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sign function


In mathematics, the sign function is an odd 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 (from , Latin for "sign").

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

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}

Properties


Any real number can be expressed as the product of its absolute value and its sign function:
      x = \sgn(x) \cdot |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} = \sgn(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 H1/2(x) thus:
      \sgn(x) = 2 H_{1/2}(x) - 1 \,,
where the 1/2 subscript of the step function means that H1/2(0) = 1/2. The signum can also be written using the Iverson bracket notation:
     \ \sgn x = -< 0 + > 0 \,.
For k \gg 0, a smooth approximation of the sign function is
     \ \sgn x \approx \tanh(kx) \,.
See Heaviside step function – Analytic approximations.

Complex signum


The signum function can be generalized to complex numbers as
     \sgn z = \frac{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,Maple V documentation. May 21 1998 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}.

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,

     \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~.)

See also


Absolute value
Heaviside function
Negative and non-negative numbers
Rectangular function

References


    
   
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