Let $S_{\alpha} = \{ z \in \mathbb C \ | \ |arg(z)| \lt \alpha \} $ with $0 \lt \alpha \leq \pi$. I have to find a conformal mapping $\phi: S_{\alpha} \to B(0,1)$.
What would be a good approach to this? Could I map the sector to, for example, the upper half plane and then to the unit ball?