Proof of integral identity (popped up in a Fourier transform)

[phagist_]

Homework Statement

Prove;
$\int_{-\infty}^{\infty} \frac{sin(\gamma)}{cosh(\lambda)-cos(\gamma)} e^{i \omega \lambda}d \lambda= 2 \pi \frac{sinh(\omega(\pi-\gamma))}{sinh(\pi \omega)}$

Homework Equations

Contour Integration/Residue Theorem?

The Attempt at a Solution

I have messed around with the exponential for a bit, but to no avail - I was thinking maybe the Residue theorem might play a part? I'm not really sure how to continue from here.

 

[phagist_]

any ideas? I am legitimately stumped on this one..