Chapter 9 Several Complex Variables

Abstract

Throughout the nineteenth century and well into the twentieth the study of complex functions of several variables posed a challenge to the experts in the function theory of a single variable. As we have seen in Chap. 6, the prospect of creating a theory of Abelian functions was one that Weierstrass continually had in mind; it was the ultimate goal of all his work. And yet a marked distinction between the theories of one and several variables persists to the present day. Almost all universities offer a mainstream course in single variable complex function theory; few, if any, present the theory of several variables as other than a specialist option. We shall see that this distinction is in the nature of the functions studied. Because this dichotomy survives in the modern syllabus, we have divided this chapter into three sections. The first is a survey of the claim that the theories of one and several variables diverge markedly. We give an indication of what was discovered about the complex function theory of several variables, but generally slight the proofs so that the section can be read by a broad audience. The second section looks at the history of the principal results about Abelian functions and theta functions which, for a long time, were the only examples known of complex functions of several variables. In the final section we reconsider the general theory, look at some of the techniques used, and seek sharpen the discussion of the opening section. The latter sections naturally place greater demands on the reader.