UnivariatePuiseuxSeriesWithExponentialSingularity(R, FE, var, cen)ΒΆ

UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent functions with essential singularities. Objects in this domain are sums, where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series. Thus, the elements of this domain are sums of expressions of the form `g(x) * exp(f(x))`, where `g`(`x`) is a univariate Puiseux series and `f`(`x`) is a univariate Puiseux series with no terms of non-negative degree.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> % if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
*: (%, UnivariatePuiseuxSeries(FE, var, cen)) -> %

from RightModule UnivariatePuiseuxSeries(FE, var, cen)

*: (Fraction Integer, %) -> % if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

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

from LeftModule UnivariatePuiseuxSeries(FE, var, cen)

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, UnivariatePuiseuxSeries(FE, var, cen)) -> % if UnivariatePuiseuxSeries(FE, var, cen) has Field

from AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associates?: (%, %) -> Boolean

from EntireRing

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

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if UnivariatePuiseuxSeries(FE, var, cen) has CharacteristicNonZero
coefficient: (%, ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)) -> UnivariatePuiseuxSeries(FE, var, cen)

from FreeModuleCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

coefficients: % -> List UnivariatePuiseuxSeries(FE, var, cen)

from FreeModuleCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

coerce: % -> %

from Algebra %

coerce: % -> OutputForm
coerce: Fraction Integer -> % if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer or UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
coerce: Integer -> %
coerce: UnivariatePuiseuxSeries(FE, var, cen) -> %

from CoercibleFrom UnivariatePuiseuxSeries(FE, var, cen)

commutator: (%, %) -> %
construct: List Record(k: ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), c: UnivariatePuiseuxSeries(FE, var, cen)) -> %

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

constructOrdered: List Record(k: ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), c: UnivariatePuiseuxSeries(FE, var, cen)) -> %

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

content: % -> UnivariatePuiseuxSeries(FE, var, cen) if UnivariatePuiseuxSeries(FE, var, cen) has GcdDomain

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

degree: % -> ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)

from AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

dominantTerm: % -> Union(Record(%term: Record(%coef: UnivariatePuiseuxSeries(FE, var, cen), %expon: ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), %expTerms: List Record(k: Fraction Integer, c: FE)), %type: String), failed)

`dominantTerm(f(var))` returns the term that dominates the limiting behavior of `f(var)` as `var -> cen+` together with a String which briefly describes that behavior. The value of the String will be `"zero"` (resp. `"infinity"`) if the term tends to zero (resp. infinity) exponentially and will `"series"` if the term is a Puiseux series.

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

from EntireRing

exquo: (%, UnivariatePuiseuxSeries(FE, var, cen)) -> Union(%, failed) if UnivariatePuiseuxSeries(FE, var, cen) has EntireRing

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

fmecg: (%, ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), UnivariatePuiseuxSeries(FE, var, cen), %) -> % if UnivariatePuiseuxSeries(FE, var, cen) has Ring

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

ground?: % -> Boolean

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

ground: % -> UnivariatePuiseuxSeries(FE, var, cen)

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

latex: % -> String

from SetCategory

leadingCoefficient: % -> UnivariatePuiseuxSeries(FE, var, cen)

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

leadingMonomial: % -> %

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

leadingSupport: % -> ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

leadingTerm: % -> Record(k: ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), c: UnivariatePuiseuxSeries(FE, var, cen))

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

limitPlus: % -> Union(OrderedCompletion FE, failed)

`limitPlus(f(var))` returns `limit(var -> cen+, f(var))`.

linearExtend: (ExponentialOfUnivariatePuiseuxSeries(FE, var, cen) -> UnivariatePuiseuxSeries(FE, var, cen), %) -> UnivariatePuiseuxSeries(FE, var, cen) if UnivariatePuiseuxSeries(FE, var, cen) has CommutativeRing

from FreeModuleCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

listOfTerms: % -> List Record(k: ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), c: UnivariatePuiseuxSeries(FE, var, cen))

from IndexedDirectProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

map: (UnivariatePuiseuxSeries(FE, var, cen) -> UnivariatePuiseuxSeries(FE, var, cen), %) -> %

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

mapExponents: (ExponentialOfUnivariatePuiseuxSeries(FE, var, cen) -> ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), %) -> %

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

minimumDegree: % -> ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

monomial?: % -> Boolean

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

monomial: (UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)) -> %

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

monomials: % -> List %

from FreeModuleCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

numberOfMonomials: % -> NonNegativeInteger

from IndexedDirectProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

plenaryPower: (%, PositiveInteger) -> %
pomopo!: (%, UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen), %) -> %

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

primitivePart: % -> % if UnivariatePuiseuxSeries(FE, var, cen) has GcdDomain

from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

retract: % -> Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer
retract: % -> Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Integer
retract: % -> UnivariatePuiseuxSeries(FE, var, cen)

from RetractableTo UnivariatePuiseuxSeries(FE, var, cen)

retractIfCan: % -> Union(Fraction Integer, failed) if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Integer
retractIfCan: % -> Union(UnivariatePuiseuxSeries(FE, var, cen), failed)

from RetractableTo UnivariatePuiseuxSeries(FE, var, cen)

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

smaller?: (%, %) -> Boolean if UnivariatePuiseuxSeries(FE, var, cen) has Comparable

from Comparable

subtractIfCan: (%, %) -> Union(%, failed)
support: % -> List ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)

from FreeModuleCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

AbelianProductCategory UnivariatePuiseuxSeries(FE, var, cen)

AbelianSemiGroup

Algebra Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer

Algebra UnivariatePuiseuxSeries(FE, var, cen) if UnivariatePuiseuxSeries(FE, var, cen) has CommutativeRing and % has VariablesCommuteWithCoefficients

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer) if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer

BiModule(UnivariatePuiseuxSeries(FE, var, cen), UnivariatePuiseuxSeries(FE, var, cen))

CancellationAbelianMonoid

CharacteristicNonZero if UnivariatePuiseuxSeries(FE, var, cen) has CharacteristicNonZero

CharacteristicZero if UnivariatePuiseuxSeries(FE, var, cen) has CharacteristicZero

CoercibleFrom Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer

CoercibleFrom Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Integer

CoercibleFrom UnivariatePuiseuxSeries(FE, var, cen)

CommutativeRing

CommutativeStar

Comparable if UnivariatePuiseuxSeries(FE, var, cen) has Comparable

EntireRing

FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

FreeModuleCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

FullyRetractableTo UnivariatePuiseuxSeries(FE, var, cen)

IndexedDirectProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

IndexedProductCategory(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))

IntegralDomain

LeftModule Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer

LeftModule UnivariatePuiseuxSeries(FE, var, cen)

Magma

MagmaWithUnit

Module Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer

Module UnivariatePuiseuxSeries(FE, var, cen) if UnivariatePuiseuxSeries(FE, var, cen) has CommutativeRing

Monoid

NonAssociativeAlgebra UnivariatePuiseuxSeries(FE, var, cen) if UnivariatePuiseuxSeries(FE, var, cen) has CommutativeRing and % has VariablesCommuteWithCoefficients

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

RetractableTo Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer

RetractableTo Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Integer

RetractableTo UnivariatePuiseuxSeries(FE, var, cen)

RightModule Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer

RightModule UnivariatePuiseuxSeries(FE, var, cen)

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

unitsKnown