# 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 http://www.rose-hulman.edu/ rickert/Classes/ma222/Wint0102/dirac.pdfon line.

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

## References

• 1 W. Rudin, , 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.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
Title Dirac delta function DiracDeltaFunction 2013-03-22 12:11:45 2013-03-22 12:11:45 PrimeFan (13766) PrimeFan (13766) 16 PrimeFan (13766) Definition msc 34L40 delta function DiracSequence DiracMeasure Distribution4