The commands p^e and Exponentiate(p,e) exponentiate the power series p by raising it to the power e.

The commands s^e and Exponentiate(s,e) exponentiate the Puiseux series s by raising it to the power e.

The commands u^n and Exponentiate(u,n) exponentiate the univariate polynomial over power series u by raising it to the power n.

Note that Puiseux series and unit power series can generally be raised to any integer power, whereas non-unit power series and univariate polynomials over power series or over Puiseux series can only be raised to non-negative integer powers.

When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series, Puiseux series, and univariate polynomials over these series. If you do, you may see invalid results.

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

$a≔\mathrm{GeometricSeries}\left(\left[x\,y\right]\right)\:$

$b≔\mathrm{Multiply}\left(a\,a\,a\,a\right)\:$

$c≔a^4$

$c≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{{\left(1-x-y\right)}^4}\mathrm{:}1+4x+4y+\mathrm{\dots }\right]$

$d≔\mathrm{Exponentiate}\left(a\,4\right)$

$d≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{{\left(1-x-y\right)}^4}\mathrm{:}1+4x+4y+\mathrm{\dots }\right]$

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

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(b\,d\,10\right)$

$\mathrm{true}$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left(z^2+z\right)\,z\right)\:$

$g≔fff$

$g≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(\mathrm{-216}\right)\mathrm{+}\left(1188+\mathrm{\dots }\right)z\mathrm{+}\left(\mathrm{-2826}+\mathrm{\dots }\right)z^2\mathrm{+}\left(3815+\mathrm{\dots }\right)z^3\mathrm{+}\left(\mathrm{-3222}+\mathrm{\dots }\right)z^4\mathrm{+}\left(1767+\mathrm{\dots }\right)z^5\mathrm{+}\left(\mathrm{-630}+\mathrm{\dots }\right)z^6\mathrm{+}\left(141+\mathrm{\dots }\right)z^7\mathrm{+}\left(\mathrm{-18}+\mathrm{\dots }\right)z^8\mathrm{+}\left(1\right)z^9\right]$

$h≔\mathrm{Exponentiate}\left(f\,3\right)$

$h≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(\mathrm{-216}\right)\mathrm{+}\left(1188+\mathrm{\dots }\right)z\mathrm{+}\left(\mathrm{-2826}+\mathrm{\dots }\right)z^2\mathrm{+}\left(3815+\mathrm{\dots }\right)z^3\mathrm{+}\left(\mathrm{-3222}+\mathrm{\dots }\right)z^4\mathrm{+}\left(1767+\mathrm{\dots }\right)z^5\mathrm{+}\left(\mathrm{-630}+\mathrm{\dots }\right)z^6\mathrm{+}\left(141+\mathrm{\dots }\right)z^7\mathrm{+}\left(\mathrm{-18}+\mathrm{\dots }\right)z^8\mathrm{+}\left(1\right)z^9\right]$

$k≔f^3$

$k≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(\mathrm{-216}\right)\mathrm{+}\left(1188+\mathrm{\dots }\right)z\mathrm{+}\left(\mathrm{-2826}+\mathrm{\dots }\right)z^2\mathrm{+}\left(3815+\mathrm{\dots }\right)z^3\mathrm{+}\left(\mathrm{-3222}+\mathrm{\dots }\right)z^4\mathrm{+}\left(1767+\mathrm{\dots }\right)z^5\mathrm{+}\left(\mathrm{-630}+\mathrm{\dots }\right)z^6\mathrm{+}\left(141+\mathrm{\dots }\right)z^7\mathrm{+}\left(\mathrm{-18}+\mathrm{\dots }\right)z^8\mathrm{+}\left(1\right)z^9\right]$

$\mathrm{ApproximatelyEqual}\left(g\,h\,10\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(g\,k\,10\right)$

$\mathrm{true}$

$s≔\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y}\,\left[x=x{y}^{\frac{1}{2}}\,y=x{y}^{-1}\right]\right)$

$s≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{x\sqrt{y}+\frac{x}{y}}{1+x\sqrt{y}+\frac{x}{y}}\mathrm{:}0+\mathrm{\dots }\right]$

$\mathrm{Exponentiate}\left(s\,3\right)$

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{{\left(x\sqrt{y}+\frac{x}{y}\right)}^3}{{\left(1+x\sqrt{y}+\frac{x}{y}\right)}^3}\mathrm{:}0+\mathrm{\dots }\right]$

$h≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right)\,\mathrm{PuiseuxSeries}\left(0\right)\,\mathrm{PuiseuxSeries}\left(x\,\left[x=x^{\frac{1}{3}}\right]\right)\,\mathrm{PuiseuxSeries}\left(y\,\left[y=y^{\frac{1}{2}}\right]\right)\,\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y}\,\left[x=x{y}^{\frac{1}{2}}\,y=x{y}^{-1}\right]\right)\right]\,z\right)$

$h≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(0\right)z\mathrm{+}\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)z^2\mathrm{+}\left(\sqrt{y}\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\right]$

$\mathrm{Exponentiate}\left(h\,2\right)$

$\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(0\right)z\mathrm{+}\left(0+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\mathrm{+}\left(0+\mathrm{\dots }\right)z^5\mathrm{+}\left(0+\mathrm{\dots }\right)z^6\mathrm{+}\left(0+\mathrm{\dots }\right)z^7\mathrm{+}\left(0+\mathrm{\dots }\right)z^8\right]$

The MultivariatePowerSeries[Exponentiate] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.

The MultivariatePowerSeries[Exponentiate] command was updated in Maple 2023.

The  and  options were introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.