Dirac delta function

Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.

The Dirac delta as the limit as $a\to 0$ (in the sense of distributions) of the sequence of zero-centered normal distributions ${\delta }_a(x)=\frac{1}{\left|a\right|\sqrt{\pi }}{e}^{-(x/a{)}^2}$

In mathematical analysis, the Dirac delta function (or Template:Mvar distribution), also known as the unit impulse,Template:Sfn is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.Template:SfnTemplate:SfnTemplate:Sfn Thus it can be represented heuristically as

$$\delta (x)=\{\begin{array}{cc}0,& x\ne 0\\ \mathrm{\infty },& x=0\end{array}$$

such that

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x)dx=1.$$

Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.

The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.

Motivation and overview

The graph of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis.Template:Rp The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar abstractions such as a point charge, point mass or electron point. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).

To be specific, suppose that a billiard ball is at rest. At time $t=0$ it is struck by another ball, imparting it with a momentum Template:Mvar, with units kg⋅m⋅s⁻¹. The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The force therefore is Template:Math; the units of Template:Math are s⁻¹.

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval Template:Nowrap That is,

$$F_{\mathrm{\Delta }t}(t)=\{\begin{array}{cc}P/\mathrm{\Delta }t& 0<t\le T,\\ 0& \text{otherwise}.\end{array}$$

Then the momentum at any time Template:Mvar is found by integration:

$$p(t)=\underset{0}{\overset{t}{\int }}{F}_{\mathrm{\Delta }t}(\tau )d\tau =\{\begin{array}{cc}P& t\ge T\\ Pt/\mathrm{\Delta }t& 0\le t\le T\\ 0& \text{otherwise.}\end{array}$$

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Template:Math, giving a result everywhere except at Template:Math:

$$p(t)=\{\begin{array}{cc}P& t>0\\ 0& t<0.\end{array}$$

Here the functions $F_{\mathrm{\Delta }t}$ are thought of as useful approximations to the idea of instantaneous transfer of momentum.

The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of pointwise convergence) $\underset{\mathrm{\Delta }t\to 0^{+}}{\lim }{F}_{\mathrm{\Delta }t}$ is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{F}_{\mathrm{\Delta }t}(t)dt=P,$$

which holds for all Template:Nowrap should continue to hold in the limit. So, in the equation Template:Nowrap it is understood that the limit is always taken Template:Em.

In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers. For example, the objects Template:Math and Template:Math are equal everywhere except at Template:Math yet have integrals that are different. According to Lebesgue integration theory, if Template:Mvar and Template:Mvar are functions such that Template:Math almost everywhere, then Template:Mvar is integrable if and only if Template:Mvar is integrable and the integrals of Template:Mvar and Template:Mvar are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions.

History

In physics, the Dirac delta function was popularized by Paul Dirac in this book The Principles of Quantum Mechanics published in 1930.Template:Sfn However, Oliver Heaviside, 35 years before Dirac, described an impulsive function called the Heaviside step function for purposes and with properties analogous to Dirac's work. Even earlier several mathematicians and physicists used limits of sharply peaked functions in derivations. An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin-Louis Cauchy.Template:Sfn Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of the Quantum Dynamics. He called it the "delta function" since he used it as a continuum analogue of the discrete Kronecker delta.

Mathematicians refer to the same concept as a distribution rather than a function.Template:Rp Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:

$$f(x)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}d\alpha f(\alpha )\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dp\cos (px-p\alpha ),$$

which is tantamount to the introduction of the Template:Mvar-function in the form:

$$\delta (x-\alpha )=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dp\cos (px-p\alpha ).$$

Later, Augustin Cauchy expressed the theorem using exponentials:

$$f(x)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{ipx}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-ip\alpha }f(\alpha )d\alpha )dp.$$

Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).

As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as

$$\begin{array}{cc}f(x)& =\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{ipx}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-ip\alpha }f(\alpha )d\alpha )dp\\ & =\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{ipx}{e}^{-ip\alpha }dp\right)f(\alpha )d\alpha =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x-\alpha )f(\alpha )d\alpha ,\end{array}$$

where the δ-function is expressed as

$$\delta (x-\alpha )=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{ip(x-\alpha )}dp.$$

A rigorous interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:

The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.

Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L²-theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945) ...", and leading to the formal development of the Dirac delta function.

Definitions

The Dirac delta function $\delta (x)$ can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

$$\delta (x)\simeq \{\begin{array}{cc}+\mathrm{\infty },& x=0\\ 0,& x\ne 0\end{array}$$

and which is also constrained to satisfy the identityTemplate:Sfn

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x)dx=1.$$

This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no extended real number valued function defined on the real numbers has these properties.Template:Sfn

As a measure

One way to rigorously capture the notion of the Dirac delta function is to define a measure, called Dirac measure, which accepts a subset Template:Mvar of the real line Template:Math as an argument, and returns Template:Math if Template:Math, and Template:Math otherwise. If the delta function is conceptualized as modeling an idealized point mass at 0, then Template:Math represents the mass contained in the set Template:Mvar. One may then define the integral against Template:Mvar as the integral of a function against this mass distribution. Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure Template:Mvar satisfies

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\delta (dx)=f(0)$$

for all continuous compactly supported functions Template:Mvar. The measure Template:Mvar is not absolutely continuous with respect to the Lebesgue measure—in fact, it is a singular measure. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\delta (x)dx=f(0)$$

holds.Template:Sfn As a result, the latter notation is a convenient abuse of notation, and not a standard (Riemann or Lebesgue) integral.

As a probability measure on Template:Math, the delta measure is characterized by its cumulative distribution function, which is the unit step function.

$$H(x)=\{\begin{array}{cc}1& \text{if }x\ge 0\\ 0& \text{if }x<0.\end{array}$$

This means that Template:Math is the integral of the cumulative indicator function Template:Math with respect to the measure Template:Mvar; to wit,

$$H(x)=\underset{𝐑}{\int }{\mathrm{𝟏}}_{(-\mathrm{\infty },x]}(t)\delta (dt)=\delta ((-\mathrm{\infty },x]),$$

the latter being the measure of this interval. Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:Template:Sfn

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\delta (dx)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dH(x).$$

All higher moments of Template:Mvar are zero. In particular, characteristic function and moment generating function are both equal to one.

As a distribution

In the theory of distributions, a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them.Template:Sfn In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" test function Template:Mvar. Test functions are also known as bump functions. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.

A typical space of test functions consists of all smooth functions on Template:Math with compact support that have as many derivatives as required. As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined byTemplate:Sfn

for every test function Template:Mvar.

