Conformal transformation rule for the Poisson kernel

For smooth planar domains and a conformal map $\Phi :D\to \Omega$ extending as a $C^1$ boundary diffeomorphism with nonzero derivative, the Poisson kernel as a density against boundary arclength satisfiesThe derivative is evaluated at the boundary point, not at the interior observation point.

To derive it, use conformal invariance of harmonic measure and change boundary variables: $d{s}_{\Phi (\zeta )}=|{\Phi }^{\prime }(\zeta )|d{s}_{\zeta }$. The two Poisson integrals must give the same harmonic extension for every continuous boundary datum, identifying the densities. Kernels defined against an angular measure instead require the corresponding measure conversion.

For $\Phi (z)=Rz$ from the unit disk to the radius-$R$ disk, the rule gives $P_{R\mathbb{D}}(Rz,R\zeta )=P_{\mathbb{D}}(z,\zeta )/R$. At the center this is $1/(2\pi R)$, whose integral around circumference $2\pi R$ is one. This normalization check detects a misplaced derivative factor.