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