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| | The name of mcdougal is Joe Ton. To lift weights is what he loves doing. She currently lives in West Virginia and he or she has everything that she needs there. Taking care of animals exactly what she engages in.<br><br>Here is my homepage [http://disqus.com/AlejandroMalcolm/ Resume] |
| |+ Selected members of the factorial [[sequence]] {{OEIS|id=A000142}}; values specified in scientific notation are rounded to the displayed precision
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| |-
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| ! ''n''
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| ! ''n''!
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| |-
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| | 0 || 1
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| |-
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| | 1 || 1
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| |-
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| | 2 || 2
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| |-
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| | 3 || 6
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| |-
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| | 4 || 24
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| |-
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| | 5 || 120
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| |-
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| | 6 || 720
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| |-
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| | 7 || {{gaps|5|040}}
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| |-
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| | 8 || {{gaps|40|320}}
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| |-
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| | 9 || {{gaps|362|880}}
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| |-
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| | 10 || {{gaps|3|628|800}}
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| |-
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| | 11 || {{gaps|39|916|800}}
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| |-
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| | 12 || {{gaps|479|001|600}}
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| |-
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| | 13 || {{gaps|6|227|020|800}}
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| |-
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| | 14 || {{gaps|87|178|291|200}}
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| |-
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| | 15 || {{gaps|1|307|674|368|000}}
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| |-
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| | 16 || {{gaps|20|922|789|888|000}}
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| |-
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| | 17 || {{gaps|355|687|428|096|000}}
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| |-
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| | 18 || {{gaps|6|402|373|705|728|000}}
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| |-
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| | 19 || {{gaps|121|645|100|408|832|000}}
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| |-
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| | 20 || {{gaps|2|432|902|008|176|640|000}}
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| |-
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| | 25 || {{val|1.5511210043|e=25}}
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| |-
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| | 42 || {{val|1.4050061178|e=51}}
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| |-
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| | 50 || {{val|3.0414093202|e=64}}
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| |-
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| | 70 || {{val|1.1978571670|e=100}}
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| |-
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| | 100 || {{val|9.3326215444|e=157}}
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| |-
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| | 450 || {{val|1.7333687331|e=1000}}
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| |-
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| | {{gaps|1|000}} || {{val|4.0238726008|e=2567}}
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| |-
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| | {{gaps|3|249}} || {{val|6.4123376883|e=10000}}
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| |-
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| | {{gaps|10|000}} || {{val|2.8462596809|e=35659}}
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| |-
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| | {{gaps|25|206}} || {{val|1.2057034382|e=100000}}
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| |-
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| | {{gaps|100|000}} || {{val|2.8242294080|e=456573}}
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| |-
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| | {{gaps|205|023}} || {{val|2.5038989317|e=1000004}}
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| |-
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| | {{gaps|1|000|000}} || {{val|8.2639316883|e=5565708}}
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| |-
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| | {{gaps|1|723|508}} || {{val|5.2900703070|e=10000001}}
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| |-
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| | {{gaps|2|000|000}} || {{val|3.7768210576|e=11733474}}
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| |-
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| | {{gaps|10|000|000}} || {{val|1.2024234005|e=65657059}}
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| |-
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| | {{gaps|14|842|907}} || {{val|2.7886629747|e=100000000}}
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| |-
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| | {{val|1.0248383838|e=98}} || [[googolplex|10<sup>{{val|e=100}}</sup>]]
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| |-
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| | [[googol|{{val|e=100}}]] || 10<sup>{{val|9.9565705518|e=101}}</sup>
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| |}
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| In [[mathematics]], the '''factorial''' of a [[non-negative integer]] ''n'', denoted by ''n''!, is the [[Product (mathematics)|product]] of all positive integers less than or equal to ''n''. For example,
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| :<math>5! = 5 \times 4 \times 3 \times 2 \times 1 = 120. \ </math>
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| The value of 0! is 1, according to the convention for an [[empty product]].<ref>Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) ''[[Concrete Mathematics]]'', Addison-Wesley, Reading MA. ISBN 0-201-14236-8, p. 111</ref> | |
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| The factorial operation is encountered in many areas of mathematics, notably in [[combinatorics]], [[algebra]] and [[mathematical analysis]]. Its most basic occurrence is the fact that there are ''n''! ways to arrange ''n'' distinct objects into a sequence (i.e., [[permutation]]s of the set of objects). This fact was known at least as early as the 12th century, to Indian scholars.<ref>N. L. Biggs, ''The roots of combinatorics'', Historia Math. 6 (1979) 109−136</ref>
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| The notation ''n''<nowiki>!</nowiki> was introduced by [[Christian Kramp]] in 1808.<ref>{{Citation |title=Number Story: From Counting to Cryptography |last=Higgins |first=Peter |year=2008 |publisher=Copernicus |location=New York |isbn=978-1-84800-000-1 |page=12 |pages= }} says Krempe though.</ref>
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| The definition of the factorial function can also be [[#Extension of factorial to non-integer values of argument|extended to non-integer arguments]], while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis.
