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Laplace transform proof

  1. May 16, 2010 #1
    Hello, I am trying to find the laplace transform of [tex]\frac{f(t)}{t}[/tex]

    http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node92.html [Broken]
    Lemma 10.3.9

    I don't understand why we are taking the limit as s goes to infinity

    Can anyone help me?
    thank you
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 16, 2010 #2


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    Look at it this way. You have this relationship:

    [tex]F(s) = -\frac d {ds} G(s) = -G'(s)[/tex]

    Take the anitiderivative of both sides from s to b

    [tex]\int_s^b F(u)\, du = -(G(b) - G(s)) = G(s)-G(b)[/tex]

    This must hold for all b, so let [itex]b\rightarrow \infty[/itex], knowing [itex]G(b)\rightarrow 0[/itex] and switching sides:

    [tex]G(s) = \int_s^\infty F(u)\, du = \int_s^\infty F(s)\, ds [/tex]

    This builds in the constant of integration in the formula.
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