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.
This uses the
operator to define the function
.
Because
was used, the definition is maintained in an unevaluated form.
When you make an assignment with the
operator, the right-hand side is evaluated immediately.
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The definition now stored is the result of the
Expand command.
When you execute
, the
Expand is performed.
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simply substitutes its argument into the already expanded form, giving a different answer.
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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
.
Here is an expression involving
.
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This defines a function whose argument is the value to be taken for
.
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Here is the result when
is taken to be
.
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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.
| f[x_]=expr | define a function which gives the value expr for any particular value of x |
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.
This evaluates
RandomReal
to find a pseudorandom number, then assigns this number to
.
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Here
RandomReal
is maintained in an unevaluated form, to be evaluated afresh each time
is used.
Here are values for
and
.
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The value of
never changes. Every time
is used, however, a new pseudorandom number is generated.
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The distinction between immediate and delayed assignments is particularly important when you set up chains of assignments.
This defines
to be
.
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Here
is evaluated to give
, and the result is assigned to be the value of
.
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Here
is maintained in an unevaluated form, to be evaluated every time the value of
is requested.
In this case,
and
give the same values.
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Now the value of
is changed.
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Now
uses the new value for
, while
keeps its original value.
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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.
The right-hand side of the delayed assignment is maintained in an unevaluated form.
This sets
to
, then finds the value of
.
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Here
is 6.
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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.
| lhs->rhs | rhs is evaluated when the rule is given |
| lhs:>rhs | rhs is evaluated when the rule is used |
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.
The right-hand side of this rule is evaluated when you give the rule.
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A rule like this is probably not particularly useful.
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Here the right-hand side of the rule is maintained in an unevaluated form, to be evaluated every time the rule is used.
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Applying the rule causes the expansion to be done.
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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.
