I don't really understand what branches of log are in complex analysis. The definition I have is that 'a branch of log $z$ in G is a continuous function $l$ in G such that, for each $z$ in G, the value $l(z)$ is a logarithm of z.
I know this definition, but somehow still have a hard time understanding the concept. I may be overthinking it... I thought a branch of log z was essentially an identical log of z, just rotated around the plane by a constant times 2π. So just the same log z, but on top or underneath the original one (if you graphed it using $\theta$ as the z axis). Is this correct? Does this mean that there is only one branch in each [0,2π] interval? (or is this interval open?)