Find if a function is the Laplace transform of a periodic function

1. Feb 2, 2010

libelec

1. The problem statement, all variables and given/known data

Could $$$F(s) = \frac{1}{{s(1 - {e^{ - s}})}}$$$ be the Laplace transform of some periodic function? Why? If so, find that periodic function

3. The attempt at a solution

If it was the Laplace transform of some periodic function, the the Laplace transform of the first wave should be 1/s, and the period T should be 1. The function whose Laplace transform is 1/s is H(t). Then the periodic function should be the stairs function, $$$f(t) = \left\{ \begin{array}{l} n,x \in [nT,(n + 1)T] \\ n + 1,x \in [(n + 1)T,(n + 2)T] \\ \end{array} \right.,n \in N + \left\{ 0 \right\}$$$

Now, the stairs function is of exponential order, since it's smaller or equal than the function t+1 for all t, and that is an exponential order function. Then it has a Laplace transform.

So far, so good. If the stairs function transforms to F(s) there, then F(s) is the Laplace transform of some periodic function.

But I don't think that's what the exercise asks me to do, since a latter question asks me to find the periodic function. I think there's something I have to prove through F(s) that allows me to say that it is the Laplace transform of a periodic function.

But I don't know what is that.

Any ideas?

2. Feb 2, 2010

LCKurtz

I'm thinking the answer is no and you should be thinking about the "why" rather than trying to find a periodic function. Your stairs function certainly isn't periodic...

3. Feb 2, 2010

libelec

Allright, it's true.

But then why couldn't it be periodic? Is it because the inverse Laplace transform of 1/s is H(t), and that isn't a periodic function? What does F(s) has to have to check if it belongs to a periodic function or not?

4. Feb 2, 2010

LCKurtz

Well, I didn't check your steps, but I assume you have the correct inverse with your staircase function. It isn't periodic and the FT is a 1-1 transform so doesn't that settle it?

5. Feb 2, 2010

libelec

Yes, but I thought there was another argument to say that. I mean, since the question seems to be theorical.

6. Feb 2, 2010

Anybody?