Let $[i,\infty)=\{it: t\geq 1\}$ be a ray. Denote as usual the unit disk $\mathbb D=\{|z|<1\}$ and the upper half-plane ${\mathbb{H}}=\{\mathrm{Im}(z) > 0\}$.
Find a biholomorphism (conformal mapping) from $$\mathbb H\cup \mathbb D \setminus [i, \infty) \setminus \{0\} $$ to the punctured disk $$\mathbb D \setminus \{0\}.$$
I know the usual tricks, but this evades me. It's not supposed to use anything too heavy (e.g. Christoffel-Schwarz).
On
it's easier to treat these sets on a riemanian sphere. Then the first one is a three-quarter-sphere and the second a half-sphere. For example via $z\mapsto z^2$ a quarter.sphere maps to a half-sphere. All you need is a map from the three-quarter to the half-sphere, i.e. $z\mapsto z^r$ with a suitable $r>0$. Then apply mobius transformations to move them around and fix the missing point.
Let $\arg z \in [0, 2 \pi)$ and $z^a = |z|^a e^{i a \arg z}$.
Start with $D_1 = \mathbb H \cup \mathbb D \setminus \{i t: t \geq 1\}$.
$(z - 1)/(z + 1)$ maps $D_1$ to $D_2 = \{r e^{i t}: r > 0 \land 0 < t < 3 \pi/2\} \setminus \{e^{i t}: 0 < t \leq \pi/2\}$.
$z^{2/3}$ maps $D_2$ to $D_3 = \mathbb H \setminus \{e^{i t}: 0 < t \leq \pi/3\}$.
$(z - 1)/(z + 1)$ maps $D_3$ to $D_4 = \mathbb H \setminus \{i t: 0 < t \leq 1/\sqrt 3\}$.
$z^2$ maps $D_4$ to $D_5 = \mathbb C \setminus \{t: t \geq -1/3\}$.
$(z + 1/3)^{1/2}$ maps $D_5$ to $\mathbb H$.
Now repeat these steps for $D_1 \setminus \{0\}$ and choose a Mobius transformation that maps $\mathbb H$ to $\mathbb D$ and maps the missing point in $\mathbb H$ to $0$.