fexpr_builtin.h

The header words of the expressions consisting of the symbols Pos, Neg, Add, Sub, Mul, Div and Pow, for fast comparisons when recognising arithmetic operations.

For(x, ...) binds the symbol x as a local variable in its parent expression; the further arguments give a range, set or point whose meaning depends on the parent operator. For(x, S) iterates over a set, optionally followed by a filter predicate P(x), as in Set(f(x), For(x, S), P(x)) (\left\{ f(x) : x \in S\,\mathbin{\operatorname{and}}\, P(x) \right\}) or Sum(f(x), For(x, S)) (\sum_{x \in S} f(x)). For(x, a, b) lets x run between endpoints, as in Sum(f(n), For(n, a, b)) (\sum_{n=a}^{b} f(n)) or Integral(f(x), For(x, a, b)) (\int_{a}^{b} f(x) \, dx). For(x, a) lets x approach a point or selects a point of differentiation, as in Limit(f(x), For(x, a)) (\lim_{x \to a} f(x)) or Derivative(f(x), For(x, a)) (f'\!\left(a\right)), and For(x, a, n) gives the order of a derivative, as in Derivative(f(x), For(x, a, n)) ({f}^{(n)}\!\left(a\right)).

Where(f(x), Def(x, a)) defines x as an alias for a within f(x), so that it is equivalent to f(a); it renders as Where(f(x), Def(x, a)) (f(x)\; \text{ where } x = a). Def(f(t), a) defines f as a function of the dummy variable t, and Def(Tuple(a, b), T) destructures a tuple. A Where expression may contain several definitions. Def occurs only inside Where.

Fun(x, expr) is the anonymous function mapping x to expr, rendered Fun(x, Pow(x, 2)) (x \mapsto {x}^{2}); x is bound within the expression.

Notation for sequences: Step(x, For(x, a, b)) (a, a + 1, \ldots, b) and Repeat(x, n) (\underbrace{x, \ldots, x}_{n \text{ times}}).

Equal(a, b) (a = b) is True if a and b represent the same object and False otherwise; with more arguments it states that all are equal. NotEqual(a, b) (a \ne b) is Not(Equal(a, b)).

Same(a, b) is a (equivalently b) if a and b represent the same object, and Undefined otherwise. It asserts or emphasises that expressions have the same value within a formula; it accepts any number of arguments.

The logical constants True (\operatorname{True}) and False (\operatorname{False}).

Logical negation Not(x) (\operatorname{not} x), conjunction And(x, y) (x \;\mathbin{\operatorname{and}}\; y) and disjunction Or(x, y) (x \;\mathbin{\operatorname{or}}\; y), both with any number of arguments, equivalence Equivalent(x, y) (x \iff y), which is Equal applied to logical values, and implication Implies(x, y) (x \;\implies\; y).

Quantifiers: Exists(f(x), For(x, S)) (f(x) \;\text{ for some } x \in S) and All(f(x), For(x, S)) (f(x) \;\text{ for all } x \in S), optionally with a filter predicate as third argument, as in All(f(x), For(x, S), P(x)) (f(x) \;\text{ for all } x \in S \text{ with } P(x)).

Piecewise definitions: Cases(Case(f(x), P(x)), Case(g(x), Otherwise)) (\begin{cases} f(x), & P(x)\\g(x), & \text{otherwise}\\ \end{cases}). Any number of cases can be given. If several predicates hold simultaneously, the corresponding values must agree; if no predicate holds and there is no Otherwise case, the value is undefined.

A tuple Tuple(a, b, c) (\left(a, b, c\right)), a list List(a, b, c) (\left[a, b, c\right]) and a finite set Set(a, b, c) (\left\{a, b, c\right\}). A set can also be given by comprehension, as in Set(f(x), For(x, S), P(x)) (\left\{ f(x) : x \in S\,\mathbin{\operatorname{and}}\, P(x) \right\}).

The element with index n of a tuple or list Item(L, n) ({L}_{n}), the length of a tuple or list Length(L) (\# L), the cardinality of a set Cardinality(S) (\# S), and the concatenation of lists Concatenation(L, M) (L \,^\frown M).

Membership Element(x, S) (x \in S) and NotElement(x, S) (x \notin S); EqualAndElement(a, b, S) states that a equals b and is an element of S, rendered EqualAndElement(a, b, S) (a = b \in S).

Set operations Union(A, B) (A \cup B), Intersection(A, B) (A \cap B), SetMinus(A, B) (A \setminus B) and the relations Subset(A, B) (A \subset B) and SubsetEqual(A, B) (A \subseteq B).

