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	C
Complexes
EulerGamma
Glaisher	A
GoldenRatio
Integers
Primes
Rationals
Reals

Mathematical Constants and Domains.

Numerical Functions

StandardForm	TraditionalForm
Abs[z]	z
Ceiling[z]	z
Floor[z]	z
FractionalPart[x]	frac (x)
Sign[z]	sgn (z)

Numerical Functions

Elementary Functions

StandardForm	TraditionalForm
ArcCos[z]	cos^-1 (z)
ArcCosh[z]	cosh^-1 (z)
ArcCot[z]	cot^-1 (z)
ArcCoth[z]	coth^-1 (z)
ArcCsc[z]	csc^-1 (z)
ArcCsch[z]	csch^-1 (z)
ArcSec[z]	sec^-1 (z)
ArcSech[z]	sech^-1 (z)
ArcSin[z]	sin^-1 (z)
ArcSinh[z]	sinh^-1 (z)
ArcTan[z]	tan^-1 (z)
ArcTanh[z]	tanh^-1 (z)
Cos[z]	cos (z)
Cos[z]^p	cos^p (z)
Cosh[z]	cosh (z)
Cosh[z]^p	cosh^p (z)
Cot[z]	cot (z)
Cot[z]^p	cot^p (z)
Coth[z]	coth (z)
Coth[z]^p	coth^p (z)
Csc[z]	csc (z)
Csc[z]^p	csc^p (z)
Csch[z]	csch (z)
Csch[z]^p	csch^p (z)
Log[z]	log (z)
Log[z]^p	log^p (z)
Log[b,z]	log_b (z)
Log[b,z]^p
Sec[z]	sec (z)
Sec[z]^p	sec^p (z)
Sech[z]	sech (z)
Sech[z]^p	sech^p (z)
Sin[z]	sin (z)
Sin[z]^p	sin^p (z)
Sinh[z]	sinh (z)
Sinh[z]^p	sinh^p (z)
Tan[z]	tan (z)
Tan[z]^p	tan^p (z)
Tanh[z]	tanh (z)
Tanh[z]^p	tanh^p (z)

Elementary Functions

Factorial Related Functions

StandardForm	TraditionalForm
Beta[a,b]	(a, b)
Beta[z,a,b]	_z (a, b)
Beta[z₀,z₁,a,b]	(z₀, z₁, a, b)
Binomial[n,m]
Gamma[z]	(z)
Gamma[a,z]	(a, z)
Gamma[a,z₁,z₂]	(a, z₁, z₂)
GammaRegularized[a,z]	Q (a, z)
GammaRegularized[a,z₀,z₁]	Q (a, z₀, z₁)
InverseBetaRegularized[s,a,b]
InverseBetaRegularized[z₀,s,a,b]
LogGamma[z]	log (z)
Multinomial[n₁,n₂,...,n_k]	(n₁+n₂+n_k+...;n₁, n₂, ..., n_k)
Pochhammer[a,n]	(a)_n
PolyGamma[z]	(z)
PolyGamma[n,z]	⁽ⁿ⁾ (z)

Factorial Related Functions

Combinatorial Functions

StandardForm	TraditionalForm
BernoulliB[n]	B_n
BernoulliB[n,z]	B_n (z)
ClebschGordan[{j₁,m₁},{j₂,m₂},{j₃,m₃}]	j₁j₂m₁m₂j₁j₂j₃m₃
EulerE[n]	E_n
EulerE[n,z]	E_n (z)
Fibonacci[n]	F_n
Fibonacci[n,z]	F_n (z)
HarmonicNumber[n]	H_n
HarmonicNumber[n,r]
PartitionsP[z]	p (z)
PartitionsQ[z]	q (z)
Signature[e₁,e₂,...]	_{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]	agm (a, b)
CarmichaelLambda[n]	(n)
DivisorSigma[k,n]	_k (n)
EulerPhi[n]	(n)
GCD[n₁,n₂,...]	gcd (n₁, n₂, ...)
JacobiSymbol[n,m]
LCM[n₁,n₂,...]	lcm (n₁, n₂, ...)
Mod[m,n]	mmodn
MoebiusMu[n]	(n)
MultiplicativeOrder[k,n]	ord_n (k)
PowerMod[a,b,n]	a^bmodn
Prime[n]	p_n
PrimePi[z]	(z)
RamanujanTau[n]	(n)
SumOfSquaresR[d,n]	r_d (n)

Number Theory

Zeta Related Functions

StandardForm	TraditionalForm
LerchPhi[z,s,a]	(z, s, a)
PolyLog[n,z]	Li_n (z)
PolyLog[n,p,z]	S_{n, p} (z)
RiemannSiegelTheta[t]	(t)
RiemannSiegelZ[t]	Z (t)
StieltjesGamma[z]	_z
Zeta[s]	(s)
Zeta[s,a]	(s, a)

