I have a problem with a complex integral for which poles are on the path of integration. For that, I took the integral and "disassembled" it into segments with an infinitesimal step from the poles, and I ended up with the principal value of the integral, consisting of three line segments, and two residues that represent pole contribution. I resolved all the integrals and residues but one: I am trying to work out the integral for the line segment between two poles, but seem to be stuck. I can't use the Residue theorem here, and since the integral is not from infinity to infinity I don't see how trying to assess limits of the integral could be helpful. I easily may be missing something, but I'm not sure what.
Could anyone provide any guidance on how this works? intuitively I know this integral should go to zero, but I don't see how/where