View source for Factorial

{{Short description|Product of all integers between 1 and the integral input of the function}}
{| class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;"
|+ Selected members of the factorial [[sequence]] {{OEIS|id=A000142}}; values specified in scientific notation are rounded to the displayed precision
|-
! {{math|''n''}}
! {{math|''n''!}}
|-
| 0 || 1
|-
| 1 || 1
|-
| 2 || 2
|-
| 3 || 6
|-
| 4 || 24
|-
| 5 || 120
|-
| 6 || 720
|-
| 7 || {{val|5040|fmt=gaps}}
|-
| 8 || {{val|40320}}
|-
| 9 || {{val|362880}}
|-
| 10 || {{val|3628800}}
|-
| 11 || {{val|39916800}}
|-
| 12 || {{val|479001600}}
|-
| 13 || {{val|6227020800}}
|-
| 14 || {{val|87178291200}}
|-
| 15 || {{val|1307674368000}}
|-
| 16 || {{val|20922789888000}}
|-
| 17 || {{val|355687428096000}}
|-
| 18 || {{val|6402373705728000}}
|-
| 19 || {{val|121645100408832000}}
|-
| 20 || {{val|2432902008176640000}}
|-
| 25 
| style="text-align:left" | {{val|1.551121004|e=25}}
|-
| 50 
| style="text-align:left" | {{val|3.041409320|e=64}}
|-
| 70 
| style="text-align:left" | {{val|1.197857167|e=100}}
|-
| 100 
| style="text-align:left" | {{val|9.332621544|e=157}}
|-
| 450 
| style="text-align:left" | {{val|1.733368733|e=1000|fmt=gaps}}
|-
| {{val|1000|fmt=gaps}} 
| style="text-align:left" | {{val|4.023872601|e=2567|fmt=gaps}}
|-
| {{val|3249|fmt=gaps}} 
| style="text-align:left" | {{val|6.412337688|e=10000}}
|-
| {{val|10000|fmt=gaps}} 
| style="text-align:left" | {{val|2.846259681|e=35659}}
|-
| {{val|25206|fmt=gaps}} 
| style="text-align:left" | {{val|1.205703438|e=100000}}
|-
| {{val|100000|fmt=gaps}} 
| style="text-align:left" | {{val|2.824229408|e=456573}}
|-
| {{val|205023|fmt=gaps}} 
| style="text-align:left" | {{val|2.503898932|e=1000004}}
|-
| {{val|1000000|fmt=gaps}} 
| style="text-align:left" | {{val|8.263931688|e=5565708}}
|-
| [[googol|{{val|e=100}}]] ||10<sup>{{val|e=101.9981097754820}}</sup>
|}

In [[mathematics]], the '''factorial''' of a positive [[integer]] {{mvar|n}}, denoted by {{math|''n''!}}, is the [[Product (mathematics)|product]] of all positive integers less than or equal to {{mvar|n}}:
:<math>n! = n  \times  (n-1)  \times (n-2)  \times  (n-3) \times \cdots \times  3 \times  2 \times  1 \,. </math>
For example,
:<math>5! = 5  \times  4  \times  3  \times  2  \times  1 = 120 \,. </math>

The value of 0! is 1, according to the convention for an [[empty product]].{{sfn|Graham|Knuth|Patashnik|1988|page=111}}

The factorial operation is encountered in many areas of mathematics, notably in [[combinatorics]], [[algebra]], and [[mathematical analysis]]. Its most basic use counts the possible distinct [[sequence]]s – the [[permutations]] – of {{mvar|n}} distinct objects: there are {{math|''n''!}}.

The factorial [[Function (mathematics)|function]] can also be [[Factorial#Factorial of non-integer values|extended to non-integer arguments]] while retaining its most important properties. This involves using [[gamma function]] to define {{math|1=''x''! = Γ(''x'' + 1)}}.  However, this extension does not work when {{mvar|x}} is a negative integer.

==History==
Factorials were used to count permutations at least as early as the 12th century, by Indian scholars.<ref>{{Cite journal |last=Biggs |first=Norman L. |author-link=Norman L. Biggs |date=May 1979 |title=The roots of combinatorics |journal=Historia Mathematica |volume=6 |issue=2 |pages=109–136 |doi=10.1016/0315-0860(79)90074-0 |issn=0315-0860 }}</ref>  In 1677, [[Fabian Stedman]] described factorials as applied to [[change ringing]], a musical art involving the ringing of many tuned bells.{{sfn|Stedman|1677|pages=6–9}}  After describing a recursive approach, Stedman gives a statement of a factorial (using the language of the original):
{{Cquote
| quote = Now the nature of these methods is such, that the changes on one number comprehends [includes] the changes on all lesser numbers ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body.{{sfn|Stedman|1677|p=8}}
}}
The [[mathematical notation|notation]] {{math|{{math|''n''!}}}} was introduced by the French mathematician [[Christian Kramp]] in 1808.<ref>{{harvnb|Higgins|2008|page=12}}</ref>

==Definition==
The factorial function is defined by the product

:<math>n!  = 1 \cdot 2 \cdot 3 \cdots (n-2) \cdot (n-1) \cdot n,</math>

for integer {{math|''n'' ≥ 1}}.  This may be written in [[Multiplication#Capital pi notation|pi product notation]] as

:<math>n!  = \prod_{i = 1}^n i.</math>

From these formulas, one may derive the [[recurrence relation]]

:<math> n! = n \cdot (n-1)! .</math>

For example, one has 
: <math>\begin{align}
 5! &=  5 \cdot  4! \\
 6! &=  6 \cdot  5! \\
50! &= 50 \cdot 49!
\end{align}</math>
and so on.

===Factorial of zero===
The factorial of {{math|0}} is {{math|1}}, or in symbols, {{math|1=0! = 1}}.

