Let f(z) = log(z) be the principle branch of logarithm in $\mathbb{C}$ $-$ ($-$$\infty$, 0], I need to
(1) find the respective taylor's series for f(z) centered at z$_0$ = $-$1$-$i and z$_0$ = $-$1+i
(2) determine their values at z = $-$1
For z$_0$ = $-$1 + i , log(z) = $\sum_{n=0}^\infty$a$_n$(z $-$($-$1 + i))$^n$, with a$_0$=log$\sqrt{2}$ + i$\frac{3\pi}{4}$ and a$_n$ = ($-$1)$^{n+1}$$\frac{e^{-3\pi in/4}}{n2^n/2}$
My question is how can we obtain this a$_n$? Also, for at z$_0$ = $-$1 $-$ i, is it true that a$_0$=log$\sqrt{2}$ + i$\frac{5\pi}{4}$ and a$_n$ = ($-$1)$^{n+1}$$\frac{e^{-5\pi in/4}}{n2^n/2}$?