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.