UnivariateLaurentSeries(Coef, var, cen)

laurent.spad line 508 [edit on github]

• Coef: Ring

• var: Symbol

• cen: Coef

Dense Laurent series in one variable UnivariateLaurentSeries is a domain representing Laurent series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring, the power series variable, and the center of the power series expansion. For example, UnivariateLaurentSeries(Integer, x, 3) represents Laurent series in (x - 3) with integer coefficients.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from Magma

*: (%, Coef) -> %

from RightModule Coef

*: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer

from RightModule Fraction Integer

*: (%, Integer) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has LinearlyExplicitOver Integer

from RightModule Integer

*: (%, UnivariateTaylorSeries(Coef, var, cen)) -> % if Coef has Field

from RightModule UnivariateTaylorSeries(Coef, var, cen)

*: (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

*: (UnivariateTaylorSeries(Coef, var, cen), %) -> % if Coef has Field

from LeftModule UnivariateTaylorSeries(Coef, var, cen)

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> % if Coef has Field

from Field

/: (%, Coef) -> % if Coef has Field

from AbelianMonoidRing(Coef, Integer)

/: (UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen)) -> % if Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

<=: (%, %) -> Boolean if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

from PartialOrder

<: (%, %) -> Boolean if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

from PartialOrder

>: (%, %) -> Boolean if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

from PartialOrder

^: (%, %) -> % 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

abs: % -> % if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

from OrderedRing

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 ^: (Coef, Integer) -> Coef and Coef has coerce: Symbol -> 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

ceiling: % -> UnivariateTaylorSeries(Coef, var, cen) if UnivariateTaylorSeries(Coef, var, cen) has IntegerNumberSystem and Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

center: % -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if % has CharacteristicNonZero and UnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field or Coef has CharacteristicNonZero or UnivariateTaylorSeries(Coef, var, cen) has CharacteristicNonZero and Coef has Field

from PolynomialFactorizationExplicit

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 CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: Symbol -> % if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Symbol and Coef has Field

from CoercibleFrom Symbol

coerce: UnivariateTaylorSeries(Coef, var, cen) -> %

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

coerce: Variable var -> %

coerce(var) converts the series variable var into a Laurent series.

commutator: (%, %) -> %

from NonAssociativeRng

complete: % -> %

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

conditionP: Matrix % -> Union(Vector %, failed) if % has CharacteristicNonZero and UnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field

from PolynomialFactorizationExplicit

construct: List Record(k: Integer, c: Coef) -> %

from IndexedProductCategory(Coef, Integer)

constructOrdered: List Record(k: Integer, c: Coef) -> %

from IndexedProductCategory(Coef, Integer)

convert: % -> DoubleFloat if UnivariateTaylorSeries(Coef, var, cen) has RealConstant and Coef has Field

from ConvertibleTo DoubleFloat

convert: % -> Float if UnivariateTaylorSeries(Coef, var, cen) has RealConstant and Coef has Field

from ConvertibleTo Float

convert: % -> InputForm if UnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo InputForm and Coef has Field

from ConvertibleTo InputForm

convert: % -> Pattern Float if UnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo Pattern Float and Coef has Field

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if UnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo Pattern Integer and Coef has Field

from ConvertibleTo Pattern Integer

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 UnivariateTaylorSeries(Coef, var, cen) has DifferentialRing and Coef has Field or Coef has *: (Integer, Coef) -> Coef

from DifferentialRing

D: (%, List Symbol) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if UnivariateTaylorSeries(Coef, var, cen) has DifferentialRing and Coef has Field or Coef has *: (Integer, Coef) -> Coef

from DifferentialRing

D: (%, Symbol) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

D: (%, UnivariateTaylorSeries(Coef, var, cen) -> UnivariateTaylorSeries(Coef, var, cen)) -> % if Coef has Field

from DifferentialExtension UnivariateTaylorSeries(Coef, var, cen)

