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a strange integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71443] a strange integral
*From*: "dimitris" <dimmechan at yahoo.com>
*Date*: Sat, 18 Nov 2006 04:41:13 -0500 (EST)
$VersionNumber
5.2
Consider the following function
f[x_] := Log[1 + 1/Sqrt[x]]*(Log[1 + x]/x^(3/2))
Plot[f[x], {x, 0, 10}, PlotPoints -> 200]
Series[f[x], {x, 0, 2}]
Limit[f[x], x -> 0, Direction -> -1]
Here is the numerical estimate
NIntegrate[Log[1 + 1/Sqrt[x]]*(Log[1 + x]/x^(3/2)), {x, 0, Infinity},
PrecisionGoal -> 20, WorkingPrecision -> 40]
3.6705983269534578050
Strangely the following command "kill" the Kernel after several
minutes...
Integrate[Log[1 + 1/Sqrt[x]]*(Log[1 + x]/x^(3/2)), {x, 0, Infinity}]
What is more strange is that previous integral can be evaluated by
Mathematica
version 4.0; I don't check it myself by I adopted this integral from
the Mathematica
Guidebook for Symbolics of M. Trott where the computations take place
in v. 4.0 .
On the contrary, for v. 5.2 Integrate must be helped a little and then
the integration becomes trivial
integrand = Simplify[f[x]*dx /. x -> y^2 /. dx -> D[y^2, y], y > 0]
(2*Log[1 + 1/y]*Log[1 + y^2])/y^2
Timing[Integrate[integrand, {y, 0, Infinity}]]
N[%[[2]], 20]
{2.4689999999999994*Second, 4*Catalan + (5*Pi^2)/12 + Pi*(-2 + Log[2])}
3.6705983269534578050
Best Regards to all
Dimitris
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