Chapter 21 - Legendre Polynomials - DSP-Book

Poularikas A. D. “Legendre Polynomials”
The Handbook of Formulas and Tables for Signal Processing.
Ed. Alexander D. Poularikas
Boca Raton: CRC Press LLC,1999
© 1999 by CRC Press LLC

21.1 Legendre Polynomials
21.1.1 Definition
21.1.2 Generating Function
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w(–t,–s) = w(t,s)
21.1.3 Rodrigues Formula
21.1.4 Recursive Formulas
1.
2.
P () t =
n
21
Legendre Polynomials
21.1 Legendre Polynomials
21.2 Legendre Functions of the Second Kind
(Second Solution)
21.3 Associated Legendre Polynomials
21.4 Bounds for Legendre Polynomials
21.5 Table of Legendre and Associate Legendre
Functions
References
[ n / 2] k n 2k
∑
k=
0
−
( −1) ( 2n−2k)! t
n
2 k!( n−k)!( n−2k)! ⎧n/
2 neven
[ n / 2]
= ⎨
⎩(
n−1)/ 2 n odd
wts (, ) =
1
1− 2st
+ s
2
⎧
⎪
⎪
= ⎨
⎪
⎪
⎩
∞
∑
n=
0
∞
∑
n=
0
n
P () t s s < 1
n
−n−1 P () t s s > 1
n
1 d 2 n
Pn() t = ( t − 1) n = 012 , , L
n n
2 n!
dt
( n+ 1) P () t − ( 2n+ 1) tP() t + nP () t = 0 n = 12 , ,
n+ 1 n n−1
L
P′ + − ′ = + ′ = ⋅
n 1() t tPn() t ( n 1) Pn() t ( P () t derivative of P() t n = 012 ,, , L
n
generating function

3.
4.
5.
6.
Figure 21.1 shows a few Legendre functions.
FIGURE 21.1
21.1.5 Legendre Differential Equation
If y = P ( x) ( n = 012L , , , ) is a solution to the second-order DE
For
tP′ () t − P′ () t = nP() t n=
12L , ,
n n−1 n
Pn′ + 1() t − Pn′ −1()
t = ( 2n+ 1) Pn() t n = 12 , ,L
2
( t − 1)
Pn′ () t = ntPn() t −nPn−1() t
P() t = 1 P() t = t
n
0 1
t = cos ϕ:
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TABLE 21.1 Legendre Polynomials
P 0
= 1
P1 t =
3 P = t −
2
2
2 1 2
5 P = t − t
3
2
3 3 2
35 P = t − t +
4
8
4 30 8
2 3 8
63 P = t − t + t
5
8
5 70 8
3 15 8
231 6 315 4 105
P = t − t + t −
6
16
16
16
2 5
16
429 7 693 5 315 3 35
P = t − t + t − t
7
16
16
16
2
( 1− t ) y′′ − 2ty′ + n( n+ 1) y = 0
1 d ⎛ dy ⎞
⎜sin
ϕ ⎟ + nn ( + 1) y=
0
sinϕ
dϕ⎝
dϕ⎠
16

