Let $S$ be the strip $S := \{ z \in \mathbb{C} \mid − \pi < \operatorname{Im}(z) < \pi\}$. Construct a biholomorphic mapping of $S$ onto $D$ where $D$ is unit open disc.
My attempt : I know that the existence of such maps is obtained by Riemann mapping theorem. I think the following holomorphic map works $ z \mapsto \frac{e^{z/2} -1}{e^{z/2} +1}$ Do I need to show this more rigorously ?
Your map is correct, but shouldn't you show the steps at least graphically? First one brings you to a thinner strip, second one brings you to the right half-plane, and the third one finishes.