# Inverse Laplace Transform

• Aug 18th 2010, 05:38 AM
raed
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
• Aug 18th 2010, 06:47 AM
yeKciM
Quote:

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 :D
just solve integral...

$\displaystyle f(t)= \frac {1}{2\pi i} \lim_{T \to \infty } \int _{\sigma - iT} ^{\sigma + iT } F(S) e^{st} ds$
• Aug 18th 2010, 08:23 AM
chisigma
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

$\chi$ $\sigma$
• Aug 18th 2010, 08:37 AM
Ackbeet
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.