# Complex Analysis in Several Variables I

## Official Course Description

## Description

Repetition of background theory for functions in one complex variable. Complex vector spaces, complex structures and the operator $frac{partial}{partialbar z}$. Holomorphic functions in several variables and their main properties. Comparison with the theory of functions in one variable will be emphasized. Integral formulas in one and several variables: Cauchy integral formula, Bochner-Martinelli formula. Introduction to integrals of Leray type. Solution to d-bar problem in one variable, introduction to d-bar problem in several variables. Hartog extension theorem. Approximation of holomorphic functions by polynomials. Runge's theorem, polynomial Convexity. Existence domains for holomorphic functions in one and several variables. Introduction to pseudoconvexity. Holomorphic and biholomorphic mappings, Riemann mapping theorem and it cannot be generalized to several variables. Introduction to automorphic groups and invariant distance.