Dirac delta function

"Delta function" redirects here. For other uses, see Delta function (disambiguation).

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.

\sqrt{\pi}} e^{-(x/a)^2}</math>

In mathematical analysis, the Dirac delta function (or δ distribution), also known as the unit impulse,<ref name="FOOTNOTEatis2013unit impulse">atis 2013, unit impulse.</ref> 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.<ref name="FOOTNOTEArfkenWeber200084">Arfken & Weber 2000, p. 84.</ref><ref name="FOOTNOTEDirac1930§22 The δ function">Dirac 1930, §22 The δ function.</ref><ref name="FOOTNOTEGelfandShilov1966–1968Volume I, §1.1">Gelfand & Shilov 1966–1968, Volume I, §1.1.</ref> Since there is no function having this property, to model 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.<ref>Zhao, Ji-Cheng (2011-05-05). Methods for Phase Diagram Determination. Elsevier. ISBN 978-0-08-054996-5.</ref> 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 <math>t=0</math> 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 {{{1}}} That is,

<math display="block">F_{\Delta t}(t) = \begin{cases} P/\Delta t& 0<t\leq T, \\ 0 &\text{otherwise}. \end{cases}</math>

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

<math display="block">p(t) = \int_0^t F_{\Delta t}(\tau)\,d\tau = \begin{cases} P & t \ge T\\ P\,t/\Delta t & 0 \le t \le T\\ 0&\text{otherwise.}\end{cases}</math>

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

<math display="block">p(t)=\begin{cases}P & t > 0\\ 0 & t < 0.\end{cases}</math>

Here the functions <math>F_{\Delta t}</math> 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) <math display="inline">\lim_{\Delta t\to 0^+}F_{\Delta t}</math> 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

<math display="block">\int_{-\infty}^\infty F_{\Delta t}(t)\,dt = P,</math>

which holds for all <math>\Delta t>0</math>, should continue to hold in the limit. So, in the equation {{{1}}} 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 (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 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

Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:<ref name=Fourier>Fourier, JB (1822). The Analytical Theory of Heat (English translation by Alexander Freeman, 1878 ed.). The University Press. p. [1]., cf. https://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA449 and pp. 546–551. Original French text.</ref>

<math display="block">f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty\ \ d\alpha \, f(\alpha) \ \int_{-\infty}^\infty dp\ \cos (px-p\alpha)\ , </math>

which is tantamount to the introduction of the δ-function in the form:<ref name= Kawai>Komatsu, Hikosaburo (2002). "Fourier's hyperfunctions and Heaviside's pseudodifferential operators". In Takahiro Kawai; Keiko Fujita (eds.). Microlocal Analysis and Complex Fourier Analysis. World Scientific. p. [2]. ISBN 978-981-238-161-3.</ref>

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty dp\ \cos (px-p\alpha) \ . </math>

Later, Augustin Cauchy expressed the theorem using exponentials:<ref name= Myint-U>Myint-U., Tyn; Debnath, Lokenath (2007). Linear Partial Differential Equations for Scientists And Engineers (4th ed.). Springer. p. [3]. ISBN 978-0-8176-4393-5.</ref><ref name=Debnath>Debnath, Lokenath; Bhatta, Dambaru (2007). Integral Transforms And Their Applications (2nd ed.). CRC Press. p. [4]. ISBN 978-1-58488-575-7.</ref>

<math display="block">f(x)=\frac{1}{2\pi} \int_{-\infty} ^ \infty \ e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp. </math>

Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).<ref name=Grattan-Guinness>Grattan-Guinness, Ivor (2009). Convolutions in French Mathematics, 1800–1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics, Volume 2. Birkhäuser. p. 653. ISBN 978-3-7643-2238-0.</ref><ref name=Cauchy>

See, for example, Cauchy, Augustin-Louis (1789-1857) Auteur du texte (1882–1974). "Des intégrales doubles qui se présentent sous une forme indéterminèe". Oeuvres complètes d'Augustin Cauchy. Série 1, tome 1 / publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique...{{cite book}}: CS1 maint: numeric names: authors list (link)</ref>

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

<math display="block">\begin{align} 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{align}</math>

where the δ-function is expressed as

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\,dp \ . </math>

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:<ref name="Mitrović">Mitrović, Dragiša; Žubrinić, Darko (1998). Fundamentals of Applied Functional Analysis: Distributions, Sobolev Spaces. CRC Press. p. 62. ISBN 978-0-582-24694-2.</ref>

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) ...",<ref name=Kracht>Kracht, Manfred; Kreyszig, Erwin (1989). "On singular integral operators and generalizations". In Themistocles M. Rassias (ed.). Topics in Mathematical Analysis: A Volume Dedicated to the Memory of A.L. Cauchy. World Scientific. p. https://books.google.com/books?id=xIsPrSiDlZIC&pg=PA553 553]. ISBN 978-9971-5-0666-7.</ref> and leading to the formal development of the Dirac delta function.

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. <ref name="FOOTNOTELaugwitz1989230">Laugwitz 1989, p. 230.</ref> 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.<ref>A more complete historical account can be found in van der Pol & Bremmer 1987, §V.4.</ref> The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of the Quantum Dynamics<ref>Dirac, P. A. M. (January 1927). "The physical interpretation of the quantum dynamics". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 113 (765): 621–641. Bibcode:1927RSPSA.113..621D. doi:10.1098/rspa.1927.0012. ISSN 0950-1207. S2CID 122855515.</ref> and used in his textbook The Principles of Quantum Mechanics.<ref name="FOOTNOTEDirac1930§22 The δ function">Dirac 1930, §22 The δ function.</ref> He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.

