SparseMultivariateTaylorSeries(Coef, Var, SMP)

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This domain provides multivariate Taylor series with variables from an arbitrary ordered set. A Taylor series is represented by a stream of polynomials from the polynomial domain SMP. The nth element of the stream is a form of degree n. SMTS is an internal domain.

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

*: (SMP, %) -> %

smp*ts multiplies a TaylorSeries ts by a monomial smp.

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, Coef) -> % if Coef has Field

from AbelianMonoidRing(Coef, IndexedExponents Var)

=: (%, %) -> Boolean

from BasicType

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

from ElementaryFunctionCategory

^: (%, 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

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

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero

from CharacteristicNonZero

coefficient: (%, IndexedExponents Var) -> Coef

from AbelianMonoidRing(Coef, IndexedExponents Var)

coefficient: (%, List Var, List NonNegativeInteger) -> %

from MultivariateTaylorSeriesCategory(Coef, Var)

coefficient: (%, NonNegativeInteger) -> SMP

coefficient(s, n) gives the terms of total degree n.

coefficient: (%, Var, NonNegativeInteger) -> %

from MultivariateTaylorSeriesCategory(Coef, Var)

coefficients: % -> Stream SMP

coefficients(s) gives stream of coefficients of s, i.e. [coefficient(s,0), coefficient(s,1), …]

coerce: % -> % if Coef has CommutativeRing

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: SMP -> %

coerce(poly) regroups the terms by total degree and forms a series.

coerce: Var -> %

coerce(var) converts a variable to a Taylor series

commutator: (%, %) -> %

from NonAssociativeRng

complete: % -> %

from PowerSeriesCategory(Coef, IndexedExponents Var, Var)

construct: List Record(k: IndexedExponents Var, c: Coef) -> %

from IndexedProductCategory(Coef, IndexedExponents Var)

constructOrdered: List Record(k: IndexedExponents Var, c: Coef) -> %

from IndexedProductCategory(Coef, IndexedExponents Var)

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

csubst: (List Var, List Stream SMP) -> SMP -> Stream SMP

csubst(a, b) is for internal use only

D: (%, List Var) -> %

from PartialDifferentialRing Var

D: (%, List Var, List NonNegativeInteger) -> %

from PartialDifferentialRing Var

D: (%, Var) -> %

from PartialDifferentialRing Var

D: (%, Var, NonNegativeInteger) -> %

from PartialDifferentialRing Var

degree: % -> IndexedExponents Var

from PowerSeriesCategory(Coef, IndexedExponents Var, Var)

differentiate: (%, List Var) -> %

from PartialDifferentialRing Var

differentiate: (%, List Var, List NonNegativeInteger) -> %

from PartialDifferentialRing Var

differentiate: (%, Var) -> %

from PartialDifferentialRing Var

differentiate: (%, Var, NonNegativeInteger) -> %

from PartialDifferentialRing Var

eval: (%, %, %) -> %

from InnerEvalable(%, %)

eval: (%, Equation %) -> %

from Evalable %

eval: (%, List %, List %) -> %

from InnerEvalable(%, %)

eval: (%, List Equation %) -> %

from Evalable %

eval: (%, List Var, List %) -> %

from InnerEvalable(Var, %)

eval: (%, Var, %) -> %

from InnerEvalable(Var, %)

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

from ElementaryFunctionCategory

exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain

from EntireRing

extend: (%, NonNegativeInteger) -> %

from MultivariateTaylorSeriesCategory(Coef, Var)

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

fintegrate(f, v, c) is the integral of f() with respect to v and having c as the constant of integration. The evaluation of f() is delayed.

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

from MultivariateTaylorSeriesCategory(Coef, Var)

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

integrate(s, v, c) is the integral of s with respect to v and having c as the constant of integration.

latex: % -> String

from SetCategory

leadingCoefficient: % -> Coef

from PowerSeriesCategory(Coef, IndexedExponents Var, Var)

leadingMonomial: % -> %

from PowerSeriesCategory(Coef, IndexedExponents Var, Var)

leadingSupport: % -> IndexedExponents Var

from IndexedProductCategory(Coef, IndexedExponents Var)

leadingTerm: % -> Record(k: IndexedExponents Var, c: Coef)

from IndexedProductCategory(Coef, IndexedExponents Var)

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 IndexedProductCategory(Coef, IndexedExponents Var)

monomial?: % -> Boolean

from IndexedProductCategory(Coef, IndexedExponents Var)

monomial: (%, List Var, List NonNegativeInteger) -> %

from MultivariateTaylorSeriesCategory(Coef, Var)

monomial: (%, Var, NonNegativeInteger) -> %

from MultivariateTaylorSeriesCategory(Coef, Var)

monomial: (Coef, IndexedExponents Var) -> %

from IndexedProductCategory(Coef, IndexedExponents Var)

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

from RadicalCategory

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: (%, Var) -> NonNegativeInteger

from MultivariateTaylorSeriesCategory(Coef, Var)

order: (%, Var, NonNegativeInteger) -> NonNegativeInteger

from MultivariateTaylorSeriesCategory(Coef, Var)

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, IndexedExponents Var, Var)

polynomial: (%, NonNegativeInteger) -> Polynomial Coef

from MultivariateTaylorSeriesCategory(Coef, Var)

polynomial: (%, NonNegativeInteger, NonNegativeInteger) -> Polynomial Coef

from MultivariateTaylorSeriesCategory(Coef, Var)

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(Coef, IndexedExponents Var)

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 SMP -> %

series(st) creates a series from a stream of coefficients.

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

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

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, IndexedExponents Var)

AbelianProductCategory Coef

AbelianSemiGroup

Algebra % if Coef has CommutativeRing

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

ElementaryFunctionCategory if Coef has Algebra Fraction Integer

EntireRing if Coef has IntegralDomain

Evalable %

HyperbolicFunctionCategory if Coef has Algebra Fraction Integer

IndexedProductCategory(Coef, IndexedExponents Var)

InnerEvalable(%, %)

InnerEvalable(Var, %)

IntegralDomain if Coef has IntegralDomain

LeftModule %

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

Magma

MagmaWithUnit

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

Monoid

MultivariateTaylorSeriesCategory(Coef, Var)

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeAlgebra Fraction Integer if Coef has Algebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

PartialDifferentialRing Var

PowerSeriesCategory(Coef, IndexedExponents Var, Var)

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

TwoSidedRecip if Coef has CommutativeRing

unitsKnown

VariablesCommuteWithCoefficients