= Herglotz pairing with an obstacle far field
{c}
Pairing a <far-field pattern> with the conjugate density of a <Herglotz wave function> converts the angular integral into boundary data:
$$
\int_{S^2}f_\infty(\widehat{\mathbf r})\overline{g(\widehat{\mathbf r})}\,dS=\frac1{4\pi}\int_{\partial D}[\psi_s\partial_n\overline{v_g}-\overline{v_g}\partial_n\psi_s]\,dS.
$$
This follows by substituting the <surface representation of an obstacle far-field pattern> and interchanging the integrals.
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