Its correct according to Mathcad
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
so there is a simple pole inside of the contour at
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