## anonymous one year ago How to get the laplace inverse for a fraction a complex poles at S-domain ?

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1. abb0t

@wio

2. anonymous

@wio

3. anonymous

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

4. anonymous

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 $$s_0>\max|p_j|$$

5. anonymous

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