ModularFunctionQSeriesInfinity C

qetamodfunqseriesinf.spad line 98 [edit on github]

• C: CommutativeRing

ModularFunctionQSeriesInfinity represents Laurent series such that if x is such a series and order(x)>0 then x=0. Quotients of Dedekind eta-functions that are modular functions that only have a pole (if any) at infinity, can be represented as such series.

0: %

from QEtaAlgebra C

1: %

from QEtaAlgebra C

*: (%, %) -> %

from QEtaAlgebra C

*: (%, C) -> %

from RightModule C

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

from RightModule Fraction Integer

*: (C, %) -> %

from QEtaAlgebra C

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

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from QEtaAlgebra C

-: % -> %

from QEtaAlgebra C

-: (%, %) -> %

from QEtaAlgebra C

/: (%, %) -> % if C has Field

from Field

/: (%, C) -> % if C has Field

from AbelianMonoidRing(C, Integer)

=: (%, %) -> Boolean

from BasicType

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

from ElementaryFunctionCategory

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

from RadicalCategory

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

from DivisionRing

^: (%, NonNegativeInteger) -> %

from QEtaAlgebra C

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

acos: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

acosh: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

acot: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

acoth: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

acsc: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

acsch: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

approximate: (%, Integer) -> C if C has ^: (C, Integer) -> C and C has coerce: Symbol -> C

from UnivariatePowerSeriesCategory(C, Integer)

asec: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

asech: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

asin: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

asinh: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

associates?: (%, %) -> Boolean if C has IntegralDomain

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

atan: % -> % if C has Algebra Fraction Integer

from ArcTrigonometricFunctionCategory

atanh: % -> % if C has Algebra Fraction Integer

from ArcHyperbolicFunctionCategory

center: % -> C

from UnivariatePowerSeriesCategory(C, Integer)

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if C has CharacteristicNonZero

from CharacteristicNonZero

choose: (PositiveInteger, NonNegativeInteger, %) -> %

from QEtaLaurentSeriesCategory C

coefficient: (%, Integer) -> C

from AbelianMonoidRing(C, Integer)

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: % -> QEtaLaurentSeries C

coerce(x) returns the element x as a QEta Laurent series.

coerce: C -> %

from Algebra C

coerce: Fraction Integer -> % if C has Algebra Fraction Integer

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

coerce: QEtaLaurentSeries C -> %

coerce(x) assumes that the QEta Laurent series x belongs to the subalgebra of series with the properties of this damain. No check is made.

coerce: QEtaTaylorSeries C -> %

from CoercibleFrom QEtaTaylorSeries C

commutator: (%, %) -> %

from NonAssociativeRng

complete: % -> %

from PowerSeriesCategory(C, Integer, SingletonAsOrderedSet)

construct: List Record(k: Integer, c: C) -> %

from IndexedProductCategory(C, Integer)

constructOrdered: List Record(k: Integer, c: C) -> %

from IndexedProductCategory(C, Integer)

cos: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

cosh: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

cot: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

coth: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

csc: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

csch: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

D: % -> % if C has *: (Integer, C) -> C

from DifferentialRing

D: (%, List Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

D: (%, NonNegativeInteger) -> % if C has *: (Integer, C) -> C

from DifferentialRing

D: (%, Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

degree: % -> Integer

from PowerSeriesCategory(C, Integer, SingletonAsOrderedSet)

differentiate: % -> % if C has *: (Integer, C) -> C

from DifferentialRing

differentiate: (%, List Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

differentiate: (%, NonNegativeInteger) -> % if C has *: (Integer, C) -> C

from DifferentialRing

differentiate: (%, Symbol) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if C has PartialDifferentialRing Symbol and C has *: (Integer, C) -> C

from PartialDifferentialRing Symbol

divide: (%, %) -> Record(quotient: %, remainder: %) if C has Field

from EuclideanDomain

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, Integer) -> C

from UnivariatePowerSeriesCategory(C, Integer)

euclideanSize: % -> NonNegativeInteger if C has Field

from EuclideanDomain

eval: (%, C) -> Stream C if C has ^: (C, Integer) -> C

from UnivariatePowerSeriesCategory(C, Integer)

exp: % -> % if C has Algebra Fraction Integer

from ElementaryFunctionCategory

exponentGcd: (%, PositiveInteger) -> Integer

from QEtaLaurentSeriesCategory C

expressIdealMember: (List %, %) -> Union(List %, failed) if C has Field

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed) if C has IntegralDomain

from EntireRing

extend: (%, Integer) -> %

from UnivariatePowerSeriesCategory(C, Integer)

