= Solution
Let $g=g_A$ and define $u(z)=\operatorname{Im}(z-g(z))$. This is <harmonic> on $\mathbb H\setminus A$, has boundary values $\operatorname{Im}z$ on the hull boundary and zero on the real boundary, and tends to zero at infinity. The representation by <harmonic measure> and <optional sampling theorem> therefore give
$$
u(z)=\mathbb E_z[\operatorname{Im}B_\tau].
$$
At $z=iy$, the hydrodynamic expansion gives
$$
u(iy)=\operatorname{Im}\left(-\frac{a}{iy}+O(y^{-2})\right)
=\frac{a}{y}+O(y^{-2}).
$$
Consequently the <Brownian representation of half-plane capacity> is
$$
\boxed{\operatorname{hcap}(A)=
\lim_{y\to\infty}y\,\mathbb E_{iy}[\operatorname{Im}B_\tau].}
$$
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