chapter6.page01 - itive numbers M,a such that | f t | ≤...

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CHAPTER 1 Laplace Transform Methods Laplace transform is a method frequently employed by engineers. By applying the Laplace transform, one can change an ordinary dif- ferential equation into an algebraic equation, as algebraic equation is generally easier to deal with. Another advantage of Laplace transform is in dealing the external force is either impulsive , (the force lasts a very shot time period such as the bat hits a baseball) or the force is on and off for some regular or irregular period of time. 1. The Laplace Transform If f ( t ) is defined over interval [0, ), the Laplace transform of f , denoted as ˆ f ( s ) , is L ( f ) = b f ( s ) = Z 0 e - st f ( t ) dt Our first theorem states when Laplace transform can be performed, Theorem 1.1 . If f ( t ) is (piecewise) continuous and there are pos-
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Unformatted text preview: itive numbers M,a such that | f ( t ) | ≤ Me at for all t ≥ c Then b f ( s ) is defined for all s > c The next result shows that Laplace transform is unique in the sense that different continuous functions will have different Laplace trans-form. Theorem 1.2 . If b f ( s ) = b g ( s ) for all s > c , then f ( t ) = g ( t ) at all t where both are continuous. Notice if f ( t ) and g ( t ) are piecewise continuous (continuous except at finite points where left and right limits exists,) their Laplace trans-forms can be same for all s > c even if they are different at the isolated discontinuous point. Since solutions of ordinary equations must be continuous, so this is of no important concern. 1...
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This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

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