4.5 Other Properties of Harmonic Functions
43
The solution (4.23) can also be represented as
ϕ(ξ, η, ζ ) = −
1
2π
lim
S→∞
∂
∂ζ
(S)
ϕ 0 (x, y)dxdy
(x − ξ) 2 + (y − η) 2 + ζ 2
.
(4.24)
4.5 Other Properties of Harmonic Functions
Let ϕ be some harmonic function defined in the area D limited by an arbitrary
surface S. Assume that in formula (4.9) ψ ≡ 1; we obtain
(S)
∂ϕ
∂n
dS = 0.
(4.25)
Calculating in formula (4.12) ≡ 0, we have
ϕ(ξ, η, ζ ) =
1
4π
(S)
ϕ
∂
∂n
1
r
−
1
r
∂ϕ
∂n
dS.
In particular, if S is a sphere with a radius R and with a center in the point (ξ, η, ζ),
we have
∂
∂n
1
r
= −
∂
∂r
1
r
r=R
=
1
R 2 , (dn = −dr),
and the previous formula for the function ϕ(ξ, η, ζ ) gives
ϕ(ξ, η, ζ ) =
1
4π 2 R 2
(S)
ϕdS −
1
4πR
(S)
∂ϕ
∂n
dS.
Taking into account formula (4.25), we obtain from the last result
ϕ(ξ, η, ζ ) =
1
4πR 2
(S)
ϕdS.
(4.26)
The expression in the right part of formula (4.26) is the average value of the
harmonic function ϕ(ξ, η, ζ ) on the sphere surface. In this manner, formula (4.26)
expresses that the value of the harmonic function in the sphere center equals its
average value on the surface of this sphere.