/(^2)(^2-^l)+...+/(fn)(6-a^„-l) (1) obviously lies between the sums S and 8 for this mode of division, since we have mr=f(^r) = Mr for each of the partial intervals. But, when the number of points of division (Xr) increases in- definitely in such a way that all the partial intervals tend to zero, the sums 8 and 8 have a common limit, namely I f(x) dx. Therefore the sum (1) has the same limit. " Thus we have shown that, for an integrable function f{x), the mm f(i,)(x^-a)+f(Q(x^-x,)+...+f(i„)(h-x^^) has the definite integral I f{x)dx for its liTnit, when the number Ja of points of division (Xr) increases indefinitely in such a way that all the partial intervals tend to zero, ii, ^2* -" in being any values of x in these partial intervals.* In particular, we may take a,x^yX^y ... x^^^, or aj^, ajg, ... ir^_i, h, for the values of ^i, ^2» ••• in- 42. Necessary and Sufficient Conditions for Integrability. Any one of the following is a necessary and sufficient condition for the integrability of the bounded function f(x) given in the interval (a, 6) : I. When any positive number e has been chosen, as small as we please, there shall be a positive number tj such that S— s8 smnall as we please, there shall be a Tnode of division of (a, b) such that S-s<:€, It has been proved in I. that this condition is sufficient Also it is necessary. For we are given /=/, as f{x) is integrable, and we have shown that in this case there are any number of modes of division, such that S — s < e. III. Let CO, cr be any 'pair of positive numbers. There shall be a mode of division of (a, b) siich that the sum of the lengths of the partial intervals in which the oscillation is greatei^ than or equal to w shall be less than o-.* This condition is sufficient. For, having chosen the arbitrary positive number €, take g g 2(if-m) 2(o-a) where M^ m are the upper and lower bounds respectively oif{x) in (a, b). Then there is a mode of division such that the sum of the lengths of the partial intervals in which the oscillation is greater than or equal to w shall be less than o-. Let the intervals (^,._i, :r,.) in which the oscillation is greater than or equal to w be denoted by Dr, and those in which it is less than w by c?^, and let the oscillation (il/',. - m,.) in (.iv_i, .r^) be denoted by (0,.. Then we have, for this mode of division, S-S = ^iOrDr + ^dyjdr <(M-m)=rj-ji x + ^TTi \{^-^) ^ ^2(M-m) 2{b-ay ^ < I + I and, by II., /(^) is integrable in (a, b). *Cf. Pierpont, Theory of Functions of Real Variables, Vol. I., §498. Digitized by VjOOQ IC 84 THE DEFINITE INTEGRAL Also the condition is necessary. For, by II., if /(.^•) is integrable in (a, 6), there is a mode of division such that S-s < oht. Using Dr, dr as above, Therefore cdo- > w2Dr , and l.Dr<(T. 43. Integrable Functions. I. Iff(x) is continuous in (a, 6), it is integrable in (a, 6). In the first place, we know that f{x) is bounded in the interval, since it is continuous in (a, 6) [cf. §31]. Next, we know that, to the arbitrary positive number e, there corresponds a positive number 17 such that the oscillation of f{x) is less than e in all partial intervals less than or equal to 1/ [cf.§3l]. Now we wish to show that, given the arbitrary positive number e, there is a mode of division such that S — s^^e [^42, II.]. . Starting with the given e, we know that for e/ib—a) there is a positive number rj such that the oscillation of f(x) is less than e/ib — a) in all partial intervals less than or equal to tj. If we take a mode of division in which the partial intervals are less than or equal to this j;, then for it we have >S-«<(fe-a), ^- =e. ^ ^b — a Therefore f(x) is integrable in (a, b). II. // f{x) is monotonic in (a, 6), it is integrable in (a, 6). In the first place, we note that the function, being given in the closed interval (a, b), and being monotonic, is also bounded. We shall take the case of a monotonic increasing function, so that we have f(a)^f(x,)^f^x,) ...^f(x„_,)^f{b) for the mode of division given by (X, X-^, CC2 , . . . iC^—i » 0, Thus we have S=f{x,Xx,-a)+f(x,)(x,--x,) ... +f(b){b-x,,_,\ 1 s=f(a) (x^-a)+f(x,){x.,''X^) ...4-/(aJ„_i)(b-a?n-i)-J Therefore, if all the partial intervals are less than or equal to 17, S-s^„lf(Jb)-f{a)l Digitized by' Google THE DEFINITE INTEGRAL 85 since f{x^)-f{a). /(^2) -/(^i). • • • /W -/K-i) are none of them negative. If we take " <7(6)3)^)' it follows that fif — 8 < c. Thus f{x) is integrable in (a, 6). The same proof applies to a monotonia decreasing function. We have seen that a monotonic function, given in (a, h), can only have ordinary discontinuities, but these need not be finite in number (cf. §34). We are thus led to consider other cases in which a bounded function is integrable, when discontinuities of the function occur in the given interval. A simple test of integrability is contained in the following theorem : III. A hounded f and ion is integrable in (a, h), when all its 'points of discontinuity in (a, h) can he enclosed in a finite nuTnber of intervals the sum of which is less than any arhitrary positive numher. Let e be any positive number, as small as we please, and let the upper bound of \f{x) \ in (a, 6) be ^. By our hypothesis we can enclose all the points of discontinuity of f{x) in a finite number of intervals, the sum of which is less than 6/4J.. The part of S—s coming from these intervals is, at most, 2 A multiplied by their sum. On the other hand, f(x) is continuous in all the remaining (closed) intervals. We can, therefore, break up this part of (ct, h) into a finite number of partial intervals such that the corresponding portion ofS-s (U2> W3 ... be an infinite sequence of positive numbers, such that Lt a)„ = 0. n— ►» Let (a, (3) be any interval contained in (a, b) such that a^a\ however small, contains a point of continuity. 45. Some Properties of the Definite Integral. We shall now establish some of the properties of I f{x) dx, the integrand being bounded in; (a, 6) and integrable. I. If f(x) is integrable in (a, 6), it i$ also integrable in any interval (a, /3) contained in (a, 6). From §42, I. we know that to the arbitrary positive number e there corresponds a positive number 17 such that the difference S— s{x) thus obtained is integrable in (a, 6), and its integral is the same as that of fix). We can enclose the points to which reference is made in a finite number of intervals, the sum of which is less than e/4i4, where e is any given positive number, and A is the upper bound of ' (x) \ in (a, 6). Digitized by VjOOQ IC 88 THE DEFINITE INTEGRAL The part of S— s for 0(a;), arising from these intervals, is at most 2A multiplied by their sum, i.e. it is less than ^e. On the other hand, f{x) and 0(a:), which is identical with /(a?) in the parts of (a, b) which are left, are integrable in each of these parts. Thus we can obtain a mode of division for the whole of them which will contribute less than ^e to S—s, and, finally, we have a mode of division of (a, 6) for which S — s < e. Therefore (f}(x) is integrable in (a, b). Further, I (f}(x)dx=\ f(x)dx, J a J a For we have seen in § 41 that I (f}{x)dx is the limit of Ja 0(fl)(^l-«) + 0(^2)(^2-^l)+.-. + 0(^n)(^-«^n-l) when the intervals (a, aj^), (x^y ajg), ... (i»„«i, b) tend to zero, and ^i» ^2» ••• in *^® ^°y values of x in these intervals. We may put /(f,), /(f,), .../(^,,) for (i,), {x) and f{x) are equal. In this way we obtain a sum of the form Lt2/(fr)(^r— ^^r-i), which is identical with I f(x) dx. J a III. It follows immediately from the definition of the integral, that iff(x) is integrable in (a, 6), so also is Cf{x\ where C is any constant Again, if fi{x) and f^ix) are integrable in (a, 6), their sitvi is also integrable. For, let S, s; S' s' ; and 2, o- be the sums corresponding to the same mode of division ior f^(x), f^(x) and /^(aj) 4-/2 (aj). Then it is clear that 5;-(r^(s-s)+(/S'-8'), ' and the result follows. Also it is easy to show that [cf(x)dx=C?f(x)dXy J a J a Jb Cb Cb {fi{^)+U{^)]dx=-\ f^{x)dx-\-\ Mx)dx. a J a J a Digitized by VjOOQ IC THE DEFINITE INTEGRAL 89 IV. The product of two integrable functions fi{x\ f^i^) is integrable. To begin with, let the functions fi(x)y /gCa?) be positive in (a, 6). Let Mr, nir ; M'r, mV ; BIr, ni^ be the upper and lower bounds of f\{^)yA{^) ^^^ f\{^)f%K^) in the partial interval (a:^_i, Xr). Let Sy s ; S\ s' and S, a- be the corresponding sums for a certain mode of division in which (aj^_i, Xr) is a partial interval. Then it is clear that M,. — nir = MrM'r — nirm'r = Mr {M 'r — ^V) + m\{Mr — m,.). AfoHiori, M,-m,^if(JI/;-mV)+MWr-m,), where M, M are the upper bounds oi f^{x), f^{x) in (a, 6). Multiplying this inequality by (ic,.— ic^_i)and adding the corre- sponding results, we have It follows that S — o- tends to zero, and the product oif^{x), f^ix) is integrable in (a, 6). If the two functions are not both positive throughout the interval, we can always add constants c^ and Cg, so that/i(a;)-t-Ci, fi{x)+C2 remain positive iu (a, h). The product ifli^) + Cl)(/2(«) + Cg) =/l («)/2(^) + (^ihi^) + ^2/1 (^) + ^1^2 is then integrable. But Cj/2(fl?) + C2/i(aj)+CiC'2 is integrable. It follows iYidii f^{x)f^{x) is integrable. On combining these results, we see that if fi{x), fii^) ••• fn{x) are integrable functions, every 'polynomial in is also an integrable function* 46. Properties of the Definite Integral {continued). I. • ^f{x)dx=^[f{x)dx. In the definition of the sums S and s, and of the definite integral I f(x)dx, we assumed that a was less than 6. This Ja restriction is, however, unnecessary, and will now be removed. * This result can be extended to any continuous function of the n functions [cf. Hobson, loc. cU:, p. 345]. Digitized by VjOOQ IC 90 THE DEFINITE INTEGRAL If ct >► 6, we take as before the set of points and we deal with the sums S=itfi(aJi-a)+ikf2(iK2-«i)+--- + ^n(^-^n-iXl (1) 8 = mi (fl?i - a) + mg (a;2 - aJi) + . . . + m„(6 - a;^-i)J The new sum S is equal in absolute value, but opposite in sign, to the sum obtained from The existence of the bounds of 8 and 8 in (1) follows, and the definite integral is defined as the common value of these bounds, when they have a common value. It is thus clear that, with this extension of the definition of §41, we have ^yix)dx=-^y(x)dx,* a, h being any points of an interval in which f(x) is bounded and integi-able, II. Let c be any point of an interval (a, 6) in which f{x) is bounded and integrable. Then [f{x) = [fix) dx + [ f{x) dx. J a . J a J c Consider a mode of division of (a, 6) which has not c for a point of section. If we now introduce c as an additional point of section, the sum S is certainly not increased. But the sums S for (a, c) and (c, 6), given by this mode of division, are respectively not less than I f(x)dx and I f(x)dx. J a Jc Thus every mode of division of (a, 6) gives a sum 8 not less than [f(x)dx+[f{x)dx. Ja J c It follows that [fix) dx^[ fix) dx + r fix) dAC. Ja Ja Jc But the modes of division of (a, c) and (c, b) together form a possible mode of division of (a, 6). And we can obtain modes * The results proved in §§ 42-45 are also applicable, in some cases with slight verbal alterations, to the Definite Integral thus generalised. Digitized by VjOOQIC THE DEFINITE INTEGRAL 91 of division of (a, c) and (c, 6), the sums for which differ from f{x)dx and I f(x)dx, respectively, by as little as we please. It follows that the sign of inequality in the above relation must be deleted, so that we have [f(x)dx= [f(x)dx+ [f{x)dx. Ja Ja Jc If c lies on (a, b) produced in either direction, it is easy to show, as above, that this result remains true, provided that f(x) is integrable in (a, c) in the one case, and (c, b) in the other. 47. If f (x) = g(x), and both fiinctions are integrable in (a, b), thenrf(x)dx^rg(x)dx. Let 4>(<^)=f(x)-g(x)^0: Then (f}{x) is integrable in (a, b), and obviously, from the sum 8, I (f>{x)dx^O, J a Therefore \ f(x) dx - f gix} dx^O. J a Ja Corollary I. If f{x) is integrable in (a, 6), then W" f{x)d0, then [f{x)dx>0. We have seen in §44 that if f{x) is integrable in (a, 6), it must have points of continuity in the interval. What is assumed here is that at one of these points of continuity f{x) is positive. Let this point c be an internal point of the interval (a, 6), and not an end-point. Then there is an interval (c\ (1\ where Digitized by VjOOQIC 92 THE DEFINITE INTEGRAL ak for every point of (c', c"), k being some positive number. Thus, since f{x) = in (a, c'), I f{x) cfo = 0. J a And, since f(x) >k\xi {c\ c"), J f{x) dx ^ /c(c" - c') > 0. Also, since/(a;)SOin(c",fc), [ f{x)dx^O. Adding these results, we have I f{x)dx'>0. The changes in the argument when c is an end-point of (a, 6) are slight. Corollary III. Let f(x)^g(x), and both be integrable in (a, 6). At a point c in (a, 6), let f(x) and g(x) both be con- rb rb tinuons, and f{c)^g{c). Then I f{x)dx^\ g(x)dx, Ja Ja This follows at once from Corollary II. by writing i>{x)^f{x)-g(x). By the aid of the theorem proved in § 44, the following simpler result may be obtained : If f{x) > g{x), and both are integrable in (a, b), then j f{x)dx> I g(x)dx. For, if f{x) and g(x) are integrable in (a, b\ we know that f{x)-g{jc) is integrable and has an infinite number of points of continuity in (a, b). At any one of these points f{x)'-g{x) is positive, and the result follows from Corollary II. 48. The First Theorem of Itlean Value. Let {x) in (a, b). Then we have, in (a, 6), m = {x)^My and multiplying by the factor ^(a;), which is not negative, mx//- {x) ^(f>{x)yjr (x) = Myfr (x). It follows from § 47 that m\ \j/{x)dx^\ (l>[x)\lr{x)dx = M\ \[r{x)dxy Ja Ja Ja since (l>{x)\lr{x) is also integrable in (a, 6). Digitized by Google THE DEFINITE INTEGRAL 93 Therefore I {x), yl^lx) are two bounded functions, integrable in (a, 6), {^)'^{^)dx = (^)\ \[r{x)dXf J a J a where ^ is some definite value of x in a.= cc = 6. Further, if €j>{x) is not continuous in (a, ft), we replace ^(^) hy JUL, where /ul satisfies the relation on = iuL = M, m, M being the hounds of (x) in (a, 6). This is usually called the First Theorem of Mean Value. As a particular case, when ^(a;) is continuous, I {^), where a = ^=6. J a It will be seen from the corollaries to the theorem in § 47 that in certain cases we can replace a^$^bhy a<^{x)) = in (a, 6).* "^^ To determine the constant (7, we use the fact that F(x) ] vanishes a,t x = a. Thus we have I f{x) dx = ff>{x)-'(p (a). Ja 50. The Second Theorem of Mean Value. We now come to a theorem of which frequent use will be made. I. Let 0(ic) be inonotonic, and therefore integrable, in (a, b) and let \lr{x) be bounded and integrable, and not change sigifi more than a finite number of times in (a, b)A * Cf. Hardy, loc, cit., p. 228. tThis restriction on ^(a;) can be removed. B'or a proof in which the only condition imposed upon \p{x) is that it is bounded and integrable in (a, ft), see Hobson, loc. ciLy p. .360, or Goursat, loc. cit., T. I., p. 182. Digitized by Google THE DEFINITE INTEGRAL 95 Then I {x)ylr{x)(lx=^ff}{a)\ \j/[x)dx + if>{]o)\ \jf(pc)dx, where ^ is some definite valtie ofxina^x^b. For clearness we shall take {x) monotonic increasing iij (a, b). The modifications in the proof for a monotonic decreasing function will be obvious. Since we assume that ^(a?) does not change sign more than a finite number of times in (a, 6), we can take a = aQ, a^, ag, ... a^.j, a^-hy such that ylf{x) keeps the same sign in the partial intervals K, ai), {a^,a^, ... (a„_i, aj. Then ^ (») V^ («) ^^ = 2 ^ (^) V^ (^) ^^• Jo 1 Ja^.i Now, by the First Theorem of Mean Value, E(l>{x)\lr{x)dx = fir\ yjr{x)dxy I .-1 J«r-1 where 0(a^.i) = /ir^^(«r). Therefore ) 0(a;)^(i»)da; = /Ar[-i^K)-^(ar-i)l where we have written F{x) =1 i/r(fl:;) c?a^. Ja Thus we have ^i,(x)y}r{x)dx='^,^r\.F{ar)-F{a,.,)-] (1) J a 1 Since ^(a)==0, we may add on the term 0(a) i^(a), and rewrite (1) in the form : J it>(p^)ylr(x)dx = F{a)[{a)'-fjL^] + + ^(6)[/Xn-0(6)] +F(h){hY The multipliers of F{a), F{a^, ... i^(6) in all but the last line must be negative or zero. (2) Digitized by Google 96 THE DEFINITE INTEGRAL We rnay therefore replace the sum of these (n + l) terms by M{[0(a)-/xJ + [M,-M2]+...+Ui-^(6)]}, ie.hy M[^(a)-^(6)], wh^re M is some definite number intermediate between the greatest and least of F(a), F{a^), ... F(a^_^)y F{b), or coinciding with one or other. But, since F{x)= I \lr(x)dx, we know that F{x) is continuous in (a, b), •'" Therefore M may be taken as F{^)y where ^ is some definite value oi X in a = x = b (cf. § 31). It follows from (2) that jV(^) n^)dx^F{i)[{a)^(b)]+F{b} 0(6) = {a)F(i) + 4>{b)[F(b)--F{i)] = 0(a) I \lr(x)dx + {b)\ \l/^{x)dxy where ^ is some definite value oi x in a = x = b. Thus the theorem stated above is proved. We have seen in § 45 that I f{x) dx is unaltered if we change Ja the value of f{x) at a finite number of points in (a, b). Now ^(x) is monotonic in (a, 6), and therefore 0(a + O), 0(6 — 0) exist. Also we may give ff>{x) these values at x = a and x = h, respectively, without altering its monotonic character or changing I (x)\l/(x)dx, Ja We thus * obtain the result : II. Let {a + 0)\ \lA(x)dx + (l}{b — 0)\ \l/(x)dx, where ^ is some definite value of x in a = x = h ♦ This and the other theorems which follow could . be obtained at once by making suitable changes in the terms (x) and \lr{x) as before, III. 0(aj) \[r{x) dx = A\ \Jr{x) dx+B\ yff{x) dx, where -4 = 0(ci+O) and 5 = 0(6 — 0), if {x) is monotonic in- creasing, and il=0(a + O), 5 = 0(6 — 0), if 0(iB) is monotonic decreasing, ^ being some definite value ofxina^x^b. The value of ^ in I., II., III. need not, of course, be the same, and in III. it will depend on the values chosen for A and B. Finally, as in III., we may take A = and 5 = 0(6), when 0(ic) = and is monotonic increasing in (a, 6). Thus, with the same limitations on ylr{x) as before, when 0(a?) = and is monotonic increasing in {a, 6), we have IV. I 0(aj)^(a;)daj = 0(6) \[f{x)dx, where ^ is some definite value ofxina = x = b. Again, when {a)\ \fr{x)dx, Ja Ja where ^ is some definite value of x in a = x^b. Theorems IV. and V. are the earliest form of the Second Theorem of Mean Value, and are due to Bonnet,* by whom they were employed in the discussion of the Theory of Fourier's Series. The other Theorems I., II., III. can be deduced from Bonnet s results. Theorem I. was given by Weierstrass in his lectures and Du Bois-Reymond,t independently of Bonnet. Theorem 11. is the form in which we shall most frequently use the Second Theorem of Mean Value. * Bruxdles, M4m. cour, Acad, roy., 23, p. 8, 1850; also J, Math., Paris, 14 p. 249, 1849. t/. McUh., Berlin, 69, p. 81, 1869; and 79, p. 42, 1875. Digitized by VjOOQIC 98 THE DEFINITE INTEGRAL INFINITE INTEGRALS. INTEGRAND BOUNDED. INTERVAL INFINITE. 51. In the definition of the ordinary integral I f(x) dx, and Ja in the preceding sections of this chapter, we have supposed that the integi*and is bounded in the interval of integration which extends from one given point a to another given point b. We proceed to extend this definition so as to include cases in which (i) the interval increases without limit, (ii) the integrand has a finite number of infinite discontinuities.* I. Integrals to +00. I f(x)dx. Let f{x) be bounded and integrable in the interval (a, b), where a is fixed and b is any nv/ntber greater than a. We define the integral I f{x) dx as Lt \ f{x) dx, when this limit existsA -^^ ""^"•'" We speak of / f{oo)doc in this case as an infrnite integral, and say that it converges. On the other hand, when / f{x)dx tends to x as x-^cc, we say that the f{x)dx diverges to «, and there is a similar defini- tion of divergence to ~ oo of / f{x) dx. r^dx= Lt I €r'dx= Lt (l-e-*)=l. p =Ltr^=Lt2(l-l)=2. fe'€lx=QD; I -.-=00. Ji kJx For [ e^dx= Lt fVda?= Lt (e*-l)=Qo. Ex. 1. For And Ex.2. *For the definition of the term ** infinite discontinuities," see §33. fit is more convenient to use this notation, but, if the presence of the < variable x in the integrand offers difficulty, we may replace these integrals by I f{t)dt and Lt l^f{t)dt Ja x->ao Ja Digitized by Google THE DEFINITE INTEGRAL 99 And r^- = Lt r^=Lt2(V.t-l) = oc. Similarly j" log i c^^= - oo ; ^* j^^ = - x . These integrals diverge to oo or — oo , as the case may be. Finally, when none of these alternatives occur, we say that the infinite integral / f{x)dx oscillates finitely or infinitely, as in §§ 16 and 25. Ex.3, / sin^rcfo? oscillates finitely. / ^sin^fltr oscillates infinitely. II. Integrals to -oo. l f(x)dx. J -00 When f{x) is bounded and integrable in the interval (a, 6), where b is fixed and a is any number less than b, we define the integral j f{x) dx as Lt I f{x) dx, when this limit exists. J -00 X-> -coJ X /b f{x)dx as an infinite integral, and say that it converges. Cb The cases in which / f(pc) dx is said to diverge to oo or to ~ oo , or to J— 00 oscillate finitely or infinitely, are treated as before. Ex. 2. / e'~^dx diverges to oo. J — X / 8inh.rc21:p diverges to -00. rsin X dx oscillates finitely. oo / ^sin xdv oscillates infinitely. /•oo III. Integrals from - oo tooo, f(x)dx. J - 00 If the infinite integrals I f(x)dx and I f{x)dx are both J -co J a convergent^ we say that the infinite integral | f(x) dx is con- vergent and is equal to their sum. Since I f{x)dx=\ f{x)dx+\ f{x)dx, a<^a<^x, Ja Ja Ja Google Digitized by ' 100 THE DEFINITE INTEGRAL it follows that, if one of the two integrals | f{po) dx or j f{x) dx converges, the other does. * Also ! f(x)dx=^\ f{x)dx+\ f{x)dx. Ja Ja Ja Similarly, I /(a5)da;=| /(a?) dec + I f{x)dx, x<^aaj' = X We have seen in § 51 that if I f{x) dx converges, then Ja /.QO |.jj /.» I f{x)dx=\ f{x)dx+\ f(x)dx, a<^a. Ja Ja Ja /•QO It follows from I. that, if ! f{x) dx converges, to the arbitrary positive number e there corresponds a positive number JT such that I f* I I f{x)dx\-a, it is clear that f(x)dx is a monotonic in- creasing function of x. Thus I f{x) dx must either converge or diverge to oo • "* I. It will converge if there is a positive number A such that Jf{x)dxa, and in this case j f{x}dx^A. a Ja It will diverge to oo if there is no such number. These statements follow from the properties of monotonic functions (§ 34). Further, there is an important " comparison test " for the con- vergence of integrals when the integrand is positive. II. Let f(x), g{x) be two functions which are positive, bounded and integrable in the arbitrary interval {a, 6). Also /•CO let g{x)=f{x) when x = a. Then, if \ f(x)dx is convergent, it /•oo J a /•<» /•« follows that I g{x)dx is convergent, and I gr(a:;)rfaj= f{x)dx, . Ja Ja Ja Digitized by VjOOQ IC 102* • ••'• y^ •-' 'i irflk-BfiFINITE INTEGRAL For from § 47 we know that g{x)dx=\ f(x)dx, when x>a. Ja J a Therefore I g{x)dx0. •'"^ We have -^ = r - {«^"'' — a^'"}, when n=^l, Cx ^jjp and I — = logic — log a, when 71=1. Ja X Thus, when >i>l, Lt f — = ^' !> i.e. I = JaX'' n-l And, when ti — 1, Lt j — = oo, u:— >* JaX . rdx ,. *•*• :;;: diverges. * Since the relative behaviour of the positive integrands f{x) and g{x) matters only as x-x» , these conditions may l)e expressed in terms of limits : When g{x)/f{x) has a limit as x-^oc , / g(x) dx converges, if / f(x) dx converges. When fj{x)lf(x) has a limit, not zero, or diverges, as a;->QO , / g{x) dx diverges, if / f{x) dx diverges. Digitized by Google THE DEFINITE INTEGRAL i/ 103 fdx 1 1 7/T i' :^\ converges, since ^ //, ^^y, < -z^t when a? g a > 0. w(r+i^)^^^''«^' «^'^^"s7(i+^)<^' dlr ,. .1 ^ diverges, since > -, when J7^2. « /""sin'o:, . sin'x^ 1 , -* ^ r^ 3. / — ^ CM? converges, since —^^-g, when ^= a >0. /•oo 54. Absolute Convergence. The integral I f{x) dx is said to be absolutely convergent xvhen f{x) is bovmded and integrable in the arbitrary interval (a, 6), and I \f{x) \ dx is convergent Ja Since I {"^ f{x) dx i ^ \^ \f{x) \ dx, for »" >x^a '•'*' I •''' ^ (cf.§47,Cor.L), it follows from § 52, II. that if I \f(x) \ dx converges, so also does /•oo J a Jo But the converse is not true. An infinite integral of this type may converge, and yet not converge absolutely. For example, consider the integral /•CO • '^dx. Jo aj The Second Theorem of Mean Value (§50) shows that this integral converges. For we have f^'sinaj , 1 f^ . , : 1 r . , Jj^ X x Jj^ X J( whereO -Ce, when «">«' = X, 4 Digitized by Google 104 THE DEFINITE INTEGRAL r sin QC Therefore I dx converges, and we shall find in § 88 that Jo "^ its value is Jx. But the integral I ' -' da? diverges. To prove this, it is only necessary to consider the integral ' I sin oj I r Jo 'cfo. lo ^ where n is any positive integer. We have [""«'°^ldx= V T ^-^^^d^vl. Jo aJ ttY"^ But the series on the right hand diverges to 00 as n->yo . Therefore Lt I ^ -dx^oo. n— >» Jo *^ But when cc^titt, Jq aj Jo i» Therefore Lt I ' ' da? = 00 . «— >oo Jo *^ When infinite integrals of this type converge, but do not con- it verge absolutely, the convergence must be due to changes of sign in the integrand as x-> . /•oo 65. The M-Test for the Convergence of f(x)dx. Ja I. Let f{x) be hounded and integrahle in the arbitrary interval (a, h) where a>0. If there is a mimber jn greater than 1 stvch Digitized by VjOOQ IC THE DEFINITE INTEGRAL 105 that a^f{x) is hounded when x^a^ then \ f(x)dx converges absolutely. * Here \x'^f{x)\0. // there is a number ji less than or equal to 1 such that x^f{x) has a positive lower bound /•QO 'when x = a, then I f{x)dx diverges to 00 . Here we have, as before, xf*^f{x)^A'^Of when x = a. It follows that -=/(«). f^dx But I — diverges to 00 when /jl = 1. Ja X^ It follows that I f{x) dx diverges to 00 . Ja III. Let f(x) be bourided and integrable in the arbitrary interval (a, 6), where a>0. // there is a number jjl less than or equal to 1 such that (c^f{x)^ has a negative upper bound when x^a, then I f{x) dx diverges to —oo x^a; also, by properly choosing the positive number Z, a^/(x) will either have a positive lower bound, when this limit is Digitized by VjOOQ IC 106 THE DEFINITE INtEGRAL positive, or a negative upper bound, when this limit is negative provided that x = X. Thus, from I.-1II., the following theorem can be immediately deduced : Let f{x) be hounded and integrable in the arbitrary interval {ayb),ivhere a^O. ; If there is a number fx greater than I svA^h that Lt (x^f(x)) {exists, then \ f{x) dx converges. Ja If there is a number jm less than or equal to 1 such /•QO that Lt {pc^f(x)) exists and is not zero, then I f{x) dx diverges ; ar— >«> J a and the same is true if x^f{x) diverges to +00 , or to -co , as x->oo . We shall make very frequent use of this test, and refer to it as the "/x-^st." It is clear that we are simply compar- f * . f * dx ing the integral I f{x)dx with the integral 1 — , and deducing J a J a *^ the convergence or divergence of the former from that of the latter. ^•/o (^^2^ diverges, since JLt^ (^.x^^^,) =1. It should be noticed that the theorems of this section do not apply to the integral / — ;— dx, /•oo 56. Farther Tests for the Convergence of f (x) dx. Ja I. // ff}{x) is bounded when x = a, and integrable in the arbitrary interval (a, 6), and \ \Jr{x)dx converges absolutely, then I (t>{x) yjrix) dx is absolutely convergent, Ja For we have \(j}{x)\{x) I \[f{x) I dx<:iA I I \lr{x) I dx, when oi'^xl^a. Since we are given that I ,^(03) Ax converges, the result follows. ** C gin X I cos A* Ex. 1. / -j^„ dxj j -j^- cLv converge absolutely, when 7t and a are positive. 2. / e~** cos 6a? (ir converges absolutely, when a is positive. 3. / g ^ c?d7 converges absolutely. II. Let {x) be monotonic and bounded when x = a. Let \lr(x) be bounded and integrable in tlie arbitrary interval (a, 6), and not change sign inore than a finite number of times in the poo interval. Also let \ \fr{x) dx converge, Ja Then I (f>{x) ^{x) dx converges, Ja This follows from the Second Theorem of Mean Value, since {x)\lr{x)dx = ^{x')\ yfrix^dx+^ix^'u' \lr(x)dx, wherea{x') I and ; ^(cc") i are each less than some definite positive number A. Also we can choose JC so that I \ff{x)dx and I \lr(x)dx are each less than €/2A, when x"'^x' = X, and c is any given positive number, as small as we please. It follows that If x'^Z, (x)\fr{x)dx and the given integral converges. Ex. 1. / e~* — - d,v conversres. Jo X ^ fCOS DC (1 - e~*) — ^ dx converges when a>0. X /Google Digitized by * 108 THE DEFINITE INTEGRAL III. Let ^[x) be monotonic arid bounded when x = a, and Lt 0(ic) = O. Let \lr{x) be bounded and integrable in the arbitrary interval (a, 6), and not change sign more than a finite number of times in the interval. Also let I \lr{x) dx be bounded when x^a, Ja Then I ^(x)\ff{x)dx is convergent. As above, in II., we know that I (x)\lr(x)dx = (l){x')\ \lr(x)dx + (l){x")\ \fr(x)dXy J x' Jx' J$ where a-a, where A is some definite positive number. And I >/r(a3)(Zaj = I \/r(ic)da? + I \fr(x)dx <2A. I f ^' Similarly I ^(a;)ciic <2^. ' Jf Also Lt 0(aj) = O. X— >oo Therefore, if e is any positive number, as small as we please, there will be a positive number X such that |0(a;)Kj-T, when x = X. It follows that )dx (x)\lr(x)( iJx' and I ^{x)\lr(x)dx converges, Ja fSlD V 1^ COS .27 ~~^^^f I — ^fl^ converge when n and a are positive. 2. / tj-— 2 sin. ^Tfl?^ converges. /Google Digitized by ' THE DEFINITE INTEGRAL 109 The Mean Valne Theorems for the Infinite Integral. 