= Poisson kernel for the upper half-space
{c}
{title2=$P_z(u,v)=z/[2\pi(u^2+v^2+z^2)^{3/2}]$}
In three dimensions the <Poisson kernel for the upper half-space> is
$$
P_z(u,v)=\frac{z}{2\pi(u^2+v^2+z^2)^{3/2}},\qquad z>0.
$$
It gives the <harmonic function> $\phi(x,y,z)=\int_{\mathbb R^2}P_z(x-\xi,y-\eta)f(\xi,\eta)\,d\xi\,d\eta$ for compatible boundary data. The <method of images> constructs the zero-boundary <Dirichlet Green function> by subtracting the reflected-source <Green function of the Laplacian> from the source kernel. On the boundary plane the outward <unit normal> is $-e_z$, and $-\partial_nG=P_z$, giving the integral formula through <Green's second identity>.
The kernel is positive and has integral one: in <polar coordinates> this is $z\int_0^\infty r(r^2+z^2)^{-3/2}dr=1$. Its mass outside any fixed disk tends to zero as $z\downarrow0$. It is therefore an <approximate identity>, recovering bounded boundary data at continuity points. At a jump across a smooth curve, the vertical limit gives the average of the two adjacent values; assigning a different value only on that curve does not change the harmonic extension.
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