For Template:Mvar to be properly a distribution, it must be continuous in a suitable topology on the space of test functions. In general, for a linear functional Template:Mvar on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer Template:Mvar there is an integer Template:Math and a constant Template:Mvar such that for every test function Template:Mvar, one has the inequalityTemplate:Sfn

$$|S[\phi ]|\le C_N\sum _{k=0}^{M_N}\underset{x\in [-N,N]}{\sup }|{\phi }^{(k)}(x)|$$

where Template:Math represents the supremum. With the Template:Mvar distribution, one has such an inequality (with Template:Math with Template:Math for all Template:Mvar. Thus Template:Mvar is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being Template:Math).

The delta distribution can also be defined in several equivalent ways. For instance, it is the distributional derivative of the Heaviside step function. This means that for every test function Template:Mvar, one has

$$\delta [\phi ]=-\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\phi }^{\prime }(x)H(x)dx.$$

Intuitively, if integration by parts were permitted, then the latter integral should simplify to

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\phi (x)H^{\prime }(x)dx=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\phi (x)\delta (x)dx,$$

and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have

$$-\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\phi }^{\prime }(x)H(x)dx=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\phi (x)dH(x).$$

In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation (Template:EquationNote) defines a Daniell integral on the space of all compactly supported continuous functions Template:Mvar which, by the Riesz representation theorem, can be represented as the Lebesgue integral of Template:Mvar with respect to some Radon measure.

Generally, when the term Dirac delta function is used, it is in the sense of distributions rather than measures, the Dirac measure being among several terms for the corresponding notion in measure theory. Some sources may also use the term Dirac delta distribution.

Generalizations

The delta function can be defined in Template:Mvar-dimensional Euclidean space Template:Math as the measure such that

$$\underset{{𝐑}^n}{\int }f(𝐱)\delta (d𝐱)=f(\mathrm{𝟎})$$

for every compactly supported continuous function Template:Mvar. As a measure, the Template:Mvar-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with Template:Math, one hasTemplate:Sfn

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.Template:Sfn However, despite widespread use in engineering contexts, (Template:EquationNote) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.Template:SfnTemplate:Sfn

The notion of a Dirac measure makes sense on any set.Template:Sfn Thus if Template:Mvar is a set, Template:Math is a marked point, and Template:Math is any sigma algebra of subsets of Template:Mvar, then the measure defined on sets Template:Math by

$${\delta }_{x_0}(A)=\{\begin{array}{cc}1& \text{if }{x}_0\in A\\ 0& \text{if }{x}_0\notin A\end{array}$$

is the delta measure or unit mass concentrated at Template:Math.

Another common generalization of the delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the differentiable structure. The delta function on a manifold Template:Mvar centered at the point Template:Math is defined as the following distribution:

for all compactly supported smooth real-valued functions Template:Mvar on Template:Mvar.Template:Sfn A common special case of this construction is a case in which Template:Mvar is an open set in the Euclidean space Template:Math.

On a locally compact Hausdorff space Template:Mvar, the Dirac delta measure concentrated at a point Template:Mvar is the Radon measure associated with the Daniell integral (Template:EquationNote) on compactly supported continuous functions Template:Mvar. At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping $x_0\mapsto {\delta }_{x_0}$ is a continuous embedding of Template:Mvar into the space of finite Radon measures on Template:Mvar, equipped with its vague topology. Moreover, the convex hull of the image of Template:Mvar under this embedding is dense in the space of probability measures on Template:Mvar.Template:Sfn

Properties

Scaling and symmetry

The delta function satisfies the following scaling property for a non-zero scalar Template:Mvar:Template:SfnTemplate:Sfn

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (\alpha x)dx=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (u)\frac{du}{|\alpha |}=\frac{1}{|\alpha |}$$

and so Template:NumBlk2

Scaling property proof: $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dxg(x)\delta (ax)=\frac{1}{a}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}d{x}^{\prime }g\left(\frac{x^{\prime }}{a}\right)\delta (x^{\prime })=\frac{1}{a}g(0).$$ where a change of variable Template:Math is used. If Template:Mvar is negative, i.e., Template:Math, then $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dxg(x)\delta (ax)=\frac{1}{-\left|a\right|}\underset{\mathrm{\infty }}{\overset{-\mathrm{\infty }}{\int }}d{x}^{\prime }g\left(\frac{x^{\prime }}{a}\right)\delta (x^{\prime })=\frac{1}{\left|a\right|}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}d{x}^{\prime }g\left(\frac{x^{\prime }}{a}\right)\delta (x^{\prime })=\frac{1}{\left|a\right|}g(0).$$ Thus, Template:Nowrap

In particular, the delta function is an even distribution (symmetry), in the sense that

$$\delta (-x)=\delta (x)$$

which is homogeneous of degree Template:Math.

Algebraic properties

The distributional product of Template:Mvar with Template:Mvar is equal to zero:

$$x\delta (x)=0.$$

More generally, $(x-a{)}^n\delta (x-a)=0$ for all positive integers $n$.

Conversely, if Template:Math, where Template:Mvar and Template:Mvar are distributions, then

$$f(x)=g(x)+c\delta (x)$$

for some constant Template:Mvar.Template:Sfn

Translation

The integral of any function multiplied by the time-delayed Dirac delta ${\delta }_T(t)=\delta (t-T)$ is

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)\delta (t-T)dt=f(T).$$

This is sometimes referred to as the sifting property or the sampling property. The delta function is said to "sift out" the value of f(t) at t = T.

It follows that the effect of convolving a function Template:Math with the time-delayed Dirac delta is to time-delay Template:Math by the same amount:

$$\begin{array}{cc}(f*{\delta }_T)(t)& \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )\delta (t-T-\tau )d\tau \\ & =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )\delta (\tau -(t-T))d\tau \text{since}\delta (-x)=\delta (x)\text{by (4)}\\ & =f(t-T).\end{array}$$

The sifting property holds under the precise condition that Template:Mvar be a tempered distribution (see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution sense)

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (\xi -x)\delta (x-\eta )dx=\delta (\eta -\xi ).$$

