This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Function

 Function[body]or body& is a pure function. The formal parameters are (or ), , etc. Functionis a pure function with a single formal parameter x. Functionis a pure function with a list of formal parameters.
• When Function[body] or body& is applied to a set of arguments, (or ) is replaced by the first argument, by the second, and so on. is replaced by the function itself.
• If there are more arguments supplied than #i in the function, the remaining arguments are ignored. »
• stands for the sequence of all arguments supplied. »
• ##n stands for arguments from number n on. »
• Function is analogous to in LISP or formal logic.
• Function has attribute HoldAll. The function body is evaluated only after the formal parameters have been replaced by arguments.
• The named formal parameters in Function are treated as local, and are renamed when necessary to avoid confusion with actual arguments supplied to the function. »
• Function constructs can be nested in any way. Each is treated as a scoping construct, with named inner variables being renamed if necessary. »
• Function represents a pure function that is to be treated as having attributes for the purpose of evaluation. »
Pure function with one parameter:
Pure function with two parameters:
Set to be a pure function:
Use the pure function:
Pure function with one parameter:
 Out[1]=
 Out[2]=
 Out[3]=

Pure function with two parameters:
 Out[1]=
 Out[2]=

Set to be a pure function:
 Out[1]=
Use the pure function:
 Out[2]=
 Scope   (12)
Map a pure function over a list:
Select with a pure function:
Use a pure function as a predicate:
Create an array from a pure function:
Sort by comparing the second part of each element:
Specify a custom comparison function in FixedPoint:
Specify a custom color function:
Provide a custom distance function:
Derivative of a pure function:
Derivative of Tan:
Solutions of differential equations may be expressed as pure functions:
Difference equations may return pure functions:
stands for all arguments:
##n stands for argument n on:
Create a pure function with attribute Listable:
stands for the whole pure function:
A recursive definition for factorial using :
 Applications   (3)
Turn a function that takes several arguments into one that takes a list of arguments:
A function that returns a function that multiplies its argument by :
Preserve arguments in unevaluated form:
uses only the first argument supplied; the rest are ignored:
Not using any arguments results in a constant pure function:
Replacements can be done inside pure functions:
Formal parameters are renamed whenever there is a possibility of confusion:
The names of the parameters do not matter:
However, reusing a name introduces a new scope:
Nested functions take their arguments one at a time:
is the same as simply f in the univariate case:
In general is the same as f:
Turn a formula involving a variable into a pure function:
Use a formula in Table:
Use the corresponding pure function in an equivalent Array expression:
Special-purpose function constructs include InterpolatingFunction:
binds more loosely than , so it usually needs parentheses in rules:
binds more loosely than , so it usually needs parentheses in pattern tests:
Function does not evaluate its body until the function is applied:
Supplying fewer than the required number of arguments generates an error:
Define the recursion operator of recursion theory []:
Use it to define the factorial function:
Newton's formula for finding a zero of a function: