# Explicit Riemann Mappings for the Complement of a Plane Curve

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.

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 $$\Phi(z)=A+B\int_1^z \frac{1}{\zeta^2}(\zeta-a_1)(\zeta-a_2)^{\frac{1}{2}}(\zeta-a_3)(\zeta-a_4)^{-\frac{1}{2}}d\zeta.$$