Section6.3

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## 6.3 Introduction to Functions

Each name in your workspace can refer to a single object. This may be any kind of object including a function. You can use interactively any function from the library or any that you define in the workspace. In the library the same name can have very many functions, but you can have only one function with a given name, although it can have any number of arguments that you choose.

If you define a function in the workspace that has the same name and number of arguments as one in the library, then your definition takes precedence. In fact, to get the library function you must package-call it (see section ugTypesPkgCall ).

To use a function in Axiom, you apply it to its arguments. Most functions are applied by entering the name of the function followed by its argument or arguments.

factor(12)

2 2 3

Type: Factored Integer

Some functions like ``+'' have infix operators as names.

3 + 4

7

Type: PositiveInteger

The function ``+'' has two arguments. When you give it more than two arguments, Axiom groups the arguments to the left. This expression is equivalent to (1+2)+7.

1 + 2 + 7

10

Type: PositiveInteger

All operations, including infix operators, can be written in prefix form, that is, with the operation name followed by the arguments in parentheses. For example, 2+3 can alternatively be written as +(2, 3). But +(2, 3, 4) is an error since + takes only two arguments.

Prefix operations are generally applied before the infix operation. Thus the form factorial 3+1 means factorial(3)+1 producing 7, and -2+5 means (-2)+5 producing 3. An example of a prefix operator is prefix ``-''. For example, -2+5 converts to (-2)+5 producing the value 3. Any prefix function taking two arguments can be written in an infix manner by putting an ampersand ``&'' before the name. Thus D(2*x, x) can be written as 2*x&D x returning 2.

Every function in Axiom is identified by a name and type. (An exception is an ``anonymous function'' discussed in ugUserAnon .) The type of a function is always a mapping of the form Source->Target where Source and Target are types. To enter a type from the keyboard, enter the arrow by using a hyphen ``-'' followed by a greater-than sign ``>'', e.g. Integer -> Integer.

Let's go back to ``+''. There are many ``+'' functions in the Axiom library: one for integers, one for floats, another for rational numbers, and so on. These ``+'' functions have different types and thus are different functions. You've seen examples of this overloading before---using the same name for different functions. Overloading is the rule rather than the exception. You can add two integers, two polynomials, two matrices or two power series. These are all done with the same function name but with different functions.

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