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Thread: The Laplace Transform of a Derivative

  1. January 29th 2009, 04:07 PM #1
    razorfever
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    The Laplace Transform of a Derivative

    show that \mathcal{L} \left \{ \frac d{dt} f(t) \right \} = sF(s) - f(0)

    how???
    Last edited by Jhevon; January 29th 2009 at 09:48 PM. Reason: Making things neater a bit
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  2. January 29th 2009, 09:46 PM #2
    Jhevon
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by razorfever View Post
    show that L [df(t) / dt ] = sF(s) - f(0)

    how???
    By definition: F(s) = \mathcal{L} \{ f(t) \} = \int_0^\infty e^{-st} f(t)~dt

    Thus, \mathcal {L} \{ f'(t) \} = \int_0^\infty e^{-st} f'(t)~dt

    Now proceed using integration by parts with u = e^{-st} and dv = f'(t)~dt
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