Through the researches of A. Brill of Tübingen, M. Nöther of Erlangen, and Ferdinand Lindemann of Munich, made in connection with Riemann-Roch's theorem and the theory of residuation, there has grown out of the theory of Abelian functions a theory of algebraic functions and point-groups on algebraic curves.
Before proceeding to the general theory of functions, we make mention of the "calculus of functions," studied chiefly by C. Babbage, J. F. W. Herschel, and De Morgan, which was not so much a theory of functions as a theory of the solution of functional equations by means of known functions or symbols.
The history of the general theory of functions begins with the adoption of new definitions of a function. With the Bernoullis and Leibniz, was called a function of , if there existed an equation between these variables which made it possible to calculate for any given value of lying anywhere between and . The study of Fourier's theory of heat led Dirichlet to a new definition: is called a function of , if possess one or more definite values for each of certain values that is assumed to take in an interval to . In functions thus defined, there need be no analytical connection between and , and it becomes necessary to look for possible discontinuities. A great revolution in the ideas of a function was brought about by Cauchy when, in a function as defined by Dirichlet, he gave the variables imaginary values, and when he extended the notion of a definite integral by letting the variable pass from one limit to the other by a succession of imaginary values along arbitrary paths. Cauchy established several fundamental theorems, and gave the first great impulse to the study of the general theory of functions. His researches were continued in France by Puiseux and Liouville. But more profound investigations were made in Germany by Riemann.
