# Morera’s theorem

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

Theorem [1] Suppose $G$ is a region in $\mathbb{C}$, and $f:G\to\mathbb{C}$ 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 (http://planetmath.org/BoundaryInTopology) 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].

## References

• 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.
Title Morera’s theorem MorerasTheorem 2013-03-22 12:58:09 2013-03-22 12:58:09 matte (1858) matte (1858) 12 matte (1858) Theorem msc 30D20