# Immediate and Delayed Definitions

You may have noticed that there are two different ways to make assignments in *Mathematica*: and . The basic difference between these forms is *when* the expression rhs is evaluated. is an *immediate assignment*, in which rhs is evaluated at the time when the assignment is made. , on the other hand, is a *delayed assignment*, in which rhs is not evaluated when the assignment is made, but is instead evaluated each time the value of lhs is requested.

lhs=rhs (immediate assignment) | rhs is evaluated when the assignment is made |

lhs:=rhs (delayed assignment) | rhs is evaluated each time the value of lhs is requested |

The two types of assignments in *Mathematica*.

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As you can see from the example above, both and can be useful in defining functions, but they have different meanings, and you must be careful about which one to use in a particular case.

One rule of thumb is the following. If you think of an assignment as giving the final "value" of an expression, use the operator. If instead you think of the assignment as specifying a "command" for finding the value, use the operator. If in doubt, it is usually better to use the operator than the one.

lhs=rhs | rhs is intended to be the "final value" of lhs (e.g. ) |

lhs:=rhs | rhs gives a "command" or "program" to be executed whenever you ask for the value of lhs (e.g. f[x_]:=Expand[1-x^2]) |

Interpretations of assignments with the and operators.

Although is probably used more often than in defining functions, there is one important case in which you must use to define a function. If you do a calculation, and get an answer in terms of a symbolic parameter , you often want to go on and find results for various specific values of . One way to do this is to use the operator to apply appropriate rules for in each case. It is usually more convenient, however, to use to define a function whose argument is .

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An important point to notice in the example above is that there is nothing special about the name that appears in the pattern. It is just a symbol, indistinguishable from an that appears in any other expression.

Defining functions for evaluating expressions.

You can use and not only to define functions, but also to assign values to variables. If you type , then value is immediately evaluated, and the result is assigned to x. On the other hand, if you type , then value is not immediately evaluated. Instead, it is maintained in an unevaluated form, and is evaluated afresh each time is used.

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The distinction between immediate and delayed assignments is particularly important when you set up chains of assignments.

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You can use delayed assignments such as to set up variables whose values you can find in a variety of different "environments". Every time you ask for , the expression rhs is evaluated using the current values of the objects on which it depends.

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In the example above, the symbol acts as a "global variable", whose value affects the value of . When you have a large number of parameters, many of which change only occasionally, you may find this kind of setup convenient. However, you should realize that implicit or hidden dependence of one variable on others can often become quite confusing. When possible, you should make all dependencies explicit, by defining functions which take all necessary parameters as arguments.

Two types of transformation rules in *Mathematica*.

Just as you can make immediate and delayed assignments in *Mathematica*, so you can also set up immediate and delayed transformation rules.

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In analogy with assignments, you should typically use when you want to replace an expression with a definite value, and you should use when you want to give a command for finding the value.