| Version 5 |
Version 4 |
| \newcommand{\signum}[0]{\mathop{\mathrm{sign}}} |
\newcommand{\signum}[0]{\mathop{\mathrm{sign}}} |
| \newcommand{\R}[0]{\mathbb{R}} |
\newcommand{\R}[0]{\mathbb{R}} |
| |
|
| The \emph{signum function} is the function $\signum :\R\to \R$ |
The \emph{signum function} is the function $\signum :\R\to \R$ |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \signum (x) &=& \left\{ \begin {array}{ll} |
\signum (x) &=& \left\{ \begin {array}{ll} |
| -1 & \mbox{when}\,\, x<0, \\ |
-1 & \mbox{when}\,\, x<0, \\ |
| 0 & \mbox{when}\,\, x=0,\\ |
0 & \mbox{when}\,\, x=0,\\ |
| 1 & \mbox{when}\,\, x>0. \\ \end{array} \right. |
1 & \mbox{when}\,\, x>0. \\ \end{array} \right. |
| \end{eqnarray*} |
\end{eqnarray*} |
| |
|
| The following properties hold: |
The following properties hold: |
| \begin{enumerate} |
\begin{enumerate} |
| \item For all $x\in \R$, $\signum(-x) = -\signum(x).$ |
\item For all $x\in \R$, $\signum(-x) = -\signum(x).$ |
| \item For all $x\in \R$, $|x|=\signum(x) x.$ |
\item For all $x\in \R$, $|x|=\signum(x) x.$ |
| \item For all $x\neq 0$, $\frac{d}{dx}|x|=\signum(x)$. |
\item For all $x\neq 0$, $\frac{d}{dx}|x|=\signum(x)$. |
| \end{enumerate} |
\end{enumerate} |
| |
|
| Here, we should point out that the signum function is often |
Here, we should point out that the signum function is often |
| defined simply as $1$ for $x>0$ and $-1$ for $x<0$. |
defined simply as $1$ for $x>0$ and $-1$ for $x<0$. |
| Thus, at $x=0$, it is left undefined. See e.g. \cite{kreyszig93}. |
Thus, at $x=0$, it is left undefined. See e.g. \cite{kreyszig93}. |
| In applications, |
In applications, |
| such as the Laplace transform, this definition is adequate since |
such as the Laplace transform, this definition is adequate since |
| the value of a function at a single point does not change the |
the value of a function at a single point does not change the |
| analysis. One could then, in fact, set $\signum(0)$ to any |
analysis. One could then, in fact, set $\signum(0)$ to any |
|
value. However, setting $\signum(0)=0$ 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 $\signum(\infty)=1$ and $\signum(-\infty)=-1$.
|
value. However, setting $\signum(0)=0$ is motivated by the above relations..
|
| |
|
| A related function is the \emph{Heaviside step function} |
A related function is the \emph{Heaviside step function} |
| defined as |
defined as |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| H(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \\ |
H(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \\ |
| 1/2 & \mbox{when}\,\, x= 0,\\ |
1/2 & \mbox{when}\,\, x= 0,\\ |
| 1 & \mbox{when}\,\, x> 0.\\ |
1 & \mbox{when}\,\, x> 0.\\ |
| \end{array} \right. |
\end{array} \right. |
| \end{eqnarray*} |
\end{eqnarray*} |
| Again, this function is sometimes left undefined at $x=0$. |
Again, this function is sometimes left undefined at $x=0$. |
| The motivation for setting $H(0)=1/2$ is that |
The motivation for setting $H(0)=1/2$ is that |
| for all $x\in\R$, we then have the relations |
for all $x\in\R$, we then have the relations |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| H (x) &=& \frac{1}{2}(\signum(x)+1), \\ |
H (x) &=& \frac{1}{2}(\signum(x)+1), \\ |
| H(-x) &=& 1-H(x). |
H(-x) &=& 1-H(x). |
| \end{eqnarray*} |
\end{eqnarray*} |
| This first relation is clear. For the second, we have |
This first relation is clear. For the second, we have |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| 1-H(x) &=& 1-\frac{1}{2}(\signum(x)+1) \\ |
