Morera's theorem

Morera's theorem states that a continuous complex-valued function on a domain is holomorphic if its integral around every closed triangle in the domain is zero.

Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. This is the regularity Morera's theorem concludes from continuity and vanishing triangle integrals.
Complex differentiability: Differentiability with respect to a complex variable, using the same limit along every direction of approach. Morera's conclusion is complex differentiability at every point.
Uniform convergence theorem for holomorphic functions: A theorem stating that a locally uniform limit of holomorphic functions is holomorphic. Its standard proof passes vanishing triangle integrals to the limit, then applies Morera's theorem.
Giacinto Morera: An Italian mathematician known for the theorem that characterizes holomorphic functions by triangle integrals. Morera gave the result its name.
Looman–Menchoff theorem: A theorem giving conditions under which a continuous function satisfying the Cauchy–Riemann equations is holomorphic. It replaces Morera's integral hypothesis with a differential condition under specific assumptions.
Cauchy's integral theorem: The theorem that a holomorphic function has zero integral around closed curves that bound suitable regions. It supplies the forward implication that Morera's theorem reverses under continuity.
Continuity: A property of a function whose values change by arbitrarily little near each point when its input changes sufficiently little. Continuity is essential in Morera's hypothesis and cannot generally be omitted.
Weierstrass theorem on locally uniform convergence: The theorem that locally uniform limits of holomorphic functions remain holomorphic. Morera's theorem gives a common proof by integrating the convergent sequence around triangles.
Augustin-Louis Cauchy: A French mathematician whose work established foundational results in complex analysis. Cauchy's integral theorem provides the converse direction central to Morera's criterion.
Goursat's theorem: A proof of Cauchy's integral theorem that does not require continuity of the derivative. Goursat proceeds from holomorphy to vanishing integrals, whereas Morera reverses that implication with continuity.

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