GetAnalyticExpression - Maple Help

MultivariatePowerSeries

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

 Calling Sequence GetAnalyticExpression(p) GetAnalyticExpression(u) GetAnalyticExpression(s)

Parameters

 p - power series generated by this package u - univariate polynomial over power series or over Puiseux series generated by this package s - Puiseux 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 or over a Puiseux series, if it is known. This is determined in the natural way from the analytic expressions of each coefficient of s.
 • The command GetAnalyticExpression(s) returns the analytic expression a Puiseux series s, if it is known. This is determined by taking the analytic expressions of its internal power series and applying the change of variables to it.
 • If an analytic expression is known for a power series or a Puiseux series, it is part of the default display of that power series. This is not the case for univariate polynomials over power series.
 • 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 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)

Create a Puiseux series in $x$ and $y$. Extract its analytic expression.

 > $s≔\mathrm{PuiseuxSeries}\left(\mathrm{PowerSeries}\left(\frac{1}{1+U}\right),\left[U={x}^{-\frac{1}{3}}{y}^{2}\right],\left[x=3,y=-4\right]\right)$
 ${s}{≔}\left[{PuisⅇuxSⅇriⅇs of}\frac{{{x}}^{{3}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}{{y}}^{{4}}}{:}\frac{{{x}}^{{3}}}{{{y}}^{{4}}}{+}{\dots }\right]$ (17)
 > $\mathrm{GetAnalyticExpression}\left(s\right)$
 $\frac{{{x}}^{{3}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}{{y}}^{{4}}}$ (18)

We can get the internal power series of $s$, get its analytic expression, apply the change of variables given by $U=\frac{{y}^{2}}{{x}^{1}{3}}}$ and multiply this by $\frac{{x}^{3}}{{y}^{4}}$.

 > $p≔\mathrm{GetPowerSeries}\left(s\right)$
 ${p}{≔}\left[{PowⅇrSⅇriⅇs of}\frac{{1}}{{1}{+}{U}}{:}{1}{+}{\dots }\right]$ (19)
 > $\frac{{x}^{3}}{{y}^{4}}\left(\mathrm{eval}\left(\mathrm{GetAnalyticExpression}\left(p\right),U={x}^{-\frac{1}{3}}{y}^{2}\right)\right)$
 $\frac{{{x}}^{{3}}}{\left({1}{+}\frac{{{y}}^{{2}}}{{{x}}^{{1}}{{3}}}}\right){}{{y}}^{{4}}}$ (20)

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

 > $h≔\mathrm{UnivariatePolynomialOverPuiseuxSeries}\left(\left[\mathrm{PuiseuxSeries}\left(1\right),\mathrm{PuiseuxSeries}\left(0\right),\mathrm{PuiseuxSeries}\left(x,\left[x={x}^{\frac{1}{3}}\right]\right),\mathrm{PuiseuxSeries}\left(y,\left[y={y}^{\frac{1}{2}}\right]\right),\mathrm{PuiseuxSeries}\left(\frac{x+y}{1+x+y},\left[x=x{y}^{\frac{1}{2}},y=x{y}^{-1}\right]\right)\right],z\right)$
 ${h}{≔}\left[{UnivariatⅇPolynomialOvⅇrPuisⅇuxSⅇriⅇs:}\left({1}\right){+}\left({0}\right){}{z}{+}\left({{x}}^{{1}}{{3}}}\right){}{{z}}^{{2}}{+}\left(\sqrt{{y}}\right){}{{z}}^{{3}}{+}\left({0}{+}{\dots }\right){}{{z}}^{{4}}\right]$ (21)

We get the analytic expression of $h$.

 > $\mathrm{GetAnalyticExpression}\left(h\right)$
 ${1}{+}{{x}}^{{1}}{{3}}}{}{{z}}^{{2}}{+}\sqrt{{y}}{}{{z}}^{{3}}{+}\frac{\left({x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}\right){}{{z}}^{{4}}}{{1}{+}{x}{}\sqrt{{y}}{+}\frac{{x}}{{y}}}$ (22)

Compatibility

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