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?)
We seek an inverse $f(z)$ to the exponential function. So we need $f \circ \exp(z)=z$. Since $\exp{z}= \exp{(z+2i\pi)}$ this identity cannot be simultaneously satisfied by $z$ and $z'$ if $z-z'$ is a multiple of $2\pi i$. Hence we have to assume that the domain contains only one representative of each equivalence class $z+2i\pi \mathbb{Z}$.