I have a finite cylinder with the solution satisfying the Laplace equation outside. Using separation of variables in cylindrical coordinates $(r, \phi, z)$ for the 3D Laplace equation, I get (with separation constants $m,s$)
\begin{align} f(r, \phi, z)&= [C_{0,0,0} + C_{0,0,1}\ln{r}][C_{0,0,2}z+C_{0,0,3}] \\ &+ \int\limits_{0}^\infty [C_{0,1}(s) J_0(sr) +C_{0,2}(s) Y_0(sr)][C_{0,3}(s)e^{sz} + C_{0,4}(s)e^{-sz}]\,\mathrm{d}s\\ &+ \sum\limits_{m=1}^\infty [C_{m,0,0}r^n + C_{m,0,1}r^{-n}][C_{m,0,2}\cos(m \phi) + C_{m,0,3} \sin(m\phi) ][C_{m,0,4}z+C_{m,0,5}] \\ &+\sum\limits_{m=1}^\infty\int\limits_{0}^\infty [C_{m,1}(s) J_m(sr) +C_{m,2}(s) Y_m(sr)][C_{m,3}(s)\cos(m \phi) + C_{m,4}(s) \sin(m\phi) ][C_{m,5}(s)e^{sz} + C_{m,6}(s)e^{-sz}]\,\mathrm{d}s \end{align}
where the constants are $C_{m,s,i}$ for the discrete terms, or $C_{m,i}(s)$ for the continuous terms. $m$ must be integer (summation) due to periodic solution in $\phi$, but $s$ is continuous (integration) as there are no bounds in $r$ or $z$. The first term is when $m=0,s=0$, the second is when $m=0,s \neq 0$ the third is for $m\neq 0 , s=0$ and the last is for $m\neq 0,s\neq 0$.
However, do I need all these terms? I believe it is possible to combine the first 3 terms into the last integral and drop the term $C_{m,2}(s) Y_m(sr)$. I think this is possible as the Bessel functions $J_m(s r)$ form a complete (orthogonal) basis for the whole space (Hankel transform) so all other terms can just be written as a combination of these Bessel functions of the first kind. I don't believe it will constrain the problem at all, as I still have an infinite number of coefficients to describe the solution.
So my question is: can I write the entire general solution as $$\sum\limits_{m=1}^\infty\int\limits_{0}^\infty [C_{m,1}(s) J_m(sr)][C_{m,3}(s)\cos(m \phi) + C_{m,4}(s) \sin(m\phi) ][C_{m,5}(s)e^{sz} + C_{m,6}(s)e^{-sz}]\,\mathrm{d}s?$$ because the Bessel functions form a complete basis (Perhaps I need to change the lower limit of the summation to start from $m=0$?)
I am not interested in the case with no $\phi$ dependence.
I am also interested if this also works for the general solution to the 3D Helmholtz equation where the $r^n$ and $r^{-n}$ terms are replaced with Bessel functions with different arguments to the term with $m\neq0,s\neq0$ see here?
