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POWER, LAURENT AND PUISEUX SERIES

 
Acknowledgements
 
Introduction
      Kinds of Series
      Puiseux Series
      Representation of Series
      Precision
      Free and Fixed Precision
      Equality
      Polynomials over Series Rings
 
Creation Functions
      Creation of Structures
      Special Options
      Creation of Elements
 
Structure Operations
      Related Structures
      Invariants
      Ring Predicates and Booleans
 
Basic Element Operations
      Parent and Category
      Arithmetic Operators
      Equality and Membership
      Predicates on Ring Elements
      Precision
      Coefficients and Degree
      Evaluation and Derivative
      Square Root
      Composition and Reversion
 
Transcendental Functions
      Exponential and Logarithmic Functions
      Trigonometric Functions and their Inverses
      Hyperbolic Functions and their Inverses
 
The Hypergeometric Series
 
Polynomials over Series Rings
 
Extensions of Series Rings
      Constructions of Extensions
      Operations on Extensions
      Elements of Extensions
      Optimized Representation
 
Bibliography







DETAILS

 
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 (H49E1)

      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, ..., ad], 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 (H49E2)

 
Transcendental Functions

      Exponential and Logarithmic Functions
            Exp(f) : RngSerElt -> RngSerElt
            Log(f) : RngSerElt -> RngSerElt
            Example RngSer_Bernoulli (H49E3)

      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 (H49E4)

 
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 (H49E5)

      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 (H49E6)

      Elements of Extensions
            Valuation(e) : RngSerExtElt -> RngIntElt
            RelativePrecision(e) : RngSerExtElt -> RngIntElt
            AbsolutePrecision(e) : RngSerExtElt -> RngIntElt
            Coefficients(e) : RngSerExtElt -> [ RngElt ]
            Example RngSer_serext-simple (H49E7)

      Optimized Representation
            OptimizedRepresentation(E) : RngSerExt -> RngSer, Map
            Example RngSer_opt-rep (H49E8)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013