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Message from discussion Wolfram Research QA process defect: Bug in Mathematica 6 - Integrate - 64 (Log, Cos, false convergence, multiple regression bug)
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David W. Cantrell  
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 More options Jan 26 2008, 10:46 am
Newsgroups: sci.math.symbolic, sci.math
From: David W. Cantrell <DWCantr...@sigmaxi.net>
Date: 26 Jan 2008 15:46:30 GMT
Local: Sat, Jan 26 2008 10:46 am
Subject: Re: Wolfram Research QA process defect: Bug in Mathematica 6 - Integrate - 64 (Log, Cos, false convergence, multiple regression bug)
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Vladimir Bondarenko <v...@cybertester.com> wrote:

[snip]

> First, in 1996, Mathematica 3.0 returns a false result.  For
> a divergent integral, it reports (-12 EulerGamma^2 + Pi^2)/24
> which is 0.244645...

> This bug persists in 2002, in Mathematica 4.2.

> Then, in 2005, in Mathematica 5.2, it is repaired.

> Now, in Mathematica 6, this functionality is broken -- AGAIN!

> ----------------------------------------------------------------

> Table[NIntegrate[Log[z] (1 - Cos[z])/z, {z, 0, a}],{a,10,50,10}]

>          {3.00932, 4.59746, 6.141, 6.97878, 7.91825}

> ----------------------------------------------------------------

>       Integrate[Log[z] (1 - Cos[z])/z, {z, 0, Infinity}]

> ----------------------------------------------------------------
> VERSION           OUTPUT                              RESOLUTION
> ----------------------------------------------------------------

> Mathematica 6.0   (-12 EulerGamma^2 + Pi^2)/24  <----------- BUG

> Mathematica 5.2   Integral of does not converge               OK

> Mathematica 4.2   (-12 EulerGamma^2 + Pi^2)/24  <----------- BUG

> Mathematica 3.0   (-12 EulerGamma^2 + Pi^2)/24  <----------- BUG

> ----------------------------------------------------------------

It's interesting to note that one can easily get version 6 to tell us that
the integral does not converge. Just set an option:

Integrate[Log[z] (1 - Cos[z])/z, {z, 0, Infinity},
 GenerateConditions -> True]

yields

Integrate::idiv: Integral of Log[z]/z-(Cos[z] Log[z])/z does not converge
on {0,\[Infinity]}. >>

But I don't understand why that works! The documentation for
GenerateConditions states "GenerateConditions is an option for Integrate
that specifies whether explicit conditions on parameters should be
generated in the results of definite integrals." and "The default setting
is GenerateConditions->Automatic, which is equivalent to a setting of True
for one-dimensional integrals."

Is there a "parameter" involved? I don't see one.
Is our integral not one-dimensional? We should have had True automatically.

Obviously, either I don't understand something or the documentation is
severely lacking.

David


 
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