Editing Series (mathematics) (section)

===Fourier series===
[[Fourier series]] were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
[[Jacob Bernoulli]] (1702) and his brother [[Johann Bernoulli]] (1701) and still
earlier by [[Franciscus Vieta|Vieta]]. Euler and [[Joseph Louis Lagrange|Lagrange]] simplified the subject,
as did [[Louis Poinsot|Poinsot]], [[Karl Schröter|Schröter]], [[James Whitbread Lee Glaisher|Glaisher]], and [[Ernst Kummer|Kummer]].

Fourier (1807) set for himself a different problem, to
expand a given function of {{tmath|x}} in terms of the sines or cosines of
multiples of {{tmath|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 general
theorem. [[Siméon Denis Poisson|Poisson]] (1820–23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for [[Augustin Louis Cauchy|Cauchy]] (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see [[convergence of Fourier series]]). Dirichlet's treatment (''[[Journal für die reine und angewandte Mathematik|Crelle]]'', 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, [[Rudolf Lipschitz|Lipschitz]], [[Ludwig Schläfli|Schläfli]], and
[[Paul du Bois-Reymond|du Bois-Reymond]]. Among other prominent contributors to the theory of
trigonometric and Fourier series were [[Ulisse Dini|Dini]], [[Charles Hermite|Hermite]], [[Georges Henri Halphen|Halphen]],
Krause, Byerly and [[Paul Émile Appell|Appell]].