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if C has Field

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if C has Field

from EuclideanDomain

factor: % -> Factored % if C has Field

from UniqueFactorizationDomain

gcd: (%, %) -> % if C has Field

from GcdDomain

gcd: List % -> % if C has Field

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if C has Field

from GcdDomain

integrate: % -> % if C has Algebra Fraction Integer

from UnivariateSeriesWithRationalExponents(C, Integer)

integrate: (%, Symbol) -> % if C has variables: C -> List Symbol and C has Algebra Fraction Integer and C has integrate: (C, Symbol) -> C

from UnivariateSeriesWithRationalExponents(C, Integer)

inv: % -> % if C has Field

from DivisionRing

latex: % -> String

from SetCategory

laurent: (Integer, QEtaTaylorSeries C) -> %

from QEtaLaurentSeriesCategory C

laurent: (Integer, Stream C) -> %

from UnivariateLaurentSeriesCategory C

lcm: (%, %) -> % if C has Field

from GcdDomain

lcm: List % -> % if C has Field

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if C has Field

from LeftOreRing

leadingCoefficient: % -> C

from PowerSeriesCategory(C, Integer, SingletonAsOrderedSet)

leadingMonomial: % -> %

from PowerSeriesCategory(C, Integer, SingletonAsOrderedSet)

leadingSupport: % -> Integer

from IndexedProductCategory(C, Integer)

leadingTerm: % -> Record(k: Integer, c: C)

from IndexedProductCategory(C, Integer)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

log: % -> % if C has Algebra Fraction Integer

from ElementaryFunctionCategory

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

from IndexedProductCategory(C, Integer)

mapn: ((C, Integer) -> C, %, Integer) -> %

from QEtaLaurentSeriesCategory C

monomial?: % -> Boolean

from IndexedProductCategory(C, Integer)

monomial: (C, Integer) -> %

from IndexedProductCategory(C, Integer)

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

from EuclideanDomain

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

from UnivariateLaurentSeriesCategory C

multiplyExponents: (%, PositiveInteger) -> %

from UnivariatePowerSeriesCategory(C, Integer)

nthRoot: (%, Integer) -> % if C has Algebra Fraction Integer

from RadicalCategory

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> Integer

from UnivariatePowerSeriesCategory(C, Integer)

order: (%, Integer) -> Integer

from UnivariatePowerSeriesCategory(C, Integer)

pi: () -> % if C has Algebra Fraction Integer

from TranscendentalFunctionCategory

plenaryPower: (%, PositiveInteger) -> %

from NonAssociativeAlgebra %

pole?: % -> Boolean

from PowerSeriesCategory(C, Integer, SingletonAsOrderedSet)

prime?: % -> Boolean if C has Field

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %) if C has Field

from PrincipalIdealDomain

qetaCoefficient: (%, Integer) -> C

from QEtaGradedAlgebra C

qetaGrade: % -> Integer

from QEtaGradedAlgebra C

qetaLeadingCoefficient: % -> C

from QEtaGradedAlgebra C

qetaTaylorRep: % -> QEtaTaylorSeries C

qetaTaylorRep(x) returns the underlying Taylor representation of the Laurent serires. This might return a Taylor series that has 0 as its constant coefficient.

