= Neumann point source on a half-space boundary
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{title2=$\Phi(x,y,z)=1/[2\pi\sqrt{x^2+y^2+z^2}]$}
In the upper half-space $z>0$, the displayed decaying <harmonic function> has outward normal derivative $\partial_n\Phi=-\partial_z\Phi=z/[2\pi(x^2+y^2+z^2)^{3/2}]$. This is the <Poisson kernel for the upper half-space>, whose integral on a horizontal plane is one and whose limit as $z\downarrow0$ is the two-dimensional <Dirac delta>. Thus $\partial_n\Phi=\delta(x)\delta(y)$ in the distributional boundary sense. The compensating flux through a large hemisphere is minus one. This illustrates why a nonzero total boundary source is possible on an unbounded <domain> even when the interior source is zero.
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