I would like to know if there is some relation between Laplace Transform and similar definite integrals. For instance, if I know that


have I some information about $\int_a^be^{-st}f(t)dt$?

It would be useful on functions defined by parts etc.

Many thanks!

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    $\begingroup$ by definition $\mathcal{L}\left\{f(t)\right\}=\int_{0}^{\infty}e^{-st}f(t)dt=F(s)$ . If I understand, you want to know if you have $F(s)$, do I have information about $f(t)$. The answer is yes, it´s called the inverse transform, $\mathcal{L}^{-1}\left\{F(s)\right\}$. $\endgroup$ – Ricardo770 Aug 10 at 18:41
  • $\begingroup$ @Ricardo770, many thanks! In fact, I have $f(t)$ and $F(s)$, but I have troubles to calculate $\int_a^be^{-st}f(t)dt$. So, I'd like to know if having $F(s)$ this integral has some property. Thank you so much. $\endgroup$ – Still_waters Aug 11 at 0:48
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    $\begingroup$ You should look for the region of convergence of the laplace transform to see when its valid. $\endgroup$ – Ricardo770 Aug 11 at 1:28
  • $\begingroup$ Say $F(s)$ exists for any $s\geq 0$. I have some information for instance to $\int_{5}^{100}e^{-st}f(t)dt$...? Many thanks! $\endgroup$ – Still_waters Aug 11 at 1:39
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    $\begingroup$ It will depend on how well behaved $f(t)$ is. $\endgroup$ – Ricardo770 Aug 11 at 1:43

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 $[a,b]\subseteq \mathbb{R}_{+}$.

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  • $\begingroup$ Many thanks. Your answer involving indicator function make me search a little about Laplace transform of step functions and this help me a lot. $\endgroup$ – Still_waters Aug 15 at 0:00
  • $\begingroup$ Thats great, sounds like a $step$ in the right direction badum tss $\endgroup$ – Tom Ariel Aug 15 at 12:57

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