There are several motivations for this definition:
* For {{math|1= ''n'' = 0}}, the definition of {{math|''n''!}} as a product involves the product of no numbers at all, and so is an example of the broader convention that the product of no factors is equal to the multiplicative identity (see [[Empty product]]).
* There is exactly one permutation of zero objects (with nothing to permute, the only rearrangement is to do nothing).
* It makes many identities in [[combinatorics]] valid for all applicable sizes. The number of ways to choose 0 elements from the [[empty set]] is given by the [[binomial coefficient]]
::<math>\binom{0}{0} = \frac{0!}{0!0!} = 1. </math>
: More generally, the number of ways to choose all {{mvar|n}} elements among a set of {{mvar|n}} is
::<math>\binom{n}{n} = \frac{n!}{n!0!} = 1. </math>
* It allows for the compact expression of many formulae, such as the [[exponential function]], as a power series:
:: <math> e^x = \sum_{n = 0}^\infty \frac{x^n}{n!}.</math>
* It extends the recurrence relation to 0.

==Applications==
Although the factorial function has its roots in [[combinatorics]], formulas involving factorials occur in many areas of mathematics.
* There are {{math|''n''!}} different ways of arranging {{mvar|n}} distinct objects into a sequence, the [[permutation]]s of those objects.<ref>{{Cite book |title=Beyond Infinity: An expedition to the outer limits of the mathematical universe |last=Cheng |first=Eugenia |date=2017-03-09 |publisher=Profile Books |isbn=9781782830818 |language=en |author-link=Eugenia Cheng}}</ref><ref name="ConwayGuy1998">{{Cite book |title=The Book of Numbers |last=Conway |first=John H. |last2=Guy |first2=Richard |date=1998-03-16 |publisher=Springer Science & Business Media |isbn=9780387979939 |language=en |author-link=John Horton Conway |author-link2=Richard K. Guy}}</ref>
* Often factorials appear in the [[denominator]] of a formula to account for the fact that ordering is to be ignored. A classical example is counting {{mvar|k}}-[[combination]]s (subsets of {{mvar|k}} elements) from a set with {{mvar|n}} elements. One can obtain such a combination by choosing a {{mvar|k}}-permutation: successively selecting and removing one element of the set, {{mvar|k}} times, for a total of
::<math>(n-0)(n-1)(n-2)\cdots\left(n-(k-1)\right) = \frac{n!}{(n-k)!} = n^{\underline k}</math>
:possibilities. This, however, produces the {{mvar|k}}-combinations in a particular order that one wishes to ignore; since each {{mvar|k}}-combination is obtained in {{math|''k''!}} different ways, the correct number of {{mvar|k}}-combinations is
::<math>\frac{n(n-1)(n-2)\cdots(n-k+1)}{k(k-1)(k-2)\cdots 1} = \frac{n^{\underline k}}{k!}=  \frac{n!}{(n-k)!k!} = \binom {n}{k}.</math>
:This number is known<ref name="Knuth1997">{{Cite book |title=The Art of Computer Programming: Volume 1: Fundamental Algorithms |last=Knuth |first=Donald E. |date=1997-07-04 |publisher=Addison-Wesley Professional |isbn=9780321635747 |language=en |author-link=Donald Knuth}}</ref> as the [[binomial coefficient]], because it is also the coefficient of {{math|''x''<sup>''k''</sup>}} in {{math|(1 + ''x'')<sup>''n''</sup>}}. The term <math>n^{\underline k}</math> is often called a [[Falling and rising factorials|falling factorial]] (pronounced "''n'' to the falling ''k''").
* Factorials occur in [[algebra]] for various reasons, such as via the already mentioned coefficients of the [[binomial formula]], or through [[averaging]] over [[permutations]] for [[symmetrization]] of certain operations.
* Factorials also turn up in [[calculus]]; for example, they occur in the denominators of the terms of [[Taylor's formula]],<ref>{{Cite web|url=https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/|title=18.01 Single Variable Calculus, Lecture 37: Taylor Series|last=|first=|date=Fall 2006|website=MIT OpenCourseWare|archive-url=https://web.archive.org/web/20121706133400/http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/|archive-date=2018-04-26|url-status=live|access-date=2017-05-03|df=}}</ref> where they are used as compensation terms due to the {{mvar|n}}th [[derivative]] of {{math|''x''<sup>''n''</sup>}} being equivalent to {{math|''n''!}}.
* Factorials are also used extensively in [[probability theory]]<ref>{{Cite book |title=Statistical Physics of Particles |last=Kardar |first=Mehran |date=2007-06-25 |publisher=Cambridge University Press |isbn=9780521873420 |pages=35–56 |language=English |chapter=Chapter 2: Probability |author-link=Mehran Kardar}}</ref> and [[number theory]] ([[Factorial#Number theory|see below]]).
* Factorials can be useful to facilitate expression manipulation. For instance the number of {{mvar|k}}-permutations of {{mvar|n}} can be written as
::<math>n^{\underline k}=\frac{n!}{(n-k)!}\,;</math>
:while this is inefficient as a means to compute that number, it may serve to prove a symmetry property<ref name="ConwayGuy1998" /><ref name="Knuth1997" /> of binomial coefficients:
::<math>\binom nk=\frac{n^{\underline k}}{k!}=\frac{n!}{(n-k)!k!} = \frac{n^{\underline{n-k}}}{(n-k)!} = \binom n{n-k}\,.</math>
* The factorial function can be shown, using the [[power rule]], to be
::<math>n! = D^n\,x^n = \frac{d^n}{dx^n}\,x^n</math>
:where {{math|''D''<sup>''n''</sup> ''x''<sup>''n''</sup>}} is [[Notation for differentiation#Euler.27s notation|Euler's notation]] for the {{mvar|n}}th [[derivative]] of {{math|''x<sup>n</sup>''}}.<ref>{{Cite web|url=https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/|title=18.01 Single Variable Calculus, Lecture 4: Chain rule, higher derivatives|last=|first=|date=Fall 2006|website=MIT OpenCourseWare|archive-url=https://web.archive.org/web/20121706133400/http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/|archive-date=2018-04-26|url-status=live|access-date=2017-05-03|df=}}</ref>

==Rate of growth and approximations for large ''{{mvar|n}}''==
[[File:Log-factorial.svg|upright=1.35|thumb|right|Plot of the natural logarithm of the factorial]]
As {{mvar|n}} grows, the factorial {{math|''n''!}} increases faster than all [[polynomial]]s and [[exponential growth|exponential functions]] (but slower than <math>n^n</math>and [[double exponential function]]s) in {{mvar|n}}.

