---
tag: 6G0079XRKK
kind: definition
name: Domain of holomorphy
msc: [32D05, 32T05]
defines: 4487CSC0MB
uses: [507C34W0ZT, 02BH3MCBQB, 0SPE8WNWFV]
sources:
  - url: https://en.wikipedia.org/wiki/Domain_of_holomorphy
    locator: Definition
---

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
Then $D$ is a *domain of holomorphy* if there are no nonempty open sets $U_1, U_2 \subseteq \C^n$ such that:

1. $U_2$ is [[connected|0SPE8WNWFV]] and $U_2 \not\subseteq D$;
2. $U_1 \subseteq D \cap U_2$;
3. for every [[holomorphic function|507C34W0ZT]] $f$ on $D$ there is a holomorphic function $g$ on $U_2$ with $f = g$ on $U_1$.

## Remarks

Example: every convex open set $D \subsetneq \C^n$ is a domain of holomorphy.
For a boundary point $p$ there is a complex linear function $\ell$ with $\operatorname{Re} \ell(z) < \operatorname{Re} \ell(p)$ for all $z \in D$, and $f_p(z) = 1/(\ell(z) - \ell(p))$ is holomorphic on $D$ and unbounded near $p$.
Suppose $U_1, U_2$ satisfied the three clauses, and let $W$ be a component of $D \cap U_2$ that meets $U_1$.
Since $U_2$ is connected and not contained in $D$, it contains a boundary point $p$ of $W$, and $p$ lies on the boundary of $D$.
The extension $g$ of $f_p$ equals $f_p$ on $W$ by the identity theorem, and $g$ is continuous at $p$.
So $f_p$ is bounded near $p$ on $W$, which is a contradiction.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$.
Take $U_2 = \set{\abs{z} < 1}$ and $U_1 = D$.
Clauses 1 and 2 hold, and clause 3 holds by [[Hartogs's extension theorem|FSGES670AZ]] with $K = \set{0}$.

Every open subset of $\C$ is a domain of holomorphy.