Argument (Complex Number)

A Complex Number may be represented as

$\begin{displaymath} z \equiv x+iy = \vert z\vert e^{i \theta}, \end{displaymath}$

(1)

where $\vert z\vert$ is called the Modulus of

, and $\theta$ is called the argument

$\begin{displaymath} \arg(x+iy) \equiv \tan^{-1}\left({y \over x}\right). \end{displaymath}$

(2)

Therefore,

$\displaystyle \arg(zw)$	$\textstyle =$	$\displaystyle \arg(\vert z\vert e^{i\theta_z}\vert w\vert e^{i\theta_w}) = \arg(e^{i\theta_z}e^{i\theta_w})$
	$\textstyle =$	$\displaystyle \arg[e^{i(\theta_z+\theta_w)}] = \arg(z)+\arg(w).$	(3)

Extending this procedure gives

$\begin{displaymath} \arg(z^n) = n \arg(z). \end{displaymath}$

(4)

The argument of a Complex Number is sometimes called the Phase.

References