Holomorphic Functions and Mappings
This chapter argues that a holomorphic function has a local convergent power series and conversely that a convergent power series represents a holomorphic function. The Cauchy integral formula for polydiscs is important for obtaining many elementary properties of holomorphic functions. The inverse of a biholomorphic mapping is holomorphic, hence continuous, and therefore takes compact sets to compact sets; thus a biholomorphic mapping is necessarily proper. In complex analysis in several variables, it is seldom the case that given domains are biholomorphically equivalent, so one also wishes to know whether there is a proper holomorphic mapping from one domain onto another. There are theorems in function theory of several complex variables stating that proper mappings between certain classes of domains are necessarily biholomorphic. The famous Weierstrass “Vorbereitungssatz” enables us to solve equations when the conditions of the inverse function theorem fail.