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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

( ) ( )=