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

Theorem [1] Suppose $G$ is a region in $\sC$ , and $f:G\to \sC$ is a continuous function. If for every closed triangle $\Delta$ in $G$ , we have $$\int_{\partial \Delta} f\, dz = 0,$$ then $f$ is analytic on $G$ . (Here, $\partial \Delta$ is the piecewise linear boundary of $\Delta$ .)

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

Bibliography

1

W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Inc., 1987.

2

E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.

3

R.A. Silverman, Introductory Complex Analysis, Dover Publications, 1972.