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| {{Redirect|Sgn|the capitalized abbreviation SGN|SGN (disambiguation)}}
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| {{for|the signature sgn(σ) of a permutation|even and odd permutations}}
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| {{distinguish|Sine function}}
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| [[Image:Signum function.svg|thumb|299px|Signum function y = sgn(x)]]
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| In [[mathematics]], the '''sign function''' or '''signum function''' (from ''[[wikt:signum#Latin|signum]]'', [[Latin language|Latin]] for "sign") is an [[Even and odd functions|odd]] [[function (mathematics)|mathematical function]] that extracts the [[sign (mathematics)|sign]] of a [[real number]]. In mathematical expressions the sign function is often represented as '''sgn'''.
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| ==Definition==
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| The signum function of a [[real number]] ''x'' is defined as follows:
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| :<math> \sgn(x) := \begin{cases}
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| -1 & \text{if } x < 0, \\
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| 0 & \text{if } x = 0, \\
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| 1 & \text{if } x > 0. \end{cases}</math>
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| ==Properties==
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| Any real number can be expressed as the product of its [[absolute value]] and its sign function:
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| :<math> x = \sgn(x) \cdot |x|\,.</math>
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| It follows that whenever ''x'' is not equal to 0 we have
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| :<math> \sgn(x) = {x \over |x|}</math>
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| Similarly, for ''any'' real number ''x'',
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| :<math> |x| = \sgn(x) \cdot x </math>
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| The signum function is the [[derivative]] of the absolute value function (up to the indeterminacy at zero): Note, the resultant power of x is 0, similar to the ordinary derivative of x. The numbers cancel and all we are left with is the sign of x.
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| :<math> {d |x| \over dx} = \sgn(x) \mbox{ for } x \ne 0</math> .
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| 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 (mathematics)|distribution theory]],
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| the derivative of the signum function is two times the [[Dirac delta function]], which can be demonstrated using the identity
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| :<math> \sgn(x) = 2 H(x) - 1 \,</math><ref>{{MathWorld |title=Sign |id=Sign}}</ref>
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| (where ''H''(''x'') is the [[Heaviside step function]] using the standard ''H''(0) = 1/2 formalism).
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| Using this identity, it is easy to derive the distributional derivative:
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| :<math> {d \sgn(x) \over dx} = 2 {d H(x) \over dx} = 2\delta(x) \,.</math><ref>{{MathWorld |title=Heaviside Step Function |id=HeavisideStepFunction}}</ref>
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| The signum can also be written using the [[Iverson bracket]] notation:
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| :<math>\ \sgn(x) = -[x < 0] + [x > 0] \,.</math>
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| For <math>k \gg 1</math>, a smooth approximation of the sign function is
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| :<math>\ \sgn(x) \approx \tanh(kx) \,.</math>
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| Another approximation is
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| :<math>\ \sgn(x) \approx \frac{x}{\sqrt{x^2 + \epsilon^2}} \,.</math>
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| which gets sharper as <math>\epsilon \to 0</math>, note that it's the derivative of <math>\sqrt{x^2 + \epsilon^2}</math>. This is inspired from the fact that the above is exactly equal for all nonzero ''x'' if <math>\epsilon = 0</math>, and has the advantage of simple generalization to higher dimensional analogues of the sign function (for example, the partial derivatives of <math>\sqrt{x^2 + y^2}</math>).
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| See [[Heaviside step function#Analytic approximations|Heaviside step function – Analytic approximations]].
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| ==Complex signum==
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| The signum function can be generalized to [[complex numbers]] as | |
| :<math>\sgn(z) = \frac{z}{|z|} </math>
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| for any ''z'' ∈ <math>\mathbb{C}</math> except ''z'' = 0. The signum of a given complex number ''z'' is the [[point (geometry)|point]] on the [[unit circle]] of the [[complex plane]] that is nearest to ''z''. Then, for ''z'' ≠ 0,
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| :<math>\sgn(z) = e^{i\arg z}\,,</math>
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| where arg is the [[Complex number#Polar form|complex argument function]].
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| 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 ''z'' = 0:
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| :<math>\operatorname{sgn}(0+0i)=0</math>
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| Another generalization of the sign function for real and complex expressions is ''csgn'',<ref>Maple V documentation. May 21, 1998</ref> which is defined as:
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| :<math> | |
| \operatorname{csgn}(z)= \begin{cases}
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| 1 & \text{if } \Re(z) > 0, \\
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| -1 & \text{if } \Re(z) < 0, \\
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| \sgn(\Im(z)) & \text{if } \Re(z) = 0
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| \end{cases}
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| </math>
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| where <math>\Re(z)</math> is the real part of z, <math>\Im(z)</math> is the imaginary part of z.
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| We then have (except for ''z'' = 0):
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| :<math>\operatorname{csgn}(z) = \frac{z}{\sqrt{z^2}} = \frac{\sqrt{z^2}}{z}. </math>
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| ==Generalized signum function==
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| At real values of <math>~x~</math>, it is possible to define a [[generalized function]]–version of the signum function, <math> \varepsilon (x),</math> such that
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| <math>~\varepsilon(x)^2 =1~</math> everywhere, including at the point <math>~x=0~</math> (unlike <math>~\sgn~</math>, for which <math>\sgn(0)^2 =0~</math>). 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 Dirac delta function<ref name="Algebra"> | |
| {{cite journal
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| |author=Yu.M.Shirokov
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| |title = Algebra of one-dimensional generalized functions
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| |journal=[[Theoretical and Mathematical Physics|TMF]]
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| |year=1979
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| |volume=39
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| |issue=3
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| |pages=471–477
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| |url=http://springerlink.metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3
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| |doi=10.1007/BF01017992
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| }}</ref>
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| :<math>\varepsilon(x) \delta(x)+\delta(x) \varepsilon(x) = 0~;</math>
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| in addition, <math>~\varepsilon(x)~</math> cannot be [[evaluate]]d at
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| <math>~x=0~</math>; and the special name, <math>\varepsilon</math> is necessary to distinguish it from the function <math>~\sgn~</math>. (<math>\varepsilon(0)</math> is not defined, but <math>~\sgn(0) = 0~</math>.)
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| ==See also==
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| * [[Absolute value]]
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| * [[Heaviside function]]
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| * [[Negative number]]
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| * [[Rectangular function]]
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| * [[Sigmoid function]]
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| * [[Three-way comparison]]
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| * [[Zero crossing]]
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| ==Notes==
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| <references/>
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| {{DEFAULTSORT:Sign Function}}
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| [[Category:Special functions]]
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Jerrie Swoboda is what you actually can call me and I totally dig who seem to name. What me and my family genuinely like is acting but I can't make it simple profession really. The job I've been taking up for years is a people manager. Guam is even I've always been living. You does find my website here: http://prometeu.net
my website ... clash of clans hack (simply click the next site)