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Math Help - Inverse Laplace Transform

  1. #1
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    Inverse Laplace Transform

    Hi All;
    I need the solution of the following question :
    Find the inverse Laplace Transform of \frac{1}{s^n-1} ;   n  is postive integer
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by raed View Post
    Hi All;
    I need the solution of the following question :
    Find the inverse Laplace Transform of \frac{1}{s^n-1} ;   n  is postive integer
    use formula for inverse Laplace transformation
    just solve integral...

     \displaystyle f(t)= \frac {1}{2\pi i} \lim_{T \to \infty } \int _{\sigma - iT} ^{\sigma + iT } F(S) e^{st} ds
    Last edited by yeKciM; August 18th 2010 at 06:02 AM.
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  3. #3
    MHF Contributor chisigma's Avatar
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    A confortable alternative is to develop F(s) in partial fractions...

    \displaystyle \frac{1}{s^{n}-1} = \sum_{k=0}^{n-1}\frac {r_{k}}{s-e^{i 2 \pi \frac{k}{n}}} = \sum_{k=0}^{n-1}\frac {e^{i 2 \pi \frac{k}{n}}}{n\ (s-e^{i 2 \pi \frac{k}{n}})} (1)

    ... so that is...

    \displaystyle \mathcal {L}^{-1} \{\frac{1}{s^{n}-1} \} =  \sum_{k=0}^{n-1}\frac {e^{i 2 \pi \frac{k}{n}}}{n}\ e^{i 2 \pi \frac{k}{n}\ t} (2)

    Kind regards

    \chi \sigma
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  4. #4
    A Plied Mathematician
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    Which, considering the contour you'd have had to integrate over, and the residue calculus you'd have had to do in order to compute the integral, probably amounts to the same underlying math.
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