= Far-field approximation for an outgoing source
{title2=$\psi(R\boldsymbol\theta)=R^{-1}e^{ikR}f_\infty(\boldsymbol\theta)+O(R^{-2})$}
For a bounded source region and fixed positive <wavenumber>, expand $|R\boldsymbol\theta-\mathbf r'|=R-\boldsymbol\theta\cdot\mathbf r'+O(R^{-1})$. In the <outgoing Green function for the three-dimensional Helmholtz equation>, this gives $G=e^{ikR}e^{-ik\boldsymbol\theta\cdot\mathbf r'}/(4\pi R)+O(R^{-2})$. Volume integration gives the outgoing field's <far-field pattern> with an $O(R^{-2})$ remainder. The estimate is an asymptotic statement for fixed source size and frequency.
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