Re: Integrate[s^s(1-s)^(1-s)Sin[Pi s],{s,0,1}]

*To*: mathgroup at smc.vnet.net*Subject*: [mg76247] Re: Integrate[s^s(1-s)^(1-s)Sin[Pi s],{s,0,1}]*From*: Peter Pein <petsie at dordos.net>*Date*: Fri, 18 May 2007 06:19:09 -0400 (EDT)*References*: <f2bs3m$ga6$1@smc.vnet.net><f2emip$2sq$1@smc.vnet.net> <f2h8pe$tb$1@smc.vnet.net>

janos schrieb: > On May 16, 12:31 pm, CKWong <CKWon... at gmail.com> wrote: >> Are you serious? We don't even know how to do >> >> Integrate[s^s, {s, 0, 1}] > > Yes, I realized (only now :() this as well. > It is mystic how the Taylor series of the original integrand around > 1/2 behaves. > It is quite nice up to 7, and fills up a few screens if you are > interested in the Taylor series up to say 9. > Any idea of the reason? > > Janos > > Hi Janos, I've seen worse expansions... << "DiscreteMath`" f[s_] = s^s*(1 - s)^(1 - s)*Sin[Pi*s]; Timing[Expand /@ Series[f[s], {s, 1/2, 12}]] Out[3]= {0.344946*Second, SeriesData[s, 1/2, {1/2, 0, 1 - Pi^2/4, 0, 5/3 - Pi^2/2 + Pi^4/48, 0, 46/15 - (5*Pi^2)/6 + Pi^4/24 - Pi^6/1440, 0, 2057/315 - (23*Pi^2)/15 + (5*Pi^4)/72 - Pi^6/720 + Pi^8/80640, 0, 4954/315 - (2057*Pi^2)/630 + (23*Pi^4)/180 - Pi^6/432 + Pi^8/40320 - Pi^10/7257600, 0, 6469598/155925 - (2477*Pi^2)/315 + (2057*Pi^4)/7560 - (23*Pi^6)/5400 + Pi^8/24192 - Pi^10/3628800 + Pi^12/958003200}, 0, 13, 1]} Regards, Peter