The question asks: 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. How would you construct such conformal map? I think you need to map the arc to part of real/imaginary axis then use square root function. How would you construct this map in details though?
Thanks
First map $z\mapsto \frac1{z+1}$ to bring $\infty$ to $0$ (because we want to keep an eye on the point at $\infty$). At the same instance, this turns the arc to a half line, ending in $\frac12$. Taking the square root centered at its end point, i.e. applying $z\mapsto \sqrt{x-\frac12}$, we obtain a half plane. You sure know how to turn this into the unit disk. Now watch where $\infty$ has gone in this process and apply a reciprocal map that brings it back to $\infty$. Now you already have a complement of some disk, and after scling and translation it becomes the unit disk.