Geoscience Reference
In-Depth Information
In particular, we have
Q
0
(
z
)=
1
2
ln
z
+1
z
1
=coth
−
1
z,
−
Q
1
(
z
)=
z
2
ln
z
+1
1=
z
coth
−
1
z
z −
1
−
−
1
,
(1-80)
Q
2
(
z
)=
3
ln
z
+1
2
z
=
3
coth
−
1
z
1
4
3
1
2
3
2
z.
4
z
2
2
z
2
−
z −
1
−
−
−
1.9 Expansion theorem and orthogonality relations
In (1-52) and (1-53), we have expanded
harmonic
functions in space into
series of
solid
spherical harmonics. In a similar way an
arbitrary
(at least in
a very general sense) function
f
(
ϑ, λ
) on the surface of the sphere can be
expanded into a series of
surface
spherical harmonics:
f
(
ϑ, λ
)=
∞
Y
n
(
ϑ, λ
)=
∞
n
[
a
nm
R
nm
(
ϑ, λ
)+
b
nm
S
nm
(
ϑ, λ
)]
,
(1-81)
n
=0
n
=0
m
=0
where we have introduced the abbreviations
R
nm
(
ϑ, λ
)=
P
nm
(cos
ϑ
)cos
mλ ,
(1-82)
S
nm
(
ϑ, λ
)=
P
nm
(cos
ϑ
)sin
mλ .
The symbols
a
nm
and
b
nm
are constant coecients, which we will now
determine. Essential for this purpose are the
orthogonality relations
.These
remarkable relations mean that the integral over the unit sphere of the prod-
uct of any two
different
functions
R
nm
or
S
nm
is zero:
⎬
⎭
R
nm
(
ϑ, λ
)
R
sr
(
ϑ, λ
)
dσ
=0
σ
S
nm
(
ϑ, λ
)
if
s
=
n
or
r
=
m
or both ;
S
sr
(
ϑ, λ
)
dσ
=0
(1-83)
σ
R
nm
(
ϑ, λ
)
S
sr
(
ϑ, λ
)
dσ
=0
in any case .
σ
For the product of two
equal
functions
R
nm
or
S
nm
,wehave
[
4
π
2
n
+1
;
R
n
0
(
ϑ, λ
)]
2
dσ
=
σ
[
R
nm
(
ϑ, λ
)]
2
dσ
=
σ
[
2
π
2
n
+1
(
n
+
m
)!
(
n
S
nm
(
ϑ, λ
)]
2
dσ
=
(
m
=0)
.
−
m
)!
σ
(1-84)
