Select Mathematica Commands and Examples

ASSIGN expression to a name

You can define the value of a symbol to be any expression, not just a number. Once defined, a definition will continue to be used whenever the symbol appears, until you explicitly change or remove the definition. The function Clear will clear the value of the symbol while the function Remove will actually remove the symbol completely from Mathematica. See Mathematica Tips, Tricks and Traps: Equal signs, assignments.

 

		name = value or expression
Example:	In[n]:=	x=3
	Out[n]=	3
	In[n+1]:=	x^3-6x
	Out[n+1]=	9
Example (expression):	In[n]:=	y = (x+1)^3 + 3 x - Abs[x]
Clear assignment:	In[n]:=	y= .

COMMENT out the input to be ignored

You may want to add some commentary text to your programs to make it easier to understand. You can add such text at any point in your code simply by enclosing it in matching (* and *).

 

Example:

In[n]:=

(* Enclose like this *)

CONSTANTS (built-in)

$e, i, \pi, \gamma, \infty$		E I Pi EulerGamma Infinity
Example: $e^{\pi i}$	In[n]:=	E^(Pi I)

CONTINUATION of input for several lines

Different text-based interfaces use slightly different schemes for letting Mathematica know when you have finished typing your input. With the notebook feature you press Shift-Return, while from command tool you press Meta-Return. In interfaces that use Shirt-Return, you can continue your input for several lines by typing Return at the end of each line. In interfaces where Meta-Return is used to signify the end of your input, expressions in a line which Mathematica detects as imcomplete will automatically continue reading successive lines until it has received a complete expression. The continuation parentheses are not required in the above example. Otherwise, the following will allow you the continuation of input for several lines.

	In[n]:=	( start with left parenthesis
		and end with right )
Example:	In[n]:=	( Plot3D[Exp[-(x^2 + y^2)],{x,-2,2},
		{y,-2,2}, HiddenSurface->False] )

DEFINE a function of a variable

		functionname[x_]:= expression with x in it
Example:	In[n]:=	f[x_]:= Exp[x] + Cos[x] Sin[x]
Show the definition of f alone:	In[n]:=	?f
(cont. from the example above)	Out[n]=	f[x_]:=Exp[x]+Cos[x]*Sin[x]
Clear definition:	In[n]:=	Clear[f]
Remove variable completely:	In[n]:=	Remove[f]

DERIVATIVE

Partial derivative $\frac{\partial f}{\partial x}$:		D[ expression, variable (s) ]
Simple derivative:	In[n]:=	D[x^n, x]
	Out[n]=	n x-1+n
Second derivative:	In[n]:=	D[Exp[I x], {x, 2}]
	Out[n]=	-EI x
Mixed partial (here x & y):	In[n]:=	D[Sin[x y] + y (Pi/4), x, y]
	Out[n]=	Cos[x y]-x y Sin[x y]

The function D[x^n, x] gives a partial derivative in which n is assumed not to depend on x. Mathematica has another function, called Dt, which finds total derivatives in which all variables are assumed to be related.

Total derivative $\frac{d f}{d x}$ :		Dt[ expression, variable]
Example:	In[n]:=	Dt[ x^n ]
	Out[n]=	n x-1+n Dt[x] +xn Dt[n] Log[x]

DO loop

Do[expr, {i, imax}]

DO loop will evaluate expr repetitively, with i varying from 1 to imax in steps of 1. When a sequence of expressions are to be evaluated, the expressions may be separated by semicolons.

Example:	In[n]:=	Do[Print[i], {i, 3}]
	Out[n]=	1
		2
		3
i varies from imin to imax in steps of di		Do[expr, {i, imin,imax, di}]
Example:	In[n]:=	Do[Print[i/2], {i,2,6,2}]
	Out[n]=	1
		2
		3
Evaluate n times		Do[expr, {n}]
Example:	In[n]:=	t=x; Do[t=1+(2 t), {3}]; t
	Out[n]=	1+2(1+2(1+2x))

EXPAND expression (products and powers)

		Expand[ expression]
Example:	In[n]:=	Expand[(x + 1)^4]
	Out[n]=	1 + 4x + 6x2 + 4x3 +x4
To multiply out:	In[n]:=	Expand[(x+2)(x-2)(x^2+4)]
	Out[n]=	-16 + x4
Apply Expand everywhere:	In[n]:=	ExpandAll[(x+2)(x-2)/(x^2+4)(x-3)]
	Out[n]=	2 3 12 4x 3x x ------ - ------ - ------ + ------ 2 2 2 2 4 + x 4 + x 4 + x 4 + x

EVALUATE to number

To six decimal digits by default.