Most approximations for ''n''! are based on approximating its [[natural logarithm]]
:<math>\ln n! = \sum_{x=1}^n \ln x \,.</math>

The graph of the function {{math|''f''(''n'') {{=}} ln ''n''!}} is shown in the figure on the right. It looks approximately [[linear function|linear]] for all reasonable values of {{mvar|n}}, but this intuition is false. We get one of the simplest approximations for {{math|ln ''n''!}} by bounding the sum with an [[integral]] from above and below as follows:

:<math> \int_1^n \ln x \, dx \leq \sum_{x=1}^n \ln x \leq \int_0^n \ln (x+1) \, dx</math>

which gives us the estimate

:<math> n\ln\left(\frac{n}{e}\right)+1 \leq \ln n! \leq (n+1)\ln\left( \frac{n+1}{e} \right) + 1 \,.</math>

Hence {{math|ln ''n''! ∼ ''n'' ln ''n''}} (see [[Big O notation#Family of Bachmann–Landau notations|Big {{mvar|O}} notation]]). This result plays a key role in the analysis of the [[computational complexity theory|computational complexity]] of [[sorting algorithm]]s (see [[comparison sort]]). From the bounds on {{math|ln ''n''!}} deduced above we get that

:<math>\left(\frac ne\right)^n e \leq n! \leq \left(\frac{n+1}e\right)^{n+1} e \,.</math>

It is sometimes practical to use weaker but simpler estimates. Using the above formula it is easily shown that for all {{mvar|n}} we have {{math|({{sfrac|''n''|3}})<sup>''n''</sup> < ''n''!}}, and for all {{math|''n'' ≥ 6}} we have {{math|''n''! < ({{sfrac|''n''|2}})<sup>''n''</sup>}}.

[[File:Mplwp factorial gamma stirling.svg|thumb|right|upright=1.35|Comparison of Stirling's approximation with the factorial]]
For large {{mvar|n}} we get a better estimate for the number {{math|''n''!}} using [[Stirling's approximation]]:

:<math>n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\,.</math>

This in fact comes from an asymptotic series for the logarithm, and {{mvar|n}} factorial lies between this and the next approximation:

:<math>\sqrt{2\pi n}\left(\frac{n}{e}\right)^n<n!<\sqrt{2\pi n}\left(\frac{n}{e}\right)^ne^{1/(12n)} \,.</math>

Another approximation for {{math|ln ''n''!}} is given by [[Srinivasa Ramanujan]] {{harv|Ramanujan|1988}}

:<math>\begin{align}
\ln n! &\approx n\ln n - n + \frac {\ln\Bigl(n\bigl(1+4n(1+2n)\bigr)\Bigr)}{6} + \frac {\ln\pi}{2} \\[6px]
\Longrightarrow \; n!&\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\left(1 +\frac 1{2n} +\frac 1{8n^2}\right)^{1/6} \,.
\end{align}</math>

Both this and Stirling's approximation give a relative error on the order of {{math|{{sfrac|1|''n''<sup>3</sup>}}}}, but Ramanujan's is about four times more accurate. However, if we use ''two'' correction terms in a Stirling-type approximation, as with Ramanujan's approximation, the relative error will be of order {{math|{{sfrac|1|''n''<sup>5</sup>}}}}:{{Citation needed|date=March 2019}}

:<math>n!\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\exp\left({\frac 1{12n}-\frac 1{360n^3}}\right) \,.</math>

==Computation==
If efficiency is not a concern, computing factorials is trivial from an algorithmic point of view: successively multiplying a variable initialized to 1 by the integers up to {{mvar|n}} (if any) will compute {{math|''n''!}}, provided the result fits in the variable. In functional languages, the recursive definition is often implemented directly to illustrate recursive functions.

The main practical difficulty in computing factorials is the size of the result. To assure that the exact result will fit for all legal values of even the smallest commonly used integral type (8-bit signed integers) would require more than 700 bits, so no reasonable specification of a factorial function using fixed-size types can avoid questions of overflow. The values 12! and 20! are the largest factorials that can be stored in, respectively, the 32-bit and 64-bit integers commonly used in [[personal computer]]s, however many languages support variable length integer types capable of calculating very large values.<ref>{{cite web|url=https://github.com/wesselbosman/nFactorial|title=wesselbosman/nFactorial|author=|date=2017-12-25|website=GitHub|access-date=26 April 2018|url-status=live|archive-url=https://web.archive.org/web/20180426001413/https://github.com/wesselbosman/nFactorial|archive-date=26 April 2018|df=}}</ref> [[Floating-point]] representation of an approximated result allows going a bit further, but this also remains quite limited by possible overflow. Most [[calculator]]s use [[scientific notation]] with 2-digit decimal exponents, and the largest factorial that fits is then 69!, because {{nowrap|69! < 10<sup>100</sup> < 70!}}.  Other implementations (such as computer software such as spreadsheet programs) can often handle larger values.

Most software applications will compute small factorials by direct multiplication or table lookup. Larger factorial values can be approximated using [[Stirling's formula]]. [[Wolfram Alpha]] can calculate exact results for the [[ceiling function]] and [[floor function]] applied to the [[binary logarithm|binary]], [[natural logarithm|natural]] and [[common logarithm]] of {{math|''n''!}} for values of {{mvar|n}} up to {{val|249999}}, and up to {{val|20000000|end=!}} for the integers.

