Analytic capacity

In complex analysis, the analytic capacity of a compact subset K of the complex plane is the number

$\gamma (K)=\sup\{|f'(\infty )|;\ f\in {\mathcal {H}}(\mathbb {C} \setminus E),\ \|f\|_{\infty }\leq 1,\ f(\infty )=0\}$

Here, <math> \mathcal{H} (U) <math> denotes the set of bounded analytic functions <math> U\to\mathbb{C} <math>, where is an open subset of \mathbb{C}.

where the supremum is taken over the set of functions f which are holomorphic on the complement of K and are bounded in absolute value by 1. This supremum is called the Ahlfors function, since it was first introduced by Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions.

The compact set K is called removable if whenever Ω is an open set containing K, every function which is bounded and holomorphic on the set Ω\K has an analytic extension to all of Ω. Then the set K is removable if and only if the analytic capacity γ(K)=0.