The Sokhotski–Plemelj theorem on the real axis states:
$$ \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi f(0) + \mathcal{P}\int_a^b \frac{f(x)}{x}\, dx $$
where $\mathcal{P}$ is Cacuhy's principal value of the integral.
I'm looking for a generalisation of the theorem, as I would like to calculate an integral like:
$$ \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{(x + i \varepsilon - x_1)(x + i \varepsilon - x_2)}\,dx $$
in terms of the principal value of the $\varepsilon=0$ integral plus some other quantity. I can easily assume $a=0$, $b=\infty$. Moreover, at least one between $x_1, x_2$ will be inside the integration region an will provide a singularity.
Is there a way of deriving something akin to the Sokhotski–Plemelj theorem for my problem? Thanks in advance!