# GeneralizedUnivariatePowerSeries(Coef, Expon, var, cen)¶

Author: Waldek Hebisch

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

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

from Field

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

from AbelianMonoidRing(Coef, Expon)

=: (%, %) -> Boolean

from BasicType

^: (%, %) -> % if Coef has Algebra Fraction Integer
^: (%, Integer) -> % if Expon has AbelianGroup and Coef has Field

from DivisionRing

^: (%, 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: (%, %) -> %
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
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

from UnivariatePowerSeriesCategory(Coef, Expon)

characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
coefficient: (%, Expon) -> Coef

from AbelianMonoidRing(Coef, Expon)

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: % -> %

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
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 *: (Expon, Coef) -> Coef

from DifferentialRing

D: (%, List Symbol) -> % if Coef has *: (Expon, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef and Coef has 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
D: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
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
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef and Coef has 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
differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (Expon, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
divide: (%, %) -> Record(quotient: %, remainder: %) if Expon has AbelianGroup and Coef has Field

from EuclideanDomain

elt: (%, %) -> %

from Eltable(%, %)

elt: (%, Expon) -> Coef

from UnivariatePowerSeriesCategory(Coef, Expon)

euclideanSize: % -> NonNegativeInteger if Expon has AbelianGroup and Coef has Field

from EuclideanDomain

eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Expon) -> Coef

from UnivariatePowerSeriesCategory(Coef, Expon)

exp: % -> % if Coef has Algebra Fraction Integer
expressIdealMember: (List %, %) -> Union(List %, failed) if Expon has AbelianGroup and Coef has Field
exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain

from EntireRing

extend: (%, Expon) -> %

from UnivariatePowerSeriesCategory(Coef, Expon)

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if Expon has AbelianGroup and Coef has Field

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if Expon has AbelianGroup and Coef has Field

from EuclideanDomain

factor: % -> Factored % if Expon has AbelianGroup and Coef has Field
gcd: (%, %) -> % if Expon has AbelianGroup and Coef has Field

from GcdDomain

gcd: List % -> % if Expon has AbelianGroup and Coef has Field

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if Expon has AbelianGroup and Coef has Field

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 Expon has AbelianGroup and Coef has Field

from DivisionRing

latex: % -> String

from SetCategory

lcm: (%, %) -> % if Expon has AbelianGroup and Coef has Field

from GcdDomain

lcm: List % -> % if Expon has AbelianGroup and Coef has Field

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if Expon has AbelianGroup and Coef has Field

from LeftOreRing

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

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
map: (Coef -> Coef, %) -> %

from IndexedProductCategory(Coef, Expon)

monomial?: % -> Boolean

from IndexedProductCategory(Coef, Expon)

monomial: (Coef, Expon) -> %

from IndexedProductCategory(Coef, Expon)

multiEuclidean: (List %, %) -> Union(List %, failed) if Expon has AbelianGroup and Coef has Field

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
plenaryPower: (%, PositiveInteger) -> % if Coef has Algebra Fraction Integer or Coef has CommutativeRing
pole?: % -> Boolean

from PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

prime?: % -> Boolean if Expon has AbelianGroup and Coef has Field
principalIdeal: List % -> Record(coef: List %, generator: %) if Expon has AbelianGroup and Coef has Field
quo: (%, %) -> % if Expon has AbelianGroup and Coef has Field

from EuclideanDomain

recip: % -> Union(%, failed)

from MagmaWithUnit

reductum: % -> %

from IndexedProductCategory(Coef, Expon)

rem: (%, %) -> % if Expon has AbelianGroup and Coef has Field

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
sech: % -> % if Coef has Algebra Fraction Integer
sin: % -> % if Coef has Algebra Fraction Integer
sinh: % -> % if Coef has Algebra Fraction Integer
sizeLess?: (%, %) -> Boolean if Expon has AbelianGroup and Coef has Field

from EuclideanDomain

squareFree: % -> Factored % if Expon has AbelianGroup and Coef has Field
squareFreePart: % -> % if Expon has AbelianGroup and Coef has Field
subtractIfCan: (%, %) -> Union(%, failed)
tan: % -> % if Coef has Algebra Fraction Integer
tanh: % -> % if Coef has Algebra Fraction Integer
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

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Expon)

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

canonicalsClosed if Expon has AbelianGroup and Coef has Field

canonicalUnitNormal if Expon has AbelianGroup and Coef has Field

CharacteristicNonZero if Coef has CharacteristicNonZero

CharacteristicZero if Coef has CharacteristicZero

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

DifferentialRing if Coef has *: (Expon, Coef) -> Coef

DivisionRing if Expon has AbelianGroup and Coef has Field

Eltable(%, %)

EntireRing if Coef has IntegralDomain

EuclideanDomain if Expon has AbelianGroup and Coef has Field

Field if Expon has AbelianGroup and Coef has Field

GcdDomain if Expon has AbelianGroup and Coef has Field

IndexedProductCategory(Coef, Expon)

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

LeftOreRing if Expon has AbelianGroup and Coef has Field

Magma

MagmaWithUnit

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

Monoid

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if Coef has IntegralDomain

PartialDifferentialRing Symbol if Coef has *: (Expon, Coef) -> Coef and Coef has PartialDifferentialRing Symbol

PowerSeriesCategory(Coef, Expon, SingletonAsOrderedSet)

PrincipalIdealDomain if Expon has AbelianGroup and Coef has Field

RightModule Coef

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

UniqueFactorizationDomain if Expon has AbelianGroup and Coef has Field

unitsKnown

UnivariatePowerSeriesCategory(Coef, Expon)

VariablesCommuteWithCoefficients