Formula Stirlinga ye nablizhennyam dlya faktorialiv pri velikih znachennyah n nazvana na chest Dzhejmsa Stirlinga Formalne tverdzhennya formuliVidnoshennya ln n do n ln n n pri n pryamuyuchomu do neskinchenosti pryamuye do 1 lim n n n n e n 2 p n 1 displaystyle lim n to infty n over n n e n sqrt 2 pi n 1 abon n n e n 2 p n n displaystyle n approx n n e n sqrt 2 pi n n to infty Zbizhnist ta pohibki RedaguvatiFormula Stirlinga otrimuyetsya iz Asimptotichnogo rozkladu Stirlinga dlya G z displaystyle Gamma z nbsp ta n displaystyle n nbsp G z e z z z 1 2 2 p 1 1 12 z 1 288 z 2 139 51840 z 3 571 2488320 z 4 O z 5 displaystyle Gamma z e z z z 1 2 sqrt 2 pi begin bmatrix 1 1 over 12z 1 over 288z 2 139 over 51840z 3 571 over 2488320z 4 O z 5 end bmatrix nbsp de a r g z lt p displaystyle begin vmatrix arg z end vmatrix lt pi nbsp ryad Stirlinga Ryad Stirlinga osoblivo korisnij dlya velikih znachen z displaystyle begin vmatrix z end vmatrix nbsp dlya dijsnih dodatnih z absolyutna pohibka mensha nizh absolyutna velichina ostannogo iz vzyatih elementiv ryadu Ryadom Stirlinga takozh nazivayetsya asimptotichnij rozklad logarifma vid n log n n log n n 1 2 log 2 p n 1 12 n 1 360 n 3 1 1260 n 5 1 1680 n 7 displaystyle log n n log n n 1 over 2 log 2 pi n 1 over 12n 1 over 360n 3 1 over 1260n 5 1 over 1680n 7 cdots nbsp Vidnosna pohibka formuli Stirlinga spadaye iz zrostannyam n cya formula chasto vikoristovuyetsya dlya obchislennya vidnoshennya dvoh faktorialiv abo gamma funkcij oskilki v comu vipadku vidnosna pohibka osoblivo vazhliva Zauvazhimo zokrema sho Formula Stirlinga ye prosto pershim nablizhennyam dlya ryadu Stirlinga Specialni formuli Redaguvatin n e n 2 p n lt n lt n n 2 p n e n 1 12 n displaystyle n n e n sqrt 2 pi n lt n lt n n sqrt 2 pi n e n 1 over 12n nbsp tan n n 2 p n e n 1 12 n 1 360 n 2 displaystyle n approx n n sqrt 2 pi n e n 1 over 12n 1 over 360n 2 nbsp pri n displaystyle n to infty nbsp Dovedennya RedaguvatiGrubo kazhuchi najprostishu versiyu formuli Stirlinga mozhna shvidko otrimati nablizhayuchi sumu ln n j 1 n ln j displaystyle ln n sum j 1 n ln j nbsp do integralu j 1 n ln j 1 n ln x d x n ln n n 1 displaystyle sum j 1 n ln j approx int 1 n ln x rm d x n ln n n 1 nbsp Povna formula razom iz tochnoyu pohibkoyu mozhe buti otrimana nastupnim chinom Zamist nablizhennya n rozglyadayetsya logarifm naturalnij oskilki vin ye funkciyeyu yaka povilno zminyuyetsya ln n ln 1 ln 2 ln n displaystyle ln n ln 1 ln 2 cdots ln n nbsp Vid pravoyi chastini rivnyannya vidnimayemo 1 2 ln 1 ln n 1 2 ln n displaystyle tfrac 1 2 ln 1 ln n tfrac 1 2 ln n nbsp i nablizhuyemo metodom trapecij integral ln n 1 2 ln n 1 n ln x d x n ln n n 1 displaystyle ln n tfrac 1 2 ln n approx int 1 n ln x rm d x n ln n n 1 nbsp Pohibka v comu nablizhenni zadayetsya formuloyu Ejlera Maklorena ln n 1 2 ln n 1 2 ln 1 ln 2 ln 3 ln n 1 1 2 ln n n ln n n 1 k 2 m 1 k B k k k 1 1 n k 1 1 R m n displaystyle begin aligned ln n tfrac 1 2 ln n amp tfrac 1 2 ln 1 ln 2 ln 3 cdots ln n 1 tfrac 1 2 ln n amp n ln n n 1 sum k 2 m frac 1 k B k k k 1 left frac 1 n k 1 1 right R m n end aligned nbsp de Bk chisla Bernulli ta Rm n zalishkovij chlen u formuli Ejlera Maklorena Perejdemo do granici lim n ln n n ln n n 1 2 ln n 1 k 2 m 1 k B k k k 1 lim n R m n displaystyle lim n to infty left ln n n ln n n tfrac 1 2 ln n right 1 sum k 2 m frac 1 k B k k k 1 lim n to infty R m n nbsp Poznachimo cyu granicyu yak y Oskilki zalishok Rm n u formuli Ejlera Maklorena zadovolnyaye R m n lim n R m n O 1 n m displaystyle R m n lim n to infty R m n O left frac 1 n m right nbsp de mi vikoristovuyemo notaciyu Landau ob yednuyuchi vishenavedeni rivnyannya otrimuyemo nablizhenu formulu v yiyi logarifmichnij formi ln n n ln n e 1 2 ln n y k 2 m 1 k B k k k 1 n k 1 O 1 n m displaystyle ln n n ln left frac n mathrm e right tfrac 1 2 ln n y sum k 2 m frac 1 k B k k k 1 n k 1 O left frac 1 n m right nbsp Vzyavshi eksponentu oboh storin i vibirayuchi bud yake naturalne m otrimuyemo formulu z nevidomoyu velichinoyu mathrm e y Dlya m 1 formula nabuvaye viglyadu n e y n n e n 1 O 1 n displaystyle n mathrm e y sqrt n left frac n mathrm e right n left 1 O left frac 1 n right right nbsp Velichina e y displaystyle mathrm e y nbsp mozhe buti znajdena yaksho v oboh storonah perejti do granici pri n displaystyle n to infty nbsp ta zastosuvavshi formulu Vallisa yaka pokazuye sho e y 2 p displaystyle mathrm e y sqrt 2 pi nbsp Takim chinom otrimayemo formulu Stirlinga n 2 p n n e n 1 O 1 n displaystyle n sqrt 2 pi n left frac n mathrm e right n left 1 O left frac 1 n right right nbsp Istoriya RedaguvatiFormulu vpershe vidkriv Abraham de Muavr u formi n c o n s t a n t n n 1 2 e n displaystyle n sim rm constant cdot n n 1 2 e n nbsp Stirling vstanoviv sho konstanta dorivnyuye 2 p displaystyle sqrt 2 pi nbsp Dzherela RedaguvatiPidkujko Sergij 2004 Matematichnij analiz T 1 Mnozhini Dijsni chisla Granicya poslidovnosti Granicya funkciyi Neperervnist funkciyi Diferencialne chislennya funkcij odniyeyi zminnoyi Lviv Galicka vidavnicha spilka s 530 ISBN 966 7893 26 HPerevirte znachennya isbn dovidka Grigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr G Korn i T Korn Spravochnik po matematike dlya nauchnih rabotnikov i inzhenerov Otrimano z https uk wikipedia org w index php title Formula Stirlinga amp oldid 40456007
