# SparseUnivariateTaylorSeries(Coef, var, cen)ΒΆ

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 LeftModule %

*: (%, Coef) -> %

from RightModule Coef

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

from LeftModule Coef

*: (Fraction Integer, %) -> % if Coef has Algebra 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
^: (%, Coef) -> % if Coef has Field

from UnivariateTaylorSeriesCategory Coef

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

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

acos: % -> % if Coef has Algebra Fraction Integer
acosh: % -> % if Coef has Algebra Fraction Integer
acot: % -> % if Coef has Algebra Fraction Integer
acoth: % -> % if Coef has Algebra Fraction Integer
acsc: % -> % if Coef has Algebra Fraction Integer
acsch: % -> % if Coef has Algebra Fraction Integer
annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
approximate: (%, NonNegativeInteger) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, NonNegativeInteger) -> Coef
asec: % -> % if Coef has Algebra Fraction Integer
asech: % -> % if Coef has Algebra Fraction Integer
asin: % -> % if Coef has Algebra Fraction Integer
asinh: % -> % if Coef has Algebra Fraction Integer
associates?: (%, %) -> Boolean if Coef has IntegralDomain

from EntireRing

associator: (%, %, %) -> %
atan: % -> % if Coef has Algebra Fraction Integer
atanh: % -> % if Coef has Algebra Fraction Integer
center: % -> Coef
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
coefficient: (%, NonNegativeInteger) -> Coef

from AbelianMonoidRing(Coef, NonNegativeInteger)

coefficients: % -> Stream Coef

from UnivariateTaylorSeriesCategory Coef

coerce: % -> % if Coef has CommutativeRing

from Algebra %

coerce: % -> OutputForm
coerce: Coef -> % if Coef has CommutativeRing

from Algebra Coef

coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
coerce: Integer -> %
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: (%, %) -> %
complete: % -> %
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
cosh: % -> % if Coef has Algebra Fraction Integer
cot: % -> % if Coef has Algebra Fraction Integer
coth: % -> % if Coef has Algebra Fraction Integer
csc: % -> % if Coef has Algebra Fraction Integer
csch: % -> % if Coef has Algebra Fraction Integer
D: % -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef

from DifferentialRing

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

from DifferentialRing

differentiate: (%, List Symbol) -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef and Coef has 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
differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (NonNegativeInteger, Coef) -> Coef and Coef has 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
eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, NonNegativeInteger) -> Coef
exp: % -> % if Coef has Algebra Fraction Integer
exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain

from EntireRing

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

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

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> NonNegativeInteger
order: (%, NonNegativeInteger) -> NonNegativeInteger
pi: () -> % if Coef has Algebra Fraction Integer
plenaryPower: (%, PositiveInteger) -> % if Coef has CommutativeRing or Coef has Algebra Fraction Integer
pole?: % -> Boolean
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)

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

sec: % -> % if Coef has Algebra Fraction Integer
sech: % -> % if Coef has Algebra Fraction Integer
series: Stream Coef -> %

from UnivariateTaylorSeriesCategory Coef

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

from UnivariateTaylorSeriesCategory Coef

sin: % -> % if Coef has Algebra Fraction Integer
sinh: % -> % if Coef has Algebra Fraction Integer
sqrt: % -> % if Coef has Algebra Fraction Integer

subtractIfCan: (%, %) -> Union(%, failed)
tan: % -> % if Coef has Algebra Fraction Integer
tanh: % -> % if Coef has Algebra Fraction Integer
terms: % -> Stream Record(k: NonNegativeInteger, c: Coef)
truncate: (%, NonNegativeInteger) -> %
truncate: (%, NonNegativeInteger, 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
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

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

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

Eltable(%, %)

EntireRing if Coef has IntegralDomain

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

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

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

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

RadicalCategory if Coef has Algebra Fraction Integer

RightModule Coef

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

unitsKnown

VariablesCommuteWithCoefficients