See contour below.
where is the holomorphic extension of the principal branch to the left helf-plane described below. Therefore
We now use the following which I believe is correct:
For polynomials, there exists a holomorphic extension of to the entire complex plane except along branch-cuts extending from each of the zeros and poles of the expression . Additionally, the difference across each branch-cut is determined by the order of the branch point such that traveling in the positive sense around the branch point, this difference is for a zero or order n and for a pole of order m.
And I'll do the red and blue branch-cut integrals at . You can do the others just like how I do this one (and what will the difference of the integrand across the branch-cut at the real axis be?) I'll let .
And of course you'll get for the one at which then combine to give you the cos term and the two at the real axis give you the other term.
This stuff is fun ain't it?
(and for the record, we really need to be addressing how the integral goes to zero on all the other parts of the contour)