Examples for the algorithmic calculation of formal Puisieux, Laurent and power series

Abstract

Formal Laurent-Puisieux series (LPS) of the form [EQUATION] are important in calculus and complex analysis. In some Computer Algebra Systems (CASs) it is possible to define an LPS by direct or recursive definition of its coefficients. Since some operations cannot be directly supported within the LPS domain, some systems generally convert LPS to finite truncated LPS for operations such as addition, multiplication, division, inversion and formal substitution. This results in a substantial loss of information. Since a goal of Computer Algebra is --- in contrast to numerical programming --- to work with formal objects and preserve such symbolic information, CAS should be able to use LPS when possible.There is a one-to-one correspondence between formal power series with positive radius of convergence and corresponding analytic functions. It should be possible to automate conversion between these forms. Among CASs only MACSYMA [5] provides a procedure powerseries to calculate LPS from analytic expressions in certain special cases, but this is rather limited.In [2]-[4] we gave an algorithmic approach for computing an LPS for a very rich family of functions. It covers e.g. a high percentage of the power series that are listed in the special series dictionary [1]. The algorithm has been implemented by the author and A. Rennoch in the CAS MATHEMATICA [7], and by D. Gruntz in MAPLE [6].In this note we present some example results of our MATHEMATICA implementation which give insight in the underlying algorithmic procedure.