Notations Listing of the Mathematical Notations used in the Mathematical Functions Website

 Functions in alphabetical order The characteristic value for even Mathieu functions with characteristic exponent and parameter , such that there exists a solution of the corresponding Mathieu differential equation that is of the form , where is an even function of with period .Identities containing MathieuCharacteristicA The Glaisher constant : Identities containing Glaisher The arithmetic‐geometric mean of and : .Identities containing ArithmeticGeometricMean The root of the equation : .Identities containing AiryAiZero The Airy function Ai: .Identities containing AiryAi The first derivative of the Airy function Ai: .Identities containing AiryAiPrime Jacobi amplitude function with module . The value of for which the elliptic integral of the first kind has the value :.Identities containing JacobiAmplitude The argument of the complex number (where ): .Identities containing Arg The characteristic value for odd Mathieu functions with characteristic exponent and parameter , such that there exists a solution of the corresponding Mathieu differential equation that is of the form , where is an odd function of with period .Identities containing MathieuCharacteristicB The Bell number: .Identities containing BellB The Bell polynomial of order in : .Identities containing BellB The Bernoulli number: .Identities containing BernoulliB The Bernoulli polynomial of order in : .Identities containing BernoulliB The Norlund polynomial B of order in : .Identities containing NorlundB The Norlund polynomial B: .Identities containing NorlundB The Kelvin function of the first kind bei: .Identities containing KelvinBei The Kelvin function of the first kind bei: .Identities containing KelvinBei The Kelvin function of the first kind ber: .Identities containing KelvinBer The Kelvin function of the first kind ber: .Identities containing KelvinBer The root of the equation : .Identities containing AiryBiZero The Airy function Bi: .Identities containing AiryBi The first derivative of the Airy function Bi: .Identities containing AiryBiPrime The Catalan constant : Identities containing Catalan The Fresnel integral C: .Identities containing FresnelC The Catalan numbers: .Identities containing CatalanNumber The cyclotomic polynomial of order in : .Identities containing Cyclotomic The renormalized form of the Gegenbauer function in : . For the nonnegative integer , the function is a polynomial in .Identities containing GegenbauerC The Gegenbauer function in for parameter : . For the nonnegative integer , the function is a polynomial in .Identities containing GegenbauerCIdentities containing GegenbauerC The Jacobi elliptic function cd: .Identities containing JacobiCD The inverse of the Jacobi elliptic function cd is the value of for which the Jacobi elliptic function cd, such that .Identities containing InverseJacobiCD The even Mathieu function with characteristic value and parameter .Identities containing MathieuC The derivative with respect to of the even Mathieu function with characteristic value and parameter : .Identities containing MathieuCPrime The hyperbolic cosine integral function: .Identities containing CoshIntegral The cosine integral function: ⩵.Identities containing CosIntegral The Jacobi elliptic function cn: .Identities containing JacobiCN The inverse of the Jacobi elliptic function cn. The value of such that .Identities containing InverseJacobiCN The cosine function: .Identities containing Cos The inverse cosine function: .Identities containing ArcCos The hyperbolic cosine function: .Identities containing Cosh The inverse hyperbolic cosine function: .Identities containing ArcCosh The cotangent function: .Identities containing Cot The inverse cotangent function: .Identities containing ArcCot The hyperbolic cotangent function: .Identities containing Coth The inverse hyperbolic cotangent function: .Identities containing ArcCoth The Jacobi elliptic function cs: .Identities containing JacobiCS The inverse of the Jacobi elliptic function cs. The value of such that .Identities containing InverseJacobiCS The cosecant function: .Identities containing Csc The inverse cosecant function: .Identities containing ArcCsc The hyperbolic cosecant function: .Identities containing Csch The inverse hyperbolic cosecant function: .Identities containing ArcCsch The Wigner ‐function: The parabolic cylinder function D: .Identities containing ParabolicCylinderD The Wigner ‐function: . The Jacobi elliptic function dc: .Identities containing JacobiDC The inverse of the Jacobi elliptic function dc. The value of such that .Identities containing InverseJacobiDC The denominator of . The list of the integers that divide .Identities containing Divisors The Jacobi elliptic function dn: .Identities containing JacobiDN The inverse of the Jacobi elliptic function dn. The value of such that .Identities containing InverseJacobiDN The Jacobi elliptic function ds: .Identities containing JacobiDS The inverse of the Jacobi elliptic function ds. The value of such that .Identities containing InverseJacobiDS The Euler exponential constant : Identities containing E Exponential function: .Identities containing Exp The values of the Weierstrass function at the half‐periods : . The values of the Weierstrass function at the half‐periods : . The complete elliptic integral of the second kind: .Identities containing EllipticE The elliptic integral of the second kind: .Identities