Page 18 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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Unit 1: Bessel s Functions

Hankel Functions: The Hankel Function, or the Bessel Functions of the third kind are defined by Notes

x
H (1) ( ) = J (x) + i Y (x)
H (2) ( ) = J (x) + i Y (x)
x
Prove that

1/2
2
J (x) = sinx ...(29)
1/2
x
Proof: J (x) is given by
n
( 1) s x 2s n
J (x) =
n (n 1 s ) (s 1) 2
s 0

( 1) s x 2s 1/2
So J (x) =
1/2 (s 3/2) (s 1) 2
s 0

1/2 s 2s
x ( 1) x
=
2 (s 3/2) (s 1) 2
s 0
Expanding

1 o 2 4
x 2 1 x 1 x 1 x
J (x) = ....
1/2 3 5 7
2 ( 2 ) (1) 2 ( 2 ) (2) 2 ( 2 ) (3) 2
1 2 4
x 2 1 1 x 1 x
or J (x) = 1 ...
1/2
2 ( 3 2 ) 3 2 (2) 2 3 2 5/2 (3) 2
1 2 4
x 2 1 x (2) 2.2 x
= 1 ...
1
2 ( 1 2 ) ( ) 3 2 2 1 2 3 5 (2) 4
2
1 2 4
x 2 1 x x
= 1 ...
2 ( 1 2 ) ( ) 2 3 1 2 3 4 5
1
2
1 3 5
x 2 2 1 x x
= x ...
2 ( 1 2 ) x 3 5
1
1 2 2
= sinx
( 1 2 ) x
Here 1 2

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