Optional and Default Arguments
Sometimes you may want to set up functions where certain arguments, if omitted, are given "default values". The pattern 
stands for an object that can be omitted, and if so, will be replaced by the default value v.
This defines a function
with a required argument
, and optional arguments
and
, with default values
and
, respectively.
The default value of
is used here.
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Now the default values of both
and
are used.
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| x_:v | an expression which, if omitted, is taken to have default value v |
| x_h:v | an expression with head h and default value v |
| x_. | an expression with a built-in default value |
Pattern objects with default values.
Some common Mathematica functions have built-in default values for their arguments. In such cases, you need not explicitly give the default value in 
, but instead you can use the more convenient notation 
in which a built-in default value is assumed.
| x_+y_. | default for  is  |
| x_y_. | default for  is  |
| x_^y_. | default for  is  |
Some patterns with optional pieces.
Here
matches the pattern
with
taken to have the default value
.
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Because Plus is a flat function, a pattern such as 
can match a sum with any number of terms. This pattern cannot, however, match a single term such as 
. However, the pattern 
contains an optional piece, and can match either an explicit sum of terms in which both 
and 
appear, or a single term 
, with 
taken to be 
.
Using constructs such as 
, you can easily construct single patterns that match expressions with several different structures. This is particularly useful when you want to match several mathematically equal forms that do not have the same structure.
The pattern matches
, but not
.
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By giving a pattern in which the exponent is optional, you can match both cases.
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The pattern
matches any linear function of
.
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Standard Mathematica functions such as Plus and Times have built-in default values for their arguments. You can also set up defaults for your own functions, as described in "Patterns".
Sometimes it is convenient not to assign a default value to an optional argument; such arguments can be specified with the help of PatternSequence[].
Optional argument without a default value.
The pattern matches an optional second argument of
, without a default value.
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