I Elementary local properties of holomorphic functions of several complex variables

Abstract

In this chapter we study the local properties of holomorphic functions of several complex variables which can be deduced directly from the classical theory of holomorphic functions in one complex variable. The basis for our work is a Cauchy formula for polydiscs which generalises the classical Cauchy formula. Most of the theorems proved in this chapter extend well-known theorems for holomorphic functions in dimension 1 (such as the open mapping theorem, the maximum principle, Montel's theorem and the local inversion theorem) to multivariable analysis. However, when we try to extend holomorphic function a phenomenon which is specific to n-dimensional space with n ⩾ 2 appears, namely Hartog's phenomenon. A special case of this phenomenon, which is studied in detail in Chapter III, is discussed at the end of this chapter.