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Re: Can Integrate[expr,{x,a,b}] give an incorrect result?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg82555] Re: Can Integrate[expr,{x,a,b}] give an incorrect result?
*From*: "David W.Cantrell" <DWCantrell at sigmaxi.net>
*Date*: Wed, 24 Oct 2007 04:33:27 -0400 (EDT)
*References*: <20071023002439.352$aK@newsreader.com> <ffkfc1$sn3$1@smc.vnet.net>
Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> On 23 Oct 2007, at 13:24, David W. Cantrell wrote:
> > Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
[snip]
> >> As for the algorithm that is implemented in "another CAS", you do not
> >> make it clear what exactly it does.
> >
> > I could provide a reference. But you said in a subsequent post that
> > "integration on the real line that has no attraction for me whatever"
> > and so I know the reference would not interest you.
>
> I am not interested enough to read it. But from your post get the
> idea that you are unable to distinguish between an algorithm (like
> the Risch algorithm) and heuristics, which is quite a different
> matter.
I don't know why you think that. In any event, it's false.
> I challenged you to provide an algorithm that would be able
> to completely replace the the Risch algorithm for problems involving
> integration on the real line.
I never claimed to know of an algorithm which would do that.
> If you can't do that than you haven't got an algorithm, you have only got
> some, possibly sophisticated and useful, heuristics.
Huh? Something can be _an_ algorithm and yet not satisfy the requirement
that it "be able to completely replace the the Risch algorithm for problems
involving integration on the real line."!
> If you can provide a "real" replacement for the Risch algorithm I
> will admit that I was quite wrong on this.
I have never suggested that I either could or desired to provide a
replacement for the Risch algorithm. Below, I had already briefly described
what the algorithm does:
> > it gets rid of the
> > unnecessary discontinuities on the real line which arise as a
> > consequence of the Weierstrass substitution.
But in case you would prefer the authors' own words, here's what they say
at the end of their abstract: "The algorithm works by first evaluating the
given integral using the Weierstrass substitution in the usual way, and
then removing any spurious discontinuities present in the antiderivative."
Mathematica and, indeed, all CASs known to me use the Weierstrass
substitution. An unnecessary discontinuity caused by it was not detected by
Mathematica as being present on the interval from 0 to 2Pi; as best I can
tell, it was that failure of detection which lead to the erroneous
evaluation of the OP's definite integral. But of course, following use of
the Weierstrass substitution, if Mathematica had removed the discontinuity
before applying Newton-Leibniz, no error would have occurred.
Since the algorithm for removing such discontinuities has been known for
over a decade, I must assume that the people at Wolfram Research have a
good reason, unknown to me, for not having chosen to implement it.
David W. Cantrell
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