# Good Computer Package for Calculating Inverse of a Formal Power Series?

This might be a question people already asked or is obvious to experts, or is not appropriate for this forum, if so, I apologize. I am trying to calculate things like $z/(e^z-1)$, or find the inverse of $x=z+ 2z+5z^2+\cdots$, expand as power series. What is the best (or favorite) software package you use to do stuff like this?

• Sage can do it: sagemath.org. (Search hint: finding the inverse function of a power series is called "Lagrange inversion", or sometimes "reversion".) – Tom Leinster Nov 3 '12 at 20:04
• Pari/GP's serreverse(). – Pasha Zusmanovich Nov 6 '12 at 16:47
• In flint library (flintlib.org) there's a bunch of functions for computing inverse series in different fields. For example, fmpz_poly_inv_series – Anton Apr 3 '13 at 2:16
• Assuming you mean the compositional inverse, fmpz_poly_revert_series. flint is a good choice if you want say 10000 terms. – Fredrik Johansson Apr 3 '13 at 22:57
• You don't say what kind of answer you're looking for. Would you be happy with a list of numerical values for a finite part of the list of coefficients or are you looking for something in a symbolic form where possible? – Dan Piponi May 25 '13 at 14:54

1. flint has the fmpz_poly_revert_series command (A web search revealed a reference pdf, but no web page). Some benchmarks seem to indicate that flint is unusually fast.

2. Maple has the reversion command.

3. Mathematica has the InverseSeries command.

4. Pari/GP has the serreverse function.

5. SAGE has the reversion command.

For the TI-nspire CAS calculator: http://www.ticalc.org/archives/files/fileinfo/424/42499.html.

Maple produces it. See http://www.maplesoft.com/support/help/Maple/view.aspx?path=powseries , especially the command http://www.maplesoft.com/support/help/Maple/view.aspx?path=powseries%2freversion for info.