Theory of Functions

The theory of functions of one complex variable has been studied since Riemann's time mainly by Karl Weierstrass of

Berlin (born 1815), Gustaf Mittag-Leffler of Stockholm (born

1846), and Poincaré of Paris. Of the three classes of such

functions (viz. functions uniform throughout, functions uniform only in lacunary spaces, and non-uniform functions) Weierstrass showed that those functions of the first class which can be developed according to ascending powers of $x$ into converging series, can be decomposed into a product of an infinite number of primary factors. A primary factor of the

species $n$ is the product $(1-{\displaystyle \frac{x}{a}})e^{P(x)}$, $P(x)$ being an entire polynomial of the $n$th degree. A function of the species $n$ is one, all the primary factors of which are of species $n$. This classification gave rise to many interesting problems studied also by Poincaré.

The first of the three classes of functions of a complex variable embraces, among others, functions having an infinite number of singular points, but no singular lines, and at the same time no isolated singular points. These are Fuchsian functions, existing throughout the whole extent. Poincaré first gave an example of such a function.

Uniform functions of two variables, unaltered by certain linear substitutions, called hyperfuchsian functions, have been studied by E. Picard of Paris, and by Poincaré.81

Functions of the second class, uniform only in lacunary spaces, were first pointed out by Weierstrass. The Fuchsian and the Kleinian functions do not generally exist, except in

the interior of a circle or of a domain otherwise bounded, and are therefore examples of functions of the second class. Poincaré has shown how to generate functions of this class, and has studied them along the lines marked out by Weierstrass. Important is his proof that there is no way of generalising them so as to get rid of the lacunæ.

Non-uniform functions are much less developed than the preceding classes, even though their properties in the vicinity of a given point have been diligently studied, and though

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 $y$ is any analytical non-uniform function of $x$, one can always find a variable $z$, such that $x$ and $y$ are uniform functions of $z$.

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