Riemann Mapping Theorem

Let $z_0$ be a point in a simply connected region $R \not= \Bbb{C}$. Then there is a unique Analytic Function $w = f(z)$ mapping $R$ one-to-one onto the Disk $\vert w\vert < 1$ such that $f(z_0) = 0$ and $f'(z_0) = 0$. The Corollary guarantees that any two simply connected regions except $\Bbb{R}^2$ can be mapped Conformally onto each other.