TaylorSeries(Coef)

mts.spad line 310 [edit on github]

TaylorSeries is a general multivariate Taylor series domain over the ring Coef and with variables of type Symbol.

* : (%, %) -> %
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, IndexedExponents(Symbol))
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
D : (%, List(Symbol)) -> %
from PartialDifferentialRing(Symbol)
D : (%, List(Symbol), List(NonNegativeInteger)) -> %
from PartialDifferentialRing(Symbol)
D : (%, Symbol) -> %
from PartialDifferentialRing(Symbol)
D : (%, Symbol, NonNegativeInteger) -> %
from PartialDifferentialRing(Symbol)
^ : (%, %) -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
^ : (%, 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
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 : (%, List(Symbol), List(NonNegativeInteger)) -> %
from MultivariateTaylorSeriesCategory(Coef, Symbol)
coefficient : (%, Symbol, NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Symbol)
coefficient : (%, IndexedExponents(Symbol)) -> Coef
from AbelianMonoidRing(Coef, IndexedExponents(Symbol))
coefficient : (%, NonNegativeInteger) -> Polynomial(Coef)

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

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 : Polynomial(Coef) -> %

coerce(s) regroups terms of s by total degree and forms a series.

coerce : Symbol -> %

coerce(s) converts a variable to a Taylor series

coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
complete : % -> %
from PowerSeriesCategory(Coef, IndexedExponents(Symbol), Symbol)
construct : List(Record(k : IndexedExponents(Symbol), c : Coef)) -> %
from IndexedProductCategory(Coef, IndexedExponents(Symbol))
constructOrdered : List(Record(k : IndexedExponents(Symbol), c : Coef)) -> %
from IndexedProductCategory(Coef, IndexedExponents(Symbol))
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 : % -> IndexedExponents(Symbol)
from PowerSeriesCategory(Coef, IndexedExponents(Symbol), Symbol)
differentiate : (%, List(Symbol)) -> %
from PartialDifferentialRing(Symbol)
differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> %
from PartialDifferentialRing(Symbol)
differentiate : (%, Symbol) -> %
from PartialDifferentialRing(Symbol)
differentiate : (%, Symbol, NonNegativeInteger) -> %
from PartialDifferentialRing(Symbol)
eval : (%, %, %) -> %
from InnerEvalable(%, %)
eval : (%, Equation(%)) -> %
from Evalable(%)
eval : (%, List(%), List(%)) -> %
from InnerEvalable(%, %)
eval : (%, List(Equation(%))) -> %
from Evalable(%)
eval : (%, List(Symbol), List(%)) -> %
from InnerEvalable(Symbol, %)
eval : (%, Symbol, %) -> %
from InnerEvalable(Symbol, %)
exp : % -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
exquo : (%, %) -> Union(%, "failed") if Coef has IntegralDomain
from EntireRing
extend : (%, NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Symbol)
fintegrate : (Mapping(%), Symbol, 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 : (%, Symbol) -> % if Coef has Algebra(Fraction(Integer))
from MultivariateTaylorSeriesCategory(Coef, Symbol)
integrate : (%, Symbol, 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(Symbol), Symbol)
leadingMonomial : % -> %
from PowerSeriesCategory(Coef, IndexedExponents(Symbol), Symbol)
leadingSupport : % -> IndexedExponents(Symbol)
from IndexedProductCategory(Coef, IndexedExponents(Symbol))
leadingTerm : % -> Record(k : IndexedExponents(Symbol), c : Coef)
from IndexedProductCategory(Coef, IndexedExponents(Symbol))
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, IndexedExponents(Symbol))
monomial : (%, List(Symbol), List(NonNegativeInteger)) -> %
from MultivariateTaylorSeriesCategory(Coef, Symbol)
monomial : (%, Symbol, NonNegativeInteger) -> %
from MultivariateTaylorSeriesCategory(Coef, Symbol)
monomial : (Coef, IndexedExponents(Symbol)) -> %
from IndexedProductCategory(Coef, IndexedExponents(Symbol))
monomial? : % -> Boolean
from IndexedProductCategory(Coef, IndexedExponents(Symbol))
nthRoot : (%, Integer) -> % if Coef has Algebra(Fraction(Integer))
from RadicalCategory
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
order : (%, Symbol) -> NonNegativeInteger
from MultivariateTaylorSeriesCategory(Coef, Symbol)
order : (%, Symbol, NonNegativeInteger) -> NonNegativeInteger
from MultivariateTaylorSeriesCategory(Coef, Symbol)
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(Symbol), Symbol)
polynomial : (%, NonNegativeInteger) -> Polynomial(Coef)
from MultivariateTaylorSeriesCategory(Coef, Symbol)
polynomial : (%, NonNegativeInteger, NonNegativeInteger) -> Polynomial(Coef)
from MultivariateTaylorSeriesCategory(Coef, Symbol)
recip : % -> Union(%, "failed")
from MagmaWithUnit
reductum : % -> %
from IndexedProductCategory(Coef, IndexedExponents(Symbol))
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
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
~= : (%, %) -> Boolean
from BasicType

Module(Fraction(Integer))

NonAssociativeAlgebra(Coef)

Module(Coef)

NonAssociativeSemiRing

BiModule(%, %)

Rng

TrigonometricFunctionCategory

ArcTrigonometricFunctionCategory

TwoSidedRecip

TranscendentalFunctionCategory

SemiRing

EntireRing

RightModule(Coef)

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

RadicalCategory

NonAssociativeRng

CharacteristicNonZero

MagmaWithUnit

AbelianProductCategory(Coef)

noZeroDivisors

Magma

InnerEvalable(%, %)

SemiGroup

IntegralDomain

LeftModule(%)

AbelianMonoidRing(Coef, IndexedExponents(Symbol))

ArcHyperbolicFunctionCategory

NonAssociativeAlgebra(%)

PartialDifferentialRing(Symbol)

CharacteristicZero

Algebra(%)

RightModule(Fraction(Integer))

CommutativeRing

NonAssociativeSemiRng

CancellationAbelianMonoid

CommutativeStar

VariablesCommuteWithCoefficients

AbelianMonoid

RightModule(%)

BiModule(Coef, Coef)

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

Monoid

HyperbolicFunctionCategory

Algebra(Fraction(Integer))

BasicType

InnerEvalable(Symbol, %)

Ring

LeftModule(Fraction(Integer))

PowerSeriesCategory(Coef, IndexedExponents(Symbol), Symbol)

AbelianSemiGroup

SetCategory

MultivariateTaylorSeriesCategory(Coef, Symbol)

IndexedProductCategory(Coef, IndexedExponents(Symbol))

LeftModule(Coef)

NonAssociativeRing

Evalable(%)

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

AbelianGroup

ElementaryFunctionCategory