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?