Half-plane capacity of a vertical slit ($\mathrm{hcap}((x,x+ih])=h^2/2$)

For a vertical slit of height $h>0$ attached at $x\in \mathbb{R}$, the mapping-out function of a compact H-hull is $g(z)=x+\sqrt{(z-x{)}^2+h^2}$, with the branch asymptotic to $z-x$ at infinity. Its half-plane capacity is $h^2/2$.