argument - Maple Programming Help

argument

complex argument function

 Calling Sequence argument(z)

Parameters

 z - algebraic expression

Description

 • The argument function returns the principal value of the argument of the complex-valued expression z.  This means that argument(z) = t specifies $z$ = $\mathrm{polar}\left(\left|z\right|,t\right)=\left|z\right|{ⅇ}^{It}$ where $-\mathrm{\pi }.
 • From the previous definition, for an arbitrary complex number $z=x+Iy$, argument(z) >= 0 if $0\le y$. Otherwise argument(z) < 0.
 For the case where $y=0$, argument(z) = 0 if $0\le x$. Otherwise argument(z) = Pi.

Examples

 > $\mathrm{argument}\left(\left|z\right|{ⅇ}^{\frac{I\cdot 2\mathrm{π}}{3}}\right)$
 $\frac{{2}}{{3}}{}{\mathrm{π}}$ (1)
 > $\mathrm{argument}\left(\mathrm{polar}\left(2,\frac{\mathrm{π}}{7}\right)\right)$
 $\frac{{1}}{{7}}{}{\mathrm{π}}$ (2)

Consider the following examples.

For $\mathrm{\Im }\left(z\right)<0$, argument(z) is always negative.

 > $\mathrm{argument}\left(\frac{1}{2.}-\frac{1I}{3}\right)$
 ${-}{0.5880026035}$ (3)
 > $\mathrm{argument}\left(-\frac{1}{2.}-\frac{1I}{3}\right)$
 ${-}{2.553590050}$ (4)

For $0<\mathrm{\Im }\left(z\right)$, argument(z) is always positive.

 > $\mathrm{argument}\left(\frac{1}{2.}+\frac{1I}{3}\right)$
 ${0.5880026035}$ (5)
 > $\mathrm{argument}\left(-\frac{1}{2.}+\frac{1I}{3}\right)$
 ${2.553590050}$ (6)

For $\mathrm{\Im }\left(z\right)=0$, argument(z) is zero or $\mathrm{\pi }$.

 > $\mathrm{argument}\left(\frac{1}{2}\right)$
 ${0}$ (7)
 > $\mathrm{argument}\left(-\frac{1}{2}\right)$
 ${\mathrm{π}}$ (8)

For exact numbers and in other situations, argument returns an expression using arctan.

 > $\mathrm{argument}\left(3+4I\right)$
 ${\mathrm{arctan}}{}\left(\frac{{4}}{{3}}\right)$ (9)