Morera’s theorem

Morera’s theorem provides the converse of Cauchy’s integral theorem.

Theorem [1]Suppose $G$ is a region in $ℂ$, and $f:G\to ℂ$ is a continuousfunction. If for every closed triangle $\mathrm{\Delta }$ in $G$, we have

$${\int }_{\partial \mathrm{\Delta }}f𝑑z=0,$$

then $f$ is analytic on $G$. (Here, $\partial \mathrm{\Delta }$ is the piecewise linearboundary (http://planetmath.org/BoundaryInTopology) of $\mathrm{\Delta }$.)

In particular, if for every rectifiable closed curve $\mathrm{\Gamma }$ in $G$, we have${\int }_{\mathrm{\Gamma }}f𝑑z=0,$then $f$ is analytic on $G$. Proofs of this can be found mostundergraduate books on complex analysis [2, 3].