How to compute $chi_1$, a Lorentzian absorption in the following paper?

Question

I have the following paper:
https://arxiv.org/abs/1508.01213
on the third page, where they take: $$chi_2(omega)= frac{-kgamma}{(omega-omega_0)^2+gamma^2},$$ I don't see how to explicitly compute $chi_1$.
Can someone please show me the way?

Philip · Answer

I realised after writing this that the OP probably doesn't need this answer any more, but anyway:
You need to use the Kramers-Kronig relations which connect the real and imaginary parts of any complex function $chi(omega) = chi_1(omega) + i chi_2(omega)$ that is analytic in the upper half-plane.
In particular, the relation you need (as described in the paper) is:
$$chi_1(omega) = frac{1}{pi} mathcal{P} int_{-infty}^infty frac{chi_2(omega')}{omega'-omega} text{d}omega',$$
where $mathcal{P}$ denotes the "Cauchy principal value".

How to compute $chi_1$, a Lorentzian absorption in the following paper?

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