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All symbol definitions

Fungrim symbol	Notation	Short description
`Entry`	$\ldots$	Entry
`ID`	$\ldots$	Entry ID
`Formula`	$\ldots$	Formula
`Variables`	$\ldots$	Declaration of variables
`Assumptions`	$\ldots$	Assumptions (domain declaration) for the variables
`References`	$\ldots$	References
`Description`	$\ldots$	Text description
`For`	$\ldots$	General-purpose generator
`ForElement`	$\ldots$	Generator for all the elements of a set
`Repeat`	$\underbrace{x, \ldots, x}_{n \text{ times}}$	Repeating sequence
`Step`	$f(a), f\!\left(a + 1\right), \ldots, f(b)$	Enumerated sequence
`Parentheses`	$\left(\ldots\right)$	Parentheses
`Brackets`	$\left[\ldots\right]$	Square brackets
`Braces`	$\left\{\ldots\right\}$	Curly braces
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Not`	$\operatorname{not} x$	Logical not
`And`	$x \;\mathbin{\operatorname{and}}\; y$	Logical and
`Or`	$x \;\mathbin{\operatorname{or}}\; y$	Logical or
`Equivalent`	$\left(x\right) \iff \left(y\right)$	Logical equivalence
`Implies`	$\left(x\right) \;\implies\; \left(y\right)$	Logical implication
`Set`	$\left\{\ldots\right\}$	Set with given elements
`Cardinality`	$\# S$	Set cardinality
`PowerSet`	$\mathscr{P}(S)$	Power set
`Union`	$S \cup T$	Set union
`Intersection`	$S \cap T$	Set intersection
`SetMinus`	$S \setminus T$	Set difference
`Element`	$x \in S$	Set membership
`NotElement`	$x \notin S$	Set non-membership
`Subset`	$S \subset T$	Strict subset
`SubsetEqual`	$S \subseteq T$	Subset
`ZZ`	$\mathbb{Z}$	Integers
`QQ`	$\mathbb{Q}$	Rational numbers
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers
`Infinity`	$\infty$	Positive infinity
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`OpenInterval`	$\left(a, b\right)$	Open interval
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Sum`	$\sum_{n} f(n)$	Sum
`Product`	$\prod_{n} f(n)$	Product
`PrimeSum`	$\sum_{p} f(p)$	Sum over primes
`PrimeProduct`	$\prod_{p} f(p)$	Product over primes
`DivisorSum`	$\sum_{k \mid n} f(k)$	Sum over divisors
`DivisorProduct`	$\prod_{k \mid n} f(k)$	Product over divisors
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`UniqueZero`	$\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$	Unique zero (root) of function
`Solutions`	$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$	Solution set
`UniqueSolution`	$\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)$	Unique solution
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum of a set or function
`Infimum`	$\mathop{\operatorname{inf}}\limits_{x \in S} f(x)$	Infimum of a set or function
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function
`Maximum`	$\mathop{\max}\limits_{x \in S} f(x)$	Maximum value of a set or function
`ArgMin`	$\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x)$	Locations of minimum value
`ArgMax`	$\mathop{\operatorname{arg\,max}}\limits_{x \in S} f(x)$	Locations of maximum value
`ArgMinUnique`	$\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x)$	Unique location of minimum value
`ArgMaxUnique`	$\mathop{\operatorname{arg\,max*}}\limits_{x \in S} f(x)$	Unique location of maximum value
`Limit`	$\lim_{x \to a} f(x)$	Limiting value
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`RealLimit`	$\lim_{x \to a} f(x)$	Limiting value, real variable
`LeftLimit`	$\lim_{x \to {a}^{-}} f(x)$	Limiting value, from the left
`RightLimit`	$\lim_{x \to {a}^{+}} f(x)$	Limiting value, from the right
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`MeromorphicLimit`	$\lim_{z \to a} f(z)$	Limiting value, allowing poles
