I'm working on a question from Ahlfors' $\textit{Complex Analysis}.$ The question reads:
Map the complement of the arc $|z| = 1, y \geq 0$ on the outside of the unit circle so that the points at $\infty$ correspond to each other.
I have multiple questions with this, but the foremost of these is that I'm not even sure I understand what region I'm mapping into. Is it the entire complement of the unit circle? And from there, I'm not sure I'd know where to go. I know how to map the region into the half plane (if I'm picturing the correct region), but I'm not even sure that is the right direction.
Any clarification would be much appreciated. Thanks!