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!

Here is a few from the top of my head.

*

*Cauchy's Formula

*Showing existence of Laurent and Taylor series

*Integral representation for binomial coefficients

*From the above, the hockey stick identity

I'll add more when I learn more.
