Factorial: Difference between revisions
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m (Text replacement  "</source>" to "</syntaxhighlight>") 

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{{BuiltinFactorial!}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:factorialfactorial]] of a nonnegative integer. Factorial takes its [[glyph]] <source lang=apl inline>!</  {{BuiltinFactorial!}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:factorialfactorial]] of a nonnegative integer. Factorial takes its [[glyph]] <source lang=apl inline>!</syntaxhighlight> from [[Comparison_with_traditional_mathematics#Prefixtraditional mathematics]] but, like all [[monadic function]]s, takes its argument on the right <source lang=apl inline>!Y</syntaxhighlight> instead of traditional mathematics' <math>Y!</math>. It shares the glyph with the dyadic arithmetic function [[Binomial]].  
== Examples ==  == Examples ==  
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×/⍳4  ×/⍳4  
24  24  
</  </syntaxhighlight>  
== Extended definition ==  == Extended definition ==  
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</  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</syntaxhighlight> is defined as <math>\Gamma(Y+1)</math>.  
<source lang=apl>  <source lang=apl>  
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!2J1 ¯2J¯1  !2J1 ¯2J¯1  
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482  0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482  
</  </syntaxhighlight>{{Works in[[Dyalog APL]]}}  
The Gamma function diverges at 0 or negative numbers, so <source lang=apl inline>!Y</  The Gamma function diverges at 0 or negative numbers, so <source lang=apl inline>!Y</syntaxhighlight> is undefined at negative integers.  
<source lang=apl>  <source lang=apl>  
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!¯1  !¯1  
∧  ∧  
</  </syntaxhighlight>{{Works in[[Dyalog APL]]}}  
In [[J]], where literal [[infinity]] is supported, negative integer factorial evaluates to positive infinity <source lang=j inline>_</  In [[J]], where literal [[infinity]] is supported, negative integer factorial evaluates to positive infinity <source lang=j inline>_</syntaxhighlight> (if the argument is odd) or negative infinity <source lang=j inline>__</syntaxhighlight> (if even). This corresponds to the positiveside limit of the Gamma function.  
<source lang=j>  <source lang=j>  
!_1 _2 _3 _4  !_1 _2 _3 _4  
_ __ _ __  _ __ _ __  
</  </syntaxhighlight>{{Works in[[J]]}}  
== External links ==  == External links == 
Revision as of 21:17, 10 September 2022
!

Factorial (!
) is a monadic scalar function which gives the factorial of a nonnegative integer. Factorial takes its glyph <source lang=apl inline>!</syntaxhighlight> from traditional mathematics but, like all monadic functions, takes its argument on the right <source lang=apl inline>!Y</syntaxhighlight> 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.
<source lang=apl>
!0 1 2 3 4
1 1 2 6 24
×/⍳4
24 </syntaxhighlight>
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 , <source lang=apl inline>!Y</syntaxhighlight> is defined as .
<source lang=apl>
!¯1.2 0.5 2.7
¯5.821148569 0.8862269255 4.170651784
!2J1 ¯2J¯1
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482
</syntaxhighlight>
The Gamma function diverges at 0 or negative numbers, so <source lang=apl inline>!Y</syntaxhighlight> is undefined at negative integers.
<source lang=apl>
!¯1
DOMAIN ERROR
!¯1 ∧
</syntaxhighlight>
In J, where literal infinity is supported, negative integer factorial evaluates to positive infinity <source lang=j inline>_</syntaxhighlight> (if the argument is odd) or negative infinity <source lang=j inline>__</syntaxhighlight> (if even). This corresponds to the positiveside limit of the Gamma function.
<source lang=j>
!_1 _2 _3 _4
_ __ _ __
</syntaxhighlight>