The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. The graph of the delta function can be thought of as following the whole x-axis and the positive y-axis.

The Dirac delta is very useful as an approximation for tall narrow spike functions. 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

1 Formal introduction 2 Fourier transform 3 Japanese definition 4 Representations of the delta function 5 See also 6 External links

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 function δ(x) with this property. The Dirac delta is not a function but 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 φ. It is a distribution with compact support (the support being {0}). Because of this definition, and the absense of a true function with the delta function's properties, its 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 \, d\delta

= f(0)<math>

for all continuous f.

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

Fourier transform

The Fourier transform of the Dirac delta is the constant function <math>\frac{1}{\sqrt{2\pi}}<math>, and the convolution of δ with any distribution S yields S.

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

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

for every test function φ. 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 <math>x_i<math> 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>

Japanese 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 function. That is, it satisfies the integral equation

<math>

\int^{x}_{-\infin} \delta (t) dt = h(x) <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}{\pi} {a \over a^2 + x^2}<math>

<math>\delta_a(x)

= \left\{ \begin{matrix} \frac{1}{2a} : -a < x < a \\ 0 : \mathrm{otherwise} \end{matrix} \right. <math>

<math>

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

<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}{\pi x}\sin\left(\frac{x}{a}\right)

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

<math>

<math>

\delta_a(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} k x-a |k|}\;dk <math>

See also

• Kronecker delta
• Dirac comb

External links

• Delta Function (http://mathworld.wolfram.com/DeltaFunction.html) on MathWorld
• Dirac Delta Function (http://planetmath.org/encyclopedia/DiracDeltaFunction.html) on PlanetMath

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