Environmental Engineering Reference
In-Depth Information
When
n
=
0
,
1
,
2
,
3
,
4
,
5, we have
1
2
(
1
2
(
3
x
2
5
x
3
P
0
(
x
)=
1
,
P
1
(
x
)=
x
,
P
2
(
x
)=
−
1
)
,
P
3
(
x
)=
−
3
x
)
,
1
8
(
1
8
(
35
x
4
30
x
2
63
x
5
70
x
3
P
4
(
x
)=
−
+
3
)
,
P
5
(
x
)=
−
+
15
x
)
.
Another particular solution
Q
n
(
and is a series of
infinite terms. It is called the
Legendre function of the second kind
.When
n
is an
integer, therefore, the general solutions of Legendre equations can be expressed by
the Legendre functions of the first and the second kind, i.e.
x
)
is linearly independent of
P
n
(
x
)
y
=
C
1
P
n
(
x
)+
C
2
Q
n
(
x
)
.
It can be shown that the convergence radius of
Q
n
(
x
)
is also 1. However,
Q
n
(
x
)
is
divergent at
x
=
±
1; thus we discuss the Legendre polynomials always in
(
−
1
,
1
)
.
A.5 Properties of Legendre Polynomials
1. The Rodrigue Expression of P
n
(
x
)
Let
u
and
v
be the
n
-th differentiable functions. By using the binomial theorem to
expand
n
k
=
0
c
n
u
(
k
)
v
(
n
−
k
)
,
x
2
n
)
(
n
)
=
(
−
1
)
or
(
uv
we can obtain
d
n
d
x
n
(
1
2
n
n
!
x
2
n
P
n
(
x
)=
−
1
)
,
which is called the
Rodrigue expression of the Legendre polynomials
.
2. Generating Function and Recurrence Formula
Consider a function of the complex variable
t
1
t
2
.
Let
r
be the root with a smaller normal of two roots of the quadratic equation 1
w
(
x
,
t
)=
√
1
−
2
xt
+
−
t
2
2
xt
+
=
0. Thus the
w
(
x
,
t
)
is analytical in the circle
|
t
| <
r
. By the theory of
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