Finding a conformal mapping to the unit disc, from the unit disc missing the x axis

Question

I am revising conformal transformations, and am confused in general as to how to find one between two sets. I know that circlines map to circlines, but not sure how I can use that to help me with this problem.

I have the set $\{z\in\mathbb{C}:|z|<1\}/ $[0,1). I want to map this conformally to the open unit disc. I can't see a way, because I don't know which 3 points would define the first set and what I would map them to.

Any help appreciated. Thanks.

Answer 1

First map your slit disk to a half-disk by $z \to \sqrt{-z}$. Then invert around $-i$ to get a quarter-plane. From there get a half-plane, and finally a disk.