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Title:  Representations of the dirac deltafunction 
Type of Resource:  Animação/simulação 
Objective:  The most rudimentary representation is a rectangular pulse: fn(x)=n for –(1/2n) ≤ x ≤ (1/2n). Then δ(x) = lim fn(x) when n→∞ as the rectangle becomes higher and narrower. The most commonly cited representation is based on the normalized Gaussian distribution (bellshaped curve): δ(x) = lim n→∞ n/(√(2π)) e^((n^2 x^2)/2). The limit n→∞ is equivalent to σ→∞, where σ is the standard deviation. Closely analogous is the Lorentzian representation: δ(x) = lim n→∞ (n/π)/(1+n^2x^2). The Lorentzian function is proportional to the derivative of the arctangent, shown as an inset. In the limit as n→∞, the arctangent approaches the unit step function (Heaviside function). Thus the deltafunction represents the derivative of a step function. In quantum mechanics, one frequently encounters the representation δ(x) = lim n→∞ (sin(nx))/(πx). The rapidly oscillating normalized sinc function gives an effective contribution of zero when x≠0 The Fourier series for the deltafunction contains unit contributions from all frequencies. A deltafunction is approached as the number of terms in the following expansion increases: δ(x) = lim n→∞ ∑e^(ikx) from k=n to n. This representation applies only in a neighborhood of the origin. More correctly, this Fourier series represents a delta comb function or shah function Ш(x2nπ)=∑δ(x2nπ) from n=∞to ∞ (after the Russian letter Ш) 
Abstract:  The "deltafunction" was invented by P. A. M. Dirac around 1930 in order to compactly express the completeness relation in quantum mechanics. (Essentially equivalent definitions appear in earlier works of Fourier, Kirchhoff, and Heaviside.) The deltafunction is the limit of a function that grows infinitely large in an infinitesimally small region, while its integral remains normalized to 1. The deltafunction is too singular to be considered a function in the usual sense. Mathematicians have, however, accepted it as a linear functional, a "generalized function", or "distribution". The deltafunction has computational significance only when it appears under an integral sign. Its defining relation can, in fact, be written ∫f(x)δ(x)dx from ∞ to ∞= f(0) or, more generally, ∫f(x)δ(xa)dx from ∞ to ∞= f(a). There are a number of representations of the deltafunction based on limits of a family of functions as some parameter approaches infinity (or zero). In this Demonstration, five of these representations are illustrated. You can select the parameter n to take values from 1 to 10, on its way toward infinity 
Observation:  This demonstration needs the "MathematicaPlayer.exe" to run. Found in http://objetoseducacionais2.mec.gov.br/handle/mec/4737 
Curriculum Component:  Educação Superior::Ciências Exatas e da Terra::Matemática 
Theme:  Educação Superior::Ciências Exatas e da Terra::Matemática::Equações Diferenciais Funcionais 
Author:  Blinder, S. M. 
Language:  English (en) 
Country:  United States (us) 
Publisher:  Wolfram Demonstration Project 
Description:  limit of a function, integral, Dirac Deltafunction 
Web Address:  http://demonstrations.wolfram.com/RepresentationsOfTheDiracDeltafunction/ 
Date:  2008 
Rightsholder:  The Wolfram Demonstrations Project & Contributors 
License:  Demonstration freeware using Mathematica Player 
Submitter:  Universidade Federal de São Carlos (UFSCAR) 
URI:  http://objetoseducacionais2.mec.gov.br/handle/mec/7992 
This Educational Object appears in the following Types of resources:  Educação Superior: Ciências Exatas e da Terra: Matemática: Animações/Simulações 