		N[ expression ]
Example:	In[n]:=	N[E]
	Out[n]=	2.71828
To 10 decimal digits:	In[n]:=	N[Pi^2/6,10]
	Out[n]=	1.644934067

FACTOR expression

		Factor[ expression]
Example:	In[n]:=	Factor[x^2 -1]
	Out[n]=	(-1+x)(1+x)

FLOOR function (Greatest integer function, TRUNCATE to an integer)

		Floor[ expression ]
Example:	In[n]:=	Floor[3.1415]
	Out[n]=	3

FOR loop
For[ start, test, increment, body]

For function will evaluate start, then repetitively evaluate body and increment, until the specified condition test fails.

Example, print first 4 numbers:
	In[n]:=	For[n=1, n<=4, n++, Print[n]]
	Out[n]=	1
		2
		3
		4

FRACTION Operations

Denominator of ratio:		Denominator[ rational expression ]
Example:	In[n]:=	Denominator[a/b]
	Out[n]=	b
Numerator of ratio:	In[n]:=	Numerator[a/b]
	Out[n]=	a
Combine fractions:	In[n]:=	Together[a/b + c/d]
	Out[n]=	bc+ad
		b d
Separate rational:	In[n]:=	Apart[(b c + a d)/(bd)]
Rationalize a real:		Rationalize[ real number, accuracy (error) ]
Example:	In[n]:=	Rationalize[3.14159265, .00001]
	Out[n]=	355
		113
		accuracy=0 gives the best possible approximation.
Example:	In[n]:=	Rationalize[3.14159265,0]
	Out[n]=	392699
		125000

HELP via ? and ??

	In[n]:=	?
Example, about command Range:	In[n]:=	?Range
with pattern matching:	In[n]:=	?Ran*
Info about a function:	In[n]:=	?f
More info about a function:	In[n]:=	??f

From a Mathematica notebook window, one may right click on the Help Option to study a wide variety of help options and examples.

IF conditions
If[ condition, then, else ]

IF function will test the condition and evaluate then if the condition is true, else if it is false

Example:
	In[n]:=	f[x_]:= If[x<=0, 0,Log[x]]

INTEGRATE

		Integrate[ expression, variable]
Example:	In[n]:=	Integrate[f[x],x]
Definite integral:		Integrate[ expr, { variable, min, max}]
Example, from 0 to $\frac{\pi}{4}$:	In[n]:=	Integrate[Sin[x], {x, 0, Pi/4}]
Double integral:		Integrate[ Integrate[ expr, { variable, min, max}],{ variable, min, max}]
Example:	In[n]:=	Integrate[ Integrate[Cos[x],{x,0,Pi}],{x,0,Pi}]
	Out[n]=	Pi
Numerical integration		NIntegrate[ expr, { variable, min, max}]
Example:	In[n]:=	NIntegrate[Sin[x], {x, -Infinity, Infinity}]

INTERRUPT COMPUTATION (ABORT execution)

INTERRUPT computation:		CTRL-C (if in cmd tool)
		CTRL-C CTRL-C or F2 F2 (if in EPOCH)
for menu after interruption:		h
Abort current calculation:		abort
Continue:		continue
Enter an interactive dialog:		inspect
Show current operation:		show
Show all operations:		trade

LIMIT

		Limit[ expression, variable -> value ]
Example, at 0:	In[n]:=	Limit[Sin[x]/x, x -> 0]
Example, at $\infty$:	In[n]:=	Limit[ArcTan[x], x -> Infinity]

TABLE, LIST and ARRAY operations (VECTOR and MATRIX operations)

Create a list:		Table[ expr, { variable, min, max, step}]
		stepsize is 1 by default
Example:	In[n]:=	Table[1 + 2^n, {n, 0, 7}]
	Out[n]=	{2,3,5,9,17,33,65,129}
Applying function to the list:	In[n+1]:=	PrimeQ[%]
	Out[n+1]=	{True,True,True,False,True,False,False,False}
Constant array: E.g., ai = 1, i=1,..9	In[n]:=	a = Table[1,{9}]
	Out[n]=	{1,1,1,1,1,1,1,1,1}
Function array: E.g., $b_i = \sqrt{i}, i=1,..9$	In[n]:=	b = Array[Abs, 9]
	Out[n]=	{1,2,3,4,5,6,7,8,9}
Dot product:	In[n]:=	a . b (* a dot b *)
	Out[n]=	45
		two vectors must have the same dimention
Create an m x n matrix a[i,j]:		Array[ expr, { rowsize, columnsize}]
Two dimensional array:	In[n]:=	arr=Table[i*j, {i,1,3},{j,2,4}]
	Out[n]=	{{2,3,4},{4,6,8},{6,9,12}}
The nth term in array:		array [[n]]
Example, Third element of arr:	In[n+1]:=	arr[[3]]
	Out[n+1]=	{6,9,12}
Example, 3rd row, 2nd column of arr:	In[n+2]:=	arr[[3,2]]
	Out[n+2]=	9
Inverse of matrix:	In[n]:=	inv = Inverse[arr]
Display row by column:	In[n]:=	MatrixForm[arr]

LOOPs

The execution of a Mathematica program involves the evaluation of a sequence of expressions. You may need some kind of "loop" to evaluate expressions several times.

