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Re: Hankel transform question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82437] Re: Hankel transform question
  • From: danl at wolfram.com
  • Date: Sat, 20 Oct 2007 05:52:16 -0400 (EDT)
  • References: <ff9t5n$6cu$1@smc.vnet.net>

On Oct 19, 4:25 am, Jim Rockford <jim.rockfo... at gmail.com> wrote:
> I'm getting some disparate (or at least different looking) results for
> a particular Hankel transform related integral in Mathematica 5.2
> versus Mathematica 6.01.
>
> The integral I'm dealing with is the order-V Hankel transform of a
> constant ( f(r) = 1, say )
>
> Define
> g[s_] = s BesselJ[V,w s]
>
> and I need  the output for
>
> Integrate[g[s],{s,0,Infinity}]
>
> Both versions of Mathematica complain about not being able to verify
> convergence.  I believe this integral should give back a Dirac delta
> function.  Here are the outputs:
>
> *****************
>    version 5.2
> *****************
> V/w^2      ( with V and w real and positive)
>
> *****************
>   version 6.01
> *****************
> Integrate::idiv:  Integral of s BesselJ[V,w s] does not converge on
> (0,Infinity) >>
>
> (1) First of all, what accounts for the differences in output between
> the two versions of Mathematica?
> (2) Second, I know that   Integrate[s BesselJ[0,s],{s,0,Infinity}]
> should give Delta(s), but Mathematica 5.2  gives back the answer 0,
> and Mathematica 6.01 again balks and says that the integral doesn't
> converge.
>
> Is there any way for me to get Mathematica to "look for" Delta
> functions, instead of trying to grind out a numerical integration?
>
> Thanks,
> Jim


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.

Daniel Lichtblau
Wolfram Research



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