Page 16 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

,
Unit 1: Bessel s Functions

Notes
2r n
( 1) r x
n
= ( 1)
(n r 1) (r 1) 2
r 0
n
or J (x) = ( 1) J (x) ...(26)
n n
Thus J (x) is not independent of J (x)
n n
,
1.3.2 Solution of Bessel s Differential Equation when n is a Non-negative
Integer
We had seen that when n is not an integer there are two independent solutions i.e. J (x) and J (x).
n n
When n is a non-negative integer
n
J (x) = ( 1) J (x) ...(26)
n n
And so it is dependent on J (x). To find a second solution we introduce Neumann Function
n
J ( )cos v J ( )
x
x
Y (x) = v v ...(29)
v sin v
,
If v is not an integer, then Y (x) and J (x) form a general solution of the Bessel s equation. If v is
v v
a non-negative integer, then from equation (26), equation (29) becomes an indeterminate form.
To calculate the limit of (29) for v n, differentiate both the numerator and denominator with
respect to v. Then setting v n, we have

sin J ( ) cos J ( ) J v ( )
x
x
x
lim
x
Y n ( ) limY ( ) = n cos
x
n
x
x
1 J ( ) ( 1) n J ( )
= ...(29a)
n n
Now from equation (21)
( 1)s x 2s
J (x) =
( s 1) s 2
s 0
J ( ) ( 1)s x 2s x 2 log x 2 ( s 1) x
x
= 2
s 2 ( s 1) (u s 1) 2
s 0
x
s
( 1) ( /2) 2 x
= log ( s 1) ]
s s 1 2
s 0
where
( s 1)
( s 1) = ...(30)
( s 1)

x
J ( ) ( 1) s x 2 n 2s x
thus lim = log (n s 1) ...(31)
n s n s 1 2
s 0

LOVELY PROFESSIONAL UNIVERSITY 9