UnivariatePowerSeriesCategory(Coef, Expon)

pscat.spad line 51 [edit on github]

• Coef : Ring
• Expon : OrderedAbelianMonoid

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.

* : (%, %) -> %

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)

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

D : % -> % if Coef has * : (Expon, Coef) -> Coef

from DifferentialRing

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

from PartialDifferentialRing(Symbol)

D : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)

from PartialDifferentialRing(Symbol)

D : (%, NonNegativeInteger) -> % if Coef has * : (Expon, Coef) -> Coef

from DifferentialRing

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

from PartialDifferentialRing(Symbol)

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

from PartialDifferentialRing(Symbol)

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

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 : 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

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

complete : % -> %

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

construct : List(Record(k : Expon, c : Coef)) -> %

from IndexedProductCategory(Coef, Expon)

constructOrdered : List(Record(k : Expon, c : Coef)) -> %

from IndexedProductCategory(Coef, Expon)

degree : % -> Expon

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

differentiate : % -> % if Coef has * : (Expon, Coef) -> Coef

from DifferentialRing

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

from PartialDifferentialRing(Symbol)

differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)

from PartialDifferentialRing(Symbol)

differentiate : (%, NonNegativeInteger) -> % if Coef has * : (Expon, Coef) -> Coef

from DifferentialRing

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

from PartialDifferentialRing(Symbol)

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

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.

latex : % -> String

from SetCategory

leadingCoefficient : % -> Coef

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

leadingMonomial : % -> %

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

leadingSupport : % -> Expon

from IndexedProductCategory(Coef, Expon)

leadingTerm : % -> Record(k : Expon, c : Coef)

from IndexedProductCategory(Coef, Expon)

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

map : (Mapping(Coef, Coef), %) -> %

from IndexedProductCategory(Coef, Expon)

monomial : (Coef, Expon) -> %

from IndexedProductCategory(Coef, Expon)

monomial? : % -> Boolean

from IndexedProductCategory(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.

plenaryPower : (%, PositiveInteger) -> % if Coef has Algebra(Fraction(Integer)) or Coef has CommutativeRing

from NonAssociativeAlgebra(Coef)

pole? : % -> Boolean

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

recip : % -> Union(%, "failed")

from MagmaWithUnit

reductum : % -> %

from IndexedProductCategory(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. Warning: If the series f has only finitely many non-zero terms, then accessing the resulting stream might lead to an infinite search for the next non-zero coefficient.

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.

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

CharacteristicNonZero

Module(Fraction(Integer))

PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

LeftModule(Fraction(Integer))

SetCategory

noZeroDivisors

RightModule(%)

Monoid

Algebra(%)

AbelianMonoid

NonAssociativeAlgebra(Fraction(Integer))

Module(Coef)

BiModule(Coef, Coef)

CancellationAbelianMonoid

MagmaWithUnit

LeftModule(Coef)

NonAssociativeRing

AbelianGroup

RightModule(Fraction(Integer))

CommutativeStar

LeftModule(%)

IndexedProductCategory(Coef, Expon)

Eltable(%, %)

Module(%)

Algebra(Coef)

AbelianProductCategory(Coef)

BiModule(%, %)

Algebra(Fraction(Integer))

Rng

CommutativeRing

IntegralDomain

TwoSidedRecip

Magma

SemiGroup

NonAssociativeRng

RightModule(Coef)

PartialDifferentialRing(Symbol)

unitsKnown

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

VariablesCommuteWithCoefficients

AbelianMonoidRing(Coef, Expon)

NonAssociativeAlgebra(%)

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

DifferentialRing

Ring

SemiRng

EntireRing

NonAssociativeSemiRng

CharacteristicZero

BasicType

NonAssociativeAlgebra(Coef)

SemiRing