ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)

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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.

* : (%, %) -> %
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))
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
< : (%, %) -> Boolean
from PartialOrder
<= : (%, %) -> Boolean
from PartialOrder
= : (%, %) -> Boolean
from BasicType
> : (%, %) -> Boolean
from PartialOrder
>= : (%, %) -> Boolean
from PartialOrder
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)
^ : (%, %) -> %
from ElementaryFunctionCategory
^ : (%, Fraction(Integer)) -> %
from RadicalCategory
^ : (%, Integer) -> %
from DivisionRing
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
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 coerce : Symbol -> FE and FE has ^ : (FE, Fraction(Integer)) -> 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 : FE -> %
from Algebra(FE)
coerce : Fraction(Integer) -> %
from Algebra(Fraction(Integer))
coerce : Integer -> %
from NonAssociativeRing
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
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
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
integrate : % -> %
from UnivariateSeriesWithRationalExponents(FE, Fraction(Integer))
integrate : (%, Symbol) -> % if FE has integrate : (FE, Symbol) -> FE and FE has variables : FE -> List(Symbol)
from UnivariateSeriesWithRationalExponents(FE, Fraction(Integer))
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 : (Mapping(FE, FE), %) -> %
from IndexedProductCategory(FE, Fraction(Integer))
max : (%, %) -> %
from OrderedSet
min : (%, %) -> %
from OrderedSet
monomial : (FE, Fraction(Integer)) -> %
from IndexedProductCategory(FE, Fraction(Integer))
monomial? : % -> Boolean
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
plenaryPower : (%, PositiveInteger) -> %
from NonAssociativeAlgebra(%)
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
~= : (%, %) -> Boolean
from BasicType

Algebra(Fraction(Integer))

Module(Fraction(Integer))

PrincipalIdealDomain

PartialOrder

OrderedAbelianSemiGroup

NonAssociativeSemiRing

BiModule(%, %)

Field

canonicalUnitNormal

Rng

NonAssociativeAlgebra(FE)

ArcTrigonometricFunctionCategory

UnivariateSeriesWithRationalExponents(FE, Fraction(Integer))

TwoSidedRecip

TranscendentalFunctionCategory

SemiRing

EntireRing

AbelianMonoidRing(FE, Fraction(Integer))

NonAssociativeAlgebra(Fraction(Integer))

CharacteristicNonZero

unitsKnown

RadicalCategory

Algebra(FE)

noZeroDivisors

Magma

SemiGroup

UnivariatePuiseuxSeriesCategory(FE)

GcdDomain

IntegralDomain

LeftModule(%)

NonAssociativeRing

Module(FE)

ArcHyperbolicFunctionCategory

NonAssociativeAlgebra(%)

PartialDifferentialRing(Symbol)

CharacteristicZero

BasicType

UniqueFactorizationDomain

Algebra(%)

CommutativeRing

DifferentialRing

OrderedAbelianMonoid

Eltable(%, %)

RightModule(FE)

NonAssociativeSemiRng

CancellationAbelianMonoid

EuclideanDomain

canonicalsClosed

Comparable

VariablesCommuteWithCoefficients

SetCategory

OrderedSet

CommutativeStar

AbelianMonoid

ElementaryFunctionCategory

RightModule(%)

IndexedProductCategory(FE, Fraction(Integer))

LeftModule(FE)

BiModule(FE, FE)

Module(%)

CoercibleTo(OutputForm)

RightModule(Fraction(Integer))

PowerSeriesCategory(FE, Fraction(Integer), SingletonAsOrderedSet)

UnivariatePowerSeriesCategory(FE, Fraction(Integer))

SemiRng

Monoid

LeftOreRing

AbelianProductCategory(FE)

DivisionRing

MagmaWithUnit

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

TrigonometricFunctionCategory

NonAssociativeRng

BiModule(Fraction(Integer), Fraction(Integer))

HyperbolicFunctionCategory

AbelianGroup