| Version 3 |
Version 2 |
| \newcommand{\signum}[0]{\mathop{\mathrm{sign}}} |
\newcommand{\signum}[0]{\mathop{\mathrm{sign}}} |
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\newcommand{\R}[0]{\mathbb{R}}
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\newcommand{\R}[0]{\mathbb{R}} |
| The \emph{signum function} is the function $\signum :\R\to \R$ |
The \emph{signum function} is the function $\signum :\R\to \R$ |
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\begin{eqnarray*}
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\begin{eqnarray*} |
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\signum (x) &=& \left\{ \begin {array}{ll}
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\signum (x) &=& \left\{ \begin {array}{ll}
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-1 & \mbox{when}\,\, x<0, \\
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-1 & \mbox{when}\,\, x<0, \\ |
| 1 & \mbox{when}\,\, x>0. \\ \end{array} \right. |
1 & \mbox{when}\,\, x>0. \\ \end{array} \right. |
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\end{eqnarray*}
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\end{eqnarray*} |
| The following properties hold: |
The following properties hold: |
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\begin{enumerate}
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\begin{enumerate} |
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\item For all $x\in \R$, $\signum(-x) = -\signum(x).$
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\item For all $x\in \R$, $\signum(-x) = -\signum(x).$ |
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\item For all $x\in \R$, $|x|=\signum(x) x.$
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\item For all $x\in \R$, $|x|=\signum(x) x.$ |
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\item For all $x\neq 0$, $\frac{d}{dx}|x|=\signum(x)$.
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\item For all $x\neq 0$, $\frac{d}{dx}|x|=\signum(x)$. |
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\end{enumerate}
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\end{enumerate} |
| defined simply as $1$ for $x>0$ and $-1$ for $x<0$. |
defined simply as $1$ for $x>0$ and $-1$ for $x<0$. |
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Thus, at $x=0$, it is left undefined. See e.g. \cite{kreyszig93}.
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Thus, at $x=0$, it is left undefined. See e.g. \cite{kreyszig93}.
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| In applications, |
In applications, |
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such as the Laplace transform, this definition is adequate since
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such as the Laplace transform, this definition is adequate since |
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the value of a function at a single point does not change the
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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 |
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value. However, setting $H(0)=0$ is motivated by the above relations.
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value. However, setting $H(0)=0$ is motivated by the above relations. |
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defined as
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defined as |
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\begin{eqnarray*}
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\begin{eqnarray*} |
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H(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \\
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H(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \\
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\end{array} \right.
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\end{array} \right. |
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\end{eqnarray*}
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\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 |
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for all $x\in\R$, we then have the relations
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for all $x\in\R$, we then have the relations |
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\begin{eqnarray*}
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\begin{eqnarray*} |
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H (x) &=& \frac{1}{2}(\signum(x)+1), \\
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H (x) &=& \frac{1}{2}(\signum(x)+1), \\ |
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H(-x) &=& 1-H(x).
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H(-x) &=& 1-H(x). |
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\end{eqnarray*}
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\end{eqnarray*} |
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This first relation is clear. For the second, we have
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This first relation is clear. For the second, we have |
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\begin{eqnarray*}
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\begin{eqnarray*} |
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1-H(x) &=& 1-\frac{1}{2}(\signum(x)+1) \\
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1-H(x) &=& 1-\frac{1}{2}(\signum(x)+1) \\ |
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\end{eqnarray*}
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\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 |
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\begin{eqnarray*}
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\begin{eqnarray*} |
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f (x) &=& \left\{ \begin {array}{ll}
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f (x) &=& \left\{ \begin {array}{ll}
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4 & \mbox{when}\,\, x\in(a,b), \\
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4 & \mbox{when}\,\, x\in(a,b), \\ |
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\end{array} \right.
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\end{array} \right. |
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\end{eqnarray*}
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\end{eqnarray*} |
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Using the Heavyside step function, we can write
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Using the Heavyside step function, we can write |
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\begin{eqnarray}
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\begin{eqnarray} |
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\label{almost}
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\label{almost} |
| \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} |
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| For a complex number $z$, the signum function is defined as \cite{bachman} |
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| \begin{eqnarray*} |
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| \signum (z) &=& \left\{ \begin {array}{ll} |
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| 0 & \mbox{when}\,\, z=0,\\ |
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| {z}/{|z|} & \mbox{when}\,\, z\neq 0. \\ \end{array} \right. |
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| \end{eqnarray*} |
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| In other words, if $z$ is non-zero, then $\signum z$ is the projection |
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| of $z$ onto the unit circle $\{z\in \mathbb{C} \mid |z| = 1\}$. |
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| Clearly, the complex signum function reduces to the real signum function |
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| for real arguments. |
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| For all $z\in \mathbb{C}$, we have |
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| $$ z |
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| \signum \overline{z} = |z|,$$ |
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| where $\overline{z}$ is the complex conjugate of $z$. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
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\bibitem {kreyszig93} E. Kreyszig,
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\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, |
\end{thebibliography} |
| \emph{Functional analysis}, Academic Press, 1966. |
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| \end{thebibliography} |
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