= Solution
For $r>0$, the <conformal map> $h(z)=r g_A(z/r)$ maps $\mathbb H\setminus rA$ onto $\mathbb H$ and has <Laurent series> $h(z)=z+r^2a/z+O(z^{-2})$. The uniqueness in <hydrodynamic normalization at infinity> gives
$$
\boxed{\operatorname{hcap}(rA)=r^2\operatorname{hcap}(A).}
$$
At $r=0$ the scaled <closure> is contained in $\mathbb R$, so the <interior> <compact H-hull> is empty and both sides are zero. This is the scaling part of <scaling and translation of half-plane capacity>.
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