
10.3. Helmholtz Equation ∆
3
w + λw = −Φ(x) 899
Relations (13) and (17) involve the eigenfunctions u
n
and eigenvalues µ
n
of the two-
dimensional second boundary value problem (10) w ith s = 1 and k = 0.
Another representation of the Green’s function:
G(x, y, z, ξ, η, ζ) =
1
a
∞
X
n=0
∞
X
m=0
ε
m
u
n
(x, y)u
n
(ξ, η) cos(q
m
z) cos(q
m
ζ)
ku
n
k
2
(µ
n
+ q
2
m
− λ)
,
q
m
=
πm
a
, ε
m
=
(
1 for m = 0,
2 for m 6= 0,
µ
0
= 0, u
0
= 1.
It is a consequence of formula (4).
⊙ Literature: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
3
◦
. The Green’s function of the three-dimensional third boundary value problem for equa-
tion (7) with the boundary conditions
∂w
∂z
−k
1
w = 0 at z = 0,
∂w
∂z
+ k
2
w = 0 at ...