Tanfalit n Stirling

Tanfalit n Stirling nɣ Tamawaẓt n Stirling tga ɣ tusnakt, yat tnfalit nna yakkan agdazal n uskir n yan umḍan ummid agaman $n$ iɣ yaẓ $n$ i wartmi:

$$\underset{n\to +\mathrm{\infty }}{\lim }\frac{n!}{\sqrt{2\pi n}{(n/\mathrm{e})}^n}=1$$nna nttafa tettyura ula s tɣarast ad:

$$n!\sim \sqrt{2\pi n}{\left(\frac{n}{\mathrm{e}}\right)}^n$$maɣ $e$ ɣid tga amḍan n tusnakt ikspununsyal.

Amzruy

Brahim du Mwavr ad akk izwarn is yufa tanfalit ad ddaw-as:

$$n!\sim C{n}^{n+\frac{1}{2}}{\mathrm{e}}^{-n}$$

maɣ $C$ 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 $C=\sqrt{2\pi n}$ fad ad aɣ tfk asbuɣlu n $ln(n!)$ i kraygat amizzwr.

Atig ɣ wartmi

Bac ad nit ntḥqqaq is nit ar ttaẓ tanfalit ad i atig n $n!$, nẓḍar ad nɛdl taflwit ad lli aɣ immaln atign n $n!$ d win $n!\sim \sqrt{2\pi n}{\left(\frac{n}{\mathrm{e}}\right)}^n$ s sin-itsn:

n	n !	$\sqrt{2\pi n}{\left(\frac{n}{\mathrm{e}}\right)}^n$	$\sqrt{2\pi n}{\left(\frac{n}{\mathrm{e}}\right)}^n(1+\frac{1}{12n})$
1	1	0,92	0,999
2	2	1,92	1,999
3	6	5,84	5,998
4	24	23,51	23,996
5	120	118,02	119,99
6	720	710,08	719,94
7	5 040	4 980,4	5 039,7
8	40 320	39 902,4	40 318,1
9	362 880	359 536,9	362 866,0
10	3 628 800	3 598 696	3 628 685
15	1 307 674 368 000	1,300 431 × $10^{12}$	1,307 665 × $10^{12}$
20	2 432 902 008 176 640 000	2,422 787 × $10^{18}$	2,432 882 × $10^{18}$
25	15 511 210 043 330 985 984 000 000	1,545 959 × $10^{25}$	1,551 113 × $10^{25}$
30	265 252 859 812 191 058 636 308 480 000 000	2,645 171 × $10^{32}$	2,652 519 × $10^{32}$
40	815 915 283 247 897 734 345 611 269 596 115 894 272 000 000 000	8,142 173 × $10^{47}$	8,159 136 × $10^{47}$

Amawal

• Timzgit = constante
• Ilaw = reel
• Amizzwr = ordre
• Tafllawt = resultat

Isaɣuln

( en ) ( fr ) Tẓḍarm ad tafim amgrad ad ɣ Wikipidya tafransist (Formule de Stirling) d Wikipidya tanglizt (Stirling's approximation) rad gisn tafim uggar n ɣaylli yad illan ɣid.

Ẓṛ uggar

• James Stirling