Transformation Rules for Functions
"Values for Symbols" discussed how you can use transformation rules of the form 
to replace symbols by values. The notion of transformation rules in Mathematica is, however, quite general. You can set up transformation rules not only for symbols, but for any Mathematica expression.
Applying the transformation rule
replaces
by
.
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You can also use a transformation rule for
. This rule does not affect
.
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is a
pattern that stands for
with any argument.
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Probably the most powerful aspect of transformation rules in Mathematica is that they can involve not only literal expressions, but also patterns. A pattern is an expression such as 
which contains a blank (underscore). The blank can stand for any expression. Thus, a transformation rule for 
specifies how the function 
with any argument should be transformed. Notice that, in contrast, a transformation rule for 
without a blank, specifies only how the literal expression 
should be transformed, and does not, for example, say anything about the transformation of 
.
When you give a function definition such as 
, all you are doing is telling Mathematica to automatically apply the transformation rule 
whenever possible.
You can set up transformation rules for expressions of any form.
| Out[4]= |  |
This uses a transformation rule for
.
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"Patterns" and "Transformation Rules and Definitions" explain in detail how to set up patterns and transformation rules for any kind of expression. Suffice it to say here that in Mathematica all expressions have a definite symbolic structure; transformation rules allow you to transform parts of that structure.
