On 22 July 2014 05:40, Waldek Hebisch <[email protected]> wrote:
> Bill Page wrote:
>>
>> 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...
>
OK I see. So the standard convention to is a branch-cut (principle
value) but FriCAS does not choose a cut, rather it chooses a
particular branch.
>>
>> > 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)/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.
>
Is it not possible to consistently and indefinitely delay the choice
of branch or branch-cuts? What problems could result from producing
x*sqrt(x^2)/2
and similar results in other cases? Then in a give context such as
definite integral, draw and limit the choice of branch or branch-cut
could be made explicitly.
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