Compute the inverse of a power series
power series generated by this package
The commands 1/p and Inverse(p) compute the multiplicative inverse of the power series p. This requires that p is invertible, that is, that p has a non-zero constant term; if that is not the case, an error is signaled.
When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series and univariate polynomials over power series. If you do, you may see invalid results.
We define a power series, a, representing a polynomial.
We can define its inverse in two equivalent ways:
b≔PowⅇrSⅇrⅈⅇs of 11−x−y : 1+…
c≔PowⅇrSⅇrⅈⅇs of 11−x−y : 1+…
We verify that the two definitions are equal, at least for the terms up to homogeneous degree 10.
A different power series represents the sine of x.
sx≔PowⅇrSⅇrⅈⅇs of sin⁡x : 0+…
Because the constant coefficient of sx is zero, we cannot invert it. (Its multiplicative inverse is a Laurent series, not a power series.)
Error, not invertible
The MultivariatePowerSeries[Inverse] command was introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
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