If the exact values of large factorials are needed, they can be computed using [[arbitrary-precision arithmetic]]. Instead of doing the sequential multiplications {{nowrap|((1 × 2) × 3) × 4...}}, a program can partition the sequence into two parts, whose products are roughly the same size, and multiply them using a [[divide-and-conquer algorithm|divide-and-conquer]] method. This is often more efficient.<ref>{{cite web|work=[[GNU MP]] Software Manual |url=http://gmplib.org/manual/Factorial-Algorithm.html |title=Factorial Algorithm |url-status=dead |archive-url=https://web.archive.org/web/20130314174653/http://gmplib.org/manual/Factorial-Algorithm.html |archive-date=2013-03-14 |access-date=2013-01-22}}</ref>

The asymptotically best efficiency is obtained by computing {{math|''n''!}} from its prime factorization. As documented by [[Peter Borwein]], prime factorization allows {{math|''n''!}} to be computed in time {{math|[[Big O notation|''O'']](''n''(log ''n'' log log ''n'')<sup>2</sup>)}}, provided that a fast [[multiplication algorithm]] is used (for example, the [[Schönhage–Strassen algorithm]]).<ref>{{cite journal|first=Peter |last=Borwein |authorlink=Peter Borwein |title=On the Complexity of Calculating Factorials |journal=Journal of Algorithms |volume=6 |issue=3 |pages=376–380 |date=1985|doi=10.1016/0196-6774(85)90006-9 }}</ref> Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a [[prime sieve]].<ref>{{cite web|first=Peter |last=Luschny |url=http://www.luschny.de/math/factorial/FastFactorialFunctions.htm |title=Fast-Factorial-Functions: The Homepage of Factorial Algorithms |url-status=dead |archive-url=https://web.archive.org/web/20050305083634/http://www.luschny.de/math/factorial/FastFactorialFunctions.htm |archive-date=2005-03-05 }}</ref>

==Number theory==
Factorials have many applications in number theory. In particular, {{math|''n''!}} is necessarily divisible by all [[prime number]]s up to and including&nbsp;{{mvar|n}}. As a consequence, {{math|''n'' > 5}} is a [[composite number]] [[if and only if]]
:<math>(n-1)! \equiv 0 \pmod n.</math>

A stronger result is [[Wilson's theorem]], which states that
:<math>(p-1)! \equiv -1 \pmod p</math>
if and only if {{mvar|p}} is prime.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref><ref>{{MathWorld|url=WilsonsTheorem.html|title=Wilson's Theorem|access-date=2017-05-17}}</ref>

[[Legendre's formula]] gives the multiplicity of the prime {{mvar|p}} occurring in the prime factorization of {{math|''n''!}} as
:<math>\sum_{i=1}^\infty \left \lfloor \frac n {p^i} \right \rfloor</math>
or, equivalently, 
:<math>\frac{n - s_p(n)}{p - 1},</math>
where {{math|''s<sub>p</sub>''(''n'')}} denotes the sum of the standard base-{{mvar|p}} digits of {{mvar|n}}.

Adding 1 to a factorial {{math|''n''!}} yields a number that is only divisible by primes that are larger than {{mvar|n}}. This fact can be used to prove [[Euclid's theorem]] that the number of primes is infinite.{{sfn|Bostock |Chandler |Rourke |2014|pages=168}}  Primes of the form {{math|''n''! ± 1}} are called [[factorial prime]]s.

==Series of reciprocals==
The [[multiplicative inverse|reciprocals]] of factorials produce a [[convergent series]] whose sum is [[e (mathematical constant)|the exponential base {{mvar|e}}]]:
:<math>\sum_{n=0}^\infty \frac{1}{n!} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24}  + \frac{1}{120} + \cdots = e\,.</math>
Although the sum of this series is an [[irrational number]], it is possible to multiply the factorials by positive integers to produce a convergent series with a rational sum:
:<math>\sum_{n=0}^\infty \frac 1 {(n+2)n!} = \frac{1}{2}+\frac{1}{3}+\frac{1}{8} + \frac{1}{30} + \frac{1}{144} + \cdots=1\,.</math>
The convergence of this series to 1 can be seen from the fact that its [[partial sum]]s are less than one by an inverse factorial.
Therefore, the factorials do not form an [[irrationality sequence]].{{sfn|Guy |2004|page=[{{google books|plainurl=yes|id=1AP2CEGxTkgC|pg=PA346}} 346]}}

==Factorial of non-integer values==

===The gamma and pi functions===
[[File:Factorial Interpolation.svg|thumb|320px|The gamma function interpolates the factorial function to non-integer values. The main clue is the [[recurrence relation]] generalized to a continuous domain.]]

{{Main|Gamma function}}

Besides nonnegative integers, the factorial can also be defined for non-integer values, but this requires more advanced tools from [[mathematical analysis]].

One function that fills in the values of the factorial (but with a shift of 1 in the argument), that is often used, is called the [[gamma function]], denoted {{math|Γ(''z'')}}. It is defined for all complex numbers {{mvar|z}} except for the non-positive integers, and given when the real part of {{mvar|z}} is positive by

:<math>\Gamma(z)=\int_0^\infty t^{z-1} e^{-t}\, dt.</math>

Its relation to the factorial is that {{math|1=''n''! = &Gamma;(''n'' + 1)}} for every nonnegative integer {{mvar|n}}.

[[Leonhard Euler|Euler's]] original formula for the gamma function was

:<math>\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\displaystyle\prod_{k=0}^n (z+k)}. </math>

[[Carl Friedrich Gauss]] used the notation {{math|Π(''z'')}} to denote the same function, but with argument shifted by 1, so that it agrees with the factorial for nonnegative integers.  This '''pi function''' is defined by

:<math>\Pi(z)=\int_0^\infty t^z e^{-t}\, dt.</math>

The pi function and gamma function are related by the formula {{math|1=&Pi;(''z'') = &Gamma;(''z'' + 1)}}.  Likewise, {{math|1=&Pi;(''n'') = ''n''!}} for any nonnegative integer {{mvar|n}}.