Example
From (21.1.4.4) and t = 1 implies 0 = nPn() 1 − nPn−1() 1 or Pn() 1 = Pn−1().
1 For n = 1, P1( 1) = P0(
1) = 1.
For n = 2, P ( 1) = P(
1) = 1 and so forth. Hence Pn () 1 = 1.
21.1.6 Integral Representation
1. Laplace integral:
2. Mehler-Dirichlet formula:
3. Schläfli integral:
1
Pn() t =
2π
j
C
2 n
( z − 1)
dz
n n+
1
2 ( z−t) where C is any regular, simple, closed curve surrounding t.
21.1.7 Complete Orthonormal System
The Legendre polynomials are orthogonal in [–1,1]
and therefore the set
is orthonormal.
2 1
21.1.8 Asymptotic Representation:
δ = fixed positive number
21.1.9 Series Expansion
If f(t) is integrable in [–1,1] then
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∫
π
1
2
n
Pn( t) = [ t+ t −1cos
ϕ] dϕ
π 0
2
Pn
(cos θ)
=
π
1
1/ 2
{[ ( 2n+ 1)]
P ( t)}
∫
2
n m
−1
∫
1
1
−1
P () t P () t dt = 0
n
θ
∫0
2 2
[ Pn( t)] dt =
n = 012 , , L
2n+ 1
2n+ 1
() P () t n = 012L ,,
2
ϕn t =
n
∫
1 cos( n + 2 ) ψ
dψ 0< θ< π,
n = 012 , , , …
2cosψ−cosθ
Pn
(cos )
sin n , n
nsin
θ
π
≅
⎛
+
⎞
θ
δ θ π δ
π θ ⎝ ⎠ +
2 ⎡ 1 ⎤
⎢
⎥ →∞, ≤ ≤ −
⎣ 2 4⎦
∞
∑
n=
0
f() t = a P () t − 1< t < 1
n n

For even f(t), the series will contain term Pn(t) of even index; if f(t) is odd, the term of odd index only.
If the real function f(t) is piecewise smooth in (–1,1) and if it is square integrable in (–1,1), then the
series converges to f(t) at every continuity point of f(t). If there is a discontinuity at t then the series
converges at [ f( t + 0) + f( t −0)]/
2.
21.1.10 Change of Range
If a function f(t) is defined in [a,b], it is sometimes necessary in the applications to expand the function
in a series in the applications to expand the function in a series of orthogonal polynomials in this interval.
Clearly the substitution
transform the interval [a,b] of the x-axis into the interval [–1,1] of the t-axis. It is, therefore, sufficient
to consider
The above equation can also be accomplished as follows:
Example
Suppose f(t) is given by
Then
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a
2n+ 1
=
2
∫
n n
1
−1
f () t P () t dt n = 012 ,, L
t
b a x
b a
b a b a
= − a b x t
−
+ ⎡ ⎤
−
< = +
⎣⎢ ⎦⎥
+
2
, ,
⎡
⎤
2 ⎣⎢ 2 2 ⎦⎥
f b a − b a
t + +
⎡
⎤
⎣⎢
⎦⎥ =
2 2
a
1
∫
∞
∑
n=
0
a P () t
n n
n
f b a + − b a
=
t + P t dt
+
2 1 ⎡
⎤
()
2 ⎣⎢ 2 2 ⎦⎥
n n
−1
f() t = a X () t
1
X () t =
n!( b−a) a
n
∞
∑
n=
0
2n+ 1
=
b−a n n
n n
a
∫
b
d ( t −a) ( t −b)
n
dt
n n n
f() t X () t dt
⎧0
−1≤ t < a
f()= t ⎨
⎩1
a < t ≤1
a
n
2n+ 1
=
2
1
∫
a
n
P ()
t dt
n

