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!
 A: You can find every conceivable (and several inconveivable) application of the residue theorem in The Cauchy method of residues: theory and applications by Mitrinović and Kečkić, Dordrecht, 1984 (ISBN: 9027716234). 
If that's not enough, there's even a second volume: The Cauchy method of residues: theory and applications, Vol. 2 by the same authors, and publisher. This one published in 1993 (ISBN: 0792323114.)
Amazon seems to carrry a one-volume book by the same authors and with a very similar title, published in 2001 by Kluwer, but I haven't seen that exact version.
A: Other then as a fantastic tool to evaluate some difficult real integrals, complex integrals have many purposes.
Firstly, contour integrals are used in Laurent Series, generalizing real power series.
The argument principle can tell us the difference between the poles and roots of a function in the closed contour $C$:
$$\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (\text{Number of Roots}-\text{Number of Poles})$$
and this has been used to prove many important theorems, especially relating to the zeros of the Riemann zeta function.
Noting that the residue of $\pi \cot (\pi z)f(z)$ is $f(z)$ at all the integers.  Using a square contour offset by the integers by $\frac{1}{2}$, we note the contour disappears as it gets large, and thus
$$\sum_{n=-\infty}^\infty f(n) = -\pi \sum \operatorname{Res}\, \cot (\pi z)f(z)$$
where the residues are at poles of $f$.
While I have only mentioned a few, basic uses, many, many others exist.
A: 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.
A: 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.
