I was teaching complex analysis today, and wondered about this question.

Let .

Compute,

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 hasinfinitely(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 beabsolutely convergentbecause 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 .