Algorithmic development of power series

Abstract

There is a one-to-one correspondence between formal power series (FPS) \(\sum\limits_{k = 0}^\infty {a_k x^k }\) with positive radius of convergence and corresponding analytic functions. Since a goal of Computer Algebra is to work with formal objects and preserve such symbolic information, it should be possible to automate conversion between these forms in Computer Algebra Systems (CASs). However, only Macsyma provides a rather limited procedure powerseries to calculate FPS from analytic expressions in certain special cases.We present an algorithmic approach to compute an FPS, which has been implemented by the author and A. Rennoch in Mathematica, and by D. Gruntz in Maple. Moreover, the same algorithm can be reversed to calculate a function that corresponds to a given FPS, in those cases when an initial value problem for a certain ordinary differential equation can be solved.Further topics of application like infinite summation, and asymptotic expansion are presented.KeywordsPower SeriesAsymptotic ExpansionRecurrence EquationFormal Power SeriesComputer Algebra SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.