Composition with a function

More generally, the delta distribution may be composed with a smooth function Template:Math in such a way that the familiar change of variables formula holds (where $u=g(x)$), that

$$\underset{ℝ}{\int }\delta (g(x))f(g(x))|g^{\prime }(x)|dx=\underset{g(ℝ)}{\int }\delta (u)f(u)du$$

provided that Template:Mvar is a continuously differentiable function with Template:Math nowhere zero.Template:Sfn That is, there is a unique way to assign meaning to the distribution $\delta \circ g$ so that this identity holds for all compactly supported test functions Template:Mvar. Therefore, the domain must be broken up to exclude the Template:Math point. This distribution satisfies Template:Math if Template:Mvar is nowhere zero, and otherwise if Template:Mvar has a real root at Template:Math, then

$$\delta (g(x))=\frac{\delta (x-x_0)}{|g^{\prime }(x_0)|}.$$

It is natural therefore to Template:Em the composition Template:Math for continuously differentiable functions Template:Mvar by

$$\delta (g(x))=\sum _i\frac{\delta (x-x_i)}{|g^{\prime }(x_i)|}$$

where the sum extends over all roots of Template:Mvar, which are assumed to be simple. Thus, for example

$$\delta (x^2-{\alpha }^2)=\frac{1}{2|\alpha |}[\delta (x+\alpha )+\delta (x-\alpha )].$$

In the integral form, the generalized scaling property may be written as

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\delta (g(x))dx=\sum _i\frac{f(x_i)}{|g^{\prime }(x_i)|}.$$

Indefinite integral

For a constant $a\in ℝ$ and a "well-behaved" arbitrary real-valued function Template:Math, $$\int y(x)\delta (x-a)dx=y(a)H(x-a)+c,$$ where Template:Math is the Heaviside step function and Template:Math is an integration constant.

Properties in n dimensions

The delta distribution in an Template:Mvar-dimensional space satisfies the following scaling property instead, $$\delta (\alpha 𝒙)=|\alpha {|}^{-n}\delta (𝒙),$$ so that Template:Mvar is a homogeneous distribution of degree Template:Math.

Under any reflection or rotation Template:Mvar, the delta function is invariant, $$\delta (\rho 𝒙)=\delta (𝒙).$$

As in the one-variable case, it is possible to define the composition of Template:Mvar with a bi-Lipschitz function Template:Math uniquely so that the following holds $$\underset{{ℝ}^n}{\int }\delta (g(𝒙))f(g(𝒙))|\det g^{\prime }(𝒙)|d𝒙=\underset{g({ℝ}^n)}{\int }\delta (𝒖)f(𝒖)d𝒖$$ for all compactly supported functions Template:Mvar.

Using the coarea formula from geometric measure theory, one can also define the composition of the delta function with a submersion from one Euclidean space to another one of different dimension; the result is a type of current. In the special case of a continuously differentiable function Template:Math such that the gradient of Template:Mvar is nowhere zero, the following identity holdsTemplate:Sfn $$\underset{{ℝ}^n}{\int }f(𝒙)\delta (g(𝒙))d𝒙=\underset{g^{-1}(0)}{\int }\frac{f(𝒙)}{|\mathrm{\nabla }g|}d\sigma (𝒙)$$ where the integral on the right is over Template:Math, the Template:Math-dimensional surface defined by Template:Math with respect to the Minkowski content measure. This is known as a simple layer integral.

More generally, if Template:Mvar is a smooth hypersurface of Template:Math, then we can associate to Template:Mvar the distribution that integrates any compactly supported smooth function Template:Mvar over Template:Mvar: $${\delta }_S[g]=\underset{S}{\int }g(𝒔)d\sigma (𝒔)$$

where Template:Mvar is the hypersurface measure associated to Template:Mvar. This generalization is associated with the potential theory of simple layer potentials on Template:Mvar. If Template:Mvar is a domain in Template:Math with smooth boundary Template:Mvar, then Template:Math is equal to the normal derivative of the indicator function of Template:Mvar in the distribution sense,

$$-\underset{{ℝ}^n}{\int }g(𝒙)\frac{\partial 1_D(𝒙)}{\partial n}d𝒙=\underset{S}{\int }g(𝒔)d\sigma (𝒔),$$

where Template:Mvar is the outward normal.Template:SfnTemplate:Sfn For a proof, see e.g. the article on the surface delta function.

In three dimensions, the delta function is represented in spherical coordinates by:

$$\delta (𝒓-{𝒓}_0)=\{\begin{array}{cc}\frac{1}{r^2\sin \theta }\delta (r-r_0)\delta (\theta -{\theta }_0)\delta (\varphi -{\varphi }_0)& x_0,y_0,z_0\ne 0\\ \frac{1}{2\pi r^2\sin \theta }\delta (r-r_0)\delta (\theta -{\theta }_0)& x_0=y_0=0,z_0\ne 0\\ \frac{1}{4\pi r^2}\delta (r-r_0)& x_0=y_0=z_0=0\end{array}$$

Derivatives

The derivative of the Dirac delta distribution, denoted Template:Math and also called the Dirac delta prime or Dirac delta derivative as described in Laplacian of the indicator, is defined on compactly supported smooth test functions Template:Mvar byTemplate:Sfn $${\delta }^{\prime }[\phi ]=-\delta [{\phi }^{\prime }]=-{\phi }^{\prime }(0).$$

The first equality here is a kind of integration by parts, for if Template:Mvar were a true function then $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\delta }^{\prime }(x)\phi (x)dx=\delta (x)\phi (x){|}_{-\mathrm{\infty }}^{\mathrm{\infty }}-\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x){\phi }^{\prime }(x)dx=-\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x){\phi }^{\prime }(x)dx=-{\phi }^{\prime }(0).$$

By mathematical induction, the Template:Mvar-th derivative of Template:Mvar is defined similarly as the distribution given on test functions by

$${\delta }^{(k)}[\phi ]=(-1{)}^k{\phi }^{(k)}(0).$$

In particular, Template:Mvar is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:Template:Sfn $${\delta }^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\delta (x+h)-\delta (x)}{h}.$$

More properly, one has $${\delta }^{\prime }=\underset{h\to 0}{\lim }\frac{1}{h}({\tau }_h\delta -\delta )$$ where Template:Mvar is the translation operator, defined on functions by Template:Math, and on a distribution Template:Mvar by $$({\tau }_{h}S)[\phi ]=S[{\tau }_{-h}\phi ].$$

In the theory of electromagnetism, the first derivative of the delta function represents a point magnetic dipole situated at the origin. Accordingly, it is referred to as a dipole or the doublet function.

The derivative of the delta function satisfies a number of basic properties, including:Template:Sfn $$\begin{array}{cc}{\delta }^{\prime }(-x)& =-{\delta }^{\prime }(x)\\ x{\delta }^{\prime }(x)& =-\delta (x)\end{array}$$ which can be shown by applying a test function and integrating by parts.

