Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$

Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$

I'm guided through this problem:

First I need to find the image of the quarter disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ by $f(z)=z^2$?

My answer to this is a semi-circle (edit:disk), with unit radius.

Now I need to find the map of the halfdisk $D=\{ |z-1|<1 :im z>0 \}$ onto the the first quadrant $\{im z>0, rez>0\}$.

I'm stuck on this bit:

I tried: rewriting the halfdisk as $1+e^{it}$ for $0\leq t \leq 2\pi$ , but I didn't get far.

Lastly, I need to somehow put the first two together, to get a map from the quarter disk onto the upper halfplane

• Your words "semi-circle" are less appropriate that "semi-disc" (in analogy to "quarter-disc"). – Lee Mosher Apr 23 '16 at 21:35
• Yes, sorry, I meant disk – GRS Apr 23 '16 at 21:40

Consider the map $f_1=z^2$, then the map $f_2=-1/2(z+z^{-1})$.
• @Ruzayqat did you mean $f_2=-\frac{1}{2} (z+z^{-1})$ or $f_2=\frac{-1}{2 (z+z^{-1})}$, at first I thought it is the first one but I couldn't show that it maps the upper half-disk into the upper half-space. – thehardyreader Jun 19 '18 at 20:54