Heaviside step functionHeavisideStepFunction2003-07-18 12:14:392008-05-15 07:28:17Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}The \emph{Heaviside step function} is the function $H:\sR\to \sR$ defined as
\begin{eqnarray*}
H(x) &=& \left\{ \begin {array}{ll} 0 & \mbox{when}\,\, x< 0, \\
1/2 & \mbox{when}\,\, x= 0,\\
1 & \mbox{when}\,\, x> 0.\\
\end{array} \right.
\end{eqnarray*}
Here, there are many conventions for the value at $x=0$. The
motivation for setting $H(0)=1/2$ is that we can then write
$H$ as a function of the signum function (see
\PMlinkname{this page}{SignumFunction}). In applications, such as
the Laplace transform, where the Heaviside function is used extensively,
the value of $H(0)$ is irrelevant.
The Fourier transform of heaviside function is
$$\mathcal{F}_0 H(t)=\frac{1}{2}\left(\delta(t)-\frac{i}{\pi t}\right)$$
where $\delta$ denotes the Dirac delta centered at $0$.
The function is named after Oliver Heaviside (1850-1925)
\cite{heaviside_bib}. However, the function was already used by
Cauchy\cite{cauchy_bib}, who defined the function as
$$ u(t) = \frac{1}{2}\big( 1 + t/\sqrt{t^2}\big)$$
and called it a \emph{coefficient limitateur} \cite{hoskins}.
\begin{thebibliography}{9}
\bibitem{heaviside_bib}
The MacTutor History of Mathematics archive,
\PMlinkexternal{Oliver Heaviside}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Heav
iside.html}.
\bibitem{cauchy_bib}
The MacTutor History of Mathematics archive,
\PMlinkexternal{Augustin Louis Cauchy}{http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Cauc
hy.html}.
\bibitem{hoskins}
R.F. Hoskins, \emph{Generalised functions},
Ellis Horwood Series: Mathematics and its applications,
John Wiley \& Sons, 1979.
\end{thebibliography}