---
tag: FX92PP3HZR
kind: concept
name: Biholomorphism
msc: [32H02]
specializes: [3MQTWKBBBT]
sources:
  - url: https://en.wikipedia.org/wiki/Biholomorphism
same_as:
  wikidata: Q377166
  wikipedia: Biholomorphism
---

## Summary

A biholomorphism is a bijective [[holomorphic map|3MQTWKBBBT]] between open subsets of $\C^n$ whose inverse is also holomorphic.
Two domains related by a biholomorphism are biholomorphically equivalent: they are the same for complex analysis.
In one variable every simply connected proper domain of $\C$ is biholomorphic to the disc, by the [[Riemann mapping theorem|02WHMEG5ZR]].
In several variables this fails: for $n \ge 2$ the [[unit ball|PDQ9KW9FAP]] and the unit [[polydisc|DJBWP7D3M8]] are [[not biholomorphic|1NF8T07H9X]].

## Explanations

### L1 human

A biholomorphism is a change of complex coordinates that can be undone.
It matches the points of two domains one to one, and it carries holomorphic functions on one domain to holomorphic functions on the other.
For example, the map $(z_1, z_2) \mapsto (2 z_1, \ z_2 + z_1^2)$ is a biholomorphism of $\C^2$ onto itself, with inverse $(w_1, w_2) \mapsto (w_1/2, \ w_2 - w_1^2/4)$.
Biholomorphic domains cannot be told apart by complex analysis.