# InnerSparseUnivariatePowerSeries CoefΒΆ

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

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

/: (%, 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: (%, %) -> %
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: (%, %, %) -> %
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
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
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, 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

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

from PowerSeriesCategory(Coef, Integer, SingletonAsOrderedSet)

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
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

AbelianGroup

AbelianMonoid

AbelianMonoidRing(Coef, Integer)

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

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

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

RightModule Coef

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if Coef has CommutativeRing

unitsKnown

VariablesCommuteWithCoefficients