Multiply - Maple Help

MultivariatePowerSeries

 Multiply
 multiply a list of power series or a list of univariate polynomial over power series or list of multivariate Puiseux series

 Calling Sequence p1 * p2 Multiply(P) u1 * u2 Multiply(U) s1 * s2 Multiply(S)

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 or over Puiseux 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 s1, s2 - Puiseux series generated by this package with compatible Puiseux series order, univariate polynomials over power series generated by this package with the same main variable, power series generated by this package, or polynomials S - sequence of Puiseux series with compatible order, univariate polynomial over power series generated by this package with the same main variable, power series generated by this package, and polynomials

Description

 • 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.

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)

Create three Puiseux series.

 > $\mathrm{s1}≔\mathrm{PuiseuxSeries}\left(\mathrm{PowerSeries}\left(\frac{1}{1+u}\right),\left[u={x}^{-\frac{1}{3}}{y}^{2}\right],\left[x=3,y=-4\right]\right)$
 ${\mathrm{s1}}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{{x}}^{{3}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}{{y}}^{{4}}}{:}\frac{{{x}}^{{3}}}{{{y}}^{{4}}}{+}{\dots }\right]$ (9)
 > $\mathrm{s2}≔\mathrm{PuiseuxSeries}\left(2+2\left(u+v\right),\left[u={x}^{-\frac{1}{2}}y,v=y\right],\left[x=3,y=2\right]\right)$
 ${\mathrm{s2}}{≔}\left[{PuisⅇuxSⅇriⅇs of}\left({2}{+}\frac{{2}{}{y}}{\sqrt{{x}}}{+}{2}{}{y}\right){}{{x}}^{{3}}{}{{y}}^{{2}}{:}{2}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{2}{}{{x}}^{{5}}{{2}}}{}{{y}}^{{3}}{+}{2}{}{{y}}^{{3}}{}{{x}}^{{3}}\right]$ (10)
 > $\mathrm{s3}≔\mathrm{PuiseuxSeries}\left(\mathrm{PowerSeries}\left(\frac{1}{1+uv}\right),\left[y,x\right],\left[u,v\right],\left[\left[1,0\right],\left[1,-\frac{1}{2}\right]\right]\right)$
 ${\mathrm{s3}}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{1}}{\frac{{{y}}^{{2}}}{\sqrt{{x}}}{+}{1}}{:}{1}{+}{\dots }\right]$ (11)

We multiply $\mathrm{s1}$ and $\mathrm{s2}$.

 > $\mathrm{s1}\mathrm{s2}$
 $\left[{PuisⅇuxSⅇriⅇs of}\frac{\left({2}{+}\frac{{2}{}{y}}{\sqrt{{x}}}{+}{2}{}{y}\right){}{{x}}^{{6}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}{{y}}^{{2}}}{:}\frac{{2}{}{{x}}^{{6}}}{{{y}}^{{2}}}{+}{\dots }\right]$ (12)

We multiply $\mathrm{s1}$ with a polynomial.

 > $\mathrm{s1}\left(1+xy\right)$
 $\left[{PuisⅇuxSⅇriⅇs of}\frac{\left({x}{}{y}{+}{1}\right){}{{x}}^{{3}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}{{y}}^{{4}}}{:}\frac{{{x}}^{{3}}}{{{y}}^{{4}}}{+}{\dots }\right]$ (13)

We can multiply $\mathrm{s1}$ and the power series $a$. The result is a Puiseux series.

 > $\mathrm{s1}a$
 $\left[{PuisⅇuxSⅇriⅇs of}\frac{{{x}}^{{3}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}\left({1}{-}{x}{-}{y}\right){}{{y}}^{{4}}}{:}\frac{{{x}}^{{3}}}{{{y}}^{{4}}}{+}{\dots }\right]$ (14)

We can also multiply $\mathrm{s1}$ and the univariate polynomial over power series $f$. The result is again a Puiseux series.

 > $\mathrm{s1}f$
 $\left[{PuisⅇuxSⅇriⅇs of}\frac{\left({x}{}{\dots }{}{\dots }{+}{y}{}{{z}}^{{2}}{+}{x}{}{z}\right){}{{x}}^{{3}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}{{y}}^{{4}}}{:}{0}{+}{\dots }\right]$ (15)

We get an error if we try to multiply $\mathrm{s1}$ and $\mathrm{s3}$, since the orders [x,y] and [y,x] are not compatible.

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

We can use the command GetPuiseuxSeriesOrder to obtain the Puiseux series order of $\mathrm{s1}$ and $\mathrm{s3}$.

 > $\mathrm{GetPuiseuxSeriesOrder}\left(\mathrm{s1}\right)$
 $\left[{x}{,}{y}\right]$ (16)
 > $\mathrm{GetPuiseuxSeriesOrder}\left(\mathrm{s3}\right)$
 $\left[{y}{,}{x}\right]$ (17)

Finally, we create a univariate polynomial over power series from a list of Puiseux series.

 > $u≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${u}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (18)

Now we multiply $u$ with $f$.

 > $uf$
 $\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({0}\right){+}\left({0}{+}{\dots }\right){}{z}{+}\left({0}{+}{\dots }\right){}{{z}}^{{2}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{5}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{6}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{7}}\right]$ (19)

Compatibility

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