---
tag: QGQPHRANWZ
kind: concept
name: Analytic capacity
msc: [30C85, "30H05"]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_capacity
same_as:
  wikidata: Q4751124
  wikipedia: Analytic capacity
---

## Summary

The analytic capacity of a compact set $K \subset \C$ measures how large a bounded holomorphic function on $\C \setminus K$ can be near infinity, see the [[definition|X252NY7CS1]].
It is zero exactly when $K$ is removable for bounded holomorphic functions.
The search for a geometric description of the sets of analytic capacity zero is the Painlevé problem.

## Explanations

### L1 human

Remove a compact set $K$ from the plane and look at holomorphic functions on what is left that stay bounded by $1$ and vanish at infinity.
If $K$ is a single point, the only such function is $0$, so the point is too small to matter.
If $K$ is a disc of radius $r$, the function $r/z$ works, and the capacity is $r$.
Analytic capacity is a number that tells how big $K$ is from the point of view of such functions.