# Real and Complex Fields

- [Introduction](introduction.md)

  - [Real Numbers in Magma](introduction.md#real-numbers-in-magma)

    - [`Example: Real Intro`](introduction.md#example-ex-5afbbc)

  - [Coercion](introduction.md#coercion)

  - [Homomorphisms](introduction.md#homomorphisms)

    - [`hom< R -> S | >: FldRe, Str → Map`](introduction.md#constructor-constructor-hom-fldre-str-map)

    - [`Example: Homomorphisms`](introduction.md#example-ex-edd37f)

  - [Special Options](introduction.md#special-options)

    - [`SetDefaultRealField(R): FldRe`](introduction.md#function-setdefaultrealfield-fldre)

    - [`GetDefaultRealField() → FldRe`](introduction.md#function-getdefaultrealfield)

    - [`AssignNames(~C, [s]): FldCom, [ MonStgElt ])`](introduction.md#function-assignnames-fldcom-monstgelt-ref)

    - [`Name(C, 1): FldCom, RngIntElt → FldComElt`](introduction.md#function-name-fldcom-rngintelt)

  - [Version Functions](introduction.md#version-functions)

    - [`GetGMPVersion()`](introduction.md#function-getgmpversion)

    - [`GetMPFRVersion()`](introduction.md#function-getmpfrversion)

    - [`GetMPCVersion()`](introduction.md#function-getmpcversion)

- [Creation Functions](creation.md)

  - [Creation of Structures](creation.md#creation-of-structures)

    - [`RealField(p): RngIntElt → FldRe`](creation.md#function-fldre-realfield)

    - [`RealField() → FldRe`](creation.md#function-realfield)

    - [`ComplexField(p): RngIntElt → FldCom`](creation.md#function-complexfield-rngintelt)

    - [`ComplexField() → FldCom`](creation.md#function-complexfield)

    - [`ComplexField(R): FldRe → FldCom`](creation.md#function-complexfield-fldre)

    - [`Example: Create Complex Field`](creation.md#example-ex-108c46)

  - [Creation of Elements](creation.md#creation-of-elements)

    - [`a . becpd: RngIntElt, RngIntElt, RngIntElt → FldReElt`](creation.md#operation-operation-rngintelt-rngintelt-rngintelt-fldreelt)

    - [`a . bEcPd: RngIntElt, RngIntElt, RngIntElt → FldReElt`](creation.md#operation-operation-rngintelt-rngintelt-rngintelt-fldreelt-2)

    - [`a.bec P d: RngIntElt, RngIntElt, RngIntElt → FldReElt`](creation.md#operation-operation-rngintelt-rngintelt-rngintelt-fldreelt-3)

    - [`a.bec p d: RngIntElt, RngIntElt, RngIntElt → FldReElt`](creation.md#operation-operation-rngintelt-rngintelt-rngintelt-fldreelt-4)

    - [`a.b E cpd: RngIntElt, RngIntElt, RngIntElt → FldReElt`](creation.md#operation-operation-rngintelt-rngintelt-rngintelt-fldreelt-5)

    - [`a.b e cpd: RngIntElt, RngIntElt, RngIntElt → FldReElt`](creation.md#operation-operation-rngintelt-rngintelt-rngintelt-fldreelt-6)

    - [`elt<R | m, n>: FldRe, FldReElt, RngIntElt → FldReElt`](creation.md#constructor-constructor-elt-fldre-fldreelt-rngintelt-fldreelt)

    - [`elt<C | x, y>: FldCom, FldReElt, FldReElt → FldComElt`](creation.md#constructor-constructor-elt-fldcom-fldreelt-fldreelt-fldcomelt)

    - [`C ! [x, y]: FldCom, [FldReElt, FldReElt] → FldComElt`](creation.md#operation-op-fldcom-fldreelt-fldreelt)

    - [`R ! a: FldRe, RngElt → FldReElt`](creation.md#operation-op-fldre-rngelt)

    - [`C ! a: FldCom, RngElt → FldComElt`](creation.md#operation-op-fldcom-rngelt)

    - [`Example: Create Elements`](creation.md#example-ex-356721)

    - [`One(R): FldRe → FldReElt`](creation.md#function-one-fldre)

    - [`One(R): FldCom → FldComElt`](creation.md#function-one-fldcom)

    - [`Identity(R): FldRe → FldReElt`](creation.md#function-identity-fldre)

    - [`Identity(R): FldCom → FldComElt`](creation.md#function-identity-fldcom)

    - [`Zero(R): FldRe → FldReElt`](creation.md#function-zero-fldre)

    - [`Zero(R): FldCom → FldComElt`](creation.md#function-zero-fldcom)

    - [`Representative(R): FldRe → FldReElt`](creation.md#function-representative-fldre)

    - [`Representative(R): FldCom → FldComElt`](creation.md#function-representative-fldcom)

- [Structure Operations](operation-structure.md)

  - [Related Structures](operation-structure.md#related-structures)

    - [`Category(R): FldRe → Cat`](operation-structure.md#function-category-fldre)

    - [`Category(R): FldCom → Cat`](operation-structure.md#function-category-fldcom)

    - [`Parent(R): FldRe → PowerStructure`](operation-structure.md#function-parent-fldre)

    - [`Parent(R): FldCom → PowerStructure`](operation-structure.md#function-parent-fldcom)

    - [`PrimeField(R): FldRe → FldRat`](operation-structure.md#function-primefield-fldre)

    - [`PrimeField(R): FldCom → FldRat`](operation-structure.md#function-primefield-fldcom)

  - [Numerical Invariants](operation-structure.md#numerical-invariants)

    - [`Characteristic(R): FldRe → RngIntElt`](operation-structure.md#function-characteristic-fldre)

    - [`Characteristic(R): FldCom → RngIntElt`](operation-structure.md#function-characteristic-fldcom)

  - [Ring Predicates and Booleans](operation-structure.md#ring-predicates-and-booleans)

    - [`IsCommutative(R): FldRe → BoolElt`](operation-structure.md#function-iscommutative-fldre)

    - [`IsCommutative(R): FldCom → BoolElt`](operation-structure.md#function-iscommutative-fldcom)

    - [`IsUnitary(R): FldRe → BoolElt`](operation-structure.md#function-isunitary-fldre)

    - [`IsUnitary(R): FldCom → BoolElt`](operation-structure.md#function-isunitary-fldcom)

    - [`IsFinite(R): FldRe → BoolElt`](operation-structure.md#function-isfinite-fldre)

    - [`IsFinite(R): FldCom → BoolElt`](operation-structure.md#function-isfinite-fldcom)