Definitions

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

<math display="block">\delta(x) \simeq \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>

and which is also constrained to satisfy the identity<ref name="FOOTNOTEGelfandShilov1966–1968Volume I, §1.1, p. 1">Gelfand & Shilov 1966–1968, Volume I, §1.1, p. 1.</ref>

<math display="block">\int_{-\infty}^\infty \delta(x) \, dx = 1.</math>

This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.<ref name="FOOTNOTEDirac193063">Dirac 1930, p. 63.</ref>

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 A of the real line R as an argument, and returns δ(A) = 1 if 0 ∈ A, and δ(A) = 0 otherwise.<ref name="Rudin 1966 loc=§1.20">Rudin 1966, §1.20</ref> 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

<math display="block">\int_{-\infty}^\infty f(x) \, \delta(dx) = f(0)</math>

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

<math display="block">\int_{-\infty}^\infty f(x)\, \delta(x)\, dx = f(0)</math>

holds.<ref name="FOOTNOTEHewittStromberg1963§19.61">Hewitt & Stromberg 1963, §19.61.</ref> 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 R, the delta measure is characterized by its cumulative distribution function, which is the unit step function.<ref>Driggers 2003, p. 2321 See also Bracewell 1986, Chapter 5 for a different interpretation. Other conventions for the assigning the value of the Heaviside function at zero exist, and some of these are not consistent with what follows.</ref>

<math display="block">H(x) = \begin{cases} 1 & \text{if } x\ge 0\\ 0 & \text{if } x < 0. \end{cases}</math>

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

<math display="block">H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta(dt) = \delta(-\infty,x],</math>

the latter being the measure of this interval; more formally, δ((−∞, x]). Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:<ref name="FOOTNOTEHewittStromberg1963§9.19">Hewitt & Stromberg 1963, §9.19.</ref>

<math display="block">\int_{-\infty}^\infty f(x)\,\delta(dx) = \int_{-\infty}^\infty f(x) \,dH(x).</math>

All higher moments of δ 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.<ref name="FOOTNOTEHazewinkel2011[httpsbooksgooglecombooksid_YPtCAAAQBAJpgPA41 41]">Hazewinkel 2011, p. 41.</ref> 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 functions on R 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<ref name="FOOTNOTEStrichartz1994§2.2">Strichartz 1994, §2.2.</ref>

1

 

 

 

 

({{{3}}})

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<ref name="FOOTNOTEHörmander1983Theorem 2.1.5">Hörmander 1983, Theorem 2.1.5.</ref>

<math display="block">\left|S[\varphi]\right| \le C_N \sum_{k=0}^{M_N}\sup_{x\in [-N,N]} \left|\varphi^{(k)}(x)\right|</math>

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 Template:Brace).

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

<math display="block">\delta[\varphi] = -\int_{-\infty}^\infty \varphi'(x)\,H(x)\,dx.</math>

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

<math display="block">\int_{-\infty}^\infty \varphi(x)\,H'(x)\,dx = \int_{-\infty}^\infty \varphi(x)\,\delta(x)\,dx,</math>

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

<math display="block">-\int_{-\infty}^\infty \varphi'(x)\,H(x)\, dx = \int_{-\infty}^\infty \varphi(x)\,dH(x).</math>

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

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

<math display="block">\int_{\mathbf{R}^n} f(\mathbf{x})\,\delta(d\mathbf{x}) = f(\mathbf{0})</math>

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<ref name="FOOTNOTEBracewell1986Chapter 5">Bracewell 1986, Chapter 5.</ref>

2

 

 

 

 

({{{3}}})

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.<ref name="FOOTNOTEHörmander1983§3.1">Hörmander 1983, §3.1.</ref> 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.<ref name="FOOTNOTEStrichartz1994§2.3">Strichartz 1994, §2.3.</ref><ref name="FOOTNOTEHörmander1983§8.2">Hörmander 1983, §8.2.</ref>

The notion of a Dirac measure makes sense on any set.<ref name="FOOTNOTERudin1966§1.20">Rudin 1966, §1.20.</ref> 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 ∈ Σ by

<math display="block">\delta_{x_0}(A)=\begin{cases} 1 &\text{if }x_0\in A\\ 0 &\text{if }x_0\notin A \end{cases}</math>

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:

3

 

 

 

 

({{{3}}})

for all compactly supported smooth real-valued functions φ on M.<ref name="FOOTNOTEDieudonné1972§17.3.3">Dieudonné 1972, §17.3.3.</ref> 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.<ref>Krantz, Steven G.; Parks, Harold R. (2008-12-15). Geometric Integration Theory. Springer Science & Business Media. ISBN 978-0-8176-4679-0.</ref> 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 <math>x_0\mapsto \delta_{x_0}</math> 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.<ref name="FOOTNOTEFederer1969§2.5.19">Federer 1969, §2.5.19.</ref>

Properties

Scaling and symmetry

The delta function satisfies the following scaling property for a non-zero scalar α:<ref name="FOOTNOTEStrichartz1994Problem 2.6.2">Strichartz 1994, Problem 2.6.2.</ref>

<math display="block">\int_{-\infty}^\infty \delta(\alpha x)\,dx =\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|} =\frac{1}{|\alpha|}</math>

and so

\alpha

 

 

 

 

(}.</math>)

Scaling property proof: <math display="block">\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') </math> where a change of variable x′ = ax is used. If a is negative, i.e., a = −|a|, then <math display="block">\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). </math> Thus, {{{1}}}

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

<math display="block">\delta(-x) = \delta(x)</math>

which is homogeneous of degree −1.

