I was reading a paper which featured the following horrendous integral(adsbygoogle = window.adsbygoogle || []).push({});

##\displaystyle\prod_{n=1}^L\oint_{C_n}\frac{dx_n}{2\pi i}\prod_{k<l}^L(x_k-x_l)\prod_{m=1}^L\frac{Q_w(x_m)}{Q^+_\theta(x_m)Q^-_\theta(x_m)}##

where ##Q^\pm_\theta(x)=\prod_{k=1}^L(u-\theta_k\pm \frac{i}{2})## and ##Q_w(x)=\prod_{r=1}^M(x-w_r)##.

From the paper, ##C_n## denotes the integration contour which encloses ##\theta_n \pm \frac{i}{2}## counterclockwise.

Obviously I can just use the residue theorem and sum over the poles to evaluate it but I'm having trouble doing that since the thing is so complicated. How do I write this as a residue? Can I just focus on the denominators and ignore the other parts?

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# Evaluating this contour integral

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