Page 216 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 13: Orthogonality of Solutions

Thus we have obtained infinity many orthogonal sets corresponding to each fixed value of n. Notes
If a function is represented by generalized Fourier Bessel series

f(x) = C J mn x , for n fixed ...(iii)
m n
m 1
b
1
then C = x f ( )J x dx , m 1,2...
x
m 2 n mn
J x
n mn a
Since p(x) = x, mn mn
a

a
2
where J n mn x = x J n 2 mn x dx ...(iv)
0
2
To bind J n mn x ,

let us proceed as follows:
),
Multiplying (i) by 2x J n ( x we have

n 2 1
1
1
2
)
2x J n ( x x J n ( x ) x 2 2x J n ( x ) ( x ) = 0
J
n
x
2 2 2 2
1
x
or x J n ( ) x n J n ( x ) = 0
Integrating over the limits 0 to a, we have
a
2
x J 1 n ( x ) = 2 2 n 2 J 2 n ( x ) dx
x
0
Integrating R.H.S. by parts, we have
a a a
2
2
x J 1 n ( x ) = 2 2 n J 2 n ( x ) 2 2 x J 2 n ( x )dx ...(v)
x
0 0
0
From the following recurrence formulas for J ( ), we have
n
d n n
µ
µ
µ J n ( ) = µ J ( )
dµ n 1
n d n 1 n
µ
µ
µ
or µ J n ( ) n µ J n ( ) = µ J n 1 ( )
dµ
Multiplying both sides by n+1
d
J n ( ) n J n ( ) = µ J ( )
µ
d n 1
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