Algebraic properties

The distributional product of δ with x is equal to zero:

<math display="block">x\,\delta(x) = 0.</math>

More generally, <math>(x-a)^n\delta(x-a) =0</math> for all positive integers <math>n</math>.

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

<math display="block">f(x) = g(x) +c \delta(x)</math>

for some constant c.<ref name="FOOTNOTEVladimirov1971Chapter 2, Example 3(d)">Vladimirov 1971, Chapter 2, Example 3(d).</ref>

Translation

The integral of the time-delayed Dirac delta is<ref>Rottwitt, Karsten; Tidemand-Lichtenberg, Peter (2014-12-11). Nonlinear Optics: Principles and Applications. CRC Press. p. [5] 276. ISBN 978-1-4665-6583-8.</ref>

<math display="block">\int_{-\infty}^\infty f(t) \,\delta(t-T)\,dt = f(T).</math>

This is sometimes referred to as the sifting property<ref>Weisstein, Eric W. "Sifting Property". MathWorld.</ref> or the sampling property.<ref>Karris, Steven T. (2003). Signals and Systems with MATLAB Applications. Orchard Publications. p. 15. ISBN 978-0-9709511-6-8.</ref> The delta function is said to "sift out" the value of f(t) at t = T.<ref>Roden, Martin S. (2014-05-17). Introduction to Communication Theory. Elsevier. p. [6]. ISBN 978-1-4831-4556-3.</ref>

It follows that the effect of convolving a function f(t) with the time-delayed Dirac delta <math> \delta_T(t) = \delta(t-T)</math> is to time-delay f(t) by the same amount:

<math display="block">\begin{align} (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{align}</math>

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)

<math display="block">\int_{-\infty}^\infty \delta (\xi-x) \delta(x-\eta) \,dx = \delta(\eta-\xi).</math>

Composition with a function

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

<math display="block">\int_{\R} \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) \left|g'(x)\right| dx = \int_{g(\R)} \delta(u)\,f(u)\,du</math>

provided that g is a continuously differentiable function with g′ nowhere zero.<ref name="FOOTNOTEGelfandShilov1966–1968Vol. 1, §II.2.5">Gelfand & Shilov 1966–1968, Vol. 1, §II.2.5.</ref> That is, there is a unique way to assign meaning to the distribution <math>\delta\circ g</math> 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)) = 0 if g is nowhere zero, and otherwise if g has a real root at x₀, then

<math display="block">\delta(g(x)) = \frac{\delta(x-x_0)}{|g'(x_0)|}.</math>

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

<math display="block">\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}</math>

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

<math display="block">\delta\left(x^2-\alpha^2\right) = \frac{1}{2|\alpha|} \Big[\delta\left(x+\alpha\right)+\delta\left(x-\alpha\right)\Big].</math>

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

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

Indefinite integral

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

Properties in n dimensions

The delta distribution in an n-dimensional space satisfies the following scaling property instead, <math display="block">\delta(\alpha\mathbf{x}) = |\alpha|^{-n}\delta(\mathbf{x}) ~,</math> so that δ is a homogeneous distribution of degree −n.

Under any reflection or rotation ρ, the delta function is invariant, <math display="block">\delta(\rho \mathbf{x}) = \delta(\mathbf{x})~.</math>

As in the one-variable case, it is possible to define the composition of δ with a bi-Lipschitz function<ref>Further refinement is possible, namely to submersions, although these require a more involved change of variables formula.</ref> g: Rⁿ → Rⁿ uniquely so that the identity <math display="block">\int_{\R^n} \delta(g(\mathbf{x}))\, f(g(\mathbf{x}))\left|\det g'(\mathbf{x})\right| d\mathbf{x} = \int_{g(\R^n)} \delta(\mathbf{u}) f(\mathbf{u})\,d\mathbf{u}</math> 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 gradient of g is nowhere zero, the following identity holds<ref name="FOOTNOTEHörmander1983§6.1">Hörmander 1983, §6.1.</ref> <math display="block">\int_{\R^n} f(\mathbf{x}) \, \delta(g(\mathbf{x})) \,d\mathbf{x} = \int_{g^{-1}(0)}\frac{f(\mathbf{x})}{|\mathbf{\nabla}g|}\,d\sigma(\mathbf{x}) </math> 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: <math display="block">\delta_S[g] = \int_S g(\mathbf{s})\,d\sigma(\mathbf{s})</math>

where σ is the hypersurface measure associated to S. This generalization is associated with the potential theory of simple layer potentials on S. If D is a domain in Rⁿ with smooth boundary S, then δ_S is equal to the normal derivative of the indicator function of D in the distribution sense,

<math display="block">-\int_{\R^n}g(\mathbf{x})\,\frac{\partial 1_D(\mathbf{x})}{\partial n}\,d\mathbf{x}=\int_S\,g(\mathbf{s})\, d\sigma(\mathbf{s}),</math>

where n is the outward normal.<ref name="FOOTNOTELange2012pp.29–30">Lange 2012, pp.29–30.</ref><ref name="FOOTNOTEGelfandShilov1966–1968212">Gelfand & Shilov 1966–1968, p. 212.</ref> 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:

<math display="block">\delta(\mathbf{r}-\mathbf{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 \ne 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 \ne 0 \\ 
   \displaystyle\frac{1}{4\pi r^2}\delta(r-r_0) & x_0=y_0=z_0 = 0  

\end{cases}</math>

Fourier transform

The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds<ref>The numerical factors depend on the conventions for the Fourier transform.</ref>

<math display="block">\widehat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi} \,\delta(x)dx = 1.</math>

Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing <math>\langle\cdot,\cdot\rangle</math> of tempered distributions with Schwartz functions. Thus <math>\widehat{\delta}</math> is defined as the unique tempered distribution satisfying

<math display="block">\langle\widehat{\delta},\varphi\rangle = \langle\delta,\widehat{\varphi}\rangle</math>

for all Schwartz functions φ. And indeed it follows from this that <math>\widehat{\delta}=1.</math>

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

<math display="block">S*\delta = S.</math>

That is to say that δ 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 δ, and once it is known, it characterizes the system completely. See LTI system theory § Impulse response and convolution.

