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Dirac delta function



 
 
The Dirac delta or Dirac's delta is a mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 construct introduced by theoretical physicist Paul Dirac
Paul Dirac

Paul Adrien Maurice Dirac, Order of Merit , Royal Society was a United Kingdom theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics....
. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 d(x) that has the value zero
0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
 everywhere except at x = 0 where its value is infinitely large
Infinity

Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
 in such a way that its total integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [ab] of the real line, the integral...
  is 1. It is a continuous analogue of the discrete
Discrete mathematics

Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense that its objects can assume only distinct, separate values, rather than a values on a continuum ....
 Kronecker delta
Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
. In the context of signal processing
Signal processing

Signal processing is the analysis, interpretation, and manipulation of signal . Signals of interest include: audio signal processing, , time-varying measurement values and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals such as radio signals, and many others....
 it is often referred to as the unit impulse function.






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The Dirac delta or Dirac's delta is a mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 construct introduced by theoretical physicist Paul Dirac
Paul Dirac

Paul Adrien Maurice Dirac, Order of Merit , Royal Society was a United Kingdom theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics....
. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 d(x) that has the value zero
0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
 everywhere except at x = 0 where its value is infinitely large
Infinity

Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
 in such a way that its total integral
Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [ab] of the real line, the integral...
  is 1. It is a continuous analogue of the discrete
Discrete mathematics

Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense that its objects can assume only distinct, separate values, rather than a values on a continuum ....
 Kronecker delta
Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
. In the context of signal processing
Signal processing

Signal processing is the analysis, interpretation, and manipulation of signal . Signals of interest include: audio signal processing, , time-varying measurement values and sensor data, for example biological data such as electrocardiograms, control system signals, telecommunication transmission signals such as radio signals, and many others....
 it is often referred to as the unit impulse function. Note that the Dirac delta is not strictly a function. While for many purposes it can be manipulated as such, formally it can be defined as a distribution
Distribution (mathematics)

In mathematical analysis, distributions are objects which generalize function s. They extend the concept of derivative to all locally integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations....
 that is also a measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
.

Overview


A Dirac function can be of any size in which case its 'strength' A is defined by duration multiplied by amplitude. The graph
Graph of a function

In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian coordinate system, together with Cartesian axes, etc....
 of the delta function is usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function
Sinc function

In mathematics, the sinc function, denoted by and sometimes as , has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
.)

Despite its name, the delta function is not truly a function, at least not a usual one with domain in reals
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
. For example, the objects f(x) = d(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
Almost everywhere

In measure theory , one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i.e....
, then f is integrable if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 g is integrable and the integrals of f and g are identical. Rigorous treatment of the Dirac delta requires measure theory or the theory of distribution
Distribution (mathematics)

In mathematical analysis, distributions are objects which generalize function s. They extend the concept of derivative to all locally integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations....
s.

The Dirac delta is very useful as an approximation for a tall narrow spike function (an impulse). It is the same type of abstraction
Abstraction

Abstraction is the process or result of generalization by reducing the information content of a concept or an observable phenomenon, typically in order to retain only information which is relevant for a particular purpose....
 as a point charge
Electric charge

Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields....
, point mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
 or electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
 point. For example, in calculating the dynamics
Dynamics (mechanics)

In physics the term dynamics customarily refers to the time evolution of physical processes. These processes may be microscopic as in particle physics, kinetic theory, and chemical reactions, or macroscopic as in the predictions of statistical mechanics and nonequilibrium thermodynamics....
 of a baseball
Baseball

Baseball is a bat-and-ball sport played between two team sport of nine players each. The goal of baseball is to score run by hitting a thrown Baseball with a baseball bat and touching a series of four markers called base arranged at the corners of a ninety-foot square, or diamond. Players on one team take turns hitting against...
 being hit by a bat, approximating the force
Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
 of the bat hitting the baseball by a delta function is a helpful trick. In doing so, one not only simplifies the equations, but one also is able to calculate the motion
Motion (physics)

In physics, motion means a constant change in the location of a body. Change in motion is the result of applied force. Motion is typically described in terms of velocity, acceleration, Displacement , and time....
 of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball.