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| ==Definition==
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| The factorial function is formally defined by the [[Multiplication#Capital Pi notation|product]]
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| :<math> n!=\prod_{k=1}^n k \!</math>
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| or by the [[recurrence relation]]
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| :<math> n! = \begin{cases}
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| 1 & \text{if } n = 0, \\
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| (n-1)!\times n & \text{if } n > 0
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| \end{cases}
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| </math>
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| The factorial function can also be defined by using the [[power rule]] as
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| :<math> n! = D^nx^n \;</math><ref>http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/lecture-notes/lec4.pdf</ref>
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| All of the above definitions incorporate the instance
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| :<math>0! = 1, \ </math>
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| in the first case by the convention that the [[empty product|product of no numbers at all]] is 1. This is convenient because:
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| * There is exactly one permutation of zero objects (with nothing to permute, "everything" is left in place).
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| * The [[recurrence relation]] {{nowrap|(''n'' + 1)! {{=}} ''n''! × (''n'' + 1)}}, valid for ''n'' > 0, extends to ''n'' = 0.
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| * It allows for the expression of many formulae, such as the [[exponential function]], as a power series:
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| :: <math> e^x = \sum_{n = 0}^{\infty}\frac{x^n}{n!}.</math>
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| * It makes many identities in [[combinatorics]] valid for all applicable sizes. The number of ways to choose 0 elements from the [[empty set]] is <math>\tbinom{0}{0} = \tfrac{0!}{0!0!} = 1</math>. More generally, the number of ways to choose (all) ''n'' elements among a set of ''n'' is <math>\tbinom nn = \tfrac{n!}{n!0!} = 1</math>.
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| The factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the [[#Extension of factorial to non-integer values of argument|section below]]. This more generalized definition is used by advanced [[calculator]]s and [[mathematical software]] such as [[Maple (software)|Maple]] or [[Mathematica]].
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| ==Applications==
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| {{unreferenced section|date=May 2013}}
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| Although the factorial function has its roots in [[combinatorics]], formulas involving factorials occur in many areas of mathematics.
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| * There are ''n''! different ways of arranging ''n'' distinct objects into a sequence, the [[permutation]]s of those objects.
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| * 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 ''k''-[[combination]]s (subsets of ''k'' elements) from a set with ''n'' elements. One can obtain such a combination by choosing a ''k''-permutation: successively selecting and removing an element of the set, ''k'' times, for a total of
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| ::<math>n^{\underline k}=n(n-1)(n-2)\cdots(n-k+1)</math>
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| :possibilities. This however produces the ''k''-combinations in a particular order that one wishes to ignore; since each ''k''-combination is obtained in ''k''! different ways, the correct number of ''k''-combinations is
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| ::<math>\frac{n^{\underline k}}{k!}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k(k-1)(k-2)\cdots1}.</math>
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| :This number is known as the [[binomial coefficient]] <math>\tbinom nk</math>, because it is also the coefficient of ''X''<sup>''k''</sup> in {{nowrap|(1 + ''X'')<sup>''n''</sup>}}.
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| * 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.
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| * Factorials also turn up in [[calculus]]; for example they occur in the denominators of the terms of [[Taylor's formula]], where they are used as compensation terms due to the ''n''-th [[derivative]] of ''x''<sup>''n''</sup> being equivalent to ''n''<nowiki>!</nowiki>.
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| * Factorials are also used extensively in [[probability theory]].
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| * Factorials can be useful to facilitate expression manipulation. For instance the number of ''k''-permutations of ''n'' can be written as
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| ::<math>n^{\underline k}=\frac{n!}{(n-k)!};</math>
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| :while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients:
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| ::<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>
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| ==Number theory==
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| {{unreferenced section|date=May 2013}}
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| Factorials have many applications in [[number theory]]. In particular, ''n''<nowiki>!</nowiki> is necessarily divisible by all [[prime number]]s up to and including ''n''. As a consequence, ''n'' > 5 is a [[composite number]] [[if and only if]]
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| :<math>(n-1)!\ \equiv\ 0 \pmod n.</math>
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| A stronger result is [[Wilson's theorem]], which states that
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| :<math>(p-1)!\ \equiv\ -1 \pmod p</math>
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| if and only if ''p'' is prime.
