---
tag: X71JE6A3ZJ
kind: concept
name: Envelope of holomorphy
msc: [32D10, 32D26]
sources:
  - url: https://en.wikipedia.org/wiki/Domain_of_holomorphy
---

## Summary

The envelope of holomorphy of a domain $D \subseteq \C^n$ is the largest domain to which every [[holomorphic function|507C34W0ZT]] on $D$ continues.
This domain may be spread over $\C^n$ in several sheets.
It exists for every domain and is unique up to isomorphism.
In general it is a [[Riemann domain over $\C^n$|A15BF6QHDR]] and not a subset of $\C^n$.
A domain is a [[domain of holomorphy|4487CSC0MB]] exactly when it is its own envelope of holomorphy.

## Explanations

### L1 human

Given a region, ask how far all of its holomorphic functions can be pushed at once.
The answer is the envelope of holomorphy: the region together with all its hidden extensions.
For the [[Hartogs figure|GHJC53RFBT]], a polydisc with a piece cut out, the envelope is the full polydisc.
For a ball with a smaller closed ball removed, the envelope is the whole ball.
In one variable nothing new appears: every domain is its own envelope.