Analytic capacity

In complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C \ K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K.

It was first introduced by Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions.

Let K ⊂ C be compact. Then its analytic capacity is defined to be

Here, ${\displaystyle {\mathcal {H}}^{\infty }(U)}$ denotes the set of bounded analytic functions U → C, whenever U is an open subset of the complex plane. Further,

(note that usually ${\displaystyle f'(\infty )\neq \lim _{z\to \infty }f'(z)}$)

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