= Square modified Helmholtz Dirichlet-to-Neumann coefficients
{title2=$c_{jn}=\int_{-1}^1q_j(s)\phi_n(s)ds$}
Let $R_{jn}=\int_{\partial\Omega}f\partial_nv_{jn}ds$ for a square sine adjoint test, and put $\rho_{jn}=e^{-\omega_n}R_{jn}$, $\delta_n=e^{-2\omega_n}$. The opposite-side normal-trace coefficients obey
$$
c_{Bn}=\frac{\rho_{Bn}-\delta_n\rho_{Tn}}{1-\delta_n^2},\qquad c_{Tn}=\frac{\rho_{Tn}-\delta_n\rho_{Bn}}{1-\delta_n^2},
$$
with the same formulas for $L,R$. When the known <Dirichlet boundary condition> is integrated exactly, these are exact coefficients, since <orthogonality> eliminates all other unknown modes.
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