7.5, Dirac Delta Function - = ∠∞ ∞-1 dt t t t This...

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Differential Equations – The Dirac Delta Function (Section 7.5) Suppose ) ( t g cy y b y a = + + where g is large during a short interval ( 29 τ , - and otherwise is 0. Define - + - = = - dt t g dt t g t I t t ) ( ) ( ) ( 0 0 . This is a measure of the strength of a forcing function. If ) ( t g is a force, ) ( I is the total impulse of the force over the interval ( 29 + - 0 0 , t t . If y = electrical current and ) ( t g = time derivative of voltage, then ) ( I = total voltage impressed on the circuit on the time interval. In particular suppose that 0 t is 0 and - < < - = = t or t t t d t g , 0 , 2 1 ) ( ) ( where is a small positive constant. Graph ) ( t d . Evaluate ) ( I for ) ( t g defined above. Now let d act over shorter and shorter time periods (i.e., __________ ). Note that the smaller gets the ________________________ 2 1 gets. 0 ________, ) ( lim 0 = t t d and since 1 ) ( = I for each 0 , it follows that = ) ( lim 0 I _____________. These two equations define an idealized unit impulse function δ . is called the Dirac Delta Function .
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i.e., δ is defined as follows: =
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Unformatted text preview: = ∫ ∞ ∞-1 ) ( , ) ( dt t t t This corresponds to a unit impulse at t = 0. We can define a unit impulse at an arbitrary point by finding ( 29 t t-. Note that there is not a nice β€œbland” function satisfying this set of equations (we didn’t see anything like this in Calculus I). This is an example of a generalized function . We will define the Laplace transform as follows: { } { } lim lim lim lim lim lim lim lim lim ) ( lim ) ( β†’ β†’ β†’ β†’ β†’ β†’ β†’ β†’ β†’ β†’ = = = = = = = = =-=-Ο„ t t d L t t L So we have { } _ __________ ( =-t t L and { } ________ __________ lim ) ( = = β†’ t t L . Example: ) ( , 1 ) ( ); 3 ( 20 = β€² =--=-β€² β€² y y t y y...
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This note was uploaded on 01/13/2012 for the course MATH 2914 taught by Professor Pamelasatterfield during the Fall '10 term at NorthWest Arkansas Community College.

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7.5, Dirac Delta Function - = ∠∞ ∞-1 dt t t t This...

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