Morera's theorem

Cauchy's theorem says: if a function is holomorphic, then its integral around every closed loop is zero. Morera's theorem is the converse, and it is surprisingly powerful: if a continuous function has the property that its integral around every closed loop in a region is zero, then it must be holomorphic there. Vanishing loop integrals are not just a symptom of holomorphy — they are enough to guarantee it.

Here is why it works in plain steps. Suppose f is continuous on a disk and the integral over the boundary of every triangle inside is zero. Then you can define a function F by integrating f from a fixed base point to z along any path — path-independence is exactly what vanishing loop integrals provide, so F is well defined. A short estimate shows F is complex-differentiable with F' = f, so F is holomorphic. But holomorphic functions are infinitely differentiable, so F', which is f, is holomorphic too. Continuity plus zero triangle integrals upgrades all the way to holomorphy.

Morera's theorem is the workhorse for proving that limits and integrals of holomorphic functions stay holomorphic. If a sequence of holomorphic functions converges uniformly on compact sets, the loop integrals pass to the limit and stay zero, so the limit is holomorphic by Morera — no need to check derivatives directly. The same trick shows that functions defined by integrals depending on a parameter are holomorphic, which is why it sits next to the analyticity-of-parameter-integrals result.