 MultivariatePowerSeries/Inverse - Maple Help

MultivariatePowerSeries

 Inverse
 Compute the inverse of a power series Calling Sequence 1/p Inverse(p) Parameters

 p - power series generated by this package Description

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right):$

We define a power series, $a$, representing a polynomial.

 > $a≔\mathrm{PowerSeries}\left(1-x-y\right)$
 ${a}{≔}\left[{PowⅇrSⅇriⅇs:}{1}{-}{x}{-}{y}\right]$ (1)

We can define its inverse in two equivalent ways:

 > $b≔\frac{1}{a}$
 ${b}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{\dots }\right]$ (2)
 > $c≔\mathrm{Inverse}\left(a\right)$
 ${c}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{\dots }\right]$ (3)

We verify that the two definitions are equal, at least for the terms up to homogeneous degree 10.

 > $\mathrm{Truncate}\left(b-c,10\right)$
 ${0}$ (4)

A different power series represents the sine of $x$.

 > $\mathrm{sx}≔\mathrm{PowerSeries}\left(d↦\mathrm{ifelse}\left(d::\mathrm{odd},\frac{{\left(-1\right)}^{\frac{d}{2}-\frac{1}{2}}\cdot {x}^{d}}{d!},0\right),\mathrm{analytic}=\mathrm{sin}\left(x\right)\right)$
 ${\mathrm{sx}}{≔}\left[{PowⅇrSⅇriⅇs of}{\mathrm{sin}}{}\left({x}\right){:}{0}{+}{\dots }\right]$ (5)

Because the constant coefficient of $\mathrm{sx}$ is zero, we cannot invert it. (Its multiplicative inverse is a Laurent series, not a power series.)

 > $\frac{1}{\mathrm{sx}}$ Compatibility

 • The MultivariatePowerSeries[Inverse] command was introduced in Maple 2021.