This question comes from the context of calculating the grand potential for a simple toy problem (a linear chain of masses connected by springs with a mass defect) using statistical field theory.
In order to compute an infinite sum, I have been following Eliashberg procedure to instead relate the sum to a contour integral. The resulting integral is:
$\displaystyle\mathcal{I} = \oint dz \frac{1}{e^{z/T}-1}\ln\left[1-a\left(\sqrt{\frac{z^2}{z^2-\omega_D^2}}-1\right)\right], $
where $a$ and $\omega_D$ are real constants and $T$ is another real variable (temperature) to be retained. The original contour, $\Gamma_1$, is shown on the left of the figure.
My thoughts (please point out errors) are that the points $ z=\pm\omega_D $ are branch points from the square root. They would therefore require a branch cuts on the real axis. We could either use one cut on $(-\omega_D, \omega_D)$ or two, one on $(-\infty, -\omega_D)$ and one on $(\omega_D, \infty)$. The second contour in the figure, $\Gamma_2$ shows a possible deformation if we choose the two infinite cuts.
I am also aware however that the logarithm itself requires a branch cut to be imposed. As far as I'm aware we may choose this to be on the real axis and it should begin at the point where the argument of the logarithm is zero, and extend to infinity. The starting point is found to be on the real axis and is $>\omega_D$. For this reason I thought it best to pick the two infinite cuts relating to the square root, since then they just overlap?
My questions are:
How would one evaluate the residues at the branch points, since they are also poles?
Instead of tackling the integral directly by calculating residues, is there a way of using contour $\Gamma_2$ to write $\mathcal{I}$ as a combination of integrals over a real variable (since I believe The large and small arcing integrals are zero in the appropriate limits) from the contributions on the real axis? ie. the parts tracing close to the branch cuts.
Help and advice would be greatly appreciated.