= Solution
Let $w_0=1$ and $w_m=2$ for $m\geq1$, 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> gives
$$
\psi_{\rm sc}\sim\frac{e^{ikr-i\pi/4}}{\sqrt{2\pi kr}}
\sum_{m=0}^\infty w_m(S_m-1)\cos(m\theta).
$$
Comparing with the prescribed <scattering amplitude> convention yields the <two-dimensional partial-wave scattering formula>,
$$
\boxed{f(\theta)=\frac1{i\sqrt{2\pi}}\sum_{m=0}^\infty w_m(e^{2i\delta_m}-1)\cos(m\theta)
=\sqrt{\frac2\pi}\sum_{m=0}^\infty w_me^{i\delta_m}\sin\delta_m\cos(m\theta).}
$$
<Orthogonality> gives $\int_0^{2\pi}1\,d\theta=2\pi$ and $\int_0^{2\pi}\cos^2(m\theta)d\theta=\pi$ for $m\geq1$, while all cross terms vanish. Hence
$$
\boxed{\lambda=\frac4k\left[\sin^2\delta_0+2\sum_{m=1}^\infty\sin^2\delta_m\right].}
$$
The multiplicities printed in this part are reversed. With those printed values and every $\delta_m=0$, the purported free wave would be $\tfrac12e^{ikr\cos\theta}+\tfrac32J_0(kr)$, 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.
Back to article page
