Page 60 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 2: Legendre’s Polynomials

Self Assessment Notes

5. Obtain the first three terms in the expansion of the function

0 1 x 0
f ( ) =
x
x 0 x 1
in terms of Legendre’s Polynomials and show that

1 3 25
f ( ) = P 0 ( ) P 1 ( ) P 2 ( ) ...
x
x
x
x
4 4 48
x
Prove that all the roots of P n ( ) 0 are distinct
x
Solution: If the roots of P n ( ) 0 are not all different, then at least two of them must be equal.
Let be their common value. Then
P n ( ) = 0 (i)

dp
and P ( ) = 0 Here P
n dx

Since P n ( ) is the solution of Legendre’s equation
x
2 d 2 dP ( )
x
(1 x ) 2 P n ( ) 2x n ( n n 1)P n ( ) = 0 ...(ii)
x
x
dx dx
Differentiating (ii) r times by Leibnitz’s theorem, we get
2 d r 2 r d n 1 r dr
x
x
x
(1 x ) P n ( ) 2x c 1 P n ( ) 2 c 2 P n ( )
dx r 2 dx n 1 dx r
d r 1 r d r dr
x
2 x P n ( ) 1.c 1 P n ( ) ( n n 1) P n ( ) = 0
x
x
dx r 1 dx r dx r
r
x
2 d r 2 d r 1 d P n ( )
x
x
or (1 x ) r 2 P n ( ) 2x r C 1 r 1 P n ( ) 2r C 2r C ( n n 1) = 0 ...(iii)
dx 1 dx 2 1 dr
Putting r 0, x
d 2 d
2
x
x
(1 ) P n ( ) 2 P n ( ) ( n n 1) ( ) = 0 ...(iv)
P
n
dx 2 dx
x x
d
Since P n ( ) 0 and P n ( ) 0,so
x
dx x
2
x
d P ( )
n
dx 2 = 0 ...(v)
x
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