Theory of Functions

four theta-functions of two variables, and researches of H. H. Weber of Marburg, F. Prym of Würzburg, Adolf Krazer, and

Martin Krause of Dresden led to broader views. Researches

on double theta-functions, made by Cayley, were extended to

quadruple theta-functions by Thomas Craig of the Johns

Hopkins University.

Starting with the integrals of the most general form and considering the inverse functions corresponding to these integrals (the Abelian functions of $p$ variables), Riemann

defined the theta-functions of $p$ variables as the sum of a $p$-tuply infinite series of exponentials, the general term depending on $p$ variables. Riemann shows that the Abelian functions are algebraically connected with theta-functions of the proper arguments, and presents the theory in the broadest form.56 He rests the theory of the multiple theta-functions upon the general principles of the theory of functions of a complex variable.

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, $y$ was called a function of $x$, if there

existed an equation between these variables which made it possible to calculate $y$ for any given value of $x$ lying anywhere between $-\infty$ and $+\infty$. The study of Fourier's theory

of heat led Dirichlet to a new definition: $y$ is called a function

of $x$, if $y$ possess one or more definite values for each of certain values that $x$ is assumed to take in an interval $x_0$ to $x_1$. In functions thus defined, there need be no analytical connection between $y$ and $x$, 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