# 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$W_{k}\!\left(z\right)$ 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$G_{q}^{\text{primitive}}$ 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC