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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
Repeatx,,xn times\underbrace{x, \ldots, x}_{n \text{ times}} Repeating sequence
Stepf(a),f ⁣(a+1),,f(b)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
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
Notnotx \operatorname{not} x Logical not
Andx  and  yx \;\mathbin{\operatorname{and}}\; y Logical and
Orx  or  yx \;\mathbin{\operatorname{or}}\; y Logical or
Equivalent(x)    (y)\left(x\right) \iff \left(y\right) Logical equivalence
Implies(x)        (y)\left(x\right) \;\implies\; \left(y\right) Logical implication
Set{}\left\{\ldots\right\} Set with given elements
Cardinality#S\# S Set cardinality
PowerSetP(S)\mathscr{P}(S) Power set
UnionSTS \cup T Set union
IntersectionSTS \cap T Set intersection
SetMinusSTS \setminus T Set difference
ElementxSx \in S Set membership
NotElementxSx \notin S Set non-membership
SubsetSTS \subset T Strict subset
SubsetEqualSTS \subseteq T Subset
ZZZ\mathbb{Z} Integers
QQQ\mathbb{Q} Rational numbers
RRR\mathbb{R} Real numbers
CCC\mathbb{C} Complex numbers
AlgebraicNumbersQ\overline{\mathbb{Q}} Algebraic numbers
Infinity\infty Positive infinity
UnsignedInfinity~{\tilde \infty} Unsigned infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
ClosedInterval[a,b]\left[a, b\right] Closed interval
OpenInterval(a,b)\left(a, b\right) Open interval
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Sumnf(n)\sum_{n} f(n) Sum
Productnf(n)\prod_{n} f(n) Product
PrimeSumpf(p)\sum_{p} f(p) Sum over primes
PrimeProductpf(p)\prod_{p} f(p) Product over primes
DivisorSumknf(k)\sum_{k \mid n} f(k) Sum over divisors
DivisorProductknf(k)\prod_{k \mid n} f(k) Product over divisors
ZeroszerosxSf(x)\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x) Zeros (roots) of function
UniqueZerozero*xSf(x)\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x) Unique zero (root) of function
SolutionssolutionsxSQ(x)\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x) Solution set
UniqueSolutionsolution*xSQ(x)\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x) Unique solution
SupremumsupxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x) Supremum of a set or function
InfimuminfxSf(x)\mathop{\operatorname{inf}}\limits_{x \in S} f(x) Infimum of a set or function
MinimumminxSf(x)\mathop{\min}\limits_{x \in S} f(x) Minimum value of a set or function
MaximummaxxSf(x)\mathop{\max}\limits_{x \in S} f(x) Maximum value of a set or function
ArgMinarg minxSf(x)\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x) Locations of minimum value
ArgMaxarg maxxSf(x)\mathop{\operatorname{arg\,max}}\limits_{x \in S} f(x) Locations of maximum value
ArgMinUniquearg min*xSf(x)\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x) Unique location of minimum value
ArgMaxUniquearg max*xSf(x)\mathop{\operatorname{arg\,max*}}\limits_{x \in S} f(x) Unique location of maximum value
Limitlimxaf(x)\lim_{x \to a} f(x) Limiting value
SequenceLimitlimnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
RealLimitlimxaf(x)\lim_{x \to a} f(x) Limiting value, real variable
LeftLimitlimxaf(x)\lim_{x \to {a}^{-}} f(x) Limiting value, from the left
RightLimitlimxa+f(x)\lim_{x \to {a}^{+}} f(x) Limiting value, from the right
ComplexLimitlimzaf(z)\lim_{z \to a} f(z) Limiting value, complex variable
MeromorphicLimitlimzaf(z)\lim_{z \to a} f(z) Limiting value, allowing poles
SequenceLimitInferiorlim infnaf(n)\liminf_{n \to a} f(n) Limit inferior of sequence
SequenceLimitSuperiorlim supnaf(n)\limsup_{n \to a} f(n) Limit superior of sequence
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
RealDerivativeddxf ⁣(x)\frac{d}{d x}\, f\!\left(x\right) Real derivative
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
ComplexBranchDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing branch cuts
MeromorphicDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing poles
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
IndefiniteIntegralEqualf(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral
RealIndefiniteIntegralEqualf(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral, real derivative
ComplexIndefiniteIntegralEqualf(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral, complex derivative
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
IsMeromorphicf(z) is meromorphic at z=cf(z) \text{ is meromorphic at } z = c Meromorphic predicate
ComplexZeroMultiplicityordz=cf(z)\mathop{\operatorname{ord}}\limits_{z=c} f(z) Multiplicity (order) of complex zero
Residueresz=cf(z)\mathop{\operatorname{res}}\limits_{z=c} f(z) Complex residue
Pathabca \rightsquigarrow b \rightsquigarrow c Line path
CurvePath(f(t),t:ab)\left(f(t),\, t : a \rightsquigarrow b\right) Path along a curve
AnalyticContinuationContinuationz:abf(z)\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z) Analytic continuation
Signsgn(z)\operatorname{sgn}(z) Sign function
Absz\left|z\right| Absolute value
Argarg(z)\arg(z) Complex argument
ReRe(z)\operatorname{Re}(z) Real part
ImIm(z)\operatorname{Im}(z) Imaginary part
Conjugatez\overline{z} Complex conjugate
Csgncsgn(z)\operatorname{csgn}(z) Real-valued sign function for complex numbers
ConstGammaγ\gamma The constant gamma (0.577...)
