# Math Help - Inverse Laplace Transform

1. ## 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

2. Originally Posted by raed
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...

$\displaystyle f(t)= \frac {1}{2\pi i} \lim_{T \to \infty } \int _{\sigma - iT} ^{\sigma + iT } F(S) e^{st} ds$

3. 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

$\chi$ $\sigma$

4. 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.