To access one of the expressions:

For example:

$5$

Using negative integers, you can select an expression from the end of a sequence.

$\sin \left(x^2\right)$

You can select multiple expressions by specifying a range using the range operator (..).

$5,\sin \left(x^2\right)$

Note: This syntax is valid for most data structures.

To perform mathematical set operations, use the set data structure.

$\left\{1\,2\,5\,6\,7\,8\right\}$

Note: The union operator is available in 1-D Math input as union.

To refer to an element in a list:

For example:

$\left[1\,0\right]$

For more information, see Accessing Elements.

Some commands accept a list (or set) of expressions.

For example, you can solve a list (or set) of equations using the context panel or the solve command.

For more information, see Solving Equations and Inequations.

For more information on sets and lists, refer to the set help page.

To define an Array, use the Array constructor.

Standard Array constructor arguments are:

For example:

$a≔\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

To access entries in an Array, use either square bracket or round bracket notation.

Square bracket notation respects the actual index of an Array, even when the index does not start at 1.

$1$

$6$

$5.5$

Error, Array index out of range

Round bracket indexing normalizes the dimensions to begin at 1. Since this method is relative, you can access the end of the array by entering $-1.$

$8$

$1.2$

Arrays can have more than one or two dimensions.  For example, the following is a simple three-dimensional Array.

The Array constructor supports other syntaxes. It also supports many options. For more information on the Array constructor and the Array data structure, refer to the Array help page. For more information on indexing methods, refer to the rtable_indexing help page.

Larger Arrays than can be practically displayed in the document display with a summary format.  For example, the multidimensional Array in  is displayed in summary format.  The following 100-entry Array is as well.  In both cases, double-clicking on the placeholder output opens a browser for the entire data structure.

To view large Arrays:

The Matrix Browser displays the Array.

In the case of a multidimensional array, you can change what slice you are viewing in the Options tab.

For more information, see Viewing Large Matrices and Vectors.

$\mathrm{\beta }$

You can also assign anything, for example, a list, to each element.

$\left[\mathrm{deux}\,\mathrm{dos}\right]$

For more information on tables, refer to the table help page.

To define a function of one or two variables:

For example, define a function that adds 1 to its input.

Note: Instead of using the palettes, you can type the definition.  To insert the right arrow, you can enter the characters ->. In 2-D Math, Maple replaces -> with the right arrow symbol $\to$. In 1-D Math, the characters are not replaced.

You can evaluate the function add1 with symbolic or numeric arguments.

$13$

$x+6$

Note: The expression $x+1$ is different from the functional operator $x\to x+1$.

Assign the functional operator $x\to x+1$ to f.

Assign the expression $x+1$ to g.

To evaluate the functional operator f at a value of x:

$23$

To evaluate the expression g at a value of x:

The following is not meaningful:

$x\left(22\right)+1$

Evaluating $g$ at $x\=22$ gives the desired result.$$

$23$

For more information on the eval command, and on using palettes and the context panel to evaluate an expression at a point, see Substituting a Value for a Subexpression.

To define a multivariate or vector function:

For example, a multivariate function:

$0$

$\mathrm{-2.008893709}$

A vector function:

$0,1,0$

$1,0,\frac{\mathrm{\pi }}{2}$

To perform an operation on a functional operator, specify arguments to the operator. For example, for the operator f, specify f(x), which Maple evaluates as an expression. See the following examples.

Plotting:

Plot a three-dimensional operator as an expression using the plot3d command.

For information on plotting, see Plots and Animations.

Integration:

Integrate a function using the int command.

$\frac{\mathrm{\pi }\mathrm{StruveH}\left(0\,\mathrm{\pi }\right)}{2}$

The result uses the Struve function $\mathrm{StruveH}\left(v\,x\right)$.

For information on integration and other calculus operations, see Calculus.  For information on mathematical functions, including accessing detailed information on the properties of a function, see Mathematical Functions and the FunctionAdvisor help page.

You can access characters in a string using brackets.

$''sequence\; of\; characters''$

The StringTools package is an advanced set of tools for manipulating and using strings.

$''MJoY2Qs0F''$

$''impress''$

$\left[''Create''\,''a''\,''list''\,''of''\,''strings''\,''from''\,''the''\,''words''\,''in''\,''a''\,''string''\right]$

A Maple type is a broad class of expressions that share common properties. Maple contains over 200 types, including:

For more information and a complete list of Maple types, refer to the type help page.

