GeneralizedUnivariatePowerSeries(Coef, Expon, var, cen)

genser.spad line 194 [edit on github]

Domain for univariate power series with variable 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
/ : (%, %) -> % if Coef has Field and Expon has AbelianGroup
from Field
/ : (%, Coef) -> % if Coef has Field
from AbelianMonoidRing(Coef, Expon)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
D : % -> % if Coef has * : (Expon, Coef) -> Coef
from DifferentialRing
D : (%, List(Symbol)) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, NonNegativeInteger) -> % if Coef has * : (Expon, Coef) -> Coef
from DifferentialRing
D : (%, Symbol) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
D : (%, Symbol, NonNegativeInteger) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
^ : (%, %) -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
^ : (%, Integer) -> % if Coef has Field and Expon has AbelianGroup
from DivisionRing
^ : (%, 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
apply_taylor : (Stream(Coef), %) -> %

apply_taylor(ts, s) applies Taylor series with coefficients ts to s, that is computes infinite sum ts(0) + ts(1)*s + ts(2)*s^2 + ... Note: s must be of positive order

approximate : (%, Expon) -> Coef if Coef has coerce : Symbol -> Coef and Coef has ^ : (Coef, Expon) -> Coef
from UnivariatePowerSeriesCategory(Coef, Expon)
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, Expon)
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
charthRoot : % -> Union(%, "failed") if Coef has CharacteristicNonZero
from CharacteristicNonZero
coefficient : (%, Expon) -> Coef
from AbelianMonoidRing(Coef, Expon)
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, Expon, SingletonAsOrderedSet)
construct : List(Record(k : Expon, c : Coef)) -> %
from IndexedProductCategory(Coef, Expon)
constructOrdered : List(Record(k : Expon, c : Coef)) -> %
from IndexedProductCategory(Coef, Expon)
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 : % -> Expon
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
differentiate : % -> % if Coef has * : (Expon, Coef) -> Coef
from DifferentialRing
differentiate : (%, List(Symbol)) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, List(Symbol), List(NonNegativeInteger)) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, NonNegativeInteger) -> % if Coef has * : (Expon, Coef) -> Coef
from DifferentialRing
differentiate : (%, Symbol) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
differentiate : (%, Symbol, NonNegativeInteger) -> % if Coef has * : (Expon, Coef) -> Coef and Coef has PartialDifferentialRing(Symbol)
from PartialDifferentialRing(Symbol)
divide : (%, %) -> Record(quotient : %, remainder : %) if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
elt : (%, %) -> %
from Eltable(%, %)
elt : (%, Expon) -> Coef
from UnivariatePowerSeriesCategory(Coef, Expon)
euclideanSize : % -> NonNegativeInteger if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
eval : (%, Coef) -> Stream(Coef) if Coef has ^ : (Coef, Expon) -> Coef
from UnivariatePowerSeriesCategory(Coef, Expon)
exp : % -> % if Coef has Algebra(Fraction(Integer))
from ElementaryFunctionCategory
expressIdealMember : (List(%), %) -> Union(List(%), "failed") if Coef has Field and Expon has AbelianGroup
from PrincipalIdealDomain
exquo : (%, %) -> Union(%, "failed") if Coef has IntegralDomain
from EntireRing
extend : (%, Expon) -> %
from UnivariatePowerSeriesCategory(Coef, Expon)
extendedEuclidean : (%, %) -> Record(coef1 : %, coef2 : %, generator : %) if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
extendedEuclidean : (%, %, %) -> Union(Record(coef1 : %, coef2 : %), "failed") if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
factor : % -> Factored(%) if Coef has Field and Expon has AbelianGroup
from UniqueFactorizationDomain
gcd : (%, %) -> % if Coef has Field and Expon has AbelianGroup
from GcdDomain
gcd : List(%) -> % if Coef has Field and Expon has AbelianGroup
from GcdDomain
gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%) if Coef has Field and Expon has AbelianGroup
from GcdDomain
infsum : Stream(%) -> %

infsum(x) computes sum of all elements of x. Degrees of elements of x must be nondecreasing and tend to infinity.

