Sign function

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"Sgn" redirects here. For the capitalized abbreviation SGN, see SGN.

For the signature sgn(σ) of a permutation, see even and odd permutations.

Not to be confused with Sine function.

Signum function

In mathematics, the sign function is a mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function (after the Latin form of "sign").

In mathematical expressions the sign function is often represented as sgn.

1 Definition
2 Properties
3 Complex Signum
4 Generalized signum function
5 See also
6 References

[edit] Definition

The signum function of a real number x is defined as follows:

$\sgn x = \begin{cases} -1 & \text{if } x < 0, \\ 0 & \text{if } x = 0, \\ 1 & \text{if } x > 0. \end{cases}$

[edit] Properties

Any real number can be expressed as the product of its absolute value and its sign function:

$x = ( \sgn x ) |x|. \qquad \qquad (1)$

From equation (1) it follows that whenever x is not equal to 0 we have

$\sgn x = {x \over |x|}\,. \qquad \qquad (2)$

The signum function is the derivative of the absolute value function (up to the indeterminacy at zero):

${d |x| \over dx} = {x \over |x|}\,.$

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under the generalised notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function,

${d \ \sgn x \over dx} = 2 \delta (x).$

The signum function is related to the Heaviside step function H_1/2(x) thus

$\sgn x = 2 H_{1/2}(x) - 1, \,$

where the 1/2 subscript of the step function means that H_1/2(0) = 1/2. The signum can also be written using the Iverson bracket notation:

$\ \sgn x = -[x < 0] + [x > 0].$

For $k \gg 0$ , a smooth approximation of the step function is

$\ \sgn x \approx \tanh(kx).$

See Heaviside step function.

[edit] Complex Signum

The signum function can be generalized to complex numbers as

$\sgn z = \frac{z}{|z|}$

for any z ∈ $\mathbb{C}$ except z = 0. The signum of a given complex number z is the point on the unit circle of the complex plane that is nearest to z. Then, for z ≠ 0,

$\sgn z = \exp(i\arg z)\,,$

where arg is the complex argument function. For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines sgn 0 = 0.

Another generalization of the sign function for real and complex expressions is csgn,^[1] which is defined as:

$\operatorname{csgn}(z)= \begin{cases} 1 & \text{if } \Re(z) > 0 \vee (\Re(z) = 0 \land \Im(z) > 0), \\ -1 & \text{if } \Re(z) < 0 \vee (\Re(z) = 0 \land \Im(z) < 0), \\ 0 & \text{if } \Re(z)=\Im(z)=0. \end{cases}$

We then have (except for z = 0):

$\operatorname{csgn}(z) = \frac{z}{\sqrt{z^2}} = \frac{\sqrt{z^2}}{z}.$

[edit] Generalized signum function

At real values of $~x~$ , it is possible to define a generalized function–version of the signum function, $\varepsilon (x),$ such that $~\varepsilon(x)^2 =1~$ everywhere, including at the point $~x=0~$ (unlike sgn, for which $\sgn(0)^2 =0~$ ). This generalized signum allows construction of the algebra of generalized functions, but the price of such generalization is the loss of commutativity. In particular, the generalized signum anticommutes with the delta-function,^[2]

$\varepsilon(x) \delta(x)+\delta(x) \varepsilon(x) = 0~;$

in addition, $~\varepsilon(x)~$ cannot be evaluated at $~x=0~$ ; and the special name, $\varepsilon$ is necessary to distinguish it from the function sgn. ( $\varepsilon(0)$ is not defined, but sgn(0) = 0.)