Applications of Residue Theorem in complex analysis? Does anyone know the applications of Residue Theorem in complex analysis? I would like to do a quick paper on the matter, but am not sure where to start. 

The residue theorem
The residue theorem, sometimes called Cauchy's residue theorem (one of many things named after Augustin-Louis Cauchy), is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem.
Illustration of the setting
The statement is as follows:
  Suppose $U$ is a simply connected open subset of the complex plane, and $a_1,\ldots,a_n$ are finitely many points of $U$ and $f$ is a function which is defined and holomorphic on $U\setminus\{a_1,\ldots,a_n\}$. If $\gamma$ is a rectifiable curve in $U$ which does not meet any of the $a_k$, and whose start point equals its endpoint, then 
  $$\oint_\gamma f(z)\,dz=2\pi i\sum_{k=1}^n I(\gamma,a_k)\mathrm{Res}(f,a_k)$$

I'm sure many complex analysis experts are very familiar with this theorem. I was just hoping someone could enlighten me on its many applications for my paper. Thank you!

Skip to the 11th page of 
this document. It uses residue calculus to prove the classical result that
$\sum_{i=1}^{\infty}1/n^{2} = \pi^{2}/6$. Plus it leaves the easy stages of the argument for you to fill in for yourself.
