Morera's Theorem

Theorem

Let $D$ be a simply connected domain in $\C$.

Let $f: D \to \C$ be a continuous function.

If, for every simply closed contour $\gamma$ in $D$:

$\displaystyle \int_\gamma f \left({z}\right) \ \mathrm d z = 0$

then $f$ is analytic on $D$.

Proof

For a fixed $z_0\in D$ and $z\in D$ we consider the function:

$\displaystyle F \left({z}\right) = \int_{z_0}^z f \left({w}\right) \ \mathrm d w$

because the integration over any closed counter in $D$ is equal to $0$, then the integration it's only depend on the beginnig and ending of any path between to points, and thus the above integration it's well defined. Now it's sufficient to use the generalized Cauchy's Integral Formula :

$f^{(k)}(z)=\frac{k!}{2\pi i}\int_{\gamma}\frac{f(\epsilon)}{(\epsilon -z)^{k+1}}d\epsilon$

for any $z$ in the interior of $\gamma$,by definition of the primitive we have $F^{\prime}=f$, then we conclude that $f$ is analytic on $D$.

Source of Name

This entry was named for Giacinto Morera.

This is the converse of Cauchy's Integral Theorem.