Factorial: Difference between revisions
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{{BuiltinFactorial!}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:factorialfactorial]] of a nonnegative integer. Factorial  {{BuiltinFactorial!}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:factorialfactorial]] of a nonnegative integer. Factorial takes its [[glyph]] <source lang=apl inline>!</source> from [[Comparison_with_traditional_mathematics#Prefixtraditional mathematics]] but, like all [[monadic function]]s, takes its argument on the right <source lang=apl inline>!Y</source> instead of traditional mathematics' <math>Y!</math>. It shares the glyph with the dyadic arithmetic function [[Binomial]].  
== Examples ==  == Examples ==  
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== Extended definition ==  == Extended definition ==  
In multiple implementations, this function has an extended definition using the [[wikipedia:Gamma functionGamma function]] Gamma(n), so that it is defined for real and [[complex]] numbers. Because Gamma(n) equals (n1)!, <source lang=apl inline>!Y</source> is defined as Gamma(Y+1).  In multiple implementations, this function has an extended definition using the [[wikipedia:Gamma functionGamma function]] <math>\Gamma(n)</math>, so that it is defined for real and [[complex]] numbers. Because <math>\Gamma(n)</math> equals <math>(n1)!</math>, <source lang=apl inline>!Y</source> is defined as <math>\Gamma(Y+1)</math>.  
<source lang=apl>  <source lang=apl> 
Revision as of 19:07, 3 June 2020
!

Factorial (!
) is a monadic scalar function which gives the factorial of a nonnegative integer. Factorial takes its glyph !
from traditional mathematics but, like all monadic functions, takes its argument on the right !Y
instead of traditional mathematics' . It 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 , so that it is defined for real and complex numbers. Because equals , !Y
is defined as .
!¯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 _ __ _ __