The Riemann mapping theorem is as follows: Let $U \neq \mathbb{C}$ be a simply connected domain and $w_{1}, w_{2} \in U$ any points. Then, there exists a unique conformal mapping $f: \mathbb{D} \rightarrow U$ such that $f^{-1}(w_{1}) = 0$ and $f^{-1}(w_{2}) > 0$ (where $\mathbb{D}$ is the unit disk).
I would like to know the reason why the Riemann mapping theorem is so important. In particular I am curious to know if it is of interest to calculate the aforementioned function $f$ and if there is a technique to do it.
Best Answer
For the importance of the Riemann mapping theorem, see Wikipedia.
The Riemann mapping theorem can be generalized to the biholomorphic classification of Riemann surfaces: these are essentially the Riemann sphere, the whole complex plane, and the open unit disc. This classification is known as the uniformization theorem.
For methods to compute $f$ that are useful in applications, see the Schwarz–Christoffel mapping.