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Unformatted text preview: Definitions Properties of the Laplace transform Applications to ODEs and systems of ODEs Chapter 6: Laplace Transforms Chapter 6: Laplace Transforms Definitions Properties of the Laplace transform Applications to ODEs and systems of ODEs 1. Definitions The Laplace transform , L ( f ), of a piecewise continuous function f (defined on [0 , )) is given by L ( f )( s ) = F ( s ) = exp(- s t ) f ( t ) dt . Clearly, the above integral only converges if f does not grow too fast at infinity . More precisely, if there exist constants M > 0 and k R such that | f ( t ) | M exp( k t ) for t large enough, then the Laplace transform of f exists for all s > k . If f has a Laplace transform F , we also say that f is the inverse Laplace transform of F , and write f = L- 1 ( F ). Chapter 6: Laplace Transforms Definitions Properties of the Laplace transform Applications to ODEs and systems of ODEs General properties s-shifting, Laplace transform of derivatives & antiderivatives Heaviside and delta functions; t-shifting Differentiation and integration of Laplace transforms 2. Properties of the Laplace transform The Laplace transform is a linear transformation , i.e. if f 1 and f 2 have Laplace transforms, and if 1 and 2 are constants, then L ( 1 f 1 + 2 f 2 ) = 1 L ( f 1 ) + 2 L ( f 2 ) ....
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- Spring '08