The Cartesian product CartesianProduct(A, B) (A \times B) and power CartesianPower(S, n) ({S}^{n}), the power set Subsets(S) (\mathscr{P}(S)) of all subsets of S, the class Sets (\operatorname{Sets}) of all sets, and Tuples (the class of all tuples), Tuples(S) (tuples with elements in S) and Tuples(S, n) (tuples of length n with elements in S).

Undefined (\mathfrak{u}) is the special value \mathfrak{u} for undefined results, for example of 0/0. Unknown (\operatorname{Unknown}) is the meta-value used by ca.h for values that could not be computed.

The constants Pi (\pi), the imaginary unit NumberI (i), the base of the natural logarithm NumberE (e), the golden ratio GoldenRatio (\varphi), Euler's constant Euler (\gamma), Catalan's constant CatalanConstant (G), Khinchin's constant KhinchinConstant (K) and Glaisher's constant GlaisherConstant (A). The verbose names NumberI and NumberE leave i, I, e and E free as variable names.

RootOfUnity(n) (\zeta_{n}) is the principal root of unity e^{2 \pi i / n}, and RootOfUnity(n, k) (\zeta_{n, k}) its k-th power.

Decimal(str) is the rational number given in decimal floating-point notation by the string str, for example Decimal("-3.25e-725"), rendered Decimal("-3.25e-725") (-3.25 \cdot 10^{-725}).

Algebraic numbers: AlgebraicNumberSerialized(List(c_0, ..., c_d), enclosure) gives the root of the polynomial with integer coefficients c_0, \ldots, c_d isolated by the ball enclosure (a RealBall, or Add(RealBall(...), Mul(RealBall(...), NumberI)) for a complex ball); PolynomialRootIndexed( List(c_0, ..., c_d), k) is the k-th root (from 1) in the order of qqbar_cmp_root_order; and PolynomialRootNearest(List(c_0, ..., c_d), x) is the root closest to the number x. These are produced and read by the qqbar_get_fexpr functions and qqbar_set_fexpr (see qqbar.h).

Notation for numerical information: a value enclosed by x, Enclosure(x) (\left(\in x \right)); an approximation Approximation(x, n); a conjectured value Guess(x) (\left(?\; x \right)); and the statement that x rounds to a given decimal string with a given number of digits, EqualNearestDecimal(x, "1.25", 3) (x = \text{``1.25''} \;\, {\scriptstyle (\text{nearest } 3 \text{ digits})}).

Arithmetic: Pos(x) (+x), Neg(x) (-x), Add(x, y) (x + y), Sub(x, y) (x - y), Mul(x, y) (x y), Div(x, y) (\frac{x}{y}) and Pow(x, y) ({x}^{y}). Add and Mul accept any number of arguments.

The principal square root Sqrt(x) (\sqrt{x}) and the principal n-th root Root(x, n).

Order relations Less(x, y) (x < y), LessEqual(x, y) (x \le y), Greater(x, y) (x > y) and GreaterEqual(x, y) (x \ge y). With more arguments they denote chains, such as Less(x, y, z) (x < y < z).

Sets of numbers: the natural numbers including 0 NN (\mathbb{N}), the integers ZZ (\mathbb{Z}), the rationals QQ (\mathbb{Q}), the reals RR (\mathbb{R}), the complex numbers CC (\mathbb{C}), the positive primes Primes (\mathbb{P}), the algebraic numbers AlgebraicNumbers (\overline{\mathbb{Q}}) and the real algebraic numbers RealAlgebraicNumbers (\overline{\mathbb{Q}}_{\mathbb{R}}).

Integer ranges: IntegersGreaterEqual(x) (\mathbb{Z}_{\ge x}) and IntegersLessEqual(x) (\mathbb{Z}_{\le x}) for an extended real number x, and Range(a, b) (\{a, a + 1, \ldots, b\}), the integers from a to b (empty if a > b).

Real intervals with extended real endpoints: Interval(a, b) (\left[a, b\right]), OpenInterval(a, b) (\left(a, b\right)), ClosedOpenInterval(a, b) (\left[a, b\right)), OpenClosedInterval(a, b) (\left(a, b\right]); and real balls with real center m and extended real radius r, closed RealBall(m, r) (\left[m \pm r\right]) or open OpenRealBall(m, r) (\left(m \pm r\right)).

Regions of the complex plane: the open disk OpenComplexDisk(m, r) (D(m, r)), the closed disk ClosedComplexDisk(m, r) (\overline{D}(m, r)) with center m and radius r, the upper half-plane UpperHalfPlane (\mathbb{H}) of numbers with positive imaginary part, the unit circle UnitCircle (\mathbb{T}), the Bernstein ellipse BernsteinEllipse(rho) (\mathcal{E}_{\rho}) with foci \pm 1 and sum of semi-axes \rho, and the lattice Lattice(a, b) (\Lambda_{(a, b)}) generated by a and b.

