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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.

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    If I haven't mistaken, there is such a map due to Riemann Mapping Theorem.2012-12-30
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    Most likely you're expected to prove that the Riemann mapping theorem applies. For this note that both domains are convex and so are simply connected. Or are you asked to give an explicit map?2012-12-30
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    As has been said, it's not apparent that you need an explicit map. But for what it's worth, I compute the intersection points as $\left(-\sqrt{1/2-\sqrt{12}/8},1/4(\sqrt{2}+\sqrt{6})\right)$ and $\left(\sqrt{1/2+1/8\sqrt{12}},1/4(\sqrt{2}-\sqrt{6})\right)$. They might have been nicer in polar coordinates. Note there are only two, not three.2012-12-30
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    @Jose27, Yes we do need an explicit map.2012-12-30
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    @KevinCarlson, I would assume they are nicer in polar coordinates, the intersection point you gave looks not very nice. But that is something anyway. Thaks2012-12-30
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    @Deepak: This seems hard with Möbius transformations since the second circle in $\Omega$ is not orthogonal to the unit disk.2012-12-31
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    Any other thoughts then?2012-12-31

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