`SequenceLimitInferior`	$\liminf_{n \to a} f(n)$	Limit inferior of sequence
`SequenceLimitSuperior`	$\limsup_{n \to a} f(n)$	Limit superior of sequence
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`MeromorphicDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing poles
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`IndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral
`RealIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, real derivative
`ComplexIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, complex derivative
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`IsMeromorphic`	$f(z) \text{ is meromorphic at } z = c$	Meromorphic predicate
`ComplexZeroMultiplicity`	$\mathop{\operatorname{ord}}\limits_{z=c} f(z)$	Multiplicity (order) of complex zero
`Residue`	$\mathop{\operatorname{res}}\limits_{z=c} f(z)$	Complex residue
`Path`	$a \rightsquigarrow b \rightsquigarrow c$	Line path
`CurvePath`	$\left(f(t),\, t : a \rightsquigarrow b\right)$	Path along a curve
`AnalyticContinuation`	$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$	Analytic continuation
`Sign`	$\operatorname{sgn}(z)$	Sign function
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg(z)$	Complex argument
`Re`	$\operatorname{Re}(z)$	Real part
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Conjugate`	$\overline{z}$	Complex conjugate
`Csgn`	$\operatorname{csgn}(z)$	Real-valued sign function for complex numbers
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Pi`	$\pi$	The constant pi (3.14...)
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`ConstI`	$i$	Imaginary unit
`Exp`	${e}^{z}$	Exponential function
`ConstE`	$e$	The constant e (2.718...)
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`Sin`	$\sin(z)$	Sine
`Sinc`	$\operatorname{sinc}(z)$	Sinc function
`Atan`	$\operatorname{atan}(z)$	Inverse tangent
`Atan2`	$\operatorname{atan2}\!\left(y, x\right)$	Two-argument inverse tangent
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`EllipticK`	$K(m)$	Legendre complete elliptic integral of the first kind
`EllipticE`	$E(m)$	Legendre complete elliptic integral of the second kind
`EllipticPi`	$\Pi\!\left(n, m\right)$	Legendre complete elliptic integral of the third kind
`IncompleteEllipticF`	$F\!\left(\phi, m\right)$	Legendre incomplete elliptic integral of the first kind
`IncompleteEllipticE`	$E\!\left(\phi, m\right)$	Legendre incomplete elliptic integral of the second kind
`IncompleteEllipticPi`	$\Pi\!\left(n, \phi, m\right)$	Legendre incomplete elliptic integral of the third kind
`CarlsonRF`	$R_F\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the first kind
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`CarlsonRJ`	$R_J\!\left(x, y, z, w\right)$	Carlson symmetric elliptic integral of the third kind
`CarlsonRC`	$R_C\!\left(x, y\right)$	Degenerate Carlson symmetric elliptic integral of the first kind
`CarlsonRD`	$R_D\!\left(x, y, z\right)$	Degenerate Carlson symmetric elliptic integral of the third kind
`CarlsonHypergeometricR`	$R_{-a}\!\left(b, z\right)$	Carlson multivariate hypergeometric function
`CarlsonHypergeometricT`	$T_{N}\!\left(b, z\right)$	Term in expansion of Carlson multivariate hypergeometric function
`LambertW`	$W\!\left(z\right)$	Lambert W-function
`LambertWPuiseuxCoefficient`	$\mu_{k}$	Coefficient in scaled Puiseux expansion of Lambert W-function
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`LCM`	$\operatorname{lcm}\!\left(a, b\right)$	Least common multiple
