Geoscience Reference
In-Depth Information
Substituting this relation into equations (1-43), we see that
V
i
(
r, ϑ, λ
)=
∞
n
r
n
[
a
nm
P
nm
(cos
ϑ
)cos
mλ
+
b
nm
P
nm
(cos
ϑ
)sin
mλ
]
,
n
=0
m
=0
(1-52)
V
e
(
r, ϑ, λ
)=
∞
n
1
r
n
+1
[
a
nm
P
nm
(cos
ϑ
)cos
mλ
+
b
nm
P
nm
(cos
ϑ
)sin
mλ
]
n
=0
m
=0
(1-53)
are solutions of Laplace's equation ∆
V
= 0; that is, harmonic functions.
Furthermore, as we have mentioned, they are very general solutions indeed:
every function which is harmonic inside a certain sphere can be expanded
into a series (1-52), where the subscript i indicates the interior, and every
function which is harmonic outside a certain sphere (such as the earth's
gravitational potential) can be expanded into a series (1-53), where the
subscript e indicates the exterior. Thus, we see how spherical harmonics can
be useful in geodesy.
1.7
Legendre's functions
In the preceding section we have introduced Legendre's function
P
nm
(cos
ϑ
)
as a solution of Legendre's differential equation (1-46). The subscript
n
denotes the
degree
and the subscript
m
the
order
of
P
nm
.
It is convenient to transform Legendre's differential equation (1-46) by
the substitution
t
=cos
ϑ.
(1-54)
In order to avoid confusion, we use an overbar to denote
g
as a function of
t
. Therefore,
g
(
ϑ
)=
g
(
t
)
,
g
(
ϑ
)=
dg
dϑ
=
dg
dt
dt
dϑ
g
(
t
)sin
ϑ,
=
−
(1-55)
g
(
ϑ
)=
g
(
t
)sin
2
ϑ
g
(
t
)cos
ϑ.
−
Inserting these relations into (1-46), dividing by sin
ϑ
, and then substituting
sin
2
ϑ
=1
− t
2
,weget
2
t g
(
t
)+
n
(
n
+1)
t
2
g
(
t
)=0
.
m
2
t
2
)
g
(
t
)
(1
−
−
−
(1-56)
1
−
The Legendre function
g
(
t
)=
P
nm
(
t
), which is defined by
2
n
n
!
(1
− t
2
)
m/
2
d
n
+
m
1
dt
n
+
m
(
t
2
−
1)
n
,
P
nm
(
t
)=
(1-57)