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| [[Adrien-Marie Legendre]] found that the multiplicity of the prime ''p'' occurring in the prime factorization of ''n''<nowiki>!</nowiki> can be expressed exactly as
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| :<math>\sum_{i=1}^{\infty} \left \lfloor \frac{n}{p^i} \right \rfloor .</math>
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| This fact is based on counting the number of factors ''p'' of the integers from 1 to ''n''. The number of multiples of ''p'' in the numbers 1 to ''n'' are given by <math>\textstyle \left \lfloor \frac{n}{p} \right \rfloor</math>; however, this formula counts those numbers with two factors of ''p'' only once. Hence another <math>\textstyle \left \lfloor \frac{n}{p^2} \right \rfloor</math> factors of ''p'' must be counted too. Similarly for three, four, five factors, to infinity. The sum is finite since ''p''<sup> ''i''</sup> can only be less than or equal to ''n'' for finitely many values of ''i'', and the [[Floor and ceiling functions|floor function]] results in 0 when applied for ''p''<sup> ''i''</sup> > ''n''.
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| The only factorial that is also a prime number is 2, but there are many primes of the form ''n''! ± 1, called [[factorial prime]]s.
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| All factorials greater than 1! are [[Parity (mathematics)|even]], as they are all multiples of 2. Also, all factorials from 5! upwards are multiples of 10 (and hence have a [[trailing zero]] as their final digit), because they are multiples of 5 and 2.
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| ==Series of reciprocals==
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| The [[multiplicative inverse|reciprocals]] of factorials produce a [[convergent series]]: (see ''[[e (mathematical constant)|e]]'')
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| :<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} + \ldots = e\,.</math>
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| 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:
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| :<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}\ldots=1\,.</math>
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| 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.
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| Therefore, the factorials do not form an [[irrationality sequence]].<ref>{{citation |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=0-387-20860-7 | zbl=1058.11001 | contribution=E24 Irrationality sequences|page=346|url=http://books.google.com/books?id=1AP2CEGxTkgC&pg=PA346 }}.</ref>
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| ==Rate of growth and approximations for large n==
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| [[File:Log-factorial.svg|300px|thumb|right|Plot of the natural logarithm of the factorial]]
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| As ''n'' grows, the factorial ''n''<nowiki>!</nowiki> increases faster than all [[polynomial]]s and [[exponential growth|exponential functions]] (but slower than [[double exponential function]]s) in ''n''.
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| Most approximations for ''n''! are based on approximating its [[natural logarithm]]
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| :<math>\log n! = \sum_{x=1}^n \log x.</math>
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| The graph of the function ''f''(''n'') = log ''n''! is shown in the figure on the right. It looks approximately [[linear function|linear]] for all reasonable values of ''n'', but this intuition is false.
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| We get one of the simplest approximations for log ''n''! by bounding the sum with an [[integral]] from above and below as follows:
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| :<math> \int_1^n \log x \, dx \leq \sum_{x=1}^n \log x \leq \int_0^n \log (x+1) \, dx</math>
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| which gives us the estimate
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| :<math> n\log\left(\frac{n}{e}\right)+1 \leq \log n! \leq (n+1)\log\left( \frac{n+1}{e} \right) + 1.</math>
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| Hence log ''n''! is Θ(''n'' log ''n'') (see [[Big O notation#Family of Bachmann–Landau notations|Big ''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 log ''n''! deduced above we get that
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| :<math>e\left(\frac ne\right)^n \leq n! \leq e\left(\frac{n+1}e\right)^{n+1}.</math>
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| It is sometimes practical to use weaker but simpler estimates. Using the above formula it is easily shown that for all ''n'' we have <math>(n/3)^n < n!</math>, and for all ''n'' ≥ 6 we have <math>n! < (n/2)^n</math>.
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| For large ''n'' we get a better estimate for the number ''n''<nowiki>!</nowiki> using [[Stirling's approximation]]:
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| :<math>n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.</math>
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| In fact, it can be proved that for all ''n'' we have
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| :<math>n! > \sqrt{2\pi n}\left(\frac{n}{e}\right)^n.</math>
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| Another approximation for {{nowrap|log ''n''!}} is given by [[Srinivasa Ramanujan]] {{harv|Ramanujan|1988}}
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| :<math>\log n! \approx n\log n - n + \frac {\log(n(1+4n(1+2n)))} {6} + \frac {\log(\pi)} {2}</math>
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| :<math>= n\log n - n + \frac {\log(1 +1/(2n) +1/(8n^2))} {6} + \frac {3\log (2n)} 6 + \frac {\log(\pi)} {2}.</math>
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| Thus it is even smaller than the next correction term <math>\tfrac 1 {12n}</math> of Stirling's formula.
