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 continuous function. 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 linear boundary (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 most undergraduate books on complex analysis [2, 3].

Title	Morera’s theorem
Canonical name	MorerasTheorem
Date of creation	2013-03-22 12:58:09
Last modified on	2013-03-22 12:58:09
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	12
Author	matte (1858)
Entry type	Theorem
Classification	msc 30D20