Page 17 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes The expression for J (x) is from (27)
n
n 1 s 2s s 2s
( 1) x ( 1) x
J (x) = ...(27)
s 0 s 1 (s 1) 2 s n s 1) (s 1 ) 2
As you know from the properties of gamma functions

(x) (1 x) = ...(32)
sin x
From (27) we obtain
2s 2s
x x
( s )sin( ) s
2 2
= ...(33)
( s 1)
Differentiating (33), we see that for 0 s n

x
d ( /2) 2s ( s )sin( ) s 1 2s
= x ( ) s 1 ( s )sin( ) s
d 2
n
x
cos( m ) 1 log( /2)sin( ) s
n
n 2m
x
= (n m )cos(n m )
2
J ( )
x
Therefore as n, v tends to
x
n 1 n n 2s s 2s
( 1) ( s )( /2) ( 1) ( /2)
x
x
log( /2) ( n s 1)
x
(s 1) s ( n s )
s 0 s n
n 1 (n s 1) x n 2s x
m
= ( 1) n ( 1) n 1 ( 1) ( 1 2 )x n 2s log (s 1) ...(34)
s 2 2
s 0 s n
Using (31) and (34) we get for Neumann Function Y (x) with n being a non-negative integer the
n
following
n 2s
2 x 1 s x (s 1) (n s 1)
x
Y (x) = J n ( )log ( 1)
n 2 2 s !(n s )!
s 0
n 1 n 2s
1 (n 1 s )! x
! s 2 ...(35)
s 0
For n = 0, the last term does not appear. Thus J (x) and Y (x) form the general solution.
n n
Thus we see that the Neumann Function Y (x) defined by the relation
n
x
x
J ( )cos J ( )
Y (x) =
n sin
converges uniformly to Y (x) given by equation (35) as v n is any bounded closed domain in
n
the complex x plane except for the origin. Formula (35) for Y (x) is known as Hankel Formula.
n
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