= Zero-boundary harmonic function with pointwise vertical decay
{title2=$v(x,y)=e^{x^2-y^2}\sin(2xy)$}
The imaginary part of the entire function $e^{z^2}$ is a <harmonic function>. On the upper half-plane it vanishes at $y=0$ and tends to zero as $y\to\infty$ for every fixed $x$, but is nonzero and grows rapidly in the horizontal direction. Adding it to a <Poisson integral> preserves those boundary and vertical-decay conditions. A boundedness or suitable transform-growth condition is therefore needed to make the half-plane <Dirichlet problem> unique.
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