Analytic capacity

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 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. 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 $U \supseteq K$.