MultivariatePowerSeries/GetAnalyticExpression - Maple Help

MultivariatePowerSeries

 GetAnalyticExpression
 Get an analytic expression for a power series or a univariate polynomial over power series

 Calling Sequence GetAnalyticExpression(p) GetAnalyticExpression(u)

Parameters

 p - power series generated by this package u - univariate polynomial over power series generated by this package

Description

 • The command GetAnalyticExpression(p) returns the analytic expression of the power series p, if it is known, or undefined if it is not known.
 • The power series that have a known analytic expression are the following:
 – those created with the commands GeometricSeries and SumOfAllMonomials;
 – those created with the command PowerSeries from a polynomial, or from a procedure while specifying the analytic expression explicitly;
 – those obtained by applying arithmetic operations (addition, multiplication, inversion, exponentiation) to power series whose arithmetic expression is known.
 • The command  GetAnalyticExpression(u) returns the analytic expression of univariate polynomial u over power series, if it is known. This is determined in the natural way from the analytic expressions of each coefficient of u.
 • If an analytic expression is known for a power series, it is part of the default display of that power series. This is not the case for univariate polynomials for power series.
 • 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 a geometric power series in $x$ and $y$. Extract its analytic expression.

 > $a≔\mathrm{GeometricSeries}\left(\left[x,y\right]\right):$
 > $\mathrm{GetAnalyticExpression}\left(a\right)$
 $\frac{{1}}{{1}{-}{x}{-}{y}}$ (1)

Create a power series for a rational function. Extract its analytic expression.

 > $b≔\mathrm{Inverse}\left(\mathrm{PowerSeries}\left(3+2x+y\right)\right):$
 > $\mathrm{GetAnalyticExpression}\left(b\right)$
 $\frac{{1}}{{3}{+}{2}{}{x}{+}{y}}$ (2)

Define the product of $a$ and $b$. Its analytic expression is known because the analytic expressions for both $a$ and $b$ are known.

 > $c≔ab$
 ${c}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{\left({1}{-}{x}{-}{y}\right){}\left({3}{+}{2}{}{x}{+}{y}\right)}{:}\frac{{1}}{{3}}{+}{\dots }\right]$ (3)
 > $\mathrm{GetAnalyticExpression}\left(c\right)$
 $\frac{{1}}{\left({1}{-}{x}{-}{y}\right){}\left({3}{+}{2}{}{x}{+}{y}\right)}$ (4)

Below, $e$ and $f$ are defined as the same power series, but $f$ knows its analytic expression and $e$ doesn't.

 > $e≔\mathrm{PowerSeries}\left(d↦\frac{{x}^{d}}{d!},\mathrm{variables}=\left\{x\right\}\right)$
 ${e}{≔}\left[{PowⅇrSⅇriⅇs:}{1}{+}{\dots }\right]$ (5)
 > $\mathrm{GetAnalyticExpression}\left(e\right)$
 ${\mathrm{undefined}}$ (6)
 > $f≔\mathrm{PowerSeries}\left(d↦\frac{{x}^{d}}{d!},\mathrm{analytic}=\mathrm{exp}\left(x\right)\right)$
 ${f}{≔}\left[{PowⅇrSⅇriⅇs of}{{ⅇ}}^{{x}}{:}{1}{+}{\dots }\right]$ (7)
 > $\mathrm{GetAnalyticExpression}\left(f\right)$
 ${{ⅇ}}^{{x}}$ (8)

If we create power series from $e$ and $f$ by arithmetic operations, then those involving $e$ do not know their analytic expressions, but those involving $f$ do (if the other power series involved know their analytic expressions). Below, $g$ and $h$ represent the same power series, but because $h$ used $f$ rather than $e$ in its definition, it knows its analytic expression.

 > $g≔a+\frac{b}{e}$
 ${g}{≔}\left[{PowⅇrSⅇriⅇs:}\frac{{4}}{{3}}{+}{\dots }\right]$ (9)
 > $\mathrm{GetAnalyticExpression}\left(g\right)$
 ${\mathrm{undefined}}$ (10)
 > $h≔a+\frac{b}{f}$
 ${h}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{-}{x}{-}{y}}{+}\frac{{1}}{\left({3}{+}{2}{}{x}{+}{y}\right){}{{ⅇ}}^{{x}}}{:}\frac{{4}}{{3}}{+}{\dots }\right]$ (11)
 > $\mathrm{GetAnalyticExpression}\left(h\right)$
 $\frac{{1}}{{1}{-}{x}{-}{y}}{+}\frac{{1}}{\left({3}{+}{2}{}{x}{+}{y}\right){}{{ⅇ}}^{{x}}}$ (12)

If we create a univariate polynomial over power series, it will know its analytic expression if each of the coefficients of the main variable knows its analytic expression. Below, $v$ knows its analytic expression, but $u$ doesn't.

 > $u≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[a,b,e\right],z\right)$
 ${u}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{y}{+}{\dots }\right){+}\left(\frac{{1}}{{3}}{+}{\dots }\right){}{z}{+}\left({1}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (13)
 > $\mathrm{GetAnalyticExpression}\left(u\right)$
 ${\mathrm{undefined}}$ (14)
 > $v≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[a,b,f\right],z\right)$
 ${v}{≔}\left[{UnivariatⅇPolynomialOvⅇrPowⅇrSⅇriⅇs:}\left({1}{+}{x}{+}{y}{+}{\dots }\right){+}\left(\frac{{1}}{{3}}{+}{\dots }\right){}{z}{+}\left({1}{+}{\dots }\right){}{{z}}^{{2}}\right]$ (15)
 > $\mathrm{GetAnalyticExpression}\left(v\right)$
 $\frac{{1}}{{1}{-}{x}{-}{y}}{+}\frac{{z}}{{3}{+}{2}{}{x}{+}{y}}{+}{{ⅇ}}^{{x}}{}{{z}}^{{2}}$ (16)

Compatibility

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