4.3 Green’s Spatial Functions
41
With some limitations applied to the behavior of the function ϕ(ξ, η, ζ ) in the
infinities, formula (4.13) can be used for unlimited surfaces S, for example, when
S coincides with the plane z = 0. The area D in this case can be considered as a
limited part of the plane z = 0 and the sphere of an unrestrictedly large radius with
the center in the point (ξ, η, ζ ). In the case of a sufficiently fast decrease in functions
ϕ and ψ, the integral over the spherical part of the surface S in formula (4.9) with
an unrestricted increase in the radius tends to zero, and only integral over the plane
z = 0 will remain in the right part of formula (4.9).
Let us consider the case when the area D includes also an infinitely distanced
point. Let us suggest that some harmonic function on the surface of S satisfies
the condition of
∂∂
∂n
= −
∂
1
r
∂n
S
.
(4.14)
With limited ϕ and in infinities, in the considered case we can use formula
(4.12) that results in
ϕ(ξ, η, ζ ) = −
1
4π
(S)
H
∂ϕ
∂n
dS,
(4.15)
where
H =
1
r
+ , (r =
(x − ξ) 2 + (y − η) 2 + (z − ζ ) 2 ).
(4.16)
Due to the property (4.14), the function H harmonic in D satisfies the condition
of
∂H
∂n
S
= 0.
(4.17)
The function H harmonic in the area D having a pole (single charge) in the point
ϕ(ξ, η, ζ ) whose normal derivative on the boundary surface S of the area D turns to
zero is referred to as the [2] Neumann function. It can be defined as a potential of
movement speeds of ideal liquid enveloping the body of a defined shape provided
there is a source in the point ϕ(ξ, η, ζ ) and a drain in infinitely distanced points.
For the case of half-space z > 0, the Neumann function can be built as a potential
of single masses located in the points ϕ(ξ, η, ζ ) ? ϕ(ξ, η, −ζ ), e. g.
H =
1
r
+
1
r ∗ ,
(4.18)
41
With some limitations applied to the behavior of the function ϕ(ξ, η, ζ ) in the
infinities, formula (4.13) can be used for unlimited surfaces S, for example, when
S coincides with the plane z = 0. The area D in this case can be considered as a
limited part of the plane z = 0 and the sphere of an unrestrictedly large radius with
the center in the point (ξ, η, ζ ). In the case of a sufficiently fast decrease in functions
ϕ and ψ, the integral over the spherical part of the surface S in formula (4.9) with
an unrestricted increase in the radius tends to zero, and only integral over the plane
z = 0 will remain in the right part of formula (4.9).
Let us consider the case when the area D includes also an infinitely distanced
point. Let us suggest that some harmonic function on the surface of S satisfies
the condition of
∂∂
∂n
= −
∂
1
r
∂n
S
.
(4.14)
With limited ϕ and in infinities, in the considered case we can use formula
(4.12) that results in
ϕ(ξ, η, ζ ) = −
1
4π
(S)
H
∂ϕ
∂n
dS,
(4.15)
where
H =
1
r
+ , (r =
(x − ξ) 2 + (y − η) 2 + (z − ζ ) 2 ).
(4.16)
Due to the property (4.14), the function H harmonic in D satisfies the condition
of
∂H
∂n
S
= 0.
(4.17)
The function H harmonic in the area D having a pole (single charge) in the point
ϕ(ξ, η, ζ ) whose normal derivative on the boundary surface S of the area D turns to
zero is referred to as the [2] Neumann function. It can be defined as a potential of
movement speeds of ideal liquid enveloping the body of a defined shape provided
there is a source in the point ϕ(ξ, η, ζ ) and a drain in infinitely distanced points.
For the case of half-space z > 0, the Neumann function can be built as a potential
of single masses located in the points ϕ(ξ, η, ζ ) ? ϕ(ξ, η, −ζ ), e. g.
H =
1
r
+
1
r ∗ ,
(4.18)