---
tag: 0419794R0T
kind: definition
name: Entire function
msc: [30D20]
defines: 0G4F0XB9X2
uses: [0DG0Z72CQD]
sources:
  - url: https://en.wikipedia.org/wiki/Entire_function
    locator: lead section
---

## Definition

An entire function is a function $f : \mathbb{C} \to \mathbb{C}$ that is [[holomorphic|0DG0Z72CQD]] on all of $\mathbb{C}$.

## Remarks

Polynomials, the [[complex exponential|054VXHMCA3]] and the complex [[sine and cosine|0A7YDKRNR6]] are entire.
Sums, products and compositions of entire functions are entire.
An entire function is equal to one power series $\sum_{n \ge 0} a_n z^n$ that converges for every $z \in \mathbb{C}$.
By [[Liouville's theorem|0EHKVRF4J6]], each bounded entire function is constant.

## Explanations

### L1 human

An entire function is a function that is holomorphic at every complex number, with no exceptions.
For example, $z^3 - 2z + 5$ and $e^z$ are entire, but $1/z$ is not, because it is not defined at $0$.