D: (%, UnivariateTaylorSeries(Coef, var, cen) -> UnivariateTaylorSeries(Coef, var, cen), NonNegativeInteger) -> % if Coef has Field

from DifferentialExtension UnivariateTaylorSeries(Coef, var, cen)

degree: % -> Integer

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

denom: % -> UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

denominator: % -> % if Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

differentiate: % -> % if UnivariateTaylorSeries(Coef, var, cen) has DifferentialRing and Coef has Field or Coef has *: (Integer, Coef) -> Coef

from DifferentialRing

differentiate: (%, List Symbol) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if UnivariateTaylorSeries(Coef, var, cen) has DifferentialRing and Coef has Field or Coef has *: (Integer, Coef) -> Coef

from DifferentialRing

differentiate: (%, Symbol) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

from PartialDifferentialRing Symbol

differentiate: (%, UnivariateTaylorSeries(Coef, var, cen) -> UnivariateTaylorSeries(Coef, var, cen)) -> % if Coef has Field

from DifferentialExtension UnivariateTaylorSeries(Coef, var, cen)

differentiate: (%, UnivariateTaylorSeries(Coef, var, cen) -> UnivariateTaylorSeries(Coef, var, cen), NonNegativeInteger) -> % if Coef has Field

from DifferentialExtension UnivariateTaylorSeries(Coef, var, cen)

differentiate: (%, Variable var) -> %

differentiate(f(x), x) returns the derivative of f(x) with respect to x.

divide: (%, %) -> Record(quotient: %, remainder: %) if Coef has Field

from EuclideanDomain

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, Integer) -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

elt: (%, UnivariateTaylorSeries(Coef, var, cen)) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has Eltable(UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen))

from Eltable(UnivariateTaylorSeries(Coef, var, cen), %)

euclideanSize: % -> NonNegativeInteger if Coef has Field

from EuclideanDomain

eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Integer) -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

eval: (%, Equation UnivariateTaylorSeries(Coef, var, cen)) -> % if UnivariateTaylorSeries(Coef, var, cen) has Evalable UnivariateTaylorSeries(Coef, var, cen) and Coef has Field

from Evalable UnivariateTaylorSeries(Coef, var, cen)

eval: (%, List Equation UnivariateTaylorSeries(Coef, var, cen)) -> % if UnivariateTaylorSeries(Coef, var, cen) has Evalable UnivariateTaylorSeries(Coef, var, cen) and Coef has Field

from Evalable UnivariateTaylorSeries(Coef, var, cen)

eval: (%, List Symbol, List UnivariateTaylorSeries(Coef, var, cen)) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has InnerEvalable(Symbol, UnivariateTaylorSeries(Coef, var, cen))

from InnerEvalable(Symbol, UnivariateTaylorSeries(Coef, var, cen))

eval: (%, List UnivariateTaylorSeries(Coef, var, cen), List UnivariateTaylorSeries(Coef, var, cen)) -> % if UnivariateTaylorSeries(Coef, var, cen) has Evalable UnivariateTaylorSeries(Coef, var, cen) and Coef has Field

from InnerEvalable(UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen))

eval: (%, Symbol, UnivariateTaylorSeries(Coef, var, cen)) -> % if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has InnerEvalable(Symbol, UnivariateTaylorSeries(Coef, var, cen))

from InnerEvalable(Symbol, UnivariateTaylorSeries(Coef, var, cen))

eval: (%, UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen)) -> % if UnivariateTaylorSeries(Coef, var, cen) has Evalable UnivariateTaylorSeries(Coef, var, cen) and Coef has Field

from InnerEvalable(UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen))

