notes 6-5 - SECTION 6.5 IMPULSE FUNCTIONS AND THE DIRAC...

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SECTION 6.5 IMPULSE FUNCTIONS AND THE DIRAC DELTA FUNCTION EXAMPLES. (1) An electrical circuit is hit by lightning. (2) A subterranean rock layer is hit by an explosion. (3) Alex Rodriguez hits a baseball. (4) A heavy swinging pendulum is hit by a sledgehammer. In each of these examples we have a very large voltage or force applied over a very short time period, that is, the forcing functions have a impulsive nature – so we call them impulse functions. We want to develop a model for such situations. If g ( t ) is a forcing function, then a measure of its strength is Z -∞ g ( t ) dt. To develop a standard impulse function whose action takes place at t = 0, we need to look at functions which are 0 except very close to t = 0 but whose integrals have value 1. Here’s such a function: d τ ( t ) = ( 1 2 τ - τ < t < τ, 0 t ≤ - τ or t τ. ) Let’s draw some graphs and check some of this stuff.
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The Dirac delta function δ is sort of but not quite given by δ ( t ) = lim τ 0 d
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This note was uploaded on 11/28/2010 for the course M 56840 taught by Professor Schurle during the Spring '10 term at University of Texas at Austin.

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notes 6-5 - SECTION 6.5 IMPULSE FUNCTIONS AND THE DIRAC...

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