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# Dirac delta function

The Dirac delta “function” $\delta(x)$, or *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

$\delta(x)=0\;\text{for}\;x\neq 0,\;\text{and}\;\int_{{-\infty}}^{\infty}\delta% (x)dx=1$ |

For any continuous function $F$:

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

or in $n$ dimensions:

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

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

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
on line.

Remarks: Distributions play important roles in Dirac’s formulation of quantum mechanics.

# References

- 1 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
- 2 L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
- 3 Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.html)

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