The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed as <math display="block">\int_{-\infty}^\infty 1 \cdot e^{2\pi i x\xi}\,d\xi = \delta(x)</math> and more rigorously, it follows since <math display="block">\langle 1, \widehat{f}\rangle = f(0) = \langle\delta,f\rangle</math> for all Schwartz functions f.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has <math display="block">\int_{-\infty}^\infty e^{i 2\pi \xi_1 t} \left[e^{i 2\pi \xi_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (\xi_2 - \xi_1) t} \,dt = \delta(\xi_2 - \xi_1).</math>

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution <math display="block">f(t) = e^{i2\pi\xi_1 t}</math> is <math display="block">\widehat{f}(\xi_2) = \delta(\xi_1-\xi_2)</math> 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 be<ref name="FOOTNOTEBracewell1986">Bracewell 1986.</ref> <math display="block"> \int_{0}^{\infty}\delta(t-a)\,e^{-st} \, dt=e^{-sa}.</math>

Derivatives

The derivative of the Dirac delta distribution, denoted δ′ 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 φ by<ref name="FOOTNOTEGelfandShilov1966–196826">Gelfand & Shilov 1966–1968, p. 26.</ref> <math display="block">\delta'[\varphi] = -\delta[\varphi']=-\varphi'(0).</math>

The first equality here is a kind of integration by parts, for if δ were a true function then <math display="block">\int_{-\infty}^\infty \delta'(x)\varphi(x)\,dx = \delta(x)\varphi(x)|_{-\infty}^{\infty} -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx = -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx = -\varphi'(0).</math>

By mathematical induction, the k-th derivative of δ is defined similarly as the distribution given on test functions by

<math display="block">\delta^{(k)}[\varphi] = (-1)^k \varphi^{(k)}(0).</math>

In particular, δ is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:<ref name="FOOTNOTEGelfandShilov1966–1968§2.1">Gelfand & Shilov 1966–1968, §2.1.</ref> <math display="block">\delta'(x) = \lim_{h\to 0} \frac{\delta(x+h)-\delta(x)}{h}.</math>

More properly, one has <math display="block">\delta' = \lim_{h\to 0} \frac{1}{h}(\tau_h\delta - \delta)</math> where τ_h is the translation operator, defined on functions by τ_hφ(x) = φ(x + h), and on a distribution S by <math display="block">(\tau_h S)[\varphi] = S[\tau_{-h}\varphi].</math>

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.<ref>Weisstein, Eric W. "Doublet Function". MathWorld.</ref>

The derivative of the delta function satisfies a number of basic properties, including:<ref name="FOOTNOTEBracewell200086">Bracewell 2000, p. 86.</ref> <math display="block"> \begin{align}

\delta'(-x) &= -\delta'(x) \\
x\delta'(x) &= -\delta(x)

\end{align} </math> 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, Liebniz 's theorem and linearity of inner product:<ref>"Gugo82's comment on the distributional derivative of Dirac's delta". matematicamente.it. 12 September 2010.</ref>

<math display="block"> \begin{align} \langle x\delta', \varphi \rangle \, &= \, \langle \delta', x\varphi \rangle \, = \, -\langle\delta,(x\varphi)'\rangle \, = \, - \langle \delta, x'\varphi + x\varphi'\rangle \, = \, - \langle \delta, x'\varphi\rangle - \langle\delta, x\varphi'\rangle \, = \, - x'(0)\varphi(0) - x(0)\varphi'(0) \\ &= \, -x'(0) \langle \delta , \varphi \rangle - x(0) \langle \delta, \varphi' \rangle \, = \, -x'(0) \langle \delta,\varphi\rangle + x(0) \langle \delta', \varphi \rangle \, = \, \langle x(0)\delta' - x'(0)\delta, \varphi \rangle \\

\Longrightarrow x(t)\delta'(t) &= x(0)\delta'(t) - x'(0)\delta(t) = -x'(0)\delta(t) = -\delta(t)

\end{align} </math>

Furthermore, the convolution of δ′ with a compactly-supported, smooth function f is

<math display="block">\delta'*f = \delta*f' = f',</math>

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

Higher dimensions

More generally, on an open set U in the n-dimensional Euclidean space <math>\mathbb{R}^n</math>, the Dirac delta distribution centered at a point a ∈ U is defined by<ref name="FOOTNOTEHörmander198356">Hörmander 1983, p. 56.</ref> <math display="block">\delta_a[\varphi]=\varphi(a)</math> for all <math>\varphi \in C_c^\infty(U)</math>, the space of all smooth functions with compact support on U. If <math>\alpha = (\alpha_1, \ldots, \alpha_n)</math> is any multi-index with <math> |\alpha|=\alpha_1+\cdots+\alpha_n</math> and <math>\partial^\alpha</math> denotes the associated mixed partial derivative operator, then the α-th derivative ∂^αδ_a of δ_a is given by<ref name="FOOTNOTEHörmander198356">Hörmander 1983, p. 56.</ref>

<math display="block">\left\langle \partial^\alpha \delta_{a}, \, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = (-1)^{| \alpha |} \partial^\alpha \varphi (x) \Big|_{x = a} \quad \text{ for all } \varphi \in C_c^\infty(U).</math>

