Problem: Find a conformal map $f$ from $ A = \left\{ z \in \mathbb{C} \mid \text{Im}(z) > 0, |z| > 1 \right\}$ into the unit disk.
Attempt: I started off with a map $F_1: z \mapsto z + \frac{1}{z}$. I was trying to visualize what this mapping actually does to the region in the complex plane. I know that the semi-circle in the upper half plane will get mapped to the interval $[-2, 2]$ on the real line. Also, a point like $2i$ gets mapped to $3i/2$.
I wish to map the region $A$ into the whole upper half plane (if that is possible), then I can easily find a map into the unit disc. Do I need to use a dilatation to rescale the $|z| > 1$ condition?
Help is appreciated.
Best Answer
Hint: With conformal mapping $$w=\left(\dfrac{z+1}{z-1}\right)^2$$ you map region $A=\{z\in\mathbb{C}:{\bf Im}\ z>0,\ |z|>1\}$ to lower half plane $\{z\in\mathbb{C}:{\bf Im}\ z<0\}$.