How to get the laplace inverse for a fraction a complex poles at S-domain ?

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How to get the laplace inverse for a fraction a complex poles at S-domain ?

Mathematics
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a fraction a complex poles at s-domain? can you clarify what you mean?

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if you have a rational expression \(F(s)=P(s)/Q(s)\) where \(P(s)=k(s-z_1)\cdots(s-z_m),Q(s)=(s-p_1)\cdots(s-p_n)\) where \(z_i,p_j\) are the zeros and poles of \(F\) then we can use Mellin's inversion formula as normal: $$f(t)=\int_{\gamma} e^{st}F(s)\ ds$$where \(\gamma\) is given by the line \((s_0-i\infty, s_0+i\infty)\) where \(s_0\) is chosen to ensure \(F\) is well-behaved along this line; in particular, it is sufficient to pick \(s_0>\max|p_j|\)
this is basically just the inverse Fourier transform taking into account that we need exponentials to tame asymptotic behavior

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