Morera's Theorem

Theorem

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

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

Let $f$ be such that:

$\displaystyle \int_\gamma \map f z \rd z = 0$

for every simple closed contour $\gamma$ in $D$

Then $f$ is analytic on $D$.

Proof

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

$\displaystyle \map F z = \int_\gamma \map f w \rd w$

where $\gamma$ is any (simple) contour starting at $z_0$ and ending at $z$.

By Primitive of Function on Connected Domain, $F$ is a primitive of $f$.

Since $F$ is analytic and $F' = f$, we conclude that $f$ is analytic as well.

$\blacksquare$

Also see

This is the converse of the Cauchy-Goursat Theorem.

Source of Name

This entry was named for Giacinto Morera.