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MultivariatePowerSeries

  

TaylorShift

  

perform a Taylor shift of power series or a univariate polynomial over power series

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

TaylorShift(p, e)

TaylorShift(u, c)

Parameters

p

-

power series generated by this package

e

-

one or more equations of the form x = c, indicating that x should be shifted by c; e can be a single equation, or a list or set of equations.

c

-

numeric value or algebraic number

u

-

univariate polynomial over power series generated by this package -- note, univariate polynomials over Puiseux series are not supported

Description

• 

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.

Examples

withMultivariatePowerSeries:

We define a power series for sinxⅇy.

sinxPowerSeriesd→ifelsed::even,0,1d12xdd!,analytic=sinx

sinxPowⅇrSⅇrⅈⅇs of sinx : 0+

(1)

expyPowerSeriesd→ydd!,analytic=ⅇy

expyPowⅇrSⅇrⅈⅇs of ⅇy : 1+

(2)

pssinxexpy

psPowⅇrSⅇrⅈⅇs of sinxⅇy : 0+

(3)

We shift x by 1 and y by 2, performing all steps three times: all at once, x first, or y first. (In practice, one would typically do both shifts at once: it is computationally more efficient.)

ps_bothTaylorShiftps,x=1,y=2

ps_bothPowⅇrSⅇrⅈⅇs of ⅇyⅇ2sinxcos1+ⅇy : ⅇ2sin1+ⅇ2cos1x+ⅇ2sin1y+ⅇ2cos1xy+ⅇ2sin1y22ⅇ2sin1x22+ⅇ2cos1xy22ⅇ2cos1x36+ⅇ2sin1y36ⅇ2sin1x2y2+ⅇ2cos1xy36ⅇ2cos1x3y6+ⅇ2sin1y424ⅇ2sin1x2y24+ⅇ2sin1x424+ⅇ2cos1xy424ⅇ2cos1x3y212+ⅇ2cos1x5120+ⅇ2sin1y5120ⅇ2sin1x2y312+ⅇ2sin1x4y24+

(4)

ps_xTaylorShiftps,x=1

ps_xPowⅇrSⅇrⅈⅇs of ⅇysinxcos1+ⅇycosx : sin1+cos1x+sin1y+cos1xy+sin1y22sin1x22+cos1xy22cos1x36+sin1y36sin1x2y2+cos1xy36cos1x3y6+sin1y424sin1x2y24+sin1x424+cos1xy424cos1x3y212+cos1x5120+sin1y5120sin1x2y312+sin1x4y24+

(5)

ps_x_yTaylorShiftps_x,y=2

ps_x_yPowⅇrSⅇrⅈⅇs of ⅇyⅇ2sinxcos1+ⅇy : ⅇ2sin1+ⅇ2cos1x+ⅇ2sin1y+ⅇ2cos1xy+ⅇ2sin1y22ⅇ2sin1x22+ⅇ2cos1xy22ⅇ2cos1x36+ⅇ2sin1y36ⅇ2sin1x2y2+ⅇ2cos1xy36ⅇ2cos1x3y6+ⅇ2sin1y424ⅇ2sin1x2y24+ⅇ2sin1x424+ⅇ2cos1xy424ⅇ2cos1x3y212+ⅇ2cos1x5120+ⅇ2sin1y5120ⅇ2sin1x2y312+ⅇ2sin1x4y24+

(6)

ps_yTaylorShiftps,y=2

ps_yPowⅇrSⅇrⅈⅇs of sinxⅇyⅇ2 : ⅇ2x+ⅇ2xy+ⅇ2xy22ⅇ2x36+ⅇ2xy36ⅇ2x3y6+ⅇ2xy424ⅇ2x3y212+ⅇ2x5120+

(7)

ps_y_xTaylorShiftps_y,x=1

ps_y_xPowⅇrSⅇrⅈⅇs of ⅇyⅇ2sinxcos1+ⅇy : ⅇ2sin1+ⅇ2cos1x+ⅇ2sin1y+ⅇ2cos1xy+ⅇ2sin1y22ⅇ2sin1x22+ⅇ2cos1xy22ⅇ2cos1x36+ⅇ2sin1y36ⅇ2sin1x2y2+ⅇ2cos1xy36ⅇ2cos1x3y6+ⅇ2sin1y424ⅇ2sin1x2y24+ⅇ2sin1x424+ⅇ2cos1xy424ⅇ2cos1x3y212+ⅇ2cos1x5120+ⅇ2sin1y5120ⅇ2sin1x2y312+ⅇ2sin1x4y24+

(8)

Let's take a look at the first few homogeneous components of ps_both.

Truncateps_both,4

ⅇ2sin1+ⅇ2cos1x+ⅇ2sin1y+ⅇ2cos1xy+ⅇ2sin1y22ⅇ2sin1x22+ⅇ2cos1xy22ⅇ2cos1x36+ⅇ2sin1y36ⅇ2sin1x2y2+ⅇ2cos1xy36ⅇ2cos1x3y6+ⅇ2sin1y424ⅇ2sin1x2y24+ⅇ2sin1x424

(9)

Now we verify that the three results, ps_both, ps_x_y and ps_y_x, are equal (up to homogeneous degree 20).

ApproximatelyEqualps_both,ps_x_y,20

true

(10)

ApproximatelyEqualps_both,ps_y_x,20

true

(11)

We define a univariate polynomial over power series.

fUnivariatePolynomialOverPowerSeriesPowerSeries1,SumOfAllMonomialsx,y,GeometricSeriesy,z

fUnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs: 1+1+x+y+z+1+y+z2

(12)

We apply a Taylor shift by 1, and then by -1 on the result.

f1TaylorShiftf,1

f1UnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs: 3++3+x+3y+z+1+y+z2

(13)

f0TaylorShiftf1,1

f0UnⅈvarⅈatⅇPolynomⅈalOvⅇrPowⅇrSⅇrⅈⅇs: 1+1+x+y+z+1+y+z2

(14)

We verify that the result is equal to the original polynomial (up to homogeneous degree 20).

ApproximatelyEqualf,f0,20

true

(15)

Compatibility

• 

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.

See Also

ApproximatelyEqual

EvaluateAtOrigin

MainVariable

PuiseuxSeries

UnivariatePolynomialOverPowerSeries