I need to find a conformal map of $\mathbb{C}-[i,2i]$ over the upper half plane, $\{z \in \mathbb{C}: Im(z)>0 \}$.
My try:
I first aply $z^2$ to take $[i,2i]$ to the real line and then i get $\mathbb{C}-[-4,-1]$ and by a translation $z+1$ i got $\mathbb{C}-[-3,0]$ so i define a transformation $z/z+3$ because $0$ goes to $0$ and $-3$ to $\infty$ and then i get $\mathbb{C}-[0,\infty)$ and so i aply $\sqrt{z}$ with the branch cut on $(-\infty,0$) to go back to the upper half plane.
Is this correct?