Regular part of the planar Dirichlet Green function ($G_D(z,y)=-\log |z-y|-G_z(y)$)

For the canonical Dirichlet Green function with logarithmic singularity, its harmonic correction satisfies $G_z(z)=-\log {\mathrm{crad}}_D(z)$. If $\varphi$ maps the unit disc to $D$ with $\varphi (0)=z$, thenThe quotient extends at zero to ${\varphi }^{\prime }(0)$. On unbounded domains the canonical Green function, rather than boundary values alone, fixes the harmonic correction.