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| ==Computation==
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| 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 2 up to ''n'' (if any) will compute ''n''<nowiki>!</nowiki>, provided the result fits in the variable. In functional languages, the recursive definition is often implemented directly to illustrate recursive functions.
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| 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. [[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 69! < 10<sup>100</sup> < 70!. Calculators that use 3-digit exponents can compute larger factorials, up to, for example, 253! ≈ 5.2{{e|499}} on [[Hewlett-Packard|HP]] calculators and 449! ≈ 3.9{{e|997}} on the [[TI-86]]. The calculator seen in [[Mac OS X]], [[Microsoft Excel]] and [[Google Calculator]], as well as the freeware Fox Calculator, can handle factorials up to 170!, which is the largest factorial whose floating-point approximation can be represented as a [[IEEE 754-1985#Double-precision 64 bit|64-bit IEEE 754 floating-point]] value. The scientific calculator in Windows 7 and Windows 8 is able to calculate factorials up to 3248!.
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| 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 ''n''<nowiki>!</nowiki> for values of ''n'' up to 249999, and up to 20,000,000! for the integers.
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| If the exact values of large factorials are needed, they can be computed using [[arbitrary-precision arithmetic]]. Instead of doing the sequential multiplications <math>((1 \times 2) \times 3) \times 4\dots</math>, 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>[[GNU MP]] software manual, [http://gmplib.org/manual/Factorial-Algorithm.html "Factorial Algorithm"] (retrieved 22 January 2013).</ref>
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| The asymptotically best efficiency is obtained by computing ''n''<nowiki>!</nowiki> from its prime factorization. As documented by [[Peter Borwein]], prime factorization allows ''n''<nowiki>!</nowiki> to be computed in time [[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>Peter Borwein. "On the Complexity of Calculating Factorials". ''Journal of Algorithms'' 6, 376–380 (1985)</ref> Peter Luschny presents source code and benchmarks for several efficient factorial algorithms, with or without the use of a [[prime sieve]].<ref>Peter Luschny, [http://www.luschny.de/math/factorial/FastFactorialFunctions.htm ''Fast-Factorial-Functions: The Homepage of Factorial Algorithms''].</ref>
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| ==Extension of factorial to non-integer values of argument==
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| ===The Gamma and Pi functions===
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| {{Main|Gamma function}}
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| [[File:Generalized factorial function.svg|thumb|right|325px|The factorial function, generalized to all real numbers except negative integers. For example, 0! = 1! = 1, (−0.5)! = √''π'', (0.5)! = √''π''/2.]]
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| Besides nonnegative integers, the factorial function 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) is called the [[Gamma function]], denoted Γ(''z''), defined for all complex numbers ''z'' except the non-positive integers, and given when the real part of ''z'' is positive by
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| :<math>\Gamma(z)=\int_0^\infty t^{z-1} e^{-t}\, \mathrm{d}t. \!</math>
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| Its relation to the factorials is that for any natural number ''n''
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| :<math>n!=\Gamma(n+1).\,</math>
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| [[Leonhard Euler|Euler's]] original formula for the Gamma function was
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| :<math>\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\displaystyle\prod_{k=0}^n (z+k)}. \!</math>
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| An alternative notation, originally introduced by [[Carl Friedrich Gauss|Gauss]], is sometimes used. The '''Pi function''', denoted Π(''z'') for real numbers ''z'' no less than 0, is defined by
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| :<math>\Pi(z)=\int_0^\infty t^{z} e^{-t}\, \mathrm{d}t\,.</math>
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| In terms of the Gamma function it is
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| :<math>\Pi(z) = \Gamma(z+1) \,.</math>
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| It truly extends the factorial in that
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| :<math>\Pi(n) = n!\text{ for }n \in \mathbf{N}\, .</math>
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| In addition to this, the Pi function satisfies the same recurrence as factorials do, but at every complex value ''z'' where it is defined
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| :<math>\Pi(z) = z\Pi(z-1)\,.</math>
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| In fact, this is no longer a recurrence relation but a [[functional equation]].
