# UnivariateLaurentSeriesCategory CoefΒΆ

- Coef: Ring

UnivariateLaurentSeriesCategory is the category of Laurent series in one variable.

- 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 Coef has Field
- from Field
- /: (%, Coef) -> % if Coef has Field
- from AbelianMonoidRing(Coef, Integer)
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, %) -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory
- ^: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer
- from RadicalCategory
- ^: (%, Integer) -> % if 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
- approximate: (%, Integer) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, Integer) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Integer)
- 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, Integer)
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
- from CharacteristicNonZero
- coefficient: (%, Integer) -> Coef
- from AbelianMonoidRing(Coef, Integer)
- 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, Integer, 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 *: (Integer, Coef) -> Coef
- from DifferentialRing
- D: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
- from DifferentialRing
- D: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- degree: % -> Integer
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- differentiate: % -> % if Coef has *: (Integer, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- divide: (%, %) -> Record(quotient: %, remainder: %) if Coef has Field
- from EuclideanDomain
- elt: (%, %) -> %
- from Eltable(%, %)
- elt: (%, Integer) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Integer)
- euclideanSize: % -> NonNegativeInteger if Coef has Field
- from EuclideanDomain
- eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Integer) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Integer)
- exp: % -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory
- expressIdealMember: (List %, %) -> Union(List %, failed) if Coef has Field
- from PrincipalIdealDomain
- exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain
- from EntireRing
- extend: (%, Integer) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)
- extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if Coef has Field
- from EuclideanDomain
- extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if Coef has Field
- from EuclideanDomain
- factor: % -> Factored % if Coef has Field
- from UniqueFactorizationDomain
- gcd: (%, %) -> % if Coef has Field
- from GcdDomain
- gcd: List % -> % if Coef has Field
- from GcdDomain
- gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if Coef has Field
- from GcdDomain
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory

- integrate: % -> % if Coef has Algebra Fraction Integer
`integrate(f(x))`

returns an anti-derivative of the power series`f(x)`

with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.

- integrate: (%, Symbol) -> % if Coef has AlgebraicallyClosedFunctionSpace Integer and Coef has Algebra Fraction Integer and Coef has TranscendentalFunctionCategory and Coef has PrimitiveFunctionCategory or Coef has Algebra Fraction Integer and Coef has integrate: (Coef, Symbol) -> Coef and Coef has variables: Coef -> List Symbol
`integrate(f(x), y)`

returns an anti-derivative of the power series`f(x)`

with respect to the variable`y`

.- inv: % -> % if Coef has Field
- from DivisionRing
- latex: % -> String
- from SetCategory

- laurent: (Integer, Stream Coef) -> %
`laurent(n, st)`

returns`xn * series st`

where`xn = monomial(1, n)`

and`series st`

stands for the power series with coefficients given by the stream st.- lcm: (%, %) -> % if Coef has Field
- from GcdDomain
- lcm: List % -> % if Coef has Field
- from GcdDomain
- lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if Coef has Field
- from LeftOreRing
- leadingCoefficient: % -> Coef
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- leadingMonomial: % -> %
- from PowerSeriesCategory(Coef, Integer, 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, Integer)
- monomial: (Coef, Integer) -> %
- from AbelianMonoidRing(Coef, Integer)
- monomial?: % -> Boolean
- from AbelianMonoidRing(Coef, Integer)
- multiEuclidean: (List %, %) -> Union(List %, failed) if Coef has Field
- from EuclideanDomain

- multiplyCoefficients: (Integer -> Coef, %) -> %
`multiplyCoefficients(f, sum(n = n0..infinity, a[n] * x^n)) = sum(n = 0..infinity, f(n) * a[n] * x^n)`

. This function is used when Puiseux series are represented by a Laurent series and an exponent.- multiplyExponents: (%, PositiveInteger) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)
- nthRoot: (%, Integer) -> % if Coef has Algebra Fraction Integer
- from RadicalCategory
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- order: % -> Integer
- from UnivariatePowerSeriesCategory(Coef, Integer)
- order: (%, Integer) -> Integer
- from UnivariatePowerSeriesCategory(Coef, Integer)
- pi: () -> % if Coef has Algebra Fraction Integer
- from TranscendentalFunctionCategory
- pole?: % -> Boolean
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- prime?: % -> Boolean if Coef has Field
- from UniqueFactorizationDomain
- principalIdeal: List % -> Record(coef: List %, generator: %) if Coef has Field
- from PrincipalIdealDomain
- quo: (%, %) -> % if Coef has Field
- from EuclideanDomain

- rationalFunction: (%, Integer) -> Fraction Polynomial Coef if Coef has IntegralDomain
`rationalFunction(f, k)`

returns a rational function consisting of the sum of all terms of`f`

of degree`<=`

`k`

.

- rationalFunction: (%, Integer, Integer) -> Fraction Polynomial Coef if Coef has IntegralDomain
`rationalFunction(f, k1, k2)`

returns a rational function consisting of the sum of all terms of`f`

of degree`d`

with`k1 <= d <= k2`

.- recip: % -> Union(%, failed)
- from MagmaWithUnit
- reductum: % -> %
- from AbelianMonoidRing(Coef, Integer)
- rem: (%, %) -> % if Coef has Field
- from EuclideanDomain
- 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

- series: Stream Record(k: Integer, c: Coef) -> %
`series(st)`

creates a series from a stream of non-zero terms, where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.- sin: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- sinh: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- sizeLess?: (%, %) -> Boolean if Coef has Field
- from EuclideanDomain
- sqrt: % -> % if Coef has Algebra Fraction Integer
- from RadicalCategory
- squareFree: % -> Factored % if Coef has Field
- from UniqueFactorizationDomain
- squareFreePart: % -> % if 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: Integer, c: Coef)
- from UnivariatePowerSeriesCategory(Coef, Integer)
- truncate: (%, Integer) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)
- truncate: (%, Integer, Integer) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)
- 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, Integer)
- zero?: % -> Boolean
- from AbelianMonoid

AbelianMonoidRing(Coef, Integer)

Algebra % if Coef has CommutativeRing

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 Coef has Field

canonicalUnitNormal if 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 *: (Integer, Coef) -> Coef

DivisionRing if Coef has Field

ElementaryFunctionCategory if Coef has Algebra Fraction Integer

Eltable(%, %)

EntireRing if Coef has IntegralDomain

EuclideanDomain if 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 Coef has Field

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 *: (Integer, Coef) -> Coef

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

PrincipalIdealDomain if Coef has Field

RadicalCategory if Coef has Algebra Fraction Integer

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

TwoSidedRecip if Coef has CommutativeRing

UniqueFactorizationDomain if Coef has Field