Holomorphic map from quarter plane to unit disk.

Question

Let $D$ be the quarter plane where $-\frac\pi4 < \arg(z) < \frac\pi4$. By Riemann mapping theorem, there exists a holomorphic bijection from the open unit disk to $ D$ where $f(0)=1$ and $f'(0)>0$. Calculate explicitly the inverse of this function.

I am thinking maybe I can use the Cayley transform, but this is half the plane not a quarter.

score 4 · Answer 1 · answered Apr 29 '13 at 18:38

4

Another Hint: Disk$\longrightarrow $ Upper half plane$\longrightarrow $ First quadrant $\longrightarrow$D

Find the maps in each step!

answered Apr 29 '13 at 18:38

Dimitris

6,984
1
30
47

score 1 · Answer 2 · answered Apr 29 '13 at 18:08

1

Hint: $z\mapsto z^2$ maps the first quadrant to the upper half plane (and in fact any quadrant to some half plane).

answered Apr 29 '13 at 18:08

Hagen von Eitzen

374,180

Holomorphic map from quarter plane to unit disk.

2 Answers2