The Dirac delta function was named after the Kronecker delta
Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
 , since it can be used 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
Heuristic

Heuristic is an adjective for methods that help in problem solving, in turn leading to learning and discovery. These methods in most cases employ experimentation and trial-and-error techniques....
 definition. The Dirac delta is not a real function, as no real function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, (where sinc is the sinc function
Sinc function

In mathematics, the sinc function, denoted by and sometimes as , has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
) behaves as a delta function in the limit as , yet this function does not approach zero for values of x  outside the origin, rather it oscillates between 1/x  and −1/x  more and more rapidly as a approaches zero.

The Dirac delta function can be rigorously defined either as a distribution
Distribution (mathematics)

In mathematical analysis, distributions are objects which generalize function s. They extend the concept of derivative to all locally integrable functions and beyond, and are used to formulate generalized solutions of partial differential equations....
 or as a measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
.

The delta function as a measure


As a measure
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
, δ(A) = 1 if 0 ∈ A, and δ(A) = 0 otherwise. Then,



for all functions ƒ.

The delta function as a distribution


As a distribution, the Dirac delta is a linear functional
Linear functional

In linear algebra, a branch of mathematics, a linear functional or linear form is a linear map from a vector space to its field of scalar s....
 on the space of test functions and is defined by
for every test function . It is a distribution with compact support (the support
Support (mathematics)

In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set. This concept is used very widely in mathematical analysis....
 being ). Because of this definition, and the absence of a true function with the delta function's properties, it is important to realize the above integral notation is simply a notational convenience
Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition ....
, and not a garden-variety (Riemann
Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an Interval ....
 or Lebesgue) integral.

Thus, the Dirac delta function may be interpreted as a probability distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
. Its characteristic function
Characteristic function (probability theory)

In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real number line it is given by the following formula, where X is any random variable with the distribution in question:...
 is then just unity, as is the moment generating function, so that all moments are zero. The cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 is the Heaviside step function
Heaviside step function

The Heaviside step function, H, also called the unit step function, is a continuous function Function whose value is 0 for negative argument and 1 for positive argument....
.

Equivalently, one may define as a distribution whose indefinite integral is the function

usually called the Heaviside step function
Heaviside step function

The Heaviside step function, H, also called the unit step function, is a continuous function Function whose value is 0 for negative argument and 1 for positive argument....
 or commonly the unit step function. That is, it satisfies the integral equation

for all real numbers x. It is important to realize this "density" interpretation is a notational convenience; if dt is Lebesgue measure, then no such density exists. However, by choosing to interpret as a singular measure giving point mass to 0, one can move beyond mere notational convenience and state something both logically coherent and actually true, namely,

Delta function of more complicated arguments


A helpful identity is the scaling property ( is non-zero),

and so

The scaling property may be generalized to:



where xi are the real root
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
s of g(x) (assumed simple roots). Thus, for example

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



In an n-dimensional space with position vector , this is generalized to:



where the integral on the right is over , the n-1  dimensional surface defined by .

The integral of the time-delayed Dirac delta is given by:

(the sifting property). The delta function is said to "sift out" the value at .

It follows that the convolution
Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operator on two function s f and g, producing a third function that is typically viewed as a modified version of one of the original functions....
:

  
 
  


means that the effect of convolving with the time-delayed Dirac delta is to time-delay by the same amount.

Fourier transform


Using Fourier transform
Fourier transform

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
s, one finds that

and therefore:

which is a statement of the orthogonality property for the Fourier kernel. Equating these non-converging improper integral
Improper integral

In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or 8 or −8 or, in some cases, as both endpoints approach limits....
s to is not mathematically rigorous. However, they behave in the same way under a definite integral. That is,

according to the definition of the Fourier transform. Therefore, the bracketed term is considered equivalent to the Dirac delta function.

Laplace transform


The direct Laplace transform
Laplace transform

In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily solvable algebraic equation from an ordinary differential equation....
 of the delta function is:



a curious identity using Euler's formula
Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematics formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function....
  allows us to find the Laplace inverse transform for the cosine

and a similar identity holds for .

Distributional derivatives


As a tempered distribution, the Dirac delta distribution is infinitely differentiable. Let U be an open subset
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 of Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 Rn and let S(U) denote the Schwartz space
Schwartz space

In mathematics, Schwartz space is the function space of rapidly decreasing functions. This space has the important property that the Fourier transform is an endomorphism on this space....
 of smooth, rapidly decaying real-valued functions on U. Let a be a point of U and let da be the Dirac delta distribution centred at a. If a = (a1, ..., an) is any multi-index and ?a denotes the associated mixed partial derivative
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 ....
 operator, then the ath derivative ?ada of da is given by

That is, the ath derivative of da is the distribution whose value on any test function f is the ath derivative of f at a (with the appropriate positive or negative sign). This is rather convenient, since the Dirac delta distribution da applied to f is just f(a). For the a=1 case this means

.

