# InnerTaylorSeries CoefΒΆ

Internal package for dense Taylor series. This is an internal Taylor series type in which Taylor series are represented by a Stream of Ring elements. For univariate series, the `Stream` elements are the Taylor coefficients. For multivariate series, the `n`th Stream element is a form of degree `n` in the power series variables.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Coef) -> %

`x*c` returns the product of `c` and the series `x`.

*: (%, Integer) -> %

`x*i` returns the product of integer `i` and the series `x`.

*: (Coef, %) -> %

`c*x` returns the product of `c` and the series `x`.

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associates?: (%, %) -> Boolean if Coef has IntegralDomain

from EntireRing

associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
coefficients: % -> Stream Coef

`coefficients(x)` returns a stream of ring elements. When `x` is a univariate series, this is a stream of Taylor coefficients. When `x` is a multivariate series, the `n`th element of the stream is a form of degree `n` in the power series variables.

coerce: % -> % if Coef has IntegralDomain

from Algebra %

coerce: % -> OutputForm
coerce: Integer -> %
commutator: (%, %) -> %
exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain

from EntireRing

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> NonNegativeInteger

`order(x)` returns the order of a power series `x`, i.e. the degree of the first non-zero term of the series.

order: (%, NonNegativeInteger) -> NonNegativeInteger

`order(x, n)` returns the minimum of `n` and the order of `x`.

plenaryPower: (%, PositiveInteger) -> % if Coef has IntegralDomain
pole?: % -> Boolean

`pole?(x)` tests if the series `x` has a pole. Note: this is `false` when `x` is a Taylor series.

recip: % -> Union(%, failed)

from MagmaWithUnit

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

series: Stream Coef -> %

`series(s)` creates a power series from a stream of ring elements. For univariate series types, the stream `s` should be a stream of Taylor coefficients. For multivariate series types, the stream `s` should be a stream of forms the `n`th element of which is a form of degree `n` in the power series variables.

subtractIfCan: (%, %) -> Union(%, failed)
unit?: % -> Boolean if Coef has IntegralDomain

from EntireRing

unitCanonical: % -> % if Coef has IntegralDomain

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if Coef has IntegralDomain

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra % if Coef has IntegralDomain

BasicType

BiModule(%, %)

CancellationAbelianMonoid

CommutativeRing if Coef has IntegralDomain

CommutativeStar if Coef has IntegralDomain

EntireRing if Coef has IntegralDomain

IntegralDomain if Coef has IntegralDomain

Magma

MagmaWithUnit

Module % if Coef has IntegralDomain

Monoid

NonAssociativeAlgebra % if Coef has IntegralDomain

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has IntegralDomain

unitsKnown