Hi All;

I need the solution of the following question :

Find the inverse Laplace Transform of $\displaystyle \frac{1}{s^n-1} ; n $ is postive integer

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- Aug 18th 2010, 04:38 AMraedInverse Laplace Transform
Hi All;

I need the solution of the following question :

Find the inverse Laplace Transform of $\displaystyle \frac{1}{s^n-1} ; n $ is postive integer - Aug 18th 2010, 05:47 AMyeKciM
- Aug 18th 2010, 07:23 AMchisigma
A confortable alternative is to develop $\displaystyle F(s)$ in partial fractions...

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

... so that is...

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - Aug 18th 2010, 07:37 AMAckbeet
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.