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 great importance in Weierstrass' theory of functions. It became necessary to prove that a trigonometric series representing a continuous function converges uniformly. This was done by Heinrich Eduard Heine (1821–1881), of Halle. Later researches on Fourier's series were made by G. Cantor and Du Bois-Reymond.
As compared with the vast development of other mathe-
