Morera's theorem

Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be continuous. Assume that $\oint_{\partial T} f(z) \, dz = 0$ for each closed triangle $T \subseteq U$, where $\partial T$ is the boundary of $T$ traveled once and the integral is a contour integral. Then $f$ is holomorphic on $U$.

Remarks

It is enough to know that $\oint_\gamma f(z) \, dz = 0$ for each closed piecewise continuously differentiable path $\gamma$ in $U$, because the boundary of a triangle is such a path. Proof idea: on each disc in $U$, $F(z) = \int_{[c, z]} f(w) \, dw$ is a primitive of $f$. So $F$ is holomorphic, and $f = F'$ is holomorphic because derivatives of holomorphic functions are holomorphic. Together with Goursat's theorem, this shows: a continuous $f$ is holomorphic on $U$ if and only if its integrals over all triangles in $U$ are zero. A main use is the theorem on uniform limits of holomorphic functions.

Explanations

L1 human

Morera's theorem is a converse of Cauchy's theorem. If the integral of a continuous function around each small triangle is zero, then the function is holomorphic.