InnerTaylorSeries(Coef)

Coef : Ring

Internal package for dense Taylor series. This is an internal Taylor series type in which Taylor series are represented by a Stream of Ring elements. For univariate series, the Stream elements are the Taylor coefficients. For multivariate series, the nth Stream element is a form of degree n in the power series variables.

* : (%, %) -> %: from Magma

* : (%, Coef) -> %: x*c returns the product of c and the series x.

* : (%, Integer) -> %: x*i returns the product of integer i and the series x.

* : (Coef, %) -> %: c*x returns the product of c and the series x.

* : (Integer, %) -> %: from AbelianGroup

* : (NonNegativeInteger, %) -> %: from AbelianMonoid

* : (PositiveInteger, %) -> %: from AbelianSemiGroup

+ : (%, %) -> %: from AbelianSemiGroup

- : % -> %: from AbelianGroup

- : (%, %) -> %: from AbelianGroup

0 : () -> %: from AbelianMonoid

1 : () -> %: from MagmaWithUnit

= : (%, %) -> Boolean: from BasicType

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

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

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

antiCommutator : (%, %) -> %: from NonAssociativeSemiRng

associates? : (%, %) -> Boolean if Coef has IntegralDomain: from EntireRing

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

characteristic : () -> NonNegativeInteger: from NonAssociativeRing

coefficients : % -> Stream(Coef): coefficients(x) returns a stream of ring elements. When x is a univariate series, this is a stream of Taylor coefficients. When x is a multivariate series, the nth element of the stream is a form of degree n in the power series variables.

coerce : % -> % if Coef has IntegralDomain: from Algebra(%)

coerce : Integer -> %: from NonAssociativeRing

coerce : % -> OutputForm: from CoercibleTo(OutputForm)

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

exquo : (%, %) -> Union(%, "failed") if Coef has IntegralDomain: from EntireRing

latex : % -> String: from SetCategory

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

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

leftRecip : % -> Union(%, "failed"): from MagmaWithUnit

one? : % -> Boolean: from MagmaWithUnit

opposite? : (%, %) -> Boolean: from AbelianMonoid

order : % -> NonNegativeInteger: order(x) returns the order of a power series x, i.e. the degree of the first non-zero term of the series.

order : (%, NonNegativeInteger) -> NonNegativeInteger: order(x, n) returns the minimum of n and the order of x.

plenaryPower : (%, PositiveInteger) -> % if Coef has IntegralDomain: from NonAssociativeAlgebra(%)

pole? : % -> Boolean: pole?(x) tests if the series x has a pole. Note: this is false when x is a Taylor series.

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

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

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

rightRecip : % -> Union(%, "failed"): from MagmaWithUnit

sample : () -> %: from AbelianMonoid

series : Stream(Coef) -> %: series(s) creates a power series from a stream of ring elements. For univariate series types, the stream s should be a stream of Taylor coefficients. For multivariate series types, the stream s should be a stream of forms the nth element of which is a form of degree n in the power series variables.