That is, the α-th derivative of δ_a is the distribution whose value on any test function φ is the α-th derivative of φ at a (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 S is any distribution on U supported on the set Template:Brace consisting of a single point, then there is an integer m and coefficients c_α such that<ref name="FOOTNOTEHörmander198356">Hörmander 1983, p. 56.</ref><ref name="FOOTNOTERudin1991Theorem 6.25">Rudin 1991, Theorem 6.25.</ref> <math display="block">S = \sum_{|\alpha|\le m} c_\alpha \partial^\alpha\delta_a.</math>

Representations of the delta function

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

<math display="block">\delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x), </math>

where η_ε(x) is sometimes called a nascent delta function. This limit is meant in a weak sense: either that

5

 

 

 

 

({{{3}}})

for all continuous functions f having compact support, or that this limit holds for all smooth functions f 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 η_ε can be constructed in the following manner. Let η be an absolutely integrable function on R of total integral 1, and define <math display="block">\eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). </math>

In n dimensions, one uses instead the scaling <math display="block">\eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). </math>

Then a simple change of variables shows that η_ε also has integral 1. One may show that (5) holds for all continuous compactly supported functions f,<ref name="FOOTNOTESteinWeiss1971Theorem 1.18">Stein & Weiss 1971, Theorem 1.18.</ref> and so η_ε converges weakly to δ in the sense of measures.

The η_ε constructed in this way are known as an approximation to the identity.<ref name="FOOTNOTERudin1991§II.6.31">Rudin 1991, §II.6.31.</ref> This terminology is because the space L¹(R) of absolutely integrable functions is closed under the operation of convolution of functions: f ∗ g ∈ L¹(R) whenever f and g are in L¹(R). However, there is no identity in L¹(R) for the convolution product: no element h such that f ∗ h = f for all f. Nevertheless, the sequence η_ε does approximate such an identity in the sense that

<math display="block">f*\eta_\varepsilon \to f \quad \text{as }\varepsilon\to 0.</math>

This limit holds in the sense of mean convergence (convergence in L¹). Further conditions on the η_ε, for instance that it be a mollifier associated to a compactly supported function,<ref>More generally, one only needs η = η₁ to have an integrable radially symmetric decreasing rearrangement.</ref> are needed to ensure pointwise convergence almost everywhere.

If the initial η = η₁ is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing η to be a suitably normalized bump function, for instance

<math display="block">\eta(x) = \begin{cases} e^{-\frac{1}{1-|x|^2}}& \text{if } |x| < 1\\ 0 & \text{if } |x|\geq 1. \end{cases}</math>

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η₁ to be a hat function. With this choice of η₁, one has

<math display="block"> \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-\left|\frac{x}{\varepsilon}\right|,0 \right) </math>

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 η₁ 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 η₁ to be any probability distribution at all, and letting η_ε(x) = η₁(x/ε)/ε as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments. For instance, if η₁ is the uniform distribution on <math display="inline">\left[-\frac{1}{2},\frac{1}{2}\right]</math>, also known as the rectangular function, then:<ref name="FOOTNOTESaichevWoyczyński1997§1.1 The "delta function" as viewed by a physicist and an engineer, p. 3">Saichev & Woyczyński 1997, §1.1 The "delta function" as viewed by a physicist and an engineer, p. 3.</ref> <math display="block"> \eta_\varepsilon(x) = \frac{1}{\varepsilon}\operatorname{rect}\left(\frac{x}{\varepsilon}\right)= \begin{cases} \frac{1}{\varepsilon},&-\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2}, \\ 0, &\text{otherwise}. \end{cases}</math>

Another example is with the Wigner semicircle distribution <math display="block">\eta_\varepsilon(x)= \begin{cases} \frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon < x < \varepsilon, \\ 0, & \text{otherwise}. \end{cases}</math>

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

Semigroups

Nascent delta functions often arise as convolution semigroups.<ref>Milovanović, Gradimir V.; Rassias, Michael Th (2014-07-08). Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava. Springer. p. 748. ISBN 978-1-4939-0258-3.</ref> This amounts to the further constraint that the convolution of η_ε with η_δ must satisfy <math display="block">\eta_\varepsilon * \eta_\delta = \eta_{\varepsilon+\delta}</math>

for all ε, δ > 0. Convolution semigroups in L¹ 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

<math display="block">\begin{cases} \dfrac{\partial}{\partial t}\eta(t,x) = A\eta(t,x), \quad t>0 \\[5pt] \displaystyle\lim_{t\to 0^+} \eta(t,x) = \delta(x) \end{cases}</math>

in which the limit is as usual understood in the weak sense. Setting η_ε(x) = η(ε, x) 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

<math display="block">\eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}}</math>

represents the temperature in an infinite wire at time t > 0, if a unit of heat energy is stored at the origin of the wire at time t = 0. This semigroup evolves according to the one-dimensional heat equation:

<math display="block">\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}.</math>

In probability theory, η_ε(x) is a normal distribution of variance ε and mean 0. It represents the probability density at time t = ε 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 Rⁿ, the heat kernel is <math display="block">\eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}},</math> and has the same physical interpretation, mutatis mutandis. It also represents a nascent delta function in the sense that η_ε → δ in the distribution sense as ε → 0.

