Find a biholomorphic map to the unit disk Let $\Omega := \{ x + 0\mathrm{i} \in \mathbb{C} : x > 1 \} \cup \{ x + y\mathrm{i} \in \mathbb{C} : x>0,\,y \neq 0 \}$.
Let $\mathbb{D} := \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk.
I'm on the search for a biholorphic function $f:\Omega \to \mathbb{D}$, but have no idea where to start. The Cayley Transform (biholomorphic function from the upper half plane to $\mathbb{D}$) may be used. Do you have an idea how such an exercise is tackled (because I think there should be some way to do this systematically and not just by guessing). I'd be thankful for guidance. :)
 A: Define the following (biholomorphic) maps:
\begin{align*}
g_1 : \mathbb{C} &\to \mathbb{C}, z \mapsto z^2-1; \\
g_2 : \mathbb{C} &\to \mathbb{C}, z \mapsto \sqrt{|z|}\exp\left(\mathrm{i}\frac{\arg(z)}{2}\right); \\
g_3 : \mathbb{C} &\to \mathbb{C}, z \mapsto \mathrm{i}z; \\
T : \mathbb{C} &\to \mathbb{C}, z \mapsto \frac{z-\mathrm{i}}{z+\mathrm{i}}.
\end{align*}
Here, $T$ is the Cayley transform.
From this, let $f:\mathbb{C} \to \mathbb{C}$ be the map, defined by
$$
f(z) := (T \circ g_3 \circ g_2 \circ g_1)(z)
$$
for all $z \in \mathbb{C}$. As composition of biholomorphic maps, $f$ is biholomorphic. Furthermore, we have indeed $$f:\Omega \to \mathbb{D},$$ since
\begin{align*}
\Omega
\stackrel{g_1}{\longrightarrow}
\mathbb{C} \setminus \{x+0\mathrm{i} : x \leq 0\}
\stackrel{g_2}{\longrightarrow}
\mathbb{U}
\stackrel{g_3}{\longrightarrow}
\mathbb{H}
\stackrel{T}{\longrightarrow}
\mathbb{D}
\end{align*}
where $\mathbb{U} := \{z \in \mathbb{C} : \operatorname{Re}(z) > 0\}$ and $\mathbb{H} := \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}$ denote the right and upper half-planes, respectively.
