.. _fexpr-builtin: **fexpr_builtin.h** -- builtin symbols =============================================================================== This module defines symbol names with a predefined meaning for use in symbolic expressions. These symbols will eventually all support LaTeX rendering as well as symbolic and numerical evaluation (where applicable). By convention, all builtin symbol names are at least two characters long and start with an uppercase letter. Single-letter symbol names and symbol names beginning with a lowercase letter are reserved for variables. For any builtin symbol name ``Symbol``, the header file ``fexpr_builtin.h`` defines a C constant ``FEXPR_Symbol`` as an index to a builtin symbol table. The symbol will be documented as ``Symbol`` below. C helper functions ------------------------------------------------------------------------ .. function:: slong fexpr_builtin_lookup(const char * s) Returns the internal index used to encode the builtin symbol with name *s* in expressions. If *s* is not the name of a builtin symbol, returns -1. .. function:: const char * fexpr_builtin_name(slong n) Returns a read-only pointer for a string giving the name of the builtin symbol with index *n*. .. function:: slong fexpr_builtin_length(void) Returns the number of builtin symbols. Variables and iteration ------------------------------------------------------------------------ Expressions involving the following symbols have a special role in binding variables. .. macro:: For Generator expression. This is a syntactical construct which does not represent a mathematical object on its own. In general, ``For(x, ...)`` defines the symbol ``x`` as a locally bound variable in the scope of the parent expression. The following arguments ``...`` specify an evaluation range, set or point. Their interpretation depends on the parent operator. The following cases are possible. Case 1: ``For(x, S)`` specifies iteration or comprehension for ``x`` ranging over the values of the set ``S``. This interpretation is used in operators that aggregate values over a set. The ``For`` expression may be followed by a filter predicate ``P(x)`` restricting the range to a subset of ``S``. Examples: ``Set(f(x), For(x, S))`` denotes `\{f(x) : x \in S\}`. ``Set(f(x), For(x, S), P(x))`` denotes `\{f(x) : x \in S \operatorname{and} P(x)\}`. ``Sum(f(x), For(x, S))`` denotes `\sum_{x \in S} f(x)`. ``Sum(f(x), For(x, S), P(x))`` denotes `\sum_{x \in S, \, P(x)} f(x)`. Case 2: ``For(x, a, b)`` specifies that ``x`` ranges between the endpoints ``a`` and ``b`` in the context of ``Sum``, ``Product``, ``Integral``, and similar operators. Examples: ``Sum(f(n), For(n, a, b))`` denotes `\sum_{n=a}^b f(n)`. The iteration is empty if `b < a`. ``Integral(f(x), For(x, a, b))`` denotes `\int_a^b f(x) dx`, where the integral follows a straight-line path from *a* to *b*. Swapping *a* and *b* negates the value. Case 3: ``For(x, a)`` specifies that ``x`` approaches the point ``a`` in the context of ``Limit``-type operator, or differentiation with respect to ``x`` at the point ``a`` in the context of a ``Derivative``-type operator. Examples: ``Derivative(f(x), For(x, a))`` denotes `f'(a)`. ``Limit(f(x), For(x, a))`` denotes `\lim_{x \to a} f(x)`. Case 4: ``For(x, a, n)`` specifies differentiation with respect to ``x`` at the point ``a`` to order ``n`` in the context of a ``Derivative``-type operator. Examples: ``Derivative(f(x), For(x, a, n))`` denotes `f^{(n)}(a)`. .. macro:: Where ``Where(f(x), Def(x, a))`` defines the symbol ``x`` as an alias for the expression ``a`` and evaluates the expression ``f(x)`` with this bound