I have an equation which is the sum of exponentials $V(t)=\sum\_{pre}\epsilon(t-t\_{pre})$ where $\epsilon(t)=\frac{\epsilon\_0}{\tau\_m}e^{\frac{-t}{\tau\_m}}$. The terms in $pre$ are evenly spaced time intervals. Is there not some identity that relates this sum to a single exponential? I feel like it is something related to the fourier series, but I am not remembering.


Any clues?



How do I analyze a sum of decaying exponentials?
How's this:


Take the times $t\_{pre}$ to be of the form $t\_{pre} = T\_{0} + iT$, where $i$ ranges over
some set of integers of the form $\{0, 1, 2, . . . , N\}$, and we allow the possibility
that $N$ is infinite and place no restriction on the sign of $T$, the interval size, though
we take $T \ne\ 0$; since the case $T = 0$ boils down to there being only one value of $t\_{pre}$, $T\_{0}$, in which case $V(t)$ trivially becomes $(\epsilon\_{0}/\tau)e^{-(t - T\_{0})/\tau}$. (I'm dropping your subscripts to $\tau$ for convenience, i.e. to make typing
slightly easier.) Then in general we have $V(t) = \sum\_{i = 0}^{N}(\epsilon\_{0}/\tau)e^{-(t-T\_{0} - iT)/\tau}$; factoring out $(\epsilon\_{0}/\tau)e^{-(t-T\_{0})/\tau}$ yields $V(t) = (\epsilon\_{0}/\tau)e^{-(t-T\_{0})/\tau}\sum\_{i = 0}^{N}e^{iT/\tau}$; in this form, $V(t)$ is a single exponential times a constant, as per your request. The sum can be cleaned up using standard formulae for geometric series (cf. <http://en.wikipedia.org/wiki/Geometric_series>), *viz* $\sum\_{i = 0}^{N}e^{iT/\tau} = ((1 - e^{(N + 1)T/\tau})/(1- e^{T/\tau}))$, so finally $V(t) = (\epsilon\_{0}/\tau)e^{-(t-T\_{0})/\tau}((1 - e^{(N + 1)T/\tau})/(1- e^{T/\tau}))$. In the event $T > 0$, the sum diverges
unless $N$ is finite; for $T < 0$, however, letting $N \to \infty$ we obtain $V(t) = (\epsilon\_{0}/\tau)e^{-(t-T\_{0})/\tau}(1/(1- e^{T/\tau}))$ or $V(t) = (\epsilon\_{0}/\tau)(1/(1- e^{T/\tau}))e^{-(t-T\_{0})/\tau}$.


It is interesting to note that, as $T < 0$ becomes smaller in magnitude, i.e. $T \to 0^{-}$, $V(t)$ explodes, as if it were the sum of an infinite number of identical terms 
$(\epsilon\_{0}/\tau)e^{-(t-T\_{0})/\tau}$; I leave a similar analysis when $T > 0$ to 
you.


I think you will find this stuff is generally covered under Laplace transforms, rather
than Fourier series.


To the boys at <http://mathoverflow.tqft.net/discussion/784/question-being-bumped-to-the-front-page/>: I did it again! Accidentally hit "Save" instead of "Preview"! Only lasted
an hour this time. Whew! Gotta be more careful with those buttons!


