The command TaylorShift(p, e) returns the power series obtained by substituting x + c for x in p for each equation x = c in e. This can only be computed from the analytic expression of p. If the analytic expression for p is not known, an error is signaled.

The command TaylorShift(u, c) returns the univariate polynomial over power series obtained by substituting v + c for v in u, where v is the main variable of u.

A typical usage is when c is a root of the polynomial returned by EvaluateAtOrigin(u). This happens, for example, in HenselFactorize.

This command is supported for univariate polynomials over power series, but not for univariate polynomials over Puiseux 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.

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

$\mathrm{sinx}≔\mathrm{PowerSeries}\left(d\mapsto \mathrm{ifelse}\left(d::\mathrm{even}\,0\,\frac{{\left(\mathrm{-1}\right)}^{\frac{d}{2}-\frac{1}{2}}\cdot x^d}{d!}\right)\,\mathrm{analytic}=\sin \left(x\right)\right)$

$\mathrm{sinx}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\sin \left(x\right)\mathrm{:}0+\mathrm{\dots }\right]$

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

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

$\mathrm{ps}≔\mathrm{sinx}\mathrm{expy}$

$\mathrm{ps}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\sin \left(x\right){\ⅇ}^y\mathrm{:}0+\mathrm{\dots }\right]$

$\mathrm{ps\_both}≔\mathrm{TaylorShift}\left(\mathrm{ps}\,\left\{x=1\,y=2\right\}\right)$

$\mathrm{ps\_both}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^y{\ⅇ}^2\sin \left(x\right)\cos \left(1\right)+{\ⅇ}^y\mathrm{\dots }\mathrm{\dots }\mathrm{\dots }\mathrm{:}{\ⅇ}^2\sin \left(1\right)+{\ⅇ}^2\cos \left(1\right)x+{\ⅇ}^2\sin \left(1\right)y+{\ⅇ}^2\cos \left(1\right)xy+\frac{{\ⅇ}^2\sin \left(1\right)y^2}{2}-\frac{{\ⅇ}^2\sin \left(1\right)x^2}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^2}{2}-\frac{{\ⅇ}^2\cos \left(1\right)x^3}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^3}{6}-\frac{{\ⅇ}^2\sin \left(1\right)x^{2}y}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^3}{6}-\frac{{\ⅇ}^2\cos \left(1\right)x^{3}y}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^4}{24}-\frac{{\ⅇ}^2\sin \left(1\right)x^2{y}^2}{4}+\frac{{\ⅇ}^2\sin \left(1\right)x^4}{24}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^4}{24}-\frac{{\ⅇ}^2\cos \left(1\right)x^3{y}^2}{12}+\frac{{\ⅇ}^2\cos \left(1\right)x^5}{120}+\frac{{\ⅇ}^2\sin \left(1\right)y^5}{120}-\frac{{\ⅇ}^2\sin \left(1\right)x^2{y}^3}{12}+\frac{{\ⅇ}^2\sin \left(1\right)x^{4}y}{24}+\mathrm{\dots }\right]$

$\mathrm{ps\_x}≔\mathrm{TaylorShift}\left(\mathrm{ps}\,x=1\right)$

$\mathrm{ps\_x}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^y\sin \left(x\right)\cos \left(1\right)+{\ⅇ}^y\cos \left(x\right)\mathrm{\dots }\mathrm{\dots }\mathrm{:}\sin \left(1\right)+\cos \left(1\right)x+\sin \left(1\right)y+\cos \left(1\right)xy+\frac{\sin \left(1\right)y^2}{2}-\frac{\sin \left(1\right)x^2}{2}+\frac{\cos \left(1\right)x{y}^2}{2}-\frac{\cos \left(1\right)x^3}{6}+\frac{\sin \left(1\right)y^3}{6}-\frac{\sin \left(1\right)x^{2}y}{2}+\frac{\cos \left(1\right)x{y}^3}{6}-\frac{\cos \left(1\right)x^{3}y}{6}+\frac{\sin \left(1\right)y^4}{24}-\frac{\sin \left(1\right)x^2{y}^2}{4}+\frac{\sin \left(1\right)x^4}{24}+\frac{\cos \left(1\right)x{y}^4}{24}-\frac{\cos \left(1\right)x^3{y}^2}{12}+\frac{\cos \left(1\right)x^5}{120}+\frac{\sin \left(1\right)y^5}{120}-\frac{\sin \left(1\right)x^2{y}^3}{12}+\frac{\sin \left(1\right)x^{4}y}{24}+\mathrm{\dots }\right]$

$\mathrm{ps\_x\_y}≔\mathrm{TaylorShift}\left(\mathrm{ps\_x}\,y=2\right)$

$\mathrm{ps\_x\_y}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^y{\ⅇ}^2\sin \left(x\right)\cos \left(1\right)+{\ⅇ}^y\mathrm{\dots }\mathrm{\dots }\mathrm{\dots }\mathrm{:}{\ⅇ}^2\sin \left(1\right)+{\ⅇ}^2\cos \left(1\right)x+{\ⅇ}^2\sin \left(1\right)y+{\ⅇ}^2\cos \left(1\right)xy+\frac{{\ⅇ}^2\sin \left(1\right)y^2}{2}-\frac{{\ⅇ}^2\sin \left(1\right)x^2}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^2}{2}-\frac{{\ⅇ}^2\cos \left(1\right)x^3}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^3}{6}-\frac{{\ⅇ}^2\sin \left(1\right)x^{2}y}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^3}{6}-\frac{{\ⅇ}^2\cos \left(1\right)x^{3}y}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^4}{24}-\frac{{\ⅇ}^2\sin \left(1\right)x^2{y}^2}{4}+\frac{{\ⅇ}^2\sin \left(1\right)x^4}{24}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^4}{24}-\frac{{\ⅇ}^2\cos \left(1\right)x^3{y}^2}{12}+\frac{{\ⅇ}^2\cos \left(1\right)x^5}{120}+\frac{{\ⅇ}^2\sin \left(1\right)y^5}{120}-\frac{{\ⅇ}^2\sin \left(1\right)x^2{y}^3}{12}+\frac{{\ⅇ}^2\sin \left(1\right)x^{4}y}{24}+\mathrm{\dots }\right]$

