A + B cos φ + C cos 2φ + ..., then A = 1∫ π 0 Φ dφ, B = 2∫ π 0 Φ cos φ dφ, &c.
π π

The second period in the development of the theory commenced in 1807, when Fourier communicated his first memoir on the Theory of Heat to the French Academy. His exposition of the present theory is contained in a memoir sent to the Academy in 1811, of which his great treatise the Théorie analytique de la chaleur, published in 1822, is, in the main, a reproduction. Fourier set himself to consider the representation of a function given graphically, and was the first fully to grasp the 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 complete proof that a function with discontinuities is representable by the series, he indicated in particular cases the method of procedure afterwards carried out by Dirichlet. As an exposition of principles, Fourier’s work is still worthy of careful perusal by all students of the subject. Poisson’s treatment of the subject, which has been adopted in English works (see the Journal de l’école polytechnique, vol. xi., 1820, and vol. xii., 1823, and also his treatise, Théorie de la chaleur, 1835), depends upon the equality

∫ π −π ƒ(α) 1 − h²dα = 1∫ π −π ƒ(α) dα + 1Σ hn ∫ π −π ƒ(α) cos n(x − α) dα
1 − 2h cos (x − α) + h² 2ππ

where 0 < h < 1; the limit of the integral on the left-hand side is evaluated when h = 1, and found to be ½ {ƒ(x + 0) + ƒ(x − 0)}, 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 works on the integration of the equation (δ²u / δx²) + (δ²u / δy²) = 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 “Mémoire sur les développements des fonctions en séries périodiques,” Mém. de l’Inst., vol. vi., also Œuvres complètes, 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 Schläfli (see Crelle’s Journal, vol. lxxii.) and by Du Bois-Reymond (Mathem. Annalen, vol. vii.), who maintained that the sum is really indeterminate. Their objection appears, however, to rest upon a misapprehension as to the meaning of the sum of the series; if x1 be the point of discontinuity, it is possible to make x approach x1, 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 = x1 first and afterwards n = ∞, 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. lxiii., 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 β, |ƒ(β + δ) − f(β)| < Bδα as δ converges to zero, B being a constant, and α a positive exponent. A somewhat wider condition is

{ƒ(β + δ) − ƒ(β)} log δ = 0,
δ = 0

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, Über 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 infinite number of discontinuities. Important contributions to the theory of the series have been published by Du Bois-Reymond (Abh. der Bayer. Akademie, vol. xii., 1875, two memoirs, also in Crelle’s Journal, vols. lxxiv. lxxvi. lxxix.), by Kronecker (Berliner Berichte, 1885), by O. Hölder (Berliner Berichte, 1885), by Jordan (Comptes rendus, 1881, vol. xcii.), by Ascoli (Math. Annal., 1873, and Annali di matematica, vol. vi.), and by Genocchi (Atti della R. Acc. 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 Brodén (Math. Annalen, vol. lii.) contains a good investigation of some of the most recent results on the subject. The scope of Fourier’s Series has been extended by Lebesgue, 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, Leçons sur les séries trigonométriques (1906); also in a memoir, “Sur les séries trigonométriques,” Annales sc. de l’école normale supérieure, series ii. vol. xx. (1903), and in a paper “Sur la convergence des séries de Fourier,” Math. Annalen, vol. lxiv. (1905).

Authorities.—The foregoing historical account has been mainly drawn from A. Sachse’s work, “Versuch einer Geschichte der Darstellung willkürlicher Functionen einer Variabeln durch trigonometrische Reihen,” published in Schlömilch’s Zeitschrift für Mathematik, Supp., vol. xxv. 1880, 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 subject from an elementary point of view in C. Neumann’s treatise, Über die nach Kreis-, Kugel- und Cylinder-Functionen fortschreitenden Entwickelungen. Jordan’s discussion of the subject in his Cours d’analyse is worthy of attention: an account of functions with limited variation is given in vol. i.; see also a paper by Study in the Math. Annalen, vol. xlvii. On the second mean-value theorem papers by Bonnet (Brux. Mémoires, vol. xxiii., 1849, Lionville’s Journal, vol. xiv., 1849), by Du Bois-Reymond (Crelle’s Journal, vol. lxxix., 1875), by Hankel (Zeitschrift für Math. und Physik, vol. xiv., 1869), by Meyer (Math. Ann., vol. vi., 1872) and by Hölder (Göttinger Anzeigen, 1894) may be consulted; the most general form of the theorem 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. Bôcher’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.)