Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 1 / c / Solution

For a nonnegative integer $m$, expand the plane wave in powers of $\rho$:The first contribution to the positive angular harmonic $e^{im\theta }$ occurs at $n=m$. Its coefficient is $(i\rho {)}^m/(2^{m}m!)$. Comparing this with $a_m{J}_m(\rho )\sim a_m{\rho }^m/(2^{m}m!)$ gives $a_m=i^m$ for $m\ge 0$.

The plane wave is even in $\theta$, so its $m$ and $-m$ Fourier coefficients agree. Therefore $a_{-m}{J}_{-m}=a_m{J}_m$, implying $a_{-m}=(-1{)}^m{i}^m=i^{-m}$. HenceThis is the cosine form of the Jacobi-Anger expansion. The factorial asymptotic stated in the source is used for nonnegative order; applying it literally to negative integers would introduce an undefined factorial. The Integer-order Bessel reflection identity is the required extension, not a separate set of divergent radial solutions.