Helmholtz Differential Equation--Circular Cylindrical Coordinates

In Cylindrical Coordinates, the Scale Factors are $h_r=1$, $h_\theta=r$, $h_z=1$ and the separation functions are $f_1(r)=r$, $f_2(\theta)=1$, $f_3(z)=1$, so the Stäckel Determinant is 1. Attempt Separation of Variables by writing

$$F(r, \theta, z) = R(r)\Theta(\theta)Z(z),$$ (1)

then the Helmholtz Differential Equation becomes

$${d^2R\over dr^2}\Theta Z + {1\over r}{dR\over dr}\Theta Z + ... ...ver d\theta^2}RZ + {d^2Z\over dz^2}R\Theta +k^2 R\Theta Z = 0.$$ (2)

Now divide by $R\Theta Z$,

$$\left({{r^2\over R}{d^2R\over dr^2}+ {r\over R}{dR\over dr}}... ...ta^2}{1\over \Theta }+ {d^2Z\over dz^2}{r^2\over Z} + k^2 = 0,$$ (3)

so the equation has been separated. Since the solution must be periodic in $\Theta$ from the definition of the circular cylindrical coordinate system, the solution to the second part of (3) must have a Negative separation constant

$${d^2\Theta\over d\theta^2}{1\over\Theta}= -(k^2+m^2),$$ (4)

which has a solution

$$\Theta(\theta) = C_me^{-i\sqrt{k^2+m^2}\,\theta}+D_me^{i\sqrt{k^2+m^2}\,\theta}.$$ (5)

Plugging (5) back into (3) gives

$${r^2\over R}{d^2R\over dr^2}+ {r\over R}{dR\over dr}-m^2 + {d^2Z\over dz^2}{r^2\over Z}= 0$$ (6)

$${1\over R}{d^2R\over dr^2}+ {1\over rR}{dR\over dr}-{m^2\over r^2} + {d^2Z\over dz^2}{1\over Z}= 0.$$ (7)

The solution to the second part of (7) must not be sinusoidal at $\pm \infty$ for a physical solution, so the differential equation has a Positive separation constant

$${d^2Z\over dz^2}{1\over Z}= n^2,$$ (8)

and the solution is

$$Z(z) = E_ne^{-nz}+F_ne^{nz}.$$ (9)

Plugging (9) back into (7) and multiplying through by $R$ yields

$${d^2R\over dr^2}+ {1\over r}{dR\over dr}+ \left({n^2 - {m^2\over r^2}}\right)R = 0$$ (10)

$${1\over n^2}{d^2R\over dr^2}+ {1\over (nr)}{1\over n}{dR\over dr} + \left[{1 - {m^2\over (nr)^2}}\right]R = 0$$ (11)

$${d^2R\over d(nr)^2}+ {1\over (nr)}{dR\over d(nr)}+\left[{1-{m^2\over(nr)^2}}\right]R = 0.$$ (12)

This is the Bessel Differential Equation, which has a solution

$$R(r) = A_{mn}J_m(nr)+B_{mn}Y_m(nr),$$ (13)

where $J_n(x)$ and $Y_n(x)$ are Bessel Functions of the First and Second Kinds, respectively. The general solution is therefore

$F(r, \theta, z) =\sum_{m=0}^\infty\sum_{n=0}^\infty [A_{mn}J_m(nr)+B_{mn}Y_m(nr)]$
$\times (C_me^{-i\sqrt{k^2+m^2}\,\theta}+D_me^{i\sqrt{k^2+m^2}\,\theta})(E_ne^{-nz}+F_ne^{nz}).\quad$	(14)

Actually, the Helmholtz Differential Equation is separable for general $k$ of the form

$$k^2(r,\theta,z)=f(r)+{g(\theta)\over r^2}+h(z)+k'^2.$$ (15)

See also Cylindrical Coordinates, Helmholtz Differential Equation

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 656-657, 1953.