I am finding the complex logarithm very hard to understand.
My text defines $G = \mathbb{C} - \{z \in \mathbb{C} : \Re(z) \leq 0, \Im(z) = 0\}$ and defines the principal logarithm to be the branch of the logarithm $\log(z) = \log|z| + i\arg\theta$ on $G$.
So is each function $log_k(z) = \log|z| + i(\arg\theta + 2\pi ik), k\in \mathbb{Z}$ also a branch of the logarithm on $G$?
Can every branch of the logarithm be defined on $G$?