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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Dirac delta distribution ( distribution), also known as the unit impulse, is a
generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
or
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
over the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, whose value is zero everywhere except at zero, and whose
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
over the entire real line is equal to one. The current understanding of the unit impulse is as a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
that maps every continuous function (e.g., f(x)) to its value at zero of its domain (f(0)), or as the weak limit of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
s (e.g., \delta(x) = \lim_ \frace^), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
as a tool for the normalization of state vectors. It also has uses in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
. Its validity was disputed until
Laurent Schwartz Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in 19 ...
developed the theory of distributions where it is defined as a linear form acting on functions. The
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
function, which is usually defined on a discrete domain and takes values 0 and 1, is the discrete analog of the Dirac delta function.


Motivation and overview

The
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of the Dirac delta is usually thought of as following the whole ''x''-axis and the positive ''y''-axis. The Dirac delta is used to model a tall narrow spike function (an ''impulse''), and other similar
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstr ...
s such as a
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
,
point mass A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
or
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
point. For example, to calculate the dynamics of a
billiard ball A billiard ball is a small, hard ball used in cue sports, such as carom billiards, pool, and snooker. The number, type, diameter, color, and pattern of the balls differ depending upon the specific game being played. Various particular ball p ...
being struck, one can approximate the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
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 In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...
of the ball by only considering the total impulse of the collision without a detailed model of all of the elastic energy transfer at subatomic levels (for instance). To be specific, suppose that a billiard ball is at rest. At time t=0 it is struck by another ball, imparting it with a
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
P, in \text^. 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 In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
therefore is P\,\delta(t). (The units of \delta(t) are \mathrm^.) To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval \Delta t = ,T/math>. That is, :F_(t) = \begin P/\Delta t& 0 Then the momentum at any time ''t'' is found by integration: :p(t) = \int_0^t F_(\tau)\,\mathrm d\tau = \begin P & t \ge T\\ P\,t/\Delta t & 0 \le t \le T\\ 0&\text\end Now, the model situation of an instantaneous transfer of momentum requires taking the limit as \Delta t\to 0, giving a result everywhere except at 0: :p(t)=\beginP & t > 0\\ 0 & t < 0.\end Here the functions F_ 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 In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it i ...
) \lim_F_ 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 :\int_^\infty F_(t)\,\mathrm t = P, which holds for all \Delta t>0, should continue to hold in the limit. So, in the equation F(t)=P\,\delta(t)=\lim_F_(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 (a weak limit) of a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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 In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
tending to zero. The Dirac delta is not truly a function, at least not a usual one with domain and range in
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. For example, the objects and are equal everywhere except at yet have integrals that are different. According to Lebesgue integration theory, if and are functions such that
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, then is integrable
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
is integrable and the integrals of and are identical. A rigorous approach to regarding the Dirac delta function as a
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical pr ...
in its own right requires
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
or the theory of
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
s.


History

Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
presented what is now called the
Fourier integral theorem In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency#Frequency_of_waves, fre ...
in his treatise ''Théorie analytique de la chaleur'' in the form:, cf. and pp. 546–551. The original French text may be found 'here'' :f(x)=\frac\int_^\infty\ \ d\alpha \, f(\alpha) \ \int_^\infty dp\ \cos (px-p\alpha)\ , which is tantamount to the introduction of the ''δ''-function in the form: :\delta(x-\alpha)=\frac \int_^\infty dp\ \cos (px-p\alpha) \ . Later,
Augustin Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
expressed the theorem using exponentials: :f(x)=\frac \int_ ^ \infty \ e^\left(\int_^\infty e^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 In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
). See, for example, As justified using the
theory of distributions A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the ''δ''-function as :\begin f(x)&=\frac \int_^\infty e^\left(\int_^\infty e^f(\alpha)\,d \alpha \right) \,dp \\ pt&=\frac \int_^\infty \left(\int_^\infty e^ e^ \,dp \right)f(\alpha)\,d \alpha =\int_^\infty \delta (x-\alpha) f(\alpha) \,d \alpha, \end where the ''δ''-function is expressed as :\delta(x-\alpha)=\frac \int_^\infty e^\,dp \ . A rigorous interpretation of the exponential form and the various limitations upon the function ''f'' necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows: : The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles. Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking ''L''2-theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945) ...", and leading to the formal development of the Dirac delta function. An
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
formula for an infinitely tall, unit impulse delta function (infinitesimal version of
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
) explicitly appears in an 1827 text of
Augustin Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
.
Siméon Denis Poisson Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
considered the issue in connection with the study of wave propagation as did
Gustav Kirchhoff Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coine ...
somewhat later. Kirchhoff and
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
also introduced the unit impulse as a limit of Gaussians, which also corresponded to
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
's notion of a point heat source. At the end of the 19th century,
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
used formal
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
to manipulate the unit impulse. The Dirac delta function as such was introduced by
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
in his 1927 paper ''The Physical Interpretation of the Quantum Dynamics'' and used in his textbook '' The Principles of Quantum Mechanics''. He called it the "delta function" since he used it as a continuous analogue of the discrete
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
.


Definitions

The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, : \delta(x) \simeq \begin +\infty, & x = 0 \\ 0, & x \ne 0 \end and which is also constrained to satisfy the identity :\int_^\infty \delta(x) \, \mathrm dx = 1. This is merely a
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties. The Dirac delta function can be rigorously defined either as a
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
or as a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
.