containing EllipticE The Euler number: Identities containing EulerE The Euler polynomial of order in : .Identities containing EulerE The exponential integral : .Identities containing ExpIntegralE The elliptic exponential function . The values such that .Identities containing EllipticExp The first derivative of the elliptic exponential function with respect to : .Identities containing EllipticExpPrime The extended greatest common divisor of the integers and : Identities containing ExtendedGCD The generalized elliptic logarithm associated with the elliptic curve : .Identities containing EllipticLog The error function: .Identities containing Erf The inverse of the error function. The value of such that .Identities containing InverseErf The generalized error function: .Identities containing Erf The inverse of the generalized error function. The value of such that .Identities containing InverseErf The complementary error function: .Identities containing Erfc The inverse of the complementary error function. The value of such that .Identities containing InverseErfc The imaginary error function: .Identities containing Erfi The exponential integral function Ei: .Identities containing ExpIntegralEi The Fibonacci number: .Identities containing Fibonacci The Fibonacci polynomial of order in : .Identities containing FibonacciIdentities containing Fibonacci The elliptic integral of the first kind: .Identities containing EllipticF The Appell hypergeometric function of two variables .Identities containing AppellF1 The generalized hypergeometric function .Identities containing HypergeometricPFQ The generalized hypergeometric function .Identities containing HypergeometricPFQ The generalized hypergeometric function .Identities containing Hypergeometric0F1 The regularized generalized hypergeometric function .Identities containing Hypergeometric0F1Regularized The Kummer confluent hypergeometric function .Identities containing Hypergeometric1F1 The regularized confluent hypergeometric function .Identities containing Hypergeometric1F1Regularized The Gauss hypergeometric function .Identities containing Hypergeometric2F1 The regularized Gauss hypergeometric function .Identities containing Hypergeometric2F1Regularized The generalized hypergeometric function .Identities containing HypergeometricPFQ The generalized hypergeometric function .Identities containing HypergeometricPFQ The generalized hypergeometric function .Identities containing HypergeometricPFQ The generalized hypergeometric function .Identities containing HypergeometricPFQ The generalized hypergeometric function .Identities containing HypergeometricPFQ The generalized hypergeometric function .Identities containing HypergeometricPFQ The regularized generalized hypergeometric function .Identities containing HypergeometricPFQRegularized The generalized hypergeometric function of two variables (Kampe de Feriet function): . The regularized generalized hypergeometric function of two variables (regularized Kampe de Feriet function): . The Lauricella function A of variables: . The Lauricella function B of variables: . The Lauricella function C of variables: . The Lauricella function D of variables: . The prime factors of the integer , together with their exponents.Identities containing FactorInteger The fractional part of number : .Identities containing FractionalPart The Meijer G function: .The infinite contour of integration separates the poles of at , from the poles of at , . Such a contour always exists in the cases .There are three possibilities for the contour :(i) runs from γ-ⅈ ∞ to γ+ⅈ ∞ (where ) so that all poles of , are to the left of , and all poles of , are to the right of ℒ. This contour can be a straight line if (then ). (In this case, the integral converges if , . If , then must be real and positive and the additional condition should be added.)(ii) is a left loop, starting and ending at -∞ and encircling all poles of ,, once in the positive direction, but none of the poles of , . In this case, the integral converges if and one of the following conditions is satisfied: ◼ or and ◼ and and and .(iii) is a right loop, starting and ending at +∞ and encircling all poles of , , once in the negative direction, but none of the poles of , . In this case, the integral converges if and one of the following conditions is satisfied: ◼ or and ◼ and and and .Identities containing MeijerG The generalized Meijer G function: For the description of the contour , see .Identities containing MeijerG The Meijer G function of two variables: For the description of the contours and , see . The invariants for Weierstrass elliptic functions corresponding to the half‐periods : .Identities containing WeierstrassInvariants The greatest common divisor of the integers .Identities containing GCD The Hankel spherical function of the first kind H1: .Identities containing SphericalHankelH1 The Hankel spherical function of the second kind H2: .Identities containing SphericalHankelH2 The harmonic number: .Identities containing HarmonicNumber The generalized harmonic number of order : .Identities containing HarmonicNumber The Hermite function in : . For nonnegative integer it is a polynomial in .Identities containing HermiteHIdentities containing HermiteH The Struve function H: .Identities containing StruveH The Hankel function of the first kind H1: .Identities containing HankelH1 The Hankel function of the second kind H2: .Identities containing HankelH2 The Fox H function: The infinite contour