MultivariatePowerSeries/Display - Maple Help

MultivariatePowerSeries

 Display
 Nicely display a power series or univariate polynomial over power series

 Calling Sequence Display(p, d) Display(u, d)

Parameters

 p - power series generated by this package u - univariate polynomial over power series generated by this package d - (optional) list of equation(s) of the form option = value where option can be maxterms, precision, or (if the argument is a univariate polynomial over power series) maxdegree, and value is a nonnegative integer or the symbol infinity

Description

 • Display(p,d) displays the power series p in the style given by d. The argument d is a list of equations which contains either  maxterms=n or precision=m, or both. In those equations n and m are of type either nonnegint or infinity.
 • If d is not provided, then the display style of p is taken from an earlier call of the form SetDisplayStyle(p,d), if any. Otherwise, the display style of p is taken from an earlier call of the form SetDefaultDisplayStyle(d), if any. Finally, if the given display style doesn't provide values for one (or both) of the settings, Maple uses the default values maxterms=50 and precision=infinity.
 • The attribute maxterms sets the maximum number of terms of p to be displayed while the the attribute precision sets the maximum degree of the displayed terms.
 • Using infinity for either n or m indicates that no limits for the number of terms, or their degree, is set.
 • Display(u,d) displays the coefficients of the univariate polynomial over power series u in in the style given by d.  The argument d can take the same entries as for Display(p, d), but additionally, the option maxdegree=n can be used: it limits the maximum degree  of a displayed term with respect to the main variable of u only. Like for power series, Display uses d if provided; or otherwise the argument to an earlier call to SetDisplayStyle(u, d), if any; or otherwise the argument to SetDefaultDisplayStyle(d), if any; or finally, the same default values maxterms=50 and precision=infinity and additionally maxdegree=infinity.
 • The limit on the number of terms enforced by the maxterms option is enforced for all terms of the coefficients of u together, the terms of the coefficient power series are not counted independently. Only nonzero terms are counted.
 • The command Display will only ever display coefficients that were computed before; it does not cause computation of further coefficients.
 • 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

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

Define a power series in $x$ and $y$. Initially, only the constant and linear terms are computed, so these are shown.

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

If we compute more terms (in this case, up to homogeneous degree 10), more terms will be displayed.

 > $\mathrm{UpdatePrecision}\left(a,10\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{4}{}{{x}}^{{3}}{}{y}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{4}{}{x}{}{{y}}^{{3}}{+}{{y}}^{{4}}{+}{{x}}^{{5}}{+}{5}{}{{x}}^{{4}}{}{y}{+}{10}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{10}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{5}{}{x}{}{{y}}^{{4}}{+}{{y}}^{{5}}{+}{{x}}^{{6}}{+}{6}{}{{x}}^{{5}}{}{y}{+}{15}{}{{x}}^{{4}}{}{{y}}^{{2}}{+}{20}{}{{x}}^{{3}}{}{{y}}^{{3}}{+}{15}{}{{x}}^{{2}}{}{{y}}^{{4}}{+}{6}{}{x}{}{{y}}^{{5}}{+}{{y}}^{{6}}{+}{{x}}^{{7}}{+}{7}{}{{x}}^{{6}}{}{y}{+}{21}{}{{x}}^{{5}}{}{{y}}^{{2}}{+}{35}{}{{x}}^{{4}}{}{{y}}^{{3}}{+}{35}{}{{x}}^{{3}}{}{{y}}^{{4}}{+}{21}{}{{x}}^{{2}}{}{{y}}^{{5}}{+}{7}{}{x}{}{{y}}^{{6}}{+}{{y}}^{{7}}{+}{{x}}^{{8}}{+}{8}{}{{x}}^{{7}}{}{y}{+}{28}{}{{x}}^{{6}}{}{{y}}^{{2}}{+}{56}{}{{x}}^{{5}}{}{{y}}^{{3}}{+}{70}{}{{x}}^{{4}}{}{{y}}^{{4}}{+}{56}{}{{x}}^{{3}}{}{{y}}^{{5}}{+}{28}{}{{x}}^{{2}}{}{{y}}^{{6}}{+}{8}{}{x}{}{{y}}^{{7}}{+}{{y}}^{{8}}{+}\left({{x}}^{{9}}{+}{9}{}{{x}}^{{8}}{}{y}{+}{36}{}{{x}}^{{7}}{}{{y}}^{{2}}{+}{84}{}{{x}}^{{6}}{}{{y}}^{{3}}{+}{126}{}{{x}}^{{5}}{}{{y}}^{{4}}{+}{\dots }\right){+}{\dots }\right]$ (3)

The following calling sequence shows up to 20 terms and up to homogeneous degree 5. For this power series, there are a total of 21 terms of homogeneous degree less than or equal to 5, so it shows all but one of them.

