ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)ΒΆ

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ExponentialOfUnivariatePuiseuxSeries is a domain used to represent essential singularities of functions. An object in this domain is a function of the form exp(f(x)), where f(x) is a Puiseux series with no terms of non-negative degree. Objects are ordered according to order of singularity, with functions which tend more rapidly to zero or infinity considered to be larger. Thus, if order(f(x)) < order(g(x)), i.e. the first non-zero term of f(x) has lower degree than the first non-zero term of g(x), then exp(f(x)) > exp(g(x)). If order(f(x)) = order(g(x)), then the ordering is essentially random. This domain is used in computing limits involving functions with essential singularities.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, FE) -> %
from RightModule FE
*: (%, Fraction Integer) -> %
from RightModule Fraction Integer
*: (FE, %) -> %
from LeftModule FE
*: (Fraction Integer, %) -> %
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> %
from Field
/: (%, FE) -> %
from AbelianMonoidRing(FE, Fraction Integer)
<: (%, %) -> Boolean
from PartialOrder
<=: (%, %) -> Boolean
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean
from PartialOrder
>=: (%, %) -> Boolean
from PartialOrder
^: (%, %) -> %
from ElementaryFunctionCategory
^: (%, Fraction Integer) -> %
from RadicalCategory
^: (%, Integer) -> %
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
acos: % -> %
from ArcTrigonometricFunctionCategory
acosh: % -> %
from ArcHyperbolicFunctionCategory
acot: % -> %
from ArcTrigonometricFunctionCategory
acoth: % -> %
from ArcHyperbolicFunctionCategory
acsc: % -> %
from ArcTrigonometricFunctionCategory
acsch: % -> %
from ArcHyperbolicFunctionCategory
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
approximate: (%, Fraction Integer) -> FE if FE has ^: (FE, Fraction Integer) -> FE and FE has coerce: Symbol -> FE
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
asec: % -> %
from ArcTrigonometricFunctionCategory
asech: % -> %
from ArcHyperbolicFunctionCategory
asin: % -> %
from ArcTrigonometricFunctionCategory
asinh: % -> %
from ArcHyperbolicFunctionCategory
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
atan: % -> %
from ArcTrigonometricFunctionCategory
atanh: % -> %
from ArcHyperbolicFunctionCategory
center: % -> FE
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if FE has CharacteristicNonZero
from CharacteristicNonZero
coefficient: (%, Fraction Integer) -> FE
from AbelianMonoidRing(FE, Fraction Integer)
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: FE -> %
from Algebra FE
coerce: Fraction Integer -> %
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
complete: % -> %
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
cos: % -> %
from TrigonometricFunctionCategory
cosh: % -> %
from HyperbolicFunctionCategory
cot: % -> %
from TrigonometricFunctionCategory
coth: % -> %
from HyperbolicFunctionCategory
csc: % -> %
from TrigonometricFunctionCategory
csch: % -> %
from HyperbolicFunctionCategory
D: % -> % if FE has *: (Fraction Integer, FE) -> FE
from DifferentialRing
D: (%, List Symbol) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if FE has *: (Fraction Integer, FE) -> FE
from DifferentialRing
D: (%, Symbol) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
degree: % -> Fraction Integer
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
differentiate: % -> % if FE has *: (Fraction Integer, FE) -> FE
from DifferentialRing
differentiate: (%, List Symbol) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if FE has *: (Fraction Integer, FE) -> FE
from DifferentialRing
differentiate: (%, Symbol) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if FE has PartialDifferentialRing Symbol and FE has *: (Fraction Integer, FE) -> FE
from PartialDifferentialRing Symbol
divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
elt: (%, %) -> %
from Eltable(%, %)
elt: (%, Fraction Integer) -> FE
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
eval: (%, FE) -> Stream FE if FE has ^: (FE, Fraction Integer) -> FE
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
exp: % -> %
from ElementaryFunctionCategory
exponent: % -> UnivariatePuiseuxSeries(FE, var, cen)
exponent(exp(f(x))) returns f(x)
exponential: UnivariatePuiseuxSeries(FE, var, cen) -> %
exponential(f(x)) returns exp(f(x)). Note: the function does NOT check that f(x) has no non-negative terms.
exponentialOrder: % -> Fraction Integer
exponentialOrder(exp(c * x ^(-n) + ...)) returns -n. exponentialOrder(0) returns 0.
expressIdealMember: (List %, %) -> Union(List %, failed)
from PrincipalIdealDomain
exquo: (%, %) -> Union(%, failed)
from EntireRing
extend: (%, Fraction Integer) -> %
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)