The positive signed infinity Infinity (\infty) and the unsigned infinity UnsignedInfinity (\hat{\infty}). Signed infinities in other directions are written as products such as Mul(NumberI, Infinity).

Sets of infinities: RealSignedInfinities (\{\pm \infty\}), equal to \{+\infty, -\infty\}, ComplexSignedInfinities (\{[e^{i \theta}] \infty\}), equal to \{e^{i \theta} \infty : \theta \in \mathbb{R}\}, and the same sets together with unsigned infinity, RealInfinities (\{\hat{\infty}, \pm \infty\}) and ComplexInfinities (\{\hat{\infty}, [e^{i \theta}] \infty\}).

Extended number systems: ExtendedRealNumbers (\overline{\mathbb{R}}), equal to \mathbb{R} \cup \{\pm \infty\}, ProjectiveRealNumbers (\hat{\mathbb{R}}_{\infty}), equal to \mathbb{R} \cup \{\tilde \infty\}, SignExtendedComplexNumbers (\overline{\mathbb{C}}_{[e^{i \theta}] \infty}), the complex numbers with all signed infinities, and ProjectiveComplexNumbers (\hat{\mathbb{C}}_{\infty}) = \mathbb{C} \cup \{\tilde \infty\} (the Riemann sphere).

The value sets of ca.h: RealSingularityClosure (\overline{\mathbb{R}}_{\text{Sing}}), equal to \mathbb{R} \cup \{\pm \infty, \tilde \infty, \mathfrak{u}\} and ComplexSingularityClosure (\overline{\mathbb{C}}_{\text{Sing}}), the complex numbers together with all signed infinities, unsigned infinity and Undefined.

Sums and products: Sum(f(n), For(n, a, b)) (\sum_{n=a}^{b} f(n)), Product(f(n), For(n, a, b)) (\prod_{n=a}^{b} f(n)), and over sets with optional filter, Sum(f(x), For(x, S), P(x)) (\sum_{\textstyle{x \in S \atop P(x)}} f(x)).

Sums and products over primes and divisors: PrimeSum(f(p), For(p)) (\sum_{p} f(p)), PrimeProduct(f(p), For(p)) (\prod_{p} f(p)), DivisorSum(f(d), For(d, n)) (\sum_{d \mid n} f(d)) and DivisorProduct(f(d), For(d, n)) (\prod_{d \mid n} f(d)); a filter predicate may follow.

Sets of zeros, solutions and poles: Zeros(f(x), For(x, S)) (\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)), Solutions(Equal(f(x), 0), For(x, S)) (\mathop{\operatorname{solutions}\,}\limits_{x \in S} f(x) = 0), Poles(f(z), For(z, S)) (\mathop{\operatorname{poles}\,}\limits_{z \in S} f(z)); UniqueZero and UniqueSolution denote the unique element of such a set.

Extrema over sets: Supremum(f(x), For(x, S)) (\mathop{\operatorname{sup}\,}\limits_{x \in S} f(x)), Infimum(f(x), For(x, S)) (\mathop{\operatorname{inf}\,}\limits_{x \in S} f(x)), Minimum(f(x), For(x, S)) (\mathop{\min\,}\limits_{x \in S} f(x)), Maximum(f(x), For(x, S)) (\mathop{\max\,}\limits_{x \in S} f(x)), the sets of minimizers ArgMin(f(x), For(x, S)) (\mathop{\operatorname{arg\,min}\,}\limits_{x \in S} f(x)) and maximizers ArgMax, and the unique minimizer or maximizer ArgMinUnique, ArgMaxUnique.

Limits: Limit(f(x), For(x, a)) (\lim_{x \to a} f(x)), the limit of a sequence SequenceLimit(a(n), For(n, Infinity)) (\lim_{n \to \infty} a(n)), limits through real arguments RealLimit, from the left LeftLimit(f(x), For(x, a)) (\lim_{x \to {a}^{-}} f(x)) or the right RightLimit, through complex arguments ComplexLimit, the limit in the sense of meromorphic functions MeromorphicLimit, and the limit inferior and superior of sequences SequenceLimitInferior, SequenceLimitSuperior.

Asymptotic equivalence: AsymptoticTo(f(n), g(n), n, Infinity) (f(n) \sim g(n), \; n \to \infty).

Derivatives: Derivative(f(x), For(x, a)) (f'\!\left(a\right)) and of higher order Derivative(f(x), For(x, a, n)) ({f}^{(n)}\!\left(a\right)); the variants specify real differentiation, complex differentiation, complex differentiation along a branch, and differentiation of meromorphic functions.

