Given the following summation over a pole-like structure (or rather a summation over Lorentzians for $\eta > 0$)
$\sum_\limits{j} \frac{a_j}{x-b_j-i\eta}$
I'm interested in the following. Say that for some $j$ we have that a bunch of $b_j$ lie very closely together, i.e. fulfill
$|b_j-b_{j+1}|< \epsilon$
I would like to approximate these close lying "poles" by something more simple, preferably
$\sum_\limits{j=1}^n \frac{a_j}{x-b_j-i\eta} = \frac{\tilde{a}}{x-\tilde{b}-i \tilde{\eta}} $
Testing around a bit with Wolfram Alpha reveals that this approximation can be pretty bad if the $a_j$ are very different from one another but also reasonable if this is not the case.
Now the question is how to best determine the tilde parameters ($\tilde{a},\tilde{b},\tilde{\eta}$). For example just using the average of all $b_j$ for $\tilde{b}$ would be possible, but I'm sure I can do better by weighting these or so?