 Quadrant modified Helmholtz Dirichlet Poisson formula

Here $B(p)=\sqrt{p^2+4\lambda }$ and $S_j$ are the Fourier sine transforms of the boundary values. The modified Helmholtz closed spectral one-form and its reflected global relations remove the unknown normal traces. Equivalently, four reflected fundamental solutions give a zero-boundary Green function in the quadrant. Its inward derivatives give the displayed formula. The two terms recover the bottom and left traces respectively; matching corner values ensure continuity at the corner.