As a measure

One way to rigorously capture the notion of the Dirac delta function is to define a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
, called
Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...
, which accepts a subset of the real line as an argument, and returns if , and otherwise. If the delta function is conceptualized as modeling an idealized point mass at 0, then represents the mass contained in the set . One may then define the integral against as the integral of a function against this mass distribution. Formally, the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
provides the necessary analytic device. The Lebesgue integral with respect to the measure satisfies : \int_^\infty f(x) \, \delta(\mathrm dx) = f(0) for all continuous compactly supported functions . The measure is not
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
with respect to the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
—in fact, it is a
singular measure In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on ...
. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property :\int_^\infty f(x)\, \delta(x)\, \mathrm dx = f(0) holds. As a result, the latter notation is a convenient
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...
, and not a standard (
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
or
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
) integral. As a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
on , the delta measure is characterized by its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
, which is the
unit step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
. :H(x) = \begin 1 & \text x\ge 0\\ 0 & \text x < 0. \end This means that is the integral of the cumulative
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
with respect to the measure ; to wit, :H(x) = \int_\mathbf_(t)\,\delta(\mathrm dt) = \delta(-\infty,x], the latter being the measure of this interval; more formally, . Thus in particular the integration of the delta function against a continuous function can be properly understood as a
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
: :\int_^\infty f(x)\,\delta(\mathrm dx) = \int_^\infty f(x) \,\mathrm dH(x). All higher moments of are zero. In particular,
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
and
moment generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
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 about how it affects other functions when "integrated" against them. 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 function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
s. 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 function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s on R with
compact support In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
that have as many derivatives as required. As a distribution, the Dirac delta is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
on the space of test functions and is defined by for every test function ''\varphi''. 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 :\left, S
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
\le C_N \sum_^\sup_ \left, \varphi^(x)\ where sup represents the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
. With the ''δ'' distribution, one has such an inequality (with with for all ''N''. Thus ''δ'' is a distribution of order zero. It is, furthermore, a distribution with compact support (the
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
being ). The delta distribution can also be defined in several equivalent ways. For instance, it is the
distributional derivative Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...
of the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
. This means that for every test function ''φ'', one has :\delta
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
= -\int_^\infty \varphi'(x)\,H(x)\,\mathrm dx. Intuitively, if
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
were permitted, then the latter integral should simplify to :\int_^\infty \varphi(x)\,H'(x)\,\mathrm dx = \int_^\infty \varphi(x)\,\delta(x)\,\mathrm dx, and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have :-\int_^\infty \varphi'(x)\,H(x)\, \mathrm dx = \int_^\infty \varphi(x)\,\mathrm dH(x). In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation () defines a
Daniell integral In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional f ...
on the space of all compactly supported continuous functions ''φ'' which, by the
Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the R ...
, can be represented as the Lebesgue integral of ''φ'' concerning some
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...
. Generally, when the term "''Dirac delta function''" is used, it is in the sense of distributions rather than measures, the
Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...
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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
R''n'' as the measure such that :\int_ f(\mathbf)\,\delta(\mathrm d\mathbf) = f(\mathbf) for every compactly supported continuous function ''f''. As a measure, the ''n''-dimensional delta function is the
product measure In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two ...
of the 1-dimensional delta functions in each variable separately. Thus, formally, with , one has The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case. However, despite widespread use in engineering contexts, () should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances. The notion of a
Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...
makes sense on any set. Thus if ''X'' is a set, is a marked point, and Σ is any
sigma algebra Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as ...
of subsets of ''X'', then the measure defined on sets by :\delta_(A)=\begin 1 &\textx_0\in A\\ 0 &\textx_0\notin A \end is the delta measure or unit mass concentrated at ''x''0. Another common generalization of the delta function is to a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
where most of its properties as a distribution can also be exploited because of the
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for dif ...
. The delta function on a manifold ''M'' centered at the point is defined as the following distribution: for all compactly supported smooth real-valued functions ''φ'' on ''M''. A common special case of this construction is a case in which ''M'' is an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
in the Euclidean space R''n''. On a locally compact Hausdorff space ''X'', the Dirac delta measure concentrated at a point ''x'' is the
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...
associated with the Daniell integral () on compactly supported continuous functions ''φ''. At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping x_0\mapsto \delta_ is a continuous embedding of ''X'' into the space of finite Radon measures on ''X'', equipped with its
vague topology In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces. Let X be a locally co ...
. Moreover, the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of the image of ''X'' under this embedding is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in the space of probability measures on ''X''.


Properties


Scaling and symmetry

The delta function satisfies the following scaling property for a non-zero scalar ''α'': :\int_^\infty \delta(\alpha x)\,\mathrm dx =\int_^\infty \delta(u)\,\frac =\frac and so Scaling property proof:\begin \delta(\alpha x) &= \lim_ \frace^ \qquad \text b \text b=\alpha c \\ &=\lim_ \frace^ \\ &=\lim_ \frac \frace^ = \frac \delta(x) \end In this proof, the delta function representation as the limit of the sequence of zero-centered
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
s \delta(x) = \lim_ \frace^ is used. This proof can be made by using other delta function representations as the limits of sequences of functions, as long as these are even functions. In particular, the delta function is an
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...
distribution (symmetry), in the sense that :\delta(-x) = \delta(x) which is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
of degree −1.


