On the web, it is given that the general solution to the Helmholtz equation ($\nabla^2u+ku=0$) in cylindrical coordinates is
$$\sum_{m=0}^\infty\sum_{n=0}^\infty{[A_{mn}\cdot J_m(\sqrt{k^2 - n^2}\cdot r) + B_{mn}\cdot Y_m(\sqrt{k^2 - n^2}\cdot r)]\cdot [C_m\cdot \cos(m\varphi) + D_m\cdot \sin(m\varphi)]\cdot[E_n\cdot e^{nz}+F_ne^{-nz}]}$$
Where $J_m, Y_m$ are the Bessel functions of the first and second kind respectively.
My problem with this: I see no reason for $n$ (one of the separation variables) to be quantized. On Wolfram MathWorld, it's just stated that some equation involving $n$ is a modified Bessel equation, and then they give the solution. But the Bessel equation, as far as I know, has a continuous range of solutions ($J_v(x), v\in\mathbb{R})$, so there is no reason for $n$ to quantize. Furthermore, at least with how I learned it in class, the linear combination of $J_v, Y_v$ is only relevant when $v$ is an integer, and otherwise you use the linear combination of $J_v, J_{-v}$.
If $n$ isn't quantized then, should we have an integral over all possible $n$'s? This isn't something I've seen anywhere, but I don't understand what makes $n$ quantized. For $m$ for example, it's the condition that $\Theta(\theta)=\Theta(\theta+2\pi)$.
Thanks.