This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 MATHEMATICA TUTORIAL

TraditionalForm differs from StandardForm, the default format for input and output. It is important to understand that TraditionalForm expressions cannot always be provided as unambiguous input to Mathematica. Therefore, while StandardForm is an input format and an output format, TraditionalForm is primarily intended as an output format.
In general, the TraditionalForm representation of a mathematical function differs from its StandardForm representation in two ways: function arguments are enclosed in parentheses rather than square brackets, and one-character variable and function names are set in italics rather than plain text.
In addition to these general differences, TraditionalForm transforms a large group of expressions into their conventionally used mathematical notation. A table listing these expressions and their special TraditionalForm representations appears later in this tutorial.
This displays a mathematical function that does not have a special notation; the input is in StandardForm and the output is in TraditionalForm.
Here is an example of a function that has its own special TraditionalForm notation.
The TraditionalForm representation of matrices is shown here.
The TraditionalForm representations of Mathematica functions and commands distinct from conventional mathematics use square brackets, as in StandardForm.
Here is the TraditionalForm representation of the Mathematica function Plot.
The following tables list the expressions that have their own specific TraditionalForm representations. Entries marked with a star () contain hidden information (using TagBox or InterpretationBox constructs or specially designed characters) and may not be suitable for unambiguous input.

## Mathematical Constants and Domains

 StandardForm TraditionalForm Algebraics Booleans Catalan ChampernowneNumber[b] * Complexes EulerGamma Glaisher GoldenRatio Integers Khinchin * Primes Rationals Reals StieltjesGamma[n] * StieltjesGamma[n,a] *

Mathematical constants and domains.

## Numerical Functions

 StandardForm TraditionalForm Abs[z] Arg[z] Ceiling[z] Conjugate[z] * Floor[z] FractionalPart[x] Max[z] Min[z] Sign[z]

Numerical functions.

## Elementary Functions

 StandardForm TraditionalForm ArcCos[z] ArcCosh[z] ArcCot[z] ArcCoth[z] ArcCsc[z] ArcCsch[z] ArcSec[z] ArcSech[z] ArcSin[z] ArcSinh[z] ArcTan[z] ArcTanh[z] Cos[z] Cos[z]p Cosh[z] Cosh[z]p Cot[z] Cot[z]p Coth[z] Coth[z]p Csc[z] Csc[z]p Csch[z] Csch[z]p Log[z] Log[z]^p Log[b,z] Log[b,z]^p Sec[z] Sec[z]p Sech[z] Sech[z]p Sin[z] Sin[z]p Sinh[z] Sinh[z]p Tan[z] Tan[z]p Tanh[z] Tanh[z]p

Elementary functions.

## Factorial-Related Functions

 StandardForm TraditionalForm Beta[a,b] Beta[z,a,b] Beta[z0,z1,a,b] Binomial[n,m] Gamma[z] Gamma[a,z] Gamma[a,z1,z2] GammaRegularized[a,z] GammaRegularized[a,z0,z1] InverseBetaRegularized[s,a,b] InverseBetaRegularized[z0,s,a,b] LogGamma[z] Multinomial[n1,n2,...,nk] Pochhammer[a,n] PolyGamma[z] PolyGamma[n,z]

Factorial-related functions.

## Combinatorial Functions

 StandardForm TraditionalForm BernoulliB[n] BernoulliB[n,z] ClebschGordan[{j1,m1},{j2,m2},{j3,m3}] EulerE[n] EulerE[n,z] Fibonacci[n] Fibonacci[n,z] HarmonicNumber[n] HarmonicNumber[n,r] PartitionsP[z] PartitionsQ[z] Signature[e1,e2,...] SixJSymbol[{j1,j2,j3},{j4,j5,j6}] StirlingS1[n,m] StirlingS2[n,m] ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}]

Combinatorial functions.

## Number Theory

 StandardForm TraditionalForm ArithmeticGeometricMean[a,b] CarmichaelLambda[n] DivisorSigma[k,n] EulerPhi[n] GCD[n1,n2,...] JacobiSymbol[n,m] LCM[n1,n2,...] LiouvilleLambda[n] * MangoldtLambda[n] * Mod[m,n] MoebiusMu[n] MultiplicativeOrder[k,n] PowerMod[a,b,n] Prime[n] PrimeNu[n] * PrimeOmega[n] * PrimeZetaP[x] * PrimePi[z] RamanujanTau[n] RiemannR[x] * SquaresR[d,n]

Number theory.

