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¶

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.

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.

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.

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 \(f^{(n)}(a)\).

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.

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.

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.

Step¶

Repeat¶

Booleans and logic¶

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.

NotEqual¶: NotEqual(a, b), signifying \(a \ne b\), is equivalent to Not(Equal(a, b)).

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.

True¶: True is a logical constant.

False¶: False is a logical constant.

Not¶: Not(x) is the logical negation of x.

And¶: And(x, y) is the logical AND of x and y. This function can be called with any number of arguments.

Or¶: Or(x, y) is the logical OR of x and y. This function can be called with any number of arguments.

Equivalent¶: Equivalent(x, y) denotes the logical equivalence \(x \Leftrightarrow y\). Semantically, this is the same as Equal called with logical arguments.

Implies¶: Implies(x, y) denotes the logical implication \(x \implies y\).

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 \(f(x) \;\text{ for some } x \in S \text{ with } P(x)\).

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 \(f(x) \;\text{ for all } x \in S \text{ with } P(x)\).

Cases¶

Cases(Case(f(x), P(x)), Case(g(x), Otherwise)) denotes:

\[\begin{split}\begin{cases} f(x), & P(x)\\g(x), & \text{otherwise}\\ \end{cases}\end{split}\]

Cases(Case(f(x), P(x)), Case(g(x), Q(x)), Case(h(x), Otherwise)) denotes:

\[\begin{split}\begin{cases} f(x), & P(x)\\g(x), & Q(x)\\h(x), & \text{otherwise}\\ \end{cases}\end{split}\]

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.

Case¶: See Cases.

Otherwise¶: See Cases.

Tuples, lists and sets¶

Tuple¶

List¶

Set¶

Item¶

Element¶

NotElement¶

EqualAndElement¶

Length¶

Cardinality¶

Concatenation¶

Union¶

Intersection¶

SetMinus¶

Subset¶

SubsetEqual¶

CartesianProduct¶

CartesianPower¶

Subsets¶: Subsets(S) is the power set \(\mathscr{P}(S)\) comprising all subsets of the set S.

Sets¶: Sets is the class \(\operatorname{Sets}\) of all sets.

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¶

Undefined¶: Undefined is the special value \(\mathfrak{u}\) (undefined).

Particular numbers¶

Pi¶: Pi is the constant \(\pi\).

NumberI¶: NumberI is the imaginary unit \(i\). The verbose name leaves i and I to be used as a variable names.

NumberE¶: NumberE is the base of the natural logarithm \(e\). The verbose name leaves e and E to be used as a variable names.

GoldenRatio¶: GoldenRatio is the golden ratio \(\varphi\).

Euler¶: Euler is Euler’s constant \(\gamma\).

CatalanConstant¶: CatalanConstant is Catalan’s constant \(G\).

KhinchinConstant¶: KhinchinConstant is Khinchin’s constant \(K\).

GlaisherConstant¶: GlaisherConstant is Glaisher’s constant \(A\).

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).

Decimal¶: Decimal(str) gives the rational number specified by the string str in ordinary decimal floating-point notation (for example -3.25e-725).

AlgebraicNumberSerialized¶

PolynomialRootIndexed¶

PolynomialRootNearest¶

Enclosure¶

Approximation¶

Guess¶

Unknown¶

Arithmetic operations¶

Pos¶

Neg¶

Add¶

Sub¶

Mul¶

Div¶

Pow¶

Sqrt¶

Root¶

Inequalities¶

Less¶

LessEqual¶

Greater¶

GreaterEqual¶

EqualNearestDecimal¶

Sets of numbers¶

NN¶: NN is the set of natural numbers (including 0), \(\mathbb{N}\).

ZZ¶: ZZ is the set of integers, \(\mathbb{Z}\).

QQ¶: QQ is the set of rational numbers, \(\mathbb{Q}\).

RR¶: RR is the set of real numbers, \(\mathbb{R}\).