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| Expressed in terms of the Gamma function this functional equation takes the form
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| :<math>\Gamma(n+1)=n\Gamma(n)\,.</math>
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| Since the factorial is extended by the Pi function, for every complex value ''z'' where it is defined, we can write:
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| :<math>z! = \Pi(z)\,</math>
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| The values of these functions at [[half-integer]] values is therefore determined by a single one of them; one has
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| :<math>\Gamma\left (\frac{1}{2}\right )=\left (-\frac{1}{2}\right )!=\Pi\left (-\frac{1}{2}\right ) = \sqrt{\pi},</math>
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| from which it follows that for ''n'' ∈ '''N''',
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| :<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 {2k - 1 \over 2} = {(2n)! \over 4^n n!} \sqrt{\pi} = {(2n-1)! \over 2^{2n-1}(n-1)!} \sqrt{\pi}.</math>
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| For example,
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| :<math>\Gamma\left (4.5 \right ) = 3.5! = \Pi\left (3.5\right ) = {1\over 2}\cdot{3\over 2}\cdot{5\over 2}\cdot{7\over 2} \sqrt{\pi} = {8! \over 4^4 4!} \sqrt{\pi} = {7! \over 2^7 3!} \sqrt{\pi} = {105 \over 16} \sqrt{\pi} \approx 11.63.</math>
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| It also follows that for ''n'' ∈ '''N''',
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| :<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 {2 \over 1 - 2k} = {(-4)^n n! \over (2n)!} \sqrt{\pi}.</math>
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| For example,
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| :<math>\Gamma\left (-2.5 \right ) = (-3.5)! = \Pi\left (-3.5\right ) = {2\over -1}\cdot{2\over -3}\cdot{2\over -5} \sqrt{\pi} = {(-4)^3 3! \over 6!} \sqrt{\pi} = -{8 \over 15} \sqrt{\pi} \approx -0.9453.</math>
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| 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 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 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, [[Jacques Hadamard|Hadamard's]] 'Gamma'-function {{harv|Hadamard|1894}} which, unlike the Gamma function, is an [[entire function]].<ref>Peter Luschny, [http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html ''Hadamard versus Euler - Who found the better Gamma function?''].</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 or Gamma function, as its rate of convergence is slow.
| |
| | |
| ===Applications of the Gamma function===
| |
| The [[volume]] of an [[Dimension|''n''-dimensional]] [[N-sphere|hypersphere]] of radius ''R'' is
| |
| | |
| :<math>V_n=\frac{\pi^{n/2}}{\Gamma((n/2)+1)}R^n.</math>
| |
| | |
| ===Factorial at the complex plane===
| |
| [[File:Factorial05.jpg|400px|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 \exp({\rm i}\varphi)=(x+{\rm i}y)!=\Gamma(x+{\rm i}y+1) </math>. Several levels of constant modulus (amplitude) <math>\rho =\rm const</math> and constant phase <math>\varphi=\rm const</math> are shown. The grid covers range
| |
| <math>~-3 \le x \le 3~</math>,
| |
| <math>~-2 \le y \le 2~</math>
| |
| with unit step. The scratched line shows the level <math>\varphi=\pm \pi</math>.
| |
| | |
| Thin lines show intermediate levels of constant modulus and constant phase. At poles <math> x+ {\rm i}y \in \rm (negative ~ integers)</math>, phase and amplitude are not defined. Equilines are dense in vicinity of singularities along negative integer values of the argument.
| |
| | |
| For <math>|z|<1</math>, 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"
| |
| |-
| |
| ! <math>n</math>
| |
| ! <math>g_n</math>
| |
| ! approximation
| |
| |-
| |
| | 0
| |
| | <math>1</math>
| |
| | <math> 1</math>
| |
| |-
| |
| | 1
| |
| | <math>-\gamma</math>
| |
| | <math>- 0.5772156649</math>
| |
| |-
| |
| | 2
| |
| | <math>\frac{\pi^2}{12}+\frac{\gamma^2}{2}</math>
| |
| | <math> 0.9890559955</math>
| |
| |-
| |
| | 3
| |
| | <math>-\frac{\zeta(3)}{3}-\frac{\pi^2\gamma}{12}-\frac{\gamma^3}{6}</math>
| |
| | <math>-0.9074790760</math>
| |
| |}
| |
| | |
| where <math>\gamma</math> is the [[Euler–Mascheroni constant|Euler constant]] and <math>\zeta</math> is the [[Riemann zeta function]]. [[Computer algebra system]]s such as [[Sage (mathematics software)]] can generate many terms of this expansion.
