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Choose the scattering-potential convention $V(\mathbf{r})=k^2[n(\mathbf{r}{)}^2-1]$. Let $G_k(\mathbf{r},{\mathbf{r}}^{\prime })=e^{ik|\mathbf{r}-{\mathbf{r}}^{\prime }|}/(4\pi |\mathbf{r}-{\mathbf{r}}^{\prime }|)$, so $(\Delta +k^2)G_k=-\delta$. Since $(\Delta +k^2)\psi =-V\psi$, the outgoing Green representation is the Lippmann-Schwinger equationIterating it produces the Born seriesFor example its first two terms areThe first term describes one scattering event driven by the incident field. The second propagates its first scattered wave to another interaction before reaching the observer. Higher terms contain more successive interactions, including repeated visits to the same region; they account for multiple scattering rather than additional incident waves.

A sufficient convergence condition on an appropriate field norm is $\Vert T\Vert =\rho <1$. Then the first-Born remainder is bounded by ${\rho }^2\Vert {\psi }_i\Vert /(1-\rho )$, while $\Vert T{\psi }_i\Vert \le \rho \Vert {\psi }_i\Vert$. A simple concrete sufficient condition on the bounded scatterer, using the supremum norm, isThis makes the field inside the inhomogeneity close to the incident field. Physically weak index contrast with small accumulated phase shift, such as $k\ell |n-1|\ll 1$ for a smooth weak scatterer of thickness $\ell$, is a common first-Born regime, provided resonant enhancement and multiple reflections are negligible. Weak local contrast alone is not sufficient for an arbitrarily thick scatterer. If the opposite sign convention for $V$ is used, the integral operator changes sign and the same consistent iteration applies.

Use the scattering potential convention $V=k_0^2(n^2-1)$. With time dependence $e^{-i\omega t}$ the outgoing Green function is $G_0(r,r^{\prime })=e^{i{k}_0|r-r^{\prime }|}/(4\pi |r-r^{\prime }|)$ and satisfies $(\Delta +k_0^2)G_0=-\delta$. The total-field Lippmann-Schwinger equation isReplacing the field in the integral by the incident field gives the first Born approximation for scalar wave scattering:For the first Rytov approximation, write $\psi ={\psi }_0{e}^{\chi }$ on a region where the incident field is nonzero and a continuous complex logarithm can be chosen. The logarithmic wave perturbation equation isThe last dot product is bilinear, not a squared modulus. Dropping it leaves a linear equation for ${\chi }_1$. Multiplication by ${\psi }_0$ shows that ${\psi }_0{\chi }_1$ satisfies the same outgoing inhomogeneous Helmholtz equation as ${\psi }_1$. HenceIntroduce a small multiplier $\lambda$ on $V$. Then ${\psi }_1=O(\lambda )$ andThe two total-field approximations agree to first order. Exponentiating the first logarithmic wave perturbation does not supply the generally missing second-order Born series contribution.

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 givesTherefore the exact algebraic relation between the two first-order approximations isIf $V=O(\epsilon )$, then ${\chi }_1=O(\epsilon )$ on a controlled fixed region, andThus 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.

Take the same $e^{-i\omega t}$ convention and an outgoing field. Define the scattering potential with the positive-contrast sign,The outgoing Green function for the three-dimensional Helmholtz equation isThus the Lippmann-Schwinger equation has a plus sign:Replacing the total field inside the integral by the incident field gives the Born approximation for scalar wave scatteringMore generally, iterating the integral equation gives the Born series ${\psi }_s={\sum }_{j\ge 1}{T}^j{\psi }_i$; its first term is the boxed expression. A sufficient convergence condition is $\Vert T\Vert <1$ on the chosen space of fields in $V$, and accurate single-scattering truncation needs the neglected terms to be small. Weak refractive contrast is useful, but for a sufficiently large or resonant scatterer it is not by itself a guarantee of negligible multiple scattering.

Use the physical refractive index convention $k=k_{0}n$, with $k_0$ the background wavenumber and time dependence $e^{-i\omega t}$. Treat $B$ as a bounded region in ${\mathbb{R}}^3$; the printed $B\in {\mathbb{R}}^3$ should be $B\subset {\mathbb{R}}^3$. Define the contrast $q=n^2-1=2n_{ϵ}+n_{ϵ}^2$ and the outgoing Green functionThe Sommerfeld radiation condition selects this sign of the outgoing wave phase. Since $(\Delta +k_0^2)\psi =-k_0^{2}q\psi$, the Lippmann-Schwinger equation isThe positive sign in this integral follows from the minus sign in the defining Green function equation. One may instead define the scattering potential with the opposite sign, provided both equations change consistently.

If $\Vert T_q\Vert <1$ on the chosen field space inside $B$, its Neumann series converges. The Born series for the scattered field isFor observation points outside $B$, restrict the intermediate factors to $B$ and use the same outgoing integral for the final factor. For example, the first two terms areThe $m$th term describes $m$ successive scattering interactions. The first Born approximation for scalar wave scattering keeps one interaction and replaces the field inside the medium by the incident field. Higher terms describe multiple scattering and the resulting feedback on the internal field.

A concrete sufficient condition follows on $L^{\infty }(B)$. If $d_B$ is the diameter of $B$, then for $\mathbf{r}\in B$ the region lies in the ball of radius $d_B$ about $\mathbf{r}$, andThus $\boxed{\rho \ll 1}$ is a conservative sufficient condition for the first Born approximation for scalar wave scattering. When $\rho <1$, the omitted internal-field terms satisfyThe bound ignores cancellation in the oscillatory Green function, so it is sufficient, not necessary. A common physical small-contrast criterion for an extended weak medium is small accumulated extra wave phase, $k_0\ell \Vert n_{ϵ}{\Vert }_{\infty }\ll 1$, together with weak scattering and no resonant internal enhancement. Small local contrast alone is not enough for an arbitrarily large or resonant object.