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A residue formula I have strong evindence to believe that the following identity holds: $$ \frac{n!}{2\pi i}\oint_{|z-1|=\epsilon} \frac{z^{a-1} \mathrm{d}z}{(z^d-1)^{n+1}} = d^{-n-1}\prod_{j=1}^{n}\...

General Question: Is there a reference where the saddle point approximation is applied to multiple contour integrals? In particular, say we have the integral $$ I_N = \frac{1}{(2\pi i)^N} \oint \left[\...