The argument function returns the principal value of the argument of the complex-valued expression z.  This means that argument(z) = t specifies $z$ = $\mathrm{polar}\left(\left|z\right|\&comma;t\right)=\left|z\right|{\&ExponentialE;}^{\mathrm{I}t}$ where $-\mathrm{\pi }<t\le \mathrm{\pi }$.

From the previous definition, for an arbitrary complex number $z=x+\mathrm{I}y$, $\mathrm{argument}\left(z\right)\&gt;0$ if $y\&gt;0$. If $y<0$, then $\mathrm{argument}\left(z\right)<0$.

For the case $y=0$, if $y$ is the special floating-point value $\mathrm{-0.}$, argument returns a floating-point approximation to -Pi. For the case where $y=0$ but $y$ is not equal to the special floating-point value $\mathrm{-0.}$, $\mathrm{argument}\left(z\right)\&equals;0$ if $x\ge 0$ and otherwise $\mathrm{argument}\left(z\right)\&equals;\mathrm{\pi }$.

For more information about the special floating-point value $\mathrm{-0.}$, see Numeric Computation in Maple.

$\mathrm{argument}\left(\mathrm{abs}\left(z\right)\exp \left(\frac{\mathrm{I}\cdot 2\mathrm{\pi }}{3}\right)\right)$

$\frac{2\mathrm{\pi }}{3}$

$\mathrm{argument}\left(\mathrm{polar}\left(2\&comma;\frac{\mathrm{\pi }}{7}\right)\right)$

$\frac{\mathrm{\pi }}{7}$

$\mathrm{argument}\left(\frac{1}{2.}-\frac{1}{3}\mathrm{I}\right)$

$\mathrm{-0.5880026035}$

$\mathrm{argument}\left(-\frac{1}{2.}-\frac{1}{3}\mathrm{I}\right)$

$\mathrm{-2.553590050}$

$\mathrm{argument}\left(\frac{1}{2.}+\frac{1}{3}\mathrm{I}\right)$

$0.5880026035$

$\mathrm{argument}\left(-\frac{1}{2.}+\frac{1}{3}\mathrm{I}\right)$

$2.553590050$

$\mathrm{argument}\left(\frac{1}{2}\right)$

$0$

$\mathrm{argument}\left(-\frac{1}{2}\right)$

$\mathrm{\pi }$

$\mathrm{argument}\left(3+4\mathrm{I}\right)$

$\arctan \left(\frac{4}{3}\right)$