conformal mapping from $\left|\arg z\right| < α < π$ onto the unit circle

Question

Find a holomorphic function $\ w = f(z)$ mapping the sector $\left|\arg z\right| < α < π$ conformally onto the unit circle $\ |w| < 1$. Describe the behavior of $\ f (z)$ near $\ z = 0$.

I know how to map the upper plane to the unit circle, but have no idea about a sector, I am thinking about first map this to the upper plane... shall I take log?

Answer 1

First do the map $z \rightarrow z^{ \frac{ \pi}{2 \alpha}}$ to send it to the positive real half plane.

Then the map \begin{eqnarray*} z \rightarrow \frac{z-1}{z+1} \end{eqnarray*} will map this to the unit circle.