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Unformatted text preview: Inverse Laplace Transform So far, we have dealt with the problem of finding the Laplace transform for a given function f(t), t > 0, L {f(t)} = F(s) = e ! st f(t)dt " # Now, we want to consider the inverse problem, given a function F(s), we want to find the function f(t) whose Laplace transform is F(s). We introduce the notation L1 {F(s)} = f(t) to denote such a function f(t), and it is called the inverse Laplace transform of F. Remark : The inverse Laplace transform is not unique: If g(t) = 1 if 0 < t < 3 ! 8 if t = 3 1 if t > 3 " # $ % $ then L {g(t)} = 1/s and L {1} = 1/s So, both functions have the same Lapalce transform, therefore 1/s has two inverse transforms. But, the only continuous function with Laplace transform 1/s is f(t) =1. A crude, but sometimes effective method for finding inverse Laplace transform is to construct the table of Laplace transforms and then use it in reverse to find the inverse transform. Example : 1) Since L {1} = 1/s, then L1 {1/s} = 1 2) Since L {t} = 1/s 2 , then L1 {1/s 2 } = t 3) Since L {cos at} = s s 2 + a 2 , then L1 { s s 2 + a 2 } = cos at The following properties will simplify our calculations: 1) Linear Property If c 1 and c 2 are constants,...
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This note was uploaded on 07/23/2011 for the course MAP 2302 taught by Professor Staff during the Fall '08 term at FIU.
 Fall '08
 STAFF

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