NationMaster - Encyclopedia: Delta distribution

**Dirac delta function**
Probability density function Schematic representation of the Dirac delta function for x₀ = 0. A line with 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	$x in [x_0; x_0]$
pdf	$delta(x-x_0),$
cdf	$H(x-x_0),$ (Heaviside)
Mean	$x_0,$
Median	$x_0,$
Mode	$x_0,$
Variance	$0,$
Skewness	$0,$
Kurtosis	(undefined)
Entropy	$-infty$
mgf	$e^{tx_0}$
Char. func.	$e^{itx_0}$

The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. The integral from minus infinity to plus infinity is 1. The discrete analog of the delta function is the degenerate distribution which is sometimes known as a delta function. Image File history File links Download high resolution version (1300x975, 59 KB) File links The following pages link to this file: Dirac delta function ... Image File history File links Download high resolution version (1300x975, 61 KB) Diagram of the Heaviside unit step function which is also the cumulative density function of the Dirac delta function. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Partial plot of a function f. ... Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ... 0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... Discrete time is non-continuous time. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ...

1 Overview
2 Formal introduction
3 Delta function of more complicated arguments
4 Fourier transform
5 The Dirac delta function as a probability density function
6 Derivatives of the delta function
7 Equivalent definition
8 Representations of the delta function
9 See also
10 External links

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 can be 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.) The sinc function sinc(x) from x=-8π to 8π. ...

Despite its name, the delta function is not a function as defined in the strictest mathematical sense. 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 ostensibly different. According to Lebesgue integration theory, if f, g are functions such that f = g almost everywhere, then f is integrable iff g is integrable and the integrals of f and g are the same. Precise treatment of the Dirac delta requires measure theory or the theory of distributions. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... 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. ... â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In mathematics, a measure is a function that assigns a number, e. ... 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. 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. // An abstraction is an idea, concept, or word which defines the phenomena which make up the concrete events or things which the abstraction refers to, the referents. ... Charge is a word with many different meanings. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ... Properties The electron is a fundamental subatomic particle that carries a negative electric charge. ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ... Baseball is a team sport in which a player on one team (the pitcher) attempts to throw a hard, fist-sized ball past a player on the other team (the batter), who attempts to hit the baseball with a tapered, smooth, cylindrical stick called a bat. ... In physics, a force is an external cause responsible for any change of a physical system. ... In physics, motion means a change in the position of a body with respect to time, as measured by a particular observer in a particular frame of reference. ...

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 Kroneckers 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. ...

Formal introduction

The Dirac delta is often introduced with the property:

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

valid for any continuous function f. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

However, there is no actual function δ(x) with this property. The Dirac delta is not a function; but it can be usefully treated as a distribution, as well as a measure. Partial plot of a function f. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, a measure is a function that assigns a number, e. ...

As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...

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

for every test function $phi$ . 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. In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

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

$int_{-infty}^infty f(x) , ddelta(x) = f(0)$

for all continuous f. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

As distributions, the Heaviside step function is an antiderivative of the Dirac delta distribution. In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

Delta function of more complicated arguments

A helpful identity is the scaling property:

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

and so

$delta(alpha x) = frac{delta(x)}{|alpha|}$

This concept may be generalized to:

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

where x_i are the roots of g(x). In the integral form it is equivalent to

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

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 $g(mathbf{r})=0$ .

Fourier transform

The continuous Fourier transform of the Dirac delta is the constant function $frac{1}{sqrt{2pi}}$ . The inverse transform of this constant function will be the Dirac delta again, yielding the orthogonality property for the Fourier kernel: In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

$frac{1}{2pi}int_{-infty}^infty e^{ikx},dx=delta(k)$

From the convolution theorem for the Fourier transform, the convolution of δ with any distribution S yields S. For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...

The Dirac delta function as a probability density function

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. In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

Derivatives of the delta function

The derivative of the Dirac delta function (also called a doublet) is the distribution δ' defined by

$delta'[phi] = -phi'(0),$

for every test function $phi$ . From this it follows that

$xdelta'(x)=-delta(x),$

The n-th derivative δ⁽ⁿ⁾ is given by

$delta^{(n)}[phi] = (-1)^n phi^{(n)}(0),$

The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomials. In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ...

Equivalent definition

The Dirac delta function $delta : mathbb{R} ni &# 0;longrightarrow delta ( &# 0;)in delta(mathbb{R})$ is a distribution $δ(ξ)$ whose indefinite integral is the function In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

$h : mathbb{R} ni &# 0;longrightarrow frac{1+{rm sgn} , &# 0;}{2} in mathbb{R},$

usually called the Heaviside step function or commonly the unit step function. That is, it satisfies the integral equation The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

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

for all real numbers x.

Representations of the delta function

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

$delta (x) = lim_{ato 0} delta_a(x),$

where $δ a (x)$ is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting sequence will be used. On the other hand the term limit needs to be made precise, as this equality holds only for some meanings of limit. 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 (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. In functional analysis, a right approximate identity in a Banach algebra A is a net (or a sequence) such that for every element of , the net (or sequence) has limit . ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... 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 set. ... For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... Illustration of a unit circle. ...

Some nascent delta functions are:

$delta_a(x)=frac{1}{asqrt{pi}} mathrm{e}^{-x^2/a^2}$	Limit of a Normal distribution
$delta_a(x) = frac{1}{pi} {a over a^2 + x^2} =frac{1}{2pi}int_{-infty}^{infty}mathrm{e}^{mathrm{i} k x-\|ak\|};dk$	Limit of a Cauchy distribution
$delta_a(x)=frac{e^{-\|x/a\|}}{2a} =frac{1}{2pi}int_{-infty}^{infty}frac{e^{ikx}}{1+a^2k^2},dk$	Cauchy $varphi$ (see note below)
$delta_a(x)= frac{textrm{rect}(x/a)}{a} =frac{1}{2pi}int_{-infty}^infty textrm{sinc}(ak/2)e^{ikx},dk$	Limit of a rectangular function
$delta_a(x)=frac{1}{pi x}sinleft(frac{x}{a}right) =frac{1}{2pi}int_{-1/a}^{1/a} cos (k x);dk$	rectangular function $varphi$ (see note below)
$delta_a(x)=partial_x frac{1}{1+mathrm{e}^{-x/a}} =-partial_x frac{1}{1+mathrm{e}^{x/a}}$
$delta_a(x)=frac{a}{pi x^2}sin^2left(frac{x}{a}right)$
$delta_a(x) = frac{1}{a}A_ileft(frac{x}{a}right)$	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

The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... The rectangular function (also known as the rectangle function or the normalized boxcar function) is defined as or in terms of the Heaviside step function The rectangular function is normalized: The Fourier transform of the rectangular function is where sinc is the sinc function. ... In mathematics, the Airy function Ai(x) is a special function, i. ... In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

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 δ_φ(a, x) can be built from its characteristic function as follows: In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ...

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

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). This result is related to the localization property of the continuous Fourier transform. In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

External links

Delta Function on MathWorld
Dirac Delta Function on PlanetMath

Category: Mathematical analysis MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

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