UnivariatePowerSeriesCategory(Coef, Expon)ΒΆ

pscat.spad line 59

UnivariatePowerSeriesCategory is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Coef) -> %
from RightModule Coef
*: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer
from RightModule Fraction Integer
*: (Coef, %) -> %
from LeftModule Coef
*: (Fraction Integer, %) -> % if Coef has Algebra Fraction Integer
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, Coef) -> % if Coef has Field
from AbelianMonoidRing(Coef, Expon)
=: (%, %) -> Boolean
from BasicType
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
approximate: (%, Expon) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, Expon) -> Coef
approximate(f) returns a truncated power series with the series variable viewed as an element of the coefficient domain.
associates?: (%, %) -> Boolean if Coef has IntegralDomain
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
center: % -> Coef
center(f) returns the point about which the series f is expanded.
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
from CharacteristicNonZero
coefficient: (%, Expon) -> Coef
from AbelianMonoidRing(Coef, Expon)
coerce: % -> % if Coef has CommutativeRing
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Coef -> % if Coef has CommutativeRing
from Algebra Coef
coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
from Algebra Fraction Integer
coerce: Integer -> %
from NonAssociativeRing
commutator: (%, %) -> %
from NonAssociativeRng
complete: % -> %
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
D: % -> % if Coef has *: (Expon, Coef) -> Coef
from DifferentialRing
D: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef
from DifferentialRing
D: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
degree: % -> Expon
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
differentiate: % -> % if Coef has *: (Expon, Coef) -> Coef
from DifferentialRing
differentiate: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef
from DifferentialRing
differentiate: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef
from PartialDifferentialRing Symbol
elt: (%, %) -> % if Expon has SemiGroup
from Eltable(%, %)
elt: (%, Expon) -> Coef
elt(f(x), r) returns the coefficient of the term of degree r in f(x). This is the same as the function coefficient.
eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Expon) -> Coef
eval(f, a) evaluates a power series at a value in the ground ring by returning a stream of partial sums.
exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain
from EntireRing
extend: (%, Expon) -> %
extend(f, n) causes all terms of f of degree <= n to be computed.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> Coef
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
leadingMonomial: % -> %
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
map: (Coef -> Coef, %) -> %
from AbelianMonoidRing(Coef, Expon)
monomial: (%, List SingletonAsOrderedSet, List Expon) -> %
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
monomial: (%, SingletonAsOrderedSet, Expon) -> %
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
monomial: (Coef, Expon) -> %
from AbelianMonoidRing(Coef, Expon)
monomial?: % -> Boolean
from AbelianMonoidRing(Coef, Expon)
multiplyExponents: (%, PositiveInteger) -> %
multiplyExponents(f, n) multiplies all exponents of the power series f by the positive integer n.
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
order: % -> Expon
order(f) is the degree of the lowest order non-zero term in f. This will result in an infinite loop if f has no non-zero terms.
order: (%, Expon) -> Expon
order(f, n) = min(m, n), where m is the degree of the lowest order non-zero term in f.
pole?: % -> Boolean
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
recip: % -> Union(%, failed)
from MagmaWithUnit
reductum: % -> %
from AbelianMonoidRing(Coef, Expon)
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
terms: % -> Stream Record(k: Expon, c: Coef)
terms(f(x)) returns a stream of non-zero terms, where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents.
truncate: (%, Expon) -> %
truncate(f, k) returns a (finite) power series consisting of the sum of all terms of f of degree <= k.
truncate: (%, Expon, Expon) -> %
truncate(f, k1, k2) returns a (finite) power series consisting of the sum of all terms of f of degree d with k1 <= d <= k2.
unit?: % -> Boolean if Coef has IntegralDomain
from EntireRing
unitCanonical: % -> % if Coef has IntegralDomain
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %) if Coef has IntegralDomain
from EntireRing
variable: % -> Symbol
variable(f) returns the (unique) power series variable of the power series f.
variables: % -> List SingletonAsOrderedSet
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Expon)

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(Coef, Coef)

BiModule(Fraction Integer, Fraction Integer) if Coef has Algebra Fraction Integer

CancellationAbelianMonoid

CharacteristicNonZero if Coef has CharacteristicNonZero

CharacteristicZero if Coef has CharacteristicZero

CoercibleTo OutputForm

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

DifferentialRing if Coef has *: (Expon, Coef) -> Coef

Eltable(%, %) if Expon has SemiGroup

EntireRing if Coef has IntegralDomain

IntegralDomain if Coef has IntegralDomain

LeftModule %

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

Magma

MagmaWithUnit

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

PartialDifferentialRing Symbol if Coef has PartialDifferentialRing Symbol and Coef has *: (Expon, Coef) -> Coef

PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

RightModule %

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown

VariablesCommuteWithCoefficients