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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
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