= Half-plane capacity of a half-disc
{title2=$\operatorname{hcap}(\{z\in\mathbb H:|z-x|\leq R\})=R^2$}
For $x\in\mathbb R$ and $R>0$, the filled half-disc $K=\{z\in\mathbb H:|z-x|\leq R\}$ is a <compact H-hull>. Its <mapping-out function of a compact H-hull> is $g_K(z)=z+R^2/(z-x)$. On the semicircle the image is real, and the inverse branch asymptotic to the identity maps the <complex upper half-plane> onto the exterior half-disc. Its <Laurent series> has coefficient $R^2/z$, so $\boxed{\operatorname{hcap}(K)=R^2}$. Using an open half-disc instead requires taking its relative <closure> first.
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