inv : % -> % if Coef has Field and Expon has AbelianGroup
from DivisionRing
latex : % -> String
from SetCategory
lcm : (%, %) -> % if Coef has Field and Expon has AbelianGroup
from GcdDomain
lcm : List(%) -> % if Coef has Field and Expon has AbelianGroup
from GcdDomain
lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %) if Coef has Field and Expon has AbelianGroup
from LeftOreRing
leadingCoefficient : % -> Coef
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
leadingMonomial : % -> %
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
leadingSupport : % -> Expon
from IndexedProductCategory(Coef, Expon)
leadingTerm : % -> Record(k : Expon, c : Coef)
from IndexedProductCategory(Coef, Expon)
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, Expon)
monomial : (Coef, Expon) -> %
from IndexedProductCategory(Coef, Expon)
monomial? : % -> Boolean
from IndexedProductCategory(Coef, Expon)
multiEuclidean : (List(%), %) -> Union(List(%), "failed") if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
multiplyExponents : (%, PositiveInteger) -> %
from UnivariatePowerSeriesCategory(Coef, Expon)
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
order : % -> Expon
from UnivariatePowerSeriesCategory(Coef, Expon)
order : (%, Expon) -> Expon
from UnivariatePowerSeriesCategory(Coef, Expon)
pi : () -> % if Coef has Algebra(Fraction(Integer))
from TranscendentalFunctionCategory
plenaryPower : (%, PositiveInteger) -> % if Coef has Algebra(Fraction(Integer)) or Coef has CommutativeRing
from NonAssociativeAlgebra(%)
pole? : % -> Boolean
from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)
prime? : % -> Boolean if Coef has Field and Expon has AbelianGroup
from UniqueFactorizationDomain
principalIdeal : List(%) -> Record(coef : List(%), generator : %) if Coef has Field and Expon has AbelianGroup
from PrincipalIdealDomain
quo : (%, %) -> % if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
recip : % -> Union(%, "failed")
from MagmaWithUnit
reductum : % -> %
from IndexedProductCategory(Coef, Expon)
rem : (%, %) -> % if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
removeZeros : (%, Expon) -> %

removeZeros(s, k) removes leading zero terms in s with exponent smaller than k

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
sizeLess? : (%, %) -> Boolean if Coef has Field and Expon has AbelianGroup
from EuclideanDomain
squareFree : % -> Factored(%) if Coef has Field and Expon has AbelianGroup
from UniqueFactorizationDomain
squareFreePart : % -> % if Coef has Field and Expon has AbelianGroup
from UniqueFactorizationDomain
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 : Expon, c : Coef))
from UnivariatePowerSeriesCategory(Coef, Expon)
truncate : (%, Expon) -> %
from UnivariatePowerSeriesCategory(Coef, Expon)
truncate : (%, Expon, Expon) -> %
from UnivariatePowerSeriesCategory(Coef, Expon)
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, Expon)
zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

Module(Fraction(Integer))

PrincipalIdealDomain

Module(Coef)

NonAssociativeSemiRing

BiModule(%, %)

Field

canonicalUnitNormal

ArcTrigonometricFunctionCategory

TwoSidedRecip

TranscendentalFunctionCategory

SemiRing

EntireRing

Rng

RightModule(Coef)

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

NonAssociativeSemiRng

CharacteristicNonZero

MagmaWithUnit

noZeroDivisors

AbelianProductCategory(Coef)

UniqueFactorizationDomain

SemiGroup

RightModule(Fraction(Integer))

Magma

GcdDomain

LeftModule(%)

NonAssociativeRing

canonicalsClosed

ArcHyperbolicFunctionCategory

PartialDifferentialRing(Symbol)

CharacteristicZero

Algebra(%)

CommutativeRing

DifferentialRing

IndexedProductCategory(Coef, Expon)

DivisionRing

Eltable(%, %)

LeftOreRing

CancellationAbelianMonoid

EuclideanDomain

VariablesCommuteWithCoefficients

NonAssociativeAlgebra(Coef)

UnivariatePowerSeriesCategory(Coef, Expon)

CommutativeStar

AbelianMonoid

ElementaryFunctionCategory

RightModule(%)

BiModule(Coef, Coef)

LeftModule(Coef)

AbelianMonoidRing(Coef, Expon)

Module(%)

CoercibleTo(OutputForm)

Algebra(Coef)

SemiRng

Monoid

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

LeftModule(Fraction(Integer))

AbelianSemiGroup

IntegralDomain

SetCategory

TrigonometricFunctionCategory

NonAssociativeRng

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

HyperbolicFunctionCategory

AbelianGroup

PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)