i am trying to get the real part of electric susceptibility using the imaginary part with Kramers-Kronig relation for a Lorentz-Drude model.I chose to ask this question in math stack exchange as im having issues with the computation.
(1)The expression i am trying to get is : $ \chi'=\chi_s \frac{1-\Omega^2}{(1-\Omega^2)^2+(\Omega\Delta)^2} $
(2)The expression i am starting with is : $ \chi''=\chi_s \frac{\Omega\Delta}{(1-\Omega^2)^2+(\Omega\Delta)^2} $
(3)And i am using Kramers-Kronig relation for only positive frequencies : $ \chi'=\frac{2}{\pi}P \int_0^\infty \frac{s\chi''(s)}{s^2-\omega^2} ds $
When putting (2) into (3) i find myself with an expression having 5 poles which is tedious to evaluate using residus theorem.I am asking if there's an easier way to compute this Principal value integral ?