MultivariatePowerSeries/Multiply - Maple Help

MultivariatePowerSeries

 Multiply
 Multiply a list of power series or a list of univariate polynomial over power series

 Calling Sequence p1 * p2 Multiply(P) u1 * u2 Multiply(U)

Parameters

 p1, p2 - power series generated by this package, polynomials, or complex constants P - sequence of power series generated by this package, polynomials, and complex constants u1, u2 - univariate polynomials over power series generated by this package with the same main variable, power series generated by this package, polynomials, or complex constants U - sequence of univariate polynomial over power series generated by this package with the same main variable, power series generated by this package, polynomials, and complex constants

Description

 • The command p1 * p2 returns the product of p1 and p2. The result is a power series.
 • The command Multiply(P) returns the product of the factors in P.
 • The command u1 * u2 returns the product of u1 and u2. The result is a univariate polynomial over power series.
 • The command Multiply(U) returns the product of the factors in U. They are converted to univariate polynomials over power series in the same variable. If this is not possible, an error is raised. This may happen if there are univariate polynomials over power series in different variables. It can also happen if the univariate polynomials over power series all have the same main variable, say x, but one of the other arguments is a power series that is not known to be expressible as a polynomial in x. The same restrictions apply to the calling sequence u1 * u2.
 • 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 two power series.

 > $a≔\mathrm{GeometricSeries}\left(\left[x,y\right]\right):$
 > $b≔\mathrm{PowerSeries}\left(1+x+y+z\right):$

Form their product.

 > $ab$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}{+}{x}{+}{y}{+}{z}}{{1}{-}{x}{-}{y}}{:}{1}{+}{2}{}{x}{+}{2}{}{y}{+}{z}{+}{\dots }\right]$ (1)

One can also multiply a power series with a "plain" polynomial.

 > $a\left(x+y\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{x}{+}{y}}{{1}{-}{x}{-}{y}}{:}{x}{+}{y}{+}{\dots }\right]$ (2)

We create another power series, and multiply several power series and "plain" polynomials together.

 > $c≔\mathrm{PowerSeries}\left(2xy+3{z}^{3}\right):$
 > $d≔\mathrm{Multiply}\left(a,b,c,1+x+y\right)$
 ${d}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{\left({1}{+}{x}{+}{y}{+}{z}\right){}\left({3}{}{{z}}^{{3}}{+}{2}{}{x}{}{y}\right){}\left({1}{+}{x}{+}{y}\right)}{{1}{-}{x}{-}{y}}{:}{0}{+}{\dots }\right]$ (3)

To inspect the truncation of this product to degree 2, we can use the Truncate command.

 > $\mathrm{Truncate}\left(d,2\right)$
 ${2}{}{x}{}{y}$ (4)

We create a univariate polynomial over power series with main variable $z$. We can multiply such a polynomial with a power series that is known to be polynomial in $z$.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(xz+y{z}^{2}+xy{z}^{3},z\right):$
 > $fb$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({0}{+}{\dots }\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (5)

We can also multiply this polynomial with another univariate polynomial over power series, if it is polynomial in $z$. The two calling sequences for this are equivalent, as suggested by the fact that $h$ and $k$ agree up to homogeneous degree 10.

 > $g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{GeometricSeries}\left(\left[x,y\right]\right)\right],z\right):$
 > $h≔fg$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({x}{+}{\dots }\right){}{z}{+}\left({y}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{3}}\right]$ (6)
 > $k≔\mathrm{Multiply}\left(f,g\right)$
 ${k}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({0}\right){+}\left({x}{+}{\dots }\right){}{z}{+}\left({y}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{3}}\right]$ (7)
 > $\mathrm{Truncate}\left(h-k,10\right)$
 ${0}$ (8)

Compatibility

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