## Zeta-Related Functions

 StandardForm TraditionalForm LerchPhi[z,s,a] PolyLog[n,z] PolyLog[n,p,z] RiemannSiegelTheta[t] RiemannSiegelZ[t] StieltjesGamma[z] Zeta[s] Zeta[s,a]

Zeta-related functions.

## Hypergeometric-Related Functions

 StandardForm TraditionalForm AiryAi[z] AiryAiPrime[z] AiryBi[z] AiryBiPrime[z] AngerJ[,x] * AngerJ[,,x] * AppellF1[a,b1,b2,c,x,y] BesselI[n,z] BesselJ[n,z] BesselK[n,z] BesselY[n,z] CosIntegral[z] CoshIntegral[z] DawsonF[x] * Erf[z] Erf[z0,z1] Erfc[z] Erfi[z] ExpIntegralE[n,z] ExpIntegralEi[z] FresnelC[z] FresnelS[z] Hypergeometric0F1[a,z] Hypergeometric0F1Regularized[a,z] Hypergeometric1F1[a,b,z] Hypergeometric1F1Regularized[a,b,z] Hypergeometric2F1[a,b,c,z] Hypergeometric2F1Regularized[a,b,c,z] HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] HypergeometricU[a,b,z] LegendreQ[n,x] LegendreQ[n,m,x] LegendreQ[n,m,a,z] LogIntegral[z] MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z] MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] SinIntegral[z] SinhIntegral[z] StruveH[,z] StruveL[,z] WeberE[,x] * WeberE[,,x] *

Hypergeometric-related functions.

## Orthogonal Polynomials

 StandardForm TraditionalForm ChebyshevT[n,x] ChebyshevU[n,x] GegenbauerC[n,x] GegenbauerC[n,m,x] HermiteH[n,x] JacobiP[n,a,b,x] LaguerreL[n,x] LaguerreL[n,a,x] LegendreP[n,x] LegendreP[n,m,x] LegendreP[n,m,a,z] SphericalHarmonicY[l,m,,]

Orthogonal polynomials.

## Inverse Functions

 StandardForm TraditionalForm InverseErf[z0,s] InverseFunction[f] ProductLog[z] ProductLog[k,z]

Inverse functions.

## Elliptic Integrals

 StandardForm TraditionalForm EllipticE[m] EllipticE[,m] EllipticF[,m] EllipticK[m] EllipticNomeQ[m] EllipticPi[n,m] EllipticPi[n,,m] JacobiZeta[,m]

Elliptic integrals.

## Elliptic Functions

 StandardForm TraditionalForm DedekindEta[t] EllipticTheta[a,u,q] EllipticThetaPrime[a,u,q] InverseEllipticNomeQ[q] InverseJacobiCD[u,m] InverseJacobiCN[u,m] InverseJacobiCS[u,m] InverseJacobiDC[u,m] InverseJacobiDN[u,m] InverseJacobiDS[u,m] InverseJacobiNC[u,m] InverseJacobiND[u,m] InverseJacobiNS[u,m] InverseJacobiSC[u,m] InverseJacobiSD[u,m] InverseJacobiSN[u,m] InverseWeierstrassP[p,{g2,g3}] JacobiAmplitude[u,m] JacobiCD[u,m] JacobiCN[u,m] JacobiCS[u,m] JacobiDC[u,m] JacobiDN[u,m] JacobiDS[u,m] JacobiNC[u,m] JacobiND[u,m] JacobiNS[u,m] JacobiSC[u,m] JacobiSD[u,m] JacobiSN[u,m] KleinInvariantJ[] ModularLambda[] NevilleThetaC[u,m] NevilleThetaD[u,m] NevilleThetaN[u,m] NevilleThetaS[u,m] WeierstrassP[u,{g2,g3}] WeierstrassPPrime[u,{g2,g3}] WeierstrassSigma[u,{g2,g3}] WeierstrassZeta[u,{g2,g3}]

Elliptic functions.

## Mathieu Functions

Mathieu functions.

