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Factorial

The factorial of a positive integer typeset structure, denoted by typeset structure, is the product of all positive integers less than or equal to n, i.e.

n != 1.2 . ...n = Underoverscript[∏, k = 1, arg3] k .(1)

The notation typeset structure was introduced by Christian Kramp (1760 - 1826) in 1808 in his Elements d'arithmétique universelle  .

The factorial may also defined recursively

n != {1, if n = 0, n · (n - 1) !, if n > 0.(2)

To compute some values of the factorial function go to .

The factorial function is closely related to the Gamma function . This gives the following integral definition of the factorial

n != Underoverscript[∫, 0, arg3] e^(-x) x^n d x .(3)

We have the Stirling’s approximation

Underscript[lim, n -> ∞] n !/((2 π n)^(1/2) (n/e)^n) = 1 (4)

This formula goes back to A. de Moivre who found that

n ! ~~ C · n^(n + 1/2)/e^n

Stirling showed that typeset structure.

Relation (4)  is a first approximation following from the series expansion

n != (2 π n)^(1/2) (n/e)^n (1 + 1/(12 n) + 1/(288 n^2) - 139/(51840 n^3) + O(1/n^4)) .(5)

Cite this web-page as:

Štefan Porubský: Factorial.


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