Sphere minus interval is upper half plane $\mathbb{C}_{\infty} - [a \ , b] \simeq \mathcal{H}$?

Question

Let $a<b$. How can I show that $\mathbb{C}_{\infty} - [a \ , b]$ is biholomorphic to upper half plane $\mathcal{H}$? Any hints or reference are appreciated. Thank you.

Answer 1

By a Mobius transformation, $\Bbb C_\infty-[a,b]$ is equivalent to $U$ which is $\Bbb C$ with the nonpositive real axis removed. Then $z\mapsto i\sqrt z$ (principal branch) maps $U$ to the upper half-plane.

Answer 2

Since $\mathbb{C}_{\infty} - [a \ , b]$ is simply connected and not all of $\mathbb{C}$, by Riemann mapping theorem, it is biholomorphic to $\mathcal{H}$