Zeta Related Functions

Hypergeometric Related Functions

StandardForm	TraditionalForm
AiryAi[z]	Ai (z)
AiryAiPrime[z]	Ai (z)
AiryBi[z]	Bi (z)
AiryBiPrime[z]	Bi (z)
AppellF1[a,b₁,b₂,c,x,y]	F₁ (a;b₁, b₂;c;x, y)
BesselI[n,z]	I_n (z)
BesselJ[n,z]	J_n (z)
BesselK[n,z]	K_n (z)
BesselY[n,z]	Y_n (z)
CosIntegral[z]	Ci (z)
CoshIntegral[z]	Chi (z)
Erf[z]	erf (z)
Erf[z₀,z₁]	erf (z₀, z₁)
Erfc[z]	erfc (z)
Erfi[z]	erfi (z)
ExpIntegralE[n,z]	E_n (z)
ExpIntegralEi[z]	Ei (z)
FresnelC[z]	C (z)
FresnelS[z]	S (z)
Hypergeometric0F1[a,z]	₀F₁ (;a;z)
Hypergeometric0F1Regularized[a,z]
Hypergeometric1F1[a,b,z]	₁F₁ (a;b;z)
Hypergeometric1F1Regularized[a,b,z]
Hypergeometric2F1[a,b,c,z]	₂F₁ (a, b;c;z)
Hypergeometric2F1Regularized[a,b,c,z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]
	_pF_q (a₁, a₂, ...;b₁, b₂, ...;z)
HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]

HypergeometricU[a,b,z]	U (a, b, z)
LegendreQ[n,x]	Q_n (x)
LegendreQ[n,m,x]
LegendreQ[n,m,a,z]
LogIntegral[z]	li (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]	Si (z)
SinhIntegral[z]	Shi (z)
StruveH[,z]	H (z)
StruveL[,z]	L (z)

Hypergeometric Related Functions

Orthogonal Polynomials

StandardForm	TraditionalForm
ChebyshevT[n,x]	T_n (x)
ChebyshevU[n,x]	U_n (x)
GegenbauerC[n,x]	C_n (x)
GegenbauerC[n,m,x]
HermiteH[n,x]	H_n (x)
JacobiP[n,a,b,x]
LaguerreL[n,x]	L_n (x)
LaguerreL[n,a,x]
LegendreP[n,x]	P_n (x)
LegendreP[n,m,x]
LegendreP[n,m,a,z]
SphericalHarmonicY[l,m,,]

Orthogonal Polynomials

Inverse Functions

StandardForm	TraditionalForm
InverseErf[z₀,s]	erf^-1 (z₀, s)
InverseFunction[f]	f^(-1)
ProductLog[z]	W (z)
ProductLog[k,z]	W_k (z)

Inverse Functions

Elliptic Integrals

StandardForm	TraditionalForm
EllipticE[m]	E (m)
EllipticE[,m]	E (m)
EllipticF[,m]	F (m)
EllipticK[m]	K (m)
EllipticNomeQ[m]	q (m)
EllipticPi[n,m]	(nm)
EllipticPi[n,,m]	(n;m)
JacobiZeta[,m]	(m)

Elliptic Integrals

Elliptic Functions

StandardForm	TraditionalForm
DedekindEta[t]	(t)
EllipticTheta[a,u,q]	_a (u, q)
EllipticThetaPrime[a,u,q]
InverseEllipticNomeQ[q]	q^-1 (q)
InverseJacobiCD[u,m]	cd^-1 (um)
InverseJacobiCN[u,m]	cn^-1 (um)
InverseJacobiCS[u,m]	cs^-1 (um)
InverseJacobiDC[u,m]	dc^-1 (um)
InverseJacobiDN[u,m]	dn^-1 (um)
InverseJacobiDS[u,m]	ds^-1 (um)
InverseJacobiNC[u,m]	nc^-1 (um)
InverseJacobiND[u,m]	nd^-1 (um)
InverseJacobiNS[u,m]	ns^-1 (um)
InverseJacobiSC[u,m]	sc^-1 (um)
InverseJacobiSD[u,m]	sd^-1 (um)
InverseJacobiSN[u,m]	sn^-1 (um)
InverseWeierstrassP[p,{g₂,g₃}]	^-1 (p;g₂, g₃)
JacobiAmplitude[u,m]	am (um)
JacobiCD[u,m]	cd (um)
JacobiCN[u,m]	cn (um)
JacobiCS[u,m]	cs (um)
JacobiDC[u,m]	dc (um)
JacobiDN[u,m]	dn (um)
JacobiDS[u,m]	ds (um)
JacobiNC[u,m]	nc (um)
JacobiND[u,m]	nd (um)
JacobiNS[u,m]	ns (um)
JacobiSC[u,m]	sc (um)
JacobiSD[u,m]	sd (um)
JacobiSN[u,m]	sn (um)
KleinInvariantJ[]	J ()
ModularLambda[]	()
NevilleThetaC[u,m]	_c (um)
NevilleThetaD[u,m]	_d (um)
NevilleThetaN[u,m]	_n (um)
NevilleThetaS[u,m]	_s (um)
WeierstrassP[u,{g₂,g₃}]	(u;g₂, g₃)
WeierstrassPPrime[u,{g₂,g₃}]	(u;g₂, g₃)
WeierstrassSigma[u,{g₂,g₃}]	(u;g₂, g₃)
WeierstrassZeta[u,{g₂,g₃}]	(u;g₂, g₃)

