= Solution
For a nonnegative integer $m$, expand the <plane wave> in powers of $\rho$:
$$
e^{i\rho\cos\theta}=\sum_{n=0}^{\infty}\frac{(i\rho)^n}{n!}
\left(\frac{e^{i\theta}+e^{-i\theta}}2\right)^n.
$$
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 $a_mJ_m(\rho)\sim a_m\rho^m/(2^m m!)$ gives $a_m=i^m$ for $m\geq0$.
The <plane wave> is even in $\theta$, so its $m$ and $-m$ <Fourier coefficients> agree. Therefore $a_{-m}J_{-m}=a_mJ_m$, implying $a_{-m}=(-1)^mi^m=i^{-m}$. Hence
$$
\boxed{a_m=i^m\quad(m\in\mathbb Z),\qquad
e^{i\rho\cos\theta}=\sum_{m=-\infty}^{\infty}i^mJ_m(\rho)e^{im\theta}.}
$$
This 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.
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