m2k_opm_distrb - Introduction to Distributions Impulses In...

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Introduction to Distributions; Impulses In physics and mechanics we quite commonly work with idealized particles or electrical entities, having a fnite non-zero mass or charge, respectively, located at a single point. The mathematical analog oF such ideas lies in the concept oF a distribution or generalized function . It is not easy (we do not say impossible!) to include these ideas in the development oF the basic calculus because the standard defnitions oF derivative and integral require some adaptation to encompass them. It is easy to deal with these ideas in the Laplace transForm context because the Laplace transForms oF the commonly used distributions are very ordinary Functions oF s . We begin our development by observing that a pulse oF strength A and duration h defned by g ( x ) = ± A, 0 < x < h ; 0 , x h has the Laplace transForm ( L g ) ( s ) = A s (1 - e - sh ) . Let g h ( x ) be the pulse Function obtained with A = 1 h . It is clear then that R b 0 g h ( x ) dx = 1 as long as 0 < h < b . As we let h 0 this uniForm distribu- tion oF strength 1 h on the interval [0 , h ] tends toward a point concentration oF strength 1 located at the point 0. Mathematically, this limit concentration is not a Function but, rather, what is called a distribution . It was introduced by the English physicist P.A.M. Dirac and is traditionally denoted by δ 0 or, in a common misuse oF Function notation, by δ ( x ). We will call it the ”Dirac distribution with support 0” but it is common in the literature to fnd it re- Ferred to as the ”Dirac delta Function”. This latter terminology is a misnomer 1
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This note was uploaded on 01/23/2012 for the course MATH 4254 taught by Professor Robinson during the Spring '10 term at Virginia Tech.

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m2k_opm_distrb - Introduction to Distributions Impulses In...

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