I've been working on this for the past hour.
let
then
let
then
which implies that
so finally ?
I didn't even think about using contour integration. The problem is actually an exercise in my complex analysis textbook.
Moo suggested using the keyhole contour. My book suggests using a semicircle in the upper-half plane with a little semicircle about the origin (because ln z is undefined for z=0). They also suggest using the branch so that ln z is analytic on and inside of the contour.
So let R be the radius of the big semicircle and r be the radius of the small semicircle
and let
then
so there is a simple pole inside of the contour at
then
So I have
now letting R go to infinity and r go to zero
I'm just assuming that the two contour integrals evaluate to zero. I'll try to offer a justification in another post. But I'll probably need some help.
Equating real parts on both sides I have
so finally