Re: Problematic family of integrals

• To: mathgroup at smc.vnet.net
• Subject: [mg109404] Re: Problematic family of integrals
• From: "Alexey Popkov" <lehin.p at gmail.com>
• Date: Sun, 25 Apr 2010 06:25:30 -0400 (EDT)
• References: <hqrje7\$m59\$1@smc.vnet.net>

```Interestingly, in Mathematica 5.2 everything is right:

In[2]:=
Table[Integrate[1/(a-x)^b^(-1),{x,-a,a}],{a,1,10},{b,2,10}]//N
Out[2]=
{{2.82843,2.3811,2.24239,2.17638,2.13816,2.11336,2.09601,2.08322,2.07341},{4.,
3.77976,3.77124,3.78929,3.80976,3.82823,3.8441,3.8576,3.86911},{4.89898,
4.95289,5.11154,5.2412,5.34122,5.41918,5.48118,5.53149,5.57306},{5.65685,
6.,6.34244,6.59754,6.78823,6.93464,7.05011,7.1433,7.22002},{6.32456,
6.96238,7.49788,7.88697,8.1755,8.39633,8.57022,8.71047,8.82587},{6.9282,
7.86222,8.59656,9.12547,9.51698,9.81656,10.0525,10.2429,10.3997},{7.48331,
8.71318,9.65017,10.3232,10.8215,11.2032,11.5041,11.7472,11.9474},{8.,
9.52441,10.6667,11.487,12.0952,12.5617,12.93,13.2276,13.473},{8.48528,
10.3024,11.6518,12.622,13.3426,13.8962,14.3336,14.6876,14.9797},{8.94427,
11.0521,12.6099,13.732,14.5671,15.2095,15.7179,16.1296,16.4697}}

"Dimitris Emmanoulopoulos" <demmanoulopoulos at hotmail.com> ÓÏÏÂÝÉÌ/ÓÏÏÂÝÉÌÁ ×
ÎÏ×ÏÓÔÑÈ ÓÌÅÄÕÀÝÅÅ: news:hqrje7\$m59\$1 at smc.vnet.net...
> 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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