DE.03.4.Literacy - :Sin@a tD, : at Differential...

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Mathematica Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©2001-2007 Publisher: Math Everywhere, Inc. Version 6.0 DE.03 Laplace Transform and Fourier Analysis LITERACY What you need to know when you're away from the machine. · L.1) Look at these calculations of Ÿ 0 E - st f @ t D t and the Laplace transform of f @ t D for three sample f @ t D 's: Clear @ f, s, t D ; f @ t _ D = Sin @ t D ; : 0 E - s t f @ t D t, LaplaceTransform @ f @ t D , t, s D> : If B Re @ s D > 0, 1 1 + s 2 , Integrate A - s t Sin @ t D , 8 t, 0, < , Assumptions Ø Re @ s D § 0 EF , 1 1 + s 2 > Clear @ f, s, t, k D ; f @ t _ D = t; : 0 E - s t f @ t D t, LaplaceTransform @ f @ t D , t, s D> : If B Re @ s D > 0, 1 s 2 , Integrate A - s t t, 8 t, 0, < , Assumptions Ø Re @ s D § 0 EF , 1 s 2 > Clear @ f, s, t, k D ; f @ t _ D = DiracDelta @ t - 5.4 D ; : 0 E - s t f @ t D t, LaplaceTransform @ f @ t D , t, s D> 9 - 5.4 s , - 5.4 s = Explain why any Laplace literate person knows in advance that the calculations were guaranteed to come out the way they did. · L.2) Here is Mathematica 's calculation of the Laplace transform of f @ t D = 2.2 DiracDelta @ t - 3.5 D + 3.8 DiracDelta @ t - 5.6 D : Clear @ f, t, s D ; f @ t _ D = 2.2 DiracDelta @ t - 3.5 D + 3.8 DiracDelta @ t - 5.6 D ; LaplaceTransform @ f @ t D , t, s D 3.8 - 5.6 s + 2.2 - 3.5 s Use the definition LaplaceTransform @ f @ t D , t, s D = Ÿ 0 E - st f @ t D t to say why you could have written this down without help from Mathematica . Without machine help, write down the formula for the Laplace transform of f @ t D = DiracDelta @ t - 1 D + 2 DiracDelta @ t - 2 D + 3 DiracDelta @ t - 3 D · L.3) When you go with a function of t such as f @ t D = 3.8 E - 0.4t Cos @ 4.7 t D , and calculate its Laplace transform F @ s D , you get: Clear @ f, F, t, s D ; f @ t _ D = 3.8 E - 0.4 t Cos @ 4.7 t D ; F @ s _ D = LaplaceTransform @ f @ t D , t, s D 3.8 H 0.4 + s L 22.09 + H 0.4 + s L 2 You began with a function f @ t D of t and learned that its Laplace transform F @ s D is a function of s . Is this a problem? · L.4) Here is a little table of Laplace transforms: Clear @ f, F, s, t, a, b D ; 9 f @ t _ D = E a t , F @ s _ D = LaplaceTransform @ f @ t D , t, s D= Clear @ f, F, s, t, a, b D ; 8 f @ t _ D = Cos @ a t D , F @ s _ D = LaplaceTransform @ f @ t D , t, s D< Clear @ f, F, s, t, a, b D ; 8 f @ t _ D = Sin @ a t D , F @ s _ D = LaplaceTransform @ f @ t D , t, s D< Clear @ f, F, s, t, a, b D ; 9 f @ t _ D = E a t Cos @ b t D , F @ s _ D = Together @ ExpandAll @ LaplaceTransform @ f @ t D , t, s DDD= Clear @ f, F, s, t, a, b D ; 9 f @ t _ D = E a t Sin @ b t D , F @ s _ D = Together @ ExpandAll @ LaplaceTransform @ f @ t D , t, s DDD= : a t , 1 - a + s > : Cos @ a t D , s a 2 + s 2 > : Sin
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DE.03.4.Literacy - :Sin@a tD, : at Differential...

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