# Notes - Complex Analysis MT23, Morera's theorem

### Flashcards

Can you state Morera’s theorem, which gives you a partial converse to Cauchy’s theorem and conditions on which a function $f : U \to \mathbb C$ will be holomorphic?

Suppose

- $f : U \to \mathbb C$ is continuous
- $U \subseteq \mathbb C$ open subset
- $\int _ \gamma f(z) \text d z = 0$ for all closed paths $\gamma$

Then

- $f$ is holomorphic

(really this is almost a restatement of the theorem that if $U$ is a domain, then $f$ has a primitive on that domain)

Quickly prove Morera’s theorem, i.e. suppose

- $f : U \to \mathbb C$ is continuous
- $U \subseteq \mathbb C$ open subset
- $\int _ \gamma f(z) \text d z = 0$ for all closed paths $\gamma$

then

- $f$ is holomorphic

Do this by appealing to a result about when $f$ is primitive, and a result about analytic functions.

We know from the stated conditions that $f$ has a primitive ($\star$), say $F$. Then $F’ = f$, so $F$ is holomorphic, but holomorphic functions are infinitely differentiable, hence $f$ is holomorphic.

($\star$): The theorem that says this actually required $U$ is a domain, since the construction used relies on a path between $z$ and an arbitrary fixed point of $U$. I think $U$ can be partioned into a set of domains (which are the connected components of $U$) and then you can apply the above argument to each domain seperately. Or maybe there is a proof of the theorem that $f$ has a primitive that doesn’t rely on $U$ being connected.