Argument

The argument is the angle of a complex number.

$\arg z = \arg (|z| cos \theta + i |z| sin \theta) = \theta$

The principal argument is within the interval $(-\pi, \pi]$, although some definitions use $[0, 2\pi)$

Properties

The argument is obtained with the $\arctan$ function, although it requires various conditions to ensure the correct value.

$\operatorname{Arg}(x + iy) = \operatorname{atan2}(y,\, x) = \begin{cases} \arctan(\frac y x) &\text{if } x > 0, \\ \arctan(\frac y x) + \pi &\text{if } x < 0 \text{ and } y \ge 0, \\ \arctan(\frac y x) - \pi &\text{if } x < 0 \text{ and } y < 0, \\ +\frac{\pi}{2} &\text{if } x = 0 \text{ and } y > 0, \\ -\frac{\pi}{2} &\text{if } x = 0 \text{ and } y < 0, \\ \text{undefined} &\text{if } x = 0 \text{ and } y = 0. \end{cases}$

The following mathematical properties apply:

$\begin{align}\arg(ab) & = \arg(a) + \arg(b)\\ \arg(a/b) & = \arg(a) - \arg(b)\\ \arg(a^n) & = n \arg (a) \end{align}$