Geoscience Reference
In-Depth Information
1.8
Legendre's functions of the second kind
The Legendre function
P
nm
(
t
) is not the only solution of Legendre's differ-
ential equation (1-56). There is a completely different function which also
satisfies this equation. It is called
Legendre's function of the second kind
,of
degree
n
and order
m
, and denoted by
Q
nm
(
t
).
Although the
Q
nm
(
t
) are functions of a completely different nature, they
satisfy relationships very similar to those satisfied by the
P
nm
(
t
).
The “zonal” functions
Q
n
(
t
)
≡
Q
n
0
(
t
)
(1-73)
are defined by
n
Q
n
(
t
)=
1
2
P
n
(
t
)ln
1+
t
1
k
P
k−
1
(
t
)
P
n−k
(
t
)
,
1
− t
−
(1-74)
k
=1
and the others by
t
2
)
m/
2
d
m
Q
n
(
t
)
dt
m
Q
nm
(
t
)=(1
−
.
(1-75)
Equation (1-75) is completely analogous to (1-65); furthermore, the func-
tions
Q
n
(
t
) satisfy the same recursion formula (1-62) as the functions
P
n
(
t
).
If we evaluate the first few
Q
n
, from (1-74) we find
Q
0
(
t
)=
1
2
ln
1+
t
1
t
=tanh
−
1
t,
−
Q
1
(
t
)=
t
2
ln
1+
t
1=
t
tanh
−
1
t
1
− t
−
−
1
,
(1-76)
Q
2
(
t
)=
3
ln
1+
t
1
2
t
=
3
tanh
−
1
t
1
4
3
1
2
3
2
t.
4
t
2
2
t
2
−
t
−
−
−
−
These formulas and Fig. 1.6 show that the functions
Q
nm
are really
quite different from the functions
P
nm
. From the singularity
1
(i.e.,
ϑ
=0or
π
), we see that it is impossible to substitute
Q
nm
(cos
ϑ
)for
P
nm
(cos
ϑ
)if
ϑ
means the polar distance, because harmonic functions must
be regular.
However, we will encounter them in the theory of ellipsoidal harmon-
ics (Sect. 1.16), which is applied to the normal gravity field of the earth
(Sect. 2.7). For this purpose we need Legendre's functions of the second
±∞
at
t
=
±
