GeneralUnivariatePowerSeries(Coef, var, cen)

gseries.spad line 1 [edit on github]

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).

* : (%, %) -> %
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 : UnivariatePuiseuxSeries(Coef, var, cen) -> %

coerce(f) converts a Puiseux series to a general power series.

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 PowerSeriesCategory(Coef, Fraction(Integer), SingletonAsOrderedSet)
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
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 CommutativeRing or Coef has Algebra(Fraction(Integer))
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
quo : (%, %) -> % if Coef has Field
from EuclideanDomain
recip : % -> Union(%, "failed")
from MagmaWithUnit
reductum : % -> %
from IndexedProductCategory(Coef, Fraction(Integer))
rem : (%, %) -> % if Coef has Field
from EuclideanDomain
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

Algebra(Fraction(Integer))

Module(Fraction(Integer))

NonAssociativeAlgebra(Coef)

Module(Coef)

NonAssociativeSemiRing

BiModule(%, %)

Field

canonicalUnitNormal

Rng

ArcTrigonometricFunctionCategory

UnivariateSeriesWithRationalExponents(Coef, Fraction(Integer))

TwoSidedRecip

TranscendentalFunctionCategory

SemiRing

EntireRing

RightModule(Coef)

NonAssociativeAlgebra(Fraction(Integer))

CharacteristicNonZero

unitsKnown

RadicalCategory

UnivariatePowerSeriesCategory(Coef, Fraction(Integer))

AbelianProductCategory(Coef)

UniqueFactorizationDomain

SemiGroup

RightModule(Fraction(Integer))

Magma

GcdDomain

IntegralDomain

LeftModule(%)

NonAssociativeRing

AbelianMonoidRing(Coef, Fraction(Integer))

ArcHyperbolicFunctionCategory

PartialDifferentialRing(Symbol)

CharacteristicZero

BasicType

Algebra(%)

PowerSeriesCategory(Coef, Fraction(Integer), SingletonAsOrderedSet)

CommutativeRing

IndexedProductCategory(Coef, Fraction(Integer))

DifferentialRing

DivisionRing

Eltable(%, %)

PrincipalIdealDomain

NonAssociativeSemiRng

CancellationAbelianMonoid

EuclideanDomain

canonicalsClosed

VariablesCommuteWithCoefficients

CommutativeStar

AbelianMonoid

ElementaryFunctionCategory

UnivariatePuiseuxSeriesCategory(Coef)

RightModule(%)

BiModule(Coef, Coef)

LeftModule(Coef)

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

Monoid

LeftOreRing

NonAssociativeAlgebra(%)

MagmaWithUnit

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

SetCategory

noZeroDivisors

TrigonometricFunctionCategory

NonAssociativeRng

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

HyperbolicFunctionCategory

AbelianGroup