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In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function. The sign function is often represented as sgn and can be defined thus:
- <math> \sgn x = \left\{ \begin{matrix}
-1 & : & x < 0 \\
0 & : & x = 0 \\
1 & : & x > 0 \end{matrix} \right. <math>
Any real number can be expressed as the product of its absolute value and its sign function:
- <math> x = ( \sgn x ) |x|. \qquad \qquad (1)<math>
From equation (1) it follows that
- <math> \sgn x = {x \over |x|} \qquad \qquad (2) <math>
but equation (2) is indeterminate when x is set to zero.
The signum function is the derivative of the absolute value function (up to the indeterminacy at zero):
- <math> {d |x| \over dx} = {x \over |x|}. <math>
Also, the derivative of the signum function is two times the Dirac delta function,
- <math> {d \ \sgn x \over dx} = 2 \delta (x). <math>
The signum function is related to the Heaviside step function h0.5(x) thus
- <math> \sgn x = 2 h_{0.5}(x) - 1, <math>
where the 0.5 subscript of the step function means that <math> h_{0.5}(0) = 0.5. <math>
Also, if the step function h0(x) is thought of as a mathematical switch, with h0(x) = 0, then the signum function can be expressed as
- <math> \sgn x = | h_0 (x) | (-1)^{h_0(-x)}. <math>
See also
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