Integral Homework: Proving the Yellow Equation

[asi123]

Homework Statement

Hey guys.

I have this problem:

http://img12.imageshack.us/img12/6729/91881544.jpg

It's like a two parts problem.
The first one was to prove the equation in red, I did that with a bit of trigo.
The second part is to use the integral of the function in green to prove the thing in yellow. I'm stuck on the path that goes from - epsilon to epsilon, that semi circle, how can I find the value of this function on the path?

Thanks.

Homework Equations

The Attempt at a Solution

 

[yyat]

Compute the http://en.wikipedia.org/wiki/Residue_(complex_analysis)" of f at zero, the use

$$2\pi iRes(f,0)=\int_{|z|=\epsilon}f(z)dz$$

and try to find the integral over the half-circle in terms of the integral over the entire circle.