I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really am at a loss. $R$ is symmetric about the imaginary axis, and it is bounded by three circles that need to be sent to the real axis. I thought maybe we could think of the unit circle geodesic as a line with two points at infinity. still no luck. I would appreciate some help. Thanks.
I was thinking maybe the map $f(z) = -z^{-1}$ sends everything to the half disk. Does this look right?