A condition for an analytic function on the punctured plane to have an antiderivative

Question

Suppose $f$ is an analytic function on the punctured plane $\Bbb C-\{0\}$. It is clear that if $f$ has an antiderivative, i.e., if $f=F'$ for some analytic function $F$, then $\int_{|z|=r} f(z)dz=0$ for all $r>0$. I want to show that the converse is also true. If I could show that for any two points $z_0,z_1 \in\Bbb C-\{0\}$ that the integral of $f$ on a path from $z_0$ to $z_1$ is independent of the choice of a path, then I would proceed similarly with Morera's theorem, but I'm not sure about this. Is this true?

Answer 1

It's the case that if $\int f(z)\,dz$ vanishes along any circle centred at $0$ then $f$ has an antiderivative. For a quick proof consider the Laurent series $f(z)=\sum_{n=-\infty}^\infty c_n z^n$. The vanishing condition ensures $c_{-1}=0$ and then the Laurent series can be integrated termwise.