Let $$f(z)=\frac{z+1-i}{z-1+i}$$ and $S=\{z\in \mathbb C | im z>rez\}$.What is the image of $f(S)$?
I sketched the region, and it corresponds to a halfplane, from $\pi/4$ to $5\pi /4$. I tried mappning individual points such as $f(i)=(1-2i)/5$, $f(-1)=(2i-1)/5$ to see the pattern, but unable to see it.
I know that the map should preserve angles, but my set is infinite, and I don't know how to parametrize it.
Rewrite $f$ in the form $$ f(z)=\frac{z-(i-1)}{z-(1-i)} $$
The set $S$ is the half-plane of points closer to $i-1$ than to $1-i$, hence $f(S)$ is the open unit disk.