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HISTORY AND LITERATURE OF THE

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Originally appearing in Volume V10, Page 758 of the 1911 Encyclopedia Britannica.
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HISTORY AND LITERATURE OF THE THEORY The history of the theory of the See also:representation of functions by See also:series of sines and cosines is of See also:great See also:interest in connexion with the progressive development of the notion of an arbitrary See also:function of a real variable, and of the peculiarities which such a function may possess; the See also:modern views on the See also:foundations of the infinitesimal calculus have been to a very considerable extent formed in this connexion (see FUNCTION). The representation of functions by these series was first considered in the 18th See also:century, in connexion with the problem of a vibrating See also:cord, and led to a controversy as to the possibility of such expansions. In a memoir published in 1747 (See also:Memoirs of the See also:Academy of See also:Berlin, vol. iii.) D'See also:Alembert showed that the See also:ordinate y at anytime t of a vibrating cord satisfies a See also:differential See also:equation of the See also:form See also:Sty = a2bx , where x is measured along the undisturbed length of the cord, and that with the ends of the cord of length I fixed, the appropriate See also:solution is y = f (at +x) -f(at-x), where f is a function such that f(x) =f(x+21) ; in another memoir in the same See also:volume he seeks for functions which satisfy this See also:condition. In the See also:year 1748 (Berlin Memoirs, vol. iv.) See also:Euler, in discussing the problem, gave f(x) = a See also:sin l +S sin2l xx + . . as a particular solution, and maintained that every See also:curve, whether See also:regular or irregular, must be representable in this form. This was objected to by D'Alembert (1750) and also by See also:Lagrange on the ground that irregular curves are inadmissible. D. See also:Bernoulli (Berlin Memoirs, vol. ix., 1753) based a similar result to that of Euler on See also:physical See also:intuition; his method was criticized by Euler (1753). The question was then considered from a new point of view by Lagrange, in a memoir on the nature and See also:propagation of See also:sound (Miscellanea Taurensia, 1759; (Euvres, vol. i.), who, while criticizing Euler's method, considers a finite number of vibrating particles, and then makes the number of them See also:infinite; he did not, however, quite fully carry out the determination of the coefficients in Bernoulli's Series. These mathematicians were hampered by the narrow conception of a function, in which it is regarded as necessarily continuous; a discontinuous function was considered only as a See also:succession of several different functions. Thus the possibility of the expansion of a broken function was not generally admitted. The first cases in which rational functions are expressed in sines and cosines were given by Euler (Subsidium calculi sinuum, Novi See also:Comm.

Petrop., vol. v., 1754-1755), who obtained the formulae =sin ¢-Z sin 2¢+3 sin 30... ,2 _ 4,2 12 4 = See also:

cos ¢-4 cos 2¢+1- cos 34, .. . In a memoir presented to the Academy of St See also:Petersburg in 1777, but not published until 1798, Euler gave the method afterwards used by See also:Fourier, of determining the coefficients in the expansions; he remarked that if'' is expansible in the form A+B cos ¢-+ cos 2rp+..., then A = J4)d4, B =-f ~ cos 440, &c. The second See also:period in the development of the theory commenced in 1807, when Fourier communicated his first memoir on the Theory of See also:Heat to the See also:French Academy. His exposition of the See also:present theory is contained in a memoir sent to the Academy in 1811, of which his great See also:treatise the Theorie analytique de la chaleur, published in 1822, is, in the See also:main, a See also:reproduction. Fourier set himself to consider the representation of a function given graphically, and was the first fully to grasp the See also:idea that a single function may consist of detached portions given arbitrarily by a graph. He had an accurate conception of the convergence of a series, and although he did not give a formally See also:complete See also:proof that a function with discontinuities is representable by the series, he indicated in particular cases the method of See also:procedure afterwards carried out by Dirichlet. As an exposition of principles, Fourier's See also:work is still worthy of careful perusal by all students of the subject. See also:Poisson's treatment of the subject, which has been adopted in See also:English See also:works (see the See also:Journal de l'ecole polytechnique, vol. xi., 182o, and vol. xii., 1823, and also his treatise, Theorie de la chaleur, 1835), 1_h2 depends upon the equalityf f (a) 1-2h cos (x-a) } hzda =2— 77,f(a)da+;h" f T f(a) cos n(x-a)da where o < h< 1; the limit of the integral on the See also:left-See also:hand See also:side is evaluated when h=1, and found to be i(f(x+o)-f-f(x—o)), the series on the right-hand side becoming Fourier's Series. The equality of the two limits is then inferred. If the series is assumed to be convergent when h=1, by a theorem of See also:Abel's its sum is continuous with the sum for values of h less than unity, but a proof of the convergency for h=1 is requisite for the validity of Poisson's proof ; as Poisson gave no such proof of convergency, his proof of the See also:general theorem cannot be accepted.