$\mathrm{ps\_y}≔\mathrm{TaylorShift}\left(\mathrm{ps}\,y=2\right)$

$\mathrm{ps\_y}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\sin \left(x\right){\ⅇ}^y{\ⅇ}^2\mathrm{:}{\ⅇ}^{2}x+{\ⅇ}^{2}xy+\frac{{\ⅇ}^{2}x{y}^2}{2}-\frac{{\ⅇ}^2{x}^3}{6}+\frac{{\ⅇ}^{2}x{y}^3}{6}-\frac{{\ⅇ}^2{x}^{3}y}{6}+\frac{{\ⅇ}^{2}x{y}^4}{24}-\frac{{\ⅇ}^2{x}^3{y}^2}{12}+\frac{{\ⅇ}^2{x}^5}{120}+\mathrm{\dots }\right]$

$\mathrm{ps\_y\_x}≔\mathrm{TaylorShift}\left(\mathrm{ps\_y}\,x=1\right)$

$\mathrm{ps\_y\_x}≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}{\ⅇ}^y{\ⅇ}^2\sin \left(x\right)\cos \left(1\right)+{\ⅇ}^y\mathrm{\dots }\mathrm{\dots }\mathrm{\dots }\mathrm{:}{\ⅇ}^2\sin \left(1\right)+{\ⅇ}^2\cos \left(1\right)x+{\ⅇ}^2\sin \left(1\right)y+{\ⅇ}^2\cos \left(1\right)xy+\frac{{\ⅇ}^2\sin \left(1\right)y^2}{2}-\frac{{\ⅇ}^2\sin \left(1\right)x^2}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^2}{2}-\frac{{\ⅇ}^2\cos \left(1\right)x^3}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^3}{6}-\frac{{\ⅇ}^2\sin \left(1\right)x^{2}y}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^3}{6}-\frac{{\ⅇ}^2\cos \left(1\right)x^{3}y}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^4}{24}-\frac{{\ⅇ}^2\sin \left(1\right)x^2{y}^2}{4}+\frac{{\ⅇ}^2\sin \left(1\right)x^4}{24}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^4}{24}-\frac{{\ⅇ}^2\cos \left(1\right)x^3{y}^2}{12}+\frac{{\ⅇ}^2\cos \left(1\right)x^5}{120}+\frac{{\ⅇ}^2\sin \left(1\right)y^5}{120}-\frac{{\ⅇ}^2\sin \left(1\right)x^2{y}^3}{12}+\frac{{\ⅇ}^2\sin \left(1\right)x^{4}y}{24}+\mathrm{\dots }\right]$

$\mathrm{Truncate}\left(\mathrm{ps\_both}\,4\right)$

${\ⅇ}^2\sin \left(1\right)+{\ⅇ}^2\cos \left(1\right)x+{\ⅇ}^2\sin \left(1\right)y+{\ⅇ}^2\cos \left(1\right)xy+\frac{{\ⅇ}^2\sin \left(1\right)y^2}{2}-\frac{{\ⅇ}^2\sin \left(1\right)x^2}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^2}{2}-\frac{{\ⅇ}^2\cos \left(1\right)x^3}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^3}{6}-\frac{{\ⅇ}^2\sin \left(1\right)x^{2}y}{2}+\frac{{\ⅇ}^2\cos \left(1\right)x{y}^3}{6}-\frac{{\ⅇ}^2\cos \left(1\right)x^{3}y}{6}+\frac{{\ⅇ}^2\sin \left(1\right)y^4}{24}-\frac{{\ⅇ}^2\sin \left(1\right)x^2{y}^2}{4}+\frac{{\ⅇ}^2\sin \left(1\right)x^4}{24}$

$\mathrm{ApproximatelyEqual}\left(\mathrm{ps\_both}\,\mathrm{ps\_x\_y}\,20\right)$

$\mathrm{true}$

$\mathrm{ApproximatelyEqual}\left(\mathrm{ps\_both}\,\mathrm{ps\_y\_x}\,20\right)$

$\mathrm{true}$

$f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left[\mathrm{PowerSeries}\left(1\right)\,\mathrm{SumOfAllMonomials}\left(\left[x\,y\right]\right)\,\mathrm{GeometricSeries}\left(y\right)\right]\,z\right)$

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

$\mathrm{f1}≔\mathrm{TaylorShift}\left(f\,1\right)$

$\mathrm{f1}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(3+\mathrm{\dots }\right)\mathrm{+}\left(3+x+3y+\mathrm{\dots }\right)z\mathrm{+}\left(1+y+\mathrm{\dots }\right)z^2\right]$

$\mathrm{f0}≔\mathrm{TaylorShift}\left(\mathrm{f1}\,-1\right)$

$\mathrm{f0}≔\left[\mathrm{Univariat\ⅇPolynomialOv\ⅇrPow\ⅇrS\ⅇri\ⅇs:}\left(1\right)\mathrm{+}\left(1+x+y+\mathrm{\dots }\right)z\mathrm{+}\left(1+y+\mathrm{\dots }\right)z^2\right]$

$\mathrm{ApproximatelyEqual}\left(f\,\mathrm{f0}\,20\right)$

$\mathrm{true}$

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

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

The p and e parameters were introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.