Re: [Axiom-mail] Defining piece-wise functions and drawing, integrating, ...

[Martin Rubey]

Dear Sumant,

"Sumant S.R. Oemrawsingh" <address@hidden> writes:


> (1) -> h(k,x)==1/2 + 1/%pi * atan(k*x)

> (2) -> f(k,x)==x**2 * (h(k,x)-h(k,-x))

> (4) -> g(x)==limit(f(k,x),k=%plusInfinity)

> What I would think is that, since g(x) is a sum of two functions, the
> integral would split up into a sum of integrals as well.

Hm, why should g(x) be a sum of two functions? My axiom gives me upon issueing
g x

 x abs(x)

> (6) -> integrate(g(x),x=-1..1,"noPole")
> integrate(g(x),x=-1..1,"noPole")
> (6) -> 
>    (6)  "failed"
>                                                 Type: Union(fail: failed,...)

This just says that axiom couldn't do the integral.  In particular, bugs aside,
it is a "proof" that it not expressible as an elementary function.  (The Risch
algorithm is more or less completely implemented in axiom, and axiom will spit
out an error message if you hit an unimplemented branch.)

> So I've been looking a bit into how this could be done in spad. But I've not
> been able to understand where and how the functionality of such special
> functions is or can be implemented, or if I somehow would have to extend the
> definition or workings of the integrate function itself.

For a start, I'd think it is easier to split up integration boundaries
semi-automatically, as I indicated in a previous email.

If you are really interested in the internals, I think that intpm.spad is for
you, it tells axiom some pattern matching rules.  In my opinion, we should go
the mupad way here, the source of axiom is a horror, while the one of mupad is
extremely easy to grasp and to extend.

But that's way beyond my (computer algebra) skills.

All the best,

Martin