The Poisson kernel

The Poisson kernel <math display="block">\eta_\varepsilon(x) = \frac{1}{\pi}\mathrm{Im}\left\{\frac{1}{x-\mathrm{i}\varepsilon}\right\}=\frac{1}{\pi} \frac{\varepsilon}{\varepsilon^2 + x^2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} \xi x-|\varepsilon \xi|}\,d\xi</math>

is the fundamental solution of the Laplace equation in the upper half-plane.<ref name="FOOTNOTESteinWeiss1971§I.1">Stein & Weiss 1971, §I.1.</ref> 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.<ref>Mader, Heidy M. (2006). Statistics in Volcanology. Geological Society of London. p. 81. ISBN 978-1-86239-208-3.</ref> This semigroup evolves according to the equation <math display="block">\frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x)</math>

where the operator is rigorously defined as the Fourier multiplier <math display="block">\mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).</math>

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,<ref name="FOOTNOTEValléeSoares2004§7.2">Vallée & Soares 2004, §7.2.</ref> is the rescaled Airy function <math display="block">\varepsilon^{-1/3}\operatorname{Ai}\left (x\varepsilon^{-1/3} \right). </math>

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 R¹⁺¹:<ref name="FOOTNOTEHörmander1983§7.8">Hörmander 1983, §7.8.</ref> <math display="block"> \begin{align} c^{-2}\frac{\partial^2u}{\partial t^2} - \Delta u &= 0\\ u=0,\quad \frac{\partial u}{\partial t} = \delta &\qquad \text{for }t=0. \end{align} </math>

The solution u 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) <math display="block">\eta_\varepsilon(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=\frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}} \cos(kx)\,dk </math>

and the Bessel function <math display="block"> \eta_\varepsilon(x) = \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right). </math>

Plane wave decomposition

One approach to the study of a linear partial differential equation <math display="block">L[u]=f,</math>

where L is a differential operator on Rⁿ, is to seek first a fundamental solution, which is a solution of the equation <math display="block">L[u]=\delta.</math>

When L 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 <math display="block">L[u]=h</math>

where h is a plane wave function, meaning that it has the form <math display="block">h = h(x\cdot\xi)</math>

for some vector ξ. Such an equation can be resolved (if the coefficients of L are analytic functions) by the Cauchy–Kovalevskaya theorem or (if the coefficients of L 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).<ref name="FOOTNOTECourantHilbert1962§14">Courant & Hilbert 1962, §14.</ref> Choose k so that n + k is an even integer, and for a real number s, put <math display="block">g(s) = \operatorname{Re}\left[\frac{-s^k\log(-is)}{k!(2\pi i)^n}\right] =\begin{cases} \frac{|s|^k}{4k!(2\pi i)^{n-1}} &n \text{ odd}\\[5pt] -\frac{|s|^k\log|s|}{k!(2\pi i)^n}&n \text{ even.} \end{cases}</math>

Then δ is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure dω of g(x · ξ) for ξ in the unit sphere Sⁿ⁻¹: <math display="block">\delta(x) = \Delta_x^{(n+k)/2} \int_{S^{n-1}} g(x\cdot\xi)\,d\omega_\xi.</math>

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ, <math display="block">\varphi(x) = \int_{\mathbf{R}^n}\varphi(y)\,dy\,\Delta_x^{\frac{n+k}{2}} \int_{S^{n-1}} g((x-y)\cdot\xi)\,d\omega_\xi.</math>

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 φ(x) from its integrals over hyperplanes. For instance, if n is odd and k = 1, then the integral on the right hand side is <math display="block"> \begin{align} & c_n \Delta^{\frac{n+1}{2}}_x\iint_{S^{n-1}} \varphi(y)|(y-x) \cdot \xi| \, d\omega_\xi \, dy \\[5pt] & \qquad = c_n \Delta^{(n+1)/2}_x \int_{S^{n-1}} \, d\omega_\xi \int_{-\infty}^\infty |p| R\varphi(\xi,p+x\cdot\xi)\,dp \end{align} </math>

where Rφ(ξ, p) is the Radon transform of φ: <math display="block">R\varphi(\xi,p) = \int_{x\cdot\xi=p} f(x)\,d^{n-1}x.</math>

An alternative equivalent expression of the plane wave decomposition is:<ref name="FOOTNOTEGelfandShilov1966–1968I, §3.10">Gelfand & Shilov 1966–1968, I, §3.10.</ref> <math display="block">\delta(x) = \begin{cases}

 \frac{(n-1)!}{(2\pi i)^n}\displaystyle\int_{S^{n-1}}(x\cdot\xi)^{-n} \, d\omega_\xi & n\text{ even} \\
 \frac{1}{2(2\pi i)^{n-1}}\displaystyle\int_{S^{n-1}}\delta^{(n-1)}(x\cdot\xi)\,d\omega_\xi & n\text{ odd}.
 \end{cases}</math>

Fourier kernels

See also: Convergence of Fourier series

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 n-th partial sum of the Fourier series of a function f of period 2π is defined by convolution (on the interval [−π,π]) with the Dirichlet kernel: <math display="block">D_N(x) = \sum_{n=-N}^N e^{inx} = \frac{\sin\left(\left(N+\frac12\right)x\right)}{\sin(x/2)}.</math> Thus, <math display="block">s_N(f)(x) = D_N*f(x) = \sum_{n=-N}^N a_n e^{inx}</math> where <math display="block">a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy.</math> A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval [−π,π] tends to a multiple of the delta function as N → ∞. This is interpreted in the distribution sense, that <math display="block">s_N(f)(0) = \int_{-\pi}^{\pi} D_N(x)f(x)\,dx \to 2\pi f(0)</math> for every compactly supported smooth function f. Thus, formally one has <math display="block">\delta(x) = \frac1{2\pi} \sum_{n=-\infty}^\infty e^{inx}</math> on the interval [−π,π].

