Factorial: Difference between revisions

{{Built-in|Factorial|!}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:factorial|factorial]] of a non-negative integer. Factorial takes its [[glyph]] <source lang=apl inline>!</source> from [[Comparison_with_traditional_mathematics#Prefix|traditional 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]].

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In multiple implementations, this function has an extended definition using the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, so that it is defined for real and [[complex]] numbers. Because <math>\Gamma(n)</math> equals <math>(n-1)!</math>, <source lang=apl inline>!Y</source> is defined as <math>\Gamma(Y+1)</math>.

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The Gamma function diverges at 0 or negative numbers, so <source lang=apl inline>!Y</source> is undefined at negative integers.

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In [[J]], where literal [[infinity]] is supported, negative integer factorial evaluates to positive infinity <source lang=j inline>_</source> (if the argument is odd) or negative infinity <source lang=j inline>__</source> (if even). This corresponds to the positive-side limit of the Gamma function.

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