= Solution
The canonical <Dirichlet Green function> on the <unit disc> with its pole at zero is $-\log|w|$. Conformal invariance of the planar Green kernel therefore gives
$$
G_D(z,\phi(w))=-\log|w|.
$$
Comparing with $G_D(z,y)=-\log|z-y|-G_z(y)$ yields
$$
G_z(\phi(w))=-\log|\phi(w)-\phi(0)|+\log|w|=-\psi(w).
$$
This also explains the boundary data: when $|w|=1$, the correction is $-\log|\phi(w)-z|$. The composition with $\phi^{-1}$ is harmonic by <conformal invariance of harmonicity>. Its apparent singularity at $z$ is removable because the quotient defining $\psi$ extends to $\phi'(0)$.
Evaluating there proves the <regular part of the planar Dirichlet Green function> formula
$$
\boxed{G_z(y)=-\psi(\phi^{-1}(y)),\qquad G_z(z)=-\log|\phi'(0)|=-\log\operatorname{crad}_D(z).}
$$
On an unbounded domain, boundary values alone need not specify a unique <harmonic function>. Here $G_z$ is the correction belonging to the canonical Dirichlet Green kernel, which fixes that ambiguity.
Back to article page