Algebraic properties

The distributional product of ''δ'' with ''x'' is equal to zero: :x\,\delta(x) = 0. Conversely, if , where ''f'' and ''g'' are distributions, then :f(x) = g(x) +c \delta(x) for some constant ''c''.


Translation

The integral of the time-delayed Dirac delta is :\int_^\infty f(t) \,\delta(t-T)\,\mathrm dt = f(T). This is sometimes referred to as the ''sifting property'' or the ''sampling property''. The delta function is said to "sift out" the value of ''f(t)'' at ''t'' = ''T''. It follows that the effect of convolving a function ''f''(''t'') with the time-delayed Dirac delta \delta_T(t) = \delta(t-T) is to time-delay ''f''(''t'') by the same amount. This is sometimes referred to as the ''shifting property'' (not to be confused with the ''sifting property''): :\begin (f * \delta_T)(t) \ &\stackrel\ \int_^\infty f(\tau)\, \delta(t-T-\tau) \, \mathrm d\tau \\ &= \int_^\infty f(\tau) \,\delta(\tau-(t-T)) \,\mathrm d\tau \qquad \text~ \delta(-x) = \delta(x) ~~ \text\\ &= f(t-T). \end Note that the ''sifting property'' finds the value of a function centered at ''T'' whereas the ''shifting property'' returns a delayed function. The shifting property holds under the precise condition that ''f'' be a
tempered distribution Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
(see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution sense) :\int_^\infty \delta (\xi-x) \delta(x-\eta) \,\mathrm dx = \delta(\eta-\xi).


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 :\int_ \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) \left, g'(x)\ \mathrm dx = \int_ \delta(u)\,f(u)\,\mathrm du provided that ''g'' is a
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
function with ''g''′ nowhere zero. That is, there is a unique way to assign meaning to the distribution \delta\circ g so that this identity holds for all compactly supported test functions ''f''. Therefore, the domain must be broken up to exclude the ''g''′ = 0 point. This distribution satisfies if ''g'' is nowhere zero, and otherwise if ''g'' has a real
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
at ''x''0, then :\delta(g(x)) = \frac. It is natural therefore to ''define'' the composition ''δ''(''g''(''x'')) for continuously differentiable functions ''g'' by :\delta(g(x)) = \sum_i \frac where the sum extends over all roots (i.e., all the different ones) of ''g''(''x''), which are assumed to be
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
. Thus, for example :\delta\left(x^2-\alpha^2\right) = \frac \Big delta\left(x+\alpha\right)+\delta\left(x-\alpha\right)\Big In the integral form, the generalized scaling property may be written as : \int_^\infty f(x) \, \delta(g(x)) \, \mathrm dx = \sum_\frac.


Properties in ''n'' dimensions

The delta distribution in an ''n''-dimensional space satisfies the following scaling property instead, :\delta(\alpha\mathbf) = , \alpha, ^\delta(\mathbf) ~, so that ''δ'' is a
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
distribution of degree −''n''. Under any
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
or
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
ρ, the delta function is invariant, :\delta(\rho \mathbf) = \delta(\mathbf)~. As in the one-variable case, it is possible to define the composition of ''δ'' with a bi-Lipschitz function uniquely so that the identity :\int_ \delta(g(\mathbf))\, f(g(\mathbf))\left, \det g'(\mathbf)\ \mathrm d\mathbf = \int_ \delta(\mathbf) f(\mathbf)\,\mathrm d\mathbf for all compactly supported functions ''f''. Using the coarea formula from
geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
, 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 Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
. In the special case of a continuously differentiable function such that the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of ''g'' is nowhere zero, the following identity holds : \int_ f(\mathbf) \, \delta(g(\mathbf)) \,\mathrm d\mathbf = \int_\frac\,\mathrm d\sigma(\mathbf) where the integral on the right is over ''g''−1(0), the -dimensional surface defined by with respect to the
Minkowski content The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth ...
measure. This is known as a ''simple layer'' integral. More generally, if ''S'' is a smooth hypersurface of R''n'', then we can associate to ''S'' the distribution that integrates any compactly supported smooth function ''g'' over ''S'': :\delta_S = \int_S g(\mathbf)\,\mathrm d\sigma(\mathbf) where σ is the hypersurface measure associated to ''S''. This generalization is associated with the
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
of
simple layer potential In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental obje ...
s on ''S''. If ''D'' is a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
in R''n'' with smooth boundary ''S'', then ''δ''''S'' is equal to the
normal derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
of the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of ''D'' in the distribution sense, :-\int_g(\mathbf)\,\frac\,\mathrm d\mathbf=\int_S\,g(\mathbf)\, \mathrm d\sigma(\mathbf), where ''n'' is the outward normal. For a proof, see e.g. the article on the surface delta function.