Using (21.1.4.4), and noting that P n(1) =1 we obtain
which leads to the expansion
Example
Suppose f(t) is given by
The function is an odd function and, therefore, f(t)P n(t) is an odd function of P n(t) with even index.
Hence a n are zero for n = 0,2,4,…. For odd index n, the product f(t)P n(t) is even and hence
Using (21.1.4.4) and setting n = 2k + 1, k = 0,1,2… we obtain
where we have used the property P n(1) =1 for all n. But
and, thus, we have
The expansion is
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1
1
a =− [ P ( a) − P ( a)], a = ( −a)
n 2 n+ 1 n−1
0
∞
∑
1
f() t ≅ ( −a) − [ Pn+ ( a) − Pn− ( a)] Pn(), t − < t <
2 1
1
1 1
1 1
2
1
n=
1
2 1
⎧−1
−1≤ t < 0
f()= t ⎨
⎩1
0 < t ≤1
∫ ∫
an =
⎛ 1
n+ ⎞
f t Pn t dt =
⎛ 1
() () 2 n+ ⎞
Pn() t dt n = 135 , , ,L
⎝ 2⎠
⎝ 2⎠
a
−1
1
∫ ∫
a = ( 4k+ 3)
P ( t) dt = [ P′ ( t) − P′ ( t)] dt
2k+ 1
2k+ 1
2k+ 2 2k
0
0
1
= [ P ( t) − P ( t)] = P ( 0) −P
( 0)
2k+ 2 2k 0 2k 2k+ 2
P
2n
1
n
2 ( 1) ( 2n)!
( 0)
= 2n2 n 2 (!) n
− ⎛ ⎞
⎜ ⎟ =
⎝ ⎠
−
k
k+
1
k
1 2k
1 2k2 1 2k
2k1 = 1
k
2k+ 2 2 2k 2
2 k 2 k 1 2 k 2k2 − − +
− =
+
−
( ) ( )! ( ) ( ) ( ) ( )! ⎡ +
+
⎤
(!) [( )!] (!) ⎣⎢ + ⎦⎥
2k+ 1 2 2
k
1 2k 4k 3
= 2k+ 1
2 k k 1
− +
( ) ( )!( )
!( + )!
f() t =
∞
∑
n=
0
1
1
0
k
( − 1) ( 2k)!( 4k + 3)
P t t
k k () − ≤ ≤
2 + 1 2 + 1 1 1
2 k!( k + 1)!

21.1.11 Expansion of Polynomials
k
If q t c x is an arbitrary polynomial, then where =
m()= k
qm() t = c0P0() t + c1P1() t + + cmPm() t L cn ∫
Example
To find P 2n(0) we use the summation
with k = 0. Hence
Example
m
∑
k=
0
1
⎛ 1
n+ ⎞
qm() t Pn() t dt = 0, n = 012 ,, L.
⎝ 2⎠
−1
1
q t P t dt = 0 m < r
∫ m() r()
, .
−1
∫
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1
n k
P t
k n k n k t
n
( 1)
( 1) ( 2 2 1)!
2n () = 2n1 2 ( 2 )!( 1)!(
)!
− − + −
−
+ − −
P
If q m(t) is a polynomial of degree m and m < r, then
To evaluate P for m ≠ 0 we must consider the two cases: m = odd and m = even.
m t dt
0
(a) m = even and m ≠ 0
()
The result is due to the orthogonality principle.
(b) m = odd and m ≠ 0. From the relation (see Table 21.2)
with t = 0 we obtain
∑
n k
k=
0
n
n
n
( 1) ( 2n1)! ( 1) 2n[( 2n 1)!]
( 1) ( 2n)!
( 0)
= n
n
n
2 ( n 1)!
n!
2 n[( n 1)!]
n!
2 (!) n
− −
=
−
− −
=
−
−
−
2n 2 1 2 2 2
Using the results of the previous example, we obtain
1
∫ 0
2k
1
1
1
1
1
1
1
1
∫ m
0 ∫ m
1 ∫ m ⋅
− −1 ∫ m
−1
P () t dt = P () t dt = P () t dt = P () t P0() t dt = 0
2
2
2
∫
1
1
P () t dt = [ P () t − P ()] t
2m+ 1
m m− 1 m+
1
−1
1
∫ 0
1
Pm() t dt = [ Pm− 1( 0) − Pm+
1(
0)]
2m+ 1
⎡
⎤
⎢ m− 1
m+
1 ⎥
2
2
1 1 1 1 1
= ⎢ ( − ) ( m − )! ( − ) ( m + )!
P 2 − ⎥
m()
t dt
2
2m+ 1⎢
1⎡
−1
1 1
2
⎛ ⎞ ⎤ ⎡ +
⎢
2
⎛ ⎞ ⎤ ⎥
m− m
m+
m
⎢
!
⎣⎝
2 ⎠ ⎥ ⎢ !
⎦ ⎣⎝
2 ⎠ ⎥ ⎥
⎣
⎦ ⎦
m−1
m−1
2
2
( −1) ( m− 1)!( 2m+ 1)(
m+
1)
( −1) ( m −1)!
=
=
m+ 1 m 1 m 1 m 1 m m 1 m 1
( 2m+ 1) 2
⎛ + ⎞
!
⎛ + ⎞ ⎛ − ⎞
! 2
⎛ + ⎞
!
⎛ − ⎞
!
⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠
m = odd

