= Solution
The <mapping-out function of a compact H-hull> is the unique <conformal bijection> from $H=\mathbb H\setminus K$ onto the <complex upper half-plane> with <hydrodynamic normalization at infinity>:
$$
\boxed{g_K:H\to\mathbb H,\qquad
g_K(z)=z+\frac{a_K}{z}+O(|z|^{-2})\quad(z\to\infty).}
$$
The nonnegative coefficient $a_K$ is the <half-plane capacity> of $K$. The leading coefficient one and constant term zero remove the affine ambiguity of a half-plane map. In the <compact H-hull> convention, $K$ is bounded and relatively closed in $\mathbb H$, and its complement is a <simply connected domain>; boundary attachment points can be included in its closure.
We will use these standard properties explicitly: $g_K$ and its inverse extend holomorphically by <Schwarz reflection principle> across real intervals outside the hull's real attachment region; their real boundary restrictions are strictly increasing; and their normalized <Laurent series> hold in a full reflected neighbourhood of infinity. In particular, when $K$ lies in the unit disc, these extensions exist across $(-\infty,-1)$ and $(1,\infty)$.
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