 MultivariatePowerSeries/Negate - Maple Help

MultivariatePowerSeries

 Negate
 Negate a power series or a univariate polynomial over power series Calling Sequence -p Negate(p) -u Negate(u) Parameters

 p - power series generated by this package u - univariate polynomial over power series generated by this package Description

 • The commands -p and Negate(p) return the additive inverse of the power series p.
 • The commands -u and Negate(u) return the additive inverse of the univariate polynomial over power series u.
 • 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 create a power series and form its additive inverse in two ways.

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

We verify that the results are the same up to homogeneous degree 10.

 > $\mathrm{ApproximatelyEqual}\left(b,c,10\right)$
 ${\mathrm{true}}$ (4)

We create a univariate polynomial over power series and form its additive inverse.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left({z}^{3}-{z}^{2}x-2xy,z\right)$
 ${f}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({-}{2}{}{x}{}{y}\right){+}\left({0}\right){}{z}{+}\left({-}{x}\right){}{{z}}^{{2}}{+}\left({1}\right){}{{z}}^{{3}}\right]$ (5)
 > $g≔-f$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}{+}{\dots }\right){+}\left({0}\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({-1}\right){}{{z}}^{{3}}\right]$ (6)

The additive inverse of $g$ should be equal to $f$.

 > $\mathrm{ApproximatelyEqual}\left(f,\mathrm{Negate}\left(g\right),10\right)$
 ${\mathrm{true}}$ (7) Compatibility

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