TITLE: Looking for some interesting complex integration contours
QUESTION [35 upvotes]: I am currently working on some tools to make contour integration in a proof assistant less painful and I'm looking for interesting examples of contours in the complex plane used in the literature. I am most interested in contours that add little epsilon detours in order to avoid singularities/branch cuts/etc.
One example of what I am looking for is the following contour that is used in the study of modular functions.
[An interesting integration contour from Apostol's "Modular Functions and Dirichlet Series in Number Theory"]

REPLY [2 votes]: See page 382 in [1], a paper I have found very enlightening.
Reference
[1] HARDY, G. H. (1905), "A method for determining the behavior of certain classes of power series
near a singular point on the circle of convergence", Proceedings of the London Mathematical Society 2, pp. 381-389, JFM 36.0474.01.