Dirac delta function
The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses.
For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta.
In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).
Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δt → 0, giving a result everywhere except at 0:
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.
[15][16] As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as
[29] 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.
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 (where
In the theory of electromagnetism, the first derivative of the delta function represents a point magnetic dipole situated at the origin.
Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support.
One may show that (5) holds for all continuous compactly supported functions f,[54] and so ηε converges weakly to δ in the sense of measures.
Further conditions on the ηε, for instance that it be a mollifier associated to a compactly supported function,[56] are needed to ensure pointwise convergence almost everywhere.
In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments.
Abstractly, if A is a linear operator acting on functions of x, then a convolution semigroup arises by solving the initial value problem
It represents the probability density at time t = ε of the position of a particle starting at the origin following a standard Brownian motion.
As a result, the nascent delta functions that arise as fundamental solutions of the associated Cauchy problems are generally oscillatory integrals.
So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations.
Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by Johann Radon, and then developed in this form by Fritz John (1955).
This is essentially a form of the inversion formula for the Radon transform because it recovers the value of φ(x) from its integrals over hyperplanes.
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.
The n-th partial sum of the Fourier series of a function f of period 2π is defined by convolution (on the interval [−π,π]) with the Dirichlet kernel:
A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval [−π,π] tends to a multiple of the delta function as N → ∞.
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.
This is a special case of the situation in several complex variables in which, for smooth domains D, the Szegő kernel plays the role of the Cauchy integral.
Given a complete orthonormal basis set of functions {φn} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as
[73] Cauchy used an infinitesimal α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δα satisfying
The article by Yamashita (2007) contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals.
Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property
The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion).
[79] In this context, the position operator has a complete set of generalized eigenfunctions,[80] labeled by the points y of the real line, given by