 > $\mathrm{Display}\left(a,\left[\mathrm{maxterms}=20,\mathrm{precision}=5\right]\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{4}{}{{x}}^{{3}}{}{y}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{4}{}{x}{}{{y}}^{{3}}{+}{{y}}^{{4}}{+}\left({{x}}^{{5}}{+}{5}{}{{x}}^{{4}}{}{y}{+}{10}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{10}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{5}{}{x}{}{{y}}^{{4}}{+}{\dots }\right){+}{\dots }\right]$ (4)

If we omit the $\mathrm{maxterms}$ parameter, its default value is 50, so the following command shows all terms of homogeneous degree less than or equal to 5.

 > $\mathrm{Display}\left(a,\left[\mathrm{precision}=5\right]\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{4}{}{{x}}^{{3}}{}{y}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{4}{}{x}{}{{y}}^{{3}}{+}{{y}}^{{4}}{+}{{x}}^{{5}}{+}{5}{}{{x}}^{{4}}{}{y}{+}{10}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{10}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{5}{}{x}{}{{y}}^{{4}}{+}{{y}}^{{5}}{+}{\dots }\right]$ (5)

We can set the $\mathrm{maxterms}$ and $\mathrm{precision}$ parameters to $\mathrm{\infty }$ to show all currently computed terms.

 > $\mathrm{Display}\left(a,\left[\mathrm{maxterms}=\mathrm{\infty },\mathrm{precision}=\mathrm{\infty }\right]\right)$
 $\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{:}{1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{4}{}{{x}}^{{3}}{}{y}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{4}{}{x}{}{{y}}^{{3}}{+}{{y}}^{{4}}{+}{{x}}^{{5}}{+}{5}{}{{x}}^{{4}}{}{y}{+}{10}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{10}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{5}{}{x}{}{{y}}^{{4}}{+}{{y}}^{{5}}{+}{{x}}^{{6}}{+}{6}{}{{x}}^{{5}}{}{y}{+}{15}{}{{x}}^{{4}}{}{{y}}^{{2}}{+}{20}{}{{x}}^{{3}}{}{{y}}^{{3}}{+}{15}{}{{x}}^{{2}}{}{{y}}^{{4}}{+}{6}{}{x}{}{{y}}^{{5}}{+}{{y}}^{{6}}{+}{{x}}^{{7}}{+}{7}{}{{x}}^{{6}}{}{y}{+}{21}{}{{x}}^{{5}}{}{{y}}^{{2}}{+}{35}{}{{x}}^{{4}}{}{{y}}^{{3}}{+}{35}{}{{x}}^{{3}}{}{{y}}^{{4}}{+}{21}{}{{x}}^{{2}}{}{{y}}^{{5}}{+}{7}{}{x}{}{{y}}^{{6}}{+}{{y}}^{{7}}{+}{{x}}^{{8}}{+}{8}{}{{x}}^{{7}}{}{y}{+}{28}{}{{x}}^{{6}}{}{{y}}^{{2}}{+}{56}{}{{x}}^{{5}}{}{{y}}^{{3}}{+}{70}{}{{x}}^{{4}}{}{{y}}^{{4}}{+}{56}{}{{x}}^{{3}}{}{{y}}^{{5}}{+}{28}{}{{x}}^{{2}}{}{{y}}^{{6}}{+}{8}{}{x}{}{{y}}^{{7}}{+}{{y}}^{{8}}{+}{{x}}^{{9}}{+}{9}{}{{x}}^{{8}}{}{y}{+}{36}{}{{x}}^{{7}}{}{{y}}^{{2}}{+}{84}{}{{x}}^{{6}}{}{{y}}^{{3}}{+}{126}{}{{x}}^{{5}}{}{{y}}^{{4}}{+}{126}{}{{x}}^{{4}}{}{{y}}^{{5}}{+}{84}{}{{x}}^{{3}}{}{{y}}^{{6}}{+}{36}{}{{x}}^{{2}}{}{{y}}^{{7}}{+}{9}{}{x}{}{{y}}^{{8}}{+}{{y}}^{{9}}{+}{{x}}^{{10}}{+}{10}{}{{x}}^{{9}}{}{y}{+}{45}{}{{x}}^{{8}}{}{{y}}^{{2}}{+}{120}{}{{x}}^{{7}}{}{{y}}^{{3}}{+}{210}{}{{x}}^{{6}}{}{{y}}^{{4}}{+}{252}{}{{x}}^{{5}}{}{{y}}^{{5}}{+}{210}{}{{x}}^{{4}}{}{{y}}^{{6}}{+}{120}{}{{x}}^{{3}}{}{{y}}^{{7}}{+}{45}{}{{x}}^{{2}}{}{{y}}^{{8}}{+}{10}{}{x}{}{{y}}^{{9}}{+}{{y}}^{{10}}{+}{\dots }\right]$ (6)

