# UnivariateLaurentSeriesCategory CoefΒΆ

UnivariateLaurentSeriesCategory is the category of Laurent series in one variable.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Coef) -> %

from RightModule Coef

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

from LeftModule Coef

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

=: (%, %) -> Boolean

from BasicType

^: (%, %) -> % if Coef has Algebra Fraction Integer
^: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer

^: (%, Integer) -> % if Coef has Field

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

acos: % -> % if Coef has Algebra Fraction Integer
acosh: % -> % if Coef has Algebra Fraction Integer
acot: % -> % if Coef has Algebra Fraction Integer
acoth: % -> % if Coef has Algebra Fraction Integer
acsc: % -> % if Coef has Algebra Fraction Integer
acsch: % -> % if Coef has Algebra Fraction Integer
annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
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
asech: % -> % if Coef has Algebra Fraction Integer
asin: % -> % if Coef has Algebra Fraction Integer
asinh: % -> % if Coef has Algebra Fraction Integer
associates?: (%, %) -> Boolean if Coef has IntegralDomain

from EntireRing

associator: (%, %, %) -> %
atan: % -> % if Coef has Algebra Fraction Integer
atanh: % -> % if Coef has Algebra Fraction Integer
center: % -> Coef

from UnivariatePowerSeriesCategory(Coef, Integer)

characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
coefficient: (%, Integer) -> Coef

from AbelianMonoidRing(Coef, Integer)

coerce: % -> % if Coef has CommutativeRing

from Algebra %

coerce: % -> OutputForm
coerce: Coef -> % if Coef has CommutativeRing

from Algebra Coef

coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
coerce: Integer -> %
commutator: (%, %) -> %
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
cosh: % -> % if Coef has Algebra Fraction Integer
cot: % -> % if Coef has Algebra Fraction Integer
coth: % -> % if Coef has Algebra Fraction Integer
csc: % -> % if Coef has Algebra Fraction Integer
csch: % -> % if Coef has Algebra Fraction Integer
D: % -> % if Coef has *: (Integer, Coef) -> Coef

from DifferentialRing

D: (%, List Symbol) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has 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
D: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
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
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has 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
differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has 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
expressIdealMember: (List %, %) -> Union(List %, failed) if Coef has Field
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
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

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.

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

integrate(f(x), y) returns an anti-derivative of the power series f(x) with respect to the variable y.

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

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

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
map: (Coef -> Coef, %) -> %

from IndexedProductCategory(Coef, Integer)

monomial?: % -> Boolean

from IndexedProductCategory(Coef, Integer)

monomial: (Coef, Integer) -> %

from IndexedProductCategory(Coef, Integer)

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

from EuclideanDomain

multiplyCoefficients: (Integer -> Coef, %) -> %

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

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
pole?: % -> Boolean

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

prime?: % -> Boolean if Coef has Field
principalIdeal: List % -> Record(coef: List %, generator: %) if Coef has Field
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
sech: % -> % if Coef has Algebra Fraction Integer
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
sinh: % -> % if Coef has Algebra Fraction Integer
sizeLess?: (%, %) -> Boolean if Coef has Field

from EuclideanDomain

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

squareFree: % -> Factored % if Coef has Field
squareFreePart: % -> % if Coef has Field
subtractIfCan: (%, %) -> Union(%, failed)
tan: % -> % if Coef has Algebra Fraction Integer
tanh: % -> % if Coef has Algebra Fraction Integer
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

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Integer)

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer 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

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

DivisionRing if Coef has Field

Eltable(%, %)

EntireRing if Coef has IntegralDomain

EuclideanDomain if Coef has Field

Field if Coef has Field

GcdDomain if Coef has Field

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

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

PrincipalIdealDomain if Coef has Field

RadicalCategory if Coef has Algebra Fraction Integer

RightModule Coef

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

UniqueFactorizationDomain if Coef has Field

unitsKnown

VariablesCommuteWithCoefficients