The question is: does there exist a linear fractional (Möbius) transformation that maps the set $U = \{z \in \Bbb{C} \mid |z-1|<1, |z-i| < 1\}$ onto the quarter plane $\operatorname{Im}z>0, \operatorname{Re}z>0$.
I thought to take the map $z \mapsto \frac{z}{z-1-i}$. The reason I chose this is because I thought to send $0$ to $0$ and send $1+i$ to $\infty$. Because the set $U$ has a right angle at $0$, its image will as well, so intuitively, it seems like the image will indeed be the first quadrant. However, I can't prove that it maps $U$ onto the first quadrant. Maybe I still need a rotation?
Can someone explain how to do so, or why I am wrong?