Geoscience Reference
In-Depth Information
P
N
¸¸
'
-
#
#
'
P
Ã
P'
Fig. 1.7. Spherical distance
ψ
1.10
Fully normalized spherical harmonics
The formulas of the preceding section for the expansion of a function into a
series of surface spherical harmonics are somewhat inconvenient to handle.
If we look at equations (1-84) and (1-88), we see that there are different
formulas for
m
=0and
m
= 0; furthermore, the expressions are rather
complicated and dicult to remember.
Therefore, it has been proposed that the “conventional” harmonics
R
nm
and
S
nm
, defined by (1-82) together with (1-57), be replaced by other func-
tions which differ by a constant factor and are easier to handle. We consider
here only the
fully normalized
harmonics, which seem to be the most conve-
nient and the most widely used.
The “fully normalized” harmonics are simply “normalized” in the sense
of the theory of real functions; we have to use this clumsy expression because
the term “normalized spherical harmonics” has already been used for other
functions, unfortunately often for some that are not “normalized” at all in
the mathematical sense.
We denote the fully normalized harmonics by
¯
¯
R
nm
and
S
nm
;theyare
defined by
R
n
0
(
ϑ, λ
)=
√
2
n
+1
≡
√
2
n
+1
P
n
(cos
ϑ
);
¯
R
n
0
(
ϑ, λ
)
⎫
⎬
⎭
R
nm
(
ϑ, λ
)=
2(2
n
+1)
(
n
m
)!
(
n
+
m
)!
R
nm
(
ϑ, λ
)
−
¯
(1-91)
(
m
=0)
.
2(2
n
+1)
(
n
m
)!
(
n
+
m
)!
S
nm
(
ϑ, λ
)
−
¯
S
nm
(
ϑ, λ
)=
The orthogonality relations (1-83) also apply for these fully normalized har-
