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Signum Function

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.

1	Point
1	Extended Real Numbers
1	Complex Number
1	Complex
1	Real Number
1	Property
1	Obvious
1	Surjective
1	Function
1	Equation
2	Relation Between Objects
2	Almost Surely
2	Topic Entry On Analysis
2	Search Problem
3	Involutary Ring
3	Complex Conjugate
4	Laplace Transform
4	Unit Disk
5	Piecewise
13	Heaviside Step Function