I am hoping to find a conformal map sending the unit disc to $\mathrm{Re} z>0$ such that the upper semicircle is mapped to $\{|z|\leq 1\mid \mathrm{Re} z>0\}$. Is this possible? I know of the $\frac{z+1}{1-z}$ one but it maps semicircles to one-fourth-planes.
2026-03-28 20:07:39.1774728459
Conformal map from open unit disc to right half plane while specifying behavior of semicircle
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Yes, $\frac{i-z}{i+z}$ maps the unit disc to $\operatorname{Re }z>0$ and the upper semicircle to $\{|z|\le 1, \operatorname{Re }z>0\}.$