CC¶: CC is the set of complex numbers, \(\mathbb{C}\).

Primes¶: Primes is the set of positive prime numbers, \(\mathbb{P}\)

IntegersGreaterEqual¶: IntegersGreaterEqual(x), given an extended real number x, gives the set \(\mathbb{Z}_{\ge x}\) of integers greater than or equal to x.

IntegersLessEqual¶: IntegersLessEqual(x), given an extended real number x, gives the set \(\mathbb{Z}_{\le x}\) of integers less than or equal to x.

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.

AlgebraicNumbers¶: The set of complex algebraic numbers \(\overline{\mathbb{Q}}\).

RealAlgebraicNumbers¶: The set of real algebraic numbers \(\overline{\mathbb{Q}}_{\mathbb{R}}\).

Interval¶: Interval(a, b), given extended real numbers a and b, gives the closed interval \([a, b]\).

OpenInterval¶: OpenInterval(a, b), given extended real numbers a and b, gives the open interval \((a, b)\).

ClosedOpenInterval¶: ClosedOpenInterval(a, b), given extended real numbers a and b, gives the closed-open interval \([a, b)\).

OpenClosedInterval¶: OpenClosedInterval(a, b), given extended real numbers a and b, gives the closed-open interval \((a, b]\).

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.

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.

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.

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.

UpperHalfPlane¶: UpperHalfPlane is the set \(\mathbb{H}\) of complex numbers with positive imaginary part.

UnitCircle¶

BernsteinEllipse¶

Lattice¶

Infinities and extended numbers¶

Infinity¶: Infinity is the positive signed infinity \(\infty\).

UnsignedInfinity¶: UnsignedInfinity is the unsigned infinity \(\tilde \infty\).

RealSignedInfinities¶: RealSignedInfinities is the set of real signed infinities \(\{+\infty, -\infty\}\).

ComplexSignedInfinities¶: ComplexSignedInfinities is the set of complex signed infinities \(\{e^{i \theta} \cdot \infty : \theta \in \mathbb{R}\}\).

RealInfinities¶: RealInfinities is the set of real infinities (signed and unsigned) \(\{+\infty, -\infty\} \cup \{\tilde \infty\}\).

ComplexInfinities¶: ComplexInfinities is the set of complex infinities (signed and unsigned) \(\{e^{i \theta} \cdot \infty : \theta \in \mathbb{R}\} \cup \{\tilde \infty\}\).

ExtendedRealNumbers¶: ExtendedRealNumbers is the set of extended real numbers \(\mathbb{R} \cup \{+\infty, -\infty\}\).

ProjectiveRealNumbers¶: ProjectiveRealNumbers is the set of projectively extended real numbers \(\mathbb{R} \cup \{\tilde \infty\}\).

SignExtendedComplexNumbers¶: SignExtendedComplexNumbers is the set of complex numbers extended with signed infinities \(\mathbb{C} \cup \{e^{i \theta} \cdot \infty : \theta \in \mathbb{R}\}\).

ProjectiveComplexNumbers¶: ProjectiveComplexNumbers is the set of projectively extended complex numbers (also known as the Riemann sphere) \(\mathbb{C} \cup \{\tilde \infty\}\).

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 \(\mathbb{R}_{\text{Sing}} = \mathbb{R} \cup \{+\infty, -\infty\} \cup \{\tilde \infty\} \cup \{ \mathfrak{u} \}\).

