Solution
= Solution
A <compact H-hull> is a bounded relatively closed set $A\subset\mathbb H$ such that $\mathbb H\setminus A$ is a <simply connected domain>. Its <mapping-out function of a compact H-hull> is the unique <conformal map> $g_A:\mathbb H\setminus A\to\mathbb H$ with <hydrodynamic normalization at infinity>
$$
g_A(z)=z+\frac a z+O(|z|^{-2}).
$$
The nonnegative coefficient $a$ is the <half-plane capacity> $\operatorname{hcap}(A)$.