Dirac Delta Function

Formal introduction

$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; \backslash ,\; \backslash delta(x)\; \backslash ,\; dx$

$\backslash delta\backslash phi=\; \backslash phi(0)\backslash ,$

$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; \backslash ,\; d\backslash delta(x)$

Fourier transform

$\backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{ikx\}\backslash ,dx=\backslash delta(k)$

The Dirac Delta function as a probability density function

Derivatives of the delta function

$\backslash delta\text{'}\backslash phi=\; -\backslash phi\text{'}(0)\backslash ,$

$\backslash delta\text{'}(x)=-\backslash frac\{\backslash delta(x)\}\{x\}$

$\backslash delta^\{(n)\}\backslash phi=\; (-1)^n\; \backslash phi^\{(n)\}(0)\backslash ,$

$\backslash delta(g(x))\; =\; \backslash sum\_\{i\}\backslash frac\{\backslash delta(x-x\_i)\}$>

/dd> where xi are the roots of g(x). In the integral form it is equivalent to $$ \int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|} Japanese definition The Dirac delta function $\backslash delta\; :\; \backslash mathbb\{R\}\; \backslash ni\; \backslash xi\; \backslash longrightarrow\; \backslash delta\; (\; \backslash xi\; )\backslash in\; \backslash delta(\backslash mathbb\{R\})$ is a distribution $\backslash delta\; (\; \backslash xi\; )$ whose indefinite integral is the function $h\; :\; \backslash mathbb\{R\}\; \backslash ni\; \backslash xi\; \backslash longrightarrow\; \backslash frac\{1+\{\backslash rm\; sgn\}\; \backslash ,\; \backslash xi\; \}\{2\}\; \backslash in\; \backslash mathbb\{R\},$ usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation $$ \int^{x}_{-\infin} \delta (t) dt = h(x) \equiv \frac{1+{\rm sgn}(x) }{2} for all real numbers x. Representations of the delta function The delta function can be viewed as the limit of a sequence of functions $$ \delta (x) = \lim_{a\to 0} \delta_a(x), where $\backslash delta\_a(x)$ is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting sequence will be used. On the other hand the term limit needs to be made precise, as this equality holds only for some meanings of limit. The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions. Some nascent delta functions are: {| border="0" cellpadding="5" cellspacing="10" align="left"
math>\delta_a(x)=\frac{1}{a\sqrt{\pi}} \mathrm{e}^{-x^2/a^2}	Limit of a Normal distribution
math>\delta_a(x) = \frac{1}{\pi} {a \over a^2 + x^2} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} k x-|ak|}\;dk	Limit of a Cauchy distribution
math>\delta_a(x)=\frac{e^{-|x/a

}{2a} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ikx}}{1+a^2k^2}\,dk |Cauchy $\backslash varphi$(see note below) |- |$\backslash delta\_a(x)=\; \backslash frac\{\backslash textrm\{rect\}(x/a)\}\{a\}\; =\backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash textrm\{sinc\}(ak/2)e^\{ikx\}\backslash ,dk$ |Limit of a rectangular function |- |$\backslash delta\_a(x)=\backslash frac\{1\}\{\backslash pi\; x\}\backslash sin\backslash left(\backslash frac\{x\}\{a\}\backslash right)$

              =\frac{1}{2\pi}\int_{-1/a}^{1/a}               \cos (k x)\;dk 

|rectangular function $\backslash varphi$(see note below) |- |$\backslash delta\_a(x)=\backslash partial\_x\; \backslash frac\{1\}\{1+\backslash mathrm\{e\}^\{-x/a\}\}$

              =-\partial_x \frac{1}{1+\mathrm{e}^{x/a}} 

| |- |$\backslash delta\_a(x)=\backslash frac\{a\}\{\backslash pi\; x^2\}\backslash sin^2\backslash left(\backslash frac\{x\}\{a\}\backslash right)$ | |- |$\backslash delta\_a(x)\; =\; \backslash frac\{1\}\{a\}A\_i\backslash left(\backslash frac\{x\}\{a\}\backslash right)$ |Limit of the Airy function |- |$$

  \delta_a(x) =   

\frac{1}{a}J_{1/a} \left(\frac{x+1}{a}\right) |Limit of a Bessel function |}
Note: If δ(a,x) is a nascent delta function which is a probability distribution over the whole real line (i.e. is always non-negative between -∞ and +∞) then another nascent delta function δ_φ(a,x) can be built from its characteristic function as follows:

$\backslash delta\_\backslash varphi(a,x)=\backslash frac\{1\}\{2\backslash pi\}~\backslash frac\{\backslash varphi(1/a,x)\}\{\backslash delta(1/a,0)\}$

where

$\backslash varphi(a,k)=\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta(a,x)e^\{-ikx\}dx$

is the characteristic function of the nascent delta function δ(a,x). This result is related to the localization property of the continuous Fourier transform.

See also

• Kronecker delta
• Dirac comb

External links

• Delta Function on MathWorld
• Dirac Delta Function on PlanetMath

 

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