Introduction to Complex Analysis in Several Variables
The theory of several complex variables is a broad, beautiful and (nowadays) classical part of mathematics. There are several excellent books available that describe the theory in full detail. However, what is often needed for students and researchers who come from other fields of mathematics is an introduction to the basic properties of functions of several complex variables without going into the full and difficult theory. This is what the reader can find here.
The book starts with the basics (definitions, the Cauchy integral formula, properties of rings of holomorphic functions, power series expansions and Reinhardt domains). Properties of the Dolbeault complex are used in the proof of a version of the Hartogs theorem. A chapter is devoted to the implicit and inverse function theorems, the Riemann mapping theorem and to properties of biholomorphic maps. Analytic continuation, domains of holomorphy, holomorphically convex domains, the Bochner theorem and the Cartan-Thullen theorems are treated in two chapters. The book also includes a description of basic properties of analytic sets and the proof of the Nullstellensatz for principal ideals. Hence basic features of the theory are introduced and illustrated in a relatively small space (of course, difficult parts of the theory are not treated here). The book contains a lot of examples and exercises