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MultivariatePowerSeries

  

Inverse

  

Compute the inverse of a power series

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

1/p

Inverse(p)

Parameters

p

-

power series generated by this package

Description

• 

The commands 1/p and Inverse(p) compute the multiplicative inverse of the power series p. This requires that p is invertible, that is, that p has a non-zero constant term; if that is not the case, an error is signaled.

• 

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

withMultivariatePowerSeries:

We define a power series, a, representing a polynomial.

aPowerSeries1xy

aPowⅇrSⅇrⅈⅇs: 1xy

(1)

We can define its inverse in two equivalent ways:

b1a

bPowⅇrSⅇrⅈⅇs of 11xy : 1+

(2)

cInversea

cPowⅇrSⅇrⅈⅇs of 11xy : 1+

(3)

We verify that the two definitions are equal, at least for the terms up to homogeneous degree 10.

Truncatebc,10

0

(4)

A different power series represents the sine of x.

sxPowerSeriesdifelsed::odd,−1d212xdd!,0,analytic=sinx

sxPowⅇrSⅇrⅈⅇs of sinx : 0+

(5)

Because the constant coefficient of sx is zero, we cannot invert it. (Its multiplicative inverse is a Laurent series, not a power series.)

1sx

Error, not invertible

Compatibility

• 

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

• 

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

See Also

ApproximatelyEqual

Divide

GeometricSeries

Inverse

IsUnit

PowerSeries