UnivariatePuiseuxSeries(Coef, var, cen)

puiseux.spad line 410 [edit on github]

Dense Puiseux series in one variable UnivariatePuiseuxSeries 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, UnivariatePuiseuxSeries(Integer, x, 3) represents Puiseux 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))
* : (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))
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
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)
^ : (%, %) -> % 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
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 : 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 : UnivariateLaurentSeries(Coef, var, cen) -> %
from UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(Coef, var, cen))
coerce : UnivariateTaylorSeries(Coef, var, cen) -> %
from CoercibleFrom(UnivariateTaylorSeries(Coef, var, cen))
coerce : Variable(var) -> %

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

coerce : % -> OutputForm
from CoercibleTo(OutputForm)
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
degree : % -> Fraction(Integer)
from UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(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 Algebra(Fraction(Integer)) and Coef has integrate : (Coef, Symbol) -> Coef and Coef has variables : Coef -> List(Symbol)
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 : % -> UnivariateLaurentSeries(Coef, var, cen)
from UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(Coef, var, cen))
laurentIfCan : % -> Union(UnivariateLaurentSeries(Coef, var, cen), "failed")
from UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(Coef, var, cen))
laurentRep : % -> UnivariateLaurentSeries(Coef, var, cen)
from UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(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 : (Mapping(Coef, Coef), %) -> %
from IndexedProductCategory(Coef, Fraction(Integer))
monomial : (Coef, Fraction(Integer)) -> %
from IndexedProductCategory(Coef, Fraction(Integer))
monomial? : % -> Boolean
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 Algebra(Fraction(Integer)) or Coef has CommutativeRing
from NonAssociativeAlgebra(Coef)
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), UnivariateLaurentSeries(Coef, var, cen)) -> %
from UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(Coef, var, cen))
quo : (%, %) -> % if Coef has Field
from EuclideanDomain
rationalPower : % -> Fraction(Integer)
from UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(Coef, var, cen))
recip : % -> Union(%, "failed")
from MagmaWithUnit
reductum : % -> %
from IndexedProductCategory(Coef, Fraction(Integer))
rem : (%, %) -> % if Coef has Field
from EuclideanDomain
retract : % -> UnivariateLaurentSeries(Coef, var, cen)
from RetractableTo(UnivariateLaurentSeries(Coef, var, cen))
retract : % -> UnivariateTaylorSeries(Coef, var, cen)
from RetractableTo(UnivariateTaylorSeries(Coef, var, cen))
retractIfCan : % -> Union(UnivariateLaurentSeries(Coef, var, cen), "failed")
from RetractableTo(UnivariateLaurentSeries(Coef, var, cen))
retractIfCan : % -> Union(UnivariateTaylorSeries(Coef, var, cen), "failed")
from RetractableTo(UnivariateTaylorSeries(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
~= : (%, %) -> Boolean
from BasicType

Module(Fraction(Integer))

NonAssociativeAlgebra(Coef)

Module(Coef)

NonAssociativeSemiRing

BiModule(%, %)

RetractableTo(UnivariateTaylorSeries(Coef, var, cen))

Field

canonicalUnitNormal

Rng

ArcTrigonometricFunctionCategory

UnivariateSeriesWithRationalExponents(Coef, Fraction(Integer))

UnivariatePuiseuxSeriesConstructorCategory(Coef, UnivariateLaurentSeries(Coef, var, cen))

TwoSidedRecip

TranscendentalFunctionCategory

SemiRing

EntireRing

RightModule(Coef)

NonAssociativeAlgebra(Fraction(Integer))

CharacteristicNonZero

CoercibleFrom(UnivariateLaurentSeries(Coef, var, cen))

unitsKnown

RadicalCategory

UnivariatePowerSeriesCategory(Coef, Fraction(Integer))

NonAssociativeRng

AbelianProductCategory(Coef)

Magma

SemiGroup

GcdDomain

IntegralDomain

LeftModule(%)

NonAssociativeRing

AbelianMonoidRing(Coef, Fraction(Integer))

ArcHyperbolicFunctionCategory

PartialDifferentialRing(Symbol)

CharacteristicZero

UniqueFactorizationDomain

PowerSeriesCategory(Coef, Fraction(Integer), SingletonAsOrderedSet)

CommutativeRing

IndexedProductCategory(Coef, Fraction(Integer))

Algebra(%)

DifferentialRing

RightModule(Fraction(Integer))

Eltable(%, %)

MagmaWithUnit

RetractableTo(UnivariateLaurentSeries(Coef, var, cen))

PrincipalIdealDomain

NonAssociativeSemiRng

CancellationAbelianMonoid

EuclideanDomain

canonicalsClosed

VariablesCommuteWithCoefficients

CommutativeStar

AbelianMonoid

UnivariatePuiseuxSeriesCategory(Coef)

RightModule(%)

BiModule(Coef, Coef)

CoercibleFrom(UnivariateTaylorSeries(Coef, var, cen))

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

Monoid

LeftOreRing

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

DivisionRing

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

noZeroDivisors

SetCategory

TrigonometricFunctionCategory

LeftModule(Coef)

BasicType

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

HyperbolicFunctionCategory

AbelianGroup

ElementaryFunctionCategory