Piπ\pi The constant pi (3.14...)
GoldenRatioφ\varphi The golden ratio (1.618...)
ConstIii Imaginary unit
Expez{e}^{z} Exponential function
ConstEee The constant e (2.718...)
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
Sinsin(z)\sin(z) Sine
Sincsinc(z)\operatorname{sinc}(z) Sinc function
Atanatan(z)\operatorname{atan}(z) Inverse tangent
Atan2atan2 ⁣(y,x)\operatorname{atan2}\!\left(y, x\right) Two-argument inverse tangent
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
AGMSequenceagmn ⁣(a,b)\operatorname{agm}_{n}\!\left(a, b\right) Convergents in AGM iteration
EllipticKK(m)K(m) Legendre complete elliptic integral of the first kind
EllipticEE(m)E(m) Legendre complete elliptic integral of the second kind
EllipticPiΠ ⁣(n,m)\Pi\!\left(n, m\right) Legendre complete elliptic integral of the third kind
IncompleteEllipticFF ⁣(ϕ,m)F\!\left(\phi, m\right) Legendre incomplete elliptic integral of the first kind
IncompleteEllipticEE ⁣(ϕ,m)E\!\left(\phi, m\right) Legendre incomplete elliptic integral of the second kind
IncompleteEllipticPiΠ ⁣(n,ϕ,m)\Pi\!\left(n, \phi, m\right) Legendre incomplete elliptic integral of the third kind
CarlsonRFRF ⁣(x,y,z)R_F\!\left(x, y, z\right) Carlson symmetric elliptic integral of the first kind
CarlsonRGRG ⁣(x,y,z)R_G\!\left(x, y, z\right) Carlson symmetric elliptic integral of the second kind
CarlsonRJRJ ⁣(x,y,z,w)R_J\!\left(x, y, z, w\right) Carlson symmetric elliptic integral of the third kind
CarlsonRCRC ⁣(x,y)R_C\!\left(x, y\right) Degenerate Carlson symmetric elliptic integral of the first kind
CarlsonRDRD ⁣(x,y,z)R_D\!\left(x, y, z\right) Degenerate Carlson symmetric elliptic integral of the third kind
CarlsonHypergeometricRRa ⁣(b,z)R_{-a}\!\left(b, z\right) Carlson multivariate hypergeometric function
CarlsonHypergeometricTTN ⁣(b,z)T_{N}\!\left(b, z\right) Term in expansion of Carlson multivariate hypergeometric function
LambertWW ⁣(z)W\!\left(z\right) Lambert W-function
LambertWPuiseuxCoefficientμk\mu_{k} Coefficient in scaled Puiseux expansion of Lambert W-function
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
LCMlcm ⁣(a,b)\operatorname{lcm}\!\left(a, b\right) Least common multiple
XGCDxgcd ⁣(a,b)\operatorname{xgcd}\!\left(a, b\right) Extended greatest common divisor
Factorialn!n ! Factorial
Binomial(nk){n \choose k} Binomial coefficient
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
FallingFactorial(z)k\left(z\right)^{\underline{k}} Falling factorial
FibonacciFnF_{n} Fibonacci number
GammaΓ(z)\Gamma(z) Gamma function
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
StirlingSeriesRemainderRn ⁣(z)R_{n}\!\left(z\right) Remainder term in the Stirling series for the logarithmic gamma function
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Loglog(z)\log(z) Natural logarithm
PartitionsPp(n)p(n) Integer partition function
HardyRamanujanAA ⁣(n,k)A\!\left(n, k\right) Exponential sum in the Hardy-Ramanujan-Rademacher formula
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
RiemannZetaZeroρn\rho_{n} Nontrivial zero of the Riemann zeta function
RiemannHypothesisRH\operatorname{RH} Riemann hypothesis
DeBruijnNewmanLambdaΛ\Lambda De Bruijn-Newman constant
KeiperLiLambdaλn\lambda_{n} Keiper-Li coefficient
StieltjesGammaγn ⁣(a)\gamma_{n}\!\left(a\right) Stieltjes constant
AiryAiAi ⁣(z)\operatorname{Ai}\!\left(z\right) Airy function of the first kind
AiryBiBi ⁣(z)\operatorname{Bi}\!\left(z\right) Airy function of the second kind
