Let's consider
. Then, clearly
is analytic except it has a simple pole at
. So, we may define a family of contours
given by
where
is the semi-circle on the
-axis going through
and
is the semi-circle on the
-axis passing through
and
are just the intervals on the real line; of course orient
counterclockwise. Note then by the Residue Theorem that
Evidently it's true that
and so it easily follows that
but the whole point to picking
was that
and thus