# InnerTaylorSeries CoefΒΆ

- 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 Magma

- *: (%, 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: (%, %) -> %
- from NonAssociativeSemiRng
- 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
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- 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`

.

- 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)
- from CancellationAbelianMonoid
- 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

noZeroDivisors if Coef has IntegralDomain