SparseUnivariateTaylorSeries(Coef, var, cen)

sups.spad line 1072 [edit on github]

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.

* : (%, %) -> %
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)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
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)
^ : (%, %) -> % 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
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 : 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.

coerce : % -> OutputForm
from CoercibleTo(OutputForm)
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
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)
exp : % -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
exquo : (%, %) -> Union(%, "failed") if Coef has IntegralDomain
from EntireRing
extend : (%, NonNegativeInteger) -> %
from UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)
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.

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 : (Mapping(Coef, Coef), %) -> %
from IndexedProductCategory(Coef, NonNegativeInteger)
monomial : (Coef, NonNegativeInteger) -> %
from IndexedProductCategory(Coef, NonNegativeInteger)
monomial? : % -> Boolean
from IndexedProductCategory(Coef, NonNegativeInteger)
multiplyCoefficients : (Mapping(Coef, Integer), %) -> %
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
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)
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
~= : (%, %) -> Boolean
from BasicType

Module(Fraction(Integer))

NonAssociativeAlgebra(Coef)

Module(Coef)

NonAssociativeSemiRing

BiModule(%, %)

Rng

ArcTrigonometricFunctionCategory

TwoSidedRecip

TranscendentalFunctionCategory

SemiRing

EntireRing

RightModule(Coef)

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

RadicalCategory

AbelianMonoidRing(Coef, NonNegativeInteger)

CharacteristicNonZero

MagmaWithUnit

AbelianProductCategory(Coef)

Magma

SemiGroup

RightModule(Fraction(Integer))

IntegralDomain

LeftModule(%)

NonAssociativeRing

ArcHyperbolicFunctionCategory

PartialDifferentialRing(Symbol)

CharacteristicZero

UnivariateTaylorSeriesCategory(Coef)

Algebra(%)

PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

CommutativeRing

DifferentialRing

Eltable(%, %)

NonAssociativeSemiRng

CancellationAbelianMonoid

IndexedProductCategory(Coef, NonNegativeInteger)

AbelianMonoid

VariablesCommuteWithCoefficients

CommutativeStar

UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

RightModule(%)

BiModule(Coef, Coef)

LeftModule(Coef)

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

Monoid

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

UnivariateSeriesWithRationalExponents(Coef, NonNegativeInteger)

BasicType

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

noZeroDivisors

SetCategory

TrigonometricFunctionCategory

NonAssociativeRng

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

HyperbolicFunctionCategory

AbelianGroup

ElementaryFunctionCategory