Consider the complex-valued function
which has simple poles at each of the fourth roots of -4. If
, then
where
Now consider a semicircular contour centered at the origin with radius
, where the diameter is affixed to the real axis. Let
denote the perimeter of the contour, with
denoting the semicircular part of the contour and
denoting the part of the contour that lies in the real axis.
and we'll be considering what happens as
. Clearly, the latter integral will be correspond exactly to the integral of
over the entire real line. Meanwhile, taking
, we have
and as
, we see that the above integral must approach 0.
Now, by the residue theorem, the value of the contour integral over the entirety of
is given by
times the sum of the residues at the poles within the region; in this case, there are only two simple poles to consider when
.
So we have
(where
are the poles surrounded by
)
Presumably, we wanted to show that
This integrand is even, so
as required.
