I have a hard time seeing, for non-continuous functions, e.g. square root, or functions with singularities, how you can confirm that a keyhole contour integral approaches zero on the smaller circle.
It's relatively easy to see how that happens on the larger circle (e.g. by looking at degree of polynomial denominator, type of trig function in the numerator), but I have trouble finding a reasonably good heuristic or a quick proof when deciding on a contour under time constraint. Appreciate any feedback.
Example: $log(z)/(1+z^3)$
I can't see what distinguishes this integral from one that does not approach zero on the smaller keyhole contour circle (although I can't easily think of an example of that myself). Or put another way: I don't see what distinctive feature should immediately signal that the smaller circle integral will converge to zero.