Introduction

      Real Numbers in Magma
            Example FldRe_RealIntro (H26E1)

      Coercion

      Homomorphisms
            Example FldRe_Homomorphisms (H26E2)

      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 (H26E3)

      Creation of Elements
            a . becpd : 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 (H26E4)

 
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 -> RngIntElt, RngIntElt
            ComplexToPolar(c) : FldComElt -> FldReElt, FldReElt
            PolarToComplex(m, a) : FldReElt, FldReElt -> FldComElt
            Argument(c) : FldComElt -> FldReElt
            Example FldRe_demonstrate-arg (H26E5)
            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 (H26E6)
            RootsNonExact(p) : RngUPolElt[FldRe] -> [ FldComElt ], [ FldComElt ]
            Example FldRe_RootsNonExact (H26E7)
            HenselLift(f, R, k) : RngUPolElt, FldReElt, RngIntElt -> FldReElt

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

      Linear and Algebraic Dependencies
            LinearRelation(q: parameters) : [ FldComElt ] -> [ RngIntElt ]
            AllLinearRelations(q,p): SeqEnum, RngIntElt -> Lat
            IntegerRelation(q): SeqEnum -> SeqEnum, FldReElt
            PowerRelation(r, k: parameters) : FldReElt, RngIntElt -> RngUPolElt
            MinimalPolynomial(r,d,N) : FldReElt, RngIntElt, RngIntElt -> RngUPolElt, FldReElt
            Example FldRe_LLLPolFact (H26E8)

 
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 (H26E9)

      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 θ and 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 (H26E10)

 
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(s, t) : 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

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

 
Numerical Integration

      Polynomial Interpolation
            Interpolation(P, V, t) : [FldReElt], [FldReElt], FldReElt -> FldReElt, FldReElt

      Discrete Fourier Transform
            DiscreteFourierTransform(E) : SeqEnum[FldComElt] -> SeqEnum[FldComElt]

      Integration of Complex Functions

            Gaussian Quadratures
                  GaussLegendreIntegrationPoints(N,D) : RngIntElt, RngIntElt -> SeqEnum[FldReElt], SeqEnum[FldReElt]
                  GaussJacobiIntegrationPoints(N,D,a,b) : RngIntElt, RngIntElt, RngReSubElt, RngReSubElt) -> SeqEnum, SeqEnum

            Clenshaw--Curtis Quadrature
                  ClenshawCurtisIntegrationPoints(N,D) : RngIntElt, RngIntElt -> SeqEnum[FldReElt], SeqEnum[FldReElt]

            Tanh--Sinh Quadrature
                  TanhSinhIntegrationPoints(N,h) : RngIntElt, FldReElt -> SeqEnum[FldReElt], SeqEnum[FldReElt], SeqEnum[FldReElt]
                  Example FldRe_num-int-ex-1 (H26E11)
                  Example FldRe_num-int-ex-2 (H26E12)
                  Example FldRe_num-int-ex-3 (H26E13)

      Romberg-Type Integration
            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 (H26E14)

 
Bibliography

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