# InnerSparseUnivariatePowerSeries CoefΒΆ

- Coef: Ring

InnerSparseUnivariatePowerSeries is an internal domain used for creating sparse Taylor and Laurent series.

- 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
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup
- /: (%, Coef) -> % if Coef has Field
- from AbelianMonoidRing(Coef, Integer)
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng
- approximate: (%, Integer) -> Coef if Coef has coerce: Symbol -> Coef and Coef has ^: (Coef, Integer) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Integer)
- associates?: (%, %) -> Boolean if Coef has IntegralDomain
- from EntireRing
- associator: (%, %, %) -> %
- from NonAssociativeRng

- cAcos: % -> % if Coef has Algebra Fraction Integer
`cAcos(f)`

computes the arccosine of the power series`f`

. For use when the coefficient ring is commutative.

- cAcosh: % -> % if Coef has Algebra Fraction Integer
`cAcosh(f)`

computes the inverse hyperbolic cosine of the power series`f`

. For use when the coefficient ring is commutative.

- cAcot: % -> % if Coef has Algebra Fraction Integer
`cAcot(f)`

computes the arccotangent of the power series`f`

. For use when the coefficient ring is commutative.

- cAcoth: % -> % if Coef has Algebra Fraction Integer
`cAcoth(f)`

computes the inverse hyperbolic cotangent of the power series`f`

. For use when the coefficient ring is commutative.

- cAcsc: % -> % if Coef has Algebra Fraction Integer
`cAcsc(f)`

computes the arccosecant of the power series`f`

. For use when the coefficient ring is commutative.

- cAcsch: % -> % if Coef has Algebra Fraction Integer
`cAcsch(f)`

computes the inverse hyperbolic cosecant of the power series`f`

. For use when the coefficient ring is commutative.

- cAsec: % -> % if Coef has Algebra Fraction Integer
`cAsec(f)`

computes the arcsecant of the power series`f`

. For use when the coefficient ring is commutative.

- cAsech: % -> % if Coef has Algebra Fraction Integer
`cAsech(f)`

computes the inverse hyperbolic secant of the power series`f`

. For use when the coefficient ring is commutative.

- cAsin: % -> % if Coef has Algebra Fraction Integer
`cAsin(f)`

computes the arcsine of the power series`f`

. For use when the coefficient ring is commutative.

- cAsinh: % -> % if Coef has Algebra Fraction Integer
`cAsinh(f)`

computes the inverse hyperbolic sine of the power series`f`

. For use when the coefficient ring is commutative.

- cAtan: % -> % if Coef has Algebra Fraction Integer
`cAtan(f)`

computes the arctangent of the power series`f`

. For use when the coefficient ring is commutative.

- cAtanh: % -> % if Coef has Algebra Fraction Integer
`cAtanh(f)`

computes the inverse hyperbolic tangent of the power series`f`

. For use when the coefficient ring is commutative.

- cCos: % -> % if Coef has Algebra Fraction Integer
`cCos(f)`

computes the cosine of the power series`f`

. For use when the coefficient ring is commutative.

- cCosh: % -> % if Coef has Algebra Fraction Integer
`cCosh(f)`

computes the hyperbolic cosine of the power series`f`

. For use when the coefficient ring is commutative.

- cCot: % -> % if Coef has Algebra Fraction Integer
`cCot(f)`

computes the cotangent of the power series`f`

. For use when the coefficient ring is commutative.

- cCoth: % -> % if Coef has Algebra Fraction Integer
`cCoth(f)`

computes the hyperbolic cotangent of the power series`f`

. For use when the coefficient ring is commutative.

- cCsc: % -> % if Coef has Algebra Fraction Integer
`cCsc(f)`

computes the cosecant of the power series`f`

. For use when the coefficient ring is commutative.

