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