= Solution
Put $x=r\cos\theta$. The function $e^{ikx}$ solves the two-dimensional <Helmholtz equation>, $(\nabla^2+k^2)e^{ikx}=0$. Expand its angular dependence as a <Fourier series>,
$$
e^{ikr\cos\theta}=\sum_{m\in\mathbb Z}F_m(r)e^{im\theta}.
$$
Substitution into the polar-coordinate <Laplacian>,
$$
\nabla^2=\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_\theta^2,
$$
gives
$$
r^2F_m''+rF_m'+(k^2r^2-m^2)F_m=0.
$$
With $\rho=kr$, this is the <Bessel differential equation>. Regularity of the original <plane wave> at the origin excludes the singular second-kind solution. Thus $F_m(r)$ is a constant multiple of the regular first-kind function, and
$$
\boxed{e^{ikr\cos\theta}=\sum_{m\in\mathbb Z}a_mJ_m(kr)e^{im\theta}.}
$$
For integer negative orders, $J_{-m}=(-1)^mJ_m$, so this indexing still supplies the regular radial solution. The radial equation fixes the form of each term; the small-radius plane-wave expansion fixes its coefficient.
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