UnivariatePuiseuxSeriesWithExponentialSingularity(R, FE, var, cen)ΒΆ

expexpan.spad line 69

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 Magma
*: (%, Fraction Integer) -> % if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
from RightModule Fraction Integer
*: (%, UnivariatePuiseuxSeries(FE, var, cen)) -> %
from RightModule UnivariatePuiseuxSeries(FE, var, cen)
*: (Fraction Integer, %) -> % if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
from LeftModule 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: (%, %) -> %
from NonAssociativeSemiRng
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
binomThmExpt: (%, %, NonNegativeInteger) -> %
from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if UnivariatePuiseuxSeries(FE, var, cen) has CharacteristicNonZero
from CharacteristicNonZero
coefficient: (%, ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)) -> UnivariatePuiseuxSeries(FE, var, cen)
from AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
coefficients: % -> List UnivariatePuiseuxSeries(FE, var, cen)
from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer or UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
coerce: UnivariatePuiseuxSeries(FE, var, cen) -> %
from Algebra UnivariatePuiseuxSeries(FE, var, cen)
commutator: (%, %) -> %
from NonAssociativeRng
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: % -> UnivariatePuiseuxSeries(FE, var, cen)
from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
ground?: % -> Boolean
from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> UnivariatePuiseuxSeries(FE, var, cen)
from AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
leadingMonomial: % -> %
from AbelianMonoidRing(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)).
map: (UnivariatePuiseuxSeries(FE, var, cen) -> UnivariatePuiseuxSeries(FE, var, cen), %) -> %
from AbelianMonoidRing(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: (UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)) -> %
from AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
monomial?: % -> Boolean
from AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
numberOfMonomials: % -> NonNegativeInteger
from FiniteAbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
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 AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
retract: % -> Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Integer
from 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
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Integer
from 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
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
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))

AbelianSemiGroup

Algebra %

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

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

EntireRing

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

FullyRetractableTo UnivariatePuiseuxSeries(FE, var, cen)

IntegralDomain

LeftModule %

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

LeftModule UnivariatePuiseuxSeries(FE, var, cen)

Magma

MagmaWithUnit

Module %

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

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

Monoid

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 %

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

RightModule UnivariatePuiseuxSeries(FE, var, cen)

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown