= Conjugate global relations for the modified Helmholtz equation
{title2=$\widetilde{\mathcal G}(\lambda)=\overline{\mathcal G(\overline\lambda)}$}
The two <modified Helmholtz adjoint plane waves> $v_\lambda$ and $\widetilde v_\lambda=\overline{v_{\overline\lambda}}$ give boundary identities $\mathcal G(\lambda)=0$ and $\widetilde{\mathcal G}(\lambda)=0$. For real solution traces, $\widetilde{\mathcal G}(\lambda)=\overline{\mathcal G(\overline\lambda)}$. Reality is needed for this conjugation argument; the parametrization separately yields the reciprocal identity $\widetilde{\mathcal G}(\lambda)=\mathcal G(1/\lambda)$.
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