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
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
$\displaystyle \chi$ $\displaystyle \sigma$