Need reference for working with branch cuts

Question

I am in a MOOC where we are using contour integration a lot, in conjunction with special functions (not an introductory or undergrad class). Working with branch cuts, deforming them, or just determining when to add which phase to multi-valued functions to compute integrals correctly, is proving to be challenging (the MOOC is: https://www.edx.org/course/complex-analysis-physical-applications-misisx-18-11x).

The class uses deformations of contours and branch cuts in a very advanced way (e.g. "double deforming" semi-infinite branch cuts to wrap around multiple branch points while preserving steepest descent directions when evaluating some integrals), without systematically presenting all the details of how to work with those deformations and phases introduced by multi-valued functions. I am left with the impression that I am not really mastering the topic. Most of the references I've found online - and there are many good ones - do not go far enough in presenting "real life" examples (usually from physics, when estimating the asymptotic behavior of integral representations that cannot be solved exactly, those representing solutions of ODEs).

Is there a comprehensive and systematic reference out there, preferably with worked out examples, that I could consult to get more mastery of multi-valued functions and branch cuts?

Answer 1

I would always recommend http://residuetheorem.com by Ron. It contains many complicated integrals. I also posted in my blog many contour integrals http://advancedintegrals.com/category/contour-integration/. Also you can search here by tag for example https://math.stackexchange.com/questions/tagged/contour-integration

You can try to find some interesting integrals.

Answer 2

I think @Zaid mentioned my blog, in which I try to post interesting integration problems, most of which are done out by contour integration. I think you raise a very interesting point from a pedagogical perspective, one about which I complain bitterly (and about which I hope to write something): the way contour integration is being taught here and in the classroom is rather disjointed. Further, it has the feel of a golden hammer - nice, but not something you would ever use in a pinch. (Why use a golden hammer to kill the fly when a nice newspaper is there for the taking?)

Here's the thing: there is a way to teach contour integration in a simple, unified way. That is, there does not need to be one lesson for a an integral with poles in the complex plane and another with branch points. Especially irritating is the "residue at infinity," which confounds so many people and is absolutely unnecessary if just a little intelligence and consistency is used in selecting contours for evaluating definite integrals.

So, without trusting me to finish my magnum opus on contour integration techniques, my advice is to forget the residue theorem (except as a time-saving device) and focus on using Cauchy's theorem to evaluate integrals. If you think about it, the residue theorem is derived by deforming a closed contour so that it detours around a pole. Heck, we do the same thing for branch points! The difference is, for poles, the integrals up to and back from the singularity always cancel, while for branch points they typically do not. The cancellation of the integrals for poles results in the residue theorem...but forgetting that, you should see that the treatment of integrals with and without branch points are the same!

Another point you raise is deriving a systematic framework for deriving contours given a definite integral to be evaluated. Many people have stated that choosing countours is an art. I agree, for now, but I think it may be possible to define contours mechanically. At this point there is simply the nub of an idea and I have yet to piece anything concrete, but I do not see why contours cannot in principle be defined parametrically such that the poles/branch points within the contour are found easily.

So for concrete, worked out examples, do visit my blog (and of course other blogs and the many many worked out examples here), but for a systematic treatment of contour integration techniques for evaluating definite integrals, I have yet to see such a treatment that results in a framework for practical, everyday use.