21.2 Legendre Functions of the Second Kind (Second Solution)
21.2.1 Second Kind:
1.
2.
3.
4.
21.2.2 Recursions
Q n(t) satisfies all the recurrence relations of P n(t).
21.2.3 Property
21.2.4 Newman Formula
21.3 Associated Legendre Polynomials
21.3.1 Definition
If m is a positive integer and –1 ≤ t ≤ 1, then
m
where P () t is known as the associated Legendre function or Ferrers’ functions.
21.3.2 Rodrigues Formula
21.3.3 Properties
1.
Q
0
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1 1+
t
= ln , t < 1;
2 1−
t
1 1 + t
Q1() t = tln
− 1, t < 1;
2 1 − t
2n+ 1 n
Qn+ 1() t = tQn() t − Qn−1(), t n = 12 , ,L
n + 1 n + 1
[ 1(
n−1)]
∑
2
2n−4k−1 Qn() t = Pn() t Q0() t −
Pn−2k−1(), t t < 1, n=
12 , , L
( 2k+ 1)(
n−k) k=
0
for [ ( n −1)]
1
see 21.1.1.
2
n
−
P () t = ( −1)
m m
n
m ( 1 − t )
Pn() t = n
2 n!
( n−m)! m
Pn() t
( n+ m)! 1
Qn() t =
2
∞
∑
1
= ( 2n+ 1)
Pn( t) Qn( x)
x − t
∫
1
−1
n=
0
Pn( x)
dx, n = 012 , , L
t − x
m
m 2 m/
2 d Pn() t
Pn() t = ( 1− t )
m = 12 , , L,
n
m
dt
2 m / 2
d
dt
n+ m
n+ m
2 n
( t − 1) , m = 12 , , L,
n; n+ m ≥ 0

2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
0
P () t = P() t
n n
m
m
m
( n− m+ 1) P () t − ( 2n+ 1) tP () t + ( n+ m) P () t = 0
21.3.4 Differential Equation
21.3.5 Schlafli Formula
where C is any regular closed curve surrounding the point t and taking it counterclockwise.
21.4 Bounds for Legendre Polynomials
21.4.1 Stieltjes Theorem
21.4.2 Second Stieltjes Theorem
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n+
1 n
n−1
2 1/ 2 m 1 m+
1 m+
1
( 1 − t ) Pn( t)
= [ Pn+ 1 ( t) − Pn−1( t)]
2n+ 1
2 1/ 2 m 1
m−1
m−1
( 1 − t ) Pn( t)
= [( n+ m)( n+ m−1) Pn−1( t) −( n− m+ 1)( n− m+ 2)
Pn+ 1 ( t)]
2n+ 1
m
2 −1/ 2 m
m−1
P () t = 2mt( 1−t ) P () t − [ n( n+ 1) −m( m−1)] P () t
n
n
m+
1 2 −1/
2
m
m
P ( t) = ( t −1) [( n−m) tP ( t) − ( n+ m) P ( t)]
n
m m
2 1/ 2 m−1
P () t = P () t + ( 2n+ 1)( t −1)
P () t
n+
1 n−1
P () t = ( t −1)
m
n
2 m/
2
1
−1
m
d Pn() t
m
dt
m m
P t P t dt = 0 k ≠ n
∫ n () k ()
1
∫
−1
m 2 2(
n+ m)!
[ Pn( t)] dt =
( 2n+ 1)(
n−m)! n
n
n−1
2 m
m
d P () () ⎡
2 ⎤
2 n t dPn t
m m
( 1−t) − 2t+ ( + 1)
− () 0
2
⎢nn
2 ⎥ Pn t =
dt dt
⎣⎢
( 1−
t )
⎦⎥
n
m 2 m/
2
∫ n n+ m+
1
C
( n m)!
Pn() t
t
jn!
( )
+
= 1 −
2π
n
2
( x − 1)
2 ( x − t)
Pn
(cos )
, , n , ,
n sin γ
4 1
≤ 2 0 < γ < π = 12L
π γ
P () t − P () t <
n n+2
4
π
n + 2
dx

