GetCoefficient - Maple Help

MultivariatePowerSeries

 GetCoefficient
 get a coefficient of a power series or a univariate polynomial over power series or over Puiseux series

 Calling Sequence GetCoefficient(p, m) GetCoefficient(u, d)

Parameters

 p - power series generated by this package m - monomial u - univariate polynomial over power series or over Puiseux series generated by this package d - non-negative integer

Description

 • The command GetCoefficient(p,m) returns the coefficient of p with respect to the multivariate monomial m. This coefficient is a complex number.
 • The command GetCoefficient(u,d) returns the coefficient of z^d in u, where z is the main variable of u. This coefficient is a power series.
 • 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 create a power series corresponding to a polynomial and find a few of its coefficients.

 > $a≔\mathrm{PowerSeries}\left(1+3x+7xy+4{x}^{2}\right):$
 > $\mathrm{GetCoefficient}\left(a,x\right)$
 ${3}$ (1)
 > $\mathrm{GetCoefficient}\left(a,xy\right)$
 ${7}$ (2)
 > $\mathrm{GetCoefficient}\left(a,x{y}^{2}\right)$
 ${0}$ (3)

We create a univariate polynomial over power series with $z$ as its main variable, corresponding to the expression $x+yz+\frac{{z}^{2}}{1-x-y}$.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(x\right),\mathrm{PowerSeries}\left(y\right),\mathrm{GeometricSeries}\left(\left[x,y\right]\right)\right],z\right):$

The coefficient of ${z}^{0}$ is the power series corresponding to $x$.

 > $\mathrm{GetCoefficient}\left(f,0\right)$
 $\left[{PowⅇrSⅇriⅇs:}{x}\right]$ (4)

The coefficient of ${z}^{0}$ is the power series corresponding to $\frac{1}{1-x-y}$.

 > $\mathrm{GetCoefficient}\left(f,2\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{\dots }\right]$ (5)

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

 > $g≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${g}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (6)
 > $\mathrm{GetCoefficient}\left(f,2\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{\dots }\right]$ (7)

Compatibility

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