= Harmonic function in a rectangle with one constant boundary value
{title2=$u(x,y)=\frac{4F}{\pi}\sum_{n\geq1,\ n\text{ odd}}\frac{\sin(n\pi x/a)\sinh(n\pi y/a)}{n\sinh(n\pi b/a)}$}
On $0<x<a$, $0<y<b$, the bounded <harmonic function> with value $F$ on the open upper edge and zero on the other three open edges has the sine-hyperbolic series in the title. <Separation of variables> gives the modes, and the <Fourier sine series> of a constant selects only odd $n$. The series converges exponentially on compact subsets of the interior, justifying its derivatives in <Laplace's equation>.
For $F\ne0$, the upper corners have incompatible one-sided boundary limits, so the solution cannot extend continuously to them. Assigning isolated corner values does not alter the bounded interior <Dirichlet problem>. For numerical evaluation, the stable mode ratio is
$$
\frac{\sinh(ky)}{\sinh(kb)}=e^{-k(b-y)}\frac{1-e^{-2ky}}{1-e^{-2kb}},\qquad k>0.
$$
This avoids overflow from separately evaluating large hyperbolic sines.
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