Page 24 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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Unit 1: Bessel s Functions

So (ii) may be written as Notes
i
e ix sin = J + 2 sin J + 2 cos 2 J + 2 i sin 3 J + ... ...(iii)
0 1 2 3
comparing real and imaginary part on both sides
we have

(a) cos (x sin ) = J + 2 cos 2 J + 2 cos 4 J + ... ...(iv)
0 2 4
(b) sin (x sin ) = 2 sin J + 2 sin 3 J + ... ...(v)
1 3
Replacing by /2 in (iv) and (v) and using sin sin ( /2 ) = cos , we get
(c) cos (x cos ) = J 2 cos 2 J + 2 cos 4 J ... ...(vi)
0 2 4
(d) sin (x cos ) = 2 cos J 2 cos 3 J + 2 cos 5 J ... ...(vii)
1 3 5
Replacing by 0 in (iv) and (vii) we get
(e) cos x = J 2 J + 2 J ... ...(viiii)
0 2 4
and

(f) sin x = 2 J 2 J + 2 J ... ...(ix)
1 3 5
1.6 On the Zeros of Bessel Functions J (x)
n
We know that Bessel function J (x) satisfies the equation
n
2
x
x
d J ( ) d J ( )
x
x n x n (x 2 n 2 ) ( ) 0
J
n
dx 2 dx
Here n is a positive integer
let us put x = ,
d J n 1 d J n
dx d
2
2
d J n 1 d J n
dx 2 2 d 2
So equation (1) becomes
2
2 d J n ( ) d J n ( ) ( 2 2 n 2 ) ( ) 0 ...(ii)
J
d 2 d n
which may be written as

d d J n ( ) n 2 2 J ( ) 0
d d n ..(iii)

n 2
let us put R = , P = , Q =

d d J n ( )
Then R Q 2 p J n ( ) ...(iv)
d d

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