= Solution
Let $\delta\psi_B=\psi_1^{(B)}-\psi_i$ be the first-order scattered correction in the <Born approximation for scalar wave scattering>, using the paper's <Green function> convention. Comparing its outgoing integral with the preceding logarithmic correction gives
$$
\chi_1=\frac{\delta\psi_B}{\psi_i}.
$$
Therefore the exact algebraic relation between the two first-order approximations is
$$
\boxed{\psi_1^{(R)}=\psi_i\exp\left(\frac{\psi_1^{(B)}-\psi_i}{\psi_i}\right)=\psi_i\exp\left(\frac{\psi_1^{(B)}}{\psi_i}-1\right).}
$$
If $V=O(\varepsilon)$, then $\chi_1=O(\varepsilon)$ on a controlled fixed region, and
$$
\psi_1^{(R)}=\psi_i(1+\chi_1)+O(\varepsilon^2)=\psi_1^{(B)}+O(\varepsilon^2).
$$
Thus \b[Born and Rytov agree to first order in the <scattering potential>], but differ as finite approximations. The <Rytov approximation> exponentiates the first logarithmic correction; the <Born approximation> adds the first field correction. The exponential's higher powers are not a calculation of all higher multiple-scattering terms in the <Born series>. This distinction explains why a smooth, appreciable <wave phase> accumulation can be represented more naturally by the <Rytov approximation> even when the corresponding linear field expansion is inaccurate.
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