Question

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

Answer 1

@wio

Answer 2

@wio

Answer 3

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

Answer 4

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 \(s_0>\max|p_j|\)

Answer 5

this is basically just the inverse Fourier transform taking into account that we need exponentials to tame asymptotic behavior