MultivariatePowerSeries/Variables - Maple Help

MultivariatePowerSeries

 Variables
 Return the variables in a power series or univariate polynomial over power series

 Calling Sequence Variables(p) Variables(u)

Parameters

 p - power series generated by this package u - univariate polynomial over power 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(u), where p is a univariate polynomial over power 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 and univariate polynomials over power 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 a $c$ 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)

Compatibility

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