Poisson extension of interval boundary data ($\psi (x,y)=\frac{c}{\pi }\left[\arctan \frac{a-x}{y}+\arctan \frac{a+x}{y}\right]$)

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.