Factorial

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' $Y!$ . 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 </syntaxhighlight>

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 $(n-1)!$ , <source lang=apl inline>!Y</syntaxhighlight> 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

</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.

     !¯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.

     !_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 ∙ Round
		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 ∙ Identity (Null, Ident)
Operators	Dyadic	Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names	Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector

Factorial

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Examples

Extended definition

External links

Documentation

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Factorial

Examples

Extended definition

External links

Documentation

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