SparseUnivariateTaylorSeries(Coef, var, cen)ΒΆ

sups.spad line 1078

Sparse Taylor series in one variable SparseUnivariateTaylorSeries 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, SparseUnivariateTaylorSeries(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 IntegralDomain
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)
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
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)
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
from UnivariateTaylorSeriesCategory Coef
integrate: (%, Symbol) -> % if Coef has Algebra Fraction Integer and Coef has integrate: (Coef, Symbol) -> Coef and Coef has variables: Coef -> List Symbol or Coef has AlgebraicallyClosedFunctionSpace Integer and Coef has Algebra Fraction Integer and Coef has TranscendentalFunctionCategory and Coef has PrimitiveFunctionCategory
from UnivariateTaylorSeriesCategory Coef
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.
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: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)
monomial: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)
monomial: (Coef, NonNegativeInteger) -> %
from AbelianMonoidRing(Coef, NonNegativeInteger)
monomial?: % -> Boolean
from AbelianMonoidRing(Coef, NonNegativeInteger)
multiplyCoefficients: (Integer -> Coef, %) -> %
from UnivariateTaylorSeriesCategory Coef
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
from UnivariateTaylorSeriesCategory Coef
polynomial: (%, NonNegativeInteger, NonNegativeInteger) -> Polynomial Coef
from UnivariateTaylorSeriesCategory Coef
quoByVar: % -> %
from UnivariateTaylorSeriesCategory Coef
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 -> %
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)
variables: % -> List SingletonAsOrderedSet
from PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, NonNegativeInteger)

AbelianSemiGroup

Algebra % if Coef has IntegralDomain

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

IntegralDomain if Coef has IntegralDomain

LeftModule %

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

Magma

MagmaWithUnit

Module % if Coef has IntegralDomain

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 *: (NonNegativeInteger, Coef) -> Coef

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

unitsKnown

UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

UnivariateTaylorSeriesCategory Coef

VariablesCommuteWithCoefficients