The first derivative of the delta function is referred to as a doublet (or the doublet function). Its schematic representation looks like that of da(t) and -da(t) superposed.

Representations of the delta function


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

where δa(x) is sometimes called a nascent delta function (and should not be confused with the Dirac's delta centered at a, denoted by the same symbol in the previous section). This limit is in the sense that

for all continuous
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
 bounded .

The term approximate identity
Approximate identity

In functional analysis, a right approximate identity in a Banach algebra, A, is a net such that for every element, a, of A, the net ...
 has a particular meaning in harmonic analysis
Harmonic analysis

Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms....
, in relation to a limiting sequence to an identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
 for the convolution
Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operator on two function s f and g, producing a third function that is typically viewed as a modified version of one of the original functions....
 operation (also on groups more general than the real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, e.g. the unit circle
Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
). There the condition is made that the limiting sequence should be of positive functions.

Some nascent delta functions are:



Note: If d(ax) is a nascent delta function which is a probability distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
 over the whole real line (i.e. is always non-negative between -8 and +8) then another nascent delta function df(ax) can be built from its characteristic function
Characteristic function (probability theory)

In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real number line it is given by the following formula, where X is any random variable with the distribution in question:...
 as follows:

where

is the characteristic function of the nascent delta function d(ax). This result is related to the localization property of the continuous Fourier transform
Continuous Fourier transform

In mathematics, the Fourier transform is an operation that Transform one complex number-valued function of a real variable into another. The new function, often called the frequency domain representation of the original function, describes which frequencies are present in the original function....
.

There are also series and integral representations of the Dirac delta function in terms of special functions, such as integrals of products of Airy functions, of Bessel functions, of Coulomb wave functions and of parabolic cylinder functions, and also series of products of orthogonal polynomials.

The Dirac comb

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb
Dirac comb

In mathematics, a Dirac comb is a periodic function Schwartz distribution constructed from Dirac delta functionsfor some given period T....
, or as the Shah distribution, creates a sampling
Sampling (signal processing)

In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of sample ....
 function, often used in digital signal processing
Digital signal processing

Digital signal processing is concerned with the representation of the signal s by a sequence of numbers or symbols and the processing of these signals....
 (DSP) and discrete time signal analysis.

Application to quantum mechanics


We give an example of how the delta function is expedient in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
. Suppose a set of orthonormal wave functions is complete, so that for any wave function ? we have

,


with . Generalizing to the continuous spectrum
Continuous spectrum

In physics, continuous wiktionary:spectrum refers to a range of values which may be graphed to fill a range with closely-spaced or overlapping intervals....
, we expect relations of the form



and

.


Substituting the first of these relations into the second and using the property of linearity of the scalar product gives us

.


From this it is apparent that

Relationship to the Kronecker delta


The Dirac delta function may be seen as a continuous analog of the Kronecker delta
Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a Function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise....
. To see this let (ai)iZ be any doubly infinite sequence. The Kronecker delta, dik, then satisfies:

Similarly, for any real or complex valued continuous function ƒ on R, the Dirac delta satisfies:

See also

  • Dirac comb
    Dirac comb

    In mathematics, a Dirac comb is a periodic function Schwartz distribution constructed from Dirac delta functionsfor some given period T....
  • Logarithmically-spaced Dirac comb
    Logarithmically-spaced Dirac comb

    Like the standard Dirac comb, the logarithmically-spaced Dirac comb consists of an infinite sequence of Dirac delta functions. In the case of the logarithmically-spaced comb, these are spaced in octave intervals, i.e., the delta functions are placed at positions , for all integers ....
  • Green's function
    Green's function

    In mathematics, a Green's function is a type of function used to solve inhomogeneous ordinary differential equation differential equations subject to boundary conditions....
  • Dirac measure
    Dirac measure

    In mathematics, a Dirac measure is a measure δx on a set X that gives the singleton set the measure 1, for a chosen element x ∈ X:...


External links

  • on MathWorld
    MathWorld

    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
  • on PlanetMath
    PlanetMath

    PlanetMath is a free content, collaborative, online mathematics encyclopedia. The emphasis is on peer review, rigour, openness, pedagogy, real-time content, interlinked content, and community....