Poisson kernel of a three-dimensional quarter-space ($P=\frac{z}{2\pi }(r_{-}^{-3}-r_{+}^{-3})$)

For the face $z=0$ of $x>0,z>0$, set $r_{\mp }^2=(x\mp \xi {)}^2+(y-\eta {)}^2+z^2$, with source boundary coordinates $\xi >0$, $\eta \in \mathbb{R}$. The Poisson kernel for data on that face and zero data on $x=0$ is the displayed positive difference. It follows by differentiating the Dirichlet Green function for a three-dimensional quarter-space in the inward source direction. Integrating it against bounded continuous face data gives the bounded harmonic solution away from compatibility issues at the edge. Without a boundedness or suitable infinity condition, uniqueness fails because the harmonic function $xz$ vanishes on both faces.

Its integral over this one face is $2\arctan (x/z)/\pi$, the harmonic measure of that face. To calculate it, integration in $\eta$ first gives $z\{[(x-\xi {)}^2+z^2{]}^{-1}-[(x+\xi {)}^2+z^2{]}^{-1}\}/\pi$; the remaining half-line integral is an elementary arctangent difference. This mass lies between zero and one: the other face carries the remaining boundary influence.