Functions and Programs—Wolfram Language Documentation [proxy]

Defining Functions	Repetitive Operations
Functions as Procedures	Transformation Rules for Functions
Manipulating Options

As a first example, consider adding a function called f which squares its argument. The Wolfram Language command to define this function is f[x_]:=x^2. The _ (referred to as "blank") on the left‐hand side is very important; what it means is discussed here. For now, just remember to put a _ on the left‐hand side, but not on the right‐hand side, of your definition.

This defines the function f. Notice the _ on the left‐hand side:

f squares its argument:

You can use f in a calculation:

This shows the definition you made for f:

f[x_]:=x^2	define the function f
?f	show the definition of f
Clear[f]	clear all definitions for f

The names like f that you use for functions in the Wolfram Language are just symbols. Because of this, you should make sure to avoid using names that begin with capital letters, to prevent confusion with built‐in Wolfram Language functions. You should also make sure that you have not used the names for anything else earlier in your session.

You can use the hump function just as you would any of the built‐in functions:

This gives a new definition for hump, which overwrites the previous one:

This clears all definitions for hump:

When you have finished with a particular function, it is always a good idea to clear definitions you have made for it. If you do not do this, then you will run into trouble if you try to use the same function for a different purpose later in your Wolfram Language session. You can clear all definitions you have made for a function or symbol f by using Clear[f].

This defines a function exprod which constructs a product of n terms, then expands it out:

Every time you use the function, it will execute the Product and Expand operations:

expr₁;expr₂;…	a sequence of expressions to evaluate
Module[{a,b,…},proc]	a procedure with local variables a, b, …

The function cex defined above is not a module, so the value of t "escapes", and exists even after the function returns:

This function is defined as a module with local variable u:

Now, however, the value of u does not escape from the function:

There are a number of functions built into the Wolfram Language which, like Plot, have various options you can set. The Wolfram Language provides some general mechanisms for handling such options.

If you do not give a specific setting for an option to a function like Plot, then the Wolfram Language will automatically use a default value for the option. The function Options[function,option] allows you to find out the default value for a particular option. You can reset the default using SetOptions[function,option->value]. Note that if you do this, the default value you have given will stay until you explicitly change it.

Options[function]	give a list of the current default settings for all options
Options[function,option]	give the default setting for a particular option
SetOptions[function,option->value,…]	reset defaults

This resets the default for the PlotRange option. The semicolon stops the Wolfram Language from printing out the rather long list of options for Plot:

Until you explicitly reset it, the default for the PlotRange option will now be All:

The graphics objects that you get from Plot or Show store information on the options they use. You can get this information by applying the Options function to these graphics objects.

Options[plot]	show all the options used for a particular plot
Options[plot,option]	show the setting for a specific option
AbsoluteOptions[plot,option]	show the absolute form used for a specific option, even if the setting for the option is Automatic or All

When a plot created with no explicit ImageSize is placed into an input cell, it will automatically shrink to more easily accommodate input.

Another approach is to use the Wolfram Language function Do, which works much like the iteration constructs in languages such as C and Fortran. Do uses the same Wolfram Language iterator notation as Sum and Product, described in "Sums and Products".

Do[expr,{i,i_max}]	evaluate expr with i running from 1 to i_max
Do[expr,{i,i_min,i_max,di}]	evaluate expr with i running from i_min to i_max in steps of di
Print[expr]	print expr
Table[expr,{i,i_max}]	make a list of the values of expr with i running from 1 to i_max

"Values for Symbols" discussed how you can use transformation rules of the form x->value to replace symbols by values. The notion of transformation rules in the Wolfram System is, however, quite general. You can set up transformation rules not only for symbols, but for any Wolfram System expression.

Applying the transformation rule x->3 replaces x by 3:

You can also use a transformation rule for f[x]. This rule does not affect f[y]:

f[t_] is a pattern that stands for f with any argument:

Probably the most powerful aspect of transformation rules in the Wolfram System is that they can involve not only literal expressions, but also patterns. A pattern is an expression such as f[t_] which contains a blank (underscore). The blank can stand for any expression. Thus, a transformation rule for f[t_] specifies how the function f with any argument should be transformed. Notice that, in contrast, a transformation rule for f[x] without a blank, specifies only how the literal expression f[x] should be transformed, and does not, for example, say anything about the transformation of f[y].

When you give a function definition such as f[t_]:=t^2, all you are doing is telling the Wolfram System to automatically apply the transformation rule f[t_]->t^2 whenever possible.

This uses a transformation rule for x^p_:

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