what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere?

Question

what would be an example of a function such that $\int_{|z|=r} f(z)=0$ for all $r>0$, but not analytic everywhere.

I cannot think of one..

Answer 1

Let $\phi\colon[0,\infty)\to\mathbb{C}$ and $f(z)=\phi(|z|)$. Then for all $r>0$ $$ \int_{|z|=r}f(z)\,dz=\phi(r)\int_{|z|=r}\,dz=0. $$

A different example is a function $f$ analytic in $\{z\ne0\}$ and a pole at $z=0$ with residue $0$.