Page 30 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

,
Unit 1: Bessel s Functions

Solution: From example above, we have Notes

1
x
J (x) = cos( sin )dx
0
0
1
ax
ax
e j (bx )dx e cos(bx sin )d dx
0
x
0 0 0
1 ax
e cos(bx sin )dx d
0 0

1 ax e ( i bx sin ) e ( i bx sin )
e dx d
2
0 0

1 (a ib sin )x ( i sin )x
b
a
e e dx d
2
0 0
ib
a
1 e (a ib sin )x e ( sin )x
d
2 (a ib sin ) (a ib sin )
0 0
1 1 1
d
2 a ib sin a ib sin
0
1 2 d
a

2 a 2 b 2 sin 2
0
/2 2
a cosec d
2. 2 2 2
b a cosec
0
/2 2
a cosec d
2.
(a 2 b 2 ) a 2 cot 2
0
/2
a 1 1 a cot
2. cot
a {(a 2 b 2 )} {(a 2 b 2 )}
0

2 1 1
cot 0 cot
{(a 2 b 2 )}

1

{(a 2 b 2 )}

LOVELY PROFESSIONAL UNIVERSITY 23