
10.3. Helmholtz Equation ∆
3
w + λw = −Φ(x) 929
10.3.6 O ther Orthogonal Curvilinear Coordinates
The homogenous three-dimensional Helmholtz equation admits separation of variables in
the eleven orthogonal systems of coordinates listed in Table 10.4.
For the parabolic cylindrical system of coordinates, the multipliers f and g are ex-
pressed in terms of the parabolic cylinder functions as
f(ξ)=A
1
D
µ−1/2
(σξ)+A
2
D
µ−1/2
(−σξ), g(η)=B
1
D
−µ−1/2
(ση)+B
2
D
−µ−1/2
(−ση),
µ=
1
2
β(k
2
−λ)
−1/2
, σ=
4(k
2
−λ)
1/4
,
where A
1
, B
1
, A
2
, and B
2
are arbitrary constants.
For the elliptic cylindrical system of coordinates, the functions f and g are determined
by the modified Mathieu equation ...