of integration separates the poles of at , from the poles of at points , . The imaginary unit : .Identities containing I The modified Bessel function of the first kind: .Identities containing BesselI The regularized incomplete beta function: .Identities containing BetaRegularized The inverse of the regularized incomplete beta function. The value of such that .Identities containing InverseBetaRegularized The generalized regularized incomplete beta function: .Identities containing BetaRegularized The inverse of the generalized regularized incomplete beta function. The value of such that .Identities containing InverseBetaRegularized The imaginary part of the number : .Identities containing Im The integer part of number : .Identities containing IntegerPart The spherical Bessel function of the first kind: .Identities containing SphericalBesselJ The root of the equation : .Identities containing BesselJZero The Klein invariant modular function: .Identities containing KleinInvariantJ The Bessel function of the first kind: .Identities containing BesselJ The Khinchin constant : Identities containing Khinchin The complete elliptic integral of the first kind: .Identities containing EllipticK The modified Bessel function of the second kind: .Identities containing BesselK The Kelvin function of the second kind kei: Identities containing KelvinKei The Kelvin function of the second kind kei: .Identities containing KelvinKei The Kelvin function of the second kind ker: Identities containing KelvinKer The Kelvin function of the second kind ker: .Identities containing KelvinKer The Lucas number: .Identities containing LucasL The Laguerre function in : . For nonnegative integer it is a polynomial in .Identities containing LaguerreLIdentities containing LaguerreL The generalized Laguerre polynomial in for parameter : . For nonnegative integer it is a polynomial in .Identities containing LaguerreLIdentities containing LaguerreL The modified Struve function: .Identities containing StruveL The least common multiple of the integers (or rational) .Identities containing LCM The logarithmic integral: .Identities containing LogIntegral The polylogarithm function of : . For it is a dilogarithm function in .Identities containing PolyLogIdentities containing PolyLog The natural logarithm: .Identities containing Log The logarithm in base : .Identities containing Log The logarithmic gamma function: .Identities containing LogGamma The Whittaker hypergeometric function M: .Identities containing WhittakerM The maximum function (the numerically largest of the real numbers ): Identities containing Max The minimum function (the numerically smallest of the real numbers ): Identities containing Min The Jacobi elliptic function nc: .Identities containing JacobiNC The inverse of the Jacobi elliptic function nc. The value of such that .Identities containing InverseJacobiNC The Jacobi elliptic function nd: .Identities containing JacobiND The inverse of the Jacobi elliptic function nd. The value of such that .Identities containing InverseJacobiND The Jacobi elliptic function ns: .Identities containing JacobiNS The inverse of the Jacobi elliptic function ns. The value of such that .Identities containing InverseJacobiNS The number of unrestricted partitions (independent of the order and with repetitions allowed) of the positive integer into a sum of strictly positive integers that add up to : .Identities containing PartitionsP The prime number (the smallest integer greater than that cannot be divided by any integer greater than 1 and smaller than itself): .Identities containing Prime The Legendre function in : . For nonnegative integer it is a polynomial in .Identities containing LegendrePIdentities containing LegendreP The associated Legendre function of the first kind of type 2: .Identities containing LegendrePIdentities containing LegendreP The associated Legendre function of the second kind of type 3: .Identities containing LegendreP The Jacobi function in for parameters and : . For nonnegative integer it is a polynomial in .Identities containing JacobiPIdentities containing JacobiP A Boolean function that tests whether the angular momentum quantum numbers are physically realizable: The angular spheroidal function of the first kind with variable and parameters , , .Identities containing SpheroidalPS The derivative with respect to of the angular spheroidal function of the first kind with variable and parameters , , : .Identities containing SpheroidalPSPrime The number of ordered partitions (independent of the order and no repetitions allowed) of the positive integer into a sum of strictly positive integers which add up to : .Identities containing PartitionsQ The elliptic nome of the module : .Identities containing EllipticNomeQ The module of the nome : .Identities containing InverseEllipticNomeQ The Legendre function of the second kind: .Identities containing LegendreQ associatedThe associated Legendre function of the second kind of type 2: Identities containing LegendreQ The associated Legendre function of the second kind of type 3: Identities containing LegendreQ The regularized incomplete gamma function: .Identities containing GammaRegularized The inverse of the regularized incomplete gamma function. The value of such that .Identities containing InverseGammaRegularized The generalized regularized incomplete gamma function: .Identities containing GammaRegularized The inverse of the generalized regularized incomplete gamma function. The