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
, with default values
The default value of
is used here.
Now the default values of both
|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.
matches the pattern
taken to have the default value
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
By giving a pattern in which the exponent is optional, you can match both cases.
matches any linear function of
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.