Dirac delta function

The Dirac delta “functionMathworldPlanetmathδ(x), or distributionDlmfPlanetmathPlanetmathPlanetmath 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 δ(x) stands for


For any continuous functionMathworldPlanetmath F:


or in n dimensions:


δ(x) can also be defined as a normalized Gaussian function (normal distributionMathworldPlanetmath) 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.


  • 1 W. Rudin, Functional AnalysisMathworldPlanetmath, 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
Canonical name DiracDeltaFunction
Date of creation 2013-03-22 12:11:45
Last modified on 2013-03-22 12:11:45
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 16
Author PrimeFan (13766)
Entry type Definition
Classification msc 34L40
Synonym delta function
Related topic DiracSequence
Related topic DiracMeasure
Related topic Distribution4