The Picard Theorems and the Riemann Mapping Theorem

We look at various interesting results from complex analysis. In particular we focus on the Picard Theorems and the Riemann Mapping Theorem. The former of these tells us that image of a holomorphic function omits at most one value under certain conditions. The Riemann Mapping Theorem gives us an equivalence relation between all proper simply connected subsets of the complex plane. We prove these as well as other interesting auxiliary results.