Fourier transform

The delta function is a
tempered distribution Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
, and therefore it has a well-defined
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
. Formally, one finds :\widehat(\xi)=\int_^\infty e^ \,\delta(x)\mathrm dx = 1. Properly speaking, the Fourier transform of a distribution is defined by imposing
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a sta ...
ness of the Fourier transform under the duality pairing \langle\cdot,\cdot\rangle of tempered distributions with
Schwartz functions In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
. Thus \widehat is defined as the unique tempered distribution satisfying :\langle\widehat,\varphi\rangle = \langle\delta,\widehat\rangle for all Schwartz functions \varphi. And indeed it follows from this that \widehat=1. As a result of this identity, the
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of the delta function with any other tempered distribution ''S'' is simply ''S'': :S*\delta = S. That is to say that ''δ'' is an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
with identity the delta function. This property is fundamental in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, as convolution with a tempered distribution is a linear time-invariant system, and applying the linear time-invariant system measures its
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
. 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 . The inverse Fourier transform of the tempered distribution ''f''(''ξ'') = 1 is the delta function. Formally, this is expressed :\int_^\infty 1 \cdot e^\,\mathrm d\xi = \delta(x) and more rigorously, it follows since :\langle 1, \widehat\rangle = f(0) = \langle\delta,f\rangle 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 :\int_^\infty e^ \left ^\right*\,\mathrm dt = \int_^\infty e^ \,\mathrm dt = \delta(\xi_2 - \xi_1). This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution :f(t) = e^ is :\widehat(\xi_2) = \delta(\xi_1-\xi_2) which again follows by imposing self-adjointness of the Fourier transform. By
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
of the Fourier transform, the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
of the delta function is found to be : \int_^\delta(t-a)\,e^ \, \mathrm dt=e^.


Derivatives of the Dirac delta function

The derivative of the Dirac delta distribution, denoted \delta^\prime and also called the ''Dirac delta prime'' or ''Dirac delta derivative'' as described in
Laplacian of the indicator In mathematics, the Laplacian of the indicator of the domain ''D'' is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the ''surface'' of ''D''. It can be viewed as the ''surface delta pr ...
, is defined on compactly supported smooth test functions \varphi by :\delta'
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
= -\delta varphi'-\varphi'(0). The first equality here is a kind of integration by parts, for if \delta were a true function then :\int_^\infty \delta'(x)\varphi(x)\,dx = -\int_^\infty \delta(x) \varphi'(x)\,dx. The k-th derivative of \delta is defined similarly as the distribution given on test functions by :\delta^
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
= (-1)^k \varphi^(0). In particular, \delta is an infinitely differentiable distribution. The first derivative of the delta function is the distributional limit of the difference quotients: :\delta'(x) = \lim_ \frac. More properly, one has :\delta' = \lim_ \frac(\tau_h\delta - \delta) where \tau_h is the translation operator, defined on functions by \tau_h \varphi(x) = \varphi(x + h), and on a distribution S by :(\tau_h S)
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
= S tau_\varphi In the theory of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, the first derivative of the delta function represents a point magnetic
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
situated at the origin. Accordingly, it is referred to as a dipole or the doublet function. The derivative of the delta function satisfies a number of basic properties, including: : \begin & \delta'(-x) = -\delta'(x) \\ & x\delta'(x) = -\delta(x) \end 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, Liebnitz's theorem and linearity of inner product: \begin \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 Furthermore, the convolution of \delta' with a compactly-supported, smooth function f is :\delta'*f = \delta*f' = f', which follows from the properties of the distributional derivative of a convolution.


Higher dimensions

More generally, on an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
U in the n-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
\mathbb^n, the Dirac delta distribution centered at a point a \in U is defined by :\delta_a
varphi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
\varphi(a) for all \varphi \in C_c^\infty(U), the space of all smooth functions with compact support on U. If \alpha = (\alpha_1, \ldots, \alpha_n) is any
multi-index Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
with , \alpha, =\alpha_1+\cdots+\alpha_n and \partial^\alpha denotes the associated mixed
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
operator, then the \alpha-th derivative \partial^\alpha \delta_a of \delta_a is given by :\left\langle \partial^\alpha \delta_, \, \varphi \right\rangle = (-1)^ \left\langle \delta_, \partial^ \varphi \right\rangle = (-1)^ \partial^\alpha \varphi (x) \Big, _ \quad \text \varphi \in C_c^\infty(U). That is, the \alpha-th derivative of \delta_a is the distribution whose value on any test function \varphi is the \alpha-th derivative of \varphi 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 In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
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
multipole A multipole expansion is a Series (mathematics), mathematical series representing a Function (mathematics), function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and Azimuth, azimuthal angles) f ...
s. 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 \ consisting of a single point, then there is an integer m and coefficients c_\alpha such that :S = \sum_ c_\alpha \partial^\alpha\delta_a.


Representations of the delta function

The delta function can be viewed as the limit of a sequence of functions :\delta (x) = \lim_ \eta_\varepsilon(x), where ''ηε''(''x'') is sometimes called a nascent delta function. This limit is meant in a weak sense: either that for all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
functions ''f'' having
compact support In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
, or that this limit holds for all
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
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 In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces. Let X be a locally co ...
of measures, and the latter is convergence in the sense of distributions.


