UnivariateLaurentSeriesCategory(Coef)

pscat.spad line 371 [edit on github]

UnivariateLaurentSeriesCategory is the category of Laurent series in one variable.

* : (%, %) -> %
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, Integer)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
D : % -> % if Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
D : (%, List(Symbol)) -> % if Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, NonNegativeInteger) -> % if Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
D : (%, Symbol) -> % if Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, Symbol, NonNegativeInteger) -> % if Coef has * : (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 : (%, Integer) -> Coef if Coef has coerce : Symbol -> Coef and Coef has ^ : (Coef, Integer) -> Coef
from UnivariatePowerSeriesCategory(Coef, 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, Integer)
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
charthRoot : % -> Union(%, "failed") if Coef has CharacteristicNonZero
from CharacteristicNonZero
coefficient : (%, Integer) -> Coef
from AbelianMonoidRing(Coef, 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 : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
complete : % -> %
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
construct : List(Record(k : Integer, c : Coef)) -> %
from IndexedProductCategory(Coef, Integer)
constructOrdered : List(Record(k : Integer, c : Coef)) -> %
from IndexedProductCategory(Coef, 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 : % -> Integer
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
differentiate : % -> % if Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
differentiate : (%, List(Symbol)) -> % if Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, NonNegativeInteger) -> % if Coef has * : (Integer, Coef) -> Coef
from DifferentialRing
differentiate : (%, Symbol) -> % if Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, Symbol, NonNegativeInteger) -> % if Coef has * : (Integer, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
divide : (%, %) -> Record(quotient : %, remainder : %) if Coef has Field
from EuclideanDomain
elt : (%, %) -> %
from Eltable(%, %)
elt : (%, Integer) -> Coef
from UnivariatePowerSeriesCategory(Coef, Integer)
euclideanSize : % -> NonNegativeInteger if Coef has Field
from EuclideanDomain
eval : (%, Coef) -> Stream(Coef) if Coef has ^ : (Coef, Integer) -> Coef
from UnivariatePowerSeriesCategory(Coef, 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 : (%, Integer) -> %
from UnivariatePowerSeriesCategory(Coef, 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, 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, Integer)
inv : % -> % if Coef has Field
from DivisionRing
latex : % -> String
from SetCategory
laurent : (Integer, Stream(Coef)) -> %

laurent(n, st) returns xn * series st where xn = monomial(1, n) and series st stands for the power series with coefficients given by the stream st.

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, Integer, SingletonAsOrderedSet)
leadingMonomial : % -> %
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
leadingSupport : % -> Integer
from IndexedProductCategory(Coef, Integer)
leadingTerm : % -> Record(k : Integer, c : Coef)
from IndexedProductCategory(Coef, 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, Integer)
monomial : (Coef, Integer) -> %
from IndexedProductCategory(Coef, Integer)
monomial? : % -> Boolean
from IndexedProductCategory(Coef, Integer)
multiEuclidean : (List(%), %) -> Union(List(%), "failed") if Coef has Field
from EuclideanDomain
multiplyCoefficients : (Mapping(Coef, Integer), %) -> %

multiplyCoefficients(f, sum(n = n0..infinity, a[n] * x^n)) = sum(n = 0..infinity, f(n) * a[n] * x^n). This function is used when Puiseux series are represented by a Laurent series and an exponent.

multiplyExponents : (%, PositiveInteger) -> %
from UnivariatePowerSeriesCategory(Coef, Integer)
nthRoot : (%, Integer) -> % if Coef has Algebra(Fraction(Integer))
from RadicalCategory
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
order : % -> Integer
from UnivariatePowerSeriesCategory(Coef, Integer)
order : (%, Integer) -> Integer
from UnivariatePowerSeriesCategory(Coef, 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, 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
rationalFunction : (%, Integer) -> Fraction(Polynomial(Coef)) if Coef has IntegralDomain

rationalFunction(f, k) returns a rational function consisting of the sum of all terms of f of degree <= k.

rationalFunction : (%, Integer, Integer) -> Fraction(Polynomial(Coef)) if Coef has IntegralDomain

rationalFunction(f, k1, k2) returns a rational function consisting of the sum of all terms of f of degree d with k1 <= d <= k2.

recip : % -> Union(%, "failed")
from MagmaWithUnit
reductum : % -> %
from IndexedProductCategory(Coef, 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 : Stream(Record(k : Integer, c : Coef)) -> %

series(st) creates a series from a stream of non-zero terms, where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.

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 : Integer, c : Coef))
from UnivariatePowerSeriesCategory(Coef, Integer)
truncate : (%, Integer) -> %
from UnivariatePowerSeriesCategory(Coef, Integer)
truncate : (%, Integer, Integer) -> %
from UnivariatePowerSeriesCategory(Coef, 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, Integer)
zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

Module(Fraction(Integer))

NonAssociativeAlgebra(Coef)

Module(Coef)

NonAssociativeSemiRing

BiModule(%, %)

Field

canonicalUnitNormal

Rng

ArcTrigonometricFunctionCategory

TwoSidedRecip

TranscendentalFunctionCategory

SemiRing

EntireRing

RightModule(Coef)

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

NonAssociativeAlgebra(Fraction(Integer))

CharacteristicNonZero

EuclideanDomain

unitsKnown

RadicalCategory

NonAssociativeRng

IndexedProductCategory(Coef, Integer)

SetCategory

MagmaWithUnit

AbelianProductCategory(Coef)

PartialDifferentialRing(Symbol)

Magma

SemiGroup

LeftModule(%)

GcdDomain

IntegralDomain

AbelianMonoidRing(Coef, Integer)

NonAssociativeRing

canonicalsClosed

UniqueFactorizationDomain

ArcHyperbolicFunctionCategory

CharacteristicZero

Algebra(%)

CommutativeRing

DifferentialRing

RightModule(Fraction(Integer))

Eltable(%, %)

PrincipalIdealDomain

NonAssociativeSemiRng

CancellationAbelianMonoid

UnivariatePowerSeriesCategory(Coef, Integer)

VariablesCommuteWithCoefficients

CommutativeStar

AbelianMonoid

RightModule(%)

BiModule(Coef, Coef)

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

UnivariateSeriesWithRationalExponents(Coef, Integer)

Monoid

LeftOreRing

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

DivisionRing

BasicType

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

noZeroDivisors

TrigonometricFunctionCategory

LeftModule(Coef)

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

HyperbolicFunctionCategory

AbelianGroup

ElementaryFunctionCategory