Geometric function theory

Riemann mapping theorem[edit]

Let ${\displaystyle z_{0}}$ be a point in a simply-connected region ${\displaystyle D_{1}(D_{1}\neq \mathbb {C} )}$ and ${\displaystyle D_{1}}$ having at least two boundary points. Then there exists a unique analytic function ${\displaystyle w=f(z)}$ mapping ${\displaystyle D_{1}}$ bijectively into the open unit disk ${\displaystyle |w|<1}$ such that ${\displaystyle f(z_{0})=0}$ and ${\displaystyle f'(z_{0})>0}$.


Although Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function. An example is given below.


In the above figure, consider ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$ as two simply connected regions different from ${\displaystyle \mathbb {C} }$. The Riemann mapping theorem provides the existence of ${\displaystyle w=f(z)}$ mapping ${\displaystyle D_{1}}$ onto the unit disk and existence of ${\displaystyle w=g(z)}$ mapping ${\displaystyle D_{2}}$ onto the unit disk. Thus ${\displaystyle g^{-1}f}$ is a one-to-one mapping of ${\displaystyle D_{1}}$ onto ${\displaystyle D_{2}}$. If we can show that ${\displaystyle g^{-1}}$, and consequently the composition, is analytic, we then have a conformal mapping of ${\displaystyle D_{1}}$ onto ${\displaystyle D_{2}}$, proving "any two simply connected regions different from the whole plane ${\displaystyle \mathbb {C} }$ can be mapped conformally onto each other."