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I'm trying to find a bijective conformal map from the half disc $\{z: |z| < 1, \Re(z)>0\}$ to the upper half plane $\{z: \Re(z) > 0\}$. Any help is appreciated. Thanks!

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    $\begingroup$ I don't think you want |z|<0. Maybe |z| < 1 ? $\endgroup$ – Dylan Yott May 31 '13 at 20:07
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Use a Möbius transform to bring one vertex of the half disk to $\infty$ and the other to $0$. Now you have a quadrant. Square to get a half plane.

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