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