I hope you are doing great!
I wanted to work on my game of complex analysis with a simple problem: the Laplace transform of a product of characteristic functions $\mathbb{1}_{[0,b]}$ and $\mathbb{1}_{[a,c]}$ with $a < b < c$. You can for instance think of those functions (distributions) as representing two overlapping rectangular waves (or pulses) traveling on a rope.
We know that the product of such functions should be just $\mathbb{1}_{[a,b]}$. Therefore the Laplace transform of their product is just, with $s \in \mathbb{C}$:
$$ \begin{align} \mathcal{L}\{\mathbb{1}_{[a,b]}(t)\} & = \int_0^\infty \mathbb{1}_{[a,b]}(t) e^{-st} \, dt \\ &= \frac{e^{-as}-e^{-bs}}{s} \end{align} $$
From the integrand above, I deduced that the region of convergence (ROC) was $\mathbb{C}$.
So far, so good. Now, I wanted to derive this result, the hard way, using the property of Laplace transform applied to products of functions:
$$ \mathcal{L}\{f(t)g(t)\} = \frac{1}{2i\pi} \int_{\gamma-i\infty}^{\gamma+i\infty} F(s) G(r-s) \, ds $$
with $r \in \mathbb{C}$, $F = \mathcal{L}\{f\}$ and $G = \mathcal{L}\{g\}$.
So, here, we would have:
$$ \begin{align} \mathcal{L}\{\mathbb{1}_{[0,b]}(t) \, \mathbb{1}_{[a,c]}(t)\} &= \frac{1}{2i\pi} \int_{\gamma-i\infty}^{\gamma+i\infty} \left( \frac{1-e^{-bs}}{s} \right)\left( \frac{e^{-a(r-s)}-e^{-c(r-s)}}{(r-s)} \right) ds \\ &= \frac{1}{2i\pi} \int_{\gamma-i\infty}^{\gamma+i\infty} \frac{1}{s(r-s)} \left( e^{-a(r-s)}-e^{-c(r-s)} - e^{-ar+(a-b)s} + e^{-cr+(c-b)s} \right) ds \end{align} $$
I see that the integrand is bounded but I got stuck at computing this integral. I see that there are different potential approaches (Expansion in Entire Series or using some properties of Exponential Integral Functions). Would you have some derivation to propose?
I would greatly appreciate explicit bounds on the integrand when a claim on an integral going to 0 on a part of contour is made (as in Jordan's lemma).