Let $C_{1}$ and $C_{2}$ be smooth curves in the complex plane, intersecting at some point $p$. Show that if the map $f: \mathbb{C} \rightarrow \mathbb{C}$ is complex differentiable, then it preserves the angle between $C_{1}$ and $C_{2}$ at $p$, provided $f^{\prime}(p) \neq 0$. Give an example that illustrates why the condition $f^{\prime}(p) \neq 0$ is important.

Show that $f(z)=z+1 / z$ is a one-to-one conformal map on each of the two regions $|z|>1$ and $0<|z|<1$, and find the image of each region.

Hence construct a one-to-one conformal map from the unit disc to the complex plane with the intervals $(-\infty,-1 / 2]$ and $[1 / 2, \infty)$ removed.