Let
$$
\Omega = \{z \in \mathbb{C} : |z| > 1, z \notin \mathbb{R}_{< -1}, z \notin \mathbb{R}_{\ge 2}\}.
$$
Find a conformal map which maps the region $\Omega$ to the upper half plane.
I would want to know what I am supposed to do first. I tried to shift by $1$ to the right $(z+1)$ then used $1/z$ but I was not sure about how the points around $z=-1$ and $z=2$ moved by these two maps if it is a good way to start this problem.
Thank you in advance.
Best Answer
To start, apply the map $f_1(z)=z+1/z$ which collapses the unit circle onto the segment $[-2,2]$; the exterior of the unit circle is mapped onto the exterior of the circle. Then you have a plane with two slits, and things should become clearer.