Geoscience Reference
In-Depth Information
monics, whereas Eqs. (1-84) are thoroughly simplified: they become
1
4
π
1
4
π
¯
2
nm
¯
2
nm
R
dσ
=
S
dσ
=1
.
(1-92)
σ
σ
This means that the
average square of any fully normalized harmonic is
unity
, the average being taken over the sphere (the average corresponds to
the integral divided by the area 4
π
). This formula now applies for any
m
,
whether it is zero or not.
If we expand an arbitrary function
f
(
ϑ, λ
) into a series of fully normalized
harmonics, analogously to (1-81),
f
(
ϑ, λ
)=
∞
n
[
a
nm
¯
R
nm
(
ϑ, λ
)+
b
nm
¯
S
nm
(
ϑ, λ
)]
,
(1-93)
n
=0
m
=0
then the coecients
a
nm
, b
nm
are simply given by
f
(
ϑ, λ
)
¯
1
4
π
a
nm
=
R
nm
(
ϑ, λ
)
dσ ,
σ
(1-94)
f
(
ϑ, λ
)
1
4
π
b
nm
=
¯
S
nm
(
ϑ, λ
)
dσ
;
σ
that is, the coecients are the average products of the function and the
corresponding harmonic
¯
¯
S
nm
.
The simplicity of formulas (1-92) and (1-94) constitutes the main ad-
vantage of the fully normalized spherical harmonics and makes them useful
in many respects, even though the functions
R
nm
or
¯
¯
R
nm
and
S
nm
in (1-91) are a
little more complicated than the conventional
R
nm
and
S
nm
.Wehave
R
nm
(
ϑ, λ
)=
P
nm
(cos
ϑ
)cos
mλ ,
¯
¯
(1-95)
S
nm
(
ϑ, λ
)=
P
nm
(cos
ϑ
)sin
mλ ,
where
r
P
n
0
(
t
)=
√
2
n
+12
−n
(2
n
−
2
k
)!
(
−
1)
k
2
k
)!
t
n−
2
k
(1-96)
k
!(
n
−
k
)! (
n
−
k
=0
for
m
=0,and
P
nm
(
t
)=
2(2
n
+1)
(
n
m
)!
(
n
+
m
)!
−
2
−n
(1
t
2
)
m/
2
−
·
(1-97)
r
(2
n
−
2
k
)!
(
−
1)
k
2
k
)!
t
n−m−
2
k
k
!(
n
−
k
)! (
n
−
m
−
k
=0