## Generalized and Related Functions

 StandardForm TraditionalForm DiracDelta[x1,x2,...] DiscreteDelta[n1,n2,...] HeavisideLambda[x] * HeavisideLambda[x1,x2,...] * HeavisidePi[x] * HeavisidePi[x1,x2,...] * KroneckerDelta[n1,n2,...] UnitBox[x] * UnitBox[x1,x2,...] * UnitStep[x1,x2,...] UnitTriangle[x] * UnitTriangle[x1,x2,...] *

Generalized and related functions.

## Matrix Operations

Matrix operations.

## Logical Operations

 StandardForm TraditionalForm And[p1,p2,...] Implies[a,b] Nand[p1,p2,...] Nor[p1,p2,...] Not[p] Or[p1,p2,...] Xor[p1,p2,...]

Logical operations.

## Calculus

 StandardForm TraditionalForm C[n] D[f[x]] D[f[x],x] D[f[x],{x,2}] D[f[x],{x,n}] Dt[f[x]] Dt[f[x],x] Dt[f[x],{x,2}] Dt[f[x],{x,n}] Derivative[1][f] Derivative[2][f] Derivative[d1,...][f] FourierTransform[expr,t,s] FourierTransform[expr,{t1,t2,...},{s1,s2,...}] Integrate[expr,x] Integrate[expr,x1,y,z] Integrate[expr,{x,a,b}] Integrate[expr,{x,a,b},{y,m,n},{z,p,q}] InverseFourierTransform[expr,s,t] InverseFourierTransform[expr,{s1,s2,...},{t1,t2,...}] InverseLaplaceTransform[expr,s,t] InverseLaplaceTransform[expr,{s1,s2,...},{t1,t2,...}] LaplaceTransform[expr,t,s] LaplaceTransform[expr,{t1,t2,...},{s1,s2,...}] Limit[f[x],x->a] Limit[f[x],x->a,Direction->+1] Limit[f[x],x->a,Direction->-1] O[x] O[x]^n O[x,a] O[x,a]^n Piecewise[{{v1,c1},{v2,c2},...}] Residue[z] Series[f[x],{x,a,0}] Series[f[x],{x,a,1}] Series[Tan[z^(2/3)],{z,0,3}]

Calculus.

## Discrete Calculus

 StandardForm TraditionalForm DifferenceDelta[f,i] * DifferenceDelta[f,{i,n}] * DifferenceDelta[f,{i,n,h}] * DifferenceDelta[f,i,j,...] * DiscreteRatio[f,i] * DiscreteRatio[f,{i,n}] * DiscreteRatio[f,{i,n,h} * DiscreteRatio[f,i,j,...] * DiscreteShift[f,i] * DiscreteShift[f,{i,n}] * DiscreteShift[f,{i,n,h}] * DiscreteShift[f,i,j,...] * InverseZTransform[exp,z,n] InverseZTransform[exp,{z1,z2,...},{n1,n2,...}] ZTransform[exp,n,z] ZTransform[exp,{n1,n2,...},{z1,z2,...}]

Discrete calculus.

## Polynomial Functions

Polynomial functions.

## q Functions

 StandardForm TraditionalForm QBinomial[n,m,q] * QFactorial[n,q] * QGamma[z,q] * QHypergeometricPFQ[{a1,...,at},{b1,...,bs},q,z] * QPochhammer[a,q,n] * QPochhammer[a,q] * QPochhammer[q] * QPolyGamma[z,q] * QPolyGamma[n,z,q] *

Q functions.