Elliptic Functions

Mathieu Functions

StandardForm	TraditionalForm
MathieuCharacteristicA[r,q]	a_r (q)
MathieuCharacteristicB[r,q]	b_r (q)

Mathieu Functions

Generalized and Related Functions

StandardForm	TraditionalForm
DiracDelta[x₁,x₂,...]	(x₁, x₂, ...)
DiscreteDelta[n₁,n₂,...]	(n₁, n₂, ...)
KroneckerDelta[n₁,n₂,...]	_{n₁, n₂, ...}
UnitStep[x₁,x₂,...]	(x₁, x₂, ...)

Generalized and Related Functions

Matrix Operations

StandardForm	TraditionalForm
Det[A]	A
Inverse[A]	A^-1
Transpose[A]	A^T

Matrix Operations

Logical Operations

StandardForm	TraditionalForm
And[p₁,p₂,...]	p₁p₂...
Implies[a,b]	ab
Nand[p₁,p₂,...]	p₁p₂...
Nor[p₁,p₂,...]	p₁p₂...
Not[p]	¬p
Or[p₁,p₂,...]	p₁p₂...
Xor[p₁,p₂,...]	p₁p₂...

Logical Operations

Calculus

StandardForm	TraditionalForm
C[n]	c_n
D[f[x]]	D[f (x)]
D[f[x],x]
D[f[x],{x,2}]
D[f[x],{x,n}]
Dt[f[x]]	f (x)
Dt[f[x],x]
Dt[f[x],{x,2}]
Dt[f[x],{x,n}]
Derivative[1][f]	f
Derivative[2][f]	f
Derivative[d₁,...][f]	f^{(d₁, ...)}
FourierTransform[expr,t,s]	_t[expr] (s)
FourierTransform[expr,{t₁,t₂,...},{s₁,s₂,...}]	_{t₁, t₂, ...}[expr] (s₁, s₂, ...)
Integrate[expr,x]	exprx
Integrate[expr,x₁,y,z]	exprzyx₁
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]	_t[expr] (s)
LaplaceTransform[expr,{t₁,t₂,...},{s₁,s₂,...}]	_{t₁, t₂, ...}[expr] (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)
O[x]^n	O (x)ⁿ
O[x,a]	O (x-a)
O[x,a]^n	O (x-a)ⁿ
Piecewise[{{v₁,c₁},{v₂,c₂},...}]
Residue[z]	res (z)
Series[f[x],{x,a,0}]	f (a)+O ( (x-a)¹)
Series[f[x],{x,a,1}]	f (a)+f (a) (x-a)+O ( (x-a)²)
Series[Tan[z^(2/3)],{z,0,3}]

Calculus

Polynomial Functions

StandardForm	TraditionalForm
Cyclotomic[n,z]	C_n (z)
PolynomialMod[poly,m]	polymodm

Polynomial Functions

Complete Alphabetical Listing

StandardForm	TraditionalForm
Abs[z]	z
AiryAi[z]	Ai (z)
AiryAiPrime[z]	Ai (z)
AiryBi[z]	Bi (z)
AiryBiPrime[z]	Bi (z)
Algebraics
And[p₁,p₂,...]	p₁p₂...
AppellF1[a,b₁,b₂,c,x,y]	F₁ (a;b₁, b₂;c;x, y)
ArcCos[z]	cos^-1 (z)
ArcCosh[z]	cosh^-1 (z)
ArcCot[z]	cot^-1 (z)
ArcCoth[z]	coth^-1 (z)
ArcCsc[z]	csc^-1 (z)
ArcCsch[z]	csch^-1 (z)
ArcSec[z]	sec^-1 (z)
ArcSech[z]	sech^-1 (z)
ArcSin[z]	sin^-1 (z)
ArcSinh[z]	sinh^-1 (z)
ArcTan[z]	tan^-1 (z)
ArcTanh[z]	tanh^-1 (z)
ArithmeticGeometricMean[a,b]	agm (a, b)