I'm trying to find a conformal map from $ G = \{z ∈ C : |z − i| < √ 2 $ and $ |z + i| > √ 2\} $ to the open unit disc. This is what I've tried so far:
I know that we can't have a mobius transformation as the disc has a boundary of one circline and $G$ has a boundary of two.
So I tried to use $f(z) = \frac{z-\alpha}{z-\beta}$, where $\alpha$ and $\beta$ are the intersection points of the two circles as a first step and I'm not sure what this maps $G$ to and where to go from here.