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I am trying to figure out properties of the following integral:

$$p(t)=\int_{0}^{t} e^{\alpha(t-t')} f(t')dt', \hspace{1 cm} t>t'$$

I would google and read more info about this integral but I do not know a proper specific name for it. It seems like an exponential smoothing, filter, Laplace transform? I am pretty sure it is widely encountered in physics. I would appreciate any link or references on any textbook of paper containing such kind of integral.

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It can be easily expressed as a Laplace transform:

$$ p(t) = e^{\alpha t} \int_0^t e^{- \alpha t'} f(t') \mathrm{d} t' = e^{\alpha t} F(\alpha)$$

where $F(\alpha)$ is, indeed, the Laplace transform of $f(t)$. Also by looking at a Laplace transform table, we can see that this is the Laplace transform of time-shifted $f(t)$, more precisely of $f(t'-t) \Theta (t'-t)$, when transforming using $t'$ variable, $\Theta \left( \# \right)$ being the Heaviside step function.

As far as the physical meaning is concerned, maybe a little bit of context is needed. What did this integral arise from?

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    $\begingroup$ I think you mean $P$ is the transform of $f$. $\endgroup$
    – Javier
    Commented Apr 27, 2014 at 23:04
  • $\begingroup$ Edited! Thanks! $\endgroup$
    – zakk
    Commented Apr 27, 2014 at 23:09

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