Page 197 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
results of this chapter. For example, the Bessel functions J v x satisfy the Sturm-Liouville
2
equation, with p(x) = x, q(x) = v /x and r(x) = x. They satisfy the orthogonality relation
a
xJ v x J v x dx 0
0
if and are distinct eigenvalues. Using the regular endpoint condition J v a = 0 and the
singular endpoint condition at x = 0, the eigenvalues, that is the zeros of J (x), can be written as
v
a = a , a..., so that = for i = 1, 2, ..., and we can write
1 1 2 i
x
f ( ) a J ( x ),
i
i
i 1
with
2 a
x
f
a i 2 2 xJ ( x ) ( )dx
i
a { ( a )} 0
J
i
Example: Show that the functions g = cos mx, m = 0, 1, 2, ... form orthogonal set of
m
functions on the interval < x > and determine the corresponding orthonormal set of functions.
Solution: We have, for m n
cos mx cos nx dx

2 cos mx cos nx dx
0
cos[(m n ) ] cos[(m n ) ] dx
x
x
0
x
x
sin[(m n ) ] sin[(m n ) ]
0
(m n ) m n
0
Hence the given functions g = cos mx, m = 0, 1, 2, .... are orthogonal set of functions.
m
Now the norm of g is
m
1 2
2
g m cos mx cos mx dx
1 2
2
2 cos mx dx
0
2 when m 0
and when m 1, 2, 3, ....
Hence the orthonormal set is
1 cos x cos 2x cos 3x
, , , ,...
2
Self Assessment

4. Show that the functions 1, cos x, sin x, cos 2x, sin 2x, ... form an orthogonal set on an interval
x and obtain the orthonormal set.

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