# Factorial

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 `!`

Factorial (`!`) is a monadic scalar function which gives the factorial of a non-negative integer. Factorial shares the glyph `!` with the dyadic arithmetic function Binomial.

## Examples

The factorial of a positive integer n is defined as the product of 1 to n inclusive.

```      !0 1 2 3 4
1 1 2 6 24
×/⍳4
24
```

## Extended definition

In multiple implementations, this function has an extended definition using the Gamma function Gamma(n), so that it is defined for real and complex numbers. Because Gamma(n) equals (n-1)!, `!Y` is defined as Gamma(Y+1).

```      !¯1.2 0.5 2.7
¯5.821148569 0.8862269255 4.170651784
!2J1 ¯2J¯1
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482
```
Works in: Dyalog APL

The Gamma function diverges at 0 or negative numbers, so `!Y` is undefined at negative integers.

```      !¯1
DOMAIN ERROR
!¯1
∧
```
Works in: Dyalog APL

In J, where literal infinity is supported, negative integer factorial evaluates to positive infinity `_` (if the argument is odd) or negative infinity `__` (if even). This corresponds to the positive-side limit of the Gamma function.

```      !_1 _2 _3 _4
_ __ _ __
```
Works in: J