Geoscience Reference
In-Depth Information
results. Assuming
r
<r
,wemaywrite
1
l
1
1
=
r
2
=
r
√
1
2
αu
+
α
2
,
(1-102)
2
rr
cos
ψ
+
r
2
−
−
where we have put
α
=
r
/r
and
u
=cos
ψ
.If
r
<r
, this can be expanded
into a power series with respect to
α
. It is remarkable that the coecients
of
α
n
are the (conventional) zonal harmonics, or Legendre's polynomials
P
n
(
u
)=
P
n
(cos
ψ
):
=
∞
1
√
1
−
2
αu
+
α
2
α
n
P
n
(
u
)=
P
0
(
u
)+
αP
1
(
u
)+
α
2
P
2
(
u
)+
···
.
(1-103)
n
=0
Hence, we obtain
=
∞
r
n
r
n
+1
P
n
(cos
ψ
)
,
1
l
(1-104)
n
=0
whichisanimportantformula.
It would still be desirable in this equation to express
P
n
(cos
ψ
)intermsof
functions of the spherical coordinates
ϑ, λ
and
ϑ
,λ
of which
ψ
is composed
according to (1-90). This is achieved by the
decomposition formula
P
n
(cos
ψ
)=
P
n
(cos
ϑ
)
P
n
(cos
ϑ
)+
2
n
(
n − m
)!
(
n
+
m
)!
[
R
nm
(
ϑ
,λ
)+
S
nm
(
ϑ
,λ
)]
.
R
nm
(
ϑ, λ
)
S
nm
(
ϑ, λ
)
m
=1
(1-105)
Substituting this into (1-104), we obtain
P
n
(cos
ϑ
)
r
n
+1
=
∞
n
1
l
(
n
m
)!
(
n
+
m
)!
·
−
r
n
P
n
(cos
ϑ
)+2
n
=0
m
=1
(1-106)
R
nm
(
ϑ, λ
)
r
n
+1
S
nm
(
ϑ
,λ
)
.
R
nm
(
ϑ
,λ
)+
S
nm
(
ϑ, λ
)
r
n
+1
r
n
r
n
The use of fully normalized harmonics simplifies these formulas. Replacing
the conventional harmonics in (1-105) and (1-106) by fully normalized har-
monics by means of (1-91), we find
n
¯
S
nm
(
ϑ
,λ
)
;
1
2
n
+1
R
nm
(
ϑ, λ
)
¯
R
nm
(
ϑ
,λ
)+
¯
S
nm
(
ϑ, λ
)
¯
P
n
(cos
ψ
)=
m
=0
(1-107)
