Page 23 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Thus in the expansion of t,
x 1 æ 1ö 2 1 3 1 n n 1
e t = J + t - ÷ J + t 2 J 2 t 3 J 3 ... ..... t (1) n J n ...
ç
2 t 0 è t ø 1 t t t
x
x
x
= J 0 ( ) t J 1 ( ) J 1 ( )] t 2 J 2 ( ) J 2 ( ) ...
x
x
n
t
= t J n ( )
n
Here we have used the result J (x) = ( 1) J (x)
n
n n
,
(A) Trigonometric Expansions involving Bessel s Functions
Show that
(a) cos (x sin ) = J (x) +2 cos 2 J +2 cos 4 J + ....
0 2 4
(b) sin (x sin ) = 2 sin J + 2 sin 3 J + ....
1 3
(c) cos (x cos ) = J 2 cos J + 2 cos 4 J ....
0 2 4
(d) sin (x cos ) = 2 cos J 2 cos 3 J + 2 cos 5 J + ...
1 3 5
(e) cos x = J 2 J + 2J 2J + ...
0 2 3 4
(f) sin x = 2 J 2 J + 2 J ....
1 3 5
Proof: We know from generating function that
x 1 n
x
e t = t J n ( )
2 t
n
n
n
x
x
= t J n ( ) t J n ( )
n 0 n 1
n
x
x
= t J n ( ) t n J n ( )
n 0 n 1
n
J
x
= J 0 (t n ( 1) t n ) ( ) {since J (x) = ( 1) J (x)}
n
n
n
n 1
Thus,
x 1
e 2 t = J (t t 1 )J (t 2 t 2 ) J (t 3 t 3 ) J ...(i)
t 0 1 2 3
i
n
Let us put t = e , t = e in
x
then (i) becomes e 2 (e i e i ) = J 0 (e i e i )J 1 (e 2i e 2i )J 2 (e 3i e 3i )J 3 ... (ii)
1 in in
Since cos n = e e
2
1 in in
sin n = e e
2i

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