I'm trying to calculate the integral $$ \frac{1}{2\pi i} \int_\mathcal{C} g(\omega) \log[1 - \chi(q,\omega)], $$ where $g(\omega) = (e^{\beta \omega}-1)^{-1}$ has an infinite number of evenly spaced poles along the imaginary axis and $\mathcal{C}$ is a curve encircling the real axis excluding the pole at $\omega = 0$ shown in the figure below. Furthermore $$\chi(q,\omega) = \frac{1 - 2f(q^2)}{2 q^2 - \omega}$$ with $f(\epsilon) = (e^{\beta \epsilon}+1)^{-1}.$
The answer I'm supposed to get is $$ - \frac{1}{\pi} \int_{-\infty}^{\infty} d\omega \; g(\omega)[ \delta(q,\omega) - \delta(q,0) ],$$ with $$\delta(q,\omega) - \text{Arg}[1 - \chi(q,\omega + i0+)].$$ I can see that this is connected to picking up the argument by going around a branch cut situated on the negative real axis as well as excluding the pole at $\omega = 0$ in the contour $\mathcal{C}$, but I cannot make the correct arguments to obtain this result.
Physics: For anyone interested the question arises from the evaluation of a Matsubara sum from the article Low Temp. Phys. 59, Nos. 3/4 (1985) by Nozieres and Schmitt-Rank.