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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!

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This is a paper for pure personal enjoyment. It is not a homework assignment of any form. –  Anthony Peter Jan 16 '13 at 2:23
Hello, Welcome to math.SE. IMO one of the main use is to find integrals (i.e, contour integration). It is pretty important for finding real definite integrals also..Search for contour inegration would yielda lot of results. –  TheJoker Jan 16 '13 at 2:27
Sorry for sounding flippant, but this is like asking if power series have applications and wanting people to tell you about them. Can you first please tell us what work you have put in to find applications yourself? A quick web search should give you several. –  KCd Jan 16 '13 at 2:27
I personally have seen the many applications when solving real integrals, but other than this, I haven't been able to find much. –  Anthony Peter Jan 16 '13 at 2:43
If you're also interested in applications in other fields, google the lecture notes of Andreas Gathmann for his Algebraic Geometry class. There, he uses the residue theorem in the proof of the Riemann-Roch theorem to show that certain sections cannot exist. –  InvisiblePanda Jan 16 '13 at 12:09
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3 Answers

up vote 5 down vote accepted

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.

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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.

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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.

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Nice answer! Those who were my teachers in complex variables speak very well in this book. He is not as conhencido as it should be in the West. –  Elias Jan 16 '13 at 12:18
@Elias, I agree, the books are fascinating, and not as well-known as they should be. –  mrf Jan 16 '13 at 13:08
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