Approximations to the identity

Typically a nascent delta function ''ηε'' can be constructed in the following manner. Let ''η'' be an absolutely integrable function on R of total integral 1, and define :\eta_\varepsilon(x) = \varepsilon^ \eta \left (\frac \right). In ''n'' dimensions, one uses instead the scaling :\eta_\varepsilon(x) = \varepsilon^ \eta \left (\frac \right). Then a simple change of variables shows that ''ηε'' also has integral 1. One may show that () holds for all continuous compactly supported functions ''f'', and so ''ηε'' converges weakly to ''δ'' in the sense of measures. The ''ηε'' constructed in this way are known as an approximation to the identity. This terminology is because the space ''L''1(R) of absolutely integrable functions is closed under the operation of
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of functions: whenever ''f'' and ''g'' are in ''L''1(R). However, there is no identity in ''L''1(R) for the convolution product: no element ''h'' such that for all ''f''. Nevertheless, the sequence ''ηε'' does approximate such an identity in the sense that :f*\eta_\varepsilon \to f \quad \text\varepsilon\to 0. This limit holds in the sense of
mean convergence In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
(convergence in ''L''1). Further conditions on the ''ηε'', for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. If the initial is itself smooth and compactly supported then the sequence is called a
mollifier In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
. The standard mollifier is obtained by choosing ''η'' to be a suitably normalized
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
, for instance :\eta(x) = \begin e^& \text , x, < 1\\ 0 & \text , x, \geq 1. \end In some situations such as
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
, a piecewise linear approximation to the identity is desirable. This can be obtained by taking ''η''1 to be a
hat function A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and base 2 in which case it is referred to as ''th ...
. With this choice of ''η''1, one has : \eta_\varepsilon(x) = \varepsilon^\max \left (1-\left, \frac\,0 \right) which are all continuous and compactly supported, although not smooth and so not a mollifier.


Probabilistic considerations

In the context of
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, it is natural to impose the additional condition that the initial ''η''1 in an approximation to the identity should be positive, as such a function then represents a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
. Convolution with a probability distribution is sometimes favorable because it does not result in overshoot or undershoot, as the output is a
convex combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...
of the input values, and thus falls between the maximum and minimum of the input function. Taking ''η''1 to be any probability distribution at all, and letting 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 ''η''1 is the uniform distribution on , also known as the
rectangular function The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname(t) = \Pi(t) = \left\{\begin{array}{r ...
, then: :\eta_\varepsilon(x) = \frac\operatorname\left(\frac\right)= \begin \frac,&-\frac Another example is with the
Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)=\sq ...
:\eta_\varepsilon(x)= \begin \frac\sqrt, & -\varepsilon < x < \varepsilon, \\ 0, & \text. \end This is continuous and compactly supported, but not a mollifier because it is not smooth.


Semigroups

Nascent delta functions often arise as convolution
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s. This amounts to the further constraint that the convolution of ''ηε'' with ''ηδ'' must satisfy :\eta_\varepsilon * \eta_\delta = \eta_ for all ''ε'', . Convolution semigroups in ''L''1 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 solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ad ...
s or
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
s to physically motivated
elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
or parabolic
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
. In the context of
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, 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 In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or oth ...
:\begin \dfrac\eta(t,x) = A\eta(t,x), \quad t>0 \\ pt\displaystyle\lim_ \eta(t,x) = \delta(x) \end in which the limit is as usual understood in the weak sense. Setting 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 In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectru ...
, defined by :\eta_\varepsilon(x) = \frac \mathrm^ 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 In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
: :\frac = \frac\frac. In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, ''ηε''(''x'') is a
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
of
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
''ε'' and mean 0. It represents the
probability density In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
at time of the position of a particle starting at the origin following a standard
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. In this context, the semigroup condition is then an expression of the
Markov property In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov propert ...
of Brownian motion. In higher-dimensional Euclidean space R''n'', the heat kernel is :\eta_\varepsilon = \frac\mathrm^, and has the same physical interpretation, ''
mutatis mutandis ''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used i ...
''. It also represents a nascent delta function in the sense that in the distribution sense as . ;The Poisson kernel The
Poisson kernel In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriv ...
:\eta_\varepsilon(x) = \frac\mathrm\left\=\frac \frac=\frac\int_^\mathrm^\,d\xi is the fundamental solution of the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
in the upper half-plane. It represents the
electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
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 The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
and Epanechnikov and Gaussian kernel functions. This semigroup evolves according to the equation :\frac = -\left (-\frac \right)^u(t,x) where the operator is rigorously defined as the Fourier multiplier :\mathcal\left left(-\frac \right)^f\right\xi) = , 2\pi\xi, \mathcalf(\xi).


Oscillatory integrals

In areas of physics such as
wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative to ...
and
wave mechanics Wave mechanics may refer to: * the mechanics of waves * the ''wave equation'' in quantum physics, see Schrödinger equation See also * Quantum mechanics * Wave equation The (two-way) wave equation is a second-order linear partial different ...
, the equations involved are
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
and so may have more singular solutions. As a result, the nascent delta functions that arise as fundamental solutions of the associated
Cauchy problem A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value proble ...
s are generally
oscillatory integral In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators fo ...
s. An example, which comes from a solution of the
Euler–Tricomi equation In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi. : u_+xu_=0. \, It is elliptic in the ha ...
of
transonic Transonic (or transsonic) flow is air flowing around an object at a speed that generates regions of both subsonic and supersonic airflow around that object. The exact range of speeds depends on the object's critical Mach number, but transonic ...
gas dynamics Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the r ...
, is the rescaled
Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solut ...
:\varepsilon^\operatorname\left (x\varepsilon^ \right). Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonneg ...
below), rather than in the sense of measures. Another example is the Cauchy problem for the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
in R1+1: : \begin c^\frac - \Delta u &= 0\\ u=0,\quad \frac = \delta &\qquad \textt=0. \end 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 In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
(used widely in electronics and telecommunications) :\eta_\varepsilon(x)=\frac\sin\left(\frac\right)=\frac\int_^ \cos(kx)\,dk and the
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
: \eta_\varepsilon(x) = \fracJ_ \left(\frac\right).


