Conformal map from upper half plane with slit to unit disc.

Question

I need to find a conformal map from $U = \{z : \text{Im } z > 0\} \setminus \{z : \text{Re } z = 0 \text{ and } 0 < \text{Im } z ≤ 1\}$ to the open unit disc $D=\{|z|<1\}$. I am not sure where to go with this. Thanks in advance

Answer 1

$z \mapsto z^2 + 1$ maps $U$ to $\mathbb C \setminus [0, \infty)$. If $\sqrt z$ is the branch which is analytic on $\mathbb C \setminus [0, \infty)$ and takes the value $i$ at $-1$, then $z \mapsto \sqrt {z^2 + 1}$ maps $U$ to the upper half-plane.