Dirac delta function

Dirac delta function
Probability density 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 Using the half-maximum convention, with x₀ = 0
Parameters	$x_0\,$ location (real)
Support	$x \in [x_0; x_0]$
Probability density function (pdf)	$\delta(x-x_0)\,$
Cumulative distribution function (cdf)	$H(x-x_0)\,$ (Heaviside)
Mean	$x_0\,$
Median	$x_0\,$
Mode	$x_0\,$
Variance	$0\,$
Skewness	(undefined)
Excess kurtosis	(undefined)
Entropy	$-\infty$
Moment-generating function (mgf)	$e^{tx_0}$
Characteristic function	$e^{itx_0}$

The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. In Statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter μ, which determines the "location" In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution. In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued In Probability theory and Statistics, kurtosis (from the Greek word κυρτός kyrtos or kurtos, meaning bulging is a measure of the "peakedness" In Probability theory and Statistics, the moment-generating function of a Random variable X is M_X(t=\operatorname{E}\left(e^{tX}\right In Probability theory, the characteristic function of any Random variable completely defines its Probability distribution. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located Informally, it is a function representing an infinitely sharp peak bounding unit area: a function δ(x) that has the value zero everywhere except at x = 0 where its value is infinitely large in such a way that its total integral is 1. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space It is a continuous analogue of the discrete Kronecker delta. Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two In the context of signal processing it is often referred to as the unit impulse function. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as 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 that is also a measure. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions 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

1 Overview
2 Definitions
3 The delta function as a measure
4 The delta function as a probability density function
5 Delta function of more complicated arguments
6 Fourier transform
7 Laplace transform
8 Distributional derivatives
9 Representations of the delta function
10 The Dirac comb
11 See also
12 External links
13 References

Overview

A Dirac function can be of any size in which case its '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. In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function. In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes )

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. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of 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 ↔ Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions. 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 In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability 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. --> Abstraction is the process or result of generalization by reducing the information Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J 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 physics the term dynamics customarily refers to the time evolution of physical processes Baseball is a Bat-and-ball Sport played between two teams of nine players each In Physics, a force is whatever can cause an object with Mass to Accelerate. 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. In Physics, motion means a constant change in the location of a body

The Dirac delta function was named after the Kronecker delta, since it can be used as a continuous analogue of the discrete Kronecker delta. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

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,

$\delta(x) = \begin{cases} \infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$

and which is also constrained to satisfy the identity

$\int_{-\infty}^\infty \delta(x) \, dx = 1.$

This heuristic definition should not be taken too seriously though. heuristic (hyu̇-ˈris-tik is a method to help solve a problem commonly an informal method The Dirac delta is not a function, as no function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, $sinc(x / a) / a$ (where sinc is the sinc function) behaves as a delta function in the limit of $a\rightarrow 0$ , 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 infinity. In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes

The defining characteristic

$\int_{-\infty}^\infty f(x) \, \delta(x) \, dx = f(0)$

where f is a suitable test function, cannot be achieved by any function, but the Dirac delta function can be rigorously defined either as a distribution or as a measure. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions 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

In terms of dimensional analysis, this definition of $δ(x)$ implies that $δ(x)$ has dimensions reciprocal to those of dx. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand

The delta function as a measure

As a measure, $δ(A) = 1$ if $0\in A$ , and $δ(A) = 0$ otherwise. 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 Then,

$\int_{-\infty}^\infty f(x) \, \delta(x) \, dx = f(0)$

for all continuous $f$ . In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

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

$\delta[\phi] = \phi(0)\,$

for every test function $\phi \$ . This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional It is a distribution with compact support (the support being {0}). In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set 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 garden-variety (Riemann or Lebesgue) 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 In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of

Thus, the Dirac delta function may be interpreted as a probability density function. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability Its characteristic function is then just unity, as is the moment generating function, so that all moments are zero. In Probability theory, the characteristic function of any Random variable completely defines its Probability distribution. In Probability theory and Statistics, the moment-generating function of a Random variable X is M_X(t=\operatorname{E}\left(e^{tX}\right The cumulative distribution function is the Heaviside step function. In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative

Equivalently, one may define $\delta : \mathbb{R} \ni \xi \longrightarrow \delta ( \xi )\in \delta(\mathbb{R})$ as a distribution $δ(ξ)$ whose indefinite integral is the function

$h : \mathbb{R} \ni \xi \longrightarrow \frac{1+{\rm sgn} \, \xi }{2} \in \mathbb{R},$

usually called the Heaviside step function or commonly the unit step function. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative That is, it satisfies the integral equation

$\int^{x}_{-\infin} \delta (t) dt = h(x) \equiv \frac{1+{\rm sgn}(x)}{2}$

for all real numbers x.

