# SparseUnivariatePuiseuxSeries(Coef, var, cen)ΒΆ

Sparse Puiseux series in one variable SparseUnivariatePuiseuxSeries is a domain representing Puiseux 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, SparseUnivariatePuiseuxSeries(Integer, x, 3) represents Puiseux 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
*: (Coef, %) -> %

from LeftModule Coef

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

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

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

from Field

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

from AbelianMonoidRing(Coef, Fraction Integer)

=: (%, %) -> Boolean

from BasicType

^: (%, %) -> % 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

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

from UnivariatePowerSeriesCategory(Coef, Fraction 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
center: % -> Coef

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
coefficient: (%, Fraction Integer) -> Coef

from AbelianMonoidRing(Coef, Fraction 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: SparseUnivariateLaurentSeries(Coef, var, cen) -> %

from UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

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

from CoercibleFrom SparseUnivariateTaylorSeries(Coef, var, cen)

coerce: Variable var -> %

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

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

from IndexedProductCategory(Coef, Fraction Integer)

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

from IndexedProductCategory(Coef, Fraction Integer)

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 Coef has *: (Fraction Integer, Coef) -> Coef

from DifferentialRing

D: (%, List Symbol) -> % if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if Coef has *: (Fraction Integer, Coef) -> Coef

from DifferentialRing

D: (%, Symbol) -> % if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
degree: % -> Fraction Integer

from UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

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

from DifferentialRing

differentiate: (%, List Symbol) -> % if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Fraction Integer, Coef) -> Coef

from DifferentialRing

differentiate: (%, Symbol) -> % if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (Fraction 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: (%, Fraction Integer) -> Coef

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

euclideanSize: % -> NonNegativeInteger if Coef has Field

from EuclideanDomain

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

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

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: (%, Fraction Integer) -> %

from UnivariatePowerSeriesCategory(Coef, Fraction 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
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

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

from UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

laurentIfCan: % -> Union(SparseUnivariateLaurentSeries(Coef, var, cen), failed)

from UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

laurentRep: % -> SparseUnivariateLaurentSeries(Coef, var, cen)

from UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

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

from GcdDomain

lcm: List % -> % if Coef has Field

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if Coef has Field

from LeftOreRing

leadingCoefficient: % -> Coef
leadingMonomial: % -> %
leadingSupport: % -> Fraction Integer

from IndexedProductCategory(Coef, Fraction Integer)

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

from IndexedProductCategory(Coef, Fraction 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, Fraction Integer)

monomial?: % -> Boolean

from IndexedProductCategory(Coef, Fraction Integer)

monomial: (Coef, Fraction Integer) -> %

from IndexedProductCategory(Coef, Fraction Integer)

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

from EuclideanDomain

multiplyExponents: (%, Fraction Integer) -> %

from UnivariatePuiseuxSeriesCategory Coef

multiplyExponents: (%, PositiveInteger) -> %

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

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

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> Fraction Integer

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

order: (%, Fraction Integer) -> Fraction Integer

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

pi: () -> % if Coef has Algebra Fraction Integer
plenaryPower: (%, PositiveInteger) -> % if Coef has CommutativeRing or Coef has Algebra Fraction Integer
pole?: % -> Boolean
prime?: % -> Boolean if Coef has Field
principalIdeal: List % -> Record(coef: List %, generator: %) if Coef has Field
puiseux: (Fraction Integer, SparseUnivariateLaurentSeries(Coef, var, cen)) -> %

from UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

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

from EuclideanDomain

rationalPower: % -> Fraction Integer

from UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(Coef, Fraction Integer)

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

from EuclideanDomain

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

from RetractableTo SparseUnivariateLaurentSeries(Coef, var, cen)

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

from RetractableTo SparseUnivariateTaylorSeries(Coef, var, cen)

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

from RetractableTo SparseUnivariateLaurentSeries(Coef, var, cen)

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

from RetractableTo SparseUnivariateTaylorSeries(Coef, var, cen)

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sec: % -> % if Coef has Algebra Fraction Integer
sech: % -> % if Coef has Algebra Fraction Integer
series: (NonNegativeInteger, Stream Record(k: Fraction Integer, c: Coef)) -> %

from UnivariatePuiseuxSeriesCategory Coef

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

from EuclideanDomain

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

squareFree: % -> Factored % if Coef has Field
squareFreePart: % -> % if Coef has Field
subtractIfCan: (%, %) -> Union(%, failed)
tan: % -> % if Coef has Algebra Fraction Integer
tanh: % -> % if Coef has Algebra Fraction Integer
terms: % -> Stream Record(k: Fraction Integer, c: Coef)

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

truncate: (%, Fraction Integer) -> %

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

truncate: (%, Fraction Integer, Fraction Integer) -> %

from UnivariatePowerSeriesCategory(Coef, Fraction 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, Fraction Integer)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(Coef, Coef)

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

CancellationAbelianMonoid

canonicalsClosed if Coef has Field

canonicalUnitNormal if Coef has Field

CharacteristicNonZero if Coef has CharacteristicNonZero

CharacteristicZero if Coef has CharacteristicZero

CoercibleFrom SparseUnivariateLaurentSeries(Coef, var, cen)

CoercibleFrom SparseUnivariateTaylorSeries(Coef, var, cen)

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

DivisionRing if Coef has Field

Eltable(%, %)

EntireRing if Coef has IntegralDomain

EuclideanDomain if Coef has Field

Field if Coef has Field

GcdDomain if Coef has Field

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

LeftOreRing 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

Monoid

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

PartialDifferentialRing Symbol if Coef has *: (Fraction Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

PrincipalIdealDomain if Coef has Field

RadicalCategory if Coef has Algebra Fraction Integer

RetractableTo SparseUnivariateLaurentSeries(Coef, var, cen)

RetractableTo SparseUnivariateTaylorSeries(Coef, var, cen)

RightModule Coef

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

UniqueFactorizationDomain if Coef has Field

unitsKnown

UnivariatePuiseuxSeriesConstructorCategory(Coef, SparseUnivariateLaurentSeries(Coef, var, cen))

VariablesCommuteWithCoefficients