Principal argument (Arg)

The one argument value in $(-\pi,\pi]$ , written $\operatorname{Arg} z$ , singled out from the infinitely many valid arguments of $z$ .

The principal argument of a nonzero complex number $z$ , written $\operatorname{Arg} z$ (capital A), is the unique value of its argument lying in the interval $(-\pi,\pi]$ . Every other valid argument of $z$ differs from $\operatorname{Arg} z$ by an integer multiple of $2\pi$ , so fixing the interval $(-\pi,\pi]$ is exactly what turns the multivalued $\arg z$ into the single, ordinary number $\operatorname{Arg} z$ .

Examples

$\operatorname{Arg}(-1)=\pi$ , not $-\pi$ , since $\pi$ is the representative that falls in $(-\pi,\pi]$ .
$\operatorname{Arg}(-i)=-\pi/2$ .