
9.1. Laplace Equation ∆
2
w = 0 791
Here, the a
(i)
n
and b
(i)
n
(i = 1, 2) are the coefficients of the Fourier series expansions of the
functions f
1
(ϕ) and f
2
(ϕ):
a
(i)
n
=
1
π
Z
2π
0
f
i
(ψ) cos(nψ) dψ, n = 0, 1, 2, . . . ,
b
(i)
n
=
1
π
Z
2π
0
f
i
(ψ) s in(nψ) dψ, n = 1, 2, 3, . . .
⊙ Literature: M. M. Smirnov (1975).
◮ Domain: R
1
≤ r ≤ R
2
. Second boundary value problem.
An annular domain is considered. Boundary conditions are prescribed:
∂
r
w = f
1
(ϕ) at r = R
1
, ∂
r
w = f
2
(ϕ) at r = R
2
.
Solution:
w(r, ϕ) = B ln r +
∞
X
n=1
r
n
(A
n
cos nϕ+B
n
sin nϕ)+
∞
X
n=1
1
r
n
(C
n
cos nϕ+D
n
sin nϕ)+K.
The coefficients B, A
n
, B
n
, C
n
, and D
n
are expressed as
B =
1
2
R
1
a
(1)
0
, A
n
=
R
n+1
2
a
(2)
n
− R
n+1
1
a
(1)
n
n(R
2n
2
− R
2n
1
)
, B
n
=
R