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Concept# Sign function

Summary

In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that returns the sign of a real number. In mathematical notation the sign function is often represented as .
The signum function of a real number is a piecewise function which is defined as follows:
Any real number can be expressed as the product of its absolute value and its sign function:
It follows that whenever is not equal to 0 we have
Similarly, for any real number ,
We can also ascertain that:
The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero. More formally, in integration theory it is a weak derivative, and in convex function theory the subdifferential of the absolute value at 0 is the interval , "filling in" the sign function (the subdifferential of the absolute value is not single-valued at 0). Note, the resultant power of is 0, similar to the ordinary derivative of . The numbers cancel and all we are left with is the sign of .
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, which can be demonstrated using the identity
where is the Heaviside step function using the standard formalism.
Using this identity, it is easy to derive the distributional derivative:
The Fourier transform of the signum function is
where means taking the Cauchy principal value.
The signum can also be written using the Iverson bracket notation:
The signum can also be written using the floor and the absolute value functions:
The signum function has a very simple definition if is accepted to be equal to 1. Then signum can be written for all real numbers as
The signum function coincides with the limits
and
as well as,
Here, is the Hyperbolic tangent and the superscript of -1, above it, is shorthand notation for the inverse function of the Trigonometric function, tangent.

Official source

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