value such that .Identities containing InverseGammaRegularized The angular spheroidal function of the second kind with variable and parameters , , .Identities containing SpheroidalQS The derivative with respect to of the angular spheroidal function of the second kind with variable and parameters , , : .Identities containing SpheroidalQSPrime The integer quotient of and: .Identities containing Quotient The number of representations of as a sum of squares of different positive or negative integers.Identities containing SquaresR The characteristic exponent of the Mathieu functions. (where has period ).Identities containing MathieuCharacteristicExponent The Zernike polynomial R in : .Identities containing ZernikeR The real part of the number : .Identities containing Re The Fresnel integral S: .Identities containing FresnelS The Nielsen generalized polylogarithm: .Identities containing PolyLog The Stirling number of the first kind: .Identities containing StirlingS1 The Stirling number of the second kind: .Identities containing StirlingS2 The radial spheroidal function of the first kind with variable and parameters , , .Identities containing SpheroidalS1 The derivative with respect to of the radial spheroidal function of the first kind with variable and parameters , , : .Identities containing SpheroidalS1Prime The radial spheroidal function of the second kind with variable and parameters , , .Identities containing SpheroidalS2 The derivative with respect to of the radial spheroidal function of the second kind with variable and parameters , , : .Identities containing SpheroidalS2Prime The Jacobi elliptic function sc: .Identities containing JacobiSC The inverse of the Jacobi elliptic function sc. The value of such that .Identities containing InverseJacobiSC The Jacobi elliptic function sd: .Identities containing JacobiSD The inverse of the Jacobi elliptic function sd. The value of such that .Identities containing InverseJacobiSD The odd Mathieu function with characteristic value and parameter .Identities containing MathieuS The derivative with respect to of the odd Mathieu function with characteristic value and parameter : .Identities containing MathieuSPrime The secant function: .Identities containing Sec The inverse secant function: .Identities containing ArcSec The hyperbolic secant function: .Identities containing Sech The inverse hyperbolic secant function: .Identities containing ArcSech The signum of the number : Identities containing Sign The hyperbolic sine integral function: .Identities containing SinhIntegral The sine integral function: .Identities containing SinIntegral The sine function: .Identities containing Sin The inverse sine function: .Identities containing ArcSin The sinc (sampling) function: .Identities containing Sinc The hyperbolic sine function: Identities containing Sinh The inverse hyperbolic sine function: .Identities containing ArcSinh The Jacobi elliptic function sn: .Identities containing JacobiSN The inverse of the Jacobi elliptic function sn. The value of such that .Identities containing InverseJacobiSN The spheroidal joining factor of degree and order appearing in the relations between radial and angular spheroidal functions.Identities containing SpheroidalJoiningFactor The spheroidal radial factor of degree and order appearing in expansions of radial spheroidal function of the first kind around .Identities containing SpheroidalRadialFactor The subfactorial function (number of complete permutations).Identities containing Subfactorial The Chebyshev function of the first kind: . For nonnegative integer it is a polynomial in .Identities containing ChebyshevTIdentities containing ChebyshevT The tangent function: .Identities containing Tan The inverse tangent function: .Identities containing ArcTan The inverse tangent function of two variables: .Identities containing ArcTan The hyperbolic tangent function: .Identities containing Tanh The inverse hyperbolic tangent function: .Identities containing ArcTanh The Chebyshev function of the second kind: . For nonnegative integer it is a polynomial in .Identities containing ChebyshevUIdentities containing ChebyshevU The Tricomi hypergeometric function : .Identities containing HypergeometricU The Gauss type hypergeometric function : The product log function on the principal sheet. The value of such that .Identities containing ProductLog The product log function on the sheet. The value of such that .Identities containing ProductLog The Whittaker hypergeometric function W: .Identities containing WhittakerW The spherical Bessel function of the second kind: .Identities containing SphericalBesselY The root of the equation : .Identities containing BesselYZero The Bessel function of the second kind: .Identities containing BesselY The spherical harmonic function of and for parameters and : .Identities containing SphericalHarmonicYIdentities containing SphericalHarmonicY The Riemann-Siegel Zeta function: .Identities containing RiemannSiegelZ The Jacobi Zeta function: .Identities containing JacobiZeta The Euler beta function: .Identities containing Beta The incomplete beta function: .Identities containing Beta The generalized incomplete beta function: .Identities containing Beta Euler gamma : Identities containing EulerGamma The Stieltjes constant: .Identities containing StieltjesGamma The Euler gamma function: .Identities containing Gamma The incomplete gamma function: .Identities containing Gamma The generalized incomplete gamma function: .Identities