Ẓṛ asagm n Tanfalit n Stirling

{{Amgrad icqqan}}[[Afaylu:Mplwp factorial gamma stirling.svg|thumb]]
'''Tanfalit n Stirling''' nɣ '''Tamawaẓt n Stirling''' tga ɣ tusnakt, yat tnfalit nna yakkan agdazal n uskir n yan umḍan ummid agaman <math>n</math> iɣ yaẓ <math>n</math> i wartmi:

<math display="block">\lim_{n \to +\infty} {n\,!\over \sqrt{2 \pi n} \; \left({n}/{\rm e}\right)^{n} } = 1</math>nna nttafa tettyura ula s tɣarast ad:

<math display="block">n\,!\sim \sqrt{2\pi n}\,\left({n \over {\rm e}}\right)^n</math>maɣ <math>e</math> ɣid tga amḍan n tusnakt [[e (amḍan n tusnakt)|ikspununsyal]].
==Amzruy==
[[Brahim du Mwavr]] ad akk izwarn is yufa tanfalit ad ddaw-as:

<math display="block">n\,!\sim C\; n^{n+\frac12}\, \mathrm e^{-n}</math>

maɣ <math>C</math> tga yat timzgit tilawt (ur igaddan d amya)

Mad ifka nɣ mad izayd Stirling i tanfalit ad iga-tt atig n timzgit, iga y'as <math>C = \sqrt{2\pi n}</math> fad ad aɣ tfk asbuɣlu n <math>ln(n!)</math> i kraygat amizzwr.

== Atig ɣ wartmi ==
Bac ad nit ntḥqqaq is nit ar ttaẓ tanfalit ad i atig n <math>n!</math>, nẓḍar ad nɛdl taflwit ad lli aɣ immaln atign n <math>n!</math> d win <math>n\,!\sim \sqrt{2\pi n}\,\left({n \over {\rm e}}\right)^n</math> s sin-itsn:
{|class="wikitable centre alternance" style="text-align:center;"
|--- 
! ''n''
! ''n'' !
! <math>\sqrt{2\pi n}\,\left({n \over {\mathrm e}}\right)^n</math>
! <math>\sqrt{2\pi n}\,\left({n \over {\mathrm e}}\right)^n \left(1 + \frac1{12\ n}\right)</math>
|---
! 1
| '''1''' || 0,92 || 0,999
|---
! 2
| '''2''' || 1,92 || 1,999
|---
! 3
| '''6''' || 5,84 || 5,998
|---
! 4
| '''24''' || '''2'''3,51 || '''2'''3,996
|---
! 5
| '''120''' || '''1'''18,02 || '''1'''19,99
|---
! 6
| '''720''' || '''7'''10,08 || '''7'''19,94
|---
! 7
| '''5 040''' || 4 980,4 || '''5 0'''39,7
|---
! 8
| '''40 320''' || 39 902,4 || '''40 3'''18,1
|---
! 9
| '''362 880''' || '''3'''59 536,9 || '''362 8'''66,0
|---
! 10
| '''3 628 800''' || '''3''' 598 696  || '''3 628''' 685
|---
! 15
| '''1 307 674 368 000''' || '''1,30'''0 431 × <math display="inline">10^{12}</math> || '''1,307 6'''65 × <math display="inline">10^{12}</math>
|---
! 20
| '''2 432 902 008 176 640 000''' || '''2,4'''22 787 × <math display="inline">10^{18}</math> || '''2,432''' 882 × <math display="inline">10^{18}</math>
|---
! 25
| '''15 511 210 043 330 985 984 000 000''' || '''1,5'''45 959 × <math display="inline">10^{25}</math> || '''1,551 1'''13 × <math display="inline">10^{25}</math>
|---
! 30
| '''265 252 859 812 191 058 636 308 480 000 000''' || '''2,6'''45 171 × <math display="inline">10^{32}</math> || '''2,652 5'''19 × <math display="inline">10^{32}</math>
|---
! 40
| '''815 915 283 247 897 734 345 611 269 596 115 894 272 000 000 000''' || '''8,1'''42 173 × <math display="inline">10^{47}</math> || '''8,159 1'''36 × <math display="inline">10^{47}</math>
|}

==Amawal==
* Timzgit = constante
* Ilaw = reel
* Amizzwr = ordre
* Tafllawt = resultat
== Isaɣuln ==
'''<small>( en ) ( fr )</small>''' Tẓḍarm ad tafim amgrad ad ɣ Wikipidya tafransist ([[:fr:Formule_de_Stirling|Formule de Stirling]]) d Wikipidya tanglizt ([[:en:Stirling's_approximation|Stirling's approximation]]) rad gisn tafim uggar n ɣaylli yad illan ɣid.
<references />

== Ẓṛ uggar ==
* [[James Stirling]]

{{Aggur:Tusnakt/Tin imgradn}}

[[Taggayt:Tusnakt]]
[[Taggayt:Tussna]]