---
tag: 0DG0Z72CQD
kind: definition
name: Holomorphic function
msc: [30A99]
defines: 0NAHXR1FRH
uses: [0D1C9Y1EXA]
sources:
  - url: https://en.wikipedia.org/wiki/Holomorphic_function
    locator: Definition
---

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function.
The function $f$ is holomorphic on $U$ if $f$ is [[complex differentiable|0D1C9Y1EXA]] at each point of $U$.
Let $z_0 \in \mathbb{C}$. A function $f$ is holomorphic at $z_0$ if $f$ is holomorphic on some open disc with center $z_0$.

## Remarks

Polynomials in $z$ are holomorphic on $\mathbb{C}$, and $z \mapsto 1/z$ is holomorphic on $\mathbb{C} \setminus \{0\}$.
Sums, products, quotients with a nonzero denominator and compositions of holomorphic functions are holomorphic.
Each holomorphic function is analytic, see [[holomorphic functions are analytic|0826SGV3R0]]. For this reason, many authors use the words holomorphic and analytic for the same functions.
A function that is holomorphic on all of $\mathbb{C}$ is an [[entire function|0419794R0T]].

## Explanations

### L1 human

A holomorphic function is a function of a complex variable that has a complex derivative at each point of an open set.
This one condition has strong consequences: the function then has derivatives of all orders and is equal to its Taylor series near each point.