| |
| | |
| ===Approximations of factorial===
| |
| For the large values of the argument,
| |
| 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). Writing ''z''! = exp(P(''z'')) where P(''z'') is
| |
| : <math> P(z) = p(z) + \log(2\pi)/2 - z + \left(z+\frac{1}{2}\right)\log(z),</math>
| |
| Stieltjes gave a continued fraction for 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 a<sub>n</sub> are<ref name="dlmf5.10" >Digital Library of Mathematical Functions, http://dlmf.nist.gov/5.10</ref>
| |
| {| class="wikitable"
| |
| |-
| |
| ! n
| |
| ! a<sub>n</sub>
| |
| |-
| |
| | 0
| |
| | 1 / 12
| |
| |-
| |
| | 1
| |
| | 1 / 30
| |
| |-
| |
| | 2
| |
| | 53 / 210
| |
| |-
| |
| | 3
| |
| | 195 / 371
| |
| |-
| |
| | 4
| |
| | 22999 / 22737
| |
| |-
| |
| | 5
| |
| | 29944523 / 19733142
| |
| |-
| |
| | 6
| |
| | 109535241009 / 48264275462
| |
| |}
| |
| | |
| There is common [[List of common misconceptions|misconception]], that <math>\displaystyle\log(z!)=P(z)</math> or <math>\log(\Gamma(z\!+\!1))=P(z)</math>
| |
| for any complex ''z'' ≠ 0. Indeed, the relation through the logarithm is valid only for specific range of values of ''z'' in vicinity of the real axis, while <math>|\Im(\Gamma(z\!+\!1))| < \pi </math>. The larger is the real part of the argument, the smaller should be the imaginary part. However, the inverse relation, ''z''! = exp(''P''(''z'')), is valid for the whole complex plane apart from zero. The convergence is poor in vicinity of the negative part of the real axis. (It is difficult to have good convergence of any approximation in vicinity of the singularities). While <math>|\Im(z)| >2 </math> or <math>\Re(z)>2</math>, the 6 coefficients above are sufficient for the evaluation of the factorial with the complex<double> precision. For higher precision more coefficients can be computed by a rational QD-scheme ([[H. Rutishauser]]'s [[QD algorithm]]).<ref>Peter Luschny, [http://www.luschny.de/math/factorial/approx/continuedfraction.html ''On Stieltjes' Continued Fraction for the Gamma Function.''].</ref>
| |
| | |
| ===Non-extendability to negative integers===
| |
| The relation ''n''! = ''n'' × (''n'' − 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>
| |
| | |
| Note, however, that this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)<nowiki>!</nowiki> would require a division by zero, and thus blocks us from computing a factorial value for every negative integer. (Similarly, the Gamma function is not defined for non-positive 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:
| |
| | |
| ===Primorial===
| |
| The [[primorial]] {{OEIS|id=A002110}} is similar to the factorial, but with the product taken only over the [[prime number]]s.
| |
| | |
| === Double factorial ===
| |
| {{main|Double factorial}}
| |
| The product of all the odd integers up to some odd positive integer ''n'' is called the '''double factorial''' of ''n'', and denoted by ''n''<nowiki>!!</nowiki>.<ref name="callan">{{citation|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|arxiv=0906.1317|year=2009}}.</ref> That is,
| |
| :<math>(2k-1)!! = \prod_{i=1}^k (2i-1) = \frac{(2k)!}{2^k k!} = \frac {_{2k}P_k} {2^k} = \frac {{(2k)}^{\underline k}} {2^k}.</math>
| |
| For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.
| |
| | |
| The sequence of double factorials for ''n'' = 1, 3, 5, 7, ... starts as
| |
| : 1, 3, 15, 105, 945, 10395, 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}}</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!!</math>), three (<math>n!!!</math>), or more. The double factorial is the most commonly used variant, but one can similarly define the triple factorial (<math>n!!!</math>) and so on. One can define the ''k''-th factorial, denoted by <math>n!^{(k)}</math>, recursively for non-negative integers as
| |
| | |
| :<math>
| |
| n!^{(k)}=
| |
| \left\{
| |
| \begin{matrix}
| |
| 1,\qquad\qquad\ &&\mbox{if }0\le n<k,
| |
| \\
| |
| n((n-k)!^{(k)}),&&\mbox{if }n\ge k\,,\quad\ \ \,
| |
| \end{matrix}
| |
| \right.
| |
| </math>
| |
| though see [[#Alternative extension of the multifactorial|the alternative definition below]].
