# Laplace Transform of a periodic function

1. May 7, 2007

### G01

1. The problem statement, all variables and given/known data
Suppose f(t) is a periodic function with period T. Show that the Laplace transform of f is:

$$L(f) = \frac{1}{1-e^{-sT}}\int_0^T f(t)e^{-st} dt$$

3. The attempt at a solution
I started with the definition of a Laplace Transform for f:

$$L(f) = \int_0^\infty f(t)e^{-st}dt$$

Using the periodicity of the function this becomes:

$$\sum_{K=0}^\infty \int_{KT}^{(K+1)T} f(t)e^{-st}dt$$

At this point I have been trying to get this in the form of a geometric series, since the fraction in the final result leads me to look for a geometric series, but this has been without success. Any hints into the right direction to move from here on out would be appreciated. Thank you for any help you can offer.

2. May 8, 2007

### TD

Try a substitution: t = nT+u with u as new variable.