= Solution
For unit <plane wave> incidence $\psi_i=e^{ik_0\widehat{\mathbf r}_0\cdot\mathbf r}$, define the scattering <wavevector> difference $\mathbf q=k_0(\widehat{\mathbf r}_0-\widehat{\mathbf r})$ and
$$
A(\widehat{\mathbf r},\widehat{\mathbf r}_0)=\frac1{4\pi}\int_D V(\mathbf r')e^{i\mathbf q\cdot\mathbf r'}\,d\mathbf r'.
$$
The <outgoing Green function for the three-dimensional Helmholtz equation> gives
$$
\boxed{\psi_B(\mathbf r)\simeq e^{ik_0\widehat{\mathbf r}_0\cdot\mathbf r}+\frac{e^{ik_0r}}r A,\qquad \psi_R(\mathbf r)\simeq e^{ik_0\widehat{\mathbf r}_0\cdot\mathbf r}\exp\left[\frac{e^{ik_0(r-\widehat{\mathbf r}_0\cdot\mathbf r)}}r A\right].}
$$
The Born <far-field pattern> is $A$, a <Fourier transform> sample of the potential. Expanding the far-field Rytov exponential in $1/r$ gives the same leading outgoing term, while the logarithmic representation preserves its accumulated phase and amplitude correction.
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