SparseUnivariateLaurentSeries(Coef, var, cen)¶

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 LeftModule %

*: (%, Coef) -> %

from RightModule Coef

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

^: (%, 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
acosh: % -> % if Coef has Algebra Fraction Integer
acot: % -> % if Coef has Algebra Fraction Integer
acoth: % -> % if Coef has Algebra Fraction Integer
acsc: % -> % if Coef has Algebra Fraction Integer
acsch: % -> % if Coef has Algebra Fraction Integer
annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
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
asech: % -> % if Coef has Algebra Fraction Integer
asin: % -> % if Coef has Algebra Fraction Integer
asinh: % -> % if Coef has Algebra Fraction Integer
associates?: (%, %) -> Boolean if Coef has IntegralDomain

from EntireRing

associator: (%, %, %) -> %
atan: % -> % if Coef has Algebra Fraction Integer
atanh: % -> % if Coef has Algebra Fraction Integer
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
charthRoot: % -> Union(%, failed) if SparseUnivariateTaylorSeries(Coef, var, cen) has CharacteristicNonZero and Coef has Field or Coef has CharacteristicNonZero or SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and % has CharacteristicNonZero and Coef has Field
coefficient: (%, Integer) -> Coef

from AbelianMonoidRing(Coef, Integer)

coerce: % -> % if Coef has CommutativeRing

from Algebra %

coerce: % -> OutputForm
coerce: Coef -> % if Coef has CommutativeRing

from Algebra Coef

coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
coerce: Integer -> %
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
coerce: Variable var -> %

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

commutator: (%, %) -> %
complete: % -> %

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

conditionP: Matrix % -> Union(Vector %, failed) if SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and % has CharacteristicNonZero and Coef has Field
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
convert: % -> Float if SparseUnivariateTaylorSeries(Coef, var, cen) has RealConstant and Coef has Field
convert: % -> InputForm if SparseUnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo InputForm and Coef has Field
convert: % -> Pattern Float if SparseUnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo Pattern Float and Coef has Field
convert: % -> Pattern Integer if SparseUnivariateTaylorSeries(Coef, var, cen) has ConvertibleTo Pattern Integer and Coef has Field
cos: % -> % if Coef has Algebra Fraction Integer
cosh: % -> % if Coef has Algebra Fraction Integer
cot: % -> % if Coef has Algebra Fraction Integer
coth: % -> % if Coef has Algebra Fraction Integer
csc: % -> % if Coef has Algebra Fraction Integer
csch: % -> % if Coef has Algebra Fraction Integer
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
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
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
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
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
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
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
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
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
expressIdealMember: (List %, %) -> Union(List %, failed) if Coef has Field
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
factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field
factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field
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 GcdDomain

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

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

from StepThrough

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

from UnivariateLaurentSeriesCategory Coef

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

from UnivariateLaurentSeriesCategory Coef

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

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

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
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

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
patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if SparseUnivariateTaylorSeries(Coef, var, cen) has PatternMatchable Integer and Coef has Field
pi: () -> % if Coef has Algebra Fraction Integer
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
principalIdeal: List % -> Record(coef: List %, generator: %) if Coef has Field
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
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
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
retract: % -> Integer if SparseUnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field
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
retractIfCan: % -> Union(Fraction Integer, failed) if SparseUnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field
retractIfCan: % -> Union(Integer, failed) if SparseUnivariateTaylorSeries(Coef, var, cen) has RetractableTo Integer and Coef has Field
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
rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sec: % -> % if Coef has Algebra Fraction Integer
sech: % -> % if Coef has Algebra Fraction Integer
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
sinh: % -> % if Coef has Algebra Fraction Integer
sizeLess?: (%, %) -> Boolean if Coef has Field

from EuclideanDomain

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

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed) if SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field
sqrt: % -> % if Coef has Algebra Fraction Integer

squareFree: % -> Factored % if Coef has Field
squareFreePart: % -> % if Coef has Field
squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial % if SparseUnivariateTaylorSeries(Coef, var, cen) has PolynomialFactorizationExplicit and Coef has Field
subtractIfCan: (%, %) -> Union(%, failed)
tan: % -> % if Coef has Algebra Fraction Integer
tanh: % -> % if Coef has Algebra Fraction Integer
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)

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

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

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

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

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

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

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

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

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 Coef

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

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

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 Coef

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

TwoSidedRecip if Coef has CommutativeRing

UniqueFactorizationDomain if Coef has Field

unitsKnown

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

VariablesCommuteWithCoefficients