= Solution
At large $r$, $|\mathbf r-\mathbf r_j|=r-\hat{\mathbf r}\cdot\mathbf r_j+O(r^{-1})$. Retain the displacement in the phase, but replace the denominator by $r$. Therefore
$$
\boxed{f(\hat{\mathbf r})=\frac1k\left[c_1e^{-ik\hat{\mathbf r}\cdot\mathbf r_1}+c_2e^{-ik\hat{\mathbf r}\cdot\mathbf r_2}\right]}.
$$
The <differential scattering cross-section> is $|f|^2$ for <elastic scattering> with this unit incident plane-wave normalization. <interference> depends on outgoing direction and centre separation as well as the incoming phases already contained in $c_j$.
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