Let $z\in\mathbb{C}\setminus\{0\}$
Then, there exists a unique $\theta \in [0,2\pi)$ and $\phi\in(-\pi,\pi]$ such that $z=|z|e^{i\theta}=|z|e^{i\phi}$.
Between $\phi$ and $\theta$, which is the usual definition of $Arg(z)$?
I know that these two are basically the same for the purpose of its use, but i'm hesitating to take which to choose as my definition of $Arg(z)$..
Usually, the $\theta$ version is used, i.e. $[0, 2\pi)$. However, in the case of functions that are locally holomorphic, but cannot be taken to be holomorphic in the whole of $\Bbb C$ (e.g. $\ln z$), we have to define a branch cut, and that is usually taken to be the negative real axis, and we are thus implicitly choosing to use arguments in the range $[-\pi, \pi)$.