Factorization of Analytic Functions of Several Variables.

Abstract

The theorems of factorization with which this paper deals are in part so obvious as to seem trivial. In part, however, they involve definitions and conceptions, the necessity for which is not evident at a glance, nor can their justification be established on any a priori grounds; one has to go over the whole field and ascertain what the situation is that is to be met. The main theorems are contained in a memoir by Weierstrass (cf. ? 3 below); but the presentation in that paper is altogether inadequate for making clear to the reader that which is essential. This theory of factorization is of much importance in practice, and it is embarrassing for a writer, in studying a subject in the theory of analytic functions of several complex variables, to be obliged to assume it, well knowing that most readers have never thought it through in detail. It is for this reason that I have ventured to offer a systematic presentation of the theory as a whole.* The reader will do well to turn at once to ? 4, for he will find there one of the chief topics with which the paper deals.