---
tag: 4487CSC0MB
kind: concept
name: Domain of holomorphy
msc: [32D05, 32T05]
sources:
  - url: https://en.wikipedia.org/wiki/Domain_of_holomorphy
same_as:
  wikidata: Q1625065
  wikipedia: Domain of holomorphy
---

## Summary

A domain of holomorphy is an open set in $\C^n$ that is the natural domain of its [[holomorphic functions|507C34W0ZT]].
No part of its boundary can be crossed by all of these functions at once.
In one variable every open set has this property, but for $n \ge 2$ many open sets do not, by the Hartogs phenomenon.
Domains of holomorphy are characterised by holomorphic convexity and, by the solution of the Levi problem, by pseudoconvexity.
They are the domains in $\C^n$ on which the global theory of holomorphic functions works best.

## Explanations

### L1 human

Some regions in $\C^n$ have hidden extensions: every holomorphic function on them automatically extends further.
A ball with a small closed ball removed from the middle is such a region in $\C^2$, since every function fills in the hole.
A domain of holomorphy has no such hidden extension: for each boundary point there is a function that really breaks down there.
The open unit ball is a domain of holomorphy; near a boundary point $p$, the function $1/(\ip{z}{p} - 1)$ blows up, where $\ip{z}{p} = \sum_j z_j \bar p_j$.