Resonant s-wave unitarity check ($\mathrm{Im}f=k|f{|}^2$)

For an isotropic elastic amplitude normalized by an incident unit plane wave and outgoing $f{e}^{ikr}/r$, total cross section is $4\pi |f{|}^2$. The optical theorem gives the displayed identity. At a resonance where the real part of $1/f$ vanishes, $f=i/k$ and the cross section is $4\pi /k^2$.