Delta function of more complicated arguments

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

$\int_{-\infty}^\infty \delta(\alpha x)\,dx =\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|} =\frac{1}{|\alpha|}$

and so

$\delta(\alpha x) = \frac{\delta(x)}{\|\alpha\|}$		(Eq. 1)

The scaling property may be generalized to:

$\delta(g(x)) = \sum_{i}\frac{\delta(x-x_i)}{|g'(x_i)|}$

and, $\delta(\alpha g(x)) = \frac{1}{|\alpha|}\delta(g(x))$

where x_i are the real roots of g(x) (assumed simple roots). Thus, for example

$\delta(x^2-\alpha^2) = \frac{1}{2|\alpha|}[\delta(x+\alpha)+\delta(x-\alpha)]$

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

$\int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}$

In an n-dimensional space with position vector $\mathbf{r}$ , this is generalized to:

$\int_V f(\mathbf{r}) \, \delta(g(\mathbf{r})) \, d^nr = \int_{\partial V}\frac{f(\mathbf{r})}{|\mathbf{\nabla}g|}\,d^{n-1}r$

where the integral on the right is over $\partial V$ , the n-1 dimensional surface defined by $g(\mathbf{r})=0$ .

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

$\int\limits_{-\infty}^\infty f(t) \delta(t-T)\,dt = f(T)$

(the sifting property). The delta function is said to "sift out" the value at $t=T\,$ .

It follows that the convolution:

$f(t) * \delta(t-T)\,$	$\ \stackrel{\mathrm{def}}{=}\ \int\limits_{-\infty}^\infty f(\tau) \cdot \delta(t-T-\tau) \ d\tau$
	$= \int\limits_{-\infty}^\infty f(\tau) \cdot \delta(\tau-(t-T)) \ d\tau$ (using (Eq. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and 1) with $α = - 1$ )
	$= f(t-T)\,$

means that the effect of convolving with the time-delayed Dirac delta is to time-delay $f(t)\,$ by the same amount.

Fourier transform

Using Fourier transforms, one finds that

$\int_{-\infty}^\infty 1 \cdot e^{-i 2\pi f t}\,dt = \delta(f)$

and therefore:

$\int_{-\infty}^\infty e^{i 2\pi f_1 t} \left[e^{i 2\pi f_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{-i 2\pi (f_2 - f_1) t} \,dt = \delta(f_2 - f_1)$

which is a statement of the orthogonality property for the Fourier kernel. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Equating these non-converging improper integrals to $δ(x)$ is not mathematically rigorous. In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified However, they behave in the same way under a definite integral. That is,

$\begin{align} \int_{-\infty}^\infty F(f) \left(\int_{-\infty}^\infty e^{-i 2\pi f t} dt\right) df &= F(0) \end{align}$

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 of the delta function is:

$\int_{0}^{\infty}\delta (t-a)e^{-st} \, dt=e^{-as}$

a curious identity using Euler's formula $2cos(a s) = e - i a s + e i a s$ allows us to find the Laplace inverse transform for the cosine

$2\frac{1}{2\pi {i}}\int_{c-i\,\infty}^{c+i\,\infty} \cos(as)e^{st} \, ds=2[\delta (t+ia) +\delta (t-ia)]$ and a similar identity holds for

sin(a s)

. 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 soluble algebraic

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. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, Schwartz space is the Function space of rapidly decreasing functions Let a be a point of U and let δ_a be the Dirac delta distribution centred at a. If α = (α₁, . . . , α_n) is any multi-index and ∂^α denotes the associated mixed partial derivative operator, then the α^th derivative ∂^αδ_a of δ_a is given by

$\left\langle \partial^{\alpha} \delta_{a}, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = \left. (-1)^{| \alpha |} \partial^{\alpha} \varphi (x) \right|_{x = a} \mbox{ for all } \varphi \in S(U).$

That is, the α^th derivative of δ_a is the distribution whose value on any test function φ is the α^th derivative of φ at a (with the appropriate positive or negative sign). The Mathematical notation of multi-indices simplifies formulae used in Multivariable calculus, Partial differential equations and the theory of distributions 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 This is rather convenient, since the Dirac delta distribution δ_a applied to φ is just φ(a). For the α=1 case this means

$\int_{-\infty}^{\infty} \delta'(x-a)f(x)dx = -f'(a)$ .

The first derivative of the delta function is referred to as a doublet (or the doublet function). [1] Its schematic representation looks like that of δ_a(t) and -δ_a(t) superposed.

Representations of the delta function

A sequence of normal distributions that in the limit approximates the Dirac delta function (see the formula in the text).

