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.

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

The following mathematical properties apply:

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