what is the argument of 0?

Question

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$.

What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?

Answer 1

As has been discussed in the comments, the argument of $0$ is taken to be either zero or undefined. It is a matter of convention.