    - [`IsOrdered(R): FldRe → BoolElt`](operation-structure.md#function-isordered-fldre)

    - [`IsOrdered(R): FldCom → BoolElt`](operation-structure.md#function-isordered-fldcom)

    - [`IsField(R): FldRe → BoolElt`](operation-structure.md#function-isfield-fldre)

    - [`IsField(R): FldCom → BoolElt`](operation-structure.md#function-isfield-fldcom)

    - [`IsEuclideanDomain(R): FldRe → BoolElt`](operation-structure.md#function-iseuclideandomain-fldre)

    - [`IsEuclideanDomain(R): FldCom → BoolElt`](operation-structure.md#function-iseuclideandomain-fldcom)

    - [`IsPID(R): FldRe → BoolElt`](operation-structure.md#function-ispid-fldre)

    - [`IsPID(R): FldCom → BoolElt`](operation-structure.md#function-ispid-fldcom)

    - [`IsUFD(R): FldRe → BoolElt`](operation-structure.md#function-isufd-fldre)

    - [`IsUFD(R): FldCom → BoolElt`](operation-structure.md#function-isufd-fldcom)

    - [`IsDivisionRing(R): FldRe → BoolElt`](operation-structure.md#function-isdivisionring-fldre)

    - [`IsDivisionRing(R): FldCom → BoolElt`](operation-structure.md#function-isdivisionring-fldcom)

    - [`IsEuclideanRing(R): FldRe → BoolElt`](operation-structure.md#function-iseuclideanring-fldre)

    - [`IsEuclideanRing(R): FldCom → BoolElt`](operation-structure.md#function-iseuclideanring-fldcom)

    - [`IsPrincipalIdealRing(R): FldRe → BoolElt`](operation-structure.md#function-isprincipalidealring-fldre)

    - [`IsPrincipalIdealRing(R): FldCom → BoolElt`](operation-structure.md#function-isprincipalidealring-fldcom)

    - [`IsDomain(R): FldRe → BoolElt`](operation-structure.md#function-isdomain-fldre)

    - [`IsDomain(R): FldCom → BoolElt`](operation-structure.md#function-isdomain-fldcom)

    - [`R eq S: FldRe, FldRe → BoolElt`](operation-structure.md#operation-op-eq-fldre-fldre)

    - [`R ne S: FldRe, FldRe → BoolElt`](operation-structure.md#operation-op-ne-fldre-fldre)

    - [`R eq S: FldCom, FldCom → BoolElt`](operation-structure.md#operation-op-eq-fldcom-fldcom)

    - [`R ne S: FldCom, FldCom → BoolElt`](operation-structure.md#operation-op-ne-fldcom-fldcom)

  - [Other Structure Functions](operation-structure.md#other-structure-functions)

    - [`Precision(R): FldCom → RngIntElt`](operation-structure.md#function-precision-fldcom)

    - [`Precision(R): FldRe → RngIntElt`](operation-structure.md#function-precision-fldre)

    - [`BitPrecision(R): FldCom → RngIntElt`](operation-structure.md#function-bitprecision-fldcom)

    - [`BitPrecision(R): FldRe → RngIntElt`](operation-structure.md#function-bitprecision-fldre)

- [Element Operations](operation-element.md)

  - [Generic Element Functions and Predicates](operation-element.md#generic-element-functions-and-predicates)

    - [`Parent(r): FldReElt → Rng`](operation-element.md#function-parent-fldreelt)

    - [`Parent(r): FldComElt → Rng`](operation-element.md#function-parent-fldcomelt)

    - [`Category(r): FldReElt → Cat`](operation-element.md#function-category-fldreelt)

    - [`Category(r): FldComElt → Cat`](operation-element.md#function-category-fldcomelt)

    - [`IsZero(r): FldReElt → BoolElt`](operation-element.md#function-iszero-fldreelt)

    - [`IsZero(r): FldComElt → BoolElt`](operation-element.md#function-iszero-fldcomelt)

    - [`IsOne(r): FldReElt → BoolElt`](operation-element.md#function-isone-fldreelt)

    - [`IsOne(r): FldComElt → BoolElt`](operation-element.md#function-isone-fldcomelt)

    - [`IsMinusOne(r): FldReElt → BoolElt`](operation-element.md#function-isminusone-fldreelt)

    - [`IsMinusOne(r): FldComElt → BoolElt`](operation-element.md#function-isminusone-fldcomelt)

    - [`IsUnit(r): FldReElt → BoolElt`](operation-element.md#function-isunit-fldreelt)

    - [`IsUnit(r): FldComElt → BoolElt`](operation-element.md#function-isunit-fldcomelt)

    - [`IsZeroDivisor(r): FldReElt → BoolElt`](operation-element.md#function-iszerodivisor-fldreelt)

    - [`IsZeroDivisor(r): FldComElt → BoolElt`](operation-element.md#function-iszerodivisor-fldcomelt)

    - [`IsIdempotent(r): FldReElt → BoolElt`](operation-element.md#function-isidempotent-fldreelt)

    - [`IsIdempotent(r): FldComElt → BoolElt`](operation-element.md#function-isidempotent-fldcomelt)

    - [`IsNilpotent(r): FldReElt → BoolElt`](operation-element.md#function-isnilpotent-fldreelt)

    - [`IsNilpotent(r): FldComElt → BoolElt`](operation-element.md#function-isnilpotent-fldcomelt)

    - [`IsIrreducible(r): FldReElt → BoolElt`](operation-element.md#function-isirreducible-fldreelt)

    - [`IsIrreducible(r): FldComElt → BoolElt`](operation-element.md#function-isirreducible-fldcomelt)

    - [`IsPrime(r): FldReElt → BoolElt`](operation-element.md#function-isprime-fldreelt)

    - [`IsPrime(r): FldComElt → BoolElt`](operation-element.md#function-isprime-fldcomelt)

  - [Comparison of and Membership](operation-element.md#comparison-of-and-membership)

    - [`a eq b: FldReElt, FldReElt → BoolElt`](operation-element.md#operation-op-eq-fldreelt-fldreelt)

    - [`a ne b: FldReElt, FldReElt → BoolElt`](operation-element.md#operation-op-ne-fldreelt-fldreelt)

    - [`a eq b: FldComElt, FldComElt → BoolElt`](operation-element.md#operation-op-eq-fldcomelt-fldcomelt)

    - [`a ne b: FldComElt, FldComElt → BoolElt`](operation-element.md#operation-op-ne-fldcomelt-fldcomelt)

    - [`a in R: FldReElt, FldRe → BoolElt`](operation-element.md#operation-op-in-fldreelt-fldre)

