# Conformal map from a trapezoid like subset of a unit disk to the upper half plane.

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!

• I think you misunderstood, and the intended domain is $\bigl\{ z : \lvert z\rvert < 1, \operatorname{Im} z > \frac{1}{2}\bigr\}$. Then it's rather standard (map to an angular sector via a Möbius transformation, then apply a suitable power). For the trapezoid-like set, I think you'll need something akin to the Schwarz-Christoffel formula, only more complicated. – Daniel Fischer Apr 22 '17 at 18:45
• I see, I know how to take care of this region. Probably the wording of the problem confused me , it is hard to understand the region in this question... thank you for your comment ! – GRE Apr 23 '17 at 16:08
• Yes, the verbal description is not clear. Considering that both regions have a simple formulaic description, it's a bad decision not to use that. – Daniel Fischer Apr 23 '17 at 16:12