Factorial: Difference between revisions

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{{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]].
{{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>!</syntaxhighlight> 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</syntaxhighlight> instead of traditional mathematics' <math>Y!</math>. It shares the glyph with the dyadic arithmetic function [[Binomial]].


== Examples ==
== Examples ==
Line 10: Line 10:
       ×/⍳4
       ×/⍳4
24
24
</source>
</syntaxhighlight>


== Extended definition ==
== Extended definition ==


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>.
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</syntaxhighlight> is defined as <math>\Gamma(Y+1)</math>.


<source lang=apl>
<source lang=apl>
Line 21: Line 21:
       !2J1 ¯2J¯1
       !2J1 ¯2J¯1
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482
0.962865153J1.339097176 ¯0.1715329199J¯0.3264827482
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


The Gamma function diverges at 0 or negative numbers, so <source lang=apl inline>!Y</source> is undefined at negative integers.
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>
Line 30: Line 30:
  !¯1         
  !¯1         
  ∧   
  ∧   
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


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.
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 positive-side limit of the Gamma function.


<source lang=j>
<source lang=j>
       !_1 _2 _3 _4
       !_1 _2 _3 _4
_ __ _ __
_ __ _ __
</source>{{Works in|[[J]]}}
</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 non-negative 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>

Works in: Dyalog APL

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>

Works in: Dyalog APL

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 positive-side limit of the Gamma function.

<source lang=j> 

     !_1 _2 _3 _4

_ __ _ __

</syntaxhighlight>

Works in: J

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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