I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side of a rectangle, with the residues inside the rectangle providing the main terms and the integral along the other three sides of the boundary going to 0 as the rectangle gets large in some direction (e.g., as it gets wider and wider). But in my case this is not what I find happening.

I have an idea of what the value of my integral should be, and the sum of the residues inside the rectangle matches what should be the dominant terms (the function I'm integrating has a parameter in it, so by "dominant" I mean when the parameter is large). So the integral around the other three sides of the rectangle should be smaller than the sum of the residues, but the integral along those three sides is *not* tending to 0 as it usually does in such calculations.

It feels like I am on the right track, but I don't know how to extract lower order terms from the integral along the other three sides of the rectangle. Has anyone seen an example of something similar to this, where the integral along the added part of a contour doesn't go to 0 as the contour grows (in some direction), but is smaller than the sum of residues inside the contour and can be computed in a reasonable way?

I prefer not to get into more details about the exact integral I am calculating (it is part of my thesis work).