I want to solve this problem:
Let $z_0\in \mathbb C$ such that $\mathrm{Re}\,z_0<0$, $\mathrm{Im}\,z_0<0$. Find the power series of $f(z)=\mathrm{Log}(-\mathrm i z)$ in terms of $z-z_0$ and find its radius of convergence.
I know the power series of $\mathrm{Log}(1+z)$, but I don't know how to use it here. Also, I don't know how to use the fact that $z_0$ is in the third quadrant.