       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=
{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

```

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