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## Re: [Axiom-developer] Complex exponentiation and 0

 From: William Sit Subject: Re: [Axiom-developer] Complex exponentiation and 0 Date: Mon, 21 Jun 2004 15:25:56 -0400

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David MENTRE wrote:
>
> Hello,
>
>
> > I my perhaps less than humble opinion: No!
> >
> > I think Axiom should *not* use Mathematica, Maple, MuPad or
> > Maxima as a guide. Axiom should only appeal to the mathematics
> > involved. In one way or another all of M^4 (and others) make
> > compromises when it comes to fundamentals. I think Axiom was
> > built with greater respect for the underlying mathematics and
> > that is something that we must retain and nurture. It is the
> > main thing that distinguises Axiom from the others.
>
> I strongly support Bill on that point. We must only refer to
> mathematical work.

I also strongly support Bill Page's point.

Also, I think the discussion on 0^0 = 1 seems to have mixed the raising of 0 to
the 0-th power (which in my opinion should be 1 in ANY domain that the base 0
belongs where the domain has an exponentiation to the NNI 0, because this is
pure algebra: -- note I am not talking about FLOAT where one would use 0.0),
with the LIMITS of the function
x^y as x AND y approaches 0 in domains in which limits make sense. In the latter
case, the eventual limit (or whether it exists) depends on the topology, and
thus should be left to the mathematics of that domain. In those cases, we should
bear in mind that the domains in Axiom dealt with a finite (though variable)
precision, trying to approximate the abstract mathematical infinite precision
object where the limit may theoretically exist. But even in those domains,
limits are distinct processes from the evaluation of the exponential function on
CONSTANT inputs.

The algebraic convention 0^0=1 is assumed for taking products over an empty
list, just as the sum over an empty list is 0. This assumption or convention is
commonly (even universally?) accepted in mathematics.

William

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