Semistrip Dirichlet-to-Neumann sine transforms ($S_0=\kappa (-G_0^s\cosh \kappa l+G_l^s+I_{-})/\sinh \kappa l$)

In the semistrip Laplace spectral global relation, reality makes $W(\pm \kappa )$ real for $\kappa >0$. Its imaginary parts at those two arguments remove the unknown vertical-side normal derivative. With $G_j^s(\kappa )={\int }_0^{\infty }\sin (\kappa x)g_j(x)dx$ and $S_j(\kappa )={\int }_0^{\infty }\sin (\kappa x)q_y(x,j)dx$, the result isThe lower outward normal derivative is $-q_y$, so its Dirichlet-to-Neumann map coefficient is $-S_0$, whereas the upper coefficient is $S_l$.

// Target: analysis.bigb