§7 Hankel Transform
[ Hankel transform and its inversion formula ]
_{}The v -order Hankel transform of
_{}
The inversion formula of the v -order Hankel transform is:
_{}
where J _{v} ( x ) is a Bessel function .
[ Hankel transformation table ]
_{} |
v |
_{} |
_{} |
>_{} |
_{} |
_{} |
0 |
_{} |
_{} |
0 |
_{} |
_{} |
>_{} |
_{} |
_{} |
>_{} |
_{} |
_{} |
0 |
_{} |
_{} |
0 |
_{} |
_{} |
1 |
_{} |
_{} |
1 |
_{} |
_{} |
1 |
_{} |
_{} |
0 |
_{} |
_{} |
0 |
_{} |
_{} |
1 |
_{} |
_{} |
0 |
_{} |
[ Finite Hankel transform and its inversion formula ]
_{}The finite Hankel transform of
_{}
where pi_{} is a root _{of} the equation ._{}
The inversion formula of the finite Hankel transform is:
_{}
where is the sum of all positive roots of the equation ._{}_{}
[ Finite Hankel transformation table ]
_{}
_{}
where is the sum of all positive roots of the equation ._{}_{}
_{} |
v |
_{} |
_{} |
>_{} |
_{} _{} |
c |
0 |
_{} _{} |
_{} |
0 |
_{} _{} |
_{} |
>_{} |
_{} _{} |
_{} |
0 |
_{} _{} |