The latter of these properties can also be demonstrated by applying distributional derivative definition, Leibniz 's theorem and linearity of inner product:

$$\begin{array}{cc}⟨x{\delta }^{\prime },\phi ⟩& =⟨{\delta }^{\prime },x\phi ⟩=-⟨\delta ,(x\phi {)}^{\prime }⟩=-⟨\delta ,x^{\prime }\phi +x{\phi }^{\prime }⟩=-⟨\delta ,x^{\prime }\phi ⟩-⟨\delta ,x{\phi }^{\prime }⟩=-x^{\prime }(0)\phi (0)-x(0){\phi }^{\prime }(0)\\ & =-x^{\prime }(0)⟨\delta ,\phi ⟩-x(0)⟨\delta ,{\phi }^{\prime }⟩=-x^{\prime }(0)⟨\delta ,\phi ⟩+x(0)⟨{\delta }^{\prime },\phi ⟩=⟨x(0){\delta }^{\prime }-x^{\prime }(0)\delta ,\phi ⟩\\ ⟹x(t){\delta }^{\prime }(t)& =x(0){\delta }^{\prime }(t)-x^{\prime }(0)\delta (t)=-x^{\prime }(0)\delta (t)=-\delta (t)\end{array}$$

Furthermore, the convolution of Template:Mvar with a compactly-supported, smooth function Template:Mvar is

$${\delta }^{\prime }*f=\delta *f^{\prime }=f^{\prime },$$

which follows from the properties of the distributional derivative of a convolution.

Higher dimensions

More generally, on an open set Template:Mvar in the Template:Mvar-dimensional Euclidean space ${ℝ}^n$, the Dirac delta distribution centered at a point Template:Math is defined byTemplate:Sfn $${\delta }_a[\phi ]=\phi (a)$$ for all $\phi \in C_c^{\mathrm{\infty }}(U)$, the space of all smooth functions with compact support on Template:Mvar. If $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ is any multi-index with $|\alpha |={\alpha }_1+\cdots +{\alpha }_n$ and ${\partial }^{\alpha }$ denotes the associated mixed partial derivative operator, then the Template:Mvar-th derivative Template:Mvar of Template:Mvar is given byTemplate:Sfn

$$⟨{\partial }^{\alpha }{\delta }_a,\phi ⟩=(-1{)}^{|\alpha |}⟨{\delta }_a,{\partial }^{\alpha }\phi ⟩=(-1{)}^{|\alpha |}{\partial }^{\alpha }\phi (x){|}_{x=a}\text{ for all }\phi \in C_c^{\mathrm{\infty }}(U).$$

That is, the Template:Mvar-th derivative of Template:Mvar is the distribution whose value on any test function Template:Mvar is the Template:Mvar-th derivative of Template:Mvar at Template:Mvar (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as double layers along the coordinate planes. More generally, the normal derivative of a simple layer supported on a surface is a double layer supported on that surface and represents a laminar magnetic monopole. Higher derivatives of the delta function are known in physics as multipoles.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If Template:Mvar is any distribution on Template:Mvar supported on the set Template:Math consisting of a single point, then there is an integer Template:Mvar and coefficients Template:Mvar such thatTemplate:SfnTemplate:Sfn $$S=\sum _{|\alpha |\le m}{c}_{\alpha }{\partial }^{\alpha }{\delta }_a.$$

Representations

Nascent delta function

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

$$\delta (x)=\underset{\epsilon \to 0^{+}}{\lim }{\eta }_{\epsilon }(x),$$

where Template:Math is sometimes called a nascent delta functionTemplate:Anchor. This limit is meant in a weak sense: either that

for all continuous functions Template:Mvar having compact support, or that this limit holds for all smooth functions Template:Mvar with compact support. The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the vague topology of measures, and the latter is convergence in the sense of distributions.

Approximations to the identity

Typically a nascent delta function Template:Mvar can be constructed in the following manner. Let Template:Mvar be an absolutely integrable function on Template:Math of total integral Template:Math, and define $${\eta }_{\epsilon }(x)={\epsilon }^{-1}\eta \left(\frac{x}{\epsilon }\right).$$

In Template:Mvar dimensions, one uses instead the scaling $${\eta }_{\epsilon }(x)={\epsilon }^{-n}\eta \left(\frac{x}{\epsilon }\right).$$

Then a simple change of variables shows that Template:Mvar also has integral Template:Math. One may show that (Template:EquationNote) holds for all continuous compactly supported functions Template:Mvar,Template:Sfn and so Template:Mvar converges weakly to Template:Mvar in the sense of measures.

The Template:Mvar constructed in this way are known as an approximation to the identity.Template:Sfn This terminology is because the space Template:Math of absolutely integrable functions is closed under the operation of convolution of functions: Template:Math whenever Template:Mvar and Template:Mvar are in Template:Math. However, there is no identity in Template:Math for the convolution product: no element Template:Mvar such that Template:Math for all Template:Mvar. Nevertheless, the sequence Template:Mvar does approximate such an identity in the sense that

$$f*{\eta }_{\epsilon }\to f\text{as }\epsilon \to 0.$$

This limit holds in the sense of mean convergence (convergence in Template:Math). Further conditions on the Template:Mvar, for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence almost everywhere.

If the initial Template:Math is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing Template:Mvar to be a suitably normalized bump function, for instance

$$\eta (x)=\{\begin{array}{cc}\frac{1}{I_n}\exp (-\frac{1}{1-|x{|}^2})& \text{if }|x|<1\\ 0& \text{if }|x|\ge 1.\end{array}$$ ($I_n$ ensuring that the total integral is 1).

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking Template:Math to be a hat function. With this choice of Template:Math, one has

$${\eta }_{\epsilon }(x)={\epsilon }^{-1}\max (1-\left|\frac{x}{\epsilon }\right|,0)$$

which are all continuous and compactly supported, although not smooth and so not a mollifier.

Probabilistic considerations

In the context of probability theory, it is natural to impose the additional condition that the initial Template:Math in an approximation to the identity should be positive, as such a function then represents a probability distribution. Convolution with a probability distribution is sometimes favorable because it does not result in overshoot or undershoot, as the output is a convex combination of the input values, and thus falls between the maximum and minimum of the input function. Taking Template:Math to be any probability distribution at all, and letting Template:Math as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, Template:Mvar has mean Template:Math and has small higher moments. For instance, if Template:Math is the uniform distribution on Template:Nowrap also known as the rectangular function, then:Template:Sfn $${\eta }_{\epsilon }(x)=\frac{1}{\epsilon }\mathrm{rect}\left(\frac{x}{\epsilon }\right)=\{\begin{array}{cc}\frac{1}{\epsilon },& -\frac{\epsilon }{2}<x<\frac{\epsilon }{2},\\ 0,& \text{otherwise}.\end{array}$$

Another example is with the Wigner semicircle distribution $${\eta }_{\epsilon }(x)=\{\begin{array}{cc}\frac{2}{\pi {\epsilon }^2}\sqrt{{\epsilon }^2-x^2},& -\epsilon <x<\epsilon ,\\ 0,& \text{otherwise}.\end{array}$$

This is continuous and compactly supported, but not a mollifier because it is not smooth.

