p1 * p2 returns the product of p1 and p2. The result is a power series.

The calling sequence Multiply(P) returns the product of the factors in P.

u1 * u2 returns the product of u1 and u2. The result is a univariate polynomial over power series.

The calling sequence 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.

s1 * s2 returns the product of the Puiseux series s1 and s2. If the orders of s1 and s2 are not compatible, an error is signaled. The result is a Puiseux series.

The calling sequence Multiply(S) returns the product of the factors in S. They are converted to Puiseux series. If this is not possible, an error is signaled.

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.

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

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

$b≔\mathrm{PowerSeries}\left(1+x+y+z\right)\:$

$ab$

$\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1+x+y+z}{1-x-y}\mathrm{:}1+2x+2y+z+\mathrm{\dots }\right]$

$a\left(x+y\right)$

$\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{x+y}{1-x-y}\mathrm{:}x+y+\mathrm{\dots }\right]$

$c≔\mathrm{PowerSeries}\left(2xy+3z^3\right)\:$

$d≔\mathrm{Multiply}\left(a\,b\,c\,1+x+y\right)$

$d≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{\left(1+x+y+z\right)\left(3z^3+2xy\right)\left(1+x+y\right)}{1-x-y}\mathrm{:}0+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(d\,2\right)$

$2xy$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(xz+y{z}^2+xy{z}^3\,z\right)\:$

$fb$

$\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(0+\mathrm{\dots }\right)z\mathrm{+}\left(0+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\right]$

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

$h≔fg$

$h≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(x+\mathrm{\dots }\right)z\mathrm{+}\left(y+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\right]$

$k≔\mathrm{Multiply}\left(f\,g\right)$

$k≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(x+\mathrm{\dots }\right)z\mathrm{+}\left(y+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\right]$

$\mathrm{Truncate}\left(h-k\,10\right)$

$0$

$\mathrm{s1}≔\mathrm{PuiseuxSeries}\left(\mathrm{PowerSeries}\left(\frac{1}{1+u}\right)\,\left[u=x^{-\frac{1}{3}}{y}^2\right]\,\left[x=3\,y=-4\right]\right)$

$\mathrm{s1}≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{x^3}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s2}≔\mathrm{PuiseuxSeries}\left(2+2\left(u+v\right)\,\left[u=x^{-\frac{1}{2}}y\,v=y\right]\,\left[x=3\,y=2\right]\right)$

$\mathrm{s2}≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\left(2+\frac{2y}{\sqrt{x}}+2y\right)x^3{y}^2\mathrm{:}2x^3{y}^2+2x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{y}^3+2y^3{x}^3\right]$

$\mathrm{s3}≔\mathrm{PuiseuxSeries}\left(\mathrm{PowerSeries}\left(\frac{1}{1+uv}\right)\,\left[y\,x\right]\,\left[u\,v\right]\,\left[\left[1\,0\right]\,\left[1\,-\frac{1}{2}\right]\right]\right)$

$\mathrm{s3}≔\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{1}{\frac{y^2}{\sqrt{x}}+1}\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{s1}\mathrm{s2}$

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{\left(2+\frac{2y}{\sqrt{x}}+2y\right)x^6}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y^2}\mathrm{:}\frac{2x^6}{y^2}+\mathrm{\dots }\right]$

$\mathrm{s1}\left(1+xy\right)$

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{\left(xy+1\right)x^3}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s1}a$

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{x^3}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)\left(1-x-y\right)y^4}\mathrm{:}\frac{x^3}{y^4}+\mathrm{\dots }\right]$

$\mathrm{s1}f$

$\left[\mathrm{Puis\ⅇuxS\ⅇri\ⅇs\; of}\frac{\left(x\mathrm{\dots }\mathrm{\dots }+y{z}^2+xz\right)x^3}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y^4}\mathrm{:}0+\mathrm{\dots }\right]$

$\mathrm{s1}\mathrm{s3}$

Error, the order of Puiseux series [x, y] and [y, x] are not compatible

$\mathrm{GetPuiseuxSeriesOrder}\left(\mathrm{s1}\right)$

$\left[x\,y\right]$

$\mathrm{GetPuiseuxSeriesOrder}\left(\mathrm{s3}\right)$

$\left[y\,x\right]$

$u≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right)\,\mathrm{PuiseuxSeries}\left(0\right)\,\mathrm{PuiseuxSeries}\left(x\,\left[x=x^{\frac{1}{3}}\right]\right)\,\mathrm{PuiseuxSeries}\left(y\,\left[y=y^{\frac{1}{2}}\right]\right)\,\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y}\,\left[x=x{y}^{\frac{1}{2}}\,y=x{y}^{-1}\right]\right)\right]\,z\right)$

$u≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(0\right)z\mathrm{+}\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)z^2\mathrm{+}\left(\sqrt{y}\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\right]$

$uf$

$\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPuis\ⅇuxS\ⅇri\ⅇs:}\left(0\right)\mathrm{+}\left(0+\mathrm{\dots }\right)z\mathrm{+}\left(0+\mathrm{\dots }\right)z^2\mathrm{+}\left(0+\mathrm{\dots }\right)z^3\mathrm{+}\left(0+\mathrm{\dots }\right)z^4\mathrm{+}\left(0+\mathrm{\dots }\right)z^5\mathrm{+}\left(0+\mathrm{\dots }\right)z^6\mathrm{+}\left(0+\mathrm{\dots }\right)z^7\right]$

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

For more information on Maple 2021 changes, see Updates in Maple 2021.

The MultivariatePowerSeries[Multiply] command was updated in Maple 2023.

The s1, s2 and S parameters were introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.