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
- References:
- integration
- From: "dimitris" <dimmechan@yahoo.com>
- integration