The type commands return true if the expression satisfies the type check. Otherwise, they return false.

Testing the Type of an Expression

To test whether an expression is of a specified type: •  Use the type command. >  $\mathrm{type}\left(\sin \left(x\right)\,\'\mathrm{trig}\'\right)$ $\mathrm{true}$ >  $\mathrm{type}\left(\sin \left(x\right)+\cos \left(x\right)\,\'\mathrm{trig}\'\right)$ $\mathrm{false}$ For information on enclosing keywords in right single quotes ('), see Delaying Evaluation. Maple types are not mutually exclusive. An expression can be of more than one type. >  $\mathrm{type}\left(3\,\'\mathrm{constant}\'\right)$ $\mathrm{true}$ >  $\mathrm{type}\left(3\,\'\mathrm{integer}\'\right)$ $\mathrm{true}$ For information on converting an expression to a different type, see Converting.

	Testing the Type of Subexpressions
	To test whether an expression has a subexpression of a specified type: •  Use the hastype command. >  $\mathrm{hastype}\left(\sin \left(x\right)+\cos \left(x\right)\,\'\mathrm{trig}\'\right)$ $\mathrm{true}$

Testing for a Subexpression

To test whether an expression contains an instance of a specified subexpression: •  Use the has command. >  $\mathrm{has}\left(\sin \left(x\+y\right)\,x\right)$ $\mathrm{true}$ >  $\mathrm{has}\left(\sin \left(x\+y\right)\,x\+y\right)$ $\mathrm{true}$ >  $\mathrm{has}\left(\sin \left(x+y\right)\,\sin \left(x\right)\right)$ $\mathrm{false}$ The has command searches the structure of the expression for an exactly matching subexpression. For example, the following calling sequence returns false. >  $\mathrm{has}\left(x+y+z\,x \+ z\right)$ $\mathrm{false}$ To return all subexpressions of a particular type, use the indets command. For more information, see Indeterminates.

To simplify an expression:

The simplify command applies simplification rules to an expression. Maple has simplification rules for various types of expressions and forms, including trigonometric functions, radicals, logarithmic functions, exponential functions, powers, and various special functions. You can also specify custom simplification rules using a set of side relations.

$35$

$1+\ln \left(2\right)+\ln \left(y\right)$

To limit the simplification, specify the type of simplification to be performed.

$1+\ln \left(2y\right)$

${\sin \left(x\right)}^2+\ln \left(2\right)+\ln \left(y\right)+{\cos \left(x\right)}^2$

You can also use the simplify command with side relations. See Substituting a Value for a Subexpression.

To factor a polynomial:

$x\left(x-2\right)\left(x-3\right)\left(x+2\right){\left(x+1\right)}^2$

$\left(y-3\right){\left(x+1\right)}^2\left(x+y\right)$

Maple can factor polynomials over the domain specified by the coefficients. You can also factor polynomials over algebraic extensions. For details, refer to the factor help page.

For more information on polynomials, see Polynomial Algebra.

To factor an integer:

${\left(3\right)}^4\left(11\right)\left(13\right)\left(17\right)$

For more information on integers, see Integer Operations.

To expand an expression:

The expand command distributes products over sums and expands expressions within functions.

$x^{3}y+x^2{y}^2-3x^3-x^{2}y+2x{y}^2-6x^2-5xy+y^2-3x-3y$

$\sin \left(x\right)\cos \left(y\right)+\cos \left(x\right)\sin \left(y\right)$

To combine subexpressions in an expression:

The combine command applies transformations that combine terms in sums, products, and powers into a single term.

$\sin \left(x+y\right)$

Recall that $a$ was previously assigned to represent a two-dimensional array (see Creating and Using Arrays).

$\left[\begin{array}{ccc}{x}^3& x^5& x^7\\ x^9& x^{11}& x^{13}\\ x^{15}& x^{17}& x^{19}\end{array}\right]$

The combine command applies only transformations that are valid for all possible values of names in the expression.

$4\ln \left(x\right)-\ln y$

To perform the operation under assumptions on the names, use the assuming command. For more information about assumptions, see Assumptions on Variables.

$\ln \left(\frac{x^4}{y}\right)$

To convert an expression:

The convert command converts expressions to a new form, type (see Expression Types), or in terms of a function. For a complete list of conversions, refer to the convert help page.

