REAL AND COMPLEX FIELDS

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REAL AND COMPLEX FIELDS

Acknowledgements

Introduction
Fixed Precision Real Numbers
Free Real Numbers
Coercion
Homomorphisms
Special Options
Zero

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 theta and Dedekind eta-functions
The j-invariant and the Discriminant
Weber's Functions

Gamma, Bessel and Associated Functions

The Hypergeometric Function

Other Special Functions

Numerical Functions
Summation of Infinite Series
Integration

Bibliography

DETAILS

Introduction

Fixed Precision Real Numbers
Example FldRe_FixedPrecision (H40E1)

Free Real Numbers

Homomorphisms
Example FldRe_Homomorphisms (H40E2)

Special Options
SetDefaultRealField(R) : FldRe ->
GetDefaultRealField() : Null -> FldPr
AssertAttribute(FldPr, {"Precision", n}) : Cat, MonStgElt, RngIntElt ->
AssertAttribute(FldPr, {"OutputPrecision", n}) : Cat, MonStgElt, RngIntElt ->
HasAttribute(FldPr, "Precision") : Cat, MonStgElt -> BoolElt, RngIntElt
HasAttribute(FldPr, {"OutputPrecision"}) : Cat, MonStgElt -> BoolElt, RngIntElt
AssignNames(~C, [s]) : FldPr, [ MonStgElt ]) ->
Name(C, 1) : FldPr, RngIntElt -> FldComElt

Zero
Example FldRe_Zero (H40E3)

Creation Functions

Creation of Structures
RealField(p) : RngIntElt -> FldRe
RealField() : Null -> FldPr
ComplexField(p) : RngIntElt -> FldCom
ComplexField() : Null -> FldPr
Example FldRe_CreateComplexField (H40E4)

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 (H40E5)

Structure Operations

Related Structures

Numerical Invariants

Ring Predicates and Booleans

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

Element Operations

Generic Element Functions and Predicates

Comparison of and Membership

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

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) : FldReElt -> RngIntElt
Floor(r) : FldReElt -> RngIntElt

Precision
Precision(r) : FldReElt -> RngIntElt
Precision(s) : FldPrElt -> RngIntElt

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

Simple Element Functions
AbsoluteValue(s) : FldPrElt-> FldPrElt
Sign(s) : FldPrElt -> RngIntElt
ComplexConjugate(s) : FldPrElt -> FldPrElt
Norm(c) : FldComElt -> FldReElt
Root(r, n) : FldReElt, RngIntElt -> FldReElt
SquareRoot(c) : FldComElt -> FldComElt

Roots
Roots(p) : RngUPolElt -> [ <FldComElt, RngIntElt> ]
Example FldRe_Roots (H40E6)
RootsNonExact(p) : RngUPolElt -> [ FldPrElt ], [ FldPrElt ]
Example FldRe_RootsNonExact (H40E7)

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

Algebraic Dependencies
LinearRelation(q: parameters) : [ FldPrElt ] -> [ RngIntElt ]
PowerRelation(r, k: parameters) : FldPrElt, RngIntElt -> RngUPolElt

Transcendental Functions

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

Trigonometric Functions
Sin(f) : RngSerElt -> RngSerElt
Sin(c) : FldComElt -> FldComElt
Cos(f) : RngSerElt -> RngSerElt
Cos(c) : FldComElt -> FldComElt
Sincos(f) : RngSerElt -> RngSerElt
Sincos(s) : FldPrElt -> FldPrElt, FldPrElt
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(s) : FldPrElt -> FldPrElt
Arccos(f) : RngSerElt -> RngSerElt
Arccos(s) : FldPrElt -> FldPrElt
Arctan(f) : RngSerElt -> RngSerElt
Arctan(s) : FldPrElt -> FldPrElt
Arctan(a, b) : FldPrElt, FldPrElt -> FldPrElt
Arccot(s) : FldPrElt -> FldPrElt
Arcsec(s) : FldPrElt -> FldPrElt
Arccosec(s) : FldPrElt -> FldPrElt

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

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

Elliptic and Modular Functions

Eisenstein Series
Eisenstein(k, z) : RngIntElt, RngSerElt -> RngSerElt
Eisenstein(k, t) : RngIntElt, FldPrElt -> FldPrElt
Eisenstein(k, L) : RngIntElt, SeqEnum -> FldPrElt
Eisenstein(k, F) : RngIntElt, QuadBinElt -> RngSerElt
Example FldRe_Eisenstein (H40E8)

Weierstrass Series
WeierstrassSeries(z, q, p) : RngElt, RngSerElt, RngIntElt -> RngSerElt
WeierstrassSeries(z, q) : RngSerElt, RngSerElt -> RngSerElt
WeierstrassSeries(z, t) : RngSerElt, FldPrElt -> RngSerElt
WeierstrassSeries(z, L) : RngSerElt, SeqEnum -> RngSerElt
WeierstrassSeries(z, F) : RngSerElt, QuadBinElt -> RngSerElt

The Jacobi theta and Dedekind eta-functions
JacobiTheta(q, z) : FldPrElt, RngSerElt[FldPr] -> RngSerElt
JacobiTheta(q, z) : FldPrElt, FldPrElt -> FldPrElt
JacobiThetaNullK(q, k) : FldPrElt, RngIntElt -> FldPr
DedekindEta(z) : RngSerElt -> RngSerElt
DedekindEta(s) : FldPrElt -> FldPrElt

The j-invariant and the Discriminant
jInvariant(q) : RngSerElt -> RngSerElt
jInvariant(s) : FldPrElt -> FldPrElt
jInvariant(L) : SeqEnum -> FldPrElt
jInvariant(F) : QuadBinElt -> FldPrElt
Delta(z) : RngSerElt -> RngSerElt
Delta(t, p) : FldPrElt, RngIntElt -> FldPrElt
Delta(L, p) : SeqEnum, RngIntElt -> RngPrElt

Weber's Functions
WeberF(s) : FldPrElt -> FldPrElt
WeberF2(g) : RngSerElt -> RngSerElt
WeberF2(s) : FldPrElt -> FldPrElt
Example FldRe_Eisenstein (H40E9)

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

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

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

Numerical Functions

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

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

Bibliography