The deficiency cannot be removed except by a See also:

process of the same nature as that afterwards applied by Dirichlet. The definite integral has been carefully studied by See also:Schwarz (see two memoirs in his collected z works on the integration of the equation 62u .x+- =0), who showed that the limiting value of the integral depends upon the manner in which the limit is approached. Investigations of Fourier's Series were also given by See also:Cauchy (see his " Memoire sur See also:les developpements See also:des fonctions en series periodiques," Mem. de l'Inst., vol. vi., also t.Ruvres completes, vol. vii.) ; his method, which depends upon a use of complex variables, was accepted, with some modification, as valid by See also:Riemann, but one at least of his proofs is no longer regarded as satisfactory. The first completely satisfactory investigation is due to Dirichlet; his first memoir appeared in Crelle's Journal for 1829, and the second, which is a See also:model of clearness, in See also:Dove's Repertorium der Physik. Dirichlet laid down certain definite sufficient conditions in regard to the nature of a function which is expansible, and found under these conditions the limiting value of the sum of n terms of the series. Dirichlet's determination of the sum of the series at a point of discontinuity has been criticized by Schlafli (see Crelle's Journal, vol. lxxii.) and by Du Bois-Reymond (Mathem. Annalen, vol. vii.), who maintained that the sum is really great n is, are each less than a fixed finite quantity. For See also:writing f(x) =fl(x) -See also:f2(x), we have f fi(x)cos nxdx =fi(-r+0) f cos nxdx +fi(r-0) f w cos nxdx -w -r hence . J J of the form ¢(c) cos nc f0 COKVdv sin nc See also:fog svKvdv which is finite, both the integrals being convergent and of known value. The other integral has a similar See also:property, and we infer that nI-Ka,,, n1-Kb. are less than fixed finite See also:numbers. The Differentiation of Fourier's Series.—If we assume that the differential coefficient of a function f(x) represented by a Fourier's Series exists, that function f'(x) is not necessarily representable by the series obtained by differentiating the terms of the Fourier's Series, such derived series being in fact not necessarily convergent. See also:Stokes has obtained general formulae for finding the series which represent f'(x), f"(x)—the successive differential coefficients of a limited function f(x).

As an example of such formulae.. consider the sine series (1) ; f(x) is represented by d E sin--J 0f(x) sinnixdx; on integration by parts we have f u f (x) sin nl xdx nn [ f (+0) = f (l-0) +E cos ni { f(a+0) f (a-0) }] If f(x) is infinite at x = c, and is of the form 0(x)g near the point (x-c) n+F f (x) cos nxdx contains portions of the form f m(—(x c) 7 cos nxdx f:-7r (b(x) (tee (x_c) K cos nxdx ; consider the first of these, and put x = c+u, it thus becomes f E4(utn) cos n(c+u)du, which is of the form fE cos n (c+u) ¢(c+Os)J o nx -du; now let nu=v, the integral becomes cos nc a cos v d sin nc " sin v 4'(c+Os) { nI-x f o vx v - n • J dv hence ni-K (•w f(x) cos nxdx becomes, as n is definitely increased, J r c, where o<K<I, the integral 758 indeterminate. Their objection appears, however, to See also:

rest upon a misapprehension as to the meaning of the sum of the series; if x, be the point of discontinuity, it is possible to make x approach xi, and n become indefinitely great, so that the sum of the series takes any assigned value in a certain See also:interval, whereas we ought to make x = xi first and afterwards n= oo, and no other way of going to the See also:double limit is really admissible. Other papers by Dircksen (Crelle, vol. iv.) and See also:Bessel (Astronomische Nachrichten, vol. xvi.), on similar lines to those by Dirichlet, are of inferior importance. Many of the investigations subsequent to Dirichlet's have the See also:object of freeing a function from some of the restrictions which were imposed upon it in Dirichlet's proof, but no complete set of necessary and sufficient conditions as to the nature of the function has been obtained. Lipschitz (" De explicatione per series trigonometricas," Crelle's Journal, vol. Ixiii., 1864) showed that, under a certain condition, a function which has an infinite number of See also:maxima and minima in the neighbourhood of a point is still expansible; his condition is that at the point of discontinuity R, f(R+b) —f(13) < BS" as S converges to zero, B being a See also:constant, and a a See also:positive exponent. A somewhat wider condition is f(13+b)—f($)} See also:log S=o, =o for which Lipschitz's results would hold. This last condition is adopted by Dini in his treatise (Sopra la serie di Fourier, &c., See also:Pisa, 1880). The modern period in the theory was inaugurated by the publication by Riemann in 1867 of his very important memoir, written in 1854, Uber See also:die Darstellbarkeit einer Function durch eine trigonometrische Reihe. The first See also:part of his memoir contains a See also:historical See also:account of the work of previous investigators; in the second part there is a discussion of the foundations of the Integral Calculus, and the third part is mainly devoted to a discussion of what can be inferred as to the nature of a function respecting the changes in its value for a continuous See also:change in the variable, if the function is capable of representation by a trigonometrical series. Dirichlet and probably Riemann thought that all continuous functions were everywhere representable by the series; this view was refuted by Du Bois-Reymond (Abh. der Bayer. Akad. vol. xii.

