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Unformatted text preview: 26 The Inverse Laplace Transform We now know how to find Laplace transforms of unknown functions satisfying various initial- value problems. Of course, its not the transforms of those unknown function which are usually of interest. Its the functions, themselves, that are of interest. So let us turn to the general issue of finding a function y ( t ) when all we know is its Laplace transform Y ( s ) . 26.1 Basic Notions On Recovering a Function from Its Transform In attempting to solve the differential equation in example 25.1, we got Y ( s ) = 4 s 3 , which, since Y ( s ) = L [ y ( t ) ] | s and 4 s 3 = L bracketleftbig 4 e 3 t bracketrightbigvextendsingle vextendsingle s , we rewrote as L [ y ( t ) ] = L bracketleftbig 4 e 3 t bracketrightbig . From this, seemed reasonable to conclude that y ( t ) = 4 e 3 t . But, what if there were another function f ( t ) with the same transform as 4 e 3 t ? Then we could not be sure whether the above y ( t ) should be 4 e 3 t or that other function f ( t ) . Fortunately, someone has managed to prove the following: Theorem 26.1 (uniqueness of the transforms) Suppose f and g are any two piecewise continuous functions on [ , ) of exponential order and having the same Laplace transforms, L [ f ] = L [ g ] . Then, as piecewise continuous functions, f ( t ) = g ( t ) on [ , ) . 521 522 The Inverse Laplace Transform (You may want to quickly review the discussion of equality of piecewise continuous func- tions on page 493.) The proof of this theorem goes beyond our abilities at this time. 1 What is important is that it assures us that, if L [ y ( t ) ] | s = L bracketleftbig 4 e 3 t bracketrightbigvextendsingle vextendsingle s , then y ( t ) = 4 e 3 t , at least for t 0 . What about for t < 0 ? Well, keep in mind that the Laplace transform of any function f , F ( s ) = L [ f ] | s = integraldisplay f ( t ) e st dt , involves integration only over the positive T axis. The behavior of f on the negative T axis has no effect on the formula for F ( s ) . In fact, f ( t ) need not even be defined for t < 0 . So, even if they exist, there can be no way to recover the values of f ( t ) on the negative T axis from F ( s ) . But that is not a real concern because we will just use the Laplace transform for problems over the positive T axis problems in which we have initial values at t = 0 and want to know what happens later . What all this means is that we are only interested in functions of t with t 0 . That was hinted at when we began our discussions of the Laplace transform (see note 3 on page 473), but we did not make an issue of it to avoid getting too distracted by technical details. Now, with the inverse transform, requiring t 0 becomes more of an issue. Still, there is no need to obsess about this any more than necessary, or to suddenly start including for t 0 with every formula of t . Let us just agree that the negative T axis is irrelevant to our discussions, and that...
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