= Regular part of the planar Dirichlet Green function
{title2=$G_D(z,y)=-\log|z-y|-G_z(y)$}
For the canonical <Dirichlet Green function> with logarithmic singularity, its harmonic correction satisfies $G_z(z)=-\log\operatorname{crad}_D(z)$. If $\phi$ maps the unit disc to $D$ with $\phi(0)=z$, then
$$
G_z(\phi(w))=-\log\left|\frac{\phi(w)-\phi(0)}w\right|.
$$
The quotient extends at zero to $\phi'(0)$. On unbounded domains the canonical Green function, rather than boundary values alone, fixes the harmonic correction.
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