Arg (mathematics)

In mathematics the arg function is a logical function that extracts the angular component of a complex number or function.An alternative definition is available at Complex argument (continued fraction). The essential difference is the treatment of arg("0" + "i0").] The angular component is also referred to as the "argument". For real numbers "x" and "y", arg("x" + "iy") is equivalent to a function used in three computer languages called atan2("y", "x"), which is constrained to the range (−π, π] . That is, for "y" ≠ 0:

:

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

And:

:

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:

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

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

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:

Therefore, arg(0 + "i" 0) is sometimes defined as 0, for the sake of uniqueness. However, solving EquationNote|Eq.1 for $phi,$ gives:

:$arg(x\; +\; iy)\; =\; phi\; =\; -ilog\_efrac\{x+iy\}\{r\},\; ,$ which is indeterminate/undefined when r=0. In this viewpoint, arg("x" + "i y") is not necessarily perceived as an angle.

Arg of rational complex numbers

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

:$argleft(frac\{z\_1\}\{z\_2\}\; ight)\; =\; [\; arg(z\_1)\; -\; arg(z\_2)\; ]\; \_\{mod\; 2pi\}.$

E.g.::$argleft(frac\{-1-i\}\{i\}\; ight)\; =\; arg(-1-i)\; -\; arg(i)\; =\; -frac\{3pi\}\{4\}\; -\; frac\{pi\}\{2\}\; =\; -frac\{5pi\}\{4\}\; stackrel\{mod\; 2pi\}\{longrightarrow\}quad\; frac\{3pi\}\{4\}.$

Notes

External links

*MathWorld|title=ComplexArgument|urlname=ComplexArgument