UnivariatePuiseuxSeriesWithExponentialSingularity(R, FE, var, cen)¶

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))

=: (%, %) -> Boolean: from BasicType

^: (%, NonNegativeInteger) -> %: from MagmaWithUnit
^: (%, PositiveInteger) -> %: from Magma

~=: (%, %) -> Boolean: from BasicType

annihilate?: (%, %) -> Boolean: from Rng

antiCommutator: (%, %) -> %: from NonAssociativeSemiRng

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: from 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 of f(var) as var -> 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.