# 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:

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

```
["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 `\_`

.

```
["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"]`

.

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

```
["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 in the domain of `Numbers`

for example, it is an
error to assign a function to it.

`Assign`

is not a pure function.

```
["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.

```
["Apply", "f", ["Apply", "g", "x"]]
```