[[File:Generalized factorial function.svg|thumb|right|upright=1.6|The factorial function, generalized to all real numbers except negative integers. For example, {{nowrap|1=0! = 1! = 1}}, {{nowrap|1=(−{{sfrac|1|2}})! = {{sqrt|{{pi}}}}}}, {{nowrap|1={{sfrac|1|2}}! = {{sfrac|{{sqrt|{{pi}}}}|2}}}}.]]

In addition to this, the pi function satisfies the same recurrence as factorials do, but at every complex value {{mvar|z}} where it is defined

:<math>\Pi(z) = z\Pi(z-1)\,.</math>

In fact, this is no longer a recurrence relation but a [[functional equation]]. Expressed in terms of the gamma function this functional equation takes the form

:<math>\Gamma(n+1)=n\Gamma(n)\,.</math>

The values of these functions at [[half-integer]] values is therefore determined by a single one of them; one has

:<math>\Gamma\left (\frac{1}{2}\right )=\left (-\frac{1}{2}\right )!=\Pi\left (-\frac{1}{2}\right ) = \sqrt{\pi}\,,</math>

from which it follows that for&nbsp;{{math|''n'' ∈ '''N'''}},

:<math>\begin{align}
& \Gamma\left (\frac{1}{2}+n\right ) = \left (-\frac{1}{2}+n\right )! = \Pi\left (-\frac{1}{2}+n\right ) \\[5pt]
= {} & \sqrt{\pi} \prod_{k=1}^n \frac{2k - 1}{2} = \frac{(2n)!}{4^n n!} \sqrt{\pi} = \frac{(2n-1)!}{2^{2n-1}(n-1)!} \sqrt{\pi}\,.
\end{align}</math>

For example,
:<math>\Gamma\left(\frac{9}{2}\right) = \frac{7}{2}! = \Pi\left(\frac{7}{2}\right) = \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{5}{2} \cdot \frac{7}{2} \sqrt{\pi} = \frac{8!}{4^4 4!} \sqrt{\pi} = \frac{7!}{2^7 3!} \sqrt{\pi} = \frac{105}{16} \sqrt{\pi} \approx 11.631\,728\ldots</math>

It also follows that for&nbsp;{{math|''n'' ∈ '''N'''}},

:<math>\Gamma\left (\frac{1}{2}-n\right ) = \left (-\frac{1}{2}-n\right )! = \Pi\left (-\frac{1}{2}-n\right ) = \sqrt{\pi} \prod_{k=1}^n \frac{2}{1 - 2k} = \frac{\left(-4\right)^n n!}{(2n)!} \sqrt{\pi}\,.</math>

For example,
:<math>\Gamma\left(-\frac{5}{2}\right) = \left(-\frac{7}{2}\right)! = \Pi\left(-\frac{7}{2}\right) = \frac{2}{-1}\cdot\frac{2}{-3}\cdot\frac{2}{-5} \sqrt{\pi} = \frac{\left(-4\right)^3 3!}{6!} \sqrt{\pi} = -\frac{8}{15} \sqrt{\pi} \approx -0.945\,308\ldots</math>

The pi function is certainly not the only way to extend factorials to a function defined at almost all complex values, and not even the only one that is [[Analytic function|analytic]] wherever it is defined. Nonetheless it is usually considered the most natural way to extend the values of the factorials to a complex function.  For instance, the [[Bohr–Mollerup theorem]] states that the gamma function is the only function that takes the value 1 at 1, satisfies the functional equation {{math|Γ(''n'' + 1) {{=}} ''n''Γ(''n'')}}, is [[meromorphic]] on the complex numbers, and is [[log-convex]] on the positive real axis.  A similar statement holds for the pi function as well, using the {{math|Π(''n'') {{=}} ''n''Π(''n'' − 1)}} functional equation.

However, there exist complex functions that are probably simpler in the sense of analytic function theory and which interpolate the factorial values.  For example, [[Hadamard's gamma function|Hadamard's 'gamma' function]] {{harv|Hadamard|1894}} which, unlike the gamma function, is an [[entire function]].<ref>{{cite web|first=Peter |last=Luschny |url=http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html |title=Hadamard versus Euler – Who found the better Gamma function? |url-status=dead |archive-url=https://web.archive.org/web/20090818171719/http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html |archive-date=2009-08-18 }}</ref>

Euler also developed a convergent product approximation for the non-integer factorials, which can be seen to be equivalent to the formula for the gamma function above:

:<math>\begin{align}
n! = \Pi(n) &= \prod_{k = 1}^\infty \left(\frac{k+1}{k}\right)^n\!\!\frac{k}{n+k} \\
&= \left[ \left(\frac{2}{1}\right)^n\frac{1}{n+1}\right]\left[ \left(\frac{3}{2}\right)^n\frac{2}{n+2}\right]\left[ \left(\frac{4}{3}\right)^n\frac{3}{n+3}\right]\cdots \end{align}</math>

However, this formula does not provide a practical means of computing the pi function or the gamma function, as its rate of convergence is slow.

===Applications of the gamma function===
The [[volume]] of an [[Dimension|{{mvar|n}}-dimensional]] [[N-sphere|hypersphere]] of radius {{mvar|R}} is

:<math>V_n=\frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)}R^n\,.</math>

===Factorial in the complex plane===
[[File:Factorial05.jpg|upright=1.8|thumb|Amplitude and phase of factorial of complex argument]]
Representation through the gamma function allows evaluation of factorial of complex argument. Equilines of amplitude and phase of factorial are shown in figure. Let
:<math> f=\rho e^{i\varphi}=(x+iy)!=\Gamma(x+iy+1) \,.</math>
Several levels of constant modulus (amplitude) {{mvar|ρ}} and constant phase {{mvar|φ}} are shown. The grid covers the range {{math|−3 ≤ ''x'' ≤ 3}}, {{math|−2 ≤ ''y'' ≤ 2}}, with unit steps. The scratched line shows the level {{math|''φ'' {{=}} ±π}}.

Thin lines show intermediate levels of constant modulus and constant phase. At the poles at every negative integer, phase and amplitude are not defined. Equilines are dense in vicinity of singularities along negative integer values of the argument.

