Residue Theory Problem

[confusedalot]

I can't figure out how to prove that the integral from -infinity to +infinity of F(t,w)dw=2(pi)t

where F(t,w)=2(1-cos wt)/w^2

[James R]

Well, the integrand has a pole at w=0 ...

I'd have to revise what I know of contour integration to help you further. Sorry. Maybe somebody else can help.

[PhysMachine]

You can build up contours around the entire domain of the function. There is a singularity at the origin, which integrates to 2 pi t because you can shrink the entire contour around the singularity by the Cauchy-Goursat theorem.