= Optical theorem at second Born order
{title2=$\operatorname{Im}f^{(2)}(0)=k\sigma^{(1)}_{\rm total}/(4\pi)$}
For a real <elastic scattering> potential the first Born forward amplitude is real, while the leading total <scattering cross-section> is quadratic in the potential. The second Born amplitude supplies an imaginary part from its outgoing <Green function>. Symmetrizing the two interaction positions makes its imaginary part equal to $k/(4\pi)$ times the integrated squared first Born amplitude. The identity therefore compares quantities at the same perturbative order; a real first Born amplitude alone cannot test it against a nonzero <scattering cross-section>.
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