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