# InnerSparseUnivariatePowerSeries CoefΒΆ

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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)

- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit

- ^: (%, PositiveInteger) -> %
from Magma

- annihilate?: (%, %) -> Boolean
from Rng

- antiCommutator: (%, %) -> %

- 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

- 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
- coerce: Integer -> %
from NonAssociativeRing

- commutator: (%, %) -> %
from NonAssociativeRng

- complete: % -> %
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

- construct: List Record(k: Integer, c: Coef) -> %
from IndexedProductCategory(Coef, Integer)

- constructOrdered: List Record(k: Integer, c: Coef) -> %
from IndexedProductCategory(Coef, Integer)

- 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 *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
- D: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
- D: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
from DifferentialRing

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

- degree: % -> Integer
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

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

- differentiate: (%, List Symbol) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
- differentiate: (%, List Symbol, List NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
- differentiate: (%, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef
from DifferentialRing

- differentiate: (%, Symbol) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has PartialDifferentialRing Symbol
- differentiate: (%, Symbol, NonNegativeInteger) -> % if Coef has *: (Integer, Coef) -> Coef and Coef has 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`

.

- 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)

- leadingSupport: % -> Integer
from IndexedProductCategory(Coef, Integer)

- leadingTerm: % -> Record(k: Integer, c: Coef)
from IndexedProductCategory(Coef, Integer)

- 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 IndexedProductCategory(Coef, Integer)

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

tests if`f`

is a single monomial.

- monomial: (Coef, Integer) -> %
from IndexedProductCategory(Coef, Integer)

- 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)

- plenaryPower: (%, PositiveInteger) -> % if Coef has CommutativeRing or Coef has Algebra Fraction Integer
from NonAssociativeAlgebra Coef

- pole?: % -> Boolean
from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

- recip: % -> Union(%, failed)
from MagmaWithUnit

- reductum: % -> %
from IndexedProductCategory(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)

- 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)

- 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

IndexedProductCategory(Coef, Integer)

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

NonAssociativeAlgebra % if Coef has CommutativeRing

NonAssociativeAlgebra Coef if Coef has CommutativeRing

NonAssociativeAlgebra Fraction Integer if Coef has Algebra Fraction Integer

noZeroDivisors if Coef has IntegralDomain

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

PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

RightModule Coef

RightModule Fraction Integer if Coef has Algebra Fraction Integer

TwoSidedRecip if Coef has CommutativeRing