Past exam of the theoretical physics II course of the University of Cambridge / 2017 / 4 / c / Solution

The incident plane wave is periodic in $\theta$, so expand it as a Fourier series ${\sum }_m{R}_m(r)e^{im\theta }$. Substitution into the Helmholtz equation, with $\rho =kr$, gives the radial Bessel differential equationRegularity at the origin selects the Bessel function of the first kind, rather than the singular second solution. ThusThe coefficients $a_m$ are independent of $r$; their values follow from matching near the origin in the next part. The completed identity is the Jacobi-Anger expansion.