    - [`a notin R: FldReElt, FldRe → BoolElt`](operation-element.md#operation-op-notin-fldreelt-fldre)

    - [`a in R: FldComElt, FldCom → BoolElt`](operation-element.md#operation-op-in-fldcomelt-fldcom)

    - [`a notin R: FldComElt, FldCom → BoolElt`](operation-element.md#operation-op-notin-fldcomelt-fldcom)

    - [`a gt b: FldReElt, FldReElt → BoolElt`](operation-element.md#operation-op-gt-fldreelt-fldreelt)

    - [`a ge b: FldReElt, FldReElt → BoolElt`](operation-element.md#operation-op-ge-fldreelt-fldreelt)

    - [`a lt b: FldReElt, FldReElt → BoolElt`](operation-element.md#operation-op-lt-fldreelt-fldreelt)

    - [`a le b: FldReElt, FldReElt → BoolElt`](operation-element.md#operation-op-le-fldreelt-fldreelt)

    - [`Maximum(a, b): FldReElt, FldReElt → FldReElt`](operation-element.md#function-maximum-fldreelt-fldreelt)

    - [`Minimum(a, b): FldReElt, FldReElt → FldReElt`](operation-element.md#function-minimum-fldreelt-fldreelt)

    - [`Maximum(Q): [FldReElt] → FldReElt`](operation-element.md#function-maximum-fldreelt)

    - [`Minimum(Q): [FldReElt] → FldReElt`](operation-element.md#function-minimum-fldreelt)

  - [Other Predicates](operation-element.md#other-predicates)

    - [`IsIntegral(c): FldReElt → BoolElt`](operation-element.md#function-isintegral-fldreelt)

    - [`IsIntegral(c): FldComElt → BoolElt`](operation-element.md#function-isintegral-fldcomelt)

    - [`IsReal(c): FldComElt → BoolElt`](operation-element.md#function-isreal-fldcomelt)

  - [Arithmetic](operation-element.md#arithmetic)

    - [`+ r: FldReElt → FldReElt`](operation-element.md#operation-operation-fldreelt-fldreelt)

    - [`+ r: FldComElt → FldComElt`](operation-element.md#operation-operation-fldcomelt-fldcomelt)

    - [`- r: FldReElt → FldReElt`](operation-element.md#operation-operation-fldreelt-fldreelt-2)

    - [`- r: FldComElt → FldComElt`](operation-element.md#operation-operation-fldcomelt-fldcomelt-2)

    - [`r + s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-op-plus-fldreelt-fldreelt)

    - [`r + s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-op-plus-fldcomelt-fldcomelt)

    - [`r - s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-op-minus-fldreelt-fldreelt)

    - [`r - s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-op-minus-fldcomelt-fldcomelt)

    - [`r * s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-op-times-fldreelt-fldreelt)

    - [`r * s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-op-times-fldcomelt-fldcomelt)

    - [`r / s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-op-div-fldreelt-fldreelt)

    - [`r / s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-op-div-fldcomelt-fldcomelt)

    - [`r ^ k: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-op-pow-fldreelt-fldreelt)

    - [`r ^ k: FldReElt, RngIntElt → FldReElt`](operation-element.md#operation-op-pow-fldreelt-rngintelt)

    - [`r ^ k: FldComElt, RngIntElt → FldComElt`](operation-element.md#operation-op-pow-fldcomelt-rngintelt)

    - [`r ^ k: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-op-pow-fldcomelt-fldcomelt)

    - [`r +:= s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-operation-fldreelt-fldreelt-fldreelt)

    - [`r +:= s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-operation-fldcomelt-fldcomelt-fldcomelt)

    - [`r -:= s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-operation-fldreelt-fldreelt-fldreelt-2)

    - [`r -:= s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-operation-fldcomelt-fldcomelt-fldcomelt-2)

    - [`r *:= s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-operation-fldreelt-fldreelt-fldreelt-3)

    - [`r *:= s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-operation-fldcomelt-fldcomelt-fldcomelt-3)

    - [`r /:= s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-operation-fldreelt-fldreelt-fldreelt-4)

    - [`r /:= s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-operation-fldcomelt-fldcomelt-fldcomelt-4)

    - [`r ^:= s: FldReElt, FldReElt → FldReElt`](operation-element.md#operation-operation-fldreelt-fldreelt-fldreelt-5)

    - [`r ^:= s: FldComElt, FldComElt → FldComElt`](operation-element.md#operation-operation-fldcomelt-fldcomelt-fldcomelt-5)

  - [Conversions](operation-element.md#conversions)

    - [`MantissaExponent(r): FldReElt → RngIntElt, RngIntElt`](operation-element.md#function-mantissaexponent-fldreelt)

    - [`ComplexToPolar(c): FldComElt → FldReElt, FldReElt`](operation-element.md#function-complextopolar-fldcomelt)

    - [`PolarToComplex(m, a): FldReElt, FldReElt → FldComElt`](operation-element.md#function-polartocomplex-fldreelt-fldreelt)

    - [`Argument(c): FldComElt → FldReElt`](operation-element.md#function-argument-fldcomelt)

    - [`Arg(c): FldComElt → FldReElt`](operation-element.md#function-arg-fldcomelt)

    - [`Example: Demonstrate Arg`](operation-element.md#example-ex-9ff75f)

    - [`Modulus(c): FldComElt → FldReElt`](operation-element.md#function-modulus-fldcomelt)

    - [`Real(c): FldComElt → FldReElt`](operation-element.md#function-real-fldcomelt)

    - [`Re(c): FldComElt → FldReElt`](operation-element.md#function-re-fldcomelt)

    - [`Imaginary(c): FldComElt → FldReElt`](operation-element.md#function-imaginary-fldcomelt)

    - [`Im(c): FldComElt → FldReElt`](operation-element.md#function-im-fldcomelt)

  - [Rounding](operation-element.md#rounding)

    - [`Round(r): FldReElt → FldReElt`](operation-element.md#function-round-fldreelt)

    - [`Round(r): FldComElt → RngQuadElt`](operation-element.md#function-round-fldcomelt)

    - [`Truncate(r): FldReElt → RngIntElt`](operation-element.md#function-truncate-fldreelt)

    - [`Ceiling(r): Infty → Infty`](operation-element.md#function-ceiling-infty)

    - [`Ceiling(r): FldReElt → RngIntElt`](operation-element.md#function-ceiling-fldreelt)

    - [`Floor(r): Infty → Infty`](operation-element.md#function-floor-infty)

    - [`Floor(r): FldReElt → RngIntElt`](operation-element.md#function-floor-fldreelt)

