Dirac delta function

In mathematical analysis, the Dirac delta function, also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as
such that
Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.
The delta function is named after physicist Paul Dirac, and has been applied routinely in physics and engineering to model point masses and concentrated loads. It is called the delta function because it is a continuous analogue of the Kronecker delta function. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.

Motivation and overview

The graph of the Dirac delta is usually thought of as following the whole -axis and the positive -axis. The Dirac delta is used to model a tall narrow spike function, and other similar abstractions such as a point charge or point mass. 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 can simplify the equations and calculate the motion of the ball by only considering the total impulse of the collision.
In applied mathematics, the delta function is often manipulated as a kind of limit of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.
The Dirac delta, given the desired properties outlined above, cannot be a function with domain and range in real numbers. 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, then is integrable if and only if is integrable and the integrals of and are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right uses measure theory or the theory of distributions.

History

As part of his development of quantum mechanics, Paul Dirac introduced the -function in a 1927 paper, subsequently popularized in his 1930 book The Principles of Quantum Mechanics. He called it the "delta function" since he used it as a continuum analog of the discrete Kronecker delta. However, it had been used by multiple mathematical scientists in the nineteenth century. Dirac biographer Graham Farmelo surmised that Oliver Heaviside was likely a direct influence on Dirac, given Dirac's background in engineering. Indeed, Heaviside introduced the -function in his work on electromagnetism and electrical engineering. In a 1963 interview, Dirac stated, "All electrical engineers are familiar with the idea of a pulse, and the -function is just a way of expressing a pulse mathematically." Mathematicians refer to the same concept as a generalized function or distribution rather than a function in the ordinary sense.
The earliest known use of the -function is in the works of Jean-Baptiste Joseph Fourier. Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:
which is tantamount to the introduction of the -function in the form:
Later, an infinitesimal formula for an infinitely tall, unit impulse delta function explicitly appears in an 1827 text of Augustin-Louis Cauchy. Cauchy expressed the theorem using exponentials:
Cauchy pointed out that in some circumstances the order of integration is significant in this result.
As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the -function as
where the -function is expressed as
Siméon Denis Poisson and Charles Hermite introduced the -function in their investigations of Fourier integrals. Gustav Kirchhoff employed it in a paper applying Green's theorem in wave optics. Kirchhoff, Hermann von Helmholtz, and William Thomson viewed it as the limit of a sequence of Gaussian functions. But it was Heaviside and Dirac who first presented the -function explicitly as an independent entity.
A rigorous interpretation of the exponential form and the various limitations upon the function necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows: The classical notion of a function is too narrow because must approach zero sufficiently rapidly at infinity in order for the Fourier integral to exist. For this reason, extending the classical Fourier transform to distributions substantially enlarges the class of objects that could be transformed. Further works on the Fourier integral included contributions by Michel Plancherel ; Norbert Wiener, and Salomon Bochner ; and finally Laurent Schwartz, who established a rigorous theory of distributions.

Definitions

The Dirac delta function 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 extended real number valued function defined on the real numbers has these properties.

As a measure

One way to rigorously capture the notion of the Dirac delta function is to define a measure, called Dirac measure, which accepts a subset of the real line as an argument, and returns if, and if 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 provides the necessary analytic device. The Lebesgue integral with respect to the measure satisfies
for all continuous compactly supported functions. The measure 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. As a result, the latter notation is a convenient abuse of notation, and not a standard integral.
As a probability measure on, the delta measure is characterized by its cumulative distribution function, which is the unit step function.
This means that is the integral of the cumulative indicator function with respect to the measure ; to wit,
the latter is the measure of this interval. Thus, in particular, the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:
All higher moments of are zero. In particular, characteristic function and moment generating function are both equal to one.

As a distribution

In the theory of distributions, a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them. 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. 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 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
for every test function.
For 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 on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer, there is an integer and a constant, such that for every test function, one has the inequality
where represents the supremum. With the distribution, one has such an inequality defines a Daniell integral on the space of all compactly supported continuous functions which, by the Riesz representation theorem, can be represented as the Lebesgue integral of 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 -dimensional Euclidean space as the measure such that
for every compactly supported continuous function. As a measure, the -dimensional delta function is the product measure 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 makes sense on any set. Thus if is a set, is a marked point, and is any sigma algebra of subsets of, then the measure defined on sets by
is the delta measure or unit mass concentrated at.
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 centered at the point is defined as the following distribution:
for all compactly supported smooth real-valued functions on. A common special case of this construction is a case in which is an open set in the Euclidean space.
On a locally compact Hausdorff space, the Dirac delta measure concentrated at a point is the Radon measure associated with the Daniell integral on compactly supported continuous functions. 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 into the space of finite Radon measures on, equipped with its vague topology. Moreover, the convex hull of the image of under this embedding is dense in the space of probability measures on.