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REAL AND COMPLEX FIELDS

 
Acknowledgements
 
Introduction
      Overview of Real Numbers in Magma
      Coercion
      Homomorphisms
      Special Options
      Version Functions
 
Creation Functions
      Creation of Structures
      Creation of Elements
 
Structure Operations
      Related Structures
      Numerical Invariants
      Ring Predicates and Booleans
      Other Structure Functions
 
Element Operations
      Generic Element Functions and Predicates
      Comparison of and Membership
      Other Predicates
      Arithmetic
      Conversions
      Rounding
      Precision
      Constants
      Simple Element Functions
      Roots
      Continued Fractions
      Algebraic Dependencies
 
Transcendental Functions
      Exponential, Logarithmic and Polylogarithmic Functions
      Trigonometric Functions
      Inverse Trigonometric Functions
      Hyperbolic Functions
      Inverse Hyperbolic Functions
 
Elliptic and Modular Functions
      Eisenstein Series
      Weierstrass Series
      The Jacobi θand Dedekind η- functions
      The j-invariant and the Discriminant
      Weber's Functions
 
Theta Functions
 
Gamma, Bessel and Associated Functions
 
The Hypergeometric Function
 
Other Special Functions
 
Numerical Functions
      Summation of Infinite Series
      Integration
      Numerical Derivatives
 
Bibliography







DETAILS

 
Introduction

      Overview of Real Numbers in Magma
            Example FldRe_RealIntro (H25E1)

      Coercion

      Homomorphisms
            Example FldRe_Homomorphisms (H25E2)

      Special Options
            SetDefaultRealField(R) : FldRe ->
            GetDefaultRealField() : -> FldRe
            AssignNames(~C, [s]) : FldCom, [ MonStgElt ]) ->
            Name(C, 1) : FldCom, RngIntElt -> FldComElt

      Version Functions
            GetGMPVersion() : ->

 
Creation Functions

      Creation of Structures
            RealField(p) : RngIntElt -> FldRe
            RealField() : -> FldRe
            ComplexField(p) : RngIntElt -> FldCom
            ComplexField() : -> FldCom
            ComplexField(R) : FldRe -> FldCom
            Example FldRe_CreateComplexField (H25E3)

      Creation of Elements
            d . eefpg : RngIntElt, RngIntElt, RngIntElt -> FldReElt
            elt<R | m, n> : FldRe, FldReElt, RngIntElt -> FldReElt
            elt<C | x, y> : FldCom, FldReElt, FldReElt -> FldComElt
            R ! a : FldRe, RngElt -> FldReElt
            C ! a : FldCom, RngElt -> FldComElt
            Example FldRe_CreateElements (H25E4)

 
Structure Operations

      Related Structures

      Numerical Invariants

      Ring Predicates and Booleans

      Other Structure Functions
            Precision(R) : FldCom -> RngIntElt
            BitPrecision(R) : FldCom -> RngIntElt

 
Element Operations

      Generic Element Functions and Predicates

      Comparison of and Membership

      Other Predicates
            IsIntegral(c) : FldReElt -> BoolElt
            IsReal(c) : FldComElt -> BoolElt

      Arithmetic

      Conversions
            MantissaExponent(r) : FldReElt -> FldReElt, RngIntElt
            ComplexToPolar(c) : FldComElt -> FldReElt, FldReElt
            PolarToComplex(m, a) : FldReElt, FldReElt -> FldComElt
            Argument(c) : FldComElt -> FldReElt
            Modulus(c) : FldComElt -> FldReElt
            Real(c) : FldComElt -> FldReElt
            Imaginary(c) : FldComElt -> FldReElt

      Rounding
            Round(r) : FldReElt -> FldReElt
            Truncate(r) : FldReElt -> RngIntElt
            Ceiling(r) : Infty -> Infty
            Floor(r) : Infty -> Infty

      Precision
            Precision(r) : FldReElt -> RngIntElt
            BitPrecision(r) : FldReElt -> RngIntElt
            Precision(L) : [FldReElt] -> RngIntElt
            ChangePrecision(r, n) : FldReElt, RngIntElt -> FldReElt

      Constants
            Catalan(R) : FldRe -> FldReElt
            EulerGamma(R) : FldRe -> FldReElt
            Pi(R) : FldRe -> FldReElt

      Simple Element Functions
            AbsoluteValue(r) : FldReElt-> FldReElt
            Sign(r) : FldReElt -> RngIntElt
            ComplexConjugate(r) : FldReElt -> FldReElt
            Norm(c) : FldComElt -> FldReElt
            Root(r, n) : FldReElt, RngIntElt -> FldReElt
            SquareRoot(c) : FldComElt -> FldComElt
            Distance(x, L) : FldReElt, [FldReElt] -> FldReElt, RngIntElt
            Diameter(L) : [FldReElt] -> FldReElt

