= Poisson extension of interval boundary data
{c}
{title2=$\psi(x,y)=\frac c\pi\left[\arctan\frac{a-x}y+\arctan\frac{a+x}y\right]$}
For boundary value $c$ on $(-a,a)$ and zero outside, integrating the <Poisson kernel for the upper half-plane> gives the displayed bounded <harmonic function> for $y>0$. It approaches the assigned values at boundary continuity points and the midpoint value $c/2$ at either jump. Changing the value at an isolated endpoint does not alter the integral. This is the unique bounded harmonic extension; pointwise decay in the vertical direction alone, without a spatial growth restriction, does not give uniqueness.
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