Convert a measurement in radians to degrees:

$180\mathrm{degrees}$

To convert measurements that use units, use the Unit Converter or the convert/units command.

$992.5211043⟦\mathrm{lb}⟧$

For information on the Unit Converter and using units, see Units.

Convert a list to a set:

$\left\{c\,d\,\mathrm{Array}\left(1..2\,2..5\,\left\{\left(1\,2\right)=1.2\,\left(1\,3\right)=4.9\,\left(1\,4\right)=6.3\,\left(1\,5\right)=7.1\,\left(2\,2\right)=9.2\,\left(2\,3\right)=5.5\,\left(2\,4\right)=2.4\,\left(2\,5\right)=1.7\right\}\right)\right\}$

Maple has extensive support for converting mathematical expressions to a new function or function class.

$\frac{{\ⅇ}^{\mathrm{I}x}}{2}+\frac{{\ⅇ}^{\mathrm{-I}x}}{2}$

Find an expression equivalent to the inverse hyperbolic cotangent function in terms of Legendre functions.

$\mathrm{LegendreQ}\left(0\,\frac{1}{5}\right)+\frac{\mathrm{\pi }\sqrt{\mathrm{-16}}}{8}$

For more information on converting to a class of functions, refer to the convert/to_special_function help page.

To normalize an expression:

The normal command converts expressions into factored normal form.

$\frac{x+y}{{\left(x-y\right)}^2}$

You can also use the normal command for zero recognition.

$0$

To expand the numerator and denominator, use the expanded option.

$\frac{x+y}{x^2-2xy+y^2}$

$\sin \left(\frac{x+1}{x}\right)$

To sort the elements of an expression:

The sort command orders a list of values or terms of a polynomial.

$\left[\mathrm{-4}\,0\,2.1\,3\,4\,43\right]$

$4x^5+9x^4-5x^3-7x^2+x+1$

$-6x{y}^2+2y^3+xy+5x-1$

For information on sorting polynomials, see Sorting Terms.

For more information on sorting, refer to the sort help page.

To evaluate an expression at a point, you must substitute a value for a variable.

To substitute a value for a variable using the context panel:

In Worksheet mode, Maple inserts the eval command calling sequence that performs the substitution. This is the most common use of the eval command.

For example, substitute $x\=3$ in the following polynomial.

$x^3+4x^2-7x+2$

$44$

To substitute a value for a variable using palettes:

For example:

$\sqrt{17}$

Substitutions performed by the eval function are syntactical, not the more powerful algebraic form of substitution.

If the left-hand side of the substitution is a name, Maple performs the substitution.

$\cos \left(\frac{\mathrm{\pi }bc}{6}\right)$

If the left-hand side of the substitution is not a name, Maple performs the substitution only if the left-hand side of the substitution is an operand of the expression.

$\frac{\sqrt{3}}{2}$

$\cos \left(abc\right)$

Maple did not perform the evaluation because $ab$ is not an operand of $\cos \left(abc\right)\.$ For information on operands, refer to the op help page.

For algebraic substitution, use the algsubs command, or the simplify command with side relations.

$\cos \left(\frac{c\mathrm{\pi }}{6}\right)$

$\cos \left(\frac{c\mathrm{\pi }}{6}\right)$

To compute an approximate numerical value of an expression:

The evalf command returns a floating-point (or complex floating-point) number or expression.

$0.8660254040$

$9.814954579x^2+x-23.14069264$

$3.141592654$

By default, Maple calculates the result to ten digits of accuracy, but you can specify any number of digits as an index, that is, in brackets ([ ]).

$3.141592653589793238462643383279502884197$

For more information, refer to the evalf help page.

See also Numerically Computing a Limit and Numeric Integration.

To evaluate a complex expression:

If possible, the evalc command returns the output in the canonical form expr1 + i expr2.

In 2-D Math input, you can enter the imaginary unit using the following two methods.

$\frac{\sqrt{2+2\sqrt{2}}}{2}+\frac{\mathrm{I}\sqrt{-2+2\sqrt{2}}}{2}$

$\sin \left(3\right)\cosh \left(5\right)+\mathrm{I}\cos \left(3\right)\sinh \left(5\right)$

In 1-D Math input, enter the imaginary unit as an uppercase i (I).

$2\cos \left(\ln \left(2\right)\right)+2\mathrm{I}\sin \left(\ln \left(2\right)\right)$