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The Dirac delta ``function'' $\delta(x)$ is not a true function because it is not uniquely defined for all values of the argument $x$. Similar to the Kronecker delta, the notation $\delta(x)$ stands for
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The Dirac delta ``function'' $\delta(x)$ is not a true function since it cannot be defined completely by giving the function value for all values of the argument $x$. Similar to the Kronecker delta, the notation $\delta(x)$ stands for
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| $$ \delta(x) = 0 \;\text{for}\; x \ne 0, \;\text{and}\; \int_{-\infty}^\infty \delta(x) dx = 1 $$ |
$$ \delta(x) = 0 \;\text{for}\; x \ne 0, \;\text{and}\; \int_{-\infty}^\infty \delta(x) dx = 1 $$ |
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| For any continuous function $F$: |
For any continuous function $F$: |
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| $$ \int_{-\infty}^\infty \delta(x) F(x)dx = F(0) $$ |
$$ \int_{-\infty}^\infty \delta(x) F(x)dx = F(0) $$ |
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| or in $n$ dimensions: |
or in $n$ dimensions: |
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| $$\int_{\mathbb{R}^n} \delta(x - s)f(s) \, d^ns = f(x)$$ |
$$\int_{\mathbb{R}^n} \delta(x - s)f(s) \, d^ns = f(x)$$ |
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| $\delta(x)$ can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width. |
$\delta(x)$ can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width. |
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| \textbf{Notes:} |
\textbf{Notes:} |
| However, the limit of the normalized Gaussian function is still meaningless as a function, but people nonetheless often write such a limit as being equal to the Dirac distribution considered above in the first paragraph. \\ |
However, the limit of the normalized Gaussian function is still meaningless as a function, but people nonetheless often 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 |
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} |
\PMlinkexternal{on line.}{http://www.rose-hulman.edu/~rickert/Classes/ma222/Wint0102/dirac.pdf} |
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| {\bf References} |
{\bf References} |
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| \begin{itemize} |
\begin{itemize} |
| \item Originally from The Data Analysis Briefbook |
\item Originally from The Data Analysis Briefbook |
| (\PMlinkexternal{http://rkb.home.cern.ch/rkb/titleA.html}{http://rkb.home.cern.ch/rkb/titleA.html}) |
(\PMlinkexternal{http://rkb.home.cern.ch/rkb/titleA.html}{http://rkb.home.cern.ch/rkb/titleA.html}) |
| \end{itemize} |
\end{itemize} |
