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Dirac delta function

Inmathematical analysis, the Dirac delta function (orδ distribution), also known as the unit impulse,^[1] 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.^[2][3][4] 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[edit]

The graph of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis.^[5]: 174 The Dirac delta is used to model a tall narrow spike function (animpulse), and other similar abstractions such as a point charge, point massorelectron 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 ${\displaystyle t=0}$ it is struck by another ball, imparting it with a momentum P, 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 P δ(t); the units of δ(t) are s⁻¹.

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval ${\displaystyle \Delta t=[0,T]}$. That is,

$${\displaystyle F_{\Delta t}($$t)={\begin{cases}P/\Delta t&0<t\leq T,\\0&{\text{otherwise}}.\end{cases}}}

Then the momentum at any time t is found by integration:

$${\displaystyle p($$t)=\int _{0}^{t}F_{\Delta t}(\tau )\,d\tau ={\begin{cases}P&t\geq T\\P\,t/\Delta t&0\leq t\leq T\\0&{\text{otherwise.}}\end{cases}}}

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δt → 0, giving a result everywhere except at 0:

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

Here the functions ${\displaystyle F_{\Delta t}}$ 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) ${\textstyle \lim _{\Delta t\to 0^{+}}F_{\Delta t}}$ 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

$${\displaystyle \int _{-\infty }^{\infty }F_{\Delta t}($$t)\,dt=P,}

which holds for all ${\displaystyle \Delta t>0}$, should continue to hold in the limit. So, in the equation ${\textstyle F($t)=P\,\delta (t)=\lim _{\Delta t\to 0}F_{\Delta t}(t)} , it is understood that the limit is always taken outside the integral.

In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (aweak 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 f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g 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[edit]

Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:^[6]

$${\displaystyle f($$x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha \,f(\alpha )\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ ,}

which is tantamount to the introduction of the δ-function in the form:^[7]

$${\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ .}$$

Later, Augustin Cauchy expressed the theorem using exponentials:^[8][9]

$${\displaystyle f($$x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp.}

Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).^[10][11]

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

$${\displaystyle {\begin{aligned}f($$x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp\\[4pt]&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(\int _{-\infty }^{\infty }e^{ipx}e^{-ip\alpha }\,dp\right)f(\alpha )\,d\alpha =\int _{-\infty }^{\infty }\delta (x-\alpha )f(\alpha )\,d\alpha ,\end{aligned}}}

where the δ-function is expressed as

$${\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ip(x-\alpha )}\,dp\ .}$$

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:^[12]

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) ...",^[13] and leading to the formal development of the Dirac delta function.

Aninfinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin-Louis Cauchy. ^[14] 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. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse.^[15] The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of the Quantum Dynamics^[16] and used in his textbook The Principles of Quantum Mechanics.^[3] He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.

Definitions[edit]

The Dirac delta function ${\displaystyle \delta ($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,

$${\displaystyle \delta ($$x)\simeq {\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}}

and which is also constrained to satisfy the identity^[17]

$${\displaystyle \int _{-\infty }^{\infty }\delta ($$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.^[18]

As a measure[edit]

One way to rigorously capture the notion of the Dirac delta function is to define a measure, called Dirac measure, which accepts a subset A of the real line R as an argument, and returns δ(A) = 1if0 ∈ A, and δ(A) = 0 otherwise.^[19] If the delta function is conceptualized as modeling an idealized point mass at 0, then δ(A) represents the mass contained in the set A. One may then define the integral against δ 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 δ satisfies

$${\displaystyle \int _{-\infty }^{\infty }f($$x)\,\delta (dx)=f(0)}

for all continuous compactly supported functions f. The measure δ 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

$${\displaystyle \int _{-\infty }^{\infty }f($$x)\,\delta (x)\,dx=f(0)}

holds.^[20] As a result, the latter notation is a convenient abuse of notation, and not a standard (RiemannorLebesgue) integral.

As a probability measureonR, the delta measure is characterized by its cumulative distribution function, which is the unit step function.^[21]

$${\displaystyle H($$x)={\begin{cases}1&{\text{if }}x\geq 0\\0&{\text{if }}x<0.\end{cases}}}

This means that H(x) is the integral of the cumulative indicator function 1_{(−∞, x]} with respect to the measure δ; to wit,

$${\displaystyle H($$x)=\int _{\mathbf {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta (dt)=\delta \!\left((-\infty ,x]\right),}

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:^[22]

$${\displaystyle \int _{-\infty }^{\infty }f($$x)\,\delta (dx)=\int _{-\infty }^{\infty }f(x)\,dH(x).}

All higher momentsofδ are zero. In particular, characteristic function and moment generating function are both equal to one.

As a distribution[edit]

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.^[23] 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 φ. 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 functionsonR 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 by^[24]

${\displaystyle \delta [\varphi ]=\varphi (0)}$

(1)

for every test function φ.