Plane wave decomposition

One approach to the study of a linear partial differential equation :L f, where ''L'' is a
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
on R''n'', is to seek first a fundamental solution, which is a solution of the equation :L \delta. 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 :L h where ''h'' is a
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
function, meaning that it has the form :h = h(x\cdot\xi) for some vector ξ. Such an equation can be resolved (if the coefficients of ''L'' are
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s) 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 Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna). Life RadonBrigitte Bukovics: ''Biography of Johan ...
, and then developed in this form by
Fritz John Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a ...
(
1955 Events January * January 3 – José Ramón Guizado becomes president of Panama. * January 17 – , the first nuclear-powered submarine, puts to sea for the first time, from Groton, Connecticut. * January 18– 20 – Battle of Yijian ...
). Choose ''k'' so that is an even integer, and for a real number ''s'', put :g(s) = \operatorname\left frac\right=\begin \frac &n \text\\ pt-\frac&n \text \end Then ''δ'' is obtained by applying a power of the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
to the integral with respect to the unit
sphere measure In mathematics — specifically, in geometric measure theory — spherical measure ''σ'n'' is the "natural" Borel measure on the ''n''-sphere S''n''. Spherical measure is often normalized so that it is a probability measure on the ...
dω of for ''ξ'' in the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
''S''''n''−1: :\delta(x) = \Delta_x^ \int_ g(x\cdot\xi)\,d\omega_\xi. The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function ''φ'', :\varphi(x) = \int_\varphi(y)\,dy\,\Delta_x^ \int_ g((x-y)\cdot\xi)\,d\omega_\xi. The result follows from the formula for the
Newtonian potential In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object ...
(the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the
Radon transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the l ...
because it recovers the value of ''φ''(''x'') from its integrals over hyperplanes. For instance, if ''n'' is odd and , then the integral on the right hand side is : \begin & c_n \Delta^_x\iint_ \varphi(y), (y-x) \cdot \xi, \, d\omega_\xi \, dy \\ pt= & c_n \Delta^_x \int_ \, d\omega_\xi \int_^\infty , p, R\varphi(\xi,p+x\cdot\xi)\,dp \end where is the Radon transform of ''φ'': :R\varphi(\xi,p) = \int_ f(x)\,d^x. An alternative equivalent expression of the plane wave decomposition, from , is : \delta(x) = \frac\int_(x\cdot\xi)^ \, d\omega_\xi for ''n'' even, and :\delta(x) = \frac\int_\delta^(x\cdot\xi)\,d\omega_\xi for ''n'' odd.


Fourier kernels

In the study of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, a major question consists of determining whether and in what sense the Fourier series associated with a
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
converges to the function. The ''n''th partial sum of the Fourier series of a function ''f'' of period 2 is defined by convolution (on the interval ) with the
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonneg ...
: :D_N(x) = \sum_^N e^ = \frac. Thus, :s_N(f)(x) = D_N*f(x) = \sum_^N a_n e^ where :a_n = \frac\int_^\pi f(y)e^\,dy. A fundamental result of elementary Fourier series states that the Dirichlet kernel tends to the a multiple of the delta function as . This is interpreted in the distribution sense, that :s_N(f)(0) = \int_ D_N(x)f(x)\,dx \to 2\pi f(0) for every compactly supported ''smooth'' function ''f''. Thus, formally one has :\delta(x) = \frac1 \sum_^\infty e^ on the interval . Despite this, the result does not hold for all compactly supported ''continuous'' functions: that is ''DN'' 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 In mathematical analysis, Cesàro summation (also known as the Cesàro mean ) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
leads to the
Fejér kernel In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lip ...
:F_N(x) = \frac1N\sum_^ D_n(x) = \frac\left(\frac\right)^2. The
Fejér kernel In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lip ...
s tend to the delta function in a stronger sense that :\int_ F_N(x)f(x)\,dx \to 2\pi f(0) 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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
on the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
L2 of
square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
s. Indeed, smooth compactly supported functions are
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in ''L''2, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of ''L''2 and to give a stronger
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
on which the delta function defines a
bounded linear functional In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector s ...
. ; Sobolev spaces The
Sobolev embedding theorem In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the R ...
for
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
s on the real line R implies that any square-integrable function ''f'' such that :\, f\, _^2 = \int_^\infty , \widehat(\xi), ^2 (1+, \xi, ^2)\,d\xi < \infty is automatically continuous, and satisfies in particular :\delta , f(0), < C \, f\, _. Thus ''δ'' is a bounded linear functional on the Sobolev space ''H''1. Equivalently ''δ'' is an element of the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
''H''−1 of ''H''1. More generally, in ''n'' dimensions, one has provided .


Spaces of holomorphic functions

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the delta function enters via
Cauchy's integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
, which asserts that if ''D'' is a domain in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
with smooth boundary, then :f(z) = \frac \oint_ \frac,\quad z\in D for all
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s ''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: :\delta_z = f(z) = \frac \oint_ \frac. Moreover, let ''H''2(∂''D'') be the
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . ...
consisting of the closure in ''L''2(∂''D'') of all holomorphic functions in ''D'' continuous up to the boundary of ''D''. Then functions in ''H''2(∂''D'') uniquely extend to holomorphic functions in ''D'', and the Cauchy integral formula continues to hold. In particular for , the delta function ''δ''''z'' is a continuous linear functional on ''H''2(∂''D''). This is a special case of the situation in
several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
in which, for smooth domains ''D'', the
Szegő kernel In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathemati ...
plays the role of the Cauchy integral.


