Jordan's lemma

From Companal

Statement

Suppose is a function (possibly with isolated singularities) on an open subset of , that contains the real axis and upper half-plane , such that has no essential singularities in the strict upper half-plane, and only finitely many poles on the real axis and in the upper half-plane. Suppose further that:

Then, if denotes the semicircle of radius centered at the origin, and if , we have:

Thus, we get:

where the sum is taken over all poles in the upper half-plane.