For δ 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 S on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer N there is an integer M_N and a constant C_N such that for every test function φ, one has the inequality^[25]

$${\displaystyle \left|S[\varphi ]\right|\leq C_{N}\sum _{k=0}^{M_{N}}\sup _{x\in [-N,N]}\left|\varphi ^{($$k)}(x)\right|}

where sup represents the supremum. With the δ distribution, one has such an inequality (with C_N = 1) with M_N = 0 for all N. Thus δ is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being {0}).

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 φ, one has

$${\displaystyle \delta [\varphi ]=-\int _{-\infty }^{\infty }\varphi '($$x)\,H(x)\,dx.}

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

$${\displaystyle \int _{-\infty }^{\infty }\varphi ($$x)\,H'(x)\,dx=\int _{-\infty }^{\infty }\varphi (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

$${\displaystyle -\int _{-\infty }^{\infty }\varphi '($$x)\,H(x)\,dx=\int _{-\infty }^{\infty }\varphi (x)\,dH(x).}

In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation (1) defines a Daniell integral on the space of all compactly supported continuous functions φ which, by the Riesz representation theorem, can be represented as the Lebesgue integral of φ 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[edit]

The delta function can be defined in n-dimensional Euclidean space Rⁿ as the measure such that

$${\displaystyle \int _{\mathbf {R} ^{n}}f(\mathbf {x} )\,\delta (d\mathbf {x} )=f(\mathbf {0} )}$$

for every compactly supported continuous function f. As a measure, the n-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with x = (x₁, x₂, ..., x_n), one has^[26]

${\displaystyle \delta (\mathbf {x} )=\delta (x_{1})\,\delta (x_{2})\cdots \delta (x_{n}).}$

(2)

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.^[27] However, despite widespread use in engineering contexts, (2) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.^[28][29]

The notion of a Dirac measure makes sense on any set.^[30] Thus if X is a set, x₀ ∈ X is a marked point, and Σ is any sigma algebra of subsets of X, then the measure defined on sets A ∈ Σby

$${\displaystyle \delta _{x_{0}}($$A)={\begin{cases}1&{\text{if }}x_{0}\in A\\0&{\text{if }}x_{0}\notin A\end{cases}}}

is the delta measure or unit mass concentrated at x₀.

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 M centered at the point x₀ ∈ M is defined as the following distribution:

${\displaystyle \delta _{x_{0}}[\varphi ]=\varphi (x_{0})}$

(3)

for all compactly supported smooth real-valued functions φonM.^[31] A common special case of this construction is a case in which M is an open set in the Euclidean space Rⁿ.

On a locally compact Hausdorff space X, the Dirac delta measure concentrated at a point x is the Radon measure associated with the Daniell integral (3) on compactly supported continuous functions φ.^[32] 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 ${\displaystyle x_{0}\mapsto \delta _{x_{0}}}$ is a continuous embedding of X into the space of finite Radon measures on X, equipped with its vague topology. Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X.^[33]

Properties[edit]

Scaling and symmetry[edit]

The delta function satisfies the following scaling property for a non-zero scalar α:^[34]

$${\displaystyle \int _{-\infty }^{\infty }\delta (\alpha x)\,dx=\int _{-\infty }^{\infty }\delta ($$u)\,{\frac {du}{|\alpha |}}={\frac {1}{|\alpha |}}}

and so

${\displaystyle \delta (\alpha x)={\frac {\delta ($x)}{|\alpha |}}.}

(4)

Scaling property proof: $${\displaystyle \int \limits _{-\infty }^{\infty }dx\ g($$x)\delta (ax)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')} where a change of variable x′ = ax is used. If a is negative, i.e., a = −|a|, then $${\displaystyle \int \limits _{-\infty }^{\infty }dx\ g($$x)\delta (ax)={\frac {1}{-\left\vert a\right\vert }}\int \limits _{\infty }^{-\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}\int \limits _{-\infty }^{\infty }dx'\ g\left({\frac {x'}{a}}\right)\delta (x')={\frac {1}{\left\vert a\right\vert }}g(0).} Thus, ${\displaystyle \delta ($ax)={\frac {1}{\left\vert a\right\vert }}\delta (x)} .

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

$${\displaystyle \delta (-x)=\delta ($$x)}

which is homogeneous of degree −1.

Algebraic properties[edit]

The distributional productofδ with x is equal to zero:

$${\displaystyle x\,\delta ($$x)=0.}

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

Conversely, if xf(x) = xg(x), where f and g are distributions, then

$${\displaystyle f($$x)=g(x)+c\delta (x)}

for some constant c.^[35]

Translation[edit]

The integral of the time-delayed Dirac delta is^[36]

$${\displaystyle \int _{-\infty }^{\infty }f($$t)\,\delta (t-T)\,dt=f(T).}

This is sometimes referred to as the sifting property^[37] or the sampling property.^[38] The delta function is said to "sift out" the value of f(t)att = T.^[39]