value of ``x``. This is equivalent to ``f(a)``. This may be rendered as `f(x) \; \operatorname{where} x = a`. ``Where(f(x), Def(f(t), a))`` defines the symbol ``f`` as a function mapping the dummy variable ``t`` to ``a``. ``Where(Add(a, b), Def(Tuple(a, b), T))`` is a destructuring assignment. .. macro:: Def Definition expression. This is a syntactical construct which does not represent a mathematical object on its own. The ``Def`` expression is used only within a ``Where``-expression; see that documentation of that symbol for more examples. ``Def(x, a)`` defines the symbol ``x`` as an alias for the expression ``a``. ``Def(f(x, y, z), a)`` defines the symbol ``f`` as a function of three variables. The dummy variables ``x``, ``y`` and ``z`` may appear within the expression ``a``. .. macro:: Fun ``Fun(x, expr)`` defines an anonymous univariate function mapping the symbol ``x`` to the expression ``expr``. The symbol ``x`` becomes locally bound within this ``Fun`` expression. .. macro:: Step .. macro:: Repeat Booleans and logic ------------------------------------------------------------------------ .. macro:: Equal ``Equal(a, b)``, signifying `a = b`, is ``True`` if ``a`` and ``b`` represent the same object, and ``False`` otherwise. This operator can be called with any number of arguments, in which case it evaluates whether all arguments are equal. .. macro:: NotEqual ``NotEqual(a, b)``, signifying `a \ne b`, is equivalent to ``Not(Equal(a, b))``. .. macro:: Same ``Same(a, b)`` gives ``a`` (or equivalently ``b``) if ``a`` and ``b`` represent the same object, and ``Undefined`` otherwise. This can be used to assert or emphasize that two expressions represent the same value within a formula. This operator can be called with any number of arguments, in which case it asserts that all arguments are equal. .. macro:: True ``True`` is a logical constant. .. macro:: False ``False`` is a logical constant. .. macro:: Not ``Not(x)`` is the logical negation of ``x``. .. macro:: And ``And(x, y)`` is the logical AND of ``x`` and ``y``. This function can be called with any number of arguments. .. macro:: Or ``Or(x, y)`` is the logical OR of ``x`` and ``y``. This function can be called with any number of arguments. .. macro:: Equivalent ``Equivalent(x, y)`` denotes the logical equivalence `x \Leftrightarrow y`. Semantically, this is the same as ``Equal`` called with logical arguments. .. macro:: Implies ``Implies(x, y)`` denotes the logical implication `x \implies y`. .. macro:: Exists Existence quantifier. ``Exists(f(x), For(x, S))`` denotes `f(x) \;\text{ for some } x \in S`. ``Exists(f(x), For(x, S), P(x))`` denotes `f(x) \;\text{ for some } x \in S \text{ with } P(x)`. .. macro:: All Universal quantifier. ``All(f(x), For(x, S))`` denotes `f(x) \;\text{ for all } x \in S`. ``All(f(x), For(x, S), P(x))`` denotes `f(x) \;\text{ for all } x \in S \text{ with } P(x)`. .. macro:: Cases ``Cases(Case(f(x), P(x)), Case(g(x), Otherwise))`` denotes: .. math:: \begin{cases} f(x), & P(x)\\g(x), & \text{otherwise}\\ \end{cases} ``Cases(Case(f(x), P(x)), Case(g(x), Q(x)), Case(h(x), Otherwise))`` denotes: .. math:: \begin{cases} f(x), & P(x)\\g(x), & Q(x)\\h(x), & \text{otherwise}\\ \end{cases} If both `P(x)` and `Q(x)` are true simultaneously, no ordering is implied; it is assumed that `f(x)` and `g(x)` give the same value for any such `x`. More generally, this operator can be called with any number of case distinctions. If the *Otherwise* case is omitted, the result is undefined if neither predicate is true. .. macro:: Case See ``Cases``. .. macro:: Otherwise See ``Cases``. Tuples, lists and sets ------------------------------------------------------------------------ .. macro:: Tuple .. macro:: List .. macro:: Set .. macro:: Item .. macro:: Element .. macro:: NotElement .. macro:: EqualAndElement .. macro:: Length .. macro:: Cardinality .. macro:: Concatenation .. macro:: Union .. macro:: Intersection .. macro:: SetMinus .. macro:: Subset .. macro:: SubsetEqual .. macro:: CartesianProduct .. macro:: CartesianPower .. macro:: Subsets ``Subsets(S)`` is the power set `\mathscr{P}(S)` comprising all subsets of the set ``S``. .. macro:: Sets ``Sets`` is the class `\operatorname{Sets}` of all sets. .. macro:: Tuples ``Tuples`` is the class of all tuples. ``Tuples(S)`` is the set of all tuples with elements in the set ``S``. ``Tuples(S, n)`` is the set of all length-``n`` tuples with elements in the set ``S``. Numbers and arithmetic ------------------------------------------------------------------------ Undefined ........................................................................ .. macro:: Undefined ``Undefined`` is the special value `\mathfrak{u}` (undefined). Particular numbers ........................................................................ .. macro:: Pi ``Pi`` is the constant `\pi`. .. macro:: NumberI ``NumberI`` is the imaginary unit `i`. The verbose name leaves ``i`` and ``I`` to be used as a variable names. .. macro:: NumberE ``NumberE`` is the base of the natural logarithm `e`. The verbose name leaves ``e`` and ``E`` to be used as a variable names. .. macro:: GoldenRatio ``GoldenRatio`` is the golden ratio `\varphi`. .. macro:: Euler ``Euler`` is Euler's constant `\gamma`. .. macro:: CatalanConstant ``CatalanConstant`` is Catalan's constant `G`. .. macro:: KhinchinConstant ``KhinchinConstant`` is Khinchin's constant `K`. .. macro:: GlaisherConstant ``GlaisherConstant`` is Glaisher's constant `A`. .. macro:: RootOfUnity ``RootOfUnity(n)`` is the principal complex *n*-th root of unity `\zeta_n = e^{2 \pi i / n}`. ``RootOfUnity(n, k)`` is the complex *n*-th root of unity `\zeta_n^k`. Number constructors ........................................................................ Remark: the rational number with numerator *p* and denominator *q* can be constructed as ``Div(p, q)``. .. macro:: Decimal ``Decimal(str)`` gives the rational number specified by the string *str* in ordinary decimal floating-point notation (for example ``-3.25e-725``). .. macro:: AlgebraicNumberSerialized .. macro:: PolynomialRootIndexed .. macro:: PolynomialRootNearest .. macro:: Enclosure .. macro:: Approximation .. macro:: Guess .. macro:: Unknown Arithmetic operations ........................................................................ .. macro:: Pos .. macro:: Neg .. macro:: Add .. macro:: Sub .. macro:: Mul .. macro:: Div .. macro:: Pow .. macro:: Sqrt .. macro:: Root Inequalities ........................................................................ .. macro:: Less .. macro:: LessEqual .. macro:: Greater .. macro:: GreaterEqual .. macro:: EqualNearestDecimal Sets of numbers ........................................................................ .. macro:: NN ``NN`` is the set of natural numbers (including 0), `\mathbb{N}`. .. macro:: ZZ ``ZZ`` is the set of integers, `\mathbb{Z}`. .. macro:: QQ ``QQ`` is the set of rational numbers, `\mathbb{Q}`. .. macro:: RR ``RR`` is the set of real numbers, `\mathbb{R}`. .. macro:: CC ``CC`` is the set of complex numbers, `\mathbb{C}`. .. macro:: Primes ``Primes`` is the set of positive prime numbers, `\mathbb{P}` .. macro:: IntegersGreaterEqual ``IntegersGreaterEqual(x)``, given an extended real number *x*, gives the set `\mathbb{Z}_{\ge x}` of integers greater than or equal to *x*. .. macro:: IntegersLessEqual ``IntegersLessEqual(x)``, given an extended real number *x*, gives the set `\mathbb{Z}_{\le x}` of integers less than or equal to *x*. .. macro:: Range ``Range(a, b)``, given integers *a* and *b*, gives the set `\{a, a+1, \ldots, b\}` of integers between *a* and *b*. This is the empty set if *a* is greater than *b*. .. macro:: AlgebraicNumbers The set of complex algebraic numbers `\overline{\mathbb{Q}}`. .. macro:: RealAlgebraicNumbers The set of real algebraic numbers `\overline{\mathbb{Q}}_{\mathbb{R}}`. .. macro:: Interval ``Interval(a, b)``, given extended real numbers *a* and *b*, gives the closed interval `[a, b]`. .. macro:: OpenInterval ``OpenInterval(a, b)``, given extended real numbers *a* and *b*, gives the open interval `(a, b)`. .. macro:: ClosedOpenInterval ``ClosedOpenInterval(a, b)``, given extended real numbers *a* and *b*, gives the closed-open interval `[a, b)`. .. macro:: OpenClosedInterval ``OpenClosedInterval(a, b)``, given extended real numbers *a* and *b*, gives the closed-open interval `(a, b]`. .. macro:: RealBall ``RealBall(m, r)``, given a real number *m* and an extended real number *r*, gives the the closed real ball `[m \pm r]` with center *m* and radius *r*. .. macro:: OpenRealBall ``OpenRealBall(m, r)``, given a real number *m* and an extended real number *r*, gives the the open real ball `(m \pm r)` with center *m* and radius *r*. .. macro:: OpenComplexDisk ``OpenComplexDisk(m, r)``, given a complex number *m* and an extended real number *r*, gives the open complex disk `D(m, r)` with center *m* and radius *r*. .. macro:: ClosedComplexDisk ``ClosedComplexDisk(m, r)``, given a complex number *m* and a real number *r*, gives the closed complex disk `\overline{D}(m, r)` with center *m* and radius *r*. .. macro:: UpperHalfPlane ``UpperHalfPlane`` is the set `\mathbb{H}` of complex numbers with positive imaginary part. .. macro:: UnitCircle .. macro:: BernsteinEllipse .. macro:: Lattice Infinities and extended numbers ........................................................................ .. macro:: Infinity ``Infinity`` is the positive signed infinity `\infty`. .. macro:: UnsignedInfinity ``UnsignedInfinity`` is the unsigned infinity `\tilde \infty`. .. macro:: RealSignedInfinities ``RealSignedInfinities`` is the set of real signed infinities `\{+\infty, -\infty\}`. .. macro:: ComplexSignedInfinities ``ComplexSignedInfinities`` is the set of complex signed infinities `\{e^{i \theta} \cdot \infty : \theta \in \mathbb{R}\}`. .. macro:: RealInfinities ``RealInfinities`` is the set of real infinities (signed and unsigned) `\{+\infty, -\infty\} \cup \{\tilde \infty\}`. .. macro:: ComplexInfinities ``ComplexInfinities`` is the set of complex infinities (signed and unsigned) `\{e^{i \theta} \cdot \infty : \theta \in \mathbb{R}\} \cup \{\tilde \infty\}`. .. macro:: ExtendedRealNumbers ``ExtendedRealNumbers`` is the set of extended real numbers `\mathbb{R} \cup \{+\infty, -\infty\}`. .. macro:: ProjectiveRealNumbers ``ProjectiveRealNumbers`` is the set of projectively extended real numbers `\mathbb{R} \cup \{\tilde \infty\}`. .. macro:: SignExtendedComplexNumbers ``SignExtendedComplexNumbers`` is the set of complex numbers extended with signed infinities `\mathbb{C} \cup \{e^{i \theta} \cdot \infty : \theta \in \mathbb{R}\}`. .. macro:: ProjectiveComplexNumbers ``ProjectiveComplexNumbers`` is the set of projectively extended complex numbers (also known as the Riemann sphere) `\mathbb{C} \cup \{\tilde \infty\}`. .. macro:: RealSingularityClosure ``RealSingularityClosure`` is