## Complete Alphabetical Listing

 StandardForm TraditionalForm Abs[z] AiryAi[z] AiryAiPrime[z] AiryBi[z] AiryBiPrime[z] Algebraics And[p1,p2,...] AngerJ[,x] * AngerJ[,,x] * AppellF1[a,b1,b2,c,x,y] ArcCos[z] ArcCosh[z] ArcCot[z] ArcCoth[z] ArcCsc[z] ArcCsch[z] ArcSec[z] ArcSech[z] ArcSin[z] ArcSinh[z] ArcTan[z] ArcTanh[z] Arg[z] ArithmeticGeometricMean[a,b] BernoulliB[n] BernoulliB[n,z] BesselI[n,z] BesselJ[n,z] BesselK[n,z] BesselY[n,z] Beta[a,b] Beta[z,a,b] Beta[z0,z1,a,b] BetaRegularized[z,a,b] BetaRegularized[z0,z1,a,b] Binomial[n,m] Booleans C[n] CarmichaelLambda[n] Catalan Ceiling[z] ChampernowneNumber[b] * ChebyshevT[n,x] ChebyshevU[n,x] ClebschGordan[{j1,m1},{j2,m2},{j3,m3}] Complexes Conjugate[z] * Cos[z] Cos[z]p Cosh[z] Cosh[z]p CosIntegral[z] CoshIntegral[z] Cot[z] Cot[z]p Coth[z] Coth[z]p Csc[z] Csc[z]p Csch[z] Csch[z]p Cyclotomic[n,z] D[f[x]] D[f[x],x] D[f[x],{x,2}] D[f[x],{x,n}] Dt[f[x]] Dt[f[x],x] Dt[f[x],{x,2}] Dt[f[x],{x,n}] DawsonF[x] * DedekindEta[t] Derivative[1][f] Derivative[2][f] Derivative[d1,...][f] Det[A] DifferenceDelta[f,i] * DifferenceDelta[f,{i,n}] * DifferenceDelta[f,{i,n,h}] * DifferenceDelta[f,i,j,...] * DiracDelta[x1,x2,...] DiscreteDelta[n1,n2,...] DiscreteRatio[f,i] * DiscreteRatio[f,{i,n}] * DiscreteRatio[f,{i,n,h} * DiscreteRatio[f,i,j,...] * DiscreteShift[f,i] * DiscreteShift[f,{i,n}] * DiscreteShift[f,{i,n,h}] * DiscreteShift[f,i,j,...] * DivisorSigma[k,n] EllipticE[m] EllipticE[,m] EllipticF[,m] EllipticK[m] EllipticNomeQ[m] EllipticPi[n,m] EllipticPi[n,,m] EllipticTheta[a,u,q] EllipticThetaPrime[a,u,q] Erf[z] Erf[z0,z1] Erfc[z] Erfi[z] EulerE[n] EulerE[n,z] EulerGamma EulerPhi[n] ExpIntegralE[n,z] ExpIntegralEi[z] Fibonacci[n] Fibonacci[n,z] Floor[z] FourierTransform[expr,t,s] FourierTransform[expr,{t1,t2,...},{s1,s2,...}] FractionalPart[x] FresnelC[z] FresnelS[z] Gamma[z] Gamma[a,z] Gamma[a,z1,z2] GammaRegularized[a,z] GammaRegularized[a,z0,z1] GCD[n1,n2,...] GegenbauerC[n,x] GegenbauerC[n,m,x] Glaisher GoldenRatio HarmonicNumber[n] HarmonicNumber[n,r] HeavisideLambda[x] * HeavisideLambda[x1,x2,...] * HeavisidePi[x] * HeavisidePi[x1,x2,...] * HermiteH[n,x] Hypergeometric0F1[a,z] Hypergeometric0F1Regularized[a,z] Hypergeometric1F1[a,b,z] Hypergeometric1F1Regularized[a,b,z] Hypergeometric2F1[a,b,c,z] Hypergeometric2F1Regularized[a,b,c,z] HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] HypergeometricU[a,b,z] Implies[a,b] Integers Integrate[expr,x] Integrate[expr,x1,y,z] Integrate[expr,{x,a,b}] Integrate[expr,{x,a,b},{y,m,n},{z,p,q}] Inverse[A] InverseBetaRegularized[s,a,b] InverseBetaRegularized[z0,s,a,b] InverseEllipticNomeQ[q] InverseErf[z0,s] InverseFourierTransform[expr,s,t] InverseFourierTransform[expr,{s1,s2,...},{t1,t2,...