Re: [Axiom-math] Curious behavior of Taylor series

[Martin Rubey]

Ralf Hemmecke <address@hidden> writes:

> Well, but how can you tell this to Axiom? It should be impossible to construct
> the domain UnivariateTaylorSeries(Expression Integer,x,0).
> I guess the Axiom designers thought that returning that domain for taylor x
> would be reasonable. I must say, I question that.
> UTS(Fraction Integer, x, 0) would have been sufficient and you wouldn't have
> the trouble.

No, I disagree. As long as we don't have domains UnivariateExpression and
MultivariateExpression that take variables as parameters, things like UTS(EXPR
INT, x, 0) are necessary.

Otherwise you cannot create series like

(67) -> series(sin(y+x), x=0)

   (67)
                        sin(y)  2   cos(y)  3   sin(y)  4   cos(y)  5
     sin(y) + cos(y)x - ------ x  - ------ x  + ------ x  + ------ x
                           2           6          24          120
   + 
       sin(y)  6   cos(y)  7   sin(y)  8   cos(y)  9    sin(y)  10      11
     - ------ x  - ------ x  + ------ x  + ------ x  - ------- x   + O(x  )
         720        5040        40320      362880      3628800
                        Type: UnivariatePuiseuxSeries(Expression Integer,x,0)

I agree that taking EXPR INT as the default domain is a bad idea though.

I think there is one important point for MMA, Maple and MuPAD to make: as long
as the mathematics of a particular topic is unclear, good expression domains
are useful and sometimes maybe even necessary. Of course, once the mathematics
is well understood, for example, as is the case with DFinite functions (as is
the case with sin(x+y), an appropriate domain should be created and used. But
for the discovery, I believe that "insecure" domains are very useful. In my
opinion, especially MuPAD got this very right.

By the way, just today I was led to use the domain SUP SUP INT. (In code, of
course)

As soon as one realizes that different things may generate the same output
(printed representation), most of the problems go away.

I had very long discusssions with William Sit about that topic on this list,
and I have to admit that I now agree with him :-)

Martin