The Riemann Mapping Theorem says that if $U$ is a simply connected open subset of $\mathbb C$ that is not $\mathbb C$, then there is a conformal isomorphism between the open unit disk $\mathbb D$ and $U$. However, it is rarely the case that the conformal isomorphism can be explicitly written.
The problem I am thinking about has the following context: I would like to consider a plane curve $\Gamma \subseteq \mathbb C$ parameterized by $\gamma \colon [0,L] \to \mathbb C$. By RMT, we know that there is a conformal isomorphism $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$.
In some cases, this conformal isomorphism can be explicitly written out. For example, if $\Gamma$ is the interval $[-2,2]$, then $\Phi(z) = z + \frac1z$ is such a conformal isomorphism.
If $\Gamma$ is a circular arc (on the unit circle), $\Phi^{-1}$ can also be explicitly constructed with the help of the Mobius transformation $z \mapsto \frac{z-i}{z+i}$, which sends the unit circle to the extended real line.
Here comes my question: is it possible to construct explicit Riemann mappings for curves other than an interval or a circular arc? I have looked at some references about Schwarz-Christoffel mapping, but I am not sure how to apply the theory in the setting of degenerate polygons in my case (for example, if I try to find a Riemann mapping for $\hat{\mathbb C} \backslash \Gamma$, where $\Gamma$ is an L-shaped curve). Any help would be appreciated.
Best Answer
Let $\Gamma \subseteq \mathbb C$ be a polygon with vertices $w_i,\, i=1,2,...,n$. There is a conformal isomorphism $ \Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$. Let $a_1,a_2,...,a_n$ be the points on $|z|=1$ mapped to vertices of $\Gamma.$ If $\Gamma$ has interior angle $\beta_i \pi$ at $w_i$, then $\Phi$ is represented by Schwarz-Christoffel integral $$ \Phi(z)=A+B\int_1^z \left(\frac{1}{\zeta^2}\prod_{i=1}^n (\zeta-a_i)^{\alpha _i-1}\right)d\zeta, $$ where $\alpha _i=2-\beta _i$ $(i=1,2,...,n)$ and $A, B$ are constants. For instance, see this, (I.45), page 14.
In the setting of an L-shaped curve, the degenerate polygon $\Gamma$ has four vertices $w_1,w_2,w_3$ and $w_4(=w_2)$ with interior angles $\beta _1\pi=0, \beta _2\pi=\frac{\pi}{2}, \beta _3\pi=0$ and $\beta _4\pi=\frac{3\pi}{2}$, respectively, as described in figure below.
Thus we have $\alpha _1=2, \alpha _2=\frac{1}{2},\alpha _3=2$ and $\alpha _4=-\frac{1}{2}$ and Schwarz-Christoffel integral will be $$ \Phi(z)=A+B\int_1^z \frac{1}{\zeta^2}(\zeta-a_1)(\zeta-a_2)^{\frac{1}{2}}(\zeta-a_3)(\zeta-a_4)^{-\frac{1}{2}}d\zeta. $$