  - [Precision](operation-element.md#precision)

    - [`Precision(r): FldReElt → RngIntElt`](operation-element.md#function-precision-fldreelt)

    - [`Precision(c): FldComElt → RngIntElt`](operation-element.md#function-precision-fldcomelt)

    - [`BitPrecision(r): FldReElt → RngIntElt`](operation-element.md#function-bitprecision-fldreelt)

    - [`BitPrecision(c): FldComElt → RngIntElt`](operation-element.md#function-bitprecision-fldcomelt)

    - [`Precision(L): [FldReElt] → RngIntElt`](operation-element.md#function-precision-fldreelt-2)

    - [`Precision(L): [FldComElt] → RngIntElt`](operation-element.md#function-precision-fldcomelt-2)

    - [`ChangePrecision(r, n): FldReElt, RngIntElt → FldReElt`](operation-element.md#function-changeprecision-fldreelt-rngintelt)

    - [`ChangePrecision(c, n): FldComElt, RngIntElt → FldComElt`](operation-element.md#function-changeprecision-fldcomelt-rngintelt)

  - [Constants](operation-element.md#constants)

    - [`Catalan(R): FldRe → FldReElt`](operation-element.md#function-catalan-fldre)

    - [`Catalan(R): FldCom → FldComElt`](operation-element.md#function-catalan-fldcom)

    - [`EulerGamma(R): FldRe → FldReElt`](operation-element.md#function-eulergamma-fldre)

    - [`EulerGamma(R): FldCom → FldComElt`](operation-element.md#function-eulergamma-fldcom)

    - [`Pi(R): FldRe → FldReElt`](operation-element.md#function-pi-fldre)

    - [`Pi(R): FldCom → FldComElt`](operation-element.md#function-pi-fldcom)

  - [Simple Element Functions](operation-element.md#simple-element-functions)

    - [`AbsoluteValue(r): FldReElt → FldReElt`](operation-element.md#function-absolutevalue-fldreelt)

    - [`AbsoluteValue(r): FldComElt → FldReElt`](operation-element.md#function-absolutevalue-fldcomelt)

    - [`Abs(r): FldReElt → FldReElt`](operation-element.md#function-abs-fldreelt)

    - [`Abs(r): FldComElt → FldReElt`](operation-element.md#function-abs-fldcomelt)

    - [`Sign(r): FldReElt → RngIntElt`](operation-element.md#function-sign-fldreelt)

    - [`ComplexConjugate(r): FldReElt → FldReElt`](operation-element.md#function-complexconjugate-fldreelt)

    - [`ComplexConjugate(c): FldComElt → FldComElt`](operation-element.md#function-complexconjugate-fldcomelt)

    - [`Conjugate(c): FldComElt → FldComElt`](operation-element.md#function-conjugate-fldcomelt)

    - [`Norm(c): FldComElt → FldReElt`](operation-element.md#function-norm-fldcomelt)

    - [`Norm(r): FldReElt → FldReElt`](operation-element.md#function-norm-fldreelt)

    - [`Root(r, n): FldReElt, RngIntElt → FldReElt`](operation-element.md#function-root-fldreelt-rngintelt)

    - [`Root(r, n): FldComElt, RngIntElt → FldComElt`](operation-element.md#function-root-fldcomelt-rngintelt)

    - [`SquareRoot(c): FldComElt → FldComElt`](operation-element.md#function-squareroot-fldcomelt)

    - [`Sqrt(c): FldComElt → FldComElt`](operation-element.md#function-sqrt-fldcomelt)

    - [`SquareRoot(r): FldReElt → FldReElt`](operation-element.md#function-squareroot-fldreelt)

    - [`Sqrt(r): FldReElt → FldReElt`](operation-element.md#function-sqrt-fldreelt)

    - [`Distance(x, L): FldReElt, [FldReElt] → FldReElt, RngIntElt`](operation-element.md#function-distance-fldreelt-fldreelt)

    - [`Distance(x, L): FldReElt, [FldComElt] → FldReElt, RngIntElt`](operation-element.md#function-distance-fldreelt-fldcomelt)

    - [`Distance(x, L): FldComElt, [FldReElt] → FldReElt, RngIntElt`](operation-element.md#function-distance-fldcomelt-fldreelt)

    - [`Distance(x, L): FldComElt, [FldComElt] → FldReElt, RngIntElt`](operation-element.md#function-distance-fldcomelt-fldcomelt)

    - [`Diameter(L): [FldReElt] → FldReElt`](operation-element.md#function-diameter-fldreelt)

    - [`Diameter(L): [FldComElt] → FldReElt`](operation-element.md#function-diameter-fldcomelt)

  - [Roots](operation-element.md#roots)

    - [`Roots(p): RngUPolElt → [ <FldComElt, RngIntElt> ]`](operation-element.md#function-fldre-roots)

    - [`Example: Roots`](operation-element.md#example-ex-04bda5)

    - [`RootsNonExact(p): RngUPolElt[FldRe] → [ FldComElt ], [ FldComElt ]`](operation-element.md#function-rootsnonexact-rngupolelt-fldre)

    - [`RootsNonExact(p): RngUPolElt[FldCom] → [ FldComElt ], [ FldComElt ]`](operation-element.md#function-rootsnonexact-rngupolelt-fldcom)

    - [`Example: Roots Non Exact`](operation-element.md#example-ex-a1670d)

    - [`HenselLift(f, R, k): RngUPolElt, FldReElt, RngIntElt → FldReElt`](operation-element.md#function-hensellift-rngupolelt-fldreelt-rngintelt)

    - [`HenselLift(f, R, k): RngUPolElt, FldComElt, RngIntElt → FldComElt`](operation-element.md#function-hensellift-rngupolelt-fldcomelt-rngintelt)

  - [Continued Fractions](operation-element.md#continued-fractions)

    - [`ContinuedFraction(r): FldRatElt → [ RngIntElt ]`](operation-element.md#function-continuedfraction-fldratelt)

    - [`ContinuedFraction(r): FldReElt → [ RngIntElt ]`](operation-element.md#function-continuedfraction-fldreelt)

    - [`BestApproximation(r, n): FldReElt, RngIntElt → FldReElt`](operation-element.md#function-bestapproximation-fldreelt-rngintelt)

    - [`Convergents(s): [ RngIntElt ] → ModMatRngElt`](operation-element.md#function-convergents-rngintelt)

  - [Linear and Algebraic Dependencies](operation-element.md#linear-and-algebraic-dependencies)

    - [`LinearRelation(q: parameters): [ FldComElt ] → [ RngIntElt ]`](operation-element.md#function-linearrelationlabel)