- cCsch: % -> % if Coef has Algebra Fraction Integer
`cCsch(f)`

computes the hyperbolic cosecant of the power series`f`

. For use when the coefficient ring is commutative.- center: % -> Coef
- from UnivariatePowerSeriesCategory(Coef, Integer)

- cExp: % -> % if Coef has Algebra Fraction Integer
`cExp(f)`

computes the exponential of the power series`f`

. For use when the coefficient ring is commutative.- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if Coef has CharacteristicNonZero
- from CharacteristicNonZero

- cLog: % -> % if Coef has Algebra Fraction Integer
`cLog(f)`

computes the logarithm of the power series`f`

. For use when the coefficient ring is commutative.- coefficient: (%, Integer) -> Coef
- from AbelianMonoidRing(Coef, Integer)
- 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
- commutator: (%, %) -> %
- from NonAssociativeRng
- complete: % -> %
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

- cPower: (%, Coef) -> % if Coef has Algebra Fraction Integer
`cPower(f, r)`

computes`f^r`

, where`f`

has constant coefficient 1. For use when the coefficient ring is commutative.

- cRationalPower: (%, Fraction Integer) -> % if Coef has Algebra Fraction Integer
`cRationalPower(f, r)`

computes`f^r`

. For use when the coefficient ring is commutative.

- cSec: % -> % if Coef has Algebra Fraction Integer
`cSec(f)`

computes the secant of the power series`f`

. For use when the coefficient ring is commutative.

- cSech: % -> % if Coef has Algebra Fraction Integer
`cSech(f)`

computes the hyperbolic secant of the power series`f`

. For use when the coefficient ring is commutative.

- cSin: % -> % if Coef has Algebra Fraction Integer
`cSin(f)`

computes the sine of the power series`f`

. For use when the coefficient ring is commutative.

- cSinh: % -> % if Coef has Algebra Fraction Integer
`cSinh(f)`

computes the hyperbolic sine of the power series`f`

. For use when the coefficient ring is commutative.

- cTan: % -> % if Coef has Algebra Fraction Integer
`cTan(f)`

computes the tangent of the power series`f`

. For use when the coefficient ring is commutative.

- cTanh: % -> % if Coef has Algebra Fraction Integer
`cTanh(f)`

computes the hyperbolic tangent of the power series`f`

. For use when the coefficient ring is commutative.- D: % -> % if Coef has *: (Integer, Coef) -> Coef
- from DifferentialRing
- D: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
- from DifferentialRing
- D: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- D: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- degree: % -> Integer
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- differentiate: % -> % if Coef has *: (Integer, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, List Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
- from DifferentialRing
- differentiate: (%, Symbol) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has PartialDifferentialRing Symbol and Coef has *: (Integer, Coef) -> Coef
- from PartialDifferentialRing Symbol
- elt: (%, %) -> %
- from Eltable(%, %)
- elt: (%, Integer) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Integer)
- eval: (%, Coef) -> Stream Coef if Coef has ^: (Coef, Integer) -> Coef
- from UnivariatePowerSeriesCategory(Coef, Integer)
- exquo: (%, %) -> Union(%, failed) if Coef has IntegralDomain
- from EntireRing
- extend: (%, Integer) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)

- getRef: % -> Reference OrderedCompletion Integer
`getRef(f)`

returns a reference containing the order to which the terms of`f`

have been computed.

- getStream: % -> Stream Record(k: Integer, c: Coef)
`getStream(f)`

returns the stream of terms representing the series`f`

.- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory

- iCompose: (%, %) -> %
`iCompose(f, g)`

returns`f(g(x))`

. This is an internal function which should only be called for Taylor series`f(x)`

and`g(x)`

such that the constant coefficient of`g(x)`

is zero.

- iExquo: (%, %, Boolean) -> Union(%, failed)
`iExquo(f, g, taylor?)`

is the quotient of the power series`f`

and`g`

. If`taylor?`

is`true`

, then we must have`order(f) >= order(g)`

.

