# UnivariatePowerSeriesCategory(Coef, Expon)ΒΆ

- Coef: Ring
- Expon: OrderedAbelianMonoid

UnivariatePowerSeriesCategory is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.

- 0: %
- from AbelianMonoid
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from Magma
- *: (%, Coef) -> %
- from RightModule Coef
- *: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer
- from RightModule Fraction Integer
- *: (Coef, %) -> %
- from LeftModule Coef
- *: (Fraction Integer, %) -> % if Coef has Algebra Fraction Integer
- from LeftModule Fraction Integer
- *: (Integer, %) -> %
- from AbelianGroup
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup
- /: (%, Coef) -> % if Coef has Field
- from AbelianMonoidRing(Coef, Expon)
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng

- approximate: (%, Expon) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, Expon) -> Coef
`approximate(f)`

returns a truncated power series with the series variable viewed as an element of the coefficient domain.- associates?: (%, %) -> Boolean if Coef has IntegralDomain
- from EntireRing
- associator: (%, %, %) -> %
- from NonAssociativeRng

- center: % -> Coef
`center(f)`

returns the point about which the series`f`

is expanded.- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
- from CharacteristicNonZero
- coefficient: (%, Expon) -> Coef
- from AbelianMonoidRing(Coef, Expon)
- coerce: % -> % if Coef has CommutativeRing
- from Algebra %
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Coef -> % if Coef has CommutativeRing
- from Algebra Coef
- coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
- from Algebra Fraction Integer
- coerce: Integer -> %
- from NonAssociativeRing
- commutator: (%, %) -> %
- from NonAssociativeRng
- complete: % -> %
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- D: % -> % if Coef has *: (Expon, Coef) -> Coef
- from DifferentialRing
- D: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef
- from DifferentialRing
- D: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- degree: % -> Expon
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- differentiate: % -> % if Coef has *: (Expon, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
- from PartialDifferentialRing Symbol
- elt: (%, %) -> % if Expon has SemiGroup
- from Eltable(%, %)

- elt: (%, Expon) -> Coef
`elt(f(x), r)`

returns the coefficient of the term of degree`r`

in`f(x)`

. This is the same as the function coefficient.

- eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Expon) -> Coef
`eval(f, a)`

evaluates a power series at a value in the ground ring by returning a stream of partial sums.- exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain
- from EntireRing

- extend: (%, Expon) -> %
`extend(f, n)`

causes all terms of`f`

of degree`<=`

`n`

to be computed.- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- latex: % -> String
- from SetCategory
- leadingCoefficient: % -> Coef
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- leadingMonomial: % -> %
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- map: (Coef -> Coef, %) -> %
- from AbelianMonoidRing(Coef, Expon)
- monomial: (%, List SingletonAsOrderedSet, List Expon) -> %
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- monomial: (%, SingletonAsOrderedSet, Expon) -> %
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- monomial: (Coef, Expon) -> %
- from AbelianMonoidRing(Coef, Expon)
- monomial?: % -> Boolean
- from AbelianMonoidRing(Coef, Expon)

- multiplyExponents: (%, PositiveInteger) -> %
`multiplyExponents(f, n)`

multiplies all exponents of the power series`f`

by the positive integer`n`

.- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid

- order: % -> Expon
`order(f)`

is the degree of the lowest order non-zero term in`f`

. This will result in an infinite loop if`f`

has no non-zero terms.

- order: (%, Expon) -> Expon
`order(f, n) = min(m, n)`

, where`m`

is the degree of the lowest order non-zero term in`f`

.- pole?: % -> Boolean
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- recip: % -> Union(%, failed)
- from MagmaWithUnit
- reductum: % -> %
- from AbelianMonoidRing(Coef, Expon)
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid

- terms: % -> Stream Record(k: Expon, c: Coef)
`terms(f(x))`

returns a stream of non-zero terms, where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents.

- truncate: (%, Expon) -> %
`truncate(f, k)`

returns a (finite) power series consisting of the sum of all terms of`f`

of degree`<= k`

.

- truncate: (%, Expon, Expon) -> %
`truncate(f, k1, k2)`

returns a (finite) power series consisting of the sum of all terms of`f`

of degree`d`

with`k1 <= d <= k2`

.- 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

- variable: % -> Symbol
`variable(f)`

returns the (unique) power series variable of the power series`f`

.- variables: % -> List SingletonAsOrderedSet
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- zero?: % -> Boolean
- from AbelianMonoid

AbelianMonoidRing(Coef, Expon)

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

BiModule(%, %)

BiModule(Coef, Coef)

BiModule(Fraction Integer, Fraction Integer) if Coef has Algebra Fraction Integer

CharacteristicNonZero if Coef has CharacteristicNonZero

CharacteristicZero if Coef has CharacteristicZero

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

DifferentialRing if Coef has *: (Expon, Coef) -> Coef

Eltable(%, %) if Expon has SemiGroup

EntireRing if Coef has IntegralDomain

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

noZeroDivisors if Coef has IntegralDomain

PartialDifferentialRing Symbol if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef

PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer