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Re: Integrate[ x^2 * Erfc[a x] *(BesselJ[1, b x])^2 ,

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100914] Re: [mg100842] Integrate[ x^2 * Erfc[a x] *(BesselJ[1, b x])^2 ,
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 18 Jun 2009 04:52:22 -0400 (EDT)
  • References: <200906170149.VAA25659@smc.vnet.net>

ralf.schaa wrote:
> Hi group,
> 
> The integral
> 
> Integrate[ x^2  * Erfc[a x] *(BesselJ[1, b x])^2 , {x, 0, Inf}]
> 
> has a solution in Mathematica (in terms of HypergeometricPFQ)
> 
> My question: how did Mathematica know?
> 
> I tried the usual suspects:
> i) for example Gradshteyn and Ryzhik (1965), p.769, 6.784 is very
> close, but not close enough
> ii) Abramowitz & Stegun hasn't got it on the menu either
> 
> Then how about expressing Erfc[a*x] in terms of...
> Erfc(z)=-1/(sqrt(pi)) * IncompleteGamma[1/2,x^2]+1 (Grads.+Ryzh. p.
> 942)
> ...and hoping to get lucky? ... no alas!
> 
> sorry, this is a cross-post (also in sci.math.symbolic)...
> 
> Help!
> -Ralf

As was suggested in the s.m.s thread, MeijerG functions are used in a 
convolution approach. Here isn your example with certain assumptions on 
parameter ranges.

Integrate[x^2  * Erfc[a*x] * BesselJ[1,b*x]^2, {x,0,Infinity},
   Assumptions->{b>0,a>0}]

In this case, up to multiplicative constants the Erfc becomes

MeijerG[{{}, {1}}, {{0, 1/2}, {}}, a^2*x^2]

and the BesselJ square becomes

MeijerG[{{1/2}, {}}, {{1}, {-1, 0}}, b^2*x^2]

The code subsequently uses these MeijerG representations in convolving 
to get an integral from 0 to infinity.

Daniel Lichtblau
Wolfram Research


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