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

$\delta (x) = \lim_{a\to 0} \delta_a(x),$

where $δ a (x)$ is sometimes called a nascent delta function. This limit is in the sense that

$\lim_{a\to 0} \int_{-\infty}^{\infty}\delta_a(x)f(x)dx = f(0) \$

for all continuous $f$ . In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

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. In Functional analysis, a right approximate identity in a Banach algebra, A, is a net (or a Sequence) \{\e_\lambda 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 In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and In Mathematics, the real numbers may be described informally in several different ways g. the unit circle). In Mathematics, a unit circle is There the condition is made that the limiting sequence should be of positive functions.

Some nascent delta functions are:

$\delta_a(x) = \frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2}$	Limit of a normal distribution
$\delta_a(x) = \frac{1}{\pi} \frac{a}{a^2 + x^2} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} k x-\|ak\|}\;dk$	Limit of a Cauchy distribution
$\delta_a(x)=\frac{e^{-\|x/a\|}}{2a} =\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ikx}}{1+a^2k^2}\,dk$	Cauchy $\varphi$ (see note below)
$\delta_a(x)= \frac{\textrm{rect}(x/a)}{a} =\frac{1}{2\pi}\int_{-\infty}^\infty \textrm{sinc} \left( \frac{a k}{2 \pi} \right) e^{ikx}\,dk$	Limit of a rectangular function
$\delta_a(x)=\frac{1}{\pi x}\sin\left(\frac{x}{a}\right) =\frac{1}{2\pi}\int_{-1/a}^{1/a} \cos (k x)\;dk$	rectangular function $\varphi$ (see note below)
$\delta_a(x)=\partial_x \frac{1}{1+\mathrm{e}^{-x/a}} =-\partial_x \frac{1}{1+\mathrm{e}^{x/a}}$	Derivative of the sigmoid (or Fermi-Dirac) function
$\delta_a(x)=\frac{a}{\pi x^2}\sin^2\left(\frac{x}{a}\right)$
$\delta_a(x) = \frac{1}{a}A_i\left(\frac{x}{a}\right)$	Limit of the Airy function
$\delta_a(x) = \frac{1}{a}J_{1/a} \left(\frac{x+1}{a}\right)$	Limit of a Bessel function
$\delta_a(x)=\begin{cases} \frac{1}{a},&-\frac{a}{2}<x<\frac{a}{2}\\ 0,&\mbox{otherwise} \end{cases}$	^[1]

Note: If δ(a, x) is a nascent delta function which is a probability distribution over the whole real line (i. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. The rectangular function (also known as the rectangle function, rect function, unit pulse, or the normalized Boxcar function) A sigmoid function is a Mathematical function that produces a sigmoid curve &mdash a curve having an "S" shape In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that In Mathematics, the Airy function Ai( x) is a Special function named after the British astronomer George Biddell Airy. In Mathematics, Bessel functions, first defined by the Mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are Canonical In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable e. is always non-negative between -∞ and +∞) then another nascent delta function δ_φ(a, x) can be built from its characteristic function as follows:

$\delta_\varphi(a,x)=\frac{1}{2\pi}~\frac{\varphi(1/a,x)}{\delta(1/a,0)}$

where

$\varphi(a,k)=\int_{-\infty}^\infty \delta(a,x)e^{-ikx}\,dx$

is the characteristic function of the nascent delta function δ(a, x). In Probability theory, the characteristic function of any Random variable completely defines its Probability distribution. This result is related to the localization property of the continuous Fourier transform. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and

The Dirac comb

Main article: Dirac comb

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the shah distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis. In Mathematics, a Dirac comb (also known as an impulse train and sampling function in Electrical engineering) is a periodic In Mathematics, a Dirac comb (also known as an impulse train and sampling function in Electrical engineering) is a periodic In Signal processing, sampling is the reduction of a Continuous signal to a Discrete signal. Digital signal processing ( DSP) is concerned with the representation of the signals by a sequence of numbers or symbols and the processing of these signals

External links

References

^ McMahon, D. In Mathematics, a Dirac comb (also known as an impulse train and sampling function in Electrical engineering) is a periodic 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 In Mathematics, Green's function is a type of function used to solve inhomogeneous Differential equations subject to boundary conditions In Mathematics, a Dirac measure is a measure &delta x on a set X (with any ''&sigma''-algebra of Subsets MathWorld is an online Mathematics reference work created and largely written by Eric W PlanetMath is a free, collaborative online Mathematics Encyclopedia. (2005-11-22). "An Introduction to State Space", Quantum Mechanics Demystified, A Self-Teaching Guide, Demystified Series (in English). New York: McGraw-Hill, pp. 108. DOI:10.1036/0071455469. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document. ISBN 0-07-145546-9. Retrieved on 2008-03-17. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 45 BC - In his last victory Julius Caesar defeats the Pompeian forces of Titus Labienus and Pompey the Younger

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