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

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if UnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field

from PolynomialFactorizationExplicit

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if UnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field

from PolynomialFactorizationExplicit

floor: % -> UnivariateTaylorSeries(Coef, var, cen) if UnivariateTaylorSeries(Coef, var, cen) has IntegerNumberSystem and Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

fractionPart: % -> % if UnivariateTaylorSeries(Coef, var, cen) has EuclideanDomain and Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

gcd: (%, %) -> % if Coef has Field

from GcdDomain

gcd: List % -> % if Coef has Field

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if Coef has Field

from PolynomialFactorizationExplicit

init: % if UnivariateTaylorSeries(Coef, var, cen) has StepThrough and Coef has Field

from StepThrough

integrate: % -> % if Coef has Algebra Fraction Integer

from UnivariateSeriesWithRationalExponents(Coef, Integer)

integrate: (%, Symbol) -> % if Coef has Algebra Fraction Integer and Coef has variables: Coef -> List Symbol and Coef has integrate: (Coef, Symbol) -> Coef

from UnivariateSeriesWithRationalExponents(Coef, Integer)

integrate: (%, Variable var) -> % 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.

inv: % -> % if Coef has Field

from DivisionRing

latex: % -> String

from SetCategory

laurent: (Integer, Stream Coef) -> %

from UnivariateLaurentSeriesCategory Coef

laurent: (Integer, UnivariateTaylorSeries(Coef, var, cen)) -> %

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

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)

leadingSupport: % -> Integer

from IndexedProductCategory(Coef, Integer)

leadingTerm: % -> Record(k: Integer, c: Coef)

from IndexedProductCategory(Coef, Integer)

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 IndexedProductCategory(Coef, Integer)

map: (UnivariateTaylorSeries(Coef, var, cen) -> UnivariateTaylorSeries(Coef, var, cen), %) -> % if Coef has Field

from FullyEvalableOver UnivariateTaylorSeries(Coef, var, cen)

max: (%, %) -> % if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

from OrderedSet

min: (%, %) -> % if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

from OrderedSet

monomial?: % -> Boolean

from IndexedProductCategory(Coef, Integer)

monomial: (Coef, Integer) -> %

from IndexedProductCategory(Coef, Integer)

multiEuclidean: (List %, %) -> Union(List %, failed) if Coef has Field

from EuclideanDomain

multiplyCoefficients: (Integer -> Coef, %) -> %

from UnivariateLaurentSeriesCategory Coef

multiplyExponents: (%, PositiveInteger) -> %

from UnivariatePowerSeriesCategory(Coef, Integer)

negative?: % -> Boolean if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

from OrderedRing

nextItem: % -> Union(%, failed) if UnivariateTaylorSeries(Coef, var, cen) has StepThrough and Coef has Field

from StepThrough

nthRoot: (%, Integer) -> % if Coef has Algebra Fraction Integer

from RadicalCategory

numer: % -> UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

numerator: % -> % if Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> Integer

from UnivariatePowerSeriesCategory(Coef, Integer)

order: (%, Integer) -> Integer

from UnivariatePowerSeriesCategory(Coef, Integer)

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if UnivariateTaylorSeries(Coef, var, cen) has PatternMatchable Float and Coef has Field

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if UnivariateTaylorSeries(Coef, var, cen) has PatternMatchable Integer and Coef has Field

from PatternMatchable Integer

pi: () -> % if Coef has Algebra Fraction Integer

from TranscendentalFunctionCategory

plenaryPower: (%, PositiveInteger) -> % if Coef has CommutativeRing or Coef has Algebra Fraction Integer

from NonAssociativeAlgebra %

pole?: % -> Boolean

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

positive?: % -> Boolean if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

from OrderedRing

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

from UnivariateLaurentSeriesCategory Coef

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

from UnivariateLaurentSeriesCategory Coef

recip: % -> Union(%, failed)

from MagmaWithUnit

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix UnivariateTaylorSeries(Coef, var, cen), vec: Vector UnivariateTaylorSeries(Coef, var, cen)) if Coef has Field

from LinearlyExplicitOver UnivariateTaylorSeries(Coef, var, cen)

reducedSystem: Matrix % -> Matrix Integer if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

from LinearlyExplicitOver UnivariateTaylorSeries(Coef, var, cen)

reductum: % -> %

from IndexedProductCategory(Coef, Integer)

