Sketch the following region $R$ and find conformal one to one map of $R$ onto the unit disk.
The exterior of the unit half disk $$R=\{z:\ |z|<1,\ {\bf Im}(z)>0\}$$ (including $\infty$) cut along $[0,−i]$.
I have been able to do the sketching but can anyone please help me with the mapping.
In the سimplest way we see:
the map $\displaystyle z+\frac{1}{z}$ maps $R$ to upper half-plane.
the map $\displaystyle iz$ maps upper half-plane to right half-plane.
the map $\displaystyle \frac{z-1}{z+1}$ maps right half-plane to unit circle.
With composit these maps we have $\displaystyle \frac{-iz^2-i-z}{-iz^2-i+z}$ that maps $R$ to unit circle.