BesselJJν ⁣(z)J_{\nu}\!\left(z\right) Bessel function of the first kind
BesselYYν ⁣(z)Y_{\nu}\!\left(z\right) Bessel function of the second kind
BesselIIν ⁣(z)I_{\nu}\!\left(z\right) Modified Bessel function of the first kind
BesselKKν ⁣(z)K_{\nu}\!\left(z\right) Modified Bessel function of the second kind
HankelH1Hν(1) ⁣(z)H^{(1)}_{\nu}\!\left(z\right) Hankel function of the first kind
HankelH2Hν(2) ⁣(z)H^{(2)}_{\nu}\!\left(z\right) Hankel function of the second kind
CoulombFF,η ⁣(z)F_{\ell,\eta}\!\left(z\right) Regular Coulomb wave function
CoulombGG,η ⁣(z)G_{\ell,\eta}\!\left(z\right) Irregular Coulomb wave function
CoulombHH,ηω ⁣(z)H^{\omega}_{\ell,\eta}\!\left(z\right) Outgoing and ingoing Coulomb wave function
CoulombCC ⁣(η)C_{\ell}\!\left(\eta\right) Coulomb wave function Gamow factor
CoulombSigmaσ ⁣(η)\sigma_{\ell}\!\left(\eta\right) Coulomb wave function phase shift
BernoulliBBnB_{n} Bernoulli number
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
StirlingCycle[nk]\left[{n \atop k}\right] Unsigned Stirling number of the first kind
StirlingS1s ⁣(n,k)s\!\left(n, k\right) Signed Stirling number of the first kind
StirlingS2{nk}\left\{{n \atop k}\right\} Stirling number of the second kind
HHH\mathbb{H} Upper complex half-plane
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Hypergeometric2F1Regularized2F1 ⁣(a,b,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) Regularized Gauss hypergeometric function
Hypergeometric0F10F1 ⁣(a,z)\,{}_0F_1\!\left(a, z\right) Confluent hypergeometric limit function
Hypergeometric0F1Regularized0F1 ⁣(a,z)\,{}_0{\textbf F}_1\!\left(a, z\right) Regularized confluent hypergeometric limit function
Hypergeometric1F11F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric function
Hypergeometric1F1Regularized1F1 ⁣(a,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) Regularized Kummer confluent hypergeometric function
HypergeometricUU ⁣(a,b,z)U\!\left(a, b, z\right) Tricomi confluent hypergeometric function
HypergeometricUStarU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
Hypergeometric2F02F0 ⁣(a,b,z)\,{}_2F_0\!\left(a, b, z\right) Tricomi confluent hypergeometric function, alternative notation
HypergeometricUStarRemainderRn ⁣(a,b,z)R_{n}\!\left(a,b,z\right) Error term in asymptotic expansion of Tricomi confluent hypergeometric function
Erferf(z)\operatorname{erf}(z) Error function
Erfcerfc(z)\operatorname{erfc}(z) Complementary error function
Erfierfi(z)\operatorname{erfi}(z) Imaginary error function
JacobiThetaθj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) Jacobi theta function
JacobiThetaPermutationSj ⁣(a,b,c,d)S_{j}\!\left(a, b, c, d\right) Index permutation in modular transformation of Jacobi theta functions
JacobiThetaEpsilonεj ⁣(a,b,c,d)\varepsilon_{j}\!\left(a, b, c, d\right) Root of unity in modular transformation of Jacobi theta functions
WeierstrassP ⁣(z,τ)\wp\!\left(z, \tau\right) Weierstrass elliptic function
WeierstrassZetaζ ⁣(z,τ)\zeta\!\left(z, \tau\right) Weierstrass zeta function
WeierstrassSigmaσ ⁣(z,τ)\sigma\!\left(z, \tau\right) Weierstrass sigma function
LatticeΛ(a,b)\Lambda_{(a, b)} Complex lattice with periods a, b
PPP\mathbb{P} Prime numbers
PrimeNumberpnp_{n} nth prime number
PrimePiπ(x)\pi(x) Prime counting function
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
PSL2ZPSL2(Z)\operatorname{PSL}_2(\mathbb{Z}) Modular group (canonical representatives)
ModularGroupActionγτ\gamma \circ \tau Action of modular group
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
