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

[Igor Khavkine]

On 8/21/06, Ralf Hemmecke <address@hidden> wrote:
> On 08/21/2006 07:18 PM, Martin Rubey wrote:

> > (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.

That is in deed a very nice way to characterize the coefficients of of
this power series. But how exactly would you coax Axiom into producing
a power series with coefficients of this type starting with sin(x+y)?
It'd also be nice if the variables s and c printed as sin(y) and
cos(y) and behaved the same under operations like, say, taking
derivatives with respect to y.

Igor