Functions

The Compute Engine Standard Library includes many built-in functions such as Add, Sin, Power, etc…

The standard library can be extended with your own functions.

Anonymous Functions

A function that is not bound to an identifier is called an anonymous function.

Anonymous functions are frequently used as arguments to other functions.

In the example below, the ["Function"] expression is an anonymous function that is passed as an argument to the ["Sum"] function.

The first argument of the ["Function"] expression is the body of the function, the remaining arguments are the name of the parameters of the function.

["Sum", ["Function", ["Multiply", "x", 2], "x"]]

To specify an anonymous function with LaTeX use the \mapsto command:

x \mapsto 2x
\[ x \mapsto 2x \]
["Function", ["Multiply", "x", 2], "x"]
(x, y) \mapsto 2x + y
\[ (x, y) \mapsto 2x + y \]
["Function", ["Add", ["Multiply", "x", 2], "y"], "x", "y"]

Anonymous Parameters

The parameters of a function can also be anonymous.

In this case, the parameters are bound to the wildcards _, _1, _2, etc… in the body of the function. The wildcard _ is a shorthand for _1, the first parameter.

In the example below, both the function and its parameters are anonymous.

["Sum", ["Multiply", "_", 2]]

Note that as a shortcut when using anonymous parameters, the ["Function"] expression can be omitted.

Anonymous parameters can also be used in LaTeX, but the anonymous parameters must be wrapped with an \operatorname command except for \_.

() \mapsto \_ + \operatorname{\_2}
\[ () \mapsto \_ + \operatorname{\_2} \]
["Function", ["Add", "_", "_2"]]

Evaluating an Anonymous Function

To apply a function to some arguments, use an ["Apply"] expression.

["Apply", ["Function", ["Add", 2, "x"], "x"], 11]
// ➔ 22
 
["Apply", ["Add", 2, "_"], 4]
// ➔ 6
 
["Apply", "Power", 2, 3]
// ➔ 8

The first argument of Apply is an anonymous function, either as an identifier, or as a ["Function"] expression. The rest of the arguments are the arguments of the anonymous function.

Operating on Functions

Function

["Function", body]

["Function", body, arg-1, arg-2, …]

Create an anonymous function, also called lambda expression.

The arg-n arguments are identifiers of the bound variables (parameters) of the anonymous function.

All the arguments have the Hold attribute set, so they are not evaluated when the function is created.

The body is a MathJSON expression that is evaluated when the function is applied to some arguments.

To apply some arguments to a function expression, use ["Apply"].

x \mapsto 2x
\[ x \mapsto 2x \]
["Function", ["Multiply", "x", 2], "x"]
(x, y) \mapsto 2x + y
\[ (x, y) \mapsto 2x + y \]
["Function", ["Add", ["Multiply", "x", 2], "y"], "x", "y"]

Assign

["Assign", id, fn]

Assign the anonymous function fn to the identifier id.

The identifier id should either not have been declared yet, or been declared as a function. If id is already defined as a Number for example, it is an error to assign a function to it.

\operatorname{double} \coloneqq x \mapsto 2x
\[ \operatorname{double} \coloneqq x \mapsto 2x \]
["Assign", "double", ["Function", ["Multiply", "x", 2], "x"]]

Apply

["Apply", function, expr-1, …expr-n]

Apply a list of arguments to a function. The function is either an identifier of a function, or a ["Function"] expression.

The following wildcards in body are replaced as indicated

  • _ or _1 : the first argument
  • _2 : the second argument
  • _3 : the third argument, etc…
  • __: the sequence of arguments, so ["Length", "__"] is the number of arguments

If body is a ["Function"] expression, the named arguments of ["Function"] are replaced by the wildcards.

["Apply", ["Multiply", "_", "_"], 3]
// ➔ 9
 
["Apply", ["Function", ["Multiply", "x", "x"], "x"], 3]
// ➔ 9

The \lhd and \rhd operators can be used to apply a function to a single argument on the left or right respectively.

f\lhd g \lhd x
\[ f\lhd g \lhd x \]
x \rhd g \rhd f
\[ x \rhd g \rhd f \]
["Apply", "f", ["Apply", "g", "x"]]