Despite this, the result does not hold for all compactly supported continuous functions: that is D_N 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 kernel<ref name="FOOTNOTELang1997312">Lang 1997, p. 312.</ref>

<math display="block">F_N(x) = \frac1N\sum_{n=0}^{N-1} D_n(x) = \frac{1}{N}\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)^2.</math>

The Fejér kernels tend to the delta function in a stronger sense that<ref>In the terminology of Lang (1997), the Fejér kernel is a Dirac sequence, whereas the Dirichlet kernel is not.</ref>

<math display="block">\int_{-\pi}^{\pi} F_N(x)f(x)\,dx \to 2\pi f(0)</math>

for every compactly supported continuous function f. 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 L², and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of L² 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 R implies that any square-integrable function f such that

<math display="block">\|f\|_{H^1}^2 = \int_{-\infty}^\infty |\widehat{f}(\xi)|^2 (1+|\xi|^2)\,d\xi < \infty</math>

is automatically continuous, and satisfies in particular

<math display="block">\delta[f]=|f(0)| < C \|f\|_{H^1}.</math>

Thus δ is a bounded linear functional on the Sobolev space H¹. Equivalently δ is an element of the continuous dual space H⁻¹ of H¹. More generally, in n dimensions, one has δ ∈ H^−s(Rⁿ) provided s > n/2.

Spaces of holomorphic functions

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

<math display="block">f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D</math>

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

<math display="block">\delta_z[f] = f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z}.</math>

Moreover, let H²(∂D) be the Hardy space consisting of the closure in L²(∂D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H²(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral formula continues to hold. In particular for z ∈ D, the delta function δ_z is a continuous linear functional on H²(∂D). This is a special case of the situation in several complex variables in which, for smooth domains D, the Szegő kernel plays the role of the Cauchy integral.<ref name="FOOTNOTEHazewinkel1995[httpsbooksgooglecombooksidPE1a-EIG22kCpgPA357 357]">Hazewinkel 1995, p. 357.</ref>

Resolutions of the identity

Given a complete orthonormal basis set of functions Template:Brace in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as <math display="block">f = \sum_{n=1}^\infty \alpha_n \varphi_n. </math> The coefficients {α_n} are found as <math display="block">\alpha_n = \langle \varphi_n, f \rangle,</math> which may be represented by the notation: <math display="block">\alpha_n = \varphi_n^\dagger f, </math> a form of the bra–ket notation of Dirac.<ref>

The development of this section in bra–ket notation is found in (Levin 2002, Coordinate-space wave functions and completeness, pp.=109ff)</ref> Adopting this notation, the expansion of f takes the dyadic form:<ref name="FOOTNOTEDavisThomson2000Perfect operators, p.344">Davis & Thomson 2000, Perfect operators, p.344.</ref>

<math display="block">f = \sum_{n=1}^\infty \varphi_n \left ( \varphi_n^\dagger f \right). </math>

Letting I denote the identity operator on the Hilbert space, the expression

<math display="block">I = \sum_{n=1}^\infty \varphi_n \varphi_n^\dagger, </math>

is called a resolution of the identity. When the Hilbert space is the space L²(D) of square-integrable functions on a domain D, the quantity:

<math display="block">\varphi_n \varphi_n^\dagger, </math>

is an integral operator, and the expression for f can be rewritten

<math display="block">f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi.</math>

The right-hand side converges to f in the L² sense. It need not hold in a pointwise sense, even when f is a continuous function. Nevertheless, it is common to abuse notation and write

<math display="block">f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, </math>

resulting in the representation of the delta function:<ref name="FOOTNOTEDavisThomson2000Equation 8.9.11, p. 344">Davis & Thomson 2000, Equation 8.9.11, p. 344.</ref>

<math display="block">\delta(x-\xi) = \sum_{n=1}^\infty \varphi_n (x) \varphi_n^*(\xi). </math>

With a suitable rigged Hilbert space (Φ, L²(D), Φ*) where Φ ⊂ L²(D) contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φ_n. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the distribution sense.<ref name="FOOTNOTEde la MadridBohmGadella2002">de la Madrid, Bohm & Gadella 2002.</ref>

Infinitesimal delta functions

Cauchy used an infinitesimal α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δ_α satisfying <math display="inline">\int F(x)\delta_\alpha(x) \,dx = F(0)</math> in a number of articles in 1827.<ref name="FOOTNOTELaugwitz1989">Laugwitz 1989.</ref> 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 Yamashita (2007) 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 F one has <math display="inline">\int F(x)\delta_\alpha(x) \, dx = F(0)</math> as anticipated by Fourier and Cauchy.

Dirac comb

Main article: Dirac comb

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T

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,

<math display="block">\operatorname{\text{Ш}}(x) = \sum_{n=-\infty}^\infty \delta(x-n),</math>

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 f is any Schwartz function, then the periodization of f is given by the convolution <math display="block">(f * \operatorname{\text{Ш}})(x) = \sum_{n=-\infty}^\infty f(x-n).</math> In particular, <math display="block">(f*\operatorname{\text{Ш}})^\wedge = \widehat{f}\widehat{\operatorname{\text{Ш}}} = \widehat{f}\operatorname{\text{Ш}}</math> is precisely the Poisson summation formula.<ref name="FOOTNOTECórdoba1988">Córdoba 1988.</ref><ref name="FOOTNOTEHörmander1983[httpsbooksgooglecombooksidaaLrCAAAQBAJpgPA177 §7.2]">Hörmander 1983, §7.2.</ref> More generally, this formula remains to be true if f is a tempered distribution of rapid descent or, equivalently, if <math>\widehat{f}</math> 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 p.v. 1/x, the Cauchy principal value of the function 1/x, defined by

