Formal power series

In this article we will describe the \Maple\ implementation of an algorithm presented in~\cite{Koe92}--\cite{Koeortho} which computes an {\em exact\/} formal power series (FPS) of a given function. This procedure will enable the user to reproduce most of the results of the extensive bibliography on series~\cite{Han}. We will give an overview of the algorithm and then present some parts of it in more detail

Full Partial Fraction Decomposition of Rational Functions

We describe a rational algorithm that computes the full partial fraction expansion of a rational function over the algebraic closure of its field of definition. The algorithm uses only gcd operations over the initial field but the resulting decomposition is expressed with linear denominators. We give examples from its Axiom and Maple implementations. Introduction The partial fraction decomposition of a rational function is a form where both the local and global behaviour of the function are easy to find. This is used when computing a primitive by hand, or any linear operation which is most easily done on a pole. An example is the efficient computation of asymptotic expansion of the solutions of a linear recurrence with constant coefficients [4]. Let f = A=D be a rational function in some field K(z). By the fundamental theorem of algebra, it is clear that f admits a partial fraction decomposition of the form f = P + X D(ff)=0 n ff X i=1 b ff;i (z \Gamma ff) i ; (1) where P is..