Information about Dirac Delta Function
Probability density function) Plot of the Dirac delta function Schematic representation of the Dirac delta function for x0 = 0. A line surmounted by an arrow is usually used to schematically represent the Dirac delta function. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead. | |
Cumulative distribution function) Plot of the Heaviside step function Using the half-maximum convention, with x0 = 0 | |
| Parameters |  location (real) |
|---|---|
| Support |  |
| Probability density function (pdf) |  |
| Cumulative distribution function (cdf) |  (Heaviside) |
| Mean | |
| Median | |
| Mode | |
| Variance |  |
| Skewness | |
| Excess kurtosis | (undefined) |
| Entropy |  |
| Moment-generating function (mgf) |  |
| Characteristic function |  |
Overview
Dirac functions can be of any size in which case their 'strength' A is defined by duration multiplied by amplitude. The graph 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.)Despite its name, the delta function is not truly a function. One reason for this is because the functions 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. Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions.
The Dirac delta is very useful as an approximation for a tall narrow spike function (an impulse). It is the same type of abstraction as a point charge, point mass or electron point. For example, in calculating the dynamics of a baseball being hit by a bat, approximating the force 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 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 , since it can be used as a continuous analogue of the discrete Kronecker delta.
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 heuristic definition should not be taken too seriously though. Firstly, the Dirac delta is not a function, as no function has the above properties. Moreover there exist descriptions which differ from the above conceptualization. For example,
(where sinc is the sinc function) behaves as a delta function in the limit of 
, yet this function does not approach zero for values of x outside the origin.
The defining characteristic
In terms of dimensional analysis, this definition of
implies that has dimensions reciprocal to those of dx.
The delta function as a measure
As a measure,
if 
, and 
otherwise. Then,
for all continuous
.
As distributions, an antiderivative of the Dirac delta distribution.
The delta function as a probability density function
As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by
. It is a distribution with compact support (the support being {0}). 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, and not a true integral.
Thus, the Dirac delta function may be interpreted as a probability density function. Its characteristic function is then just unity, as is the moment generating function, so that all moments are zero. The cumulative distribution function is the Heaviside step function.
Equivalently, one may define
as a distribution 
whose indefinite integral is the function
usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation
for all real numbers x.
Delta function of more complicated arguments
A helpful identity is the scaling property:
and so
The scaling property may be generalized to:
where xi are the real roots 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-shifted Dirac delta is given by:
Thus, the delta function is said to "shift out" the function
at the value 
, when integrated over all time.Similarly, the convolution:
means that the effect of convolving with the time-shifted Dirac delta is to time-shift
by the same amount.
Fourier transform
Using Fourier transforms, one has
and therefore:
which is a statement of the orthogonality property for the Fourier kernel.
Laplace transform
The direct Laplace transform of the delta function is:
a curious identity using Euler's formula
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 of Euclidean space Rn and let S(U) denote the Schwartz space of smooth, rapidly decaying real-valued functions on U. Let a be a point of U and let δa be the Dirac delta distribution centred at a. If α = (α1, ..., αn) is any multi-index and ∂α denotes the associated mixed partial derivative operator, then the αth derivative ∂αδa of δa is given by
That is, the αth derivative of δa is the distribution whose value on any test function φ is the αth derivative of φ at a (with the appropriate positive or negative sign). This is rather convenient, since the Dirac delta distribution δa applied to φ is just φ(a). For the α=1 case this means
.
Representations of the delta function
The delta function can be viewed as the limit of a sequence of functions
where
is sometimes called a nascent delta function. This limit is in the sense that
for all continuous
.
The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (also on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions.
Some nascent delta functions are:
Limit of a Normal distribution 
Limit of a Cauchy distribution 
Cauchy 
(see note below)
Limit of a rectangular function 
rectangular function (see note below)
Derivative of the sigmoid (or Fermi-Dirac) function 
Limit of the Airy function 
Limit of a Bessel function
Note: If δ(a, x) is a nascent delta function which is a probability distribution over the whole real line (i.e. is always non-negative between -∞ and +∞) then another nascent delta function δφ(a, x) can be built from its characteristic function as follows:
where
is the characteristic function of the nascent delta function δ(a, x). This result is related to the localization property of the continuous Fourier transform.The Dirac comb
- Main article: Dirac comb
See also
External links
- Delta Function on MathWorld
- Dirac Delta Function on PlanetMath
- The Dirac delta measure is a hyperfunction
- We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure
- Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure.
location parameter, since its value determines the "location" of the probability distribution.
In other words, when you graph the function, the location parameter determines where the origin will be located.
..... Click the link for more information.In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
..... Click the link for more information.In mathematics, a support of a function f from a set X to the real numbers R is a subset Y of X such that f (x) is zero for all x in X and outside Y.
..... Click the link for more information.In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.
Formally, a probability distribution has density f, if f
..... Click the link for more information.In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,[1] completely describes the probability distribution of a real-valued random variable X.
..... Click the link for more information.Heaviside step function, H, also called unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument. It seldom matters what value is used for H(0), since is mostly used as a distribution.
..... Click the link for more information.expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
..... Click the link for more information.median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
..... Click the link for more information.In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.
..... Click the link for more information.variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
..... Click the link for more information.skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.Introduction
Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.
..... Click the link for more information.kurtosis (from the Greek word kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent
..... Click the link for more information.Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable.
Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must
..... Click the link for more information.In probability theory and statistics, the moment-generating function of a random variable X is
wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.
..... Click the link for more information.In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:
..... Click the link for more information.Motto
"Dieu et mon droit" [2] (French)
"God and my right"
Anthem
"God Save the Queen" [3]
..... Click the link for more information.Paul Dirac
Paul Adrien Maurice Dirac
Born July 8 1902
Bristol, England
Died September 20 1984 (aged 82)
..... Click the link for more information.function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
..... Click the link for more information.The word 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.
..... Click the link for more information.0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
..... Click the link for more information.INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
..... Click the link for more information.Discrete time is non-continuous time. Sampling at non-continuous times results in discrete-time samples. For example, a newspaper may report the price of crude oil once every 24 hours.
..... Click the link for more information.In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. So, for example, , but .
..... Click the link for more information.distributions (also known as generalized functions) are objects which generalize functions and probability distributions. They extend the concept of derivative to all integrable functions and beyond, and are used to formulate generalized solutions of partial differential
..... Click the link for more information.In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the
..... Click the link for more information.graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc.
..... Click the link for more information.In mathematics, the sinc function, denoted by , has two definitions, sometimes distinguished as the normalized sinc function and unnormalized sinc function:- In digital signal processing and information theory, the normalized sinc function
..... Click the link for more information.In measure theory (a branch of mathematical analysis), 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. is a set with measure zero.
..... Click the link for more information.“Iff” redirects here. For other uses, see IFF.If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements
..... Click the link for more information.In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the
..... Click the link for more information.
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