Mathematica Tutorial

Entering Formulas

character	short form	long form	symbol
	Esc p Esc	\[Pi]	Pi
	Esc inf Esc	\[Infinity]	Infinity
°	Esc deg Esc	\[Degree]	Degree

Special forms for some common symbols.

This is equivalent to Sin[60Degree].

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Here is the long form of the input.

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You can enter the same input like this.

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Here the angle is in radians.

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special characters	short form	long form	ordinary characters
x≤y	x Esc <= Esc y	x \[LessEqual] y	x <= y
x≥y	x Esc >= Esc y	x \[GreaterEqual] y	x >= y
x≠y	x Esc Esc y	x \[NotEqual] y	x y
xy	x Esc el Esc y	x \[Element] y	Element[x,y]
x→y	x Esc -> Esc y	x \[Rule] y	x -> y

Special forms for a few operators. "Operator Input Forms" gives a complete list.

Here the replacement rule is entered using two ordinary characters, as ->.

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This means exactly the same.

As does this.

When you type the ordinary-character form for certain operators, the front end automatically replaces them with the special-character form. For instance, when you type the last three examples, the front end automatically substitutes the → character for ->.

The special arrow form → is by default also used for output.

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special characters	short form	long form	ordinary characters
x ÷ y	x Esc div Esc y	x \[Divide] y	x / y
x × y	x Esc * Esc y	x \[Times] y	x * y
x y	x Esc cross Esc y	x \[Cross] y	Cross[x,y]
x = y	x Esc Esc y	x \[Equal] y	x y
x y	x Esc l= Esc y	x \[LongEqual] y	x y
x y	x Esc && Esc y	x \[And] y	x && y
x y	x Esc \|\| Esc y	x \[Or] y	x \|\| y
¬ x	Esc ! Esc x	\[Not] x	! x
x y	x Esc => Esc y	x \[Implies] y	x => y
x y	x Esc un Esc y	x \[Union] y	Union[x,y]
x y	x Esc inter Esc y	x \[Intersection] y	Intersection[x,y]
xy	x Esc , Esc y	x \[InvisibleComma] y	x , y
fx	f Esc @ Esc x	f \[InvisibleApplication] x	f @ x or f[x]
x	x Esc + Esc	x \[ImplicitPlus]	x + y / z

Some operators with special forms used for input but not output.

Mathematica understands ÷, but does not use it by default for output.

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Many of the forms of input discussed here use special characters, but otherwise just consist of ordinary one-dimensional lines of text. Mathematica notebooks, however, also make it possible to use two-dimensional forms of input.

two-dimensional	one-dimensional
x^y	x^y	power
	x/y	division
	Sqrt[x]	square root
	x^(1/n)	n^th root
f	Sum[f,{i,i_min,i_max}]	sum
f	Product[f,{i,i_min,i_max}]	product
f x	Integrate[f,x]	indefinite integral
	Integrate[f,{x,x_min,x_max}]	definite integral
_xf	D[f,x]	partial derivative
_{x, y}f	D[f,x,y]	multivariate partial derivative
z^*	Conjugate[x]	complex conjugate
m^	Transpose[m]	transpose
m^	ConjugateTranspose[m]	conjugate transpose
expr_{[[i, j, ...]]}	Part[expr,i,j,...]	part extraction

Some two-dimensional forms that can be used in Mathematica notebooks.

You can enter two-dimensional forms using any of the mechanisms discussed in "Entering Two-Dimensional Input". Note that upper and lower limits for sums and products must be entered as overscripts and underscripts—not superscripts and subscripts.

This enters an indefinite integral. Note the use of Esc dd Esc to enter the "differential d".

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Here is an indefinite integral that can be explicitly evaluated.

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Here is the usual Mathematica input for this integral.

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short form	long form
Esc sum Esc	\[Sum]	summation sign
Esc prod Esc	\[Product]	product sign
Esc int Esc	\[Integral]	integral sign
Esc dd Esc	\[DifferentialD]	special for use in integrals
Esc pd Esc	\[PartialD]	partial derivative operator
Esc co Esc	\[Conjugate]	conjugate symbol *
Esc tr Esc	\[Transpose]	transpose symbol 
Esc ct Esc	\[ConjugateTranspose]	conjugate transpose symbol 
Esc [[ Esc	\[LeftDoubleBracket]	part brackets

Some special characters used in entering formulas. "Mathematical and Other Notation" gives a complete list.

You should realize that even though a summation sign can look almost identical to a capital sigma it is treated in a very different way by Mathematica. The point is that a sigma is just a letter; but a summation sign is an operator which tells Mathematica to perform a Sum operation.

Capital sigma is just a letter.

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A summation sign, on the other hand, is an operator.

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Much as Mathematica distinguishes between a summation sign and a capital sigma, it also distinguishes between an ordinary d, the "partial d"

that is used for taking derivatives, and the special "differential d"

that is used in the standard notation for integrals. It is crucial that you use the differential

—entered as Esc dd Esc—when you type in an integral. If you try to use an ordinary d, Mathematica will just interpret this as a symbol called d—it will not understand that you are entering the second part of an integration operator.

This computes the derivative of xⁿ.

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Here is the same derivative specified in ordinary one-dimensional form.

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This computes the third derivative.

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Here is the equivalent one-dimensional input form.

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