Re: Curious behavior of Taylor series

"Igor Khavkine" <[email protected]> writes:

> Can someone explain the following behavior of Taylor series in Axiom?
> 
> (113) -> y := taylor x
>    (113)  x
>                          Type: UnivariateTaylorSeries(Expression Integer,x,0)
> (114) -> x*y
>    (114)  x x
>                          Type: UnivariateTaylorSeries(Expression Integer,x,0)
> (115) -> coefficient(%,1)
>    (115)  x
>                                                      Type: Expression Integer

The reason is that Axiom cannot really know whether you meant x in (114) to be
an element of the coefficient Ring EXPR INT, or to be a univariate Taylor
series. In case of doubt, it usually chooses the wrong possibility :-)

Thus, you should help Axiom by saying

 monomial(1,1)$UTS(EXPR INT,x,0) * y

> I've been trying to write a routine that takes some expression f(x,y), but
> possibly containing other symbolic constants, and then solves the implicit
> equation f(x,y)=0 for y as a Taylor series in x. The above behavior has
> proved to be an obstacle.

I think such a routine already exists - at least if f(x, y) involves only
powers and derivatives of y. Look at seriesSolve. If you need more help or
examples, ask again. If I'm wrong, tell me so :-)

Martin