\[1<\left|z\right|<2\sqrt{2}\]
\[f\left(z\right)=\ \sum_{k=1}^{\infty}(-1)^k\left(z-2\right)^{-k}+\frac{i}{1+i}\sum_{k=0}^{\infty}\left(\frac{-1}{2+2i}\right)^k(z-2)^k\]