Page 62 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 2: Legendre’s Polynomials

Notes
P
x
x
x
 Rodrigue’s formula for Legendre polynomials help us to find a few P n ( ) i.e. P 0 ( ), ( ),
1
x
P 2 ( ),..... .
P
x
x
 Orthogonal properties of P n ( ) are obtained. It is seen that { ( )}n 0, 1, ... form a
n
complete set in the range 1 x 1.
 Just as Fourier series we show that a function in the range 1 x 1 is expanded in terms
of P n ( )’ .
s
x
2.8 Keywords
Regular singular points of Legendre equations are x 1 and x .

x
n
Legendre polynomial P n ( ) is a terminating series with highest power of x as x .
n
Generating function of the Legendre polynomial is (1 2hx h 2 ) 1 h P n ( )
x
n 0
x
Rodrigue’s formula has been obtained and certain properties of P n ( ) are obtained in a straight
forward manner.
Recurrence relations between various Legendre’s polynomials obtained are useful in expressing
x
x
higher polynomials in terms of P 0 ( ) and P 1 ( ).
Orthogonality properties of the Legendre Polynomials obtained, help us in evaluating certain
integrals easily.

2.9 Review Questions

Show that

1. P n ( ) P n 2 ( ) (2n 1)P n 1 ( )
x
x
x
1
2n
x
2. x P n ( ) P n 1 ( )dx
x
4n 2 1
1
3. x P 9 ( ) P 8 ( ) 9 P 9 ( )
x
x
x
x
4. Show that all the roots of P n ( ) 0 are real and lie between 1 and +1.
5. Prove that
8 6
4
2
x 3x x P 4 ( ) P 2 ( ) P 1 ( )
x
x
x
35 35

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