14 - Morera's Theorem

Morera's Theorem

Suppose that $f:U\to \mathbb{C}$ is a continuous function on an open subset $U\subseteq \mathbb{C}$

If for any closed path $\gamma :[a,b]\to U$ where ${\int }_{\gamma }f(z)dz=0$ then

$$f\text{ is holomorphic}$$

Proof

By Existence of Primitives Theorem then $f$ has primitive $F:U\to \mathbb{C}$
But as $F$ is holomorphic on $U$ then it is infinitely differentiable on $U$ so

$$f=F^{\prime }\text{ is also holomorphic}$$