Resolutions of the identity

Given a complete
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
set of functions in a separable Hilbert space, for example, the normalized
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of a compact self-adjoint operator, any vector ''f'' can be expressed as :f = \sum_^\infty \alpha_n \varphi_n. The coefficients are found as :\alpha_n = \langle \varphi_n, f \rangle, which may be represented by the notation: :\alpha_n = \varphi_n^\dagger f, a form of the bra–ket notation of Dirac. The development of this section in bra–ket notation is found in Adopting this notation, the expansion of ''f'' takes the dyadic form: :f = \sum_^\infty \varphi_n \left ( \varphi_n^\dagger f \right). Letting ''I'' denote the
identity operator Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), a ...
on the Hilbert space, the expression :I = \sum_^\infty \varphi_n \varphi_n^\dagger, is called a resolution of the identity. When the Hilbert space is the space ''L''2(''D'') of square-integrable functions on a domain ''D'', the quantity: :\varphi_n \varphi_n^\dagger, is an integral operator, and the expression for ''f'' can be rewritten :f(x) = \sum_^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi. The right-hand side converges to ''f'' in the ''L''2 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 :f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, resulting in the representation of the delta function: :\delta(x-\xi) = \sum_^\infty \varphi_n (x) \varphi_n^*(\xi). With a suitable
rigged Hilbert space In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study s ...
where 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 Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
sense.


Infinitesimal delta functions

Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
used an infinitesimal ''α'' to write down a unit impulse, infinitely tall and narrow Dirac-type delta function ''δα'' satisfying \int F(x)\delta_\alpha(x) \,dx = F(0) in a number of articles in 1827. Cauchy defined an infinitesimal in ''
Cours d'Analyse ''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in ...
'' (1827) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and
Lazare Carnot Lazare Nicolas Marguerite, Count Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician. He was known as the "Organizer of Victory" in the French Revolutionary Wars and Napoleonic Wars. Education and early ...
's terminology.
Non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
allows one to rigorously treat infinitesimals. The article by contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the
hyperreals In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
. Here the Dirac delta can be given by an actual function, having the property that for every real function ''F'' one has \int F(x)\delta_\alpha(x) \, dx = F(0) as anticipated by Fourier and Cauchy.


Dirac comb

A so-called uniform "pulse train" of Dirac delta measures, which is known as a
Dirac comb In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and the ...
, or as the Sha distribution, creates a sampling function, often used in
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
(DSP) and discrete time signal analysis. The Dirac comb is given as the
infinite sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, whose limit is understood in the distribution sense, :\operatorname(x) = \sum_^\infty \delta(x-n), which is a sequence of point masses at each of the integers. Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if f is any Schwartz function, then the
periodization In historiography, periodization is the process or study of categorizing the past into discrete, quantified, and named blocks of time for the purpose of study or analysis.Adam Rabinowitz. It's about time: historical periodization and Linked Ancie ...
of f is given by the convolution :(f * \operatorname)(x) = \sum_^\infty f(x-n). In particular, :(f*\operatorname)^\wedge = \widehat\widehat = \widehat\operatorname is precisely the
Poisson summation formula In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a ...
. More generally, this formula remains to be true if f is a tempered distribution of rapid descent or, equivalently, if \widehat is a slowly growing, ordinary function within the space of tempered distributions.


Sokhotski–Plemelj theorem

The
Sokhotski–Plemelj theorem The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it ( see below) is often used in physics, although rarely referred to by nam ...
, important in quantum mechanics, relates the delta function to the distribution p.v. 1/''x'', the
Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of singularity in the integrand , ...
of the function 1/''x'', defined by :\left\langle\operatorname\frac, \varphi\right\rangle = \lim_\int_ \frac\,dx. Sokhotsky's formula states that :\lim_ \frac = \operatorname\frac \mp i\pi\delta(x), Here the limit is understood in the distribution sense, that for all compactly supported smooth functions ''f'', :\lim_ \int_^\infty\frac\,dx = \mp i\pi f(0) + \lim_ \int_\frac\,dx.


Relationship to the Kronecker delta

The
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
''δij'' is the quantity defined by :\delta_ = \begin 1 & i=j\\ 0 &i\not=j \end for all integers ''i'', ''j''. This function then satisfies the following analog of the sifting property: if (a_i)_ is any
doubly infinite sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
, then :\sum_^\infty a_i \delta_=a_k. Similarly, for any real or complex valued continuous function ''f'' on R, the Dirac delta satisfies the sifting property :\int_^\infty f(x)\delta(x-x_0)\,dx=f(x_0). This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.


