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[Axiom-developer] [Axiom-mail] beginner question about sum(...)

From: Page, Bill
Subject: [Axiom-developer] [Axiom-mail] beginner question about sum(...)
Date: Mon, 24 Jan 2005 18:53:42 -0600


On Monday, January 24, 2005 9:19 AM you wrote:
> Thanks very much for the elucidation.  By way of background, 
> I have been using Mathematica for about 10 years, and I'm
> contemplating a switch to Axiom.

I have been using Maple for more than 10 years and I am also
contemplating a switch to Axiom. Of course you know that I have
been helping to make Axiom available, but I continue to use
Maple for most research. I find that there are many things I
know immediately how to do in Maple but discovering how to do
them in Axiom takes time. However I remain convinced that my
time spent with Axiom will be beneficial to me (and hopefully
to others) in the long term.

> It seems that Axiom is, let's say, "conceptually", very 
> advanced in the sense of its domains, types, categories, etc.
> Perhaps we could call that its abstract mathematical knowledge.
> This is notably absent from Mathematica, until at least v5.0, 
> which is almost current.

I agree. In it's approach to the formal representation of
mathematics Axiom is probably "conceptually" superior to any
other computer algebra system.

> But I was disappointed by the sum(1/k^2, k=1..n) example.  I 
> saw that Gosper's method is implemented in sum.spad.pamphlet,
> but this (rather simple) sum needs symbolic manipulation of
> gamma and psi functions, which is not there.  More generally,
> special functions seem to be handled only numerically.  At
> least for my prospective use of Axiom, this points to a rather
> big "hole".

I agree. And now comes my pitch: Axiom is open source. This
means that having identified and documented this deficiency

(The best place to do this is:

anyone with sufficient knowledge and motivation can extend Axiom
to solve these problems. These extensions will become part of
the free Axiom distribution. That's why creating the environment
where such contributions are encouraged is so important.

> And I wonder how many others of this sort there are.

I am sure there are many. For reasons that have nothing at all
to do with Axiom's "conceptually" advanced mathematics, Axiom
has remained essentially static since about 1995, while during
the last 10 years both Mathematica and Maple have made steady
advances. (But perhaps not as much as we would like. :)

On the other hand, there are also some things which Axiom
implements which have not found there way into any commerical
system yet.

> I also tried sum(1/(k*(k+a)), k=1..n).  That was also returned
> unevaluated, although Gosper's method should handle it.  Why
> is that?

I don't know, but I think this is exactly the kind of thing
that should be reported at

> I don't mind reading the book, so if somewhere there is
> a discussion of (limitations of) symbolic evaluation, please 
> point me there.

I am not aware of any discussions on the limitations of symbolic
evaluation in Axiom except here, on the axiom-developer email
list and on MathAction:

Bill Page.

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