Conformal mapping to upper half plane

Question

I need to find conformal mapping of $U = \{z \in \mathbb{C}:Im(z) >0\}\setminus\{ it:t\in [1;\infty)\}$ to upper half plane. I tried to square $U$ and then use inverse of $w = \frac{1}{2}(z + \frac{1}{z})$ and on paper it seems like i am on the right way, but cannot understand what's wrong. Any hints?

Answer 1

By this answer the map $$z\mapsto\sqrt{\frac{z^2}{z^2+1}}$$ achieves the desired objective. It is the successive application of

• $z^2$: maps $U$ to $U_1=\mathbb C\setminus((-\infty,-1]\cup[0,\infty))$
• $\frac z{z+1}$: maps $U_1$ to $U_2=\mathbb C\setminus[0,\infty)$
• $\sqrt z$: maps $U_2$ to upper half-plane