23.1 Separability in Three Dimensions
265
(7) Oblate Spheroidal Coordinates
x = f ζ
1 − η 2 cos φ
1 ≤ ζ < ∞
(23.21a)
y = f ζ
1 − η 2 sin φ
− 1 ≤ η ≤ 1
(23.21b)
z = f η
ζ 2 − 1
− π ≤ φ < π, f > 0 and constant. (23.21c)
(8) Parabolic Cylindrical Coordinates
x = ζ η
− ∞ < η < ∞
(23.22a)
y =
1
2
(ζ
2
− η
2 )
0 ≤ ζ < ∞
(23.22b)
z = z
− ∞ < z < ∞,
(23.22c)
and the Laplace–Beltrami operator
Δ ηζ z =
1
η 2 + ζ 2
∂ 2
∂η 2 +
∂ 2
∂ζ 2
+
∂ 2
∂z 2 .
(23.22d)
(9) Paraboloidal Coordinates
x = 2c cosh α cos β sinh γ
0 ≤ α < ∞
(23.23a)
y = 2c sinh α sin β cosh γ
0 ≤ γ < ∞
(23.23b)
z =
c
2
(cosh 2α + cos 2β − cosh 2γ )
− π ≤ β < π,
(23.23c)
c = 0 and constant.
(10) Ellipsoidal Coordinates
x = ±
(μ − a)(ν − a)(ρ − a)
a(a + 1)
a ≤ μ < ∞
(23.24a)
y = ±
(μ − 1)(ν − 1)(ρ − 1)
1 − a
1 ≤ ν ≤ a
(23.24b)
z = ±
μνρ
a
0 ≤ ρ ≤ 1, a > 1 and constant.
(23.24c)