1-H(x) &=& 1-\frac{1}{2}(\signum(x)+1) \\ |
| &=& \frac{1}{2}(1-\signum(x)) \\ |
&=& \frac{1}{2}(1-\signum(x)) \\ |
| &=& \frac{1}{2}(1+\signum(- x)) \\ |
&=& \frac{1}{2}(1+\signum(- x)) \\ |
| &=& H(-x). |
&=& H(-x). |
| \end{eqnarray*} |
\end{eqnarray*} |
| |
|
| {\bf Example} Let $a<b$ be real numbers, and let $f:\R\to\R$ be the |
{\bf Example} Let $a<b$ be real numbers, and let $f:\R\to\R$ be the |
| piecewise defined function |
piecewise defined function |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| f (x) &=& \left\{ \begin {array}{ll} |
f (x) &=& \left\{ \begin {array}{ll} |
| 4 & \mbox{when}\,\, x\in(a,b), \\ |
4 & \mbox{when}\,\, x\in(a,b), \\ |
| 0 & \mbox{otherwise.} \\ |
0 & \mbox{otherwise.} \\ |
| \end{array} \right. |
\end{array} \right. |
| \end{eqnarray*} |
\end{eqnarray*} |
| Using the Heaviside step function, we can write |
Using the Heaviside step function, we can write |
| \begin{eqnarray} |
\begin{eqnarray} |
| \label{almost} |
\label{almost} |
| f(x) &=& 4\big(H(x-a) - H(x-b)\big) |
f(x) &=& 4\big(H(x-a) - H(x-b)\big) |
| \end{eqnarray} |
\end{eqnarray} |
| almost everywhere. |
almost everywhere. |
| Indeed, if we calculate $f$ using equation \ref{almost} we obtain |
Indeed, if we calculate $f$ using equation \ref{almost} we obtain |
| $f(x)=4$ for $x\in(a,b)$, $f(x)=0$ for $x\notin[a,b]$, |
$f(x)=4$ for $x\in(a,b)$, $f(x)=0$ for $x\notin[a,b]$, |
| and $f(a)=f(b)=2$. Therefore, equation \ref{almost} |
and $f(a)=f(b)=2$. Therefore, equation \ref{almost} |
| holds at all points except $a$ and $b$. |
holds at all points except $a$ and $b$. |
| $\Box$ |
$\Box$ |
|
|
| \section{Signum function for complex arguments} |
\section{Signum function for complex arguments} |
| For a complex number $z$, the signum function is defined as \cite{bachman} |
For a complex number $z$, the signum function is defined as \cite{bachman} |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \signum (z) &=& \left\{ \begin {array}{ll} |
\signum (z) &=& \left\{ \begin {array}{ll} |
| 0 & \mbox{when}\,\, z=0,\\ |
0 & \mbox{when}\,\, z=0,\\ |
| {z}/{|z|} & \mbox{when}\,\, z\neq 0. \\ \end{array} \right. |
{z}/{|z|} & \mbox{when}\,\, z\neq 0. \\ \end{array} \right. |
| \end{eqnarray*} |
\end{eqnarray*} |
| In other words, if $z$ is non-zero, then $\signum z$ is the projection |
In other words, if $z$ is non-zero, then $\signum z$ is the projection |
| of $z$ onto the unit circle $\{z\in \mathbb{C} \mid |z| = 1\}$. |
of $z$ onto the unit circle $\{z\in \mathbb{C} \mid |z| = 1\}$. |
| Clearly, the complex signum function reduces to the real signum function |
Clearly, the complex signum function reduces to the real signum function |
| for real arguments. |
for real arguments. |
| For all $z\in \mathbb{C}$, we have |
For all $z\in \mathbb{C}$, we have |
|
$$ z \signum \overline{z} = |z|,$$
|
$$ z \signum \overline{z} = |z|,$$
|
| where $\overline{z}$ is the complex conjugate of $z$. |
where $\overline{z}$ is the complex conjugate of $z$. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {kreyszig93} E. Kreyszig, |
\bibitem {kreyszig93} E. Kreyszig, |
| \emph{Advanced Engineering Mathematics}, |
\emph{Advanced Engineering Mathematics}, |
| John Wiley \& Sons, 1993, 7th ed. |
John Wiley \& Sons, 1993, 7th ed. |
| \bibitem{bachman} G. Bachman, L. Narici, |
\bibitem{bachman} G. Bachman, L. Narici, |
| \emph{Functional analysis}, Academic Press, 1966. |
\emph{Functional analysis}, Academic Press, 1966. |
| \end{thebibliography} |
\end{thebibliography} |
