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Fourier series

Fourier series were being investigatedas the result of physical considerations at the same time thatGauss, Abel, and Cauchy were working out the theory of infiniteseries. Series for the expansion of sines and cosines, of multiplearcs in powers of the sine and cosine of the arc had been treated by Jacob Bernoulli (1702) and his brother Johann Bernoulli (1701) and stillearlier by Vieta. Euler and Lagrange simplified the subject,as did Poinsot, Schröter, Glaisher, and Kummer.

Fourier (1807) set for himself a different problem, toexpand a given function of x in terms of the sines or cosines ofmultiples of x, a problem which he embodied in his Théorie analytique de la chaleur (1822). Euler had already given the formulas for determining the coefficients in the series;Fourier was the first to assert and attempt to prove the generaltheorem. Poisson (1820–23) also attacked the problem from adifferent standpoint. Fourier did not, however, settle the questionof convergence of his series, a matter left for Cauchy (1826) toattempt and for Dirichlet (1829) to handle in a thoroughlyscientific manner (see convergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement byRiemann (1854), Heine, Lipschitz, Schläfli, and du Bois-Reymond. Among other prominent contributors to the theory oftrigonometric and Fourier series were Dini, Hermite, Halphen,Krause, Byerly and Appell.