<math display="block">\left\langle\operatorname{p.v.}\frac{1}{x}, \varphi\right\rangle = \lim_{\varepsilon\to 0^+}\int_{|x|>\varepsilon} \frac{\varphi(x)}{x}\,dx.</math>

Sokhotsky's formula states that<ref name="FOOTNOTEVladimirov1971§5.7">Vladimirov 1971, §5.7.</ref>

<math display="block">\lim_{\varepsilon\to 0^+} \frac{1}{x\pm i\varepsilon} = \operatorname{p.v.}\frac{1}{x} \mp i\pi\delta(x),</math>

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

<math display="block">\int_{-\infty}^{\infty}\lim_{\varepsilon\to0^{+}}\frac{f(x)}{x\pm i\varepsilon}\,dx=\mp i\pi f(0)+\lim_{\varepsilon\to0^{+}}\int_{|x|>\varepsilon}\frac{f(x)}{x}\,dx.</math>

Relationship to the Kronecker delta

The Kronecker delta δ_ij is the quantity defined by

<math display="block">\delta_{ij} = \begin{cases} 1 & i=j\\ 0 &i\not=j \end{cases} </math>

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

<math display="block">\sum_{i=-\infty}^\infty a_i \delta_{ik}=a_k.</math>

Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property

<math display="block">\int_{-\infty}^\infty f(x)\delta(x-x_0)\,dx=f(x_0).</math>

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.<ref name="FOOTNOTEHartmann1997pp. 154–155">Hartmann 1997, pp. 154–155.</ref>

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 f(x) of a discrete distribution consisting of points x = Template:Brace, with corresponding probabilities p₁, ..., p_n, can be written as

<math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

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

<math display="block">f(x) = 0.6 \, \frac {1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} + 0.4 \, \delta(x-3.5).</math>

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 Y = g(X) is a continuous differentiable function, then the density of Y can be written as

<math display="block">f_Y(y) = \int_{-\infty}^{+\infty} f_X(x) \delta(y-g(x)) \,dx. </math>

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 B(t) is given by <math display="block">\ell(x,t) = \int_0^t \delta(x-B(s))\,ds</math> and represents the amount of time that the process spends at the point x in the range of the process. More precisely, in one dimension this integral can be written <math display="block">\ell(x,t) = \lim_{\varepsilon\to 0^+}\frac{1}{2\varepsilon}\int_0^t \mathbf{1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds</math> where <math>\mathbf{1}_{[x-\varepsilon,x+\varepsilon]}</math> is the indicator function of the interval <math>[x-\varepsilon,x+\varepsilon].</math>

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 L² 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:Brace of wave functions is orthonormal if they are normalized by

<math display="block">\langle\varphi_n \mid \varphi_m\rangle = \delta_{nm}</math>

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

<math display="block"> \psi = \sum c_n \varphi_n, </math>

with c_n = Template:Bra-ket. 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, as above, this equality implies the resolution of the identity:

<math display="block">I = \sum |\varphi_n\rangle\langle\varphi_n|.</math>

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an observable may be continuous rather than discrete. An example is the position observable, Qψ(x) = xψ(x). 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: that is, to replace the Hilbert space of quantum mechanics with an appropriate rigged Hilbert space.<ref name="FOOTNOTEIsham1995§6.2">Isham 1995, §6.2.</ref> In this context, the position operator has a complete set of eigen-distributions, labeled by the points y of the real line, given by

<math display="block">\varphi_y(x) = \delta(x-y).</math>

The eigenfunctions of position are denoted by φ_y = |y⟩ in Dirac notation, and are known as position eigenstates.

Similar considerations apply to the eigenstates of the momentum operator, or indeed any other self-adjoint unbounded operator P on the Hilbert space, provided the spectrum of P is continuous and there are no degenerate eigenvalues. In that case, there is a set Ω of real numbers (the spectrum), and a collection φ_y of distributions indexed by the elements of Ω, such that

<math display="block">P\varphi_y = y\varphi_y.</math>

That is, φ_y are the eigenvectors of P. If the eigenvectors are normalized so that

<math display="block">\langle \varphi_y,\varphi_{y'}\rangle = \delta(y-y')</math>

in the distribution sense, then for any test function ψ,

<math display="block"> \psi(x) = \int_\Omega c(y) \varphi_y(x) \, dy</math>

where c(y) = ⟨ψ, φ_y⟩. That is, as in the discrete case, there is a resolution of the identity

<math display="block">I = \int_\Omega |\varphi_y\rangle\, \langle\varphi_y|\,dy</math>

where the operator-valued integral is again understood in the weak sense. If the spectrum of P 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 I at time t = 0 can be written

<math display="block">m \frac{d^2 \xi}{dt^2} + k \xi = I \delta(t),</math>

where m is the mass, ξ is the deflection, and k is the spring constant.

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

<math display="block">EI \frac{d^4 w}{dx^4} = q(x),</math>

where EI is the bending stiffness of the beam, w is the deflection, x is the spatial coordinate, and q(x) is the load distribution. If a beam is loaded by a point force F at x = x₀, the load distribution is written

<math display="block">q(x) = F \delta(x-x_0).</math>

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 F at a distance d apart. They then produce a moment M = Fd acting on the beam. Now, let the distance d approach the limit zero, while M is kept constant. The load distribution, assuming a clockwise moment acting at x = 0, is written

<math display="block">\begin{align} q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\[4pt] &= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\[4pt] &= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\[4pt] &= M \delta'(x). \end{align}</math>

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

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External links

• Media related to at Wikimedia Commons
• "Delta-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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. Archived 2008-03-07 at the Wayback Machine