Usage of the term $\arg(z)$

Question

Consider the complex number $z = -1 - i$. Is it mathematically correct to say that $\arg(z) = 5\pi/4$? Sure, $5\pi/4$ is not the principle argument of $z$, but it is an element of the set $\arg(z)$. That being said, is $\arg(z) = 5\pi/4$ actually correct, or is it an incorrect use of the term?

Answer 1

Usually I define $\arg(z)=-3 \pi/4$ as the principal value, so that the function is well defined (single valued), but really you can define it as any branch. Priestley's notation is useful:

$$[\arg(z)]=\{\arg(z)+2 n \pi : n \in \mathbb{Z} \}$$