Geoscience Reference
In-Depth Information
for
m
= 0. This corresponds to (1-67); here, as in (1-67),
r
is the greatest
integer
≤
(
n − m
)
/
2.
There are relations between the coecients
a
nm
and
b
nm
for fully normal-
ized harmonics and the coecients
a
nm
and
b
nm
for conventional harmonics
that are inverse to those in (1-91):
a
n
0
√
2
n
+1
;
a
n
0
=
⎫
⎬
1
2(2
n
+1)
(
n
+
m
)!
(
n
a
nm
=
m
)!
a
nm
(1-98)
−
(
m
=0)
.
⎭
1
2(2
n
+1)
(
n
+
m
)!
(
n
b
nm
=
m
)!
b
nm
−
1.11 Expansion of the reciprocal distance into zonal
harmonics and decomposition formula
The distance
l
between two points with spherical coordinates
P
(
r, ϑ, λ
)
,P
(
r
,ϑ
,λ
)
(1-99)
is given by
2
rr
cos
ψ,
(1-100)
where
ψ
is the angle between the radius vectors
r
and
r
(Fig. 1.8), so that,
from (1-90),
l
2
=
r
2
+
r
2
−
cos
ψ
=cos
ϑ
cos
ϑ
+sin
ϑ
sin
ϑ
cos(
λ
− λ
)
(1-101)
P
l
r
P'
r'
Ã
O
Fig. 1.8. The spatial distance
l