Semigroups

Nascent delta functions often arise as convolution semigroups. This amounts to the further constraint that the convolution of Template:Mvar with Template:Mvar must satisfy $${\eta }_{\epsilon }*{\eta }_{\delta }={\eta }_{\epsilon +\delta }$$

for all Template:Math. Convolution semigroups in Template:Math that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction.

In practice, semigroups approximating the delta function arise as fundamental solutions or Green's functions to physically motivated elliptic or parabolic partial differential equations. In the context of applied mathematics, semigroups arise as the output of a linear time-invariant system. Abstractly, if A is a linear operator acting on functions of x, then a convolution semigroup arises by solving the initial value problem

$$\{\begin{array}{c}\frac{\partial }{\partial t}\eta (t,x)=A\eta (t,x),t>0\\ \underset{t\to 0^{+}}{\lim }\eta (t,x)=\delta (x)\end{array}$$

in which the limit is as usual understood in the weak sense. Setting Template:Math gives the associated nascent delta function.

Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.

The heat kernel

The heat kernel, defined by

$${\eta }_{\epsilon }(x)=\frac{1}{\sqrt{2\pi \epsilon }}{\mathrm{e}}^{-\frac{x^2}{2\epsilon }}$$

represents the temperature in an infinite wire at time Template:Math, if a unit of heat energy is stored at the origin of the wire at time Template:Math. This semigroup evolves according to the one-dimensional heat equation:

$$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{{\partial }^{2}u}{\partial x^2}.$$

In probability theory, Template:Math is a normal distribution of variance Template:Mvar and mean Template:Math. It represents the probability density at time Template:Math of the position of a particle starting at the origin following a standard Brownian motion. In this context, the semigroup condition is then an expression of the Markov property of Brownian motion.

In higher-dimensional Euclidean space Template:Math, the heat kernel is $${\eta }_{\epsilon }=\frac{1}{(2\pi \epsilon {)}^{n/2}}{\mathrm{e}}^{-\frac{x\cdot x}{2\epsilon }},$$ and has the same physical interpretation, Template:Lang. It also represents a nascent delta function in the sense that Template:Math in the distribution sense as Template:Math.

The Poisson kernel

The Poisson kernel $${\eta }_{\epsilon }(x)=\frac{1}{\pi }\mathrm{I}\mathrm{m}\left\{\frac{1}{x-\mathrm{i}\epsilon }\right\}=\frac{1}{\pi }\frac{\epsilon }{{\epsilon }^2+x^2}=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\mathrm{e}}^{\mathrm{i}\xi x-|\epsilon \xi |}d\xi$$

is the fundamental solution of the Laplace equation in the upper half-plane.Template:Sfn It represents the electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the Cauchy distribution and Epanechnikov and Gaussian kernel functions. This semigroup evolves according to the equation $$\frac{\partial u}{\partial t}=-{(-\frac{{\partial }^2}{\partial x^2})}^{\frac{1}{2}}u(t,x)$$

where the operator is rigorously defined as the Fourier multiplier $$ℱ\left[{(-\frac{{\partial }^2}{\partial x^2})}^{\frac{1}{2}}f\right](\xi )=|2\pi \xi |ℱf(\xi ).$$

Oscillatory integrals

In areas of physics such as wave propagation and wave mechanics, the equations involved are hyperbolic and so may have more singular solutions. As a result, the nascent delta functions that arise as fundamental solutions of the associated Cauchy problems are generally oscillatory integrals. An example, which comes from a solution of the Euler–Tricomi equation of transonic gas dynamics,Template:Sfn is the rescaled Airy function $${\epsilon }^{-1/3}\mathrm{Ai}\left(x{\epsilon }^{-1/3}\right).$$

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the Dirichlet kernel below), rather than in the sense of measures.

Another example is the Cauchy problem for the wave equation in Template:Math:Template:Sfn $$\begin{array}{cc}{c}^{-2}\frac{{\partial }^{2}u}{\partial t^2}-\mathrm{\Delta }u& =0\\ u=0,\frac{\partial u}{\partial t}=\delta & \text{for }t=0.\end{array}$$

The solution Template:Mvar represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin.

Other approximations to the identity of this kind include the sinc function (used widely in electronics and telecommunications) $${\eta }_{\epsilon }(x)=\frac{1}{\pi x}\sin \left(\frac{x}{\epsilon }\right)=\frac{1}{2\pi }\underset{-\frac{1}{\epsilon }}{\overset{\frac{1}{\epsilon }}{\int }}\cos (kx)dk$$

and the Bessel function $${\eta }_{\epsilon }(x)=\frac{1}{\epsilon }{J}_{\frac{1}{\epsilon }}\left(\frac{x+1}{\epsilon }\right).$$

Plane wave decomposition

One approach to the study of a linear partial differential equation $$L[u]=f,$$

where Template:Mvar is a differential operator on Template:Math, is to seek first a fundamental solution, which is a solution of the equation $$L[u]=\delta .$$

When Template:Mvar is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form $$L[u]=h$$

where Template:Mvar is a plane wave function, meaning that it has the form $$h=h(x\cdot \xi )$$

for some vector Template:Mvar. Such an equation can be resolved (if the coefficients of Template:Mvar are analytic functions) by the Cauchy–Kovalevskaya theorem or (if the coefficients of Template:Mvar are constant) by quadrature. So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations.

Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by Johann Radon, and then developed in this form by Fritz John (1955).Template:Sfn Choose Template:Mvar so that Template:Math is an even integer, and for a real number Template:Mvar, put $$g(s)=\mathrm{Re}\left[\frac{-s^k\log (-is)}{k!(2\pi i{)}^n}\right]=\{\begin{array}{cc}\frac{|s{|}^k}{4k!(2\pi i{)}^{n-1}}& n\text{ odd}\\ -\frac{|s{|}^k\log |s|}{k!(2\pi i{)}^n}& n\text{ even.}\end{array}$$

Then Template:Mvar is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure Template:Mvar of Template:Math for Template:Mvar in the unit sphere Template:Math: $$\delta (x)={\mathrm{\Delta }}_x^{(n+k)/2}\underset{S^{n-1}}{\int }g(x\cdot \xi )d{\omega }_{\xi }.$$

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function Template:Mvar, $$\phi (x)=\underset{{𝐑}^n}{\int }\phi (y)dy{\mathrm{\Delta }}_x^{\frac{n+k}{2}}\underset{S^{n-1}}{\int }g((x-y)\cdot \xi )d{\omega }_{\xi }.$$

