Conformal mapping of plane with slit onto annulus

Question

I want to prove that the region $\mathbb{C}\setminus\{x+i0:-1\le x\le1\}$ is conformally equivalent to an annulus. I am trying to use a linear fractional transformation to map this region onto a disc or something with a hole, but am having trouble finding the appropriate map. Any suggestions?

Answer 1

The function $z\mapsto z+i\sqrt{1-z^2}$ maps your slit to a circle, with branch points at $-1$ and $1$. By choosing appropriate branches you can get it to map your region to all of $\mathbb C$ with a circular hole. Then you're half done.