ExponentialOfUnivariatePuiseuxSeries(FE, var, cen)¶

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

from LeftModule FE

*: (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

^: (%, %) -> %
^: (%, Fraction Integer) -> %

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

acos: % -> %
acosh: % -> %
acot: % -> %
acoth: % -> %
acsc: % -> %
acsch: % -> %
annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
approximate: (%, Fraction Integer) -> FE if FE has ^: (FE, Fraction Integer) -> FE and FE has coerce: Symbol -> FE
asec: % -> %
asech: % -> %
asin: % -> %
asinh: % -> %
associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %
atan: % -> %
atanh: % -> %
center: % -> FE
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if FE has CharacteristicNonZero
coefficient: (%, Fraction Integer) -> FE

from AbelianMonoidRing(FE, Fraction Integer)

coerce: % -> %

from Algebra %

coerce: % -> OutputForm
coerce: FE -> %

from Algebra FE

coerce: Fraction Integer -> %
coerce: Integer -> %
commutator: (%, %) -> %
complete: % -> %
construct: List Record(k: Fraction Integer, c: FE) -> %
constructOrdered: List Record(k: Fraction Integer, c: FE) -> %
cos: % -> %
cosh: % -> %
cot: % -> %
coth: % -> %
csc: % -> %
csch: % -> %
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
D: (%, List Symbol, List NonNegativeInteger) -> % if FE has *: (Fraction Integer, FE) -> FE and FE has 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
D: (%, Symbol, NonNegativeInteger) -> % if FE has *: (Fraction Integer, FE) -> FE and FE has PartialDifferentialRing Symbol
degree: % -> Fraction Integer
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
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if FE has *: (Fraction Integer, FE) -> FE and FE has 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
differentiate: (%, Symbol, NonNegativeInteger) -> % if FE has *: (Fraction Integer, FE) -> FE and FE has PartialDifferentialRing Symbol
divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, Fraction Integer) -> FE
euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

eval: (%, FE) -> Stream FE if FE has ^: (FE, Fraction Integer) -> FE
exp: % -> %
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)
exquo: (%, %) -> Union(%, failed)

from EntireRing

extend: (%, Fraction Integer) -> %
extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)

from EuclideanDomain

factor: % -> Factored %
gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from GcdDomain

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

integrate: % -> %
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
inv: % -> %

from DivisionRing

latex: % -> String

from SetCategory

lcm: (%, %) -> %

from GcdDomain

lcm: List % -> %

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)

from LeftOreRing

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

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

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

from OrderedSet

min: (%, %) -> %

from OrderedSet

monomial?: % -> Boolean
monomial: (FE, Fraction Integer) -> %
multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

multiplyExponents: (%, Fraction Integer) -> %
multiplyExponents: (%, PositiveInteger) -> %
nthRoot: (%, Integer) -> %

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> Fraction Integer
order: (%, Fraction Integer) -> Fraction Integer
pi: () -> %
pole?: % -> Boolean
prime?: % -> Boolean
principalIdeal: List % -> Record(coef: List %, generator: %)
quo: (%, %) -> %

from EuclideanDomain

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %
rem: (%, %) -> %

from EuclideanDomain

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sec: % -> %
sech: % -> %
series: (NonNegativeInteger, Stream Record(k: Fraction Integer, c: FE)) -> %
sin: % -> %
sinh: % -> %
sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

sqrt: % -> %

squareFree: % -> Factored %
squareFreePart: % -> %
subtractIfCan: (%, %) -> Union(%, failed)
tan: % -> %
tanh: % -> %
terms: % -> Stream Record(k: Fraction Integer, c: FE)
truncate: (%, Fraction Integer) -> %
truncate: (%, Fraction Integer, Fraction Integer) -> %
unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

variable: % -> Symbol
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra FE

ArcHyperbolicFunctionCategory

ArcTrigonometricFunctionCategory

BasicType

BiModule(%, %)

BiModule(FE, FE)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero if FE has CharacteristicZero

CommutativeRing

CommutativeStar

Comparable

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

DivisionRing

ElementaryFunctionCategory

Eltable(%, %)

EntireRing

EuclideanDomain

Field

GcdDomain

HyperbolicFunctionCategory

IntegralDomain

LeftOreRing

Magma

MagmaWithUnit

Module FE

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedSet

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

PartialOrder

PrincipalIdealDomain

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TranscendentalFunctionCategory

TrigonometricFunctionCategory

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown

VariablesCommuteWithCoefficients