from EuclideanDomain
extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)
from EuclideanDomain
factor: % -> Factored %
from UniqueFactorizationDomain
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
integrate: % -> %
from UnivariatePuiseuxSeriesCategory FE
integrate: (%, Symbol) -> % if FE has integrate: (FE, Symbol) -> FE and FE has variables: FE -> List Symbol or FE has TranscendentalFunctionCategory and FE has PrimitiveFunctionCategory and FE has AlgebraicallyClosedFunctionSpace Integer
from UnivariatePuiseuxSeriesCategory FE
inv: % -> %
from DivisionRing
latex: % -> String
from SetCategory
lcm: (%, %) -> %
from GcdDomain
lcm: List % -> %
from GcdDomain
lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)
from LeftOreRing
leadingCoefficient: % -> FE
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
leadingMonomial: % -> %
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
log: % -> %
from ElementaryFunctionCategory
map: (FE -> FE, %) -> %
from AbelianMonoidRing(FE, Fraction Integer)
max: (%, %) -> %
from OrderedSet
min: (%, %) -> %
from OrderedSet
monomial: (%, List SingletonAsOrderedSet, List Fraction Integer) -> %
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
monomial: (%, SingletonAsOrderedSet, Fraction Integer) -> %
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
monomial: (FE, Fraction Integer) -> %
from AbelianMonoidRing(FE, Fraction Integer)
monomial?: % -> Boolean
from AbelianMonoidRing(FE, Fraction Integer)
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
multiplyExponents: (%, Fraction Integer) -> %
from UnivariatePuiseuxSeriesCategory FE
multiplyExponents: (%, PositiveInteger) -> %
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
nthRoot: (%, Integer) -> %
from RadicalCategory
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
order: % -> Fraction Integer
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
order: (%, Fraction Integer) -> Fraction Integer
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
pi: () -> %
from TranscendentalFunctionCategory
pole?: % -> Boolean
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
prime?: % -> Boolean
from UniqueFactorizationDomain
principalIdeal: List % -> Record(coef: List %, generator: %)
from PrincipalIdealDomain
quo: (%, %) -> %
from EuclideanDomain
recip: % -> Union(%, failed)
from MagmaWithUnit
reductum: % -> %
from AbelianMonoidRing(FE, Fraction Integer)
rem: (%, %) -> %
from EuclideanDomain
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
sec: % -> %
from TrigonometricFunctionCategory
sech: % -> %
from HyperbolicFunctionCategory
series: (NonNegativeInteger, Stream Record(k: Fraction Integer, c: FE)) -> %
from UnivariatePuiseuxSeriesCategory FE
sin: % -> %
from TrigonometricFunctionCategory
sinh: % -> %
from HyperbolicFunctionCategory
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
smaller?: (%, %) -> Boolean
from Comparable
sqrt: % -> %
from RadicalCategory
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
tan: % -> %
from TrigonometricFunctionCategory
tanh: % -> %
from HyperbolicFunctionCategory
terms: % -> Stream Record(k: Fraction Integer, c: FE)
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
truncate: (%, Fraction Integer) -> %
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
truncate: (%, Fraction Integer, Fraction Integer) -> %
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
variable: % -> Symbol
from UnivariatePowerSeriesCategory(FE, Fraction Integer)
variables: % -> List SingletonAsOrderedSet
from PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(FE, Fraction Integer)

AbelianSemiGroup

Algebra %

Algebra FE

Algebra Fraction Integer

ArcHyperbolicFunctionCategory

ArcTrigonometricFunctionCategory

BasicType

BiModule(%, %)

BiModule(FE, FE)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero if FE has CharacteristicNonZero

CharacteristicZero if FE has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

DifferentialRing if FE has *: (Fraction Integer, FE) -> FE

DivisionRing

ElementaryFunctionCategory

Eltable(%, %)

EntireRing

EuclideanDomain

Field

GcdDomain

HyperbolicFunctionCategory

IntegralDomain

LeftModule %

LeftModule FE

LeftModule Fraction Integer

LeftOreRing

Magma

MagmaWithUnit

Module %

Module FE

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedSet

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

PartialOrder

PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)

PrincipalIdealDomain

RadicalCategory

RightModule %

RightModule FE

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TranscendentalFunctionCategory

TrigonometricFunctionCategory

UniqueFactorizationDomain

unitsKnown

UnivariatePowerSeriesCategory(FE, Fraction Integer)

UnivariatePuiseuxSeriesCategory FE

VariablesCommuteWithCoefficients