I was teaching complex analysis today, and wondered about this question.
I formulated a conjecture, which agrees with my intuition.
Let be meromorphic in and it has finitely many poles in a contour then:
Now suppose that has infinitely (but countably many) it seems that the correct result would be:
However, the expression on the right has two problems.
1)The sequence might now converge.
2)Even if convergence it needs to be absolutely convergent because otherwise it is not well-defined by the Riemann rearragnement theorem.
Here is the conjecture.
Let be meromorphic on an open set containint the interior of the contour with countably infinite many poles at . If the sequence is absolutely convergent with limit then the round integral over of is .