    - [`LinearRelation(v: parameters): ModTupRngElt → ModTupRngElt`](operation-element.md#function-linearrelation-modtuprngelt)

    - [`AllLinearRelations(q,p): SeqEnum, RngIntElt → Lat`](operation-element.md#function-alllinearrelations-seqenum-rngintelt)

    - [`IntegerRelation(q): SeqEnum → SeqEnum, FldReElt`](operation-element.md#function-integerrelation-seqenum)

    - [`IntegerRelation(q,N): SeqEnum, RngIntElt → SeqEnum, FldReElt`](operation-element.md#function-integerrelation-seqenum-rngintelt)

    - [`PowerRelation(r, k: parameters): FldReElt, RngIntElt → RngUPolElt`](operation-element.md#function-powerrelation-fldreelt-rngintelt)

    - [`PowerRelation(r, k: parameters): FldComElt, RngIntElt → RngUPolElt`](operation-element.md#function-powerrelation-fldcomelt-rngintelt)

    - [`MinimalPolynomial(r,d,N): FldReElt, RngIntElt, RngIntElt → RngUPolElt, FldReElt`](operation-element.md#function-minimalpolynomial-fldreelt-rngintelt-rngintelt)

    - [`MinimalPolynomial(r,d,N): FldComElt, RngIntElt, RngIntElt → RngUPolElt, FldReElt`](operation-element.md#function-minimalpolynomial-fldcomelt-rngintelt-rngintelt)

    - [`MinimalPolynomial(r,d): FldReElt, RngIntElt → RngUPolElt -> FldReElt`](operation-element.md#function-minimalpolynomial-fldreelt-rngintelt)

    - [`MinimalPolynomial(r,d): FldComElt, RngIntElt → RngUPolElt -> FldReElt`](operation-element.md#function-minimalpolynomial-fldcomelt-rngintelt)

    - [`Example: LLL Pol Fact`](operation-element.md#example-ex-b60a49)

- [Transcendental Functions](transcendental.md)

  - [Exponential, Logarithmic and Polylogarithmic Functions](transcendental.md#exponential-logarithmic-and-polylogarithmic-functions)

    - [`Exp(f): RngSerElt → RngSerElt`](transcendental.md#function-exp-rngserelt)

    - [`Exp(c): FldComElt → FldComElt`](transcendental.md#function-exp-fldcomelt)

    - [`Exp(r): FldReElt → FldReElt`](transcendental.md#function-exp-fldreelt)

    - [`Log(f): RngSerElt → RngSerElt`](transcendental.md#function-log-rngserelt)

    - [`Log(r): FldReElt → FldReElt`](transcendental.md#function-log-fldreelt)

    - [`Log(c): FldComElt → FldComElt`](transcendental.md#function-log-fldcomelt)

    - [`Log(b, r): FldReElt → FldReElt`](transcendental.md#function-log-fldreelt-2)

    - [`Dilog(s): FldComElt → FldComElt`](transcendental.md#function-dilog-fldcomelt)

    - [`Dilog(s): FldReElt → FldReElt`](transcendental.md#function-dilog-fldreelt)

    - [`Polylog(m, f): RngIntElt, RngSerElt → RngSerElt`](transcendental.md#function-polylog-rngintelt-rngserelt)

    - [`Polylog(m, s): RngIntElt, FldComElt → FldComElt`](transcendental.md#function-polylog-rngintelt-fldcomelt)

    - [`PolylogD(m, s): RngIntElt, FldComElt → FldComElt`](transcendental.md#function-polylogd-rngintelt-fldcomelt)

    - [`PolylogDold(m, s): RngIntElt, FldComElt → FldComElt`](transcendental.md#function-polylogdold-rngintelt-fldcomelt)

    - [`PolylogP(m, s): RngIntElt, FldComElt → FldComElt`](transcendental.md#function-polylogp-rngintelt-fldcomelt)

  - [Trigonometric Functions](transcendental.md#trigonometric-functions)

    - [`Sin(f): RngSerElt → RngSerElt`](transcendental.md#function-sin-rngserelt)

    - [`Sin(c): FldComElt → FldComElt`](transcendental.md#function-sin-fldcomelt)

    - [`Sin(r): FldReElt → FldReElt`](transcendental.md#function-sin-fldreelt)

    - [`Cos(f): RngSerElt → RngSerElt`](transcendental.md#function-cos-rngserelt)

    - [`Cos(c): FldComElt → FldComElt`](transcendental.md#function-cos-fldcomelt)

    - [`Cos(r): FldReElt → FldReElt`](transcendental.md#function-cos-fldreelt)

    - [`Sincos(f): RngSerElt → RngSerElt`](transcendental.md#function-sincos-rngserelt)

    - [`Sincos(s): FldReElt → FldReElt, FldReElt`](transcendental.md#function-sincos-fldreelt)

    - [`Sincos(s): FldComElt → FldComElt, FldComElt`](transcendental.md#function-sincos-fldcomelt)

    - [`Tan(f): RngSerElt → RngSerElt`](transcendental.md#function-tan-rngserelt)

    - [`Tan(c): FldComElt → FldComElt`](transcendental.md#function-tan-fldcomelt)

    - [`Tan(r): FldReElt → FldReElt`](transcendental.md#function-tan-fldreelt)

    - [`Cot(f): RngSerElt → RngSerElt`](transcendental.md#function-cot-rngserelt)

    - [`Cot(c): FldComElt → FldComElt`](transcendental.md#function-cot-fldcomelt)

    - [`Cot(r): FldReElt → FldReElt`](transcendental.md#function-cot-fldreelt)

    - [`Sec(f): RngSerElt → RngSerElt`](transcendental.md#function-sec-rngserelt)

    - [`Sec(c): FldComElt → FldComElt`](transcendental.md#function-sec-fldcomelt)

    - [`Sec(r): FldReElt → FldReElt`](transcendental.md#function-sec-fldreelt)

    - [`Cosec(f): RngSerElt → RngSerElt`](transcendental.md#function-cosec-rngserelt)

    - [`Cosec(c): FldComElt → FldComElt`](transcendental.md#function-cosec-fldcomelt)

    - [`Cosec(r): FldReElt → FldReElt`](transcendental.md#function-cosec-fldreelt)

  - [Inverse Trigonometric Functions](transcendental.md#inverse-trigonometric-functions)

    - [`Arcsin(f): RngSerElt → RngSerElt`](transcendental.md#function-arcsin-rngserelt)

    - [`Arcsin(r): FldReElt → FldReElt`](transcendental.md#function-arcsin-fldreelt)

