How to map this area conformally to the upper half-plane?

Question

Could you help me, please? How to map this area conformally to the upper half-plane? I am so tired of solving this simple task, but I still cannot find the way to solve it. $$\{z: \alpha< \text{arg}(z)< \pi-\alpha\}\setminus \{z=iy, y\geq 1\}, $$ with $0< \alpha< \pi/2$

Answer 1

First apply $z\mapsto z^{\pi/\alpha}$ to obtain the plane minus two opposing slits.

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You may already know that $z\mapsto z+\frac1z$ maps $\Bbb D-\{0\}\to \Bbb C-[-2,2]$, hence $z\mapsto \frac1{z+\frac1z}$ maps $\Bbb D\to \Bbb C-[\frac12,+\infty)-(-\infty,-\frac12]$, which is also a plane with two opposing slits.

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Combine ...