Exponentiate - Maple Help

MultivariatePowerSeries

 Exponentiate
 exponentiate a power series or a Puiseux series or a univariate polynomial over power series or over Puiseux series

 Calling Sequence p^e s^e u^n Exponentiate(p, e) Exponentiate(s, e) Exponentiate(u, n)

Parameters

 p - power series generated by this package s - Puiseux series generated by this package e - integer u - univariate polynomial over power series or over Puiseux series generated by this package n - integer

Description

 • 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.

Examples

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

We define a power series, $a$.

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

We can define ${a}^{4}$ in three different ways: using multiplication, using the exponentiation operator, or using the Exponentiate command.

 > $b≔\mathrm{Multiply}\left(a,a,a,a\right):$
 > $c≔{a}^{4}$
 ${c}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{\left({1}{-}{x}{-}{y}\right)}^{{4}}}{:}{1}{+}{4}{}{x}{+}{4}{}{y}{+}{\dots }\right]$ (1)
 > $d≔\mathrm{Exponentiate}\left(a,4\right)$
 ${d}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{\left({1}{-}{x}{-}{y}\right)}^{{4}}}{:}{1}{+}{4}{}{x}{+}{4}{}{y}{+}{\dots }\right]$ (2)

We verify that the homogeneous components of $b$, $c$, and $d$ of degree at most 10 are the same.

 > $\mathrm{ApproximatelyEqual}\left(b,c,10\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{ApproximatelyEqual}\left(b,d,10\right)$
 ${\mathrm{true}}$ (4)

We define a univariate polynomial over power series, $f$.

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

Again, we can define ${f}^{3}$ in three different ways. We verify that they give the same result (at least for degrees at most 10).

 > $g≔fff$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({-216}\right){+}\left({1188}{+}{\dots }\right){}{z}{+}\left({-2826}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({3815}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({-3222}{+}{\dots }\right){}{{z}}^{{4}}{+}\left({1767}{+}{\dots }\right){}{{z}}^{{5}}{+}\left({-630}{+}{\dots }\right){}{{z}}^{{6}}{+}\left({141}{+}{\dots }\right){}{{z}}^{{7}}{+}\left({-18}{+}{\dots }\right){}{{z}}^{{8}}{+}\left({1}\right){}{{z}}^{{9}}\right]$ (5)
 > $h≔\mathrm{Exponentiate}\left(f,3\right)$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({-216}\right){+}\left({1188}{+}{\dots }\right){}{z}{+}\left({-2826}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({3815}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({-3222}{+}{\dots }\right){}{{z}}^{{4}}{+}\left({1767}{+}{\dots }\right){}{{z}}^{{5}}{+}\left({-630}{+}{\dots }\right){}{{z}}^{{6}}{+}\left({141}{+}{\dots }\right){}{{z}}^{{7}}{+}\left({-18}{+}{\dots }\right){}{{z}}^{{8}}{+}\left({1}\right){}{{z}}^{{9}}\right]$ (6)
 > $k≔{f}^{3}$
 ${k}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({-216}\right){+}\left({1188}{+}{\dots }\right){}{z}{+}\left({-2826}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({3815}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({-3222}{+}{\dots }\right){}{{z}}^{{4}}{+}\left({1767}{+}{\dots }\right){}{{z}}^{{5}}{+}\left({-630}{+}{\dots }\right){}{{z}}^{{6}}{+}\left({141}{+}{\dots }\right){}{{z}}^{{7}}{+}\left({-18}{+}{\dots }\right){}{{z}}^{{8}}{+}\left({1}\right){}{{z}}^{{9}}\right]$ (7)
 > $\mathrm{ApproximatelyEqual}\left(g,h,10\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{ApproximatelyEqual}\left(g,k,10\right)$
 ${\mathrm{true}}$ (9)

Now we define a Puiseux series s and compute ${s}^{3}$.

 > $s≔\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)$
 ${s}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{{1}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{:}{0}{+}{\dots }\right]$ (10)
 > $\mathrm{Exponentiate}\left(s,3\right)$
 $\left[{PuisⅇuxSⅇriⅇs of}\frac{{\left({x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}\right)}^{{3}}}{{\left({1}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}\right)}^{{3}}}{:}{0}{+}{\dots }\right]$ (11)

Finally, we create a univariate polynomial over power series from a list of Puiseux series.

 > $h≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (12)

We compute ${h}^{2}$.

 > $\mathrm{Exponentiate}\left(h,2\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{5}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{6}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{7}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{8}}\right]$ (13)

Compatibility

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