For {{math|{{abs|''z''}} < 1}}, the Taylor expansions can be used:
:<math>z!=\sum_{n=0}^\infty g_n z^n\,.</math>
The first coefficients of this expansion are
{| class="wikitable"
|-
! {{mvar|n}}
! {{mvar|g<sub>n</sub>}}
! Approximation
|-
| 0
| 1
| 1
|-
| 1
| {{math|−''γ''}}
| {{val|-0.5772156649}}
|-
| 2
| {{math|{{sfrac|π<sup>2</sup>|12}} + {{sfrac|''γ''<sup>2</sup>|2}}}}
| {{val|0.9890559955}}
|-
| 3
| {{math|−{{sfrac|''ζ''(3)|3}} − {{sfrac|π<sup>2</sup>|12}} − {{sfrac|''γ''<sup>3</sup>|6}}}}
| {{val|-0.9074790760}}
|}

where {{mvar|γ}} is the [[Euler–Mascheroni constant]] and {{math|''ζ''}}  is the [[Riemann zeta function]]. [[Computer algebra system]]s such as [[SageMath]] can generate many terms of this expansion.

===Approximations of the factorial===
For the large values of the argument, the factorial can be approximated through the integral of the [[digamma function]], using the [[continued fraction]] representation. This approach is due to [[T. J. Stieltjes]] (1894).{{citation needed|date=November 2018}} Writing {{math|''z''! {{=}} ''e''<sup>''P''(''z'')</sup>}} where {{math|''P''(''z'')}} is
: <math> P(z) = p(z) + \frac{\ln 2\pi}{2} - z + \left(z+\frac{1}{2}\right)\ln(z) \,,</math>
Stieltjes gave a continued fraction for {{math|''p''(''z'')}}:
: <math>
p(z)=\cfrac{a_0}{z+
\cfrac{a_1}{z+
\cfrac{a_2}{z+
\cfrac{a_3}{z+\ddots}}}}
</math>
The first few coefficients {{math|''a''<sub>''n''</sub>}} are<ref name="dlmf5.10">{{cite web |url=http://dlmf.nist.gov/5.10 |title=5.10 |work=Digital Library of Mathematical Functions |access-date=2010-10-17 |url-status=live |archive-url=https://web.archive.org/web/20100529224939/http://dlmf.nist.gov/5.10 |archive-date=2010-05-29 |df= }}</ref>
:{| class="wikitable"
|-
! {{math|''n''}}
! {{math|''a''<sub>''n''</sub>}}
|-
| 0
| {{sfrac|1|12}}
|-
| 1
| {{sfrac|1|30}}
|-
| 2
| {{sfrac|53|210}}
|-
| 3
| {{sfrac|195|371}}
|-
| 4
| {{sfrac|{{val|22999}}|{{val|22737}}}}
|-
| 5
| {{sfrac|{{val|29944523}}|{{val|19733142}}}}
|-
| 6
| {{sfrac|{{val|109535241009}}|{{val|48264275462}}}}
|}

There is a misconception that {{math|ln ''z''! {{=}} ''P''(''z'')}} or {{math|ln Γ(''z'' + 1) {{=}} ''P''(''z'')}} for any complex {{math|''z'' ≠ 0}}.{{citation needed|date=February 2015}} Indeed, the relation through the logarithm is valid only for a specific range of values of {{mvar|z}} in the vicinity of the real axis, where {{math|−π < Im(Γ(''z'' + 1)) < π}}. The larger the real part of the argument, the smaller the imaginary part should be.  However, the inverse relation, {{math|''z''! {{=}} ''e''<sup>''P''(''z'')</sup>}}, is valid for the whole complex plane apart from {{math|''z'' {{=}} 0}}. The convergence is poor in the vicinity of the negative part of the real axis;{{citation needed|date=February 2015}} it is difficult to have good convergence of any approximation in the vicinity of the singularities.  When {{math|{{abs|Im ''z''}} > 2}} or {{math|Re ''z'' > 2}}, the six coefficients above are sufficient for the evaluation of the factorial with complex double precision. For higher precision more coefficients can be computed by a rational QD scheme (Rutishauser's QD algorithm).<ref>{{cite web|first=Peter |last=Luschny |url=http://www.luschny.de/math/factorial/approx/continuedfraction.html |title=On Stieltjes' Continued Fraction for the Gamma Function |url-status=dead |archive-url=https://web.archive.org/web/20110514230817/http://www.luschny.de/math/factorial/approx/continuedfraction.html |archive-date=2011-05-14 }}</ref>

===Non-extendability to negative integers===
The relation {{math|''n''! {{=}} ''n'' × (''n''&nbsp;−&nbsp;1)!}} allows one to compute the factorial for an integer given the factorial for a smaller integer.  The relation can be inverted so that one can compute the factorial for an integer given the factorial for a larger integer:

:<math>(n-1)! = \frac{n!}{n} .</math>

However, this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)! would require a [[division by zero|division of a nonzero value by zero]], and thus blocks us from computing a factorial value for every negative integer. Similarly, the [[gamma function]] is not defined for zero or negative integers, though it is defined for all other complex numbers.

==Factorial-like products and functions==
There are several other integer sequences similar to the factorial that are used in mathematics:

=== Double factorial ===
{{main|Double factorial}}
The product of all the odd integers up to some odd positive integer {{mvar|n}} is called the '''double factorial''' of {{mvar|n}}, and denoted by {{math|''n''!!}}.<ref name="callan">{{citation|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|arxiv=0906.1317|year=2009|bibcode=2009arXiv0906.1317C}}.</ref> That is,
:<math>(2k-1)!! = \prod_{i=1}^k (2i-1) = \frac{(2k)!}{2^k k!} = \frac {_{2k}P_k} {2^k} = \frac {\left(2k\right)^{\underline k}} {2^k}\,.</math>
For example, {{nowrap|1=9!! = 1 × 3 × 5  × 7 × 9 = 945}}.

