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Encyclopedia Britannica - Main :: HIG-HOR |
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HISTORY AND LITERATURE OF THE THEORY The history of the theory of the representation of functions by series of sines and cosines is of great interest
series was first considered in the 18th century, in connexion with the problem of a vibrating cord, and led to a controversy as to the possibility of such expansions. In a memoir published in 1747 (Memoirs of the Academy of Berlin, vol. iii.) D'Alembert showed that the ordinate y at anytime t of a vibrating cord satisfies a differentialequation of the form 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 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 volume he seeks for functions which satisfy this condition. In the year 1748 (Berlin Memoirs, vol. iv.) Euler, in discussing the problem, gave f(x) = a sin l +S sin2l xx + . . as a particular solution, and maintained that every curve, whether regular or irregular, must be representable in this form. This was objected to by D'Alembert (1750) and also by Lagrange on the ground that irregular curves are inadmissible. D. Bernoulli (Berlin Memoirs, vol. ix., 1753) based a similar result to that of Euler on physical 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 propagation of 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 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 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 Comm. Petrop., vol. v., 1754-1755), who obtained the formulae =sin -Z sin 2+3 sin 30... ,2 _ 4,2 12 4 = cos -4 cos 2+1- cos 34, .. . In a memoir presented to the Academy of St Petersburg
Fourier
A+B cos -+ cos 2rp+..., then A = J4)d4, B =-f ~ cos 440, &c. The second period in the development of the theory commenced in 1807, when Fourier
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 left-hand side is evaluated when h=1, and found to be i(f(x+o)-f-f(xo)), 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 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 general theorem cannot be accepted. The deficiency cannot be removed except by a process of the same nature as that afterwards applied by Dirichlet. The definite integral has been carefully studied by 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 Cauchy (see his " Memoire sur les developpements 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 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 model of clearness, in 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 writing f(x) =fl(x) -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 fog svKvdv which is finite, both the integrals being convergent and of known value. The other integral has a similar property, and we infer that nI-Ka,,, n1-Kb. are less than fixed finite 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. 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 indeterminate. Their objection appears, however, to 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 interval, whereas we ought to make x = xi first and afterwards n= oo, and no other way of going to the double limit is really admissible. Other papers by Dircksen (Crelle, vol. iv.) and 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 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 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 constant, and a a positive exponent. A somewhat wider condition is f(13+b)f($)} 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., 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 die Darstellbarkeit einer Function durch eine trigonometrische Reihe. The first part of his memoir contains a historical 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 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 call attention to the importance of the theory of uniform convergence of series in connexion with Fourier's Series was Stokes, in his memoir " On the 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. 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 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.. 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 Jordan (Comptes rendus, 1881, vol. xcii.), by Ascoli (Math. Annal., 1873, and Annali di matematica, vol. vi.), and by Genocchi (Atli della R. Ace. di Torino, vol. x., 1875). 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 good investigation of some of the most recent
AuraoRIT11s.The foregoing historical account has been mainly drawn from A. Sachse's work, " Versuch einer Geschichte der Darstellung willkfirlicher Functionen einer Variabeln durch trigonometrische Reihen," published in Schlomilch's Zeitschrift fur Mathematik, Stipp., vol. xxv. 188o, and from a paper by G. A. Gibson " On the History of the Fourier Series " (Proc. Ed. Math. 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. Neumann's treatise, Uber die nach Kreis-, Kugel- and Cylinder-Functionen fortschreitenden Entwirkelnn,gen. Jordan's discussion of the subject in his Cours d'analy.cc is worthy of at t cnt ion; an 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 Bonnet (Brux. Memoires, vol. 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 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. 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 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 information on the general theory of series. Bother's " Introduction to the Theory of Fourier's Series," 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 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. H.) End of Article: HISTORY AND LITERATURE OF THE If you wish, you can link directly to this article.
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