    - [`Arcsin(r): FldComElt → FldComElt`](transcendental.md#function-arcsin-fldcomelt)

    - [`Arccos(f): RngSerElt → RngSerElt`](transcendental.md#function-arccos-rngserelt)

    - [`Arccos(r): FldReElt → FldReElt`](transcendental.md#function-arccos-fldreelt)

    - [`Arccos(r): FldComElt → FldComElt`](transcendental.md#function-arccos-fldcomelt)

    - [`Arctan(f): RngSerElt → RngSerElt`](transcendental.md#function-arctan-rngserelt)

    - [`Arctan(r): FldReElt → FldReElt`](transcendental.md#function-arctan-fldreelt)

    - [`Arctan(r): FldComElt → FldComElt`](transcendental.md#function-arctan-fldcomelt)

    - [`Arctan(x, y): FldReElt, FldReElt → FldReElt`](transcendental.md#function-arctan-fldreelt-fldreelt)

    - [`Arctan2(x, y): FldReElt, FldReElt → FldReElt`](transcendental.md#function-arctan2-fldreelt-fldreelt)

    - [`Arccot(r): FldReElt → FldReElt`](transcendental.md#function-arccot-fldreelt)

    - [`Arccot(r): FldComElt → FldComElt`](transcendental.md#function-arccot-fldcomelt)

    - [`Arcsec(r): FldReElt → FldReElt`](transcendental.md#function-arcsec-fldreelt)

    - [`Arcsec(r): FldComElt → FldComElt`](transcendental.md#function-arcsec-fldcomelt)

    - [`Arccosec(r): FldReElt → FldReElt`](transcendental.md#function-arccosec-fldreelt)

    - [`Arccosec(r): FldComElt → FldComElt`](transcendental.md#function-arccosec-fldcomelt)

  - [Hyperbolic Functions](transcendental.md#hyperbolic-functions)

    - [`Sinh(f): RngSerElt → RngSerElt`](transcendental.md#function-sinh-rngserelt)

    - [`Sinh(s): FldComElt → FldComElt`](transcendental.md#function-sinh-fldcomelt)

    - [`Sinh(r): FldReElt → FldReElt`](transcendental.md#function-sinh-fldreelt)

    - [`Cosh(f): RngSerElt → RngSerElt`](transcendental.md#function-cosh-rngserelt)

    - [`Cosh(r): FldReElt → FldReElt`](transcendental.md#function-cosh-fldreelt)

    - [`Cosh(r): FldComElt → FldComElt`](transcendental.md#function-cosh-fldcomelt)

    - [`Tanh(f): RngSerElt → RngSerElt`](transcendental.md#function-tanh-rngserelt)

    - [`Tanh(r): FldReElt → FldReElt`](transcendental.md#function-tanh-fldreelt)

    - [`Tanh(r): FldComElt → FldComElt`](transcendental.md#function-tanh-fldcomelt)

    - [`Coth(r): FldReElt → FldReElt`](transcendental.md#function-coth-fldreelt)

    - [`Coth(r): FldComElt → FldComElt`](transcendental.md#function-coth-fldcomelt)

    - [`Sech(r): FldReElt → FldReElt`](transcendental.md#function-sech-fldreelt)

    - [`Sech(r): FldComElt → FldComElt`](transcendental.md#function-sech-fldcomelt)

    - [`Cosech(r): FldReElt → FldReElt`](transcendental.md#function-cosech-fldreelt)

    - [`Cosech(r): FldComElt → FldComElt`](transcendental.md#function-cosech-fldcomelt)

  - [Inverse Hyperbolic Functions](transcendental.md#inverse-hyperbolic-functions)

    - [`Argsinh(f): RngSerElt → RngSerElt`](transcendental.md#function-argsinh-rngserelt)

    - [`Argsinh(r): FldReElt → FldReElt`](transcendental.md#function-argsinh-fldreelt)

    - [`Argsinh(r): FldComElt → FldComElt`](transcendental.md#function-argsinh-fldcomelt)

    - [`Argcosh(f): RngSerElt → RngSerElt`](transcendental.md#function-argcosh-rngserelt)

    - [`Argcosh(r): FldReElt → FldReElt`](transcendental.md#function-argcosh-fldreelt)

    - [`Argcosh(r): FldComElt → FldComElt`](transcendental.md#function-argcosh-fldcomelt)

    - [`Argtanh(f): RngSerElt → RngSerElt`](transcendental.md#function-argtanh-rngserelt)

    - [`Argtanh(s): FldReElt → FldReElt`](transcendental.md#function-argtanh-fldreelt)

    - [`Argtanh(r): FldComElt → FldComElt`](transcendental.md#function-argtanh-fldcomelt)

    - [`Argsech(s): FldReElt → FldReElt`](transcendental.md#function-argsech-fldreelt)

    - [`Argsech(r): FldComElt → FldComElt`](transcendental.md#function-argsech-fldcomelt)

    - [`Argcosech(s): FldReElt → FldReElt`](transcendental.md#function-argcosech-fldreelt)

    - [`Argcosech(r): FldComElt → FldComElt`](transcendental.md#function-argcosech-fldcomelt)

    - [`Argcoth(s): FldReElt → FldReElt`](transcendental.md#function-argcoth-fldreelt)

    - [`Argcoth(r): FldComElt → FldComElt`](transcendental.md#function-argcoth-fldcomelt)

- [Elliptic and Modular Functions](elliptic-modular.md)

  - [Eisenstein Series](elliptic-modular.md#eisenstein-series)

    - [`Eisenstein(k, z): RngIntElt, RngSerElt → RngSerElt`](elliptic-modular.md#function-eisenstein-rngintelt-rngserelt)

    - [`Eisenstein(k, t): RngIntElt, FldComElt → FldComElt`](elliptic-modular.md#function-eisenstein-rngintelt-fldcomelt)

    - [`Eisenstein(k, L): RngIntElt, SeqEnum → FldComElt`](elliptic-modular.md#function-eisenstein-rngintelt-seqenum)

    - [`Eisenstein(k, F): RngIntElt, QuadBinElt → RngSerElt`](elliptic-modular.md#function-eisenstein-rngintelt-quadbinelt)

    - [`Example: Eisenstein`](elliptic-modular.md#example-ex-db3e10)

  - [Weierstrass Series](elliptic-modular.md#weierstrass-series)

    - [`WeierstrassSeries(z, q): RngSerElt, RngSerElt → RngSerElt`](elliptic-modular.md#function-weierstrassseries-rngserelt-rngserelt)

