Applications of Residue Theorem in complex analysis?

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.

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.