
9.3. Helmholtz Equation ∆
2
w + λw = −Φ(x) 831
◮ Elliptic coordinate system.
In the elliptic coordinates that are introduced by the relations
x = a cosh u cos v, y = a sinh u sin v (0 ≤ u < ∞, 0 ≤ v < 2π, a > 0),
the Helmholtz equation is expressed as
∂
2
w
∂u
2
+
∂
2
w
∂v
2
+ a
2
λ(cosh
2
u − cos
2
v)w = 0.
Setting w = F (u)G(v), we arrive at the following linear ordinary differential equations
for F = F (u) and G = G(v):
F
′′
+
1
2
a
2
λ cosh 2u −k
F = 0, G
′′
−
1
2
a
2
λ cos 2v − k
G = 0,
where k is the separation constant. The solutions of these equations periodic in v are given
by
F (u) =
(
Ce
n
(u, q),
Se
n
(u, q),
G(v) =
(
ce
n
(v, q),
se
n
(v, q),
q =
1
4
a
2
λ,
where Ce
n
(u, q) and Se
n
(u, q) are ...