MATHEMATICA TUTORIAL

TraditionalForm Reference Information

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.

In[1]:=

Out[1]//TraditionalForm=

Here is an example of a function that has its own special TraditionalForm notation.

In[2]:=

Out[2]//TraditionalForm=

The TraditionalForm representation of matrices is shown here.

In[3]:=

Out[3]//TraditionalForm=

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.

In[4]:=

Out[4]//TraditionalForm=

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,...,n_k]
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,...,a_p},{b₁,...,b_q},z]

HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]

HypergeometricU[a,b,z]
LegendreQ[n,x]
LegendreQ[n,m,x]
LegendreQ[n,m,a,z]
LogIntegral[z]
MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z]

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},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

StandardForm	TraditionalForm
MathieuCharacteristicA[r,q]
MathieuCharacteristicB[r,q]

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

StandardForm	TraditionalForm
Det[A]
Inverse[A]
Transpose[A]

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

StandardForm	TraditionalForm
Cyclotomic[n,z]
PolynomialMod[poly,m]

Polynomial functions.

q Functions

StandardForm	TraditionalForm
QBinomial[n,m,q]		*
QFactorial[n,q]		*
QGamma[z,q]		*
QHypergeometricPFQ[{a₁,...,a_t},{b₁,...,b_s},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,...,a_p},{b₁,...,b_q},z]
HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},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,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z]
MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z,r]
Min[z]
Mod[m,n]
ModularLambda[]
MoebiusMu[n]
Multinomial[n₁,n₂,...,n_k]
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,...,a_t},{b₁,...,b_s},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.