      Roots
            Roots(p) : RngUPolElt -> [ <FldComElt, RngIntElt> ]
            Example FldRe_Roots (H25E5)
            RootsNonExact(p) : RngUPolElt[FldRe] -> [ FldComElt ], [ FldComElt ]
            Example FldRe_RootsNonExact (H25E6)
            HenselLift(f, R, k) : RngUPolElt, FldReElt, RngIntElt -> FldReElt

      Continued Fractions
            ContinuedFraction(r) : FldRatElt -> [ RngIntElt ]
            BestApproximation(r, n) : FldReElt, RngIntElt -> FldReElt
            Convergents(s) : [ RngIntElt ] -> ModMatRngElt

      Algebraic Dependencies
            LinearRelation(q: parameters) : [ FldComElt ] -> [ RngIntElt ]
            AllLinearRelations(q,p): SeqEnum, RngIntElt -> Lat
            PowerRelation(r, k: parameters) : FldReElt, RngIntElt -> RngUPolElt

 
Transcendental Functions

      Exponential, Logarithmic and Polylogarithmic Functions
            Exp(f) : RngSerElt -> RngSerElt
            Exp(c) : FldComElt -> FldComElt
            Log(f) : RngSerElt -> RngSerElt
            Log(r) : FldReElt -> FldReElt
            Log(b, r) : FldReElt -> FldReElt
            Dilog(s) : FldComElt -> FldComElt
            Polylog(m, f) : RngIntElt, RngSerElt -> RngSerElt
            Polylog(m, s) : RngIntElt, FldComElt -> FldComElt
            PolylogD(m, s) : RngIntElt, FldComElt -> FldComElt

      Trigonometric Functions
            Sin(f) : RngSerElt -> RngSerElt
            Sin(c) : FldComElt -> FldComElt
            Cos(f) : RngSerElt -> RngSerElt
            Cos(c) : FldComElt -> FldComElt
            Sincos(f) : RngSerElt -> RngSerElt
            Sincos(s) : FldReElt -> FldReElt, FldReElt
            Tan(f) : RngSerElt -> RngSerElt
            Tan(c) : FldComElt -> FldComElt
            Cot(f) : RngSerElt -> RngSerElt
            Cot(c) : FldComElt -> FldComElt
            Sec(f) : RngSerElt -> RngSerElt
            Sec(c) : FldComElt -> FldComElt
            Cosec(f) : RngSerElt -> RngSerElt
            Cosec(c) : FldComElt -> FldComElt

      Inverse Trigonometric Functions
            Arcsin(f) : RngSerElt -> RngSerElt
            Arcsin(r) : FldReElt -> FldReElt
            Arccos(f) : RngSerElt -> RngSerElt
            Arccos(r) : FldReElt -> FldReElt
            Arctan(f) : RngSerElt -> RngSerElt
            Arctan(r) : FldReElt -> FldReElt
            Arctan(x, y) : FldReElt, FldReElt -> FldReElt
            Arccot(r) : FldReElt -> FldReElt
            Arcsec(r) : FldReElt -> FldReElt
            Arccosec(r) : FldReElt -> FldReElt

      Hyperbolic Functions
            Sinh(f) : RngSerElt -> RngSerElt
            Sinh(s) : FldComElt -> FldComElt
            Cosh(f) : RngSerElt -> RngSerElt
            Cosh(r) : FldReElt -> FldReElt
            Tanh(f) : RngSerElt -> RngSerElt
            Tanh(r) : FldReElt -> FldReElt
            Coth(r) : FldReElt -> FldReElt
            Sech(r) : FldReElt -> FldReElt
            Cosech(r) : FldReElt -> FldReElt

      Inverse Hyperbolic Functions
            Argsinh(f) : RngSerElt -> RngSerElt
            Argsinh(r) : FldReElt -> FldReElt
            Argcosh(f) : RngSerElt -> RngSerElt
            Argcosh(r) : FldReElt -> FldReElt
            Argtanh(f) : RngSerElt -> RngSerElt
            Argtanh(s) : FldReElt -> FldReElt
            Argsech(s) : FldReElt -> FldReElt
            Argcosech(s) : FldReElt -> FldReElt
            Argcoth(s) : FldReElt -> FldReElt

 
Elliptic and Modular Functions

      Eisenstein Series
            Eisenstein(k, z) : RngIntElt, RngSerElt -> RngSerElt
            Eisenstein(k, t) : RngIntElt, FldComElt -> FldComElt
            Eisenstein(k, L) : RngIntElt, SeqEnum -> FldComElt
            Eisenstein(k, F) : RngIntElt, QuadBinElt -> RngSerElt
            Example FldRe_Eisenstein (H25E7)

