Tanfalit n Stirling

Amnni ad icqqa ad t tfhmt ! Amnni ad gis kra n iwalwn nna ur iẓḍaṛ yan ad tn ifhm iɣ ur issin ɣ umawal iẓlin n tutlayt taclḥiyt tatrart. Mad igan afssay ? Fad ad tfhmt mad gis illan tzḍart ad tawst i yixf nnk s umawal lli illan ɣ izddar akkʷ n tasna. Iɣ t ur tufit, tzḍart ad nn taggʷt ɣ imawaln d isgzawaln.

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:

$$\lim _{n\to +\infty }{n\,! \over {\sqrt {2\pi n}}\;\left({n}/{\rm {e}}\right)^{n}}=1$$

nna nttafa tettyura ula s tɣarast ad:

$$n\,!\sim {\sqrt {2\pi n}}\,\left({n \over {\rm {e}}}\right)^{n}$$

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 $n\,!\sim {\sqrt {2\pi n}}\,\left({n \over {\rm {e}}}\right)^{n}$ s sin-itsn:

n	n !	${\sqrt {2\pi n}}\,\left({n \over {\mathrm {e} }}\right)^{n}$	${\sqrt {2\pi n}}\,\left({n \over {\mathrm {e} }}\right)^{n}\left(1+{\frac {1}{12\ n}}\right)$
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 × ${\textstyle 10^{12}}$	1,307 665 × ${\textstyle 10^{12}}$
20	2 432 902 008 176 640 000	2,422 787 × ${\textstyle 10^{18}}$	2,432 882 × ${\textstyle 10^{18}}$
25	15 511 210 043 330 985 984 000 000	1,545 959 × ${\textstyle 10^{25}}$	1,551 113 × ${\textstyle 10^{25}}$
30	265 252 859 812 191 058 636 308 480 000 000	2,645 171 × ${\textstyle 10^{32}}$	2,652 519 × ${\textstyle 10^{32}}$
40	815 915 283 247 897 734 345 611 269 596 115 894 272 000 000 000	8,142 173 × ${\textstyle 10^{47}}$	8,159 136 × ${\textstyle 10^{47}}$

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