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Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to compute if only within the scope of real calculus. There are a great many techniques designed for this purpose, for instance, both the choice of the complex variable function and the choice of an appropriate contour is critical to our success of computation.

I want to study these techniques systematically, so I want recommendations for tutorials that cover them in a systematical way, the more inclusive the better. In a word, I want to learn as many those techniques as possible, so I think such a tutorial is a must.

Any advice is welcome. Thanks!

Ps: the tutorial I currently have at hand is Stein's Complex Analysis, it is good but covers too few exercises about such techniques.

Vim
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  • what for ? and I'm not convinced there are systematical ways (yet), maybe 5 main methods, and many (many) tricks useful only for very special cases. – reuns May 25 '16 at 03:56
  • @user1952009 to be honest, the exam is drawing near... And I'd love to learn about the "five main methods". – Vim May 25 '16 at 03:59
  • and you know that for finding primitives, computer algebra systems use things like https://en.wikipedia.org/wiki/Meijer_G-function ? and you didn't say "what for" ? – reuns May 25 '16 at 04:07
  • @user1952009 it seems that some methods are indeed useful not only for some special cases but also for classes of cases they give rise to. So I suspect that quite a lot problems that are doable using contour integral techniques are actually interconnected. For instance, the method used to compute $$\int_{(0,\infty)} \frac{\sin x}x dx$$ is also workable for general cases like $$\int_{(0,\infty)} R(x){\sin x} dx$$ where $R(x)$ is a rational function of appropriate decay. – Vim May 25 '16 at 04:07
  • I don't think you need to know all the methods systematically, simply search contour integral on the forum. and https://en.wikipedia.org/wiki/Methods_of_contour_integration is already quite nice. for example at your exam I can tell there will be something close to https://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28III.29_.E2.80.93_trigonometric_integrals with $\cos t = \frac{z+1/z}{2}$ on the unit circle – reuns May 25 '16 at 04:10
  • @user1952009 thanks for the wonderful source! I simply didn't expect Wikipedia page to contain so many examples. – Vim May 25 '16 at 04:11

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The most comprehensive treatment of residue-based techniques that I know is the two volume set:

  • The Cauchy method of residues: theory and applications by Mitrinović and Kečkić, Dordrecht, 1984 (ISBN: 9027716234).

  • The Cauchy method of residues: theory and applications, Vol. 2 by the same authors, and publisher. This one published in 1993 (ISBN: 0792323114.)

Amazon carries 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.

These books cover more or less every imaginable (and many unimaginable) application of residues.

mrf
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