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MathGroup Archive 2007

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integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72738] integration
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Wed, 17 Jan 2007 06:40:43 -0500 (EST)

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


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