Sign function
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
Definition
The signum function of a real number
Properties
Any real number can be expressed as the product of its absolute value and its sign function:
It follows that whenever
Similarly, for any real number
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 [2]
The Fourier transform of the signum function is[4]
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:
For
See Heaviside step function § Analytic approximations.
Complex signum
The signum function can be generalized to complex numbers as:
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, for
Another generalization of the sign function for real and complex expressions is
We then have (for
Generalized signum function
At real values of
Generalization to matrices
Thanks to the Polar decomposition theorem, a matrix
In the special case where
See also
- Absolute value
- Heaviside function
- Negative number
- Rectangular function
- Sigmoid function (Hard sigmoid)
- Step function (Piecewise constant function)
- Three-way comparison
- Zero crossing
- Polar decomposition
Notes
- ↑ Jump up to: 1.0 1.1 "Signum function - Maeckes". http://www.maeckes.nl/Signum%20functie%20GB.html.
- ↑ Weisstein, Eric W.. "Sign". http://mathworld.wolfram.com/Sign.html.
- ↑ Weisstein, Eric W.. "Heaviside Step Function". http://mathworld.wolfram.com/HeavisideStepFunction.html.
- ↑ Burrows, B. L.; Colwell, D. J. (1990). "The Fourier transform of the unit step function". International Journal of Mathematical Education in Science and Technology 21 (4): 629–635. doi:10.1080/0020739900210418.
- ↑ Maple V documentation. May 21, 1998
- ↑ Yu.M.Shirokov (1979). "Algebra of one-dimensional generalized functions". Theoretical and Mathematical Physics 39 (3): 471–477. doi:10.1007/BF01017992. http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3.
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