rem: (%, %) -> % if Coef has Field

from EuclideanDomain

removeZeroes: % -> %

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

removeZeroes: (Integer, %) -> %

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

retract: % -> Fraction Integer if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

from RetractableTo Fraction Integer

retract: % -> Integer if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

from RetractableTo Integer

retract: % -> Symbol if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Symbol and Coef has Field

from RetractableTo Symbol

retract: % -> UnivariateTaylorSeries(Coef, var, cen)

from RetractableTo UnivariateTaylorSeries(Coef, var, cen)

retractIfCan: % -> Union(Fraction Integer, failed) if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed) if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

from RetractableTo Integer

retractIfCan: % -> Union(Symbol, failed) if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Symbol and Coef has Field

from RetractableTo Symbol

retractIfCan: % -> Union(UnivariateTaylorSeries(Coef, var, cen), failed)

from RetractableTo UnivariateTaylorSeries(Coef, var, cen)

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) -> %

from UnivariateLaurentSeriesCategory Coef

sign: % -> Integer if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

from OrderedRing

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

smaller?: (%, %) -> Boolean if UnivariateTaylorSeries(Coef, var, cen) has Comparable and Coef has Field

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if UnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field

from PolynomialFactorizationExplicit

sqrt: % -> % if Coef has Algebra Fraction Integer

from RadicalCategory

squareFree: % -> Factored % if Coef has Field

from UniqueFactorizationDomain

squareFreePart: % -> % if Coef has Field

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if UnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field

from PolynomialFactorizationExplicit

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

tan: % -> % if Coef has Algebra Fraction Integer

from TrigonometricFunctionCategory

tanh: % -> % if Coef has Algebra Fraction Integer

from HyperbolicFunctionCategory

taylor: % -> UnivariateTaylorSeries(Coef, var, cen)

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

taylorIfCan: % -> Union(UnivariateTaylorSeries(Coef, var, cen), failed)

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

taylorRep: % -> UnivariateTaylorSeries(Coef, var, cen)

from UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

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)

wholePart: % -> UnivariateTaylorSeries(Coef, var, cen) if UnivariateTaylorSeries(Coef, var, cen) has EuclideanDomain and Coef has Field

from QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Integer)

AbelianProductCategory Coef

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

Algebra UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

ArcHyperbolicFunctionCategory if Coef has Algebra Fraction Integer

ArcTrigonometricFunctionCategory if Coef has Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(Coef, Coef)

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

BiModule(UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen)) if Coef has Field

CancellationAbelianMonoid

canonicalsClosed if Coef has Field

canonicalUnitNormal if Coef has Field

CharacteristicNonZero if UnivariateTaylorSeries(Coef, var, cen) has CharacteristicNonZero and Coef has Field or Coef has CharacteristicNonZero

CharacteristicZero if UnivariateTaylorSeries(Coef, var, cen) has CharacteristicZero and Coef has Field or Coef has CharacteristicZero

CoercibleFrom Fraction Integer if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

CoercibleFrom Integer if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

CoercibleFrom Symbol if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Symbol and Coef has Field

CoercibleFrom UnivariateTaylorSeries(Coef, var, cen)

CoercibleTo OutputForm

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

Comparable if UnivariateTaylorSeries(Coef, var, cen) has Comparable and Coef has Field

ConvertibleTo DoubleFloat if UnivariateTaylorSeries(Coef, var, cen) has RealConstant and Coef has Field

ConvertibleTo Float if UnivariateTaylorSeries(Coef, var, cen) has RealConstant and Coef has Field

ConvertibleTo InputForm if UnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo InputForm and Coef has Field

ConvertibleTo Pattern Float if UnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo Pattern Float and Coef has Field

ConvertibleTo Pattern Integer if UnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo Pattern Integer and Coef has Field

DifferentialExtension UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

DifferentialRing if UnivariateTaylorSeries(Coef, var, cen) has DifferentialRing and Coef has Field or Coef has *: (Integer, Coef) -> Coef

