signum function

The signum function is the function $sgn:ℝ\to ℝ$

$sgn(x)$$=$

The following properties hold:

1. For all $x\in ℝ$, $sgn(-x)=-sgn(x).$

2. For all $x\in ℝ$, $|x|=sgn(x)x.$

3. For all $x\ne 0$, $\frac{d}{dx}|x|=sgn(x)$.

Here, we should point out that the signum functionis often defined simply as $1$ for $x>0$ and $-1$ for .Thus, at $x=0$, it is left undefined. See for example [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 $sgn(0)$ to any value.However, setting $sgn(0)=0$ is motivated by the above relations.On a related note, we can extend the definition to the extended real numbers$\overline{ℝ}=ℝ\cup \{\mathrm{\infty },-\mathrm{\infty }\}$by defining $sgn(\mathrm{\infty })=1$ and $sgn(-\mathrm{\infty })=-1$.

A related function is the Heaviside step functiondefined as

$H(x)$$=$

Again, this function is sometimes left undefined at $x=0$.The motivation for setting $H(0)=1/2$ is thatfor all $x\in ℝ$, we then have the relations

$H(x)$$=$${\displaystyle \frac{1}{2}}(sgn(x)+1),$

$H(-x)$$=$$1-H(x).$

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

$1-H(x)$$=$$1-{\displaystyle \frac{1}{2}}(sgn(x)+1)$

$=$${\displaystyle \frac{1}{2}}(1-sgn(x))$

$=$${\displaystyle \frac{1}{2}}(1+sgn(-x))$

$=$$H(-x).$

Example Let be real numbers, and let $f:ℝ\to ℝ$ be thepiecewise defined function

$f(x)$$=$

Using the Heaviside step function, we can write

$f(x)$$=$$4\left(H(x-a)-H(x-b)\right)$(1)

almost everywhere.Indeed, if we calculate $f$ using equation 1 we obtain$f(x)=4$ for $x\in (a,b)$, $f(x)=0$ for $x\notin [a,b]$,and $f(a)=f(b)=2$. Therefore, equation 1holds at all points except $a$ and $b$.$\mathrm{\square }$

1 Signum function for complex arguments

For a complex number $z$, the signum function is defined as [2]

$sgn(z)$$=$

In other words, if $z$ is non-zero, then $sgnz$ is the projectionof $z$ onto the unit circle $\{z\in ℂ\mid |z|=1\}$.Clearly, the complex signum function reduces to the real signum functionfor real arguments.For all $z\in ℂ$, we have

$$zsgn\overline{z}=|z|,$$

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