We define a univariate polynomial over power series. Its coefficients have very few terms precomputed, so Display doesn't show much.

 > $f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{SumOfAllMonomials}\left(\left[x\right]\right),\mathrm{GeometricSeries}\left(\left[x,y\right]\right)\right],z\right):$
 > $\mathrm{Display}\left(f\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{\dots }\right){+}\left({1}{+}{x}{+}{y}{+}{\dots }\right){}{z}\right]$ (7)

If we increase the precision (computing more terms), then Display will show more terms.

 > $\mathrm{UpdatePrecision}\left(f,10\right):$
 > $\mathrm{Display}\left(f\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{{x}}^{{2}}{+}{{x}}^{{3}}{+}{{x}}^{{4}}{+}{{x}}^{{5}}{+}{{x}}^{{6}}{+}{{x}}^{{7}}{+}{{x}}^{{8}}{+}{{x}}^{{9}}{+}{{x}}^{{10}}{+}{\dots }\right){+}\left({1}{+}{x}{+}{y}{+}{{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{{x}}^{{3}}{+}{3}{}{{x}}^{{2}}{}{y}{+}{3}{}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{4}{}{{x}}^{{3}}{}{y}{+}{6}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{4}{}{x}{}{{y}}^{{3}}{+}{{y}}^{{4}}{+}{{x}}^{{5}}{+}{5}{}{{x}}^{{4}}{}{y}{+}{10}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{10}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{5}{}{x}{}{{y}}^{{4}}{+}{{y}}^{{5}}{+}{{x}}^{{6}}{+}{6}{}{{x}}^{{5}}{}{y}{+}{15}{}{{x}}^{{4}}{}{{y}}^{{2}}{+}{20}{}{{x}}^{{3}}{}{{y}}^{{3}}{+}{15}{}{{x}}^{{2}}{}{{y}}^{{4}}{+}{6}{}{x}{}{{y}}^{{5}}{+}{{y}}^{{6}}{+}{{x}}^{{7}}{+}{7}{}{{x}}^{{6}}{}{y}{+}{21}{}{{x}}^{{5}}{}{{y}}^{{2}}{+}{35}{}{{x}}^{{4}}{}{{y}}^{{3}}{+}{35}{}{{x}}^{{3}}{}{{y}}^{{4}}{+}{21}{}{{x}}^{{2}}{}{{y}}^{{5}}{+}{7}{}{x}{}{{y}}^{{6}}{+}{{y}}^{{7}}{+}\left({{x}}^{{8}}{+}{8}{}{{x}}^{{7}}{}{y}{+}{28}{}{{x}}^{{6}}{}{{y}}^{{2}}{+}{\dots }\right){+}{\dots }\right){}{z}\right]$ (8)

The $\mathrm{maxdegree}$ option cuts the display off after displaying the constant coefficient of the main variable, $z$.

 > $\mathrm{Display}\left(f,\left[\mathrm{maxdegree}=0,\mathrm{precision}=5\right]\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{{x}}^{{2}}{+}{{x}}^{{3}}{+}{{x}}^{{4}}{+}{{x}}^{{5}}{+}{\dots }\right){+}{\dots }\right]$ (9)

The SetDisplayStyle command makes future calls to Display use a given set of parameters.

 > $\mathrm{SetDisplayStyle}\left(f,\left[\mathrm{maxterms}=5\right]\right)$
 $\left[{\mathrm{maxterms}}{=}{5}\right]$ (10)
 > $\mathrm{Display}\left(f\right)$
 $\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{{x}}^{{2}}{+}{{x}}^{{3}}{+}{{x}}^{{4}}{+}{\dots }\right){+}{\dots }\right]$ (11)

Compatibility

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