---
tag: X252NY7CS1
kind: definition
name: Analytic capacity
msc: [30C85, "30H05"]
defines: QGQPHRANWZ
uses: [0DG0Z72CQD, 0QPHW703PV, 0BTMQ04556, 0BSQDZWF34, 0MB48XDBY3]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_capacity
---

## Definition

Let $K \subset \C$ be a [[compact|0QPHW703PV]] set.
Let $\mathcal{A}(K)$ be the set of [[holomorphic|0DG0Z72CQD]] functions $f \colon \C \setminus K \to \C$ with $\abs{f(z)} \le 1$ for all $z$ and $f(z) \to 0$ as $z \to \infty$.
For $f \in \mathcal{A}(K)$ put $f'(\infty) = \lim_{z \to \infty} z f(z)$; this limit exists.
The *analytic capacity* of $K$ is
$$\gamma(K) = \sup \set{\abs{f'(\infty)} : f \in \mathcal{A}(K)}.$$

## Remarks

The limit $f'(\infty)$ exists because $w \mapsto f(1/w)$ has a removable singularity at $w = 0$ with value $0$.
Example: for the closed disc $K = \set{\abs{z} \le r}$, the function $r/z$ lies in $\mathcal{A}(K)$ with $f'(\infty) = r$.
For any $f \in \mathcal{A}(K)$, the function $g(w) = f(r/w)$ extends to the unit disc with $g(0) = 0$ and $g'(0) = f'(\infty)/r$.
The [[Schwarz lemma|0MB48XDBY3]] gives $\abs{g'(0)} \le 1$, so $\gamma(K) = r$.
Non-example of positive capacity: for a single point $K = \set{a}$, each $f \in \mathcal{A}(K)$ extends to a bounded entire function by [[Riemann's theorem on removable singularities|0BSQDZWF34]].
By Liouville's theorem it is constant, hence $0$, so $\gamma(\set{a}) = 0$.
A segment of length $\ell$ has analytic capacity $\ell/4$.
Ahlfors proved that $\gamma(K) = 0$ if and only if every bounded holomorphic function on $U \setminus K$ extends holomorphically to $U$, for each open $U \supseteq K$.