ModularJj(τ)j(\tau) Modular j-invariant
PrimitiveReducedPositiveIntegralBinaryQuadraticFormsQD\mathcal{Q}^{*}_{D} Primitive reduced positive integral binary quadratic forms
HilbertClassPolynomialHD ⁣(x)H_{D}\!\left(x\right) Hilbert class polynomial
DedekindEtaη(τ)\eta(\tau) Dedekind eta function
EulerQSeriesϕ(q)\phi(q) Euler's q-series
DedekindEtaEpsilonε ⁣(a,b,c,d)\varepsilon\!\left(a, b, c, d\right) Root of unity in the functional equation of the Dedekind eta function
DedekindSums ⁣(n,k)s\!\left(n, k\right) Dedekind sum
EisensteinGGk ⁣(τ)G_{k}\!\left(\tau\right) Eisenstein series
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
ModularLambdaλ(τ)\lambda(\tau) Modular lambda function
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
DirichletGroupGqG_{q} Dirichlet characters with given modulus
PrimitiveDirichletCharactersGqPrimitiveG^{\text{Primitive}}_{q} Primitive Dirichlet characters with given modulus
DirichletCharacterχq.\chi_{q \, . \, \ell} Dirichlet character
ConreyGeneratorgpg_{p} Conrey generator
DiscreteLoglogb ⁣(x)modq\log_{b}\!\left(x\right) \bmod q Discrete logarithm
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
GeneralizedBernoulliBBn,χB_{n,\chi} Generalized Bernoulli number
DirichletLZeroρn,χ\rho_{n,\chi} Nontrivial zero of Dirichlet L-function
GeneralizedRiemannHypothesisGRH\operatorname{GRH} Generalized Riemann hypothesis
DirichletLambdaΛ ⁣(s,χ)\Lambda\!\left(s, \chi\right) Completed Dirichlet L-function
GaussSumGq ⁣(χ)G_{q}\!\left(\chi\right) Gauss sum
BetaFunctionB ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
IncompleteBetaBx ⁣(a,b)\mathrm{B}_{x}\!\left(a, b\right) Incomplete beta function
IncompleteBetaRegularizedIx ⁣(a,b)I_{x}\!\left(a, b\right) Regularized incomplete beta function
Totientφ(n)\varphi(n) Euler totient function
LandauGg(n)g(n) Landau's function
ConstCatalanGG Catalan's constant
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
DigammaFunctionZeroxnx_{n} Zero of the digamma function
SloaneAA00000X ⁣(n)\text{A00000X}\!\left(n\right) Sequence X in Sloane's OEIS
MultiZetaValueζ ⁣(s1,,sk)\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right) Multiple zeta value (MZV)
BellNumberBnB_{n} Bell number
BarnesGG(z)G(z) Barnes G-function
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
LogBarnesGRemainderRN ⁣(z)R_{N}\!\left(z\right) Remainder term in asymptotic expansion of logarithmic Barnes G-function
HalphenConstantΛ\Lambda Halphen's constant (one-ninth constant) 0.10765...
Coscos(z)\cos(z) Cosine
SquaresRrk ⁣(n)r_{k}\!\left(n\right) Sum of squares function
LiouvilleLambdaλ(n)\lambda(n) Liouville function
DivisorSigmaσk ⁣(n)\sigma_{k}\!\left(n\right) Sum of divisors function
MoebiusMuμ(n)\mu(n) Möbius function
KroneckerDeltaδ(x,y)\delta_{(x,y)} Kronecker delta
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
LegendrePolynomialZeroxn,kx_{n,k} Legendre polynomial zero
GaussLegendreWeightwn,kw_{n,k} Gauss-Legendre quadrature weight
HermitePolynomialHn ⁣(z)H_{n}\!\left(z\right) Hermite polynomial
BernsteinEllipseEρ\mathcal{E}_{\rho} Bernstein ellipse with foci -1,+1 and semi-axis sum rho
UnitCircleT\mathbb{T} Unit circle
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
LogIntegralli(z)\operatorname{li}(z) Logarithmic integral
RiemannXiξ(s)\xi(s) Riemann xi-function
PowerSeriesK[[x]]K[[x]] Formal power series
LaurentSeriesK( ⁣(x) ⁣)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.

2021-03-15 19:12:00.328586 UTC