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Re: Problematic family of integrals

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109394] Re: Problematic family of integrals
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Sat, 24 Apr 2010 04:04:23 -0400 (EDT)

MANY definite Integrals are faulty in Mathematica, probably in any CAS,  
and they probably always will be. It's a fact of life.

Hence, the need to double-check them with NIntegrate or other means.

Bobby

On Fri, 23 Apr 2010 02:49:34 -0500, Dimitris Emmanoulopoulos  
<demmanoulopoulos at hotmail.com> wrote:

> Dear Mathgroup
>
>
> I would
> like to draw your attention to the following family of Integrals that  
> are faulty
> evaluated  from Mathematica 7.0 1.0 for
> Microsoft Windows (32-bit) (February 18, 2009).
>
>
> A special
> case
>
> Table[Integrate[1/(a-x)^b^(-1),{x,-a,a}],{a,1,10},{b,2,10}]
> (*Wrong  results*)
>
> Table[NIntegrate[1/(a-x)^b^(-1),{x,-a,a}],{a,1,10},{b,2,10}]
> (*Correct results*)
>
>
> The general
> cases
>
> Table[Integrate[1/(a-b
> x)^c^(-1),{x,-a/b,a/b}],{a,1,5},{b,1,5},{c,2,10}](*Wrong  results*)
>
> Table[NIntegrate[1/(a-b
> x)^c^(-1),{x,-a/b,a/b}],{a,1,5},{b,1,5},{c,2,10}](*Correct results*)
>
>
> Take a look
> also to the next integral giving:
>
> Integrate[(4-x)^a^(-1),{x,-2,2}]/.a->10^3(*Correct result*)
>
> Which gives
> the correct result which is equal to
>
> Integrate[(4-x)^(10^3)^(-1),{x,-2,2}](*Correct result*)
>
> Now
> estimate the following case having very small exponent
>
> Integrate[(4-x)^a^(-1),{x,-2,2}]/.a->10^10(*Correct result*)
>
> The last integral also
> gives the correct result BUT setting the exponent directly inside the  
> Integral to the same very small value
> causes kernel-failure
>
> Integrate[(4-x)^(10^10)^(-1),{x,-2,2}](*kernel-failure*)
>
>
> It would be
> useful to check if these =93bugs=94 occur under other operating systems  
> and /or
> versions of Mathematica. It would be also very interesting to find out,  
> whether
> or not there are even more general expressions for this family of  
> integrations being also problematic.
> Naturally, WolframAlpha also gives wrong results for all these cases  
> which you
> can readily cross check by the visual representation of the integral  
> which is
> produced automatically. For the case of the kernel-failure-Integral, the  
> WolframAlpha
> server seems not to respond.
>
>
> I have
> already reported the aforementioned =93bugs=94 to the technical support  
> and they
> will be fixed in the future release version.
>
>
> Best Regards,
>
> Dimitris


-- 
DrMajorBob at yahoo.com


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