Geoscience Reference
In-Depth Information
t
2
)
m/
2
=sin
m
ϑ
and from a
satisfies (1-56). Apart from the factor (1
−
constant, the function
P
nm
is the (
n
+
m
)th derivative of the polynomial
(
t
2
1)
n
. It can, thus, be evaluated. For instance,
P
11
(
t
)=
(1
− t
2
)
1
/
2
2
−
2
1
2=
1
d
2
1) =
1
dt
2
(
t
2
t
2
t
2
=sin
ϑ.
(1-58)
−
−
·
−
·
1
The case
m
= 0 is of particular importance. The functions
P
n
0
(
t
)areoften
simply denoted by
P
n
(
t
). Then (1-57) gives
d
n
1
2
n
n
!
dt
n
(
t
2
1)
n
.
P
n
(
t
)=
P
n
0
(
t
)=
−
(1-59)
Because
m
= 0, there is no square root, that is, no sin
ϑ
. Therefore, the
P
n
(
t
) are simply polynomials in
t
. They are called
Legendre's polynomials
.
We give the Legendre polynomials for
n
=0through
n
=5:
P
0
(
t
)=1
,
P
3
(
t
)=
2
t
3
3
−
2
t,
P
4
(
t
)=
3
8
t
4
15
4
t
2
+
8
,
P
1
(
t
)=
t,
−
(1-60)
P
2
(
t
)=
2
t
2
1
5
(
t
)=
6
8
35
4
t
3
+
1
8
t
5
−
2
,
−
t.
Remember that
t
=cos
ϑ.
(1-61)
The polynomials may be obtained by means of (1-59) or more simply by the
recursion formula
n
−
1
P
n−
2
(
t
)+
2
n
−
1
P
n
(
t
)=
−
tP
n−
1
(
t
)
,
(1-62)
n
n
by which
P
2
can be calculated from
P
0
and
P
1
,P
3
from
P
1
and
P
2
,etc.
Graphs of the Legendre polynomials are shown in Fig. 1.4.
The powers of cos
ϑ
can be expressed in terms of the cosines of multiples
of
ϑ
,suchas
cos
2
ϑ
=
2
cos 2
ϑ
+
2
,
cos
3
ϑ
=
4
cos 3
ϑ
+
4
cos
ϑ.
(1-63)
Therefore, we may also express the
P
n
(cos
ϑ
) in this way, obtaining
P
2
(cos
ϑ
)=
4
cos 2
ϑ
+
4
,
P
3
(cos
ϑ
)=
8
cos 3
ϑ
+
8
cos
ϑ,
(1-64)
P
4
(cos
ϑ
)=
3
64
cos 4
ϑ
+
5
9
16
cos 2
ϑ
+
64
,
P
5
(cos
ϑ
)=
63
128
cos 3
ϑ
+
1
64
cos
ϑ,
35
128
cos 5
ϑ
+
···
=
···
.
