the homology form of Cauchy's theorem

The simple Cauchy theorem assumes a simply connected domain; the homotopy version asks loops to be contractible. The homology form is the most general and flexible statement, freeing you from the shape of the domain entirely. It works on any open set and asks only an arithmetic condition: a closed contour (or a combination of them, a 'cycle') gives a zero integral for every holomorphic f exactly when it winds zero times around every point outside the domain.

Precisely: let f be holomorphic on an open set U, and let C be a cycle in U that is null-homologous in U, meaning the winding number n(C, w) = 0 for every point w not in U. Then the integral of f over C is zero. The intuition: the only way a loop can produce a nonzero integral is by encircling something missing from U — a hole or a removed point. If the loop winds zero times around every absent point, it encircles nothing problematic, and the integral collapses to zero just as in the simply connected case. Simple connectivity is the special case where U has no outside points to worry about, so the condition is automatic.

This is the version that powers the general residue theorem. For a cycle C null-homologous in U minus its singularities, the integral of f over C equals 2 pi i times the sum over the singular points a of n(C, a) times the residue of f at a. The winding numbers do the bookkeeping, counting each singularity with its multiplicity and sign. The homology form is the clean, final word: it tells you a contour integral depends only on holomorphy plus how the contour winds around what is missing, with no assumption about the global shape of the region.