quo: (%, %) -> % if C has Field

from EuclideanDomain

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

from UnivariateLaurentSeriesCategory C

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

from UnivariateLaurentSeriesCategory C

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(C, Integer)

rem: (%, %) -> % if C has Field

from EuclideanDomain

removeZeroes: % -> %

from QEtaLaurentSeriesCategory C

removeZeroes: (Integer, %) -> %

from QEtaLaurentSeriesCategory C

retract: % -> QEtaTaylorSeries C

from RetractableTo QEtaTaylorSeries C

retractIfCan: % -> Union(QEtaTaylorSeries C, failed)

from RetractableTo QEtaTaylorSeries C

revert: % -> %

from QEtaLaurentSeriesCategory C

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from MagmaWithUnit

sec: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

sech: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

series: Stream Record(k: Integer, c: C) -> %

from UnivariateLaurentSeriesCategory C

sin: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

sinh: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

sizeLess?: (%, %) -> Boolean if C has Field

from EuclideanDomain

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

from RadicalCategory

squareFree: % -> Factored % if C has Field

from UniqueFactorizationDomain

squareFreePart: % -> % if C has Field

from UniqueFactorizationDomain

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

tan: % -> % if C has Algebra Fraction Integer

from TrigonometricFunctionCategory

tanh: % -> % if C has Algebra Fraction Integer

from HyperbolicFunctionCategory

terms: % -> Stream Record(k: Integer, c: C)

from UnivariatePowerSeriesCategory(C, Integer)

traceout: NonNegativeInteger -> % -> OutputForm

from QEtaAlgebra C

truncate: (%, Integer) -> %

from UnivariatePowerSeriesCategory(C, Integer)

truncate: (%, Integer, Integer) -> %

from UnivariatePowerSeriesCategory(C, Integer)

unit?: % -> Boolean if C has IntegralDomain

from EntireRing

unitCanonical: % -> % if C has IntegralDomain

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if C has IntegralDomain

from EntireRing

variable: % -> Symbol

from UnivariatePowerSeriesCategory(C, Integer)

zero?: % -> Boolean

from QEtaAlgebra C

AbelianGroup

AbelianMonoid

AbelianMonoidRing(C, Integer)

AbelianProductCategory C

AbelianSemiGroup

Algebra %

Algebra C

Algebra Fraction Integer if C has Algebra Fraction Integer

ArcHyperbolicFunctionCategory if C has Algebra Fraction Integer

ArcTrigonometricFunctionCategory if C has Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(C, C)

BiModule(Fraction Integer, Fraction Integer) if C has Algebra Fraction Integer

CancellationAbelianMonoid

canonicalsClosed if C has Field

canonicalUnitNormal if C has Field

CharacteristicNonZero if C has CharacteristicNonZero

CharacteristicZero if C has CharacteristicZero

CoercibleFrom QEtaTaylorSeries C

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

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

DivisionRing if C has Field

ElementaryFunctionCategory if C has Algebra Fraction Integer

Eltable(%, %)

EntireRing if C has IntegralDomain

EuclideanDomain if C has Field

Field if C has Field

GcdDomain if C has Field

HyperbolicFunctionCategory if C has Algebra Fraction Integer

IndexedProductCategory(C, Integer)

IntegralDomain if C has IntegralDomain

LeftModule %

LeftModule C

LeftModule Fraction Integer if C has Algebra Fraction Integer

LeftOreRing if C has Field

Magma

MagmaWithUnit

Module %

Module C

Module Fraction Integer if C has Algebra Fraction Integer

Monoid

NonAssociativeAlgebra %

NonAssociativeAlgebra C

NonAssociativeAlgebra Fraction Integer if C has Algebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if C has IntegralDomain

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

PowerSeriesCategory(C, Integer, SingletonAsOrderedSet)

PrincipalIdealDomain if C has Field

QEtaAlgebra C

QEtaGradedAlgebra C

QEtaLaurentSeriesCategory C

RadicalCategory if C has Algebra Fraction Integer

RetractableTo QEtaTaylorSeries C

RightModule %

RightModule C

RightModule Fraction Integer if C has Algebra Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TranscendentalFunctionCategory if C has Algebra Fraction Integer

TrigonometricFunctionCategory if C has Algebra Fraction Integer

TwoSidedRecip

UniqueFactorizationDomain if C has Field

unitsKnown

UnivariateLaurentSeriesCategory C

UnivariatePowerSeriesCategory(C, Integer)

UnivariateSeriesWithRationalExponents(C, Integer)

VariablesCommuteWithCoefficients