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Expand the constant right-edge value in a Fourier sine series. Its coefficients areThe superposition principle combines the normalized modes, givingThis is a harmonic function in a rectangle with one constant boundary value, with the coordinate roles interchanged. The series and its derivatives converge uniformly on compact subsets of $0<x<\pi$, so it satisfies Laplace's equation there. It vanishes on the other three edges and approaches one at every interior point of the right edge. At the two right corners the one-sided prescribed values disagree; no continuous extension can satisfy both. The stated open-edge condition avoids that incompatibility, and isolated corner assignments do not change the bounded interior Dirichlet problem.

The zero values at $x=0,a$ select $X_n=\sin (n\pi x/a)$, and the zero value at $y=0$ selects $Y_n=\sinh (n\pi y/a)$. Expanding the constant upper-edge value as a Fourier sine series gives coefficients $b_n=2F[1-(-1{)}^n]/(n\pi )$. The resulting harmonic function in a rectangle with one constant boundary value isEach term satisfies Laplace's equation and the three zero boundary values. At $y=a$, the series equals $F$ for $0<x<a$. On compact subsets of the interior it and its derivatives converge exponentially, which justifies the termwise equation.

For $F\ne 0$, the top-corner boundary prescriptions conflict: a corner cannot be both zero on a vertical edge and $F$ on the upper edge. The bounded interior solution has the specified limits along the open sides; it has no single continuous extension at those two corners. Values assigned only at the isolated corners do not change the interior solution.