}] InverseFunction[f] InverseJacobiCD[u,m] InverseJacobiCN[u,m] InverseJacobiCS[u,m] InverseJacobiDC[u,m] InverseJacobiDN[u,m] InverseJacobiDS[u,m] InverseJacobiNC[u,m] InverseJacobiND[u,m] InverseJacobiNS[u,m] InverseJacobiSC[u,m] InverseJacobiSD[u,m] InverseJacobiSN[u,m] InverseLaplaceTransform[expr,s,t] InverseLaplaceTransform[expr,{s1,s2,...},{t1,t2,...}] InverseWeierstrassP[p,{g2,g3}] InverseZTransform[exp,z,n] InverseZTransform[exp,{z1,z2,...},{n1,n2,...}] JacobiAmplitude[u,m] JacobiCD[u,m] JacobiCN[u,m] JacobiCS[u,m] JacobiDC[u,m] JacobiDN[u,m] JacobiDS[u,m] JacobiNC[u,m] JacobiND[u,m] JacobiNS[u,m] JacobiSC[u,m] JacobiSD[u,m] JacobiSN[u,m] JacobiP[n,a,b,x] JacobiSymbol[n,m] JacobiZeta[,m] Khinchin * KleinInvariantJ[] KroneckerDelta[n1,n2,...] LaguerreL[n,x] LaguerreL[n,a,x] LegendreP[n,x] LegendreP[n,m,x] LegendreP[n,m,a,z] LaplaceTransform[expr,t,s] LaplaceTransform[expr,s,t] LCM[n1,n2,...] LegendreQ[n,x] LegendreQ[n,m,x] LegendreQ[n,m,a,z] LerchPhi[z,s,a] Limit[f[x],x->a] Limit[f[x],x->a,Direction->+1] Limit[f[x],x->a,Direction->-1] LiouvilleLambda[n] * Log[z] Log[b,z] Log[z]^p Log[b,z]^p LogGamma[z] LogIntegral[z] MangoldtLambda[n] * MathieuCharacteristicA[r,q] MathieuCharacteristicB[r,q] Max[z] MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z] MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] Min[z] Mod[m,n] ModularLambda[] MoebiusMu[n] Multinomial[n1,n2,...,nk] MultiplicativeOrder[k,n] Nand[p1,p2,...] NevilleThetaC[u,m] NevilleThetaD[u,m] NevilleThetaN[u,m] NevilleThetaS[u,m] Nor[p1,p2,...] Not[p] O[x] O[x]^n O[x,a] O[x,a]^n Or[p1,p2,...] PartitionsP[z] PartitionsQ[z] Piecewise[{{v1,c1},{v2,c2},...}] Pochhammer[a,n] PolyGamma[z] PolyGamma[n,z] PolyLog[,z] PolyLog[,p,z] PolynomialMod[poly,m] PowerMod[a,b,n] Prime[n] PrimeNu[n] * PrimeOmega[n] * PrimePi[z] PrimeZetaP[x] * Primes ProductLog[z] ProductLog[k,z] QBinomial[n,m,q] * QFactorial[n,q] * QGamma[z,q] * QHypergeometricPFQ[{a1,...,at},{b1,...,bs},q,z] * QPochhammer[a,q,n] * QPochhammer[a,q] * QPochhammer[q] * QPolyGamma[z,q] * QPolyGamma[n,z,q] * RamanujanTau[n] Rationals Reals Residue[z] RiemannR[x] * RiemannSiegelTheta[t] RiemannSiegelZ[t] Sec[z] Sec[z]p Sech[z] Sech[z]p Series[f[x],{x,a,0}] Series[f[x],{x,a,1}] Series[Tan[z^(2/3)],{z,0,3}] Sign[z] Signature[e1,e2,...] Sin[z] Sin[z]p Sinh[z] Sinh[z]p SinIntegral[z] SinhIntegral[z] SixJSymbol[{j1,j2,j3},{j4,j5,j6}] SphericalHarmonicY[l,m,,] SquaresR[d,n] * StieltjesGamma[n] StieltjesGamma[n,a] * StirlingS1[n,m] StirlingS2[n,m] StruveH[,z] StruveL[,z] Tan[z] Tan[z]p Tanh[z] Tanh[z]p ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}] Transpose[A] UnitBox[x] * UnitBox[x1,x2,...] * UnitStep[x1,x2,...] UnitTriangle[x] * UnitTriangle[x1,x2,...] * WeberE[,x] * WeberE[,,x] * WeierstrassP[u,{g2,g3}] WeierstrassPPrime[u,{g2,g3}] WeierstrassSigma[u,{g2,g3}] WeierstrassZeta[u,{g2,g3}] Xor[p1,p2,...] Zeta[s] Zeta[s,a] ZTransform[exp,n,z] ZTransform[exp,{n1,n2,...},{z1,z2,...}]

Complete alphabetical listing.