Page 45 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Self Assessment

1. Find
x
x
x
P 1 ( ), P 2 ( ), P 3 ( )
from Rodrigue formula
2.3 Generating Function for Legendre Polynomials

In the following we will show that P n ( ) is the coefficient of h in the expansion of
n
x
1
(1 2xh h 2 ) 2

for | | |,| | |
h
x
1
n
x
i.e. (1 2xh h 2 ) 2 = h P n ( ) ...(A)
n 0
1 1
Now (1 2hx h 2 ) 2 = [1 h (2x h )] 2
1 1 3 1 2 2
= 1 ( h )(2x h ) . h (2x h ) ...
2 2 2 2
1.3...(2n 3) n 1 n 1
... h (2x h )
2.4.6(2n 2)
1.3. ...(2n 1) n n
h (2x h ) ...
n
2.4.6....(2 )
n
Therefore the coefficients of h are
1.3.5...(2n 1) 2 1.3.5. ...(2n 3) n 2 1.3.5...(2n 5) n 4
= (2 ) (2 ) n 1 C n 2 (2 ) ...(B)
x
x
x
C
n
2.4.6...(2 ) 2.4.6 ...(2n 2) 1 2.4.6...(2n 4) 2
n
1.3.5...(2n 1) n 2n x n 2 (2 )(2n 2)(n 2)(n 3)
= x (n 1) 2 4 ...
n 2n 1 2 (2n 1)(2n 3) 2 2
1.3.5...(2n 1) n ( n n 1) n 2 ( n n 1)(n 2)(n 3) n 4
= x x x ...
n 2(2n 1) 2.4.(2n 1)(2n 2)
= P n ( ) ...(C)
x
Thus
1 h P ( )
n
x
(1 2xh h 2 ) 2 = n
n 0
Where P n ( ) is given by
x
1.3.5...(2n 1) n ( n n 1) n 2 (n 2)(n 1) n (n 3) n 4
P n ( ) = n x 2(2n 1) x 2.4.(2n 1)(2n 2) x ... ...(D)
x

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