21.4.3
21.4.4
21.5 Table of Legendre and Associate Legendre Functions
TABLE 21.2 Properties of Legendre and Associate Legendre Functions [P n(t) = Legendre Functions,
= Associate Legendre Functions, Q n(t) = Legendre Functions of the Second Kind]
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
dPn t
dt
()
<
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2 n
π 1 − t
2
, t < 1, n = 12 , , L
6 2 1 1
Pn+ 1()
t + Pn() t <
,
π n 1 − t
1
1− 2tx+
x
P () t =
n
P ()= t 1
0
P
2n
P n+
2
∞
∑ n
n=
0
n
= P () t x t ≤ 1 x < 1
[ n / 2] k n 2k
∑
k=
0
−
( −1) ( 2n−2k)! t
n
2 k!( n−k)!( n−2k)! n
1 ( 1) ( 2n)!
( 0)
= 2
2n2 n 2 (!) n
− ⎛ ⎞
⎜ ⎟ =
⎝ ⎠
−
2 1( 0) = 0
P ( − t) = P ( t) P ( − t) = −P
( t)
2n 2n 2n+ 1 2n+ 1
n
P ( − t) = ( −1 ) P ( t)
n
n
P() 1 = 1 n= 012 ,, L ; P(
− 1) = ( −1)
n n
n
1 d 2 n
Pn() t = ( t − 1)
= Rodrigues formula, n n
2 n!
dt
( n+ 1) Pn+ 1() t − ( 2n+ 1) tPn() t + nPn−1 () t = 0
P′ () t − 2tP′ () t + P′ () t − P () t = 0
n+ 1 n n−1 n
P′ () t = P () t + 2tP′
() t − P′ () t
n− 1 n n n+
1
P′ () t = P () t + 2tP′
() t − P′ () t
n+ 1 n n n−1
P′ () t − tP′ () t = ( n+ 1)
P () t
n+ 1 n n
tP′ () t − P′ () t = nP() t
n n−1 n
Pn′ + 1() t − Pn′ −1()
t = ( 2n+ 1)
Pn () t
2
( 1 − t ) P′ () t = nP () t −ntP
() t
n n−1 n
P () t < 1, − 1< t < 1
n
∑
n k
n
( 1)
( 1) ( 2n 2k 1)!
P2n() t = t
2n1 2 ( 2k)!( n k 1)!(
n k)!
− − + −
−
+ − −
k=
0
2
( 1− t ) Pn′ ( t) = ( n+ 1)[
tPn ( t) − Pn + ( t)]
1
∫
−1
Pn t dt ()
= 0
1
t < 1
n
[ n / 2]
= n = even ; [ n/ 2] = ( n− 1)/ 2 n = odd
2
n
2k
n = 12 , ,L
n = 012 ,, L
n = 012 ,, L
n = 012 ,, L
n = 012 ,, L
n = 012 ,, L
n = 12 , ,L
n = 12 , ,L
n = 12 , ,L
n = 12 , ,L
n = 012 ,, L
n = 012 ,, L
n = 12 , ,L
Pn ()≤ t 1 t ≤ 1
m
Pn() t

