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Re: Hankel transform question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82496] Re: Hankel transform question
  • From: Jim Rockford <jim.rockford1 at gmail.com>
  • Date: Mon, 22 Oct 2007 05:40:44 -0400 (EDT)
  • References: <ff9t5n$6cu$1@smc.vnet.net><ffcjm9$scg$1@smc.vnet.net>

On Oct 20, 6:01 am, d... at wolfram.com wrote:
> What I get in the development Mathematica kernel is simply an
> unevaluated Integrate.
>
> In[12]:= Integrate[s*BesselJ[v,w*s], {s,0,Infinity},
>   Assumptions->{v>0,w>0}]
>
> Out[12]= Integrate[s BesselJ[v, s w], {s, 0, Infinity},
>   Assumptions -> {v > 0, w > 0}]
>
> But the integral is in fact divergent. To see this, check the series
> expansion at infinity.
>
> In[13]:= InputForm[Series[s*BesselJ[v,w*s], {s,Infinity,2},
>   Assumptions->{v>0,w>0}]]
>
> Out[13]//InputForm=
> Cos[Pi/4 + (Pi*v)/2 - s*w]*
>   SeriesData[s, Infinity, {Sqrt[2/Pi]/Sqrt[w]}, -1, 3, 2] +
>   SeriesData[s, Infinity,
>     {(-1 + 4*v^2)/(4*Sqrt[2*Pi]*w^(3/2))}, 1, 3, 2]*
>     Sin[Pi/4 + (Pi*v)/2 - s*w]
>
> Notice that the "main" term is, up to scaling and phase change, of the
> form Cos[s]*Sqrt[s]. This diverges in classical integration.
>
> To get a regularized result, you can do as below.
>
> In[14]:= InputForm[Integrate[s*BesselJ[v,w*s], {s,0,Infinity},
>   Assumptions->{v>0,w>0}, GenerateConditions->False]]
>
> Out[14]//InputForm= v/w^2
>
> Offhand I do not see any reason to expect a delta function result.
> Certainly DiracDelta[s] would be incorrect, because s is the integral
> variable of (definite) integration, hence cannot appear in the result.


Thanks Daniel.  Let me add a few comments.  First, I was not expecting
an integral of f(s) to converge either, with f(s) of the form

f(s) = s^a BesselJ(v,w s)       (v is the Bessel order, w is a
transform variable)

if a > 1/2 , since we know that the Bessel function only decays like 1/
sqrt(s)  as s goes to infinity.

However, it was Mathematica itself (version 5.2) that suggested this
integral might make sense, given some sort of regularization (what is
this regularization, by the way?  Principal Value?)

In fact, from the mathworld page
http://mathworld.wolfram.com/HankelTransform.html

one can see that the delta function   Delta[w]   can be written as the
Hankel Transform of unity  (I'm ignoring 2*pi conventions):

Delta[w] = Integrate[s BesselJ[0,w s],{s,0,Infinity}]

I'm certain that this is simply the analog of the usual representation
of the Delta function as a Fourier Transform (the Hankel version is
simply in polar coordinates).

In any case, it doesn't seem that Mathematica recognizes this variant
of the Delta function.

J






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