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Poularikas A. D. Legendre Polynomials
The Handbook of Formulas and Tables for Signal Processing.
Ed. Alexander D. PoularikasBoca Raton: CRC Press LLC,1999
1999 by CRC Press LLC
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1999 by CRC Press LLC
21Legendre Polynomials21.1 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 LegendreFunctions
References
21.1 Legendre Polynomials
21.1.1 Definition
21.1.2 Generating Function
w(t,s) = w(t,s)
21.1.3 Rodrigues Formula
21.1.4 Recursive Formulas
1.
2.
P tn k t
k n k n k
nn n
n n
n
k
n k n k
n( )
( ) ( )!
!( )!( )!
[ / ]/
( ) /
[ / ]
=
=
=
0
2 21 2 2
2 2
22
1 2
even
odd
w t sst s
P t s s
P t s s
n
n
n
n
n
n
( , )
( )
( )
= +
=
=
=
1
1 2
1
1
2
0
1
0generating function
P tn
d
dtt n
n n
n
n
n( )!
( ) , ,= =1
21 0 1 22 L
( ) ( ) ( ) ( ) ( ) , ,n P t n t P t n P t nn n n
+ + + = =+ 1 2 1 0 1 21 1 L
= + = =+P t t P t n P t P t P t n
n n n11 0 1 2( ) ( ) ( ) ( ) ( ( ) ( ) , , ,derivative of L
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3.
4.
5.
6.
Figure 21.1 shows a few Legendre functions.
21.1.5 Legendre Differential Equation
If is a solution to the second-order DE
For
TABLE 21.1 Legendre Polynomials
FIGURE 21.1
t P t P t n P t nn n n = =( ) ( ) ( ) , ,1 1 2 L
= + =+ P t P t n P t nn n n1 1 2 1 1 2( ) ( ) ( ) ( ) , ,L
( ) ( ) ( ) ( )t P t nt P t n P t n n n
2
11 =
P t P t t 0 11( ) ( )= =
P0
1=
P t1
=
P t2
32
2 12=
P t t3
52
3 32=
P t t4
358
4 308
2 38= +
P t t t 5 638 5 708 3 158= +
P t t t 6
23116
6 31516
4 10516
2 516= +
P t t t t 7
42916
7 69316
5 31516
3 3516= +
y P x nn
= =( ) , , , )( 0 1 2 L
( ) ( )1 2 1 02 + + =t y ty n n y
t dd
dyd
n n y= + + =cos :
sinsin ( )
1 1 0
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Example
From (21.1.4.4) and t= 1 implies For
For and so forth. Hence
21.1.6 Integral Representation
1. Laplace integral:
2. Mehler-Dirichlet formula:
3. Schlfli integral:
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.
21.1.8 Asymptotic Representation:
= fixed positive number
21.1.9 Series Expansion
Iff(t) is integrable in [1,1] then
0 1 1 1 11 1
= = nP nP P Pn n n n( ) ( ) ( ) ( ).or n P P= = =1 1 1 11 0, ( ) ( ) .n P P= = =2 1 1 1
2 1, ( ) ( ) P
n( ) .1 1=
P t t t d n
n( ) [ cos ]= + 1 102
Pn
d nn(cos )
cos( )
cos cos, , , ,
= +
< < = 2 2 0 0 1 2012
P tj
z
z tdz
n
C
n
n n( )
( )
( )=
+
1
2
1
2
2
1
{[ ( )] ( )}/121 22 1n P t
n+
=
= + =
1
1
1
1
2
0
22 1
0 1 2
P t P t dt
P t dt n
n
n m
n
( ) ( )
[ ( )] , ,L
n n
tn
P t n( ) ( ) , ,=+
=2 1
20 1 2L
Pn
n nn(cos )
sinsin ,
+
+
2 1
2 4,
f t a P t tn n
n
( ) ( )=