Integrals Integral(f(x), For(x, a, b)) (\int_{a}^{b} f(x) \, dx) along the straight segment, or along a path given as a polygon Path(a, b, c) (a \rightsquigarrow b \rightsquigarrow c) or a parametrised curve CurvePath(f(t), For(t, a, b)) (\left(f(t),\, t : a \rightsquigarrow b\right)) in place of the endpoints.

The residue Residue(f(z), For(z, a)) (\mathop{\operatorname{res}}\limits_{z=a} f(z)) and the order of a zero ComplexZeroMultiplicity(f(z), For(z, a)) (\mathop{\operatorname{ord}}\limits_{z=a} f(z)) at a point.

Predicates IsHolomorphicOn(f(z), For(z, S)) (f(z) \text{ is holomorphic on } z \in S) and IsMeromorphicOn, and the analytic continuation of f along a path, AnalyticContinuation(f(z), For(z, a, b)) (\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)).

Matrices: Matrix(Row(a, b), Row(c, d)) (\displaystyle{\begin{pmatrix}a & b \\c & d\end{pmatrix}}), also written Matrix(List(Row(a, b), Row(c, d))), the row and column vectors RowMatrix(a, b, c) (\displaystyle{\begin{pmatrix}a & b & c\end{pmatrix}}) and ColumnMatrix(a, b, c) (\displaystyle{\begin{pmatrix}a \\ b \\ c\end{pmatrix}}), the diagonal matrix DiagonalMatrix(a, b, c) (\displaystyle{\begin{pmatrix}a & & \\ & b & \\ & & c\end{pmatrix}}) and Matrix2x2(a, b, c, d) (\displaystyle{\begin{pmatrix}a & b \\ c & d\end{pmatrix}}). Column is reserved for columns as Row is used for rows.

Special n \times n matrices: ZeroMatrix(n) (0_{n}), IdentityMatrix(n) (I_{n}) and the Hilbert matrix HilbertMatrix(n) (H_{n}).

The determinant Det(A) (\operatorname{det}(A)), the set of eigenvalues Spectrum(A) (\operatorname{spec}(A)) and the singular values SingularValues(A) (\sigma(A)).

Sets of matrices: Matrices(CC, n) (\operatorname{M}_{n}(\mathbb{C})) and Matrices(CC, m, n) (\operatorname{M}_{m \times n}(\mathbb{C})), the modular groups SL2Z (\operatorname{SL}_2(\mathbb{Z})) and PSL2Z (\operatorname{PSL}_2(\mathbb{Z})), and the groups SpecialLinearGroup(n, ZZ) (\operatorname{SL}_{n}\!\left(\mathbb{Z}\right)) and GeneralLinearGroup(n, ZZ) (\operatorname{GL}_{n}\!\left(\mathbb{Z}\right)).

Polynomial(List(c_0, ..., c_n), x) is the polynomial c_0 + c_1 x + \cdots + c_n x^n in x (the variable may be omitted, as in the output of ca_poly_get_fexpr); Coefficient(f, x, n) ([{x}^{n}] f) is the coefficient of x^n and PolynomialDegree(f) (\deg(f)) the degree.

Placeholders for polynomials and series in formulas: Pol(x) (\operatorname{Pol}(x)), Ser(x) (\operatorname{Ser}(x)).

Rings: Polynomials(ZZ, x) (\mathbb{Z}[x]), the field of fractions PolynomialFractions(QQ, x) (\mathbb{Q}(x)), formal power series FormalPowerSeries(ZZ, x) (\mathbb{Z}[[x]]), formal Laurent series FormalLaurentSeries(ZZ, x) (\mathbb{Z}(\!(x)\!)) and formal Puiseux series FormalPuiseuxSeries(ZZ, x) (\mathbb{Z}\!\left\langle\!\left\langle x \right\rangle\!\right\rangle).

The zero Zero(R) (0_{R}) and one One(R) (1_{R}) of a ring and its characteristic Characteristic(R) (\operatorname{char}(R)).

The classes of rings Rings, commutative rings CommutativeRings and fields Fields; the quotient ring QuotientRing(R, I); and the finite field FiniteField(q) with q elements.

Notation from Fungrim formulas: QSeriesCoefficient(f, q, n), the coefficient of q^n in a q-series; EqualQSeriesEllipsis, which states that a q-series begins with given terms; and IndefiniteIntegralEqual(f(x), g(x), x) (\int f(x) \, dx = g(x) + \mathcal{C}).

Function application: Call(f, x) (f(x)) is f applied to x, and CallIndeterminate(f, x, v) applies f to the indeterminate v, rendered CallIndeterminate(f, x, v) (f(v)).

