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.

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

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

$\mathrm{GetAnalyticExpression}\left(a\right)$

$\frac{1}{1-x-y}$

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

$\mathrm{GetAnalyticExpression}\left(b\right)$

$\frac{1}{3+2x+y}$

$c≔ab$

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

$\mathrm{GetAnalyticExpression}\left(c\right)$

$\frac{1}{\left(1-x-y\right)\left(3+2x+y\right)}$

$e≔\mathrm{PowerSeries}\left(d\mapsto \frac{x^d}{d!}\,\mathrm{variables}=\left\{x\right\}\right)$

$e≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}1+\mathrm{\dots }\right]$

$\mathrm{GetAnalyticExpression}\left(e\right)$

$\mathrm{undefined}$

$f≔\mathrm{PowerSeries}\left(d\mapsto \frac{x^d}{d!}\,\mathrm{analytic}=\exp \left(x\right)\right)$

$f≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^x\mathrm{:}1+\mathrm{\dots }\right]$

$\mathrm{GetAnalyticExpression}\left(f\right)$

${\ⅇ}^x$

$g≔a+\frac{b}{e}$

$g≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs:}\frac{4}{3}+\mathrm{\dots }\right]$

$\mathrm{GetAnalyticExpression}\left(g\right)$

$\mathrm{undefined}$

$h≔a+\frac{b}{f}$

$h≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x-y}+\frac{1}{\left(3+2x+y\right){\ⅇ}^x}\mathrm{:}\frac{4}{3}+\mathrm{\dots }\right]$

$\mathrm{GetAnalyticExpression}\left(h\right)$

$\frac{1}{1-x-y}+\frac{1}{\left(3+2x+y\right){\ⅇ}^x}$

$u≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[a\,b\,e\right]\,z\right)$

$u≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1+x+y+\mathrm{\dots }\right)\mathrm{+}\left(\frac{1}{3}+\mathrm{\dots }\right)z\mathrm{+}\left(1+\mathrm{\dots }\right)z^2\right]$

$\mathrm{GetAnalyticExpression}\left(u\right)$

$\mathrm{undefined}$

$v≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[a\,b\,f\right]\,z\right)$

$v≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1+x+y+\mathrm{\dots }\right)\mathrm{+}\left(\frac{1}{3}+\mathrm{\dots }\right)z\mathrm{+}\left(1+\mathrm{\dots }\right)z^2\right]$

$\mathrm{GetAnalyticExpression}\left(v\right)$

$\frac{1}{1-x-y}+\frac{z}{3+2x+y}+{\ⅇ}^x{z}^2$

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

$s≔\left[\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]$

$\mathrm{GetAnalyticExpression}\left(s\right)$

$\frac{x^3}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y^4}$

$p≔\mathrm{GetPowerSeries}\left(s\right)$

$p≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1+U}\mathrm{:}1+\mathrm{\dots }\right]$

$\frac{x^3}{y^4}\left(\mathrm{eval}\left(\mathrm{GetAnalyticExpression}\left(p\right)\,U=x^{-\frac{1}{3}}{y}^2\right)\right)$

$\frac{x^3}{\left(1+\frac{y^2}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y^4}$

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

$h≔\left[\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]$

$\mathrm{GetAnalyticExpression}\left(h\right)$

$1+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{z}^2+\sqrt{y}{z}^3+\frac{\left(x\sqrt{y}+\frac{x}{y}\right)z^4}{1+x\sqrt{y}+\frac{x}{y}}$

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

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

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

The s parameter was introduced in Maple 2023.

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