It follows that the effect of convolving a function f(t) with the time-delayed Dirac delta ${\displaystyle \delta _{T}($t)=\delta (t-T)} is to time-delay f(t) by the same amount:

$${\displaystyle {\begin{aligned}(f*\delta _{T})($$t)\ &{\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }f(\tau )\,\delta (t-T-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )\,\delta (\tau -(t-T))\,d\tau \qquad {\text{since}}~\delta (-x)=\delta (x)~~{\text{by (4)}}\\&=f(t-T).\end{aligned}}}

The sifting property holds under the precise condition that f 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)

$${\displaystyle \int _{-\infty }^{\infty }\delta (\xi -x)\delta (x-\eta )\,dx=\delta (\eta -\xi ).}$$

Composition with a function[edit]

More generally, the delta distribution may be composed with a smooth function g(x) in such a way that the familiar change of variables formula holds, that

$${\displaystyle \int _{\mathbb {R} }\delta {\bigl (}g($$x){\bigr )}f{\bigl (}g(x){\bigr )}\left|g'(x)\right|dx=\int _{g(\mathbb {R} )}\delta (u)\,f(u)\,du}

provided that g is a continuously differentiable function with g′ nowhere zero.^[40] That is, there is a unique way to assign meaning to the distribution ${\displaystyle \delta \circ g}$ so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g′ = 0 point. This distribution satisfies δ(g(x)) = 0ifg is nowhere zero, and otherwise if g has a real rootatx₀, then

$${\displaystyle \delta (g($$x))={\frac {\delta (x-x_{0})}{|g'(x_{0})|}}.}

It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions gby

$${\displaystyle \delta (g($$x))=\sum _{i}{\frac {\delta (x-x_{i})}{|g'(x_{i})|}}}

where the sum extends over all roots (i.e., all the different ones) of g(x), which are assumed to be simple. Thus, for example

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

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

$${\displaystyle \int _{-\infty }^{\infty }f($$x)\,\delta (g(x))\,dx=\sum _{i}{\frac {f(x_{i})}{|g'(x_{i})|}}.}

Indefinite integral[edit]

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

Properties in n dimensions[edit]

The delta distribution in an n-dimensional space satisfies the following scaling property instead, $${\displaystyle \delta (\alpha {\boldsymbol {x}})=|\alpha |^{-n}\delta ({\boldsymbol {x}})~,}$$ so that δ is a homogeneous distribution of degree −n.

Under any reflectionorrotation ρ, the delta function is invariant, $${\displaystyle \delta (\rho {\boldsymbol {x}})=\delta ({\boldsymbol {x}})~.}$$

As in the one-variable case, it is possible to define the composition of δ with a bi-Lipschitz function^[41] g: Rⁿ → Rⁿ uniquely so that the following holds $${\displaystyle \int _{\mathbb {R} ^{n}}\delta (g({\boldsymbol {x}}))\,f(g({\boldsymbol {x}}))\left|\det g'({\boldsymbol {x}})\right|d{\boldsymbol {x}}=\int _{g(\mathbb {R} ^{n})}\delta ({\boldsymbol {u}})f({\boldsymbol {u}})\,d{\boldsymbol {u}}}$$ for all compactly supported functions f.

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 g : Rⁿ → R such that the gradientofg is nowhere zero, the following identity holds^[42] $${\displaystyle \int _{\mathbb {R} ^{n}}f({\boldsymbol {x}})\,\delta (g({\boldsymbol {x}}))\,d{\boldsymbol {x}}=\int _{g^{-1}(0)}{\frac {f({\boldsymbol {x}})}{|{\boldsymbol {\nabla }}g|}}\,d\sigma ({\boldsymbol {x}})}$$ where the integral on the right is over g⁻¹(0), the (n − 1)-dimensional surface defined by g(x) = 0 with respect to the Minkowski content measure. This is known as a simple layer integral.

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

where σ is the hypersurface measure associated to S. This generalization is associated with the potential theoryofsimple layer potentialsonS. If D is a domaininRⁿ with smooth boundary S, then δ_S is equal to the normal derivative of the indicator functionofD in the distribution sense,

$${\displaystyle -\int _{\mathbb {R} ^{n}}g({\boldsymbol {x}})\,{\frac {\partial 1_{D}({\boldsymbol {x}})}{\partial n}}\,d{\boldsymbol {x}}=\int _{S}\,g({\boldsymbol {s}})\,d\sigma ({\boldsymbol {s}}),}$$

where n is the outward normal.^[43][44] 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:

$${\displaystyle \delta ({\boldsymbol {r}}-{\boldsymbol {r}}_{0})={\begin{cases}\displaystyle {\frac {1}{r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})\delta (\phi -\phi _{0})&x_{0},y_{0},z_{0}\neq 0\\\displaystyle {\frac {1}{2\pi r^{2}\sin \theta }}\delta (r-r_{0})\delta (\theta -\theta _{0})&x_{0}=y_{0}=0,\ z_{0}\neq 0\\\displaystyle {\frac {1}{4\pi r^{2}}}\delta (r-r_{0})&x_{0}=y_{0}=z_{0}=0\end{cases}}}$$