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
$$\delta (x)=0\text{for}x\ne 0,\text{and}{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\delta (x)\mathit{d}x=1$$ |
For any continuous function^{} $F$:
$${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\delta (x)F(x)\mathit{d}x=F(0)$$ |
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, 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.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 |