Why is the contour integral in upper plane different from the lower plane in this case?
where Im[a] and Im[b] are negative and p is real. Besides, Re[a] and Re[b], and p are positive.
The the poles in complex plane are shown below: enter image description here
I get these residues by just putting $a$ into the other three denominators $(k+a)$, $(p-k-b)$, $(p-k+b)$. and I also use the same procedure to the other three poles $p+b, -a, p-b$ (since each pole is a simple pole). I think the summation of these four poles should give zero. What's the problem?