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.