Consider the change of coordinates
which yields a Jabobian determinant of
Then the double integral is equivalent to
As a function over the positive reals, this will only converge if
. We can find the value of the integral by considering the complex-valued function,
and integrate it over the contour
consisting of a circle
of radius
connected to a smaller circle
of radius
(both centered at the origin) by two line segments parallel and close to the positive real axis (but not touching it), oriented in the opposite direction relative to one another and denoted
and
, respectively. (See attachment)
By the residue theorem, the value of the contour integral will be the sum of the residues at the poles of
multiplied by
. We have only one simple pole at
, which has residue
So we have
By the ML lemma and the restriction of
, we have as
and
that
We're left with
Note that the integral along
corresponds to the integral we wanted to compute in the first place, so we can replace
. For the other, we write the numerator of
as
, to account for the fact that we're considering a particular branch of
. As
and
, we're left with
as required.
