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Package for Bessel Transforms

• To: mathgroup at yoda.physics.unc.edu
• Subject: Package for Bessel Transforms
• From: Robert.Prus at fuw.edu.pl
• Date: Fri, 13 May 94 18:03:44 +0200

Hi,

Is there a package for Mathematica which supports Bessel transforms, i.e.
integral transforms with Bessel functions J,Y,I,K as the kernel?

Here is a simple example.

When I tried to calculate some Bessel transforms with MMA 2.2 I got strange
results:

BT[f_,x_,y_]:=Integrate[Sqrt[x] Sqrt[y] BesselJ[0,x y] f,{x,0,Infinity}]
BT[1/(Sqrt[x] Sqrt[a^2+x^2]) (Sqrt[a^2+x^2]+x),x,y]

The result should be Sqrt[y] (1+Exp[-a y]) (F.Oberhettinger, Tables of Bessel
Transforms or A.Erdelyi, et.al., Tables of Integral Transforms). MMA 2.2
produces:
                                     2    2
                          (x + Sqrt[a  + x ]) BesselJ[0, x y]
Out[2]= Sqrt[y] Integrate[-----------------------------------,
                                           2    2
                                     Sqrt[a  + x ]

>     {x, 0, Infinity}]

If I set a=0 and next try again to calculate Bessel transform:

a=0
BT[1/(Sqrt[x] Sqrt[a^2+x^2]) (Sqrt[a^2+x^2]+x),x,y]//PowerExpand

I get result, which is differ from one given by Oberhettinger:

           2
Out[4]= -------
        Sqrt[y]

For other a's MMA cannot calculate Bessel transform.

However, because of fact, that in that case Bessel transform is self reciprocal,
i.e. BT[BT[f[x],x,y],y,z] should be equal f[z], I can check if function given
by Oberhettinger is valid:

BT[Sqrt[y] (1+Exp[-a y]),y,x]-1/(Sqrt[x] Sqrt[a^2+x^2]) 
(Sqrt[a^2+x^2]+x)//.(Sqrt[1+x^2/a^2]->Sqrt[a^2+x^2]/a)//Together//Simplify

produces result:
                2      3    2       2    2     2       2    2
         a x - a  x - x  - a  Sqrt[a  + x ] - x  Sqrt[a  + x ]
Out[21]= -----------------------------------------------------
                                   2    2 3/2
                         Sqrt[x] (a  + x )

but

BT[1/Sqrt[y] (1+Exp[-a y]),y,x]-1/(Sqrt[x] Sqrt[a^2+x^2]) 
(Sqrt[a^2+x^2]+x)//.(Sqrt[1+x^2/a^2]->Sqrt[a^2+x^2]/a)//Together//PowerExpand

produces:

Out[25]= 0

So, I found an error in Tables of Bessel Transforms of F.Oberhettinger, but
I cannot still calculate some Bessel transforms.

There are standard packages supporting Fourier Transforms and Laplace
Transforms but I didn't find any package supporting Bessel Transforms even
at MathSource.

Thank you in advance for any help.

Robert Prus
robert at fuw.edu.pl
Institute of Theoretical Physics
Warsaw University
Poland