`XGCD`	$\operatorname{xgcd}\!\left(a, b\right)$	Extended greatest common divisor
`Factorial`	$n !$	Factorial
`Binomial`	${n \choose k}$	Binomial coefficient
`RisingFactorial`	$\left(z\right)_{k}$	Rising factorial
`FallingFactorial`	$\left(z\right)^{\underline{k}}$	Falling factorial
`Fibonacci`	$F_{n}$	Fibonacci number
`Gamma`	$\Gamma(z)$	Gamma function
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`StirlingSeriesRemainder`	$R_{n}\!\left(z\right)$	Remainder term in the Stirling series for the logarithmic gamma function
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Log`	$\log(z)$	Natural logarithm
`PartitionsP`	$p(n)$	Integer partition function
`HardyRamanujanA`	$A\!\left(n, k\right)$	Exponential sum in the Hardy-Ramanujan-Rademacher formula
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`RiemannZetaZero`	$\rho_{n}$	Nontrivial zero of the Riemann zeta function
`RiemannHypothesis`	$\operatorname{RH}$	Riemann hypothesis
`DeBruijnNewmanLambda`	$\Lambda$	De Bruijn-Newman constant
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`AiryAi`	$\operatorname{Ai}\!\left(z\right)$	Airy function of the first kind
`AiryBi`	$\operatorname{Bi}\!\left(z\right)$	Airy function of the second kind
`BesselJ`	$J_{\nu}\!\left(z\right)$	Bessel function of the first kind
`BesselY`	$Y_{\nu}\!\left(z\right)$	Bessel function of the second kind
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`BesselK`	$K_{\nu}\!\left(z\right)$	Modified Bessel function of the second kind
`HankelH1`	$H^{(1)}_{\nu}\!\left(z\right)$	Hankel function of the first kind
`HankelH2`	$H^{(2)}_{\nu}\!\left(z\right)$	Hankel function of the second kind
`CoulombF`	$F_{\ell,\eta}\!\left(z\right)$	Regular Coulomb wave function
`CoulombG`	$G_{\ell,\eta}\!\left(z\right)$	Irregular Coulomb wave function
`CoulombH`	$H^{\omega}_{\ell,\eta}\!\left(z\right)$	Outgoing and ingoing Coulomb wave function
`CoulombC`	$C_{\ell}\!\left(\eta\right)$	Coulomb wave function Gamow factor
`CoulombSigma`	$\sigma_{\ell}\!\left(\eta\right)$	Coulomb wave function phase shift
`BernoulliB`	$B_{n}$	Bernoulli number
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`StirlingCycle`	$\left[{n \atop k}\right]$	Unsigned Stirling number of the first kind
`StirlingS1`	$s\!\left(n, k\right)$	Signed Stirling number of the first kind
`StirlingS2`	$\left\{{n \atop k}\right\}$	Stirling number of the second kind
`HH`	$\mathbb{H}$	Upper complex half-plane
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`Hypergeometric2F1Regularized`	$\,{}_2{\textbf F}_1\!\left(a, b, c, z\right)$	Regularized Gauss hypergeometric function
`Hypergeometric0F1`	$\,{}_0F_1\!\left(a, z\right)$	Confluent hypergeometric limit function
`Hypergeometric0F1Regularized`	$\,{}_0{\textbf F}_1\!\left(a, z\right)$	Regularized confluent hypergeometric limit function
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`Hypergeometric2F0`	$\,{}_2F_0\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function, alternative notation
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function
`Erf`	$\operatorname{erf}(z)$	Error function
`Erfc`	$\operatorname{erfc}(z)$	Complementary error function
`Erfi`	$\operatorname{erfi}(z)$	Imaginary error function
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`JacobiThetaPermutation`	$S_{j}\!\left(a, b, c, d\right)$	Index permutation in modular transformation of Jacobi theta functions
`JacobiThetaEpsilon`	$\varepsilon_{j}\!\left(a, b, c, d\right)$	Root of unity in modular transformation of Jacobi theta functions
