I would like to know if there is some relation between Laplace Transform and similar definite integrals. For instance, if I know that
$$\mathcal{L}\{f(t)\}(s)=F(s),$$
have I some information about $\int_a^be^{-st}f(t)dt$?
It would be useful on functions defined by parts etc.
Many thanks!
Do you know that $\mathcal{L}(f(t)g(t)) = lim_{T \rightarrow \infty} \frac{1}{2\pi i} \int_{c - iT}^{c+iT}{F(\sigma)G(\tau - \sigma) d\sigma}$ (where the integration is done along a line in the complex plane which is contained in the domain of convergence of $F,G$)? If you know $F(s)$ you could plug in $g$ as the indicator function $1_{[a,b]}$ which has a simple transform, assuming $[a,b]\subseteq \mathbb{R}_{+}$.