= Solution
A <compact H-hull> is a bounded subset $A\subset\mathbb H$, closed relative to the <complex upper half-plane>, such that $\mathbb H\setminus A$ is a <simply connected domain>. The word compact refers to its bounded closure in $\overline{\mathbb H}$; a nonempty hull need not be a compact subset of the open half-plane.
The <Riemann mapping theorem> and normalization at infinity give a unique <mapping-out function of a compact H-hull> $g_A:\mathbb H\setminus A\to\mathbb H$ with $g_A(z)-z\to0$ as $z\to\infty$. This <hydrodynamic normalization at infinity> yields a <Laurent series>
$$
g_A(z)=z+\frac{a_A}{z}+O(|z|^{-2}),\qquad a_A\geq0.
$$
The coefficient is the <half-plane capacity>:
$$
\boxed{\operatorname{hcap}(A)=a_A=\lim_{y\to\infty}iy\bigl(g_A(iy)-iy\bigr).}
$$
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