• DO loop
• FOR loop
• WHILE loop

MODULE

The most common way that modules are used is to set up temporary or intermediate variables inside functions you define. It is important to make sure that such variables are kept local. If they are not, then you will run into a problem whenever their names happen to coincide with the names of other variables.

Specify the symbols x,y.. in expr are local	In[n]:=	Module[{x,y,..}, expr]
Define the global x	In[n]:=	x=Pi
Specify the x inside the Module is local	In[n+1]:=	Module[{x}, x=3; x]
	Out[n+1]=	3
The global x is unchanged	In[n+2]:=	t
	Out[n+2]=	Pi

PLOT (GRAPH)

Plot is the function used to plot one or more functions of a single variable on the same axis. Plot[expr, returns the plot of the expr as a function of variable on the interval from min to max. Specific examples follow.

		Plot[function, {variable, min, max}]
Example, from 2 to $\pi$:	In[n]:=	Plot[x^3+Sin[x]-1, {x,2, Pi}]
Two functions:	In[n]:=	Plot[{Gamma[x], Zeta[x]}, {x,2, 3}]
Three dimensions:	In[n]:=	( Plot3D[Sin[x y],{x,-2,2},{y,-2,2}] )
Contour plot:	In[n]:=	( ContourPlot[(x y)/(x^2 + y^2),
		{x, -1, 1}, {y, -1, 1}] )
Density plot:	In[n]:=	( DensityPlot[Cos[Sin[x y]],
		{x,-1,1}, {y,-1,1}] )

Type ??Plot to view the numerous plot options. A few commonly used options are listed below. For more detail on the options, you may have to check the current manual.

Plot3D usually colors surfaces using a simulated lighting model. Lighting $\rightarrow$ False switches off the simulated lighting, and instead shades surfaces with gray levels determined by height.

Example:	In[n]:=	( Plot3D[Sin[x y],{x,-2,2},{y,-2,2},
		Lighting->False] )

Mesh $\rightarrow$ False shows the surface without the mesh drawn.

Example:	In[n]:=	( Plot3D[Sin[x y],{x,-2,2},{y,-2,2},
		Mesh->False] )

HiddenSurface $\rightarrow$ False will show the surface as solid.

Example:	In[n]:=	( Plot3D[Sin[x y],{x,-2,2},{y,-2,2},
		HiddenSurface->False] )

PlotPoints defines the number of points in each direction at which to sample the function (15 by default). You need to do this to get good pictures of functions that wiggle a lot.

Example:	In[n]:=	( Plot3D[Sin[x y],{x,-2,2},{y,-2,2},
		PlotPoints -> {40, 40}] )

ViewPoint option allows you to specify the point x,y,z in space from which you view a surface.

Example, directly in front:	In[n]:=	( Plot3D[Sin[x y],{x,-2,2},{y,-2,2},
		Viewpoint -> {0,-2,0}] )

ContourPlot is a topographic map of a function. The contours join points on the surface that have the same height, shading in such a way that regions with higher z values are lighter. ContourShading switches off shading in contour plots.

Example:	In[n]:=	( ContourPlot[(x y)/(x^2 + y^2),
		{x, -1, 1}, {y, -1, 1}, ContourShading-> False] )

DensityPlot shows the values of a function at a regular array of points. Lighter regions are higher. Mesh $\rightarrow$False will remove the mesh.

Example:	In[n]:=	( DensityPlot[Cos[Sin[x y]],
		{x,-1,1}, {y,-1,1}, Mesh->False] )

k-th PREVIOUS output

kth PREVIOUS results:		%%..% k times
Example:	In[n]:=	Expand[(x+1)^4]
The last result:	In[n+1]:=	Factor[%]
The next-to-last result:	In[n+2]:=	ExpandAll[%%]
By line (eg., output of In[4]):	In[n]:=	%4

PRINT graphic objects

With text-based Mathematica interfaces, the command PSPrint is typically set up to print a piece of graphics (hard copy).