Complex parts: Abs(z) (\left|z\right|), the complex sign Sign(z) (\operatorname{sgn}(z)), equal to z/|z| for z \ne 0; the real-valued sign extension Csgn(z) (\operatorname{csgn}(z)); the absolute value of a real number RealAbs(x) (\left|x\right|), Re(z) (\operatorname{Re}(z)), Im(z) (\operatorname{Im}(z)), the argument Arg(z) (\arg(z)) and the complex conjugate Conjugate(z) (\overline{z}).

Max(a, b) (\max(a, b)) and Min(a, b) (\min(a, b)), with any number of arguments, and Floor(x) (\left\lfloor x \right\rfloor), Ceil(x) (\left\lceil x\right\rceil).

Integer predicates and operations: KroneckerDelta(i, j) (\delta_{(i,j)}), IsOdd(n) (n \text{ odd}), IsEven(n) (n \text{ even}), CongruentMod(a, b, m) (a \equiv b \pmod {m }), Divides(d, n) (d \mid n) and the remainder Mod(a, n) (a \bmod n).

GCD(a, b) (\gcd(a, b)) and LCM(a, b) (\operatorname{lcm}(a, b)), with any number of arguments, and the extended GCD XGCD(a, b) (\operatorname{xgcd}(a, b)).

IsPrime(n) (n \text{ prime}), the n-th prime Prime(n) (p_{n}) and the prime-counting function PrimePi(x) (\pi(x)).

Arithmetic functions: the divisor function DivisorSigma(k, n) (\sigma_{k}\!\left(n\right)), the Möbius function MoebiusMu(n) (\mu(n)), Euler's totient function EulerPhi(n) (\varphi(n)), the Liouville function LiouvilleLambda(n) (\lambda(n)) and the number of representations as a sum of squares SquaresR(k, n) (r_{k}\!\left(n\right)).

LegendreSymbol(a, p) (\left(\frac{a}{p}\right)), JacobiSymbol(a, n) (\left(\frac{a}{n}\right)), KroneckerSymbol(a, n) (\left(\frac{a}{n}\right)), and the discrete logarithm DiscreteLog(n, b, p) ((\epsilon : {b}^{\epsilon} \equiv n \text{ mod }p)), the least \epsilon \ge 0 with b^\epsilon \equiv n \pmod p.

The exponential function Exp(z) (e^{z}) and the principal branch of the natural logarithm Log(z) (\log(z)).

Trigonometric functions Sin(z) (\sin(z)), Cos(z) (\cos(z)), Tan(z) (\tan(z)), Cot(z) (\cot(z)), Sec(z) (\sec(z)), Csc(z) (\csc(z)), and the sinc function Sinc(z) (\operatorname{sinc}(z)), equal to \sin(z)/z.

Hyperbolic functions Sinh(z) (\sinh(z)), Cosh(z) (\cosh(z)), Tanh(z) (\tanh(z)), Coth(z) (\coth(z)), Sech(z) (\operatorname{sech}(z)) and Csch(z) (\operatorname{csch}(z)).

Principal branches of the inverse trigonometric functions Asin(z) (\operatorname{asin}(z)), Acos(z) (\operatorname{acos}(z)), Atan(z) (\operatorname{atan}(z)), Acot(z) (\operatorname{acot}(z)), Asec(z) (\operatorname{asec}(z)), Acsc(z) (\operatorname{acsc}(z)), and the two-argument arctangent Atan2(y, x) (\operatorname{atan2}(y, x)).

Principal branches of the inverse hyperbolic functions Asinh(z) (\operatorname{asinh}(z)), Acosh(z) (\operatorname{acosh}(z)), Atanh(z) (\operatorname{atanh}(z)), Acoth(z) (\operatorname{acoth}(z)), Asech(z) (\operatorname{asech}(z)) and Acsch(z) (\operatorname{acsch}(z)).

The Lambert W function: the principal branch LambertW(z) (W(z)) and the branch with index k, LambertW(z, k) (W_{k}(z)).

Factorial(n) (n!), DoubleFactorial(n) (n!!), the binomial coefficient Binomial(n, k) ({n \choose k}), and the rising and falling factorials RisingFactorial(z, k) (\left(z\right)_{k}) and FallingFactorial(z, k) (\left(z\right)^{\underline{k}}).

The harmonic number HarmonicNumber(n) (H_{n}), the Fibonacci number Fibonacci(n) (F_{n}), the Bell number BellNumber(n) (\operatorname{B}_{n}), the partition function PartitionsP(n) (p(n)) and Landau's function LandauG(n) (g(n)).

Bernoulli numbers BernoulliB(n) (B_{n}) and polynomials BernoulliPolynomial(n, x) (B_{n}\!\left(x\right)), and Euler numbers EulerE(n) (E_{n}) and polynomials EulerPolynomial(n, x) (E_{n}\!\left(x\right)).