- integrate: % -> % if Coef has Algebra Fraction Integer
`integrate(f(x))`

returns an anti-derivative of the power series`f(x)`

with constant coefficient 0. Warning: function does not check for a term of degree`-1`

.- latex: % -> String
- from SetCategory
- leadingCoefficient: % -> Coef
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- leadingMonomial: % -> %
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- leftPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
- from Magma
- leftRecip: % -> Union(%, failed)
- from MagmaWithUnit

- makeSeries: (Reference OrderedCompletion Integer, Stream Record(k: Integer, c: Coef)) -> %
`makeSeries(refer, str)`

creates a power series from the reference`refer`

and the stream`str`

.- map: (Coef -> Coef, %) -> %
- from AbelianMonoidRing(Coef, Integer)
- monomial: (%, List SingletonAsOrderedSet, List Integer) -> %
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- monomial: (%, SingletonAsOrderedSet, Integer) -> %
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- monomial: (Coef, Integer) -> %
- from AbelianMonoidRing(Coef, Integer)

- monomial?: % -> Boolean
`monomial?(f)`

tests if`f`

is a single monomial.

- multiplyCoefficients: (Integer -> Coef, %) -> %
`multiplyCoefficients(fn, f)`

returns the series`sum(fn(n) * an * x^n, n = n0..)`

, where`f`

is the series`sum(an * x^n, n = n0..)`

.- multiplyExponents: (%, PositiveInteger) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)
- one?: % -> Boolean
- from MagmaWithUnit
- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- order: % -> Integer
- from UnivariatePowerSeriesCategory(Coef, Integer)
- order: (%, Integer) -> Integer
- from UnivariatePowerSeriesCategory(Coef, Integer)
- pole?: % -> Boolean
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- recip: % -> Union(%, failed)
- from MagmaWithUnit
- reductum: % -> %
- from AbelianMonoidRing(Coef, Integer)
- rightPower: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
- from Magma
- rightRecip: % -> Union(%, failed)
- from MagmaWithUnit
- sample: %
- from AbelianMonoid

- series: Stream Record(k: Integer, 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.

- seriesToOutputForm: (Stream Record(k: Integer, c: Coef), Reference OrderedCompletion Integer, Symbol, Coef, Fraction Integer) -> OutputForm
`seriesToOutputForm(st, refer, var, cen, r)`

prints the series`f((var - cen)^r)`

.- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid

- taylorQuoByVar: % -> %
`taylorQuoByVar(a0 + a1 x + a2 x^2 + ...)`

returns`a1 + a2 x + a3 x^2 + ...`

- terms: % -> Stream Record(k: Integer, c: Coef)
- from UnivariatePowerSeriesCategory(Coef, Integer)
- truncate: (%, Integer) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)
- truncate: (%, Integer, Integer) -> %
- from UnivariatePowerSeriesCategory(Coef, Integer)
- 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, Integer)
- variables: % -> List SingletonAsOrderedSet
- from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)
- zero?: % -> Boolean
- from AbelianMonoid

AbelianMonoidRing(Coef, Integer)

Algebra % if Coef has CommutativeRing

Algebra Coef if Coef has CommutativeRing

Algebra Fraction Integer if Coef has Algebra Fraction Integer

BiModule(%, %)

BiModule(Coef, Coef)

BiModule(Fraction Integer, Fraction Integer) if Coef has Algebra Fraction Integer

CharacteristicNonZero if Coef has CharacteristicNonZero

CharacteristicZero if Coef has CharacteristicZero

CommutativeRing if Coef has CommutativeRing

CommutativeStar if Coef has CommutativeRing

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

Eltable(%, %)

EntireRing if Coef has IntegralDomain

IntegralDomain if Coef has IntegralDomain

LeftModule Coef

LeftModule Fraction Integer if Coef has Algebra Fraction Integer

Module % if Coef has CommutativeRing

Module Coef if Coef has CommutativeRing

Module Fraction Integer if Coef has Algebra Fraction Integer

noZeroDivisors if Coef has IntegralDomain

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

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer