Given $U\subseteq \mathbb{C}$ open, $z_0\in U$, and $f:U\to \mathbb{C}$ holomorphic. Then $z_0$ is a zero of $f$ of infinite order if 1;;$f^{(n)}(z_0)=0$ for all $n\in {\mathbb{Z}}_{\ge \mathbb{0}}$.