Optional (:)

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Optional[pattern, default]

pattern : default

is a pattern matching pattern; when pattern is omitted, default is substituted for pattern.

Optional is used to specify optional arguments in function signatures.

Set up a default value of 1 for the pattern y_ in function f:

• • f[x_, y_:1] := {x, y}

Above, we put no spaces before or after :, but they can be added. So:

• • f[x_, y_: 1] := {x, y}

is the same as the above.

When we specify a value for the y parameter, it has the value provided:

• • f[a, 2]

But if the y parameter is missing, we replace the parameter using the default given in the delayed assignment above:

• • f[a]

Both Optional and Pattern use : as their operator symbol. And both operators are used to represent a pattern.

The way to disambiguate which of the two is used is by the first or left operand. When this is a symbol, like y, the : operator indicates a Pattern:

• • y : 1 // FullForm

In contrast, we have a pattern to the left of the colon, like y_ we have an Optional expression:

• • y_ : 1 // FullForm

The special form y_. is equivalent to Optional[y_]:

• • FullForm[y_.]

In this situation, when the is y parameter omitted, the value comes from Default:

• • Default[g] = 4

• • g[x_, y_.] := {x, y}

• • g[a]

Note that the Optional operator binds more tightly than the Pattern. Keep this in mind when there is more than one colon, juxtaposed, each representing different operators:

• • x : _+y_ : d // FullForm