The sequence of double factorials for {{math|''n'' {{=}} 1, 3, 5, 7,...}} starts as
: 1, 3, 15, 105, 945, {{val|10395}}, {{val|135135}},... {{OEIS|id=A001147}}

Double factorial notation may be used to simplify the expression of certain [[List of integrals of trigonometric functions|trigonometric integrals]],<ref>{{citation
 | last = Meserve | first = B. E.
 | doi = 10.2307/2306136
 | issue = 7
 | journal = The American Mathematical Monthly
 | mr = 1527019
 | pages = 425–426
 | title = Classroom Notes: Double Factorials
 | volume = 55
 | year = 1948| jstor = 2306136
 }}</ref> to provide an expression for the values of the [[gamma function]] at half-integer arguments and the volume of [[hypersphere]]s,<ref>{{citation|title=Some dimension problems in molecular databases|first=Paul G.|last=Mezey|year=2009|journal=Journal of Mathematical Chemistry|volume=45|issue=1|pages=1–6|doi=10.1007/s10910-008-9365-8}}.</ref> and to solve many [[enumerative combinatorics|counting problems in combinatorics]] including counting [[rooted binary tree|binary trees]] with labeled leaves and [[perfect matching]]s in [[complete graph]]s.<ref name="callan"/><ref>{{citation
 | last1 = Dale | first1 = M. R. T.
 | last2 = Moon | first2 = J. W.
 | doi = 10.1016/0378-3758(93)90035-5
 | issue = 1
 | journal = Journal of Statistical Planning and Inference
 | mr = 1209991
 | pages = 75–87
 | title = The permuted analogues of three Catalan sets
 | volume = 34
 | year = 1993}}.</ref>

===Multifactorials===<!-- This section is linked from [[Catalan number]] -->
A common related notation is to use multiple exclamation points to denote a '''multifactorial''', the product of integers in steps of two ({{math|''n''!!}}), three ({{math|''n''!!!}}), or more (see [[double factorial#Generalizations|generalizations of the double factorial]]). The double factorial is the most commonly used variant, but one can similarly define the triple factorial ({{math|''n''!!!}}) and so on. One can define the {{mvar|k}}-tuple factorial, denoted by {{math|''n''!<sup>(''k'')</sup>}}, recursively for positive integers as
:<math> n!^{(k)} = \begin{cases}
n & \text{if } 0 < n \leq k, \\
n\left({(n-k)!}^{(k)}\right) & \text{if } n > k.
\end{cases}</math>
In addition, similarly to {{nowrap|0! {{=}} {{sfrac|1!|1}} {{=}} 1}}, one can define:
:<math> {n!}^{(k)} = 1 \quad \text{if } -k < n \leq 0</math>

For sufficiently large {{math|''n'' ≥ 1}}, the ordinary single factorial function is expanded through the multifactorial functions as follows:

:<math>\begin{align}
n! & = n!! \cdot (n-1)!!\,, &n &\geq 1 \\ [5px]
   & = n!!! \cdot (n-1)!!! \cdot (n-2)!!!\,, &n &\geq 2 \\ [5px]
   & = \prod_{i=0}^{k-1} (n-i)!^{(k)},\quad \text{ for } k \in \mathbb{Z}^{+}\,, &n &\geq k-1\,. 
\end{align}</math>

In the same way that {{math|''n''!}} is not defined for negative integers, and {{math|''n''!!}} is not defined for negative even integers, {{math|''n''!<sup>(''k'')</sup>}} is not defined for negative integers divisible by {{mvar|k}}.

===Primorial===
{{Main|Primorial}}
The '''primorial''' of a natural number {{mvar|n}} {{OEIS|id=A002110}}, denoted {{Math|''n''#}}, is similar to the factorial, but with the product taken only over the [[prime number]]s less than or equal to {{mvar|n}}. That is,

:<math>n\# = \prod_{p \leq n} p,</math>

where {{mvar|p}} ranges over the prime numbers less than or equal to {{mvar|n}}.  For example, the primorial of 11 is

:<math>11\#=11\times7\times5\times3\times2=2310.</math>

===Superfactorial===
{{See also|Large numbers}}{{redirect|N$|the currency|Namibian dollar}}
[[Neil Sloane]] and [[Simon Plouffe]] defined a '''superfactorial''' in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first {{mvar|n}} factorials.  So the superfactorial of 4 is

:<math> \operatorname{sf}(4)=1! \times 2! \times 3! \times 4!=288\,. </math>

In general

:<math>\begin{align}
\operatorname{sf}(n) =\prod_{k=1}^n k! &=\prod_{k=1}^n k^{n-k+1}\\
 & =1^n\cdot2^{n-1}\cdot3^{n-2}\cdots(n-1)^2\cdot n^1\,.
\end{align} </math>

Equivalently, the superfactorial is given by the formula
:<math> \operatorname{sf}(n) =\prod_{0 \le i < j \le n} (j-i) </math>
which is the [[determinant]] of a [[Vandermonde matrix]].

The sequence of superfactorials starts (from {{math|''n'' {{=}} 0}}) as

:1, 1, 2, 12, 288, {{val|34560}}, {{val|24883200}}, {{val|125411328000}},... {{OEIS|id=A000178}}

By this definition, we can define the {{mvar|k}}-superfactorial of {{mvar|n}} (denoted {{math|sf<sub>''k''</sub>(''n'')}}) as:

:<math> \operatorname{sf}_k(n) = \begin{cases}
n & \text{if } k=0\\
\prod_{r=1}^n \operatorname{sf}_{k-1}(r) & \text{if } k \ge 1
\end{cases}</math>

The 2-superfactorials of {{mvar|n}} are

:1, 1, 2, 24, {{val|6912|fmt=gaps}}, {{val|238878720}}, {{val|5944066965504000}}, {{val|745453331864786829312000000}},... {{OEIS|id=A055462}}

The 0-superfactorial of {{mvar|n}} is {{mvar|n}}.

