Re: Curious behavior of Taylor series

On 08/21/2006 07:18 PM, Martin Rubey wrote:
> Ralf Hemmecke <[email protected]> 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)

Looking at this thing I would say that if you take

R = Q[s,c]       -- polynomial ring in two variables over rationals
I = (s^2+c^2-1)R -- ideal in R
A = R/I          -- factor structure
S = A[[x]]       -- formal power series

then S would be a perfect candidate for the result type of the above 
expression. And there is no "Expression Integer".
While constructing the result of "series", Axiom should try hard to get 
a reasonable (in some sense minimal) type for the result.
Note that you get "PositiveInteger if you just type 1 on the axiom 
prompt. Would you be happy, if Axiom returned 1@Expression(Integer)?

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

I think, we are on the same side anyway. ;-)

> 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.

Good point and I agree wholeheatedly. But Axiom allows after 
understanding the structure of things to encode even that structure 
(into the type system). The latter is THE feature of Axiom and should 
not be forgotten due to laziness.

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

What is so special about that? It is just Z[x][y], only that the names 
of x an y are not given. If you, however, think of a univariate 
polynomial ring over a ring R as the collection of functions N -> R with 
finite support, you don't even see the x or y.

Ralf