What is the mathematical definition of the principal argument of a complex number?

Question

I am seeking a succinct mathematical definition of the "principal argument" of a complex number.

Principal argument of a complex number, denoted $\operatorname{Arg}(z)$ means the argument of a complex number $\arg(z)$ within the range $(-\pi, \pi]$.

How do I represent this mathematically?

I've tried using $\operatorname{Arg}(z) = \arg(z) \in (-\pi, \pi]$, but this doesn't make sense.

There is also a very complicated one on Wikipedia, $\operatorname{Arg}(z) = \{\arg(z) - 2\pi k \mid k \in \mathbb{Z} \wedge -\pi < \operatorname{Arg}(z) \leq \pi \}$

There definition makes it seem as if there can be multiple values of $\operatorname{Arg}(z)$.

Answer 1

For each complex number $z\in\mathbb C\setminus\{0\}$ there is a unique $r>0$ and $\theta\in(-\pi,\pi]$ such that $z=r\,\exp(i\theta)$, where $\exp\colon\mathbb C\to\mathbb C$ is the complex exponential function. You can now define the principal argument as $\operatorname{Arg}(z) = \theta$. The multivalued argument assigns to $z$ all possible $\theta\in\mathbb R$ such that $z=r\,\exp(i\theta)$ which amounts to $\operatorname{Arg}(z)+2\pi n$ where $n$ ranges over all integers.

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The definition of the principal argument in terms of the multivalued argument as stated in the Wikipedia article you referenced is circular and doesn't make any sense. It could be corrected as

$\operatorname{Arg}(z)$ is the unique element $\theta\in\operatorname{arg}(z)$ such that $\theta\in(-\pi,\pi]$.