# GeneralizedUnivariatePowerSeries(Coef, Expon, var, cen)ΒΆ

- Coef: Ring
- Expon: Join(OrderedAbelianMonoid, SemiRing)
- var: Symbol
- cen: Coef

undocumented

- 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
- /: (%, %) -> % if Expon has AbelianGroup and Coef has Field
- from Field
- /: (%, Coef) -> % if Coef has Field
- from AbelianMonoidRing(Coef, Expon)
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, %) -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory
- ^: (%, Integer) -> % if Expon has AbelianGroup and Coef has Field
- from DivisionRing
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- acos: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- acosh: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- acot: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- acoth: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- acsc: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- acsch: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng

- apply_taylor: (Stream Coef, %) -> %
`apply_taylor(ts, s)`

applies Taylor series with coefficients`ts`

to`s`

, that is computes infinite sum`ts`

(0) +`ts`

(1)`*s`

+`ts`

(2)`*s^2`

+ ... Note:`s`

must be of positive order- approximate: (%, Expon) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, Expon) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Expon)
- asec: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- asech: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- asin: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- asinh: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- associates?: (%, %) -> Boolean if Coef has IntegralDomain
- from EntireRing
- associator: (%, %, %) -> %
- from NonAssociativeRng
- atan: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- atanh: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- center: % -> Coef
- from UnivariatePowerSeriesCategory(Coef, Expon)
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
- from CharacteristicNonZero
- coefficient: (%, Expon) -> Coef
- from AbelianMonoidRing(Coef, Expon)
- coerce: % -> % if Coef has IntegralDomain
- 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)
- cos: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- cosh: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- cot: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- coth: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- csc: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- csch: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- 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
- divide: (%, %) -> Record(quotient: %, remainder: %) if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain
- elt: (%, %) -> %
- from Eltable(%, %)
- elt: (%, Expon) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Expon)
- euclideanSize: % -> NonNegativeInteger if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain
- eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Expon) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Expon)
- exp: % -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory
- expressIdealMember: (List %, %) -> Union(List %, failed) if Expon has AbelianGroup and Coef has Field
- from PrincipalIdealDomain
- exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain
- from EntireRing
- extend: (%, Expon) -> %
- from UnivariatePowerSeriesCategory(Coef, Expon)
- extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain
- extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain
- factor: % -> Factored % if Expon has AbelianGroup and Coef has Field
- from UniqueFactorizationDomain
- gcd: (%, %) -> % if Expon has AbelianGroup and Coef has Field
- from GcdDomain
- gcd: List % -> % if Expon has AbelianGroup and Coef has Field
- from GcdDomain
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if Expon has AbelianGroup and Coef has Field
- from GcdDomain
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory

- infsum: Stream % -> %
`infsum(x)`

computes sum of all elements of`x`

. Degrees of elements of`x`

must be nondecreasing and tend to infinity.- inv: % -> % if Expon has AbelianGroup and Coef has Field
- from DivisionRing
- latex: % -> String
- from SetCategory
- lcm: (%, %) -> % if Expon has AbelianGroup and Coef has Field
- from GcdDomain
- lcm: List % -> % if Expon has AbelianGroup and Coef has Field
- from GcdDomain
- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if Expon has AbelianGroup and Coef has Field
- from LeftOreRing
- 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
- log: % -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory
- 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)
- multiEuclidean: (List %, %) -> Union(List %, failed) if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain
- multiplyExponents: (%, PositiveInteger) -> %
- from UnivariatePowerSeriesCategory(Coef, Expon)
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- order: % -> Expon
- from UnivariatePowerSeriesCategory(Coef, Expon)
- order: (%, Expon) -> Expon
- from UnivariatePowerSeriesCategory(Coef, Expon)
- pi: () -> % if Coef has Algebra Fraction Integer
- from TranscendentalFunctionCategory
- pole?: % -> Boolean
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- prime?: % -> Boolean if Expon has AbelianGroup and Coef has Field
- from UniqueFactorizationDomain
- principalIdeal: List % -> Record(coef: List %, generator: %) if Expon has AbelianGroup and Coef has Field
- from PrincipalIdealDomain
- quo: (%, %) -> % if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain
- recip: % -> Union(%, failed)
- from MagmaWithUnit
- reductum: % -> %
- from AbelianMonoidRing(Coef, Expon)
- rem: (%, %) -> % if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain

- removeZeros: (%, Expon) -> %
`removeZeros(s, k)`

removes leading zero terms in`s`

with exponent smaller than`k`

- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid
- sec: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- sech: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- sin: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- sinh: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- sizeLess?: (%, %) -> Boolean if Expon has AbelianGroup and Coef has Field
- from EuclideanDomain
- squareFree: % -> Factored % if Expon has AbelianGroup and Coef has Field
- from UniqueFactorizationDomain
- squareFreePart: % -> % if Expon has AbelianGroup and Coef has Field
- from UniqueFactorizationDomain
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid
- tan: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- tanh: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- terms: % -> Stream Record(k: Expon, c: Coef)
- from UnivariatePowerSeriesCategory(Coef, Expon)
- truncate: (%, Expon) -> %
- from UnivariatePowerSeriesCategory(Coef, Expon)
- truncate: (%, Expon, Expon) -> %
- from UnivariatePowerSeriesCategory(Coef, Expon)
- 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
- from UnivariatePowerSeriesCategory(Coef, Expon)
- variables: % -> List SingletonAsOrderedSet
- from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
- zero?: % -> Boolean
- from AbelianMonoid

AbelianMonoidRing(Coef, Expon)

Algebra % if Coef has IntegralDomain

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

ArcHyperbolicFunctionCategory if Coef has Algebra Fraction Integer

ArcTrigonometricFunctionCategory if Coef has Algebra Fraction Integer

BiModule(%, %)

BiModule(Coef, Coef)

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

canonicalsClosed if Expon has AbelianGroup and Coef has Field

canonicalUnitNormal if Expon has AbelianGroup and Coef has Field

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

DivisionRing if Expon has AbelianGroup and Coef has Field

ElementaryFunctionCategory if Coef has Algebra Fraction Integer

Eltable(%, %)

EntireRing if Coef has IntegralDomain

EuclideanDomain if Expon has AbelianGroup and Coef has Field

Field if Expon has AbelianGroup and Coef has Field

GcdDomain if Expon has AbelianGroup and Coef has Field

HyperbolicFunctionCategory if Coef has Algebra Fraction Integer

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

LeftOreRing if Expon has AbelianGroup and Coef has Field

Module % if Coef has IntegralDomain

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)

PrincipalIdealDomain if Expon has AbelianGroup and Coef has Field

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

TranscendentalFunctionCategory if Coef has Algebra Fraction Integer

TrigonometricFunctionCategory if Coef has Algebra Fraction Integer

UniqueFactorizationDomain if Expon has AbelianGroup and Coef has Field

UnivariatePowerSeriesCategory(Coef, Expon)