57. The First Theorem of Mean Valne. Let {x) he hounded when x^a^ and integrable in the arhitrary interval («, h). Let V^(^) keep the same sign in a7~a, and j '^(x)dx converge. Then f {x) ylr(x) d.v = ^ T ylr(a^) dv, where m^fi^M^ the upper and lower hounds of {x) in x^a heing M and m. We have m^(x)^M, when x^a, and, if •^(^)gO, m ip{x) ^ (x) yjr (x) ^ Mylr{x). Therefore ml ylr{x)dx^ j (x)-^{x)dx^M\ •\lr{x)dx, when x^a. But, by. §56, I., / {x)ylr(x)dx converges, and we are given that rJa ^{x) dx converges. Thus we have from these inequalities m^ yp{x)dx:^r {x) ^(^) dx^Mfif^ix) dx, Ja Ja Ja In other words, / (x) \f/{x)dx=ixl \p (x) dx, Ja Ja where m^/A^M, 58. The Second Theorem of Mean Valne. f{x) dxhea convergent integral, and F{x)= j f(x) dx{x^a). Then F{x) is continuous when x^a, and bounded in the interval (a, x ). Also it takes at least once in that interval evert/ value hetween its upper and lower hounds, these heing included. The continuity of F{x) follows from the equation F{x+h)-F{x)= - r^y(x)dx. Further, Lt F{x) exists and is zero. ar— >-ao It follows from § 32 that F(x) is. bounded in the interval (a, oo ), as defined in that section, and, if M, m are its upper and lower bounds, it takes at least once in (a, oo ) the values M and m and every value between J/, m. Let {x) he hounded and monotonic whe^i x^a. Let ^{x) he hounded and integrahle in the arhitrary interval (a, 6), and not change sign more than a finite numher of times in the interval. Also let f\l/{x)dx converge. Digitized by VjOOQ IC 110 THE DEFINITE INTEGRAL Theii I {a-\-0)\ ■^{x)dx-\-<^{)\ \j/{x)dx, where a ^ ^ ^ oo .* Suppose ^(.v) to be monotonic increasing. We apply the Second Theorem of Mean Value to the arbitrary interval («, fe). Then we have [ ^{x)y^{x)dx=.{h-0)[ yfr(x)dv, where a^^^b. Add to both sides 5= ^(oo ) I \lr{x)dxy observing that ^(oo) exists, since « Ja Then B+\ ^{x)\l^{x)dx = <^(a+0)| \p{x)dx-\-ij>{h-0)[ ^(a?)c?a7+(oo)j \f^{x)dx = ^(a+0)rj ^(^)rf^-| ^(a7)rf^1 + ^(6-0)r[ \f^(x)dx-\ ip{x)dxl + ^(oo)l \l/{x)dx = <^(a + 0) f xl^{x)dv-\- U+ V, (1) where ^={<^(6-0)-<^(a+0)}l \P(x)dr, and V={{oo)-'{b-0)}{ if^{x)dx. Now we know from the above Lemma that I \f/{x)dxia bounded in («, oc ). Let M, m be its upper and lower bounds. Then m^\ \P(x)dx^M, and »wgl xl/{x)dx^M, Therefore {4>{h-0)-{a + 0)\m^ U^{{b-0)-{a + 0)}M, {{cc ) -4>{b- 0)} m ^ rg {<^(oo ) - (co)-(a-\-0)}M, Therefore ^+ F=ft{<^(oo)-c/)(a+0)}, where m^jx^M, Insert this value for U-h V in (1), and proceed to the limit when 6->qo . Then I (a:)\P(x)cLc=^(a + 0)\ \^(^)i^+/A'{<^(oo)-^(a+0)}, where /a'= Lt /x. ft— >•« This limit must exist, since the other terms in (1) have limits when 6->ao . Also, since m^fi ^M, it follows that m^fi^M. But I \f/{x)cLv takes the value /x' at least once in the interval (a, oo ). i»co Thus we may put ft'= I \f/{x) da}, where a^^^co. Therefore we have finally I {a;)\f^{x)da!={oo)\ \l/{x)dx, where « ^ ^' ^ oo . It is clear that we might have used the other forms I. and III., § 50, of the Second Theorem of Mean Value and obtained corresponding results. INFINITE INTEGRALS. INTEGRAND INFINITE. 59. I f(x)dx. In the preceding sections we have dealt with the infinite integrals I f{x) dx, I f(x) dx and I f(x) dx, when Ja J -co J -co ^ the integrand f(x) is bounded in any arbitrary interval, however large. ^ A further extension of the definition of the integral is required so as to include the case in which f{x) has a finite number of infinite discontinuities (cf. § 33) in the interval of integration. First we take the case when a is the only point of infinite discontinuity in (a, 6). The integrand f{x) is supposed bounded and integrable in the arbitrary interval {a + ^, b), where ct0, we define the infinite integral I f(x) dx as Lt I f{x) dx. Digitized by VjOOQ IC 112 THE DEFINITE INTEGRAL Similarly, when the point h is the only point of infinite dis- continuity in (a, b), and f{x) is bounded and integrabU in the arbitrary interval (a, 6 — ^), where a<^b'-^<^by we define the in,- finite integral I f(x)dx as Lt 1 f(x)dx, when this limit exists. Again, when a and b are both points of infinite discontinuity, we define the infinite integral I f{x)dx a,s the sum of the Ja infinite integrals I f{x)dx and I f(x)dx, when these integrals J a Jc exist, as defined above, c being a point between a and b. This definition is independent of the position of c between a and 6, since we have f(x)dx=\ f(x)dx+\ f(x)dx, Ja Ja Jd where aaj„. Let a = ajiOJa+f when this limit exists. It follows at once, from the definition, that : I. The integral I f(x) dx is convergent and has the value I Ja when, any positive number e having been chosen^ as small as we please, there is a positive number rj such that /—I f(x)dx in the second ca^e. And finally, V. Letf(x) be bounded and integrable in the arbitrary interval (a + f, b), where 0<^<6 — a. If there is a number /x between and 1 such that Lt {X'-a)'^f(x) exists, then I f{x) dx converges absolutely. Ja If there is a number jul greater than or equal to 1 such that Lt {x — a)f^f{x) exists and is not zero, then I f{x)dx diverges; a;->a+0 J a and the same is true if {X'-aYf(x) tends to +oo , or to -^oo ,a^ x->a+0. We shall speak of this test as the /x-test for the infinite integral I f[x)dx, when 05 = a is a point of infinite discontinuity. It is Ja clear that in applying this test we are simply asking ourselves the order of the infinity that occurs in the integrand. The results can be readily adapted to the case when the upper limit 6 is a point of infinite discontinuity. Also, it is easy to show that VI. If (f>(x) .is bounded and integrable in (a, b), and I '^{x)dx converges absolutely, then I 0 » rdx' z 5— r-. ("> ^* /(-)=^)- Then Lt xf{x)^\. Therefore the integral diverges by the same test. sin X . -3:;^ dx converges, when < « < 1. X The integral is an ordinary finite integral if «^0. Also Lt ^"f^Vl- Therefore the integral converges when 00, ^ -^ H (^"^^^^ positive integer), and r! But whatever n may be, we can choose r so that r-n-\-\>\. It follows that, whatever n may be, / e'-'af^-^dx converges. (ii) LetO<7i0, the integral is an ordinary integral, X— ►© w^hen 71 > 1 . Also we know that / log :i7(ia7=Lr (log 07-1) =x{\-\ogx)-\. It follows that / log 0? c?jF = Lt {x{\ - log a?) - 1} = - 1 . Again, Lt {xi*-xx^'^ log x) = Lt (^/*+*» - 1 log a;) = 0, if /a > 1 - w. X— ►© . x->0 And when 0<7i< 1, we can choose a positive number /a less than 1 which satislies this condition. Therefore / ^*-^log x dx converges, when < » ^ 1. Finally, we have Lt (a7X:i;"-^|loga;|)= Lt u.-"|loga7| = oo, when w^O. Therefore / af^^Xogxdx diverges, when «^0. Jo KEFEEENCES. Pk la VALLiB PoussiN, toc, cit, T. I. (3« ed.), Ch. VI, ; T. 11. (2« ed.), Ch. II. DiNi, Lezimii di Aiudisi Infinitesimale^ T. II., 1* Parte, Cap. I. and VII., Pisa, 1909. GouRSAT, loc, ciU, T. I. (3* ^d.), Ch. IV. and XIV. KowALEWSKi, GrundzUge det' Diferentiod' u, Integrcdrechnung^ Kap. XIV., Leipzig, 1909. Lkbesoue, Legons sur V Integration^ Ch. I. and II., Paris, 1904. OsoooD, loc, cit.^ Bd. I., Tl. I., Kap. III. PiBRPONT, loc, cit.^ Vol. L, Ch. V. Stolz, Orundziige der inferential- u, Integ'ralrechnungj Bd. I., Absch. X., Leipzig, 1893. And Brunel, " Bestimmte Integrale," Enc, d. math. Wise., Bd. II., Tl. I., Leipzig, 1899. * This integral can be reduced to the Gamma Function integral, and its con- vergence or divergence follows from Ex. 5. Also see below, Ex. 5, p. 120. Digitized by Google 120 THE DEFINITE INTEGRAL EXAMPLES ON CHAPTER IV. 1. Show that the following integrals converge : Jo l+cosor+c* Ji l+a^+x^+smx Jo r6-«^co8h6^d^, r^^dv, ri^da.^ Jo Jo 1+^ Jo 1-0.'^ 2. Discuss the convergence or divergence of the following integrals : f/ w/A V r^^» r^rf^, where 0sJ{o - X) Jo x+l Jo JC-l r^dv, r ^dx, P'sm'^ecos^ede. Jo x+l ^ Jo x-l Jo 3. Show that the following integrals are absolutely convergent : f sin- -tt;, j e-"^"^ cos bxdx, j e-^'^^arsinnxdv (m>0), and r?^dx, • where P{x) is a polynomial of the m^^ degree, and Q{x) a polynomial of the w*"* degree, n^m-\-2, and a is a number greater than the largest root of 4. Let/{:i) be defined in the interval 00 and 7i> -I. And by means of a similar substitution, show that r ar-\\ogx)"dv converges, provided that ?/K0 and 7i > - 1. - r. Tj^p^ converges when /x>0 and that it diverges when /i ~ 0, the lower limit a of the integral being some number greater than unity. Deduce that if there is a number /x > 0, such that Lt {:r(log xy+^f(x) } exists, then / f{x) dx converges, and give a corresponding test for the divergence of Ja. Show that / 7 . - ■ ^ ,, ;^cLc converses. and / -. r4^^-i diverges. J2 {x-{-s\fi^x)\ogx 7. On integrating / cos x log xdxhy parts, we obtain / cosa7log.rc^'=sina;loga:- / dx. Deduce that / cos x log x dx oscillates infinitely. Also show that / cos^log^rfo; converges, and is equal to - / — - dx, 8. On integrating / cosx^dx by parts, we obtain / C08a;*cM;=^rT/8inji: 2-— _8in.i;2 + - / — n— cU', Jtf 2af 2x 2 Jjcf X* where x">af> 0. Deduce the convergence of / cos x^dx. 9. Let f{x) and g{x) be bounded and integrable in (a, 6), except at a certain number of points of infinite discontinuity, these points being different for the two functions. Prove that / f{x)g{x)dx converges, if I \f{x)\dx and / \g{x)\dx converge. -^^ -^c Ja 10. Let /(a:) be monotonic when x^a^ and Lt f{x)=0. Then the series f(a) +f(a + 1) +/(« + 2) + . . . is convergent or divergent according as i /(a;) cLv converges or diverges. Prove that for all values of the positive integer n, AlBo show that _^_ + -i^+_i_... converges to a value between i(7r + 1) and |7r. Digitized by Google y^ CHAPTER V THE THEORY OF INFINITE SERIES, WHOSE TERMS ARE FUNCTIONS OF A SINGLE VARIABLE 62. We shall now consider some of the properties of series whose terms are functions of x. We denote such a series by and the terms of the series are supposed to be given for values of X in some interval, e,g, (a, 6).* When we speak of the sum of the infinite series it is to be understood : t (i) that we settle for what value of x we wish the sum of the series ; (ii) that we then insert this value of x in the different terms of the series ; (iii) that we then find the sum — Sn{x) — of the first n terms ; and (iv) that we then find the limit of this sum as 7i->30 , keeping x all the time at the value settled upon. On this understanding, the series u^{x) + Uc^{x) + u^(x)+... is said to be convergent for the value x, and to have f{x) for its sum J if this value of x having been first inserted in the different * As mentioned in § 24, when we say that x lies in the inUrvcU (a, b) we mean that a^x^b. In some of the results of this chapter the ends of the interval are excluded from the range of x. When this is so, the fact that we are dealing with the open interval {a < x0. Again, in the case of the definite integi-al / f(x^ a)dx, it is understood that we insert in f{x, a) the particular value of a for which we wish the integral before we proceed to the summation and limit involved in the integration. We shall write, as before, f{x)Sn{x)^Rn{x), where f{x) is the sum of the series, and we shall call ii„(x) the remainder after n terms. As we have seen in § 19, Rn{!x) is the sum of the series Also we shall write pRn{x) = /Sn+p(x) - Sn(x), and call this a partial remainder. With this notation, the two conditions for convergence are (i) \Rn{x)\<,€, when n^v\ (ii) li^n(^)l<^' when n = j/, for every positive integer p, A series may converge for every value of x in the open interval a <^'< 6 and not for the end-points a or 6. E.g. the series 1 -\-x-{-x^-\-.,. converges and has for its sum, when - 1 < .r < 1. 1 —X When a?=l, it diverges to -f-oo ; when a;= -1, it oscillates finitely. Digitized by VjOOQ IC 124 THEORY OF INFINITE SERIES, WHOSE TERMS 63. The Sum of a Series whose Terms are Continuous Func- tions of X may be discontinuous. Until Abel * pointed out that the periodic function of x given by the series -^ -=- 2(8ina; — |-sin2a;+isin3a!J— ...), which represents x in the interval — 7r<;aj<7r, is discontinuous at the points aj = (2r+l)'T, r being any integer, it was supposed that a function defined by a convergent series of functions, continuous in a given interval, must itself be continuous in that interval. Indeed Cauchy f distinctly stated that this was the case, and later writers on Fourier s Series have, sometimes tried to escape the diflSculty by asserting that the sums of these trigonometrical series, at the critical values of x, passed con- tinuously from the values just before those at the points of discontinuity to those just after. J This mistaken view of the sum of such series was due to two different errors. The first consisted in the assumption that, as n increases, the curves y = Sn{x) must approach more and more nearly to the curve y=f(x), when the sum of the series is f(x), an ordinary function capable of graphical representation. These curves y = Sn(x) we shall call the approximation curves for the series, but we shall see that cases may arise where the approxi- mation curves, even for large values of n, differ very considerably from the curve y =f(x). It is true that, in a certain sense, the curves ;iy (i) y = Sn{x) and (ii) y=f{x) approach towards coincidence ; but the sense is that, if we choose any particular value of x in the interval, and the arbitrary small positive number e, there will be a positive integer p such that, for this value of x^ the absolute value of the difference of the ordi- nates of the curves (i) and (ii) will be less than e when n = p. Still this is not the same thing as saying that the curves coincide geometrically. They do not, in fact, lie near to each other in the neighbourhood of a point of discontinuity of f(x)] and they may not do so, even where f{x) is continuous. *Abel, J. Math,, Berlin, 1, p. 316, 1826. t Cauchy, Cours d' Analyse, 1" Partie, p. 131, 1821. Also OSuiTes de Gawihy, (Ser. 2), T. III., p. 120. } Cf. Saohse, loc, ciL ; Donkin, Acoustics, p. 63, 1870. Digitized by VjOOQIC- ARE FUNCTIONS OF A SINGLE VARIABLE 126 The following examples and diagmiuH illustrate these points : Ex 1. Consider the series .r+1 (j^'+l)(2.r+l) 1 :0. w„(.r) = 1 («-l).r4-l w.r+l' i Sn(-r) = l- Lt /S;(.r)=0, since ^„(0)=0. Here and Thus, when .r>0, when ^ = 0, The curve y =/(^), when .v ^ 0, consists of the part of the line y = 1 f or which .r > 0, and the origin. The sura of the series is discontinuous at .r=0. Now examine the approximation curves This equation may be written As n increases, this rectangular hyperbola (cf. Fig. 10) approaches more and more closely to the lines y=l, .v=0. If we reasoned from the shape of the Fig. 10. approximate curves, we should expect to find that part of the axis of y for which 0=H^ has a maximum at> (1/w, \) and a minimum at ( - 1/n, - J) (cf. Fig. 1 1). The points on the axis of x just below the maximum and minimum move in towards the origin as n increases. And if we reasoned from the shape of the curves y=Sn{x\ we should expect to find the part of the axis of y from — J to ^ appearing as a portion of the curve y=f{x). Ex. 3. Consider the series where Here and Lt *S'„(^)=0 for all values of x. The sum of the series is again continuous, but the approximation curves (cf. Fig. 12), which have a maximum at (l/V^i^ W*0 ^^^ ^ minimum at Digitized by VjOOQ iC ARE. FUNCTIONS OF A SINGLE VARIABLE 127 (- 1/Vrt^ - W^)> differ very greatly from the curve y='f(x) in the neigh- l)ourhood of the origin. Indeed they would suggest that the whole of the axis of y should appear as part of y=f{r). 64. Repeated Limits. These remarks dispose of the assump- tion referred to at the beginning of the previous section that the approximation curves y=Sn(x)y when n is large, must approaph closely to the curve y=f(x\ where /(a?) is the sum of the series. The second error alluded to above arose from neglect of the convention implied in the definition of the sum of an infinite series whose terms are functions of x. The proper method of finding the sum has been set out in § 62, but the mathematicians to whom reference is now made proceeded in quite a different manner. In finding the sum for a value of cc, say x^, at which a discontinuity occurs, they replaced a? by a function of n, which converges to ic^ as 'H increases. Then they took the limit when 71^00 of Sn{x) in its new form. In this method x and n approach their limits concurrently, and the value of Lt £fn(a?)may quite well diflTer from the actual sum for x = Xq, Indeed, by choosing the function of n suitably, this double limit may be made to take any value between /(oj^^+O) and/(a?j^ — 0). Digitized by Google 128 THEORY OF INFINITE SERIES, WHOSE TERMS For instance, in the series of § 63, Ex. 1, [J we have seen that a; = is a point of discontinuity. If we put x=p/n in the expression for Sn{x), and* then let 7i->x, p remaining fixed, we can make Lt 8n(x) j^ke any value between and 1, according to our choice of p, /f¥or we have Sn{p/n)= ^ , which is independent of n, and j P~T ■*• ^ ^ which passes from to 1 as p increases from to oo . t It will be seen that the matter at issue was partly a question of words and a definition. The confusion can also be traced, in some cases, to ignorance of the care which must be exercised in any operation involving repeated limits, for we are really dealing here with two limiting processes. If the series is convergent and its sum is f[x\ then f{x)= Lt S^(x\ n — >-co - , and the limit oif(x) as x tends to x^, assuming that there Is. such a limit, is given by Uf[x)=U [Ltfif„(a^)] (1) If we may use the curve as an illustration, this is the ordinate of the point towards which we move as we proceed along the curve y = f(x)y the abscissa getting nearer and nearer to Xq, but not quite reaching Xq, According as x approaches Xq from the right or left, the limit given in (1) will hef{xQ+0) or /(Xq—O). Now/(aJo), the sum of the series for x^x^y is, by definition, Lt [S„(x,)l n— >oo and since we are now dealing with a definite number of con- tinuous functions, Sn{x) is a continuous function of x in the interval with which we are concerned. Thus 8n{x^)= Lt SJx). X-^Xq Digitized by Google ARE FUNCTIONS OF A SINGLE VARIABLE 129 ThereforlJjhe sum of the series for x = Xq may be written >^ Lt [Lt Sn{x)] (2) |^?\ n-»oo x~>Xo The twh^xpressions in (1 ) and (2) need not be the same. They are so onljA^hen f{x) is continuous at Xq. 65. UnUbrm Convergence.'^ When the question of changing the order attwo limiting processes arises, the principle of uniform convergeMfc, ^hich we shall now explain for the case of infinite series wl^e terms are functions of x, is fundamental. What is involved^ this principle will be seen most clearly by returning to the sefiete x t 6x + l^{x+l)(2x+l)^"" ^-''• ries Sn(x) = 1 --:= , Lt Sn{x) = l, when a;>0. n— ><» Als<^J[a;) = — —y, when x>0, and jRn(0)=0. If tha^rbitrary positive number c is chosen, less than unity, and soD^positive x is taken, it is clear that l/(nx+l) V J — 1 %.i«hi^- If A'=0-1, 0-01, 0001, ... , 10-^ respectively, l/(w^ + l)<€ only when n>lO\ 10^, 106, ..., io(i>+3). And when €= , and .r=10~^, n must be greater than 10p+« if i/(«^+i)<£. ^^-^^ As we approach the origin we have to take more and more terms of the series to make the sura of w terms differ from the sum of the series by less than a given number. When .r=10~^, the first million terms do not contribute 1 % of the sum. 1-1 The inequality n >► — X shows that when e is any given positive number less than unity, * The simplest treatment of uniform convergence will be found in a paper by Osgood, Bull. Amer. Math. Soc, 3, 1896. C.I T Digitized by VjOOQ IC 130 THEORY OF INFINITE SERIES, WHOSE TERMS and X approaches nearer and nearer to zero, the smallest positive integer which will make Rn{x)y R^^^^(x)y ... all less than e increases without limit. There is no positive integer v which will make Rv{x), R^+i(x), ... all less than this c in a? ^ 0, the same v serving for all values of x in this range. On the other hand there is a positive integer v which will satisfy this condition, if the range of x is given by a; = a, where a is some definite positive number. Such a value of v would be the integer next above i — ^)/^- Our series is said to converge uniformly in a; = a, but it does not converge uniformly in a? = 0. We turn now to the series u^{x)+u^{x)+u^{x)+.,. , and define uniform convergence * in an interval as follows : Let the series u^[x)'{'U^(x)+u^{x) + :,. converge for all values of x in the interval a = x = b and its sum be f{x). It is said to converge uniformly in that interval, if any positive number e having been chosen^ as small as we please, there is a positive integer v such that, for all values 'of x in the interval, \f(x)-'Sn(x)\<€, when n^pA It is true that, if the series converges, \Rn{ixi)\<^€ for each x in (a, 6) when n^v. The additional point in the definition of uniform convergence is that, any positive number e having been chosen, as small as we please, the same value of v is to serve for all the vahves of X in the interval. For this integer v we must have \R,{x)l |i?.+i(aj)|, ... all less than e, no matter where x lies in (a, 6). * The property of uniform convergence was discovered independently by Stokes (cf. Cambridge, Tram. Phil. Soc., 8, p. 533, 1847) and Seidel (cf. Mimcheii, Abh. Ak. WiB8., 5, p. 381, 1848). See also Hardy, Cambridge, Proc. Phil. 8oc. 19, p. 148, 1920. t We can also have uniform convergence in the open interval a < a; < 6, or the half -open intervals a00 Therefore S,{x)-^^f{x)^S,{x)+l. But \S,,{x)-f(x)\'^\S„{x)-8^{x)\ + \SM)-f(x)\. It follows that, when n is greater than or equal to the value v specified above, M^e^^e <^ and this holds for all values of x in (a, b). Thus the series converges uniformly in this interval. Digitized by Google 134 THEORY OF INFINITE SERIES, WHOSE TERMS 67. Weierstrass's M-Test for Uniform Convergence. The following simple test for uniform convergence is due to Weierstrass : The aeries u^{x)+ u^{x) ■\-u^{x) + ... will converge uniformly in (a, 6), if there is a convergent series of positive constants such that, no matter what value x may have in (a, b), I Un{x) I = Mn for every positive integer n. Since the series J/j + Jlf g + J/g + . . . is convergent, with the usual notation, when n = Vy for every positive integer p. But \plin{x)\^\u^,+,{x)\ + \u^+^(x)\+... + \u,+^{x)\. Thus \pRn{x)\^M,^, + M,^,+ .,.+M,^, <€, when 71 = J/, for every positive integer p^ the inequality holding for all values of x in (a, b). Thus the given series is uniformly convergent in {a, b). For example, we know that the series is convergent, when a is any given positive number less than unity^ It follows that the series l + 2x + 3x^+... is uniformly convergent in the interval ( — a, a). ^ Ex. 1. Show that the series ;rcos0+A'2co8 2^+^ cos 3^+... is uniformly convergent for any interval (a'q, a'i), where -l0, for in both cases \Rn(x)\<—^~-, when x^a>0. ' ^ ' nx na Thus \Rn{x)\0, IV. If the terma of the series are continuous in the closed interval (a, 6), and the series converges uniformly in an^v, (1) the same v serving for every x in this open interval. Let m, n be any two positive integers satisfying this relation. Since the terms of the series are continuous in the closed interval (a, 6), there are positive numbers fj^ and i/g, say, such that \Sm(x)-8,n{a)\;i, ''''^ i^n(aj)-Sn(a)|<|, when 0^(x--a)^rj^. Choose a positive number tj not greater than jy^ or rj^, and letO^(a;-.a)^^. Digitized by Google 138 THEORY OF INFINITE SERIES, WHOSE TERMS Then \8„,{a)^8n{a)\ ^ <€, when m>n = i/ (2) A similar argument shows that \Sm{b)-8n{b)\n^v (3) From (2) and (3) we see that the series converges for x = a and ic = 6, and, combining (1), (2) and (3), we see that the con- dition for uniform convergence in the closed interval (a, 6) is satisfied. If the terms of the series are continuous ki (a, b), and the series converges uniformly in every interval (a, )3), where an^v, the same v serving for all values of x in the interval. Ill particular, | SmM - Sp{a;) \ < €, when m>v, and we shall suppose v the smallest positive integer which will satisfy this condition for the given € and every x in the interval. Plot the curve y=Sv{x) and the two parallel curves y=Sv{x)±€, forming a strip o- of breadth 2€, whose central line \»y=Sv{xy, (Fig. 13.) Fig. 18. All the approximation curves y = Sm{x\ m>v, lie in this strip, and the curve y=f{x\ where /(a;) is the sum of the series, also lies within the strip* or at most reaches its boundaries. (Cf. § 66 (ii).) Next choose c' less than €, and let the corresponding smallest positive integer satisfying the condition for uniform convergence be v'. Then v' is greater thaa or equal to v. The new curve y='Svf{x) thus lies in the first strip, and the new strip & of breadth 2c', formed as before, if it goes outside the first strip in any part, can have this portion blotted out, for we are concerned only with the region in which the approximation curves may lie as m increases from the value v. In this way, if we take the set of positive numbers €>€'>€"..., where Lt e(''>=0, »c->co and the corresponding positive integers we obtain the set of strips 0-, Digitized by Google 140 THEORY OF INFINITE SERIES, WHOSE TERMS Any strip lies within, or at most reaches the boundary of the preceding one, and their bi*eadth tends to zero as their number increases. Further, the curve y=zf{x) lies within, or at most reaches, the boundary of the strips. This construction, therefore, not only establishes the continuity of the sum of the series of continuous functions, in an interval of uniform convergence, but it shows tha,t the approximation curves, as the number of the terms increase, may be used as a guide to the shape of the curve for the sum right through the interval.* 70. A sufficient Condition for Term by Term Integration of a Series whose Terms ar« Continuous Functions of x. When the series of continuous functions v>^{x) + u^(x) + u^(x)+ ... is uniformly convergent in the interval (a, 6), we have seen that its sum, f(x), is continuous in (a, h). It follows that f{x) is integrable between Xq and ar^, when a'^x^l. We know that ;/ — cos .V 1 —2i/cos.v+i/' when |y|> by ^ and ^ respectively. Digitized by Google ARE FUNCTIONS OF A SINGLE VARIABLE 143 Also the series (1) converges uniformly for any interval of y within (-1,1) (§67). Therefore j[''r3^"^^+^!i«'y= -|c"8'«(''y"-'<'y. ^hen |.y| 1. 71. A sufficient Condition for Term by Term Differentiation. If the series u^(x)+u^(x)+u^(x)+ ... converges in (a, 6) arid each of its terms has a differential coefficient, continuous in (a, 6), and if the series of differential coefficients u/(aj)+<(aj)+<(ic)+... converges uniformly in (a, b), then f(x), the sum of the original series, has a differential coefficient at every point of (a, 6), and Let (i)Ax=f(x + Ax)--f(xl where x^^=x + Ax, Therefore ^(^)JA^±^lM, . But Lt 0(f) = 0(aj), since 0(a3) is continuous in (a, 6). Therefore f{x) has a diiFerential coefficient /'(a?) in (a, 6), and f(^) = {^) • = u^'(x) + u^'(x) + u^'{x)+ ..,. It must be remembered that the conditions for continuity, for term by term differentiation and integration, which we have obtained are only sufficient conditions. They are not necessary conditions. We have imposed more restrictions on the functions than are required. But no other conditions of equal simplicity have yet been found, and for that reason these theorems are of importance. It should also be noted that in these sections we have again been dealing with repeated limits (cf. § 64), and we have found that in certain cases the order in which the limits are taken may be reversed without altering the result. In term by term integration, we have been led to the equality, in certain cases, of I Lt Sn{x)dx and Lt | Sn{x)dx. Similarly in term by term differentiation we have found that, in certain cases, Lt r u (§i^(a±IpS^))-\ and Lt \u fi(^+M-^»('^))1 are equal. 72. The Power Series. The properties of the Power Series are so important, and it offers so simple an illustration of the results we have just obtained, that a separate discussion of this series will now be given. Digitized by VjOOQ IC ARE FUNCTIONS OF A SINGLE VARIABLE 145 I. If the series cIq + a j^* + a^^v^ + . . . is convergent foi^ x=jl'q^ it u ahsolutehj convergent for every value of x such that \x\<\xq\. Since the series is convergent for a?=a7o, there is a positive number M such that I a„.ro" | < M, when n ^ 0. But \a„7f^\ = \a^^-\x\?^ ". Therefore if — =c< 1, the terms of the series are less than the corresponding terms of the convergent series and our tfieorem follows. II. If the series does not conveiye for .^=^q, it does not converge for any value of X mch that | ^ | > | ^o I •- This follows from I., since if the series converges for a value of x^ such that I ^ I > I a?o I, it must converge for x=Xq. III. It follows from I. and II. that only the following three cases can occur: (i) The series converges for x=0 and no other value of x. E.g. l + \\x+2\x'+..., l+^+22jp2 + .... (ii) The series converges for all values of x. E.g. l+;p+g+.... (iii) There is some positive number p such that, when \x\ p, the series does not converge. E.g. x-\a^^}ia^-.... The interval —p when n^i\ But \a^-\ + \a„^^x»^' \ + .,. is less than the above for every point in M'M, including the ends M\ M, It follows that our series is absolutely and uniformly convergent in the closed interval {M\ M),* And the sum of the series is continuous in this closed interval. It remains to examine the behaviour of the series at the ends of the interval of convergence, and we shall now prove Abel's Theorem : t VI. If the series converges for either of the ends of the interval of converge^ice, the interval of uniform convergence extends up to and includes that pointy atul the continuity off(x\ the sum of the senes^ extends up to and includes that point. Let the series converge for x=p. Then, with the usual notation, pRn{p) = anp*'+an+ip'''-^ + ... + an+p-ip"-'^-', and |p/?