TITLE: Why does a branch need to be defined in complex analysis?
QUESTION [5 upvotes]: $\newcommand{\arg}{\operatorname{arg}}$Say we have the principal branch, $\arg_\tau(z)$.  This is defined so that $\arg_\tau(z) \in (-\pi,\pi]$.  Why is it necessary to define the limits on the argument of $z$ at all?
Thanks!

REPLY [5 votes]: Essentially because $e^{i\theta}$ is a periodic function.
The Euler formula say us that $e^{i\theta}=\cos\theta +i\sin\theta$ so that $e^{i \theta}=e^{i \theta+2k\pi}$ for $k \in \mathbb{N}$.
Thanks to this formula we can write any complex number $z=a+ib$ in a polar form $z=\rho e^{i\theta}$ where $\rho = |z|=\sqrt{a^2+b^2}$ (the positive rooth) is univocally defined, but the argument $\theta$ is not and this cause troubles in defining an inverse function $\log z$ or to extend exponentiation to any complex basis $z^{i\theta}$. So, to have a one to one correspondence we define a principal brunch for $\theta$, usually chosing the interval $(-\pi,\pi]$.