====Pickover’s superfactorial====
In his 1995 book ''Keys to Infinity'', [[Clifford Pickover]] defined a different function {{math|''n''$}} that he called the superfactorial.  It is defined by
:<math>n\$\equiv \begin{matrix} \underbrace{ n!^{{n!}^{{\cdot}^{{\cdot}^{{\cdot}^{n!}}}}}} \\ n! \mbox{ copies of } n! \end{matrix}. </math>
This sequence of superfactorials starts
:<math>\begin{align}
1\$&=1\,,\\
2\$&=2^2=4\,,\\
3\$&=6^{6^{6^{6^{6^6}}}}\,.
\end{align}</math>
(Here, as is usual for compound [[exponentiation]], the grouping is understood to be from right to left: {{math|''a<sup>b<sup>c</sup></sup>'' {{=}} ''a''<sup>(''b<sup>c</sup>'')</sup>}}.)

This operation may also be expressed as the [[tetration]] 
:<math>n\$={}^{n!}(n!)\,,</math>
or using [[Knuth's up-arrow notation]] as
:<math>n\$=(n!)\uparrow\uparrow(n!)\,. </math>

===Hyperfactorial===
Occasionally the '''hyperfactorial''' of ''n'' is considered. It is written as {{math|''H''(''n'')}} and defined by

:<math>\begin{align}
H(n)  &=\prod_{k=1}^n k^k \\
& =1^1\cdot2^2\cdot3^3\cdots(n-1)^{n-1}\cdot n^n.
\end{align} </math>

For {{math|''n'' {{=}} 1, 2, 3, 4,...}} the values of {{math|''H''(''n'')}} are 1, 4, 108, {{val|27648}},... {{OEIS|id=A002109}}.

The asymptotic growth rate is

:  <math>H(n) \sim A n^{(6n^2 + 6n + 1)/12} e^{-n^2/4}</math>

where {{mvar|A}} = 1.2824... is the [[Glaisher–Kinkelin constant]].<ref>{{MathWorld | urlname=Glaisher-KinkelinConstant | title=Glaisher–Kinkelin Constant}}</ref> {{math|''H''(14)}} ≈&nbsp;{{val|1.8474e99}} is already almost equal to a [[googol]], and {{math|''H''(15)}} ≈&nbsp;{{val|8.0896e116}} is almost of the same magnitude as the [[Shannon number]], the theoretical number of possible chess games. Compared to the Pickover definition of the superfactorial, the hyperfactorial grows relatively slowly.

The hyperfactorial function can be generalized to [[complex number]]s in a similar way as the factorial function. The resulting function is called the [[K-function|{{mvar|K}}-function]].

==See also==
{{Portal|Arithmetic|Mathematics}}
{{div col|colwidth=30em}}
* [[Alternating factorial]]
* [[Bhargava factorial]]
* [[Digamma function]]
* [[Exponential factorial]]
* [[Factorial number system]]
* [[Factorion]]
* [[List of factorial and binomial topics]]
* [[Pochhammer symbol]], which gives the falling or rising factorial
* [[Subfactorial]]
* [[Trailing zeros#Factorial|Trailing zeros]] of factorial
* [[Triangular number]], the additive analogue of factorial
{{div col end}}

== References ==
{{Reflist}}

===Sources===
{{Refbegin}}
*{{Citation |last=Bostock |first=Linda |title=Further Pure Mathematics |date=2014-11-01 |publisher=Nelson Thornes |language=en |isbn=9780859501033 |last2=Chandler |first2=Suzanne |author2-link= Sue Chandler |last3=Rourke |first3=C.}}
*{{citation|first1=Ronald L.|last1=Graham|author1-link= Ronald Graham |first2=Donald E.|last2=Knuth|author2-link=Donald Knuth|first3=Oren|last3=Patashnik|author3-link=Oren Patashnik|date=1988|title=Concrete Mathematics|publisher=Addison-Wesley|location=Reading, MA|isbn=0-201-14236-8|title-link=Concrete Mathematics}}
*{{Citation |last=Guy |first=Richard K. |title=Unsolved problems in number theory |url={{google books|plainurl=yes|id=1AP2CEGxTkgC}} |year=2004 |contribution=E24 Irrationality sequences |edition=3rd |publisher=[[Springer-Verlag]] |isbn=0-387-20860-7 |zbl=1058.11001 |author-link=Richard K. Guy}}
*{{Citation |last=Higgins |first=Peter |title=Number Story: From Counting to Cryptography |year=2008 |place=New York |publisher=Copernicus |isbn=978-1-84800-000-1}}
*{{citation|last=Stedman|first=Fabian|authorlink=Fabian Stedman|title=Campanalogia|year=1677|place=London}}  The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the [[Ancient Society of College Youths|Society of College Youths]], to which society the "Dedicatory" is addressed.
{{Refend}}

===Further reading===
{{Refbegin}}
* {{citation |first=M. J. |last=Hadamard
|chapter=Sur L’Expression Du Produit 1·2·3· · · · ·(''n''−1) Par Une Fonction Entière
|title=Œuvres de Jacques Hadamard |publisher=Centre National de la Recherche Scientifiques |location=Paris (1968)
|chapter-url=http://www.luschny.de/math/factorial/hadamard/HadamardFactorial.pdf|year=1894 |language=French}}
* {{citation|first=Srinivasa|last=Ramanujan|title=The Lost Notebook and Other Unpublished Papers
|publisher=Springer Berlin |page=339 |year=1988|isbn=3-540-18726-X}}
{{Refend}}

==External links==
{{Commons category|Factorial (function)}}
* {{Britannica|200040|Factorial (mathematics)}}
* {{springer|title=Factorial|id=p/f038080}}
* {{MathWorld | urlname=Factorial | title=Factorial}}
* {{PlanetMath | urlname=Factorial | title=Factorial}}
{{Calculus topics}}
{{Series (mathematics)}}

[[Category:Combinatorics]]
[[Category:Gamma and related functions]]
[[Category:Factorial and binomial topics]]
[[Category:Unary operations]]