The result follows from the formula for the Newtonian potential (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the Radon transform because it recovers the value of Template:Math from its integrals over hyperplanes. For instance, if Template:Mvar is odd and Template:Math, then the integral on the right hand side is $$\begin{array}{cc}& c_n{\mathrm{\Delta }}_x^{\frac{n+1}{2}}\underset{S^{n-1}}{\iint }\phi (y)|(y-x)\cdot \xi |d{\omega }_{\xi }dy\\ & =c_n{\mathrm{\Delta }}_x^{(n+1)/2}\underset{S^{n-1}}{\int }d{\omega }_{\xi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|p|R\phi (\xi ,p+x\cdot \xi )dp\end{array}$$

where Template:Math is the Radon transform of Template:Mvar: $$R\phi (\xi ,p)=\underset{x\cdot \xi =p}{\int }f(x)d^{n-1}x.$$

An alternative equivalent expression of the plane wave decomposition is:Template:Sfn $$\delta (x)=\{\begin{array}{cc}\frac{(n-1)!}{(2\pi i{)}^n}\underset{S^{n-1}}{\int }(x\cdot \xi {)}^{-n}d{\omega }_{\xi }& n\text{ even}\\ \frac{1}{2(2\pi i{)}^{n-1}}\underset{S^{n-1}}{\int }{\delta }^{(n-1)}(x\cdot \xi )d{\omega }_{\xi }& n\text{ odd}.\end{array}$$

Fourier transform

The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds

$$\hat{\delta }(\xi )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-2\pi ix\xi }\delta (x)dx=1.$$

Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing $⟨\cdot ,\cdot ⟩$ of tempered distributions with Schwartz functions. Thus $\hat{\delta }$ is defined as the unique tempered distribution satisfying

$$⟨\hat{\delta },\phi ⟩=⟨\delta ,\hat{\phi }⟩$$

for all Schwartz functions Template:Mvar. And indeed it follows from this that $\hat{\delta }=1.$

As a result of this identity, the convolution of the delta function with any other tempered distribution Template:Mvar is simply Template:Mvar:

$$S*\delta =S.$$

That is to say that Template:Mvar is an identity element for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an associative algebra with identity the delta function. This property is fundamental in signal processing, as convolution with a tempered distribution is a linear time-invariant system, and applying the linear time-invariant system measures its impulse response. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for Template:Mvar, and once it is known, it characterizes the system completely. See Template:Section link.

The inverse Fourier transform of the tempered distribution Template:Math is the delta function. Formally, this is expressed as $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}1\cdot e^{2\pi ix\xi }d\xi =\delta (x)$$ and more rigorously, it follows since $$⟨1,\hat{f}⟩=f(0)=⟨\delta ,f⟩$$ for all Schwartz functions Template:Mvar.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on Template:Math. Formally, one has $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{i2\pi {\xi }_{1}t}{\left[e^{i2\pi {\xi }_{2}t}\right]}^{*}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-i2\pi ({\xi }_2-{\xi }_1)t}dt=\delta ({\xi }_2-{\xi }_1).$$

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution $$f(t)=e^{i2\pi {\xi }_{1}t}$$ is $$\hat{f}({\xi }_2)=\delta ({\xi }_1-{\xi }_2)$$ which again follows by imposing self-adjointness of the Fourier transform.

By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to beTemplate:Sfn $$\underset{0}{\overset{\mathrm{\infty }}{\int }}\delta (t-a)e^{-st}dt=e^{-sa}.$$

Fourier kernels

Template:See also In the study of Fourier series, a major question consists of determining whether and in what sense the Fourier series associated with a periodic function converges to the function. The Template:Mvar-th partial sum of the Fourier series of a function Template:Mvar of period Template:Math is defined by convolution (on the interval Template:Closed-closed) with the Dirichlet kernel: $$D_N(x)=\sum _{n=-N}^N{e}^{inx}=\frac{\sin \left((N+\frac{1}{2})x\right)}{\sin (x/2)}.$$ Thus, $$s_N(f)(x)=D_N*f(x)=\sum _{n=-N}^N{a}_n{e}^{inx}$$ where $$a_n=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}f(y)e^{-iny}dy.$$ A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval Template:Closed-closed tends to a multiple of the delta function as Template:Math. This is interpreted in the distribution sense, that $$s_N(f)(0)=\underset{-\pi }{\overset{\pi }{\int }}{D}_N(x)f(x)dx\to 2\pi f(0)$$ for every compactly supported Template:Em function Template:Mvar. Thus, formally one has $$\delta (x)=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{inx}$$ on the interval Template:Closed-closed.

Despite this, the result does not hold for all compactly supported Template:Em functions: that is Template:Math does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of summability methods to produce convergence. The method of Cesàro summation leads to the Fejér kernelTemplate:Sfn

$$F_N(x)=\frac{1}{N}\sum _{n=0}^{N-1}{D}_n(x)=\frac{1}{N}{\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)}^2.$$

The Fejér kernels tend to the delta function in a stronger sense that

$$\underset{-\pi }{\overset{\pi }{\int }}{F}_N(x)f(x)dx\to 2\pi f(0)$$

for every compactly supported Template:Em function Template:Mvar. The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point.

Hilbert space theory

The Dirac delta distribution is a densely defined unbounded linear functional on the Hilbert space L² of square-integrable functions. Indeed, smooth compactly supported functions are dense in Template:Math, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of Template:Math and to give a stronger topology on which the delta function defines a bounded linear functional.

Sobolev spaces

The Sobolev embedding theorem for Sobolev spaces on the real line Template:Math implies that any square-integrable function Template:Mvar such that

$$\Vert f{\Vert }_{H^1}^2=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|\hat{f}(\xi ){|}^2(1+|\xi {|}^2)d\xi <\mathrm{\infty }$$

is automatically continuous, and satisfies in particular

$$\delta [f]=|f(0)|<C\Vert f{\Vert }_{H^1}.$$

Thus Template:Mvar is a bounded linear functional on the Sobolev space Template:Math. Equivalently Template:Mvar is an element of the continuous dual space Template:Math of Template:Math. More generally, in Template:Mvar dimensions, one has Template:Math provided Template:Math.

