laplace.xform - The Laplace Transform Calculus Jonathan L.F...

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The Laplace Transform : Calculus Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 11 February, 2009 (at 12:56 ) Abstract: This first gives a defn of exponential order which is better adapted to convolution. Following, is a discussion of the “tapping on a bell” problem; one text called this “soldiers marching on a bridge”. Both interpretations need a grain-of-salt. . . Recall. For a fnc f : [ 0 , ) R and real number s we define the Laplace transform of f , evaluated at s , b f ( s ) = L ( f ) ( s ) := Z 0 e - st f ( t ) d t , 1 : whenever this integral exists. For a real R , say that f has exponential order R , written f Ord( R ) , if f is locally-integrable and for each number Q > R : lim t →∞ | f ( t ) | / e Q · t = 0 . : Exercise: One can replace ( ) by the limsup t →∞ | f ( t ) | / e Q · t < , : which is only superficially weaker. 2 : Lemma. Consider an f Ord( R ) . Then b f ( s ) exists for each s > R . Proof. Fix s , then pick Q with s > Q > R . The integrand in (1) is eventually bounded by product e - st · e Qt , i.e by e - [ s - Q ] t . Since s - Q
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