containing Gamma The Dirac delta function: .Identities containing DiracDelta The multidimensional Dirac delta function: .Identities containing DiracDelta The discrete delta function: Identities containing DiscreteDelta The multidimensional discrete delta function: Identities containing DiscreteDelta The Kronecker delta function: Identities containing KroneckerDelta The signature of the permutation needed to place the list elements in canonical order.Identities containing Signature The Riemann zeta function: .Identities containing Zeta The generalized Riemann zeta function: .Identities containing Zeta The generalized classical Riemann zeta function: .Identities containing Zeta The regularized generalized classical Riemann zeta function: Identities containing Zeta The Weierstrass elliptic zeta function: Identities containing WeierstrassZeta The Dedekind eta modular function: .Identities containing DedekindEta The values of the Weierstrass zeta function at the half-periods : . The unit step function: .Identities containing UnitStep The multidimensional unit step: Identities containing UnitStep The Riemann‐Siegel theta function: .Identities containing RiemannSiegelTheta The first elliptic theta function: .Identities containing EllipticTheta The first derivative with respect to of the first elliptic theta function: .Identities containing EllipticThetaPrime The second elliptic theta function: .Identities containing EllipticTheta The first derivative with respect to of the second elliptic theta function: .Identities containing EllipticThetaPrime The third elliptic theta function: .Identities containing EllipticTheta The first derivative with respect to of the third elliptic theta function: .Identities containing EllipticThetaPrime The fourth elliptic theta function: .Identities containing EllipticTheta The first derivative with respect to of the fourth elliptic theta function: .Identities containing EllipticThetaPrime The Neville elliptic theta function C: .Identities containing NevilleThetaC The Neville elliptic theta function D: .Identities containing NevilleThetaD The Neville elliptic theta function N: .Identities containing NevilleThetaN The Neville elliptic theta function S: .Identities containing NevilleThetaS The Siegel theta function with symmetric Riemann modular matrix with positive definite imaginary part and vector is defined through , where means transposed to matrix (or vector) and ranges over all possible vectors in the -dimensional integer lattice: Identities containing SiegelTheta The Siegel theta function with characteristic , symmetric Riemann modular matrix with positive definite imaginary part and vector is defined through , where means transposed to matrix (or vector) and ranges over all possible vectors in the -dimensional integer lattice: Identities containing SiegelTheta The Carmichael lambda function: the smallest integer such that for any with the congruence holds.Identities containing CarmichaelLambda The lambda modular function: .Identities containing ModularLambda The eigenvalue of the spheroidal wave functions (the spheroidal eigenvalue of degree and order of the corresponding Sturm‐Liouville wave differential equation ).Identities containing SpheroidalEigenvalue The Möbius function : . Identities containing MoebiusMu The constant pi: Identities containing Pi The number of primes less than or equal to : .Identities containing PrimePi The complete elliptic integral of the third kind: .Identities containing EllipticPi The incomplete elliptic integral of the third kind: .Identities containing EllipticPi The nontrivial zero of the Riemann's zeta function on the critical half‐line : .Identities containing ZetaZero The sum of the powers of the divisors of : .Identities containing DivisorSigma The elliptic Weierstrass sigma function: Identities containing WeierstrassSigma The associated elliptic Weierstrass sigma function: Identities containing WeierstrassSigma The Ramanujan tau function of : .Identities containing RamanujanTau The Ramanujan tau L function: .Identities containing RamanujanTauL The Ramanujan tau Zeta function: .Identities containing RamanujanTauZ The Ramanujan tau theta function: .Identities containing RamanujanTauTheta The Lerch function: Identities containing LerchPhi The Lerch classical transcendent phi function: .Identities containing LerchPhi The Lerch classical regularized transcendent phi function: Identities containing LerchPhi The golden ratio : Identities containing GoldenRatio The number of positive integers less than and relatively prime to (the Euler totient function): .Identities containing EulerPhi The digamma function : .Identities containing PolyGamma The derivative of the digamma function: .Identities containing PolyGamma The half‐periods for Weierstrass elliptic functions corresponding to the invariants : Identities containing WeierstrassHalfPeriods The half‐periods for Weierstrass elliptic functions corresponding to the invariants : The Weierstrass elliptic function ℘: .Identities containing WeierstrassP The derivative with respect to of the Weierstrass elliptic function P: .Identities containing WeierstrassPPrime The inverse of the Weierstrass elliptic function . The value of such that : .Identities containing InverseWeierstrassP The inverse of the Weierstrass function . The value of such that and : .Identities containing InverseWeierstrassP The generalized Dirac comb function Ш(x): .

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