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Problematic family of integrals

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109362] Problematic family of integrals
  • From: Dimitris Emmanoulopoulos <demmanoulopoulos at hotmail.com>
  • Date: Fri, 23 Apr 2010 03:49:34 -0400 (EDT)

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


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