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I recently took a complex analysis class and obviously studied a lot about contour integration, but I wonder if there's more to it. I mean, usually taking the integral comes down to choosing a branch cut (99% of them are standard), contour (again, 99% of them are standard) and applying the Cauchy's Residue theorem. In Real analysis however, there are many interesting ideas, such as

  • Leibniz's differentiation under the integral sign

  • Taking the sum of different integrals, so that the resulting integrable function is much easier

  • Making the integral double, switching the order using the Fubini's theorem and blah blah blah

and many others.

So I wonder if there's any book that covers non standard techniques. For example, in mathematical physics class, we crossed the branch cut a couple of times while studying Hankel functions - I'm talking about this type of ideas.

I would also love to hear about interesting ideas you applied while taking contour integrals!

J W
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Daniel Neskorodov
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    Apostol's book has many interesting results using contour integrals, e.g., the keyhole contour, also known as Hankel path, for the analytic continuation of the zeta function and related topics. – Dietrich Burde Jun 06 '25 at 18:46
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    See here for a lot of references on good complex analysis books in general. – Jean Marie Jun 06 '25 at 20:07

2 Answers2

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A really good and important and entertaining book is Ron Gordon’s Complex Integration. It complements standard treatments.

A rural reader
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Look this book The Cauchy method of residues: Theory and applications, by Dragoslav Mitronivić and Jovan Kečkić.

Delta
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