Editing Analytic capacity (section)

==Definition==
Let ''K'' ⊂ '''C''' be [[compact space|compact]]. Then its analytic capacity is defined to be

:<math>\gamma(K) = \sup \{|f'(\infty)|;\ f\in\mathcal{H}^\infty(\mathbf{C}\setminus K),\ \|f\|_\infty\leq 1,\ f(\infty)=0\}</math>

Here, <math>\mathcal{H}^\infty (U) </math> denotes the set of [[bounded function|bounded]] analytic [[Function (mathematics)|functions]] ''U'' → '''C''', whenever ''U'' is an [[open set|open]] subset of the [[complex plane]]. Further,

:<math> f'(\infty):= \lim_{z\to\infty}z\left(f(z)-f(\infty)\right) </math>
:<math> f(\infty):= \lim_{z\to\infty}f(z) </math>

Note that <math>f'(\infty) = g'(0)</math>, where <math>g(z) = f(1/z)</math>. However, usually <math> f'(\infty)\neq \lim_{z\to\infty} f'(z)</math>.

Equivalently, the analytic capacity may be defined as<ref>{{eom| title = Capacity| last = Solomentsev| first = E. D.}}</ref>
:<math>\gamma(K)=\sup \left|\frac1{2\pi} \int_C f(z)dz\right|</math>
where ''C'' is a contour enclosing ''K'' and the supremum is taken  over ''f'' satisfying the same conditions as above: ''f'' is bounded analytic outside ''K'', the bound is one, and <math>f(\infty)=0.</math>

If ''A'' ⊂ '''C''' is an arbitrary set, then we define

:<math>\gamma(A) = \sup \{ \gamma(K) : K \subset A, \, K \text{ compact} \}.</math>