Two answer the question if a Logarithm of a holomorphic function exists I know the following creterions:
a) If $G\subset \mathbb{C}$ is a simple connected domain and $f$ holomorphic with $f(z)\ne 0$ in $G$. Then it exists a Logarithm of $f$ in $G$.
b) If for the holomorphic function on $G \subset \mathbb{C}$ it is $f(G)\subset \mathbb{C}\setminus \mathbb{R}_{\leq 0}$, then $\log(f(z))$ is a Logarithm of $f$.
Now I have problems with the follow exercise:
Let $\Omega = \mathbb{C}\setminus[-1,1]$. Then there are two functions given:
Show, that on $\Omega$ there does not exist a holomorphic Logarithm of $f(z)=\frac{1}{z^2-1}$.
Show that on $\Omega$ there exists a holomorphic Logarithm of $h(z)=i\frac{z+1}{z-1}$.
But as $\Omega$ is not simple connected domain, I cannot use a). And b) in general is not very practible, I think.
Can anyone please help me? Thank you! :)