TABLE 21.2 Properties of Legendre and Associate Legendre Functions [P n(t) = Legendre Functions,
= Associate Legendre Functions, Q n(t) = Legendre Functions of the Second Kind] (continued)
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
∫
−1
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1
1
∫
−1
1
2
1
2
1
∫
−1
Pn() t Pm() t dt = 0
n≠m 2 2
[ Pn ( t)] dt =
2n+ 1
1
∫
−1
1
∫
−1
m
mm−2 m− s+
2
t Ps t dt =
m+ s+ 1 m+ s− 1 m+
1
()
( ) L(
)
( )( ) L(
)
m
m−1 m−3 m− s+
2
t Ps t dt =
m+ s+ 1 m+ s− 1 m+
2
()
( )( ) L(
)
( )( ) L(
)
2n
tPn() t Pn−1() t dt=
2
4n−1 1
P ′ ∫ n() t Pn+ 1()
t dt =
−1
1
2n
tP′ = ∫ n() t Pn() t dt
2n+ 1
−1
1
−1
2
n = 012 ,, L
ms , = even
ms , = odd
n = 12 , ,L
n = 012 ,, L
n = 012 ,, L
2
( 1− t ) P′ () t P′ () t dt = 0
∫ n k k ≠ n
1
−1/
2
( 1 − t) P ( ) = ∫ n t dt
−1
1
∫
−1
2 2
2n+ 1
2
2nn ( + 1)
t Pn+ 1() t Pn−1() t dt = 2
( 4n − 1)( 2n+ 3)
1
2
( t − 1)
P ∫ n+ 1()
t Pn ′ () t dt =
−1
1
∫
−1
1
∫
−1
n+
1 2
n 2 (!) n
t Pn() t dt =
( 2n+ 1)!
2nn ( + 1)
( 2n+ 1)( 2n+ 3)
2 2
2 2 2 ⎡(
n + 1)
n ⎤
t [ Pn( t)] dt = ⎢ +
2
⎥
( 2n+ 1)
⎣ 2n+ 3 2n−1⎦ m
m 2 m/
2 d
Pn() t = ( 1 −t
) P t m n ()
dt
m
Pn() t
t
n
n!
n m
m d
n m
dt
t
( ) [( ) ]
1 2
= 1− 2
+
/ 2
+
2
−1
n
−m
m ( n−m)! m
Pn() t = ( −1)
Pn() t
( n+ m)! 0
P () t = P(), t
m
P () t = 0 for m> n
n n n
m
m
m
( n− m+ 1) P () t − ( 2n+ 1) tP () t + ( n+ m) P () t = 0
n+
1 n
n−1
2 1/ 2 m 1 m+
1 m+
1
( 1−
t ) Pn( t)
= [ Pn+ 1 ( t) − Pn−1( t)]
2n+ 1
n = 012 ,, L
n = 12 , ,L
n = 12 , ,L
n = 012 ,, L
n = 012 ,, L
m > 0
m+ n ≥ 0
m
Pn() t

