# Inverse Laplace Transform of a multiple 2nd order pole

Hi all,
I'm not able to find the inverse laplace transform of a multiple 2nd order pole:
(LaTeX notation) \frac{ s + c }{ ( s^2 + a s + b )^n }
Thank you
-- Daniele Pallastrelli
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What an interesting system you must be working on!
It'll have the form (At^(n-1)+Bt^(n-2)+etc)(e^etc + cos(etc + etc*t)). Your handbook won't cover this because multiple same-frequency resonances in a control system are quite rare, so you need to find the answer with one of the same ways that you can find the transform of a single 2nd-order pole pair:
- Factor the pole into it's complex conjugate pair, to give you (s + c) / ((s + q + jp)^n (s + q - jp)^n) - Find the partial fraction expansion of the expression. Your math package will probably choke. - Find the inverse transform using the expression for multiple 1st-order poles - Convert the various complex conjugates back to real numbers using Euler's identity - Verify your arithmatic by seeing that all the imaginary numbers cancel - you'll probably have to go back to the partial fraction expansion step once or twice - Once everything checks out you're done! - Clean all the scattered paper from the floor, - retrieve the documentation for your math package from the trash, - and your computer if necessary, - shave, - eat something nutritious, - let your friends know you're alive and sane, - have a beer.
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Well, actually I'm not working on a system. I'm writing a paper ;-) and I'm trying to study the transfer function as much general as possible. As you can understand, I can't apply the method you suggest. I have to invert the general transfer function I wrote, not a particular instance of it.
Thank you Daniele.

Your
package
Euler's