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Re: integrate log*sinc

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109210] Re: integrate log*sinc
  • From: sashap <pavlyk at gmail.com>
  • Date: Sat, 17 Apr 2010 06:04:01 -0400 (EDT)
  • References: <hq9c10$qe2$1@smc.vnet.net>

On Apr 16, 4:52 am, pimeja <sed.n... at gmail.com> wrote:
> Hi All,
>
> For Integrate[Log[x] Sin[x]/x, {x, 0, \[Infinity]}] Mathematica
> returns -EulerGamma \[Pi].
> How to proof this analytical?

In[8]:= Limit[D[Integrate[x^(s - 1)*Sin[x], {x, 0, Infinity},
       Assumptions -> -1 < s < 1], s], s -> 0]

Out[8]= -((EulerGamma*Pi)/2)

>
> I've tried to use residue theory but it seems not suitable since
> integrand has pool of second order in zero (for Jordan lema should be
> first order only). Substitution x=Exp[y] returns even more strange
> result.
>
> Thanks in advance.



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