Sgn

$\begin{figure}\begin{center}\BoxedEPSF{Sgn.epsf scaled 700}\end{center}\end{figure}$

Also called Signum. It can be defined as

$\begin{displaymath} \mathop{\rm sgn}\nolimits \equiv \cases{ -1 & $x < 0$\cr 0 & $x = 0$\cr 1 & $x > 0$\cr} \end{displaymath}$

(1)

$\begin{displaymath} \mathop{\rm sgn}\nolimits (x) = 2H(x)-1, \end{displaymath}$

(2)

where

is the Heaviside Step Function. For $x\not=0$ , this can be written

$\begin{displaymath} \mathop{\rm sgn}\nolimits (x) \equiv {x\over\vert x\vert}. \end{displaymath}$

(3)