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All symbol definitions

Fungrim symbol Notation Short description
Notnot(x) \operatorname{not} \left(x\right) Logical not
Andxandyx \,\mathbin{\operatorname{and}}\, y Logical and
Orxoryx \,\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
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
CardinalityS\left|S\right| Set cardinality
PowerSetP ⁣(S)\mathscr{P}\!\left(S\right) 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
ConstIii Imaginary unit
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
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
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
SupremumsupP(x)f ⁣(x)\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right) Supremum of a set or function
InfimuminfP(x)f ⁣(x)\mathop{\operatorname{inf}}\limits_{P\left(x\right)} f\!\left(x\right) Infimum of a set or function
MinimumminP(x)f ⁣(x)\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right) Minimum value of a set or function
MaximummaxP(x)f ⁣(x)\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right) Maximum value of a set or function
ArgMinarg minP(x)f ⁣(x)\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right) Locations of minimum value
ArgMaxarg maxP(x)f ⁣(x)\mathop{\operatorname{arg\,max}}\limits_{P\left(x\right)} f\!\left(x\right) Locations of maximum value
ArgMinUniquearg min*P(x)f ⁣(x)\mathop{\operatorname{arg\,min*}}\limits_{P\left(x\right)} f\!\left(x\right) Unique location of minimum value
ArgMaxUniquearg max*P(x)f ⁣(x)\mathop{\operatorname{arg\,max*}}\limits_{P\left(x\right)} f\!\left(x\right) Unique location of maximum value
Signsgn ⁣(z)\operatorname{sgn}\!\left(z\right) Sign function
Absz\left|z\right| Absolute value
Argarg ⁣(z)\arg\!\left(z\right) Complex argument
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
Conjugatez\overline{z} Complex conjugate
ConstGammaγ\gamma The constant gamma (0.577...)
ConstPiπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
ConstEee The constant e (2.718...)
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
Sinsin ⁣(z)\sin\!\left(z\right) Sine
Atanatan ⁣(z)\operatorname{atan}\!\left(z\right) Inverse tangent
Atan2atan2 ⁣(y,x)\operatorname{atan2}\!\left(y, x\right) Two-argument inverse tangent
LambertWWk ⁣(z)W_{k}\!\left(z\right) Lambert W-function
LambertWPuiseuxCoefficientμk{\mu}_{k} Coefficient in scaled Puiseux expansion of Lambert W-function
GCDgcd ⁣(n,k)\gcd\!\left(n, k\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
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
LogGammalogΓ ⁣(z)\log \Gamma\!\left(z\right) Logarithmic gamma function
StirlingSeriesRemainderRn ⁣(z)R_{n}\!\left(z\right) Remainder term in the Stirling series for the logarithmic gamma function
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
PartitionsPp ⁣(n)p\!\left(n\right) 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
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
BesselJDerivativeJν(r) ⁣(z)J^{(r)}_{\nu}\!\left(z\right) Differentiated Bessel function of the first kind
BesselYDerivativeYν(r) ⁣(z)Y^{(r)}_{\nu}\!\left(z\right) Differentiated Bessel function of the second kind
BesselIDerivativeIν(r) ⁣(z)I^{(r)}_{\nu}\!\left(z\right) Differentiated modified Bessel function of the first kind
BesselKDerivativeKν(r) ⁣(z)K^{(r)}_{\nu}\!\left(z\right) Differentiated 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
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
BellNumberBnB_{n} Bell number
HHH\mathbb{H} Upper complex half-plane
Hypergeometric0F10F1 ⁣(a,z)\,{}_0F_1\!\left(a, z\right) Confluent hypergeometric limit function
Hypergeometric0F1Regularized0F~1 ⁣(a,z)\,{}_0{\tilde 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
Hypergeometric1F1Regularized1F~1 ⁣(a,b,z)\,{}_1{\tilde 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
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\!\left(x\right) Prime counting function
RiemannHypothesisRiemannHypothesis\operatorname{RiemannHypothesis} Riemann hypothesis
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\!\left(\tau\right) Modular j-invariant
EulerQSeriesϕ ⁣(q)\phi\!\left(q\right) Euler's q-series
DedekindEtaη ⁣(τ)\eta\!\left(\tau\right) Dedekind eta function
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
GCDgcd ⁣(n,k)\gcd\!\left(n, k\right) Greatest common divisor
DivisorSigmaσ ⁣(n)\sigma\!\left(n\right) Sum of divisors function
MoebiusMuμ ⁣(n)\mu\!\left(n\right) 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
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Hypergeometric2F1Regularized2F~1 ⁣(a,b,c,z)\,{}_2{\tilde F}_1\!\left(a, b, c, z\right) Regularized Gauss hypergeometric function
Erferf ⁣(z)\operatorname{erf}\!\left(z\right) Error function
Erfcerfc ⁣(z)\operatorname{erfc}\!\left(z\right) Complementary error function
Erfierfi ⁣(z)\operatorname{erfi}\!\left(z\right) Imaginary error function
JacobiTheta1θ1 ⁣(z,τ)\theta_1\!\left(z, \tau\right) Jacobi theta function
JacobiTheta2θ2 ⁣(z,τ)\theta_2\!\left(z, \tau\right) Jacobi theta function
JacobiTheta3θ3 ⁣(z,τ)\theta_3\!\left(z, \tau\right) Jacobi theta function
JacobiTheta4θ4 ⁣(z,τ)\theta_4\!\left(z, \tau\right) Jacobi theta function
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
LogIntegralli ⁣(z)\operatorname{li}\!\left(z\right) Logarithmic integral
FormalPowerSeriesK[[x]]K[[x]] Formal power series
FormalLaurentSeriesK( ⁣(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.

2019-05-23 08:00:13.607731 UTC