Spaces of holomorphic functions

In complex analysis, the delta function enters via Cauchy's integral formula, which asserts that if Template:Mvar is a domain in the complex plane with smooth boundary, then

$$f(z)=\frac{1}{2\pi i}{\oint }_{\partial D}\frac{f(\zeta )d\zeta }{\zeta -z},z\in D$$

for all holomorphic functions Template:Mvar in Template:Mvar that are continuous on the closure of Template:Mvar. As a result, the delta function Template:Math is represented in this class of holomorphic functions by the Cauchy integral:

$${\delta }_z[f]=f(z)=\frac{1}{2\pi i}{\oint }_{\partial D}\frac{f(\zeta )d\zeta }{\zeta -z}.$$

Moreover, let Template:Math be the Hardy space consisting of the closure in Template:Math of all holomorphic functions in Template:Mvar continuous up to the boundary of Template:Mvar. Then functions in Template:Math uniquely extend to holomorphic functions in Template:Mvar, and the Cauchy integral formula continues to hold. In particular for Template:Math, the delta function Template:Mvar is a continuous linear functional on Template:Math. This is a special case of the situation in several complex variables in which, for smooth domains Template:Mvar, the Szegő kernel plays the role of the Cauchy integral.Template:Sfn

Another representation of the delta function in a space of holomorphic functions is on the space $H(D)\cap L^2(D)$ of square-integrable holomorphic functions in an open set $D\subset {ℂ}^n$. This is a closed subspace of $L^2(D)$, and therefore is a Hilbert space. On the other hand, the functional that evaluates a holomorphic function in $H(D)\cap L^2(D)$ at a point $z$ of $D$ is a continuous functional, and so by the Riesz representation theorem, is represented by integration against a kernel $K_z(\zeta )$, the Bergman kernel. This kernel is the analog of the delta function in this Hilbert space. A Hilbert space having such a kernel is called a reproducing kernel Hilbert space. In the special case of the unit disc, one has $${\delta }_w[f]=f(w)=\frac{1}{\pi }\underset{|z|<1}{\iint }\frac{f(z)dxdy}{(1-\overline{z}w{)}^2}.$$

Resolutions of the identity

Given a complete orthonormal basis set of functions Template:Math in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector Template:Mvar can be expressed as $$f=\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{\phi }_n.$$ The coefficients {α_n} are found as $${\alpha }_n=⟨{\phi }_n,f⟩,$$ which may be represented by the notation: $${\alpha }_n={\phi }_n^{†}f,$$ a form of the bra–ket notation of Dirac. Adopting this notation, the expansion of Template:Mvar takes the dyadic form:Template:Sfn

$$f=\sum _{n=1}^{\mathrm{\infty }}{\phi }_n\left({\phi }_n^{†}f\right).$$

Letting Template:Mvar denote the identity operator on the Hilbert space, the expression

$$I=\sum _{n=1}^{\mathrm{\infty }}{\phi }_n{\phi }_n^{†},$$

is called a resolution of the identity. When the Hilbert space is the space Template:Math of square-integrable functions on a domain Template:Mvar, the quantity:

$${\phi }_n{\phi }_n^{†},$$

is an integral operator, and the expression for Template:Mvar can be rewritten

$$f(x)=\sum _{n=1}^{\mathrm{\infty }}\underset{D}{\int }({\phi }_n(x){\phi }_n^{*}(\xi ))f(\xi )d\xi .$$

The right-hand side converges to Template:Mvar in the Template:Math sense. It need not hold in a pointwise sense, even when Template:Mvar is a continuous function. Nevertheless, it is common to abuse notation and write

$$f(x)=\int \delta (x-\xi )f(\xi )d\xi ,$$

resulting in the representation of the delta function:Template:Sfn

$$\delta (x-\xi )=\sum _{n=1}^{\mathrm{\infty }}{\phi }_n(x){\phi }_n^{*}(\xi ).$$

With a suitable rigged Hilbert space Template:Math where Template:Math contains all compactly supported smooth functions, this summation may converge in Template:Math, depending on the properties of the basis Template:Math. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator (e.g. the heat kernel), in which case the series converges in the distribution sense.Template:Sfn

Infinitesimal delta functions

Cauchy used an infinitesimal Template:Mvar to write down a unit impulse, infinitely tall and narrow Dirac-type delta function Template:Mvar satisfying $\int F(x){\delta }_{\alpha }(x)dx=F(0)$ in a number of articles in 1827.Template:Sfn Cauchy defined an infinitesimal in Cours d'Analyse (1827) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology.

Non-standard analysis allows one to rigorously treat infinitesimals. The article by Template:Harvtxt contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals. Here the Dirac delta can be given by an actual function, having the property that for every real function Template:Mvar one has $\int F(x){\delta }_{\alpha }(x)dx=F(0)$ as anticipated by Fourier and Cauchy.

Dirac comb

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of Template:Mvar

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Sha distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis. The Dirac comb is given as the infinite sum, whose limit is understood in the distribution sense,

$$\text{Ш}(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x-n),$$

which is a sequence of point masses at each of the integers.

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if Template:Mvar is any Schwartz function, then the periodization of Template:Mvar is given by the convolution $$(f*\text{Ш})(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}f(x-n).$$ In particular, $$(f*\text{Ш}{)}^{\wedge }=\hat{f}\widehat{\text{Ш}}=\hat{f}\text{Ш}$$ is precisely the Poisson summation formula.Template:SfnTemplate:Sfn More generally, this formula remains to be true if Template:Mvar is a tempered distribution of rapid descent or, equivalently, if $\hat{f}$ is a slowly growing, ordinary function within the space of tempered distributions.

Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem, important in quantum mechanics, relates the delta function to the distribution Template:Math, the Cauchy principal value of the function Template:Math, defined by

$$⟨\mathrm{p.v.}\frac{1}{x},\phi ⟩=\underset{\epsilon \to 0^{+}}{\lim }\underset{|x|>\epsilon }{\int }\frac{\phi (x)}{x}dx.$$

Sokhotsky's formula states thatTemplate:Sfn

$$\underset{\epsilon \to 0^{+}}{\lim }\frac{1}{x\pm i\epsilon }=\mathrm{p.v.}\frac{1}{x}\mp i\pi \delta (x),$$

Here the limit is understood in the distribution sense, that for all compactly supported smooth functions Template:Mvar,

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{\epsilon \to 0^{+}}{\lim }\frac{f(x)}{x\pm i\epsilon }dx=\mp i\pi f(0)+\underset{\epsilon \to 0^{+}}{\lim }\underset{|x|>\epsilon }{\int }\frac{f(x)}{x}dx.$$

Relationship to the Kronecker delta

The Kronecker delta Template:Mvar is the quantity defined by

$${\delta }_{ij}=\{\begin{array}{cc}1& i=j\\ 0& i=j\end{array}$$

for all integers Template:Mvar, Template:Mvar. This function then satisfies the following analog of the sifting property: if Template:Mvar (for Template:Mvar in the set of all integers) is any doubly infinite sequence, then

$$\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i{\delta }_{ik}=a_k.$$

Similarly, for any real or complex valued continuous function Template:Mvar on Template:Math, the Dirac delta satisfies the sifting property

