Elementary properties of functions of several complex variables

Abstract

In Section 1.1 we define holomorphic functions of several complex variables and prove simple properties of these functions. In Section 1.2, by means of a simple extension of the Cauchy-Green formula to several variables, we solve the inhomogeneous Cauchy-Riemann equations for some special cases. As a consequence we obtain a theorem of Hartogs which gives examples of open sets in ℂ n (n ≥ 2) where all holomorphic functions can be continued holomorphically to a larger open set. Open sets for which this is not possible are called domains of holomorphy. These are investigated in Section 1.3. Section 1.4 is devoted to continuous plurisubharmonic and strictly plurisubharmonic C 2 -functions. In Section 1.5, by means of these functions, pseudoconvex and strictly pseudoconvex open sets are introduced. We prove that every domain of holomorphy is pseudoconvex, but the converse (Levi’s problem) is left to Section 2.7. Sections 1.6–1.12 are devoted to integral representation formulas for functions as well as for differential forms n ℂ n . These formulas form the basic tool for the methods developed in the present book.