UnivariateTaylorSeriesCategory Coef¶

UnivariateTaylorSeriesCategory is the category of Taylor series in one variable.

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

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

^: (%, 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

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

from Algebra Coef

coerce: Fraction Integer -> % if Coef has Algebra Fraction Integer
coerce: Integer -> %
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
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) -> %
hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

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

integrate(f(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.

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

integrate(f(x), y) returns an anti-derivative of the power series f(x) with respect to the variable y.

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, %) -> %

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) -> %
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
pole?: % -> Boolean
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
sech: % -> % if Coef has Algebra Fraction Integer
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
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

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

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