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.