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or
1
π
τ
(
M
)=
K
(
M
,
P
)
τ
(
P
)
d
s
−
f
(
M
)
,
(7.174)
C
where the
K
is defined in Eq. (7.171).
Remark 2
. Similar to three-dimensional Poisson equations, two-dimensional Pois-
son equations
(
M
,
P
)
Δ
u
=
F
(
x
,
y
)
,
(
x
,
y
)
∈
D
have a particular solution
1
2
1
u
∗
(
(
x
,
y
)=
−
F
(
ξ
,
η
)
ln
2
d
ξ
d
η
.
(7.175)
π
x
−
ξ
)
2
+(
y
−
η
)
D
u
∗
(
Thus a function transformation of
u
(
M
)=
ω
(
M
)+
M
)
will transform Dirichlet
internal problems of Poisson equations
Δ
u
=
F
(
M
)
,
M
∈
D
,
(7.176)
u
|
C
=
f
(
M
)
.
into Dirichlet internal problems of Laplace equations
Δω
=
0
,
M
∈
D
,
(7.177)
u
∗
|
C
.
ω
|
C
=
f
(
M
)
−
Similarly, the other three problems of Poisson equations can also be transformed
into those of Laplace equations.
Remark 3
. Neumann problems in a plane can also be transformed into the Dirichlet
problems by using the relation between analytical and harmonic functions. Suppose
that a Neumann problem of the Laplace equation has solution
u
(
x
,
y
)
in a simply-
connected plane domain
D
.The
u
(
x
,
y
)
and its first partial derivatives are assumed to
be continuous in a closed domain
D
=
D
∪
∂
D
. There must exist a harmonic function
v
(
x
,
y
)
, conjugate to
u
, such that
u
and
v
satisfy the Cauchy-Riemann equation
∂
u
x
=
∂
v
∂
u
y
=
−
∂
v
x
.
(7.178)
∂
∂
y
∂
∂
The
v
that satisfies Eq. (7.178) is also unique up to an arbitrary constant.
Let
n
and
(
x
,
y
)
τ
be the external normal and positive tangent of
∂
D
, respectively. A
counter-clockwise rotation of
n
by 90
◦
will thus arrive at
τ
. By the relation between
directional and partial derivatives, we have, for
(
x
,
y
)
∈
∂
D
,
∂
u
n
=
∂
u
)+
∂
u
x
cos
(
n
,
x
y
cos
(
n
,
y
)
∂
∂
∂
=
∂
v
)
−
∂
v
y
cos
(
n
,
x
x
cos
(
n
,
y
)
∂
∂
=
∂
v
)+
∂
v
)=
∂
v
∂τ
(
τ
,
(
τ
,
x
cos
x
y
cos
y
∂
∂
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