= Two-dimensional partial-wave scattering formula
{title2=$f(\theta)=\sqrt{2/\pi}\sum_{m\geq0}w_me^{i\delta_m}\sin\delta_m\cos(m\theta)$}
{title2=$\sigma=(4/k)\sum_{m\geq0}w_m\sin^2\delta_m$}
For a real central potential and the outgoing convention $e^{i\pi/4}f(\theta)e^{ikr}/\sqrt{kr}$, the <Partial-wave S-matrix> is $S_m=e^{2i\delta_m}$. The cosine form of the <Jacobi-Anger expansion> fixes $w_0=1$ and $w_m=2$ for $m\geq1$. Comparing the outgoing part of each phase-shifted radial cosine with the incident <plane wave> gives the displayed <scattering amplitude>. Integrating $|f|^2/k$ gives the displayed <scattering cross-section>, since the constant angular mode has squared norm $2\pi$ and each positive-order cosine has squared norm $\pi$. Reversing the multiplicities would already fail to reproduce the free <plane wave> at zero <scattering phase shift>.
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