An Introduction to complex analysis in several variables: Riemann mapping and Bergman spaces

Abstract

The main goal of this work is to give an introduction of the fundamental concepts in complex analysis in several variables.

It starts by introducing holomorphic functions of several complex variables, their representation via power series, and fundamental results like the Cauchy integral formula.

Then it follows by the Riemann mapping theorem, a cornerstone result that guarantees the existence of conformal mappings between simply connected domains and the unit disc in $\mathbb{C}$. We show also that the Riemann mapping theorem cannot be extended to $\mathbb{C}^n$.

Finally, the last part of the report delves into Bergman spaces, studying their kernels and their connection to the Riemann Mapping Theorem.