UnivariateTaylorSeriesCategory(Coef)

pscat.spad line 161 [edit on github]

UnivariateTaylorSeriesCategory is the category of Taylor series in one variable.

* : (%, %) -> %
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

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
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 : 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 : % -> 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)
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)
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), %) -> %

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

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
~= : (%, %) -> 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)

NonAssociativeRng

CharacteristicNonZero

MagmaWithUnit

AbelianProductCategory(Coef)

Magma

SemiGroup

IntegralDomain

LeftModule(%)

NonAssociativeRing

ArcHyperbolicFunctionCategory

PartialDifferentialRing(Symbol)

CharacteristicZero

Algebra(%)

PowerSeriesCategory(Coef, NonNegativeInteger, SingletonAsOrderedSet)

CommutativeRing

DifferentialRing

RightModule(Fraction(Integer))

Eltable(%, %)

NonAssociativeSemiRng

CancellationAbelianMonoid

IndexedProductCategory(Coef, NonNegativeInteger)

AbelianMonoid

VariablesCommuteWithCoefficients

CommutativeStar

UnivariatePowerSeriesCategory(Coef, NonNegativeInteger)

RightModule(%)

BiModule(Coef, Coef)

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

Monoid

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

UnivariateSeriesWithRationalExponents(Coef, NonNegativeInteger)

BasicType

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

SetCategory

noZeroDivisors

TrigonometricFunctionCategory

LeftModule(Coef)

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

HyperbolicFunctionCategory

AbelianGroup

ElementaryFunctionCategory