
10.3. Helmholtz Equation ∆
3
w + λw = −Φ(x) 919
Here,
G(r, ϕ, z, ξ, η, ζ) =
1
π
∞
X
n=0
∞
X
m=1
A
n
µ
2
nm
J
n
(µ
nm
r)J
n
(µ
nm
ξ) cos[n(ϕ − η)]
(µ
2
nm
R
2
+ k
2
1
R
2
− n
2
)J
2
n
(µ
nm
R)
F
nm
(z, ζ),
F
nm
(z, ζ) =
exp(−β
nm
z)[β
nm
cosh(β
nm
ζ) + k
2
sinh(β
nm
ζ)]
β
nm
(β
nm
+ k
2
)
for z > ζ,
exp(−β
nm
ζ)[β
nm
cosh(β
nm
z) + k
2
sinh(β
nm
z)]
β
nm
(β
nm
+ k
2
)
for ζ > z,
β
nm
=
p
µ
2
nm
− λ, A
n
=
(
1 for n = 0,
2 for n 6= 0,
where the J
n
(ξ) are Bessel functions and the µ
nm
are positive roots of the transcendental
equation
µJ
′
n
(µR) + k
1
J
n
(µR) = 0.
◮ Domain: 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π, 0 ≤ z < ∞. Mixed boundary value problem.
A semiinfinite circular cylinder is considered. Boundary conditions are prescribed:
w = f
1
(ϕ, z) at r = R, ∂
z
w = f