Introduction

      Kinds of Series

      Puiseux Series

      Representation of Series

      Precision

      Free and Fixed Precision

      Equality

      Polynomials over Series Rings

 
Creation Functions

      Creation of Structures
            PowerSeriesRing(R) : Rng -> RngSerPow
            LaurentSeriesRing(R) : Rng -> RngSerLaur
            PuiseuxSeriesRing(R) : Rng -> RngSerPuis
            Example RngSer_Creation (H51E1)

      Special Options
            AssertAttribute(S, "DefaultPrecision", n) : RngSer, MonStgElt, RngIntElt ->
            HasAttribute(S, "DefaultPrecision") : RngSer, MonStgElt -> BoolElt, RngIntElt
            AssignNames(~S, ["x"]) : RngSer, [ MonStgElt ] ->
            Name(S, 1) : RngSer, RngIntElt -> RngSerElt

      Creation of Elements
            R . 1 : RngSer, RngInt -> RngSerElt
            elt< R | v, [ a₁, ..., a_d], p > : RngIntElt, SeqEnum, RngIntElt -> RngSerElt
            R ! s : RngSer, SeqEnum -> RngSerElt
            BigO(f) : RngSerElt -> RngIntElt

 
Structure Operations

      Related Structures
            BaseRing(R) : RngSer -> Rng
            IntegerRing(R) : RngSer -> RngSerPow
            FieldOfFractions(R) : RngSer -> RngSerLaur
            ChangePrecision(R, r) : RngSer, Any -> RngSer
            ChangeRing(R, C) : RngSer, Rng -> RngSer, Map
            ResidueClassField(R) : RngSer -> Rng, Map

      Invariants
            Precision(R) : RngSer -> ExtReElt

      Ring Predicates and Booleans

 
Basic Element Operations

      Parent and Category

      Arithmetic Operators

      Equality and Membership

      Predicates on Ring Elements
            IsWeaklyZero(f) : RngSerElt -> BoolElt
            IsWeaklyEqual(f, g) : RngSerElt, RngSerElt -> BoolElt
            IsIdentical(f, g) : RngSerElt, RngSerElt -> BoolElt

      Precision
            AbsolutePrecision(f) : RngSerElt -> RngIntElt
            RelativePrecision(f) : RngSerElt -> RngIntElt
            ChangePrecision(f, r) : RngSerElt, RngIntElt -> RngSerElt

      Coefficients and Degree
            Coefficients(f) : RngSerElt -> [ RngElt ], RngIntElt, RngIntElt
            Coefficient(f, i) : RngSerElt, RngElt -> RngElt
            LeadingCoefficient(f) : RngSerElt -> RngElt
            LeadingTerm(f) : RngSerElt -> RngElt
            Truncate(f) : RngSerElt -> RngSerElt
            ExponentDenominator(f) : RngMSerElt -> RngElt
            Degree(f) : RngSerElt -> RngIntElt
            Valuation(f) : RngSerElt -> RngIntElt
            ExponentDenominator(f) : RngSerElt -> RngIntElt

      Evaluation and Derivative
            Derivative(f) : RngSerElt -> RngSerElt
            Derivative(f, n) : RngSerElt, RngIntElt -> RngSerElt
            Integral(f) : RngSerElt -> RngSerElt
            Evaluate(f, s) : RngSerElt, RngElt -> RngElt
            Laplace(f) : RngSerElt -> RngSerElt

      Square Root
            SquareRoot(f) : RngSerElt -> RngSerElt

      Composition and Reversion
            Composition(f, g) : RngSerElt, RngSerElt -> RngSerElt
            Reversion(f) : RngSerElt -> RngSerElt
            Convolution(f, g) : RngSerElt, RngSerElt -> RngSerElt
            Example RngSer_CompositionReversion (H51E2)

 
Transcendental Functions

      Exponential and Logarithmic Functions
            Exp(f) : RngSerElt -> RngSerElt
            Log(f) : RngSerElt -> RngSerElt
            Example RngSer_Bernoulli (H51E3)

      Trigonometric Functions and their Inverses
            Sin(f) : RngSerElt -> RngSerElt
            Cos(f) : RngSerElt -> RngSerElt
            Sincos(f) : RngSerElt -> RngSerElt
            Tan(f) : RngSerElt -> RngSerElt
            Arcsin(f) : RngSerElt -> RngSerElt
            Arccos(f) : RngSerElt -> RngSerElt
            Arctan(f) : RngSerElt -> RngSerElt

      Hyperbolic Functions and their Inverses
            Sinh(f) : RngSerElt -> RngSerElt
            Cosh(f) : RngSerElt -> RngSerElt
            Tanh(f) : RngSerElt -> RngSerElt
            Argsinh(f) : RngSerElt -> RngSerElt
            Argcosh(f) : RngSerElt -> RngSerElt
            Argtanh(f) : RngSerElt -> RngSerElt

 
The Hypergeometric Series
      HypergeometricSeries(a,b,c, z) : RngElt, RngElt, RngElt, RngElt -> RngElt

 
Polynomials over Series Rings
      HenselLift(f, L) : RngUPolElt[RngSer], SeqEnum[RngUPolElt] -> [RngUPolElt]
      Factorization(f) : RngUPolElt[RngSerPow[FldFin]] -> [ < RngUPolElt[RngSerPow], RngIntElt > ], RngSerPowElt
      Example RngSer_series_poly_fact (H51E4)

 
Extensions of Series Rings

      Constructions of Extensions
            UnramifiedExtension(R, f) : RngSerPow[FldFin], RngUPolElt -> RngSerExt
            TotallyRamifiedExtension(R, f) : RngSerPow[FldFin], RngUPolElt -> RngSerExt
            ChangePrecision(E, r) : RngSerExt, RngIntElt -> RngSerExt
            FieldOfFractions(E) : RngSerExt -> RngSerExt
            Example RngSer_extensions_eg (H51E5)

      Operations on Extensions
            Precision(E) : RngSerExt -> RngIntElt
            CoefficientRing(E) : RngSerExt -> Rng
            DefiningPolynomial(E) : RngSerExt -> RngUPolElt
            InertiaDegree(E) : RngSerExt -> RngIntElt
            RamificationIndex(E) : RngSerExt -> RngIntElt
            ResidueClassField(E) : RngSerExt -> FldFin
            UniformizingElement(E) : RngSerExt -> RngSerExtElt
            IntegerRing(E) : RngSerExt -> RngSerExt
            E1 eq E2 : RngSerExt, RngSerExt -> BoolElt
            E . i : RngSerExt, RngIntElt -> RngSerExtElt
            AssignNames(~E, S) : RngSerExt, [ MonStgElt ] ->
            Example RngSer_ext-ops (H51E6)

      Elements of Extensions
            Valuation(e) : RngSerExtElt -> RngIntElt
            RelativePrecision(e) : RngSerExtElt -> RngIntElt
            AbsolutePrecision(e) : RngSerExtElt -> RngIntElt
            Coefficients(e) : RngSerExtElt -> [ RngElt ]
            Example RngSer_serext-simple (H51E7)

      Optimized Representation
            OptimizedRepresentation(E) : RngSerExt -> RngSer, Map
            Example RngSer_opt-rep (H51E8)

 
Bibliography

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