Find a conformal map 2

Question

$\textbf{Exercise}$ Find a conformal mapping which maps the domain $D$ onto the open unit disc, where $D$ is the intersection of $\vert z \vert <1 $ and $\vert z-1 \vert <1 $.

I knew that Mobius Transformation is a self-conformal map on the open unit disc. However, I don't know how to find a conformal map when two regions intersect...

Any help is appreciated...

Thank you!

Answer 1

Note that $D$ has two corners that somehow have to be straightened by the map. Construct your map in several steps:

• The two circles intersect in two points $p$, $q$. Find a Moebius transformation $T$ that maps these two points to $0$ and $\infty$. Such a $T$ will map the two circles to a pair of lines through $0$, hence $D$ to a wedge. Determine the opening angle of this wedge.
• A second map of the form $w\mapsto w^\lambda$ with a suitable $\lambda>0$ will map the wedge to a half plane.
• Map this half plane to the unit disc by a suitable Moebius transformation.