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¶

Sum¶

Product¶

PrimeSum¶

PrimeProduct¶

DivisorSum¶

DivisorProduct¶

Solutions and zeros¶

Zeros¶

UniqueZero¶

Solutions¶

UniqueSolution¶

Extreme values¶

Supremum¶

Infimum¶

Minimum¶

Maximum¶

ArgMin¶

ArgMax¶

ArgMinUnique¶

ArgMaxUnique¶

Limits¶

Limit¶

SequenceLimit¶

RealLimit¶

LeftLimit¶

RightLimit¶

ComplexLimit¶

MeromorphicLimit¶

SequenceLimitInferior¶

SequenceLimitSuperior¶

AsymptoticTo¶

Derivatives¶

Derivative¶

RealDerivative¶

ComplexDerivative¶

ComplexBranchDerivative¶

MeromorphicDerivative¶

Integrals¶

Integral¶

Complex analysis¶

Path¶

CurvePath¶

Poles¶

IsHolomorphicOn¶

IsMeromorphicOn¶

Residue¶

ComplexZeroMultiplicity¶

AnalyticContinuation¶

Matrices and linear algebra¶

Matrix¶

Row¶

Column¶

RowMatrix¶

ColumnMatrix¶

DiagonalMatrix¶

Matrix2x2¶

ZeroMatrix¶

IdentityMatrix¶

Det¶

Spectrum¶

SingularValues¶

Matrices¶

SL2Z¶

PSL2Z¶

SpecialLinearGroup¶

GeneralLinearGroup¶

HilbertMatrix¶

Polynomials, series and rings¶

Pol¶

Ser¶

Polynomial¶

Coefficient¶

PolynomialDegree¶

Polynomials¶

PolynomialFractions¶

FormalPowerSeries¶

FormalLaurentSeries¶

FormalPuiseuxSeries¶

Zero¶

One¶

Characteristic¶

Rings¶

CommutativeRings¶

Fields¶

QuotientRing¶

FiniteField¶

EqualQSeriesEllipsis¶

IndefiniteIntegralEqual¶

QSeriesCoefficient¶

Call¶

CallIndeterminate¶

Special functions¶

Number parts and step functions¶

Abs¶

Sign¶

Re¶

Im¶

Arg¶

Conjugate¶

Csgn¶

RealAbs¶

Max¶

Min¶

Floor¶

Ceil¶

KroneckerDelta¶

Primes and divisibility¶

IsOdd¶

IsEven¶

CongruentMod¶

Divides¶

Mod¶

GCD¶

LCM¶

XGCD¶

IsPrime¶

Prime¶

PrimePi¶

DivisorSigma¶

MoebiusMu¶

EulerPhi¶

DiscreteLog¶

LegendreSymbol¶

JacobiSymbol¶

KroneckerSymbol¶

SquaresR¶

LiouvilleLambda¶

Elementary functions¶

Exp¶

Log¶

Sin¶

Cos¶

Tan¶

Cot¶

Sec¶

Csc¶

Sinh¶

Cosh¶

Tanh¶

Coth¶

Sech¶

Csch¶

Asin¶

Acos¶

Atan¶

Acot¶

Asec¶

Acsc¶

Asinh¶

Acosh¶

Atanh¶

Acoth¶

Asech¶

Acsch¶

Atan2¶

Sinc¶

LambertW¶

Combinatorial functions¶

SloaneA¶

SymmetricPolynomial¶

Cyclotomic¶

Fibonacci¶

BernoulliB¶

BernoulliPolynomial¶

StirlingCycle¶

StirlingS1¶

StirlingS2¶

EulerE¶

EulerPolynomial¶

BellNumber¶

PartitionsP¶

LandauG¶

Gamma function and factorials¶

Factorial¶

Binomial¶

Gamma¶

LogGamma¶

DoubleFactorial¶

RisingFactorial¶

FallingFactorial¶

HarmonicNumber¶

DigammaFunction¶

DigammaFunctionZero¶

BetaFunction¶

BarnesG¶

LogBarnesG¶

StirlingSeriesRemainder¶

LogBarnesGRemainder¶

Orthogonal polynomials¶

ChebyshevT¶

ChebyshevU¶

LegendreP¶

JacobiP¶

HermiteH¶

LaguerreL¶

GegenbauerC¶

SphericalHarmonicY¶

LegendrePolynomialZero¶

GaussLegendreWeight¶

Exponential integrals¶

Erf¶

Erfc¶

Erfi¶

UpperGamma¶

LowerGamma¶

IncompleteBeta¶

IncompleteBetaRegularized¶

LogIntegral¶

ExpIntegralE¶

ExpIntegralEi¶

SinIntegral¶

SinhIntegral¶

CosIntegral¶

CoshIntegral¶

FresnelC¶

FresnelS¶

Bessel and Airy functions¶

AiryAi¶

AiryBi¶

AiryAiZero¶

AiryBiZero¶

BesselJ¶

BesselI¶

BesselY¶

BesselK¶

HankelH1¶

HankelH2¶

BesselJZero¶

BesselYZero¶

CoulombF¶

CoulombG¶

CoulombH¶

CoulombC¶

CoulombSigma¶

Hypergeometric functions¶

Hypergeometric0F1¶

Hypergeometric1F1¶

Hypergeometric1F2¶

Hypergeometric2F1¶

Hypergeometric2F2¶

Hypergeometric2F0¶

Hypergeometric3F2¶

HypergeometricU¶

HypergeometricUStar¶

HypergeometricUStarRemainder¶

Hypergeometric0F1Regularized¶

Hypergeometric1F1Regularized¶

Hypergeometric1F2Regularized¶

Hypergeometric2F1Regularized¶

Hypergeometric2F2Regularized¶

Hypergeometric3F2Regularized¶

Zeta and L-functions¶

RiemannZeta¶

RiemannZetaZero¶

RiemannHypothesis¶

RiemannXi¶

HurwitzZeta¶

LerchPhi¶

PolyLog¶

MultiZetaValue¶

DirichletL¶

DirichletLZero¶

DirichletLambda¶

DirichletCharacter¶

DirichletGroup¶

PrimitiveDirichletCharacters¶

GeneralizedRiemannHypothesis¶

ConreyGenerator¶

GeneralizedBernoulliB¶

StieltjesGamma¶

KeiperLiLambda¶

GaussSum¶

Elliptic integrals¶

AGM¶

AGMSequence¶

EllipticK¶

EllipticE¶

EllipticPi¶

IncompleteEllipticF¶

IncompleteEllipticE¶

IncompleteEllipticPi¶

CarlsonRF¶

CarlsonRG¶

CarlsonRJ¶

CarlsonRD¶

CarlsonRC¶

CarlsonHypergeometricR¶

CarlsonHypergeometricT¶

Elliptic, theta and modular functions¶

JacobiTheta¶

JacobiThetaQ¶

DedekindEta¶

ModularJ¶

ModularLambda¶

EisensteinG¶

EisensteinE¶

DedekindSum¶

WeierstrassP¶

WeierstrassZeta¶

WeierstrassSigma¶

EllipticRootE¶

HilbertClassPolynomial¶

EulerQSeries¶

DedekindEtaEpsilon¶

ModularGroupAction¶

ModularGroupFundamentalDomain¶

ModularLambdaFundamentalDomain¶

PrimitiveReducedPositiveIntegralBinaryQuadraticForms¶

JacobiThetaEpsilon¶

JacobiThetaPermutation¶

Nonsemantic markup¶

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.

Parentheses¶: Parentheses(x) semantically represents x, but renders with parentheses (\(\left(x\right)\)) when converted to LaTeX.

Brackets¶: Brackets(x) semantically represents x, but renders with brackets (\(\left[x\right]\)) when converted to LaTeX.

Braces¶: Braces(x) semantically represents x, but renders with braces (\(\left\{x\right\}\)) when converted to LaTeX.

AngleBrackets¶: AngleBrackets(x) semantically represents x, but renders with angle brackets (\(\left\langle x\right\rangle\)) when converted to LaTeX.

Logic¶: Logic(x) semantically represents x, but forces logical expressions within x to be rendered using symbols instead of text.

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.

Subscript¶