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what is the best book or website to study contour integration ?

I find in some question answer using contour integration but I can't understand how they do that

so is there any help ?

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Contour integration is studied in the context of complex analysis. The answer to your question depends on what you already know. Did you study general topology, real analysis or measure theory before? – Ayman Hourieh Jul 7 '13 at 22:35
@AymanHourieh : I can say yes but not correctly 100% – mhd.math Jul 7 '13 at 22:36
@AymanHourieh :but for complex analysis I know alot of general information – mhd.math Jul 7 '13 at 22:37
Stein & Shakarchi and Lang are good books with plenty of examples and exercises. Also check out this question. – Ayman Hourieh Jul 7 '13 at 22:41
Try to study and work out as many examples as possible. Finding the right contour is an art and the best way to learn it is practice. – Ayman Hourieh Jul 7 '13 at 22:49
up vote 3 down vote accepted

While I'm not entirely clear on your background, I believe you'll find a systematic exposure to complex analysis is really the best way since you'll then gain an appreciation for Cauchy's integral theorem and formula, and the residue theorem, enabling you (in conjunction with doing exercises) to develop insight for the typical techniques and why they are useful. The tools complex analysis offers (as you may have seen to prompt your question) are powerful and versatile enough for this to be worthwhile, and since the basic theory has been honed well by now, the prerequisites are often at a minimum (at the level of basic real analysis).

A concise but insightful primer is Junjiro Noguchi's Introduction to Complex Analysis. Its approach and detail towards the integral formula (via homotopy) and residues especially might make it a good text for your requirements, but it may read slightly terse given the inclination towards rigour. However, there are hints and answers included for the exercises.

If your background in real analysis is a little stronger (perhaps with a bit of general topology), I recommend the text of Freitag and Busam (the first of a two-volume set; I am told the second English edition is a considerable improvement on the translation). Quickly enough (the the first three chapters) one finds a very thorough development of the calculus of holomorphic functions, at a level of completeness and sophistication that facilitates the applications of the machinery in elliptic functions and modular forms, and analytic number theory.

Alternatives to these choices that I've perused include Priestley (one of the friendliest choices), Needham (highly intuitive, but takes a while getting there) and Stein and Shakarchi.

I also came across a stand-alone monograph by G N Watson entitled Complex Integration and Cauchy's Theorem republished by Dover. This might be worth looking up, especially since it's affordable, but its going to be better to know some complex function theory before coming to it.

Some free textbook resources may be found here. There are notes for courses online which might be helpful too, like these.

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