vanishing of integrals around closed loops

A closed contour starts and ends at the same point — a loop. Asking whether the integral of f around every such loop is zero turns out to be one of the most informative questions you can ask about f on a given domain. If the answer is yes for all loops, f is extraordinarily well-behaved there; if no, f is detecting some obstruction, usually a hole or a singularity.

The criterion sits inside the great equivalence: on a connected open set, the integral of a continuous f around every closed contour is zero if and only if f has a primitive there, which is in turn equivalent to f's integrals being path-independent. One direction is the fundamental theorem (a primitive forces loops to vanish since start equals end). The other builds the primitive by integrating from a fixed base point — and loops vanishing is exactly what makes that integral independent of the chosen path, hence well-defined.

So 'loops vanish' is a compact litmus test you can sometimes check directly. The recurring counterexample is f(z) = 1/z on any region encircling the origin: its loop integral around the origin is 2 pi i, not zero, signaling that no single-valued primitive exists there. Cauchy's theorem is precisely the powerful sufficient condition that makes loops vanish: holomorphy plus a simply connected domain. When those hold, every loop integral is automatically zero, and you may stop checking.