Stirling numbers: unsigned of the first kind StirlingCycle(n, k) (\left[{n \atop k}\right]), signed of the first kind StirlingS1(n, k) (s\!\left(n, k\right)) and of the second kind StirlingS2(n, k) (\left\{{n \atop k}\right\}).

The elementary symmetric polynomial SymmetricPolynomial(k, List(x, y, z)) (e_{k}\!\left(\left[x, y, z\right]\right)), the cyclotomic polynomial Cyclotomic(n, x) (\Phi_{n}\!\left(x\right)), and the n-th term of an OEIS sequence SloaneA("A000001", n) (\text{A000001}\!\left(n\right)).

The gamma function Gamma(z) (\Gamma(z)), the log-gamma function LogGamma(z) (\log \Gamma(z)) (with the branch cut of the principal logarithm of \Gamma continued analytically from the positive real axis), the digamma function DigammaFunction(z) (\psi(z)) and its derivatives (polygamma functions) DigammaFunction(z, n) ({\psi}^{(n)}\!\left(z\right)), and the zeros of the digamma function DigammaFunctionZero(n) (x_{n}).

The beta function BetaFunction(a, b) (\mathrm{B}(a, b)), the Barnes G-function BarnesG(z) (G(z)) and its logarithm LogBarnesG(z) (\log G(z)).

Remainders of asymptotic series after n terms: of the Stirling series for \log \Gamma(z), StirlingSeriesRemainder(n, z) (R_{n}\!\left(z\right)), and of the asymptotic series for \log G(z), LogBarnesGRemainder(n, z).

Orthogonal polynomials and related functions: Chebyshev polynomials ChebyshevT(n, x) (T_{n}\!\left(x\right)) and ChebyshevU(n, x) (U_{n}\!\left(x\right)), Legendre polynomials (and functions) LegendreP(n, x) (P_{n}\!\left(x\right)), Jacobi polynomials JacobiP(n, a, b, x), Hermite polynomials HermiteH(n, x) (H_{n}\!\left(x\right)), Laguerre polynomials LaguerreL(n, x) and LaguerreL(n, a, x), Gegenbauer polynomials GegenbauerC(n, lamda, x), and spherical harmonics SphericalHarmonicY(ell, m, theta, phi).

The k-th zero LegendrePolynomialZero(n, k) (x_{n, k}) of the Legendre polynomial P_n, and the corresponding Gauss–Legendre quadrature weight GaussLegendreWeight(n, k) (w_{n, k}).

The error function Erf(z) (\operatorname{erf}(z)), the complementary error function Erfc(z) (\operatorname{erfc}(z)) and the imaginary error function Erfi(z) (\operatorname{erfi}(z)).

The upper and lower incomplete gamma functions UpperGamma(s, z) (\Gamma(s, z)) and LowerGamma(s, z) (\gamma(s, z)), and the incomplete beta function IncompleteBeta(x, a, b) (\mathrm{B}_{x}\!\left(a, b\right)) and its regularised version IncompleteBetaRegularized(x, a, b) (I_{x}\!\left(a, b\right)).

Exponential and trigonometric integrals: the logarithmic integral LogIntegral(z) (\operatorname{li}(z)), the generalised exponential integral ExpIntegralE(s, z) (E_{s}\!\left(z\right)), the exponential integral ExpIntegralEi(z) (\operatorname{Ei}(z)), the sine and cosine integrals SinIntegral(z) (\operatorname{Si}(z)), CosIntegral(z) (\operatorname{Ci}(z)), their hyperbolic counterparts SinhIntegral(z) (\operatorname{Shi}(z)), CoshIntegral(z) (\operatorname{Chi}(z)), and the Fresnel integrals FresnelC(z) (C(z)) and FresnelS(z) (S(z)).

The Airy functions AiryAi(z) (\operatorname{Ai}(z)) and AiryBi(z) (\operatorname{Bi}(z)), with derivatives AiryAi(z, 1) (\operatorname{Ai}'\!\left(z\right)), and their n-th zeros on the negative real axis AiryAiZero(n) (a_{n}) and AiryBiZero(n) (b_{n}); a second argument 1 selects the zeros of the derivatives.

Bessel functions of the first and second kind BesselJ(nu, z) (J_{\nu}\!\left(z\right)) and BesselY(nu, z) (Y_{\nu}\!\left(z\right)), modified Bessel functions BesselI(nu, z) (I_{\nu}\!\left(z\right)) and BesselK(nu, z) (K_{\nu}\!\left(z\right)), with derivatives BesselJ(nu, z, r) ({J}^{(r)}_{\nu}\!\left(z\right)), and the Hankel functions HankelH1(nu, z) (H^{(1)}_{\nu}\!\left(z\right)) and HankelH2(nu, z) (H^{(2)}_{\nu}\!\left(z\right)).