„(p)|<€, when ^^v, for every positive integer jt?. *When the interval of convergence extends to infinity, the series will be absolutely convergent for every value of x, but it need not be uniformly con- vergent in the infinite interval. However, it will be uniformly convergent in auj'^ interval ( - 6, 6), where h is fixed, but may be fixed as large as we please. E.g. the exponential series converges uniformly in any fixed interval, which may be arbitrarily great, but not in an infinite interval [of. § 65]. f/. Math., Berlin, 1, p. 311, 1826. Digitized by Google VM+|»-1 ARE FUNCTIONS OF A SINGLE VARIABLE 147 But p/?„Cr) = a„.t» + cf„^i.i-'*+i-4- ... + «„+^_i.t-»-^^-i and the factors (- J , ('-) ,...(-) are all positive and decreasing, when 0<.r)[c"+'>-2-c»+'->] +,fl„(/))c-+'-'. But t^-c'^^, c'+'-c"*^, ... c"+*-' are all positive, and WRnW, U«»(p)|,...|,/f„(p)| are all less than c, when w ^ v. Therefore, when n^v^ I p/?„(.r) I < € [(c« - c"+^) + (c«+^ - c«+2) + . . . (c«+P-2 _ c«+p-i) + c"+^-i] < c, provided that < a? < p. But we started with \pR„(p) | < c, when m ^ v. Thus we have shown that the series is uniformly convergent in the interval Combining this result with V., we see that the series is now uniformly convergent in the interval -p+S^^^p, where 8 is any assigned positive number less than p. And it follows that f(^), the sum of the series, is con- tinuous in this closed interval. In particular, when the series converges at .v=pj Lt /(^)=/(p) a;— >p-0 = ao + «iP+«2P^+---- In the case of the logarithmic series, the interval of convergence is - 1 0. Hence Lt (l+.r)«=l+m+??^P^^ + ..., I.e. 2'^=l+m+—^ — -'+..., in both these cases. On the other hand, if we put ^= 1 in the series for (1 +^)~*, we get a series which does not converge. The uniformity of the convergence of the series is for the interval -l^x^l^ where I is any given positive number less than 1. VII. Term hy term differentiation and integration of the Power Series. We have seen in V. that the Power Series is uniformly convergent in any closed interval M\ i^[-/3+S^a?^/5-8] contained within its interval of convergence Z', Z [ - p < ^ < /i]. It follows from § VO that the series may be integrated term hy terra in the interval if, M ; and the process may he repeated any numher of times. We shall now show that a corresponding result holds for differentiation. From the theorem proved in § 71, it is clear that we need only show that the interval of convergence {-pp. When 1^1 l«2lp'^ i«3ip^--» *Cf. Chrystal, Algehra, Vol. 11. p. 131. Digitized by VjOOQ IC ARE FUNCTIONS OF A SINGLE VARIABLE 149 which are all less than some fixed number, since (by IV.) the series KI+l«ilp'+l«2lA>''+..- is convergent, it is clear that the series which we thus obtain, namely ,ai| + 2|aj^|+3|aja721 + ..., is convergent when | ^ | < p. We have yet to show that this last series diverges when \x\>p. If this series were convergent for je= | ^j |, where | :rj | > p, the same would hold. for the series ia,a;,\+2\a^,^\+S\a^,^\ + ..., and also for the series | a^^i | + | a^^i^ \ + | a^^^ I + . . • , since the terms of the latter are not greater than those of the former. But this is impossible, since we are given that the interval of convergence of the original series is - p < :f < p. Thus the series Oo+ai^ +a2^^ +a^ + ,., and the series '«! + 2a^ + 3a^^ + . . . , obtained by differentiating it term by term, have the same interval of con- vergence,* and it follows from § 71 that If /(.r)=ao+ai^+a2^+..., /(^)== ai + 2a2.r+..., when X is any point in the open intei'val -p Lt /(^) = ao + ai + <^2+---- «— »-i-o The above theorem of Abel's is a special case of the following : 00 * J^t the series S^h converge^ and o^, aj, a^^ ,..he a sequence of positive numbers stick tfiat ^ oq < ttj < cuj . . . . Then the series 2 cLrfi " **"* i^ uniformly convergent^ when /^ 0, and iff{t)=^ a„c- *»»«, we have Lt f(t) = 2«n • «->+0 „ Consider the partial remainder pRn(t) for the series 2 a„e-»r*'. * If we know that Lt -^ exists, this result follows immediately from the ratio-test for convergence, since in each series Lt 1'-^^|<1, if |a;l< Lt 1-^1. Digitized by Google 150 THEORY OF INFINITE SERIES, WHOSE TERMS Then , R„(t) = a^e - *n« + ce„+ie - *«+!« + . . . + a„+p_i 0. It follows that, when ^>0, < c, when n ^ v, for every positive integer p. And this also holds when <=0. Therefore the series 2 «ne - «»»* is uniformly convergent When t^O, Let its sum be f{t). Then Lt /(0=/(0) = 2 1/„.* <^+o 00 II. In Abel's Theorem and the extension proved above, the series 2 ctn ai*© supposed convergent. We proceed to prove Bromwich's Theorem dealing with series which need not converge.t In this discussion we shall adopt the following notation : «n = ^0 + 0^1 + <^2+ • •• + <^n > and we write *S„ for the Arithmetic Mean of the first 7i terms of the sequence *0) *1) *2J ••• • Thus 0^0 + ^1 + ^2+... +^n-1^0n-l^ n n 00 It can be shown (cf. § 102) that, if the series Za,. converges and its sum is S^ then, with the above notation, Lt Sn = S, But the converse does not hold. ♦If a©, Oj , .. . are functions of x and the series ]S a** converges uniformly to F{x) in 00 a given interval. Then it follows from the above argument that Lt 1ane-<^nt «->+0 converges uniformly to F{x) in this interval. iMath. Ann. , Leipzig, 65» p. '350, 1908. See also a paper by C. N. Moore in Btdl. Amer. Math, Soc.y 25> p. 258, 1919. Digitized by Google ARE FUNCTIONS OF A SINGLE VARIABLE 151 The sequence of Arithmetic Means may converge, while the sum 2 «n fails to converge.* Bromwich's Theorem. Let the sequence of Arithmetic Means S„ for the 00 se ries 2«« convetye to S, A Iso let Unbea ftmction of t with the folio wing proper- ties, token t>0'. (a) S w I A^w I < iTt (^' ^ ^"^ positive integers ; K, a positive\ p " \number independe^it of p^ q and t /' (/?) Lt «tt„=0, M— ♦CO (y) Lt ?*„ = !. 00 CO Then ttie series ^a„u„ conve^yes when t>0, and Lt 2a«««=«S- <->+0 We have 0*3 — 20*2 + 0"i = «3 — «2 = ^3» ©tC. n Thus 2 «„w„ = o-oWo + (o^i - 2(ro)Wi + (o-j - 2o-i + o-o) Wa + . • • + ((r„ - 2 ^2 J *3> ••• converges, and Lt Sn=iS. It follows that there is a number C, not less than \S\f such that |o"n| < (w + 1)(7 for every integer n. Also from {/3) it is clear that hi {a-ntin+i)= Jjt ((T„u„+2)= Lt (a-„_i?«„+i) = (2) Further, the series2(^ + l)|A^?^„] converges, since, from (a), the series 00 2 ^i I A^^n ! converges. ♦If «„ = (-.!)", n^O, it is obvious that Lt «S^„ = i, but the series 2 a,» is not con- it— > 00 eo vergent. But see Hardy's Theorem, § 102, II. When Lt 5„=/S\ the series 2an is n->oo often said to be " summahle (CI)" and its sum (CI) is said to be S. For a discussion of this method of treating series, due to Ces^ro, reference may be made to Whittakcr and Watson, Course of Modem Analysis, p. 155, 1920. Also see below, §§ 101-103 and § 108. t A^« is written for (tt„ - 2«„+i + tt„+2). Since all the terms in the series 00 ^n\Ah(n\ are positive, this condition (a) implies the convergence of this series. Digitized by VjOOQIC 152 THEORY OF INFINITE SERIES, WHOSE TERMS Also I (Tn^^n \ < C(u + l)'\ A'^l^n |. Therefore the series 2 a-n^^Un converges absolutely. It follows from (1) and (2) that 2a„w«=2 «i = ^2 — • • • = ^> we have (r„=;i + l and 2^=2 (?i + l)A%„ (4) Thus, from (3) and (4), ian«„-^tt0=i(o-n-(w+l)lS)A2w„ '....; (5) Now U -^ = S. Therefore, to the arbitrary positive number e, there corresponds a positive integer v such that |n-|-l I 4K^ — Thus I o-„ - (» + 1) ^S" I < j^ (m + 1), when ?igv. Also \(r„\<{n + iyC, for every positive integer, and \S\^C. It follows, from these inequalities and (5), that |2an«n-^Wo|^rf(cr„-(w+l)>S)A2w„| + |i(o-„-(n-Hl)^)A2w„| V <2C2'(»+l)|A%,.l+j^^i(«+l)|A%„| (6) 4A V But i (7i + l)|A2wn|<2 2n|A2M„| <2Jr,by(a) (7) And Lt A-Mh=0, since Lt t^«=l,by(y). «-*+0 t— t'+O It follows that, V being fixed, there is a positive number tj such that l^^""l'^ 2v(v + l)C^ > when 0< t^rj and n^v- 1 (8) Thus from (6), (7) and (8), we see that eo 1 2 a«Wn - Sicq] < ^€ 4- Jc < f, when 0< t^Tj. Therefore Lt (2a«Wn-^w«)=0. t-y+O 00 And, finally, Lt 2«nW«=^, e->+0 since, from (y), Lt ^0=1- <->+o „ III. Let the sequence of Arithmetic Means for the series 2«n converge to Sy and let Un be e-"' {w* e~"^). Then the series ^o.nU„ converges^ when t>0, and 00 Lt 2«nW« = AS^. <->.+0 Digitized by Google ARE FUNCTIONS OF A SINGLE VARIABLE 153 This follows at once from Bromwich's Theorem above, if e-^ (or e""**) satisfy the conditions (a), (/?) and (y) of that theorem. It is obvious that (j8) and (y) are satisfied, so it only remains to establish that (a) is satisfied. (i) Let t^„=e-"', t>0. Then A*m„ = w„ - 2m„+i + Wh+2 =^-«'(l-e-7'*. Therefore l^hi^ is positive. Also 2w| A^t^nl « 2w A2w„ I 1 =w«„+2-(n+l)w„+i4-Wi. 00 Therefore 2^|A%„| -e~\ and the condition (a) is satisfied. 1 (ii) Let Wn=e-^"% ^>0. In this case A%„=e-<«+'''* (A:(n+dff - 2t\ whereO<0<2.* Therefore the sign of A%„ depends upon that of (4(n-^0)H^ - 2/). It follows that it is positive or negative according as Also 0^+^)>;7i)' -^^'^ ^>^y and (,,4-0)<-^,when.4-2<-^. Therefore A%„ cannot change sign more than three times for any positive value of t. But it follows at once from the equation wA2i*„= we-<«+e)'^' (4(w 4- eyt'^ - 2t) that a positive number, independent of ^, can be assigned such that wjA^t/,,! is less than this number for all values of n. Hence K can be chosen so that the condition (a) is satisfied, provided that the sum of any sequence of terms, all of the same sign, that we can choose from 2wA%„, is less in absolute value than some fixed positive number for all values of t. $ Let 2*iA^w» be the sum of such a set of consecutive terms. r Then we have tnAhin = re~''' - (r - l)e-^r+m - (« + 1) e-(«+i)»< + se-^'+^y\ ♦ This follows from the fact that where 0<^<2, provided that f{x), /'(ic), f"{x) are continuous from xtox+2h. (Cf. Goursat, loc. cit., T. I., §22.) Digitized by Google 154 THEORY OF INFINITE SERIES, WHOSE TERMS which differs from ;.(e-ra< _ g-(r+l)3t) _ (^ ^ l)(e-C+l)»« - g-('+a>*«) by at most unity. But, when ?i is a positive integer and ^>0, 00. This follows at once by making slight changes in the argument of II. The theorems proved in this section will be found useful in the solution of problems in Applied Mathematics, when the differential equation, which corresponds to the problem, is solved by series. The solution has to satisfy certain initial and boundary conditions. What we really need is that, as we approach the boundary, or as the time tends to zero, our solution shall have the given value as its limit. What happens upon the boundaries, or at the instant ^=0, is not discussed. (See below § 123.) 74. Integration of Series. Infinite Integrals. Finite Interval. In the discussion of § 70 we dealt only with ordinary finite integrals. We shall now examine the question of term by term integration, both when the integrand has points of infinite discontinuity in the interval of integration, supposed finite, and when the integrand is bounded in any finite interval, but the interval of integration itself extends to infinity. In this section we shall deal with the first of these forms, and it will be sufficient to confine the discussion to the case when the infinity occurs at one end of the interval («, 6), say a:=b. ' GO I. Let Wi(^), ti2(^), ...be continuous in (a, b) and the series ^nn{x) converge I uniformly tof(a;) in {a, b). ^ i Digitized by Google ARE FUNCTIONS OF A SINGLE VARIABLE 155 A Uo let g {x) have an infinite discontinuity atx'=b and \ g (a) dx he absolutely convergent* * rb 00 n Then / f(x)g{x)dx='2l \in{x)g{x)dx, J a 1 Ja 00 From the uniform convergence of 2 ^nW in (<»> ^)> w© know that its sum f{a;) is continuous in (a, h\ and thus bounded and integrable. Also / f(x)g(x)dx is absolutely convergent, since l g(x)dxisso (§61, VI.). liet / \g{x)\dx=A, Ja Then, having chosen the positive number c, as small as we please, we may put Ax) = Sn(a:)+Rn{^\ where | Rn{^) I < t » when n ^ v, the same v serving for all values of x in (a, b). But / f(x)g(x)dx and / Sn{^)g{x)dxhoi}i exist, Ja Ja It follows that / Rn{3o)g{x)dx also exists, and that Ja / f(^)9(^)f^= Sn(x)g(x)dx+j Rn{x)g(x)dx, Ja Ja Ja Thus \Ke see that \ rb H rb I 1 r* I 1 / f{^)9{^)d^-^\ Ur{x)g{x)dx\ = \ \ R„{x)g{x)dx\ \Ja 1 Ja \ \Ja " <€, when n-^Vy » rb which proves that the series 2 / Un(x)g(x)dx is convergent and that its sum is / f{x)g(x)dx. Ex. This case is illustrated by j^ log X log (l+x)dx=f,(- 1)"-!^ ^ log X dx =?4h:i?' '^'^ Jo -^^^^^^^^^-C^hTi)-.' T^ ^ L/i w+1 (n + lfJ = 2-2 log 2 - - TT^, using the series for ja'^^'^ * It is clear that this proof also applies when g{x) has a finite number of infinite discontinuities in (a, h) and / g{x)dx is absolutely convergent. t Cf. Oarslaw, Plane Trigonometry {*2nd Ed,), p. 279. Digitized by Google 156 THEORY OF INFINITE SERIES, WHOSE TERMS Here the series for log(l +.r) converges uniformly in 0^;t'= 1, and converges absolutely (as a matter of fact log^ is always of the same sign in 0<^^1), while |logA'|->QO as a;->0. On the other hand, we may still apply term by term integration in certain cases when the above conditions are not satisfied, as will be seen from the following theorems : 00 II. Let Ui{a;\ «2(^)> ••• ^ continuous and positive and the series S^C-^^) converge uniformly/ to f{x) in the arbitrary interval (a, a), tohere a < a<6. Further, let g{x) he positive, hounded and integrdble in (a, a). /•ft 00 rt Then / f{x)g{x)dx=^\ tin(^)g(x)dv, provided that either the integral I f{x)g{x)dx or the series 2/ Un{x)g{x)dx converges. Let us suppose that / f{x)g{x)dx converges. Ja In other words, we are given that the repeated limit r6-f n Lt [ Lt ^Ur{x)'\g{x)dx exists. f->0 J a n->oo 1 Since the functions u^(x\ u^ix), ... are all positive, as well as g{^\ in (a, a), from the convergence of / f(x)g(x)dx there follows at once the convergence e Ja \ Ur{x)g{x)dx (r=l, 2, ...). Ja Again, let f{x) = u^ {x) + %(:r) -H . . . + Un{x) 4- i2,»(A). Then / Rn{x)g{x)dx also converges, and for every positive integer n Ja rf(x)g{x)dx-''2rur(x)g{x)dx=^ {' Rn{x)g(x)dx, (1) Jck 1 J a •'» But from the convergence of / f(x)g{x)dx it follows that, when the arbitrary positive number € has been chosen, as small as we please, there will be a positive number ^ such that 4 fortiori, 0

0.* 75. Integration of Series. Infinite Integrals. Interval Infinite. For the second form of infinite integral we have results corresponding to the theorems proved in § 74. I. Let Ui(x\ U2{x\ ... 6c continuous and hounded in oc^a^ and let the series 00 ^Un{x) converge uniformly tof{x) in .r^a. Further^ let g{x) he hounded and integrahle in the arbitrary interval (a, a), where a0, Theorem III. can be used at once. If 0>p> - 1, the interval has to be broken up into two parts (0, a) and (a, 1). In the first we use Theorem I. and in the second Theorem III. Or we may apply §72, VI. Digitized by VjOOQ IC ARE FUNCTIONS OF A SINGLE VARIABLE 159 But it is often necessary to justify term by term integration when either I Ii7(-*^)l ^'** is divergent or 2wnW t^an only be shown to converge uniformly /« 1 in the arbitrary interval (a, a), where a can be taken as large as we please. Many important cases are included in the following theorems, which correspond to II. and III. of § 74 : 00 II. Let Wi(ar), 1*2 W> ••• ^ continuow and positive and the series ^u„(x) con- 1 verge unifoi^mly tof{x) in the arbitrary interval (a, a), where a may be taken as large as we please. Also let g{x) be positive^ bounded and integrate in (a, a). Then rf{^)g(^) dv==2 run{x)g{x) dx, J a iJa f{x)g(x)dx or the series'^ I Un{x)g{x)dx converge. Let us suppose that / f{x)g{x)dx converges. In other words we are given that the repeated limit /*a n Lt / [ Lt '^Ur{x)'\g{x)dx Qx\s>ta, a— ►x a n-^oo 1 Since the terms of the series ^Ur{x) are all positive, as well as g{x\ in A'^a, from the convergence of / f(x)g(x)dx there follows at once that of jur(x)g(x)dx (r=l, 2, ...). Again let /(^)= ^W + W2(^) + • • • + «n W + //«(^). Then / R„(x)g(x)dv also converges, and for every positive integer n / /(•2^)5^Wfl^-2/ Ur{x)g{x)dx= R„{x)g{x)dr (1) Ja 1 Ja J a But from the convergence of / f(x)g(x)dx, it follows that, when the Ja arbitrary positive number € has been chosen, as small as we please, there will be a positive number a such that r € A fortiori, < / Rn(x)g{x) dx < 3> and this holds for all positive integers 7i. With this choice of a, let the upper bound of g{x) in (a, a) be M. The series ^Un(x) converges uniformly in (a, a). Digitized by Google 160 THEORY OF INFINITE SERIES, WHOSE TERMS Keeping the number c we have chosen above, there will be a positive integer v such that the same v serving for all values of a? in (a, a). Thus 0« Also we know that Lt Sn{x) exists, and we denote it by 0{n). We shall now show that Lt F{x) and Lt 0{n) both exist, and that the two limits are equal. * ""^^ From this result our theorem as to term by term integration will follow at once. I. To prove Lt F{x) exists. Since Lt *S^«(^) converges uniformly to F{x) in .r^a, with the usual notation, we have |^(,)_^„(,)|<1, ^^en n^v, the same v serving for every x greater than or equal to a. Choose some value of n in this range. Then we have Lt Sn{x) = 0{n). Therefore we can choose X so that l^«(.r")-^„(^)|<|, when x">af^X>a. C.I L Digitized by Google 162 THEORY OF INFINITE SERIES, WHOSE TERMS But \F(x')-'F(x')\ <€, when af'>af^X>a. Thus F(x) has a limit as .r->oo . II. To prove Lt Q{n) exists. Since Lt Sn{x) converges uniformly to F{x) in x^a^ n— >-ao \Sn"{x)-'Sn'{x)\<\, when w">w'^v, 3 the same v serving for every x greater than or equal to a. But Lt Sn{x)=G{n), Therefore we can choose Xi, X^ such that I Q{rt!) - Snt(pc) I < ^ , when ;r ^ ^i > a, a. Then, taking a value of x not less than X^ or Zj, ^3^3^3 * <€, when n">n'^v. Therefore Lt (?(n) exists. III. To prove Lt J^(^)= Lt (7(w). Since Lt (?(») exists, we can choose Vj so that 2 I t('r{x)dx <-, when w^Vp In+ija 3 Again 2 I Ur(x)dv converges uniformly to F{x) in ^ga. 1 Ja Therefore we can choose V2 so that I 00 Tx ^ I 2 I Ur{x}dx <-, when w = V2, in+ijo 3 the same Vj serving for every x greater than or equal to a. Choose V not less than Vj or Vg- poo „ From the convergence of the integral I 2 Ur(x) dx, we can choose X so that Ja 1 If**' I € I X'u>r(x)dx\<-f when x^X>a. \jx 1 I 3 Digitized by Google ARE FUNCTIONS OF A SINGLE VARIABLE 163 But 2 I Ur(a;)dv-F(a;)\ I 1 Ja I \jx 1 v+\Ja v+lja \C'^y I I ** f ** I I 00 C« ^1 2w^(.^)e^.r + 2 I Ur{x)dx\-\-\ 2 I t«^(.r)c/.r \}x 1 I Iv+lja i U+lJa ^3 +3 "^3' < €, when x^X>a. Therefore Lt J^(^)=2| Ur{x)dx= Lt 0{n). IV. But we are given that the series converges uniformly tof(x) in any arbitrary interval (o, a). Therefore we have / f(x)dx=\ tCi(x)dv+ I U2(x)dv+,,, in .rga. Ja Ja Ja Thus, with the above notation, F{x)=jj{x)dx. It follows from I. that / f(x)dx converges, and from III. that Ja rf{.v) dv= j Ui(x) dx+ I U2(x) dx+.., . Ja Ja REFERENCES. Bromwich, loc. city Ch. VII.- VIII. and App. IIL, 1908. De la ValliSe Poussin, loc, cit.^ T. I. (3* 6d.), Ch. XI. DiNi, Lezioni di Analisi Infinitenmale^ T. II., 1* Parte, Cap. VIII., Pisa, 1909. GouRSAT, loc, cit., T. I. (3' ^.), Ch. II. HoBsoN, loc. ciL, Ch. VI. KowALEWSKi, loc. cit.y Kap. VII., XV. Osgood, loc. cit., Bd. I., Tl. I., Kap. III. PiERPONT, loc. dt.y Vol. II., Ch. V. Stolz, loc. cit.^ Bd. I., Absch. X. And Prinoshbim, "Grundlagen der allgemeinen Funktionenlehre," Enc. d, imk. Wiss., Bd. II., Tl. I., Leipzig, 1899. - Digitized by Google 164 THEORY OF INFINITE SERIES, WHOSE TERMS EXAMPLES ON CHAPTEB V. UNIFORM CONVERGENCE. 1. Examine the convergency of the series ?(?w: + l)[(« + l)^+l][(n + 2).r-Hl]' ^ = ^' and by its means illustrate the effect of non-uniform convergence upon the continuity of a function of x represented by an infinite series. 2. Prove that the series is convergent for all values of x, but is not uniformly convergent in an interval including the origin. 3. Determine whether the series « 1 y t 1 /l3 + W*^2 is uniformly convergent for all values of x, 4. Find for what values of x the series 2 w« converges where of* — .r~**~* «« = (x^^x-^)(ar^^+xr^^) 1 1 (a?-l)(a?"+^-»*) (^-l)(a?»+H:r-t»*i>)' Find also whether the series is (i) uniformly convergent through an interval including +1 ; (ii) continuous when x passes through the value +1. 5. Discuss the uniformity or non-uniformity of the convergence of the series whose general term is ^ l-(l4-:g)" 1-(1+^)"-^ 6. Let «o+^i + *" be an absolutely convergent aeries of constant terms, and let /o(^))/i(^),... be a set of functions each continuous in the interval a~.^=j8, and each comprised between certain fixed limits, A^fM^B, 1 = 0,1,..., where il, B are constants. Show that the series «o/o(^) + «i/i(^)+... represents a continuous function of x in the interval a^x^j3. Digitized by VjOOQIC ARE FUNCTIONS OF A SINGLE VARIABLE 165 7. Show that the function defined by the series I ^ is finite and continuous for all values of x. Examine whether the series is uniformly convergent for all such values. 8. Show that if ^to(^) + Wi (^) + . . . is a series of functions each continuous and having no roots in the interval a^a;^by and if h»+iM|0, and if the series satisfies the if-test (§67), then the original series may be integrated term by term from a to 00 . 23. Show that the series can be integrated term by term between any two positive finite limits. Can this series be integrated term by term between the limits and 00 ? Show that the function defined by the series cannot be integrated between these limits. 24. Show that the function defined by the series (1+^)3^(2+^)3 can be integrated from to 00 , and that its value is given by the term by term integration of the series. ' Ex. 21-27 depend upon the theorems of §§ 74-76. . Digitized by VjOOQ IC 168 THEORY OF INFINITE SERIES 26. Prove that 'r^cla;=- 1- + -1--... (a>0). Jo 1+0? a a+l a+2 ^ Explain the nature of the difficulties involved in your proof, and justify the process you have used. 26. By expansion in powers of a, prove that, if | a | < 1, re-(l-6— )^=log(l+a), JO * r do: tan~*(asina?)-; — =i7rsinh~^a, 'o 'sino? ^ ' examining carefully the legitimacy of term by term integration in each case. .( 27. Assuming that Jn(^')=^' ,'M'J)- (nlf r 1 show that j^ e-^J^(hx)dx=-j^^^^-^y when a > 0. Digitized by Google CHAPTER VI. DEFINITE INTEGEALS CONTAINING AN ARBITRARY PARAMETER. 77. Continuity of the Integral. Finite Interval. In the ordinary definite integral I (x, y)dx let a, a be constants. Then the integral will be a function of y* The properties of such integrals will be found to correspond very closely to those of infinite series whose terms are functions of a single variable. Indeed this chapter will follow almost the same lines as the preceding one, in which such infinity series were treated. I. // 0(05, y) is a continuous function of {x, y) in the region a^x^a\ b^y^b\ then I (p{x,y)dx is a continuous function ofy in the interval Ja (b,b'). Since 0(a;, y) is a continuous function of {x, j/)!, as defined in § 37, it is also a continuous function of x and a continuous function of y. Thus (x,y + ^y)-'4>(x,y)\{x,y + ^y)dx. Then f{y+Ay)-Ay)= f" [ y) ^(^) d^ is <^ continuous function of y in (6, 6'). Ja Let /(2/)= (p{x,y)\lr{x)dx. Ja The integral exists, since the product of two integrable func- tions is integrable. Also, with the same notation as in I., Ay+^y)-Ay)= \ [4>{^yy+i^y)-i>{^,y)\^{^)dx, Ja Let M be the upper bound of | ^{x) \ in (a, a'). Then |/(t/ + A3/)-/(2/)|(Xy y) dxy wftere (x, y) is a continuous func- Ja tion of (x,y) in a = x = a\ b = y = fc', and ^ exists and satisfies the same condition. *This follows from the theorem on the uniform continuity of a continuous function (cf . § 37, p. 75). Digitized by Google AN ARBITRARY PARAMETER 171 Then f\y) exists and is equal to I ^ dx. Since ^ is a continuous function of (x, y) in the given region, to any positive number e, chosen as small as we please, there cor- responds a positive number tj, such that, with the usual notation. ^0(^>y + Aj/) d(f>(x,y) dy dy <€, when \Ay\^fi, the same rj serving for all values of x in (a, a'). Let Ay satisfy this condition. Then f(y+Ay)'-f(y) ^ p' 0(^>y + Ay)-^(a;,y) ^ j^^o Ay J a Ay Ja oy ^ p' M^da: + f p^(^>J/+^Aj/) d(x,y) l ^^^ ia 'dy JaL dy dy J ' Thus we have | /(y+Ay)-/(y) ^^^^^1 I pT 30fa,y + eA2/) ^0(^>y) l^^ Ay Ja^j^ ! iJaL 32/ 3y J <(a'— a)e, when \Ay\ = ri. And this establishes that Lt / /(y + Ay )::/(y)| ^^.^^g ^^^^ j^ f^'3<^ Ay->0 I At/ J equal to I ^ daj at any point in (6, 6'). H. Let f(y)'= I (x,y)\lr(x)dx, where ^(x, y) and -^ are cw? J a <^2/ in I., and y{/{x) is hounded and integrable in (a, a'). Then f(y) exists and is equal to I ^\[A(x)dx. Let the upper bound of | V^(i») I in (a, a') be if. Then we find, as above, that |teM=/iL)_£-^^,.,^ (x,y) is a continuous J a function of (x,y) in a'^x = a\ b = y = 6', and \[r(x) is bounded and integrdble in (a, a'). Then f(y)(iy=\ d(x,y)dy. Then we know that — = ^(a;, y) [§ 49], and it is easy to show that ^(x, y) is a continuous function of (x, y) in the region a^x = a\ b = y = b\ Now let g{y) = I *(x, y) \{r{x) dx. From § 78, we know that : 'C= 4>{x,y)ylr(x)dM, Also g\y) is continuous in the interval (6, 6') by § 77. > Therefore I g\y)dy=\ dy \ ^{x,y)\lr{x)dx, where j/q, y are any two points in (fc, b'). /Google Digitized by * AN ARBITKARY PARAMETER 173 Thus I dy \ (p{Xy y) ^(a?) dx =o{y)-9{yo) = [*(aJ,2/)-*(iK,2/o)]V^(^)^ Ja = daj (l>{x,y)\ff{x)dy. Thus we have shown that we may invert the order of integra- tion with respect to x and y in the repeated integral \ dx \ F{x,y)dy, when the integrand satisfies the above conditions; and in par- ticular, since we may put t^(a5) = l, when F{Xy y) is a continuous function of {x, y) in the region with which the integral deals. 80. In the preceding sections of this chapter the intervals (a, a") and (6, 6') have been supposed finite, and the integrand bounded in a^x^a\ b^y^b\ The argument employed does not apply to infinite integrals. For example, the infinite integral /(3/) = J ye-'^dx converges when y = 0, but it is discontinuous at y = 0, since /(y) = l when 3/>0, and /(0) = 0. Similarly sin Try e - ^ **"' "^ dx Jo converges for all values of y, but it is discontinuous for every positive and negative integral value of i/, as well as for 2/ = 0. Under what conditions tlien, it may be asked, will the infinite integrals pw z.^' I F{x, y) dx and I F{x, y) dx, J a J a if convergent when b = y = b\ define continuous functions of y in (6, 6') ? And when can we differentiate and integrate under the sign of integration ? Digitized by VjOOQ IC 174 DEFmiTE INTEGRALS CONTAINING In the case of infinite series, we have met with the same questions and partly answered them [of. §§68, 70, 71]. We proceed to discuss them for both types of infinite integral. The discussion requires the definition of the form of convergence of infinite integrals which corresponds to uniform convergence in infinite series. 81. Uniform Oonvergence of Infinite Integrals. We deal first with the convergent infinite integral /•OO F(x,y)dx, Ja where F{x,y) is bounded in the region a^x'^a\ b^y^b\ the number a' being arbitrary. /•OO I. The integral I F{x, y) dx is said to converge uniformly to its Ja value f{y) in the interval (6, b"), if, any positive number e having been chosen, as small as we please, there is a positive number X such that f(y)-\''F{x,y)dx Ja x' = X, the same X serving for every y in (b, b'). The proof that II. forms a necessary and sufficient condition for the uniform convergence of the integral, as defined in I., follows exactly the same lines as the proof in § 66 for the corre- sponding theorem in infinite series. Digitized by Google AN ARBITRARY PARAMETER 175 Further, it will be seen that if I F(Q^y y)dx converges um- fomdy in (6, b% to the arbitrary positive number e there corre- sponds a positive number X such that ir F{{c, y) dx <€, when x = Xy the same X serving for every y in (b, b'). The definition and theorem given above correspond exactly to those for the series u^(x)+u^(x)+.., , uniformly convergent in a = aj = 6; namely, I-Kn(a5)| when O^y^b. ^ ^^ Jo s/{x+yy ^ (ii) I xy^e-^dx, when OSi/ Ja the same 9j serving for every y in (b, b'). And, from this definition, the following test for uniform con- vergence can be established as before : II. A necessary and sufficient condition for tlie uniform con- vergence of the integral I F{x, y) dx in the interval (6, 6') is tliat, Ja if any positive number e has been chosen, as small as we please, there shall be a positive number tj such that \['^ F{x,y)dx <€, when 0<^a?' = a and b = y = b\ I F(x, y)dx\=\ fi(x) dx, and, from (iii), there is a positive number X such that 1 fji{x)dxa;'=Z. These conditions will be satisfied if x^^F(x, y) is bounded when aj = «, and b^y' = b' for some constant n greater than 1. C. I M Digitized by VjOOQ IC 178 DEFINITE INTEGRALS CONTAINING Corollary. Let F{x,y) = ff>(x,y)ylr{x\ where 0• e ^dx converges uniformly in 0<^^ F, where Fis an arbitrary positive number. Ex.3. r?