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Unformatted text preview: The Laplace Transform : Calculus Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA firstname.lastname@example.org 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 : [ , ) R and real number s we define the Laplace transform of f , evaluated at s , b f ( s ) = L ( f ) ( s ) := Z 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...
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.
- Fall '07