= Semistrip Dirichlet-to-Neumann sine transforms
{title2=$S_0=\kappa(-G_0^s\cosh\kappa l+G_l^s+I_-)/\sinh\kappa l$}
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 is
$$
S_0=\frac{\kappa[-\cosh(\kappa l)G_0^s+G_l^s+\int_0^l\sinh(\kappa(l-y))h(y)\,dy]}{\sinh(\kappa l)},\qquad
S_l=\frac{\kappa[-G_0^s+\cosh(\kappa l)G_l^s-\int_0^l\sinh(\kappa y)h(y)\,dy]}{\sinh(\kappa l)}.
$$
The 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
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