Page 218 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 13: Orthogonality of Solutions

and is therefore a Sturm-Liouville equation with Notes
x
R ( ) = 1 x 2 , ( ) 1 and ( ) 0Q x
P
x
Here no boundary conditions are needed to form a Sturm-Liouville problem on the internal (
1, 1) since R = 0 when x = 1.
Further we know that Legendre Polynomials
x
P ( ), (n 0,1,2,...)
n
are the solutions of the problem, hence they are the eigenfunctions and since they have continuous
derivatives, therefore it follows that {P (x)}, n = 0, 1, 2, ... are orthogonal on the interval 1,
n
x 1 with respect to the weight function

1
p = 1, i.e., P m ( ) P n ( ) dx 0 if (m n )
x
x
1
1
2 2 1
x
and P = P ( ) dx , m 0,1,2,...
m m
2m 1
1
If g (x), g (x), ..... are eigenfunctions which are orthogonal on the interval a x c with respect to
0 1
the weight function p(x), and if a given function f(x) can be represented by a generalised Fourier
series
x
f(x) = C g n ( )
n
n 1
b
1
x
x
x
f
then, c = p ( ) ( )g ( )dx (m 0,1,2,...)
n 2 m
g n a
b
2 2
x
where g m = p ( ) g m ( )dx
x
a
(c) Orthogonality of Hermite Polynomials
The Hermite polynomials H (x), given by
n
n
2 d e x 2
n x
H (x) = ( 1) e
n n
dx
are orthogonal with respect to the weight function ( )p x e x 2 on the interval x .
n
H ( )H ( )e x 2 dx = ( 1) n H ( ) d e x 2 dx
x
x
x
m n m n
dx
d n 1 e x 2
n
x
= ( 1) H m ( ) n 1
dx

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