---
tag: 1S20A3R659
kind: statement
role: theorem
name: Arakelyan's theorem
msc: ["30E10", 30D20]
uses: [0G5AV9Q4RR, 0P27W9RW0G, 0DG0Z72CQD, 0C38S0KMJK, 0419794R0T, 0CDMXGPSES, 0D939CMME7]
about: [0419794R0T]
sources:
  - url: https://en.wikipedia.org/wiki/Arakelyan%27s_theorem
same_as:
  wikidata: Q25304227
  wikipedia: Arakelyan's theorem
---

## Statement

Let $E \subseteq \C$ be a [[closed set|0G5AV9Q4RR]], and let $\hat\C = \C \cup \set{\infty}$ be the [[Riemann sphere|0CDMXGPSES]].
The following are equivalent.

1. Let $f \colon E \to \C$ be any function that is [[continuous|0P27W9RW0G]] on $E$ and [[holomorphic|0DG0Z72CQD]] on the [[interior|0C38S0KMJK]] of $E$, and let $\varepsilon > 0$ be arbitrary.
Then there is an [[entire function|0419794R0T]] $g$ with $\abs{f(z) - g(z)} < \varepsilon$ for all $z \in E$.
2. The set $\hat\C \setminus E$ is [[connected|0D939CMME7]] and locally connected.

## Remarks

For compact $E$, the set $\hat\C \setminus E$ is an open subset of the sphere, so it is automatically locally connected, and connectedness of $\hat\C \setminus E$ is the same as connectedness of $\C \setminus E$.
So for compact $E$ the theorem reduces to [[Mergelyan's theorem|5KKCSZJC29]], since entire functions can be approximated by polynomials on compact sets.
The new condition for unbounded $E$ is local connectedness of $\hat\C \setminus E$ at the point $\infty$.

Example: let $E = \R$.
Then $\hat\C \setminus \R$ is the union of the two open half-planes and the point $\infty$, which is connected and locally connected.
So every continuous function on $\R$ is a uniform limit on all of $\R$ of entire functions, a result first proved by Carleman.

Arakelyan also proved a version on an arbitrary open set $\Omega$ in place of $\C$, with $E$ closed in $\Omega$.
There the approximants are holomorphic on $\Omega$, and the one-point compactification of $\Omega$ replaces $\hat\C$.

## Explanations

### L1 human

Mergelyan's theorem tells when functions on a compact set can be approximated by polynomials.
Arakelyan's theorem does the same for closed sets that may go off to infinity, with entire functions instead of polynomials.
The set must have no holes, and in addition it must not wind around near infinity in a way that traps parts of its complement.
For the real line, the conclusion is that any continuous function can be uniformly approximated by entire functions on the whole line.

### L2 human

The theorem is the final answer to uniform holomorphic approximation on closed subsets of the plane.
It is used to construct entire functions with prescribed behaviour along unbounded sets, for example functions that tend to $0$ along every ray but are not identically $0$.
Its proof combines Mergelyan-type estimates with a careful exhaustion of $E$ by compact pieces, where local connectedness at $\infty$ controls how the pieces fit together.