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

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 $$\operatorname{Arg}(z)$$.

• what's the mathematical definition of $arg(z)$ ? Commented Sep 20, 2018 at 20:05
• Let there be a circle on a Cartesian plane with center ($0,0$). Let ($a, b$) be a point on the circle arg(a,b) is the number of radians of the circular arc between ($a, b$) and ($\sqrt{a^2+b^2}, 0$) Commented Feb 14 at 1:59

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.
$$\operatorname{Arg}(z)$$ is the unique element $$\theta\in\operatorname{arg}(z)$$ such that $$\theta\in(-\pi,\pi]$$.
• If we have $Arg(z^5)$ where $z = 1 + i$ for example, what happens to our principal argument? Does it remain unchanged or does it become $-5\pi< \theta < 5\pi$ Commented May 19 at 0:51