Let $\Omega$ be the region enclosed between the two circles $C_{1}$ and $C_{2}$, where

$$C_{1}=\{z \in \mathbf{C}:|z-i|=1\}, \quad C_{2}=\{z \in \mathbf{C}:|z-2 i|=2\}$$

Find a conformal mapping that maps $\Omega$ onto the unit disc.

[Hint: you may find it helpful first to map $\Omega$ to a strip in the complex plane. ]