    - [`WeierstrassSeries(z, t): RngSerElt, FldComElt → RngSerElt`](elliptic-modular.md#function-weierstrassseries-rngserelt-fldcomelt)

    - [`WeierstrassSeries(z, L): RngSerElt, SeqEnum → RngSerElt`](elliptic-modular.md#function-weierstrassseries-rngserelt-seqenum)

    - [`WeierstrassSeries(z, F): RngSerElt, QuadBinElt → RngSerElt`](elliptic-modular.md#function-weierstrassseries-rngserelt-quadbinelt)

  - [The Jacobi $\theta$ and Dedekind $\eta$-functions](elliptic-modular.md#the-jacobi-theta-and-dedekind-eta-functions)

    - [`JacobiTheta(q, z): FldReElt, RngSerElt[FldRe] → RngSerElt`](elliptic-modular.md#function-jacobitheta-fldreelt-rngserelt-fldre)

    - [`JacobiTheta(q, z): FldComElt, RngSerElt[FldCom] → RngSerElt`](elliptic-modular.md#function-jacobitheta-fldcomelt-rngserelt-fldcom)

    - [`JacobiTheta(q, z): FldReElt, FldReElt → FldReElt`](elliptic-modular.md#function-jacobitheta-fldreelt-fldreelt)

    - [`JacobiTheta(q, z): FldComElt, FldComElt → FldComElt`](elliptic-modular.md#function-jacobitheta-fldcomelt-fldcomelt)

    - [`JacobiThetaNullK(q, k): FldReElt, RngIntElt → FldReElt`](elliptic-modular.md#function-jacobithetanullk-fldreelt-rngintelt)

    - [`DedekindEta(z): RngSerElt → RngSerElt`](elliptic-modular.md#function-dedekindeta-rngserelt)

    - [`DedekindEta(s): FldComElt → FldComElt`](elliptic-modular.md#function-dedekindeta-fldcomelt)

  - [The $j$-Invariant and the Discriminant](elliptic-modular.md#the-j-invariant-and-the-discriminant)

    - [`jInvariant(q): RngSerElt → RngSerElt`](elliptic-modular.md#function-jinvariant-rngserelt)

    - [`jInvariant(s): FldComElt → FldComElt`](elliptic-modular.md#function-jinvariant-fldcomelt)

    - [`jInvariant(L): SeqEnum → FldComElt`](elliptic-modular.md#function-jinvariant-seqenum)

    - [`jInvariant(F): QuadBinElt → FldComElt`](elliptic-modular.md#function-jinvariant-quadbinelt)

    - [`Delta(z): RngSerElt → RngSerElt`](elliptic-modular.md#function-delta-rngserelt)

    - [`Delta(t): FldComElt → FldComElt`](elliptic-modular.md#function-delta-fldcomelt)

    - [`Delta(L): SeqEnum → FldComElt`](elliptic-modular.md#function-delta-seqenum)

  - [Weber’s Functions](elliptic-modular.md#weber-s-functions)

    - [`WeberF(s): FldComElt → FldComElt`](elliptic-modular.md#function-weberf-fldcomelt)

    - [`WeberF2(g): RngSerElt → RngSerElt`](elliptic-modular.md#function-weberf2-rngserelt)

    - [`WeberF1(s): FldComElt → FldComElt`](elliptic-modular.md#function-weberf1-fldcomelt)

    - [`WeberF2(s): FldComElt → FldComElt`](elliptic-modular.md#function-weberf2-fldcomelt)

    - [`Example: Eisenstein`](elliptic-modular.md#example-ex-3718fd)

- [Theta Functions](theta.md)

  - [`Theta(char, z, tau): Mtrx, Mtrx, Mtrx → FldComElt`](theta.md#function-theta-mtrx-mtrx-mtrx)

  - [`Theta(char, z, A): Mtrx, Mtrx, AnHcJac → FldComElt`](theta.md#function-theta-mtrx-mtrx-anhcjac)

- [Gamma, Bessel and Associated Functions](gamma-bessel.md)

  - [`Gamma(f): RngSerElt → RngSerElt`](gamma-bessel.md#function-gamma-rngserelt)

  - [`Gamma(r): FldReElt → FldReElt`](gamma-bessel.md#function-gamma-fldreelt)

  - [`Gamma(r): FldComElt → FldComElt`](gamma-bessel.md#function-gamma-fldcomelt)

  - [`Gamma(s, t): FldReElt, FldReElt → FldReElt`](gamma-bessel.md#function-gamma-fldreelt-fldreelt)

  - [`GammaD(s): FldReElt → FldReElt`](gamma-bessel.md#function-gammad-fldreelt)

  - [`LogGamma(f): RngSerElt → RngSerElt`](gamma-bessel.md#function-loggamma-rngserelt)

  - [`LogGamma(r): FldReElt → FldReElt`](gamma-bessel.md#function-loggamma-fldreelt)

  - [`LogGamma(r): FldComElt → FldComElt`](gamma-bessel.md#function-loggamma-fldcomelt)

  - [`LogDerivative(s): FldReElt → FldReElt`](gamma-bessel.md#function-logderivative-fldreelt)

  - [`LogDerivative(s): FldComElt → FldComElt`](gamma-bessel.md#function-logderivative-fldcomelt)

  - [`Psi(s): FldReElt → FldReElt`](gamma-bessel.md#function-psi-fldreelt)

  - [`Psi(s): FldComElt → FldComElt`](gamma-bessel.md#function-psi-fldcomelt)

  - [`BesselFunction(n, r): RngIntElt, FldReElt → FldReElt`](gamma-bessel.md#function-besselfunction-rngintelt-fldreelt)

  - [`BesselFunctionSecondKind(n, r): RngIntElt, FldReElt → FldReElt`](gamma-bessel.md#function-besselfunctionsecondkind-rngintelt-fldreelt)

  - [`JBessel(n, s): RngIntElt, FldReElt → FldReElt`](gamma-bessel.md#function-jbessel-rngintelt-fldreelt)

  - [`JBessel(n, s): FldReElt, FldReElt → FldReElt`](gamma-bessel.md#function-jbessel-fldreelt-fldreelt)

  - [`KBessel(n, s): FldReElt, FldReElt → FldReElt`](gamma-bessel.md#function-kbessel-fldreelt-fldreelt)

  - [`KBessel(n, s): FldComElt, FldReElt → FldComElt`](gamma-bessel.md#function-kbessel-fldcomelt-fldreelt)

  - [`KBessel2(n, s): FldReElt, FldReElt → FldReElt`](gamma-bessel.md#function-kbessel2-fldreelt-fldreelt)