| |
| | |
| Some mathematicians have suggested an alternative notation of <math>n!_2</math> for the double factorial and similarly <math>n!_k</math> for other multifactorials, but this has not come into general use.
| |
| | |
| The factorial operation is encountered in many different areas of mathematics, notably in [[combinatorics]], [[algebra]] and [[mathematical analysis]]. Its most basic occurrence is the fact that there are ''n''! ways to arrange ''n'' distinct objects into a sequence (i.e., [[permutation]]s of the set of objects). This fact was known to Indian scholars at least as early as the 12th century.
| |
| | |
| In the same way that <math>n!</math> is not defined for negative integers, and <math>n!!</math> is not defined for negative even integers, <math>n!^{(k)}</math> is not defined for negative integers divisible by <math>k</math>.
| |
| | |
| ====Alternative extension of the multifactorial====
| |
| Alternatively, the multifactorial ''z''!<sup>(''k'')</sup> can be extended to most real and complex numbers ''z'' by noting that when ''z'' is one more than a positive multiple of ''k'' then
| |
| :<math>z!^{(k)} = z(z-k)\cdots (k+1)
| |
| = k^{(z-1)/k}\left(\frac{z}{k}\right)\left(\frac{z-k}{k}\right)\cdots \left(\frac{k+1}{k}\right)
| |
| = k^{(z-1)/k} \frac{\Gamma\left(\frac{z}{k}+1\right)}{\Gamma\left(\frac{1}{k}+1\right)}\,.</math>
| |
| This last expression is defined much more broadly than the original; with this definition, ''z''!<sup>(''k'')</sup> is defined for all complex numbers except the negative real numbers evenly divisible by ''k''. This definition is consistent with the earlier definition only for those integers ''z'' satisfying ''z'' ≡ 1 mod ''k''.
| |
| | |
| In addition to extending ''z''!<sup>(''k'')</sup> to most complex numbers ''z'', this definition has the feature of working for all positive real values of ''k''. Furthermore, when ''k'' = 1, this definition is mathematically equivalent to the Π(''z'') function, described above. Also, when ''k'' = 2, this definition is mathematically equivalent to the [[Double factorial#Complex arguments|alternative extension of the double factorial]].
| |
| | |
| ===Quadruple factorial===
| |
| {{sources|section|date=September 2013}}
| |
| The quadruple factorial is not the multifactorial ''n''!<sup>(4)</sup>; it is a much larger number given by (2''n'')!/''n''!, starting as
| |
| | |
| :1, 2, 12, 120, 1680, 30240, 665280, ... {{OEIS|id=A001813}}.
| |
| | |
| It is also equal to
| |
| | |
| : <math>
| |
| \begin{align}
| |
| 2^n\frac{(2n)!}{n!2^n} & = 2^n \frac{(2\cdot 4\cdots 2n) (1\cdot 3\cdots (2n-1))}{2\cdot 4\cdots 2n} \\[8pt]
| |
| & = (1\cdot 2)\cdot (3 \cdot 2) \cdots((2n-1)\cdot 2)=(4n-2)!^{(4)}.
| |
| \end{align}
| |
| </math>
| |
| | |
| ===Superfactorial===
| |
| {{Main|Large numbers}}
| |
| {{redirect|N$|the currency|Namibian dollar}}
| |
| [[Neil Sloane]] and [[Simon Plouffe]] defined a '''superfactorial''' in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first <math>n</math> factorials. So the superfactorial of 4 is
| |
| | |
| :<math> \mathrm{sf}(4)=1! \times 2! \times 3! \times 4!=288. \,</math>
| |
| | |
| In general
| |
| | |
| :<math>
| |
| \mathrm{sf}(n)
| |
| =\prod_{k=1}^n k! =\prod_{k=1}^n k^{n-k+1}
| |
| =1^n\cdot2^{n-1}\cdot3^{n-2}\cdots(n-1)^2\cdot n^1.
| |
| </math>
| |
| | |
| Equivalently, the superfactorial is given by the formula
| |
| :<math>
| |
| \mathrm{sf}(n)
| |
| =\prod_{0 \le i < j \le n} (j-i)
| |
| </math>
| |
| which is the [[determinant]] of a [[Vandermonde matrix]].