^rf.^, r'^^:?cf.r, f^-?i^c^r, r?i5-^rf.r converge uniformly for all values of y, where n > 0. II. Let (j> (x, y) be bounded inx = aj b = y^b\ and a monotonic function of x for every y in (&, &'). Also let ^{^(x) be bounded and not change sign more than a finite number of tirnes in /•oo the arbitrary interval (a, a')* and let I \f/(x) dx exist Ja Then I (Xy y) V^(aj) dx converges uniformly in (6, 6'). Ja This follows immediately from the Second Theorem of Mean Value, which gives, subject to the conditions named above, {x, y) ylr{x) dx = 0(aj', y) '^{x) dx + (/>{x'\ y) yj^ix) dx, Jx' Jx' J$ where ^ satisfies a < a;' = ^ = x'\ /•oo But ^(x, y) is bounded in cc = a and b = y = b\ and I \[r(x) dx converges. "* Thus it follows from the relation If (x,y)'^{x)dx\ = (x'\y)\ I {'yfr(x)dx \ \ Jx' I I J f /•oo that I ^(Xy y) \lr{x) dx converges uniformly in (6, 6'). Ja * This condition is borrowed from the enunciation in the Second Theorem of Mean Value, as proved in §50. If a more general proof had been given, a corresponding extension of II. would have been treated here. Digitized by Google AN ARBITRARY PARAMETER 179 It is evident that 'y[r(x) in this theorem may be replaced by V^(^> y)y if I yl/'ix, y) dx converges uniformly in (6, 6'). J a fsill 3S l^ COS J7 e~** dx^ \ e-** '- d.v(a>0) converge uniformly in y ^ 0. III. Let ^(x, y) be a nfionotonic function of x for each y in . (6, 6'), and tend uniforTuly to zero as x in^creasesy y being kept constant Also let \fr{x) be bounded and integraljle in the arbi- trary interval (a, a'), and not change sign more than a finite number of times in such an interval. Further^ let I ^{^{x) dx be hounded in x^a, without converging as x-xx) . " /•« Then I {x,y)^{x)dx is uniformly convergent in (6, 6'). This follows at once from the Second Theorem of Mean Value, as in II. Also it will be seen that >/^(cc) may be replaced by ylr(x,y)yii I '^(x,y)dx is bounded in x = a and b = y = b'J^ Ex. 1. / e~^Binxd.v, l e~*»' cos .rdlr converge uniformly in y^yo> 0- fsin rv /** v sin xv — ~d.v and / frr/ ^^ both converge uniformly in y^yQ>0 and y'S-yQ<0, It can be left to the reader to enunciate and prove similar theorems for the second type of infinite integral I F{x,y)dx. Ja The most useful test for uniform convergence in this case is that corresponding to I. above. Ex.1./ a*~^dxy I ^p^^'^e'^ci?^ converge uniformly in l>yg^o >0- -^r^ dv converges uniformly in 0{x, y) is continuous as above, and yf/^ix) is bounded^ and integrable in the arbitrary interval (a, a'). /•oo Also let I F{x,'y)dx converge uniformly in (6, 6'). Jo Then f{y) is a continuous function of y in (6, b'). Let the positive number e be chosen, as small as we please. Then to c/3 there corresponds a positive number X such that ii: F(x,y)dx when x = JC, i the same X serving for every y in (6, 6'). But we have proved in § 77 that, under the given conditions, X F(Xy y) dx is continuous in y in (6, 6'). Therefore, for some positive number i;, F{x,y + £^y)dx-\ i^(ir,2/)da; <^, when |Ay| = i;. Also f{y)= I ^{^^ y)dx+[ F(x, y)dx. Thus f(y + Ay) ^f(y) = [ | V(aj, y + Ay) dx - 1 V(a:, y) dx'] + 1 F(x,y + Ay)dx-^ F{x,y)dx. I poo I F{x, y)dx <|, Jx o and F{x,y + Ay)d^ -Qo J yo J a provided we can show that the limit on the right-hand exists. But dy F{x,y)dx=\ dy F{x,y)dx"\ dy F{x,y)dx. Thus we have only to show that Lt \ dy\ F{x,y)dx = 0. Of course we cannot reverse the order of these limiting pro- cesses and write this as \ dyU I F{x,y)dx, JVo X— >-oo Jx for we have not shown that this inversion would not alter the result. /•CO But we are given that I F{x,y)dx is uniformly convergent in (6, 6'). * Let the positive number e be chosen, as small as we please. Then take €/(6' — &). To this number there corresponds a positive number JT such that l£ F(x, y) dx the same X serving for every y in (6, 6'). (x,y)\lr(x), where (x,y) is continuous as above, and \]^{x) is bounded and integrable in the arbitrary interval (a, a'). Also let F(x, y) have a j)artial differential coefficient ^ which satis- fies tJce same conditions. Then, if the integral I F(x, y) dx converges to f(y), and the integral I — dx converges uniformly in (6, 6'), f{y) has a dif- j a ^y ferentictl coefficient at every point in (6, 6'), and dF /'(2/)=r Ja ^ dx, -dy We know from § 84 that, on the assumption named above, ?)F " dy is a continuous function of y in (b, V). a ^ Let g{y)=\_^fyd^. Then, by §85, Tj^y)dy=\y^f^fydy, where 2/0 > Vi are any two points in (6, 6'). Let 1/0 = 3/ and y^^y + ^y. Then g{y)dy=\ [F(x,y + Ay)^F(x,y)]dx. Jy J a Therefore gH) Ay =f(y + Ay) -f(yl poo where y^^:Sy + Ay and f{y)=\ F{x,y)dx, Ja Digitized by VjOOQIC AN ARBrrRARY PARAMETER 183 Thus g^^yJ^y+Avlzm. if But Lt gr(^) =gf (2/), since gr(fl5) is continuous. Ay->0 It follows that f(y) is difFerentiable, and that where y is any point in (6, 6'). 87. Properties of the Infinite Integral rF(x,y)dx. The J a results of §§ 84-86 can be readily extended to the second type of infinite integral. It will be sufficient to state the theorems without proof. The steps in the argument are in each case parallel to those in the preceding discussion. As before, the region with which we deal is a^x^a\ b^y^b\ I. Continuity of I F(x,y)dx. Ja Let f(y)=\ F{x,y)dXy where F(x,y) has points of infinite Ja discontinuity on certain lines (e.g.,x = aj^, a.^, ... a.n) between x=a andx = a\ and is either a continuous function of (x, y), or the product of a continuous function (f>(Xy y) and a bounded and integrable function V^(aj), eaxept in the neighbourhood of the said lines. Then, if I F{Xf y)dx is uniformly convergent in (6, 6'), f{y) is J a a continuous function of y in (6, 6'). fa' II. Integration of the Integral I F{^,y)dx. Ja Let F(x, y) satisfy the same conditions as in I. Cy fa' ftt' fy TJien dy\ F{x,y)dx=\ dx\ F(x,y)dy, JVq Ja J a J 1/0 where y^, y are any two points in (6, 6'). III. Differentiation of the Integral I F{x, y) dx. Ja fa' Let f{y) = I F(x, y) dx, where F(x, y) has points of infinite J a Digitized by VjOOQ IC 184 DEFINITE INTEGRALS CONTAINING discontinuity on certain lines (e.gr., a; = ai, ttg, ... a^) between x = a and x=a\ and is either a continuous function of {x, y), or the product of a continuous function (f>{x, y) and a bounded ctnd integrable function ^(x), except in the neighbourhood of the said lines. Also let F(Xy y) have a partial differential coefficient^ , which satisfies the same conditions. Further, let I F(Xy y) dx converge, and \ —dx converge uniformly in (b, 6'). ThenfXy) exists and is equal to \ K-dx in (6, 6'). 88. Applications of the preceding Theorems. • Ex.1. To prove. C^^dv^^. Jo a? 2 (i) Let F{a)=j^e'<^^-^€Lv (agO). This integfral converges uniformly when a^O. (Cf. § 83, III.) For e - ^jx is a monotonic function of x when x > 0. Thus, by the Second Theorem of Mean Value, jx' X X' Jx' X' i^ where < a?' ^ ^ ^ x". Therefore / ^\nxdx\ — - ,-\ I siTLxdx\-\ ^ / ^inxax] \jx' X \— X ■ \Jx'. \ X \J^ I < 4 ^ , since | sin a; g?^; ^ 2 for all values of p and q, of \Jp I <— , since a^O. X It follows that I {"^'e - «^ ^Hif dx\<€, when x"> x' g X, \Jx' X \ provided that X> 4/€, and this holds for every a greater than or equal to 0. This establishes the uniform continuity of the integral, and from § 84 F{a) must be continuous in a^O. ,sin^^, Thus F{0)= Lt / .-- ^^ A X Jo X a-^Jo X /Google Digitized by * AN ARBITRARY PARAMETER 185 (ii) Again, the integral j e-^sina: dx 18 uniformly convergent in a ^ a^ > 0. This follows as above, and is again an example of § 83, III. Thus, by § 86, when a > 0, JQ Oa\' X J = - j e-<*^smxdx. d ■ But ^e-«^(co8A'+asina;)= -(a2+ !)€-«•« sin a*. Therefore f e-<^^nxdx= --^,. Jo a'*+l Thus F'U\=^-1 And j^(a)=-tan-ia+|, '2' ^=1- since Lt F{a)=0* a— XJo It follows from (i) that / < 'Jo X Ex. 2. To prove rcoscuF , TT , r* sin cur , tt ^^ \ / -^r^x * If a formal proof of this is required, we might proceed as follows : Let the arbitrary positive e be chosen, as small as we please. Since / e-"* ^^^^dx converges uniformly in a^O, there is a positive number f such that \re---^-}^dx\p>0 (cf . p. 202). Therefore I / ^ e-«« ^^^ dx I < 2ire-*'». I Jxo X I Thus we can choose A so large that >o a* I 3 It follows that I re-«-?^dx| /Google . Digitized by * 186 DEFINITE INTEGRALS CONTAINING The integral is uniformly convergent for every a, so that, by § 84, /(a) is continuous for all values of a, and we can integrate under the sign of integration (§85). (ii) Let ^(^)— I f{o)da. Then <^(a) is a continuous function of a for all values of a (§ 49). Also <^(a)=fx'^\y rx" X x' f^ x" f^' I ,-T— wsincur(i^=:r-; — tt, / sin ax dx+:r- — tto I aiaaxdx, where x^^^^x". It follows that / ---- A sm ax dx = — rr— — ^ / sin x dx+ —t—- — jy^. / sm x dx. X Therefore / , ^ .^ sinaxdx r+^^^^"' ^x' = aO+x''^y Thus / -"^ sin cur 0?^ is uniformly convergent when a ^ao>0, and (iv) Now <^(a)= [*/(a)rfa. Therefore <^'(«)=/(«) and (l>"(a) =/(a) = - j^ y^2 ^^^ «^ ^- r" sin 0U7 , . r* sin cur , = -/ dx+ I ,. . ^. dx k X k J7(l+ar*) This result has been established on the understanding that a > 0. (v) From (iv) we have <^(a)=^e« + i?e-» + -, when a>0. But <^(a) is continuous in a^O, and <^(0)=0. Digitized by VjOOQIC AN ARBITRARY PARAMETER 187 Therefore Lt <^(a)=0, a— H) and ^+5+1=0. Also ^{p.) is continuous in a = 0, and XO)=^' Therefore A-B-^=0. It follows that il=0 and 5=--. Thus «^(a)=^(l-e-»), when a>0. And /(a)=<^'(«)=|-^"*> when a>0. Both these results obviously hold for a=0 as well. Ex. 3. The Gamina Function V{n)=\ e-'af-^dx, n>0, and its derivatives, Jo (i) To prove T(n) is uniformly convergent when iV^ ^ w ^ Wo > 0, however large N may he and however near zero n^ may he. When w^ 1, the integral / e-'af'-^dx has to be examined for convergence only at the upper limit. When 00, Therefore e-'x^^ ^ e-*;27"«-i jf n^nQ> 0. It follows from the theorem which corresponds to § 83, I. that / e'^x"^^ dx converges uniformly when n^nQ>0. Again consider / e~*^"~^ dx, n > 0. When x>l, af'-^^x^-\ if 00, however large N may be and however near zero n^) may be. (ii) To prove T'{7i)= I e-'x*'-^ log x dx, n>0. We know that Lt (^'■loga7)=0, when r>0, x->0 80 that the integrand has an infinity at ^=0 for positive values of n only when 00. Therefore «-*:c"-^|log:p|^^«-^|log.v|, if n^nQ>0, And we have seen [Ex. 6, p. 119] that / ar*o-^\ogxdx converges when It follows as above that / e~'af*^^\ogxda; is uniformly convergent when Also for / 6~'j7"-^ log X dxy we proceed as follows : When ^ x>\, sf^^:^x^-\ \i 0l, X But / e~'x'^dx is convergent. Therefore / e~^x^^ log xdxia uniformly convergent when 00y however large N may be and however near zero Uq may be. We are thus able to state, relying on §§ 86, 87, that F(n)=/ e-^x^^logxdx for n>0. It can be shown in the same way that the successive derivatives of T(n) can be obtained by differentiating under the integral sign. Ex. 4. (i) To prove j log (I- 2?/ cos x +y^) dx is uniformly cmivergent for aiiy -«• interval ofy {e.g. h^y^h')\ and (ii) to dedtLce that j^ logsinxdx= -^ir log 2. Jo (i) Since l-2yc(}sx + i/^=={i/-cosxy-k-ain^x, this expression is positive for all values of Xy y, unless when x=nnr and y =( - 1)*", m=0, ±1, ±2, etc., and for these values it is zero. It follows that the integrand becomes infinite at ^=0 and x^ir \ in the one case when y= 1, and in the other when y = - 1. We consider first the infinity at x=0. As the integrand is bounded in any strip O^x^X^ where A'' j d.v converges uniformly for | A | ^ a. [Cf. § 83, I.] And therefore / log ( 1 - 2y cos x + y^) dv Jo converges uniformly for any interval (6, b') of y. The infinity at ^=7r can be treated in the same way, and the unifori^ convergence of the integral I log ( 1 - 2y cos X + y 2) dx is thus established for any interval (6, b') of y. (ii) Let f(i/) = I log ( 1 - 2^ cos x + y^) dx. We know from § 70 that /(y)=0, when |2^|<1, and f(2/)=Tr\ogy^, when \y\>\> But we have just seen that the integral converges uniformly for any finite interval of y. It follows from § 87, 1., that /(1)= Lt /(y) = and /(-1)=? Lt /(y)=0. -1 Digitized by Google 190 DEFINITE INTEGRALS CONTAINING But /•(!)= r log 2(1 -cofix)d.v = 27r log 2 + 2 Tlog sin ^cLr Jo Ii = 27r log 2 + 4 / 'log sin xdx. Thus / ' log sin xdx=—\TF log 2.* Frora/( - 1)=0, we find in the same way that / 'log cos X dx= - \ir log 2, a result which, of course, could have been deduced from the preceding. 89. The Repeated Integral j dxj^ f(x, y)dy. It is not easy to deter- mine general conditions under which the equation / dx\ f{^,!/)dt/= di/ f{oc,y)dr Ja Jb Jb Ja is satisfied. The problem is closely analogous to that of term by term integration of an infinite series between infinite limits. We shall discuss only a case some- what similar to that in infinite series given in § 76. Let f(x, y) he a continvxms function of (.r, y) in x^a, y^ b, and let the integrals /•* /•* (0 I /(^, y) dx, (ii) ( fix, y) dy, respectively^ converge nnifcmnly in the arbitrary intervals b^y'^b\ a^x^a\ Also let the integral (iii) I dx f(^,y)dy converge nnifomdy in y ^ b. Then the integrals <^^\ A^^y)^y and / dy\ f(x,y)dx _ Jb Jb Ja exist and are equal. Since we are given that / f{^,y)dx converges uniformly in the arbitrary interval b^y^b', we know from § 85 that / ^y/ f{^\y)d'V=\ dxj f{x,y)dy, when y>b. Jb Ja Ja Jb It follows that If^yf f(^',y)dx= Lt / dxjf{xyy)dy, » * »'a ff—*^Ja Jb provided that the limit on the right-hand side exists. * This integral was obtained otherwise in Ex. 4, p. 118. Digitized by VjOOQIC AN ARBITRARY PARAMETER 191 To prove the existence of this limit, it is sufficient to show that to the arbitrary positive number c there corresponds a positive number Y such that |£ cLvj' f{a:,y)di/\<€, when />ygr. But from the uniform convergence of i^ Sf = ^i we can choose the positive number X such that \j dxj /(^,y)c?y|<^, when x^X, (1) the same X serving for every y greater than or equal to 6. Also we are given that / f{x^ y) dy Jb is uniformly convergent in the arbitrary interval (a, a'). Therefore we can choose the positive number Y so that |j[/(^,y)rfy|<3(-^'_„y when />ysr, the same F serving for every x in (a, X). Thus we have I T dx\ f{x,y)dy < |, when y">y'^ Y. (2) But it is clear that \ dx\ f{x,y)dy=\ dx f(x,y)dy+ dx f{x,y)dy- dxj f{x,y)dy. Ja .V *'« *V -'X Jb Jx Jb Therefore from (1) and (2) we have < €, when y" >y^^Y, We have thus shown that r dyC f{x,y)dx= Lt C dx(\f{x,y)dy (3) Jb J a p—^aoJa Jb It remains to prove that Lt / dx f(x,y)dy= dx f(x,y)dy. y— XJDv'a Jb Ja Jb Jb Let the limit on the left-hand side be I. Then € being any positive number, as small as we please, there is a positive number Yi such that \l- I dxj f{^,y)dy\<^, when .yg F, (4) I Ja Jb I o Also, from the uniform convergence of J a y* /('^»y)^^» "^^^^ y=^^ Digitized by VjOOQIC 192 DEFINITE INTEGRALS CONTAINING we know that there is a positive number X such that \[dx'^^ f(ic,y)d^f- Qdx\y(x,y)dy\<^-, when X'gX, (5) ' the same X serving for every y greater than or equal to h. Choose a number X' such that X'^X>a. Then, from the uniform convergence of / f(x,y)dy in any arbitrary interval, we know that there is a positive number Fg ^^^ ^^**' I ^V(-^> y)dy-j^ /(.r, y)dy\^< ^^jgf^^y when y g Fg, the same Fg serving for every a: in (a, X'). Tbus Ij^'dvj' f{x,y)dy- jy.vfy(.v\y)dy <'^, when y^Y^ (6) Now take a number F greater than Fj and Fg. Equations (4), (5) and (6) hold for this number F. But \l- j dx\ f{x,y)dy\ \ ' a - b 1 ^h- rdxrf(x,y)dy\ + \ rdJ'f{x,y)dy- f dx^ fi_x,y)dy\ + 1 / dxj f{x,y)dy- \ dx\ f{x,y)dy \Ja -'b •'n •'6 <| + |+|, from (4), (5) and (6), <€. This i-esult holds for every number JT' greater than or equal to X. Thus we have shown that rx r« /•« /-« 1= Lt / dx f{jc,y)dy=\ dx f{x,y)dy. ar— ►« Ja Jb J a ^ b Also, from (3), we have / rf^l f{x,y)dy=\ dy f(x,y)dx Ja Jb Jb Ja, under the conditions stated in the theorem. It must be noticed that the conditions we have taken are suficienty but not necessary. For a more complete discussion of the conditions under which the integrals T" , T" .. v , T" , /*" ../ / dx] f(x,y)dy, \ dy f(x,y)dx, Ja Jb Jb -'a when they both exist, are equal, reference should be made to the works of de la Vall6e Poussin,* to whom the above treatment is due. A valuable dis- * His investigations are contained in three memoirs, the first in BruxelleSf Ann. Soc. scient.f 17 ; the second in J. 7nath.j Paris, (Ser. 4) 8) 1892 ; and the third in ./. math. , Paris, (S^r. 5) 5. See also Broinwich, London, Proc. Math. Soc. (Ser. 2), 1, 1903. Digitized by Google AN ARBITRARY PARAMETER 193 cussion of the whole subject is also given in Pieqwnt's Theory of Functions ' of a Heal Variable. The question is dealt with in Hobson's Theory of *F^U7ictuyn8 of a Real Variable, but from a more difficult standpoint. REFERENCES. Bromwich, loc. ciL, App. III. De la Vall^b Foussin, loc. cit, T. II. (2* 6d.), Ch. II. DiNi, Lezioni di Analisi Infinitesimale, T. II. 1' Parte, Cap. IX., X., Pisa, 1909. GouRSAT, loc. cit., T. I. (3' 6d.X Ch. IV.-V. Osgood, loc. cit., Bd. I., Tl. I., Kap. III. PiBRPONT, loc. cit., Vol. I., Ch. XIII.-XV. Stolz, loc. cit., Bd. I., Absch. X. And Brunel, " Bestimmte Integrale,'* Enc. d. math. Wisa., Bd. II., Tl. I., Leipzig, 1899. EXAMPLES ON CHAPTER VL /•« 1. Prove that / e'^^dx is uniformly convergent in a^a^>0, and that / u , 2 ^s uniformly convergent in a=0, when b >0. /•OO 2. Prove that / e'*^ — —dx is uniformly convergent in y^yo>0, and •'o y that / e""* — -^ dx is uniformly convergent in y^O, when a>0. 3. Prove that / e~^3if*-^ co^dx is uniformly convergent in the interval 0, when 7i = l, and in the interval a^O, when 00, when w^O, and in the interval a^O, when -l0. 8. Show that differentiation under the integral sign is allowable in the following integrals, and hence obtain the results that are given opposite each : Wjo a:2_,_«-2^^' «^^' Jo (.t'2+a)»+i~2 * 2«7i! a«+i ' (iii) j„ ^<^=;^l. «>-! : j„ *"(-l'>«-fr'^''=(-„4:iy^.i- (iv) f" ^rf^=^^. 00- (ii) / e-^cosb.vcLv ; interval « = ao>0, or any interval of b. (iii) / e~**sin bxdx ; interval a^ao>0, or any interval of b. (iv) / afdv; interval a^ ao> - 1. 10. Assuming that / e-'^ainbxdx^-j-j^j «>0, show that T— '^^sin6jrc?a:=tan-if-tan-i-^, g>f>0. \ Jo X •^ • -sin6^G?a7=tan~^T- X o . \ (ii)jr"^*rf:r=i7r, 6>0. 11. Show that the integrals ra f^ b e-^ cos bxdx==-^-r«y / e-'"ambxdx = -K—j%i a>0, a^ + b^ Jo a^ + b^ can be differentiated under the integral sign, either with regard to a or 6, and hence obtain the values of .-ao /•« / xe~^'' COS bx dx, I xe~^ sin bxdx, I a^e~^*QOsbxdx, I x^e"^' ain bx dx. Digitized by VjOOQIC AN ARBITRARY PARAMETER 195 12. Let m^n-'^'^^e-'d.i: Jo ^V Show that/'(y)= / siiiJL'f/ e~*dx for all values of y, and deduce that /Cy)=iiog(i+2^2). 13. Let f(y)= r e-'^coa^Ji^i/djc. Jo Show thRtf(9/)= - 2 j .re-** sin 2^^ dx for all values of y. On integrating by parts, it will be found that /(y)+2y/(y)=o. From this result, show th&t fQ/)=^^ir e-'^, assuming that r(J)=^/7r. Also show that / dx I e-'^cos 2xy di/= j f(y) dy^ and deduce that / g-'-^*'* ^^ dx = s]Tr\ e-^dy. 14. Let 6'= I e-^af-'^coahxdx, V= j e-'^'x^'^ ain bx dx, ^where a>0, n>0. Make the following substitutions : a=r cos ^, 6=r8in^, where -i7r<^0 Jo And deduce that ■n/ \ ^^^ -n/ \ • f^TT ('>i, ^^''•'= — Z? <"\/o ^'-»'^= F — ' /...x r°cosx J iTT /**sin.r , (in) / — dx=J-= —J— dx. [Compare Gibson, Treatise oii the Calculus, p. 471.] 15. Prove that / dx\ e'"^ sin X dy = j dyj e~'^amxdx, where b is any positive number. Digitized by Google CHAPTER VII FOURIER'S SERIES 90. Trigonometrical Series and Fourier's Series. We have already discussed some of the properties of infinite series whose terms are functions of cc, confining our attention chiefly to those whose terms are continuous functions. The trigonometrical series, aQ+{a-^^co8X + b^8mx)'\T{a2Cos2x + b2sm2x)+ .,. , (1) where a^, a^, fc^, etc., are constants, is a special type of such series. Let f{x) be given in the interval { — 'rr, tt). If bounded, let it be integrable in this interval; if unbounded, let the infinite integral I f{x) dx^he absolutely convergent. Then I f{x')cosnx'dx' and I f{x')sinnx'dx' J -IT J -IT exist for all values oi n. (§61, VI.) The trigonometrical series (1) is called a Fourier 8 Series, when the coefficients a^, a^^b^, etc., are given by ^o = ^~ f{x-)tmvi^dx'y ' '^^•'-'^ ' I ...(2) 1 fir 1 fir 1 an = —\ f{x') COS nx' dx\ bn = -\ f{x') sin nx' dx\ \ and these coefficients are called Fourier's Constants for the function f{x). The important thing about the Fourier's Series is that, when f{x) satisfies very general conditions in the interval ( — tt, x), the sum of this series is equal to /(ic), or in special eases to i[/(^ + ^)+/(^~'^)]» when x lies in this interval. Digitized by VjOOQ IC FOURIER'S SERIES 197 If we assume that the arbitrary function f{x), given in the interval ( — x, tt), can be expanded in a trigonometrical series of the form (1), and that the series may be integrated term by term after multiplying both sides by cos nx or sin tix, we obtain these values for the coefficients. For, multiply both sides of the equation /(a;) = ao+(«icpsaj + 6isinaj) + (a2cos 20^ + 62'^^^ 20?)+.. . , -ir^x^ir, (3) by cos nx, and integrate from — tt to tt. Then I f{x) cos nxdx = ira^ , fir •'-* fir J . since I cos mx cos nx dx = I sin nix cos nxdx = 0, \ (. '. when 771, n are different integers, and I cos^ njxdx = Tr, ''^'^ ~ I Thus we have •'"'^ ^^ » If- ^ f^n = - 1 fip^') COS nx'dx'y when 7^ ^ 1. And similarly, If' 6n = — 1 f{x') sin nx'dx\ 1 f'^ ^0 = 2:^] J{x')dx\ Inserting these values in the series (3), the result may be written /(a?) = 2:^ J f{x') dx + - 2 J fix') cos n{x - x) dx\ — Tr^x = Tr (4) This is the Fourier's Series for f{x). If the arbitrary function, given in ( — tt, tt), is an even function — in other words, if f(x)=f( — x) when 0x . In this way we shall show that when f(x) satisfies very general conditions, the Fourier's Series iov f{x) converges to /(a?) at every point in ( — tt, tt), where f(x) is continuous ; that it converges to '2[/('^ + 0)+/(^""^)] ^^ every point of ordinary discontinuity; also that it converges to ^[f( — Tr+())+f{ir-'{)y\ at a;=dt'7r, when these limits exist. Since the series is periodic in x with period 27r, when the sUm is known in ( — tt, tt), it is also known for every value of x. If it is more convenient to take the interval in which f{x) is defined as (0, 27r), the values of the coefficients in the correspond- ing expansion would be ^^0 = 2^ J f{^')dx\ a„ = - I f{x') cos nx'dx\ hn=-\ f(x) sin nxdx\ n^l. It need hardly be a^ed that the function f{x) can have diflferent analytical exn/essions in different parts of the given interval. And in partfcular we can obtain any number of such / I Digitized by VjOOQ IC 200 FOURIER'S SERIES expansions which will hold in the interval (0, tt), since we can give f{x) any value we please, subject to the general conditions we shall establish, in the interval ( — tt, 0) The following discussion of the possibility of the expansion o£ an arbitrary function in the corresponding Fourier's Series depends upon a modified form of the integrals by means of which Dirichlet * gave the first rigorous proof that, for a large class of functions, the Fourier's Series converges to fix). With the help of the Second Theorem of Mean Value the sum of the series can be deduced at once from these integrals, which we shall call Dirichlet's Integrals. 91. Dirichlet's Integrals (First Form). fL->ao Jo *^ ^ fL-> CO J a '*' where 0 I sin x dx, where h' = ^=c\ Thus W'^J^dxUiil+V) \h' ^ . \b cJ It follows that the integral ,pertorinm der Pliysik, Bd. I., p. 152, 1837. Digitized by Google FOURIER'S SERIES 201 It follows that I ^ siu ulor I sin QC Lt I - —dx=\ dx = ^7r, 0-oo Jo X L Lt [*/(a;)?HL^(te = 0, 0-Q0 Jo ^ fl-^-CO In the discussion of this theorem we shall, first of all, assume that f{x) satisfies the conditions we have imposed upon (x) in the statement of the Second Theorem of Mean Value (§ 50) ; viz., it is to be bounded and monotonic (and therefore integrable) in the interval with which we are concerned. It is clear that ^— satisfies the conditions imposed upon X \fr(x) in that theorem. It is bounded and integrable, and does not change sign more than a finite number of times in the interval. We shall remove some of the restrictions placed upon f(x) later. I. Consider the integral r Jo X From the Second Theorem of Mean Value where ^ is some definite value qi x in a = x = b. Since f(x) is monotonic in a = x^b, the limits /(a + O) and /(6-0) exist. And we have seen that the limits of the integrals on the right- hand are zero as /i->oo . /' It follows that, under the conditions named above, Lt \'f(x)^^j^dx = 0, when 0(x)+f( + 0). The limit /(+0) exists, since f{x) is monotonic in 0^x = a. Then {x) is monotonic and ^( + 0) = 0. Also As /jL^ 00 the first integral on the right-hand has the limit Jtt. We shall now show that the second integral has the limit zero. To prove this, it is sufficient to show that, to the arbitrary- positive number e, there corresponds a positive number v such that I ^(aj) — ^ dx {x) — ;p- dx X <|, when /jL^v, If", sin /xic , I <: I f * ^ / \ sin ua; , i , , f " , , . sin ixa; , IJo X I'Jo X I Ja . X Therefore \i , , , sin ux , ^ e , € Thus Lt I d>(x)^~^-^' dx=0. •< e, when iji = v. And, finally, under the conditions named above, Lt f/(^)?i^oo Jo •*' ^ 92. In the preceding section we have assumed that f{x) is bounded and monotonic in the intervals (0, a) and (a, 6). We shall now show that these restrictions may be somewhat relaxed. I. Dirichlefs Integrals still hold when f{x) is bounded, and the interval of integration can be broken up into a finite number of open partial inter^jals, in each of which f(x) is monotonia* This follows at once from the fact that under these conditions we may write j^/ \ -ny/ \ n/ \ ^ f{x) = F{x)-G{x\ * This condition is sometimes, but less exactly, expressed by the phrase : f{x) shall have only a finite number of maxima and minima in the interval. Digitized by Google 204 FOURIER'S SERIES where F{x) and G{x) are positive, bounded, and monotonic in- creasing in the interval with which we are concerned [cf . § 36]. This result can be obtained, as follows, without the use of the theorem of § 36 : Let the interval (0, a) be broken up into the n open intervals, (0, «i), («!, ^2)1 ••• > (<*n-l, «)» in each of which f{x) is bounded and monotonic. Then, writing ao=0 and a„=a, we have The first integral in this sum has the limit ^/(+0), and the others have the limit zero when ft->oo . It follows that, under the given conditiong, Lt /7(.r)!i^d:r=|/(+0), 0f^<^^ \'Axf^O^,+ r/(-)""^^> a<» J a X Digitized by VjOOQ IC 206 FOURIER'S SERIES Further, we have assumed that ic = is not a point of infinite discontinuity of f{x). Thus the interval (0, a) can be broken up into two intervals, (0, a) and (a, a), where f{x) is bounded in (0, a), and satisfies the conditions given in I. of this section in (0, a). It follows that and we have just shown that Therefore, under the conditions stated above in II., Lt [" m'^dx^if i+0). /x-^oo Jo •*' ^ 93. Dirichlet's Conditions. The results which we have ob- tained in §§ 91, 92 can be conveniently expressed in terms of what we shall call Dirichlet's Conditions. A function f{x) will be said to satisfy Dirichlet's Conditions in an interval {a, h), in which it is defined, when it is subject to one of the two following conditions : (i) f(x) is hounded in (a, 6), and the interval can he broken up into a finite number of open partial intervals, in each of which f{x) is monotonia (ii) f{x) has a finite number of points of infinite discontinuity in the interval, hut, when arbitrarily small neighbour- hoods of these points are excluded, fix) is bounded- in the remainder of the interval, and this can be broken up into a finite number of open partial intervals, in each of which f(x) is monotonia Further, the infinite integral I f(x) dx is to he absolutely cmivergent Ja We may now say that : Whenf{x) satisfies Dirichlet's Conditions in the intervals (0, a) and {a, b) respectively, where 0-cc J a X Digitized by Google FOURIER'S SERIES 207 It follows from the properties of monotonic functions (cf. § 34) that except at the -points, if any, where f{x) becomes infirute, or oscillates infinitely, a function which satisfies DiricKlet's Con- ditions, as defined above, can only have ordinary discontinuities.* But we have not assumed that the function f(x) shall have only a finite number of ordinary discontinuities. A bounded function which is monotonic in an open interval can have an infinite number of ordinary discontinuities in that interval [cf.§34]. Perhaps it should be added that the conditions which Dirichlet himself imposed upon the function f(x) in a given interval (a, b) were not so general as those to which we have given the name Dirichlefs Conditions, He contemplated at first only bounded functions, continuous, except at a finite number of ordinary dis- continuities, and with only a finite number of maxima and minima. Later he extended his results to the case in which there are a finite number of points of infinite discontinuity in the interval, provided that the infinite integral I f{x) dx is absolutely convergent. If the somewhat difficult idea of a function of bounded variatioUy due to Jordan, is introduced, the statement of Dirichlet's Conditions can be simpli- fied. But, at least in this place, it seems uriadvisable to complicate the discussion by further reference to this class of function. 