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