# Finding a conformal map to the upper half plane

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.

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.
@YeonjooYoo You can see a plot here. Also, what you really want to know is where the slits $[-\infty,-1]$ and $[2,+\infty]$ go. Both lie on the real line, so it's really a question about real function $x+1/x$ which you can plot easily, e.g., by typing "plot x+1/x" into Google search. And by the way, 2+1/2 is 5/2. not 3/2. – user31373 Jun 28 '12 at 18:52