Delta function, Dirac

The Dirac delta function is not really a function because its value is not defined for all x. The delta function S(x — xq) can be considered to be the limit of the unit pulse with width 2 and height jy at the position x = xo as - 0 (Bronstein et al. 2001 Horn 1986). 

Consider the unit step function as the limit of the sequence u (x).  

The area under the unit pulse is 1. Hence, we have 

This sequence of functions Se defines the unit impulse or Dirac delta function. 

Color Constancy M. Ebner 2007 John Wiley Sons, Ltd 

As a consequence of this definition, if /(x) is an arbitrary function which is well-defined at X = 0, then integration of /(x) with the delta fiinction selects out the value of /(x) at the origin 

The integration is taken over the range of x for which /(x) is defined, provided that the range ineludes the origin. It also follows that 

The follotving properties of the Dirac delta funetion can be demonstrated by multiplying both sides of each expression hy /(x) and observing that, on integration, eaeh side gives the same result 

As defined above, the delta function by itself lacks mathematical rigor and has no meaning. Only when it appears in an integral does it have an operational meaning. 

That two integrals are equal does not imply that the integrands are equal. However, for the sake of convenience we often write mathematical expressions involving (3(x) such 

As defined above, the delta function by itself lacks mathematical rigor and has no meaning. Only when it appears in an integral does it have an operational meaning. That two integrals are equal does not imply that the integrands are equal. However, for the sake of convenience we often write mathematical expressions involving d(x) such 

The concept of the Dirac delta function can be made more mathematically rigorous by regarding d(x) as the limit of a function which becomes successively more peaked at the origin when a parameter approaches zero. One such function is 

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The Boltzmann constant is ks and T the absolute temperature. — is the Dirac delta function. Below we assume for convenience (equation (5)) that the delta function is narrow, but not infinitely narrow. The random force has a zero mean and no correlation in time. For simplicity we further set the friction to be a scalar which is independent of time or coordinates. 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

Dirac delta function S Elution volume, exclusion Vo... 

Dirac delta function, an ideal pulse change ... 

Fig, 7,2 Spin-glass overlap probability Pspin-giaes(5) versus overlap q (equation 7,57) 6 x) is the Dirac-Delta function. 

Fig. 3-3. Some Important Probability Density Functions and Their Corresponding Distribution Functions. Arrows are used to indicate Dirac delta functions with the height of the arrow indicating the area under the delta function.

Xq) denotes a unit Dirac delta function located at x x0,... 

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

In most cases of interest, this n + m order derivative can be written as an ordinary n + mth order derivative and some Dirac delta functions. Situations do exist in which this is not true, but they do not seem to have any physical significanpe and we shall ignore them. In any event, all difficulties of this nature could be avoided by replacing integrals involving probability density functions by their corresponding Lebesque-Stieltjes integrals. 

It is often important to be able to extend our present notion of conditional probability to the case where the conditioning event has probability zero. An example of such a situation arises when we observe a time function X and ask the question, given that the value of X at some instant is x, what is the probability that the value of X r seconds in the future will be in the interval [a,6] As long as the first order probability density of X does not have a Dirac delta function at point x, P X(t) = x = 0 and our present definition of conditional probability is inapplicable. (The reader should verify that the definition, Eq. (3-159), reduces to the indeterminate form in this case.)... 

The statistical matrix may be written in the system of functions in which the coordinate x is diagonal. In one dimension, the eigenfunction of is the Dirac delta function. The expansion of (x) in terms of it is... 

The right side of this is the Dirac delta function.3... 

Mathematically,/(l) can be determined from F t) or W t) by differentiation according to Equation (15.7). This is the easiest method when working in the time domain. It can also be determined as the response of a dynamic model to a unit impulse or Dirac delta function. The delta function is a convenient mathematical artifact that is usually defined as... 

The Fourier representation of the Dirac delta function leads then to the result... 

The integral over k may be expressed in terms of the Dirac delta function through equation (C.6) in Appendix C, so that we have... 

This relation may be obtained by the same derivation as that leading to equation (B.28), using the integral representation (C.7) for the three-dimensional Dirac delta function. 

We may also evaluate the Fourier transform <5( ) of the Dirac delta function... 

The Dirac delta function may be readily generalized to three-dimensional space. If r represents the position vector with components x, y, and z, then the three-dimensional delta function is... 

Dirac delta function, transition state trajectory, white noise, 203-207... 

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