---
tag: A3R53S95KP
kind: concept
name: Runge domain
msc: ["32E30"]
sources:
  - url: https://en.wikipedia.org/wiki/Runge%27s_theorem
---

## Summary

A Runge domain is an open set in $\C^n$ on which every [[holomorphic function|507C34W0ZT]] can be approximated by polynomials, uniformly on each compact subset.
The name comes from [[Runge's theorem|6DG4CTGXJD]], which describes such open sets in the plane: they are the open sets whose connected components are simply connected.
In several variables, convex domains and balls are Runge domains, and the [[Oka–Weil theorem|7AWZJNQGT2]] is the main tool to recognise them.

## Explanations

### L1 human

Polynomials are the simplest holomorphic functions, so one asks on which regions they are enough to approximate all holomorphic functions.
On a disc, Taylor polynomials do the job.
On an annulus they do not: the function $1/z$ cannot be approximated by polynomials on a circle around the hole.
A Runge domain is a region where polynomials always suffice, at least on each compact part.