SparseUnivariateLaurentSeries(Coef, var, cen)

sups.spad line 1446 [edit on github]

• Coef: Ring

• var: Symbol

• cen: Coef

Sparse Laurent series in one variable SparseUnivariateLaurentSeries 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, SparseUnivariateLaurentSeries(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 SparseUnivariateTaylorSeries(Coef, var, cen) has LinearlyExplicitOver Integer and Coef has Field

from RightModule Integer

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

from RightModule SparseUnivariateTaylorSeries(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

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

from LeftModule SparseUnivariateTaylorSeries(Coef, var, cen)

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

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

from Field

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

from AbelianMonoidRing(Coef, Integer)

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

from QuotientFieldCategory SparseUnivariateTaylorSeries(Coef, var, cen)

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

from PartialOrder

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

from PartialOrder

=: (%, %) -> Boolean

from BasicType

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

from PartialOrder

>: (%, %) -> Boolean if SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(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 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

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

from QuotientFieldCategory SparseUnivariateTaylorSeries(Coef, var, cen)

center: % -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

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

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 CoercibleFrom Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

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

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

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

from CoercibleFrom Symbol

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 Coef has Field and SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit

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 SparseUnivariateTaylorSeries(Coef, var, cen) has RealConstant and Coef has Field

from ConvertibleTo DoubleFloat

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

from ConvertibleTo Float

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

from ConvertibleTo InputForm

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

from ConvertibleTo Pattern Float

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

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

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

from DifferentialExtension SparseUnivariateTaylorSeries(Coef, var, cen)

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

from DifferentialExtension SparseUnivariateTaylorSeries(Coef, var, cen)

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

degree: % -> Integer

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

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

from QuotientFieldCategory SparseUnivariateTaylorSeries(Coef, var, cen)

denominator: % -> % if Coef has Field

from QuotientFieldCategory SparseUnivariateTaylorSeries(Coef, var, cen)

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

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

from DifferentialExtension SparseUnivariateTaylorSeries(Coef, var, cen)

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

from DifferentialExtension SparseUnivariateTaylorSeries(Coef, var, cen)

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from Eltable(SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(Coef, var, cen)) -> % if SparseUnivariateTaylorSeries(Coef, var, cen) has Evalable SparseUnivariateTaylorSeries(Coef, var, cen) and Coef has Field

from Evalable SparseUnivariateTaylorSeries(Coef, var, cen)

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

from Evalable SparseUnivariateTaylorSeries(Coef, var, cen)

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

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

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

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

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

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

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

from InnerEvalable(Symbol, SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field

from PolynomialFactorizationExplicit

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

from PolynomialFactorizationExplicit

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

from QuotientFieldCategory SparseUnivariateTaylorSeries(Coef, var, cen)

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

from QuotientFieldCategory SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(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 integrate: (Coef, Symbol) -> Coef and Coef has variables: Coef -> List Symbol

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, SparseUnivariateTaylorSeries(Coef, var, cen)) -> %

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

laurent: (Integer, Stream Coef) -> %

from UnivariateLaurentSeriesCategory Coef

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

from FullyEvalableOver SparseUnivariateTaylorSeries(Coef, var, cen)

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

from OrderedSet

min: (%, %) -> % if SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(Coef, var, cen) has OrderedIntegralDomain and Coef has Field

from OrderedRing

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

from StepThrough

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

from RadicalCategory

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

from QuotientFieldCategory SparseUnivariateTaylorSeries(Coef, var, cen)

numerator: % -> % if Coef has Field

from QuotientFieldCategory SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(Coef, var, cen) has PatternMatchable Float and Coef has Field

from PatternMatchable Float

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

from LinearlyExplicitOver Integer

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

from LinearlyExplicitOver SparseUnivariateTaylorSeries(Coef, var, cen)

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

from LinearlyExplicitOver Integer

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

from LinearlyExplicitOver SparseUnivariateTaylorSeries(Coef, var, cen)

reductum: % -> %

from IndexedProductCategory(Coef, Integer)

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

from EuclideanDomain

removeZeroes: % -> %

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

removeZeroes: (Integer, %) -> %

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

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

from RetractableTo Fraction Integer

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

from RetractableTo Integer

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

from RetractableTo SparseUnivariateTaylorSeries(Coef, var, cen)

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

from RetractableTo Symbol

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

from RetractableTo Fraction Integer

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

from RetractableTo Integer

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

from RetractableTo SparseUnivariateTaylorSeries(Coef, var, cen)

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

from RetractableTo Symbol

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 SparseUnivariateTaylorSeries(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 Coef has Field and SparseUnivariateTaylorSeries(Coef, var, cen) has Comparable

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(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: % -> SparseUnivariateTaylorSeries(Coef, var, cen)

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

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

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

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

from UnivariateLaurentSeriesConstructorCategory(Coef, SparseUnivariateTaylorSeries(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: % -> SparseUnivariateTaylorSeries(Coef, var, cen) if SparseUnivariateTaylorSeries(Coef, var, cen) has EuclideanDomain and Coef has Field

from QuotientFieldCategory SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(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(SparseUnivariateTaylorSeries(Coef, var, cen), SparseUnivariateTaylorSeries(Coef, var, cen)) if Coef has Field

CancellationAbelianMonoid

canonicalsClosed if Coef has Field

canonicalUnitNormal if Coef has Field

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

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

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

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

CoercibleFrom SparseUnivariateTaylorSeries(Coef, var, cen)

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

CoercibleTo OutputForm

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

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

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

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

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

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

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

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

EntireRing if Coef has IntegralDomain

EuclideanDomain if Coef has Field

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

Field if Coef has Field

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

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

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

GcdDomain if Coef has Field

HyperbolicFunctionCategory if Coef has Algebra Fraction Integer

IndexedProductCategory(Coef, Integer)

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

InnerEvalable(Symbol, SparseUnivariateTaylorSeries(Coef, var, cen)) if SparseUnivariateTaylorSeries(Coef, var, cen) has InnerEvalable(Symbol, SparseUnivariateTaylorSeries(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 SparseUnivariateTaylorSeries(Coef, var, cen) if Coef has Field

LeftOreRing if Coef has Field

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

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

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

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

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

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

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

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

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

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

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

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

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

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

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

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

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

PrincipalIdealDomain if Coef has Field

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

RadicalCategory if Coef has Algebra Fraction Integer

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

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

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

RetractableTo SparseUnivariateTaylorSeries(Coef, var, cen)

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

RightModule %

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

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

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

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough if SparseUnivariateTaylorSeries(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, SparseUnivariateTaylorSeries(Coef, var, cen))

UnivariatePowerSeriesCategory(Coef, Integer)

UnivariateSeriesWithRationalExponents(Coef, Integer)

VariablesCommuteWithCoefficients