NationMaster - Encyclopedia: Sign function

In mathematics, the sign function is a mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function (after the Latin form of "sign"). Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ... In mathematics, the permutations of a finite set (i. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about functions in mathematics. ... A negative number is a number that is less than zero, such as âˆ’3. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... For other uses, see Latin (disambiguation). ...

In mathematical expressions the sign function is often represented as sgn.

1 Definition
2 Properties
3 Complex Signum
4 Generalized signum function
5 See also
6 References

Definition

The signum function is defined as follows:

$sgn x = left{ begin{matrix} -1 & text{if} & x < 0, 0 & text{if} & x = 0, 1 & text{if} & x > 0. end{matrix} right.$

Properties

Any real number can be expressed as the product of its absolute value and its sign function: In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

From equation (1) it follows that whenever x is not equal to 0 we have

The signum function is the derivative of the absolute value function (up to the indeterminacy at zero): This article is about derivatives and differentiation in mathematical calculus. ...

${d |x| over dx} = {x over |x|},.$

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, In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...

The signum function is related to the Heaviside step function H_1/2(x) thus The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

where the 1/2 subscript of the step function means that H_1/2(0) = 1/2. The signum can also be written using the Iverson bracket notation: In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is defined as follows where P is a proposition. ...

For $k gg 0$ , a smooth approximation of the step function is

See Heaviside step function. The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

Complex Signum

The signum function can be generalized to complex numbers as $sgn z = frac{z}{|z|}$ for any z ∈ $mathbb{C}$ except z = 0. The signum of a given complex number z is the point on the unit circle of the complex plane that is nearest to z. Then, for z ≠ 0, The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Illustration of a unit circle. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

where arg is the 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 sgn 0 = 0. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

Another generalization of the sign function for real and complex expressions is csgn,^[1] which is defined as:

We then have (except for z = 0):

Generalized signum function

At real values of $~x~$ , it is possible to define a generalized function–version of the signum function, $varepsilon (x),$ such that $~varepsilon(x)^2 =1~$ everywhere, including at the point $~x=0~$ (unlike sgn, for which $sgn(0)^2 =0~$ ). 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 delta-function,^[2] In mathematics, generalized functions are objects generalizing the notion of functions. ... Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...

in addition, $~varepsilon(x)~$ cannot be evaluated at $~x=0~$ ; and the special name, $varepsilon$ is necessary to distinguish it from the function sgn. ( $varepsilon(0)$ is not defined, but sgn(0) = 0.)

References

^ Maple V documentation. May 21 1998
^ Yu.M.Shirokov (1979). "Algebra of one-dimensional generalized functions". TMF 39 (3): 471-477.

Categories: Elementary special functions

Results from FactBites:

New Mappings · The Sign Function in Science Fiction (3278 words)
	It is precisely this sign function which helps create the cognitive estrangement typically seen as a defining feature of sf.
	In this way, the footnote functions as a form of reality effect, in Barthes sense, to convince the reader of the real status of this text, even though it is clearly fictional.
	Thereby, the sign refers exclusively to a conventionalised architext where the illusion of reality (although absent) is maintained with the same bold confidence in language and referentiality which realist fiction exhibits.

Continuous function - Wikipedia, the free encyclopedia (1432 words)
	In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output.
	Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper.
	Again consider a function f that maps a set of real numbers to another set of real numbers, and suppose c is an element of the domain of f.

More results at FactBites »

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Contents

Definition GA_googleFillSlot("encyclopedia_square");

Properties

Complex Signum

Generalized signum function

See also

References

Definition