      Weierstrass Series
            WeierstrassSeries(z, q) : RngSerElt, RngSerElt -> RngSerElt
            WeierstrassSeries(z, t) : RngSerElt, FldComElt -> RngSerElt
            WeierstrassSeries(z, L) : RngSerElt, SeqEnum -> RngSerElt
            WeierstrassSeries(z, F) : RngSerElt, QuadBinElt -> RngSerElt

      The Jacobi θand Dedekind η- functions
            JacobiTheta(q, z) : FldReElt, RngSerElt[FldRe] -> RngSerElt
            JacobiTheta(q, z) : FldReElt, FldReElt -> FldReElt
            JacobiThetaNullK(q, k) : FldReElt, RngIntElt -> FldReElt
            DedekindEta(z) : RngSerElt -> RngSerElt
            DedekindEta(s) : FldComElt -> FldComElt

      The j-invariant and the Discriminant
            jInvariant(q) : RngSerElt -> RngSerElt
            jInvariant(s) : FldComElt -> FldComElt
            jInvariant(L) : SeqEnum -> FldComElt
            jInvariant(F) : QuadBinElt -> FldComElt
            Delta(z) : RngSerElt -> RngSerElt
            Delta(t) : FldComElt -> FldComElt
            Delta(L) : SeqEnum -> FldComElt

      Weber's Functions
            WeberF(s) : FldComElt -> FldComElt
            WeberF2(g) : RngSerElt -> RngSerElt
            WeberF1(s) : FldComElt -> FldComElt
            Example FldRe_Eisenstein (H25E8)

 
Theta Functions
      Theta(char, z, tau) : Mtrx, Mtrx, Mtrx -> FldComElt
      Theta(char, z, A) : Mtrx, Mtrx, AnHcJac -> FldComElt

 
Gamma, Bessel and Associated Functions
      Gamma(f) : RngSerElt -> RngSerElt
      Gamma(r) : FldReElt -> FldReElt
      Gamma(r, s) : FldReElt, FldReElt -> FldReElt
      GammaD(s) : FldReElt -> FldReElt
      LogGamma(f) : RngSerElt -> RngSerElt
      LogGamma(r) : FldReElt -> FldReElt
      LogDerivative(s) : FldReElt -> FldReElt
      BesselFunction(n, r) : RngIntElt, FldReElt -> FldReElt
      BesselFunctionSecondKind(n, r) : RngIntElt, FldReElt -> FldReElt
      JBessel(n, s) : RngIntElt, FldReElt -> FldReElt
      KBessel(n, s) : FldReElt, FldReElt -> FldReElt

 
The Hypergeometric Function
      HypergeometricSeries(a,b,c, z) : RngElt, RngElt, RngElt, RngElt -> RngElt
      HypergeometricU(a, b, s) : FldReElt, FldReElt, FldReElt -> FldReElt

 
Other Special Functions
      ArithmeticGeometricMean(x, y) : RngSerElt, RngSerElt -> RngSerElt
      ArithmeticGeometricMean(x, y) : FldReElt, FldReElt -> FldReElt
      BernoulliNumber(n) : RngIntElt -> FldRatElt
      BernoulliApproximation(n) : RngIntElt -> FldReElt
      DawsonIntegral(r) : FldReElt -> FldReElt
      ErrorFunction(r) : FldReElt -> FldReElt
      ComplementaryErrorFunction(r) : FldReElt -> FldReElt
      ExponentialIntegral(r) : FldReElt -> FldReElt
      ExponentialIntegralE1(r) : FldReElt -> FldReElt
      LogIntegral(r) : FldReElt -> FldReElt
      ZetaFunction(s) : FldReElt -> FldReElt

 
Numerical Functions

      Summation of Infinite Series
            InfiniteSum(m, i) : Map, RngIntElt -> FldReElt
            PositiveSum(m, i) : Map, RngIntElt -> FldReElt
            AlternatingSum(m, i) : Map, RngIntElt -> FldReElt

      Integration
            Interpolation(P, V, x) : [FldReElt], [FldReElt], FldReElt -> FldReElt, FldReElt
            RombergQuadrature(f, a, b: parameters) : Program, FldReElt, FldReElt -> FldReElt
            SimpsonQuadrature(f, a, b, n) : Program, FldReElt, FldReElt, RngIntElt -> FldReElt
            TrapezoidalQuadrature(f, a, b, n) : Program, FldReElt, FldReElt, RngIntElt -> FldReElt

      Numerical Derivatives
            NumericalDerivative(f, n, z) : UserProgram, RngIntElt, FldComElt -> FldComElt
            Example FldRe_NumericalDerivative (H25E9)

 
Bibliography

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