The n-th positive zeros BesselJZero(nu, n) (j_{\nu, n}) and BesselYZero(nu, n) (y_{\nu, n}) of J_\nu and Y_\nu.

Coulomb wave functions: the regular and irregular functions CoulombF(ell, eta, z) (F_{\ell,\eta}(z)) and CoulombG(ell, eta, z) (G_{\ell,\eta}(z)), the combination CoulombH(omega, ell, eta, z) (H^{\omega}_{\ell,\eta}(z)), equal to G \pm i F for \omega = \pm 1, the normalising constant CoulombC(ell, eta) (C_{\ell}\!\left(\eta\right)) and the Coulomb phase shift CoulombSigma(ell, eta) (\sigma_{\ell}\!\left(\eta\right)).

Generalised hypergeometric functions {}_pF_q with the parameters followed by the argument, such as Hypergeometric0F1(a, z) (\,{}_0F_1(a, z)), Hypergeometric1F1(a, b, z) (\,{}_1F_1(a, b, z)), Hypergeometric2F1(a, b, c, z) (\,{}_2F_1(a, b, c, z)) and Hypergeometric3F2(a, b, c, d, e, z) (\,{}_3F_2(a, b, c, d, e, z)); {}_2F_0 is understood in the sense of an asymptotic series (equivalently via U).

The regularised hypergeometric functions, with each lower parameter b contributing a factor 1/\Gamma(b), such as Hypergeometric2F1Regularized(a, b, c, z) (\,{}_2{\textbf F}_1(a, b, c, z)).

The confluent hypergeometric function of the second kind HypergeometricU(a, b, z) (U(a, b, z)), the scaled function HypergeometricUStar(a, b, z) (U^{*}(a, b, z)), equal to z^a U(a, b, z), and the remainder of its asymptotic series after n terms HypergeometricUStarRemainder(n, a, b, z) (R_{n}\!\left(a, b, z\right)).

The Riemann zeta function RiemannZeta(s) (\zeta(s)) and its derivatives RiemannZeta(s, r) ({\zeta}^{(r)}\!\left(s\right)), its n-th nontrivial zero in the upper half-plane RiemannZetaZero(n) (\rho_{n}), the Riemann xi function RiemannXi(s) (\xi(s)), and the proposition RiemannHypothesis (\operatorname{RH}).

The Hurwitz zeta function HurwitzZeta(s, a) (\zeta(s, a)), the Lerch transcendent LerchPhi(z, s, a) (\Phi(z, s, a)), the polylogarithm PolyLog(s, z) (\operatorname{Li}_{s}\!\left(z\right)) and multiple zeta values MultiZetaValue(s, t) (\zeta(s, t)).

The Stieltjes constants StieltjesGamma(n) (\gamma_{n}), the generalised constants StieltjesGamma(n, a) (\gamma_{n}\!\left(a\right)) of the Hurwitz zeta function, and the Keiper–Li coefficients KeiperLiLambda(n) (\lambda_{n}).

Dirichlet characters in Conrey's labelling: the character DirichletCharacter(q, ell) (\chi_{q \, . \, \ell}) and its value DirichletCharacter(q, ell, n) (\chi_{q \, . \, \ell}(n)), the group of characters modulo q DirichletGroup(q) (G_{q}), the set of primitive characters PrimitiveDirichletCharacters(q) (G^{\text{Primitive}}_{q}), the generator used by Conrey's labelling ConreyGenerator(q) (g_{q}), and the Gauss sum GaussSum(q, chi) (G_{q}\!\left(\chi\right)).

The Dirichlet L-function DirichletL(s, chi) (L(s, \chi)), its zeros DirichletLZero(n, chi) (\rho_{n, \chi}), the completed L-function DirichletLambda(s, chi) (\Lambda(s, \chi)), the generalised Bernoulli numbers GeneralizedBernoulliB(n, chi) (B_{n, \chi}), and the proposition GeneralizedRiemannHypothesis (\operatorname{GRH}).

The arithmetic-geometric mean AGM(a, b) (\operatorname{agm}(a, b)) and the n-th terms of the AGM iteration AGMSequence(n, a, b) (\operatorname{agm}_{n}\!\left(a, b\right)).

Legendre's complete elliptic integrals with parameter m = k^2, EllipticK(m) (K(m)), EllipticE(m) (E(m)), EllipticPi(n, m) (\Pi(n, m)), and the incomplete integrals IncompleteEllipticF(phi, m) (F(\phi, m)), IncompleteEllipticE(phi, m) (E(\phi, m)) and IncompleteEllipticPi(n, phi, m) (\Pi(n, \phi, m)).

