UnivariateTaylorSeries(Coef, var, cen)ΒΆ

taylor.spad line 134 [edit on github]

Dense Taylor series in one variable UnivariateTaylorSeries is a domain representing Taylor series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring, the power series variable, and the center of the power series expansion. For example, UnivariateTaylorSeries(Integer, x, 3) represents Taylor series in (x - 3) with Integer coefficients.

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

from UnivariateTaylorSeriesCategory Coef

^: (%, 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

from UnivariateTaylorSeriesCategory Coef

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

coerce: UnivariatePolynomial(var, Coef) -> %

coerce(p) converts a univariate polynomial p in the variable var to a univariate Taylor series in var.

coerce: Variable var -> %

coerce(var) converts the series variable var into a Taylor series.

commutator: (%, %) -> %

from NonAssociativeRng

complete: % -> %

from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

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

from IndexedProductCategory(Coef, NonNegativeInteger)

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

from IndexedProductCategory(Coef, NonNegativeInteger)

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

differentiate: (%, Variable var) -> %

differentiate(f(x), x) computes the derivative of f(x) with respect to x.

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, NonNegativeInteger) -> Coef

from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, NonNegativeInteger) -> Coef

from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

evenlambert: % -> %

evenlambert(f(x)) returns f(x^2) + f(x^4) + f(x^6) + .... f(x) should have a zero constant coefficient. This function is used for computing infinite products. If f(x) is a Taylor series with constant term 1, then product(n=1..infinity, f(x^(2*n))) = exp(evenlambert(log(f(x)))).

exp: % -> % if Coef has Algebra Fraction Integer

from ElementaryFunctionCategory

exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain

from EntireRing

extend: (%, NonNegativeInteger) -> %

from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

generalLambert: (%, Integer, Integer) -> %

generalLambert(f(x), a, d) returns f(x^a) + f(x^(a + d)) + f(x^(a + 2 d)) + ... . f(x) should have zero constant coefficient and a and d should be positive.

integrate: % -> % if Coef has Algebra Fraction Integer

from UnivariateSeriesWithRationalExponents(Coef, NonNegativeInteger)

integrate: (%, Symbol) -> % if Coef has Algebra Fraction Integer and Coef has integrate: (Coef, Symbol) -> Coef and Coef has variables: Coef -> List Symbol

from UnivariateSeriesWithRationalExponents(Coef, NonNegativeInteger)

integrate: (%, Variable var) -> % if Coef has Algebra Fraction Integer

integrate(f(x), 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.

invmultisect: (Integer, Integer, %) -> %

invmultisect(a, b, f(x)) substitutes x^((a+b)*n) for x^n and multiples by x^b.

lagrange: % -> %

lagrange(g(x)) produces the Taylor series for f(x) where f(x) is implicitly defined as f(x) = x*g(f(x)).

lambert: % -> %

lambert(f(x)) returns f(x) + f(x^2) + f(x^3) + .... f(x) should have zero constant coefficient. This function is used for computing infinite products. If f(x) is a Taylor series with constant term 1, then product(n = 1..infinity, f(x^n)) = exp(lambert(log(f(x)))).

latex: % -> String

from SetCategory

leadingCoefficient: % -> Coef

from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

leadingMonomial: % -> %

from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

leadingSupport: % -> NonNegativeInteger

from IndexedProductCategory(Coef, NonNegativeInteger)

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

from IndexedProductCategory(Coef, NonNegativeInteger)

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 IndexedProductCategory(Coef, NonNegativeInteger)

monomial?: % -> Boolean

from IndexedProductCategory(Coef, NonNegativeInteger)

monomial: (Coef, NonNegativeInteger) -> %

from IndexedProductCategory(Coef, NonNegativeInteger)

multiplyCoefficients: (Integer -> Coef, %) -> %

from UnivariateTaylorSeriesCategory Coef

multiplyExponents: (%, PositiveInteger) -> %

from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

multisect: (Integer, Integer, %) -> %

multisect(a, b, f(x)) selects the coefficients of x^((a+b)*n+a), and changes this monomial to x^n.

nthRoot: (%, Integer) -> % if Coef has Algebra Fraction Integer

from RadicalCategory

oddlambert: % -> %

oddlambert(f(x)) returns f(x) + f(x^3) + f(x^5) + .... f(x) should have a zero constant coefficient. This function is used for computing infinite products. If f(x) is a Taylor series with constant term 1, then product(n=1..infinity, f(x^(2*n-1)))=exp(oddlambert(log(f(x)))).

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

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

from NonAssociativeAlgebra Coef

pole?: % -> Boolean

from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

polynomial: (%, NonNegativeInteger) -> Polynomial Coef

from UnivariateTaylorSeriesCategory Coef

polynomial: (%, NonNegativeInteger, NonNegativeInteger) -> Polynomial Coef

from UnivariateTaylorSeriesCategory Coef

quoByVar: % -> %

from UnivariateTaylorSeriesCategory Coef

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(Coef, NonNegativeInteger)

revert: % -> %

revert(f(x)) returns a Taylor series g(x) such that f(g(x)) = g(f(x)) = x. Series f(x) should have constant coefficient 0 and invertible 1st order coefficient.

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 -> %

from UnivariateTaylorSeriesCategory Coef

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

from UnivariateTaylorSeriesCategory Coef

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

univariatePolynomial: (%, NonNegativeInteger) -> UnivariatePolynomial(var, Coef)

univariatePolynomial(f, k) returns a univariate polynomial consisting of the sum of all terms of f of degree <= k.

variable: % -> Symbol

from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, NonNegativeInteger)

AbelianProductCategory Coef

AbelianSemiGroup

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

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 *: (NonNegativeInteger, Coef) -> Coef

ElementaryFunctionCategory if Coef has Algebra Fraction Integer

Eltable(%, %)

EntireRing if Coef has IntegralDomain

HyperbolicFunctionCategory if Coef has Algebra Fraction Integer

IndexedProductCategory(Coef, NonNegativeInteger)

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

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeAlgebra Fraction Integer if Coef has Algebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

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

PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

RadicalCategory if Coef has Algebra Fraction Integer

RightModule %

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TranscendentalFunctionCategory if Coef has Algebra Fraction Integer

TrigonometricFunctionCategory if Coef has Algebra Fraction Integer

TwoSidedRecip if Coef has CommutativeRing

unitsKnown

UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

UnivariateSeriesWithRationalExponents(Coef, NonNegativeInteger)

UnivariateTaylorSeriesCategory Coef

VariablesCommuteWithCoefficients