# UnivariateTaylorSeriesCategory CoefΒΆ

- Coef: Ring

UnivariateTaylorSeriesCategory is the category of Taylor series in one variable.

- 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, NonNegativeInteger)
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, %) -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory

- ^: (%, Coef) -> % if Coef has Field
`f(x) ^ a`

computes a power of a power series. When the coefficient ring is a field, we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.- ^: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer
- from RadicalCategory
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- acos: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- acosh: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- acot: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- acoth: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- acsc: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- acsch: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng
- approximate: (%, NonNegativeInteger) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, NonNegativeInteger) -> Coef
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- asec: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- asech: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- asin: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- asinh: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- associates?: (%, %) -> Boolean if Coef has IntegralDomain
- from EntireRing
- associator: (%, %, %) -> %
- from NonAssociativeRng
- atan: % -> % if Coef has Algebra Fraction Integer
- from ArcTrigonometricFunctionCategory
- atanh: % -> % if Coef has Algebra Fraction Integer
- from ArcHyperbolicFunctionCategory
- center: % -> Coef
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
- from CharacteristicNonZero
- coefficient: (%, NonNegativeInteger) -> Coef
- from AbelianMonoidRing(Coef, NonNegativeInteger)

- coefficients: % -> Stream Coef
`coefficients(a0 + a1 x + a2 x^2 + ...)`

returns a stream of coefficients:`[a0, a1, a2, ...]`

. The entries of the stream may be zero.- 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, NonNegativeInteger, SingletonAsOrderedSet)
- cos: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- cosh: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- cot: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- coth: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- csc: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- csch: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- D: % -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef
- from DifferentialRing
- D: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef
- from DifferentialRing
- D: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- degree: % -> NonNegativeInteger
- from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)
- differentiate: % -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (NonNegativeInteger, Coef) -> Coef
- from PartialDifferentialRing Symbol
- elt: (%, %) -> %
- from Eltable(%, %)
- elt: (%, NonNegativeInteger) -> Coef
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, NonNegativeInteger) -> Coef
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- exp: % -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory
- exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain
- from EntireRing
- extend: (%, NonNegativeInteger) -> %
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory

- integrate: % -> % if Coef has Algebra Fraction Integer
`integrate(f(x))`

returns an anti-derivative of the power series`f(x)`

with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.

- integrate: (%, Symbol) -> % if Coef has AlgebraicallyClosedFunctionSpace Integer and Coef has Algebra Fraction Integer and Coef has TranscendentalFunctionCategory and Coef has PrimitiveFunctionCategory or Coef has Algebra Fraction Integer and Coef has integrate: (Coef, Symbol) -> Coef and Coef has variables: Coef -> List Symbol
`integrate(f(x), y)`

returns an anti-derivative of the power series`f(x)`

with respect to the variable`y`

.- latex: % -> String
- from SetCategory
- leadingCoefficient: % -> Coef
- from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)
- leadingMonomial: % -> %
- from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit
- log: % -> % if Coef has Algebra Fraction Integer
- from ElementaryFunctionCategory
- map: (Coef -> Coef, %) -> %
- from AbelianMonoidRing(Coef, NonNegativeInteger)
- monomial: (Coef, NonNegativeInteger) -> %
- from AbelianMonoidRing(Coef, NonNegativeInteger)
- monomial?: % -> Boolean
- from AbelianMonoidRing(Coef, NonNegativeInteger)

- multiplyCoefficients: (Integer -> Coef, %) -> %
`multiplyCoefficients(f, sum(n = 0..infinity, a[n] * x^n))`

returns`sum(n = 0..infinity, f(n) * a[n] * x^n)`

. This function is used when Laurent series are represented by a Taylor series and an order.- multiplyExponents: (%, PositiveInteger) -> %
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- nthRoot: (%, Integer) -> % if Coef has Algebra Fraction Integer
- from RadicalCategory
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- order: % -> NonNegativeInteger
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- order: (%, NonNegativeInteger) -> NonNegativeInteger
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- pi: () -> % if Coef has Algebra Fraction Integer
- from TranscendentalFunctionCategory
- pole?: % -> Boolean
- from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

- polynomial: (%, NonNegativeInteger) -> Polynomial Coef
`polynomial(f, k)`

returns a polynomial consisting of the sum of all terms of`f`

of degree`<= k`

.

- polynomial: (%, NonNegativeInteger, NonNegativeInteger) -> Polynomial Coef
`polynomial(f, k1, k2)`

returns a polynomial consisting of the sum of all terms of`f`

of degree`d`

with`k1 <= d <= k2`

.

- quoByVar: % -> %
`quoByVar(a0 + a1 x + a2 x^2 + ...)`

returns`a1 + a2 x + a3 x^2 + ...`

Thus, this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.- recip: % -> Union(%, failed)
- from MagmaWithUnit
- reductum: % -> %
- from AbelianMonoidRing(Coef, NonNegativeInteger)
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid
- sec: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- sech: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory

- series: Stream Coef -> %
`series([a0, a1, a2, ...])`

is the Taylor series`a0 + a1 x + a2 x^2 + ...`

.

- series: Stream Record(k: NonNegativeInteger, c: Coef) -> %
`series(st)`

creates a series from a stream of non-zero terms, where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.- sin: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- sinh: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- sqrt: % -> % if Coef has Algebra Fraction Integer
- from RadicalCategory
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid
- tan: % -> % if Coef has Algebra Fraction Integer
- from TrigonometricFunctionCategory
- tanh: % -> % if Coef has Algebra Fraction Integer
- from HyperbolicFunctionCategory
- terms: % -> Stream Record(k: NonNegativeInteger, c: Coef)
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- truncate: (%, NonNegativeInteger) -> %
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- truncate: (%, NonNegativeInteger, NonNegativeInteger) -> %
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- 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
- from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
- zero?: % -> Boolean
- from AbelianMonoid

AbelianMonoidRing(Coef, NonNegativeInteger)

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

ArcHyperbolicFunctionCategory if Coef has Algebra Fraction Integer

ArcTrigonometricFunctionCategory if Coef has Algebra Fraction Integer

BiModule(%, %)

BiModule(Coef, Coef)

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

CharacteristicNonZero if Coef has CharacteristicNonZero

CharacteristicZero if Coef has CharacteristicZero

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

ElementaryFunctionCategory if Coef has Algebra Fraction Integer

Eltable(%, %)

EntireRing if Coef has IntegralDomain

HyperbolicFunctionCategory if Coef has Algebra Fraction Integer

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

noZeroDivisors if Coef has IntegralDomain

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

PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

RadicalCategory if Coef has Algebra Fraction Integer

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

TranscendentalFunctionCategory if Coef has Algebra Fraction Integer

TrigonometricFunctionCategory if Coef has Algebra Fraction Integer

TwoSidedRecip if Coef has CommutativeRing