SparseUnivariatePuiseuxSeries(Coef, var, cen)

sups.spad line 1638 [edit on github]

• Coef: Ring

• var: Symbol

• cen: Coef

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

from RightModule Fraction Integer

*: (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

+: (%, %) -> %

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

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

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: (%, 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

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

center: % -> Coef

from UnivariatePowerSeriesCategory(Coef, Fraction Integer)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero

from CharacteristicNonZero

coefficient: (%, Fraction Integer) -> Coef

from AbelianMonoidRing(Coef, Fraction 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 Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

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: (%, %) -> %

from NonAssociativeRng

complete: % -> %

from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)

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

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from 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

from PartialDifferentialRing Symbol

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

from 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

from PartialDifferentialRing Symbol

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

from 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

from PartialDifferentialRing Symbol

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

from ElementaryFunctionCategory

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

from PrincipalIdealDomain

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

from UniqueFactorizationDomain

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

from UnivariateSeriesWithRationalExponents(Coef, Fraction Integer)

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

from UnivariateSeriesWithRationalExponents(Coef, 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

from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)

leadingMonomial: % -> %

from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)

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

from ElementaryFunctionCategory

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

from RadicalCategory

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

from TranscendentalFunctionCategory

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

from NonAssociativeAlgebra %

pole?: % -> Boolean

from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)

prime?: % -> Boolean if Coef has Field

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %) if Coef has Field

from PrincipalIdealDomain

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

from TrigonometricFunctionCategory

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

from HyperbolicFunctionCategory

series: (NonNegativeInteger, Stream Record(k: Fraction Integer, c: Coef)) -> %

from UnivariatePuiseuxSeriesCategory Coef

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

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

from RadicalCategory

squareFree: % -> Factored % if Coef has Field

from UniqueFactorizationDomain

squareFreePart: % -> % if Coef has Field

from UniqueFactorizationDomain

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

from CancellationAbelianMonoid

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

from TrigonometricFunctionCategory

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

from HyperbolicFunctionCategory

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

AbelianMonoidRing(Coef, Fraction Integer)

AbelianProductCategory Coef

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

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

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)

CoercibleTo OutputForm

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

DivisionRing if Coef has Field

ElementaryFunctionCategory if Coef has Algebra Fraction Integer

Eltable(%, %)

EntireRing if Coef has IntegralDomain

EuclideanDomain if Coef has Field

Field if Coef has Field

GcdDomain if Coef has Field

HyperbolicFunctionCategory if Coef has Algebra Fraction Integer

IndexedProductCategory(Coef, Fraction Integer)

IntegralDomain if Coef has IntegralDomain

LeftModule %

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

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

NonAssociativeAlgebra Fraction Integer if Coef has Algebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

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

PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)

PrincipalIdealDomain if Coef has Field

RadicalCategory if Coef has Algebra Fraction Integer

RetractableTo SparseUnivariateLaurentSeries(Coef, var, cen)

RetractableTo SparseUnivariateTaylorSeries(Coef, var, cen)

RightModule %

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

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

UnivariatePowerSeriesCategory(Coef, Fraction Integer)

UnivariatePuiseuxSeriesCategory Coef

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

UnivariateSeriesWithRationalExponents(Coef, Fraction Integer)

VariablesCommuteWithCoefficients