Various equivalent definitions of holomorphic functions in C^n; power series development(s), the domain of
convergence of a power series, circular and Reinhardt domains; characterizations of domains of convergence of power series and Laurent series.
Analytic continuation: basic theory and contrasts with the one-variable theory; introduction to the Hartogs
phenomenon and domains of holomorphy. Role of convexity in analytic continuation, holomorphic convexity,
plurisubharmonic functions, pseudoconvexity.
Characterizations of domains of holomorphy; the Levi problem and the role of the d-bar equation in this
characterization. The d-bar equation: review of distribution theory and an overview of Hormander’s method of
solution and applications.
Zeros of holomorphic functions: Weierstrass's Preparation Theorem, analytic varieties and some of their local and global properties; holomorphic maps; the inequivalence of the unit ball and the unit polydisc.