---
tag: 0CEGEASY49
kind: definition
name: Zero of a holomorphic function
msc: [30C15]
defines: 00GC8ZQP4Q
uses: [0DG0Z72CQD]
sources:
  - url: https://en.wikipedia.org/wiki/Zeros_and_poles
    locator: Definitions
---

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] 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|054VXHMCA3]] has no zeros.
Each zero has an [[order|064N002TX4]], a positive integer or $\infty$.
If $f$ is not the zero function on a [[domain|097KNJFJ7V]], its zeros are [[isolated|0K83KEJ43P]].

## 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 $-i$.