Re: [Axiom-mail] Dirac Delta

[Ondrej Certik]

> Oddly enough, I was just looking at Dirac Deltas (from a physics
> standpoint, a CAS with a good handle on the Dirac Delta is a Good
> Thing).
>
> It looks like a mathematically rigorous definition of the Dirac Delta
> would require generalized functions (a.k.a distributions).  I don't
> know much about them and it looks like a rather advanced branch of
> mathematics.  Whether it would fit into Axiom (or whether it might
> already be there in some form) I don't know, but certainly it's a Topic
> Of Interest - someday we will want to deal with the Dirac Delta and
> when we do it the treatment needs to be up to Axiom's standards for
> mathematical correctness.  Mathematica has a function dealing with it
> but I'm not sure how their definition relates to the latest research.
>
> It looks like the place to start would be the work of Laurent Schwartz.
>  Apparently there are theoretical problems with multiplying
> distributions.   Colombeau algebra may be of some interest there.
> Prime candidates for literate programs ;-).

I was also thinking how to implement delta functions. I know the
"distribution" way of doing it, however, I don't see any advantage of
using the

<delta, phi> = phi(0)

formalism instead of writing directly delta(0)phi(x) and of course
understand that in order to give a precise meaning to it, one needs to
integrate it and thus arriving at the <delta, phi> formalism, which is
mathematically precise, but to me it is as precise as writing the
delta(0)phi(x) directly. For a research in mathematics it can be maybe
useful to use <delta, phi> formalism, but I don't know of any
application of delta functions (for example in quantum mechanics or
the quantum field theory) where the simpler formalism delta(0)phi(x)
would give less precise (or less well-defined) results.

So the way it is done in Mathematica is the way I like and I'll do the
same in SymPy, when I find some time.

Ondrej