Two-dimensional partial-wave scattering formula ($f(\theta )=\sqrt{2/\pi }{\sum }_{m\ge 0}{w}_m{e}^{i{\delta }_m}\sin {\delta }_m\cos (m\theta )$, $\sigma =(4/k){\sum }_{m\ge 0}{w}_m{\sin }^2{\delta }_m$)

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 $m\ge 1$. 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.