Carlson's symmetric elliptic integrals CarlsonRF(x, y, z) (R_F(x, y, z)), CarlsonRG(x, y, z) (R_G(x, y, z)), CarlsonRJ(x, y, z, p) (R_J(x, y, z, p)), CarlsonRD(x, y, z) (R_D(x, y, z)) and CarlsonRC(x, y) (R_C(x, y)), and Carlson's multivariate hypergeometric functions CarlsonHypergeometricR and CarlsonHypergeometricT.

Jacobi theta functions \theta_j, j = 1, 2, 3, 4, in terms of the lattice parameter \tau, JacobiTheta(j, z, tau) (\theta_{j}\!\left(z, \tau\right)), with derivatives JacobiTheta(j, z, tau, r) ({\theta}^{(r)}_{j}\!\left(z, \tau\right)), or in terms of the nome q, JacobiThetaQ(j, z, q) (\theta_{j}\!\left(z, q\right)).

Modular forms and functions: the Dedekind eta function DedekindEta(tau) (\eta(\tau)), the j-invariant ModularJ(tau) (j(\tau)), the modular lambda function ModularLambda(tau) (\lambda(\tau)), the Eisenstein series EisensteinG(k, tau) (G_{k}\!\left(\tau\right)) and their normalised versions EisensteinE(k, tau) (E_{k}\!\left(\tau\right)), and the Euler function EulerQSeries(q) (\phi(q)), equal to \prod_{k \ge 1} (1 - q^k).

Quantities in modular transformation formulas: the Dedekind sum DedekindSum(h, k) (s(h, k)), the root of unity DedekindEtaEpsilon(a, b, c, d) (\varepsilon(a, b, c, d)) in the transformation of \eta under a matrix \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right), and the root of unity JacobiThetaEpsilon(j, a, b, c, d) (\varepsilon_{j}\!\left(a, b, c, d\right)) and index permutation JacobiThetaPermutation(j, a, b, c, d) (S_{j}\!\left(a, b, c, d\right)) in the transformation of \theta_j.

The action of a matrix on the upper half-plane ModularGroupAction(gamma, tau) (\gamma \circ \tau), and the fundamental domains ModularGroupFundamentalDomain (\mathcal{F}) of the modular group and ModularLambdaFundamentalDomain (\mathcal{F}_{\lambda}) of the modular lambda function.

The Weierstrass elliptic functions for the lattice with periods 1 and \tau, WeierstrassP(z, tau) (\wp(z, \tau)), WeierstrassZeta(z, tau) (\zeta(z, \tau)) and WeierstrassSigma(z, tau) (\sigma(z, \tau)), and the roots EllipticRootE(j, tau) (e_{j}\!\left(\tau\right)) of the cubic polynomial associated with \wp.

The Hilbert class polynomial HilbertClassPolynomial(D, x) (H_{D}\!\left(x\right)) and the set of primitive reduced positive definite binary quadratic forms of discriminant D, PrimitiveReducedPositiveIntegralBinaryQuadraticForms(D) (\mathcal{Q}^{*}_{D}).

Parentheses(x), Brackets(x), Braces(x) and AngleBrackets(x) denote x but render it inside delimiters: Parentheses(x) (\left(x\right)), Brackets(x) (\left[x\right]), Braces(x) (\left\{x\right\}), AngleBrackets(x) (\left\langle x\right\rangle).

Ellipsis (\ldots) renders as an ellipsis, for example to indicate omitted arguments; it has no builtin meaning.

Subscript(x, n) ({x}_{n}) renders x with a subscript.

Logic(x) denotes x, but logical connectives within x are rendered with symbols instead of words, as with FEXPR_LATEX_LOGIC: Logic(And(P, Not(Q))) (P \,\land\, \neg Q).

ShowExpandedNormalForm(x) denotes x, but renders its expanded normal form (see fexpr_expanded_normal_form) instead of x itself. This triggers a computation that can be expensive.

Lowercase Greek letters and \ell, such as alpha (\alpha), epsilon (\varepsilon), lamda (\lambda), spelled without b, and ell (\ell).

Subscripted lowercase Greek letters, such as alpha_(n) (\alpha_{n}) or ell_(i, j) (\ell_{i, j}).

Uppercase Greek letters, such as GreekGamma (\Gamma), Delta (\Delta) and Omega (\Omega). GreekGamma and GreekPi are named so as not to clash with the functions Gamma and the constant Pi.

Subscripted uppercase Greek letters, such as Delta_(n) (\Delta_{n}).

XX(n) (\text{x}_{n}) renders as a subscripted upright x, for indexed variables \text{x}_1, \text{x}_2, \ldots.