Laplace transform of $\frac{f(t)}{1-e^{-at}}$

Question

I would like to know if there is a way to calculate the Laplace transform for a given $f(t)$:

$$\dfrac{f(t)}{1-e^{-at}}$$

Thanks!

Answer 1

Assuming $a>0$, get

$$\begin{align}\int_0^{\infty} dt \frac{f(t)}{1-e^{-a t}} e^{-s t} &=\int_0^{\infty} dt\, f(t) \sum_{k=0}^{\infty} e^{-(s + a k)t} \\ &= \sum_{k=0}^{\infty} \hat{f}(s+k a) \end{align}$$

where we reversed the order of summation and integration, and

$$\hat{f}(s) = \int_0^{\infty} dt \,f(t) e^{-s t}$$