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

$$\delta (x)=0\text{for}x\ne 0,\text{and}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)𝑑x=1$$

For any continuous function $F$:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)F(x)𝑑x=F(0)$$

or in $n$ dimensions:

$${\int }_{{ℝ}^n}\delta (x-s)f(s)d^{n}s=f(x)$$

$\delta (x)$ 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.

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