Let $U\subseteq \mathbb{C}$ be open, $z_0\in U$, $f:U\subseteq \mathbb{C}$ be holomorphic. Then $z_0$ is a zero of $f$ of order $m$ (of finite order) if 1;;$z_0$ is not of infinite order, and $f^{(m)}(z_0)$ is the first derivative not vanishing at $z_0$