2). It was shown by Riemann that the convergence or non-convergence of the series at a particular point x depends only upon the nature of the function in an arbitrarily small neighbourhood of the point x. The first to See also:

call See also:attention to the importance of the theory of See also:uniform convergence of series in connexion with Fourier's Series was Stokes, in his memoir " On the See also:Critical Values of the Sums of Periodic Series " (Camb. Phil. Trans., 1847; Collected Papers, vol. i.). As the method of determining the coefficients in a trigonometrical series is invalid unless the series converges in general uniformly, the question arose whether series with coefficients other than those of Fourier exist which represent arbitrary functions. See also:Heine showed (Crelle's Journal, vol. lxxi., 1870, and in his treatise Kugelfunctionen, vol. i.) that Fourier's Series is in general uniformly convergent, and that if there is a uniformly convergent series which represents a function, it is the only one of the See also:kind. G. Cantor then showed (Crelle's Journal, vols. lxxii. lxxiii.) that even if uniform convergence be not demanded, there can be but one convergent expansion for a function, and that it is that of Fourier. In the Math.. See also:Ann. vol. v., Cantor extended his investigation to functions having an in-finite number of discontinuities. Important contributions to the theory of the series have been published b-; Du Bois-Reymond (Abh. der Bayer.

Akademie, vol. xii., 18/5, two memoirs, also in Crelle's Journal, vols. lxxiv. lxxvi. lxxix.), by Kronecker (Berliner Berichte, 1885), by O. Holder (Berliner Berichte, 1885), by See also:

Jordan (Comptes rendus, 1881, vol. xcii.), by See also:Ascoli (Math. Annal., 1873, and Annali di matematica, vol. vi.), and by Genocchi (Atli della R. See also:Ace. di Torino, vol. x., 1875). See also:Hamilton's memoir on " Fluctuating Functions " (Trans. R.I.A., vol. xix., 1842) may also be studied with profit in this connexion. A memoir by Broden (Math. Annalen, vol. lii.) contains a See also:good investigation of some of the most See also:recent results on the subject. The See also:scope of Fourier's Series has been extended by I.ebesgue, who introduced a conception of integration wider than that due to Riemann. Lebesgue's work on Fourier's Series will be found .in his treatise, Lecons sur les series trigonometriques (1906) ; also in a memoir, " Sur les series trigonometriqucs," Annales sc. de l'ecole normale superieure, series ii. vol. xx. (1903), and in a See also:paper " Sur la convergence des series de Fourier," Math. Annalen, vol. lxiv.

(1905). AuraoRIT11s.—The foregoing historical account has been mainly See also:

drawn from A. Sachse's work, " Versuch einer Geschichte der Darstellung willkfirlicher Functionen einer Variabeln durch trigonometrische Reihen," published in Schlomilch's Zeitschrift See also:fur Mathematik, Stipp., vol. See also:xxv. 188o, and from a paper by G. A. See also:Gibson " On the History of the Fourier Series " (Proc. Ed. Math. See also:Soc. vol. xi.). Reiff's Geschichte der unendlichen Reihen may also be consulted, and also the first part of Riemann's memoir referred to above. Besides Dini's treatise already referred to, there is a lucid treatment of the subejct from an elementary point of view in C. See also:Neumann's treatise, Uber die nach Kreis-, Kugel- and See also:Cylinder-Functionen fortschreitenden Entwirkelnn,gen.

Jordan's discussion of the subject in his Cours d'analy.cc is worthy of at t cnt See also:

ion; an See also:acron nt of fq nr11Qn5 with limited variation is given in vol. i.; see also a paper by Studyin the Math. Annalen, vol. xlvii. On the second mean-value theorem papers by See also:Bonnet (Brux. Memoires, vol. See also:xxiii., 1849, Lionville's Journal, vol. xiv., 1849), by Du Bois-Reymond (Crelle's Journal, vol. lxxix., 1875), by Hankel (Zeitschrift fur Math. and Physik, vol. xiv., 1869), by See also:Meyer (Math. Ann., vol. vi., 1872) and by Holder (Gbltinger Anzeigen, 1894) may be consulted; the most general form.of the thebrem has been given by Hobson (Proc. See also:London Math. Soc., Series II. vol. vii., 1909). On the theory of uniform convergence of series, a memoir by W. F. Osgood (Amer. Journal of Math. xix.) may be with See also:advantage consulted. On the theory of series in general, in relation to the functions which they can represent, a memoir by Baire (Annali, di matematica, Series III. vol. iii.) is of great importance.

Bromwich's Theory of Infinite Series (1908) contains much See also:

information on the general theory of series. Bother's " Introduction to the Theory of Fourier's Series," See also:Annals of Math., Series II. vol. vii., 1906, will be found useful. See also Carslaw's Introduction to the Theory of Fourier's Series and Integrals, and the Mathematical Theory of the See also:Conduction of Heat (1906). A full account of the theory will be found in Hobson's treatise On the Theory of Functions of a Real- Variable and on the Theory of Fourier's Series (1907)• (E. W.

End of Article: HISTORY AND LITERATURE OF THE

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