Can Laplace Transform Converge for Periodic Functions on [0,infty)?

[amcavoy]

I need to show that for f(t)=f(t+T) on [0,infty), that the Laplace Transform is:

$$\mathcal{L}\left[f(t)\right]=\frac{\int_0^Te^{-st}f(t)\,dt}{1-e^{-sT}}.$$

The first thing I did was to write the transform as:

$$\mathcal{L}\left[f(t)\right]=\sum_{n=0}^{\infty}\int_{nT}^{\left(n+1\right)T}e^{-st}f(t)\,dt.$$

Am I on the right track here? It looks like the formula given to me (that I need to show) is an infinite geometric series multiplied by the integral in the numerator. However, I am unable to get what I have into something of that form. Any ideas?

Thank you.

 

[Physics Monkey]

Yes, you are practically done already. Make a good change of variables, use the periodicity of f, and you're home free.

 

[amcavoy]

Could you please elaborate on that a bit more? Thank you.

 

[Physics Monkey]

Sure, you want each term to give $$\int^T_0 e^{-st} f(t) dt$$ times the geometric series part, right? So why not try to make a change of variable in each term to see if you can get this out? Make the limits look right for each term and see where that leads you.