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.

* : (%, %) -> %

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))

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

< : (%, %) -> 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

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

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

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

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

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

from DifferentialRing

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)

^ : (%, %) -> % 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

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 : 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.

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

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

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

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

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

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

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

from DifferentialRing

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 : (%, 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), %)

elt : (%, Integer) -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

euclideanSize : % -> NonNegativeInteger if Coef has Field

from EuclideanDomain

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))

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

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 : (Mapping(Coef, Coef), %) -> %

from IndexedProductCategory(Coef, Integer)

map : (Mapping(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 : (Coef, Integer) -> %

from IndexedProductCategory(Coef, Integer)

monomial? : % -> Boolean

from IndexedProductCategory(Coef, Integer)

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

from EuclideanDomain

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

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(%) -> 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))

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))

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

~= : (%, %) -> Boolean

from BasicType

EntireRing

Ring

Algebra(%)

Eltable(%, %)

CancellationAbelianMonoid

CommutativeStar

ConvertibleTo(Pattern(Float))

Field

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

UnivariateSeriesWithRationalExponents(Coef, Integer)

LinearlyExplicitOver(Integer)

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

SemiGroup

LinearlyExplicitOver(SparseUnivariateTaylorSeries(Coef, var, cen))

OrderedSet

UnivariateLaurentSeriesCategory(Coef)

ConvertibleTo(Float)

VariablesCommuteWithCoefficients

ConvertibleTo(DoubleFloat)

ArcTrigonometricFunctionCategory

PartialOrder

UnivariatePowerSeriesCategory(Coef, Integer)

NonAssociativeAlgebra(Coef)

Monoid

OrderedAbelianSemiGroup

LeftModule(%)

BiModule(Fraction(Integer), Fraction(Integer))

noZeroDivisors

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

RightModule(SparseUnivariateTaylorSeries(Coef, var, cen))

TrigonometricFunctionCategory

Module(%)

TranscendentalFunctionCategory

Module(SparseUnivariateTaylorSeries(Coef, var, cen))

LeftOreRing

NonAssociativeSemiRing

RetractableTo(Fraction(Integer))

ConvertibleTo(InputForm)

Rng

BiModule(SparseUnivariateTaylorSeries(Coef, var, cen), SparseUnivariateTaylorSeries(Coef, var, cen))

SemiRing

NonAssociativeAlgebra(%)

SetCategory

CharacteristicNonZero

Eltable(SparseUnivariateTaylorSeries(Coef, var, cen), %)

IndexedProductCategory(Coef, Integer)

CoercibleFrom(SparseUnivariateTaylorSeries(Coef, var, cen))

PolynomialFactorizationExplicit

EuclideanDomain

CoercibleFrom(Fraction(Integer))

NonAssociativeRing

RealConstant

CoercibleFrom(Symbol)

QuotientFieldCategory(SparseUnivariateTaylorSeries(Coef, var, cen))

NonAssociativeRng

Module(Coef)

Algebra(Coef)

Comparable

PrincipalIdealDomain

canonicalsClosed

DivisionRing

Evalable(SparseUnivariateTaylorSeries(Coef, var, cen))

BiModule(Coef, Coef)

DifferentialExtension(SparseUnivariateTaylorSeries(Coef, var, cen))

ArcHyperbolicFunctionCategory

ElementaryFunctionCategory

unitsKnown

AbelianSemiGroup

canonicalUnitNormal

LeftModule(Fraction(Integer))

PatternMatchable(Float)

AbelianProductCategory(Coef)

IntegralDomain

FullyPatternMatchable(SparseUnivariateTaylorSeries(Coef, var, cen))

RightModule(Integer)

Algebra(SparseUnivariateTaylorSeries(Coef, var, cen))

NonAssociativeAlgebra(Fraction(Integer))

FullyLinearlyExplicitOver(SparseUnivariateTaylorSeries(Coef, var, cen))

RadicalCategory

NonAssociativeSemiRng

GcdDomain

CharacteristicZero

NonAssociativeAlgebra(SparseUnivariateTaylorSeries(Coef, var, cen))

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

CommutativeRing

PartialDifferentialRing(Symbol)

Patternable(SparseUnivariateTaylorSeries(Coef, var, cen))

Algebra(Fraction(Integer))

HyperbolicFunctionCategory

AbelianGroup

OrderedRing

DifferentialRing

BasicType

RetractableTo(SparseUnivariateTaylorSeries(Coef, var, cen))

TwoSidedRecip

MagmaWithUnit

CoercibleFrom(Integer)

AbelianMonoid

PatternMatchable(Integer)

CoercibleTo(OutputForm)

BiModule(%, %)

LeftModule(Coef)

StepThrough

OrderedCancellationAbelianMonoid

SemiRng

RetractableTo(Symbol)

RightModule(Coef)

RightModule(Fraction(Integer))

OrderedAbelianMonoid

OrderedAbelianGroup

UniqueFactorizationDomain

FullyEvalableOver(SparseUnivariateTaylorSeries(Coef, var, cen))

OrderedIntegralDomain

ConvertibleTo(Pattern(Integer))

Module(Fraction(Integer))

AbelianMonoidRing(Coef, Integer)

LeftModule(SparseUnivariateTaylorSeries(Coef, var, cen))

RightModule(%)

RetractableTo(Integer)

Magma