Zero of a holomorphic function

Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be holomorphic on $U$. A zero of $f$ is a point $z_0 \in U$ with $f(z_0) = 0$.

Remarks

The zeros of $\sin$ are the numbers $k\pi$ with $k \in \mathbb{Z}$. The complex exponential has no zeros. Each zero has an order, a positive integer or $\infty$. If $f$ is not the zero function on a domain, its zeros are isolated.

Explanations

L1 human

A zero of a function is a point where the function takes the value $0$, as for real functions. For example, $z^2 + 1$ has the zeros $i$ and $-i$.