How do I find a conformal map from $W=\{Im(z)>0,|z|>1\}$, that is, the upper-half plane with semi-disk removed, onto the unit disk.
My first thought involves applying $\phi(z)=\frac{1}{z^2}$ or maybe the usual map from the upper-half plane to the unit disk $\psi(z)=\frac{z-i}{z+i}$. But I'm not really sure how to proceed. Any hints would be appreciated.
Best Answer
If you map $z=1, -1$ to $w=0, \infty$ then all boundary parts of $W$ become straight line segments, and it is not difficult to see that $T(z) = (z-1)/(z+1)$ maps $W$ to the first (upper right) quadrant.
Then continue with a mapping to the upper half plane and from there to the unit disk.