# Making Definitions for Functions

"Defining Functions" discusses how you can define functions in

*Mathematica*. In a typical case, you would type in

to define a function

. (Actually, the definitions in

"Defining Functions" use the

operator, rather than the

one.

"Immediate and Delayed Definitions" explains exactly when to use each of the

and

operators.)

The definition

specifies that whenever

*Mathematica* encounters an expression that matches the pattern

, it should replace the expression by

. Since the pattern

matches all expressions of the form

f[anything], the definition applies to functions

with any "argument".

Function definitions like

can be compared with definitions like

for indexed variables discussed in

"Making Definitions for Indexed Objects". The definition

specifies that whenever the

*particular* expression

occurs, it is to be replaced by

. But the definition says nothing about expressions such as

, where

appears with another "index".

To define a "function", you need to specify values for expressions of the form

f[x], where the argument

x can be anything. You can do this by giving a definition for the pattern

, where the pattern object

stands for any expression.

f[x]=value | definition for a *specific* *expression* x |

f[x_]=value | definition for *any expression*, referred to as x |

The difference between defining an indexed variable and a function.

Making definitions for

or

can be thought of as being like giving values to various elements of an "array" named

. Making a definition for

is like giving a value for a set of "array elements" with arbitrary "indices". In fact, you can actually think of any function as being like an array with an arbitrarily variable index.

In mathematical terms, you can think of

as a

*mapping*. When you define values for, say,

and

, you specify the image of this mapping for various discrete points in its domain. Defining a value for

specifies the image of

on a continuum of points.

This defines a transformation rule for the

*specific* *expression* .

Out[1]= | |

When the specific expression

appears, it is replaced by

. Other expressions of the form

f[argument] are, however, not modified.

Out[2]= | |

This defines a value for

with

*any expression* as an "argument".

Out[3]= | |

The old definition for the

*specific* *expression* is still used, but the new general definition for

is now used to find a value for

.

Out[4]= | |

This removes all definitions for

.

*Mathematica* allows you to define transformation rules for any expression or pattern. You can mix definitions for specific expressions such as

or

with definitions for patterns such as

.

Many kinds of mathematical functions can be set up by mixing specific and general definitions in

*Mathematica*. As an example, consider the factorial function. This particular function is in fact built into

*Mathematica* (it is written

n!). But you can use

*Mathematica* definitions to set up the function for yourself.

The standard mathematical definition for the factorial function can be entered almost directly into

*Mathematica*, in the form

. This definition specifies that for any

n,

f[n] should be replaced by

, except that when

n is

,

should simply be replaced by

.

Here is the value of the factorial function with argument 1.

Out[6]= | |

Here is the general recursion relation for the factorial function.

Now you can use these definitions to find values for the factorial function.

Out[8]= | |

The results are the same as you get from the built-in version of factorial.

Out[9]= | |