UnivariatePuiseuxSeriesConstructorCategory(Coef, ULS)ΒΆ

puiseux.spad line 1

This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair [r, f(x)], where r is a positive rational number and f(x) is a Laurent series. This pair represents the Puiseux series f(x^r).

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, 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: ULS -> %
coerce(f(x)) converts the Laurent series f(x) to a Puiseux series.
commutator: (%, %) -> %
from NonAssociativeRng
complete: % -> %
from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)
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 PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if Coef has *: (Fraction Integer, Coef) -> Coef
from DifferentialRing
D: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
degree: % -> Fraction Integer
degree(f(x)) returns the degree of the leading term of the Puiseux series f(x), which may have zero as a coefficient.
differentiate: % -> % if Coef has *: (Fraction Integer, Coef) -> Coef
from DifferentialRing
differentiate: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Fraction Integer, Coef) -> Coef
from DifferentialRing
differentiate: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Fraction Integer, Coef) -> Coef
from PartialDifferentialRing Symbol
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
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
integrate: % -> % if Coef has Algebra Fraction Integer
from UnivariatePuiseuxSeriesCategory Coef
integrate: (%, Symbol) -> % if Coef has Algebra Fraction Integer and Coef has integrate: (Coef, Symbol) -> Coef and Coef has variables: Coef -> List Symbol or Coef has AlgebraicallyClosedFunctionSpace Integer and Coef has Algebra Fraction Integer and Coef has TranscendentalFunctionCategory and Coef has PrimitiveFunctionCategory
from UnivariatePuiseuxSeriesCategory Coef
inv: % -> % if Coef has Field
from DivisionRing
latex: % -> String
from SetCategory
laurent: % -> ULS
laurent(f(x)) converts the Puiseux series f(x) to a Laurent series if possible. Error: if this is not possible.
laurentIfCan: % -> Union(ULS, failed)
laurentIfCan(f(x)) converts the Puiseux series f(x) to a Laurent series if possible. If this is not possible, “failed” is returned.
laurentRep: % -> ULS
laurentRep(f(x)) returns g(x) where the Puiseux series f(x) = g(x^r) is represented by [r, g(x)].
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)
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 AbelianMonoidRing(Coef, Fraction Integer)
monomial: (%, List SingletonAsOrderedSet, List Fraction Integer) -> %
from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)
monomial: (%, SingletonAsOrderedSet, Fraction Integer) -> %
from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)
monomial: (Coef, Fraction Integer) -> %
from AbelianMonoidRing(Coef, Fraction Integer)
monomial?: % -> Boolean
from AbelianMonoidRing(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
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, ULS) -> %
puiseux(r, f(x)) returns f(x^r).
quo: (%, %) -> % if Coef has Field
from EuclideanDomain
rationalPower: % -> Fraction Integer
rationalPower(f(x)) returns r where the Puiseux series f(x) = g(x^r).
recip: % -> Union(%, failed)
from MagmaWithUnit
reductum: % -> %
from AbelianMonoidRing(Coef, Fraction Integer)
rem: (%, %) -> % if Coef has Field
from EuclideanDomain
retract: % -> ULS
from RetractableTo ULS
retractIfCan: % -> Union(ULS, failed)
from RetractableTo ULS
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)
variables: % -> List SingletonAsOrderedSet
from PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Fraction Integer)

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

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

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

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

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

PowerSeriesCategory(Coef, Fraction Integer, SingletonAsOrderedSet)

PrincipalIdealDomain if Coef has Field

RadicalCategory if Coef has Algebra Fraction Integer

RetractableTo ULS

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

UniqueFactorizationDomain if Coef has Field

unitsKnown

UnivariatePowerSeriesCategory(Coef, Fraction Integer)

UnivariatePuiseuxSeriesCategory Coef

VariablesCommuteWithCoefficients