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Re: A wrong definite integral in 5.0?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg77319] Re: A wrong definite integral in 5.0?
*From*: dh <dh at metrohm.ch>
*Date*: Wed, 6 Jun 2007 07:32:06 -0400 (EDT)
*References*: <f43hau$301$1@smc.vnet.net>
Hi,
you will remember from calculus that Log is a multivalued function:
principal value + 2Pi n I. Therefore, the results are correct (but not
as consistent as you would like it).
Hint: in version 6 you get -1 for both integrals.
hope this help, Daniel
bolud-el-kotur wrote:
> I get this result in version 5.0,
>
>> Integrate[Log[1 - 4 x(1 - x)], {x, 0, 1}]
>> -2 + I Pi
>
> and the same thing if I "declare" the singularity with {x,0,1/2,1}.
>
> Another way to look at the problem is computing,
>
>> Integrate[Log[1 - 4 x(1 - x)], {x, 0, 1/2}]
>> -1
>
> and
>
>> Integrate[Log[1 - 4 x(1 - x)], {x, 1/2, 1}]
>> -1 + I Pi
>
> Since the integrand is symmetric about x=1/2, the result should have
> been the same one (-1) in both cases, and the integral over [0,1]
> should yield -2.
>
> A numerical approach,
>
>> NIntegrate[Log[1 - 4 x(1 - x)], {x, 0, 1/2, 1},
> MaxRecursion -> 100, SingularityDepth -> 20]
>> -1.9999997086422834`
>
> gives the correct result, within the numerical accuracy required.
>
>
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