Difficult questions arose in the study of Fourier's series.79
Cauchy was the first who felt the necessity of inquiring into
its convergence. But his mode of proceeding was found by Dirichlet to be unsatisfactory. Dirichlet made the first
thorough researches on this subject (Crelle, Vol. IV.). They culminate in the result that whenever the function does not become infinite, does not have an infinite number of discontinuities, and does not possess an infinite number of maxima and minima, then Fourier's series converges toward the value of that function at all places, except points of discontinuity, and there it converges toward the mean of the two boundary values. Schläfli of Bern and Du Bois-Reymond
expressed doubts as to the correctness of the mean value, which were, however, not well founded. Dirichlet's conditions are sufficient, but not necessary. Lipschitz, of
Bonn, proved that Fourier's series still represents the function when the number of discontinuities is infinite, and
established a condition on which it represents a function having an infinite number of maxima and minima. Dirichlet's
belief that all continuous functions can be represented by Fourier's series at all points was shared by Riemann and
H. Hankel, but was proved to be false by Du Bois-Reymond
and H. A. Schwarz.
Riemann inquired what properties a function must have, so that there may be a trigonometric series which, whenever
it is convergent, converges toward the value of the function. He found necessary and sufficient conditions for this. They do not decide, however, whether such a series actually represents the function or not. Riemann rejected Cauchy's definition
of a definite integral on account of its arbitrariness, gave a new definition, and then inquired when a function has an integral. His researches brought to light the fact that continuous functions need not always have a differential coefficient. But this property, which was shown by Weierstrass to
belong to large classes of functions, was not found necessarily to exclude them from being represented by Fourier's series. Doubts on some of the conclusions about Fourier's series were thrown by the observation, made by Weierstrass, that the integral of an infinite series can be shown to be equal to the
sum of the integrals of the separate terms only when the series converges uniformly within the region in question. The subject of uniform convergence was investigated by Philipp Ludwig
Seidel (1848) and G. G. Stokes (1847), and has assumed