UnivariatePuiseuxSeriesWithExponentialSingularity(R, FE, var, cen)ΒΆ
expexpan.spad line 69 [edit on github]
R: Join(Comparable, RetractableTo Integer, LinearlyExplicitOver Integer, GcdDomain)
FE: Join(AlgebraicallyClosedField, TranscendentalFunctionCategory, FunctionSpace R)
var: Symbol
cen: FE
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))
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- 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
- 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
from CoercibleTo OutputForm
- coerce: Fraction Integer -> % if UnivariatePuiseuxSeries(FE, var, cen) has RetractableTo Fraction Integer or UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
from CoercibleFrom Fraction Integer
- coerce: Integer -> %
from NonAssociativeRing
- coerce: UnivariatePuiseuxSeries(FE, var, cen) -> %
from CoercibleFrom UnivariatePuiseuxSeries(FE, var, cen)
- commutator: (%, %) -> %
from NonAssociativeRng
- 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 off(var)
asvar -> 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))
returnslimit(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) -> %
from NonAssociativeAlgebra %
- 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
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
- 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
AbelianMonoidRing(UnivariatePuiseuxSeries(FE, var, cen), ExponentialOfUnivariatePuiseuxSeries(FE, var, cen))
AbelianProductCategory UnivariatePuiseuxSeries(FE, var, cen)
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
BiModule(%, %)
BiModule(Fraction Integer, Fraction Integer) if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
BiModule(UnivariatePuiseuxSeries(FE, var, cen), UnivariatePuiseuxSeries(FE, var, cen))
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)
Comparable if UnivariatePuiseuxSeries(FE, var, cen) has Comparable
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))
LeftModule Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
LeftModule UnivariatePuiseuxSeries(FE, var, cen)
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
NonAssociativeAlgebra Fraction Integer if UnivariatePuiseuxSeries(FE, var, cen) has Algebra Fraction Integer
NonAssociativeAlgebra UnivariatePuiseuxSeries(FE, var, cen) if UnivariatePuiseuxSeries(FE, var, cen) has CommutativeRing and % has VariablesCommuteWithCoefficients
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)