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\delta (x-x_0)dx=f(x_0).$$

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.Template:Sfn

Applications

Probability theory

In probability theory and statistics, the Dirac delta function is often used to represent a discrete distribution, or a partially discrete, partially continuous distribution, using a probability density function (which is normally used to represent absolutely continuous distributions). For example, the probability density function Template:Math of a discrete distribution consisting of points Template:Math, with corresponding probabilities Template:Math, can be written as

$$f(x)=\sum _{i=1}^n{p}_i\delta (x-x_i).$$

As another example, consider a distribution in which 6/10 of the time returns a standard normal distribution, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution). The density function of this distribution can be written as

$$f(x)=0.6\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}+0.4\delta (x-3.5).$$

The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If Template:Math is a continuous differentiable function, then the density of Template:Mvar can be written as

$$f_Y(y)=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{f}_X(x)\delta (y-g(x))dx.$$

The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion). The local time of a stochastic process Template:Math is given by $$\ell (x,t)=\underset{0}{\overset{t}{\int }}\delta (x-B(s))ds$$ and represents the amount of time that the process spends at the point Template:Mvar in the range of the process. More precisely, in one dimension this integral can be written $$\ell (x,t)=\underset{\epsilon \to 0^{+}}{\lim }\frac{1}{2\epsilon }\underset{0}{\overset{t}{\int }}{\mathrm{𝟏}}_{[x-\epsilon ,x+\epsilon ]}(B(s))ds$$ where ${\mathrm{𝟏}}_{[x-\epsilon ,x+\epsilon ]}$ is the indicator function of the interval $[x-\epsilon ,x+\epsilon ].$

Quantum mechanics

The delta function is expedient in quantum mechanics. The wave function of a particle gives the probability amplitude of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space Template:Math of square-integrable functions, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set Template:Math of wave functions is orthonormal if

$$⟨{\phi }_n\mid {\phi }_m⟩={\delta }_{nm},$$

where Template:Mvar is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function Template:Math can be expressed as a linear combination of the Template:Math with complex coefficients:

$$\psi =\sum c_n{\phi }_n,$$

where Template:Math. Complete orthonormal systems of wave functions appear naturally as the eigenfunctions of the Hamiltonian (of a bound system) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the spectrum of the Hamiltonian. In bra–ket notation this equality implies the resolution of the identity:

$$I=\sum |{\phi }_n⟩⟨{\phi }_n|.$$

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an observable can also be continuous. An example is the position operator, Template:Math. The spectrum of the position (in one dimension) is the entire real line and is called a continuous spectrum. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well, i.e., to replace the Hilbert space with a rigged Hilbert space.Template:Sfn In this context, the position operator has a complete set of generalized eigenfunctions,Template:Sfn labeled by the points Template:Mvar of the real line, given by

$${\phi }_y(x)=\delta (x-y).$$

The generalized eigenfunctions of the position operator are called the eigenkets and are denoted by Template:Math.Template:Sfn

Similar considerations apply to any other (unbounded) self-adjoint operator with continuous spectrum and no degenerate eigenvalues, such as the momentum operator Template:Mvar. In that case, there is a set Template:Math of real numbers (the spectrum) and a collection of distributions Template:Mvar with Template:Math such that

$$P{\phi }_y=y{\phi }_y.$$

That is, Template:Mvar are the generalized eigenvectors of Template:Mvar. If they form an "orthonormal basis" in the distribution sense, that is:

$$⟨{\phi }_y,{\phi }_{y^{\prime }}⟩=\delta (y-y^{\prime }),$$

then for any test function Template:Mvar,

$$\psi (x)=\underset{\mathrm{\Omega }}{\int }c(y){\phi }_y(x)dy$$

where Template:Math. That is, there is a resolution of the identity

$$I=\underset{\mathrm{\Omega }}{\int }|{\phi }_y⟩⟨{\phi }_y|dy$$

where the operator-valued integral is again understood in the weak sense. If the spectrum of Template:Mvar has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum and an integral over the continuous spectrum.

The delta function also has many more specialized applications in quantum mechanics, such as the delta potential models for a single and double potential well.

Structural mechanics

The delta function can be used in structural mechanics to describe transient loads or point loads acting on structures. The governing equation of a simple mass–spring system excited by a sudden force impulse Template:Mvar at time Template:Math can be written

$$m\frac{d^2\xi }{d{t}^2}+k\xi =I\delta (t),$$

where Template:Mvar is the mass, Template:Mvar is the deflection, and Template:Mvar is the spring constant.

As another example, the equation governing the static deflection of a slender beam is, according to Euler–Bernoulli theory,

$$EI\frac{d^{4}w}{d{x}^4}=q(x),$$

where Template:Mvar is the bending stiffness of the beam, Template:Mvar is the deflection, Template:Mvar is the spatial coordinate, and Template:Math is the load distribution. If a beam is loaded by a point force Template:Mvar at Template:Math, the load distribution is written

$$q(x)=F\delta (x-x_0).$$

As the integration of the delta function results in the Heaviside step function, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise polynomials.

Also, a point moment acting on a beam can be described by delta functions. Consider two opposing point forces Template:Mvar at a distance Template:Mvar apart. They then produce a moment Template:Math acting on the beam. Now, let the distance Template:Mvar approach the limit zero, while Template:Mvar is kept constant. The load distribution, assuming a clockwise moment acting at Template:Math, is written

$$\begin{array}{cc}q(x)& =\underset{d\to 0}{\lim }(F\delta (x)-F\delta (x-d))\\ & =\underset{d\to 0}{\lim }(\frac{M}{d}\delta (x)-\frac{M}{d}\delta (x-d))\\ & =M\underset{d\to 0}{\lim }\frac{\delta (x)-\delta (x-d)}{d}\\ & =M{\delta }^{\prime }(x).\end{array}$$

Point moments can thus be represented by the derivative of the delta function. Integration of the beam equation again results in piecewise polynomial deflection.

See also

• Atom (measure theory)
• Laplacian of the indicator
• Uncertainty principle

References

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External links

• Template:Commons category-inline
• Template:Springer
• KhanAcademy.org video lesson
• The Dirac Delta function, a tutorial on the Dirac delta function.
• Video Lectures – Lecture 23, a lecture by Arthur Mattuck.
• The Dirac delta measure is a hyperfunction
• We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure
• Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure. Template:Webarchive