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 LeftModule %

*: (%, 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)

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

from IndexedProductCategory(FE, Fraction Integer)

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

from IndexedProductCategory(FE, Fraction Integer)

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

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 variables: FE -> List Symbol and FE has integrate: (FE, Symbol) -> FE 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)

leadingSupport: % -> Fraction Integer

from IndexedProductCategory(FE, Fraction Integer)

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

from IndexedProductCategory(FE, Fraction Integer)

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

log: % -> %

from ElementaryFunctionCategory

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

from IndexedProductCategory(FE, Fraction Integer)

max: (%, %) -> %

from OrderedSet

min: (%, %) -> %

from OrderedSet

monomial?: % -> Boolean

from IndexedProductCategory(FE, Fraction Integer)

monomial: (FE, Fraction Integer) -> %

from IndexedProductCategory(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 IndexedProductCategory(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)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(FE, Fraction Integer)

AbelianProductCategory FE

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

IndexedProductCategory(FE, Fraction Integer)

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

PartialOrder

PowerSeriesCategory(FE, Fraction Integer, SingletonAsOrderedSet)

PrincipalIdealDomain

RadicalCategory

RightModule %

RightModule FE

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TranscendentalFunctionCategory

TrigonometricFunctionCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown

UnivariatePowerSeriesCategory(FE, Fraction Integer)

UnivariatePuiseuxSeriesCategory FE

VariablesCommuteWithCoefficients