`WeierstrassP`	$\wp\!\left(z, \tau\right)$	Weierstrass elliptic function
`WeierstrassZeta`	$\zeta\!\left(z, \tau\right)$	Weierstrass zeta function
`WeierstrassSigma`	$\sigma\!\left(z, \tau\right)$	Weierstrass sigma function
`Lattice`	$\Lambda_{(a, b)}$	Complex lattice with periods a, b
`PP`	$\mathbb{P}$	Prime numbers
`PrimeNumber`	$p_{n}$	nth prime number
`PrimePi`	$\pi(x)$	Prime counting function
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)
`ModularGroupAction`	$\gamma \circ \tau$	Action of modular group
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`ModularJ`	$j(\tau)$	Modular j-invariant
`PrimitiveReducedPositiveIntegralBinaryQuadraticForms`	$\mathcal{Q}^{*}_{D}$	Primitive reduced positive integral binary quadratic forms
`HilbertClassPolynomial`	$H_{D}\!\left(x\right)$	Hilbert class polynomial
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`EulerQSeries`	$\phi(q)$	Euler's q-series
`DedekindEtaEpsilon`	$\varepsilon\!\left(a, b, c, d\right)$	Root of unity in the functional equation of the Dedekind eta function
`DedekindSum`	$s\!\left(n, k\right)$	Dedekind sum
`EisensteinG`	$G_{k}\!\left(\tau\right)$	Eisenstein series
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`ModularLambda`	$\lambda(\tau)$	Modular lambda function
`ModularLambdaFundamentalDomain`	$\mathcal{F}_{\lambda}$	Fundamental domain of the modular lambda function
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ConreyGenerator`	$g_{p}$	Conrey generator
`DiscreteLog`	$\log_{b}\!\left(x\right) \bmod q$	Discrete logarithm
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`GeneralizedBernoulliB`	$B_{n,\chi}$	Generalized Bernoulli number
`DirichletLZero`	$\rho_{n,\chi}$	Nontrivial zero of Dirichlet L-function
`GeneralizedRiemannHypothesis`	$\operatorname{GRH}$	Generalized Riemann hypothesis
`DirichletLambda`	$\Lambda\!\left(s, \chi\right)$	Completed Dirichlet L-function
`GaussSum`	$G_{q}\!\left(\chi\right)$	Gauss sum
`BetaFunction`	$\mathrm{B}\!\left(a, b\right)$	Beta function
`IncompleteBeta`	$\mathrm{B}_{x}\!\left(a, b\right)$	Incomplete beta function
`IncompleteBetaRegularized`	$I_{x}\!\left(a, b\right)$	Regularized incomplete beta function
`Totient`	$\varphi(n)$	Euler totient function
`LandauG`	$g(n)$	Landau's function
`ConstCatalan`	$G$	Catalan's constant
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`DigammaFunctionZero`	$x_{n}$	Zero of the digamma function
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`BellNumber`	$B_{n}$	Bell number
`BarnesG`	$G(z)$	Barnes G-function
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`HalphenConstant`	$\Lambda$	Halphen's constant (one-ninth constant) 0.10765...
`Cos`	$\cos(z)$	Cosine
`SquaresR`	$r_{k}\!\left(n\right)$	Sum of squares function
`LiouvilleLambda`	$\lambda(n)$	Liouville function
`DivisorSigma`	$\sigma_{k}\!\left(n\right)$	Sum of divisors function
`MoebiusMu`	$\mu(n)$	Möbius function
`KroneckerDelta`	$\delta_{(x,y)}$	Kronecker delta
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`HermitePolynomial`	$H_{n}\!\left(z\right)$	Hermite polynomial
`BernsteinEllipse`	$\mathcal{E}_{\rho}$	Bernstein ellipse with foci -1,+1 and semi-axis sum rho
`UnitCircle`	$\mathbb{T}$	Unit circle
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`LogIntegral`	$\operatorname{li}(z)$	Logarithmic integral
`RiemannXi`	$\xi(s)$	Riemann xi-function
`PowerSeries`	$K[[x]]$	Formal power series
`LaurentSeries`	$K(\!(x)\!)$	Formal Laurent series
`List`	$\left[\ldots\right]$	List with given elements
`Tuple`	$\left(\ldots\right)$	Tuple with given elements