Dirac delta function

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Template:Probability distribution

The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral is one. The graph of the delta function can be usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function.)

Despite its name, the delta function is not a function as defined in the strictest mathematical sense. One reason for this is because the functions f(x) = δ(x) and g(x) = 0 are equal everywhere except at x=0 yet have integrals that are ostensibly different. According to Lebesgue integration theory, if f, g are functions such that f = g almost everywhere, then f is integrable iff g is integrable and the integrals of f and g are the same. Precise treatment of the Dirac delta requires measure theory or the theory of distributions.

The Dirac delta is very useful as an approximation for a tall narrow spike function (an impulse). It is the same type of abstraction as a point charge, point mass or electron point. For example, in calculating the dynamics of a baseball being hit by a bat, approximating the force of the bat hitting the baseball by a delta function is a helpful trick. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball.

The Dirac delta function was named after the Kronecker delta, since it can be used as a continuous analogue of the discrete Kronecker delta.

Contents

Formal introduction

The Dirac delta is often introduced with the property:

<math>\int_{-\infty}^\infty f(x) \, \delta(x) \, dx

= f(0)<math>

valid for any continuous function f.

However, there is no actual function δ(x) with this property. The Dirac delta is not a function; but it can be usefully treated as a distribution, as well as a measure.

As a distribution, the Dirac delta is defined by

<math>\delta[\phi] = \phi(0)\,<math>

for every test function <math>\phi \ <math>. It is a distribution with compact support (the support being {0}). Because of this definition, and the absence of a true function with the delta function's properties, it is important to realize the above integral notation is simply a notational convenience, and not a true integral.

As a measure, <math>\delta (A)=1<math> if <math>0\in A<math>, and <math>\delta (A)=0<math> otherwise. Then,

<math>\int_{-\infty}^\infty f(x) \, d\delta(x)

= f(0)<math>

for all continuous f.

As distributions, the Heaviside step function is an antiderivative of the Dirac delta distribution.

Fourier transform

The continuous Fourier transform of the Dirac delta is the constant function <math>\frac{1}{\sqrt{2\pi}}<math>. The inverse transform of this constant function will be the Dirac delta again, yielding the orthogonality property for the Fourier kernel:

<math>\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx}\,dx=\delta(k)<math>

From the convolution theorem for the Fourier transform, the convolution of δ with any distribution S yields S.

The Dirac delta function as a probability density function

The Dirac delta function may be interpreted as a probability density function. Its characteristic function is then just unity, as is the moment generating function, so that all moments are zero. The cumulative distribution function is the Heaviside step function.

Derivatives of the delta function

The derivative of the Dirac delta is the distribution δ' defined by

<math>\delta'[\phi] = -\phi'(0)\,<math>

for every test function <math>\phi \ <math>. From this it follows that

<math>\delta'(x)=-\frac{\delta(x)}{x}<math>

The n-th derivative δ(n) is given by

<math>\delta^{(n)}[\phi] = (-1)^n \phi^{(n)}(0)\,<math>

The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomials.

A helpful identity is

<math>\delta(g(x)) = \sum_{i}\frac{\delta(x-x_i)}{|g'(x_i)|}<math>

where xi are the roots of g(x). In the integral form it is equivalent to

<math>

\int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|} <math>

Equivalent definition

The Dirac delta function <math>\delta : \mathbb{R} \ni \xi \longrightarrow \delta ( \xi )\in \delta(\mathbb{R})<math> is a distribution <math>\delta ( \xi )<math> whose indefinite integral is the function

<math>h : \mathbb{R} \ni \xi \longrightarrow \frac{1+{\rm sgn} \, \xi }{2} \in \mathbb{R}, <math>

usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation

<math>

\int^{x}_{-\infin} \delta (t) dt = h(x) \equiv \frac{1+{\rm sgn}(x) }{2} <math>

for all real numbers x.

Representations of the delta function

The delta function can be viewed as the limit of a sequence of functions

<math>

\delta (x) = \lim_{a\to 0} \delta_a(x), <math>

where <math>\delta_a(x)<math> 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:

<math>\delta_a(x)=\frac{1}{a\sqrt{\pi}} \mathrm{e}^{-x^2/a^2}<math> 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 <math>

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<math>

Cauchy <math>\varphi<math>(see note below)
<math>\delta_a(x)= \frac{\textrm{rect}(x/a)}{a}

=\frac{1}{2\pi}\int_{-\infty}^\infty \textrm{sinc}(ak/2)e^{ikx}\,dk <math>

Limit of a rectangular function
<math>

\delta_a(x)=\frac{1}{\pi x}\sin\left(\frac{x}{a}\right)

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

<math>

rectangular function <math>\varphi<math>(see note below)
<math>

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

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

<math>

<math>

\delta_a(x)=\frac{a}{\pi x^2}\sin^2\left(\frac{x}{a}\right) <math>

<math>

\delta_a(x) = \frac{1}{a}A_i\left(\frac{x}{a}\right) <math>

Limit of the Airy function
<math>
\delta_a(x) =  

\frac{1}{a}J_{1/a} \left(\frac{x+1}{a}\right) <math>

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 δφ(a,x) can be built from its characteristic function as follows:

<math>\delta_\varphi(a,x)=\frac{1}{2\pi}~\frac{\varphi(1/a,x)}{\delta(1/a,0)}<math>

where

<math>\varphi(a,k)=\int_{-\infty}^\infty \delta(a,x)e^{-ikx}dx<math>

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

External links

es:Delta de Dirac fr:Fonction δ de Dirac he:פונקציית דלתא של דיראק it:Delta di Dirac ja:ディラックのデルタ関数 pl:Delta Diraca sl:Porazdelitev delta sv:Diracs delta-funktion

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