TABLE 21.2 Properties of Legendre and Associate Legendre Functions [P n(t) = Legendre Functions,
= Associate Legendre Functions, Q n(t) = Legendre Functions of the Second Kind] (continued)
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
2 1/ 2 m 1
m−1
( 1 − t ) Pn( t)
= [( n+ m)( n+ m−1) Pn−1( t)
2n+ 1
m−1
−( n− m+ 2)
P ( t)]
© 1999 by CRC Press LLC
n+
1
m+
1 2 −1/ 2 m
m−1
P () t = 2mt( 1−t) P () t − [ n( n + 1) −m( m −1)]
P () t
1
∫
n
−1
1
∫
−1
n
m m
P () t P () t dt = 0
k ≠ n
n
k
m 2 2 ( n+ m)!
[ Pn( t)] dt =
2n+ 1 ( n−m)! m n+ m m
P ( − t) = ( −1)
P ( t)
n
P n
n
m
( ± 1) = 0 m > 0
n
1
1 ( − 1) ( 2n+ 1)!
P2n( 0) = 0 P2n+
1(
0)
= 2n2 2 (!) n
P n
P
m
( 0) = 0
n+ m = odd
( 0) = ( −1)
m ( n−m)/ 2
n
1
∫
−1
m k
Pnt Pnt 2
− t
−1
dt =
( n+ m)!
n
2 [( n− m) / 2]![( n+ m)
/ 2]!
() ()( 1 ) 0
k ≠ m
1
2 −1/
2
( 1 − t ) P = ∫ 2m
( t) dt
−1
1
2 −1
2
t( 1 − t ) P ∫ 2m+ 1(
t) dt =
−1
1
∫
t
1
0
1 ⎡ Γ(
2 + m)
⎤
⎢ ⎥
⎣ m!
⎦
1
3
/ Γ( 2 + m) Γ(
2 + m)
m!( m+
1)!
1
Pn () t dt = [ Pn− 1() t − Pn+ 1()]
t
2n+ 1
n k
( − 1) Γ ( n+ k + 1)
q
t P t dt = q+
1 ∫ n () Γ(
) ∑ k
2 k! Γ ( n− k + 1)( q+ k + 2)
1
1 2
∫
0
− /
1
∫
0
Q
0
k=
0
⎧ n / 2
⎪2(
−1)
⎪ 2n+ 1
t Pn() t dt = ⎨
⎪
⎪2(
−1)
⎩
⎪
2n+ 1
1/ 2
t Pn() t dt
1 1+
t
= ln ,
2 1−
t
( n−1)/
2
⎧
( n+
2)/ 2
⎪ 2( −1)
⎪
⎪
( 2n− 1)( 2n+ 3)
= ⎨
⎪
( n+
3)/ 2
⎪ 2( −1)
⎪
⎩(
2n− 1)( 2n+ 3)
t < 1
2
n = even
n = odd
n = even
n = odd
n
n+ m = even
q >−1
m
Pn() t

TABLE 21.2 Properties of Legendre and Associate Legendre Functions [P n(t) = Legendre Functions,
= Associate Legendre Functions, Q n(t) = Legendre Functions of the Second Kind] (continued)
58.
59.
60.
61.
62.
63.
64.
65.
References
1 1+
t
Q1() t = tln
− 1= tQ0() t − 1, t < 1
2 1−
t
2n+ 1 n
Qn+ 1() t = tQn() t − Qn−1(), t n = 12 , ,L
n + 1 n + 1
Q () t = P() t Q () t −
n n
Abramowitz, M. and I. S. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New
York, NY, 1965.
Andrews, L. C., Special Functions for Engineers and Applied Mathematicians, MacMillan Publishing
Co. New York, NY. 1985.
Hochstadt, H., The Functions of Mathematical Physics, Dover Publications Inc., New York, NY, 1986.
McLachlan, N. W., Bessel Functions for Engineers, 2nd Edition, Oxford University Press, London, 1961.
Sansone, G., Orthogonal Functions, Interscience Publishers, New York, NY., 1959.
© 1999 by CRC Press LLC
0
[ 1 ( n−1)]
2
∑
k=
0
1 1 Pn( x)
Qn() t =
dx
2 ∫−1
t−x n = 012 ,, L
Q′ () t − 2tQ′ () t + Q′ () t − Q () t = 0
n+ 1 n n−1 n
Qn′ + 1() t − tQn′ () t − ( n+ 1) Qn() t = 0
Qn′ + 1() t − Qn′ −1()
t = ( 2n+ 1)
Qn() t
2n−4k−1 Pn−2k−1(), t t < 1
( 2k+ 1)(
n−k) n+
1
Q ( − t) = −Q ( t), Q ( − t) = ( − 1) Q ( t), n=
12 , , L
0 0
n
n
m
Pn() t