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MultivariatePowerSeries

 Variables
 return the variables in a power series or a Puiseux series or univariate polynomial over power series or over Puiseux series

 Calling Sequence Variables(p) Variables(s) Variables(u)

Parameters

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

Description

 • The command Variables(p), where p is a power series object created by the MultivariatePowerSeries package, returns the variables occurring in p.
 • The command Variables(s), where s is a Puiseux series object created by the MultivariatePowerSeries package, returns the variables occurring in s.
 • The command Variables(u), where p is a univariate polynomial over power series or over Puiseux series created by the MultivariatePowerSeries package, returns the variables occurring in u (its main variable and the variables occurring in all coefficients).
 • Not every variable name returned by the Variables command needs to actually occur in p or u, however, every variable that occurs in p or u is returned by the Variables command.
 • 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

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

Create a power series from a polynomial. The variables in a are the variables in the polynomial.

 > $a≔\mathrm{PowerSeries}\left(1+x+xy+{x}^{2}\right)$
 ${a}{≔}\left[{PowⅇrSⅇriⅇs:}{1}{+}{x}{+}{{x}}^{{2}}{+}{x}{}{y}\right]$ (1)
 > $\mathrm{Variables}\left(a\right)$
 $\left\{{x}{,}{y}\right\}$ (2)

Create a power series from a procedure. The variables in b are specified manually.

 > bproc := proc(d) local s; s := 2*x + y; return expand(s^d); end proc:
 > $b≔\mathrm{PowerSeries}\left(\mathrm{bproc},\mathrm{variables}=\left\{x,y\right\}\right)$
 ${b}{≔}\left[{PowⅇrSⅇriⅇs:}{1}{+}{\dots }\right]$ (3)
 > $\mathrm{Variables}\left(b\right)$
 $\left\{{x}{,}{y}\right\}$ (4)

Create a power series from a procedure. The variables in c are the variables occurring in the analytic expression.

 > $c≔\mathrm{PowerSeries}\left(d↦\frac{{x}^{d}}{d!},\mathrm{analytic}=\mathrm{exp}\left(x\right)\right)$
 ${c}{≔}\left[{PowⅇrSⅇriⅇs of}{{ⅇ}}^{{x}}{:}{1}{+}{\dots }\right]$ (5)
 > $\mathrm{Variables}\left(c\right)$
 $\left\{{x}\right\}$ (6)

If we define a power series as $c$ times its inverse, then the power series is equal to 1, so no variables actually occur in it. However, the Variables command will still return the variable $x$ occurring in $c$.

 > $d≔c\mathrm{Inverse}\left(c\right)$
 ${d}{≔}\left[{PowⅇrSⅇriⅇs:}{1}\right]$ (7)
 > $\mathrm{Variables}\left(d\right)$
 $\left\{{x}\right\}$ (8)

Create a univariate polynomial over power series. The variables in u are the variables in its coefficients, together with the main variable.

 > $u≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[a,b,c\right],z\right)$
 ${u}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{{x}}^{{2}}{+}{x}{}{y}\right){+}\left({1}{+}{\dots }\right){}{z}{+}\left({1}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (9)
 > $\mathrm{Variables}\left(u\right)$
 $\left\{{x}{,}{y}{,}{z}\right\}$ (10)

Create a univariate polynomial over Puiseux series from a list of Puiseux series.

 > $g≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (11)

We verify the variables of g.

 > $\mathrm{Variables}\left(g\right)$
 $\left\{{x}{,}{y}{,}{z}\right\}$ (12)

Finally, we define a Puiseux series s and verify its variables.

 > $s≔\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)$
 ${s}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{{1}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}{:}{0}{+}{\dots }\right]$ (13)
 > $\mathrm{Variables}\left(s\right)$
 $\left\{{x}{,}{y}\right\}$ (14)

Compatibility

 • The MultivariatePowerSeries[Variables] command was introduced in Maple 2021.
 • For more information on Maple 2021 changes, see Updates in Maple 2021.
 • The MultivariatePowerSeries[Variables] command was updated in Maple 2023.