Morera's Theorem
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Theorem
Let $D$ be a simply connected domain in $\C$.
Let $f: D \to \C$ be a continuous function.
Let $f$ be such that:
- $\ds \int_\gamma \map f z \rd z = 0$
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:
- $\ds \map F z = \int_\gamma \map f w \rd w$
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.
Sources
- 1977: Serge Lang: Complex Analysis