| |
| | |
| The sequence of superfactorials starts (from <math>n = 0</math>) as
| |
| | |
| :1, 1, 2, 12, 288, 34560, 24883200, 125411328000, ... {{OEIS|id=A000178}}
| |
| | |
| ====Alternative definition====
| |
| [[Clifford Pickover]] in his 1995 book ''Keys to Infinity'' used a new notation, ''n$'', to define the superfactorial
| |
| :<math>n\$\equiv \begin{matrix} \underbrace{ n!^{{n!}^{{\cdot}^{{\cdot}^{{\cdot}^{n!}}}}}} \\ n! \end{matrix}, \,</math>
| |
| or as,
| |
| :<math>n\$=n!^{(4)}n! \,</math>
| |
| where the <sup>(4)</sup> notation denotes the [[Tetration|hyper4]] [[operator (mathematics)|operator]], or using [[Knuth's up-arrow notation]],
| |
| :<math>n\$=(n!)\uparrow\uparrow(n!). \,</math>
| |
| This sequence of superfactorials starts:
| |
| :<math>1\$=1 \,</math>
| |
| :<math>2\$=2^2=4 \,</math>
| |
| :<math>3\$=6\uparrow\uparrow6={^6}6=6^{6^{6^{6^{6^6}}}}.</math>
| |
| Here, as is usual for compound [[exponentiation]], the grouping is understood to be from right to left: | |
| :<math>a^{b^c}=a^{(b^c)}.\,</math>
| |
| | |
| ===Hyperfactorial===
| |
| Occasionally the '''hyperfactorial''' of ''n'' is considered. It is written as ''H''(''n'') and defined by
| |
| | |
| :<math>
| |
| H(n)
| |
| =\prod_{k=1}^n k^k
| |
| =1^1\cdot2^2\cdot3^3\cdots(n-1)^{n-1}\cdot n^n.
| |
| </math>
| |
| | |
| For ''n'' = 1, 2, 3, 4, ... the values ''H''(''n'') are 1, 4, 108, 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 ''A'' = 1.2824... is the [[Glaisher–Kinkelin constant]].<ref>{{MathWorld | urlname=Glaisher-KinkelinConstant | title=Glaisher–Kinkelin Constant}}</ref> ''H''(14) = 1.8474...×10<sup>99</sup> is already almost equal to a [[googol]], and ''H''(15) = 8.0896...×10<sup>116</sup> 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]].
| |
| | |
| ==See also==
| |
| {{div col|colwidth=30em}}
| |
| * [[Alternating factorial]]
| |
| * [[Digamma function]]
| |
| * [[Exponential factorial]]
| |
| * [[Factorial number system]]
| |
| * [[Factorial prime]]
| |
| * [[Factorion]]
| |
| * [[Gamma function]]
| |
| * [[List of factorial and binomial topics]]
| |
| * [[Pochhammer symbol]], which gives the falling or rising factorial
| |
| * [[Stirling's approximation]]
| |
| * [[Subfactorial]]
| |
| * [[Trailing zeros#Factorial|Trailing zeros]] of factorial
| |
| * [[Triangular number]], the additive analogue of factorial
| |
| {{div col end}}
| |
| | |
| ==Notes==
| |
| {{Reflist|2}}
| |
| | |
| ==References==
| |
| * {{citation |first=M. J. |last=Hadamard
| |
| |title=Sur L’Expression Du Produit 1·2·3· · · · ·(n−1) Par Une Fonction Entière
| |
| |publisher=''OEuvres de Jacques Hadamard'', Centre National de la Recherche Scientifiques, Paris, 1968
| |
| |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}}
| |
| | |
| ==External links==
| |
| * [http://factorielle.free.fr/index_en.html All about factorial notation n!]
| |
| * {{springer|title=Factorial|id=p/f038080}}
| |
| * {{MathWorld | urlname=Factorial | title=Factorial}}
| |
| * {{PlanetMath | urlname=Factorial | title=Factorial}}
| |
| * [http://www.docstoc.com/docs/5606124/Double-Factorials-Selected-Proofs-and-Notes "Double Factorial Derivations"]
| |
| * [http://www.gfredericks.com/main/sandbox/arith/factorial Animated Factorial Calculator]: shows factorials calculated as if by hand using common elementary school algorithms
| |
| * [http://www.luschny.de/math/factorial/FastFactorialFunctions.htm Fast Factorial Functions (with source code in Java, C#, C++, Scala and Go)]
| |
| * [http://www.gnu.org/software/gsl/manual/html_node/Factorials.html C functions to calculated factorials in the GNU Scientific Library (GSL)]
| |
| | |
| {{Series (mathematics)}}
| |
| | |
| [[Category:Integer sequences]]
| |
| [[Category:Combinatorics]]
| |
| [[Category:Number theory]]
| |
| [[Category:Gamma and related functions]]
| |
| [[Category:Factorial and binomial topics]]
| |