Dirac delta function
In mathematics, the Dirac delta function (δ function) is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one.[1][2][3] As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero.[4][5] The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.
In engineering and signal processing, the delta function, also known as the unit impulse symbol,[6] may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable. The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.
Motivation and overview
The graph of the delta function is usually thought of as following the whole xaxis and the positive yaxis. The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar abstractions such as a point charge, point mass or electron point. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a delta function. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball by only considering the total impulse of the collision without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).
To be specific, suppose that a billiard ball is at rest. At time it is struck by another ball, imparting it with a momentum P, in . The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The force therefore is . (The units of are .)
To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval . That is,
Then the momentum at any time t is found by integration:
Now, the model situation of an instantaneous transfer of momentum requires taking the limit as , giving
Here the functions 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 ordinary calculus) is zero everywhere but a single point, where it is infinite. To make proper sense of the delta function, we should instead insist that the property
which holds for all , should continue to hold in the limit. So, in the equation , it is understood that the limit is always taken outside the integral.
In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.
Despite its name, the delta function is not truly a function, at least not a usual one with range in real numbers. For example, the objects f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions.
History
Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:[7]
which is tantamount to the introduction of the δfunction in the form:[8]
Later, Augustin Cauchy expressed the theorem using exponentials:[9][10]
Cauchy pointed out that in some circumstances the order of integration in this result is significant (contrast Fubini's theorem).[11][12]
As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δfunction as
where the δfunction is expressed as
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:[13]
 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) in order 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) ...",[14] and leading to the formal development of the Dirac delta function.
An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy.[15] Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse.[16] The Dirac delta function as such was introduced as a "convenient notation" by Paul Dirac in his influential 1930 book The Principles of Quantum Mechanics.[17] He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.
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,
and which is also constrained to satisfy the identity
This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.[17] The Dirac delta function can be rigorously defined either as a distribution or as a measure.
As a measure
One way to rigorously capture the notion of the Dirac delta function is to define a measure, which accepts a subset A of the real line R as an argument, and returns δ(A) = 1 if 0 ∈ A, and δ(A) = 0 otherwise.[19] If the delta function is conceptualized as modeling an idealized point mass at 0, then δ(A) represents the mass contained in the set A. One may then define the integral against δ as the integral of a function against this mass distribution. Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure δ satisfies
for all continuous compactly supported functions f. The measure δ is not absolutely continuous with respect to the Lebesgue measure — in fact, it is a singular measure. Consequently, the delta measure has no Radon–Nikodym derivative — no true function for which the property
holds.[20] As a result, the latter notation is a convenient abuse of notation, and not a standard (Riemann or Lebesgue) integral.
As a probability measure on R, the delta measure is characterized by its cumulative distribution function, which is the unit step function[21]
This means that H(x) is the integral of the cumulative indicator function 1_{(−∞, x]} with respect to the measure δ; to wit,
the latter being the measure of this interval; more formally, Thus in particular the integral of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:[22]
All higher moments of δ are zero. In particular, characteristic function and moment generating function are both equal to one.
As a distribution
In the theory of distributions, a generalized function is considered not a function in itself but only in relation to 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 functions. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.
A typical space of test functions consists of all smooth functions on R with compact support that have as many derivatives as required. As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by[23]

(1)
for every test function .
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[24]
With the δ distribution, one has such an inequality (with C_{N} = 1) with M_{N} = 0 for all N. Thus δ is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being {0}).
The delta distribution can also be defined in a number of equivalent ways. For instance, it is the distributional derivative of the Heaviside step function. This means that, for every test function φ, one has
Intuitively, if integration by parts were permitted, then the latter integral should simplify to
and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case one does have
In the context of measure theory, the Dirac measure gives rise to a distribution by integration. Conversely, equation (1) defines a Daniell integral on the space of all compactly supported continuous functions φ which, by the Riesz representation theorem, can be represented as the Lebesgue integral of φ with respect to some Radon measure.
Generally, when the term "Dirac delta function" is used, it is in the sense of distributions rather than measures, the Dirac measure being among several terms for the corresponding notion in measure theory. Some sources may also use the term Dirac delta distribution.
Generalizations
The delta function can be defined in ndimensional Euclidean space R^{n} as the measure such that
for every compactly supported continuous function f. As a measure, the ndimensional delta function is the product measure of the 1dimensional delta functions in each variable separately. Thus, formally, with x = (x_{1}, x_{2}, ..., x_{n}), one has[6]

(2)
The delta function can also be defined in the sense of distributions exactly as above in the onedimensional case.[25] However, despite widespread use in engineering contexts, (2) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.[26]
The notion of a Dirac measure makes sense on any set.[19] Thus if X is a set, x_{0} ∈ X is a marked point, and Σ is any sigma algebra of subsets of X, then the measure defined on sets A ∈ Σ by
is the delta measure or unit mass concentrated at x_{0}.
Another common generalization of the delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the differentiable structure. The delta function on a manifold M centered at the point x_{0} ∈ M is defined as the following distribution:

(3)
for all compactly supported smooth realvalued functions φ on M.[27] A common special case of this construction is that in which M is an open set 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 associated with the Daniell integral (3) on compactly supported continuous functions φ.[28] 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 is a continuous embedding of X into the space of finite Radon measures on X, equipped with its vague topology. Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X.[29]
Properties
Scaling and symmetry
The delta function satisfies the following scaling property for a nonzero scalar α:[30]
and so

(4)
In particular, the delta function is an even distribution, in the sense that
which is homogeneous of degree −1.
Algebraic properties
The distributional product of δ with x is equal to zero:
Conversely, if xf(x) = xg(x), where f and g are distributions, then
for some constant c.[31]
Translation
The integral of the timedelayed Dirac delta is
This is sometimes referred to as the sifting property[32] or the sampling property. The delta function is said to "sift out" the value at t = T.
It follows that the effect of convolving a function f(t) with the timedelayed Dirac delta is to timedelay f(t) by the same amount:
(using (4): )
This holds under the precise condition that f be a tempered distribution (see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution sense)
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
provided that g is a continuously differentiable function with g′ nowhere zero.[33] That is, there is a unique way to assign meaning to the distribution so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g′ = 0 point. This distribution satisfies δ(g(x)) = 0 if g is nowhere zero, and otherwise if g has a real root at x_{0}, then
It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions g by
where the sum extends over all roots of g(x), which are assumed to be simple.[33] Thus, for example
In the integral form the generalized scaling property may be written as
Properties in n dimensions
The delta distribution in an ndimensional space satisfies the following scaling property instead,
so that δ is a homogeneous distribution of degree −n.
Under any reflection or rotation ρ, the delta function is invariant,
As in the onevariable case, it is possible to define the composition of δ with a biLipschitz function[34] g: R^{n} → R^{n} uniquely so that the identity
for all compactly supported functions f.
Using the coarea formula from geometric measure theory, one can also define the composition of the delta function with a submersion from one Euclidean space to another one of different dimension; the result is a type of current. In the special case of a continuously differentiable function g: R^{n} → R such that the gradient of g is nowhere zero, the following identity holds[35]
where the integral on the right is over g^{−1}(0), the (n − 1)dimensional surface defined by g(x) = 0 with respect to the Minkowski content measure. This is known as a simple layer integral.
More generally, if S is a smooth hypersurface of R