94. Dirichlet's Integrals (Second Form). Lt fV(^)?l!L^da. = ^/( + 0), Lt ?f(xf^?J^dx^O. where 0»Ja Since Let us suppose that f{x) satisfies the first of the two conditions given in § 93 as Dirichlet's Conditions : f{x) is bounded, and the intervals (0, a) and (a, b) can be broken up into a finite number of open partial intervals, in each of which f{x) is monotonic. Then, by § 36, we can write f(x) = F{x)--0{xl where F{x)f G{x) are positive, bounded and monotonic increasing in the interval with which we are concerned. ^, J,. .sin/jiX [', . X n/ \ ^ Isin/xaj Then fix) . ^ = F(x) -, 0(x)^ ^ . -^^ ^ smx L ^ sm a; ^ ' sm a? J x But xjmix is bounded, positive and monotonic increasing in (0, a) or (a, 6), when 0<(X Jo Sin x we need only, as before, break up the interval (0, a) into (0, a) and (a, a), where f{x) is bounded in (0, a), and from the results we have already obtained in this section the limit is found as stated. If it is desired to obtain the second form of Dirichlet's Integrals without the use of the theorem of § 36, the reader may proceed as follows : (i) Let/(^) be positive, bounded and monotonic increasing in (0, a) and (a, 6). Then - — is so also, and M=fM - — is so also, 0>ao JO A ^ ^ Therefore Also fV(^)?«L/*?d:,= ff^.^fj^d,- f'f^^f^d.^^. Ja sm:p jo-'^ ' sin^ jo ^^ sin:r Therefore Lt C flxf.^^^dx=0. (ii) Let /(or) be positive, bounded and monotonic decreasing. Then for some value of c the function c-f(x) is positive, bounded and monotonic increasing. C.I o Digitized by VjOOQ IC 210 FOURIER'S SERIES Also jo -^ ^ 'J sm X Jo smx jo sm .c Using (i), the result follows, (iii) If f{x) is bounded and raonotonic increasing, but not positive all the time, by adding a constant we can make it positive, and proceed as in (ii); and a similar remark applies to the case of the monotoinc decreasing function, (iv) When f{x) is bounded and the interval can be broken up into a finite number of open partial intervals in which it is monotonic, the result follows from (i)-(iii). (v) And if f{x) has a finite number of points of infinite discontinuity, as stated in the second of Dirichlet*s Conditions, so far as these points are concerned the proof is sintilar to that given above. ^ 96. Proof of the Convergence of Fourier's Series. In the opening sections of this chapter we have given the usual elementary, but quite incomplete, argument, by means of which the coefficients in the expansion /(a?) = a^ + (a^ cos a; + ?^j sin a?) + (ttg cos 2a3 + 62 sin 2ir ) + . . . :;^ — 7r = ajS7r are obtained. We now return to this question, which we approach in quite a diflFerent way. We take the Fourier's Series ^o+(^i cos 03 + 61 sin aj)4-(a2 cos 2x + b2 sin 2x)+, . . , where the coefficients are given by a„ = - 1 /(aj') cos nx dx\ 6« = - 1 f{x') sin nx dx\ We find the sum of the terms of this series up to cosTiaj and sinna;, and we then examine whether this sum has a limit as w->oo. We shall prove that, when f(x) is given in the interval | ( — TT, 7r), and satisfies Dirichlet's Conditions in thai intei^val, this sum has a limit as n-> 00. It is equal to f(x) at any point in — 7r0, provided that at the point x with which we are concerned f(x + 0) and f{x—0) exist. It follows from § 94 that, when x lies between — tt and ir and /(aj — 0) and f(x + 0) both exist, =h[A^-o)+f{x+o)i giving the value f{x) at a point where f{x) is continuous. We have yet to examine the cases cc = ± tt. In finding the sum of the series for x = tt, we must insert this value for x in 8n{x) before proceeding to the limit. Thus ^.(^)=MV(^-2a) ^^"<^.^ + ^^% 7«, since the second integral in (1) is zero. Digitized by Google 212 FOURIER'S SERIES It follows that o/ \ If'"^// o .8in(27i + l)a , TrJ^.f*'^ sma ttJo sma -L^f^// . o , sin(2n+l)a ^ ttJo sma where ^ is any number between and tt. We can apply the theorem of § 94 to these integrals, if f(x) satisfies Dirichlet's Conditions in ( — tt, tt), and the limits /(^r — 0), /(-TT+O) exist. Thus we have Lt Sn(7r) = K/("^ + 0)+/(7r-0)]. A similar discussion gives the same value for the sum at aj= — TT, which is otherwise obvious since the series has a period 27r. Thus we have shown that when the arbitrary function f{x) satisfies Dirichlefs Conditions in the interval ( — tt, tt), and 1 f w 1 f '^ a„ = - I fix') cos nx' dx\ 6» = - I f{x') sin nx' dx\ IT J -It IT J -It - tlie Fourier's Series aQ+ (»! cos x + h^ sin x) + (a^ cos 2x+\ sin 2aj) + . . . converges to j[/(aj+0)+/(aj-0)] at every point in '-'Tr according to the side from which we approach the point. On the other hand, if we insert the value Xq in the terms of the series and then sum the series, we obtain \ [/(x^ + 0) +f(xQ - 0)]. We have already pointed out more than once that when we sj^eak of the sum of the series for any value of x, it is understood that we first insert this value of X in the terms of the series, then find the sum of n terms, and finally obtain the limit of this sum. Ex. 1. Find a sel'ies of sines and cosines of multiples of x which will represent j ^ ^ . i^-c* in the interval -7r<.t?<7r. What is the sum of the series for x='± it ? TT , ^ 1 Here /W = 2 sinb -e* and 2 sinh TT. /e* cos nx dx, n^l. Fig. 16. Integrating by parts, ( 1 + ^2) / e* cos nx dx = („ = — / {x-{- x^) sin ?u' ci?:i' = - 1 .t" sin nx dx, TTJ-Tr TTJO Google 2 which reduces to h„ = ( — 1)"~^ Digitized by ' FOURIER'S SERIES 215 Therefore when -Trn=-^sin~ • /w= - .^„ sin 3x+—„ sin bx - . Fig. 26 contains the lines y = Jtt^, O^X^^TTy' ix^TT, i and the approximation curves ^=sin^, y = sin ^ — "2 sin 3jp, sinS.r+r^jsin bx. 3^ O^.r^TT. Digitized by Google 224 FOURIER'S SERIES It will be noticed that the last of these curves approaches the given lines closely, except at the sharp comer, right through the interval. (1) i| i 1 !fn :| mc ^t 21 J' 1 ;i |:| : 1 1 m "R 1 11 'h fttj; ™ ■4 1 \h 35:1 tr.'i ri: ■.\:ii ^ iffl d£ aim utR 4l±t: y , ,.a++u w (3) PlO. 25. For unrestricted values of .r the series represents the ordinates of the lines ' shown in Fig. 26, the part from -tt to +7r being repeated indefinitely in- both directions. The sum is continuous for all values of x. Pig. 26. Ex. 3. Find a aeries of sines of multiples of ^ which will represent /(:r) in the interval (0, tt), where /(.r)=0, 0^^| Digitized by Google Here FOURIER'S SERIES 6„= / sin rue dx 225 4( cos — - cos mr 1 =-siii— r-^sin-r. n 4 4 I O^^^TT. Therefore 6„ vanishes when n is a multiple of 4. And /(:r)=sin;r-8inar+Jsin3j7+j8in5ar- Jsin6j7+... O'^x'^w. Fig. 27 contains the graph of the given function, and the approximation curves y=sina:, y=8ina? — 8in2^, y = sin a: - sin 2:p+ J sin 3a;, y =sin .r- sin 2.x: + J sin 3a?+ J sin 5^?, The points x=^ and a?=7r are points of discontinuity in the sum of the series. The behaviour of the approximation curves for large values of n at these points will be examined in Chapter IX. Ex. 4. Find a series of sines of multiples of x which will represent f{x) in the interval (0, tt), where /(^) = j7r, 0 c, I p ^ a; ^ TT, Digitized by Google 226 FOURIER'S SERIES 0) (2) (3) (4) I 1+*' mrc i I t r± : i t ' i tpi i; 1 ! 1:1: : 1 ■'U ;: :::: ■ ■ 1 ^ u I ■■| : ■i i).. i i : h ;::S 1 ::: i f ' [' i " t "^ 1 ^iS ^ ^ * I T ut f}[ '' r'g^ 1 \ 'm li i*f Iff i;: H fl i ^i ?4 i 11 iii ■ Ftg. 27. Digitized by Google (I) (2) (3) BMIH f 11 ^ im :MM N:m| ;| i--'iii ■' ■ 1 iifllilpW "" ^iH W tL :Jp:ii-^^^^^^iii ; :;;|l || i I ■■'"• m ::i \mm i "■ li i ::L ;;: : = ; =:;;i 1 ; :i t^^'^ i- h'\l ' '■ 1 :^| :ji:|:i 1 'tI ■ If- IMilH: MNlijl: -pM |; ^^^mili nii:l!i m\\ r (4) Digitized by Google 228 FOURIER'S SERIES 98. Other Forms- of Fourier's Series. When the arbitrarj function is given in the interval ( — i, 0* ^^ can change this interval to ( — x, x) by the substitution u = Trx/l. In this way we may deduce the following expansions from those already obtained : ^f /(^')^'+7i;f f{x')cos-^{x'^x)dx\ ^l^x^l, ...(1) U' fix') dx'+j"^ cos^x[ fix') cos^x'dx\ O^x^l, ...(2) j-^Qin—x^ fix')smj^x'dx\ O^x^l (3) When/(a5) satisfies Dirichlet's Conditions in ( — i, I), the sum of the series (1) is equal to ^[f(x+0)+f(x — Oy] at every point in — ii)sm — sin-^ 4. (2^2 - v^ - Vg) cos — ^ | is equal to v^ when - ll, the conclusion he sought to draw is invalid until it is shown that the series does converge when r = 1, and this, in fact, is the real difficulty. In accordance with Abel's Theorem on the Power Series (§72), if the series converges when r = l, its sum is continuous up to and including r= 1. In other words, if we write I Cw 1 * f*^ F(r, ic) = s- fi^l dx -h- V r** I fix') cos n{x - x) dx\ we know that, if -F(l, x) converges, then Lt F{r,x)^F(l,xy But we have no right to assume, from the convergence of UF(r,x\ that F{1, x) does converge. Poisson's method, however, has a definite value in the treatment of Fourier's Series, and we shall now give a presentation of it on the more exact lines which we have followed in the discussion of series and integi^als in the previous pages of this book. 100. Poisson's Integral. The integral — r r- ^ ^7f X ofi^')<^A \r\{x') be defined when -w'^x'^Tr by the equation W)=f(x')-i Lt [/(a; + 0+/(^-0]. Then F{r,x)-i Lt [/(x+tHfix-t)] = ~- r i— ^ ^-'?— rx 2*W^^' (2) 27rj-»l-2rcos(^* -.r)4-H ^ But we are given that Lt [f(x+t) +f{x - 1)] exists. Let the arbitrary positive number c be chosen, as small as we please. Then to c/2 there will correspond a positive number rj such that \Ax+t)+f{x-t)~ Lt [/(a.- + 0+/(^'-0]l<|, (3) The number r) fixed upon will be such that i^^-rj, x+rj) does not go beyond ( - tt, tt). Then 27rL,r^2rco¥(Z^)T^2<^("^^^^ • -2if i-2'c"or.^.2 ^-^(-^^>^-^(-^>j-^^ It follows that |27rj^^ l-2rcos(a,'-^) + r2^^^^'**^ I 47rJo 1 -2rcos^+r2'^' ' 47rJ - ff 1 - 2r cos ^ +r2 "^ 47rJ-irl-? • i4-i-8i„3|' It follows that 2I ( t\ " + r+J \^.rLi-x)^r^ *<^> ''•^■' if l>r> 1 . ...; (6) 1+isin^l Combining (4) and (6), it will be seen that when any positive number € has been chosen, as small as we please, there is a positive number p such that I ¥{3; x)-\ Lt [/(^+0+/(-'--0]i<«, <->0 when p= r < 1, provided that for the value of x considered Lt [/(a; -H +/(-^ ~ 0] exists. We have thus established the following theorem : Let f(x\ given in the interval ( - tt, tt), he hounded and integrahle, or have an absolutely convergent infinite integral^ in this range. Then for any value of xin -n 1 uniformly to the value f(x) in any interval (a, h) in which f(x) is continuous* This last theorem has an important application in connection with the approximate representation of functions by finite trigonometrical series.t 101. Fej^r's Theorem, t Let f{x) be given in the interval { — tt, tt). If hounded, let it he integrable; if unbounded, let the infinite integral I f(x)dx be absolutely convergent Denote by Sn the swin of the {n + 1) terms i fir 1 '^ f*^ .§^] J{x) + ~^] J{x')^sn{x'-x)dx'. Aho let Sn{x) = «0 + »l+--+8n-l , n *It is assumed in this that /(a -(>)=/(«)=/(« + 0) and /(6-0)=/(6)=/(6 + 0). Also /(a;) is subject to the conditions given at the beginning of this section. Cf. § 107. t Cf. Picard, Trait6 d' Analyse, (2« ^d.), T. I., p. 275, 1905 ; Bdcher, Ann. Math., Princeton, N.J, (Ser. 2), 7, p. 102, 1906; Hobson, loc. cit., p. 722. jCf. Math. Ann.. Leipzig, 58, p. 51, 1904. Digitized by Google FOURIER'S SERIES 235 Then at every point x in tlie interval — 7r<[a5<7r at which /(oj + O) and f{x — 0) exist, Lt Sn(x)=^^[f(x+O)+f(x^0)l With the above notation, 1 f' /•/ ,^cosn(x''-x) — cos(n+l)(x'^x) , , 2'7ri./^ ^ 1— cos(a;— ic) Therefore ^ ^ 2'?17rJ _„^^ ^ l — Q08{x—x) = /(^) • 2 1// T^^ =.^r.:><»'>i;^^ii^'*'' <■> if f(x) is defined outside the interval ( — tt, tt) by the equation f{x + 2'7r)=f{x). Dividing the range of integration into ( — tt+x^x) and {x, TT+x), and substituting x' = x — 2a in the first, and x' = x+2a in the second, we obtain 'Sn(i») = — I /fa-2«) . » rfaH 1 /(a:; + 2a) . o aa (2) Now suppose that x is a point in ( — tt, tt) at which f(x+0) and/(a5 — 0) exist. Let e be any positive number, chosen as small as we please. Then to e there corresponds a positive number tj chosen less than Jtt such that |/(aj + 2a)l-/(aj + 0)|<€ when 0l <«> Digitized by Google FOURIER'S SERIES 237 Combining these results, it follows from (3) that <4'+SSHr,U+l/(»+0>l). Now let 1/ be a positive integer such that ^^{/+|/(a:+0)|}<. (7) Then 'C.e, when n — v. In other words, when /(aj+0) exists. In precisely the same way we find that n-^oo^'TTjo sin'^a when /(a: — 0) exists. Then, returning to (2), we have Lt 8n(x)=^[f{x + 0)+f(x^0)l n->oo when f(xdzO) exist. This proof applies also to the points a; = zt tt, when /(tt— 0) aiKi /( — TT+O) exist. Since we have defined f(x) outside the interval ( — tt, tt) by the equation f(x + 27r)=f(x)y it is clear that /(-'7r + 0)=/(7r + 0) and /(-7r-0)=/(7r- 0). In this way we obtain Lt fifn(±7r) = K/(-^ + 0)+/(7r-0)], n->oo when /( — x + O) and /(tt — O) exist. Corollary. If f{x) is continuous in a ^^^6, including the end-points, when the arbitrary positive number c is chosen, the same r; will do for all values of a? from a to h, including the end-points. Then, from (7), it follows that the sequence of arithmetic means ^li ^2J *^3> ••• converges vniformly to the sum f{x) in the interval (a, b). Digitized by Google 238 FOURIER'S SERIES It is assumed in this statement that f{x) is continuous at x=a and x=h as well as in the interval («, h)\ i.e. /(a-0)=/(a)=/(a+0), and /(6-0)=/(6)=/(6+0). 102. Two Thearems on the Arithmetic Means. Before applying this very important theorem to the discussion of Fourier's Series, we shall prove two theorems regarding the sequence of arithmetic means for any series In this connection we adopt the notation «n = Wl + M2+...+M„, On = • n Theorem I. If the series ^1+^2+^3+... converges and its sum is «, theni Lt /S„= Lt «„=*. n — ►« »l— >«) We are given that the series ttl + M2 + W34-... converges, and that its sum is s. Thus, to the arbitrary positive number c there corresponds a positive integer v such that 1 5 - «„ , < c, when n S v. Thus, with the usual notation, I pRn I = 1 8n+p - «« I < 2€, when n ^ v, for every positive integer p. Also it follows, from the definition of v, 5,-[«.+t«,(i-i)+...+«.(i-^)] =«.+,(i-^)+«.+,(i-?^)+...+«„(i-'^^). (-=)■ (•-^'). ■■■(■—') are all positive and decreasing ; and are all less than 2€ when n>v. It follows, as in § 50, that Thus Ls'„--{m, + ?(2(i--) + ---+M1-'— -)f <2«, when n>v. Therefore |,$,-.,|<2. + l^"±-^^»> -;;+^'"^>"-' , when n>v. But V being fixed, we can choose the positive integer N so that —^ ^^ — < c, when n -Z N> v. n Digitized by Google FOURIER'S SERIES 239 Therefore | iS„ - «,. | < 3c, when n^N>v, But \8-Sv\<€. And l/S'n-il-rl^^-^.^l + l*-*,.!. Therefore t iS'„ - « | < 4c, when n^N>v. Thus Lt Sn=8. «— >■« Theorem II.* Let the sequence of arithmetic means Sn of the series Wi + Wa + Wg-H (1) converge to S. Then, if a positive integer Uq exists such that \Un\Wo, where K is some positive number independent of w, the series (1) converges and its sum is S. With the same notation as above, let <„=«„-/S. We have to prove that Lt / — If tn has not the limit zero as ?i->oo, there must be a positive number h such that there are an infinite number of the terms tn which satisfy eithe7* (\)t„>h or (ii) tn<-h. We shall show that neither of these hy{)otheses can be admitted. Take the former to be true, and let o-„ = fi + «2 + '- + ^«- Then o-„/w=>S'„-^, and Lt (r„/w = 0. n— x» Thus, to the arbitrary positive number c, there corresponds a positive integer Wj such that |o-„/?i|h. Then, when r = 0, 1, 2, . . . , | u^+r I < ^A^ J" FlO. 29. Now plot the points Pry whose coordinates are (r, t„+r) in a Cartesian diagram. Since ^n+r+i-^n+r=w„+r+i, the slope of the line PrPr+i is less ♦ This theorem was published by Hardy in London, Proc. Math. Soc. (Ser. 2), 8, pp. 302-304, 1910. This proof, due to Littlewood, is given in Whittaker and Watson, Modern Analysis, p. 157, 1920. Of. also de la Valine Poussin, loc. ci^. T. II.,§151. Digitized by Google 240 FOURIER'S SERIES ill absolute value than taii-i/r/?i. Therefore the points Fq, Pj, Pj, ... He above the line t/=h-A't&n6j where tsmO^^K/n. Let Fk be the last of the points i*o, Pj, Pg? ••• > which lies to the left of, or upon, the line x=hcot 6, so that k ^ Acot 0=hn/K. Draw rectangles as shown in the diagram. The area of these rectangles exceeds the area of the triangle bounded by the axes and y= A-ortan 6. Thus we have 0-„+, - (Tn-l=-tn + tn^i + ... + ^„+, > JA^cot ^= }A%/iL. But |o-„+^-o-n-i|S|(r„+,| + |o-„_i|<[(w + K) + («-l)]€, siuce »>Wi. Therefore (k + 2n - 1) € > ^hhi/K. But K^hcotO==kn/K. Thus An/ZS K > ^A^n/ JTc - (2» - 1 ). Therefore n [h/K-ih^IKe + 2] > 1. But A/jr-iAVA^€4-2<0. But we have assumed that there are an infinite number of values of n such that t„>h. The hypothesis (i) thus leads to a contradiction ; and a precisely similar argument shows that the hypothesis (ii) does the same. Thus Lt ^„=0, n— >>ao and it follows that Lt «„= Lt Sn=S. Corollary. Let Wi(^) + M2(.2^) + W3(^)+ (1) be a series whose terms are fu7ictions of Xy and let the sequence of arithmetic means Sn(x) for the series converge uniformly to S(x) in an interval (a, h). Then^ if a positive integer n^ exists such that ' I Unix) I < Kjn, when n>nQy where K is independent of n and x, and the same n^ s&rves for all volumes of x in the interval, the series (1) converges uniformly to S(x) in (a, 6). For, with the notation of the theorem just proved, if tn(x) does not tend uniformly to zero through (a, 6), there must be a positive number A, inde- pendent of X and n, such that an infinite sequence of values of n can be found for which ^«(.r„)>A, or tn(Xn)<-h for some point x^ in the interval; the value of Xn depends on the particular value of n under consideration. We then find, as in the original theorem, that both these hypotheses are inadmissible. 103. Fej^r's Theorem and Fourier's Series.* We shall now use Fej^r's Theorem to establish the convergence of Fourier's Series under the limitations imposed in our previous discussion ; that is we shall show that : *Cf. Whittaker and Watson, loc. cit., p. 167, 1915, and the note on p. 263 below. /Google Digitized by ' FOURIER'S SERIES 241 \Vhenf{x) satisfies Dirichlet^a Conditions in the interval ( — tt, tt), and ^0 = ST / /(^'') ^'j ^n= I f{x') cos nx' dx\ ^TTJ -IT TTJ -ir bn=- I f(x') sin nx'dx' (n = 1 ), TTJ -IT M« sum of the series «o + (^1 cos a? + 6i sin x) + (ag cos 2^ + 62 sin 2.r) + . . . is i[/(^+^)+/(-^"^)] ^^ c^e»y poiW in -7rl. Then, forming the arithmetic means for the series a^ + (a/ cos X + 6/ sin x) + (ag'cos 2.r + 63' sin 2.r) + . . . , c, I Q Digitized by VjOOQIC 24,2 FOURIER'S SERIES we have, with the notation of § 101, Sn(^)-- where i(.r-a) and ^(6-^) are each positive and less than ^tt. But it will be seen that the argument used in Fej^r'a Theorem with regard to the integrals 1 T** ^/ ox sin^wa , mrJo •'^ sm^a applies equally well when the upper limits of the integrals are positive and less than Jtt.* Therefore, in this case, Lt >S„(.^)=l[/(.r+0)+/(^-0)]. n — ►» And, as the terms (a„'cos nx + 6„'sin n.r) satisfy the condition of Theorem II. of § 102, it follows that the series ao'+(a/cos.r+6i'sin;p) + (a2'cos2j?+Vsi"2.r)+... converges and that its sum is Lt ^„(.r). «— ►90 But (ao - Uq) + S{(«n - «n ) cos Tix + (6„ - b„') sin n.v} 7rJi(«-a) sma 7rjj(6-x) sma By § 94 both of these integrals vanish in the limit as n-^oo. It follows that the series so («o - «o') + 2{(an - ctn) cos Hx + (6„ - 6„')sin 7i:r} converges, and that its sum is zero. But we have already shown that 00 Gq + 2(«n'cos ^i.r + 6„'sin nx) converges, and that its sum is i[/(-^ + 0)+/(^-0)]. * Cf. footnote, p. 235. t Outside the interval ( - t, ir)/(x) is defined by the equation /(a? + 2ir)=y*(x). Digitized by VjOOQ IC FOURIER'S SERIES 243 It follows, by adding the two series, that «o + 2 («nCos 1UV + 6„sin nx) converges, and that its sum is i[/(^+o)+/(^-o)] at any point between - tt and ir at which these limite exist. When the limits /( -tt + O) and/(7r-0) exist, we can reduce the discussion of the sum of the series for x= ±17 to the above argument, using the equ ation /(ar + 2ir ) =f(oc). We can then treat ;p= ±7r as inside an interval (a, 6), as* above. REFERENCES. B6cHEK, "Introduction to the Theory of Fourier's Series," Ann. Math.^ Princeton, N.J, (Ser. 2), 7, 1906. De la Valli^e Poussin, loc. city T. II. (2* 6d.), Ch. IV. DiNi, 8e^*xe di Fourier e altre rappresentazioni analytiche delle funzioni di ima variahile reali, Cap. IV., Pisa, 1880. Fourier, Th^orie analytique de la chaleur, Ch. III. GouRSAT, loc. cit., T. l! (3* 6d.), Ch. IX. HoBSON, loc. cit.y Ch. VIT. Jordan, Cours d'Anali/sCy T. II. (3« 6d.), Ch. V. Lebesque, Lemons sur les s&ies trigonom^triques, Ch. II., Paris, 1906. Neumann, Uher d. nach Kreis-, Kuget- u. Cylinder- Functionen fort- schreitenden Eiitmckelungeny Kap. II., Leipzig, 1881. RiEMANN, loc. cit. Weber-Riemann, Die partiellen Differential- gleichungen der mathematischen Physiky Bd. I. (2. Aufl.), Braunschweig, 1910. Whittaker and Watson, Modern Analysis^ Ch. IX., Cambridge, 1915 and 1920. EXAMPLES ON CHAPTER VII. 1. In the interval < .r < ^ , f(x) = l-Xy I 3 and in the interval ^ when ^tt^^^tt. Show that -, . 6 S,sini(2?i-l)7rsin(2?i-l).r , ,. ^ ^ •^<-^> = ^? {in-Xf ' . when 0^.r5.x. Digitized by Google 244 FOURIER'S SERIES 3. Expand /(^) in a aeries of sines of multiples of Tr.i/a, given that f(.v) = 7)U'y when O^a-^^ok, f{j;) = m(a-a:)y when \a'^x'Sa. 4. Prove that . 2u7ra' I « sin -7- i^-^=- 2 — J when 0<.r<^, TT T » and (j;_.r)!=?*^+ J|__ji., when O^x^l. 5. Obtain an expansion in a mixed series of sines and cosines of multiples of X which is zero between - ir and 0, and is equal to e' between and tt, and gives its values at the three limits. 6. Show that between the values -tt and +ir of ^ the, following expansions hold : 2 . / sin^ 2 sin 207 . SsinS^r \ 8mwi.r=- 8m?/i7rl Ys «-^i9 a + ^ia o + ... )> IT \l*-m2 22-wi2 Z^-m- J 2 . /I mco&x 97icos2;r 971 cos 3^ \ cos7W;r=-sm?»7rl ;i— + T2 ^--^ 2 +"5^? 2 -...)> IT \2m l^-m^ 22-m2 Z^-m^ / cosh m,v _2 f 1 wi cos ^ 7nco82^ _ m cos 3j7 \ 8inhw7r~7r \2w~ 12 + ^2"^ 22 + wi2 " "32+^ "^ * ' * A 7. Express a^ for values of x between - w and tt as the sum of a constant and a series of cosines of multiples of .r. Prove that the locus represented by ^ y- 5 — sin nx sin ny=0 is two systems of lines at right angles dividing the plane of x, y into squares of area tt^. 8. Prove that .r represents a series of circles of radius c with their centres on the axis of ;r at distances 2c? apart, and also the portions of the axis exterior to the circles, one circle having its centre at the origin. 9. A polygon is inscribed in a circle of radius a, and is such that the alternate sides beginning at ^ = subtend angles a and fi at the centre of the circle. Prove that the first, third, ... pairs of sides of the polygon may be represented, except at angular points, by the polar equation . ttO , . 27r^ , r=a,sin^^+a,sin^^-^+..., Digitized by Google FOURIER'S SERIES 245 a , aJa-^B) . mra a f^ mr(h ,,, where ~j — ^-^ = sm ^ — — gr cos - 1 cos —r-n sec d

+ Tbcosl2(9 + ..., where a is the side of the square. 15. On the sides of a regular pentagon remote from the centre are described segments of circles which contain angles equal to that of the pentagon ; prove that the equation to the cinquefoil thus obtained is r=5atan^Ll-22(-ir25;^nrii a being the radius of the circle circumscribing the pentagon. 16. In the interval 0S'i, ^2> a"*^ *^3 separately for values of x lying within the assigned interval. [Cf. Ex. 1, p. 230.] n* Tc rt \ ^ ( ' si" ^ . sin ^x \ 24. If /W= -2 (si" •^--32- + -52 -j .2 / . sin 2a; , sin 3a; \ show that /(a;) is continuous between and tt, and that /(ir-0)=l. Also 2 TT show that /'(.r) has a sudden change of value - at the point _. [See Ex. 1,2, pp. 222-3.] 25. Let /•/ \_ V s'" 3(2?i- l)a; „ ^sin (2?t - l).f , 6 5, sin \{^n - l)7rsin(2w - \)x when 0^a;^7r. Show that /(+0)=/(7r -0)= - Jtt, /aT + 0)-/(j7r-0)=-j7r, /(S^ + 0)-/(ii7r-0) = i7r; also that /(O) =/( Jtt) =/(|f tt) =/(7r) = 0. Draw the graph of f(x) in the interval (0, tt). See Ex. 1, p. 230, and Ex. 2 above.] Digitized by Google CHAPTER VIII THE NATUKE OF THE CONVERGENCE OF FOURIER'S SERIES 104. The Order of the Terms. Before entering upon the discussion of the nature of the convergence of the Fourier's Series for a function satisfying Dirichlet's Conditions, we shall show that in certain cases the order of the terms may be determined easily. I. If f{x) is bounded and otherwise satisfies Dirichlefs Con- ditions in the interval ( — tt, tt), the coejficients in the Fourier s Series for f{x) are less in absolute value than K/v, where K is some positive number independent of n. Since Tra^ = I f{x) cos nx dx, and the interval may be broken up into a finite number of open partial intervals (c^, c^+j) in which f(x) is monotonic, it follows, from the Second Theorem of Mean Value, that ln=x[^^'f{x)i = 2|/(c^+0)l \o^7ixdx+f{crj^^ — 0)\ ""^^Go^nxdxV where ^r is some definite number in (c^, c^+i). Thus x|an|{xQ±0) do not both vanish. In this case the condition for absolute convergence is satisfied. Then, in determining a„ and 6„, where 7ra„= I f{x)QO%nxdx and 7r6„= / f{x)»mtixdx^ we break up the interval into ( - TT, a), (a, ^o)» (^o» ^)» and (/?, tt), where (a, p) is the interval in which f{x) has the given form. In ( - tt, a) and (^, ir) it is supposed that f{x) is bounded and otherwise satisfies Dirichlet's Conditions, and we know from § 98 that these partial intervals give to an and hn contributions of the order 1/w. The remainder of the integral, e.g, in a„, is given by the sum of Lt / . ^^ \ cosy?a;cf^ and Lt -^^^^^-cosn^rdr, these limits being known to exist. We take the second of these integrals, and apply to it the Second Theorem of Mean Value. Thus we have Jxo-hB{x-X^)y Jxo+6 (.r - A'o) -^^ \^'-^o) where A'y -H 8 ^ ^ ^ ^. Putting n(x-XQ)=t/, we obtain <^(^) r»P C08(y + »u:o) . But f^-?i(2;±^«)rf^=cos,^„f 5^rfy-sin«../!^ Also when a, 6 are positive. ''-^rfy|. \?"^dy\ Ja }r I •-'a y are both less than definite numbers independent of a and i, when < v < 1. Thus, whatever positive integer n may be, and whatever value S may have, subject to 0< 8<^- Xq, I [^ _i(£L cos nx dx I < K'/ni - ^ 1^0+8(^-^0)' I ' ' where K' is some positive number independent of w and ^. Digitized by Google i: FOURIER'S SERIES 253 It follows that I Lt (^ -^^- cos nxd,v\, sin ma -. I i^(aj + 2a)— ^ da + \ F(x — 2a)—, da. Jo ^ sma Jo sin a Let JUL be any number such that O^^u^^tt. Then J*'' ET/ .CI \ sin ma , j^. , ... f*'" sin7rta , + £{^(a. + 2a)-^(a. + ())}?^cZa + jV(«^+2a)-^(x+0)}^-^da = 7i+/2+73,say (2) We can replace F{x + 0) by F(x), since J^(cc) is continuous at x. Now we know that I — . — - da = ix, Jo sma ^ since on or 2ri. + 1 is an odd positive integer. Thus A = i'^TO (3) Also {F{x+2a)'-F{x)} is bounded, positive and monotonic increasing in any interval ; and 4. — is also bounded, positive and monotonic increasing in 0la , I,= {F(x + 2^.)-Fix)}^J^^-^da, where = ^—/*. But we know that I f" sin via , _ f^ sin a , } J^ sma m ^ ^^ It follows that 4 < — cosec 0. 771 < : > (o) m,sin/i w^here iT is some positive number, independent of m, and de- pending on the upper bound of \f(x) | in ( — tt, tt). Combining (3), (4) and (5), we see from (2) that I If*'' „ I^Jo , ^ . sin ?na , i m \\ <:{F{x + 2f,)-^F(x)} .^-+ ^-^ (6) ^ ^ r-/ V /^sm^ mTT sin /i ^ ' Digitized by Google 256 THE NATURE OF THE CONVERGENCE OF A similar argument applies ix) the integral f^"^ n/ -» . sin7na , but in this case it has to be remembered that F{X'-2a) is mono- tonic decreasing as a increases from to ^tt. The corresponding result for this integral is that 1 1 f*' IT/ o \ sin ma , , it/ \ -I F(x — 2a) —. da — hF(x) ttJo ^ sina - ^ ^ <:\F(x--2^)^F(x)\-J^+ ^^ , (7) K as before being some positive number independent of rriy and depending on the upper bound of \f(x) | in ( — tt, tt). Without loss of generality we can take K the same in (6) and (7). From (6) and (7) we obtain at once i^r F(x + 2af^^^^da-F(x)\ <-A-{\F(x + 2fi)^F(x)\ + \F(x^2fjL)^F{x)\}+ ^^ . (8) Similarly we find that - I G(aj + 2a) -. da — 0(x) < .^ {|(?(a; + 2^)-g(a;)| + |g(ce~2M)~g(a^)|}+ ^^ . (9) Thus, from (1), i^]_^y(x + 2a)-j^^-da-/(x) OlU JUL + \G(x+2^)-G{x)\ + \G(x-2iul)-G(x)\} +^^ (10) mir sm /i Now ^(aj) and 0{x) are continuous in a ti^. Then it follows from (10) that \8n{x)-'f{x)\l. Digitized by Google FOURIER'S SERIES 259 Then we know, from the corollary to Theorem II. § 102, that the series OP Uq + 2(<*n COS nx + hn sin nx) 1 converges uniformly to f{x) in (a, 6). Let us now suppose .r to be any point in the interval (a + 8, 6-8) lying within the interval (a, 6)., With the usual notation 2TraQ=r f{x')dx\ J -IT TTdn— I f(x')C0S7LV'd.v\ J -n Trbn = / f(a/) sin luv' dx'. J -tr J It follows, as in § 103, that (ao - O + 2{(«n - «n ) COS w;r + (6n - 6«') sin tt^} 1 is equal to irMx-a) sin a TrJ^^b-x) ' sma f(x) being defined outside the interval (-tt, tt) by the equation fix+2^)=f{x). Now f(x) is supposed to have not more than a finite number of points (say m) of infinite discontinuity in (-tt, tt), and j \f(x')\dx^ converges. We can therefore take intervals 2yi, 2y2, ... 