I Integration above or below axis

[TheCanadian]

If you are trying to find the integral of a function with a complex pole on the real axis (i.e. $[-\infty,\infty]$) via analytic continuation, does it matter if you integrate along a contour either in the upper-half plane or lower-half plane? I was under the assumption that either approach was equivalent. For example, if you have a pole at $a + ib$, where $a$ and $b$ are positive constants, and take your contour to be a semicircle with radius out to infinity, so that by Jordan's lemma that component of the integral goes to 0, then shouldn't either approach (i.e. top half or bottom half of a circle) give you the same answer? It seems that I am missing or incorrectly assuming something obvious here.

(I've attached an image that attempts to depict what I tried to explain above.)

 

[mfb]

The integral over your arcs doesn’t vanish with that function unless f(x) has some special properties (that will lead to additional poles).

1/x (here: 1/r) is not sufficient as the length of the arcs scales with r and r/r=1 doesn’t converge to 0 for r to infinity.

 

[Orodruin]

To be more specific, Jordan’s lemma applies to functions that can be written on the form $f(z) = e^{iaz} g(z)$ with $a$ real and non zero. The sign of $a$ determines which half circle you close and the behaviour of $g$ determines whether or not the integral over that half circle vanishes.