= 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 equation>
$$
\rho^2R_m''+\rho R_m'+(\rho^2-m^2)R_m=0.
$$
Regularity at the origin selects the <Bessel function of the first kind>, rather than the singular second solution. Thus
$$
\boxed{e^{ikr\cos\theta}=\sum_{m=-\infty}^\infty a_mJ_m(kr)e^{im\theta}.}
$$
The 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>.
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