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signum function

(Definition)

The signum function is the function $\mathop{\mathrm{sign}}:\mathbb{R}\to \mathbb{R}$

$\displaystyle \mathop{\mathrm{sign}}(x)$

$\displaystyle =$

$\displaystyle \left\{ \begin {array}{ll} -1 & \mbox{when}\,\, x<0, \ 0 & \mbox{when}\,\, x=0,\ 1 & \mbox{when}\,\, x>0. \\ \end{array} \right.$

The following properties hold:

For all $x\in \mathbb{R}$ , $\mathop{\mathrm{sign}}(-x) = -\mathop{\mathrm{sign}}(x).$
For all $x\in \mathbb{R}$ , $\vert x\vert=\mathop{\mathrm{sign}}(x) x.$
For all $x\neq 0$ , $\frac{d}{dx}\vert x\vert=\mathop{\mathrm{sign}}(x)$ .

Here, we should point out that the signum function is often defined simply as for and for . Thus, at , it is left undefined. See e.g. [1]. In applications, such as the Laplace transform, this definition is adequate since the value of a function at a single point does not change the analysis. One could then, in fact, set $\mathop{\mathrm{sign}}(0)$ to any value. However, setting $\mathop{\mathrm{sign}}(0)=0$ is motivated by the above relations. On a related note, we can extend the definition to the extended real numbers $\overline{\mathbb{R}}=\mathbb{R}\cup\{\infty,-\infty\}$ by defining $\mathop{\mathrm{sign}}(\infty)=1$ and $\mathop{\mathrm{sign}}(-\infty)=-1$ .

A related function is the Heaviside step function defined as

$\displaystyle H(x)$

$\displaystyle =$

$\displaystyle \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \ 1/2 & \mbox{when}\,\, x= 0,\ 1 & \mbox{when}\,\, x> 0.\ \end{array} \right.$

Again, this function is sometimes left undefined at

. The motivation for setting

is that for all $x\in\mathbb{R}$ , we then have the relations

$\displaystyle H (x)$	$\displaystyle =$	$\displaystyle \frac{1}{2}(\mathop{\mathrm{sign}}(x)+1),$
$\displaystyle H(-x)$	$\displaystyle =$	$\displaystyle 1-H(x).$

This first relation is clear. For the second, we have

$\displaystyle 1-H(x)$	$\displaystyle =$	$\displaystyle 1-\frac{1}{2}(\mathop{\mathrm{sign}}(x)+1)$
	$\displaystyle =$	$\displaystyle \frac{1}{2}(1-\mathop{\mathrm{sign}}(x))$
	$\displaystyle =$	$\displaystyle \frac{1}{2}(1+\mathop{\mathrm{sign}}(- x))$
	$\displaystyle =$	$\displaystyle H(-x).$

Example Let be real numbers, and let $f:\mathbb{R}\to\mathbb{R}$ be the piecewise defined function

$\displaystyle f (x)$

$\displaystyle =$

$\displaystyle \left\{ \begin {array}{ll} 4 & \mbox{when}\,\, x\in(a,b), \ 0 & \mbox{otherwise.} \ \end{array} \right.$

Using the Heaviside step function, we can write

$\displaystyle f(x)$

$\displaystyle =$

$\displaystyle 4\big(H(x-a) - H(x-b)\big)$

(1)

almost everywhere. Indeed, if we calculate

using equation 1 we obtain

for $x\in(a,b)$ ,

for $x\notin[a,b]$ , and

. Therefore, equation 1 holds at all points except

and

. $\Box$

Signum function for complex arguments

For a complex number

, the signum function is defined as [2]

$\displaystyle \mathop{\mathrm{sign}}(z)$

$\displaystyle =$

$\displaystyle \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, z=0,\ {z}/{\vert z\vert} & \mbox{when}\,\, z\neq 0. \\ \end{array} \right.$

In other words, if

is non-zero, then $\mathop{\mathrm{sign}}z$ is the projection of

onto the unit circle $\{z\in \mathbb{C} \mid \vert z\vert = 1\}$ . Clearly, the complex signum function reduces to the real signum function for real arguments. For all $z\in \mathbb{C}$ , we have

$\displaystyle z \mathop{\mathrm{sign}}\overline{z} = \vert z\vert,$

where $\overline{z}$ is the complex conjugate of

Bibliography

1: E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.
2: G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.

"signum function" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Also defines:

Heavyside step function, step function

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Cross-references: complex conjugate, arguments, complex, unit circle, onto, projection, complex number, equation, calculate, almost everywhere, piecewise, real numbers, clear, Heaviside step function, extended real numbers, relations, Laplace transform, applications, point, properties, function
There are 9 references to this entry.

This is version 5 of signum function, born on 2003-05-05, modified 2007-05-09.
Object id is 4243, canonical name is SignumFunction.
Accessed 36377 times total.

Classification:

AMS MSC:	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	30-00 (Functions of a complex variable :: General reference works )

Pending Errata and Addenda

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