POWER, LAURENT AND PUISEUX SERIES

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 (H63E1)

      Special Options
            AssertAttribute(S, "Precision", n) : RngSer, MonStgElt, RngIntElt ->
            HasAttribute(S, "Precision") : 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, [ a_1, ..., a_d], p > : RngIntElt, SeqEnum, RngIntElt -> RngSerElt
            R ! s : RngSer, SeqEnum -> RngSerElt
            BigO(f) : RngSerElt -> RngIntElt

 
Structure Operations

      Related Structures
            BaseRing(R) : RngSer -> Rng

      Invariants
            Precision(R) : RngSer -> Rng

      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

      Coefficients and Degree
            Coefficients(f) : RngSerElt -> [ RngElt ], RngIntElt, RngIntElt
            Coefficient(f, i) : RngSerSerElt, RngElt -> RngElt
            LeadingCoefficient(f) : RngSerElt -> RngElt
            LeadingTerm(f) : RngSerElt -> RngElt
            Truncate(f) : RngSerElt -> RngSerElt
            ExponentDenominator(f) : RngMSerElt -> RngElt
            Degree(f) : RngMSerElt -> RngIntElt
            Valuation(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 (H63E2)

 
Transcendental Functions

      Exponential and Logarithmic Functions
            Exp(f) : RngSerElt -> RngSerElt
            Log(f) : RngSerElt -> RngSerElt
            Example RngSer_Bernoulli (H63E3)

      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

 
Bibliography