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Notations

Listing of the Mathematical Notations used in the Mathematical Functions Website












Notations





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 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 GegenbauerC

Identities 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 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 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 generalized harmonic number of order : .

Identities containing HarmonicNumber

The Hermite function in : . For nonnegative integer it is a polynomial in .

Identities containing HermiteH

Identities 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 LaguerreL

Identities containing LaguerreL

The generalized Laguerre polynomial in for parameter : . For nonnegative integer it is a polynomial in .

Identities containing LaguerreL

Identities 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 PolyLog

Identities 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 LegendreP

Identities containing LegendreP

The associated Legendre function of the first kind of type 2: .

Identities containing LegendreP

Identities 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 JacobiP

Identities 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 Legendre function of the second kind: .

Identities containing LegendreQ

associated

The 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 ChebyshevT

Identities 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 ChebyshevU

Identities 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 SphericalHarmonicY

Identities 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

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 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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