en>NathanReading: /* Properties */ added comma and one word to clarify reference to the later description of the chaos game.

2015-01-09T13:22:58Z

Properties: added comma and one word to clarify reference to the later description of the chaos game.

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	~~{{Redirect\|Sgn\|the capitalized abbreviation SGN\|SGN (disambiguation)}}~~		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<br><br>my website ... clash of clans hack ([http://prometeu.net simply click the next site])
	~~{{for\|the signature sgn(σ) of a permutation\|even and odd permutations}}~~
	~~{{distinguish\|Sine function}}~~

	~~[[Image:Signum function.svg\|thumb\|299px\|Signum function y = sgn(x)]]~~

	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'''.

	~~==Definition==~~
	~~The signum function of a [[real number]] ''x''~~ is ~~defined as follows:~~

	~~:<math> \sgn(x) := \begin{cases}~~
	~~-1 & \text{if } x < 0, \\~~
	~~0 & \text{if } x = 0, \\~~
	~~1 & \text{if } x > 0. \end{cases}</math>~~

	~~==Properties==~~
	~~Any real number~~ can ~~be expressed as the product of its [[absolute value]]~~ and ~~its sign function:~~
	~~:<math> x = \sgn(x) \cdot \|x\|\,.</math>~~
	~~It follows that whenever ''x'' is not equal~~ to ~~0 we have~~
	~~:<math> \sgn(x) = {x \over \|x\|}</math>~~
	~~Similarly, for ''any'' real number ''x'',~~
	~~:<math> \|x\| = \sgn(x) \cdot x </math>~~

	~~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.~~
	~~:<math> {d \|x\| \over dx} = \sgn(x) \mbox{ for } x \ne 0</math> .~~

	~~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]],~~
	~~the derivative of the signum function is two times the [[Dirac delta function]], which~~ can ~~be demonstrated using the identity~~
	~~:<math> \sgn(x) = 2 H(x) - 1 \,</math><ref>{{MathWorld \|title=Sign \|id=Sign}}</ref>~~
	~~(where~~ '~~'H''(''x'') is the [[Heaviside step function]] using the standard ''H''(0) = 1/2 formalism).~~
	~~Using this identity,~~ it ~~is easy to derive the distributional derivative:~~
	~~:<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>~~

	~~The signum can also be written using the [[Iverson bracket]] notation:~~
	~~:<math>\ \sgn(x) = -[x < 0] + [x > 0] \,.</math>~~
	~~For <math>k \gg 1</math>, a smooth approximation of the sign function is~~
	~~:<math>\ \sgn(x) \approx \tanh(kx) \,.</math>~~
	~~Another approximation is~~
	~~:<math>\ \sgn(x) \approx \frac{x}{\sqrt{x^2 + \epsilon^2}} \,.</math>~~
	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>).~~

	~~See [[Heaviside step function#Analytic approximations\|Heaviside step function – Analytic approximations]]~~.

	~~==Complex signum==~~
	The ~~signum function can be generalized to [[complex numbers]] as~~
	~~:<math>\sgn(z) = \frac{z}{\|z\|} </math>~~
	~~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,~~
	~~:<math>\sgn(z) = e^{i\arg z}\,,</math>~~
	~~where arg~~ is ~~the [[Complex number#Polar form\|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, for ''z'' = 0:~~
	~~:<math>\operatorname{sgn}(0+0i)=0</math>~~

	~~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~~:
	:~~<math>~~
	~~\operatorname{csgn}(z)= \begin{cases}~~
	~~1 & \text{if } \Re(z) > 0, \\~~
	~~-1 & \text{if } \Re(z) < 0, \\~~
	~~\sgn(\Im(z)) & \text{if } \Re(z) = 0~~
	~~\end{cases}~~
	</~~math>~~
	~~where <math>\Re(z)</math> is the real part of z, <math>\Im(z)<~~/~~math> is the imaginary part of z~~.

	~~We then have (except for ''z'' = 0):~~
	~~:<math>\operatorname{csgn}(z) = \frac{z}{\sqrt{z^2}} = \frac{\sqrt{z^2}}{z}. </math>~~

	~~==Generalized signum function==~~
	~~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~~
	<~~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~~
	~~\|author=Yu~~.~~M.Shirokov~~
	~~\|title = Algebra~~ of ~~one-dimensional generalized functions~~
	~~\|journal=[~~[~~Theoretical and Mathematical Physics\|TMF]]~~
	~~\|year=1979~~
	~~\|volume=39~~
	~~\|issue=3~~
	~~\|pages=471–477~~
	~~\|url=~~http://~~springerlink~~.~~metapress.com/content/w3010821x8267824/?p=5bb23f98d846495c808e0a2e642b983a&pi=3~~
	~~\|doi=10.1007/BF01017992~~
	~~}}</ref>~~
	~~:<math>\varepsilon(x) \delta(x)+\delta(x) \varepsilon(x) = 0~;</math>~~
	~~in addition, <math>~\varepsilon(x)~</math> cannot be [[evaluate~~]~~]d at~~
	<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>.)

	~~==See also==~~
	* [[Absolute value]]
	* [[Heaviside function]]
	* [[Negative number]]
	* [[Rectangular function]]
	* [[Sigmoid function]]
	* [[Three-way comparison]]
	* [[Zero crossing]]

	~~==Notes==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Sign Function}}~~
	~~[[Category:Special functions]]~~

en>KylieTastic: Removed missing (red-linked) image

2013-11-19T00:12:14Z

Removed missing (red-linked) image

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188.26.22.131: /* Definition */ [[]]

2012-03-06T12:40:09Z

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Iterated function system - Revision history

en>NathanReading: /* Properties */ added comma and one word to clarify reference to the later description of the chaos game.

en>KylieTastic: Removed missing (red-linked) image

188.26.22.131: /* Definition */ [[]]