/sci/ - Science & Math

Any idea how to prove this integral equals zero using complex analysis? I can do the finite cases simply by invoking the residue theorem, with a contour that includes the poles at z = 0+ni, but I'm not certain how to do it with this infinite product in the denominator.

I'm pretty certain it should evaluate out to zero. I need to use complex analysis here, I can't just do the partial fraction decomposition method sadly.