# Thread: Laplace transform involving Heaviside functions

1. ## Laplace transform involving Heaviside functions

Hello again. I'm stumped by this excercise:

Determine the function $\displaystyle f(t)$ for which
$\displaystyle f(t) = L^{-1}\left\{ \frac{2(1-e^{-s})}{s(1-e^{-3s})}\right\}$
Sketch the waveform and express the function in analytical form.

I've been getting the hang of Laplace transforms involving unit step functions the last few days, but I'm just stumped on how to start this one. I just need a pointer on how to rearrange it into a form that makes sense. I can't see it working with the Heaviside shift in the denominator...

2. ## Re: Laplace transform involving Heaviside functions

I'm thinking it has something to do with the power series of $\displaystyle e^{-3s}$. If I separate it out:

$\displaystyle (\frac{2}{s} - \frac{2e^{-s}}{s}) . \frac{1}{1-e^{-3s}}$

Then I get

$\displaystyle (\frac{2}{s} - \frac{2e^{-s}}{s}) + (\frac{2e^{-3s}}{s} - \frac{2e^{-4s}}{s}) + (\frac{2e^{-6s}}{s} - \frac{2e^{-7s}}{s})...$

The inverse Laplace of which is:

$\displaystyle 2u(t)-2u(t-1)+2u(t-3)-2u(t-4)+2u(t-6)-2u(t-7)...$

I think that could be it