Dirac delta functionDiracDeltaFunction2002-01-19 23:04:472009-01-07 20:59:03Definition\usepackage{amssymb}
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%\usepackage{xypic}The Dirac delta ``function'' $\delta(x)$, or \emph{distribution} is not a true function because it is not uniquely defined for all values of the argument $x$. Similar to the Kronecker delta symbol, the notation $\delta(x)$ stands for
$$ \delta(x) = 0 \;\text{for}\; x \ne 0, \;\text{and}\; \int_{-\infty}^\infty \delta(x) dx = 1 $$
For any continuous function $F$:
$$ \int_{-\infty}^\infty \delta(x) F(x)dx = F(0) $$
or in $n$ dimensions:
$$\int_{\mathbb{R}^n} \delta(x - s)f(s) \, d^ns = f(x)$$
$\delta(x)$ can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width.
\textbf{Notes:}
However, the limit of the normalized Gaussian function is still meaningless as a function, but some people still write such a limit as being equal to the Dirac distribution considered above in the first paragraph. \\
An example of how the Dirac distribution arises in a physical, classical context is available
\PMlinkexternal{on line.}{http://www.rose-hulman.edu/~rickert/Classes/ma222/Wint0102/dirac.pdf}
{\bf Remarks:}
Distributions play important roles in Dirac's formulation of quantum mechanics.
\begin{thebibliography}{9}
\bibitem{rudin_fap}
W. Rudin, {\em Functional Analysis},
McGraw-Hill Book Company, 1973.
\bibitem{hormander}
L. H\"ormander, {\em The Analysis of Linear Partial Differential Operators I,
(Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990.
\bibitem{TDAB}
Originally from The Data Analysis Briefbook
(\PMlinkexternal{http://rkb.home.cern.ch/rkb/titleA.html}{http://rkb.home.cern.ch/rkb/titleA.html})
\end{thebibliography}