PlanetMath: Dirac delta function

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Dirac delta function

(Definition)

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 . Similar to the Kronecker delta, the notation $\delta(x)$ stands for

$\displaystyle \delta(x) = 0 \;$ for $\displaystyle \; x \ne 0, \;$ and $\displaystyle \; \int_{-\infty}^\infty \delta(x) dx = 1$

For any continuous function :

$\displaystyle \int_{-\infty}^\infty \delta(x) F(x)dx = F(0)$

or in dimensions:

$\displaystyle \int_{\mathbb{R}^n} \delta(x - s)f(s) \, d^ns = f(x)$

$\delta(x)$ can also be defined as a normalized Gaussian function (normal distribution) in the limit of zero width.

References

Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)

"Dirac delta function" is owned by PrimeFan. [ full author list (2) | owner history (2) ]

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