Applications


Probability theory

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the Dirac delta function is often used to represent a
discrete distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, or a partially discrete, partially
continuous distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, using a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
(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 = , with corresponding probabilities ''p''1, ..., ''pn'', can be written as :f(x) = \sum_^n p_i \delta(x-x_i). As another example, consider a distribution in which 6/10 of the time returns a standard
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete
mixture distribution In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection a ...
). The density function of this distribution can be written as :f(x) = 0.6 \, \frac e^ + 0.4 \, \delta(x-3.5). The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If ''Y'' = g(''X'') is a continuous differentiable function, then the density of ''Y'' can be written as :f_Y(y) = \int_^ f_X(x) \delta(y-g(x)) d x. The delta function is also used in a completely different way to represent the
local time Local time is the time observed in a specific locality. There is no canonical definition. Originally it was mean solar time, but since the introduction of time zones it is generally the time as determined by the time zone in effect, with daylight s ...
of a
diffusion process In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
(like
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
). The local time of a stochastic process ''B''(''t'') is given by :\ell(x,t) = \int_0^t \delta(x-B(s))\,ds 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 :\ell(x,t) = \lim_\frac\int_0^t \mathbf_(B(s))\,ds where is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of the interval .


Quantum mechanics

The delta function is expedient in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
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 ''L''2 of
square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
s, 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 of wave functions is orthonormal if they are normalized by :\langle\varphi_n \mid \varphi_m\rangle = \delta_ where \delta is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function , \psi\rangle can be expressed as a linear combination of the with complex coefficients: : \psi = \sum c_n \varphi_n, with c_n = \langle \varphi_n , \psi \rangle . Complete orthonormal systems of wave functions appear naturally as the
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of the Hamiltonian. In bra–ket notation, as above, this equality implies the resolution of the identity: : I = \sum , \varphi_n\rangle\langle\varphi_n, . Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
may be continuous rather than discrete. An example is the position observable, . 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 In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study s ...
. In this context, the position operator has a complete set of eigen-distributions, labeled by the points ''y'' of the real line, given by :\varphi_y(x) = \delta(x-y). The eigenfunctions of position are denoted by \varphi_y = , y\rangle in Dirac notation, and are known as position eigenstates. Similar considerations apply to the eigenstates of the
momentum operator In quantum mechanics, the momentum operator is the operator (physics), operator associated with the momentum (physics), linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case o ...
, or indeed any other self-adjoint
unbounded operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The ter ...
''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 :P\varphi_y = y\varphi_y. That is, ''φ''''y'' are the eigenvectors of ''P''. If the eigenvectors are normalized so that :\langle \varphi_y,\varphi_\rangle = \delta(y-y') in the distribution sense, then for any test function ψ, : \psi(x) = \int_\Omega c(y) \varphi_y(x) \, dy where : c(y) = \langle \psi, \varphi_y \rangle. That is, as in the discrete case, there is a resolution of the identity :I = \int_\Omega , \varphi_y\rangle\, \langle\varphi_y, \,dy 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 In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it t ...
models for a single and double potential well.


Structural mechanics

The delta function can be used in
structural mechanics Structural mechanics or Mechanics of structures is the computation of deformations, deflections, and internal forces or stresses (''stress equivalents'') within structures, either for design or for performance evaluation of existing structures. It ...
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 Impulse or Impulsive may refer to: Science * Impulse (physics), in mechanics, the change of momentum of an object; the integral of a force with respect to time * Impulse noise (disambiguation) * Specific impulse, the change in momentum per uni ...
''I'' at time ''t'' = 0 can be written :m \frac + k \xi = I \delta(t), where ''m'' is the mass, ξ the deflection and ''k'' the
spring constant In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
. As another example, the equation governing the static deflection of a slender
beam Beam may refer to: Streams of particles or energy *Light beam, or beam of light, a directional projection of light energy **Laser beam *Particle beam, a stream of charged or neutral particles **Charged particle beam, a spatially localized grou ...
is, according to Euler–Bernoulli theory, :EI \frac = q(x), where ''EI'' is the
bending stiffness The bending stiffness (K) is the resistance of a member against bending deformation. It is a function of the Young's modulus E, the second moment of area I of the beam cross-section about the axis of interest, length of the beam and beam boundary c ...
of the beam, ''w'' the
deflection Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflection ...
, ''x'' the spatial coordinate and ''q''(''x'') the load distribution. If a beam is loaded by a point force ''F'' at ''x'' = ''x''0, the load distribution is written :q(x) = F \delta(x-x_0). As the integration of the delta function results in the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s. 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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
zero, while ''M'' is kept constant. The load distribution, assuming a clockwise moment acting at ''x'' = 0, is written :\begin q(x) &= \lim_ \Big( F \delta(x) - F \delta(x-d) \Big) \\ pt&= \lim_ \left( \frac \delta(x) - \frac \delta(x-d) \right) \\ pt&= M \lim_ \frac\\ pt&= M \delta'(x). \end Point moments can thus be represented by the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the delta function. Integration of the beam equation again results in piecewise
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
deflection.


See also

*
Atom (measure theory) In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless. Definition Given a measurable s ...
*
Laplacian of the indicator In mathematics, the Laplacian of the indicator of the domain ''D'' is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the ''surface'' of ''D''. It can be viewed as the ''surface delta pr ...


Notes


References

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External links

* *
KhanAcademy.org video lessonThe Dirac Delta function
a tutorial on the Dirac delta function.
Video Lectures – Lecture 23
a lecture by
Arthur Mattuck Arthur Paul Mattuck (June 11, 1930 – October 8, 2021) was an emeritus professor of mathematics at the Massachusetts Institute of Technology. He may be best known for his 1998 book, ''Introduction to Analysis'' () and his differential equations ...
.
The Dirac delta measure is a hyperfunctionWe show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure
{{good article Fourier analysis Generalized functions Measure theory Digital signal processing
Delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...