Order of a zero

Definition

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be holomorphic on $U$, and let $z_0 \in U$. Assume that $f$ is not identically zero on any open disc with center $z_0$. The order of $f$ at $z_0$ is the smallest integer $m \ge 0$ with $f^{(m)}(z_0) \neq 0$, where $f^{(m)}$ is the $m$-th complex derivative of $f$. If $f(z_0) = 0$, this order $m \ge 1$ is the order of the zero $z_0$ of $f$. A zero of order $1$ is a simple zero. If $f$ is identically zero on an open disc with center $z_0$, the order of $f$ at $z_0$ is $\infty$.