2y„, enclosing these points, the intervals being so small that i I f(x') I dx' < 2€ sin ^a, [r = 1, 2, . . . wi] J2yr c being any given positive number. Consider the integral /•*' /(^-.2a) ""<^!' + ^>° .fa, ./^(x-a)-'^ '^ sma ^, as already stated, being a point in (a + 8, 6 - 8). As a passes from ^{x-a) to ^tt, we may meet some or all of the m points of discontinuity of the given function iu/(^- 2a). Let these be taken as the centres of the corresponding intervals yj, yg, ... ym. Also the smallest value of (x — a) is 8. Thus Jj(.-^af^I^m^da\<^^JjA.-2a)\" rfa will at most consist of (m + 1) separate integrals (7^). [r = 1 , 2, . . . m + 1.] In each of these integrals (7^) we can take /(^*- 2a) as the difference of two bounded, positive and monotonic increasing functions F{x-2a) and 0{x-2a). Then, confining our attention to (7^), we see that I r 8in(2» + l)a^ I . j ^ ^ sm a I = I / (F- G'){cosec 8/2 - (cosec 8/2 - cosec a)} sin (2n + l)a da I where Ji = cosec 8/2 jF sin. (2/1 + 1 ) a rfa, J^ = cosec 8/2 jo sin (2n -M )a da, J^= f ^{ cosec 8/2 - cosec a} sin (2n + 1 )a rfa, J^= I O {cosec 8/2 - cosec a} sin (2n + l)a da. But we can apply the Second Theorem of Mean Value to each of these integrals, since the factor in each integmnd which multiplies sin(2n + l)a is monotonic. It follows that I /r I < 2^^ cosec ^8, where K is some positive number independent of 7i and x, and depending only on the values of /(:»') in (-x, tt), when the intervals 2yi, 2y2, ... have been removed from that interval. Thus ! / fix -2a) ^. ^»a A' cosec ^8. Since this choice of n is independent of .r, the integral converges uniformly to zero as 7i->oo , when x lies in the interval (a + 8, h - 8). Similarly we find that J^ib-x) ' sma converges uniformly to zero when x lies in this interval. Thus the series («o-0 + 2{(«»-««)cos7?,r + (6„-6„')sinn^} converges uniformly to zero in {a-^-Ki 6-8). Digitized by VjOOQ IC FOURIER'S SERIES 261 But we have shown that the series «o' + 2(«n'cos nx + 6n sin nx) converges uniformly to /(a?) in (a, h). Since the sum of two uniformly convergent series converges uniformly, we see that . ^, , , . v «o + Z,\OLn cos IIX + On SlU UX) 1 converges uniformly iof(.v) in (a +5, 6-5). 109. Differentiation and Integration of Fourier's Series. We have seen that, when f(x) satisfies Dirichlet's Conditions, the Fourier's Series {orf{x) is uniformly convergent in the interior of any interval in which f{x) is continuous. We may therefore integrate the series term by term within such an interval, and equate the result to the integral of f{x) between the same limits. And this operation may be repeated any number of times. With regard to term by term differentiation, a similar statement would obviously be untrue. We have shown that in certain general cases the order of the terms is l/n or I/71K In the first alternative, it is clear that the series we would obtain by term by term differentiation would not converge ; and the same remark applies to the other, when second differential coefiicients are taken. This difiiculty does not occur in the application of Fourier's Series to the Conduction of Heat. In this case the terms are multiplied by a factor (e.g. e-«n%««/a2)^ which may be called a couvergency factor, as it increases the rapidity of the convergence of the series, and allows term by term differentia- tion both with regard to x and t. If f{x) is continuotcs in ( -tt, tt) and f{ -7r)=/(7r), while /'(x) having only a finite number of points of infinite discontiniUty is absolutely integrable, the Fourier's Series fo^ f\^\ whether it converges or noty is giv&n by tertn to term differentiation of that corresponding tof(x). Let ao, «!, 61,..., ao'» «i'> V»'-^® ^^^ Fourier's Constants for f{x) and f\x). Then 27rao= / f{x)cL\ 2'n'aQ — j f'(x)dXf 'n-an= I f(x) cos nxdx, 7ra,/= / fXx) cos nxdi\ J -IT J ~ir . etc. Integrating the expression for a,/ by parts, we see that an = - {/(tt) -/( - tt)} cos wtt + — / f(x') sin nx' dx' = nbn, since we are given /(tt) =/( -tt). Also ao'=Q, Similarly, starting with 6„' and integrating by parts, we see that 6 '=-wa„. Digitized by VjOOQ IC 262 THE NATURE OF THE CONVERGENCE OF Thus the Fourier's Series for f\x) is that which we obtain on differentiating the series for/(^) term by term. On the other hand, with the same assumptions, except that /(tt) ^f /( - tt), the terms in the series for /'(a;) are given by ««'= {/(7r)-/(-7r)}cos7?7r + w6„, TT 6,/= -nan. Thus the Fourier's Series for f\x) does not agree with that which we obtain on term by term differentiation of the series for/(^).* 110. More General Theory of Fourier's Series. The questions treated in this chapter and the' preceding may be looked at from another point of view. . We may start with the Fourier's Series ao + (a, cosa: + 6isinA')4-(«2COs2^ + 62sin2^)4-... , (1) where 27rao = (" f(x') dx\ ^«n= / f(x^)co^iix'dx\ irhn— \ f{af)mu7ix'dx\ the only condition imposed at this stage being that, if the arbitrary function is bounded, it shall be integrable in (-tt, tt), and, if unbounded, the infinite integral / f(x)dx shall be absolutely convergent. J -It We then proceed to examine under what conditions the series (1) converges for 37 = ^0. It is not diflficult to show that-^Ae behaviour of the series at Xq depends only on the nature of the function f{x) in the neighbourhood ofxQ. Also a number of criteria have been obtained for the convergence of the series at Xq to i[f{xQ + 0)+f{xQ-0)], when these limits exist.t Further, it has been shown that, if (a, 6) be any interval contained in ( - TT, tt), such that f{x) is continuous in (a, 6), including the end-points^ the answer to the question whether the Fourier^s Series convei^ges uniformly in (a, 6), or not, depends only upon the nature off{x) in an interval {a\ 6'), which includes (a, b) in its interiory and exceeds it in length by an arbitrarily small amount. Again it can be shown that, when f(x) has only a finite number of points of infinite discontinuity ^ while I f(x)dx converges absolutely, then j f{x)dxy J—w Ja where -7r = a<^^7r, is represe^ited by the convergent series obtained by ♦See also Gibson, Edinburgh, Proc. Math. 5oc., 12, p. 47, et seq,, 1S94. t Cf. de la Vallee Poussin, loc. cit., T. II., §§ 137-143. Digitized by Google FOURIER'S SERIES 263 integrating the Fowier^s Series (1) term hy termi^ no assumption being made as regards the convergence of the original Fourier^ s Series. The reader who wishes to pursue the study of Fourier's Series on these lines is referred to the treatises of Hobson and de la Valine Poussin. REFERENCES. B6cHER, loc, cit. ' De la Vall^e Poussin, loc. cit,, T. II. (2' 6d.), Ch. IV. • HoBSON, loc. cit., Ch. VII. LiEBESGUE, L^^mis sur les s&ies tngonom^triques, Ch. III., Paris, 1906. Whittaker and Watson, loc. cit., Ch. IX. Note. — The treatment of the uniform convergence of Fouriei-'s Series in § 108 is founded on the proof given by Whittaker and' Watson on'pp. 169-170 of the 1915 edition of their Modem Analysis. In the 1920 edition the authors have made considerable changes both in the proof of Fourier's Theorem (9. 42) and in the discussion of the uniform convergence of Fourier's Series (9. 44). Digitized by Google CHAPTER IX THE APPROXIMATION CURVES AND GIBBS'S PHENOMENON IN FOURIER'S SERIES* 111. We have seen in § 104 that, when f{x) is bounded and continuous, and otherwise satisfies Diriehlet s Conditions in — '7rco In Fig. 32, the line y = x and the curve 2/ = 2 (sin a; — 1 sin 2a; + \ sin 3aj — \ sin 4a; + 1 sin bx) . are drawn, and the diagram might seem to confirm the above view of the matter — namely, that there will be a steep descent Fig. 32. near one end of the line, from the point ( — tt, 0) to near the point ( — TT, — tt), and a corresponding steep descent near the other end of the line. But it must be remembered that the convergence of this series is slow, and that n = b would not count as a large value of n. Digitized by Google 268 THE APPROXIMATION CURVES AND / 113. In -1899 Gibbs, in a letter to Nature,'^ pointed out that the approximation curves for this series do, in fact, behave in quite a different way at the points of discontinuity iTr in the sum. He stated, in effect, that the curve y = £f„(aj), for large values of n, falls from the point ( — tt, 0) at a steep gradient to a point very nearly at a depth 2 1 dx below the axis of x, then Jo ^ oscillates above and below y = x close to this line until x ap- proaches TT, when it rises to a point very nearly at a height 2 1 -dx above the axis of x and then falls rapidly to (tt, 0). Jo ^ ^ The approximation curves, for large values of n^ would thus approach closely to the continuous curve in Fig. 33, instead of the straight lines in Fig. 32, for the interval ( — x, x). Fig. 33. His statement was not accompanied by any proof. Though the remainder of the correspondence, of which his letter formed a part, attracted considerable attention, this remarkable observa- tion passed practically unnoticed for several years. In 190C Bocher returned to the subject in a memoir f on Fourier's Series, ♦ Nature, 59, p. 606, 1899. \Ann. Math., Princeton^ N.J, (Ser. 2), 7, 1906. See also a recent paper by Bocher in J, Math,, Berlin, 144, 1914, entitled **0n Gibbs's Phenomenon." Reference should also be made to Runge, Theorie u. Praxis der Beiken, pp. 170-180, Leipzig, 1904. A certain series is there discussed, and the nature of the jump in the approximation curves described ; but no reference is made to Gibbs, and the example seems to have been regarded aer quite an isolated one. Digitized by VjOOQIC GIBBS'S PHENOMENON IN FOURIER'S SERIES 269 and greatly extended Gibbs's result. He showed, among other things, that the phenomenon which Gibbs had observed iii the ease of this particular Fourier's Series holds in general at ordinary points of discontinuity. To quote his own words : * If f(x) has the period 27r and in any finite interval has no discontinuities other than a finite number of finite jumps, and if it has a derivative which in any finite interval has no dis- continuities other than a finite number of finite discontinuities, then as n becomes infinite the approximation curve y = Sn{x) approaches uniformly the continuous curve made up of (a) the discontinuous cu/rvey^fix), (b) an infinite number of straight lines of finite lengths parallel to the axis of y and passing through the points a^, ag. . . . on the axis of x where the discontinuities of f(x) occur. If a is any one of these points, the line in question extends between the two points whose ordinates are DP DP TT TT where D is the magnitude of the jump in f{x) at a,t and J It since x daj= -0-2811. a^+2Tr FIO. 34. This theorem is illustrated in Fig. 34, where the amounts of the jumps at a^, a^ are respectively negative and positive. Until *loc. a^,p. 131. tt.e. Z>=/(a + 0)-/(«-0). Digitized by Google 270 THE APPROXIMATION CURVES AND Gibbs made his remark, it was supposed that the vertical lines extend merely between the points whose ordinates are/(azfcO). The series in the examples of Chapter VII. converge so slowly that the approximation curves of Figs. 21-23, 27, 28 are of little use in this connection. 114. The series on which Bocher founded his demonstration of this and other extensions of Gibbs's theorem is sinflj+i^sin2a;+^sin3a;+... , which, in the interval (0, 27r) represents the function f{x) defined as follows : ^ /(O) =f{2ir) = 0, f{x)^\{ir-x\ 0oo bourhood of a point where the sum of the series is discontinuous, and they also may do so in the neighbourhood of a point where the sum is continuous. This question we have already examined in Chapter III. But, in passing, it may be remarked that when f{x) satisfies Dirichlet's Conditions, the approximation curves for the corresponding Fourier's Series will, for large values of n, differ only slightly from the curve y =f(x) in the neighbourhood of points at which f(x) is continuous, for we have shown that the Fourier's Series converges uniformly in any interval contained within an interval in which f{x) is continuous. The existence of maxima (or minima) of Sn(x)y the abscissae of which tend towards a as ti increases (a being a value of x for which the sum of the series is discontinuous), while their ordinates remain at a finite distance from f(a+0) and/(a — 0), is the chief feature of Gibbs's Phenomenon in Fourier's Series. And it is most remarkable that its occurrence in Fourier's Series remained undiscovered till so recent a date.* The Trigonometrical Sum Sn(x) = 2 (sin 0? + ^ sin Sx+,,. + ^ _ sin (271 — 1) x) • 115. If we define /(a;) by the equations /(-7r)=/(0)=/(7r) = 0, ] /(a;) = j7r..., 0 ^4 = ^> • • • ' ^(»-i) = -^'^ (mimma). We have f ^ sin 2na ' , dy sin 2nx y = I — ; day and -j^ = — -. ^ J (J sin a dx smo; The result follows at once. Digitized by VjOOQIC GIBBS'S PHENOMENON IN FOURIER'S SERIES 273 IV. As we proceed from aj = to aj = j7r, the heights of the maxima aontinually diminish, and the heights of the minima continually increase, n being kept fixed. i^Eiiili .^0. The curve y = *S^„ W, when n = 6. Fia. 35. Consider two consecutive maxima in the interval < aj ~ ^tt, namely, Snl—^ '^j and Sni — ^ xj, m being a positive integer less than or equal to ^(n — l). We have « /2m-l \ CI /2m+l \ 1 f(2»*-i)' sina , sm 2n /.(2m+l), gJn^ I — — da I 2mv Sin 2n The denominator in both integrands is positive and it con- tinually increases in the interval (2m — ]) tt^ a = (2m-f- 1) tt ; also the numerator in the first is continually negative and in the second continually positive, the absolute values for elements at equal distances from (2m — l)'7r and 2m7r being the same. Digitized by Google 274 THE APPROXIMATION CURVES AND Thus the result follows. Similarly for the minima, we have to examine the sign of where m is a positive integer less than or equal to ^n. V. The first maximum to the right of x — is at ^ = 9- ^^''^^^^ its height cmitimuilly diminishes as n increases. When n tends to infinity, its limit is p, gjj^ ^ The curves y=Sn{x\ when n = l, 2, 3, 4, 5 and 6. Fig. 36. We have cosec ^ da. Thus c, / '7r\ f2*t sin 2na ^ If. \2n/ Jq sin a 2^13 ^ f ' . / 1 a 1 a \ , = 1 sm a ( ^r— cosec 5 ^ — .-^ cosec ^ — —^ I da, Jo \2n 2n 2n + 2 2n + 2/ Since a/sin a continually increases from 1 to 00 , as a passes from to TT, it is clear that in the interval with which we have to deal 1 a 1 a ^ ^ i cosec — — r^ >► 0. 5— cosec „ ^ , ^, 2n 2n 2n + 2 271+2' Digitized by VjOOQ IC GIBBS'S PHENOMENON IN FOURIER'S SERIES 275 Thus ^«©-^-.>(2(^)>0- But, from (I), Sn{x) is positive when 0<» \^7fr/ Jo "^ Jo '^ Jo '*' n— >oo VI. The result obtained in (V) foi> the first wave is a special case of the following : 2r— 1 The r** mdxiTnv/ni to the right 0/ a; = is at aJ2r-i=~9 ''''» and its heipht continually diminiahes as n increases, r being kept constant. When n tends to infinity y its limit is ■(2r-l),rginaj Jo X ■dXy "which is greater than Jtt. r The r** rrtinimum to the right of x = is at x^^—tt^ and its height continually increases as n increases, r being kept constant. When n tends to infinity , its limit is I dx, which is less than Jtt. ® To prove these theorems we consider first the integral a 1 cosec 2^ -^ — -7, cosec w 2n 2n 27i+2 27i + T2)^«' * Ann. MatJi.y Princeton, N.J. (Ser. 2), 7, p. 124. Also Hobson, loc. cit., p. 649. Digitized by Google 276 THE APPROXIMATION CURVES AND m being a positive integer less than or equal to 2n — 1, so that 0<2--0 in this interval. (Cf. (V).) Further, Tjift \1 ^ 9^ 1 a ^ a ^ \P) = /6 — Tow COS ^ — -"5 cosec^ ^ — —^ - TK-^ COS jr- cosec^ ^- (2n + 2)2 2n + 2 2?i + 2 (2n)2 27i 2^1 = a-2{02 cQg cosec^ ^ - -^^ ^^g ^ cosec^ i/r}, where = a/(2n + 2) and i/r = a/2ti. But ;7;a(^^ ^^^ ^ cosec^ 0) = — cosec^ 0[^( J^ Tcos 0)^± 2 cos 0(0=Fsin 0)]. And the right-hand side of the equation will be seen to be negative, choosing the upper signs for O<0<|7r and the lower for^x<0<'7r. Therefore 0^cos cosec^^* diminishes as increases from to -tt. It follows, from the expression for F\a), that J''(a)>0,. and F{a) increases with a in the interval of integration. The curve thus consists of a succession of waves of length tt, alternately above and below the axis, and the absolute values of the ordi- nates at points at the same distance from the beginning of each wave continually increase. It follows that, when m is equal to 2, 4, .,.,2(ti— 1), the integral f«*' . /I a 1 a \j J '^^ <2^ ^^^^^ 2^ - 2^H:2 ^^^^^ 2^H:2r« is negative ; and, when m is equal to 1, 3, ..., 27i — 1, this integral is positive. Returning to the maxima and minima, we have, for the ?'*** maximum to the right of aj = 0, Digitized by Google GIBBS'S PHENOMENON IN FOURIER'S SERIES 277 2r-l 2 r-l f 2n ' sin2na , f2(H+i)' sin2(7i + l)a , = 1 . -aa— 1 ^ —da } Q sm a J Q sin a f(2r-i), . /I a 1 a \, = 1 sm a ( ^ cosec ^^ — -^cosec ^ — -^ ) aa. Jo \2n 2n 2n + 2 2n+2j Therefore, from the above argument, S^CTg^-i) >^«+i(i»2r-i)- Also for the r^^ minimum to the right of cc = 0, we have and S^(x^) < 8n+i (x^,.)- By an argument similar to that at the close of (V) we have Jo Lt SJXr)=\ '?^^dx* CO It is clear that these limiting values are all greater than Jtt for the maxima, and positive and less than ^tt for the minima. GiBBs's Phenomenon for the Series 2 (sin 0? + ^ sin 3ii;+isin5a;+ ...). 116. From the Theorems I.- VI. of § 115 all the features of Gibbs's Phenomenon for the series 2(sinaJ + ^sin3iic+^sin5iC+...).,., —Tr^oj^'Tr, follow imnuediately. It is obvious that we need only examine the interval ^ a? ^ tt, and that a discontinuity occurs at a? = 0. For large values of ri, the curve y=Sn{x), where Snix) = 2 (sin a; + ^ sin 3a? + . . . + ^ _.. sin (2ti — 1 ) aj j, rises at a steep gi'adient from the origin to its first maximum, which 0, dx j (§ 115, v.). The curve, then, falls at a steep gradient, without reaching the axis of aj(§ 115, II.), to its first minimum, which is very near, but below, the point (o, 1 dxj{^ 115, VI.). It then oscillates *For the values of j^—dx, see Ann. Math,, Princeton, N.J. (Ser. 2f), 7, p. 129. J' ^ Digitized by VjOOQ IC 278 THE APPROXIMATION CURVES AND above and below the line y = \'7r, the heights (and depths) of the waves continually diminishing as we proceed from a; = to ^ = I'T (§ 115, ly.); and from x=^Tr to aj = x, the procedure is reversed, the curve in the interval O^aj^Tr being symmetrical about 05 = Jtt (§ 115, 1.). The highest (or lowest) point of the r^^ wave to the right of a? = will, for large values of n, be at a point whose abscissa is vtt ^ (§ 115, III.) and whose ordinate is very nearly 'sma; X dx (§ 115, VI.). By increasing n the curve for = close as we please to the lines x = 0, 0 when n = r, behaves as follows : It rises at a steep gradient from the origin to its first maximum, I' sin QK dx and within the rectangle ^ 0/r(a + 0)=i/r(a — 0) = i/r(a) = 0, and y}r(x) is continuous at x = a. The following distinct steps in the argument are numbered for the sake of clearness : (i) Since ^(x) is continuous at a; = a and >/r(a) = 0, if e is a positive number, as small as we please, a number fj exists such that e I yjr{x)\ {x-a) that \J/{x) does so also. Digitized by VjOOQ IC GIBBS'S PHENOMENON IN FOURIER'S SERIES 281 Let Sn(x)y ^ni^—a) and (rn(x) be the sums of the terms up to and including those in sin nx and cos nx in the Fourier's Series for f(x)y r ?^d.. TT Jo ^ Digitized by Google 282 THE APPROXIMATION CURVES This may be written /(a+0)-A«+0)-/(«-0) f ?i^ dx, IT J w ^ and, from Bocher's table, referred to in § 115, we have r«iE^rf^=_0.2811. in X It then oscillates about y = f(x} till x reaches a + 1;, the character of the waves being determined by the function 0n(a?— ct), since the term , an assumption which, of course, would have to be defended if the proof were to be regarded .as in any degree complete, its value would be 1 f " f°^ - I da\ f(x') cos a {x' — x) dx\ ^0a, such that \[ f¥f^^^^~^dx' <%, when A'^A>a, (2) the same A serving for every value of q. But we know from § 94 that q-^tn Ja-X - =0, since f{n+x) satisfies Dirichlet's Conditions in the interval (a — 05, 4 — flj), and both these numbers are positive. Digitized by Google 286 FOURIER'S INTEGRALS Thus we can choose the positive number Q so thut |jy^'"> x'-x ^ ^<2' ^hen q^Q....: (3) It follows 'from (2) and (3) that when q^Q, Thus • itr/(a.')«>^^-^>da.'=0, poo poo and from (1), rfa f{x')coaa{x'''X)dx' = (4) But, by §87, I da\ f{x') cos a{x'-x)dx'= \ dx'^ f{x') cob a(x' — x) da jy^^,^ sinqjx'-x) ^^, = rf(n+xf^dn. Letting g->oo , we have j da^J{x')cosa(x'-x)dx' = '^f(x + Ol (5) when /(«:+()) exists. , Adding (4) and (5), we have da^J(x')coBa(x' ''X)dx' ^^f{x + Ol (6) when f{x + 0) exists. Similarly, under the given conditions, J c?aj f{x')cosa{x'-^x)dx' = '^f(x-Ol (7) when f{x — 0) exists. Adding (6) and (7), we obtain Fourier's Integral in the form 1 «»c» poo - da\ f{x')co8a{x' -x)dx' = \[f{x-\-0)+f{x-0)\ " J Q J -co for every point in — oq -ao Consider the integral / f{x') cos a (^ - ^) dx'. By the Second Theorem of Mean Value, I f(^) cos a(x' - x)dx^ ^fi-^') j , cos tt(^ - .r) dx^ +f(A") I cos a(x^ - x)dx', J A J A Jg where a< 4'^^^ A". Thus / f(x') cos a(x'-x) daf —^-^ — ' I cos u du H-* ^ ^ 1 cos u du. I f^^'fi^') cos a(x' - x) dx' I < i\JMll for a ^ ^o > 0. But we are given that Lt f{x)=0. /•OO It follows that / f{x') cos a{x' - x) dx' is uniformly convergent for a = q^>0, and this integral represents a con- tinuous function of a in a'^q^J*^ Also, by § 85, \ da I f{x') cos a{x' - x) dx = / dx' \ f(x') cos a{x' - x) da Jqo Ja Ja Jqo = rf(a/)l^^ g(^' -^) _8in go(x' -.r)| ^^ ^^^ J a \. X — X X — X J But x' - x^ a - x> iu the interval x' ^ a, since we have chosen a greater than X. And [A-^f^^^^^^^, [a^)'^^^¥^'^ both converge. * These extensions are due to Pringsheim. Cf. Math, Ann., Leipzig y 68, p. 367, 1910, and 71, p. 289, 1911. Reference should also be made to a paper by W. H. Young in Edinburgh, Proc. R. Soc, 31, p. 559, 1911. Digitized by Google 288 FOURIER'S INTEGRALS Tlierefore, from (1), / da I f(a;')cosa(,v -.i')cLv' Jgo Ja = fj(--r-flr^d.rf-[m''^^^^^d.r' (2) Now consider tlie integral From the Second Theorem of Mean Value, =f(A')f^:^^^ (^> wherea<2l'^fS^". AlBo P ^1^M'^-J^d.7f= r^^"*^5Mrfu, JA' X -^ hoW-x) u the limits of the integral being both positive. Therefore I Y^ sin yo(^ - ^) ^ J ^ (Cf. §91.) \Ja' X —X I And similarly, , | /'''' "" ^"^^"""^ d^\<^- Thus, from (3), |/>.^o ""^;y,:"^ ^^i<2H/(^T (4) It follows that r f(a!)^^^^^M^^^^d,rf J a X -X is uniformly convergent for 9o = ^^ ^^'^ by § 84 it is continuous in this range. Thus Lt r/(y)?^oiflz5.),ip' = 0, (5) since the integral vanishes when go=^- Also from (2), ^rfa^/(.^•')cosa(y-.r)c^.r'= C f(affy^^^^^^ d.x' (6) ./O }fi Jo, X X But we have already shown that the integral on the right-hand side of (6) is uniformly convergent for q^O, Proceeding as in § 119, (2) and (3),* it follows that Lt rf{^f^^\^-^-dx'^o. Thus, from (6), f daf f{x')co&a{x' -x)dx'.= (7) .'0 Ja * Or we might use the Second Theorem of Mean Value as proved in § 58 for the Infinite Integral. Digitized by Google FOURIER'S INTEGRALS 289 But we know from § 119 that ^ r daT f(a;')co%a{a/-x)dx'='^f{a; + 0), (8) when this limit exists, /•OO /.OO .(9) Thus ^-Tdaf f(x')QO^a(x' -x)daf =^y(x-\-0\ ^yO Jx when this limit exists. Similarly, under the given conditions, we find that -TdaT /(^')cosa(^'-.r) = i/(.^-0), (10) when this limit exists. Adding (9) and (10), our formula is proved for every point at which f(x±0) exist. 121. Other conditions for f(x). We shall now show that Fourier's Integral formula also holds when the conditions at ± oo of the previous section are replaced by the following : (I.) For some positive number and to the right of it, and for some negative number and to the left of it, f(x') is of the forHi g(x')cos{)^ -^-i*), where g{a/) is bounded and monotonic in these intet^als and has the limit zero as od-^ ± oo . Also,{\\.) r^dx' and j_^ ^ dx' converge. We have shown in § 120 that when g(x') satisfies the conditions named above in (I.), there will be a positive number a greater than x such that /•oo i»00 / dal g{x^)co8a{x'-x)dx^=0. Jo Ja But, if k is any positive number, /•oo /-co / da g(x')coaa(a/-x)dx' Jo Ja = / da g(x^)cosa{x^-x)dx^+ da g(x')co»a(x'-x)dx' -0 Ja Jk Ja da I g{x') cos a(x^ -x)dx'+\ dal g(x') cos a(x' - x) dx' -k Ja •'A J a = \r daT g{xf)co8a(x'-x)dx'-\-\j daj g{x')cosa{x' -x)d:v' ^iTda Cgix') cos (a + X)(x' - x) daf + ^ /* da( g( .r/jcos (a -^ A)(.«^ - x)dv'. Therefore /•oo /•oo / dal g{x')co8a(x' -x) dx' /.ao /•oo = / daj g{x')cosX{x'-x)co8a{x'-'X)dx'^0 (1) Jo Ja Digitized by VjOOQIC 290 FOURIER'S INTEGRALS Again we know, from Ji 120, that / cr(^) sin a{.v' - .v) dx' Ja is uniformly convergent for a ~ Ao > 0. It follows that / da j g{:tf)Bma{a/ -x)da/ Jfio Ja exists, for Aj > Aq > 0. Also / da I g {a;') Bin a{.'t/ — x) da/ = / cLv' I g{a/) sin a{a/ - x) da (by § 84) Ja J\q =/;y(^)gg^-^w-/;<,(y) °^y5^-^w , ...(2) since both integrals converge. But we are givei^ that / ^^^-— dx' converges ; and it follows that \ ^,^— ' dcxf Ja X — X also converges, so that we know that is uniformly convergent for A^ ^ 0, and therefore continuous. Thus Lt r^(.r-)^^« w-^)d^-= rsi^dx\ Ao-^O Ja x-x Ja X -X It follows from (2) that, when A > 0, Pdaj%{x')sma(.it/-'X)d7/=^ f^ p^dv' ^ j^ g(x'f-^^^^^ dx'. (3) Also, as before, we find that, with the conditions imposed upon g{x^\* Lt rg{a/)SS^M^^dv'=0. Therefore, from (2) and (3), r daT g(x')sma(x^-x)dv'= H gix'f^^^'"^^ dx', (4) Jk Ja Ja X —X and r daT g{x')mna(x'-x)dx'= P p^dx' (5) * This can be obtained at once from the Second Theorem of Me^n Value, as proved in § 58 for the Infinite Integral ; but it is easy to establish the result, as in § 119, without this theorem. Digitized by Google that FOURIER'S mrfiORALS 291 Again, since L ^^1 3 ("^^ ^^^ " (^' ~ ^*) ^'^' c?a/ 5r(y)8ina(:t'-^)rf^=0 (6) c?a / g(x') sin a(.i?' - ;f) cbf^ it follows at -^ •'^ • c?a/ ^(ar')sina(y-ar)c^- / rfal g{x')%iiiia(3i/ -x)cLxf—0, Thus mQO moo /*ao /•^ / c?a/ (7(y)sin(a + A)(^'-j?)(ir'- / da I g(x')9\i\(a- k)(a/ - x)daf =0. Therefore \ daj g{x')Bin \{x^-x)coBa(x'-x)cb/=0 (7) Multiply {I) hy COB (Xx+fi) and (7) by sin(A-r+/x) and subtract. It follows that /•OO i»0O / dal g {a/) COB {Xa/ + fi) COS a(a/-'X)dx^=0 (8) Jo Jet And in the same way, with the conditions imposed upon g{x')y we have / dal g{xl)coB(ko(/+ii)coBa(a/'-x)dx'=0 (9) These results, (8) and (9), may be written /.ao /•OO > / ^^\ f(^) COS a(x^ - x) dot! = 0, Jo Jo> I daj '^ flx^) COB a(x^^x)dx^=Oy Jo J-to ♦ J .(10) when /(.r')=5r(^) COS (A:t'' + /ui) in (a, oo ) and ( - oo , -a'). But we know that, when f(x) satisfies Dirichlet's Conditions in ( - a\ a), j^ darj(x')coBa(x' ■-x)daf = ~[f{x+0)-{-f{x-0)l (11) when these limits exist. [Cf. § 119 (5).] Adding (10) and (11), we see that Fourier's Integral formula holds, when the arbitrary function satisfies the conditions imposed upon it in this section. It is clear that the results just established still hold if we replace cob(\x -\'Im) in (I.) by the sum of a number of terms of the type a„cos(A„^'+/A„). It can be proved * that tlie theorem is also valid when this sum is replaced by an infinite series „ 2an cos (A„:r' +/!„), 1 when 2«n converges absolutely and the constants A„, so far arbitrary, tend to infinity with n. *Cf. Pringsheim, McUh. Ann., Leipzig, 68, p. 399. Digitized by VjOOQIC 292 FOURIER'S INTEGRALS 122. Fourier's Cosine Integral and Sine Integral. In the case when f{x) is given only for positive values of a?, there are two forms of Fourier's Integral which correspond to the Cosine Series and Sine Series respectively. I. In the first place, consider the result of §119, when j\x) has the same values for negative values of aj as for the corre- sponding positive values of x\ i.e. f('-x)=f(x\ a;> 0. Then - c?a I f(x')coQa(x—x)dx' 1 f«' f* = -l da\ f(x')[co9a(x'+x)+cosa(x'—x)]dx '^ Jo Jo 2 r** Too = - I da\ f(x) cos ax cos ax'dx\ It follows from §119 that wJien f{x) is defined for positive values of x, and satisfies Dirichlet's Conditions in any finite interval, while I f{x)dx converges absolutely, then Jo 2 poo pCO " ^« I /(^') COS ax cos axdx' = J [f(x+ 0) +f(x - 0)], '^ J Jo at every point where f(x+0) and /(x— 0) exist, and when x=0 the value of the integral is f( + 0), if this limit exists. Also it follows from §§120 and 121 that the condition at infinity may be replaced by either of the following: (i) For soTue positive number and to the right of it, f{x') shall be bounded and monotonia and Lt f(x')=0] or, (ii) For some positive number and to the right of it, f(x) shall be of the form g{x') cos(\x' + jii), where g{x') is bounded and monotonic and Lt g(x) = 0. Also I ^ -,-- ax must converge. II. In the. next place, by taking f{—x)=^f{x), iK>0, we see that, when f(x) is defined for positive values of x, and satisfies Dirichlefs Conditions in any finite interval, while f{x) dx converges absolutely, then ^ 2 p» r» ; cZa f{x')9\nax9max'dx' = l[f{x+0)+f(x-'0)'\, J J 'TT , Digitized by Google FOURIER'S INTEGRALS 293 at every point where f(x+G) and /(« — 0) exist, and when x = the integral is zero. Also it follows from ^120 and 121 that the condition at infinity may be replaced by one or other of those given under I. It should be noticed that, when we express the arbitrary function f(x) by any of Fourier's Integrals, we must first decide for what value of x we wish the value of the integral, and that this value of x must be inserted in the integrand before the integi-ations indicated are carried out (cf. § 62). 