Factorial

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Factorial (!) is a monadic scalar function which gives the factorial of a non-negative integer. Factorial 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 Gamma(n), so that it is defined for real and complex numbers. Because Gamma(n) equals (n-1)!, !Y 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
Works in: Dyalog APL

The Gamma function diverges at 0 or negative numbers, so !Y is undefined at negative integers. 

      !¯1
DOMAIN ERROR
 !¯1        
 ∧
Works in: Dyalog APL

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 positive-side limit of the Gamma function. 

      !_1 _2 _3 _4
_ __ _ __
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)
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
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