Bill Page wrote:
>
> Hmmm... so the context of a differential field makes sqrt(x^2) equivalent to
> x?
>
> D(integrate(sqrt(x^2),x),x) = x
>
> and we have
>
> integrate(sqrt(x^2),x=-2..2,"noPole") = 0
>
> The latter result looks especially strange. Both Maple and
> Mathematica return 4. If we were being very pedantic we might admit
> that there is some uncertainty in the sign. It seems very unnatural
> to choose one branch for the lower limit and a different branch for
> the upper limit or to argue that such uncertainties somehow cancel
> out.
There are two branches: x and -x. "Real" result is obtained
by changing branch at 0. In other words Maple and
Mathematica choose one branch for the lower limit and a different branch for
the upper limit...
>
> > We could try to preserve uncertainity about choice of
> > factor. This requires essentially the same code as
> > support for signum.
> >
>
> Yes, trying to preserve "uncertainty" seems like the right idea to me.
> Perhaps a better word might be "ambiguity". In that sense
>
> x*sqrt(x)/2
>
> has the same ambiguity as integrate(sqrt(x^2),x).
>
> It is not clear to me how this might amount to the same thing as
> supporting signum.
Well, in both cases we have somewhat unpredictable factor
equal either to 1 or to -1. To do the integral we need to
consider all combination of signs, compute integral for each
and then combine results.
--
Waldek Hebisch
[email protected]
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