123. Fourier's Integrals. Sommerfeld's Discussion. In many of the problems of Applied Mathematics in the solution of which Fourier's Integrals occur, they appear in a slightly different form, with an exponential factor (e.g. c-««^«) added. In these cases we are concerned with the limiting value as t->0 of the integral {t) = - [ da( f (a;') COS a(a/-j;)e-'^'^^i daf, TT Jq Ja and, so far as the actual physical problem is concerned, the val\ie of the integral for ^=0 is not required. It was shown, first of all by Sommerfeld,* that, when the limit on the right-hand side exists, Lt i f da( f(x')Qo^a(x'-x)e'-'^«^tdaf=\[f(x^O)-\'f(x-0)\ t—*fi TT Jo J a when a0), prove^that r asinour ^ , Jo ^?+F 2"^ '^^ and in the same way, from the Cosine Integral, prove that /•" cos CUP , TT . 3. Show that the expression is equal to a;* when ^.r a, 4. Show that - / sin go;-! - + tan a?^5J!?JL?i5-2? I dq is the ordinate of a broken line running parallel to the axis of x from 07=0 to j:=a, and from ar=6 to ^= oo , and inclined to the axis of x at an angle a between x^a and x=^h, 6. Show ihsX f{x)=-j-\ — i satisfies the conditions of §120 for Fourier's Integral, and verify independently that 2 6. Show that /(^) = satisfies the conditions of §121 for Fourier's / da I cos ow; coBax^—r-,=-r when x > 0. Jo Jo v*^ v-^ Xx)= satisfies tb ^ ^ X Integral, and verify independently that , , .dx' 8in.r Google TTJo J-' , , J .c?.r 8in.r sm X cos a\x - x) —r = • ^ x^ X Digitized t)y* APPENDIX I PRACTICAL HARMONIC ANALYSIS AND PERIODOGRAM ANALYSIS 1. Let y=f(x) be a given periodic function, with period 27r. We have seen that, for a very general class of functions, we may represent f{x) by its Fourier's Series Uq+u^ cos aj+ag cos 2x+ . . .1 + b^Bmx+b^ sin 2x+..J' where aQ, a^, a^y ... fej, ftg* ••• ^^^ ^^^^ Fourier's Constants for /(a;). We may suppose the range of x to be O^x^iir. If the period is a, instead of 2-^, the terms . ® nx are replaced by . Imrxja, and the range becomes O — aj—a. However, in many practical applications, y is not known analytically as a function of a;, but the relation between the dependent and independent variables is given in the form of a curve obtained by continuous observations. Or again, we may only be given the values of y corresponding to isolated values of X, the observations having been made at definite intervals. In the latter case we may suppose that a continuous curve is drawn through the isolated points in the plane of cc, y. And in both cases the Fourier's Constants for the function^ can be obtained by mechanical means. One of the best known machines for the purpose is Kelvin's Harmonic Analyser.* 2. The practical questions referred to above can also be treated * 8uoh mechanical methods are descril)ed in the handbook entitled Modem Instnimtivta and Methods of GalctUation, published by Bell & Sons in connection with the Napier Tercententary Meeting of 1914. 295 Digitized by Google 296 APPENDIX by substituting for Fourier's Infinite Series a trigonometrical series with only a limited number of terms. Suppose the value of the function given at the points 0, a, 2a, ... (m — l)a, where ma = 27r. Denote these points on the interval (0, 27r) by Xqj X^y X^f ... ^m-iy and the corresponding values of y by Vo* Vv 2^2 > ••• 2/w-i' Let S„(aj) = a© + a^ cos oj+ctg cos 2a; + ... +a„coswx| + 6i sina; + fe2sin2a;+ ... +&,,sinwa;J* If 2n+l=m, we can determine these 2n+\ constants so that Sn(Xs) = yg, when s = 0, 1, 2, ... 2w. The 2n+l equations giving the values of ao, a^, ... a„, &i, are as follows : ao+ai+... +ar+... +a„ =2/o aQ + a^C08Xi + ,.. +«r cos ra;i+... + an cos wa^ \ _ +h^8inxi +... +brsinrXi +.., +bnSinnoc^ f hny ■yi Oo+ai cos a«jH+...+ar cos ra2n+...+On cos wa^gn) _,, + 6i sm QC.^+.., + hr sin ra;2n+ ... +6n sm nx^n) Adding these equations, we see that 2n (2w+l)ao=S3/«> since 1 + cos ra+ cos 2ra+ ... +cos2wra = 0, and sin ra + sin 2ra+ ... +sin2wra = 0, when (2w+l)a = 27r. Further, we know that 1 + cos ra cos sa + cos 2ra cos 2sa + . . . + cos 2nra cos 2wsa = 0, s=^r, cos ra sin sa + cos 2ra sin 2sa f?' = 1, 2, . . . nl + ...+cos2nrasin2wsa = 0, U = l,2, ... n] And 1 +cos2 ra + cos^ 2ra + . . . + cos^ 2wra = ^ (2n+ 1). It follows that, if we multiply the second equation by cosrXi, the third by cos rx^, etc., and add, we have J(2n+l)a;.= 22/«cos rsa. Digitized by Google APPENDIX 297 Similarly, we find that 2m «=i A trigonometrical series of (2w+l) terms has thus been formed, whose sum takes the required values at the points 0, a, 2a, . . . 2na, where {2n+l)a = 27r. It will be observed that as w-^oo the values of a©, «!,... and 6i, ftj* ••• reduce to the integral forms for the coefficients, but as remarked in § 90, p. 199, this passage from a finite number of equations to an infinite number requires more careful handling if the proof is to be made rigorous. 3. For purposes of calculation, there are advantages in taking an even number of equidistant points instead of an odd number. Suppose that to the points 0, a, 2a, ... (2w— l)a, where Wa = '7r, we have the corresponding values of y, 2/o»2/i»2/2>---2/2«-i. V In this case we can obtain the values of the 2n constants in y^the expression a^+a^ cosx+a.2 cos 2x+ . . . +a„_i cos (n—l)x+an cos nx] + b^ sinx+b^ sin 2x+ . . . + 6n_i 8in{n—l)x f* so that the sum shall take the values j/q, J/p ... y^n-i at these 2n points in (0, 27r). It will be found that 1 ^'*"^ 2n 1 'i ^»- i 2 2n-l 2n ;._-o 2 2n-l ^r= „ ^ ys sin rsa r/n. Runge * gave a convenient scheme for evaluating these con- stants in the case of 12 equidistant points. This and a similar table devised by Whittaker for the case of 24 equidistant points * Zs, Math.y Leipzig, 48, 1903, and 52, 190i>. Also Theorie u. Praxis der Beihen, pp. 147-164, Leipzig, 1904. Digitized by VjOOQIC 298 APPENDIX will be found in Tract No. 4 of the Edinburgh Mathematical Tracts* 4. This question may be looked at from another point of view. Suppose we are given the values of y, viz. yo» 2/p ^2, •••2/m-i, corresponding to the points 0, a, 2a, ...{m— l)a, where ma = 27r. Denote these values of x, as before, by a?o> ^^ ^> '•• ^m~V Let Sn(x) — aQ+aiCosx+a2COB2x+ ... +a,jCos7ia;^ + 6i sin x+ 62 sin 2a; + ... +bn sin nxf For a given value of n, on the understanding that m>2n + l,f the 271 + 1 constants a^^ a^, ...On, fe^, ...6^ are to be determined so that Sn{x) shall approximate as closely as possible to 2/o» Vv" Vm-i at a^o. ajj, ... x^.^. The Theory of Least Squares shows that the closest approxi- mation will be obtained by making the function «=o regarded as a function of aQ, a^, ... a^ b^y ... &„, a minimum. The conditions for a minimum, in this case, are : S (ys-SAx,))^o S (y>-Sn{x,))cospx.=0 \p = l, 2,...n. «=0 m-1 2 (2/» — 'Sfn(a;,))sinpa;, = It will be found, as in § 2 above, that these equations lead to the following values for the coefficients : ♦ Carse and Shearer, A Course in Fourier's AncUysia and Periodogram Analysis, Edinburgh, 1916. tif m <2)i + l, we can choose the constants in any number of ways so that Sn{x) shall be equal to yo, yi, ... y«-i at Xq, arj, ... a:^_i, for there are more con- stants than equations. And if m = 2?i + 1, we can choose the constants in one way so that this condition is satisfied. Digitized by Google «=0 APPENDIX 299 m-l r=l, 2. ..71, m odd. imar=^y^coar8a »=o TO-l But if m is even, the coefficient Or (when r=^i7n) is given by m-l the others remaining as above. In some cases^ it is sufficient to find the terps up to cos a; and sin je, viz. a© + ^1 cos <^ + ^1 sin X. The values of ag, a^ and 6|, which will make this expression approximate most closely to .yo, .Vl, ^2) '-^m-l at 0, OL, 2oL, ...(»i-l)oL, when wa.=27r, are then given by : m-l tn-1 Jwuij = 2 y« COS «a, «=o' m-l Tables for evaluating the coefficients in such cases have been constructed by Turner.* 6. In the preceding sections we have been dealing with a set of observations known to have a definite period. The graph for the observations would repeat itself exactly after the lapse of the period ; and the function thus defined could be decomposed into simple undulations by the methods just described. But when the graph of the observations is not periodic, the function may yet be represented by a sum of periodic terms whose periods are incommensurable with each other. The gravitational attractions of the heavenly bodies, to which the tides are due, are made up of components whose periods are not commensurable. But in the tidal graph of a port the component * Tables for Harmonic Aivodyais, Oxford University Press, 1913. Digitized by VjOOQ IC 300 APPENDIX due to each would be resolvable into simple undulations. A method of extracting these trains of simple waves from the record would allow the schedule of the tidal oscillations at the port to be constructed for the future so far as these components are concerned. The usual method of extracting from a graph of length L, a part that repeats itself periodically in equal lengths X is to cut up the graph into segments of this length, and superpose th^m by addition or mechanically. If there are enough segments, the sum thus obtained, divided by the number of the segments, approximates to the periodic part sought ; the other oscillations of different periods may be expected to contribute a constant to the sum, namely the sum of the mean part of each. 6. The principle of this method is also used in searching for hidden periodicities in a set of observations taken over a considerable time. Suppose that a period T occurs in these observations and that they are taken at equal intervals, there being n observations in the period T. Arrange the numbers in rows thus : i^o. Uj, Ug, ... «n-2, «»-i. -l^n, ^»+l. '^n+2, ... «^-!, «2»-l. %»-!)«. '^(«i-l)»+l. '^(m-l)»f2, ... W'mH-J. ««„-!. le vertical columns, and let the sums be U,. C^i, v^, ... U.-.. £/•„-. In the sequence C/q, U^, ?72, ... ?7„_i, Z7„ the component of period T will be multiplied 7>i-fold, and the variable parts of the other components may be expected to disappear, as these will enter with different phases into the horizontal rows, and the rows are supposed to be numerous. The difference between the greatest and least of the numbers Z/q, V^, Z/j, ... Z7„_2, TJr^-i furnishes a rough indication of the amplitude of the component of period T, if such exists ; and the presence of such a period is indicated by this difference being large. Let y denote the difference between the greatest and least of the numbers Uq, TJ^yV^^y ... U^.^y ^n-i corresponding to the trial period x. If y is plotted as a function of x, we obtain a "curve Digitized by Google APPENDIX 301 of periods." This curve will have peaks at the values of x corresponding to the periodicities which really exist. When the presence of such periods is indicated by the curve, the statistics are then analysed by the methods above described. This method was devised by Whittaker for the discussion of the periodicities entering into the variation of variable stars.* It is a modification of Schuster's work, applied by him to the discussion of the statistics of sunspots and other cosmical phenomena.t To Schuster, the term " periodogram analysis " is due, but the "curve of periods" referred to above is not identical with that finally adopted by Schuster and termed periodograph (or periodogram). For numerical examples, and for descriptions of other methods of attacking this problem, reference may be made to the Edinburgh Mathematical Tract, No. 4, already cited, and to Schuster's papers. * Monthly Notices, If. A, 8., 71, p. 686, 1911. See also a paper by Gibb, "The Periodogram Analysis of the Variation of SS Cygni," ibid., 74, p. 678, 1914. t The following papers may be mentioned : Cambridge, Traiia. Phil. Sac, 18, p. 108, 1900. London, Phil. Trans. B. Soc., 206 (A), p. 69, 1906. London, Proc. R. Soc., 77 (A), p. 136, 1906. Digitized by Google APPENDIX II BIBLIOGRAPHY* FOURIER S SERIES AND INTEGRALS TREATISES Fourier. Th^orie analytique de la chaleur. Paris, 1822. English Translation (Freeman). Cambridge, 1872. German Translation (Weinstein). Berlin, 1884. PoissoN. Traits de M6eanique (2* 6d.). Paris, 1833. Th6orie math^matique de la chaleur. Paris, 1835. DiNi. Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale. Pisa, 1880. Neumann. tJber die nach Kreis-, Kugel-, und Cylinder-Functionen fort- schreitenden Entwickelungen. Leipzig, 1881. Beau. Analytische Untersuchungen im Gebiete der trigonometrischen Reihen und der Fourier'schen Integrale. Halle, 1884. JgSfiBj^Y. An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. Boston, 1893. FoiNCARifi. Th6orie analytique de la propagation de la chaleur. Paris, 1895. Frischauf. Vorlesungen Uber Kreis- und Kugel-Functionen Reiheu. Leipzig, 1897. BuRKHARDT. Entwicklungen nach oscillirenden Functionen. Jahresber. D. Math. Ver., Leipzig, Bd. X., Heft. 11., 1901. Carslaw. Fourier's Series and Integrals and the Mathematical Theory of the Conduction of Heat. London, 1906. Lebesgue. Lemons sur les series trigonometriques. Paris, 1906. Ho T{«^oM- The Theory of Functions of a Real Variable and the Theory of Fourier's Series. Cambridge, 1907. *The abbreviated titles for the Journals are taken from the International GcUalogiie of Scientific Literature. 302 Digitized by Google APPENDIX 303 Carsb and Shearer. A course in Fourier's Analysis and Periodogram Analysis. (Edinburgh Mathematical Tracts, No. 4.) Edinburgh, 1915. In many other works considerable space is given to the treatment of Fourier's Series and Integrals. The following may be specially mentioned : Db la Vall^e Poussin. Cours d'Analyse, T. II. {2* 6d.). Pari?, 1912. GouRSAT. Cours d' Analyse, T. I. (3» 6d.). Paris, 1918. Jordan. Cours d'Analyse, T. II. (3* 6d.). Paris, 1913. RuNGB. Theorie und Praxis der Eeihen. Leipzig, 1904. Serret-Harnack. Lehrbuch der Differential- und Integral-Rechnung, Bd. II. (6 u. 7 Aufl.). Leipzig, 1915. Weber. Die partiellen Differential-gleichungen der mathematischen Physik, Bd. I. (2 Aufl.). Braunschweig, 1910. [This work is also known as the fifth edition of Biemanu's lectures entitled Partielle Differential-gleichungen und dere^i Anwendungen auf physikalischen Fragen, Braunschweig, 1869. It is referred to in the text as Weber-Biemann.] Whittaker and Watson. Course of Modern Analysis. Cambridge, 1920. [This is the third edition of Whittaker's work with the same title.] MEMOIRS AND PAPERS. Fourier. Fourier's first paper was communicated to the Academy in 1807. A fuller discussion of the subject was presented in his M4moire mir la 'propagation de la chaleur, communicated in 1811, and not printed till 1824-6 (Paris, J/^?«. Acad, sci,, 4, 1825, and 6, 1826). This memoir is practically contained in his treatise (1822). Many other papers bearing upou this subject and upon the Mathematical Theory of the Conduction of Heat were published by him about the same time. A list of these writings is given in Freeman's Edition, pp. 10-13. See also (Euvres de Fourier, Paris, 1888. PoissoN. (i) Sur la mani^re d'exprimer les fonctions par les s6ries de quantit6s p^riodiques, et sur I'usage de cette transformation dans les resolutions de diff§rents probl^mes. J. 6c. poly tech., Paris, 11, 1820. (ii) M6moire sur les int6grales d^finies et sur la sommation des s6ries. J. 6c. polytechn., Paris, 12, 1823. (Cf. also : Sur les integrates d6finies. J. 6c. polytechn., 9, 10, and 11.) Cauchy. (i) Sur le dev61oppement des fonctions en s6ries periodiques. Paris, Mem. Acad, sci., 6, 1826. (ii) Sur les r6sidus de fonctions exprim6e3 par int6grale8 d6finie8. 1827. Cf. (Euvres de Cauchy (S6r. 2), T. 7. (iii) M6moire sur I'application du Calcal des Besidus aux questions de physique math6matique. 1827. Cf. (Euvres de Caiichy (S6r. 2), T. 15. (iv) Second m6moire sur I'application du Calcul des B6sidus aux questions de physique math6matique. Paris, M6m. Acad, sci., 7, 1827. Digitized by VjOOQ IC 304 APPENDIX PoissoN. Siir le calcul num6rique des int6grales d^finies. Paris, M6ro. Acad, sci., 7, 1827. DiRKSEN. Uber die Darstellung beliebiger Functionen mittelst Reihen die iiach den Sinussen und Cosinussen der Vielfachen eines Winkels fort- schreiten. Berlin, Abh. Ak. Wiss., 1827. DiRiCHLET. Sur la convergence des series trigonometriques qui servent a repr^senter une fonction arbitraire entre des Hmites donn^es. J. Math., Berlin, 4, 1829. DiRKSEN (i). Uber die Convergenz einer nach den Sinussen und Cosinussen der Vielfachen eines Winkels fortschreitenden Reihe. J. Math., Berlin, 4, 1829. (ii). Uber die Summe einer nach den Sinussen und Cosinussen der Vielfachen eines Winkels fortschreitenden Beihe. Berlin, Abh. Ak. Wiss., 1829. DiRiCHLBT. (i) Die Darstellung ganz willkurlicher Functionen durch Sinus- und Cosinus-Reihen. Dove's Repertorium der Physik., Bd. I., 1837. (ii) Sur les series dont le term g6n6ral depend de deux angles, et qui servent h exprimer des fonctions arbitraires entre des Hmites donn6es. J. Math., Berlin, 17, 1837. (iii) Sur I'usage des int6grales d^finies dans la sommation des series finies oh infinies. J. Math., Berlin, 17, 1837. Hamilton. On Fluctuating Functions. Dublin, Trans. R. Irish Acad., 19, 1843. Bonnet. Demonstration simple du th^r^me de Fourier. Nouv. ann. math., Paris, 8, 1844. Boole. On the Analysis of Discontinuous Functions. Dublin, Trans. B. Irish Acad., 21, 1848. Stokes. On the Critical Values of the Sums of Periodic Functions. Gam- bridge, Trans. PhiL See., 8, 1849. Bonnet, (i) Sur quelques int^grales d^finies. J. math., Paris, 14, 1849. (ii) Sur la throne g^n^rale des series. Bruxelles, M6m. cour. Acad, roy., 23, 1860. DuHAMEL. Sur la discontinuit6 des valeurs des s6riee, et sur les moyens de la reconnaitre. J. math., Paris, 19, 1854 RiEMANN. t)ber die Darstellbarkeit einer Function durch eine trigonome- trische Reihe. Gottingen, Hab. Schrift, 1864. (Also Gdttingen, Abh. Ges. Wiss., 18, 1868. Lipschitz. De explicatione per series trigonometricas instituenda functi- onum unius variabilis arbitrarium et praecipue earum quae per variabilis spatium finitum valorum maximorum et minimorum numerum habent infinitum, disquisitio. J. Math., Berlin, 68, 1864. Digitized by Google APPENDIX 305 Du Bois-Beymond. t)ber die aUgemeinen Eigensohaften der Klasse von Doppel-Integralen zu weloher das Fourier'sche Doppel-Integral gehort. J. Math., Berlin, 69, 1878. Heine, t^ber trigonometiische Beihen. J. Math., Berlin, 71, 1870. Cantor, (i) t^ber einen die tngonometrischen Beihen betreffenden Lehrsatz. J. Math., Berlin, 72, 1870. (ii) Beweis dass eine fiir jeden reellen Werth von x durch eine trigono- metrische Beihe gegebene Function sioh nur auf einzige Weise in dieser Form darstellen lasst. J. Math., Berlin, 72, 1870. (iii) Notiz zu diesem Aufsatze. J. Math., Berlin, 73, 1871. (iv) t^r trigonometrische Beihen. Math. Ann., Leipzig, 4, 1871. Du Bois-Beymond. Die Theorie der Fourier*schen Integrale and Formeln. Math. Ann., Leipzig, 4, 1871. Prym. Ziir Integration der Diflferential-Gleichung ^-i+%^=0. J. Math. Berlin, 78, 1871. "^^ "^ Cantor. "Dber die Ausdehnnng eines Satzes aus der Theorie der tngono- metrischen Beihen. Math. Ann., Leipzig, 5, 1872. ScHWARz. t^er die Integration der Differential Gleiohung At^= 0. J. Math., Berlin, 74, 1872. Thomae. Bemerkung iiber Fourier'sche Beihen. Zs. Math., Leipzig, 17, 1872. AscoLi. t^er trigonometrische Beihen. Math. Ann., Leipzig, 6, 1873. Du Bois-Beymond. tTber die Fourierschen Beihen. G5ttingen, Nachr. Ges. Wiss. 1873. Glaisher. On Fourier's Double Integral Theorem. Mess. Math., Cam- bridge, 2, 1873. Du Bois-Beymond. tJber die sprungweisen Werthanderungen analytischer Functionen. Math. Ann., Leipzig, 7, 1874. ScHLAFLi. Einige Zweifel an der allgemeinen Darstellbarkeit einer will- klirlicher periodischer Function einer Variablen durch eine trigono- metrische Beihe. Bern. Universitats-Program, 1874. (Cf. also. tJber die partielle Differential-Gleichung -^ =k;^-2. J. Math., Berlm, 72, 1870.) AscoLi. Sulla serie di Fourier. Ann. mat., Milano (Ser. 2), 6, 1875. Du Bois-Beymond. Allgemeine Lehrsatze liber den Giiltigkeitsbereioh der Integralformeln die zur Darst/ellung willkiirhcher Functionen dienen. J. Math., Berlin, 79, 1876. Genoochi. Intorno ad alcune sorie. Torino, Atti Ace. sc, 10, 1875. Du Bois-Beymond. (i) Beweis dass die coefficienten der tngonometrischen Beihe f{x) = 2 {<^p cos px + hp sm j5^), p=Q U Digitized by Google 306 ' APPENDIX die Werthe 1 /•' 1 r' 1 /•' . aQ=— I /(a) da, (ip=- I /(«) cos pa da, bp=- I /(a) sin pa da haben, jedesmal wenn diese Integrale endlich und bestimmt sind. Miin- chen, Abh. Ak. Wiaa., 12, 1876. (ii) Untersuchungen iiber die Convergenz and Divergenz der Fourier' - schen DarsteUungsformeln. Miinclien, Abh. Ak. Wiss., 12, 1876. (iii) Zusatze zu dieser Abhandlung. Math. Ann., Leipzig, 10, 1876. TdPLER. Noidz fiber eine bemerkenswerthe Eigenschaft der periodischen Beihen. Wien, Anz. Ak. Wiss., 18, 1876. Glaisher. Note on Fourier's Theorem. Mess. Math., Cambridge, 6, 1877. AscoLi. (i) Nuove rioerche sulla serie di Fourier. Boma, Bend. Ace. Lincei (Ser 3), 2, 1878. (ii) Sulla rappresentabilita di una funzione a due variabili per serie doppia trigonometrica. Boma, Bend. Ace. Lincei (Ser. 3), 4, 1879. Appel. Sur une th^or^me concemant les series trigonom^triques. Arch. Math., Leipzig, 64, 1879. Bonnet. Sur la formule qui sert de fondement h. une th^orie des series trigonom^triques. Bui. sci. math., Paris (S4r. 3), 3, 1879. Du Bois-Beymond. Determination de la valeur limite d'une integrale. Bui. scL math., Paris (S6r. 3), 3, 1879. HoppE. Bemerkung liber trigonometrischen Beihen. Arch. Math., Leipzig, 64, 1879. Sachse. Versuch einer Geschichte der Darstellung willkiirlicher Functionen einer Variablen durch trigonometrischen Beihen. Diss., Gottingen, 1879. Also Zs. Math., Leipzig, 25, 1880, and in French, Bui. sci. math., Paris (S6r. 3), 4, 1880. AscoLi. Sulla serie trigonometriche a due variabili. Boma, Bend. Ace. Lincei (Sei*. 3), 8, 1880. Cantoiu (i) Bemerkung iiber trigonometrische Beihen. Math. Ann., Leipzig, 16, 1880. (ii) Femere Bemerkung liber trigonometrische Beihen. Math. Ann., Leipzig, 16, 1880. Du Bois-Beymond. Zur Geschichte der trigonometrisohen Beihen. Tub- ingen, 1880. Boussinesq. Coup d'oeil sur la th^orie des series trigonom^triques et sur une raison naturelle de leur convergence. J. math., Paris (Sdr. 3), 7, 1881. Du Bois-Betmond, t}ber Darstellimgsfunctionen. Math. Ann., Leipzig, 18, 1881. Harnack. Uber die trigonometrische Beihe und die Darstellung willldir- licher Functionen. Math. Ann., Leipzig, 17, 1881. Jordan. Sur la s6rie de Fourier. Paris, C. B. Acad, sci., 92, 1881. Digitized by VjOOQIC APPENDIX 307 Sharp. On Fourier's Theorem. Q. J. Math., London, 17, 1881. AscoLi. Una osiservanza relativa ad un teorema contenuto nella sua memoria ; Sulla rappresentabilita di una funzione a due variabili per serie doppia trigonometrica. Milano, Bend. 1st. lomb., 15, 1882. Halphen. Sut la s^rie de Fourier. Paris, C.R. Acad, sci., 05, 1882. Harnack. (i) Th6orie de la s^rie de Fourier. Bui. sci. math., Paris (S6r. 3), 6, 1882. (ii) Vereinfachimg der Beweise in der Theorie der Fourier'schen Reihen. Math. Ann., Leipzig, 19, 1882. HoBSON. On Fourier's Theorems. Mess. Math., Cambridge, 11, 1882. Lindbmann. Ober das Verhalten der Fourier'schen Beihe an Sprung- stellen. Math. Ann., Leipzig, 19, 1882. PoiNCAR^. Sur les series trigonom^triques. Paris, C.B. Acad, sci., 95, 1882. Veltmann. Die Fouriersche Beihe. Zs. Math., Leipzig, 27, 1882. Cantor. Sur les series trigonom^triques. Acta Math., Stockholm, 2, 1883. Du Bois-Betmond. tJber die Integration der trigonometrischen Beihe. Math. Ann., Leipzig, 22, 1883. Gilbert. Demonstration simplifi6e des formules de Fourier. Bruxelles, Ann. Soc. scient., 8, 1884. Holder. Zur Theorie der trigonometrischen Beihen. Math. Ann., Leipzig, 24, 1884. Alexander, (i) Failing Cases of Fourier's Double Integral Theorem. Edinburgh, Proc. Math. Soc., 8, 1885. (ii) Boole's and other proofs of Fourier's Double Integral Theorem. Edinburgh, Proc. Math. Soc., 3, 1885. Holder. Ober eine neue hinreichende Bedingung fur die Darstellbarkeit einer Function durch die Fourier'sche Beihe. Berlin, SitzBer. Ak. Wiss., 1885. Kroneckbr. t)ber das Dirichlet'sche Integral. Berlin, SitzBer. Ak. Wiss., 1885. PoiNCAR^. Sur les series trigonom^triques. Paris, C.B. Acad, sci., 101, 1885. Weierstrass. tJTjer die Darstellbarkeit sogenannter willkiirlicher Func- tionen durch die Fouriersche Beihe. Berlin, SitzBer. Ak. Wiss., 1885. Alexander. A symbob'c proof of Fourier's Double Integral Theorem. Mess. Math., Cambridge, 15, 1886. Arzela. Sopra una certa estensione di un teorema relative alle serie trigono- metriche. Boma, Bend. Ace. Lincei (Ser. 4), 1, 1886. Alexander, (i) A proof of Fourier's Series for Periodic Functions. Mess. Math., Cambridge, 16, 1887. (ii) Extensions of Fourier's trigonometric Series Theorem. Mess. Math., Cambridge, 16, 1887. C.I u2 Digitized by Google 308 APPENDIX Johnson. A proof of Fourier's Series Theorem. Mess. Math., Cambridge, 16, 1887. Harnack. tJber Caachy's zweiten Beweis fiir die Convergenz der Fourier' - schen Beihen und eine damit verwandte altere Methode von Poisson. Math. Ann., Leipzig, 32, 1888. BouRLET. Sur la multiplication des series trigonom^triques. Bui. soi math. , Paris (S6r. 2), 18, 1889. PiCARD. Sur la representation approch^ des fonctions. Paris, O.K. Acad. sd., 112, 1891. SoMMERFELD. Die willkurlicheu Functionen in der mathematischen Physik. Diss., Konigsbergy 1891. De la Vall^ Poussin. Sur une demonstration des formules de Fourier g^neralis^. Bruxelles, Ann. Soo. scient., 15, 1892. Sequier. Sur la s^rie de Fourier. Nouv. ann. math., Paris (S^r. 3), 11, 1892. Beau. Einige Mittheilimgen aus dem Gebiete der trigonometrischen Beihen und der Fourier'schen Integrale. Prog. Gymn. Sorau, No. 87, 1893. De la Vall^e Poussin. Sur quelques applications de Tint^grale de Poiason. Bruxelles, Ann. Soc. scient., 17, 1893. Gibson, (i) On the History of the Fourier's Series. Edinburgh, Proc. Math. Soc, 11, 1893. (ii) A proof of the Uniform Convergence of Fourier's Series, with notes on the Differentiation of the Series. Edinburgh, Proc. Math. Soc., 12, 1894. Berqer. Sur le d6veloppement de quelques fonctions discontinues en series de Fourier. Upsala, Soc. Scient. Acta (S6r. 3), 16, 1895. Williams. On the Convergence of Fourier's Series. Phil. Mag., London (Ser. 5), 41, 1896. Poincar:^. Les rapports de I'analyse et de la physique math6matique. K«v. g6n. sci., Paris, 18, 1897. Preston. On the general extension of Fourier's Theorem. Phil. Mag., London (Ser. 5), 43, 1897. Brod^n. tJher das Dirichlet'sche Integral. Math. Ann., Leipzig, 52, 1899. Marolli. Di una classe di funzioni finite e continue non sviluppibabili in serie di Fourier nel pimto x=0, Giom. mat., Napoli, 87, 1899. VoiGT. Demonstrazione semplice delle sviluppibilitd in serie di Fourier di una funzione angolare finita e ad una sol valore. Roma, Bend. Ace. Lincei (Ser. 5), 8, 1899. Fej^r. Sur les fonctions bom^es et int^grables. Paris, C.B. Acad, sci., 181, 1900. Lerch. Bemarque sur la s6rie de Fourier. Bui. sci, math., Paris (S6r. 2), 24,1900. Digitized by Google APPENDIX 309 Ford. Dini's method of showing the convergence of Fourier's series and of other allied developments. New York, N.Y., Bull. Amer. Math. Soc., 7, 1901. HuRWiTz. Sur les series de Fourier. Paris, C.B. Acad, sci., 132, 1901. Stackbl. (i) tJher das Dirichlet*sche Integral. Leipzig, Ber. Ges. Wiss., 53, 1901. (ii) tTber die Convergenz der trigonometrischen Beihen. Arch. Math., Leipzig (3. Beihe), 2, 1901. Fej£r. Sur la differentiation de la s6rie de Fourier. Paris, C.B. Acad, sci., 134, 1902. HuRwiTz, Sur quelques applications gtom^triques des series de Fourier. Ann. sci. Ec. norm., Paris (S^r. 3), 19, 1902. Stekloff. (i) Sur certaines 6galit6s remarquables. Paris, C,|t. Acad, sci., 135, 1902. (ii) Sur la rapr6sentation approch^ des fonctions. Paris, C.B. Acad. sci., 135, 1902. (iii) Sur quelques consequences de certains d^veloppements en series analogues aux d6veloppements trigonom^triques. Paris, C.B. Acad. sci., 135, 1902. (iv) Bemarque relative k ma Note : Sur la representation approch^e des fonctions. Paris, C. Bi Acad, sci., 135, 1902. HuRwiTz. tJher die Fourier'schen Konstanten integrierbarer Functionen. Math. Ann., Leipzig, 57, 1903. Krause. "Ober Fouriersche Beihen mit zwei veranderlichen Grossen. Leipzig, Ber. Ges. Wiss., 55, 1903. Lebesoue. Sur les s6ries trigonom^triques. Ann. sci. £c. norm., Paris (Ser. 3), 20, 1903. Macdonald. Some Applications of Fourier's Theorem. London, Proc. Math. Soc., 35, 1903. Stephenson. An extension of the Fourier Method of Expansion in Sine Series. Mess. Math., Cambridge, 23, 1903. BiERMANN. Cber das Bestglied trigonometrischer Beihen. Wien, SitzBer. Ak. Wiss., 113, 1904. Fischer. Zwei neue Beweise fiir den Fundamental-Satz der Fourier'schen Konstanten. MonHfte. Math. Phys., Wien, 15, 1904. Fejj^r. Untersuchungen uber Fourier'sche Beihen. Math. Ann., Leipzig, 58, 1904. Knesbr. (i) Die Fourier'sche Beihe und die angenaherte Darstellung perio- discher Fimctionen durch endliche trigonometrische Beihen. Arch. Math., Leipzig (3. Beihe), 7, 1904. (ii) Untersuchungen liber die Darstellung willkiirlicher Functionen in der mathematischen Physik. Math. Ann., Leipzig, 58, 1904. Digitized by Google 310 APPENDIX Stekloff. Sur certaines 6galit^ g^n^rales communes k plusieurs series de fonctions souvents employees dans Tanalyse. St. Petersburg, M6m. Ac. Sc. (S6r. 8), 16, 1904. HoBsoN. On the failure of the convergence of Fourier's