the Calcium singularity closure for real functions, encompassing real numbers, signed infinities, unsigned infinity, and *undefined* (u). This set is defined as `\mathbb{R}_{\text{Sing}} = \mathbb{R} \cup \{+\infty, -\infty\} \cup \{\tilde \infty\} \cup \{ \mathfrak{u} \}`. .. macro:: ComplexSingularityClosure ``ComplexSingularityClosure`` is the Calcium singularity closure for complex functions, encompassing complex numbers, signed infinities, unsigned infinity, and *undefined* (u). This set is defined as `\mathbb{C}_{\text{Sing}} = \mathbb{C} \cup \{e^{i \theta} \cdot \infty : \theta \in \mathbb{R}\} \cup \{\tilde \infty\} \cup \{ \mathfrak{u} \}`. Operators and calculus ------------------------------------------------------------------------ Sums and products ........................................................................ .. macro:: Sum .. macro:: Product .. macro:: PrimeSum .. macro:: PrimeProduct .. macro:: DivisorSum .. macro:: DivisorProduct Solutions and zeros ........................................................................ .. macro:: Zeros .. macro:: UniqueZero .. macro:: Solutions .. macro:: UniqueSolution Extreme values ........................................................................ .. macro:: Supremum .. macro:: Infimum .. macro:: Minimum .. macro:: Maximum .. macro:: ArgMin .. macro:: ArgMax .. macro:: ArgMinUnique .. macro:: ArgMaxUnique Limits ........................................................................ .. macro:: Limit .. macro:: SequenceLimit .. macro:: RealLimit .. macro:: LeftLimit .. macro:: RightLimit .. macro:: ComplexLimit .. macro:: MeromorphicLimit .. macro:: SequenceLimitInferior .. macro:: SequenceLimitSuperior .. macro:: AsymptoticTo Derivatives ........................................................................ .. macro:: Derivative .. macro:: RealDerivative .. macro:: ComplexDerivative .. macro:: ComplexBranchDerivative .. macro:: MeromorphicDerivative Integrals ........................................................................ .. macro:: Integral Complex analysis ........................................................................ .. macro:: Path .. macro:: CurvePath .. macro:: Poles .. macro:: IsHolomorphicOn .. macro:: IsMeromorphicOn .. macro:: Residue .. macro:: ComplexZeroMultiplicity .. macro:: AnalyticContinuation Matrices and linear algebra ------------------------------------------------------------------------ .. macro:: Matrix .. macro:: Row .. macro:: Column .. macro:: RowMatrix .. macro:: ColumnMatrix .. macro:: DiagonalMatrix .. macro:: Matrix2x2 .. macro:: ZeroMatrix .. macro:: IdentityMatrix .. macro:: Det .. macro:: Spectrum .. macro:: SingularValues .. macro:: Matrices .. macro:: SL2Z .. macro:: PSL2Z .. macro:: SpecialLinearGroup .. macro:: GeneralLinearGroup .. macro:: HilbertMatrix Polynomials, series and rings ------------------------------------------------------------------------ .. macro:: Pol .. macro:: Ser .. macro:: Polynomial .. macro:: Coefficient .. macro:: PolynomialDegree .. macro:: Polynomials .. macro:: PolynomialFractions .. macro:: FormalPowerSeries .. macro:: FormalLaurentSeries .. macro:: FormalPuiseuxSeries .. macro:: Zero .. macro:: One .. macro:: Characteristic .. macro:: Rings .. macro:: CommutativeRings .. macro:: Fields .. macro:: QuotientRing .. macro:: FiniteField .. macro:: EqualQSeriesEllipsis .. macro:: IndefiniteIntegralEqual .. macro:: QSeriesCoefficient .. macro:: Call .. macro:: CallIndeterminate Special functions ------------------------------------------------------------------------ Number parts and step functions ........................................................................ .. macro:: Abs .. macro:: Sign .. macro:: Re .. macro:: Im .. macro:: Arg .. macro:: Conjugate .. macro:: Csgn .. macro:: RealAbs .. macro:: Max .. macro:: Min .. macro:: Floor .. macro:: Ceil .. macro:: KroneckerDelta Primes and divisibility ........................................................................ .. macro:: IsOdd .. macro:: IsEven .. macro:: CongruentMod .. macro:: Divides .. macro:: Mod .. macro:: GCD .. macro:: LCM .. macro:: XGCD .. macro:: IsPrime .. macro:: Prime .. macro:: PrimePi .. macro:: DivisorSigma .. macro:: MoebiusMu .. macro:: EulerPhi .. macro:: DiscreteLog .. macro:: LegendreSymbol .. macro:: JacobiSymbol .. macro:: KroneckerSymbol .. macro:: SquaresR .. macro:: LiouvilleLambda Elementary functions ........................................................................ .. macro:: Exp .. macro:: Log .. macro:: Sin .. macro:: Cos .. macro:: Tan .. macro:: Cot .. macro:: Sec .. macro:: Csc .. macro:: Sinh .. macro:: Cosh .. macro:: Tanh .. macro:: Coth .. macro:: Sech .. macro:: Csch .. macro:: Asin .. macro:: Acos .. macro:: Atan .. macro:: Acot .. macro:: Asec .. macro:: Acsc .. macro:: Asinh .. macro:: Acosh .. macro:: Atanh .. macro:: Acoth .. macro:: Asech .. macro:: Acsch .. macro:: Atan2 .. macro:: Sinc .. macro:: LambertW Combinatorial functions ........................................................................ .. macro:: SloaneA .. macro:: SymmetricPolynomial .. macro:: Cyclotomic .. macro:: Fibonacci .. macro:: BernoulliB .. macro:: BernoulliPolynomial .. macro:: StirlingCycle .. macro:: StirlingS1 .. macro:: StirlingS2 .. macro:: EulerE .. macro:: EulerPolynomial .. macro:: BellNumber .. macro:: PartitionsP .. macro:: LandauG Gamma function and factorials ........................................................................ .. macro:: Factorial .. macro:: Binomial .. macro:: Gamma .. macro:: LogGamma .. macro:: DoubleFactorial .. macro:: RisingFactorial .. macro:: FallingFactorial .. macro:: HarmonicNumber .. macro:: DigammaFunction .. macro:: DigammaFunctionZero .. macro:: BetaFunction .. macro:: BarnesG .. macro:: LogBarnesG .. macro:: StirlingSeriesRemainder .. macro:: LogBarnesGRemainder Orthogonal polynomials ........................................................................ .. macro:: ChebyshevT .. macro:: ChebyshevU .. macro:: LegendreP .. macro:: JacobiP .. macro:: HermiteH .. macro:: LaguerreL .. macro:: GegenbauerC .. macro:: SphericalHarmonicY .. macro:: LegendrePolynomialZero .. macro:: GaussLegendreWeight Exponential integrals ........................................................................ .. macro:: Erf .. macro:: Erfc .. macro:: Erfi .. macro:: UpperGamma .. macro:: LowerGamma .. macro:: IncompleteBeta .. macro:: IncompleteBetaRegularized .. macro:: LogIntegral .. macro:: ExpIntegralE .. macro:: ExpIntegralEi .. macro:: SinIntegral .. macro:: SinhIntegral .. macro:: CosIntegral .. macro:: CoshIntegral .. macro:: FresnelC .. macro:: FresnelS Bessel and Airy functions ........................................................................ .. macro:: AiryAi .. macro:: AiryBi .. macro:: AiryAiZero .. macro:: AiryBiZero .. macro:: BesselJ .. macro:: BesselI .. macro:: BesselY .. macro:: BesselK .. macro:: HankelH1 .. macro:: HankelH2 .. macro:: BesselJZero .. macro:: BesselYZero .. macro:: CoulombF .. macro:: CoulombG .. macro:: CoulombH .. macro:: CoulombC .. macro:: CoulombSigma Hypergeometric functions ........................................................................ .. macro:: Hypergeometric0F1 .. macro:: Hypergeometric1F1 .. macro:: Hypergeometric1F2 .. macro:: Hypergeometric2F1 .. macro:: Hypergeometric2F2 .. macro:: Hypergeometric2F0 .. macro:: Hypergeometric3F2 .. macro:: HypergeometricU .. macro:: HypergeometricUStar .. macro:: HypergeometricUStarRemainder .. macro:: Hypergeometric0F1Regularized .. macro:: Hypergeometric1F1Regularized .. macro:: Hypergeometric1F2Regularized .. macro:: Hypergeometric2F1Regularized .. macro:: Hypergeometric2F2Regularized .. macro:: Hypergeometric3F2Regularized Zeta and L-functions ........................................................................ .. macro:: RiemannZeta .. macro:: RiemannZetaZero .. macro:: RiemannHypothesis .. macro:: RiemannXi .. macro:: HurwitzZeta .. macro:: LerchPhi .. macro:: PolyLog .. macro:: MultiZetaValue .. macro:: DirichletL .. macro:: DirichletLZero .. macro:: DirichletLambda .. macro:: DirichletCharacter .. macro:: DirichletGroup .. macro:: PrimitiveDirichletCharacters .. macro:: GeneralizedRiemannHypothesis .. macro:: ConreyGenerator .. macro:: GeneralizedBernoulliB .. macro:: StieltjesGamma .. macro:: KeiperLiLambda .. macro:: GaussSum Elliptic integrals ........................................................................ .. macro:: AGM .. macro:: AGMSequence .. macro:: EllipticK .. macro:: EllipticE .. macro:: EllipticPi .. macro:: IncompleteEllipticF .. macro:: IncompleteEllipticE .. macro:: IncompleteEllipticPi .. macro:: CarlsonRF .. macro:: CarlsonRG .. macro:: CarlsonRJ .. macro:: CarlsonRD .. macro:: CarlsonRC .. macro:: CarlsonHypergeometricR .. macro:: CarlsonHypergeometricT Elliptic, theta and modular functions ........................................................................ .. macro:: JacobiTheta .. macro:: JacobiThetaQ .. macro:: DedekindEta .. macro:: ModularJ .. macro:: ModularLambda .. macro:: EisensteinG .. macro:: EisensteinE .. macro:: DedekindSum .. macro:: WeierstrassP .. macro:: WeierstrassZeta .. macro:: WeierstrassSigma .. macro:: EllipticRootE .. macro:: HilbertClassPolynomial .. macro:: EulerQSeries .. macro:: DedekindEtaEpsilon .. macro:: ModularGroupAction .. macro:: ModularGroupFundamentalDomain .. macro:: ModularLambdaFundamentalDomain .. macro:: PrimitiveReducedPositiveIntegralBinaryQuadraticForms .. macro:: JacobiThetaEpsilon .. macro:: JacobiThetaPermutation Nonsemantic markup ........................................................................ .. macro:: Ellipsis ``Ellipsis`` renders as `\ldots` in LaTeX. It can be used to indicate missing function arguments for display purposes, but it has no predefined builtin semantics. .. macro:: Parentheses ``Parentheses(x)`` semantically represents ``x``, but renders with parentheses (`\left(x\right)`) when converted to LaTeX. .. macro:: Brackets ``Brackets(x)`` semantically represents ``x``, but renders with brackets (`\left[x\right]`) when converted to LaTeX. .. macro:: Braces ``Braces(x)`` semantically represents ``x``, but renders with braces (`\left\{x\right\}`) when converted to LaTeX. .. macro:: AngleBrackets ``AngleBrackets(x)`` semantically represents ``x``, but renders with angle brackets (`\left\langle x\right\rangle`) when converted to LaTeX. .. macro:: Logic ``Logic(x)`` semantically represents ``x``, but forces logical expressions within *x* to be rendered using symbols instead of text. .. macro:: ShowExpandedNormalForm ``ShowExpandedNormalForm(x)`` semantically represents ``x``, but displays the expanded normal form of the expression instead of rendering the expression verbatim. Warning: this triggers a nontrivial (potentially very expensive) computation. .. macro:: Subscript .. raw:: latex \newpage