  - [`KBessel2(n, s): FldComElt, FldReElt → FldComElt`](gamma-bessel.md#function-kbessel2-fldcomelt-fldreelt)

- [The Hypergeometric Function](hypergeometric.md)

  - [`HypergeometricSeries(a, b, c, z): RngElt, RngElt, RngElt, RngElt → RngElt`](hypergeometric.md#function-hypergeometricseries-rngelt-rngelt-rngelt-rngelt)

  - [`HypergeometricU(a, b, s): FldReElt, FldReElt, FldReElt → FldReElt`](hypergeometric.md#function-hypergeometricu-fldreelt-fldreelt-fldreelt)

- [Other Special Functions](special.md)

  - [`ArithmeticGeometricMean(x, y): RngSerElt, RngSerElt → RngSerElt`](special.md#function-arithmeticgeometricmean-rngserelt-rngserelt)

  - [`AGM(f, g): RngSerElt, RngSerElt → RngSerElt`](special.md#function-agm-rngserelt-rngserelt)

  - [`ArithmeticGeometricMean(x, y): FldReElt, FldReElt → FldReElt`](special.md#function-arithmeticgeometricmean-fldreelt-fldreelt)

  - [`AGM(x, y): FldReElt, FldReElt → FldReElt`](special.md#function-agm-fldreelt-fldreelt)

  - [`BernoulliNumber(n): RngIntElt → FldRatElt`](special.md#function-bernoullinumber-rngintelt)

  - [`BernoulliApproximation(n): RngIntElt → FldReElt`](special.md#function-bernoulliapproximation-rngintelt)

  - [`DawsonIntegral(r): FldReElt → FldReElt`](special.md#function-dawsonintegral-fldreelt)

  - [`ErrorFunction(r): FldReElt → FldReElt`](special.md#function-errorfunction-fldreelt)

  - [`Erf(r): FldReElt → FldReElt`](special.md#function-erf-fldreelt)

  - [`ComplementaryErrorFunction(r): FldReElt → FldReElt`](special.md#function-complementaryerrorfunction-fldreelt)

  - [`Erfc(r): FldReElt → FldReElt`](special.md#function-erfc-fldreelt)

  - [`ExponentialIntegral(r): FldReElt → FldReElt`](special.md#function-exponentialintegral-fldreelt)

  - [`ExponentialIntegralE1(r): FldReElt → FldReElt`](special.md#function-exponentialintegrale1-fldreelt)

  - [`LogIntegral(r): FldReElt → FldReElt`](special.md#function-logintegral-fldreelt)

  - [`ZetaFunction(s): FldReElt → FldReElt`](special.md#function-zetafunction-fldreelt)

  - [`ZetaFunction(R, n): FldRe, RngIntElt → FldReElt`](special.md#function-zetafunction-fldre-rngintelt)

- [Summation of Infinite Series](infinite-summation.md)

  - [`InfiniteSum(m, i): Map, RngIntElt → FldReElt`](infinite-summation.md#function-infinitesum-map-rngintelt)

  - [`PositiveSum(m, i): Map, RngIntElt → FldReElt`](infinite-summation.md#function-positivesum-map-rngintelt)

  - [`AlternatingSum(m, i): Map, RngIntElt → FldReElt`](infinite-summation.md#function-alternatingsum-map-rngintelt)

- [Numerical Integration](integration1.md)

  - [Polynomial Interpolation](integration1.md#polynomial-interpolation)

    - [`Interpolation(P, V, t): [FldReElt], [FldReElt], FldReElt → FldReElt, FldReElt`](integration1.md#function-interpolation-fldreelt-fldreelt-fldreelt)

  - [Discrete Fourier Transform](integration1.md#discrete-fourier-transform)

    - [`DiscreteFourierTransform(E): SeqEnum[FldComElt] → SeqEnum[FldComElt]`](integration1.md#function-discretefouriertransform-seqenum-fldcomelt)

  - [Integration of Complex Functions](integration1.md#integration-of-complex-functions)

    - [Gaussian Quadratures](integration1.md#gaussian-quadratures)

      - [`GaussLegendreIntegrationPoints(N,D): RngIntElt, RngIntElt → SeqEnum[FldReElt], SeqEnum[FldReElt]`](integration1.md#function-gausslegendreintegrationpoints-rngintelt-rngintelt)

      - [`GaussJacobiIntegrationPoints(N,D,a,b): RngIntElt, RngIntElt, RngReSubElt, RngReSubElt) → SeqEnum, SeqEnum`](integration1.md#function-gaussjacobiintegrationpoints-rngintelt-rngintelt-rngresubelt-rngresubelt)

    - [Clenshaw–Curtis Quadrature](integration1.md#clenshaw-curtis-quadrature)

      - [`ClenshawCurtisIntegrationPoints(N,D): RngIntElt, RngIntElt → SeqEnum[FldReElt], SeqEnum[FldReElt]`](integration1.md#function-clenshawcurtisintegrationpoints-rngintelt-rngintelt)

    - [Tanh–Sinh Quadrature](integration1.md#tanh-sinh-quadrature)

      - [`TanhSinhIntegrationPoints(N,h): RngIntElt, FldReElt → SeqEnum[FldReElt], SeqEnum[FldReElt], SeqEnum[FldReElt]`](integration1.md#function-tanhsinhintegrationpoints-rngintelt-fldreelt)

      - [`Example: Num Int Ex 1`](integration1.md#example-ex-fd7870)

      - [`Example: Num Int Ex 2`](integration1.md#example-ex-9327e9)

      - [`Example: Num Int Ex 3`](integration1.md#example-ex-1222c3)

  - [Romberg-Type Integration](integration1.md#romberg-type-integration)

    - [`RombergQuadrature(f, a, b: parameters): Program, FldReElt, FldReElt → FldReElt`](integration1.md#function-rombergquadrature-program-fldreelt-fldreelt)

    - [`SimpsonQuadrature(f, a, b, n): Program, FldReElt, FldReElt, RngIntElt → FldReElt`](integration1.md#function-simpsonquadrature-program-fldreelt-fldreelt-rngintelt)

    - [`TrapezoidalQuadrature(f, a, b, n): Program, FldReElt, FldReElt, RngIntElt → FldReElt`](integration1.md#function-trapezoidalquadrature-program-fldreelt-fldreelt-rngintelt)

  - [Numerical Derivatives](integration1.md#numerical-derivatives)

    - [`NumericalDerivative(f, n, z): UserProgram, RngIntElt, FldComElt → FldComElt`](integration1.md#function-numericalderivative-userprogram-rngintelt-fldcomelt)

    - [`Example: Numerical Derivative`](integration1.md#example-ex-c82809)