The commands p^e and Exponentiate(p,e) exponentiate the power series p 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 power series can be raised to any integer power, whereas univariate polynomials over power series can only be raised to nonnegative integer powers.

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.

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

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

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

$c≔a^4$

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

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

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$g≔fff$

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

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

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

$k≔f^3$

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

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

$\mathrm{true}$

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

$\mathrm{true}$

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

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