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Integrate[ x^2 * Erfc[a x] *(BesselJ[1, b x])^2 , {x, 0, Inf}]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100842] Integrate[ x^2 * Erfc[a x] *(BesselJ[1, b x])^2 , {x, 0, Inf}]
  • From: "ralf.schaa" <ralf.schaa at gmail.com>
  • Date: Tue, 16 Jun 2009 21:49:41 -0400 (EDT)

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


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