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MathGroup Archive 2007

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Re: integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72762] Re: [mg72738] integration
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 18 Jan 2007 05:50:25 -0500 (EST)
  • References: <200701171140.GAA04733@smc.vnet.net>

dimitris wrote:
> Hello.
> 
> Consider the following divergent integral
> 
> Block[{Message}, Integrate[Cos[x]/x, {x, 0, Infinity}]]
> Infinity
> 
> There is a non-integrable singularity at x=0
> 
> Series[Cos[x]/x, {x, 0, 3}]
> SeriesData[x, 0, {1, 0, -1/2, 0, 1/24}, -1, 4, 1]
> 
> In the Hadamard sense the integral converges to -EulerGamma. Indeed
> 
> Integrate[Cos[x]/x, {x, 0, Infinity}, GenerateConditions -> False]
> -EulerGamma
> 
> or
> 
> Integrate[Cos[x]/x, {x, e, Infinity}, Assumptions -> e > 0]
> (Series[#1, {e, 0, 3}] & )[%]
> (DeleteCases[#1, (a_)*Log[e], Infinity] & )[%]
> (Limit[#1, e -> 0, Direction -> -1] & )[%]
> 
> -CosIntegral[e]
> SeriesData[e, 0, {-EulerGamma - Log[e], 0, 1/4}, 0, 4, 1]
> SeriesData[e, 0, {-EulerGamma, 0, 1/4}, 0, 4, 1]
> -EulerGamma
> 
> Next, consider the function
> 
> f = x^4/(1 + Exp[-x]);
> 
> The integral does not exist in the Riemann sense. One way to get the
> Hadamard finite part is by directly removing the divergent term
> 
> Integrate[f - x^4, {x, 0, Infinity}]
> N[%]
> NIntegrate[f - x^4, {x, 0, Infinity}]
> 
> -((45*Zeta[5])/2)
> -23.33087449072582
> -23.330874489932825
> 
> So, I wonder if there is any possibility settings like below to ever
> work?
> 
> Integrate[Cos[x]/x - 1/x, {x, 0, Infinity}]
> NIntegrate[Cos[x]/x - 1/x, {x, 0, Infinity}]
> 
> Any ideas?
> Thanks!
> 
> Dimitris

Of course not, at least for Integrate. It now has a logarithmic 
divergent term at infinity. So one would have to remove that singular 
part instead of the one at the origin.

Daniel Lichtblau
Wolfram Research


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