Does there exist a conformal map from the region $\Omega = \{z :|z|<1\} \cap \{z: |z- \frac{1+i}{\sqrt2}|<1\}$ onto the region $\{z: |z|<1, \operatorname{Im}z>0\}$?
I think I need to find at least three intersection point of the two circles and mapped them to real axis using the formula of fractional linear transformation. I even have difficulty finding the intersection points. I would really appreciate if someone do this rigorously. This is not a homework problem. This is from the collection of previous qual exams.
Best Answer
Here is the standard way to map any domain bounded by two circular arcs, or a circular arc and a line segment, to a half-plane. Doing this for both of these domains and composing one map with the inverse of the other gives the desired map.
Find a linear fractional transformation mapping the intersection points to $0$ and $\infty$. The two boundary arcs will be mapped to rays from $0$ to $\infty$, so the image will be a sector of some opening angle $\alpha$ (which is the angle at which the two circles or circle/line segment intersect.) Then compose with a power map $z \mapsto z^\beta$ such that $\alpha \beta = \pi$, which maps the sector to a half-plane.
(You might have to rotate the half-planes to get the same for both of your domains, depending on the choice of the LFT in the first step.)