Factorial
!

Factorial (!
) is a monadic scalar function which gives the factorial of a nonnegative 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 (n1)!, !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
The Gamma function diverges at 0 or negative numbers, so !Y
is undefined at negative integers.
!¯1 DOMAIN ERROR !¯1 ∧
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 positiveside limit of the Gamma function.
!_1 _2 _3 _4 _ __ _ __