In mathematics the arg function is a logical function that extracts the angular component of a complex number or function. The angular component is also referred to as the argument. For real numbers x and y, arg(x + iy) is equivalent to the function atan2(y, x), which is constrained to the range (−π, π]. That is, for y ≠ 0:

$arg(x+iy)\; =\; begin\{cases\}$

phicdot sgn(y) & qquad x > 0 frac{pi}{2}cdot sgn(y) & qquad x = 0 (pi - phi)cdot sgn(y) & qquad x < 0 end{cases}

where $phi,$ is the angle in [0,π/2) such that:  $tan(phi)\; =\; left|\; frac\{y\}\{x\}right|.,$  And sgn is the sign function.

And:

$arg(x+i0)\; =\; begin\{cases\}$

0 & qquad x > 0 text{undefined} & qquad x = 0 pi & qquad x < 0 end{cases}

This produces results in the range (−π, π], which can be mapped to [0, 2π) by adding 2π to the negative values.

arg is also used less formally to represent an unconstrained angle. For instance, when:

• $phi(t),$ is a continuous function of time (such as $omega\; t),$,
• and $z(t)\; =\; r,mathrm\{e\}^\{i\; phi(t)\},$  (called exponential form),
• or $z(t)\; =\; r,(cos\; phi(t)\; +\; isin\; phi(t)),$  (called trigonometric form),

arg(z(t)) often denotes the continuous function, $phi(t).,$

Alternative implementation

If $r\; =\; sqrt\{x^2+y^2\}$ is readily available, a potentially simpler implementation of arg(x + iy) is also available.

For y ≠ 0:

$arg(x\; +\; iy)\; =\; theta\; cdot\; sgn(y),,$

where $theta,$ is the angle in [0,π) such that:  $cos(theta)\; =\; frac\{x\}\{r\}.,$

And  $arg(x+i0),$  is defined as before.

arg(0 + i 0)

When x and y are both zero,  $r\; =\; 0,,$  and any angle $phi,$ satisfies:

{{NumBlk|:|$x+iy\; =\; r,mathrm\{e\}^\{i\; phi\}.,$|}}

Therefore, arg(0 + i0) is sometimes defined as 0, for the sake of uniqueness. However, solving   for $phi,$ gives:

$$

arg(x + iy) = phi = -ilog_efrac{x+iy}{r}, , which is indeterminate/undefined when r=0. In this viewpoint, arg(x + iy) is not necessarily perceived as an angle.

Arg of rational complex numbers

If z₁ and z₂≠0 are two complex numbers then:

$$

argleft(frac{z_1}{z_2}right) = [arg(z_1) - arg(z_2) ]_{mod 2pi}.

E.g.:

$$

argleft(frac{-1-i}{i}right) = arg(-1-i) - arg(i) = -frac{3pi}{4} - frac{pi}{2} = -frac{5pi}{4} stackrel{mod 2pi}{longrightarrow}quad frac{3pi}{4}.

Notes

External links