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[Axiom-math] Curious behavior of Taylor series


From: Igor Khavkine
Subject: [Axiom-math] Curious behavior of Taylor series
Date: Sun, 20 Aug 2006 17:11:47 -0400

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 same behavior does not occur if the coefficient ring is changed to
say the rationals:

(118) -> y := (taylor x) :: UTS(FRAC INT,x,0)
  (118)  x
                          Type: UnivariateTaylorSeries(Fraction Integer,x,0)
(119) -> x*y
          2
  (119)  x
                          Type: UnivariateTaylorSeries(Fraction Integer,x,0)
(120) -> coefficient(%,2)
  (120)  1
                                                      Type: Fraction Integer

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.

Thanks.

Igor




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