much light has been thrown on them by the use of Riemann's surfaces. With the view of reducing their study to that of uniform transcendents, Poincaré proved that if is any analytical non-uniform function of , one can always find a variable , such that and are uniform functions of .
Weierstrass and Darboux have each given examples of continuous functions having no derivatives. Formerly it had been generally assumed that every function had a derivative. Ampère was the first who attempted to prove analytically (1806) the existence of a derivative, but the demonstration is not valid. In treating of discontinuous functions, Darboux established rigorously the necessary and sufficient condition that a continuous or discontinuous function be susceptible of integration. He gave fresh evidence of the care that must be exercised in the use of series by giving an example of a series always convergent and continuous, such that the series formed by the integrals of the terms is always convergent, and yet does not represent the integral of the first series.[87]
The general theory of functions of two variables has been investigated to some extent by Weierstrass and Poincaré.
H. A. Schwarz of Berlin (born 1845), a pupil of Weierstrass, has given the conform representation (Abbildung) of various surfaces on a circle. In transforming by aid of certain substitutions a polygon bounded by circular arcs into another also bounded by circular arcs, he was led to a remarkable differential equation , where is the expression which Cayley calls the "Schwarzian derivative," and which led Sylvester to the theory of reciprocants. Schwarz's developments on minimum surfaces, his work on hypergeometric series, his inquiries on the existence of solutions to important partial differential equations under prescribed conditions, have secured a prominent place in mathematical literature.
