Holomorphic function mapping unit disc to the "pacman" $U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]\}$

Question

Find an injective and surjective holomorphic function, that transfers the unit disc $\Bbb D = \{|z|<1\}$ to the domain $U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]$}.

I tried to begin with a branch of $\log$, and after applying $z \mapsto z- \frac{\pi}{4}$ and $z \mapsto \frac{2}{3}z$ I got to the domain $\{z \in \Bbb C :\mathrm{Re}(z)<0, 0<\mathrm{Im}(z)<\pi \}$. Next I thought about getting to half a plane, and using Caley transform $z \mapsto \frac{z-i}{z+i}$ and be done, but I'm having trouble with this part. Any ideas of such a mapping/something else?

Answer 1

You can do it in the following steps:

1. The function $z\mapsto {1\over {i}}{z-1 \over z+1}$ maps the unit disc to the upper half plane.
2. The function $z\mapsto \sqrt{z}$ maps the upper half plane to the first quadrant.
3. The function $z\mapsto {z-1\over z+1}$ maps the first quadrant to the semidisc $\{w = u+{i}v : |w|<1, v>0\}$.
4. The function $z\mapsto z^{3/2}$ maps the semidisc to a $3\over 4$-disc.
5. Finally, the function $z\mapsto e^{{i}\pi/4}z$ maps the $3\over 4$-disc to the Pac-Man.

Answer 2

You're missing only a single ingredient:

Hint You can map an appropriate half-strip to a half-plane with $z \mapsto \sin z$.