I'm having some trouble with finding a conformal map for this problem.
Find a holomorphic one-to-one map from the open set bounded by the unit semicircle $|z| = 1$, ${\bf Im}\, z > 0$ and the line $y =\dfrac{1}{2}$ and the upper half plane.
So I have set it up and obtained a trapezoid-like region with vertices $(1,-1,-\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}, \dfrac{\sqrt{3}}{2}+\dfrac{i}{2})$. So now I have four points I need to take care of and I ust can't see how can I map them. I only know how to map 3 points to $1,0,\infty$.
Thank you!