Page 19 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Self Assessment

1. Prove that

1
2 2
J 1 ( )x = cosx
2 x
Show that when n is any integer positive or negative
n
J ( x) = ( 1) J (x)
n n
The expression for J (x) is given by
n
( 1) s x n 2s
J (x) =
n s !(n s 1)! 2
s 0
Case I let n be a positive integer. Replacing x x in the above equation we have

( 1) s x n 2s
J ( x) =
n s !(n s 1)! 2
s 0
s
( 1) ( 1) n 2s x n 2s
=
s !(n s 1)! 2
s 0
s
( 1) ( 1) 2s x n 2s
n
= ( 1)
s !(n s 1)! 2
s 0
n
x
= ( 1) J n ( ) as ( 1) 2s 1
n
Thus J ( x) = ( 1) J (x)
n n
1.4 Recurrence Formulas for J (x)
n

Some of the recurrence relations involving Bessel functions are as follows:
I. x J (x) = n J (x) x J (x),
n n n + 1
d
where J (x) = J (x)
n dx n
To prove the above relation, we start from the series expansion of J (x) as follows:
n
( 1) s x n 2s
J (x) =
n s !(n s )! 2
s 0
Differentiating it w.r.t. x and multiplying by x on both sides, we have

s
s
( 1) (n 2 ) x n 2s
x J (x) = n 2s
n s !(n s )! 2
s 0

12 LOVELY PROFESSIONAL UNIVERSITY