Series. London, Proc. Math. Soc. (Ser. 2), 3, 1905. Kneser. Beitrago sur Theorie der Sturm-Liouville'schen Darstellung will- kiirlicher Funktionen. Math. Ann., Leipzig, 61, 1905. Lebesoue. (i) Sur une condition de convergence des series de Fourier. Paris, CB. Acad, sci., 140, 1905. (ii) Sur la divergence et la convergence non-uniforme dcs series de Fourier. Paris, C.R. Acad, sci., 141, 1906. (iii) Recherches sur la convergence des series de Fourier. Math. Ann., Leipzig, 61, 1905. Severini. Sulla serie di Fourier. Venezia, Atti 1st. ven., 64, 1905. BOcHER. Introduction to the theory of Fourier's Series. Ann. Math., Prince- ton, N.J. (Ser. 2), 7, 1906. Fatou. (i) Sur le d^veloppement en s^rie trigonom6trique des fonctions non- int^grables. Paris, C.R. Acad, sci., 142, 1906. (ii) Series trigonom^triques et series de Taylor. Acta Math., Stock- hohn, 30, 1906. Fej^r. Sur la s6rie de Fourier. Paris, C.R. Acad, sci., 142, 1906. (Also paper in Hungarian in Math. Termt. £rt., Budapest, 24, 1906.) JouRDAiN. The development of the theory of the transfinite numbers. Arch. Math., Leipzig (3. Reihe), 10, 1906. (See also 14, 1909 ; 16, 1910 ; and 22, 1913.) Lees. On an extension of the Fourier method of expanding a function in a series of sines and cosines. Mess. Math., Cambridge, 35, 1906. Brencke. On the convergence and differentiation of trigonometric series. Ann. Math., Princeton, N.J. (Ser. 3), 8, 1907. Buhl. Sur les nouvelles formules de sommabilit^. Bui. sci. math., Paris (S^r. 2), 31, 1907. Fej^r. tJber die Fouriersche Reihe. Math. Ann., Leipzig, 64, 1907. HoBsoN. On the imiform convergence of Fourier's Series. London, Proc. Math. Soc. (Ser. 2), 5, 1907. Moore. Note on Fourier's constants. New York, N.Y., Bull. Amer. Math. Soc., 13, 1907. Pringsheim. tJber die Fouriersche Integraltheorem. Jahresber. D. Math. Ver., Leipzig, 16, 1907. Riesz. Les series trigonom^triques. Paris, C.R. Acad, sci., 145, 1907. Buhl. Sur la sommabilit^ des series de Fourier. Paris, C.R. Acad, sci., 146, 1908. Digitized by Google APPENDIX 311 De la Vall^e Poussin. Sur Fapproxiiuation des fonctions d*une variable resile et de leurs d6riv6es par des polynomes et des suites limittoi de Fourier. Bruxelles, Bui. Acad, roy., 1908. Hardy. Some points in the integral calculus, XXIII. -V. On certain oscillating cases of Dirichlet's integral. Mess. Math., Cambridge, 38, 1908. Orlando. Sulla formula integrale di Fourier. Roma, Bend. Ace. lincei (Ser. 5), 17, 1908. Young. A note on trigonometrical series. Mess. Math., Cambridge, 38, 1908. Bromwich. Note on Fourier's Theorem. Math. Gaz., London, 5, 1909. Fej^r. (i) Beispiele stetiger Funktionen mit divergenter Fourierreihen. J. Math., BerUn, 187, 1909. (ii) Eine stetige Fnnktion dereu Fouriersche Reihe divergiert. Palermo, Rend. Circ. mat., 28, 1909. Graziani. (J) Sulla formula integrale di Fourier. Roma, Rend. Ace. Lincef>6er. 5), 18, 1909. (ii) Funzioni rappresentabili con la formula integrale di Fourier. Roma, Rend. Ace. Lincei (Ser. 5), 18, 1909. Orlando. iJuove osservazioni sulla formula integrale di Fourier. Roma, Rend. Ace. Lincei (Ser. 5), 18, 1909. De la Vall^e Poussin. Un nouveau cas de convergence des series de Fourier. Palermo, Rend. Circ. mat., 81, 1910. Fej^r. Lebesguesche Konstanten und divergente Fourierreihen. J. Math., Berlin, 138, 1910. Lebesgue. Sur la representation trigonom6trique approch6e des fonctions satisfaisant k une condition de Lipschitz. Paris, Bui. soc. math., 38, 1910. Plancherel. Contribution k F^tude de la representation d'une fonction arbitrarie par des int6grales d^finies. Palermo, Rend. Circ. mat., 30, 1910. Prasad. On the present state of the theory and application of Fourier's Series. Calcutta, Bull. Math. Soc, 2, 1910. pRiNGSHEiM. tJber neue Giiltigkeitsbedingungen fiir die Fouriersche Integralformel. Math. Ann., Leipzig, 68, 1910. RiEsz. Untersuchungen iiber Systeme integrierbarer Funktionen. Math. Ann., Leipzig, 69, 1910. Rossi. Intomo ad un caso notevole negli integrali doppi di Fourier. Rir. fis. mat. sc. nat., Pavia, 21, 1910. Young, (i) On the conditions that a trigonometrical series should have the Fourier form. London, Proc. Math. Soc. (Ser. 2), 9, 1910. (ii) On the integration of Fourier Series. London, Proc. Math. Soc. (Ser. 2), 9, 1910. Digitized by Google 312 APPENDIX Carath^odory. t^er den Variabilitatsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen. Palermo, Bend. Giro, mat., 32, 1911. Dixon. The multiplication of Fourier Series. Cambridge, Proc. Phii. Soc, 16, 1911. Fej^r. Sur le singularit^s de la s^rie de Fourier des fonctions continues. Ann. sci. Eo. norm., Paris (S6r. 3), 28, 1911. Hardy. Fourier's double integral and the theory of divergent integrals. Cambridge, Trans. Phil. Soc, 21, 1911. Neumann. Zur Theorie des logarithmischen Potentials. Aufsatz VIII. Leipzig, Ber. Ges. Wiss., 63, 1911. pRiNQSHEiM. Nachtrag zu der Abhandlimg : Neue Gtiltigkeitsbedingungen fiir die Fouriersche Integralformel. Math. Ann., Leipzig, 71, 1911. ScHBCHTER. Ubcr die Summation divergenter Fourierscher Beihen. Mon- Hfte Math. Phys., Wien, 22, 1911. Thompson. Nouvelle m^thode d'analyse harmonique par la summation algdbrique d'ordon^s d^termin^. Paris, C.R. Acad, sci., 153, 1911. ToPLiTZ. tJber die Fouriersche Entwickelung positiver Funktionen. Palermo, Rend. Circ. mat., 32, 1911. Young, (i) On the convergence of a Fourier Series and of its allied series. London, Proc. Math. Soc. (Ser. 2), 10, 191L (ii) On the nature of the successions formed by the coefficients of a Fourier Series. London, Proc. Math. Soc. (Ser. 2), 10, 1911. (iii) On the Fourier Constants of a Function. London, Proc. R. Soc., 86, 1911. (iv) On a class of parametric integrals and their application in the theory of Fourier Series. London, Proc. R. Soc., 85, 1911. (v) On a mode of generating Fourier Series. London, Proc. R. Soc. 85, 1911. (vi) On Fourier's repeated integral. Edinburgh, Proc. R. Soc., 31, 1911. (vii) Ou Somraerf eld's form of Fourier's repeated integral. Edin burgh, Proc. R. Soc., 31, 1911. (viii) tJher eine Summationsmethode fiir die Fouriersche Reihe. Leipzig, Ber. Ges. Wiss., 63, 1911. (ix) Konvergenzbedingungen fiir die verwandte Reihe einer Fourier- sohen Reihe. Miinchen, SitzBer. Ak. Wiss., 41, 1911. Chapman. On the general theory of summability, with applications to Fourier's and other series. Q.J. Math., London, 43, 1912. Gronwall. (i) t)ber die Gibbs'sche Erscheinimg und die trigonometrischen Summen sin a; + J sin 2a; + ...+- sin nz. Math. Ann., Leipzig, 72, 1912. (ii) On a theorem of Fej^r's and an analogon to Gibbs' phenomenon. New York, N.Y., Trans. Amer. Math. Soc, 13, 1912. Digitized by Google APPENDIX 313 Hardy. Oscillating Dirichlet's Integrals. Q.J. Math., London, 43, 1912. Jackson. On approximation by trigonometric sums and polynomials. New York, N.Y., Trans. Amer. Math. Soc., 18, 1912. LiCHTENSTEiN. tJber das Poissonsche Integral und iiber die partiellen Ablei- tmigen zweiter Ordnung des logarithmischen Potentials. J. Math., Berlin, 141, 1912. Neumann. Zur Theorie des logarithmischen Potentials. Aufsatz IX. Leipzigy Ber. Ges. Wiss., 64, 1912. Katlbiqh. Remarks concerning Fourier's Theorem as applied to physical problems. Phil. Mag., London (Ser. 6), 24, 1912. Thomas. Cber die Konvergenz einer Fouriersch^n Reihe. Gottingen, Nachr. Ges. Wiss., 1.912. Young, (i) On successions of integrals and Fourier Series. London, Proc. Math. Soc. (Ser. 2), 11, 1912. (ii) On multiple Fourier Series. London, Proc. Math. Soc. (Ser. 2), 11, 1912. (iii) On a certain series of Fourier. London, Proc. Math. Soc. (Ser. 2), 11, 1912. (iv) Note on a certain functional reciprocity in the theory of Fourier Series. Mess. Math., Cambridge, 41, 1912. (v) Sur la generalisation du th^or^me de Parsival. Paris, C.R. Acad, sci., 166, 1912. (vi) On the convergence of certain series involving the Fourier constants of a function. London, Proc. R. Soc, 87, 1912. (vii) On classes of summable functions and their Fourier Series. London, Proc. R. Soc, 87, 1912. (viii) On the multipUcation of successions of Fourier Constants. Ix)ndon, Proc. R. Soc, 87, 1912. Fej^r. (i) Sur les polynomes trigonom6triques. Paris, C.R. Acad, sci., 157, 1913. (ii) t)ber die Bestimmung des Sprunges der Funktion aus ihrer Fourier- reihc J. Math., Berlin, 142, 1913. (Also paper in Himgarian in Math. Termt. £rt., Budapest, 31, 1913.) FiLON. On a symbolic proof of Fourier's Theorem. Math. Gaz., London, 7, 1913. Hardy, (i) Oscillating Dirichlet's Integrals, an essay in the " Infinitar- calcul " of Paul du Bois-Reymond. Q. J. Math., London, 44, 1913. (ii) On the sununability of Fourier's Series. London, Proc. Math. Soc (Ser. 2), 12, 1913. Hardy and Littlewood. Sur la s6rie de Fourier d'une fonction k carr6 sommable. Paris, C.R. Acad, sci., 166| 1913. Digitized by Google 314 APPENDIX Jackson, (i) On the approximate representation of an indefinite integral and the degree of convergence of related Fourier's Series. New York, N.Y., Trans. Amer. Math. Soc., 14, 1913. (ii) On the accuracy of trigonometric interpolation. New York, N.Y., Trans. Amer. Math. Soc., 14, 1913. KusTERMANN. t)ber Fouriersche Doppelreihen und das Poissonsche Doppel- integral. Diss., Miinchen, 1913. LusiN. Sur la convergence des s6ries trigonom6triques de Fourier. Paris, C.R. Acad, sci., 156, 1913. Moore, (i) On convergence fat;tors in double series and the double Fourier Series. New York, N.Y., Trans. Amer. Math. Soc., 14, 1913. (ii) On the summability of the double Fourier's Series of discontinuous functions. Math. Ann., Leipzig, 74, 1913. Neumann. Zur Theorie der Fourierschen Beihen. Leipzig, Ber. Ges. Wiss., 65, 1913. Orlando. Sopra un nuovo aspetto della formula integrate de Fourier. Roma, Bend. Ace. Lincei (Ser. 5), 22, 1913. Steinhaus. Sur le d^veloppement du produit de deux fonctions en une s6rie de Fourier. Krakow, Bull. Intern. Acad., 1913. Van Vleck. The influence of Fourier's Series upon the development of mathematics. Amer. Ass. Adv. Sc. (Atlanta), 1913. Young, (i) On the Fourier Series of bounded functions. London, Proc. Math. Soc. (Ser. 2), 12, 1913. (ii) On the determination of the summability of a function by means of its Fourier constants. London, Proc. Math. Soc. (Ser. 2), 12, 1913. (iii) On the mode of oscillation of a Fourier Series and of its allied series. London, Proc. Math. Soc. (Ser. 2), 12, 1913. (iv) On the usual convergence of a class of trigonometric series. London, Proc. Math. Soc. (Ser. 2), 13, 1913. (v) On the formation of usually convergent series. London, Proc. R. Soc., 88, 1913. (vi) On Fourier Series and functions of bounded variation. London, Proc. R. Soc, 88, 1913. (vii) On the condition that a trigonometrical series should have a certain form. London, Proc. R. Soc, 88, 1913. Young, W. H. and G. C. On the theorem of Riesz-Fischer. Q.J. Math., London, 44, 1913. Bernstein. Sur la convergence absolue des series trigonomdtriques. Paris, C.R. Acad, sci., 158, 1914. BocHER. On Gibbs's Phenomenon. J. Math., Berlin, 144, 1914. Camp. Lebesgue Integrals containing a parameter, with applications. New York, N.Y., Trans. Amer. Math. Soc, 16, 1914. Faton. Sur la convergence absolue des series trigonom6triques. Paris, Bui. soc. math., 41, 1914. Digitized by Google APPENDIX 315 FejiSr. tJber konjugierte trigonometrische Reihen. J. Math., Berlin, 144, 1914. Gronwall. (i) Sur quelques m^thodes de sommation et leur application k la s^rie de Fourier. Paris, C.R. Acad, sci., 158, 1914. (ii) On the summability of Fourier's Series. New York, N.Y., Bull. Amer. Math. Soc , 20, 1914. Krtloff. Sur . T^quation de fermeture pour les s6ries trigonom^triques. Nouv. ann. math., Paris, 14, 1914. Neumann. t)ber die Dirichletsche Theorie der Fourierschen Reihen. Leipzig, Abh. Ges. Wiss., 33, 1914. Pol. Sur les transformation de fonctions qui font converger leur s6ries de Fourier. Paris, C.R. Acad, sci., 168, 1914. Young. On trigonometrical series whose Ces&ro partial summations oscillate finitely. London, Proc. R. Soc., 89, 1914. Fekete. Uutersuchung Uber Fouriersche Reihen. (Hungarian.) Math. Termt. Ert., Budapest, 83, 1915. Gronwall. On Approximation by trigonometric sums. New York, N.Y., BuU. Amer. Math. Soc, 21, 1915. Gross. Zur Poisaonschen Summierung. Wien, SitzBer. Ak. Wiss., 124, 1915. KUsTERMANN. Proof of the convergence of Poisson's integral for non- absolutely integrable functions. New York, N.Y., Bull. Amer. Math. Soc, 21, 1916. Rayleigh. Some calculations in illustration of Fourier's Theorem. London, Proc R. Soc, 90, 1916. BosE. Fourier Series and its influence on some of the developments of mathematical analysis. Calcutta, Bull. Math. Soc, 8, 1916. Brown. Fourier's Integral. Edinburgh, Proc. Math. Soc, 34, 1916. Frank. Cber das Vorwiegen des ersten Koefl&cienten in der Fourierent- wickelung einer konvexen Funktion. Math. Ann., Leipzig, 77, 1916. Gronwall. tJber einige Sunimationsmethoden und ihre Anwendung auf die Fouriersche Reihe. Jahresber. D. Math. Ver., Leipzig, 26, 1916. Hahn. Uber Fejers Summierung der Fourierschen Reihe. Jahresber. D. Math. Ver., Leipzig, 25, 1916. Privaloff. (i) Sur la derivation des series de Fourier. Palermo, Rend. Circ mat., 41, 1916. (ii) Sur la con vergence des series trigonometriques conjugu^es. Paris, C.R. Acad, sci., 162, 1916. SiERPiNSKi. Un exemple 616mentaire d'une function croissante qui a presque partout une deriv6e nulle. Giom* mat., Napoli, 64, 1916. Digitized by VjOOQ IC 316 APPENDIX Young, (i) Sar la convergence des series de Fourier. Paris, C.R Acad. scL, 163, 1916. (ii) Les series trigonom^triques et les moyennes de Ceskro. Paris, C.K. Acad, sci., 168, 1916. (iii) Sur les conditions de convergence des s^iies de Fourier. Paris, C.E. Acad, sci., 168, 1916. Angelbsco. Sur un proc6d6 de sommation des series trigonom6triques. Paris, C.R. Acad, sci., 165, 1917. Camp. Multiple integrals over infinite fields and the Fourier multiple integral. Amer. J. Math., Baltimore, Md., 39, 1917. Carslaw. a trigonometrical sum and the Gibbs' Phenomenon in Fourier's Series. Amer. J. Math., Baltimore, Md., 89, 1917. 00 Dim, Sugli sviluppi in serie Jao+2(«nC08 An-^+^nsiu A„2) dovele X^ sono 1 radici delle equazioni trascendente F (z) cos wz-^F^iz) sin Trz=0, Ann. mat., Milano (Ser. 3), 26, 1917. Fej^r. Fourierreihe und Potenzreihe. MonHfte. Math. Phys.,Wien, 28, 1917. Hardy. Notes on some points in the integral calculus. XLV. On a point in the theory of Fourier Series. Mess. Math., Cambridge, 46, 1917. Hardy and Littlewood. Sur la convergence des series de Fourier et des series de Taylor. Paris, C.R. Acad, sci., 166, 1917. Hart. On trigonometric series. Ann. Math., Princeton, N.J. (Ser. 2), 18, 1917. Jackson. Note on representations of the partial sum of a Fourier's Series. Ann. Math., Princeton, N.J. (Ser. 2), 18, 1917. JouRDAiN. The Influence of Fourier's Theory of the Conduction of Heat on the development of pure mathematics. Scientia, Bologna (Ser. 2), 22,1917. KiJSTERMANN. Fourier's Constants of functions of several variables. Amer. J. Math., Baltimore, Md., 89, 1917. LusiN. Sur la notion de I'int^grale. Ann. mat., Milano (Ser. 3), 26, 1917. Pollard. On the deduction of criteria for the convergence of Fourier's Series from Fej^r's theorem concerning their summability. London, Proc. Math. Soc. (Ser. 2), 15, 1917. Van Vleck. Current tendencies of mathematical research. New York, N.Y., BuU. Amer. Math. Soc, 28, 1917. Young, (i) On the order of magnitude of the coefficients of a Fourier Series. London, Proc. R. Soc, 94, 1917. (ii) On ordinary convergence of restricted Fourier Series. London, Proc. R. Soc, 94, 1917. (iii) On the mode of approach to zero of the coefficients of a Fourier Series. London, Proc. R. Soc, 94, 1917. Digitized by Google APPENDIX 317 (iv) Sur une nouvelle suite de conditions pour la convergence dee series de Fourier. Paris, C.R. Acad, sci., 164, 1917. (y) Sur la th^rie de la convergence des series de Fourier. Paris, C.E. Acad, sci., 164, 1917. (vi) Sur la th6orie des series trigonom^triques. Paris, C.B. Acad. sci., 165, 1917. Carath^odort. VJher die Fourierschen Koefficienten der nach Riemann integrierbaren Funktionen. Math. Zs., Berlin, 1, 1918. Cablbman. tJber die Fourierkoefl&cienten einer stetigen Funktion. Acta Math., Stockholm, 41, 1918. Geirinoer. Trigonometrische Doppelreihen. MonHfte. Math. Phys., Wien, 29, 1918. Larmor. The Fourier Harmonic Analysis ; its practical scope, with optical illustrations. London, Proc. Math. Soc. (Ser. 2), 16, 1918. KiEsz. Uber die Fourierkoefficienten einer stetigen Funktion von be- schrankter Schwankung. Math. Zs., Berlin, 2, 1918. Young, (i) On non-harmonic Trigonometrical Series. London, Proc. R. Soc, 95, 1918. (ii) On the Cesaro convergence of restricted Fourier Series. London, Proc. R. Soc, 95, 1918. Landau. Bemerkungen zu einer Arbeit von Herrn Carleman : tJber die Fourierkoefficienten einer stetigen Funktion. Math. Zs., Berlin, 5, 1919. Moore. Applications of the theory of summability to developments of orthogonal functions. New York, N.Y., Bull. Amer. Math. Soc, 25, 1919. YouNO. (i) On the convergence of the derived series of Fourier Series. London, Proc. Math. Soc (Ser. 2) 17, 1919. (ii) On restricted Fourier Series and the convergence of power series. London. Proc. Math. Soc (Ser. 2), 17, 1919. (iii) On the connection between Legendre Series and Fourier Series. London, Proc Math. Soc. (Ser. 2), 18, 1919. (iv) On non-harmonic Fourier Series. London, Proc. Math. Soc. (Ser. 2), 18, 1919. Jackson. Note on a method of proof in the theory of Fourier's Series, New York, N.Y., Bull. Amer. Math. Soc, 27, 1920. Neder. tJber die Fourierkoefficienten der Funktionen von beschrankter Schwankung. Math. Zs., Berlin, 8, 1920. Stbinhaus. (i) Bemerkungen zu der Arbeit der H^rrn Neder : tJber die Fourierkoefficienten der Funktionen von beschrankter Schwankung. Math. Zs., Berlin, 8, 1920. (ii) On Fourier's coefficients of bounded functions. London, Proc. Math. Soc. (Ser. 2), 19, 1920. Digitized by Google LIST OF AUTHORS QUOTED The numbers refer to pages. Abel, 124, 146. Baker, 122. Bernoulli, 1, 3. Bliss, 77. B6cher, 8, 234, 243, 263, 268-271, 275, 282. Bonnet, 97. Boussinesq, 7. Bromwich, 28, 48, 150, 157, 163, 165, 192, 193. Brunei, 119, 193. Burkhardt, 2, 15. Byerly, 198. Cantor, 12, 14, 16, 26, 27, 29. Carse and Shearer, 298. Carslaw, 156, 264. • Cauchy, 8, 9, 77, 124. Ces^ro, 12, 151. Chapman, 13. Chiystal, 148. Clairaut, 3, 5. D'Alembert, 1, 3. Darboux, 4, 6, 7. Dedekind, 18, 19, 23, 25-28. Delambre, 6. De la Valine Poussin, 10, 12, 28, 48, 75, 77, 119, 163, 192, 193, 239, 243, 262, 263. Descartes, 26. Dini, 12, 28, 112, 119, 163, 193, 243. Dirichlet, 4, 8, 9, 11, 77, 200, 208. Donkin, 124. Du Bois-Reymond, 12, 34, 77, 97, 112. Euclid, 25, 26. Euler, 1-5. Fejer, 13, 234, 268. Fourier, 2-9, 243, 294. Gibb, 301. Gibbs, 268-270, 282. Gibson, 16, 195, 262. Gmeiner (see Stolz and Gmeiner). Goursat, 28, 48, 59, 75, 77, 94, 119, 153, 163, 165, 193, 243. Gronwall, 270. Hardy, 13, 19, 20, 48, 76, 77, 94,1130,239. Hamack, 112. Heine, 11, 12, 14, 26. Hildebrandt, 77. Hobson, 12, 17, 28, 69, 89, 94, 138, 163, 193, 234, 243, 263, 275. Jackson, 270. Jordan, 12, 207, 243, 294. Jourdain, 14. Kelvin, Lord, 6. Kneser, 14, Kowalewski, 119, 163. Lagrange, 2, 3, 5, 6, 198. Laplace, 6. Lebesgue, 10, 12, 77, 112, 119, 243, 263, 294. Legendre, 6. Leibnitz, 26. Lipschitz, 12. Littlewood, 239. Moore, 150. Neumann, 243, 294. Newton, 26. Osgood, 45, 58, 75, 119, 129, 163, 193. Picard, 14, 234. Pierpont, 75, 83, 86, 93, 110, 119, 163, 193. Poincare, 7. Poisson, 7, 8, 230, 231, 293. Pringsheim, 16, 26, 28, 34, 48, 75, 163, 287, 291. 318 Digitized by Google LIST OF AUTHORS 319 Raabe, 165. Riemann, 5, 9-11, 15, 77, 112, 243. Runge, 268, 297. Russell, 26, 28. Sachse, 15, 124. Schuster, 301. Seidel, 11, 130. Shearer (see Carse and Shearer). Sommerfeld, 293. Stekloff, 14. Stokes, 11, 130. Stolz, 119, 163, 193. Stolz and Gmeiner, 16, 28, 48. Tannery, 28. Topler, 14. Turner, 299. Van Vleck, 14. Watson (see Whittaker and Watson). Weber-Riemann, 243, 294. Weierstrass, 14, 26, 27, 77, 97, 134. Weyl, 271. Whittaker, 297, 301. Whittaker and Watson, 151, 239, 240, 243, 263. Young (Grace Chishohn), 29, 77. Young, W. H., 10, 12, 14, 29, 77, 287, 293. Digitized by Google GENERAL INDEX The numbers refer to pages. Abel'i theorem on the power series, 146 ; extensions of, 149. Absolute convergence, of series, 44 ; of integrals, 103. Absolute value, 32. Aggregate, general notion of, 29 ; bounded above (or on the right), 29 ; bounded below (or on the left), 29 ; bounded, 30 ; upper and lower bounds of, 30 ; limiting points of, 31 ; Weierstrass's theorem on limiting points of, 32. Approximation carves for a series, 124. See also Oibbs'e phenomenon. B6cher's treatment of Oibbs's phenomenon, 268. Bounds (upper and lower), of an aggregate, 30 ; of f{x) in an interval, 50 ; of f{Xf y) in a domain, 72. .^ Bromwich's theorem, 151. Ces&ro's method of summing series (CI), 161, 238-240. Change of order of terms, in an absolutely convergent series, 45 ; in a conditionally convergent series, 47. Closed interval, definition of, 49. Conditional convergence of series, definition of, 45. Continuity, of functions, 59 ; of the sum of a uniformly convergent series of continuous functions, 135 ; of the power series (Abel's theorem), 146 ; of I' I f(z)dx when f(x) is bounded and integrable, 93 ; of ordinary integrals involving a single parameter, 169 ; of infinite integrals involving a single parameter, 179, 183. Continuous functions ; theorems on, 60 ; integrability of, 84 ; of two variables, 73 ; non-differentiable, 77. Continuum, arithmetical, 25 ; linear, 25. Convergence, of sequences, 33 ; of series, 41 ; of functions, 51 ; of integrals, 98. See also AbsoltUe convergence, Conditional convergence, and Uniform convergence. Cosine integral (Fourier's integral), 284, 292. Cosine series (Fourier's series), 197, 215. Darbouz's theorem, 79. Dedekind's axiom of continuity, 24. Dedekind's sections, 20. Dedekind's theory of irrational numbers, 18. Dedekind's theorem on the system of real numbers, 23. Definite integrals containing an arbitrary parameter (Chapter VI.) ; ordinary integrals, 169 ; continuity, integration and differentiation of, 169 ; infinite integrals, 173 ; uniform convergence of, 174 ; continuity, integration and differentiation of, 179. 320 Digitized by VjOOQ IC GENERAL INDEX 321 Definite integrals, ordinary (Chapter IV.); the sums S and s, 77; Darboux's theorem, 79 ; definition of upper and lower integrals, 81 ; definition of, 81 necessary and sufiicient conditions for existence, 82 ; some properties of, 87 first theorem of mean value, 92 ; considered as functions of the upper limit, 93 second theorem of mean value, 94. See also DiricJUefs integrals, Fourier's integrals. Infinite integrals and Poisson's integral. Differentiation, of series, 143 ; of power series, 148 ; of ordinary integrals, 170 ; of infinite integrals, 182 ; of Fourier's series, 261. Dirichlet's conditioni, definition of, 206. Dirichlef s integrals, 200. Discontinuity, of functions, 64 ; classification of, 65. See also Infinite discontinuity and PoirUs of infinite discontinuity. Divergence, of sequences, 37 ; of series, 41 ; of functions, 51 ; of integrals, 98. Fej6r's theorem, 234. Fej6r'8 theorem and the convergence of Fourier's series, 240, 258. Fourier's constants, definition of, 196. Fourier's integrals (Chapter X.); simple treatment of, 284; more general con- ditions for, 287 ; cosine and sine integrals, 292 ; Sommerf eld's discussion of, 293. Fourier's series (Chapters VII. and VIII.) ; definition of, 196 ; Lagrange's treat- ment of, 198 ; proof of convergence of, under certaHn conditions, 210 ; for even functions (the cosine series), 216; for odd functions (the sine series), 220 ; for intervals other than ( - tt, tt), 228 ; Poisson's discussion of, 230 ; Fej6r's theorem, 234, 240 ; order of the terms in, 248 ; uniform convergence of, 253 ; differentiation and integration of, 261 ; more general theory of, 262. Functions of a single variable, definition of, 49 ; bounded in an interval, 50 ; upper and lower bounds of, 50 ; oscillation in an interval, 50 ; limits of, 50 ; con- tinuous, 59 ; discontinuous, 64 ; monotonic, 66 ; inverse, 68 ; integrable, 84 ; of bounded variation, 207. Functions of several variables, 71. General principle of convergence, of sequences, 34 ; of functions, 56. Gibbs's phenomenon in Fourier's series (Chapter IX.) ; 264. Hardy's theorem, 239. Harmonic analyser (Kelvin's), 295. Harmonic analysis (Appendix I.), 295. Improper integrals, definition of, 113. ' Infinite aggregate. See Aggregate. Infinite discontinuity. See Points of infinite discontinuity. Infinite integrals (integrand function of a single variable), integrand bounded and interval infinite, 98 ; necessary and sufficient condition for convergence of, 100 ; with positive integrand, 101 ; absolute convergence of, 103 ; /x-test for convergence of, 104 ; other tests for convergence of, 106 ; mean value theorems for, 109. Infinite integrals (integrand function of a single variable), integrand infinite. 111 ; /i-test and other tests for convergence of, 114 ; absolute convergence of, 115. Infinite integrals (integrand function of two variables), definition of uniform con- vergence of, 174 ; tests for uniform convergence of, 174 ; continuity, in- tegration and differentiation of, 179. Infinite sequences and series. See Sequences and Series. Infinity of a function, definition of, 65. Integrable functions, 84. Digitized by Google 322 GENERAL INDEX Integration of intesrali : ordinary, 172 ; infinite, 180, 183, 190. Integration of series (ordinary integrals), 140 ; power series, 148 ; Fourier's series, 261 ; (infinite integrals), 154. Interval : open, closed, open at one end and closed at the other, 49. Inverse functions. 68. Irrational numbers. See Numbers. Limits, of sequences, 33 ; of functions, 50 ; of functions of two variables, 72 ; repeated, 127. Limiting points of an aggregate, 31. Lower integrals, definition of, 81. Mean value theorems of the integral calcolus ; first theorem (ordinary integrals), 92 ; (infinite integrals), 109 ; second theorem (ordinary integrals), 94 ; (infinite integrals), 109. Modulus. See Ahaol'ide. value. Monotonic functions, 66 ; admit only ordinary discontinuities, 67 ; integrability of, 84. Monotonic in the stricter sense, definition of, 39. M-test for convergence of series, 134. /t-test for convergence of integrals, 104, 116. Neighbourhood of a point, definition of, 52. Numbers (Chapter I.) ; rational, 16 ; irrational, 17 ; Dedekind's theory of irrational, 18 ; real, 21 ; Dedekind's theorem on the system of real, 23 ; development of the system of real, 25. See also Dedekind's axiom of continuity, and Dedekind^s sections. Open interval, definition of, 49. Ordinary or simple discontinuity, definition of, 65. Oscillation of a function in an interval, 50 ; of a function .of two variables in a domain, 72. Oscillatory, sequences, 38 ; series, 41 ; functions, 52 ; integrals, 99. Partial remainder (pRn)* definition of, 42 ; (pRn(x))y definition of, 123. Periodogram analysis, 299. Points of infinite discontinuity, definition of, 66. Points of oscillatory discontinuity, definition of, 66. Poisson's discussion of Fourier's series, 230. Poisson's integral, 231. Power series ; interval of conyergence of, 145 ; nature of convergence of, 146; AbePs theorem on, 146 ; integration and differentiation of, 148. Proper integrals, definition of, 113. Rational numbers and real numbers. See Numbers. Remainder after n terms (i2„), definition of, 43 ; (R.^(x)), 123. Repeated limits, 127. Repeated integrals, (ordinary), 172 ; (infinite), 180, 183, 190. Sections. See DedeTdnd^s sections. Sequences ; convergent, 33 ; limit of, 33 ; necessary and sufficient condition for convergence of (general principle of convergence), 34 ; divergent and oscil- latory, 37 ; monotonic, 39. Digitized by Google GENERAL INDEX 323 Series : definition of sum of an infinite, 41 ; convergent, 41 ; divergent and oscil- latory, 41 ; necessary and sufficient condition for convergence of, 42 ; with positive terms, 43 ; absolute and conditional convergence of, 44 ; definition of sum, when terms are functions of a single variable, 122; uniform con- vergence of, 129 ; necessary and sufficient condition for uniform convergence of, 132 ; Weierstrass's jtf-test for imiform convergence of, 134 ; unuorm convergence and continuity of, 135 ; term by term differentiation and integra- tion of, 140. See also Differentiation of series, Fourier's series. Integration of series. Power series and Trigonometrical series. Simple (or ordinary) discontinaity, definition of, 65. Sine integral (Fourier's integral), 284, 2«(2. Sine series (Fourier's series), 197, 220. Summable series (CI), definition of, 151. Sums S and s, definition of, 77. Trigonometrical series, 196. Uniform continuity of a function, 62. Uniform convergence, of series, 129; of integrals, 174. Upper integrals, definition of, 81. . ' Weierstrass's non-differentiable coBtinuous function, 77. Weierstrass's if-test for uniform convergence, 134. t- Weierstrass's theorem on limiting points to a bounded aggregate, 32. GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD. Digitized by VjOOQIC Digitized by Google -t > Digitized by Google 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. 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