# InnerTaylorSeries CoefΒΆ

taylor.spad line 1 [edit on github]

Coef: Ring

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

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- annihilate?: (%, %) -> Boolean
from Rng

- antiCommutator: (%, %) -> %

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

- associator: (%, %, %) -> %
from NonAssociativeRng

- characteristic: () -> NonNegativeInteger
from NonAssociativeRing

- 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
from CoercibleTo OutputForm

- coerce: Integer -> %
from NonAssociativeRing

- commutator: (%, %) -> %
from NonAssociativeRng

- 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
from NonAssociativeAlgebra %

- 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

Algebra % if Coef has IntegralDomain

BiModule(%, %)

CommutativeRing if Coef has IntegralDomain

CommutativeStar if Coef has IntegralDomain

EntireRing if Coef has IntegralDomain

IntegralDomain if Coef has IntegralDomain

Module % if Coef has IntegralDomain

NonAssociativeAlgebra % if Coef has IntegralDomain

noZeroDivisors if Coef has IntegralDomain

TwoSidedRecip if Coef has IntegralDomain