DivisionRing if Coef has Field

ElementaryFunctionCategory if Coef has Algebra Fraction Integer

Eltable(%, %)

Eltable(UnivariateTaylorSeries(Coef, var, cen), %) if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has Eltable(UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen))

EntireRing if Coef has IntegralDomain

EuclideanDomain if Coef has Field

Evalable UnivariateTaylorSeries(Coef, var, cen) if UnivariateTaylorSeries(Coef, var, cen) has Evalable UnivariateTaylorSeries(Coef, var, cen) and Coef has Field

Field if Coef has Field

FullyEvalableOver UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

FullyLinearlyExplicitOver UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

FullyPatternMatchable UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

GcdDomain if Coef has Field

HyperbolicFunctionCategory if Coef has Algebra Fraction Integer

IndexedProductCategory(Coef, Integer)

InnerEvalable(Symbol, UnivariateTaylorSeries(Coef, var, cen)) if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has InnerEvalable(Symbol, UnivariateTaylorSeries(Coef, var, cen))

InnerEvalable(UnivariateTaylorSeries(Coef, var, cen), UnivariateTaylorSeries(Coef, var, cen)) if UnivariateTaylorSeries(Coef, var, cen) has Evalable UnivariateTaylorSeries(Coef, var, cen) and Coef has Field

IntegralDomain if Coef has IntegralDomain

LeftModule %

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

LeftModule UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

LeftOreRing if Coef has Field

LinearlyExplicitOver Integer if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has LinearlyExplicitOver Integer

LinearlyExplicitOver UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

Magma

MagmaWithUnit

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

Module UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

Monoid

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeAlgebra Fraction Integer if Coef has Algebra Fraction Integer

NonAssociativeAlgebra UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

OrderedAbelianGroup if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

OrderedAbelianMonoid if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

OrderedAbelianSemiGroup if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

OrderedCancellationAbelianMonoid if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

OrderedIntegralDomain if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

OrderedRing if UnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

OrderedSet if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

PartialDifferentialRing Symbol if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has PartialDifferentialRing Symbol or Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

PartialOrder if UnivariateTaylorSeries(Coef, var, cen) has OrderedSet and Coef has Field

Patternable UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

PatternMatchable Float if UnivariateTaylorSeries(Coef, var, cen) has PatternMatchable Float and Coef has Field

PatternMatchable Integer if UnivariateTaylorSeries(Coef, var, cen) has PatternMatchable Integer and Coef has Field

PolynomialFactorizationExplicit if UnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

PrincipalIdealDomain if Coef has Field

QuotientFieldCategory UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

RadicalCategory if Coef has Algebra Fraction Integer

RealConstant if UnivariateTaylorSeries(Coef, var, cen) has RealConstant and Coef has Field

RetractableTo Fraction Integer if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

RetractableTo Integer if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field

RetractableTo Symbol if UnivariateTaylorSeries(Coef, var, cen) has RetractableTo Symbol and Coef has Field

RetractableTo UnivariateTaylorSeries(Coef, var, cen)

RightModule %

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

RightModule Integer if Coef has Field and UnivariateTaylorSeries(Coef, var, cen) has LinearlyExplicitOver Integer

RightModule UnivariateTaylorSeries(Coef, var, cen) if Coef has Field

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if UnivariateTaylorSeries(Coef, var, cen) has StepThrough and Coef has Field

TranscendentalFunctionCategory if Coef has Algebra Fraction Integer

TrigonometricFunctionCategory if Coef has Algebra Fraction Integer

TwoSidedRecip if Coef has CommutativeRing

UniqueFactorizationDomain if Coef has Field

unitsKnown

UnivariateLaurentSeriesCategory Coef

UnivariateLaurentSeriesConstructorCategory(Coef, UnivariateTaylorSeries(Coef, var, cen))

UnivariatePowerSeriesCategory(Coef, Integer)

UnivariateSeriesWithRationalExponents(Coef, Integer)

VariablesCommuteWithCoefficients