 Solution

Let $w_0=1$ and $w_m=2$ for $m\ge 1$, as required by the cosine form of the Jacobi-Anger expansion. For a central real potential, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. Splitting each shifted radial cosine into incoming and outgoing exponentials leaves its incoming coefficient unchanged and multiplies its outgoing coefficient by $S_m$. Subtracting the incident plane wave givesComparing with the prescribed scattering amplitude convention yields the two-dimensional partial-wave scattering formula,Orthogonality gives ${\int }_0^{2\pi }1d\theta =2\pi$ and ${\int }_0^{2\pi }{\cos }^2(m\theta )d\theta =\pi$ for $m\ge 1$, while all cross terms vanish. HenceThe multiplicities printed in this part are reversed. With those printed values and every ${\delta }_m=0$, the purported free wave would be $\frac{1}{2}{e}^{ikr\cos \theta }+\frac{3}{2}{J}_0(kr)$, rather than the required incident plane wave. The extra $J_0$ contains an incoming cylindrical wave, which cannot be absorbed into $f$. The formulas above use the corrected multiplicities and therefore satisfy the specified scattering boundary condition.