Create graphics object:	In[n]:=	Plot[Cosh[x],{x,-3,3}]
Print graphics object to the default printer	In[n+1]:=	PSPrint[%]
Print output n to the default printer	In[n+2]:=	PSPrint[% n]

PRODUCT

PRODUCT: $\prod_{var=min}^{max}$ f		Product[ f, { var, min, max, step}]
Example: $\prod_{n=2}^{1000} \frac{n^2}{n^2 - 1}$	In[n]:=	Product[n^2/(n^2 - 1), {n, 2, 1000}]
Numerical approx: $\prod_{n=1}^{\infty} \frac{n^2 + 1}{n^2}$	In[n]:=	NProduct[(n^2 + 1)/n^2, {n,Infinity}]
	Out[n]=	3.67608

READ in (Get) a file

You may find that you often need to use certain functions that are not built into Mathematica. You may find a Mathematica package that contains the functions you need. Mathematica packages consist of collections of Mathematica definitions which teach Mathematica about particular application areas. You must read the package into Mathematica in order to use functions from a particular package.

		<< filename
Example:	In[n]:=	<<DataAnalysis/DescriptiveStatistics.m

SAVE symbol definition

You may save definitions for variables and functions using Save.

in file filename:		Save[" filename", symbol ]
Example:(define function f)	In[n]:=	f[x_] :=Sin[a x]
(gives a some value)	In[n+1]:=	a=Pi/4
(save the definition of f in the file Functions.m")	In[n+2]:=	Save["Functions.m",f]
Save several symbols:	In[n]:=	Save["Functions.m",s,h]

SIMPLIFY an expression

		Simplify[ expression ]
Example:	In[n]:=	ArcSinh[(Exp[2] - 1)/(2 E)]
	In[n+1]:=	Simplify[%]
Simplest form	In[n+2]:=	FullSimplify[%]
		Find the simplest form by applying transformations

SOLVE

SOLVE equation:		Solve[ lhs==rhs, variable ]
Example:	In[n]:=	Solve[ArcSinh[x] == 1, x]
Example:	In[n]:=	Solve[x^2-6 x == -8, x]
	Out[n]=	{{x->2},{x->4}}
		Treat the output as 1 x 2 matrix
Get the roots	In[n+1]:=	x/. %
	Out[n+1]=	{2,4}
Get n-th root (n=2 here)	In[n+2]:=	x/. %%[[2]]
	Out[n+2]=	4
Linear system:	In[n]:=	Solve[{x - y == 0, x + y == 1}, {x, y}]
Numerically:		FindRoot[ equation,{ var, initial values}]
Example:	In[n]:=	FindRoot[x^2 == Sin[x], {x, .8}]
For non-differentiables:	In[n]:=	FindRoot[Gamma[x]==Zeta[x],{x,{2,3}}]

STOP session (EXIT Mathematica) from cmdtool

In[n]:=

Quit

SUBSTITUTE into expression

Replace variable by value in the expression

		expression /. variable -> value
Example: x=3, y=1-a	In[n]:=	(x+y)(x-y)^2 /. {x->3, y->1-a}
Example: $i = \sqrt{-1}$	In[n]:=	Sqrt[x] /. x -> -1

SUM

		Sum[ expr ,{ var, min, max, step}]
		min, step =1 by default
Example: $\sum_{i=1}^{10} i^3$	In[n]:=	Sum[i^3, {i,10}]
Example: $\sum_{i=1}^{10} 2i$	In[n]:=	Sum[i, {i,2, 20, 2}]
Numerical approximation: $\sum_{i=1}^{\infty} \frac{1}{i^2}$	In[n]:=	NSum[1/(i^2), {i, Infinity}]
	Out[n]=	1.664493

Taylor SERIES

		Series[ expr, { var, center, order}]
Example, at 0, degree 5:	In[n]:=	Series[Sin[x], {x,0, 5}]
Several variables:	In[n]:=	Series[Sin[x + y], {x, 0, 5}, {y, 0, 4}]
Convert to polynomial:	In[n+1]:=	Normal[%]

TIME required for computation

		Timing[ command ]
Example: $\sum_{i=1}^{\infty} \frac{1}{i^2}$	In[n]:=	Timing[NSum[1/i^2, {i,Infinity}]]
	Out[n]=	{1.17 Second, 1.64493}

WHILE loop
While[ test, body ]

WHILE function will evaluate body so long as test is TRUE.

Example, while $n! \leq 10^{10}$ do:	In[n]:=	n=1 (* supply starting value *)
print n!, increment n by 1:	In[n]:=	While[n!<=10^10,Print[n!];n++]

 

Additional Mathematica commands and examples can be found at
Mathematica Tips, Tricks and Traps