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MathGroup Archive 1998

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Challenging Integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg12747] Challenging Integration
  • From: quartier at umnw.ethz.ch (Robin Quartier)
  • Date: Wed, 10 Jun 1998 03:04:10 -0400
  • Organization: ETH Zurich
  • Sender: owner-wri-mathgroup at wolfram.com

Good Morning everybody.

I have the following distribution:

epsilon[x_]:=(a*b/(b^2 - a^2))*(b*Exp[-a*x] - a*Exp[-b*x])

a,b >0

This function is kind of symmetric, in the sens that it is stable for
the permutation of a and b.
It is a distribution, normalised on the positive half-axis

The integration I want to make is:

Integrate[epsilon[x]*Log[epsilon[x]],{x,0,Infinity}]

Mathematica does not want to do it.

I can get a primitive, 

Integrate[epsilon[x]*Log[epsilon[x]],x] 

But I don't think that the output I get is correct, because it is not
symmetricalfor the permutation of a and b anymore. Namely, it contains
an Hypergeometric function that diverges for a>b, and converges for
a<b, which can not be.

Ideas, questions and comments welcomed!

Best regards

Robin Quartier


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