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:


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

By definition of a primitive we have $F^\prime = f$.

From Cauchy's Integral Formula it follows that $f$ is analytic on $D$.

This is the converse of Cauchy's Integral Theorem.